A. C. Brańka

Percolation threshold of hard-sphere fluids in between the soft-core and hard-core limits

A study of the soft-core to hard-core percolation properties of the hard-sphere fluid as a function of hard core (sphere) packing fraction, ζ, has been made using the Molecular Dynamics computer simulation method. The interparticle separation between the soft shells to achieve percolation, σ p , exhibits a monotonic decrease with ζ from the permeable spheres limit to approximately the glass transition, where σ p tends to the hard-sphere radius, σ. σ p was fitted to a semi-empirical function of ζ, which has the exact low density limiting behaviour. The ζ-dependence of the corresponding packing fraction of the permeable shells, is discussed. The local coordination number at the percolation threshold showed a transition between the soft-core and hard-core limits from ca. 2.74 to 1.51, as found in previous studies. A reasonably accurate simple analytic expression is given for the packing fraction dependence of the coordination number up to the percolation distance. Various key length scales of the hard-sphere system are compared with σ p as a function of ζ. The hard-sphere percolation distance dependence on packing shows a similar behaviour to that of rescaled Weeks–Chandler–Andersen (WCA) fluids, but not the same, which is consistent with the conclusions of that previous study.

Static properties and time correlation functions of fluids with steeply repulsive potentials

There has been much interest in recent years in the properties of fluids composed of particles that interact through potentials with a ‘tunable’ softness, particularly for highly repulsive particles close to the hard-sphere limit. Much of this interest is driven by the granular media and colloid communities, but there are fundamental issues concerning the physics of fluids in general that can be addressed with such model systems. In this report we continue our series of investigations into the properties of an example of such a fluid, the so-called inverse power or soft-sphere fluid which is composed of particles interacting through a pair potential, ϕ (r)=ϵ (σ/r) n , where n measures the steepness or stiffness of the potential. We review the current state of our knowledge of the properties of such fluids and point out some still unresolved areas. We present the results of new computations for a range of n values and densities. Interest is focused on the time correlation functions as exemplified by the force autocorrelation function, CF (t), and the shear stress correlation function, C s(t). Powles and Heyes [POWLES, J. G. and HEYES, D. M., 2000, Molec. Phys., 98, 917.] showed that at short times the shear stress autocorrelation function fits quite closely to the analytic form, where x is a reduced time that incorporates the effects of n and temperaure, x = (T)*1/2 nt*. T* and T* are the reduced temperature and time in particle units consisting of particle mass and the pair potential parameters, ϵ and σ. We give further supporting evidence for this functional form, both for C s and CF . However, we still lack a theoretical explanation for this behaviour. A formal time expansion of CF (t) with similar approximations to those used in our previous publications for the O(t 2) coefficient predicts a Gaussian analytic form for C(t), therefore posing an unresolved paradox at present. We present a general procedure by which we can derive formally exact expressions for static properties (such as the interaction energy, pressure and infinite frequency elastic moduli) that can be conveniently reduced to relatively simple approximate expressions involving n and an equivalent hard-sphere density and equation of state, which become more accurate in the hard-sphere limit. We also derive a simple (approximate) analytic formula for the mean square force on a particle in terms of n and the hard-sphere equation of state, which is accurate in the hard-sphere limit. We examine the radial distribution functions of these fluids for various n and density, and compare their forms close to contact with the hard-sphere radial distribution functions with appropriate diameters.

Density and pressure dependence of the equation of state and transport coefficients of soft-sphere fluids

Molecular Dynamics, MD, simulations were used to compute physical properties of model fluids in which the particles interacted via the soft-sphere or inverse power pair potential, φ(r) = ε(σ/r) n . n dictates the steepness or stiffness of the potential, and ε and σ are a characteristic energy and distance, respectively. A wide range of n values were considered, from the hard-sphere (n → ∞) limit down to (the latter for the first time). A linear isotherm relationship for dense fluids observed by Parsafar and Mason for supercritical compressed gases [J. Phys. Chem. 97, 9048 (1993)], was found to apply to the data for n ∼ 12 (values typical of simple fluids). For smaller n, there is a change in sign of the slope, and the data exhibited more curvature. The self-diffusion coefficient, D, and shear viscosity, ηs, were also calculated. At intermediate to high densities, D −1 and ηs depend to a very good approximation linearly on pressure, as was found by van der Gulik [Physica A 256, 39 (1998)] on treatment of experimental shear viscosity data for simple molecules. Values for D and ηs at fluid–solid coexistence are given as a function of n. We refine further simple formulae for D and ηs proposed in our previous publication [Phys. Chem. Chem. Phys. 10, 4036 (2008)]. The glass transition packing fraction and pressure for the fluid are estimated by extrapolation of the self-diffusion coefficient data. In contrast to the n = 12 case, the shear stress correlation function correlation time shows only a weak density dependence near coexistence for the very soft interactions (e.g. ). It is shown that for the very soft interactions close to n = 3, the increase in viscosity is largely determined by the infinite frequency shear modulus rather than the relaxation time, which hardly changes with density at high density.

Monte Carlo simulations of hard cyclic pentamers in two dimensions: Translational-rotational coupling in the orientationally disordered phase

The orientationally disordered phase of the two-dimensional hard cyclic pentamer system has been studied by Monte Carlo simulations. Various structural quantities have been calculated at several densities. The singlet distribution function, studied in detail, reveals considerable coupling between translational and rotational molecular motions. The coupling, stronger at higher densities, is present also at lower densities, down to melting, and cannot be neglected in any structural analysis of the system. An orientational transition connected with a change observed in the translational-orientational coupling is suggested.

Mechanical simulation of two-dimensional systems of hard dimers and hard cyclic trimers

Mechanical simulations of two-dimensional systems of hard dimers (consisting of two hard discs stuck together) and hard cyclic trimers (consisting of three hard discs stuck together) showed in both cases that at high densities the molecules form specific non-crystalline solid structures: the discs are arranged in a hexagonal lattice but the molecules themselves, i.e. their axes and centres of mass, do not show any order. At low densities both systems are in the fluid state. Phase coexistence observed in the melting region of the dimers suggests a first-order phase transition. Interpretation of the transition region between solid and fluid in the system of trimers remains an open question; the existence of an intermediate phase cannot be excluded.

Non-equilibrium phase behavior and friction of confined molecular films under shear: a non-equilibrium molecular dynamics study

The phase behavior of a confined liquid at high pressure and shear rate, such as is found in elastohydrodynamic lubrication, can influence the traction characteristics in machine operation. Generic aspects of this behavior are investigated here using Non-equilibrium Molecular Dynamics (NEMD) simulations of confined Lennard-Jones (LJ) films under load with a recently proposed wall-driven shearing method without wall atom tethering [C. Gattinoni et al., Phys. Rev. E 90, 043302 (2014)]. The focus is on thick films in which the nonequilibrium phases formed in the confined region impact on the traction properties. The nonequilibrium phase and tribological diagrams are mapped out in detail as a function of load, wall sliding speed, and atomic scale surface roughness, which is shown can have a significant effect. The transition between these phases is typically not sharp as the external conditions are varied. The magnitude of the friction coefficient depends strongly on the nonequilibrium phase adopted by the confined region of molecules, and in general does not follow the classical friction relations between macroscopic bodies, e.g., the frictional force can decrease with increasing load in the Plug-Slip (PS) region of the phase diagram owing to structural changes induced in the confined film. The friction coefficient can be extremely low (∼0.01) in the PS region as a result of incommensurate alignment between a (100) face-centered cubic wall plane and reconstructed (111) layers of the confined region near the wall. It is possible to exploit hysteresis to retain low friction PS states well into the central localization high wall speed region of the phase diagram. Stick-slip behavior due to periodic in-plane melting of layers in the confined region and subsequent annealing is observed at low wall speeds and moderate external loads. At intermediate wall speeds and pressure values (at least) the friction coefficient decreases with increasing well depth of the LJ potential between the wall atoms, but increases when the attractive part of the potential between wall atoms and confined molecules is made larger.

The second virial coefficient and critical point behavior of the Mie Potential.

Aspects of the second virial coefficient, b2, of the Mie m : n potential are investigated. The Boyle temperature, T0, is shown to decay monotonically with increasing m and n, while the maximum temperature, Tmax, exhibits a minimum at a value of m which increases as n increases. For the 2n : n special case T0 tends to zero and Tmax approaches the value of 7.81 in the n → ∞ limit which is in quantitative agreement with the expressions derived in Rickayzen and Heyes [J. Chem. Phys. 126, 114504 (2007)] in which it was shown that the 2n : n potential in the n → ∞ limit approaches Baxters sticky-sphere model. The same approach is used to estimate the n - dependent critical temperature of the 2n : n potential in the large n limit. The ratio of T0 to the critical temperature tends to unity in the infinite n limit for the 2n : n potential. The rate of convergence of expansions of b2 about the high temperature limit is investigated, and they are shown to converge rapidly even at quite low temperatures (e.g., 0.05). In contrast, a low temperature expansion of the Lennard-Jones 12 : 6 potential is shown to be an asymptotic series. Two formulas that resolve b2 into its repulsive and attractive terms are derived. The convergence at high temperature of the Lennard-Jones b2 to the m = 12 inverse power value is slow (e.g., requiring T ≃ 10(4) just to attain two significant figure accuracy). The behavior of b2 of the ∞ : n and the Sutherland potential special case, n = 6, is explored. By fitting to the exact b2 values, a semiempirical formula is derived for the temperature dependence of b2 of the Lennard-Jones potential which has the correct high and low temperature limits.

CALCULATION OF NANOCOLLOIDAL LIQUID TIME SCALES BY MOLECULAR DYNAMICS SIMULATIONS

Molecular dynamics, MD, simulations have been used to calculate the translational and rotational relaxation dynamics of model atomistically rough spherical nanocolloidal particles in solution at infinite dilution by immersing a single Lennard-Jones cluster in a molecularly discrete solvent. Key time scales characterizing colloidal particle dynamical relaxation were computed from time correlation functions. For translational motion these were τv, the colloidal velocity relaxation time, τf, the hydrodynamic relaxation time and the time scale for significant particle displacement, τd. We show that τv ≃ τf when the relative mass density of the colloidal particle divided by the bulk density of the solvent is ca. ρ∗ = 20, in agreement with theoretical predictions. Preliminary evidence from the velocity autocorrelation functions, VACF, of the nanocolloidal particle also supports the theoretical treatments that the transition from the Liouville to Fokker—Planck description (evident by exponential decay in the VAC...

The effects of particle softness on the dynamics of molecular and colloidal systems

We extend previous studies of the dynamical relaxation in liquids consisting of particles interacting with the soft-sphere or inverse power potential, , where ε and σ set the energy and length scales, respectively, and n is the steepness or stiffness parameter. In previous work we have simulated these systems using molecular dynamics (MD) simulation. Here we include model colloidal particles interacting with this potential, executing the position Langevin equation of motion as implemented in the Brownian dynamics (BD), method. A formal statistical mechanical expansion of the force autocorrelation function for such particles was carried out in both cases. Using the (molecular) Liouville operator this gives at short time a scaling with time, t, in terms of a reduced time, , where d depends on the state point, and for the colloidal case, the Smoluchowski operator gives the reduced time, . The first term accounts for the leading two-body contributions to the relaxation, and the second term, in d, represents additional two-body, and the leading three-body contribution. d depends mainly on the packing fraction and only weakly on n for ca. n ≥ 18. In the colloidal case, because of the greater sensitivity to n, the simulations were limited to relatively small values (in the range, 18–72). In this range of n values, the term in d is shown to make a significant contribution and is necessary to produce collapse at short times for both the MD and BD data. We also show that in this n range there is a collapse of the correlation functions at short times with an alternative (empirical) definition of the reduced time ∼ n α t, with α < 1 for the molecular case and α < 2 for the colloids. This applies not only to the force autocorrelation function but also for the shear stress and deviatoric pressure autocorrelation functions. In the 18–72 range, the α were typically 10% lower than the large n limiting value (1 and 2 for MD and BD, respectively). The optimum value of α for the various time correlation functions depended on the packing fraction and choice of correlation function. The shear stress correlation functions for these finite n have α values closer to 1 (MD) and 2 (BD) than the pressure and force autocorrelation functions.

Equation of state of inverse power fluids

A new equation of state for the inverse power, r − n potential, fluid is proposed. It is derived on the basis of the local scaling behaviour of its structural properties, without referring to any perturbative scheme, and therefore recourse to an effective hard sphere diameter. It is shown that the general formula for the compressibility factor can be expressed as the product of three functions. The first represents the hard-sphere equation of state at the same packing fraction, and the other two incorporate the effects of the potential softness, again as functions of density. Using computer simulation results, explicit forms for these soft parts have been established, to give an approximate analytic expression for the r − n fluid equation of state. Two different regions, characterized by positive and negative softness ‘compressibility’ have been found.A new equation of state for the inverse power, r − n potential, fluid is proposed. It is derived on the basis of the local scaling behaviour of its structural properties, without referring to any perturbative scheme, and therefore recourse to an effective hard sphere diameter. It is shown that the general formula for the compressibility factor can be expressed as the product of three functions. The first represents the hard-sphere equation of state at the same packing fraction, and the other two incorporate the effects of the potential softness, again as functions of density. Using computer simulation results, explicit forms for these soft parts have been established, to give an approximate analytic expression for the r − n fluid equation of state. Two different regions, characterized by positive and negative softness ‘compressibility’ have been found.