Barrier Membranes

Journal of Membrane Science, 38 (1988) 161-174 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

161

E.L. CUSSLER, STEPHANIE E. HUGHES, WILLIAM J. WARD, III3 and RUTHERFORD ARIS

Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 (U.S.A.) 2Department of Chemical Engineering, Stanford University, Palo Alto, CA (U.S.A.) General Electric Company, Research and Development Center, Schenectady, NY 12301 (U.S.A.)

(Received July 31,1987; accepted in revised form December 9,1987)

Summary

Membranes which contain impermeable flakes or lamellae can show permeabilities much lower than conventional membranes, and hence can serve as barriers for oxygen, water and other solutes. This paper develops and verifies theories predicting the properties of these barrier membranes. In particular, the theories predict the variation of permeability with the concentration and the aspect ratio of the flakes.

Introduction

Most membrane research concentrates on separations. This research has been successful: membranes for purifying water are now big business, and membranes for gas separations are expanding rapidly.

Another area of membrane research uses membranes as barriers. For ex- ample, polymer film is used for food wrapping, and paint is common for metal protection. Still, many polymer barriers are less effective than desired.

This paper will discuss an alternative type of barrier membrane. The mem- brane consists of a thin polymer film filled with flakes aligned with the plane of the film. Micrographs of such a film would look like a bed of wet leaves imbedded in a continuous polymer phase. Material diffuses through the poly- mer but only by a very tortuous path around the impermeable flakes. The result is a membrane whose permeability can be orders of magnitude less than that through the polymer alone.

This paper will discuss both models and experiments for this type of mem- brane. The models, which are highly simplified, lead to two main predictions. First, they predict the variation of membrane permeability with the volume fraction of flakes. Interestingly, this prediction is almost independent of the detailed geometries that are assumed. It is verified experimentally.

0376-7388/88/$03.50 0 1988 Elsevier Science Publishers B.V.

162

The second prediction is of the variation of permeability with the aspect ratio of the flakes. Here the models are less definitive and different geometries lead to different predictions. The differences between these predictions can be used to characterize the available experiments. The results help us to think more clearly about this type of barrier membrane.

Theory

Four models of barrier membrane, shown in Fig. 1, contain impermeable flakes aligned with the plane of the membrane. The four models differ in the geometry assumed for the flakes. The most realistic model, shown in Fig. 1 (a), has flakes which are randomly shaped and randomly distributed throughout the plane of the film. The impermeable flakes impede solute transport across the film by creating a tortuous path for diffusion. Clearly, this model is too complex for simple analysis.

To make this simple analysis, we idealize the model in Fig. 1 (a) in two ways. First, we assume that the flakes are not randomly located in the film, but rather occur periodically in a discrete number of planes within the film. We will make this assumption in all three models detailed here.

Second, we assume a particular shape and spacing for the flakes. We assume three such geometries. Most simply, we assume that the flakes are rectangles

A.

B.

C.

D.

slits _;;_

a

random flakes

Fig. 1. Models of barrier membranes. The first drawing is a sketch of the actual membrane. In the second and third drawings, diffusion occurs through regularly spaced slits or pores. In the last, it occurs through randomly spaced slits.

163

of uniform size but great width, regularly spaced like the bricks in a wall. In such an idealization, shown in Fig. 1 (b ) , diffusion will occur through the slits between the bricks. Alternatively, we can assume that each layer of flakes is a single flake perforated with regularly spaced pores. In this extreme idealiza- tion, shown in Fig. 1 (c), diffusion takes place through pores rather than slits. Finally and more realistically, we can assume that the flakes are randomly sized rectangles randomly located in the discrete planes. We will discuss all three geometries in the paragraphs which follow.

We begin the discussion for the slit model in Fig. 1 (b) by considering a unit cell of area (2dW). The total flux J,, through this unit cell when no flakes are present is [ 1 ] :

J 0

=D(2dW) AC

1 (1)

where 1 is the total thickness of the membrane and dC is the concentration difference across it. For later convenience, we rearrange this result as a resis- tance across the membrane

DLIC 1 -=- Jo 2dW

(2)

This resistance is proportional to the membrane thickness and inversely pro- portional to the area through which diffusion occurs. It forms a reference for later discussion.

Next we turn to the case of a membrane with one barrier. Now, diffusing solute can not pass through the membrane without necking down to pass through one of the periodic slits. The resistance in this case is approximately given by [2]:

DAC 1 -= J,

-+bln 2dW dW

(3)

in which J1 is the flux through a unit cell of area 2dW. The first term on the right-hand side of eqn. (3 ) is the resistance without flakes, just as in eqn. (2 ). The second term represents the constriction into and out of the slit; and the third term is the resistance of the slit itself. This result is approximate because we are essentially counting part of the resistance to diffusion across the mem- brane twice, once in the first term and once in the third. Still, while this result is exact only when 1 + d % s, it will not dramatically alter the results for a membrane with many layers. Exact results when a=0 can be obtained from the solution by elliptic functions [ 31.

The resistance of such a multilayer membrane can be found by extending these results. In such a membrane, solute must diffuse to the slit in the first layer and diffuse through this first slit. It then diffuses to one of the two sym-

164

metrically placed slits in the second layer of flakes. (One may show that small displacements from symmetry have no effect.) The distance to each of these next slits is d; the area through which this diffusion occurs is that between the flakes b IV. Thus the resistance for diffusion across a membrane with N flakes is

DAC 1 --+bln d +&+i (N-I)&

JN- 0 2dW dW 2s (4)

As before, the first term on the right-hand side is the resistance of the layer without flakes, and the second term is the resistance of the constriction into the first layer of flakes and out of the last layer of flakes. These terms are the same as those in eqn. (3) because there is no additional constriction; once in the membrane, diffusion must follow its narrow, tortuous path. The third term is the resistance of the N slits through which solute must pass to cross the membrane. This term is just N times the final term in eqn. (3 ), for now we have N layers instead of just one layer. The fourth term on the right-hand side of eqn. (4) reflects the tortuosity: (N- 1) wiggles each d long. The factor of f in front of this term represents the reduced resistance due to the periodic array of flakes: solute can diffuse into each slit either from the left or from the right. Equation (4) is more useful if it is divided by eqn. (2 ) and rearranged:

(5)

This is the key result for the first model in Fig. 1 (b) . We now consider the limit of a membrane containing many layers, each of

which contains flakes which block diffusion across much of the layer. If there are many layers, N, (N- 1) and (N+ 1) are virtually the same. As a result, the total membrane thickness 1 equals N(a+ b). If each layer is almost filled with impermeable flakes, then the volume fraction of flakes, $, equals a/ (a + b). This volume fraction is called the loading. We also define two new variables. The flake aspect ratio CI! ( =d/a) is a measure of the flake shape. The pore aspect ratio CJ (=s/a) characterizes the pore shape. With these changes, eqn. (5 ) becomes

da d2

s(u+b)+b(a+b) (6)

The second term on the right-hand side of eqn. (5 ) has dropped out because it is proportional to N -l.

Equation (6) shows some features in common with a sound, earlier theory

165

published by Brydges et al. [4] for glass-ribbon reinforced composites. This theory is developed in terms of fluid flow, but the basic approach is similar. Both eqn. (6) and the Brydges theory contain a value of one as the lead term on the right-hand side. Both include the same variation with aspect ratio in the second term, but the Brydges theory contains no variation with loading @ Both theories predict the same variation with loading in the third term, but the Brydges theory contains a geometrical factor which one may show always equals four.

We are especially interested in two cases of eqn. (6) which can be checked experimentally. First, we consider the case where u/a! 4 1. In this case, the wiggles within the films are dominant, and eqn. (6) becomes

(7)

Second, we consider the case where a/a! % 1. Now, eqn. (6) becomes

(8)

Diffusion is limited not by the wiggles but by the slits themselves. Equations (7) and (8) are the desired results for the two-dimensional model

shown in Fig. 1 (b). They give the change in the flux caused by the flakes in the membrane. This change is a function of three variables which can be al- tered experimentally, the loading $, the flake aspect ratio CX, and the slit aspect ratio 0.

We turn next to the pore model shown in Fig. 1 (c). This model has the same, multilayered structure as the slit model studied above. However, the gaps be- tween layers are no longer wide thin slits, but regularly spaced pores. Diffusion from the pores in one layer to those in the next is a multidimensional process, a significant change from the previous case.

As before, we can write the resistance across a membrane containing no flakes in terms of the flux JO:

DAC 1

Jo =4d2 (9)

This resistance is a close parallel to eqn. (2)) except that the unit cell of area 2dW is now replaced by a unit cell of area 4d2. In a similar way, we can write the resistance of a composite of N layers as

DAC 1 ln(dl,.b) -=$+;+s+ (N-l) nb JN

(10)

which is a parallel of eqn. (4). The first term on the right-hand side is the resistance without flakes. The second is the resistance due to the constriction

166

into the top layer of holes, and out of the bottom layer of holes [ 51. The third of these terms is the resistance of the N holes - each a long and of area 7~s - through which solute must diffuse in transversing the membrane [ 11. The fourth term on the right-hand side of eqn. (10) represents the (N - 1) wiggles which the solute makes [6]. Strictly speaking, the logarithm in this term should be the inverse hyperbolic cosine, but this function is almost identical to the logarithm when d/s $1, as is true here.

As before, we are most interested in the limit of many layers. When we divide eqn. (10) by eqn. (9 ) , we find for multilayered limit:

(11)

in which (x, o, and $ are the flake aspect ratio (d/a), the pore aspect ratio (s/ a), and the loading a/ (a + b), analogous to eqn. (6). The first term on the right hand side is the resistance of flake free membrane, the second team is the resistance of the pores, and the third term is the effect of tortuosity. The effect of the constriction has dropped out in the limit of many layers, just as it dropped out of eqns. (7) and (8).

Finally, we turn to the third model suggested in Fig. 1 (d), that of random flakes. The development of this model, detailed in the Appendix, is a general- ization of the first case discussed. The key result for a many-layered membrane is

(12)

This close parallel to eqn. (7) differs only by the unknown factor p, a combined geometric factor characteristic of the random porous media.

Equations (7)) (8)) (11)) and (12) are the core of the analysis developed in this paper. Each predicts the change in membrane permeability as a function of the loading and the aspect ratio. We test the predicted variation with loading using data in the literature. We test the predicted variation with the aspect ratio by means of the experiments described next.

Experimental

We have no literature data to test predicted variations of flux with the aspect ratio. To make diffusion experiments with variable aspect ratios would be both difficult and tedious. As an alternative, we chose to study electrical conduct- ance of aqueous solutions of reagent grade potassium chloride (Aldrich), using a modified Jones bridge [ 71. In these solutions, the equivalent ionic conduct-

167

antes of K+ and Cl- are almost equal, so the equivalent conductance of this salt is almost equal to the diffusion coefficient times a dimensional conversion factor [ 11. As a result, the quantity D&/J is within ten percent of the elec- trical resistance divided by the resistivity. Resistivities for these solutions are known [8].

We used the two poly (methyl methacrylate) devices shown in Fig. 2 to per- form these conductance experiments. Each consists of three flat plates. In the first, the two outer plates, which are 1.2 cm thick, are pierced by platinum wires 0.051 cm in diameter bent to give the equivalent of a slit source 2.5 cm long. The third, central plate, which is 1.2 cm thick, contains an open slit which is 0.16 cm wide and 2.5 cm long. The plates are separated by spacers 0.051 cm thick. The steady-state resistance from one wire to the other in this device is given by

DAC a +dl +d, -=- - J3 2sW Wb

(13)

Note that d, and dz are not necessarily equal to each other or to the flake size 2d used above. The first term on the right-hand side of eqn. (13) is the resis- tance of the slit, and the second that of the two wiggles. This result parallels eqn. (4) for the special case in Fig. 2. The second device in Fig. 2 is similar, except that there is a pore instead of a slit. The outer two discs, each 1.2 cm thick, are pierced by platinum wires 0.080 cm in diameter to give the equivalent of a point source. The central plate, 1.2 cm thick, has a single 0.16 cm diameter

Fig. 2. Apparatus to study aspect ratio. Each device consists of three plates whose alignments are easily altered. The electrical resistance of salt solutions in these devices is used to mimic the tortuosity in the barrier membranes.

168

pore; the three plates are separated by spacers 0.025 cm thick. The resistance in this second device is:

DAC a -=---+

ln(d,d2/2s2)

J3 m2 nb (14)

The first term on the right-hand side is now the resistance of the pores, and the second that of the two wiggles. Experiments with both devices are reported in the section which follows.

Results

The theory developed above predicts the variation of flux across a flake- filled polymer membrane with the loading and the aspect ratio of the flakes. In this section, we test the predicted variation with loading first. Literature data basic to this discussion are available for two types of membranes, those with many small flakes and those with a few large lamellae. Each type of mem- brane merits separate consideration.

Membranes with many small flakes are made by adding an inorganic mate- rial such as mica to a polymer melt or solution. Films are formed from this mixture either by extrusion or by casting. Microscopic cross-sections of these films show highly anisotropic flakes pressed together like a pile of wet leaves but immersed in a polymer continuum.

The theories of barrier membranes presented above predict that the diffu- sion flux can vary with flake loading @ in two ways. First, if wiggles through the flakes are paramount, then (Jo/J,- 1) is proportional to $/ (1 - @) (cf. eqns. 7, 11, 12). Second, if diffusion through gaps between flakes is slowest, then (Jo/J,-1) varies with @ (cf. eqn. 8 or 11).

Data for flake-filled membranes are consistent with the prediction that wig- gles are paramount, as shown in Fig. 3 [ g-111. The diffusing solute is oxygen, water, or carbon dioxide. The flakes are of mica or polyamide, and the polymer continuum is of polycarbonate, polyester, or polyethylene. We can use plots like this to determine an effective aspect ratio if we are willing to assume a more specific model. Values inferred from these models are shown in Table 1. We can find d/a from eqn. (7 ) without assumption. We can find d/a from eqn. (11) only by two additional assumptions. First, we take the limit of eqn. (11) in which the wiggles dominate; i.e., we neglect the second term of the right- hand side of this relation. Second, we assume that 0 equals one. This second assumption reflects two-dimensional vs. three-dimensional transport: when the wiggles dominate, J,,/J, is independent of s in slits but not in pores.

While the flux varies with (1 - @) /c$ for membranes fitted with many layers of flakes, it does not do so for membranes containing a few layers of poorly permeable lamellae. It may be that the membranes made by Subramanian [ 121 are of this type. These membranes are made by incompletely mixing polyam-

169

Loading

Fig. 3. The flux across flake-filled membranes. The flux, plotted as (Jo/J,- 1 ), is proportional to $/ ( 1 - $), where $ is the flake loading. Note the flakes can reduce the flux hundreds of times. o, Ward [ll]; q ,Kamaletal. [lo]; A,Okuda [9].

Fig. 4. Flux vs. loading in membranes with lamellae. The flux is inversely proportional to loading, as predicted by eqn. (8). These data are not linear when correlated in a fashion parallel to Fig. 3.

TABLE 1

Effective aspect ratios inferred from changes with loading

Membrane Ref. d/o (from eqn. 7)

dlo (from eqn. 11 assuming a = s )

Mica flakes in polycarbonate (~~,,/~=3.6/1.2) 11

Mica flakes in phenolic polyester (PflelJP=3.6/1.35) 9

Polyamide flakes in polyethylene 10 Polyamide lamellae in polyethylene 12

The aspect ratio (d/s), estimated from eqn. (8).

30 18

30 18 4 4

330 -

ide, which is relatively impermeable, in polyethylene, which is much more permeable. Micrographs of films made by this incomplete mixing look like thin sheets of marble cake. These membranes may have a permeability dominated by diffusion through a few slits or pores, and not by many tortuous wiggles in the flake-filled membrane. They should be described by relations like eqn. (81, rather than by eqn. (7).

That eqn. (8) is consistent with these data is shown in Fig. 4. The altered flux Jo/J, is proportional to the loading $, rather than to @ / ( 1 - @). However, this figure does not prove that pores dominate the diffusion, for there are other

170

physical models which also give a linear variation with @ For example, if we assume that there are no pores or slits, diffusion may still occur through alter- nating layers of the two materials. The diffusion flux across these alternating layers will be linear in the loading @.

Thus the literature data show a variation with loading consistent with the theory developed above. Unfortunately, these data do not include a systematic variation of aspect ratio which permits a similar test of this second important variable. To make this second test, we turn from eqns. (7)) (8)) and (11) to eqns. (13) and (14).

The predictions of eqns. (13) and (14) are justified by the results in Figs. 5 and 6. In Fig. 5, we show that the value of Jo/J,, plotted as the resistance divided by the resistivity, is proportional to dl + d,. In physical terms, this means that the flux wiggling through a slit is proportional to the length of the wiggles. Note that the data in this figure do include a five-fold variation of KC1 con- centration. Moreover, the slope of these data is 4.0 cme2, close to the value of 5.0 cme2 predicted for these solutions [S].

In Fig. 6, we show that the flux oozing through a pore is proportional to the logarithm of the product of the distances to reach the pore, consistent with eqn. (14). However, repeated experiments at different concentrations show scatter. While we are unsure why this scatter occurs, we suspect that it is due to slight changes in the separation between the plates. Still, the data clearly show Jo/J,, plotted as the resistance divided by the resistivity, is linear in In

500 r

dl+ d2 (cm)

200 I I 1 -2 -1 0 1

Log W,+)

Fig. 5. Flux vs. aspect ratio in a slit. The flux, as resistance divided by resistivity, is proportional to the total length of the wiggles, as predicted by eqn. (13).

Fig. 6. Flux vs. aspect ratio through a pore. The flux, as resistance divided by resistivity, varies inversely with the logarithm of the product of the distances to the slit, as predicted by eqn. (14). The circles, squares, and triangles refer to 0.01,0.05, and 0.10 M, respectively.

171

C&C&: the flux in pores varies with the logarithm of the product of distances traveled.

Discussion

At this point, we need to review what the theory and experiments have shown. Most importantly, the theory correctly predicts that for flake-filled mem- branes, the diffusion flux varies with (1 - $J) /#. This correct prediction occurs in spite of the major approximations in idealizing the geometry of these mem- branes. In the actual membrane - cf. Fig. 1 (a) - the flakes are of random size, and randomly arrayed. In our models - cf. Figs. 1 (b)-1 (d) - the flakes are collapsed into discrete layers, which in turn are organized to give slits or pores.

In both the actual and the idealized membranes, the diffusion is retarded by the tortuous paths around the flakes. In the actual membrane, most diffusion will occur around the nearest boundary. The solute diffuses around this bound- ary and across the membrane until it meets the next random flake. In the idealizations, the diffusion is much more symmetric and periodic. Geometrical complexities have been avoided by approximations.

These approximations are probably the reason that the aspect ratios in- ferred from our models are so small. As shown in Table 1, these ratios range from 5 to 30; direct measurements of the flakes imply values around 100. There are several reasons why this might be so. One concerns the adhesion between the polymer and the flakes. If the adhesion where poor, then diffusing solute might move rapidly through the resulting gaps. Anecdotal reports of such rapid transport are frequent, although we have seen no published evidence.

An alternative reason for these inconsistencies in aspect ratio is the flake geometry itself. To see how this geometry might be responsible, consider dif- fusion around a single flake. Each point on the edge of the flake represents a possible pathway for diffusion. These pathways are in parallel, and the shortest pathways will be prefered. As a result, we suspect that the effective aspect ratio represents a type of harmonic average biassed towards the shortest paths. Such an average will be less than that measured geometrically.

Even with the successes of Figs. 3 and 4, we have not been able to use vari- ations of flux with loading to distinguish between the three models in Fig. 1. This is because all three models predict an identical variation of flux with tortuosity. On the one hand, this identity is frustrating, for we would like to know if diffusion in barrier membranes occurs through gaps which are like slits or like pores. On the other hand, the identity is reassuring, for it says that the flux will vary in the same way with loading independent of the detailed geometry.

The theory also predicts how the flux varies with aspect ratio, and the results in Figs. 5 and 6 support these predictions. This support is incomplete. We do find the variation of flux with aspect ratio expected for membranes containing many flakes which behave like slits or pores in series. We have actually made

172

experiments only for one pore or slit, and we know of no data systematically varying lamellae shape.

Our results represent an extension of earlier studies of transport phenomena in composite media. These studies began with Maxwell [ 13 1, who considered the thermal conductivity of a periodic array of conducting spheres, and were improved by Rayleigh [ 141, who focussed on the transmission of sound. More recent studies have included diffusion in cylindrical arrays [ 151 and in random porous media [ 161. Our efforts seem the first for composites of flakes oriented across the path of diffusion.

The results in this paper raise other interesting questions. First, what would be the behavior of flake-filled membranes of other properties? For example, imagine the flakes were of zeolite, and hence selectively permeable to linear alkanes. Could such a membrane separate linear and branched alkanes? As a second example, imagine the flakes were of gas, large flat bubbles in a polymer continuum. Would such a membrane be a good thermal insulator?

A second set of interesting questions comes from the fact that all the discus- sion in this paper is related to the steady state. Ironically, barrier membranes are sometimes used in unsteady situations: for wrapping food, for coating, or for protecting transistors. What is the unsteady diffusion into a barrier mem- brane as a function of loading and aspect ratio? We know no answers to these questions, but we anticipate their resolution.

Acknowledgements

This work was supported by the National Science Foundation (grants 8408999 and 8611646), by the General Electric Corporation, and by Questar, Inc.

List of symbols

N PFZ s W

flake thickness distance between flakes solute concentration half flake size wiggle lengths (eqns. 13 and 14) diffusion coefficient total flux across a membrane of n layers of flakes total membrane thickness number of flakes or lamellae probability of encountering n of N flakes in crossing a membrane half slit size or pore radius width

173

flake aspect ratio, d/a geometric factor pore aspect ratio, s/a volume fraction flakes in membrane ( = a/ (a + b) ) .

References

8

9

10

11 12

13

14

15

16

17

E.L. Cussler, Diffusion, Cambridge University Press, Cambridge, 1984. W.A. Wakeham and E.A. Mason, Diffusion through multiperforate laminae, Ind. Eng. Chem., 18 (1979)301-305. R. Aris, On the permeability of membranes with parallel but interconnected pathways, Math. Biosci., 77 (1985) 5-16. W.T. Brydges, S.T. Gulati and G. Baum, Permeability of glass ribbon-reinforced composites, J. Mater. Sci., 10 (1975) 2044-2049. H.C. Berg, Random Walks in Biology, Princeton University Press, Princeton, NJ, 1983. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd edn., Oxford University Press, Oxford, 1959. D.F. Evans and M.A. Matesich, Measurement and interpretation of electrolytic conductance, in: E. Yaeger and A.J. Salkind (Eds.), Techniques of Electrochemistry, Vol. 2, Wiley, New York, NY, 1972, pp. l-85. J.A. Dean (Ed.), Langes Handbook of Chemistry, 12th edn., McGraw-Hill, New York, NY, 1979, pp. 6-36. S. Okuda, On the performance of flake filled coatings for the permeation of water, in: G.D. Parfitt and A.V. Patsis (Eds.), Organic Coatings: Science and Technology, Vol. 7, Academic Press, New York, NY, 1982, pp. 285-298. M.R. Kamal, I.A. Jinnah and L.A. Utracki, Permeability of oxygen and water vapor through polyethylene/polyamide films, Polym. Eng. Sci., 24 (1984) 1337-1346. W.J. Ward, unpublished data, 1986. P.M. Subramanian, Permeability barriers by controlled morphology of polymer blends, Po- lym. Eng. Sci., 25 (1985) 483-487. J. Clerk Maxwell, A Treatise on Electricity and Magnetism, Vol. I, Clarendon Press, London, 1881, p. 440 et seq. J.W. Strut (Lord Reyleigh), On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag., 34 (series 5) (1892) 481-502. P. Stroeve and K. Eagle, Analysis of diffusion in a medium containing dispersed reactive cylinders, Chem. Eng. Commun., 3 (1979) 1899198. K.K. Mohanty, J.M. Ottino and H.T. Davis, Reaction and transport in disordered composite media, Chem. Eng. Sci., 37 (1982) 905-924. R. Aris, On a problem in hindered diffusion, Arch. Rat. Mech. Anal., 95 (1986) 83-91.

Appendix

In the above, we developed expressions for the flux through membranes con- taining periodically spaced slits, as suggested by Fig. 1 (a). We now want to extend these arguments to very thin randomly spaced flakes of the same size d, as suggested by Fig. 1 (c ) .

To do so, we assume a membrane containing N layers, where N is large.

174

Diffusion across this membrane will miss a flake in (N-n) layers, and hit a flake in n layers. As a result, this process can take place via (N+ 1) parallel modes of probabilityp,,n = 0 ,...,N. The probabilityp, of hitting n flakes is [ 171:

pn= ff @(l--$)N- 0

(A-1)

where 4 is the loading in the membrane. The path for diffusion is (a + b) for each layer where a flake is missed. The

path for each layer where a flake is hit is more difficult: it is increased by ,ud, where ,u is a geometric factor. However, the area available for this transport is proportional to Wb rather than Wd. Thus the effective path for each layer where a flake is hit is (a + b +pd /b) . The total path through a membrane with n hits is thus [N(a+b) +npd2/b].

The average flux JN across an area d 2 of a membrane like this is

JN= $ in d 2DAC IZ=O N(a+b) +npd2/b 1 C-4-2)

Equations (A-l ) and (A-2 ) can be combined and rearranged to give

(A-3)

in which 1( = N(a+ b) ) is the total thickness of the membrane. Because N is large, this binomial distribution of probabilities is close to the Gaussian; if N is very large, the only significant probability is the mean ri. Thus

DAC pd2 ii

-==+ (a+b)bN Jd (A-4)

@2 =1+@!2- I-$

This is identical to eqn. (12) in the text.