Hard-sphere Fluid Research Articles

Gas-liquid phase transition in a binary mixture with an interaction that creates constant density profiles.

If, in a hard sphere fluid, a single (test) particle is fixed, the other particles display a density profile that possesses long-ranged oscillations. Surprisingly, one can show via classical density functional theory that it takes a simple, purely repulsive (external) potential with a finite range in addition to the fixed hard sphere that forces these oscillations to vanish completely. This can give rise to interesting phenomena; however, it gained little attention in the past. In this work, we use the potential in question as an inter-component interaction in a binary hard-sphere mixture, where it is shown that the effective interaction induced by one component resembles qualitatively the well-known Asakura-Oosawa-Vrij potential and can lead to a liquid-gas phase transition in the other component.

From soft- to hard-sphere fluids: Crossover evidenced by high-frequency elastic moduli.

The conventional (Zwanzig-Mountain) expressions for instantaneous elastic moduli of simple fluids predict their divergence as the limit of hard-sphere (HS) interaction is approached. However, elastic moduli of a true HS fluid are finite. Here we demonstrate that this paradox reveals the soft-to-hard-sphere crossover in fluid excitations and thermodynamics. With extensive in silico study of fluids with repulsive power-law interactions (∝r^{-n}), we locate the crossover at n≃10-20 and develop a simple and accurate model for the HS regime. The results open prospects to deal with the elasticity and related phenomena in various systems, from simple fluids to melts and glasses.

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.

Phase behaviour of a simple fluid confined in a periodic porous material

Using density functional theory for a simple fluid of hard spheres with a square-well attraction, we study the phase behaviour of the fluid confined in triply periodic porous material, where the pore space is bounded by a bicontinuous minimal surface. We combine our numerical results with the morphometric thermodynamics, an ansatz that expresses thermodynamic quantities as a linear combination of four additive geometrical terms, and analyse the dependency of the transition point between a liquid and a gas on the geometrical properties of the confining space. By demonstrating that the morphometric approach can be employed in order to account for the thermodynamic behaviour of a simple fluid in a complex porous material, we make the first step towards a better understanding of experimental observations that can help to characterise the pore space.

Testing the Stokes-Einstein relation with the hard-sphere fluid model.

The Stokes-Einstein (SE) relation has been widely applied to quantitatively describe the Brownian motion. Notwithstanding, here we show that even for a simple fluid, the SE relation may fail over a wide range of the Brownian particle's size. Namely, although the SE relation could be a good approximation for a large enough Brownian particle, a significant error may appear when decreasing the Brownian particle's size down to several hundred times the size of the fluid molecules, and the error increases with the decrease of the Brownian particle's size. The cause is rooted in the fact that the kinetic contribution to the diffusion coefficient is inversely proportional to the squared radius of the Brownian particle. After excluding the kinetic contribution, we show that the applicable range of the SE relation is expanded significantly.

Interacting hard-sphere fluids in an external field.

We present a method for studying equilibrium properties of interacting fluids in an arbitrary external field. The fluid is composed of monodisperse spherical particles with hard-core repulsion and additional interactions of arbitrary shape and limited range. Our method of analysis is exact in one dimension and provides demonstrably good approximations in higher dimensions. It can cope with homogeneous and inhomogeneous environments. We derive an equation for the pair distribution function. The solution, to be evaluated numerically, in general, or analytically for special cases, enters expressions for the entropy and free energy functionals. For some one-dimensional systems, our approach yields analytic solutions, reproducing available exact results from different approaches.

Thermodynamic Properties of the Parabolic-Well Fluid

The thermodynamic properties of the parabolic-well fluid are considered. The intermolecular interaction potential of this model, which belongs to the class of the so-called van Hove potentials, shares with the square-well and the triangular well potentials the inclusion of a hard-core and an attractive well of relatively short range. The analytic second virial coefficient for this fluid is computed explicitly and an equation of state is derived with the aid of the second-order thermodynamic perturbation theory in the macroscopic compressibility approximation and taking the hard-sphere fluid as the reference system. For this latter, the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation of state as derived within the rational function approximation method, is employed. The results for the reduced pressure of the parabolic-well fluid as a function of the packing fraction and two values of the range of the parabolic-well potential at different temperatures are compared with Monte Carlo and Event‐driven molecular dynamics simulation data. Estimates of the values of the critical temperature are also provided.

Carnahan-Starling type equations of state for stable hard disk and hard sphere fluids

The well-known Carnahan-Starling (CS) equation of state (EoS) [N.F. Carnahan and K.E. Starling. J. Chem. Phys. 51 (2), 635–636 (1969). doi:10.1063/1.1672048] for hard sphere (HS) fluid was derived from a quadratic relationship between the integer portions of the virial coefficients, , and their orders, . In this paper, the method is extended to cover the full virial coefficients, , for the general D-dimensional case. We propose a (D-1)th order polynomial for the virial coefficients starting from the 4th order and EoS’s are derived from it. For the hard rod (D = 1) case, the exact solution is obtained. For the stable hard disk fluid (D = 2), the most recent virial coefficients, up to the 10th order, [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0] and accurate compressibility data [J. Kolafa and M. Rottner. Mol. Phys. 104 (22–24), 3435–3441 (2006). doi:10.1080/00268970600967963; J.J. Erpenbeck and M. Luban. Phys. Rev. A. 32 (5), 2920–2922 (1985). doi:10.1103/PhysRevA.32.2920] are employed to construct and to test the EoS. For the stable hard sphere (D = 3) fluid, a new CS-type EoS is obtained by using the most recent virial coefficients [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0; A.J. Schultz and D.A. Kofke. Physical Review E. 90 (2) (2014). doi:10.1103/PhysRevE.90.023301], up to the 11th order, along with highly-accurate compressibility data [S. Pieprzyk et al. Phys. Chem. Chem. Phys. 21 (13), 6886–6899 (2019). doi:10.1039/C9CP00903E; M.N. Bannerman et al. J. Chem. Phys. 132 (8), 084507 (2010). doi:10.1063/1.3328823; J. Kolafa, et al. Phys. Chem. Chem. Phys. 6 (9), 2335–2340 (2004). doi:10.1039/B402792B]. The simple new EoS’s prove to be as accurate as the Padé approximations based on all available virial coefficients, which significantly improve the accuracy of the CS-type EoS in the hard sphere case. This research also reveals that as long as the virial coefficients obey a polynomial function, any EoS derived from it will diverge at the non-physical packing fraction, .

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.

Free volume power law for transport properties of hard sphere fluid

This paper presents a study on the relationship between transport properties and geometric free volume for a hard sphere (HS) system in a dense fluid region. First, a generic free volume distribution function is proposed based on recent simulation results on the HS geometric free volume by Maiti and Sastry [J. Chem. Phys. 141(4), 044510 (2014)] and Maiti et al. [Eur. Phys. J. E 36(1), 5 (2013)]. Combining the new distribution function with a local particle transportation model, we obtain a power law for the HS transport properties. Then, a relation between the geometric free volume and thermodynamic free volume is established, which makes it possible to use well-developed equations of state (EoS) for the expressions of the geometric free volume. The new power law models are tested with molecular dynamic simulation results for HS viscosity, diffusivity and thermal conductivity, respectively, and the results are very satisfactory. Moreover, using the power law, we are able to reproduce several equations obtained from different approaches, such as the entropy scaling laws [Bell et al., J. Phys. Chem. B 123(29), 6345–6363 (2019]), mode coupling theory [Barrat et al., J. Phys. Condens. Matter 1, 7163–7170 (1989)], or empirical correlations [Sigurgeirsson and Heyes, J. Mol. Phys. 101(3), 469–482 (2003)]. In particular, a long-standing controversy regarding the well-known Cohen–Turnbull–Doolittle free volume model [Cohen and Turnbull, J. Chem. Phys. 31(3), 1164–1169 (1959); Doolittle, J. Appl. Phys. 22(12), 1471–1475 (1951)] is resolved by using the power law combined with the Heyes and Woodcock EoS [Heyes and Woodcock, Mol. Phys. 59(6), 1369–1388 (1986)].

Density functional study of non-isothermal hard sphere fluids

ABSTRACT Using a version of dynamical density functional theory, we explore a microscopic structure of hard-sphere fluids in the presence of a temperature gradient. When combined with the assumption of local equilibrium, this approach predicts the density profile in confinement and in bulk both in very good overall agreement with the results from the reverse non-equilibrium molecular dynamics simulation, which we used to impose a temperature gradient. Thus, the assumption of local equilibrium is found to be surprisingly accurate down to a microscopic scale even under a large temperature gradient. An oscillatory density profile, indicating the layering of particles, was observed in bulk as well as in confinement. This behaviour in the latter is well known and results from packing of hard spheres next to a wall. In the former, the oscillation is seen to emanate from the point of sharp change in temperature. However, our theoretical predictions greatly exaggerate the amplitude of oscillation when the temperature exhibits a sharp change over a very small distance.

Clustering effects on the diffusion of patchy colloids in disordered porous media

Enskog theory is extended for the description of the self-diffusion coefficient of patchy colloidal fluid in disordered porous media. The theory includes the contact values of fluid-fluid and fluid-matrix pair distribution functions that are modified to include the dependence from the so-called probe particle porosity, φ, in order to correctly describe the effects of trapping the fluid particles by a matrix. The proposed expressions for the modified contact values of fluid-fluid and fluid-matrix pair distribution functions include three terms. Namely, a hard sphere contribution obtained by us in the previous work [Holovko M. F., Korvatska M. Ya., Condens. Matter Phys., 2020, 23, 23605], the depletion contribution connected with the cluster-cluster and cluster-matrix repulsion and the intramolecular correlation inside the cluster. It is shown that the last term leads to a remarkable decrease of the self-diffusion coefficient at a low fluid density. With a decreasing matrix porosity, this effect becomes weaker. For intermediate fluid densities, the depletion contribution leads to an increase of the self-diffusion coefficient in comparison with the hard sphere fluid. For a sufficiently dense fluid, the self-diffusion coefficient strongly decreases due to a hard sphere effect. The influence of the cluster size and the type of clusters as well as of the parameters of porous media is investigated and discussed in detail.

Interpretation of the pressure-induced Raman frequency shift of the ν1 stretching bands of CH4 and N2 within CH4-CO2, N2-CO2 and CH4-N2 binary mixtures.

The relationships between the frequency shift of the ν1 stretching bands of CH4 and N2 with pressure (or density) and composition have been previously provided in the literature as reliable parameters for accurate empirical barometers and densimeters for the direct determination of the pressure or density of gas mixtures. However, the latter results still remain a pure description of the experimental data without any interpretation of the physical mechanisms hidden behind the variation trend of the observed peak position. The present paper is devoted to interpreting the origin of the pressure-induced vibrational frequency shifts of the ν1 stretching bands of CH4 and N2 within CH4-CO2, N2-CO2 and CH4-N2 binary mixtures at the molecular level. Two different theoretical models (i.e., the Lennard-Jones 6-12 potential approximation - LJ, and the generalized perturbed hard-sphere fluid - PHF) are used to intuitively and qualitatively assess the variation trend as well as the magnitude of the frequency shift of the CH4 and N2ν1 bands for an in-depth understanding. Thereby, the contribution of the attractive and repulsive solvation-mean forces to the variation of the Raman frequency shift as a function of pressure and composition is assessed. A predictive model of the variation trend of the frequency shift of the CH4ν1 band as a function of pressure (up to 3000 bars), density and composition within CH4-N2 and CH4-CO2 binary mixtures is then provided.

The role of collective elasticity on activated structural relaxation, yielding, and steady state flow in hard sphere fluids and colloidal suspensions under strong deformation.

We theoretically study the effect of external deformation on activated structural relaxation and aspects of the nonlinear mechanical response of glassy hard sphere fluids in the context of elastically collective nonlinear Langevin equation theory. This microscopic force-based approach describes activated relaxation as a coupled local-nonlocal event involving caging and longer range collective elasticity, with the latter becoming more important and ultimately dominant with increasing packing fraction under equilibrium conditions. The central new question we address is how this physical picture of activated relaxation, and the relative importance of local caging vs collective elasticity physics, depends on external deformation. Theoretical predictions are presented for deformation-induced enhancement of mobility, the onset of relaxation speed up at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady state structural relaxation time and viscosity, a non-vanishing activation barrier in the shear thinning regime, an apparent Herschel-Bulkley form of the rate dependence of the steady state shear stress, exponential growth of different measures of a dynamic yield or flow stress with the packing fraction, and reduced fragility and dynamic heterogeneity under deformation. The results are contrasted with experiments and simulations, and qualitative or better agreement is found. An overarching conclusion is that deformation strongly reduces the importance of longer range collective elastic effects relative to the local caging aspect for most, but not all, physical questions, with deformation-dependent fragility and dynamic heterogeneity phenomena being qualitatively sensitive to collective elasticity. Overall, nonlinear rheology is predicted to be a more local problem than quiescent structural relaxation, albeit with deformation-modified activated processes still important.

Determination of Young's Modulus of Active Pharmaceutical Ingredients by Relaxation Dynamics at Elevated Pressures.

A new approach is theoretically proposed to study the glass transition of active pharmaceutical ingredients and a glass-forming anisotropic molecular liquid at high pressures. We describe amorphous materials as a fluid of hard spheres. Effects of nearest neighbor interactions and cooperative motions of particles on glassy dynamics are quantified through a local and collective elastic barrier calculated using the elastically collective nonlinear Langevin equation theory. Inserting two barriers into Kramer's theory gives the structural relaxation time. Then, we formulate a new mapping based on the thermal expansion process under pressure to intercorrelate particle density, temperature, and pressure. This analysis allows us to determine the pressure and temperature dependence of α relaxation. From this, we estimate the effective elastic modulus of amorphous materials and capture the effects of conformation on the relaxation process. Remarkably, our theoretical results agree well with experiments.

Equation of state of hard-sphere fluid in nanoporous media

The present work is devoted to the study of fluid equations of state in nanoporous media. The modelling is carried out in the framework of the molecular dynamics method using the potential of hard spheres. The dependences of the compressibility factor on packing fraction are studied for various characteristic pore sizes, porosities, and types of grains packing of a porous medium. It is shown that for all cases considered, deviations of the obtained equations of state from the Carnahan-Starling equation of state are insignificant. These deviations may be due to the presence of restricted zones in a porous medium inaccessible to fluid molecules that is also discussed in the work.

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.

Dynamic properties of quasi-confined colloidal hard-sphere liquids near the glass transition

The complex behavior of confined fluids arising due to a competition between layering and local packing can be disentangled by considering quasi-confined liquids, where periodic boundary conditions along the confining direction restore translational invariance. This system provides a means to investigate the interplay of the relevant length scales of the confinement and the local order. We provide a mode-coupling theory of the glass transition (MCT) for quasi-confined liquids and elaborate an efficient method for the numerical implementation. The nonergodicity parameters in MCT are compared to computer-simulation results for a hard-sphere fluid. We evaluate the nonequilibrium-state diagram and investigate the collective intermediate scattering function. For both methods, nonmonotonic behavior depending on the confinement length is observed.

Confinement-induced demixing and crystallization

We simulate a strongly size-disperse hard-sphere fluid confined between two parallel, hard walls. We find that confinement induces crystallization into n-layered hexagonal lattices and a novel honeycomb-shaped structure, facilitated by fractionation. The onset of freezing prevents the formation of a stable glass phase and occurs at much smaller packing fraction than in bulk. Varying the wall separation triggers solid-to-solid transitions and a systematic change of the size-distribution of crystalline particles, which we rationalize using a semi-quantitative theory. We show that the crystallization can be exploited in a wedge geometry to demix particles of different sizes.