Isotropic-nematic phase transition of uniaxial variable softness prolate and oblate ellipsoids.

Onsager's theory of the isotropic-nematic phase separation of rod shaped particles is generalized to include particle softness and attractions in the anisotropic interparticle force field. The procedure separates a scaled radial component from the angular integral part, the latter being treated in essentially the same way as in the original Onsager formulation. Building on previous treatments of more idealised hard-core particle models, this is a step toward representing more realistic rod-like systems and also allowing temperature (and in principle specific chemical factors) to be included at a coarse grained level in the theory. The focus of the study is on the coexisting concentrations and associated coexistence properties. Prolate and oblate ellipsoids are considered in both the small and very large aspect ratio limits. Approximations to the terms in the angular integrals derived assuming the very large (prolate) and very small (oblate) aspect ratios limits are compared with the formally exact treatment. The approximation for the second virial coefficient matches the exact solution for aspect ratios above about 20 for the prolate ellipsoids and less than ca. 0.05 for the oblate ellipsoids from the numerical evaluation of the angular integrals. The temperature dependence of the coexistence density could be used to help determine the interaction potential of two molecules. The method works at temperatures above a certain threshold temperature where the second virial coefficient is positive.

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 Baxter's 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.

Binary mixtures of asymmetric continuous charge distributions: molecular dynamics simulations and integral equations.

An investigation is carried out of the association and clustering of mixtures of Gaussian charge distributions (CDs) of the form ∼Qexp(-r(2)/2α(2)), where Q is the total charge, r is the separation between the centers of charge and α governs the extent of charge spreading (α → 0 is the point charge limit). The general case where α and Q are different for the positive and negatives charges is considered. The Ewald method is extended to treat these systems and it is used in Molecular Dynamics (MD) simulations of electrically neutral CD mixtures in the number ratios of 1:1 and 1:4 (or charge ratio 4:1). The MD simulations reveal increased clustering with decreasing temperature, which goes through a state in which each large CD is overlapped by four of the oppositely signed CD in the 1:4 case. At very low reduced temperatures, these mini-clusters progressively coalesce into much larger tightly bound clusters. This is different from the 1:1 mixture case, where the low temperature limit is a random distribution of neutral dimers. At higher temperatures, the MD radial distribution functions g(r) agree well with those from the hypernetted chain solution of the Ornstein-Zernike integral equation, and (at not too high densities) a previously introduced mean field approximation extended to these charge distribution systems.

Continuous distributions of charges: extensions of the one component plasma.

The electrostatic interaction between finite charge distributions, ρ(r), in a neutralizing background is considered as an extension of the one component plasma (OCP) model of point charges. A general form for the interaction potential is obtained which can be applied to molecular theories of many simple charged fluids and mixtures and to the molecular dynamics (MD) simulation of such systems. The formalism is applied to the study of a fluid of Gaussian charges in a neutralizing background by MD simulation and using hypernetted-chain integral equation theory. The treatment of these interactions is extended to a periodic system using a Fourier Transform formulation and, for a rapidly decaying charge distribution, an application of the Ewald method. The contributions of the self-energy and neutralizing background to the system's energy are explicitly included in the formulation. Calculations reveal differences in behavior from the OCP model when the Wigner-Seitz radius is of order and less than the Gaussian charge density decay length. For certain parameter values these systems can exhibit a multiple occupancy crystalline phase at high density which undergoes re-entrant melting at higher density. An exploration of the effects of the various length scales of the system on the equation of state and radial distribution function is made.

The force distribution probability function for simple fluids by density functional theory.

Classical density functional theory (DFT) is used to derive a formula for the probability density distribution function, P(F), and probability distribution function, W(F), for simple fluids, where F is the net force on a particle. The final formula for P(F) ∝ exp(-AF(2)), where A depends on the fluid density, the temperature, and the Fourier transform of the pair potential. The form of the DFT theory used is only applicable to bounded potential fluids. When combined with the hypernetted chain closure of the Ornstein-Zernike equation, the DFT theory for W(F) agrees with molecular dynamics computer simulations for the Gaussian and bounded soft sphere at high density. The Gaussian form for P(F) is still accurate at lower densities (but not too low density) for the two potentials, but with a smaller value for the constant, A, than that predicted by the DFT theory.

Single particle force distributions in simple fluids.

The distribution function, W(F), of the magnitude of the net force, F, on particles in simple fluids is considered, which follows on from our previous publication [A. C. Branka, D. M. Heyes, and G. Rickayzen, J. Chem. Phys. 135, 164507 (2011)] concerning the pair force, f, distribution function, P(f), which is expressible in terms of the radial distribution function. We begin by discussing the force on an impurity particle in an otherwise pure fluid but later specialize to the pure fluid, which is studied in more detail. An approximate formula, expected to be valid asymptotically, for W(F) referred to as, W(1)(F) is derived by taking into account only binary spatial correlations in the fluid. It is found that W(1)(F) = P(f). Molecular dynamics simulations of W for the inverse power (IP) and Lennard-Jones potential fluids show that, as expected, W(F) and P(f) agree well in the large force limit for a wide range of densities and potential forms. The force at which the maximum in W(F) occurs for the IP fluids follows a different algebraic dependence with density in low and high density domains of the equilibrium fluid. Other characteristic features in the force distribution functions also exhibit the same trends. An exact formula is derived relating W(F) to P(x)(F(x)), the distribution function of the x-cartesian components of the net force, F(x), on a particle. W(F) and P(x)(F(x)) have the same analytical forms (apart from constants) in the low and high force limits.

Pair force distributions in simple fluids.

Analytic expressions are derived for the frequency distribution, P(f), of pair forces, f, and those of their α-Cartesian component, f(α), or P(f(α)), for some typical model simple fluids, expressed in terms of the radial distribution function and known constants. For strongly repulsive inverse power (IP), exponential and Yukawa purely repulsive potentials, P(f) diverges at the origin approximately as ∼f(-1), but with different limiting analytic forms. P(f(α)) is also shown to diverge as ∼f(-1) as f → 0 for the IP fluid. For the Lennard-Jones potential fluid, P(f) is finite for all f ≥ 0 but has two singularities for negative f, corresponding to the zero force limit (i.e., f → 0(-)) and the point of inflection in the potential. The corresponding component force distribution is singular as f(α) → 0 from both positive and negative force sides. The large force limit of P(f), which originates from the close neighbor interactions, is nearly exponential for the IP and LJ fluids, as is also found for granular materials. A more complete picture of force distributions in off-lattice particulate systems as a function of force law and state point (particularly the extent of "thermalization" of the particles) is provided.

Parametrization of linear dielectric response.

The quantum linear response of a dielectric to an external electric field yields expressions for the dielectric susceptibility and the associated impulse response function. These are measured properties that, during analysis, are often "curve-fitted" to diverse forms of parametric functional forms that shall herein be referred to as fit-functions. The main purpose of this paper is to show, from a very general linear response formalism that encompasses virtually all microscopic models of dielectric response, that there are constraints on the forms that the susceptibilities must obey and to examine common parametrizations of the dielectric function in light of these constraints. Naturally these constraints should, whenever possible, be in-built into the fit-functions employed. The linear response approach due to Madden and Kivelson [Adv. Chem. Phys. 56, 467 (1984)], where the cause is considered to be a uniform external field, E(ext)(t), is utilized as it affords a much more straightforward interaction term, viz., -M·E(ext)(t), (M being the system's total electric dipole moment operator) than would be the case if the mean internal field (or "Maxwell field") were taken as the cause. It is shown that this implies definite relations between the quasipermittivity, ζ(ω), of the Madden-Kivelson approach and the normal permittivity, χ(ω)≡ε(ω)-ε(0). These relations indicate a condition for the divergence of the normal susceptibility, which, arguably, marks the onset of a ferroelectric transition in "sufficiently polar" dielectrics. Finally, some common parametric "fit-function" forms are investigated as to whether they comply with the constraints that the formalism imposes, and examples are given of their associated Cole-Cole plots in typical cases involving one or more relaxation times.

Rounded stretched exponential for time relaxation functions.

A rounded stretched exponential function is introduced, C(t)=exp{(tau(0)/tau(E))(beta)[1-(1+(t/tau(0))(2))(beta/2)]}, where t is time, and tau(0) and tau(E) are two relaxation times. This expression can be used to represent the relaxation function of many real dynamical processes, as at long times, t>>tau(0), the function converges to a stretched exponential with normalizing relaxation time, tau(E), yet its expansion is even or symmetric in time, which is a statistical mechanical requirement. This expression fits well the shear stress relaxation function for model soft soft-sphere fluids near coexistence, with tau(E)<

The autocorrelation functions of a fluid of molecules interacting through steep attractive potentials.

In a previous paper [G. Rickayzen and D. M. Heyes, J. Chem. Phys. 126, 234503 (2007)] we investigated by theory and molecular dynamics (MD) simulation the force and velocity autocorrelation functions of a fluid of molecules interacting through steeply varying potentials of the form phi(r) = 4epsilon[z(2)(r)-z(r)], where z(r) = (sigma r)(n), epsilon and sigma set the energy and length scale of the interaction, respectively, and n is an adjustable exponent (n = 72 and 144 were considered in that work). Discrepancies between the theory and simulation were found except at the shortest times for some of the state points. In order to identify the origin(s) of these discrepancies, we have investigated here another fluid, in which the particles interact via the Morse potential in which z(r) = exp(-kappa(r-sigma)sigma), where kappa is now the "steepness" parameter. The parameter kappa is the analog of n, and this potential form is used in order to compare with previous results and establish better the origin of the differences between theory and simulation. It is shown in a further development of the theory that the actual form of the potential in the steep and short-ranged attractive limit is immaterial, and there exists a law of corresponding states for such potentials. This conclusion is confirmed by the MD simulations with the two potential forms for kappa = n = 144. The difference between the theory and simulation correlation functions increases with density, and it is concluded that these differences probably originate in many-body effects in time, which are absent in the theory. For packing fractions below about 0.2 the agreement between the theory and MD simulation force and velocity autocorrelation functions is nevertheless very good at all accessible times.

Monte Carlo simulations of fluids whose particles interact with a logarithmic potential.

Monte Carlo simulations of a model fluid in which the particles interact via a continuous potential that has a logarithmic divergence at a pair separation of sigma, which we introduced in J. G. Powles et al., Proc. R. Soc. London, Ser. A 455, 3725 (1999), have been carried out. The potential has the form, phi(r)= -epsilon ln(fr), where epsilon sets the energy scale and fr=1-(sigma/r)m. The value of m chosen was 12 but the qualitative trends depend only weakly on the value of m, providing it is greater than 3. The potential is entirely repulsive and has a logarithmic divergence as approximately -ln(r/sigma-1) in the r-->sigma limit. Predictions of the previous paper that the internal energy can be computed at all temperatures using the standard statistical mechanics formula for continuous potentials are verified here. The pressure can be calculated using the usual virial expression for continuous potentials, although there are practical limitations in resolving the increasingly important contribution from the r-->sigma limit at reduced temperatures greater than approximately 5. The mean square force F2 and infinite frequency shear Ginfinity and bulk Kinfinity moduli are only finite for T*=kBT/epsilon<1. The logarithmic fluid's physical properties become increasingly more like that of the hard sphere fluid with increasing temperature, showing a sharp transition in the behavior of the mean square force and infinite frequency elastic constants at T*=1. The logarithmic fluid is shown to exhibit a solid-fluid phase transition.

A configurational temperature for molecules with hard-core or discontinuous interactions.

We extend the usual formula for a configurational temperature so that it applies to condensates in which the molecules interact through hard-core or discontinuous potentials. The new formula involves extra terms which may be calculated during the course of a simulation. The formula is tested by its application to a number of systems with discontinuous or hard-core potentials in thermodynamic equilibrium. Metropolis Monte Carlo simulations were performed on these systems in a canonical ensemble and the configurational temperature is compared with the input temperature. The two are in agreement to within less than 0.1%.

Autocorrelation functions of a fluid of molecules interacting through steeply repulsive and attractive forces.

A new method is presented for an extension of Enskog's approximation for the evaluation of the autocorrelation functions of a fluid, and this approach is used to evaluate these functions when the interaction between the molecules includes both steeply repulsive and steeply attractive forces. Consequently the correlation functions depend upon the temperature in a nontrivial way. As an example, the method is applied to calculate the velocity and force autocorrelation functions of a fluid when the molecules interact through the specific potential, V(r)=4epsilon[(sigma/r)2n-(sigma/r)n] when the parameter n is large. There is a relationship between this model and the "sticky sphere" one which is exploited in the theoretical computations. The results obtained from the theory are compared with molecular dynamics simulation for n=72 and 144 and for a range of temperatures from T=epsilon/kB down to epsilon/3kB. The two approaches agree very well for a range of state points, especially at short times. At later times the theory predicts a more oscillatory behavior than the simulation especially at very low reduced temperatures.

Stability of separation-shifted Lennard-Jones fluids.

Ruelle's thermodynamic stability criteria are applied to the separation-shifted Lennard-Jones (SSLJ) fluid, and the domain of its parameters giving "normal" thermodynamic stability in the thermodynamic limit is established. Fluids interacting with the SSLJ both conforming to and breaking these stability criteria were modeled using molecular dynamics computer simulation. For system sizes typical of most simulations, the transition between the two patterns of behavior was found to be smeared out over a range of parameter values. Thermodynamic instability is marked by a collapse of the system into a small "ball" or volume. The collapsed state nevertheless has some statistical mechanical properties typical of systems exhibiting normal thermodynamics (e.g., the kinetic and configurational temperatures were found to be the same in the collapsed state within statistics).

(2n,n) potential and sticky-sphere fluids.

The authors investigate the behavior of a model fluid for which the interaction energy between molecules at a separation r is of the form 4epsilon[(sigma/r)2n-(sigma/r)n], where epsilon and sigma are constants and n is a large integer. The particular properties they study are the pressure p, the mean square force F2, the elastic shear modulus at infinite frequency Ginfinity, the bulk modulus at infinite frequency Kinfinity, and the potential energy per molecule u. They show that if n is sufficiently large it is possible to derive the properties of the system in terms of two parameters, the values of the cavity function and of its derivative at the position r=sigma. As an example they examine in detail the cases with n=144 and n=72 for three different temperatures and they test the theory by comparison with a computer simulation of the system. They use the simulated pressure and the average mean square force to determine the two parameters and use these values to evaluate other properties; it is found that the theory produces results which agree with computer simulation to within approximately 3%. It is also shown that the model, when the parameter n is large, is equivalent to Baxter's sticky-sphere model with the strength of the adhesion determined by the value of n and the temperature. They use Baxter's solution of the Percus-Yevick equations for the sticky-sphere model to determine the cavity function and from that the values of the same properties. In this second approach there are no free parameters to determine from simulation; all properties are completely determined by the theory. The results obtained agree with computer simulation only to within approximately 6%. This suggests that for this model one needs a better approximation to the cavity function than that provided by the Percus-Yevick solution. Nevertheless, the model looks promising for the study of (typically small) colloidal liquids where the range of attraction is short but finite when compared to its diameter, in contrast to Baxter's sticky-sphere limit where the attractive interaction range is taken to be infinitely narrow. The continuous function approach developed here enables important physical properties such as the infinite shear modulus to be computed, which are finite in experimental systems but are undefined in the sticky-sphere model.

The stability of many-body systems.

We apply the pair interaction stability criteria of Fisher and Ruelle (1966 J. Math. Phys. 7 260) to establish the range of thermodynamic stability for a number of simple analytic potential forms used for condensed matter theory and modelling in the literature. We identify the ranges of potential parameters where, for a given potential, the system is thermodynamically stable, unstable and of uncertain stability. This was further explored by carrying out molecular dynamics simulations on the double Gaussian potential in the stable and unstable regimes. We show that, for example, the widely used exponential-6 and Born-Mayer-Huggins alkali halide potentials produce many-particle systems that are thermodynamically unstable.

Memory function for a fluid of molecules interacting through steeply repulsive potentials.

Previous studies of the properties of fluids of molecules interacting through steeply repulsive central potentials are extended to the investigation of the memory function. It is assumed that collisions are dominated by binary collisions and a general formula previously derived by Miyazaki, Srinivas, and Bagchi [J. Chem. Phys. 114, 6276 (2001)] is applied to the present problem. It is shown that the equations of motion of a pair of molecules can be solved explicitly and substitution of the result into the formula leads to a closed explicit expression for the memory function which is easily evaluated for any given state. In the limit of hard spheres this result leads to Enskog's equation and represents a generalization of that formula to fluids with softer potentials. The results obtained from the formula are compared with those derived from the molecular dynamics simulation. The velocity autocorrelation function was calculated using the generalized soft sphere potential, phi(r) = epsilon (sigma/r)(n), where epsilon and sigma set the energy and size of the molecule, and the exponent, n, is a variable. The two approaches agree very well for a range of state points for n large, especially at short times.