SLLOD algorithm

Here, pa,- is the bead momentum vector and u(rm. f) = iyrV is the linear streaming velocity profile, where y = dux/dy is the shear strain rate. Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)). 

Both the Doll s and SLLOD algorithms are correct in the limit of zero-shear rate. However, for finite shear rates, the SLLOD equations are exact but Doll s tensor algorithm begins to yield incorrect results at quadratic order in the strain rate, since the former method has succeeded in transforming the boundary condition expressed in the form of the local distribution function into the form of a smooth mechanical force, which appears as a mechanical perturbation in the equation of motion (Equation (12)) (Evans and Morriss, 1990). To thermostat the... 

SLLOD algorithm, the thermostat Gaussian multiplier j/ is introduced ... 

C. J. Mundy, J. I. Siepmann, and M. L. Klein, /. Chem. Phys., 103, 10192 (1995). Decane Under Shear—A Molecular Dynamics Study Using Reversible NVT-SLLOD and NPT-SLLOD Algorithms. Ibid., 104, 7797 (1996). Correction. 

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]. 

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] ... 

In order to describe shear flow effects in a velocity field of the from v (r) = yvy, Kindt and Briels [136] proposed the use of the SLLOD algorithm [140]. Starting with a Langevin equation, such a method results in the following expression for the blob dynamics ... 

Note that the dependence of the shear viscosity on shear rate has been explicitly noted. Alternatively, a non-equilibrium MD simulation can be conducted in which the response of the system to an external perturbation have been considered. The most widely used nonequilibrium approach for viscosity calculations is called "SLLOD" algorithm[146] in which a shear rate is imposed on the system and the resulting stress is computed. The shear viscosity is found at a given shear rate from where Pij is an off-diagonal component of the stress tensor, i is the direction of flow caused by the imposed shear rate, and j is the direction normal to the flow. For the SLLOD algorithm, special "sliding brick" boundary conditions are also typically used, which require modification of the Ewald sum if charged systems are simulated. [147]... 

We thus conclude the section on the numerical implementation of SLLOD dynamics for two very important and useful ensembles. However, our work is not yet complete. The use of periodic boundary conditions in the presence of a shear field must be reconsidered. This is explained in detail in the next section. Furthermore, one could imagine a situation in which SLLOD dynamics is executed in conjunction with constraint algorithms for the internal degrees of freedom and electrostatic interactions. An immediate application of this extension would be the simulation of polar fluids (e.g., water) under shear. This extension has been performed, and the integrator is discussed in detail in Ref. 42. 

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. ... 

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