# Effect of interdiffusion regions on penetrant flux through multilayer films.

Shilpa Y. Sankhe (*)

Douglas E. Hirt (*)

David A. Zumbrunnen (+)

INTRODUCTION

Polymer films are used in a variety of applications to reduce the permeation of a gas or liquid species. Koros and others have summarized several barrier enhancement techniques for films (1-4). For example, a polymer laminate consisting of 3-7 layers, with core layers having an inherently low permeability, could be formed for improving barrier performance. Also, blends with dispersed impermeable material (inorganic or interfacially stabilized immiscible polymer), popularly known as nanocomposites could be used for barrier enhancement. Yet another method is to reactively modify a film surface or coat it with a less permeable material to impart high barrier properties to the film. In other studies, layer multiplication has been used to produce microlayer films of alternating polymers. For example, Nazarenko et al. have produced films with up to 2048 layers and layer thicknesses down to 150 nm (5). Such layered composites consisting of hundreds or thousands of alternating layers reveal a ten-fold enhancement in physical properties (mechanical, optical, etc.) when the size of the layers is reduced from the macroscale (tens of microns) to the microscale (several microns) (6). The dramatic improvement has been attributed to the fundamental changes that occur in the composite due to decreased layer thicknesses. Recently, very highly multilayered films, which consist of many thousands of unbroken individual layers, have been produced by a new process (7). The layers were formed in situ by inducing chaotic mixing in melts containing two or more polymers. Recursive stretching and folding that is characteristic of chaotic mixing yielded a multilayered blend morphology that was extruded (see Fig. 1). This process is amenable to control so films with a desired number of layers and layer thicknesses can be selected on-line (8).

Because of the availability of layer multiplication and the new chaotic mixing film process, this paper focuses on the potential to enhance barrier performance by increasing the number of layers in film products. The goal was to explore the effect of various parameters on the flux of material through a multilayer film, with a particular emphasis on the interdiffusion regions that are present at layer-layer boundaries (Fig. 2) in a film containing any number of specified layers. Therefore, a mathematical model was developed to predict the number of layers needed to change permeant flux by a certain amount for a variety of generalized conditions.

Jabbari and Peppas have presented a model that can be used to predict the polymer-concentration profiles Within an interdiffusion region (9). Possible concentrations in terms of the volume fraction of polymer y, [phi].sub.y], are shown in Fig. 3 in the vicinity of a layer-layer boundary, where z is the distance from the boundary and [L.sub.xo] is the thickness of polymer x. Jabbari and Peppas showed that, depending on the relative mobilities of polymers x and y, concentration profiles across an interdiffusion region could be balanced (i.e., polymers interpenetrating one another equally) or unbalanced (i.e., polymers interpenetrating to different extents). It seems reasonable to expect that the diffusivity of a penetrant would depend on the polymer concentration in the interdiffusion region, so that dependence was built into the model. Moreover, owing to the thinness of an interdiffusion region, it also seems reasonable that many layers would be required before the interdiffusion regions influenced flux. The i nvestigation reported here is a first step in understanding the effect of interdiffusion regions on the permeation of species through a multilayer film where the number of layers is very large.

MATHEMATICAL MODEL

A continuum approach is used to model species diffusion through an n-layer film at steady state. Traditionally, the problem of determining flux through a multilayer film is solved assuming a discontinuity in concentration at a polymer-polymer interface (10), which arises as a result of a difference in penetrant solubilities ([S.sub.x] and [S.sub.y]), in the adjacent polymers. Discontinuities are shown in Fig. 4 for a multilayer film with steady, linear concentration profiles in each layer. As depicted, alternating polymers x and y constitute a multilayer film of total thickness [z.sub.n]. The vertical solid lines represent the layer-layer boundaries and [L.sub.1], [L.sub.2], ... [L.sub.n] represent the thicknesses of each layer between solid lines (the [z.sub.i]'s in the region 0 < z < [z.sub.n] indicate the coordinate positions of the solid lines). The penetrant diffusivities in polymers x and y are designated [D.sub.x] and [D.sub.y] and are assumed to be constant.

One-dimensional Fickian species conservation equations are given as:

[partial]C/[partial]t = [partial]/[partial]z (D [partial]C/[partial]z) (variable diffusivity) (1a)

[partial]C/[partial]t = D [[partial].sup.2]C/[partial][z.sup.2] (constant diffusivity) (1b)

where C is the penetrant concentration dependent on time, t, and position, z. Solution to Eq 1b at steady state yields the concentration profile of a penetrant in any layer q:

[C.sub.q] = [a.sub.q]z + [b.sub.q] q = 1, 2,...,n (2)

where [a.sub.q] and [b.sub.q] are constants. The boundary conditions that apply to this system are:

[C.sub.1] = [C.sub.[alpha]] at z = 0 (3a)

[C.sub.n] = [C.sub.[beta]] at z = [z.sub.n] (3b)

[C.sub.j]/[C.sub.j+1] = [S.sub.j]/[S.sub.j+1] / [D.sub.j] [partial][C.sub.j]/[partial]z = [D.sub.j+1] [partial][C.sub.j+1]/[partial]z }at z = [z.sub.j] where j = 1, 2,...,n-1 (3c, d)

Solving for the constants [a.sub.q] and [b.sub.q], general expressions for penetrant concentration in any of the n layers are obtained as:

[C.sub.[alpha]] - [C.sub.q]/[C.sub.[alpha]] - [MC.sub.[beta]] = (z - [summation over (j=1).sup.q][L.sub.j]/[L.sup.*]) + 1/[L.sup.*] [summation over (j=1).sup.q] [D.sub.1/j][L.sub.j]/[S.sub.j/q] (q odd) (4a)

[C.sub.[alpha]]/[S.sub.x/y] - [C.sub.q]/[C.sub.[alpha]] - [MC.sub.[beta]] = [D.sub.x/y](z - [summation over (j=1).sup.q][L.sub.j]/[L.sup.*]) + 1/[L.sup.*] [summation over (j=1).sup.q][D.sub.1/j][L.sub.j]/[S.sub.j/q] (q even) (4b)

with

[L.sup.*] = [summation over (n/k=1 odd)][L.sub.xk] + ([S.sub.x/y][D.sub.x/y]) [summation over (n/j=2 even)][L.sub.yj] = [L.sub.xo] + ([S.sub.x/y][D.sub.x/y])[L.sub.yo]

In Eq 4, M = 1 (for odd values of n) or [S.sub.x/y] (for even values of n); [S.sub.x/y] = [S.sub.x]/[S.sub.y]; [D.sub.x/y] = [D.sub.x]/[D.sub.y]; and [L.sub.xo] and [L.sub.yo] are the total thicknesses of all polymer x and y layers, respectively. The advantage of writing [L.sup.*] as above is that the product of the solubility and the diffusivity is equivalent to the permeability, P (i.e., [S.sub.x/y][D.sub.x/y] = ([S.sub.x]/[S.sub.y])([D.sub.x]/[D.sub.y]) = [P.sub.x]/[P.sub.y]). The flux of diffusing species through a film with alternating layers is obtained from Eq 4 as:

[N.sub.a] = - [D.sub.q] d[C.sub.q]/dz

[N.sub.a] = [D.sub.x]([C.sub.[alpha]] - [MC.sub.[beta]])/[L.sup.*] = [D.sub.x]([C.sub.[alpha]] - [MC.sub.[beta]])/[L.sub.xo] + ([S.sub.x/y][D.sub.x/y])[L.sub.yo] (5)

According to Eq 5, the steady state flux through a film with a concentration discontinuity at each layer interface is not influenced by the number of layers, as long as the total masses of each polymer remain constant. In many situations, however, the polymers interpenetrate at the interface such that a gradual change in polymer concentration occurs across a very thin interdiffusion region. Hence, in cases where polymer-polymer interdiffusion occurs, there is also a gradual change in solubility through the interdiffusion region that leads to a continuous penetrant concentration profile across the interface (rather than a discontinuity). The subsequent analysis includes an interdiffusion region at each polymer-polymer interface, which is on the order of a few nanometers thick while the film thickness typically ranges from 1-50 [micro]m. Figure 5 shows a multilayer film where the vertical solid lines represent fictitious layer-layer boundaries in the absence of interdiffusion. Interdiffusion regions are defined by the vertical dashed lines, which are a distance 2[L.sub.i] apart. To simplify the analysis, each interdiffusion region is taken to have the same thickness and is positioned symmetrically about each solid line. As shown below, the diffusivity in the interdiffusion region ([D.sub.i]) will be treated in two ways, as spatially constant and as a function of polymer concentration.

Since diffusivities, [D.sub.x] and [D.sub.y], of the layers away from the interdiffusion regions are assumed to be constant, the penetrant concentration profile in the polymer layers is still given by Eq 2. If C' is the penetrant concentration in the interdiffusion region, the boundary conditions that apply to this system in addition to Eq 3a and Eq 3b are:

[C.sub.j] = [C'.sub.j] }

[D.sub.j] [partial][C.sub.j]/[partial]z = [D.sub.i] [partial][C'.sub.j]/[partial]z }

at z = [z.sub.j] - [L.sub.i] where j = 1, 2,..., n - 1 (6a, b)

[C.sub.j+1] = [C'.sub.j] }

[D.sub.j+1] [partial][C.sub.j+1]/[partial]z = [D.sub.i] [partial][C'.sub.j]/[partial]z }

at z = [z.sub.j] + [L.sub.i] where j = 1, 2,..., n - 1 (6c, d)

Thus, continuity in molar flux and concentration are specified at every dashed line in Fig. 5. Now, two cases for diffusivity in the interdiffusion region, [D.sub.i], are considered.

Case I: Constant [D.sub.i]

In the case of constant diffusivity of the penetrant in the interdiffusion region, [D.sub.i], the penetrant concentration profile in the interdiffusion region is similar to Eq 2:

[C'.sub.r] = [a'.sub.r]z + [b'.sub.r] r = 1, 2,..., n - 1 (7)

where C' represents the penetrant concentration in the interdiffusion region and r signifies each interdiffusion region (e.g., r = 1 represents the interface between layers 1 and 2). The penetrant flux, [N.sub.[alpha]]. can be derived following the analogous heat-transfer problem through a composite wall (11) to give:

[N.sub.[alpha]] = [C.sub.[alpha]] - [C.sub.[beta]] /[[L.sub.xo] - (n - 1)[L.sub.i]/[D.sub.x] + 2(n - 1)[L.sub.i]/[D.sub.i] + [L.sub.yo] - (n - 1)[L.sub.i] / [D.sub.y]] (8)

In Eq 8, [L.sub.xo] and [L.sub.yo] are the total thicknesses of all polymer x and y layers, where [L.sub.t] = [L.sub.xo] + [L.sub.yo]. The film layer thicknesses are effectively reduced owing to the interdiffusion regions as n increases.

The following parameters are defined for convenience:

[DELTA]C = [C.sub.[alpha]] - [C.sub.[beta]]; [L.sub.rx] = [L.sub.xo]/[L.sub.t]; [L.sub.ri] = 2[L.sub.i]/[L.sub.t]; [L.sub.ry] = [L.sub.yo]/[L.sub.t]; [L.sub.x/y] = [L.sub.xo]/[L.sub.yo]; [N.sup.*.sub.[alpha]] = [N.sub.[alpha]]/[[DELTA]C[D.sub.y]/[L.sub.t]]; [D.sub.i/y] = [D.sub.i]/[D.sub.y]; [D.sub.x/y] = [D.sub.x]/[D.sub.y] (9)

With the aid of these parameters Eq 8 can be expressed as Eq 10:

[N.sup.*.sub.[alpha]] = 1 / [L.sub.ry] + [L.sub.rx]/[D.sub.x/y] + [1/[D.sub.i/y] - 1/2 - 1/2[D.sub.x/y]](n - 1)[L.sub.ri] (10)

Whether the flux increases or decreases with respect to the total number of layers, n, depends on the value inside the square brackets. For the flux to decrease as n increases, with the other parameters remaining constant, 1/[D.sub.i/y] - 1/2 - 1/2[D.sub.x/y] > 0. The conditions below can thereby be determined:

[D.sub.i/y] < 2/1 + 1/[D.sub.x/y] (11 a)

[D.sub.i] < 2/1/[D.sub.x] + 1/[D.sub.y] (11 b)

These define the eritical [D.sub.i/y] (or [D.sub.i]) at which [N.sup.*.sub.[alpha]] transitions from an increasing to a decreasing function of n. The physical significance of Eq 11a and Eq 11b will be discussed later when the model results are presented.

Case II: [phi]-dependent [D.sub.i]

In an interdiffusion region, two polymers intermingle across an interface, as shown in Fig. 2. Therefore, the diffusivity of the penetrant in this region is likely to be intermediate between those of the pure components. This hypothesis is supported by a study done by George et al. on the transport of aromatic solvents in isotactic-polypropylene (PP)/acrylonitrile-co-butadiene ruber (NBR) blends (12). They showed that in creasing the content of NBR from 0 from that is pure PP to one in pure NBR. As a first approximation, the rule of mixtures is used to represent [D.sub.i], where [[phi].sub.x] and [[phi].sub.y] are the volume fractions of polymers x and y.

[D.sub.i] = [[phi].sub.x][D.sub.x] + [[phi].sub.y][D.sub.y] (12)

Many studies have shown that the polymer concentration varies across the interdiffusion region (9, 13, 14). Depending on the polymer pair, the polymer-concentration profiles can be balanced or unbalanced about the interface. Figure 3 shows a comparison of three types of concentration profiles for polymer y across the interdiffusion region, viz, balanced - linear, balanced - non-linear, and unbalanced. A balanced profile is obtained when the adjacent polymers interpenetrate equally, while an unbalanced profile is obtained when one polymer interdiffuses to a greater extent than the other does. For this study, a balanced profile is assumed. For simplicity, the polymer concentrations [[phi].sub.x] and [[phi].sub.y] vary linearly with z. Thus, for the first interdiffusion region,

[[phi].sub.x] = [z.sub.1] + [L.sub.i] - z / 2[L.sub.t] [z.sub.1] - [L.sub.i] < z < [z.sub.1] + [L.sub.i] (13)

Substituting the above expression in Eq 12, [D.sub.i] is obtained as a function of z. Solution to Eq 1a with a variable diffusivity in the interdiffusion regions is given as:

[C'.sub.r] [a'.sub.r] ln(z + [A.sub.r]) + [b'.sub.r] r = 1,2,..., n - 1 (14)

In Eq 14, [A.sub.r] = [L.sub.i] ([D.sub.r+1] + [D.sub.r]) / ([D.sub.r+1] - [D.sub.r]) - [z.sub.r]

such that, [D.sub.j] = [D.sub.x ] ([for all] j [equivalent to] odd) and [D.sub.j] = [D.sub.y] ([for all] j [equivalent to] even) where j = 1, 2, ..., n-1.

The boundary conditions remain the same as those for Case I (Eqs 3a, 3b and 6). Solving for the constants, the concentration profile for the nth layer in an n-layer film is given by Eq 15:

[C.sub.n] - [C.sub.[beta]] / [DELTA]C = - 1/[A.sup.*.sub.n] [D.sub.y]/[D.sub.n] (z - [z.sub.n]);

where [A.sup.*.sub.n] = [D.sub.y]/[D.sub.x] (2[L.sub.x0]/n - [L.sub.i]) + (2[L.sub.y0]/n - [L.sub.i]) + 2(n - 1)[L.sub.i] / ([D.sub.x]/[D.sub.y] - 1) ln ([D.sub.x]/[D.sub.y]) + (n - 2) ([L.sub.y0]/n - [L.sub.i]) + ( n - 2) [D.sub.y]/[D.sub.x] ([L.sub.x0]/n - [L.sub.i]) (15)

The molar flux [N.sup.*.sub.a], is determined from the above concentration profile, Eq 16, and Eq 9:

[N.sub.a] = - [D.sub.n] d[C.sub.n]/dZ = [D.sub.y]/[A.sup.*.sub.n] [DELTA]C (16)

[N.sup.*.sub.a] = 1 / [1/[D.sub.x/y] (2[L.sub.rx]/n - [L.sub.ri]/2) + (2[L.sub.ry]/n - [L.sub.ri]/2) + (n-1)[L.sub.ri]/([D.sub.x/y] - 1) ln ([D.sub.x/y]) + (n - 2) ([L.sub.ry]/n - [L.sub.ri]/2) + (n - 2) 1/[D.sub.x/y] ([L.sub.rx]/n - [L.sub.ri]/2)] (17)

Note that the above expression was derived for a multilayer film (xyxy...) assuming an even number of layers in the film with same alternate layer thickness.

RESULTS AND DISCUSSION

The objective of this study was to predict the number of layers needed to cause a certain change in penetrant flux in a multilayer film and also to study the effects of various parameters on the number of layers required for a desired steady state flux. Two cases were considered for the diffusivity in the interdiffusion region, [D.sub.i], and are discussed below.

Case I: Constant [D.sub.i]

Equation 10 was obtained for the dimensionless flux when [D.sub.i] was assumed to be constant. Figures 6 a-c show dimensionless flux, [N.sup.*.sub.a], vs. number of layers, n, for different values of the diffusivity ratio [D.sub.i/y]. The solid curves represent model results for [D.sub.x/y] = 10 and the dashed curves for [D.sub.x/y] = 100. These values are typical of gas-diffusivity ratios for various polymers. For example. Compan et al. report [O.sub.2] and [CO.sub.2] diffusivities in LLDPE to be ~2 X [10.sub.-7] at 30[degrees]C (15). Diffusivity values of [O.sub.2] in polypropylene have been found to be the same order of magnitude (16). On the other hand, the diffusivity of [O.sub.2] in a polyamide is about 4 X [10.sub.-9] at 29[degrees]C (16). Thus, if LLDPE is assigned as polymer x and polyamide as polymer y, [D.sub.x]/[D.sub.y] ~ 50. The interdiffusion thickness ratio, [L.sub.ri], was assumed equal to 0.0001, which was determined assuming 2[L.sub.i] = 5 nm (17) and [L.sub.t] = 50 [mu]m. Last, the polymer contents in the film were assigned as [L.sub.x0]/[L.sub.y0] = 0.333, 1, and 3 in Figs. 6a, b, and c, respectively. Note that the y-axis scales are different and, depending on the value of [D.sub.i/y], the flux either increases above or decreases below its initial value at n = 2 (corresponding to a bilayer).

For [D.sub.i/y] values below the critical value defined in Eq 11, the flux decreases with increasing number of layers (e.g., the critical value of is 1.82 for [D.sub.x/y] = 10). As shown in Fig. 6a, the flux decreases with increasing n for [D.sub.i/y] values of 1 and 0.1. Also, the further the value is below critical [D.sub.i/y], fewer layers are required to significantly reduce the flux. However, for a [D.sub.i/y] = 0.1, the diffusivity in the interdiffusion region would be less than both [D.sub.y] ([D.sup.i] = 0.1 [D.sub.y]) and [D.sub.x] ([D.sub.i] = 0.01[D.sub.x]) for the given value of [D.sub.x/y] (e.g., 10). The case of [D.sub.i/y] = 0.1 is unrealistic as [D.sub.i] is expected to be at least equal to if not greater than the lesser of [D.sub.x] and [D.sub.y] ([D.sub.y] in this case). Thus, for [D.sub.i/y] values greater than 1 and less than 1.82, the flux decreases with increasing n. For values of [D.sub.i/y] above 1.82, the flux increases with increasing number of layers, as is the case for [D.sub.i/y] = 10 in Fig. 6. In Fig. 6, it is also evident that the flux is greater when = 100 (dashed curves) compared to = 10 (solid curves), as expected. Moreover, as the content of polymer x increases (i.e., [L.sub.x/y] increases), the flux increases at a given n. This behavior is expected because the results are based on cases where [D.sub.x] > [D.sub.y].

Figures 7 and 8 show the effect of increasing thickness ratios on the number of layers required for a desired flux reduction. In particular, Fig. 7 shows the minimum number of layers required to reduce the dimensionless flux, [N.sub.a.sup.*], to an arbitrarily chosen value of 1 as the content of x increases, assuming [D.sub.i/y] = 1. [D.sub.x/y] = 10, and maintaining the interdiffusion thickness ratio, [L.sub.ri] at 0.0001. The dimensionless flux is 1.82 for a bilayer film with the above parameters at [L.sub.x/y] = 1. Since polymer x is a higher diffusivity material, increasing the amount of x increases the flux with increasing For example, the flux increases for a bilayer film from 1.82 to 5.5 when [L.sub.x/y] increases from 1 to 10. Hence, the number of layers required to reduce the flux to a certain value increases as the content of polymer x increases, as shown in Fig. 7.

The effect of increasing interdiffusion thickness ratio on the number of layers required for a desired steady state flux is shown in Fig. 8. Results correspond to the number of layers required to achieve [N.sub.a.sup.*] = 1 for increasing interdiffusion thickness ratio. Three curves corresponding to different polymer content ratios of [L.sub.x/y] = 0.333. 1 and 3 are shown. All other parameters remain the same as in the case above. Increasing the interdiffusion region thickness decreases the number of layers required to obtain [N.sub.a.sup.*] = 1. This result is expected for [D.sub.i]/[D.sub.y] = 1 because [D.sub.y] is lower than [D.sub.x] and adding thicker interdiffusion regions with [D.sub.i] = [D.sub.y] yields a film with more low-diffusivity material, thereby reducing the overall flux. Also, at a given interdiffusion thickness ratio the number of layers required increases as the polymer content of x increases from 0.333 to 3 (also shown in Fig. 7 for [L.sub.ri] = 0.0001). Moreover, beyond [L.sub.ri] ~ 0. 001, the number of layers required for a desired steady state flux is less sensitive to an increase in the interdiffusion thickness ratio. Assuming a constant [L.sub.i] higher values of [L.sub.ri] correspond to thinner films (smaller [L.sub.t]). Therefore, the interdiffusion regions constitute a larger fraction of the overall thickness and fewer layers are required to achieve the desired [N.sub.a.sup.*]

Case II: [phi]-dependent [D.sub.i]

Equation 17 was derived for the dimensionless flux, [N.sub.a.sup.*], when [D.sub.i] was assumed to be given by a mixture rule depending on the polymer volume fractions. Fig. 9 illustrates the effect of increasing the number of layers on the dimensionless flux, [N.sub.a.sup.*] relative to that of a bilayer film, [N.sub.a2.sup.*], for the case of [phi]-dependent diffusivity ([N.sub.a.sup.*] was normalized by [N.sub.a2.sup.*] because a bilayer is the simplest multilayer film). As observed, the flux increases with increasing n and the trend is true for each value of shown. Since n appears in many places in the denominator of Eq 17, it is difficult to identify the trend in flux with increasing n at first glance. However, by calculating an average [D.sub.i] and applying the criterion given in Eq 11, one can predict whether the flux would decrease or increase with increasing n. For the assumed linear polymer-concentration profiles that are balanced in the interdiffusion region,

[D.sub.i] = 1/2[L.sub.t] [[integral].sup.[z.sub.1] + [L.sub.i].sub.[z.sub.1] - [L.sub.t]] D(z)dz = [D.sub.x] + [D.sub.y]/2 > 2/1/[D.sub.x] + 1/[D.sub.y]

or

[D.sub.i/y] = 1 + [D.sub.x/y]/2 > 2/1 + 1/[D.sub.x/y] (18)

As indicated by Eq 18. the flux would always increase with increasing number of layers, and Fig. 9 supports this prediction (e.g., the average [D.sub.i/y] value for the case of [D.sub.x/y] = 10 is 5.5, which is greater than the critical [D.sub.i/y] value of 1.82). The inequality in Eq 18 is true for any balanced profile with [D.sub.i] given by Eq 12, because the average would always be an algebraic mean of [D.sub.x] and [D.sub.y] as shown above.

CONCLUSIONS

A model was developed to predict the steady-state flux of a species through a multilayer film consisting of alternating polymers. The major objective of this study was to investigate the effect of interdiffusion regions on the permeation of species through multilayer films. The model incorporated species diffusion through interdiffusion regions at each polymer-polymer interface. It was assumed that each interdiffusion region was balanced, that is, the adjacent polymers interpenetrated one another equally. The species diffusivity was assumed to be constant in each polymer and two cases were considered for the diffusivity in the interdiffusion regions. In case one, the diffusivity was constant and a criterion was defined to predict whether the flux would decrease as the number of layers increased. Results showed that thousands of layers were required to achieve appreciable flux reduction. The relative amounts of the two polymers also greatly influenced the flux, with the incorporation of more low-diffusivity po lymer yielding a lower overall flux. The effect of increasing interdiffusion thickness on the number of layers required for a desired steady state flux was also studied. Introducing thicker interdiffusion regions with diffusivity ([D.sub.i]) reduced the overall flux and the number of layers required to achieve a given flux, as long as [D.sub.i] was less than a critical diffusivity defined by Eq 11. For case two, the species diffusivity was assumed to depend linearly on the polymer volume fraction in the interdiffusion region. In all instances where a balanced interdiffusion region was considered, the flux increased with the number of layers.

ACKNOWLEDGMENTS

This work was supported in part by the ERC Program of the National Science Foundation under Award Number EEC-9731680. The authors also gratefully acknowledge financial support from ILC Dover, Inc.

Department of Chemical Engineering (*)

Department of Mechanical Engineering (+), and Center for Advanced Engineering Fibers and Films

Clemson University

Clemson, SC 29634

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[Figure 3 omitted]

[Figure 6 omitted]

[Figure 7 omitted]

[Figure 8 omitted]

[Figure 9 omitted]

Douglas E. Hirt (*)

David A. Zumbrunnen (+)

INTRODUCTION

Polymer films are used in a variety of applications to reduce the permeation of a gas or liquid species. Koros and others have summarized several barrier enhancement techniques for films (1-4). For example, a polymer laminate consisting of 3-7 layers, with core layers having an inherently low permeability, could be formed for improving barrier performance. Also, blends with dispersed impermeable material (inorganic or interfacially stabilized immiscible polymer), popularly known as nanocomposites could be used for barrier enhancement. Yet another method is to reactively modify a film surface or coat it with a less permeable material to impart high barrier properties to the film. In other studies, layer multiplication has been used to produce microlayer films of alternating polymers. For example, Nazarenko et al. have produced films with up to 2048 layers and layer thicknesses down to 150 nm (5). Such layered composites consisting of hundreds or thousands of alternating layers reveal a ten-fold enhancement in physical properties (mechanical, optical, etc.) when the size of the layers is reduced from the macroscale (tens of microns) to the microscale (several microns) (6). The dramatic improvement has been attributed to the fundamental changes that occur in the composite due to decreased layer thicknesses. Recently, very highly multilayered films, which consist of many thousands of unbroken individual layers, have been produced by a new process (7). The layers were formed in situ by inducing chaotic mixing in melts containing two or more polymers. Recursive stretching and folding that is characteristic of chaotic mixing yielded a multilayered blend morphology that was extruded (see Fig. 1). This process is amenable to control so films with a desired number of layers and layer thicknesses can be selected on-line (8).

Because of the availability of layer multiplication and the new chaotic mixing film process, this paper focuses on the potential to enhance barrier performance by increasing the number of layers in film products. The goal was to explore the effect of various parameters on the flux of material through a multilayer film, with a particular emphasis on the interdiffusion regions that are present at layer-layer boundaries (Fig. 2) in a film containing any number of specified layers. Therefore, a mathematical model was developed to predict the number of layers needed to change permeant flux by a certain amount for a variety of generalized conditions.

Jabbari and Peppas have presented a model that can be used to predict the polymer-concentration profiles Within an interdiffusion region (9). Possible concentrations in terms of the volume fraction of polymer y, [phi].sub.y], are shown in Fig. 3 in the vicinity of a layer-layer boundary, where z is the distance from the boundary and [L.sub.xo] is the thickness of polymer x. Jabbari and Peppas showed that, depending on the relative mobilities of polymers x and y, concentration profiles across an interdiffusion region could be balanced (i.e., polymers interpenetrating one another equally) or unbalanced (i.e., polymers interpenetrating to different extents). It seems reasonable to expect that the diffusivity of a penetrant would depend on the polymer concentration in the interdiffusion region, so that dependence was built into the model. Moreover, owing to the thinness of an interdiffusion region, it also seems reasonable that many layers would be required before the interdiffusion regions influenced flux. The i nvestigation reported here is a first step in understanding the effect of interdiffusion regions on the permeation of species through a multilayer film where the number of layers is very large.

MATHEMATICAL MODEL

A continuum approach is used to model species diffusion through an n-layer film at steady state. Traditionally, the problem of determining flux through a multilayer film is solved assuming a discontinuity in concentration at a polymer-polymer interface (10), which arises as a result of a difference in penetrant solubilities ([S.sub.x] and [S.sub.y]), in the adjacent polymers. Discontinuities are shown in Fig. 4 for a multilayer film with steady, linear concentration profiles in each layer. As depicted, alternating polymers x and y constitute a multilayer film of total thickness [z.sub.n]. The vertical solid lines represent the layer-layer boundaries and [L.sub.1], [L.sub.2], ... [L.sub.n] represent the thicknesses of each layer between solid lines (the [z.sub.i]'s in the region 0 < z < [z.sub.n] indicate the coordinate positions of the solid lines). The penetrant diffusivities in polymers x and y are designated [D.sub.x] and [D.sub.y] and are assumed to be constant.

One-dimensional Fickian species conservation equations are given as:

[partial]C/[partial]t = [partial]/[partial]z (D [partial]C/[partial]z) (variable diffusivity) (1a)

[partial]C/[partial]t = D [[partial].sup.2]C/[partial][z.sup.2] (constant diffusivity) (1b)

where C is the penetrant concentration dependent on time, t, and position, z. Solution to Eq 1b at steady state yields the concentration profile of a penetrant in any layer q:

[C.sub.q] = [a.sub.q]z + [b.sub.q] q = 1, 2,...,n (2)

where [a.sub.q] and [b.sub.q] are constants. The boundary conditions that apply to this system are:

[C.sub.1] = [C.sub.[alpha]] at z = 0 (3a)

[C.sub.n] = [C.sub.[beta]] at z = [z.sub.n] (3b)

[C.sub.j]/[C.sub.j+1] = [S.sub.j]/[S.sub.j+1] / [D.sub.j] [partial][C.sub.j]/[partial]z = [D.sub.j+1] [partial][C.sub.j+1]/[partial]z }at z = [z.sub.j] where j = 1, 2,...,n-1 (3c, d)

Solving for the constants [a.sub.q] and [b.sub.q], general expressions for penetrant concentration in any of the n layers are obtained as:

[C.sub.[alpha]] - [C.sub.q]/[C.sub.[alpha]] - [MC.sub.[beta]] = (z - [summation over (j=1).sup.q][L.sub.j]/[L.sup.*]) + 1/[L.sup.*] [summation over (j=1).sup.q] [D.sub.1/j][L.sub.j]/[S.sub.j/q] (q odd) (4a)

[C.sub.[alpha]]/[S.sub.x/y] - [C.sub.q]/[C.sub.[alpha]] - [MC.sub.[beta]] = [D.sub.x/y](z - [summation over (j=1).sup.q][L.sub.j]/[L.sup.*]) + 1/[L.sup.*] [summation over (j=1).sup.q][D.sub.1/j][L.sub.j]/[S.sub.j/q] (q even) (4b)

with

[L.sup.*] = [summation over (n/k=1 odd)][L.sub.xk] + ([S.sub.x/y][D.sub.x/y]) [summation over (n/j=2 even)][L.sub.yj] = [L.sub.xo] + ([S.sub.x/y][D.sub.x/y])[L.sub.yo]

In Eq 4, M = 1 (for odd values of n) or [S.sub.x/y] (for even values of n); [S.sub.x/y] = [S.sub.x]/[S.sub.y]; [D.sub.x/y] = [D.sub.x]/[D.sub.y]; and [L.sub.xo] and [L.sub.yo] are the total thicknesses of all polymer x and y layers, respectively. The advantage of writing [L.sup.*] as above is that the product of the solubility and the diffusivity is equivalent to the permeability, P (i.e., [S.sub.x/y][D.sub.x/y] = ([S.sub.x]/[S.sub.y])([D.sub.x]/[D.sub.y]) = [P.sub.x]/[P.sub.y]). The flux of diffusing species through a film with alternating layers is obtained from Eq 4 as:

[N.sub.a] = - [D.sub.q] d[C.sub.q]/dz

[N.sub.a] = [D.sub.x]([C.sub.[alpha]] - [MC.sub.[beta]])/[L.sup.*] = [D.sub.x]([C.sub.[alpha]] - [MC.sub.[beta]])/[L.sub.xo] + ([S.sub.x/y][D.sub.x/y])[L.sub.yo] (5)

According to Eq 5, the steady state flux through a film with a concentration discontinuity at each layer interface is not influenced by the number of layers, as long as the total masses of each polymer remain constant. In many situations, however, the polymers interpenetrate at the interface such that a gradual change in polymer concentration occurs across a very thin interdiffusion region. Hence, in cases where polymer-polymer interdiffusion occurs, there is also a gradual change in solubility through the interdiffusion region that leads to a continuous penetrant concentration profile across the interface (rather than a discontinuity). The subsequent analysis includes an interdiffusion region at each polymer-polymer interface, which is on the order of a few nanometers thick while the film thickness typically ranges from 1-50 [micro]m. Figure 5 shows a multilayer film where the vertical solid lines represent fictitious layer-layer boundaries in the absence of interdiffusion. Interdiffusion regions are defined by the vertical dashed lines, which are a distance 2[L.sub.i] apart. To simplify the analysis, each interdiffusion region is taken to have the same thickness and is positioned symmetrically about each solid line. As shown below, the diffusivity in the interdiffusion region ([D.sub.i]) will be treated in two ways, as spatially constant and as a function of polymer concentration.

Since diffusivities, [D.sub.x] and [D.sub.y], of the layers away from the interdiffusion regions are assumed to be constant, the penetrant concentration profile in the polymer layers is still given by Eq 2. If C' is the penetrant concentration in the interdiffusion region, the boundary conditions that apply to this system in addition to Eq 3a and Eq 3b are:

[C.sub.j] = [C'.sub.j] }

[D.sub.j] [partial][C.sub.j]/[partial]z = [D.sub.i] [partial][C'.sub.j]/[partial]z }

at z = [z.sub.j] - [L.sub.i] where j = 1, 2,..., n - 1 (6a, b)

[C.sub.j+1] = [C'.sub.j] }

[D.sub.j+1] [partial][C.sub.j+1]/[partial]z = [D.sub.i] [partial][C'.sub.j]/[partial]z }

at z = [z.sub.j] + [L.sub.i] where j = 1, 2,..., n - 1 (6c, d)

Thus, continuity in molar flux and concentration are specified at every dashed line in Fig. 5. Now, two cases for diffusivity in the interdiffusion region, [D.sub.i], are considered.

Case I: Constant [D.sub.i]

In the case of constant diffusivity of the penetrant in the interdiffusion region, [D.sub.i], the penetrant concentration profile in the interdiffusion region is similar to Eq 2:

[C'.sub.r] = [a'.sub.r]z + [b'.sub.r] r = 1, 2,..., n - 1 (7)

where C' represents the penetrant concentration in the interdiffusion region and r signifies each interdiffusion region (e.g., r = 1 represents the interface between layers 1 and 2). The penetrant flux, [N.sub.[alpha]]. can be derived following the analogous heat-transfer problem through a composite wall (11) to give:

[N.sub.[alpha]] = [C.sub.[alpha]] - [C.sub.[beta]] /[[L.sub.xo] - (n - 1)[L.sub.i]/[D.sub.x] + 2(n - 1)[L.sub.i]/[D.sub.i] + [L.sub.yo] - (n - 1)[L.sub.i] / [D.sub.y]] (8)

In Eq 8, [L.sub.xo] and [L.sub.yo] are the total thicknesses of all polymer x and y layers, where [L.sub.t] = [L.sub.xo] + [L.sub.yo]. The film layer thicknesses are effectively reduced owing to the interdiffusion regions as n increases.

The following parameters are defined for convenience:

[DELTA]C = [C.sub.[alpha]] - [C.sub.[beta]]; [L.sub.rx] = [L.sub.xo]/[L.sub.t]; [L.sub.ri] = 2[L.sub.i]/[L.sub.t]; [L.sub.ry] = [L.sub.yo]/[L.sub.t]; [L.sub.x/y] = [L.sub.xo]/[L.sub.yo]; [N.sup.*.sub.[alpha]] = [N.sub.[alpha]]/[[DELTA]C[D.sub.y]/[L.sub.t]]; [D.sub.i/y] = [D.sub.i]/[D.sub.y]; [D.sub.x/y] = [D.sub.x]/[D.sub.y] (9)

With the aid of these parameters Eq 8 can be expressed as Eq 10:

[N.sup.*.sub.[alpha]] = 1 / [L.sub.ry] + [L.sub.rx]/[D.sub.x/y] + [1/[D.sub.i/y] - 1/2 - 1/2[D.sub.x/y]](n - 1)[L.sub.ri] (10)

Whether the flux increases or decreases with respect to the total number of layers, n, depends on the value inside the square brackets. For the flux to decrease as n increases, with the other parameters remaining constant, 1/[D.sub.i/y] - 1/2 - 1/2[D.sub.x/y] > 0. The conditions below can thereby be determined:

[D.sub.i/y] < 2/1 + 1/[D.sub.x/y] (11 a)

[D.sub.i] < 2/1/[D.sub.x] + 1/[D.sub.y] (11 b)

These define the eritical [D.sub.i/y] (or [D.sub.i]) at which [N.sup.*.sub.[alpha]] transitions from an increasing to a decreasing function of n. The physical significance of Eq 11a and Eq 11b will be discussed later when the model results are presented.

Case II: [phi]-dependent [D.sub.i]

In an interdiffusion region, two polymers intermingle across an interface, as shown in Fig. 2. Therefore, the diffusivity of the penetrant in this region is likely to be intermediate between those of the pure components. This hypothesis is supported by a study done by George et al. on the transport of aromatic solvents in isotactic-polypropylene (PP)/acrylonitrile-co-butadiene ruber (NBR) blends (12). They showed that in creasing the content of NBR from 0 from that is pure PP to one in pure NBR. As a first approximation, the rule of mixtures is used to represent [D.sub.i], where [[phi].sub.x] and [[phi].sub.y] are the volume fractions of polymers x and y.

[D.sub.i] = [[phi].sub.x][D.sub.x] + [[phi].sub.y][D.sub.y] (12)

Many studies have shown that the polymer concentration varies across the interdiffusion region (9, 13, 14). Depending on the polymer pair, the polymer-concentration profiles can be balanced or unbalanced about the interface. Figure 3 shows a comparison of three types of concentration profiles for polymer y across the interdiffusion region, viz, balanced - linear, balanced - non-linear, and unbalanced. A balanced profile is obtained when the adjacent polymers interpenetrate equally, while an unbalanced profile is obtained when one polymer interdiffuses to a greater extent than the other does. For this study, a balanced profile is assumed. For simplicity, the polymer concentrations [[phi].sub.x] and [[phi].sub.y] vary linearly with z. Thus, for the first interdiffusion region,

[[phi].sub.x] = [z.sub.1] + [L.sub.i] - z / 2[L.sub.t] [z.sub.1] - [L.sub.i] < z < [z.sub.1] + [L.sub.i] (13)

Substituting the above expression in Eq 12, [D.sub.i] is obtained as a function of z. Solution to Eq 1a with a variable diffusivity in the interdiffusion regions is given as:

[C'.sub.r] [a'.sub.r] ln(z + [A.sub.r]) + [b'.sub.r] r = 1,2,..., n - 1 (14)

In Eq 14, [A.sub.r] = [L.sub.i] ([D.sub.r+1] + [D.sub.r]) / ([D.sub.r+1] - [D.sub.r]) - [z.sub.r]

such that, [D.sub.j] = [D.sub.x ] ([for all] j [equivalent to] odd) and [D.sub.j] = [D.sub.y] ([for all] j [equivalent to] even) where j = 1, 2, ..., n-1.

The boundary conditions remain the same as those for Case I (Eqs 3a, 3b and 6). Solving for the constants, the concentration profile for the nth layer in an n-layer film is given by Eq 15:

[C.sub.n] - [C.sub.[beta]] / [DELTA]C = - 1/[A.sup.*.sub.n] [D.sub.y]/[D.sub.n] (z - [z.sub.n]);

where [A.sup.*.sub.n] = [D.sub.y]/[D.sub.x] (2[L.sub.x0]/n - [L.sub.i]) + (2[L.sub.y0]/n - [L.sub.i]) + 2(n - 1)[L.sub.i] / ([D.sub.x]/[D.sub.y] - 1) ln ([D.sub.x]/[D.sub.y]) + (n - 2) ([L.sub.y0]/n - [L.sub.i]) + ( n - 2) [D.sub.y]/[D.sub.x] ([L.sub.x0]/n - [L.sub.i]) (15)

The molar flux [N.sup.*.sub.a], is determined from the above concentration profile, Eq 16, and Eq 9:

[N.sub.a] = - [D.sub.n] d[C.sub.n]/dZ = [D.sub.y]/[A.sup.*.sub.n] [DELTA]C (16)

[N.sup.*.sub.a] = 1 / [1/[D.sub.x/y] (2[L.sub.rx]/n - [L.sub.ri]/2) + (2[L.sub.ry]/n - [L.sub.ri]/2) + (n-1)[L.sub.ri]/([D.sub.x/y] - 1) ln ([D.sub.x/y]) + (n - 2) ([L.sub.ry]/n - [L.sub.ri]/2) + (n - 2) 1/[D.sub.x/y] ([L.sub.rx]/n - [L.sub.ri]/2)] (17)

Note that the above expression was derived for a multilayer film (xyxy...) assuming an even number of layers in the film with same alternate layer thickness.

RESULTS AND DISCUSSION

The objective of this study was to predict the number of layers needed to cause a certain change in penetrant flux in a multilayer film and also to study the effects of various parameters on the number of layers required for a desired steady state flux. Two cases were considered for the diffusivity in the interdiffusion region, [D.sub.i], and are discussed below.

Case I: Constant [D.sub.i]

Equation 10 was obtained for the dimensionless flux when [D.sub.i] was assumed to be constant. Figures 6 a-c show dimensionless flux, [N.sup.*.sub.a], vs. number of layers, n, for different values of the diffusivity ratio [D.sub.i/y]. The solid curves represent model results for [D.sub.x/y] = 10 and the dashed curves for [D.sub.x/y] = 100. These values are typical of gas-diffusivity ratios for various polymers. For example. Compan et al. report [O.sub.2] and [CO.sub.2] diffusivities in LLDPE to be ~2 X [10.sub.-7] at 30[degrees]C (15). Diffusivity values of [O.sub.2] in polypropylene have been found to be the same order of magnitude (16). On the other hand, the diffusivity of [O.sub.2] in a polyamide is about 4 X [10.sub.-9] at 29[degrees]C (16). Thus, if LLDPE is assigned as polymer x and polyamide as polymer y, [D.sub.x]/[D.sub.y] ~ 50. The interdiffusion thickness ratio, [L.sub.ri], was assumed equal to 0.0001, which was determined assuming 2[L.sub.i] = 5 nm (17) and [L.sub.t] = 50 [mu]m. Last, the polymer contents in the film were assigned as [L.sub.x0]/[L.sub.y0] = 0.333, 1, and 3 in Figs. 6a, b, and c, respectively. Note that the y-axis scales are different and, depending on the value of [D.sub.i/y], the flux either increases above or decreases below its initial value at n = 2 (corresponding to a bilayer).

For [D.sub.i/y] values below the critical value defined in Eq 11, the flux decreases with increasing number of layers (e.g., the critical value of is 1.82 for [D.sub.x/y] = 10). As shown in Fig. 6a, the flux decreases with increasing n for [D.sub.i/y] values of 1 and 0.1. Also, the further the value is below critical [D.sub.i/y], fewer layers are required to significantly reduce the flux. However, for a [D.sub.i/y] = 0.1, the diffusivity in the interdiffusion region would be less than both [D.sub.y] ([D.sup.i] = 0.1 [D.sub.y]) and [D.sub.x] ([D.sub.i] = 0.01[D.sub.x]) for the given value of [D.sub.x/y] (e.g., 10). The case of [D.sub.i/y] = 0.1 is unrealistic as [D.sub.i] is expected to be at least equal to if not greater than the lesser of [D.sub.x] and [D.sub.y] ([D.sub.y] in this case). Thus, for [D.sub.i/y] values greater than 1 and less than 1.82, the flux decreases with increasing n. For values of [D.sub.i/y] above 1.82, the flux increases with increasing number of layers, as is the case for [D.sub.i/y] = 10 in Fig. 6. In Fig. 6, it is also evident that the flux is greater when = 100 (dashed curves) compared to = 10 (solid curves), as expected. Moreover, as the content of polymer x increases (i.e., [L.sub.x/y] increases), the flux increases at a given n. This behavior is expected because the results are based on cases where [D.sub.x] > [D.sub.y].

Figures 7 and 8 show the effect of increasing thickness ratios on the number of layers required for a desired flux reduction. In particular, Fig. 7 shows the minimum number of layers required to reduce the dimensionless flux, [N.sub.a.sup.*], to an arbitrarily chosen value of 1 as the content of x increases, assuming [D.sub.i/y] = 1. [D.sub.x/y] = 10, and maintaining the interdiffusion thickness ratio, [L.sub.ri] at 0.0001. The dimensionless flux is 1.82 for a bilayer film with the above parameters at [L.sub.x/y] = 1. Since polymer x is a higher diffusivity material, increasing the amount of x increases the flux with increasing For example, the flux increases for a bilayer film from 1.82 to 5.5 when [L.sub.x/y] increases from 1 to 10. Hence, the number of layers required to reduce the flux to a certain value increases as the content of polymer x increases, as shown in Fig. 7.

The effect of increasing interdiffusion thickness ratio on the number of layers required for a desired steady state flux is shown in Fig. 8. Results correspond to the number of layers required to achieve [N.sub.a.sup.*] = 1 for increasing interdiffusion thickness ratio. Three curves corresponding to different polymer content ratios of [L.sub.x/y] = 0.333. 1 and 3 are shown. All other parameters remain the same as in the case above. Increasing the interdiffusion region thickness decreases the number of layers required to obtain [N.sub.a.sup.*] = 1. This result is expected for [D.sub.i]/[D.sub.y] = 1 because [D.sub.y] is lower than [D.sub.x] and adding thicker interdiffusion regions with [D.sub.i] = [D.sub.y] yields a film with more low-diffusivity material, thereby reducing the overall flux. Also, at a given interdiffusion thickness ratio the number of layers required increases as the polymer content of x increases from 0.333 to 3 (also shown in Fig. 7 for [L.sub.ri] = 0.0001). Moreover, beyond [L.sub.ri] ~ 0. 001, the number of layers required for a desired steady state flux is less sensitive to an increase in the interdiffusion thickness ratio. Assuming a constant [L.sub.i] higher values of [L.sub.ri] correspond to thinner films (smaller [L.sub.t]). Therefore, the interdiffusion regions constitute a larger fraction of the overall thickness and fewer layers are required to achieve the desired [N.sub.a.sup.*]

Case II: [phi]-dependent [D.sub.i]

Equation 17 was derived for the dimensionless flux, [N.sub.a.sup.*], when [D.sub.i] was assumed to be given by a mixture rule depending on the polymer volume fractions. Fig. 9 illustrates the effect of increasing the number of layers on the dimensionless flux, [N.sub.a.sup.*] relative to that of a bilayer film, [N.sub.a2.sup.*], for the case of [phi]-dependent diffusivity ([N.sub.a.sup.*] was normalized by [N.sub.a2.sup.*] because a bilayer is the simplest multilayer film). As observed, the flux increases with increasing n and the trend is true for each value of shown. Since n appears in many places in the denominator of Eq 17, it is difficult to identify the trend in flux with increasing n at first glance. However, by calculating an average [D.sub.i] and applying the criterion given in Eq 11, one can predict whether the flux would decrease or increase with increasing n. For the assumed linear polymer-concentration profiles that are balanced in the interdiffusion region,

[D.sub.i] = 1/2[L.sub.t] [[integral].sup.[z.sub.1] + [L.sub.i].sub.[z.sub.1] - [L.sub.t]] D(z)dz = [D.sub.x] + [D.sub.y]/2 > 2/1/[D.sub.x] + 1/[D.sub.y]

or

[D.sub.i/y] = 1 + [D.sub.x/y]/2 > 2/1 + 1/[D.sub.x/y] (18)

As indicated by Eq 18. the flux would always increase with increasing number of layers, and Fig. 9 supports this prediction (e.g., the average [D.sub.i/y] value for the case of [D.sub.x/y] = 10 is 5.5, which is greater than the critical [D.sub.i/y] value of 1.82). The inequality in Eq 18 is true for any balanced profile with [D.sub.i] given by Eq 12, because the average would always be an algebraic mean of [D.sub.x] and [D.sub.y] as shown above.

CONCLUSIONS

A model was developed to predict the steady-state flux of a species through a multilayer film consisting of alternating polymers. The major objective of this study was to investigate the effect of interdiffusion regions on the permeation of species through multilayer films. The model incorporated species diffusion through interdiffusion regions at each polymer-polymer interface. It was assumed that each interdiffusion region was balanced, that is, the adjacent polymers interpenetrated one another equally. The species diffusivity was assumed to be constant in each polymer and two cases were considered for the diffusivity in the interdiffusion regions. In case one, the diffusivity was constant and a criterion was defined to predict whether the flux would decrease as the number of layers increased. Results showed that thousands of layers were required to achieve appreciable flux reduction. The relative amounts of the two polymers also greatly influenced the flux, with the incorporation of more low-diffusivity po lymer yielding a lower overall flux. The effect of increasing interdiffusion thickness on the number of layers required for a desired steady state flux was also studied. Introducing thicker interdiffusion regions with diffusivity ([D.sub.i]) reduced the overall flux and the number of layers required to achieve a given flux, as long as [D.sub.i] was less than a critical diffusivity defined by Eq 11. For case two, the species diffusivity was assumed to depend linearly on the polymer volume fraction in the interdiffusion region. In all instances where a balanced interdiffusion region was considered, the flux increased with the number of layers.

ACKNOWLEDGMENTS

This work was supported in part by the ERC Program of the National Science Foundation under Award Number EEC-9731680. The authors also gratefully acknowledge financial support from ILC Dover, Inc.

Department of Chemical Engineering (*)

Department of Mechanical Engineering (+), and Center for Advanced Engineering Fibers and Films

Clemson University

Clemson, SC 29634

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[Figure 3 omitted]

[Figure 6 omitted]

[Figure 7 omitted]

[Figure 8 omitted]

[Figure 9 omitted]

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Author: | Sankhe, Shilpa Y.; Hirt, Douglas E.; Zumbrunnen, David A. |
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Publication: | Polymer Engineering and Science |

Article Type: | Statistical Data Included |

Geographic Code: | 1USA |

Date: | Dec 1, 2001 |

Words: | 4766 |

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