Statistical Mechanics of Fusion
TL;DR: A statistical mechanical theory of fusion based upon the use of local free energies is presented in this paper. But this theory is restricted to the case of a set of parameters and is not suitable for all parameters.
Abstract: A statistical mechanical theory of fusion based upon the use of local free energies is presented. An integral equation is formulated for the distribution function of average density in a region occupied by a system of molecules. Periodic solutions characteristic of a crystalline phase are found for certain ranges of values of a set of parameters depending upon temperature and volume. When the parameters decrease below certain critical values, all terms of the Fourier series representing the distribution function vanish with the exception of the constant term. A uniform density distribution characteristic of a fluid phase is then obtained. The melting parameters of argon at several pressures are calculated with the aid of the theory and compared with experiment.
Citations
More filters
TL;DR: In this paper, the authors reviewed new perspectives on the phase field approach in modeling deformation and fracture at the fundamental defect level, including the ability to predict fundamental properties of individual defects such as size, formation energy, saddle point configuration and activation energy of defect nuclei, and the micromechanisms of their mutual interactions.
375 citations
TL;DR: A comprehensive overview of phase transition studies can be found in this article, where the authors identify the essential key concepts and points of difficulty associated with the study of phase transitions and discuss the most widely used experimental techniques for measuring these transition properties.
308 citations
TL;DR: In this article, an integral equation for the radial distribution function for pairs in a liquid, and an approximate solution is effected for a system of ''hard spheres'' is derived. But the form of the function depends on a single parameter λ which can be related to certain observed physical properties of the liquid and to the diameter of closest approach.
Abstract: In accordance with the general methods of an earlier paper (reference 1) an integral equation is evolved for the radial distribution function for pairs in a liquid, and an approximate solution is effected for a system of ``hard spheres.'' The form of the function depends on a single parameter λ which can be related to certain observed physical properties of the liquid and to the diameter of closest approach. The theoretical function has been calculated for a value of λ appropriate to liquid argon at 90°K, and compared to experimental radial distribution functions derived from x‐ray scattering data.
305 citations
TL;DR: It is shown quantitatively that shape drives the phase behavior of systems of anisotropic particles upon crowding through DEFs, and the mechanism that generates directional entropic forces is the maximization of entropy by optimizing local particle packing.
Abstract: Entropy drives the phase behavior of colloids ranging from dense suspensions of hard spheres or rods to dilute suspensions of hard spheres and depletants. Entropic ordering of anisotropic shapes into complex crystals, liquid crystals, and even quasicrystals was demonstrated recently in computer simulations and experiments. The ordering of shapes appears to arise from the emergence of directional entropic forces (DEFs) that align neighboring particles, but these forces have been neither rigorously defined nor quantified in generic systems. Here, we show quantitatively that shape drives the phase behavior of systems of anisotropic particles upon crowding through DEFs. We define DEFs in generic systems and compute them for several hard particle systems. We show they are on the order of a few times the thermal energy ([Formula: see text]) at the onset of ordering, placing DEFs on par with traditional depletion, van der Waals, and other intrinsic interactions. In experimental systems with these other interactions, we provide direct quantitative evidence that entropic effects of shape also contribute to self-assembly. We use DEFs to draw a distinction between self-assembly and packing behavior. We show that the mechanism that generates directional entropic forces is the maximization of entropy by optimizing local particle packing. We show that this mechanism occurs in a wide class of systems and we treat, in a unified way, the entropy-driven phase behavior of arbitrary shapes, incorporating the well-known works of Kirkwood, Onsager, and Asakura and Oosawa.
232 citations
Additional excerpts
...Proc Natl Acad Sci USA 107(23):10348–10353....
[...]
TL;DR: In this paper, the Fisher-Widom (FW) line was introduced to define the divergence point between pure exponential from exponentially damped oscillatory decay of the radial distribution function g(r) at a liquid-vapour interface.
Abstract: Recent work has highlighted the existence of a unified theory for the asymptotic decay of the density profile ρ(r) of an inhomogeneous fluid and of the bulk radial distribution function g(r). For a given short-ranged interatomic potential ρ(r) decays into bulk in the same fashion as g(r), i.e. with the same exponential decay length (α0/-1) and, for sufficiently high bulk density (ρb) and/or temperature (T), oscillatory wavelength (2π/α1). The quantities α0 and α1 are determined by a linear stability analysis of the bulk fluid; they depend on only the bulk direct correlation function. In this paper we reintroduce the concept of the Fisher-Widom (FW) line. This line was originally introduced, in say the (ρb, T plane, as that which separates pure exponential from exponentially damped oscillatory decay of g(r). We explore the relevance of the FW line for the form of the density profile at a liquid-vapour interface. Using a weighted density approximation (WDA) density functional theory we locate the FW line fo...
191 citations
References
More filters
TL;DR: In this paper, the chemical potentials of the components of gas mixtures and liquid solutions are obtained in terms of relatively simple integrals in the configuration spaces of molecular pairs, and the molecular pair distribution functions appearing in these integrals are investigated in some detail, in their dependence upon the composition and density of the fluid.
Abstract: Expressions for the chemical potentials of the components of gas mixtures and liquid solutions are obtained in terms of relatively simple integrals in the configuration spaces of molecular pairs. The molecular pair distribution functions appearing in these integrals are investigated in some detail, in their dependence upon the composition and density of the fluid. The equation of state of a real gas mixture is discussed, and an approximate molecular pair distribution function, typical of dense fluids, is calculated. Applications of the method to the theory of solutions will be the subject of a later article.
2,946 citations
Book•
27 Jan 2005
1,152 citations
TL;DR: In this article, the free energy of a rigid body is calculated as a function of temperature, and of the six homogeneous strain components, for a regular (cubic) lattice.
Abstract: The Helmholtz free energy, A, of a rigid body is a function of temperature, and of the six homogeneous strain components. If the crystal is to be rigid, three inequalities must be satisfied for the derivatives of A with respect to the six strain components, for a regular (cubic) lattice. This enables one to limit the pressure‐temperature range for which the crystal is stable. The violation of the condition c44>0, that the crystal resist shearing, is interpreted as leading to melting. From a knowledge of the forces between the molecules the phase integral, and therefore the free energy, may be calculated as a function of T, V, and the six strain components. The numerical calculations are carried out for a body‐centered cubic lattice. The product of all the frequencies is calculated directly, so that the assumption that the Debye equation for the frequency distribution holds, is not necessary. The melting curve, pressure against temperature, is then determined.
583 citations
TL;DR: In this paper, it was shown that the van der Waals equation is not valid for gases at high densities such as obtain in the neighbourhood of the critical point, and that the usual method of representing isotherms as simple functions of density or pressure ceases to be useful.
Abstract: The exact measurements of the isotherms of gases have proved extremely valuable in the determination of interatomic forces. For this purpose it has been found necessary to express the pv values of a gas as a finite power series in the density or in the pressure, and the coefficients so obtained have been compared with theoretical expressions in terms of interatomic fields. Many accounts of the method have been given and it is not necessary to give further details here (cf. Lennard-Jones 1931). While these methods are valid for gases at low densities where binary encounters are predominant, they fail for gases at high densities such as obtain in the neighbourhood of the critical point. Michels and his collaborators (Michels and others 1937) have recently studied the isotherms of gases at pressures as high as 3000 atm., and they find that the usual method of representing isotherms as simple functions of density or pressure ceases to be useful. The equation of state of van der Waals was astonishingly successful in accounting for the critical phenomena of gases and the form of the isotherms for temperatures below the critical temperature. Other empirical equations of state, for example that of Dieterici, were even more successful in reproducing the observed relations between the critical pressure, volume and temperature, and their very success has often obscured the fact that they were not logical theories of critical phenomena in gases, based as they were on arguments which were valid only for gases of low concentration. Thus the van der Waals equation, valuable as it has been and useful as it still is, implies that the internal energy of a vapour and its liquid phase is proportional only to the first power of the density, and this cannot be true for gases or vapours at densities comparable with those of liquids. The problem still remains of explaining why gases exhibit critical properties and of correlating the observed values of the critical temperature with the forces which atoms or molecules exert on each other.
447 citations
272 citations