Biomedical subjects

Frank H Stillinger

Publications and source records attributed to Frank H Stillinger.

At least 19 recordsLinked to original sources

PubMed2006

Scaled particle theory for hard sphere pairs. I. Mathematical structure.

We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to a prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the large-cavity regime where macroscopic properties become relevant. Closure conditions emerge which produce a nonlinear integral equation that must be satisfied by the pair correlation function. This integral equation has a structure which straightforwardly generates a solution that is a power series in density. The results of this series replicate the exact second and third virial coefficients for the hard sphere system via the contact value of the pair correlation function. The predicted fourth virial coefficient is approximately 0.6% lower than the known exact value. Detailed numerical analysis of the nonlinear integral equation has been deferred to the subsequent paper.

Journal Article↗

PubMed2006

Scaled particle theory for hard sphere pairs. II. Numerical analysis.

We use the extension of scaled particle theory presented in the accompanying paper [F. H. Stillinger et al., J. Chem. Phys. 125, 204504 (2006)] to calculate numerically the pair correlation function of the hard sphere fluid over the density range 0< or =rhosigma(3)< or =0.96. Comparison with computer simulation results reveals that the new theory is able to capture accurately the fluid's structure across the entire density range examined. The pressure predicted via the virial route is systematically lower than simulation results, while that obtained using the compressibility route is lower than simulation predictions for rhosigma(3)< or =0.67 and higher than simulation predictions for rhosigma(3)> or =0.67. Numerical predictions are also presented for the surface tension and Tolman length of the hard sphere fluid.

Journal Article↗

PubMed2006

Computational probes of molecular motion in the Lewis-Wahnstrom model for ortho-terphenyl.

We use molecular dynamics simulations to investigate translational and rotational diffusion in a rigid three-site model of the fragile glass former ortho-terphenyl, at 260 K< or =T< or =346 K and ambient pressure. An Einstein formulation of rotational motion is presented, which supplements the commonly used Debye model. The latter is shown to break down at supercooled temperatures as the mechanism of molecular reorientation changes from small random steps to large infrequent orientational jumps. We find that the model system exhibits non-Gaussian behavior in translational and rotational motion, which strengthens upon supercooling. Examination of particle mobility reveals spatially heterogeneous dynamics in translation and rotation, with a strong spatial correlation between translationally and rotationally mobile particles. Application of the Einstein formalism to the analysis of translation-rotation decoupling results in a trend opposite to that seen in conventional approaches based on the Debye formalism, namely, an enhancement in the effective rate of rotational motion relative to translation upon supercooling.

Journal Article↗

PubMed2006

Packing hyperspheres in high-dimensional Euclidean spaces.

We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be phi(MRJ) approximately 0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form phi(MRJ)=c1/2(d)+(c2d)/2d, where c1=-2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed "decorrelation principle," which, among other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d-->infinity. As in three dimensions (where phi(MRJ) approximately 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent approximately 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, phi(F) approximately 0.32 and phi(M) approximately 0.39, respectively, for d=4, and phi(F) approximately 0.20 and phi(M) approximately 0.25, respectively, for d=5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.

Journal Article↗

PubMed2006

Collective coordinate control of density distributions.

Real collective density variables C(k) [cf. Eq. 1 3 ] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S(k) , where k denotes the wave vector. Our objective is to prescribe C(k) and then to find many-particle configurations that correspond to such a target C(k) using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control S(k) in the neighborhood of |k|=0. The optimization method employed generates multiparticle configurations for which S(k) proportional, |k|alpha, |k|<or=K, and alpha=1, 2, 4, 6, 8, and 10. The case alpha=1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid 4He, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.

Journal Article↗

PubMed2006

Self-assembly of the simple cubic lattice with an isotropic potential.

Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures [M. C. Rechtsman, Phys. Rev. Lett. 95, 228301 (2005); Phys. Rev. E 73, 011406 (2006)], we present an isotropic pair potential V(r) for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems for the purpose of achieving targeted self-assembly. The purpose of these methods is to design interparticle interactions that cause self-assembly of technologically important target structures for applications in photonics, catalysis, separation, sensors, and electronics. We also show that standard approximate integral-equation theories of the liquid state that utilize pair correlation function information cannot be used in the reverse mode to predict the correct simple cubic potential. We report in passing optimized isotropic potentials that yield the body-centered-cubic and simple hexagonal lattices, which provide other examples of non-close-packed structures that can be assembled using isotropic pair interactions.

Journal Article↗

PubMed2006

Global phase diagram for the honeycomb potential.

We calculate the global phase diagram using classical statistical mechanics for an isotropic pair potential that has been previously [Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005)] shown to produce the low-coordinated two-dimensional honeycomb crystal as the ground-state structure. Low-coordinated crystals are of practical interest because they have desirable photonic band-gap properties. The phase diagram is obtained from Helmholtz free energies calculated using thermodynamic integration and Monte Carlo simulations. Our results show that the honeycomb crystal remains stable in the global phase diagram even after temperature effects are taken fully into account. Other stable phases in the phase diagram are high and low density triangular phases and a fluid phase. We find no evidence of gas-liquid or liquid-liquid phase coexistence.

Journal Article↗

PubMed2006

Do binary hard disks exhibit an ideal glass transition?

We demonstrate that there is no ideal glass transition in a binary hard-disk mixture by explicitly constructing an exponential number of jammed packings with densities spanning the spectrum from the accepted amorphous glassy state to the phase-separated crystal. Thus the configurational entropy cannot be zero for an ideal amorphous glass, presumed distinct from the crystal in numerous theoretical and numerical estimates in the literature. This objection parallels our previous critique of the idea that there is a most-dense random (close) packing for hard spheres [Torquato, Phys. Rev. Lett. 84, 2064 (2000)10.1103/PhysRevLett.84.2064].

Journal Article↗

PubMed2005

Generating inherent structures of liquids: comparison of local minimization algorithms.

Properties of minima obtained by two different local minimization algorithms, the conjugate gradient and the limited-memory Broyden-Fletcher-Goldfarb-Shanno, are compared for a liquid whose particles interact via a modified Lennard-Jones pair interaction. The average properties of inherent structures obtained by the two minimization procedures are shown to agree within statistical error, though for a given starting configuration the two algorithms may not quench to identical minima.

Journal Article↗

PubMed2005

Optimized interactions for targeted self-assembly: application to a honeycomb lattice.

We devise an inverse statistical-mechanical methodology to find optimized interaction potentials that lead spontaneously to a target many-particle configuration. Target structures can possess varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. For illustration purposes, our computational technique is applied to yield an optimized isotropic (nondirectional) pair potential that spontaneously yields the three-coordinated honeycomb lattice as the ground state structure in two dimensions. This target choice is motivated by its three-dimensional analog, the diamond lattice, which is known to possess desirable photonic band gap properties.

Biomechanical Phenomena↗

PubMed2005

Novel computational probes of diffusive motion.

Self-diffusion constants, D, and the atomic-level processes that produce them have been investigated numerically for the binary-mixture Lennard-Jones (BMLJ) model and for liquid silica as described by the Van Beest-Kramer-Van Santen interaction model. The primary conceptual tool for this study is the joint probability distribution for single particles as a function of initial velocity and positional displacement at a given later instant. Self-diffusion constants can be expressed exactly in terms of this probability function. The numerical simulations for the BMLJ case reveal an unusual temperature effect; in contrast to the high-temperature behavior, particles with high initial velocities experience disproportionate retardation in forward displacement. In the silica modeling simulations, diffusive processes have been compared at constant-temperature "isodiffusive" pairs of states, demonstrating a significant role played by the amount of local tetrahedral order that is present in the medium.

Journal Article↗

PubMed2005

Unexpected density fluctuations in jammed disordered sphere packings.

We computationally study jammed disordered hard-sphere packings as large as a million particles. We show that the packings are saturated and hyperuniform, i.e., that local density fluctuations grow only as a logarithmically augmented surface area rather than the volume of the window. The structure factor shows an unusual nonanalytic linear dependence near the origin, S(k) approximately |k|. In addition to exponentially damped oscillations seen in liquids, this implies a weak power-law tail in the total correlation function, h(r) approximately -r(-4), and a long-ranged direct correlation function c(r).

Journal Article↗

PubMed2005

Experiments on random packings of ellipsoids.

Recent simulations indicate that ellipsoids can pack randomly more densely than spheres and, remarkably, for axes ratios near 1.25:1:0.8 can approach the densest crystal packing (fcc) of spheres, with a packing fraction of 74%. We demonstrate that such dense packings are realizable. We introduce a novel way of determining packing density for a finite sample that minimizes surface effects. We have fabricated ellipsoids and show that, in a sphere, the radial packing fraction phi(r) can be obtained from V(h), the volume of added fluid to fill the sphere to height h. We also obtain phi(r) from a magnetic resonance imaging scan. The measurements of the overall density phi(avr), phi(r) and the core density phi(0) = 0.74 +/- 0.005 agree with simulations.

Journal Article↗

PubMed2005

Alternative view of self-diffusion and shear viscosity.

By performing an elementary transformation, the conventional velocity autocorrelation function expression for the temperature and density dependent self-diffusion constant D(T,rho) has been reformulated to emphasize how initial particle momentum biases final mean displacement. Using collective flow variables, an analogous expression has been derived for 1/eta(T,rho), the inverse of shear viscosity. The Stokes-Einstein relation for liquids declares that D and T/eta should have a fixed ratio as T and rho vary, but experiment reveals substantial violations for deeply supercooled liquids. Upon analyzing the self-diffusion and viscous flow processes in terms of configuration space inherent structures and kinetic transitions between their basins, one possible mechanism for this violation emerges. This stems from the fact that interbasin transitions become increasingly Markovian as T declines, and though self-diffusion is possible in a purely Markovian regime, shear viscosity in the present formulation intrinsically relies on successive correlated transitions.

Journal Article↗

PubMed2005

Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings.

We study the approach to jamming in hard-sphere packings and, in particular, the pair correlation function g(2) (r) around contact, both theoretically and computationally. Our computational data unambiguously separate the narrowing delta -function contribution to g(2) due to emerging interparticle contacts from the background contribution due to near contacts. The data also show with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic: i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces P(f) (f) , which we measure computationally, and the contact contribution to g(2) . We verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state. We clearly observe a maximum in P(f) and a nonzero probability of zero force, shedding light on long-standing questions in the granular-media literature. We computationally observe an unusual power-law divergence in the near-contact contribution to g(2) , persistent even in the jamming limit, with exponent -0.4 clearly distinguishable from previously proposed inverse-square-root divergence. Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of g(2) and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. Finally, we present the computational data on the contact contribution to g(2) for vacancy-diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. We find that the contact network remains isostatic even when ordering is present. Unlike previous studies, we find that ordering has a significant impact on the shape of P(f) for small forces.

Journal Article↗

PubMed2004

Epitaxial frustration in deposited packings of rigid disks and spheres.

We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered fixed line of larger disks, where the diameter ratio of movable to fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations is 0.7, where the fixed substrate has the structure of a (001) plane of a face-centered cubic array. Results obtained for both disks and spheres exhibit various forms of density-reducing packing frustration next to the incommensurate substrate, including some cases displaying disorder that extends far from the substrate. The disk system calculations strongly suggest that the most efficient (highest-density) packings involve configurations that are periodic in the lateral direction parallel to the substrate, with substantial geometric disruption occurring only near the substrate. Some evidence has also emerged suggesting that for the sphere systems a corresponding structure doubly periodic in the lateral directions would yield the highest packing density; however, all of the sphere simulations completed thus far produced some residual "bulk" disorder not obviously resulting from substrate mismatch. In view of the fact that the cases studied here represent only a small subset of all that eventually deserve attention, we end with discussion of the directions in which first extensions of the present simulations might profitably be pursued.

Journal Article↗

PubMed2004

Constraints on collective density variables: two dimensions.

Collective density variables rho (k) have proved to be useful tools in the study of many-body problems in a variety of fields that are concerned with structural and kinematic phenomena. In spite of their broad applicability, mathematical understanding of collective density variables remains an underexplored subject. In this paper, we examine features associated with collective density variables in two dimensions using numerical exploration techniques to generate particle patterns in the classical ground state. Particle pair interactions are governed by a continuous, bounded potential. Our approach involves constraining related collective parameters C (k) , with wave vector k magnitudes at or below a chosen cutoff, to their absolute minimum values. Density fluctuations for those k 's thus are suppressed. The resulting investigation distinguishes three structural regimes as the number of constrained wave vectors is increased-disordered, wavy crystalline, and crystalline regimes-each with characteristic distinguishing features. It should be noted that our choice of pair potential can lead to pair correlation functions that exhibit an effective hard core and thus signal the formation of a hard-disk-like equilibrium fluid. In addition, our method creates particle patterns that are hyperuniform, thus supporting the notion that structural glasses can be hyperuniform as the temperature T-->0 .

Journal Article↗

PubMed2004

Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder".

Phys. Rev. E. 68, 011306 (2003)] claim that a special point J of a "jamming phase diagram" (in density, temperature, stress space) is related to random close packing of hard spheres and that it represents, for their suggested definitions of jammed and random, the recently introduced maximally random jammed state. We point out several difficulties with their definitions and question some of their claims. Furthermore, we discuss the connections between their algorithm and other hard-sphere packing algorithms in the literature.

Comment↗