Experimental and Theoretical Study of Enhanced Vapor Transport

Apr 27, 2016 - The pressure-driven flow of condensable and permanent gases was studied experimentally for anodic alumina membranes with channel ...
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Experimental and Theoretical Study of Enhanced Vapor Transport through Nanochannels of Anodic Alumina Membranes in a Capillary Condensation Regime Dmitrii I. Petukhov,*,†,‡ Mikhail V. Berekchiian,‡ Evgenii S. Pyatkov,§ Konstantin A. Solntsev,‡,∥ and Andrei A. Eliseev†,‡ †

Department of Chemistry, Lomonosov Moscow State University, Leninskie Hills 1-3, Moscow 119991, Russia Department of Materials Science, Lomonosov Moscow State University, Leninskie Hills, Moscow 119991, Russia § OJSC “OC “Rosneft”, Sofiyskaya Embankment 26/1, Moscow 115035, Russia ∥ Baikov Institute of Metallurgy and Materials Science RAS, Leninskii Avenue 49, Moscow 119991, Russia ‡

S Supporting Information *

ABSTRACT: The pressure-driven flow of condensable and permanent gases was studied experimentally for anodic alumina membranes with channel diameters ranging from 20 to 90 nm depending on feed and permeate pressures. A substantial permeability rise for condensable gases was detected under capillary condensation conditions. A selfconsistent theoretical model for mixed liquid−gas transport through the nanopore was suggested based on the obtained experimental results. The model was used for the evaluation of the pressure drop in a liquid defined by menisci curvatures and determination of the optimal conditions to enhance the membrane performance. The highest pressure difference in a liquid can be obtained by decreasing the entrance meniscus curvature with a simultaneous increase in the curvature of the evaporating meniscus. This case was realized in asymmetric membranes with multiple branching of channels into 5 nm nanocapillaries, resulting in up to 10× enhancement of isobutane permeability (up to 450 m3/(m2·bar·h)) in the capillary condensation regime. Combined with a serious selectivity rise for condensable and permanent components of gas mixtures, this enables mesoporous membranes to be successfully utilized in the self-controlled removal of water and hydrocarbon vapors for the conditioning of natural and associated petroleum gas.

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INTRODUCTION

temperatures and atmospheres. Limited utilization of these membranes for gas separation is explained by low permeability of microporous membranes (such as zeolites), operating as highly selective molecular sieves, and low selectivity of mesoporous membranes functioning in the Knudsen or molecular flow regime. However, selectivity of mesoporous membranes can be drastically improved for gas mixtures containing at least one condensable component guided by adsorption and capillary condensation of vapors in the mesopores.6−8 The flow of individual vapors and vapor containing mixtures in the capillary condensation regime was described previously by a number of authors.8−18 The first demonstration of the capillary condensation effect was given by Rhim and Hwang.9 They provide the permeance measurments of a Vycor porous glass membrane with mean pore diameter of 4 nm for ethane, n-butane, and carbon dioxide at various temperatures and pressures. It was shown that

The interest in utilization of mesoporous and microporous inorganic membranes in different fields of technology has considerably increased during the past few years. Inorganic membranes have been extensively studied due to their potential application in micro-, ultra-, and nanofiltration, pervaparation, etc.1 On the other hand, gas separation technologies generally omit the use of microporous and mesoporous inorganic membranes. Over the last half century, porous inorganic membranes have only been utilized on large-scale in the process of uranium isotope enrichment.2 Due to reproducibility in the manufacturing process and acceptable separation performance, polymeric membranes have dominated the field of industrial gas separation.3 However, current polymer membrane materials have nearly attained the limit in their permeability/selectivity trade-off. Moreover, high pressures of hydrocarbons or CO2 typically induce membrane material plasticization, which seriously diminishes membrane separation efficiency.4,5 On the contrary, mesoporous ceramic membranes are free of these disadvantages and possess high chemical stability at different © 2016 American Chemical Society

Received: March 22, 2016 Revised: April 25, 2016 Published: April 27, 2016 10982

DOI: 10.1021/acs.jpcc.6b02971 J. Phys. Chem. C 2016, 120, 10982−10990

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distance from 50 to 500 nm; porosity 10−30%).23−25 Moreover, the thickness of anodic aluminum oxide (AAO) films can be regulated precisely using conventional electrochemistry methods by controlling total electric quantity.24 Variation of anodization voltage during AAO film formation enables preparation of asymmetric membranes with hierarchical porous structures.26 Earlier anodic alumina films were successfully applied as membranes for gas separation,27,28 flow-through membrane catalysts,29 ultrafiltration,30 membrane emulsification,31 or electroosmotic pumping.32 Thus, in the present study we focused on experimental determination of condensable and noncondensable gas transport parameters through anodic alumina membranes within the capillary condensation regime and theoretical description of the process of condensable gas permeation.

the vapor permeance increases with increasing mean pressure until it goes through a maximum and then decreases before the saturated vapor pressure is attained. On the contrary, the permeance for permanent gases gave no significant variation with mean pressure. Lee and Hwang have extended this study by measuring the permeance of a Vycor membrane for Freon-113 and water vapors.11 With reference to experimental data, Rhim and Hwang suggested and Lee modified a theoretical model proposing six different ways to form a capillary condensate in a pore, depending on the upstream and downstream pressures. The prospect of using capillary condensation for separation of gaseous mixtures containing condensable and noncondensable gases is shown in ref 10. A separation factor of 27 was obtained for 5 μm thick γ-Al2O3 supported membrane for C3H6−N2 mixture with the highest propylene permeance of 241.8 m3/(m2· bar·h). This ultimate value is 3 orders of magnitude higher than propylene permeance through a 200 μm PDMS membrane at −20 °C: 0.43 m3/(m2·bar·h) with a C3H6/N2 separation factor of 200.19 Moreover, a capillary condensation process was suggested for application in the dehumidification of gases, desaturation from heavy hydrocarbons,8,15 removal of methanol from methanol−hydrogen mixtures,16 and even for separation of lower hydrocarbons from natural gas20 and associated petroleum gas conditioning.21 Today the generally accepted model of capillary condensation is based on the assumption that the condensation of vapors in the pore occurs according to the Kelvin equation. At pressures below capillary condensation pressure, only gas diffusion and surface flow contribute to an overall flux. When capillary condensation occurs, a part of the pore length is filled with condensate and the overall flux of vapor can be calculated from the equality of matter flux through the liquid phase (using the Poiseuille equation) and through the gas phase (using the Knudsen equation). Detailed description of vapor transport through Vycor, γ-alumina, or track-etched membranes in steady state or dynamic regimes can be found in the literature.10,12−14,17,22 Generally, the results on gas separation in the capillary condensation regime were reported for Vycor glass membranes. The porous structure of these membranes is represented by a system of interconnected spherical voids formed through controlled etching of boron−silica glass. Generally, the average pore diameter of Vycor can be varied between 4 and 200 nm, with the porosity of membrane ranging from 10% to 30%. However, the pore size distribution in the glass membrane is quite wide, which complicates the description of vapor penetration within the model of transport through cylindrical pores. Moreover, the formation of the liquid phase meniscus occurs in the narrowest part of the pore, while curvature distribution impedes meniscus movement in the pores with pressure growth. This negatively affects both the capillary condensation pressure and permeability in the capillary condensation regime. On the other hand, for experimental study of the capillary flow regime, one can suggest a number of porous systems having cylindrical pores with narrow pore size distribution, including track-etched membranes, anodic aluminum oxide films, or stuctured mesoporous and zeolite materials. Among these systems, the simplest control of porous structure parameters is achieved for anodic alumina membranes. These membranes possess a through porosity represented by a closely packed system of cylindrical channels that can be easily formed by anodic oxidation of metallic aluminum. Anodic oxidation enables control over channel diameter and interpore distance in a wide range of values (pore diameter from 15 to 200 nm; interpore

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EXPERIMENTAL METHODS Sample Preparation. Anodic alumina membranes with controlled diameters were prepared according to a procedure reported elsewhere.24 Prior to anodization, aluminum foils (99.999%, 0.5 mm thick, Goodfellow) were annealed at 450 °C for 24 h in air and polished mechanically and electrochemically. The anodization was carried out in a two-electrode cell in 0.3 M H2C2O4 (98%, Aldrich) at constant voltages ranging from 20 to 80 V. To achieve maximal through porosity for preparation of asymmetrical membranes with branched channels, the first porous layer was formed at highest voltage.26 Then the anodization voltage was reduced by a defined factor to create a selective layer with narrow pores. During anodization, the electrolyte was pumped through the cell by a peristaltic pump, and its temperature was kept constant (2 °C). Membrane thickness was carefully adjusted to 100 μm by measuring total electric quantity and assuming a current efficiency of 90%.24 After anodization, the oxide films were separated from the metallic substrates by selective metal dissolution with 0.5 M CuCl2 in 5 vol % HCl. Subsequently, the pore bottoms were opened by chemical etching in 25 vol % H3PO4 aqueous solution with electrochemical detection of the pore opening point. Sample Characterization. Scanning electron microscopy (SEM) images were recorded on a Supra 50VP instrument (LEO). The statistical analysis of obtained micrographs for determination of pore size distribution, porosity, and pore density was performed using ImageJ software. Porosity characteristics were also determined from adsorption−desorption isotherms (Quantachrome Nova 4200e) using N2 at 77 K and i-C4H10 at 262 K as working gases. Isotherms were treated using BET and BJH models for pore size and size distribution analysis. The permeability of individual permanent gases and vapors was studied using a differential scheme with flow registered by mass flow controllers SLA5850 (Brooks, England) at 25 °C. The pressure from the feed and permeate side was controlled by Carel SPKT00E3R pressure transducers. The measurement technique is reported in the literature in detail.33 The typical duration to achieve steady-state conditions for single-point measurement did not exceed 1 h. To minimize temperature variations, the apparatus was embedded into a thermostat unit (Huber).

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RESULTS AND DISCUSSION Characterization of Membranes with Symmetrical Porous Structure. Detailed microstructure characterization of membranes was performed using SEM, capillary adsorption of 10983

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Table 1. Average Pore Diameter (defined from SEM and BJH, and calculated from permeance and capillary condensation pressure), Effective Porosity Taking into Account the Presence of Dead-End Pores and Gas Permeance in the Knudsen Regime and Specific Surface Area for Membranes Utilized in the Present Study pore diameter, nm calculated

sample name

anodization voltage (V)

top surface (SEM)

AAo_20nm AAo_40nm AAo_45nm AAo_70nm AAo_90nm

20 30 40 60 80

28 ± 7 45 ± 8 38 ± 10 40 ± 12 89 ± 14

in-volume (BJH) 22 ± 5 37 ± 9 65 ± 33 84 ± 36

bottom surface (SEM)

from permeance

from condensation pressure

22 ± 6 37 ± 10 44 ± 7 67 ± 6 71 ± 9

21 41 50 67 87

13 17 32 53 65

effective permeance (i-C4H10, 1 bar feed porosity, % pressure), m3/m2·bar·h 17.2 19.5 29.5 27.8 26.7

13.04 28.66 49.78 71.59 93.39

specific surface area, m2/g 15 9.1 − 5 2.7

Figure 1. Adsorption−desorption isotherm measured for AAo_40nm using N2 at 77K and i-C4H10 at 262 K (a) and measured for AAo_20nm, AAo_40nm, and AAo_90nm using N2 at 77K (b). Obtained pore size distribution for membranes AAo_20nm (c), AAo_40nm (d), AAo_70mn (e), and AAo_90nm (f).

regular cylinder shape of the pores,34 pore diameters on both top and bottom surfaces of membranes was measured. An average diameter of the pores within the membrane volume was also determined using the BJH procedure from adsorption− desorption isotherms. Capillary condensation experiments

N2 and i-C4H10, and permeance measurements. The data for constant-diameter membranes are summarized in Table 1. Pore diameters obtained from SEM for all AAO samples were found narrowly distributed and nearly proportional to the oxidation voltage. As AAOs generally possess slight conical deviation from 10984

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The Journal of Physical Chemistry C were carried out both using N2 at 77 K and C4H10 at 273 K as working gases. No significant difference was observed between isotherms obtained for different gases. Generally, both gases illustrate type IV isotherms and exhibit a hysteresis loop for all channel diameters (Figure 1a,b). At low pressures, a slow increase of adsorbate amount is observed corresponding to multilayer adsorption. However, specific surface area (Table 1) and a quantity of surface adsorbate are seen as insignificant to give a considerable contribution of surface diffusion to a total condensate flux through AAO membranes. At higher pressures (P/P0 = 0.8−0.97), a sharp step appears in the adsorption− desorption isotherms, illustrating capillary condensation of working gases within the pores of AAO membranes. Condensation pressure increases with increasing AAO channel diameters (Figure 1b). Butane desorption isotherms obtained under carefully achieved thermostatic conditions were used for accurate determination of AAO channel diameters and distribution according the BJH procedure (Figure 1c−f, Table 1). Permeability of AAO membranes was measured toward permanent (He, Ar, CO2) and condensable (i-C4H10) gases. All the membranes demonstrated a Knudsen diffusion mechanism at low pressures as supported by linear dependences of membrane permeances from an inverse square root of the gas molecular weight (Figure 2). Permeances for all gases were found

Figure 3. Dependence of permeance on the normalized feed pressure for studied membranes.

Theoretical Analysis of Vapor Transport in the Capillary Condensation Regime. Permeance growth after the condensation point is obviously governed by the change of the transport mechanism. Indeed, the condensation of gases and appearance of a meniscus in the nanopore introduces colossal growth of the pressure gradient. Despite the linear pressure slope inside the nanopore of constant diameter, a steplike pressure drop or steplike increase should appear at the meniscus position depending on its shape and liquid/wall contact angle. According to the Laplace equation, the pressure drop/increase at the concave/convex meniscus should attain ∼40 bar for the isobutane meniscus radius of ∼5 nm. This value notably exceeds the pressure gradient provided in typical membrane permeance experiments. Earlier theoretical models introduced negative pressures in the liquid phase under a meniscus defined by the pore curvature according to the Young−Laplace equation:35 Pc* = Pc − 2σ /rc

(1)

where rc is the curvature of the meniscus, and Pc is the pressure of the gaseous phase above the meniscus within the pore. Besides the meniscus at the liquid/vapor interface within a pore volume, another meniscus (entrance meniscus) is also forming at the pore entrance. The curvature of this meniscus depends strongly on the feed pressure and transmembrane flux. At a feed pressure of Pc,0, its curvature is equal or higher than the thermodynamically stable value (depending on nanopore diameter), while increasing the feed pressure results in flattening of the meniscus and its possible disappearance at Pin = P0 (condensation pressure over the flat surface). The pressure in the liquid phase under an entrance meniscus is strictly dependent on the feed pressure (Pin) and entrance meniscus curvature (rin):

Figure 2. Dependence of the gas permeance on the inverse square root of the gas molecular weight for studied membranes.

nearly proportional to pore diameters, which also complies with the Knudsen diffusion mechanism. The values of Knudsen diffusivities for isobutane (obtained by interpolating of permeance−pressure dependences to an ordinate axis) were used to determine average pore diameters from permeability data (Table 1). Obtained values are in good agreement with other data (Table 1). Increase of gas pressure leads to the appearance of intermolecular collisions in nanopore volume and growth of viscous flow contribution. Gas flow through nanopores of anodic alumina in the transitional flow region is reviewed in the literature in detail.33 Further growth of pressure in the case of condensable gases leads to a sudden uprise of permeance after the condensation point (Figure 3). The absolute values of condensation pressures for different membranes stay generally in line with the Kelvin equation, while evaluated channel diameters were found slightly lower than those obtained by other techniques, which can be ascribed to a deviation from the equilibrium value due to intrachannel transport of vapors or surface roughness of AAO channels (Table 1).

Pin* = Pin − 2σ /rin

(2)

The absolute values of both Pin* and Pc* are lower than Pin and can sink to negative values in the case of high curvatures. Notably, the absolute pressure in the liquid phase below the equilibrium condensation pressure would provide tensile forces which can cause cavitation and liquid phase discontinuity. Despite the forming bubbles compressing and further dissolving in the liquid phase at a distance of an advancing meniscus, their presence can significantly affect the permeance of nanochannel and liquidfilling kinetics.36−38 This effect is not completely understood at the moment and needs separate and detailed investigation which is, however, beyond the range of the current study. A more crucial and understandable effect is provided by the change of the meniscus curvature affected by the liquid flow in 10985

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The Journal of Physical Chemistry C the nanochannel. At the beginning of imbibition of the nanochannel (about several nanoseconds after meniscus formation), the velocity profile of the condensate exhibits a plug shape which indicates inviscid flow. However, after several nanoseconds, the velocity profile in liquid assumes a parabolic distribution as predicted by the Pouseille law:39 υ(r ) =

1 2 dP (R − r 2 ) 4η dx

JL =

εd pore 2 ρ (Pin* − Pc*) 32ηLx M

(4)

where ε is the membrane porosity, dpore is the pore diameter, ρ is liquid density, M is the molecular weight, η is the viscosity of condensate, Lx is the thickness of the pore filled with liquid, and P*in and P*c are the pressures under entrance (condensing) and in-pore (evaporating) menisci, respectively. (3) The flux of substance through the gas-filled part of the pore can be calculated using the Knudsen equation with a correction to the contribution of the viscous flow:

(3)

where R is the pore radius, η is the liquid viscosity, and dP/dx is the pressure gradient. Maximum liquid velocity appears at the pore center while minimal or even no flow is observed at the liquid/wall interface. On the other hand, an evaporation rate over an undisturbed meniscus at equilibrium (having a constant curvature) can be considered equal within the pore section. This leads to insufficient liquid inflow input to the meniscus close to the nanopore walls, which should cause liquid flow redistribution at the meniscus. This obviously changes the curvature of the meniscus from a thermodynamically stable concave spherical shape to a complicated profile, which hardly can be deduced empirically. An example of apparent meniscus shape in a nanochannel has been reported in the literature.40 Obviously, the shape of the meniscus depends on liquid viscosity, velocity, and the pressure above the meniscus. The higher the flux through the pore, the higher the meniscus curvature achievable at the same conditions. As the shape of the meniscus inside the nanopore at nonequilibrium conditions cannot be easily solved analytically, we have provided an algorithm for experimental determination of the menisci curvatures and pressure drop in the liquid phase. The calculations were performed using permeance data of the specific condensing gas (i-C4H10) for the anodic alumina membranes with different pore diameters. The following assumptions and statements have been used for the calculations: (1) At feed pressures exceeding the capillary condensation pressure within the pore (Pc,0), the part of the nanopore with length Lx is filled with condensate (Figure 4). The other part of the pore (L − Lx) is filled with gas phase. The possible presence of bubbles in the liquid phase is neglected. (2) The flux through the liquid-filled part of the nanopore can be calculated using the Poiseuille equation:

JG =

εd pore(Pc − Pout) 3RT (L − Lx)

8RT (1 + K (Pc + Pout)/2) πM

(5)

where R is the universal gas constant, T is the gas temperature, and K is the viscous flow correction factor, evaluated from the permeance data at pressures below capillary condensation. (4) At steady-state conditions, the fluxes through the liquidand gas-filled parts of pore are equal and equal to the total flux through the membrane: J = JL = JG

(6)

(5) Menisci radii (rin and rc) are related to gas pressure over menisci (Pin and Pc) according to the Kelvin equation: ln

Pin −2σM = P0 ρrinRT

ln

Pc −2σM = P0 ρrcRT

(7)

where σ is the surface tension of condensate. (6) Pressures under menisci (Pin* and Pc*) can be evaluated from gas pressure over menisci (Pin and Pc) and menisci curvatures (rin and rc) using the Young−Laplace equations (eqs 1 and 2). Expressions 1, 2, 4−7 represent a self-containing system of equations with the only solution. The easiest way to obtain an analytical expression for flux within the nanochannel involves Taylor expansion of the logarithmic function for the pressure drop in the liquid phase (ΔP*): ⎛1 1⎞ ΔP* = Pin* − Pc* = (Pin − Pc) − 2σ ⎜ − ⎟ rc ⎠ ⎝ rin = (Pin − Pc) − (Pin − Pc)

⎞ ⎛ ρRT Pin ρRT = ⎜1 + ln ⎟ M Pc M ·(Pin + Pc)/2 ⎠ ⎝ (8)

From eqs 4 and 5 using the following system of equations, we can calculate the pressure above the downstream meniscus and the thickness of the liquid phase in the pores for given values of flux through the membrane, upstream (Pin) and downstream (Pout) pressures:

Figure 4. Scheme of the pore partly filled with condensate and corresponding pressure distribution across the pore. 10986

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Figure 5. Dependence of the liquid phase thickness, Pc, and pressure difference in the liquid phase (a) and input and output menisci curvature (b) on the normalized feed pressure for the AAo_40nm membrane.

⎧ εd 2 ⎛ ⎞ ρRT ⎪ pore ρ ⎜1 + ⎟(Pin − Pc) ⎪ 32ηLx M ⎝ M(Pin + Pc)/2 ⎠ ⎪ εd (P − Pout) 8RT ⎪ = pore c ⎪ πM 3RT (L − Lx) ⎪ ⎨ (1 + K (Pin + Pout)/2) ⎪ J 3RT (L − Lx) ⎪ ⎪ Pc = Pout + ε + K (Pin + Pout)/2) d (1 pore ⎪ πM ⎪ ⎪ ⎩ 8RT

radius of the nanochannels. On the other hand, the rather low maximal ΔP*max illustrates the experimental similarity of the entrance and evaporating menisci curavtures to all the other pressure range and growth values of experimental Pc with Pin. Strong deviation of the evaporating meniscus curvature from the equilibrium value in the nanopores gives us evidence for limitation of the evaporation rate by heat transfer from the entrance meniscus to the evaporating meniscus, which limits the total achievable flux within the nanochannel. Notably, the heat flux in the liquid phase depends only on heat transfer properties of the system and temperature gradients in the membrane and cannot be easily affected by any of external experimental parameters, except the operating temperature. An analysis of membrane permeance on the output pressure illustrated a great divergence of P(Pout) dependences with external flow restriction (Figure 6). The obtained picture

(9)

For numerical evaluation of the regime parameters, we tabulated the physical constants of the gas and liquid phases of isobutane (Table S1, Supporting Information).41 The general solution for Lx, Rin, Pc, and ΔP*, as a function of the reduced input pressure for the AAo_40nm membrane with a pore diameter of ∼40 nm in a constant flow resistance regime (Pout ∝ J−1), is represented in Figure 5. All the essential parameters for membranes with pore diameters of 20−90 nm are summarized in Table 2. Full numerical data are given in Table S2, Supporting Information. Table 2. Starting Capillary Condensation Pressures (Pc,min/ P0) and Extreme Values of Pressure under Liquid Meniscus (P*c), In-Pore Menisci Curvature Radius (rc), Pressure Drop in the Liquid Phase (ΔP*max), and the Thickness of Liquid Phase in the Membrane (Lx,max) for Membranes with Different Nanochannel Diameters membrane

dpore, nm

Pc,min/P0

P*c, bar

rc, nm

ΔP*max, bar

Lx,max, μm

AAo_20nm AAo_40nm AAo_45nm AAo_70nm AAo_90nm

21 41 50 67 87

0.8777 0.912 0.931 0.983 0.975

−29.1 −19.57 −14.5 −3.01 −2.23

6.1 8.7 11.2 32.5 37.6

1.894 1.048 0.787 0.429 0.464

55.0 30 24.3 26.2 28.1

Figure 6. Dependence of membrane permeance normalized to permeance in the Knudsen regime on the permeate side pressure measured for membrane AAo_40nm in conditions of external flow restriction.

principally resembles absorption−desorption isotherms of AAOs shown in Figure 1. Likely this originates from the contact angle hysteresis for an advancing/receding meniscus. Notably, partial condensation in pores occurs for all points over Pout > 0.15 on Figure 6. However, no permeance rise is observed until high Pout in the case of strong external flow restriction (fluxes limited to 45−150 m3/(m2·h)). Thus, one can conclude that the transport through the liquid phase in the case of low fluxes (close to equilibrium conditions) is self-adjusted by the menisci form and position to fit transport through the gas phase. A contrary high transmembrane pressure difference encourages menisci shape and position change, which promotes high pressure

Shallow analysis of the obtained data indicates that both menisci curvatures (1/rin and 1/rc) generally decrease with an increasing input pressure, while the nanopore filling level (Lx), pressure drop in the liquid phase (ΔP*), and Pc simultaneously increase with an increase in feed pressure. Maximal flux through the nanopores is achieved at a feed pressure close to the saturated vapor pressure (P0). Decrease of the nanochannel diameter makes the effect more pronounced. The resulting minimal rc (achievable near Pc,0) is about twice lower as compared to the 10987

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most effect in the experimentally observed permeance rise in nanopores is dictated by Pc change and the meniscus curvature. Unfortunately, Pc cannot be directly measured in nanochannels. On the other hand, we can derive the value from eq 9. According to calculations, Pc closely follows Pin over the entire pressure range, with a nearly linear decrease of all Pc/Pin ratio with the increase in normalized feed pressure (Figure 7). The

difference in the liquid with corresponding permeance rise. From an application point of view, only maximal fluxes through membranes in the capillary condensation regime make sense. Therefore, hereafter we focus on the data obtained with no flow restriction and a permeate side pressure close to zero. Further examination of the permeance rise in the capillary condensation regime involved determination of the explanatory variables. Analysis of the left part of eq 9 allows us to conclude that transport through the liquid phase is mainly dictated by the pressure difference (Pin − Pc) and the meniscus position Lx. The same is true for (Pc − Pout) and (L − Lx) in the right part of the eq 9. Among these variables, only Pc can be directly governed by the external factors such as P0 or the nanochannel diameter; thus, this parameter was chosen as an argument for further evaluation. Consequently Lx(Pc) dependence was derived by the transformation of eq 9: JL − JG = 0

(10)

FG F (Pc − Pout) − L (Pin − Pc) = 0 L − Lx Lx

(11) Figure 7. Dependence of pressure above the output meniscus (Pc) normalized to feed pressure (Pc) on the normalized feed pressure (Pin/ P0).

where FG and FL correspond to the gaseous (Knudsen and viscous) and liquid (Poiseuille) phase permeabilities, calculated as FG =

FL =

εd pore 3RT

8RT (1 + K (Pin + Pout)/2) πM

⎞ εd pore 2 ρ ⎛ ρRT ⎜1 + ⎟ 32η M ⎝ M(Pin + Pc)/2 ⎠

bending point corresponds well to the equilibrium condensation pressure in the pores of the membrane and the meniscus radius of ∼1/4 dpore. The slope of all curves was found nearly equal for all the membranes independent of their pore diameter. Fitting of all obtained experimental data gives an average value of slope equal to −0.060 ± 0.003. Thus, to estimate all permeability rise in nanochannels in the condensation regime, one can derive the expression for Pc:

(12)

(13)

Therefore, the relative thickness of the liquid phase can be expressed as Lx FL(Pin − Pc) = L FL(Pin − Pc) + FG(Pc − Pout) Pin − Pc ≈ Pin − Pc + FG/FL(Pin − Pout)

⎛ ⎛ ⎞⎞ ⎛ P 8σM ⎞⎟⎟⎟ Pc = Pin⎜⎜1 − 0.060⎜⎜ in − exp⎜⎜ − ⎟⎟⎟ ⎝ ρd poreRT ⎠⎠⎠ ⎝ P0 ⎝

Exponential growth of Pin − Pc difference with inverse pore diameter and temperature indicates a relative increase of the effect with a decrease in both nanochannel size and temperature, which is however limited by the slope coefficient, and leveled off by the Y(dpore,T,gas) function in eq 16. Probably the derived coefficient is mostly affected by condensate properties and membrane heat transfer parameters, and those would be the subject of further studies. The suggested theoretical model was used to fit the obtained experimental data (Figure 8). It was shown that the calculated values of membrane permeance in the capillary condensation regime are in a good accordance with the measured values. Slight difference between experimental and theoretical results can be explained by the nonzero permeate pressure during permeance measurements, whereas the theoretical equations were derived on the assumption of no external flow restriction, which means that the permeate side pressure is close to zero. Gas Permeance of Membranes with Asymmetric Porous Structure. From an application point of view, membrane permeance in the condensation regime is in most demand in natural gas sweetening or condensate desaturation of technological gases. For these issues, condensate pressures below P0 are of major importance. In this case, the membranes with nanometer-sized channels are necessary to promote capillary condensation below P0. Contrary to both Poiseuille and Knudsen equations, both flows would be greatly diminished at such low

(14)

With substituting FG/(FL·Pin) by Y (d pore ,T , gas) = FG/(FLPin) ≈

64 3

3/2 η 2 ⎛⎜ M ⎞⎟ π ⎝ RT ⎠ ρ2 d pore

(15)

one can obtain the following equation for condensable gas permeance (P) through the nanochannel membrane in the capillary condensation regime: P = PG

⎞ ⎛ 1 − Pc/Pin L ⎟ = PG⎜⎜1 + L − Lx Y (d pore ,T , gas)ΔP ⎟⎠ ⎝

(16)

where PG can either be measured experimentally or calculated, assuming a Knudsen diffusion mechanism with viscous flow contribution: PG =

FG L

(18)

(17)

Notably, the Y(dpore,T,gas) function is inversely proportional −5/2 to both d−1 (as soon as η ∼ exp(T−1) close to pore and ∼ T condensation point), diminishing the absolute effect of capillary transport onto the total mass flow through the membrane. This function is weakly dependent on the external pressures; thus, the 10988

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conditions for enhanced mass transport in asymmetric membranes. We believe that the present findings can benefit a better understanding of both gas and liquid transport in nanocapillaries, especially in dynamic condensation or pervaporation regimes. Moreover, capillary condensation and liquid separation of feed and permeate sides of membranes provide the conditions for the colossal selectivity rise for condensable and permanent components of gas mixtures, which enables microporous membranes to be successfully utilized in self-controlled natural and associated petroleum gas sweetening for removal of water and hydrocarbon vapors.

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ASSOCIATED CONTENT

S Supporting Information *

Figure 8. Comparison of experimental data and results calculated according to the suggested model.

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b02971. Physical properties of gas and liquid isobutane used for calculation and numerical experimental data and derived calculated parameters for isobutane transport in the capillary condensation regime through anodic alumina membranes with different pore diameters (PDF)

channel diameters. However, an extraordinary increase in the pressure drop in a liquid, provided by the evaporating meniscus with a decreasing channel diameter, allows compensating flow reduction. From a theoretical point of view, to provide a maximal pressure difference in a liquid one necessitates a decrease in entrance meniscus curvature with a simultaneous increase in the curvature of the evaporating meniscus. This could be realized by the preparation of asymmetric membranes with multiple branching of large channels into the nanocapillaries. Such membranes are easily formed by anodic oxidation of aluminum with a gradual decrease of anodization voltage.26 Recently, a theoretical study on permeability of this kind of structure in the capillary condensation regime was reported, illustrating asymmetric transport and Knudsen flow limitation at the end of the channel.42 Here we have tested the permeance of the asymmetric membrane with multiple branching of 40 nm channels down to 5 nm. The membrane operating in the condensation regime indicated 10× enhancement of i-butane permeance (Figure 9) which notably exceeded the permeance of CH4 and even He.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work is partially supported by Rosneft Oil Company, Russian Scientific Foundation (grant no. 14-13-00809), RF President grant MK-5085.2015.3, Russian Foundation of Basic Research grant no. 16-29-05285, and Lomonosov Moscow State University Development Programme.

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REFERENCES

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Figure 9. Permeance of asymmetric membrane AAo_40nm_5nm vs relative feed pressure.

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CONCLUSIONS In this paper we first present a self-consistent theoretical and experimental model for mixed liquid−gas transport of condensable gases in nanochannels. We provide the relations for the experimental and theoretical evaluation of the pressure drop in a liquid defined by menisci curvatures and report the 10989

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