Two-particle Distribution Function of a Non-ideal ... - Science Direct

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ScienceDirect Physics Procedia 71 (2015) 364 – 368

18th Conference on Plasma-Surface Interactions, PSI 2015, 5-6 February 2015, Moscow, Russian Federation and the 1st Conference on Plasma and Laser Research and Technologies, PLRT 2015, 18-20 February 2015

Two-particle distribution function of a non-ideal molecular system near a hard surface Yu. Agrafonov*, I. Petrushin Irkutsk State University, 664003, Karl Marx str., 1, Irkutsk, Russia

Abstract Surface forces in boundary layers or thin films of classic molecular systems should be considered when describing different physical-chemical phenomena (adsorption, wetting). In this case a molecular system has axial symmetry. We obtained the solution for two-particle distribution function at low densities. It is shown that the solution describes the transition from axial to spherical symmetry while each particle goes to infinity 2015The TheAuthors. Authors. Published by Elsevier ©©2015 Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the N ational R esearch Nuclear U niversity M E P hI (M oscow E ngineering Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) P hys ics Institute)

. Keywords: fluids; Ornstein-Zernike equation; distribution functions

1. Introduction We should take into account surface forces boundary layers and thin films of classic molecular systems when describing phenomena (adsorption, wetting) in near-surface layers. In such cases a molecular system has axial symmetry, so we may apply a model of a liquid near a hard surface. Other cases for such a model are phases contact for liquid-steam, liquid-crystal, where there is big difference between concentration of contacting phases. These cases well fit the model of a liquid near hard surface. When far

* Corresponding author. Tel.: +7-902-5669153; fax: +7-3952-242194. E-mail address: [email protected]

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) doi:10.1016/j.phpro.2015.08.353

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from the surface contact, a molecular system is heterogeneous and isotropic and that’s why we should take into account the boundary condition for transition from axial to spherical symmetry. Moreover, properties of the system are described by pair molecular correlations, decreasing with the distance between particles as interaction potential U(r), usually r-6. In quantum systems the situation is more complex, where pair molecular correlations exist even in ideal gas systems, and interaction potential U(r) decreases as r-4. In case, when the size of the system is much more than correlation length l0, details of molecular interaction are incidental. That’s why we should carry out renormalization of potential in Fermi-liquid theory, if l0 is comparable with the size of the system (thin films, liquid nanodrops). In this paper we will focus on description of classic molecular systems by model of a liquid near a hard surface. 2. Common equations Statistics considering molecular system is based on Born-Green-Yvon-equations system (also known as BGY hierarchy) for l-particle functions Gl ,},i  rl ,}, ri for ensemble of identical particles. These particles interact with each other through potential )ij rij , where rij ri  rj is the distance between particles i and j centers. The diameter of each particle is σ. The BGY-equations system may be transformed to equations for one- and twoparticle distribution functions, which may be written as Ornstein-Zernike Martynov (2008), Martynov (1992), Apfelbaum et al. (2007):



Z1 n ³ G2C121 d  2  ln a

h12

;

C12 2  n ³ C13 2 h23d  3

(1)

Here we integrating on coordinates of i-particle: d  i { dri , n – density, G1 exp(  )i  kT  Z1 ) oneparticle distribution function, which describes particle position; ) i - potential energy in external field; Zi - oneparticle thermal potential; a – activity coefficient, which is defined by condition of passing to isotropic system. 1 hij [exp(  )ij  kT  ˟ij )  1] pair correlation function, which is connected with two particle distribution function by expression: Gij G j Gi 1  hij ; ˟ij two-particle thermal potential, which takes into account indirect k interaction of two particles; Cij - direct correlation functions: 1



Cij1







hij  Zij  1 hij Zij  M ij1 ; 2

Cij 2

hij  Zij  M ij 2

(2)

Functions G1(r1) and G12(r1,r2) are critical ones while they describe internal structure and let us obtain 1  2 thermodynamic parameters of the system. Equations (1) and (2) are difficult to solve because M ij and M ij contains infinite series of distribution functions. To use the following equations in practice one should approximate these series by simple expressions (closures). Thus we can obtain approximated equations for high density systems. Most known of them are hyperchain, the Percus-Yevick and Martynov-Sarkisov approximations. Isotropic systems, such as spatial liquids when far from surface, are of special interest. In these systems G1  r { 1 and Z1  r { 0 . As a result, the first equation of the system (1-2) simplifies to

P ln a n ³ C121  r12 dr12

(3)

Second equation may be formed as

h12

C12 2  n ³ C13 2 h23d  3

It defines the unknown function two-particle distribution function

(4)

h120 G12 0  r12  1 . The structure and thermodynamic parameters evaluates by G12  r12 , which depends on density n V 3 NV 1 .

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Space-heterogeneous systems (such as a liquid near a hard surface) are described by one- and two-particle distribution functions: G1(r1) and G12(r1,r2). Boundary condition for these equations is a transition from a hard surface to a liquid. 3. Singlet approximation Equations (1-2) are very hard for direct solution. In order to simplify this problem we may substitute function  0 G12(r1,r2) by its boundary value G12 r12 for spatial liquid. We call such substitution as singlet approximation. As a result we obtain the equation for one-particle distribution function G1(r1), which describes local density profile n(r1)=nG1(r1) near a hard surface. Numerical solution for this equation is written by Tikhonov et al. (1999). We cannot estimate the precision of singlet approximation because of neglecting the close order change near surface. The possible approach to estimate the precision is to compare the result of calculation with the result of numerical modeling (numeric experiment). In the following work we increase the precision of a singlet approximation by taking into account the close order change near surface for functions G 1(r1) and G12(r1,r2). Let’s form the equations for molecular system near hard surface. Computing origin is located in the center of the particle, which contacts with hard surface. The Z axis is perpendicular to the surface, thus the whole liquid is placed in upper half-space ( z t 0 ). Bottom half-space ( z  0 ) is unavailable for the particles (Fig. 1). Such a system has axial symmetry,



G1  r1 G1  z1 exp Z1  z1

G12  r1 , r2 G12  z1 , z2 , r12

where r12 is measured in particle diameter units, for G1(r1) and G12 (r1,r2) are defined as follows

Z1  z1 o 0, z1 o f

G1  z1 o 1 z1 o f

(5)

zi t 0 – particle distance from the surface. Boundary conditions

G12 0  r12

lim

r12

z1 of z2 of r1 r2 const

G12  z1 , z2 , r12

(6)

System (1-2) with a singlet approximation takes the form

Z1 n ³ G2C121,0 d  2  ln a h12 0

0 C12 2,0  n ³ C13 2,0 h23 d  3

One-particle potential may be obtained from equation (7), where direct correlation function  0 correlation function h12 are considered to be known.

(7) (8)

C121,0  r12 and pair

4. Going beyond singlet approximation Close order in a molecular system changes even at low densities. Let’s take, for example of rarefied gas. Thermal potential is obtained through series expansion by powers of density. At the first order we get:

Z1  z1 nZ11  z1 , Z12  r1 , r2 nZ121  r1 , r2 Substituting (9) into (7, 8) we obtain expression for factoring coefficients

(9)

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³ f12 d  2  P 1

Z11

(10)

Z121 ³ f13 f 23d  2

(11)

Where fij – Mayer’s function. Integrating is carried out through the whole upper half-space. Constants μ(i) are derived from border condition at infinity. Interaction between particles, and between particles and hard surface is set through hard-sphere potential. When evaluating the integral (10) we place the origin at the hard surface (Fig. 1). Perpendicular for the surface goes through the center of the particle. All upper half-space is reachable for the second particle. By integrating in cylindrical system we get:

Z11  z1

S

z 3

3 1



 3z1  2 T 1  z1

(12)

z 2 φ12

r12

1 z1

Fig. 1. Axial symmetry of one-particle distribution function near hard surface

When integrating (11) origin is placed in the center of particles’ center. We pass a plane (which one is perpendicular to hard surface) through the centers of first and second particles (Fig. 2). Intersection field of Mayer’s functions (11) is limited by two hemispheres with radii Δ1 and Δ2. When integrating at each quadrant due to symmetry of replacing the particles on each other:









f13 f 23 T 1  r132 T 1  r232 { T 1  r132 where

T х S

T 1  ( U32  R122  2U3 R12cos-3 )

(13)

– Heavyside function. In particular range of integration, specified in Fig. 2 is the following T1

S /2

³dM ³ sin - d- ³U d U T 1  ( U 3

0



3

0

2 3

3

3

2 3

 R122  2 U3 R12 cos -3 )



(14)

0

Doing integration by whole lower half-sphere (taking into account boundary condition at infinity (6)), we get

S 3

3 T  '12  R122  1 T 1  R122  R12  3R12  2

Integration in upper half-sphere may be done the same way. As a result we obtain

(15)

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Yu. Agrafonov and I. Petrushin / Physics Procedia 71 (2015) 364 – 368

Z121 '1







z0 z sin M12 , 0

z1  z2 R12 2 ,

1 1  T '12  R122  1 T ' 22  R122  1 Z12(1)  R12 2



z0 '1 cos M12 ,

where z1, z2 - distance from the particles to the surface, Heavyside function. We need to stress that

Z12(1)  R12

2S 3 R12  3R12  2 3



(16)

r12 2

M12

(17) – radius-vector angle to the z-axis,



T х

– the

(18)

'1

'2 2 R12

0 -3

R12 M12

r23

U3 r13

3

1

'1

z0

'2

Fig. 2. Axial symmetry of two-particle distribution function near hard surface

Is two-particle distribution function of volume liquid when far from the surface. When z1 , z2 goes to infinity (1) 2 1 converges to zero, and Z12 to volume value of Z12 ( R12 )T (1  R12 ) . Thus, the solution obtained describes transition from axial to spherical symmetry while each particle goes to infinity.

Z11  z1

Acknowledgements We are grateful to prof. A. P. Menushenkov and Dr. A. V. Kuznetsov for their interest to our work. Following work is supported by Russian Basic Research Fund (grant No. 15-02-08204a). References Apfelbaum, E.M., Vorobev, V.S., Martynov, G.A.,2007. J. Chem. Phys. 127, 064507 . Мartynov, G. A. 1992. Fundamental Theory of Liquids; Method of Distribution Functions, Adam Hilger, NY. Martynov, G. A., 2008. J. Chem. Phys. 129, 244-509. Tikhonov, D. A. , Kiselyov, O.E., Martynov, G.A., Sarkisov, G.N., 1999. J. Mol. Liquid. 82, 3-17.