Effect of volume changes due to absorption in polymer membranes

Journal of Membrane Science 169 (2000) 249–254

Effect of volume changes due to absorption in polymer membranes M.S. Vicente ∗ , J.C. y Gottifredi INIQUI (CONICET), Facultad de Ingenieria, Universidad Nacional de Salta, Buenos Aires 177–(4400) Salta, Argentina Received 21 July 1999; received in revised form 21 October 1999; accepted 25 October 1999

Abstract A mathematical model to simulate the diffusion phenomenon associated with change of volume in a rubbery polymeric membrane is developed. The model is based on a concentration-dependent diffusion coefficient. The continuity equation is integrated, by assuming that the thickness varies linearly with the mass absorbed. Solutions for small and large values of time are achieved. Then, these are matched to obtain the general solution, for the whole range of time. The resulting matching expression is valid for rigid solid and for those that suffer a great change of volume due to swelling. The results predicted by this equation are compared with experiments reported by different authors for these cases, finding very good coincidence. As will be shown the so-called sigmoidal behavior of the uptake curve can also be due to membrane shrinking. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Diffusion; Swelling; Hydrogels; Anomalous behavior

1. Introduction The mass transfer process of low molecular weight species in solid polymers shows a broad variety of behaviors, depending on polymer–penetrant interactions, operating conditions, particularly temperature and penetrant activity, and prior treatment of the polymer sample [1,2]. In many systems the diffusion cannot be described adequately by Fick’s law, especially when diffusing species cause an extensive swelling of the polymer or when relaxation phenomena are strongly associated [3]. Alfrey et al. [4] proposed a useful classification according to the relative rates of diffusion and polymer relaxation. Three classes are distinguished:

∗ Corresponding author. Tel./fax: +54-387-425-1006. E-mail address: [email protected] (M.S. Vicente).

• Case I or Fickian diffusion, in which the rate of diffusion is much smaller than that of relaxation. In this case the system is controlled by a diffusion phenomenon and the amount sorbed or desorbed is directly proportional to the square root of time during the first stages of the process. • Case II is the other extreme, in which the diffusion process is very fast compared to the relaxation process. The controlling step is the velocity of an advancing front, which forms the boundary between a swollen gel and a glassy core. The amount sorbed in this case varies linearly with time. • Case III, Non-Fickian or anomalous diffusion describes those cases where the diffusion and relaxation rates are comparable. These systems lie between Cases I and II and the amount sorbed vs. time curve can be represented by a power law relation with an exponent between 0.5 and 1. Two or more parameters are needed to describe the interacting diffusion and relaxation effects inherent to it.

0376-7388/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 9 9 ) 0 0 3 4 2 - 7

250

M.S. Vicente, J.C. y Gottifredi / Journal of Membrane Science 169 (2000) 249–254

There are several models reported in the literature trying to describe these situations [5–7] but none of these take into account changes in the size of the samples due to swelling. However, Kuipers et al. [5] reported an increase in diameter of a starch granule due to water sorption of 12%. Also, Sarti [1] found that in the absorption of alcohol in unstretched films of PMMA, the cross-sectional area relaxes towards final equilibrium value, which is larger than the initial one. On other hand, it has been stated [8] that the time dependence of fractional release of theophylline from cross-linked poly(vinyl alcohol) changes from a linear dependence with t1/2 to an anomalous diffusion case when the polymer has been dried prior to release and consequently suffers a great swelling. However, cases in which swelling causes an appreciable change in the initial dimensions of the material are of increasing interest. At the present, gels and superabsorbent gels are widely used as absorbents in the pharmaceutical industry or in the release of controlled quantities of active agents to their surroundings, such as drugs, pesticides or herbicides. In this paper a model is developed, which takes into account the variation of thickness due to swelling or shrinking and assuming a concentration-dependent diffusion coefficient.

2. Theoretical model

where the following dimensionless variables are defined: c(x 0 , t) − c0 cs − c0

D(C) =

D(c) D(cs )

tD(cs ) x02

(4)

x=

x0 x0

(5)

c being the concentration of A while the subindex s and 0 are to denote equilibrium and initial values, t the actual time, D the diffusion coefficient and x0 the half thickness of the polymer when t = 0. Since the driving force used in Eq. (1) is the concentration instead of the chemical potential gradient, D results are concentration-dependent. The boundary conditions are: C=0

τ =0

∀x

(6)

C=1

τ ≥0

x = χ(τ )

(7)

∂C =0 τ ≥0 x=0 ∂x where χ (t) is the time-dependent thickness. Now, by introducing the new variable x η= χ(τ )

(8)

(9)

the differential equation can be rewritten as follows:     ∂C 1 ∂C dχ 2 ∂ ∂C 2 − η = D(C) (10) χ (τ ) ∂τ 2 ∂η dτ ∂η ∂η and the boundary conditions are:

A regular plane sheet of polymer of small thickness is considered. The penetrant species, A, causes a change of volume in such a way that the thickness becomes a function of time. By neglecting the effect of convective terms the usual dimensionless continuity equation for species A within the polymer film can be written as:   ∂ ∂C ∂C = D(C) (1) ∂τ ∂x ∂x

C=

τ=

(2)

(3)

C=0

τ =0

0≤η≤1

(11)

C=1

τ ≥0

η=1

(12)

∂C =0 ∂η

τ ≥0

η=0

(13)

It can be seen that now boundary conditions are as usual in problems of diffusion in solid bodies but mass balance is similar to the case of an unsteady boundary layer with a convection term normal to the solid surface. The total amount of diffusing species uptaken by the sheet after time t, is, in dimensionless form, given as follows:  Z τ ∂C m(t) − m0 dτ = D(C) M(τ ) = m∞ − m0 ∂x 0 Z τ ∂C 1 dτ (14) = ∂η χ 0 η=1

M.S. Vicente, J.C. y Gottifredi / Journal of Membrane Science 169 (2000) 249–254

where m0 and m∞ are the initial and the equilibrium mass of A in the film respectively. It will be assumed that χ(τ ) will be a linear function of M. In actual fact this assumption will be valid only during the first step of the absorption process since the sorbed molecules will be distributed along the axis of diffusion. When the process approaches equilibrium this assumption may no longer be valid but the effect of volume change is no longer important as it will be shown below. Thus: χ = 1 + KM (τ )

(15)

Regarding the possible range of K, from Eq. (15), it is clear that when K > 0 membrane thickness increases with uptake and when K < 0 the opposite will be observed. However, K must always be greater than –1. Usually when an important effect is noticed the expansion or contraction is to the order of 50%. In these cases one would expect that –0.5 ≤ K ≤ 0.5 is a reasonable assumption. It should be noticed that K is wholly independent of the rate of diffusion. By defining the average concentration CAV as: Z 1 Cdη (16) CAV = 0

χ2

dCAV dτ

∂C 1 dχ 2 = (1 − CAV ) + 2 dτ ∂η η=1

(17)

(18)

or

∂C 1 dχ 2 =K 2 dτ ∂η η=1

Replacing in Eq. (17)   dCAV χ 2K 1 dχ 2 = 2 dτ 1 + K (1 − CAV ) dτ

This very simple expression clearly shows the effect of K. When K = 0 the term in brackets is equal to one. On the other hand when K 6= 0 this term plays a changing role for small and large values of τ depending upon the sign of K. To solve this differential equation first approximate solutions valid for small (τ → 0) and large values (τ  1) will be sought. Then they will be appropriately matched. For τ → 0, the mass uptake is confined to a region immediately adjacent to the external surface of the film. By assuming that the boundary layer integral method can be applied within this region of thickness δ (δ  1), the concentration profile can be given by the following expression: C = a(τ ) + b(τ )η + d(τ )η2 , (1 − δ) ≤ η ≤ 1 (23a) C=0

(19)

0 ≤ η ≤ (1 − δ)

(23b)

where a, b and d must be determined by fitting the following boundary conditions: η=1

∂C =0 ∂η C=0

and from Eq. (15)

dM 1 ∂C dχ =K =K dτ dτ χ ∂η η=1

By replacing Eqs. (19) and (21) in Eq. (17) yields: " # dCAV [1 + K (1 − CAV )]3 ∂C (22) = dτ ∂η η=1 (1 + K)2

C=1

Eq. (10) can be rewritten as:

251

η = (1 − δ) η ≤ (1 − δ)

(24a) (24b) (24c)

Thus, the following set of algebraic expressions can be deduced: a+b+d =1

(25a)

b + 2d(1 − τ ) = 0

(25b)

a + b(1 − δ) + d(1 − δ)2 = 0

(25c)

which can be solved for a, b and d to give: (20)

Now, by integrating this equation, a relationship between χ and CAV is obtained:   1+K (21) χ= 1 + K (1 − CAV )

a=

(1 − δ)2 δ2

(26a)

b=

−2(1 − δ) δ2

(26b)

d=

1 δ2

(26c)

252

M.S. Vicente, J.C. y Gottifredi / Journal of Membrane Science 169 (2000) 249–254

With these coefficients, the following relations can be obtained: Z 1 1 C dη = δ (27) CAV = 3 0 2 2 ∂C = = (28) ∂η η= 1 δ 3 CAV Using these results, Eq. (22) can now be rewritten as: (1 + K)2 CAV 3 2 [1 + K (1 − CAV )]

3

dCAV =1 dτ

(29)

Taking into account that for small times CAV  1, the term in bracket can be developed by the binomial formula, resulting: 3 1 K 3 C3 + . . . C2 + τ≈ 4 (1 + K) AV 2 (1 + K)2 AV

To determine α, the following concentration profile and boundary conditions can be assumed: C = 1 − γ ε f (η) + 0(ε2 )

(38)

η=0

∂C =0 ∂η

(39a)

η=1

C=1

The simplest proposal for f (η) is f (η) = 1 − η2

which after being introduced into Eq. (40), yields:

and

(31)

and the final result is: (32)

For large τ , (1−CAV )  1, and a new variable can be defined: (33)

With this assumption, Eq. (22) can be expressed as:   dε 1 + 3Kε + · · · ∂C dCAV =− ≈ (34) dτ dτ ∂η η=1 (1 + K)2

(35)

αε dε ≈ dτ (1 + K)2

(42a)

α=3

(42b)

Thus, the asymptotic expression for CAV when τ → ∞ is   3 τ (43) CAV = 1 − exp − (1 + K)2 With the scope to produce a unique expression valid in the whole range of τ -values, the following expression for CAV is proposed: CAV = 1 − 

1 + (θ

exp(−λτ )  − βτ )exp(−λτ )

τ 1/2

(44)

In fact, by expanding Eq. (44) when τ → ∞, it is found that: (45)

while (36)

After integration, ατ lnε = − (1 + K)2

3 2

CAV = 1 − exp(−λτ )

Equating terms of equal order in ε: −

(41)

(30)

CAV ≈ Aτ 1/2 + Bτ + · · ·

By assuming: ∂C ≈ αε + βε2 + . . . ∂η η=1

(39b)

0

γ =

ε (τ ) = (1 − CAV )

f (η) = 0

Since CAV = 1−ε (τ ), γ f (η) must also fit the following condition: Z 1 γ f (η)dη = 1 (40)

which suggests that CAV must fit the following series for small τ :

2 4 CAV ≈ √ (1 + K)1/2 τ 1/2 − Kτ 3 3

f 0 (η) = 0

(37)

CAV = θ τ 1/2 − (β + θ 2 − λ)τ + 0(τ 3/2 )

(46)

when τ → 0. By comparing Eqs. (45) and (46) with Eqs. (43) and (32) respectively, the coefficients λ, β and θ are deduced by the following relationships:

M.S. Vicente, J.C. y Gottifredi / Journal of Membrane Science 169 (2000) 249–254

253

Fig. 1. •••• Crank expression; — Eq. (48).

λ=

3 (1 + K)2

         

2 θ = √ (1 + K)1/2  3     2  9 − 4(1 + K)    β= 3(1 + K)2

Fig. 2.

(47)

Then, from Eqs. (14), (21) and (22), the expression for the mass uptake can be written as:   CAV (τ ) (48) M(τ ) = 1 + K(1 − CAV (τ ))

3. Results 3.1. Rigid boundary (K=0) In this case, results predicted by expression (48) are compared with those obtained by Crank [3]: ∞  2  X (1 − (−1)n ) 4 2π τ) exp(−n M(τ ) = 1 − 4 n2 π2

K = −0.5; 䊉 K = 0, + K = 0.5.

noticed, for negatives K-values, the curves show sigmoidal behavior. This has been reported by Yoshida et al, [10], in experiments with temperature responsive hydrogels. With the purpose of studying situations where an apparent case of non-Fickian diffusion is taking place, an example of drug-releasing in gels is analyzed. Korsmeyer and Peppas [8], reported that when the gel is pre-dyed, drug release behavior is like an anomalous diffusion case, and they used Good’s expression [9] to fit. However, if expression (48) is used, with the diffusion values reported, an almost perfect adjustment is achieved (see Fig. 3) with K = −0.45. Thus it is shown that the anomalous behavior can be explained by taking into account simultaneous membrane-shrinking with Fickian diffusion. In this case there is no need to assume the presence of relaxation phenomena to explain observed sigmoidal uptake curve. Another interesting conclusion of this work is related with the concentration dependence of the diffu-

n=1

(49) The results are compared in Fig. 1. As can be seen, the two expressions are fairly coincident. However, the expression deduced in this work is simpler to use and the fit produced by Eq. (48) is extremely good. Maximum deviations are in all cases bellow 2.2 %. 3.2. Appreciable change of volume The numerical simulation of expression (48), for different K-values, is presented in Fig. 2. It can be

Fig. 3. •••• Korsmeyer and Peppas data, K = −0.45.

Eq. (48), with

254

M.S. Vicente, J.C. y Gottifredi / Journal of Membrane Science 169 (2000) 249–254

sion coefficient. It is clearly shown that concentration dependence does not produce any effect provided the resulting diffusion coefficient is that corresponding to equilibrium concentration. In other words, the analysis would produce the same results for the uptake considering constant diffusivity provided its value is that given by the equilibrium concentration. Thus the reported values of diffusivity are not mean values as reported repeatedly but rather values corresponding to equilibrium concentration.

4. Conclusions Our very simple analytical analysis simulating a diffusion phenomenon associated with change of volume is shown to be useful to predict mass uptake even when the diffusion coefficient is an arbitrary function of sorbed concentration. It is very interesting to note that, in the absence of volume change, a measurement of diffusion coefficient would result in a value corresponding to the surface or equilibrium concentration and not a mean value as consistently reported [11]. This is in agreement with a previous finding of Goodman [12] investigating heat transfer transient phenomena with temperature-dependent thermal conductivity. It should be also stressed that the resulting matching expression is much simpler to use than the classical series that requires a great number of terms at short values of τ . With the expression deduced in this contribution there is no need to neglect those experimental values obtained for values of τ where the dependence on the square root of τ is no longer valid. The effect of volume-shrinking can produce sigmoidal behavior of classical uptake vs. time curve even in the absence of relaxation phenomena. Although more experimental results would be needed to verify the goodness of the model presented

in this work, most of the results show reasonable behavior of the classical uptake curve. The model can also be used to get a better understanding of relaxation phenomena associated with diffusion as will be shown in future contributions.

References [1] G. Sarti, F. Doghieri, Non-Fickian transport of alkyl alcohols through prestretched poly-methyl methacrylate, Chem. Eng. Sci. 49 (5) (1994) 733. [2] G. Sarti, G. Gostoli, L. Masoni, Diffusion of alcohols and relaxation in poly(methyl methacrylate). Effect of thermal history, J. Membr. Sci. 15 (1983) 181. [3] J. Crank, The Mathematics of Diffusion, 2nd Edition, Clarendon, Oxford. 1975. [4] T. Alfrey, E.F. Gurnee, W.G. Lloyd, Diffusion in glassy polymers, J. Polymer Sci. C-12 (1966) 249. [5] N.J. Kuipers, A.A. Beenackers, Non-Fickian diffusion with chemical reaction in glassy polymers with swelling induced by the penetrant. A mathematical model, Chem. Eng. Sci. 48 (16) (1993) 2957. [6] N. Peppas, L. Peppas, Water diffusion and sorption en amorphous macromolecular systems and foods, J. Food Eng. 22 (1994) 189. [7] P.P. Roussis, Diffusion of water vapour in polymethyl methacrylate, J. Membr. Sci. 16 (1983) 141. [8] R. Korsmeyer, N. Peppas, Effect of the morphology of hydrophilic polymeric matrices on the diffusion and release of water soluble drugs, J. Membr. Sci. 9 (1981) 211. [9] W. Good, R. Kostelnik, Diffusion of water soluble drugs from initially dry hydrogels, in: Polymeric Delivery Systems, Gordon and Breach, New York, 1976. [10] R. Yoshida, K. Okuyama, T. Sakai, T. Okano, Y. Sakurai, Sigmoidal swelling profiles for temperature responsive poly (N-isopropylacrylamide-co-butyl methacrylate) hydrogels, J Membr. Sci. 89 (1994) 267. [11] H. Odani, M. Uchikura, Y. Ogino, M. Kurata, Diffusion and solution of methanol vapour in poly(2-vinylpiridine)–blockpolyisoprene and poly(2-vinylpyridine)–block-polystyrene, J. Membr. Sci. 15 (1983) 193. [12] T.R. Goodman, Application of integral methods to transient nonlinear heat transfer, in: T.F. Irvine, J.P. Hartnett, (Ed.), Advances in Heat Transfer, Academic Press, New York, 1964, pp. 52–122.