Structure and dynamics of model colloidal clusters with short-range attractions
aa r X i v : . [ c ond - m a t . s o f t ] J a n Structure and dynamics of model colloidal clusters with short-range attractions
Robert S. Hoy ∗ Department of Physics, University of South Florida, Tampa, FL, 33620 (Dated: May 10, 2018)We examine the structure and dynamics of small isolated N -particle clusters interacting via short-ranged Morse potentials. “Ideally preprared ensembles” obtained via exact enumeration studies ofsticky hard sphere packings serve as reference states allowing us to identify key statistical-geometricalproperties and to quantitatively characterize how nonequilibrium ensembles prepared by thermalquenches at different rates ˙ T differ from their equilibrium counterparts. Studies of equilibriumdynamics show nontrival temperature dependence: nonexponential relaxation indicates both glassydynamics and differing stabilities of degenerate clusters with different structures. Our results shouldbe useful for extending recent experimental studies of small colloidal clusters to examine bothequilibrium relaxation dynamics at fixed T and a variety of nonequilibrium phenomena. I. INTRODUCTION
Understanding how varying the shape and strength ofa pair potential affects the energy landscape and dynam-ics of systems composed of several particles interactingvia that potential lies at the heart of theoretical clusterphysics [1]. Variable-shape potentials are of particularutility in understanding common features of apparentlydisparate systems. For example, varying the dimension-less range parameter αD of the Morse potential U Morse ( α ; r ) = ǫ [exp( − α ( r − D )) − − α ( r − D ))] , (1)yields accurate models for clusters formed by constituentsranging from alkali-metal atoms to buckyballs to micron-sized colloids [2]. Studies of colloidal clusters are partic-ularly valuable in this context since individual particlepositions can be tracked. Most valuable are “model” sys-tems with precisely controllable interparticle interactionsand cluster size N . These systems are a veritable play-ground for studies of few-body statistical mechanics, andcan (through the universality evident in cluster physics)provide insights into the behavior of their more micro-scopic counterparts.Manoharan and collaborators have recently attractedgreat interest by characterizing the structure and dynam-ics [3, 4] of colloids interacting via hard-core-like repul-sive and (variably) short-ranged attractive interactions.While published experimental studies and related theo-retical modeling [2, 3, 5–13] of these systems have fo-cused on equilibrium phenomena, rapid advances in ex-perimental particle-tracking techniques [4, 14, 15] suggestthat much of their nonequilibrium physics may soon be-come experimentally observable. For example, the room-temperature transition rate between the two degenerateground-state clusters (GSC) of N = 6 particles is of or-der 10 − − s − . Since their longest relaxation timesshould increase dramatically with increasing N and de-creasing temperature T , it seems plausible that these ∗ Electronic address: [email protected] model colloidal systems could soon be utilized for fun-damental studies of nonequilibrium few-body statisticalmechanics.In this paper, we provide theoretical guidance for suchstudies by elucidating the statistical-geometrical proper-ties and several key equilibrium and nonequilibrium phe-nomena in small ( N ≤
13) clusters that mimic the sys-tems studied in experiments [3, 4, 16]. First we performexact-enumeration studies that extend the work of Refs.[5–8] by obtaining all minimally [17] mechanically sta-ble packings of N ≤
13 sticky hard spheres. Then we usethe “ideally prepared ensembles” of ground-state clustersgenerated by these studies as initial conditions for molec-ular dynamics simulations of N -particle model colloidalclusters. These simulations focus on identifying note-worthy features in their equilibrium relaxation dynam-ics and their preparation-protocol-dependent, nonequi-librium structure that should be observable in particle-tracking experiments.Our key results are that: (i) the fraction of “off-pathway nuclei” that are mechanically stable yet incom-patible with close-packed crystallization grows rapidlywith increasing N ; (ii) fast temperature quenches pro-duce ensembles retaining memory of equilibrium ensem-bles at higher T , e.g. favoring structures that are morestable against excitation because they lie in deeper en-ergy wells: and (iii) systems exhibit nonexponential re-laxation indicative of both glassy dynamics and differingstabilities of degenerate clusters with different structures.In addition to being directly relevant for experimentalstudies of small clusters, these results may also improveour understanding of the role such clusters play in con-trolling kinetic arrest in bulk systems. II. MODEL AND METHODS
The interaction potential for sticky hard spheres withdiameter D is [18]: U ss ( r ) = ( ∞ , r < D − ǫ , r = D , r > D, (2)where ǫ is the energy at contact. A key feature of stickyhard-sphere clusters is that their isoenergetic, isocon-tacting states are in general highly degenerate. Theset of all possible arrangements of N hard spheres with N c pair contacts consists of M ( N, N c ) nonisomorphic“macrostates.” M is an integer for isostatic ( N c = 3 N − ≡ N ISO ) and hyperstatic ( N c > N −
6) clusters [5, 19]wherein each sphere contacts at least 3 others, and dif-ferent macrostates have different “shapes”, i.e. distinctsets of interparticle distances { r ij } ( i, j ∈ [1 , N ]) thatcorrespond to distinguishable inherent structures [20].We determine M ( N, N c ) and find the structure of eachmacrostate using an updated version of the numericalprocedure described at great length in Ref. [8]. The maindifferences are that here: (a) we consider adjacency ma-trices { ¯ A } of arbitrary rather than “polymeric” topol-ogy; (b) rather than performing a sequential [8] passover all distinct { ¯ A } , we (following Arkus et. al. [5, 6])use NAUTY [21] to generate complete sets of noniso-morphic { ¯ A } . Note that (a) precludes the possibility offailing to detect clusters that do not possess Hamilto-nian paths, and that implementing (b) yields an orders-of-magnitude decrease in the computer time (relative tothat reported in [8]) required to perform exact enumera-tion of M ( N, N c ).Systems interacting via sticky-hard-sphere potential(Eq. 2) are well known to exhibit anomalous thermody-namics [22, 23]. In order to simulate the T -dependentstructure and dynamics of “model” (but realistic) col-loidal clusters, a continuous and finite-ranged interactionpotential must be introduced. We perform MD simu-lations using a modified Morse potential U MM ( r ) withshape and range (Figure 1) similar to the effective inter-actions between colloids in systems with micellar deple-tants [3, 4]; U MM ( α, b ; r ) = ( U Morse ( α ; r ) − c ( α, b )1 + c ( α, b ) , r ≤ r c ( α, b )0 , r > r c ( α, b ) . (3)The structure and dynamics of Morse clusters with large αD are contact-dominated [2]. In particular, rearrange-ments can be understood in terms of contact breakingand reformation. However, defining “contact” is am-biguous for potentials that decrease smoothly to zero.One advantage of using U MM ( r ) rather than U Morse ( r ) isthat it facilitates contact identification and concomitantanalyses of transitions between macrostates; F MM ( r ) = − dU MM /dr remains finite at r c , allowing us to definecontact as finite-force interaction. The shift/stretchterm c ( α, b ) is defined to make U MM continuous at r = r c , i.e. c ( α, b ) ≡ U Morse ( α ; r c ( α, b )). We define b to produce a well controlled approximation in which lim b →∞ c ( α, b ) = 0 and hence lim b →∞ U MM ( α, b ; r ) = U Morse ( α ; r ). Choosing r c ( α, b ) /D = 1 + b ( r ∗ − | dU Morse /dr | is maximal at r ∗ /D = ( α + log(2)) /α , gives c ( α, b ) = − [4 − b (2 b +1 − ǫ .Here we study systems with αD = 150 and b = α/ (30 log(2)), yielding r c ( α, b ) /D = 31 /
30. We haveverified both that this U MM is long-ranged enough toavoid the thermodynamic and dynamic anomalies thatare known to arise in the α → ∞ “Baxter” limit [22], andthat replacement of U Morse (150; r ) with this U MM hasminimal effects on the structural and dynamic propertiesof interest here. Our results should thus be scalable toboth larger α and smaller α using (for example) the “ge-ometrical” free energy landscape techniques of Holmes-Cerfon et. al. [24] or the Noro-Frenkel extended law ofcorresponding states [25]. A preliminary attempt at ap-plying the latter method is reported in the Appendix. FIG. 1: Standard (red) and modified (blue) Morse potentialsfor αD = 150. The inset highlights differences between U Morse and U MM for r ≃ r c . For r < ∼ r ∗ ≃ . D , U MM and U Morse are essentially indistinguishable.
Another advantage of using this U MM ( r ) is that itallows us to use well-defined “ideally prepared ensem-bles” (IPE) as initial conditions for our MD simula-tions. We define IPE as follows: Suppose a given po-tential has M ( N ) strain-free, energetically degenerate N -particle ground state clusters (GSC) with permuta-tional entropies ω k . Statistical mechanics predicts thatthe equilibrium population fraction of each GSC at T = 0is ω k / Ω, where Ω = Ω( N ) ≡ M ( N ) X i =1 ω k ( N ) . (4)An IPE is an ensemble of molecules containing all of(and only the) M ( N ) GSC, such that the populationfraction of every GSC is equal to ω k / Ω. Our exact-enumeration studies yield the structures of these GSC[26]; values of ω k are obtained by evaluating the sym-metry of their associated adjacency matrices [6]. We useIPE of N m = f ( N )Ω( N ) N -particle molecules as initial( T = 0) conditions for MD. Here f ( N ) is chosen to be suf-ficiently large to give good statistics yet sufficiently smallfor computational tractability; for the N = 13 systemsstudied below we employ f (13) = 1 / N m (13) = Ω(13) / f ( N ) = 1simply corresponds to multiplying a system’s partitionfunctions by a constant; its value should not (apart fromstatistical error) alter any results.MD simulations are performed using an in-house codethat employs per-cluster parallelization. All particleshave mass m and diameter D . Each cluster is confinedto a cubic cell with hard reflecting walls and side length L ( N ) chosen to give a particle number density ρ in the di-lute limit [27]: here ρ = N/L ( N ) = . D − . Thus whileall particles in a given molecule interact via U MM ( r ),different clusters do not interact with each other. Thischoice of simulation protocol and boundary conditions ismotivated by the experiments [3, 4], which also examinedensembles of isolated N -colloid systems in dilute solution.MD integration is performed using the velocity-Verlet al-gorithm with a timestep δt = . τ /α , where the unit oftime is τ = p mD /ǫ [28]. Temperature is controlledusing a strong Langevin thermostat (with damping time τ Lang = τ ) that mimics the strong damping experiencedby colloids in a solvent. Comparing to experimental val-ues ǫ ≃ . D = 1 µm , and m ≃ − kg [4] gives τ ≃ − s . Our simulations extend as long as 2 . · τ ;this maps to 25 s , which is comparable to the duration of atypical experiment [4]. In Section III, all energies, times,and temperatures are respectively expressed in units of ǫ , τ , and ǫ/k B .To set up our studies, IPEs are heated from T = 0to T = 2 . T eqh = 10 − /τ . “Snapshots” from this heating run aretaken at various T = T i and are further equilibrated atthese T i ; these equilibrated samples are used as initialconditions for our studies of fixed- T dynamics. We char-acterize dynamical relaxation phenomena at these T byexamining the traversal of clusters through their variousGSCs using the function f mad ( t ) = 1 N m N m X j =1 h R ( F j ( t ′ ) , F j ( t ′′ − t ′ ); t ′ , t ′′ ) i . (5)Here f mad ( t ) is the probability that a randomly cho-sen cluster will not execute a transition to a differentmacrostate within a time interval t . It is calculated bytracking the structure of each cluster over an “experimen-tal” time interval t = t ′′ − t ′ , and then averaging resultsover all clusters and all “start times” t ′ . In Equation 5, F j ( t ′′′ ) is the index of the macrostate in which the j thcluster resides at time t ′′′ . The self-correlation function R ( A, B ; t ′ , t ′′ ) = 1 if A = B for all internediate times t ′′′ between t ′ and t ′′ , and zero otherwise. Thus f mad ( t )decays monotonically from one to zero as the ensembleof clusters transition out of their initial states.Preparation-protocol-dependence studies are per-formed by taking the T = 2 . T = 2 . τ inorder to obtain a thoroughly equilibrated high- T fluidstate, and cooling systems back to T = 0 at three rates: | ˙ T | = 10 − /τ, − /τ , and 10 − /τ . During these coolingruns we monitor such quantities as the potential energy of clusters U = N − X i =1 N X j = i +1 U MM ( r ij ) (6)and the population fractions of clusters F k /N m that cor-respond to each GSC. The latter are identified by com-paring their adjacency matrices (assuming particles i, j contact if r ij < r c ) to those of the N c = N maxc packings.In all cases, finite- T structures correspond either to ex-actly one zero-temperature GSC, or to an excited statewith N c < N maxc . III. RESULTS
The bulk ground states of the sticky-hard-sphere po-tential (Eq. 2) are the (infinitely degenerate) set S formedby all possible stackings of perfect hexagonal planes intoa close-packed crystal. Local ordering within these statesmay be FCC, HCP, or mixed FCC/HCP. Barlow pack-ings [29] are finite- N “grains” (subsets) of any mem-ber of S . Since they correspond to “on-pathway” nucleithat can grow into defect-free members of S , they areexpected [8] to be be critical to understanding crystalnucleation and growth in systems with hard-core-like re-pulsions and short-range attractions. It is important tofind all such nuclei that can form (as opposed to thosethat do form under specific conditions); this is most con-veniently achieved via exact enumeration of sticky-hard-sphere packings.In Table I, we report the total number of macrostates M , as well as the the numbers of macrostates M X possessing structural features X such as Barlow order,stacking faults, and five-fold symmetric defects. Thelatter three structural motifs are shown in Figure 2(a-c), preclude Barlow order, and thus correspond to “off-pathway” nuclei incompatible with close-packed crystal-lization. Here M X ( N, N c ) = M ( N,N c ) X k =1 G k ( X ) , (7)where G k ( X ) is 1 if structure of the k th macrostatematches the pattern X and 0 otherwise.We find that the fraction of macrostates possessingBarlow order increases rapidly with increasing hyper-staticity H = N c − N ISO , where isostatic packings have N ISO = 3 N − N considered here, many packings retain non-Barlow orderfor H as large as three. Many of these possess stackingfaults; M stack − fault decreases with increasing H but re-mains nonzero for H up to three. Fivefold-symmetricmotifs are highly prevalent in isostatic packings, andwhile their prevalence decreases rapidly with increasing H , they are still relevant motifs in these more-stable,lower-energy nuclei. TABLE I: Numbers of macrostates M , macrostates with Bar-low order M Barlow , stack faults M stack − fault , and fivefold-symmetric substructures M fivefold . Results include both me-chanically stable and floppy packings. Stable packings cor-respond to zero-dimensional points in configuration space.Floppy packings occupy finite “volumes” in configurationspace [24], but we have verified that all reported here aredisconnected from one another, and thus are “macrostates”as defined above. Results for for N ≤
11 were reported inRef. [8], and values of M agree with those reported in Ref.[13]. N N c M M
Barlow M stack − fault M fivefold
12 30 11638 339 8420 665712 31 174 77 88 1612 32 8 4 4 012 33 1 1 0 013 33 95799 1070 69897 5326513 34 1318 363 859 24813 35 96 42 46 813 36 8 5 3 0 (a) (b) (c)
FIG. 2: Ordered and disordered motifs in monodisperseSHS packings. (Top) A stack-faulted structure (a), and twofivefold-symmetric structures (b-c). (Bottom) The populationfractions of packings with Barlow order ( f Barlow ; red), five-fold order ( f fivefold ; green), and stacking faults ( f stack − fault ;blue). Line types are solid for isostatic, dashed for H = 1,dotted for H = 2, and dash-dotted for H = 3. The abovementioned trends are further reinforced byconsidering the fractions f X of microstates with thesemotifs: f X ( N, N c ) = Ω − M ( N,N c ) X k =1 ω k G k ( X ) , (8)where { ω } and Ω are given by Equation 4. Note that f X is the fraction of clusters in IPEs possessing motif X . Figure 2 (bottom panel) shows f Barlow , f stack − fault and f fivefold for 7 ≤ N ≤
13 and 0 ≤ H ≤
3. Notably, f Barlow for isostatic nuclei decreases monotonically withincreasing N to only about 1% for N = 13. This meansthat 99% of the highest-energy mechanically stable N =13 nuclei are off-pathway, and nucleation of structureswith Barlow order is likely to be a rare event. While f Barlow is far higher for hyperstatic (
H >
0) nuclei, thesame trend of decrease with increasing N persists.Most non-Barlow nuclei possess stacking faults or five-fold defects; for isostatic nuclei with 8 ≤ N ≤ f stack − fault and f fivefold are in the 50 −
80% range.While they decrease sharply with increasing H , theystill increase in hyperstatic systems to large values withincreasing N . Both stack-faulted and fivefold symmet-ric structures are known to play key roles in inhibit-ing crystallization in bulk particulate systems by pro-moting dynamical arrest and glass formation [30, 31].Since the energy barriers for transitions between off-pathway and Barlow-ordered nuclei are generally large[12, 24], the very low values of f Barlow and high val-ues of f stack − fault and f fivefold reported here provide apotential microscopic explanation for the propensity ofsticky-hard-sphere-like systems to jam and glass-form inboth simulations and experiments (e.g. [23, 31, 32]).In the remainder of this paper, we focus on N = 13clusters, and in particular on their nucleation and growthduring cooling from high T to T = 0, as well as on theirrelaxation dynamics at fixed T . The top panel of Figure3 shows the eight degenerate GSCs for N = 13. Twoare core-shell structures (respectively HCP- and FCC-ordered) wherein a single center sphere contacts twelveneighbors, and the rest are irregularly shaped Barlowand stack-faulted clusters. Labels above the structuresindicate ordering (FCC, HCP, Barlow, or stack-faulted)and numbers below them indicate their relative permu-tational entropies (ratios of their ω k ) in the IPE.The left-bottom panel of Figure 3 shows results for theevolution of the average molecular energy h U ( T ) i duringcooling from T = 2 . T = 0 with quench rates | ˙ T | that vary over a factor of 100. Results for all | ˙ T | fallon a common curve above T melt ≃ . T dynamics are very fast. Below T melt , h U i begins to drop,indicating the onset of cluster formation. For the lowertwo | ˙ T | , as T continues to decrease, h U i drops sharplyas clusters grow and merge, then flattens out as particlescoalesce into single clusters. A narrow range of small (cid:10) ∂ U/∂T (cid:11) indicates a T regime where clusters have co-alesced bur continue exploring their energy landscape via hcp fcc sf Barlow sf Barlow Barlow sf1 4 12 8 24 8 24 48 FIG. 3: Top: The eight N = 13 , N c = 36 ground states. States 1-8 are depicted from left to right. Labels above the structuresindicate ordering (FCC, HCP, Barlow, or stack-faulted) and numbers below them indicate their relative permutational entropies(ratios of their ω k ). Bottom: Results from MD simulations for N = 13 Morse clusters: (Left) Mean cluster energy vs. reducedtemperature for fast, medium, and slow quench rates from top to bottom. (Middle) The population fractions ( F i /N m ) ofground states (1-8) and excited states ((1 − P i =1 F i ) /N m ) over the course of a slow quench (here F i is the number of clustersin macrostate i .). Line colors match those of the circles on the left edge, which indicate these states’ equilibrium populationfractions ( ω i / Ω) at T = 0. (Right) Temperature dependence of cluster relaxation dynamics in thermodynamic equilibrium, asmeasured by the macrostate decorrelation f mad ( t ) (Eq.5). inter-macrostate transitions. h ∂U/∂T i converges as clus-ter rearrangement ceases and clusters proceed down theharmonic basins of their energy landscapes. However, h U i /ǫ remains above −
36 = − N maxc even at T = 0,indicating that many clusters freeze into mechanicallystable excited states rather than GSCs. Results for thefastest quench rate (10 − ) are markedly different: h U ( T ) i decreases much more gradually and remains well above − ( N ISO ) ǫ even at T = 0, indicating that systems oftenfreeze into multiple clusters (that do not merge by theend of the cooling runs) rather than single clusters [33].The middle-bottom panel shows the population frac-tions of the GSCs and of excited states as a function of T during the | ˙ T | = 10 − /τ quench. Even at this low coolingrate, about 2% of clusters remain in (mechanically sta-ble) excited states at T = 0. The left edge of this panelcompares the values of F i /N m at the end of the quenchto their equilibrium T = 0 counterparts ( ω i / Ω from theIPE). The FCC and HCP clusters populate the quenchedensemble in excess at low T because they form at slightlyhigher T , and as described below, rearrange more slowly.Conversely, the other clusters’ populations are somewhatlower than equilibrium predictions, showing that for thisslow quench rate, clusters inhabiting deep, narrow wellson the potential energy landscape are favored, that is,on-pathway crystal growth is favored.Higher quench rates (not shown) reverse these trends. Clusters are more likely to freeze into less-ordered statesthat are favored at high T because of their larger vibra-tional entropy [3], and deviations of the final populationfractions from equilibrium T = 0 values are much larger.To understand these results, it is useful to recall thatthe key parameter controlling the growth of ordered crys-talline nuclei is the ratio of the particle attachment rate r a to the cluster reorganization rate r r [34]. When r a /r r is large, the larger entropy [3, 8] of disordered(yet mechanically stable) nuclei lacking close-packed or-der should promote growth of amorphous clusters. Con-versely, when r a /r r is small, enthalpy should rule, andclose-packed nuclei should experience stable growth.Our results are consistent with and reinforce theseideas. For our fastest quenches, systems often freeze intomultiple clusters because | ˙ T | > r a even at high T . In con-trast, for | ˙ T | = 10 − /τ , the sharp, first-order-like dropin h U ( T ) i is characteristic of the | ˙ T | ≪ r a regime wheresingle clusters form within a narrow range of T ≃ T melt ,and the rest of this curve is consistent with | ˙ T | remain-ing above r r down to the T at which h ∂U/∂T i converges.Results in the middle-bottom panel illustrate how r r grows with decreasing T and increases well beyond | ˙ T | at T ≃ . r r varies with T and macrostate isone key to developing principles for controlled nonequilib-rium self-assembly of these systems. Towards this end,we now turn to examining their equilibrium relaxationdynamics. The right-bottom panel of Figure 3 showsresults for the decorrelation f mad ( t ) of macrostates viastate-to-state transitions (Eq. 5). Results are shown fora range of temperatures over which characteristic r r varyby several orders of magnitude. At high T , excitationsfrom GSCs are very common, energy barriers are easilyovercome, and relaxation is nearly exponential. As T decreases, clear shoulders develop in f mad ( t ), and relax-ation becomes very clearly non-exponential. One reasonfor this is that different GSCs possess different stabil-ity (i.e. lie in potential energy wells of different depths),and so decay at different rates, i.e. possess different r r .Highly ordered N = 13 clusters such as HCP and FCCnuclei (states 1-2) are most stable, and have the lowest r r , because every atom in these clusters is bonded to atleast five others. In contrast, states 7-8 have a “loose”atom possessing only three bonds, and rearrange muchfaster. Another potential reason for the complex shapesof f mad observed at lower T is that short-ranged Morseclusters possess glassy dynamics [2]; this will be furtherexamined in forthcoming work. IV. DISCUSSION AND CONCLUSIONS
In this paper, we characterized the equilibrium andprepraration-protocol-dependent structure and dynam-ics of small clusters interacting via hard-core-like repul-sions and short-range attractions. Our results provide atheoretical framework for extending recent experimentalstudies [3, 4, 16] of small colloidal clusters to examineboth equilibrium relaxation dynamics at fixed T and avariety of nonequilibrium phenomena. In particular, theyshould be relevant to understanding the factors control-ling nonequilibrium self-assembly of such clusters, andshould be testable using plausible extensions of currentlyavailable experimental techniques [4, 14, 15].We extended recent exact enumeration studies of stickyhard sphere packings [5–8] to N = 13. This is an impor-tant advance because N = 13 clusters can form completecore-shell structures (i.e. HCP and FCC crystallites); ourwork will aid experimental studies of core-shell structureswhere observation of the inner-core particles is difficult.We then employed these complete sets of packings as“ideally-prepared-ensemble” (IPE) initial conditions forMD simulations of colloids interacting via a short-rangedmodified Morse potential, focusing on N = 13 clusters.It is important to note that the results presented hereare strictly valid only for systems interacting via “steep”(short-ranged) pair potentials. Softer, longer-ranged in-teractions dramatically alter the lower regions of smallclusters’ energy landscapes [12, 35]. However, the short-ranged limit considered here is experimentally accessible,e.g. in systems of micron-sized colloids and micellar de-peletants [4]. To aid experimental tests of our results,we include an Appendix containing a Noro-Frenkel anal-ysis [25] that can be used for mapping them to systems interacting via other pair potentials.We gratefully acknowledge Miranda Holmes-Cerfon forsharing preliminary results for N ≥
12 packings [13, 36],and Miranda Holmes-Cerfon, David Wales, and PaddyRoyall for helpful discussions.
Appendix A: Noro-Frenkel Analysis
Our results can be used to make predictions for sys-tems interacting via other short-ranged pair potentials -including experimental systems (see e.g. Ref. [32]) - us-ing Noro and Frenkel’s extension [25] of the law of cor-responding states. Both thermodynamical and dynami-cal results can be effectively compared by “temperature-matching” different systems at the same value of “freevolume concentration” c p = πρσ eff / B ∗ ( T ) = 32 σ eff ( T ) Z r c ( a,b )0 [1 − exp ( − U MM ( r ) /k B T )] r dr. (A1)Here the temperature-dependent effective hard-sphere di-ameter [37] is σ eff ( T ) = Z [1 − exp ( − U MM ( r )] dr. (A2)Values of σ eff ( T ) and B ∗ ( T ) for the temperatures ex-amined in the lower-right panel of Fig. 3 are given inTable II. The variation of B ∗ with T is small becausefor the steep, short-ranged interaction potential U MM used in this study, the integrand in Eq. A1 is close tounity except in a very narrow range δr ∼ ( r c −
1) about r = 1. However, our study of dynamical relaxation inequilibrium systems suggests that the timescales as wellas the character of relaxation in real systems with simi-larly short-ranged interactions can vary very sharply overa narrow range of B ∗ . Future work will consider widerranges of N , c p and B ∗ in order to allow comparison topublished results for phenomena such as dynamical ar-rest in individual clusters [16, 38] and bulk systems [32],as well as guiding future experiments. TABLE II: Values of σ eff ( T ) and B ∗ ( T ) (Eqs. A1-A2) for thetemperatures examined in the lower-right panel of Fig. 3. T σ eff B ∗ [1] D. J. Wales, Energy Landscapes: Applications to Clus-ters, Biomolecules and Glasses (Cambridge MolecularScience, 2004).[2] F. Calvo, J. P. K. Doye, and D. J. Wales, Nanoscale ,1085 (2012).[3] G. Meng, N. Arkus, M. P. Brenner, and V. N. Manoha-ran, Science , 560 (2010).[4] R. W. Perry, G. Meng, T. G. Dimiduk, J. Fung, andV. N. Manoharan, Faraday Discuss. , 211 (2012).[5] N. Arkus, V. N. Manoharan, and M. P. Brenner, Phys.Rev. Lett. , 118303 (2009).[6] N. Arkus, V. N. Manoharan, and M. P. Brenner, SIAMJ. Discrete Math. , 1860 (2011).[7] R. S. Hoy and C. S. O’Hern, Phys. Rev. Lett. , 068001(2010).[8] R. S. Hoy, J. Harwayne-Gidansky, and C. S. O’Hern,Phys. Rev. E , 051403 (2012).[9] D. J. Wales, ChemPhysChem , 2491 (2010).[10] S. J. Khan, O. L. Weaver, C. M. Sorensen, andA. Chakrabarti, Langmuir , 16015 (2012).[11] C. L. Klix, K. Murata, H. Tanaka, S. R. Williams, A. Ma-lins, and C. P. Royall, Sci. Rep. , 2072 (2013).[12] J. W. R. Morgan and D. J. Wales, Nanoscale , 10717(2014).[13] M. Holmes-Cerfon; http://arxiv.org/abs/1407.3285.[14] J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoha-ran, J. Quant. Spect. Rad. Trans. , 2482 (2012).[15] J. Fung and V. N. Manoharan, Phys. Rev. E , 020302(2013).[16] A. Malins, S. R. Williams, J. Eggers, H. Tanaka, andC. P. Royall, J. Phys. Cond. Matt. , 425103 (2009).[17] D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. ,4051 (1995).[18] S. B. Yuste and A. Santos, Phys. Rev. E , 4599 (1993).[19] M. R. Hoare and J. McInnes, Faraday Discuss. Chem.Soc. , 12 (1976).[20] F. H. Stillinger, Science , 1935 (1995).[21] B. D. McKay and A. Piperno, J. Symbolic Computation , 94 (2013).[22] G. Stell, J. Stat. Phys. , 1203 (1991).[23] G. Foffi, E. Zaccarelli, F. Sciortino, and P. Tartaglia, J. Stat. Phys. , 363 (2000).[24] M. C. Holmes-Cerfon, S. J. Gortler, and M. P. Brenner,Proc. Natl. Acad. Sci. USA , E5 (2013).[25] M. G. Noro and D. Frenkel, J. Chem. Phys. , 2941(2000).[26] This procedure is simple only for potentials with short-ranged attractive tails. For example, U MM is sufficientlyshort-ranged for α > α conv , where α conv as the in-teraction range such that r c ( α, b ) is greater than allGSC’s minimum second-nearest-neighbor distance r for α < α conv and less than r for α < α conv . For N ≤ α conv ( N ) < , 760 (2011).[28] We have verified that this value of δt is sufficiently smallby checking that the velocity autocorrelation function isinsensitive to varying δt over the range [ . τ /α, . τ /α ]in systems at k B T /ǫ = 1.[29] W. Barlow, Nature , 186 (1883).[30] F. C. Frank, Proc. Roy. Soc. London. Ser. A , 43(1952).[31] C. P. Royall, S. R. Williams, T. Ohtsuka, and H. Tanaka,Nature Mat. , 556 (2008).[32] P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield,F. Sciortino, and D. A. Weitz, Nature , 499 (2008).[33] The h U ( T ) i > − N ISO ǫ condition may be understood byexamining the maximum number of pair contacts pos-sessed by multiple clusters. For example, two clusters of N and N atoms with N + N ≤
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