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In the ideal systems considered in the previous chapters the Hamiltonian Hn(q, p; α) and the Hamiltonian operator \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{N} \left( \alpha \right)\) are, respectively, a sum of one-particle dynamical functions and of one-particle operators (kinetic energy and harmonic oscillators). Real systems are characterized by the fact that, besides the kinetic energy, Hn(q, p; α) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{N} \left( \alpha \right)\) also include the potential energy, which describes how the particles, atoms, or molecules, interact with each other. As has been shown throughout the text, ideal quantum systems are more complex than classical ideal systems. This complexity is also greater in systems with interaction and so from here onward only classical systems will be considered. Since, in general, the interaction potential in real systems is not known exactly, it is rather common to introduce the so-called reference systems in which the potential energy of interaction is relatively simple, while their thermodynamic properties are nevertheless similar to those of real systems, and so they provide a qualitative description of the latter. Along this chapter some approximate methods for the determination of the free energy of an interacting system are considered. In the last section, a brief summary of the so-called numerical simulation methods is provided.

In this chapter, the grand canonical ensemble describing a system that exchanges energy and particles with the external world is introduced. The density fluctuations at the critical point and the ideal quantum boson and fermion gases are presented as key applications of this ensemble.

The objective of statistical physics is to derive the macroscopic (thermodynamic) properties of matter from the laws of mechanics that govern the motion of its microscopic constituents. Note that in this context microscopic may have different meanings. Indeed, although it is common to consider that matter is made of atoms or molecules (something known as the atomic description), this is not the only possible description. For instance, one might consider from the outset that matter is made of electrons and nuclei (subatomic description) that interact through Coulomb forces. Clearly, the derivation of the thermodynamic properties of matter is much more involved when, instead of the atomic description, one uses the subatomic one. The reason is that, in this case, one should analyze successively how electrons and nuclei form atoms, how these atoms constitute molecules, and, finally, how the macroscopic properties of matter may be derived from the molecular description. As a matter of fact, the subject of going from the subatomic description to the atomic one does not truly belong to the realm of statistical physics (rather it belongs to atomic and molecular physics) which usually takes as starting point the atomic and molecular interactions. There is yet a third description in the thermodynamic study of matter (supramolecular or mesoscopic description), which has sometimes been used to study systems having such a complex molecular architecture that it is very difficult to derive the intermolecular interactions from the atomic interactions. The properties of mesoscopic systems are nevertheless similar to those of atomic systems, but they differ in the relevant length scales: the angstrom in atomic systems (microscopic) and the micron in the supramolecular ones (mesoscopic). In the last few years, there has been a spectacular development of research in mesoscopic systems in what is called soft condensed matter (liquid crystals, colloidal dispersions, polymers, etc.), some examples of which are considered later in this chapter. Due to the existence of an interaction potential between the constituents of a non-ideal system, the latter may be found in different structures or phases whose relative stability depends on the thermodynamic state, such as the pressure and the temperature. When these variables are changed, a phase transition may occur in which the structure of the system changes. These transitions will be studied in Chap. 9. In the present chapter a summary is provided of some common structures of matter.

As was seen in Chap. 8 many bulk states of matter exhibit a, partial or full, orientational and/or translational order. Until now this order has always been considered to be perfect although in practice, many bulk phases exhibit defects, i.e., domains where this order departs from the one originally postulated. Such defects are usually very stable, because the free energy cost to remove them easily exceeds the thermal energy and prevent the system to reach its true defect-free equilibrium state. In some cases they are induced by the system's boundary conditions and are hence, unavoidable; i.e., they correspond to the true equilibrium state given the applied boundary conditions.

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 locali...

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.

It is shown that the density functional theory (DFT) of soft matter is able to capture, in a semi-quantitative way, the essential properties of the phase behaviour in these complex systems. This is illustrated here for the case of the fractionation and segregation of polydisperse fluids of spherical (colloidal) particles and for the case of the self-assembling of monodisperse fluids of non-spherical particles. Both homogeneous and inhomogeneous situations will be considered and described within the same DFT.

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.

We propose a density-functional theory for the study of the phase behavior of discotic molecules interacting via a Gay–Berne potential. The attractions are treated perturbatively in a van der Waals-type manner while the virial series of the repulsions is resummed approximately with the inclusion of the exact second-order Onsager result. On this basis we have studied the isotropic (I)–nematic (N)–columnar (Co) phase behavior for various aspect ratios of the discotic molecules. For small aspect ratios (i.e., large nonsphericities) the phase diagram is found to involve a I–N–Co triple point where the two I–N and N–Co first-order transitions are transformed into a direct I–Co first-order transition. For larger aspect ratios the domain where the N-phase is stable shrinks and disappears above a critical value of the aspect ratio. When compared to the available simulation data it is found that the theory underestimates this critical aspect ratio.

It is shown that the van der Waals free-energy of polydisperse fluids, as introduced previously (L. Bellier-Castella, H. Xu and M. Baus, {J. Chem. Phys.} {113}, 8337 (2000)), predicts that for certain thermodynamic states (e.g. low temperatures and large polydispersities) the ordinary two-phase coexistences become metastable relative to a fractionation of the system into three phases, reducing thereby the polydispersity of each of the coexisting phases., Comment: 12 pages, 8 figures, to appear in J.Chem.Phys

The pressure profiles across a liquid–vapor interface introduced previously [J. Chem. Phys. 106, 635 (1997)] have been evaluated with the aid of molecular dynamics simulations for a system of particles interacting via a (truncated and shifted) Lennard-Jones potential. This investigation extends earlier results [J. Chem. Phys. 106, 645 (1997)] to spherical interfaces. Further evidence is found that, for the range of curvatures investigated, the surface tension is curvature independent while the investigation of larger curvatures is prevented by the considerable noise found on the liquid side of the interface.

A van der Waals-based description of a system that approximates a Lennard-Jones fluid is presented as a density functional theory. Numerical calculations of the density and tension profiles for the planar liquid-vapor interface, the wall-fluid interface, and the wall-liquid-vapor interface are made for this system. The forces associated with capillary rise are determined.

The van der Waals theory for the thermodynamic properties of equilibrium fluids is usually presented as an example of a mean field approximation to a simple interaction model. We use density functional arguments to show that the van der Waals description can also be viewed as an exact consequence of a more complex interaction model. The relation to standard statistical mechanical quantities is constructed and the theory is used to determine the tension profile at a vapor–liquid–wall interface.

The absence of demixing in the Percus–Yevick theory of fluid mixtures of additive hard-spheres is related to the fact that this theory predicts incorrect virial coefficients Bn for n>3. Incorporation of the exact Bn for 1⩽n⩽5 into a rescaled virial expansion is shown instead to lead to demixing for any size asymmetry between the spheres. This demixing is however thermodynamically metastable relative to freezing of the mixture into a partially ordered solid phase. This conclusion is reached on the basis of a density functional estimate of the free-energy of a nonuniform phase in which the large spheres form a face-centered cubic lattice whereas the small spheres remain disordered.

It is shown that the region of the phase diagram where the ferromagnetic liquid phase of a Heisenberg fluid (with long-ranged exchange interactions) is thermodynamically stable (relative to the solid phases) remains confined to a fairly small temperature–density domain also in the presence of nematic interactions. The present analysis is carried out on the basis of a van der Waals theory for both translationally and/or orientationally ordered and disordered phases.

We use a simple van der Waals theory, suitably extended to the solid phase and to anisotropic interactions, to study the phase behavior of a system of particles with magnetic exchange interactions. A very rich phase behavior is found which indicates, in particular, that the ferromagnetic liquid phase is favored by increasing the range of both the magnetic exchange interactions and of the nonmagnetic interactions. This could well explain why it turns out to be difficult to find such a phase in simulations which use interactions which are cut off at finite range.

Equilibrium in a two-phase system, a liquid in equilibrium with its vapor, for example, is characterized by the constancy throughout the system of the temperature and the chemical potential and a relation between the pressures of the phases, the Laplace relation of macroscopic thermodynamics. We give a molecular expression for this latter relation in terms of a local pressure defined as the thermodynamic response to a local volume change and we show that this local pressure is the same as the local force distribution in an interface. The macroscopic characteristics of the two-phase system, including the bulk pressures, the surface tension and the location of the surface of tension are all determined by this local pressure function. As shown in a companion paper this function can easily be determined by numerical simulation.

The equilibrium force distribution, or the local pressure in an interface as defined in a companion paper, has been determined from molecular dynamics simulations of a Lennard-Jones fluid. Both a cylindrical and a planar interface are considered. Limited evidence is found that the surface tension could be independent of the curvature of the interface.

The Wilson-Frenkel growth law of colloidal crystals formed in charge-stabilized dispersions of latex spheres is investigated theoretically on the basis of variational free-energy estimates. We find, in agreement with recent experimental results, that the reduced growth velocity is a universal function of some rescaled density variable.

Colloidal dispersions allow various order-disorder phase transitions to be observed by optical means. These transitions can be understood in terms of free-volume and cohesion-energy considerations provided these concepts are defined not only for the disordered phase, as in the Van der Waals theory, but also for the ordered phase.

A thermodynamic definition is given for the spatial distribution of the force in a curved interface that manifests itself macroscopically as the ‘‘surface tension.’’ A general thermodynamic characterization of a curved interface requires a complete knowledge of this force distribution. While a thermodynamic description in terms of a ‘‘surface tension’’ acting on a ‘‘surface of tension’’ is always incomplete, we identify circumstances under which this description is useful and we show how, in these circumstances, the ‘‘surface tension’’ and the location of the ‘‘surface of tension’’ can be derived from the force distribution. Using external field methods, we derive molecular level expressions for the force distribution in planar, cylindrical, and spherical liquid–vapor interfaces in terms of the intermolecular pair potential and the one‐ and two‐particle distribution functions.

In analogy with the well-known theory for fluids, a van der Waals theory for solids is proposed. It is shown that, in agreement with recent predictions, the competition between the van der Waals loop of the fluid and the van der Waals loop of the solid can produce three different types of phase diagrams for a simple fluid. This could be of relevance to the phase behavior of colloidal dispersions.

The b.c.c.-f.c.c.-fluid triple point of the Yukawa model for colloidal suspensions has been relocated using precise Monte Carlo simulations in order to determine the free energies.

The interpretation which Rowlinson and Xiao give to the free energy density is at odds with the interpretation we have offered. They claim to show that our picture is incorrect “by means of counter-examples”. To clarify the issue, we indicate how our picture differs from that attributed to us, we give an expanded demonstration that our free energy density does do what we claim, and we discuss how the total free energy could be calculated from our free energy density.

The stress tensor has been touted for over forty years as providing the most sophisticated description of fluid interfaces. We review the physical interpretation of the stress tensor, the questions which the stress tensor is purported to answer, and the problems which make the standard usage of questionable value. The same physical properties are then formulated in the language of density functional theory. We show that the formulation is straightforward, that the physical ambiguities of the stress tensor approach can be avoided, and that the identification of molecular level expressions for observables is equally straightforward.

The surface tension is a thermodynamic force localized in the interfacial region between two phases. We show that the spatial distribution of this force on a molecular scale can be given a clear thermodynamic definition. In fact, two distinct force distributions are identified, a wall distribution associated with an actual system boundary and an intrinsic distribution associated with an interface unperturbed by (i.e., physically remote from) a wall. While mechanical definitions of the intrinsic distribution are ambiguous on a molecular scale because there is no unique localization of the intermolecular forces, we show that formal density‐functional theory admits a natural, operational definition of the intrinsic distribution in terms of a many‐body external field. The result is a unified phenomenological definition of these two force distributions from which expressions in terms of the molecular distribution functions are readily developed. The surface tension and location of the surface of tension deduced from the intrinsic force distribution agree with established results. These quantities are also shown to be independent of the details of the structure of a wall as long as the force exerted by the wall is normal to the wall. This verifies at a molecular level the original hypothesis of Young.

JOURNAL ARTICLE, info:eu-repo/semantics/published

It is shown that a simple perturbation expansion around the free energy of a (BCC/FCC) hard-sphere crystal leads to accurate predictions for the (BCC/FCC) freezing of soft spheres interacting via the inverse-power potential, V(r)= epsilon ( sigma /r)n, for all values of n including the extreme case of the one-component plasma (n=1). In particular, the authors find that for 1 6 the BCC phase is unstable and only the fluid-FCC transition survives, in agreement with the computer simulations.

The basic advantages of the density functional theory over the other statistical mechanical theories of freezing are outlined. The feasibility of a density functional based calculation is illustrated in the case of the hard sphere freezing. The difficulties encountered in the extension of these results to continuous potentials are discussed and a possible solution is presented.

A simple work‐of‐deformation calculation is shown to produce a family of equivalent expressions for the pressure of an equilibrium fluid adjacent to a mathematical wall. One of these expressions is a statement of the so‐called wall theorem which asserts that the density of a fluid at contact with a wall is related to the pressure and temperature of the bulk fluid by the ideal gas equation of state. The work‐of‐deformation route provides a clear connection between macroscopic and microscopic quantities without requiring the singular limiting procedure used in earlier proofs. The standard expressions for the pressure and surface tension of the planar interface follow from particular interpretations of this family of expressions.

The local stress tensor is defined as the field thermodynamically conjugate to the local strain tensor. Any free-energy change resulting from a strain can then be expressed either in terms of the stresses or in terms of the equivalent single-particle external force field. Using this equivalence, one obtains a well-defined expression for the local stress tensor in terms of this force field. On this basis we obtain unambiguous expressions for the surface tension and for the location of the surface of tension of a planar interface.

A short overview of the density-functional theory of the liquid-solid coexistence and related phenomena is provided for the particular case of the hard-sphere freezing.

SCOPUS: ar.j, info:eu-repo/semantics/published

We show how a van der Waals theory of simple fluids with an interaction potential whose range has a nonmonotonic density dependence can lead to a liquid-liquid transition in close analogy to recent observations and simulations.

In analogy with the well-known Van der Waals theory for disordered fluid phases we propose a simple analytic expression for the Helmholtz free energy of an ordered crystalline phase. It is based on a free-volume approximation to the entropic contribution due to the hard-core repulsions and a static lattice energy approximation to the contribution of the attractions. In this way we are able to describe in a simple way phase transitions between two disordered phases, between two ordered phases and between an ordered and a disordered phase in systems of any spatial dimensionality.

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 compressibility equation of a fluid with an effective, state-dependent, pair-potential is derived. It is shown that the thermodynamic states for which the infinite wavelength limit of the static structure factor diverges are not consistent with the thermodynamic states for which the compressibility diverges whenever the effective potential is density-dependent. This result is in agreement with some recent numerical simulations.

We consider a system of spherical colloidal particles with a size polydispersity and use a simple van der Waals description in order to study the combined effect of both the polydispersity and the spatial non-uniformity induced by a planar interface between a low-density fluid phase (enriched in small particles) and a high-density fluid phase (enriched in large particles). We find a strong adsorption of small particles at the interface, the latter being broadened with respect to the monodisperse case. We also find that the surface tension of the polydisperse system results from a competition between the tendancy of the polydispersity to lower the surface tension and its tendancy to raise the critical-point temperature (i.e. its tendancy to favor phase separation) with the former tendancy winning at low temperatures and the latter at the higher temperatures., 9 pages, 8 figures, PRE (to appear)