Barrier membranes with different sizes of aligned flakes

Barrier membranes with different sizes of aligned flakes

Journal of Membrane Science 254 (2005) 21–30 Barrier membranes with different sizes of aligned flakes Jonathan P. DeRocher, Brian T. Gettelfinger, Ju...

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Journal of Membrane Science 254 (2005) 21–30

Barrier membranes with different sizes of aligned flakes Jonathan P. DeRocher, Brian T. Gettelfinger, Junshan Wang, Eric E. Nuxoll, E.L. Cussler ∗ Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue, Minneapolis, MN 55455, USA Received 19 August 2004; received in revised form 8 December 2004; accepted 17 December 2004 Available online 24 March 2005

Abstract The permeability of polymer films can be reduced dramatically with many layers of thin mineral flakes aligned parallel to the film’s surface. For a film with a volume fraction φ of flakes of aspect ratio α, the permeability reduction is expected to be proportional to αφ for the dilute limit (φ  1 and αφ < 1) but proportional to (αφ)2 in the semi-dilute limit (φ  1 but αφ > 1). Permeabilities of hydrochloric acid and sodium hydroxide across films containing mica or vermiculite in polyvinyl alcohol agree with the second, semi-dilute prediction. Permeabilities of helium across films of montmorillonite in polyethylene glycol do as well. These improvements in barrier properties are independent of flake size, permeant and polymer chemistry. © 2005 Elsevier B.V. All rights reserved. Keywords: Permeability; Polyvinyl alcohol; Flake

Many membranes are designed to be selectively permeable. Such thin films are commonly intended for use in membrane separations. Other membranes are intended to be impermeable barriers to protect a surface or a product. Paint, food wrap, and electronic packaging are examples. For these barrier membranes, we seek polymers through which solutes like water, oxygen, and chloride permeate slowly, rather than rapidly. Often we are successful, but equally often, we want still lower permeation with uncompromised mechanical properties. We have been developing barrier membranes containing aligned mineral flakes [1–5]. Because these mineral flakes are crystalline, they have very low permeabilities and can reduce the permeability of the composite film. However, the mechanism of reduction remains controversial. Many expect that the permeability change will vary linearly with the volume fraction of flakes; others, including us, think that it will vary more nearly with the square of this volume fraction. There ∗

Corresponding author. Tel.: +1 612 625 1596; fax: +1 612 626 7246. E-mail address: [email protected] (E.L. Cussler).

0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2004.12.025

are similar disagreements about how the permeability will vary with flake shape, usually expressed as an aspect ratio. Interestingly, most working in the area expect that the permeability is independent of flake size, even though there seems to be little experimental evidence for this belief. In this paper, we want to begin to resolve these ambiguities, primarily by measuring diffusion of gases or aqueous acids and bases across rubbery polymer membranes containing flakes of mica, vermiculite, and montmorillonite. We begin by reviewing past theories which predict these effects, and then describe how our measurements are made. We conclude by reporting the measurements themselves, and by discussing their implications for practical barrier coatings. 1. Theory The study of permeability in composite materials gets its initial foundation from calculations by Clerk Maxwell [6], who investigated transport in periodic layers of neutrally buoyant spheres immersed in a continuum. Two limits of his result are relevant to this paper. First, for a suspension of

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impermeable spheres, the relative permeability is 1 + φ/2 P0 = P 1−φ

(1)

where P0 is the permeability of the continuum, P that of the composite, and φ the volume fraction of spheres. Second, for the opposite case of highly permeable spheres, the result is P0 1−φ = P 1 + 2φ

(2)

Others have extended these results to different geometries. For example, Strutt and Rayleigh [7] showed that for periodic layers of impermeable cylinders aligned perpendicularly to the diffusion, P0 1+φ = P 1−φ

(3)

where φ is now the volume fraction of cylinders, not of spheres. These results have proved to be remarkably successful, even at concentrations higher than those for which the theories are expected to be accurate. The results in Eqs. (1)–(3) share some remarkable features. First, the relative permeability changes caused by either cylinders or spheres are independent of the permeability of the continuum. In other words, the right-hand side of these equations is not a function of the permeability of the continuum P0 . A second feature is that the permeability changes depend on the volume fraction of the particles, but not on their size. For example, it does not matter if the particles are 10 nm or 10 ␮m in diameter: at the same volume fraction, they are predicted to have the same effect on permeability. A third characteristic of Eqs. (1)–(3) is that both spheres and cylinders do not have a major effect on the permeability. For example, if the volume fraction of spheres φ is 0.1, the permeability for a composite of impermeable spheres is 0.86P0 , and that for completely permeable spheres is 1.33P0 . In other words, a change from completely impermeable spheres to rapidly permeable spheres changes the composite permeability by only 50%. Similarly, the effects on permeability of either impermeable spheres or cylinders are modest, as shown in Fig. 1. The results for layers of impermeable flakes are significantly different, partially because flakes cannot be described

Fig. 1. Permeability changes caused by impermeable particles. The changes caused by aligned flakes of aspect ratio 30 are much greater than those effected by periodic spheres or aligned cylinders.

with a single dimension—a diameter—but require at least two—a width and a thickness for a ribbon-shaped flake, for example. As was the case for the spheres or cylinders, the flakes must be aligned periodically. The direction of alignment is important. If the flakes are aligned perpendicular to the barrier surface, they have a minor effect; if they are aligned parallel to the surface, their effect is much larger. We are interested in the latter case, when the flakes are parallel to the membrane’s surface. In addition, flakes now imply a different definition for what a “dilute suspension” is [8]. In every case, a dilute suspension of flakes must have a small volume fraction (φ  1). Now, however, there are two other limits which depend on the aspect ratio α, which is defined as half the flake width divided by its thickness. When φ  1 and αφ < 1, the suspension’s permeability is predicted to be [9]: P0 = 1 + αφ (4) P which is a close echo for Eqs. (1) and (3) above. However, when φ  1 but αφ > 1, the suspension, now termed semidilute, has a relative permeability equal to [1,10] P0 α2 φ 2 =1+µ P 1−φ

(5)

where µ is a geometric parameter. The dependence on the square of volume fraction φ and aspect ratio α is a consequence of the increased tortuosity and of the reduced area available for diffusion. More complete analyses of flake-filled barrier membranes give more complex results reflecting different transport mechanisms like slow diffusion through the gaps between flakes, or increased transport resistance to enter this narrow gap [1,8,11,12]. However, the two mechanisms of tortuosity and area appear from experiment to be most important. The geometric factor µ which appears in Eq. (5) is not completely understood. Two limits have been discussed: periodic flakes and random flakes. If the flakes are periodic and ribbon-like so their length is much greater than their width, then µ equals 1. If the flakes are periodic hexagons, then µ equals 4/9 [3]. If the flakes are random, then estimates of µ range from 2/27 to π2 /16 ln2 α [3,8]. If the flakes are polydisperse in size, there is little effect, both theoretically and experimentally [5]. The permeability changes for aligned ribbon-like flakes suggested by Eq. (5) are typically much larger than spheres or cylinders, as shown in Fig. 1. For 10% impermeable spheres, Eq. (1) and Fig. 1 suggest a 14% decrease in permeability. For the same concentration of cylinders described by Eq. (3) and in Fig. 1, the decrease is 18%. For the same concentration of aligned flakes of aspect ratio 30, the decrease given by Eq. (5) and in Fig. 1 is 91%. Flakes can be much more effective. We will use Eqs. (4) and (5) to analyze the measurements in this paper. We are particularly interested in the prediction of these equations that the relative permeability P0 /P is independent of the continuum permeability P0 and of the flake size. We want to test whether the relative permeability varies

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with the volume fraction to the first or second power. These comparisons of theory and experiment should improve the rationale for developing better barrier membranes.

2. Experimental 2.1. Materials Polyvinyl alcohol (PVA; DuPont Elvanol, grade 71-30, molecular weight 138,000–146,500) and polyethylene glycol (PEG; Sigma Aldrich lot 122K0094, molecular weight 1450) were used as received. Mica (King’s Mountain Hi Mod 270, Oglebay Norton) was exfoliated by heating the mica at 800 ◦ C for 2 h, and then quenching it in hot saturated aqueous Na2 CO3 [13]. After filtration, it was washed overnight with concentrated HCl and dried in air. Vermiculite (Microlite 963, W.R. Grace) and montmorillonite (Cloisite Na+ , Southern Clay Products) were used as received. Hydrochloric acid (39%, Mallinckrodt) was diluted to 0.1N with distilled water. Sodium hydroxide pellets (Mallinckrodt) were dissolved in distilled water to form a 0.1N solution. Helium (Minneapolis Oxygen) and nitrogen (Airgas) were at least high-purity grade. 2.2. Membrane preparation Membranes were cast from blends of aqueous polymer solutions and aqueous flake suspensions and then spread on a Teflon block with a doctor blade (Mitutoyo). The resulting films were air-dried and sometimes heat-treated to encourage crosslinking. Details are given below. Crosslinked polyvinyl alcohol (PVA) membranes with and without various flakes were used for measurements of acid and base permeation. A typical preparation for a pure PVA membrane was begun by dissolving 4 g of PVA in 36 mL of water preheated to 70 ◦ C. The mixture was covered and stirred at least 1 h, until the PVA was completely dissolved. The mixture was then placed under vacuum to remove dissolved air. The degassed solution was cooled to room temperature and cast on the Teflon block. The membrane was allowed to dry for at least 1 day. While drying, it was covered to prevent contamination by dust and to ensure the slow drying which helps to produce a flat film of uniform thickness. Membranes which were dried too fast could warp or develop dimples. The membrane was then heated for 135 min at 150 ◦ C to induce crosslinking. PVA membranes containing mica were typically made by dissolving 4.5 g of PVA in 60 mL of water at 70 ◦ C. The mixture was covered and stirred until the PVA was completely dissolved. In a separate beaker, varying quantities of mica flakes were suspended in 20 mL of water and heated to 70 ◦ C. After the PVA solution had been degassed under vacuum, the mica slurry was added. The suspension was stirred, uncovered, to allow some of the water to evaporate. When the volume of the solution was 50–60 mL, the heat was reduced

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and the suspension was allowed to cool while stirring. When the suspension was cool, it was again put under vacuum for about 15 min, allowing most of the dissolved air to escape. Any remaining bubbles were scooped off the top with a spatula. The PVA/mica suspension was cast on the Teflon block which had been heated to 75 ◦ C. Preheating the block appeared to reduce settling of the mica flakes. The membrane was allowed to dry for at least 24 h. PVA membranes containing vermiculite are prepared in a similar fashion, but dried for at least 2 days. Vermiculite membranes tend to stick to the Teflon block, but the flakes settle less than mica because they are smaller. Films containing graphite (Asbury) and synthetic hydrotalcite clay [14] were prepared similarly to the vermiculite films. All flake-filled membranes were heat-treated to encourage crosslinking in the same way as the pure PVA films. Membranes were then stored swollen in distilled water. Membranes of polyethylene glycol (PEG) and montmorillonite, used for gas permeation experiments, were made differently. An aqueous solution of PEG was made by preheating distilled water to nearly boiling, and then adding about 10% polymer with vigorous stirring. A suspension of around 5% clay was made separately, also with hot water and stirring, which helps prevent clumping of the clay. The clay slurry was then added to the polymer solution. This combined suspension was stirred at least 2 h and heated without a cover in a fume hood to remove excess water. The suspension was removed from the heat when the fluid viscosity reached about 5 cp, and was cooled and degassed under vacuum. It was cast on the Teflon block and dried for a minimum of 12 h. Once dry, the films were peeled from the blocks and placed in labeled plastic bags for storage. Note that these membranes are water-soluble. Both PVA and PEG membranes were characterized by scanning electron microscopy. Micrographs of the crosssection of the films are useful in qualitatively determining the degree of alignment and settling. Samples for microscopy were prepared by fracturing a small piece of the film under liquid nitrogen. They were mounted on a sample stub, coated with a 5-nm layer of platinum, and visualized with a JEOL 6500 FEG-SEM. PVA membranes containing mica and various clays were also studied by X-ray diffraction, using a Bruker-AXS Microdiffractometer with a Hi-Star 2-D Area Detector. The accelerating voltage was 45 kV and the filament current was 40 mA. A perfectly aligned structure will result in a single dot at the diffraction angle; a completely random structure will result in a ring at the diffraction angle. Integration along the ring will give a quantitative measure of the distribution of angles present in the sample. An aligned structure will look like a sharp peak, but a random structure will look like a flat line. 2.3. Permeability Transport across the PVA and PEG membranes was measured in a diaphragm cell [15]. The PVA measurements used

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a cell with a membrane of area 5.2 cm2 mounted between two silicone gaskets. The membrane separated two stirred compartments, as shown in Fig. 2a. One “donating” compartment, with a volume of 15.0 cm3 , normally contained 0.1 M HCl but occasionally 0.1 M NaOH. The second “receiving” compartment, also of 15.0 cm3 , initially contained pure water and an Orion 420 A pH probe, used to measure the acid or base concentration. All experiments were made at room temperature, which was typically 21 ± 1 ◦ C over an experiment. For experiments with liquid solutions, temperature had little effect; for experiments with gases, pressures can be corrected as described elsewhere [16]. Because measurements were made at short times when the concentration in the receiving compartment was less than 5% of that initially in the donating compartment c10 , this receiving concentration varied linearly with time, with slope m. This linearity was observed after a lag time tL . The membrane’s permeability P was found from the relation P=

mV c10 A

(6)

where V is the volume of the receiving compartment, and A and are the membrane’s area and thickness, respectively. Occasionally, the diffusion coefficient D in the membrane was estimated from the lag time tL : 2 D= 6tL

(7)

However, scatter in the data can compromise the accuracy of this estimate. Transport measurements across the PEG membranes used a different diaphragm cell, which is shown in Fig. 2b. This cell used a membrane of area of 0.79 cm2 clamped between compartments of 24.1 and 11.7 cm3 . Occasionally, a brass plug was added to reduce the volume of the receiving compartment and thus allow a shorter experiment. In a typical experiment, both compartments were first filled with about 120 kPa nitrogen. After an hour, both compartments were refreshed with more of the same gas. Then an additional 100 kPa of helium was added to the larger (donating) compartment, and the pressure difference between the compartments was measured with two Cole Palmer model 07345-00 pressure transducers as a function of time. The permeability P was calculated from these differences by means of the equation [15] P=

At



1 V

+

1 V 

 ln

p0 p

(8)

where p0 and p are the pressure differences initially and at time t, respectively, and V and V are the volumes of the two compartments, respectively.

3. Results This research continues development of barrier membranes containing impermeable, oriented flakes. It shows how flake alignment can be assessed, and that alignment parallel to

Fig. 2. Diaphragm cells. For PVA membranes, acid and base diffusion is measured with a pH meter as shown in (a). For PEG membranes, gas transport is recorded as pressure changes illustrated in (b).

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Fig. 3. Micrographs of PVA membranes containing vermiculite and graphite. The vermiculite in (a) is aligned and dispersed, but the graphite in (b) forms clumps which settle out during membrane preparation.

the membrane surfaces gives significantly lower permeabilities. The research tests theories for these increased barrier properties and for the membranes’ altered mechanical properties. The measurements of flake alignment depend on micrographs and X-ray scattering, as shown in Figs. 3 and 4. The SEM pictures show that vermiculite (Fig. 3a) is well dispersed in PVA, with flakes aligned parallel to the membrane surface. In this case, the alignment is probably largely due to evaporation, although the shear in casting the membrane may also have an effect [4]. In contrast, the micrograph for graphite (Fig. 3b) shows that graphite flakes have settled to the bottom of the PVA film. These flakes are not aligned or dispersed, in spite of having been treated with potassium metal in a process said to cause exfoliation [17]. In this experiment, graphite does not give the structure that we seek.

The X-ray scattering data shown in Fig. 4 give a measure of alignment over a larger sample than that covered by the micrographs. As a result, they are a more definitive test. The data in Fig. 4a, again for vermiculite in PVA, show the sharp spots characteristic of good alignment. The distribution of flakes versus angle is like that observed for mica in separate experiments [4]. However, the data for the synthetic hydrotalcite shown in Fig. 4b show little evidence of orientation. On the basis of these data, we expect that the vermiculite should reduce the permeability, but that the hydrotalcite should not do so. This expectation is supported by the data discussed later in this section. We measured permeability changes across membranes in aqueous solutions by measuring pH changes and in gas mixtures by recording pressure changes. Typical data for hydrochloric acid diffusing across a variety of PVA membranes

Fig. 4. X-ray scattering of flake-filled membranes. Vermiculite in PVA (a) shows good alignment, but synthetic hydrotalcite (b) does not.

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J.P. DeRocher et al. / Journal of Membrane Science 254 (2005) 21–30 Table 1 Ion permeabilities of flake-filled membranes Film

Fig. 5. Typical permeability data for PVA membranes. When the flakes are dispersed and well oriented, the permeability is small and the lag time is large.

are shown in Fig. 5. For pure PVA, the acid concentration in the receiving compartment of the diaphragm cell starts to rise almost immediately. For a PVA membrane containing 11 vol.% vermiculite clay, no acid appears for about 15 min, and then the acid concentration rises more slowly. For a PVA membrane containing 14 vol.% mica, the lag before acid appears is around half an hour, and then the acid concentration rises more slowly still. These increasing acid concentrations are the basis for calculating the permeability from Eq. (6). Typical data for helium diffusion, shown in Fig. 6, are similar but less reproducible. The pressure in the receiving compartment does not change for a brief time and then rises linearly; this rise is slower for more flakes with a larger aspect ratio. We calculate the permeability from Eq. (8), although for the cases shown, we could have equally well used Eq. (6). The poorer reproducibility is because small pressure changes are harder to measure than small pH changes. The altered permeabilities for the range of films studied here are summarized in Table 1. The first column in this table gives the polymer and flake used, and the second column gives the membrane thickness. The third column gives the flake volume fraction φ. The fourth column is the measured

Fig. 6. Typical permeability data for PEG membranes. These results, based on pressure changes, are less accurate than the PVA results, based on pH measurements.

(␮m) φ

P (10−7 cm2 /s)

PVA

170

0.00

40

PVA + mica (α = 25)

300 350 290 290

0.0456 0.0456 0.0540 0.137

16 15 14 2.8

PVA

200

0.00

PVA + vermiculite (α = 30)

1+

α2 φ 2 1−φ

P0 /P

1.0

1.0

2.4 2.4 2.9 15

2.5 2.7 2.9 14

30

1.0

1.0

2.6

2.7

2.5 1.4 2.6 4.4 6.7 9.4 12

3.0 1.6 3.4 6.3 3.7 7.9 7.9

250

0.0413

19a

230 200 230 250 250 270 330

0.0401 0.0216 0.0413 0.0595 0.0764 0.0919 0.106

17a 19 8.8 4.8a 8.1 3.8 3.8

Films were tested with 0.1 M HCl in the donating compartment except as noted. Note: 1 cm2 /s = 9.04 × 10−6 cm3 (STP) cm/cm2 Pa s. a 0.1 M NaOH in the donating compartment.

permeability, given in the units of 10−7 cm2 /s because these occur naturally from the analysis in Eqs. (6) and (8). Conversions are given in a footnote to the table. These values, which are not unusually low, reflect our choice of highly permeable PVA and PEG. We chose these materials not because they are potentially good barriers, but because they are highly permeable. Thus we could easily measure their reduced permeability in short experiments and test the predictions developed above. The values in PVA are consistent with those reported earlier [3,5], but the values in PEG show smaller permeability changes than reported elsewhere [18]. Eq. (5) is tested by the fifth and sixth columns in Table 1. The fifth column gives the factor (1 + α2 φ2 /(1 − φ)), expected to be the key variable. Note that this assumes that the geometric factor µ is 1. Column 6 gives the relative permeability, which correlates closely with the factors in column 5. More details of this correlation are given in Fig. 7a. The open symbols, which are results of this work, include the two polymers PVA and PEG: the three flakes mica, vermiculite, and montmorillonite; and the three solutes HCl, NaOH, and helium. The closed symbols, which are the results of others [1–3,19,20], include flakes of mica, nylon, and clay, and the polymers polydimethylsiloxane and polycarbonate. The correlation with (1 + α2 φ2 /(1 − φ)) is successful. We should stress that this correlation, while impressive, is not definitive. In particular, because the solid PEG is changed to a rubbery film by the addition of montmorillonite, we cannot measure a permeability P0 for pure PEG. We have used a value inferred from the correlation in Fig. 7a. More seriously, the values of α for each of the three data sets are inferred from other, independent measurements, but these are really estimates. Independent of these ambiguities, the data do show a variation with φ2 , and not with φ.

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Fig. 7. Permeability vs. flake concentration and aspect ratio. The results in part (a) are consistent with the semi-dilute limit in Eq. (5). The results in part (b) show that the data are not described as well by the dilute limit in Eq. (4). Data from Refs. [1–3,25,26] are included.

We expect that the permeabilities in Fig. 7 will be correlated by Eq. (5) because we are in the semi-dilute region, when αφ > 1. We would not expect Eq. (4) to apply because we are not in the dilute region, when both φ  1 and αφ < 1. The data in Fig. 7b, which are the same as those in Fig. 7a, show that Eq. (4) does seem to be less successful. Thus the change in permeability does seem to vary with φ2 for the cases considered here. However, we recognize that this may not be true if the flakes are not well aligned or not periodically arrayed. We speculate about this in the discussion below.

4. Discussion This research uses aligned mineral flakes to reduce the permeability of flexible polymer films. The results given above show some success in exfoliating the flakes, consistent with earlier research. They show that these flakes can be aligned by casting a suspension of flakes in a polymer solution, so that both shear and evaporation contribute to alignment. The results show that X-ray scattering can give evidence of this alignment. These conclusions apply to different flakes whose largest dimension is greater than 1 ␮m. We have found that the diaphragm cell is a reliable though tedious method of measuring these effects. Diaphragm cell results for liquids are more reliable than those for gases, and measurements of permeability are more successful than measurements of lag time. Characteristically, we can easily get a permeability drop of a factor of 10. When we try to get larger decreases in permeability, we compromise the mechanical properties of the barrier films. They become brittle and crack easily. Our data strongly support the prediction that permeability varies with the square of (αφ). In other words, they are consistent with Eq. (5), but not with Eq. (4). This is true for the new results for mica, vermiculite, and montmorillonite presented in this work. It is also true for other, older data as plotted in Fig. 7. These results on different chemical systems reinforce the argument that good barriers should be semi-dilute solutions, not dilute solutions. In either case, the volume fraction φ is much less than 1. However, in a semi-

dilute solution the flakes overlap: while φ  1, αφ > 1. Thus diffusion through the composite film involves a lot of wiggles, which increase the path length for diffusion by a factor proportional to αφ. Diffusion is also through a reduced crosssectional area, which is proportional to αφ/(1 − φ). It is this combination of increased path length and reduced area which is responsible for the variation predicted by Eq. (5). We are uncertain about the value of the geometric factor µ which occurs in Eq. (5). If the flakes are regularly spaced ribbons this factor should equal 1. If the flakes are randomly spaced hexagons, the factor may be as low as 2/27 [3]; if the flakes are random both in position and in shape, then µ may be a logarithmic function of α [8]. For our data, we do not feel that we know α accurately enough to make a definitive test. We do believe, however, that the value of µ is not that different than 1. While we are not sure, we suspect that this may be due to two factors. First, the flakes in any given layer seem to be independent of those in an adjacent layer. This is apparently why a barrier made of long and short flakes is not seriously compromised by the short ones [5]. Second, when molecules of a diffusing solute encounter a flake, they will be biased towards always taking the shortest route around no matter what the orientation for the flakes is [12]. This may mean the picture of ribbon-shaped flakes works better than we would first expect. We must stress that every flake-filled system will not be correlated by Eq. (5). In many cases, the disagreement will be due to experiments that do not produce the desired lamellar architecture. In particular, every system which we attempted to study did not work well, as detailed in Table 2. We suspect that the three failures in this table are for different reasons. The montmorillonite available from Nanocor is surfacetreated, just like our other sample of the same clay. We believe that this surface treatment was not appropriate for the hydrophilic PVA which we tried to use. We suspect that we did not adequately exfoliate the graphite sample, which certainly clumped and settled in our films. We are not sure why our experiments on the hydrotalcite are unsuccessful. Indeed, we are most frustrated by these experiments since we worked harder on this than on any other system. Our only hypothesis is that our efforts to achieve alignment were compromised

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Table 2 Gas permeabilities of flake-filled membranes Film

(␮m) φ

PEG

0.00

PEG + montmorillonite 270 (α = 9) 240 210 150 200 170 150 240 130 200 240

0.227

P (1 × 10−7 cm2 /s)

1+

α2 φ 2 1−φ

P0 /P

100a

1.0

1.0

20

6.4

5.0

0.325 0.351 0.365 0.387 0.405 0.418 0.440 0.458 0.486 0.506

9.0 5.4 6.1 5.7 3.3 7.6 5.6 2.7 2.0 2.6

13 16 18 21 23 25 29 32 38 43

11 19 16 18 31 13 18 37 49 38

Films were tested with helium in the donating compartment except as noted. Note: 1 (cm2 /s) = 9.04 × 10−6 (cm3 (STP) cm/cm2 Pa s). a Best value inferred from extrapolation.

by rotational diffusion, but we were unable to verify this hypothesis by measuring changes in alignment versus time. Even in our more successful experiments, we believe that exfoliation of all our flakes is incomplete, be they of mica, clay, or any other material. We suspect that this is normally true not only in permeability measurements but also in many measurements of reinforced composites (e.g. [21–24]). After all, the aspect ratios which we measure typically imply that the flake thickness is on the order of a micron, but the known crystal structure of these materials dictates a flake thickness of at least 100 times thinner. We do not have individual flakes in our experiments; we have decks of flakes (see Table 3). While we recognize this, we are handicapped by the vocabulary used to describe exfoliation. Normally, in studying dispersion of lamellar materials like clays, we speak first of intercalation, when a solvent or surfactant molecule forces its way between two lamellae. As this process continues, other molecules, including polymers, force the lamellae apart. These processes of intercalation and exfoliation can be tracked by X-ray scattering, with results like those shown in Fig. 4. However, we want not only intercalation and increased lamellae spacing, but also the complete separation of lamellae and rearrangement either in an end-to-end structure or

in a staggered, brick wall-like structure. Such rearrangement probably requires more chemistry than we have brought to bear in this research. Such rearrangement would increase α at least 10 times, and so improve the barrier properties 100 times or more. In other cases, we may have flakes in the desired lamellar architecture but still not get the variation of the permeability on the inverse square of the product αφ. Most obviously, the permeability will vary linearly in the dilute case, when the flakes do not overlap (i.e., when αφ < 1 [8]). Even when the flakes do overlap, we may observe a linear variation in at least two other cases. First, when the ends of the flakes fit tightly together, transport between the flakes will be rate limiting, and transport by tortuous wiggles between adjacent layers of flakes will be relatively fast. In this first case, the permeability will vary with φ−1 , not with φ−2 [1,11]. To our knowledge, this case has not been experimentally observed. Second, when the flakes are not neutrally buoyant, they may settle into a more compact aligned region. In this second case, both theory and experiment show that the apparent permeability varies linearly with (αφ)−1 [25]. We also recognize that the variation of permeability in concentrated films (when φ ∼ = 1) has not been carefully studied. Our measurements in Fig. 7 do have values of φ as high as 0.5, well beyond the dilute range (φ  1) where the theory is best established [8], but well short of volume fractions of 0.8 or more, where extremely large effects are possible. On the one hand, films with φ > 0.5 are very hard to construct experimentally without some additional method of flake selfassembly. On the other hand, nothing in the arguments leading to Eq. (5) should prevent its successful use in concentrated solution [1,10,11]. We do hope to study this point in the future. Finally, we want to discuss the changes in permeability versus changes in mechanical properties. We are getting smaller permeabilities by means of a composite material. Any composite material tries to capture the best properties of each of its components. In other words, a composite is an effort to let us “have our cake and eat it too”. In this case, we are putting impermeable but brittle flakes into a permeable but flexible polymer continuum. We want to get a less permeable but still flexible film. In this paper, we have shown that we can get a less permeable film, but we also know that it is less flexible. To see how

Table 3 Flake-filled films studied in this work Flake

Continuum

Better barrier?

Preparation

Size

Probable difficulty

Remarks

Mica (Oglebay Norton) Cloisite Na+ (Southern Clay) MicroLite 963 (W.R. Grace) Nanomer PGV (Nanocor) Graphite (Asbury)

PVA PEG PVA PVA PVA

Yes Yes Yes No No

Heat, Na2 CO3 As received As received As received Exfoliate with K metal

40 ␮m × 1 ␮m 5–10 ␮m 10–100 ␮m Nanometer size 50 ␮m

None Hard to disperse None Low aspect ratio Clumping settling

Best system to date Largest effects to date Used commercially

Hydrotalcite Clay

PVA

No

Synthesized

100 nm

Too small to align

The first three systems worked well, but the second three did not.

May work with proper surfactants Reactive barrier for HCl; poor results for NaOH

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much the flexibility is compromised, we turn to Halpin-Tsai Equations [26,27] E 4αηφ =1+ E0 1 − ηφ

(9)

where E and E0 are the Young’s moduli of the composite and the polymer continuum, respectively, α and φ are as before the aspect ratio and volume fraction of the aligned flakes, and η is given by   (EF /E0 ) − 1 η= (10) (EF /E0 ) + 4α where EF is the Young’s modulus of the flakes themselves. Two limits of these equations are interesting. First, when the flakes have an extremely large aspect ratio, then α EF /E0 1, η  1, and Eqs. (9) and (10) reduce to EF φ E =1+ E0 E0

(11)

This says that the composite is very brittle indeed, with a Young’s modulus closely related to that of the flakes themselves. Second, when the aspect ratio is more modest but the flakes themselves are still brittle, EF /E0 α 1, η is about 1, and Eqs. (9) and (10) become E 4αφ =1+ E0 1−φ

(12)

This relation, a rough parallel to Eqs. (1) and (4), says that the composite is certainly less elastic than the continuum, but not dramatically so. We can now compare the result for permeability in Eq. (5) with that expected for Young’s modulus from Eqs. (11) and (12). Eq. (5) says that we want the aspect ratio α to be as big as possible. However, Eqs. (11) and (12) imply that we want α to be smaller than EF /E0 . We must compromise. For the systems we are studying here, EF /E0 is about 106 [26], so α should be less than perhaps 104 . This is still much larger than the value for α of around 30 typical of our experiments. Thus we should be able to get still less permeable films without seriously compromising their mechanical properties. We plan to continue experiments which seek this goal.

Acknowledgments This work was largely supported by the Air Force Office of Scientific Research (grant F49620-01-1-0333). Other support came from the Department of Energy (grant DE-FG0702ER63509), the Petroleum Research Fund (grant 39083AC9), and the National Science Foundation (grant CTS 0322882).

29

Nomenclature A c10 D E E0 EF m p p0 P P0 tL V

membrane area (cm2 ) initial permeant concentration (M) diffusion coefficient (cm2 /s) Young’s modulus (GPa) Young’s modulus of pure polymer (GPa) Young’s modulus of flake (GPa) membrane thickness (cm) breakthrough curve slope (M/s) pressure difference at time t (kPa) initial pressure difference (kPa) permeability of composite film (cm2 /s) permeability of pure film (cm2 /s) lag time (s) volume of diffusion cell compartment (cm3 )

Greek symbols α flake aspect ratio η intermediate Halpin-Tsai parameter µ generic geometric factor φ volume fraction of flakes

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