The effect of attractions on the local structure of liquids and colloidal fluids
Jade Taffs, Alex Malins, Stephen R. Williams, C. Patrick Royall
aa r X i v : . [ c ond - m a t . s o f t ] J a n The effect of attractions on the local structure of liquids andcolloidal fluids
Jade Taffs
School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK
Alex Malins
School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK andBristol Centre for Complexity Science,University of Bristol, Bristol, BS8 1TS, UK
Stephen R. Williams
Research School of Chemistry, The AustralianNational University, Canberra, ACT 0200, Australia
C. Patrick Royall
School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK [email protected] ∗ (Dated: August 22, 2018) Abstract
We revisit the role of attractions in liquids and apply these concepts to colloidal suspensions.Two means are used to investigate the structure; the pair correlation function and a recentlydeveloped topological method. The latter identifies structures topologically equivalent to groundstate clusters formed by isolated groups of 5 ≤ m ≤
13 particles, which are specific to the systemunder consideration. Our topological methodology shows that, in the case of Lennard-Jones, theaddition of attractions increases the system’s ability to form larger ( m ≥
8) clusters, although pair-correlation functions are almost identical. Conversely, in the case of short-ranged attractions, paircorrelation functions show a significant response to adding attraction, while the liquid structureexhibits a strong decrease in clustering upon adding attractions. Finally, a compressed, weaklyinteracting system shows a similar pair structure and topology.
PACS numbers: 82.70.Dd; 82.70.Gg; 64.75.+g; 64.60.My . INTRODUCTION Among the cornerstones of our understanding of the structure of bulk simple liquids isthat it is dominated by the repulsive core. This leads to the idea that hard spheres form asuitable basic model of the liquid state. The liquid pair structure may then be accuratelycalculated using, for example the Percus-Yevick closure to the Ornstein-Zernike equation forhard spheres, and treating the remainder of the interaction as a perturbation .Although in principle colloidal dispersions are rather complex multicomponent systems,the spatial and dynamic asymmetry between the colloidal particles (10 nm-1 µ m) and smallermolecular and ionic species has enabled the development of schemes where the smallercomponents are formally integrated out . This leads to a one-component picture, whereonly the effective colloid-colloid interactions need be considered. The behaviour in theoriginal complex system may then be faithfully reproduced by appealing to liquid statetheory and computer simulation . Since the shape of the particles is typically spherical,and the effective colloid-colloid interactions may be tuned, it is often possible to use modelsof simple liquids to accurately describe colloidal dispersions.In colloidal systems, due to the mesoscopic length- and longer time-scales, one mayalso determine the structure in real space in 2D and 3D at the single particle level usingoptical microscopy and optical tweezers . This may be done with sufficient precision thatinteraction potentials can be accurately determined both for purely repulsive systems and for systems with attractive interactions .It has been conjectured as far back as the 1950s that the structures formed by clustersof small groups of particles in isolation might be prevalent in liquids . More recently ithas been demonstrated that for spherically symmetric attractive interactions, the structureof clusters of size m > . This brings a natural question: if the structures defined by these clusters areindeed prevalent in liquids, and they depend upon the range of the interaction, then mightliquids with differing interaction ranges exhibit differing cluster populations? Moreover,while removing the attractive component of Lennard-Jones [Fig. 2(b)] has little effect onthe pair structure , what is the effect on any cluster population?We have recently developed a novel means to identify structure in simple liquids. Inisolation, small groups of particles form clusters with well-defined topologies. These have2 igure 1: (color online) Clusters found in bulk systems using the topological cluster classification. For m ≤ , where m is the number of particles in a cluster, all studied ranges of theMorse potential equation (3) form clusters of identical topology. In the case of larger m the clustertopology depends on the interaction range. Here we follow the nomenclature of Doye et. al. where A corresponds to long-ranged potentials and B. ... to minimum energy clusters of shorter-rangedpotentials. been identified for the Lennard-Jones potential and for the Morse potential, which hasa variable range . We identify clusters relevant to the Lennard-Jones and Morse Poten-tials in bulk liquids, with a method we term the Topological Cluster Classification (TCC) .Here we use this scheme as a highly sensitive probe of the liquid structure. It is our inten-tion to use the TCC to investigate possible differences in structure between the Lennard-Jones liquid and that resulting from the repulsive part of the Lennard-Jones interaction, theWeeks-Chandler-Andersen (WCA) potential . Although we have argued that some colloidalliquids are well described by a short-ranged Morse potential , the structure of clustersof adhesive hard spheres has very recently been shown to exhibit some degeneracy, withmultiple cluster topologies having the same number of bonds in the limit of short-ranged3 a) (b) long-rangedMorse (c) Lennard-Jones short-rangedMorse
Figure 2: (color online) Interaction potentials used. (a) Long-ranged potentials: Morse (dark green)and truncated Morse (bright green) with range parameter ρ = 4 .
0. (b) Lennard-Jones (red) andWCA (pink). (c) Short-ranged potentials: Morse (blue) and truncated Morse (turquoise) withrange parameter ρ = 25 .
0. Dashed cyan line in (c) denotes the hard sphere interaction. σ EF F denotes the effective hard sphere diameter as defined in equation (7) and listed in table I. attractions . However, minimising the second moment (or radius of gyration) of clustersof hard spheres shows a strong similarity with the short-ranged Morse system . OurTCC methodology has some similarities to the common neighbour analysis introduced byAndersen , however here we focus on clusters rather than bonds.Since the tunability of colloidal systems allows a wide range of potentials to be realised,including long-ranged interactions relevant to metals , we also consider long-ranged(Morse) potentials, in addition to the Lennard-Jones interaction and short-ranged Morsepotential, along with their purely repulsive counterparts. We further compare with hardspheres. In these systems, we study the groups of particles topologically equivalent toground state clusters found in isolation.The canonical model of colloid-polymer mixtures of Asakura and Oosawa, assumes hardsphere colloid-colloid and colloid-polymer interactions, while the polymer-polymer interac-tion is ideal . A one-component description , accurate for small polymer-colloid sizeratios leads to a hard core with a short-range attraction. We have recently shown that,for the parameters we shall consider here, the continuous Morse potential provides a reason-ably accurate description of this system . Meanwhile, longer interaction ranges correspondto metals and purely repulsive long-ranged interactions are relevant to soft matter sys-tems such as charged colloids, star polymers , star polyelectrolytes and colloidal microgel4articles.This paper is organised as follows. In section II we describe the simulation methodologyand our approach for comparing different interaction potentials, our results are presentedin section III and we conclude with a discussion in section IV. Our main results can besummarised as follows. Although Lennard-Jones shows very little change in the radial dis-tribution function g ( r ) upon adding attractions, the topology is significantly altered: addingattractions promotes the formation of larger clusters. Conversely, short-ranged systems showthe opposite behaviour: adding attractions strongly decreases clustering, while the first peakof g ( r ) shows some increase upon adding attractions. II. SIMULATIONS AND INTERACTION POTENTIALS
We use standard Monte-Carlo (MC) simulation in the NVT ensemble with N = 2048particles. Each simulation run is equilibrated for 2 × MC moves and run for up to afurther 10 moves. In all cases, we confirmed that the system was in equilibrium on thesimulation timescale by monitoring the potential energy. A. Interaction potentials
We seek to compare systems with different interactions, under similar conditions. Weeks,Chandler and Andersen provided a protocol by which the Lennard-Jones potential couldbe compared with the so-called WCA potential (Lennard-Jones without attractions). Thepair interaction u ( r ) is separated into two parts: u ( r ) = u ( r ) + w ( r )where r is the separation between particles, u ( r ) is the reference (repulsive) interaction and w ( r ) is the perturbative attraction. In the Lennard-Jones case, βu LJ ( r ) = 4 βε LJ (cid:20)(cid:16) σr (cid:17) − (cid:16) σr (cid:17) (cid:21) (1)where β = 1 /k B T where k B is Boltzmann’s constant and T is temperature. Here ε LJ = 1 /T is the well depth. WCA thus define the reference potential as5 u W CA ( r ) = βε LJ h(cid:0) σr (cid:1) − (cid:0) σr (cid:1) i + βε LJ for r ≤ / σ, r > / σ. (2)2nd order perturbation theories allow accurate prediction of the pair structure. However,here we are interested in a particle based analysis that probes the structure at a levelbeyond the two body distribution function and restrict ourselves to the interactions givenby equations (1) and (2).In the case of the longer and shorter ranged interactions, we use the Morse potentialwhich reads βu M ( r ) = βε M e ρ ( σ − r ) ( e ρ ( σ − r ) −
2) (3)where ρ is a range parameter and βε M is the potential well depth. We set ρ = 25 . ρ = 4 . βu T M ( r ) = βε M e ρ ( σ − r ) ( e ρ ( σ − r ) −
2) + βε M for r ≤ σ, r > σ, (4)This truncated Morse potential is thus similar to hard spheres for ρ = 25 . / σ for WCA and σ for the Morsepotential. In the case of the attractive systems, we truncate and shift both Lennard-Jonesand Morse ( ρ = 25 .
0) at 2 . σ and the long-ranged Morse ( ρ = 4 .
0) at 4 . σ . B. Comparing different systems
We have outlined a means by which we can compare systems with and without attraction,by removing the attractive part of the interaction. In order to match state points betweensystems with differing interaction ranges, we use the extended law of corresponding statesintroduced by Noro and Frenkel . Specifically, this requires two systems to have identicalreduced second virial coefficients B ∗ where 6 ∗ = B πσ EF F (5)where σ EF F is the effective hard sphere diameter and the second virial coefficient B = 2 π ∞ Z drr [1 − exp ( − βu ( r ))] . (6)The effective hard sphere diameter is defined as σ EF F = ∞ Z dr [1 − exp ( − βu REP ( r ))] (7)where the repulsive part of the potential u REP is described above in section II A. Thus wecompare different interactions by equating B ∗ and σ EF F . The latter condition leads to aconstraint on number density ρ EF F = N πσ EF F V (8)where V is the volume of the simulation box. We fix ρ EF F to a value equivalent to theLennard-Jones triple point ( ρ LJ = 0 .
85) throughout. In the case of hard spheres, this valueis ρ HS ≈ . φ HS = πρ HS / ≈ . φ is the packing fraction. Details of statepoints investigated are given in table I. C. The Topological Cluster Classification
To analyse the structure, we first identify the bond network using a modified Voronoiconstruction with a maximum bond length r c = 1 . σ and four-membered ring parameter f c = 0 .
82 (TCC) paper . Having identified the bond network, we use the TCC to determinethe nature of the cluster. This analysis identifies all the shortest path three, four and fivemembered rings in the bond network. We use the TCC to find clusters which are globalenergy minima of the Lennard-Jones and Morse potentials. We identify all topologicallydistinct Morse clusters, of which the Lennard-Jones clusters form a subset (the Lennard-Jones and Morse interactions are similar in the case that the range parameter ρ = 6 . > . We therefore consider ground state clusters for each system and,separately, calculate all topologically distinct Morse clusters for m <
14. For more detailssee .To compare the various fluids we study here, we proceed as follows. Comparing systemswith and without attractions, we consider the ground state clusters (of the attractive sys-tem). If a particle is a member of more than one cluster, it is taken to ‘belong’ to the largercluster. Thus, the total cluster population ≤ N the total number of particles. However,when we seek to compare different potentials, we need to account for the fact that thesemay have different ground state clusters. If the particle is part of two clusters which aredifferent in size, we choose to count it as the larger cluster, but if the particle is part of twoclusters of the same size, it is counted as the cluster corresponding to the shorter-rangedinteraction. In this case, the total number of particles counted as belonging to a cluster canexceed the number of particles in the simulation. D. Systems studied
The different systems considered are listed in Table I. In addition to the state point( ε, ρ ), we list the reduced second virial coefficient B ∗ and effective hard sphere diameter σ EF F . Some comments upon the use of clusters in the case of repulsive systems are in order.Clearly, isolated clusters require cohesive forces, however here we compare the WCA andthe truncated and shifted Morse potential with their cohesive counterparts and we assumeit is appropriate to consider the same clusters. Given the similarity of the truncated Morsepotential to hard spheres [Fig. 2(c)], it is instructive to include these also.We are motivated to consider the Lennard-Jones triple point, as we expect clusters to bemore prevalent at lower temperature. However, mapping the short-ranged Morse potential( ρ = 25 .
0) to the Lennard-Jones triple point leads to a system unstable to crystallisation.We found that at higher temperature the Morse system was stable (on the timescales ofthese simulations) against crystallisation for βε M = 2 .
0. Thus we compare high-temperatureLennard-Jones (T=2.284) with Morse ρ = 25 . ρ = 4 .
0. 8 able I: State points studied. LJ high temp. and triple correspond to the two temperatures atwhich Lennard-Jones and WCA simulations were carried out. Trunc. Morse denotes the truncatedMorse interaction (equation 4).System B ∗ βε ρ σ EF F
LJ high temp. -0.2325 0.4447 0.9776 0.9839WCA high temp. 2.013 0.4447 0.9776 0.9839Morse ρ = 25 . ρ = 25 . ρ = 4 . ρ = 4 . III. RESULTS AND DISCUSSIONA. The Lennard-Jones Triple Point: Long-ranged interactions
We take as our starting point for the analysis of these data the result that for denseliquids, the WCA potential readily captures the pair structure of the Lennard-Jones liquid .The radial distribution functions g ( r ) for the Lennard-Jones liquid at the triple point andthe corresponding WCA system are plotted in Figure 3(b). The effectiveness of WCA indescribing the pair structure is clear. The same observation holds for the longer-ranged( ρ = 4 .
0) Morse and truncated Morse systems shown in Figure 3(a).Turning to the TCC analysis, in the WCA-Lennard-Jones system we see a somewhatdifferent story, as shown in Fig. 4(b). Note the logarithmic scale in this plot: clusterpopulations vary over three orders of magnitude, clear relative differences are seen betweenLennard-Jones and WCA. Unlike the g ( r ) which are very similar, there is a clear trendin the cluster populations N c /N . Larger clusters are more prevalent in the Lennard-Jonessystem, compared to the WCA. The differences are emphasized in Fig. 5(b) which plots theratio of the cluster populations in Fig. 4(b). One might argue that smaller clusters mayreadily be formed simply by compressing spheres. The addition of attractions promotes the9 (cid:136) Figure 3: (color online) Pair-correlation functions. (a) Long-ranged potentials: Morse ρ = 4 . βε M = 0 . βε LJ = 1 .
471 (the triple point). formation of larger clusters which require more organisation and co-operativity. We remarkthat the difference in structure revealed by the TCC is rather significant, given that theradial distribution functions are so similar. For example, there is a twofold difference inthe triangular bipyramid 5A, one of the most popular clusters. As for the 13A icosahedron,its population is quadrupled by adding attractions. Note also that in these equilibriumliquids, we find a small but measurable number of particles with local crystalline topology,even though there is no sign of splitting in the second peak of g ( r ) (Fig. 3), which is oftentaken to be a sign that the liquid is close to crystallising . Of the smaller clusters, the 7Apentagonal bipyramid is found in limited quantities. However, it is found also as part of allthe larger clusters which form minima for the Lennard-Jones except HCP and FCC so ourcounting methodology counts some 7A ( D h ) particles as members of larger clusters.The longer-ranged Morse ( ρ = 4 .
0) system on the other hand shows very little differenceupon adding attractions. Due to its softness, the long-ranged Morse system has a rathersmall value of σ EF F (table I). Thus, matching Lennard-Jones requires a higher density,which leads to some overlap of the particles. For example the mean inter-particle spacing d m = ρ − / ≈ . σ . The Morse potential has its minimum located at σ . However, such isthe long ranged nature of Morse ρ = 4 . . σ remains within the attractive10 b Figure 4: (color online) Population of particles in a given cluster. N c is the number of particlesin a given cluster, N the total number of particles sampled. Here we consider only ground stateclusters for each system. (a) Morse ( ρ = 4 .
0) (dark green) truncated Morse (bright green). (b)Lennard-Jones at the triple point (red) and corresponding WCA (pink). Note the semi-log scale. well, although there is some compression. Thus both with and without attractions, thesystem is compressed, which may dominate the local structure. Furthermore, the value of βε M = 0 . C v ) if it is a member of more than one m = 11 clusters of which one is an 11F. While this may inflate the populations of such11 b Figure 5: (color online) Ratio of cluster populations in systems mapped to the Lennard-Jones triplepoint. (a) Morse and truncated Morse ( ρ = 4 .
0) and (b) Lennard-Jones and WCA. These plotthe same data as Fig. 4 expressed to emphasize the difference between the systems. clusters, we argue that systems with differing ground states are compared in an unbiasedway. One result of considering all Morse clusters is that Lennard-Jones has by far the largestnumber of 13A icosahedra, although the population of the 13B decahedral cluster is larger.
B. High-Temperature systems : Short-ranged interactions
For shorter-ranged interactions relevant to colloid-polymer mixtures , to avoid crystalli-sation we used an attractive well depth of βε M = 2 .
0, which corresponds via equation (6)to a Lennard-Jones well depth of βε LJ ≈ . ρ = 25 . reduced population of par-ticles in a locally crystalline environment upon adding attractions (Fig. 8). However, weare unaware of an equilibrium phase diagram for the Morse ρ = 25 . g ( r ) is rather higher in the case of the Morseinteraction, compared to the purely repulsive truncated Morse and hard-sphere interactions.We note also that there is little difference between the truncated Morse and hard sphere12 oth both both M both both both bothLJ LJM Figure 6: (color online) Population of particles in a given cluster at parameters mapped to theLennard-Jones triple point. N c is the number of particles in a given cluster, N the total number ofparticles sampled. Here we consider ground state clusters for all ranges of the Morse potential .Colours are Lennard-Jones (red), corresponding WCA (pink), Morse ( ρ = 4 .
0) (bright green) andtruncated Morse (dark green). Those clusters which are ground states are labelled as ‘both’ whenboth potentials share the same ground state, and ‘LJ’ and ‘M’ corresponding to the Lennard-Jonesand Morse cases accordingly. Note the semi-log scale. pair correlation functions. This suggests that the truncated Morse ρ = 25 . continuous approximation to the hard sphere system. We remark that the idea ofthe pair structure being dominated by the hard core appears less satisfactory here.We now turn our attention to the cluster populations in the Lennard-Jones-WCA systems[Fig. 8(a)]. As before, we consider clusters that are ground states for Lennard-Jones. At thishigher temperature, compared to Fig. 4(b), relatively little difference is seen between WCAand Lennard-Jones, consistent with the concept that in dense liquids, it is the repulsionsthat are responsible for the structure and that attractive interactions have less effect athigher temperature. However, the same trend is apparent as was found at the triple point[Fig. 4(b)]: Lennard-Jones shows a tendency to form larger clusters than WCA, which seemsreasonable given its cohesive energy and that these clusters minimise the energy of isolatedsystems. However, as shown by the ratio [ N c /N ] LJ / [ N c /N ] W CA in Fig. 9(a), the differencein population is slight. 13 a Figure 7: (color online) Pair-correlation functions. (a) Long-ranged potentials: Lennard-Jones(red) and WCA (pink) for a well depth of βε LJ = 0 . βε M = 2 . a b Figure 8: (color online) Population of particles in a given ground state cluster. N c is the numberof particles in a given cluster, N the total number of particles sampled. (a) Lennard-Jones with βε LJ = 0 .
440 (red) and corresponding WCA (pink). (b) Morse ( ρ = 25 .
0) (turquoise) truncatedMorse (light blue) and hard sphere (dark blue). Note the semi-log scale.
For the truncated Morse and hard spheres, the cluster populations are not tremendouslydifferent to the Lennard-Jones case, in fact for smaller clusters the differences are compa-rable to those between WCA and Lennard-Jones. In particular, hard spheres show a very14 (cid:136)
Figure 9: (color online) Ratio of cluster populations in high temperature systems. (a) Lennard-Jones and WCA. (b) Truncated Morse and Morse. This plot is the same data as Fig. 8 expressed toemphasize the difference between the two systems. Note that in (b) we invert the ratio to considerthe truncated Morse divided by the attractive Morse potential and plot on a different scale. similar population to the truncated Morse, further suggesting that the latter might make areasonable approximation to hard spheres. However, upon adding attractions, the popula-tion of clusters drops dramatically. In Fig. 9(b) we plot the ratio [ N c /N ] T M / [ N c /N ] M whichis inverted with respect to Figs. 9(a) and 5 in the sense that the truncated system formsthe numerator and the attractive system forms the denominator. In the truncated Morsesystem, the population of clusters of size m ≥
10, is at least eight times greater than theattractive system. We return to the possible origins of this discrepancy in the next section.We note that the attractive Morse g ( r ) exhibits a split second peak which the truncatedMorse and hard spheres g ( r ) do not. Yet, contrary to the notion that the split second peakis associated with crystallisation , in Fig. 8(b) we see precisely the opposite trend: thatthe split second peak is apparently associated with less crystallinity.We now plot the population of all identified clusters, noting that it is only for m ≥ C v ), 12B ( C v ) and 13A icosahedronwhereas for the short-ranged Morse the ground states are 11F ( C v ), 12E ( D h ) and 13B( D h ). Excluding the attractive Morse system there is rather little variation between thedifferent systems. In other words, it appears that for the other higher-temperature systems,the topological bond structure may be dominated by the hard core, which is matched in all15 igure 10: (color online) Population of particles in a given cluster, at parameters mapped to theMorse potential ( βǫ M = 2 . ρ = 25 . N c is the number of particles in a given cluster, N thetotal number of particles sampled. Here we consider ground state clusters for all ranges of theMorse potential . Note the semi-log scale. cases. In particular, although the Lennard-Jones and Morse potentials have different groundstates, at these high temperatures this is little reflected in the structure. IV. DISCUSSION AND CONCLUSIONS
We have analysed the pair structure and performed a topological cluster classification ona range of liquids. The pair structure of Lennard-Jones and longer-ranged liquids is entirelyconsistent with the well-known result that repulsive interactions dominate the local packingin dense liquids. Shorter-ranged potentials exhibit a strong response in the g ( r ) upon theaddition of attractions. However, one expects that these will be accounted for by the useof perturbation theory . We note that as ρ → g ( r ) → exp[ − βu ( r )] so a short-rangedattraction leads to a strong peak at contact as we see in Figure 7(b).Although the pair structure of Lennard-Jones and WCA is very similar, we are nonethelessable to identify clear differences using the TCC. We find that Lennard-Jones is more able toform higher-order clusters than the purely repulsive WCA. These differences become muchmore significant upon cooling to the triple point. Applying the extended law of corresponding16tates to compare with a weakly interacting longer-ranged system, little effect on the clusterpopulation, or pair correlation function is found upon adding attractions. Conversely, inshort-ranged systems, the radial distribution function is influenced by attractions and thecluster population is strongly enhanced upon removing attraction. That we see differenttrends in the short-ranged system is rather curious, and will be investigated further in thefuture. One comment we can make at this stage is that short-ranged attractive systemsexhibit non-monotonic dynamics as a function of attraction at high densities, in the form ofa re-entrant glass transition . Whether it is truly appropriate to expect the behaviourof short-ranged attractive systems to be similar to long-ranged Lennard-Jones type liquidsis perhaps an open question.We rationalise these three scenarios as follows. The long-ranged Morse is weakly inter-acting and compressed. Together, these lead to little response either of the g ( r ) (the spatialdistribution of particles) or the topology upon adding attractions. In the Lennard-Jones casecompression is less important, adding attraction promotes organisation and clustering, how-ever the interactions are sufficiently long-ranged that the repulsive core dominates the pairstructure for both Lennard-Jones and WCA. In the short-ranged case, the hard spheres (andpresumably the truncated Morse) are close to freezing (here the packing fraction φ = 0 . g ( r ), and can open up freevolume. However, considering the second Virial coefficents B ∗ in table I, these short-rangedsystems are quite weakly interacting, and there is apparently insufficient cohesive energy topromote organisation into clusters.Returning to the non-monotonic dynamics of short-ranged systems, one expects thatthe attractive Morse system might exhibit faster dynamics than hard spheres (and perhapsthe truncated Morse system). Although the hard sphere packing is far from dynamicalarrest, even so some kind of slowing is expected relative to a dilute fluid. This couldthen be reduced by the short-ranged attraction. Now we have correlated the clusters withdynamical arrest and found that arrested states have a high cluster population and thatit is biased towards higher-order clusters. As a function of density, hard sphere fluids showa similar trend . Thus we speculate that one possible underlying cause may be related todynamics. However, the hard sphere packing fraction is very close to freezing, and we notea substantial quantity of locally crystalline particles in Fig. 8(b). Now the colloid-polymer17iterature would tend to suggest that adding short-ranged attractions widens the fluid-crystal coexistence region. Since the hard sphere state point is so close to freezing, and thetruncated Morse seems similar to hard spheres, it is possible that the equilibrium state forMorse ( ρ = 25 . , βε M = 2 .
0) is crystal-fluid coexistence. It is interesting to note that thispossibly metastable fluid has a population of HCP and FCC structures around a factor of30 less than the stable hard sphere fluid.Among the key underlying ideas of clusters in liquids is that they represent energeticallylocally favoured structures . The most famous of these, the icosahedron, appears onlyin small quantities in this analysis, although it is most prevalent in Lennard-Jones. It wouldbe most interesting to investigate whether particles in these clusters are in fact in a lowenergy environment. It would also be interesting to seek a link between structure and dy-namics, particularly concerning the recent observation of very different dynamical behaviourbetween the WCA and Lennard-Jones systems and the observation that power-law repul-sive interactions seem to recapture the original Lennard-Jones behaviour . Moreover, othermappings have been proposed for example between Lennard-Jones and WCA. Here one canplace more emphasis upon the dynamics, albeit at some expense in the accuracy with whichthe radial distribution function is matched .Here we have focused on the ground state clusters for each system. Furthermore, liquidsare by definition at finite temperature, and it may be appropriate to consider the structureof clusters at higher temperature in addition to the ground states we have investigated sofar . Conversely, further quenching might favour the ground states beyond the trendswe have so far seen. Recently, we found we needed around 10 k B T of attraction to formisolated clusters .A final point for discussion is the link between attractions and reciprocal space structure.The static structure factor S ( q ) measures compressibility at wavevector q = 0. While allstate points sampled show no indication of sny phase transition, one nonetheless expectssome hint of attractions at low q . In particular, around the Lennard-Jones triple point,for q → S ( q ) between WCA and LJ . Thisdifference in S ( q ) upon adding attractions is predicted to be most prevalent at a value of qσ ≈ πσ in real space. This is a rather larger lengthscale than the clusters weprobe, and in fact Stell and Weis show that by qσ ≈
3, at the length scales we considerhere, this effect is much reduced. It would be most interesting to extend the TCC to larger18lusters, such that the qσ ≈ q behaviour. However, we note thatthe TCC is a particle-based methodology, and may not be sensitive to such delicate changesin long-ranged structure. Acknowledgements
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