Green-Kubo relation simulations

Sindzinger and Gillan have calculated the thermal conductivity for NaCl and KCl melts as well as for sohds on the basis of MD simulations in Ml thermal equilibrium using the Green-Kubo relations (Table 17). In a single molten salt system, the local fluxes jz and of charge and energy... 

Computer simulations of transport properties using Green-Kubo relations... 

In the molecular dynamics calculations the trajectories of methane molecules in the pore are followed using the equation of motion with appropriate temperature control. A diffuse reflection condition is applied at the pore wall. For the EMD simulations a collective transport coefficient obtained from autocorrelation of the fluctuating axial streaming velocity via a Green-Kubo relation [S]... 

Liquid simulation studies have been essential in assessing the applicability of various fluctuation relations to real physical systems. These are important relations in nonequilibrium statistical mechanics that are valid far from equilibrium and can be used to derive Green-Kubo relations for transport coefficients.223,224 They show how thermodynamic irreversibility emerges from... 

Equilibrium MD simulations of self-diffusion coefficients, shear viscosity, and electrical conductivity for C mim][Cl] at different temperatures were carried out [82] The Green-Kubo relations were employed to evaluate the transport coefficients. Compared to experiment, the model underestimated the conductivity and self-diffusion, whereas the viscosity was over-predicted. These discrepancies were explained on the basis of the rigidity and lack of polarizability of the model [82], Despite this, the experimental trends with temperature were remarkably well reproduced. The simulations reproduced remarkably well the slope of the Walden plots obtained from experimental data and confirmed that temperature does not alter appreciably the extent of ion pairing [82],... 

The structure of a water/silicon interface was studied (Ursenbach et al, 1997), in addition to a water/copper interface (Halley et al, 1998) and a water/palladium interface (Klesingeza/., 1998). Finally, two studies have used Car-Parrinello simulations in conjunction with the Green-Kubo relations to calculate viscosities in liquid metals (Alfe and Gillan, 1998 Stadler et al, 1999). 

Self-diffusion coefficients in supercritical water can be determined from MD simulations through the molecular mean-square displacement analysis, or through the velocity autocorrelation functions (Eqn. 20) with the help of the Green-Kubo relation (Allen and Tildesley 1987) ... 

Viscosity. The molecular modeling techniques used for determining the other transport coefficients, viscosity and thermal conductivity, in addition to dif-fusivity that was discussed above, fall into two categories. In the first category, the coefficients are calculated from Green-Kubo relations, with the required correlation fimctions being evaluated with equilibrium simulations at the desired state point. Indeed, this was the approach followed traditionally. However, the calculation of correlation functions is known to yield large statistical uncertainties. This... 

As stated in the introduction, molecular simulations can be used as complementary tools to characterize ionic liquids from a modelling approach. The transport coefficients can be calculated through the corresponding Green-Kubo relations (Allen Tildesley, 1987 Frenkel Smit, 2002). Within this formalism, the self-diffusion coefficient (D) of each ion is calculated from its velocity autocorrelation function (vacf) through the following expression ... 

For the flexible model simulations two different approaches were used when calculating those properties the first one was performing the simulations in the NVT ensemble, as has been done in several previous works, even they do not used the Green-Kubo relations the calculation but other statistical equivalent methods, and the second one was by simulating the system in the NVE ensemble, as previously done by some authors (Rey-Castro Vega, 2006 Rey-Castro et al., 2007). 

Fig. 10. Diffusion coefficient for the [C2-rriim]+ (circles) and Ch (squares) ions calculated by the Green-Kubo relations using the rigid model (full symbols) and the flexible model from the NVE ensemble (open symbols). Results obtained for the flexible model in the NVT ensemble were very similar to the NVE ensemble and are not presented here. The symbols represent MD simulation results while the lines are just guides to the eyes.

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

The gas diffusion coefficient D may be extracted from a simulation in various ways. In a Molecular Dynamics simulation, if the hydrodynamk limit is reached (Le., the simulation time is long enough), D may be calculated either from the evolution of the penetrants instantaneous velocities by means of the Green-Kubo formalism or, alternatively, with the aid of the Einstein relation from their spatial positions. For TSA, there is only the Hnstein-route, via the mean-square displacement =< r(t) — r(0) > which is plotted against time, best in a log-log plot. The portion of the curve that represents oc t is used to fit a least-squares line, the slope of which (in a vs t plot) or the ptmtion of which (in a log [Pg.211]

Figure 9 Molecular dynamics simulation of a Lennard-Jones, bead-spring model, (a) Slip length, 8, as a function of the strength, [ an, of attraction between a hard, corrugated substrate and liquid for temperature, kgT/ [ =. 2. The solid line with circles is obtained from the Couette and Poiseuille profiles (NEMO) according toeqn [37], whereas the dashed line with squares, from the Green-Kubo (GK) relation, eqn [39]. The curve marks the behavior 1/ [ jii in accord with eqn [40]. The inset illustrates the velocity profiles of the Couette and Poiseuille flows, from which the slip length has been estimated for [mii = 0.6, measured in units of the Lennard-Jones parameter, [. Adapted from Servantie, J. Muller, M. Phys. Rev. Lett. 2008, 101,...

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