Calculations of the diffusion coefficients in gases and liquids by the molecular dynamics method

In this paper the calculation results of the diffusion coefficients in gases and liquids by the method of molecular dynamics are presented. The modified intermolecular Lennard-Jones potential has been used. The diffusion coefficients dependences on temperature and density have been calculated. Using the diffusion coefficients dependence on temperature it has been found, that the collisional mechanism in rarefied and dense gases changed for the collective one in liquids. In the field of the thermodynamics parameters of two phase coexistence, drops of liquid occur in the vapour or bubbles occur in the liquid. The comparison of the experimental data with our calculations results in a satisfactory agreement.


Introduction
The molecular dynamics (MD) method is a direct numeric simulation method and is applied for solving problems of the statistical physics and thermodynamics.It appears with the rise of computers and is developing quickly as the computation abilities of the contemporary computers are increasing.
The molecular dynamics method was developed in B. Alder's and T. Wainwright's papers in 1959 -1960 [1,2].The initial algorithm of the method used the absolutely smooth hard elastic spheres model.A very effective algorithm was devised, which afforded to solve model problems of the physics of liquids.Then the same authors published a number of articles in The Journal of Chemical Physics with the calculations of thermodynamic properties and transport coefficients of simple liquids.
Later the MD algorithm with continuous interaction of molecules with each other was devised [3].It afforded to solve more real problems of the physics of liquids.With the increase of the computers power the MD method possibilities increased.Earlier calculations dealt with a system of some tens of particles.Nowadays the typical MD calculations include tens of thousands and even millions of the molecules.At present the molecular dynamics method is widely used in numerous fields of science: physics, chemistry, biology.
The main idea of the MD method is as follows.A system of N particles (atoms or molecules) is placed in a certain volumea cell of the MD method.Coordinates and velocity are set for each particle.Laws of the particles interaction and particlesboundaries interactions (boundary conditions) are also set.As a rule the intermolecular potentials serve for such laws.The choice of the intermolecular potential is the most important part of the method, because it determines a success of the further calculations.

The calculation procedure
The modified Lennard-Jones potential is used in our calculations.The usual Lennard-Jones potential with the parameters  and  is cut at

U r r rr
The coefficients of the spline a and b are chosen from the matching conditions: the values of the potential and the spline and their derivatives values are equal to each other.This potential guarantees absence of the molecules interaction for a long intermolecular distances ( 5 r

 
).At the same time such cutting method of the potential gives a smooth dependence of the intermolecular forces on the distance and improves the MD calculations accuracy.
For the calculations the reduced units are used: are the reduced distance and the reduced time, is the reduced density, is the reduced energy.A for the interaction potential between argon and krypton atoms [15].Our papers [9][10] contain some details of the MD method we used.

Results and discussion
At first the velocity autocorrelation functions (VACF) are calculated for argon and krypton atoms Here v ( ) i t is the velocity of an atom at a moment of time t.Then the diffusion coefficients D md are calculated using the Green -Kubo formula Also the reduced self-diffusion coefficients D md /D 0 for argon atoms and the reduced diffusion coefficients D md /D 12 for krypton atoms in argon are calculated.Here 32 16 are the Chapman -Enskog diffusion coefficients in the rarified gases kinetic theory [15].Our calculations have been carried out for the systems with the reduced densities from 0.00349 to 0.888 in a wide temperature range.
In figure 1 the krypton atoms velocity autocorrelation functions in argon (figure 1a and 1b) and the argon atoms liquid argon (figure 1c) are shown.At a low system density (ρ < 0.1) the VACF have the exponential time dependence (see figure 1a).At such density of the system the calculations of the diffusion coefficients of the argon atoms and the krypton atoms in argon by the formula (3) give the values close to theoretical ones, expressed by the formulae (4).The diffusion coefficient temperature dependence in this case is fitted satisfactory by the power function (5) for the argon atoms at ρ = 0.00347 (see figure 3a) or (6) for the krypton atoms in argon at ρ = 0.00349 (see figure 3c).00 , 40.9, 0.88.
The VACF of the atoms at a middle density of the system (0.1 ≤ ρ ≤ 0.61) and high temperatures are non-exponential and have long power tails proportional to 32 t (see figure 1b).In this case the diffusion coefficients dependence on temperature is similar to the theoretical temperature dependence of the diffusion coefficients in the Chapman -Enskog theory (see formulae (4)).Thus, the universal dependence of the diffusion coefficients on temperature at a low and middle density of the system is observed.This dependence is expressed by the approximation (7) for the krypton atoms in argon and (8) for the argon atoms in argon.
It is evident, that the diffusion mechanism for the low and middle density systems is the same.Such mechanism can be called a collisional mechanism, because the atom velocity relaxation takes place mainly by the pair collisions of atoms with one another.
Two phases can form in the system at a middle density and low temperatures (Т ≤ 1).For example, they can be nanodrops of liquid in a gas phase (see figure 2a) or gas bubbles in a liquid (see figure 2a).In this case the atoms concentrate mainly in a liquid phase.The velocity autocorrelation functions of the argon atoms in argon for the dense system (ρ > 0.61) are presented in figure 1c.Here the distinctive for liquids negative tails of the velocity autocorrelation functions are observed.The diffusion coefficients dependence calculated by formula (3) on the temperature is linear with high accuracy (figure 3b and 3d).This linear approximation is expressed by formula (9) for the argon atoms in argon (ρ = 0.84) and (10) for the krypton atoms in argon (ρ = 0.845).
, 0.0132, 0.0688.The coefficients α and β depend on density.Such change of the diffusion coefficients temperature dependence shows the changing of the diffusion mechanism caused by the transition from the low and middle density systems (gases) to the high density systems (liquids).Evidently, a new diffusion mechanism related to the collective movements of atoms occurs in liquids.Apparently, according to this mechanism an atom moves together with its environment and vibrates inside the shell consisting of the nearest neighbors.Thus, groups of strongly interacting atoms appear in the liquid.The atoms velocity relaxation results from the interaction of the whole group with the rest of the liquid.Here, the diffusion coefficient of an atom is defined by the diffusion of the group of the atoms, connected with this atom.

Conclusions
At present the molecular dynamics method is being developed actively.As the computation speed of the contemporary computers systems increases the MD method acquires the ability to solve more complicated problems, which could not be solved earlier with the help of other methods.For instance the interesting results of the study of the nanoparticles diffusion in liquids and gases have been acquired in our research [21][22][23].The diffusion coefficient in liquids taking into account molecular rotational degrees of freedom [24] and the fluctuations of thermodynamics quantities in liquids and gases [25,26] have been also calculated.As the method of molecular dynamics can be applied for the rarified gases calculations, one can hope that it may be used for simulations of complicated and interesting phenomena such as the light-induced drift [27].The solid curves are the fits ( 6) and (10).

1  , 1  1 m 12 
are the argon atom Lennard-Jones potential parameters and is the mass of argon atom.The dimensional variables are marked by subscript d.The following parameters of the Lennard-Jones potential are used: = 3.514

Figure 2 .
Snapshots of the particles positions distribution in the cell at ρ = 0.1 and T = 0.75 (a); ρ = 0.61 and T = 1 (b).The positions of the argon atoms are shown by the small points, the bold points are the krypton atoms positions.The comparison of the experimental data and our calculations results shows satisfactory agreement.For instance, in[16] the argon atoms diffusion coefficients in gaseous argon are measured.For density ρ d = 8.139•10 25 1/m -3 and Т d = 90.15K the diffusion coefficient is D d = 0.0178 cm 2 /s.Our calculation using formulae (4) and (8) results in D d = 0.0180 cm 2 /s.In[17] the diffusion coefficients experimental data in the gaseous argon at high pressures are given.For density ρ d = 2.74•1027 1/m -3 and Т d = 322.4K the diffusion coefficient is D d = 1.61•10 -3 cm 2 /s.Our calculation results in D d = 1.63•10 -3 cm 2 /s.In [18] the diffusion coefficients experimental data in liquid argon are given.For density ρ d = 2.126•10 28 1/m -3 and Т d = 84.56K the diffusion coefficient is D d = 1.53•10 -5 cm 2 /s.Our calculation using formula (9) results in D d = 1.85•10 -5 cm 2 /s.In [19] the experimental data on the krypton atoms diffusion coefficients in liquid argon are given.For density ρ d = 2.121•10 28 1/m -3 and Т d = 85.44 K the diffusion coefficient is D d = 1.35•10 -5 cm 2 /s.Our calculation using formula (10) results in D d = 1.58•10 -5 cm 2 /s.In [20] the newer diffusion coefficients experimental data of the krypton atoms in liquid argon are obtained.For density ρ d = 2.101•10 28 1/m -3 and Т d = 87.6K the krypton atoms diffusion coefficient equals D d = 1.58•10 -5 cm 2 /s.Our calculation gives the value D d = 1.64•10 -5 cm 2 /s.

Figure 3 .
The temperature dependences of the diffusion coefficient md D for the argon atoms at ρ = 0.00347 (a) and ρ = 0.84 (b).The solid curves are the fits (5) and (9).The temperature dependences of the diffusion coefficient md D for the krypton admixture in argon for ρ = 0.00349 (c) and ρ = 0.845 (d).