Soft-sphere Fluids Research Articles

Quasiuniversal behavior of shear relaxation times in simple fluids.

We calculate the shear relaxation times in four important simple monatomic model fluids: Lennard-Jones, Yukawa, soft-sphere, and hard-sphere fluids. It is observed that in properly reduced units, the shear relaxation times exhibit quasiuniversal behavior when the density increases from the gaslike low values to the high-density regime near crystallization. They first decrease with density at low densities, reach minima at moderate densities, and then increase toward the freezing point. The reduced relaxation times at the minima and at the fluid-solid phase transition are all comparable for the various systems investigated, despite more than ten orders of magnitude difference in real systems. Important implications of these results are discussed.

Noether invariance theory for the equilibrium force structure of soft matter

We give details and derivations for the Noether invariance theory that characterizes the spatial equilibrium structure of inhomogeneous classical many-body systems, as recently proposed and investigated for bulk systems (Sammüller et al 2023 Phys. Rev. Lett. 130 268203). Thereby an intrinsic thermal symmetry against a local shifting transformation on phase space is exploited on the basis of the Noether theorem for invariant variations. We consider the consequences of the shifting that emerge at second order in the displacement field that parameterizes the transformation. In a natural way the standard two-body density distribution is generated. Its second spatial derivative is thereby balanced by two further and different two-body correlation functions, which respectively introduce thermally averaged force correlations and force gradients in a systematic and microscopically sharp way into the framework. Separate exact self and distinct sum rules express this balance. We exemplify the validity of the theory on the basis of computer simulations for the Lennard–Jones gas, liquid, and crystal, the Weeks–Chandler–Andersen fluid, monatomic Molinero–Moore water at ambient conditions, a three-body-interacting colloidal gel former, the Yukawa and soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay–Berne particles. We describe explicitly the derivation of the sum rules based on Noether’s theorem and also give more elementary proofs based on partial phase space integration following Yvon’s theorem.

Stress fluctuations and adiabatic speed of sound in liquids: a simple way to estimate it from ab initio simulations

One of the fundamental quantities in dynamics of the liquid state, the adiabatic speed of sound c_s, is extremely difficult to predict from computer simulations, especially in ab initio simulations. Here we derive an expression for the instantaneous correlator of fluctuations of longitudinal component of stress tensor, which contains c_s along with others quantities easy accessible via classical and ab initio computer simulations. We show that the proposed methodology works well in the case of Lennard-Jones and soft-sphere simple fluids, Kr–Ar liquid mixture in connection with simulations with effective pair interactions as well as for liquid Sb, fluid Hg and molten NaCl from ab initio simulations.

Continuous nonequilibrium transition driven by heat flow

We discovered an out-of-equilibrium transition in the ideal gas between two walls, divided by an inner, adiabatic, movable wall. The system is driven out-of-equilibrium by supplying energy directly into the volume of the gas. At critical heat flux we have found a continuous transition to the state with a low-density, hot gas on one side of the movable wall and a dense, cold gas on the other side. Molecular dynamic simulations of the soft-sphere fluid confirm the existence of the transition in the interacting system. We introduce a stationary state Helmholtz-like function whose minimum determines the stable positions of the internal wall. This transition can be used as a paradigm of transitions in stationary states and the Helmholtz-like function as a paradigm of the thermodynamic description of these states.

Topological extension of the isomorph theory based on the Shannon entropy.

Isomorph theory is one of the promising theories for understanding the quasiuniversal relationship between thermodynamic, dynamic, and structural characteristics. Based on the hidden scale invariance of the inverse power law potentials, it rationalizes the excess entropy scaling law of dynamic properties. This work aims to show that this basic idea of isomorph theory can be extended by examining the microstructural features of the system. Using the topological framework in conjunction with the entropy calculation algorithm, we demonstrate that Voronoi entropy, a measure of the topological diversity of single atoms, provides a scaling law for the transport properties of soft-sphere fluids, which is comparable to the frequently used excess entropy scaling. By examining the relationship between the Voronoi entropy and the solidlike fraction of simple fluids, we suggest that the Frenkel line, a rigid-nonrigid crossover line, be a topological isomorphic line at which the scaling relation qualitatively changes.

Topological generalization of the rigid-nonrigid transition in soft-sphere and hard-sphere fluids.

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 in our earlier work is applied to soft-sphere and hard-sphere fluids. The dependence of the classification result on the bulk density is understood based on the theory of fluid polyamorphism. The percolation behavior of solid-like clusters is described based on the fraction of solid-like molecules in an integrated manner. The crossover densities are obtained by examining the percolation of solid-like clusters. The resultant crossover densities of soft-sphere fluids converge to that of hard-sphere fluid. Hence the topological method successfully highlights the generality of the Frenkel line.

Liquid-like and gas-like features of a simple fluid: An insight from theory and simulation

A soft-sphere system is one of the simplest and most studied models of many-body problems. In contrast to real substances, it does not have a cohesive state and boiling line. For this reason, the fluid phase has been considered to be the same everywhere on the phase diagram with no difference between the gas and liquid phase. Here, we show that the soft-sphere fluid can in fact be demarcated into two states with liquid-like and gas-like properties even though these states are not separated by a thermodynamic phase transition. We find that particle dynamics and excitation spectra are qualitatively different in the two states. This difference applies to other system properties such as viscosity, diffusion and speed of sound. We consider different repulsion exponents of the soft-sphere system as well as the limiting case of the hard-sphere system and comment on their implications for demarcating the fluid phase into liquid-like and gas-like states.

Thermodynamic curvature of soft-sphere fluids and solids.

The influence of the strength of repulsion between particles on the thermodynamic curvature scalar R for the fluid and solid states is investigated for particles interacting with the inverse power (r^{-n}) potential, where r is the pair separation and 1/n is the softness. Exact results are obtained for R in certain limiting cases, and the R behavior determined for the systems in the fluid and solid phases. It is found that in such systems the thermodynamic curvature can be positive for very soft particles, negative for steeply repulsive (or large n) particles across almost the entire density range, and can change sign between negative and positive at a certain density. The relationship between R and the form of the interaction potential is more complex than previously suggested, and it may be that R is an indicator of the relative importance of energy and entropy contributions to the thermodynamic properties of the system.

A geometrical criterion for glass transition in soft-sphere fluids.

As glass-forming fluids become colder and denser, structural rearrangements become slow and eventually cease. For hard-sphere fluids, percolation of particles unable to change neighbors (T1-inactive particles) signals the glass transition. To investigate this geometrical criterion for mobility in soft-sphere systems, we simulate monodisperse fluids interacting with a generalized Weeks-Chandler-Andersen (WCA) potential in metastable equilibrium, using our previously developed crystal-avoiding method. We find that the vanishing diffusivity as the glass transition is approached can be described by a power law below the onset temperature of super-Arrhenius behavior. By mapping the soft spheres to hard spheres based on mean collision energy, we find that the diffusivity versus effective volume fraction curves collapse onto the hard-sphere curve for all systems studied. We find that the onset of super-Arrhenius behavior and the MCT dynamic glass transition correlate well with temperature when the T1-inactive particles form clusters of two particles on average and when the T1-inactive clusters percolate the entire system, respectively. Our findings provide new insight into the structural origin of glassy dynamics.

Enthalpy-based equation of state for highly porous materials employing modified soft sphere fluid model

Enthalpy-based equation of state based on a modified soft sphere model for the fluid phase, which includes vaporization and ionization effects, is formulated for highly porous materials. Earlier developments and applications of enthalpy-based approach had not accounted for the fact that shocked states of materials with high porosity (e.g., porosity more than two for Cu) are in the expanded fluid region. We supplement the well known soft sphere model with a generalized Lennard-Jones formula for the zero temperature isotherm, with parameters determined from cohesive energy, specific volume and bulk modulus of the solid at normal condition. Specific heats at constant pressure, ionic and electronic enthalpy parameters and thermal excitation effects are calculated using the modified approach and used in the enthalpy-based equation of state. We also incorporate energy loss from the shock due to expansion of shocked material in calculating porous Hugoniot. Results obtained for Cu, even up to initial porosities ten, show good agreement with experimental data.

De Gennes Narrowing and Hard-Sphere Approach

The energy width Δω of the quasielastic coherent dynamic structure factor S(Q, ω) for a simple liquid exhibits the oscillating dependence on wavenumber Q with the sharp minimum at Qmax corresponding to the maximum of the structure factor S(Q). The only known expression for Δω(Q) was derived for a dense hard-sphere (HS) fluid (Cohen et al., Phys. Rev. Lett. 1987, 59, 2872). Though this expression has been frequently used for the analysis of the experimental data obtained for liquid metals, until now, it has never been tested against a true HS fluid. A test performed by means of HS molecular dynamic simulations reveals a considerable discrepancy between the simulations results and the examined model. The main output of the analysis is the finding that the ΔωHS(Q) behavior is defined in terms of the average cage size, ⟨Lc⟩, rather than of the HS diameter, σHS. The simulated ΔωHS(Q) has been compared with the results for the soft-spherical potential. The microscopic dynamics of the soft-sphere fluid shows significant difference in comparison to the HS system. Nevertheless, the diffusive mobility of soft spheres can be characterized within the HS approximation using an effective diameter, σeff, and this parameter can be found from Δω(Q) at Q ≈ Qmax. A similar result has been obtained for the neutron scattering data measured for liquid Rb.

Note: Sound velocity of a soft sphere model near the fluid-solid phase transition.

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.

Simulations of dipolar fluids using effective many-body isotropic interactions.

The partition function of a system with pairwise-additive anisotropic dipole-dipole interactions is equal to that of a hypothetical system with many-body isotropic interactions [G. Stell, Phys. Rev. Lett. 32, 286 (1974)]. The effective many-body interactions contain n-body contributions of all orders. Each contribution is known as an expansion in terms of the particle-particle distances r, and the coefficients are temperature dependent. The leading-order two-body term is the familiar -r(-6) attraction, and the leading-order three-body term is equivalent to the Axilrod-Teller interaction. In this work, a fluid of particles with the leading-order two-body and three-body interactions is compared to an equivalent dipolar soft-sphere fluid. Molecular simulations are used to determine the conditions under which the effective many-body interactions reproduce the fluid-phase structures of the dipolar system. The effective many-body interaction works well at moderately high temperatures but fails at low temperatures where particle chaining is expected to occur. It is shown that an adjustment of the coefficients of the two-body and three-body terms leads to a good description of the structure of the dipolar fluid even in the chaining regime, due primarily to the ground-state linear configuration of the three-body Axilrod-Teller interaction. The vapor-liquid phase diagrams of systems with different Axilrod-Teller contributions are determined. As the strength of the three-body interaction is increased, the critical temperature and density both decrease and disappear completely above a threshold strength, where chaining eventually suppresses the condensation transition.

Equilibrium and nonequilibrium dynamics of soft sphere fluids.

We use computer simulations to test the freezing-point scaling relationship between equilibrium transport coefficients (self-diffusivity, viscosity) and thermodynamic parameters for soft sphere fluids. The fluid particles interact via the inverse-power potential (IPP), and the particle softness is changed by modifying the exponent of the distance-dependent potential term. In the case of IPP fluids, density and temperature are not independent variables and can be combined to obtain a coupling parameter to define the thermodynamic state of the system. We find that the rescaled coupling parameter, based on its value at the freezing point, can approximately collapse the diffusivity and viscosity data for IPP fluids over a wide range of particle softness. Even though the collapse is far from perfect, the freezing-point scaling relationship provides a convenient and effective way to compare the structure and dynamics of fluid systems with different particle softness. We further show that an alternate scaling relationship based on two-body excess entropy can provide an almost perfect collapse of the diffusivity and viscosity data below the freezing transition. Next, we perform nonequilibrium molecular dynamics simulations to calculate the shear-dependent viscosity and to identify the distinct role of particle softness in underlying structural changes associated with rheological properties. Qualitatively, we find a similar shear-thinning behavior for IPP fluids with different particle softness, though softer particles exhibit stronger shear-thinning tendency. By investigating the distance and angle-dependent pair correlation functions in these systems, we find different structural features in the case of IPP fluids with hard-sphere like and softer particle interactions. Interestingly, shear-thinning in hard-sphere like fluids is accompanied by enhanced translational order, whereas softer fluids exhibit loss of order with shear. Our results provide a systematic evaluation of the role of particle softness in equilibrium and nonequilibrium transport properties and their underlying connection with thermodynamic and structural properties.

Collective excitations in soft-sphere fluids

Despite that the thermodynamic distinction between a liquid and the corresponding gas ceases to exist at the critical point, it has been recently shown that reminiscence of gaslike and liquidlike behavior can be identified in the supercritical fluid region, encoded in the behavior of hypersonic waves dispersion. By using a combination of molecular dynamics simulations and calculations within the approach of generalized collective modes, we provide an accurate determination of the dispersion of longitudinal and transverse collective excitations in soft-sphere fluids. Specifically, we address the decreasing rigidity upon density reduction along an isothermal line, showing that the positive sound dispersion, an excess of sound velocity over the hydrodynamic limit typical for dense liquids, displays a nonmonotonic density dependence strictly correlated to that of thermal diffusivity and kinematic viscosity. This allows rationalizing recent observation parting the supercritical state based on the Widom line, i.e., the extension of the coexistence line. Remarkably, we show here that the extremals of transport properties such as thermal diffusivity and kinematic viscosity provide a robust definition for the boundary between liquidlike and gaslike regions, even in those systems without a liquid-gas binodal line. Finally, we discuss these findings in comparison with recent results for Lennard-Jones model fluid and with the notion of the "rigid-nonrigid" fluid separation lines.

Probing the pair structure of supercooled fluids by integral equations: Evidence for an equilibrium liquid-ideal glass transition?

We solve the hypernetted-chain (HNC) and Rogers-Young (RY) integral equations for the pair structure of a system of two weakly coupled replicæ of a soft-sphere fluid, in order to detect signatures of an equilibrium transition to an “ideal glass”. Both integral equations predict, in addition to the supercooled-liquid branch, the appearance of a new disordered state characterized by different pair correlations, over a range of temperatures. The ideal-glass transition temperature , identified by a discontinuous jump in the inter-replica pair correlation function, is predicted to be substantially higher with RY compared to HNC theory. The internal energy is predicted to undergo a weak discontinuity at .

“Liquid-Gas” Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics

Recently, we have proposed a new dynamic line on the phase diagram in the supercritical region, the Frenkel line. Crossing the line corresponds to the radical changes of system properties. Here, we focus on the dynamics of model Lennard-Jones and soft-sphere fluids. We show that the location of the line can be rigorously and quantitatively established on the basis of the velocity autocorrelation function (VAF) and mean-square displacements. VAF is oscillatory below the line at low temperature, and is monotonically decreasing above the line at high temperature. Using this criterion, we show that the crossover of particle dynamics and key liquid properties occur on the same line. We also show that positive sound dispersion disappears in the vicinity of the line in both systems. We further demonstrate that the dynamic line bears no relationship to the existence of the critical point. Finally, we find that the region of existence of liquidlike dynamics narrows with the increase of the exponent of the repulsive part of interatomic potential.

Thermodynamic scaling and corresponding states for the self-diffusion coefficient of non-conformal soft-sphere fluids

In this work, we explore transport properties of a special type of repulsive spheres that exhibit remarkable scaling of their thermodynamic properties. In order to accomplish that we propose a new way to derive and express effective hard-sphere diameters for transport properties of simple fluids. The procedure relies on mapping the system's transport properties, in the low density limit, to the hard-sphere fluid. We have chosen a set of soft-sphere systems characterised by a well-defined variation of their softness. These systems represent an extension of the repulsive Lennard-Jones potential widely used in statistical mechanics of fluids and are an accurate representation of the effective repulsive potentials of real systems. The self-diffusion coefficient of the soft-sphere fluids is obtained by equilibrium molecular dynamics. The soft-sphere collision integrals of different systems are shown to follow quite simple relationships between each other. These collision integrals are incorporated, through the definition of the effective hard-sphere diameter, in the resulting equation for the self-diffusion coefficient. The approach followed exhibits a density rescaling that leads to a single master curve for all systems and temperatures. The scaling is carried through to the level of the mean-squared displacement.

Glassy dynamics in a confined monatomic fluid

Molecular dynamic simulations of a strongly inhomogeneous system reveals that a single-component soft-sphere fluid can behave as a fragile glass former due to confinement. The self-intermediate scattering function, F(s)(k,t), of a Lennard-Jones fluid confined in slit-shaped pores, which can accommodate two to four fluid layers, exhibits a two-step relaxation at moderate temperatures. The mean-squared displacement data are found to follow time-temperature superposition and both the self-diffusivity and late α relaxation times exhibit power-law divergences as the fluid is cooled. The system possesses a crossover temperature and follows the scalings of mode coupling theory for the glass transition. The temperature dependence of the self-diffusivity can be expressed using the Vogel-Fulcher-Tammann equation, and estimates of the fragility index of the system indicates a fragile glass former. At lower temperatures, signatures of additional relaxation processes are observed in the various dynamical quantities with a three-step relaxation observed in the F(s)(k,t).

Transport coefficients of soft sphere fluid at high densities

Molecular dynamics computer simulation has been used to compute in wide pressure-temperature region the self-diffusion coefficient and shear viscosity of soft-sphere fluids, in which the particles interact through the inverse power pair potential. The limitations of the previous empirical approaches and models for the description of the kinetic coefficients are shown. On the basis of the simulations the universal functions for diffusion coefficient and shear viscosity of soft spheres are constructed which allow to compute these quantities for any density and temperature. We show that the behavior of shear viscosity at high temperatures is more complex than the one of diffusion coefficient and give quantitative explanation of this result.