Identification of Structure in Condensed Matter with the Topological ClusterClassification

Alex MalinsBristol Centre for Complexity Sciences, University of Bristol, Bristol, BS8 1TS, UK and

School of Chemistry, University of Bristol, Cantock’s Close, Bristol, BS8 1TS, UK

Stephen R. WilliamsResearch School of Chemistry, The Australian National University, Canberra, ACT 0200, Australia

Jens EggersSchool of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

C. Patrick RoyallHH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK

School of Chemistry, University of Bristol, Cantock’s Close, Bristol, BS8 1TS, UK andCentre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol, BS8 1FD, UK∗

(Dated: July 12, 2018)

We describe the topological cluster classification (TCC) algorithm. The TCC detects local struc-tures with bond topologies similar to isolated clusters which minimise the potential energy for anumber of monatomic and binary simple liquids with m ≤ 13 particles. We detail a modified Voronoibond detection method that optimizes the cluster detection. The method to identify each cluster isoutlined, and a test example of Lennard-Jones liquid and crystal phases is considered and criticallyexamined.

PACS numbers: 61.43.Fs, 61.20.Ja, 64.70.Q-, 02.70.Ns

I. INTRODUCTION

Research into the structure of disordered systems, suchas liquids, glasses, liquid-vapor interfaces and cluster flu-ids, has a rich history. Early studies focused on pairdistribution functions as these were readily accessiblefrom experiments using scattering techniques. The ex-perimental results inspired the development of theorieswith the capability to predict the radial distribution func-tions for disordered states. The most famous example ofsuch a theory is perhaps the Percus-Yevick closure of theOrnstein-Zernike relation which provides the radial dis-tribution function for the hard sphere fluid1.

Efforts to understand the structure of liquids in greaterdetail than at the level of pair correlations of the par-ticle density were pioneered by Bernal, Rahman andFinney2–4. These authors studied the properties ofVoronoi cells in liquids, which are convex polyhedra thatcontain all points in space closer to one particle thanany other. The geometrical characteristics of each poly-hedron, such as the number and shape of the faces, de-pends on the position of the particle and its immediateneighbors. As such, Voronoi cells provide informationon higher-degree correlations of the particle density thantwo-body distribution functions.

These studies into Voronoi polyhedra were among someof the earliest that sought to characterize the structureof disordered systems in terms of the shapes formed bysmall clusters of particles. Many subsequent efforts havebeen made in this direction, and there are now severalmethods available for quantifying structural correlations

in disordered systems in terms of the shapes of clusters ofparticles. We designate methods as topological methodsif they identify clusters from topological features of the“bond network” formed by the particle locations and theconnections between neighboring particles.

Two topological methods that see frequent use are thecommon neighbor analysis (CNA) introduced by Honey-cutt and Andersen5 and the Voronoi face analysis (VFA)by Tanemura et al.6. The CNA identifies the struc-tural ordering around pairs of particles in terms of shared(common) neighboring particles. The VFA technique isa generalisation of the early Voronoi cell studies. Thestructure around a single particle is characterized by thearrangement of its nearest neighbors and the bonds be-tween them. This information is encoded by the shapesof the faces of its Voronoi cell. Both of these methodshave been used to study phenomena including the melt-ing and freezing of clusters5, structure in supercooledliquids on cooling towards the glass transition7–12, andcrystallisation6,13,14.

An alternative approach is the bond orientational or-der parameters, introduced by Steinhardt et al. to char-acterize the arrangement of neighboring particles arounda central particle15,16. A series of bond orientation order(BOO) parameters are defined that quantify the similar-ity between the directions from a particle to its neighborsand the spherical harmonic solutions to the Laplace equa-tion, in a method that has been described as a kind ofshape spectroscopy16,17. BOO parameters have been ap-plied for the study of a variety of phenomena. Their orig-inal application was to study structural ordering in su-

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percooled liquids15,16, where depending upon the systemunder consideration, both icosahedral-18 and crystalline-ordering19 has been found to develop on cooling towardsthe glass transition. Other researchers have employedBOO parameters to study the structure at interfaces20,21

and of large clusters22. Arguably the most successfulapplication of BOO parameters however is as order pa-rameters for crystallisation23–28.

Recent advances in experimental techniques on col-loidal dispersions have delivered methods that allow mea-surement of real space configurational data for colloids.In particular, particle tracking software allows the posi-tions to be extracted from images taken with 3D confo-cal microscopy29–31. This development means that thestructure of experimental systems that display Brownianmotion can now be analysed with higher-order structuraldetection algorithms, with interest in, for example, col-loidal gels32, glasses29 and critical fluctuations33.

Here we describe a recent addition to the collectionof high-order structural detection algorithms, the topo-logical cluster classification (TCC) algorithm. This algo-rithm has been used to investigate higher-order structurein colloidal32,34 and molecular35 gels, colloid-polymermixtures36, colloidal clusters37–39, simple liquids60,liquid-vapor interfaces40, supercooled liquids41–43 andcrystallising fluids44. The method shares some character-istics with other topological methods (e.g. the VFA andCNA methods). However, rather than detecting clustersaround a central or pair of particles, it is based on the de-tection of minimum energy clusters with a range of sizes.By minimum energy cluster, we refer to those structureswhich minimize the potential energy of a given number ofparticles in isolation. Here we consider spherically sym-metric interactions, whose minimum energy configura-tions can be calculated by optimization methods such asin the GMIN software45.

The philosophy of the TCC algorithm draws heavilyupon an idea proposed by Sir Charles Frank in a sem-inal paper in 195246. When discussing the ability tostabilize metallic liquids below the melting temperature,Frank proposed that collections of particles may pref-erentially adopt certain energy-minimising structures onshort lengthscales. He inferred that if the symmetry ofthe structures formed is incommensurate with that of thebulk crystal symmetry, the structures would inhibit crys-tallisation and stabilize the undercooled melt. The basisfor this was that there must be a substantial rearrange-ment of the structures for crystallisation to proceed, andthis operation would carry an energetic cost.

To demonstrate his idea Frank cited the example of theminimum energy clusters formed by 13 Lennard-Jonesparticles in isolation. Arranging the particles at the cen-tre and the vertices of a regular icosahedron yields a clus-ter with 8.4% lower interaction energy than for compactFCC or HCP crystal clusters of 13 particles. Assumingthat the energy of the icosahedral arrangement would stillbe lowest for the three arrangements in the disorderedliquid environment, he inferred that the icosahedral for-

mations would be more prevalent than FCC and HCP ar-rangements in the supercooled liquid. This turned out tobe a reasonable assumption, the icosahedron was shownto have lower energy than either crystal arrangement ina mean-field description of the Lennard-Jones liquid47.

While Frank considered minimum energy clusters of 13Lennard-Jones particles for his example, as a natural ex-tension, the TCC algorithm takes the minimum energyclusters of other numbers of particles as well. The algo-rithm also considers models with interactions other thanthe Lennard-Jones potential. The TCC algorithm worksby searching a configuration for arrangements of m par-ticles whose bond network is similar to that found in theminimum energy clusters of a given model. By identi-fying minimum energy clusters of the model of interest,the method therefore provides a direct link between theinteractions in the system under consideration and anystructural ordering that is found. Moreover, the TCCsimultaneously finds clusters for all models incorporated.Thus an analysis of a given coordinate set is not limitedto the minimum energy clusters which correspond to themodel with which the set was generated.

The TCC favors no specific “origin” particles for theclusters, i.e. a central particle or a bonded pair. Thismeans that routines can be devised to detect clusters withdisparate shapes and sizes where there are no such “ori-gin” particles. This stance differentiates the TCC frommany of the other structure detection methods. The algo-rithm presented here includes detection routines for min-imum energy clusters of up to 13 particles for the modelsthat are considered. Extending the algorithm to includedetection routines for larger minimum energy clusters ispossible. This paper is organized as follows. In sectionII, we give a high-level overview of the method, and pro-ceed, section III, to detail the Voronoi construction de-vised to optimize the structure detection algorithm. Insection IV and the appendix, we provide details of theTCC detection routines for specific clusters. In sectionV, we demonstrate the TCC algorithm by identifying thestructure in Lennard-Jones crystals. In section VI wecritically analyse its performance on four phases of theLennard-Jones system before concluding in section VII.

II. OVERVIEW OF THE TOPOLOGICALCLUSTER CLASSIFICATION

A high-level overview of the TCC algorithm is as fol-lows:

1. The neighbors of each particle are identified.

2. The network formed by the particles and connec-tions to their neighbors is searched for shortest-path rings of 3, 4 and 5 particles.

3. From the shortest-path rings, a set of structuresknown as the “basic clusters” are identified. Thebasic clusters are distinguished by the number of

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additional particles that are common neighbors ofall the particles in a shortest-path ring.

4. Larger clusters are then identified by combining ba-sic clusters together, sometimes with the additionof one or two separately bonded particles, accord-ing to a set of predefined rules. The method yieldsstructures with bond networks with similar topol-ogy to the bond networks of the minimum energyclusters.

Throughout we endeavor to highlight the strengths andweaknesses of the TCC method. Although the focus ofthe TCC is on disordered systems, the performance ofdifferent structure detection algorithms for identifyingcrystalline order will provide a helpful benchmark for theaccuracy and efficiency of the algorithm. These tests areuseful because the ordering of particles in a crystal isknown a priori, whereas this is not necessarily the casefor a disordered system.

III. DETECTING NEIGHBORS WITHVORONOI TESSELLATIONS

Although simple and intuitive, the main disadvan-tage of using a cut-off length to determine a particle’sneighbors is the sensitivity of the result to the chosenlength and the consequential difficulty in deciding whichlength yields the most valid result. A method that canidentify the neighbor network independently of an ad-justable length parameter is therefore appealing. Onesuch method is the Voronoi tessellation which decom-poses space into regions of non-intersecting domains withdistinct boundaries48. Each region contains the positionof a single particle and all points in space closer to thatparticle than any other. In three dimensions these regionsare convex polyhedra that are termed Voronoi cells. Twoparticles are neighbors if their Voronoi cells share a face.Two particles with Voronoi cells that share only an edgeor a vertex are not considered bonded. It is importantto make this clear as later on we will consider the detec-tion of four-membered rings of particles with the Voronoimethod. In practice fluctuations of the particle positionsin thermal systems mean that edge- and vertex-sharingpolyhedra are rare.

The properties of Voronoi cells for disordered systemswere first studied by Bernal2, Rahman3 and Finney4.Subsequently the method of using a Voronoi tessella-tion to define neighbors in a network was popularizedin studies of crystallisation6 and glass-formers49. Theearly studies of crystallisation noted that the detectionof the crystal-order when using the Voronoi method washighly sensitive to thermal fluctuations of the particlepositions6,13,50. The Voronoi cell of a particle in a T = 0FCC crystal contains six vertices that are each commonto four faces. Small thermal vibrations lead to these ver-tices being split into two vertices joined by a new edge.This adds a new common face between two of the Voronoi

cells of the neighboring particles, and consequently addsa new bond to the system. The additional bond compli-cates the detection of the crystalline order, as there are awide variety of Voronoi polyhedra in the thermal systemcorresponding to a single crystal structure.

This problem is equivalent to the unwanted bond form-ing across a square of particles as discussed in sectionIII A below. Poor performance is often found when try-ing to detect structures with four-fold symmetries (e.g.octahedral structures) when employing a structure detec-tion method upon the Voronoi neighbor network. For thisreason numerous authors have proposed modifications tothe Voronoi method in order that the detected bond net-work is more robust against change by thermal fluctu-ations of the particles13,14,51–53. The sensitivity of theVoronoi method to thermal fluctuations can be demon-strated by examining the number of first-nearest neigh-bors it detects for FCC and HCP crystals. Frequentlysecond-nearest neighbors are misdetected as first-nearestneighbors14,53,54, meaning that the number of the first-nearest neighbors is inflated55.

In the case of the TCC, one of us (SRW), proposeda modification for the Voronoi method when introduc-ing the TCC structure algorithm to address this issue56.The method adds a dimensionless parameter that sets themaximum amount of distortion that a four-memberedring in a plane can undergo before a bond forms be-tween particles at opposite vertices. Good performancewas found for detection of octahedral structures in thehard sphere liquid56, and here also we demonstrate thatdetection of FCC and HCP order is improved using themodified Voronoi bond detection method.

The Voronoi method is attractive in that the neighbornetwork can be identified without having to select anadjustable parameter. It provides a natural and intuitiveway to determine the neighbors for each particle. Themethod has problems when thermal vibrations destroydegenerate vertices and this scenario is especially relevantfor the detection of crystalline order if the number of first-nearest neighbors for each particle is over-estimated.

A. A modified Voronoi method

Our modified Voronoi method makes two modificationsto the original Voronoi method (hereafter referred to asthe standard Voronoi method). The modifications areintroduced to improve the detection of the four- andfive-membered rings of particles56. The first modifica-tion ensures that particles are neighbors only if theirVoronoi cells share a face and the line that connectsthe particle positions intersects the shared face. Pairsthat obey this condition are known as “direct Voronoineighbors”57–59. Conversely neighbors within the stan-dard Voronoi method for which this condition is notobeyed are termed “indirect Voronoi neighbors”. Thiscondition is depicted in Fig. 1(a) for a 2D system, wherethe connections between direct Voronoi neighbors are

4

i

j

k

k

j

i

k

j

i

θ

θ

ri ri

rjrj

rkrk

rkj

rij

rik

rkj

rij

rik

|rij| < |rik|

rij · rkj ≤ 0 rij · rkj > 0

(a) (b) (c)i, k not neighbours i, k neighbours

Figure 1. (a) The lines that connect particle centres must also intersect the face that is shared between the two Voronoi cellsfor the particles to be bonded. Particles i and k are not bonded therefore, as highlighted by the red line connecting theirpositions. Conversely all other particles are considered neighbors of particle i (blue lines - direct Voronoi neighbors). Particlej is closer to i than k, and the volume of its Voronoi cell is shielding particle k from particle i. (b) and (c) Identification ofdirect Voronoi neighbors. Particle k is further from i than j in both cases. In (b) particle k is not bonded to i as there existsj that is closer to i and the angle subtended by vectors rij and rkj is greater than π/2 radians. Conversely k is bonded to i in(c) as the bond there is no shielding by j.

highlighted with blue lines, and with red line for theindirect Voronoi neighbors. The second modification isto introduce a dimensionless parameter fc that controlsthe maximum degree of asymmetry of a four-fold ring ofneighbors before it is detected instead as two three-foldrings of neighbors.

The first condition has two effects when compared tothe standard Voronoi method:

1. It strictly reduces the number of bonds derivedfrom the Voronoi tessellation. The removed bondsoccur when the volume of the Voronoi cell of a thirdparticle overlaps with the line connecting the posi-tions of two particles that share a Voronoi face. Thethird particle shields the bond between the otherpair and the bond present in the standard Voronoimethod is removed [Fig. 1(a)]. The removed bondstend to be “long bonds” that cause misdetection ofsecond-nearest neighbors as first-nearest neighbors.

2. Algorithms to detect neighbors that enforce thiscondition are generally more straightforward to im-plement than algorithms which detect all neighborsof the standard Voronoi method.

The direct Voronoi neighbors of particle i are identifiedby first finding all particles within some cut-off distancerc, which can be made as large as necessary. These parti-cles are ordered in increasing distance from i. A particlek is a direct Voronoi neighbor of i if and only if for all jcloser to i than k is the angle subtended by the vectorsrij and rkj is less than π/2 radians. Or equivalently theinequality

rij · rkj > 0 , (1)

(a) (b)i

k

j

w/2

h

r

r

θ

r

r

Figure 2. Detection of four-membered rings of particles. (a)Particles are positioned at the vertices of a rhombus. In thecase that h = w, i.e. a square, there is no bond between par-ticles i and k. If h < w then a bond forms between i andk across the rhombus if using the standard Voronoi method.For a given rhombus defined by h/w or θ, the existence ofthe bond between i and k in the modified Voronoi methoddepends on the value of fc. (b) The detection of an octa-hedral cluster with the CNA and TCC methods relies in theintegrity of its four-membered rings, e.g. the ring highlightedin red. Thermal fluctuations may cause this structure to bedetected incorrectly if using the standard Voronoi algorithmto determine the neighbors of each particle.

holds ∀j, where |rij | < |rik|. Figs. 1(b) and (c) show caseswhere this inequality is not and is satisfied respectively.

The second modification to the standard Voronoimethod improves detection of four-membered rings ofparticles. Consider particles placed at the vertices of arhombus, as in Fig. 2(a). If the rhombus were a perfectsquare there would be a total of four bonds detected bythe standard Voronoi method, each running along the

5

edges of the square. If the square is distorted by anysmall amount to form a rhombus, a bond forms betweentwo particles on opposite vertices thus creating two three-membered rings of particles. This bond breaks the de-tection of any cluster based on the integrity of a four-membered ring, e.g. the octahedral cluster shown in Fig.2(b), when any small fluctuation of the particle positionscauses a distortion to the ring.

A dimensionless parameter fc, known as the four-membered ring parameter60, is introduced that deter-mines the maximum amount of asymmetry that a four-membered ring of particles can display before it is iden-tified as two three-membered rings. Bonds between par-ticles i and k [Fig. 2(a)] will be removed if there ex-ists a particle j that is both bonded to i and closerto i than k is, and that shields k from i. We con-sider the plane perpendicular to rij that contains a pointrp = ri+fc(rj−ri). If fc < 1 this corresponds to movingthe plane perpendicular to rij and containing rj towardsri [Fig. 3(a)]. Adapting Eq. 1, the condition for k to bebonded to i is if ∀j where |rij | < |rik|:

rip · rkp > 0 . (2)

Re-expressing the inequality in terms of the position ofparticle j gives (fc 6= 0)

fc(ri·ri+rj ·rj−2ri·rj) > ri·ri+rj ·rk−ri·rj−ri·rk . (3)

Eq. 3 is not invariant to swapping indices i and k. Theconsequence of this is that i may be bonded to k but notvice versa. It is therefore necessary to consider bondingfrom the point of view of particle k. The plane perpen-dicular to rkj containing rj is moved towards k such thatit contains the point rq = rk+fc(rj−rk). From the view-point of particle k, particles i and k are neighbors if ∀jwhere |rkj | < |rik|:

rkq · riq > 0 . (4)

The inequalities in Eqs. 2 and 4 are depicted geomet-rically in Figs. 3(a) and (b). Both inequalities are notsatisfied as the angles θ are obtuse. Adding togetherEqs. 2 and 4 yields a single, symmetric, criterion for iand k to be bonded. If ∀j with |rij | < |rik|

rip · rkp + rkq · riq > 0 , (5)

i and k are said to be neighbors in the modified Voronoimethod.

Eq. 5 is invariant to inversion of the indices i and k,therefore the resulting modified Voronoi method neigh-bor network is necessarily symmetric. A symmetric bondnetwork is a required in order to use either the CNA orTCC algorithms to identify structure. Expanding Eq. 5in j and setting fc = 1 recovers the definition of all thedirect Voronoi neighbors (Eq. 1). However if fc < 1, themodified Voronoi method neighbors (Eq. 5) are only asubset of the direct Voronoi neighbors.

k

j

iθ

ri

rj

rk

rkp

rp=ri+fcrij

rik

rip · rkp ≤ 0

(a)

rip

k

j

i

θ

ri

rj

rk

rkq

riq

rik

(b)

riq · rkq ≤ 0

rq=rk+fcrkj

Figure 3. The modified Voronoi four-membered ring param-eter parameter fc for the scenario in Fig. 1(c). Particles iand k are direct Voronoi neighbors (as per Fig. 1(c)), how-ever with the chosen value of fc they are not considered asneighbors by the modified Voronoi method. This is becauseinequality (5) from the main text is not satisfied. In (a) theparticle k lies on the far side of the plane perpendicular to ripand containing rp from particle i. Likewise in (b), i lies onthe far side of the plane perpendicular to rkq and containingrq from k. Therefore rip ·rkp +rkq ·riq must be less than zero.

To demonstrate the effect of fc we reconsider the rhom-bus of particles in Fig. 2(a). The parameter fc determinesthe maximum asymmetry of the rhombus (h/w) before abond forms between particles i and k on opposite vertices.Expanding and rearranging Eq. 5 gives an inequality fora bond to exist between i and k in terms of the particleseparations:

fc >r2ik

r2ij + r2jk

. (6)

Using rij = rjk (side lengths of a rhombus are equal),rik = h and Pythagoras’ theorem gives this condition interms of the lengths h and w:

fc >2

1 + (w/h)2. (7)

If fc is greater than this value then there is a bond be-tween i and k and the rhombus of particles is identified astwo three-membered rings. Conversely if fc is less thanor equal to this value then i and k are not bonded andthe rhombus may be a four-membered ring subject thestatus of j and l. This is found by swapping w and h inEq. 7. In terms of the angle θ subtended by the bondsconnecting j to i and k, the condition for i and k to bebonded is

fc >2

1 + [tan(θ/2)]−2. (8)

For angles θ less than that where Eq. 8 is an equality,i and k are bonded. If fc = 1, the angle θ = π/2 and

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Particle i

Set Si containingparticles < rc from i

Ordered set Si indistance from i

j from Si

k > j from Si

fc > rikʹ

2

rijʹ + rjʹkʹ 2 2 no

Eliminate k from Si

next k

next k

next j

next i

yes

Figure 4. Algorithm for detecting neighbors using the mod-ified Voronoi method. The apostrophes on j and k indicatethe particles referenced by the jth and kth elements of the setSi.

for fc = 0.82 we have θ ≈ 1.39 (≈ 80◦). The limit forvalidity of Eq. 8 is θ ≤ π/3 (θ ≤ 60◦), i.e. valid choicesfor fc are in the range (0.5, 1].

The algorithm to identify the modified Voronoi neigh-bor network is therefore as follows (see Fig. 4):

1. Loop over all N particles with index i.

2. Find all particles within rc of ri, where rc is to belonger than the longest bond in the network, andadd to a set Si.

3. Order the particles in Si by increasing distancefrom particle ri.

4. Loop over all elements j in Si, i.e. particles in in-creasing distance from particle i.

5. For each j loop over all k > j in Si and eliminatekth particle from Si if inequality 6 is not satisfied.

The particles left in the sets Si on completion are themodified Voronoi method neighbors of particle i. Thevalue of rc can be used to set the maximum bond length

in the system. Doing so would be useful if studying gelsor cluster fluids, for example, where long bonds are notdesired between gaseous particles or particles in separateclusters.

B. Comparison of the neighbor detection methods

We compare the neighbor detection methods by con-sidering a Lennard-Jones system at fixed temperatureT = 0.92 and pressure P = 5.68 as studied by ten Woldeet al.62. Here we use reduced units where Boltzmann’sconstant kB = 1 and the Lennard-Jones diameter σ = 1.The simulations are Monte-Carlo in the constant NPTensemble and the potential is truncated at rtr = 2.5.This state point corresponds to roughly 20% undercool-ing of the liquid phase. Four phases are considered: asupercooled liquid with N = 864, a FCC crystal withN = 864, a HCP crystal with N = 1000 and a BCCcrystal N = 1024. The supercooled liquid is initializedby random insertion with an overlap separation rol = 0.9,and the crystals by particles on the lattice sites of a crys-tal filling the simulation box. These samples are used toassess the performance of the TCC in section VI, but firstwe compare the cut-off, standard and modified Voronoibond detection methods. Before proceeding we note thathere the stable phase is the FCC crystal, the others aremetastable to varying degrees. However, for our param-eters (system size and run time) we saw no change in thestate or crystallization to FCC in the metastable states.We choose to simulate one state point in this way so asto provide a fair a test as possible of the TCC’s abilityto detect different structures.

In Fig. 5(a) the radial distribution function for eachphase is plotted about the first minimum. The posi-tion of the first minimum for the liquid and BCC phases,rmin ≈ 1.5, is different to that of the FCC and HCP crys-tal phases, rmin ≈ 1.36. A single cut-off length does nottherefore determine the neighbors consistently betweenthe phases with respect to rmin. This may cause prob-lems if interfaces exist in the system, or if nucleation andgrowth is occurring63.

In Fig. 5(b) we plot the distribution of bond lengthsbetween neighbors for the under-cooled liquid as identi-fied by the different detection methods. If all particleswere neighbors the distribution of bond lengths is an un-normalized plot of 4πρr2g(r). All the data coincide upto the first peak indicating that these neighbors are de-tected by all the different methods for bond detection.The data for the simple cut-off method drop abruptly tozero at the cut-off length rc = 1.4, whereas the Voronoimethods show a smooth transition to zero. The standardVoronoi method gives the largest number of neighbors intotal and includes a number of pairs with separationsgreater than rmin = 1.5 for this phase.

The effect of only considering direct Voronoi neighborsas bonded with the modified Voronoi method is clearwhen comparing the fc = 1.0 data to that of the stan-

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1.0 1.2 1.4 1.6 1.80

1

2

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4Liquid

frequ

ency

x10

5

bond length

rc=1.4Voro++fc=1.0fc=0.95fc=0.9fc=0.85fc=0.82fc=0.8fc=0.75

1.0 1.2 1.4 1.6 1.8 2.0

0

1

2

3

4

g(r)

r

BCCFCCHCPLiquid

(a) (b)

Figure 5. Four phases of a Lennard-Jones system comparing neighbor detection methods. (a) The radial distribution functionsfor four phases of a Lennard-Jones system at state point T = 0.92 and P = 5.68. Note that the position of the first minima ofg(r) for the FCC and HCP phases is different from that of the BCC and liquid phases. A single cut-off length cannot determinethe neighbors consistently across all the phases. (b) Frequency of bond lengths for different neighbor detection routines. Thesimple cut-off method is the dotted line (rc = 1.4) and the standard Voronoi method is the black line

61. The colored linesare for the modified Voronoi method. The effect of reducing fc is to remove neighbor pairs that have large separations (longbonds) from the standard Voronoi neighbors.

dard Voronoi method. The difference between the graphsindicates that the indirect Voronoi neighbors have rela-tively long bonds. The effect of reducing fc is to furtherreduce the number of neighbors with longer separations.The ideal choice for fc will be a function of the structuredetection method that is being utilized. Typically clusterdetection algorithms, such as the TCC, are sensitive tothe inclusion of “long bonds” as these may cause misde-tection of structures. BOO parameters on the other handare less sensitive to the inclusion of long bonds provid-ing second nearest neighbors are not included. HoweverBOO parameters are themselves sensitive to fluctuationsthat lead to bonds being broken and reformed64. A valuefc = 0.82 for the TCC method was proposed, based ona study of the hard sphere fluid56. We also employ thisvalue in the Lennard-Jones test case in section V. Thechoice for fc is discussed further in the sections below.

IV. THE TOPOLOGICAL CLUSTERCLASSIFICATION ALGORITHM

The topological cluster classification algorithm identi-fies clusters within a bulk system that are defined by theminimum energy clusters of m isolated particles inter-acting with a pair-wise potential. The algorithm worksby searching a neighbor network for all three-, four- andfive-membered shortest path rings of particles. The ringsare categorized by the number of common neighbors toall particles in the ring, forming a set of “basic clusters”.Larger clusters are then identified as concatenations ofthe basic clusters and additional particles.

This brief summary of the TCC algorithm allows twoimportant features of the method to be highlighted. (i)Because the minimum energy clusters form the candidatestructures that are searched for within the bulk system

by the TCC algorithm, in the case that the bulk systemis one of the models incorporated in the TCC, qualita-tive relationships can be drawn between minimum energyclusters identified and the interactions of the particles inthe system. (ii) There is no fixed size for the structuralordering that is considered by the algorithm.

This section contains a description of the methodologyof the TCC algorithm. We proceed to define some piecesof terminology used repeatedly throughout the method-ology. If two sets of particles are being considered, forexample two clusters or two shortest-path rings, we saytwo particles are common if they are members of bothsets, and distinct or uncommon otherwise. Additionalparticles are particles which are not members of the im-mediate set under consideration.

A. Models

The TCC identifies minimum energy clusters of dis-tinct topology. Thus far, it incorporates a number ofmodels, as described here, but can be adapted to includeclusters whose bond topology is not yet included. Modelscovered include the variable range Morse potential (Fig.6), which reads

uM(r) = εMeρ0(σ−r)(eρ0(σ−r) − 2) , (9)

where εM is the depth of the potential well and σ is aparticle diameter. The range of the potential is set byρ0

65,66. For ρ0 = 6, the minimum energy clusters form ≤ 13 considered are identical to those of the Lennard-Jones model. We also include them = 6 minimum energycluster of the Dzugutov potential67,68. As shown in Fig.6, the Dzugutov potential is distinguished by a repulsion

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0.5 1.0 1.5 2.0

-1

0

1

2u(r)

r

Morse ρ0=3Morse ρ0=25Lennard-JonesDzugutov

Figure 6. Some potentials considered in the TCC. Black long-ranged Morse (ρ0 = 3.0), red short-ranged Morse (ρ0 = 25.0),green Lennard-Jones, and blue is the Dzugutov potential.

at around r = 1.5, introduced to suppress crystallisation.We also include two binary Lennard-Jones glassform-

ers, the Wahnström69 and Kob-Andersen70 models. Inthe Wahnström model two equimolar species of Lennard-Jones particles interact with a pair-wise potential,

uLJ(r) = 4εαβ

[(σαβrij

)12−(σαβrij

)6], (10)

where α and β denote the atom types A and B, andrij is the separation. The (additive) energy, length andmass values are εAA = �AB = εBB , σBB/σAA = 5/6,σAB/σAA = 11/12 and mA = 2mB respectively. TheKob-Andersen binary mixture is composed of 80% large(A) and 20% small (B) particles possessing the same massm70. The nonadditive Lennard-Jones interactions be-tween each species, and the cross interaction, are given byσAA = σ, σAB = 0.8σ, σBB = 0.88σ, �AA = �, �AB = 1.5�,and �BB = 0.5�.

B. Shortest-path rings

The starting point of the TCC algorithm is a neighbornetwork. The algorithm identifies all the shortest-pathrings of particles71. Shortest-path rings are closed loopsof bonded particles in the neighbor network where theshortest distance in terms of bonds between any two par-ticles in the ring can be achieved by traversing only bondsbetween the ring particles. Using more formal notation, ifthe neighbor network is a graph G consisting of verticesV representing the particles and edges E representingbonds between them, a shortest-path ring is defined asa subgraph g ⊆ G where for all particles i and j withing the equation dg(i, j) = dG(i, j) holds (dg/G(i, k) is theminimum number of bonds connecting i and j within thegraph). The TCC method considers shortest-path rings

(a) sp3 (b) sp4 (c) sp5

Figure 7. neighbor networks for shortest-path rings of (a)three, (b) four and (c) five particles. Particles are denoted byblack spheres and the bonds connecting them by black lines.

of three, four and five particles. The neighbor networksfor these shortest-path rings are shown in Fig. 7. Thethree-, four- and five-membered shortest path rings aredenoted sp3, sp4 and sp5 respectively.

Shortest-path rings consisting of six or more particlesare rarely found in systems with short-range isotropicinteractions, unless the particle size disparity is ratherhigher than the systems we consider (section IV A)72.Thus so far the TCC is limited to five-membered rings.It would be possible to extend the algorithm for systemswith larger shortest-path rings, such as network liquidsand glasses, and patchy particles.

The algorithm used to detect the shortest-path rings issimilar to that described by Franzblau71 and is depictedin Fig. 8. Starting from the neighbor network G, a loopproceeds over all the particles, index i. The three-, four-and five-membered clusters containing i are generatedby the backtracking method73. A depth-first search pro-ceeds over the nb(i) neighbors of i with index j > i. Ifj has a neighbor that is also bonded to i and with indexk > j we have found an sp3 ring. The search then pro-ceeds through the neighbors of j and subsequently backthrough indices j then i. If no sp3 ring is identified forindex combination ijk, the depth-first search proceeds toseek out sp4 rings ijkl where j < l, or sp5 rings ijklmwhere j < m. If no sp4 or sp5 ring is found the depth-first search halts and returns to the next neighbor j of i.The inequality conditions stated for the particle indicesensure that each ring is detected only once when usingthis algorithm. The statistics of shortest-path rings havebeen used as measures for local structure in their ownright, for example in silica glasses71,74 and hard spherecrystals53.

C. The basic clusters

The basic clusters are identified upon the shortest-pathrings. The shortest-path rings are divided into three fur-ther categories, denoted by spma, spmb or spmc, wherem is the number of particles in the ring. For spma clus-ters there are no additional particles in the system thatare bonded to all m members of the spm ring. For spmbthere is a single additional particle that is a neighbor of

9

Particle i

Found sp3 ring

ijk

Found sp4 ring

ijkl

Found sp5 ring

ijklm

Neighbour of iwith index j > i

Particle j

Particle k

Particle l

Particle m

Neighbour of jwith index k > i

Neighbour of kwith index l > i & l ≠ j

Neighbour of l withindex m > j & m ≠ k

k bonded to i with k > j

l bonded to i with l > j

m bonded to i

next k

next l

next m

next l

next k

next j

next i

next m

Figure 8. Schematic flow diagram depicting the algorithmused to detect three-, four- and five-membered shortest-pathrings.

all members for the spm ring. For spmc there are twoadditional neighbors for all the m ring particles. The ad-ditional particles bonded to the ring particles are termedspindle particles.

It is assumed that the maximum number of particlesthat can be bonded to any spm ring is two, and thatif we have an spmc cluster that the two additional par-ticles are located either side of the approximate planedefined by the shortest-path ring particles. These twoassumptions are non-trivial and depend on the nature ofthe interactions, the state point and the method used toderive the neighbor network. The assumptions generallyhold true to a good approximation if the interaction po-tential has a strongly repulsive core at short ranges suchthat there is a steric minimum distance for approach rstbetween the particles that is in practice not breached.The second requirement is that the maximum length ofthe bonds is only a small multiple of rst. The validity ofthese conditions is a function of the interaction potentialand the method to detect the neighbors. It is prudentto check that the number of common neighbors to thering particles is not frequently more than two in order tohave confidence that the results obtained from the TCC

analysis have physical significance.For the spmc clusters, no condition is specified as to the

bonding between the spindles, i.e. they may be bondedor not bonded. In practice the spindles are rarely bondedfor sp3c clusters, rarely not bonded for sp5c, and bondedin around half of all instances of sp4c clusters.

In total nine basic clusters are defined, as shown in Fig.9. The basic clusters include the first five minimum en-ergy clusters of the Morse potential consisting of three ormore particles. We follow the naming scheme of Doye etal.66 and term the minimum energy clusters as 3A trimer(sp3a), 4A tetrahedron (sp3b), 5A triangular bipyramid(sp3c), 6A octahedron (sp4c) and 7A pentagonal bipyra-mid (sp5c). These are also the minimum energy clustersfor the Lennard-Jones model.

D. Compound clusters

Larger clusters are formed of one or more particlesbound to one or more basic clusters, or of collectionsof basic clusters that are bonded to each other. We termthese compound clusters. Here and in the appendix thedetection routines for compound clusters that are iden-tified starting from the basic clusters are described. Foreach compound cluster a number of conditions are givenfor detection. These conditions are expressed in termsof how the constituent particles, either spindle, ring, oradditional, are connected by bonds. The conditions arein general satisfied by a number of subtly different bondnetworks that we term the TCC bond networks for aminimum energy cluster. The TCC bond networks aresimilar to the bond network of the minimum energy con-figuration of the cluster. Detecting structure in terms ofmultiple bond networks allows for some degree of ther-mally induced distortion of the cluster in the bulk relativeto the minimum energy structure.

The aims for a detection routine for a compound clus-ter are that it is (i) accurate in detecting a given typeof order and (ii) simple to implement computationally.If the conditions that a routine imposes for a particularcluster to exist are overly restrictive, thermal fluctua-tions may mean that detection of the structure in a bulksystem is rare. Likewise if the conditions are not suffi-ciently prescriptive, the order may be detected spuriouslyor conflicting types of order detected in the same locality.

For the first of the stated aims, the accuracy of de-tection will be a function of the system under study, thestate point and the method used to detect the bonds.The following three general principles are followed rela-tive to the minimum energy bond network in order tomaximize the accuracy of the routines:

1. The shortest-path rings of particles in basic clustersmust lie approximately in a plane in the minimumenergy configuration of the cluster.

2. The bonds that are required for the structure to ex-ist should all be of similar in length in the minimum

10

sp3

sp5

sp4

a b c

3A 4A 5A

6A

7A

Figure 9. The basic clusters identified from shortest-path rings. If a basic cluster is a minimum energy cluster for m Morseparticles its TCC name is shown below the cluster. Two representations are shown for the basic clusters: renderings of thecluster where ring particles are gray, bonds between ring particles are white and spindle particles are yellow, and neighbornetworks as in Fig. 7.

energy bond network.

3. The ratio between the shortest bond that if in placewould break the detection of the cluster to thelongest bond required for the cluster to exist shouldbe maximized (where the bond lengths are takenfrom the minimum energy configuration).

Simplicity of the routines is achieved by utilising thelargest sub-clusters where possible from the minimumenergy bond network for the cluster. If multiple sub-clusters are going to be required to detect the cluster,the overlap between the sub-clusters should be mini-mized. These guidelines minimize the number of con-ditions needed to detect the cluster.

A summary of how the detection routines for the com-pound TCC clusters are devised is as follows:

1. A configuration of particle positions for a clusteris found by identifying a global or local minimumof the potential energy landscape for a given po-tential. The minimisation is performed using theGMIN method45.

2. The bond network is found for this configurationusing a neighbor detection method. In the case ofthe Morse, Lennard-Jones and Wahnström models,this is the modified Voronoi method with fc = 0.82.The value fc = 1.0 is used for the Kob-Andersenmodel.

3. The basic clusters and other smaller compoundclusters are identified from within the bond net-work.

4. A set of sub-clusters that contain all the particles(perhaps with the inclusion of one or two addi-tional particles) is chosen and the detection routineis based upon these sub-clusters.

5. The sub-clusters chosen tend to be the largest avail-able in the bond network, whilst ensuring that thethree previously stated principles are obeyed asclosely as possible.

6. Conditions are imposed on the particles shared be-tween sub-clusters, the bonds required between thesub-clusters and additional particles for the struc-ture to exist. The bonds are selected using princi-ples 2 and 3 stated above.

Detection of clusters is hierarchical: smaller sub-clustersmust be detected prior to larger compound clusters.

The naming scheme for the TCC clusters follows fromDoye et al.66,75 for the minimum energy clusters of theMorse potential. Each name mX describes the numberof particles in the cluster (m) and a character X denotesthe potential for which the cluster is a minimum energyconfiguration. The characters X=A,B,C. . . indicate therange ρ0 of the Morse potential for which the cluster is aminimum energy, where A is for the longest ranges, andB, C, · · · indicate increasingly short range attractions(i.e. increasing ρ0). The shortest range Morse potentialconsidered in reference75 is ρ0 = 25.

The 6Z cluster is the minimum energy cluster for sixparticles of the Dzugutov potential68 and only a localminimum of the Morse interaction potential for 6 parti-cles. Its detection routine is included as it is a free energy

11

minimum cluster for six colloids with depletion-mediatedattractions37,76,77. Minimum energy clusters specific tobinary KA and Wahnström Lennard-Jones potentials aredenoted by characters “K” and “W” respectively.

E. Summary of clusters

A summary of all the TCC clusters is given in table I.The model used to generate a configuration for a min-imum energy cluster is listed. The detection methodfor each cluster is devised from the bond network whichcorresponds to the minimum energy cluster in isolation.This bond network necessarily depends on the methodused to detect the particles that are neighbors. It ispossible that if another neighbor detection method isemployed to detect the bond network of the minimumenergy cluster, that a different bond network would beobtained. It is therefore not guaranteed that the TCCalgorithm would identify a cluster if a neighbor detec-tion method is used that differs from the method used toidentify the network for the minimum energy cluster anddefine the TCC routine for the detection of the cluster.

In practice, if using the simple cut-off or modifiedVoronoi fc = 0.82, 1 methods, there are few cases when acluster would go undetected if using a neighbor networkdifferent to that used to define the detection algorithm.Specifically the Morse clusters 8A, 9A, 10A and 11B failto satisfy one or more of the conditions for the clustersto exist when additional long bonds are identified usingthe modified Voronoi method with fc = 1 on the mini-mum energy configuration. Conversely the KA clusters7K and 8K are not found using fc = 0.82 as the detectionroutine relies on the existence of bonds identified whenfc = 1.

We consider the stability of the TCC cluster detectionto fluctuations in bond lengths by examining the lengthsof the bonds in the minimum energy bond networks foreach cluster. Specifically we consider the lengths of theshortest rs and longest bonds rl that are required to ex-ist for the reference minimum energy configuration of acluster to be correctly detected as that structure. If theratio rl/rs is large a bond detection method that includesbonds longer than rl is necessary in order to successfullydetect the cluster.

As a rule of thumb, the Morse minimum energy clus-ters have rl/rs ≈ 1 meaning that the bonds utilized bythe TCC detection routines for these clusters are all ofsimilar length. Exceptions to this rule are the 11A, 11Band 12B clusters, where better detection performance isobtained by using a neighbor detection method that in-cludes bonds with a wider range of lengths, for examplethe modified Voronoi method with fc → 1. The KA andWahnström clusters also have relatively large ratios forrl/rs and a careful choice for the bond detection methodis necessary to determine structural order accurately rel-ative to the minimum energy clusters for these systems.We also consider the shortest length rb of any additional

0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

/N

fc

FCC-TCCHCP-TCCFCC-VFA(0,12)HCP-VFA(0,12)

C

Figure 10. Comparison of the detection of FCC and HCPcrystalline order with the TCC and VFA methods. The num-ber of FCC, HCP and VFA-(0,12) clusters is NC.

bond that if included in the bond network would not sat-isfy one of the conditions for the cluster to exist. Theratio rb/rs indicates the relative lengthscale of this addi-tional bond.

The ratio rb/rl contains information on the robustnessof the detection routine for each cluster. A large value ofrb/rl indicates that there is a large separation betweenthe longest bond required for the cluster to be detected,and the shortest bond that if included results in the clus-ter going undetected. Clusters with low values of rb/rl,for example 8A, 10W, 11B, 12A, and 12K, are difficultto identify reliably in thermal systems.

Some of the minimum energy clusters with detectionroutines implemented by the TCC have similar structuresto the clusters identified by the CNA and VFA methods.Specifically these are the clusters based around a pair ofparticles or are a central particle and its complete firstcoordination shell of neighbors. For these clusters themost similar CNA and VFA clusters are indicated in thefinal column of table I.

V. LENNARD-JONES FCC AND HCPCRYSTALS

We demonstrate the TCC algorithm on the simulationsof the four Lennard-Jones phases mentioned in sectionIII B. Fig. 10 shows the ensemble average of the numberof FCC and HCP crystal clusters NC versus modifiedVoronoi parameter fc for the FCC and HCP phases. Forcomparison the number of (0,12) clusters is shown for thetwo phases as obtained with the VFA method. For allvalues of fc the TCC algorithm finds more of the crystalclusters than the VFA method because the TCC detec-tion routines allow for a greater distortion of the order bythermal fluctuations. For both methods the number ofclusters tends to zero as fc → 1, because of the tendency

12

Cluster Model fc = 0.82 fc = 1 rl/rs rb/rs rb/rl CNA / VFA

3A ρ0 = 6 • • 14A ρ0 = 6 • • 15A ρ0 = 6 • • 1 CNA-23316A ρ0 = 6 • • 1 1.41 1.41 CNA-1441 / 24416Z ρ0 = 6 • • 1 1.63 1.637A ρ0 = 6 • • 1.01 1.62 1.61 CNA-15517K KA mA = 4 (7A) • 1.26 1.78 1.41 CNA-15518A ρ0 = 6 • 1.02 1.28 1.268B ρ0 = 6 • • 1.01 1.64 1.628K KA mA = 5 • 1.29 1.91 1.48 CNA-16619A ρ0 = 6 • 1.01 1.41 1.49B ρ0 = 6 • • 1.01 1.62 1.599K KA mA = 6 • • 1.26 1.75 1.3910A ρ0 = 6 • 1.01 1.41 1.410B ρ0 = 6 • • 1.03 1.65 1.5910K KA mA = 7 • • 1.29 1.76 1.3710W Wahn mA = 9 • • 1.16 1.49 1.28 VFA-(0,0,9)11A ρ0 = 3 • • 1.27 1.79 1.41 VFA-(0,2,8)11B ρ0 = 3.6 • 1.18 1.3 1.11 VFA-(0,2,6,2)11C ρ0 = 6 • • 1.06 1.67 1.5811E ρ0 = 6 • • 1.03 1.65 1.611F ρ0 = 6 • • 1.01 1.42 1.4111W Wahn mA = 8 • • 1.24 1.78 1.44 VFA-(0,1,6,3)12A ρ0 = 2.5 • 1.2 1.36 1.13 VFA-(1,0,6,4)12B ρ0 = 6 • • 1.07 1.69 1.5812D ρ0 = 6 • • 1.04 1.66 1.612E ρ0 = 6 • • 1.01 1.42 1.4112K KA mA = 8 • • 1.26 1.4 1.1213A ρ0 = 6 • • 1.05 1.7 1.62 VFA-(0,0,12)13B ρ0 = 14 • • 1.02 1.43 1.41 VFA-(0,10,2)13K KA mA = 7 • • 1.29 1.76 1.36FCC Bulk FCC • 1 1.41 1.41 VFA-(0,12)HCP Bulk HCP • 1 1.41 1.41 VFA-(0,12)9X Bulk BCC • • 1.15 1.63 1.41

Table I. Clusters detected by the TCC algorithm. The second column contains details for the reference potential used to makea configuration of the minimum energy cluster. mA denotes the number of A-species particles in the minimum-energy clusterin the case of the binary systems. The bullets in the fc columns indicate if the TCC algorithm successfully detects the clusterin its minimum configuration. The lengths rs, and rl are the shortest and longest bonds required in the minimum energy bondnetwork in order that the TCC algorithm successfully detects the cluster. The length rb is for the shortest bond that couldform and result in the cluster no longer being detected. The final column contains suggestions for CNA and VFA clusters withstructure similar to the TCC clusters.

of the bond detection method to over-estimate the truenumber of nearest neighbors by including long bonds.The number of clusters identified also tends to zero forlow values of fc as the bond detection method does notfind all of the nearest neighbors. The optimal value offc for best detection of the crystalline order across bothphases and methods is close to the value of fc = 0.82

originally proposed by SRW56.

VI. FOUR PHASES OF THE LENNARD-JONESSYSTEM

We proceed to consider the detection of structure inall four phases of the Lennard-Jones system – BCC, FCC

13

--

BCCFCCHCPLiquid

r ij -

r ijC

UC/

NC

- U/N

NC/

N

5A 6A 7A 8B 9B 10B 11C 12B 13A FCC HCP 9X

1.0

0.8

0.6

0.4

0.2

0.0

0.4

0.2

0.0

-0.2

-0.45A 6A 7A 8B 9B 10B 11C 12B 13A FCC HCP 9X

5A 6A 7A 8B 9B 10B 11C 12B 13A FCC HCP 9X

0.01

0.00

-0.01

Cluster

Cluster

Cluster

(a)

(b)

(c)

Figure 11. Detection of the Lennard-Jones minimum energy clusters with the TCC algorithm. The bond detection methodis modified Voronoi method with fc = 0.82. (a) Fraction of particles detected within each cluster type. (b) Potential energydifference of particles within each cluster type. (c) Deviation of bond lengths between particles in each cluster type r̄Cij fromthe mean bond length r̄ij .

and HCP crystals, and liquid – for fc = 0.82. Here thestate point is the same as that in section III B, thus thestable phase is FCC and the liquid is supercooled. InFig. 11(a) the fraction of all particles detected withineach cluster type is shown for each phase, where NC isthe number of particles detected to be within each clustertype (as opposed to the number of clusters NC).

We begin by analysing the detection of crystalline FCCand HCP clusters in their respective bulk phases, as thisoffers a test for identification of phases with known local

structure. All particles are found within FCC and HCPclusters, i.e. NC/N ≈ 1, for the respective crystallinephases. We note that NC ≤ NC for FCC and HCP clus-ters (cf. Fig. 10), as the former measure is equivalent tocounting the number of particles at the centre of the crys-tal clusters while the latter also includes particles in theshells of crystal clusters even if they are not themselvesat the centre of another crystal cluster.

For the FCC crystal phase, 6A and 9X clusters are de-tected in large quantities. This result is expected as these

14

clusters form part of the minimum energy structure of theFCC crystal. Similarly 5A and 6A are part of the min-imum energy HCP crystal structure. The 5A and otherclusters found in trace quantities are not part of the min-imum energy FCC crystal structure. These clusters arisedue to thermal fluctuations of the particle positions cre-ating transient bonds between particles that do not existin the minimum energy configuration. The 9X cluster isan example of this phenomenon for the HCP phase. Forthe BCC crystal phase 5A, 6A and 9X are structures thatare found universally throughout the phase as these arepart of the minimum energy crystal structure. Quantitiesof FCC and HCP are identified along with trace amountsof the other clusters due to thermal fluctuations.

In the liquid phase the 5A structure is most frequentlyseen and other structures are found in varying quanti-ties. The are trace amounts of FCC and HCP crystallineorder, and the 13A icosahedron is the least frequentlyseen structure. This result is contrary to Frank’s originalhypothesis46 and has been shown in earlier studies60,78–80

and is attributed to thermal fluctuations leading to dis-ruption of the bond network, which become more signif-icant for larger clusters. Analysing data following steep-est descent quenches to yield inherent structures wouldtherefore be expected to enhance the number of icosahe-dra identified. A sizeable fraction of particles within 9Xstructural environments are found in the liquid phase,confirming the statement in the appendix that this clus-ter cannot be used to uniquely distinguish crystalline or-der. Other clusters are seen in various quantities.

In Fig. 11(b) the difference in potential energy betweenparticles identified within each cluster type and the all-particle average is shown. The energy UC is the totalpotential energy of all NC particles identified within acertain cluster type. The energy UC/NC is therefore themean potential energy of the particles that participate inclusters of type C. This is compared to with the meanpotential energy of any particle, U/N .

For the FCC and HCP phases the formation of anystructural order not found in the minimum energy crys-tal by thermal fluctuations is associated with an increasein potential energy, relative to the average, for those par-ticipating particles. Trace amounts of FCC order areseen in the HCP phase and, although the FCC crystal isthe free energy minimum phase for this state point81, theincrease in potential energy arises due to the interfacialenergy penalty of the boundary introduced between thetwo crystalline structures.

For the BCC phase the formation of clusters not asso-ciated with either BCC, FCC or HCP crystalline orderresults in an increase in potential energy for the partic-ipating particles. When FCC and HCP order is formedthe participating particles see a drop in potential energy.This result is an indication of the greater stability of theFCC and HCP phases even at the expense of introducingan interface.

For the liquid phase, the particles that participate inclusters (of any type) have lower potential energy than

the mean. The exception is for the 5A structure wherealmost all the particles participate in that type of cluster(N5A/N ≈ 1), and UC/NC tends to U/N . There is amoderate anti-correlation between NC/N and the dropin potential energy for the minimum energy clusters (5Ato 13A).

Figure 11(c) displays the effect of participating in clus-ters on the local packing of particles. The length r̄ij isthe mean bond length of the neighbor network, and r̄Cij isthe mean length of bonds between particles participatingin a particular structure type. The difference r̄Cij − r̄ijindicates the change in bond lengths for particles form-ing a particular structural order. The FCC and HCPphases show an increase in bond lengths for structuresnot associated with the pure crystalline order for eachphase. This result implies that the formation of any non-crystalline order is associated with local fluctuations ofthe density. As the mean bond length in uniform sys-tems is proportional to ρ−1/3 to a good approximation,the magnitude of any density fluctuation is necessarilysmall due to the low relative fluctuations in the meanlength between the regions of different structure. Forthe BCC phase the formation of clusters associated withFCC and HCP order results in a local increase in density.There is a local decrease in density for any order foundthat is not associated with crystallinity.

In the case of the liquid phase only very small changesin the mean bond lengths are seen on formation of theminimum energy clusters 5A to 13A. Again this indi-cates that the formation of this order is not associatedwith any local fluctuations in density, but instead causedby different orientational arrangements of the particlesbeing explored. There is an increase in local density onformation of FCC and HCP clusters, however the magni-tude of the increase is small compared to the difference indensities of the liquid and crystalline phases at this statepoint. Related behavior has recently been observed incrystal nucleation in hard spheres, in the formation of acrystal nucleus by structural ordering prior to an increasein density28.

For completeness Figs. 12 and 13 contain detection re-sults for all the other structures that are included in theTCC algorithm within the four Lennard-Jones phases.The results are broadly similar to those described abovefor the minimum energy clusters and the FCC, HCP and9X clusters.

VII. SUMMARY

This paper details the topological cluster classification(TCC) algorithm. This method identifies groups of par-ticles whose bond network is similar to that found inminimum energy clusters of a number of monatomic andbinary simple liquids. We have described the modifica-tion to the Voronoi construction such that 4-memberedrings are more robust to thermal fluctuation. These 4-membered rings are required for identification of a num-

15

BCCFCCHCPLiquid

r ij -

r ijC

UC/

NC

- U/N

NC/

N

6Z 7K 8A 8K 9A 9K 10A 10K 10W

1.0

0.8

0.6

0.4

0.2

0.0

0.4

0.2

0.0

-0.2

-0.4

0.01

0.00

-0.01

Cluster

Cluster

Cluster

(a)

(b)

(c) 0.02

6Z 7K 8A 8K 9A 9K 10A 10K 10W

6Z 7K 8A 8K 9A 9K 10A 10K 10W

--

Figure 12. Detection of the other m = 6 to 10 particle clusters with the TCC algorithm. Bond detection method is modifiedVoronoi with fc = 0.82.

ber of clusters. Then, the algorithm was described andits implementation for 33 different clusters was discussed.The TCC was then tested on a bulk one-componentLennard-Jones system in FCC and HCP crystal phaseswhere known local structure was subject to distortion bythermal fluctuations. The test then proceeded to a super-cooled liquid phase, where it was identified that particlesalign into structures with topologies similar to that ofa number of minimum energy clusters for the Lennard-Jones and other models. The particles that participatein these structures were found to reduce their potentialenergy by doing so, and that this occurred not due to

density fluctuations but to due to their local changes inarrangement.

Acknowledgements

We thank Peter Harrowell for introducing C. P. R.to the CNA and Hajime Tanaka for many stimulatingconversations. A. M. is funded by EPSRC grant codeEP/E501214/1. C. P. R. thanks the Royal Society forfunding. This work was carried out using the compu-tational facilities of the Advanced Computing Research

16

--

BCCFCCHCPLiquid

r ij -

r ijC

UC/

NC

- U/N

NC/

N

11A 11B 11E 11F 11W 12A 12D 12E 12K 13B 13K

1.0

0.8

0.6

0.4

0.2

0.0

0.4

0.2

0.0

-0.2

-0.4

0.01

0.00

-0.01

Cluster

Cluster

Cluster

(a)

(b)

(c)

11A 11B 11E 11F 11W 12A 12D 12E 12K 13B 13K

11A 11B 11E 11F 11W 12A 12D 12E 12K 13B 13K

Figure 13. Detection of the other m = 11 to 13 particle clusters with the TCC algorithm. Bond detection method is modifiedVoronoi with fc = 0.82.

Centre, University of Bristol.

∗ paddy.royall@bristol.ac.uk1 J.-P. Hansen and I. R. McDonald, Theory of simple liquids,

3rd ed. (Academic Press, 2006).2 J. D. Bernal, Nature (London), 183, 141 (1959).3 A. Rahman, J. Chem. Phys., 45, 2585 (1966).4 J. L. Finney, Proc. R. Soc. London, Ser. A, 319, 479

(1970).

5 J. D. Honeycutt and H. C. Andersen, J. Phys. Chem., 91,4950 (1987).

6 M. Tanemura, Y. Hiwatari, H. Matsuda, T. Ogawa,N. Ogita, and A. Ueda, Progress of Theoretical Physics,58, 1079 (1977).

7 H. Jónsson and H. Andersen, Phys. Rev. Lett., 60, 2295(1988).

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17

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19

sc1

sc2

sd1

sd2

(a) 6Z (b) 7K

rc1

rc2

sc

sd2

sd1

rc2rc1

Figure 14. The (a) 6Z and (b) 7A clusters. The two sp3 ringsof the 5A clusters are shown in white (5Ai) and blue (5Aj).

(a) 8A (a) 8B

s1

s2

rc1

rc2 rc3

rc4

rd1

rd2

s1

r1

r2

a1

Figure 15. The (a) 8A and (b) 8B clusters. For 8A the secondsp5 ring is shown in blue. For 8B a red additional particle isbonded to the labeled particles in the 7A cluster.

APPENDIX: DETECTION ROUTINES FOR THECLUSTERS

In the following the symbol s is used to denote the in-dex of a spindle particle in a basic cluster, r to denote aring particle, and a an additional particle. The subscripts“c” and “d” on a particle index denote common and dis-tinct particles respectively. Sub-clusters are referencedby the indices i, j, k, . . . .

m = 6 to m = 9 compound clusters

6Z. — The 6Z cluster is the minimum energy for sixparticles interacting with the Dzugutov potential68 [Fig.14(a)]. It is also a local minimum of the Morse potential.See table II for details.

7K. — The 7K cluster is the minimum energy of KALennard-Jones for seven particles [Fig. 14(b)]. In theminimum energy there are four of the larger A-speciesparticles. We give a detection routine based on the bondnetwork of the minimum energy cluster as detected whenusing the modified Voronoi method with fc = 1.0. Thecluster is similar to a 7A cluster, but one of the sp5 ringparticles is drawn towards the centre of the ring. Theratio of the longest to the shortest bond in neighbor net-work of the minimum energy cluster is 1.26. Both thesebond types must be in place in the bulk system for thecluster to be detected by the TCC algorithm. See tableII for details.

8A. — The 8A cluster is the minimum energy of theMorse potential for ranges ρ0 < 5.28 [Fig. 15(a)]. It con-sists of two sp5b/c clusters and three detection routines

(a) 8K (b) 9A

sc1 sc2

sc3rd3 rd2

rd1

rc1

rc2sd1 sd2

sd3

rc1

rc2

rd1 rd3

rd4rd2

Figure 16. The (a) 8K and (b) 9A clusters. Pink bondsindicate the shortest-path ring in the kth sub-cluster involvedin the detection of 8K and 9A.

(a) 9B (b) 9Krc1

rc2

rc1

rc2

scsd1 sd2

sc

sd1 sd2

rd1

rd2

rd3

rd4

rd1

rd2

rd3

rd4

Figure 17. The (a) 9B and (b) 9K clusters.

are necessary as detailed in Table II.8B. — The 8B cluster is the minimum energy of the

Morse potential for ranges 5.28 ≤ ρ0 < 25 [Fig. 15(b)].See table II for details.8K. — The 8K cluster is the minimum energy of KA

Lennard-Jones for eight particles, and contains five ofthe larger A-species particles [Fig. 16(a)]. See table IIfor details.9A. — The 9A cluster is the minimum energy for

the Morse potential for range parameter ρ0 < 3.42 [Fig.16(b)]. See table III for details.9B. — The 9B cluster is the minimum energy for the

Morse potential for ranges 3.42 ≤ ρ0 < 25 [Fig. 17(a)].See table III for details.9K. — The 9K cluster is the minimum energy of KA

Lennard-Jones for nine particles, where six of the parti-cles are the larger A-species [Fig. 17(b)]. See table III fordetails.10A. — The 10A cluster is the minimum energy for

the Morse potential for range parameter ρ0 < 2.28 [Fig.18(a)]. See table III for details.10B. — The 10B cluster is the minimum energy for the

Morse potential for ranges 2.28 ≤ ρ0 < 25 [Fig. 18(b)].The detection routine presented decomposes 10B as a 9Band a 7A cluster. The 10B cluster is the first compoundcluster where its detection routine relies on a sub-clusterthat is also a compound cluster (9B). It is equally validto formulate the detection of 10B in terms of three 7Aclusters if prior detection of 9B clusters is not of interest.See table III for details.10K. — The 10K cluster is the minimum energy of KA

Lennard-Jones mixture for ten particles, where seven ofthe particles are the A-species [Fig. 19(a)]. See table III

20

(a) 10A (b) 10B

sd2

sd1

sd2

sd1

sd3

rc1 rc2

rc3

rd1

rd2 rd3

rd1

rd2rd3

sc

Figure 18. The (a) 10A and (b) 10B clusters. For the 10Acluster the three labeled sp4 particles indicate the bondingbetween the two sp4b clusters.

(a) 10K (b) 10W

a

sc sc

Figure 19. The (a) 10K and (b) 10W clusters. (a) The com-mon spindle particle of 9K and the additional particle arelabeled. (b) The common spindle particle of the sp5b clustersconstituting 10W is shown.

for details.10W. — The 10W cluster is only a local minimum

of the Wahnström Lennard-Jones potential for ten par-ticles with nine A-species [Fig. 19(b)]. It is included forcompleteness such that all the energy minimum clustersfor each composition of mA A-species for m Wahnströmparticles are included in the TCC algorithm. See tableIII for details.

11A. — The 11A cluster is the minimum energy foreleven Morse potential for range parameter ρ0 < 3.40[Fig. 20(a)]. The minimum energy cluster for the Morsepotential ρ0 = 3 is considered to understand the nature

(a) 11A (b) 11B

sc

sd1

sd2

rd1

rd2 rd3

a1

a2

sc

rd1

rd2

rd3

rd4

Figure 20. The (a) 11A and (b) 11B clusters. (a) The bondingbetween sp4 ring particles in each 6A is shown by the labeledring particles. (b) The additional particle a1 is bonded to rd1and rd3, and a2 is bonded to rd2 and rd4.

(a) 11C (b) 11E

scsd1 sd2

rd1 rd2

rd3 rd4

rd5 rd6rc1

rc2

scsd1 sd2

sd3r

Figure 21. The (a) 11C and (b) 11E clusters. (a) Particle rd1is a neighbor of rd2, and rd3 is a neighbor of rd4. Particlesrd5 and rd6 are not neighbors.

of the bonds required to be in place in order that 11Ais detected by this routine. The ratio of the lengths ofthe longest bond to the shortest bond required is 1.27.The longest bonds are those bonds between the sp4 ringsof the 6A clusters. The shortest bonds are between thecommon spindle and the sp4 ring particles. Relativelyfew 11A clusters are detected when using a bond networkthat discounts long bonds, e.g. when using a short cut-offlength or a low value of fc, as thermal fluctuations causethe bonds between the sp4 ring particles to break andthe cluster to go undetected.

For the KA liquid the 11A cluster has previously beenidentified as relevant for the glassy behavior11,42,43,82.The KA minimum energy cluster is topologically equiva-lent to 11A, i.e. it has the same bond network, but the rel-ative lengths are different due to its composition in termsof eight A- and three B-species. The ratio of the longestto the shortest bonds is 1.23 and this is large enoughto cause poor detection of the 11A cluster when usingthe modified Voronoi method with fc = 0.82. Thereforethe value fc = 1.0 is used to determine the neighborsfor when studying the KA system42,82. See table IV fordetails.11B. — The 11B cluster is the minimum energy for

eleven Morse potential for ranges 3.40 ≤ ρ0 < 3.67 [Fig.20(b)]. See table IV for details.11C and 11D. — There are two minimum energy clus-

ters of eleven particles for the Morse potential for ranges3.67 ≤ ρ0 < 13.57, namely 11C and 11D [Fig. 21(a)].Both these clusters have identical minimum energy bondnetworks, so are not distinguished between by the TCCalgorithm. We term both clusters 11C. See table IV fordetails.11E. — The 11E cluster is the minimum energy for

eleven Morse particles with the range of the potential in13.57 ≤ ρ0 < 20.60 [Fig. 21(b)]. See table IV for details.

11F. — The 11F cluster is the minimum energy foreleven Morse particles with the range of the potential in20.60 ≤ ρ0 < 25 [Fig. 22(a)]. See table V for details.

11W. — The 11W cluster is the minimum energy ofWahnström Lennard-Jones mixture for eleven particles[Fig. 22(b)]. Nine of the particles are the larger A-species.See table V for details.

12A. — The 12A cluster is the minimum energy for

21

(a) 11F (b) 11Wst1 st2

sb1 sb2

rcrd1 rd2 sc

a

Figure 22. The (a) 11F and (b) 11W clusters. (a) The whiteand blue sp3 rings are indicate the 5A clusters, and the pinkand green sp4 rings indicate the 6A clusters. Only the spindlesof the 5A clusters are highlighted in yellow. (b) The additionalneighbor a of the common spindle sc in the 10B cluster ishighlighted.

(a) 12A (b) 12B

a

sc

rd5 rd6sc

sds1

s2 s3

s4

s5

Figure 23. The (a) 12A and (b) 12B clusters. (a) The ad-ditional particle is bonded only sc, rd5 and rd6. (b) Five 7Aclusters have spindles sc and s1 to s5, the latter being bondedto sd of the central 7A cluster.

twelve Morse particles with the range of the potential inρ0 < 2.63 [Fig. 23(a)]. See table V for details.

12B/C. — There are two minimum energy clustersof twelve particles for the Morse potential for ranges2.63 ≤ ρ0 < 12.15, namely 12B and 12C [Fig. 23(b)].Both these clusters have identical minimum energy bondnetworks and so cannot be distinguished between by theTCC algorithm. We term both clusters 12B. See table Vfor details.

(a) 12D (b) 12E

sd2

sd3sd1

sc

rc1

rc2st3

st1

st2

sb1sb2sb3

rd1rd2

Figure 24. The (a) 12D and (b) 12E clusters. (a) The 7Acluster added to the 11E cluster has its sp5 ring highlightedwith green bonds. (b) The additional 5A cluster to the 11Fcluster has its sp3 ring highlighted with white bonds and itsspindles are st3 and sb3.

(a) 12K (b) 13A

rd1

rd2

rd3

a

sc

sd

sd2

Figure 25. The (a) 12K and (b) 13A clusters. (a) The addi-tional particle a is bonded to particles rd1, rd2 and rd3 in thesp4 rings of the 11A cluster.

(a) 13B (b) 13K

sc

sd1

sd2

st1

sb1sb2

st2 rca1

a2

Figure 26. The (a) 13B and (b) 13K clusters. (a) Each sp5ring particle from one 7A clusters is bonded to a single sp5ring particle in the other 7A cluster. (b) The sp3 rings of theadditional 5A clusters are not shown.

12D. — The 12D cluster is the minimum energy fortwelve Morse particles with the range of the potential in12.15 ≤ ρ0 < 17.08 [Fig. 24(a)]. See table V for details.12E. — The 12E cluster is the final Morse cluster for

twelve particles [Fig. 24(b)]. It is the minimum energyfor ranges 17.08 ≤ ρ0 < 25. See table V for details.12K. — The 12K cluster is the minimum energy of KA

Lennard-Jones mixture for twelve particles, where eightof the particles are the larger A-species [Fig. 25(a)]. Seetable VI for details.13A. — The 13A cluster is the minimum energy for

thirteen Morse particles with the range of the potentialin ρ0 < 14.76 [Fig. 25(b)]. This cluster is topologi-cally equivalent to the icosahedral cluster discussed byFrank46. See table VI for details.13B. — The 13B cluster is the minimum energy for

thirteen Morse particles with the range parameter of thepotential in 14.76 ≤ ρ0 < 25 [Fig. 26(a)]. See table VIfor details.13K. — The 13K cluster is the minimum energy of

KA Lennard-Jones mixture for thirteen particles, whereseven of the particles are the larger A-species [Fig. 26(b)].See table VI for details.

CRYSTAL CLUSTERS

In order to detect crystalline structure and to test theperformance of the TCC algorithm against phases with

22

(a) FCC (b) HCP

sd1

sd2

sd3

sc

sb1sb2 sb3

st1

st3st2

rc

Figure 27. The (a) FCC and (b) HCP clusters. (a) The six-membered ring consists of the gray particles in the horizontalplane around sc. (b) The six-membered ring consists of thegray particles in the horizontal plane around rc.

9X

sc

Figure 28. The 9X cluster. Each particle in the sp4 rings isbonded to a single particle in the other sp4 ring.

known structure, detection routines are implemented forcrystalline clusters.FCC. — The m = 13 FCC cluster is a particle and

its first-nearest neighbors as taken from a FCC crystallattice [Fig. 27(a)]. See table VII for details.HCP. — The m = 13 HCP cluster is a particle and its

neighbors taken from a HCP crystal lattice [Fig. 27(b)].See table VII for details.9X. — The 9X cluster is found in the minimum energy

FCC and BCC crystal lattices [Fig. 28]. However the sizeof the cluster means that it does not uniquely determinecrystalline order (in the same manner that tetrahedralorder, although found in all bulk crystallin