Perturbation Theory for Classical Solids

Solid-solid transitions induced by repulsive interactions revisited

We revisit a problem already studied 15 years ago by us in collaboration with Stell and Hemmer: the isostructural solid-solid transitions induced by repulsive particle interactions exhibited by classical systems interacting via the Stell-Hemmer potentials. The full phase diagram in the crystal region is obtained by applying a perturbation theory for classical solids used during our collaboration with Stell. Also, the performance of such a theory is now tested by comparing the perturbative phase diagram with that obtained from computer simulations. The latter was calculated using a recently refined method to obtain the free-energy of crystals by means of Monte Carlo simulations. The perturbation theory captures the correct topology and correctly identifies the stable, fcc and bcc, phases. In addition, the theory predicts the occurrence of special points: a point where the two stable structures coexist at the same density, and two critical points terminating the corresponding isostructural phase transitions for fcc and bcc phases. The location of some of these features in the phase diagram is predicted almost quantitatively. However, phase boundaries involving the non-compact bcc phase are much less accurate, a problem that can be traced to the poor representation used for the bcc phase of the reference, hard-sphere, system.

Hard-body models of bulk liquid crystals

Hard models for particle interactions have played a crucial role in the understanding of the structure of condensed matter. In particular, they help to explain the formation of oriented phases in liquids made of anisotropic molecules or colloidal particles and continue to be of great interest in the formulation of theories for liquids in bulk, near interfaces and in biophysical environments. Hard models of anisotropic particles give rise to complex phase diagrams, including uniaxial and biaxial nematic phases, discotic phases and spatially ordered phases such as smectic, columnar or crystal. Also, their mixtures exhibit additional interesting behaviours where demixing competes with orientational order. Here we review the different models of hard particles used in the theory of bulk anisotropic liquids, leaving aside interfacial properties and discuss the associated theoretical approaches and computer simulations, focusing on applications in equilibrium situations. The latter include one-component bulk fluids, mixtures and polydisperse fluids, both in two and three dimensions, and emphasis is put on liquid-crystal phase transitions and complex phase behaviour in general.

Theoretical calculation of the melting curve of Cu-Zr binary alloys

Helmholtz free energies of the dominant binary crystalline solids found in the Cu-Zr system at high temperatures close to the melting curve are calculated. Our theoretical approach combines fundamental measure density functional theory (applied to the hard-sphere reference system) and a perturbative approach to include the attractive interactions. The studied crystalline solids are Cu(fcc), Cu_{51}Zr_{14}(β), CuZr(B2), CuZr_{2}(C11b), Zr(hcp), and Zr(bcc). The calculated Helmholtz free energies of crystalline solids are in good agreement with results from molecular-dynamics (MD) simulations. Using the same perturbation approach, the liquid phase free energies are calculated as a function of composition and temperature, from which the melting curve of the entire composition range of this system can be obtained. Phase diagrams are determined in this way for two leading embedded atom method potentials, and the results are compared with experimental data. Theoretical melting temperatures are compared both with experimental values and with values obtained directly from MD simulations at several compositions.

Perturbation theory of solid-liquid interfacial free energies of bcc metals

A perturbation theory is used to calculate bcc solid-liquid interfacial free energies of metallic systems with embedded-atom model potentials. As a reference system for bcc crystals we used a single-occupancy cell, hard-sphere bcc system. Good agreements between the perturbation theory results and the corresponding results from simulations are found. The strategy to extract hard-sphere bcc solid-liquid interfacial free energies may have broader applications for other crystal lattices.

Phonon dispersions of cluster crystals

We analyze the ground states and the elementary collective excitations (phonons) of a class of systems, which form cluster crystals in the absence of attractions. Whereas the regime of moderate-to-high temperatures in the phase diagram has been analyzed in detail by means of density functional considerations (Likos et al 2007 J. Chem. Phys. 126 224502), the present approach focuses on the complementary regime of low temperatures. We establish the existence of an infinite cascade of isostructural transitions between crystals with different lattice site occupancies at T = 0 and we quantitatively demonstrate that the thermodynamic instabilities are bracketed by mechanical instabilities arising from long-wavelength acoustical phonons. We further show that all optical modes are degenerate and flat, giving rise to almost perfect realizations of Einstein crystals. We calculate analytically the complete phonon spectrum for the whole class of models as well as the Helmholtz free energy of the systems. On the basis of the latter, we demonstrate that the aforementioned isostructural phase transitions must terminate at an infinity of critical points at low temperatures, brought about by the anharmonic contributions in the Hamiltonian and the hopping events in the crystals.

Perturbation theory for solid-liquid interfacial free energies

A perturbation theory is developed to calculate solid-liquid interfacial free energies, including anisotropy. The method is applied to systems with inverse-power and Lennard-Jones pair potentials as well as to metal systems with embedded-atom model potentials. The results are in reasonable agreement with the corresponding ones obtained from molecular dynamics simulations.

Theoretical studies of the correlations in moderately asymmetric binary hard-sphere solid mixtures

A theoretical approach to calculate the correlation functions in binary hard-sphere (HS) solid mixtures is developed. The method is motivated by the theory of correlations in one-component HS solids [Rascon, Phys. Rev. E 54, 1261 (1996)] and the Kirkwood-Buff theory for solutions. Combined with the fundamental measure density functional theory the approach provides correlation functions in good agreement with the numerical simulations results for the moderately asymmetric mixtures.

Noncompact crystalline solids in the square-well potential

We reexamine the phase diagram of the square-well potential, using both theoretical and computer-simulation techniques, for not too short ranges of the potential. The phase diagram turns out to contain a variety of crystalline structures, both compact and, interestingly, also noncompact. The latter result from a large increase in negative energy when pairs of particles come at distances within the interaction range, which more than compensates the entropy loss associated with reduced packing. Transitions between these crystalline structures give rise to a surprisingly rich phase diagram.

Phase behavior of short-range square-well model

Various Monte Carlo techniques are used to determine the complete phase diagrams of the square-well model for the attractive ranges lambda = 1.15 and lambda = 1.25. The results for the latter case are in agreement with earlier Monte Carlo simulations for the fluid-fluid coexistence curve and yield new results for the liquidus-solidus lines. Our results for lambda = 1.15 are new. We find that the fluid-fluid critical point is metastable for both cases, with the case lambda = 1.25 being just below the threshold value for metastability. We compare our results with prior studies and with experimental results for the gamma(II)-crystallin.

Calculations of free energies in liquid and solid phases: fundamental measure density-functional approach

In this paper, a theoretical description of the free energies and correlation functions of hard-sphere (HS) liquid and solid phases is developed using fundamental measure density-functional theory. Within the framework of Weeks-Chandler-Andersen perturbation theory, free energies of liquid and solid phases with many interaction potentials can be obtained from these characteristics of the HS system within a single theoretical description. An application to the Lennard-Jones system yields liquid-solid coexistence results in good agreement with the ones from simulations.

Theoretical description of phase coexistence in model C60

We have investigated the phase diagram of a pair interaction model of C60 fullerene [L. A. Girifalco, J. Phys. Chem. 96, 858 (1992)], in the framework provided by two integral equation theories of the liquid state, namely, the modified hypernetted chain (MHNC) implemented under a global thermodynamic consistency constraint, and the self-consistent Ornstein-Zernike approximation (SCOZA), and by a perturbation theory (PT) with various degrees of refinement, for the free energy of the solid phase. We present an extended assessment of such theories as set against a recent Monte Carlo study of the same model [D. Costa, G. Pellicane, C. Caccamo, and M. C. Abramo, J. Chem. Phys. 118, 304 (2003)]. We have compared the theoretical predictions with the corresponding simulation results for several thermodynamic properties such as the free energy, the pressure, and the internal energy. Then we have determined the phase diagram of the model, by using either the SCOZA, the MHNC, or the PT predictions for one of the coexisting phases, and the simulation data for the other phase, in order to separately ascertain the accuracy of each theory. It turns out that the overall appearance of the phase portrait is reproduced fairly well by all theories, with remarkable accuracy as for the melting line and the solid-vapor equilibrium. All theories show a more or less pronounced discrepancy with the simulated fluid-solid coexistence pressure, above the triple point. The MHNC and SCOZA results for the liquid-vapor coexistence, as well as for the corresponding critical points, are quite accurate; the SCOZA tends to underestimate the density corresponding to the freezing line. All results are discussed in terms of the basic assumptions underlying each theory. We have then selected the MHNC for the fluid and the first-order PT for the solid phase, as the most accurate tools to investigate the phase behavior of the model in terms of purely theoretical approaches. It emerges that the use of different procedures to characterize the fluid and the solid phases provides a semiquantitative reproduction of the thermodynamic properties of the C60 model at issue. The overall results appear as a robust benchmark for further theoretical investigations on higher order C(n>60) fullerenes, as well as on other fullerene-related materials, whose description can be based on a modelization similar to that adopted in this work.

Confirmation of anomalous dynamical arrest in attractive colloids: a molecular dynamics study

Previous theoretical simulation and experimental studies have indicated that particles with a short-ranged attraction exhibit a range of dynamical arrest phenomena. These include very pronounced reentrance in the dynamical arrest curve, a logarithmic singularity in the density correlation functions, and the existence of "attractive" and "repulsive" glasses. Here we carry out extensive molecular dynamics calculations on dense systems interacting via a square-well potential. This is one of the simplest systems with the required properties, and may be regarded as canonical for interpreting the phase diagram, and now also the dynamical arrest. We confirm the theoretical predictions for reentrance, logarithmic singularity, and give a direct evidence of the existence, independent of theory, of two distinct glasses. We now regard the previous predictions of these phenomena as having been established.

Phase equilibria and glass transition in colloidal systems with short-ranged attractive interactions: application to protein crystallization

We have studied a model of a complex fluid consisting of particles interacting through a hard-core and short-range attractive potential of both Yukawa and square-well form. Using a hybrid method, including a self-consistent and quite accurate approximation for the liquid integral equation in the case of the Yukawa fluid, perturbation theory to evaluate the crystal free energies, and mode-coupling theory of the glass transition, we determine both the equilibrium phase diagram of the system and the lines of equilibrium between the supercooled fluid and the glass phases. For these potentials, we study the phase diagrams for different values of the potential range, the ratio of the range of the interaction to the diameter of the repulsive core being the main control parameter. Our arguments are relevant to a variety of systems, from dense colloidal systems with depletion forces, through particle gels, nanoparticle aggregation, and globular protein crystallization.

Global equation of state of two-dimensional hard sphere systems

Hard sphere systems in two dimensions are examined for arbitrary density. Simulation results are compared to the theoretical predictions for both the low- and the high-density limit, where the system is either disordered or ordered, respectively. The pressure in the system increases with the density, except for an intermediate range of volume fractions 0.65< or =nu< or =0.75, where a disorder-order phase transition occurs. The proposed global equation of state (which describes the pressure for all densities) is applied to the situation of an extremely dense hard sphere gas in a gravitational field and shows reasonable agreement with both experimental and numerical data.

Liquid-solid transition of hard spheres under gravity

We investigate the liquid-solid transition of two-dimensional hard spheres in the presence of gravity. We determine the transition temperature and the fraction of particles in the solid regime as a function of temperature via event-driven molecular-dynamics simulations and compare them with the theoretical predictions. We then examine the configurational statistics of a vibrating bed from the viewpoint of the liquid-solid transition by explicitly determining the transition temperature and the effective temperature T of the bed, and present a relation between T and the vibration strength.

Complex phase behavior induced by repulsive interactions

For a solid in which the interactions have a hard core plus a simple soft repulsive tail we show, using a perturbation theory, that the possible stable crystalline structures give rise to a rich phase behavior. We find two concomitant critical points each corresponding to phase transitions separating bcc and fcc structures, respectively, and the occurrence of a transition between fcc and bcc phases without change in density. This novel phenomenology may be relevant to the behavior of some metallic systems, colloids, and to water.

Solid-phase structures of the dzugutov pair potential

In recent computer simulations of a simple monatomic system interacting via the Dzugutov pair potential, freezing of the fluid into an equilibrium dodecagonal quasicrystal has been reported [M. Dzugutov, Phys. Rev. Lett. 70, 2924 (1993)]. Here, using a combination of molecular dynamics simulation and thermodynamic perturbation theory, we conduct a detailed analysis of the relative stabilities of solid-phase structures of the Dzugutov-potential system. At low pressures, the most stable structure is found to be a bcc crystal, which gives way at higher pressures to a fcc crystal. Although a dodecagonal quasicrystal and a sigma-phase crystal compete with the bcc crystal for stability, they always remain metastable.

Phase behavior of binary hard-sphere mixtures from perturbation theory

Using a first-order perturbation theory, we have studied the phase diagram of a binary mixture of hard spheres for different values of the size ratio. Recent models for the two-body depletion potential between large spheres are used to take into account the role of the small spheres. The theory predicts a complex phase diagram including a fluid-solid transition at high packing fraction of small spheres, metastability of fluid-fluid demixing, an isostructural solid-solid transition at high packing fraction of the large spheres for sufficiently small values of the size ratio q of the spheres, and the tendency to sticky-sphere behavior in the limit q-->0. The agreement with recent simulation results is quite good. We also show that this phenomenology was already implicit in the pioneering work of Asakura and Oosawa.