2.2 Director constraint algorithm

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 equations [21],

$$I\dot{\mathbf{o}}\_i = \Gamma\_i + \lambda\_\mathbf{x} \frac{\partial \Omega\_\mathbf{x}}{\partial \mathbf{o}\_i} + \lambda\_\mathbf{y} \frac{\partial \Omega\_\mathbf{y}}{\partial \mathbf{o}\_i},\tag{3}$$

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 the same.

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics… DOI: http://dx.doi.org/10.5772/intechopen.80977
