Thermodynamic and dynamical properties of the hard sphere system revisited by molecular dynamics simulation

Models to predict configurational adiabats of Lennard-Jones fluids and their transport coefficients

A comparison is made between three simple approximate formulas for the configurational adiabat (i.e., constant excess entropy, sex) lines in a Lennard-Jones (LJ) fluid, one of which is an analytic formula based on a harmonic approximation, which was derived by Heyes et al. [J. Chem. Phys. 159, 224504 (2023)] (analytic isomorph line, AIL). Another is where the density is normalized by the freezing density at that temperature (freezing isomorph line, FIL). It is found that the AIL formula and the average of the freezing density and the melting density ("FMIL") are configurational adiabats at all densities essentially down to the liquid-vapor binodal. The FIL approximation departs from a configurational adiabat in the vicinity of the liquid-vapor binodal close to the freezing line. The self-diffusion coefficient, D, shear viscosity, ηs, and thermal conductivity, λ, in macroscopic reduced units are essentially constant along the AIL and FMIL at all fluid densities and temperatures, but departures from this trend are found along the FIL at high liquid state densities near the liquid-vapor binodal. This supports growing evidence that for simple model systems with no or few internal degrees of freedom, isodynes are lines of constant excess entropy. It is shown that for the LJ fluid, ηs and D can be predicted accurately by an essentially analytic procedure from the high temperature limiting inverse power fluid values (apart from at very low densities), and this is demonstrated quite well also for the experimental argon viscosity.

Crossover from gas-like to liquid-like molecular diffusion in a simple supercritical fluid

According to textbooks, no physical observable can be discerned allowing to distinguish a liquid from a gas beyond the critical point. Yet, several proposals have been put forward challenging this view and various transition boundaries between a gas-like and a liquid-like behaviour, including the so-called Widom and Frenkel lines, and percolation line, have been suggested to delineate the supercritical state space. Here we report observation of a crossover from gas-like (Gaussian) to liquid-like (Lorentzian) self-dynamic structure factor by incoherent quasi-elastic neutron scattering measurements on supercritical fluid methane as a function of pressure, along the 200 K isotherm. The molecular self-diffusion coefficient was derived from the best Gaussian (at low pressures) or Lorentzian (at high pressures) fits to the neutron spectra. The Gaussian-to-Lorentzian crossover is progressive and takes place at about the Widom line intercept (59 bar). At considerably higher pressures, a liquid-like jump diffusion mechanism properly describes the supercritical fluid on both sides of the Frenkel line. The present observation of a gas-like to liquid-like crossover in the self dynamics of a simple supercritical fluid confirms emerging views on the unexpectedly complex physics of the supercritical state, and could have planet-wide implications and possible industrial applications in green chemistry.

Revised Enskog theory and molecular dynamics simulations of the viscosities and thermal conductivity of the hard-sphere fluid and crystal

Hard-sphere (HS) shear, longitudinal, cross, and bulk viscosities and the thermal conductivity are obtained by molecular dynamics (MD) simulations, covering the entire density range from the dilute fluid to the solid crystal near close-packing. The transport coefficient data for the HS crystal are largely new and display, unlike for the fluid, a surprisingly simple behavior in that they can be represented well by a simple function of the density compressibility factor. In contrast to the other four transport coefficients (which diverge), the bulk viscosity in the solid is quite small and decreases rapidly with increasing density, tending to zero in the close-packed limit. The so-called cross viscosity exhibits a different behavior to the other viscosities, in being negative over the entire solid range, and changes sign from negative to positive on increasing the density in the fluid phase. The extent to which the viscosity tensor and thermal conductivity of the HS crystal can be represented by revised Enskog theory (RET) is investigated. The RET expressions are sums of an instantaneous (I), a kinetic (K), and a so-called α part. The I part of the transport coefficients evaluated directly by MD are statistically indistinguishable from those of the corresponding kinetic theory (Enskog and RET) expressions. For the K part the integral over the spatial two-particle distribution function at contact was determined and the α part was estimated using the direct correlation function and density functional theory approximations. All three parts were determined in this work which allowed the accuracy of RET for solid systems to be assessed rigorously. It is found that in the case of the thermal conductivity the predictions of RET are in excellent agreement with the MD results. Also, for the shear viscosity the agreement over the entire solid phase is quite good but is considerably worse for the three remaining viscosities in the solid phase.

Freezing density scaling of transport coefficients in the Weeks-Chandler-Andersen fluid

It is shown that the transport coefficients (self-diffusion, shear viscosity, and thermal conductivity) of the Weeks-Chandler-Andersen (WCA) fluid along isotherms exhibit a freezing density scaling (FDS). The functional form of this FDS is essentially the same or closely related to those in the Lennard-Jones fluid, hard-sphere fluid, and some liquefied noble gases. This proves that this FDS represents a quasi-universal corresponding state principle for simple classical fluids with steep interactions. Some related aspects, such as a Stokes-Einstein relation without a hydrodynamic diameter and gas-to-liquid dynamical crossover, are briefly discussed. Simple fitting formulas for the transport coefficients of the dense WCA fluid are suggested.

Hierarchical Framework for Predicting Entropies in Bottom-Up Coarse-Grained Models

The thermodynamic entropy of coarse-grained (CG) models stands as one of the most important properties for quantifying the missing information during the CG process and for establishing transferable (or extendible) CG interactions. However, performing additional CG simulations on top of model construction often leads to significant additional computational overhead. In this work, we propose a simple hierarchical framework for predicting the thermodynamic entropies of various molecular CG systems. Our approach employs a decomposition of the CG interactions, enabling the estimation of the CG partition function and thermodynamic properties a priori. Starting from the ideal gas description, we leverage classical perturbation theory to systematically incorporate simple yet essential interactions, ranging from the hard sphere model to the generalized van der Waals model. Additionally, we propose an alternative approach based on multiparticle correlation functions, allowing for systematic improvements through higher-order correlations. Numerical applications to molecular liquids validate the high fidelity of our approach, and our computational protocols demonstrate that a reduced model with simple energetics can reasonably estimate the thermodynamic entropy of CG models without performing any CG simulations. Overall, our findings present a systematic framework for estimating not only the entropy but also other thermodynamic properties of CG models, relying solely on information from the reference system.

Comment on 'Pseudo hard-sphere viscosities from equilibrium molecular dynamics'

In a recent article, Nicasio-Collazoet al(2023J. Phys.: Condens. Matter35425401) explore the viscosity of the pseudo-hard-sphere (PHS) model. In this comment, we highlight some discrepancies with expected behavior, and compare their results to new simulations of the same model as well as to true hard spheres. In contrast to the results of Nicasio-Collazoet al, our results follow the relation between shear, bulk, and longitudinal viscosity expected for isotropic fluids. Moreover, we observe clear differences in behavior between PHS and true hard sphere, and encourage future hard-sphere studies to focus on the true hard sphere model whenever possible.

Computing Accurate True Self-Diffusion Coefficients and Shear Viscosities Using the OrthoBoXY Approach

In a recent paper [Busch, J.; Paschek, D. J. Phys. Chem. B 2023, 127, 7983-7987], we have shown that for molecular dynamics (MD) simulations using orthorhombic periodic boundary conditions with "magic" box length ratios of Lz/Lx = Lz/Ly = 2.7933596497, the self-diffusion coefficients Dx and Dy in x- and y-directions are independent of the system size. They both represent the true self-diffusion coefficient D0 = (Dx + Dy)/2, while the shear viscosity can be calculated from diffusion coefficients in x-, y-, and z-directions, using η = kBT·8.1711245653/[3πLz(Dx + Dy - 2Dz)]. In this contribution, we test this "OrthoBoXY" approach by its application to a variety of different systems: liquid water, dimethyl ether, methanol, triglyme, water/methanol mixtures, water/triglyme mixtures, and imidazolium-based ionic liquids. The chosen systems range from small-sized molecular liquids to complex mixtures and ionic liquids, while spanning a viscosity range of almost 3 orders of magnitude. We assess the efficiency of the method for computing true self-diffusion and viscosity data and provide simple formulas for estimating the required MD simulation lengths and sizes for delivering reliable data with targeted uncertainty levels. Our analysis of the system size dependence of statistical uncertainties for both the viscosity and the self-diffusion coefficient leads us to the conclusion that it is preferable to extend the simulation length instead of increasing the system size. MD simulations consisting of 768 molecules or ion pairs seem to be perfectly adequate.

System Size Dependence of the Diffusion Coefficients in MD Simulations: A Simple Correction Formula for Pure Dense Fluids

A practical correction formula relating the self-diffusion coefficient of dense liquids from molecular dynamics (MD) simulations with periodic boundary conditions to the self-diffusion coefficient in the thermodynamic limit is discussed. This formula applies to pure dense fluids and has a very simple form D = D0(1 - γN-1/3), where D0 is the self-diffusion coefficient in the thermodynamic limit and N is the number of particles in the simulation. The numerical factor γ is dependent on the geometry of the simulation cell. Remarkably, γ ≃ 1.0 is the most popular cubic geometry. The success of this formula is supported by results from MD simulations, including very recent simulations with a "magic" simulation geometry.

Geometric conjecture about phase transitions

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach the steady state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and ε-mixing time of the configuration spaces of hard-disk and hard-sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems and numerical evidence suggests that a discontinuity in the ε-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.

Departures from perfect isomorph behavior in Lennard-Jones fluids and solids

Isomorphs are lines on a fluid or solid phase diagram along which the microstructure is invariant on affine density scaling of the molecular coordinates. Only inverse power (IP) and hard sphere potential systems are perfectly isomorphic. This work provides new theoretical tools and criteria to determine the extent of deviation from perfect isomorphicity for other pair potentials using the Lennard-Jones (LJ) system as a test case. A simple prescription for predicting isomorphs in the fluid range using the freezing line as a reference is shown to be quite accurate for the LJ system. The shear viscosity and self-diffusion coefficient scale well are calculated using this method, which enables comments on the physical significance of the correlations found previously in the literature to be made. The virial–potential energy fluctuation and the concept of an effective IPL system and exponent, n′, are investigated, particularly with reference to the LJ freezing and melting lines. It is shown that the exponent, n′, converges to the value 12 at a high temperature as ∼ T−1/2, where T is the temperature. Analytic expressions are derived for the density, temperature, and radius derivatives of the radial distribution function along an isomorph that can be used in molecular simulation. The variance of the radial distribution function and radial fluctuation function are shown to be isomorph invariant.

From Atoms to Colloids: Does the Frenkel Line Exist in Discontinuous Potentials?

The Frenkel line has been proposed as a crossover in the fluid region of phase diagrams between a "nonrigid" and a "rigid" fluid. It is generally described as a crossover in the dynamical properties of a material and as such has been described theoretically using a very different set of markers from those with which is it investigated experimentally. In this study, we have performed extensive calculations using two simple yet fundamentally different model systems: hard spheres and square-well potentials. The former has only hardcore repulsion, while the latter also includes a simple model of attraction. We computed and analyzed a series of physical properties used previously in simulations and experimental measurements and discuss critically their correlations and validity as to being able to uniquely and coherently locate the Frenkel line in discontinuous potentials.

Dependence on the thermodynamic state of self-diffusion of pseudo-hard-sphere and Lennard-Jones potentials

Self-diffusion D in a system of particles that interact with a pseudo-hard-sphere or a Lennard-Jones potential is analyzed. Coupling with a solvent is represented by a Langevin thermostat, characterized by the damping time t_{d}. The hypotheses that D=D_{0}φ is proposed, where D_{0} is the small concentration diffusivity and φ is a thermodynamic function that represents the effects of interactions as concentration is increased. Molecular dynamics simulations show that different values of the noise intensity modify D_{0}, but do not have an effect on φ. This result is consistent with the assumption that φ is a thermodynamic function since the thermodynamic state is not altered by the presence of damping and noise.

Bulk viscosity of hard sphere fluids by equilibrium and nonequilibrium molecular dynamics simulations

The bulk viscosity, η_b, of the hard sphere (HS) fluid is computed by equilibrium and nonequilibrium molecular dynamics (NEMD) simulations, the latter using an adaptation of the time-stepping method for continuous potential systems invented by Hoover et al. [Phys. Rev. A 21, 1756 (1980)], which employs an imposed cyclic density variation on the system by affine scaling of the particle coordinates. The time-stepping method employed for HS is validated against exact event-driven hard sphere methodology for a series of equilibrium quantities over a wide density range, including the pressure, singular parts of the hard sphere viscosities, and the nonsingular parts of the shear viscosity time correlation functions. The time steps used are typically only a little smaller than those employed in continuous potential simulations. Exact pressure tensor fluctuation expressions are derived for the singular (or infinite limiting frequency) equilibrium parts of the viscosities, which were employed in the simulations. The values obtained agree well with the predictions of the Enskog theory for all densities considered. The bulk viscosity obtained by NEMD is shown to be noticeably frequency dependent for densities in excess of ∼0.8, decaying approximately exponentially to the Enskog and equilibrium simulation values at all densities considered for frequencies in excess of ∼5 in hard sphere units. Temperature profiles during the cycle and the effects of strain amplitude on the computed frequency dependent bulk viscosity are presented. The bulk viscosity increases with the maximum density amplitude.

Aggregation behavior of nanoparticles: Revisiting the phase diagram of colloids

Surface functionalization of metal nanoparticles (NPs), e.g., using peptides and proteins, has recently attracted a considerable attention in the field of design of therapeutics and diagnostics. The possibility of diverse functionalization allows them to selectively interact with proteins, while the metal core ensures solubility, making them tunable therapeutic agents against diseases due to mis-folding or aggregation. On the other hand, their action is limited by possible self-aggregation, which could be, however, prevented based on the full understanding of their phase diagram as a function of the environmental variables (temperature, ionic strength of the solution, concentration) and intrinsic characteristics (size, charge, amount, and type of functional groups). A common modeling strategy to study the phase behavior is to represent the NPs as spheres interacting via effective potentials implicitly accounting for the solvation effects. Their size put the NPs into the class of colloids, albeit with particularly complex interactions including both attractive and repulsive features, and a consequently complex phase diagram. In this work, we review the studies exploring the phases of these systems starting from those with only attractive or repulsive interactions, displaying a simpler disperse-clustered-aggregated transitions. The phase diagram is here interpreted focusing on the universal aspects, i.e., those dependent on the general feature of the potentials, and available data are organized in a parametric phase diagram. We then consider the potentials with competing attractive short range well and average-long-range repulsive tail, better representing the NPs. Through the proper combination of the attractive only and repulsive only potentials, we are able to interpret the appearance of novel phases, characterized by aggregates with different structural characteristics. We identify the essential parameters that stabilize the disperse phase potentially useful to optimize NP therapeutic activity and indicate how to tune the phase behavior by changing environmental conditions or the NP chemical–physical properties.

Tethered hard spheres: A bridge between the fluid and solid phases

The thermodynamics of hard spheres tethered to a Face-Centered Cubic (FCC) lattice is investigated using event-driven molecular-dynamics. The particle-particle and the particle-tether collision rates are related to the phase space geometry and are used to study the FCC and fluid states. In tethered systems, the entropy can be determined by at least two routes: (i) through integration of the tether collision rates with the tether length $r_T$ or (ii) through integration of the particle-particle collision rates with the hard-sphere diameter $\sigma$ (or, equivalently, the density). If the entropy were an entirely analytic function of $r_T$ and $\sigma$, these two methods for calculating the entropy should lead to the same results; however, a non-analytic region exists as an extension of the solid-fluid phase transition of the untethered hard-sphere system, and integration paths that cross this region will lead to values for the entropy that depend on the particular path chosen. The difference between the calculated entropies appears to be related to the communal entropy, and the location of the non-analytic region appears to be related to conditions where the regions of phase space associated with the FCC configuration become separated from those associated with the disordered fluid. The non-analytic region is finite in extent, vanishing below $r_T/a\approx0.55$, where $a$ is the lattice spacing, and there are many continuous paths that connect the fluid and solid phases that can be used to determine the crystal free energy with respect to the fluid.

Comparing four hard-sphere approximations for the low-temperature WCA melting line

By combining interface-pinning simulations with numerical integration of the Clausius–Clapeyron equation, we accurately determine the melting-line coexistence pressure and fluid/crystal densities of the Weeks–Chandler–Andersen system, covering four decades of temperature. The data are used for comparing the melting-line predictions of the Boltzmann, Andersen–Weeks–Chandler, Barker–Henderson, and Stillinger hard-sphere approximations. The Andersen–Weeks–Chandler and Barker–Henderson theories give the most accurate predictions, and they both work excellently in the zero-temperature limit for which analytical expressions are derived here.

Chemical potential and surface free energy of a hard spherical particle in hard-sphere fluid over the full range of particle diameters

The excess chemical potential $\mu^\mathrm{ex}(\sigma,\eta)$ of a test hard spherical particle of diameter $\sigma$ in a fluid of hard spheres of diameter $\sigma_0$ and packing fraction $\eta$ can be computed with high precision using Widom's particle insertion method [J.~Chem.~Phys.~{\bf 39}, 2808 (1963)] for $\sigma$ between 0 and just larger than 1 and/or small $\eta$. Heyes and Santos [J.~Chem.~Phys.~{\bf 145}, 214504 (2016)] showed analytically that the only polynomial representation of $\mu^\mathrm{ex}$ consistent with the limits of $\sigma$ at zero and infinity has a cubic form. On the other hand, through the solvation free energy relationship between $\mu^\mathrm{ex}$ and the surface free energy $\gamma$ of hard-sphere fluid at a hard spherical wall, we can obtain precise measurements of $\mu^\mathrm{ex}$ for large $\sigma$, extending up to infinity (flat wall) [J.~Chem. Phys.~{\bf 149}, 174706 (2018)]. Within this approach, the cubic polynomial representation is consistent with the assumptions of Morphometric Thermodynamics. In this work, we present measurements of $\mu^\mathrm{ex}$ that combine the two methods to obtain high-precision results for the full range of $\sigma$ values from zero to infinity, which show statistically significant deviations from the cubic polynomial form. We propose an empirical functional form for $\mu^\mathrm{ex}$ dependence on $\sigma$ and $\eta$ which better fits the measurement data while remaining consistent with the analytical limiting behaviour at zero and infinite $\sigma$.

Freezing Temperature and Density Scaling of Transport Coefficients

It is demonstrated that the freezing density scaling of transport coefficients in fluids, similar to the freezing temperature scaling, originates from the quasi-universal excess entropy scaling approach proposed by Rosenfeld. The freezing density scaling has a considerably wider applicability domain on the phase diagram of Lennard-Jones and related systems. As an illustration of its predictive power, we show that it reproduces with an excellent accuracy the shear viscosity coefficients of saturated liquid argon, krypton, xenon, and methane.

Configuration spaces of hard spheres

Hard sphere systems are often used to model simple fluids. The configuration spaces of hard spheres in a three-dimensional torus modulo various symmetry groups are comparatively simple and could provide valuable information about the nature of phase transitions. Specifically, the topological changes in the configuration space as a function of packing fraction have been conjectured to be related to the onset of first-order phase transitions. The critical configurations for 1 to 12 spheres are sampled using a Morse-theoretic approach, and they are available in an online, interactive database. Explicit triangulations are constructed for the configuration spaces of the two sphere system, and their topological and geometric properties are studied. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter as measured by the commute time and diffusion distances. The number of such critical configurations around the packing fraction of the solid-liquid phase transition increases exponentially with the number of spheres, suggesting that the onset of the first-order phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter.

Application of cell models to the melting and sublimation lines of the Lennard-Jones and related potential systems

Harmonic cell models (HCMs) are shown to predict the melting line of the Lennard-Jones (LJ) but not the sublimation line. In addition, even for the melting line, the HCMs are found to be physically unrealistic for inverse power potential systems near the hard-sphere limit, and for the Weeks-Chandler-Andersen system at extremely low temperatures. Despite this, the HCM accurately predicts the LJ mean-square displacement (MSD) from molecular-dynamics (MD) simulations along both lines after simple scaling corrections, to include the effects of anharmonicity and correlated dynamics of the atoms, are applied. Single caged atom molecular dynamics and Monte Carlo simulations provide further quantitative characterization of these additional effects, which go beyond harmonicity. The melting indicator and a modification of the cell model in a similar form are shown to be approximately constant along the melting line, which indicates an isomorph. The less well studied LJ sublimation line is shown not to be an isomorph, yet it still can be represented analytically very accurately by the relationship k_{B}T=aρ^{4}+bρ^{2}, where a and b are constants (k_{B} is Boltzmann's constant, T is the temperature, and ρ is the number density). This relationship has been found previously for the melting line, but the two constants have opposite signs for the sublimation and melting lines. This simple formula is also predicted using a nonharmonic static lattice expression for the pressure. The probability distribution function of the melting factor indicates departures from harmonic or Gaussian behavior in the lower wing. Nevertheless, the mean melting factor is shown to follow a simple MSD Debye-Waller factor dependence along both the melting and sublimation lines. This work combining HCM and MD simulations provides a comparison of the melting and sublimation lines of the LJ system, which could provide the foundations for a more unified statistical mechanical description of these two solid boundary lines.

Excess entropy and Stokes-Einstein relation in simple fluids

The Stokes-Einstein (SE) relation between the self-diffusion and shear viscosity coefficients operates in sufficiently dense liquids not too far from the liquid-solid phase transition. By considering four simple model systems with very different pairwise interaction potentials (Lennard-Jones, Coulomb, Debye-Hückel or screened Coulomb, and the hard sphere limit) we identify where exactly on the respective phase diagrams the SE relation holds. It appears that the reduced excess entropy s_{ex} can be used as a suitable indicator of the validity of the SE relation. In all cases considered the onset of SE relation validity occurs at approximately s_{ex}≲-2. In addition, we demonstrate that the line separating gaslike and liquidlike fluid behaviours on the phase diagram is roughly characterized by s_{ex}≃-1.

Molecular dynamics study of six-dimensional hard hypersphere crystals

Six-dimensional hard hypersphere systems in the A₆, D₆, and E₆ crystalline phases have been studied using event-driven molecular dynamics simulations in periodic, skew cells that reflect the underlying lattices. In all the simulations, the systems had sufficient numbers of hyperspheres to capture the first coordination shells, and the larger simulations also included the complete second coordination shell. The equations of state, for densities spanning the fluid, metastable fluid, and solid regimes, were determined. Using molecular dynamics simulations with the hyperspheres tethered to lattice sites allowed the computation of the free energy for each of the crystal lattices relative to the fluid phase. From these free energies, the fluid-crystal coexistence region was determined for the E₆, D₆, and A₆ lattices. Pair correlation functions for all the examined states were computed. Interestingly, for all the states examined, the pair correlation functions displayed neither a split second peak nor a shoulder in the second peak. These behaviors have been previously used as a signature of the freezing of the fluid phase for hard hyperspheres in two to five dimensions.

Tethered-particle model: The calculation of free energies for hard-sphere systems

Two methods for computing the entropy of hard-sphere systems using a spherical tether model are explored, which allow the efficient use of event-driven molecular-dynamics simulations. An intuitive derivation is given, which relates the rate of particle collisions, either between two particles or between a particle and its respective tether, to an associated hypersurface area, which bounds the system's accessible configurational phase space. Integrating the particle-particle collision rates with respect to the sphere diameter (or, equivalently, density) or the particle-tether collision rates with respect to the tether length then directly determines the volume of accessible phase space and, therefore, the system entropy. The approach is general and can be used for any system composed of particles interacting with discrete potentials in fluid, solid, or glassy states. The entropies calculated for the liquid and crystalline hard-sphere states using these methods are found to agree closely with the current best estimates in the literature, demonstrating the accuracy of the approach.

Thermodynamic stability of hard sphere crystals in dimensions 3 through 10

Although much is known about the metastable liquid branch of hard spheres-from low dimension d up to [Formula: see text]-its crystal counterpart remains largely unexplored for [Formula: see text]. In particular, it is unclear whether the crystal phase is thermodynamically stable in high dimensions and thus whether a mean-field theory of crystals can ever be exact. In order to determine the stability range of hard sphere crystals, their equation of state is here estimated from numerical simulations, and fluid-crystal coexistence conditions are determined using a generalized Frenkel-Ladd scheme to compute absolute crystal free energies. The results show that the crystal phase is stable at least up to [Formula: see text], and the dimensional trends suggest that crystal stability likely persists well beyond that point.

Dynamic Crossover in Fluids: From Hard Spheres to Molecules

We propose a simple and generic definition of a demarcation reconciling structural and dynamic frameworks when combined with the entropy scaling framework. This crossover line between gas- and liquid-like behaviors is defined as the curve for which an individual property, the contribution to viscosity due to molecules' translation, is exactly equal to a collective property, the contribution to viscosity due to molecular interactions. Such a definition is shown to be consistent with the one based on the minima of the kinematic viscosity. For the hard sphere, this is shown to be an exact solution. For Lennard-Jones spheres and dimers and for some simple real fluids, this relation holds very well. This crossover line passes nearby the critical point, and for all studied fluids, it is well captured by the critical excess entropy curve for atomic fluids, emphasizing the link between transport properties and local structure.

Characterization of MIPS in a suspension of repulsive active Brownian particles through dynamical features

We study a two-dimensional system composed by Active Brownian Particles (ABPs), focusing on the onset of Motility Induced Phase Separation (MIPS), by means of molecular dynamics simulations. For a pure hard-disk system with no translational diffusion, the phase diagram would be completely determined by their density and Péclet number. In our model, two additional effects are present: translational noise and the overlap of particles; we study the effects of both in the phase space. As we show, the second effect can be mitigated if we use, instead of the standard Weeks-Chandler-Andersen potential, a stiffer potential: the pseudo-hard sphere potential. Moreover, in determining the boundary of our phase space, we explore different approaches to detect MIPS and conclude that observing dynamical features, via the non-Gaussian parameter, is more efficient than observing structural ones, such as through the local density distribution function. We also demonstrate that the Vogel-Fulcher equation successfully reproduces the decay of the diffusion as a function of density, with the exception of very high densities. Thus, in this regard, the ABP system behaves similar to a fragile glass.

Transport properties of Lennard-Jones fluids: Freezing density scaling along isotherms

It is demonstrated that properly reduced transport coefficients (self-diffusion, shear viscosity, and thermal conductivity) of Lennard-Jones fluids along isotherms exhibit quasi-universal scaling on the density divided by its value at the freezing point. Moreover, this scaling is closely related to the density scaling of transport coefficients of hard-sphere fluids. The Stokes-Einstein relation without the hydrodynamic diameter is valid in the dense fluid regime. The lower density boundary of its validity can serve as a practical demarcation line between gaslike and liquidlike regimes.

Diffusion in dense supercritical methane from quasi-elastic neutron scattering measurements

Methane, the principal component of natural gas, is an important energy source and raw material for chemical reactions. It also plays a significant role in planetary physics, being one of the major constituents of giant planets. Here, we report measurements of the molecular self-diffusion coefficient of dense supercritical CH₄ reaching the freezing pressure. We find that the high-pressure behaviour of the self-diffusion coefficient measured by quasi-elastic neutron scattering at 300 K departs from that expected for a dense fluid of hard spheres and suggests a density-dependent molecular diameter. Breakdown of the Stokes-Einstein-Sutherland relation is observed and the experimental results suggest the existence of another scaling between self-diffusion coefficient D and shear viscosity η, in such a way that Dη/ρ=constant at constant temperature, with ρ the density. These findings underpin the lack of a simple model for dense fluids including the pressure dependence of their transport properties.

Prandtl Number in Classical Hard-Sphere and One-Component Plasma Fluids

The Prandtl number is evaluated for the three-dimensional hard-sphere and one-component plasma fluids, from the dilute weakly coupled regime up to a dense strongly coupled regime near the fluid-solid phase transition. In both cases, numerical values of order unity are obtained. The Prandtl number increases on approaching the freezing point, where it reaches a quasi-universal value for simple dielectric fluids of about ≃1.7. Relations to two-dimensional fluids are briefly discussed.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Integration through transients approach to the μ(I) rheology

This work generalizes the granular integration through transients formalism introduced by Kranz et al. [Phys. Rev. Lett. 121, 148002 (2018)10.1103/PhysRevLett.121.148002] to the determination of the pressure. We focus on the Bagnold regime and provide theoretical support to the empirical μ(I) rheology laws that have been successfully applied in many granular flow problems. In particular, we confirm that the interparticle friction is irrelevant in the regime where the μ(I) laws apply.

A comprehensive study of the thermal conductivity of the hard sphere fluid and solid by molecular dynamics simulation

This work reports a new set of hard sphere (HS) thermal conductivity coefficient, λ, data obtained by Molecular Dynamics (MD) computer simulation, over a density range covering the dilute fluid to near the close-packed solid, and for a large number of particles (up to N = 13 1072) and long simulation times. The N-dependence of the thermal conductivity is shown to be proportional to N-2/3 to a good approximation over a wide range of system sizes, which enabled λ values in the thermodynamic limit to be predicted accurately. The fluid and solid λ can be represented well by the Enskog theory (ET) formula, λE, times a density-dependent correction term, which is close to unity for the fluid and practically constant for the solid. The convergence of the MD λ data back towards ET in the metastable fluid starts just above the freezing density. For the HS solid and dense fluid it was found that the thermal conductivity is nearly linear in pressure, as has been observed experimentally for a number of solids. Simple excess entropy scaling over the higher density fluid phase region was found, and Rosenfeld's exponential relationship can be fitted to the simulation data for the solid to a high degree of accuracy. The simulation analysis has revealed a number of new trends in the behaviour of the HS thermal conductivity which could be useful in building more accurate models for heat conduction in experimental systems.

Performance of the asymptotic expansion method to derive equations of state for hard polyhedron fluids

The asymptotic expansion method is used to derive analytical expressions for the equations of state of 14 hard polyhedron fluids such as cube, octahedron, rhombic dodecahedron, etc., by knowing the values of only the first eight virial coefficients.

Predicting maximally random jammed packing density of non-spherical hard particles via analytical continuation of fluid equation of state

We developed a formalism for accurately predicting the density of MRJ packing state of a wide spectrum of congruent non-spherical hard particles in 3D via analytical fluid EOS.

Elastic properties of dense hard-sphere fluids

A new analysis of elastic properties of dense hard-sphere (HS) fluids is presented, based on the expressions derived by Miller [J. Chem. Phys. 50, 2733 (1969)JCPSA60021-960610.1063/1.1671437]. Important consequences for HS fluids in terms of sound waves propagation, Poisson's ratio, Stokes-Einstein relation, and generalized Cauchy identity are explored. Conventional expressions for high-frequency elastic moduli for simple systems with continuous and differentiable interatomic interaction potentials are known to diverge when approaching the HS repulsive limit. The origin of this divergence is identified here. It is demonstrated that these conventional expressions are only applicable for sufficiently soft interactions and should not be applied to HS systems. The reported results can be of interest in the context of statistical physics, physics of fluids, soft condensed matter, and granular materials.

Equations of the state of hard sphere fluids based on recent accurate virial coefficients B5–B12

A review on the numerical virial coefficients, compressibility factor, fluid–solid phase transition point and equations of the state of hard sphere fluids.