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Restructuring effects in the rain model for randomdeposition

P. Meakin, R. Jullien

To cite this version:P. Meakin, R. Jullien. Restructuring effects in the rain model for random deposition. Journal dePhysique, 1987, 48 (10), pp.1651-1662. �10.1051/jphys:0198700480100165100�. �jpa-00210603�

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Restructuring effects in the rain model for random deposition

P. Meakin (1) and R. Jullien (2)

(1) Central Research and Development Department, E.I. du Pont de Nemours and Company, Wilmington,DE 19898 U.S.A.

(2) Laboratoire de Physique des Solides, bâtiment 510, Université Paris-Sud, Centre d’Orsay, 91405 Orsay,France

(Reçu le 24 mars 1987, accepté le 17 juin 1987)

Résumé. 2014 On étudie deux sortes de mécanismes de restructuration dans le modèle de la pluie, sans supportde réseau, qui décrit le dépôt de particules sur une ligne en deux dimensions où les particules tombent une àune le long de trajectoires verticales aléatoires et se collent irréversiblement à un dépôt. Dans un premier cas,la particule qui tombe peut rouler sur la particule de contact jusqu’à ce qu’un deuxième contact soit établi.Dans un deuxième cas, la particule peut rouler autant de fois qu’il est nécessaire jusqu’à ce qu’elle rejoigne unminimum local d’énergie potentielle. Dans les deux cas, on étudie numériquement la densité du dépôt ainsique les lois d’échelle qui gouvernent le comportement de l’épaisseur de la surface. On trouve que le

comportement n’est pas affecté par le premier type de restructuration. Dans le deuxième cas, même les très

grosses simulations donnent des résultats ambigus mais une version simplifiée, sur réseau, de ce modèle donnedes resultats différents de ceux généralement rencontrés dans le cas ordinaire de déposition à deux dimensions(l’épaisseur de la surface croit comme la puissance 1/4 de la hauteur du dépôt).

Abstract. 2014 Two kinds of restructuring mechanisms are investigated in the off-lattice « rain » (ballisticdeposition) model for random deposition of particles on a line in two dimensions in which particles fall alongrandom vertical lines and irreversibly stick to a growing deposit. In the first case, the falling particle is allowedto rotate about the first contacting particle in the deposit until a second contact is obtained. In the second case,the particle is allowed to rotate as often as necessary until it finally reaches a local minimum position. In bothcases, the mean density of the deposit and the scaling behaviour of its surface thicknesses have been

numerically investigated. It is found that the scaling properties are not affected by the first type of

restructuring. For the second type of restructuring even very large scale simulations give ambiguous results buta simple lattice version of this model gives results which are quite different from those found in ordinary 2dballistic deposition (the surface thickness (03C3) grows as the 1/4 power of the deposit thickness).

J. Physique 48 (1987) 1651-1662 OCTOBRE 1987, :

Classification

Physics Abstracts81.90 - 46.60

1. Introduction.

In recent years considerable interest has developedin a variety of nonequilibrium growth and aggrega-tion phenomena [1]. Much of this interest is a resultof the progress which has been made in the computermodeling of aggregation processes. Two basic mod-els for diffusion limited aggregation have been

introduced. The diffusion limited particle cluster

aggregation model of Witten and Sander [2] demon-strated that simple aggregation models could be usedto help understand the origin of certain types ofcomplexes (frequently fractal [3]) structures. The

development of this model stimulated much of the

subsequent interest in fractal aggregates and hasbeen shown to provide a basis for describing a wide

variety of phenomena [1]. The diffusion limited

cluster-cluster aggregation model [4] provided a

physically realistic basis for understanding both thestructure and kinetics of fast colloidal aggregationprocesses.The main motivation for the work described here

is the deposition of particles in the presence of agravitational field. We assume that the addition ofparticles is slow (i.e., the deposition of one particleon the surface is completed before a second particlearrives in its vicinity and irreversible. Under theseconditions the process can be represented by a

particle cluster aggregation model with a « strip »geometry. If the gravitational field is strong and/orthe particles are large, the particle trajectories willapproach the limit of vertical ballistic paths.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100165100

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The role of the nature of the trajectory has beenextensively studied in the particle-cluster model.While Brownian trajectories lead to fractal deposits,even in a strip geometry [5], linear trajectories,either randomly oriented in space, as in the randomballistic model [6] or parallel to each other, as in therain model [7], both lead to homogeneous deposits,i.e., compact structures (their fractal dimension

being equal to the dimension of space). Anothervery simple model, the Eden model [8], in which

particles are added equiprobably on surface sites,also leads to a compact deposit [9]. A modifiedversion of the Eden model (the Eden model withfrozen holes) might be applied to random deposition,but this implies that there exists an extremely lowsticking probability, so that the falling particles caninvestigate all the landing possibilities before choos-ing one at which it will stick permanently. This is notphysically realistic in the case of strong fields.

If the bulk properties of the deposit appear to bequite trivial in these three basic models (ballistic,rain, Eden) for random deposition, the rough charac-ter of the surface has been the subject of manyrecent numerical investigations [10-13]. It appears,as a general result, that the surface thickness, 0", of

the deposit, usually calculated as the standard devia-tion of the vertical coordinates of the surface points,varies with the transverse dimension, L, of the strip,and the mean height of the whose deposit, (h),according to the following scaling relation, firstintroduced by Family and Vicsek [10] :

with f(x) --+ Cst. for x -> oo and f (x) - x ° for

x --+ 0, so that the scaling behaviour of a is differentwith L for large (h) : 0" - L" than with (h) for

large L : 0" - (h) ø. The most recent estimations ofa and B, in two dimensions are very close to

a = 1/2 and j8 = 1/3 [11-13], in the three models.These are the exact values derived analytically byKardar et al. [14], using a continuous model, whichis an extension of the random deposition model ofEdwards and Wilkinson [15] by including an extranonlinear term in the surface growth equation.For all three of the models discussed above

(random ballistic, rain and Eden) no restructuring isallowed during or after the deposition process. Formost real aggregation and deposition processes, thissimplification is, in most cases, unrealistic and leadsto structures which do not closely resemble thoseobserved experimentally. Some attempts to includesimple restructuring mechanisms have been exploredin cluster-cluster [16] and particle-cluster [17] aggre-gation models. In some cases these modificationslead to much better agreement between simulatedstructures and those observed experimentally [18].Much of the work on surface deposition models

has been motivated by the use of a variety of

deposition technologies to manufacture electronic

and optical devices. The simple rain and randomballistic deposition models lead to structures whichare uniform on all but very short length scales.

However, the densities of the deposits generated bythese models is much smaller than those found

experimentally. This has led to a number of 2d and3d simulations in which structural readjustment is

included [19]. The structural readjustment (or relax-ation) mechanisms used in these models are quitesimilar to those employed in our work. However, wehave carried out simulations on a considerably largerscale and we are, for the most part, concerned withdifferent aspects of the deposit morphology.

In this paper, we consider the off-lattice version of

the rain model, in the two-dimensional strip geomet-ry (hereafter called model I) and we introduce twopossible restructuring mechanisms in this model. Inthe first process (model II) only a local readjustmentis allowed while in the second process (model III)the readjustment may eventually extend to a largerscale. We also introduce a very simple lattice versionof this last model (model IV). The aim of the work isto determine if off-lattice deposition and/or differentkinds of restructuring could affect the general scalingproperties of the surface thickness which has, up tonow, only been studied on lattices and without anyrestructuring. A motivation for investigating mod-els III and IV is that it is generally believed that longrange effects could affect the scaling properties.Family [20] has investigated a two dimensional

lattice model for random deposition in which parti-cles are added to the top of the column in which theyare dropped. Since the growth of each column is anindependent random event, the exponent j8 in

equation (1) has a value of 1/2. Restructuring wasincluded in this model (to represent the effects ofsurface diffusion) by adding the particle to the top ofthe lowest column at positions in the range i - f toi + f when the particle was dropped in the i-th

column. Family found that even for f = 1 the

exponent was changed from 1/2 to 1/4. Model IV isquite similar to that of Family and also gives a valueof 1/4 for B. However, it differs in two importantaspects. In model IV the value of hi - hi + 1 is

always 1 where hi is the height of the column atposition i whereas in Family’s model there are norestrictions on his. In model IV the particle is depo-sited in the nearest local minimum in the surface

(however far away that local minimum might be)whereas in the Family model the particle is depositedon the lowest column within a range of ± f latticeunits even if there are minima in the surface at a

shorter distance.

In addition to the surface thickness, we havecalculated the averaged density of the deposit and itsvariation with the angle between the basal line and

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the horizontal. The paper is divided as follows. Themodels are described in part 2, numerical calcula-tions are presented in part 3 and a short discussion isgiven in part 4.

2. Models.

2.1 MODEL I : NO RESTRUCTURING. - This modelis a straightforward extension of the standard latticeversion of the rain model in a strip geometry.Particles are considered as hard disks of unit

diameter. The deposit is grown on a basal line,making a given angle 0 with the horizontal, within asemi-infinite vertical strip of width L particle diame-ters, with periodic boundary conditions at the edgesof the strip. Particles are added one at a time to thedeposit. At each step, a random vertical trajectory isselected by choosing its horizontal coordinatebetween 0 and L, at random. The vertical coordinatehn of the centre of the n-th particle is determined asbeing the maximum vertical coordinate among allthe possibilities where this particle stays centred onthe trajectory and contacts a particle of the deposit(case A in Fig. la). If no contacting particle is found(case B), the new particle is deposited on the basalline. For this model deposition onto a basal lineinclined at an angle of 0* from the horizontal is

equivalent to deposition onto a horizontal basal linewith a fixed angle of incidence of 0’ for the trajec-tory.

2.2 MODEL II : PARTIAL RESTRUCTURING. - This

is a model in which the falling particle, after its firstcontact, searches, with the help of « gravity », for amore favorable situation where another contact is

realized. The simulation proceeds as in model I

except that, when the contacted particle in the

deposit is found, the newly added particle is allowedto rotate about the centre of this contacted particle,in the direction which reduces its vertical coordinate,until it either contacts a second particle (case A inFig. 1b), the basal line (case B) or hangs verticallybelow the contacted particle (case C). When nocontacted particle is found, the particle is depositedon the basal line. If the basal line is not horizontal,the falling particle sticks at the point where it firstcontacts the surface (it is not allowed to roll « downhill » on the basal line). For this model depositiononto an inclined basal line is no longer equivalent todeposition onto a horizontal basal line with a non-normal incidence of incoming particles.

2.3 MODEL III : MULTIPLE RESTRUCTURING. -

This is a quite realistic representation of what couldhappen in a very strong gravity field. The fallingparticle, after its first contact with the deposit, canrotate as often as necessary (always decreasing its

potential energy), to finally reach the nearest local

Fig. 1. - The deposition models used in this work.

Figure la shows the standard off-lattice deposition model(model I) with vertical particle trajectories. The particlesstick at either their first contact with the deposit (A) orwith the basal line (B). Figure Ib show the single readjust-ment model (model II). After contacting a particle in thedeposit the deposited particle rolls « down hill » maintain-ing contact with the first contacted particle in the deposituntil it contacts a second particle in the deposit (A),contacts the basal line (B) or hangs vertically below thefirst contacted particle in the deposit (C). If no contact ismade with the growing deposit, the particle is depositedonto the basal line without lateral displacement (D).Figure lc shows the multiple readjustment model. In thismodel the deposited particle moves along the surface ofthe deposit maintaining contact with the deposit and

reducing its potential energy until it reaches on local

minimum in contact with two of the particles on thedeposit (A) or reaches the basal line. Figure 1d showsdeposition onto a linear base of tangent disks instead ofonto a line. This model will give a close packed structure.This process can be presented by the lattice model shownin figure 2.

minimum. The procedure differs from modelII,only in case A and if the horizontal coordinate of thecentre of the added particle, after its first restruc-

turing, lies outside the interval IX, 1, Xc 2 1, wherexc 1 and 2 are the horizontal coordinates of the two

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contacted particles. In that case, the particle is

allowed to rotate again and so on... until either thecondition that xi lies in the interval [xcp’ xcp + 1] is

fulfilled (case A in Fig. 1c) or it reaches the basal

line (case B). It can easily be demonstrated that anew contact is always found as long as repeatedrotation is allowed and thus, in this version, theadded particle will never hang below any contactedparticle.

2.4 MODEL IV : LATTICE VERSION OF MODEL III.- A lattice version of the preceeding model, in thecase 0 = 0, can, in principle, be recovered byconsidering, at the beginning of a simulation, a

periodic alignment of L tangent disks instead of abasal line (to fulfill the periodic boundary conditionsL must then necessarily be an integer). In that case,even if continuously positioned random vertical

trajectories are used, the above defined processmust necessarily lead to a deposit whose particles areall centred on the discrete sites of a triangular lattice.For practical reasons, one of which is explained lateron, it is much easier to discretize the trajectories aswell. A simple lattice model which is sketched in

figure 2 is then obtained. The disks are replaced byvertical « bricks » of base 1 and height 2. One startswith a « castle wall » like arrangement of L bricks.The horizontal coordinate of the i-th brick is chosenas being a random integer between 1 and L. If it fitsa local minimum, it stays there (case A). If it falls ona stair-like local arrangement, it then steps down thestairs until it reaches the nearest minimum (case B).If it falls on a local maximum, it moves either left orright, with the same probability and then behaves asabove (case C).

It can be seen that this model is equivalent to therandom deposition of hard disks on a triangularlattice, by normalizing dimensions of the bricks from(1, 2) to (1/2, B/3) (see Fig.1d). This lattice modelis illustrated in figure 2. The simulation is started

with each of the positions on the basal line having aheight of 0 (for even positions) and 1 for odd

positions. Because of the way this model is construc-ted, the height difference between neighbouringpositions on the surface (hi - hi + 1) must alwayshave a value of ± 1 and this model is closely relatedto the « single step » ballistic deposition model [13]and to the model of Family [20] discussed in theintroduction.

3. Results of the numerical calculations.

3.1 GENERAL FEATURES. - To visualize the effectof restructuring we show in figures 3, 4 and 5 typicalexamples of simulations carried out with models Iand II, for different 0 values, and with model III for0 = 0. In these figures, the increase of the density ofthe deposit from model I to model III can be directly

Fig. 2. - A lattice model for close packed deposition(Fig. lc). The particles (which are presented by two latticesites) are deposited at a random position on the surface.The neighbouring positions on the surface are examinedand if one of them is at a lower level, the deposited particleis moved to that position. The particle continues to move« down stairs » until it reached a local minimum in the

deposit surface. If (as in case C) the deposited particle isplaced at a position with two lower neighbours, one ofthese is selected at random with a probability of 1/2 foreach. The simulation is started off with each position onthe basal line having a height of 0 (for odd positions) or 1(for even positions). Periodic boundary conditions are

used in the simulation.

seen in the case 0 = 0. It can also be seen that the

density decreases when increasing 0. Precise num-

bers for the densities will be given below, but it is

important to notice that the disordered amorphous-like structure obtained with model III, which appearsto be quite compact to the naked eye, has a

definitely smaller density (p = 0.820 ) than the mostcompact arrangement (triangular lattice) in two

dimensions, whose density is w / (2 B/3)= 0.9069.Moreover, we have observed that such disorderedstructure is remarkably stable against any change inthe initial conditions. For example, if a simulation isstarted out using a line of contacting disks, the firstfew layers in the deposit have a density very close tothe theoretical value of 7r/(2 B/3). However, as thedeposit grows, it could happen that, when a particlerolls down to a single-site vacancy of the triangularlattice, it does not exactly fit such position, due tothe round-off errors of the computer. This mighthappen because, in our program, the particle is

stopped when two contacts are realized in a stableposition, while in such single-site deep, three con-tacts should occur simultaneously. Even using a

double-precision program, this is enough to causesome infinitesimal irregularities in the packing.These irregularities are quickly amplified and, aftera few rows, the structure deviates from the regulartriangular arrangement to reach the disordered struc-

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Fig. 3. - Deposition without restructuring. Figure 3ashows deposition onto a horizontal surface, 3b shows

deposition onto a basal line inclined at 45° to the horizontaland figure 3c shows deposition onto a surface inclined at75°. In all cases the « rain » of particles was vertical. Thesefigures show small sections from larger scale simulations.

ture with p = 0.82. This result is of real practicalinterest since one can never realize an experimentwith rigorously monodisperse particles. Our dis-ordered structure is thus the one which is alwaysnaturally obtained (as long as one stays within theapproximation of the model which neglects multi-particle restructuring and particle deformation).

3.2 DEPOSIT DENSITIES. - The deposit densities pobtained from our simulations are shown in figure 6and in tables I and II. For model I (no structuralreadjustment) 4 x 107 particles were deposited ontoa basal line with a horizontal projection (L) of4 096 diameters using periodic boundary conditions.The densities were obtained only for those regionswell removed from the basal line and the uppersurface. For deposition onto a horizontal basal line,a density of 0.3568 ± 0.0002 was obtained. For suchlarge systems finite size effects are quite small anddensities of 0.3570 were obtained for L = 2 048 and0.3568 was obtained for L = 8 192. Consequently,the results shown in table I are believed to be closeto the asymptotic (large system size) densities.

Table II shows a similar set of results obtainedfrom model II (the single adjustment model). Again4 x 107 particles were deposited at each angle andthe horizontal projection of the basal line was

4 096 particle diameters. For deposition onto a hori-zontal line a mean density (p) of 0.7229 ± 0.0001 wasobtained. For a simulation in which L was increasedto 8 192 particle diameters a mean density of

0.7230 ± 0.0001 was obtained using 8 x 107 particles.

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Fig. 4. - Typical structures obtained using model II

(deposition with one restructuring rotation). Figures 4a,

The results shown in figures 3, 4 and 6 and in tables Iand II show that the inclined angle (0) can have animportant effect on the deposit density. For smallangles the density is almost independent of 0 but

4b, 4c and 4d show the results obtained with basal linesinclined at angles of 0°, 45°, 70° and 80° respectively.

For model III (the multiple readjustment model)densities of 0.8210, 0.8170, 0.8177 and 0.8180 wereobtained for L = 256, 1 024, 4 096 and 16 384 parti-cle diameters respectively. For each value of L p wasobtained from a simulation in which 5 x 107 particles

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Fig. 5. - Part of a deposit generated using model III (themultiple restructuring model).

Table I. - Mean density ( p ) and effective value ofthe exponent f3 obtained from deposition simulationswithout structural readjustment (model I) on surfacesinclined at an angle 0 from the horizontal. All of theparticle trajectories are vertical.

Table II. - Mean densities obtained from the 2dsingle readjustment model (model II) as a function ofthe incline (0). In all cases the simulations were

carried out using a horizontally projected surfacelength of 4 096 diameters.

Fig. 6. - Dependence of the deposit density (p ) on theangle 9 between the basal line and a horizontal line formodel I (Fig. 6a) and model II (Fig. 6b).

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were deposited. The dependence of the density onthe incline angle for the basal line was not exploredfor this model.

3.3 SURFACE (ACTIVE ZONE) STRUCTURE. - Thethickness of the surface (a) has been calculated, inthe case of models I, II, III using the definition

where hn is the height (or minimum distance fromthe basal plane for 0 # 0) at which the n-th particleis deposited. The scaling a - L ", with « = 1/2, hasbeen reconfirmed with a quite good accuracy in

these three models.For model IV every site on the lattice is filled and

the density is 1. In this case the location of all of thesurface sited is known and the surface thickness canbe defined more directly. Two surface thicknessesdefined by :

have been measured.In the case of the « active zone » (Cra) the sum is

over the sites which could be filled during the nextaddition step, while in the case of the actual surface

«(T s)’ the surface thickness is measured using thehighest occupied lattice site corresponding to each ofthe sites of the original surface.

In the case of model IV, we could grow very longdeposits (L = 220 = 1 048 756 lattice units). Fivesimulations were carried out and in each simulationmore than 5 x 109 pairs of lattice sites were filled.Figure 7b shows the dependence of (T a and (T s on

(h) a and (h) s. As in ordinary ballistic deposition ona lattice [11], the corrections to the asymptoticbehaviour are smaller for 0-a than for (T s’ From the

dependence of 0-a on (h) a for heights in the range100 (h) ,10 000 lattice units, a value of

j8 = 0.248 ± 0.003 is obtained for the correspondingexponent {3 a’ In a separate simulation 1.7 x 1010

pairs of lattice sites were deposited onto a base of22° lattice sites and a value of 0.252 was obtained forthe exponent 0 for heights in the rangez * (h) * 34 000 lattice units where z = 1, 10, 100or 1 000. The same value (0.252) was obtained forheights in the range 1 (h) ,1000 lattice units.These results suggest that the asymptotic value for jSmight be 1/4. For a = 1/2 and P = 1/4 the exponenty = « /B would be equal to 2. This would mean thatfor a strip of width L = 1 024 lattice units we wouldhave to go to heights (h) > 106 lattice units to be inthe region where a is controlled only by L. However,a comparison of figures 7a and 7b indicate that the« amplitudes » are such that this is not necessary

Fig. 7. - Dependence of the width of the active zone(ga) and the surface thickness (gs) on the strip width (L,Fig. 7a) and on the mean deposit height (ha or hs, Fig. 7b)obtained from model IV (the lattice model).

since Qis larger for (A) = 104 and L > (h) than it isfor L = 1 024 and (h) L. Figure 7a shows thedependence of these two quantities as a function ofthe width L of the strip, from L = 16 to 1 024, underthe condition that the heights of the deposit are verylarge. For the case L = 1 024, over 4 x 109 pairs oflattice sites were added and the surface thicknesseswere measured for deposit heights greater than

400 000 lattice units. The results are consistent withthe idea that the corresponding exponents a anda a both have a value of 1/2.

Using the same definition of a as above, the

exponent j6 has been estimated from the behaviourof Q versus (h) for very large L values (up toL = 8 192, 16 384 and 16 384 particle diameters inthe case of model I, II, III, respectively).

The estimated values of Q, for different 0 values,in the case of model I and II are given together withthe deposit densities in tables I and II. The resultsshown in these tables are consistent with a value of

1/3 for the exponent B (consistent with the resultsobtained from lattice models for ballistic de-

position [10-13]). For model III 8 simulations werecarried out (with a zero inclination angle 0) in which5 x 107 particles were deposited onto a surface

16 384 particle diameters in length (i.e., a total of

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4 x 10g particles were deposited) the exponent j3 hasan effective value between 1/4 and 1/3 for (h) less

than about 500 diameters but increases to a value ofabout 1/3 at larger heights. Eventually, the surfacethickness will be controlled by the strip width

(L ) and the surface thickness will saturate (13 --+ 0).Figure 8 shows similar results obtained from mod-

el III. The results in this figure were obtained from37 simulations in each simulation 4 x 107 particleswere deposited into a « surface » with a length of16 384 diameters. These results suggest a transitionfrom a slope of about 1/4 at intermediate height to1/3 at larger heights. About 90 hours of IBM 3090CPU time were required to obtain these results sothat it would be difficult to obtain a more definitivevalue four 13 in the limit 1 h L.

Fig. 8. - Dependence of the variance in the surface

height (a-,) and the variance in the deposition height(a2) on the mean heights for deposits grown on strips ofwidth 16 344 particle diameters using model IV (the multi-ple restructuring model). The thickness Q1 is equal to( j hrt a, 2 - hn I) where hn is the height at which the n-thparticle is deposited. The surface thickness a2 is the

variance in the maximum heights in each interval of oneparticle diameter measured parallel to the basal line.

3.4 RESULTS FROM A LARGE NUMBER OF SMALLER

SCALE SIMULATIONS. - In another series of cal-

culations, we have calculated (h) and 0’2 by averag-ing the height of the n-th particle over a greatnumber of samples Ns:

when s labels the samples. This is, in principle, thebest way to calculate the surface thickness. However,to get sufficiently good results, we are obliged to runNS ~ 1000 samples and consequently the largestrealistic sizes are smaller by a factor of one

thousand. In these calculations, the density has beencalculated by :

We have considered strips from size L = 128 up tosizes L = 1 024, 512, 2 048 and, for each sample, wehave grown a deposit up to N = aL 2, with a = 4, 1and 1, in models II, III and IV, respectively. In allcases, we have taken N, = 1 000.The extrapolations of p for LIN -+ 0 do not

depend on L and give p = 0.732 ± 0.002 and

p = 0.820 ± 0.005 for models II and III respectively.These values are reported in table III where they arecompared with the corresponding large scale results.The results for model III are in excellent agreementwith those obtained from the larger scale

simulations. For model II the density obtained fromthese simulations is slightly larger than that obtainedfrom the larger scale simulations. We think this is

due to size effects which affect the extrapolationfrom such small sizes.

Table III. - Mean density and exponent 13 obtained with a large number of small scale calculations (SS),compared with those obtained with large scale simulations (LS). The two different values for the P exponent inthe case of LS simulations on model IV correspond to different methods of determination (see text).

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The estimations of the exponent 8 from the slopesof the curves In or versus In (N/L ) have beenreported as a function of 1/L in figure 9. While inthe case of model II the estimated values seem to

converge nicely to 1/3, in the case of model III theconvergence is slower. This slow convergence moti-

vated the extensive large scale simulations for mod-el III described above but these simulations did notremove the uncertainties. In view of the results

discussed above, a value of 1/3 for the exponent Bseems most unlikely for model IV. The extrapolatedvalue for B is slightly higher than 1/4 for model IV. Itis closer to 1/4 than to 1/3.

Fig. 9. - Dependence of the effective value of the expo-nent li on the strip width L for models II (x), III (0) andIV (0).

4. Discussion.

There are many subjects for discussion in the

results presented above. Here we emphasize twopoints.

(1) It appears that model III leads to a quasicom-pact structure with a very interesting defect structure(which will be the subject of additional work). Thewell defined density (= 0.82) is lower than the mostcompact two-dimensional structure (= 0.907). De-nsities of about 0.82 have been associated withrandom packing of hard discs in a variety of twodimensional computer simulations [21, 22], experi-ments [23-25], and theoretical studies [26]. Berryman[21] proposed a value of 0.82 ± 0.02 based on theresults from a number of sources. Kaush et al. [22]obtained a value of 0.821 ± 0.002. As is described

above for our simulations they found that a density

of about 0.82 was obtained even if the simulation

was started with a close-packed « seed ».

(2) Our results from model II indicate that localrestructuring does not affect the scaling properties ofthe surface thickness. The situation with model IIIand the corresponding lattice model (model IV) is

more ambiguous. Even if there could be some

renormalization of the exponents due to long rangeeffects, it is difficult to understand why they aredifferent in the lattice version than in the off-latticeversion. One could invoke the random character ofthe surface which is certainly more pronounced (dueto the local random defects) in the off-lattice caseand would inhibit the multiple readjustingmechanism. This point is important because it is veryrare that there is so much difference between the off-lattice and lattice version of the same model. To our

knowledge the only other example is the scaling ofthe surface in the Eden model (without restruc-

turing) when the surface of the holes are included inthe active zone [27]. The explanation of the differ-ence was based on the nature and distribution of

holes which were different in the off-lattice and

lattice region. A similar difference exists here. Thereare some random defects in the off-lattice version

while there are, in principle, no holes in the latticeversion. Based on the results presented here, wecannot exclude the possibility that the asymptoticvalues of the exponent 13 are the same for both thelattice and nonlattice versions of model IV. In thisevent it seems most probable that j8 ~ 1/4 since thelattice model simulations were carried out on a verylarge scale.

If the values of 1/3 and 1/4 for Q in the nonlatticeand lattice models respectively are reliable, then it isinteresting to ask why these values fit so well withthe results of the continuous model for random

deposition [13, 14], with and without nonlinear

terms, respectively. We can only give a qualitativeargument that nonlinear terms are linked with thelocal rugosity of the surface, which is certainly moreimportant in the off-lattice case, due to randomness.

Finally, we would like to compare our results withthose obtained from a lattice model for depositionwith restructuring introduced by Family [20]. Thismodel has been briefly described in the introduction.From relatively small scale simulations with a re-structuring range (or « surface diffusion » range off = 1 lattice unit, Family obtained a value of 1/4 forthe exponent B in equation (1). We have carried outa single large scale simulation in which more than101° sites were deposited onto a base of 220 latticeunits (the thickness of the completely dense depositexceeded 10 000 lattice units) and obtained a valuefor B very close to 1/4, thus confirming the earlierresults. Although model IV differs from this modelin several important details, it seems that they both

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belong to the same universality class in the sense thatthe exponents a and b and their fractal dimen-sionalities (D = 2) are the same.

Also in recent work [28], Racz and Plischke

studied some models for random deposition on alattice in which irreversibility can be introduced

gradually. They find that the exponent changesfrom 1/3 to 1/4 when going from irreversible to

reversible growth. One could be tempted to make acomparison with our results in the case of multi-

restructuring. However, we cannot find any seriousargument to explain why the on-lattice model shouldbe time-reversible and the off-lattice model not

time-reversible.

5. Conclusion.

In this paper we have introduced two kinds of

restructuring mechanisms in the rain model for

random deposition in two dimensions. We have

quantitatively estimated the change of the meandensity due to restructuring and the scaling proper-

ties of the surface thickness of the deposit. Theextension of this study to three dimensions, which isunder progress, is very interesting for many reasons.First, from a theoretical point of view, there mightbe some logarithmic behaviour of the surface thick-ness [9-12] and it is interesting to know how thisbehaviour is influenced by readjusting. Second,comparison with experiments could then be perfor-med, especially for the mean density of the deposit.The three dimensional equivalent of model III willcertainly be very helpful modelling amorphous struc-tures obtained by deposition.

Acknowledgments.

One of us (R.J.) would like to acknowledgeR. Botet and C. Petipas for interesting discussions.Calculations done at Orsay were performed at

CIRCE (Centre Inter-Regional de Calcul Electroni-que). P.M. would like to acknowledge the hospitalityof the IBM Science Center, Bergen, Norway wheresome of these simulations were carried out and an

informative discussion with George Onoda.

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