Liquid-crystal transitions: a first-principles multiscale approach

A rigorous theory of liquid-crystal transitions is developed starting from the Liouville equation. The starting point is an all-atom description and a set of order-parameter field variables that are shown to evolve slowly via Newton's equations. The separation of time scales between that of atomic collision or vibrations and the order-parameter fields enables the derivation of rigorous equations for stochastic order-parameter field dynamics. When the fields provide a measure of the spatial profile of the probability of molecular position, orientation, and internal structure, a theory of liquid-crystal transitions emerges. The theory uses the all-atom/continuum approach developed earlier to obtain a functional generalization of the Smoluchowski equation wherein key atomic details are embedded. The equivalent nonlocal Langevin equations are derived, and the computational aspects are discussed. The theory enables simulations that are much less computationally intensive than molecular dynamics and thus does not require oversimplification of the system's constituent components. The equations obtained do not include factors that require calibration and can thus be applicable to various phase transitions which overcomes the limitations of phenomenological field models. The relation of the theory to phenomenological descriptions of nematic and smectic phase transitions, and the possible existence of other types of transitions involving intermolecular structural parameters are discussed.