On the calculation of the absolute grand potential of confined smectic-A phases

Abstract

We determine the absolute grand potential Λ along a confined smectic-A branch of a calamitic liquid crystal system enclosed in a slit pore of transverse area A and width L, using the rod–rod Gay–Berne potential and a rod–wall potential favouring perpendicular orientation at the walls. For a confined phase with an integer number of smectic layers sandwiched between the opposite walls, we obtain the excess properties (excess grand potential Λ^exc, solvation force f _s and adsorption Γ) with respect to the bulk phase at the same μ (chemical potential) and T (temperature) state point. While usual thermodynamic integration methods are used along the confined smectic branch to estimate the grand potential difference as μ is varied at fixed L, T, the absolute grand potential at one reference state point is obtained via the evaluation of the absolute Helmholtz free energy in the (N, L, A, T) canonical ensemble. It proceeds via a sequence of free energy difference estimations involving successively the cost of localising rods on layers and the switching on of a one-dimensional harmonic field to keep layers integrity coupled to the elimination of inter-layers and wall interactions. The absolute free energy of the resulting set of fully independent layers of interacting rods is finally estimated via the existing procedures. This work opens the way to the computer simulation study of phase transitions implying confined layered phases.

Keywords:

• liquid crystal
• smectic phase
• confined
• free energy
• Monte Carlo simulation

Acknowledgements

C.-C. Huang and J.-P. Ryckaert thank S. Ramanchandran for useful discussions.

This work has been inspired by the historical Hansen-Verlet work on the determination of the Lennard-Jones phase diagram [28] where new methods for the free energy determination have led these authors to predict, in 1969, a standard phase diagram, essentially without approximations (for finite size systems), for the archetype of continuous pair potential.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The referee pointed out that according to Sheu, Mou and Lovett (Phys. Rev. E, 51, R3795(1995)), it could be more efficient in our case (to minimise statistical errors) to switch on both fields at the same time, choosing μ₀ and λ₀ values by requiring that the centre of mass distribution (along z) and the distribution of rod orientations are similar at the beginning (in the absence of the two fields) and at the end of the path (as both fields are fully operative). This suggestion would be worth trying but its application would again require an ad hoc approach, given the persistence of strong short range intra-layer interactions between rods, as the interactions between layers are progressively switched off and fields turned on.

Additional information

Funding

C.-C. Huang and J.-P. Ryckaert acknowledge financial support from the ARC program of the Communauté Française de Belgique [grant number 20061].