Liquid Crystal Observables: Static and Dynamic Properties

In this Chapter we introduce the description of single and pair particle static properties of liquid crystals, and discuss their calculation from computer simulations. We also briefly describe the calculation of dynamic properties from molecular dynamics simulations using Linear Response theory. 1. Single particle properties We consider a system of N molecules at certain specified thermodynamic conditions. Typically we shall consider that volume V and temperature T are fixed together with N (canonical conditions), but we shall also refer to the case where pressure P is fixed together with T (isobaric conditions). We assume the molecules to be classical, rigid particles with centre of mass at position r and orientation ω given for instance by a set of three Euler angles (α, β, γ), or only two angles (α, β) as illustrated in fig. 1 if we can assume that the molecules have cylindrical symmetry [1]. We shall discuss the calculation of observables in liquid crystals [2] in fairly general terms, but adopting a rather special point of view, that of computer simulations. As will be clear from the contributions in this book, computer simulations techniques [3] actually generate configurations of the system, i.e. sets of positions ri = (xi, yi, zi) and orientations ωi of all the particles. In particular the Monte Carlo (MC) method generates equilibrium configurations, albeit non necessarily in the proper time order, while Molecular Dynamics (MD) actually generates configurations time step after time step in their natural time sequence. In this last case configurations consists not only of positions and orientations, but also of the full set of linear and angular velocities.