Colloidal gelation with variable attraction energy
CColloidal Gelation with Variable Attraction Energy
Alessio Zaccone ∗ , J´erˆome J. Crassous and Matthias Ballauff Cavendish Laboratory, University of Cambridge,JJ Thomson Avenue, Cambridge CB3 0HE, U.K. Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, 22100 Lund, Sweden and Helmholtz Zentrum f¨ur Materialien und Energie, D-14109 Berlin, Germany,and Department of Physics, Humboldt-University, Berlin, Germany (Dated: November 7, 2018)We present an approximation scheme to the master kinetic equations for aggregation and gelationwith thermal breakup in colloidal systems with variable attraction energy. With the cluster fractaldimension d f as the only phenomenological parameter, rich physical behavior is predicted. Theviscosity, the gelation time and the cluster size are predicted in closed form analytically as a functionof time, initial volume fraction and attraction energy by combining the reversible clustering kineticswith an approximate hydrodynamic model. The fractal dimension d f modulates the time evolutionof cluster size, lag time and gelation time and of the viscosity. The gelation transition is stronglynonequilibrium and time-dependent in the unstable region of the state diagram of colloids where theassociation rate is larger than the dissociation rate. Only upon approaching conditions where theinitial association and the dissociation rates are comparable for all species (which is a condition forthe detailed balance to be satisfied) aggregation can occur with d f = 3. In this limit, homogeneousnucleation followed by Lifshitz-Slyozov coarsening is recovered. In this limited region of the statediagram the macroscopic gelation process is likely to be driven by large spontaneous fluctuationsassociated with spinodal decomposition. PACS numbers:
I. INTRODUCTION
Colloidal suspensions gel if there is an attractive inter-actions of sufficient strength between the particles. Thisgelation transition has been the focus of intense researchduring the last decade since it plays an important rolein many practical applications as e.g. processing of poly-mers or food technology. In spite of intensive efforts in the past aimed at clarifying the nature of the gelationtransition, the basic mechanism by which a fluid colloidalsuspension turns into solid remains unclear. Many nu-merical studies have been proposed over the last decadeswhich have brought a wealth of phenomenological infor-mation about the connection between microscopic attrac-tion and the gelation process . However, analyticalmodels are lacking, and therefore it is difficult to eluci-date the basic mechanisms and to extract scaling laws inanalytical form.Some time ago, the idea has been proposed that thegelation transition may be interpreted as a ”renormal-ized” glass transition where the growing colloidal clustersoccupy an increasingly larger volume fraction up to thepoint at which their motions become governed by glassycorrelation, the clusters become caged by their neighborsand the system becomes solid by interconnection or ran-dom packing of clusters. This scenario is different fromwhat one observes in chemical gels where the bonding ispermanent (in contrast with colloidal bonds that can bebroken up by thermal energy) and percolation providesan excellent description of chemical gelation . With col-loidal gels, however, simulations have established thatthe dynamics is strikingly different from that of chemi-cal gels and colloidal gelation cannot be understood with percolation concepts alone. The concept of colloidal gela-tion as a cluster-jamming transition has clearly broughtprogress in the modeling of the static structure-elasticityrelation of dense colloidal gels . However, it has notbeen implemented in an analytical model of the gela-tion transition that can be tested in comparison withexperiments. The main problem resides in the diffi-culty of bridging the macroscopic mechanical response(the viscosity) with the mesoscopic level of the clustersand ultimately with the underlying microscopic associa-tion/dissociation kinetics of individual colloidal particles.Here we present an analytical model of colloidal gela-tion with variable attraction energy. The model pro-vides a framework which connects the level of the pair-attraction energy V with the mesoscopic level of theclusters and finally with the macroscopic mechanical re-sponse. Analytical laws can be extracted for the viscosity,the gelation time and the cluster size. These laws pro-vide a theoretical explanation to several observations inthe past for which no theoretical description is available.This investigation was prompted by our finding thatwell-defined attractive interaction can be induced in sus-pensions of thermosensitive microgels by raising the tem-perature above the volume transition of these systems. Figure 1 shows these particles in a schematic fash-ion: A dense network of the thermosensitive poly(N-isopropylacrylamide) (PNIPAM) is grafted to solidpolystyrene core. Immersed in cold water the shell ofthese particles will swell. Above the volume transition at32-33 o C most of the water will be expelled from the net-work because water becomes a poor solvent under theseconditions. As a consequence, the shrunken shells willbecome mutually attractive and the strength of the en- a r X i v : . [ c ond - m a t . s o f t ] A p r δ V
24 28 32 36 40024681012 V m i n / k B T T [ o C] V V / k B T FIG. 1: (color online). Schematic representation of the ther-moresponsive nanoparticles used in the gelation experiments.The dependence of the attraction on temperature (Eq. (31))is also shown in comparison with the experimental data ofRef. . V ∞ = 12 k B T is the highest attraction energy reach-able with this system. suing attractive interaction can be adjusted precisely. The advantages of this system for the study of gelationare at hand: Well-defined short-range attraction can beinduced without adding an additional component by justraising the temperature. Moreover, the fluid state canbe recovered by simply lowering the temperature below32 o C . The sharp transition observed experimentally canbe traced back to the predicted strongly nonlinear de-pendence of the viscosity on V .This article is structured as follows. First we presentthe analytical theory of colloidal aggregation and gela-tion with variable attraction energy. Then we discussthe predictions in terms of scaling laws and a state dia-gram extracted from the theory. Then we describe thegelation experiments with thermosensitive colloidal par-ticles. Finally, we present the comparison between theoryand experiments. II. KINETICS OF AGGREGATION ANDGELATION WITH VARIABLE ATTRACTIONENERGYA. Assumptions and steps in the derivation
The model is based upon the following steps and as-sumptions. (i) Any two colloidal particles interact via arectangular-well attractive potential of width δ and depth V . (ii) The clustering process is described by a masterkinetic equation with an effective association rate whichaccounts for bond dissociation. (iii) The association anddissociation rates between two particles are evaluatedfrom steady-state solutions to the Smoluchowski diffusionequations for the two-body dynamics in a mutual attrac-tion potential. (iv) The so obtained analytical solution of the master kinetic equation is used to obtain analyticalexpressions of the time-dependent cluster size distribu-tion and of the time-dependent volume fraction occupiedby the clusters in the system. (v) Since the attraction isshort-range and the hydrodynamics is screened from theclusters interior, clusters are assumed to behave hydro-dynamically like hard-spheres and the time evolution vis-cosity of the system is calculated using the time-evolutionof the cluster population as input to the hydrodynamicdescription. (vi) The gelation time at which the gelation(fluid-solid) transition occurs is calculated analyticallyas the time at which the clusters connect into a randomclose packing and the low-shear viscosity diverges. B. Derivation Master kinetic equation for aggregation withreversible bonds
Let us consider the master kinetic equation which gov-erns the time evolution of the concentrations of clustersof any size present in the system as a result of the micro-scopic two-body association and dissociation processes: dN k dt = 12 i + j = k (cid:88) i,j =1 K + ij N i N j − N k ∞ (cid:88) i =1 K + ik N i − K − k N k + ∞ (cid:88) i = k +1 K − ik N i (1)where N i is the number concentration of aggregates with i particles in each of them. K + ij is the rate of associationbetween two aggregates, one with mass i and the otherwith mass j , while K − ij is the rate of dissociation of a j + i aggregate into two aggregates i and j . The first term ex-presses the ”birth” of clusters with mass k , the secondexpresses the ”death” of clusters with mass k due to ag-gregation with another aggregate. The last two termsexpress the ”death” and ”birth” of k -aggregates due toaggregate breakup, respectively. Instead of consideringthe two dissociation terms in the master equation explic-itly, we can account for dissociation in an effective way byreplacing the association constant with an effective size-independent rate constant K eff and dropping the breakupterms in the master equation. If association is controlledby diffusion, as we are going to see in the next section,the rate of association is in good approximation indepen-dent of the sizes of the two colliding clusters. Further,we also assume that dissociation is also independent ofthe clusters. These simplifications are indeed justifiedif we assume that two clusters aggregate by forming abond between two particles protruding on the respectivesurfaces, such that the association/dissociation kineticsbetween any two clusters can be effectively described bymeans of K eff . The new master equation under thesesimplifications reads as: dN k dt = 12 K eff i + j = k (cid:88) i,j =1 N i N j − K eff N k ∞ (cid:88) i =1 N i (2)Upon discrete-Laplace transforming this equation , theanalytical solution for the time evolution of the clustermass distribution (CMD) reads: N k = N ( t/θ ) k − (1 + t/θ ) k +1 . (3) N denotes the number per unit volume of monomer par-ticles in the colloidal sol at t = 0. θ is the characteristicaggregation time or lag time and is equal to: θ = 2 N K eff (4)During this lag time, aggregation is slow because of bondbreakage, and the formation of large clusters is unfa-vorable because N k ∼ ( t/θ ) k − with t/θ (cid:28)
1. Hence,for t/θ < θ the formation of stable bonds is possible. This has tobe interpreted in a stochastic sense as the probability oflarge fluctuations around the average dissociation rateincreases with time thus making possible the stochasticformation of long-lived bonds over a long time.In the next section we derive an analytical expressionfor θ exploiting the fact that for t < θ only monomersand dimers are present in the system. Effective association rate accounting fordissociation
We start by considering the kinetics of reversible asso-ciation between two colloidal particles to form a dimer monomer + monomer (cid:10) dimer. (5)The association rate be denoted by k + and the dissocia-tion rate by k − . If we denote with n ( t ) the concentrationof monomers at time t and with N the total concentra-tion of monomers at t = 0, the evolution of n is governedby: dn ( t ) dt = − k + n ( t ) − k − n ( t ) + 12 k − N (6)where we made use of the conservation condition: n ( t ) =( N − n ( t )) /
2, with n the concentration of dimers. Withthe initial condition n (0) = N , Eq.(6) has the followingsolution: n ( t ) = − k − k + + √A k + (cid:34) tanh( √A t/
2) + B / √A B / √A ) tanh( √A t/ (cid:35) (7) with A = k − ( k − + 4 k + N ) and B = k − + 2 k + N . Oneshould note that k − has dimensions of an inverse time,while k + has dimensions of [ volume/time ] because it isthe rate constant of a bimolecular second-order reaction,whereas k − is the rate constant of a unimolecular or first-order reaction.With only monomers and dimers as for t < θ , the tem-poral evolution of the average hydrodynamic radius ofthe system a ( t ) as measured by dynamic light scattering(DLS) is given by a ( t ) = I n ( t ) /a + I n ( t ) /a I n ( t ) + I n ( t ) (8)where I is the intensity of light scattered by a populationof pure monomers of radius a , and I is the intensity oflight scattered by a population of pure dimers having ahydrodynamic radius a . a ( t ) can be calculated by sub-stituting Eq. (7) together with the conservation relation n ( t ) = ( N − n ( t )) / t , theresulting expression reads as: a ( t ) ∝ N k + ) ( I /I ) a (1 − a /a )[2 k − + 4 k + N − k − ( I /I )( a /a )] t. (9)Upon taking the derivative and rearranging terms we ob-tain the standard form1 a da ( t ) dt = I I (cid:18) − a a (cid:19) N K eff (10)The truncation to first-order in time implies that weare neglecting the equilibrium plateau that ultimately isreached according to the law of mass action. Rigorously,this approximation is valid for k + N > k − as discussedin the Appendix A. While keeping this in mind, it is in-structive to consider its predictions also outside the rig-orous regime of validity. By comparing the previous twoexpressions, we are now able to obtain the effective asso-ciation rate accounting for bond-dissociation: K eff = 16 k N [2 k − + 4 k + N − k − α ] . (11)where α = ( I /I )( a /a ). Since I /I = 1 +sin(2 a q ) / a q , at the particle length scale one has I /I (cid:39)
1, while at the same time a /a = 1 .
38 forspheres. Hence, in good approximation and to make ourformulae more transparent, in our model we take α = 1,which is not going to change our results neither quali-tatively nor quantitatively. With this result we can alsoidentify the lag time: θ = 2 N K eff = (4 k + N + k − ) k + N ) . (12)This framework allows us to account for dissociation inthe kinetics of aggregation between colloidal particles, inan effective way. It contains indeed both the associationrate k + and the dissociation rate k − .These rates can be estimated by solving the station-ary equation of diffusion for the two particles in theframe of one of the two taken to be the origin. In thecase of association, upon assuming stick-upon-contact asfor short-range attraction one has: ∇ n = 0 with theboundary conditions n = 0 at r = 2 a and n = const at r → ∞ . At steady-state, the solution for the rateof collision per unit volume or flux J follows upon in-tegration as: J = 4 π (2 a )(2 D ) n , which, upon usingthe Stokes-Einstein relation, leads to the Smoluchowskirate k + = (8 / k B T /µ . The association rate is thereforeindependent of the size of the two particles or clustersthat aggregate. This fact implies that larger particles (orclusters) aggregate at the same rate as smaller particlesbecause the lowering of the diffusivity brought about bythe larger size is exactly compensated by the increase inthe collisional cross-section. When the attraction range δ cannot be neglected, one has to solve the diffusion equa-tion for two particles in the field of force of a rectangularwell of depth V and width δ . The result is : k + = 4 πD (cid:16) a − a + δ (cid:17) e − V/k B T + a + δ (13)where D = k B T / πµa is the mutual diffusion coefficientof the particles. For short-range potentials δ (cid:28) a onerecovers the Smoluchowski rate which we are going to usethroughout this work. We should also mention that hy-drodynamic interactions and elastic deformation effectsof polymer-functionalized surfaces might play a role aswell in the very short ranged limit. Since we cannot ac-curately model the latter effect we choose here to usethe classic Smoluchowski rate theory where the slowingdown of the rate brought about by hydrodynamics nearthe surface cancels, approximately, with the speeding upbrought about by the finiteness of the attraction range.For a discussion of this effect see Ref. .In a similar fashion, the dissociation rate k − can be es-timated by solving the steady diffusion equation for thetwo bonded particles in the field of the attraction poten-tial. At steady-state one calculates the (Kramers) rateof escape of one particle from the attraction well whichdrives the dissociation event. What changes with respectto the association process is obviously the boundary con-dition, since now it is assumed that the two particles startin a quasi-equilibrium steady state in the attraction well.In good approximation, the result for the average disso-ciation rate is: k − = ( D/δ ) e − V/k B T . (14)In Ref. a theoretical justification for this formula andits derivation can be found. Here we chose this simpli-fied form and neglect non-essential prefactors in order tobe consistent with our previous characterization of thecolloidal systems under study . Eq.(14) is not accurate,however, as soon as one deals with shallow attractionenergies V ∼ k B T . Even in the limit V = 0 this for-mula predicts a finite dissociation rate k − > V / k B T k - [ s - ] / ~ B V k T e − ~ V β− FIG. 2: (color online). Comparison between the Kramersdissociation rate (continuous line) given by Eq.14 and theinterpolation formula Eq. 15 which interpolates between theKramers formula and the V → it is clear that no bond can be present to start with be-cause the particles are hard-spheres and dissociation istherefore instantaneous as the spheres collide. This ar-tifact is due to the approximation under which Eq.(14)is derived, whereby Kramers assumed in its derivationthat the attraction well must be significant in order forthe two particles to be in a quasi-equilibrium steady-state in the well . To overcome this problem rigor-ously, one should solve the full time-dependent diffusionequation which however would undermine the analytic-ity of our approach. Hence here we propose the followingsemi-empirical formula which interpolates between theKramers formula for V (cid:29) k B T and the limit k − = 0 at V = 0: k − = ( D/δ ) e − V/k B T + Λ / ( V /k B T ) β . (15)In our calculations below, we are going to use Λ = 10 s − and β = 20, With this choice, the dissociation rate isequal to the Kramers formula for all attractions down to V /k B T (cid:38) V = 0. This interpolation formula is plotted in Fig.2together with the Kramers formula for comparison.With these identifications, the expression of the lagtime, with the explicit dependence on T , is given by: θ = [( k B T /µ ) N + ( D/δ ) e − V/k B T + Λ( V /k B T ) − β ] k B T /µ ) N ] . (16)The lag time is thus a function of the competition be-tween microscopic association and dissociation kinetics.In the limit controlled by association k − (cid:28) k + N , thelag time is set by the time of diffusive transport as forirreversible diffusion-limited aggregation: θ = 2 / ( k + N ).In the opposite limit where dissociation is controlling, k − (cid:29) k + N , the lag time goes as θ = k − / (2 k + N ) ∼ D e − V/k B T . Finally, when the condition k + N = k − , isexactly satisfied, which fixes V , such that the initial in-dividual frequencies of the forward process (association)and of the backward one (dissociation) are equal, the lagtime scales as: θ ∼ / (8 k + N ) ∼ µ/ ( k B T N ) , (17)and it is inversely proportional to T through the inverseof the Smoluchowski aggregation rate. This is a physi-cally meaningful outcome because in this regime an in-crease of T causes the speeding up of the diffusive trans-port which reduces the lag time. The physical meaning ofthis result is that, in the regime of equilibrium aggrega-tion, the kinetics is controlled by the activated stochasticjump of the particles out of the attractive well which isthe kinetically limiting process. Upon reducing the at-traction, the lag time increases because the formationof bonds requires stochastically a longer time. Vicev-ersa, upon increasing the attraction the lag time getsreduced because it becomes stochastically more likely toform long lived bonds on shorter times. Although it istempting to identify the condition k + N = k − with thecondition of detailed balance straightaway, this is not arigorous identification. As discussed in the Appendix Awhere we refer to Tolman’s definition , this conditionrather corresponds to the microscopic reversibility in theearly time limit.Finally, it should be noted that, once that t > θ ,the aggregation kinetics enters an extremely fast regimewhich is independent of the lag time and hence is thesame for all (finite) attractions. During this fast regimethe kinetics is relatively insensitive to the microscopicdetails of cluster aggregation and it is very difficult todetect the effect of the microscopic details in the macro-scopic properties which evolve very rapidly towards thesolid state. Analytical solution for the clustering kinetics withreversible bonds
We now have a connection between the mesoscopictime-evolution of the clusters and the microscopic in-teractions between colloidal particles which can also beallowed to vary with time. The CMD can be used toderive the time-evolution of quantities such as the aver-age cluster size R and the effective volume fraction inthe system (cid:101) φ defined as the fraction of volume occupiedby the clusters at time t . In general, the radius of thesphere enclosing the cluster is given by: R k = a k /d f where d f is the cluster fractal dimension or mass-scalingexponent: k = q ( R k /a ) d f , where q is a prefactor of orderunity. Using Eq. (3) for N k , the average cluster radius can be readily calculated:¯ R ( t ) = ∞ (cid:80) k =1 aN k k /d f ∞ (cid:80) k =1 N k = a (cid:18) θt (cid:19) Li − /d f (cid:18) tt + θ (cid:19) . (18)Here, Li n ( x ) denotes the polylogarithm of order n ofthe variable x . Closed form expressions for the polylog-arithm of negative order are known only for the specialcase Li − ( x ) = x/ (1 − x ) which corresponds to d f = 1.Using this form we have that ( θ/t )Li − ( tt + θ ) = 1 + ( t/θ ),which is a significant simplification. Extrapolating thisresult to higher values of d f we obtain a good closed-formapproximation:¯ R ( t ) = a (cid:18) θt (cid:19) Li − /d f (cid:18) tt + θ (cid:19) (cid:39) a [1 + ( t/θ ) /d f ] . (19)One can check that this approximation is reasonablequantitatively and it is always qualitatively correct in thewhole range of d f and θ . This formula implies that in theaggregation process there is a lag time of order θ , duringwhich aggregation events are stochastically rare becauseof the low rate of successful collisions leading to bond-formation, where successful collisions are those which donot result in immediate thermally-activated bond rup-ture. If θ is a small number, meaning that the lag timeis short as for irreversible aggregation, then the kineticstransitions after short time to the ¯ R ∼ ( t/θ ) /d f growthlaw that has been reported experimentally for irreversiblecolloidal aggregation in the past . In the limit d f = 3of equilibrium aggregation where microscopic reversibil-ity is satisfied (see section IV.a and the Appendices forthe connection between thermodynamic equilibrium and d f = 3), by combining the above expression with Eq.(16),this treatment gives:¯ R ( t ) (cid:39) a [1 + δ / ( Dt ) / ] . (20)which in the asymptotic limit correctly recovers the wellknown Lifshitz-Slyozov scaling ¯ R ∼ t / for the growthrate in the coalescence (coarsening) regime of phase sepa-ration following nucleation under equilibrium conditions.The link between nucleation and phase separation is dis-cussed more in detail in section IV.a. Hence Eq.(19) isimportant because it covers all limits of colloidal aggrega-tion kinetics, from irreversible aggregation to nucleationat equilibrium, and provides theoretical justification tomany experimental observations in the past.Similarly, the effective cluster volume fraction is givenby: (cid:101) φ = 43 π a V ∞ (cid:80) k =1 N k k /d f ∞ (cid:80) k =1 N k . (21)This definition of the effective cluster volume fraction isthe most used in the colloidal gelation literature . Analternative would be the mass-weighted average volumefraction which is obtained by introducing a factor k in thesum in the numerator and is more used in the context ofpolymer gelation where the molecular weight is the keyparameter in the gelation process. The mass-averagingis however less used with colloidal gelation as here thekey parameter which controls the gelation process is thecluster size (and, of course, its cube which is the volume)rather than the mass. Using V = vN/φ , where φ is thevolume fraction of the colloidal gas and v the volume ofa single particle, and Eq.(3) for the CMD, we obtain: (cid:101) φ = φ ( θ/t )Li − /d f (cid:18) tt + θ (cid:19) . (22)Eq.(22) gives the effective cluster volume fraction as afunction of the time-dependent interaction (accountingalso for dissociation) embedded in the characteristic ag-gregation time τ . The polylogarithm of order − /d f with 1 < d f < − /d f ( x ) ≈ x ( x + 1) − /d f ) / (1 − x ) − − (3 /d f ) . Thenthe volume fraction occupied by clusters after some ma-nipulation becomes: (cid:101) φ = φ [1 + ( t/θ )][1 + 2( t/θ )] (3 − d f ) /d f . (23) Linking the clustering kinetics with themacroscopic viscosity
Consistent with our main approximation of treatingthe clusters as renormalized spheres occupying an effec-tive rescaled volume fraction (cid:101) φ , we now describe the ef-fective viscosity of the system as a function of (cid:101) φ . Thistreatment applies to fractal clusters as well in the regime d f ∼ . The viscos-ity of the system can be estimated by treating the clus-ters as effective hard spheres since their hydrodynamicbehavior is very close to that of hard spheres even forfractal clusters and the short-range attraction has littleeffect on the hydrodynamic viscous dissipation. Underthese assumptions, it is possible to obtain analytical ex-pressions in closed for for the viscosity over the entire ˜ φ range up to the cluster close packing where the systemarrests. This approach has the advantage of providinganalytical scaling laws for the gelation time as a functionof the controlling parameters.The differential effective medium theory allows us tocalculate the viscosity of a dense suspension starting fromEinstein’s method for calculating the viscosity of a dilutesuspension of spheres. Because of the assumption of di-lute and non-interacting particles, one first obtains a lin-ear dependence, i.e. the Einstein formula η = µ (1 + ˜ φ )which accounts for the hydrodynamic dissipation of asingle cluster . Upon introducing a small increment ofparticles in the system and accounting for their mutual correlations, it is possible to account for the many-bodyhydrodynamic interactions as well as for excluded vol-ume effects in an effective way, leading to the followingexpression : η = µ (cid:32) − ˜ φ − [(1 − ˜ φ c ) / ˜ φ c ] ˜ φ (cid:33) − / . (24)This equation is a key result of this work. Here, ˜ φ c (cid:39) . and on the particle interactions ,in the range 0 . ÷ .
67. For our scope these differ-ences are irrelevant and we have checked that they donot minimally affect the qualitative predictions of ourmodel. Hence, consistent with our hard-sphere approx-imation in the viscosity calculation, we take ˜ φ c = 0 . φ (cid:38) .
5, the system undergoes a glassy dynamical arrestwhere the clusters become caged by their neighboringclusters. Within this regime the viscosity still increases,as a power-law of ˜ φ according to Mode-Coupling theo-ries and with an exponential dependence according toAdam-Gibbs theories , before diverging at the randomclose packing. These scenarios could be implemented inour frameworks, in future work, to provide a more de-tailed description in the glassy regime. Here we are in-terested in the overall kinetics of the process and deferthis type of detailed analysis of the glassy regime to fu-ture work.Upon replacing this expression in the above expressionfor the viscosity, we obtain a closed-form expression forthe viscosity as a function of the microscopic interaction,of the volume fraction, and of time: η = µ (cid:18) − φ [1 + ( t/θ )][1 + 2( t/θ )] (3 − d f ) /d f − c φ [1 + ( t/θ )][1 + 2( t/θ )] (3 − d f ) /d f (cid:19) − / (25)with c = (1 − ˜ φ c ) / ˜ φ c . (26)Predictions of Eq.(25) for the viscosity as a functionof time are shown in Fig.3 for different values of the at-traction energy. It is seen how the viscosity diverges at awell defined gelation time t g , which is the time at whichthe effective volume fraction occupied by clusters reachesthe random close packing fraction ˜ φ c . For shallow attrac-tions, it is evident that the viscosity remains constantand not too much above the value of the initial colloidalgas for a long time before it starts to increase. Thisobservation has very important implications. In mostexperimental studies in the past the separation betweengelling and non-gelling systems has been defined some-what arbitrarily. Indeed it is customary in the experi-mental practice to establish that a system does not gel t [s] η [ P a s ] V FIG. 3: (color online). Low shear viscosity calculated as afunction of time with varying attraction energy V accordingto Eq. 25 and using Eq. 23 for the cluster packing fraction forthe case d f = 2. The existence of a lag time for all attractions,after which the aggregation is very fast for all attractions, isevident. a = 85 nm and φ = 0 . V = 1 k B T , V = 1 . k B T , V = 2 k B T , V = 3 k B T , V = 10 k B T . under certain conditions if it remains in a fluid state anddoes not aggregate significantly over a chosen period oftime. Clearly, in this way the choice of the time spanmight be such that the observation time is shorter thanthe lag time, i.e. t obs < θ and states that would gel aftera time ∼ θ might be improperly classified as non-gelling.The expressions reported here can therefore be of help toexperimentalists in establishing a more rigorous criterionfor drawing experimental state diagrams of colloids. III. GELATION TIME
We define the gelation time as the time at which η = ∞ , where η is the zero frequency viscosity. Of course,there are many criteria that have been proposed in or-der to identify the gelation point. One that is widelyused is the Winter-Chambon rheological criterion basedon the elastic moduli , which requires a thorough char-acterization of the frequency response. Since here we areinterested in the low-frequency behavior and will presentexperiments done in this limit, the η = ∞ is a strin-gent criterion to identify the liquid to solid transition.In the present context, since we are focusing on the low-frequency behavior and we can deal with the viscosity inclosed form, we choose the η = ∞ criterion which im-plies that at the gelation point the system is connectedand does not flow at least over a long time scale.From Eq.(25) it is possible to determine the gelationtime t g as a function of the other microscopic parameters,such as the attraction, the aggregation rate and the vol-ume fraction. By setting the content of the main bracketin Eq.(25) equal to zero, in theory one could solve forthe gelation time t g for an arbitrary d f . In practice, the equation cannot be solved analytically for an arbitrary d f . By studying the solution for a few special cases suchas d f = 3 and d f = 2, and dropping high-order terms in φ , we obtain the following approximate formula: t g = θ c ) φ/ d f / (27)Hence the T dependence of t g is the same as for θ givenin Eq. (16). As one could expect, also from the con-sideration of Fig. 3, the gelation time is proportional tothe lag time θ . Using Eq. (12), we can relate the gela-tion time to the microscopic association and dissociationprocesses: t g = (4 k + ( φ/v )+ k − ) k + ( φ/v )) c ) φ/ d f / . (28)Here we wrote N = φ/v to make the full dependenceon φ explicit. Let us discuss the various limits of thisformula as a function of the attraction energy first. When V /k B T (cid:29) k − (cid:28) k + ( φ/v ). In this limit we get a gelationwhich depends only on the diffusive transport and on φ : t g = 116 k + ( φ/v )[(1 + c ) φ/ d f / ∼ φ − − ( d f / . (29)In the opposite limit of shallow attraction, k − (cid:29) k + ( φ/v ), we get: t g = (cid:16) ( D/δ ) e − V/k B T + Λ( V/k B T ) β (cid:17) k + ( φ/v )) [(1 + c ) φ/ d f / ∼ k − k φ − − ( d f / . (30)Finally, when the early-time microscopic reversibilitycondition is satisfied, we have that the gelation time fol-lows an Arrhenius law as a function of the attraction: t g ∼ e − V/k B T . This law implies that the gelation time,in this regime, increases upon increasing T because thebonds become more short-lived at higher T which slowsdown the aggregation process. Therefore, it is clear thatthe temperature affects the gelation in a very differentway depending on the strength of the binding energy V .In particular, in the regime of strong binding close todiffusion-limited aggregation, the T dependence of thegelation time is governed by Eq. (17) and the gela-tion time decreases upon increasing T because the dif-fusive transport is enhanced at higher T . In the oppositelimit of lower V , instead, the gelation time obeys Arrhe-nius behavior and increases upon increasing T becauseof the slowing down of aggregation caused by the en-hanced thermal breakup of the bonds. Hence, both thesepredicted behaviors appear physically meaningful in thetwo opposite regimes.The gelation time as a function of the colloid fraction φ is shown in Fig.4 for d f = 2. Upon increasing theattraction, the power-law decay with the exponent − − φ t g [ s ] V ~ f d φ − − ~ f d φ − − FIG. 4: (color online). Gelation time calculated as a functionof the colloid fraction φ for different values of the attractionenergy V . d f = 2. From top to bottom: V = 1 k B T , V =2 k B T , V = 4 k B T , V = 10 k B T . t g [ s ] V / k B T φ FIG. 5: (color online). Gelation time calculated as a functionof the attraction energy V for different values of the colloidfraction φ . d f = 2. From top to bottom: φ = 1 · − , φ = 1 · − , φ = 0 . φ = 0 . ( d f /
3) predicted in the limit of weak attraction graduallydecreases and melds into the limiting power-law φ ∼ − − ( d f /
3) at high attraction.The behavior of the gelation time as a function of theattraction is plotted in Fig.5. Three different regimes canbe identified. At low attraction,
V /k B T (cid:46)
2, the gela-tion time decays very rapidly from its asymptotic valueat V = 0 and melds into a second regime where the de-cay with the attraction is exponential. Clearly the firstregime is dominated by the scaling t g ∼ [Λ / ( V /k B T ) β ] ,whereas the second regime is controlled by the scaling t g ∼ [( D/δ ) e − V/k B T ] , according to Eq.(30). This pre-dicted exponential scaling regime is confirmed by ear-lier experimental observations . In both these regimesthe dissociation rate dominates over the association rate,i.e. k − (cid:29) k + ( φ/v ). Upon further increasing the at- traction, however, the dissociation rate becomes increas-ingly smaller in comparison with the association rate as4 k + ( φ/v ) (cid:29) k − and the exponential behavior flattensout into a plateau where the gelation time is independentof V . This latter regime recovers the diffusion-limitedirreversible aggregation which is characterized by per-manent bonds since dissociation is now infinitely slowercompared to association. IV. STATE DIAGRAM OF ATTRACTIVECOLLOIDAL MATTER
It is possible to summarize the model predictions in astate diagram of attractive colloids. In order to be con-sistent with previous studies, we introduce the effectivetemperature τ customarily defined as : τ = 112 (cid:18) σ + δδ (cid:19) exp (cid:18) − Vk B T (cid:19) (31)where σ = 2 a is the colloid diameter. Clearly, low valuesof τ correspond to high attraction energies, and there isalso a close relation with our dissociation rate, with lowvalues of τ corresponding to low values of the dissociationrate k − . The state diagram is plotted in Fig.6. Contin-uous lines are calculated for specified values of t g andrefer to nonequilibrium boundaries. In practice, everycontinuous line separates systems that undergo gelationon a time scale t < t g , at higher φ (i.e. to the right ofthe curve), from systems that undergo gelation at t > t g ,at lower φ (i.e. to the left of the curve). All curves areplotted for d f = 2 since most experimental observationsof gelation in this regime report values of d f close to thisvalue, at least for φ < .
2. We have checked that chang-ing d f in the range 1 . ÷ . t g , gelation can-not be observed and the system appears liquid-like and atmost composed of freely diffusing clusters. On the otherhand, if the time of observation is long compared to thegelation time, a transition from a liquid-like material intoa solid-like one will appear. Hence, in light of our results,the observation of so called ”equilibrium” clusters in theabsence of gelation in purely attractive colloids mighthave been due to the time scale of the experiment beingshort compared to the theoretical time scale of gelationfor those conditions (and in fact the attractions reportedin Ref. lie well in the lower regions of our diagram).The situation might be different, however, in the case ofcharged colloids where the electrostatic repulsion playsa major role giving rise to further effects that are notconsidered in our analysis. The role of the time coor-dinate on the gelation transition has been neglected in FIG. 6: (color online). State diagram of attractive colloids.Continuous lines represent gelation lines (see text for expla-nation). From left to right: t g = 0 . t g = 0 . t g = 0 . t g = 0 . t g = 1, t g = 10, t g = 100, t g = 1000, t g = 1 · , t g = 1 · . MR (dashed) line: this line represents the set ofpoints on the diagram for which the early-time microscopicreversibility condition k + N = k − holds exactly. B (dotted)line: binodal line. S (dotted) line: spinodal line. many previous studies of colloidal gelation, both exper-imental and computational, despite being a key controlparameter in all nonequilibrium transitions.In the state diagram we have also plotted the early-time microscopic reversibility condition (DB line in Fig.6)represented by the equality between the rates of the mi-croscopic forward and backward process, in this case as-sociation and dissociation, respectively. The early-timemicroscopic reversibility condition is expressed by theequality: k + N = k − . Regions of the state diagram ly-ing much below this line are certainly away from ther-modynamic equilibrium and in fact it is below the linethat we observe the most striking dependence on time.Above this line, the gelation lines at different t g tend tobecome more closely spaced together until they almostmerge together in the top part of the diagram. Inter-estingly, the region where all gelation lines (correspond-ing to gelation times separated by up to 10 orders ofmagnitude) practically merge, coincides with the binodal(liquid-liquid) phase-separation line calculated here usinga simple mean-field Bragg-Williams approach. Hence,from the point of view of kinetic theory, gelation in theupper region of the diagram above the binodal line, is anextremely unlikely occurrence even over extremely longtimes. From the point of view of equilibrium thermody-namics, no gelation can happen above the binodal be-cause here the free energy of the system is controlled bythe unfavorable entropy of mixing which favors the solstate (that is obviously more ”mixed up” than any ag-gregated state) and makes macroscopic aggregation ther-modynamically forbidden. A. The metastable region: nucleation andliquid-liquid phase separation
In Fig. 6 we have also plotted the spinodal line and themetastable region (delimited by the binodal and by thespinodal) deserves careful analysis. Although we haveplotted our nonequilibrium gelation lines also in this re-gion, they should be taken as purely indicative becausein the metastable region, gelation is replaced by nucle-ation leading to liquid-liquid phase separation. Indeed,the metastable region appears to be centered upon theearly-time microscopic reversibility line which is a nec-essary condition for the system to be close to the mi-croscopic equilibrium between association and dissoci-ation and for detailed balance to be satisfied (see alsoZeldovich for the detailed balance principle within thecontext of phase separation). Under these conditions,Eq.(1) leads straightforward to homogeneous nucleation,as shown in the Appendix. Nucleation leads to the for-mation of compact d f = 3 clusters which can be seenwith a simple calculation.Being close to the microscopic reversibility line wheredetailed balance can be satisfied (see the Appendix), wecan write down the free energy for the formation of anucleus or cluster. As in nucleation theory, the nucleusis treated as a macroscopic object which allows us to for-mulate the free energy of a single cluster. The Gibbsfree energy contains two contributions. One is the vol-ume enthalpy arising from the bonds that are formed:Φ v = − ( zk/ V = − ( z ( R k /a ) d f / V where k denotesthe number of particles in the cluster and V the bondenergy, as usual, and z is the mean number of near-est neighbors. The other term is the energy spent tocreate the interface between the cluster and the solvent.For a fractal the interface is intrinscally discrete and theeffective surface can be estimated as the surface occu-pied by the particles in the outmost shell of the cluster:4 πa d f k ( d f − /d f , where we used that the number of par-ticles in the outermost shell of a fractal cluster is equalto d f k ( d f − /d f . In the limit d f = 3 one recovers 4 πR k for the cluster surface.Therefore, the surface energy is equal to: Φ s =4 πγa d f k ( d f − /d f , where γ is the surface tension. Thefree energy of the cluster is given by: Φ = Φ v +Φ s = − ( z ( R k /a ) d f /
2) + 4 πγa d f k ( d f − /d f . The clustergrowth at equilibrium occurs along the path of minimalfree energy. For all values of the parameters involved inthe free energy, minimization of Φ with respect to d f in3 D gives d f = 3. We did not consider the d f -dependenceof z . However, since z is a monotonically increasing func-tion of d f (because the average density of a cluster in-creases upon increasing d f at a fixed size), it is clear thataccounting for its dependence on d f leaves this result un-changed.0 B. The unstable region: nonequilibriumaggregation
Hence we have shown that close to thermodynamicequilibrium the clusters are compact objects with d f = 3which is in agreement with many experimental and sim-ulation results presented in the past. As a consequencein the metastable region in between the binodal and thespinodal line the system is more likely to undergo ag-gregation into compact aggregates with the growth law R ∼ ( Dt ) / derived in section II.b.3, and there is nocompetition with gelation.It is possible, however, that a solid-like state is formedfollowing spinodal decomposition if the volume fractionin the dense phase reaches the critical volume fractionfor the attractive glass transition predicted by Mode-Coupling theories, according to a mechanism that hasbeen recently discussed in several studies . Al-though this mechanism is certainly a good candidate toexplain a fluid-solid transition in the proximity of thespinodal line, its application is however limited to theregion of the phase diagram close to the microscopic re-versibility line. For deeper quenches well below this line,we have k − (cid:28) k + N and the time scale associated withthe rearrangement of the bonds on the way to equilibrium(i.e. along the minimization of Φ) is thus longer thanthe time scale on which new particles join the cluster toform new bonds. The incoming particles stick irreversiblyonto the particles in the outer layers of the clusters whichleads to d f < d f = 3. With this input, our analytical solution to themaster kinetic equation for aggregation gives the growthlaw R ∼ t / , in agreement with the coarsening kineticsof spinodally decomposing systems . Upon departingfrom the equilibrium conditions towards higher attrac-tions, the relaxation time over which the cluster mini-mizes its free energy becomes long compared to the timescale of bond-formation due to the microscopic imbalancebetween association and dissociation. As a consequence, d f < R ∼ t /d f accordingto Eq.(19). Hence, under these conditions deep insidethe unstable region in the state diagram, gelation cannotbe driven by spinodal decomposition because the kinet-ics associated with the spinodal-like coarsening is slowerthan the kinetics associated with nonequilibrium fractalaggregation. As we are going to see below, this is alreadythe case with a relatively mild attraction energy such as12 k B T . V. COMPARISON WITH EXPERIMENTSA. Complex viscosity
The model for the steady shear viscosity of the sys-tem as a function of the attraction presented in sectionII.b.4 can be used within a generalized hydrodynamicsapproach which bridges the gap between the hydrody-namic (small ω ) and the kinetic (large ω ) regimes andthus provides predictions for the rheological response inthe whole frequency spectrum. According to generalizedhydrodynamics, the constitutive relation is written in thefollowing Maxwell form : (cid:18) η + 1 G ∞ ∂∂t (cid:19) σ xy = − ∂∂t (cid:18) ∂r x ∂y + ∂r y ∂x (cid:19) (32)where r x and r y are the x and y components, respectively,of the microscopic displacement field, and σ xy is the shearstress. Upon Laplace-transforming the above equationwe obtain the complex viscosity as: η ∗ ( ω ) = G ∞ − iω + 1 /τ M (33)where G ∞ represents the instantaneous ( ω → ∞ ) shearmodulus and τ M = η/G ∞ is the Maxwell relaxation time.Below the transition where the system is fluid, we useexperimentally measured values of G ∞ reported in pre-vious studies for all calculations. As η → ∞ , G ∞ isthe one of the solidified system and can be evaluated asthe affine contribution to the shear modulus of a disor-dered lattice of harmonically-bonded particles. The lat-ter, consistent with our picture, are the clusters presentin the system at the time at which (cid:101) φ = 0 . . Infact, whilst the low- ω shear modulus is strongly affectedby nonaffinity , the high- ω one reduces to the affineshear modulus given by the following mesoscopic the-ory : G ∞ ≡ G A = N c z c κσ c . All the parameters inthis formula can be evaluated as follows. The harmonicspring constant is κ (cid:39) V /δ , where V is given by Eq.(35),and δ (cid:39)
10 nm is the hydrophobic attraction range. Thelattice constant is given by the average cluster diameter.This can be calculated using the CMD Eq.(3) evaluatedat the time at which (cid:101) φ = 0 .
64, and the result is σ c (cid:39) µm . N c is the number density of clusters at the random closepacking and is given by N c = (cid:102) φ c / ( πσ c / z c (cid:39)
6. Hence, there are no nontrivial adjustableparameters.The observable quantity which is measured in the ex-periments is the modulus of the complex viscosity, de-fined as | η ∗ | ≡ (cid:112) η (cid:48) + η (cid:48)(cid:48) which then gives: | η ∗ | = (cid:115)(cid:18) G ∞ /ηG ∞ /η + ω (cid:19) + (cid:18) ωG ∞ /η + ω /G ∞ (cid:19) (34)1 | η * | [ P a s ] T [ o C] V FIG. 7: (color online). Model calculation of the complex vis-cosity of thermosensitive colloids using the theory presentedhere [Eq.(34)]. See text for the estimate of G ∞ . φ = 0 . a = 85 · − nm , ω = 1 s − . The attraction energy is cal-culated using Eq. (31) and T c = 33 . o C . As in the experi-ment, the temperature is a ramp function of time accordingto T = 30 . t/ . o C . From right to left: V = 0 . k B T , V = 1 . k B T , V = 2 k B T , V = 6 k B T , V = 12 k B T . From this expression we can extract the most interestinglimit which gives | η ∗ | = G ∞ /ω when η = ∞ , i.e. at thefluid-solid critical point.Eq.(34), together with the microscopic model for thecluster evolution as a function of time and attraction, canbe used to study the rheological response of the systemupon varying the microscopic parameters. Fig.7 displaysthe results of model calculations. The theory predictsa sharp rheological fluid-solid transition as a function oftime upon varying the attraction V ( T ) provided that thefinal attraction strength V ∞ is larger than a threshold,i.e. V ∞ cr (cid:39) k B T for the attraction range used here δ =10 nm , typical of hydrophobic attraction . For V ∞ The colloidal particles used here are the same onescharacterized in detail in Ref. . They consist of a 52 nm radius solid polystyrene core onto which a polymericnetwork of crosslinked PNIPAM with T -dependent thick-ness ( (cid:39) T (cid:46) o C and (cid:39) T (cid:38) o C )is affixed. All electrostatic interactions are fully screenedby the addition of 5.10 − M potassium chloride. Attrac- 30 32 34 36 38 40 42 4410 -4 -3 -2 -1 η ∗ [ P a s ] T [ o C] FIG. 8: (color online). Comparison between the complex vis-cosity from the theoretical predictions of the model (lines)and the complex viscosity measured experimentally (sym-bols). Volume fraction: ( (cid:52) ) 0 . (cid:3) ) 0 . φ = 0 . tion is induced by the hydrophobic effect as the particleshell upon collapsing becomes hydrophobic upon increas-ing T . The bonds due to hydrophobic attraction be-come increasingly long-lived upon increasing T becausethe rate of bond dissociation k − decreases with V , ac-cording to Eq.(15). As for two-level systems, V is a sig-moidal (Fermi-type) function of T and goes from zero upto a plateau value V ∞ across T c according to the follow-ing equation derived in Ref. : V ( T ) V ∞ = (cid:18) 11 + exp[ − ( T c − T )∆ S/k B T ] (cid:19) (35)where ∆ S ≈ − k B is the entropy change across thetransition . The latter parameter is an experimentalinput which is fixed by the surface chemistry of the par-ticles used in the experiments and it controls the rate ofvariation of V with T . The behavior of Eq.(35) for theexperimental system used here is shown in Fig.1.The complex viscosity as a function of T was measuredat ω = 1 s − in a stress-controlled rotational rheometerMCR 301 (Anton Paar) where T is varied using Peltierelements with accuracy of ± . o C . The modulus of thecomplex viscosity | η ∗ | was measured at a strain γ = 0 . T at a rate of 0 . o C/min . The solidfractions of the dispersions investigated are in the range6 − 12% which corresponds to initial volume fractionsoccupied by the particles before the onset of the gelation φ < . C. Comparison In the theoretical calculation we take d f = 2 in agree-ment with previous studies . The comparison between2the theory and the experiments is shown in Fig.8. Itis seen that the complex viscosity is constant up to T (cid:39) o C which is the T value at which the attractiveinteraction sets in . Within this regime, the response,according to our viscoelastic model is completely dom-inated by the dissipative part, | η ∗ | ≈ η . The theory isable to capture the very sharp jump of | η ∗ | which growsby several orders of magnitude within a fraction of degreeKelvin shortly after the onset of attraction. Thus, giventhe fact that no nontrivial adjustable parameter has beeninduced, the agreement of theory and experiment can beregarded as excellent. VI. CONCLUSION In conclusion, we proposed a theoretical, fully analyt-ical model that bridges the microscopic physics of col-loidal interactions and physical bond-formation and dis-sociation, with the macroscopic rheology and the col-loidal gelation transition. According to the model pre-dictions, for shallow attraction energy V , growth oc-curs with a lag time θ ∼ De − V/k B T . The size growsas R ∼ [1 + ( t/θ )] /d f , asymptotically recovering theLifshitz-Slyozov law R ∼ ( Dt ) / in the equilibrium limit d f = 3. The gelation time t g decays as a power lawof the volume fraction with an exponent which is bothattraction-dependent and d f -dependent. In the limit ofweak attraction, t g ∼ φ − − ( d f / , whilst t g ∼ φ − − ( d f / is found in the infinite attraction limit. The theoryalso yields a state diagram of attractive colloids. Inthe metastable region, an equilibrium-like nucleation sce-nario can be recovered as long as detailed balance holds,at least approximately. Although it is possible that gela-tion is driven by spinodal decomposition close to thespinodal line, identifying a gelation boundary which isindependent of time for deeper quenches seems unfea-sible and nonequilibrium aggregation with d f < d f = 3. Ourresults are supported by experiments on a model suspen-sion of thermosensitive colloids in which V is increasedfrom zero in the same sample by raising T . The compari-son shows that gelation already for V = 12 k B T is a sharpnonequilibrium fluid-solid transition with d f = 2, andculminates with a solid random packing of clusters withinfinite viscosity. The solidification is irreversible unlessone switches off the attraction by reverting T , which is aunique feature of the experimental system under study.We have also presented experimental observations ofcolloidal gelation in a system of thermosensitive colloidalparticles where the attraction is varied from zero up to amaximum in the same system by simply varying T . Thecomplex viscosity as a function of T can be quantitativelymodeled with the theory presented here and exhibits asharp gelation transition with d f = 2. The comparisonbetween theory and experiments also indicates that al-ready with a mild attraction energy of 12 k B T short-rangeattractive colloids undergo fast nonequilibrium gelation with d f = 2 and with no detectable hallmarks of spin-odal decomposition. Our analysis suggests that the lattermechanism is more likely to play a role and affect gelationfor weaker attractions close to the spinodal line where therates of two-particle association and dissociation are closein value and aggregation occurs with d f = 3.Clearly, the framework presented here opens theunprecedented possibility of engineering functionalmaterials that turn into solid on a desired time-scale andthat can be switched in a fully controlled and reversibleway between fluid and solid states. APPENDIX: MICROSCOPIC REVERSIBIL-ITY, DETAILED BALANCE, AND THE ON-SET OF NONEQUILIBRIUM AGGREGATION Let us consider one-step association and dissociation pro-cesses by which one particle joins a cluster and disso-ciates from a cluster, respectively. According to thedefinition given by Tolman , the principle of micro-scopic reversibility is satisfied when for every cluster j the absolute frequencies of particle attachment (to a j − j cluster): k j − n = k j − , here n represents the number of monomers in the system.Clearly, in the early stage of the aggregation process,if the latter is sufficiently slow, n can be taken as a con-stant and equal to the initial concentration of monomers N . Then, upon neglecting cluster size dependencies, werecover what we called the early-time microscopic re-versibility condition k + N = k − . When this conditionis strongly violated and the association process to formdimers is initially much faster than the dimer dissocia-tion process, k + N (cid:29) k − , then the formation of trimersetc. becomes a faster process. As a consequence of thefast trimer formation, the equilibrium concentration ofdimers given by n eq = ( N − n ( t → ∞ )) / n ( t ) isgiven by Eq. (7), cannot be reached. Then, the condi-tion k + ( n eq ) = k − ( n eq ) for j = 2 cannot be satisfied.In turn, this fact implies that also the detailed balancecondition k j − ( n eq ) = k − ( n eqj ) which has to be satis-fied ∀ j , cannot hold since it is already violated at leastfor j = 2. In view of this argument, it is clear that thecondition k + N = k − that we encountered multiple timesand plays a crucial role in the development of our theory,is tightly connected with the detailed balance principleand it may be regarded as a necessary (although prob-ably not sufficient) condition for detailed balance to besatisfied. Hence, this argument also provides a simplebut robust quantitative criterion to assess whether theaggregation process is nonequilibrium or not, with theinequality k + N > k − marking the crossover from theequilibrium aggregation (nucleation) dynamics into thenonequilibrium one.Finally, we should interpret the approximation under-lying Eq.(9) within this framework. The truncation tofirst-order in time is equivalent to neglecting the kineticequilibrium plateau for the dimers and to restricting theanalysis to the initial growth stage only which occurs3before the plateau. The time at which the plateau isreached can be easily estimated from Eq.(7). Since thetanh reaches a plateau when its argument is (cid:39) 2, thisgives t eq = 4 (cid:112) k − ( k − + 4 k + N ) for the time required toestablish the law-of-mass action kinetic equilibrium forthe dimers. If the arrival frequency of the third particlesis k + N , when k + N > t eq then only the linearly grow-ing part of a ( t ) from Eqs.(7)-(8) is necessary to arrive at K eff . Then the inequality to be satisfied for our approxi-mation to be fully justified is k + N > (cid:112) k − ( k − + 4 k + N )which can be rewritten as k + N > k − . Hence, any re-sult presented here in the regime k + N < k − N shouldbe regarded as merely illustrative. This fact is howevernot worrisome because in that regime the crossover intonucleation takes place and our kinetic theory anywayshas to be replaced by nucleation and coarsening at somepoint. APPENDIX: NUCLEATION KINETICS In this Appendix we shall see how the classical homoge-neous nucleation is recovered when the detailed balancecondition is satisfied and the aggregation occurs with d f = 3 by minimizing the free energy of clustering Φaccording to the mechanism discussed in section IV.a.Recall that the master kinetic equation for aggregationand breakup reads as: dN k dt = 12 i + j = k (cid:88) i,j =1 K + ij N i N j − N k ∞ (cid:88) i =1 K + ik N i − K − k N k + ∞ (cid:88) i = k +1 K − ik N i (36)Let us assume that detailed balance is satisfied such thataggregation occurs under equilibrium conditions by min-imizing the free energy Φ = Φ v +Φ s = − ( z ( R k /a ) d f / πγa d f k ( d f − /d f . Minimization gives d f = 3, mainlyto maximize the mean coordination number z . Then onlydetachments of individual colloidal particles are relevantwhereas body-fragmentation plays no role because it in-volves the breakup of many bonds, which is a much slowerprocess. Furthermore, clusters form as a result of spon-taneous fluctuations and the growth of each cluster isindependent of the behavior of the others. Clearly, un-der these conditions, the above equation can be rewrittenaccounting only for one-step association and dissociationprocesses: dN k dt = K + k − , N k − − K + k, N k − K − k N k + K − k +1 , N k +1 . (37)where we have incorporated the constant factor N = N inside the association rate constants, as the monomerconcentration is assumed to be constant for the slow pro-cess. To shorten the notation we put K + k − , ≡ K + k − etc.and rewrite this equation as: dN k dt = K + k − N k − − K k N k − K − k N k + K − k +1 N k +1 . (38)Since the attraction is relatively weak and thermal dis-sociation is important, the principle of detailed balance is applicable in this limit. Hence, we now introduce theequilibrium or steady-state concentration of aggregatesof size k N eqk which is a Boltzmann function of the min-imum available work (free energy) Φ needed to form anaggregate of size k : N eqk ∼ exp( − Φ /kT ). Upon applyingthe principle of detailed balance we have: N eqk K k = N eqk +1 K − k +1 N eqk − K k − = N eqk K − k (39)These relations allow us to eliminate from Eq. (38) thequantities K − k and K − k +1 and Eq. (38) becomes: dN k dt = K k (cid:2) − N k + N k +1 N eqk /N eqk +1 (cid:3) + K k − (cid:2) N k − − N k N eqk − /N eqk (cid:3) = K k N eqk (cid:2) N k +1 /N eqk +1 − N k /N eqk (cid:3) − K k − N eqk − (cid:2) N k /N eqk − N k − /N eqk − (cid:3) . (40)Let us now transform the discrete distribution N k into acontinuous one N ( x ) where x ∼ R is a continuous vari-able expressing the cluster size. Denoting by λ the spac-ing along the x axis between the neighboring sizes k and k + 1, we have N k = λN ( x ), N k +1 = λN ( x + λ ) etc.Then we get: dN k dt = K ( x ) N eq ( x ) [ N ( x + λ ) /N eq ( x + λ ) − N ( x ) /N eq ( x )] − K ( x − λ ) N eq ( x − λ )[ N ( x ) /N eq ( x ) − N ( x − λ ) /N eq ( x − λ )] . (41)Since λ is constant and N , N eq and K vary little withinthe length λ , one can do an expansion in power series of λ up to the first non-vanishing terms, under the assumptionthat N varies slowly and hence N > λN (cid:48) > λ N (cid:48)(cid:48) . TheTaylor expansion is done on the terms in ( x + λ ) and( x − λ ) and is centered on x : ∂N ( x ) ∂ t = K ( x ) N eq ( x ) (cid:20) λ ∂∂x N ( x ) N eq ( x ) (cid:21) − K ( x − λ ) N eq ( x − λ ) (cid:20) λ ∂∂x N ( x − λ ) N eq ( x − λ ) (cid:21) (42)We now expand the second term on the r.h.s.: K ( x − λ ) N eq ( x − λ ) (cid:20) λ ∂∂x N ( x − λ ) N eq ( x − λ ) (cid:21) = K ( x ) N eq ( x ) (cid:20) λ ∂∂x N ( x ) N eq ( x ) (cid:21) − λ ∂∂x (cid:26) K ( x ) N eq ( x ) (cid:20) λ ∂∂x N ( x ) N eq ( x ) (cid:21)(cid:27) (43)Upon replacing Eq.(43) in Eq. (42), cancelation of termsleads to: ∂N ( x ) ∂ t = λ ∂∂x (cid:26) K ( x ) N eq ( x ) (cid:20) ∂∂x N ( x ) N eq ( x ) (cid:21)(cid:27) . (44)4The product in the braces can be rewritten as: KN eq ( x ) (cid:20) ∂∂x (cid:18) N ( x ) N eq ( x ) (cid:19)(cid:21) == K ∂N ( x ) ∂x + KN ( x ) ∂ ln N eq ( x ) ∂x . (45)The equilibrium distribution is given by the Boltzmannform: N eq ( x ) ∼ e − Φ( x ) /kT . Inserting this and upon fi-nally replacing in Eq.(44) we obtain: ∂N ( x ) ∂ t = D eff ∂∂x (cid:26) ∂N ( x ) ∂x + Φ (cid:48) ( x ) kT · N ( x ) (cid:27) (46)where D eff = λ K is an effective diffusion coefficient in size space. Thus, we obtained a diffusion equation in sizespace for the growth kinetics. This equation can solvedwith Kramers saddle-point approximation to recover thestandard nucleation rate for systems close to equilibrium: I = N K (cid:18) Φ (cid:48)(cid:48) ( x ∗ )2 πk B T (cid:19) exp (cid:18) − Φ( x ∗ ) k B T (cid:19) (47)where x ∗ denotes the critical nucleus corresponding tothe maximum of Φ, i.e. the size at which the surfaceterm and the enthalpy term in Φ balance. ∗ [email protected] H.J. Herrmann, Physics Reports 136,153 (1986). W.C.K. Poon and M.D. Haw, Adv. Colloid Interface Sci.73, 71 (1997). V. Trappe and P. Sandkuehler, Curr. Opin. Colloid Inter-face Sci. 8, 494 (2004). S. Romer, F. Scheffold and P. Schurtenberger, Phys. Rev.Lett. , 4890 (2000). P. N. Segr`e, V. Prasad, A. B. Schofield, and D. A. Weitz,Phys. Rev. Lett. , 6042 (2001). V. Trappe et al., Nature , 772 (2001). E. Zaccarelli, J. Phys.: Condensed Matter , 323101(2007). P. J. Lu et al. Nature , 499 (2008). S. Buzzaccaro, R. Rusconi, and R. Piazza, Phys. Rev. Lett. , 098301 (2008). M. J. Solomon and P. Varadan, Phys. Rev. E , 051402. H. Tanaka, S. Jabbari-Farouji, J. Meunier, and D. Bonn,Phys. Rev. E , 021402 (2005). T. Gibaud and P. Schurtenberger, J. Phys.: CondensedMatter , 322201 (2009). A.P.R. Eberle, N.J. Wagner, R. Castaneda-Priego, Phys.Rev. Lett. 106, 105704 (2011). W. Y. Shih, I. A. Aksay, and R. Kikuchi, Phys. Rev. A 36,5015 (1987). M.D. Haw, M. Sievwright, W.C.K. Poon, P.N. Pusey, Adv.Colloid Interface Sci. 61, 1 (1995). M.A. Miller and D. Frenkel, Phys. Rev. Lett. 90, 135702(2003). S.A. Shah, Y.L. Chen, K.S. Schweizer, and C.F. Zukoski,J. Chem. Phys. 119, 8747 (2003). J. Bergenholtz, W. C. K. Poon and M. Fuchs, Langmuir , 4493 (2003). G. Foffi, C. De Michele, F. Sciortino, and P. Tartaglia,Phys. Rev. Lett. 94, 078301 (2005). E. Del Gado and W. Kob, Phys. Rev. Lett. 98, 028303(2007). K. Kroy, M. E. Cates, and W. C. K. Poon, Phys. Rev.Lett. , 148302 (2004). H.H. Winter and F. Chambon, J. Rheol. 30, 367 (1986). E. Del Gado, A. Fierro, L. de Arcangelis, and A. Coniglio,Europhys. Lett. 63, 1 (2003). A. Zaccone, H. Wu and E. Del Gado, Phys. Rev. Lett. , 208301 (2009); M. Laurati et al., J. Chem. Phys. , 134907 (2009). A. Zaccone, J. J. Crassous, B. Beri and M. Ballauff, Phys.Rev. Lett. , 168303 (2011). V. I. Agoshkov, P. B. Dubovski, V. P. Shutyaev, Methodsfor Solving Mathematical Physics Problems (CambridgeInt. Science Publishing, Cambridge, 2006). H. Holthoff et al. , Langmuir , 5541 (1996). A. Zaccone and E. M. Terentjev, Phys. Rev. Lett. 108,038302 (2012). C. Schneider, A. Jusufi, R. Farina, P. Pincus, M. Tirrell,and M. Ballauff, Phys. Rev. E , 011401 (2010). H. A. Kramers, Physica , 284 (1940). R.P. Tolman, The Principles of Statistical Mechanics , (Ox-ford University Press, Oxford, 1938). M. Abramowitz and I. A. Stegun, Handbook of Mathemat-ical Functions with Formulas, Graphs, and MathematicalTables (Dover, New York, 1972). D. Asnaghi, M. Carpineti, M. Giglio, and M. Sozzi, Phys.Rev. A 45, 1018 (1992). E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics (Perg-amon Press, OPxford, 1981). P. Sandkuehler, J. Sefcik, and M. Morbidelli, J. Phys.Chem. B , 20105-20121 (2004). A. Zaccone, D. Gentili, H. Wu, M. Morbidelli, E. Del Gado,Phys. Rev. Lett. , 138301 (2011) P. G. Debenedetti, Metastable Liquids (Princeton Univer-sity Press, Princeton NJ, 1996). P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca NY, 1979) p.176. L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Butterworth-Heinemann, Oxford, 1987). C.I. Mendoza and I. Santamara-Holek, J. Chem. Phys. 130,044904 (2009). W. Schaertl and H. Sillescu, J. Stat. Phys. 77, 1007, (1994). G. Lois, J. Blawzdziewicz, and C. S. O’Hern, Phys. Rev.Lett. , 028001 (2008). W. Goetze and L. Sjoegren, Z. Phys. B 65, 415 (1987). Z. Cheng, J. Zhu, P. M. Chaikin, S.-E. Phan, and W. B.Russel, Phys. Rev. E , 041405 (2002). W.C.K. Poon , et al. Faraday Discuss. 112, 143 (1999). P.J. Lu et al., Phys. Rev. Lett. 96, 028306 (2006). A. Stradner et al., Nature 432, 492 (2004). Ya.B. Zeldovich, J. Phys. USSR 12, 525 (1942). S. Manley, et al. Phys. Rev. Lett. 95, 238302 (2005). J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 2006). I. Deike, M. Ballauff, N. Willenbacher, and A. Weiss, J.Rheol. , 709 (2001) A. Zaccone and E. Scossa-Romano, Phys. Rev. B ,184205 (2011). C. S. O’Hern, et al. Phys. Rev. E , 011306 (2003). A. Fernandez-Nieves et al. , Langmuir , 1841 (2001); M.Rasmusson, A. Routh, and B. Vincent, Langmuir , 3536(2004); M. Ballauff and Y. Lu, Progr. Polym. Sci.36