Researchers: Salvatore Torquato, Thomas M. Truskett, Pablo Debenedetti
Description: Despite its long history, there are many issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing the new concept of a maximally random jammed state, which can be made precise.
Researchers: Chris Yeong and Salvatore Torquato
Description: The reconstruction of the structure of random heterogeneous media, such as porous and composite media, from a knowledge of limited morphological information (correlation functions) is an intriguing inverse problem. An effective reconstruction procedure enables one to generate accurate structures at will, and subsequent analysis can be performed on the image to obtain desired macroscopic properties (e.g. transport, electromagnetic and mechanical properties) of the media. This provides a non-destructive means of estimating the macroscopic properties: a problem of important technological relevance. We formulate a procedure to reconstruct the structure of general random media from limited morphological information. The procedure has the advantages that it is simple to implement and generally applicable to multidimensional, multiphase, and anisotropic structures. Furthermore, an extremely useful feature is that it can incorporate any type and number of correlation functions in order to provide as much morphological information as is necessary for accurate reconstruction. We have used our algorithm to construct heterogeneous media from specified hypothetical correlation functions as well as physically unrealizable ones. We also used the algorithm to carry out an investigation concerning the utilization of morphological information obtained from a two-dimensional slice (thin section) of a random medium (sandstone) to reconstruct the full three-dimensional medium.
Researchers: Anuraag Kansal, Tom Deisboeck, and Salvatore Torquato
Description: We have developed a novel cellular automata model, which simulates the 3D proliferative growth of an MTS brain tumor. This model predicts important clinical data over time in agreement with published clinical and experimental data for a tumor growing over three orders of magnitude in radius. Further research has modeled tumors comprised of two separate subpopulations. The likelihood of a small subpopulation emerging from a larger one has been quantified. In addition, the importance of understanding clonal composition in forming medical prognosis has been underscored. Other work is underway to model the dynamics of invasive tumor growth.
Researchers: Obioma Uche, Salvatore Torquato, and Frank Stillinger
Description: In contrast to the increased attention to 3-dimensional packings, many questions remain regarding the nature of rigid, binary disk systems. This project aims to explore high density structures via a molecular dynamics study of hard disks using the Lubachevsky-Stillinger compression algorithm. We intend to use a recursive packing method to obtain additional dense, bidisperse disk systems. Finally, we will compute correlation functions and coordination number histograms for the acquired systems.
Researchers: Aleksandar Donev, Salvatore Torquato, Frank Stillinger and Robert Connelly
Description: Hard disk and sphere packings can be locally, collectively or strictly jammed. If they are not jammed, then there exist motions of the spheres, possibly accompanied by global deformations of the boundary, that lead to unjamming. In this work we have developed efficient numerical algorithms based on linear programming to test for jamming in sphere packings for each of the jamming categories above. In addition, the algorithm finds representative unjamming motions for packings which are not jammed. We have applied the algorithm to ordered periodic packings and to random packings. We have found that the random packings in 3D are typically strictly jammed, but in 2D they are usually only collectively jammed. Numerous further algorithmic and theoretical developments are awaiting exploration, in particular, the identification of the MRJ state for each of the three jamming categories in both 2 and 3D. See http://atom.Princeton.EDU/donev/Packing for more information.
Researchers: Anuraag R. Kansal, Salvatore Torquato, and Frank H. Stillinger
Description:
Recently the convential 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.
This critical evaluation of existing order metrics has identified significant flaws in
each metric we have employed. Our investigation shows that, even in the large system
limit, jammed systems of hard spheres
can be generated with a wide range ofpacking fractions from
~ 0.52 to the FCC limit ( ~ 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 ( ~ 0.63).
Researchers: Salvatore Torquato and Frank H. Stillinger
Description:
Questions surrounding the spatial disposition of particles in various
condensed-matter systems continue to
pose many theoretical challenges. This paper explores the geometric
availability of amorphous many-particle configurations
that conform to a given pair correlation function g(r). Such a study isrequired to
observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k).
The hard sphere case receives special attention,
to help identify what qualitative features play significant roles in
determining upper limits to maximum amorphous packing
densities. For that purpose, a five-parameter test family of g's has
been considered, which incorporates the known
features of core exclusion, contact pairs, and damped oscillatory
short-range order beyond contact. Numerical optimization
over this five-parameter set produces a maximum-packing value for the
fraction of covered volume, and about 5.8 for the mean
contact number, both of which are within the range of previous
experimental and simulational packing results. However, the
corresponding maximum-density g(r) and S(k) display some
unexpected characteristics. A byproduct of our investigation is a lower
bound on the maximum density for random sphere packings in d dimensions,
which issharper than a well-known lower bound for regular lattice packings for
d 
3.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: Our collision-driven MD algorithm for hard-particle systems was described in detail in a two-part series of papers[Event_Driven_HE]. In the first part, we presented a collision-driven molecular dynamics algorithm for a system of nonspherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. The algorithm extends previous event-driven molecular dynamics algorithms for spheres. We presented a novel partial-update near-neighbor list (NNL) algorithm that is superior to previous algorithms at high densities, without compromising the correctness of the algorithm. This efficiency of the algorithm was further increased for systems of very aspherical particles by using bounding sphere complexes (BSC). These techniques will be useful in any particle-based simulation, including Monte Carlo and time-driven molecular dynamics. Additionally, the algorithm allows for a non-vanishing rate of deformation of the boundary, which can be used to model macroscopic strain and also alleviate boundary effects for small systems.
In the second part, we specialized the algorithm to systems of ellipses and ellipsoids. The theoretical machinery needed to treat such particles, including the overlap potentials, was developed in full detail. We described an algorithm for predicting the time of collision for two moving ellipses or ellipsoids. We presented performance results for our implementation of the algorithm, demonstrating that, for dense systems of very aspherical ellipsoids, the novel techniques of using neighbor lists and bounding sphere complexes offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event-driven molecular dynamics algorithms. The practical utility of the algorithm was demonstrated by presenting several interesting physical applications, including the generation of jammed packings inside spherical containers, the study of contact force chains in jammed packings, and melting the densest-known equilibrium crystals of prolate spheroids.
Linux executables implementing this algorithm can be found here.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: Packing problems, 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 of about 0.74. It is also well-known that the corresponding maximally random jammed (MRJ) packings [as first defined by Torquato et al. in 2000] have density of about 0.64. We used our collision-driven algorithm to study random (disordered) packings of hard ellipsoids by growing the particles as they under go collisions until the "jammed." We showed experimentally and computationally that ellipsoids can randomly pack more densely; up to 0.68-0.71 for spheroids with an aspect ratio close to that of M&Ms, and even approach 0.74 for general ellipsoids. We suggested that the higher density relates directly to the higher number of degrees of freedom per particle and supported this claim by measurements of the number of contacts per particle Z, obtaining Z~10 for our spheroids as compared to Z=6 for spheres.
The simulations predicted the ellipsoid shape which gives the highest random packing density, namely,
that ellipsoids with axes ratios near 1:25:1:0:8 form amorphous packings as dense as the densest
crystal packing (FCC) of spheres. Subsequent experimental work confirmed the predictions.
We demonstrated that such dense packings are realizable by manufacturing ellipsoids
using stereolitography and packing them inside spherical containers. The analysis of the
packings required better understanding of finite-size effects, and we used both simulations
and novel experimental methods to understand and minimizes surface effects and to obtain
good estimates of bulk packing densities. We showed that, in a sphere, the radial packing
fraction 
(r) can be obtained from V(h), the volume of added fluid to fill the
sphere to height h. We also obtained (r) from a magnetic resonance
imaging (MRI) scan. The measurements agreed well with simulations. This verified that
idealized computer packings of frictionless hard particles are relevant to the quantitative
understanding of real-life packings of frictional macroscopic particles in a gravitational field.
Future work will extend these studies to packings of superellipsoids, which are generalized ellipsoidal shapes capable of interpolating between smooth shapes such as spheres and shapes with flat edges and sharp corners such as cylinders and cubes. Also interesting and challenging is the study of ellipsoids of large aspect ratios.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: Extensive experience with spheres has shown that for reasonably large packings, sufficiently slowing down the growth of the density, so that the hard-particle system remains close to the equilibrium solid branch of the equation of state, leads to packings near the FCC lattice. This however requires impractically long simulation times for large ellipsoid packings. By running the MD packing algorithm for very small unit cells, from 4 to 16 particles per unit cell, we were able to identify crystal packings of ellipsoids significantly denser than the FCC lattice. Subsequent analytical calculations suggested by the simulation results led us to discover ellipsoid packings with a remarkably high density of 0.7707. The family of new packings we discovered are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids, and always surpass the densest lattice packing. The maximum density of 0.7707 is achieved for both prolate and oblate ellipsoids with aspect ratios of 31/2 and 1/31/2, respectively, and each ellipsoid has 14 touching neighbors. These results do not exclude the possibility that even denser crystal packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids. Additionally, similar studies are planned for superellipsoids.
Researchers: Monica Skoge, Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: We present the first study of disordered jammed hard-sphere packings in four and five dimensions using event-driven molecular dynamics. We give the first estimates for the densities of the maximally random jammed states in four and five dimensions and calculate the pair correlation function g2(r) and structure factor S(k) for these states. We find that both g2(r) and S(k) are significantly damped in four and five dimensions compared to three dimensions, consistent with a decorrelation principle recently proposed Torquato et al. 2005, stating that correlations diminish in very high dimensions. We verify that, as in three dimensions, the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact. Additionally, we obtain estimates of the freezing and melting densities for the liquid-solid transition.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: We studied the approach to jamming in hard-sphere packings, and, in particular, the pair correlation function g2(r) around contact, both theoretically and computationally. Our computational data unambiguously separated the narrowing delta-function contribution to g2(r) due to emerging interparticle contacts from the background contribution due to near contacts. We also showed 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 derived a theoretical connection between the probability distribution of interparticle forces P_f(f) and the contact contribution to g2, and verified this relation computationally. We observed a clear maximum in P_f and a nonzero probability of zero force, as well as an unusual power-law divergence in the near-contact contribution to g2, persistent even in the jamming limit. Additionally, we presented high-quality numerical data on the two discontinuities in the split-second peak of g2, and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. We also investigated partially crystallized packings along the transition from maximally disordered to fully ordered packings.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description: Continuing on previous theoretical investigations of local density fluctuations in atomic systems by Torquato and Stillinger in 2003, we computationally studied jammed disordered hard-sphere packings of as many as one-million particles. We reported on the decay of the pair correlation function, demonstrating that it is consistent with exponentially damped oscillatory but without ruling out the possibility of a weak (quasi)long-ranged tail. We showed that local density fluctuations are suppressed from volume to surface ones, i.e., the generated packings are hyperuniform, with structure factor which vanishes at the origin to within 10-3. The numerical results suggested a strange non-analytic linear dependence of the structure factor around the origin, while the direct pair correlation function showed a significant long-range tail outside the core, unlike the stable liquid. Finally, we demonstrated that the generated packings are saturated and all voids in the mechanically rigid backbone of the packing are filled with rattlers. Our results illuminated many open questions about density fluctuations in glassy atomic systems.
Researchers: Aleksandar Donev, Frank H. Stillinger and Salvatore Torquato
Description:
We have developed an MD algorithm for measuring the free energy of nearly jammed packings of
hard particles. We applied this algorithm to a model glass former, namely, binary disk mixtures
with large-to-small disk stoichiometry of 1:2 and diameter ratio 
= 1.4.
We have produced bidisperse hard disk glasses by compressing liquids at a wide range of compression
rates using event-driven MD. We observe that the density at which the liquid falls out of equilibrium
and becomes glassy increases with decreasing compression rates, and find that the jamming density of
the glass itself continuously increases as the liquid is given more time to equilibrate. Even at the
slowest compression rates equilibration seems unattainable beyond density of about
g~0.8 with classical methods. Our free energy calculations
give a freezing point of F~0.775, and also show that the estimated
configurational entropy near g is very close
to the entropy of mixing, a fact seen in numerous other studies in the literature. Furthermore, the
equilibrated liquids slightly below g show micro-clustering of
the large particles indicating that supercooled liquids at higher densities would show increased
demixing of the two disk species. The configurational entropy has been postulated to go to zero at
the density of an amorphous ideal glass, however, our numerical observations suggest that the
predicted ideal glass is not an amorphous structure but rather a phase-separated crystal. In fact,
we provide an explicit construction (based on discretized leveled random Gaussian fields) of an exponential
number of binary jammed packings with density ranging from that of the MRJ state to the crystal. This
construction clearly demonstrates that there is not a maximally dense amorphous packing,
but rather a continuum of structures from most disordered to most ordered, consistent with
[Torquato et al, Phys. Rev. Lett., 84, 2064 (2000)]. This work shows that the very premise of the
proposed theories of an ideal glass transition underlying the kinetic one is flawed.
Researchers: Mikael C. Rechtsman, Salvatore Torquato, and Frank H. Stillinger
Description: We formulate statistical-mechanical inverse methods in order to determine optimized interparticle interactions that spontaneously produce target many-particle configurations. Motivated by advances that give experimentalists greater and greater control over colloidal interaction potentials, we have developed two computational algorithms that search for optimal potentials for self-assembly of a given target configuration. The first optimizes the potential near the ground state and the second near the melting point. We have applied these techniques to assembling open structures in two dimensions (square and honeycomb lattices) using only circularly symmetric pair interaction potentials ; we have demonstrated that the algorithms do indeed cause self-assembly of the target lattice. Our approach is distinguished from previous work in that we consider (i) lattice sums, (ii) mechanical stability (phonon spectra), and (iii) annealed Monte Carlo and Molecular Dynamics simulations.
Researchers: Obioma U. Uche, Salvatore Torquato, and Frank H. Stillinger
Description:
The determination of the maximal packing arrangements of two-dimensional, binary
hard disks of radii RS and RL (with RS
RL) for sufficiently small RS amounts to
finding the optimal arrangement of the small disks within a tricusp: the
nonconvex cavity between three close-packed large disks. We present a particle-growth
Monte Carlo algorithm for the generation of geometric packings of equi-sized
hard disks within such a tricusp. The first nineteen members of an infinite
sequence of maximal density structures thus produced are reported. In addition,
the Monte Carlo algorithm is applied to the geometric packing of disks within a
flat-sided equilateral triangle and compared to published results for the packing
problem. We perform an analysis of geometric properties, e.g. packing fraction of
structures confined to both containers. Interestingly, we find a non-monotonic increase
in the packing fraction with increasing number of disks packed within both the flat-sided
triangle and tricusp. It is important to note that for disk packings within a flat-sided
equilateral triangle, this non-monotonic behaviour of the packing fraction had not been
reported in previously published works. For the flat-sided equilateral triangle, local
maxima occur at the triangular integers NS = 1, 3, 6, 10, 15 ... where
NS is the number of disks in each packing. However, local maxima for
packings within the tricusp exist at NS = 1, 3, 6, 10, 18 .... Finally, we
analyze the asymptotic approach to the upper bound on the packing fraction of the infinite
sequence of maximal structures of disks confined to the tricusp.
Researchers: Obioma U. Uche, Salvatore Torquato, and Frank H. Stillinger
Description:
Collective density variables 
(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. We examine features associated
with collective density variables in two and three 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 structural regimes, as the number of constrained wave vectors
is increased, 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. An additional challenge involves understanding the extent
to which these real collective variables at small wavevectors k are controllable. Our method
creates particle patterns that are hyperuniform, thus supporting the notion that structural glasses can be
hyperuniform as the temperature T approaches 0.
Researchers: Obioma U. Uche, Salvatore Torquato, and Frank H. Stillinger
Description: The pair correlation function
g2(r) provides a basic geometric descriptor for
many-particle systems. It must obey two necessary conditions: (i)
non-negativity for all distances r, and (ii) non-negativity of
its associated structure factor S(k) for all k. Here we
utilize an improved stochastic construction algorithm for particle
configurations to establish conditions in which (i) and (ii) are also
sufficient, i.e. g2(r) is in fact realizable. Two
types of target pair correlation functions have been investigated in
one, two, and three dimensions for hard-core particles, specifically a
unit step function, and a contact 
plus step
pair correlation function. Results indicate that the former target
function is realizable up to a terminal density set by necessary
condition (ii), at which the particle core packing fraction equals
2-d in d dimensions. Furthermore, results
are consistent with the proposition that for d > 1 the contact
plus step function is realizable up to a
terminal density due to condition (ii) at which the packing fraction
of cores is d+2/2d+1.
Researchers: Jana Gevertz and Salvatore Torquato
Description: We have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1 mm^2 in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that antiangiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.
Researchers: Jana Gevertz and Salvatore Torquato
Description: Description: We propose a novel biologically constrained three-phase model of the brain microstructure. Designing a realistic model is tantamount to a packing problem and, for this reason, a number of techniques from the theory of random heterogeneous materials can be brought to bear on this problem. Our analysis strongly suggests that previously developed two-phase models in which brain cells are packed in the extracellular space are insufficient representations of the brain microstructure. These models either do not preserve realistic geometric and topological features of brain tissue, or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. Our findings highlight the importance of modeling the brain as three-phase random heterogeneous material, where the extracellular matrix is the key third phase that is unaccounted for in conventional two-phase models. In light of the highly connected nature of three-dimensional space, we hypothesize that the extracellular matrix is one of the dominant mechanisms that determine the diffusivity of brain tissue. Using accurate first-passage-time techniques, we support this hypothesis by showing that extracellular matrix incorporation into a biologically constrained model gives the reduction in the diffusion coefficient necessary for the three-phase model to be a valid representation of the brain microstructure.
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