Density Distribution of Binary Liquid in Cylindrical Pore Under Influence of External Field

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1 WDS'09 Proceedings of Contributed Papers, Part III, 30 35, 009. ISBN MATFYZPESS Density Distribution of Binary Liquid in Cylindrical Pore Under Influence of External Field P.I. Gordiichuk, and A.N. Vasilev,3 Taras Shevchenko Kiev University, Physics Faculty, Department of Theoretical Physics, 64 Vladimirskaja str., Kiev, 060, Ukraine. Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands. 3 National Technical University of Ukraine KPI, Kiev 03056, 6/ Jangelja str., Faculty of Medical Engineering, Department of Biomedical Engineering, Ukraine. Abstract. We solve the problem of calculation of the density distribution of binary liquid mix in a cylindrical pore under an influence of an external field. We use the approximation method of slow inhomogeneity for phenomenological representation of the Helmholtz Free energy of a system. This method gives us possibilities to find solutions for systems with different geometries. The cases of homogeneous gravity potential and inhomogeneous wall potential are analyzed. Introduction The behaviour of binary liquid mix essentially depends on an influence of an outside field and a size of the system. For the first time, this idea was presented by Fisher et al. [97] for systems near a critical point. Further, this idea had been father developed by Privman et al, [990] for the area which is far from a critical point with the external field influence. We can observe interesting behaviours of the confined liquid system if external special fields are presented. Different binary component interactions with the external field and different interactions among components within liquid can produce non-equilibrium density distribution of the binary system with different spatial effects which are not obvious. This problem is very important in medicine for the process of obtaining natural oils from medicinal plants. We can model an extraction of natural oil from a porous material as binary liquid mixed process of oil and a resolvent. The task of finding a binary density distribution in a pore with a presence of the external field is very important for the development of modern nanomedicine and nanotechnology. There is no completely analytical solution of this problem and that is a reason why this problem is quite interesting. There are many theoretical approaches for solving one component density distribution in a confined system with a presence of an external potential. We can see two approaches for determination of the density distribution in one component system which is confined by two parallel walls [Bulavin et al., 007]. These authors use as a function representation of the external field the Taylor expansion by the density deviation with saving main terms of the expansion. Chalyi et al. [993] use asymptotic correlation functions as Green functions of the Helmholtz operator. These approaches have a complicated character in the confined system with a special geometry. The strong analysis of the density distribution in different liquid systems was made by numerical simulations [Brovchenko et al., 003]. The descriptions of experimental and theoretical investigations of liquid mixed systems by slow neutron scattering are in the book of Antonchenko et al. [99]. The goal of this paper is to describe a behaviour of the binary liquid system and to obtain analytical results for a specific geometry. Our mathematical solution combines following approaches: one component approach [Bulavin et al., 007] and the method of matrix formalism which is used for finding couple correlation functions in multicomponent systems [Vasilev, 003]. The matrix formalism gives an important simplification in describtion of the density distribution in a many component liquid system. Observations Our cylindrical pore can be represented as an infinite pore along the z axis as it is presented in Fig.. The inhomogeneous external field has a direction perpendicular to the longer axis of the cylindrical pore. The most interesting external fields are the gravitational field and an inhomogeneous 30

2 GODIICHUK AND VASILEV: DENSITY DISTIBUTION OF BINAY LIQUID Figure. A geometry of the in vestigating system: binary liquid system under an external inhomogeneous field in the cylindrical pore. The outside field has a perpendicular direction to the main axis of the pore. field at a wall of the system. The external wall potential produces a density deviation from the average value which differs along the radius of pore. That is why we should observe our functions of the density deviation, ρ( r ) and ρ( r ) as functions of coordinates. We suppose that an external field reaches a small value and that we can represent our density deviation as small supplements to the average value. In this representation, the density functions can be expressed as: ρ( r) = ρ + δρ( r) and ρ( r) = ρ + δρ( r) where ρ and ρ are average density values of two components and supplements δρ( r ) and δρ( r ) are density deviations of two components from equilibrium v alues. We do not know the view of the Helmholtz Free energy for our equilibrium system because this is a very difficult many particle task. The external field produces density deviations from the equilibrium. In this case, the Helmholtz Free energy of the nonequilibrium system depends on the density deviation and its gradient. This is the main simplification in slow inhomogeneity approximation. A non-equilibrium part of the Helmholtz Free energy can be presented as a sum of squares of the density deviation and density gradient deviation: δ Φ= ( aijδρδρ i j + bij δρi δρj ) + hi( r) δρ i dv, () i= j= i= where h,( r ) are external fields for respective components of the mixed liquid system. The phenomenological coefficients bij describe inner components of interaction between density deviations, and δφ0 aij = ( ij=,,) are respective coefficients of inner component interactions. δρ δρ i j We are finding a solution of our system after minimization of the Helmholtz s Free energy () with addition volume mass saving conditions for two components: δρ,( rdv ) = 0. () This task is known as isoperimetric [Metiyz-et-al., 97]. We can find solution of the system by the method of indeterminate Lagrange s multipliers. The density deviation of the system depends on the polar angle, ϕ and radius of the pore, r. After minimization of the equation (), we obtain the matrix equation for t he density profile in the following form: ˆ ˆ ( Aδρ BΔ δρ = λ + h ( r, ϕ )), (3) a a b where Δ is the Laplace operator in cylindrical coordinates, ˆ b A = and ˆ are a a B = b b symmetric matrices of interactions, δρ δρ = is a vector of density deviation of two components, δρ h h = is a vector of external fields for either component, λ h λ = is a vector of the Lagrange s λ m ultipliers. Since the particles cannot penetrate the walls, we can write for two components: δρ = 0, (4) r r= where is a radius of the cylindrical pore. 3

3 GODIICHUK AND VASILEV: DENSITY DISTIBUTION OF BINAY LIQUID General solution We can obtain the Lagrange multipliers from equations (), () and (4): π λ = dϕ h(, r ϕ)rdr. (5) π 0 0 It is more convenient to make a following substitution for simplification of future calculations: π ˆ A ξ(, r ϕ) = δρ ( r, ϕ). (6) π dϕ h(, r ϕ) rdr 0 0 We have obtained equation for profile ξ(, Âξ r ϕ) in the following form: Bˆ Δ ξ = h(, r ϕ), (7) with an appropriate boundary condition: ξ. (8) = 0 r r= A solution of the equation (7) with the boundary condition (8) can be represented as: + ( m ) μn r ξ(, r ϕ) = ξnmj m exp( imϕ), (9) n= m= where ( m ) μ n are eigenvalues of the Bessel function which can be found from the equation ( m ) J m ( μ n ) = 0. External fields are expanded by functions of Bessel with coefficients: + ( m ) μ r n hr (, ϕ) = hnmj m exp( imϕ), (0) n= m= where the coefficients of the expansion are: h nm = π ( m ) μ n r dϕ h(, r ϕ rj m exp( imϕ) dr m ( m ) π J ( m ) m n μ ( μ ) n ) 0 0. () W e have expanding coefficients of densities in this form: ξ ˆ ˆB nm =Θnm h nm. () ( m ) ˆ ˆ ˆ ˆ ˆ μ, (3) ˆ n θ E θ Θ ˆ nm = B A E + ( m ) = + ( ) μ n μ m κ κ n + + where E ˆ is a unit matrix and matrix ˆθ is a spectrum matrix [Metiyz-et-al., 97] which has the next form ˆ ˆ ˆ ˆ Eκ B A θ =, (4) κ κ where κ, are eigenvalues of the matrix ˆB Aˆ. We have obtained a general solution of our system in equations (9) - (4). The next step is investigations of different kinds of external fields. Gravitational Field The external gravitational field of a system can be expressed as hr (, ϕ) = egrcos( ϕ), where ϕ is a polar angle, e = is a unit can present the solution of such system in the following form: (5) vector and g is a gravitational acceleration constant. We 3

4 GODIICHUK AND VASILEV: DENSITY DISTIBUTION OF BINAY LIQUID μnr ξ(, r ϕ) = ξnj ( ) cos( ϕ), (6) n = where μ n are ei genvalues of the Bessel function, J ( μ n ) = 0. The vectors of expanding coefficients have the following view: ( ˆ ˆ ) ˆ g B A + E μn B e. (7) ξn = 3 μnj μn μ n In our specific case, we have the next conditions: ( )( ) π 0 0. The solution of our system can dϕ h(, r ϕ ) rdr = 0 be then written as: ˆ ˆ ˆ μnr ( ) ˆ θ E θ J B e. (8) δρ(, r ϕ) = g cos( ϕ) + n = κ + + μn μn 3 κ μ nj ( μn) μn where Ê is a unit matrix, ˆθ is a spectrum matrix (4), is a radi us of the cylindrical pore, and, are eigenvalues of the matrix, κ ˆB Aˆ. adial Wall-potential There are many physical systems where an external field depends not only on the radius of a pore but also on another variable. We can observe a wall potential which has also periodic dependence along the z axis. In this case, we use a similar representation of the solution in view (0), (). After calculation, we have coefficients of the solution in forms ()-(5). There are many model potentials of interaction between components and a wall of the system. The potential kl u() r = Be chkr (9) where B is an amplitude of interaction a, k is a return effective radius of the external potential (9) is most important. We can find the solution of the system as in previous cases from the first equation for the density profile (3): μmr ξn() r = { ξn, mj0( )}, (0) where coefficients of the solution are: ξ m= μmr () r = { ξ J0( )}, () n n, m m= ˆ, () B h nm, ξ nm, = ˆ πn ( ) ˆ + + B A L ˆ, (3) B h nm, ξ nm, = πn ˆ ( ) + + Bˆ A L where L is a surface inhomogeneity constant. Other coefficients have following forms: L πnz μmr, (4) h nm, = h(, r z)cos( ) dzj0( ) dr L L J 0 ( μm ) 0 0 L πnz μmr. (5) h nm, = h(, r z)sin( ) dzj0( ) dr J L L 0 ( μm )

5 GODIICHUK AND VASILEV: DENSITY DISTIBUTION OF BINAY LIQUID esults Figure. Density distributions of the binary liquid system with the external gravitational field for two different phenomenological coefficients: (left side) a[,]=5, b[,]=, a[,]=a[,]=0, b[,]= b[,]=0, (right side) a[,]=5, b[,]=, [,]=0, a[,]=0, b[,]=0, b[, ]=0. Only the interactions between different components were considered. We use dimensionless representations of variables. Figure 3. Density profiles for homogeneous wall-potential with hydrophobic and hydrophilic interactions between components and wall of the pore with 3D visualization. The phenomenological coefficients are: a[,]=5, b[,]=, a[,]=,b[,]=0.5. We use dimensionless representations of variables. Figure 4. Areas of density distributions of the binary liquid system with inhomogeneous wall potential along the z axis and radius with modelling potential (9). The phenomenological coefficients: (left side) a[,]=, b[,]=0., a[,]=0, b[,]=0, b[,]=0., a=[,]=. (right side) a[,]=, b[,]=, a[,]=0, b[,]=0. 34

6 GODIICHUK AND VASILEV: DENSITY DISTIBUTION OF BINAY LIQUID Figure 5. The density distribution of the binary liquid mix along a radius of the system. The external wall potential according to (9) was used. The phenomenological coefficients of left side are: a[,]=0, b[,]=0, a[,]=0, b[,]=0, a[,]=, b[,]=0.. The right side shows the results of Bulavin et al. [007]. Discussion We have found a new way for description of the binary liquid mix in the confined system. This approach uses approximations which were mentioned earlier in the paper. With help of this theory, we can obtain analytical results for the density distribution in the cylindrical pore with the external interaction potential. We have investigated the case of external homogeneity gravitational field with different phenomenological coefficients of interaction. We have also reviewed the case of inhomogeneous wall potential which depends only on the radius of the pore. Furthermore, we have investigated the case where the external potential depends on the radius and variable z. We have m odeled different interactions between components which provide a wide space for modeling of different kinds of possible interactions. Conclusion We have obtained a general solution for the binary liquid density distribution in the system with bounded geometry. Our theoretical approach gives us possibilities of easy calculation of the density profiles in systems with different geometries. This method avoids complicated statistical calculations which are almost not solvable for many particle inhomogeneous systems. Our results have a high precision and satisfy compare theoretical and experimental data. We can vary phenomenological coefficients for different modelling systems important for simulation of complicated biological interactions in matter. The approximation of slow inhomogeneity has limitation area of activity near a critical point. That is why this is a good background for further theory improvements and developments. eferences Antonchenko, V. Y., A. S. Davudov, V. V. Ilin, Physical fundamentals of water. Kyiv: Science Opinion, 99. Binder, K., Ann. ev. Phys. Chem. 99. N43, P. 33. Brovchenko, I., A. Geiger, A. Oleinikova, D. Paschek. Eur. Phys. J. E v., P. 69. Bulavin, L. A., D.A Gavryushenko, V.M. Sysoev. Spatial distribution of components of a binary mixture in abounded system, ISSN Ukr. J. Phys V 5, N 0. Chalyi, A. V., Critical phenomena in finite-size systems, J. Mol. Liquids Vol. 58. P Fisher, M. E., In Critical Phenomena. Proceeding of the International School of Physics Enrico Fermi. Vol.5. / Ed. Green M.S. New York: Academic.-97 Metiyz, Dz., Р. Yoker. Mathematical methods of physics. 97, 39 P. Privman, V., Finite Size Scaling and Numerical Simulation of Statistical System / Ed. Singapore: World Scientific Vasilev, O. N., Theoretical Physics T. 35,, P