Dynamics of nonequilibrium deposition

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 165 (2000) 231–240

Dynamics of nonequilibrium deposition

Vladimir PrivmanDepartment of Physics, Clarkson Uni6ersity, Potsdam, NY 13699-5820, USA

Abstract

In this work we survey selected theoretical developments for models of deposition of extended particles, with andwithout surface diffusion, on linear and planar substrates, of interest in colloid, polymer, and certain biologicalsystems. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Nonequilibrium dynamical models; Random sequential adsorption; Colloids

www.elsevier.nl/locate/colsurfa

1. Introduction

Dynamics of important physical, chemical, andbiological processes, e.g. [1,2], provides examplesof strongly fluctuating systems in low dimensions,D=1 or 2. These processes include surface ad-sorption, for instance of colloid particles orproteins, possibly accompanied by diffusional orother relaxation (such as detachment), for whichthe experimentally relevant dimension is that ofplanar substrates, D=2, or that of large collec-tors. The surface of the latter is also semi-two-di-mensional owing to their large size as comparedto the size of the deposited particles.

For reaction–diffusion kinetics, the classicalchemical studies were for D=3. However, recentemphasis on heterogeneous catalysis generated in-terest in D=2. Actually, for both deposition andreactions, some experimental results exist even inD=1 (literature citations will be given later).

Finally, kinetics of ordering and phase separation,largely amenable to experimental probe in D=3and 2, attracted much recent theoretical effort inD=1, 2.

Theoretical emphasis on low-dimensional mod-els has been driven by the following interestingcombination of properties. Firstly, models in D=1, and sometimes in D=2, allow derivation ofanalytical results. Secondly, it turns out that allthree types of model: deposition–relaxation, reac-tion–diffusion, phase separation, are interrelatedin many, but not all, of their properties. Thisobservation is by no means obvious, and in fact itis model-dependent and can be firmly establishedand explored only in low dimensions, especially inD=1, see, e.g. [1,2].

It turns out that for systems with stochasticdynamics without the equilibrium state, importantregimes, such as the large-time asymptotic behav-ior, are frequently governed by strong fluctuationsmanifested in power-law rather than exponentialtime dependence, etc. However, the upper criticaldimension above which the fluctuation behavior is

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V. Pri6man / Colloids and Surfaces A: Physicochem. Eng. Aspects 165 (2000) 231–240232

described by the mean-field (rate equation) ap-proximation, is typically lower than in the morefamiliar and better studied equilibrium models. Asa result, attention has been drawn to low dimen-sions where the strongly fluctuating non-mean-field behavior can be studied.

Low-dimensional nonequilibrium dynamicalmodels pose several interesting challenges theoret-ically and numerically. While many exact, asymp-totic, and numerical results are already availablein the literature, as reviewed in [1,2], this fieldpresently provides examples of properties (such aspower-law exponents) which lack theoretical ex-planation even in 1D. Numerical simulations arechallenging and require large scale computationaleffort already for 1D models. For more experi-mentally relevant 2D cases, where analytical re-sults are scarce, difficulty in numerical simulationshas been the ‘bottle neck’ for understanding manyopen problems.

The purpose of this work is to provide anintroduction to the field of nonequilibrium surfacedeposition models of extended particles. No com-prehensive survey of the literature is attempted.Relation of deposition to other low-dimensionalmodels mentioned earlier will be only referred toin detail in few cases. The specific models andexamples selected for a more detailed exposition,i.e. models of deposition with diffusional relax-ation, were biased by author’s own work.

The outline of the review is as follows. The restof this introductory section is devoted to definingthe specific topics of surface deposition to besurveyed. Section 2 describes the simplest modelsof random sequential adsorption (RSA). Section 3is devoted to deposition with relaxation, withgeneral remarks followed by definition of thesimplest, 1D models of diffusional relaxation forwhich we present a more detailed description ofvarious theoretical results. Multilayer depositionis also addressed in Section 3. More numericallybased 2D results for deposition with diffusionalrelaxation are surveyed in Section 4, along withconcluding remarks.

Surface deposition is a vast field of study. In-deed, dynamics of the deposition process is gov-erned by substrate structure, substrate–particleinteractions, particle–particle interactions, and

transport mechanism of particles to the surface.Furthermore, deposition processes may be accom-panied by particle motion on the surface and bydetachment. Our emphasis here will be on thosedeposition processes where the particles are ‘large’as compared to the underlying atomic and mor-phological structure of the substrate and as com-pared to the range of the particle–particle andparticle–substrate interactions. Thus, colloids, forinstance, involve particles of submicron to severalmicron size. We note that 1 mm=10 000 A, ,whereas atomic dimensions are of order 1 A, ,while the range over which particle–surface andparticle–particle interactions are significant ascompared to kT, is typically of order 100 A, orless.

Extensive theoretical study of such systems isrelatively recent and it has been motivated byexperiments where submicron-size colloid, poly-mer, and protein ‘particles’ were the depositedobjects; see [3–18] for a partial literature list, aswell as other articles in this issue. It is usuallyassumed that the main mechanism by which parti-cles ‘talk’ to each other is the exclusion effect dueto their size. In contrast, deposition processesassociated, for instance, with crystal growth, e.g.[19], involve atomic-scale interactions and whilethe particle–particle exclusion is always an impor-tant factor, its interplay with other processeswhich affect the growth dynamics is quitedifferent.

Perhaps the simplest and the most studiedmodel with particle exclusion is RSA. The RSAmodel, to be described in detail in Section 2,assumes that particle transport (incoming flux)onto the surface results in a uniform depositionattempt rate R per unit time and area. In thesimplest formulation, one assumes that onlymonolayer deposition is allowed. This could cor-respond, for instance, to repulsive particle–parti-cle and attractive particle–substrate forces.Within this monolayer deposit, each new arrivingparticle must either ‘fit in’ in an empty areaallowed by the hard-core exclusion interactionwith the particles deposited earlier, or the deposi-tion attempt is rejected.

As mentioned, the basic RSA model will bedescribed shortly, in Section 2. More recent work

V. Pri6man / Colloids and Surfaces A: Physicochem. Eng. Aspects 165 (2000) 231–240 233

has been focused on its extensions to allow forparticle relaxation by diffusion, Sections 3 and 4,to include detachment processes, and to allowmultilayer formation. The latter two extensionswill be briefly surveyed in Section 3. Many otherextensions will not be discussed, such as for in-stance ‘softening’ the hard-core interactions[13,20] or modifying the particle transport mecha-nism, etc. [21,22].

2. Random sequential adsorption

The irreversible RSA process [21,22] modelsexperiments of colloid and other, typically, submi-cron, particle deposition [4–16] by assuming aplanar 2D substrate and, in the simplest case,continuum (off-lattice) deposition of sphericalparticles. However, other RSA models have re-ceived attention. In 2D, noncircular cross-sectionshapes as well as various lattice-deposition modelswere considered [21,22]. Several experiments onpolymers [3] and attachment of fluorescent unitson DNA molecules [18] (the latter, in fact, isusually accompanied by motion of these units onthe DNA ‘substrate’ and detachment), suggestconsideration of the lattice-substrate RSA pro-cesses, in 1D. RSA processes have also foundapplications in traffic problems and certain otherfields and they were reviewed extensively in theliterature [21,22]. Our presentation in this sectionaims at defining some RSA models and outliningcharacteristic features of their dynamics.

Fig. 1 illustrates the simplest possible mono-layer lattice RSA model: irreversible deposition ofdimers on the linear lattice. An arriving dimer willbe deposited if the underlying pair of lattice sitesare both empty. Otherwise, it is discarded. Thus,the deposition attempt of a will succeed. How-ever, if the arriving particle is at b then thedeposition attempt will be rejected unless there issome relaxation mechanism such as detachmentof dimers or monomers, or diffusional hopping.For instance, if c first hops to the left then laterdeposition of b can succeed. For d, the depositionis, again, not possible unless detachment and/ormotion of monomers or whole dimers clear theappropriate landing sites marked by e.

Let us consider the irreversible RSA withoutdetachment or diffusion. Note that once a at-taches, in Fig. 1, the configuration is fully jammedin the interval shown. The substrate is usuallyassumed to be empty initially, at t=0. In thecourse of time t, the coverage, r(t), increases andbuilds up to order 1 on the time scales of order(RV)−1, where R was defined earlier as the depo-sition attempt rate per unit time and ‘area’ of theD-dimensional surface, while V is the particlevolume. The latter is D-dimensional; for deposi-tion of spheres on a planar surface, V is actuallythe cross-sections area.

At large times the coverage approaches thejammed-state value where only gaps smaller thanthe particle size were left in the monolayer. Theresulting state is less dense than the fully ordered‘crystalline’ (close-packed) coverage. For the D=1 deposition shown in Fig. 1 the fully orderedstate would have r=1. The variation of the RSAcoverage is illustrated by the lower curve in Fig. 2.

At early times the monolayer deposit is notdense and the deposition process is largely uncor-related. In this regime, mean-field like low-densityapproximation schemes are useful [23–26]. Depo-sition of k-mer particles on the linear lattice in 1Dwas in fact solved exactly for all times [3,27,28]. InD=2, extensive numerical studies were reported[26,29–40] of the variation of coverage with timeand large-time asymptotic behavior which will bediscussed shortly. Some exact results for correla-tion properties are also available, in 1D [27].

Fig. 1. Deposition of dimers on the 1D lattice. Once thearriving dimer a attaches, the configuration shown will be fullyjammed in the interval displayed. Further deposition can onlyproceed if dimer or monomer diffusion (hopping) and/ordetachment are allowed. Letter labels b, c, d, e are referred toin the text.


Fig. 2. Schematic variation of the coverage fraction r(t) withtime for lattice deposition without (lower curve) and with(upper curve) diffusional or other relaxation. The ‘ordered’density corresponds to close-packing. Note that the short-timebehavior deviates from linear at times of order 1/(RV). (Quan-tities R, V are defined in the text.)

r(�)−r(t)�[ln(RVt)]D−1

RVt, (2.2)

while for spherical particles,

r(�)−r(t)� (−RVt)−1/D. (2.3)

For the linear surface, the D=1 cubes and spheresboth reduce to the deposition process of segmentsof length V. As mentioned earlier, this 1D processis exactly solvable [27].

The D\1 expressions (2.2) and (2.3), and simi-lar relations for other particle shapes, etc. areactually empirical asymptotic laws which havebeen verified, mostly for D=2, by extensive nu-merical simulations [29–40]. The most studied 2Dgeometries are circles (corresponding to the depo-sition of spheres on a plane) and squares. Thejamming coverages are [29–31,39,40]

rsquares(�)#0.5620 and

rcircles(�)#0.544–0.550. (2.4)

For square particles, the crossover to continuumin the limit k�� and l�0, with fixed V1/D=klin deposition of k×k×…×k lattice squares, hasbeen investigated in some detail [40], both analyt-ically (in any D) and numerically (in 2D).

The correlations in the large-time ‘jammed’ stateare different from those of the equilibrium random‘gas’ of particles with density near r(�). In fact,the two-particle correlations in continuum deposi-tion develop a weak singularity at contact, andcorrelations generally reflect the infinite memory(full irreversibility) of the RSA process [27,31,42].

3. Deposition with relaxation

Monolayer deposits may ‘relax’ (i.e. exploremore configurations) by particle motion on thesurface, by their detachment, etc. In fact, detach-ment has been experimentally observed in deposi-tion of colloid particles which were otherwise quiteimmobile on the surface [7]. Theoretical interpre-tation of colloid particle detachment data hasproved difficult, however, because binding to thesubstrate once deposited, can be different fordifferent particles, whereas the transport to the

The large-time deposit has several characteristicproperties that have attracted much theoreticalinterest. For lattice models, the approach to thejammed-state coverage is exponential [40–42].This was shown to follow from the property thatthe final stages of deposition are in few sparse, wellseparated surviving ‘landing sites.’ Estimates ofdecrease in their density at late stages suggest that

r(�)−r(t)�exp(−RlDt), (2.1)

where l is the lattice spacing. The coefficient inEq. (2.1) is of order lD/V if the coverage is definedas the fraction of lattice units covered, i.e. thedimensionless fraction of area covered, alsotermed the coverage fraction, so that the coverageas density of particles per unit volume would beV−1r. The detailed behavior depends of the sizeand shape of the depositing particles as comparedto the underlying lattice unit cells.

However, for continuum off-lattice deposition,formally obtained as the limit l�0, the approachto the jamming coverage is power-law. This inter-esting behavior [41,42] is due to the fact that forlarge times the remaining voids accessible to parti-cle deposition can be of sizes arbitrarily close tothose of the depositing particles. Such voids arethus reached with very low probability by thedepositing particles, the flux of which is uniformlydistributed. The resulting power-law behavior de-pends on the dimensionality and particle shape.For instance, for D-dimensional cubes of volumeV,


substrate, i.e. the flux of the arriving particles inthe deposition part of the process, typically byconvective diffusion, is more uniform. Detach-ment also plays role in deposition on DNAmolecules [18]. Theoretical interpretation of thelatter data, which also involves hopping motion onDNA, was achieved by mean-field type modeling[43].

Recently, more theoretically motivated studiesof the detachment relaxation processes, in someinstances with surface diffusion allowed as well,have lead to interesting model studies [44–50].These investigations did not always assume de-tachment of the original units. For instance, in the1D dimer deposition shown in Fig. 1, each dimeron the surface could detach and open up a ‘land-ing site’ for future deposition. However, in orderto allow deposition in the location representedschematically by the dimer particle d, twomonomers could detach (marked by e) which wereparts of different dimers. Such models of ‘recom-bination’ prior to detachment, of k-mers in D=1,were mapped onto certain spin models and sym-metry relations identified which allowed derivationof several exact and asymptotic results on the cor-relations and other properties [44–50]. We notethat deposition and detachment combine to drivethe dynamics into a steady state, rather thanjammed state as in ordinary RSA. These studieshave been largely limited thus far to 1D models.

We now turn to particle motion on the surface,in a monolayer deposit, which was experimentallyobserved in deposition of proteins [17] and also indeposition on DNA molecules [18,43]. From nowon, we focus on diffusional relaxation (randomhopping in the lattice case). Consider the dimer de-position in 1D; see Fig. 1. The configuration inFig. 1, after particle a is actually deposited, isjammed in the interval shown. Hopping of particlec one site to the left would open up a two-site gapto allow deposition of b. Thus, diffusional relax-ation allows the deposition process to reachdenser, in fact, ordered (close-packed) configura-tions. For short times, when the empty area isplentiful, the effect of the in-surface particle mo-tion will be small. However, for large times, thedensity will exceed that of the RSA process, as il-lustrated by the upper curve in Fig. 2.

Further investigation of this effect is much sim-pler in 1D than in 2D. Let us therefore considerthe 1D case first, postponing the discussion of 2Dmodels to the next section. Specifically, considerdeposition of k-mers of fixed length V. In order toallow limit k�� which corresponds to contin-uum deposition, we take the underlying latticespacing l=V/k. Since the deposition attempt rateR was defined per unit area (unit length here), ithas no significant k-dependence. However, theadded diffusional hopping of k-mers on the 1Dlattice, with attempt rate H and hard-core or simi-lar particle interaction, must be k-dependent. In-deed, we consider each deposited k-mer particle asrandomly and independently attempting to moveone lattice spacing to the left or to the right withrate H/2 per unit time. Of course, particles cannotrun over each other so some sort of hard-core in-teraction must be assumed, i.e. in a dense statemost hopping attempts will fail. However, if leftalone, each particle would move diffusively forlarge time scales. In order to have the resulting dif-fusion constant D finite in the continuum limitk��, we put

H8D/l2=Dk2/V2. (3.1)

which is only valid in 1D.Each successful hopping of a particle results in

motion of one empty lattice site (see particle c inFig. 1). It is useful to reconsider the dynamics ofparticle hopping in terms of the dynamics of thisrearrangement of empty area fragments [51–53].Indeed, if several of these empty sites are combinedto form large enough voids, deposition attemptscan succeed in regions of particle density whichwould be ‘frozen’ or ‘jammed’ in ordinary RSA. Interms of these new ‘particles’ which are empty lat-tice sites of the deposition problem, the process isin fact that of reaction–diffusion. Indeed, k reac-tants (empty sites) must be brought together by dif-fusional hopping in order to have finite probabilityof their annihilation, i.e. disappearance of a groupof consecutive nearest-neighbor empty sites due tosuccessful deposition. Of course, the k-group canalso be broken apart due to diffusion. Therefore,the k-reactant annihilation is not instantaneous inthe reaction nomenclature. Such k-particle reac-tions are of interest on their own [54–59].


The simplest mean-field rate equation for anni-hilation of k reactants describes the time depen-dence of the coverage, r(t), in terms of thereactant density 1−r,

dr

dt=G(1−r)k, (3.2)

where G is the effective rate constant. Note thatwe assume that the close-packing dimensionalcoverage is 1 in 1D. There are two problems withthis approximation. Firstly, it turns out that fork=2 the mean-field approach breaks down. Dif-fusive–fluctuation arguments for non-mean-fieldbehavior have been advanced for reactions[54,56,60,61]. In 1D, several exact calculationssupport this conclusion [62–68]. The asymptoticlarge-time behavior turns out to be

1−r�1/t (k=2, D=1), (3.3)

rather than the mean-field prediction �1/t. Thecoefficient in Eq. (3.3) is expected to be universal,when expressed in an appropriate dimensionlessform by introducing single-reactant diffusionconstant.

The power law (3.3) was confirmed by extensivenumerical simulations of dimer deposition [69]and by exact solution for one particular value ofH [70] for a model with dimer dissociation. Thelatter work also yielded some exact results forcorrelations. Specifically, while the connected par-ticle–particle correlations spread diffusively inspace, their decay it time is nondiffusive; see [70]for details. Series expansion studies of models ofdimer deposition with diffusional hopping of thewhole dimers or their ‘dissociation’ into hoppingmonomers, have confirmed the expected asymp-totic behavior and also provided estimates of thecoverage as a function of time [71].

The case k=3 is marginal with the mean-fieldpower law modified by logarithmic terms. Thelatter were not observed in Monte Carlo studiesof deposition [52]. However, extensive results areavailable directly for three-body reactions [56–59], including verification of the logarithmic cor-rections to the mean-field behavior [57–59].

The second problem with the mean-field rateequation was identified in the continuum limit ofoff-lattice deposition, i.e. for k��. Indeed, the

mean-field approach is essentially the fast diffu-sion approximation assuming that diffusional re-laxation is efficient enough to equilibratenonuniform density profile fluctuations on thetime scales fast as compared to the time scales ofthe deposition events. Thus, the mean-field resultsare formulated in terms of the uniform properties,such as the density. It turns out, however, that thesimplest, k th-power of the reactant density form(3.2) is only appropriate for times t�ek−1/(RV).

This conclusion was reached [51] by assumingthe fast-diffusion, randomized (equilibrium) hard-core reactant system form of the inter-reactantdistribution function in 1D (essentially, an as-sumption on the form of certain correlations).This approach, not detailed here, allowsGinzburg-criterion-like estimation of the limits ofvalidity of the mean-field results and it correctlysuggests mean-field validity for k=4, 5,…, withlogarithmic corrections for k=3 and completebreakdown of the mean-field assumptions for k=2. However, this detailed analysis yields themodified mean-field relation

dr

dt=

gRV(1−r)k

1−r+k−1r(D=1), (3.4)

where g is some effective dimensionless rate con-stant. This new expression applies uniformly ask��. Thus, the continuum deposition is alsoasymptotically mean-field, with the essentially sin-gular ‘rate equation’

dr

dt=g(1−r) exp[−r/(1−r)]

(k=�, D=1). (3.5)

The approach to the full, saturation coverage forlarge times is extremely slow,

1−r(t):1

ln(t ln t)(k=�, D=1). (3.6)

Similar predictions for k-particle reactions can befound in [55].

When particles are allowed to attach also ontop of each other, with possibly some rearrange-ment processes allowed as well, multilayer de-posits will be formed. It is important to note thatthe large-layer structure of the deposit and fluctu-


ation properties of the growing surface will bedetermined by the transport mechanism of parti-cles to the surface and by the allowed relaxations(rearrangements). Indeed, these two characteris-tics determine the screening properties of the mul-tilayer formation process which in turn shape thedeposit morphology, which can range from fractalto dense, and the roughening of the growingdeposit surface. There is a large body of researchstudying such growth, with recent emphasis onthe growing surface fluctuation properties.

However, the feature characteristic of the RSAprocess, i.e. the exclusion due to particle size,plays no role in determining the universal, large-scale properties of ‘thick’ deposits and their sur-faces. Indeed, the RSA-like jamming will be onlyimportant for detailed morphology of the first fewlayers in a multilayer deposit. However, it turnsout that RSA-like approaches (with relaxation)can be useful in modeling granular compaction[72].

In view of the above remarks, multilayer depo-sition models involving jamming effects were rela-tively less studied. They can be divided into twogroups. Firstly, structure of the deposit in the firstfew layers is of interest [73–75] since they retain‘memory’ of the surface. Variation of density andother correlation properties away from the wallhas structure on the length scales of particle size.These typically oscillatory features decay awaywith the distance from the wall. Numerical MonteCarlo simulation aspects of continuum multilayerdeposition (ballistic deposition of 3D balls) were

reviewed in [75]. Secondly, few-layer depositionprocesses have been of interest in some experi-mental systems. Mean-field theories of multilayerdeposition with particle size and interactions ac-counted for were formulated [76] and used to fitsuch data [12,14–16].

4. Two-dimensional deposition with diffusionalrelaxation

We now turn to the 2D case of deposition ofextended objects on planar substrates, accompa-nied by diffusional relaxation (assuming mono-layer deposits). We note that the availabletheoretical results are limited to few studies[38,77–79]. They indicate a rich pattern of neweffects as compared to 1D. In fact, there existsextensive literature, e.g. [81], on deposition withdiffusional relaxation in other models, in particu-lar those where the jamming effect is not presentor plays no significant role. These include, e.g.deposition of ‘monomer’ particles which alignwith the underlying lattice without jamming, aswell as models where many layers are formed(mentioned in the preceding section).

The 2D deposition with relaxation of extendedobjects is of interest in certain experimental sys-tems where the depositing objects are proteins[17]. Here we focus on the combined effect ofjamming and diffusion, and we emphasize dynam-ics at large times. For early stages of the deposi-tion process, low-density approximation schemescan be used. One such application was reported in[38] for continuum deposition of circles on aplane.

In order to identify features new to 2D, let usconsider deposition of 2×2 squares on the squarelattice. The particles are exactly aligned with the2×2 lattice sites as shown in Fig. 3. Furthermore,we assume that the diffusional hopping is alongthe lattice directions 9x and 9y, one latticespacing at a time. In this model dense configura-tions involve domains of four phases as shown inFig. 3. As a result, immobile fragments of emptyarea can exist. Each such single-site vacancy (Fig.3) serves as a meeting point of four domain walls.By ‘immobile’ we mean that the vacancy cannot

Fig. 3. Fragment of a deposit configuration in the depositionof 2×2 squares. Illustrated are one single-site frozen vacancyat which four domain walls converge (indicated by heavylines), as well as one dimer vacancy which causes a kink in oneof the domain walls.


Fig. 4. Illustration of deposition of 2×2 particles on thesquare lattice. Diffusional motion during time interval from t1

to t2 can rearrange the empty area ‘stored’ in the domain wallto open up a new landing site for deposition. This is illustratedby the shaded particles.

the shaded particles diffusively rearrange to openup a deposition slot which can be covered by anarriving particle. Numerical simulations [78,79]find behavior reminiscent of the low-temperatureequilibrium ordering processes [83–85] driven bydiffusive evolution of the domain-wall structure.For instance, the remaining uncovered area van-ishes according to

1−r(t)�1

t. (4.1)

This quantity, however, also measures the lengthof domain walls in the system (at large times).Thus, disregarding finite-size effects and assumingthat the domain walls are not too convoluted (asconfirmed by numerical simulations), we concludethat the power law (4.1) corresponds to typicaldomain sizes growing as �t, reminiscent of theequilibrium ordering processes of systems withnonconserved order parameter dynamics [83–85].

We now turn again to the 2×2 model of Fig.3. The equilibrium variant of this model corre-sponds to hard-squares with both nearest andnext-nearest neighbor exclusion [82,86,87]. It hasbeen studied in lesser detail than the two-phasehard-square model described in the precedingparagraphs. In fact, the equilibrium phase transi-tion has not been fully classified (while it wasIsing for the simpler model). The ordering at lowtemperatures and high densities was studied in[86]. However, many features noted, for instancelarge entropy of the ordered arrangements, re-quire further study. The dynamical variant (RSAwith diffusion) of this model was studied numeri-cally in [77]. The structure of the single-site frozenvacancies and associated network of domain wallsturns out to be boundary-condition sensitive. Forperiodic boundary conditions the density ‘freezes’at values 1−r�L−1, where L is the linear sys-tem size.

Preliminary indications were found [77] that thedomain size and shape distributions in such afrozen state are nontrivial. Extrapolation L��indicates that the power law behavior similar toEq. (4.1) is nondiffusive: the exponent 1/2 is re-placed by �0.57. However, the density of thesmallest mobile vacancies, i.e. dimer kinks in do-main walls, one of which is illustrated in Fig. 3,

move due to local motion of the surroundingparticles. For it to move, a larger empty-areafragment must first arrive, along one of the do-main walls. One such larger empty void is shownin Fig. 3. Note that it serves as a kink in thedomain wall.

Existence of immobile vacancies suggests possi-ble ‘frozen,’ glassy behavior with extremely slowrelaxation, at least locally. In fact, the full charac-terization of the dynamics of this model requiresfurther study. The first numerical results [77] doprovide some answers which will be reviewedshortly. We first consider a simpler model de-picted in Fig. 4. In this model [78,79] the extendedparticles are squares of size 2×2 They arerotated 45° with respect to the underlying squarelattice. Their diffusion; however, is along the ver-tical and horizontal lattice axes, by hopping onelattice spacing at a time. The equilibrium variantof this model (without deposition, with fixed par-ticle density) is the well-studied hard-squaremodel [82] which, at large densities, phase sepa-rates into two distinct phases. These two phasesalso play role in the late stages of RSA withdiffusion. Indeed, at large densities the empty areais stored in domain walls separating ordered re-gions. One such domain wall is shown in Fig. 4.Snapshots of actual Monte Carlo simulation re-sults can be found in [78,79].

Fig. 4 illustrates the process of ordering whichessentially amounts to shortening of domainwalls. In Fig. 4, the domain wall gets shorter after


does decrease diffusively. Further studies areneeded to fully clarify the ordering process associ-ated with the approach to the full coverage ast�� and L�� in this model.

Even more complicated behaviors are possiblewhen the depositing objects are not symmetricand can have several orientations as they reachthe substrate. In addition to translational diffu-sion (hopping), one has to consider possible rota-tional motion. The square-lattice deposition ofdimers, with hopping processes including one lat-tice spacing motion along the dimer axis and 90°rotations about a constituent monomer, was stud-ied in [80]. The dimers were allowed to depositvertically and horizontally. In this case [80] thefull close-packed coverage is not achieved at allbecause the frozen vacancy sites can be embeddedin, and move by diffusion in, extended structuresof different ‘topologies.’ These structures areprobably less efficiently ‘demolished’ by the mo-tion of mobile vacancies than the elimination oflocalized frozen vacancies in the model of Fig. 3.

In summary, we reviewed the deposition pro-cesses involving extended objects, with jammingand its interplay with diffusional relaxation yield-ing interesting new dynamics of approach to thelarge-time state. While significant progress hasbeen achieved in 1D, the 2D systems requirefurther investigations. Mean-field and low-densityapproximations can be used in many instances forlarge enough dimensions, for short times, and forparticle sizes larger than few lattice units. Addeddiffusion allows formation of denser deposits andleads to power-law large-time tails which, in 1D,were related to diffusion-limited reactions, whilein 2D, associated with evolution of domain-wallnetwork and defects, reminiscent of equilibriumordering processes.

References

[1] V. Privman, Trends Stat. Phys. 1 (1994) 89.[2] V. Privman (Ed.), Nonequilibrium Statistical Mechanics

in One-Dimension, Cambridge University Press, London,1997.

[3] E.R. Cohen, H. Reiss, J. Chem. Phys. 38 (1963) 680.[4] J. Feder, I. Giaever, J. Colloid Interface Sci. 78 (1980)

144.

[5] A. Schmitt, R. Varoqui, S. Uniyal, J.L. Brash, C. Pusiner,J. Colloid Interface Sci. 92 (1983) 25.

[6] G.Y. Onoda, E.G. Liniger, Phys. Rev. A 33 (1986) 715.[7] N. Kallay, B. Biskup, M. Tomic, E. Matijevic, J. Colloid

Interface Sci. 114 (1986) 357.[8] N. Kallay, M. Tomic, B. Biskup, I. Kunjasic, E. Mati-

jevic, Colloids Surf. 28 (1987) 185.[9] J.D. Aptel, J.C. Voegel, A. Schmitt, Colloids Surf. 29

(1988) 359.[10] Z. Adamczyk, Colloids Surf. 35 (1989) 283.[11] Z. Adamczyk, Colloids Surf. 39 (1989) 1.[12] C.R. O’Melia, in: W. Stumm (Ed.), Aquatic Chemical

Kinetics, Wiley, New York, 1990, p. 447.[13] Z. Adamczyk, M. Zembala, B. Siwek, P. Warszynski, J.

Colloid Interface Sci. 140 (1990) 123.[14] N. Ryde, N. Kallay, E. Matijevic, J. Chem. Soc. Faraday

Trans. 87 (1991) 1377.[15] N. Ryde, H. Kihira, E. Matijevic, J. Colloid Interface Sci.

151 (1992) 421.[16] L. Song, M. Elimelech, Colloids Surf. A73 (1993) 49.[17] J.J. Ramsden, J. Stat. Phys. 73 (1993) 853.[18] C.J. Murphy, M.R. Arkin, Y. Jenkins, N.D. Ghatlia,

S.H. Bossmann, N.J. Turro, J.K. Barton, Science 262(1993) 1025.

[19] V.M. Glazov, S.N. Chizhevskaya, N.N. Glagoleva, Liq-uid Semiconductors, Plenum, New York, 1969.

[20] P. Schaaf, A. Johner, J. Talbot, Phys. Rev. Lett. 66 (1991)1603.

[21] Review: M.C. Bartelt, V. Privman, Intern. J. Mod. Phys.B5 (1991) 2883.

[22] Review: J.W. Evans, Rev. Mod. Phys. 65 (1993) 1281.[23] B. Widom, J. Chem. Phys. 44 (1966) 3888.[24] B. Widom, J. Chem. Phys. 58 (1973) 4043.[25] P. Schaaf, J. Talbot, Phys. Rev. Lett. 62 (1989) 175.[26] R. Dickman, J.-S. Wang, I. Jensen, J. Chem. Phys. 94

(1991) 8252.[27] J.J. Gonzalez, P.C. Hemmer, J.S. Høye, Chem. Phys. 3

(1974) 228.[28] J.W. Evans, J. Phys. A 23 (1990) 2227.[29] J. Feder, J. Theor. Biol. 87 (1980) 237.[30] E.M. Tory, W.S. Jodrey, D.K. Pickard, J. Theor. Biol.

102 (1983) 439.[31] E.L. Hinrichsen, J. Feder, T. Jøssang, J. Stat. Phys. 44

(1986) 793.[32] E. Burgos, H. Bonadeo, J. Phys. A 20 (1987) 1193.[33] G.C. Barker, M.J. Grimson, J. Phys. A 20 (1987) 2225.[34] R.D. Vigil, R.M. Ziff, J. Chem. Phys. 91 (1989) 2599.[35] J. Talbot, G. Tarjus, P. Schaaf, Phys. Rev. A 40 (1989)

4808.[36] R.D. Vigil, R.M. Ziff, J. Chem. Phys. 93 (1990) 8270.[37] J.D. Sherwood, J. Phys. A 23 (1990) 2827.[38] G. Tarjus, P. Schaaf, J. Talbot, J. Chem. Phys. 93 (1990)

8352.[39] B.J. Brosilow, R.M. Ziff, R.D. Vigil, Phys. Rev. A 43

(1991) 631.[40] V. Privman, J.-S. Wang, P. Nielaba, Phys. Rev. B 43

(1991) 3366.


[41] Y. Pomeau, J. Phys. A 13 (1980) L193.[42] R.H. Swendsen, Phys. Rev. A 24 (1981) 504.[43] S.H. Bossmann, L.S. Schulman, p. 443 in Ref. [2].[44] M. Barma, M.D. Grynberg, R.B. Stinchcombe, Phys.

Rev. Lett. 70 (1993) 1033.[45] R.B. Stinchcombe, M.D. Grynberg, M. Barma, Phys.

Rev. E 47 (1993) 4018.[46] M.D. Grynberg, T.J. Newman, R.B. Stinchcombe, Phys.

Rev. E 50 (1994) 957.[47] M.D. Grynberg, R.B. Stinchcombe, Phys. Rev. E 49

(1994) R23.[48] G.M. Schutz, J. Stat. Phys. 79 (1995) 243.[49] P.L. Krapivsky, E. Ben-Naim, J. Chem. Phys. 100 (1994)

6778.[50] M. Barma, D. Dhar, Phys. Rev. Lett. 73 (1994) 2135.[51] V. Privman, M. Barma, J. Chem. Phys. 97 (1992) 6714.[52] P. Nielaba, V. Privman, Mod. Phys. Lett. B 6 (1992) 533.[53] B. Bonnier, J. McCabe, Europhys. Lett. 25 (1994) 399.[54] K. Kang, P. Meakin, J.H. Oh, S. Redner, J. Phys. A 17

(1984) L665.[55] S. Cornell, M. Droz, B. Chopard, Phys. Rev. A 44 (1991)

4826.[56] V. Privman, M.D. Grynberg, J. Phys. A 25 (1992) 6575.[57] D. ben-Avraham, Phys. Rev. Lett. 71 (1993) 3733.[58] P.L. Krapivsky, Phys. Rev. E 49 (1994) 3223.[59] B.P. Lee, J. Phys. A 27 (1994) 2533.[60] K. Kang, S. Redner, Phys. Rev. Lett. 52 (1984) 955.[61] K. Kang, S. Redner, Phys. Rev. A 32 (1985) 435.[62] Z. Racz, Phys. Rev. Lett. 55 (1985) 1707.[63] M. Bramson, J.L. Lebowitz, Phys. Rev. Lett. 61 (1988)

2397.[64] D.J. Balding, N.J.B. Green, Phys. Rev. A 40 (1989) 4585.[65] J.G. Amar, F. Family, Phys. Rev. A 41 (1990) 3258.[66] D. ben-Avraham, M.A. Burschka, C.R. Doering, J. Stat.

Phys. 60 (1990) 695.[67] M. Bramson, J.L. Lebowitz, J. Stat. Phys. 62 (1991) 297.[68] V. Privman, J. Stat. Phys. 69 (1992) 629.

[69] V. Privman, P. Nielaba, Europhys. Lett. 18 (1992) 673.[70] M.D. Grynberg, R.B. Stinchcombe, Phys. Rev. Lett. 74

(1995) 1242.[71] C.K. Gan, J.-S. Wang, Phys. Rev. E55 (1997) 107.[72] M.J. de Oliveira, A. Petri, J. Phys. A 31 (1998) L425.[73] R.-F. Xiao, J.I.D. Alexander, F. Rosenberger, Phys. Rev.

A 45 (1992) R571.[74] B.D. Lubachevsky, V. Privman, S.C. Roy, Phys. Rev. E

47 (1993) 48.[75] B.D. Lubachevsky, V. Privman, S.C. Roy, J. Comp. Phys.

126 (1996) 152.[76] V. Privman, H.L. Frisch, N. Ryde, E. Matijevic, J. Chem.

Soc. Faraday Trans. 87 (1991) 1371.[77] J.-S. Wang, P. Nielaba, V. Privman, Physica A 199 (1993)

527.[78] J.-S. Wang, P. Nielaba, V. Privman, Mod. Phys. Lett. B

7 (1993) 189.[79] E.W. James, D.-J. Liu, J.W. Evans, Relaxation Effects in

Random Sequential Adsorption: Application toChemisorption Systems, this volume.

[80] S.A. Grigera, T.S. Grigera, J.R. Grigera, Phys. Lett A226 (1997) 124.

[81] J.A. Vernables, G.D.T. Spiller, M. Hanbucken, Rept.Prog. Phys. 47 (1984) 399.

[82] L.K. Runnels, in: C. Domb, M.S. Green (Eds.), PhaseTransitions and Critical Phenomena, vol. 2, AcademicPress, London, 1972, p. 305.

[83] J.D. Gunton, M. San Miguel, P.S. Sahni, in: C. Domb,J.L. Lebowitz (Eds.), Phase Transitions and Critical Phe-nomena, vol. 8, Academic Press, London, 1983, p. 267.

[84] O.G. Mouritsen, in: M.G. Lagally (Ed.), Kinetics andOrdering and Growth at Surfaces, Plenum, New York,1990, p. 1.

[85] A. Sadiq, K. Binder, J. Stat. Phys. 35 (1984) 517.[86] K. Binder, D.P. Landau, Phys. Rev. B 21 (1941) 1980.[87] W. Kinzel, M. Schick, Phys. Rev. B 24 (1981) 324.

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Dynamics of nonequilibrium deposition

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