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Unsteady diffusion in barrier membranes
Journal of Membrane Science, 44 (1989) 305-311 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
305
Short Communication
UNSTEADY
DIFFUSION
IN BARRIER
MEMBRANES
DIANA PERRY’, WILLIAM J. WARD’ and E.L. CUSSLER’ ‘Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 (U.S.A.) 2General Electric Company, Research and Deoelopment Center, Schenectady, NY 12301 (U.S.A.) (Received May 13,1988; accepted in revised form September 26,1988)
Membranes containing mica flakes have a dramatically lower permeability than membranes without mica flakes [ 1,2]. For example, carbon dioxide diffuses across a polycarbonate membrane containing 30 vol.% mica about two hundred times more slowly than across a pure polycarbonate membrane of the same thickness. Such membranes, examples of a growing family of barrier films, may find applications in packaging. In a recent paper [ 21, we developed and verified a theory for the steady-state diffusion across such flake-filled membranes. The most important special case of the theory idealized the membrane as containing regularly spaced lamellae with the flakes arranged in each lamella much like bricks in a wall. Solute was assumed to diffuse in the gaps between the bricks, but could not diffuse through the bricks. The steady-state flux across such a lamellar membrane is predicted to vary with the inverse square of the aspect ratio of the flakes. It is also predicted to be proportional to [ (1- $)/@J’], where @ is the “loading”, i.e., the volume fraction of mica in the membrane. Both predictions are supported by experiment. In this note, we turn to unsteady diffusion across such a flake-filled membrane. We do so because membranes like these are so impermeable that they can take a considerable time to reach steady state. To study the unsteady diffusion, we use a diaphragm cell consisting of two well-stirred compartments separated by the membrane. Each compartment contains the same total pressure of carbon dioxide; one compartment initially contains some 14C0,. We measure the number of counts n in the second compartment as a function of time. At very small times, no carbon dioxide has penetrated through the membrane, and there are no counts above the background. At somewhat larger times, penetration begins, and the counts above the background in the second compartment become an elaborate function of time [3]. At still larger times, the
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers B.V.
306
counts follow the simpler relation
n A++,)
[ 4, p. 781:
(1)
-=
no
in which no is the initial number of counts above background in the first compartment, A is the membrane area, D is the diffusion coefficient in the polymer, H is the partition coefficient between polymer and gas, V is one cell compartment volume, 1 is the effective membrane thickness, and t is the time. The lag time to, a measure of the time required to reach this simple relation, is given by
l2
to==
(2)
These two equations form the basis of this note. We want to estimate the effect of the mica flakes both on the lag time to and on the transport coefficient (ADHI W). To do so, we use the same model of brick-like flakes that was successful for steady-state diffusion [ 231. This considers the geometry in more detail than earlier, more approximate analyses [ 6,7]. In this model, shown in Fig. 1, flakes of thickness a are regularly spaced a distance b apart. The width of the flakes is (2d), and their length is W. When the flakes are impermeable, diffusion occurs through the polymer, between the flakes, and through the slits between flakes. Our previous studies show that this diffusion is essentially one-dimensional but along a tortuous path between the flakes. They also show that diffusion through the slits is relatively rapid. We will find it easiest to compare unsteady diffusion without flakes to that with flakes. In making this comparison, we will emphasize cases where the effect of the added flakes is large, because these cases are those of greatest practical interest. This means that we will always be concerned with cases where the lag time and the permeability of the flake filled membrane are far different from those of the flake-free membrane. We begin this comparison with the lag time to. As eqn. (2 ) suggests, this lag time depends on the square of the thickness. For a membrane with flakes having exactly the same permeability as the polymer, this thickness is just (a + b)N,
I t
Fig. 1.Model for barrier membrane. The flakes are regularly spaced like bricks. Diffusion occurs in the polymer between the flakes, but not across the flakes themselves, as suggested by the heavy line.
307
where N is the number of layers of flakes in the membrane. For a membrane with impermeable flakes, the thickness is dominated by the membrane with impermeable flakes, the thickness is dominated by the membrane tortuosity, and hence is (2d)N In fact, the effective thickness is only half this value, to allow for diffusion around both ends of the flakes. The ratio ;1of the length for diffusion with impermeable flakes to that for flakes having the same permeability as the polymer is thus
/l=d a+b
(3)
where (x ( = d/a) is the aspect ratio of the flakes and $ ( = a/ (a+ b) ) is the loading. Thus from eqns. (2 ) and (3 ) , we expect to (with flakes) =I_@* to (without flakes)
(4)
Doubling the concentration of flakes should increase the lag four times. We next turn to the transport coefficient (ADH/VZ). To focus our discussion, we consider a unit area of 2dW. When no flakes are present, diffusion takes place through this entire area. However, when flakes are present, diffusion is through the area between the plates, equal to b W. Again, because solute encountering a plate can diffuse both ways, the total area must be double this, or 2bW. When we combine these changes in area and the changes in length, we find (ADH/lV) (without flakes) (ADH/ZV) (with flakes) d* =b(a+b)
(5)
-~ a*qP - (1-q) We must check the predictions in eqns. (4) and (5) by experiment. Before making this check, we should recognize that eqns. (4) and (5 ) predict that the ratio of lag times and of permeabilities in absence of flakes are zero. This limit is incorrect: the correct limiting value of these ratios is one. This discrepancy, a consequence of our focus on cases where the flakes have a large effect, is explored in the earlier, more complex analysis [ 21. Experimental
Because the diaphragm cell experiments are detailed elsewhere [8,9], only a synopsis is given here. The cell contained two compartments, each 15 cm3,
308
0
40 Time
Fig. 2. Typical
80 t,
experimental
120
min
results. The data shown are consistent
with eqn.
(1 ), the starting
point of our analysis.
separated by a barrier membrane of 15.5 cm2 area. One compartment contained a few milligrams of Ba14C0s in a small stainless steel cup. When dilute sulfuric acid was injected into this cup, 14C0, was uniformly distributed throughout the compartment within a minute. The second compartment was equipped with a Geiger-Mueller tube tuned to detect the beta decay of 14C0, permeating through the membrane. The tube was connected to a counter which recorded the total number of radioactive decays detected. The number of decays per time was a measure of the concentration of 14C0, in the second compartment. The membranes were made by spreading suspensions of mica in a solution of 10% polycarbonate-90% chloroform on plate glass, allowing the solvent to evaporate, and peeling off the membrane. Typical data, shown in Fig. 2, do follow the form suggested by eqn. (1). Results and discussion
The experiments described above give results consistent with eqns. (4) and (5). In particular, the logarithm of the ratio of lag times, shown on the lefthand scale of Fig. 3, is proportional to the logarithm of the loading, shown on the upper scale of that figure. The solid line drawn through the data has a slope of two, consistent with eqn. (4). In the same sense, the logarithm of the ratio of permeabilities, shown on the right-hand scale, is proportional to the logarithm of $“/ (1- @), shown on the bottom scale. The line drawn through these data has a slope of one, consistent with eqn. (5). Interestingly, fragmentary data on commercially available films are also consistent with eqn. (5) [lo]. The success of these results justifies the major idealizations in the model in Fig. 1. These idealizations include assuming that the flakes are regularly spaced and that they are infinitely long. The regular assumption of spacing can be replaced with one of random flakes, which still results in equivalent equations [ 2 1. The assumption of infinite length may not be severe, because diffusion will always be dominated by the shortest route around the flakes. As a result, we believe that the aspect ratio cy will be a harmonic average of flake shape,
WY
Loading 0.1
101001 ’
0.2
$
0.5
I
I
0.02
0.05
I
0.10
110
PW-@)
Fig. 3. Permeability and lag time vs. loading. The line through the squares, of slope two, is consistent with eqn. (4). The line through the circles, of slope one, supports eqn. (5).
more strongly influenced by the minimum dimensions than by the maximum ones. At the same time, we should stress that these results are not completely consistent. In particular, the aspect ratio inferred from Fig. 3 using eqn. (4) is almost fifty, but the aspect ratio inferred from Fig. 3 and eqn. (5) is only fifteen. We do not understand this discrepancy. It may be due to the equivalent of dead-end pores, which serve as a solute sink in the unsteady experiments, but which have little effect on the steady-state experiments. We have speculated on the effect of chemically reactive flakes. For example, if such flakes reacted rapidly and reversibly with oxygen, they could produce an improved oxygen barrier. Such reactive flakes would alter the unsteady state diffusion by replacing the polymer diffusion coefficient D with a new value [D/ (1 +K) 1, where K is the equilibrium constant of the solute-flake reaction [ 4, p. 381. As a result, we expect that the lag time would increase by (1 + K). However, we believe that such reactions would have little effect on the steady-state permeability. These results may find applications in a variety of packaging, including that for electronics, food wrap, and coated beverage containers. As an example, we imagine a layer of polyethyleneterephthalate (PET) as part of the wall of a beverage bottle. We assume that the thickness of this bottle is such that the lag time for carbon dioxide transport is about 30 days. If we add to this wall 30 vol.% mica flakes of an aspect ratio of twenty, we can estimate the lag time to from eqn. (4): to =30 days (20)2(0.3)2 = 3 years
(6)
310
This increase in lag does not depend on the polymer used. It does depend on orienting the flakes and maintaining their high aspect ratio during processing. Orientation can be accomplished by extruding a polymer film and then by stretching the film. Acknowledgements
This research is a contribution from the Center for Interfacial Engineering at the University of Minnesota. It was partially supported by the National Science Foundation (grants 8408999 and 8611646 ), and by Hoechst-Celanese. William J. Ward benefited from a Coolidge Fellowship from the General Electric Company. List of symbols
a
flake thickness membrane area b polymer thickness between flakes d half flake width D diffusion coefficient in polymer H partition coefficient between polymer and surroundings K equilibrium constant of reaction between solute and flakes 1 effective membrane thickness n scintillation counts per time in initially 14C-free compartment n, scintillation counts per time in initially 14C-containing compartment t time t, lag time V cell compartment volume W long dimension of flakes
A
Greek symbols cy flake aspect (d/a)
R @
length ratio (eqn. 3 ) loading [a/(a=b)]
References 1
2 3
S. Okuda, On the performance Parfitt and A.V. Patsis (Eds.), 285-298. E.L. Cussler, S.E. Hughes, W.J. (1988) 161-174. J. Crank, The Mathematics of
of flake filled coatings for the permeation of water, in: G.D. Organic Coatings Science and Technology, Vol. 7, 1982, pp. Ward and R. Aris, Barrier membranes,
J. Membrane
Sci., 38
Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, p. 50.
311 4 5 6 7 8 9 10
E.L. Cussler, Diffusion, Cambridge University, Press, Cambridge, 1984. W.T. Brydges, S.T. Gulati and G. Baum, Permeability of glass ribbon-reinforced composites, J. Mater. Sci., 10 (1975) 2044-2049. A.S. Michaels and R. Bixler, Solubility of gases in polyethylene, J. Polym. Sci., 50 (1961) 393-412. L.E. Nielsen, Models for permeability of filled polymer systems, J. Macromol. Sci., Al (5) (1967) 929-942. M.M. Alger and W.J. Ward, Measurement of COBdiffusion in polymer films, 1986 Polymers Laminations and Coatings Conference, Vol. 2, TAPPI, Atlanta, GA, 1986, pp. 329-332. M.M. Alger, W.J. Ward and T. Stanley, ‘Y!O, and CO, transport in polycarbonate: Measurement of time lag and permeability, J. Polym. Sci., (1988) accepted. DuPont data sheet, SELAR OH Barrier Resins, E.I. DuPont Company, Wilmington, DE, 1988.
305
Short Communication
UNSTEADY
DIFFUSION
IN BARRIER
MEMBRANES
DIANA PERRY’, WILLIAM J. WARD’ and E.L. CUSSLER’ ‘Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 (U.S.A.) 2General Electric Company, Research and Deoelopment Center, Schenectady, NY 12301 (U.S.A.) (Received May 13,1988; accepted in revised form September 26,1988)
Membranes containing mica flakes have a dramatically lower permeability than membranes without mica flakes [ 1,2]. For example, carbon dioxide diffuses across a polycarbonate membrane containing 30 vol.% mica about two hundred times more slowly than across a pure polycarbonate membrane of the same thickness. Such membranes, examples of a growing family of barrier films, may find applications in packaging. In a recent paper [ 21, we developed and verified a theory for the steady-state diffusion across such flake-filled membranes. The most important special case of the theory idealized the membrane as containing regularly spaced lamellae with the flakes arranged in each lamella much like bricks in a wall. Solute was assumed to diffuse in the gaps between the bricks, but could not diffuse through the bricks. The steady-state flux across such a lamellar membrane is predicted to vary with the inverse square of the aspect ratio of the flakes. It is also predicted to be proportional to [ (1- $)/@J’], where @ is the “loading”, i.e., the volume fraction of mica in the membrane. Both predictions are supported by experiment. In this note, we turn to unsteady diffusion across such a flake-filled membrane. We do so because membranes like these are so impermeable that they can take a considerable time to reach steady state. To study the unsteady diffusion, we use a diaphragm cell consisting of two well-stirred compartments separated by the membrane. Each compartment contains the same total pressure of carbon dioxide; one compartment initially contains some 14C0,. We measure the number of counts n in the second compartment as a function of time. At very small times, no carbon dioxide has penetrated through the membrane, and there are no counts above the background. At somewhat larger times, penetration begins, and the counts above the background in the second compartment become an elaborate function of time [3]. At still larger times, the
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers B.V.
306
counts follow the simpler relation
n A++,)
[ 4, p. 781:
(1)
-=
no
in which no is the initial number of counts above background in the first compartment, A is the membrane area, D is the diffusion coefficient in the polymer, H is the partition coefficient between polymer and gas, V is one cell compartment volume, 1 is the effective membrane thickness, and t is the time. The lag time to, a measure of the time required to reach this simple relation, is given by
l2
to==
(2)
These two equations form the basis of this note. We want to estimate the effect of the mica flakes both on the lag time to and on the transport coefficient (ADHI W). To do so, we use the same model of brick-like flakes that was successful for steady-state diffusion [ 231. This considers the geometry in more detail than earlier, more approximate analyses [ 6,7]. In this model, shown in Fig. 1, flakes of thickness a are regularly spaced a distance b apart. The width of the flakes is (2d), and their length is W. When the flakes are impermeable, diffusion occurs through the polymer, between the flakes, and through the slits between flakes. Our previous studies show that this diffusion is essentially one-dimensional but along a tortuous path between the flakes. They also show that diffusion through the slits is relatively rapid. We will find it easiest to compare unsteady diffusion without flakes to that with flakes. In making this comparison, we will emphasize cases where the effect of the added flakes is large, because these cases are those of greatest practical interest. This means that we will always be concerned with cases where the lag time and the permeability of the flake filled membrane are far different from those of the flake-free membrane. We begin this comparison with the lag time to. As eqn. (2 ) suggests, this lag time depends on the square of the thickness. For a membrane with flakes having exactly the same permeability as the polymer, this thickness is just (a + b)N,
I t
Fig. 1.Model for barrier membrane. The flakes are regularly spaced like bricks. Diffusion occurs in the polymer between the flakes, but not across the flakes themselves, as suggested by the heavy line.
307
where N is the number of layers of flakes in the membrane. For a membrane with impermeable flakes, the thickness is dominated by the membrane with impermeable flakes, the thickness is dominated by the membrane tortuosity, and hence is (2d)N In fact, the effective thickness is only half this value, to allow for diffusion around both ends of the flakes. The ratio ;1of the length for diffusion with impermeable flakes to that for flakes having the same permeability as the polymer is thus
/l=d a+b
(3)
where (x ( = d/a) is the aspect ratio of the flakes and $ ( = a/ (a+ b) ) is the loading. Thus from eqns. (2 ) and (3 ) , we expect to (with flakes) =I_@* to (without flakes)
(4)
Doubling the concentration of flakes should increase the lag four times. We next turn to the transport coefficient (ADH/VZ). To focus our discussion, we consider a unit area of 2dW. When no flakes are present, diffusion takes place through this entire area. However, when flakes are present, diffusion is through the area between the plates, equal to b W. Again, because solute encountering a plate can diffuse both ways, the total area must be double this, or 2bW. When we combine these changes in area and the changes in length, we find (ADH/lV) (without flakes) (ADH/ZV) (with flakes) d* =b(a+b)
(5)
-~ a*qP - (1-q) We must check the predictions in eqns. (4) and (5) by experiment. Before making this check, we should recognize that eqns. (4) and (5 ) predict that the ratio of lag times and of permeabilities in absence of flakes are zero. This limit is incorrect: the correct limiting value of these ratios is one. This discrepancy, a consequence of our focus on cases where the flakes have a large effect, is explored in the earlier, more complex analysis [ 21. Experimental
Because the diaphragm cell experiments are detailed elsewhere [8,9], only a synopsis is given here. The cell contained two compartments, each 15 cm3,
308
0
40 Time
Fig. 2. Typical
80 t,
experimental
120
min
results. The data shown are consistent
with eqn.
(1 ), the starting
point of our analysis.
separated by a barrier membrane of 15.5 cm2 area. One compartment contained a few milligrams of Ba14C0s in a small stainless steel cup. When dilute sulfuric acid was injected into this cup, 14C0, was uniformly distributed throughout the compartment within a minute. The second compartment was equipped with a Geiger-Mueller tube tuned to detect the beta decay of 14C0, permeating through the membrane. The tube was connected to a counter which recorded the total number of radioactive decays detected. The number of decays per time was a measure of the concentration of 14C0, in the second compartment. The membranes were made by spreading suspensions of mica in a solution of 10% polycarbonate-90% chloroform on plate glass, allowing the solvent to evaporate, and peeling off the membrane. Typical data, shown in Fig. 2, do follow the form suggested by eqn. (1). Results and discussion
The experiments described above give results consistent with eqns. (4) and (5). In particular, the logarithm of the ratio of lag times, shown on the lefthand scale of Fig. 3, is proportional to the logarithm of the loading, shown on the upper scale of that figure. The solid line drawn through the data has a slope of two, consistent with eqn. (4). In the same sense, the logarithm of the ratio of permeabilities, shown on the right-hand scale, is proportional to the logarithm of $“/ (1- @), shown on the bottom scale. The line drawn through these data has a slope of one, consistent with eqn. (5). Interestingly, fragmentary data on commercially available films are also consistent with eqn. (5) [lo]. The success of these results justifies the major idealizations in the model in Fig. 1. These idealizations include assuming that the flakes are regularly spaced and that they are infinitely long. The regular assumption of spacing can be replaced with one of random flakes, which still results in equivalent equations [ 2 1. The assumption of infinite length may not be severe, because diffusion will always be dominated by the shortest route around the flakes. As a result, we believe that the aspect ratio cy will be a harmonic average of flake shape,
WY
Loading 0.1
101001 ’
0.2
$
0.5
I
I
0.02
0.05
I
0.10
110
PW-@)
Fig. 3. Permeability and lag time vs. loading. The line through the squares, of slope two, is consistent with eqn. (4). The line through the circles, of slope one, supports eqn. (5).
more strongly influenced by the minimum dimensions than by the maximum ones. At the same time, we should stress that these results are not completely consistent. In particular, the aspect ratio inferred from Fig. 3 using eqn. (4) is almost fifty, but the aspect ratio inferred from Fig. 3 and eqn. (5) is only fifteen. We do not understand this discrepancy. It may be due to the equivalent of dead-end pores, which serve as a solute sink in the unsteady experiments, but which have little effect on the steady-state experiments. We have speculated on the effect of chemically reactive flakes. For example, if such flakes reacted rapidly and reversibly with oxygen, they could produce an improved oxygen barrier. Such reactive flakes would alter the unsteady state diffusion by replacing the polymer diffusion coefficient D with a new value [D/ (1 +K) 1, where K is the equilibrium constant of the solute-flake reaction [ 4, p. 381. As a result, we expect that the lag time would increase by (1 + K). However, we believe that such reactions would have little effect on the steady-state permeability. These results may find applications in a variety of packaging, including that for electronics, food wrap, and coated beverage containers. As an example, we imagine a layer of polyethyleneterephthalate (PET) as part of the wall of a beverage bottle. We assume that the thickness of this bottle is such that the lag time for carbon dioxide transport is about 30 days. If we add to this wall 30 vol.% mica flakes of an aspect ratio of twenty, we can estimate the lag time to from eqn. (4): to =30 days (20)2(0.3)2 = 3 years
(6)
310
This increase in lag does not depend on the polymer used. It does depend on orienting the flakes and maintaining their high aspect ratio during processing. Orientation can be accomplished by extruding a polymer film and then by stretching the film. Acknowledgements
This research is a contribution from the Center for Interfacial Engineering at the University of Minnesota. It was partially supported by the National Science Foundation (grants 8408999 and 8611646 ), and by Hoechst-Celanese. William J. Ward benefited from a Coolidge Fellowship from the General Electric Company. List of symbols
a
flake thickness membrane area b polymer thickness between flakes d half flake width D diffusion coefficient in polymer H partition coefficient between polymer and surroundings K equilibrium constant of reaction between solute and flakes 1 effective membrane thickness n scintillation counts per time in initially 14C-free compartment n, scintillation counts per time in initially 14C-containing compartment t time t, lag time V cell compartment volume W long dimension of flakes
A
Greek symbols cy flake aspect (d/a)
R @
length ratio (eqn. 3 ) loading [a/(a=b)]
References 1
2 3
S. Okuda, On the performance Parfitt and A.V. Patsis (Eds.), 285-298. E.L. Cussler, S.E. Hughes, W.J. (1988) 161-174. J. Crank, The Mathematics of
of flake filled coatings for the permeation of water, in: G.D. Organic Coatings Science and Technology, Vol. 7, 1982, pp. Ward and R. Aris, Barrier membranes,
J. Membrane
Sci., 38
Diffusion, 2nd edn., Clarendon Press, Oxford, 1975, p. 50.
311 4 5 6 7 8 9 10
E.L. Cussler, Diffusion, Cambridge University, Press, Cambridge, 1984. W.T. Brydges, S.T. Gulati and G. Baum, Permeability of glass ribbon-reinforced composites, J. Mater. Sci., 10 (1975) 2044-2049. A.S. Michaels and R. Bixler, Solubility of gases in polyethylene, J. Polym. Sci., 50 (1961) 393-412. L.E. Nielsen, Models for permeability of filled polymer systems, J. Macromol. Sci., Al (5) (1967) 929-942. M.M. Alger and W.J. Ward, Measurement of COBdiffusion in polymer films, 1986 Polymers Laminations and Coatings Conference, Vol. 2, TAPPI, Atlanta, GA, 1986, pp. 329-332. M.M. Alger, W.J. Ward and T. Stanley, ‘Y!O, and CO, transport in polycarbonate: Measurement of time lag and permeability, J. Polym. Sci., (1988) accepted. DuPont data sheet, SELAR OH Barrier Resins, E.I. DuPont Company, Wilmington, DE, 1988.