The scattering functions of ideal tri-functional comb and dendrimer polymers.

The scattering functions of nine generations of ideal tri-functional comb and dendrimer polymers are computed by novel graph techniques. The properties of polymers having from 9 to 3069 branches are explored. The g-ratios and the scattering functions indicate that as the number of branches increases, comb polymers behave more and more like linear polymers with half the number of branches, whereas dendrimers become more like spherical objects.

Molecular dynamics study of six-dimensional hard hypersphere crystals.

Six-dimensional hard hypersphere systems in the A6, D6, and E6 crystalline phases have been studied using event-driven molecular dynamics simulations in periodic, skew cells that reflect the underlying lattices. In all the simulations, the systems had sufficient numbers of hyperspheres to capture the first coordination shells, and the larger simulations also included the complete second coordination shell. The equations of state, for densities spanning the fluid, metastable fluid, and solid regimes, were determined. Using molecular dynamics simulations with the hyperspheres tethered to lattice sites allowed the computation of the free energy for each of the crystal lattices relative to the fluid phase. From these free energies, the fluid-crystal coexistence region was determined for the E6, D6, and A6 lattices. Pair correlation functions for all the examined states were computed. Interestingly, for all the states examined, the pair correlation functions displayed neither a split second peak nor a shoulder in the second peak. These behaviors have been previously used as a signature of the freezing of the fluid phase for hard hyperspheres in two to five dimensions.

Fluid-solid demixing in four and five dimensional asymmetric binary hard hypersphere mixtures.

Additive asymmetric binary mixtures of hard hyperspheres in four and five dimensions are investigated by Monte Carlo simulations. These investigations probe systems with diameter ratios of 0.4 and 0.5 in which the larger hyperspheres are dominant at a mole fraction of 3/4. At the lower densities, the equations of state compare well with molecular dynamics data and a variety of theoretical predictions. When the mixture enters the metastable, two-phase regime, the smaller hyperspheres exhibit unusual phenomena as the system density increases. To understand this behavior, the mean-square displacement at equilibrium from initial lattice positions, the various pair correlation functions, and occupancy numbers are calculated. In addition, the characteristics of an initially demixed system are studied.

Five dimensional binary hard hypersphere mixtures: A Monte Carlo study.

Additive binary mixtures of five dimensional hyperspheres were investigated by Monte Carlo simulations. Both equal packing fraction and equal mole fraction systems with diameter ratios of 0.4 and 0.5 were examined. A range of total densities were studied, spanning low to moderate density fluids. The pair correlation functions and the equations of state were determined and compared with molecular dynamics data and a variety of theoretical predictions. A significant result of the equal packing fraction simulations was the discovery of how quickly the larger hyperspheres reorganized into a dense fluid after a random initial placement. In the equal mole fraction case, the pair correlation functions for the larger hypersphere agree with the pair correlation function of a pure fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.

The shapes of simple three and four junction comb polymers.

A scheme originally proposed by G. Wei [Physica A 222, 152 (1995); Physica A 222, 155 (1995)] is redesigned to produce numerical shape parameters of arbitrary tree-branched polymers based on the Kirchhoff matrix eigenvalue spectrum. This method and two different Monte Carlo techniques (pivot and growth) are employed to investigate the asphericity of three and four junction comb polymers in both the ideal and excluded volume regimes. It is found that the extrapolated g-ratio and asphericity values obtained by all of these methods are in excellent agreement with each other and the available theory in the ideal regime and that polymers with a complete set of interior branches display a more sphere-like shape.

Phase transitions in four-dimensional binary hard hypersphere mixtures.

Previous Monte Carlo investigations of binary hard hyperspheres in four-dimensional mixtures are extended to higher densities where the systems may solidify. The ratios of the diameters of the hyperspheres examined were 0.4, 0.5, and 0.6. Only the 0.4 system shows a clear two phase, solid-liquid transition and the larger component solidifies into a D4 crystal state. Its pair correlation function agrees with that of a one component fluid at an appropriately scaled density. The 0.5 systems exhibit states that are a mix of D4 and A4 regions. The 0.6 systems behave similarly to a jammed state rather than solidifying into a crystal. No demixing into two distinct fluid phases was observed for any of the simulations.

Monte Carlo study of four dimensional binary hard hypersphere mixtures.

A multithreaded Monte Carlo code was used to study the properties of binary mixtures of hard hyperspheres in four dimensions. The ratios of the diameters of the hyperspheres examined were 0.4, 0.5, 0.6, and 0.8. Many total densities of the binary mixtures were investigated. The pair correlation functions and the equations of state were determined and compared with other simulation results and theoretical predictions. At lower diameter ratios the pair correlation functions of the mixture agree with the pair correlation function of a one component fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.

The fluid to solid phase transition of hard hyperspheres in four and five dimensions.

Molecular dynamics and Monte Carlo simulations are performed for four- and five-dimensional hard hyperspheres at a variety of densities, ranging from the fluid state to the solid regime of A(4), D(4), D(4)*, and D(5) lattices. The equation of state, the radial distribution functions, and the average number of hyperspheres in a coordination layer are determined. The equations of state are in excellent agreement with values obtained from both theoretical approaches and other simulations. The results for the average number of hyperspheres in a coordination layer are in agreement with the theoretical predictions for the different lattices. The radial distribution function gives better insight about the fluid to solid transition than the equation of state.

The form factor of H-comb polymers.

A Monte Carlo pivot algorithm is employed to investigate the form factor of continuum, tangent hard sphere H-comb polymers in both the ideal and excluded volume regimes. The simulated form factors for 241 and 931 "bead" ideal H-combs are essentially the same. The results for these polymers are in excellent agreement with the theoretical prediction. There is only a slight difference in the form factor between the ideal and excluded volume regimes at larger values of distance.

The shapes of H-comb polymers.

The Monte Carlo pivot algorithm is employed to investigate the shapes of continuum, tangent hard sphere H-comb polymers in both the ideal and excluded volume regimes. Polymers with a number of units ranging from 241 to 931 have been simulated. It is found that the extrapolated asphericity values are in excellent agreement with theory in the ideal regime and that, somewhat, higher values occur in the excluded volume regime.

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.

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.

Molecular dynamics study of the thermodynamics and transport coefficients of hard hyperspheres in six and seven dimensions.

Molecular dynamics (MD) simulations are performed for six- and seven-dimensional hard-hypersphere fluids. The equation of state, velocity autocorrelation function, self-diffusion coefficient, shear viscosity, and thermal conductivity are determined as a function of density. The molecular dynamics results for the equation of state are found to be in excellent agreement with values obtained from theoretical approaches and previous MD simulations in seven dimensions. The short-time behavior of the velocity autocorrelation function is well described by the Enskog exponential approximation. The Enskog predictions for the self-diffusion coefficient and the viscosity agree fairly well with the simulation data at low densities, but underestimate these quantities at higher densities. Data for the thermal conductivity are in fine agreement with Enskog theory for all densities and dimensions studied.

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.

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.

The structure of hyperspherical fluids in various dimensions.

The structure of hard, hyperspherical fluids in dimension one, two, three, four, and five has been examined by calculating the pair correlation function using a Monte Carlo simulation. The pair correlation functions match known results in one, two, and three dimensions. The contact value of the pair correlation functions in all the different dimensions agrees well with the theory of Song, Mason, and Stratt [J. Phys. Chem. 93, 6916 (1989)]. The decrease in ordering as the dimension is increased is readily apparent in the structure of the pair correlation function.

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 behavior of the form factor in two dimensions for linear polymers in different regimes.

Off-lattice Monte Carlo simulations employing the PIVOT algorithm are used to generate ideal and excluded volume linear polymers in two dimensions. The form factor at small and large wave vectors is calculated from the resulting configurations and compared to the exact equation for ideal chains and to both scaling and renormalization group predictions for excluded volume chains. It is found that using the des Cloizeaux form for the distance distribution function in an analytic calculation of the form factor leads to close agreement with the Monte Carlo data and that simple expressions for both the small and large wave vector expansions reproduce the essential features of the form factor.