Global order parameters for particle distributions on the sphere
GGlobal order parameters for particle distributions on the sphere
A. Božič, S. Franzini, and S. Čopar Department of Theoretical Physics, Jožef Stefan Institute, Ljubljana, Slovenia a)2)
Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia (Dated: February 26, 2021)
Topology and geometry of a sphere create constraints for particles that lie on its surface which they otherwise do notexperience in Euclidean space. Notably, the number of particles and the size of the system can be varied separately,requiring a careful treatment of systems with one or several characteristic length scales. All this can make it difficultto precisely determine whether a particular system is in a disordered, fluid-like, or crystal-like state. Here, we showhow order transitions in systems of particles interacting on the surface of a sphere can be detected by changes in twohyperuniformity parameters, derived from spherical structure factor and cap number variance. We demonstrate theiruse on two different systems—solutions of the thermal Thomson problem and particles interacting via ultra-soft GEM-4 potential—each with a distinct parameter regulating their degree of ordering. The hyperuniformity parameters arenot only able to detect the order transitions in both systems, but also point out the clear differences in the ordereddistributions in each due to the nature of the interaction leading to them. Our study shows that hyperuniformity analysisof particle distributions on the sphere provides a powerful insight into fluid- and crystal-like order on the sphere.
INTRODUCTION
Geometrically frustrated assemblies are ubiquitous in bio-logical, soft, and condensed matter , yet even the influenceof spherical geometry—perhaps the simplest closed, curvedsurface—on crystallization and ordering of particles remainspoorly understood . Crystal-like order and defects havebeen studied in viruses , metazoan epithelia , and colloidalcapsules , where different ways of construction have beenshown to lead to different degrees of order . It is importantto understand and determine the degree of (dis)order in spher-ical structures, as it can lead to different optic , elastic ,and dynamic properties. Order of the underlying spheri-cal lattice can also significantly influence the orientations ofanisotropically interacting particles positioned on it .Nonetheless, it can be difficult to characterize the degree oforder in a distribution of particles on the sphere, particularlygiven the numerous defects and topological scars present evenin the most ordered structures . Often, local bond order pa-rameters are used to detect the presence and onset of order inparticle distributions on the sphere , but they tend to bebased on the expected order and the prevalent 6-fold characterof a locally-ordered crystal-like particle distributions. Other(“order-agnostic” ) measures such as mesh ratio and energyare used to distinguish between different types of sphericalstructures , but there are important exceptions where neitherthese nor local bond order parameters can provide a good an-swer . Recently, however, some progress has been madeby extending the notion of hyperuniformity, thoroughly ex-plored in Euclidean space , to the sphere and other curvedsurfaces , which introduces a more global view of the or-der on the sphere.In our previous work , we have shown that by generaliz-ing the notion of hyperuniformity to spherical geometry, it ispossible to derive two parameters which together indicate the a) Electronic mail: [email protected] degree of order in scale-free distributions of particles on thesphere. Here, we generalize this notion to systems with oneor more length scales, which are more difficult to tackle, asboth the number of particles and the size of the system haveto be treated as independent parameters . Nonetheless, wedemonstrate that the hyperuniformity parameters can be usedto detect the degree of order even in those systems where usualapproaches fail. Hyperuniformity on a sphere could thus pro-vide a good framework for a consistent definition of fluid- andcrystal-like order on the sphere.
METHODSSpherical structure factor and cap number variance
We describe an arbitrary distribution of N particles on thesurface of a sphere with radius R with a surface density distri-bution ρ ( Ω ) = R N ∑ k = δ ( Ω − Ω k ) = R ∑ (cid:96), m ρ (cid:96) m Y (cid:96) m ( Ω ) , (1)where Ω k are the positions of the particles in spherical coordi-nates ( ϑ , ϕ ) . The coefficients ρ (cid:96) m , used to expand the distri-bution in terms of spherical harmonics Y (cid:96) m ( Ω ) , further definethe spherical structure factor , S N ( (cid:96) ) = N π (cid:96) + ∑ m | ρ (cid:96) m | = N N ∑ i , j = P (cid:96) ( cos γ i j ) . (2)Here, P n ( x ) are the Legendre polynomials and γ i j is the spher-ical distance between particles i and j . Spherical structurefactor is tightly related to the pair correlation function and should reflect the interaction potential of the system. Wecan connect the spherical structure factor to another measure,the cap number variance σ N ( θ ) , which gives the variance ofthe number of particles contained in a spherical cap with an a r X i v : . [ c ond - m a t . s o f t ] F e b opening angle θ : σ N ( θ ) = N ∞ ∑ (cid:96) = S N ( (cid:96) ) [ P (cid:96) + ( cos θ ) − P (cid:96) − ( cos θ )] (cid:96) + . (3)In practice, σ N ( θ ) is obtained by covering the sphere with aseries of randomly-positioned spherical caps with an openingangle θ , determining the number of particles in each, and cal-culating their variance. Hyperuniformity on the sphere
We have shown previously for scale-free particle distribu-tions on the sphere that the form of the cap number variancein Eq. (3) can be approximated by σ N ( θ ) = A N N θ + B N √ N √ θ , (4)with an additional (small) residual, relevant only in the caseof ordered distributions. The form of cap number variance inEq. (4) can be considered a spherical analogue of the asymp-totic form of the number variance in Euclidean space, used todetermine the degree of hyperuniformity in such systems .Furthermore, the two parameters in Eq. (4), A N and B N , turnout to be particularly good measures of order in scale-free par-ticle distributions on the sphere. For a completely randomdistribution, we can show that A N = B N =
0; this cor-responds to a uniform structure factor where S N ( (cid:96) ) = ∀ (cid:96) .With a gradual onset of order in a system of particles, first alow- (cid:96) gap starts to appear in the structure factor, and simulta-neously A N starts to diminish while B N increases. In the limitof A N →
0, equivalent to the onset of hyperuniformity in Eu-clidean space, a particle distribution on the sphere becomesordered with a series of pronounced peaks in its structure fac-tor and can be, in principle, characterized by its value of B N —which depends not only on the type of distribution but also onany symmetries present in it (for details, see Ref. 31). Distributions of particles on the sphere
In this work, we generalize these results to particle distri-butions with one or more internal length scales. To do this,we study two completely different systems: (i) solutions ofthe thermal Thomson problem, where temperature introducesa length scale into an otherwise scale-free system; and (ii) asystem of particles interacting via ultra-soft GEM-4 potential,which exhibits an ordered phase of cluster crystals dependingon both the number of particles and the size of the system.In the first case, (i) , particles interact via long-range elec-trostatic potential, just as in the classical Thomson problem,but we additionally introduce a reduced temperature T intothe system (scaled with the electrostatic potential). Startingat a high T and gradually lowering it, we sample 250 differ-ent configurations of N particles at each T . This is achievedby virtue of MC simulations , where at each T , we per-form a series of random displacements of individual particles, drawn from a spherical Gaussian (von Mises-Fisher) distribu-tion centred around a particle , with the width parameter of λ = √ N / T ; this choice ensures a good acceptance rate also atlow T and large N . After a burn-in phase, configurations aresampled every 4 N moves until 250 different configurations areobtained, which are then used to obtain ensemble-averagedspherical structure factor and cap number variance.In the second case, (ii) we study particles interacting viaa generalized exponential model of order 4 (GEM-4 poten-tial): a bounded, purely repulsive soft pair potential of theform w ( r ) = ε exp ( − ( r / δ ) ) . Here, ε and δ determine theenergy and length scales of the model. We use the former torescale the temperature T of the system, while the latter in-troduces a length scale to the system, δ / R , which controls itsphase behaviour. While the GEM-4 system is also simulatedat a finite T , this is the least interesting variable in the system;we will thus study the system at T =
1, unless specified oth-erwise, and explore its behaviour with respect to both N and δ / R . We sample 50 configurations at each point in the phasespace to generate ensemble-averaged spherical structure fac-tor and cap number variance. Further simulation details canbe found in Ref. 27. RESULTSThermal Thomson problem
Solutions of the Thomson problem are distributions of par-ticles minimizing their electrostatic interaction . Knownminimum energy distributions are often characterized by ahigh symmetry and a locally triangular mesh where each par-ticle has 6 neighbours, with the exception of 12 5-fold defectsowing to the topology of the sphere; at high N , pairs of defectsin the form of topological scars are also common . Whentemperature (measured relative to the interaction energy) isintroduced into the system, the order disappears and differentkinds of local defects are ubiquitous. Temperature is also theonly length scale in the system—note that changing the radiusof the sphere is equivalent to changing the scale of interactionenergy and thus the scale of the reduced temperature. As it islowered, the solutions of the thermal Thomson problem con-verge towards the known minima of the Thomson problem. Structure factor and number variance
The temperature-dependent order transition in a thermalThomson system can be easily observed when we take a lookat the (ensemble-averaged) spherical structure factor [Eq. (2)].Figure 1a shows S N ( (cid:96) ) in the T – (cid:96) plane for distributions of N =
120 particles. At high T , the structure factor is essen-tially indistinguishable from that of a random distribution, S N ( (cid:96) ) = ∀ (cid:96) . As T is lowered, S N ( (cid:96) ) becomes progressivelymore defined: first, a gap appears at low (cid:96) , growing with de-creasing temperature, and as it approaches (cid:96) ≈ π √ N / √ ,the first peak of the structure factor appears at (cid:96) (Fig. 1b).Its position does not change as T is lowered further; on the − − − T ‘ (a) S N ( ‘ )048 1 50 100 ‘ S N ( ‘ ) (b) θ/π σ N ( θ ) (c) Figure 1. (a)
Ensemble-averaged spherical structure factor for distri-butions of N =
120 particles in the T – (cid:96) plane. Ensemble-averagedspherical structure factor (b) and cap number variance (c) of dis-tributions at three different temperatures, marked by dotted linesin panel (a). Dashed gray lines show the expected behaviour fora completely random distribution: S N ( (cid:96) ) = ∀ (cid:96) [panel (a)] and σ N ( θ ) = N sin θ / other hand, higher- (cid:96) peaks do not initially appear at the exactpositions of the crystal-like (minimum energy) state, but shiftslightly with decreasing T . At the very lowest T , the form of S N ( (cid:96) ) is completely defined and approaches the form of theknown minimum solutions of the Thomson problem . Insome cases, discrepancies remain: these structures, while or-dered and crystal-like, are trapped in local minima.Spherical structure factor is directly related to cap numbervariance [Eq. (4)], the variance in the number of particles con-tained in spherical caps with opening angle θ . As the temper-ature of the system is lowered and S N ( (cid:96) ) becomes more de-fined, the angular dependence of σ N ( θ ) goes from ∝ sin θ ,characteristic of a random distribution, to ∝ sin θ , typical ofcrystal-like distributions . Furthermore, when the order inthe distribution becomes crystal-like, σ N ( θ ) also starts to ex-hibit a modulation on top of its general θ -dependence, whoseform is related to (cid:96) (and thus to N ) and is another conse-quence of ordering . Hyperuniformity parameters
Changes in spherical structure factor and cap number vari-ance can be summarized by fitting σ N ( θ ) to the form givenby Eq. (4), which yields two hyperuniformity parameters A N and B N . As already mentioned, it has been shown previously that for a completely random distribution, A N = B N = A N = B N (cid:46)
1. This, of course, holds in the average sense, particu-larly for randomly-generated distributions.Figure 2 shows the hyperuniformity parameters A N and B N of the thermal Thomson distributions in the N - T plane. Thefits of Eq. (4) are performed on ensemble-averaged curves σ N ( θ ) for each N and T , as fits to individual ensemble sam-ples do not yield reliable results due to the large degree ofrandomness present in the system. We can observe several
50 150 250 N − − − T (a) (b) A N − − T A N
50 150 250 N − − − T B N − − T B N Figure 2. Hyperuniformity parameters A N (a) and B N (b) of thermalThomson distributions in the N - T plane, obtained as fits of Eq. (4) toensemble averages of cap number variance σ N ( θ ) . Insets show thetemperature dependence of the two parameters for three different val-ues of N , marked in the main plots with dotted lines. Black contourline in panel (a) shows the critical temperature T c where A N (cid:54) . T c ∝ N . things: at high T when the system is disordered, A N (cid:46) B N ≈
0, close to the values pertaining to random distribu-tions (albeit not completely, as even at highest T the systemis not completely random due to the interactions involved; seeFig. 1). Furthermore, higher N have lower values of A N athigh T , which is understandable since the energy of the sys-tem also increases with N . As T is lowered, A N → . The transition is gradual, and the critical temperatureshows a slight dependence on N —for higher N , the transitionoccurs at higher T . The approximate range of temperatureswhere the transition occurs is T ∼ . c so that A N (cid:54) c . We know that B N already has a peak when A N is of the order of magnitude of a few tenths (Fig. 2). Anythreshold choice above c (cid:38) − shows almost exact propor-tionality of the critical temperature to the number of particles, T c ∝ N (Fig. 2a shows the example of c = . B N is dominated by noise at high T , since A N is the dominant parameter there. As T is lowered, B N typi-cally crosses a “barrier” in the temperature range where A N first starts to decrease, the height of which increases with N .This increase in B N as A N is lowered could thus indicate someparticular property of the interactions in the system. At low T where A N →
0, values of B N start to converge to very similarvalues regardless of N , B N ∼ .
9, which is characteristic ofordered distributions on the sphere in general and minimumsolutions of the Thomson problem in particular . The van-ishing of parameter A N thus clearly signals a transition froma disordered to an ordered distribution. The parameter B N , onthe other hand, becomes relevant only when A N vanishes—then, B N carries some information about the nature of the or-der. GEM-4 potential
We can apply the same analysis as we did for the thermalThomson problem to particle distributions resulting from theGEM-4 interaction potential. At a given T (note that the tem-perature related to the GEM-4 potential has a different scalethan the one pertaining to the thermal Thomson problem), thesystem of GEM-4 particles is known to undergo an orderingtransition from a homogeneous fluid to a cluster crystal phase,depending on both the number of particles N and the (scaled)radius of the sphere δ / R . At high density, particles aggre-gate into clusters at sites which are distributed on the spherein a highly ordered manner (Fig. 3a). However, the internalstructure of such clusters remains disordered, as particles ran-domly move inside the potential well. The number of clustersis a function of δ / R but not N —an increase in the number ofparticles at a fixed sphere size will only lead to each clusterhaving more particles. Structure factor and number variance
Different types of order in systems of GEM-4 particles canbe clearly seen both in their spherical structure factor and intheir cap number variance (Fig. 3). When the system is in thehomogeneous fluid phase, S N ( (cid:96) ) exhibits only a shallow firstpeak while σ N ( θ ) shows no modulation related to a shell-likestructure, typical for ordered systems. However, when the sys-tem is in the cluster crystal phase, S N ( (cid:96) ) exhibits several pro- ‘ S N ( ‘ ) (a)(b) δ/R = 0 . δ/R = 0 . δ/R = 0 . . . . θ/π σ N ( θ ) (c) Figure 3. (a)
Voronoi tesselations of distributions of N =
200 GEM-4particles at T = δ / R = .
40, 0 .
65, and 0 .
90 (from left to right).Shown are also the ensemble averages of the spherical structure fac-tor (b) and cap number variance (c) for the same values of N , T , and R as in panel (a). nounced peaks and σ N ( (cid:96) ) now shows the characteristic modu-lations related to structural order. Notable is the overall scaleof both measures compared to their form: the modulationsin σ N ( θ ) (Fig. 3c) is characterized by the number of clusters N ∗ and not the total number of particles N , as was the casein the thermal Thomson distributions. At the same time, thelarge magnitude of the peaks in the spherical structure factor(Fig. 3b) when compared to those observed for thermal Thom-son distributions (Fig. 1b) is due to the fact that each of the N ∗ clusters is composed of N / N ∗ particles on average. Hyperuniformity parameters
While both S N ( (cid:96) ) and σ N ( θ ) show the transition of a sys-tem of GEM-4 particles from a homogeneous fluid to a clus-ter crystal phase, this transition is difficult to capture usingstandard order parameters due to the disordered nature of par-ticles within each cluster. However, the difference betweenthe two phases is immediately apparent if we take a look atthe hyperuniformity parameters A N and B N , again obtainedby fitting Eq. (4) to the ensemble-averaged σ N ( θ ) . Figure 4shows that the value of parameter A N clearly separates the twophases in the N - δ / R plane. In the homogeneous fluid phase, A N is always larger than zero. It also never reaches the valueof A N =
1, indicating that the system is never completely ran-dom, which is expected due to the strong interactions betweenthe particles. When the system transitions to the cluster crys-tal phase, A N suddenly vanishes, A N (cid:46) − (shown by theblack region in Fig. 4a). This is in stark contrast to the ordertransition in the thermal Thomson system, where A N slowlydecreased to zero as the temperature was lowered.When A N vanishes, B N again starts to increase. Unlike whatwe observed in the thermal Thomson problem, or what waspreviously observed in scale-free distributions , B N can takeon extremely large values. The reason is that in the orderedstate with N ∗ clusters, the distribution is closer to a hyperuni-form distribution of N ∗ particles with larger particle weights,
100 200 300 400 500 N δ / R (a) (b) A N
100 200 300 400 500 N δ / R B N Figure 4. Hyperuniformity parameters A N (a) and B N (b) in the N - δ / R plane for distributions of GEM-4 particles at reduced tem-perature T =
1. The parameters were obtained as fits of Eq. (4) toensemble-averaged cap number variance. Black contour line in panel(a) shows where the parameter A N vanishes, A N (cid:46) − . while the expression in Eq. (4) is normalized only with N , as N ∗ is not known in advance. Rescaling the structure factorshows that B N should scale as a power of the number of parti-cles per cluster, ( N / N ∗ ) / . Indeed, this scaling helps explainthe observed pattern for the number of clusters in the N - T plane , where the number of clusters, and thus the averagenumber of particles per cluster, changes with δ / R but not with N . In the ordered state, the scaling of the parameter B N showsthe same pattern; however, it also includes an unknown pref-actor, which we are currently unable to predict theoretically.Nonetheless, the parameter B N clearly shows the potential tobe used for assessing finer aspects of order, such as clustering.The observation that A N vanishes suddenly with the appear-ance of cluster crystal phase can be exploited to separate the N - δ / R plane into two regions corresponding to homogeneousfluid and cluster crystal phases. This is shown in Fig. 5 forfive different temperatures of the system. By observing when A N ≤ − , it is easy to see that the cluster crystals span alarger part of the phase diagram at lower T . Moreover, in-creasing the temperature appears to shift the phase curve to-wards larger N while maintaining its position in the δ / R di-rection. While these observations have been made previouslyby Franzini et al. in their original study, they did not use anorder parameter to delineate the regions of the phase space.Our results demonstrate that A N and B N can be used as globalorder parameters to construct the phase diagram of the sys-tem, something which cannot be done using standard orderparameters on the sphere.We also note that there is a larger uncertainty in determin-ing the phase line at low δ / R : the likely reason is that in thisregime, a very high number of clusters is formed ( N ∗ (cid:38)
100 200 300 400 500 N . . . . . . δ / R homogeneousfluid clustercrystals T = 0 . T = 0 . T = 1 . T = 1 . T = 1 . Figure 5. Curves of vanishing A N in the N - δ / R plane for five dif-ferent temperatures of the system. The curves are defined as pointswhere A N ≤ − and mark the transition from a homogeneous fluidin the left part of the phase diagram to a cluster crystal phase in theright part of the diagram. phase space our predictions also differ from the observationsof Franzini et al. ; specifically, we predict that low δ / R leadto the onset of cluster crystal phase at much higher N thanoriginally thought. DISCUSSION
Hyperuniformity has only recently been extended to non-Euclidean geometries, and the known notions from Euclideanspace have been shown to extend to spherical geometry aswell. Nonetheless, there are several notable differences be-tween the two, related to the restrictions that topology and ge-ometry of the sphere dictate. At the moment, there seem to beno distinct hyperuniformity classes on the sphere—unlike inthe Euclidean case–as the hyperuniformity parameters A N and B N derived from cap number variance can change in a contin-uous manner. However, just as it is already known for scale-free systems of particles on the sphere, we have shown herethat these two parameters can be used to consistently detectand study order transitions in systems of particles involvingone or more internal length scales.By studying two such systems—thermal Thomson prob-lem and particles interacting via GEM-4 potential—we haveshown that the parameter A N is a good measure of disorderin the system. While A N is finite, A N (cid:54)
1, the system is ineither completely disordered ( A N ≈
1) or fluid-like ( A N < A N vanishes, A N →
0, the system un-dergoes an order transition. Once this happens, the parameter B N becomes relevant and in a crystal-like state takes on a con-stant value. As A N gradually vanishes, B N might also cross abarrier, increasing at first before assuming this value. Both ofthese observations and the scale of B N are likely related to thedetails of the systems, particularly regarding the interactionsinvolved, although further theoretical insights into this are cur-rently still lacking. Ideally, these would connect the exact na-ture of the interaction potential to an improved approximationof cap number variance, currently given by Eq. (4).When the positions of the particles are known, the fit ofEq. (4) is easy to carry out not only in simulations but also inexperimental realizations of spherical assemblies. However,based on our analysis, we also see that it might be possible toderive some proxies for disorder and the parameter A N whichmight be quicker to determine. Such candidates are the dipolemoment of the spherical structure factor, S N ( (cid:96) = ) , or thehemispherical cap number variance, σ N ( θ = π / A N and B N correctly detects the phase transitions of the system. ACKNOWLEDGMENTS
This work was funded by Slovenian Research AgencyARRS (Research Core Funding No. P1-0055 (A.B.) andNo. P1-0099 (S. ˇC.) and research Grant No. J1-9149), andis associated with the COST Action EUTOPIA (Grant No.CA17139). S.F. acknowledges “Laboratorio di Calcolo e Mul-timedia (LCM)” of the University of Milan for providing ma-chine time on their cluster.
DATA AVAILABILITY
Data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
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