Entropy driven phase transition in hard core lattice gas models in three dimensions[HBNI Th170]

Abstract

A phase transition or a transformation is a sudden change from one phase to another when some thermodynamic parameter is varied. Typically, as temperature is increased, systems undergo a phase transition from an energetically favoured low-temperature ordered state to an entropically favoured high temperature disordered state. This typically involves a gain in entropy which competes with the rise in internal energy. This competition between the internal energy and entropy drives the phase transition. Examples of such energy-driven transitions are ferromagnetic transitions in spin systems, liquid-gas transition of water, etc. However there exists systems for which the ordered phase has more entropy than the disordered phase with no appreciable difference in internal energy. These transitions are primarily driven by entropy and thus called entropy driven transitions. Example systems in which entropy-driven phase transitions occur include freezing transition of hard spheres [1], phase separation in binary hard-core mixtures [2], gas adsorption on metallic surfaces [3], transitions between nematic, smectic and cholesteric phases in liquid crystals [4], nanotube gels [5], transitions to a nematic phase in aqueous solutions of tobacco mosaic viruses [6], emergence of cholesteric phases in fd viruses [7], isotropic-nematic transitions in rod-like boehmite particles [8], nematic phases in rodlike silica particles with varying aspect ratio [9], emergence of biaxial nematic and smectic phases in banana shaped liquid crystals under pressure [10] and emergence of geometrical frustration in triangular cells under biaxial compression [11].

In this thesis we study two problems on a cubic lattice: (1) hard rods of length k and (2) hard cubes of size 2 × 2 × 2. We obtain the detailed phase diagram and characterize the nature of the phase transitions for both these models. The results that we have obtained are summarized below.

The main results in this thesis are: • There are no phase transitions when k ≤ 4. • For k = 5, 6, the system undergoes a single transition from a disordered phase to a layered-disordered phase. In the layered-disordered phase, the fractional number of rods of two orientations are roughly equal, whereas the number of rods in the third orientation is suppressed. • A nematic phase is observed for k ≥ 7. • There exists four phases for k ≥ 7 as a function of density: disordered, nematic, layered-nematic and layered-disordered. • In the layered-nematic phase, each plane has two dimensional nematic order, but there is no overall bulk nematic order. • We argue that the layered-nematic phase is a finite-size artefact which is observed in simulations and will be unstable in the thermodynamic limit. This can be explained using a perturbative expansion about a pure bulk nematic phase at small defect densities. • The disordered to layered-disordered transition in rods of size k = 5, 6 is shown to be first order. • The critical values for the disordered-layered transition are: µc(5) ≈ 3.82 and ρc(5) ≈ 0.874 and µc(6) ≈ 1.0 and ρc(6) ≈ 0.68. • The disordered-nematic transition for k = 7 is expected to be very weakly first order from symmetry considerations, but no signature of first-order nature was seen as it requires simulations of large system sizes. • The critical values for the disordered-nematic transition for k = 7 is µc ≈ −0.23, corresponding to ρc ≈ 0.556.