Hard-sphere Fluid Research Articles

Contribution to the Statistical Mechanics of Static Triplet Correlations and Structures in Fluids with Quantum Spinless Behavior

The current developments in the theory of quantum static triplet correlations and their associated structures (real r-space and Fourier k-space) in monatomic fluids are reviewed. The main framework utilized is Feynman’s path integral formalism (PI), and the issues addressed cover quantum diffraction effects and zero-spin bosonic exchange. The structures are associated with the external weak fields that reveal their nature, and due attention is paid to the underlying pair-level structures. Without the pair, level one cannot fully grasp the triplet extensions in the hierarchical ladder of structures, as both the pair and the triplet structures are essential ingredients in the triplet response functions. Three general classes of PI structures do arise: centroid, total continuous linear response, and instantaneous. Use of functional differentiation techniques is widely made, and, as a bonus, this leads to the identification of an exact extension of the “classical isomorphism” when the centroid structures are considered. In this connection, the direct correlation functions, as borrowed from classical statistical mechanics, play a key role (either exact or approximate) in the corresponding quantum applications. Additionally, as an auxiliary framework, the traditional closure schemes for triplets are also discussed, owing to their potential usefulness for rationalizing PI triplet results. To illustrate some basic concepts, new numerical calculations (path integral Monte Carlo PIMC and closures) are reported. They are focused on the purely diffraction regime and deal with supercritical helium-3 and the quantum hard-sphere fluid.

Long ranged stress correlations in the hard sphere liquid.

The smooth emergence of shear elasticity is a hallmark of the liquid to glass transition. In a liquid, viscous stresses arise from local structural rearrangements. In the solid, Eshelby has shown that stresses around an inclusion decay as a power law r-D, where D is the dimension of the system. We study glass-forming hard sphere fluids by simulation and observe the emergence of the unscreened power-law Eshelby pattern in the stress correlations of the isotropic liquid state. By a detailed tensorial analysis, we show that the fluctuating force field, viz., the divergence of the stress field, relaxes to zero with time in all states, while the shear stress correlations develop spatial power-law structures inside regions that grow with longitudinal and transverse sound propagation. We observe the predicted exponents r-D and r-D-2. In Brownian systems, shear stresses relax diffusively within these regions, with the diffusion coefficient determined by the shear modulus and the friction coefficient.

Density functional theory for responsive hard-sphere fluids

We investigate the equilibrium properties of responsive hard-sphere fluids, where the particle size is a coarse-grained property that fluctuates within an internal free-energy landscape, responding to changes in concentration or the application of external fields. For this purpose, we employ a generalised density functional theory for responsive colloids based on the fundamental measure theory for polydisperse hard-sphere mixtures. We find that increasing particle softness yields a substantial reduction in mean particle size, volume fraction and pressure. We also examine the density profiles and size segregation of responsive hard-sphere fluids subjected to three representative external potentials: a planar hard wall, gravitational and Archimedes buoyant fields. We observe that increasing stiffness or particle concentration enhances density oscillations, converging to the behaviour of a monodisperse fluid. In a gravitational field, size segregation occurs, with smaller particles accumulating at the bottom due to sedimentation. This effect is more pronounced when the Archimedes force is included, causing larger colloids to accumulate at the top, leading to density oscillations in this region that intensify with particle softness, an effect not observed in monodisperse systems. Finally, we demonstrate that responsiveness leads to distinct behaviour compared to conventional polydisperse non-responsive fluids, in which particle compression is not allowed.

Theoretical analysis of the structure, thermodynamics, and shear elasticity of deeply metastable hard sphere fluids.

The structure, thermodynamics, and slow activated dynamics of the equilibrated metastable regime of glass-forming fluids remain a poorly understood problem of high theoretical and experimental interest. We apply a highly accurate microscopic equilibrium liquid state integral equation theory, in conjunction with naïve mode coupling theory of particle localization, to study in a unified manner the structural correlations, thermodynamic properties, and dynamic elastic shear modulus in deeply metastable hard sphere fluids. Distinctive behaviors are predicted including divergent inverse critical power laws for the contact value of the pair correlation function, pressure, and inverse dimensionless compressibility, and a splitting of the second peak and large suppression of interstitial configurations of the pair correlation function. The dynamic elastic modulus is predicted to exhibit two distinct exponential growth regimes with packing fraction that have strongly different slopes. These thermodynamic, structural, and elastic modulus results are consistent with simulations and experiments. Perhaps most unexpectedly, connections between the amplitude of long wavelength density fluctuations, dimensionless compressibility, local structure, and the dynamic elastic shear modulus have been theoretically elucidated. These connections are more broadly relevant to understanding the slow activated relaxation and mechanical response of colloidal suspensions in the ultradense metastable region and deeply supercooled thermal liquids in equilibrium.

Hole statistics of equilibrium 2D and 3D hard-sphere crystals.

The probability of finding a spherical "hole" of a given radius r contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions GV(r), have been well studied for hard-sphere fluids in d-dimensional Euclidean space Rd. However, little is known about these functions for hard-sphere crystals for values of r beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of r that far extend those that could be previously explored. We discover that GV(r) in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to GV(r) that monotonically increases with r for fluid states. The oscillations in GV(r) for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in GV(r) for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of GV(r) as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution f(ϕl) of Delaunay cells and find that, for d ≤ 3, the excess kurtosis of f(ϕl) switches signat a certain transitional global packing fraction. Our accurate methods to access hole statistics in hard-sphere crystals at the challenging intermediate length scales reported here can be applied to understand the important problem of solvation and hydrophobicity in water at such length scales.

Thermodynamic properties of quasi-one-dimensional fluids.

We calculate thermodynamic and structural quantities of a fluid of hard spheres of diameter σ in a quasi-one-dimensional pore with accessible pore width W smaller than σ by applying a perturbative method worked out earlier for a confined fluid in a slit pore [Franosch et al. Phys. Rev. Lett. 109, 240601 (2012)]. In a first step, we prove that the thermodynamic and a certain class of structural quantities of the hard-sphere fluid in the pore can be obtained from a purely one-dimensional fluid of rods of length σ with a central hard core of size σW=σ2-W2 and a soft part at both ends of length (σ - σW)/2. These rods interact via effective k-body potentials veff(k) (k ≥ 2). The two- and the three-body potential will be calculated explicitly. In a second step, the free energy of this effective one-dimensional fluid is calculated up to leading order in (W/σ)2. Explicit results for, e.g., the perpendicular pressure, surface tension, and the density profile as a function of density, temperature, and pore width are presented and partly compared with results from Monte-Carlo simulations and standard virial expansions. Despite the perturbative character of our approach, it encompasses the singularity of the thermodynamic quantities at the jamming transition point.

Semiclassical thermodynamic geometry.

In this work the thermodynamic geometry (TG) of semiclassical fluids is analyzed. We present results for two models. The first one is a semiclassical hard-sphere (SCHS) fluid whose Helmholtz free energy is obtained from path-integral Monte Carlo simulations. It is found that, due to quantum contributions in the thermodynamic potential, the anomaly found in TG for the classical hard-sphere fluid related to the sign of the scalar curvature is now avoided in a considerable region of the thermodynamic space. The second model is a semiclassical square-well fluid, described by a SCHS repulsive interaction coupled with a classical attractive square-well contribution. The behavior of the semiclassical curvature scalar as a function of the thermal de Broglie wavelength λ_{B} is analyzed for several attractive-potential ranges. A description of the semiclassical R Widom lines, defined by the maxima of the curvature scalar, is also obtained and results are compared with the corresponding classical systems for different square-well ranges.

Spatial distribution of reduced density of hard spheres near a hard-sphere dimer: Results from three-dimensional Ornstein–Zernike equations coupled with several different closures and from grand canonical Monte Carlo simulation

We calculated the spatial distribution function, which is the one-body reduced density distribution of solvent particles around a nonspherical solute, using the three-dimensional Ornstein–Zernike equations coupled with closures. The solvent was a hard-sphere fluid, and the contact dimer of the solvent particles was the nonspherical solute. Two traditional closures, the Percus–Yevick and hypernetted-chain approximations, and three closures with bridge functions (BFs) were examined. These spatial distribution functions were compared with the results obtained by grand canonical Monte Carlo (GCMC) simulations. Three closures with BFs, such as the modified hypernetted-chain (MHNC) closure using a bridge function proposed by Kinoshita, gave precise spatial distribution functions. These results were much more accurate than those obtained by the traditional closures. The deviations from the spatial distribution function obtained by the GCMC simulation appeared only near the concave surface of the solute dimer. However, the deviations were minor compared with those between the simulation and the predictions of the two traditional closures. By contrast, the three-body correlation functions for the closures with BFs were not more accurate than those obtained by the traditional closures.

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.

Transport properties of polydisperse hard-sphere fluid: effect of distribution shape and mass scaling

Transport properties of polydisperse hard-sphere fluid: effect of distribution shape and mass scaling

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.

Density and confinement effects on fluid velocity slip

Understanding the velocity slip of fluids in channels of molecular length scales is essential for accurately manipulating their flow in engineering applications, such as in water desalination or, by reversing the process, osmotic power generation. This study addresses two major open questions, namely (a) the range of validity of the Navier-Stokes equations with slip boundary conditions in describing fluid flows within channels of molecular confinement, and (b) the effect of fluid density and confinement as well as surface properties, such as microscopic roughness and curvature, on fluid slippage along walls. Using a hard-sphere fluid model and describing the fluid-wall interactions by the Maxwell scattering kernel, an event-driven molecular dynamics simulation study was performed in planar and cylindrical channels of different sizes and accommodation coefficients with varying fluid densities. The results show that the well-known criterion for the slip solution, which is valid in rarefied conditions, also holds for a dense gas, i.e., the predictive accuracy improves with decreasing Knudsen numbers, as long as the channel size is larger than about ten molecular diameters. We also find that the key quantity influencing the friction coefficient of a fluid is the peak density at the walls, rather than the nominal density, the fluid confinement, or the channel curvature, as found in previous studies. In particular, higher nominal densities lead to higher density peaks, and therefore a decrease in slip, while tighter confinements lead to lower density peaks, and therefore an increase in slip, regardless of channel geometry. Furthermore, the velocity slip depends on the microscopic roughness via the Smoluchowski factor. These findings are a stepping stone towards a deeper understanding of the molecular mechanisms underlying fluid velocity slip in molecular-scale channels. Published by the American Physical Society 2024

Effect of the nature of the solid substrate on spatially heterogeneous activated dynamics in glass forming supported films.

We extend the force-level elastically collective nonlinear Langevin equation theory to treat the spatial gradients of the alpha relaxation time and glass transition temperature, and the corresponding film-averaged quantities, to the geometrically asymmetric case of finite thickness supported films with variable fluid-substrate coupling. The latter typically nonuniversally slows down motion near the solid-liquid interface as modeled via modification of the surface dynamic free energy caging constraints that are spatially transferred into the film and which compete with the accelerated relaxation gradient induced by the vapor interface. Quantitative applications to the foundational hard sphere fluid and a polymer melt are presented. The strength of the effective fluid-substrate coupling has very large consequences for the dynamical gradients and film-averaged quantities in a film thickness and thermodynamic state dependent manner. The interference of the dynamical gradients of opposite nature emanating from the vapor and solid interfaces is determined, including the conditions for the disappearance of a bulk-like region in the film center. The relative importance of surface-induced modification of local caging vs the generic truncation of the long range collective elastic component of the activation barrier is studied. The conditions for the accuracy and failure of a simple superposition approximation for dynamical gradients in thin films are also determined. The emergence of near substrate dead layers, large gradient effects on film-averaged response functions, and a weak non-monotonic evolution of dynamic gradients in thick and cold films are briefly discussed. The connection of our theoretical results to simulations and experiments is briefly discussed, as is the extension to treat more complex glass-forming systems under nanoconfinement.

Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids

Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression which provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on an approach proposed for the 3D RDF. The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure and the isothermal compressibility constraints are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for the 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results.

Radial Distribution Function of Hard Spheres at Different Densities

The model of hard sphere system has important part in modern theories of liquids. Hard sphere fluid is considered as the most intensively studied system among all model fluids. The thermodynamic and structural properties of real fluids can be studied with the help of hard spheres fluid. The exact solution of Ornstein-Zernike (OZ) equation with Percus-Yevick (PY) along with the equation of state provides an approximate analytical expression for radial distribution function (RDF). This work intends to study the radial distribution function of Hard sphere (HS) using the PY approximation for different densities ranging from low to high to find out their structural properties, packing behavior, phase transitions, and thermodynamics. We use FORTRAN program for this purpose. At low densities, the RDF has a single peak at a distance corresponding to the hard sphere diameter, indicating that the particles are well- separated and not interacting with each other like gaseous. As density increases, the peak becomes broader and shifts to smaller distances, indicating that the particles are coming into closer contact and interacting more strongly with each other. The height of the peak also increases, indicating that there is a greater probability of finding a particle at a certain distance from another particle.

Calculation of the entropy for hard-sphere from integral equation method

The Ornstein-Zernike integral equation method has been employed for a single-component hard sphere fluid in terms of the Percus-Yevick (PY) and Martynov-Sarkisov (MS) approximations. The virial equation of state, excess chemical potential, and free energy have been computed in both approximations. The excess entropy has been obtained with both thermodynamic relation and free energy expression for reduced densities of 0.1 to 0.9. It has been shown that the entropy values from thermodynamic relation in the MS approximation are better than those from the PY approximation, especially for high densities, and presents a reasonable comparison with available data in the literature, while the values from free energy expression in both approximations are not only close to each other but also comparable to accurate ones.

Neural functional theory for inhomogeneous fluids: Fundamentals and applications.

We present a hybrid scheme based on classical density functional theory and machine learning for determining the equilibrium structure and thermodynamics of inhomogeneous fluids. The exact functional map from the density profile to the one-body direct correlation function is represented locally by a deep neural network. We substantiate the general framework for the hard sphere fluid and use grand canonical Monte Carlo simulation data of systems in randomized external environments during training and as reference. Functional calculus is implemented on the basis of the neural network to access higher-order correlation functions via automatic differentiation and the free energy via functional line integration. Thermal Noether sum rules are validated explicitly. We demonstrate the use of the neural functional in the self-consistent calculation of density profiles. The results outperform those from state-of-the-art fundamental measure density functional theory. The low cost of solving an associated Euler-Lagrange equation allows to bridge the gap from the system size of the original training data to macroscopic predictions upon maintaining near-simulation microscopic precision. These results establish the machine learning of functionals as an effective tool in the multiscale description of soft matter.

On the Heat Capacity of Quantum Hard Sphere Fluid

On the Heat Capacity of Quantum Hard Sphere Fluid

Analysis of probability of inserting a hard spherical particle with small diameter in hard-sphere fluid

The probability of inserting, without overlap, a hard spherical particle of diameter σ in a hard-sphere fluid of diameter σ0 and packing fraction η determines its excess chemical potential at infinite dilution, μex(σ, η). In our previous work [R. L. Davidchack and B. B. Laird, J. Chem. Phys. 157, 074701 (2022)], we used Widom’s particle insertion method within molecular dynamics simulations to obtain high precision results for μex(σ, η) with σ/σ0 ≤ 4 and η ≤ 0.5. In the current work, we investigate the behavior of this quantity at small σ. In particular, using the inclusion-exclusion principle, we relate the insertion probability to the hard-sphere fluid distribution functions and thus derive the higher-order terms in the Taylor expansion of μex(σ, η) at σ = 0. We also use direct evaluation of the excluded volume for pairs and triplets of hard spheres to obtain simulation results for μex(σ, η) at σ/σ0 ≤ 0.2247 that are of much higher precision than those obtained earlier with Widom’s method. These results allow us to improve the quality of the small-σ correction in the empirical expression for μex(σ, η) presented in our previous work.

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.