On a convective condition in the diffusion of a solvent into a polymer

On a convective condition in the diffusion of a solvent into a polymer

Marcos Gaudiano^I; Tomás Godoy^II; Cristina Turner^{I, II}

^IFAMAF-UNC-CIEM-CONICET

^IIFAMAF-UNC. Córdoba, Argentina

E-mails: [e-mail] / [e-mail] / [e-mail]

ABSTRACT

We studied a one-dimensional free boundary problem arising in the polymer industry, which solution has an interesting asymptotic behavior when a convective boundary condition is imposed. We show the asymptotic behavior of the free boundary and of the concentration of the solvent in the domain, for large t. Exact estimates and numerical results are obtained.

Mathematical subject classification: 35K05, 35K60.

Key words: free boundary problems, diffusion, convective case, asymptotic behavior.

1 Introduction

In this paper we consider a free boundary problem arising from a model for sorption of solvents into glassy polymers.

This model was proposed in [1] by Astarita and Sarti. They assumed that the sorption process can be described using a free boundary model to simulate a sharp morphological discontinuity observed in the material between a penetrated zone, with a relatively high solvent content, and an glassy region where the solvent concentration is negligibly small (and actually taken to be zero in the model).

The solvent is supposed to diffuse in the penetrated zone according to Fick's law. Moreover the penetrating zone moves into the glassy zone driven by chemical and mechanical effects that are taken into account by an empirical law relating the speed of penetration to the concentration of solvent near the front. This law must account for two main facts observed in the penetration experiences: (i) there exists a threshold value for the solvent concentration under which no penetration occurs; (ii) above such value the speed of the front increases with the concentration near the front itself. A typical form is v = a|u - q|^m where v is the front speed, u is the value of the concentration at the front, q > 0 is the threshold value and a and m are positive constants ([1]).

An additional condition on the free boundary is obtained imposing mass conservation, i.e., equating the mass density current to the product of solvent concentration and the velocity of the free boundary.

This model has been the object of a number of papers. In [2], it has been studied with the condition of constant concentration at the boundary. In [3], it has been investigated assuming that the polymer is in perfect contact with a well-stirred bath. In [4], its authors were interested in the case of a slab of non-homogeneous polymer. In [5], it has been studied assuming a flux condition at the fixed boundary. Here we are interested in a convective case, where it is supposed that there is a flux of solvent through the left side of a slab proportional to the difference between the solvent concentration at x = 0 and a given function of the time which represents an external solvent concentration (h > 0 is the proportionality constant). Denoting by c(x, t) the normalized concentration and by x = s(t) the location of the front in the slab the mathematical problem can be stated as follows:

Problem PS. Find a triple (T, s, c) such that: T > 0, s Î C¹[0, T], c Î C^2,1(D_T) Ç C(_T), where D_T= {(x, t): 0 < t < T, 0 < x < s(t)}, and satisfying

The function g(t) is positive and the quantity q+g(t) represents the external concentration. In order to assure a stable process we suppose that g Î C¹[0,T], "T > 0, g¢(t) < 0 and G º g(t) dt < ¥. Throughout the paper the function f will be supposed to satisfy f Î C¹(0, 1], f¢(c) > 0 for c Î (0, 1] and f(0) = 0 (empirically, the function f is observed to be a power law). We note that there exists F = f^-1 which has the same properties as f.

2 Existence and some global estimates

Equating (1.2) to (1.4) for t = 0, we have that c^*

c(0, 0) is the unique solution of f(c^*)(c^* + q) = -h(c^* - 1). The solution satisfies 0 < c^* < 1.

The existence for PS is accomplished as follows: Let r Î C¹[0, T] Ç C²(0, T) be such that

and consider the problem (PA) of finding c Î C^2,1(D) Ç C(), c_x continuous up to x = r(t), t Î (0, T), such that

Thus, we note that PA differs from PS by the fact that the curve r(t) is given. In [7], it is shown that the transformation

has a unique fixed point for T > 0 small enough, which is actually the desired curve s(t) of PS. The global existence and uniqueness for PS is also established in [7]. Now we prove:

Proposition 2.1. Assume s, c solve problem PS for a given T < +¥. Then

Proof. Using the Hopf's lemma we can assume that c attains its maximum value on x = 0 since c_x = -f(c)(F() + q) < 0 on x = s(t). Let be c(0, t₀) = max_DTc. If t₀ = 0, then max_DT c = c^* < 1. Otherwise we have 0 > c_x(0, t₀) = h(c(0, t₀) - g(t₀)), which implies max_DT c < g(t₀) < 1. Let be c(x₁, t₁) = min_DT c. If x₁ = 0 then there occurs either min_DT c = c^* > 0 or min_DT c = g(t₁) + 1/(h) c_x(0, t₁) > g(t₁) > 0. Moreover, if min_DT c = c(s(t₁), t₁) = 0 (with t₁ > 0) then c_x(s(t₁), t₁) = 0, contradicting the boundary point principle. Thus, (2.14) holds and c_x(0, t) < h(1 - g(t)) for all t. Finally, let us suppose that min_DT c_x = c_x(s(t₂), t₂) with t₂> 0. If t₂ > 0 then

which is a contradiction. Then min_DT c_x = c_x(0, 0) = h(c^* - 1).

Proposition 2.2. Under the assumptions above, the following estimate holds

with

Proof. Note from (1.2) that

Moreover, from (1.3) and (1.4) we have c_x = -f(c)(c + q) at x = s(t) and deriving with respect to t we get

The proposition follows by using the Hopf's lemma.

3 The numerical method

In this section will be shown a numerical scheme based on the method introduced in [6] for one-dimensional parabolic free boundary problems with arbitrary implicit or explicit free boundary conditions.

In this method the continuous problem is time discretized and solved at successive time levels as a sequence of free boundary problems for ordinary differential equations. Specifically, at time level t = t_n with t_n - t_n-1 = Dt the solution {C_n(x), S_n} is computed as the exact solution of the discretized equations

In (3.21) the function C_n-1(x) is supposed to be defined over [0, +¥), and S_n-1 supposed to be known as well. We write (3.21) as a first order system over (0, S_n)

and exploit the observation that C_n and V_n are related through the Riccati transformation

where

The function W_n is solution of well defined initial value problem and may be considered available. The free boundary S_n is determined such that the triple C_n, V_n, S_n simultaneously satisfies (3.23), (3.24) and (3.27). Elimination of C_n and V_n from (3.24) and (3.27) shows that S_n must be a root of the scalar equation

Given S_n, we set

so that

and

Thus, the triple {C_n(S_n), V_n(S_n), S_n} is an exact solution of (3.23), (3.24) and (3.27). We remark that depending on Dt the functional s_n(x) may have a root smaller than S_n-1. Such a root would correspond to a negative concentration C_n(S_n) and is not admissible. We shall therefore agree to choose for S_n the smallest root of s_n(x) = 0 on (S_n-1, ¥). Such a root will be soon to exist.

Once S_n has been determined, one can find V_n by integrating backward over [0, S_n) the equation

with V_n(S_n) given by (3.33). The concentration C_n(x) at time level t_n is obtained from (3.27). Finally, C_n(x) is extended over [S_n, ¥) as C¹ linear function, because C_n+1(x) will be computed in [0, S_n+1], with S_n+1 > S_n, as the solution of an ODE depending on C_n. For the initial concentration we shall use

C₀(x) = –h(1 – c*)x + c*.

Lemma 3.1. There exists a solution S_n of (3.30) on (S_n-1, ¥) and S_n - S_n-1 < f(1) Dt. The function C_n satisfies 0 < C_n< 1 on [0, S_n] and C¢_n< 0 on [S_n, ¥).

Proof. We note that C₀(S₀) = c^*Î (0, 1) and C¢₀ = -h(1 - c^*) < 0. Let us proceed by induction and assume the result valid for n - 1. Integrating (3.29) we have

since C_n-1 < 1 by assumption, we get

moreover W_n(S_n-1) > 0. Hence the function

is less than one and positive on some interval (S_n-1, x₀), vanishing on x₀. Then,

what is more

s_n(S_n – 1) = –f(W_n(S_n–1)) < 0,

thus there must be a point S_n Î (S_n-1, x₀) where s_n(S_n) = 0, 0 < C_n(S_n) < 1 and C¢_n(S_n) < 0. Integrating (3.27) we obtain:

Finally, from (3.31) and (3.32) we conclude that S_n - S_n-1 = f(C_n(S_n))Dt < f(1)Dt.

Remark.S_n-1 can not be an accumulation point of the set of points that satisfy s_n(x) = 0 since s_n is a continuous function and s_n(S_n-1) < 0.

4 Asymptotic behavior

In this section we show some results about the behavior of the free boundary s(t) when t goes to infinity.

Let s(t), c(x, t) solve problem PS. Using Green's identity we get:

which holds for every solution v = v(x, t) of v_xx + v_t = 0 in D_t. Thus, taking v(x, t) = 1 we obtain

which gives

and then

so

This upper bound is independent of f, and since (t) = f(c(s(t),t)) > 0 there exists

The following numerical result shows these facts for the case q = .3, h = 10, g(t) = e^-2t and several functions f ( = 16.7). We solved the equations (3.29) and (3.34) using Runge-Kutta method implemented in Matlab as ODE45, with the step size and tolerance established by default.

The Figure 1 shows that all the free boundaries are bounded nearly by 1.6, so appears to be a very large bound, and we can look for a better one. In order to do it, we will obtain two additional equations for s and c ( (4.40) and (4.41) below). First, taking v(x, t) = x in (4.35) we have

0 =

c(x, t)x dx + (xc_x(x, t) – c(x, t)) dt,

it gives

Similarly, taking v(x, t) = t - we get:

thus

and so

Lemma 4.1. The following equation holds:

Proof. From (4.41) we have

To prove the lemma it is enough that all these terms go to zero as t ® ¥. In order to do it, note that from (4.37),

then, using (4.40)

thus

On the other hand, from (4.36)

and from (1.2)

And integration by parts gives

Where in the last equality we have used (4.43), (4.44) and the L'Hôpital's rule.

Lemma 4.2. Assume that f(c) = ac, a > 0. Then the explicit formula holds:

Proof. Cancelling c(0, t) dt from (4.37) and (4.40) we obtain

and since c(s(t), t) = (t), we get

observe that

and then by (4.42) we get

Theorem 4.1. The following statement is true

The supremum is taken on the set of the functions f belonging to C¹(0, 1] satisfying f¢(c) > 0 in (0, 1] and f(0) = 0.

Proof. Proceeding as in (4.46), we have

as t ® ¥ we get

thus

and (4.47) follows taking a ® ¥ in (4.45).

5 Conclusions and final comment

In this paper the main result is the asymptotic behavior of the free boundary. We remark that the upper bound (4.47) should be very useful for real applications, where the function f is a priori unknown and a estimate of s_¥ is needed. From the physical point of view we emphasize that the bound of the free boundary does not depend on the function f. That means that this behavior of the free boundary holds for all kind of homogeneous polymers with constant diffusivity. For the case of two dimensional space variable we expect to have bounds for the free boundary that do not depend on f. This will be the subject of future work.

Acknowledgement. This paper has been sponsored by the grant Secyt-UNC and CONICET and Marcos Gaudiano was sponsored by an scholarship of CONICET.

Received: 01/III/07. Accepted: 21/VI/07.

#670/06.

• [1] G. Astarita and G.C. Sarti, A Class of Mathematical Models for Sorption of Swelling Solvents in Glassy Polymers Polymer Engineering and Science, 18(5) (1978), 388-395.
• [2] A. Fasano, G.H. Meyer and M. Primicerio, On a Problem in the Polymer Industry: Theoretical and Numerical Investigation of Swelling S.I.A.M., 17(4) (1986), 945-960.
• [3] A. Comparini and R. Ricci, On the Swelling of a Glassy Polymer in Contact with a Well-stirred Solvent Mathematical Methods in the Applied Sciences, 7 (1985), 238-250.
• [4] A. Comparini, R. Ricci and C. Turner, Penetration of a solvent into a non-homogeneous polymer Meccanica, 23 (1988), 75-80.
• [5] D. Andreucci and R. Ricci, A free boundary problem arising from sorption of solvent in glassy polymers Quarterly of applied mathematics, 44 (1987), 649-657.
• [6] G.H. Meyer, One-Dimensional Parabolic Free Boundary Problems S.I.A.M. Review, 19(1) (1977), 17-33.
• [7] M. Gaudiano, C. Turner and T. Godoy, Technical Report: Free boundary problems in Industry: Diffusion of a solvent into a polymer. (http://www.famaf.unc.edu. ar/series/AMat2005.htm).

Publication Dates

• Publication in this collection
  22 Nov 2007
• Date of issue
  2007

History

• Accepted
  21 June 2007
• Received
  01 Mar 2007