As a system is de ned by the collection of a large number of particles, so the “ensembles” can be de ned as a collection of a number macroscopically identical but essentially independent systems. Here the term macroscopically identical means, as, each of the systems constituting an ensemble satis es the same macroscopic conditions, like Volume, Energy, Pressure, Temperature and Total number of particles etc. Here again, the term essentially independent means the system (in the ensemble) being mutually non-interacting to others, i.e., the systems di er in microscopic conditions like parity, symmetry, quantum states etc.

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 Λexc, solvation force fs 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.

The numerical results for the twist angle profile xi(z) across a slab-shaped nematic cell obtained from a density functional theory (DFT) are compared to the predictions of the macroscopic Frank-Oseen theory. The latter theory predicts that xi"(z)=0, and this is also seen to be the case for the DFT results. These do, however, verify the Frank-Oseen relation, lambda+/-W+/-=K2, between the de Gennes extrapolation length (lambda+/-), the anchoring energy per unit area of the (+/-) cell wall (W+/-), and the elastic constant of the nematic for twist deformations (K2), only if W+/- is nonlinearly related to the amplitude of the anchoring term of the DFT.

Il est actuellement bien connu que les molecules non spheriques peuvent former des mesophases (ou cristaux liquides), c'est-a-dire des phases dont les proprietes sont intermediaires entre celles des liquides et celles des cristaux. La mesophase la plus connue est la phase nematique. Il s'agit d'une phase caracterisee par une distribution aleatoire des centres de masse des molecules, mais dans laquelle l'orientation des molecules presente une direction preferentielle, designee par un vecteur unite appele le directeur du nematique. Une telle phase possede donc la fluidite d'un liquide tout en presentant, tel un cristal, une birefringence. C'est cette derniere propriete qui est exploitee dans les applications technologiques, principalement dans les dispositifs d'affichage.Dans un tel dispositif, le liquide nematique est contenu dans une cellule (il y a une cellule par pixel), et son directeur est manipule a l'aide d'un champ exterieur, electrique ou magnetique. Pour une bonne comprehension du fonctionnement de ce dispositif, il est essentiel de connaitre le profil du directeur a travers la cellule en l'absence de champ exterieur. Dans le cadre de ce travail, nous avons etudie un nematique torsade, c'est-a-dire dont le directeur decrit une helice a travers la cellule. Ce profil est determine principalement par les proprietes d'ancrage du liquide nematique sur les parois solides de la cellule. En effet, celles-ci peuvent posseder une direction d'ancrage privilegiee, qui favorise l'alignement du directeur dans une direction particuliere. Nous avons considere ici le cas de directions d'ancrage planaires, c'est-a-dire que le directeur est dans le plan des parois. Alors que l'ajout de parois identiques dans le systeme induit toujours une non-uniformite spatiale dans la densite du nematique (en comparaison avec un nematique en coeur de phase), l'utilisation de directions d'ancrage differentes induit une non-uniformite orientationnelle dans le directeur du nematique; dans notre cas une torsion. C'est principalement ce profil de directeur torsade qui nous interesse ici. L'objectif general de ce travail consiste donc a etudier les proprietes d'ancrage d'une phase nematique confinee et torsadee, d'une part par une theorie microscopique (theorie de la fonctionnelle de la densite), et d'autre part sur le plan de simulations de Monte Carlo, en particulier dans le cas ou les molecules ont la forme de disques (discotiques).

This is a textbook which gradually introduces the student to the statistical mechanical study of the different phases of matter and to the phase transitions between them. Throughout, only simple models of both ordinary and soft matter are used but these are studied in full detail. The subject is developed in a pedagogical manner, starting from the basics, going from the simple ideal systems to the interacting systems, and ending with the more modern topics. The latter include the renormalisation group approach to critical phenomena, the density functional theory of interfaces, the topological defects of nematic liquid crystals and the kinematic aspects of the phase transformation process. This textbook provides the student with a complete overview, intentionally at an introductory level, of the theory of phase transitions. References include suggestions for more detailed treatments and four appendices supply overviews of the mathematical tools employed in the text. © Marc Baus, Carlos F. Tejero 2008.

The freezing of equally charged hard spheres embedded in a uniform neutralizing background is investigated theoretically using a recently developed density functional theory. We find a fluid-f.c.c. solid transition which naturally extends the uncharged hard sphere transition studied previously. As in all similar cases investigated hitherto in the literature, the bcc solid has been found to be metastable relative to the fluid. Several possible reasons for this rather surprising result are discussed in detail.

We establish the condition under which the broken symmetry equation of Lovett-Mou-Buff and Wertheim is equivalent to the first Yvon-Born-Green equation. On this basis we show that the Kirkwood-Monroe theory of freezing is missing a term related to the density derivative of the pair correlation function. This may explain the bad results obtained from the latter theory.

The identification of the force distribution in curved interfaces as a thermodynamic force [Baus and Lovett, J. Chem. Phys. 101, 377 (1995)] can be interpreted as a relation between the force distribution and the grand canonical free energy difference between two distinct systems. Using this interpretation, molecular expressions are developed for the force distribution in cylindrical and spherical interfaces that remain valid for very highly curved interfaces.

The freezing of hard-sphere mixtures of arbitrary polydispersity is studied within a van der Waals-type free-volume approximation. The present theory is simple enough to allow for a thorough numerical investigation of all the polydispersity effects on the order–disorder transition of hard spheres. Within this context we have studied the influence on the order–disorder transition of the initial preparation, the subsequent fractionation, and the possible terminal polydispersity. It is found that the order–disorder transition occupies a finite domain of the initial density–initial polydispersity plane and the frontier of this domain is determined. Considerable variation within this domain is found with respect to the initial density, while the influence of the specific form of the initial size-distribution is found to be only marginal.

Most colloids usually exhibit one or several polydispersities. A natural framework for the theoretical description of polydisperse systems is provided by the extension of density functional theory to 'continuous mixtures. This will be illustrated here by the study of both the bulk and interfacial properties of a simple van der Waals model for a polydisperse colloidal fluid.