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EFFECTIVE DIFFUSION COEFFICIENTS FOR TOLUE E IN CALCIUM CARBONATE FILLED POLY(VI YL ACETATE) FROM QUARTZ CRYSTAL MICROBALA CE SORPTION EXPERIMENTS By STEVEN LEE WILLOUGHBY Bachelor of Science Cameron University Lawton, Oklahoma 1992 Master of Science Oklahoma State University Stillwater, Oklahoma 1994 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1998 EFFECTIVE DIFFU 10 COEFFI IE T FOR TOLUE E I CALCI MCARBO T FILLED POLY(VI YL A ETATE) FROM QUARTZ CRYSTAL MICROB LA SORPTIO EXPERIME TS Thesis Approved: Thesis Advis r LJ 6 &cd! ~of the Graduate College 11 PREFACE A quartz crystal microbalance (QCM) was used to obtain effective diffusion coefficients of toluene diffusing into CaC03 filled poly(vinyl acetate). Experimental diffusion data were obtained at 60°C and 80°C for weight percents of 0,3.3,4.9, and 10% CaC03 at concentrations of toluene below 0.15 weight fraction at 60°C and at toluene weight fraction below 0.10 at 80°C. Effective diffusion coefficients were also calculated from a freevolume equation modified to account for the filler by fitting experimental effective diffusion coefficients to this equation. Several other models were also examined for their ability to accurately represent the experimental effective diffusion coefficient data. I would like to express sincere appreciation to my advisor, Dr. Martin S. High, and committee members, Dr. Robert 1. Robinson, Jr., and Dr. D. Alan Tree, for their advice, support, and interest in this work. I want to thank Mr. Mark C. Drake and Rexam Graphics for financial support and for suggesting this study. I also want to thank the students I've met at asu for their h.elp and friendship. Finally, I want to express appreciation to my wife, Sheena, for her support and encouragement, and for her help and company in many classes. III Chapter TABLE OF CONTE TS Page I. II. INTRODUCTION . BACKGROUND ON DIFFUSION OF PENETRA TS IN POLYMERS . 1 3 Experimental Methods for Obtaining Diffusion Coefficients 3 Fundamental Diffusion Equations .. . . . . . . . . 3 General Sorption Methods for Studying Diffusion. 4 The Quartz Crystal Microbalance Sorption Apparatus 7 Methods of Analyzing Data Obtained Using the Quartz Crystal Microbalance. . . . . . . . . . . . . 11 Obtaining a 'Sorption Curve from Frequency Data 13 Evaluating Diffusion Coefficients Using the HalfTime Method 15 Evaluating Diffusion Coefficients Using the Initial Slope Method . . . . . . . . . 17 Evaluating Diffusion Coefficients Using the Limiting Slope Method . . . . . . . . 19 Evaluating Diffusion Coefficients Using the Moment Method 19 Evaluating the Weight Fraction of Penetrant in the Polymer Film 22 Diffusion of Organic Penetrants into Polymers at Temperatures above the Glass Transition Temperature, Tg . • • • • . . . . • • • • • • • • • 22 Free Volume Theory . . . . . . . . . . . . . . . . 25 Anomalous Diffusion of Penetrants into Polymers 28 Mass Transfer in Heterogeneous Systems 30 Diffusion in Filled Polymers . . . 30 Diffusion in Crystalline Polymers 38 Structure Factor. . . . . . . . . . 41 Treatment of Adsorption as a Chemical Reaction. 43 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 44 III. MODIFICATIONS OF EQUIPMENT AND PROCEDURES 45 IV. EXPERIME TAL RESULTS. . . . . . . . . . . . . . 51 lV Chapter V. DISCUSSION OF RESULTS . Page 67 Comparison of Methods for Determining Diffusion Coefficients 79 Correlating the Experimental Diffusion Coefficient Da.ta .. 82 Regressions Using a Modified FreeVolume Equation. 82 Regressions Using Equations Derived by Barrer and Chio . . . . . . . . . . . . . . 93 Regressions Using an Equation Derived from Reaction Principles . . . . . . . . . . 98 Numerical Simulation and Regression of Sorption Curves 100 VI. CONCLUSIONS AND RECOMMENDATIONS. 104 Conclusions . . . . 104 Recommendations . 106 BIBLIOGRAPHY 108 APPENDICES . . 112 APPENDIX A  Experimental Procedures . 113 Preparing a Polymer Solution 113 Coating a Quartz Crystal with a Polymer Film. 114 Evacuating the Quartz Crystal Microbalance 114 Operating the Quartz Crystal Microbalance Sorption Apparatus. 116 Film Thickness Calculations . . . . . . . . . 119 APPENDIX B  Derivation of the Moment Method Formula for Evaluating Diffusion Coefficients 121 APPENDIX C  Error Analysis. . . . . 124 Diffusion Coefficient. . . . 124 Penetrant Weight Fraction 126 Sample Calculation . . . . 127 APPENDIX D  Results of Numerical Simulations and Correlations 128 APPENDIX E  Computer Programs 145 v Table LIST OF TABLES Page I Solubility Data and Diffusion Coefficients of Toluene In PVAC with 0.0% CaC03 at 60°C . . . . . . . . . . .. ..... 53 II Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 3.3% CaC03 at 60°C . . . . . . . . . . .. ..... 54 III Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 4.9% CaC03 at 60°C . . . . . . . . . . .. ..... 55 IV Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 10% CaC03 at 60°C '" .. 56 V Solubility Data and Diffusion Coefficients of Toluene III PVAC with 0.0% CaC03 at 80°C . . . . . . . . . . .. .,. .. 57 VI Solubility Data and Diffusion Coefficients of Toluene in PVAC with 3.3% CaC03 at 80°C . . . . . . . . . . .. ..... 58 VII Solubility Data and Diffusion Coefficients of Toluene III PVAC with 4.9% CaC03 at 80°C . . . . . . . . . . .. ..... 59 VIII Solubility Data and Diffusion Coefficients of Toluene III PVAC with 10% CaC03 at 80°C 60 IX X Diffusion Coefficients of Toluene in Neat PVAC from Mikkilineni . Diffusion Coefficients of Toluene in Neat PVAC from Hou . 70 71 XI Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C . . . . . . . . . . . . . .. 81 XII Parameters of the Modified FreeVolume Equation for PVACToluene . . . . . . . . . . . . . . . . . . . . . . 94 XIII Analytical Solution of the Diffusion Equation with Constant D . 130 XIV Numerical Solution of the Diffusion Equation with Constant D . 131 VI Table XV Page FreeVolume Equation Parameters For PVACToluene at 60 °C . . . 144 VB Figure 1 LIST OF FIGURES A Typical Reduced Sorption Curve Obtained from a StepChange Sorption Experiment . . . . . . . . . . . . . . . Page 6 2 3 4 5 Schematic Diagram of the Quartz Crystal Microbalance Frequency Response of a Coated Quartz Crystal to a Step Change in Chamber Pressure . . . . . , . . . . . . . . . , Sorption Curve Obtained from Frequency Measurements Calculation of Diffusion Coefficients Using the Initial Slope Method . 9 14 16 18 6 Calculation of Diffusion Coefficients Using the Limiting Slope Method 20 7 Calculation of Diffusion Coefficients Using the Moment Method 21 8 Solubility Coefficient, (J", of Propane in Natural Rubber Filled with ZnO Vs Volume Fraction Filler. Curve (a),' (J" = O'pvp + (J"JVJ with (J"J = 0.0305; Curve (b), (]' = O"pVp , O'J = o. . .. , . . . . .. 37 9 Schematic Diagram of the Quartz Crystal Microbalance Modified Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 46 10 Sorption Curve Before the Sample Cylinder was Installed 47 11 Sorption Curve After the Sample Cylinder was Installed . 49 12 Effective Diffusion Coefficients of Toluene in CaCOa Filled PVAC at 60°C. . . . .. 61 13 Effective Diffusion Coefficients of Toluene in CaCOa Filled PVAC at 80°C. 62 14 Solubility Data of Toluene in CaCOa Filled PVAC at 60°C, 63 vlll Figure 15 Solubility Data of Toluene in CaC03 Filled PVAC at 80°C. Page . 64 16 Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 60°C.. '. . . . . . . . . . . . . . . . . . . . . . . . .. 65 17 Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 80°C . 18 Diffusion Coefficients of Toluene in Neat PVAC at 60°C 19 Diffusion Coefficients of Toluene in Neat PVAC at 80°C 66 68 69 20 An Example of a Sorption Curve that was Rejected Due to Very Few Data in the Initial Stages of Diffusion 73 21 An Example of a Sorption Curve that was Rejected Due to a Hump in the Curve . . . . . . . . . . . . . . . . . . . . . . . 75 22 Frequency Curve for PolybutadieneEthylbenzene System at 80°C 78 23 Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C . . . . . . . . . . . . . . . . . . .. 80 24 Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 60°C for Various Toluene Weight Fractions . . . . . . . . . . . . . . . . . . . . . . . .. 83 25 Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 80°C for Various Toluene Weight Fractions . . . . . . . . . . . . . . . . . . . . . . . .. 84 26 Comparison of Experimental Solubility Data with the Flory Huggins Equation for PVACToluene at 60°C. . . . . . . . . . . . .. 86 27 Comparison of Experimental Solubility Data with the Flory Huggins Equation for PVACToluene at 80°C. . . . . . . . . . . . .. 87 28 Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for Toluene Diffusing into Neat PVAC (vJ = 0) at 60°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 29 Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for CaC03 Filled PVAC with a Weight of 10% CaC03 (vJ = 0.0428) at 60°C .. . . . . . . . . . . . . . . .. 90 lX Figure Page 30 Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 60°C . . . . 91 31 32 33 Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 80°C ... Comparisons of Experimental Diffusion Coefficients with a Simple Structure Factor Equation for CaC03 Filled PVAC at 60°C . Comparisons of Experimental Diffusion Coefficients with Barrer and Chio's Equation for CaC03 Filled PVAC at 60°C . . 92 96 97 34 Comparisons of Experimental Diffusion Coefficients with an Equation Derived from Ileaction Principles for CaC03 Filled PVAC at 60°C . 99 35 Comparison of Numerical and Analytical Solutions of a Diffusion Equation with Constant Diffusion Coefficient 132 36 Comparison of Numerical and Analytical Solutions for the Fractional Mass Uptake with Constant Diffusion Coefficient 133 37 Comparison of Diffusion Coefficients Calculated from a Moment Method Evaluation of Sorption Data with Diffusion Coefficients Calculated from a Constitutive Equation 135 38 Comparison of Simulated Sorption Data with Constant Diffusivity, D = 1 X 108 , to Mass Uptake Ratios with the Diffusivity Changed to D = 5 X 108 ...•................• 137 39 Regression of Simulated Sorption Data with Constant Diffusivity, D = 1 X 108 138 40 Regression of Simulated Sorption Data with a Simple Concentration Dependent Diffusivity, D = a +be 139 41 Regression of Experimental Sorption Data for Neat PVACToluene at 60°C Using a FreeVolume Equation Theory Constitutive Equation 141 42 Comparison of Experimental Diffusion Coefficients with a Diffusion Coefficient Curve Calculated from the FreeVolume Equation for Neat PVAC at 60°C 142 x ~~ p~ 43 Comparison of Experimental Sorption Curve with a Simulated Sorption Curve Using the FreeVolume Constitutive Equation for Neat PVAC at 60 °C 143 Xl CHAPTER I INTRODUCTION Heterogeneous materials such as filled polymers are used extensively in industrial processes and commercial applications. Fillers are important constituents of many elastomers, plastics and coatings. They come in the form of small particles of different shapes. Fillers may be crystalline or amorphous. They have a significant influence upon the properties of materials and can be used to increase the mechanical durability, improve elastic properties, or give certain color or optical properties to the polymeric materials. Fillers also modify the sorption and permeability to diffusants. Diffusion and transport of matter in filled polymers are important in processes such as the drying of coatings and adhesives; air or moisture permeability of paint films; and the use of membranes in separations processes. Of particular interest is the process that led to this study, viz., drying of polymer based coatings on paper in which a filled polymer solution is coated on a continually moving web of paper. The web passes through a dryer where the solvents are removed from the surface of the coating with the help of nozzles blowing air at a controlled velocity and temperature. Solvents move to the surface of the coating by molecular diffusion, which is often the rate controlling step of solvent removal. Controlling the drying process is important in order to obtain a coating free of defects. Appropriate modeling of the drying process facilitates the selection of optimum operating conditions to produce a dry coating without defects. Drying models are benificial in finding drying conditions for new products, in productivity improvements in which line speed increases, and in tlle design of new drying equipment. Models [Vrentas et al., 1994] have been developed to describe the drying 1 2 process of coatings on continuous webs. An important parameter in drying models is the diffusion coefficient for solvent diffusing through the coated film. The diffusion coefficient is a function of the solvent concentration in the film, the amount of filler in the film, and the temperature of the film and surrounding air. Mass transfer in filled polymers is a complicated process which involves not only diffusion in the continuous polymer phase. Other factors such as adsorption of penetrant on the surface of fillers and small gaps in the structure, which are filled by diffusing gases, affect the overall transfer of penetrant in a filled polymer. An application of the classical solution of Fick's diffusion equation to sorption data for filled polymers leads to an "effective" diffusion coefficient which in most cases is smaller than the true diffusivity in the polymer phase due to immobilization of gases from the presence of fillers. This subject is discussed in the next chapter. The objective of this work was to measure sorption data on a filled polymer using a piezoelectric quartz crystal microbalance and to evaluate effective diffusion coefficients from the data. The diffusion coefficient was evaluated for various solvent concentrations, filler contents and temperatures. Acrylic polymers are used for most paper coating applications, and typical solvents are toluene or water. The polymer chosen for this study was poly(vinyJ acetate) (PVAC) and the solvent was toluene. This choice showed similarities in properties with a typical coating for paper, and the results for neat PVAC could be compared with experimental results of previous researchers in this field [Mikkilineni et al., 1995]. Coatings utilize a variety of fillers, additives, and optical brighteners. Calcium carbonate is a typical filler and it was chosen for use in this work. The results of this work were compared with known models for mass transfer in filled polymers. CHAPTER II BACKGROUND ON DIFFUSION OF PENETRANTS IN POLYMERS The first section of this chapter contains a discussion of experimental methods for obtaining sorption data, with particular emphasis on the quartz crystal microbalance method used in this study. The next section contains a discussion of methods of analysis for the quartz crystal microbalance in order to obtain a diffusion coefficient. This is followed by several sections which contain discussions of various behaviors inherent to diffusion in polymers. Major topics. discussed are Fickian and anomalous diffusion, the free volume theory of polymersolvent diffusion, and mass transfer in heterogeneous materials. Experimental Methods for Obtaining Diffusion Coefficients Experimental methods for studying the diffusion of gases in polymers have been discussed by Felder and Huvard [1980}. Their historical overview dates back almost 170 years and contains discussions of many diffusion phenomena and methods. Another source, which contains more detailed demonstrations of the theory, is the work of Crank and Park [1968}. This section focuses on methods of greatest relevance to this work. Fundamental Diffusion Equations Diffusion of a gas in any material, rubber or plastic, amorphous or crystalline, neat or filled, involves transport of the gas from one part of the material to another. Just as the viscosity is a coefficient that describes the transport of momentum in a gas or liquid in the familiar Newton's law of viscosity, and the 3 4 heat transfer coefficient describes the transport of thermaJ energy in a materiaJ according to Fourier's law of heat conduction, the diffusion coefficient describes the transport of mass in a material in accordance with Fick's law of diffusion, J = _DoC ox' (1 ) where J is the rate of transfer of diffusing gas per unit area of a specified sectional area, C is the concentration of diffusing substance, x is the space coordinate measured normal to the section, and D is the diffusion coefficient, or diffusivity. This equation is referred to as Fick's first law of diffusion. The diffusion coefficient has dimensions of length2 time1 and in this work, has units cm2 secI. From a mass balance on an element of volume containing material, the differential equation of diffusion takes the form oaCt = ~ox (DOoxC) +~oy (DOoyC) +~oz (DOoCz ) , where D can be a function of C. If D is constant then equation 2 becomes (2) (3) If the diffusion is in one direction r equations 2 and 3 simplify to what is commonly referred to as Fick's second law of diffusion, (4) and, if the diffusion coefficient is constant, (5) The equations for onedimensional diffusion are valid for this work since thin polymer films were used. General Sorption Methods for Studying Diffusion In general, there are three experimental techniques from which diffusion coefficients are obtained. These are: sorption into or out of a polymer, permeation 5 through a membrane into a closed chamber, and permeation through a membrane into a flowing stream. Only sorption is discussed in detail in this thesis since this study used a sorption method. Sorption methods are mainly used in measurements of equilibrium solubil· ities, twostage sorption processes, slow processes, studies of anomalous diffusion, highpressure measurements and studies of cracking and crazing [Felder and Buvard, 1980]. Categories of sorption measurements include integral sorption, integral desorption, interval sorption, and interval desorption. In integral sorption, the polymer is abruptly exposed to a step change in penetrant concentration at the boundaries of the polymer. In integral desorption, the polymer is initially equilibrated with a penetrant, and abruptly exposed to a penetrantfree atmosphere, usually a vacuum, until no penetrant remains in the polymer. Interval sorption and desorption are the same as integral sorption and desorption, respectively, except that the initial and final penetrant concentrations are' greater than zero. In all four categories of sorption measurements, the mass of the penetrant in the polymer is measured as a function of time, either directly, as in gravimetric techniques, or indirectly by measuring a change in properties such as pressure or volume of the atmosphere surrounding the polymer. The method used in this work involved determining the mass of the penetrant in the polymer indirectly by measuring a change in the frequency of a piezoelectric quartz crystal as a function of time.. For both direct and indirect methods, the data usually appear as shown in Figure 1. The variable, MtiMoo ! is plotted on the ordinate, where Mt is the instantaneous mass sorbed or desorbed and Moo is the mass sorbed or desorbed at equilibrium. Time, t or ,fi is plotted on the absisca. In the integral sorption method, Moo is the ultimate final mass of the penetrant in the polymer. In the integral desorption method, Moo is the initial mass of the penetrant in the polymer. In the interval sorption or desorption method, Moo is the difference between the initial and the final mass of the penetrant in the polymer. 6 1.0 ,=_ 0.8 0.6 0.4 0.2 0.0 Figure 1. A Typical Reduced Sorption Curve Obtained from a StepChange Sorption Experiment. 7 If a sorption experiment was followed by a desorption experiment to the initial penetrant level, one might expect that the sorption and desorption curves would coincide. However, this is only true for Fickian diffusion with a constant diffusion coefficient, D. If D increases with the penetrant concentration, such as toluene in PVAC as is used in this work, the sorption curve lies above the desorption curve. If D decreases with the penetrant concentration, the sorption curve lies below the desorption curve. If swelling is significant and stress relaxation of the polymer controls the penetration rate, the sorption curve is sigmoidal, but desorption from the swollen polymer is Fickian and initially relatively rapid. If relaxation occurs slowly, two stage sorption occurs. Fickian and NonFickian diffusion are discussed in a later section. The Quartz Crystal Microbalance Sorption Apparatus All data in this work were' obtained with a quartz crystal microbalance (QCM). The QCM used in this work was developed by Deshpande [1993] to measure the diffusion characteristics of a penetrant in a polymer film. The QCM was later modified by Mikkilineni to reduce operational difficulties, increase the accuracy of the data, and increase the ease of operation of the apparatus [Mikkilineni, 1995]. This subsection discusses the QCM apparatus as it existed in the work of Mikkilineni [1995]. The principal component of the QCM was a 0.5 inch circular gold coated piezoelectric quartz crystal. A mechanical stress, when applied to a piezoelectric quartz crystal, induces an electric potential across the crystal, or vice versa, an electrical potential applied across a piezoelectric quartz crystal induces a mechanical strain in the crystal. This property of the quartz crystal facilitated extremely accurate measurements of the rate of diffusion of solvents in polymers. By including the quartz crystal in a circuit, a small electric potential was created across the crystal and the crystal vibrated at its resonant frequency. 8 A 6 MHz gold coated ATcut piezoelectric quartz crystal was used in the experiments. The crystal was coated with a polymer solution. A penetrant diffusing into this polymer coating decreased the resonant frequency of the crystal. The resonant frequency of the piezoelectric quartz crystal was found to be extremely sensitive to changes in mass on the surface of the crystal. A load as low as 1 nanogram could cause a detectable change in the frequency of the quartz crystal [Mikkilineni, Tree, and High, 1995]. Since the quartz crystal was so sensitive to mass changes, a small change in the concentration of the solvent in the polymer caused a measurable change in the frequency, which allowed diffusion data to be obtained at low and closely spaced solvent concentrations. The bulk of the QCM was constructed from a stainless steel six way cross chamber (Kurt J. Lesker Co., C60600) that had entrances at all six ports. Figure 2 shows a schematic diagram of the QCM. The chamber was operable from 1011 torr to slight positive pressures and up to a maximum temperature of 500°C [Mikkilineni, 1995]. A standard quartz crystal sensor (Leybold Inficon, 750207Gl) which housed and provided a circuit for the quartz crystal was placed in the right port of the chamber (see Figure 2). A thinfilmdeposition controller (Leybold Inficon, XTC/2), interfaced to a computer, monitored the frequency of the crystal. A computer program recorded frequency as a function of time as the penetrant diffused into the polymer. The QCM was enclosed in a box made of plywood lined with insulating material to accurately control the chamber and polymer film temperature. A heating element mounted on a metallic stand was used to heat the box. A fan was used to circulate the hot air in the box. The fan motor was mounted on the outside of the box to prevent overheating of the motor due to the high operating temperature in the box [Mikkilineni, 1995]. The motor was connected to the fan inside the box by an aluminum shaft and was mounted onto one side of the box. A temperature controller (LFE Instruments, 2004) was used to regulate the temperature inside the box. A resistance temperature device was also mounted in the back port of the chamber (not shown in Figure 2) to measure the temperature 9 To Computer! Interface . \ Solvent Flask Needle Valve ~ RightAngle Valve ~ Insulated Box Six Way Cross ~ Bottom Port~ Liquid Nitrogen Trap Hea~ CO;I~ Figure 2. Schematic Diagram of the Quartz Crystal Microbalance. 10 inside the chamber. Several resistance temperature devices were installed in the box and surface mounted onto the cross chamber to check for temperature gradients in the box and chamber [Mikkilineni, 1995]. The chamber was evacuated to remove any contaminants, such as air or penetrant from previous experiments. The bottom port of the chamber led to two vacuum pumps. A rotary vane pump (Leybold Inficon, Trivac A 4A) was used to obtain pressures of the order of 102 torr, and a turbomolecular pump (Leybold Inficon, Turbovac 50) was used to reduce pressures further, to the order of 107 torr. A liquid nitrogen trap (Kurt J. Lesker Co., LNF 1000) was used in line to keep solvent vapors from entering the pumps and oil vapors from entering the chamber. A gate valve (Kurt J. Lesker Co., SG0400) was connected to the turbomolecular pump and a right angle valve (Huntington Laboratories, EV 150) was connected to the rotary vane pump through the liquid nitrogen trap'. The two valves were required to connect the chamber to the rotary vane pump or to the turbomolecular pump, so that vapors would not enter the turbomolecular pump during initial evacuation of the chamber with the rotary vane pump. To prevent the turbomolecular pump from overheating, cold water was circulated through a heat exchanger mounted on the pump casing. A Pirani gauge (Leybold Inficon, PG3; sensor, TR901) and a hotcathode ionization gauge (Leybold Inficon, IG3; sensor, 850675G5) were connected on the outside of the box to the pipe which connected the chamber with the turbomolecular pump. The gauges measured pressure in the chamber and were used only during the evacuation process. The Pirani gauge measured low vacuum (,....., 1000 _104 torr absolute) and the hotcathode ionization gauge measured high vacuum ("" 102  1010 torr absolute). A pressure gauge (Druck Instruments, DPI 265) was also mounted in the top port of the chamber to measure pressure in the chamber after the penetrant was introduced. This gauge can be used to measure pressures of solvent vapors in the chamber over the range of 0  19.999 psia and can withstand temperatures up to 300°C [Mikkilineni, 1995]. 11 The front port of the chamber was used to introduce solvent into the chamber. A jacketed metallic flask containing the penetrant was connected to the chamber using 1/4 in. stainless tubing. Water could be circulated through the flask to heat or cool the solvent to a desired temperature. A toggle valve was used to control the flow of penetrant into the chamber. A needle valve was used to isolate the flask. A quick connect coupling was used in the tubing connecting the solvent flask and the chamber to make solvent change in the flask easy [Mikkilineni, 1995]. A magnetic stirrer was used to ensure that the solvent was weD mixed throughout the experiment. Further modifications, made to the QCM for this work, are discussed in the next chapter. A detailed, stepwise experimental procedure is given in Appendix A. Methods of Analyzing Data Obtained Using the Quartz Crystal Microbalance Diffusion coefficients for polymerpenetrant systems are typically highly concentration dependent. However, all of the methods discussed in the literature assume that the diffusion coefficients are concentration independent. In order to calculate the concentration depen'dence, the sorption experiments are performed over small step changes in pressure driving the sorption.' Over these small changes in driving force, the concentrations do not change significantly and the methods employing constant diffusion coefficients are valid. Crank [1956] derived solutions for onedimensional diffusion in a plane sheet. These solutions apply to diffusion into a sheet of polymer assuming that the edges of the sheet can be ignored and diffusion can be considered as taking place only through the surface of the sheet. Since the polymer films used in this work were about 3 to 5 pm in thickness and about 5 to 8 mmin diameter, this assumption is valid. The derivation starts with the onedimensional diffusion equation (6) 12 The initial and boundary conditions for this equation are C(x,O) = Gl , (7) C(L, t) = Co, (8) oC (0 t) = O. (9) ox The initial condition indicates that at t = 0 the entire polymer film is at a uniform concentration of C1 . The first boundary condition indicates that at any time after the beginning of diffusion into the polymer film, the concentration at the surface of the film, x = L, is maintained constant at Co. This boundary condition emphasizes the importance of introducing the penetrant into the quartz crystal microbalance chamber instantaneously and, during the sorption experiment, maintaining a constant concentration at the surface of the polymer film. This requires the pressure in the chamber to be constant. Since the QCM is mainly used for step change absorption experiments, Co is greater than Cl . The second boundary condition indicates that there is no transport of penetrant through the bottom of the polymer film at x = O. The exact solution of equation 6 subject to equations 7, 8 and 9 is given by Crank [1956] as C  C1 ~( )n f (2n + I)L  x ~(  = L 1 er c +L  1)"erfc (2n + I)L + x . Go  C1 n=O 2..j(Dt) n=O 2vfl5ij (10) (11) The concentration of penetrant in the polymer film according to equation 10 is a function of time and position inside the film. The mass of penetrant in the polymer film at a specified time could be determined by integrating the concentration profile of equation 10 over the thickness of the slab at that time. From this procedure, the ratio of the mass sorbed by the polymer film at time t to the mass sorbed by the film at equilibrium (t = 00) is [Crank, 1956] Mt = _~~ 1 ex {D(2n+l f ll. 2t}. Moo 1 7r2 L (2n + 1)2 P 4L2 n=O The corresponding solution that is useful for small times is [Crank, 1956] M (Dt)1/2 { 00 nL } _t = 2 2 7rl / 2 + 22":( 1)"ierfc /(15t) . Moo L n=l (Dt) (12) 13 The important assumptions made in the derivation of equations 11 and 12 are that 1. Diffusion of penetrant into the polymer film occurs in one dimension. 2. Concentration of the penetrant at the surface of the film is constant. 3. The diffusion coefficient is a constant. The first assumption holds well since the polymer films used in this work had a small thickness compared to the surface area coated on the quartz crystal. The thicknesses of the films were about 3 to 5 p.m, while the diameters of the films were about 5 to 8 rom. The second assumption holds well in this work since the pressure of the penetrant in the chamber was increased quickly and leaks in the chamber were small. Since diffusion of a penetrant into a polymer is dependent on the concentration of penetrant in the film, the third assumption can only be satisfied approximately if the sorption experiment is carried out over a small step of change in concentration. This was achieved by introducing small amounts of penetrant into the chamber for a sorption run and performing a series of these small step change absorption experiments over the penetrant concentration range of interest. Obtaining a Sorption Curve from Frequency Data As the penetrant diffuses into the polymer fiim, a computer records the frequency as a function of time. Figure 3 shows a plot of a typical response of crystal frequency to a step change in pressure of the penetrant in the chamber where the polymer coated quartz crystal is housed. From the frequency responce data, graphs of MtiMoo versus t or yfi are generated. Theoretical details of piezoelectric quartz crystals are described in Mikkilineni's thesis [1995]. The essence of the theory is that the change of mass, 6.m, on the quartz crystal is proportional to the frequency change, .6.f, of the crystal, .6.m = k6.f, (13) 14 5967240 .____, 5967220 5967200 .. 5967180 N ::r: '" ;>.. u 5967160 s= C1.) ;:s 0"' C1.) e.t:: 5967140 5967120 5967100 5967080 +r,r,l o 10 20 30 40 50 Figure 3. Frequency Response of a Coated Quartz Crystal to a Step Change in Chamber Pressure. 15 where k is a proportionality constant. The mass change of penetrant in the polymer film at time, t, after the step change in penetrant pressure, is Mt = k(ft  il), (14) where 11 is the frequency of the coated crystal before the penetrant is allowed into the chamber, and It is the frequency of the coated crystal at time t after the penetrant is allowed into the chamber. If only the first step of the sorption experiment is considered, then 11 is the frequency of the polymer coated, solventfree crystal and Mt is the total mass of penetrant in the polymer at time t. If the nth step is considered (n > 1) 1 then II is the frequency of the polymer coated crystal before the step change and Mt is the change in mass of the penetrant in the polymer from the onset of the step change to time t after the step change in chamber pressure. After sufficiently large time (I'V 15 minutes for unfilled polymer and up to 6 hours for filled polymer), the penetrant comes to equilibrium with the polymer film and the mass of penetrant in the film approaches a constant value. Polymersolvent equilibrium is established, and equation 14 is still valid as t + 00. The ratio of the mass sorbed at time t, Mt , to the mass sorbed at equilibrium, Moo, can be expressed as Mt It  II  =, (15) Moo 100  il where I (XI is the frequency of the coated crystal at polymersolvent equilibrium. Figure 4 shows a plot of Mt/Moo obtained from equation 15 versus .,ft. Such a plot is called a reduced sorption curve. Once the Mt/Moo data are obtained, the diffusion coefficients can be estimated. Evaluating Diffusion Coefficients Using the HalfTime Method The simplest technique for estimating the diffusion coefficient is the halftime method. Equation 11 is used, and the value of tf £7. for which Mt/Moo = 1/2 is approximately given as (16) 16 , 1.0 0.9 0.8 0.7 0.6 B ~ 0.5 ~ 0.4 0.3 0.2 0.1 0.0 0 10 20 30 40 50 ..Jt (s) Figure 4. Sorption Curve Obtained from Frequency Measurements. 17 with an error of about 0.001 per cent [Crank, 1956]. After performing the necessary arithmetic, the time t 1/ 2 for which MtiMoo equals 1/2 for a plane sheet is given by D = 0.19675L 2 (17) t 1/ 2 Evaluating Diffusion Coefficients Using the Initial Slope Method The initial slope method was initially used for evaluating the diffusion coefficients from data taken on the quartz crystal microbalance at Oklahoma State University. All the sorption curves obtained by Deshpande [1993] and Mikkilineni [1995] were analyzed with the initial slope method. During the initial stages of the sorption experiment, the polymer film behaves as a semiinfinite medium and the ratio of the mass uptake at time t to the mass uptake at time t = 00 increases linearly with 0, often to as much as 50 per cent of MtiMoo [Crank and Park, 1968]. At small times, the summation term in equation 12 can be neglected, and the equation becomes Mt 2 (Dt)1/2 Moo = ft L2 (18) According to equation 18, the average diffusion coefficient can be calculated as (19) where Ri is the initial slope of the MtlMoo vs. 0 curve. If the sorption curve is approximately linear up to MtiMoo = 1/2, equation 17 and equation 19 would yield about the same diffusion coefficient [Crank, 1956]. Figure 5 shows an exam· pIe of an evaluation of the diffusion coefficient by the initial slope method. The fractional solvent uptake was plotted as a function of the square root of time. Linear regression was used to determine the slope of the initial portion of the sorption, from MdMoo = 0 to 0.5. Equation 19 was used to calculate the diffusion coefficient. The sorption curve in this example is linear well above 50 per cent of MtiMoo. Most of the experimental sorption curves obtained in this work were linear to approximately 25 per cent. The length of the initial linear portion of the sorption curve will vary between sorption experiments depending on the concentration dependence of the diffusion coefficient [Crank and Park, 19681· 18 1.0 0.9 0.8 0.7 0.6 Ri=~(M/McxY~(tl/2) i 0.5 ~(tll2) ~ 0.4 0.3 ~(M/Mco) 0.2 0.1 0.0 0 5 10 15 20 25 30 35 40 .../t (8) Figure 5. Calculation of Diffusion Coefficients Using the Initial Slope Method. 19 Evaluating Diffusion Coefficients Using the Limiting Slope Method The limiting slope method [Balik, 1996; Palekar, 1995] uses only the first term of the series in equation 11. At large times, a plot of In {I  Mt/Moo } versus time approaches a straight line, and the average diffusion coefficient can be calculated as (20) R1 is the limiting slope of the In {I  Mt/Moo } vs. t curve. Figure 6 shows an example of an evaluation of the diffusion coefficient by the limiting slope method. In {I  MtiMoo} was plotted as a function of time. Linear regression was used to determine the limiting slope, from MtiMoo = 0.99 to 0.996. Equation 20 was used to calculate the diffusion coefficient. Evaluating Diffusion Coefficients Using the Moment Method In this work, the moment method was used to evaluate the diffusion coefficients. The moment method [Felder and Huvard, 1980; Palekar, 1995J is advantageous since the entire sorption curve is used instead of just the initial or final portion as in the methods discussed above. The quantity (21 ) (22) is first calculated by numerical integration, where T, is the first moment of the monotonically increasing curve, MtiMoo versus time. Then, the average diffusion coefficient is calculated by L2 D=. 3Ts Details of the derivation of equation 22 are given in Appendix B. Figure 7 shows an example of the evaluation of the diffusion coefficient by the moment method. To calculate Ts , a plot was made of 1  :::~ vs t as shown in Figure 7. The first moment, Ts , is the area under the curve of Figure 7 and was found by numerical integration. The trapezoidal rule was used in this work to find the area under the curve. 20 0...._. 1 2 ~ 8~ 3 ....I.. ..~.... 4 5 6 o 200 400 t (8) 600 800 1000 Figure 6. Calculation of Diffusion Coefficients Using the Limiting Slope Method. 21 1.0 0.9 0.8 •• 0.7 •••• ' . 0.6 ••• 8 • ~ • 0.5 •• ~ •• ..I •• 0.4 •• . . ••• 0.3 \ 0.2 0.1 0.0 0 200 400 600 800 1000 1200 1400 t (s) Figure 7. Calculation of Diffusion Coefficients Using the Moment Method. 22 Evaluating the Weight Fraction of Penetrant in the Polymer Film Since the diffusion coefficient is a function of the concentration of the penetrant in the polymer film, equations 17, 19, 20 and 22 give average diffusion coefficients over the concentration range. Vrentas et al. [1977] has concluded that the average diffusion coefficient obtained from a stepchange sorption experiment is equal to the diffusion coefficient at a concentration which is 0.7 of the way across the concentration interval, with less than 5% error. (See Equation 25.) The equilibrium weight fraction of the penetrant in the polymer film, w~q , is the ratio of the mass of the penetrant at equilibrium, Moo, to the total mass of the polymersolvent mixture at equilibrium, M~t, eq Moo ) WI = Mtot' (23 00 Since, according to equation 13, the change of mass on a quartz crystal is proportional to the change in crystal frequency, equation 23 becomes eq = 100  iI (24) WI 100  fo' where fo is the frequency of the bare, uncoated crystal. The weight fraction which corresponds to the average diffusivity can be calculated by (25) where Wli is the weight fraction of penetrant before introducing more penetrant, and WI! is the weight fraction of penetrant at equilibrium after introducing the penetrant. Diffusion of Organic Penetrants into Polymers at Temperatures above the Glass Transition Temperature, Tg Diffusion in polymerpenetrant systems do not follow the laws of the classical theory of molecular diffusion [Crank and Park, 1968]. There has been considerable interest in understanding the anomalous behavior of diffusion in polymerpenetrant systems [Vrentas et aL, 1986]. However, it is also of interest to discover 23 the circumstances under which polymerpenetrant systems can be analyzed using classical or Fickian theory. Generally, Fickian theory can be used to describe diffusion in polymerpenetrant systems which have high temperatures and concentrations, since the polymerpenetrant system behaves as a purely viscous fluid under these conditions. Diffusion at low concentrations and temperatures below the glass transition temperature can also be analyzed with the classical theory since the system has properties of an elastic solid [Vrentas et al., 1986]. At intermediate temperatures and concentrations, diffusion in the polymerpenetrant system can have anomalous or nonFickian behavior which are caused by viscoelastic effects [Vrentas et aI., 1986]. This section discusses the mechanisms of diffusion of penetrants into polymers at temperatures above the glass transition temperature, and discusses typical Fickian sorption features. Diffusion of simple gases such as hydrogen, argon, nitrogen and carbon dioxide requires a limited rotational oscillation of only one or two monomer units in order to translate from one position to a neighboring one since the molecular size of such gases is small compared to the monomer unit of a polymer, however if a gas has a molecular size comparable or larger than the monomer unit of the polymer \ a cooperative movement by the Brownian motion of several monomer units, i.e. a polymer segment, must take place during diffusion [Fujita, 1968]. Organic vapors, such as toluene which was used in this work, are among such large molecular size penetrants. The diffusion of organic vapors in polymers exhibit different features in the regions above and below the glass transition temperature, Tg , of a polymer. These features are simple at temperatures above Tg and complex at temperatures below Tg [Fujita, 1968]. At temperatures well above the glass transition temperature, polymers are in a rubbery state and diffusion of penetrants in polymers usually follow Fick's law. The presence of the penetrant weakens the molecular interaction between adjoining chains, increasing the magnitude of Brownian motion within the polymer and therefore increases the rate of penetrant diffusion [Felder and Huvard, 1980]. This effect increases with the amount of penetrant present, so the diffusion 24 coefficient of organic vapors in rubbery polymers usually exhibits a concentration dependence [Felder and Huvard, 1980}. In Fickian diffusion in a polymer film during sorption, the distribution of penetrant, and the change of penetrant concentration with time, are governed by Fick's onedimensional, differential equation for diffusion. The space coordinate is taken in the direction normal to the polymer film. The solution of this equation depends on the initial and boundary conditions for the penetrant concentration, C1 and on how the diffusion coefficient, D, varies with C. Solutions for constant D were shown earlier. Solutions for a concentration dependent D are given by Crank [1956]. Mathematical studies of Crank and coworkers have developed the following summary of sorption features which are Fickian or normal type [Fujita, 1968]. 1. Both absorption and desorption curves are linear in the initial stage. For absorption, the linear region extends over 60% or more of Moo, where Moo is the amount of vapor absorbed per gram of dry polymer until the sorption equilibrium is reached. For D(C) increasing with C the absorption curve is linear almost up to the final sorption equilibrium. 2. Above the linear portions both absorption and desorption curves are concave to the abscissa axis, irrespective of the form of D(C). 3. When the initial concentration Ci and the final concentration Cf are fixed, a series of absorption curves for films of different thicknesses are superposable to a single curve if each curve is replotted in the form of a reduced curve, i.e. Mt is plotted against t 1/ 2/ L. This same applies to the corresponding series of desorption curves. 4. The reduced absorption curve so obtained always lies above the corresponding reduced desorption curve if D is an increasing function of C between Ci and Cf. Both reduced curves coincide over the entire range of t when D is 25 constant in this concentration interval. The divergence of the two curves becomes more marked as D increases more strongly with C in the concentration range considered. 5. For absorptions from a fixed Cj to different O/s, the initial slope of the reduced curve becomes larger as the concentration increment Cf  Cj becomes larger, provided that D increases monotonically with 0 in the range considered. This same applies to the reduced desorption curves which start from different Ci's to a fixed Cf. Criteria 1, 2 and 3 are independent of how D varies with C. Therefore, these three criteria are checked to determine whether a given polymerpenetrant system exhibits Fickian diffusion. Criteria 1 and 2 can be checked easily by inspecting the appearance of the sorption curves. Since criterion 3 requires sorption measurements to be made with films of different thicknesses, a system is often regarded as Fickian if experimentally determined sorption curves have appearances consistent with criteria 1 and 2 [Fujita, 1968]. Free Volume Theory A theory that has been well accepted for describing the temperature and concentration dependence of the mutual diffusion coefficient in polymersolvent systems is the freevolume theory of transport developed by Vrentas and Duda [Duda et aI., 1982]. Vrentas and Duda [Vrentas and Duda, 1976; Vrentas et al., 1985] proposed the following equation for the solvent selfdiffusion coefficient, D1 , in a polymersolvent system, D D ( E ) ("Y(WIVt + W2e~·)) 1 = 0 exp  RT exp   , VFH (26) (27) (28) 26 In equations 26 and 27, Do is a constant preexponential factor, E is the energy per mole of molecules to overcome intermolecular attractive forces, R is the ideal gas constant, T is the temperature, WI is the mass fraction of component 1 (I = 1 for solvent; 1 = 2 for polymer), Vt is the specific critical hole free volume of component 1 required for a jump, ~ is the ratio of the critical molar volume of the solvent jumping unit to the critical molar volume of the polymer jumping unit, Kll and K 21 are freevolume parameters for the solvent, K l2 and K 22 are freevolume parameters for the polymer, I is an overlap factor which is introduced because the same free volume is available to more than one molecule, VFH is the average hole free volume per gram of mixture, and TgI is the glass transition temperature of component 1. Since this theory presents an expression for the selfdiffusion coefficient, Db expressing the mutual diffusjon coefficient, D (measured in the sorption experimental apparatus, and required for process calculatIons), in terms of D1 is desirable. Duda et al. [1979] proposed an approximation for low solvent concentrations which couples D to the selfdiffusion coefficient for polymer solvent systems, D = D1W IW2 (8J1.1) . RT 8WI T,p In this equation, J1.1 is the chemical potential of the solvent. The FloryHuggins theory was used to determine the concentration dependence of the solvent chemical potential [Duda et aI., 1982]. The FloryHuggins equation can be written as (29) where ¢h is the solvent volume fraction in the solution and X is the polymersolvent interaction parameter, which is assumed to be independent of temperature. Introduction of equations 26, 27, and 29 into equation 28 yields the following expression for the mutual diffusion coefficient, D, in a rubbery polymersolvent system [Duda et al., 1982] (30) (34) (35) 27 D D ( E ) (,(WIVt +W2eo;*)) 1 = 0 exp  RT exp  .. (31 ) VFH .. VFH = Kn K12 wI(K2I + T  TgI ) + w2(K22 + T  Tg2 ). (32) , I I A.. I = ..WI t:;.o .. (33) 'fI TEO TEO' WI vI +W2 V2 where Vp is the specific volume of pure component l. In this version of the free volume theory, there are 13 independent parameters to be evaluated. Grouping some of them together leaves only 10 parameters which are required to determine the mutual diffusion coefficient, D: Kul'l K21  TgI, Kl2 /" Kn  Tg2 , t:;.*, O;*l Do, E, eand x· The parameters ,t:;.*/K u and K 21  Tg1 can be determined from data for' solvent viscosity as a function of temperature, "11 (T) 1 by using a nonlinear regression to correlate the viscosity data with b~* /Kll ) In 171 = In Al + (K T) T ' 21  gl + where the parameters, AI, , ~* / Ku and K2I  TgI are assumed to be independent of temperature [Duda, 1983]. The parameters IV;*IK12 and K22  Tg2 can be determined from polymer melt viscosity data as a function of temperature, T/2(T), by using a nonlinear regression to correlate the data with [Duda, 1983] hV2*jK I2 ) In "12 =In A2 + (K T) T 22  g2 + The critical volumes, Vt and V:t, can be estimated to be the specific volumes of the solvent and polymer at 0 K. Molar volumes at aK can be estimated using group contribution methods developed by Sugden and Biltz [Duda, 19831· The parameter X can be determined from solubility data in which the equilibrium weight fraction of the penetrant in the polymer is known as a function of the solvent vapor pressure, PI, using the FloryHuggins equation [Duda, 1983], (36) 28 where p~ is the penetrant saturation vapor pressure. Lumping the parameters Do and E into the parameter DOl gives DOl = Doexp (.ff:r) , (37) which is the Arrhenius form of temperature dependence. The parameters DOl and ~ can be.determined by correlating diffusion coefficient data with a nonlinear regression analysis of equation 30 [Duda, 1983]. Anomalous Diffusion of Penetrants into Polymers Deviations from the Fickian type of process are generally described as anomalous or "nonFickian" and are almost always observed when the polymer is studied at temperatures below Tg , or within roughly lOoe above Tg [Felder and Huvard, 1980]. In general, the' polymers in which anomalies are observed are hard and glassy while normal sorption is observed in soft and rubbery materials. Polymeric materials in the rubbery state respond rapidly to changes in condition, but polymers in the glassy state take longer to come to equilibrium. For example, as the concentration in a polymer solution changes with time, the system must adjust to new conformations consistent with new values of concentration. Anomalous mass transfer processes are associated with the sluggish relaxation of large polymer molecules; however, in rubbery polymers which follow a Fickian diffusion process, relaxation of polymer molecules is fast compared to the diffusion process [Vrentas and Duda, 1979]. Fickian diffusion at temperatures above the glass transition, Tg , has been designated as case I diffusion and D depends only on concentration. As the glass transition temperature, Tg , of the polymer is approached, D begins to depend on time explicitly as well as on concentration. At moderate penetrant activities when swelling is appreciable and the temperature is less than, but within about 10°e of Tg , the mechanism of penetration may change from Fickian diffusion to a stress relaxationcontrolled process in which the penetrant advances in a sharply 29 defined front at a nearly uniform velocity. This mechanism is designated as case II transport [Felder and Huvard, 1980]. The two modes of transport are easily distinguished from the sorption results. In both modes at small times t, (38) where Mt is the cumulative mass absorbed at time t. For case I (Fickian) transport n = 1/2, and for case II transport n = 1. On a plot of Mt vs. 0, case I transport would be linear at small times and case II transport would be sigmoidal. A mode designated "super~case II transport" has also been observed [Felder and Huvardl which has a sorption curve convex to the time axis at large times on a plot of Mt vs. t, where a similar plot would be linear for case II transport and concave for case I transport. Another anomaly which can occur is two stage sorption in which swelling penetrants are sorbed by glassy polymers and a rapid approach to an apparant equilibrium state, followed by a gradual shift to the true equilibrium state is observed. This phenomenon has been attributed to a gradual relaxation of the elastic cohesive force in the polymer, and to a timevarying surface concentration of penetrant [Felder and Huvard, 1980]. Diffusion anomalies also occur in polymers at temperatures well above Tg when crystallites or fillers are present [Park, 19681. Fillers can lead to differences between diffusion coefficients obtained by steadystate and transient methods. Such differences have been reported for filled rubber [Barrer et al., 1963] at temperatures well above the glass transition temperature (Tg + 100°C) [Park, 1968]. These anomalies are not due to time effects, but are due to the complicated effect of microheterogeneities discussed in the next section. In crystalline polymers, sigmoidal sorption curves are obtained at temperatures well above Tg and the sigmoidal character is more marked at higher crystallineamorphous ratios. The effects in crystalline polymers are thought to result from slow responses to external changes in the crystalline regions, which could lead to a timedependence in the crystallineamorphous ratio and so produce timedependent diffusion coefficients leading to sigmoidal sorption curves [Park, 1968]. 30 Mass Transfer in Heterogeneous Systems Diffusion in Filled Polymers This discussion of diffusion in filled polymers begins with the effect of fillers in elastomers, since rubber products have been the most frequently studied filled polymer system [Barrer et al., 1962; Carpenter and Twiss, 1940; Morris, 1931; Smith, 1953; van Amerongen, 1947]. This discussion is, therefore, appropriate to introduce the present work since at the temperatures at which the experiments in this study were performed, the polymer, PVAC, is an elastomer. Most rubber compounds contain considerable amounts of fillers [van Amerongen, 1964]. The fillers may be spherical or nonspherical, and they may be reinforcing or nonreinforcing. Also, certain fillers could have a major effect on the diffusivity, solubility and permeability. Diffusion as mentioned above is the transport of a molecule from one part of a material to another. The solubility of foreign molecules in an elastomer is defined by a state of equilibrium between the molecules inside and outside of the polymer [van Amerongen, 1964]. In the equilibrium state the polymer has taken up as many penetrant molecules by dissolution as can be expected thermodynamically. As long as equilibrium has not been reached, the elastomer continues to absorb or desorb the foreign molecules, which involves transport by diffusion. Permeation is more complex than diffusion since it involves the absorption of the gas on one side of a membrane, diffusion of the gas to the other side of the membrane, and finally evaporation or extraction from the other side of the membrane. The simplest solutiondiffusion theory would be if the sorption of the gas at the surface of the membrane exposed to the gas obeyed Henry's law C = Clp, (39) where C is the dissolved species concentration in equilibrium with a gas whose partial pressure is p and CI is the solubility coefficient, and that the absorbed gas diffuses through the membrane in accordance with Fick's law J = D'lC, (40) 31 where J is the flux of gas through the membrane and D is the diffusion coefficient or diffusivity. In these equations, the solubility and diffusion coefficients (j and D are assumed independent of concentration. If the above equations are valid, the steadystate permeation rate per unit area through a membrane of thickness L is (41) where 8p is the partial pressure difference across the membrane, and the product P = aD is the permeability of the membrane to the gas. Since permeation is experimentally easier to study than diffusion [van Amerongen, 19641, much of the work on filled rubbers has focussed on the process of permeation instead of pure diffusion. A good discussion on the subject of diffusion and permeation in heterogeneous media is given by Barrer [1968]. Barrer describes the derivation of differential diffusion equations taking into account the volume fractions of polymer, filler and vacuoles. These equations and the concept of vacuoles will be discussed later. Most of the early studies of diffusion and permeation of gases in filled elastomers used gases with low molecular weight such as hydrogen, nitrogen, oxygen and air [Morris, 1931; Smith, 1953; van Amerongen, 1947, 1955]. This was a logical choice since the production of automotive tires is a major use of rubber and the permeability of air in tires was of great concern. Also, experimentation using gases with low molecular weight was easier since solute condensation was not a significant problem as when using high molecular weight condensable vapors. Van Amerongen [1955, 1964], showed that the diffusion coefficients of hydrogen, nitrogen and oxygen in natural rubber were greatly modified by the presence of fillers. Many mineral fillers, lamellar fillers and carbon black fillers were used in van Amerongen's work [1955, 1964]. All of the rubber mixtures contained about 20% filler by volume. Mineral fillers such as whiting, aluminum oxide and barium sulfate reduced the diffusivity by about 10 to 15% at a temperature of 25°C and 15 to 25% at a temperature of 50°C. Mineral fillers such as hisil and durosil 32 reduced the diffusivity by about 40 to 65% at temperatures of 25°C and 50°C. Fillers such as aerosil and tegN reduced the diffusivity by about 30 to 40% at the same temperatures. Lamellar fillers such as aluminum powder and mica powder reduced the diffusivity by about 60 to 75% at the same temperatures. At the same two temperatures, carbon black fillers reduced the diffusivity approximately 15 to 30% for thermax and P 33, 50 to 80% for slatex K and Vulcan 3, and 75 to 90% for spheron 9 and spheron 4. One explanation for a reduction in diffusivity with the addition of fillers was that the fillers behave as a geometric obstruction to the path of gas through the rubber. The average diffusion path length was increased by the presence of the filler particles and the localized direction of flow was in general not normal to the geometrical cross section of the membrane. While the tortuosity of diffusion paths might account for some reduction in diffusivity, some of the reductions noted above were quite large, which implied that the phenomenon was more than just a geometrical effect of impermeable fillers. As well as a modification in the diffusion coefficients, the values of the solubility coefficients, (J, in van Amerongen's work [1955] were considerably increased by the addition of filler. The sO},,lbility coefficients were found from the ratios of permeability and diffusion coefficients (P = (JD), and iq. some cases by direct measurements. The increase in solubility due to the added filler could have meant that a part of the gas diffusing into the polymer was adsorbed by the filler particles. Assuming that transport was restricted to the polymer phase only, the gas once adsorbed by the filler particles was rendered immobile and no longer participated in the diffusion process. The measured effective diffusion coefficient was then an average value between the diffusivity in the polymer phase and the zero diffusivity of the gas adsorbed on the filler particles. Barrer, Barrie, and Raman [1962] studied the diffusion of higher molecular weight gases such as nbutane, isobutane, npentane and neopentane in silica filled silicone rubbers. The filler used was Santocel CS, a relatively porous amorphous form of silica. Studies were performed over a temperature range of 30 to 70°C in 33 the neat rubber and at volumes of filler of 5.6, 10.6, 14.9 and 19.1%. Reported reductions in diffusivity with addition of filler were 15 to 38% for 5.6% filler, 23 to 39% for 10.6% filler, 32 to 53% for 14.9% filler, and 32 to 61% for 19.1% filler. The percent reduction in diffusivity due to the added filler was found to increase with temperature and size of the penetrant molecule. Barrer, Barrie, and Rogers [1963J studied the diffusion of propane and benzene in membranes of natural rubber with zinc oxide filler. The volumes of filler were 0, 5, 10, 20, 30, and 40% for diffusion of propane, and 0, 10, and 40% for diffusion of benzene. For diffusion of propane at 40°C, the reduction in diffusivity ranged from 3 to 26% over the volumes of filler given. The results of the filler reducing the diffusion of benzene in the membranes were reported for three temperatures, 30, 40, and 50°C, for 10% ZnO, and for one temperature, 40°C, for 40% ZnO. Results were als,O reported for various concentrations of benzene up to 0.10 volume fraction. For the membrane with 10% ZnO, the reduction in diffusivity was about 16 to 57% at 30°C, 22 to 56% at 40°C, and 19 to 26% at 50°C. For the membrane with 40% ZnO, the reduction in diffusivity was about 42 to 72% at 40°C. A filled polymer may contain two disperse phases, the filler and small vacuoles. Vacuoles are small gaps in the structure which are filled with the diffusing gas and usually occur at very high volume fractions of filler (above 50%) where incomplete wetting of the filler by the polymer is extensive [Barrer, 1968J. The following derivation of an effective diffusion coefficient which takes into account these disperse phases was developed by Barrer and Chio [1965]. The solubility of a gas in the polymerfiller system can be considered to be composed of three factors. The first is related to the solubility in the polymer, CTp . The second is related to the gas adsorption by the filler, CTf. The third is related to filling of gas pockets or vacuoles, CTl). If the distribution of gas in each phase obeys Henry's law, C = CTp, where p is the pressure of the gas at the surface of the 34 polymer, then the three factors are defined as C' ,...  p. vp, p C' Uf=.1.; p (42) where C;, Cf and C~ are the concentrations in molecules per cm3 of pure polymer, of filler particles and of vacuoles, respectively, and the solubility of the gas in the polymerfiller system becomes (43) where Vp, vf and Vv are volume fractions of polymer, filler and vacuoles, respectively. If there were no vacuoles present and the filler was nonadsorbing or fully wetted, then equation 43 would become (44) If equation 44 is valid, solubility should decrease linearly with increasing filler volume fraction, which is not the case for some polymerfiller systems studied [Barrer et al., 1962, 1963]. Barrer [1965, 1968] states that for one dimensional flow in the x direction, a differential equation of diffusion can be written as acp BCf BCv _ ( D ) fPCp D 02CJ D 02Cv at + Bt + ot  K, P Bx2 + J Bxz + v Bx2 ' (45) where, Cp , Cf and Cv are the numbers of molecules of diffusant per cm3 of membrane which are present in the polymer, on or in the fiUer and in the vacuoles, respectively. The terms K,Dp , Df and Dv are the effective diffusion coefficients in the polymer, on or in the filler and across the vacuoles, respectively. Dp is the diffusion coefficient in the pure polymer. The value K in the polymer effective diffusion coefficient is a structure factor which takes into account the tortuosity of diffusion paths. Defining C as the total number of molecules per cm3 of filled polymer gi ves C = Cp + CJ + Cll = up. Combining equations 42, 43, 45 and 46 gives BC = ((K,Dp)uPVp+Dfufvf + Dvuvvv) BZC. ot upvp+UfVf + UvVv Bxz (46) (47) 35 The term in brackets is the overall effective diffusion coefficient, DelI, in the filled polymer. Barrer [1968] also defines a diffusion coefficient, D~ff' by the flux, J, through unit area and the concentration gradient in the polymer phase only, J =  D'elfdedl'' x The effectivediffusion coefficients are related by If transport is restricted to the polymer phase (DI = 0, Dv diffusivity reduces to In the absence of vacuoles, equation 50 reduces to (48) (49) 0), the effective (50) (51) For zero sorption by the filler ((J' f = 0), equation 51 can be reduced to (52) Equations 50, 51 and 52 are the forms frequently found in literature [Barrer et al., 1962, 1963; van Amerongen, 1964]. Of interest in the study of diffusion in filled polymers is the extent to which the polymer wets the filler. Assuming no vacuoles are present, two extreme cases for the solubility of a polymerfiller system are (53) and (54) In equation 53, (J'j has the value for the unwetted filler powder. In equation 54, (J' f has a value of zero implying that the filler is completely wetted by the polymer. The parameter (J'f usually has values between the two limits of equation 53 and 54 36 since the filler must be at least partly wetted by the polymer and is by no means always fully wetted [Barrer, 1968; Barrer, Barrie, and Raman, 1962]. To illustrate this, Figure 8 shows solubility data taken from Barrer, Barrie, and Rogers [1963] The solubility coefficients are for propane in natural rubber filled with ZnO. To compare the data to the two extreme cases mentioned above, equation 53 and 54 were plotted. Curve (a) is a plot of equation 53 with O'p = 0.0495 and 0'1 = 0.0305, solubility coefficients of propane in rubber with no filler and in bulk filler, respectively. Curve (b) is a plot of equation 54 with Up = 0.0495 and O'f = 0, which would impl? complete wetting of the filler by the rubber. At low filler volume fractions the solubility coefficients follow a slope closer to that of curve (b) (complete wetting of filler), but at higher filler volume fractions the solubility coefficients are closer to curve (a) (no wetting of filler). As mentioned above, expectations are that the filler in the rubber have solubilities less than that of a free filler since the polymer should at least partially wet the filler, and that the polymer will not always fully wet the filler. This would lead to solubility coefficients lying between curves (a) and (b). To complicate matters, vacuoles may also be present, which was neglected here for simplification. Since the rate of decrease of solubility falls off with increasing filler content in Figure 8, more filler surface is probably available for sorption of penetrant. A possible explanation of this and the drastic reductions in diffusivity with added filler is nonuniform dispersion of the filler particles [Barrer, Barrie, and Rogers, 1963]. If particle conglomerates are formed, gaps are created between some of the particles and two processes could occur which reduce diffusivity. Gas may occupy the void volume created and be rendered immobile. Moreover, since the filler surface in the conglomerates will not be completely wetted by polymer, a larger fraction of the conglomerate is available for gas adsorption. 37 0.050 ., 0.045 . bO ~eu ..... 0.040 a ~eu ..... ci.. ....: CI.i ~eu '' 0.035 b 0.030 0.0 0.1 0.2 0.3 b 0.4 0.5 0.6 Figure 8. Solubility Coefficient, u, of Propane in Natural Rubber Filled with ZnO Vs Volume Fraction Filler. Curve (a), U = UpVp + UfVf with (Jj = 0.0305; Curve (b), U = UpVp , u1 = O. Taken from Barrer, Barrie, and Rogers [1963]. 38 Diffusion in Crystalline Polymers A description of some of the work done in the area of diffusion in crystalline polymers is included in this section since the effect of crystals in crystalline polymers is similar to that of fillers in filled polymers. Solution of a penetrant in perfectly crystalline regions is not to be expected [Barrer, 1968], and usually, a decreases linearly with increasing crystalline fraction, V c (subscript c and a will replace f and p, respectively since the subject IS now diffusion in crystalline polymers which have a crystal and an amorphous phase) [Michaels et aI., 1964] as (55) where Va. is the volume fraction of amorphous polymer and Va. + Vc = 1. The diffusion coefficient of a penetrant in a polymer crystal is expected to be very small; therefore, crystals act similarly to impermeable filler particles [Barrer, 1968]. The difference is that the degree of crystallinity may be changed by heating, cooling and annealing and that the crystals should always be fully wetted by the polymer chains in the amorphous regions [Barrer, 1968]. Crystals only act as a geometrical obstruction which increases the diffusion path length. The effective diffusion coefficient can be expressed similarly as was for filled polymers, (56) (57) where Da. is the diffusion coefficient in the completely amorphous polymer. However, most forms [Hedenqvist and Gedde, 1996] use a tortuosity factor, T, such as the relationship proposed by Michaels and Bixler [1961], who interpreted f'i, as the product of a geometrical impedance factor, T, and a chain immobilization factor, (3 [Barrer, 1968], Da. Dell = Tf3" Initially, suggestions were made that f3 described the reduced segmental mobility of polymer molecules in the vicinity of crystalline surfaces [Michaels and Bixler, 39 1961]. If the crystalline surface area is large, then 13 may be large.. Later, findings were that 13 depends more on the size of the penetrant molecule and only weakly on the crystallinity [Michaels et al., 1964]. Peterlin [1975, 1984] has a slightly different form for the effective diffusivity (58) where 'I/; is a detour factor describing the physical obstruction of the crystallites, which takes val ues between a and 1, B (~ 1) is a blocking factor, and D~ is the diffusion coefficient in the "relaxed" region of the amorphous phase of the crystalline polymer [Hedenqvist and Gedde, 19961. The amorphous phase is a complex network of tortuous very thin and broad fiat channels, and the amorphous chains are restrained in mobility by their ends fixed in the adjacent crystals [Peterlin, 1975]. Therefore, diffusion in the amorphous phase is anisotropic and the effective diffusion coefficient Def f of the crystalline polymer will be smaller than D;. The effect of the restraint on chain mobility increases with a smaller thickness of amorphous layers. An increase in the amorphous component implies an increased thickness of such a layer and an increase in the diffusion coefficient D~ [Peterlin, 1975]. Therefore, D: is strongly dependent on crystallinity. The blocking factor, B, in equation 58 describes the geometrical blocking that qccurs when the penetrant molecules are too large to be able to enter the amorphous interlayers. The immobilization factor, {3, in equation 57 is analogous to the blocking factor, B. It takes into account the constraining effect of the crystals on the amorphous phase and is included in the crystallinity dependence of D~ in equation 58. The detour factor, '1/;, and the tortuosity factor, T, are both purely geometrical factors [Hedenqvist and Gedde, 1996]. Several other models have been used to describe the diffusion in crystalline polymers [Hedenqvist and Gedde, 1996]. Some are empirical and some are based on physical and chemical processes involving the penetrant and the polymer. The models listed above (equations 57 and 58) do not include a dependence on the 40 concentration of the penetrant. In most polymers, the diffusivity is greatly dependent on concentration. This concentration dependence is sometimes expressed as [Hedenqvist and Gedde, 1996] Deff  Doe"fC , (59) where Do is the diffusivity at zero penetrant concentration and / is a constant. A linear dependence of diffusivity on concentration was used for poly(ethylene terephthalate) and 4nitro4'hydroxyazobenzene in an aqueous solution [Iijima and Chung, 1973] Deff = Do(l +/C). (60) A polynomial was used to fit the diffusivity of water in PA6 [Hanspach and Pinno, 1992] (61) where /1 and /2 are constants. Equations based on the free volume theory have also been used to explain diffusion in crystalline polymers. In simple terms the free volume theory is expressed as [Hedenqvist and Gedde, 1996] (62) where B is a constant and I is the fractional free volume of the system. An equation based on the free volume theory was used to fit the diffusivity of various gases into polyethylene [Kulkarni et al., 1983; Stern et al., 1972, 1983, 1986] (63) where Ad is a constant which depends on the size of the penetrant, and fa is the amorphous fractional free volume, (64) The quantities VI and V2 are the volume fractions of the penetrant and the polymer in the amorphous phase, respectively. The values 11 and h are the corresponding (66) 41 fractional free volumes. DT in equation 63 is the thermodynamic diffusivity and is related to the effective mutual diffusion coefficient, Delh through [Hedenqvist and Gedde, 1996] Dell = D T 8(lnaI), 1  VI 8(ln vd (65) where al is the activity of the penetrant. The quantity Va in equation 63 corrects the effective diffusivity for crystallinity. A similar correction is shown in this work for filler content. Another equation related to the free volume theory which has been used to fit diffusivity data is [Horas and Rizzotto, 1989] DT Dell = D' 2~ !ITA (67) where A and B are functions of crystallinity. Structure Factor The structure factor, K, takes into account the tortuosity of the diffusion path around the filler particles (or crystals). If the filler or polymer crystals are impermeable and have no vacuoles then Peff DefJ p = KVp ; D = K, (68) p p where the subscript a replaces p for crystalline polymers. The value K, can be found from a plot of PelI IPp or DelI ID p versus the volume fraction of filler V I (or crystals vc ). Many mathematical forms have been derived which give the dependence of K, on the volume fraction of filler. The problem is analogous to the electrical conductance of a heterogeneous medium composed of a dispersion of particles in a continuous medium of different conductivity [Barrer et al., 1962, 1963; van Amerongen, 1964]. Maxwell [1891] considered a suspension of spherical particles so dilute in the medium that the spheres had no effect on one another [van Arnerongen, 1964]. Fricke [1931] extended the study to include oblate and prolate spheroids. The expression for K, for these approaches is y K.= , Y +vf (69) 42 where Y is a shape factor which is 2 for random spheres and decreases to 1.1 as the shape changes from a sphere to an oblate spheroid with axial ratio 4 : 1 [Barrer, 1968; Barrer, Barrie, and Rogers, 1963]. Lord Rayleigh [1892] and Runge [1925] considered a cubic array of uniform spheres for treating a more concentrated dispersion of particles [Barrer, Barrie, and Rogers, 1963]. The equations for K are K  1 [1 _ 3vf] (70)  (1  Vj) 2 +Vj  O.392v}O/3 ' for a cubic lattice of spheres, and K= (1~Vf) [1 2+VfO.3~:~jO.0134V~]' (71) for a cubic lattice of cylinders normal to the direction of flow. Barrer, Barrie, and Raman, [1962] have found in a study of diffusion of hydrocarbon penetrants in various silicafilled rubbers that the influence of the filler in reducing the diffusion coefficients is greater than would b.e .expected for a regular dispersion of nonconducting spheres. They have suggested that a filled polymer is a considerably more complex medium, and if trends of diffusivity reduction with temperature and size of penetrant are significant then these simple models must be modified. For the crystalline polymers polyethylene terephthalate, polyethylene and Nylon, Lasoski and Cobbs [1959] found that water vapor permeabilities followed the relationship Pej j 2 ( ) p~ =Va' 72 If the effective diffusion equation is of the form of equation 56, then up to V c = 0.4 [Barrer, 1968] (73) In this equation, K decreases much more rapidly than would be expected from Maxwell's or Rayleigh's equations. Refering now to equation 57, Michaels and Bixler [1961] determined that the tortuosity factor for a series of polyethylenes with different crystallinities using He as a penetrant followed the equation n T = va , (74) 43 where n is a constant which takes different values for different polymers. The immobilization for helium was assumed to be small, hence (3 = 1 [Michaels and Bixler, 1961]. Treatment of Adsorption as a Chemical Reaction The problem of adsorption of a gas on filler particles may also be treated as diffusion combined with a chemical reaction [van Amerongen, 1964]. Physical adsorption of a gas by the filler has a similar effect on diffusion as chemical reaction. In either case the gas is rendered inactive and has the misleading effect of increasing the apparant solubility without increasing the permeability. Permeability is related to diffusivity by P = DO', (75) so the diffusivity will decrease. A differential equation which could describe diffusion with adsorption is [Crank, 1956] (76) where CI is the concentration of penetrant which is adsorbed on the filler particles, and C is the concentration of penetrant which is free to diffuse. In the simplest case, the concentration of the immobilized penetrant, CII is directly proportional to C, Cj =RC. (77) This equation is referred to as a linear adsorption isotherm [Crank, 1956]. Substituting for Cj from equation 77 into equation 76 gives ac D a2c at = R + 1 ax2 ' (78) which is the usual form of a diffusion equation with an effective diffusion coefficient D Deff = R+l' (79) 44 If tortuosity is also a factor, the effective diffusion coefficient becomes (80) where K. is the tortuosity factor and Dp is the diffusivity in the pure polymer. Summary This work uses several methods and equations which were discussed in this chapter. Diffusion coefficients for CaC03 filled PVAC were evaluated from frequency data obtained with a QCM. Sorption data were calculated by analyzing frequency data using equation 15. Diffusion coefficients were calculated by the moment method, equation 22, and penetrant weight fractions were calculated by equations 24 and 25. The freevolume equation 30 was used to fit the experimental diffusion coefficient data for toluene diffusion in unfilled PVAC. Several equations were fit to the experimental data for toluene diffusion in filled PYAC, including Barrer's equations 51 and 52, and equation 80 which was derived from reaction principles. The freevolume equation 30 was used for Dp , the unfilled polymer diffusivity, in these equations. Also, fits to filled polymer diffusion coefficient data were done using the structure factor equations 69 and 70 in combination with equation 52. CHAPTER III MODIFICATIONS OF EQUIPMENT AND PROCEDURES The equipment in this work was designed to measure diffusion coefficients of penetrants diffusing into polymers. A description of the experimental apparatus was given in Chapter II and also in the works of Deshpande :[1993] and Mikkilineni [1995] from the development stage of the QCM through several modifications. This chapter contains a discussion of additional modifications made to the equipment, changes in the data analysis for determining diffusion coefficients, and procedures for sample preparation. A schematic of the QCM with modifications is in Figure 9. A sample cylinder was installed between the solvent flask and the sixway cross chamber. The sample cylinder allows an instantaneous step change of pressure in the chamber. The toggle valve and the needle valve described in Chapter II were removed. Three ball valves were used, one to isolate the solvent flask, one to isolate the sample cylinder, and one between the sample cylinder and the sixway cross chamber. ew 3/16 inch stainless steel tubing was installed between the solvent flask and the sixway cross chamber to increase the flow rate of vapor. Previously, some of the tubing was 1/4 inch. The ball valves allowed the sample cylinder to be filled with penetrant vapor, then introduced into the chamber when needed. Before installing the sample cylinder, the penetrant was introduced into the chamber too slowly to adequately approximate a step change. Figure 10 shows a typical sorption curve for the experimental setup without the sample cylinder. The rate at which the penetrant was introduced into the chamber is reflected by the slow rate of mass sorption in the initial portion of the curve. This severely affected the accuracy of data analysis since the initial slope method uses the initial portion of the curve and the moment method uses the entire curve. Installing the sample cylinder 45 46 Insulated Box To Computer Interface ~~~~\ Liquid Nitrogen Trap Solvent Flask Rotary Vane Pump RightAngle Valve L..r.......J t To Vent Hood~ Bottom Port Hea~ COil~ Figure 9. Schematic Diagram of the Quartz Crystal Microbalance Modified Setup. 0.9 • 47 1.0 .::=_=__~ ••• •••••••• • •• 0.8 0.7 0.6 i 0.5 ::E 0.4 0.3 0.2 • • 0.1 0.0 .....~..,....__r____r__.______1 o 5 10 15 20 25 Figure 10. Sorption Curve Before the Sample Cylinder was Installed. 48 solved this problem. The sample cylinder allows the penetrant to be introduced into the chamber almost instantaneously. Figure 11 shows a sorption curve for the experimental set up with the sample cylinder. otice that the initial portion of the curve has a shorter tail since the pressure was increased almost instantaneously. The next change to the QCM was not a modification of the equipment, but a replacement of the gaskets in both the right angle valve and the gate valve. The gaskets were made of rubber and were deteriorated due to the organic vapors used in the QCM chamber. The leakage from the valves led to unreliable data. The plastic tubing which was used for circulating cold water to the turbomolecular pump was replaced with 1/4 inch copper tubing, decreasing the chances of a water leak. A water circulation line constructed of 1/4 inch copper tubing was connected to the solvent flask. A centrifugal pump was used to circulate water through the solvent flask from a water bath heated by a constant circulation immersion heater. The computer program recording the frequency data from the deposition monitor was modified to record the data at smaller time intervals. The previously used IBM XT computer was replaced by a Gateway computer with a 33 MHz 486 CPU. This also allowed for data collection at smaller time intervals. The moment method, discussed in Chapter II, was used to analyze the experimental sorption data in this work. Previously [Deshpande, 1993; Mikkilineni, 1995], the initial slope method, also discussed in Chapter II, was used to analyze the sorption data from the QCM. The initial slope method is very sensitive to the position on the curve at which the slope is calculated. Also, diffusion is sometimes so fast that there are a limited amount of data in the initial portion of the sorption curve, making analysis with the initial slope method even more difficult. The moment method uses the entire sorption curve; thus, is not as sensitive to the initial portion of the sorption curve. An Excel spreadsheet was developed to evaluate the area under the sorption curve using the trapeziodal rule. In this work, samples were prepared by dissolving a known weight of polymer (PVAC) and filler (CaC03 ) in a known volume of solvent (tetrahydrofuran) Figure 11. Sorption Curve After the Sample Cylinder was Installed. 49 50 to make a very dilute solution ('" 3% polymer and '" 0.1% filler). The crystals were coated from a micropipette by dropping a known volume (typically 7j.d) of the polymer solution onto the crystal surface and spreading it with the tip of the micropipette. After drying, the diameter of the polymer film was measured with a ruler. The thickness of the polymer film was calculated from the surface area of the film, the volume of the solution used to coat the crystal, and the weights and densities of the polymer, filler and solvent used to make the polymer solution. Weights were recorded with a mass balance; of the empty bottle, after adding polymer, after adding filler, and after adding solvent. The diameter of the film was measured at several locations. The average diameter was used to calculate the area of the polymer film by using the formula for the area of a circle. Detailed procedures for preparing polymer solutions, coating crystals, and calculating film thickness are given in Appendix A. Since the design phase of the QCM, many improvements have been made. Continual improvement of the equipment and procedures will lead to more accurate data, more ease of operation, and perhaps more uses for the QCM. Suggested improvements are below. All of the monitors for measuring pressure, temperature, and frequency are separate devices. Also, data must be copied from the data acquisition computer and analyzed as a separate step. To make the apparatus and data analysis less cumbersome, a graphical instrumentation program such as LabVIEW should be set up on the data acquisition computer. This program could be used to imitate the appearance and operation of actual instruments, such as pressure and temperature monitors. All of the measured properties, pressure, temperature, and frequency, could be monitored at one location, the computer. Such a program can also produce live graphs of data during an experimental run. All calculations involved in determining the diffusion coefficient could be performed by the graphical instrumentation program. The box surrounding the QCM is made of insulated plywood, but this does not stop all of the heat loss. The quartz crystal microbalance should be fitted inside an insulated oven to better control the temperature of the ambient air and the chamber. CHAPTER IV EXPERIMENTAL RESULTS This chapter presents the results of the sorption experiments performed with the quartz crystal microbalance to obtain effective diffusion coefficients of toluene diffusing into CaC03 filled PVAC. The diffusion coefficients of toluene diffusing into PVAC with 0.0% CaC03 (neat PVAC) are compared with the results of other researchers [Mikkilineni, 1995 and Hou, 1986] in Chapter V. The effective diffusion coefficients of toluene diffusing into CaC03 filled PVAC were measured at temperatures of 60°C and 80°C for 0, 3.3, 4.9, and 10% (by weight) CaC03 . The diffusion coefficients were obtained at low concentrations of toluene, less than 0.15 weight fraction for 60°C data and less than 0.10 weight fraction for 80°C data. Effective diffusion coefficients were calculated by using the moment method. Effective diffusion coefficients and toluene weight fractions are given in Tables I through VIII. The tabulated diffusion coefficients are plotted as a function of toluene weight fraction in Figure 12 for the 60°C data and Figure 13 for the 80°C data. To verify the reproducibility, each set of experimental data was replicated using the same quartz crystal and polymer film. Equilibrium sorption solubility data for toluene in CaC03 filled PVAC were also measured at temperatures of 60°C and 80°C for 0, 3.3, 4.9, and 10% (by weight) CaC03 . Equilibrium toluene vapor pressures, PI, were measured, and solubility data in the form of pdp~, where p~ is the toluene saturation vapor pressure, were calculated. The solubility data and equilibrium toluene weight fractions are given in Tables I through VIII. The solubility data are plotted as a function of toluene weight fraction in Figure 14 for the 60°C data and Figure 15 for the 80°C data. The solubility data are also plotted on a "filler free basis" in Figure 16 for the 60°C data and Figure 17 for the 80°C data, in which the toluene 51 52 weight fraction is calculated on a filler free basis, i.e., Ms/(Ms +Mp), where Ms is the mass of the toluene solvent and Mp is the mass of the PVAC polymer. 53 TABLE I SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 0.0% CaC03 AT 60°C eq PI/P~ wC11J D X 109 (ern2 WI 1 / s) Run #1 0,0121 0,0737 0,0085 0,218 0,0253 0.1406 0.0213 0.536 0.0380 0.2028 0.0342 1.26 0.0506 0.2593 0,0468 2,82 0,0634 0,3114 0.0596 4,83 0,0754 0,3583 0,0718 8,81 0,0878 0,4022 0.0841 14.1 ~ 0,1000 0,4431 0.0963 24,1 ~ ~ 0.1122 0.4811 0.1085 34.2 ,. 0.1243 0,5168 0.1207 44,1 i 0,1380 0.5540 0.1339 53,7 Run #2 0.0081 0.0904 0.0057 0,170 0.0176 0.1585 0.0147 0.353 0,0291 0.2221 0,0257 0.687 0,0404 0.2776 0,0370 1.57 0.0519 0.3285 0.0485 2,89 0.0631 0.3754 0,0597 5,38 0.0744 0.4190 0,0710 9,23 0.0855 0.4599 0,0822 17.2 0.0963 0.4964 0.0931 25,7 0.1239 0.5834 0.1157 24.8 0.1375 0.6210 0,1335 44,8 54 TABLE II SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 3.3% CaC03 AT 60°C eq P1jp~ wav D x109 (cm2 WI I js)  Run #1 0.0082 0.0766 0.0058 0.882 0.0148 0.1459 0.0129 1.40 0.0218 0.2050 0.0198 2.02 0.0289 0.2530 0,0268 3.54 0.0364 0.2951 0,0341 5.65 0.0452 0.3416 0,0425 7.34 ... 0.0538 0.3855 0.0512 ]1.0 2 0.0624 0.4242 0.0599 16.8 .. 0.0802 0.4975 0.0777 35.5 i 0.0880 0.5295 0.0857 65.7 Run #2 0.0086 0.0696 0.0060 1.57 0.0168 0.1306 0.0144 1.58 0.0254 0.1909 0.0228 2.69 0.0341 0.2426 0.0315 4.52 0.0428 0.2913 0.0402 7.41 0.0521 0.3397 0.0493 8.46 0.0607 0.3814 0.0581 13.9 0.0706 0.4443 0.0677 21.9 0.0788 0.4856 0.0764 29.2 0.0867 0.5224 0.0843 44.3 55 TABLE III SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 4.9% CaC03 AT 60°C WIeq PI/P~ wav D xl0g (cm2 /s) 1 Run #1 0.0086 0.0681 0.0060 0.392 0.0172 0.1325 0.0146 0.538 0.0258 0.1950 0.0232 1.16 0.0341 0.2534 0.0316 1.82 0.0422 0.3125 0.0397 3.16 0.0504 0.3624 0.0479 4.25 0.0586 0.4033 0.0561 6.70 0.0669 0.4472 0.0644 8.62 0.0874 0.5231 0.0847 17.3 0.0956 0.5548 0.0931 24.6 0.1047 0.5871 0.1019 33.1 0.1133 0.6176 0.1107 40.5 0.1213 0.6418 0.1189 54.9 i 0.1296 0.6664 0.1271 64.5 Run #2 0.0080 0.0841 0.0056 0.325 0.0154 0.1522 0.0132 0.505 0.0230 0.2125 0.0207 0.917 0.0311 0.2619 0.0286 1.27 0.0398 0.3085 0.0372 2.09 0.0487 0.3546 0.0461 3.20 0.0575 0.3937 0.0549 4.53 0.0667 0.4305 0.0639 6.15 0.0767 0.4737 0.0737 8.91 0.0865 0.5090 0.0836 13.1 0.0965 0.5451 0.0935 16.9 0.1058 0.5756 0.1030 27.5 0.1148 0.6020 0.1121 33.0 0.1238 0.6270 0.1211 43.5 0.1334 0.6541 0.1305 64.0 56 TABLE IV SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 10% CaC03 AT 60°C eq Pl/P~ wall D x109 (cm2 WI 1 /s) Run #1 0.0114 0.0633 0.0080 0.104 0.0245 0.1213 0.0206 0.133 0.0366 0.1797 0.0330 0.259 0.0482 0.2337 0.0447 0.448 0.0602 0.2820 0.0566 0.662 0.0741 0.3308 . 0.0699 0.918 .C..2. ,." 0.0895 0.3747 0.0849 1.60 . 0.1065 0.4190 0.1014 1.89 ~.. 0.1267 0.4573 0.1206 4.59 i 0.1403 0.4900 0.1362 6.17 0.1536 0.5243 0.1496 12.0 Run #2 0.0085 0.0703 0.0059 0.287 0.0213 0.1343 0.0175 0.134 0.0342 0.1890 0.0303 0.264 0.0473 0.2355 0.0434 0.390 0.0650 0.2832 0.0597 0.676 0.0818 0.3271 0.0768 0.981 0.1064 0.3702 0.0990 1.76 0.1229 0.4085 0.1179 2.85 0.1397 0.4491 0.1346 4.26 0.1530 0.4867 0.1490 10.8 57 TABLE V SOLUBILITY DATA AND DIFFl:"SION COEFFICIENTS OF TOLUENE IN PVAC WITH 0.0% CaC03 AT 80°C eq PI/P~ wl1V D X 109 (em2 WI 1 / S) Run #1 0.0071 0.0559 0.0050 0.624 0.0143 0.1186 0.0121 1.04 :'" .. 0.0206 0.1660 0.0187 1.23 .... ':.J 0.0275 0.2113 0.0254 1.48 :::. :~ 0.0345 0.2535 0.0324 2.24 ....... 0.0413 0.2917 0.0392 3.54 . 0.0477 0.3242 0.0458 3.86 0.0537 0.3526 0.0519 5.44 0.0604 0.3821 0.0584 7.18 Run #2 0.0077 0.0439 0.0054 0.600 0.0149 0:0859 0.0128 0.961 0.0218 0.1225 0.0197 1.38 0.0293 0.1612 0.0270 1.71 0.0379 0.2058 0.0353 2.45 0.0456 0.2420 0.0433 3.00 0.0532 0.2752 0.0509 3.47 0.0605 0.3071 0.0583 5.57 0.0681 0.3380 0.0658 4.80 58 TABLE VI SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 3.3% CaC03 AT 80°C eq Pl/P~ wClV D X 109 WI 1 (em2/ s) ,.. .... Run #1 :! 0.0057 0.0245 0.0040 0.540 0.0156 0.0698 0.0126 1.19 ....... 0.0263 0.1120 0.0231 1.76 ,oil Ii..' 0.0397 0.1605 0.0356 2.42 ... ': 0.0515 0.2006 0.0479 4.24 ~ 0.0624 0.2377 0.0591 9.46 " Run #2 0.0069 0.0302 0.0048 0.972 0.0158 0.0666 0.0131 1.27 0.0274 0.1111 0.0239 1.74 0.0388 0.1511 0.0353 2.67 0.0509 0.1909 0.0473 4.15 0.0615 0.2280 0.0583 10.2 59 TABLE VII SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 4.9% CaC03 AT 80°C eq PIjp~ wav D x109 (cm2 WI I js) Run #1 0.0059 0.0311 0.0041 0.534 0.0135 0.0659 0.0112 0.632 0.0211 0.1019 0.0188 0.901 0.0319 0.1466 0.0286 1.14 0.0420 0.1853 0.0389 1.67 .::1 ~ 0.0563 0.2367 0.0490 3.04 .•. ~"i 0.0698 0.2825 0.0657 5.11 ,,...... 0.0797 0.3158 0.0767 10.6 c.:' 0.0906 0.3524 0.0874 18.8 0.0907 0.3869 0.0960 45.5 Run #2 0.0060 0.0291 0.0042 0.551 0.0148 0.0685 0.0121 0.643 0.0247 0.1120 0.0217 0.839 0.0351 0.1571 0.0320 1.16 0.0464 0.2026 0.0430 1.89 0.0566 0.2406 0.0536 2.90 0.0663 0.2736 0.0634 4.40 0.0758 0.3052 0.0729 7.07 0.0848 0.3361 0.0821 16.1 60 TABLE VIn SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 10% CaC03 AT 80°C ,,' eq PI/P~ wl1V D X 109 (em2 / S) "Ill WI 1 ..II ~I ':1 Run #1 ...,. 0.0113 0.0282 0.0079 0.463 ,I..I 0.0282 0.0692 0.0232 1.10 0.0410 0.1087 0.0371 1.48 0.0562 0.1573 0.0516 2.59 Run #2 0.0164 0.0369 0.0114 0.559 0.0341 0.0820 0.0288 1.28 0.0509 0.1307 0.0459 1.88 0.0638 0.1715 0.0599 3.78 61 1e7 o o • o ":J • •o 1e8  ~ o~ ~ · o· f o. !II It. 0 • • b. o. o It. • 1e9  b. ~It. o. o•f oo. . 0 0 • • It.~ 0+ • o ~ o. .£) .~ 0 ... 1e10~~· o. ~ «) 1e11  0 open symbols Roo #1 solid symbols Roo #2 o 0 %CaC03 ~ 3.3 % CaC03 o 4.9 % CaC03 o 10 % CaC03 III ." JI 11 :1 ..,. .jl 0.00 I 0.05 I 0.10 I 0.15 I 0.20 I 0.25 0.30 Toluene Weight Fraction Figure 12. Effective Diffusion Coefficients of Toluene in CaC03 Filled PVAC at 60°C. 62 1e7 o o • o 0 • • 1e8  • ~o .0 •° 1e10  0.00 o 1e9  .to • ~ • 0 ! 0. ~ o· ". A ." 0+ JI 0. I '1 ~ .... open symbols Run #1 .! • solid symbols Run #2 0 ':: • 0 0 % CaC03 • 0 3.3 % CaC03 I::> 0 ° 4.9 % CaC03 • 10 % CaC00 0 3 I I 0.05 0.10 0.15 Toluene Weight Fraction Figure 13. Effective Diffusion Coefficients of Toluene in CaC03 Filled PVAC at 80°C. 63 0.7 r, o % CaC03 3.3 % CaC03 4.9 % CaC03 10 % CaC03 o o o 0 o. 0• • • ~ • 0 0 0 0 0 • 0 • 0 0 • • ,.. .. ,I' 11 'I ., open symbols Run #1 .. I! solid symbols Run #2 . .. o o o o • 0.1  Ie 0.2  0.5  0.3  0.4  0.6  0.0 I I I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Toluene Weight Fraction Figure 14. Solubility Data of Toluene in CaC03 Filled PYAC at 60°C. 64 0.45 0.40  0 0 0.35  0 0 • • 0 0 0.30  • • 0 • .0 0.25  0 0 • ~ /), e ... 0.. 0 .. 0.20  • ../.)., I'\' 0 I 0 • " •• 6. .. 0.15  o ... 0 open symbols Run #1 I! • solid symbols Run #2 .' 0 • " 0.10  ~ 0 0 0 0 % CaC0• • 3 ~ 0 6. 3.3 % CaC03 0.05  0• • 0 4.9 % CaC03 ~ 0 0 10 %CaC03 0.00 I I 1 1 0.00 0.02 0.04 0.06 0.08 0.10 Toluene Weight Fraction (Solvent Free Basis) Figure 15. Solubility Data of Toluene in CaC03 Filled PVAC at 80°C. 65 0.7 ,, 6~ 0 • o • 6·~ 0 ~ open symbols Run #1 Lt solid symbols Run #2 ., 0 ~ ·0 ~~ 0 0 %CaC03 ~ 0 6 3.3 % CaC03 \0 ° 4.9 % CaC03 0 10 % CaC03 o • • 0. ° 0• • • <+  ~ 0 c 0 0 0 • 0 • 0 o o o o • 0.4  0.5  0.2  0.1  0.3  0.6  0.0 I I I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Toluene Weight Fraction (Solvent Free Basis) Figure 16. Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 60°C. 66 0.45 0.40  0 0 0.35  0 0 • • 0 0 0.30  • • 0 • .0 0.25  0 0 • ~ 6 e A 0.. 0.20  0 • • 6 0 A 0 • 0.15  • o • A6 0 • open symbols Run # 1 0 • solid symbols Run #2 0.10  4lJt. 0 0 0 0 % CaC0• • 3 ~ 0 I::> 3.3 % CaC03 0.05  0• • '0 4.9 % CaC0~ 0 3 0.00 0 10 % CaC03 I I I I 0.00 0.02 0.04 0.06 0.08 0.10 Toluene Weight Fraction Figure 17. Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 80a C. CHAPTER V DISCUSSION OF RESULTS In this chapter, the experimental data presented in Chapter IV are discussed. The diffusion coefficients of toluene diffusing into PVAC with 0.0% CaC03 (neat PVAC) are compared with the results of other researchers [Mikkilineni, 1995 and Hou, 1986] and with the freevolume theory correlation of Vrentas and Duda. The methods discussed in Chapter II for determining diffusion coefficients are compared by analyzing a set of sorption curves with each method. The experimental data for filled PVAC are compared with a freevolume theory correlation modified to account for the added filler and to other correlative equations discussed in Chapter II. Diffusion coefficients for toluene diffusing into neat PVAC were compared to the data of Mikkilineni [1995] and Hou [1986] at temperatures of 60°C and 80°C in Figures 18 and 19. The diffusion coefficients obtained in this work compare well with the data of both Mikkilineni and Hou. The data shown are from Table I and V of Chapter IV for this work, Table IX for Mikkilineni, and Table X for Hou. The largest deviations between the data obtained in this work and the data obtained by Hou occur at the lowest weight fractions for both the 60°C and 80° C data. The apparatus used by Hou for obtaining diffusion coefficients was a quartz spring balance. The quartz crystal microbalance is more sensitive to changes in low values of concentration of toluene (rv 1 ng of toluene) than the quartz spring balance, while the quartz spring balance could give data points at higher values of concentration of the penetrant by using thicker polymer films than can be used by the quartz crystal microbalance. 67 'I, '11 " " ·.1, I..i. '1 '...(I 1e6 o 68 1e7  Co 0 .. ;l 0 o ~ ~ .. .. ....t...i..l. o·., N 0 S 1e8  (.) ''" Cl t~ P • 0 0 0 .,t:J cP 1e9  ..081 .. This Work 0'" ~o 0 Hou (1986) ~ ~ 0 Mikkilineni (1995) 1e10 I I I I 0.00 0.05 0.10 0.15 0.20 0.25 Toluene Weight Fraction Figure 18. Diffusion Coefficients of Toluene in Neat PVAC at 60°C. 'I, 'I' .·1:1 ":i '10 , 69 1e6  o o o o 1e7  0.20 I 0.15 .. This Work o Rou (1986) o Mikkilineni (1995) 1 0.10 o I 0.05 o o 0.00 rnAA ... f;~O ... AA A 1e8  to Toluene Weight Fraction Figure 19. Diffusion Coefficients of Toluene in Neat PVAC at 80°C. 70 TABLE IX DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC FROM MIKKILINENI Toluene Weight Fraction D x 109 (em2 / S) 60°C 0.0091 0.164 0.0264 0.468 0.0508 1.62 0.0680 4.47 0.0963 37.2 0.0136 0.220 0.0402 0.922 0.0746 5.19 " 0.1088 36.8 41 '. 0.0116 0.200 il 0.0321 0.549 H 0.0677 2.14 0.0941 13.0 0.1382 71.9 0.0144 0.429 0.0397 1.45 0.0138 0.349 0.0363 0.877 0.0654 3.16 0.0283 1.06 0.0795 31.7 0.0541 1.70 0.0380 0.994 0.0928 24.2 0.1756 56.3 80°C 0.0142 31.5 0.0364 36.5 0.0586 74.8 0.0871 163 0.1305 179 TABLE X DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC FROM HOD 71 Toluene Weight Fraction D x109 (cm2 js) '.41 '. 60°C ;1 ~l 0.0234 2.05 :.l ~. 0.0755 16.0 .•.• 0.1320 82.9 :) 0.2199 446 80°C 0.0140 4.20 0.0404 23.5 0.0734 84.2 0.1202 319 0.1903 921 0.2589 1190 72 Mikkilineni's data were obtained with the same apparatus as used in this work; however, the methods of analyzing the sorption data were different in this work. Mikkilineni used the initial slope method, and the moment method was used in this work. Also, in this work, the apparatus was modified to include a sample cylinder for quick delivery of the penetrant into the chamber. The use of a sample cylinder was critical for preventing errors which occured from the slow entry of the solvent vapor. The diffusion coefficients for neat PVAC show greater scatter at larger weight fractions shown, as can be seen in Figures 12 and 13. The reason for increased scatter of data at larger weight fractions was because the diffusion became too fast for enough data to be collected in the beginning stages of diffusion. The faster the diffusion, the fewer the number of data points that were collected in the beginning stages of diffusion. Also, since diffusion at 80°C was faster than diffusion at 60°C, data at 80°C were not obtained at as high of toluene weight fraction. Figure 20 shows an example of a sorption curve that was rejected due to very few data in the initial stages of diffusion. Diffusion coefficients calculated from such a curve would be unreliable since very few data points were collected at the beginning of the curve. Another phenomena which leads to unreliable diffusion data is the appearance of a hump in the sorption curves for larger toluene weight fractions. Diffusion coefficients obtained from analysis of sorption curves after the appearance of a hump are not shown in Figures 12 and 13 as the scatter of data is too great and the data is considered to be unreliable. The cause of the hump is unknown, but is speculated to be related to "stress effects." Normally, the changes in resonant frequency of the quartz crystal is related to the change in mass on the exposed surface. However, if there is stress in the polymer film on the quartz crystal surface, there is a net force per unit width acting across the polymer film/quartz interface that stress biases the quartz. This stress bias causes the frequency changes referred to as "stress effects" [EerNisse, 1984]. Adding mass or adding stress decreases the resonant frequency of the quartz crystal. Speculations are that, when the coated "~l ":1:4 ·l ".~ .. 73 Figure 20. An Example of a Sorption Curve that was Rejected Due to Very Few Data in the Initial Stages of Diffusion. "•• ":1 :~ .~ I. '...:~, .1 74 polymer solution drys, stress builds in the polymer film. During a sorption experiment, adding solvent to the polymer film decreases the resonant frequency of the quartz crystal. The added solvent relaxes the polymer chains, possibly reducing stress at the polymer/crystal interface, which would increase the resonant frequency of the quartz crystal and produce a hump in the sorption curve. Figure 21 shows an example of a sorption curve that was rejected due to a hump in the sorption curve. Most of the trends of the effective diffusion coefficient data in Figures 12 and 13 are as expected. The diffusion coefficients increase with weight fraction toluene and temperature as is typical of the PVACtoluene system. Diffusion coefficients decrease with weight percent of CaC03 , which is typical for diffusion in filled rubbery polymers. The diffusion coefficient data are expected to be concave towards the toluene weight fraction axis. The 60°C data appear to follow this trend better than the 80°C data. The data show some upward trends and inflection points, especially for filled PVAC at the larger of the weight fractions. This could be from error in the analysis of the data, due to inaccurate selection of the equilibrium mass uptake. In slower diffusion processes, such as diffusion in filled PVAC compared to diffusion in neat PYAC, the sorption experiment takes a long time to reach equilibrium, and a small error in the equilibrium mass uptake would cause a large error in the first moment calculation of the sorption curve which is used to determine the diffusion coefficient. There could also be errors due to "stress effects" as discussed previously. Although, if "stress effects" are present, they are not noticable enough to form a hump in the sorption data which were analyzed. Replicate experimental runs of each set of data were performed with the same crystal and polymer film for each set in order to verify reproducibility of data. As can be seen from Figures 12 and 13, the data reproduce well. Some scatter is seen, caused by factors such as limited amount of sorption data and inaccurate selection of equilibrium mass uptake, as described above. ~ "~I ':i '4 ~ '. "I ~ '~'. " '",..,. 75 1.2 1.0 0.8 '""'L.7_"b"7I11ldll1~~U....J..t,n.....r"4'·".,117"'S.mlll._7 8 E 0.6 ~ 'I. 'II 'il " ':1 '" 0.4 • '" ',. • " • ;. • '. •• 0.2 II • 'r: """ """ '" 0.0 II: Ih' , 0 10 20 30 40 50 60 "1 ~t (8) Figure 21. An Example of a Sorption Curve which was Rejected Due to a Hump in the Curve. 76 The vapor pressure data in Figure 14 through Figure 17 shows the effect of the filler on changing the solubility in the polymer films. The replicate experimental runs compare well for PVAC with 3.3, 4.9 and 10% (by weight) CaC03 at both 60°C and BO°C. However, there is more error between the replicate experimental runs for neat PVAC at both 60°C and BO°C. The vapor pressure data were plotted against the toluene weight fraction calculated on a filler free basis in Figure 16 and Figure 17 in order to determine effects of adsorption of toluene solvent by the filler. If Henry's law holds, the concentration, C, of the solvent in the polymer film is related to vapor pressure by, C = C!p. Therefore, the vapor pressure, PI, is directly proportional to toluene weight fraction, and is indirectly proportional to the solubility coefficient, C!. If PI decreases, C! increases. If PI increases, C! decreases. The data shown extrapolate to zero vapor pressure at zero toluene weight fraction, as they should. The data also follow different slopes for the different filler contents, as would be expected with different solubility coefficients. Following discussions given in Chapter II, in the absence of vacuoles and for nonadsorbing filler, C! = C!pVp , the solubility coefficient would decrease linearly with increasing filler volume fraction. With the exception of the neat PVAC data at 60°C, the general trend in Figures 16 and 17 is a decrease in the vapor pressure with increasing filler content. Therefore, the general trend of the solubility coefficient is to increase with added filler, which means that penetrant adsorption and/or filling of vacuoles with penetrant does occur. Due to the sensitivity of the quartz crystal microbalance to small changes in concentration of penetrant, the QCM has advantages over other sorption apparatus such as the quartz spring balance since closely spaced diffusion coefficient data can be obtained. The QCM has disadvantages, however, such as the inability to obtain diffusion data at as high of penetrant concentrations as the quartz spring balance, for reasons such as limited sorption data at beginning times and humps in sorption curves, as mentioned above. Yet, these limitations do not pose a problem in the present study, as requirements for model development are satisfied with "I I "I, " '" "I ,,, ,,",, "I 77 low concentration diffusion data and diffusion coefficients with higher penetrant concentrations can be estimated from models. Another disadvantage of the QCM is the inability to obtain data for highly viscous polymers. As an example, Figure 22 shows a quartz crystal frequency response curve for ethylbenzene diffusing into polybutadiene. The crystal frequency was very unstable during this run. This is due to the viscous nature of polybutadiene. Stress effects are presumed to be great with viscous polymers and the quartz crystal oscillations are damped causing instabilities in the resonant frequency. More examples of trial studies with highly viscous polymers and discussion of the stress effects in piezoelectric quartz crystals are given by Mikkilineni [1995], including interesting examples of polymers so viscous that the quartz crystals ceased to oscillate when coated with the polymer. The most significant cause of experimental error in this work is believed to be the measurement of the thickness of the polymer film. A possible error is in the measurement of the diameter of the film. The coated films are not perfectly symmetrical; therefore, several measurements of diameter are taken at different orientations and averaged to give the diameter used in the calculations. Another possible error is in the volume measurements with the micropipette. Some of the liquid could adhere to the inner surface of the micropipette tube. Since such a small volume of solution is used, significant error could result. Error also exists due to nonuniformity in the thickness of the polymer films. Visual inspection of the coated films show that the outer edges of the films are usually thicker than the center portion. This is thought to be due to surface tension in the solution while drying. Adhesion of the solution to the surface of the crystal creates forces which move solution to the outer edges of the film. For this same reason, filler particles also tend to be swept to the outer edges of the film. To minimize these effects, the solution is stirred with the tip of the micropipette while being coated. Appendix C gives and estimate of the error in the diffusion coefficient and penetrant weight fraction due to uncertainties in the thickness of the polymer film I '.I :~ • " 78 5979500 ,, 5978800 +.r,,~ o 50 100 150 200 250 5978900 5979400 5979300 N ::r:: 5979200 '" G' ~ ~ ;:j c:r 5979100 ~ <1:: . ", 5979000 ·1 Figure 22. Frequency Curve for PolybutadieneEthylbenzene System at 80°C. 19 and crystal frequencies. The error in the diffusion coefficient was estimated to be 65% and the error in the penetrant weight fraction was estimated to be 15%. Comparison of Methods for Determining Diffusion Coefficients There are four well known methods of analyzing sorption data in order to determine diffusion coefficients. The four methods are the halftime method, the initial slope method, the limiting slope method, and the moment method, and have been previously discussed in Chapter II. The moment method was chosen to analyze the sorption data in this work. This section compares the four methods of determining diffusion coefficients by using each method to analyze a set of sorption data. Sorption data for toluene diffusing into neat PVAC at 60°C was analyzed by the four methods, the halftime method(HT), the initial slope method(ISM), the limiting slope method(LSM), and the moment method(MM). The diffusion coefficients determined by the four methods are shown in Table XI, and compared in Figure 23. The diffusion coefficients determined from the four methods compare well. The diffusion coefficients determined from the limiting slope method has the most scatter. Mass uptake ratios ranging from 0.9 to 1 were used in the limiting slope method, resulting in more random error due to a limited number of data points in such a small range. Mass uptake ratios ranging from 0 to 0.5 or more were used in the initial slope method. The diffusion coefficient data determined from the initial slope method, the halftime method, and the moment method formed smooth curves. Discrepancies in the four methods were more evident at the larger weight fractions shown, since the sorption curves analyzed did not have enough data at the beginning of the curve. " 1e7  • • • 80 ~ . •• 1e8  1e9  1e10 0.00 .t. • o & .o... • • • o A • • • ~ . • I 0.05 • • •0 0 .~ • I • ~ I 0.10 0 Initial Slope Method IA HalfTime Method • Moment Method • Limiting Slope Method I I I 0.15 0.20 0.25 0.30 ·0 I I • I ·1 '~r ." 'I Toluene Weight Fraction Figure 23. Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C. 81 TABLE XI COMPARISON OF METHODS FOR DETERMINING DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC AT 60°C D X 109 (cm2/s) wav HT ISM L5M MM 1 0.0085 0.303 0.300 0.158 0.218 0.0213 0.761 0.854 0.289 0.536 0.0342 1.78 2.11 0.720 1.26 ':1 0.0468 3.45 4.48 1.84 2.82 "'1 ", 0.0596 6.22 7.35 1.74 4.83 ., ":l 0.0718 10.1 8.89 4.28 8.81 w· 0.0841 11.7 13.3 7.57 14.1 'I' 'I " 0.0963 16.0 16.5 18.4 24.1 '" JI 0.1085 18.3 29.4 26.3 34.2 "1 94.0 ., 0.1207 20.7 34.3 44.1 a., 0.1339 23.5 41.0 76.1 53.7 " 82 Correlating the Experimental Diffusion Coefficient Data Regressions Using a Modified FreeVolume Equation Experimental data from this work were compared with effective diffusion coefficient curves obtained from a freevolume equation modified to account for filled polymers. Plots of D vs. volume fraction of filler, Vj, gave insight into how diffusion varies with filler volume fraction. Graphs of D vs. vf, in semilog form, are shown in Figure 24 for 60°C data and Figure 25 for 80°C data. These plots are linear, within experimental error; thus, log D varies linearly with Vf' Models have been developed which describe the penetrant concentration dependence of diffusivity [Duda et al., 1982]. These models are also exponential in form. Combining a model which fits D vs. vf with a model which fits D vs. WI would give a model which d~scribes the dependence of D on both penetrant concentration and filler volume fraction. A linear equation which fits the log D vs. vI data may be written as (81 ) "I " '. where Deif is the effective mutual diffusion coefficient measured in the experiments, Dp is the diffusivity of neat polymer, and m is a positive constant. The negative sign signifies that diffusivity decreases with volume fraction of filler. With vI =0, this equation simplifies to Deli = Dp , as it should. Solving for Deif in equation 81 gives the following equation for diffusion of penetrants in filled polymers, ,. '! " ii 'I (82) An equation which has worked well for many polymerpenetrant systems in describing the concentration dependence of the mutual diffusion coefficient is Vrentas and Duda's [Duda et al., 1982] freevolume theory of transport which was discussed in Chapter II. For this reason, this equation was chosen for Dp . The effective diffusion coefficient in full form can then be given as (83) 83 " :1 "" 'It 0.00 0.01 0.02 0.03 0.04 0.05 1e11 1e10 w, • 0.01 1e7 • 0.02 .... 0.03 0.04 0.05 1e8 0.06 0.07 .. 0.08 CI.l N 0.09 E(,) 1e9 0.10 "" Q Figure 24. Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 60°C for Various Toluene Weight Fractions. 84 1e7 WI 0.01 0.02 0.03 1e8 0.04 0.05 0.06 .. tI:l N E 1e9 u '" Cl 1e10 , ,I 1e11 0.00 0.01 0.02 0.03 0.04 0.05 Figure 25. Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 80°C for Various Toluene Weight Fractions. Dp = Dl (1 4>1)2(1  2x4>d, D  D (/,(Wl ~ +W2eV2)) 1  01 exp .... , VFH .... VFH = Kll K12 wl(K21 + T  Tg1 ) +w2(K22 +T  Tg2 ). /' /' /' 85 (84) (85) (86) The preexponential constant, Do, mentioned in Chapter II is lumped in along with the Arrhenius form of temperature dependence into the term DOl' The free volume parameters Kn//'l K21  Tgl, K 12 //'l K 22  Tg2 , ~ and V2 were taken from Hou [1986]. The parameters DOl, e, X, and m were obtained from correlations with experimental data from this work, as explained in the following paragraphs. The interaction parameter, X, was obtained by fitting the FloryHuggins equation (87) to solubility data in which the equilibrium weight fraction of the penetrant in the polymer was known as a function of penetrant vapor pressure, Pl. The pressure p~ is the penetrant saturation vapor pressure. Table I and Table V of Chapter IV shows the data used for this regression. Figure 26 shows a comparison of solubility data for neat PVAC at 600 e with a theoretical curve calculated by equation 87. Figure 27 shows a similar comparison for 80°C data. The solubility data calculated by equation 87 compare reasonably well with experimental solubility data, with more scatter appearing in the 800 e data. The interaction parameter, X, is given in Figures 26 and 27 as 0.42 and 0.75 for 600 e and 800 e data, respectively. The parameters DOl and ewere obtained by fitting equation 83 to experimental diffusion coefficient data for neat PVAC (vf = 0). Figure 28 shows a comparison of experimenta} diffusion coefficient data for neat PVAe at 600 e with curves calculated by equation 83 with vf = o. The parameters DOl and eare given in this figure. The diffusion coefficients calculated by equation 83 compare well with experimental diffusion coefficient data. The parameter m was obtained by fitting equation 83 to experimental diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaC03 I 'I 86 0.6 ,., 0.5 0.4 x= 0.42 oe 0.3 0.. 0.08 0.10' 0.12 • experimental  theoretical 0.02 0.04 0.06 0.2 o.0 +orr,orr,......,,,r,.,,,,.,..,.....,r"""T""""""'Tr.....rr.........T,! 0.00 0.1 Toluene Weight Fraction Figure 26. Comparison of Experimental Solubility Data with the FloryHuggins Equation for PYACToluene at 60°C. 87 0.45 0.40 0.35 0.30 X= 0.75 0.25 0.eo.. 0.20 • • 0.15 • 0.10 • experimental theoretical 0.05 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Toluene Weight Fraction Figure 27. Comparison of Experimental Solubility Data with the FloryHuggins Equation for PVACToluene at 80°C. 88 1e6 1e7 • • 1e8 DOl = 5.48 X 105 ~ = 0.63 0.25 0.30 0.35 • experimental theoretical 0.05 0.10 0.15 0.20 1e10 1e9 1e11 0.00 Toluene Weight Fraction Figure 28. Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for Toluene Diffusing into Neat PYAC (vf = 0) at 60°C. 89 (V j = 0.0428). Figure 29 shows a comparison of the diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaC03 at 60°C with curves calculated by equation 83 with VI = 0.0428. The parameter m is given in this figure. The fit of equation 83 to the experimental diffusion coefficient data in this figure is reasonable, but these data show more scatter than the data for neat PYAC. Figure 30 shows both of the previous comparisons of the data of neat PVAC and CaC03 filled PVAC with a weight of 10% CaC03 , as well as comparisons of the diffusion coefficient data of CaC03 filled PVAC with weights of 3.3% (VI = 0.0137) and 4.9% (Vj = 0.0205) CaC03 at 60°C to predicted curves calculated by equation 83. The curves for data with 3.3% and 4.9% CaC03 are purely predictive. Only the volume fraction of filler, vj, in the equation was changed. All other parameters were obtained from previous correlations. Figure 31 shows similar regressions of neat PVAC data and data of CaC03 filled PVAC with a weight of 10% CaC03 using equation 83 as well as predictions of the diffusivity data of CaC03 filled PVAC with weights of 3.3% and 4.9% CaC03 at 80°C. As before, the parameters DOl and ~ were obtained by fitting the experimental diffusion coefficient data of neat PYAC to equation 83, and the parameter m was obtained by fitting the experimental diffusion coefficient data of CaC03 filled PVAC with a weight of 10% CaC03 to equation 83. The predictive curves of the diffusion coefficients of CaC03 filled PYAC with weights of 3.3% and 4.9% were calculated by equation 83 by changing only the volume fraction of filler, V j. The diffusion coefficient curves calculated with equation 83 compare very well with the experimental diffusion coefficient data of neat PVAC and PVAC with 10% CaC03 . Also, the diffusion coefficient curves calculated with equation 83 for PVAC with 3.3% and 4.9% CaC03 at 60°C and 80°C show a reasonable prediction of the experimental data. The predictions of the diffusion coefficients for PYAC with 4.9% CaC03 are the better of the two samples. The data for PVAC with 3.3% CaC03 shows more scatter, showing low predictions of data at 60°C and high predictions of data at 80°C. 90 1e6 1e7 1e8 ~ r;r.; .......... m= 96.7 N S c..> """ Cl 1e9 • • • • , 1e10 'I • ) • experimental 43 theoretical ~ :> ~ 1e11 :~ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ., Toluene Weight Fraction Figure 29. Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for CaC03 Filled PVAC with a Weight of 10% CaC03 (Vf = 0.0428) at 60°C. 91        .  DOl =5.48 X 105 ~ = 0.63 m=96.7 1e8 1e6 1e7 ... CI'J N 1e9 Eu "" 0 open symbols Run #1 0 .*J solid symbols Run #2 1e10 o· 0 0 % CaC03 D. 3.3 % CaC03 1e11 0 4.9% CaC03 •) 10 % CaC03 • 0 ~.. Regression ) "l 1e12  Predicted ~ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 , Toluene Weight Fraction Figure 30. Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 60°C. 92 1e6 DOl = 1.18 X 104 ~ = 0.59 m= 120.5 1e7    .....    /    ~o ..... 1e8 40/ .. / ~ (/) / N £~ E ~c. (,) '' 0 /~~ open symbols Run #1 1e9 A ~ ~ solid symbols Run #2 0 0 %CaC03 D. 3.3 % CaC03 1e10 0 4.9 01<, CaC03 • 0 10 % CaC03 ~ ~ Regression ) '~" 1e11   Predicted ~.. :> 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Toluene Weight Fraction Figure 31. Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 80°C. 93 The freevolume parameters used in this subsection are shown in Table XII. The solubility data regressions show the parameter X to increase with temperature with a 56% difference between 60°C and 80°C. The parameter, ~, is usually assumed to be temperature independent [Duda, 1983]. Regressions of diffusion data at 60°C and 80°C show this agreement to within 7%. The parameter, DOll is temperature dependent and is assumed to follow the Arrhenius equation. For positive activation energy, E, the parameter, DOl, increases with temperature. Regressions of data verify the increase of DOl with temperature. The DOl at 80°C was 73% greater than DOl at 60°C. The parameter, m, increases with temperature with a 22% difference between the two temperatures. In a study of diffusion of hydrocarbons in silicone rubber, Barrer [1962] reported that the percentage difference in the diffusion coefficient between filled and unfilled rubber increased with temperature. Therefore, from the mathematical form of equation 83, m is expected to increase with temperature. Regressions Using Equations Derived by Barrer and Chio Experimental diffusion coefficient data for toluene diffusing into CaC03 filled PYAC at 60°C was used to test the correlative capabilities of the equations derived by Barrer and Chio [1965], as described in Chapter II. The freevolume equation was used for the unfilled diffusion coefficient, Dp , with parameters as given in the previous subsection. The first equation used was the simple form in which only the tortuosity of the diffusion paths around the filler particles was considered where K. is the structure factor. The expression of Fricke, y K.= , Y +vf (88) (89) r TABLE XII PARAMETERS OF THE MODIFIED FREEVOLUME EQUATIO FOR PVACTOLUENE 94 Parameter Vt (em3 /g) V2* (em3 /g) Kn/, (em3 jg K) Kl2 /, (em3 jg K) K21  Tg1 (K) K22  Tg2 (K) FreeVolume Parameters from Hou [1986] Regre~sed FreeVolume Parameters Parameter 60°C 80°C Value 0.917 0.728 1.57 x 103 4.33 X 104 90.5 256 41• >• 0.42 0.63 5.48 x105 96.7 0.75 0.59 1.18 x104 120.5 where Y Runge, 95· 2 for random spheres was used in equation 89. The expression of 1 [ 3vf ] (90) K = 1  1~ 3 ' (1 vf) 2 +vf  O.392vf / for a cubic lattice of spheres was also used. Both of these expressions, equation 89 and equation 90, gave the same value for K. The smallest value of the structure factor, K = 0.979, was for polymer with 10% filler (Vf = 0.0428). This value led to only a 2% decrease in the diffusion coefficient. Figure 32 shows this comparison, as well as the previously obtained regression of neat PVAC. The fit using equation 88 is very poor. The values of K calculated from equations 89 and 90 are used in subsequent calculations. The next equation used was the more complicated form which takes into account two separate phases, the polymer phase and the filler phase, (91 ) where O'p is a parameter related to the solubility in the polymer and 0'1 is a parameter related to the gas adsorption by the filler. The ratio 0'J/0'p was obtained by fitting equation 91 to experimental diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaCOi (vf = 0.0428). Figure 33 shows this regression, as well as the previously obtained regression of neat PVAC, and the predictions of the diffusivity data of CaC03 filled PVAC with weights of 3.3% and 4.9% CaC03 at 60°C. The diffusion coefficients obtained from the regression using equation 91 compare well with the experimental diffusion coefficients of toluene in 10% CaC03 filled PVAC, however, equation 91 does not predict well the experimental diffusion coefficients of toluene in 3.3% CaC03 and 4.9% CaC03 filled PVAC. The ratio 0'f /O'p obtained in the regression using equation 91 is much larger than unity, 1,351.5. However, even assuming independent sorption by filler and rubber phases, Barrer et al. [1962] reported values of the ratio 0'J/O'p less than unity for hydrocarbons diffusing into silica filled silicone rubber. Therefore, the ratio 0'f /O'p is unlikely to be as large as that obtained by the regression. Diffusion of toluene in CaC03 filled PYAC is possibly too complex to model with a diffusion "~l. 96 1e8 c::, ~ ~.. ~~ .. .& <>~ til ~ ~. N 1e9 c::,
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Title  Effective Diffusion Coefficients for Toluene in Calcium Carbonate Filled Poly(Vinyl Acetate) from Quartz Crystal Microbalance Sorption Experiments 
Date  19980501 
Author  Willoughby, Steven Lee 
Document Type  
Full Text Type  Open Access 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  EFFECTIVE DIFFUSION COEFFICIENTS FOR TOLUE E IN CALCIUM CARBONATE FILLED POLY(VI YL ACETATE) FROM QUARTZ CRYSTAL MICROBALA CE SORPTION EXPERIMENTS By STEVEN LEE WILLOUGHBY Bachelor of Science Cameron University Lawton, Oklahoma 1992 Master of Science Oklahoma State University Stillwater, Oklahoma 1994 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1998 EFFECTIVE DIFFU 10 COEFFI IE T FOR TOLUE E I CALCI MCARBO T FILLED POLY(VI YL A ETATE) FROM QUARTZ CRYSTAL MICROB LA SORPTIO EXPERIME TS Thesis Approved: Thesis Advis r LJ 6 &cd! ~of the Graduate College 11 PREFACE A quartz crystal microbalance (QCM) was used to obtain effective diffusion coefficients of toluene diffusing into CaC03 filled poly(vinyl acetate). Experimental diffusion data were obtained at 60°C and 80°C for weight percents of 0,3.3,4.9, and 10% CaC03 at concentrations of toluene below 0.15 weight fraction at 60°C and at toluene weight fraction below 0.10 at 80°C. Effective diffusion coefficients were also calculated from a freevolume equation modified to account for the filler by fitting experimental effective diffusion coefficients to this equation. Several other models were also examined for their ability to accurately represent the experimental effective diffusion coefficient data. I would like to express sincere appreciation to my advisor, Dr. Martin S. High, and committee members, Dr. Robert 1. Robinson, Jr., and Dr. D. Alan Tree, for their advice, support, and interest in this work. I want to thank Mr. Mark C. Drake and Rexam Graphics for financial support and for suggesting this study. I also want to thank the students I've met at asu for their h.elp and friendship. Finally, I want to express appreciation to my wife, Sheena, for her support and encouragement, and for her help and company in many classes. III Chapter TABLE OF CONTE TS Page I. II. INTRODUCTION . BACKGROUND ON DIFFUSION OF PENETRA TS IN POLYMERS . 1 3 Experimental Methods for Obtaining Diffusion Coefficients 3 Fundamental Diffusion Equations .. . . . . . . . . 3 General Sorption Methods for Studying Diffusion. 4 The Quartz Crystal Microbalance Sorption Apparatus 7 Methods of Analyzing Data Obtained Using the Quartz Crystal Microbalance. . . . . . . . . . . . . 11 Obtaining a 'Sorption Curve from Frequency Data 13 Evaluating Diffusion Coefficients Using the HalfTime Method 15 Evaluating Diffusion Coefficients Using the Initial Slope Method . . . . . . . . . 17 Evaluating Diffusion Coefficients Using the Limiting Slope Method . . . . . . . . 19 Evaluating Diffusion Coefficients Using the Moment Method 19 Evaluating the Weight Fraction of Penetrant in the Polymer Film 22 Diffusion of Organic Penetrants into Polymers at Temperatures above the Glass Transition Temperature, Tg . • • • • . . . . • • • • • • • • • 22 Free Volume Theory . . . . . . . . . . . . . . . . 25 Anomalous Diffusion of Penetrants into Polymers 28 Mass Transfer in Heterogeneous Systems 30 Diffusion in Filled Polymers . . . 30 Diffusion in Crystalline Polymers 38 Structure Factor. . . . . . . . . . 41 Treatment of Adsorption as a Chemical Reaction. 43 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 44 III. MODIFICATIONS OF EQUIPMENT AND PROCEDURES 45 IV. EXPERIME TAL RESULTS. . . . . . . . . . . . . . 51 lV Chapter V. DISCUSSION OF RESULTS . Page 67 Comparison of Methods for Determining Diffusion Coefficients 79 Correlating the Experimental Diffusion Coefficient Da.ta .. 82 Regressions Using a Modified FreeVolume Equation. 82 Regressions Using Equations Derived by Barrer and Chio . . . . . . . . . . . . . . 93 Regressions Using an Equation Derived from Reaction Principles . . . . . . . . . . 98 Numerical Simulation and Regression of Sorption Curves 100 VI. CONCLUSIONS AND RECOMMENDATIONS. 104 Conclusions . . . . 104 Recommendations . 106 BIBLIOGRAPHY 108 APPENDICES . . 112 APPENDIX A  Experimental Procedures . 113 Preparing a Polymer Solution 113 Coating a Quartz Crystal with a Polymer Film. 114 Evacuating the Quartz Crystal Microbalance 114 Operating the Quartz Crystal Microbalance Sorption Apparatus. 116 Film Thickness Calculations . . . . . . . . . 119 APPENDIX B  Derivation of the Moment Method Formula for Evaluating Diffusion Coefficients 121 APPENDIX C  Error Analysis. . . . . 124 Diffusion Coefficient. . . . 124 Penetrant Weight Fraction 126 Sample Calculation . . . . 127 APPENDIX D  Results of Numerical Simulations and Correlations 128 APPENDIX E  Computer Programs 145 v Table LIST OF TABLES Page I Solubility Data and Diffusion Coefficients of Toluene In PVAC with 0.0% CaC03 at 60°C . . . . . . . . . . .. ..... 53 II Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 3.3% CaC03 at 60°C . . . . . . . . . . .. ..... 54 III Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 4.9% CaC03 at 60°C . . . . . . . . . . .. ..... 55 IV Solubility Data and Diffusion Coefficients of Toluene ln PVAC with 10% CaC03 at 60°C '" .. 56 V Solubility Data and Diffusion Coefficients of Toluene III PVAC with 0.0% CaC03 at 80°C . . . . . . . . . . .. .,. .. 57 VI Solubility Data and Diffusion Coefficients of Toluene in PVAC with 3.3% CaC03 at 80°C . . . . . . . . . . .. ..... 58 VII Solubility Data and Diffusion Coefficients of Toluene III PVAC with 4.9% CaC03 at 80°C . . . . . . . . . . .. ..... 59 VIII Solubility Data and Diffusion Coefficients of Toluene III PVAC with 10% CaC03 at 80°C 60 IX X Diffusion Coefficients of Toluene in Neat PVAC from Mikkilineni . Diffusion Coefficients of Toluene in Neat PVAC from Hou . 70 71 XI Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C . . . . . . . . . . . . . .. 81 XII Parameters of the Modified FreeVolume Equation for PVACToluene . . . . . . . . . . . . . . . . . . . . . . 94 XIII Analytical Solution of the Diffusion Equation with Constant D . 130 XIV Numerical Solution of the Diffusion Equation with Constant D . 131 VI Table XV Page FreeVolume Equation Parameters For PVACToluene at 60 °C . . . 144 VB Figure 1 LIST OF FIGURES A Typical Reduced Sorption Curve Obtained from a StepChange Sorption Experiment . . . . . . . . . . . . . . . Page 6 2 3 4 5 Schematic Diagram of the Quartz Crystal Microbalance Frequency Response of a Coated Quartz Crystal to a Step Change in Chamber Pressure . . . . . , . . . . . . . . . , Sorption Curve Obtained from Frequency Measurements Calculation of Diffusion Coefficients Using the Initial Slope Method . 9 14 16 18 6 Calculation of Diffusion Coefficients Using the Limiting Slope Method 20 7 Calculation of Diffusion Coefficients Using the Moment Method 21 8 Solubility Coefficient, (J", of Propane in Natural Rubber Filled with ZnO Vs Volume Fraction Filler. Curve (a),' (J" = O'pvp + (J"JVJ with (J"J = 0.0305; Curve (b), (]' = O"pVp , O'J = o. . .. , . . . . .. 37 9 Schematic Diagram of the Quartz Crystal Microbalance Modified Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 46 10 Sorption Curve Before the Sample Cylinder was Installed 47 11 Sorption Curve After the Sample Cylinder was Installed . 49 12 Effective Diffusion Coefficients of Toluene in CaCOa Filled PVAC at 60°C. . . . .. 61 13 Effective Diffusion Coefficients of Toluene in CaCOa Filled PVAC at 80°C. 62 14 Solubility Data of Toluene in CaCOa Filled PVAC at 60°C, 63 vlll Figure 15 Solubility Data of Toluene in CaC03 Filled PVAC at 80°C. Page . 64 16 Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 60°C.. '. . . . . . . . . . . . . . . . . . . . . . . . .. 65 17 Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 80°C . 18 Diffusion Coefficients of Toluene in Neat PVAC at 60°C 19 Diffusion Coefficients of Toluene in Neat PVAC at 80°C 66 68 69 20 An Example of a Sorption Curve that was Rejected Due to Very Few Data in the Initial Stages of Diffusion 73 21 An Example of a Sorption Curve that was Rejected Due to a Hump in the Curve . . . . . . . . . . . . . . . . . . . . . . . 75 22 Frequency Curve for PolybutadieneEthylbenzene System at 80°C 78 23 Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C . . . . . . . . . . . . . . . . . . .. 80 24 Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 60°C for Various Toluene Weight Fractions . . . . . . . . . . . . . . . . . . . . . . . .. 83 25 Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 80°C for Various Toluene Weight Fractions . . . . . . . . . . . . . . . . . . . . . . . .. 84 26 Comparison of Experimental Solubility Data with the Flory Huggins Equation for PVACToluene at 60°C. . . . . . . . . . . . .. 86 27 Comparison of Experimental Solubility Data with the Flory Huggins Equation for PVACToluene at 80°C. . . . . . . . . . . . .. 87 28 Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for Toluene Diffusing into Neat PVAC (vJ = 0) at 60°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 29 Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for CaC03 Filled PVAC with a Weight of 10% CaC03 (vJ = 0.0428) at 60°C .. . . . . . . . . . . . . . . .. 90 lX Figure Page 30 Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 60°C . . . . 91 31 32 33 Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 80°C ... Comparisons of Experimental Diffusion Coefficients with a Simple Structure Factor Equation for CaC03 Filled PVAC at 60°C . Comparisons of Experimental Diffusion Coefficients with Barrer and Chio's Equation for CaC03 Filled PVAC at 60°C . . 92 96 97 34 Comparisons of Experimental Diffusion Coefficients with an Equation Derived from Ileaction Principles for CaC03 Filled PVAC at 60°C . 99 35 Comparison of Numerical and Analytical Solutions of a Diffusion Equation with Constant Diffusion Coefficient 132 36 Comparison of Numerical and Analytical Solutions for the Fractional Mass Uptake with Constant Diffusion Coefficient 133 37 Comparison of Diffusion Coefficients Calculated from a Moment Method Evaluation of Sorption Data with Diffusion Coefficients Calculated from a Constitutive Equation 135 38 Comparison of Simulated Sorption Data with Constant Diffusivity, D = 1 X 108 , to Mass Uptake Ratios with the Diffusivity Changed to D = 5 X 108 ...•................• 137 39 Regression of Simulated Sorption Data with Constant Diffusivity, D = 1 X 108 138 40 Regression of Simulated Sorption Data with a Simple Concentration Dependent Diffusivity, D = a +be 139 41 Regression of Experimental Sorption Data for Neat PVACToluene at 60°C Using a FreeVolume Equation Theory Constitutive Equation 141 42 Comparison of Experimental Diffusion Coefficients with a Diffusion Coefficient Curve Calculated from the FreeVolume Equation for Neat PVAC at 60°C 142 x ~~ p~ 43 Comparison of Experimental Sorption Curve with a Simulated Sorption Curve Using the FreeVolume Constitutive Equation for Neat PVAC at 60 °C 143 Xl CHAPTER I INTRODUCTION Heterogeneous materials such as filled polymers are used extensively in industrial processes and commercial applications. Fillers are important constituents of many elastomers, plastics and coatings. They come in the form of small particles of different shapes. Fillers may be crystalline or amorphous. They have a significant influence upon the properties of materials and can be used to increase the mechanical durability, improve elastic properties, or give certain color or optical properties to the polymeric materials. Fillers also modify the sorption and permeability to diffusants. Diffusion and transport of matter in filled polymers are important in processes such as the drying of coatings and adhesives; air or moisture permeability of paint films; and the use of membranes in separations processes. Of particular interest is the process that led to this study, viz., drying of polymer based coatings on paper in which a filled polymer solution is coated on a continually moving web of paper. The web passes through a dryer where the solvents are removed from the surface of the coating with the help of nozzles blowing air at a controlled velocity and temperature. Solvents move to the surface of the coating by molecular diffusion, which is often the rate controlling step of solvent removal. Controlling the drying process is important in order to obtain a coating free of defects. Appropriate modeling of the drying process facilitates the selection of optimum operating conditions to produce a dry coating without defects. Drying models are benificial in finding drying conditions for new products, in productivity improvements in which line speed increases, and in tlle design of new drying equipment. Models [Vrentas et al., 1994] have been developed to describe the drying 1 2 process of coatings on continuous webs. An important parameter in drying models is the diffusion coefficient for solvent diffusing through the coated film. The diffusion coefficient is a function of the solvent concentration in the film, the amount of filler in the film, and the temperature of the film and surrounding air. Mass transfer in filled polymers is a complicated process which involves not only diffusion in the continuous polymer phase. Other factors such as adsorption of penetrant on the surface of fillers and small gaps in the structure, which are filled by diffusing gases, affect the overall transfer of penetrant in a filled polymer. An application of the classical solution of Fick's diffusion equation to sorption data for filled polymers leads to an "effective" diffusion coefficient which in most cases is smaller than the true diffusivity in the polymer phase due to immobilization of gases from the presence of fillers. This subject is discussed in the next chapter. The objective of this work was to measure sorption data on a filled polymer using a piezoelectric quartz crystal microbalance and to evaluate effective diffusion coefficients from the data. The diffusion coefficient was evaluated for various solvent concentrations, filler contents and temperatures. Acrylic polymers are used for most paper coating applications, and typical solvents are toluene or water. The polymer chosen for this study was poly(vinyJ acetate) (PVAC) and the solvent was toluene. This choice showed similarities in properties with a typical coating for paper, and the results for neat PVAC could be compared with experimental results of previous researchers in this field [Mikkilineni et al., 1995]. Coatings utilize a variety of fillers, additives, and optical brighteners. Calcium carbonate is a typical filler and it was chosen for use in this work. The results of this work were compared with known models for mass transfer in filled polymers. CHAPTER II BACKGROUND ON DIFFUSION OF PENETRANTS IN POLYMERS The first section of this chapter contains a discussion of experimental methods for obtaining sorption data, with particular emphasis on the quartz crystal microbalance method used in this study. The next section contains a discussion of methods of analysis for the quartz crystal microbalance in order to obtain a diffusion coefficient. This is followed by several sections which contain discussions of various behaviors inherent to diffusion in polymers. Major topics. discussed are Fickian and anomalous diffusion, the free volume theory of polymersolvent diffusion, and mass transfer in heterogeneous materials. Experimental Methods for Obtaining Diffusion Coefficients Experimental methods for studying the diffusion of gases in polymers have been discussed by Felder and Huvard [1980}. Their historical overview dates back almost 170 years and contains discussions of many diffusion phenomena and methods. Another source, which contains more detailed demonstrations of the theory, is the work of Crank and Park [1968}. This section focuses on methods of greatest relevance to this work. Fundamental Diffusion Equations Diffusion of a gas in any material, rubber or plastic, amorphous or crystalline, neat or filled, involves transport of the gas from one part of the material to another. Just as the viscosity is a coefficient that describes the transport of momentum in a gas or liquid in the familiar Newton's law of viscosity, and the 3 4 heat transfer coefficient describes the transport of thermaJ energy in a materiaJ according to Fourier's law of heat conduction, the diffusion coefficient describes the transport of mass in a material in accordance with Fick's law of diffusion, J = _DoC ox' (1 ) where J is the rate of transfer of diffusing gas per unit area of a specified sectional area, C is the concentration of diffusing substance, x is the space coordinate measured normal to the section, and D is the diffusion coefficient, or diffusivity. This equation is referred to as Fick's first law of diffusion. The diffusion coefficient has dimensions of length2 time1 and in this work, has units cm2 secI. From a mass balance on an element of volume containing material, the differential equation of diffusion takes the form oaCt = ~ox (DOoxC) +~oy (DOoyC) +~oz (DOoCz ) , where D can be a function of C. If D is constant then equation 2 becomes (2) (3) If the diffusion is in one direction r equations 2 and 3 simplify to what is commonly referred to as Fick's second law of diffusion, (4) and, if the diffusion coefficient is constant, (5) The equations for onedimensional diffusion are valid for this work since thin polymer films were used. General Sorption Methods for Studying Diffusion In general, there are three experimental techniques from which diffusion coefficients are obtained. These are: sorption into or out of a polymer, permeation 5 through a membrane into a closed chamber, and permeation through a membrane into a flowing stream. Only sorption is discussed in detail in this thesis since this study used a sorption method. Sorption methods are mainly used in measurements of equilibrium solubil· ities, twostage sorption processes, slow processes, studies of anomalous diffusion, highpressure measurements and studies of cracking and crazing [Felder and Buvard, 1980]. Categories of sorption measurements include integral sorption, integral desorption, interval sorption, and interval desorption. In integral sorption, the polymer is abruptly exposed to a step change in penetrant concentration at the boundaries of the polymer. In integral desorption, the polymer is initially equilibrated with a penetrant, and abruptly exposed to a penetrantfree atmosphere, usually a vacuum, until no penetrant remains in the polymer. Interval sorption and desorption are the same as integral sorption and desorption, respectively, except that the initial and final penetrant concentrations are' greater than zero. In all four categories of sorption measurements, the mass of the penetrant in the polymer is measured as a function of time, either directly, as in gravimetric techniques, or indirectly by measuring a change in properties such as pressure or volume of the atmosphere surrounding the polymer. The method used in this work involved determining the mass of the penetrant in the polymer indirectly by measuring a change in the frequency of a piezoelectric quartz crystal as a function of time.. For both direct and indirect methods, the data usually appear as shown in Figure 1. The variable, MtiMoo ! is plotted on the ordinate, where Mt is the instantaneous mass sorbed or desorbed and Moo is the mass sorbed or desorbed at equilibrium. Time, t or ,fi is plotted on the absisca. In the integral sorption method, Moo is the ultimate final mass of the penetrant in the polymer. In the integral desorption method, Moo is the initial mass of the penetrant in the polymer. In the interval sorption or desorption method, Moo is the difference between the initial and the final mass of the penetrant in the polymer. 6 1.0 ,=_ 0.8 0.6 0.4 0.2 0.0 Figure 1. A Typical Reduced Sorption Curve Obtained from a StepChange Sorption Experiment. 7 If a sorption experiment was followed by a desorption experiment to the initial penetrant level, one might expect that the sorption and desorption curves would coincide. However, this is only true for Fickian diffusion with a constant diffusion coefficient, D. If D increases with the penetrant concentration, such as toluene in PVAC as is used in this work, the sorption curve lies above the desorption curve. If D decreases with the penetrant concentration, the sorption curve lies below the desorption curve. If swelling is significant and stress relaxation of the polymer controls the penetration rate, the sorption curve is sigmoidal, but desorption from the swollen polymer is Fickian and initially relatively rapid. If relaxation occurs slowly, two stage sorption occurs. Fickian and NonFickian diffusion are discussed in a later section. The Quartz Crystal Microbalance Sorption Apparatus All data in this work were' obtained with a quartz crystal microbalance (QCM). The QCM used in this work was developed by Deshpande [1993] to measure the diffusion characteristics of a penetrant in a polymer film. The QCM was later modified by Mikkilineni to reduce operational difficulties, increase the accuracy of the data, and increase the ease of operation of the apparatus [Mikkilineni, 1995]. This subsection discusses the QCM apparatus as it existed in the work of Mikkilineni [1995]. The principal component of the QCM was a 0.5 inch circular gold coated piezoelectric quartz crystal. A mechanical stress, when applied to a piezoelectric quartz crystal, induces an electric potential across the crystal, or vice versa, an electrical potential applied across a piezoelectric quartz crystal induces a mechanical strain in the crystal. This property of the quartz crystal facilitated extremely accurate measurements of the rate of diffusion of solvents in polymers. By including the quartz crystal in a circuit, a small electric potential was created across the crystal and the crystal vibrated at its resonant frequency. 8 A 6 MHz gold coated ATcut piezoelectric quartz crystal was used in the experiments. The crystal was coated with a polymer solution. A penetrant diffusing into this polymer coating decreased the resonant frequency of the crystal. The resonant frequency of the piezoelectric quartz crystal was found to be extremely sensitive to changes in mass on the surface of the crystal. A load as low as 1 nanogram could cause a detectable change in the frequency of the quartz crystal [Mikkilineni, Tree, and High, 1995]. Since the quartz crystal was so sensitive to mass changes, a small change in the concentration of the solvent in the polymer caused a measurable change in the frequency, which allowed diffusion data to be obtained at low and closely spaced solvent concentrations. The bulk of the QCM was constructed from a stainless steel six way cross chamber (Kurt J. Lesker Co., C60600) that had entrances at all six ports. Figure 2 shows a schematic diagram of the QCM. The chamber was operable from 1011 torr to slight positive pressures and up to a maximum temperature of 500°C [Mikkilineni, 1995]. A standard quartz crystal sensor (Leybold Inficon, 750207Gl) which housed and provided a circuit for the quartz crystal was placed in the right port of the chamber (see Figure 2). A thinfilmdeposition controller (Leybold Inficon, XTC/2), interfaced to a computer, monitored the frequency of the crystal. A computer program recorded frequency as a function of time as the penetrant diffused into the polymer. The QCM was enclosed in a box made of plywood lined with insulating material to accurately control the chamber and polymer film temperature. A heating element mounted on a metallic stand was used to heat the box. A fan was used to circulate the hot air in the box. The fan motor was mounted on the outside of the box to prevent overheating of the motor due to the high operating temperature in the box [Mikkilineni, 1995]. The motor was connected to the fan inside the box by an aluminum shaft and was mounted onto one side of the box. A temperature controller (LFE Instruments, 2004) was used to regulate the temperature inside the box. A resistance temperature device was also mounted in the back port of the chamber (not shown in Figure 2) to measure the temperature 9 To Computer! Interface . \ Solvent Flask Needle Valve ~ RightAngle Valve ~ Insulated Box Six Way Cross ~ Bottom Port~ Liquid Nitrogen Trap Hea~ CO;I~ Figure 2. Schematic Diagram of the Quartz Crystal Microbalance. 10 inside the chamber. Several resistance temperature devices were installed in the box and surface mounted onto the cross chamber to check for temperature gradients in the box and chamber [Mikkilineni, 1995]. The chamber was evacuated to remove any contaminants, such as air or penetrant from previous experiments. The bottom port of the chamber led to two vacuum pumps. A rotary vane pump (Leybold Inficon, Trivac A 4A) was used to obtain pressures of the order of 102 torr, and a turbomolecular pump (Leybold Inficon, Turbovac 50) was used to reduce pressures further, to the order of 107 torr. A liquid nitrogen trap (Kurt J. Lesker Co., LNF 1000) was used in line to keep solvent vapors from entering the pumps and oil vapors from entering the chamber. A gate valve (Kurt J. Lesker Co., SG0400) was connected to the turbomolecular pump and a right angle valve (Huntington Laboratories, EV 150) was connected to the rotary vane pump through the liquid nitrogen trap'. The two valves were required to connect the chamber to the rotary vane pump or to the turbomolecular pump, so that vapors would not enter the turbomolecular pump during initial evacuation of the chamber with the rotary vane pump. To prevent the turbomolecular pump from overheating, cold water was circulated through a heat exchanger mounted on the pump casing. A Pirani gauge (Leybold Inficon, PG3; sensor, TR901) and a hotcathode ionization gauge (Leybold Inficon, IG3; sensor, 850675G5) were connected on the outside of the box to the pipe which connected the chamber with the turbomolecular pump. The gauges measured pressure in the chamber and were used only during the evacuation process. The Pirani gauge measured low vacuum (,....., 1000 _104 torr absolute) and the hotcathode ionization gauge measured high vacuum ("" 102  1010 torr absolute). A pressure gauge (Druck Instruments, DPI 265) was also mounted in the top port of the chamber to measure pressure in the chamber after the penetrant was introduced. This gauge can be used to measure pressures of solvent vapors in the chamber over the range of 0  19.999 psia and can withstand temperatures up to 300°C [Mikkilineni, 1995]. 11 The front port of the chamber was used to introduce solvent into the chamber. A jacketed metallic flask containing the penetrant was connected to the chamber using 1/4 in. stainless tubing. Water could be circulated through the flask to heat or cool the solvent to a desired temperature. A toggle valve was used to control the flow of penetrant into the chamber. A needle valve was used to isolate the flask. A quick connect coupling was used in the tubing connecting the solvent flask and the chamber to make solvent change in the flask easy [Mikkilineni, 1995]. A magnetic stirrer was used to ensure that the solvent was weD mixed throughout the experiment. Further modifications, made to the QCM for this work, are discussed in the next chapter. A detailed, stepwise experimental procedure is given in Appendix A. Methods of Analyzing Data Obtained Using the Quartz Crystal Microbalance Diffusion coefficients for polymerpenetrant systems are typically highly concentration dependent. However, all of the methods discussed in the literature assume that the diffusion coefficients are concentration independent. In order to calculate the concentration depen'dence, the sorption experiments are performed over small step changes in pressure driving the sorption.' Over these small changes in driving force, the concentrations do not change significantly and the methods employing constant diffusion coefficients are valid. Crank [1956] derived solutions for onedimensional diffusion in a plane sheet. These solutions apply to diffusion into a sheet of polymer assuming that the edges of the sheet can be ignored and diffusion can be considered as taking place only through the surface of the sheet. Since the polymer films used in this work were about 3 to 5 pm in thickness and about 5 to 8 mmin diameter, this assumption is valid. The derivation starts with the onedimensional diffusion equation (6) 12 The initial and boundary conditions for this equation are C(x,O) = Gl , (7) C(L, t) = Co, (8) oC (0 t) = O. (9) ox The initial condition indicates that at t = 0 the entire polymer film is at a uniform concentration of C1 . The first boundary condition indicates that at any time after the beginning of diffusion into the polymer film, the concentration at the surface of the film, x = L, is maintained constant at Co. This boundary condition emphasizes the importance of introducing the penetrant into the quartz crystal microbalance chamber instantaneously and, during the sorption experiment, maintaining a constant concentration at the surface of the polymer film. This requires the pressure in the chamber to be constant. Since the QCM is mainly used for step change absorption experiments, Co is greater than Cl . The second boundary condition indicates that there is no transport of penetrant through the bottom of the polymer film at x = O. The exact solution of equation 6 subject to equations 7, 8 and 9 is given by Crank [1956] as C  C1 ~( )n f (2n + I)L  x ~(  = L 1 er c +L  1)"erfc (2n + I)L + x . Go  C1 n=O 2..j(Dt) n=O 2vfl5ij (10) (11) The concentration of penetrant in the polymer film according to equation 10 is a function of time and position inside the film. The mass of penetrant in the polymer film at a specified time could be determined by integrating the concentration profile of equation 10 over the thickness of the slab at that time. From this procedure, the ratio of the mass sorbed by the polymer film at time t to the mass sorbed by the film at equilibrium (t = 00) is [Crank, 1956] Mt = _~~ 1 ex {D(2n+l f ll. 2t}. Moo 1 7r2 L (2n + 1)2 P 4L2 n=O The corresponding solution that is useful for small times is [Crank, 1956] M (Dt)1/2 { 00 nL } _t = 2 2 7rl / 2 + 22":( 1)"ierfc /(15t) . Moo L n=l (Dt) (12) 13 The important assumptions made in the derivation of equations 11 and 12 are that 1. Diffusion of penetrant into the polymer film occurs in one dimension. 2. Concentration of the penetrant at the surface of the film is constant. 3. The diffusion coefficient is a constant. The first assumption holds well since the polymer films used in this work had a small thickness compared to the surface area coated on the quartz crystal. The thicknesses of the films were about 3 to 5 p.m, while the diameters of the films were about 5 to 8 rom. The second assumption holds well in this work since the pressure of the penetrant in the chamber was increased quickly and leaks in the chamber were small. Since diffusion of a penetrant into a polymer is dependent on the concentration of penetrant in the film, the third assumption can only be satisfied approximately if the sorption experiment is carried out over a small step of change in concentration. This was achieved by introducing small amounts of penetrant into the chamber for a sorption run and performing a series of these small step change absorption experiments over the penetrant concentration range of interest. Obtaining a Sorption Curve from Frequency Data As the penetrant diffuses into the polymer fiim, a computer records the frequency as a function of time. Figure 3 shows a plot of a typical response of crystal frequency to a step change in pressure of the penetrant in the chamber where the polymer coated quartz crystal is housed. From the frequency responce data, graphs of MtiMoo versus t or yfi are generated. Theoretical details of piezoelectric quartz crystals are described in Mikkilineni's thesis [1995]. The essence of the theory is that the change of mass, 6.m, on the quartz crystal is proportional to the frequency change, .6.f, of the crystal, .6.m = k6.f, (13) 14 5967240 .____, 5967220 5967200 .. 5967180 N ::r: '" ;>.. u 5967160 s= C1.) ;:s 0"' C1.) e.t:: 5967140 5967120 5967100 5967080 +r,r,l o 10 20 30 40 50 Figure 3. Frequency Response of a Coated Quartz Crystal to a Step Change in Chamber Pressure. 15 where k is a proportionality constant. The mass change of penetrant in the polymer film at time, t, after the step change in penetrant pressure, is Mt = k(ft  il), (14) where 11 is the frequency of the coated crystal before the penetrant is allowed into the chamber, and It is the frequency of the coated crystal at time t after the penetrant is allowed into the chamber. If only the first step of the sorption experiment is considered, then 11 is the frequency of the polymer coated, solventfree crystal and Mt is the total mass of penetrant in the polymer at time t. If the nth step is considered (n > 1) 1 then II is the frequency of the polymer coated crystal before the step change and Mt is the change in mass of the penetrant in the polymer from the onset of the step change to time t after the step change in chamber pressure. After sufficiently large time (I'V 15 minutes for unfilled polymer and up to 6 hours for filled polymer), the penetrant comes to equilibrium with the polymer film and the mass of penetrant in the film approaches a constant value. Polymersolvent equilibrium is established, and equation 14 is still valid as t + 00. The ratio of the mass sorbed at time t, Mt , to the mass sorbed at equilibrium, Moo, can be expressed as Mt It  II  =, (15) Moo 100  il where I (XI is the frequency of the coated crystal at polymersolvent equilibrium. Figure 4 shows a plot of Mt/Moo obtained from equation 15 versus .,ft. Such a plot is called a reduced sorption curve. Once the Mt/Moo data are obtained, the diffusion coefficients can be estimated. Evaluating Diffusion Coefficients Using the HalfTime Method The simplest technique for estimating the diffusion coefficient is the halftime method. Equation 11 is used, and the value of tf £7. for which Mt/Moo = 1/2 is approximately given as (16) 16 , 1.0 0.9 0.8 0.7 0.6 B ~ 0.5 ~ 0.4 0.3 0.2 0.1 0.0 0 10 20 30 40 50 ..Jt (s) Figure 4. Sorption Curve Obtained from Frequency Measurements. 17 with an error of about 0.001 per cent [Crank, 1956]. After performing the necessary arithmetic, the time t 1/ 2 for which MtiMoo equals 1/2 for a plane sheet is given by D = 0.19675L 2 (17) t 1/ 2 Evaluating Diffusion Coefficients Using the Initial Slope Method The initial slope method was initially used for evaluating the diffusion coefficients from data taken on the quartz crystal microbalance at Oklahoma State University. All the sorption curves obtained by Deshpande [1993] and Mikkilineni [1995] were analyzed with the initial slope method. During the initial stages of the sorption experiment, the polymer film behaves as a semiinfinite medium and the ratio of the mass uptake at time t to the mass uptake at time t = 00 increases linearly with 0, often to as much as 50 per cent of MtiMoo [Crank and Park, 1968]. At small times, the summation term in equation 12 can be neglected, and the equation becomes Mt 2 (Dt)1/2 Moo = ft L2 (18) According to equation 18, the average diffusion coefficient can be calculated as (19) where Ri is the initial slope of the MtlMoo vs. 0 curve. If the sorption curve is approximately linear up to MtiMoo = 1/2, equation 17 and equation 19 would yield about the same diffusion coefficient [Crank, 1956]. Figure 5 shows an exam· pIe of an evaluation of the diffusion coefficient by the initial slope method. The fractional solvent uptake was plotted as a function of the square root of time. Linear regression was used to determine the slope of the initial portion of the sorption, from MdMoo = 0 to 0.5. Equation 19 was used to calculate the diffusion coefficient. The sorption curve in this example is linear well above 50 per cent of MtiMoo. Most of the experimental sorption curves obtained in this work were linear to approximately 25 per cent. The length of the initial linear portion of the sorption curve will vary between sorption experiments depending on the concentration dependence of the diffusion coefficient [Crank and Park, 19681· 18 1.0 0.9 0.8 0.7 0.6 Ri=~(M/McxY~(tl/2) i 0.5 ~(tll2) ~ 0.4 0.3 ~(M/Mco) 0.2 0.1 0.0 0 5 10 15 20 25 30 35 40 .../t (8) Figure 5. Calculation of Diffusion Coefficients Using the Initial Slope Method. 19 Evaluating Diffusion Coefficients Using the Limiting Slope Method The limiting slope method [Balik, 1996; Palekar, 1995] uses only the first term of the series in equation 11. At large times, a plot of In {I  Mt/Moo } versus time approaches a straight line, and the average diffusion coefficient can be calculated as (20) R1 is the limiting slope of the In {I  Mt/Moo } vs. t curve. Figure 6 shows an example of an evaluation of the diffusion coefficient by the limiting slope method. In {I  MtiMoo} was plotted as a function of time. Linear regression was used to determine the limiting slope, from MtiMoo = 0.99 to 0.996. Equation 20 was used to calculate the diffusion coefficient. Evaluating Diffusion Coefficients Using the Moment Method In this work, the moment method was used to evaluate the diffusion coefficients. The moment method [Felder and Huvard, 1980; Palekar, 1995J is advantageous since the entire sorption curve is used instead of just the initial or final portion as in the methods discussed above. The quantity (21 ) (22) is first calculated by numerical integration, where T, is the first moment of the monotonically increasing curve, MtiMoo versus time. Then, the average diffusion coefficient is calculated by L2 D=. 3Ts Details of the derivation of equation 22 are given in Appendix B. Figure 7 shows an example of the evaluation of the diffusion coefficient by the moment method. To calculate Ts , a plot was made of 1  :::~ vs t as shown in Figure 7. The first moment, Ts , is the area under the curve of Figure 7 and was found by numerical integration. The trapezoidal rule was used in this work to find the area under the curve. 20 0...._. 1 2 ~ 8~ 3 ....I.. ..~.... 4 5 6 o 200 400 t (8) 600 800 1000 Figure 6. Calculation of Diffusion Coefficients Using the Limiting Slope Method. 21 1.0 0.9 0.8 •• 0.7 •••• ' . 0.6 ••• 8 • ~ • 0.5 •• ~ •• ..I •• 0.4 •• . . ••• 0.3 \ 0.2 0.1 0.0 0 200 400 600 800 1000 1200 1400 t (s) Figure 7. Calculation of Diffusion Coefficients Using the Moment Method. 22 Evaluating the Weight Fraction of Penetrant in the Polymer Film Since the diffusion coefficient is a function of the concentration of the penetrant in the polymer film, equations 17, 19, 20 and 22 give average diffusion coefficients over the concentration range. Vrentas et al. [1977] has concluded that the average diffusion coefficient obtained from a stepchange sorption experiment is equal to the diffusion coefficient at a concentration which is 0.7 of the way across the concentration interval, with less than 5% error. (See Equation 25.) The equilibrium weight fraction of the penetrant in the polymer film, w~q , is the ratio of the mass of the penetrant at equilibrium, Moo, to the total mass of the polymersolvent mixture at equilibrium, M~t, eq Moo ) WI = Mtot' (23 00 Since, according to equation 13, the change of mass on a quartz crystal is proportional to the change in crystal frequency, equation 23 becomes eq = 100  iI (24) WI 100  fo' where fo is the frequency of the bare, uncoated crystal. The weight fraction which corresponds to the average diffusivity can be calculated by (25) where Wli is the weight fraction of penetrant before introducing more penetrant, and WI! is the weight fraction of penetrant at equilibrium after introducing the penetrant. Diffusion of Organic Penetrants into Polymers at Temperatures above the Glass Transition Temperature, Tg Diffusion in polymerpenetrant systems do not follow the laws of the classical theory of molecular diffusion [Crank and Park, 1968]. There has been considerable interest in understanding the anomalous behavior of diffusion in polymerpenetrant systems [Vrentas et aL, 1986]. However, it is also of interest to discover 23 the circumstances under which polymerpenetrant systems can be analyzed using classical or Fickian theory. Generally, Fickian theory can be used to describe diffusion in polymerpenetrant systems which have high temperatures and concentrations, since the polymerpenetrant system behaves as a purely viscous fluid under these conditions. Diffusion at low concentrations and temperatures below the glass transition temperature can also be analyzed with the classical theory since the system has properties of an elastic solid [Vrentas et al., 1986]. At intermediate temperatures and concentrations, diffusion in the polymerpenetrant system can have anomalous or nonFickian behavior which are caused by viscoelastic effects [Vrentas et aI., 1986]. This section discusses the mechanisms of diffusion of penetrants into polymers at temperatures above the glass transition temperature, and discusses typical Fickian sorption features. Diffusion of simple gases such as hydrogen, argon, nitrogen and carbon dioxide requires a limited rotational oscillation of only one or two monomer units in order to translate from one position to a neighboring one since the molecular size of such gases is small compared to the monomer unit of a polymer, however if a gas has a molecular size comparable or larger than the monomer unit of the polymer \ a cooperative movement by the Brownian motion of several monomer units, i.e. a polymer segment, must take place during diffusion [Fujita, 1968]. Organic vapors, such as toluene which was used in this work, are among such large molecular size penetrants. The diffusion of organic vapors in polymers exhibit different features in the regions above and below the glass transition temperature, Tg , of a polymer. These features are simple at temperatures above Tg and complex at temperatures below Tg [Fujita, 1968]. At temperatures well above the glass transition temperature, polymers are in a rubbery state and diffusion of penetrants in polymers usually follow Fick's law. The presence of the penetrant weakens the molecular interaction between adjoining chains, increasing the magnitude of Brownian motion within the polymer and therefore increases the rate of penetrant diffusion [Felder and Huvard, 1980]. This effect increases with the amount of penetrant present, so the diffusion 24 coefficient of organic vapors in rubbery polymers usually exhibits a concentration dependence [Felder and Huvard, 1980}. In Fickian diffusion in a polymer film during sorption, the distribution of penetrant, and the change of penetrant concentration with time, are governed by Fick's onedimensional, differential equation for diffusion. The space coordinate is taken in the direction normal to the polymer film. The solution of this equation depends on the initial and boundary conditions for the penetrant concentration, C1 and on how the diffusion coefficient, D, varies with C. Solutions for constant D were shown earlier. Solutions for a concentration dependent D are given by Crank [1956]. Mathematical studies of Crank and coworkers have developed the following summary of sorption features which are Fickian or normal type [Fujita, 1968]. 1. Both absorption and desorption curves are linear in the initial stage. For absorption, the linear region extends over 60% or more of Moo, where Moo is the amount of vapor absorbed per gram of dry polymer until the sorption equilibrium is reached. For D(C) increasing with C the absorption curve is linear almost up to the final sorption equilibrium. 2. Above the linear portions both absorption and desorption curves are concave to the abscissa axis, irrespective of the form of D(C). 3. When the initial concentration Ci and the final concentration Cf are fixed, a series of absorption curves for films of different thicknesses are superposable to a single curve if each curve is replotted in the form of a reduced curve, i.e. Mt is plotted against t 1/ 2/ L. This same applies to the corresponding series of desorption curves. 4. The reduced absorption curve so obtained always lies above the corresponding reduced desorption curve if D is an increasing function of C between Ci and Cf. Both reduced curves coincide over the entire range of t when D is 25 constant in this concentration interval. The divergence of the two curves becomes more marked as D increases more strongly with C in the concentration range considered. 5. For absorptions from a fixed Cj to different O/s, the initial slope of the reduced curve becomes larger as the concentration increment Cf  Cj becomes larger, provided that D increases monotonically with 0 in the range considered. This same applies to the reduced desorption curves which start from different Ci's to a fixed Cf. Criteria 1, 2 and 3 are independent of how D varies with C. Therefore, these three criteria are checked to determine whether a given polymerpenetrant system exhibits Fickian diffusion. Criteria 1 and 2 can be checked easily by inspecting the appearance of the sorption curves. Since criterion 3 requires sorption measurements to be made with films of different thicknesses, a system is often regarded as Fickian if experimentally determined sorption curves have appearances consistent with criteria 1 and 2 [Fujita, 1968]. Free Volume Theory A theory that has been well accepted for describing the temperature and concentration dependence of the mutual diffusion coefficient in polymersolvent systems is the freevolume theory of transport developed by Vrentas and Duda [Duda et aI., 1982]. Vrentas and Duda [Vrentas and Duda, 1976; Vrentas et al., 1985] proposed the following equation for the solvent selfdiffusion coefficient, D1 , in a polymersolvent system, D D ( E ) ("Y(WIVt + W2e~·)) 1 = 0 exp  RT exp   , VFH (26) (27) (28) 26 In equations 26 and 27, Do is a constant preexponential factor, E is the energy per mole of molecules to overcome intermolecular attractive forces, R is the ideal gas constant, T is the temperature, WI is the mass fraction of component 1 (I = 1 for solvent; 1 = 2 for polymer), Vt is the specific critical hole free volume of component 1 required for a jump, ~ is the ratio of the critical molar volume of the solvent jumping unit to the critical molar volume of the polymer jumping unit, Kll and K 21 are freevolume parameters for the solvent, K l2 and K 22 are freevolume parameters for the polymer, I is an overlap factor which is introduced because the same free volume is available to more than one molecule, VFH is the average hole free volume per gram of mixture, and TgI is the glass transition temperature of component 1. Since this theory presents an expression for the selfdiffusion coefficient, Db expressing the mutual diffusjon coefficient, D (measured in the sorption experimental apparatus, and required for process calculatIons), in terms of D1 is desirable. Duda et al. [1979] proposed an approximation for low solvent concentrations which couples D to the selfdiffusion coefficient for polymer solvent systems, D = D1W IW2 (8J1.1) . RT 8WI T,p In this equation, J1.1 is the chemical potential of the solvent. The FloryHuggins theory was used to determine the concentration dependence of the solvent chemical potential [Duda et aI., 1982]. The FloryHuggins equation can be written as (29) where ¢h is the solvent volume fraction in the solution and X is the polymersolvent interaction parameter, which is assumed to be independent of temperature. Introduction of equations 26, 27, and 29 into equation 28 yields the following expression for the mutual diffusion coefficient, D, in a rubbery polymersolvent system [Duda et al., 1982] (30) (34) (35) 27 D D ( E ) (,(WIVt +W2eo;*)) 1 = 0 exp  RT exp  .. (31 ) VFH .. VFH = Kn K12 wI(K2I + T  TgI ) + w2(K22 + T  Tg2 ). (32) , I I A.. I = ..WI t:;.o .. (33) 'fI TEO TEO' WI vI +W2 V2 where Vp is the specific volume of pure component l. In this version of the free volume theory, there are 13 independent parameters to be evaluated. Grouping some of them together leaves only 10 parameters which are required to determine the mutual diffusion coefficient, D: Kul'l K21  TgI, Kl2 /" Kn  Tg2 , t:;.*, O;*l Do, E, eand x· The parameters ,t:;.*/K u and K 21  Tg1 can be determined from data for' solvent viscosity as a function of temperature, "11 (T) 1 by using a nonlinear regression to correlate the viscosity data with b~* /Kll ) In 171 = In Al + (K T) T ' 21  gl + where the parameters, AI, , ~* / Ku and K2I  TgI are assumed to be independent of temperature [Duda, 1983]. The parameters IV;*IK12 and K22  Tg2 can be determined from polymer melt viscosity data as a function of temperature, T/2(T), by using a nonlinear regression to correlate the data with [Duda, 1983] hV2*jK I2 ) In "12 =In A2 + (K T) T 22  g2 + The critical volumes, Vt and V:t, can be estimated to be the specific volumes of the solvent and polymer at 0 K. Molar volumes at aK can be estimated using group contribution methods developed by Sugden and Biltz [Duda, 19831· The parameter X can be determined from solubility data in which the equilibrium weight fraction of the penetrant in the polymer is known as a function of the solvent vapor pressure, PI, using the FloryHuggins equation [Duda, 1983], (36) 28 where p~ is the penetrant saturation vapor pressure. Lumping the parameters Do and E into the parameter DOl gives DOl = Doexp (.ff:r) , (37) which is the Arrhenius form of temperature dependence. The parameters DOl and ~ can be.determined by correlating diffusion coefficient data with a nonlinear regression analysis of equation 30 [Duda, 1983]. Anomalous Diffusion of Penetrants into Polymers Deviations from the Fickian type of process are generally described as anomalous or "nonFickian" and are almost always observed when the polymer is studied at temperatures below Tg , or within roughly lOoe above Tg [Felder and Huvard, 1980]. In general, the' polymers in which anomalies are observed are hard and glassy while normal sorption is observed in soft and rubbery materials. Polymeric materials in the rubbery state respond rapidly to changes in condition, but polymers in the glassy state take longer to come to equilibrium. For example, as the concentration in a polymer solution changes with time, the system must adjust to new conformations consistent with new values of concentration. Anomalous mass transfer processes are associated with the sluggish relaxation of large polymer molecules; however, in rubbery polymers which follow a Fickian diffusion process, relaxation of polymer molecules is fast compared to the diffusion process [Vrentas and Duda, 1979]. Fickian diffusion at temperatures above the glass transition, Tg , has been designated as case I diffusion and D depends only on concentration. As the glass transition temperature, Tg , of the polymer is approached, D begins to depend on time explicitly as well as on concentration. At moderate penetrant activities when swelling is appreciable and the temperature is less than, but within about 10°e of Tg , the mechanism of penetration may change from Fickian diffusion to a stress relaxationcontrolled process in which the penetrant advances in a sharply 29 defined front at a nearly uniform velocity. This mechanism is designated as case II transport [Felder and Huvard, 1980]. The two modes of transport are easily distinguished from the sorption results. In both modes at small times t, (38) where Mt is the cumulative mass absorbed at time t. For case I (Fickian) transport n = 1/2, and for case II transport n = 1. On a plot of Mt vs. 0, case I transport would be linear at small times and case II transport would be sigmoidal. A mode designated "super~case II transport" has also been observed [Felder and Huvardl which has a sorption curve convex to the time axis at large times on a plot of Mt vs. t, where a similar plot would be linear for case II transport and concave for case I transport. Another anomaly which can occur is two stage sorption in which swelling penetrants are sorbed by glassy polymers and a rapid approach to an apparant equilibrium state, followed by a gradual shift to the true equilibrium state is observed. This phenomenon has been attributed to a gradual relaxation of the elastic cohesive force in the polymer, and to a timevarying surface concentration of penetrant [Felder and Huvard, 1980]. Diffusion anomalies also occur in polymers at temperatures well above Tg when crystallites or fillers are present [Park, 19681. Fillers can lead to differences between diffusion coefficients obtained by steadystate and transient methods. Such differences have been reported for filled rubber [Barrer et al., 1963] at temperatures well above the glass transition temperature (Tg + 100°C) [Park, 1968]. These anomalies are not due to time effects, but are due to the complicated effect of microheterogeneities discussed in the next section. In crystalline polymers, sigmoidal sorption curves are obtained at temperatures well above Tg and the sigmoidal character is more marked at higher crystallineamorphous ratios. The effects in crystalline polymers are thought to result from slow responses to external changes in the crystalline regions, which could lead to a timedependence in the crystallineamorphous ratio and so produce timedependent diffusion coefficients leading to sigmoidal sorption curves [Park, 1968]. 30 Mass Transfer in Heterogeneous Systems Diffusion in Filled Polymers This discussion of diffusion in filled polymers begins with the effect of fillers in elastomers, since rubber products have been the most frequently studied filled polymer system [Barrer et al., 1962; Carpenter and Twiss, 1940; Morris, 1931; Smith, 1953; van Amerongen, 1947]. This discussion is, therefore, appropriate to introduce the present work since at the temperatures at which the experiments in this study were performed, the polymer, PVAC, is an elastomer. Most rubber compounds contain considerable amounts of fillers [van Amerongen, 1964]. The fillers may be spherical or nonspherical, and they may be reinforcing or nonreinforcing. Also, certain fillers could have a major effect on the diffusivity, solubility and permeability. Diffusion as mentioned above is the transport of a molecule from one part of a material to another. The solubility of foreign molecules in an elastomer is defined by a state of equilibrium between the molecules inside and outside of the polymer [van Amerongen, 1964]. In the equilibrium state the polymer has taken up as many penetrant molecules by dissolution as can be expected thermodynamically. As long as equilibrium has not been reached, the elastomer continues to absorb or desorb the foreign molecules, which involves transport by diffusion. Permeation is more complex than diffusion since it involves the absorption of the gas on one side of a membrane, diffusion of the gas to the other side of the membrane, and finally evaporation or extraction from the other side of the membrane. The simplest solutiondiffusion theory would be if the sorption of the gas at the surface of the membrane exposed to the gas obeyed Henry's law C = Clp, (39) where C is the dissolved species concentration in equilibrium with a gas whose partial pressure is p and CI is the solubility coefficient, and that the absorbed gas diffuses through the membrane in accordance with Fick's law J = D'lC, (40) 31 where J is the flux of gas through the membrane and D is the diffusion coefficient or diffusivity. In these equations, the solubility and diffusion coefficients (j and D are assumed independent of concentration. If the above equations are valid, the steadystate permeation rate per unit area through a membrane of thickness L is (41) where 8p is the partial pressure difference across the membrane, and the product P = aD is the permeability of the membrane to the gas. Since permeation is experimentally easier to study than diffusion [van Amerongen, 19641, much of the work on filled rubbers has focussed on the process of permeation instead of pure diffusion. A good discussion on the subject of diffusion and permeation in heterogeneous media is given by Barrer [1968]. Barrer describes the derivation of differential diffusion equations taking into account the volume fractions of polymer, filler and vacuoles. These equations and the concept of vacuoles will be discussed later. Most of the early studies of diffusion and permeation of gases in filled elastomers used gases with low molecular weight such as hydrogen, nitrogen, oxygen and air [Morris, 1931; Smith, 1953; van Amerongen, 1947, 1955]. This was a logical choice since the production of automotive tires is a major use of rubber and the permeability of air in tires was of great concern. Also, experimentation using gases with low molecular weight was easier since solute condensation was not a significant problem as when using high molecular weight condensable vapors. Van Amerongen [1955, 1964], showed that the diffusion coefficients of hydrogen, nitrogen and oxygen in natural rubber were greatly modified by the presence of fillers. Many mineral fillers, lamellar fillers and carbon black fillers were used in van Amerongen's work [1955, 1964]. All of the rubber mixtures contained about 20% filler by volume. Mineral fillers such as whiting, aluminum oxide and barium sulfate reduced the diffusivity by about 10 to 15% at a temperature of 25°C and 15 to 25% at a temperature of 50°C. Mineral fillers such as hisil and durosil 32 reduced the diffusivity by about 40 to 65% at temperatures of 25°C and 50°C. Fillers such as aerosil and tegN reduced the diffusivity by about 30 to 40% at the same temperatures. Lamellar fillers such as aluminum powder and mica powder reduced the diffusivity by about 60 to 75% at the same temperatures. At the same two temperatures, carbon black fillers reduced the diffusivity approximately 15 to 30% for thermax and P 33, 50 to 80% for slatex K and Vulcan 3, and 75 to 90% for spheron 9 and spheron 4. One explanation for a reduction in diffusivity with the addition of fillers was that the fillers behave as a geometric obstruction to the path of gas through the rubber. The average diffusion path length was increased by the presence of the filler particles and the localized direction of flow was in general not normal to the geometrical cross section of the membrane. While the tortuosity of diffusion paths might account for some reduction in diffusivity, some of the reductions noted above were quite large, which implied that the phenomenon was more than just a geometrical effect of impermeable fillers. As well as a modification in the diffusion coefficients, the values of the solubility coefficients, (J, in van Amerongen's work [1955] were considerably increased by the addition of filler. The sO},,lbility coefficients were found from the ratios of permeability and diffusion coefficients (P = (JD), and iq. some cases by direct measurements. The increase in solubility due to the added filler could have meant that a part of the gas diffusing into the polymer was adsorbed by the filler particles. Assuming that transport was restricted to the polymer phase only, the gas once adsorbed by the filler particles was rendered immobile and no longer participated in the diffusion process. The measured effective diffusion coefficient was then an average value between the diffusivity in the polymer phase and the zero diffusivity of the gas adsorbed on the filler particles. Barrer, Barrie, and Raman [1962] studied the diffusion of higher molecular weight gases such as nbutane, isobutane, npentane and neopentane in silica filled silicone rubbers. The filler used was Santocel CS, a relatively porous amorphous form of silica. Studies were performed over a temperature range of 30 to 70°C in 33 the neat rubber and at volumes of filler of 5.6, 10.6, 14.9 and 19.1%. Reported reductions in diffusivity with addition of filler were 15 to 38% for 5.6% filler, 23 to 39% for 10.6% filler, 32 to 53% for 14.9% filler, and 32 to 61% for 19.1% filler. The percent reduction in diffusivity due to the added filler was found to increase with temperature and size of the penetrant molecule. Barrer, Barrie, and Rogers [1963J studied the diffusion of propane and benzene in membranes of natural rubber with zinc oxide filler. The volumes of filler were 0, 5, 10, 20, 30, and 40% for diffusion of propane, and 0, 10, and 40% for diffusion of benzene. For diffusion of propane at 40°C, the reduction in diffusivity ranged from 3 to 26% over the volumes of filler given. The results of the filler reducing the diffusion of benzene in the membranes were reported for three temperatures, 30, 40, and 50°C, for 10% ZnO, and for one temperature, 40°C, for 40% ZnO. Results were als,O reported for various concentrations of benzene up to 0.10 volume fraction. For the membrane with 10% ZnO, the reduction in diffusivity was about 16 to 57% at 30°C, 22 to 56% at 40°C, and 19 to 26% at 50°C. For the membrane with 40% ZnO, the reduction in diffusivity was about 42 to 72% at 40°C. A filled polymer may contain two disperse phases, the filler and small vacuoles. Vacuoles are small gaps in the structure which are filled with the diffusing gas and usually occur at very high volume fractions of filler (above 50%) where incomplete wetting of the filler by the polymer is extensive [Barrer, 1968J. The following derivation of an effective diffusion coefficient which takes into account these disperse phases was developed by Barrer and Chio [1965]. The solubility of a gas in the polymerfiller system can be considered to be composed of three factors. The first is related to the solubility in the polymer, CTp . The second is related to the gas adsorption by the filler, CTf. The third is related to filling of gas pockets or vacuoles, CTl). If the distribution of gas in each phase obeys Henry's law, C = CTp, where p is the pressure of the gas at the surface of the 34 polymer, then the three factors are defined as C' ,...  p. vp, p C' Uf=.1.; p (42) where C;, Cf and C~ are the concentrations in molecules per cm3 of pure polymer, of filler particles and of vacuoles, respectively, and the solubility of the gas in the polymerfiller system becomes (43) where Vp, vf and Vv are volume fractions of polymer, filler and vacuoles, respectively. If there were no vacuoles present and the filler was nonadsorbing or fully wetted, then equation 43 would become (44) If equation 44 is valid, solubility should decrease linearly with increasing filler volume fraction, which is not the case for some polymerfiller systems studied [Barrer et al., 1962, 1963]. Barrer [1965, 1968] states that for one dimensional flow in the x direction, a differential equation of diffusion can be written as acp BCf BCv _ ( D ) fPCp D 02CJ D 02Cv at + Bt + ot  K, P Bx2 + J Bxz + v Bx2 ' (45) where, Cp , Cf and Cv are the numbers of molecules of diffusant per cm3 of membrane which are present in the polymer, on or in the fiUer and in the vacuoles, respectively. The terms K,Dp , Df and Dv are the effective diffusion coefficients in the polymer, on or in the filler and across the vacuoles, respectively. Dp is the diffusion coefficient in the pure polymer. The value K in the polymer effective diffusion coefficient is a structure factor which takes into account the tortuosity of diffusion paths. Defining C as the total number of molecules per cm3 of filled polymer gi ves C = Cp + CJ + Cll = up. Combining equations 42, 43, 45 and 46 gives BC = ((K,Dp)uPVp+Dfufvf + Dvuvvv) BZC. ot upvp+UfVf + UvVv Bxz (46) (47) 35 The term in brackets is the overall effective diffusion coefficient, DelI, in the filled polymer. Barrer [1968] also defines a diffusion coefficient, D~ff' by the flux, J, through unit area and the concentration gradient in the polymer phase only, J =  D'elfdedl'' x The effectivediffusion coefficients are related by If transport is restricted to the polymer phase (DI = 0, Dv diffusivity reduces to In the absence of vacuoles, equation 50 reduces to (48) (49) 0), the effective (50) (51) For zero sorption by the filler ((J' f = 0), equation 51 can be reduced to (52) Equations 50, 51 and 52 are the forms frequently found in literature [Barrer et al., 1962, 1963; van Amerongen, 1964]. Of interest in the study of diffusion in filled polymers is the extent to which the polymer wets the filler. Assuming no vacuoles are present, two extreme cases for the solubility of a polymerfiller system are (53) and (54) In equation 53, (J'j has the value for the unwetted filler powder. In equation 54, (J' f has a value of zero implying that the filler is completely wetted by the polymer. The parameter (J'f usually has values between the two limits of equation 53 and 54 36 since the filler must be at least partly wetted by the polymer and is by no means always fully wetted [Barrer, 1968; Barrer, Barrie, and Raman, 1962]. To illustrate this, Figure 8 shows solubility data taken from Barrer, Barrie, and Rogers [1963] The solubility coefficients are for propane in natural rubber filled with ZnO. To compare the data to the two extreme cases mentioned above, equation 53 and 54 were plotted. Curve (a) is a plot of equation 53 with O'p = 0.0495 and 0'1 = 0.0305, solubility coefficients of propane in rubber with no filler and in bulk filler, respectively. Curve (b) is a plot of equation 54 with Up = 0.0495 and O'f = 0, which would impl? complete wetting of the filler by the rubber. At low filler volume fractions the solubility coefficients follow a slope closer to that of curve (b) (complete wetting of filler), but at higher filler volume fractions the solubility coefficients are closer to curve (a) (no wetting of filler). As mentioned above, expectations are that the filler in the rubber have solubilities less than that of a free filler since the polymer should at least partially wet the filler, and that the polymer will not always fully wet the filler. This would lead to solubility coefficients lying between curves (a) and (b). To complicate matters, vacuoles may also be present, which was neglected here for simplification. Since the rate of decrease of solubility falls off with increasing filler content in Figure 8, more filler surface is probably available for sorption of penetrant. A possible explanation of this and the drastic reductions in diffusivity with added filler is nonuniform dispersion of the filler particles [Barrer, Barrie, and Rogers, 1963]. If particle conglomerates are formed, gaps are created between some of the particles and two processes could occur which reduce diffusivity. Gas may occupy the void volume created and be rendered immobile. Moreover, since the filler surface in the conglomerates will not be completely wetted by polymer, a larger fraction of the conglomerate is available for gas adsorption. 37 0.050 ., 0.045 . bO ~eu ..... 0.040 a ~eu ..... ci.. ....: CI.i ~eu '' 0.035 b 0.030 0.0 0.1 0.2 0.3 b 0.4 0.5 0.6 Figure 8. Solubility Coefficient, u, of Propane in Natural Rubber Filled with ZnO Vs Volume Fraction Filler. Curve (a), U = UpVp + UfVf with (Jj = 0.0305; Curve (b), U = UpVp , u1 = O. Taken from Barrer, Barrie, and Rogers [1963]. 38 Diffusion in Crystalline Polymers A description of some of the work done in the area of diffusion in crystalline polymers is included in this section since the effect of crystals in crystalline polymers is similar to that of fillers in filled polymers. Solution of a penetrant in perfectly crystalline regions is not to be expected [Barrer, 1968], and usually, a decreases linearly with increasing crystalline fraction, V c (subscript c and a will replace f and p, respectively since the subject IS now diffusion in crystalline polymers which have a crystal and an amorphous phase) [Michaels et aI., 1964] as (55) where Va. is the volume fraction of amorphous polymer and Va. + Vc = 1. The diffusion coefficient of a penetrant in a polymer crystal is expected to be very small; therefore, crystals act similarly to impermeable filler particles [Barrer, 1968]. The difference is that the degree of crystallinity may be changed by heating, cooling and annealing and that the crystals should always be fully wetted by the polymer chains in the amorphous regions [Barrer, 1968]. Crystals only act as a geometrical obstruction which increases the diffusion path length. The effective diffusion coefficient can be expressed similarly as was for filled polymers, (56) (57) where Da. is the diffusion coefficient in the completely amorphous polymer. However, most forms [Hedenqvist and Gedde, 1996] use a tortuosity factor, T, such as the relationship proposed by Michaels and Bixler [1961], who interpreted f'i, as the product of a geometrical impedance factor, T, and a chain immobilization factor, (3 [Barrer, 1968], Da. Dell = Tf3" Initially, suggestions were made that f3 described the reduced segmental mobility of polymer molecules in the vicinity of crystalline surfaces [Michaels and Bixler, 39 1961]. If the crystalline surface area is large, then 13 may be large.. Later, findings were that 13 depends more on the size of the penetrant molecule and only weakly on the crystallinity [Michaels et al., 1964]. Peterlin [1975, 1984] has a slightly different form for the effective diffusivity (58) where 'I/; is a detour factor describing the physical obstruction of the crystallites, which takes val ues between a and 1, B (~ 1) is a blocking factor, and D~ is the diffusion coefficient in the "relaxed" region of the amorphous phase of the crystalline polymer [Hedenqvist and Gedde, 19961. The amorphous phase is a complex network of tortuous very thin and broad fiat channels, and the amorphous chains are restrained in mobility by their ends fixed in the adjacent crystals [Peterlin, 1975]. Therefore, diffusion in the amorphous phase is anisotropic and the effective diffusion coefficient Def f of the crystalline polymer will be smaller than D;. The effect of the restraint on chain mobility increases with a smaller thickness of amorphous layers. An increase in the amorphous component implies an increased thickness of such a layer and an increase in the diffusion coefficient D~ [Peterlin, 1975]. Therefore, D: is strongly dependent on crystallinity. The blocking factor, B, in equation 58 describes the geometrical blocking that qccurs when the penetrant molecules are too large to be able to enter the amorphous interlayers. The immobilization factor, {3, in equation 57 is analogous to the blocking factor, B. It takes into account the constraining effect of the crystals on the amorphous phase and is included in the crystallinity dependence of D~ in equation 58. The detour factor, '1/;, and the tortuosity factor, T, are both purely geometrical factors [Hedenqvist and Gedde, 1996]. Several other models have been used to describe the diffusion in crystalline polymers [Hedenqvist and Gedde, 1996]. Some are empirical and some are based on physical and chemical processes involving the penetrant and the polymer. The models listed above (equations 57 and 58) do not include a dependence on the 40 concentration of the penetrant. In most polymers, the diffusivity is greatly dependent on concentration. This concentration dependence is sometimes expressed as [Hedenqvist and Gedde, 1996] Deff  Doe"fC , (59) where Do is the diffusivity at zero penetrant concentration and / is a constant. A linear dependence of diffusivity on concentration was used for poly(ethylene terephthalate) and 4nitro4'hydroxyazobenzene in an aqueous solution [Iijima and Chung, 1973] Deff = Do(l +/C). (60) A polynomial was used to fit the diffusivity of water in PA6 [Hanspach and Pinno, 1992] (61) where /1 and /2 are constants. Equations based on the free volume theory have also been used to explain diffusion in crystalline polymers. In simple terms the free volume theory is expressed as [Hedenqvist and Gedde, 1996] (62) where B is a constant and I is the fractional free volume of the system. An equation based on the free volume theory was used to fit the diffusivity of various gases into polyethylene [Kulkarni et al., 1983; Stern et al., 1972, 1983, 1986] (63) where Ad is a constant which depends on the size of the penetrant, and fa is the amorphous fractional free volume, (64) The quantities VI and V2 are the volume fractions of the penetrant and the polymer in the amorphous phase, respectively. The values 11 and h are the corresponding (66) 41 fractional free volumes. DT in equation 63 is the thermodynamic diffusivity and is related to the effective mutual diffusion coefficient, Delh through [Hedenqvist and Gedde, 1996] Dell = D T 8(lnaI), 1  VI 8(ln vd (65) where al is the activity of the penetrant. The quantity Va in equation 63 corrects the effective diffusivity for crystallinity. A similar correction is shown in this work for filler content. Another equation related to the free volume theory which has been used to fit diffusivity data is [Horas and Rizzotto, 1989] DT Dell = D' 2~ !ITA (67) where A and B are functions of crystallinity. Structure Factor The structure factor, K, takes into account the tortuosity of the diffusion path around the filler particles (or crystals). If the filler or polymer crystals are impermeable and have no vacuoles then Peff DefJ p = KVp ; D = K, (68) p p where the subscript a replaces p for crystalline polymers. The value K, can be found from a plot of PelI IPp or DelI ID p versus the volume fraction of filler V I (or crystals vc ). Many mathematical forms have been derived which give the dependence of K, on the volume fraction of filler. The problem is analogous to the electrical conductance of a heterogeneous medium composed of a dispersion of particles in a continuous medium of different conductivity [Barrer et al., 1962, 1963; van Amerongen, 1964]. Maxwell [1891] considered a suspension of spherical particles so dilute in the medium that the spheres had no effect on one another [van Arnerongen, 1964]. Fricke [1931] extended the study to include oblate and prolate spheroids. The expression for K, for these approaches is y K.= , Y +vf (69) 42 where Y is a shape factor which is 2 for random spheres and decreases to 1.1 as the shape changes from a sphere to an oblate spheroid with axial ratio 4 : 1 [Barrer, 1968; Barrer, Barrie, and Rogers, 1963]. Lord Rayleigh [1892] and Runge [1925] considered a cubic array of uniform spheres for treating a more concentrated dispersion of particles [Barrer, Barrie, and Rogers, 1963]. The equations for K are K  1 [1 _ 3vf] (70)  (1  Vj) 2 +Vj  O.392v}O/3 ' for a cubic lattice of spheres, and K= (1~Vf) [1 2+VfO.3~:~jO.0134V~]' (71) for a cubic lattice of cylinders normal to the direction of flow. Barrer, Barrie, and Raman, [1962] have found in a study of diffusion of hydrocarbon penetrants in various silicafilled rubbers that the influence of the filler in reducing the diffusion coefficients is greater than would b.e .expected for a regular dispersion of nonconducting spheres. They have suggested that a filled polymer is a considerably more complex medium, and if trends of diffusivity reduction with temperature and size of penetrant are significant then these simple models must be modified. For the crystalline polymers polyethylene terephthalate, polyethylene and Nylon, Lasoski and Cobbs [1959] found that water vapor permeabilities followed the relationship Pej j 2 ( ) p~ =Va' 72 If the effective diffusion equation is of the form of equation 56, then up to V c = 0.4 [Barrer, 1968] (73) In this equation, K decreases much more rapidly than would be expected from Maxwell's or Rayleigh's equations. Refering now to equation 57, Michaels and Bixler [1961] determined that the tortuosity factor for a series of polyethylenes with different crystallinities using He as a penetrant followed the equation n T = va , (74) 43 where n is a constant which takes different values for different polymers. The immobilization for helium was assumed to be small, hence (3 = 1 [Michaels and Bixler, 1961]. Treatment of Adsorption as a Chemical Reaction The problem of adsorption of a gas on filler particles may also be treated as diffusion combined with a chemical reaction [van Amerongen, 1964]. Physical adsorption of a gas by the filler has a similar effect on diffusion as chemical reaction. In either case the gas is rendered inactive and has the misleading effect of increasing the apparant solubility without increasing the permeability. Permeability is related to diffusivity by P = DO', (75) so the diffusivity will decrease. A differential equation which could describe diffusion with adsorption is [Crank, 1956] (76) where CI is the concentration of penetrant which is adsorbed on the filler particles, and C is the concentration of penetrant which is free to diffuse. In the simplest case, the concentration of the immobilized penetrant, CII is directly proportional to C, Cj =RC. (77) This equation is referred to as a linear adsorption isotherm [Crank, 1956]. Substituting for Cj from equation 77 into equation 76 gives ac D a2c at = R + 1 ax2 ' (78) which is the usual form of a diffusion equation with an effective diffusion coefficient D Deff = R+l' (79) 44 If tortuosity is also a factor, the effective diffusion coefficient becomes (80) where K. is the tortuosity factor and Dp is the diffusivity in the pure polymer. Summary This work uses several methods and equations which were discussed in this chapter. Diffusion coefficients for CaC03 filled PVAC were evaluated from frequency data obtained with a QCM. Sorption data were calculated by analyzing frequency data using equation 15. Diffusion coefficients were calculated by the moment method, equation 22, and penetrant weight fractions were calculated by equations 24 and 25. The freevolume equation 30 was used to fit the experimental diffusion coefficient data for toluene diffusion in unfilled PVAC. Several equations were fit to the experimental data for toluene diffusion in filled PYAC, including Barrer's equations 51 and 52, and equation 80 which was derived from reaction principles. The freevolume equation 30 was used for Dp , the unfilled polymer diffusivity, in these equations. Also, fits to filled polymer diffusion coefficient data were done using the structure factor equations 69 and 70 in combination with equation 52. CHAPTER III MODIFICATIONS OF EQUIPMENT AND PROCEDURES The equipment in this work was designed to measure diffusion coefficients of penetrants diffusing into polymers. A description of the experimental apparatus was given in Chapter II and also in the works of Deshpande :[1993] and Mikkilineni [1995] from the development stage of the QCM through several modifications. This chapter contains a discussion of additional modifications made to the equipment, changes in the data analysis for determining diffusion coefficients, and procedures for sample preparation. A schematic of the QCM with modifications is in Figure 9. A sample cylinder was installed between the solvent flask and the sixway cross chamber. The sample cylinder allows an instantaneous step change of pressure in the chamber. The toggle valve and the needle valve described in Chapter II were removed. Three ball valves were used, one to isolate the solvent flask, one to isolate the sample cylinder, and one between the sample cylinder and the sixway cross chamber. ew 3/16 inch stainless steel tubing was installed between the solvent flask and the sixway cross chamber to increase the flow rate of vapor. Previously, some of the tubing was 1/4 inch. The ball valves allowed the sample cylinder to be filled with penetrant vapor, then introduced into the chamber when needed. Before installing the sample cylinder, the penetrant was introduced into the chamber too slowly to adequately approximate a step change. Figure 10 shows a typical sorption curve for the experimental setup without the sample cylinder. The rate at which the penetrant was introduced into the chamber is reflected by the slow rate of mass sorption in the initial portion of the curve. This severely affected the accuracy of data analysis since the initial slope method uses the initial portion of the curve and the moment method uses the entire curve. Installing the sample cylinder 45 46 Insulated Box To Computer Interface ~~~~\ Liquid Nitrogen Trap Solvent Flask Rotary Vane Pump RightAngle Valve L..r.......J t To Vent Hood~ Bottom Port Hea~ COil~ Figure 9. Schematic Diagram of the Quartz Crystal Microbalance Modified Setup. 0.9 • 47 1.0 .::=_=__~ ••• •••••••• • •• 0.8 0.7 0.6 i 0.5 ::E 0.4 0.3 0.2 • • 0.1 0.0 .....~..,....__r____r__.______1 o 5 10 15 20 25 Figure 10. Sorption Curve Before the Sample Cylinder was Installed. 48 solved this problem. The sample cylinder allows the penetrant to be introduced into the chamber almost instantaneously. Figure 11 shows a sorption curve for the experimental set up with the sample cylinder. otice that the initial portion of the curve has a shorter tail since the pressure was increased almost instantaneously. The next change to the QCM was not a modification of the equipment, but a replacement of the gaskets in both the right angle valve and the gate valve. The gaskets were made of rubber and were deteriorated due to the organic vapors used in the QCM chamber. The leakage from the valves led to unreliable data. The plastic tubing which was used for circulating cold water to the turbomolecular pump was replaced with 1/4 inch copper tubing, decreasing the chances of a water leak. A water circulation line constructed of 1/4 inch copper tubing was connected to the solvent flask. A centrifugal pump was used to circulate water through the solvent flask from a water bath heated by a constant circulation immersion heater. The computer program recording the frequency data from the deposition monitor was modified to record the data at smaller time intervals. The previously used IBM XT computer was replaced by a Gateway computer with a 33 MHz 486 CPU. This also allowed for data collection at smaller time intervals. The moment method, discussed in Chapter II, was used to analyze the experimental sorption data in this work. Previously [Deshpande, 1993; Mikkilineni, 1995], the initial slope method, also discussed in Chapter II, was used to analyze the sorption data from the QCM. The initial slope method is very sensitive to the position on the curve at which the slope is calculated. Also, diffusion is sometimes so fast that there are a limited amount of data in the initial portion of the sorption curve, making analysis with the initial slope method even more difficult. The moment method uses the entire sorption curve; thus, is not as sensitive to the initial portion of the sorption curve. An Excel spreadsheet was developed to evaluate the area under the sorption curve using the trapeziodal rule. In this work, samples were prepared by dissolving a known weight of polymer (PVAC) and filler (CaC03 ) in a known volume of solvent (tetrahydrofuran) Figure 11. Sorption Curve After the Sample Cylinder was Installed. 49 50 to make a very dilute solution ('" 3% polymer and '" 0.1% filler). The crystals were coated from a micropipette by dropping a known volume (typically 7j.d) of the polymer solution onto the crystal surface and spreading it with the tip of the micropipette. After drying, the diameter of the polymer film was measured with a ruler. The thickness of the polymer film was calculated from the surface area of the film, the volume of the solution used to coat the crystal, and the weights and densities of the polymer, filler and solvent used to make the polymer solution. Weights were recorded with a mass balance; of the empty bottle, after adding polymer, after adding filler, and after adding solvent. The diameter of the film was measured at several locations. The average diameter was used to calculate the area of the polymer film by using the formula for the area of a circle. Detailed procedures for preparing polymer solutions, coating crystals, and calculating film thickness are given in Appendix A. Since the design phase of the QCM, many improvements have been made. Continual improvement of the equipment and procedures will lead to more accurate data, more ease of operation, and perhaps more uses for the QCM. Suggested improvements are below. All of the monitors for measuring pressure, temperature, and frequency are separate devices. Also, data must be copied from the data acquisition computer and analyzed as a separate step. To make the apparatus and data analysis less cumbersome, a graphical instrumentation program such as LabVIEW should be set up on the data acquisition computer. This program could be used to imitate the appearance and operation of actual instruments, such as pressure and temperature monitors. All of the measured properties, pressure, temperature, and frequency, could be monitored at one location, the computer. Such a program can also produce live graphs of data during an experimental run. All calculations involved in determining the diffusion coefficient could be performed by the graphical instrumentation program. The box surrounding the QCM is made of insulated plywood, but this does not stop all of the heat loss. The quartz crystal microbalance should be fitted inside an insulated oven to better control the temperature of the ambient air and the chamber. CHAPTER IV EXPERIMENTAL RESULTS This chapter presents the results of the sorption experiments performed with the quartz crystal microbalance to obtain effective diffusion coefficients of toluene diffusing into CaC03 filled PVAC. The diffusion coefficients of toluene diffusing into PVAC with 0.0% CaC03 (neat PVAC) are compared with the results of other researchers [Mikkilineni, 1995 and Hou, 1986] in Chapter V. The effective diffusion coefficients of toluene diffusing into CaC03 filled PVAC were measured at temperatures of 60°C and 80°C for 0, 3.3, 4.9, and 10% (by weight) CaC03 . The diffusion coefficients were obtained at low concentrations of toluene, less than 0.15 weight fraction for 60°C data and less than 0.10 weight fraction for 80°C data. Effective diffusion coefficients were calculated by using the moment method. Effective diffusion coefficients and toluene weight fractions are given in Tables I through VIII. The tabulated diffusion coefficients are plotted as a function of toluene weight fraction in Figure 12 for the 60°C data and Figure 13 for the 80°C data. To verify the reproducibility, each set of experimental data was replicated using the same quartz crystal and polymer film. Equilibrium sorption solubility data for toluene in CaC03 filled PVAC were also measured at temperatures of 60°C and 80°C for 0, 3.3, 4.9, and 10% (by weight) CaC03 . Equilibrium toluene vapor pressures, PI, were measured, and solubility data in the form of pdp~, where p~ is the toluene saturation vapor pressure, were calculated. The solubility data and equilibrium toluene weight fractions are given in Tables I through VIII. The solubility data are plotted as a function of toluene weight fraction in Figure 14 for the 60°C data and Figure 15 for the 80°C data. The solubility data are also plotted on a "filler free basis" in Figure 16 for the 60°C data and Figure 17 for the 80°C data, in which the toluene 51 52 weight fraction is calculated on a filler free basis, i.e., Ms/(Ms +Mp), where Ms is the mass of the toluene solvent and Mp is the mass of the PVAC polymer. 53 TABLE I SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 0.0% CaC03 AT 60°C eq PI/P~ wC11J D X 109 (ern2 WI 1 / s) Run #1 0,0121 0,0737 0,0085 0,218 0,0253 0.1406 0.0213 0.536 0.0380 0.2028 0.0342 1.26 0.0506 0.2593 0,0468 2,82 0,0634 0,3114 0.0596 4,83 0,0754 0,3583 0,0718 8,81 0,0878 0,4022 0.0841 14.1 ~ 0,1000 0,4431 0.0963 24,1 ~ ~ 0.1122 0.4811 0.1085 34.2 ,. 0.1243 0,5168 0.1207 44,1 i 0,1380 0.5540 0.1339 53,7 Run #2 0.0081 0.0904 0.0057 0,170 0.0176 0.1585 0.0147 0.353 0,0291 0.2221 0,0257 0.687 0,0404 0.2776 0,0370 1.57 0.0519 0.3285 0.0485 2,89 0.0631 0.3754 0,0597 5,38 0.0744 0.4190 0,0710 9,23 0.0855 0.4599 0,0822 17.2 0.0963 0.4964 0.0931 25,7 0.1239 0.5834 0.1157 24.8 0.1375 0.6210 0,1335 44,8 54 TABLE II SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 3.3% CaC03 AT 60°C eq P1jp~ wav D x109 (cm2 WI I js)  Run #1 0.0082 0.0766 0.0058 0.882 0.0148 0.1459 0.0129 1.40 0.0218 0.2050 0.0198 2.02 0.0289 0.2530 0,0268 3.54 0.0364 0.2951 0,0341 5.65 0.0452 0.3416 0,0425 7.34 ... 0.0538 0.3855 0.0512 ]1.0 2 0.0624 0.4242 0.0599 16.8 .. 0.0802 0.4975 0.0777 35.5 i 0.0880 0.5295 0.0857 65.7 Run #2 0.0086 0.0696 0.0060 1.57 0.0168 0.1306 0.0144 1.58 0.0254 0.1909 0.0228 2.69 0.0341 0.2426 0.0315 4.52 0.0428 0.2913 0.0402 7.41 0.0521 0.3397 0.0493 8.46 0.0607 0.3814 0.0581 13.9 0.0706 0.4443 0.0677 21.9 0.0788 0.4856 0.0764 29.2 0.0867 0.5224 0.0843 44.3 55 TABLE III SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 4.9% CaC03 AT 60°C WIeq PI/P~ wav D xl0g (cm2 /s) 1 Run #1 0.0086 0.0681 0.0060 0.392 0.0172 0.1325 0.0146 0.538 0.0258 0.1950 0.0232 1.16 0.0341 0.2534 0.0316 1.82 0.0422 0.3125 0.0397 3.16 0.0504 0.3624 0.0479 4.25 0.0586 0.4033 0.0561 6.70 0.0669 0.4472 0.0644 8.62 0.0874 0.5231 0.0847 17.3 0.0956 0.5548 0.0931 24.6 0.1047 0.5871 0.1019 33.1 0.1133 0.6176 0.1107 40.5 0.1213 0.6418 0.1189 54.9 i 0.1296 0.6664 0.1271 64.5 Run #2 0.0080 0.0841 0.0056 0.325 0.0154 0.1522 0.0132 0.505 0.0230 0.2125 0.0207 0.917 0.0311 0.2619 0.0286 1.27 0.0398 0.3085 0.0372 2.09 0.0487 0.3546 0.0461 3.20 0.0575 0.3937 0.0549 4.53 0.0667 0.4305 0.0639 6.15 0.0767 0.4737 0.0737 8.91 0.0865 0.5090 0.0836 13.1 0.0965 0.5451 0.0935 16.9 0.1058 0.5756 0.1030 27.5 0.1148 0.6020 0.1121 33.0 0.1238 0.6270 0.1211 43.5 0.1334 0.6541 0.1305 64.0 56 TABLE IV SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 10% CaC03 AT 60°C eq Pl/P~ wall D x109 (cm2 WI 1 /s) Run #1 0.0114 0.0633 0.0080 0.104 0.0245 0.1213 0.0206 0.133 0.0366 0.1797 0.0330 0.259 0.0482 0.2337 0.0447 0.448 0.0602 0.2820 0.0566 0.662 0.0741 0.3308 . 0.0699 0.918 .C..2. ,." 0.0895 0.3747 0.0849 1.60 . 0.1065 0.4190 0.1014 1.89 ~.. 0.1267 0.4573 0.1206 4.59 i 0.1403 0.4900 0.1362 6.17 0.1536 0.5243 0.1496 12.0 Run #2 0.0085 0.0703 0.0059 0.287 0.0213 0.1343 0.0175 0.134 0.0342 0.1890 0.0303 0.264 0.0473 0.2355 0.0434 0.390 0.0650 0.2832 0.0597 0.676 0.0818 0.3271 0.0768 0.981 0.1064 0.3702 0.0990 1.76 0.1229 0.4085 0.1179 2.85 0.1397 0.4491 0.1346 4.26 0.1530 0.4867 0.1490 10.8 57 TABLE V SOLUBILITY DATA AND DIFFl:"SION COEFFICIENTS OF TOLUENE IN PVAC WITH 0.0% CaC03 AT 80°C eq PI/P~ wl1V D X 109 (em2 WI 1 / S) Run #1 0.0071 0.0559 0.0050 0.624 0.0143 0.1186 0.0121 1.04 :'" .. 0.0206 0.1660 0.0187 1.23 .... ':.J 0.0275 0.2113 0.0254 1.48 :::. :~ 0.0345 0.2535 0.0324 2.24 ....... 0.0413 0.2917 0.0392 3.54 . 0.0477 0.3242 0.0458 3.86 0.0537 0.3526 0.0519 5.44 0.0604 0.3821 0.0584 7.18 Run #2 0.0077 0.0439 0.0054 0.600 0.0149 0:0859 0.0128 0.961 0.0218 0.1225 0.0197 1.38 0.0293 0.1612 0.0270 1.71 0.0379 0.2058 0.0353 2.45 0.0456 0.2420 0.0433 3.00 0.0532 0.2752 0.0509 3.47 0.0605 0.3071 0.0583 5.57 0.0681 0.3380 0.0658 4.80 58 TABLE VI SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 3.3% CaC03 AT 80°C eq Pl/P~ wClV D X 109 WI 1 (em2/ s) ,.. .... Run #1 :! 0.0057 0.0245 0.0040 0.540 0.0156 0.0698 0.0126 1.19 ....... 0.0263 0.1120 0.0231 1.76 ,oil Ii..' 0.0397 0.1605 0.0356 2.42 ... ': 0.0515 0.2006 0.0479 4.24 ~ 0.0624 0.2377 0.0591 9.46 " Run #2 0.0069 0.0302 0.0048 0.972 0.0158 0.0666 0.0131 1.27 0.0274 0.1111 0.0239 1.74 0.0388 0.1511 0.0353 2.67 0.0509 0.1909 0.0473 4.15 0.0615 0.2280 0.0583 10.2 59 TABLE VII SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 4.9% CaC03 AT 80°C eq PIjp~ wav D x109 (cm2 WI I js) Run #1 0.0059 0.0311 0.0041 0.534 0.0135 0.0659 0.0112 0.632 0.0211 0.1019 0.0188 0.901 0.0319 0.1466 0.0286 1.14 0.0420 0.1853 0.0389 1.67 .::1 ~ 0.0563 0.2367 0.0490 3.04 .•. ~"i 0.0698 0.2825 0.0657 5.11 ,,...... 0.0797 0.3158 0.0767 10.6 c.:' 0.0906 0.3524 0.0874 18.8 0.0907 0.3869 0.0960 45.5 Run #2 0.0060 0.0291 0.0042 0.551 0.0148 0.0685 0.0121 0.643 0.0247 0.1120 0.0217 0.839 0.0351 0.1571 0.0320 1.16 0.0464 0.2026 0.0430 1.89 0.0566 0.2406 0.0536 2.90 0.0663 0.2736 0.0634 4.40 0.0758 0.3052 0.0729 7.07 0.0848 0.3361 0.0821 16.1 60 TABLE VIn SOLUBILITY DATA AND DIFFUSION COEFFICIENTS OF TOLUENE IN PVAC WITH 10% CaC03 AT 80°C ,,' eq PI/P~ wl1V D X 109 (em2 / S) "Ill WI 1 ..II ~I ':1 Run #1 ...,. 0.0113 0.0282 0.0079 0.463 ,I..I 0.0282 0.0692 0.0232 1.10 0.0410 0.1087 0.0371 1.48 0.0562 0.1573 0.0516 2.59 Run #2 0.0164 0.0369 0.0114 0.559 0.0341 0.0820 0.0288 1.28 0.0509 0.1307 0.0459 1.88 0.0638 0.1715 0.0599 3.78 61 1e7 o o • o ":J • •o 1e8  ~ o~ ~ · o· f o. !II It. 0 • • b. o. o It. • 1e9  b. ~It. o. o•f oo. . 0 0 • • It.~ 0+ • o ~ o. .£) .~ 0 ... 1e10~~· o. ~ «) 1e11  0 open symbols Roo #1 solid symbols Roo #2 o 0 %CaC03 ~ 3.3 % CaC03 o 4.9 % CaC03 o 10 % CaC03 III ." JI 11 :1 ..,. .jl 0.00 I 0.05 I 0.10 I 0.15 I 0.20 I 0.25 0.30 Toluene Weight Fraction Figure 12. Effective Diffusion Coefficients of Toluene in CaC03 Filled PVAC at 60°C. 62 1e7 o o • o 0 • • 1e8  • ~o .0 •° 1e10  0.00 o 1e9  .to • ~ • 0 ! 0. ~ o· ". A ." 0+ JI 0. I '1 ~ .... open symbols Run #1 .! • solid symbols Run #2 0 ':: • 0 0 % CaC03 • 0 3.3 % CaC03 I::> 0 ° 4.9 % CaC03 • 10 % CaC00 0 3 I I 0.05 0.10 0.15 Toluene Weight Fraction Figure 13. Effective Diffusion Coefficients of Toluene in CaC03 Filled PVAC at 80°C. 63 0.7 r, o % CaC03 3.3 % CaC03 4.9 % CaC03 10 % CaC03 o o o 0 o. 0• • • ~ • 0 0 0 0 0 • 0 • 0 0 • • ,.. .. ,I' 11 'I ., open symbols Run #1 .. I! solid symbols Run #2 . .. o o o o • 0.1  Ie 0.2  0.5  0.3  0.4  0.6  0.0 I I I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Toluene Weight Fraction Figure 14. Solubility Data of Toluene in CaC03 Filled PYAC at 60°C. 64 0.45 0.40  0 0 0.35  0 0 • • 0 0 0.30  • • 0 • .0 0.25  0 0 • ~ /), e ... 0.. 0 .. 0.20  • ../.)., I'\' 0 I 0 • " •• 6. .. 0.15  o ... 0 open symbols Run #1 I! • solid symbols Run #2 .' 0 • " 0.10  ~ 0 0 0 0 % CaC0• • 3 ~ 0 6. 3.3 % CaC03 0.05  0• • 0 4.9 % CaC03 ~ 0 0 10 %CaC03 0.00 I I 1 1 0.00 0.02 0.04 0.06 0.08 0.10 Toluene Weight Fraction (Solvent Free Basis) Figure 15. Solubility Data of Toluene in CaC03 Filled PVAC at 80°C. 65 0.7 ,, 6~ 0 • o • 6·~ 0 ~ open symbols Run #1 Lt solid symbols Run #2 ., 0 ~ ·0 ~~ 0 0 %CaC03 ~ 0 6 3.3 % CaC03 \0 ° 4.9 % CaC03 0 10 % CaC03 o • • 0. ° 0• • • <+  ~ 0 c 0 0 0 • 0 • 0 o o o o • 0.4  0.5  0.2  0.1  0.3  0.6  0.0 I I I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Toluene Weight Fraction (Solvent Free Basis) Figure 16. Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 60°C. 66 0.45 0.40  0 0 0.35  0 0 • • 0 0 0.30  • • 0 • .0 0.25  0 0 • ~ 6 e A 0.. 0.20  0 • • 6 0 A 0 • 0.15  • o • A6 0 • open symbols Run # 1 0 • solid symbols Run #2 0.10  4lJt. 0 0 0 0 % CaC0• • 3 ~ 0 I::> 3.3 % CaC03 0.05  0• • '0 4.9 % CaC0~ 0 3 0.00 0 10 % CaC03 I I I I 0.00 0.02 0.04 0.06 0.08 0.10 Toluene Weight Fraction Figure 17. Solubility Data on a Filler Free Basis of Toluene in CaC03 Filled PVAC at 80a C. CHAPTER V DISCUSSION OF RESULTS In this chapter, the experimental data presented in Chapter IV are discussed. The diffusion coefficients of toluene diffusing into PVAC with 0.0% CaC03 (neat PVAC) are compared with the results of other researchers [Mikkilineni, 1995 and Hou, 1986] and with the freevolume theory correlation of Vrentas and Duda. The methods discussed in Chapter II for determining diffusion coefficients are compared by analyzing a set of sorption curves with each method. The experimental data for filled PVAC are compared with a freevolume theory correlation modified to account for the added filler and to other correlative equations discussed in Chapter II. Diffusion coefficients for toluene diffusing into neat PVAC were compared to the data of Mikkilineni [1995] and Hou [1986] at temperatures of 60°C and 80°C in Figures 18 and 19. The diffusion coefficients obtained in this work compare well with the data of both Mikkilineni and Hou. The data shown are from Table I and V of Chapter IV for this work, Table IX for Mikkilineni, and Table X for Hou. The largest deviations between the data obtained in this work and the data obtained by Hou occur at the lowest weight fractions for both the 60°C and 80° C data. The apparatus used by Hou for obtaining diffusion coefficients was a quartz spring balance. The quartz crystal microbalance is more sensitive to changes in low values of concentration of toluene (rv 1 ng of toluene) than the quartz spring balance, while the quartz spring balance could give data points at higher values of concentration of the penetrant by using thicker polymer films than can be used by the quartz crystal microbalance. 67 'I, '11 " " ·.1, I..i. '1 '...(I 1e6 o 68 1e7  Co 0 .. ;l 0 o ~ ~ .. .. ....t...i..l. o·., N 0 S 1e8  (.) ''" Cl t~ P • 0 0 0 .,t:J cP 1e9  ..081 .. This Work 0'" ~o 0 Hou (1986) ~ ~ 0 Mikkilineni (1995) 1e10 I I I I 0.00 0.05 0.10 0.15 0.20 0.25 Toluene Weight Fraction Figure 18. Diffusion Coefficients of Toluene in Neat PVAC at 60°C. 'I, 'I' .·1:1 ":i '10 , 69 1e6  o o o o 1e7  0.20 I 0.15 .. This Work o Rou (1986) o Mikkilineni (1995) 1 0.10 o I 0.05 o o 0.00 rnAA ... f;~O ... AA A 1e8  to Toluene Weight Fraction Figure 19. Diffusion Coefficients of Toluene in Neat PVAC at 80°C. 70 TABLE IX DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC FROM MIKKILINENI Toluene Weight Fraction D x 109 (em2 / S) 60°C 0.0091 0.164 0.0264 0.468 0.0508 1.62 0.0680 4.47 0.0963 37.2 0.0136 0.220 0.0402 0.922 0.0746 5.19 " 0.1088 36.8 41 '. 0.0116 0.200 il 0.0321 0.549 H 0.0677 2.14 0.0941 13.0 0.1382 71.9 0.0144 0.429 0.0397 1.45 0.0138 0.349 0.0363 0.877 0.0654 3.16 0.0283 1.06 0.0795 31.7 0.0541 1.70 0.0380 0.994 0.0928 24.2 0.1756 56.3 80°C 0.0142 31.5 0.0364 36.5 0.0586 74.8 0.0871 163 0.1305 179 TABLE X DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC FROM HOD 71 Toluene Weight Fraction D x109 (cm2 js) '.41 '. 60°C ;1 ~l 0.0234 2.05 :.l ~. 0.0755 16.0 .•.• 0.1320 82.9 :) 0.2199 446 80°C 0.0140 4.20 0.0404 23.5 0.0734 84.2 0.1202 319 0.1903 921 0.2589 1190 72 Mikkilineni's data were obtained with the same apparatus as used in this work; however, the methods of analyzing the sorption data were different in this work. Mikkilineni used the initial slope method, and the moment method was used in this work. Also, in this work, the apparatus was modified to include a sample cylinder for quick delivery of the penetrant into the chamber. The use of a sample cylinder was critical for preventing errors which occured from the slow entry of the solvent vapor. The diffusion coefficients for neat PVAC show greater scatter at larger weight fractions shown, as can be seen in Figures 12 and 13. The reason for increased scatter of data at larger weight fractions was because the diffusion became too fast for enough data to be collected in the beginning stages of diffusion. The faster the diffusion, the fewer the number of data points that were collected in the beginning stages of diffusion. Also, since diffusion at 80°C was faster than diffusion at 60°C, data at 80°C were not obtained at as high of toluene weight fraction. Figure 20 shows an example of a sorption curve that was rejected due to very few data in the initial stages of diffusion. Diffusion coefficients calculated from such a curve would be unreliable since very few data points were collected at the beginning of the curve. Another phenomena which leads to unreliable diffusion data is the appearance of a hump in the sorption curves for larger toluene weight fractions. Diffusion coefficients obtained from analysis of sorption curves after the appearance of a hump are not shown in Figures 12 and 13 as the scatter of data is too great and the data is considered to be unreliable. The cause of the hump is unknown, but is speculated to be related to "stress effects." Normally, the changes in resonant frequency of the quartz crystal is related to the change in mass on the exposed surface. However, if there is stress in the polymer film on the quartz crystal surface, there is a net force per unit width acting across the polymer film/quartz interface that stress biases the quartz. This stress bias causes the frequency changes referred to as "stress effects" [EerNisse, 1984]. Adding mass or adding stress decreases the resonant frequency of the quartz crystal. Speculations are that, when the coated "~l ":1:4 ·l ".~ .. 73 Figure 20. An Example of a Sorption Curve that was Rejected Due to Very Few Data in the Initial Stages of Diffusion. "•• ":1 :~ .~ I. '...:~, .1 74 polymer solution drys, stress builds in the polymer film. During a sorption experiment, adding solvent to the polymer film decreases the resonant frequency of the quartz crystal. The added solvent relaxes the polymer chains, possibly reducing stress at the polymer/crystal interface, which would increase the resonant frequency of the quartz crystal and produce a hump in the sorption curve. Figure 21 shows an example of a sorption curve that was rejected due to a hump in the sorption curve. Most of the trends of the effective diffusion coefficient data in Figures 12 and 13 are as expected. The diffusion coefficients increase with weight fraction toluene and temperature as is typical of the PVACtoluene system. Diffusion coefficients decrease with weight percent of CaC03 , which is typical for diffusion in filled rubbery polymers. The diffusion coefficient data are expected to be concave towards the toluene weight fraction axis. The 60°C data appear to follow this trend better than the 80°C data. The data show some upward trends and inflection points, especially for filled PVAC at the larger of the weight fractions. This could be from error in the analysis of the data, due to inaccurate selection of the equilibrium mass uptake. In slower diffusion processes, such as diffusion in filled PVAC compared to diffusion in neat PYAC, the sorption experiment takes a long time to reach equilibrium, and a small error in the equilibrium mass uptake would cause a large error in the first moment calculation of the sorption curve which is used to determine the diffusion coefficient. There could also be errors due to "stress effects" as discussed previously. Although, if "stress effects" are present, they are not noticable enough to form a hump in the sorption data which were analyzed. Replicate experimental runs of each set of data were performed with the same crystal and polymer film for each set in order to verify reproducibility of data. As can be seen from Figures 12 and 13, the data reproduce well. Some scatter is seen, caused by factors such as limited amount of sorption data and inaccurate selection of equilibrium mass uptake, as described above. ~ "~I ':i '4 ~ '. "I ~ '~'. " '",..,. 75 1.2 1.0 0.8 '""'L.7_"b"7I11ldll1~~U....J..t,n.....r"4'·".,117"'S.mlll._7 8 E 0.6 ~ 'I. 'II 'il " ':1 '" 0.4 • '" ',. • " • ;. • '. •• 0.2 II • 'r: """ """ '" 0.0 II: Ih' , 0 10 20 30 40 50 60 "1 ~t (8) Figure 21. An Example of a Sorption Curve which was Rejected Due to a Hump in the Curve. 76 The vapor pressure data in Figure 14 through Figure 17 shows the effect of the filler on changing the solubility in the polymer films. The replicate experimental runs compare well for PVAC with 3.3, 4.9 and 10% (by weight) CaC03 at both 60°C and BO°C. However, there is more error between the replicate experimental runs for neat PVAC at both 60°C and BO°C. The vapor pressure data were plotted against the toluene weight fraction calculated on a filler free basis in Figure 16 and Figure 17 in order to determine effects of adsorption of toluene solvent by the filler. If Henry's law holds, the concentration, C, of the solvent in the polymer film is related to vapor pressure by, C = C!p. Therefore, the vapor pressure, PI, is directly proportional to toluene weight fraction, and is indirectly proportional to the solubility coefficient, C!. If PI decreases, C! increases. If PI increases, C! decreases. The data shown extrapolate to zero vapor pressure at zero toluene weight fraction, as they should. The data also follow different slopes for the different filler contents, as would be expected with different solubility coefficients. Following discussions given in Chapter II, in the absence of vacuoles and for nonadsorbing filler, C! = C!pVp , the solubility coefficient would decrease linearly with increasing filler volume fraction. With the exception of the neat PVAC data at 60°C, the general trend in Figures 16 and 17 is a decrease in the vapor pressure with increasing filler content. Therefore, the general trend of the solubility coefficient is to increase with added filler, which means that penetrant adsorption and/or filling of vacuoles with penetrant does occur. Due to the sensitivity of the quartz crystal microbalance to small changes in concentration of penetrant, the QCM has advantages over other sorption apparatus such as the quartz spring balance since closely spaced diffusion coefficient data can be obtained. The QCM has disadvantages, however, such as the inability to obtain diffusion data at as high of penetrant concentrations as the quartz spring balance, for reasons such as limited sorption data at beginning times and humps in sorption curves, as mentioned above. Yet, these limitations do not pose a problem in the present study, as requirements for model development are satisfied with "I I "I, " '" "I ,,, ,,",, "I 77 low concentration diffusion data and diffusion coefficients with higher penetrant concentrations can be estimated from models. Another disadvantage of the QCM is the inability to obtain data for highly viscous polymers. As an example, Figure 22 shows a quartz crystal frequency response curve for ethylbenzene diffusing into polybutadiene. The crystal frequency was very unstable during this run. This is due to the viscous nature of polybutadiene. Stress effects are presumed to be great with viscous polymers and the quartz crystal oscillations are damped causing instabilities in the resonant frequency. More examples of trial studies with highly viscous polymers and discussion of the stress effects in piezoelectric quartz crystals are given by Mikkilineni [1995], including interesting examples of polymers so viscous that the quartz crystals ceased to oscillate when coated with the polymer. The most significant cause of experimental error in this work is believed to be the measurement of the thickness of the polymer film. A possible error is in the measurement of the diameter of the film. The coated films are not perfectly symmetrical; therefore, several measurements of diameter are taken at different orientations and averaged to give the diameter used in the calculations. Another possible error is in the volume measurements with the micropipette. Some of the liquid could adhere to the inner surface of the micropipette tube. Since such a small volume of solution is used, significant error could result. Error also exists due to nonuniformity in the thickness of the polymer films. Visual inspection of the coated films show that the outer edges of the films are usually thicker than the center portion. This is thought to be due to surface tension in the solution while drying. Adhesion of the solution to the surface of the crystal creates forces which move solution to the outer edges of the film. For this same reason, filler particles also tend to be swept to the outer edges of the film. To minimize these effects, the solution is stirred with the tip of the micropipette while being coated. Appendix C gives and estimate of the error in the diffusion coefficient and penetrant weight fraction due to uncertainties in the thickness of the polymer film I '.I :~ • " 78 5979500 ,, 5978800 +.r,,~ o 50 100 150 200 250 5978900 5979400 5979300 N ::r:: 5979200 '" G' ~ ~ ;:j c:r 5979100 ~ <1:: . ", 5979000 ·1 Figure 22. Frequency Curve for PolybutadieneEthylbenzene System at 80°C. 19 and crystal frequencies. The error in the diffusion coefficient was estimated to be 65% and the error in the penetrant weight fraction was estimated to be 15%. Comparison of Methods for Determining Diffusion Coefficients There are four well known methods of analyzing sorption data in order to determine diffusion coefficients. The four methods are the halftime method, the initial slope method, the limiting slope method, and the moment method, and have been previously discussed in Chapter II. The moment method was chosen to analyze the sorption data in this work. This section compares the four methods of determining diffusion coefficients by using each method to analyze a set of sorption data. Sorption data for toluene diffusing into neat PVAC at 60°C was analyzed by the four methods, the halftime method(HT), the initial slope method(ISM), the limiting slope method(LSM), and the moment method(MM). The diffusion coefficients determined by the four methods are shown in Table XI, and compared in Figure 23. The diffusion coefficients determined from the four methods compare well. The diffusion coefficients determined from the limiting slope method has the most scatter. Mass uptake ratios ranging from 0.9 to 1 were used in the limiting slope method, resulting in more random error due to a limited number of data points in such a small range. Mass uptake ratios ranging from 0 to 0.5 or more were used in the initial slope method. The diffusion coefficient data determined from the initial slope method, the halftime method, and the moment method formed smooth curves. Discrepancies in the four methods were more evident at the larger weight fractions shown, since the sorption curves analyzed did not have enough data at the beginning of the curve. " 1e7  • • • 80 ~ . •• 1e8  1e9  1e10 0.00 .t. • o & .o... • • • o A • • • ~ . • I 0.05 • • •0 0 .~ • I • ~ I 0.10 0 Initial Slope Method IA HalfTime Method • Moment Method • Limiting Slope Method I I I 0.15 0.20 0.25 0.30 ·0 I I • I ·1 '~r ." 'I Toluene Weight Fraction Figure 23. Comparison of Methods for Determining Diffusion Coefficients of Toluene in Neat PVAC at 60°C. 81 TABLE XI COMPARISON OF METHODS FOR DETERMINING DIFFUSION COEFFICIENTS OF TOLUENE IN NEAT PVAC AT 60°C D X 109 (cm2/s) wav HT ISM L5M MM 1 0.0085 0.303 0.300 0.158 0.218 0.0213 0.761 0.854 0.289 0.536 0.0342 1.78 2.11 0.720 1.26 ':1 0.0468 3.45 4.48 1.84 2.82 "'1 ", 0.0596 6.22 7.35 1.74 4.83 ., ":l 0.0718 10.1 8.89 4.28 8.81 w· 0.0841 11.7 13.3 7.57 14.1 'I' 'I " 0.0963 16.0 16.5 18.4 24.1 '" JI 0.1085 18.3 29.4 26.3 34.2 "1 94.0 ., 0.1207 20.7 34.3 44.1 a., 0.1339 23.5 41.0 76.1 53.7 " 82 Correlating the Experimental Diffusion Coefficient Data Regressions Using a Modified FreeVolume Equation Experimental data from this work were compared with effective diffusion coefficient curves obtained from a freevolume equation modified to account for filled polymers. Plots of D vs. volume fraction of filler, Vj, gave insight into how diffusion varies with filler volume fraction. Graphs of D vs. vf, in semilog form, are shown in Figure 24 for 60°C data and Figure 25 for 80°C data. These plots are linear, within experimental error; thus, log D varies linearly with Vf' Models have been developed which describe the penetrant concentration dependence of diffusivity [Duda et al., 1982]. These models are also exponential in form. Combining a model which fits D vs. vf with a model which fits D vs. WI would give a model which d~scribes the dependence of D on both penetrant concentration and filler volume fraction. A linear equation which fits the log D vs. vI data may be written as (81 ) "I " '. where Deif is the effective mutual diffusion coefficient measured in the experiments, Dp is the diffusivity of neat polymer, and m is a positive constant. The negative sign signifies that diffusivity decreases with volume fraction of filler. With vI =0, this equation simplifies to Deli = Dp , as it should. Solving for Deif in equation 81 gives the following equation for diffusion of penetrants in filled polymers, ,. '! " ii 'I (82) An equation which has worked well for many polymerpenetrant systems in describing the concentration dependence of the mutual diffusion coefficient is Vrentas and Duda's [Duda et al., 1982] freevolume theory of transport which was discussed in Chapter II. For this reason, this equation was chosen for Dp . The effective diffusion coefficient in full form can then be given as (83) 83 " :1 "" 'It 0.00 0.01 0.02 0.03 0.04 0.05 1e11 1e10 w, • 0.01 1e7 • 0.02 .... 0.03 0.04 0.05 1e8 0.06 0.07 .. 0.08 CI.l N 0.09 E(,) 1e9 0.10 "" Q Figure 24. Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 60°C for Various Toluene Weight Fractions. 84 1e7 WI 0.01 0.02 0.03 1e8 0.04 0.05 0.06 .. tI:l N E 1e9 u '" Cl 1e10 , ,I 1e11 0.00 0.01 0.02 0.03 0.04 0.05 Figure 25. Diffusion Coefficients of Toluene in CaC03 Filled PVAC as a Function of CaC03 Volume Fraction at 80°C for Various Toluene Weight Fractions. Dp = Dl (1 4>1)2(1  2x4>d, D  D (/,(Wl ~ +W2eV2)) 1  01 exp .... , VFH .... VFH = Kll K12 wl(K21 + T  Tg1 ) +w2(K22 +T  Tg2 ). /' /' /' 85 (84) (85) (86) The preexponential constant, Do, mentioned in Chapter II is lumped in along with the Arrhenius form of temperature dependence into the term DOl' The free volume parameters Kn//'l K21  Tgl, K 12 //'l K 22  Tg2 , ~ and V2 were taken from Hou [1986]. The parameters DOl, e, X, and m were obtained from correlations with experimental data from this work, as explained in the following paragraphs. The interaction parameter, X, was obtained by fitting the FloryHuggins equation (87) to solubility data in which the equilibrium weight fraction of the penetrant in the polymer was known as a function of penetrant vapor pressure, Pl. The pressure p~ is the penetrant saturation vapor pressure. Table I and Table V of Chapter IV shows the data used for this regression. Figure 26 shows a comparison of solubility data for neat PVAC at 600 e with a theoretical curve calculated by equation 87. Figure 27 shows a similar comparison for 80°C data. The solubility data calculated by equation 87 compare reasonably well with experimental solubility data, with more scatter appearing in the 800 e data. The interaction parameter, X, is given in Figures 26 and 27 as 0.42 and 0.75 for 600 e and 800 e data, respectively. The parameters DOl and ewere obtained by fitting equation 83 to experimental diffusion coefficient data for neat PVAC (vf = 0). Figure 28 shows a comparison of experimenta} diffusion coefficient data for neat PVAe at 600 e with curves calculated by equation 83 with vf = o. The parameters DOl and eare given in this figure. The diffusion coefficients calculated by equation 83 compare well with experimental diffusion coefficient data. The parameter m was obtained by fitting equation 83 to experimental diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaC03 I 'I 86 0.6 ,., 0.5 0.4 x= 0.42 oe 0.3 0.. 0.08 0.10' 0.12 • experimental  theoretical 0.02 0.04 0.06 0.2 o.0 +orr,orr,......,,,r,.,,,,.,..,.....,r"""T""""""'Tr.....rr.........T,! 0.00 0.1 Toluene Weight Fraction Figure 26. Comparison of Experimental Solubility Data with the FloryHuggins Equation for PYACToluene at 60°C. 87 0.45 0.40 0.35 0.30 X= 0.75 0.25 0.eo.. 0.20 • • 0.15 • 0.10 • experimental theoretical 0.05 0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Toluene Weight Fraction Figure 27. Comparison of Experimental Solubility Data with the FloryHuggins Equation for PVACToluene at 80°C. 88 1e6 1e7 • • 1e8 DOl = 5.48 X 105 ~ = 0.63 0.25 0.30 0.35 • experimental theoretical 0.05 0.10 0.15 0.20 1e10 1e9 1e11 0.00 Toluene Weight Fraction Figure 28. Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for Toluene Diffusing into Neat PYAC (vf = 0) at 60°C. 89 (V j = 0.0428). Figure 29 shows a comparison of the diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaC03 at 60°C with curves calculated by equation 83 with VI = 0.0428. The parameter m is given in this figure. The fit of equation 83 to the experimental diffusion coefficient data in this figure is reasonable, but these data show more scatter than the data for neat PYAC. Figure 30 shows both of the previous comparisons of the data of neat PVAC and CaC03 filled PVAC with a weight of 10% CaC03 , as well as comparisons of the diffusion coefficient data of CaC03 filled PVAC with weights of 3.3% (VI = 0.0137) and 4.9% (Vj = 0.0205) CaC03 at 60°C to predicted curves calculated by equation 83. The curves for data with 3.3% and 4.9% CaC03 are purely predictive. Only the volume fraction of filler, vj, in the equation was changed. All other parameters were obtained from previous correlations. Figure 31 shows similar regressions of neat PVAC data and data of CaC03 filled PVAC with a weight of 10% CaC03 using equation 83 as well as predictions of the diffusivity data of CaC03 filled PVAC with weights of 3.3% and 4.9% CaC03 at 80°C. As before, the parameters DOl and ~ were obtained by fitting the experimental diffusion coefficient data of neat PYAC to equation 83, and the parameter m was obtained by fitting the experimental diffusion coefficient data of CaC03 filled PVAC with a weight of 10% CaC03 to equation 83. The predictive curves of the diffusion coefficients of CaC03 filled PYAC with weights of 3.3% and 4.9% were calculated by equation 83 by changing only the volume fraction of filler, V j. The diffusion coefficient curves calculated with equation 83 compare very well with the experimental diffusion coefficient data of neat PVAC and PVAC with 10% CaC03 . Also, the diffusion coefficient curves calculated with equation 83 for PVAC with 3.3% and 4.9% CaC03 at 60°C and 80°C show a reasonable prediction of the experimental data. The predictions of the diffusion coefficients for PYAC with 4.9% CaC03 are the better of the two samples. The data for PVAC with 3.3% CaC03 shows more scatter, showing low predictions of data at 60°C and high predictions of data at 80°C. 90 1e6 1e7 1e8 ~ r;r.; .......... m= 96.7 N S c..> """ Cl 1e9 • • • • , 1e10 'I • ) • experimental 43 theoretical ~ :> ~ 1e11 :~ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ., Toluene Weight Fraction Figure 29. Comparison of Experimental Diffusion Coefficient Data with a FreeVolume Equation for CaC03 Filled PVAC with a Weight of 10% CaC03 (Vf = 0.0428) at 60°C. 91        .  DOl =5.48 X 105 ~ = 0.63 m=96.7 1e8 1e6 1e7 ... CI'J N 1e9 Eu "" 0 open symbols Run #1 0 .*J solid symbols Run #2 1e10 o· 0 0 % CaC03 D. 3.3 % CaC03 1e11 0 4.9% CaC03 •) 10 % CaC03 • 0 ~.. Regression ) "l 1e12  Predicted ~ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 , Toluene Weight Fraction Figure 30. Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 60°C. 92 1e6 DOl = 1.18 X 104 ~ = 0.59 m= 120.5 1e7    .....    /    ~o ..... 1e8 40/ .. / ~ (/) / N £~ E ~c. (,) '' 0 /~~ open symbols Run #1 1e9 A ~ ~ solid symbols Run #2 0 0 %CaC03 D. 3.3 % CaC03 1e10 0 4.9 01<, CaC03 • 0 10 % CaC03 ~ ~ Regression ) '~" 1e11   Predicted ~.. :> 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Toluene Weight Fraction Figure 31. Comparisons of Experimental Diffusion Coefficients with a FreeVolume Equation for CaC03 Filled PVAC at 80°C. 93 The freevolume parameters used in this subsection are shown in Table XII. The solubility data regressions show the parameter X to increase with temperature with a 56% difference between 60°C and 80°C. The parameter, ~, is usually assumed to be temperature independent [Duda, 1983]. Regressions of diffusion data at 60°C and 80°C show this agreement to within 7%. The parameter, DOll is temperature dependent and is assumed to follow the Arrhenius equation. For positive activation energy, E, the parameter, DOl, increases with temperature. Regressions of data verify the increase of DOl with temperature. The DOl at 80°C was 73% greater than DOl at 60°C. The parameter, m, increases with temperature with a 22% difference between the two temperatures. In a study of diffusion of hydrocarbons in silicone rubber, Barrer [1962] reported that the percentage difference in the diffusion coefficient between filled and unfilled rubber increased with temperature. Therefore, from the mathematical form of equation 83, m is expected to increase with temperature. Regressions Using Equations Derived by Barrer and Chio Experimental diffusion coefficient data for toluene diffusing into CaC03 filled PYAC at 60°C was used to test the correlative capabilities of the equations derived by Barrer and Chio [1965], as described in Chapter II. The freevolume equation was used for the unfilled diffusion coefficient, Dp , with parameters as given in the previous subsection. The first equation used was the simple form in which only the tortuosity of the diffusion paths around the filler particles was considered where K. is the structure factor. The expression of Fricke, y K.= , Y +vf (88) (89) r TABLE XII PARAMETERS OF THE MODIFIED FREEVOLUME EQUATIO FOR PVACTOLUENE 94 Parameter Vt (em3 /g) V2* (em3 /g) Kn/, (em3 jg K) Kl2 /, (em3 jg K) K21  Tg1 (K) K22  Tg2 (K) FreeVolume Parameters from Hou [1986] Regre~sed FreeVolume Parameters Parameter 60°C 80°C Value 0.917 0.728 1.57 x 103 4.33 X 104 90.5 256 41• >• 0.42 0.63 5.48 x105 96.7 0.75 0.59 1.18 x104 120.5 where Y Runge, 95· 2 for random spheres was used in equation 89. The expression of 1 [ 3vf ] (90) K = 1  1~ 3 ' (1 vf) 2 +vf  O.392vf / for a cubic lattice of spheres was also used. Both of these expressions, equation 89 and equation 90, gave the same value for K. The smallest value of the structure factor, K = 0.979, was for polymer with 10% filler (Vf = 0.0428). This value led to only a 2% decrease in the diffusion coefficient. Figure 32 shows this comparison, as well as the previously obtained regression of neat PVAC. The fit using equation 88 is very poor. The values of K calculated from equations 89 and 90 are used in subsequent calculations. The next equation used was the more complicated form which takes into account two separate phases, the polymer phase and the filler phase, (91 ) where O'p is a parameter related to the solubility in the polymer and 0'1 is a parameter related to the gas adsorption by the filler. The ratio 0'J/0'p was obtained by fitting equation 91 to experimental diffusion coefficient data for CaC03 filled PVAC with a weight of 10% CaCOi (vf = 0.0428). Figure 33 shows this regression, as well as the previously obtained regression of neat PVAC, and the predictions of the diffusivity data of CaC03 filled PVAC with weights of 3.3% and 4.9% CaC03 at 60°C. The diffusion coefficients obtained from the regression using equation 91 compare well with the experimental diffusion coefficients of toluene in 10% CaC03 filled PVAC, however, equation 91 does not predict well the experimental diffusion coefficients of toluene in 3.3% CaC03 and 4.9% CaC03 filled PVAC. The ratio 0'f /O'p obtained in the regression using equation 91 is much larger than unity, 1,351.5. However, even assuming independent sorption by filler and rubber phases, Barrer et al. [1962] reported values of the ratio 0'J/O'p less than unity for hydrocarbons diffusing into silica filled silicone rubber. Therefore, the ratio 0'f /O'p is unlikely to be as large as that obtained by the regression. Diffusion of toluene in CaC03 filled PYAC is possibly too complex to model with a diffusion "~l. 96 1e8 c::, ~ ~.. ~~ .. .& <>~ til ~ ~. N 1e9 c::, 



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