NEMD algorithms

The topic of this article is the study of transport properties of liquid crystal model systems by various molecular dynamics simulations techniques. It will be shown how GK relations and NEMD algorithms for isotropic liquids can be generalised to liquid crystals. It is intended as a complement to the texts on transport theory such as the monograph "Statistical Mechanics of Nonequilibrium liquids [8] by Evans and Morriss and "Recent Developments in Non-Newtonian Molecular Dynamics [9] by Sarman, Evans and Cummings and textbooks on liquid crystals such as "The physics of liquid crystals" [2] by de-Gennes and Frost and "Liquid Crystals" [3] by Chandrasehkar. 

The article is organised as follows In Section 2 we review the basic theory, in Section 3 we describe NEMD-algorithms for the evaluation of the thermal conductivity and the viscosity, in Section 4 we discuss flow properties of liquid crystals, in Section 5 we present results of flow simulations of liquid crystals and finally in section 6 there is a conclusion. 

It will be shown that the restriction above allows one to consider the linear response of a phase variable coupled to nearly all the NEMD algorithms in the literature to date. The one known exception is the thermal conductivity algorithm due to Gillan and Dixon. Thus, the present formulation will provide a template for the computation of the linear response of any phase variable coupled to an external field of the form given by Eqs. [91] under the conditions specified by Eq. [94]. 

Beside the Green-Kubo and the Einstein formulations, transport properties can be calculated by non-equilibrium Ml) (NEMD) methods. These involve an externally imposed field that drives the system out of the equilibrium. Similar to experimental approaches, the transport properties can be extracted from the longtime response to this imposed perturbation. E.g., shear flow and energy flux perturbations yield shear viscosity and thermal conductivity, respectively. Numerous NEMD algorithms can be found in the literature, e.g., the Dolls tensor [221], the Sllod algorithm [222], or the boundary-driven algorithm [223]. A detailed review of several NEMD approaches can be found, e.g., in [224]. 

Section 9.3 reviews briefly the computer simulation of liquids by NEMD and outlines how the viscosity and thermal conductivity may be evaluated. Two examples, which demonstrate how NEMD algorithms are tools to understand transport phenomena better, are given (i) a calculation of the non-Newtonian viscosity of a simple liquid, and (ii) the density dependence of the contribution to the thermal conductivity from internal degrees of freedom. 

The general NEMD algorithm (Evans Morriss 1990) introduces a known but (usually) fictitious applied field, X, into the equations of motion and hence generates a dissipative thermodynamic flux, J. The dissipative flux is defined as the work performed on the system per unit time, by the applied field, X, where... 

Since the early 1970s a number of NEMD algorithms have been proposed to calculate the shear viscosity (Ashurst Hoover 1972 Lees Edwards 1972 Gosling eta/. 1973 Ciccotti et al. 1979 Evans 1979). The early algorithms were ad hoc. and deterministic thermostats were unknown, but Hoover et al. (1980) derived their so-called DOLLS tensor algorithm for shear viscosity and proved it was exact in the linear limit. Hoover et al. (1982) and Evans (1982) independently, but simultaneously, proposed the first... 

The NEMD algorithm for computing the thermal conductivity of simple fluids was developed by Evans (1982). The driving force in the algorithm is a fictitious heat field vector, Fg, which gives rise to the heat flux Jg. The equations of motion for the system are... 

Not surprisingly, the SLLOD method plays a central role among the NEMD algorithms. All the other algorithms can be studied at strong external fields but the justification of the results obtained is tightly connected to the zero-field limit... 

It has turned out to be impossible thus far to derive simple NEMD algorithms with (mechanical) external fields analogous to the equilibrium case to determine self-diffusion or heat flow coefficients superimposed on the SLLOD dynamics of the system. ... 

The first successful MD calculation of the Dufour and Soret coefficients was carried out by McGowan and Evans for ideal mixtures. The restriction of NEMD algorithms to ideal mixtures was removed by Evans and Cummings. ... 

In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. 

Comparison of the thermal conductivities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (3.3). The entries for finite field have been obtained by applying the heat flow algorithm (3.5). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 

We have presented EMD and NEMD simulation algorithms for the study of transport properties of liquid crystals. Their transport properties are richer than those of isotropic fluids. For example, in a uniaxially symmetric nematic liquid crystal the thermal conductivity has two independent components and the viscosity has seven. So far the different algorithms have been applied to various variants of the Gay-Beme fluid. This is a very simple model but the qualitative features resembles those of real liquid crystals and it is useful for the development of molecular dynamics algorithms for transport coefficients. These algorithms are completely general and can be applied to more realistic model systems. If the speed of electronic computers continues to increase at the present rate it will become possible to study such systems and to obtain agreement with experimental measurements in the near future. 

Equilibrium molecular dynamics was put on a firm theoretical ground with Andersen s seminal paper on extended system dynamics (to be discussed in great detail later). NEMD found its first success in this area with the advent of the so-called DOLL s (not an acronym) algorithm by Hoover and coworkers. ... 

An immediate use for this conserved quantity is obvious it can (and should) be used to check the NEMD code for algorithmic and programming errors. It is also possible to use the conserved energy in obtaining a knowledge of the phase space. The approach proceeds in the same fashion as presented in the section on equilibrium molecular dynamics. Let F denote the full phase space of the variables, p , q,, ri,, I. We now make the assumption of equal a priori probability for each of the microstates F with energy H. This assumption has traditionally been applied to equilibrium systems only. In the isolated system we consider, this assumption is the most obvious one to make. Thus, one can write the phase space distribution function /(F) as... 

Next, we consider the special case of a plane Couette flow with the velocity given by v(r) = 7J/e, where e is a unit vector in ar-direction and 7 = dvx/dy = const is the shear rate. Furthermore, for simplicity, the motion of the particle is restricted to the xy-plane. Then the equations of motion correspond to the (two-dimensional version) of the SLLOD algorithm used in NEMD simulation studies of the viscous properites of fluids [10] ... 

Though NEMD methods have been known since the 1980s, only recent theoretical and algorithmic developments (Evans and Morriss 1990 Sarman et al. 1998) have made it possible to use it in many applications (McCabe et al. 2001 Moore et al. 1997,1999, 2000a, 2000b, 2000c). 

A detailed theoretical background and the techniques of NEMD calculations are given in the monograph of Evans and Morriss. More recent algorithms and applications can be found in review articles. Useful information about NEMD is also provided in the monograph by Hoover. The basic methods of equilibrium MD (e.g., periodic boundary conditions, numerical integrators) are described in the monograph by Allen and Tildesley. ... 

The following algorithms represent the basic methods of NEMD simulations. Mechanical external fields play the role of the generalized thermodynamic forces (e.g., strain rate in the case of shear viscosity). It should be noted that in most of the cases the thermodynamic fluxes (e.g., shear stress in the case of shear viscosity) are equally easily constrained, then the output of the simulation is the corresponding external field. We do not describe these possibilities in this article the interested reader is referred to the monograph of Evans and Morriss. ... 

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