A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of ∼64%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the “freezing point” in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. Our results predict the experimental conditions necessary for the formation of a jammed crystal by calculating an analogue of the “entropy of fusion”. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.     

The GMSA for a hard-sphere fluid revisited: evaluation of the Henderson-Blum approximation[*

The Generalized Mean Spherical Approximation for a hard-sphere fluid developed by Waisman [E. Waisman, Mol. Phys. 25, 45 (1973)] over forty years ago is revisited. The relatively recent [C. F. Tejero and M. López de Haro, Mol. Phys. 105, 2999 (2007)] completely analytical results for the direct correlation function of such approximation are used to assess the simplification introduced by Henderson and Blum [D. Henderson and L. Blum, Mol. Phys. 32, 1627 (1976)]. The conclusion is reached that such simplification is a rather valuable compromise between simplicity and accuracy.     

Mode-coupling theory of the glass transition for colloidal liquids in slit geometry

ABSTRACT

We provide a detailed derivation of the mode-coupling equations for a colloidal liquid confined by two parallel smooth walls. We introduce irreducible memory kernels for the different relaxation channels thereby extending the projection operator technique to colloidal liquids in slit geometry. Investigating both the collective dynamics as well as the tagged-particle motion, we prove that the mode-coupling functional assumes the same form as in the Newtonian case corroborating the universality of the glass-transition singularity with respect to the microscopic dynamics.     

The free energy and the equation of state of a hard-sphere system for homogeneous stable and metastable phases

A new free energy equation is deduced; it serves as a basis for the derivation of an equation for the homogeneous phase of a hard-sphere system. It exactly reproduces all the known virial coefficients and agrees with the molecular dynamics data within their accuracy. The absence of singular points during transition from the stable region to the metastable region is shown. According to the properties of the statistical integral, the free energy and the equation of state have a singularity at close packing.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

Application of the thermodynamics of curved boundary layers to the scaled particle theory of hard-sphere fluids

The thermodynamics of curved boundary layers, with the assumption that the distance between the surface of a fluid cavity and its surface of tension is a quadratic function of the cavity radius, is applied to the exact thermo-dynamic expression forG, the central function of scaled particle theory. The coefficients in the quadratic representation are determined so as to satisfyall five of the known exact conditions onG valid for cavity radii between one-half and one molecular diameter. The results of the calculation are displayed as the hard-sphere equation of state, the boundary tension associated with the surface of tension, and the distance between the cavity surface and the surface of tension. Although the hard-sphere equation of state obtained by this method using all five conditions onG is more accurate than in the case where only two or three conditions are used, the original scaled particle theory, in whichG itself was represented simply by a quadratic function of inverse powers of cavity radius, still yields the more accurate equation of state. Nevertheless, the present approach limits approximations to the distance between the cavity surface and the surface of tension, a small quantity in itself. The path to a still more improved theory remains well defined, contingent upon the discovery of additional exact conditions, and does not depend, as in the original scaled particle theory, on a form forG arrived at in a somewhat ad hoc manner.Research supported under NSF Grant #GP-12408.     

Molecular simulation study of the glass transition in a soft primitive model for ionic liquids

In this paper, we present a molecular dynamics study of the glass transition for a soft-core primitive model for ionic liquids, in which cations are fully flexible chains of tangent soft spherical monomers, being the positively charged monomer at one of the ends of the chain, and anions as charged soft spheres. We have monitored transport coefficients such as the self-diffusion coefficients and the shear viscosity, as well as correlation functions such as the mean-square displacement, the self-intermediate scattering function, and probes of heterogeneous dynamics such as the van Hove distribution function and the four-points susceptibility. The analysis of these properties indicates that, for a given pressure, the glass transition shows a weak temperature dependence on the cation length, occurring first for short-chain than for long-chain ionic liquids.     

Pressure dependence of the superconducting state parameters of metallic glass superconductor Mg70Zn30

Explicit expressions have been derived for the volume dependence of electron-phonon coupling strength (λ) and the Coulomb pseudopotential (μ^*) considering the variation of Fermi momentum (κ _F) and Debye temperature (θ _D) with volume. Ashcroft’s model pseudopotential and RPA form of dielectric screening have been used for obtaining pressure dependence of transition temperature (T _C) and the logarithmic volume derivative (Φ) of the effective interaction strength (N ₀ V) for metallic glass superconductor Mg₇₀Zn₃₀. It has been observed that T _C of the metallic glass Mg₇₀Zn₃₀ decreases rapidly with increase of pressure and the superconducting phase disappears at about 30% decrease of volume, for which the μ^* curve shows a minimum and an elbow is formed in the Φ graph.     

Perturbation and variational approach for the equation of state for hard-sphere and Lennard-Jones fluids

The present work uses the concept of a scaled particle along with the perturbation and variation approach, to develop an equation of state (EOS) for a mixture of hard sphere (HS), Lennard–Jones (LJ) fluids. A suitable flexible functional form for the radial distribution function G(R) is assumed for the mixture, with R as a variable. The function G(R) has an arbitrary parameter m and a different equation of state can be obtained with a suitable choice of m. For m = 0.75 and m = 0.83 results are close to molecular dynamics (MD) result for pure HS and LJ fluid respectively.     

Density nonlinearities and a field theory for the dynamics of simple fluids

We study the role of the Jacobian arising from a constraint enforcing the nonlinear relationg=V, where ,g, andV are the mass density, the momentum density, and the local velocity field, respectively, in the field-theoretic formulation of the nonlinear fluctuating hydrodynamics of simple fluids. By investigating the Jacobian directly and by developing a field-theoretic formulation without the constraint, we find that no changes in dynamics result as compared to the previous formulation developed by Das and Mazenko (DM). In particular, the cutoff mechanism discovered by DM is shown to be a consequence of the 1/ nonlinearity in the problem, not of the constraint. The consequences of this result for the static properties of the system are also discussed.     

Perturbation and variational approach for the equation of state for hard-sphere and Lennardben Jones fluids

The present work uses the concept of a scaled particle along with the perturbation and variation approach, to develop an equation of state (EOS) for a mixture of hard sphere (HS), Lennard-Jones (LJ) fluids. A suitable flexible functional form for the radial distribution function G(R) is assumed for the mixture, with R as a variable. The function G(R) has an arbitrary parameter m and a different equation of state can be obtained with a suitable choice of m. For m= 0.75 and m= 0.83 results are close to molecular dynamics (MD) result for pure HS and LJ fluid respectively.     

Flow curves of colloidal dispersions close to the glass transition

The flow curves, viz. the curves of stationary stress under steady shearing, are obtained close to the glass transition in dense colloidal dispersions using asymptotic expansions in the schematic -model of mode coupling theory. The shear thinning of the viscosity in fluid states and the yielding of glassy states is discussed. At the transition between fluid and shear-molten glass, simple and generalized Herschel-Bulkley laws are derived with power law exponents that can be computed for different particle interactions from the equilibrium structure factor.     

Solving the initial condition of the string relaxation equation of the string model for glass transition: part-I

The string model for the glass transition can quantitatively describe the universal α-relaxation in glassformers,including the average relaxation time,the distribution function of the relaxation time,and the relaxation strength as functions of temperature.The string relaxation equation(SRE) of the model,at high enough temperatures,simplifies to the well-known single particle mean-field Debye relaxation equation,and at low enough temperatures to the well-known Rouse-Zimm relaxation equation that describes the relaxation dynamics of linear macromolecules.However,its initial condition,necessary to the further model predictions of glassy dynamics,has not been solved.In this paper,the special initial condition(SIC) of the SRE,i.e.for straight strings and the dielectric spectrum technique that is one of the most common methods to measure the glassy dynamics,was solved exactly.It should be expected that the obtained SIC would benefit the solution of the general initial condition of the SRE of the string model,i.e.for stochastically spatially configurating strings,as will be described in separate publications.     

Solving the initial condition of the string relaxation equation of the string model for glass transition:part-Ⅱ

The string model for the glass transition can quantitatively describe the universal α-relaxation in glassformers. The string relaxation equation (SRE) of the model simplifies the well-known Debye and Rouse--Zimm relaxation equations at high and low enough temperatures, respectively. However, its initial condition, necessary to the further model predictions of glassy dynamics, has not been solved. In this paper, the general initial condition of the SRE for stochastically spatially configurative strings is solved exactly based on the obtained special initial condition of the SRE for straight strings in a previous paper (J. L. Zhang et al. 2010 Chin. Phys. B 19, 056403).     

Changes in the state of polarization of a random electromagnetic beam propagating through tissue

Based on the recently proposed unified theory of coherence and polarization of random electromagnetic beams, we have derived formulae describing changes in the state of polarization of a random electromagnetic beam propagating through tissue. A so-called electromagnetic Gaussian Schell-model beam is used to illustrate the theory. The results may find possible applications in tissue imaging.     

Mean-field kinetic theory of a classical electron gas in a periodic potential. I. Formal solution of the Vlasov equation

We study the static and dynamic behavior of a classical electron gas in the periodic potential created by an ionic lattice. Using the well-known Vlasov approximation, we derive a mean-field kinetic equation for the density-response function of the electrons. This equation is formally solved in terms of the trajectories of one electron in the mean-field equilibrium potential which determines the local electronic density. The mean-field expressions of the static and dynamic structure factors are then obtained through the fluctuation-dissipation theorem. These expressions are used to show that within the mean-field approximation the system is a conductor at all temperatures and for all dimensions.     

Effect of quasiparticle renormalization on the localization of the excitations of a one-dimensional Bose-Einstein condensate in a random potential

We investigate the effects of renormalization on the localization of the quasiparticle excitations of one-dimensional Bose-Einstein condensate in a random potential. Starting with a set of linearized equations of motion for the phases of superfluid grains coupled by Josephson interactions, we use mode-counting techniques to calculate the inverse localization length for large (10⁸) arrays. Employing distributions for the interaction parameters that are the same as the initial (pre-renormalization) distributions used by Gurarie et al. (Phys. Rev. Lett. 101 (2008) 170407), we compare the initial-interaction results for the localization length with those obtained using renormalization group techniques.     

Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.     

Pressure dependence of the superconducting state parameters of a binary Ca<Subscript>70</Subscript>Mg<Subscript>30</Subscript> metallic glass superconductor

Theoretical computation of the pressure dependence of superconducting state parameters of a binary Ca₇₀Mg₃₀ metallic glass has been performed using the model potential formalism. Explicit expressions have been derived for the volume dependence of the electron-phonon coupling strength λ and the Coulomb pseudopotential μ*, considering the variation of the Fermi momentum k _F and Debye temperature θ_D with volume. Well-known Ashcroft’s empty core model pseudopotential and five different types of the local-field correction functions, namely, Hartree, Taylor, Ichimaru-Utsumi, Farid et al. and Sarkar et al. have been used for obtaining pressure dependence of transition temperature T _C and the logarithmic volume derivative Φ of the effective interaction strength N ₀ V for the metallic glass superconductor. It has been obtained that T _C of Ca₇₀Mg₃₀ metallic glass decreases rapidly with increasing pressure up to 60% decrease in the volume, for which the μ* and Φ curves show a linear nature. The superconducting phase disappears at about 60% decrease in the volume.