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Physica A 356 (2005) 190–195

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Off-lattice Dynamic Monte Carlo simulations ofaggregation in one dimension

Rafael Salazara,b,�, Lev D. Gelba

aDepartment of Chemistry and Center for Materials Innovation, Washington University in St. Louis,

St. Louis, MO 63130, USAbUniversidad Nacional Mayor de San Marcos. Fac. Fısica A.P. 14-0149, Lima 14, Peru

Available online 8 June 2005

Abstract

We present an approach to simulate aggregation and gelation based on the extension of the

Dynamic Monte Carlo method for chemical reactions to off-lattice inhomogeneous systems.

Diffusion is explicitly introduced into the Dynamic Monte Carlo method through the use of

first collision time distributions. The resulting method is as fast as diffusion limited cluster

aggregation simulations but is more precise, producing results in good agreement with

expensive Brownian dynamics simulations. We present results for one-dimensional aggrega-

tion using both hard-sphere and continuous potentials. Extensions to more dimensions are

discussed.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Aggregation; First collision time; DLCA; Brownian dynamics; Dynamic Monte Carlo

Sol–gel materials are produced by a polymerization process [1]. In the synthesis ofsilica sol–gels under basic conditions, a precursor solution of multifunctionalmonomers reacts relatively quickly to produce a solution of small spherical particles(the ‘‘sol’’), which then slowly aggregate to form a fully connected structure, the‘‘gel’’. This process occurs on time-scales as long as hours. In order to simulate thisprocess to produce and study realistic gel structures, this time-scale must be

see front matter r 2005 Elsevier B.V. All rights reserved.

.physa.2005.05.035

nding author.

dress: rafael@wuchem.wustl.edu (R. Salazar).

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R. Salazar, L.D. Gelb / Physica A 356 (2005) 190–195 191

somehow accommodated without losing a microscopic description of the gellingsolution. Molecular dynamics simulations of the solvent and gel precursor solutecannot access time-scales much longer than several tens of nanoseconds [2]. Since thesolvent is not itself of interest, one may coarse-grain such that the primary object ofsimulation is a sol particle and then use an approximate dynamics to includefluctuating solvent forces. Langevin dynamics is the most detailed such descriptioncommonly used [3]. The Langevin equations of motion for the coordinate xi of aparticle i are

mi €xi ¼ f i � Z _xi þ aiðtÞ , (1)

where mi is the mass of the particle and f i ¼ �dðP

juijÞ=dxi is the force com-ponent acting on the particle i due to its interaction with the other particles j viathe potential uij . Z is the viscosity coefficient and aiðtÞ is the random force com-ponent due to the interaction with the solvent environment. The random processaiðtÞ is a gaussian stochastic process (white noise), with the statistical properties,haiðtÞi ¼ 0, and haiðt0Þajðt0 þ tÞi ¼ 2mikBTZdðtÞdij , where kBT is the thermal energy.The distribution of probability of the position coordinates of one particle initiallyat the origin when f i ¼ 0 is a gaussian distribution Pðx; tÞ with zero mean andvariance s2xðtÞ ¼ kBTð2Zt � 3þ 4 exp½�Zt � exp½�2ZtÞ=ðmZ2Þ. For high viscosity,Zb1, and long times, Ztb1, one can adiabatically eliminate the velocitiesfrom Eq. (1). This produces equations of motion referred to as ‘‘Browniandynamics’’ (BD) [3]: _xi ¼ xif i þ biðtÞ, where xi ¼ 1=ðmiZÞ and the random processbiðtÞ ¼ xiaiðtÞ is still a white noise. The probability distribution of the positioncoordinates of one free particle following BD when f i ¼ 0 is still a gaussiandistribution Pðx; tÞ, but now with s2xðtÞ ¼ 2Dit, where Di ¼ xikBT is the diffusioncoefficient. In the general case f ia0, a numerical stochastic integration schememust be used, such as the first-order Euler algorithm xiðt þ dtÞ ¼

xiðtÞ þ xidtf i þ Gffiffiffiffiffiffiffiffiffiffiffiffi2Didt

p, where G is a gaussian random number of zero mean and

unit variance. The efficiency of numerical BD simulations is still limited, because theintegration time step dt should be small enough to guarantee the precision of theintegration scheme.An alternative approach used to study gelation is to approximately coarse-grain

the time evolution, as in the diffusion limited cluster aggregation (DLCA) algorithm[4]. DLCA simulates the free diffusion of particles and their (irreversible)aggregation at collision to form clusters, which eventually results in gel-likespanning structures. In this method one randomly selects a cluster to move (singleparticles are clusters of size one) based on a normalized probability Ps ¼ Ds=Dmax,where Ds�sa and s is the size of the cluster. The selected cluster is moved a distanceD0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2D0dt

pin a random direction, with dt as large as wanted. D0 is the diffusion

coefficient of one particle. If it makes contact with another cluster along thistrajectory, the clusters are joined at the point of impact to became a single biggercluster. The process can be approximately mapped onto real time by incrementing atime index by the amount dt=ðNhDsiÞ, where N is the number of clusters in thesystem at a given time. The use of DLCA is reasonable when the inter-particleinteractions uij are short-ranged and weak.

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R. Salazar, L.D. Gelb / Physica A 356 (2005) 190–195192

It would be useful to have a simulation method intermediate between BD andDLCA, which does not use a constant small time-step but also retains the effects ofthe interaction potential uij . In this paper we introduce such a method, based on theDynamic Monte Carlo (DMC) method introduced by Gillespie [5] to simulate thestochastic dynamics of chemical reactions in homogeneous systems. In the DMCmethod one writes the probability distribution of the reaction times for the reactionm, PmðtÞ ¼ km expð�kmtÞ, where km is the corresponding reaction rate. A DMCcalculation proceeds as follows. First, generate all the possible random reactiontimes tm according to the corresponding probability distribution PmðtÞ. This is doneby transforming the probability distribution PmðtÞ to a distribution of uniformrandom numbers U, by U ¼ F ðtÞ ¼

R t0 PmðtÞdt, from which we can derive

analytically tm ¼ F�1ðUÞ ¼ logð1=UÞ=km. Second, determine the next reaction tooccur by finding the minimum reaction time, tmin ¼ minftmg. Finally, ‘‘do’’ thereaction m and evolve the time by t ¼ t þ tmin. DMC has been previously applied byother groups to gelation and polymerization [6] by simulating a large number ofmonomers and tracking different clusters as different chemical species in order toobtain the time evolution of the distribution of cluster sizes. However, thesesimulations assumed a homogeneous solution at all times (implicit in the Gillespieformalism) and therefore were not able to produce realizations of late-stage gelstructures. We now introduce explicit particle coordinates in a one-dimensionalsystem; extensions of this approach to two and three dimensions are underdevelopment. For the geometry shown in Fig. 1 we compute the first collision timedistribution Gðdij ; tij ;DijÞ defined by the conditional probability relation [7]:

Pðdij ; tij ;DijÞ ¼

Z tij

0

Pðdij � d; tij � t;DijÞGðd; t;DijÞdt , (2)

where Pðd; t;DijÞ is the probability distribution of the relative distances as a functionof time, with a relative diffusion constant Dij . We will initially approximatePðd; t;DijÞ by pure Brownian motion as described by a gaussian distribution withs2d ¼ 2Dijt. We discuss later a better approximation. Eq. (2) can be solved forGðdij ; tij ;DijÞ by using Laplace transforms

Gðdij ; tij ;DijÞ ¼dij

t3=2ij

ffiffiffiffiffiffiffiffiffiffiffi4pDij

p exp�d2

ij

4Dijtij

!(3)

Gðdij ; tij ;DijÞ describes the random collision times between particles (eventuallyclusters) due to diffusion. To generate random times tij, one transforms (3) to adistribution of uniform random numbers U. The analytical solution is

i jdij

tij

Fig. 1. One-dimensional model of aggregation.

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R. Salazar, L.D. Gelb / Physica A 356 (2005) 190–195 193

tij ¼ F�1ðUÞ ¼ d2ij=ð4Dij½erf

�1ð�UÞ2Þ. For the pure Brownian approximation to

cluster movement, we use Dij ¼ D0ðs�1i þ s�1j Þ, where si; sj are the number of particles

in each cluster. The interaction potential uij describes the potential energy surface ofthe particle–particle reaction, and it is introduced by including an additional randomreaction time described by Pm from the Gillespie standard prescription. Anapproximate value for the corresponding rate constant km can be obtained from

uij with Kramer’s equation [7], 2pkm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu00

ijðAÞju00ijðBÞj

qexp½�ðuijðBÞ � uijðAÞÞ=ðkBTÞ

where A;B label the potential well below the reaction barrier and at the top of thereaction barrier, respectively. One cycle of the calculation proceeds as follows. First,

we compute all inter-particle distances dij , and then compute tij ¼ tdij þ tr

ij , where the

diffusive tdij and reactive tr

ij times are generated according to Gðdij ; tdij ;DijÞ and Pmðt

rijÞ

respectively:

tij ¼d2

ij

4Dij ½erf�1ð�UÞ2

þlogð1=UÞ

km. (4)

Next, find the next collision to occur, tmin ¼ minðt0; ftijgÞ, where t0 is a preset

maximum time-step. If tminat0, ‘‘do’’ the reaction. Then, diffuse all clusters byrandomly displacing them by an amount Dxi chosen from the truncated gaussian

distribution PðDxiominðfdigÞ; tmin;DiÞ, where Di ¼ D0s�1i . The t0 parameter ensures

that these diffusion distances are short enough to satisfy DxiominðfdigÞ withouttruncation in cases where no reaction is likely to occur in short time. Finally, evolvethe time t ¼ t þ tmin.We test our DMC method against BD and DLCA in two kind of systems:

a system with non-interacting particles which aggregates at contact (hard-sphere potential) and a system with a soft inter-particle potential, uij ¼

1=r12ij � 1=r6ij þ 0:1 exp½�1000ðrij � 1:6Þ2, which has two wells and a energy barrierin between to represent reactive aggregation (reactive potential). Both systems arecomposed of N ¼ 5000 particles at a reduced density r ¼ 0:1, and at temperatureT ¼ 0:01. The parameters are: for BD dt ¼ 10�3, for DLCA a ¼ �1=2 and dt ¼ 10,and for DMC t0 ¼ 108, and km ¼ 0:05 (estimated by Kramer’s equation). Fig. 2shows the results (averaged over five independent simulations) of each method forthe time dependence of the mean cluster size hsizeðtÞi and the distribution of clustersizes NðsizeÞ at two later times indicated by arrows in the upper plots. In the DLCAsimulation of the reactive potential, we include a probability p to merge two clustersafter collision. The value p ¼ 0:1 was adjusted to reproduce the initial slope of theBD hsizeðtÞi curve. Clearly, our modified DMC simulation is in much betteragreement with BD results than is the DLCA simulation in both curves and bothcases. The important point is that the cost (in CPU-time) of our DMC simulation isstill low, tDMC�tDLCA�1 min, while tBD�6 h. This method therefore could be usedto simulate long-time gelation phenomena without the approximations inherent inDLCA and similar algorithms.To further improve the realism of the approach, instead of approximating the

relative motion of clusters as purely Brownian, we can introduce a bias due to an

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-2 0 2 4ln[t]

0

0.2

0.4

0.6

0.8

ln[<

size

(t)>

]

BDDMCDLCA

Hard-sphere potential

-2 0 2 4ln[t]

Reactive potential

4 8 12size

0

0.1

0.2

N(s

ize)

4 8 12size

1 2 3rij

-0.2

0

u ij

Fig. 2. Two cases: hard-sphere interaction plus aggregation (left) and an inter-particle reactive potential

interaction (right). Mean cluster size hsizeðtÞi and the distribution of cluster sizes NðsizeÞ at two times

indicated by the arrows are shown.

R. Salazar, L.D. Gelb / Physica A 356 (2005) 190–195194

(assumed constant) force xijF ij ¼ xj f j � xif i between the particles, now described bythe gaussian distribution Pðdij � xijF ijtij ; tij ;DijÞ. We can again compute analyticallythe first collision time Gðdij ; tij ;Dij ;FijÞ using Laplace transforms

Gðdij ; tij ;Dij ;FijÞ ¼dij

t3=2ij

ffiffiffiffiffiffiffiffiffiffiffi4pDij

p exp�ðdij � 2xijF ijtijÞ

2

4Dijtij

!. (5)

The transformation from GðtijÞ to the uniform distribution of random numbers U isno longer analytically soluble and must be done numerically, though should still becomputationally inexpensive. Finally, extension of this approach to two- and three-dimensional simulations will require a number of enhancements, the most significantof which are the inclusion of rotational motion, intra-cluster relaxation and reactions(perhaps using short intervals of BD simulations) and improved calculation ofreaction rate constants.This work has been supported by a grant from the National Science Foundation

(CHE-0241005).

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References

[1] C.J. Brinker, G.W. Scherer, Sol–Gel Science, Academic Press, San Diego, 1990.

[2] N.Z. Rao, L.D. Gelb, J. Phys. Chem. B 108 (2004) 12418.

[3] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York,

1987.

[4] P. Meakin, J. Sol–Gel Sci. Technol. 15 (1999) 97.

[5] D.T. Gillespie, J. Computational Phys. 22 (1976) 403.

[6] S.E. Rankin, L.J. Kasehagen, A.V. KcCormick, C.W. Macosko, Macromolecules 33 (2000) 7639.

[7] C.W. Gardiner, Handbook of Stochastic Methods, Springer-Verlag, Berlin, 2004.