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Effect of pore location and pore size of the support membrane on the permeance of composite membranes
Journal of Membrane Science 594 (2020) 117465
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Effect of pore location and pore size of the support membrane on the permeance of composite membranes
T
Pingjiao Hao∗∗,1, J.G. Wijmans∗, Zhenjie He, Lloyd S. White Membrane Technology and Research, Inc., 39630 Eureka Drive, Newark, CA, 94560, USA
ABSTRACT
Composite membranes typically consist of a nonporous layer on top of a porous support. The porous support restricts diffusion in the top layer because molecules can exit the layer only where a pore is present and this reduces the permeance of the membrane. Previously, we have used computational fluid dynamics (CFD) to simulate this geometric restriction for supports with uniform pore configurations and an empirical correlation was developed that accurately quantifies the reduction in permeance. In this work we present permeance data of composite membranes with top layers of different thicknesses coated on a support prepared by the common phase inversion process. The permeances are lower than predicted by the correlation because the support is not uniform in pore location and pore size. CFD modeling with non-uniform surface structures confirms that non-uniformity reduces the permeance. This effect can be incorporated into the empirical correlation through a uniformity coefficient. This coefficient is smaller than one for non-uniform surfaces and can be considered independent of the thickness of the top layer.
1. Introduction Composite membranes are widely used in liquid permeation, gas separation and pervaporation applications. These membranes typically consist of a nonporous top layer supported by a porous structure. The nonporous layer is the selective layer responsible for the separation performance, and the porous support structure often is called the support membrane (but also is referred to as the microporous or porous support) and provides mechanical support for the top layer. As discussed in many publications over the last few decades, the surface pore structure of the support membrane plays a critical role in the preparation of thin top layers. In the current paper we will focus on one specific aspect of the support membrane: the effect surface membrane pore size and porosity on the diffusion process in the top layer of the formed composite membrane. This effect has garnered increased attention in the last decade as the thickness of the top layers is being reduced to 100 nm and below, and is becoming a major limitation in the quest for ever higher membrane permeances [1–10]. In a composite membrane transport through the top layer is by diffusion from the feed-side interface to the interface with the support membrane. In most cases the support is made of a material with a low permeability, which means that molecules permeating the top layer can only exit through the porous areas of the support interface, as is illustrated in Fig. 1. As early as in 1971 Lonsdale et al. [1] recognized that this restriction on the diffusion pathway will increase the diffusion
resistance of the top layer and thus will reduce membrane permeance. Ramon et al. [2] in 2012 were the first to use Computational Fluid Dynamics (CFD) simulations to calculate the magnitude of the effect. The effect can be accounted for by applying a restriction factor, Ψ, to the calculation of the permeance:
Permeance =
Pi H
(1)
where Pi is the permeability coefficient and H is the thickness of the top layer. In 2015 we reported that the dependence of the restriction factor on porosity, pore size and top layer thickness is very accurately described by Ref. [3]:
=
+ 1.6 NR1.1 1 + 1.6 NR1.1 with NR =
H R 1
=
1
(2)
where ɸ is the support top surface porosity, R is the surface pore radius, τ is normalized thickness of the top layer (divided by the pore radius) and NR is a dimensionless number equal that quantifies the support influence. It should be noted that equation (2) is a correlation based on our CFD analysis and is not an equation derived from first principles. Fig. 2 illustrates how the permeance of a composite membrane varies with the top layer thickness. For thick top layers the restriction factor equals unity and the permeance equals the permeability divided
Corresponding author. Corresponding author. E-mail addresses: [email protected] (P. Hao), [email protected] (J.G. Wijmans). 1 Current affiliation: NICE America Research, Inc., 2091 Stierlin Court, Mountain View, CA 94,043, USA. ∗
∗∗
https://doi.org/10.1016/j.memsci.2019.117465 Received 17 June 2019; Received in revised form 7 September 2019; Accepted 9 September 2019 Available online 11 September 2019 0376-7388/ © 2019 Elsevier B.V. All rights reserved.
Journal of Membrane Science 594 (2020) 117465
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Table 1 Carbon Dioxide permeance measured for composite membranes with polydimethylsiloxane (PDMS) top layers of different thicknesses coated onto the same support membrane. The intrinsic permeances are added for comparison and are calculated by dividing the carbon dioxide permeability of 3400 Barrer by the top layer thickness. Fig. 1. Cross-section of a composite membrane. Only the upper layer of the porous support in direct contact with the nonporous top layer is shown. The support material is assumed to be non-permeable. Diffusion of molecules in the top layer is restricted because all molecules have to exit through the pore openings of the support.
by the thickness. For thinner top layers the restriction factor decreases and the permeance is less than inversely proportional with thickness. For very thin top layers the restriction factor equals the porosity and the permeance again becomes inversely proportional with thickness [3]. 2. Experimental evidence of the support influence
Top layer thickness (micron)
Experimental permeance (GPU)
Intrinsic permeance (GPU)
0.082 0.085 0.087 0.16 0.43 0.85 3.2 6.5 12
6100 5600 5900 4700 3400 2200 910 480 280
40,000 39,000 38,000 21,000 7700 3900 1000 510 280
(Filmetrics Model F20 ellipsometer, KLA Corporation, San Diego, USA). The carbon dioxide/nitrogen selectivities measured for the three thinnest membranes deviate from the intrinsic selectivity, which equals twelve, and this indicates the presence of small defects. These defects are less significant for the carbon dioxide permeance because of the higher carbon dioxide permeability. For the three thinnest membranes the defects contribute between 2% and 5% to the measured carbon dioxide permeances, which is within the experimental accuracy. Table 1 lists the carbon dioxide permeances as a function of the thickness of the PDMS layer. The data are plotted in Fig. 3 and show the deviation from the intrinsic permeance at the lower thicknesses. The experimental data follow the trend embedded in equation (2) and can be fitted to that correlation by minimizing the root mean square deviation. The best fit, at a deviation of 6.2%, is obtained for a support porosity of 4.2% and a pore radius of 39 nm. The top surface of the MTR support membrane was examined using a high-resolution scanning electron microscope, see Fig. 4 for an SEM photo. A public domain image processing program ImageJ [12] was used to analyze the SEM image and to estimate the average pore radius
A number of recent publications by Lin et al. [4,5] and Ghadimi et al. [7] have provided clear experimental evidence that the porous supports of composite membranes reduce the permeance, even if the transport resistance of the porous structure on its own is negligible compared to the resistance of the top layer. Here we present additional experimental data obtained by systematic variation of the top layer thickness of polydimethylsiloxane (PDMS) composite membranes [6]. Composite membranes were prepared using a porous support membrane made by MTR and with PDMS as the top layer material. The porous support membrane was made via the standard non-solvent induced phase separation process and consists of a commercially available engineering polymer. A series of coating solutions with different PDMS concentrations were used, resulting in membranes with different PDMS layer thicknesses. In the coating process special care was taken to prevent penetration of the coating solution into the surface pores of the support. Pure carbon dioxide and nitrogen permeances of the membranes were determined using a gas permeation test cell and the thickness of the PDMS layer was measured using ellipsometry
Fig. 2. Permeance of a composite membrane as a function of the thickness of the top layer. Permeance is expressed in GPU, which equals Barrer/micron. The upper perforated line represents the intrinsic permeance calculated by dividing the permeability by the top layer thickness using 100 Barrer as the permeability of the top layer. The solid line is the permeance as calculated using equations (1) and (2) where the pore radius is 50 nm (0.05 μm), porosity is 5%. The lower perforated line is the intrinsic permeance multiplied by the surface porosity.
2
Journal of Membrane Science 594 (2020) 117465
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Fig. 3. Carbon dioxide permeance of PDMS composite membranes versus the PDMS layer thickness. The data points are the experimental permeances which deviate from the intrinsic permeance (equal to permeability divided by thickness) when the PDMS layer is thinner than a few microns. The measured permeances follow the trend captured by correlations (1) and (2) and a best fit yields a support porosity of 4.2% and a pore radius of 39 nm.
stronger than as predicted by equation (2). This is consistent with the observations made by Lin et al. [4,5] and Ghadimi et al. [7]. Both ourselves [6] and Ghadimi et al. [7] have suggested that the reason for the deviation is that equation (2) has been derived for an “ideal” uniform surface where all pores have the same diameter and are distributed in a regular square or hexagonal pattern. Support membranes produced by the traditional phase inversion process do not have uniform surfaces. 3. Taking into account non-uniform support surfaces The restriction factor correlation (2) has been developed on the basis of CFD simulations for pore distributions that are uniform in both pore location and pore size. These distributions do not describe the current support membranes used to fabricate composite membranes. The question therefore is how to account in the correlation for nonuniform support surfaces. We [6] and Ghadimi et al. [7] have suggested that this is done by adding a coefficient to the correlation:
= Fig. 4. SEM picture of the top surface of the support membrane used to prepare the ten PDMS composite membranes. Analysis using the ImageJ software yields a surface porosity in the range of 2%–6%, and a pore radius in the range of 4 nm–10 nm.
+ 1.6 NR1.1 with NR = 1 + 1.6 NR1.1
1
(3)
where σ is the uniformity coefficient, which will have a value of one for uniform support surfaces and is expected to have a value below one for non-uniform support surfaces. Ghadimi et al. [7] call the coefficient the pattern coefficient and consider the coefficient a modification for the normalized thickness, which is the same as considering it a modification for the pore radius. Consistent with correlation (2), correlation (3) obeys the four limit cases for the restriction coefficient described in Ref. [3]:
and porosity of the support. This analysis has significant uncertainty because it requires the operator to select a pore/no pore threshold variable, but it does support a porosity value in the range of 2%–6%, which agrees with the porosity value obtained from the permeance versus thickness data. However, the ImageJ analysis also yields an average pore radius in the range of 4 nm–10 nm, which is significantly lower than the 39 nm value obtained from the permeance versus thickness data. Using the smaller pore radius suggested by the SEM picture in equation (2) gives higher permeances than the permeances actually measured: the restriction on the permeance imposed by the support is
lim
=0
(no permeance if support has zero porosity)
lim
=1
(intrinsic permeation if support has 100% porosity)
0 1
lim lim
0
=1 =
(intrinsic permeance if top layer is very thick) (for very thin top layers permeation takes place only above the pore openings)
(4) which is a requirement as these limit cases also apply to non-uniform 3
Journal of Membrane Science 594 (2020) 117465
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Fig. 5. Distributions of one hundred pores with three distinct geometries. Average porosity is the same for each geometry and equal to 9%. Geometry (a) is a uniform square distribution of pores that all have the same pore size. Geometry (b) is a uniform square distribution of two sets of pores, each with its own uniform pore size. Geometry (c) is a random distribution of pores that all have the same pore size. The random distribution of 100 pores was obtained using a web-based random generator (www. random.org) to generate the coordinates of each pore within a square grid.
generator (www.random.org). A characteristic feature of a non-uniform geometry is that one can identify areas with higher than average porosity and areas with lower than average porosity, see Fig. 5(c). Fig. 6 shows the concave dependence of the restriction factor on porosity: the restriction factor decreases faster at lower porosities. This means that a surface with a distribution in porosity is expected to have a lower restriction factor, and thus a lower permeance, compared to a second surface with a uniform porosity equal to the average porosity of the first surface. This will result in a uniformity coefficient that is smaller than one for nonuniform surfaces. Thus, our hypothesis is that for all non-uniform surfaces the uniformity coefficient values will be less than one. One possible exception that we can think of is when there is a significant variation in the thickness of the top layer across the surface. The intrinsic permeance of such a layer will be larger than the permeance calculated from the mean thickness, and hence, analysis based on the mean top layer thickness can result in a uniformity coefficient larger than one. Ghadimi et al. [7] report uniformity coefficients that are significantly larger than one for a number of asymmetric supports. However, these coefficients are calculated using values for the surface porosity and pore size that were obtained from the Dusty Gas model [11]. This model is not suited for asymmetric membranes and does not give reliable results for asymmetric structures, as discussed by Zhu et al. [4,5] and Ghadimi et al. [7]. The model is well suited to polycarbonate track-etch membranes, and the uniformity coefficients obtained by Ghadimi et al. for these supports all are smaller than one.
Fig. 6. Restriction factor calculated from correlation (2) as a function of porosity. The dependence is concave and the restriction factor decreases faster at lower porosities. This indicates that surfaces with non-uniform porosity will have a lower restriction factor, and thus a lower permeance, than a uniform surface with the same average porosity.
surfaces. The experimental data in Table 1 are described accurately by correlation (3) when using a value for the uniformity coefficient of 0.15, which is the ratio of the pore radius obtained from the SEM analysis (about 6 nm, see Fig. 4) and the pore radius obtained from the curve fit (about 40, see Fig. 3). Fig. 5 shows three different surface geometries with one hundred pores, which have the same overall porosity but differ in pore size distribution and pore location distribution. Geometry 5(a) is a uniform square distribution of pores of equal size, geometry 5(b) is a uniform square distribution of two different pore sizes, and geometry 5(c) is a non-uniform distribution of pores of equal size. The pore locations in geometry 5(c) were generated with a public domain random number
4. Characteristics of random distributions Uniform distributions in a two dimensional surface are either in a square or in a hexagonal configuration. In the square configuration, the distance between a pore and each of its four closest neighboring pores is
Fig. 7. Ratio of Largest Distance to Smallest Distance for the four nearest pores for each of the 100 pores in a random distribution. Results shown for ten different distributions (1000 points total). For each distribution the pores are sorted from the smallest ratio (which cannot be smaller than one) to the largest ratio. 4
Journal of Membrane Science 594 (2020) 117465
P. Hao, et al.
Fig. 8. Average distance to the four nearest pores for each of the 100 pores in a random distribution. Results shown for ten different distributions (1000 points total). For each distribution the pores are sorted from the smallest average distance to the largest average distance.
Fig. 9. (a), (b), (c) and (d): Distribution of 16 pores at constant porosity of 5% and a ratio of top layer thickness to pore radius equal to 5. (e): Distribution of 4 pores at porosity of 5% at the same top layer thickness, which means that the ratio of top layer thickness to pore radius equals 2.5. CO2 permeances calculated for composite membranes with a 0.5 μm thick PDMS top layer. The uniformity coefficients are calculated via equation (3) using the restriction factors obtained from the CFD simulations. Table 2 Restriction factors and uniformity coefficients obtained from CFD simulations for composite membranes with varying normalized top layer thicknesses (top layer thickness/pore radius) and with different surface pore patterns of the porous support. Surface porosity of all support membranes is 5%.
Pattern #2
Pattern #3
Pattern #4
Normalized Top Layer Thickness (−)
Restriction Factor from CFD (−)
1 2 5 10 20 1 2 5 10 20 1 2 5 10 20
0.097 0.150 0.287 0.444 0.615 0.096 0.145 0.270 0.423 0.594 0.093 0.133 0.244 0.387 0.548
Uniformity Coefficient from Eq. (3) (−)
Restriction Factor (−) at Uniformity Coefficient Equal to
0.844 0.886 0.911 0.906 0.878 0.825 0.841 0.833 0.834 0.878 0.770 0.734 0.720 0.720 0.677
0.101 0.153 0.287 0.445 0.624
0.91
0.83
0.096 0.144 0.270 0.423 0.602
0.72
Fig. 10. Restriction factors obtained from CFD simulations for various composite membrane configurations, compared to the restriction factors obtained from equation (3), see Table 2. For each pattern the uniformity coefficient is taken to be independent of the top layer thickness, yielding excellent agreement between the CFD simulations and equation (3).
0.090 0.131 0.244 0.387 0.564
to quantify the difference between “non-uniform” and “uniform”. We created ten different random distributions of 100 pores in the same way the random distribution in Fig. 5(c) was created and
the same. In the hexagonal configuration, the distance between a pore and each of its six closest neighboring pores is the same. This will not be true for random and/or non-uniform distributions and this offers a way 5
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the four nearest pores. For the uniform square and hexagonal distributions this average is the same for all pores and we designate this distance to be one in arbitrary units. In a random distribution, the distance will be smaller than one for some pores and larger than one for others. Fig. 8 gives the distances for each pore in ten different random distributions of 100 pores each and shows that the random distribution is distinctly different from uniform distributions. In the random distributions about 70% of the pores are in areas with a “lower than average” porosity, whereas about 30% are in areas with a “larger than average” porosity. An interesting feature of both Figs. 7 and 8 is that the ten different random distributions have very similar profiles. This suggests that even for only one hundred pores an individual random distribution regresses towards an “average” random distribution. Considering that even a low porosity membrane has in excess of one billion pores per cm2 membrane area, we can be assured that the random nature of the pore distribution in typical support membranes will not result in membrane stamp to stamp variations, let alone in variations in commercial membrane modules.
Fig. 11. Restriction factors obtained from CFD simulations for various composite membrane configurations, compared to the restriction factors obtained from equation (3), see Table 3. The uniformity coefficient is taken to be independent of the porosity, yielding reasonably good agreement between the CFD simulations and equation (3).
5. Simulations for non-uniform pore distributions We carried out a number of CFD simulations to investigate the effect of non-uniform pore distribution on the restriction factor. We do not have access to sufficient computation power to simulate random distributions of 100 or more pores, so we limited ourselves to distributions that, while not uniform, are not random and form repeatable patterns for which CFD simulations can be performed more easily.
Table 3 Restriction factors and uniformity coefficients obtained from CFD simulations for composite membranes with pore pattern #4 and with varying surface pore size (and thus varying porosity) of the porous support. Normalized thickness is 5 in all cases. Porosity (%)
Restriction Factor from CFD (−)
Uniformity Coefficient from Eq. (3) (−)
Restriction Factor at Uniformity Coefficient Equal to 0.72 (−)
0.5 1 2 5 10
0.033 0.063 0.116 0.244 0.390
1.03 0.945 0.849 0.720 0.597
0.024 0.050 0.101 0.244 0.431
5.1. Effect of pore location CFD simulations were performed for the five pore patterns shown in Fig. 9. Distributions (a) and (e) are uniform square distributions of pores; the pore radius in (e) is double the radius in (a), and both are covered by equation (2). Fig. 9 gives the relative permeances for the five distributions, as well as the uniformity coefficient calculated via equation (3). The sequence (a) through (d) demonstrates that a less uniform distribution results in lower permeances, which is reflected in lower values of the uniformity coefficient. Distribution (e) is the logical end point of the sequence where every set of four pores has merged into one pore with double radius, which makes it once again a fully uniform distribution. CFD simulations were performed for patterns 2, 3 and 4 for six different relative top layer thicknesses. The data in Table 2 and Fig. 10 show that the restriction factors obtained from the CFD simulations closely match the restriction factors obtained from equation (3) using a constant value of the uniformity coefficient for each pattern. We thus conclude that the uniformity coefficient can be considered independent of the thickness of the top layer (see Fig. 11).
calculated for each pore the distances to its four closest neighbors. We then calculated the ratio of the larger distance to the smaller distance. In any uniform distribution this ratio is equal to one for every pore. Fig. 7 shows the ratios for the ten different random distributions; for each distribution the pores are sorted from the smallest ratio (which cannot be smaller than one) to the largest ratio. The largest values for the ratio are quite large, in excess of ten. The difference between the random distributions and the uniform distribution is striking. This allows us to define a “mostly uniform distribution” as one where, for a significant majority of all pores, the ratio shown in Fig. 7 is less than a specific threshold, for example less than a value of two. An alternative measure of non-uniformity is the average distance to
Fig. 12. Restriction factor obtained through CFD for a square distribution of pores at 5% surface porosity. (a) all pores have the same diameter, (b) 50% of the pores have twice the diameter of the other pores, (c) 25% of the pores have a diameter 4 times the smallest pores, 25% of the pores have a diameter 3 times the smallest pores, 25% of the pores have a diameter 2 times the smallest pores. All permeances are CO2 permeances calculated for composite membranes with a 0.5-μm thick PDMS top layer.
6
Journal of Membrane Science 594 (2020) 117465
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Also possible, and perhaps very likely, is that the permeances of the lower porosity membranes are lowered by penetration of the top layer material into the pores. We used CFD simulations to determine the restriction factors for a number of cases with varying degrees of pore penetration. The results are given in Table 5 and Fig. 13 gives the concentration profiles for two different degrees of pore penetration. Even small pore penetration depths reduce the permeance of the composite membranes and the effect is stronger at lower porosities. In addition to the CFD analysis we used a resistances-in-series model to calculate the permeances of composite membranes if the top layer penetrates into the surface pores. In this approach we simply add the resistance of the penetration layer to the resistance of the top layer:
Table 4 Restriction factors reported Zhu et al. [4] for composite membranes prepared with track-etch polycarbonate support membranes, plus the uniformity coefficients calculated by Ghadimi et al. [7]. Support
Pore Radius (nm)
Porosity (%)
Top Layer Thickness (nm)
Restriction Factor [4]
Uniformity Coefficient [7]
PC-50
51
2.7
25
3.3
PC-15
14
1.1
PC-7.5
5.8
0.6
0.14 0.15 0.13 0.18 0.029 0.032 0.037 0.009 0.010
0.828
PC-25
160 220 99 200 76 100 120 82 90
0.520 0.256
1 Permeance
0.0483
H Pi
=
Permeance=
(
Hp
+ H Pi
Pi Hp
+
(
Pi
)
1
=
Hp / H
Pi H
)
(
1
+
Hp / H
)
1
1
5.2. Variation in porosity
Restriction Factor=
Additional CFD simulations were performed for pattern #4 with a range of values for the surface porosity, which were obtained by changing the pore radius values. The normalized thickness of the top layer was kept constant at 5. As expected, the restriction factor increases rapidly with increasing porosity, see Table 3. The uniformity coefficient starts at a value very close to one for the lowest porosity of 0.5% and decreases with increasing porosity. However, Fig. 11 shows that good agreement between equation (3) and the CFD results is obtained at a constant uniformity coefficient, at least, within the range of relatively low porosities considered here.
where Hp is the penetration depth of the top layer material into the pore. This equation is surprisingly accurate when compared to the CFD results, see Fig. 13. The explanation for this can be found in Fig. 14, which shows that the concentration profiles in the pore are close to be being horizontal, which indicates that diffusion through the cross-
1
+
(5)
Table 5 Restriction factors obtained from CFD simulations for composite membranes. Included in the simulations are three different pore penetration depths for the top layer.
5.3. Non-uniform pore size In our previous paper [2] we performed CFD simulations for a number of square distributions of pores with varying diameter sizes. At constant overall porosity and at constant relative top layer thickness (normalized to the average pore diameter), the restriction factors obtained through CFD were within 7% of each other. We therefore tentatively concluded that pore size distributions did not affect the restriction effect. However, at that time we had overlooked the subtle, but consistent, trend that all distributions of pores with varying pore consistently have smaller restriction factors than distributions of pores of equal diameter, see Fig. 10 for some examples. Our revised conclusion is that a distribution in pore sizes leads to lower restriction factors, and thus to lower permeances. However, based on Figs. 5 and 12, we can conclude that the effect of non-uniform pore sizes is small compared to the effect of a non-uniform distribution in pore location.
Porosity
Top Layer Thickness/ Pore Radius
Pore Penetration Depth/Top Layer Thickness
Pore Penetration Depth/Pore Radius
Restriction Factor (CFD Simulation)
Restriction Factor (Equation (5))
2%
2
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.064 0.058 0.053 0.046 0.133 0.115 0.097 0.077 0.234 0.185 0.143 0.105
0.064 0.060 0.055 0.048 0.133 0.113 0.099 0.079 0.234 0.188 0.146 0.107
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.163 0.147 0.133 0.116 0.306 0.271 0.231 0.184 0.469 0.389 0.311 0.235
0.163 0.153 0.140 0.123 0.306 0.273 0.235 0.190 0.469 0.396 0.320 0.242
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.311 0.286 0.260 0.228 0.510 0.465 0.405 0.334 0.678 0.594 0.501 0.399
0.311 0.293 0.269 0.238 0.510 0.464 0.407 0.338 0.678 0.598 0.507 0.405
5
10
5%
6. Comparison with data for track-etch support membranes
2
5
Ghadimi et al. [7] have analyzed the permeances of composite membranes for a number of different support membranes. Most of the support membranes considered by them are asymmetric structures prepared by phase inversion, but they also analyzed data reported by Zhu et al. [4] for track-etch polycarbonate support membranes. Here we will focus on the data obtained with track-etch support membranes because only for these supports are reliable pore size and porosity data available, which are essential for the calculation of the uniformity coefficient. The data are given in Table 4 and show a significant decrease in uniformity coefficient for lower porosities. Our CFD analysis actually predicts the opposite: an increase in uniformity coefficient with decreasing porosity, but this assumes the distribution in pore location remains the same. Thus, one explanation for the trend shown in Table 4 is that for this family of support membranes the reduction in pore size coincides with a significant reduction in uniformity of the pore location distribution.
10
10%
2
5
10
7
Journal of Membrane Science 594 (2020) 117465
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Fig. 13. Restriction factors obtained from CFD simulations compared with the restriction factors obtained from correlation (2) and the resistances-in-series equation (5).
Fig. 14. Concentration profiles for a top layer that penetrates into the pore opening of the support membrane. Penetration depth, Hp, is 5% (a) and 20% (b) of the top layer thickness, H.
section of the pore is distributed reasonably even, which then validates the use of the resistances-in-series model. The distribution will be the least even for small penetration depths, but in that case the contribution of pore penetration to the overall resistance will be small and equation (5) still will be accurate. From equation (5) we see that even a pore penetration depth that is much smaller than the top layer thickness will significantly reduce the overall permeance if the porosity is small. For example, the track-etch support PC-7.5 has a porosity of only 0.6%, which means that a pore penetration depth of 1 nm adds a permeation resistance equivalent to the intrinsic resistance of a 160 nm top layer.
and an empirical correlation was developed to accurately quantify the reduction in permeance. Experimental permeance data were obtained of composite membranes with top layers of different thicknesses coated on a support prepared by the common phase inversion process. The permeances follow the trend predicted by the CFD simulations: permeances are less than inversely proportional with the top layer thickness. However, the permeances are even lower than predicted by the correlation. This is because the support is not uniform in surface pore location and surface pore size, which was an assumption, made in the previous CFD simulation effort. The new CFD simulations presented in this work with non-uniform surface geometries confirm that non-uniformity reduces the permeance. This effect can be incorporated into the empirical correlation through a uniformity coefficient. Our analysis strongly suggests that the uniformity coefficient is always smaller than one for non-uniform surfaces. The CFD results also show that the uniformity coefficient is a function of porosity, but at constant porosity is relatively independent of the normalized thickness of the top layer and of the distribution in pore size.
7. Conclusions The porous support restricts diffusion in the top layer of composite membranes because molecules can exit the layer only where a pore is present. Previously, we have used computational fluid dynamics (CFD) to simulate this geometric restriction for uniform pore configurations 8
Journal of Membrane Science 594 (2020) 117465
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Declarations of interest
References
None.
[1] H.K. Lonsdale, R.L. Riley, C.R. Lyons, D.P. Carosella, Transport in composite reverse osmosis membranes, in: M. Bier (Ed.), Membrane Processes in Industry and Biomedicine, Plenum Press, New York, 1971, pp. 101–121. [2] G.Z. Ramon, M.C.Y. Wong, E.M.V. Hoek, Transport through composite membrane, part I: is there an optimal support membrane? J. Membr. Sci. 415 (2012) 298–305. [3] J.G. Wijmans, P. Hao, Influence of the porous support on diffusion in composite membranes, J. Membr. Sci. 494 (2015) 78–85. [4] L. Zhu, W. Jia, M. Kattula, K. Ponnuru, E.P. Furlani, H. Lin, Effect of porous supports on the permeance of thin film composite membranes: Part I. Track-etched polycarbonate supports, J. Membr. Sci. 514 (2016) 684–695. [5] L. Zhu, M. Yavari, W. Jia, E.P. Furlani, H. Lin, Geometric restriction of gas permeance in ultrathin film composite membranes evaluated using an integrated experimental and modeling approach, Ind. Eng. Chem. Res. 56 (1) (2017) 351–358. [6] P. Hao, L.S. White, Z. He, D. Nguyen, T.C. Merkel, J.G. Wijmans, Effect of support pore distribution on the permeance of composite membranes, ICOM San Francisco (2017) August 3, 2017. [7] A. Ghadimi, S. Norouzbahari, H. Lin, H. Tabiee, B. Sadatnia, Geometric restriction of microporous supports on gas permeance efficiency of thin film composite membranes, J. Membr. Sci. 563 (2018) 643–654. [8] M. Kattula, K. Ponnuru, L. Zhu, W. Jia, H. Lin, E.P. Furlani, Designing ultrathin film composite membranes: the impact of a gutter layer, Sci. Rep. 5 (2015) 15016. [9] G.Z. Ramon, E.M.V. Hoek, Transport through composite membranes, part 2: roughness and its effect on permeability and fouling, J. Membr. Sci. 425 (2013) 141–148. [10] M.C.Y. Wong, L. Lin, O. Coronell, E.M.V. Hoek, G.Z. Ramon, Impact of liquid-filled voids within the active layer on transport through thin-film composite membranes, J. Membr. Sci. 500 (2016) 124–135. [11] E.A. Mason, A.P. Malinauskas, Gas Transport in Porous Media: the Dusty-Gas Model, Elsevier, Amsterdam, 1983. [12] Image processing and analysis, https://imagej.nih.gov/ij.
Acknowledgement The authors acknowledge Dung Nguyen for preparing the PDMS composite membranes and measuring their gas permeances. We also thank the U.S. Department of Energy, National Energy Technology Laboratory for financial support under contract DE-FE0031596. Nomenclature H Hp Np NR P R σ
Thickness of top layer (nanometer) Penetration depth of the top layer material into the pore (nanometer) Number of pores per surface area Restriction Number, equation (2) Permeability coefficient (Barrer = GPU.micron) Radius of pore (nanometer) uniformity coefficient, equation (3) Restriction Factor, Equation (2) Porosity Normalized thickness, equal to H/R
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Effect of pore location and pore size of the support membrane on the permeance of composite membranes
T
Pingjiao Hao∗∗,1, J.G. Wijmans∗, Zhenjie He, Lloyd S. White Membrane Technology and Research, Inc., 39630 Eureka Drive, Newark, CA, 94560, USA
ABSTRACT
Composite membranes typically consist of a nonporous layer on top of a porous support. The porous support restricts diffusion in the top layer because molecules can exit the layer only where a pore is present and this reduces the permeance of the membrane. Previously, we have used computational fluid dynamics (CFD) to simulate this geometric restriction for supports with uniform pore configurations and an empirical correlation was developed that accurately quantifies the reduction in permeance. In this work we present permeance data of composite membranes with top layers of different thicknesses coated on a support prepared by the common phase inversion process. The permeances are lower than predicted by the correlation because the support is not uniform in pore location and pore size. CFD modeling with non-uniform surface structures confirms that non-uniformity reduces the permeance. This effect can be incorporated into the empirical correlation through a uniformity coefficient. This coefficient is smaller than one for non-uniform surfaces and can be considered independent of the thickness of the top layer.
1. Introduction Composite membranes are widely used in liquid permeation, gas separation and pervaporation applications. These membranes typically consist of a nonporous top layer supported by a porous structure. The nonporous layer is the selective layer responsible for the separation performance, and the porous support structure often is called the support membrane (but also is referred to as the microporous or porous support) and provides mechanical support for the top layer. As discussed in many publications over the last few decades, the surface pore structure of the support membrane plays a critical role in the preparation of thin top layers. In the current paper we will focus on one specific aspect of the support membrane: the effect surface membrane pore size and porosity on the diffusion process in the top layer of the formed composite membrane. This effect has garnered increased attention in the last decade as the thickness of the top layers is being reduced to 100 nm and below, and is becoming a major limitation in the quest for ever higher membrane permeances [1–10]. In a composite membrane transport through the top layer is by diffusion from the feed-side interface to the interface with the support membrane. In most cases the support is made of a material with a low permeability, which means that molecules permeating the top layer can only exit through the porous areas of the support interface, as is illustrated in Fig. 1. As early as in 1971 Lonsdale et al. [1] recognized that this restriction on the diffusion pathway will increase the diffusion
resistance of the top layer and thus will reduce membrane permeance. Ramon et al. [2] in 2012 were the first to use Computational Fluid Dynamics (CFD) simulations to calculate the magnitude of the effect. The effect can be accounted for by applying a restriction factor, Ψ, to the calculation of the permeance:
Permeance =
Pi H
(1)
where Pi is the permeability coefficient and H is the thickness of the top layer. In 2015 we reported that the dependence of the restriction factor on porosity, pore size and top layer thickness is very accurately described by Ref. [3]:
=
+ 1.6 NR1.1 1 + 1.6 NR1.1 with NR =
H R 1
=
1
(2)
where ɸ is the support top surface porosity, R is the surface pore radius, τ is normalized thickness of the top layer (divided by the pore radius) and NR is a dimensionless number equal that quantifies the support influence. It should be noted that equation (2) is a correlation based on our CFD analysis and is not an equation derived from first principles. Fig. 2 illustrates how the permeance of a composite membrane varies with the top layer thickness. For thick top layers the restriction factor equals unity and the permeance equals the permeability divided
Corresponding author. Corresponding author. E-mail addresses: [email protected] (P. Hao), [email protected] (J.G. Wijmans). 1 Current affiliation: NICE America Research, Inc., 2091 Stierlin Court, Mountain View, CA 94,043, USA. ∗
∗∗
https://doi.org/10.1016/j.memsci.2019.117465 Received 17 June 2019; Received in revised form 7 September 2019; Accepted 9 September 2019 Available online 11 September 2019 0376-7388/ © 2019 Elsevier B.V. All rights reserved.
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Table 1 Carbon Dioxide permeance measured for composite membranes with polydimethylsiloxane (PDMS) top layers of different thicknesses coated onto the same support membrane. The intrinsic permeances are added for comparison and are calculated by dividing the carbon dioxide permeability of 3400 Barrer by the top layer thickness. Fig. 1. Cross-section of a composite membrane. Only the upper layer of the porous support in direct contact with the nonporous top layer is shown. The support material is assumed to be non-permeable. Diffusion of molecules in the top layer is restricted because all molecules have to exit through the pore openings of the support.
by the thickness. For thinner top layers the restriction factor decreases and the permeance is less than inversely proportional with thickness. For very thin top layers the restriction factor equals the porosity and the permeance again becomes inversely proportional with thickness [3]. 2. Experimental evidence of the support influence
Top layer thickness (micron)
Experimental permeance (GPU)
Intrinsic permeance (GPU)
0.082 0.085 0.087 0.16 0.43 0.85 3.2 6.5 12
6100 5600 5900 4700 3400 2200 910 480 280
40,000 39,000 38,000 21,000 7700 3900 1000 510 280
(Filmetrics Model F20 ellipsometer, KLA Corporation, San Diego, USA). The carbon dioxide/nitrogen selectivities measured for the three thinnest membranes deviate from the intrinsic selectivity, which equals twelve, and this indicates the presence of small defects. These defects are less significant for the carbon dioxide permeance because of the higher carbon dioxide permeability. For the three thinnest membranes the defects contribute between 2% and 5% to the measured carbon dioxide permeances, which is within the experimental accuracy. Table 1 lists the carbon dioxide permeances as a function of the thickness of the PDMS layer. The data are plotted in Fig. 3 and show the deviation from the intrinsic permeance at the lower thicknesses. The experimental data follow the trend embedded in equation (2) and can be fitted to that correlation by minimizing the root mean square deviation. The best fit, at a deviation of 6.2%, is obtained for a support porosity of 4.2% and a pore radius of 39 nm. The top surface of the MTR support membrane was examined using a high-resolution scanning electron microscope, see Fig. 4 for an SEM photo. A public domain image processing program ImageJ [12] was used to analyze the SEM image and to estimate the average pore radius
A number of recent publications by Lin et al. [4,5] and Ghadimi et al. [7] have provided clear experimental evidence that the porous supports of composite membranes reduce the permeance, even if the transport resistance of the porous structure on its own is negligible compared to the resistance of the top layer. Here we present additional experimental data obtained by systematic variation of the top layer thickness of polydimethylsiloxane (PDMS) composite membranes [6]. Composite membranes were prepared using a porous support membrane made by MTR and with PDMS as the top layer material. The porous support membrane was made via the standard non-solvent induced phase separation process and consists of a commercially available engineering polymer. A series of coating solutions with different PDMS concentrations were used, resulting in membranes with different PDMS layer thicknesses. In the coating process special care was taken to prevent penetration of the coating solution into the surface pores of the support. Pure carbon dioxide and nitrogen permeances of the membranes were determined using a gas permeation test cell and the thickness of the PDMS layer was measured using ellipsometry
Fig. 2. Permeance of a composite membrane as a function of the thickness of the top layer. Permeance is expressed in GPU, which equals Barrer/micron. The upper perforated line represents the intrinsic permeance calculated by dividing the permeability by the top layer thickness using 100 Barrer as the permeability of the top layer. The solid line is the permeance as calculated using equations (1) and (2) where the pore radius is 50 nm (0.05 μm), porosity is 5%. The lower perforated line is the intrinsic permeance multiplied by the surface porosity.
2
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Fig. 3. Carbon dioxide permeance of PDMS composite membranes versus the PDMS layer thickness. The data points are the experimental permeances which deviate from the intrinsic permeance (equal to permeability divided by thickness) when the PDMS layer is thinner than a few microns. The measured permeances follow the trend captured by correlations (1) and (2) and a best fit yields a support porosity of 4.2% and a pore radius of 39 nm.
stronger than as predicted by equation (2). This is consistent with the observations made by Lin et al. [4,5] and Ghadimi et al. [7]. Both ourselves [6] and Ghadimi et al. [7] have suggested that the reason for the deviation is that equation (2) has been derived for an “ideal” uniform surface where all pores have the same diameter and are distributed in a regular square or hexagonal pattern. Support membranes produced by the traditional phase inversion process do not have uniform surfaces. 3. Taking into account non-uniform support surfaces The restriction factor correlation (2) has been developed on the basis of CFD simulations for pore distributions that are uniform in both pore location and pore size. These distributions do not describe the current support membranes used to fabricate composite membranes. The question therefore is how to account in the correlation for nonuniform support surfaces. We [6] and Ghadimi et al. [7] have suggested that this is done by adding a coefficient to the correlation:
= Fig. 4. SEM picture of the top surface of the support membrane used to prepare the ten PDMS composite membranes. Analysis using the ImageJ software yields a surface porosity in the range of 2%–6%, and a pore radius in the range of 4 nm–10 nm.
+ 1.6 NR1.1 with NR = 1 + 1.6 NR1.1
1
(3)
where σ is the uniformity coefficient, which will have a value of one for uniform support surfaces and is expected to have a value below one for non-uniform support surfaces. Ghadimi et al. [7] call the coefficient the pattern coefficient and consider the coefficient a modification for the normalized thickness, which is the same as considering it a modification for the pore radius. Consistent with correlation (2), correlation (3) obeys the four limit cases for the restriction coefficient described in Ref. [3]:
and porosity of the support. This analysis has significant uncertainty because it requires the operator to select a pore/no pore threshold variable, but it does support a porosity value in the range of 2%–6%, which agrees with the porosity value obtained from the permeance versus thickness data. However, the ImageJ analysis also yields an average pore radius in the range of 4 nm–10 nm, which is significantly lower than the 39 nm value obtained from the permeance versus thickness data. Using the smaller pore radius suggested by the SEM picture in equation (2) gives higher permeances than the permeances actually measured: the restriction on the permeance imposed by the support is
lim
=0
(no permeance if support has zero porosity)
lim
=1
(intrinsic permeation if support has 100% porosity)
0 1
lim lim
0
=1 =
(intrinsic permeance if top layer is very thick) (for very thin top layers permeation takes place only above the pore openings)
(4) which is a requirement as these limit cases also apply to non-uniform 3
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Fig. 5. Distributions of one hundred pores with three distinct geometries. Average porosity is the same for each geometry and equal to 9%. Geometry (a) is a uniform square distribution of pores that all have the same pore size. Geometry (b) is a uniform square distribution of two sets of pores, each with its own uniform pore size. Geometry (c) is a random distribution of pores that all have the same pore size. The random distribution of 100 pores was obtained using a web-based random generator (www. random.org) to generate the coordinates of each pore within a square grid.
generator (www.random.org). A characteristic feature of a non-uniform geometry is that one can identify areas with higher than average porosity and areas with lower than average porosity, see Fig. 5(c). Fig. 6 shows the concave dependence of the restriction factor on porosity: the restriction factor decreases faster at lower porosities. This means that a surface with a distribution in porosity is expected to have a lower restriction factor, and thus a lower permeance, compared to a second surface with a uniform porosity equal to the average porosity of the first surface. This will result in a uniformity coefficient that is smaller than one for nonuniform surfaces. Thus, our hypothesis is that for all non-uniform surfaces the uniformity coefficient values will be less than one. One possible exception that we can think of is when there is a significant variation in the thickness of the top layer across the surface. The intrinsic permeance of such a layer will be larger than the permeance calculated from the mean thickness, and hence, analysis based on the mean top layer thickness can result in a uniformity coefficient larger than one. Ghadimi et al. [7] report uniformity coefficients that are significantly larger than one for a number of asymmetric supports. However, these coefficients are calculated using values for the surface porosity and pore size that were obtained from the Dusty Gas model [11]. This model is not suited for asymmetric membranes and does not give reliable results for asymmetric structures, as discussed by Zhu et al. [4,5] and Ghadimi et al. [7]. The model is well suited to polycarbonate track-etch membranes, and the uniformity coefficients obtained by Ghadimi et al. for these supports all are smaller than one.
Fig. 6. Restriction factor calculated from correlation (2) as a function of porosity. The dependence is concave and the restriction factor decreases faster at lower porosities. This indicates that surfaces with non-uniform porosity will have a lower restriction factor, and thus a lower permeance, than a uniform surface with the same average porosity.
surfaces. The experimental data in Table 1 are described accurately by correlation (3) when using a value for the uniformity coefficient of 0.15, which is the ratio of the pore radius obtained from the SEM analysis (about 6 nm, see Fig. 4) and the pore radius obtained from the curve fit (about 40, see Fig. 3). Fig. 5 shows three different surface geometries with one hundred pores, which have the same overall porosity but differ in pore size distribution and pore location distribution. Geometry 5(a) is a uniform square distribution of pores of equal size, geometry 5(b) is a uniform square distribution of two different pore sizes, and geometry 5(c) is a non-uniform distribution of pores of equal size. The pore locations in geometry 5(c) were generated with a public domain random number
4. Characteristics of random distributions Uniform distributions in a two dimensional surface are either in a square or in a hexagonal configuration. In the square configuration, the distance between a pore and each of its four closest neighboring pores is
Fig. 7. Ratio of Largest Distance to Smallest Distance for the four nearest pores for each of the 100 pores in a random distribution. Results shown for ten different distributions (1000 points total). For each distribution the pores are sorted from the smallest ratio (which cannot be smaller than one) to the largest ratio. 4
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Fig. 8. Average distance to the four nearest pores for each of the 100 pores in a random distribution. Results shown for ten different distributions (1000 points total). For each distribution the pores are sorted from the smallest average distance to the largest average distance.
Fig. 9. (a), (b), (c) and (d): Distribution of 16 pores at constant porosity of 5% and a ratio of top layer thickness to pore radius equal to 5. (e): Distribution of 4 pores at porosity of 5% at the same top layer thickness, which means that the ratio of top layer thickness to pore radius equals 2.5. CO2 permeances calculated for composite membranes with a 0.5 μm thick PDMS top layer. The uniformity coefficients are calculated via equation (3) using the restriction factors obtained from the CFD simulations. Table 2 Restriction factors and uniformity coefficients obtained from CFD simulations for composite membranes with varying normalized top layer thicknesses (top layer thickness/pore radius) and with different surface pore patterns of the porous support. Surface porosity of all support membranes is 5%.
Pattern #2
Pattern #3
Pattern #4
Normalized Top Layer Thickness (−)
Restriction Factor from CFD (−)
1 2 5 10 20 1 2 5 10 20 1 2 5 10 20
0.097 0.150 0.287 0.444 0.615 0.096 0.145 0.270 0.423 0.594 0.093 0.133 0.244 0.387 0.548
Uniformity Coefficient from Eq. (3) (−)
Restriction Factor (−) at Uniformity Coefficient Equal to
0.844 0.886 0.911 0.906 0.878 0.825 0.841 0.833 0.834 0.878 0.770 0.734 0.720 0.720 0.677
0.101 0.153 0.287 0.445 0.624
0.91
0.83
0.096 0.144 0.270 0.423 0.602
0.72
Fig. 10. Restriction factors obtained from CFD simulations for various composite membrane configurations, compared to the restriction factors obtained from equation (3), see Table 2. For each pattern the uniformity coefficient is taken to be independent of the top layer thickness, yielding excellent agreement between the CFD simulations and equation (3).
0.090 0.131 0.244 0.387 0.564
to quantify the difference between “non-uniform” and “uniform”. We created ten different random distributions of 100 pores in the same way the random distribution in Fig. 5(c) was created and
the same. In the hexagonal configuration, the distance between a pore and each of its six closest neighboring pores is the same. This will not be true for random and/or non-uniform distributions and this offers a way 5
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the four nearest pores. For the uniform square and hexagonal distributions this average is the same for all pores and we designate this distance to be one in arbitrary units. In a random distribution, the distance will be smaller than one for some pores and larger than one for others. Fig. 8 gives the distances for each pore in ten different random distributions of 100 pores each and shows that the random distribution is distinctly different from uniform distributions. In the random distributions about 70% of the pores are in areas with a “lower than average” porosity, whereas about 30% are in areas with a “larger than average” porosity. An interesting feature of both Figs. 7 and 8 is that the ten different random distributions have very similar profiles. This suggests that even for only one hundred pores an individual random distribution regresses towards an “average” random distribution. Considering that even a low porosity membrane has in excess of one billion pores per cm2 membrane area, we can be assured that the random nature of the pore distribution in typical support membranes will not result in membrane stamp to stamp variations, let alone in variations in commercial membrane modules.
Fig. 11. Restriction factors obtained from CFD simulations for various composite membrane configurations, compared to the restriction factors obtained from equation (3), see Table 3. The uniformity coefficient is taken to be independent of the porosity, yielding reasonably good agreement between the CFD simulations and equation (3).
5. Simulations for non-uniform pore distributions We carried out a number of CFD simulations to investigate the effect of non-uniform pore distribution on the restriction factor. We do not have access to sufficient computation power to simulate random distributions of 100 or more pores, so we limited ourselves to distributions that, while not uniform, are not random and form repeatable patterns for which CFD simulations can be performed more easily.
Table 3 Restriction factors and uniformity coefficients obtained from CFD simulations for composite membranes with pore pattern #4 and with varying surface pore size (and thus varying porosity) of the porous support. Normalized thickness is 5 in all cases. Porosity (%)
Restriction Factor from CFD (−)
Uniformity Coefficient from Eq. (3) (−)
Restriction Factor at Uniformity Coefficient Equal to 0.72 (−)
0.5 1 2 5 10
0.033 0.063 0.116 0.244 0.390
1.03 0.945 0.849 0.720 0.597
0.024 0.050 0.101 0.244 0.431
5.1. Effect of pore location CFD simulations were performed for the five pore patterns shown in Fig. 9. Distributions (a) and (e) are uniform square distributions of pores; the pore radius in (e) is double the radius in (a), and both are covered by equation (2). Fig. 9 gives the relative permeances for the five distributions, as well as the uniformity coefficient calculated via equation (3). The sequence (a) through (d) demonstrates that a less uniform distribution results in lower permeances, which is reflected in lower values of the uniformity coefficient. Distribution (e) is the logical end point of the sequence where every set of four pores has merged into one pore with double radius, which makes it once again a fully uniform distribution. CFD simulations were performed for patterns 2, 3 and 4 for six different relative top layer thicknesses. The data in Table 2 and Fig. 10 show that the restriction factors obtained from the CFD simulations closely match the restriction factors obtained from equation (3) using a constant value of the uniformity coefficient for each pattern. We thus conclude that the uniformity coefficient can be considered independent of the thickness of the top layer (see Fig. 11).
calculated for each pore the distances to its four closest neighbors. We then calculated the ratio of the larger distance to the smaller distance. In any uniform distribution this ratio is equal to one for every pore. Fig. 7 shows the ratios for the ten different random distributions; for each distribution the pores are sorted from the smallest ratio (which cannot be smaller than one) to the largest ratio. The largest values for the ratio are quite large, in excess of ten. The difference between the random distributions and the uniform distribution is striking. This allows us to define a “mostly uniform distribution” as one where, for a significant majority of all pores, the ratio shown in Fig. 7 is less than a specific threshold, for example less than a value of two. An alternative measure of non-uniformity is the average distance to
Fig. 12. Restriction factor obtained through CFD for a square distribution of pores at 5% surface porosity. (a) all pores have the same diameter, (b) 50% of the pores have twice the diameter of the other pores, (c) 25% of the pores have a diameter 4 times the smallest pores, 25% of the pores have a diameter 3 times the smallest pores, 25% of the pores have a diameter 2 times the smallest pores. All permeances are CO2 permeances calculated for composite membranes with a 0.5-μm thick PDMS top layer.
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Also possible, and perhaps very likely, is that the permeances of the lower porosity membranes are lowered by penetration of the top layer material into the pores. We used CFD simulations to determine the restriction factors for a number of cases with varying degrees of pore penetration. The results are given in Table 5 and Fig. 13 gives the concentration profiles for two different degrees of pore penetration. Even small pore penetration depths reduce the permeance of the composite membranes and the effect is stronger at lower porosities. In addition to the CFD analysis we used a resistances-in-series model to calculate the permeances of composite membranes if the top layer penetrates into the surface pores. In this approach we simply add the resistance of the penetration layer to the resistance of the top layer:
Table 4 Restriction factors reported Zhu et al. [4] for composite membranes prepared with track-etch polycarbonate support membranes, plus the uniformity coefficients calculated by Ghadimi et al. [7]. Support
Pore Radius (nm)
Porosity (%)
Top Layer Thickness (nm)
Restriction Factor [4]
Uniformity Coefficient [7]
PC-50
51
2.7
25
3.3
PC-15
14
1.1
PC-7.5
5.8
0.6
0.14 0.15 0.13 0.18 0.029 0.032 0.037 0.009 0.010
0.828
PC-25
160 220 99 200 76 100 120 82 90
0.520 0.256
1 Permeance
0.0483
H Pi
=
Permeance=
(
Hp
+ H Pi
Pi Hp
+
(
Pi
)
1
=
Hp / H
Pi H
)
(
1
+
Hp / H
)
1
1
5.2. Variation in porosity
Restriction Factor=
Additional CFD simulations were performed for pattern #4 with a range of values for the surface porosity, which were obtained by changing the pore radius values. The normalized thickness of the top layer was kept constant at 5. As expected, the restriction factor increases rapidly with increasing porosity, see Table 3. The uniformity coefficient starts at a value very close to one for the lowest porosity of 0.5% and decreases with increasing porosity. However, Fig. 11 shows that good agreement between equation (3) and the CFD results is obtained at a constant uniformity coefficient, at least, within the range of relatively low porosities considered here.
where Hp is the penetration depth of the top layer material into the pore. This equation is surprisingly accurate when compared to the CFD results, see Fig. 13. The explanation for this can be found in Fig. 14, which shows that the concentration profiles in the pore are close to be being horizontal, which indicates that diffusion through the cross-
1
+
(5)
Table 5 Restriction factors obtained from CFD simulations for composite membranes. Included in the simulations are three different pore penetration depths for the top layer.
5.3. Non-uniform pore size In our previous paper [2] we performed CFD simulations for a number of square distributions of pores with varying diameter sizes. At constant overall porosity and at constant relative top layer thickness (normalized to the average pore diameter), the restriction factors obtained through CFD were within 7% of each other. We therefore tentatively concluded that pore size distributions did not affect the restriction effect. However, at that time we had overlooked the subtle, but consistent, trend that all distributions of pores with varying pore consistently have smaller restriction factors than distributions of pores of equal diameter, see Fig. 10 for some examples. Our revised conclusion is that a distribution in pore sizes leads to lower restriction factors, and thus to lower permeances. However, based on Figs. 5 and 12, we can conclude that the effect of non-uniform pore sizes is small compared to the effect of a non-uniform distribution in pore location.
Porosity
Top Layer Thickness/ Pore Radius
Pore Penetration Depth/Top Layer Thickness
Pore Penetration Depth/Pore Radius
Restriction Factor (CFD Simulation)
Restriction Factor (Equation (5))
2%
2
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.064 0.058 0.053 0.046 0.133 0.115 0.097 0.077 0.234 0.185 0.143 0.105
0.064 0.060 0.055 0.048 0.133 0.113 0.099 0.079 0.234 0.188 0.146 0.107
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.163 0.147 0.133 0.116 0.306 0.271 0.231 0.184 0.469 0.389 0.311 0.235
0.163 0.153 0.140 0.123 0.306 0.273 0.235 0.190 0.469 0.396 0.320 0.242
0 0.02 0.05 0.1 0 0.02 0.05 0.1 0 0.02 0.05 0.1
0 0.04 0.1 0.2 0 0.1 0.25 0.5 0 0.2 0.5 1
0.311 0.286 0.260 0.228 0.510 0.465 0.405 0.334 0.678 0.594 0.501 0.399
0.311 0.293 0.269 0.238 0.510 0.464 0.407 0.338 0.678 0.598 0.507 0.405
5
10
5%
6. Comparison with data for track-etch support membranes
2
5
Ghadimi et al. [7] have analyzed the permeances of composite membranes for a number of different support membranes. Most of the support membranes considered by them are asymmetric structures prepared by phase inversion, but they also analyzed data reported by Zhu et al. [4] for track-etch polycarbonate support membranes. Here we will focus on the data obtained with track-etch support membranes because only for these supports are reliable pore size and porosity data available, which are essential for the calculation of the uniformity coefficient. The data are given in Table 4 and show a significant decrease in uniformity coefficient for lower porosities. Our CFD analysis actually predicts the opposite: an increase in uniformity coefficient with decreasing porosity, but this assumes the distribution in pore location remains the same. Thus, one explanation for the trend shown in Table 4 is that for this family of support membranes the reduction in pore size coincides with a significant reduction in uniformity of the pore location distribution.
10
10%
2
5
10
7
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Fig. 13. Restriction factors obtained from CFD simulations compared with the restriction factors obtained from correlation (2) and the resistances-in-series equation (5).
Fig. 14. Concentration profiles for a top layer that penetrates into the pore opening of the support membrane. Penetration depth, Hp, is 5% (a) and 20% (b) of the top layer thickness, H.
section of the pore is distributed reasonably even, which then validates the use of the resistances-in-series model. The distribution will be the least even for small penetration depths, but in that case the contribution of pore penetration to the overall resistance will be small and equation (5) still will be accurate. From equation (5) we see that even a pore penetration depth that is much smaller than the top layer thickness will significantly reduce the overall permeance if the porosity is small. For example, the track-etch support PC-7.5 has a porosity of only 0.6%, which means that a pore penetration depth of 1 nm adds a permeation resistance equivalent to the intrinsic resistance of a 160 nm top layer.
and an empirical correlation was developed to accurately quantify the reduction in permeance. Experimental permeance data were obtained of composite membranes with top layers of different thicknesses coated on a support prepared by the common phase inversion process. The permeances follow the trend predicted by the CFD simulations: permeances are less than inversely proportional with the top layer thickness. However, the permeances are even lower than predicted by the correlation. This is because the support is not uniform in surface pore location and surface pore size, which was an assumption, made in the previous CFD simulation effort. The new CFD simulations presented in this work with non-uniform surface geometries confirm that non-uniformity reduces the permeance. This effect can be incorporated into the empirical correlation through a uniformity coefficient. Our analysis strongly suggests that the uniformity coefficient is always smaller than one for non-uniform surfaces. The CFD results also show that the uniformity coefficient is a function of porosity, but at constant porosity is relatively independent of the normalized thickness of the top layer and of the distribution in pore size.
7. Conclusions The porous support restricts diffusion in the top layer of composite membranes because molecules can exit the layer only where a pore is present. Previously, we have used computational fluid dynamics (CFD) to simulate this geometric restriction for uniform pore configurations 8
Journal of Membrane Science 594 (2020) 117465
P. Hao, et al.
Declarations of interest
References
None.
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Acknowledgement The authors acknowledge Dung Nguyen for preparing the PDMS composite membranes and measuring their gas permeances. We also thank the U.S. Department of Energy, National Energy Technology Laboratory for financial support under contract DE-FE0031596. Nomenclature H Hp Np NR P R σ
Thickness of top layer (nanometer) Penetration depth of the top layer material into the pore (nanometer) Number of pores per surface area Restriction Number, equation (2) Permeability coefficient (Barrer = GPU.micron) Radius of pore (nanometer) uniformity coefficient, equation (3) Restriction Factor, Equation (2) Porosity Normalized thickness, equal to H/R
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