Experimental verification of a predictive model of penetrant transport in glassy polymers

Pergamon Chemical En9ineerino Science, Vol. 51, No. 21, pp. 4827 4841, 1996 Copyright < 1996 Elsevier Science Ltd

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EXPERIMENTAL VERIFICATION OF A PREDICTIVE M O D E L OF P E N E T R A N T T R A N S P O R T IN GLASSY POLYMERS

D.-J. KIM,* J. M. CARUTHERS and N. A. PEPPAS t School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.

(Received 14 March 1994; accepted 5 November 1995)

Abstract--The validity of the Lustig et al. (Chem. En,qn9 Sci. 47, 3037-3057. 1992) model for penetrant transport in glassy polymers was investigated by comparing experimental results of dodecane transport in crosslinked polystyrene samples with the model. All parameters of the model were measured independently. The time-dependent concentration profiles at the dodecane/polystyrene interface were predicted at various temperatures. The time-dependent concentration and the polymer stress profiles within the dodecane/ polystyrene system indicate a wide range of transport mechanisms at different temperatures. The dodecane transport kinetics was affected by dodecane diffusion coefficient, the discrete viscoelastic relaxation time and the modulus of the dodecane/polystyrene system. The significance of the viscoelastic relaxation contribution on the overall penetrant transport was evaluated by analyzing the temperature dependence of the diffusional exponent, n, of a simple exponential dependence of the dodecane uptake as a function of transport time. The diffusional exponent and the resulting penetrant transport kinetics were correlated to the mean diffusion Deborah number. In general, the transport process was closely characterized as Fickian diffusion far above or far below the glass transition temperature of the system and as anomalous transport close to the glassy/rubbery transition Copyright ~:~ 1996 Elsevier Science Ltd

Keywords: Penetrant transport in glassy polymers, non-Fickian diffusion, anomalous transport, stresses.

1. INTRODUCTION Penetrant transport in glassy polymers is characterized by rearrangements of the macromolecular chains towards new conformations where the rate of chain relaxation depends on the local penetrant concentration. The relative rates of penetrant transport and macromolecular chain relaxation determine the type of the transport process such as Fickian, anomalous (non-Fickian), case II and super case II transport.

Various aspects of non-Fickian transport have been described by numerous models based on Fick's law, linear irreversible thermodynamics (LIT), and rational thermodynamics. Fickian diffusion models of penetrant transport in glassy polymers have been relatively useful because of their easy solutions by analytical or numerical methods. The adjustable material properties are generally determined by fitting the experimental data to the proposed Fickian model. Non-Fickian transport processes have been often described by models considering a convective term in the penetrant flux (Frisch et aL, 1969; Wang and Kwei, 1973; Peppas and Sinclair, 1983); by models incorporating changes in the polymer morphology resulting in a variable penetrant diffusion coefficient (Crank, 1953, 1975; Lustig and Peppas, 1987; Petro- poulos and Roussis, 1978); and by models with non- Fickian propagation of the swelling front (Peppas and Sinclair, 1983; Peterlin, 1979; Astarita and Sarti, 1978;

*Corresponding author. *Present address: Department of Chemical Engineering,

Sung Kyun Kwan University, Suwon, Kyunggi-do, Korea.

Sarti, 1979; Gostoli and Sarti, 1982). The key difficulty with these models is that penetrant uptake data are used to fit the model constants; thus, it is not clear if these models capture the real physics or if they are just potential ways to parameterize the penetrant uptake data.

Linear irreversible thermodynamics suggests that the fluxes can be expressed in terms of linear combina- tions of chemical potential, and temperature gradients as well as stress distributions near the equilibrium state. Several investigators who have proposed models applying the LIT theory (Thomas and Windle, 1980, 1981, 1982; Cox and Cohen, 1989; Kim and Neogi, 1984) have claimed prediction of anomalous transport albeit with limited agreement with experiments. Since LIT is valid only for small perturbations away from the equilibrium state, these theories are probably not appropriate to describe diffusion processes in glassy materials which possess highly non- equilibrium character. Moreover, the incorporation of viscoelastic relaxation into the LIT framework is somewhat problematic. As discussed previously (Lus- tig et al., 1992), the entropy generation rate employed in LIT depends only upon macroscopic gradients, and thus it does not acknowledge dissipation that occurs as a result of viscoelastic relaxation in a mixture where the composition is spatially homogeneous.

Rational thermodynamics combines both mechanical and thermodynamic approaches without limita- tions on the magnitude of the departure from the equilibrium state. This framework satisfies the Clausius-Duhem entropy inequality as well as the

4827

4828

conservation of mass, linear momentum and energy for each component. The entropy inequality places restrictions on the constitutive equations for all admissible thermodynamic processes (Truesdell, 1984; Coleman and Noll, 1963). Application of these ad- missibility criteria to materials with memory (Cole- man, 1964a, b) allows us to address the viscoelastic behavior in penetrant/polymer mixtures. Transport models applying rational thermodynamics are parti- cularly useful in describing the viscoelastic behavior of glassy to rubbery transition during penetrant transport since their applicability encompasses both equilibrium and non-equilibrium characters. Based on rational thermodynamics, Lustig et al. (1992) developed a self-consistent transport model which is briefly summarized in the next section. The purpose of the present contribution is to examine critically the predictions of this new penetrant transport framework with experimental data. Since all the material parameters used in this new theory can be determined independently from the penetrant uptake data, it will be possible to compare critically the predictions of the new theory with experimental data. Thus, the analysis reported in this study should provide a true test of the underlying assumptions in the new theory, rather than just a parameterization of penetrant uptake data as is often the case in the comparison of theory vs experiments for diffusion in polymers.

The remainder of the communication is organized as follows: in the next section the defining equations for the diffusion model will be given, whereas in Sec- tion 3 a summary of the experimental methods used to determine the material parameters needed in the model will be presented. In Section 4 the principal results will be given, including the formulation of the problem for one-dimensional planar diffusion, determination of the material parameters for the polystyrene-dodecane system, and predictions of the model. Finally, in Section 5 a critical discussion of the viscoelastic diffusion model and its prediction will be given.

2. M O D E L

The comprehensive derivation of the continuum thermodynamic theory of diffusion in a viscoelastic medium developed by Lustig et al. (1992) has already been given in the original reference and will not be repeated here; however, a very abbreviated description will now be provided. The viscoelastic diffusion theory of Lustig et al. (1992) employs the theory of mixtures developed by Truesdell (1984), which as- sumes that each component in the mixture is also a continuum that satisfies the following postulates: (i) each mixture component satisfies partial mass, momentum, and energy balances although mass, momentum, and energy can be exchanged between the various components; (ii) the sum of all the components conserves the total mass, momentum, and energy; and (iii) the total mixture satisfies the Clausius-Duhem entropy inequality. Lustig et al. (1992) extended the original mixture theory to the situation where one of the components of the mixture

D.-J. KIM et al.

was a viscoelastic solid that relaxed on a material timescale that depended upon the local temperature and concentration history during the diffusion process. Although the defining equations were derived as partial mass, momentum, and energy balances, they were formally 'inverted' to obtain the individual species mass flux as a function of various driving forces, which includes the traditional gradient of the chemical potential as well as the divergence of the stress tensor. The effect of viscoelastic relaxation on penetrant diffusion is naturally incorporated via the divergence of the stress tensor driving force. We believe that the theory developed by Lustig et al. (1992) is the simplest theory that (i) satisfies all the requirements of continuum physics; (ii) reduces to the traditional multicomponent diffusion equations if there is no viscoelastic solid; and (iii) results in the appropriate three-dimensional viscoelastic description of the polymer-penetrant system when the diffusion process is complete.

The theory of diffusion in a viscoelastic medium developed by Lustig et al. (1992) predicts the penetrant flux into polymers is enhanced by several driving forces such as temperature gradient, species' inertial and body forces acting on the components as well as chemical potential and stress gradients. For an isothermal, binary mixture under the condition of constant body forces and negligible inertial contribution, solvent mass transport is given by

j l P t w ~ M 1 D 1 2 ( Vpl + I V ' T 2 (1)

where Jl is the mass flux of the penetrant, pl is the penetrant density, w2 is the polymer weight fraction, M1 is the molecular weight of penetrant, D12 is the mutual diffusion coefficient, p~ is the penetrant chemical potential and T2 is the polymer stress. The first term in eq. (1) is the traditional mass flux due to a chemical potential (i.e. concentration) gradient, while the second term is the augmentation of chemical potential driving force by the mechanical relaxation.

Continuum thermodynamics requires that the Clausius-Duhem inequality (i.e. the second law of the thermodynamics) be satisfied, formally requiring that the rate of local entropy production be non-negative for all admissible processes. It is this principle that leads to the form of eq. (1); in addition, the Clausius-Duhem inequality places the following restrictions on the constitutive equations for a simple mixture under isothermal conditions in the absence of chemical reactions (Lustig et al., 1992):

T1 = - p~ 0pl (2)

8ud2 T2 = 2p2F2 "~-~2" F~ (3)

where qJ~ is a partial Helmholtz free energy, u~ is the diffusion velocity of the component relative to the mass average velocity, and F2 is the relative deformation gradient of the polymer phase. The penetrant

Penetration transport

chemical potential is also defined in terms of the penetrant-free energy,

a(p l t I~ l ) p: - 3 p 1 (4)

Although not required by continuum thermodynamics, the total free energy of the mixture assumed to be as the sum of the pure components' free energies, p,q~o plus the free energy of mixing, P¢M,

2

pqfl = ~ p=tlJ° + p'qfl M (5) :l 1

where ~Pu is given by Flory-Huggins model. The polymer-free energy, u/2°, contains two contributions, a purely elastic, equilibrium contribution, u?~2~) , and a history-dependent, non-equilibrium contribution determined by the derivative of the second

6a~po~ Frechet-Helmholtz free energy, ½ 2, = o,

p ~vlo) _ w~ooj + ½~2wO2=o. (6) 2 1 2 ~ p , 2 t 2

Lustig et al. (1989) postulated a specific form for the equilibrium free energy, pe~P2 (oo), that consisted of the contributions from the pure component, the hydros- tatic compressibility, and the multiaxial elasticity (Ak- lonis and MacKnight, 1983; Flory, 1975; Treloar, 1975). The compressibility contribution to the penetrant and polymer-free energy was described by the Tait equation (Quach and Simha, 1971). The multiaxial elastic contribution to the polymer-free energy was described by Flory's statistical theory (Aklonis and MacKnight, 1983; Flory, 1975; Treloar, 1975) of elasticity. Finally, the non-equilibrium contribution, ½62 o~ W2,,=o, was formally expanded in terms of a set of correlation integrals that satisfy the principle of fading memory.

Thus, the total polymer-free energy was represented by the following equation:

(" (" 1 × [I - 3 - in [m33 + ~2 j_~ J_ooRK~

dI(¢) dI(() d{ d¢ × [42 ( t ) , t* - ~*, t* - ~*] ~

+,~2 f ' -~* , t*-~*] f ' :,o~ Ga [~2(t), t*

FdC(~) de(() ldI(~) ] • d l ( ~ ) . . . . X ._ d~ d# 3 d~ -~ ~ug a~" (7)

Here, ve is the effective concentration of crosslinked units (Flory, 1975; Treloar, 1975; Ferry, 1980), C is the universal constant, B is the Tait parameter, and I and III are the first and third strain invariants of the left Cauchy-Green deformation tensor, B(t), respectively.

in glassy polymers 4829

Three material properties have been introduced to account for the time-dependent, non-equilibrium response of the polymer: the material time shift function, a, the shear relaxation function, G6, and the bulk relaxation function, K~.

The free energy of mixing, PqJM, w a s described by the Flory-Huggins expression under the assumption of no change of volume upon mixing:

pU/M =-R-~T [InO, + ( i -- ~ i ) + Z I ( I - -~ ) i ) 2 ] (8)

where M1 is the penetrant molecular weight and Zl is the polymer-solvent interaction parameter.

Using eqs (2)-(4), the partial stress tensors and the solvent chemical potential were derived by Lustig et al. (1992) and are given in eqs (9) ( 11):

- R T T 1 - [ln~bi + 1 - ~b: + ZI(1 - ~b~)2]l ~9)

V~

K~[~b2(t), t* - + - ~ _ ] ~ ~_

f + 42 c,~ [4,2(ti, t* - ~*] - t .

x [B(t). dC, t#) B(t) - 1 dl(¢) 1 5 B ( t ) ~ - f f j d# 110)

RT ,u, = M--I-[Ingbl + I -gb, +ZI ( I - ~b,)2]. 111)

Here, B is the left Cauchyq3reen deformation tensor, C is the right Cauchy-Green deformation tensor and E = B - I. The terms Ge and KE are the time-independent shear and bulk moduli in the rubbery state, respectively. The material times t* and ~* are defined a s

t7 t* = d~ ,o a[T(#.) ~ PI((,)] (12a)

~* i ' d~ - Jo a I T ( d - p , (~)] (12b)

where a[T, pl] is the generalization of the well- known time-temperature shift factor to consider the effect of penetrant concentration on the relaxation processes.

The theoretical framework represented in eqs (1) and (9)-(11) has been developed to describe a wide variety of mass transport phenomena in penetrant/ polymer systems explicitly, and to address the inter- play between penetrant diffusion and the polymer's mechanical relaxation• It is the purpose of the present work to conduct a series of experiments in order to verify this model. Upon critical evaluation of various penetrant/polymer pairs it was decided to use the system dodecane/crosslinked polystyrene, because of the low volatility of dodecane which allowed for an

4830

accurate determination of the mechanical behavior of the ensuing polymer/penetrant system. Various material properties required by the model have already been measured for the polystyrene/dodecane system: linear viscoelastic properties (Kim et al., 1993b); bulk comprehensibility (Kim et al., 1994b); and diffusion coefficients in the absence of chemical potential and stress gradients (Kim et al., 1994a). The solvent uptake behavior for dodecane in polystyrene and various temperatures in the glass transition region has also been measured (Kim et al., 1993a). Thus, the purpose of this publication will be to critically analyze the predictions of this theory, using the various material properties that have been independently measured.

3. EXPERIMENTAL

3.1. Sample preparation and characterization Crosslinked polystyrene was prepared by bulk poly-

merization of styrene (Aldrich Chemical Co., Mil- waukee, WI) vacuum-distilled at 38°C/15 mmHg. It was mixed with desired amounts of the crosslinking agent, divinylbenzene (DVB, Aldrich Chemical Co., Milwaukee, WI) at crosslinking ratios, X, of 0.005 mol DVB/mol styrene, and reacted at 125°C for 48 h and in a vacuum oven at 125'~C for an additional 12 h. Dodecane/polystyrene systems were produced by mixing dodecane with the ensuing polystyrene sam- pies.

The physical properties of pure polystyrene and various dodecane/polystyrene systems were characterized using various techniques. The uniformity of the dodecane concentration distribution throughout the dodecane-containing polystyrene sample was examined by transmission FTIR spectrometry (Model 1600, Perkin Elmer, Norwalk, CT) (Kim et al., 1994b). Glass transition temperatures were measured by differential scanning calorimetry (Model 910 DSC, DuPont, Wilmington, DE) with a heat scanning rate of 10°C/min at a nitrogen flow rate of 140 ml/min. Thermogravimetric analysis (Model 951 TGA, DuPont, Wilmington, DE) was used to determine the thermal stability of samples. The glass transition temperature and the onset point of degradation of dodecane-free polystyrene were 100.5 and 250°C, respectively. Sorption experiments with cyclohexane were performed at 45°C to determine the molecular weight between consecutive crosslinks, .~¢. The re- suiting value of Mc was 5100, as calculated from equilibrium-swollen samples using an equation developed by Lucht and Peppas (Peppas, 1986).

3.2. Dynamic swellin 9 experiments Dynamic swelling experiments of crosslinked poly-

styrene were performed with dodecane at various temperatures, 70, 90, 105 and 120°C to investigate the temperature effect on the swelling behavior and to determine the interaction parameter, Z1, of the dodecane/crosslinked polystyrene systems (Kim et al., 1993a). Samples used for dynamic swelling experiments were cut in squares using a preheated scalpel;

D.-J. K1M et al.

their thickness was 0.6-0.7 ram. All of the samples had an aspect ratio (length over thickness) greater than 10, which ensured application of one-dimensional diffusional equations for analysis of the transport data.

3.3. Material properties 3.3.1. Equilibrium bulk modulus. The temperature

and concentration dependence of the equilibrium bulk moduli for dodecane/polystyrene systems were determined from PVT measurements (Kim et al., 1994b).

Samples were prepared with dodecane at five different dodecane concentrations of 2.4, 5.2, 10.1, 13.7, 19.6 wt%. The PVT behavior of dodecane/polystyrene systems was investigated using a PVT apparatus (Gnomix, Boulder, CO). This apparatus allowed the determination of the sample-specific volume under controlled temperature and pressure. The detailed procedure was reported before by Kim et al. (1994b).

The sample was isobarically cooled from well above T o to well below Tg at a constant rate, 1.5~C/min. This isobaric run was done at 10, 100, 200 MPa. The specific volume of a sample at the reference condition was measured using the pycnometer. Measurements were performed eight times and the average value was taken as a reference specific volume. The accuracy in this reference specific volume measurement was + 0.003 cm3/g. The actual specific volume was deter-

mined by adding the specific volume change measured by the PVT apparatus to the reference specific volume.

3.3.2. Viscoelastic properties. The temperature and frequency dependence of shear moduli of dodecane/ polystyrene systems were determined by dynamic mechanical analysis (Kim et al., 1993b).

Samples prepared with dodecane at seven different dodecane concentrations of 2.5, 5.0, 7.5, 10.0, 12.4, 16.8, and 20 wt% based on the total mixture as well as pure polystyrene samples were cut into rectangular specimens of typical dimensions of 50 x 10 x 2.5 mm. The specimens had uniform faces and square edges. Each sample was coated with silicone oil to prevent dodecane from evaporating from the sample at high temperatures.

Experiments were performed using a dynamic mechanical analyzer (Model 983 DMA, DuPont, Wil- mington, DE) in a vertical clamp mode. The oscilla- tion amplitude was 0.4 mm. To investigate the temperature dependence of the complex shear moduli, G*(e~), the experiments were performed at 1 Hz with a heating rate of 3°C/min. Liquid nitrogen was used to provide cooling in measuring moduli below the room temperature. For the study of frequency dependence of complex shear moduli, the experiments were performed at isothermal conditions. Isotherms were measured from 30"C below the glass transition temperature to 30°C above the glass transition temperature at intervals of 3°C. The frequencies chosen for this experiment were 0.03, 0.05, 0.1, 0.2, 0.5, 1, 2, and 4 Hz.

Penetration transport in glassy polymers

3.3.3. Diffusion coefficient. The temperature and concentration dependence of the dodecane self-diffusion coefficient in crosslinked polystyrene was measured using the pulsed gradient spin echo nuclear magnetic resonance spectroscopy (Kim et al., 1994a). Five dodecane/polystyrene systems containing 0.025, 0.050, 0.100, 0.150 and 0.200 weight fraction of dodecane were prepared by allowing varying amounts of dodecane to diffuse into the polymer sheets.

A cork borer was used to punch out a stack of prepared crosslinked polystyrene samples of the same crosslinking ratio and the same dodecane content. The final samples had diameters of 6 mm and thickness ranging from 0.7 to 0.8 mm. The discs containing a fixed amount of dodecane were placed in a flat bot tom-NMR tube using a glass rod. Six NMR tubes containing (~20 wt% dodecane were then sealed with Teflon tape to prevent dodecane evaporation. The optimum height of the sample stack was about 8 mm.

The pulse gradient spin echo (PGSE) NMR tech- nique was used by applying a large gradient of magnetic field, G, twice for duration 6 after the 90 pulse and after the 180 pulse. A steady field gradient, Go, of much smaller magnitude was also employed to nar- row the echo sufficiently to facilitate measurement of the baseline and to increase time and phase stability of the echo. The pulsed field gradient was used to reduce the considerable heat dissipation in the coils produced by the steady magnetic field gradient and to broaden the spin echo to measure its height more accurately.

In our experiments, measurements were performed at temperatures of 105, 120 and 140cC, using a Spin- Lock CPS-2 pulse NMR spectrometer operating at 33 MHz for protons. PGSE experiments were con- ducted at fixed Go and G and at varying 6. The time separation between gradient pulses, r, was less than 25 ms. The steady and pulsed field gradients, Go and G, were 0.85 and 121.8 G/cm or 304.5 G/cm, depending on the concentration of the system and the duration of the gradient pulse. The duration of field gradient pulse, ft, varied from 0.5 to 13 ms.

4. RESULTS

4.1. Viscoelastic diffusion model for one-dimensional swelling

In order to predict solvent uptake data using the previously developed model, we experimented with a thin, planar polymer sample in which polystyrene was in free contact with dodecane allowing one-dimensional swelling. In this case, the components of deformation and stress tensors in a Cartesian coordinate system reduce to the following:

1 = = 1 0 , F = - -

F ~ 0 1 4,2

[o ] T, 0 0 T = T± 0 ,

0 Tz

(13a)

4831 00] T 0 = _ P o 1 0 (13b)

0 1

where 4'2 is the polystyrene volume fraction of system, Ttt and T± denote the total mixture stresses parallel and perpendicular to the exposed sample interface unit vector n, respectively, and po is the solvent pressure exerting on the sample surface.

In a planar swelling geometry the constitutive equation for the parallel component of polystyrene stress, T2. , were obtained from eq. (10) as

T2.1 = ~ [a3GE + KE] (F 2 - 1)

+ ~ {~ G~ C4,:(t), t* - 4*]

dF2(O + KA[4,2(t),t* - ~2]} ~ d 4 (14)

where the time-dependent polystyrene deformation gradient, F, was determined from the polystyrene composition and velocity fields. In the absence of chemical reactions, the equation of continuity for polystyrene produced the following kinematic rela- tionships (Lustig et al., 1992; Malvern, 1969):

1 4,2(E, r) = - - t15)

F(E, r)

D2 F(E, r) 8f2 (E, r)

Dz ? - 16)

where ~ is a dimensionless material coordinate im- bedded in the polymer and D2/Dt is a substantial derivative holding the modified coordinate constant. Using the partial mass and momentum balances and eq. (1), the polymer's velocity field was determined to be given by

if2 (~,, r) = D,2 4,1 w2 I ~ x ~ 1 . {17)

Equations (14)-(17) define the one-dimensional planar swelling in a viscoelastic fluid, specifying both the time-dependent concentrations profile 4,2 and stress profile T2.11. In order to affect a solution the appropriate initial and boundary conditions are required. Initially, it is assumed that the entire slab is in an undeformed configuration; thus,

F(E,z = 0) = 1. (18}

One-dimensional swelling will be considered for the case where the polymer is place at E = _+ 1 at t = 0; thus, it is assumed that the sample's centerline re- mains stationary while the net motion due to swelling is directed symmetrically into the solvent, resulting in

t72( ~' = O, z~ = O. (19)

4832 D.-J. KIM et al.

Dodecane exerts only atmospheric pressure on the mixture at the dodecane/polystyrene; thus

T,I(E = 1, Q = TI.II(E = 1, z) + T2.11(E = 1, z) = 0 .

(2o)

The initial condition given by eq. (18) and the two boundary conditions given by eqs (19) and (2) are sufficient for the solution of the one-dimensional solvent uptake problem.

The time-dependent composition, stress, and chemical potential profiles were given by the numerical solutions of eqs (16)-(20) using the appropriate finite- element methods. Details of the measured methods used to solve these equations are given elsewhere (Lustig, 1989).

The composition, strain, and stress profiles which were determined in material coordinates, E, were then transformed into profiles in spatial coordinates, (, by integrating the polymer deformation gradient,

d~(E, z) - - - F ( E , z ) , ~(Z=O,z)=O. (21)

dE

The stationary boundary condition at the polymer's centerline is consistent with the boundary condition (19). Experimental results are traditionally reported in terms of the fractional penetrant mass uptake M t / M ~ , and this was calculated from the numerical solution of the model by numerical integration of the dodecane concentration profiles within the polymeric system over the entire sample volume as given in eq. (22):

Mt f f f p l d V 1 1 o

(a)

~Z

i 1.2

1.0

~ 0.8

"~ 0.6

t.u 0.4

0.2 e O Z

I I I I

I I I I

30 60 90 120 150 180 Temperature (°C)

(b)

0

e'-

E .2_

Lt.l

. - -

0 Z

0.35 , I L I

0.30

0.25

0.20

0.15

0.10

0.05

0.00 30 60 90 120 150

Temperature (°C) 80

Fig. 1. Normalized equilibrium bulk (a) and shear (b) moduli as a function of temperature in dodecane/polystyrene systems containing 0.0, 5.2, 10.3, 13.7 and 19.6wt% of

dodecane (curves from right to left).

111 If? = L--~ ~ - 1 ~bt (X, t) F(X, t) dX (22)

where Lo is the initial half-width of the polymer specimens. Experimental M t / M ~ data are often parameterized by the power law expression kt" and the solution of the viscoelastic model given. Equation (22) can also be parameterized in this form.

concentration dependences of K~, were expressed by the following Tait equations (Kim et al., 1994b):

V(~_P~ V(dp,, P, T) [P + B(~b,, T) K e ( ~ , T ) = - ~"~/r, Eq = V(c~,,O, T ) C

B(0, , T)

C (23)

4.2. Determination of material parameters Verification of the Lustig et al. transport model

required that all material parameters (e.g. G, K, D) employed in eqs (1), (11) and (14) be measured or calculated from independent experiments. Figures l(a) and (b) show the normalized equilibrium bulk and shear moduli as a function of temperature and concentration. Clearly, the transition of normalized moduli shifted to lower temperatures with increasing penetrant concentration.

Equilibrium bulk and shear moduli had two limiting values GEe and GE,, Keg and KE,, at temperatures well above and well below T o . The temperature and

at P = 0, where

B(q~l, T) = Bt(q~)exp [--B2(~bl) T ] (24a)

Bl(~bl) = Bl l + Bleq51 (24b)

and

B2(~bl) = (B21 + B22 ~bl) × 10- 3. (24c)

Here, the value of C is C = 0.0894, the values of B11, B12 are 223.7 and -96.76 MPa and the values of B21 and B22 are 3.7801 and 2.904°C - 1, respectively.

The limiting values of Keg and GEg were determined at 3.5 and 1 GPa, respectively. The magnitude of GE,


was constant at about ~106 MPa (Ferry, 1980), as it 0.0 was negligibly small compared to that of Ke, (~109 MPa). -0.5

Consideration of the moduli at transition between - 1.0 two limiting equilibrium moduli suggested that the I~ following models could describe the normalized limit- - 1.5 J ing moduli:

-2.0 K~. o - K r , , (25a) K,F.(T, dpl)= KE,, q a(T, dpl)_l/2 + 1 -2.5

Ge o - GE, GE(T, ~bl) = GE,, + ' ' (25b) -3 .0 a(T, dpl) -Uz + 1" -1 2

Clearly, these equations require knowledge of the values of the shift factors a(T, q~l) -1/2. The temperature and concentration dependence of the shift factor above the glass transition temperature (Kim et al., 1993b) was expressed by the WLF equation as represented in eq. (26a), whereas the shift factor below glass transition temperature was linearly correlated with temperature as in eq. (26b):

log a(T, q51) = - C a ( T - T o )

(C2 + T - To)

loga(T, q~,) = - C3(T - To)

for T > T o (26a)

for T < T o . (26b)

Here, the values of Ct, C2 and C3 are 14.8, 48.3°C and 0.15, respectively, and the reference temperature, To, is the glass transition temperature of each system.

These glass transition temperatures were determined from the dodecane concentration dependence of glass transition temperature, Tg, which was correlated with eq. (27) (Kim et al., 1993b),

T~ = 99.81 -- 492.01~bt + 877.33q5 z. (27)

The discrete relaxational shear spectrum, G~ vs 2~, representing the shear relaxation function was obtained (Kim et al., 1993b) by analyzing the experimentally determined frequency-dependent shear moduli:

aa(t, ~bt) = Giexp ~ . (28a) i = 1

The effects of temperature and concentration on the relaxation time were given by a shift function as in eq. (12a). The normalized discrete shear relaxation moduli, di, vs normalized discrete relaxation time, 2i, are shown in Fig. 2 for dodecane-free polystyrene at various temperatures. The values of 2~ (=2iDeq/Lo) corresponding to the maximum d~ are shifted to higher values with increasing temperature, due to the increase in the equilibrium mutual diffusion coefficient.

To express the discrete relaxational bulk spectrum, K~vs2i, a similar expression to that proposed by Wang (1992) for epoxy resins was used, since the magnitude of the bulk modulus of epoxy resins is the same as polystyrene,

i=i \ zoi /

I [ I r I I

I I I I 1 I I

- 1 0 -8 -6 -4 -2 0 2 Log

4833

Fig. 2. Normalized shear relaxation spectra of dodecane/ crosslinked polystyrene slabs at 70, 90, 105 and 120 C

curves from right to left).

0.0

-0.1

-0 .2

I v -0 .3

8' ..a -0.4

-0.5

-0.6

-0.7 -12

I I I I I

- 1 0 - 8 -6 -4 -2 0 Log k

I

2 4

Fig. 3. Normalized bulk relaxation spectra of dodecane/ crosslinked polystyrene slabs at 70, 90, 105 and 120'C

(curves from right to left).

The bulk modulus, K(t), was determined from the relationship between the bulk modulus, the longitudinal bulk modulus, M(t), and the shear modulus, G(t), (Ferry, 1980; Marvin and McKinney, 19641. The dynamic longitudinal bulk modulus was determined from an acoustic experiment (Wang, 1992; Piche et al., 1987, 1988) by measuring the longitudinal velocity and attenuation coefficient of an ultrasonic wave transmitting through the sample. The resulting normalized discrete bulk relaxation spectra are shown in Fig. 3. The temperature dependence of the shift factor was involved in normalizing the relaxation time.

4.3. Diffusion coefficients The temperature and concentration dependences of

the mutual diffusion coefficient, D12 (Kim et al.. 1994a) were expressed by first expressing the self-diffusion coefficient of dodecane, Dx, using the Vrentas and Duda free volume model:

D~ = Do, exp I ?(~°' I7" + °)2 ¢ l~*I 1 - 9 F u J (29)

4834

where

and

D.-J. KIM et al,

VFH = (K11/7)~o1(K21 + T - TG,)

T

+(Ki2/)')o)2(K22 + T - Ta ) . (31)

The temperature dependence of the specific volume of dodecane and polystyrene, I? 1 and I72, respectively, were determined from the PVT measurements (Kim et al., 1994b):

l~l = 171(P = O, T) = A ~ + Aaz + At3 T 2 (32a)

Vz= Vz(P=O, T ) = A21+ A2z T. (32b)

Here, All = 1.3081 cm3/g, A12 = 1.294x 10-3/gK, A13 = 1.822 x 10-6cm3/gK 2, A21 = 0.9166 cm3/g and A22 = 5.8419 x 10-4cm3/gK.

Thus, the value of D~ was calculated from eqs (29) to (32) using the following parameters: I71 + l~ ° (0) = 1.071 cm3/g, 1~' = I ?° (0) = 0.85 cm3/g, K12/7 = 5.16×lO-4cm3/gK, K 2 2 - Ty2 = - 324.7 K, Ku/7 = 0.001325 cm3/g K, K21 - Tot = - 79.03 cm3/g K,

Do1 = 4.6 x 10 -5 cm2/s, and ~ = 0.4l. Then, the mutual diffusion coefficient was deter-

mined from the self-diffusion coefficient according to

D,2 = D1p2 V2p~ ( gkq'] (33) R T \~Pa/r,e"

The concentration and temperature dependences of the diffusion coefficient below the glass transition temperature were calculated by estimating a parameter characterizing the volume contraction at- tributed to the glass transition. The resulting value of the diffusion coefficient below the glass transition temperature was 3.16 x 10- ~ ~ cm2/s.

Figure 4 shows the concentration and temperature dependence of the normalized mutual diffusion

1 I I

0.05 0.1 0.15 Dodecane Weight Fraction

0 . 2

q:T 1

"~ 0.8 O O

o 0 .6 t . - .o

0.4

-o 0 . 2 N

0 O z

Fig. 4. Normalized mutual diffusion coefficient in dodecane/ polystyrene systems as a function of dodecane weight fraction at 70 (curve 1), 90 (curve 2), 105 (curve 3) and 120°C

(curve 4).

coefficient in the rubbery and glassy regions. The temperature dependence was more significant at 70 or 90°C than at 105 or 12OC, as the transition from glassy to rubbery state occurred during transport at 70 and 90°C.

4.4. Surface kinetics during swelling Once all material parameters had been determined,

the time-dependent surface concentration profiles could be determined by substituting the constitutive eqs (9) and (10) into the surface momentum balance of eq. (20):

lnq51 + (1 - 4,) + z d l - (t)l) 2 - I - ~ [~ G~(T, ~b,)

(" F4 "g

×exp - L a ( Y , ~ i +~ K~

×exp[-f' de

The non-integral term in eq. (34) indicates the pure elastic contribution that governs the equilibrium surface composition after the time-dependent viscoelastic contribution represented in the integral term vanishes. Equation (34) was solved numerically by applying the initial condition (18).

Figure 5 shows the time-dependent dodecane volume fraction at the dodecane/polystyrene interface at 70, 90, 105 and 120°C. The equilibrium dodecane concentration at the dodecane/polystyrene interface increased with experimental temperature. The initial plateau in the polymer surface composition resulted from a finite limit of the viscoelastic integrals at time t = 0 ÷. The sharp transition between the initial and equilibrium concentration plateaus corresponded to the glassy/rubbery transition. The shape of the

'~ 0.25

=

.o_ 0.2 o

u_ 0.15 o E

0.1

,~ 0.05 CD " ID O

D 0

I I I

) I I I I 4 -12 -10 -8 -6 -4

Log x

I I e--

I ;

- 2

Fig. 5. Time-dependent dodecane volume fraction profiles at the polystyrene/dodecane interface at 70, 90, 105 and

120°C (curves from right to left).


transition was determined by the width of the relaxation spectrum. In our case, the scale of discrete relaxation time, )~, was so small that the relaxation in the glass transition occurred very fast. This phenomenon was more significant when the shift factor was decreased at elevated temperatures. It should be noted that the actual transition time scale is much higher at lower temperatures than at higher temperatures, as the time scale in Fig. 5 is logarithmic.

4.5. Model-predicted concentration, chemical potential and stress profiles

Figure 6 shows the experimental results of dodecane mass uptake, M/Mp, as a function of time. The transport rate (as determined from the slopes of these curves) and the equilibrium mass uptake in- (b) creased rapidly with temperature. Above the glass ~ 0.25 transition temperature of dodecane-free polystyrene ff (100 C), the initial transport rate was not affected very .9 0.2 much by temperature. It was observed that the time necessary to reach the maximum mass uptake de- " 0.15 creased with increasing temperature.

Figures 7(a), (b), (c) and (d) show the time-depen- -6 0.1 > dent dodecane concentration profiles within the poly- ~= meric systems at 120, 105, 90 and 70C , respectively, o ~ 0.05 The volume of the polymeric systems increased by 15 x~'

o 23% at the equilibrium point, depending upon the o 0 temperature.

Above the glass transition temperature of the dodecane-free polystyrene, the relaxation occurred so fast that the transport process was governed purely by (c) the concentration gradient within the system. At 120 ~,~ 0.2 and 105 C, the concentration profiles were qualita- d tively similar to those predicted by the solution of the ._o chtssical Fickian diffusion equation, as profiles were ~ 0.15 smoothly concave from the polymer interface until " a significant penetrant concentration was attained at E 0.1 the sample center. As the temperature decreased from -6 120 to 70 C, the transport behavior changed from > Fickian to case II, because of the significant concen- ~ 0.05 tration dependence of the diffusion coefficient and the v

O increase in the mechanical relaxation contribution ~ ^

0.24

0.20

~- 0.16

~ 0.12 Q .

m 0.08

o m 0.04

- o o C3

0.00 0

# Z~AAAA

OOOO O

>[3 []

>[3

I i i I

"% A Z~

O O ¢

[] [3 [] [] [] []

O O O

oOO o °

I I ] I I

10 20 30 40 50 60

Time (hr)

Fig. 6. Dodecane uptake per polystyrene dry weight, M,,.Mp, in crosslinked polystyrene slabs as a function of

diffusion time 70 (O), 90 (O), 105 (O) and 120:'C (zM


(a)

0.25

~" 0.2

if" 0.15

~ 0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Normalized Position,

I I I I l

I I

0 0.2 0.4 0.6 0.8 1 Normalized Position,

I

I

1.2 1.4

I I I I I

0 0.2 0.4 0.6 0.8 1 Normalized Position,

I

1.2 1.4

(d)

0.2 I l I I [

.o

< LI_

E 0.1 O >

0.05

O

0 0 0.2 0.4 0.6 0.8 1 1.2

Normalized Position, ~

Fig. 7. Time-dependent dodecane volume fraction profiles within polystyrene systems at 120 (a), 105 (b), 90 (c] and 7OC (d).

4836

induced by dodecane plasticization of the system. At 70 and 90°C the concentration profiles showed relatively sharp fronts at the swelling interface, as the penetrant diffusion rate was reduced in the unswollen, glassy region.

Figures 8(a), (b), (c) and (d) show the time-dependent polymer stress profiles within the polymeric systems at 120, 105, 90 and 70°C, respectively. The stress profiles at 120 and 105°C were continuous and similar to those of Fickian diffusion. Above the glass transition temperature of the dodecane-free polymer, the polymer stress, T2, was governed only by the equilibrium stress exerted by the polymer network structure.

The most interesting phenomenon in this stress behavior was observed at experimentation temperatures 20 to 50°C below the glass transition temperature of the dodecane-free polymer as the glassy to rubbery transition occurred during the transport process at these temperatures. The polymer stress profiles at 90 and 70°C exhibited maxima and minima which were moving toward the center of slab as diffusion proceeded. As the mechanical modulus in the glassy state was much higher than that in the rubbery state, even a minute swelling strain led to a high stress. Thus, the maxima indicated the swelling fronts where the glassy/rubbery state transition occurred. Once the transition had occurred, an abrupt decrease in the polymer mechanical moduli led to a sharp decrease in polymer stress. Thus, the minima following the sharply decreasing polymer stresses indicated the outer limit of swelling fronts. Once the swelling fronts reach- ed the center of the polymer slab, the polymer stresses achieved their equilibrium values since the entire system attained the rubbery state. The maximum values of the polymer stresses at 70°C were higher than those at 9ff'C due to the increase in the mechanical moduli.

In order to understand how chemical potential and polymer stress gradients contribute to the overall transport behavior of the system, we plotted the normalized time-dependent chemical potential profiles of the dodecane/polystyrene system at 70°C as shown in Fig. 9. These correspond to the concentration profiles of Fig. 7(d). The chemical potential decreased monotonically with distance and exhibited sharply decreasing fronts due to its logarithmic decrease at low penetrant concentration range. Thus, the chemical potential gradient had a positive contribution to the polymer swelling velocity and local penetrant uptake.

4.6. Model-predicted mass uptake behavior The time-dependent fractional dodecane uptake as

predicted by the model (solid curves) was compared with that experimentally determined (filled circles) at 120, 105, 90 and 70°C. Good agreement in the sorption rates was obtained as shown in Figs 10(a)-(d).

The theoretical transport kinetics were interpreted by eq. (22) and were compared with the experimental observations. Table 1 presents values of the diffusional exponent, n, for these two cases. The resulting transport processes were dependent on the system

D.-J. KIM et al.

(a)

ii._~ 0.12

0.1 ==

0 ,08

E 0 ,06

0 .04

0 .02

Z 0 0 0.2 0,4 0.6 0.8 1 1.2 .4

Normalized Position,

( b }

t1~ 0 .12 .... , t K

o.1

0 .08

0.06

0.04

0 .02

Z 0 0 0.2 0.4 0.6

(c)

i~_~. 0.12

~ 0.1

0.06

E ~ 0.06 0

"o 0 .04 N

0.02

O z 0

I 1

7 I I I I

0.8 1 1.2 1.4 Normal ized Posit ion,

I I t I I

I I

0 0.2 0.4 0.6 0.8 1 Normal ized Position, {

I

1.2 .4

(dl

i i ~ 0.3 i i i i

1 0.2 " ~ " " ~ x

_ 0 .15 O

13.

"o 0.1 N

0 .05 E O z 0

0 0.2 0.4 0.6 0.8 1 1.2 Normalized Position,

Fig. 8. Time-dependent normalized polymer stress profiles, ~rz, within polystyrene systems at 120 (a), 105 (b), 90 (c) and

70°C (d).


-d (a)

0 1.2 no 1

- 2 0 .8

~D -4 m o= o.o

o - 6 0.4 -~ .9

8 -8 o.2

- i o o E 0 0.2 0 .4 0 .6 0 .8 1 1.2 0 z Normalized Position,

Fig. 9. Time-dependent normalized chemical potential profiles within the polystyrene system at 70°C. (b)

1.2 g

1 temperature. The values of n determined from the --- experimental sorption data were relatively lower than ;~-A: 0.8 those predicted theoretically from the model. At 90, -~ 105, and 120' C, the theoretical values were within the ~ 0.6 upper 95% confidence limit of n as determined from experimental data. Although the dodecane transport ~ 0.4

O

behavior at 120 and 105C was near-Fickian, the 0.2 value of n was not exactly 0.5 due to the concentrat ion u.

dependence of diffusion coefficient. 0 -

There was a significant difference between experi- 0 mental and theoretical values of n for diffusion at 70~C, probably due to inaccurate description of the material properties, especially below the glass transition temperature of the system. In the analysis of experimental mass uptake data, all correlation values of k and n were determined by the nonlinear least- squares analysis for log[Mt/M,~] as a function of log t for initial 60% of weight gain.

4.7. Transport kinetics Figure 11 shows the diffusional exponent, n, as

a function of the temperature for the dodecane/polystyrene system. Above the glass transition temperature of dodecane-free polystyrene, the transport kinetics approached that of Fickian diffusion. The dodecane flux was mainly driven by the dodecane concentration gradient. Increasing temperatures led to decreasing values of n as the concentration dependence of the diffusion coefficient decreased. Fickian behavior was observed at temperatures well above the glass transition temperature where the concentration dependence of the diffusion coefficient became negligible. The relaxation contribution to the transport kinetics was significant below 100°C, i.e. the glass transition temperature of dodecane-free polystyrene.

in glassy polymers

(c) 1.2

1

~" 0 .8

o. 0.6

"~ 0.4 O

k ~

0.2

0 o

(d) 1.2

Fig. 10. Fractional dodecane mass uptake measured from experiments (filled circles) compared with that determined from prediction model (solid curves) as a function of diffusion time at various temperatures of 120 (a), 105 (b), 90 (c),

and 70°C (d).

~ 1

.~ 0.8

~" 0.6

"~ 0.4

'~ 0.2 , t -

w

1 2 3 4 Time (hr)

4837

@ @

I I I I I

5 10 15 20 25 30 Time (hr)

t I I I I I I

f I I I I I I t

1 2 3 4 5 6 7 Time (hr)

I I I ! I

50 100 150 200 Time (hr)

250 300

4838 D.-J. KIM et al.

Table 1. Temperature-dependent transport kinetics, predicted from model or experimental data fitted to eq. (22)

Diffusional exponent, n

Analyzed from experimental data Temperature Predicted (°C) from theory Value Lower 95% Upper 95%

70 0.86 0,69 0.67 0.71 90 0.71 0,65 0.57 0.72

105 0.70 0,62 0.43 0.82 120 0.67 0,58 0.48 0.67

Table 2. System parameters for normalization of data and solution of model

Temperature (°C)

Half-thickness of Equilibrium diffusion Equilibrium dodecane polystyrene slab Lo (cm) coefficient D12,¢q (cm2/s) concentration 4~l,eq

70 0.030 2.140 × 10 -9 0.1548 90 0.030 2.005 × 10 -8 0.1744

105 0.031 6.026 x 10 -8 0.2083 120 0.033 1.230 × 10 -7 0.2294

= 0.9 ~ ~ 0.8

~ o.7 W

0 . 6 s

s 0.5 ~'~ i i i 0 50 100 150 200

Temperature (°C)

Fig. 11. Exponent, n, as a function of temperature for the transport process (filled circles) driven by both chemical potential gradient and polymer stress gradient or for the transport process (open circles) driven by only chemical

potential gradient.

At temperatures between the glass transition temperature of dodecane-free polystyrene and the dodecane/ polystyrene systems, discontinuous swelling fronts were formed. The viscoelastic relaxation effect was most important in the temperature range 55-70°C.

4.8. Diffusional Deborah number analysis The normalized relaxation time involving the tem-

perature- and concentration-dependent shift factors influences significantly the transport behavior. The normalized viscoelastic relaxation time, 21 = 2~Deq/L2o, could be used in the diffusional Deborah number, De, defined as the ratio of the characteristic relaxation time, 2, to the characteristic diffusion time, 0 = L~/D12. By replacing the concentration-dependent relaxation time, 2, and the diffusion coefficient,

D12, by the mean viscoelastic relaxation time, 2,., and the mean diffusion coefficient, D 12,,,, the temperature dependence of transport kinetics could be analyzed by simply using the mean Deborah number, De,,, defined a s

,~"Ol2,m De,. = L2 (35)

Here, the temperature-dependent mean characteristic relaxation time of the polymer/penetrant system, 2m, was approximated by multiplying the temperature- dependent shift factor, aT, and the geometric mean of two concentration-dependent shift factors evaluated at the initial and equilibrium concentration, ac~, to the original mean characteristic relaxation time of dodecane-free polymer, 2°:

)~,"(T, C) = 2 ° ar acre. (36)

The characteristic relaxation time of dodecane-free polymer, 2 ° , was determined as

Io 2 G(2) d2 (37) 2 ° - io O(2)d2

Here, the function G(2) was obtained from the viscoelastic relaxation spectra of a dodecane-free polymer system.

The main diffusion coefficient, D12.", was also approximated with the geometric mean of two diffusion coefficients evaluated at the initial and equilibrium concentration of dodecane uptake.

Figure 12 shows the temperature dependence of the mean diffusional Deborah number. Above 80°C, the mean diffusional Deborah number was much lower

E

1 o s

1 0 3 .Q E 1 01 7 10_ 1

-g 0.3 S 1 ~, 0.5 1

1 0 .7

--~ 1 0 -9 0-11 I

10 "13

¢ 1 z; 2 O


i i i i

140 170

I I I

5 0 80 110

Temperature (°C)

Fig. 12. Mean diffusion Deborah number as a function of temperature.

than unity, the rate of viscoelastic relaxation was much faster than that of pure diffusion, and the polymer's mechanical response was elastic. At temperatures below 40~'C, the mean diffusion Deborah number was much larger than unity and the rate of viscoelastic relaxation was much slower than that of pure diffusion. At temperature ranges from 50 to 70C, the mean diffusion Deborah number was near unity. The fact that the viscoelastic relaxation and diffusion occurred on the same time scale led to the large deviation from the Fickian transport characteristics as shown in Fig. 11.


cess, discontinuous stress profiles were formed. The polymer stress gradient provided the negative driving forces to the penetrant transport against the chemical potential gradient and this led to an anomalous transport behavior.

The theoretically predicted behavior was compared with that experimentally observed for the dodecane/ polystyrene system. Several material properties measured using various techniques were used in the model. The fractional mass uptake behavior predicted from the model agreed fairly well with that observed from experiments. Several types of transport behavior were predicted at varying system temperatures. In general, Fickian characteristics were observed well above and well below the glass transition temperature. The transport mechanism deviated significantly from l-'ickian, when the diffusion rate and relaxation rate were of the same scales. These observations were verified b.~ the mean diffusion Deborah number.

Acknowledgements This work was supported in part by a grant from the National Science Foundation. We wish to thank Dr S. Lustig for allowing us to use the numerical algorithms for the solution of the Lustig et aL (1992) model.

acm

(/1" B B

5. C O N C L U S I O N S

The purpose of this work was to investigate a self- C consistent framework describing the transport behav- C ior of polymer/penetrant systems in both equilibrium and dynamic processes. The continuum field theory C,(s) provided a unified and detailed description of penetrant transport in polymeric systems by incorporating D~ fundamental driving forces: polymer relaxation, ther- Deq modynamic mixing and momentum conservation.

The viscoelastic polymer nature was characterized D12 by the observable thermodynamic difference between D12.m

the material's instantaneous and long-term proper- D~/D ties. The time scale of viscoelastic relaxation was gov- /)12 erned by penetrant concentration and temperature. The time to attain equilibrium composition at the De polymer/fluid interface was governed by the comple- De,. tion of the time-dependent viscoelastic relaxation and E was more significant with decreasing temperature and F penetrant concentration as the relaxation rate was decelerated. The nature of the transport process was F also governed by the relaxation time scale. Above T o , the mechanical response of polymeric materials is GE essentially elastic. Thus, polymer stresses increase Ge monotonically with the strain associated with increasing penetrant concentration. Both the chemical po- G~ tential and polymer stress gradients enhanced the G'~ penetrant mass uptake and provided the Fickian nature of the transport behavior. When the glassy to G~ rubbery state transition occurred in a transport pro-

N O T A T I O N

concentration-dependent shift factor mean concentration-dependent shift factor temperature-dependent shift factor Tait parameter for equation of state left Cauchy-Green strain tensor ( = F . F I ) universal constant for the Tait equation right Cauchy Green strain tensor ( = F r . Ft relative right Cauchy Green strain ten- s o r [ = F J i t ) ' C ( s ) ' F - l ] self-diffusion coefficient for component mutual diffusion coefficient at equilibrium penetrant concentration mutual diffusion coefficient mean mutual diffusion coefficient material time derivative operator normalized mutual diffusion coefficient

(=Dt2/D12.eq} diffusional Deborah number mean diffusional Deborah number strain tensor (=B - I) component of the deformation gradient tensor deformation gradient tensor of a component, d x / d X elastic shear modulus normalized equilibrium shear modulus (=GI~V2/RT} discrete shear relaxation magnitude normalized discrete shear modulus (=Gi122 RT) time-dependent shear relaxation modulus

4840

I unit tensor j~ diffusion flux for a component ct k sorption rate constant Ke elastic bulk modulus /(E normalized equilibrium shear modulus

( = K E V 2 / R T ) K~ discrete bulk relaxation modulus /(~ normalized discrete bulk modulus

( = Ki V2/R T) KA time-dependent bulk relaxation modulus Lo half-thickness of polymer slab Ma penetrant molecular weight _M~ average molecular weight between cross-

links M, molecular weight of repeating unit M~ equilibrium solvent mass uptake n sorption rate exponent R gas constant t time t* material time scale T temperature To reference temperature Tg glass transition temperature T~ partial stress tensor of component ct T2,1I normalized polymer stress (= T2,1I M1172/

R T ) u~ diffusion velocity, v~ -- v v mass average velocity, w~v~ + w2v2 v~ component velocity l?~ specific volume of component ct ~52 normalized polymer velocity (=v2Lo/

D12,eq) w, weight fraction of component alpha, p~/p x space fixed coordinates X material fixed coordinates

Greek letters 2i discrete viscoelastic relaxation time 2i normalized discrete relaxation time

(=Dx2.eq2i/L 2) }~m mean viscoelastic relaxation time for

penetrant /polymer system 2 ° mean viscoelastic relaxation time for

penetrant-free polymer #~ chemical potential per unit mass of com-

ponent ct ~1 normalized solvent chemical potential

( = I ~ M 1 / R T ) Ve concentrat ion of effective chains in poly-

mer network normalized distance in the spatial coordinate (= x/Lo)

~* material time scale E normalized distance in the material coor-

dinate ( = X/Lo) p total mass density p~ component mass density z normalized time ( = D 12, cq t~ L2) I) 2 equilibrium polymer volume fraction in

polymer/penetrant mixture

D.-J. KIM et al.

Z1 q,

qJM

%o

volume fraction of component a in mixture polymer-solvent interaction parameter total Helmholtz free energy per unit volume mixture Helmholtz free energy of mixing per unit volume of mixture equilibrium Helmholtz free energy per unit volume of component 2 pure Helmholtz free energies per unit volume of component a

Others I

III

first strain invariant of Cauchy-Green deformation tensor third strain invariant of Cauchy-Green deformation tensor

Note added in proof--The unrelaxed glassy moduli, KE, and G~,, as welt as the equilibrium rubbery moduli Ke, and G~, indicated in eqs (25a) and (25b) are approximately linear functions of temperature and concentration, at least for the temperatures and concentrations of current interest; thus/(E and Ge should formally also be linear functions of temper- ture and concentration. However, the functional form employed in eqs (25) exhibits a smooth nonlinear transition between the glassy modulus and the rubbery modulus, both of which are nearly linear functions of temperature and solvent concentration. This form of the limiting moduli, which closely resembles the K' (T, q~, to = constant) and G' (T, ~o, co = constant) response, was chosen so that the numerical solution would not have to account for the very fast relaxation processes which would substantially increase the computational difficulty of the solution. Specifically, we have focused on the relaxation processes for times longer than 1 Hz, which are times of interest in any practical swelling experiment.

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Experimental verification of a predictive model of penetrant transport in glassy polymers