The Stokes-Einstein relation for simple fluids: From hard-sphere to Lennard-Jones via WCA potentials.

The Stokes-Einstein (SE) relation is examined for hard-sphere (HS) and Weeks-Chandler-Andersen (WCA) fluids by the molecular dynamics method on temperatures and densities corresponding to the saturated vapor line of Lennard-Jones (LJ) liquids. While the self-diffusion coefficient, D, and shear viscosity, η sv, increases and decreases, respectively, with increasing steepness in interaction potentials, the same SE relation holds for HS and WCA fluids as that obtained for LJ liquids, i.e., Dη sv = (k B T/C)(N/V)1/3, where k B is the Boltzmann constant, T is the temperature, and N is the particle number included in the system volume V. The coefficient C is almost constant at about 6 to 2π for η > 0.3, where η is the packing fraction. The results show that the SE relation for simple liquids and fluids does not need to bear any concepts of both the hydrodynamic particle size and the boundary condition. In light of this SE relation, the Enskog, Eyring-Ree, and Zwanzig theories are quantitatively tested. In addition, the cause of deviation from unity of the exponent in the fractional SE relation for simple fluids is clearly accounted for. The present results show that applying both the original and the fractional SE relations to simple liquids and fluids does not lead to any meaningful discussions.

Breakdown of the Stokes-Einstein relation in pure Lennard-Jones fluids: From gas to liquid via supercritical states.

We have examined the conditions under which the breakdown of the Stokes-Einstein (SE) relation occurs in pure Lennard-Jones (LJ) fluids over a wide range of temperatures and packing fractions beyond the critical point. To this end, the temperature and packing-fraction dependence of the self-diffusion coefficient, D, and the shear viscosity, η_{sv}, were evaluated for Xe using molecular dynamics calculations with the Green-Kubo formula. The results showed good agreement with the experimental values. The breakdown was determined in light of the SE equation which we have recently derived for pure LJ liquids: Dη_{sv}=(k_{B}T/2π)(N/V)^{1/3}, where k_{B} is the Boltzmann constant, T is the temperature, and N is the particle number included in the system volume V. We have found that the breakdown occurs in the lower range of the packing fraction, η<0.2, and derived the SE relation in its broken form as Dη_{sv}=0.007(1-η)^{-5}η^{-4/3}(k_{B}T/ε)^{n}k_{B}T(N/V)^{1/3}, where n increases from 0 up to 1 with the decreasing η. The equation clearly shows that the breakdown mainly occurs because the packing-fraction dependence does not cancel out between D and η_{sv} in this region, which is attributed to the gaseous behavior in the packing-fraction dependence of the shear viscosity under a constant number density. In addition, the gaseous behavior in the temperature dependence of the shear viscosity also partially causes the breakdown.

Molecular insights into the boundary conditions in the Stokes-Einstein relation.

In order to mimic the Brownian particle in liquid, molecular dynamics calculations of dilute solutions of spherical fullerene molecules with various sizes in liquid Ar were performed. To establish the scaling equation for self-diffusion coefficient, D, of the fullerenes, the dependence of D was examined on the mass ratio of solute to solvent and on the energy-parameter ratio used in the Lennard-Jones potentials. The dependence on the energy-parameter ratio remains up to C_{540}, whereas D rapidly becomes independent of the mass ratio with increasing mass of the solute. The product of the scaling equations obtained for the D of the solute and for shear viscosity, η_{sv}, for the solvent gives a relation which replaces the Stokes-Einstein relation based on the hydrodynamics. The present expression does not need both the boundary conditions and the hydrodynamic particle size, but instead the energy-parameter ratio, packing fraction of solvent, and bare size of solute. From the viewpoint of the tackiness at the boundary, the cage correlation function around the diffusing particle was examined; it was found that the decay time of the function depends mainly on the the energy-parameter ratio. Therefore, the energy-parameter ratio accounts for the main part of both the boundary conditions and the hydrodynamic particle size in the Stokes equation, which have so far been ill-defined in any molecular theories.

Explicit expressions of self-diffusion coefficient, shear viscosity, and the Stokes-Einstein relation for binary mixtures of Lennard-Jones liquids.

Explicit expressions of the self-diffusion coefficient, D(i), and shear viscosity, η(sv), are presented for Lennard-Jones (LJ) binary mixtures in the liquid states along the saturated vapor line. The variables necessary for the expressions were derived from dimensional analysis of the properties: atomic mass, number density, packing fraction, temperature, and the size and energy parameters used in the LJ potential. The unknown dependence of the properties on each variable was determined by molecular dynamics (MD) calculations for an equimolar mixture of Ar and Kr at the temperature of 140 K and density of 1676 kg m(-3). The scaling equations obtained by multiplying all the single-variable dependences can well express D(i) and η(sv) evaluated by the MD simulation for a whole range of compositions and temperatures without any significant coupling between the variables. The equation for Di can also explain the dual atomic-mass dependence, i.e., the average-mass and the individual-mass dependence; the latter accounts for the "isotope effect" on Di. The Stokes-Einstein (SE) relation obtained from these equations is fully consistent with the SE relation for pure LJ liquids and that for infinitely dilute solutions. The main differences from the original SE relation are the presence of dependence on the individual mass and on the individual energy parameter. In addition, the packing-fraction dependence turned out to bridge another gap between the present and original SE relations as well as unifying the SE relation between pure liquids and infinitely dilute solutions.

Explicit expression for the Stokes-Einstein relation for pure Lennard-Jones liquids.

An explicit expression of the Stokes-Einstein (SE) relation in molecular scale has been determined for pure Lennard-Jones (LJ) liquids on the saturated vapor line using a molecular dynamics calculation with the Green-Kubo formula, as Dη(sv)=kTξ(-1)(N/V)(1/3), where D is the self-diffusion coefficient, η(sv) the shear viscosity, k the Boltzmann constant, T the temperature, ξ the constant, and N the particle number included in the system volume V. To this end, the dependence of D and η(sv) on packing fraction, η, and T has been determined so as to complete their scaling equations. The equations for D and η(sv) in these states are m(-1/2)(N/V)(-1/3)(1-η)(4)ε(-1/2)T and m(1/2)(N/V)(2/3)(1-η)(-4)ε(1/2)T(0), respectively, where m and ε are the atomic mass and characteristic parameter of energy used in the LJ potentials, respectively. The equations can well describe the behaviors of D and η(sv) for both the LJ and the real rare-gas liquids. The obtained SE relation justifies the theoretical equation proposed by Eyring and Ree, although the value of ξ is slightly different from that given by them. The difference of the obtained expression from the original SE relation, Dη(sv)=(kT/2π)σ(-1), where σ means the particle size, is the presence of the η(1/3) term, since (N/V)(1/3)=(6/π)(1/3)σ(-1)η(1/3). Since the original SE relation is based on the fluid mechanics for continuum media, allowing the presence of voids in liquids is the origin of the η(1/3) term. Therefore, also from this viewpoint, the present expression is more justifiable in molecular scale than the original SE relation. As a result, the η(1/3) term cancels out the σ dependence from the original SE relation. The present result clearly shows that it is not necessary to attribute the deviation from the original SE relation to any temperature dependence of particle size or to introduce the fractional SE relation for pure LJ liquids. It turned out that the η dependence of both D and η(sv) is similar to that in the corresponding equations by the Enskog theory for hard sphere (HS) fluids, although the T dependence is very different, which means that the difference in the behaviors of D and η(sv) between the LJ and HS fluids are traceable simply to their temperature dependence. Although the SE relation for the HS fluids also follows Dη(sv)=kTξ(-1((N/V)(1/3), the value of ξ is significantly different from that for the LJ liquids.

Thermal conductivity of simple liquids: temperature and packing-fraction dependence.

The thermal conductivity of rare gases in liquid and dense fluid states has been evaluated using molecular dynamics simulation with the Lennard-Jones (LJ) potentials and the Green-Kubo (GK) formula. All the calculated thermal conductivities are in very good agreement with experimental results for a wide range of temperature and density. Special attention was paid to temperature and packing-fraction dependence which is nontrivial from dimensional analysis on the LJ potentials and the GK formula. First, the temperature dependence of T(1/4) was determined from the calculations at constant densities. Secondly, in order to obtain the dependence on packing fraction from that on number density separately, a scaling method of particle and/or cell size was introduced. The number density dependence of (N/V)(2/3) which is expected from the dimensional analysis of the GK formulas was confirmed and the packing-fraction dependence of η(3/2) was determined by using the scaling method. It turned out that the summarized functional form of m(-1/2)(N/V)(2/3)η(3/2)T(1/4) can well express both the calculated and experimental thermal conductivities for Ar, Kr, and Xe, where m is the atomic mass. The scaling method has also been applied to molten NaCl and KCl so that it has been found that the thermal conductivity has the packing-fraction dependence of η(2/3) which is much weaker than that of the simple LJ liquids.

Thermal conductivity of simple liquids: origin of temperature and packing fraction dependences.

The origin of both weak temperature dependence and packing fraction dependence of T(1/4)η(3/2) in the thermal conductivity of the simple Lennard-Jones (LJ) liquid is explored. In order to discuss the relative contributions from attractive or repulsive part of the interaction potential separately, the thermal conductivity of a series of Weeks-Chandler-Anderson (WCA) fluids is calculated by molecular dynamics simulations. The results show that the repulsive part plays the main role in the heat conduction, while the attractive part has no direct effect on the thermal conductivity for a given packing fraction. By investigating WCA fluids with potentials of varying softness, we explain the difference observed between the LJ liquids such as argon and Coulombic liquids such as NaCl.

Thermal conductivity of molten alkali metal fluorides (LiF, NaF, KF) and their mixtures.

The thermal conductivities of molten alkali fluorides (LiF, NaF, and KF) and their mixtures (LiF-NaF, LiF-KF, and NaF-KF binaries and LiF-NaF-KF ternary) are predicted using molecular dynamics simulation with the Green-Kubo method. A polarizable ion model is used to describe the interionic interactions. All the systems except LiF-KF and LiF-NaF-KF mixtures follow a scaling law: it is proportionnal to mA (-1/2)(N/V)(2/3), where mA is the arithmetic average of the ionic species masses in a given melt and N is the total number of ions included in the system volume V. In LiF-KF and LiF-NaF-KF mixtures a significant departure from the scaling law is observed. By examining separately the effects of the cation mass and size asymmetry in LiF-KF mixtures, we show that both of them account for half of the deviation. Finally, we observe that the temperature dependence of the thermal conductivity is very small in these molten fluorides.

Thermal conductivity of ionic systems from equilibrium molecular dynamics.

Thermal conductivities of ionic compounds (NaCl, MgO, Mg(2)SiO(4)) are calculated from equilibrium molecular dynamics simulations using the Green-Kubo method. Transferable interaction potentials including many-body polarization effects are employed. Various physical conditions (solid and liquid states, high temperatures, high pressures) relevant to the study of the heat transport in the Earth's mantle are investigated, for which experimental measures are very challenging. By introducing a frequency-dependent thermal conductivity, we show that important coupled thermoelectric effects occur in the energy conduction mechanism in the case of liquid systems.

Calculations of the thermal conductivities of ionic materials by simulation with polarizable interaction potentials.

Expressions for the energy current of a system of charged, polarizable ions in periodic boundary conditions are developed in order to allow the thermal conductivity in such a system to be calculated by computer simulation using the Green-Kubo method. Dipole polarizable potentials for LiCl, NaCl, and KCl are obtained on a first-principles basis by "force matching" to the results of ab initio calculations on suitable condensed-phase ionic configurations. Simulation results for the thermal conductivity, and also other transport coefficients, for the melts are compared with experimental data and with results obtained with other interaction potentials. The agreement with experiment is almost quantitative, especially for NaCl and KCl, indicating that these methodologies, perhaps with more sophisticated forms for the potential, can be used to predict thermal conductivities for melts for which experimental determination is very difficult. It is demonstrated that the polarization effects have an important effect on the energy current and are crucial to a predictive scheme for the thermal conductivity.

Thermal conductivity of molten alkali halides: Temperature and density dependence.

The thermal conductivities of a series of molten alkali halides have been evaluated by using molecular dynamics simulation within the framework of Fumi-Tosi potential models. Although the calculated results showed 0%-50% larger values than experimental results depending on system, they are in agreement with each other in showing both negative temperature and ionic mass dependence. In order to clarify the cause of the negative temperature dependence in more detail, the thermal conductivity under constant temperature or constant density was evaluated for all alkali chlorides and all sodium halides. The calculations reveal that the thermal conductivity depends strongly on density but only weakly on temperature. While the integrated value of the autocorrelation function for energy current increases with temperature, this is canceled out by the reciprocal temperature factor in relation to the thermal conductivity. With increasing density the integrated value increases, and this dominates the behavior of the thermal conductivity. By repeating the calculations with different ionic masses, we have concluded that the thermal conductivity is a function of m(-1/2)(N/V)(2/3), where m is the geometric mean of ionic mass between anion and cation and N/V is the number density.

Application of Brillouin scattering to the local anisotropy and birefringence measurements of thin layers.

Brillouin scattering is an efficient nondestructive and noncontact measurement method to obtain the wave properties of thin layers at hypersonic frequencies. The reflection induced ThetaA (RIThetaA) scattering geometry enables the simultaneous measurement of the phonons, which propagate in each direction of wave vectors of q(Theta A) (propagation in the film plane) and q(180) (back scattering). Using this scattering geometry, we could observe the refractive indices and birefringence of the piezoelectric poly-vinylidene fluoride (PVDF) film as a function of temperature. By introducing the microscopic technique, the elastic anisotropy and refractive index measurements in the minute area of polycrystalline ZnO films were also performed.