SLLOD

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 Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)).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 ... 

The most immediate way of calculating viscosities and studying flow properties by molecular dynamics is to simulate a shear flow. This can be done by applying the SLLOD equations of motion [8]. In angular space they are the same as the ordinary equilibrium Euler equations. In linear space one adds the streaming velocity to the thermal motion,... 

Fluctuation relations for the shear viscosities and the twist viscosities were originally derived by Forster [28] using projection operator formalism and by Sarman and Evans analysing the linear response of the SLLOD equations [24]. They were very complicated, i. e. rational functions of TCFI s. The reason for this is that the conventional canonical ensemble was used. In this ensemble one... 

It is also possible to calculate the shear viscosities and the twist viscosities by applying the SLLOD equations of motion for planar Couette flow, Eq. (3.9). If we have a velocity field in the x-direction that varies linearly in the z-direction the velocity gradient becomes Vu=ye ej, see Fig. 3. Introducing a director based coordinate system (Cj, C2, 63) where the director points in the e3-direction and the angle between the director and the stream lines is equal to 0, gives the following expression for the strain rate in the director based coordinate system. 

The first attempt to evaluate the viscosities of a liquid crystal model system by computer simulation was made by Baalss and Hess [31]. They mapped a perfectly ordered liquid crystal onto a soft sphere fluid in order to simplify the interaction potential and thereby make the simulations faster. The three Mies-owicz were evaluated by using the SLLOD equations of motion. Even though the model system was very idealised, the relative magnitudes of the various viscosities were fairly similar to experimental measurements of real systems. 

Comparison of the Miesowicz viscosities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kulw relation (4.4)-(4.6). The entries for finite field have been obtained by applying the SLLOD equations (3.9). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 

We have thus demonstrated that Newton s law of viscosity, an inherently macroscopic result, can be obtained via linear response theory as the nonequilibrium average in the steady state. Furthermore, the distribution function for the steady state average is determined by microscopic equations of motion. Hence, the SLLOD equations, in the linear regime, reduce to the linear phenomenological law proposed by Newton. Moreover, all the quantities that are needed to compute the shear viscosity can be obtained from a molecular dynamics simulation. 

The SLLOD equations of motion presented in Eqs. [123] are for the specific case of planar Couette flow. It is interesting to consider how one could write a version of Eqs. [123] for a general flow. One way to do this is introduce a general strain tensor, denoted by Vu. For the case of planar Couette flow, Vu = j iy in dyadic form, where 1 and j denote the unit vector in the x and y directions, respectively. The matrix representation is... 

With Vu = j ly, the SLLOD equations in the unthermostated form become... 

It is interesting to analyze Eqs. [129] further to point out interesting features of SLLOD dynamics. 

However, it is clear that for a general tensor Vu, trajectory analysis based on the SLLOD dynamics in Eqs. [129] will yield incorrect results. Equation [132] has an extra term in the force, which is equivalent to saying that the momenta in Eqs. [129] are not peculiar with respect to a general flow (indeed, Eqs. [129] yield peculiar velocities for the case of planar Couette flow), and therefore the flow profile produced will not be q Vu as expected. Equations [129] also lead to problems when one is considering definitions of pressure... 

The distribution of Eq. [137] is canonical in laboratory momentum and positions for a general strain rate tensor Vu this is the expected form for a system subject to an external field. Equation [137] is the first distribution function to be derived for SLLOD-type dynamics and has provided impetus for studies concerning the nature of the distribution function in the nonequilibrium steady state. 

SLLOD Dynamics for Planar Couette Flow in the Canonical Ensemble... 

In the earlier subsection on the Dynamical Generation of the NPT Ensemble, we introduced equations of motion to perform equilibrium MD under constant temperature and pressure conditions. These equations of motion can be augmented with terms involving the shear rate from the SLLOD equations and can be written as follows ... 

The full time evolution to = e( L JPT-sLLOD)A p ij summarized as follows ... 

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. 

This exercise, which validatts Eq. [192], can also be performed to see the validity of the evolution of h in the NPT ensemble, (Eqs. [84]). To simulate flows in bulk systems, Eq. [192] must be used in conjunction with SLLOD or GSLLOD equations. We call Eq. [192] the box dynamics method. 

We have thus far discussed the basic foundations of nonequilibrium molecular dynamics, its methodology, and the details of numerically integrating the equations of motion for SLLOD dynamics. The next section presents applications of these methods. 

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