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.
`
`Speck-1009
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`162
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`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
`
`contain impermeable
`Four models of barrier membrane, shown in Fig.
`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.
`
`slits
`
`_;;_
`
`a
`
`C.
`
`D.
`
`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.
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`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 ] :
`
`=D(2dW)
`
`J
`
`0
`
`1
`
`AC
`
`(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
`1
`
`DLIC
`-=-
`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]:
`
`1
`-+bln
`2dW dW (3)
`
`DAC
`-=
`J,
`The first term on the
`is the flux through a unit cell of area
`in which
`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
`it will not dramatically alter the results for a
`is exact only when 1 + d % s,
`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-
`
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`J1
`2dW.
`

`
`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
`JN-
`
`1
`--+bln
`2dW
`
`dW
`
`d +&+i
`2s
`0
`
`(N-I)&
`
`(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
`s(u+b)+b(a+b)
`
`d2
`
`(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
`
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`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
`
`Second, we consider the case where a/a! % 1. Now, eqn. (6) becomes
`
`(7)
`
`(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 $, the flake aspect ratio CX, and the slit aspect
`ratio 0.
`We turn next to the pore model shown in Fig. 1 (c). This model has the same,
`multilayered structure as the slit model studied above. However, the gaps be-
`tween layers are no longer wide thin slits, but regularly spaced pores. Diffusion
`from the pores in one layer to those in the next is a multidimensional process,
`a significant change from the previous case.
`As before, we can write the resistance across a membrane containing no
`flakes in terms of the flux JO:
`
`DAC 1 Jo =4d2
`This resistance is a close parallel to eqn.
`is now replaced by a unit cell of area
`the resistance of a composite of N layers as
`
`(9)
`
`except that the unit cell of area
`In a similar way, we can write
`
`(N-l)
`DAC 1 ln(dl,.b) -=$+;+s+
`JN
`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
`
`nb
`
`(10)
`
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`(2))
`2dW
`4d2.
`

`
`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-
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`
`167
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`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
`is within ten percent of the elec-
`factor [ 11. As a result, the quantity D&/J
`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
`-=-
`J3
`
`a +dl +d,
`-
`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.
`
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`
`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 $“/ (1 - @) (cf.
`eqns. 7, 11, 12). Second, if diffusion through gaps between flakes is slowest,
`then (Jo/J,-1)
`varies with @ (cf. eqn. 8 or 11).
`Data for flake-filled membranes are consistent with the prediction that wig-
`gles are paramount, as shown in Fig. 3 [ g-111. The diffusing solute is oxygen,
`water, or carbon dioxide. The flakes are of mica or polyamide, and the polymer
`continuum is of polycarbonate, polyester, or polyethylene. We can use plots
`like this to determine an effective aspect ratio if we are willing to assume a
`more specific model. Values inferred from these models are shown in Table 1.
`We can find d/a from eqn. (7 ) without assumption. We can find d/a from eqn.
`(11) only by two additional assumptions. First, we take the limit of eqn. (11)
`in which the wiggles dominate; i.e., we neglect the second term of the right-
`hand side of this relation. Second, we assume that 0 equals one. This second
`assumption reflects two-dimensional vs. three-dimensional transport: when
`the wiggles dominate,
`is independent of s in slits but not in pores.
`While the flux varies with
`@) /c$” 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-
`
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`J,,/J,
`(1 -
`

`
`169
`
`1 ), is proportional to
`Fig. 3. The flux across flake-filled membranes. The flux, plotted as (Jo/J,-
`$‘/ ( 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.
`
`Loading
`
`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)
`Mica flakes in phenolic polyester
`(PflelJP=3.6/1.35)
`Polyamide flakes in polyethylene
`Polyamide lamellae in polyethylene
`
`11
`
`9
`10
`12
`
`30
`
`30
`4
`330”
`
`18
`
`18
`4
`-
`
`“The aspect ratio
`
`estimated from eqn. (8).
`
`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
`@). However,
`is proportional to the loading $, rather than to @ “/ ( 1 -
`this figure does not prove that pores dominate the diffusion, for there are other
`
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`
`(d/s),
`Jo/J,
`

`
`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
`
`-2
`
`I
`-1
`
`I
`0
`
`Log W,+)
`
`1
`1
`
`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.
`
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`
`

`
`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
`$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.
`(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
`
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`1
`

`
`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
`
`(grants
`This work was supported by the National Science Foundation
`8408999 and 8611646), by the General Electric Corporation, and by Questar,
`Inc.
`
`List of symbols
`
`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
`
`N
`
`PFZ
`s
`
`Speck-1009
`
`W
`

`
`flake aspect ratio, d/a
`geometric factor
`pore aspect ratio,
`volume fraction flakes in membrane ( = a/ (a + b) ) .
`
`173
`
`References
`
`E.L. Cussler, Diffusion, Cambridge University Press, Cambridge, 1984.
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`1979, pp. 6-36.
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`
`8
`
`9
`
`10
`
`11
`12
`
`13
`
`14
`
`15
`
`16
`
`17
`
`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.
`
`Speck-1009
`
`s/a
`

`
`174
`
`layers, and hit a
`Diffusion across this membrane will miss a flake in (N-n)
`flake in
`layers. As a result, this process can take place via (N+ 1) parallel
`modes of probabilityp,,n = 0 ,...,N. The probabilityp, of hitting
`flakes is [ 171:
`
`pn=
`
`ff @“(l--$)N-”
`0
`
`(A-1)
`
`where 4 is the loading in the membrane.
`The path for diffusion is (a + b) for each layer where a flake is missed. The
`path for each layer where a flake is hit is more difficult: it is increased by
`where ,u is a geometric factor. However, the area available for this transport is
`proportional to Wb rather than
`Thus the effective path for each layer
`where a flake is hit is (a + b
`‘/b) . The total path through a membrane with
`hits is thus [N(a+b)
`across an area
`of a membrane like this is
`The average flux
`
`1
`
`JN=
`
`IZ=O
`Equations
`
`in
`
`(A-l ) and (A-2 ) can be combined and rearranged to give
`
`C-4-2)
`
`(A-3)
`
`is the total thickness of the membrane. Because N is
`in which
`large, this binomial distribution of probabilities
`is close to the Gaussian; if N
`is very large, the only significant probability is the mean ri. Thus
`
`Jd
`DAC pd2 ii -==+ (a+b)bN
`@2
`I-$
`This is identical to eqn. (12) in the text.
`
`=1+@!2-
`
`(A-4)
`
`Speck-1009
`
`n
`n
`,ud,
`Wd.
`+pd
`n
`+npd2/b].
`JN
`d 2
`$
`d 2DAC
`N(a+b) +npd2/b
`1( = N(a+ b) )