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Slow dynamics in gelation phenomena: From chemical gels to colloidal glasses

Emanuela Del Gado,a,d, Annalisa Fierro,b,d, Lucilla de Arcangelis, c,d and Antonio Coniglio b,d

aLaboratoire des Verres, Universite Montpellier II, 34095 Montpellier, FrancebDipartimento di Scienze Fisiche, Universita di Napoli ”Federico II”,

Complesso Universitario di Monte Sant’Angelo, via Cintia 80126 Napoli, ItalycDipartimento di Ingegneria dell’Informazione, Seconda Universita di Napoli,

via Roma 29, 81031 Aversa (Caserta), Italy andd INFM Udr di Napoli and Gruppo coordinato SUN

(Dated: February 2, 2008)

We here discuss the results of 3d MonteCarlo simulations of a minimal lattice model for gellingsystems. We focus on the dynamics investigated by means of the time autocorrelation functionof the density fluctuations and the particle mean square displacement. We start from the case ofchemical gelation, i.e. with permanent bonds, and characterize the critical dynamics as determinedby the formation of the percolating cluster, as actually observed in polymer gels. By opportunelyintroducing a finite bond lifetime τb, the dynamics displays relevant changes and eventually theonset of a glassy regime. This has been interpreted in terms of a crossover to dynamics more typicalof colloidal systems and a novel connection between classical gelation and recent results on colloidalsystems is suggested. By systematically comparing the results in the case of permanent bonds tofinite bond lifetime, the crossover and the glassy regime can be understood in terms of effectiveclusters.

I. INTRODUCTION

The gelation transition transforms a viscous liquid intoan elastic disordered solid. In general this is due to theformation in the liquid phase of a spanning structure,which makes the system able to bear stresses.In polymer systems, this is due to chemical bonding, thatcan be induced in different ways [1, 2], producing a poly-merizations process. As firstly recognized by Flory, thechange in the viscoelastic properties is directly relatedto the constitution inside the sol of a macroscopic poly-meric structure, that characterizes the gel phase. In ex-periments [3] the viscosity coefficient grows as a powerlaw as function of the relative difference from the criticalpolymer concentration with a critical exponent k. Theonset of the elastic response in the system, as functionof the same control parameter, displays a power law in-creasing of the elastic modulus with a critical exponent f .As implicitly suggested in the work of Flory and Stock-mayer [1], the percolation model is considered as the basicmodel for the chemical gelation transition and the macro-molecular stress-bearing structure in these systems is apercolating network [2, 4, 5]. In experiments the gellingsolution typically displays slow dynamics: the relaxationfunctions present a long time stretched exponential de-

cay ∼ e−( tτ0

)β

as the gelation threshold is approached.In particular at the gel point the relaxation process be-comes critically slow, and the onset of a power law decayis observed [6].

In many other physical systems where aggregation pro-cesses and structure formation take place, gelation phe-nomena can be observed. Typically, these are colloidalsystems, i.e. suspensions of mesoscopic particles inter-acting via short range attraction. These systems areintensively investigated due to their relevance in manyresearch fields (from proteins studies to food industry).

Due to the possibility in experiments of opportunely tun-ing the features of the interactions, they also play the roleof model systems.

In these systems strong attraction gives rise to a diffu-sion limited cluster-cluster aggregation process and mayproduce gel formation (colloidal gelation) at very lowdensity as a spanning structure is formed [7]. The latteris generally quite different from the polymer gels case [8]however, for what the viscoelastic behavior is concerned,this gelation transition closely resembles the chemicalone, observed in polymer systems [9]. With a weaker at-traction at higher densities a gelation characterized by aglass-like kinetic arrest [10, 11] may be observed. Therelaxation patterns closely recall the ones observed inglassy systems and are well fitted by the mode-couplingtheory predictions for super-cooled liquids approachingthe glass transition [12]. On the theoretical side the ap-plication of the mode-coupling theory to systems withshort range attractive interactions [13, 14, 15] (attrac-tive glasses) has been recently considered and the con-nection with the colloidal glass transition has been pro-posed. The short range attraction enhances the cagingmechanism characteristic of glassy regimes in hard spheresystems and produces a glassy behavior at lower densi-ties, depending on the attraction strength.

At lower densities, the role of the structure formation,as directly observed in some systems [16], might be rel-evant in the dynamics but it has not been clarified yet.Also the eventual crossover to the glassy regime [9], asthe density is increased, is not completely understood. Inthis paper we investigate the connection among colloidalgelation, colloidal glass transition and chemical gelation.Some preliminary studies have been reported in [17].

We have considered a minimal model for gelling sys-tems and performed extensive numerical simulations on3d cubic lattices. In Sect. II we give the details of themodel and the simulations. In Sect.s III and IV the re-

2

sults relative to relaxation and diffusion properties arepresented and discussed. In the last section some con-cluding remarks are given.

II. DESCRIPTION OF THE MODEL

A. Permanent bonds

Our model consists in a solution of monomers diffus-ing on a cubic lattice. As in most experimental polymergels, we choose the monomers to be tetrafunctional. Eachmonomer occupies a lattice elementary cell, and thereforeeight vertices on the lattice. To take into account theexcluded volume interaction, two monomers cannot oc-cupy nearest neighbors and next nearest neighbors cellson the lattice, i.e. nearest neighbor monomers cannothave common sites. At t = 0 we fix the fraction φ ofpresent monomers with respect to the maximum num-ber allowed on the lattice, and randomly quench bondsbetween them. This actually corresponds to the typicalchemical gelation process that can be obtained by irra-diating the monomeric solution. We form at most fourbonds per monomer, randomly selected along lattice di-rections and between monomers that are nearest neigh-bors and next nearest neighbors (namely bond lengthsl = 2, 3). Once formed, the bonds are permanent.

For each value of φ there is an average value Nb(φ)of the fraction of formed bonds with respect to all thepossible ones, obtained by averaging over different initialconfigurations.

Varying φ the system presents a percolation transitionat φc = 0.718 ± 0.005 [18]. The critical exponents foundat the transition agree with the random percolation pre-dictions [19] (e.g. for the mean cluster size γ ≃ 1.8±0.05and for the correlation length ν ≃ 0.89± 0.01 in 3d [18]).

The monomers diffuse on the lattice via random lo-cal movements and the bond length may vary but notbe larger than l0 according to bond-fluctuation dynam-ics (BFD) [20], where the value of l0 is determined bythe self-avoiding walk condition. On the cubic lattice wehave l0 =

√10 in lattice spacing units and the allowed

bond lengths are l = 2,√

5,√

6, 3,√

10 [21]. We let themonomers diffuse to reach the stationary state and thenstudy the system for different values of the monomer con-centration.

This lattice model with permanent bonds has been in-troduced to study the critical behavior of the viscoelasticproperties at the gelation transition[22]. The relaxationtime is found to diverge at the percolation threshold φc

with a power law behavior [18]. The elastic response inthe gel phase has been studied by means of the fluctu-ations in the free energy and goes to zero at φc with apower law behavior as well [23].

B. Bonds with finite lifetime

Colloidal gelation is due to a short range attraction andin general the particles are not permanently bonded. Totake into account this crucial feature we introduce a novelingredient in the previous model by considering a finitebond lifetime τb and study the effect on the dynamics.

The features of this model with finite τb can be real-ized in a microscopic model: a solution of monomers in-teracting via an attraction of strength −E and excludedvolume repulsion. Due to monomer diffusion the aggrega-tion process eventually takes place. The finite bond life-time τb is related to the attractive interaction of strength−E, as τb ∼ eE/KT .

We start with the same configurations of the previouscase, with a fixed φ where the bonds have been randomlyquenched as described above. During the monomer diffu-sion with BFD at every time step we attempt to breakeach bond with a frequency 1/τb. Between monomersseparated by a distance less than l0 bonds are thenformed with a frequency fb [24]. In order to obtainmonomers configurations that are similar to the ones withpermanent bonds, for each value of τb we fix fb so thatthe fraction of present bonds coincides with its averagevalue in the case of permanent bonds, Nb(φ) [26].

With respect to the case of permanent bonds we no-tice that, as the finite bond lifetime τb corresponds to anattractive interaction of range l0, it actually introducesa correlation in the bond formation and may eventuallylead to a phase separation between a low density and ahigh density phase: There is no evidence of phase separa-tion for the values of τb and fb considered in this paper.This is evident in Fig.1, where typical equilibrium con-figurations with φ = 0.6 are shown in two different cases:in Fig.1(a) we have the case considered in this paper,obtained with τb = 100 and fb = 0.02; in Fig.1(b) withτb = 2 and fb = 1 the phase separation may occur. Thechoice of monomers of functionality 4, also in this caseof finite bond lifetime, may correspond to a directionaleffect in the interaction [27].

The case of extremely large τb gives rise to different sit-uations depending on the initial condition and the bondcreation process. We consider two extreme cases: I)Start with the initial configuration where the monomersare randomly distributed, and the bonds are randomlyquenched. At later times the frequency of forming bondsis zero. This case corresponds to the permanent bondcase (chemical gelation), described in section II A. II)Start with a random configuration of monomers and letthe monomers diffuse and form bonds with infinite life-time and frequency fb. This phenomenon of irreversibleaggregation, with the occurrence of gelation after a span-ning cluster is formed, corresponds to cluster-cluster ag-gregation class of models for fb = 1 [28], with a tendencytowards cluster-cluster reaction limited aggregation pro-cess [29] for fb < 1. This out of equilibrium phenomenoncan be representative of colloidal gelation and will notbe considered here. In chemical gelation and colloidal

3

gelation the formation of the critical cluster should pro-duce the slow dynamics. The main difference is due tothe fact that the critical density and the temperature aremuch lower in colloidal gelation than in chemical gela-tion, moreover the fractal dimension is related to cluster-cluster aggregation models and not to random percola-tion.

III. RELAXATION FUNCTIONS

In order to investigate the dynamic behavior we study,for both the permanent bond and the finite bond lifetimecases, the equilibrium density fluctuation autocorrelationfunctions, f~q(t), given by

f~q(t) =< ρ~q(t + t′)ρ−~q(t

′) >

< |ρ~q(t′)| >2(1)

where ρ~q(t) =∑N

i=1 e−i~q·~ri(t), ~ri(t) is the position of thei− th monomer at time t, N is the number of monomersand the average 〈...〉 is performed over the time t′. Dueto periodic boundary conditions the values of the wavevector ~q on the cubic lattice are ~q = 2π

L (nx, ny, nz) withnx, ny, nz = 1, ..., L/2 integer values (in our simulationsa cubic lattice of size L = 16 has been considered) [30]. Inthe following we discuss the data obtained for q ∼ 1.36(~q = (π/4, π/4, π/4 )). Qualitatively analogous behav-iors have been observed for (~q = (π/2, π/2, π/2 ) and(π, π, π)). Undoubtly, due to structure formation overdifferent length scales, a detailed study of the geometricproperties and the dynamics for different wave vectorsmight be relevant [31].

In the case of permanent bonds the system is consid-ered at equilibrium when both the diffusion coefficienthas reached its asymptotic limit, and the autocorrelationfunctions have gone to zero. For φ < φc we are alwaysable to thermalize the system, instead for φ > φc it re-mains out of equilibrium, and it is possible that it is inan aging regime [31]. In Fig.2, f~q(t) is plotted as functionof the time for different values of the monomer concen-tration. The data have been averaged over 40 differentinitial configurations. The different curves correspond todifferent values of φ, ranging from 0.5 to 0.85. At lowconcentrations the system completely relaxes within thesimulation time; the relaxation process becomes sloweras the concentration is increased and above the percola-tion threshold, φc, the system is kinetically arrested, inthe sense that the relaxation functions do not go to zerowithin the time scale of the simulations.

We analyze more quantitatively the long time decayof f~q(t) in Fig.3: As the monomer concentration, φ, ap-proaches the percolation threshold, φc, f~q(t) displays along time decay well fitted by a stretched exponential

law ∼ e−(t/τ)β

with a β ∼ 0.30 ± 0.05. Intuitively, thisbehavior can be related to the cluster size distributionclose to the gelation threshold, which produces relaxationprocesses taking place over different length scales. At

the percolation threshold the onset of a power law decay(with an exponent c) is observed as shown by the doublelogarithmic plot of Fig.3 [6]. As the monomer concen-tration is increased above the percolation threshold, i.e.in the gel phase, the long time power law decay of therelaxation functions can be fitted with a decreasing expo-nent c, varying from c ∼ 1.0 at φc to c ∼ 0.2 well aboveφc, where a nearly logarithmic decay appears. This sug-gests that the growth of the relaxation time is driven bythe formation of the critical cluster, that actually deter-mines the kinetic arrest. On the whole, this behaviorwell agrees with the one observed in gelling systems in-vestigated in the experiments of refs.[6]. It is interestingto notice that this kind of decay with a stretched expo-nential and a power law reminds the relaxation behaviorfound in spin-glasses [32]. Many analogies in the dynam-ics of gels and spin-glasses have been recently pointed out[33], but the underlying physics is rather unclear.

In the model with finite lifetime bonds, the equili-bration time is an increasing function of τb. The sys-tem is considered at equilibrium when both the averagenumber of bonds has reached its asymptotic limit andthe autocorrelation functions have gone to zero [34]. Inthis case very different behaviors are observed. In Fig.4f~q(t) is plotted as function of time for a fixed value ofτb = 10, 100, 1000 (respectively Fig.4a, 4b and 4c) for in-creasing values of the monomer concentration (φ varieson the same range as the permanent bond case). At lowconcentrations, the behavior of the autocorrelation func-tion f~q(t) is apparently very similar to the one observed inthe case of permanent bonds: the system completely re-laxes within the simulation time scale and the relaxationtime increases with the concentration φ. At high con-centrations, a two-step decay appears, closely resemblingthe one observed in super-cooled liquids. This qualita-tive behavior is observed for many different values of thebond lifetime, τb. As shown in Fig. 4, the two step decayis more pronounced for higher values of τb.

As we can see in Fig.5, where the long time decay off~q(t) for τb = 100 is shown, the long time decays arewell fitted by stretched exponentials. The exponent β(β ∼ 0.7 for the case of Fig.5) does not seem to vary sig-nificantly as the concentration varies, and this has beenobserved for all the values of τb studied. It instead de-creases as τb increases: for very small τb one recovers along time exponential decay whose behavior becomes lessand less exponential as the bond lifetime increases. Thissuggests that the stretched exponential decay is due tothe presence of long living structures.

For high monomer concentrations we fit the f~q(t)curves using the mode-coupling β-correlator [12], corre-

sponding to a short time power law ∼ f +(

tτs

)

−a

and

a long time von Schweidler law ∼ f −(

tτl

)b

. In Fig.6

we show the agreement between the fit (the full lines)and the data for τb = 1000 in the range of concentrationsφ = 0.8−0.9. The exponents obtained are a ∼ 0.33±0.01

4

and b ∼ 0.65±0.01. At long times the different curvesobtained for different φ collapse onto a unique mastercurve by opportunely rescaling the time via a factor τ(φ)(Fig.7). The master curve is well fitted by a stretched ex-ponential decay with β ∼ 0.50±0.06. The characteristictime τ(φ) diverges at a value φg ∼ 0.963±0.005 with theexponent γ ∼ 2.33±0.06 (Fig.8). This value well agreeswith the mode-coupling prediction γ = 1/2a + 1/2b [12].

The same behavior and the same level of agreementbetween the data and the mode coupling predictionshave been obtained for different values of τb (τb =100, 200, 400, 1000, 3000) . Neither the exponents a andb obtained by the β-correlator nor the exponent β of thestretched exponential vary significantly as function of φand of τb. The value of φg , where the characteristic timeτ(φ) apparently diverges, seems instead to vary with τb,but it is always close to φg = 1.

This glassy relaxation pattern suggests that also inthis case the relaxation takes place by means of a cagingmechanism: particles are trapped in a cage formed bythe surrounding ones, the first relaxation step is due tomovements within this cage, whereas the final relaxationis possible due to cage opening and rearrangement. Wenotice that, contrary to the usual behavior observed insuper-cooled liquids and predicted by the Mode-CouplingTheory, the value of the plateau of the relaxation func-tions, which is typically related to the size of the cage,strongly depends on the monomer concentration, φ. Thiseffect will be explained later in terms of effective clusters.

IV. THE RELAXATION TIME AND THE ROLE

OF EFFECTIVE CLUSTERS

We study now the relaxation times that can be ob-tained from the f~q(t), as the time τ such that f~q(τ) ∼ 0.1.In Fig.9 the relaxation time τ is plotted as function ofthe monomer concentration, φ, for the permanent bondsand for the finite lifetime bonds case at different valuesof τb. In the figure one finds the data for the permanentbond case on the left, and then from left to right the datafor decreasing values of bond lifetime, τb.

In the case of permanent bonds (chemical gelation),τ(φ) displays a power law divergence at the percolationthreshold φc. For finite bond lifetime the relaxation timeinstead increases following the permanent bond case, upto some value φ∗ and then deviates from it. The longerthe bond lifetime the higher φ∗ is. For higher φ the fur-ther increase of the relaxation time corresponds to theonset of the glassy regime in the relaxation behavior dis-cussed in the previous section. This truncated criticalbehavior followed by a glassy-like transition has been ac-tually detected in some colloidal systems in the viscositybehavior [35, 36].

In both cases of permanent bonds and finite lifetimebonds, clusters of different sizes are present in the sys-tem. In the permanent bond case, a cluster of radiusR diffuses in the medium formed by the other percola-

tion clusters with a characteristic relaxation time τ(R).At the percolation threshold the connectedness lengthcritically grows in the system and so does the overall re-laxation time. In the case of a finite bond lifetime τb, itwill exist a cluster size R∗ so that τb < τ(R∗). That is,clusters of size R ≥ R∗ will break and lose their identityon a time scale shorter than τ(R) and their full size willnot contribute to the enhancement of the relaxation timein the system. We can say that the finite bond lifetimeactually introduces an effective cluster size distributionwith a cut-off and keeps the macroscopic viscosity finitein the system [37].

At high concentrations the system approaches a glassyregime and the relaxation time increases. In order tofurther investigate the high concentration regime, inFig.10 we directly compare f~q(t) at fixed φ = 0.85 forτb = 10, 100, 400, 1000, and the permanent bond case.We observe that at a fixed value of the monomer concen-tration, φ, the relaxation functions calculated for finitelifetime bonds coincide with the permanent bond case upto times of the order of τb. This suggests that on timescales smaller than τb the relaxation process must be onthe whole the same as in the case of permanent bonds,where permanent clusters are present in the system, andgives an interpretation in terms of effective clusters forthe two step glassy behavior of the relaxation functions:The first step is due to the relaxation of a cluster withinthe cage formed by the other clusters, whereas the sec-ond step is due to the breaking of clusters. This secondrelaxation is the analog of the cage opening in an ordi-nary supercooled liquid. In conclusion, on a time scaleof the order of τb, the effective clusters play the role ofsingle molecules in an ordinary supercooled liquid, or ina colloidal hard sphere system.

Using this picture of effective clusters, we are able toexplain the increase of the plateau in fq(t). In fact, sincedifferent values of the monomer concentration correspondto different effective cluster size distributions, for eachvalue of φ one has a different glassy liquid of effectiveclusters. This will change the first relaxation and shouldcorrespond to a change in the plateau of the relaxationfunctions (Fig.4 and 6). In particular we find that forhigher φ one has a higher plateau, that is the first de-cay (the motion of clusters within the cages) produces asmaller relaxation in the system.

V. DIFFUSION PROPERTIES

In order to obtain further information on the dynam-ics we calculate the mean square displacement of all the

particles, 〈~r2(t)〉 = 1N

∑Ni=1〈(~ri(t + t′) − ~ri(t

′))2〉. In themodel with finite lifetime bonds, clusters continuouslyevolve in time and therefore the diffusion coefficient of asingle cluster cannot be defined.

In the model with permanent bonds the mean squaredisplacement of the particles 〈~r2(t)〉 presents a long-timediffusive behavior, and the diffusion coefficient decreases

5

but remains finite also above φc. This is due to the factthat the infinite cluster can be viewed as a net with alarge mesh size, through which monomers and small clus-ters can diffuse.

In previous papers [18] the diffusion coefficient of clus-ters with a fixed size was studied. We found that clusters,whose size is comparable with the connectedness length,present a diffusion coefficient going to zero at φc (withthe same exponent as the relaxation time), whereas singlemonomers present a finite diffusion coefficient also in thegel phase. As we have already noticed, this is due the factthat small clusters are able to escape through the perco-lating cluster having a structure with holes over many dif-ferent length scales close to the percolation threshold. Itis therefore clear that the behavior here observed for themean square displacement is mainly due to the diffusionof single monomers and small clusters. In Fig.11, 〈~r2(t)〉is plotted as function of time in a double logarithmic plotfor φ approaching φc in the case of permanent bonds:particles can still diffuse, and the diffusion coefficient ap-parently decreases with the monomer concentration. Athigh concentrations the sub-diffusive regime stays longer,and the long time diffusive behavior is hardly recovered.

In Fig.12, we plot the data obtained with τb = 1000and φ = 0.8, 0.82, 0.85, 0.9. According to the results justdiscussed, for low concentrations 〈~r2(t)〉 shows a simplediffusive behavior. The diffusion does not change signifi-cantly close to φc and for high concentrations the behav-ior observed, characterized by a plateau, is similarly toglass forming systems. This onset of a glassy regime hasbeen obtained for different values of τb, and again it indi-cates a caging mechanism in the dynamics. The asymp-totic diffusion coefficient goes to zero as φ approaches φg

(inset of Fig.12), as a power law, with an exponent closeto γ (section IV) in agreement with the Mode CouplingTheory predictions.

As already done for the relaxation functions we directlycompare the mean square displacement obtained in thecases of permanent bonds and finite bond lifetime. Fig.13shows for a fixed value of the concentration, φ = 0.85,that the two quantities coincide up to time scales of theorder of τb. For longer times in the system with nonpermanent bonds the final diffusive regime is recovered.These results are coherent with the behavior of the re-laxation functions discussed in the previous section. Thefirst regime is apparently related to the diffusion of effec-tive clusters. Here again the value of the plateau in thediffusion pattern, which is related to the size of the cage,varies with the concentration φ. For higher values of themonomer concentration, the size of the cage apparentlydecreases. This corresponds to larger effective clusters,which have less free space compared to their size. Bymeans of the qualitative argument used in section IV,

one expects that for a longer τb the condition τb < τ(R∗)will be fulfilled by a larger size R∗, and on average largerclusters will persist. For the same value of the concen-tration, the size of the cage should be the same, whereasthe particles of this glassy system (i.e. the effective clus-ters) are longer trapped in the cage as the bond lifetimeincreases (Fig.13).

VI. DISCUSSION AND CONCLUSION

We have studied a model for gelling systems both inthe case of permanent bonds and finite bond lifetime.The study of the dynamics shows that when bonds arepermanent (chemical gelation) the divergence of the re-laxation time is due to the formation of a macroscopiccritical cluster and the decay of the relaxation functionsis related to the relaxation of such cluster. In the caseof finite τb there is an effective cluster size distribution,with a size cutoff. Note that the clusters cannot be easilydefined as in the case of chemical gelation: The effectiveclusters do not coincide with pairwise bonded particles.A cluster can be identified in a statistical sense as a groupof monomers which keeps its identity (i.e. the bondsare unbroken) when diffusing a distance of the order ofits diameter. The formation of effective clusters leadsto an apparent divergence of the relaxation time whichis characterized by exponents corresponding to the caseof random permanent bonds (random percolation). Asthe monomer density increases the presence of effectiveclusters further slows down the dynamics, until a glasstransition is reached.

In the case τb → ∞, starting with a random config-uration of unbonded monomers one obtains an out ofequilibrium state as in cluster-cluster aggregation mod-els, which can be representative of colloidal gelation. Ide-ally this out of equilibrium system is connected to the twolines described above, the pseudo percolation line and theglassy line. The pseudo percolation line can be detectedif the effective cluster size is large enough and it is quitedistinct from the glassy line. However both lines inter-fere at low densities and low temperatures with the phasecoexistence curve.

We would like to thank K. Dawson, A. de Candia,G. Foffi, W. Kob, F. Mallamace, N. Sator, F. Sciortino,P. Tartaglia and E. Zaccarelli for many interesting dis-cussions. This work has been partially supported bya Marie Curie Fellowship of the European Communityprogramme FP5 under contract number HPMF-CI2002-01945, by MIUR-PRIN 2002, MIUR-FIRB 2002, CRdC-AMRA, and by the INFM Parallel Computing Initiative.

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Phys.Rev.E 65 041803 (2002)[24] Taking into account both the interaction energy by means

of τb and the frequency fb for the bonds corresponds tohave a free energy barrier in the bond formation [25]:Consider a pair of particles which can be in Ω+1 config-urations. Ω (unbonded) corresponds to zero energy and 1(bonded) corresponds to energy −E. Breaking of a bondcorresponds to go from a state of energy −E to a state of

entropy S = K ln Ω, with a variation in the free energy∆F = E −KT lnΩ. The lifetime of the bonding configu-ration is proportional to eE/KT , and the frequency to gofrom one of the unbonding configurations to the bondingconfiguration is given by fb = 1

Ω.

[25] A. Coniglio, H.E. Stanley, W. Klein Phys.Rev.B 25

(1982) 6805[26] We consider one starting configuration with a fixed φ

as obtained in the case of permanent bonds. For eachvalue of the bond lifetime τb we perform test simulationsfor different values of fb. The fraction of formed bondsrespect to all the possible ones varies during an initialtransient and then fluctuates around an average valuewhich, for a fixed φ, depends on τb and fb. For the studyof the dynamics, for each value of τb we take the fb sothat the average value of the fraction of present bondsduring the simulations is the same as its value at thesame φ with permanent bonds.

[27] The role of directional effects in the attractive interactionin colloidal gels is not clear and is worth to be furtherinvestigated.

[28] R. Jullien and A. Hasmy Phys. Rev. Lett. 74 4003 (1995)[29] M. Kolb and R. Jullien, J. Phys. (Paris) Lett. 45, L977

(1984); F. Family, P. Meakin, and T. Vicsek, J. Chem.Phys. 83, 4144 (1985).

[30] After having thermalized the system at a given value ofthe monomer concentration, φ, the autocorrelation func-tions of the density fluctuations, f~q(t), are calculated bytime averages over runs of ∼ 107MCstep/particle. Theerrors are calculated as the fluctuations on the statisticalensemble (∼ 20 − 30 different realizations of the systemhave been considered). Where not explicitly shown, theerror bars are of the order of the symbol size in the fig-ures.

[31] E. Del Gado, A. Fierro, L. de Arcangelis and A. Coniglioin preparation.

[32] A. T. Ogielski Phys. Rev. B 32 7384 (1985).[33] A. Parker and V. Normand cond-mat/0306056.[34] Although with these two conditions we cannot exclude

that we are in an aging regime, we do not expect to findaging, as we are not in the glassy region [31].

[35] F. Mallamace, S.H. Chen, Y. Liu, L. Lobry and N. MicaliPhysica A 266 (1999); F. Mallamace, R. Beneduci, P.Gambadauro, D. Lombardo and S.H. Chen, Physica A302 202 (2001)

[36] F. Lafleche, D. Durand and T. Nicolai Macromolecules36 1331 (2003)

[37] A. Coniglio J. Phys. : Condensed Matter 13 9039 (2001)

7

FIG. 1: Two typical configurations obtained for φ = 0.6with τb = 100 and fb = 0.02 (a), where there is no evidenceof phase separation, and with τb = 2, fb = 1 and monomersof valence 6 (b), where the phase separation occurs.

8

FIG. 2: The relaxation functions for permanent bonds f~q(t)as function of the time for q ∼ 1.36 and, from left to right,φ = 0.5, 0.6, 0.68, 0.718, 0.75, 0.8, 0.85.

FIG. 3: Double logarithmic plot of the autocorrelationfunctions f~q(t) as function of the time for q ∼ 1.36 andφ = 0.6, 0.718, 0.8, 0.87. For φ < φc the long time decay is

well fitted by a function (full line) ∼ e−(t/τ)β

with β ∼ 0.3.At the percolation threshold and in the gel phase in the longtime decay the data are well fitted by a function ∼ (1+ t

τ ′ )−c.

9

10

FIG. 4: f~q(t) as function of the time for q ∼ 1.36calculated on a cubic lattice of size L = 16: for φ =0.6, 0.7, 0.8, 0.85, 0.87, 0.9 (from left to right) and τb =10MCstep/particle (a); τb = 100MCstep/particle (b); τb =1000MCstep/particle (c); the dotted lines are a guide to theeye.

FIG. 5: The long time decay of f~q(t) in a log-log plot forq ∼ 1.36. It has been calculated on a cubic lattice of sizeL = 16 for τb = 100MCstep/particle (from left to rightφ = 0.65, 0.68, 0.718). The data are fitted using a stretchedexponential function with β ∼ 0.7 (full lines).

11

FIG. 6: f~q(t) as function of the time for q ∼1.36 calculated on a cubic lattice of size L = 16 forτb = 1000MCstep/particle. From left to right φ =0.8, 0.85, 0.87, 0.9. The full lines correspond to the fit withthe β-correlator.

FIG. 7: f~q(t) obtained for q ∼ 1.36, τb = 1000, and φ =0.85, 0.87, 0.9, 0.91, 0.92: by opportunely rescaling them by aquantity τ (φ) they collapse into a unique master curve, wellfitted by a stretched exponential function with β ∼ 0.5±0.06.

12

FIG. 8: Log-log plot of the characteristic time τ (φ) obtainedby the rescaling of the relaxation functions. The points havebeen fitted (full line) with the function 0.006(φg − φ)−2.33,where φg ∼ 0.96 ± 0.01 .

FIG. 9: The average relaxation time as function of thedensity; from left to right: the data for the permanentbond case diverge at the percolation threshold with a powerlaw (the full line); the other data refer to finite τb =3000, 1000, 400, 100MCstep/particle decreasing from left toright (the dotted lines are a guide to the eye). The apparentdivergence of the relaxation time, τ , is observed at the perco-lation threshold of the permanent bond case, φc = 0.718, forall the values of τb.

13

FIG. 10: f~q(t) obtained for φ = 0.85 and q ∼ 1.36: thedifferent curves refer to τb = 10, 100, 400, 1000, compared tothe permanent bond case (from left to right).

FIG. 11: The mean-square displacement 〈~r2(t)〉 of the par-ticles as function of the time in a double logarithmic plot forpermanent bonds: from top to bottom φ = 0.4, 0.5, 0.6, 0.7,approaching φc.

14

FIG. 12: The mean-square displacement 〈~r2(t)〉 of the par-ticles as function of the time in a double logarithmic plotfor τb = 1000MCstep/particle: from top to bottom φ =0.8, 0.82, 0.85, 0.87, 0.9, approaching φg(τb). In the inset, thediffusion coefficient: the full line is the fit with the function∼ (0.963 − φ)−2.3.

FIG. 13: The mean-square displacement 〈~r2(t)〉 of the par-ticles as function of the time in a double logarithmic plot,obtained at φ = 0.85. The different curves, from top to bot-tom, refer to τb = 10, 100, 1000 and the case of permanentbonds.