In this work we present a numerical study, based on molecular dynamics simulations, to estimate the freezing point of hard spheres and hypersphere systems in dimension D = 4, 5, 6 and 7. We have studied the changes of the Radial Distribution Function (RDF) as a function of density in the coexistence region. We started our simulations from crystalline states with densities above the melting point, and moved down to densities in the liquid state below the freezing point. For all the examined dimensions (including D = 3) it was observed that the height of the first minimum of the RDF changes in an almost continuous way around the freezing density and resembles a second order phase transition. With these results we propose a numerical method to estimate the freezing point as a function of the dimension D using numerical fits and semiempirical approaches. We find that the estimated values of the freezing point are very close to previously reported values from simulations and theoretical approaches up to D = 6 reinforcing the validity of the proposed method. This was also applied to numerical simulations for D = 7 giving new estimations of the freezing point for this dimensionality.
The solid-solid coexistence of a polydisperse hard sphere system is studied by using the Monte Carlo simulation. The results show that for large enough polydispersity the solid-solid coexistence state is more stable than the single-phase solid. The two coexisting solids have different composition distributions but the same crystal structure. Moreover, there is evidence that the solid-solid transition terminates in a critical point as in the case of the fluid-fluid transition.
In this note we investigate using basic free energy considerations the location of the gas-liquid critical point with respect to solidification for narrow attractive interactions down to the Baxter limit. Possible experimental and simulation realizations leading to a stable critical point are briefly discussed.
A new Monte Carlo approach is proposed to investigate the fluid-solid phase transition of the polydisperse system. By using the extended ensemble, a reversible path was constructed to link the monodisperse and corresponding polydisperse system. Once the fluid-solid coexistence point of the monodisperse system is known, the fluid-solid coexistence point of the polydisperse system can be obtained from the simulation. The validity of the method is checked by the simulation of the fluid-solid phase transition of a size-polydisperse hard sphere colloid. The results are in agreement with the previou…
Density Profiles and Direct Correlation Functions from Density Functional Theory in Binary Hard-Sphere Crystals: Substitutional Solid and Interstitial Solid Solution
Institute of Applied Physics, University of Tübingen（蒂宾根大学应用物理研究所）
At low temperature, spheres with a very short-ranged attraction exist as a close-packed solid coexisting with an infinitely dilute gas. We find that the ratio of the interfacial tension between these two phases to the thermal energy diverges as the range of the attraction goes to zero. The large tensions when the interparticle attractions are short-ranged may be why globular proteins only crystallise over a narrow range of conditions.
In a recent letter, Buhot proposes that entropy driven phase-separation in hard-core binary mixtures is directly related to a bond-percolation transition. However, at least for binary hard-sphere mixtures, calculations based on an accurate approximation to g_{ll}(r) demonstrate that n_b varies widely along the phase-boundaries calculated directly by simulations, implying that bond-percolation is unrelated to the phase-separation in these systems.
The Ginzburg-Landau free energy functional for hard-spheres is constructed using the Fundamental Measure Theory approach to Density Functional Theory as a starting point. The functional is used to study the liquid-fcc solid planer interface and the properties of small solid clusters nucleating within a liquid. The surface tension for planer interfaces agrees well with simulation and it is found that the properties of the solid clusters are consistent with classical nucleation theory.
It has been shown by Stell [J. Stat. Phys. 63, 1203 (1991)] that at low temperature monodisperse sticky spheres collapse to form coexisting close-packed solid and infinitely dilute gases. We show that polydisperse sticky spheres also collapse and calculate the collapse temperature. The polydisperse spheres separate into fractions with narrower polydispersities which can then solidify. This is the first example of a single-peaked polydisperse mixture phase separating. It implies that a mixture of polydisperse large hard spheres with much smaller hard spheres does not show fluid--fluid coexisten…
Colloidal systems observed in video microscopy are often analysed using the displacements correlation matrix of particle positions. In non-thermal systems, the inverse of this matrix can be interpreted as a pair-interaction potential between particles. If the system is thermally agitated, however, only an effective interaction is accessible from the correlation matrix. We show how this effective interaction differs from the non-thermal case by comparing with high-statistics numerical data from hard-sphere crystals.
We study a model amorphous solid that is subjected to repeated athermal cyclic shear deformation. It has previously been demonstrated that the memory of the amplitudes of shear deformation the system is subjected to (or trained at) is encoded, and can be retrieved by subsequent deformation cycles that serve as read operations. Here we consider different read protocols and measurements and show that single and multiple memories can be robustly retrieved through these different protocols. We also show that shear deformation by a larger amplitude always erases the stored memories. These observati…
Colloidal glasses form from hard spheres, nearly hard spheres, ellipsoids and platelets or their attractive variants have been studied in detail. Complementing and checking theoretical approaches and simulations, the many different types of model systems have significantly advanced our understanding of the glass transition in general. Despite their early prediction, however, no experimental charged sphere glasses have been found at low density, where the competing process of crystallization prevails. We here report the formation of a transient amorphous solid formed from charged polymer sphere…
The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non…
We used a thermodynamic integration scheme, which is specifically designed for disordered systems, to compute the interfacial free energy of the solid-liquid interface in the hard-sphere model. We separated the bulk contribution to the total free energy from the interface contribution, performed a finite-size scaling analysis and obtained for the (100)-interface $γ=0.591(11)k_{B}Tσ^{-2}$.
A simple model of dimerizing hard spheres with highly nontrivial fluid-solid phase behaviour is proposed. The model is studied using the recently proposed resummed thermodynamic perturbation theory for central force (RTPT-CF) associating potentials. The phase diagram has the fluid branch of the fluid-solid coexistence curve located at a temperatures lower than those of the solid branch. This unusual behaviour is related to the strong dependence of the system excluded volume on the temperature, which for the model at hand decreases with increasing temperature. This effect can be also seen for a…
Using laser tweezers and fluorescence confocal polarizing microscopy, we study colloidal interactions of solid microspheres in the nematic bulk caused by elastic distortions around the particles with strong tangential surface anchoring. The particles aggregate into chains directed at about 30 degrees to the far field director and, at higher concentrations, form complex kinetically trapped structures. We characterize the distance and angular dependencies of the colloidal interaction forces.
The quasilocalized charge approximation is applied to estimate the sound velocity of simple soft sphere fluid with the repulsive inverse-power-law interaction. The obtained results are discussed in the context of the sound velocity of the hard-sphere system and of liquid metals at the melting temperature.
We present an experimental study of the motion of a solid sphere falling through a wormlike micellar fluid. While smaller or lighter spheres quickly reach a terminal velocity, larger or heavier spheres are found to oscillate in the direction of their falling motion. The onset of this instability correlates with a critical value of the velocity gradient scale $Γ_{c}\sim 1$ s$^{-1}$. We relate this condition to the known complex rheology of wormlike micellar fluids, and suggest that the unsteady motion of the sphere is caused by the formation and breaking of flow-induced structures.
The freezing transition of hard spheres has been well described by various versions of density-functional theory (DFT). These theories should possess the close-packed crystal as a special limit, which represents an extreme testing ground for the quality of such liquid-state based theories. We therefore study the predictions of DFT for the structure and thermodynamics of the hard-sphere crystal in this limit. We examine the Ramakrishnan-Yussouff (RY) approximation and two variants of the fundamental-measure theory (FMT) developed by Rosenfeld and coworkers. We allow for general shapes of the de…
A fluid particle changes its dynamics from diffusive to oscillatory as the system density increases up to the melting density. Hence, the notion of the Frenkel line was introduced to demarcate the fluid region into rigid and nonrigid liquid subregions based on the collective particle dynamics. In this work, we apply a topological framework to locate the Frenkel lines of the soft-sphere and the hard-sphere models relying on the system configurations. The topological characteristics of the ideal gas and the maximally random jammed state are first analyzed, then the classification scheme designed…
We calculate the friction force acting on a hard cylinder or spherical ball rolling on a flat surface of a viscoelastic solid. The rolling friction coefficient depends non-linearly on the normal load and the rolling velocity. For a cylinder rolling on a viscoelastic solid characterized by a single relaxation time Hunter has obtained an exact result for the rolling friction, and our result is in very good agreement with his result for this limiting case. The theoretical results are also in good agreement with experiments of Greenwood and Tabor. We suggest that measurements of rolling friction o…
The response of amorphous solids to an applied shear deformation is an important problem, both in fundamental and applied research. To tackle this problem, we focus on a system of hard spheres in infinite dimensions as a solvable model for colloidal systems and granular matter. The system is prepared above the dynamical glass transition density, and we discuss the phase diagram of the resulting glass under compression, decompression, and shear strain, expanding on previous results [P. Urbani and F. Zamponi, Phys.Rev.Lett. 118, 038001 (2017)]. We show that the solid region is bounded by a "shea…
We present a simple model for bacteria like \emph{Escherichia coli} swimming near solid surfaces. It consists of two spheres of different radii connected by a dragless rod. The effect of the flagella is taken into account by imposing a force on the tail sphere and opposite torques exerted by the rod over the spheres. The hydrodynamic forces and torques on the spheres are computed by considering separately the interaction of a single sphere with the surface and with the flow produced by the other sphere. Numerically, we solve the linear system which contains the geometrical constraints and the …
The free energies of the FCC, BCC, HCP and Simple Cubic phases for hard spheres are calculated as a function of density using the Fundamental Measure Theory models of Rosenfeld et al (PRE 55, 4245 (1997)), Tarazona (PRL 84, 694 (2001)) and Roth et al (J. Phys.: Cond. Matt. 14, 12063 (2002)) in the Gaussian approximation. For the FCC phase, the present work confirms the vanishing of the Lindemann parameter (i.e. vanishing of the width of the Gaussians) near close packing for all three models and the results for the HCP phase are nearly identical. For the BCC phase and for packing fractions abov…
Monte Carlo simulations at constant pressure are performed to study coexistence and interfacial properties of the liquid-solid transition in hard spheres and in colloid-polymer mixtures. The latter system is described as a one-component Asakura-Oosawa (AO) model where the polymer's degrees of freedom are incorporated via an attractive part in the effective potential for the colloid-colloid interactions. For the considered AO model, the polymer reservoir packing fraction is eta_p^r=0.1 and the colloid-polymer size ratio is q=sigma_p/σ=0.15 (with sigma_p and sigma the diameter of polymers and co…
We probe the Ioffe-Regel limits of glasses with repulsions near the zero-temperature jamming transition by measuring the dynamical structure factors. At zero temperature, the transverse Ioffe-Regel frequency vanishes at the jamming transition with a diverging length, but the longitudinal one does not, which excludes the existence of a diverging length associated with the longitudinal excitations. At low temperatures, the transverse and longitudinal Ioffe-Regel frequencies approach zero at the jamming-like transition and glass transition, respectively. As a consequence, glasses between the glas…
Hard spheres with a polydispersity above approximately 8% are shown to crystallise into two phase-separated solid phases. A polydispersity above 8% is too large to be tolerated by a single solid phase but phase separation produces two fractions with polydispersities sufficiently narrow to allow them to crystallise. It may not be possible to observe this in experiment due to the intervention of a glass transition.
Complex phenomena incorporating several physical properties are abundant while they are occasionally revealing the variation of power-law behavior depending on the scale. In this present work, the global scaling-behavior of dynamical impact of solid sphere onto elastic surface is described. Its fundamental dimensionless function was successfully obtained by applying the dimensional analysis combined with the solution by energy conservation complementally. It demonstrates that its power-law behavior is given by the competition between two power-law relations representing inertial and elastic pr…
Hydrodynamic interactions play an important role in many areas of soft matter science. In simulations with implicit solvent, various techniques such as Brownian or Stokesian dynamics explicitly include hydrodynamic interactions a posteriori by using hydrodynamic diffusion tensors derived from the Stokes equation. However, this equation assumes the interaction to be instantaneous which is an idealized approximation and only valid on long time scales. In the present paper, we go one step further and analyze the time-dependence of hydrodynamic interactions in a compressible fluid on the basis of …
The acoustic radiation force exerted on a small sphere located at the focus of an ultrasonic beam is measured in a soft gel. It is proved to evolve quadratically with the local amplitude of the acoustic field. Strong oscillations of the local pressure are observed and attributed to an acoustic Fabry-P{é}rot effect between the ultrasonic emitter and the sphere. Taking this effect into account with a simple model, a quantitative link between the radiation force and the acoustic pressure is proposed and compared to theoretical predictions in the absence of dissipation. The discrepancy between exp…
We propose a simple Monte Carlo method to calculate the interfacial free energy between the substrate and the material. Using this method we investigate the interfacial free energys of the hard sphere fluid and solid phases near a smooth hard wall. According to the obtained interfacial free energys of the coexisting fluid and solid phases and the Young equation we are able to determine the contact angle with high accuracy, cos$θ$ = 1:010(31), which indicates that a smooth hard wall can be wetted completely by the hard sphere crystal at the interface between the wall and the hard sphere fluid.
The penetration by a gravity driven impact of a solid sphere into a granular medium is studied by two-dimensional simulations. The scaling laws observed experimentally for both the final penetration depth and the stopping time with the relevant physical parameters are here recovered numerically without the consideration of any solid friction. Collisional processes are thus found as essential in explaining the physics of the qualitatively observed phenomena whereas frictional processes can only be considered as secondary effects in the granular penetration by impact.
A numerical model is proposed to compute the eigenmodes and the forced response of multilayered elastic spheres. The main idea is to describe analytically the problem along the angular coordinates with spherical harmonics and to discretize the radial direction with one-dimensional finite elements. The proper test function must be carefully chosen so that both vector and tensor spherical harmonics orthogonality relationships can be used. The proposed approach yields a general one-dimensional formulation with a fully analytical description of the angular behaviour, suitable for any interpolating…
This paper presents a three-dimensional analytical study of the intrinsic free vibration of an elastic multilayered hollow sphere interacting with an exterior non-Newtonian fluid medium. The fluid is assumed to be characterized by a compressible linear viscoelastic model accounting for both the shear and compressional relaxation processes. For small-amplitude vibrations, the equations governing the viscoelastic fluid can be linearized, which are then solved by introducing appropriate potential functions. The solid is assumed to exhibit a particular material anisotropy, i.e. spherical isotropy,…
We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element (FEM) simulations of the Stokes equations for assemblies of 8 …
We study numerically the influence of confinement on the solid fraction and on the structure of three-dimensional random close packed (RCP) granular materials subject to gravity. The effects of grain shape (spherical or polyhedral), material polydispersity and confining wall friction on this dependence are investigated. In agreement with a simple geometrical model, the solid fraction is found to decrease linearly for increasing confinement no matter the grain shape. Furthermore, this decrease remains valid for bidisperse sphere packings although the gradient seems to reduce significantly when …
