2. Basic theory

### 2.1 Order parameter, director, and director angular velocity

In order to describe transport properties of a liquid crystal, we must first define the order parameter, the director, and the director angular velocity. In an axially symmetric system such as a nematic or a smectic A liquid crystal, the order parameter, S, is given by the largest eigenvalue of the order tensor,

$$\mathbf{Q} = \frac{3}{2} \left( \frac{1}{N} \sum\_{i=1}^{N} \hat{\mathbf{u}}\_i \hat{\mathbf{u}}\_i - \frac{1}{3} \mathbf{1} \right), \tag{1}$$

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

$$\mathbf{Q} = \frac{3}{2}\mathbf{S}\left(\mathbf{nn} - \frac{1}{3}\mathbf{1}\right). \tag{2}$$

3. Shear flow

and

stant of motion,

Figure 2.

127

permission of AIP Publishing.

3.1 The SLLOD equations of motion for shear flow

DOI: http://dx.doi.org/10.5772/intechopen.80977

equations are expressed in the following way:

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

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

<sup>r</sup>\_<sup>i</sup> <sup>¼</sup> <sup>p</sup><sup>i</sup> m

<sup>α</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup>

p\_ <sup>i</sup> ¼ F<sup>i</sup> � γpzie<sup>x</sup> � αp<sup>i</sup>

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 con-

> <sup>i</sup>¼<sup>1</sup> <sup>F</sup><sup>i</sup> � <sup>p</sup><sup>i</sup> � <sup>γ</sup>pixpiz ∑<sup>N</sup> <sup>i</sup>¼<sup>1</sup>p<sup>2</sup> i

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

þ γrzie<sup>x</sup> (4a)

, (4b)

: (5)

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 the whole system.
