Pervaporation from a Dense Membrane - American Chemical Society

diffusion type) models of pervaporation, suggests an important role for the membrane-permeant interactions in nanopores that can lower the equilibrium...
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Pervaporation from a Dense Membrane: Roles of Permeant-Membrane Interactions, Kelvin Effect, and Membrane Swelling Ashutosh Sharma,* Sumesh P. Thampi, Satyanarayana V. Suggala, and Prashant K. Bhattacharya Department of Chemical Engineering, Indian Institute of Technology, Kanpur-208016, India Received February 1, 2004. In Final Form: March 18, 2004 Dense polymeric membranes with extremely small pores in the form of free volume are used widely in the pervaporative separation of liquid mixtures. The membrane permeation of a component followed by its vaporization on the opposite face is governed by the solubility and downstream pressure. We measured the evaporative flux of pure methanol and 2-propanol using dense membranes with different free volumes and different affinities (wettabilities and solubilities) for the permeant. Interestingly, the evaporative flux for different membranes vanished substantially (10-75%) below the equilibrium vapor pressure in the bulk. The discrepancy was larger for a smaller pore size and for more wettable membranes (higher positive spreading coefficients). This observation, which cannot be explained by the existing (mostly solutiondiffusion type) models of pervaporation, suggests an important role for the membrane-permeant interactions in nanopores that can lower the equilibrium vapor pressure. The pore sizes, as estimated from the positron annihilation, ranged from 0.2 to 0.6 nm for the dry membranes. Solubilities of methanol in different composite membranes were estimated from the Flory-Huggins theory. The interaction parameter was obtained from the surface properties measured by the contact angle goniometry in conjunction with the acid-base theory of polar surface interactions. For the membranes examined, the increase in the “wet” pore volume due to membrane swelling correlates almost linearly with the solubility of methanol in these membranes. Indeed, the observations are found to be consistent with the lowering of the equilibrium vapor pressure on the basis of the Kelvin equation. Thus, a higher solubility or selectivity of a membrane also implies stronger permeant-membrane interactions and a greater retention of the permeant by the membrane, thus decreasing its evaporative flux. This observation has important implications for the interpretation of existing experiments and in the separation of liquid mixtures by pervaporation.

1. Introduction Pervaporation is a membrane separation process applied to separate liquid mixtures. The separation is carried out using dense membranes. The feed liquid diffuses through the membrane and vaporizes on the other side, where the applied pressure is lower than the saturation vapor pressure. Recently, pervaporation has received considerable attention from both the fundamental and applied aspects because of the tremendous upsurge in the developments of newer homogeneous membranes.1 The mass transport mechanism through the membrane is complex because of the high interaction between the liquid feed components and the membrane, resulting in substantial swelling of the membrane. The solution-diffusion model is the widely accepted mass transport mechanism for pervaporation.2-4 As expected, the driving force (and flux) for vaporization at the downstream side should vanish at a downstream vapor pressure equaling the saturation vapor pressure of the permeating component.5 However, the existing experimental results and their interpretations2-11 have been somewhat unequivocal on this simple * Corresponding author. Tel.: +91-512-2597026. Fax: +91-5122590104. E-mail: [email protected]. (1) Bru¨schke, H. E. A. In Membrane Technology in the Chemical Industry; Nunes, S. P., Peinemann, K.-V., Eds.,Wiley-VCH: Weinheim, Germany, 2001. (2) Lee, C. H. J. Appl. Polym. Sci. 1975, 19, 83. (3) Mulder, M. H. V.; Smolders, C. A. J. Membr. Sci. 1984, 17, 289. (4) Kataoka, T.; Tsuru, T.; Nakao, S.; Kimura, S. J. Chem. Eng. Jpn. 1991, 24, 326. (5) Greenlaw, F. W.; Shelden, R. A.; Thompson, E. V. J. Membr. Sci. 1977, 2, 141. (6) Okada, T.; Matsuura, T. J. Membr. Sci. 1991, 59, 133.

expectation because, operationally, it is the total downstream pressure (air plus vapor), rather than the partial vapor pressure, that is maintained by a vacuum pump. To complicate the situation, air leaks are invariably present. Recently, Vallieres et al.12 systematically showed the importance of air leaks in a pervaporation experimental setup by passing inert gas at the downstream side of the membrane. To have experiments that can differentiate between rival models and their interpretations, it is, therefore, vital to correctly estimate the downstream partial vapor pressure which also takes into account the possibility of air leaks, because vapor flux can be quite small. The experiments reported here have been designed and interpreted with special attention to these details. On the theoretical side, none of the existing models of pervaporation from dense membranes have considered the possibility of the equilibrium vapor pressure itself being reduced as a result of the permeant-membrane interactions. Of course, it is clear that a greater degree of interaction (also reflected in a greater solubility, affinity, selectivity, and wettability of the membrane) should also increase the propensity of the membrane to retain the permeant, thus decreasing its flux. In anthropomorphic (7) Wu, W. S.; Lau, W. W. Y.; Rangaiah, G. P.; Sourirajan, S. J. Colloid Interface Sci. 1993, 160, 502. (8) Yoshikawa, M.; Handa, Y. P.; Cooney, D.; Matsuura, T. Makromol. Chem. Rapid Commun. 1990, 11, 387. (9) Tyagi, R. K.; Fouda, A. E.; Matsuura, T. Chem. Eng. Sci. 1995, 50, 3105. (10) Wessling, M.; Werner, U.; Hwang, S. T. J. Membr. Sci. 1991, 57, 257. (11) Feng, X.; Huang, R. Y. M. Can. J. Chem. Eng. 1995, 73, 833. (12) Vallieres, C.; Favre, E.; Roizard, D.; Bindelle, J.; Sacco, D. Ind. Eng. Chem. Res. 2001, 40, 1559.

10.1021/la049725x CCC: $27.50 © 2004 American Chemical Society Published on Web 04/22/2004

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terms, stronger bonds of friendship engender greater pangs of separation! Put another way, higher solubility and wettability of permeant in highly confining a membrane pores should correlate with a lower equilibrium vapor pressure. In this study, we propose to quantify the above ideas by the use of the Kelvin equation,13 applied for the first time to the pervaporation process:

[

]

2Viγi cos θ 0 Pim ) Poi exp RTr

(1)

where θ, γ, V, R, and T are the equilibrium contact angle of the permeating liquid i with the membrane, permeant surface tension, permeant molar volume, universal gas 0 constant, and temperature, respectively. Further, Pim 0 and Pi are the equilibrium vapor pressure of permeant in the membrane m and bulk vapor pressure of the permeant, respectively. The pore radius r is in the swelled state. Homogeneous membranes have “pores” in the form of irregular free spaces in the polymer matrix.14 The size of such pores should increase in a good solvent as a result of the swelling of the polymer.15 The Kelvin equation predicts a decrease in the equilibrium vapor pressure (compared to its bulk value) for wettable membranes (θ < π/2) that show high solubility and affinity for the permeant. Indeed, the experiments reported in this study show a clear and quantifiable decrease in the downstream vapor pressure where the evaporative flux becomes 0. Of course, this decrease depends on the degree of confinement (pore radius, r), which, admittedly, cannot be directly measured for the dense membranes having sub-nano-size pores in the swelled state. We have, therefore, characterized the pores by positron annihilation in the dry membranes and further found a good correlation of the “wet” pore volume (which appears in the Kelvin equation) with solubility because of the greater propensity of a membrane to swell when in a better solvent. Therefore, the experimental objective of the present work is to carry out single component pervaporation experiments at controlled downstream pressures and wellcharacterized partial vapor pressures (by accounting for the air leaks) to find the depression in the equilibrium vapor pressure due to the membrane-permeant interactions. On the theoretical side, we explore the possibility of interpreting the experiments in the framework of the Kelvin equation. Toward the above ends, methanol was chosen as the feed component with a variety of dense membranes differing in their pore sizes, solubilities, and wettabilities. The pore sizes and contact angles were estimated using the positron annihilation technique and contact angle goniometry, respectively. The enlargement of the pore size due to membrane swelling is correlated with the solubility of methanol in the membrane. As noted by Gonza´liz and Uribe,16 it is not practical to directly measure the amount of sorption in composite membranes because of the difficulty to discriminate between the active layer and its porous support layer. Thus, in this study, solubility is estimated by the modified Flory-Huggins theory, which accounts for the membrane elasticity. The Flory-Huggins (13) Hunter, R. J. Foundations of colloidal science; Clarendon press: Oxford, 1987; Vol. 1. (14) Sourirajan, S.; Matsuura, T. Reverse Osmosis/Ultrafiltration Process Principles; National Research Council Canada: Ottawa, 1985. (15) Fujita, H. Fortschr. Hochpolym.-Forsch. 1961, 3, 1. (16) Gonza´lez, B. G.; Uribe, I. O. Ind. Eng. Chem. Res. 2001, 40, 1720.

Table 1. Base Polymers and Densities of Membranes

membrane

base polymer

HR98PP polypropylene CA PERVAP 2256 likely PVA with incorporation of polar moleculesa PERVAP 1070 PDMS with silicalite-filled zeolite PERVAP 2201 PVA highly cross-linked

molecular density of weight of amorphous monomer portion (g/cm3) 56 44

0.85 1.32 1.26

74

0.98

44

1.26

a Other membranes of the PERVAP 22 series are known to be PVA-based.21

interaction parameter is evaluated from the polar acidbase surface tension components obtained from the contact angle goniometry using the standard probe liquids.17-20 For a laboratory cast cellulose acetate (CA) membrane without a support layer, direct measurement of the solubility was possible and was found to be in good agreement with the estimated value, as discussed later in Table 6. 2. Experimental Methods and Materials 2.1. Materials. Analytical grade methanol (Merck, India), 2-propanol (Qualigens, India), acetone (Ranbaxy, India), formamide (Loba Chemie, India), ethylene glycol (E-Merck, India), diodomethane (SD fine, India), and CA (Jams Chemicals, Bombay) were used for the work. Double distilled water was used for the preparation of the calibration curve and contact angle measurements. The commercial composite membrane PERVAP was obtained from Sulzer Chemtech, Germany. The dense pervaporation membrane was a very thin (0.5-2 µm) separating layer on top of a porous support (70-100 µm), which in turn rested on top of a polymer fleece (nonwoven fabric of thickness 100 µm) providing the structural support. Another composite membrane HR-98-PP was obtained from Danish Separation Systems, Denmark. Characteristics of the membranes (nature of the polymers)21 along with density values22 are reported in Table 1. One type of CA pervaporation membrane was also cast in our laboratory without the support layers as detailed below. All the membranes used in this study are dense pervaporation membranes with the exception of HR-98-PP, which is a reverse-osmosis/nanofiltration membrane. 2.2. Membrane Preparation. CA polymer (17 g) was dissolved in a mixture of acetone (68 g) and formamide (15 g) and kept overnight for the removal of entrapped air. The casting of the membrane was carried out on a glass plate using a modified thin film applicator (ACME-make, India). After around 48 h of solvent evaporation at room temperature, the membrane together with the base glass plate was placed in a vacuum oven for another 4 h for the removal of residual traces of solvent. Finally, the membrane was gently peeled from the glass plate. 2.3. Sorption Measurements. A preweighed dry membrane of CA was kept in a conical flask that contained methanol for sorption purposes. The flask was kept in a water shaker bath (model SW-23, Julabo, Germany) under 200 rpm for 6-7 days at 20 °C. The membrane in the conical flask was taken out at regular intervals and wiped with tissue papers for the removal of the adhered liquid. The wet weight of the membrane was measured. The procedure was repeated until three consecutive readings of wet membrane weights were found to be within 1% of each other. The percent sorption is based on the permeant absorbed divided by the dry weight. (17) van Oss, C. J. Interfacial Forces in Aqueous Media; Marcel Dekker: New York, 1994. (18) Leo´n, V.; Tusa, A.; Araujo, Y. C. Colloids Surf., A 1999, 155, 131. (19) Rankl, M.; Laib, S.; Seeger, S. Colloids Surf., B 2003, 30, 177. (20) Kwok, D. Y. Colloids Surf., A 1999, 156, 191. (21) Jonquie`res, A.; Cle´ment, R.; Lochon, P.; Ne´el, J.; Dresch, M.; Chre´tien, B. J. Membr. Sci. 2002, 206, 87. (22) Van Krevelen, D. W. Properties of Polymers; Elsevier: Amsterdam, 1990.

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Figure 1. Schematic diagram of the pervaporation experimental setup. 2.4. Positron Annihilation Lifetime (PAL) Measurements. The PAL measurements were carried out using a fastfast system having a resolution of 300 ps (full width at halfmaximum for the 60Co prompt γ-rays, under 22Na window settings). The positron source was prepared by depositing around 2 µCi aqueous 22NaCl on a thin aluminum foil (thickness ∼ 12 µm) and covering it with an identical foil. The source was sandwiched between 13 layers (on each side) of the polymeric membrane, which were stacked together. The separating layer portion (dense membrane plus the porous support) was peeled out from the woven fabric base of the commercial membranes. Membranes prepared in our laboratory were sufficiently thick to absorb 99.9% of the positrons. The source-sample sandwich was placed between two NE111 scintillators coupled to RCA 8575 tubes. The anode signals were processed in ORTEC constant fraction differential discriminators, and an ORTEC time-to-pulse height converter generated the lifetime distribution spectra, which were recorded in a multichannel analyzer. All the measurements were made at room temperature (24 °C). Approximately 1 million counts were collected in each spectrum, and four spectra were measured for each sample. 2.5. Contact Angle Measurements. Membranes were thoroughly dried in a vacuum oven and stored in a vacuum desiccator until their use. Advancing equilibrium contact angles of methanol and the common probe liquids of the acid-base analysis (water, ethylene glycol, and diiodomethane) on the membranes were measured by the sessile drop method using a contact angle goniometer (Rame-Hart Imaging System, U.S.A.). Flat pieces of membranes were mounted in a stainless steel holder. A glass syringe with a stainless steel needle was used to place liquid drops on the membrane. The contact angles of the probe liquids at room temperature (∼20-25 °C) were measured by capturing the image with a video camera starting immediately after depositing the drop within a few seconds. Within the first 30 s of deposition, about 20 contact angles were measured at intervals of about 1 s, and their average was obtained. All of the probe liquids gave contact angles that were stable within the first minute of their deposition and changed only by a few degrees over this time. Further, four or five such measurements were made for each liquid on the same membrane, and the average value was used in calculations. All the membranes were completely wetted by methanol droplets, which spread rapidly and completely on the membranes (zero contact angle). Indeed, as will be shown later in Table 7, complete wettability by methanol is consistent with the independent predictions based on the surface tension components of membranes, which give positive spreading coefficients on all the membranes. It may be noted that the contact angle measurements are performed at room temperature even though the membrane flux was measured at a slightly elevated temperature (∼40 °C). This is necessitated by the fact that the apolar and acid-base surface tension components of the probe liquids (as given in Table 4 later) are not known with any certainty at higher temperatures. 2.6. Pervaporation Experimental Setup. Figure 1 shows the experimental setup designed and fabricated in our laboratory for the pervaporation measurements. The pervaporation test cell was made of glass and had specially designed flanges to secure

the membrane with an effective membrane area of 50.6 cm2. The membrane was kept on a highly porous stainless steel support with the shiny dense polymeric layer (for commercial membranes) facing the feed solution. Initially, a fixed volume of feed component was taken in the feed chamber. To avoid nonisothermal operation23 around the cell, both the upstream and the downstream sides of the cell were heated separately by providing the heating mantels on the cell. The temperatures on both sides were controlled by connecting heating mantels through a proportionalintegral-differential controller device (Fuji, Japan). The membrane upstream side was kept at the atmospheric pressure, and a partial vacuum was maintained at the downstream side by a vacuum pump (Vacuum Techniques, Bangalore). Total downstream pressure was regulated with an air inlet using a stainless steel microneedle valve, located between the condensers and the vacuum pump. Downstream pressure was measured near the cell as well as near the pump using Meclod/ Pirani/Capillary columns. The condenser system consisted of two traps that could be used alternately, allowing for the collection of the permeated stream continuously without interruption of the operation during the weight measurements. The permeated vapor was condensed in one of the traps, which were kept in Dewar flasks filled with the liquid nitrogen. The frozen permeant was collected periodically. The cold traps were brought to room temperature for measuring its weight using a five-decimal balance to determine the mass flux. The experimental measurements were started only after keeping the cell for 12-15 h under the maximum attainable vacuum (about 0.1 mm of Hg) to observe the leakage of air. Before switching over to a new trap by changing the vapor (Figure 1), the valves near the cell were closed. The completely evacuated condenser was then kept in the Dewar flask and the valve was opened. This procedure ensures that no residual moisture is present in the line as well as in the condenser at the time of starting the experiment. For each downstream pressure, the steady-state flux was usually reached within 3-5 h, and the measurements were completed within about 3-12 h, depending on the flux (amount collected). As explained above, the presence of water in permeant (air plus vapor) was expected and, hence, it was analyzed using a gas chromatograph (Nuchon, India) with a thermal conductivity detector. Chromosorb-102 and Puropack-Q were used as the main and reference columns, respectively. Hydrogen was used as a carrier gas. The oven, detector, and injector temperatures were 116, 160, and 160 °C for methanol and 200, 210, and 210 °C for 2-propanol to get distinct chromatographic peaks. To minimize the measurement errors, an average of three consecutive readings were taken after reaching the steady state for each operating downstream pressure. For each permeant sample, a minimum of three water concentrations were measured and averaged. The average errors in the total permeation mass flux and water concentration were around (2 and (5%, respectively. Further, the wet and dry bulb temperatures were tracked to obtain the room humidity. (23) Hillaire, A.; Favre, E. Ind. Eng. Chem. Res. 1999, 38, 211.

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Table 2. Free Volume Parameters and Pore Radii of Membranes

Table 4. Surface Tension Components of Permeants and Probe Liquids

membrane

τ3 (ns)

I3 (%)

r (nm),

component

HR98PP CA PERVAP 2256

1.96 ( 0.01 2.16 ( 0.02 1.73 ( 0.06 6.19 ( 0.36 2.32 ( 0.28, τ4: 4.53 ( 0.48 1.64 ( 0.04

15.2 ( 0.2 18.8 ( 0.3 8.8 ( 0.2 1.1 ( 0.1 10.6 ( 1.5 I4: 5.1 ( 2.0 8.3 ( 0.4

0.324 0.345 0.298 0.606 0.360 0.520 0.288

water ethylene glycol diiodomethane methanol

PERVAP 1070 PERVAP 2201

Table 3. Advancing Contact Angles of Probe Liquids on Various Membranes membrane

water

ethylene glycol

diiodomethane

HR98PP PERVAP 1070 PERVAP 2256 PERVAP 2201 CA

67.4 83.3 80.1 53.4 43.6

37.6 72.5 51.7 35.8 31

46.6 74.0 33.6 43.0 30.7

γLW (mJ/m2) γ+ (mJ/m2) γ- (mJ/m2) γ (mJ/m2) 21.8 29.0 50.8 18.2

25.5 1.9 0.0 0.06

25.5 47.0 0.0 77.0

72.8 48.0 50.8 22.5

Table 5. Estimated Surface Tension Components of the Membranes membrane HR98PP PERVAP 1070 PERVAP 2256 PERVAP 2201 CA

γLW (mJ/m2) γ+ (mJ/m2) γ- (mJ/m2) γ (mJ/m2) 36.126 20.677 42.644 38.048 43.922

0.691 0.070 0.005 0.165 0.001

12.858 12.795 5.476 29.225 39.157

42.091 22.567 42.963 42.447 44.395

Table 6. Flory-Huggins Interaction Parameter and Solubility of Methanol in Various Membranes

3. Results and Discussion

membrane

γim

χim

φpm

solubility (g/100 g of polymer)

3.1. Estimation of Dry Membrane Pore Sizes. The lifetime data were analyzed using PATFIT-88 programs.24 Source correction was done for all spectra. PAL spectra were analyzed in terms of the following three lifetime components: para-positronium (p-Ps) annihilation, τ1; free positron and positron-molecular species annihilation, τ2; and o-Ps annihilation, τ3. While τ1 and τ2 are always of the order of a few hundred picoseconds and are rather immaterial for the calculation of pore sizes, τ3 is of the order of nanoseconds and relates to the polymer free volume or pore size. Each lifetime has an intensity I, corresponding to the fraction of annihilations taking place with the respective lifetimes. The parameters τ3 and I3 corresponding to the decay of o-Ps provide the size-specific information for free volumes and pores. The commercial PERVAP 1070 and PERVAP 2256 membranes provided very poor fits with a monomodal τ3, but a bimodal τ3 fit was entirely satisfactory, indicating a bimodal pore size. The o-Ps lifetimes and intensities thus obtained are reported in the Table 2. The following expression was used to relate the o-Ps pick-off lifetime, τ3, and an equivalent cylindrical pore radius r (Å):25

HR98PP PERVAP1070 PERVAP2256 PERVAP2201 CA

-3.048 -0.121 7.404 2.528 6.618

-0.301 -0.012 0.732 0.250 0.654

0.538 0.560 0.452 0.774 0.605a

79.611 63.096 75.782 18.37 39.03b

τ3 )

1 2rπ 1 1 - r/(r + ∆r) + sin 2 2π r + ∆r

[

( ) (

-1

)]

(2)

where ∆r ()0.196 nm) is the electron layer thickness.26 The pore radii thus obtained are also reported in Table 2. The sub-nanometer pore sizes obtained for dry membranes are in general agreement with the range of values reported for the dense membranes.1 Also, as indicated by the bimodal fits to the commercial membranes, PERVAP 1070 indeed seems to contain two pore sizes.21 Although the chemical structure of PERVAP 2256 is confidential,21 the positron lifetimes similarly indicate the presence of bimodal pores. 3.2. Estimation of Solubilities. The measured advancing equilibrium contact angles of the probe liquids on the various membranes are reported in the Table 3. Because variation of the contact angles within the first 30 s of droplet deposition were not statistically significant, mean values are reported. The contact angles of the probe (24) Kirkegaard, P.; Pedersen, N. J.; Eldrup, M. PATFIT-88: a data processing system for positron annihilation spectra on mainframe and personal computers; Riso National Laboratory: Roskilde, Denmark, 1989. (25) Tao, S. J. J. Chem. Phys. 1972, 56, 5499. (26) Ciesielski, K.; Dawidowicz, A. L.; Goworek, T.; Jasinska, B.; Wawryszczuk, J. Chem. Phys. Lett. 1998, 289, 41.

b a Based on eq 4 because χ im ∼ 0.5. Direct experimental measurement by weighing gives a solubility of 38.37 g/100 g of polymer.

liquids with the membrane were used to calculate the surface tension components (Lifshitz-van der Waals, LW; acid-base; and total) of the membrane by the method of van Osset al.,27,28 as summarized in the appendix (eqs A.1, A.2, and A.6). The surface tension components of the probe liquids were taken from a recent paper19,29 (Table 4). The values of the surface tension components for different membranes thus obtained are reported in Table 5. For the CA membrane, surface tension component values are in agreement with the literature value.17 Some difference may be attributed to the membrane roughness and membrane preparation conditions (e.g., solvent used, degree of actylation). As is usually the case for most polymeric substrates,17,27-29 electron donor parameters (γ-) for all the membranes in Table 5 are found to be significantly higher than their electron acceptor parameters. The Flory-Huggins interaction parameters between methanol and the membranes were calculated using eq A.5 (χim ) -∆GimiSC/kT, where ∆Gimi ) -2γim). The minimum contactable surface area, SC, for the methanol is reported to be 0.2 nm2.30 The contactable surface area was estimated by extrapolating its value in comparison with the SC of a related alkane. The interaction parameters thus calculated are reported in Table 6. The required surface tension parameters of methanol are taken from the literature and are reported in Table 4.17,31 The following Flory-Huggins equation was used to calculate the volume fraction of swollen polymer from the interaction parameter, χip, when χip is substantially larger than 0.5, say greater than 0.7:32 (27) van Oss, C. J.; Good, R. J.; Chaudhury, M. K. Langmuir 1988, 4, 884. (28) van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927. (29) van Oss, C. J.; Giese, R. F.; Good, R. J. J. Dispersion Sci. Technol. 2002, 23 (4), 455. (30) van Oss, C. J. Personal communication. (31) Jasper, J. J. J. Phys. Chem. Ref. Data 1972, 1, 841. (32) Flory, P. J. Principles of polymer chemistry; Cornell University Press: New York, 1953.

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χip ) -

ln(1 - φpm) + φpm φpm2

Sharma et al.

(3)

where φpm is the volume fraction of polymer, p, in the swollen membrane. The above equation, eq 3, is applicable only for the interaction parameter values substantially higher than 0.5.32 In the case of strong feed-membrane interactions (χ ∼ 0.5 or smaller), excessive membrane swelling occurs resulting in an elastic restoring force by the polymer network. In this case, the original Flory-Huggins equation is modified to include the elastic free energy:32

ln(1 - φpm) + φpm + χip(φpm)2 +

ViF [(φ )1/3 Mc pm 0.5φpm] ) 0 (4)

where F is the density of the swollen membrane [)Fpφpm + Fm(1 - φpm)], Mc is the monomer molecular weight, and Fm and Fp are the densities of methanol and polymer, respectively.33 Equation 4 is valid for interaction parameter values of the order of or less than 0.5, including negative values. Finally, solubility is obtained by solubility (g/100 g of polymer) ) 100(1/φpm - 1)(Fm/Fp). The volume fraction of polymer, φpm, and the solubilities of methanol in various membranes were estimated using eq 3 or 4 with the help of data given in the Table 1. Results are summarized in Table 6. Direct sorption experiments measured the solubility to be 38.37 g of methanol/100 g of CA. In the case of the CA membrane, an excellent match between the solubility calculated on the basis of direct weight measurement and on the basis of the contact angle measurements (together with the Flory-Huggins equation) points to the basic soundness of the thermodynamic approach employed here. Because direct measurements of solubility are not possible for the other composite membranes, the surface thermodynamic approach employed here is promising, at least in semiquantitative determinations and especially in comparing the properties of different membranes within a consistent framework. 3.3. Pervaporation Flux. Single-component pervaporation experiments were carried out with methanol and 2-propanol using different membranes at different temperatures. Initially, feed liquid (methanol) was filled in the cell and kept under a complete vacuum for 10 h without collection of permeant. Then, the valves between the membrane and the condenser were closed, leading to accumulation of vapor on the downstream side. The downstream pressure rapidly rose to the saturation vapor pressure of methanol within 3 h as a result of the permeation of methanol, and thereafter, there was a slow increase of pressure. This slow increase may be attributed to the air leaks in the experimental setup. This air leakage occurs as a result of the permeation of air through the free volumes of tubes and, possibly, through the joints, even though all care was taken to make them tight. The same situation was also observed by Valiries et al.12 In view of this and, especially, as shown below, when the suction due to the vacuum pump is turned on, the possible presence of air leaks must be accounted for in the analysis and interpretation of the results. Figure 2 shows the variation of the partial methanol flux against the partial downstream pressure of methanol for the PERVAP 2256, HR98PP, PERVAP 1070, CA, and PERVAP 2201 membranes. In Figure 2, the methanol

flux should be read as the value multiplied by 10-4 for HR98PP and PERVAP 2256, for PERVAP1070 and CA, it is 10-5, and for PERVAP 2201 it is 10-6. The following procedure was used to calculate the partial flux of methanol and the partial downstream pressure of methanol. Concentration of water in permeant was measured by a gas chromatograph, and, hence, the amount of water in permeant was calculated. Air humidity (g of water/g of dry air) was noted from the humidity chart34 with the help of dry and wet bulb temperatures. Thus, the amount of air permeation was calculated, which finally provided the partial fluxes and partial downstream pressure values for the permeant after correcting for the air leaks. Diffusivity of methanol in all the membranes used, except HR98PP, showed an ideal behavior in that the diffusivity (or permeability) is independent of the concentration, because of which the permeant flux decays linearly with the downstream partial pressure, as predicted by the solution-diffusion model with a constant diffusivity.5 A different behavior was obtained for the HR98PP membrane, where a nonlinear decrease in the flux corresponds to a concentration dependent diffusivity.12 The already very low experimental values of the flux in Figure 2 were extrapolated to zero flux conditions, which provided the corresponding equilibrium partial vapor pressures for different membranes. These equilibrium vapor pressures are summarized in Table 7. The equilibrium partial pressures are well below the bulk saturation vapor pressure of the methanol at the temperatures noted in the table. As is argued below, this significant depression of the equilibrium vapor pressure is due to excellent wettability of the membranes by methanol (positive spreading coefficients), which can substantially decrease its equilibrium vapor pressure in tight pores (see the Kelvin equation, eq 1). Similarly, Figure 3 depicts the influence of the 2-propanol downstream pressure on the 2-propanol flux for PERVAP 1070 and HR98PP. Similar to the HR98PPmethanol system, the PERVAP 1070-2-propanol system provided a nonlinear decrease of flux with pressure. PERVAP 1070 is a zeolite-filled polydimethylsiloxane (PDMS) membrane, and its flux behavior is also in agreement with the literature values for the PDMS-2propanol system.12 The bulk saturation pressures of

(33) Meuleman, E. E. B.; Bosch, B.; Mulder, M. H. V.; Strathmann, H. AIChE J. 1999, 45, 2153.

(34) Treybal, R. E. Mass-Transfer Operations, 3rd ed.; McGraw-Hill Publishers: New York, 1986.

Figure 2. Variation of the methanol flux with its own partial downstream pressure for the membranes HR98PP, PERVAP 1070, PERVAP 2201, PERVAP 2256, and CA.

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Table 7. Equilibrium Vapor Pressure and Spreading Coefficient of Methanol and Pore Radii of the Membranes membrane

temp (°C)

p0i (mmHg)

0 (mmHg) pi,pore

Vi (10-5 m3/mol)

γi (mJ/m2)

Sim (mJ/m2)a

rKelvin (nm)

rpositron (nm)

HR98PP PERVAP 1070 PERVAP 2256 PERVAP 2201 CA

40 33 40 55 40

265.8 190.2 265.9 516.3 265.9

196.0 116.0 135.0 58.0 42.0

4.14 4.11 4.14 4.22 4.14

20.98 21.53 20.98 19.80 20.98

24.159 1.158 14.579 20.119 16.797

4.715 1.482 1.670 0.565 0.651

0.324 0.360; 0.520 0.298; 0.606 0.288 0.345

a The calculated spreading coefficient of methanol on all the membranes is positive, signifying complete wetting (zero contact angle), as was indeed observed.

Figure 3. Variation of the 2-propanol flux with its partial downstream pressure for the membranes HR98PP and PERVAP 1070.

2-propanol at 33 and 50 °C are 72.5 and 179.8 mmHg, respectively. The equilibrium partial vapor pressures for 2-propanol, which correspond to the zero flux condition, are again substantially below the bulk saturation pressure (Figure 3) and depend on the membrane material. These results clearly indicate that the permeant-membrane interactions have to be considered during pervaporation, and these should be accounted for in the models used for predicting the flux and selectivity. 3.4. Estimation of the Swelled Membrane Pore Size and Its Comparison with the Pore Size of the Dry Membrane. Pore radii of the membranes can be estimated using a modified Kelvin equation, which is also valid for the case when the contact angle is zero, and, therefore, Young’s equation cannot be used. The fundamental form of the Kelvin equation written in terms of the spreading coefficient, Sim, is

[

0 Pim ) P0i exp -

]

2Vi(γi + Sim) RTr

(5)

where

Sim ) γm - γim - γi

(6)

where γm and γim (eq A.4) are the surface tension of the membrane and its interfacial tension, respectively. The spreading coefficient can also be represented in terms of the equilibrium contact angle, whenever it is nonzero, by Young’s equation:

cos θ ) 1 +

Sim γi

(7)

Thus, for a nonzero contact angle, a more familiar form of the Kelvin equation, eq 1, is readily obtained.

Pore radii were estimated using the Kelvin equation, eq 5, together with the equilibrium vapor pressures and the spreading coefficients reported in Table 7. Spreading coefficients were calculated from eq 6 by using the surface tension parameters given in Table 5. The molar volumes and surface tension values of methanol at different operating temperatures were taken from the literature35 and are also reported in Table 7. The pore sizes thus calculated from the Kelvin equation range from 0.57 to 1.67 nm for different pervaporation membranes in their swelled state (Table 7). The lone exception was the HR98PP reverse-osmosis membrane with a much bigger pore size (∼5 nm). A comparison of pore sizes obtained from the Kelvin equation and positron annihilations (Table 7) shows that both are of the same order, except in the case of HR98PP. There are several issues that need to be considered in these comparisons. The positron annihilation technique provides an average pore size, whereas the Kelvin equation is weighted heavily in favor of larger pore sizes. This is because pervaporation from the smaller pores is ineffective as a result of their very low equilibrium vapor pressure, and the flux is mainly through larger pores. Further, the positron annihilation data and the spreading coefficient measurements were obtained at 20-25 °C, whereas pervaporation data were at higher temperatures (∼40 °C). Thus, temperature effects may be one source of discrepancy between the Kelvin radius of the wet membrane and the positron annihilation radius of the dry membrane. It is known that, as the temperature increases, the thermal motion of the polymer chain segments increases, and this may provide more empty space between polymer molecules. The single most important factor, however, is that the pore size estimated by the positron annihilation technique is only for dry membranes, whereas the Kelvin equation estimates the pore size for wet, swelled membranes with increased free volume/pore size.15 Clearly, a correlation between the swelled pore size and solubility may be anticipated. In what follows, we test the hypothesis that the swelled pore sizes are correlated to the solubility of the permeant in the membrane. 3.5. Relationship between Solubility and the Wet Pore Radius. The swelled volume in a membrane may have a correlation with the solubility of permeant in the membrane. If one assumes the expansion of the free volume, modeled as Kelvin’s cylindrical pores, to be in the radial direction, the pore expansion volume is Vp ∼ [rKelvin2 - rpositron2]. Figure 4 shows the variation of the swelled pore volume, Vp, with solubility. An excellent positive correlation of the swelled pore volume with solubility of the permeant in the membrane is evident. With the exception of the reverse osmosis/nanofiltration membrane (HR98PP) where Vp is very high, a linear correlation for all the pervaporation membranes produces a good fit (solid line in Figure 4). If the data for HR98PP are also included, an exponential correlation, log Vp ∼ solubility, is obtained (35) Daubert, T. E.; Danner, R. P. Physical and thermodynamic properties of pure chemicals; Hemisphere Publishers: New York, 1989.

4714

Langmuir, Vol. 20, No. 11, 2004

Sharma et al.

Acknowledgment. Very useful discussions with Carel J. van Oss are gratefully acknowledged. We also thank Harish C. Verma and V. S. Subrahmanyam for the positron experiments. A.S. acknowledges the support of DST, India, under its Nanosciences program. Appendix The surface tension of the membrane surface γm is composed of an apolar or LW component and a polar or Lewis acid-base (AB) component.17,36,37 AB γm ) γLW m + γm

Figure 4. Relationship between the increase in the membrane volume due to swelling and methanol solubility in the membrane. The filled symbols fitted by a linear correlation are for various pervaporation membranes, whereas the open symbols fitted by a semilog correlation also include the reverse osmosis/ nanofiltration membrane data (0), which shows much more swelling.

as a better fit (broken line on the semilog plot of Figure 4). These results indeed confirm the hypothesis that the swelled pore sizes obtained from the Kelvin equation depend on solubility and may be quite different from the dry pore sizes, except in the membrane-permeant pairs of low solubility. In summary, reduction of the equilibrium vapor pressure (corresponding to the zero flux) due to trapping of the permeant in the nanopores of pervaporative membranes can be predicted by the Kelvin equation if the swelling of the membrane pores due to solubilization of the permeant is also taken into account.

Further, the following equation may be used for the acidbase surface tension component: + γAB m ) 2xγm γm

(A.2)

where γ+ m and γm are the electron acceptor and electron donor surface tension components of the membrane. The following equation36 may be used to estimate the interfacial tension, γim, between the membrane and the permeating component:

LW 2 + + γim ) (xγLW m - xγi ) + 2(xγm γm + xγi γi -

xγ+m γ-i - xγ-m γ+)i )

(A.3)

The change in the free energy per unit area, ∆Gimi, for the process (in which molecules of liquid are initially present in polymer) is given by

∆Gimi ) -2γim

4. Conclusions Influence of downstream pressure on single-component permeation by pervaporation is studied for a variety of membrane-permeant pairs. Because pervaporative flux is usually small, at least in a laboratory experimental setting, air leaks must be quantified for a proper analysis and interpretation of experiments. After accounting for the air leaks, it is shown that the equilibrium vapor pressure, corresponding to the zero flux condition in the pervaporation, can be substantially lower than the bulk saturation vapor pressure of the feed component. The depression in the equilibrium vapor pressure varies systematically with the wettability of the membrane, its surface properties, and its wet pore size. The results can be quantitatively interpreted with the help of the Kelvin equation if the swelled pore size is known, which depends on the solubility of permeant. The pore sizes obtained from the positron annihilation for dry membranes are of sub-nanometer size, as expected for dense pervaporation membranes. In contrast, the pore sizes for wet, swelled membranes obtained from the Kelvin equation are larger, in the range of sub-nanometer to nanometer. As expected, the pore sizes of the dry and wet membranes show better agreement for the membranes in which the solubility of the feed component is less. Interestingly, an almost linear relationship is observed between the solubility and the increase in the membrane volume because of swelling for all the pervaporation membranes studied. The results show the importance of membrane-permeant interfacial interactions leading to a lowering of the equilibrium vapor pressure, which needs to be incorporated in the models of pervaporation flux.

(A.1)

(A.4)

Accordingly, the Flory-Huggins interaction parameter χ,32 which provides the solubility of the feed component in the membrane, in relation to Gibbs free energy,17 may be calculated from the following equation:

χim )

-∆GimiSC kT

(A.5)

where SC and k are the contactable surface area between the permeant molecules and the Boltzmann constant (1.380 48 × 10-23 J/K), respectively. Further, the use of eq A.3 requires the surface tension components of the membrane, and these can be estimated by measuring contact angle θ, using the following version of the YoungDupre´ equation:17 LW + - + (1 + cos θ)γLi ) 2(xγLW m γLi + xγm γLi + xγm γLi) (A.6)

Equation A.6 can be used to evaluate three unknown surface tension parameters by measuring the contact angles of three probe liquids of known properties: diiodomethane (an apolar liquid), water, and ethylene glycol (or glycerol or formamide) were used for this purpose. LA049725X (36) Lee, L. H. J. Colloid Interface Sci. 1999, 214, 64-78. (37) van Oss, C. J.; Good, R. J. J. Macromol. Sci., Chem. 1989, A26 (8), 1183.