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Mass transport through swelling membranes Quan Liu a, Xuefeng Wang b, Daniel De Kee a

a,*

Department of Chemical and Biomolecular, Engineering, Tulane Institute for Macromolecular Engineering and Science (TIMES), Tulane University, New Orleans, LA 70118, United States b Mathematics Department, Tulane University, New Orleans, LA 70118, United States Received 20 January 2005; received in revised form 9 April 2005; accepted 25 May 2005

(Communicated by K.R. RAJAGOPAL)

Abstract This paper deals with non-Fickian mass transport through polymeric membranes. The process is described via continuum mechanics. We introduced an appropriate function relating the stress to concentration and time, such that the model predicts a realistic stress distribution at equilibrium also. The eﬀect of diﬀerent dimensionless groups is illustrated and quantitative agreement with experimental data is shown for transport of organic solvents through PVC. Ó 2005 Published by Elsevier Ltd.

1. Introduction Polymeric materials, due to their inherent viscoelastic properties, are used in application such as gas separations, protective clothing, drug delivery and in a variety of other mass transport processes. FickÕs laws are usually used to describe diﬀusion processes [1,2]. However, many experiments and applications are associated with ‘‘non-Fickian behavior’’, requiring modiﬁcations to FickÕs laws [3–5]. The principles leading to non-Fickian diﬀusion are still being discussed, but it is widely accepted that polymer swelling plays an important role. Non-Fickian diﬀusion processes have been studied by many groups [6–16]. Because most polymers swell when in contact with certain solvents, some authors used FickÕs laws with modiﬁed boundary conditions and/or a generalized diﬀusion coeﬃcient to address the non-Fickian behavior [17–19]. The viscoelastic properties of the polymeric membrane are believed to be important in analyzing non-Fickian behavior. A few researchers have considered the permeation process through polymeric materials driven by both molecular diffusion and by a stress associated with swelling [6,10,11,20]. Rajagopal and Tao [21] used a mixture theory to study diﬀerent diﬀusion problems. Via continuum mechanics, they provided detailed information about the swelling and the stress within the polymer [13–15]. During the swelling process, the polymer is deformed *

Corresponding author. E-mail address: [email protected] (D. De Kee).

0020-7225/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.ijengsci.2005.05.010

Q. Liu et al. / International Journal of Engineering Science 43 (2005) 1464–1470

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because of an osmotic contribution to the stress. An elastic force balances the osmotic stress, to ﬁnally reach an equilibrium state. For non-Fickian diﬀusion, a sharp front, which separates the dry polymer from the swollen polymer, is assumed to move linearly with time [3]. Here, we start with the Cohen model [22–24] to describe the mass transport through swelling polymers via a continuum mechanics. That is to say: compared to the mixture theory [21], assumptions are introduced to simplify the problem while still succeeding to address the occurrence of swelling. 2. Mathematical model In the classical FickÕs laws, the chemical potential l is assumed to be a function of the concentration only. In swollen polymers, the solvent in the polymeric materials cause an increase in the systemÕs internal energy and in its chemical potential. The chemical potential is assumed to be a function of concentration and stress also. The expression for the ﬂux then becomes: oc or F ¼ D þE ð1Þ ox ox and the conservation of mass leads to: oc ¼ Dr2 c þ Er2 r ot

ð2Þ

Diﬀerent assumptions and solution methods have been made to relate the stress r to concentration and time [11,25–27]. A common assumption is: or oc o2 c þ bðcÞr ¼ f c; ; 2 ð3Þ ot ot ot , and Chan Man Fong et al. [11] and Hinestroza Durning [25] assumed that the function f is just a function of oc ot 2 o c [27] assumed that the function f is a linear function of oc and . However, those assumptions lead to a zero ot ot2 stress at equilibrium (which means that the osmotic pressure is zero), which is not realistic. The equilibrium stress must be related to the concentration. Here, we will apply the assumptions used in CohenÕs model [22,28]. We assume that the viscoelastic relaxation equation for r follows a Jeﬀereys’ type expression. or oc b 1 þ r ¼ m c þ b2 ð4Þ ot ot where D is the molecular diﬀusion coeﬃcient and E is a stress-driven diﬀusion coeﬃcient. In JeﬀereysÕ model, b1 is a relaxation time and b2 is a retardation time. Here, we assume that the equilibrium stress is a linear function of the concentration of the solvent. Eq. (4) can also be used to express the relaxation of r when the polymeric membrane is under external deformation by deﬁning r 0 = r r0, where r0 is the initial stress applied to the membrane. Next we apply a continuum mechanics approach that has been used in Chan Man Fong et al. [11] and Hinestroza [27] to solve Eqs. (2) and (4). We introduce the following dimensionless quantities: c ¼ c=cs ; c1 ¼ Em=D;

x ¼ x=L;

r ¼ r=ðcs mÞ;

b c2 ¼ 2 1 ; ðL =DÞ

t ¼

t ðL =DÞ 2

b c3 ¼ 2 2 ðL =DÞ

ð5Þ

Using the previous dimensionless parameters, Eqs. (2) and (4) in dimensionless form become: oc o2 c o2 r ¼ 2 þ c1 2 ot ox ox or oc c 2 þ r ¼ c þ c 3 ot ot

ð6Þ ð7Þ

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The initial and boundary conditions for Eqs. (6) and (7) are also given by c ðx ; 0Þ ¼ 0;

c ð0; t Þ ¼ 1;

c ð1; t Þ ¼ 0;

r ðx ; 0Þ ¼ 0

ð8Þ

In the subsequent text, we will omit the superscript *. That is it is understood that we work with nondimensionless quantities. Diﬀerentiating Eq. (6) with respect to time and diﬀerentiating Eq. (7) twice with respect to position yields, o2 c 1 oc c2 þ c1 c3 o3 c c 1 þ c 2 o2 c ¼ þ þ ot2 c2 ot otox2 c2 c2 ox2

ð9Þ

The solution to Eq. (9) is assumed to be given by the following form: cðx; tÞ ¼ uðx; tÞ þ f ðxÞ

ð10Þ

Substituting Eq. (10) into Eq. (9) yields, o2 u 1 ou c2 þ c1 c3 o3 u c þ c 2 o2 u c 1 þ c 2 o2 f ¼ þ þ 1 þ 2 2 ot c2 ot otox c2 c2 ox2 c2 ox2

ð11Þ

Eliminating all terms that are function of time and applying the boundary conditions, we obtain f ðxÞ ¼ 1 x

ð12Þ

We assume that u(x, t) can be expressed in terms of a series solution of the form 1 X ½Aek1 t þ Bek2 t sinðspxÞ uðx; tÞ ¼

ð13Þ

s¼1

where A, B, k1 and k2 are functions of the summation index s. Thus, the expression for c(x, t) becomes 1 X cðx; tÞ ¼ ð1 xÞ þ ½Aek1 t þ Bek2 t sinðspxÞ ð14Þ s¼1

In order to derive the expression for the stress in the membrane, multiplying Eq. (7) by et=c2 and integrating yields: Z t s=c2 e oc t=c2 r¼e c þ c3 ds ð15Þ os c2 0 Substituting Eq. (14) into Eq. (15), we obtain: 1 X Að1 k1 c3 Þ k1 t Bð1 k2 c3 Þ k2 t ðe et=c2 Þ þ ðe et=c2 Þ sinðspxÞ r ¼ ð1 xÞð1 et=c2 Þ þ 1 k1 c 2 1 k2 c 2 s¼1

ð16Þ

Combining Eqs. (6), (14) and (17) and comparing similar terms yields the expressions for A, B, k1 and k2 as follows: 2 ð1 k1 c2 Þð1 k2 c3 Þ ps ðc2 c3 Þðk1 k2 Þ 2 ð1 k1 c3 Þð1 k2 c2 Þ B¼ ps ðc2 c3 Þðk1 k2 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ W þ W 2 4c2 V k1 ¼ 2c2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ W W 2 4c2 V k1 ¼ 2c2 A¼

ð17Þ ð18Þ ð19Þ ð20Þ

where W ¼ 1 þ s2 p2 ðc2 þ c1 c3 Þ 2 2

V ¼ s p ð1 þ c1 Þ

ð21Þ ð22Þ

Q. Liu et al. / International Journal of Engineering Science 43 (2005) 1464–1470

The concentration distribution c is then obtained as: 1 X 2 ð1 k1 c2 Þð1 k2 c3 Þ k1 t 2 ð1 k1 c3 Þð1 k2 c2 Þ k2 t e e cðx; tÞ ¼ ð1 xÞ þ sinðspxÞ ps ðc2 c3 Þðk1 k2 Þ ps ðc2 c3 Þðk1 k2 Þ s¼1

1467

ð23Þ

and the stress distribution is: r ¼ ð1 xÞð1 e

t=c2

1 X 2 ð1 k1 c3 Þð1 k2 c3 Þ k1 t k2 t ðe Þþ e Þ sinðspxÞ ps ðc2 c3 Þðk1 k2 Þ s¼1

ð24Þ

which conﬁrms that the stress is ﬁnite at equilibrium. The dimensionless form of Eq. (1), providing the ﬂux F* at x* = 1, is given by oc or F ¼ þ c1 ox ox x ¼1

¼ 1 þ c1 ð1 et =c2 Þ 1 X ð1 k1 c2 þ c1 k1 c1 c3 Þð1 k2 c3 Þ k1 t ð1 k2 c2 þ c1 k2 c1 c3 Þð1 k1 c3 Þ k2 t e e 2ð1Þs ðc2 c3 Þðk1 k2 Þ ðc2 c3 Þðk1 k2 Þ s¼1 ð25Þ Introducing the dimensions, for the purpose of the discussion, Eq. (25) becomes: 1 X F 2 ¼ 1 þ c1 ð1 etD=L c2 Þ 2ð1Þs qDcs =L s¼1 ð1 k1 c2 þ c1 k1 c1 c3 Þð1 k2 c3 Þ k1 tD=L2 ð1 k2 c2 þ c1 k2 c1 c3 Þð1 k1 c3 Þ k2 tD=L2 e e ðc2 c3 Þðk1 k2 Þ ðc2 c3 Þðk1 k2 Þ ð26Þ 3. Discussion Fig. 1 shows the eﬀect of c1 on the dimensionless ﬂux versus dimensionless time proﬁle predicted by Eq. (25). The dimensionless time c3 is set to be zero, which corresponds to a Maxwell type model for Eq. (4) (b2 = 0). Fig. 1 illustrates that the steady state ﬂux decreases with decreasing c1. Note that for c1 to be negative, E 6 0. We proved using a mesoscopic approach that polymer swelling leads to a negative convective ﬂux [20]. Note that this negative ﬂux is responsible for the fact that the curves in Fig. 1 do not necessary level at

1.0

0.8

F

*

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

*

t

Fig. 1. Eﬀect of c1 on the diﬀusion process (c2 = 1, c3 = 0): (j): c1 = 0, (d): c1 = 0.1, (m): c1 = 0.5.

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1.0

0.8

F

*

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

*

t

Fig. 2. Eﬀect of c2 on the diﬀusion process (c1 = 0.1, c3 = 0): (d): c2 = 0.1, (j): c2 = 1, (m): c2 = 10.

F* = 1 (see Eq. (25)). The decrease of c1 (increase in the magnitude of c1) indicates that the polymerÕs relaxation process gains importance. Under speciﬁc conditions (for example, c1 = 0.5, c2 = 1), a signiﬁcant overshoot appears in the ﬂux proﬁle. The eﬀect of c2 on the predictions of Eq. (25) is shown in Fig. 2. The dimensionless time c3 is also set to be zero. The dimensionless time c2, similar to the Deborah number De in [20], is a parameter related to the rate of polymer relaxation. When c2 is very small (the polymer relaxes very fast compared to the diﬀusion time scale), the process approaches Fickian diﬀusion. However, if c2 is of order one (the polymer relaxation time is of the same order of magnitude as the diﬀusion time), polymer structural rearrangements accompany the diﬀusion process, which results in non-Fickian behavior (overshoot in Fig. 2). When c2 is very large, the relaxation time is less than the diﬀusion time. At short times, the relaxation of the polymer may not be able to aﬀect the diﬀusion process, again generating Fickian behavior. Fig. 2 shows that as c2 changes from 0.1 to 10, the ﬂux curve initially shifts to the right, reﬂecting a larger breakthrough time. In Eq. (4), we introduced a retardation time b2. In polymer viscoelasticity, a retardation time is associated with creep and a relaxation time is associated with stress relaxation. Such times usually are of the same magnitude [29,30]. In Eq. (4), b1 is the stress relaxation time and b2 is the time scale for the swelling process (similar to a creep process). The retardation time b2 is always considered to be zero since we assume that the polymer swells instantly following contact with a solvent. In reality, b2 is usually ﬁnite because of the viscoelastic

1.0

0.8

F

*

0.6

0.4

0.2

0.0 0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

*

t

Fig. 3. Eﬀect of c2 on the diﬀusion process (c1 = 0.5, c2 = 1): (j): c3 = 0, (m): c3 = 0.4, (d): c3 = 0.8.

Q. Liu et al. / International Journal of Engineering Science 43 (2005) 1464–1470

1469

3

2

flux (µg/cm s)

4

2

1

0 0

20

40

60

80

100

120

time (min) Fig. 4. Permeation of organic solvents through PVC at 303 K. (j): dichloromethane (DCM), (m): trichloroethylene (TCE), (d): benzene, (—): predictions of Eq. (26).

Table 1 Fitting parameters for the permeation of organic solvents through PVC at 303 K

c1 c2 c3

DCM

TEC

Benzene

0.766 0.228 0.1996

1.050 0.445 0.3569

0.621 1.210 1.0202

properties of the polymer. However, b2 is always smaller than b1. Fig. 3 illustrates the eﬀect of b2 on the diffusion process. As b2 increases, the ﬂux curve shifts to the right, reﬂecting that more time is required to reach steady state. This is due to the extra time required by the polymer swelling and the associated build up of the solvent concentration. Fig. 4 shows the application of Eq. (26) to model the permeation of organic solvents through a PVC membrane [31]. It clearly shows that Eq. (26) can be used to quantitatively describe the mass transport process, and successfully predict the overshoot (non-Fickian diﬀusion behavior). The curves have been shifted to the right for clarity of presentation. The ﬁtting parameters are presented in Table 1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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