I n d . Eng. Chem. Res. 1990,29, 1194-1204
1194
Shita, T. An Electron Paramagnetic Resonance Study of Alcohol Oxidation by Fenton’s Reagent. J . Phys. Chem. 1965, 69, 3805-3814. Skinner, J. F.; Glasel, A.; Hsu, L.; Funt, B. L. Rotating Ring Disk Electrode Study of the Hydrogen Peroxide Oxidation of Fe(I1) 1980,127, and Cu(1) in Hydrochloric Acid. J . Electrochem. SOC. 3 15-324. Sobkowiak, A. The Influence of pH on the Catalytic Current of the Fe(III)/H202System. Electrochim. Acta 1987, 32, 1251-1252. Sobkowiak, A.; Fleszar, B. The Equation of Catalytic Current of the Fe3+-H707System with Regard to Hydroxylation. Electrochim. Acta I!%< 26, 847-850. Walling, C. F.; El-Taliawi, G. M. Fenton’s reagent. 11. Reactions of Carbonvl ComDounds and 0.D-Unsaturated Acids. J . Am. Chem. Soc. 1973a, 95 844-847.
Walling, C. F.; El-Taliawi, G . M. Fenton’s Reagent. 111. Addition of Hydroxyl Radicals to Acetylenes and Redox Reactions of Vinyl 1973b, 95, 848-850. Radicals. J . Am. Chem. SOC. Walling, C.; Kato, S. The Oxidation of Alcohols by Fenton’s Reagent. 1971,93,4275-4281. The Effect of Copper Ion. J. Am. Chem. SOC. Wells, C. F.; Salam, M. A. Complex Formation between Fe(I1) and Inorganic Anions. Part 1.-Effect of Simple and Complex Halide Ions on the Fe(II)+H202Reaction. Trans. Faraday Soc. 1967,63, 620-629. Yen, S.; Chapman, T. W. Indirect Electrochemical Processes at a 1985, 132, 2149. Rotating Disk Electrode. J. Electrochem. SOC. Received for review October 31, 1988 Revised manuscript received September 21, 1989 Accepted November 1, 1989
MATERIALS AND INTERFACES Coupled Deformation and Mass-Transport Processes in Solid Polymers Ruben G. Carbonell* Chemical Engineering Department, North Carolina S t a t e University, Raleigh, North Carolina 27695-7905
Giulio C. Sarti Dipartimento di Ingegneria Chimica, Universitci di Bologna, 40136 Bologna, Italy
A theory is developed for the effect of stress on the diffusion of a swelling penetrant into a linear viscoelastic solid. A constitutive equation for the Helmholtz free energy of the mixture is used to obtain expressions for the dependence of the stress tensor on the history of the deformation and concentration fields. For cases in which the solid has only a single relaxation time, the stress constitutive equation is consistent with the standard linear viscoelastic model for solid deformation. The expression for the free energy is also used to obtain an equation for the difference in the chemical potential between the solute and the polymer as a function of the trace of the stress and concentration histories; as a result, the theory is thermodynamically consistent. As an example, the analysis is applied to the case of solute transport into a thin rectangular slab, resulting in a new expression for the diffusive flux. Introduction The transport of low molecular weight solute species through a polymeric matrix is of significant importance in a variety of applications (e.g, food packaging, microelectronics, drug release, etc.). Research in this field has been quite active in the last decade. The aim has been to obtain a clear understanding of the factors influencing the diffusion rate, as well as a suitable mathematical description of the physics of the process. The problem is rather complex since by changing the polymer-penetrant pair, or simply the operating conditions, several different types of transport behavior have been observed (Hartley, 1946; Crank and Park, 1951; Crank, 1953; Bagley and Long, 1955; Newns, 1956; Long and Richman, 1960; Fujita, 1961; Hopfenberg, 1970; Alfrey et al., 1966; Jacques et al. 1973, 1976; Nicolais et al., 1977; Berens and Hopfenberg, 1978; Enscore et al., 1980; Thomas and Windle, 1981; Durning and Russel, 1985) for which not even a systematic nomenclature seems to be established. On the basis of integral sorption (desorption) tests, solute transport through a rubbery polymer is typically
* Author t o whom correspondence should be addressed. 0888-5885 19012629-1194$02.50 /O
Fickian with a diffusion coefficient that is dependent on concentration. Similar behavior is also observed for the diffusion of a very dilute species in a glassy polymer. In these cases, after the external chemical potential of the sorbate is suddenly changed, a slab sample with an initial uniform concentration exchanges mass with the surroundings in such a way that the ratio of the mass of solute absorbed at a given time to the mass of solute absorbed at equilibrium, M,/M,, is proportional to the square root of the diffusion time, tl/*. For different sample thicknesses, L, all sorption (desorption) data reduce to a single master curve by using a dimensionless time, tD/L2,where D is the diffusion coefficient. This behavior is consistent with a mathematical description in which the species balance equation is solved for the case in which Fick’s law holds and where the convective contribution to solute transport is neglected. The assumption is usually made that the external surface of the sample is kept at a constant composition of solute for all time t 2 0. In contrast with this behavior, mass transport is usually non-Fickian in cases in which the solute is below its critical temperature and has a limited solubility in the polymer matrix, and when the polymer is initially an amorphous glass, virtually insoluble in the external environment. B 1990 American Chemical Societv
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1195 Indeed, the weight uptake during sorption usually is not proportional to the square root of time but can show several possible anomalies. (i) There can be a two-stage or relaxation behavior (Long and Kokes, 1953; Newns, 1956; Long and Richman, 1960; Fujita, 1961; Berens and Hopfenberg, 1978) in which an initial Fickian-like diffusion is observed followed by an abrupt decrease in the rate of mass uptake, after which there is a more or less slow drift toward the final equilibrium solute content. An apparent equilibrium intermediate between the two stages has also been observed for several systems (Newns, 1956; Kishimoto et al., 1960; Berens and Hopfenberg, 1978). It has been recognized that the rate at the beginning of the second stage depends on a creep mechanism and that the entire sorption is a combined diffusion-relaxation process (Newns, 1956). (ii) It is also possible that the solute transport exhibits what is often called Case I1 behavior, in which the weight uptake is linear in time, until equilibrium is reached (Hartley, 1946; Mandelkern and Long, 1951; Alfrey et al., 1966; Hopfenberg, 1970; Thomas and Windle, 1981). (iii) The rate of mass exchange can also exhibit dependencies on time that are intermediate between the Fickian and Case I1 behavior, namely, of the form tn where n is a number between 0.5 and 1.0 (Nicolais et al., 1977; Thomas and Windle, 1978; Berens, 1985). (iv) Super case I1 behavior refers to a sudden acceleration in the rate of mass uptake, preceded by an initial case I1 or anomalous diffusion behavior for short times (Jacques et al., 1973, 1976; Berens, 1985). (v) An overshoot in the weight uptake with respect to the final equilibrium value has also been observed in several cases (Baird et al., 1971; Franson and Peppas, 1983; Durning and Russel, 1985), followed by oscillations in the weight uptake in time during the approach to equilibrium. Even though at the very early stages of sorption a rather rapid weight uptake can be encountered (Newns, 1956; Hopfenberg, 1970; Jacques et al., 1976), as is the case with polystyrene-n-alkane pairs, other systems exhibit an induction time during which no mass transfer is observed. This was reported, for instance, in the absorption of methanol into poly(methylmethacry1ate) (Thomas and Windle, 1981). A limited but interesting set of experimental data is available on the concentration profiles within the sample (Long and Richman, 1960; Gumee, 1967; Kwei and Zupko, 1969; Hopfenberg, 1970; Thomas and Windle, 1978, 1981). It is possible either to obtain a continuous concentration field without any singularities or to have abrupt jumps or discontinuities in the concentration profile within the sample. Case I1 behavior is often associated with a constant shock or wave in the concentration field, traveling at a constant speed (Kwei and Zupko, 1969; Hopfenberg, 1970; Thomas and Windle, 1978). Even though the experimental evidence is not definitive, a Fickian tail in the concentration profile is believed to exist ahead of the concentration shock, and its existence has been used to explaii Super Case I1 behavior (Jacques et al., 1976). Anomalous diffusion is believed to be associated with shocks in the concentration profile whose magnitude and propagation rate decrease with time (Thomas and Windle, 1981). While usually a “Fickian” behavior in the overall weight gain is associated with a Fickian smooth concentration profile, there are cases in which the penetrant composition shows a jump, just as for the anomalous diffusion case (Kwei and Zupko, 1969). In the two-stage or relaxation behavior, the second stage of sorption takes place with concentration profiles that are rather uniform in space (Newns, 1956; Long and Richman,
1960; Fujita, 1961). The penetrant content increases almost uniformly throughout the sample, following the concentration increase at the external surface. The existence of a shock in the concentration profile has been reported for cases in which a sufficiently large increase in the penetrant chemical potential was instantaneously applied at the sample surface. During transient diffusion, there are thus regions in which the polymer is glassy (low solute concentration) as well as regions in which the polymer is above the glass transition temperature (high solute concentration). Although non-Fickian transport is not uniquely related to the occurrence of a glass transition (e.g., Long and Kokes, 1953; Long and Richman, 1960), it seems well established that the location of the concentration shock marks the interface between glassy and rubbery or swollen polymer regions (e.g., Hopfenberg, 1970; Berens, 1985). Swelling and deformation data provide additional valuable information on the transport process. Typical swelling and weight uptake kinetics follow a similar time behavior (Nicolais et al., 1977; Thomas and Windle, 1981) during sorption. Large penetrant contents usually imply large degrees of swelling. The large deformations that follow usually are not isotropic, insofar as the inner unswollen (or less swollen) regions are more rigid than the outer swollen part of the sample and offer a high resistance against stretching perpendicular to the diffusion direction. The polymer matrix then swells primarily along the direction of diffusion, giving rise to orientation effects in the polymer chains (Hartley, 1946; Thomas and Windle, 1981). A stress profile is built up since the unswollen region exerts a compressive force on the outer swollen layers and the swollen polymer exerts a tensile stress on the inner core. The swelling tends to increase the cross-sectional area of the sample (Crank, 1953; Fujita, 1961; Alfrey et al., 1966). The stress level reached can even cause some failure in the polymer, such as crazing (Baird et al., 1971; Sarti et al., 1984). In the presence of steep concentration and swelling profiles, the existence of diffusion-induced stresses can be very visible. After the concentration shocks have met at the sample midplane and the internal glassy core has disappeared, a rearrangement in the sample shape takes place. There is an increase in the lateral cross-sectional area and a simultaneous decrease of the sample thickness in the direction of diffusion. These changes in dimension are initially rather fast and subsequently show a slow relaxation over a broad period of time (Thomas and Windle, 1981; Sarti et al., 1983). Remarkably, the time scale associated with the advancing front has been found to be the same as the time scale associated with the above-mentioned relaxation in the cross sectional area (Sarti et al., 1983). The presence of differential swelling stresses has also been associated (Crank, 1953; Ware and Cohen 1980) with some interesting features observed upon changes in the sample thickness alone; indeed, during initial sorption times, thinner samples contain more penetrant than do thicker samples (Mandelkern and Long, 1951; Park, 1953; Ware and Cohen, 1980). We must remember here that a decrease in the sample thickness can lead to an increase in the penetrant flux and to a Super Case I1 behavior (Jacques et al., 1976). Various mathematical theories for mass transport through amorphous glassy polymers have been proposed, aiming at a satisfactory description of the diffusion, relaxation, and mechanical processes which simultaneously take place. Indeed, some of them have proved to be rather
1196 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
successful in accommodating several non-Fickian features (Petropoulos and Roussis, 1978; Sarti, 1979; Gostoli and Sarti, 1982, 1983; Thomas and Windle, 1982; Petropoulos, 1984a,b; Sarti et al., 1986). However, they still appear to suffer from the presence of some severe simplifying assumptions. Without aiming at a complete review of all the proposed mathematical models, we point out that the theories accounting for the coupling between mechanical and transport processes have followed several different approaches. Crank (1953) introduced a stress-dependent diffusion coefficient, and Newns (1956), who first introduced the osmotic swelling stress into the analysis, was able to calculate isotropic deformations during swelling through an elastic-like constitutive equation. Alfrey et al. (1966) accounted for the differential swelling stress, perpendicular to the diffusion direction, for the case of a linearly elastic solid in which the deformation field is directly expressed in terms of penetrant concentration. Later, Petropoulos and Roussis (1978) were able to describe several nonFickian features by considering a stress- and concentration-dependent diffusion coefficient. The differential swelling stress perpendicular to the diffusion direction was calculated through a linearly viscoelastic constitutive equation, while the deformation was directly expressed in terms of penetrant concentration. The model was also highly improved by accounting for stress-dependent (Petropoulos, 1984a)and time-dependent (Petropoulos, 1984b) solubility coefficients. Various non-Fickian behaviors were also accommodated in the moving boundary problem analyzed by Gostoli and Sarti (1982, 1983) and by Sarti et al. (1986) in which the presence of a concentration shock is assumed as a starting hypothesis. Its velocity was given through the kinetics of mechanical failure, after the solvent osmotic stress (Sarti, 1979) and the differential swelling stress (Gostoli and Sarti, 1982,1983) had been computed. Recently the model was improved to calculate also the actual swelling of the sample in the direction of penetration (Sarti et al., 1986). In all cases, however, the solubility was kept constant with time. Rather successfully, Thomas and Windle (1982) followed Newns’ procedure by using the solvent osmotic pressure in a viscous constitutive equation in which the deformation rate is given directly by the time derivative of the penetrant volume fraction. In the above models, the relaxation properties, if any, are associated with the relaxation of the swelling stress and/or with the relaxation in the penetrant solubility. Other different formulations have been recently considered, in which no mechanical stresses are explicity calculated, while the relaxation behavior is lumped into a viscoelastic type constitutive equation for the diffusive flux (Neogi, 1983; Camera-Roda and Sarti, 1986). In this way, the hyperbolic features of many non-Fickian effects are directly accounted for by the species balance equation. One of the more complete models proposed so far is due to Durning and Tabor (1986). By using the transient network model and the reptation model, expressions for the chemical potential of the solute in the polymer were obtained. The entropy inequality was used to show that the diffusive flux is proportional to the gradient of the chemical potential for small driving forces and small strain histories. In all the cases in which the mechanical problem has been addressed, the solution has been obtained by using constitutive equations together with simplifying assumptions that relate the deformation field to the existing penetrant concentration. Indeed those assumptions are not needed when the typical procedure used in solid mechanics is followed (Malvern, 1969; Larch6 and Cahn,
1982). On the other hand, the existence of a stress and deformation field affects not only the diffusivities but all the physicochemical properties of the polymer as well as the penetrant chemical potential. This should result in both a non-Fickian constitutive equation for the diffusive flux and in a stress- and time-dependent solubility. Our aim here is to present a consistent formulation of the coupled mechanical and transport problem in which the concentration as the well as deformation and stress are unknown fields. A general linear viscoelastic solid will be considered. For the sake of simplicity, only the one-dimensional transport problem will be treated in detail. In all cases, the assumption of small deformations will be invoked. However, an important feature of this formulation is that the expressions used for the chemical potential and the stress constitutive equation are thermodynamically consistent, since they arise from a common formula for the Helmholtz free energy. This is a departure from previous approaches in which the chemical potential and the stress formulations are treated separately. The final form of the flux expressions resulting from this analysis have features that are quite different from those proposed in previous studies. Problem Formulation Mass Continuity. A polymeric material is placed in a fluid containing a component that can dissolve into the polymer and ultimately diffuse through a sample. The external fluid is at a constant pressure, P, which is equal to the ambient pressure of the solid prior to immersion in the solution. As a result, we can consider the reference configuration of the polymer as being the original sample at ambient pressure, with no solute present. This is the zero stress condition. As the solute dissolves into the polymer, stresses will be generated that can cause polymer deformation, as well as influence subsequent diffusion into the polymer. A coupling between the deformation and the diffusive flux can occur if there is an effect of stress on the chemical potential of the solute. Furthermore, the material can be subjected to external forces and torques while in the presence of the solute, and these can again influence the rate of solute transport. In this section we provide a general formulation of problems of this type for the case of small deformations of the solid. This will help us to illustrate the nature of the coupling between diffusion and solid deformations. A particular example in which the sample is assumed to be very thin is worked out in detail in the next section for the case of linear viscoelastic materials. We begin by writing the continuity equations for the solute and the polymer ap,/at + V . ~ , V , = o (1) app/at
+ V.~,V,
=
o
(2)
Here ps and pp are the mass concentrations of solute and polymer and v, and vp are the corresponding species velocities. Since the motion of the solute is primarily by diffusion, it is convenient to write eq 1in terms of the mass average velocity, v , and the diffusive flux relative to the mass average velocity (Bird et al., 1960)
ap,/at
+ V.P,V
= -v.j,
(3)
The motion of the polymer is primarily induced by internally generated stresses. As a result, the polymer velocity will be governed by a force balance on the polymer, and no advantage is gained by rewriting eq 2 in terms of a polymer diffusive flux. Using the definition of the mass average velocity, it is easy to express it in terms of the
Ind. Eng. Chem. Res., Vol. 29, NO. 7, 1990 1197 polymer velocity and the solute diffusive flux (Bird et al., 1960) v = vp
+ p;'jS
(4)
The overall continuity equation for the mixture can be used to write the final form of eq 3 in terms of the weight fraction of solute p [ d w / d t + v V w ] = - 0-j, (5) where P = Ps
+ Pp
N
De = -0-q - V.Chiji
PDt
i=l
+TD
(12)
Here e is the internal energy per unit mass of the system, q is the conductive heat flux vector, hi is the enthalpy per
= Pp(l -
(6)
A t first sight, it might seem that for a problem in solid diffusion the convective term in eq 5 should be negligible. However, since the velocity is primarily caused by diffusive fluxes, there are cases where the convective term can be of the same order of magnitude as the diffusive flux term (Sarti et al., 1986). It should be noted that eq 2 will serve to solve for the polymer concentration, eq 6 is useful to compute the penetrant weight fraction, and eq 4 is an equation for the mass average velocity. The polymer velocity is obtained from the force balance on the solid and kinematic considerations, as shown below. Force Balance on the Polymer. We can write a species momentum balance for the polymer that takes a very simple form that is consistent with the usual formulation of problems in solid mechanics (Malvern, 1969) 0.T = 0 (7) Here T is the stress tensor for the polymer. We have neglected all body forces, inertial terms (because of slow deformations), and frictional forces due to inter-species diffusion. The stress in the polymer will be dominated by the polymer deformation and concentration histories. The constitutive equations for the stress must satisfy all thermodynamic constraints, and it should be consistent with expressions used to describe the chemical potential of the solute in the polymer. These issues are discussed in the next section. Since the constitutive equation for the stress will depend on the strain history of the material, this strain field can be used to calculate the polymer displacement vector, u, through the relations (Malvern, 1969) u=r-R (8) and 1 E = -(VU + VuT) (9) 2 Equation 8 defines u as being the difference between the position of a polymer particle at time t and the position it had in the reference configuration (t = 0). The relationship between the strain tensor and the deformation given in eq 9 is valid only for small deformations. There are six equations implied by eq 9 for only three unknown components in u. In order for these six equations to give rise to a unique deformation vector field, the strain field must satisfy the St. Venant compatibility conditions, which can be written as (Malvern, 1969) v A (v A E ) = O (10) Again, this is an expression valid for small deformation. Equation 10 will play a crucial role in the calculation of the deformation field. Once the deformation of the material a t a given instant of time has been evaluated, the velocity of the polymer can be obtained from its kinematic definition vp =
which is again consistent with the small-deformation approximation. Thermodynamic Constraints. The internal energy equation for a multicomponent system neglecting radiation effects can be written in the form (Slattery, 1972)
Dpu z -au Dt dt
unit mass of a given component, and ji is its mass diffusive flux. The product T : Dis the rate of conversion of mechanical to internal energy through the deformation power, where D is the rate of strain tensor. For the small strain approximation used in eqs 9 and 11, it is adequate to express D in terms of the strain as D=D - E z-dE Dt dt Since most of the mechanical effects are due to polymer deformation, we treat T and D in eq 12 as being the stress and rate of strain of the polymer. The internal energy can be related to the Helmholtz free energy and the entropy of the system (per unity mass) by
e=a+Ts (14) The enthalpy per unit mass of component i can be related to its chemical potential and entropy per unit mass by (15) pi = hi - T s ~ Here the enthalpy and entropy are partial mass quantities. There are only two components in our system, solute and polymer. The diffusive mass fluxes of the components sum up to zero and the sum of the weight fractions of the two components is one. Substitution of eqs 13-15 into eq 12 results in an expression of the form
Here j, is the solute diffusion flux as in eq 2 and p is defined as the difference between the solute and polymer chemical potentials per unit mass The right-hand side of eq 16 represents the rate of production of entropy per unit volume of continuum, s,,and we adopt as a postulate of the Second Law of Thermodynamics that this term be positive
s, I O
(18)
What remains now is to postulate a dependence of the Helmholtz free energy on the deformation and concentration histories. It is then possible to choose constitutive equations for the stress, chemical potential, and diffusive flux that do not violate the inequality of eq 18. In this way, one is assured that the expressions for solute fluxes, chemical potential, and stresses are thermodynamically consistent. Of course, in this analysis we will restrict ourselves to isothermal deformations so that neither the temperature nor its history will appear explicitly in the expression for the Helmholtz free energy. For viscoelastic materials, the Helmholtz free energy can be considered to be a function of the past histories and instantaneous values of the strain and concentration of solute solute (Coleman and Noll, 1960)
1198 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 a =
A(Et,E(t);wt,w(t)) + A,(w)
(19)
where A,(w) is the free energy contribution due to mixing of solute and polymer. Taking the material derivative of a, one can use the chain rule for functionals (Coleman, 1964) and write
3
u = Dt =
(&):E+
(E)&+
sa[bt,Et] (20)
Here the overhead dots represent material derivatives and 6a is the Frechet differential of the functional with respect to both the strain and concentration past histories. The partial derivatives in eq 20 are understood to mean derivatives of the Helmholtz free energy holding all other variables constant, including the past histories of both strain and concentration. Substitution of eq 20 into the right-hand side of eq 16 results in an expression for the entropy inequality 1
S , = --j8.Vp T
Linear Viscoelasticity On the basis of a truncated functional expassion, Christensen and Naghdi (1967) derived an expression of the Helmholtz free energy of a linear viscoelastic solid subject to thermal swelling. Following their procedure for the case of isothermal swelling due to concentration changes, it is possible to write the analogous expansion of eq 19, which will represent the expression for the Helmholtz free energy of a linear viscoelastic solid in terms of the strain and concentration history a = A,(w)
+
t dEij t 1-DJt-7) aT d7 - S__y(t - T )
aw d7 +
+ ($)&[T-p(%)]-$3a20
(21)
The equation above can be simplified by making use of the definition of the chemical potential as the derivative of the Helmholtz free energy per unit mass with respect to the instantaneous value of the weight fraction of solute = aa/au
(22)
By following the same procedure used by Coleman (1964), one finds the stress tensor as the change in Helmholtz free energy with respect to the instantaneous deformation
T = p(aa/aE)
(23)
After substitution of eqs 22 and 23 into inequality 21, what remains is 1
P
S, = --j,.Vp - -6a L o (24) T T This equation embodies the dissipation principle for the polymer mixture being considered. It is worth noting that, when the penetrant mass fraction history tends to the zero value history, the right-hand side of eq 24 reduces to its second term only, which represents the well-known result holding for pure polymers (Coleman, 1964). Since it is possible to conceive situations where the term 6a can be arbitrarily small without altering the first term on the right-hand side of eq 25, one is led to the classical conclusion that j,-Vp I0 (25) This constraint confirms that even for viscoelastic materials, the usual form of the constitutive equation for the diffusive flux, j, = -aVp (26) is consistent with the Second Law of Thermodynamics so long as the proportionality constant, a,is nonnegative. Of course, since the free energy depends on the history of the deformation and concentration, then T,1,and j, will also depend on the same histories. The exact form taken will be determined by the specific constitutive relation chosen between the free energy and the strain and concentration histories. As shown in the Appendix, the results of eqs 24-26 apply even to the case of finite deformations. In the next section, we consider the case of linear viscoelastic solids.
The memory functions Dij, y, Gijkl, &,, and m are mechanical properties of the material that are assumed to take on a value of zero for negative values of their arguments and to be properly monotonically decreasing functions of the arguments t = 7, t - q. They are also assumed to be differentiable. The notation adopted here is the same as Christensen and Naghdi's, apart from the sign of the memory function, m(t-7). To apply eq 27 to the derivation of constitutive equations for p and T according to eqs 22 and 23, we need to express the free energy in terms of the instantaneous values of E ( t ) and o(t). This can be done by integrating each term in eq 27 by parts. For the double integrals, this must be done twice. One can then differentiate this expression for the free energy with respect to the instantaneous strain components according to eq 23. The result can be written as
5P =
(z)
= Dij(0)
+
The constant Dij(0) can be taken to be zero since this is the stress associated with the reference configuration. For an isotropic solid, the fourth-order tensor relaxation function can be written as ~Gijkl(t-s,O)= 1 'Ci(t-T)[6ik6jl + 6$jk] + - [ G ~ ( t - 7 )- Gl(t-7)]6ij6kl (29) 2 3 while the second-order tensor takes the form ~ 4 i j ( O , t - ~=) #~(t-7)6ij (30) Substitution into the expression for the stress yields the final form
Taking the trace of this expression, one finds
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1199 If the deformation is caused only by swelling, i.e., if the stress are always maintained at zero values, eq 32 indicates that
difference between the solute and the polymer may be written as
(33)
where E" is the strain due to swelling only, in the absence of mechanical stress. Rearranging eq 33, it is possible to write 3M-7) -atrE" =(34) aw G2(t-7) Since the left-hand side of eq 34 is only a function of 7 and the right-hand side is a function only of t - 7 , then 34(t-7) = PGz(t-7) (35) where @ is a proportionality constant. This coincides with the equation used by Larch6 and Cahn (1982) in their analysis of elastic solid deformation. Equation 35 indicates that the relaxation function for isotropic mechanical strains is proportional to the relaxation function for isotropic swelling strains. This is consistent with a swelling strain that is isotropic and directly proportional to the instantaneous value of the solute concentration
E" = P w I
(36)
By using eq 33 and eq 35, then eq 31 may be written in terms of the difference between the actual strain and the strain due to stress free swelling only
where
&=E-&'"
(38)
It is interesting to note that if one assumes simple forms for the relaxation functions GI and G2in terms of a single relaxation time, X Gl(t-7) = G 7 [Gf - G T ] C ( t - r ) / A Gz(t-7)
+ = G i + [Gj - G~]C('-')/'
(39)
then eq 37 can be recognized as the solution for the stress in a standard linear elastic solid model for a compressible viscoelastic material
G - -1( G $ - GY)tr($)I+ ?'+ x1 T = CY(&) + --E+ x 3 1 1 -(G i - G ; ) i t r ( h ) I
-
3
(40)
The case of a compressible linear elastic solid is recovered ~0 in eq 40 by simply letting X T = GYE+ s1( G q - G f ) t r ( & ) I
(41)
The expression for the free energy can be differentiated with respect to w according to eq 22 to obtain an expression for the chemical potential
In this expression, p,(w) is the mixing contribution to the chemical potential. By using eq 30, which is valid for an isotropic solid, and eqs 36 and 38, the chemical potential
Integrating the last term by parts, and taking the trace of eq 37, this result can be written as
where
and
The quantity po(w) gives the explicit dependency of p on the instantaneous value of the solute concentration. For the case of a linear elastic solid, both memory functions, 4 - 7 ) and G2(t-7), are constants in time so that eq 44 reduces to
This is the result obtained for linear elastic solids by Larch6 and Cahn (1982). Equation 44 is the extension of this expression for the case of linear viscoelastic materials. It is interesting to note that the chemical potential depends on the instantaneous value of the stress. However, since the stress depends on the histories of the deformation and concentration, p can be considered to be an implicit functional of these histories. Given the constitutive equations for p and T i n eqs 44 and 37, it is now possible to focus attention on the calculation of the stress field in a deforming linear viscoelastic solid and the subsequent effects on the solute flux. For this purpose, the case of solute diffusion into a thin rectangular solid is considered.
Diffusion into a Thin Rectangular Solid Consider the solid shown in Figure 1, which is very thin along the major diffusion direction ( z direction). At the edges, both a net force and a torque per width of sample are applied externally. The solute concentration will be considered to be a function of z and t only. The local swelling is related to the local value of the solute concentration. Since the local volume change in the polymer is equal to the trace of the strain tensor, we can consider all the on-diagonal elements of the strain to be functions of z and t only, just like the concentration field Ex, Erx(z,t) E y y = Eyy(z,t) E,, = Ez,(z,t) (47) These components of the strain tensor can be related to the components of the displacement vector (see Figure 2) by the equations E,, = aux/ax Eyy
=
1200 Ind. Eng. Chem Res., Vol. 29, No. 7, 1990
TC
in the x and y directions be linear in z, with coefficients that are time dependent. This result is subject only to the constraint that the strains are small, but the conclusion is independent of the type of material composing the solid. The off-diagonal strain tensor components are given in eq 50, and with the help of eqs 52, 36, 38, and 37, it is possible to find the corresponding off-diagonal stress tensor components T I , = Tyx= 0
I I I
I
I
I
I
I I I 1
y I
il. I
l 1
w
These can now be substituted into the stress equations, eq 7, along the diffusion direction
a = w(z,t)
I
!
I
Integrating and using the boundary condition T,, = 0 at
I
ff
z = L , one finds
I
LZ=-L
z=o z=L1
T,,= G(t)(L- Z )
1
A
F
Figure 1. Geometry for the study of solute diffusion in thin rectangular solids.
Integrating the first two equations above while making use of the condition that a t the center of the sample the displacement vector components are all zero results in u, = E,,(z,t)x u, = E,,(z,t)y u, = u,(z,t)
(55)
To get the strain along the diffusion direction, one can apply eq 37, using the result of eq 55. In doing this, it must be realized that = E,, - ow
e,,
and t r E = E,,
+ 2[A(t) + B(t)z]
(56)
according to eq 52. The governing equation for E,, takes the form
(49)
Ex, = Eyx = 0
3 (50)
With the strain tensor components in eqs 47 and 50, the St. Venant compatibility conditions (Malvern,1969) reduce to a2EX,/az2 = o and
a2EY,/az2 = o
(51)
Integrating eq 51 and considering the sample to be symmetric in the x and y directions, we find that the x and y elements of the strain tensor take the form E,, = E,, = A(t) + B(t)z (52) Strictly speaking, since the strain field should be symmetric about z = 0, eq 52 should be written with the absolute value of the coordinate z, i.e., 121. This would eventually result in the proper signs of displacements, stresses, etc., in both the positive and negative z directions. Unfortunately, this gets quite cumbersome, so that from now on we obtain expressions valid only in the positive z axis and take the results for the negative z axis from the symmetry of the problem. It is clear from eq 52 that the St. Venant compatibility conditions require that the strains
St-- [GZ(t-T) - Gl(t-7)][A(7)+
d7 (57)
&T)z]
This equation can be solved by the use of Laplace transforms. The solution is of the form
t
JtrAt-7) [A(7) + B ( ~ ) zd7 ] + ( L - z ) l m I ' 3 ( 7 ) dT (58) where rl, r2, and r3 are the inverses of the Laplace transforms of ratios of G I , G2, and G
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1201 By using eq 58, it is possible to deduce that the strain along the z direction is not only a function of the concentration field o(z,t) but its magnitude also depends on the deformations along x and y (through the A and B terms) and the stress along the z direction. The functions rl,r2,and r3are the relaxation functions for the strain field. The terms A(t) and B(t) may be evaluated by applying boundary conditions on the net stress and torque per unit width at the edges of the sample, namely
where F and T are given, depending on the constraints imposed at the edges of the material. The trace of the stress tensor is needed to calculate Taking the trace of eq 37, one finds
Sample at time t=O
Edge of sample at time t
)
Figure 2. Deformations caused in the sample due to swelling.
Substituting eq 63 and using eq 46, it is possible to show that the diffusive flux can be written as
This can be integrated by parts to yield
Using eq 58, it is possible to obtain an expression for the trace of the strain tensor t r 8 = -3@~(z,+ t ) PJtrl(t-7)W ( Z , T ) dT -m
~:I'&-T)[A(T) + B ( T ) z ]dT + 2[A(t) + B ( t ) z ]+
where
For the case of a linear elastic solid, all the memory functions can be treated as constants, for example
m(t-7) = mo
Substituting into eq 59, one can shown that the time-dependent functions rl and rzare in fact delta functions in this case
rl(t+
=
G! 2 -GP 3
+ j1G 4 2 -G
r2(t+ =
2 -GP
3
The solute diffusive flux along the z direction is given by = -a a~cc/dz
B(t-7)
8
1 + -G'$ 3
b(t-7)
(68)
From eq 54, we also see that G ( t ) = GPB(t) for a linear elastic solid so that from eq 59
which along with the aid of eq 44 results in
and from eq 66
1202 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 0
coelastic solid with a single relaxation time
PO aw
(jJ2= -a - - + I d w az
Equations 67-70 can be substituted into eq 65, making use of the relationship between the shear and bulk moduli (C and G ",, Young's modulus, and the Poisson ration for an elastic solid GP =
E l + U
This results in an expression for the diffusive flux that matches exactly the equation derived by Larch6 and Cahn (1982) b(t P
As a result, eq 65 can be viewed as a true extension of the Larch6 and Cahn results to the case of a linear viscoelastic solid. If the linear viscoelastic solid exhibits relaxation behavior that can be described by using a single relaxation time, then the memory functions appearing in eq 65 can be written as
m(t) = m"
+ (mo- m")pfiA
G,(t) = G
+ (G p - G :)a'/*
G,(t) = G,"
+ (C$ - G,")a'/'
(73)
If these functions are substituted into the expression for rl, rz,and r3in eq 59, one can show that
where
and xG-l
= ax-'
(76)
Substitution of these results into eq 65 for the diffusive flux provides an expression for the case of a linear vis-
This equation should be contrasted with similar expressions proposed recently for diffusive fluxes in viscoelastic polymers (Neogi, 1983; Neogi et al., 1986; Durning and Tabor, 1986). These can normally be written in the form
-
where Do and D" are the diffusion coefficients before and a, respectively). The after relaxation ( t = 0 and t method of arriving at this type of expression varies, but in all cases the details of the stress field variation with position has been neglected in the estimation, and the chemical potential expression has not included an explicit history dependence. The derivation presented in this paper is based on a more detailed analysis of the deformation and stress in the sample, and on eq 44 for w, which takes into account the explicit dependence of the chemical potential on the history of the concentration field. A comparison of eqs 77 and 78 reveals several major differences. The first is that there are in reality two time constants for the relaxation of the flux. The first time constant is A, which was introduced in eq 73 as the time constant for the relaxation of the polymer and which describes the relaxation of the mechanical moduli GI and G,, as well as m, the memory function for the chemical potential. The second time constant is hG,which is equal to XI4 according to eq 76. This time constant reflects the modification of the time constant by X by the changes in the mechanical properties of the material between their initial and fiial values. The second difference is that even though portions of eq 77 can be recast in the form of eq 78, for example, the second and third terms, other parts of the expression cannot be written as prescribed by eq 78. The very last contribution is a double integral of the concentration gradient weighted by both the memory function with time constant X and the second relaxation function with time constant XG. Even though calculations based on eq 77 have not yet been made, there are no a priori reasons for discounting the effects of this term on the flux. The term involving r in eq 77 is also not present in eq 78. This quantity depends solely on B(t),a function
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1203 whose value depends on the total amount of solute adsorbed, the sample width, and any applied forces and torques. Larch6 and Cahn have evaluated B(t)for the case of linear elastic solids. Of course, eq 65 can be used for any type of relaxation function desired. It is restricted only to linear viscoelastic materials, but not to systems with a single relaxation time. In general the quantities m, GI, and Gz may be expressed by using a whole spectrum of relaxation times. Substitution of these functions into eq 65 would result in different forms for the dependence of the flux on the history of the concentration gradient. Acknowledgment We are grateful for the financial support provided by NATO Research Grant 8410023. The work has also been partially supported by the Italian Ministry of Education. Nomenclature a = Helmholtz free energy per unit mass of mixture A = free energy functionals of mixture A ( t ) = function used to calculate the strain B ( t ) = function used to calculate the strain D = diffusion coefficient in Fick's law e = internal energy per unit mass of mixture E = Young's modulus E,, E, Et = strain tensor, strain history Ejiw,E" = swelling strain due to solute penetration F = force per unit width acting at the membrane edges GI,G1, G 3 = memory functions j, = mass diffusive flux of solute relative to the mass average velocity L = sample thickness at time t R = position of a particle of polymer at t = 0 r = position of a particle of polymer at time t s = entropy per unit mass of mixture t = time TijLT,T t = stress tensor, stress history T - temperature T = applied torque u = displacement vector v = mass average velocity v p = species velocity for the polymer v, = species velocity for the solute x , y, z = spatial rectangular coordinates Greek Letters a = phenomenological coefficient in diffusive flux /3 = linear expansion coefficient for the swelling process p = density of the mixture pp = mass concentration of polymer p s = mass concentration of solute w = weight fraction of solute p = chemical potential difference between solute and polymer v = Poisson's ratio X = relaxation time for stress
Appendix The thermodynamic result of major concern in this work is embodied by eq 25. It is worth noting that it was derived under the small-deformation hypothesis, which is used in the mechanical analysis of the present work. However the same result holds true under the more general conditions of finite deformations as well. Indeed, in that case one can rewrite eq 16 as
PP Do -+ -1E D -
T Dt
T
P Da - - 20 ( A l )
T dt
Under the isothermal conditions used thus far, eqs 19 and 20 now take the form a = Ai=-, [F(t),Ft(7),w(t),wt(r)l u
Da aa - = -:&+ Dt
da
+ 6a[dt,Pt]
-&
aFT
dw
(A2) (A3)
where F(T)is the deformation gradient at time t and FYT) is its past history for all 7 between --a3 and t. The use of this expression for the Helmholtz free energy results in a more general statement of the entropy inequality
S,
1 --j8.vp
T
P +-
'[
T -[
DU
aa
p
--
a],
~
Dt+
T - pF * ] : V v -
T
aFT
-6a P I O (A4)
T
By using the deformation of g and the same procedure used by Coleman (1964), one obtains ~1 =
aa/aw
(A5)
aa
T=pF-
dFT
while the entropy inequality becomes again of the same form as eq 24: 1
P
S, = --js.Vp - -6a LO T T
(A7)
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Receiued f o r review November 27, 1989 Revised manuscript receiued February 22, 1990 Accepted March 4, 1990
