Higher virial coefficients of four and five dimensional hard hyperspheres

The seventh and eighth virial coefficients for hard hyperspheres are calculated by Monte Carlo techniques. It is found that B(7)/B(2) (6)=0.001 43+/-0.000 13 and 0.000 44+/-0.000 12 in four and five dimensions, respectively, and that B(8)/B(2) (7)=0.000 414+/-0.000 20 in four dimensions. These values are used to investigate various proposed equations of state. Comparisons against the molecular dynamics calculations of Luban and Michels show that their proposed semiempirical form is excellent at higher densities. Moreover, we confirm Santos observation in five dimensions that a suitable linear combination of the Percus-Yevick compressibility and virial equations of state fits the molecular dynamics data nearly as well as any other proposed form.     

The equation of state of hard hyperspheres in four and five dimensions

The equation of state of hard hyperspheres in four and five dimensions is calculated from the value of the pair correlation function at contact, as determined by Monte Carlo simulations. These results are compared to equations of state obtained by molecular dynamics and theoretical approaches. In all cases the agreement is excellent.     

Structure factor for hard hyperspheres in higher dimensions

The structure factor for hard hyperspheres in two to eight dimensions is computed by Fourier transforming the pair correlation function obtained by computer simulation at a variety of densities. The resulting structure factors are compared to the known Percus-Yevick equations for odd dimensions and to the model proposed by Leutheusser [J. Chem. Phys. 84, 1050 (1986)] and Rosenfeld [J. Chem. Phys. 87, 4865 (1987)] in even dimensions. It is found that there is fine agreement among all these approaches at low to moderate densities but that the accuracy of the analytical models breaks down as the freezing transition is approached. The structure factor gives another insight into the decrease in the ordering of the hyperspheres as the dimension is increased.     

Virial series for fluids of hard hyperspheres in odd dimensions

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients B(j) are expressed in terms of the solution of a set of (d-1)/2 coupled algebraic equations which become nonlinear for d>/=5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d>/=5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2eta(1/d), where eta is the packing fraction, is wholly consistent with the limiting value of 1 for d-->infinity. Finally, the values for B(j) predicted by the virial and compressibility routes in the Percus-Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)].     

The equation of state of hard hyperspheres in nine dimensions for low to moderate densities

The equation of state of hard hyperspheres in nine dimensions is calculated both from the values of the first ten virial coefficients and from a Monte Carlo simulation of the pair correlation function at contact. The results are in excellent agreement. In addition, we find that the virial series appears to be dominated by an unphysical singularity or singularities on or near the negative density axis, in qualitative agreement with the recently solved Percus-Yevick equation of state in nine dimensions.     

Loop-erased self-avoiding random walks in four and five dimensions

Monte Carlo simulations of loop-erased self-avoiding random walks in four and five dimensions were performed, using two distinct algorithms. We find consistency between these methods in their estimates of critical exponents. The upper critical dimension for this phenomenon is four, and it has been shown that the mean square end-to-end distance grows as n(log n)^α. It has recently been established that the mean square end-to-end distance is asymptotically bounded by n(log n)^1/3 (see Ref. 21). Our results show that asymptotic convergence to n(log n)^1/3 in fact obtains and does so rather quickly. In five dimensions we examine the rate of asymptotic convergence to the mean-field model. © 1996 by John Wiley & Sons, Inc.     

The eighth virial coefficient of four- and five-dimensional hard hyperspheres

The eighth virial coefficient for hard hyperspheres is calculated by Monte Carlo techniques. It is found that B8/B(7)2=0.000 274+/-0.000 014 and -0.000 115+/-0.000 012 in four and five dimensions, respectively. The results are in good agreement with the findings of Clisby and McCoy (e-print arXiv:cond-mat0410511), and confirm that B8 is negative in five dimensions.     

van der Waals-Tonks-type equations of state for hard-hypersphere fluids in four and five dimensions

Recently, we developed accurate van der Waals-Tonks-type equations of state for hard-disk and hard-sphere fluids by using the known virial coefficients. In this paper, we derive the van der Waals-Tonks-type equations of state. We further apply these equations of state to hard-hypersphere fluids in four and five dimensions. In the low-density fluid regime, these equations of state are in good agreement with the simulation results and existing equations of state.     

Perturbation expansion of macromolecules with realistic pair interactions in three and four dimensions

We calculated the first-order perturbation expansion of the mean-square end-to-end distance of polymers in three and four space dimensions. The segments of the chain were assumed to interact via a pair potential with a short-range repulsive and an additional attractive part. For a purely repulsive potential, the well-known results of the δ-function-pseudopotential approach are recovered, whereas the inclusion of an attractive part leads to a non-vanishing contribution even for a vanishing binary cluster integral.     

Chain dimensions and transport coefficients of sodium poly(isoprenesulfonate) in aqueous sodium chloride

Sodium poly(isoprenesulfonate) (NaPIS) fractions consisting of 1,4‐ and 3,4‐isomeric units (0.44:0.56) and ranging in molecular weight from 4.9 × 10³ to 2.0 × 10⁵ were studied by static and dynamic light scattering, sedimentation equilibrium, and viscometry in aqueous NaCl of a salt concentration (C_s) of 0.5‐M at 25 °C. Viscosity data were also obtained at C_s = 0.05, 0.1, and 1 M. The measured z‐average radii of gyration 〈S²〉_z^1/2, intrinsic viscosities [η], and translational diffusion coefficients D at C_s = 0.5‐M showed that high molecular weight NaPIS in the aqueous salt behaves like a flexible chain in the good solvent limit. On the assumption that the distribution of 1,4‐ and 3,4‐isomeric units in the NaPIS chain is completely random, the [η] data for high molecular weights at C_s = 0.5 and 1 M were analyzed first in the conventional two‐parameter scheme to estimate the unperturbed dimension at infinite molecular weight and the mean binary cluster integral. By further invoking a coarse‐graining of the NaPIS molecule, all the [η] and D data in the entire molecular weight range were then analyzed on the basis of the current theories for the unperturbed wormlike chain combined with the quasi‐two‐parameter theory. It is shown that the experimental 〈S²〉_z, [η], and D are explained by the theories with a degree of accuracy similar to that known for uncharged linear flexible homopolymers. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2071–2080, 2001     

Scalar fundamental measure theory for hard spheres in three dimensions: application to hydrophobic solvation

Hard-sphere mixtures provide one a solvable reference system that can be used to improve the density functional theory of realistic molecular fluids. We show how the Kierlik-Rosinberg's scalar version of the fundamental measure density functional theory of hard spheres [E. Kierlik and M. L. Rosinberg, Phys. Rev. A 42, 3382 (1990)], which presents computational advantages with respect to the original Rosenfeld's vectorial formulation or its extensions, can be implemented and minimized in three dimensions to describe fluid mixtures in complex environments. This implementation is used as a basis for defining a molecular density functional theory of water around molecular hydrophobic solutes of arbitrary shape.     

Collisions, caging, thermodynamics, and jamming in the barrier hopping theory of glassy hard sphere fluids

An ultralocal limit of the microscopic single particle barrier hopping theory of glassy dynamics is proposed which allows explicit analytic expressions for the characteristic length scales, energy scales, and nonequilibrium free energy to be derived. All properties are shown to be controlled by a single coupling constant determined by the fluid density and contact value of the radial distribution function. This parameter quantifies an effective mean square force exerted on a tagged particle due to collisions with its surroundings. The analysis suggests a conceptual basis for previous surprising findings of multiple inter-relationships between characteristics of the transient localized state, the early stages of cage escape, non-Gaussian or dynamic heterogeneity effects, and the barrier hopping process that defines the alpha relaxation event. The underlying physical picture is also relevant to fluids of nonspherical molecules and sticky colloidal suspensions. The possibility of a unified view of liquid dynamics is suggested spanning the range from dense gases to the zero mobility jammed state.     

Electrostatic potentials in systems periodic in one, two, and three dimensions

We consider the electrostatic potential in a unit cell containing N point charges Q(j) with positions r(j) inside the cell. The cell is replicated periodically in one, two, and three dimensions. The purpose is to give representations for the potential which contain only lattice sums which are absolutely convergent and uniformly convergent in the sampling position r. These representations are derived using variants of the Ewald method and are primarily intended for use in evaluating the accuracy of any algorithm to evaluate electrostatic energies and forces in simulations of dense matter, rather than necessarily for use of themselves in simulations. In reduced dimensionality the Ewald representations can be numerically inefficient and other representations are also provided with careful specification which allows two forms to be used for the potential functions in order to improve numerical performance. These mixed representations may be satisfactory in simulations.     

Integrin clustering in two and three dimensions

Integrins are transmembrane proteins that allow cells to bind to their external environment. They are the primary regulators of cell-matrix interactions, with direct roles in cell motility and signaling, which in turn regulate numerous physiological processes. Under common experimental conditions, integrins tend to cluster for sturdy and effective binding to extracellular matrix molecules. These clusters often evolve into focal adhesions, which regulate downstream signaling. However, integrin clusters are more pronounced and have longer lifetimes in two-dimensional assays than in more realistic three-dimensional environments. While a number of models and theoretical approaches have focused on integrin binding and diffusion, the reasons for the differences between two- and three-dimensional clustering have remained elusive. In this study, we model an individual cluster attached to a two-dimensional collagen film and attached to collagen fibers of various sizes in three-dimensional matrices. We then discuss how our results explain differences in size and lifetime, and how they hint at reasons for other differences between the two environments. Further, we make predictions regarding the stability of clusters based on different overall intracellular conditions. Our results show good agreement with experiments and provide a quantitative basis for understanding how matrix dimensionality and structure regulate integrin behavior in environments that mimic in vivo conditions.     

On the relation between virial coefficients and the close-packing of hard disks and hard spheres

The question of whether the known virial coefficients are enough to determine the packing fraction η(∞) at which the fluid equation of state of a hard-sphere fluid diverges is addressed. It is found that the information derived from the direct Pade? approximants to the compressibility factor constructed with the virial coefficients is inconclusive. An alternative approach is proposed which makes use of the same virial coefficients and of the equation of state in a form where the packing fraction is explicitly given as a function of the pressure. The results of this approach both for hard-disk and hard-sphere fluids, which can straightforwardly accommodate higher virial coefficients when available, lends support to the conjecture that η(∞) is equal to the maximum packing fraction corresponding to an ordered crystalline structure.     

Ionization of five aqueous fluorobenzoic acids: Conductance and thermodynamics

The three monofluorobenzoic acids together with 2,4-difluoro and 2,6-difluorobenzoic acids in aqueous solution are the subject of precision conductance measurements. The experimental data are analyzed to give ionization constants and limiting conductances at temperatures from 0 to 100°C. Walden products for the acid anions are derived from the limiting conductances while the ionization consatants are fitted by statistical methods to the function pK _a (m)=A+B/T+ C logT+DT. Only the 2,6- acid requires the fourth term of the function to fit the data to a precision of better than 0.03%. Mathematical analysis of the pK function gives the standard changes in enthalpy, entropy, and heat capacity. All the acids studied are more acidic than the parent, benzoic acid, as well as more acidic than the isoelectronic methylbenzoic acids. In general the increased acidity is tied to decreases in enthalpy while entropy changes on ionization differn little from those found for the parent acid.     

Irreversible thermodynamics of transport across charged membranes : Part III - efficiency of energy conversion in separation processes with nafion 120 membrane from phenomenological transport coefficients

In separation processes with charged membranes, as in electrodialysis units or electrochemical cells, the efficiency of conversion of electrical energy into the concentration gradient of an electrolyte is depressed by the immediate flow of water in the direction opposite to the solute flow. The equations for energy conversion in transport of ions and water across a cation-exchange membrane are derived in the present paper treating the system as a three-flow process and employing phenomenological transport equations. With these equations, the two-flow (q_1E, q_wE, q_1w and overall (q̃_E) degrees of coupling and the total (η) and component (η_1E,η_wE) efficiencies of energy conversion have been computed for the system sodium chloride/Nafion 120 membrane at temperatures of 298 and 333 K and for solute concentrations between 0.05 and 4 M. Considering the continuous separation processes, the so-called “driving region” and the energy requirement to keep the concentration difference in the adjacent compartments constant (static head) have been calculated and discussed.     

Investigation of fundamental transport properties and thermodynamics in diglyme-salt solutions

Ionic mobility, the thermodynamics of ionic association, and the structure of associated species are studied in solutions of diglyme containing either lithium triflate or tetrabutylammonium triflate. Infrared spectroscopic, PFG NMR, thermodynamic, and crystallographic data suggest that the solute species existing in diglyme-lithium triflate are "free" ions, contact ion pairs, and dimers. Equilibrium constants, S(o), deltaH(o), and deltaG(o) are calculated for processes occurring between these species. In particular, the equilibrium constant, corrected for nonideality using a modified Debye-Hückel expression, is calculated for the dissociation of contact ion pairs into "free" cations and anions. A second equilibrium constant for the formation of dimers from contact ion pairs is also calculated; these constants do not significantly vary with salt concentration up to about 1.3 x 10(-3) mol cm(-3). The measured temperature dependence of equilibrium constants was used to calculate deltaH(o) and deltaS(o) for the two processes. The value of deltaS(o) = -102 J mol(-1) K(-1) for the dissociation of contact ion pairs shows that the large entropy decrease due to cation solvation outweighs the entropy increase due to dissociation of a contact ion pair. Ionic mobilities are calculated in lithium triflate-diglyme solutions using conductivity data in conjunction with information about the nature and concentrations of solute species obtained from IR spectroscopy. Mobilities in tetrabutlyammonium triflate-diglyme solutions are calculated directly from conductivity data. It was concluded that the concentration dependence of the molar conductivity is due in large part to the variation of the ion mobilities with concentration.     

Determination of transport coefficients in microporous solids

Macroscopic transient methods are reviewed with respect to their applicability to the investigation of molecular transport in microporous sorption systems. Various levels of sophistication of data evaluation for nonequilibrium sorption results obtained by means of batch methods are identified and characterised. Special attention is paid to the characterisation ofFickian (intracrystalline) diffusion as well as to the identification and quantification of additional rate mechanisms that, in general, may simultaneously occur in molecular sieve systems. A state-of-art determination of transport coefficients is exemplified for the systems benzene/microporous gallosilicate of MFI-type, n-hexane/silicalite-I and p-ethyltoluene/ZSM-5. Their sorption rate behaviour can be understood either byFickian diffusion or byFickian diffusion and intracrystalline molecular immobilisation/mobilisation and surface barrier penetration, respectively. To analyse complex sorption rate patterns in microporous systems, the method oftotal curve fitting with full parameter region consideration becomes mandatory.     

Determination of concentration-dependent transport coefficients in nanofiltration: Experimental evaluation of coefficients

To characterize solute transport in nanofiltration (NF) the Spiegler–Kedem equation requires that two coefficients be determined for two-component solutions (a solute in water), solute permeability ω and reflection coefficient σ. For salts both coefficients strongly and in a complex way depend on concentration, which greatly complicates their evaluation from experiments. For this reason, the parameters are usually assumed constant for a given feed and the concentration dependence is assessed from flux–rejection curves for several feeds. This procedure however ignores the fact that the solute concentration and hence the coefficients significantly vary across the membrane. One way to overcome this inconsistency and address concentration dependence is to use physical models explicitly introducing exclusion mechanism(s) and fitting relevant membrane-specific parameters, such as fixed charge or dielectric properties. This procedure often fails to produce unique values of parameters for a given membrane and different salts. In the present study a new phenomenological approach is proposed and critically analyzed, based on the assumption of a similar concentration dependence of ω and 1 − σ, previously shown to be valid under fairly general conditions, thereby the Peclét coefficient A = (1 − σ)/ω may be assumed to be independent of concentration. The coefficients and their concentration dependence for a given solute may be directly and consistently evaluated by fitting flux–rejection data for several feeds and fluxes to numeric solution of the modified transport equations without the need to invoke specific physical models. The values of transport parameters deduced in this way for representative membranes and salts allow important conclusions regarding the transport mechanism. In particular, the roles of different mechanisms in overall salt exclusion could be addressed directly from the variation of ω or 1 − σ with concentration. On the other hand, the value of the Peclét coefficient, free of the effect of salt partitioning, may be analyzed in terms of hindered transport. Using the proposed method, this value was found to be very small for studied thin-film composite membranes, which may significantly simplify the transport equations.