J. Phys. Chem. C 2008, 112, 17231–17234
17231
Molecular-Dynamics Investigation of Phase Equilibrium and Surface Tension in Argon-Neon System Vladimir G. Baidakov* and Sergey P. Protsenko Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, Amundsen Street 106, 620016 Ekaterinburg, Russia ReceiVed: June 24, 2008; ReVised Manuscript ReceiVed: August 17, 2008
The molecular-dynamics method has been used for calculating the density profiles of the mixture argon-neon, surface tension, and adsorption in systems of 4096-8957 Lennard-Jones particles at a reduced temperature T* ) kBT/εR=0.7 ( 0.01 and a pressure up to p* ) pσR3 /εR ) 0.6, where σR and εR are parameters of the argon potential. The results of calculating phase-equilibrium parameters are compared with data for a singlefluid model of a solution. The paper discusses the applicability of an extended version of the van der Waals capillarity theory to the description of the surface tension of a gas-saturated liquid. Introduction The dissolution of a volatile component in a liquid may be accompanied by intense adsorption at the liquid-gas interface, which leads to an essential change in the surface tension. This phenomenon has found practical use in a number of technological processes.1 Experimental investigations of surface phenomena in gas-saturated liquids2-4 reveal a rapid increase in the adsorption of the volatile component with decreasing temperature. The maximum effect should be expected at temperatures close to the temperature of the triple point of a pure dissolvent. The present work is devoted to molecular-dynamics (MD) investigation of phase equilibrium and surface tension in argon-neon solution, in which argon is a liquid dissolvent and neon acts as a gas-saturating component. The argon-neon system displays increased solubility of the light component (an order higher than, for instance, an argon-helium system), which results in essential changes in its properties. For example, dissolving ∼25% of neon in liquid argon causes a shift of the * liquid-phase spinodal (T* ) 0.7) from psp ) -1.0 (pure argon) * 5 to psp ) -0.4. The surface tension in argon-neon solution is investigated in ref 4. In ref 6, the phase diagram is determined in the framework of a single-fluid model of solution with the use of the equation of state of the Lennard-Jones fluid common to the liquid and the vapor phase. The properties of the solution interface at low temperatures have been calculated in the framework of an extended version on the van der Waals capillarity theory.7 The structure of liquid-vapor and liquidliquid interfaces in Lennard-Jones fluids and mixtures have been studied using integral equations.8 Molecular-Dynamics Model. The base MD cell had the form of a rectangular parallelepiped. The cell size along the axis z Lz was 4.2× larger than in the directions x and y (Lx ) Ly). The two-phase system was a liquid film situated at the cell center perpendicularly to the axis z and surrounded on two sides by the vapor phase. Periodic boundary conditions were imposed on the cell boundaries. * To whom correspondence should be addressed. Fax: +7 343 267 8800. E-mail:
[email protected].
Figure 1. Density profiles in the cell. (1) Pure argon, (2) solution of neon in argon, cl ) 0.073, (3) argon partial density, and (4) neon partial density.
The cell contained particles of two kinds (R and β). The particles interacted by means of the Lennard-Jones cut pairadditive potential φ(r). The parameters of the potential for particles of kind R corresponded to argon: εR/kB ) 119.8 K, σR ) 0.3405 nm, of kind β- to neon: εβ/kB ) 35.05 K, σβ ) 0.275 nm. In calculations of cross interactions, use was made of modified combined Berthelot-Lorentz rules:
σRβ ) a(σR + σβ) ⁄ 2
(1)
εRβ ) b(εR · εβ)1/2
(2)
The correcting coefficients a and b are determined in ref 6 in building up the equation of state of the Lennard-Jones solution. Thermodynamic properties were calculated in reduced units. The parameter σR was taken as the length unit, εR as the energy unit, and the mass of an argon atom as the mass unit. The reduced temperature was determined as T* ) kBT/εR, pressure as p* ) pσR3 /εR, density as F* ) FσR3 , surface tension as γ* ) γσR2 /εR, and adsorption Γ* ) Γσ2. The cutoff radius of the potential was taken equal to r*c ) 6.58. As shown in ref 9, such
10.1021/jp805566g CCC: $40.75 2008 American Chemical Society Published on Web 10/10/2008
17232 J. Phys. Chem. C, Vol. 112, No. 44, 2008
Baidakov and Protsenko
Figure 2. Relative adsorption of volatile component at the liquid-gas interface. The dot and solid lines are calculated by eq 3, and the dashed line by eq 9.
a choice of r*c ensures an adequate representation of both volume and surface properties. The molecular dynamics simulations are performed in the microcanonical (NR, Nβ, V, E) ensemble, where NR and Nβ are the numbers of argon and neon particles, V is the system volume, and E is the internal energy. The calculations were made at a temperature T* = 0.7 ( 0.01. At this temperature, the values of the correcting coefficients in eqs 1 and 2 are a ) 1.01903, b ) 0.89554. The equations of particle motion were integrated with a time step ∆t ) 10-14 s. The number of particles in the system and the cell size were chosen in such a way as to ensure the existence of homogeneous regions of the liquid and the gas phase in the whole concentration range under investigation. In simulating pure argon and solutions of neon in argon with a molar fractions of neon in the liquid phase up to cl ) 0.073 the cell contained 4096 Lennard-Jones particles. An increase in the concentration was accompanied by a transition of neon particles from the liquid into the gas phase, a decrease in the thickness of the liquid film and disappearance of the homogeneity core at its center. Therefore, calculations at cl > 0.073 were made with increasing the number of particles in the cell to 6295 and 8957. The process of a two-phase system going into equilibrium comprised from 2 × 105 to 1 × 106 time steps. Thermodynamic properties of a two-phase system were determined by averaging over no less than 106 time steps. Results of Calculations. To calculate the distribution of the density of the mixture components in the system, the cell was divided along the axis z into 1160 layers 0.05σR thick. In connection with the nonuniformity of evaporation of particles from interfaces the liquid film shifted in the process of the system evolution. Therefore, the position of the liquid film in
the course of a computer experiment was controlled, and the beginning of the count of density profiles was related to the position of the liquid-phase symmetry axis. The ortobaric densities of liquid and gas phases and also partitial densities of species were calculated as mean values on parts of F(z) dependences which did not include interface layer as are shown in Figure 1. The relations F*β,l(g)/F*l(g) were used while calculating molar fractions of neon in liquid and gas phases: cl and cg. The results of calculations of the local densities profiles for pure argon and argon-neon solution with a molar concentration cl ) 0.073 are shown in Figure 1. As is seen from Figure 1, the thickness of the interfacial layer is (2-4)σR. An increase in the concentration of neon, if the volume and the total number of particles in the cell are invariable, leads to a decrease in the thickness of the liquid film. In this case, the effective thickness of the interface changes only slightly. A considerable increase in the density of the gas phase is ensured by neon particles. The concentration of neon in the liquid phase is in this case (4-9)× lower than in the gas phase. The local extrema on the partial density profile of the neon in the interfacial layer point to its considerable adsorption in this layer. A further increase in the concentration of neon is that the system results in the convergence of the densities of the liquid and the gas phase. At a concentration cl ) 0.209 (N ) 8957), the densities of the liquid and the gas phase differ by approximately 13%. The relative adsorption of neon at the equimolecular separating surface of argon is determined by the expression9
Γβ(R) )
∫0z
e,R
(Fβ(z) - Fβ,l) dz + ∫ze,R (Fβ(z) - Fβ,g) dz (3) Lz/2
Here, ze,R is the position of the equimolecular separating surface of the component R in the interfacial layer, Fβ,l, Fβ,g are the partial densities of the component β in the liquid, l, and in the gas, g, phase. The concentration dependence of Γ*β(R) is presented in Figure 2 and in Table 1. The growth of excess adsorption stops at cl > 0.12. The phase-equilibrium pressure was determined by the results of calculating the normal component of the Irving-Kirkwood pressure tensor:11
pN(zn) ) 〈F(zn)〉kBT -
1 A
〈
∑ i,j i
