SLLOD equations

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... 

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... 

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 ... 

Figure 15 Schematic of the method employed to calculate friction coefficient. The corrugated surfaces are immobile, and a shear flow is generated in the confined fluid using SLLOD equations of motion. The difference in momentum between the fluid and the surfaces results in a frictional force, which is the response function.

In the absence of shearing periodic boundary conditions (of the type introduced earlier) the system is totally isolated that is, all the degrees of freedom of the system are explicitly accounted for in the equations of motion. In this case, it is possible to obtain a conserved quantity for field-driven dynamics in general and SLLOD in particular. The approach we employ is similar to that introduced in the section on Molecular Dynamics and Equilibrium Statistical Mechanics. The SLLOD equations of motion are... 

Figure 16 Velocity profile generated in the confined fluid by SLLOD equations. The circles correspond to simulation results, and the line is the expected behavior. The simulation was performed at a shear rate, y = 0.05 ps , with a corrugation amplitude, a = 0.02 a, and the zero of the shear field, q° = -28 A.

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - " are ... 

Alternatively, one could use SLLOD equations to do direct simulations, such as shear a system under planar Couette flow and measure the shear stress. As we have already discussed, this approach has been used successfully to calculate a host of transport properties. It is important to remember, however, that direct simulation is often unable to simulate realistic materials at experimentally accessible shear rates. At low shear rates, the nonequilibrium response becomes small compared to the magnitude of the equilibrium fluctuations that naturally arise. The extremely small signal-to-noise ratio would demand prohibitively long simulations before any meaningful answers could be obtained. 

At this point, we will assume that we are shearing a system in the canonical ensemble using the SLLOD equations of motion. Hence,... 

They show that for elongational flow the SLLOD equations are identical to Newton s equations of motion with the inclusion of an additional external force that must exist in order to sustain a steady elongational flow. Their derivation shows that SLLOD is the correct set of equations to use when performing NEMD simulations of elongational flow. No doubt the issue will continue to be debated in the literature for some time to come. 

The most detailed simulation study of the orientational ordering of simple dipolar fluids undergoing planar Couette flow at a constant shear rate y has been presented in a series of papers by McWhirter and Patey [205-208]. hi their work the translational motion of the particles is obtained from the so-called SLLOD equations given by [209]... 

A validation of the p-SLLOD equations of motion for homogeneous steady-state flows. J. Chem. Phys., 124,... 

The equations of motion for all of these shearing and shear-free flows are the so-called SLLOD equations. 

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