Geometry and energy effects on the structure of hard sphere fluids

The density profile of molecules is an interesting property in the theoretical study of inhomogeneous fluids. In this work, the density profile of a hard sphere fluid around various hard and soft spheres is studied using a well-established version of the density functional theory called “modified fundamental measure theory”. The results obtained from this study show that an increase in the size of the central molecule leads to both an enhanced density profile at the contact point and an amplified layering structure. Similar effects can be observed when attraction between original and bulk molecules increases. Another finding of the present study is that increasing the size of the central molecule has a quasi-attractive role of an entropic origin called ‘depletion potential’. Since the molecule is not very large, the depletion potential can be mapped in a hard core with an attractive Yukawa tail. Increasing molecule softness, however, has an opposite effect and causes the height of the first peak of the density profile to diminish and the inhomogeneity of the structure to lighten.     

Transport properties of the rough hard sphere fluid

Results are presented of a systematic study of the transport properties of the rough hard sphere fluid. The rough hard sphere fluid is a simple model consisting of spherical particles that exchange linear and angular momenta, and energy upon collision. This allows a study of the sole effect of particle rotation upon fluid properties. Molecular dynamics simulations have been used to conduct extensive benchmark calculations of self-diffusion, shear and bulk viscosity, and thermal conductivity coefficients. As well, the validity of several kinetic theory equations have been examined at various levels of approximation as a function of density and translational-rotational coupling. In particular, expressions from Enskog theory using different numbers of basis sets in the representation of the distribution function were tested. Generally Enskog theory performs well at low density but deviates at larger densities, as expected. The dependence of these expressions upon translational-rotational coupling was also examined. Interestingly, even at low densities, the agreement with simulation results was sometimes not even qualitatively correct. Compared with smooth hard sphere behaviour, the transport coefficients can change significantly due to translational-rotational coupling and this effect becomes stronger the greater the coupling. Overall, the rough hard sphere fluid provides an excellent model for understanding the effects of translational-rotational coupling upon transport coefficients.     

Orientational order in high density dipolar hard sphere fluids

Taking advantage of recent estimates, by one of us, of the critical temperature of the isotropic-ferroelectric transition of high density dipolar hard spheres, we performed new Monte Carlo simulations in the close vicinity of these estimates and applied histogram reweighting methods to obtain refined values of the critical temperatures from the crossing of the fourth-order cumulant for different system sizes. The ferroelectric line is determined in the density range rho*=0.80-0.95, and the onset of columnar ordering is located.     

Thermodynamics of d-dimensional hard sphere fluids confined to micropores

We derive an analytical expression of the second virial coefficient of d-dimensional hard sphere fluids confined to slit pores by applying Speedy and Reiss' interpretation of cavity space. We confirm that this coefficient is identical to the one obtained from the Mayer cluster expansion up to second order with respect to fugacity. The key step of both approaches is to evaluate either the surface area or the volume of the d-dimensional exclusion sphere confined to a slit pore. We, further, present an analytical form of thermodynamic functions such as entropy and pressure tensor as a function of the size of the slit pore. Molecular dynamics simulations are performed for d = 2 and d = 3, and the results are compared with analytically obtained equations of state. They agree satisfactorily in the low density regime, and, for given density, the agreement of the results becomes excellent as the width of the slit pore gets smaller, because the higher order virial coefficients become unimportant.     

Molecular dynamics study of translational and rotational diffusion in liquid ortho-terphenyl

NVT molecular dynamics simulations were performed on liquid o-terphenyl as a function of temperature in the range 320-480 K. Computed translational diffusion coefficients displayed the non-Arrhenius behavior expected of a fragile glass-forming liquid and were in good, semiquantitative agreement with experimental results. Rotational correlation functions calculated for various vectors within the molecule exhibited a very short time (0-1 ps) initial decay, followed by a reversal, which corresponds to free reorientation within the "solvent" cage prior to collision with a wall. Rotational correlation times of three orthogonal vectors fixed on the central benzene were close to equal at all temperatures, indicating nearly isotropic overall molecular reorientation. The average correlation times exhibited a non-Arrhenius temperature dependence and were in very good agreement with experimental values derived from 2D and 1H NMR relaxation times. Correlation times of vectors located on the lateral phenyl rings were used to calculate the "spinning" internal rotation diffusion coefficients, which were approximately twice as great as the overall rotational diffusion constants, indicating rapid internal rotation of the phenyl side groups over wide ranges of angle in the liquid.     

Computer simulation study of the closure relations in hard sphere fluids

We study, using Monte Carlo simulations, the cavity and the bridge functions of various hard sphere fluids: one component system, equimolar additive, and nonadditive binary mixtures. In particular, we numerically check the assumption of local dependency of the bridge functions from the indirect correlation functions, on which most of the existing integral equation theories hinge. We find that this condition can be violated either in the region around the first and second neighbors shell, or inside the hard core, for the systems here considered. The violations manifest themselves clearly in the so-called Duh-Haymet plots of the bridge functions versus the indirect correlation functions and become amplified as the coupling of the system increases.     

Clustering phenomena in adhesive hard sphere fluids. Theory of freezing

We develop a version of the physical cluster theory of the equation of state based on a definition of a “cluster” which depends both on particle positions and relative momenta, namely we choose to call a group of i molecules a physical cluster if their total energy (kinetic energy plus interactions) does not exceed the translation energy of the centre-of-mass of the whole group. This definition in addition with the “microcrystal” model of a cluster allows us to calculate the properties of individual clusters, to consider the clustering phenomena and to write down an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken into consideration by using the adhesive hard sphere model and the Percus-Yevick approximation. Based on the relations obtained we develop the theory of freezing and perform a numerical investigation of the equilibrium behavior of the system at various temperatures and densities. In particular, the theory enables us to describe the regular phase diagram which includes (in coordinates pressure-temperature) three branches intersecting at the triple point. It is interesting that the coexistence curve of the fluid-solid as well as liquid-gas phase transition has an end point.     

Activated hopping and dynamical fluctuation effects in hard sphere suspensions and fluids

Single particle Brownian dynamics simulation methods are employed to establish the full trajectory level predictions of our nonlinear stochastic Langevin equation theory of activated hopping dynamics in glassy hard sphere suspensions and fluids. The consequences of thermal noise driven mobility fluctuations associated with the barrier hopping process are determined for various ensemble-averaged properties and their distributions. The predicted mean square displacements show classic signatures of transient trapping and anomalous diffusion on intermediate time and length scales. A crossover to a stronger volume fraction dependence of the apparent nondiffusive exponent occurs when the entropic barrier is of order the thermal energy. The volume fraction dependences of various mean relaxation times and rates can be fitted by empirical critical power laws with parameters consistent with ideal mode-coupling theory. However, the results of our divergence-free theory are largely a consequence of activated dynamics. The experimentally measurable alpha relaxation time is found to be very similar to the theoretically defined mean reaction time for escape from the barrier-dominated regime. Various measures of decoupling have been studied. For fluid states with small or nonexistent barriers, relaxation times obey a simple log-normal distribution, while for high volume fractions the relaxation time distributions become Poissonian. The product of the self-diffusion constant and mean alpha relaxation time increases roughly as a logarithmic function of the alpha relaxation time. The cage scale incoherent dynamic structure factor exhibits nonexponential decay with a modest degree of stretching. A nearly universal collapse of the different volume fraction results occurs if time is scaled by the mean alpha relaxation time. Hence, time-volume fraction superposition holds quite well, despite the presence of stretching and volume fraction dependent decoupling associated with the stochastic barrier hopping process. The relevance of other origins of dynamic heterogeneity (e.g., mesoscopic domains), and comparison of our results with experiments, simulations, and alternative theories, is discussed.     

Effects of translational and rotational diffusion on the association in kinetic model of nucleation

Kinetics of an association and dissociation of single elements with the effects of translational and rotational diffusion and angular limitations is discussed. Separated clusters embedded in a solution of orientable single elements are considered.Steady-state positional and angular distribution of single elements is calculated from the equation of translational-rotational diffusion and the boundary conditions proposed for orientation-limited association. Although spherical orientable elements are assumed, the model can be used for non-spherical particles with aspect ratios close to unity.Diffusion-limited rate constants of association and dissociation are proposed which depend on translational and rotational diffusion constants of single elements, the tolerance angle of the association, and the cluster size.Effective concentration of single elements and effective rate constants are expressed by the equilibrium and diffusion-limited rate constants. Effects of finite diffusion rates and finite tolerance angle are discussed.The equations of the kinetic model of nucleation are modified due to the diffusion-limited rate of the association.     

Molecular dynamics study of anisotropic translational and rotational diffusion in liquid benzene

Equilibrium NPT and NVT molecular dynamics simulations were performed on liquid benzene over an extended range of temperature (from 260 to 360 K) using the COMPASS force field. Densities and enthalpies of vaporization (from cohesive energy densities) were within 1% of experiment at all temperatures. tumbling and spinning rotational diffusion coefficients, D(perpendicular) and D(parallel), computed as a function of temperature, agreed qualitatively with the results of earlier reported experimental and computational investigations. Generally, it was found that D(parallel)/D(perpendicular) approximately 1.4-2.5 and the activation energy for tumbling was significantly greater than for spinning about the C6 axis [Ea(D(perpendicular)) = 8.1 kJ mol(-1) and Ea(D(parallel)) = 4.5 kJ mol(-1)]. Calculated translational diffusion coefficients were found to be in quantitative agreement with experimental values at all temperatures [deviations were less than the scatter between different reported measurements]. In addition, translational diffusion coefficients were computed in the molecule-fixed frame to yield values for Dxy (diffusion in the plane of the molecule) and Dz (diffusion perpendicular to the plane). It was found that the ratio Dxy/Dz approximately 2.0, and that the two coefficients have roughly equal activation energies. This represents the first atomistic molecular dynamics study of translational diffusion in the molecular frame.     

Separating the contribution of translational and rotational diffusion to protein association

The association of two proteins is preceded by a mutual diffusional search in solution. The role of translational and rotational diffusion in this process has been studied theoretically for many years. However, systematic experimental verification of theoretical results is still lacking. We report here measurements of association rates of the proteins beta-lactamase (TEM) and beta-lactamase inhibitor protein (BLIP) in solutions of glycerol and poly(ethylene glycol) of increasing viscosity. We also measured translational and rotational diffusion in the same solutions, using fluorescence correlation spectroscopy and fluorescence anisotropy, respectively. It is found that in glycerol both translational and rotational diffusion rates are inversely dependent on viscosity, as predicted by the classical Stokes-Einstein relations, while the association rate depends nonlinearly on viscosity. In contrast, the association rate depends only weakly on the viscosity of the polymer solutions, which results in a similar weak dependence of k(on) on viscosity. The data are modeled using the theory of diffusion-limited association. Deviations from the theory are explained by a short-range solute-induced repulsion between the proteins in glycerol solution and an attractive depletion interaction generated by the polymers. These results open the way to the creation of a unified framework for all nonspecific effects involved in the protein association process, as well as to better theoretical understanding of these effects. Further, they reflect on the complex factors controlling protein association within the crowded environment of cells and suggest that a high concentration of macromolecules does not significantly impede protein association.     

Long-time tails of translational and rotational Brownian motion in a suspension of hard spheres

The long-time translational and rotational Brownian motion of a sphere in a suspension of hard spheres is studied on the basis of the linearized Navier-Stokes equations and the fluctuation-dissipation theorem. It is shown that for the rotational long-time coefficient an effective medium conjecture is incorrect. There are short-range velocity correlations that decay at the same rate as the macroscopic flow pattern used in the effective medium conjecture. An estimate of the short-range correction is made on the basis of the pair term in the cluster expansion of the rotational admittance.     

Roles of translational and reorientational modes in translational diffusion of high-pressure water: comparison with soft-core fluids

The dynamics of two soft-core fluids that show the increase in diffusivity with isothermal compression is studied with the mode-coupling theory (MCT). The anomalous density dependence of the diffusivity of these fluids is reproduced by the theory, and it is ascribed to the decrease in the first peak of the structure factor. The mechanism is quite different from that of high-pressure water revealed by MCT on molecular liquids described by the interaction-site model [T. Yamaguchi, S.-H. Chong, and F. Hirata, J. Chem. Phys., 119, 1021 (2003)]. The structures used in that study, calculated by the reference interaction-site model integral equation theory, showed the increase in the height of the first peak of the structure factor between oxygen atoms, whereas the structure obtained by molecular dynamics (MD) simulations shows the decrease in the peak height. In this work, calculations with MCT are performed on the simple fluids whose structure factor is the same as that between oxygen atoms of water from MD simulation, in order to clarify the role of translational structure on the increase in diffusivity with compression. The conclusion is that both the translational and reorientational modes contribute to the increase in diffusivity, and the effect of the latter is indispensable for the anomaly alone at least above freezing temperature.     

Imaging translational and rotational diffusion of single anisotropic nanoparticles with planar illumination microscopy

Here we demonstrated a simple yet powerful method, planar illumination microscopy, to directly track the rotational and translational diffusion dynamics of individual anisotropic nanoparticles in solution and living cells. By illuminating gold nanorods (GNRs) with two orthogonal sheets of light and resolving the polarized scattering signal with a birefringent crystal, we readily achieved three-dimensional angular resolving capability for single GNRs in noisy surroundings. The rotational dynamics of individual GNRs dispersed in glycerol/water mixtures with different chemical modification were tracked, and the measured rotational diffusion coefficient was well fitted to a previously reported theoretical model (Torre, J. G. d. l.; Martinez, M. C. L. Macromolecules 1987, 20, 661-666; Tirado, M. M.; Torre, J. G. d. l. J. Chem. Phys. 1980, 73, 1986-1993). In addition, the translational and rotational movements of individual GNRs transported by kinesin motor protein on microtubules inside living cells were directly imaged. Compared to its motion in free solution, a GNR attached to motor-protein did not rotate significantly while moving forward. Our method can be further generalized to allow determination of three-dimensional orientation of single dipoles using many different illumination modes.     

A practical integral equation for the structure and thermodynamics of hard sphere Coulomb fluids

A closure for the Ornstein-Zernike equation is presented, applicable for fluids of charged, hard spheres. From an exact, but intractable closure, we derive the radial distribution function of nonlinearized Debye-Hückel theory by subsequent approximations, and use the information to formulate a new closure by an extension of the mean spherical approximation. The radial distribution functions of the new closure, coined Debye-Hückel-extended mean spherical approximation, are in excellent agreement with those resulting from the hyper-netted chain approximation and molecular dynamics simulations, in the regime where the latter are applicable, except for moderately dilute systems at low temperatures where the structure agrees at most qualitatively. The method is numerically more efficient, and more important, convergent in the entire temperature-density plane. We demonstrate that the method is accurate under many conditions for the determination of the structural and thermodynamic properties of homogeneous, symmetric hard-sphere Coulomb systems, and estimate it to be a valuable basis for the formulation of density functional theories for inhomogeneous or highly asymmetric systems.     

Exploiting rotational and translational invariance of the energy in derivative calculations in quantum chemistry

It is shown how the invariance of the Born—Oppenheimer potential energy to overall translations and rotations of a molecule can be used to reduce the computational labor required for derivative evaluations at various orders.     

Structure of nonuniform hard sphere fluids from shifted linear truncations of functional expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.     

Phase equilibria and plate-fluid interfacial tensions for associating hard sphere fluids confined in slit pores

The excess Helmholtz free energy functional for associating hard sphere fluid is formulated by using a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)]. Within the framework of density functional theory, the thermodynamic properties including phase equilibria for both molecules and monomers, equilibrium plate-fluid interfacial tensions and isotherms of excess adsorption, average molecule density, average monomer density, and plate-fluid interfacial tension for four-site associating hard sphere fluids confined in slit pores are investigated. The phase equilibria inside the hard slit pores and attractive slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal and the plate-fluid interfacial tensions at equilibrium states are predicted consequently. The influences of association energy, fluid-solid interaction, and pore width on phase equilibria and equilibrium plate-fluid interfacial tensions are discussed.