The SLLOD Equations of Motion

As stated in the earlier section on Generalized Equations of Motion, we would ultimately like to find a set of equations of motion in the form of Eq. [91] to compute transport coefficients such as the shear viscosity and self diffusion 

y is the shear rate, and i is a unit vector in the x direction. Thus, Eqs. [123] describe a system coupled to a shear field in the x direction with a gradient in the shear field along the y. Note that there is no explicit time dependence from the external field [fg t) = y = constant], and the compressibility due to the coupling to the velocity field satisfies Eq. [94]. Equations [123] are not derivable from an underlying Hamiltonian however, one can provide some physical insight by examining the individual equations of motion comprising Eqs. [123]. 

It is also interesting to examine Eqs. [123] within the linear response theory developed in the preceding subsection. Thus, we would like to treat the coupling to the velocity field as perturbation and examine the consequences of this. We then invoke Eq. [120]. As our phase variable B, we will choose to examine P y, the xy component of the pressure tensor. Thus, Eq. [120] reads 

It should be clear that at equilibrium, Pxy)o = 0- We can compare the result of Eq. [125] to the Green-Kubo expression for the shear viscosity given by Eq. [121]. We can equate the left-hand side of Eq. [125] to r), and we then obtain the following remarkable result  

Making use of the fact that the stress is equal to the negative of the pressure, we finally obtain 

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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,... 

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. 

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... 

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... 

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

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.

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

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... 

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. 

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 ... 

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. 

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 ... 

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. 

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] ... 

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

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]... 

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