Hypostatic Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids

Aleksandar Donev,1, 2 Robert Connelly,3 Frank H. Stillinger,4 and Salvatore Torquato1, 2, 4, 5, ∗

1Program in Applied and Computational Mathematics, Princeton University, Princeton NJ 085442PRISM, Princeton University, Princeton NJ 08544

3Department of Mathematics, Cornell University, Ithaca NY 148534Department of Chemistry, Princeton University, Princeton NJ 08544

5Princeton Center for Theoretical Physics, Princeton University, Princeton NJ 08544

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids[Donev et al., Science, vol. 303, 990-993 ] we consider jamming in packings of smooth strictly convexnonspherical hard particles. We explain why the isostatic conjecture, which states that for largedisordered jammed packings the average contact number per particle is twice the number of degreesof freedom per particle (Z = 2df ), does not apply to nonspherical particles. We develop first- andsecond-order conditions for jamming, and demonstrate that packings of nonspherical particles can bejammed even though they are hypostatic (Z < 2df ). We apply an algorithm using these conditionsto computer-generated hypostatic ellipsoid and ellipse packings and demonstrate that our algorithmdoes produce jammed packings, even close to the sphere point. We also consider packings thatare nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally,we consider the jamming conditions for nearly spherical particles and explain quantitatively thebehavior we observe in the vicinity of the sphere point.

I. INTRODUCTION

Jamming in disordered hard-sphere packings has beenstudied intensely in past years [1–3], and recently pack-ings of non-spherical particles have been investigated aswell [4, 5]. Computer simulations and experiments per-formed for packings of hard ellipsoids in Ref. [4] showedthat asphericity, as measured by the deviation of the as-pect ratio α from unity, dramatically affects the proper-ties of jammed packings. In particular, it was observedthat for frictionless particles the packing fraction (den-sity) at jamming φJ and the average coordination (con-tact) number per particle Z increase sharply from thetypical sphere values φJ ≈ 0.64 and Z = 6 when movingaway from the sphere point α = 1. If one views φJ andZ as functions of the particle shape, they have a cusp(i.e., they are non-differentiable) minimum at the spherepoint.

There have been claims [6–10] that large disorderedjammed packings of hard frictionless particles are iso-static, meaning that the number of contacts is equal tothe number of degrees of freedom. We refer to these state-ments as the isostatic conjecture because of the existenceof counter-examples. Most of previous discussions of iso-staticity have been in the context of spheres, for whichthe isostatic conjecture has been verified computationally[2, 3]. For a general particle shape, the obvious general-ization of the conjecture would produce the expectationZ = 2df , where df is the number of degrees of freedomper particle (df = 2 for disks, df = 3 for ellipses, df = 3for spheres, df = 5 for spheroids, and df = 6 for generalellipsoids). Since df increases discontinously with the in-troduction of rotational degrees of freedom as one makes

∗Electronic address: torquato@electron.princeton.edu

the particles non-spherical, the isostatic prediction wouldbe that Z would have a jump at α = 1. Such a disconti-nuity was not observed in Ref. [4], rather, it was observedthat ellipsoid packings are hypostatic, Z < 2df , near thesphere point, and only become close to isostatic for largeaspect ratios (but still remain hypostatic). Those thatbelieve strongly in the isostatic conjecture might havedoubted that our packings were really (translationallyand rotationally) jammed, as we suggested based on ourextensive experience with sphere packings.

In this paper, we generalize our previous theoreticaland computational investigations of jamming in spherepackings [2, 11] to packings of nonspherical particles,and in particular, packings of hard ellipsoids. We gen-eralize the mathematical theory of rigidity of tensegrityframeworks [12, 13] to packings of nonspherical particles,and demonstrate rigorously that the ellipsoid packingswe studied in Ref. [4] are jammed even very close to thesphere point. Armed with this theoretical understandingof jamming, we also obtain a quantitative understandingof the cusp-like behavior of φJ and Z around the spherepoint. Specifically, we:

• Explain why the isostatic conjecture does not applyto nonspherical particles.

• Develop first- and second-order conditions for jam-ming, and demonstrate that packings of nonspher-ical particles can be jammed even though they arehypostatic.

• Design an algorithm that uses the jamming con-ditions to test whether computer-generated hypo-static ellipsoid and ellipse packings are jammed,and demonstrate numerically that our algorithmdoes produce jammed packings, even close to thesphere point.

2

• Study the thermodynamics of packings that arenearly jammed and draw connections to packingsof deformable (but stiff) particles.

• Develop first-order expansions for nearly sphericalparticles and explain quantatively the behavior weobserve in the vicinity of the sphere point.

A. Random Jammed Packings of Hard Ellipsoids

The packing-generation algorithm we employ general-izes the Lubachevsky-Stillinger (LS) sphere-packing al-gorithm [14] and is described in detail in Ref. [15]. Themethod is a hard-particle molecular dynamics (MD) algo-rithm for producing dense disordered packings. Initially,small particles are randomly distributed and randomlyoriented in a box with periodic boundary conditions andwithout any overlap. The particles are given velocities(including angular velocities) and their motion followedas they collide elastically and also expand uniformly. Asthe density approaches the jamming density, the colli-sion rate diverges. In the jamming limit, the particlestouch to form the contact network of the packing, exert-ing compressive forces on each other but not being ableto move despite thermal agitation (shaking). If the rateof particle growth, or expansion rate γ, is initially suffi-ciently large to supress crystallization, and small enoughclose to jamming to allow for local relaxation necessaryfor true jamming, the final packings are disordered andrepresentative of the maximally random jammed (MRJ)state [16] (corresponding to the least ordered among alljammed packings). Note that the computational method-ology presented in Ref. [2] applies to ellipsoids as well andwe do not repeat those details here.

In Fig. 1 we show newer results than those in Ref.[4] for the jamming density φJ and contact number Zof jammed monodisperse packings of hard ellipsoids inthree dimensions. The ellipsoid semiaxes have ratiosa : b : c = 1 : αβ : α where α > 1 is the aspect ratio (forgeneral particle shapes, α is the ratio of the radius of thesmallest circumscribed to the largest inscribed sphere),and 0 ≤ β ≤ 1 is the “oblateness”, or skewness (β = 0corresponds to prolate and β = 1 to an oblate spheroid).It is seen that the density rises as a linear function ofα − 1 from its sphere value φJ ≈ 0.64, reaching den-sities as high as φJ ≈ 0.74 for the self-dual ellipsoidswith β = 1/2. The jamming density eventually decreasesagain for higher aspect ratios, however, we do not inves-tigate that region in this work. The contact number alsoshows a rapid rise with α − 1, and then plateaus at val-ues somewhat below isostatic, Z ≈ 10 for spheroids, andZ ≈ 12 for nonspheroids. In Section IX we will need torevert to two dimensions (ellipses) in order to make someanalytical calculations possible. We therefore also gen-erated jammed packings of ellipses, and show the resultsin Fig. 2. Since monodisperse packings of disks alwayscrystallize and do not form disordered jammed packings,

1 1.5 2 2.5 3Aspect ratio α

0.64

0.66

0.68

0.7

0.72

0.74

φ J

β=1 (oblate)β=1/4β=1/2β=3/4β=0 (prolate)

1 2 3α

6

8

10

12

Z

Figure 1: Jamming density and average contact number (in-set) for packings of N = 10000 ellipsoids with ratios betweenthe semiaxes of 1 : αβ : α [c.f. Fig. (2) in Ref. [4]]. The iso-static contact numbers of 10 and 12 are shown as a reference.

1 1.25 1.5 1.75 2 2.25Aspect ratio α

4

4.5

5

5.5

6

Cont

act n

umbe

r Z

γ=1Ε−5γ=1Ε−4Theory

1 1.2 1.4 1.6α

0.85

0.86

0.87

0.88

0.89

0.9

Den

sity

φJ

Figure 2: Average contact number and jamming density (in-set) for bi-disperse packings of N = 1000 ellipses with ratiosbetween the semiaxes of 1 : α, as produced by the MD al-gorithm using two different expansion rates γ (affecting theresults only slightly). The isostatic contact number is 6. Theresults of the leading-order (in α − 1) theory presented inSection IX are shown for comparison.

we used a binary packing of particles with one third ofthe particles being 1.4 times larger than the remainingtwo thirds. The ellipse packings show exactly the samequalitative behavior as ellipsoids.

B. Non-Technical Summary of Results

In this Section, we provide a non-technical summaryour theoretical results and observations discussed in themain body of the paper. This summary is intended togive readers an intuitive feeling for the mathematical for-

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malism developed in this work and demonstrate the phys-ical meaning and relevance of our results. We will referthe interested reader to appropriate sections to find ad-ditional details.

One aim of this paper is to explain the numerical re-sults presented in Section I A. In particular, we will ex-plain why jammed disordered packings of ellipsoids arestrongly hypostatic near the sphere point, and also why,even far from the sphere point, ellipsoid packings are hy-postatic rather than isostatic as are sphere packings. Bya jammed packing we mean a packing in which any mo-tion of the particles, including collective combined trans-lational and rotational displacements, introduces over-lap between some particles. Under appropriate qualifi-cations, a jammed packing can also be defined as a rigidpacking, that is, a packing that can resolve any externallyapplied forces through interparticle ones.

1. Hypostatic Packings of Nonspherical Particles Can beJammed

As explained in Section IV, the isostatic property isusually justified in two steps. First, non-degeneracy isinvoked to demonstrate the inequality Z ≤ 2df , then,the converse inequality Z ≥ 2df is invoked to demon-strate the equality Z = 2df . The inequality Z ≥ 2df

is usually justified by claiming that a packing cannot bejammed without having more contacts (impenetrabilityconstraints) than degrees of freedom. A hypostatic pack-ing necessarily has “floppy” or zero modes, which are col-lective motions of the particles that preserve the interpar-ticle distances to first order in the magnitudes of the par-ticle displacements. It is claimed that such floppy modesare not blocked by the impenetrability constraints andtherefore a hypostatic packing cannot be jammed. Alter-natively, it is claimed that externally applied forces thatare in the direction of such floppy modes cannot be re-sisted (sufficiently) by the interparticle forces and there-fore the packing cannot be rigid. We will now explain,through an example, why these claims are wrong and, infact, why a hypostatic packing can be jammed/rigid ifthe curvature of the particles at the point of contact issufficiently flat in order to block the floppy modes.

Consider an isostatic jammed packing of hard circulardisks, as illustrated in Fig. 3. In reality, the disks wouldbe elastic (soft) but stiff, and let’s imagine the system isunder a uniform state of compression, so that the par-ticles are exerting compressive forces on each other. Ifthere are no additional external forces, the interparti-cle forces would be in force equilibrium. The packing istranslationally jammed, and the disk centroids are im-mobile; however, the (frictionless) disks can freely rotatewithout introducing any additional overlap. That is, ifwe take into account orientational degrees of freedom,the disk packing would not be jammed. It would possessfloppy modes consisting of particles rotating around theirown centroids. These floppy modes are however trivial

Figure 3: A jammed packing of hard disks (colored yellow)is converted into a jammed packing of nonspherical particlesby converting the disks to polygons (colored in different col-ors), without changing the contact network or contact forces.This preserves the jamming property since the floppy modescomposed of pure particle rotations are blocked by the flatcontacts. Jamming would also be preserved if the disks swellbetween the original shape and the polygonal shape, so thatthe curvature of the particle surfaces at the point of contactis sufficiently flat.

at the circle (sphere) point in that they do not actuallychange the packing configuration.

Now imagine making the particles non-circular (or non-spherical in three dimensions), and in particular, makingthem polygons, so that the point contacts between thedisks become (extended) contacts between flat sides ofthe polygons. The floppy modes still remain, in the sensethat rotations of the polygons, to first order, simply leadto the two tangent planes at the points of contact slidingalong each other without leading to overlap. However, itis clear that this is only a first-order approximation. Inreality, the polygons cannot be rotated because such ro-tation leads to overlap in the extended region of contactaround the point of contact. To calculate the amount ofoverlap, one must use second-order terms, that is, con-sider not only the tangent planes at the point of contactbut also the curvature of the particles at the point ofcontact. Low curvature, that is, “flat” contacts, block ro-tations of the particles. It should be evident that even ifthe radius of curvature is not infinite, but is sufficientlyhigher than the radius of the (original) hard disks, thefloppy modes would in fact be blocked and the packingwould be jammed despite being hypostatic. In fact, thepacking has exactly as many contacts as the original diskpacking.

It is important to note that contact curvature cannotblock purely translational particle displacements unless

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one of the particles is curved outward, i.e., is concave(e.g., imagine a dent in a table and a sphere resting in it,not being able to slide translationally). If the particlesshapes are convex, a packing cannot have less contactsthen there are translational degrees of freedom, that is,Z ≥ 2d. This explains why hypersphere packings areindeed isostatic. It is only when considering rotationaldegrees of freedom that jammed packings can be hypo-static.

Those that prefer to think about rigidity (forces) wouldconsider applying external forces and torques on the par-ticles in the example from Fig. 3. The forces wouldclearly be resisted just like they were in the jammed diskpacking. However, at first sight, it appears that torqueswould not be resisted. In fact, it would seem that torquescannot be resisted by interparticle forces since, for eachof the particles, the normal vectors at the points of con-tact all intersect at a single point (the center of the harddisks) and therefore the net torque is identically zero.This argument, however, neglects an important physicalconsideration: the deformability of the particles. Namely,no matter how stiff the particles are, they will deformslightly under an applied load. In particular, upon appli-cation of torques, the particles will rotate and the nor-mal vectors at the points of contact would change and nolonger intersect at a single point, and the packing will beable to resist the applied torques. One may be concernedabout the amount of rotation necessary to resist the ap-plied load. If the packing needs to deform significantlyto resist applied loads, should it really be called rigid?

To answer such concerns, one must calculate the par-ticle displacements needed to resist the load. Such a cal-culation, carried out for deformable particles in SectionVIII, points to the importance of the pre-existing (i.e.,internal) contact forces. This is easy to understand phys-ically. If the packing is under a high state of compression,the interparticle forces would be large and even a smallchange in the packing geometry (deformation) would re-sist large torques. If, on the other hand, the internalforces (stresses) are small, the particles would have todeform sufficiently to both induce sufficiently large con-tact forces and to change the normal vectors sufficiently.This kind of stability, requiring sufficiently large internalstresses, is well-known for engineering structures calledtensegrities. These structures are built from elastic ca-bles and struts, and are stabilized by stretching the cablesso as to induce internal stresses. Beautiful and intriguingstructures can be built that are rigid even though theyappear not to be sufficiently braced (as bridges or otherstructures would have to be).

While the above discussion focused on packings ofmacroscopic elastic particles, similar arguments applyalso to systems such as glasses. For such systems,floppy modes are manifested as zero-frequency vibra-tional modes, that is, zero eigenvectors of the dynamicalmatrix. The calculations in Section VIII show that fornonspherical particles, the dynamical matrix contains aterm proportional to the internal forces and involving the

contact curvatures. If the system is at a positive pressure,the forces will be nonzero and this term contributes tothe overall dynamical matrix. In fact, it is this term thatmakes the dynamical matrix positive definite, i.e., thateliminates zero-frequency modes despite the existence offloppy modes.

2. Translational Versus Rotational Degrees of Freedom

Having explained that hypostatic packings of non-spherical particles can be jammed if the inteparticle con-tacts are sufficiently flat, we now try to understand whypackings of nearly spherical particles are hypostatic. Theanalysis will also demonstrate why packings of hard el-lipsoids are necessarily denser than the correspondingsphere packings.

The first point to note is that disordered isostaticpackings of nearly spherical ellipsoids are hard to con-struct. In particular, achieving isostaticity near thesphere point requires such high contact numbers [specif-ically, Z = d(d + 1)] that translational ordering will benecessary. Translationally maximally random jammed(MRJ) sphere packings have Z = 2d, and even if oneconsiders the observed multitude of near contacts [2],they fall rather short of Z = d(d + 1). It seems intuitivethat translational crystallization would be necessary inorder to raise the contact number that much. In otherwords, in order to gain isostaticity, one would have tosacrifice translational disorder. Furthermore, there is lit-tle reason to expect packings of nearly spherical particlesto be rotationally jammed. Near the jamming point, itis expected that particles can rotate significantly eventhough they will be translationally trapped and rattleinside small cages, until of course the actual jammingpoint is reached, at which point rotational jamming willalso come into play. Therefore, it is not suprising thatnear the sphere point, the translational structure of thepackings changes little.

Mathematically, jamming is analyzed by Taylor ex-panding the interparticle distances in the particle dis-placements. At the first-order level, this expansion con-tains first-order terms coming from translations and fromrotations and involving the contact points and contactnormals. The expansion also contains second-order termsfrom translations, rotations and combined motions, in-volving additionally the contact curvatures. And ofcourse, there are even more complicated higher-orderterms. One should be careful of such a Taylor expan-sion for two reasons. First, the expansion assumes thatterms coming from translations and rotations are of thesame order. This is clearly not true for neither the caseof perfectly spherical particles, when rotational terms areidentically zero, nor for the case of rods or plates, whereeven a small rotation can cause very large overlap. Sec-ond, the expansion assumes that various quantities re-lated to the particle and contact geometry (for example,the contact curvature radii) are of similar order. This

5

fails, for example, for the case of planar (flat) contacts,where even a small rotation of the particles leads to sig-nificant overlap far from the point of contact. Thesesubtle points arise only when considering aspherical par-ticles and should caution one from blindly generalizingthe mathematical formalism of jamming developed andtested only within the context of sphere packings.

In Section IX, we will consider packings of nearlyspherical ellipsoids as a perturbation of jammed spherepackings in which the particles, following a slight changeof the particle shape away from perfect spherical sym-metry, translate and rotate in order to re-establish con-tacts and jamming. While the necessary particles’ trans-lations are small, the particle rotations are large. Infact, rotational symmetry is broken, and particles mustorient themselves correctly, so that contacts can be re-established, and also so that forces and torques becomebalanced. This symmetry breaking is the cause of thecusp-like non-analyticity of the density as a function ofparticle shape [4]. We will see that the particle orienta-tions in the final jammed packing of nearly spherical ellip-soids are not random, but rather, they are determined bythe structure of the initial sphere packing. Of course, asaspect ratio increases, rotations become more and moreon equal footing with translations, and the packings be-come both truly translationally and orientationally dis-ordered.

This picture of jamming in the vicinity of the spherepoint also explains why the density rises sharply near thesphere point for ellipsoids. Start with a jammed spherepacking and apply an affine transformation to obtain analigned (nematic) packing with exactly the same density.This packing will not be rotationally jammed, and bydisplacing the particles one will be able to open up freevolume between them and therefore increase the density.We will show that in fact the maximal increase in thedensity is obtained for the choice of particle orientationsthat balances the torques on the particles in addition tothe forces. Therefore, the jammed disordered ellipsoidpackings we obtain near the sphere point are the densestperturbation of the corresponding sphere packings. Theadded rotational degrees of freedom allow one to increasethe density beyond that of the aligned (nematic) packing,which for ellipsoids has exactly the same density as thesphere point.

In conclusion, near the sphere point, there is a compe-tition between translational and rotational jamming andalso between translational and rotational disorder. Atthe sphere point α = 1, and in this neighborhood, trans-lational degrees of freedom win. As one moves away fromthe sphere point, however, translational and rotationaldegrees of freedom start to play an equal role. For verylarge aspect ratios, α 1, it is expected that rotationaldegrees of freedom will dominate, although we do notinvestigate that region here.

C. Contents

Before proceeding, we give an overview of our nota-tion in Section II. In Section III, we discuss the non-overlap conditions between convex hard particles. In Sec-tion IV, we define jamming and investigate the reasonsfor the failure of the isostatic conjecture for nonspher-ical particles. In Section V, we develop the first- andsecond-order conditions for jamming in a system of non-spherical particles, and then design and use a practicalalgorithm to test these conditions in Section V C. In Sec-tion VII, we consider the thermodynamical behavior ofhypostatic packings that are close to, but not quite at,the jamming point. In Section VIII, we discuss the con-nections between jammed packings of hard particles andstrict energy minima for systems of deformable particles.In Section IX, we focus on packings of nearly sphericalellipsoids, and finally, offer conclusions in Section X.

It is important to note that Sections III, V, and V C arehighly technical, and may be either skipped or skimmedby readers not interested in the mathematical formalismof jamming. Readers interested in specific examples ofhypostatic packings are referred to Section IVB 2 andAppendix A.

II. NOTATION

We have tried to develop a clear and consistent nota-tion, however, in order to avoid excessive indexing andnotation complexity we will often rely on the context forclarity. The notation is similar to that used in Ref. [15]and attempts to unify two and three dimensions when-ever possible. We refer to reader to Ref. [15] or Ref.[17] for details on representing particle orientations androtations in both two and three dimensions.

We will use matrix notation extensively, and denotevectors and matrices with bolded letters, and capitalizematrices in most cases. Infinite-dimensional or discretequantities such as sets or graphs will typically be denotedwith script letters. We will often capitalize the letter de-noting a vector to denote a matrix obtained from thatvector. Matrix multiplication is assumed whenever prod-ucts of matrices or a matrix and a vector appear. Weprefer to use matrix notation whenever possible and donot carefully try to distinguish between scalars and ma-trices of one element. We denote the dot product a · bwith aT b, and the outer product a ⊗ b with abT . Wedenote a vector with all entries unity by e = 1, so that∑

i ai = eT a. We consider matrices here in a more gen-eral linear operator sense, and they can be of order higherthan two (i.e., they do not necessarily have to be a rect-angular two-dimensional array). We refer to differentialsas gradients even if they are not necessarily differentialsof scalar functions. Gradients of scalars are consideredto be column vectors and gradients of vectors or matri-ces are matrices or matrices (linear operators) of higherrank.

6

A. Particle Packings

A jammed particle packing has a contact network in-dicating the touching pairs of particles i, j. We willsometimes talk about a particular particle i or a particu-lar contact i, j ≡ ij and we will usually let the contextdetermine what specific particle or contact is being re-ferred to, or, if deemed necessary, put subscripts suchas i or ij to make it specific what particle or contact isbeing referred to. The contact ji is physically the sameundirected contact as ij, but the two directed contactsare considered distinct.

There are two primary kinds of vectors x, particlevectors X = (xi) = (x1, . . . ,xN ), which are obtainedby concatenating together the vectors xi (typically ofsize of the order of the space dimensionality d) corre-sponding to each of the N particles, and contact vectorsy = (yij) = (y1, . . . , yM ), obtained by concatenating to-gether the (typically scalar) values yij corresponding toeach of the M contacts (numbered in arbitrary orderfrom 1 to M). Note the capitalization of particle vec-tors, which we will often do implicitly, to indicate thatone can view X as a matrix where each row correspondsto a given particle. If a contact vector agglomerates avector quantity attached to each contact, for example,the common normal vector n at the point of contact oftwo particles, it too would be capitalized, e.g., N = (nij).

1. Packing Configuration

A packing is a collection of N hard particles inRd suchthat no two particles overlap. Each particle i has df con-figurational degrees of freedom, for a total of Nf = Ndf

degrees of freedom. A packing Q = (Q, φ) is charac-terized by the configuration Q = (q1, . . . ,qN ) ∈ RNf ,determining the positions of the centroid and the ori-entations of each particle, and the packing fraction (den-sity), φ, determining the size of the particles. For spheresQ ≡ R corresponds to only the positions of the centroids,and df = d. For nonspherical particles without any axesof symmetry there are an additional d(d − 1)/2 rota-tional degrees of freedom, for a total of df = d(d + 1)/2degrees of freedom. In actual numerical codes particleorientation is represented using unit quaternions, whichare redundant representations in the sense that they used(d− 1)/2 + 1 coordinates to describe orientation. Herewe will be focusing on displacements of the particles ∆Qfrom a reference jammed configuration QJ , and thereforewe will represent particle orientations as a rotational dis-placement from a reference orientation ∆ϕ. In two di-mensions ∆ϕ = ∆ϕ simply denotes the angle of rotationin the plane, and in three dimensions the direction of ∆ϕgives the axis of rotation and its magnitude determinesthe angle of rotation. For simplicity, we will sometimesbe sloppy and not specifically separate centroid positionsfrom orientations, and refer to qi as (a generalized) po-sition; similarly, we will sometimes refer to both forces

and torques as (generalized) forces.

2. Rigidity Matrix

For the benefit of readers not interested in the mathe-matical formalism, we briefly introduce the concepts andnotation developed in more detail in Section III.

We denote the distance, or gap, between a pair of hardparticles with ζ. When considering all of the M contactstogether, the gradient of the distance function ζ = (ζij)with respect to the positions (i.e., displacements) of theparticles is the rigidity matrix A = ∇Qζ. This linear op-erator connects, to first order, the change in the interpar-ticle gaps to the particle displacements, ∆ζ = AT ∆Q.We denote the magnitudes of the compressive (positive)interparticles forces carried by the particle contacts withf = (fij), fij ≥ 0, where it is assumed that the force vec-tors are directed along the normal vectors at the pointof contact (since the particles are frictionless). The totalforces and torques exerted on the particles B (alterna-tively denoted by ∆B if thought of as force imbalance)are connected to the interparticle forces via a linear op-erator that can be shown to be the conjugate (transpose)of the rigidity matrix, B = AT f .

B. Cross Products

In three dimensions, the cross product of two vectorsis a linear combination of them that can be thought ofas matrix-vector multiplication

a× b = Ab = −b× a = −Ba (1)

where

A = |a|× =

0 −az ay

az 0 −ax

−ay ax 0

= −AT

is a skew-symmetric matrix which is characteristic of thecross product and is derived from a vector. We will sim-ply capitalize the letter of a vector to denote the corre-sponding cross product matrix (like A above correspond-ing to a), or use |a|× when capitalization is not possible.In two dimensions, there are two “cross products”. Thefirst one gives the velocity of a point r in a system whichrotates around the origin with an angular frequency ω(which can also be considered a scalar ω),

v = ω r =[−ωry

ωrx

]= Ωr, (2)

where

Ω =[

0 −ωω 0

]= −ΩT

7

is a cross product matrix derived from ω. The secondkind of“cross product”gives the torque around the originof a force f acting at a point (arm) r,

τ = f × r = −r× f = [fxry − fyrx] = FLr, (3)

where

FL =[−fy fx

]= −

(FR

)T

is another cross product matrix derived from a vector(the L and R stand for left and right multiplication, re-spectively). Note that in three dimensions all of thesecoincide, FL = FR = F, and also ≡ ×, while in twodimensions they are related via a b = Ab = −BRa.

III. NONOVERLAP CONSTRAINTS ANDINTERPARTICLE FORCES

In this section we will discuss hard-particle overlap po-tentials used to measure the distance between a pair ofhard particles. These potentials will be used to developanalytic expansions of the non-overlap conditions in thedisplacements of the particles. This section is technicaland may be skipped or skimmed by readers not inter-ested in the mathematical formalism of jamming. Inter-ested readers can find additional technical details on thematerial summarized in this Section in Chapter 2 of Ref.[17].

A. Overlap Potentials

The nonoverlap condition between a pair of particlesA and B can be thought of as an inequality between thepositions and orientations of the particles. For this pur-pose, we measure the distance between the two ellipsoidsusing the overlap potential ζ (A,B) = ζ(qA,qB), whosesign not only gives us an overlap criterion, ζ (A,B) > 0 if A and B are disjoint

ζ (A,B) = 0 if A and B are externally tangentζ (A,B) < 0 if A and B are overlapping,

but which is also at least twice continuously differentiablein the positions and orientations of A and B. An addi-tional requirement is that ζ(A,B) be defined and easy tocompute for all positions and orientations of the parti-cles.

We define and compute the overlap conditions usinga procedure originally developed for ellipsoids by Per-ram and Wertheim [18]. This procedure is easily general-ized to any convex particle shape given by the inequalityζ (r) ≤ 1, where the shape function ζ is strictly convexand defined through

ζ (r) = [µ (r)]2 − 1,

where µ is the scaling factor by which the particle needsto be resized in order for the point r to lie on its sur-face. The unnormalized normal vector to the surface ata given point r, if the particle is rescaled so that it passesthrough it, is n(r) = ∇ζ(r). Define also the displace-ment between the particle centroids rAB = rA− rB , andthe unit vector joining the two particle centroids withuAB = rAB/ ‖rAB‖.

The Perram and Wertheim (PW) overlap potential isdefined through

ζ = µ2 − 1 = max0≤λ≤1

minrC

[λζA (rC) + (1− λ) ζB (rC)] .

For every multiplier λ, the solution of the inner optimiza-tion over rC is unique due to the strict convexity of rC ,and satisfies the gradient condition

λnA (rC) = − (1− λ)nB (rC) ,

which shows that the normal vectors are parallel (withopposite directions). The solution of the outer optimiza-tion problem over λ is given through the condition

ζ = ζA (rC) = ζB (rC) ,

which means that when the particles are rescaled by acommon scaling factor µ = 1 + ∆µ =

√1 + ζ they are

in external tangency, sharing a common normal direc-tion n = nA/ ‖nA‖ (i.e., normalized to unit length anddirected from A to B), and sharing a contact point rC .When focusing on one particle we can measure rC withrespect to the centroid of the particle, or otherwise specif-ically denote rAC = rC − rA and rBC = rC − rB . Thisis illustrated for ellipses in Fig. 4. If the particles aretouching then µ = 1 and the procedure described abovegives us the geometric contact point and therefore thecommon normal vector. In the case of spheres of radiusO the PW overlap potential simply becomes

ζAB =(rA − rB)T (rA − rB)

(OA + OB)2− 1 =

l2AB

(OA + OB)2− 1,

(4)which avoids the use of square roots in calculating thedistance between the centers of A and B, lAB , and iseasily manipulated analytically.

8

Figure 4: Illustration of the common scaling µ that bringstwo ellipses (dark gray) into external tangency at the contactpoint rC .

1. Derivatives of the Overlap Potentials

We will frequently need to consider derivatives of theoverlap function with respect to the (generalized) posi-tions of the particles, either first order,

∇qiζ = ∇iζ =(

∂ζ

∂qi

),

or second order

∇2qiqj

ζ = ∇2ijζ =

[∂2ζ

∂qi∂qj

].

To first order, the particles can be replaced by their (par-allel) tangent planes at the point of contact and the firstorder derivatives can be expressed in terms of quantitiesrelating to the two tangent planes. To second order, theparticles can be replaced by paraboloids that have thesame tangent plane, as well as the same principal cur-vature axes and the same radii of curvatures as the twoparticles at the point of contact. It is therefore possibleto derive general expressions for the derivatives in termsof quantities relating to the normal vectors and surfacecurvatures of the particles at the point of contact.

The first order derivatives can easily be expressed interms of the position of the contact point rC and the (nor-malized and outwardly-directed) contact normal vectorn. For this purpose, it is easier to measure the distancebetween two particles in near contact via the Euclid-ian interparticle gap h giving the (minimal) surface-to-surface distance between the particles along the normalvector. Moving one of the particles by ∆q = (∆r,∆ϕ)displaces the contact point by ∆rC = ∆r + ∆ϕ rC

and therefore changes the gap by ∆h = −nT ∆rC =−nT ∆r− (rC × n)T ∆ϕ, giving the gradient

∇qh = −[

nrC × n

].

The relation between the (small) Euclidian gap h and the(small) gap as measured by the PW overlap potential ζcan be seen by observing that scaling an ellipsoid by afactor µ displaces the contact point by ∆rC = ∆µrC .Therefore, the scaling factor needed to close the inter-particle gap is

µ ≈ ζ

2≈ h

(rBC − rAC)T n=

h

rTABn

,

giving the gradient of the overlap potential ∇qζ =2 (∇qh) /

(rT

ABn),

∇A/Bζ = ∓ 2rT

ABn

[n

r(A/B)C × n

].

For spheres the cross product is identically zero and ro-tations can be eliminated from consideration.

The second-order derivatives are not as easily evalu-ated for a general particle shape. In two dimensions, orin three dimensions when the principal radii of curva-tures at the point of contact are equal, one can replacethe particle around the point of contact with a sphereof the appropriate position and radius. However, whenthe radii of curvatures are different this is not as easy todo. We will give explicit expressions for the second-orderderivatives of ζ for ellipsoids in Section VIA 2. Relatedfirst- and second-order geometric derivatives have beenderived for general particle shapes (i.e., using the normalvectors and curvature tensors of the particles at the pointof contact) in the granular materials literature in moregeneral contexts [19, 20]; here we specialize to the caseof hard frictionless ellipsoids.

B. The Rigidity Matrix

When considering all of the M contacts together, thegradient of the overlap potential ζ = (ζij) is the impor-tant rigidity matrix

A = ∇Qζ.

This [Nf ×M ] matrix connects, to first order, the changein the interparticle gaps to the particle displacements,∆ζ = AT ∆Q. It may sometimes be more convenientto work with surface-to-surface interparticle gaps, ∆h =AT

E∆Q (the subscript E stands for Euclidian), especiallyif second-order terms are not considered [11]. The rigiditymatrix is sparse and has two blocks of df non-zero entriesin the column corresponding to the particle contact i, j,namely, ∇iζij in the block row corresponding to particlei and ∇jζij in the block row corresponding to particlej (unless one of these particles is frozen). Representedschematically:

9

A =

i, j↓

i →

j →

...∇iζij

...∇jζij

...

.

C. Interparticle Forces

Hard particles in contact can exert a compressive (pos-itive) contact force f = fn, f ≥ 0, directed along thenormal vector (for frictionless particles). The total ex-cess force and torque exerted on a given particle i by thecontacts with its neighbors N (i) is

∆bi = −∑

j∈N (i)

fij

[nij(

rijiC × nij

) ]=

∑fij (∇ihij) ,

or, considering all particles together

∆B = AEf .

The fact that the matrix (linear operator) connectingforce imbalances to contact forces is the transpose of therigidity matrix is well-known and can also be derived byconsidering the work done by the contact forces to dis-place the particles

W = ∆bT ∆Q =(Af

)T

∆Q = fT(AT ∆Q

)=

= fT ∆h = fT(AT

E∆Q),

showing that A = ATE . In this work we will use forces

f that are a rescaled version of the physical forces fE ,fij =

(rT

ijnij

)fE

ij /2, so that Af = AEfE . This scalingis more natural for our choice of overlap potential, anddoes not affect any of the results.

In static packings, the contact forces must be balanced,i.e., the force/torque equilibrium condition

Af = 0 and f ≥ 0

must be satisfied. The actual magnitudes of the forcesare determined by external loads (for example the ap-plied pressure for a system of deformable particles), his-tory of the packing preparation, etc. However, the re-lation between the forces at different contacts is deter-mined by the packing geometry, or more specifically, byA. Typically forces are rescaled to a mean value of unity,eT f = M , and it is has been observed that the distribu-tion of rescaled contact forces has some universal fea-tures, for example, there is an exponential tail of con-tacts carrying a large force, and also a large number ofcontacts supporting nearly zero force [2, 21]. We will seelater that these force chains, or internal stresses, are anessential ingredient of jamming for hard particles.

IV. THE ISOSTATIC CONJECTURE

In the granular materials literature special attention isoften paid to so-called isostatic packings, and it has beenpostulated several times that realistic packings of hardparticles are isostatic. There are several different defini-tions of isostaticity, and most of the discussions in the lit-erature are specifically applied to mechanical structurescomposed of elastic bars, to packings of hard spheres, orto packings of frictional particles. In this section we sum-marize several relevant definitions of and arguments forisostaticity and generalize them to nonspherical particles.Our arguments will arrive to the following definition ofisostaticity, which we believe is the correct generaliza-tion to systems of nonspherical particles. A packing isisostatic if the number of constraints (contacts) is equalto the total number of degrees of freedom

Nc = Nf + 1,

where for jammed packings one should count the densityφ as a single degree of freedom, in addition to the degreesof freedom due to the particles and boundary Nf , as dis-cussed further in Sections and IVA 2. Packings with lesscontacts than isostatic are called hypostatic, and packingswith more contacts than isostatic are hyperstatic. Theisostatic conjecture states that large disordered packingsof hard particles are isostatic. Defining what precisely ismeant by a disordered packing is difficult in itself [16, 22].Intuitively, in a disordered packing there is only the min-imal degree of correlations between particles, as necessi-tated by the constraints of impenetrability and jamming.Therefore, it is expected that in a certain sense disorderedpackings are“generic” 1, and that“special”configurationswith geometric degeneracies will not appear. Note thatfor large systems the majority of the degrees of freedomcome from the particles themselves, Nf ≈ Ndf , and themajority of constraints come from contacts shared be-tween two particles, Nc ≈ M = NZ/2, giving the iso-static property

Z = 2df . (5)

Equation (5) has been verified to very high accuracy forjammed hard-sphere packings [2]. However, disorderedpackings of hard ellipsoids are always hypostatic and thuscontradict the isostatic conjecture [4].

In this Section we attempt to deconstruct previous dis-cussions of the isostatic conjecture and jamming in hard-particle packings, and we hope that through our discus-sions it will become clear why previous“proofs”of the iso-static conjecture do not apply to nonspherical particles,

1 The mathematically formal meaning of the term“generic” is usedin rigidity theory for configurations of points [23]. However, thatrigorous meaning of the term almost never applies to packingsof monodisperse particles. We use the term “generic” merely tomean “not special”, i.e., typical.

10

or to put it the other way around, what makes disorderedsphere packings isostatic.

A. Jamming, Rigidity and Stability

An essential initial step is defining more precisely whatis meant by a stable, rigid, or jammed packing. All ofthese terms have been used in the literature, and in factwe equate each of them with a particular perspective onjamming:

Kinematic A packing is jammed if none of the parti-cles can be displaced in a non-trivial way withoutintroducing overlap between some particles.

Static A packing is rigid if it can resolve any externallyapplied forces through interparticle ones, withoutchanging the packing configuration.

Perturbation A packing is stable if the structure of thepacking changes smoothly for small perturbationsof the packing.

We will consider each of these approaches separately. Itwill shortly become clear that all of them are closely re-lated, and under certain mild conditions they are actuallyequivalent. We will use the term jamming as an umbrellaterm, and later give our preferred definition of jamming,which is based on the kinematic perspective. We notethat it is important to precisely specify the boundaryconditions applied regardless of the view used in con-sidering jamming; different boundary conditions lead todifferent jamming categories, specifically local, collectiveor strict jamming [11, 24]. Here, we will sometimesuse local jamming in simple examples but mostly focuson collective jamming; all collective particle motions areblocked by the impenetrability constraints subject to pe-riodic boundary conditions with fixed lattice vectors. Inorder to eliminate trivial uniform translations of the sys-tems, we can freeze the centroid of one of the particles,to obtain a total of

Nf = Ndf − d

internal degrees of freedom. The exact boundary con-ditions affect the counting of constraints and degrees offreedom, however, the correction is not extensive in Nand therefore is negligible for large system when consid-ering per-particle quantities such as Z.

An important point to note is that the above defini-tions of jamming treat all degrees of freedom identically,in particular, translational motion (forces) is treated onthe same footing as rotational motion (torques). Thisis not necessarily the most appropriate definition, as iseasily seen by considering the case of spheres, which canrotate in place freely even though they are (translation-ally) jammed. This distinction between translations androtations will become important in Section VII when con-sidering packings that are nearly, but not quite jammed.

It should also be mentioned that jammed random par-ticle packings produced experimentally or in simulationstypically contain a small population of rattlers, i.e., par-ticles trapped in a cage of jammed neighbors but freeto move within the cage. For present purposes we shallassume that these have been removed before consideringthe (possibly) jammed remainder. This idea of exclud-ing rattlers can be further extended to rattling clustersof particles, i.e., groups of particles that can be displacedcollectively even though the remainder of the packing isjammed. In fact, we will consider any packing which hasa jammed subpacking (called backbone) to be jammed.

1. Kinematic View

The kinematic perspective considers a packing jammedif it is not possible to continuously displace the particlesin a non-trivial way without introducing overlap. Wehave focused on this perspective in our work, see Refs.[11, 24]. That is, the impenetrability conditions pre-clude any motion of the particles. Here trivial motionsare those that do not change the distances between anytwo particles, such as global translations when periodicboundary conditions are used. We can assume that suchtrivial motions have been eliminated via some artificialconstraint, such as fixing the centroid of one particle ex-ternally when using periodic boundary conditions.

Mathematically, for any continuous motion ∆Q (t)there exists a T > 0 such that at least one of the impene-trability constraints between a touching pairs of particles

ζ [QJ + ∆Q (t)] ≥ 0 (6)

is violated for all 0 < t < T . A motion ∆Q (t) suchthat for all 0 < t < T none of the constraints are vio-lated is an unjamming motion. One can in fact restrictattention to analytic paths ∆Q (t), and also show thata jammed packing is in a sense isolated in configurationspace, since the only way to get to a different packing isvia a discontinuous displacement ‖∆Q‖ > 0 [12].

A similar definition of jamming was used by Alexanderin Ref. [6]. He considers a packing to be geometricallyrigid if it cannot be “deformed continuously by rotatingand translating the constituent grains without deformingany of them and without breaking the contacts betweenany two grains”. This definition implies that a packingin which particles can be moved so as to break contacts(for example, imagine a pebble resting on other pebblesin gravity, and moving it upward away from the floor) isjammed. Later in the manuscript Alexander talks aboutadding constraints to block motions that break contacts.We in fact have in a certain sense a choice in the matter,determining whether we work with inequality or equal-ity constraints. We choose to work with inequality con-straints, since this is the natural choice for frictionlesshard particles; there is no cohesion between the particlesmaintaining contacts. In effect, when counting degreesof freedom for packings, we count the density φ (i.e., the

11

possible collective rescaling of the particle shapes neces-sary to maintain contacts) as a single degree of freedom,as discussed further in Section IV A 2.

2. Static View

The static perspective considers a packing rigid if itcan resolve any applied forces through interparticle ones.This is sometimes referred to as static rigidity, to be con-trasted with kinematic rigidity as defined in the previ-ous section. For hard particles, there is no scale for theforces, and so the actual magnitude of the forces does notmatter, only the relative magnitudes and the directions.The particles do not deform, but can exert an arbitrarypositive contact force.

Mathematically, we consider the existence of a solutionto the force-equilibrium equations

Af = −B, where f ≥ 0, (7)

for all resolvable external loads B. The space of resolv-able loads is determined by the boundary conditions: cer-tain forces such as pulling on the walls of a containercannot be resolved by packings and need to be excluded.This is similar to the definition used by Witten in Ref.[8]: A packing is mechanically stable “if there is a nonzeromeasure set of external forces which can be balancedby interbead ones.” The problem with this definition ofrigidity and in particular Eq. (7) is that it does nottake into account the fact that the geometry of the pack-ing, i.e., the rigidity matrix A, changes when an externalload is applied on the packing. Physically, forces ariseonly through deformation, and this deformation, how-ever small, together with the pre-existing forces in thepacking, may need to be taken into account. Forces arein essence Lagrange multipliers associated with the im-penetrability constraints in Eq. (6); the very existenceof such Lagrange multipliers may require a change in thepacking configuration.

The above formulation also neglects the existence ofsmall interparticle gaps, which cannot be neglected whenanalyzing the response of packings to applied loads, espe-cially for granular materials [10, 11]. While mathemat-ically we talk about ideal jammed packings, where geo-metric contacts are perfect, in reality one should reallyanalyze packings that are almost jammed, i.e., where thecontacts are almost closed. This is more appropriate forgranular materials, where there is typically some roomfor the particles to move freely. Alternatively, one shouldanalyze packings where all the contacts are indeed closed,however, the system is under some form of global com-pression. This is appropriate for glassy systems under auniform external pressure. When interparticle gaps arepresent, particles must displace slightly to close the gapsso that they can exert a positive contact forces on oneanother and resist the applied load. The set of contactsi, j that are closed (i.e., have a positive force fij) isthe set of active contacts. Different applied loads will be

supported by different active contact networks, and forsufficiently small interparticle gaps finding the active setrequires solving a linear program, as discussed in SectionV D1.

Various counting arguments related to force equilib-rium constraints, starting with the seminal work ofMaxwell, have appeared in the engineering literature onmechanical structures [25]. There are, however, some im-portant differences between elastic structures and pack-ings of hard particles. Most significantly, the non-negativity of the contact forces is an added condition, andit effectively adds +1 to the number of contacts needed toensure static rigidity, i.e., adds a single degree of freedomin various counting arguments. For classical structures ofelastic bars, an isostatic framework is such that it has ex-actly as many bars, i.e., unknown internal bar forces, asthere are force-equilibrium equations, M = Nf . Thatis, the rigidity matrix is square and the solution to theforce-equilibrium equations is f = −A−1B. Finding theinternal forces therefore does not require knowing any-thing about the specific elastic properties of the bars:the structure is statically determinate 2. On the otherhand, a jammed isostatic packing, as we have defined it,has M = Nf +1 contacts, and the additional one contactis needed in order to ensure that any applied load canbe resolved by non-negative interparticle forces in theactive contact network. For such a packing, if one ap-plies a specific load, only Nf of the contacts will actuallybe active, and one contact will be broken and will carryno force. Different contacts will be broken for differentloads, however, once it is known which contact is broken(see Section V D1) the active contact network is isostaticin the classical structural mechanics sense and the forcescan be determined, f = −A−1

activeB, without resorting toconstitutive elastic equations for the contacts.

3. Perturbation View

The perturbation perspective considers a packing to bestable if the structure of the packing changes smoothlyfor small perturbations of the packing. In particular,the structure of the packing includes the positions of theparticles and the contact force network. Perturbations tobe considered should include changes in the grain inter-nal geometry (deformation), strain, and stress (externalforces due to shaking, vibration, or a macroscopic load).In great generality we can restrict our perturbations tosmall perturbations of the distances between contacting

2 A hyperstatic packing is statically underdetermined, since thereare multiple ways to resolve almost any applied load. In thiscase constitutive (elastic) laws need to be invoked to determinethe forces. A hypostatic packing on the other hand is staticallyoverdetermined, and as such is considered unstable in the litera-ture on mechanical structures.

12

particles combined with small perturbations of the ap-plied forces. Such a perspective on jamming was recentlypresented in Ref. [19]. In this work, however, only per-turbations of the applied forces were considered; How-ever, it is realized in Ref. [19] that deformations of theboundary conditions can easily be incorporated withoutchanging the stability conditions. In fact, arbitrary ex-ternal perturbations of the geometry of the contacts canbe considered in addition to the applied load perturba-tions without any significant complication.

Mathematically, we consider the sensitivity of the con-figuration and force chains to all perturbations of theinterparticle gaps ∆ζ and applied forces ∆B away fromzero, i.e., we look for solutions of the coupled system ofequations of preserving contacts and maintaining forceequilibrium:

[A (Q + ∆Q)] (f + ∆f) = −ε∆Bζ (Q + ∆Q)−∆ζµ = −ε∆ζ

eT ∆f = 0, (8)

where ε > 0 is a small number and we have assumedf > 0. Note that in Ref. [19], ∆f are called the “basicstatical unknowns” and ∆Q are called the “basic kine-matical unknowns.”

Similarly to the external forces, the space of resolvablegap perturbations is determined by the boundary condi-tions: global expansions will lead to gaps that cannot allbe closed unless the particles grow by a certain scalingfactor µ = 1 + ∆µ. It is therefore convenient to include∆ζµ ≈ 2∆µ as an additional variable. An added con-straint is that the normalization eT f = M be maintained.It is important to note that we explicitly account for thedependence of the rigidity matrix on the configuration inthe force-balance equation. Notice that when we com-bine perturbations of the geometry and forces together,the total number of variables is M + Nf , and the totalnumber of constraints is also M + Nf (here we includethe global particle rescaling ∆ζµ as a degree of freedom).Therefore there are no underdetermined (linear) systemsas found in counting arguments that consider geometryand forces separately, as is typically done in the litera-ture.

B. Isostaticity

In this section we will attempt to deconstruct previ-ous arguments in justification of an isostatic conjecture,mostly in the context of sphere packings, and try to iden-tify the problems when the same arguments are appliedto nonspherical particles.

The isostatic conjecture (property) is usually justifiedin two steps. First, an inequality Z ≤ 2df is demon-strated, then, the converse inequality Z ≥ 2df is invokedto demonstrate the equality Z = 2df . We will demon-strate that it is the second of these steps that fails for

non-spherical particles, however, first we recall some typ-ical justifications for the inequality Z ≤ 2df . Isostaticityis also extensively discussed by Roux [10], and althoughthere are close connections between our discussion hereand the one in Ref. [10], we will not discuss the similari-ties or differences due to space limitations.

1. Why Z ≤ 2df applies

A packing with Z > 2df is overconstrained, and in acertain sense geometrically degenerate and thus not“ran-dom”. It can be argued that such a packing is not sta-ble against small perturbations of the packing geometry,since all contacts cannot be maintained closed withoutdeforming some of the particles. For example, Witten[8] considers hard-sphere packings with a small polydis-persity, so that particles have slightly different sizes, toconclude that “the creation of a contact network withcoordination number higher than 2d occurs with proba-bility zero in an ensemble of spheres with a continuousdistribution of diameters.” Moukarzel [7, 26] considershow the actual stiffness modulus of deformable particlesaffects the interparticle forces and concludes that mak-ing the particles very stiff will eventually lead to negativeforces and thus breaking of contacts, until the remainingcontact network has Z ≤ 2df

3: “The contact network ofa granular packing becomes isostatic when the stiffness isso large that the typical self-stress...would be much largerthan the typical load-induced stress...granular packingswill only fail to be isostatic if the applied compressiveforces are strong enough to close interparticle gaps es-tablishing redundant contacts.” A similar argument ismade by Sir Edwards in Ref. [9] for frictional grains: “ifz > 4 then there is a solution with no force on z− 4 con-tacts, and there is no reason why other solutions wouldhave validity.”

These arguments apply also to nonspherical particles,however, it is important to point out that they specif-ically only apply to truly hard-particle packings or topackings of deformable particles in the limit of zero ap-plied pressure (f → 0). In real physical systems particleswill have a finite stiffness and the applied forces will benon-negligible, and such packings will have more contactsthan the idealized hard-particle construction.

3 Specifically, a packing of stiff particles will only have as manycontacts closed as there are degrees of freedom, M ≤ Nf . Theadditional single degree of freedom due to the density φ does notcount unless the packing is compressed to close the additionalone contact that is discussed in Section IVA2. Closing morethan M = Nf + 1 contacts will require further compression andsignificantly larger deformation of the particles.

13

Figure 5: (Color online) A mobile ellipse (green) jammed be-tween three fixed ellipses (yellow). All ellipses are of the samesize and have an aspect ratio α = 2. This packing was pro-duced by a Lubachevsky-Stillinger type algorithm, where thethree particles were kept fixed by giving them infinite massand no initial velocities. The normal vectors at the pointsof contact intersect at a common point I, as is necessary toachieve torque balance. The number of force balance con-straints here is two, and the number of torque constraints isone, giving a total of three constraints. However, due to thegeometric degeneracy there are only two independent equa-tions of mechanical equilibrium (but recall to count plus dueto the non-negativity of the forces). The number of unknownforces is 3.

2. Why Z ≥ 2df does not apply

The converse inequality, stating that a minimum of 2df

contacts is necessary for jamming (rigidity), does not ap-ply to nonspherical particles. We can demonstrate thisvividly with a simple example of an ellipse jammed be-tween three other stationary (fixed) ellipses, as shown inFig. 5. Jamming a disk requires at least three touchingdisks; the additional rotational degree of freedom of theellipse would seem to indicate that four touching ellipseswould be needed in order to jam an ellipse. However,this is not true: if the normal contact vectors intersect ata single point, three ellipses can trap another ellipse, asshown in Fig. 5. We will shortly develop tools that canbe used to demonstrate rigorously that this example isindeed jammed. Another simple example demonstratingthat Z ≥ 2df does not apply is the rectangular lattice ofellipses, which is collectively jammed even though Z = 4,the minimum necessary even for discs. This exampleis discussed in Appendix A, where we also demonstratethat, in fact, any jammed isostatic (i.e., Z ≈ 2d) packingof spheres can be converted into a jammed packing ofnonspherical particles.

The above example shows that the claim of Ref. [6]that “One requires 4(= 3 + 1) contacts to fix the DOF[degrees of freedom]...of an ellipse in the plane” is wrong.

Similarly, it shows that the argument in Ref. [7], namely,that the minimum number of contacts needed for a pack-ing of N spheres in d dimensions to be rigid is dN , cannotbe generalized to nonspherical particles by simply replac-ing d with df . Claims that the number of constraintsmust be larger than the number of degrees of freedomhave been made numerous times within the kinematicperspective on jamming, for example, in Ref. [9]. Ourcareful analysis of the conditions for jamming in the nextsection will elucidate why this is correct for spheres butnot necessarily correct for nonspherical particles, and un-der what conditions a hypostatic packing can be jammed.

The example in Fig. 5 is a geometrically-degenerateconfiguration which would usually be dismissed as aprobability-zero configuration. However, we will explainin later sections why such apparently non-generic (degen-erate) configurations must appear for sufficiently smallaspect ratios for a variety of realistic packing protocols.In the structural mechanics literature geometrically-peculiar examples such as this one are well-known, how-ever, they are considered to be in unstable equilibrium[25], i.e., stable only under special types of loading. Thistype of argument, made within the static perspective onjamming [c.f. Eq. (7)], is given by Witten in the con-text of granular materials [8]: “The number of equilib-rium equations Nd should not exceed the number of forcevariables Nc; otherwise these forces would be overdeter-mined.” The example in Fig. 5 demonstrates why thisargument cannot be applied to nonspherical grains. Sincethe normal vectors at the points of contact intersect ata point, a torque around that point cannot be resolvedby any set of normal forces between the particles. Yetthe packing is jammed, and if built in the laboratory itwill resist the torque by slight deformations of the parti-cles, so that the normal vectors no longer intersect in onepoint and the contact forces can resist the applied torque.The connection between the geometry of the contact net-work, i.e., A, and the packing configuration Q, as wellas the pre-existing stresses (forces) in the packing, mustbe taken into account when considering the response ofhypostatic packings to external perturbations. This im-portant observation was also recently pointed out inde-pendently in Ref. [19], and we elaborate on it in the nextsection.

V. CONDITIONS FOR JAMMING

In this Section we develop first and second order con-ditions for jamming, using a kinematic approach. Statics(forces) will emerge through the use of duality theory.The discussion here is an adaptation of the theory of first-order, pre-stress, and second-order rigidity developed fortensegrities in Ref. [12]. This section is technical andmay be skipped or skimmed by readers not interested inthe mathematical formalism of jamming. In Section VIIIthe rigorous hard-particle results are explained more sim-ply by considering the conditions for local (stable) energy

14

minima in soft-particle systems.We consider an analytic motion of the particles

∆Q (t) = Qt + Qt2

2+ O(t3),

where Q are the velocities, and Q are the accelerations.Expanding the distances between touching particles tosecond-order, and taking into account that ζ (QJ) = 0,gives

ζ(t) ≈ AT Qt +[QTHQ + AT Q

] t2

2= ζt + ζ

t2

2, (9)

where the Hessian H = ∇2Qζ = ∇QA can be thought of

as a higher-rank symmetric matrix.

A. First-Order Terms

Velocities Q 6= 0 for which ζ = AT Q ≥ 0 representa first-order flex (using the terminology of Ref. [12]).If we can find an unjamming motion Q such that ζ > 0(note the strict inequality), then the packing is first-orderflexible, and there exists a T > 0 such that none of theimpenetrability conditions [c.f. Eq. (6)] are violated for0 ≤ t < T . We call such a Q a strict first-order flex. If onthe other hand for at least one constraint ζ < 0 for everyQ, then the packing is jammed, since every non-trivialmovement of the particles violates some impenetrabilitycondition for all 0 < t < T for some T > 0. We call such apacking first-order jammed. Finally, a Q such that ζ = 0is a null first-order flex, often referred to as zero or floppymode in the physics literature.

A packing is first-order jammed if and only if thereare no (non-trivial) first order flexes. A packing is first-order flexible if there exists a strict first-order flex. Somepackings are neither first-order jammed nor first-orderflexible; One must consider higher-order terms to accesswhether such packings are jammed, and if they are not,to identify an unjamming motion. We will consider thesecond-order terms later; in this section we develop con-ditions and algorithms to verify first-order jamming andidentify first-order flexes if they exist. The algorithmsare closely based on work in Ref. [11].

1. Strict Self-Stresses

Let us first focus on a single contact i, j, and askwhether one can find a first order flex that is strict onthat contact, i.e.,

ζij =(AT Q

)ij

=(AT Q

)T

eij = (Aeij)T Q > 0,

where eij denotes a vector that has all zero entries otherthan the unit entry corresponding to contact i, j. If

it exists, such a flex can be found by solving the linearprogram (LP)

maxQ

(Aeij)T Q

AT Q ≥ 0. (10)

If this LP has optimal objective value of zero, then thereis no first-order flex that is strict on the contact in ques-tion. Otherwise, the LP is unbounded, with an infiniteoptimal objective value. The dual LP of (10) is a feasi-bility problem

A(f + eij

)= 0

f ≥ 0, (11)

where the contact forces f are the Lagrange multiplierscorresponding to the impenetrability constraints AT Q ≥0. If the dual LP (11) is feasible, then the primal LP (10)is bounded. If we identify f = f + eij ≥ 0, fij ≥1, weare naturally led to consider the existence of non-trivialsolutions to the force-equilibrium equations

Af = 0 and f ≥ 0. (12)

A set of non-negative contact forces f 6= 0 that are inequilibrium as given by Eq. (12) is called a self-stress4. In Ref. [12] these are called proper self-stresses, asopposed self-stresses which are not required to be non-negative. Self-stresses can be scaled by an arbitrary pos-itive factor, so we will often add a normalization con-straint that the average force be unity, eT f = M . Aself-stress that is strictly positive on a given contact isstrict on that contact. A self-stress f > 0 is a strict-selfstress. The existence of a (strict) self-stress can be testedby solving the linear program

maxf ,ε

ε

Af = 0f ≥ εe

eT f = M (13)

and seeing whether the optimal value is negative (no self-stress exists), positive (a strict self-stress exists), or zero(a self-stress exists). What we showed above using linearduality is that if there is a self stress that is strict ona given contact, there is no flex strict on that contact.In particular, this means that packings that have a strictself-stress can only have null first-order flexes.

We can also show that there is a first-order flex thatis strict on all contacts that do not carry a force in anyself-stress (i.e., no self-stress is strict on them). To this

4 Note that in our notation a self-stress has dimensions of force,rather than force per unit area as in the engineering literature.

15

end, we look for a first-order flex that is strict on a givensubset of the contacts, as denoted by the positions of theunit entries in the vector e

maxQ,ε

ε

AT Q ≥ εe. (14)

The dual program is the feasibility problem

Af = 0eT f = 1

f ≥ 0, (15)

which is infeasible if there is no self-stress that is pos-itive on at least on the contacts under consideration,since eT f ≡ 0. Therefore the primal problem (13) is un-bounded, that is, one can find a self-stress that is strict(since ε → ∞) on the given set of contacts. This showsthat packings that do not have a self-stress are first-orderflexible. In other words, the existence of force chains ina packing is a necessary criterion for jamming.

In summary, if a packing has no self-stress, it is notjammed, and one can easily find a strict first-order flex bysolving a linear program [11]. The analysis is simplifiedif the packing has a strict self-stress, since in that caseall first-order flexes are null, i.e., they are solutions of alinear system of equalities AT Q = 0. This is the caseof practical importance to jammed packings, so we willfocus on it henceforth.

2. Floppy Modes

The linear system AT Q = 0 has Nfloppy = Nf − r

solutions, where r = M − Nstresses is the rank of therigidity matrix, and Nstresses is the number of (not nec-essarily proper) self-stresses (more precisely, the dimen-sionality of the solution space of Af = 0). We know thatNstresses ≥ 1 for a jammed packing. If the packing is nothypostatic, or more precisely, if the number of contactsis sufficiently large

M = Nf + Nstresses ≥ Nf + 1,

then there are no non-trivial null first-order flexes (floppymodes), Nfloppy = 0. Therefore, a packing that hasa strict self-stress and a rigidity matrix of full-rank is(first-order) jammed. We will later show that this suffi-cient condition for jamming is also necessary for spherepackings, that is, jammed sphere packings are never hy-postatic.

However, we will see that jammed ellipsoid packingsmay be hypostatic, M < Nf + 1. Such a hypostaticpacking always has floppy modes,

Nfloppy = Nf + Nstresses −M ≥ Nf + 1−M.

Every floppy mode can be expressed as a linear combi-nation of a set of Nfloppy basis vectors, i.e.,

Q = Vx for some x, (16)

where the matrix V is a basis for the null-space of AT .To determine whether any of the null first-order flexescan be extended into a true unjamming motion, we needto consider second-order terms, which we do next.

B. Second-Order Terms

Consider a given null first-order flex AT Q = 0. Wewant to look for accelerations Q that make the second-order term in the expansion (9) non-negative, i.e.,

AT Q ≥ −QTHQ. (17)

If we cannot find such a Q for any first-order flex, thenthe packing is second-order jammed. If we find a Q suchthat all inequalities in (17) are strict, than we call theunjamming motion

(Q, Q

)a strict second-order flex, and

the packing is second-order flexible, since there exists aT > 0 such that none of the impenetrability conditions[c.f. Eq. (6)] are violated for 0 ≤ t < T . If for all first-order flexes Q at least one of the inequalities in (17) hasto be an equality, then we need to consider even third- orhigher-order terms, however, we will see that for sphereand ellipsoid packings this is not necessary.

1. The Stress Matrix

In order to find a strict second-order flex, we need tosolve the LP

maxQ,ε

ε

AT Q ≥ εe− QTHQ, (18)

the dual of which is

minf

(QTHQ

)T

f

Af = 0eT f = 1

f ≥ 0, (19)

where the common optimal objective function is

ε∗ =(QTHQ

)T

f = QT (Hf) Q = QT HQ,

where H = Hf is a form of reduced Hessian that in-corporates information about the contact force and thecurvature of the touching particles. The [Nf ×Nf ] ma-trix H plays an essential role in the theory of jamming

16

for hypostatic ellipsoid packing and we will refer to it asthe stress matrix following Ref. [12].

The stress-matrix has a special block structure, whereall of the blocks are of size [df × df ], and both the block-rows and the block-columns correspond to particles. Theblock entry corresponding to the pair of particles (i, j) isnonzero if and only if there is a contact between them.Written explicitly, the stress matrix is a force-weightedsum of contributions from all the contacts

H =∑i,j

fijHij ,

where the contribution from a given contact i, j is

i · · · j↓ · · · ↓

Hij =i →

...j →

∇2iiζij · · · ∇2

jiζij

.... . .

...∇2

ijζij · · · ∇2jjζij

.(20)

If QT HQ < 0 then ε∗ < 0 and therefore the first-orderflex Q cannot be extended into a second-order flex. Wesay that the stress matrix blocks the flex Q. If on theother hand QT HQ > 0, then ε∗ > 0 and by solvingthe LP (18) we can find an unjamming motion, i.e., thepacking is second-order flexible. Therefore, finding anunjamming motion at the second-order level essentiallyconsists of looking for a null first-order flex (floppy mode)Q, AT Q = 0, that is also a positive curvature vector forthe stress matrix.

Recalling that every floppy mode can be expressed asQ = Vx [c.f. Eq. (16)], we see that

QT HQ = xT(VT HV

)x = xT HV x.

If the matrix HV is negative-definite, than the packingis second-order jammed. In Ref. [12] such packings arecalled pre-stress stable, since the self-stress f rigidifies thepacking (i.e., blocks all of the floppy modes). If HV isindefinite, than the packing is second-order flexible sinceany of the positive-curvature directions can be convertedinto a strict self-stress by solving the LP (18).

If a packing has more than one (proper) self-stress,than it is not clear which one to use in the stress-matrix.One can try to find a self-stress that provides for jamming(pre-stress stability) by looking for a solution to Eq. (13)such that HV 0 (i.e., HV is negative-semidefinite).This is known as semidefinite programming (SDP), andis a powerful generalization of linear programming thathas received lots of attention recently [27]. It is howeverpossible that different self-stresses are needed to blockdifferent portions of the space of floppy modes, and thisgeneral case of a second-order jammed packing is difficultto test for algorithmically. In our study of disorderedsphere and ellipsoid packings, we will see that in practicethe jammed packings only have one strict self-stress. Inthis case, testing for jamming reduces to calculating the

smallest eigenvalue of HV . We will discuss actual numer-ical algorithms designed for ellipsoid packings in subse-quent sections, but first we explain what makes spherepackings special.

2. The Stress Matrix for Hard Spheres

For hard spheres it is easy to write down the explicitform for Hij since the overlap function is given explicitlyby Eq. (4) and its second-order derivatives are trivial,

∇2iiFij = ∇2

jjFij = −∇2ijFij = −∇2

jiFij =2Id

(Oi + Oj)2 ,

where Id is the [d× d] identity matrix. This implies thatHij is a positive-definite matrix, since

RT HijR = (ri − rj)T (ri − rj) ≥ 0.

Therefore, any first-order flex in fact represents a trueunjamming motion, since QTHQ ≥ 0 and we can triv-ially use Q = 0 in Eqs. (18). In other words, a spherepacking is jammed if and only if it is first-order jammed,and therefore it cannot be hypostatic. To test for jam-ming in hard-sphere packings we need only focus on thevelocities of the sphere centroids and associated linearprograms in Section V A. This important conclusion wasdemonstrated using a simple calculation in Ref. [11].

For general particle shapes, however, Hij may be in-definite for some contacts, and testing for jamming mayrequire considering second-order terms. If one consid-ers general convex particle shapes but freezes the orien-tations of the particles, the packing will behave like ahard-sphere packing. In particular, a jammed packing ofnonspherical particles must have at least as many con-tacts as the corresponding isostatic packing of sphereswould, that is,

Z ≥ 2d

for any large jammed packing of convex hard particles.

C. Testing for Jamming

We now summarize the theoretical conditions for jam-ming developed in this section in the form of a procedurefor testing whether a given packing of non-spherical par-ticles is jammed. We assume that the contact network ofthe packing is known and available as input. For sphericalparticles, as already discussed, second-order terms neverneed to be considered, and testing for jamming can bedone by solving one or two linear programs, as discussedin detail in Ref. [11]. In the formulation below, we avoidsolving linear programs unless necessary, but rather usebasic linear algebra tools whenever possible.

17

1. Find a basis F for the null-space of the rigiditymatrix A, i.e., find Nstresses linearly independentsolutions to the linear system of equations Af = 0,normalized to mean of unity. This can be done, forexample, by looking for zero eigenvalues and theassociated eigenvectors of the matrix AT A. If

(a) Nstresses = 0,(b) Nstresses = 1 but the unique self-stress is not

non-negative, or(c) Nstresses > 1 but the linear feasibility pro-

gram (13) is infeasible,

then declare the packing not jammed (first-orderflexible), optionally identify an unjamming motionby solving the linear feasibility program AT Q ≥e, and terminate the procedure. Otherwise, if theidentified self-stress f is not strict, declare the testinconclusive and terminate.

2. If Nfloppy = Nf + Nstresses − M = 0, then de-clare the packing (first-order) jammed and termi-nate the procedure. Otherwise, find a basis Vfor the null-space of AT , i.e. Nfloppy linearly-independent solutions to the linear system of equa-tions AT ∆Q = 0. Compute the stress matrix Husing the previously-identified strict self-stress f ,and compute its projection HV on the space of nullfirst-order flexes.

3. Compute the smallest eigenvalue λmin and asso-ciated eigenvector xmin of the matrix HV . Ifλmin < 0, declare the packing (second-order)jammed and terminate the procedure. If λmin > 0and Nstresses = 1 declare the packing not jammed(second-order flexible), optionally compute an un-jamming motion by solving the LP (18) with Q =Vxmin, and terminate the procedure. Otherwise,declare the test inconclusive and terminate.

We will discuss the actual numerical implementationof this algorithm later, and see that in practice we donot need to solve linear programs to test for jammingin hypostatic ellipsoid packings. Essentially, the pack-ings we encounter in our work with disordered packingsof hard ellipsoids always have a single strict self-stressand a negative-definite HV . The rectangular lattice ofellipses offers a different kind of example, namely, onewith simple regular geometry but multiple self-stresses,and we analyze this example theoretically in AppendixA.

D. Outside the Kinematic Perspective

It is worthwhile to briefly consider the connections be-tween the jamming criteria developed above using thekinematic approach to jamming, and the static and per-turbation approaches.

1. Static View

We have already seen that forces appear naturallyas Lagrange multipliers corresponding to impenetrabilityconstraints, in the form of a strict self-stress f > 0. Inthe static view, we ask whether a packing can support agiven applied external force B by a set of non-negative in-terparticle forces. The key observation is that we can addan arbitrary positive multiple of a self-stress to any set ofinterparticle forces that support B in order to make themnon-negative, without affecting force balance. Therefore,if the rigidity matrix A is of full-rank, as it has to be forjammed sphere packings, any (supportable) load B canbe balanced with non-negative interparticle forces, andkinematic and static rigidity become equivalent [28].

The addition of arbitrary multiples of the self-stress tof is, however, a product of the mathematical idealizationof the packing. In fact, each specific applied load in anisostatic packing with M = Nf + 1 contacts will be sup-ported by a well-defined f . The self-stress is only physicalif all Nf + 1 contacts are active, which requires that thepacking already be compressed by some pre-existing ap-plied load. Otherwise, the density will be slightly smallerthan the jamming density and upon application of an ex-ternal load one of the contacts will break and only Nf ofthe contacts will be active. In general, finding the activeset of contacts requires solving the linear program [11]

minf eT f for virtual worksuch that Af = −B for equilibrium (21)

f ≥ 0 for repulsion only.

At the solution, modulo degenerate situations, only Nf

of the forces will be positive, the remaining ones will bezero.

For jammed hypostatic ellipsoid packings, such as theone in Fig. 5, supporting some loads may require a smalldeformation of the packing, such as a slight rotation ofthe mobile ellipse in the example in Fig. 5. After thissmall deformation, the normal vectors at the points ofcontact will change slightly and the interparticle forces fcan support the applied force B. The larger the magni-tude of the forces is, the smaller the deformation neededto support the load is. Therefore every jammed packingcan support any applied force in a certain generalizedsense. Another way to look at this is to observe that, ifthe interparticle forces are much larger than the appliedones, the applied load will act as a small perturbationto the packing and the static view becomes equivalent tothe perturbation view (with ∆ζ = 0). We consider theperturbation view next and show how the stress matrixappears in the response of the packing to perturbations.

2. Perturbation View

In the perturbation view we consider how the config-uration and the contact forces respond to perturbations

18

consisting of small changes of the contact geometry andsmall applied forces. Counting geometric and force con-straints separately, as done in the literature, is incorrectwhen f > 0: There is coupling between the particle po-sitions and the interparticle forces as represented by theHessian H = Hf .

With this in mind, we can expand Eq. (8) to first orderin ‖∆Q‖ , ‖∆f‖, to get the linear system of equations

A −H 00 AT −2ee 0 0

∆f∆Q∆µ

= −ε

∆B∆ζ0

. (22)

It can be demonstrated that if the reduced Hessian HV isdefinite, this system will have a solution for any ∆B and∆ζ. Furthermore, if HV is negative-definite the responseto perturbations will be stable, in the sense that appliedforces will do a positive work in order to perturb the pack-ing. This is explained in greater detail in Ref. [19], wherethe conditions ‖∆Q‖ = O(‖∆B‖) and ∆BT ∆Q < 0 arestated in a more general setting, and then a lineariza-tion of the response of the packing to perturbations isconsidered (recall that in that work ∆ζ ≡ 0).

Equation (22) can be used to find the jamming pointstarting with a packing that is nearly jammed, i.e., apacking that has nonzero interparticle gaps ε∆ζ and aself-stress that has a small imbalance ε∆B = Af . Thisworks well for small packings, however, for large disor-dered packings, the force chains are very sensitive tosmall changes in the geometry and the linearization ofthe perturbation response is not a good approximationeven for packings very close to the jamming point. Ad-ditionally, we note that to first order in ε, the solution toEq. (22) has ∆µ/ε = fT ∆ζ/2M = fT

Eh/2M , which canbe used to quickly estimate the jamming gap of a nearly-jammed packing from just the interparticle gaps ∆ζ = ζand the interparticle forces, without knowing the actualjamming point [2].

VI. NUMERICALLY TESTING FOR JAMMINGIN HYPOSTATIC ELLIPSOID PACKINGS

In this section we will apply the criteria for jammingand the algorithm to test for jamming from Section VCto our computationally-generated hypostatic packings ofellipsoids. This section is technical and may be skippedor skimmed by readers not interested in the mathematicalformalism of jamming. The numerical results show thatthe packings are indeed second-order jammed, even veryclose to the sphere point. Before discussing the numericaldetails of the algorithm, we need to calculate the firstand second-order derivatives of the overlap potential forellipsoids.

A. Overlap Potentials for Ellipsoids

Numerical algorithms for calculating the PW overlappotential ζ = µ2 − 1 for ellipsoids are presented in thesecond part of Ref. [15]. Here we review the essentialnotation and give the first and second-order derivativesof the overlap potential, necessary to build the rigidityand stress matrices for a given packing.

An ellipsoid is a smooth convex body consisting of allpoints r that satisfy the quadratic inequality

(r− r0)T X (r− r0) ≤ 1, (23)

where r0 is the position of the center (centroid), and Xis a characteristic ellipsoid matrix describing the shapeand orientation of the ellipsoid,

X = QT O−2Q, (24)

where Q is the rotational matrix describing the orien-tation of the ellipsoid, and O is a diagonal matrix con-taining the major semi-axes of the ellipsoid along thediagonal. Consider two ellipsoids A and B and denote

Y = λX−1B + (1− λ)X−1

A , (25)

where λ is defined in Section III. The contact point rC

of the two ellipsoids is

rC = rA + (1− λ)X−1A n = rB − λX−1

B n, (26)

where

n = Y−1rAB (27)

is the unnormalized common normal vector at the pointof contact.

In principle the overlap potential is a function of thenormalized quaternions describing the particle orienta-tions, and derivatives of ζ need to be projected onto theunit quaternion sphere. This projection can be avoidedif we do not do a traditional Taylor series in the quater-nions, namely an additive perturbation ∆q, but ratherconsider a multiplicative perturbation to the quaternionsin the form of a small rotation from the current configu-ration ∆ϕ.

1. First-Order Derivatives

The gradient of the overlap potential, which enters inthe columns of the rigidity matrix, can be shown to be

∇Bζ = −∇Aζ =[

∇rBζ

∇ϕBζ

]= 2λ (1− λ)

[n

rBC × n

],

as we derived in Section IIIA 1 for a general convexparticle shape by using the normalized normal vector

19

n = n/ ‖n‖ [note that ζ = λ (1− λ) rTABn − 1 = 0].

Additionally, it is useful to know the derivatives of λ,

∇rBλ = − 2

fλλn,

where

fλλ = 2rT

BCY−1rAC

λ (1− λ)< 0,

n = λnB + (1− λ)nA = λY−1rAC + (1− λ)Y−1rBC ,

and

∇ϕBλ = − 2

fλλ[MBnA − λ (rBC × n)] ,

where

MB = λNLX−1B + RL

CB .

2. Second-Order Derivatives

The explicit expressions for the Hessian of the overlappotential are

∇2rB

ζ = 2λ (1− λ)Y−1 − 4fλλ

(nnT

) 0

∇2ϕBrB

ζ = 2λ (1− λ)MBY−1 + 2[(

∇ϕBλ)nT

]and finally

∇2ϕB

ζ = −fλλ

[(∇ϕB

λ) (

∇ϕBλ)T

]+ 2λ (1− λ) ·

[12

(rBCnT + nrT

BC

)−

(rT

BCn)I]

+

λNLX−1B NR + MBY−1MT

B.

The derivatives with respect to the position and orienta-tion of particle A can be obtained by simply exchangingthe roles of particles A and B, however, there are alsomixed derivatives involving motion of both particles

∇2ϕBrA

ζ = −∇2ϕBrB

ζ

∇2ϕArB

ζ = −∇2ϕArA

ζ

∇2ϕBϕA

ζ = −∇2ϕB

ζ +(∇2

ϕBrBζ)RR

AB −12

∣∣∇ϕBζ∣∣× .

The stress-matrix is built from these blocks as given inEq. (20), where each of the four blocks ∇2

αβζ ( α andβ denote either A or B) involves both translations androtations,

∇2αβζ =

[∇2

rαζ ∇2

ϕαrβζ

∇2rαϕβ

ζ ∇2rβ

ζ

].

B. Numerically Testing for Jamming

The numerical implementation of the algorithm givenin Section V C poses several challenges. The most impor-tant issue is that that algorithm was designed for idealpackings, that is, it was assumed that the true contactnetwork of the packing is known. Packings produced bythe MD algorithm, although very close to jamming (i.e.,very high pressures), are not ideal. In particular, it isnot trivial to identify which pairs of particles truly touchat the jamming point. Disordered packings have a multi-tude of near contacts that play an important role in therigidity of the packing away from the jamming point [29],and these near contacts can participate in the backbone(force-carrying network) even very close to the jammingpoint. Additionally, not including a contact in the con-tact network can lead to the identification of spuriousunjamming motions, which are actually blocked by thecontact that was omitted in error.

For hard spheres, the algorithms can use linear pro-gramming to handle the inclusion of false contacts [11].For ellipsoids, we look at the smallest eigenvalues ofAT A, i.e., the least-square solution to Af = 0. The solu-tion will be positive if we have identified the true contactnetwork, f > 0, but the inclusion of false contacts willlead to small negative forces on those false contacts. Theproblem comes about because the calculation of the self-stress by just looking at the rigidity matrix does not takeinto account the actual proximity to contact between theparticles. One way to identify the true contact networkof the packing is to perform a long molecular dynamicsrun at a fixed density at the highest pressure reached,and record the list of particle neighbors participating incollisions as well as average the total transfer of colli-sional momentum between them in order to obtain the(positive) contact forces [2].

Once the contact network is identified, we want to lookfor null-vectors of the rigidity matrix. This can be doneusing specialized algorithms that ensure accurate answers[30], however, we have found it sufficient in practice tosimply calculate the few smallest eigenvalues of the semi-definite matrix AT A. We used MATLAB’s sparse lin-ear algebra tools to perform the eigenvalue calculation(internally MATLAB uses the ARPACK library, whichimplements the Implicitly Restarted Arnoldi Method).We consistently found that the smallest eigenvalue isabout 3−6 orders of magnitude smaller than the second-smallest eigenvalue, indicating that there is a near linear-dependency among the columns of A in the form of aself-stress. The self-stress, which is simply the eigen-vector corresponding to the near-zero eigenvalue, was al-ways strictly positive; in our experience, disordered pack-ings of ellipsoids have a unique strict self-stress f . Thismeans that there are Nfloppy = Nf +1−M solutions toAT ∆Q = 0, Nf −M of which are exact, and one whichis approximate (corresponding to the approximate self-stress). This can be seen, for example, by calculating theeigenvalues of AAT , since Nf −M will be zero to numer-

20

ical precision, one will be very small, and the remainingones will be orders of magnitude larger.

1. Verification of Second-Order Jamming

Once a strict self-stress is known, second-order jam-ming or flexibility can be determined by examining thesmallest eigenvalue of HV , which requires finding a ba-sis for the linear space of floppy modes. However, it iscomputationally demanding to find a basis for the null-space of AT due to the large number of floppy modes,and since sparsity is difficult to incorporate in null-spacecodes. There are algorithms to find sparse basis for thisnull-space [30], however, we have chosen a different ap-proach.

Namely, we calculate the smallest eigenvalues of

Hk = kAAT −H,

which as we saw in Section VIII B is the Hessian of the po-tential energy for a system of deformable ellipsoids wherethe stiffness coefficients are all k. For very large k (weuse k = 106), any positive eigenvalue of AAT is stronglyamplified and not affected by H, and therefore only thefloppy modes can lead to small eigenvalues of Hk, de-pending on how they are affected by H. We have foundthat MATLAB’s eigs function is not able to converge thesmallest eigenvalues of Hk for large stiffnesses k, however,the convergence is quick if one asks for the eigenvaluesclosest to zero or even closest to −1. This typically re-veals any negative eigenvalues of Hk and the correspond-ing floppy modes.

It is also possible to perform a rigorous numer-ical test for positive-definiteness of Hk using prop-erly rounded IEEE machine arithmetic and MATLAB’s(sparse) Cholesky decomposition of a numerically re-conditioned Hk [31]. We have used the code described inRef. [31] to show that indeed for our packings Hk 0and therefore the packings are second-order jammed. Forspheroids, that is, ellipsoids that have an axes of sym-metry, there will be trivial floppy modes correspondingto rotations of the particles around their own centroid.These can be removed most easily by penalizing any com-ponent of the particle rotations ∆ϕ that is parallel to theaxis of symmetry. For example, one can add to every di-agonal block of Hk corresponding to the rotation of anellipsoid with axes of symmetry u a penalization term ofthe form kuuT .

We have not performed a detailed investigation of avery wide range of samples since our goal here was to sim-ply demonstrate that under appropriate conditions thepackings we generate using the modified Lubachevsky-Stillinger algorithm are indeed jammed, even though theyare very hypostatic near the sphere point. In this work wehave given the fundamentals of the mathematics of jam-ming in these packings. A deeper understanding of themechanical and dynamical properties of nearly-jammedhypostatic ellipsoid packings is a subject for future work.

VII. NEARLY JAMMED PACKINGS

So far we have considered ideal jammed packings,where particles are exactly in contact. Computer-generated packings however always have a packing frac-tion φ slightly lower than the jamming packing fractionφJ , and the particles can rattle (move continuously) to acertain degree if agitated thermally or by shaking [2]. Wecan imagine that we started with the ideal jammed pack-ing and scaled the particle sizes by a factor µ = 1−δ < 1,so that the packing fraction is lowered to φ = φJ (1− δ)d.We call δ the jamming gap or distance to jamming.

It can be shown that if δ is sufficiently small the rattlingof the particles does not destroy the jamming property,in the sense that the configuration point Q = QJ + ∆Qremains trapped in a small jamming neighborhood or jam-ming basin J∆Q ⊂ RNf around QJ , which can be shownrather generally using arguments similar to those in Ref.[13] for tensegrities. In the limit δ → 0 the set of acces-sible configurations J∆Q → QJ, and in fact this is thedefinition of jamming used by Salsburg and Wood in Ref.[32]. Rewritten to use our terminology, this definition is:“A configuration is stable if for some range of densitiesslightly smaller than φJ , the configuration states acces-sible from QJ lie in the neighborhood of QJ . More for-mally, if for any small ε > 0 there exists a δ > 0 such thatall points Q accessible from QJ satisfy ‖Q−QJ‖ < ε

provided φ ≥ φJ (1− δ)d.” We call this the trapping viewof jamming, most natural one when considering the ther-modynamics of nearly jammed hard-particle systems [33].Note that the trapping definition of jamming is in factequivalent to our kinematic definition of jamming [13].

To illustrate the influence of the constraint curvatureon jamming, we show in Fig. 6 four different cases withtwo constraints in two dimensions. In all cases a self-stress exists since the normals of the two constraints areboth horizontal. If both constraint surfaces are concave(have negative or outward curvature), as constraints al-ways are for hard-spheres, two constraints cannot closea bounded region J∆Q around the jamming point. Oneneeds at least three constraints and in that case J∆Q

will be a curved triangle. If however at least one of theconstraints is convex (has positive curvature), two con-straints can bound a closed jamming basin. Specifically,if the sum of the radii of curvatures of the two constraintsat the jamming point R1 +R2 is positive, there is no un-jamming motion. On the other hand, if it is negative thenthere is an unjamming motion in the vertical (floppy)direction. This is equivalent to looking at the smallesteigenvalue of the stress matrix in higher dimensions.

The jamming basin J∆Q(δ) for a given jamming gap δis the local solution to the relaxed impenetrability equa-tions

ζ (∆Q) ≥ −ζδ = 1−(

11− δ

)2

.

One way to determine J∆Q(δ) for a wide range of δ’s is

21

Figure 6: (Color online) The feasible region (yellow) arounda jamming point (black circle) for two curved constraints intwo dimensions (black circles). The region of the plane for-bidden by one of the constraints is colored red, and coloredblue for the other constraint. The region forbidden by bothconstraints is purple. The distance from the jamming pointto the constraints is approximately δ and chosen small. Fourcases are shown, going from left to right: (a) Both constraintsare concave, and the yellow region is not bounded. Movingalong the vertical direction unjams the system (this is typ-ical of hard spheres). (b) Both constraints are convex, andthe yellow region is closed, even though it is very elongatedalong the vertical direction (of order

√δ). This is an exam-

ple of pre-stress stability (second-order jamming). (c) Oneof the constraints is convex, but not enough to block the un-jamming motion in the vertical direction. The motion has tocurve to avoid the convex constraint, i.e., a nonzero acceler-ation is needed to unjam the system (second-order flexible).(d) Only one of the constraints is convex, but enough to closethe yellow region (second-order jammed). If the radii of cur-vatures are very close in magnitude, this region can becomea very elongated banana-like shape.

to consider the function of the particle displacements

δ (∆Q) =√

1 + min [ζ (∆Q)]− 1, (28)

that is, to calculate by how much the particles need to beshrunk to make a given particle displacement ∆Q feasible(preserving non-overlapping). The contours (level-sets)of the function δ (∆Q) denote the boundaries of J∆Q(δ),

that is, J∆Q(δ) =

∆Q | δ (∆Q) ≤ δ

.

A. First-Order Jammed Packings

As a simple but illustrative example, we will consider asingle mobile disk jammed between three other stationarydisk, as shown in Fig. 7, an analog of the ellipse examplefrom Fig. 5. This packing is first-order jammed, andthe figure also shows a color plot of the function δ (∆Q)along with its contours. It is seen that for small δ thejamming basin J∆Q is a closed curved triangle.

These observations are readily generalized to higherdimensions. For sufficiently small δ, the jamming basinapproaches a convex jamming polytope (a closed polyhe-dron in arbitrary dimension) P∆Q. For spheres all con-straint surfaces are concave and therefore P∆Q ⊆ J∆Q

[32, 34]. The jamming polytope is determined from thelinearized impenetrability equations

AT ∆Q ≥ −ζδ ≈ −2δ, (29)

and we can see that its volume, which determines the(non-equilibrium) free-energy, scales like δNf . This leadsto the free-volume divergence of the pressure in the jam-ming limit

p =PV

NkT≈ df

1− φ/φJ, (30)

which has been verified numerically for disordered iso-static hard sphere packings [2].

B. Second-Order Jammed Packings

The ellipse analog from Fig. 5 has three degrees offreedom, two translational and one orientational. If wefix the orientation of the (mobile) ellipse, that is, we takea planar cut through δ (∆Q), the situation is identical tothat for the disk example above: For small δ the jam-ming basins J∆Q are closed curved triangles. However,the range of accessible orientations is rather large, on theorder of

√δ, since even for a small δ the ellipse can rotate

significantly. This is a consequence of the rotation of theellipse being a floppy mode, and only being blocked bysecond-order effects as given by the curvature of the im-penetrability constraints. In a certain sense, the packingis trapped to a greater extent in the subspace of configu-ration space perpendicular to the space of floppy modesthan it is in the space of floppy modes. This is illustratedin Fig. 8.

C. Pressure Scaling for Hypostatic JammedEllipsoid Packings

The observations in Fig. 8 are readily generalized tohigher dimensions, however, it is no longer easy to deter-mine the volume of J∆Q (and thus the free energy) in thejamming limit. If we consider the simple two-constraintexample in Fig. 6, we find that the area A of the feasible(yellow) region scales like δ3/2 instead of δ2,

A =163

√R1R2

R1 + R2δ3/2.

An obvious generalization of this result to higher dimen-sions can be obtained by assuming that the jammingbasin J∆Q has extent

√δ along all Nfloppy ≈ Nf−M di-

rections corresponding to floppy modes, where as it hasextent δ along all other perpendicular directions. Thevolume would then scale as

|J∆Q| ∼ δMδ(Nf−M)/2 = δN(df /2+Z/4) = δNdf (1+s)/2,

where we quantify the hypostaticity of the packing bys = Z/2df . The corresponding scaling of the pressure in

22

Figure 7: (Color online) (Left) An example of a mobile disk (green) jammed between three fixed disks (yellow). This is

analogous to the ellipse packing shown in Fig. 5. (Right) A color plot of the function δ (∆Q) for this disk packing along withits contours (level sets).

the jamming limit is

p =PV

NkT≈ df (1 + s)/2

1− φ/φJ.

However, as δ becomes very small, the jamming regionbecomes so elongated along the space of floppy modesthat the time-scales for rattling along the elongated di-rections becomes much larger than the time for rattlingin the perpendicular directions. This manifests itself as aremarkably large and regular oscillation of the “instanta-neous” pressure (as measured over time intervals of tensof collisions per particle) during molecular-dynamics runsat a fixed δ, as illustrated in Fig. 9. The oscillationsare more dramatic the smaller δ is, and can span six ormore orders of magnitudes of changes in the instanta-neous pressure. The period of oscillation, as measured innumbers of collisions per particle, is dramatically affectedby the moment of inertia of the ellipsoids I, most natu-rally measured in units of mO2, where m is the particlemass and O is the (say smallest) ellipsoid semiaxis.

We do not understand the full details of these pressureoscillations, however, it is clear that dynamics near thejamming point for the hypostatic ellipsoid packings is notergodic on small time-scales. In particular, as a packing iscompressed during the course of the packing algorithm,the time-scale of the compression may be shorter thanthe time-scale of exploring the full jamming basin. Overshorter time scales the packing can only explore the di-rections perpendicular to the floppy modes, and in this

case we expect that the pressure would scale as

p ≈ dfs

1− φ/φJ.

In Fig. 10 we show C = p(1 − φ/φJ) as a function ofthe jamming gap for compressions of systems of ellipsesof different aspect ratios close to unity. The compressionstarted with a dense liquid and the particles were grownslowly at an expansion rate γ = 10−5 to a high pressure(jamming) p = 109. The figure shows for each aspect ra-tio the lower bound CL = dfs = 3s and the upper boundCU = df (1 + s)/2 = 1.5(1 + s), where s was calculatedby counting the almost perfect contacts at the highestpressure [2]. As expected from the arguments above, wesee that very close to the jamming point C ≈ CL, how-ever, further away from jamming C ≈ CU . For packingsthat are not hypostatic CL = CU = df , and for disksCU = CL = 2.

VIII. ENERGY MINIMA IN SYSTEMS OFDEFORMABLE PARTICLES

In this section we consider the connections betweenjamming in hard particle packings and stable (local) en-ergy minima (inherent structures [35]) for systems ofdeformable (soft) particles. This has a two-fold pur-pose. Firstly, in physical systems particles are always de-formable, and therefore it is important to establish thatthe hard-particle conditions for jamming we establishedin Section V are relevant to systems of deformable par-ticles. We expect that if the particles are sufficiently

23

Figure 8: (Color online) (Left) A plot of the function δ (∆Q) for the packing from Fig. 5. The horizontal axes correspondto the translational degrees of freedom, and the vertical to the rotational degree of freedom (the rotation angle of the majoraxes). The ∆Q = 0 cut is also shown (horizontal colored plane), to be compared to the right part of Fig. 7. We also show thejamming basin J∆Q(δ = 0.0035) (dark blue region), illustrating that this region is shaped like a banana, elongated along the

direction of the floppy mode. (Right) Several contours (iso-surfaces) of δ (∆Q), bounding the banana-shaped regions J∆Q(δ).

0 2000 4000 6000Number of collisions (100s per particle)

1e+09

1e+10

1e+11

1e+12

p

I=1, δ=10−12

I=0.01, δ=10−12

I=1, δ=10−11

I=0.01, δ=10−11

I=1, δ=10−10

I=0.01, δ=10−10

Figure 9: The “instantaneous” reduced pressure for a jammedhypostatic packing of three-dimensional ellipsoids with semi-axes ratio 1.025−1 : 1 : 1.025, at different (estimated) dis-tances from the jamming point δ. Molecular dynamics runsusing a natural moment of inertia of the particles as well asones using a much smaller moment of inertia are shown. Thepressure oscillations are sustained for very long periods oftime, however, it is not clear whether they eventually dissi-pate.

stiff, to be made more quantitative shortly, the behav-ior of the soft-particle system will approach that of thecorresponding hard-particle packing. Secondly, consid-

1e-06 1e-05 0.0001 0.001 0.011−φ/φJ

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

C=p(

1−φ/

φ J)

α=1α=1.01α=1.025α=1.05α=1.075α=1.1

Figure 10: The pressure scaling coefficient C = p(1−φ/φJ) assystems of hard ellipses are compressed from a dense liquid tothe jamming point. The value of C is not constant howeverit seems to remain between the bounds CL (shown with adashed line in the same color as C) and CU (shown with asolid line).

ering the conditions for the existence of a stable energyminimum will enable us to derive in a simpler fashionand better understand the jamming conditions from theprevious section.

We consider systems with short-ranged continuous in-terparticle potentials that are a monotonically decreasing

24

function E of the overlap between particles,

Uij = E [ζ (qi,qj)] . (31)

That is, we assume that the elastic behavior of the parti-cles is such that the interaction energy only depends onthe distance between the particles as measured by theoverlap potential ζ. An example of such an elastic po-tential is an inverse power-law

E(ζ) = (1 + ζ)−ν, (32)

which in the limit p → ∞ approaches a hard-particleinteraction

EH(ζ) =

0 if ζ > 0∞ if ζ < 0 .

For sufficiently large power exponents p the interactionis localized around particles in contact and the overallenergy

U =∑ij

Uij → maxij

Uij =(

1 + minij

ζij

)−ν

=(1 + δ2

)−ν

is dominated by the most overlapping pair of particles[see Eq. (28) for the definition of δ]. Additionally, as pgrows the interparticle potential becomes stiff in the sensethat small changes in the distance between the particlescause large changes of the interparticle force

f = −dE

dζ≥ 0,

and the stiffness coefficient

k =d2E

dζ2≥ 0

becomes very large and positive. This indicates a physi-cal interpretation of the hard-particle interaction poten-tial: It is the limit of taking an infinite stiffness coeffi-cient while the force between particles is kept at somenon-negative value, which can be tuned as desired by in-finitesimal changes in the distance between the particles(but note that the forces in different contacts are cor-related since the motion of particles affects all of themsimultaneously).

A. Stable Energy Minima Correspond to JammedPackings

Assume that we have a packing of hard particles andthat we can find a set of interparticle interaction poten-tials Uij for the geometric contacts such that the configu-ration is a stable energy minimum. This means that anymotion of the particles leads to increasing the energy U ,i.e., to overlap of some pair of particles. Therefore, thepacking of hard particles is jammed. This gives a simple

way to prove that a given packing is jammed: Find a setof interparticle potentials that makes the configuration astable energy minimum [12, 13]. We examine the condi-tions for a stable energy minimum when the interactionpotentials are twice differentiable next.

The converse is also true, in the sense that arbitrarilynear a jammed packing there is an energy minimum for asufficiently“hard”interaction potential (in some cases thepotential energy U may have to be discontinuous at theorigin [12]). We demonstrate this on the examples fromFigs. 7 and 8 for a power-law interaction potential withincreasing exponent ν in Figs. 11 and 12, respectively. Itis clear that in the limit p → ∞, the contours of the in-teraction potential become those of δ(∆Q) and are thusclosed near the origin, i.e., the energy has a minimum.The higher the exponent p is, however, the more anhar-monic the interaction potential becomes and the contoursare no longer ellipsoidal near the energy minimum.

It should be emphasized that the energy minima insoft-particle systems have a variable degree of overlapbetween neighboring particles and therefore do not cor-respond to hard-particle packings. In particular, at largepressures or applied forces the deformability of the parti-cles becomes important and the energy minima no longerhave the geometric structure of packings. However, in thelimit of no externally-applied forces, i.e., f → 0, the onlyinteracting particles are those that barely overlap, i.e.,that are nearly touching. Therefore energy minima forpurely-repulsive interaction potentials and a finite cutoffcorrespond to jammed packings of hard particles in thelimit of zero external pressure (alternatively, one can keepthe applied forces constant and make the grains infinitelystiff [7]). Therefore, the packings of soft particles studiedin Ref. [3] very slightly above the “jamming treshold” φc

are closely related to collectively jammed ideal packingsof spheres of diameter D = σ (polydispersity is trivial toincorporate) [36].

B. Hessian Eigenvalues and Jamming

It is well-known that for smooth interactions a givenconfiguration is a stable energy minimum if the gradientof the energy is zero and the Hessian is positive-definite,and the converse is also true if positive-definite is re-placed with positive-semidefinite. This has been used asa criterion for jamming in systems of deformable particles[3, 36].

The gradient of U =∑

ij Uij is

∇QU =∑ij

dE

dζij(∇Qζij) = (∇Qζ)

(∇ζE

)= A

(∇ζE

)= −Af .

The first-order necessary condition for a stable energyminimum is therefore exactly the force/torque balance

25

Figure 11: The total interaction energy U(∆Q) for the example in Fig. 7 when the disks are deformable and interact via apower-law potential. We show U as a color plot with overlayed contours for power exponents ν = 12, 25, and 100 (going from

left to right). Compare the ν = 100 case to the contours of δ(∆Q) in Fig. 7.

Figure 12: The contours (iso-surfaces) of the total interaction energy U(∆Q) for the example in Fig. 8 when the ellipses aredeformable and interact via a power-law potential. Going from left to right, we show ν = 12 and 25, as well as the hard ellipsoidδ(∆Q), corresponding to the limit ν →∞.

condition

Af = 0 and f ≥ 0,

as we derived using linear programming and duality the-ory for hard-particle packings. The Hessian is

∇2QU =

[∇Q

(∇ζE

)]AT +

(∇2

Qζ) (

∇ζE)

=[A

(∇2

ζE)]

AT + (∇QA)(∇ζE

)= AKAT −Hf = AKAT −H,

where K = ∇2ζε = Diag kij is an [M ×M ] diagonal

matrix with the stiffness coefficients along the diagonal,and H = ∇QA = ∇2

Qζ is the Hessian of the overlap con-straints. Note that more careful notation with derivatives

of vectors and matrices can be developed and should inprinciple be employed in calculations to avoid confusionsabout the order of matrix multiplications and transposi-tions [37].

The Hessian

HU = ∇2QU = AKAT −H

consists of two terms, the stiffness matrix HK = AKAT ,and the stress matrix H that we already encountered inthe second-order expansion of the impenetrability con-straints. The importance of not neglecting the stress ma-trix is also noted independently in Ref. [19], where alsoexpressions are given for this matrix for certain types ofcontact geometry.

The second-order sufficient condition for a strict en-

26

ergy minimum is

HU 0.

Since K > 0, the stiffness matrix HC is positive-semidefinite: For any vector ∆Q that is not a floppymode, ∆QT HK∆Q > 0, while ∆QT HK∆Q = 0 if ∆Qis a floppy mode (i.e., AT ∆Q = 0). Therefore, for anydirection of particle motion that is not a floppy mode, onecan make the stiffness coefficients large enough to make∆QT HK∆Q > 0, regardless of the value of ∆QT H∆Q.Floppy modes, however, correspond to negative curva-ture directions of the Hessian HU if they are positive-curvature directions of the stress matrix, ∆QT H∆Q > 0.Therefore, the energy minimum is strict if and only if thestress matrix is negative-definite on the space of floppymodes. This is exactly the same result as the second-order condition for jamming we derived in Section V us-ing duality theory.

For deformable particles, the stiffness coefficients are fi-nite. Therefore, for sufficiently large interparticle forces,the stress matrix may affect the eigenspectrum of theHessian HU and therefore the stability of potential en-ergy minima. For spheres, as we derived earlier, H 0and therefore interparticle forces may only destabilizepackings: This is the well known result that increas-ing the interparticle forces leads to buckling modes insphere packings [6]. Jamming in systems of soft spheresis therefore considered in the limit of f → 0, i.e., thepoint when particles first start interacting [3, 29]. Forellipsoids however, the forces can, and in practice theydo, provide stability against negative or zero-frequencyvibrational modes. The magnitude of the forces becomesimportant, and will determine the shape of the densityof states (DOS) spectrum [29] for small vibrational fre-quencies. To quote from Ref. [6], “The basic claim...isthat one cannot understand the mechanical properties ofamorphous materials if one does not explicitly take intoaccount the direct effect of stresses.”

C. An Example of Pre-Stress Stability

Figure 13 shows a very simple example in which pre-stressing, i.e., pre-existing forces, stabilize a structure.Although the example is not a packing, it illustrates wellsome of the essential features. First, the geometry of thesystem is degenerate, since the two springs are exactlyparallel. This degeneracy insures that a self-stress ex-ists, since one can stretch/compress both springs by anidentical amount and still maintain force balance.

Observe that geometrically the change in the positionof the joint ∆x causes a quadratic change in the lengthof each spring ∆l ≈ ∆x2. To balance an applied force F ,the force inside each spring f needs to be f∆x = F . Ifthe system is not pre-stressed, then the potential energyis quartic around the origin, ∆U = 1

2k∆l2 ≈ 12k∆x4, and

the applied force causes a very large deformation of thestructure ∆x = (F/k)1/3. The structure is stable (i.e.,

Figure 13: An example of a pre-stress stable system. Twoelastic springs of stiffness k and length l are connected viaa joint that can move in the horizontal direction under theinfluence of an external force F .

corresponds to a jammed packing), however, its responseto perturbations is not harmonic. If however there is aninitial force f in the springs, then the potential energyis quadratic around the origin ∆U ≈ f∆l = f∆x2 andthe deformation is linear in the applied force ∆x = F/f .If f < 0, then the system is unstable and will buckle,and if f > 0 the system is stable and its response toperturbations is harmonic. This is exactly the form ofstability that hypostatic ellipsoid packings have.

IX. PACKINGS OF NEARLY SPHERICALELLIPSOIDS

In this section we will consider nearly spherical el-lipsoids, that is, ellipsoids with aspect ratio α close tounity. In particular, we try to understand why thesepackings are hypostatic and to quantitatively explain thesharp rise in the density and contact numbers of disor-dered packings as asphericity is introduced. We proposethat the packings of nearly spherical ellipsoids should belooked at as continuous perturbations of jammed disor-dered disk packings, and establish the leading order termsin the expansion around the sphere point.

A. Rotational and Translational Degrees ofFreedom Are Not Equal

One might at first sight expect a discontinuous changein the contact number, and therefore the structure, asasphericity is introduced. After all, the number of de-grees of freedom jumps suddenly from df = d to (fornon-spheroids) df = d(d + 1)/2 > d. However, suchan expectation is not reasonable. Firstly, the number of

27

degrees of freedom is df = d(d + 1)/2 even for spheres,since spheres can rotate too. This rotation does not affectthe non-overlap conditions and therefore is not coupledto translational degrees of freedom. If the ellipsoids arenearly spherical, particle rotation is only mildly coupledto particle translations and rotation only affects the non-overlap conditions very close to the jamming point. Thisis seen, for example, through a violation of the equiparti-tion theorem in non-equilibrium MD simulations of hardellipsoids, depending on the moment of inertia of the par-ticles and the time-scale of the system evolution. Wetherefore expect that thermodynamically and kinetically,at least at the level of translations, systems of nearlyspherical ellipsoids will behave identically to systems ofspheres until the interparticle gaps become comparableto the difference between the semiaxes. It is thereforenot really surprising that the properties of the jammedpackings such as φJ or Z change continuously with α.

What is somewhat surprising however is that φJ andZ are not differentiable functions of particle shape. Inparticular, starting with a unit sphere and changing agiven semiaxes by +ε 1 increases the density linearly inε, and changing it by −ε also increases the density by thesame amount, ∆φJ ∼ |ε|. As we will show through ourcalculations, this non-differentiability is a consequence ofthe breaking of rotational symmetry at the sphere point.The particle orientations themselves are not differentiablefunctions of particle shape and change discontinuously asthe sphere point is crossed.

Finally, there is little reason to expect packings ofnearly spherical particles to be rotationally jammed. Af-ter all, sphere packings are never rotationally jammed,since the spheres can rotate in place arbitrarily. Simi-larly, near the jamming point, it is expected that par-ticles can rotate significantly even though they will betranslationally trapped and rattle inside small cages, un-til of course the actual jamming point is reached, at whichpoint rotational jamming will also come into play. It istherefore not surprising that near the sphere point, theparameters inside the packing-generation protocol, suchas the moment of inertia of the particles and the expan-sion rate of the particles, can significantly affect the finalresults. In particular, using fast particle expansion ortoo large of a moment of inertia leads to packings thatare clearly not rotationally jammed, since the torques arenot balanced, however, they are translationally jammedand have balanced centroid forces. We do not have a fullunderstanding of the dynamics of our packing-generationalgorithm, even near the jamming point.

In this paper we will focus on packings that are alsorotationally jammed. In general one may need to dis-tinguish between translational and rotational jamming.For example, the ellipsoid packing produced by simplystretching the crystal packing of spheres along a certainaxes by a scaling factor of α is translationally but notrotationally (strictly [11]) jammed. This is because bychanging the axes along which the stretch is performedone gets a whole family of ellipsoid packings with exactly

the same density. Therefore, it is possible to shear thepacking by changing the lattice vectors used in the peri-odic boundary conditions, without changing the density,as illustrated in Fig. 14 in two dimensions.

Figure 14: The triangular packing of ellipses is not rotation-ally jammed since one can shear the packing continuously,without introducing overlap or changing the density. The fig-ure shows a sequence of snapshots as this shearing motionproceeds. The packing is however (strictly [11]) translation-ally jammed.

1. Isostatic Packings are Translationally Ordered

As we already demonstrated, in order for a hypostaticpacking of ellipsoids to be jammed, the packing geometrymust be degenerate. The existence of a self-stress f re-quires that the orientations of particles be chosen so thatthe torques are balanced in addition to the forces on thecentroids. This leads to a loss of “randomness” in a cer-tain sense, since the number of jammed configurations isreduced greatly by the fact that geometrically ”special”(not generic) configurations are needed to balance thetorques.

However, it is also important to point out that dis-ordered isostatic packings of nearly spherical ellipsoidsare hard to construct. In particular, achieving isostatic-ity near the sphere point requires translational ordering.In two dimensions, the average number of contacts perparticle needed is Z = 6, however, the maximal kissingnumber near the sphere point is also Zmax = 6. There-fore the only possibility is that every particle have exactlyZ = 6 contacts. This inevitably leads to translational or-dering on a triangular lattice. In other words, the onlyisostatic packing of ellipses in the limit α → 1 is the harddisk triangular crystal. Similarly, in three dimensions,Z = Zmax = 12 for non spheroids, and therefore everyparticle must have exactly Z = 12 neighbors. While itnot rigorously known what are the sphere packings withall particles having twelve neighbors, it is likely that onlystacking variants of the FCC/HCP lattice achieve thatproperty. For spheroids, the isostatic number of contactsis Z = 10 and the results in Fig. 1 indicate that thisvalue is nearly reached for sufficiently large aspect ratios.For nonspheroids, however, we only observe a maximumof 11.4 contacts per particle, consistent with the fact thatachieving the isostatic value requires more translationalordering.

28

B. Two Near-Spheres (Nearly) Touching

In what follows we will need first-order approximationsof the impenetrability constraints between two nearlyspherical ellipsoids. Assume there are two spheres Aand B of radius OA/B touching. Transform the spheresinto ellipsoids with semiaxes OI + ∆O, and orienta-tion described by the rotation matrix Q, and denoteεO = O−1∆O. Finally, define the matrix

T = QT εOQ,

which in the case of turning a disk into an an ellipse withsemiaxes O and O(1 − ε), i.e., aspect ratio α = 1 + ε,ε 1, becomes

T = −ε

[sin2 φ − sinφ cos φ

− sinφ cos φ cos2 φ

]= −εTφ,

where θ is the angle of orientation of the ellipse. It can beshown that to first order in ε the new distance betweenthe ellipsoids is

∆ζ = 2uTABSuAB ,

where

S =OA

OA + OBTA +

OB

OA + OBTB .

The torque exerted by the contact force f = fn on agiven particle, to first order in asphericity ε, comes aboutbecause the normal vector no longer passes through thecentroid of the particle (as it does for spheres). One canignore the small changes in the magnitude of the normalforce or the change in the contact point rC , and onlyconsider the change in the normal vector

n ≈ Xu ≈ (I− 2T)u = u− 2Tu,

giving a torque

τ = rC × f ≈ 2Of (Tu)× u.

C. Maintaining Jamming Near the Sphere Point

Assume now that we have a collectively jammed iso-static sphere packing with density φS

J and that we wantto make the disks slightly ellipsoidal by shrinking themalong a given set of axes, while still preserving jamming.Keeping orientations fixed, one can expand each near-sphere by a scaling factor ∆µ and displace each centroidby ∆r, so that all particles that were initially in contactare still in contact. Note that because the matrix S isproportional to ε, so will ∆µ and ∆R. In other words,the change in the density will be linear in asphericity.However, the value of the slope depends on the choice oforientations of the ellipsoids. Referring back to SectionV D2 we see that to first order in ε, ∆µ is

∆µ =1M

fT ∆ζ =1M

∑i,j

fijuTijSijuij

=1

2M

∑i

∑j∈N (i)

fijuTijTiuij ,

giving a new jamming density

φJ/φSJ = (1 + ∆µ)d

d∏k=1

(1 + εOi ) ≈ 1 + d∆µ + eT εO.

Keeping all ellipsoids aligned produces an affine defor-mation of the sphere packing that has the same jammingdensity, but is not (first-order) jammed. Therefore, thetrue jamming density must be higher, φJ ≥ φS

J . Thisexplains why the jamming density increases with aspectratio near the sphere point. The added rotational degreesof freedom allow one to increase the density beyond thatof the aligned (nematic) packing, which for ellipsoids hasexactly the same density as the sphere point.

Can we find a set of orientations for the ellipsoids sothat the resulting packing is jammed? The first condi-tion for jamming is that there exist a self-stress that bal-ances both forces and torques on each particle. Just fromthe force-balance condition, one can already determinethe magnitudes of the interparticle forces f . These willchange little as one makes the particles slightly aspheri-cal, because the normal vectors barely change. Therefore,the self-stress is already known a priori, without regardto the choice of particle orientations. The orientationsmust be chosen so that the torques are also balanced. Asshown above, to first order in asphericity ε, the torquebalance condition for particle i is∑

j∈N (i)

fij (Tiuij)× uij =∑

j

fijUijTiuij = 0. (33)

This gives for each particle a set of possible orientations,given the contact network of the isostatic sphere packing.The torque balance condition (33) is in fact the first-orderoptimality condition for maximizing the jamming density,as expected. It is worth pointing out that for a randomassignment of orientations to ellipses the expected changein density is identically zero; in order to get an increasein the density one must use orientations correlated withthe translational degrees of freedom.

1. Ellipses

In two dimensions, for a particular contact with u =〈cos θ, sin θ〉 we have the simple expressions

uTφu = sin2(φ− θ)

u× (Tφu) =12

sin [2(φ− θ)] .

29

Considering 2φ as the variable, one easily finds the solu-tion to Eq. (33)

2φ = arctan(±∑

i

fi sin 2θi,±∑

i

fi cos 2θi). (34)

If we calculate the second derivative for the density in-crease we find that

d2

dφ2

[∑i

fi sin2(φ− θi)

]= ±1,

and therefore in order to maximize the jamming densitywe need to choose the minus signs in Eq. (34). Once wefind the unique orientation of each ellipse that ensurestorque balance, we can calculate the jamming density

φJ/φSJ ≈ 1 + sφε, (35)

where

sφ = 2

∑i

∑j∈N (i) fij

(uT

ijTφi uij

)∑

i

∑j∈N (i) fij

− 1.

We have calculated the slope sφ for disordered binarydisk packings (with φS

J ≈ 0.84) numerically, and find avalue sφ ≈ 0.454. We compare this theoretical valuewith numerical calculations in Fig. 15. The first com-parison is directly to the packing fractions obtained us-ing the Lubachevsky-Stillinger algorithm, which do nothave anything to do with perturbing a sphere packing.Although the simulation jamming densities are not lin-ear over a wide range of aspect ratios, near α = 1 theyare and the slope is close to the theoretically-predictedsφ. We also compare to results obtained by perturbinga jammed disk packing using MD. Specifically, we startwith a jammed disk packing at a relatively high pressure(p = 1000) and assign an orientation according to Eq.(34) to every disk, and then we start growing the largesemiaxes slowly while performing a form of constant-pressure MD. The density changes automatically to keepthe pressure constant, and from the instantaneous den-sity we estimate the jamming density using Eq. (30). InFig. 15 we show how the (estimated) jamming densitychanges with aspect ratio. If we freeze the orientations(i.e., use an infinite moment of inertia), we obtain resultsthat follow the theoretical slope prediction closely. Verygood agreement with the results from the LS algorithm isobtained over a wide range of α if we start with the cor-rect orientations and then allow the ellipse orientationsto change dynamically. For comparison, in the inset weshow that the packing density actually decreases if weuse the LS algorithm and freeze orientations at their ini-tial (random) values, demonstrating that balancing thetorques and (maximally) increasing the density requiresa particular value for the particle orientations.

For ellipses, there are unique orientations that guar-antee the existence of self-stresses near a given isostaticjammed disk packing. Do these orientations actually lead

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1Aspect ratio α

0.85

0.855

0.86

0.865

0.87

0.875

Estim

ated

φJ

MD algorithmFree orientationsFrozen orientationsTheory

1 1.2 1.4 1.60.84

0.85

0.86

0.87

0.88

0.89

0.9

Frozen random

Figure 15: The estimated jamming density near the disk pointfor binary packings of hard ellipses, as obtained from the LSpacking algorithm, from perturbing the disk packing usingconstant-pressure MD, and from the first-order perturbationtheory. The inset shows some of the data over a larger rangeof aspect ratio and also shows the packing densities obtainedwhen the ellipses have infinite moment of inertia in the LSalgorithm.

to jammed packings, that is, are the second-order condi-tions for jamming also satisfied? If one starts with ajammed disk packing and transforms the disks into el-lipses of aspect ratio sufficiently close to unity, the pack-ing will remain translationally jammed [13]. Subsequentincrease in the size of the particles must eventually leadto a packing of maximal density. It is not however apriori whether this packing is rotationally and transla-tionally jammed or has some kind of peculiar unjammingmotions that preserve the density, such as the ones shownin Fig. 14. For small disk packings, we have found theperturbed ellipse packings to be second-order jammedsufficiently close to the sphere point. For larger systems,even for very small asphericities, it is difficult numericallyto perturb a given disk packing into an ellipse packingwithout leading to new contacts or breaking of old ones,as discussed shortly. An analytical investigation may beable to prove that the perturbed packings are actuallysecond-order jammed, and therefore prove that there ex-ist (large) jammed ellipse packings with Z = 4, the ab-solute minimum contact number possible for a jammedpacking.

Finally, we note that in three dimensions the torquebalance equations (33) involve quaternions and are quar-tic, and it does not seem an analytical solution is possibleas it is in two dimensions. We however expect that thecalculations performed here in d = 2 can be generalizedto higher dimensions as well. One interesting question toanswer theoretically in d = 3 is whether the middle axes(β) affects the slope of the density sφ or whether only theratio of the largest to the smallest semiaxes (α), matters.In Ref. [4] we proposed that the rapid increase in packingfraction could be attributed to the need to increase the

30

contact numbers, since forming more contacts requires adenser packing of the particles. This is supported by theobservation that the maximal packing density is achievedfor the most aspherical shape (β = 1/2). However, nu-merical results very close to the sphere point are consis-tent with a slope sφ independent of β. The arguments ofthis section indicate that the density rise is independentof the rise of the coordination number, at least near thesphere point.

D. Contact Number Near the Sphere Point

In our perturbation approach to ellipsoid packings nearthe sphere point, we assumed that the contact networkremains that of the disk packing even as the aspect ratiomoves away from unity. However, as the aspect ratio in-creases and the packing structure is perturbed more andmore, some new contacts between nearby particles willinevitably close, and some of the old contacts may break.In Fig. 16 we show a system that the linear perturba-tion prediction produces at α = 1.025. While the originalcontacts in the jammed disk packing are maintained rel-atively well, we see that many new overlaps form thatwere not contacts in the disk packing. This means thatthe contact number will increase from Z = 4 as aspheric-ity is introduced.

Figure 16: Overlaps introduced at α = 1.025 by the naivelinear perturbation theory which only takes into account theoriginal contact network of the disk packing (black lines). Wesee many overlaps forming between particles that were nearlytouching when α = 1.

These observations suggest a way to calculate the lead-ing order term of Z(α)−2d: We simply count the overlaps

introduced by orienting and displacing the centroids ofthe ellipsoids according to the linear perturbation theory.It is well-known that jammed disordered sphere packingshave an unusual multitude of nearly-touching particles,as manifested by a nearly inverse-square-root divergencein the pair correlation function near contact [2]. Thesenear contacts will close to form true contacts and causethe rapid increase in Z(α), and we expect that the growthwill be of the form

Z(α)− 2d ≈ Zα

√α− 1. (36)

A more rigorous analysis is difficult since we do not re-ally have an understanding of the geometry of the nearcontacts. We have numerically estimated the coefficientZα and plotted the prediction of Eq. (36) in Fig. 2. Itis seen that the prediction matches the actual simulationresults well sufficiently close to the sphere point.

X. CONCLUSIONS

In this paper we presented in detail the mathemati-cal theory of jamming for packings of nonspherical par-ticles and tried to understand the properties of jammedpackings of nonspherical particles of aspect ratio close tounity, focusing on hard ellipses and ellipsoids. In thissection we summarize our findings and also point to di-rections for future investigation.

Mathematically, understanding jamming in hard-particle packings is equivalent to understanding the be-havior of large systems of nonlinear inequalities as givenby the impenetrability conditions. These inequalities canbe written explicitly by introducing a continuously differ-entiable overlap potential whose sign determines whethertwo particles overlap. In Section III we generalized theoverlap potential proposed by Perram and Wertheim forhard ellipsoids to arbitrary smooth strictly convex parti-cle shapes and determined its first order derivatives.

In Section IV, we discussed the conjecture that largedisordered jammed packings of hard particles are iso-static, i.e., that they have an equal number of constraintsand degrees of freedom, Z = 2df . It is not possible tomake this conjecture into a theorem since the term“disor-dered” is highly nontrivial to define [16]. However, argu-ments have been made in the literature in support of theisostatic property. We showed that this conjecture canbe supported with reasonable arguments only for spheres,where particle rotations are not considered. In particular,while it is expected that Z ≤ 2df for “random” packings,the converse inequality Z ≥ 2df only applies to spheres.Packings of nonspherical particles can be jammed andhave less than 2df contacts per particle, i.e., be hypo-static. A minimally rigid ellipsoid packing, i.e., a pack-ing that has the minimal number of contacts needed forjamming, satisfied only the inequality Z ≥ 2d, since atleast 2d contacts per particle are needed to block particletranslations. Particle rotations, however, and combined

31

rotation/translation motions, can be blocked by the cur-vature of the particle surfaces at the point of contact. Inessence, if the radii of curvatures at the point of contactare sufficiently large, i.e., the particle contact is suffi-ciently “flat”, rotation of the particles is blocked. Thiscan be visualized by considering the limit of infinite radiiof curvatures, when have a contact between two flat sur-faces. Such contacts, in a certain sense, count as several“contact points” and block several degrees of freedom.

In Section V, we generalized the mathematics of firstand second-order rigidity for tensegrity frameworks de-veloped in Ref. [12] to packings of nonspherical parti-cles. We proved that in order for a packing to be jammedthere must exist a set of (nonzero) non-negative interpar-ticle forces that are in equilibrium, i.e., the packing musthave a self-stress. Furthermore, we considered second-order terms for hypostatic packings that do have a self-stress but also have floppy modes, that is, particle mo-tions that preserve interparticle distances to first order.The second-order analysis showed that jammed packingsof strictly convex particles cannot have less than 2d con-tacts per particle. We found that floppy modes involvingparticle rotations can be blocked (rigidified) by the stress-matrix, which includes second-order information aboutthe particle surfaces at the point of contact. We proposedthat this is exactly the type of jamming found in dis-ordered ellipsoid packings near the sphere point, and inSection VI we presented a numerical algorithm for testinghypostatic ellipsoid packings for jamming and applied itto some computer-generated samples. We demonstratedthat the packings are indeed jammed even very close tothe sphere point, where they have close to 2d contactsper particle.

In Section VII we considered the thermodynamics ofpackings that are close to, but not exactly at, the jam-ming point, so that particles have some room to rattle(free volume). We found that for hypostatic packingsthe jamming basin J∆Q, which is localized around thejamming point in configuration space, is very elongatedalong the space of floppy modes. For iso- or hyper-staticpackings, as jammed sphere packings always are, the jam-ming basin approaches a polytope in the jamming limit,whereas for hypostatic packings it approaches a (hyper)banana. The latter leads to very large oscillations of theinstantaneous pressure near the jamming point and a vi-olation of the asymptotic free-volume equation of state(pressure scaling).

Real packings are always made from deformable (al-beit very stiff) particles, i.e., particles that interact viasome elastic interaction potential. The analog of ajammed hard-particle packing for deformable particlesare strict energy minima (inherent structures), i.e., struc-tures where any motion of the particles costs energy(quadratic in the displacements). In Section VIII we an-alyzed the first- and second-order conditions for a strictenergy minimum for twice-differentiable interaction po-tentials. We found that the first-order condition is ex-actly the requirement for the existence of a self-stress,

and that the second-order condition is exactly the condi-tion that the stress-matrix blocks the floppy modes. Thisdeep analogy between jamming in hard-particle packingsand energy minima in soft-particle packings is not unex-pected since a “soft” potential can approximate the sin-gular hard-particle potential arbitrarily closely. As thepotential becomes stiffer, the energy minimum will be-come highly anharmonic and its shape will closely resem-ble that of the jamming basin J∆Q (even at very smalltemperatures).

Finally, in Section IX we developed a first-order pertur-bation theory for packings of nearly spherical ellipsoids,expanding around the sphere point. The theory is basedon the idea that packings of ellipsoids with aspect ratioα = 1 + ε near unity have the same contact network asa nearby jammed isostatic packing of hard spheres. Inorder for the ellipsoid packing to also be jammed, theorientations of the ellipsoids must be chosen so as to bal-ance the torques on each particle. These orientationsalso maximize the jamming density, increasing it beyondthat of the disk packing, and we analytically calculatedthe linear slope of the density increase with ε for binaryellipse packings. The calculated coefficient is in goodagreement with numerical results. The perturbation ofthe sphere packing also leads to a rapid increase in theaverage particle coordination Z, which we attributed tothe closing of the multitude of near contacts present indisordered disk packings. The predicted Z ∼

√ε is also

in good agreement with numerical observations.The observed peculiar behavior of packings of non-

spherical particles near the sphere point is a consequenceof the breaking of rotational symmetry. Near the spherepoint the coupling between particle positions and orien-tations is weak and translations dominate the behaviorof the system. In this sense sphere packings are a goodmodel system, and particle shapes close to spherical canbe treated as a continuous perturbation of sphere pack-ings. However, even for aspect ratios relatively closeto unity, the perturbation changes the properties of thesystem such as density and contact number in a sharpfashion, making sphere packings a quantitatively unreli-able reference point for packings of more realistic parti-cle shapes. Furthermore, even qualitative understandingof jamming and mechanical rigidity for packings of non-spherical particles requires consideration of phenomenathat simply do not have a sphere equivalent.

Future work should consider the mathematics of jam-ming for packings of hard particles that are convex, butnot necessarily smooth or strictly convex. In particular,particles with sharp corners and/or flat edges are of in-terest, such as, for example, cylinders. We also believethat jamming in frictional packings, even for the case ofspheres, is not understood well-enough. It is also impor-tant to consider packings of soft ellipsoids and in particu-lar develop algorithms to generate them computationallyand to study their mechanical properties. Investigationsof the thermodynamics of nearly jammed ellipsoid pack-ings also demand further attention.

32

Figure 17: The rectangular lattice of ellipses (i.e., affinelystretched square lattice of hard disks) with“hard-wall”bound-ary conditions created by freezing the ellipses on the bound-ary. This packing is jammed since the curvature of the flatcontacts blocks the rotations (including collective ones) of theellipses.

Acknowledgments

A.D. and S.T. were supported in part by the NationalScience Foundation under Grant No. DMS-0312067.R.C. was supported in part by the National Science Foun-dation under Grant No. DMS-0510625. We thank PaulChaikin for many inspiring discussions of ellipsoid pack-ings.

Appendix A: THE RECTANGULAR LATTICE OFELLIPSES

In this Appendix we consider a simple example of ajammed hypostatic packing of ellipses having Z = 4, theminimum necessary for jamming even for disks. Namely,the rectangular lattice of ellipses, i.e., the stretched ver-sion of the square lattice of disks, is collectively jammed,and in particular, it is second-order jammed. More specif-ically, freezing all but a finite subset of the particles, theremaining packing is second-order jammed. An illustra-tion is provided in Fig. 17. At first glance, it appears thatone can rotate any of the ellipses arbitrarily without in-troducing overlap. However, this is only true up to firstorder, and at the second-order level the “flat” contactsbetween the ellipses, that is, the contacts whose normalsare along the small ellipse semiaxes, block this rotationthrough the curvature of the particles at the point ofcontact.

The set of first-order flexes, i.e., particle motions which

preserve contact distances to first order, can easily beconstructed in this example due to the simple geometry.Namely, a basis vector for this set is a single ellipse rotat-ing around its centroid, giving the total number of first-order flexes Nf = N [17]. The basis formed by these first-order flexes is not orthogonal. However, its advantage isthat it is easier to calculate the stress matrix, or morespecifically, the matrix HV ; we only need to consider el-lipsoid rotations without considering translations. Thesame observation applies whenever one takes a jammedsphere packing and makes the particles nonspherical butdoes not change the normal vectors at the point of con-tact. This can be done, for example, by simply takinga jammed sphere packing and swelling the particles tobe nonspherical, without changing the geometry or con-nectivity of the contact network. If the particles swellenough to make all of the contacts sufficiently flat, thenew packing will be jammed, since all of the first-orderflexes consist of particle rotations only and are blockedby the flat curvature of the contacts.

The fact that“flat” (the contacts among vertical neigh-bors in Fig. 17) contacts block rotations can easily beseen analytically by considering the case of one ellipsejammed among four fixed ellipses (two horizontally, twovertically). Specifically, any self-stress for which the con-tact force in the “flat” contacts is larger than the force inthe“curved”contacts, fflat > fcurv, makes the mobile el-lipse jammed, more specifically, pre-stress rigid [17]. Thesame result can be shown to apply to the square latticeof ellipses for an arbitrary number of ellipses. If the el-lipses are not hard but rather deformable, the packingwould not support a compression along the curved con-tacts, but it would along the flat contacts. This is avery intuitive result: If one takes a smooth ellipsoid andpresses it against a table with its most curved tip, it willbuckle and the only stable configuration is one where theflat tip presses against the table. Note however that thehard-ellipse equivalent is jammed and can resist any finiteexternal forces, including a compression along the curvedcontacts. The anharmonicity of the hard-sphere potentialbecomes essential in this example, since the packing canchoose the correct internal (self) stresses (forces) neededto provide mechanical rigidity. In (realistic) systems ofdeformable particles, the internal stresses are fixed anddetermined by the state of compression.

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