Dynamical arrest in low density dipolar colloidal gels · Dynamical arrest in low density dipolar colloidal gels Mark A. Miller,1,a Ronald Blaak,2 Craig N. Lumb,1 and Jean-Pierre

Dynamical arrest in low density dipolar colloidal gelsMark A. Miller,1,a� Ronald Blaak,2 Craig N. Lumb,1 and Jean-Pierre Hansen1,3

1University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom2Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf,Universitätsstrasse 1, D-40225 Düsseldorf, Germany3Laboratoire des Liquides Ioniques et Interfaces Chargées, Université Pierre et Marie Curie,75252 Paris Cedex 05, France

�Received 5 November 2008; accepted 6 February 2009; published online 18 March 2009�

We report the results of extensive molecular dynamics simulations of a simple, but experimentallyachievable model of dipolar colloids. It is shown that a modest elongation of the particles anddipoles to make dipolar dumbbells favors branching of the dipolar strings that are routinely observedfor point dipolar spheres �e.g., ferrofluids�. This branching triggers the formation of a percolatingtransient network when the effective temperature is lowered along low packing fraction isochores��0.1�. Well below the percolation temperature the evolution of various dynamical correlationfunctions becomes arrested over a rapidly increasing period of time, indicating that a gel has formed.The onset of arrest is closely linked to ongoing structural and topological changes, which wemonitor using a variety of diagnostics, including the Euler characteristic. The present system,dominated by long-range interactions between particles, shows similarities to, but also somesignificant differences from the behavior of previously studied model systems involving short-rangeattractive interactions between colloids. In particular, we discuss the relation of gel formation tofluid–fluid phase separation and spinodal decomposition in the light of current knowledge of dipolarfluid phase diagrams. © 2009 American Institute of Physics. �DOI: 10.1063/1.3089620�

I. INTRODUCTION

Recent evidence from experiment, theory, and simula-tion has revealed not only striking similarities but also sig-nificant differences between two classes of disordered mate-rials, namely, glasses and gels. Both resist external stress, butare characterized by very different packing �or volume� frac-tions � of the constituent particles.

Glasses are commonly observed both in molecular andcolloidal systems at high packing fractions ��0.5�. Thebasic mechanism for glass formation is caging or jamming ofparticles by their neighbors, and dynamical arrest is partlyunderstood in terms of mode-coupling theory,1 which alsoapplies to the so-called attractive glasses where physicalbonding between neighboring particles plays a role.2 Gels,on the other hand, are typical soft matter systems �polymericor colloidal systems with much lower packing fractions, typi-cally ��0.1� and gelation is invariably associated with theformation of open, space-spanning networks.

Depending on the relative time scales of the bondingbetween particles in the network and of the experimentalprobe, one usually distinguishes between irreversible chemi-cal gels and weak physical gels. The former involve perma-nent bonds between network particles and are well describedby percolation kinetics.3 Physical gels involve transientbonds between particles, with bonding energies of the orderof a few times the thermal energy kBT, allowing the bonds tobe formed or broken on microscopic time scales. Such physi-cal gels are generally observed in colloidal systems and

colloid-polymer mixtures, where the range and strength ofthe depletion attraction can be tuned by varying the polymersize and concentration,4 and have been the focus of muchrecent experimental and theoretical investigation.5 Upon in-creasing the bonding lifetime within a transient network byvarying the strength of the effective attraction between col-loids, the network can undergo dynamical arrest, similar tothat observed in structural glasses at much higher packingfractions, and resulting in a stress-resistant gel.6,7 Physicalgelation can be driven by a number of mechanisms, includ-ing arrested spinodal decomposition,8–10 the jamming of pre-formed fractal clusters,11,12 or the action of strongly direc-tional bonds, as may be achieved by “patchy” particles �i.e.,particles with highly localized bonding sites on theirsurface�.13,14

So far, colloidal gelation has generally been associatedwith very short-ranged depletion-induced attractions betweencolloids �“sticky spheres”�, which lead to cluster formationor phase separation,8–10,15 with highly directional forces16 orwith a “valence saturation” constraint,13,14 which favor thespontaneous formation of open, space-spanning networks.However, network formation at low packing fractions mayalso be induced by long-range dipolar interactions betweencolloidal particles �as in ferrofluids� which favor head-to-tailconcatenation.17

Dipolar hard spheres form stringlike clusters at low tem-peratures, and branching18 or crosslinking of such transientstrings may result in a percolating network. This tendencymay be enhanced by a slight elongation of the dipolarparticles.18,19 Moreover, and contrary to earlierspeculations,20 there is now compelling evidence of phasea�Electronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 130, 114507 �2009�

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separation in fluids of dipolar spheres, spherocylinders, anddumbbells at low temperatures and packing fractions.21–24

The tendency for string and network formation, as well asthe existence of an equilibrium fluid-fluid phase transition,makes it likely that gelation of dipolar particles will occur atsufficiently low temperatures.

The present paper provides strong evidence for the exis-tence of a low density dipolar gel arising from reversiblephysical bonds, based on extensive molecular dynamics�MD� simulations of a system of slightly elongated dipolarparticles. It will be shown that the anisotropic dipolar inter-actions induce the formation of a percolating, transient net-work at a density-dependent threshold temperature, and thatthis network undergoes gelation, characterized by dynamicalarrest, at a much lower temperature.

The model and methodology will be introduced in Sec.II. The percolation threshold and its finite-size scaling prop-erties will be determined in Sec. III. The structural and topo-logical characteristics of the network are presented in Sec.IV. Section V will describe dynamical behavior and low tem-perature gelation, diagnosed using dynamical correlationfunctions, while concluding remarks will be presented inSec. VI. Part of the present work was briefly reported in apreliminary publication.19

II. DIPOLAR DUMBBELLS

The model investigated in this paper is a system of �=N /V dipolar dumbbells per unit volume. Each dumbbell ismade up of two interpenetrating soft spheres carrying oppo-site charges �q at their centers, separated by a fixed distanced.19 The resulting dipole moment is �=qd and a point dipolemoment would be recovered in the limit d→0, q→� atfixed �. The interaction potential between sites on differentdumbbells is the sum of a soft-sphere repulsion and aCoulomb interaction,

v++�r� = v−−�r� =c

r12 +q2

40�r, �1a�

v+−�r� =c

r12 −q2

40�r, �1b�

where r is the site-site distance, while 0 and � are thepermittivity of free space and the relative permittivity of thesurrounding medium �i.e., the solvent�.

A convenient length scale is the distance � between thecenters of two dumbbells in the head-to-tail configuration oflowest total potential energy,

vmin = �2v++�� + v+−�� + d� + v+−�� − d�� . �2�

The natural energy scale is set by the combination of param-eters u=�2 /40��3, while the natural time unit is t0

= �m�2 /u�1/2, where m is the mass of a dumbbell. For givenvalues of � and u, the soft-core repulsion coefficient c isfixed at c=0.0208u�12. Convenient reduced temperature,density, and time are defined as

T� = kBT/u ,

�� = ��3, � = ��/6,

t� = t/t0.

For typical colloidal particles in water ��=78�, ��102 nm, m�10−18 kg, and q�102 proton charges.Throughout this work the ratio �=d /� was taken equal to0.217, so that the temperature scale turns out to be u /kB

�103 K and the reduced room temperature would beT��0.3. The corresponding characteristic time scale ist0�10−7 s.

Related models which have been considered in the lit-erature include dipolar dumbbells of hard spheres,22,24 hardspheres with extended dipoles,25 and dipolar hardspherocylinders.18,21 In the dipolar hard sphere dumbbellmodel, the soft-sphere repulsion in Eq. �1� is simply replacedby a hard sphere repulsion of diameter s. The vapor-liquidphase diagram of the latter model has recently been deter-mined by Monte Carlo simulations for several values of theratio d /s.24 An appropriate mapping of the soft sphere ontoan effective hard sphere repulsion would provide a guide ofthe position of the thermodynamic state points explored inthe MD simulations relative to the coexistence boundary.

The finite extension �d 0� of the dipole enhances stringformation,25 while the dumbbell geometry facilitates branch-ing of chains, which can thus interconnect into a genuinethree-dimensional space-spanning network when the tem-perature is lowered at fixed �. The branching tendency fol-lows from a competition between Coulombic and soft-corerepulsion interactions between dumbbells. For a givencenter-to-center distance r, the Coulomb interaction favorsthe head-to-tail configuration, where parallel dipoles arealigned along the center-to-center vector r, while the soft-sphere repulsion between dumbbells is lower when the par-allel or antiparallel dipoles �and hence the dumbbell orienta-tions� are perpendicular to r �side-by-side configuration�.Examples of the total interaction energy of a pair of dipolardumbbells as a function of the center-to-center distance r areplotted in Fig. 1 for several relative orientations of r and thedipole moment vectors on the two particles.

0.5 1 1.5 2 2.5 3center-to-center separation, r/σ

-2

0

2

4

6

8

10

redu

ced

pair

ener

gy

head-to-tailhead-to-headperpendicular and coplanar (T-shape)side-by-side, dipoles parallelside-by-side, dipoles antiparallelorthogonal to each other and to r

FIG. 1. �Color online� Potential energy in units of u of two dipolar dumb-bells with center-to-center separation r in various orientations with respectto each other and to the center-to-center vector r.

114507-2 Miller et al. J. Chem. Phys. 130, 114507 �2009�


Dipole elongation also favors bundling of the parallelstrings of head-to-tail dumbbells. The total interaction energyper particle U of two strings of n=20 dumbbells shifted by� /2 along the parallel axes is plotted in Fig. 2 as a functionof the distance x between the two axes for several values ofthe elongation parameter �=d /�. The energy is seen to gothrough a minimum at x /��0.9 for d /��0.2 and the mini-mum deepens with increasing �, while chains of contracteddipoles �small �� always repel in these configurations. Thesame tendency is observed for all chain lengths n�5.

Constant temperature MD simulations were carried outon periodic samples of N=1000 dipolar dumbbells along twoisochores ��=0.0745 and �=0.0219� and along one iso-therm �T�=0.176�. The temperature was controlled by theBerendsen thermostat26 and the classical equations of motionwere integrated with the standard Verlet leap-frogalgorithm,27 as implemented in the GROMACS package.28

Long-range Coulomb interactions were dealt with by theparticle-mesh Ewald method.27 The time step was chosen tobe �t�=0.0008 and most simulation runs extended overroughly 106 time steps after initial equilibration, althoughruns an order of magnitude longer were used for some of thelow temperature states, where the dynamics slows down dra-matically. The Newtonian dynamics used throughout thiswork neglects the stochastic �Brownian� and hydrodynamicforces induced by the solvent. No single simulation tech-nique fully captures both these effects, and Newtonian dy-namics remains appealing because of its efficiency and lackof ambiguity and because it preserves the exact static equi-librium properties of the simulated system, which are notaffected by velocity-dependent forces. Moreover, it has beenshown, at least in the case of denser systems, that long-timedynamical evolution is not affected by the precise short-timedynamics.29,30

III. PERCOLATION

The structural, topological, and dynamical evolution ofthe dipolar dumbbell model was characterized by a numberof static and dynamical diagnostics and monitored by pro-

gressively lowering the reduced temperature T� from T�

�0.8 to T��0.02 along the two isochores �=0.0745 and�=0.0219 and by progressively increasing the packing frac-tion � from ��0.018 to ��0.15 along the isotherm T�

=0.176. The system was found to be very responsive tochanges in temperature, but much less so to changes in den-sity. A reproducible three-step scenario was observed whencooling the system along either of the two isochores. Theformation of transient strings of particles is observed as soonas T� is lowered below T��0.5, a typical behavior of allpolar fluids. Branching and interconnection of the chains setin upon further lowering of the temperature, as illustrated inFig. 3. The connectivity of two given particles was deter-mined by the simple geometric criterion that a pair of oppo-site charges—one on each dumbbell—lie closer than rcut=�.For systems with transient bonds the percolation threshold is

0 0.5 1 1.5 2 2.5 3separation of chains, x/σ

−0.1

0

0.1

0.2

ener

gype

rpa

rtic

le,U

/vm

in 0.0640.1560.2170.305

dumbbell extension δ = d/σ

x

σ/2 σ

FIG. 2. �Color online� Interaction energy per particle U of two parallelidealized chains of 20 head-to-tail dipoles, shifted with respect to each otherby � /2 along their axes, as a function of the separation x of the chain axes.The four curves correspond to different dumbbell extensions, �=d /�. Eachcurve has been scaled in length by the equilibrium pair separation � and inenergy by the minimum pair energy vmin �see Eq. �2�� for the relevant valueof �. The thick line corresponds to the � adopted in the present work. (a)

FIG. 3. �Color� Snapshot configurations at �=0.0219 and temperatures of�a� T�=0.164, above the percolation threshold, and �b� T�=0.123 �just belowthe percolation threshold�. Particles belonging to a given cluster in the pe-riodic system are shown in the same color.

114507-3 Dipolar colloidal gels J. Chem. Phys. 130, 114507 �2009�


defined in the thermodynamic limit as the locus of statepoints �� ,T�� where the average cluster size diverges. In asimulation this divergence is detected by the appearance of acluster that spans the sample, i.e., a cluster in which eachparticle is connected to its own periodic images by othermembers of the same cluster. Because of finite-size effects,the percolation transition is rounded, and not discontinuousas one would expect for an infinite system.

In view of the importance of the percolation threshold inthe network formation, we have carried out a finite-size scal-ing analysis of the percolation probability �i.e., the fractionof percolating configurations observed in the course of asimulation� along the isotherm T�=0.176. This is illustratedin Fig. 4. The inset shows the variation in the percolationprobability with volume fraction � for three different linearsystem sizes L. For the scaling analysis, the volume fractionwas controlled by changing the number of particles N atfixed volume V=L3, rather than vice versa as in the rest ofthe paper. As expected, the sigmoidal curve sharpens as Lincreases, and the three curves intersect close to a commonpoint. The infinite system percolation threshold �inf is esti-mated from the crossing point of the percolation curves forthe two larger systems �L=19.16� and L=28.74��, yielding�inf=0.075. The three curves collapse onto a mastercurve when plotted as functions of the scaling variable31

x= ��−�inf�L1/�, where �=0.88 is the universal exponent forconnectivity percolation in three dimensions.31 Figure 4shows that the collapse is excellent for the two larger sys-tems, but that increasing deviations are seen with decreasingdensity for the smallest system �L=12.77��, indicating thatthe scaling regime has not yet been reached. This may beunderstood by noting that for the smallest system at the low-est densities, the periodic boundary conditions favor configu-rations where all the particles join a single highly anisotropiccluster that spans the box in a particular direction �rather thanspreading out in all directions�, leading to an artificially highpercolation probability and even nonmonotonic behavior.

The nearly common crossing point of the percolationcurves in the inset of Fig. 4 lies just below a percolationprobability of 50%. Hence, in our finite-sized simulations, it

is an excellent approximation to adopt a practical definitionof the percolation threshold as the locus of �T� ,�� statepoints where the fraction of percolating configurations is50%. The remainder of the results in this paper are calculatedalong the isochores �=0.0745, where the percolation tem-perature is Tp

� =0.177 and �=0.0219, where it is Tp� =0.135.

Just below the percolation temperature, the space-spanning network is transient due to the finite lifetime of thephysical bonds between neighboring particles and to the factthat the breaking of a single bond may be enough to discon-nect the cluster. However, the bond lifetime increases rapidlyas the temperature drops and at the same time, clusters be-come more ramified. During this process, the single-particleand collective dynamics of the network slows down and at acritical temperature gelation occurs, characterized by dy-namical arrest on the time scale of the simulation. Quantita-tive evidence for the successive stages of this scenario ispresented in Secs. IV and V.

IV. NETWORK STRUCTURE AND TOPOLOGY

The dominant stringlike structure of the dipolar dumb-bells over a wide range of temperatures is immediately evi-dent from an inspection of the radial pair distribution g�r�, asshown in Fig. 5 for �=0.0219. As the temperature is low-ered, a clear pattern of sharp peaks separated by a distance�r��, typical of a one-dimensional chain structure,emerges. The amplitude of the quasiequidistant peaks growsrapidly with decreasing T�. At the lowest temperature�T�=0.020� g�r� develops a slowly decaying tail at large r,indicative of the proximity of a spinodal line which may beassociated with a fluid-fluid phase separation.24 We willreturn to this important issue in the concluding discussion�Sec. VI�. The variation in g�r� with temperature is qualita-tively similar along the higher density isochore �=0.0745�cf., Fig. 2 of our preliminary report19�, except that no long-range correlations appear even at the lowest temperaturestudied at the higher density.

The mean coordination number, based on the charge-charge neighbor criterion described above, is of the order of2 or somewhat larger at the lower temperatures, as expectedfor a stringlike structure with some crosslinking. A finer

−2 −1 0 1(φ − φ

inf)L1/ν

0

0.2

0.4

0.6

0.8

1

perc

o lat

ion

prob

abili

ty

L = 12.77σL = 19.16σL = 28.74σ

0 0.05 0.1 0.15volume fraction, φ

0

0.2

0.4

0.6

0.8

1

FIG. 4. �Color online� Percolation probability �fraction of percolating con-figurations� along the isochore T�=0.176. The inset shows the probability asa simple function of � for three linear system sizes L. The main plot showsthe same data as a function of the shifted and scaled density �see text�.

0 2 4 6 8 10 12 14 16r / σ

0

2

4

6

8

10

radi

aldi

stri

butio

nfu

nctio

n,g(

r)

0 1 2 3 4 5 6 7coordination number

0

0.2

0.4

0.6

0.8

1

prob

abili

ty

0.0200.0410.1640.2040.409

reduced temperature, T*

FIG. 5. �Color online� Radial distribution function and coordination numberdistribution �inset� at packing fraction �=0.0219. The lines in the inset area guide to the eye.



breakdown of coordination numbers is provided by the prob-ability distribution of the number of neighbors per particle,as obtained by averaging over a large number of configura-tions generated during the MD runs. This distribution isshown in the inset of Fig. 5. Above the percolation threshold�Tp

� =0.135� most particles have a single, or no neighbor,while below Tp

� almost all particles are doubly coordinated,and a significant fraction have three neighbors, pointing tothe existence of branched chains, which are crucial for net-work formation. At the lowest temperature illustrated in Fig.5, there is a splitting of the peaks in g�r�. The doubling of theintrachain peaks described above arises from interchain cor-relations when segments of two chains bundle together andrun in parallel over some distance. Each particle gains twoneighbors in the adjacent chain in addition to the two in itsown, and the resulting increase in fourfold connectivity atthe lowest temperature is clear in the inset of Fig. 5.

In Fourier space of wavenumbers k, the pair structure ischaracterized by the static structure factor S�k�, which can becalculated directly in MD simulations by the correlationfunction of the Fourier components �k and �−k of the micro-scopic density32 or by Fourier transforming the pair distribu-tion function g�r�. The finite linear size L of the simulationbox puts a lower limit on the accessible wavenumber, k�2 /L. Examples of S�k� for several temperatures areshown in Fig. 6. Except at the highest temperatures, S�k�exhibits a pronounced low k peak, indicative of clustering,similar to that observed for other models of gelation.16 Ex-trapolation of the MD data to k=0 is difficult due to thesusceptibility of S�k� to statistical and finite-size errors at lowk, but provides an estimate of the isothermal compressibility,

�T = �T�0� lim

k→0S�k� ,

where �T�0� is the compressibility of the corresponding ideal

gas of noninteracting particles. The amplitude S�k=0� 1implies that the gel is significantly more compressible thanthe ideal gas at the same temperature and density. Along theisochore �=0.0745, S�k=0� appears to grow slowly as T�

drops, but always remains finite. On the contrary, at thelower density �=0.0219, S�k=0� appears to diverge at thelowest temperature investigated, as expected from the slowly

decaying tail in g�r� in Fig. 5. This divergence is the well-known signature of spinodal instability. The pressure isreadily calculated from the virial theorem and drops rapidlywith T� due to clustering. The fact that divergence of S�k=0� is observed only in the very lowest temperature andlowest density simulations undertaken19 indicates that, ex-cept under those extreme conditions, phase separation is notinterfering with the structure and dynamics of the system.Hence, the majority of state points studied can be ap-proached along a reversible thermodynamic path. Neverthe-less, the low temperatures do require extra care to be takenfor equilibration. We have extended the duration of bothequilibration and accumulation runs by more than a factor of10 at the lowest temperatures. In some cases we have alsotaken an ensemble average over completely independent ini-tial conditions. Even so, if S�k=0� diverges we cannot com-pletely rule out a slow evolution of properties as the systemages over time scales much longer than the simulations.

The formation of chains and a network has a dramaticinfluence on equilibrium fluctuations, as illustrated in Fig. 7for the static dielectric permittivity. This quantity is related tothe fluctuations of the total dipole moment M of the periodicsystem by the following Kirkwood relation appropriate for“metallic” boundary conditions at infinity,25,33

= 1 +4

3

��2

kBT� �M�2

N�2� . �3�

Reliable estimates of require averages over long MDtrajectories.25 As seen from Fig. 7, first increases as thetemperature is lowered due to the 1 /T factor in Eq. �3�. Itthen drops rapidly beyond the percolation threshold to reachvalues close to 1 at the lowest temperatures due to thestrongly reduced dipolar fluctuations within the increasinglyrigid network. It is most difficult to obtain just below thepercolation threshold, where the network inhibits but doesnot completely suppress major reorganizations of the struc-ture on the time scale of the simulations. The rapidly increas-ing sluggishness of fluctuations in M at lower temperaturesmeans that the apparent permittivity measured over a shorttime interval can change on longer time scales, as illustratedby the line for T�=0.123 in Fig. 7.

0 5 10 15 20kσ

0

5

10

15

20

stat

icst

ruct

ure

fact

or,S

(k)

0.0200.0410.0820.1640.2040.409

0 5 10 15 20kσ

1

10

100

S(k)


FIG. 6. �Color online� Static structure factor S�k� at packing fraction �=0.0219. The inset shows the same data on a semi-log plot.

0 200 400 600 800reduced time, t*

0

2

4

6

8

10

diel

ectr

icpe

rmitt

ivity

,ε

0.0200.0820.1230.2040.2860.4090.818

T*

FIG. 7. �Color online� Convergence of the dielectric permittivity with theprogress of simulations at packing fraction �=0.0745 and the reduced tem-peratures T� indicated.



The morphology of the three-dimensional network, andits evolution with temperature, may be characterized by anumber of topological measures and invariants. In a first in-stance we have analyzed the morphology of the network at acontinuously variable level of coarse graining by associatinga notional sphere of diameter DE with the center of each ofthe N dipolar dumbbells. The morphology of the resultingdisconnected or connected surface � may be quantified usingconcepts of algebraic topology.34 A key invariant of a surfaceis the Euler characteristic �EC� ��, whose evolution withincreasing diameter DE of the spheres provides a quantitativecharacterization of the network and how it changes withlength scale. A similar analysis was recently used in twodimensions to characterize the clustering of binary colloidalsystems confined to a planar surface.35

The EC � of a collection of objects is defined as the sumof the number of vertices and the number of faces minus thenumber of edges. The EC of any closed surface �such as aconvex polyhedron, which is homomorphic with a sphere� is�=2, that of a surface with n holes �e.g., a torus with n=1� is2−2n, while that of a surface bounding a volume enclosing ncavities is 2+2n. The total EC of a collection of discon-nected surfaces is simply the sum of the EC of the individualobjects. When analyzing our system of N dumbbells, it isconvenient to deal with the normalized total EC �=� /2N,which tends to unity as DE→0 �since all spheres decoratingthe network are then disconnected� and tends to 0 as DE

→� �since all spheres have then merged, leaving no surfacein the periodic system�. An efficient procedure for calculat-ing � in the present context is described in the Appendix.

As DE is increased from 0, � first starts to drop belowunity around DE=�, where spheres associated with neigh-boring dumbbells overlap, leading to a reduction in the num-ber of disconnected surfaces and hence to a decrease in theEC. This drop accelerates when long strings of dumbbellsform. Near chain junctions, where more highly coordinatedparticles are found, small toroidal holes arise between thespheres associated with three mutually neighboring dumb-bells, causing � to become negative �since the EC of a torusis negative�. A small increase in DE leads to closure of thesetori, and � passes through a minimum. This process is illus-trated in Fig. 1 of the EPAPS supplementary material.36 Theclosing of tori on a larger scale, characteristic of the networkmesh �also illustrated in Fig. 1 of the EPAPS supplementarymaterial36�, produces a more gradual rise in � at larger DE.The continual increase in DE from zero also leads to theformation and subsequent closure of enclosed cavities. Thelatter topological event leads to a decrease in �.

This generic scenario is indeed observed when analyzingthe MD-generated dumbbell configurations. However, thedetailed shape of the � versus DE curves depends sensitivelyon temperature, as shown in Fig. 8 for �=0.0745 and �=0.0219. The behavior at the two densities shares somequalitative features: at high temperatures � exhibits a broadminimum around DE=��−1/3, as one would expect for a sys-tem of noninteracting particles. As the temperature is de-creased the formation of small, transient, chainlike clusterscan be detected in the EC by a faster initial drop of � and ashallower minimum. The initial drop becomes decisively

steeper around the percolation temperature. At this point, thelocation of the minimum is sensitive to the balance betweenfurther merging, involving slightly more distant particles,which decreases the EC, and the closure of small tori, whichincreases the EC. As a result, the EC profiles at reducedtemperatures in the range of 0.082�T��0.16 have ratherdifferent shapes at the two densities depicted in Fig. 8. Aquantitative difference between the two densities remains atthe lowest temperatures studied, with the higher density ex-hibiting a deeper sharp minimum near DE=� due to thehigher average coordination number at the higher density.The qualitatively changing shape of the EC profile as thetemperature and density are varied makes it a sensitive indi-cator of the gradual morphological transformation of the sys-tem. In Sec. V we will diagnose gelation of the network bythe arrest of various dynamical correlation functions. Being apurely topological quantifier, the EC profile alone thereforedoes not make it possible to conclude whether or not thenetwork is a gel. However, the EC certainly detects the factthat the structure of the network continuously evolves as thetemperature is lowered first through the percolation thresholdand then into the regime of dynamical arrest. The changes inthe profile of the EC make it possible to interpret these struc-tural changes in terms of a sequence of topological events onincreasing length scale, making the EC a valuable tool forlinking structure and dynamics, including gelation.

An associated indicator of network morphology is pro-vided by the asphericity of the Voronoi polyhedra con-structed from the bisecting planes of the vectors joining the

0 1 2 3 4 5 6 7 8sphere diameter, D

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FIG. 8. �Color online� Profile of the normalized EC �=� /2N as a functionof length scale DE at packing fractions �a� �=0.0745 and �b� �=0.0219 atthe temperatures indicated in �a�.



center of each dumbbell to its nearest neighbors. TheVoronoi analysis is performed in the process of calculatingthe EC, as described in the Appendix. In a dense fluid theVoronoi polyhedra are quasispherical, but for a network ofinterconnected strings one would expect the polyhedra to bemore elongated in directions orthogonal to the string axes.The shape of the polyhedra may be conveniently character-ized by the asphericity parameter �=RA /RV, where the char-acteristic radii RA and RV are related to the surface area Acell

and volume Vcell of the Voronoi cell by Acell=4RA2 and

Vcell=4RV3 /3, respectively. Clearly, RA=RV for a perfectly

spherical cell, while the ratio � exceeds unity for any non-spherical cell and rises for increasingly elongated polyhedra.

From a Voronoi analysis of a large number of configu-rations, we have extracted the probability density Pcell�� offinding polyhedra of asphericity � for various temperatures.Pcell�� is fairly sharply peaked at �=1.09 at the highest tem-peratures, where particles are roughly uniformly distributedinside the volume of the simulation cell. As T� drops, thepeak broadens and shifts to somewhat higher asphericitiesaround �=1.19 �the plot is given in Fig. 2 of the EPAPSsupplementary material36�. The temperature dependence ofPcell�� does not seem to be very dramatic, but it is fastestaround the percolation temperature, signaling the formationof a “filamentary” network. If the Voronoi polyhedra are lik-ened to flat cylinders �“pill boxes”� of radius R and height h,the most probable asphericity observed at the lowest tem-peratures would correspond to an aspect ratio h /R=0.5,compared to the “most spherical” cylinder characterizedby an aspect ratio h /R=2 and an asphericity parameter�=1.07.

The space-spanning network of particles defines a net-work of voids of variable size. The latter may be character-ized by the distribution of diameters DV of the largest spherethat can be inserted at a random point in the system withoutimpinging on the center of a particle. Examples of the cor-responding probability densities Pvoid�DV� as determinedfrom configurations generated in the MD simulations areshown in Fig. 9 for a range of temperatures along the isoch-ore �=0.0745. Pvoid�DV� is seen to increase quadraticallywith DV for small DV and to peak at roughly DV=2� for alltemperatures. As T� is lowered, a tail at larger DV develops

gradually, signaling the occurrence of voids in the network.While the tail appears to stabilize around and below the per-colation temperature, it suddenly extends to much larger val-ues of DV at the lowest temperature, which is associated withthe “bundling” of quasiparallel strings of the network, gen-erating larger cavities or pores. Similar bundling has recentlybeen reported in event-driven MD simulations of a model ofdipolar colloids based on a highly discretized representationof dipolar interactions.37 There is a clear-cut qualitativechange in Pvoid�DV� at the lowest temperature explored in ourMD simulations, which may be interpreted as a static indi-cation of gelation.

Gelation, like the glass transition, is, however, more gen-erally characterized by dynamical diagnostics to which weturn in Sec. V.

V. DYNAMICAL ARREST

The space-spanning network that forms at the percola-tion threshold is a transient structure because the lifetime �b

of the physical bonds linking each dipolar dumbbell to itsneighbors is finite, allowing continual rearrangement of par-ticles, strings, and crosslinks within the network. However,as the temperature is lowered, the bond lifetime increasesrapidly so that the network becomes increasingly permanentand will eventually turn into a long-lived gel, where the de-cay of correlations becomes arrested. Following the neighborcriterion based on the charge-charge distances as describedin Sec III, we define a function bij�t� which is unity whenparticles i and j are bonded at time t, and zero otherwise.One may then construct a bond correlation function

B�t� = b�t�b�0� ,

averaged both over the pairs i , j and along the MD-generatedphase space trajectory.

B�t� eventually reaches a regime of exponential decayover a wide range of temperatures, allowing a bond-breakingrate constant kb=1 /�b to be defined unambiguously. Thevariation in the reduced rate constant kb

�=kbt0 with tempera-ture is shown in the Arrhenius plot of Fig. 10. The rate con-stant is seen to decrease by four orders of magnitude betweenT��0.4 and T��0.02, but it does so in several stages. Athigh temperatures, where the fluid is not chainlike, contacts

0 2 4 6 8 10maximum sphere diameter, D

V/ σ

0

0.1

0.2

0.3

0.4

0.5

0.6

prob

abili

ty,P

void

(DV) 0.020

0.0410.1230.2040.4090.818


FIG. 9. �Color online� Distribution of void sizes at �=0.0745 measured bythe probability of the maximum diameter of a sphere that can be inserted ata random point in space without overlapping with a particle.

0 10 20 30 40 50reciprocal temperature, 1 / T*

10−6

10−5

10−4

10−3

10−2

10−1

100

101

redu

ced

cons

tant

bond breaking constant, kb*

diffusion constant, D*

FIG. 10. �Color online� Arrhenius plots of the self-diffusion constant D� andthe rate constant for bond breaking k� at packing fraction �=0.0219. Thelines joining the symbols are a guide to the eye.



between particles are highly transient and it is not yet mean-ingful to think of them as bonds. kb

� enters a regime ofArrhenius temperature dependence once the network is prop-erly established, 0.03�T��0.1. However, at the lowest tem-peratures, kb

� is found to be significantly higher than extrapo-lation of the Arrhenius dependence would predict. Therelative ease of breaking bonds here can be understood bynoting that a dipolar particle can only make two energeticallyoptimal �head-to-tail� bonds. The increasing average coordi-nation number at low temperatures necessarily means thatmany particles are forming bonds with frustrated geometries�e.g., at chain bifurcations�, which require less thermal en-ergy to break.

The bond-breaking rate constant may be expected tohave a strong effect on the self-diffusion constant D of theparticles. The latter is readily computed from the meansquare displacement �MSD� of the particles at sufficientlylong times through Einstein’s relation,32

D = limt→�

1

6t�r�t� − r�0��2 ,

where, again, averaging is over all initial times and all par-ticles. Examples of the reduced MSD plotted against reducedtime t� on a double logarithmic scale are shown in Fig. 11 forseveral temperatures, along the isochore �=0.0219. At thehigher temperatures the MSD goes over directly from theinitial ballistic regime ��t2� to the linear Einstein regime atlonger times. Below the percolation temperature a subdiffu-sive regime, characterized by a clear inflection point, sets inat intermediate times. This regime extends over a rapidlyincreasing interval as T� drops, and at the lowest temperaturethe normal diffusive regime is barely reached within the timescale of even our longest simulations �1.8�107 MD steps�.In other words, the system becomes dynamically arrestedover much of the accessible time scale, covering an intervalof several decades. This is the first dynamical diagnostic thatallows us to designate the system as a gel under these con-ditions. We note that a plateau in the MSD is really a lowerbound to the temperature at which gelation sets in, sinceother gel characteristics such as an increased elastic moduluscan arise in a network before all single-particle motion be-comes arrested.

The temperature dependence of the reduced diffusionconstant D�=Dt0 /�2, derived from the slope of the MSD

with respect to time in the linear regime, is illustrated in theArrhenius plot of Fig. 10. Arrhenius behavior is observed forT��0.1, i.e., below the percolation temperature. Since weare employing Newtonian dynamics, the diffusion constantcan also be estimated from the integral of the normalizedvelocity autocorrelation function,32

Z�t� =m

3kBTv�t� · v�0� ,

where v�t�=dr�t� /dt is the center-of-mass velocity of a par-ticle. Examples of Z�t� as a function of reduced time t� forseveral temperatures along the isochore �=0.0219 areshown in Fig. 12. As expected, Z�t� decays increasinglyquickly with decreasing temperature. Below the percolationtemperature, Z�t� exhibits marked oscillations. Careful scru-tiny indicates that a faster oscillation is superimposed on thelow-frequency oscillations. This is clearly evident from the

power spectrum �i.e., the Fourier transform� Z�f�� of the ve-locity autocorrelation function shown in the inset of Fig. 12,which exhibits two well separated peaks. An elementarycalculation19 shows that the high frequency may be associ-ated with the quasiharmonic, one-dimensional oscillation ofa particle within a chain of dipolar dumbbells. The low-

frequency component of Z�f� is most likely due to the lateralvibrations of the chains within the network.38,39 The overallbehavior of Z�t� remains very much the same at the higherpacking fraction,19 �=0.0745, with a ratio of the two char-acteristic frequencies of the order of 4.5 at both packingfractions.

A finer, spatially resolved analysis of the single-particlemotion is provided by the self-part of the van Hove correla-tion function and its spatial Fourier transform, the self-intermediate scattering function,29

Fs�k,t� =1

N�j=1

N

exp�ik · �r j�t� − r j�0�� ,

where k is a wavevector compatible with the periodic bound-ary conditions used in the MD simulations. Upon varying k= �k�, Fs�k , t� probes individual particle motion on differentlength scales. While in high density glasses ��0.5�, domi-

100

101

102

103

104

105

reduced time

10-3

10-2

10-1

100

101

102

103

104

105

redu

ced

mea

nsq

uare

disp

lace

men

t

0.0200.0330.0610.1640.409


slope 2

slope 1

FIG. 11. �Color online� MSD of particles at packing fraction �=0.0219.

0 1 2 3 4 5frequency, f*

0

0.2

0.4

0.6

0.8

Four

ier

tran

sfor

m

0 10 20 30 40reduced time, t*

0

0.2

0.4

0.6

0.8

1

velo

city

auto

corr

elat

ion

func

tion,

Z(t

*)

0.0200.0410.0820.1640.2040.409


FIG. 12. �Color online� Velocity autocorrelation function Z�t�� as a function

of reduced time and its Fourier transform Z�f�� as a function of reducedfrequency at packing fraction �=0.0219.



nated by packing effects and caging, the most relevant lengthscale is the nearest neighbor distance, which is of the orderof the particle size �, in network-forming systems the keylength scale is the network mesh size, which is of the orderof the size L of the simulation cell, and the correspondingwavenumber is k�=2� /L. We have hence focused our at-tention on the smallest wavenumbers compatible with thesystem size, i.e., k��0.22 for �=0.0219. Examples of therelaxation of Fs�k , t� with time for several temperatures areshown in Fig. 13�a� for �=0.0219 and k�=0.22. As antici-pated, the relaxation slows down dramatically as T� drops. Atthe lowest temperature, the dynamics appears to be arrestedon the time scale of the simulation �t�=14 000�. The enlarge-ment of Fs�k , t� at this lowest temperature, plotted in theinset of Fig. 13�a�, shows a narrow plateau in the decay ofFs�k , t� extending over less than a decade, which separatestwo relaxation regimes which one may be tempted to identifywith the familiar � and � regimes predicted by mode-coupling theory of the kinetic glass transition.1 Similar pla-teaus are observed at larger wavenumbers and at larger pack-ing fractions, as shown in Fig. 13�b�. The observed plateausare reminiscent of those reported by Zaccarelli et al.13,14 fora valence-limited colloidal gel, but occur at greater heights�f 0.9� compared to the earlier work. It is expected that, ontime scales far longer than those accessible to the simulation,Fs�k , t� will decay to zero for all k, since no matter how long

lived the connections between dumbbells may be, they arenever truly permanent at finite temperature.

At the higher temperatures, where Fs�k , t� decays to zerowithin the duration of the MD simulations, the slowing downmay be characterized by the relaxation time,

�s�k� = �0

�

Fs�k,t�dt .

Plots of Fs�k , t� versus ln�t /�s�k��, similar to Fig. 5 in Ref.19, show that the shape of Fs�k , t /�s�k�� changes continu-ously from nearly Gaussian at the highest temperature tostretched exponential-like at the lowest temperatures, al-though no single stretching exponent can be associated withthe relaxation of Fs�k , t� over the whole accessible time win-dow.

The dynamical slowing down of jammed or network-forming systems generally involves dynamical heterogeneity,characterized by the coexistence, within the same material,of two populations of particles with high and low averagemobilities.40–42 Such dynamical heterogeneity can be diag-nosed from an analysis of the self-part of the van Hove cor-relation function along a given direction,

Gs�x,t� =1

N�i=1

N

� �x + xi�0� − xi�t�� ,

where xi is a Cartesian component of the particle positionvector ri. If the motion of all particles is diffusive andhomogeneous, Gs�x , t� reduces to the Gaussian form32

exp�−x2 /4Dt� /�4Dt. We have calculated the van Hovefunction over a wide range of temperatures and time inter-vals, averaging over the x, y, and z directions to improvestatistics in each case. At very low temperatures, Gs�x , t�shows marked deviations from a Gaussian distribution �seeFig. 3 of the EPAPS supplementary material36 for an ex-ample�. A two-Gaussian fit, attempting to describe a mixtureof “fast” and “slow” particles, is more accurate but at thelowest temperatures it still fails to capture the tail of largedisplacements, where the variation with x appears to be ex-ponential. Hence, although it is tempting to identify the slowparticles with the strings and fast particles with networkjunctions, where bonds are weaker, this classification is am-biguous and it is perhaps more realistic to think of a spec-trum of dynamical heterogeneities rather than two discretepopulations.

The extent to which Gs�x , t� is non-Gaussian can bequantified using the parameter

��t� =�x�t� − x�0��4

3�x�t� − x�0��22 − 1,

which vanishes if x follows a Gaussian distribution. Theshape of ��t� evolves in a nonmonotonic way with tempera-ture, as shown by the logarithmic time plot of Fig. 14 forpacking fraction �=0.0219. Individual MD simulations pro-duce ��t� curves with considerable variation, and the datapresented in Fig. 14 are averages over curves calculated forfour independent runs of length 2400t0 at each temperature.The large run-to-run fluctuations in ��t� suggest that the non-Gaussian parameter is dominated by a minority of particles,

10−1 100 101 102 103 104

reduced time, t*

0

0.2

0.4

0.6

0.8

1

Fs(k

,t*)

0.0200.0330.0410.0820.1640.204

100

101

102

103

104

0.97

0.98

0.99

1


100

101

102

103

reduced time, t*

0.9

0.92

0.94

0.96

0.98

1

Fs(k

,t*)

0.02190.07450.149

packing fraction, φ

(b)

(a)

FIG. 13. �Color online� Self-part of the intermediate scattering function,Fs�k , t�. �a� Packing fraction �=0.0219 and wavenumber k�=0.22 at se-lected temperatures as marked. Inset: expanded plot of the T�=0.020 curve.�b� Wavenumber k�=0.56, temperature T�=0.020, and three packing frac-tions, as marked.



whose behavior is sensitive to the precise structure of thenetwork.

At sufficiently high temperature, as we have seen, thedynamics goes over smoothly from the ballistic to the diffu-sive regime. Since both regimes produce a Gaussian distri-bution of displacements over a given time interval, ��t� re-mains small at all times, as shown by the T�=0.409 line inFig. 14. On lowering the temperature, however, even beforethe percolation threshold is reached, the peak in ��t� growsdramatically �T�=0.204,0.164�. The peak may reflect thedifferent environments in which individual particles findthemselves in a system of short, disconnected, transientchains. A further decrease in temperature produces a qualita-tive change in ��t�. The peak reduces in height, presumablydue to the formation of a more uniform fluid of percolatingchains, and moves to longer times. Simultaneously, anotherpeak develops at much shorter times t� on the order of unity.This is the time scale of intrachain oscillations �cf. Fig. 12�,a nonballistic, nondiffusive rattling. Finally, at very low tem-peratures �not shown in Fig. 14�, ��t� rises very sharply andstays high, indicating pronounced and long-lived dynamicalheterogeneities.

VI. DISCUSSION AND CONCLUSIONS

The detailed analysis of the extensive MD simulationdata presented in this paper provides strong evidence for atwo-step gelation scenario in dilute dispersions of dipolardumbbells. The model investigated here differs significantlyfrom more widely studied colloidal systems and modelswhich involve short-ranged isotropic or patchy attractive in-teractions between particles, since the dipolar interactions areboth long ranged and anisotropic. Nevertheless, a system ofdipolar dumbbells exhibits structural and dynamical behaviorreminiscent of that reported for systems with short-range at-tractions.

The scenario which emerges from the results reported inthe previous sections involves two successive transitionsupon cooling a dilute system of dipolar dumbbells along anisochore. A percolation transition from a string phase to aspace-spanning network occurs at an intermediate reducedtemperature Tp

� which decreases with packing fraction �. Thenetwork is highly transient, with a bond lifetime �b of neigh-boring particles which is short just below Tp

�, but increases

rapidly as the temperature is lowered further. When the tem-perature drops to a value nearly an order of magnitude lowerthan Tp

�, the network gels. Gelation is signaled by a numberof diagnostics characteristic of dynamical arrest on very longtime scales and of dynamical heterogeneity. These diagnos-tics include the time dependence of the MSD, which ishighly subdiffusive and barely reaches the linear diffusionregime over the time scale of our longest simulations �t�

104�; a pronounced slowing down of the self-intermediatescattering function Fs�k , t� at small wavenumbers k��1,characterized by a two-step relaxation separated by a clear-cut, albeit narrow plateau at a height f 0.9; and the con-comitant highly non-Gaussian behavior of the van Hovefunction, which points to significant dynamical heterogene-ity.

These dynamical indicators of gelation are supplementedby a number of structural and topological diagnostics, in-cluding a qualitative change in the variation of the EC withspatial resolution and variations of the asphericity parametersof the Voronoi cells and of the void size probability distribu-tion, which points to the bundling of dipolar strings upongelation.

For systems undergoing phase separation into low andhigh density fluid phases, interrupted spinodal decompositionis the most likely mechanism leading to gelation.8,9 It ishence highly desirable to relate the state points of the dipolardumbbell system explored in the present MD simulations tothe phase diagram of the same model system. The phasediagram is not available for the model of soft dipolar dumb-bells considered in this paper, but the phase diagram of theclosely related model of hard dipolar dumbbells has recentlybeen established by Ganzenmüller and Camp for several di-polar elongations.24 In their model, the soft-core repulsionvarying as r−12 in Eq. �1� is replaced by a hard core repulsionwith a hard sphere diameter s. Since in this case the lowestenergy head-to-tail configuration is achieved for coaxialdumbbells at contact, a crude mapping of the soft dumbbellonto the hard dumbbell model amounts to the identificationof the lowest energy spacings,

� = s + d . �4�

Interpolating the critical densities and temperatures listed inRef. 24, we arrive at the following estimates of the criticalparameters of the soft dipolar dumbbell model with �=d /�=0.217: Tc

�=0.243, �c�=0.176 �i.e., �=0.092�. The estimated

critical temperature lies well above the percolation tempera-tures Tp

� along the two isochores explored in our MD simu-lations. Moreover, these two isochores lie to the low densityside of the critical isochore in a �T� ,�� phase diagram.However, a divergence of the structure factor S�k� at small kis only observed at the lowest temperature �T�=0.02� alongthe �=0.0219 isochore in our model. Thus, there seems to bea contradiction between the estimated critical parameters andthe amplitude of the long wavelength density fluctuationsobserved in our MD simulations. However, several caution-ary remarks are in order, which might resolve the apparentcontradiction. First of all, the fluid–fluid coexistence curvesreported in Ref. 24 are extremely flat, and the critical param-eters are determined there by extrapolation of lower tempera-

10−1 100 101 102 103

reduced time, t*

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

non-

Gau

ssia

npa

ram

eter

,α 0.0820.1230.1640.2040.409


FIG. 14. �Color online� Non-Gaussian parameter � of the self-part of thevan Hove correlation function Gs�x , t� at packing fraction �=0.0219.



ture coexistence data. The true critical densities could wellbe shifted to significantly lower values, bracketed by the iso-chores studied in the present work. Second, the spinodal line,for which no estimate is presently available, could lie wellinside the binodal line �except, of course at the critical pointitself�, contrary to the case of systems with short-ranged at-tractive interactions. This would imply that a system of di-polar dumbbells would have to be quenched to very lowtemperatures before the spinodal line is reached, as sug-gested by the structure factors calculated in our simulations.Finally, the binodal and spinodal lines of the dipolar dumb-bells may be very sensitive to minute details of the short-range repulsion, such that the simple mapping embodied inEq. �4� may be quantitatively unreliable. The physicalmechanism of the gelation process thus remains an openproblem, as long as the spinodal line for the present softdumbbell model has not been systematically determined bysimulations or reliable approximate liquid state theories.

Apart from the location of the spinodal line, future workshould focus on a number of dynamical aspects beyond thoseexamined in this paper. The first extension must be the col-lective dynamics, as probed by the full density autocorrela-tion function F�k , t�, and not just its self-part. This wouldrequire significantly longer MD runs to gather more statisticsnecessary for an accurate determination of the autocorrela-tion function of a collective variable. To gain more insightinto dynamical heterogeneity, one would require four-pointdensity autocorrelation functions, which are even more com-putationally demanding to converge. Another instructivephysical property would be the response to external stress tocharacterize the elasticity of the dipolar network. Finally, itwould be of interest to investigate the gelation of the presentmodel within mode-coupling theory.1

It would obviously be desirable to confront the complexbehavior of the present model, as revealed by our MD simu-lations, with laboratory experiments on a corresponding col-loidal system. Spherical dipolar colloids are routinely syn-thesized for experimental studies of ferrofluids. However, itis the moderate elongation of the particles and dipoles thatstrongly favors network formation. An experimental realiza-tion of the model system investigated in this paper may wellbe possible by synthesizing “diatomic colloids”43,44 with op-posite charges on the two constituents. It is hoped that thepresent work will encourage experiments along such lines.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Philip Camp for helpfuldiscussions on the phase diagrams of dipolar systems.M.A.M. thanks EPSRC �U.K.� �contract EP/D072751/1� forfinancial support.

APPENDIX: EULER CHARACTERISTIC

In Sec IV, the topology of the gel network was describedusing the EC � of a collection of spheres of diameter DE,centered on the dumbbells. The EC of an arbitrary surfacecan be obtained by approximating the enclosed volume usingspace-filling polyhedra such as cubes.45 It can also be foundexactly using a careful tessellation of the surface into poly-

gons. In our application, however, the surface consists onlyof overlapping spheres, and this specialization can be ex-ploited to devise a more efficient exact algorithm. The keypoint is that the topology and, therefore, � change discretelyat well defined values of DE where two or more spheres firsthave a common intersection. We must therefore identify thetouching events that produce a change in topology. There arethree types of topologically relevant event.

The first type of topological event occurs when a pair ofspheres first touch. For sufficiently small DE, all the spheresare disconnected and �=2N. When two disconnected spherestouch they become a single object, and � is reduced by 2. Afurther reduction by 2 occurs each time two spheres firsttouch, unless the point of contact lies within a third sphere.In the latter case, the touching spheres were already con-nected by other spheres, and the topology is not altered bythe new contact. Whether or not the point of contact lieswithin another sphere can be ascertained from a Voronoianalysis of the particle positions. The point of contact be-tween two neighboring spheres lies on the plane bisecting thevector that joins their centers. The Voronoi cells of the twoparticles at the centers of the spheres share a polygonal facelying in the same plane. If and only if the point of contactlies inside the shared face, then there is no other sphere closeenough to overlap with the point, and the contact constitutesa topological event. Note that it is perfectly possible for twoparticles to be neighbors in a Voronoi analysis, i.e., for theirVoronoi cells to share a face, without the interparticle vectorpassing through the shared face.

In the process of spheres touching and merging, loopswith the topology of a torus will naturally emerge. Furtherincrease in DE leads to the closure of the toroidal hole, whichincreases � by 2 and constitutes the second type of topologi-cal event that must be accounted for. A torus closes when thethree circles of contact between three mutually overlappingspheres first meet at a point inside the triangle defined by thecenters of the spheres. A torus may initially arise from a loopof more than three particles, but as DE increases the holeshrinks and the point where it disappears will involve onlythree spheres �see Fig. 1 of the EPAPS supplementarymaterial36�, except in very unlikely ordered geometries, suchas four particles in a perfect square. In such cases, care mustbe taken to count the torus closure only once. Just as the firsttype of topological event described above, the topologychanges if and only if the point of contact at the closure ofthe torus does not lie within another sphere. This is equiva-lent to requiring that the point of contact lies on the commonedge shared by the Voronoi polyhedra of the three particlesin question and also lies within the triangle spanned by them.

Merging of spheres can also lead to enclosed cavities.With increasing DE, the cavities disappear, decreasing � by 2and giving the third type of topological event. Closure of acavity occurs when four spheres meet at the shared vertex ofthe corresponding Voronoi polyhedra. The contact leads to achange in topology if and only if the vertex lies within thetetrahedron whose vertices are the four particles in question.Again, degenerate cases involving more than four particlesare possible but highly unlikely in a disordered system andshould not be double counted.



The fact that all topological events take place on theboundaries between adjacent Voronoi cells means that it isnot necessary to keep track of all pairs, triplets, and quadru-plets of spheres to identify changes in � �a task that wouldscale as N4 in effort�. It is only necessary to consider com-binations of particles that are mutual neighbors in theVoronoi tessellation. We may therefore summarize the pro-cedure for calculating � as a function of increasing DE withthe following steps.

�1� Make a Voronoi construction for the positions of the Nspheres. For DE=0 we know �=2N.

�2� For each pair of neighboring points calculate the diam-eter D for which the spheres touch. If the contact lieswithin the shared face of the two Voronoi cells, reduce� by 2 at DE=D.

�3� For each triplet forming an acute triangle, compute thelocation where the corresponding ring �if any� will dis-appear and at what diameter D. If the point lies on thecommon Voronoi cell edge, increase � by 2 at DE=D.

�4� Determine whether each Voronoi vertex lies within thetetrahedron defined by the centers of the four adjacentcells. If so, increase � by 2 at the diameter D of thespheres at mutual contact.

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