ISSN 1028-3358, Doklady Physics, 2008, Vol. 53, No. 3, pp. 118–121. © Pleiades Publishing, Ltd., 2008. Original Russian Text © I.M. Kurchatov, N.I. Laguntsov, V.N. Tronin, V.I. Uvarov, I.P. Borovinskaya, 2008, published in Doklady Akademii Nauk, 2008, Vol. 419, No. 1, pp. 38–40.
PHYSICS
On the Mechanism of Asymmetric Gas Transport in Anisotropic Porous Media I. M. Kurchatova, N. I. Laguntsovb, V. N. Troninb, V. I. Uvarovc, and I. P. Borovinskayac Presented by Academician A.G. Merzhanov June 24, 2007 Received July 6, 2007
PACS numbers: 47.56.+r, 51.10.+y DOI: 10.1134/S1028335808030026
Recently, processes occurring in anisotropic nanoporous media whose characteristics (e.g., pore radius and porosity) vary along a given direction in accordance with a certain law have attracted particular interest [1–4]. For these media, phenomena of asymmetric gas transport have been discovered, namely, anisotropic gas permeability [1–3] and catalytic activity [1], gaspermeability hysteresis [3], and the variation of the temperature dependences for gas permeability with a change in the gas-flow direction [1, 4]. As our analysis has shown, all effects of asymmetric gas transport share a common feature. We imply that these phenomena are observed in the region of pressures and pore sizes for which the free molecular-flow regime predominates (the Knudsen number is Kn ≥ 1). This is the basis for the assumption that the effects of asymmetric gas transport are associated with the character of the interaction of gas molecules with the internal pore-wall surface of a porous medium. In this paper, we propose a theoretical approach to the description of gas flows in a two-layer membrane. We have determined the dependence of the asymmetric gas transport on parameters of the interaction of gas molecules with rough membrane-pore walls. We consider the porous medium to be similar to a set of channels with randomly shaped walls having the
average hydraulic radius ρ (the ratio of the doubled volume of pores to the area of their wall surface). Then, in the case of the motion of a molecule in this channel, the angle θ of reflection of the molecule from the channel walls has both the mirror component and the random component, the latter being stipulated by the pore-wall roughness. We now also assume that the velocity of molecules is equal to the average thermal velocity Vh = 8RT ----------- , where M is the molar mass, T is temperature, πM and R is the universal gas constant. Insofar as the directions of the molecule motion before and after the collision are uncorrelated, the deviation of the angle from its mirror value in the case of the reflection from a rough wall can be represented as the result of a random process ξ(t). This process determines the random variation of the molecule recoil from the rough surface. Under these assumptions, we can write out the Langevin equation [5] for the directing angle of motion 2θV V dθ ------ = – ------------h + ------h ξ ( t ). ρ dt ρ The function ξ(t) is the white noise 〈 ξ ( t )〉 = 0,
a
Moscow Engineering Physics Institute (State University), Kashirskoe sh. 31, Moscow, 115409 Russia b JSC “Aquaservis,” Kashirskoe sh. 31, Moscow, 115409 Russia c Institute of Structural Macrokinetics and Materials Science, Russian Academy of Sciences, ul. Institutskaya 8, Chernogolovka, Moscow Region, 142432 Russia e-mail:
[email protected];
[email protected];
[email protected]
(1a)
〈 ξ ( t )ξ ( t' )〉 = σ δ ( t – t' ), 2
(1b)
where σ is the amplitude of the white-noise correlator that characterizes the interaction of molecules with the surface. The skew bracket implies averaging over the ensemble of all the molecules. The quantity σ represents the mean angular deviation from the mirrorreflection angle.
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Langevin equation (1) corresponds to the Fokker– Planck equation [5] for the distribution function w(θ, t) ∂w that determines the probability density ------- for mole∂t cules to have at the instant of time t the angle θ between the direction of motion and the normal to the surface: 2 2 2 ∂ ⎛ 2θV h ⎞ σ ∂ ⎛ V h ⎞ ∂w ------- = – ------ – ------------w + ----- --------2 ⎜ -----2- w⎟ . ⎠ 2 ∂θ ⎝ ρ ⎠ ∂θ ⎝ ρ ∂t
w(θ) 4 3 3
2 4
(2)
2
The time-independent solution to Eq. (2) within the angular range from –π/2 to +π/2, which is normalized to the variation region for angles of reflection from the pore wall, can be written out in the form w(θ) =
A exp ( – Aθ ) --- --------------------------- . π π erf ⎛ --- A⎞ ⎝2 ⎠
1
1 −π/2
0
π/2 θ
2
(3)
The function w(θ) is the even function of θ and depends 2ρ - whose value determines the on the parameter A = ----------2 σ Vh width of the angular distribution for molecules. The shape of distribution (3) for different values of the parameter A is presented in Fig. 1. Figure 1 exhibits the character of the variation of the distribution w(θ) for different values of the parameter A. For A 1, the distribution is omnidirectional (curve 1), which corresponds to the diffusive model of the interaction of a molecule with the surface (complete accommodation). Nearby, the distribution (curve 4) that corresponds to the model of “mixing billiards” is plotted [6]. According to this model, the surface is represented in the form of repeated parabolic hollows with the mirror reflection from them. We now analyze a two-layer membrane with the gas flow in it occurring in the free molecular regime. The first membrane layer is considered to be a porous medium with an anisotropic (with respect to directions of motion) distribution of gas molecules. We suppose that the second layer has a smaller permeability than the first one. When the gas flow attains the first layer, gas molecules pass through the first layer, arrive at the region of the interface between the porous layer and low-permeability layer, and then a fraction of the molecules penetrates the second layer. The other part of the molecules after their reflection from the boundary has the velocity of motion 〈Vref〉 along the axis of the channels, which exceeds the velocity of molecules penetrating without reflection through the boundary from the first layer into DOKLADY PHYSICS
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Fig. 1. Distributions of molecules over their directions of motion. The curves correspond to the following values of the parameter A: (1) A = 0.01; (2) 3; (3) 10. Curve 4 is the prediction of the model of mixing billiards.
the second one (their average velocity along the axis is 〈Vx1〉). Under the assumption that the pressures are P1 > P2, the balance equation at the interface between the layers can be written out as 0.32P 1 J = ----------------V h – n 1 〈 V x1〉 , RT J = n *1 〈 V x1〉 – 0.32ξn *2 V h – ( 1 – ξ )n *1 〈 V ref 〉 ,
(4)
P J = 0.32ξ ⎛ n 2 – -------2 ⎞ V h . ⎝ RT ⎠ Here, P1 is the pressure at the input of the first layer; n1 is the concentration of molecules in the first layer at the boundary with the external medium; n *1 is the concentration in the first layer at the boundary with the second layer; n *2 is the concentration in the second layer at the boundary with the first one; n2 is the concentration in the second layer at the boundary with the external medium; P2 is the pressure at the output of the memε brane; and ξ = ----2 is the ratio of the porosities of the secε1 ond and the first layers, respectively. In deriving the set of Eqs. (4), we assumed that molecules moved in the plane passing through the channel axis and that, in the second layer of the membrane, gas molecules are distributed omnidirectionally.
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Permeability, 10–4 cm3/(cm2 s atm) 2.5 6 2.0
1
1.5
2 7
3
1.0
4 5
0.5 0
1
3
5
7
9
11 P, atm
Fig. 2. Pressure dependence for the two-layer membrane permeability: (1) calculation curves (virtually coinciding with each other for A = 1, 1.5, and 2.5) in the case of gas arrival at the finely porous layer; (2), (3), (4), and (5) are the calculated permeability curves in the case of gas arrival at the first layer (A = 1, 1.5, 2, 2.5, respectively); (6) and (7) are experimental data in the case of gas arrival at the finely porous layer and at the layer with anisotropic pore distribution, respectively.
In the case of the free molecular regime of gas flow, the gas-transport equations in the membrane layers are of the form [8] πρ 1 V T⎞ n *1 – n 1 ----------------- , J 1 = – ε 1 g 1 ⎛ --------------⎝ 32 ⎠ l 1 πρ 2 V T⎞ n *2 – n 2 ----------------- , J 2 = – ε 2 g 2 ⎛ --------------⎝ 32 ⎠ l 2
(5)
where l1 and l2 are the thicknesses of the first and second layers, respectively, and g1 and g2 are the coefficients taking into account the roughness and twisting of channels. We now consider the flow through the membrane in the case when the deviation from equilibrium is small and the directional distribution of molecules obeys formula (3). Then, the average projections 〈Vx1〉 and 〈Vref〉 of the velocities for molecules moving along the direction of the flow motion and reflected by the interlayer boundary can be written out, respectively, in the form π/2
〈 V x1〉 = V h
∫ w ( θ ) sin θ dθ, 0 π/2
〈 V ref 〉 = V h
∫ w ( θ ) cos θ dθ. 0
For the given P1, P2, T, ρ1, ρ2, l1, l2, k1, k2, ε1, ε2, and ξ and also under the conditions of the free molecularflow regime when we may ignore surface flows, set (4)–(6) is the complete set of equations, which allows the quantities n1, n2, n *1 , n *2 , and J to be determined. The set of equations obtained makes it possible to allow for the effect of parameters of the porous medium on the flow through the two-layer membrane and to find the conditions for the appearance of the asymmetric gas transport. The analysis of set of Eqs. (4)–(6) has shown that when the condition 〈Vx1〉) < (〈Vref〉 is valid, the flow from the membrane layer in which the anisotropic distribution of molecules over directions of motion had been realized was several times lower than the incoming flow from the low-permeability layer. In Fig. 2, the results of solving the set of Eqs. (4)–(6) are presented for different values of the parameter A. As is seen, the anisotropy (the ratio of the permeabilities) rises with an increase in the parameter A, and in the case of the arrival at the selective layer, the flow is also virtually independent of the parameter A. The calculation parameters were chosen in correspondence with the ceramic two-layer membrane, whose permeability has been studied experimentally. The membrane’s first porous layer was made of boron nitride (white graphite) by employing the technology based on the self-propagating high-temperature synthesis, and it had the following parameters: the porosity was 38%; the pore diameter was 50 nm; and the specific surface was 50 m2 g–1. The second layer had pores of about 3 nm in diameter. Comparison of the experimental and calculation data shows that the model used describes adequately the experimental data for A ≈ 1–1.5. Thus, in the present study, the mechanism of asymmetric gas transport in a two-layer membrane has been proposed. The effect of anisotropic gas permeability in the two-layer membrane produced by the technology of self-propagating high-temperature synthesis has been investigated. We have also shown that the calculation results correspond adequately to the experimental data obtained. ACKNOWLEDGMENTS The authors are grateful to V.D. Borman for fruitful discussions. This work was supported by the Russian Foundation for Basic Research, project no. 07-08-00461-a.
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ON THE MECHANISM OF ASYMMETRIC GAS TRANSPORT 2. I. M. Kurchatov, N. I. Laguntsov, G. I. Pisarev, et al., in Proceedings of XXI International Symposium on Physicochemical Methods of Separations “ARS SEPARATORIA 2006,” Torun, 2006, p. 55.
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3. I. M. Kurchatov, N. I. Laguntsov, V. N. Tronin, et al., in Proceedings of XXII International Symposium on Physicochemical Methods of Separations “ARS SEPARATORIA 2007,” Szklarska Poreba, 2007, p. 137.
5. R. Balesku, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975). 6. G. M. Zaslavskiœ, Chaos in Dynamical Systems (Nauka, Moscow, 1984; Harwood, Chur, 1985). 7. E. M. Lifshitz and L. P. Pitaevskiœ, Physical Kinetics (Nauka, Moscow, 1979; Pergamon, Oxford, 1980). 8. R. M. Barrer and D. Nicholson, Br. J. Appl. Phys. 17, 1091 (1966).
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Translated by G. Merzon
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