Meniscus Curvature Effect on the Asymmetric Mass Transport through

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C: Physical Processes in Nanomaterials and Nanostructures

Meniscus Curvature Effect on Asymmetric Mass-Transport through Nanochannels in Capillary Condensation Regime Dmitrii I. Petukhov, Mikhail V Berekchiian, and Andrei A. Eliseev J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b08289 • Publication Date (Web): 03 Dec 2018 Downloaded from http://pubs.acs.org on December 3, 2018

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The Journal of Physical Chemistry

Meniscus Curvature Effect on the Asymmetric Mass-Transport through Nanochannels in Capillary Condensation Regime

M.V. Berekchiian §, D.I. Petukhov §,#.*, A.A. Eliseev §,# §

- Department of Materials Science, Lomonosov Moscow State University, Moscow 119991

Leninskie hills, Russia #

- Department of Chemistry, Lomonosov Moscow State University, Moscow 119991 Leninskie

hills 1-3, Russia Corresponded author Dmitrii .I. Petukhov, e-mail: [email protected]

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract This study reports on experimental evidence for large mass flux difference in opposite flow directions through asymmetric membranes in capillary condensation regime. Anodic alumina membranes with pore diameters of 10-80 nm in the selective layer and 40-80 nm in the supporting layer were inspected for permeance variation depending on the feed and permeate pressure conditions. Starting pressure of capillary condensation was found to follow ThompsonKelvin equation for pore diameters at the upstream side of a membrane. The experimental ratio between the permeances in the upstream and downstream membrane orientations attained ~2 at the feed pressure close to condensation pressure for membranes containing pores of small diameters. The experimental data was treated assuming Knudsen and viscous vapor transport in the gaseous state and Poiseuille flow in liquid condensate. It is shown that condensate transport through the nanochannels in the capillary condensation regime is mostly governed by the condensing/evaporating menisci curvatures deviating from the thermodynamic equilibrium value and gradually changing with pressure conditions. Coincidently the curvature of meniscus is controlled by the liquid influx velocity and velocity distribution in nanochannels, giving rise to a self-regulating regime of flow. At high liquid intake velocities, the curvature radius of meniscus significantly increases, being accompanied by a serious drop of liquid evaporation efficiency. The effect is ascribed to a sequential evaporation of liquid in the pore center and vapor condensation close to the pore walls, generating gas-liquid rotating flow.


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1. Introduction Recent progress in membrane science and quick implementation of scientific achievements in the separation technologies have resulted in the development of a number of novel membrane materials and techniques guided by continuous need for more effective separation performance. Among other developing separation techniques, the most uncommon and exciting results are provided for a vapor transport through microporous membranes operating in the capillary condensation regime. Capillary condensation enables a colossal growth of membrane permeance induced by an additional capillary pressure under liquid phase meniscus. This effect has been established for a number of porous systems

1–4

and has already been utilized for associated

petroleum gas conditioning 5. Another important feature of the capillary transport regime involves large mass flux difference for opposite flow directions through asymmetric microporous membranes as it was theoretically predicted recently by Uchytil and Loimer 6. According to calculations, the difference can attain an order of magnitude in the membrane performance and grows with decreasing the pore size of the selective layer. In a basic physical description, the effect arises from the difference of flow mechanisms in micro- and macroporous layers of the membrane and a large difference in layer permeabilities for gaseous and liquid phases. Typically, microporous membranes operate in Knudsen diffusion regime in the case of gas transport and exhibit Poiseuille flow for liquids. According to Knudsen and Poiseuille equations, the difference strictly depends on nanochannel diameters. For example, a membrane with 10% porosity consisting of 100-nm-diameter cylindrical nanochannels should exhibit a permeability of ~3.6·10-3 m3·m/(m2·atm·h) for gaseous isobutane, while the permeability for liquid isobutane increases up to ~1.7·10-2 m3·m/(m2·atm·h). Contrary, for a membrane with 10% porosity and 10-nm-diameter channels, the value for gas permeability (~3.6·10-4 m3·m /(m2·atm·h)) exceeds that for liquid ~1.7·10-4 m3·m/(m2·atm·h). This difference leads to serious pressure gradient redistribution in the asymmetric membrane layers filled with gas and liquid phases. Moreover, the presence of curved menisci in case of



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capillary condensation generates additional capillary pressure strongly affecting the transport in liquid phase. As soon as a meniscus curvature radius depends on the pore sizes, an asymmetry in condensate transport for layered membranes is promoted by the difference in the pore sizes on the opposite sides of the membrane. These two effects are of major importance for a number of applications where the condensation of vapors or evaporation of liquids can take place in nanochannels

7,8.

Technological uses for asymmetric transport include, but are not limited to

diaphragm pumps, passive pumping devices or pulsating heat pumps 9. Moreover, this phenomenon provides the waymarks for developing a very new area of autonomous pumps utilizing ambient heat to transport particles against concentration gradient 10. To prove the asymmetric transport effect in the capillary condensation regime, here we provide experimental results for isobutane vapor transport through asymmetric mesoporous membranes with microstructures similar to those described in the theoretical study 6. Anodic alumina membranes with a hexagonal closely-packed system of cylindrical channels possessing low tortuosity and uniform pore-size distribution served as an excellent model system for asymmetric transport experiments. Anodization of aluminum enables an easy control over channel diameter, interpore distance and thickness of anodic aluminum oxide (AAO) films and allows simple preparation of asymmetric membranes with desired porous structure illustrate high gas permeance

12

11.

AAO membranes

and exhibit a sharp rise in vapors permeance beyond the

condensation point 13. Description of permanent gas and vapors transport through anodic alumina membranes with different nanochannel diameters have recently been reviewed in details 14. 2. Experimental section Anodic alumina membranes were obtained using an earlier described procedure

11,12.

The

anodization of electropolished aluminium foils was performed in a two-electrode cell in 0.3 M H2C2O4 (98%, Aldrich) at voltages ranged from 10 to 80 V. Electrolyte temperature of 2 °C was kept during anodization. To achieve maximal through porosity of asymmetrical membranes with a macroporous layer branched into smaller channels, anodization was started at the highest


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voltage 11. Then, the anodization voltage was reduced to a lower value in the range of 10-60V to create a selective layer with narrow pores. The thickness of selective and supporting layers was controlled by measuring total electric quantity. For preparation of flow-through membranes, the remaining aluminum was selectively dissolved in 0.5 M CuCl2 in 5 vol % HCl followed by removal of a barrier layer by chemical etching in 25 vol % H3PO4 aqueous solution with electrochemical detection of the pores opening 15. Characterization of the membrane pore diameter distribution, porosity and pore density involved statistical analysis of scanning electron microscopy images (Supra 50VP, LEO) using ImageJ software. Microstructural characteristics for all the membranes are summarized in table 1. Isobutane permeance for the obtained membranes was measured by registering gaseous flux by mass flow controllers SLA5850 (Brooks, USA) in fully open valve regime, when the flowcontrollers have the lowest flow resistance, and measuring pressure at the feed and permeate side by Carel SPKT00E3R pressure transducers. This measurement technique was previously described in details in Refs

14,16.

Typical time to obtain the steady flow conditions after the

change of feed and permeate side pressures did not exceed 60 min. The experimental cell was placed into thermostat (Huber) to minimize the temperature variations during measurements. Unfortunately, direct implementation of the experiment suggested in

6

was not possible due to

physical limitations of the setup, resulting in the necessity of too long equilibration times for feedback coupling of transmembrane pressure and permeate flow. Therefore, all experiments were carried out by controlling feed pressure (Pin) and measuring permeate flux, while Pout was allowed to achieve steady state without external flow restrictions (under dynamic vacuum applied after the permeate flow controller). Flow resistance of membranes exceeded that of the flow controller at least 10 times, resulting in typical Pout values below 1.0 bar. 3. Experimental Results and Data Treatment We started an asymmetric transport studies from the AAO membrane with a constant pore diameter of 40 nm (AAO_40 sample, figure 1a). The pores in AAO structure exhibit slight



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conical deviation from a regular cylinder shape due to chemical etching in the course of membrane preparation

17.

This leads to an increase in pore diameter on the top surface of a

membrane (oxide/electrolyte interface) compared to pore diameter near the bottom surface (metal/oxide interface). Thus, pore diameters both on top and bottom surfaces of the membrane were accurately measured (table 1). Permeance measurements for the AAO_40 membrane indicated a slight difference in permeance-pressure dependences after the condensation point (figure 2). Notably the permeance of membrane with smaller channel diameters positioned towards the feed flow (separation layer upstream: flow direction A in notation given in 6) after the condensation point are somewhat higher than the values achieved for opposite flow direction (separation layer downstream: flow direction B in the same notation). Moreover, condensation point also shifts slightly towards higher pressures in case of flow direction B, which stays in agreement with the predictions given in 6.

Figure 1. Cross-sectional SEM micrographs of (a) symmetrical membrane with straight pores obtained at 40V (AAO_40) and (b) asymmetrical membrane obtained by decreasing anodization voltage from 40V to 10V (AAO_40_10).


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Figure 2. Dependence of permeance on the normalized feed pressure for the membrane with straight pores AAO_40 measured with bottom surface upstream (flow direction A) and bottom surface downstream (flow direction B).



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Figure 3. Dependence of permeance on the normalized feed pressure measured in flow direction A and B for the membrane AAO_40_10 with 12 nm pores in the selective layer and 40 nm pores in the supporting layer (a). Dependences of the meniscus position (b), input and output menisci curvature radii (c) on the normalized feed pressure measured in flow direction A and B for the AAO_40_10 membrane.


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Table 1. Anodization voltage, pore diameter, porosity, thickness and Knudsen permeability (flow resistivity) of selective and supporting layers for

15

25

19.2

0.17

15 15 15 15 15 15

21.8 46.0 66.0 110.0 181.4 220.6

0.28 0.22 0.30 0.26 0.33 0.33

40

80

Viscous flow coefficient (b)

13.8

100

20

0.23

40 85

32 82 52 66 74 77 72 76

15 19

16 17.5

75 100

19

18.1

90

31 54 62 62 62 62 63 62

0.23 0.29 0.30 0.30 0.30 0.30 0.25 0.33

85

Knudsen permeance, m3/m2∙bar∙h

16.2

As estimated using Knudsen equation for membrane permeance using the values of gas permeance and pore diameter determined from SEM and

thickness of the layer. ***

Thickness, µm***

36

Effective porosity**

44

SEM*

Viscous flow coefficient (b)

Knudsen permeance, m3/m2∙bar∙h

13.8

AAO_40_10 10 12 11 12.5 11 AAO_80 AAO_80_10 10 12 18 8 7.5 AAO_80_20 20 19 24 11 10 AAO_80_30 30 26 28 14.2 10.5 AAO_80_40 40 35 42 15.0 13 AAO_80_50 50 50 58 15.0 15 AAO_80_60 60 57 63 17.8 16 * Evaluated by statistical analysis of SEM images **

Thickness, µm***

Effective porosity**

SEM*

Capillary condensation 32

Gas transport characteristics****

Capillary condensation

40

Gas transport characteristics****

Supporting layer Pore Porosity, % diameter, nm

SEM*

AAO_40

Selective layer Pore Porosity, % diameter, nm

SEM*

Membrane

Anodization voltage, V

asymmetric membranes utilized in the present study.

Anodization voltage, V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46


Determined by total electric quantity spent at an appropriate anodization step (constant voltage) during the synthesis

****

Estimated by solving the equation system (1).



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The effect is much better pronounced in asymmetric membrane having ~12 nm pores in the selective layer (25 µm thickness) and 40 nm pores in the supporting layer (75 µm thickness) (figure 1b). Condensation point in the case of flow direction A (selective layer upstream) corresponds well to the anticipated value of ~0.85 P0 for ~11 nm pores (figure 3a). In the case of the opposite flow direction, condensation point shifts up to ~0.95 P0. This suggests rather a low permeance of the supporting layer, leading to a high pressure drop and an insufficient pressure of isobutane in downstream separation layer to induce capillary condensation. The same shift was established in the theoretical studies in case of high transmembrane pressures (0.5-1 bar) 6. Unlike the condensation point, experimental permeances of the membrane in asymmetric transport experiment contradict theoretical calculations. In a whole pressure range over the condensation point, the permeance of asymmetric membrane stays higher in case of direct flow (flow direction A) than for opposite flow direction (flow direction B). A deviation from model 6 concerning the higher permeance in the case of flow direction A can be attributed to a decreased permeate side pressure in the real experiment as compared to the modeled conditions (preventing condensation in the separation layer for flow direction B). However an absence of saturation plateau at high Pin in both the membrane orientations disagrees modeling results 6. Membrane permeance grows extensively with Pin after the condensation point giving no plateau or kinks in permeance-pressure dependence. To calculate meniscus position and output meniscus curvature we have modified the model suggested earlier in Ref.13 to account for two layers with different pore diameter (figure 4). The calculations were performed using original permeance data and basing on the following assumptions: 1) First, the contribution of Knudsen and viscous flows for selective and supporting layers were extracted from experimental permeance aquired in the transitional flow regime (below condensation point, figure 4a,d) using the following equation:


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  ( P  P* )  ( P  P* )  *  Pin  P*  F2  1  K 2 out  P  Pout  J F1  1  K1 in 2 2    









(1)

where J – gas flux trough the membrane, F1 –Knudsen permeance of the selective layer, K1 – viscous flow correction factor for the selective layer, F2 and K2 correspond to Knudsen permeance and the correction factor for the supporting layer, respectively, Pin, Pout and P* – are the feed stream, permeate side pressure and a pressure at the pore branching point, respectively. To solve this equation we used linear regression of J(Pin, Pout) data obtained below the condensation point. 2) Extracted permeances of the upstream and downstream layers (F1 and F2) and viscuous correction factors (K1 and K2) were used for determination of the meniscus position (Lx) and pressure above the output meniscus (Pc) using the system of equations, including those equalizing liquid and gas fluxes through the pores and fluxes through selective and supporting layers and an equation allowing us to determine the cappilary pressure in the liquid phase (see eq. 8 in 13 for details):

  1  d pore,1 2   ( Pс , 0  P * )  L1   Pс , 0  P *  Pliquid  F1   1  K1    L1  Lx  2  32  Lx M   *  ( Pс , 0  P )  L1    Pс , 0  P *  J  F1  L  L  1  K 1  2 1 x    (2)  * *   ( P  P )    L1 с,0  Pс , 0  P *  F2  1  K 2 ( Pout  P )  P *  Pout  1  K 1  F1      L1  Lx  2 2       RT Pliquid  Pin  Pc   RT ln Pin  Pin  Pc 1   M P  P  / 2   M Pc in c  

















where dpore,1 – pore diameter in the upstream layer, ε1 – the upstream layer porosity, ρ – condensate density,  - condensate viscosity, M – molecular weight, Lx – meniscus position, L1 – thickness of the upstream layer, L2 – thickness of the downsteam layer. Pc,0 – equilibrium condensation pressure in the pores with diameter dpore,1, calculated according to the Kelvin equation:



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  4M   (3) Pc , 0  P0  exp  RT  d  pore , 1  

where  - is surface tension of condensate, P0 – condensation pressure on the flat surface. Here we assumed that the pressure in the gas phase drops sharply from Pc to the equlibrium condensation pressure in the pore Pc,0 over condensate/gas interface. Gaseous pressure over meniscus deviating from Pc,0 will result in condensation/evaporation equillibrium shift with either advancing or receeding meniscus movement. It should be noted that system (2) can be utilized in either flow direction by changing selective and supporting layer contributions. At the same time it is valid when only one layer is filled with condensate (figure 4b,e). In the case of condensate penetration into the second layer (figure 4c,f) the equation system should be rewriten as:  ( P  Pout )   1 L2  Pс , 0  Pout   Pliquid  F2   1  K 2 с , 0  ( Lx  L1 ) 32M L1 L  ( L  L ) 2   2 x 1   2 2   1  d pore,1  2  d pore, 2  RT Pin  ln Pliquid  Pin  Pc   M Pc   ( P  Pout )   L2  F2  Pс , 0  Pout   J  1  K 2 с , 0 L2  ( Lx  L1 )  2   

(4) where dpore,2 – is pore diameter in the downstream layer, ε2 – the downstream layer porosity. The permeance data treatment with the above protocol allows easy extraction of the meniscus position at any pressure conditions. The results (figure 3b) illustrate pinning of the meniscus to the level of channels branching (figure 3b) followed by extension of liquid into supporting layer as soon as condensation in large diameter pores becomes possible (in the case of flow direction A). Maximum filling level at Pin≈P0 attains ~50 of 100 μm, which is limited likely by the pressure gradient in the liquid. In case of flow direction B, the liquid spreads quickly to the separation layer once the condensation appears in the supporting layer due to low flow resistivity of large-diameter channels towards liquid transport.


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Figure 4. Scheme of pressure distribution in the case of direct flow and opposite flow direction for membrane operating in transitional flow regime (a,d) and capillary condensation regime with part of the pore filled with condensate (b,c,e,f).

A careful analysis of menisci position on normalized feed pressure indicates the only quasistable region during pinning of the meniscus to the pore branching point (Pin = 0.91..0.95P0). According to the analysis given in

13

the absence of the permeance plateau due to meniscus

pinning can originate from a significant deviation of the meniscus curvature from the thermodynamically stable value and its gradual change with input pressure conditions (figure 3c). Indeed the curvature radius of menisci ( rmeniscus ) calculated from Pc according to Kelvin equation: rmeniscus 

 2M P RT  ln c P0

(5)

varies strongly with the membrane permeance but is only slightly affected by channels diameter (fig. 3c). Moreover, at certain conditions the curvature radius of evaporating meniscus exceeds



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substantially the radius of the pores. This corresponds to an intensive evaporation of liquid from the meniscus, where the pressure above liquid exceeds the equilibrium condensation pressure within the pore of a given diameter (Pc,0), followed by rapid condensation of evaporated molecules at the pore walls in the nearest vicinity of the meniscus. Earlier, the effect was detected and ascribed to nonhomogeneous velocity distribution of liquid flow within the pore and insufficient liquid inflow input to the meniscus close to nanopore walls 13. Notably, all calculations of menisci curvature radii provided in the paper are performed assuming isothermal conditions. Accounting for non-isothermal models will lead to superficial results to some extent, especially considering unknown temperature variation across the membranes in operando conditions. However, since condensation of gas leads to heat emission, the curvature radius of the condensing meniscus can be considered as an overestimated value, while the curvature radius of evaporating meniscus, where the heat is absorbed, can be considered as underestimated. To give more comprehensive picture of flow peculiarities in the capillary condensation regime for the pores of different sizes we have also provided the measurements for membranes with separation layers of different pore diameters within 10-60 nm range (fig. 5a,b). Microstructural parameters for these membranes are summarized in table 1. Сontributions of Knudsen and viscous flow for the selective and supporting layers, listed in the table 1, were calculated in the same manner as decribed previously for the membrane AAO_40_10 using equation (1). Experimental Knudsen permeances were compared with Knudsen permeances estimated from microstructural parameters (pore diameter and porosity) determined from SEM. After that, the porosity values were refined to fit experimental results, as soon as the structure of AAO contains a portion of terminated channels, decreasing the membranes permeance 12. For all the membranes, an asymmetry of vapor transport is well seen. In case of separation layer upstream orientation, the condensation point shifts with increasing minimal pore diameter according to Kelvin equation. In case of the opposite membrane orientation, the condensation


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point is achieved always at higher pressures. For all the membranes the permeance in capillary condensation regime in case of flow direction A exceeds substantially that for the opposite flow direction. Liquid-filled part of pores at Pin ≈ P0 generally increases with increasing membrane pore diameters (fig. 5c). For membranes with low permeance of the separation layer (AAO_80_10 - AAO_80_30) liquid-vapor interface stays within separation layer up to Pin ≈ P0, while for the other membranes it extends slightly to the supporting layer (fig. 5c). In case of flow direction B, condensate spreads into the separation layer only for the AAO_80_10 membrane at the pressures exceeding Pc,0 in the upstream layer. No condensation in separation layer occurs for AAO_80_20 – AAO_80_60 membranes. Thus, the gas flow resistance of the separation layer limits the permeance of the membranes, while the pressure drop in the supporting layer restricts condensation in the separation layer until high Pin.



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Figure 5. Dependence of the permeance (a,b) on the normalized feed pressure for asymmetric membranes with the selective layer formed at anodization voltages from 10 to 60V and the supporting layer formed at 80V measured with flow direction A and B. Extreme liquid phase thickness (Lx) at Pin ≈ P0 form membranes with different pore diameter in the selective layer (c) calculated from the experimental data for flow directions A and B.


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The difference in reduced mass flux for opposite flow directions is growing strictly with decreasing separation layer pore diameters, and attains ~1.5 – 2-fold difference at Pin ≈ P0 for membranes with selective layers having 10-30 nm pores. This difference depends obviously on liquid-filled fraction of the pores and is limited by the separation layer flow resistance to liquid and gaseous transport. The curvature radii of input and output menisci for both flow directions reproduce the trends observed for the AAO_40_10 membrane with a divergence point shifting towards higher pressures with increase in pore diameters (figure 6). The difference in the meniscus curvatures, calculated according to the eqn. 2 and 4 also diminishes at larger pore diameters of the selective layer. In the case of membranes with low permeance of the separation layer (AAO_80_10) we observed a huge difference between input and output meniscus curvature in the case of the flow direction A and a small difference in the case of the flow direction B. Notably, the meniscus curvature radius extracted for the first point of condensation (P/P0 = 0.967) in the flow direction B is smaller, than the pore radius, which can be a sign for Joule-Thomson cooling of the membrane by expanding gas

18,19.

At the same time, for the membranes with highly permeable

separation layer the curvatures of input and output meniscus are close to each other, providing low pressure gradient in a liquid in both cases.



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Figure 6. Dependence of input and output menisci curvature on the normalized feed pressure for AAo_80_10 (a) and AAo_80_60 membrane (b)

4. Discussion The most important feature of mass-transport in the capillary condensation regime concerns meniscus position/curvature relation and the dependence of these parameters on pressure conditions. A careful analysis of equations 2 and 4, indicates monotonic increase of permeance with advancing meniscus position, governed by an increasing feed pressure. Analytical description gives no permeance extremum or limitation throughout the pore exit. It results from the monotonic increase of pressure difference in a liquid phase with increasing Pin and decreasing Pc at Pin > Pc (see the fourth relation in system 2). Thus, according to the left part of the first an increase in the feed pressure leads to the growth of the flux through the liquid phase. This requires the same increase of the flux through the part of pore filled with gas in accordance


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with the right part of the same relation. This can be realized by spreading liquid phase in the channels by dl, which results in flux growth proportionally to

L  Lx . Thus, spreading of L  Lx  dl

liquid phase into the pore volume should result in higher membrane permeance according to:

F Lx  dl   F Lx 

L  Lx L  Lx  dl

(6)

This should lead to the advancing of the meniscus to an utmost position with minimum possible curvature radius (corresponding to Pc,0) in all the experiments and any Pin exceeding Pc,0. At the same time, experimentally we observe a growth of output meniscus curvature radius with increasing of feed pressure and well-defined pressure dependence of the meniscus coordinate indicating a limitation of permeance by an inexplicable effect. Resolving this limitation appears as a key problem for mathematical description of the permeability of nanochannels and gasliquid interface behavior in capillary condensation regime. Interestingly, a very similar problem rises in the description of spreading/evaporating meniscus of liquid droplets on solid substrates 20.

Rather a strong experimental correlation of flux with meniscus curvature and position stays

for the necessity of further model improvement linking permeation through nanochannel with pressure conditions. Several possible explanations were proposed earlier to describe the effect. Loimer has suggested that the heat transport from condensing to evaporating meniscus is responsible for the flux limitation:

J  H vap   AAO  (1   )   isobu tan e    

(7)

where J – substance flux, H vap – isobutane vaporization enthalpy, ΛAAO and Λisobutane – membrane and isobutane thermal conductivity, respectively 21. However, adopting heat transfer restriction with thermal conductivity of AAO material of 1.5 W/(m*K)

22

and evaporation

enthalpy of isobutane equal to 21.3 KJ/mol results in an ultimate 0.071 mmol/(s·m·K) specific isobutane permeability. For 50 µm liquid-filled pores this value corresponds to the limitation of mass-transfer rate of 113.6 m3/(h·m2·K), and maximum attained intramembrane temperature



Page 20 of 48

gradients below 2 K for experimentally measured permeate fluxes. It stays in agreement with a theoretical description given in 19 where the authors suggested the temperature drop from feed to permeate side along the membrane of ~3K. Moreover, heat transfer limitations cannot be accounted for the meniscus position dependence on the input pressure. Another reason for the flux limitation governed by heat transfer can appear on the intake meniscus boundary restricting liquefaction efficiency by condensing medium heat release. Accounting for heat conductance of anodic alumina and maximal experimental isobutane flux, one can expect ~2 K overheat at AAO/gas interface. This unlikely can limit condensation efficiency, if only because the curvature of the intake meniscus simplifies vapor condensation. Moreover, the limitation of condensation efficiency should lead to equal condensate permeances for all the membranes in series AAO_80_10 to AAO_80_60 in the case of flow direction B, when the condensation occurs in the supporting layer. This justification is also disposed by the fact of increasing permeance with increasing feed pressure. The third possible explanation concerns the discontinuity of the liquid phase. As soon as high curvature of evaporating meniscus results in large Laplace negative pressure attaining few MPa, cavitation of condensate is conceivable in nanochannel. At the same time, evaluation of the critical radius of cavitation (Rc) according to 23:

Rc 

2  2 Pc , 0  rmeniscus

(8)

indicates that cavitation cannot occur inside the channels with radius of 5 nm until YoungLaplace pressure of isobutane will reach ~ -3.7 MPa. This value corresponds to a critical curvature of ~5.4 nm, which appears only at starting points of condensation in small-diameter pores. Despite cavitation is possible at certain experimental points, it unlikely can result in limitation of intrachannel flux, as soon as decreasing meniscus curvature with growing Pin leads to exceedance of the critical cavitation radius over the pore radius and diminishing of cavitation probability.


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Finally, the flux limitation can be governed by the deviation of the meniscus curvature from stationary conditions noted earlier. Indeed, higher liquid flux requires higher velocity of the liquid, which will necessarily disturb the meniscus shape according to that suggested by Poiseuille velocity distribution in the liquid and will decrease its curvature in the pore center. This will assuredly affect Laplace pressure under meniscus limiting the liquid influx. Among the noticed reasons, the role of meniscus shape seems to be most adequate for governing mass transport in nanochannels in the capillary condensation regime. Moreover, the meniscus shape can take even more complicated form due to Marangoni flows appearing at the sides of meniscus. Thus we have first analyzed possible heat transfer limitations of the evaporation rate from the meniscus and possible presence of perpendicular velocity component due to thermo-capillary stress at the interface. For this reason, we have evaluated characteristic temperature gradients assuming conductive heat transfer from the pore wall through liquid isobutane. Calculations were performed for individual pore using experimental diameter and porosity values of selective layers and extreme membrane performance parameters. Heat power necessary to evaporate liquid isobutane in the smallest (~11 nm) and in the largest (85 nm) pores with maximum experimental fluxes of 125 m3/(m2·h) and 325 m3/(m2·h) equaled to 9.0·10-11 W and 5.8·10-9 W, correspondingly. Heat transfer from the pore wall with effective heat transfer area of 2𝜋𝑟2 to the central point of the meniscus was used for an upper estimate of the temperature gradient. Those calculations resulted in ΔT = 0.015K for the smallest pores and ΔT = 0.12K for the largest pores, illustrating clearly no driving force for convective heat transport appearance and no necessity for liquid intermixing in laminar flow close to the meniscus.



Page 22 of 48

Figure 7. The dependence of output meniscus curvature radius on the maximum liquid velocity for the flow direction A (a) and flow direction B (b). Experimental dependence of an average curvature radius on condensate velocity for AAo_80_10 membrane measured in the flow direction A compared to theoretical results obtained in the frame of the models of average liquid influx, Poiseuille liquid influx, gas-liquid rotating flow and elliptical meniscus shape (see text for details, c). On the inset the meniscus shapes calculated according to elliptical model are presented. To reveal a dependence of the meniscus shape on liquid influx we have plotted the meniscus radii derived from experimental data with eqn.2 and eqn.4 as a function of maximum liquid velocity in the channels (figure 7). Experimental condensate velocities were calculated from isobutane flux assuming Poiseuille velocity distribution in the channels. An obvious tendency is well seen for all the tested membranes in both flow directions. Moreover, in case of the flow direction A the curves merge into a single dependence for different membranes, indicating the presence of a well-defined relation of meniscus curvature with flow velocity in a liquid. The ACS Paragon Plus Environment

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difference in trends of the curves in case of the flow direction B can be attributed to vapor flow restrictions by low-permeable selective layers. To our opinion the effect is similar to one published earlier and originates possibly from the contact angle hysteresis for an advancing/receding meniscus 13. Despite numerous works elucidating menisci shape in stationary conditions or receding menisci case 20,24,25, we failed to find an analytical solution for the shape of an advancing meniscus with liquid substance intake. Moreover, the known models assuming liquid redistribution under meniscus due to surface tension gradients do not provide regulating terms for meniscus curvature radius on liquid velocity, thus resulting in an overestimate of meniscus curvature radius and mass fluxes

20,26.

Thus we proceeded with deducing local menisci curvatures accounting for liquid

influx and liquid-gas phase transformation of permeate. All models discussed below suggest isothermal conditions with temperature on input and output meniscus equal to an external temperature. Despite involving thermal effects can significantly affect the results, here we neglect thermal effects due to serious complications introduced to the calculations. The first approximation involved balancing the amount of substance transferred through a liquid phase with the amount of substance evaporating from the meniscus with fixed curvature radius (the model of average liquid influx):

u

2  d pore

Mr



4

1  2MRT

  d 2    P0  exp  2M   Pc , 0   pore  r    4  meniscus RT   

(9)

where u represents liquid velocity, and the left part of the equation governs evaporation from meniscus with curvature radius rmeniscus into the pore volume in accordance with theory proposed by Schrage 27. Then, the meniscus curvature can be extracted as: rmeniscus 

 2M   u Mr ln   

 2MRT  Pc , 0   RT  P0  

(10)



Page 24 of 48

The dependence of an average curvature radius vs. condensate velocity is plotted on fig. 7c with blue triangles. Meniscus curvature radii evaluated with eqn. 10 are seriously underestimated as compared to experimental results, those are believed to be evaluated assuredly. Indeed the possible error in the determination of the meniscus curvature from the experimental permeation data is rather small due to robustness of the solution for the Laplace pressure with growing permeate flux. A careful analysis of the meniscus shape changes with pressure conditions (fig. 7a, b) indicates that growth of its curvature radius (over 10 pore diameters in several cases) cannot appear at the pore circumference due to a decisive contribution of the channel diameter to a mean curvature of the meniscus surface. Thus, the only solution involves lowering of the meniscus curvature at the pore centrum. Taking into consideration the above, the second approximation involved Poiseuille velocity distribution of liquid intake and evaporation from the meniscus assuming no intermixing of a laminar flow. The balance of the amount of condensate transferred through the layer with a thickness dr distant from the pore center on r and the amount of substance evaporated from the same layer of meniscus at the stationary conditions was formulated as (Poiseuille liquid influx model):

 4r 2   1 u 0 1  2    2rdr   d  M 2MRT pore  r 

     2M  P0  exp   Pc , 0   2rdr  r    meniscus (r ) RT   

(11)

The presence of Poiseuille velocity distribution in the liquid leads to the dependence of local meniscus curvature on the distance from the pore center. The local curvature radius of meniscus can be than found using modified eqn. 10: rmeniscus (r ) 

 2M 2      u 1  4r   2MRT  P  c,0 2 M  0  d pore   ln  RT P0      

(12)


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With this assumption, meniscus curvature radius in the pore center can exceed the equilibrium meniscus curvature radius corresponding to inpore condensation pressure of Pc,0, see also figure 7c. However, a simple analytical solution of the expression 12 with Poiseuille distribution of fluid velocity results in the meniscus shape with a nonzero wetting contact angle of pore surface. On the other hand, at the inpore pressure of Pc,0 the pore wall surface should be covered with the layer of isobutane molecules, which requires zero contact angle of pore interior. This indicates an appearance of a discontinuity of curvature at the pore edges for the proposed model. To get rid of the discontinuity point and obey boundary condition of wetting contact angle equal to zero, the resulting local curvatures should be reduced to give an interpolation of the meniscus shape onto the pore radius 28. This should result in an effective evaporation of liquid at the pore center and vapor condensation at pore circumference. This situation corresponds to a gas-liquid rotating flow involving consecutive evaporation and condensation steps. The meniscus shape and in-pore distribution of the meniscus curvature for this case is schematically shown on the figure 8a. This situation controverts conventional meniscus shape in microchannels accustomed for stationary wetting, which can occur at small substance intake and low heat transfer in the condensate (figure 8b)

25.

At the same time, such meniscus geometries are well known for

advancing water menisci in microchannels 9,29.

Figure 8. A schematic representation of meniscus shape and curvature radii distribution for the cases of a) high liquid intake and b) low liquid intake or stationary wetting. Flux lines are represented by arrows.



Page 26 of 48

The presence of aforementioned rotating flow should necessarily reduce overall evaporation efficiency over meniscus with a curvature radius exceeding the radius of pores. Experimental evaporation efficiencies were calculated from in-pore flow rates by normalizing them to maximum evaporation efficiency derived from the kinetic theory:

2 pore

u0  d   2 M 4



2  d pore 2  rmeniscus       4  P 0 exp  2M   P c , 0 r 2 1    meniscus   rmeniscus 2MRT   rmeniscus RT      2 u0  d pore  2 M 4

2MRT

2  d pore 2  r  meniscus     4  P 0 exp  2M   P c , 0 r 2 1   r  meniscus   RT r meniscus  meniscus      


      

(14)

      

(13)

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Figure 9. The dependence of evaporation efficiencies in different membranes on relative meniscus curvature for the flow direction A (a) and flow direction B (b).

The resulting evaporation efficiency values were found seriously below unity for any external conditions (figure 9), whereas evaporation efficiency close to 1 was expected for liquefied nonpolar isobutane from theoretical predictions 30. The drop of experimental evaporation efficiencies can be explained by the reduction of an evaporation hemisphere over the curved meniscus surface (see figure S1). However, the estimate for pore diameter of ~10 nm gives evaporation suppression of below ~25%, which is significantly lower compared to experimental results. Thus, the reduction of evaporation efficiency was ascribed to the condensation of evaporated liquid at the sites with the lowest curvature radius of the meniscus, confirming the assumption of gas-liquid rotating flows. To explain nonmonotonic dependence of evaporation efficiency on the meniscus curvature, the following model has been suggested: with liquid intake the shape of meniscus deviates from an equilibrium profile providing higher curvature radii at the pore center (enhancing evaporation) and lowering curvature radii at the pore circumference (favoring condensation); those conditions give rise to peripheral gas-liquid rotating flows and result in initial Poiseuille velocity redistribution in the meniscus vicinity (figure 10). This can be described as narrowing of the channel cross-section to a certain site at the pore centrum (Sevap). Meniscus curvature radius at this site can exceed significantly the pore radius, providing evaporation of substance intake. Increasing liquid intake to the meniscus appears as follows. At low or zero liquid intake the curvature radius of meniscus at the pore centrum is equal or lower than the pore radius providing an evaporation efficiency close to unity. Growing of the input liquid velocity disturbs the meniscus, increasing the curvature radius at the pore center and decreasing it at the pore circumference. Condensation of vapors close to the pore walls (exponentially depending on the local meniscus curvature) reduces overall evaporation efficiency with growth of meniscus



Page 28 of 48

curvature radius at the pore centrum. However further warping of the meniscus increases evaporation site and reduces the condensation area until saturation is reached, governed likely by in-pore vapor transport or local heat transfer at the meniscus. With this assumption, general tendency of decreasing evaporation efficiency with decreasing nanochannel diameters is explained by more unsmooth meniscus profile and larger difference between the equilibrium pressure and local pressure over meniscus.

Figure 10. The schematic representation of flow redistribution in the meniscus vicinity at high substance intake. Flux lines are represented by arrows.

Mathematical description of the proposed gas-liquid rotating flow involving evaporation from meniscus sites with high curvature radius, in-gas transport and condensation at the meniscus sites with lower curvature radius necessitates incorporating an additional intermixing term to liquid velocity distribution reflecting medium exchange between the layers. The most prominent way for the determination of the velocity distribution in the system is solving incompressible Navier– Stokes equation, taking into account the difference in Laplace pressure under meniscus and evaporation at the interface. However analytical derivation of the velocity field in the vicinity of meniscus of unequal curvature is strongly impeded by the presence of radial component of liquid velocity and inhomogeneous field of body forces provided by the local curvatures of the meniscus. Thus, here we proceed with the problem formulation and derivation of an approximate analytical solution. ACS Paragon Plus Environment

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As soon as the meniscus in a cylindrical nanochannel should possess an axisymmetric shape with a symmetry axis aligned along the pore center, the shape of meniscus can be defined in polar coordinates with the function: z (r )  f u (r ) 

(15)

where 𝑢(𝑟) corresponds to a liquid velocity distribution function under meniscus. Accounting for mass balance of liquid intake and evaporation from the meniscus as well as continuity and smoothness of both meniscus curvature and input velocity functions with zero contact angle at the pore walls we can write the following system of equations:

  d pore    0  F  2      dF 0  dr r  d pore 2    1        H ( r )   2M     P  exp      u r   2 rdr   P 0  r  c , 0   2rdr  Mr ( r ) RT 2MRT   meniscus     d  1  pore     2 2 2  H (r )   P  exp  2M  H (r )   P rdr  u  d pore 0 0 2MRT  0  RT  c,0   Mr 4 

(16)

where H(r) function describes the mean local curvature, defined as:

    2F   2 1 F 1  r  (17) H (r )   2    F  2 3 / 2 r   F  2 1/ 2     1   r 1       r       r       

 1   – is the probability of evaporation or condensation of molecule in the point with and    H (r )  local curvature H(r) (see supporting information SI1 for details).



Page 30 of 48

Here we have disregarded disjoining pressure in the calculations as providing negligible effect on the meniscus curvature as compared to intake velocities. Indeed the maximum disjoining pressure for isobutane for the lowest pore diameter of 10 nm was estimated as ~7500 Pa, while the changes in Laplace pressure within the meniscus attained ~1.4∙106 Pa (see supporting information SI2 for details). Unfortunately, the solution to the system remains parametric as the velocity function cannot be deduced analytically from the available set of data. Thus, the system has been solved numerically by introducing the velocity field involving Poiseuille flow intake and convective gas-liquid flows according to the following equation:

 4r 2   u ( r )  u0 1  2  u (r ) (18)  d  c pore   where 𝑢𝑐(𝑟) describes liquid velocity changes under meniscus due to gas-liquid rotating flow.

 d pore    0 and This term should obey boundary conditions: uc   2 

d pore 2

 u (r )dr  0 . c

Resulting

0

negative values of 𝑢(𝑟) correspond to condensation of vapors on the site distant on r from the pore center, positive value – to evaporation. Experimental evaporation efficiencies were employed as fidelity criteria in the evaluations. The resulting meniscus curvatures at the pore center are plotted on fig. 7c with green rhombs (gasliquid rotating flow model). The results obtained with solution of equation system (16) provide much better fit to the experimental results, which can be regarded as an indirect prove for the proposed model. An easier (but more rough) approach for the determination of the meniscus shape in case of liquid intake to the meniscus is direct employment of an axisymmetric elliptical shape function:

d  r2  z (r )  pore  1  2  2  a 

n

(19)

and balancing liquid intake velocity with evaporation rate from the defined shape using eqn. system (15). This approach provides much faster treatment of data and gives a suitable fit to


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experimental results (elliptical meniscus model with n < 0.5 on the figure 7c). Notably the proposed shape function is unsuitable for describing data at high liquid intake velocities as it cannot describe locally negative curvature of the meniscus. On the other hand, for the intermediate region (low Laplace pressure difference), it can be utilized as an explicit estimate of the meniscus curvature distribution function for further analysis. In summary, a deviation of the meniscus shape from stationary conditions with liquid influx is considered as a main reason governing mass-transport characteristics of nanochannel in the capillary condensation regime affecting strongly both pressures in the liquid and vapor phases in the vicinity of the meniscus. Increasing the curvature radius of meniscus with liquid intake leads to both increase of the vapor pressure over the meniscus and decrease of the pressure difference in the liquid. While the first has a very limited effect on the permeance due to inpore condensation of vapors, the last significantly limits liquid intake. Thus the meniscus shape both warps under the liquid flow and controls liquid velocity in nanochannels, which should be considered as self-regulating regime of flow. At high liquid intake velocities the curvature radius of meniscus significantly increases, being accompanied by a serious drop of liquid evaporation efficiency. The effect has been ascribed to a sequential evaporation of liquid in the pore center and vapor condensation close to the pore walls, generating gas-liquid rotating flow. Despite the proposed model marks out the way for direct relation of mass transport efficiency through the meniscus to influx velocity in a liquid, a lot of issues remain underway towards the analytical expression, including pressure-velocity profile relation and radial velocity contribution in a liquid. However, we believe that present findings and proposed concept of gas-liquid convection importantly benefit an understanding of gas-liquid transport phenomena in nanochannels and nanocapillaries. Further improvement of the model with introducing thermal effects, will allow more comprehensive and precise description of capillary transport.

5. Conclusions



Page 32 of 48

Based on the above experimental results, the following principals for large mass flux differences for opposite flow directions of a condensable gas through an asymmetric porous membrane can be elucidated: 1)

Considerable mass flux differences for opposite flow directions of a condensable gas

through an asymmetric porous membrane are experimentally proved; Membrane permeance in capillary condensation regime grows extensively with Pin giving no plateau or kinks in experimental permeance-pressure dependence. In case of the separation layer upstream orientation, the liquid-vapor interface can suffer pinning to the separation layer/supporting layer boundary, followed by spreading of liquid into supporting layer as soon as condensation in large diameter pores becomes possible. For the separation layer downstream orientation, liquid spreads quickly to the separation layer once the condensation in the supporting layer becomes possible. 2)

Mass flux difference for opposite flow directions in the asymmetric membranes operating

in the capillary condensation regime is mostly governed by the separation layer flow resistance to liquid and gaseous transport and restrictions of condensation in the separation layer due to flow resistance of the supporting layer. 3)

Flow resistance of the layers to liquid transport is mostly governed by the

condensing/evaporating menisci curvatures deviating from thermodynamic equilibrium value and gradually changing with pressure conditions; Knowledge of menisci curvatures is prerequisite for quantitative predictions of membrane permeance in the capillary condensation regime. Coincidently, the curvature of meniscus is controlled by liquid influx velocity and velocity distribution in a nanochannel, giving rise to a self-regulating regime of flow. At high liquid intake velocities, the curvature radius of meniscus significantly increases, being accompanied by a serious drop of liquid evaporation efficiency. The effect is ascribed to a sequential evaporation of liquid in the pore center and vapor condensation close to the pore walls, generating gas-liquid rotating flow.


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4)

Heat transfer parameters can result in flux limitations for heat-insulating membrane

materials but unlikely to provide flux limitations in heat-conductive membranes, such as anodic aluminum oxides.

6. Supporting Information Description Calculation of probability of evaporation or condensation of molecule in the point with different local curvatures (SI1) and calculation of disjoining pressure in nanochanel filled with isobutane (SI2) as given. Numerical experimental data and derived calculated parameters for i-butane transport in both membrane orientations for membranes with different pore diameters (SI3).

7. Acknowledgement The work is supported by the Ministry of education and science of the Russian Federation within a Federal Targeted Programme for “Research and Development in Priority Areas of Development of the Russian Scientific and Technological Complex for 2014−2020” (Agreement No. 14.604.21.0177, unique Project Identification RFMEFI60417X0177).

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9. TOC Graphics



Figure 1. Cross sectional SEM micrographs of (a) symmetrical membrane with straight pores obtained at 40V (AAO_40) and (b) asymmetrical membrane obtained by decreasing anodization voltage from 40V to 10V (AAO_40_10). 177x67mm (300 x 300 DPI)


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Figure 2. Dependence of permeance on the normalized feed pressure for the membrane with straight pores AAO_40 measured with bottom surface upstream (flow direction A) and bottom surface downstream (flow direction B).



Figure 3. Dependence of permeance on the normalized feed pressure measured in flow direction A and B for the membrane AAO_40_10 with 12 nm pores in the selective layer and 40 nm pores in the supporting layer (a). Dependences of the meniscus position (b), input and output menisci curvature radii (c) on the normalized feed pressure measured in flow direction A and B for the AAO_40_10 membrane.


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Figure 4. Scheme of pressure distribution in the case of direct flow and opposite flow direction for membrane operating in transitional flow regime (a,d) and capillary condensation regime with part of the pore filled with condensate (b,c,e,f).



Figure 5. Dependence of the permeance (a,b) on the normalized feed pressure for asymmetric membranes with the selective layer formed at anodization voltages from 10 to 60V and the supporting layer formed at 80V measured with flow direction A and B. Extreme liquid phase thickness (Lx) at Pin ≈ P0 form membranes with different pore diameter in the selective layer (c) calculated from the experimental data for flow directions A and B.


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Figure 6. Dependence of input and output menisci curvature on the normalized feed pressure for AAo_80_10 (a) and AAo_80_60 membrane (b)



Figure 7. The dependence of output meniscus curvature radius on the maximum liquid velocity for the flow direction A (a) and flow direction B (b). Experimental dependence of an average curvature radius on condensate velocity for AAo_80_10 membrane measured in the flow direction A compared to theoretical results obtained in the frame of the models of average liquid influx, Poiseuille liquid influx, gas-liquid rotating flow and elliptical meniscus shape (see text for details, c). On the inset the meniscus shapes calculated according to elliptical model are presented.


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Figure 8. A schematic representation of meniscus shape and curvature radii distribution for cases of a) high liquid intake and b) low liquid intake or stationary wetting 177x42mm (300 x 300 DPI)



Figure 9. The dependence of evaporation efficiencies in different membranes on relative meniscus curvature for the flow direction A (a) and flow direction B (b).


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Figure 10. The schematic representation of flow redistribution in the meniscus vicinity at high substance intake. Flux lines are represented by arrows. 82x37mm (300 x 300 DPI)



TOC image


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Meniscus Curvature Effect on the Asymmetric Mass Transport through

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