short- and long-range topological correlations in two-dimensional aggregation of dense colloidal...

Short- and long-range topological correlations in two-dimensional aggregationof dense colloidal suspensions

J. C. Fernández-Toledano,1 A. Moncho-Jordá,1 F. Martínez-López,1 A. E. González,2 and R. Hidalgo-Álvarez11Departamento de Física Aplicada, Facultad de Ciencias, Campus Fuentenueva S/N, 18071 Granada, Spain

2Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, MéxicosReceived 11 November 2004; revised manuscript received 28 January 2005; published 7 April 2005d

We have studied the average properties and the topological correlations of computer-simulated two-dimensionals2Dd aggregating systems at different initial surface packing fractions. For this purpose, thecenters of mass of the growing clusters have been used to build the Voronoi diagram, where each cellrepresents a single cluster. The number of sidessnd and the areasAd of the cells are related to the size of theclusters and the number of nearest neighbors, respectively. We have focused our paper in the study of thetopological quantities derived from number of sides,n, and we leave for a future work the study of thedependence of these magnitudes on the area of the cells,A. In this work, we go beyond the adjacent clustercorrelations and explore the organization of the whole system of clusters by dividing the space in concentriclayers around each cluster: the shell structure. This method allows us to analyze the time behavior of thelong-range intercluster correlations induced by the aggregation process. We observed that kinetic and topo-logical properties are intimately connected. Particularly, we found a continuous ordering of the shell structurefrom the earlier stages of the aggregation process, where clusters positions approach a hexagonal distributionin the plane. For long aggregation times, when the dynamic scaling regime is achieved, the short- and long-range topological properties reached a final stationary state. This ordering is stronger for high particle densities.Comparison between simulation and theoretical data points out the fact that 2D colloidal aggregation in theabsence of interactionssdiffusion-limited cluster aggregation regimend is only able to produce short-rangecluster-cluster correlations. Moreover, we showed that the correlation between adjacent clusters verifies theAboav-Weaire law, while all the topological properties for nonadjacent clusters are mainly determined by onlytwo parameters: the second central moment of number-of-sides distributionm2=oPsndsn−6d2 and the screen-ing factora sdefined through the Aboav-Weaire equationd. We also found that the values ofm2 anda calculatedfor two-dimensional aggregating system are related through a single universal common forma~m2

−0.89, whichis independent of the particle concentration.

DOI: 10.1103/PhysRevE.71.041401 PACS numberssd: 82.70.Dd, 02.50.2r

I. INTRODUCTION

The formation of colloidal monolayers is especially inter-esting due to the ability of colloidal particles to influence thestability of emulsions, foams, and interfacial properties. Par-ticularly, the study of aggregation in colloidal systems con-fined in two dimensions has drawn wide attention due to theinteresting observed effects, such as fractal cluster growthf1–3g or mesostructure formationf4,5g. At this respect, thestructure of the clusters and the kinetic properties of the co-agulation process have been investigated by means of bothexperiments and simulationsf1–3,6g. Theoretically, the suc-cess of technics for treating phase transition phenomena hasmotivated the application of these ideas to the analysis of thescaling properties of the cluster-size distribution in aggregat-ing systemsf7,8g. The results show that the morphology ofcolloidal clusters is fractal scale invariant, and it is related tothe time evolution of the cluster-size distribution.

Some years ago, other interesting aspects of colloidal co-agulation have been applied to the study of dense colloidalsuspension in two and three dimensions: intercluster spatialf9–14g and topological orderingf15–17g. These recent pic-tures complement the ones given by the kinetics and fractalgrowth description, and provide information about the orga-nization of the colloidal clusters in the space. One of the

most significant findings is the appearance of a maximum inthe structure factorSsqd at finite q for long aggregationtimes, which corresponds to a characteristic length scale ofthe system connected with the typical cluster-cluster separa-tion. Moreover, the structure factor obtained at differenttimes in the scaling regime can be expressed into a singletime-independent common form. Also the topological prop-erties have been found to reach a final stationary values. Allthese facts imply that the aggregation process induce struc-turing not only inside the fractal clusters, but also in theintercluster correlations, and indicate the necessity of beingincluded in order to have a complete self-consistent theory ofthe aggregation phenomenon.

A very useful tool to determine the correlations and or-dering in space-filling systems is the one based on the topo-logical properties. Topology is the mathematical study of theproperties that are preserved through deformation, twistings,and stretchings of objects. The study of the topological prop-erties of such systems can be achieved with the help of theVoronoi construction, which creates a division of the wholespace in irregular partitionsscellsd. Such a method has beensuccessfully used to describe the evolution of many naturalsystems, such as soap frothsf18–22g, propagation of defectsin foamsf23,24g, metallurgical grainsf25g, biological tissuesf26g, crack patterns in ceramicsf27g, etc. Even though these

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systems differ in the way they evolvesmitosis for biologicaltissues, diffusion of gas between neighbor cells for soapfroths, etc.d, it is found that most of them arrive at a finalstationary state where topological properties remain constant.Moreover, this state is described by the same common laws,which come from maximum entropy predictionsf28,29g.However, all these works are essentially mean-field studiesand only account for correlations between adjacent cells. Inrecent times, the description of the long-range topologicalordering and the correlation between non-nearest neighborshave attracted interest and new theoretical methods based onshell structure have been applied to the understanding of theevolution of cellular systemsf30–35g.

In this work we focus our study on the topological prop-erties of two-dimensional aggregating systems. For this pur-pose, growing clusters are substituted by convex nonoverlap-ping polygonal domainsscellsd that form a tessellation of thespace where aggregation takes place. The number of sidesand area of each cell denote the number of nearest neighborsand area filled by the real cluster together with its surround-ing depletion region, respectively. In our study we go furtherto the nearest-neighbor clusters and investigate the short- andlong-range ordering created by the aggregation process andhow this structuring is related to the surface packing fractionof particles.

The paper is organized as follows. Section II describes thebasic theoretical aspects of aggregation kinetic and topology,while Sec. III focuses on the long-range topological proper-ties. We tackle the results and discuss the most importantremarks in Sec. IV. Finally, Sec. V details the conclusions.

II. KINETICS AND TOPOLOGY OF COLLOIDALAGGREGATION

The kinetics of aggregation processes are usually de-scribed in terms of the cluster-size distributionhnistdj, de-fined as the number of cluster formed byi individual colloi-dal particles. The time evolution of the cluster-sizedistribution is strongly connected to the particle-particle in-teractions, the spacial dimensionality, and the geometry ofthe clusters. One of the most studied properties derived fromhnistdj, is the weight-average cluster sizeSw:

Swstd = oi=1

i2nistdYoi=1

inistd. s1d

Analogously, the number-average cluster size is definedas Snstd=oi=1inistd /oi=1nistd. For long aggregation times, ithas been observed experimentally and by simulations thatthe weight-average cluster size develops a power-law behav-ior, Swstd, tz, where z is the so-called kinetic exponent.Then, if one represents the functionSwstd2nistd versus a nor-malized cluster size,i /Swstd, we see that all the curves can bescaled into a single time-independent scaling functionC(i /Swstd)=Swstd2nistd∀ t@1 f7,36g. The shape of this mas-ter curve depends on the aggregation regime.

One of the most studied aggregation regimes is thediffusion-limited cluster aggregationsDLCAd f37g, whichcan be considered a reference process because of its simplic-

ity. In this regime, colloidal particles freely move by Brown-ian diffusionswithout interparticle interactionsd and they be-come irreversibly stuck after collision. Therefore, theaggregation process is totally determined by the time in-volved in the diffusion of the clusters before they collide toform a new larger cluster. The clusters formed under thisconditions have an open structure, with a fractal dimensionof 1.45 for two dimensionsf38g and 1.79 for three dimen-sionsf39,40g, while the master curve is bell shaped in bothcasesf7,8g, which means that the monomers and small-sizedspecies are rapidly removed during the aggregation processto form larger clusters.

Another limit for aggregation is described by the reaction-limited cluster-cluster aggregationsRLCAd model. In thismodel aggregation is prevented by using a low sticking prob-ability for two colliding aggregatesf3g. Here, the kineticproperties and the cluster population are mainly determinedby the cluster-cluster reaction time, which becomes muchlarger than the diffusion time. In this regime, the mastercurve shows a different shape: for two-dimensional aggrega-tion the bell-shaped form broadens significantlyf41g, whilefor three dimensions the master curve becomes a monoto-nous decreasing function of the cluster sizef8g. That meansthat reaction-controlled cluster-cluster coagulation leads tomore polydisperse cluster-size distributions and, therefore, tomore disordered systems.

The kinetic aspects are essential to understand theaffinitybetween clusters with different sizes and to give an estimateof the total aggregation rate. However, the cluster-size distri-bution nistd only gives information about the population ofthe different species formed during the aggregation process,but not about the structure and distribution of these clustersin space. In order to understand the short- and long-rangestructure of the whole system of clusters confined in twodimensions, a topological description is needed. By means ofthe Voronoi construction, each cluster is replaced first by apoint representing its center of mass, and then, the spacearound each point is partitioned into convex, irregular poly-gons, calledcells. What we obtain after this mapping is adisordered tessellation orfroth, where the original clustersare represented by nonoverlapping cells. Figure 1 illustratesa typical snapshot of the colloidal clusters and the froth de-rived from this method during the coagulation process.

The number of sides of a cell,n, informs about the num-ber of closest neighboring clusters, while the area of the cell,A, is related to the total area filled by the colloidal cluster andits depletion region. The relative distance between two non-neighboring clusters is now represented by the topologicaldistancej , defined as the minimum number of cells that wemust go through from a starting cell to a given final cellsseeFig. 2d. The topological distance allows us to consider thefroth as a collection of concentric shells which are at thesame topological distance from a centralseed. The main ad-vantage of using the topological distancej instead of themetric distancer is the fact thatj gives an unambiguous andtime-independent measurement of the coordination shell atwhich a cluster is localized in relation to a central one. For asystem of growing clusters, however, the typical metric dis-tance of a certain coordination shell is not a constant quan-tity, but it increases as the aggregation proceeds. The topo-

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logical distance also gives a good characterization of thecorrelation between any couple of cells of the frothsi.e.,correlation between two colloidal clustersd. Indeed, anymodification in the shape or the number of sides of a particu-lar central cell produced by the motion and/or aggregation ofthe colloidal clusters will have more influence on the cellsplaced at small topological distances. For largej the statisti-

cal properties of the cells become independent of the centralone. In fact, as we will see later, the cell-cell correlations arenegligible for j .2.

In two-dimensional coagulation the evolution of the cel-lular pattern is ruled by two independent processes. The firstone is the diffusive motion of the clusters in the plane: everytime a cluster moves a large enough distance, a new rear-rangement of the cells is found in terms of a local sideswitchingsalso called T1 processd. The second process takesplace when two clusters coagulate together to form a largercluster; here, one of the cells disappears and the surroundingcells are accommodated to the new configurationsT2 pro-cessd f21g. Both topological processes induce a topologicalrestructuring of the two-dimensional froth that is connected,in general, to the physical conditions of the aggregating sys-tem and, more specifically, to the aggregation regime and thepacking fraction of particlesfs.

Since the aggregation process induces a disordered frothwith irregular cells, the number of sides and area of the cellswill be statistical properties distributed according toPsn,Ad,defined as the probability of finding a cell ofn sides and areaA. Analogously, we can obtain the probability of finding ann-sided cell asPsnd=oAPsn,Ad. The topology of cellularpatterns imposes some constraints on the distributionPsnd.Defining the average area of ann-sided cell asAsnd=oAAPsn,Ad, we have that

on=3

PsndAsnd =A0

Nc, s2d

whereA0 is the area of the whole froth andNc is the totalnumber of cellssi.e., colloidal clustersd. One of the otherimportant constraints which holds for the case of Voronoidiagrams formed by trivalent cell’s vertexes on a Euclideantwo-dimensional space is the so-called Euler theoremf42g. Itstates that the average number of sides of a cell is 6:

knl = on=3

Psndn = 6. s3d

One of the most interesting quantities derived from theprobability distributionPsnd is its second central moment,m2=kn2l−knl2=on=3sn−6d2Psnd, which can be understoodas an estimate of the degree of the disorder in the froth.

In principle,n andA are both essential to give a completedescription of the froth. However, only one of them is reallynecessary since these quantities are in fact strongly corre-lated through the Lewis’ law, which states that the mean areaof the n-sided cells shows a linear dependence onn, Asnd=kstdsn−n0d f43g. Here,kstd increases with the aggregationtime whereasn0 is found to be a constant valuef16g. Hence,the number of sidesn is sufficient to give a good represen-tation of the froth and the cluster-cluster correlations.

The study of correlations between the shapes of twoneighboring cells in the froth can be easily done with thehelp of m1snd, defined as the average number of sides of theadjacent cells ofn-sided cells. From space-filling conditionsfollows the result that large cellssi.e., with many sidesd tendto be surrounded by small cells. This intuitive property canbe expressed in a semiempirical law called theAboav-Weaire

FIG. 1. Typical snapshot of a two-dimensional system of aggre-gating clusters and the subsequent Voronoi diagram built up fromtheir center of masses.

FIG. 2. Cell structure and defects around a given central cell.The number inside the cells is the topological distance from thecentral cell. The grey cells represent topological defects. The “1”“2” symbol represents the convexsconcaved vertexes of the firstlayer.

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law f44,45g, which states that on averagem1snd is linear in1/n:

m1snd < s6 − ad +6a + m2

n, s4d

wherem2 is again the second central moment anda is thescreening factor. This law can be exactly deduced from theevolution of an ideal froth where the T2 transformations arerestricted to the disappearance of three-sided cells, whichgives a screening parameter ofa=1 f28g. In many naturalfroths a is of the order of 1f45g. For two-dimensional co-agulation, however, the T2 processes are due to the aggrega-tion between pair of clusters. That allows the removing ofcells with more than three sides, so the screening factor cantake, in principle, values different to 1 too.

III. LONG-RANGE TOPOLOGICAL CORRELATIONS OFTHE VORONOI CONSTRUCTION

As we have seen in the previous section, the analysis oftwo-dimensional cellular patterns can be performed bymeans of several laws regarding the evolution of the prob-ability Psnd and the correlations between adjoining cellssAboav-Weaire lawd. However, these descriptions are mean-field treatments and do not include the correlations betweennonadjoining cells. In order to describe the correlations be-tween any pair of aggregating clusters in the whole system,we need the topological distance. In general, any trivalentfroth can be studied by means of a shell structure around acentral cell. The first layerswith j =1d corresponds to thenearest-neighbor cells, the second layer to the cells adjoiningthe first layer, etc.

This decomposition of the plane in concentric regions al-lows us to systematically explore the long-range orderinginduced by the coagulation process and to measure the topo-logical distances at which a colloidal cluster “feel” the pres-ence of a second cluster. From a topological point of view,what we are looking for is to describe how the fact of havingann-sided cell affects the whole froth and to give an estimateof the topological distancej =j where the correlations van-ish. If j is large, then the cells in the froth are stronglycorrelated and mean-field models break down. On the con-trary, for j=1 only the adjoining cells are influenced by thecentral seed, and forj=0 we find a totally uncorrelated sys-tem stopological gasd.

A. Definitions and basic properties

Some j-dependent topological quantities are required inorder to describe the structure of concentric shells. Here wewill follow the notation introduced in Ref.f32g:

Kjsnd: average number of cells in thej layer around ann-sided central cell.

mjsnd: average number of sides of the cells in thej layeraround ann-sided central cell.

q=6−n: topological charge of ann-sided cell.Qjsnd=oiø jf6−misndgKisnd: total topological charge of

the cells that go from the central seed to thej layer.

Cjsn,md: probability of finding anm-sided cell at a dis-tancej from ann-sided cell.

Vj+snd: average number of convex vertexes going from the

j layer to thes j +1d layer around ann-sided central cellsrep-resented as1 symbols in Fig. 2d.

Vj−snd: average number of concave vertexes going from

the j layer to thes j −1d layer around ann-sided central cellsrepresented as2 symbols in Fig. 2d.

As we will see later, most of this quantities are connectedamong them. In particular, it is specially interesting to writedown the relationship between the topological charge and thenumber of convex and concave vertexes, which readsf32g

Qjsnd = 6 −Vj+snd + Vj

−snd. s5d

We can define the correspondingn averages ofKjsnd,mjsnd, andQjsnd, which no longer depend on the number ofsides of the central cell—i.e.,kKjl=on=3KjsndPsnd, kmjl=on=3mjsndPsnd, and kQjl=on=3QjsndPsnd. We can intro-duce the fluctuations of the topological charge as

G jsnd = Qjsnd − kQjl. s6d

In the particular case of a froth where the Aboav-Weaire lawholds, it can be shown that the fluctuation at the first layerj =1 is given by

G1snd = s1 − ads6 − nd. s7d

By definition, a froth is completely uncorrelated when theprobability Cjsn,md factorizes asf34g

Cjunsn,md = PsndPsmd

KjsndKjsmdkKjl2 . s8d

Hence,b jn,m=Cjsn,md /Cj

unsn,md is a measure of the correla-tions between cells. Ifb j

n,m is lower than 1 for some particu-lar value of j , it means that these kind of cells “repel” eachother in a topological sense. On the contrary, forb j

n,m.1 wefind an attraction of affinity betweenn andm cells at a dis-tancej . In particular, forj =1 the Aboav-Weaire law imposesan attraction between dissimilar cells and repulsion betweencells with the same number of sidessn=md. At large enoughtopological distancess j @1d the correlations vanish and sob j→`

n,m →1.In any two-dimensional Euclidean system the number of

cells per layer,Kjsnd, tends to be a linear function of thetopological distance forj @1:

Kjsnd = Cj + Bsnd, j @ 1. s9d

Since at such large distance the froth is also uncorrelateds j .jd, it follows thatC must be a constant number indepen-dent of the number of sides of the central cell. Analogously,the shape of the central cell is irrelevant at largej , and so theaverage number of sides of thej-layer cellsfmjsndg is simplygiven by the average number of sides of the whole system:i.e.,

mjsnd = knl = 6, j @ 1. s10d

An important statistical property is the so-called general-ized sum rule of Weaire, which states that


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ksmjsnd − ndKjsndl = 0. s11d

This rule holds for any value of the topological distancejand for any kind of froth. By using this general property andassuming that we are in aj layer where the cells are uncor-related from the central cells j .jd, one finally obtains thefollowing expression for the topological fluctuations:

G jsnd = G j−1 + ks6 − ndKjsndlFKjsndkKjl

− 1G, j . j.

s12d

B. Topological defects

We can distinguish two different types of cells in anyjlayer of the shell structure. The cells connecting neighborsfrom the layersj −1 and j +1 belong to theskeletonof thefroth. The cells that have neighbors in thes j −1d layer but notin the j +1 are calleddefects. Figure 2 illustrates an exampleof a shell structure with topological defectssgray cellsd. Thetotal cells of aj layer will be separated into skeletonsKj

skdand defectiveKj

d types—i.e.,Kjsnd=Kjsksnd+Kj

dsnd.Topological defects are in fact folds of the shell structure

and, asm2, give a rough estimate of the degree of disorder inthe froth. Indeed, they are not present in perfectly orderedfroths sas hexagonal or rectangular latticesd, but only in suchsystems where the Voronoi diagram is not constituted byregular cells. In our case, the Voronoi construction derivedfor a time-dependent two-dimensional system of aggregatingclusters is definitively a nonordered process, so the quantifi-cation of the number of defects during the coagulation willgive a characterization of the evolution of the cluster-clustercorrelations.

Without defects the outgoing convex vertexes from thejlayer are equal to the incoming concave vertexes to thes j+1d layer—i.e.,Vj

+snd=Vj+1− snd. However, if topological de-

fects are present in the froth, part of the outgoing vertexesare trapped in the defective cells. In this case, the previousrelation does not hold any more and a correction must beincluded in order to consider the lost of outgoing vertexes.One obtains

Vj+1− = Vj

+ − hKj+1d , s13d

whereh is the average number of sides lost in one defect.Note that we have explicitly omitted the dependence ofh onthe number of sides of the central seedsnd and the topologi-cal distancej since the value of this quantity is found to berather constant for all the defects in the whole system andclose to 1.

C. Particular case: j=2

If we have a system where the adjoining cells verify theAboav-Weaire law and where the cell-cell correlations van-ish after the second layerj=2, then it is possible to findsimple expressions for the most important topological quan-tities. Using the exact relation

Vj−snd = Kj

sksnd = Kjsnd − Kjdsnd, s14d

together with Eqs.s5d, s6d, s11d, ands13d, and following thedemostration given by Asteet al. in Ref. f32g, we find thatthe number of cells of the second and third shellsfK2snd andK3sndg can be approximated by

K2snd = 12 +m2 + s2 − adsn − 6d + s1 − hdK2dsnd,

K3snd = 18 +s4 − adm2 + S3 − 2a +s2 − ad2m2

12 +m2Dsn − 6d

− hK2dsnd + s1 − hdK3

dsnd. s15d

From these equations, we can obtain an explicit expres-sion for the asymptoic value of the average topologicalcharge,kQjl. Using the definition of the topological chargeand the generalized sum rule of WeairefEq. s11dg, we obtain

kQjl − kQj−1l = kf6 − mjsndgKjsndl = ks6 − ndKjsndl.

s16d

In the asymptotic limit,Kjsnd is given by Eq.s9d. Hence,

kQjl − kQj−1l = ks6 − ndlCj + ks6 − ndBsndl

= ks6 − ndBsndl, j ù j. s17d

As can be seen, the differenceskQjl−kQj−1ld is indepen-dent of the topological distance. In other words, the topologi-cal charge inside a layer is constant. Assuming now thatj=2 and using Eq.s15d, we have

kQjl − kQj−1l = ks6 − ndK2sndl = − s2 − adm2, j ù 2,

s18d

where the contribution of the defects has been neglected. Byiterating last expression, it follows that

kQjl = − s2 − adm2s j − 1d + kQ1l, s19d

wherekQ1l is the average topological charge contained up tothe first layer. It is given bykQ1l=k6−nl+kf6−m1sndgnl=ks6−ndnl=−m2. The final form forkQjl is then

kQjl = − m2s2 − ad j + s1 − adm2. s20d

By combining Eqs.s5d, s13d, ands14d, the following ex-pression for the average topological charge in thej layer isfound:

kQjl = k6 − Vj+ + Vj

−l

= 6 + k− Vj+1− − hKj+1

d + Vj−l

= 6 + kKj − Kj+1 − Kjd + s1 − hdKj+1

d l. s21d

We can deduce an analytic expression forC, the slope ofKjsnd for large j , by taking the asymptotic limitj @1 andusing Eqs.s9d and s20d in Eq. s21d:

− s2 − adm2j = 6 −C + s1 − hdKj+1d − Kj

d. s22d

It is appropriate to define the proportion of defects in thej layer as the ratioL j =kKj

dl / kKjl. With this definition andusing Eq.s9d, the last equation can be written as


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− s2 − adm2j = fs1 − hdL j+1 − L jgCj + s6 − Cd

+ s1 − hdL j+1sB + Cd − L jB. s23d

In the asymptotic limit, the proportion of defects convergesto a constant value independent ofj , lim j→` L j ;L. Hence,

− s2 − adm2j = hLCj + const, s24d

where const is a value independent ofj . The matching ofboth terms forj @1 imposes the following identity for theslopeC:

C =s2 − adm2

hL, s25d

which predicts thatC is directly correlated to the short- andlong-range disorder of the frothsm2 and L, respectivelyd.Finally, if kKjl shows a linear dependence already forj .2,then the average number of sides lost in defectsh can beapproximated byf32g

h =m2s2 − ad

f6 + s3 − adm2gL. s26d

In the next section, the short- and long-range topologicalproperties derived fromPsn; td and the correlations betweenneighboring and non-neighboring cells of the shell structurewill be studied for two-dimensional aggregation processes.Since colloidal coagulation is a time-dependent process, boththe probability and cell-to-cell correlations will be a functionof the aggregation time. Therefore,Psn; td will be the prob-ability of finding ann-sided cell at timet, Kjsn; td will be theaverage number of cells in thej layer around ann-sided cellat time t, and so on.

IV. RESULTS AND DISCUSSION

The purpose of this work is to study how the aggregationprocess affects the topological properties of a system of col-loidal clusters. The topological correlations will be then adirect consequence of the spatial structuring of the space-filling system of clusters. We have neglected the particle-particle interactions in order to pick up solely the effects ofthe pure diffusive aggregationsDLCAd and the monomerconcentration on the behavior of the topological properties.Brownian dynamics off-lattice DLCA simulations were per-formed, in a square box of sideL by considering a totalnumber of monomers ofN0=30 000 with a particle diameterd=735 nm and for five different packing fractions:fs=N0pr2/L2=0.005, 0.01, 0.03, 0.06, and 0.1. In the initialstate, monomers were placed at random avoiding particle-particle overlapping. Periodic boundary conditions are im-posed at the boundaries of the simulation surface. The simu-lations were stopped when the number of clusters wasaround 1500 aggregates. For the entire surface packing frac-tions considered this condition is enough to reach the scalingtime and it is a warranty to have good statistics for all ag-gregation times.

The diffusion coefficientsDd of a single cluster is ob-tained from its radius of gyrationsRgd assuming Stokes’ law

D,1/Rg. To move the aggregates they are selected one byone for a movement test. A random numberj uniformly dis-tributed inf0;1g is generated. IfjøD /Dmax=sRgdmin/Rg, thecluster is moved in a random direction. Ifj.D /Dmax, thecluster does not move.Dmax is the largest diffusion coeffi-cient andsRgdmin the smallest radius gyration in the wholesystem of clusters. Whether the cluster moves or not, thetime is increased by

Dt =1

NDmax. s27d

A collision is considered to occur when a moved aggre-gate overlaps another one. Then, the position of the movedcluster is corrected backwards along the direction of themovement as far as the surfaces of both clusters are in con-tact. These contacting clusters are joined to form a largercluster that will continue the diffusive motion in the follow-ing time step. For DLCA every collision produces coagula-tion. More technical details of the simulations can be foundin Ref. f16g.

A. Topological ordering between nearest neighbor clusters inthe scaling regime

In Fig. 3, the weight-average cluster size is plotted as afunction of the aggregation time for the five surface packingfractions.Swstd monotonically increases as time goes on, andafter the so called scaling timetsc ssee cross symbolsd, itreaches the scaling behavior given by a power-law growth:Swstd, tz. For higher particle concentrations the aggregationprocess is faster, and consequently the scaling region ismanifested at shorter times with a larger kinetic exponentzf46g ssee Table Id.

After this brief description of the kinetic properties, wewill focus on the topological aspects of the aggregation pro-cess. For all the studied situationssi.e., different times andinitial particle concentration conditionsd, the probability ofnumber of sidesPsnd was a centered distribution around themaximum value atn=6 and the first moment verified the

FIG. 3. Time evolution of the weight-average cluster size ob-tained from DLCA simulations for five surface packing fractions. Atlong times,Swstd shows a power-law behaviorsscalingd. The crosssymbols represent the scaling timetsc.


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Euler theoremsknl=6d. The second central momentm2 de-termines the dispersion ofPsnd, so it can be regarded as thekey quantity to characterize the global topological disorderof the system of clusters. As shown in Fig. 4,m2 decreaseswith time for all cases for short times. However, after a cer-tain time which corresponds to the scaling timetsc it finallyreaches a roughly time-independent value. This long-timebehavior has been previously observed by several authorsf15,17,19g. Indeed, experiments on soap froths indicate that,as time goes on, there is a final state where the topologicalproperties reach stable valuesf32g. Some experiments oftwo-dimensional colloidal coagulation performed by Earn-shaw and Robinson also show similar topological results fort. tsc f15g.

The decrease ofm2 reveals that the clusters formed in theDLCA regimen tend to adopt a hexagonal-like structure asthe aggregation proceeds; i.e., the aggregation process in-duces a topological ordering that is intimately related to thespatial organization of the clusters in the plane. As soon asthe scaling of the cluster-size distribution is manifested, thetopological ordering ends and the system of aggregatingclusters reaches a final stationary state with constantm2.Therefore, the dynamic scaling can be understood as a “sta-tionary” state in a topological point of view. Our simulationresults also show that the final stationary value ofm2 de-creases as the particle concentration increasessfrom m2=1.42 for fs=0.005 tom2=1.07 for fs=0.1d, which means

that aggregation induces stronger topological ordering fordense colloidal suspensions than for the dilute ones.

In terms of the intercluster structure, this ordering islinked to the growth of the colloidal clusters and the exis-tence of free space between them. Clusters are noncompactfractal-type structures with a special ability to fill the spaceeven for a relatively small number of particles. The typicaldistance between the surfaces of two nearest-neighbor clus-ters frsstdg is given by

rsstd = rccstd − 2Rstd, s28d

where rccstd is the average distance between centers of theclusters andRstd is the characteristic cluster radius.rs is ameasure of the size of the depletion zone around a cluster.The cluster growth is achieved at the expense of the sur-rounding particles and clusters, which are mopped up afterdiffusion through this depletion distance. When the dynamicscaling is established, the fractal dimension of the clustersdfis a well-defined quantity andRstd scales with the number-average cluster size asRstd,Snstd1/df. The center-to-centerdistance is, however, given byrccstd,Snstd1/2. Sincedf ,2,we observe thatRstd grows faster thanrccstd. At very lowparticle concentrations, this effect is not really importantsince the average cluster-cluster distance is much larger thanthe cluster radius at any aggregation time. However, forhigher particle densities, the free space available between acluster and its nearest neighbors becomes rapidly small com-pared to the effective surface filled by the fractal clusters.Then, the cluster structure is able to induce spatial order inthe cluster localization: the surrounding clusters around acentral one tend to adopt a more closely packed two-dimensional structuresthe more particle density we have, thestronger is the intercluster orderingd. Topologically, it yieldsthe increase of six-sided cell probability and the subsequentreduction ofm2 for more dense systems and long aggregationtimes. Since this effect is a consequence of the trend of frac-tal clusters to fill up the whole space, we do not expect thesame spatial organization when clusters are not fractal. Thisis the case of aggregation into a finite energy minimum,which allows the colloidal particles to rearrange inside thecluster and to form more compact structuresf47g.

The final stationary value form2 obtained during the scal-ing regime seems to be an intrinsic property of the self-assembly of the clusters linked to the value of the averagefractal dimension of the cluster. In fact, we expect departuresfrom our results form2 at very high particle densities andlong times, when the aggregating system is close to undergogelationsrs<0d, since the clusters lose their individual frac-tal character and become homogeneous structuresswith df=2d.

A similar steady-state behavior is found for screening fac-tor a obtained using the Aboav-Weaire law and the averagenumber of sides of the nearest-neighbor cells of ann-sidedcell, m1snd. The inset in Fig. 5 showsm1snd as a function of1/n for the particular case offs=0.01 after the scaling timessimilar curves are found for the rest of particle concentra-tions and aggregation timesd. This means that large clustersswith many number of sidesd tend to be surrounded by small

TABLE I. Kinetic exponentz and scaling timetsc for the fivesurface packing fractions studied in this work.

fs z s±0.01d tscssd

0.005 0.60 101.7

0.01 0.61 55.8

0.03 0.63 17.1

0.06 0.68 7.4

0.10 0.76 3.8

FIG. 4. Second central moment ofPsnd as a function of theaggregation time for five surface packing fractions. After the scalingtime m2 reaches a final stationary valuesdashed lines are onlyguides for the eyesd.


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onesswith low nd. Together with the simulated data, we rep-resent the fit using the Aboav-Weaire law given by Eq.s4dssolid lined. As observed, simulated and theoreticalm1sndagree well, except for the cells with large number of sidessn.9d where departures from the Aboav-Weaire predictionare found due to statistical uncertainties.

The screening factora obtained from the fits ofm1snd isnot an independent property but it is also strongly correlatedto the topological order of the system. In fact, for most of thefroths with weak and moderate disorders,a is a positivequantity and grows asm2 decreasesf48g. This behavior isalso found for the particular case of two-dimensional colloi-dal coagulation. Indeed, as observed from Fig. 5, the param-eter a first increases for times below the scaling time andarrives at a rather stationary state when the dynamic scalingis established. The final value ofa increases with particledensity, going froma=0.91 for fs=0.005 toa=1.07 for fs=0.1. Moreover, if we represent the simulated values ofa/m2for all times and for the five different surface packing frac-tions of particles as a functionm2, the results lie in a univer-sal curvessee Fig. 6d, given bya=Dsm2dw=1.17sm2d−0.89. Asimilar universal behavior is found for other natural orcomputer-simulated froth systems but with different valuesof D andw f35,48g. Since the exponentw may be consideredas a universality class feature of the froth evolution, we de-duce that the topological structuring induced by two-dimensional aggregation processes corresponds to a uniqueuniversality classsw=−0.89d which no longer depends onthe monomer density, but it is a general property linked tothe space-filling aspects of the cluster growth.

B. Correlations between nonadjacent clusters

This section extends the study of the topological proper-ties beyond the first nearest-neighbor cells. First we investi-gate the average number of cells around ann-sided cell for

the two nearest nonadjacent layers with respect to the centralseed:K2snd and K3snd. Figure 7 shows an example of thedependence of the simulatedK2snd andK3snd on the numberof sides ssymbolsd together with the theoretical predictionslinesd deduced for froths where the cell-cell topological cor-relations vanish after the second layerfsee Eq.s15dg andwhere the first cell is correlated according to the Aboav-Weaire law. As observed, this theoretical model reproducesfairly well the simulation data, although some departures arefound for cells withnù9 as a consequence of the failure ofthe Aboav-Weaire law for such many-sided cellsssee theinset of Fig. 5.d Although the complete set of results is notshown, similar agreement between theory and simulationwas found for all aggregation times and particle densities,supporting the idea that the aggregation process is incapableof producing correlations forj .2. Moreover, the values ofK2snd andK3snd, which characterize the shape of the secondand third layers of the shell structure, are basically given by

FIG. 5. Screening factora calculated by fittingm1snd accordingto the Aboav-Weaire lawfcf. Eq. s4dg. After the scaling timea alsoreaches a stationary valuesdashed lines are only guides for theeyesd. Inset: average number of sides of the cells adjacent to ann-sided cell calculated forfs=0.01 after the scaling timest=893 sd. The straight solid line represents the fit using the Aboav-Weaire law.

FIG. 6. The simulatedm2 anda/m2 data at any time and for thefive studied particle concentrations lie in a universal common form:a/m2=1.17sm2d−1.89. In this plot, them2 axis is representative of thetime axis, but in the opposite direction.

FIG. 7. Dependence of the average number of cells in the clos-est nonadjacent layers to the central one,K2snd and K3snd, on thesides numbern for DLCA simulations withfs=0.01 after the scal-ing time. The solid lines represent the theoretical expectations ob-tained using Eqs.s15d. Similar fittings are obtained for allsimulations.


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topological properties related to single-cell averages andfirst-layer correlations, as the second central momentm2 anda.

In the next step we go beyond the first three nearest layerswith the aim of studying the long-range topological proper-ties and their asymptotic limit. Using the quantities definedin Sec. III, we can describe the long-range ordering of thefroth induced by the aggregation process. First of all, westart our analysis with the average number of cellskKjl andthe average topological chargekQjl. Figures 8 and 9 show atypical example of the dependence of both quantities on thetopological distance for the particular case offs=0.06. Onthe one hand,kKjl shows a linear growth with the topologicaldistance after the second layers j .2d. On the other hand, thetopological charge decreases also linearly withj . Moreover,kQjl is well described by the theoretical predictionkQjl=−m2s2−ad j given in Eq.s20d for froths where the cell-cell correlations vanish after the second layer. Similar resultsare found for all times and other particle densities.

Both results corroborate an already mentioned fact: thespatial and topological correlations between two clusters are

negligible if the topological distance between them is largerthan 2. This conclusion has been independently confirmed bymeasuring the probabilitiesCjsn,md for several values ofnandm snot shownd. Indeed, the amplitude of the oscillationsof b j

n,m=Cjsn,md /Cjunsn,md is only important forj =1,2 and

becomes insignificant forj .2. Furthermore, since Eq.s20dholds for all particle concentrations and aggregation timessbefore and after the scalingd, we conclude that the aggrega-tion process is only able to induce short-range interclustercorrelationssup to j =j=2d even at high particle packingfraction: clusters in the third layer do not feel the propertiesof a given central one.

We now turn to the study of another important aspect ofthe froth: the topological defects. Their presence in the shellstructure has a dramatic influence on the cell assembly. Infact, for froths with large values of the proportion of defects,L j, the cell-cell correlations drop off very fast with the topo-logical distancej . It means that cluster-cluster correlationsbeyond nearest neighbors are mainly dominated by the per-centage of defects.

Figure 10 shows the proportion of defects as a function ofj for several aggregation times for the particular case offs=0.005ssimilar curves are found for the other particle den-sitiesd. In all casesL j increases withj , and for large dis-tances it reaches an asymptotic constant valueL, which fea-tures the long-range shell structure. As the coagulationprogresses, the whole curves go down and reach lower val-ues ofL. This decrease of the defect concentration as timeevolves clearly indicates that the aggregation process inducesordering not only between nearest neighborss j =1d, but alsobetween well-removed clusterss j @1d. It is remarkable that,for times beyond the scaling timestsc<100 sd, the curvessaturate and the topological ordering of the froth arrives at astationary state.

This behavior can be better observed if we represent theasymptotic value of the proportion of defectsL as a functionof time ssee Fig. 11d. Unfortunately, our simulated data go upto topological distancesj ø15 and we do not have valuableinformation at larger distances. This is mainly because of thepoor statistics that we have for such large distances, espe-

FIG. 8. Dependence of the average number of cellskKjl on thetopological distancej for DLCA simulation withfs=0.06 after thescaling time.kKjl grows linearly forj .2.

FIG. 9. Dependence of the average topological chargekQjl onthe topological distancej for DLCA simulations withfs=0.06 afterthe scaling timessquare symbolsd. The straight line is the theoreticalpredictionkQjl=−m2sa−2d j .

FIG. 10. Proportion of defects as a function of the topologicaldistance obtained for a DLCA simulation withfs=0.005. In thescaling regime st. tsc<100 sd, the curves become timeindependent.


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cially for long aggregation times, where the number of clus-ters is small and the analysis becomes affected by the bound-aries of the simulation box. Nevertheless, although the defectconcentration shown in Fig. 10 does not reach completelythis asymptotic value, there is a clear trend that points out theexistence of a constant value for large enough topologicaldistances. This constant value is rather logical since there isno difference between any shell and the next one for largevalues j . We refer the reader to Ref.f32g, where a similarbehavior for the proportion of defects is observed and a finalasymptotic stabilization is found. In this respect, ourasymptotic value is in fact an extrapolation of the simulateddata for largej .

In order to estimate this parameter, the simulated datahave been fitted according to the following empirical law:

L j = L +L1 − L

1 +S j − 1

j0Dp , s29d

whereL is the asymptotic value forj @1, L1 is the percent-age of defects for the first layer, andp and j0 are fittingparameters. The regression coefficient wasr .0.99 in all fit-tings and the value for the parameterp was 1.6±0.1 for alltimes and surface packing fractions.

What we obtain is a plot which resembles the one ob-tained form2 scf. Fig. 4d; i.e., L decreases with the aggrega-tion time for t, tsc and becomes constant fort. tsc. Oncemore the arrangement provoked by the aggregation is moreimportant for higher particle concentrations: the averagevalue found in the scaling regime decreases as we increasethe surface packing fraction up tofs=0.1. As discussed inSec. IV A for m2 anda parameters, the decrease of the num-ber of defects at all topological distances also accounts forthe fact that fractal cluster growth at high particle concentra-tions leads to stronger density modulation of the system andso to more structured close-packed structuressan ideal hex-agonal froth hasm2=0, a=0 andL=0d.

By using the simulated values ofL, m2, and a into ex-pressions26d we can calculate theoretically the average num-ber of sides lost in the defective cells,h. The behavior ofhdoes not show clear tendencies with aggregation time or par-ticle density. On the contrary, it remains roughly constant forall the studied situations and is given byh=1.2±0.1.

Finally, we study the behavior of the slopeC, obtainedfrom the fitting of kKjl in the asymptotic limit. The resultsare shown in Fig. 12 for the five studied particle densitiesand times before and after the scaling. Again we observedifferent behavior between the early stages of the aggrega-tion swhereC decreases monotonically withtd and the finalstationary value found in the scaling regime. Particularly,Ctakes values close to 11 at the beginning of the coagulationand decays to a final value that is smaller at higher particleconcentrationssfrom C=9.57 for fs=0.005 toC=8.82 forfs=0.1d.

The quantityCj can be understood as the average perim-eter of a j layer. For perfect spherical layers the value ofCshould be 2p<6.28. However, theC values calculated fromthe DLCA computer simulations are always higher at all ag-gregation times and particle densities, which is evidence forthe roughness of the shell structure, due not only to the factthat the cells do not form a smooth circle, but also to thepresence of topological defects. Indeed, shell structures withmany defective cells will have layers of larger perimeter andhigher values ofC. This fact establishes a connection be-tweenC and the defect concentrationL sC decrease withLd.Therefore, the long-range topological order induced by theaggregation process is also the responsable of the decrease inC with t during the prescaling regime.

The comparison between the simulated results forC andthe theoretical prediction given in Eq.s25d for froths uncor-related after the second layer is generally in good agreement.Although some departures are observed between simulateddata and theory, they are mainly due to statistical errors inthe determination in the parameters involved in such expres-sion. It proves thatC is not an independent variable, but it isattached to the values ofm2, a, andL. Similar dependenceshave been observed throughout this work for several topo-logical quantities, asK2snd, K3snd, and kQjl. Since the

FIG. 11. Asymptotic value of the proportion of defects as afunction of the aggregation time for the five studied surface packingfractions. After the scaling timeL reaches a final stationary valuesdashed lines are only guides for the eyesd.

FIG. 12. Time evolution of the slopeC of kKjl for the fivestudied surface packing fractions. After the scaling timeC also hasa constant value.


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asymptotic value of the defect concentrationL is also linkedto the global disorder in the frothsset bym2d, we concludethat all the short- and long-range topological properties in thesystem of aggregating clusters are controlled by the screen-ing factora and the second central momentm2. Both param-eters are mean-field properties that characterize the globaldisorder and the correlation between nearest-neighbor clus-ters, but do not inform anything about the long-range struc-ture. Thus, we find the surprising fact that the long-rangeorder is determined by average properties involving onlysingle-cell and nearest-neighbor correlations.

In our opinion, the fact that the long-range correlationsare determined by average properties as the second centralmomentm2 and the parametera is basically a consequenceof the lack of cell-cell correlations for topological distancesbeyond the second concentric layer,j .2. It allows us toexpress the topological properties forj .2 as functions ofthe properties forj ø2, which are mainly given bym2 anda.One good example of this may be seen in the study of theaverage topological chargekQjl fsee Eqs.s18d, s19d, ands20dg, where the lack of cell-cell correlations forj .2 leadsto an expression forkQjl only in terms of the properties ofcells with j ø2, asK2snd andkQ1l, which, in turn, are givenin terms ofm2 anda. For such froths where the topologicalcorrelations reach larger distances, this simple scheme willnot hold any more and a more sophisticated description willbe called for.

It should be note that these results obtained for two-dimensionals2Dd cluster-cluster aggregation cannot be, inprinciple, extrapolated to other types of space-filling struc-tures. For instance, the correlations between cells can be im-portant even forj .2, and then all the theoretical expressionsshown above will break down.

V. CONCLUSIONS

We studied the time evolution and topological propertiesof computer-simulated two-dimensional aggregating systemsfor several surface packing fractions of particles. We as-sumed the simplest case of freely Brownian diffusive stick-ing particles without interactionssDLCAd. The topologicalproperties have been measured with the help of the Voronoiconstruction, which replaces the original system of aggregat-ing clusters by an equivalent system of convex nonoverlap-ping regionsscellsd that completely tessellate the plane, eachcell representing a single cluster of the system and its sur-rounding depletion zones. The study of the correlations be-tween nonadjacent cells was achieved by analyzing the sys-tem as structured in concentric layers around a given centralone. This method allows us to go beyond the internal fractalstructure of the clusters and to study the external interclusterproperties. Indeed, it determines the short- and long-rangespatial organization and the cluster-cluster “interactions” in-duced by the mutual competition between neighboring deple-tion regions that occurs when the fractal-like clusters fill upthe whole space.

We found that the topological aspects of the aggregationare strongly connected to the kinetic ones. In all cases, weobtained a continuous ordering of the cell structure as aggre-

gation proceeds. Specifically, clusters tend to adopt a moreordered hexagonal organization in the plane, which topologi-cally means a reduction in both the second central momentof the number of sides distributionm2 and the proportion ofdefectsL. This ordering is more important for higher particledensities, where the already small free space existing be-tween growing clusters becomes rapidly shrunk due to thefractal growth. This ordering finishes as soon as the scalingof the cluster-size distribution is reachedst. tscd. Since allthe studied topological propertiesfPsnd, m2, a, kKjl, kQjl,and Lg remain constant in this final scaling limit, it can beregarded as a topological-invariant state. We also found thatthe whole coagulation processsafter and before the dynamicscalingd lies in a universal topological class independent ofthe particle concentration, given by the general relationa,m2

−0.89.After the comparison of the simulated data and the theo-

retical expressions, we concluded that colloidal aggregationin two dimensions is only able to produce short-range inter-cluster correlationssup to the second layer of clusters arounda central one,j =2d even for the very high packing fraction ofparticles. However, the ordering is manifested in the wholeshell structure at any topological distance in terms of a de-crease of the proportion of defective cellsL j. Finally, weobtained that the main topological property that controls thedegree of structuring at short and long intercluster distancesis the second central momentm2, which is a measurement ofthe global order in the whole system.

Future investigations will involve the study of the areadistribution of the cellsPsAd fdefined asPsAd=on=3

` Psn,Adgof a system of clusters under coagulation in two dimensions.Particularly, it would be interesting to know the time evolu-tion and scaling properties in the long-time scaling state.Moreover, since each cell corresponds to a colloidal clusterformed by a certain number of particlesi, one could con-struct the distributionPsn,A, id, as the probability of findingan n-sided cell with areaA that contains a clusters of sizei.The correlation between these three variables will give us adirect relation between kinetic featuressthat comes throughid and the topological onessn and Ad. Finally, in a logicalstep forward, it is highly interesting to extend the topologicaland structural description to three-dimensional aggregatingsystems of dense colloidal suspensions and to colloidal sys-tems with interacting particles as those with long-rangeparticle-particle repulsive interactions. Note the differencebetween our results and earlier results in 3Df13,14g, where itis found that, after the depletion region, the particle-particlecorrelation function tends to 1 after reaching the nearest-neighbor clusters, without further oscillations in this functionsbesides those coming for the statistical uncertaintiesd. It ispossible that the clusters in 3D, due their ability to interpen-etrate more easily without touchingf46g, “feel a lower repul-sion between them,” while in 2D the greater “repulsion”would make the clusters to become more ordered, increasingin this way thej to a value equal 2. Or perhaps the absenceof further oscillations is due to having considered theparticle-particle correlations function and not the cell-cellcorrelations as in the present work. All this merit a similarstudy for the three-dimensional case, as stated above.


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ACKNOWLEDGMENTS

The authors acknowledge the financial support fromSpanish “Ministerio de Educación y Ciencia, Plan Nacional

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short- and long-range topological correlations in two-dimensional aggregation of dense colloidal...

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