Biomedical subjects

Frank H Stillinger

Publications and source records attributed to Frank H Stillinger.

24 records · Page 2Linked to original sources

PubMed2004

Unusually dense crystal packings of ellipsoids.

In this Letter, we report on the densest-known packings of congruent ellipsoids. The family of new packings consists of crystal arrangements of spheroids with a wide range of aspect ratios, and with density phi always surpassing that of the densest Bravais lattice packing phi approximately equal to 0.7405. A remarkable maximum density of phi approximately equal to 0.7707 is achieved for maximal aspect ratios larger than sqrt[3], when each ellipsoid has 14 touching neighbors. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

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PubMed2004

Improving the density of jammed disordered packings using ellipsoids.

Packing problems, such as how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction phi=pi/18 approximately 0.74. It is also well known that certain random (amorphous) jammed packings have phi approximately 0.64. Here, we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely-up to phi= 0.68 to 0.71 for spheroids with an aspect ratio close to that of M&M's Candies-and even approach phi approximately 0.74 for ellipsoids with other aspect ratios. We suggest that the higher density is directly related to the higher number of degrees of freedom per particle and thus the larger number of particle contacts required to mechanically stabilize the packing. We measured the number of contacts per particle Z approximately 10 for our spheroids, as compared to Z approximately 6 for spheres. Our results have implications for a broad range of scientific disciplines, including the properties of granular media and ceramics, glass formation, and discrete geometry.

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PubMed2003

Local density fluctuations, hyperuniformity, and order metrics.

Questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges. The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d. Our specific objectives are to study the variance in the number of points contained within a regularly shaped window Omega of arbitrary size, and to further illuminate our understanding of hyperuniform systems, i.e., point patterns that do not possess infinite-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area (rather than the volume) of the window. We derive two formulations for the number variance: (i) an ensemble-average formulation, which is valid for statistically homogeneous systems, and (ii) a volume-average formulation, applicable to a single realization of a general point pattern in the large-system limit. The ensemble-average formulation (which includes both real-space and Fourier representations) enables us to show that a homogeneous point pattern in a hyperuniform state is at a "critical point" of a type with appropriate scaling laws and critical exponents, but one in which the direct correlation function (rather than the pair correlation function) is long ranged. We also prove that the non-negativity of the local number variance does not add a new realizability condition on the pair correlation. The volume-average formulation is superior for certain computational purposes, including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance. We prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one-dimensional hyperuniform patterns. We also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one, two, and three dimensions for spherical windows, enabling us to rank-order the spatial patterns. Our results suggest that the local variance may serve as a useful order metric for general point patterns. Contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension, we show that for d=3, the body-centered cubic lattice has a smaller variance than the face-centered cubic lattice. Finally, for certain hyperuniform disordered point patterns, we evaluate the direct correlation function, structure factor, and associated critical exponents exactly.

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PubMed2003

Phase transitions, Kauzmann curves, and inverse melting.

Walter Kauzmann's classic 1948 review of liquid supercooling and glass formation drew attention to the temperatures at which (by extrapolation) enthalpies and entropies of liquid and crystal phases would appear to become equal. In the temperature-pressure (T, p) plane, the collection of such 'Kauzmann temperatures' generate characteristic curves. The present study examines the connection of those Kauzmann loci to equilibrium inverse melting phenomena, i.e. cases where isobaric heating causes freezing of the liquid. Such cases are associated with local minima or maxima in the melting curve p(m)(T), and we point out the possible relevance of melting curve maxima to the thermodynamics of protein folding. Both equal-enthalpy and equal-entropy Kauzmann curves must pass through melting curve extrema. Three thermodynamic identities have been obtained to describe the vicinity of these points; they involve, respectively, the slopes of the two Kauzmann curves, and the second temperature derivative of the melting pressure. The second of these three equations is formally identical to the first Ehrenfest relation for second-order phase transitions, but carries no phase-transition implication. For purposes of specific numerical illustration, the inverse-melting behavior displayed by (3)He at low temperature has been analyzed in detail.

Helium↗

PubMed2002

Duality relations for elastic constants of the classical Gaussian core model.

The many-body Gaussian core model involves a potential energy function that consists of a sum of repelling pair interactions, each of which is a simple Gaussian function of distance. This paper examines the linear elastic response of the model for its stable lattices at absolute zero temperature, in D=1, 2, and 3 dimensions. Owing to the fact that the Gaussian function is self-similar under Fourier transformation, exact relations exist connecting each elastic constant at number density rho to a partner at dual density rho(') in the reciprocal lattice, where rhorho(')=pi(-D). By using these identities, it has been possible to show that shear elastic constants in D=2 and 3 tend strongly to zero in the asymptotic high density limit.

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PubMed2002

Diversity of order and densities in jammed hard-particle packings.

Recently the conventional notion of random close packing has been supplanted by the more appropriate concept of the maximally random jammed (MRJ) state. This inevitably leads to the necessity of distinguishing the MRJ state among the entire collection of jammed packings. While the ideal method of addressing this question would be to enumerate and classify all possible jammed hard-sphere configurations, practical limitations prevent such a method from being employed. Instead, we generate numerically a large number of representative jammed hard-sphere configurations (primarily relying on a slight modification of the Lubachevsky-Stillinger algorithm to do so) and evaluate several commonly employed order metrics for each of these packings. Our investigation shows that, even in the large-system limit, jammed systems of hard spheres can be generated with a wide range of packing fractions from phi approximately 0.52 to the fcc limit (phi approximately 0.74). Moreover, at a fixed packing fraction, the variation in the order can be substantial, indicating that the density alone does not uniquely characterize a packing. Interestingly, each order metric evaluated yielded a relatively consistent estimate for the packing fraction of the maximally random jammed state (phi(MRJ) approximately 0.63). This estimate, however, is compromised by the weaknesses in the order metrics available, and we propose several guiding principles for future efforts to define more broadly applicable metrics.

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