3. Shear flow

where N is the number of particles, and u^<sup>i</sup> f g ; 1≤i≤N is some characteristic vector of the molecule; in the case of bodies of revolution, it can be taken to be parallel to the axis of revolution but in a more realistic all atom model some other vector in the molecule has to be defined as u^i, and 1 is the unit second rank tensor. When the molecules are perfectly aligned in the same direction, the order parameter is equal to unity, and when the orientation is completely random, it is equal to zero. The eigenvector corresponding to the order parameter is defined as the director, n, and it is a measure of the average

<sup>S</sup> nn � <sup>1</sup>

The director angular velocity is given by Ω ¼ n � n\_. In a macroscopic system, the order tensor and the order parameter are functions of the position in space, but in a small system such as a simulation cell with dimensions of the order of some ten molecular lengths, there is only one director and one order parameter for

Since the molecules studied in the work presented in this chapter are modeled by the Gay-Berne potential, which can be regarded as a Lennard-Jones potential generalized to elliptical molecular cores, see Appendix 2, they are rigid bodies. Therefore, the Euler equations are applied in angular space. Moreover, since the purpose often is to find the stable orientations of the director relative to an external dissipative field, it is interesting to calculate the torque exerted on the liquid crystal, when the director attains various fixed angles relative to this field. This can be done by adding Gaussian constraints to the Euler equa-

Iω\_ <sup>i</sup> ¼ Γ<sup>i</sup> þ λ<sup>x</sup>

∂Ω<sup>x</sup> ∂ω<sup>i</sup> þ λ<sup>y</sup>

where I is the moment of inertia around the axes perpendicular to the axis of revolution, ω<sup>i</sup> is the angular velocity of molecule i, Γ<sup>i</sup> is the torque exerted on molecule i by the other molecules, Ω<sup>x</sup> and Ω<sup>y</sup> are the x- and y-components of the director angular velocity, and λ<sup>x</sup> and λ<sup>y</sup> are Gaussian constraint multipliers keeping the x- and y-components of the director angular acceleration equal to zero. These multipliers are determined in such a way that the director angular acceleration becomes a constant of motion. Then if the initial director angular acceleration and angular velocity are equal to zero, the director will remain fixed in space for all subsequent times and the time averages of the constraint multipliers will be equal to the torque exerted on the director by the external field. Finally, note that the difference between the director angular velocity, Ω, and the molecular angular velocities ω<sup>i</sup> f g ; 1≤i≤N ; the director angular velocity can be regarded as the angular velocity of the average orientation of the molecules. If the director angular velocity is constrained to be zero by applying Eq. (3), the molecular angular velocities are still nonzero and the right hand side of Eq. (3) is nonzero. The Gaussian constraint simply forces the molecules to rotate in such a way that average orientation stays

∂Ω<sup>y</sup> ∂ω<sup>i</sup>

3 1 

: (2)

, (3)

orientation of the molecules in the system. The order tensor can also be

<sup>Q</sup> <sup>¼</sup> <sup>3</sup> 2

expressed as

the whole system.

tions [21],

the same.

126

2.2 Director constraint algorithm

Non-Equilibrium Particle Dynamics

### 3.1 The SLLOD equations of motion for shear flow

In order to study shear flow and to calculate the viscosity and director alignment angles relative to the streamlines, it is convenient to apply the SLLOD equations of motion [22]. The name SLLOD stems from the similarity to the Dolls equation of motion derived from the Dolls tensor Hamiltonian. They are synthetic equations of motion that can be used to calculate the viscosity in the linear regime. On the other hand, the idea behind the SLLOD equations of motion is very simple: The velocity of the molecules is divided into the streaming velocity and the thermal velocity. The thermal velocity is related to the temperature, and the streaming velocity is the macroscopic external velocity. The SLLOD equations of motion are an exact description of adiabatic planar Couette flow and a very good approximation of shear flow at constant temperature both in the linear and nonlinear regime. The SLLOD equations are expressed in the following way:

$$
\dot{\mathbf{r}}\_i = \frac{\mathbf{p}\_i}{m} + \gamma r\_{xi} \mathbf{e}\_{\mathbf{x}} \tag{4a}
$$

and

$$
\dot{\mathbf{p}}\_i = \mathbf{F}\_i - \gamma p\_{xi} \mathbf{e}\_x - a \mathbf{p}\_i,\tag{4b}
$$

where r<sup>i</sup> and p<sup>i</sup> are the position and peculiar momentum, that is, the momentum relative to the streaming velocity, of molecule <sup>i</sup>, <sup>m</sup> is the molecular mass, <sup>γ</sup> <sup>¼</sup> <sup>∂</sup>ux=∂<sup>z</sup> is the shear rate, that is, there is a streaming velocity ux in the x-direction varying linearly in the z-direction, see Figure 2, e<sup>x</sup> is the unit vector in the x-direction, F<sup>i</sup> is the force exerted on molecule i by the other molecules, and α is a thermostatting multiplier given by the constraint that the linear peculiar kinetic energy is a constant of motion,

$$a = \frac{\sum\_{i=1}^{N} \left[ \mathbf{F}\_i \cdot \mathbf{p}\_i - \gamma p\_{ix} p\_{ix} \right]}{\sum\_{i=1}^{N} \mathbf{p}\_i^2}. \tag{5}$$

### Figure 2.

Planar Couette flow or shear flow arises when there is a streaming velocity in the x-direction, varying linearly in the <sup>z</sup>-direction, <sup>u</sup> <sup>¼</sup> <sup>γ</sup>zex, where <sup>γ</sup> <sup>¼</sup> <sup>∂</sup>ux=∂<sup>z</sup> is the shear rate or velocity gradient. The expression for the relation between the velocity gradient and the pressure tensor becomes simpler by using a director-based coordinate system ð Þ e1; e2; e<sup>3</sup> , where the director n points in the e3-direction, obtained by rotating the ordinary laboratorybased coordinate system ð Þ e1; e2; e<sup>3</sup> with an angle θ around the e<sup>y</sup> ¼ e2-axis. Reproduced from Ref. [6] with the permission of AIP Publishing.

This expression is obtained by applying Gauss's principle of least constraint [22]. This principle is essentially the same as the Lagrange's method for handling constraints. However, Gauss's principle is more general in that it in addition to constraints involving the molecular coordinates also allows handling of some constraints involving the molecular velocities. This is very useful because it makes it possible to keep the kinetic energy constant whereby the temperature also will be constant. It is possible to show that the ensemble averages of the phase functions and the time correlation functions are essentially canonical when a Gaussian thermostat is applied.
