Physica A 175 (1991) 211-221 North-Holland

BALLISTIC DEPOSITION WITH STICKY AND NON-STICKY PARTICL~

Paul MEAKIN Central Research and Development. E.I. du Pont de Nemour~ and (~mpany. Wilmmgt+m.

g DE lot '~1-0356, USA

Remi JULLIEN Laboratoire de Physiqtte des Sol(des. B?uiment ,~;I0. Univer~it~ Pari,~-Sud. ('~'m~ d'O~+y~ 91J905 Orsay. France

Received 22 March I~1

Simple ( ! + i )- and (2 + I )-dimensional models ba~,~-tl on the indcwn~nt column m ~ f ~ ~hc deposition of mixtures of sticky and non-sticky (sliding) partick~ ha~c ~ c n investigated The dependence of the surface width or perpendK-ular o~rtelation length 1() on the time ~i~ :+~6 fraction of sliding particles ~m be represented by the ~aling form ~(t :) ~ t + :,[It:" ), The cxI~+n,¢at A has a value of 2 for models in which the deposited particles may slick only while lhey ate ~ ing deposited and a value of .1 for models in which the particles remain stickx afIcr ~hc~ have ~+c~ incorporated into the growing deposit.

1. Introduction

Simple ballistic deposition models have becn used for more than ~hirty ~car', (beginning with the pioneering work of Vold [1-3.1) to stimulate pnenome ~ na such as sedimentation, thin film deposition and the grov, ah of rough surfaces. Until a few years ago. interest was focussed primarily on the internal structure of the deposits generated by deposition models. At the present time the high level of interest in ballistic deposition is concerned with the seif-affine [4.5~ fractal [6] scaling properties associated with the deposit surfaces [7t. As a consequence, interest has also transferred from off-lattice models to less realistic lattice models that allow much larger scale simulations to bc carried out in order to explore the asymptotic (long Icngth scale) geometric scaling properties.

The growth of random surfaces can be described by a Langevin equation for the evolution of the surface height (h(x, t), the height at time t and lateral position x) [8, 9]. In many cases this Langevin equation has the form

d[h(x , t ) ] dt

= a V2d h(x, t) + b[Vdh(x, t)] z +n(x,t) (I)

0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

212 P. Meakin, R. Jullien / Deposition with (non)-sticky particles

introduced by Kardar, Parisi and Zhang (KPZ). Here r/(x, t) represents the fluctuations associated with the particle deposition density. In the most simple case ~/(x, t) has neither spatial nor temporal correlations.

In most simulations the growth process starts with a fiat d-dimensional substrate so that h(x, 0 ) = 0 for all x. In eq. (1) V 2 represents the d- dimensional Laplacian and this equation describes the growth of a d-dimen- sional surface in a (d + 1)-dimensional space.

A quite wide variety of surface growth models [10, 11] (and presumably the growth of some real rough surfaces, though this is not yet well established) can be described in terms of eq. (1) (the KPZ or Burgers [12] equation). The growth in the roughness of a statistically self-affine surface can be described in terms of the correlation lengths E_t and Ell perpendicular and parallel to the substrate (with respect to some smoothed mean surface),

x i ~ t t~ , (2a)

x l l - t l'~ (2b)

The correlation length ~_L is a measure of the width of the rough surface and Ell is a measure of the lateral distance over which fluctuations in the surface height persist. It follows from eqs. (1) and (2) that

"-- ( 3 )

where a is the Hurst exponent for the self-affine fractal. This rapidly develop- ing field has recently been reviewed by Krug and Spohn [11].

The effect of particle motion on the structure of the deposit have been of interest since the earliest work of Vold [3], who studied an off-lattice model in which spherical particles followed a path of steepest descent on the surface of the growing deposit after first contacting the surface. Each time the deposited particle contacts a new particle in the deposit, it sticks irreversibly with a probability x (0 < x < 1) or continues to move on the surface with a probability of 1 - x. In the 1970's ballistic deposition models that included such restructur- ing were developed to generate morphologies that were more similar to those observed in low temperature vapor deposition than the very low density deposits generated by ballistic deposition without such restructuring [13-18]. Recently, off-lattice models with repositioning after deposition have also been used to explore the scaling behavior associated with surface roughness [19, 20].

Lattice models have also been used to invcstigatc the effects of surface mobility and restructuring. The most simple of these is based on the "in- dependent column model " [21] in which the columns of a d-dimensional lattice

P. Meakin, R. Juilien / Deposition with (non)-sticky particles 213

are selected randomly and their heights are incremented by one lattice unit. In this process only the noise term (rl(x, t)) survives on the right-hand side of eq. (1). Under these conditions the exponent /3 in eq. (2a) has a value of 1/2 and there are no lateral correlations (a = 0). If, in the (1 + 1)-dimensional case, the ith column is selected randomly and the height of the lowest of the 3 columns at positions i - 1, i or i + 1 is incremented [21], then the scaling properties change completely to

= 1 / 2 , (4a)

/3 = 1/4. (4b)

A similar result is obtained with more extensive restructuring in which the deposited particle moves "down hill" on the surface of the deposit until it reaches a local minimum. In terms of eq. (1) the first and third terms survive. but the second (non-linear) term is absent since the growth velocity does not depend on the angle of inclination. In the (2 + l)-dimensional case it has been shown analytically [8] that

_~ ~ [log(t)] ' '2 (5a)

for L >> t and

~ ~ [log(L)] ' 'z (5b)

for t >> L where L is the system size. The purpose of the work described below was to investigate the crossover

from deposition without restructuring to deposition with complete restructuring using models with a mixture of sticky and non-sticky (sliding) particles.

2. (1 + l)-dimensionai models

Simulations were carried out using four different (1 + 1)-dimensional mod- els. The simulations were carried out using lattices with a width of 2 ''~; (262144) lattice units with periodic boundary conditions in the lateral direction. In most cases the simulations were continued until an average of 2-°-, 0O0 sites had been filled in each column of the lattice (i .e. , a total of about 5 x 10 ~ sites were

filled during each simulation). In model A the particles are sticky (or non-sliding) with probability 1 - z

and non-sticky (sliding) with probability z (0 < z < 1). Each time a "bparticle'" is

214 P. Meakin, R. Jullien I Deposition with (non)-sticky particles

deposited, two random numbers are generated; one to select a column of the lattice and the other to determine if the deposited particle is sticky or non-sticky. If the particle is sticky, the height of the selected column is increased by one lattice unit and the column is labelled to indicate that it has a sticky top. If the deposited particle is non-sticky and the top of the selected column is sticky, then the height of the selected column is increased by one lattice unit and relabeUed as a non-sticky column. If the deposited particle is non-sticky and it is deposited on the top of a non-sticky column, then it moves "down hill" on the surface of the deposit (from column top to column top) until it reaches the top of a sticky column or a local minimum. In either event, the non-sticky particle comes to rest on top of the sticky column or in the local minimum, the height of the corresponding column is incremented by one lattice unit and the column is labelled as being "non-sticky".

Fig. 1 shows some of the results obtained from the (1 + 1)-dimensional version of model A. For small values of z (the fraction of sliding particles) the model growth of the surface roughness (perpendicular correlation length, ~) is expected to be given by

~ t 1/2 (6)

and our simulation results are in very good agreement with eq. (6). Similarly, for z---> 1 our simulations give results that are in good agreement with the expected behavior [21]

~: ~ t t/4 • (7)

Fig. la shows the dependence of ~ on t where the "time" t is proportional to the average number of deposition events per column (t was increased by 1 for each 106 sites added to the deposit). Here we show the dependence of In(~:t -1~2) on In(t) so that growth according to eq. (6) would lead to a horizontal line. The results shown in fig. ia suggest that the curves shown in fig. la can be represented by the scaling form

~tt/2f(tza) (8)

and fig. lb shows that a good data collapse can be obtained using this scaling form with a scaling exponent (h) of 4. The scaling function f(x) has the form f(x) = const, for x <~ 1 (so that ~ --- t !'2 for a large fraction of sticky particles and for early times). The scaling fimction f(x) has the form f(x) -- x- 1/4 for x >> 1 so that ~ - t 1'4 at late times and/or small fractions of sticky particles.

Model B is quite similar to model A except that sticky particles do not

P. Meakin, R. Jullien / Deposition with (non).sticky particles 215

1.00 , ,

0.75 (a)

0.5(3

0.00 ' - -0.25

-0.50

-0.75 -1.00 Z = 0.6.0.65, 0.7, 0.75, 0.8, 0.85, 0.9

-1,25 d = 1 + 1 (MODELA) I t I I I

-1.500 1 2 3 4 5 6 7 8

In (t)

1.00 i i ! ! | i t i i i i i i ~ i

0.75 (b)

0.50

0.25

0.00

-0.25

-0.50 -0.75

-1.00

1.25 d = 1 + 1 (

"1"5010 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

In (tZ 4)

Fig. 1. (a) shows the dependence of In(t l:E) on In(t) obtained from ( i + 1)-dimensional simulations using model A in which the sticky particles remain sticky after incorporation into the deposit. (b) shows the data collapse obtained for the results shown in (a) using the scaling form given in eq. (8) with a scaling exponent (A) of 4.

remain sticky after they have become part of the growing deposit. Conse- quently, if the deposited particle is sticky, the height of the selected column is increased by one lattice unit and if the particle is non-sticky, the height in a neighboring local minimum is increased by one lattice unit. (In all (1 + 1)- dimensional models if a particle is deposited on a local maximum and does not stick, it moves with equal probability in either of the two possible directions.) Fig. 2 shows the results obtained using model B with 9 different values for t,he sliding probability (z). Fig. 2b shows that the dependence of the surface width (~ ) on t and z can be represented quite well by the scaling form given in eq. (7) with a scaling exponent A of 2. However, the quality of the data collapse is not as good in fig. 2b as it is in fig. lb.

The results shown in figs. 1 and 2 suggest that in general the dependence of ~: on t and z can be represented by the scaling form given in eq. (8) with a scaling

216 P. Meakin, R. Jullien I Deposition with (non)-sticky particles

A 0 .1

~ lJ r*

v -1.0 ¢:

1 . 0 i , i ! i | i

0 . 5 ~

0.0 0.025

-0.5

-1.5

-2.0

; , , , ; , -2.5 1 2 3 4 6 7 9

In (t)

1.0 , , , , , , , ,

A

0.5

0.0

-0.5 =

-1.0 (b)

-1.5 d = 1 + 1 (MODELB)

13 I I I I I I I I I I I

"2""-6 -5 -4 °3 -2 -1 0 1 2 3 4 5

In (z2t)

Fig. 2. Results obtained from the (1 + 1)-dimensional version of model B. This model is similar to model A illustrated in fig. 1 except that the particles do not remain "sticky" after they have been incorporated into the deposit. (a) shows the dependence of In(t-l'2~ :) on In(t) obtained for 9 values of the sliding probability (z = 0.025, 0.05, 0.1, 0.2, 0.3, 0.7 and 0.9). (b) shows the results of an attempt to scale the data shown in (a) using the scaling form given in eq. (8) and a scaling exponent (a) of 2.0.

exponent (A) of 2 if only the deposited particles can be sticky and 4 if the sticky particles remain sticky while the/ are part of the deposit. To explore the generality of this result, we carried out simulations using the model of Family [21] with a mixture of sticky and non-sticky sites. In this model the height of the lowest of the columns at positions i, i - i and i + i is incremented by one lattice unit after the ith column of the lattice has been selected randomly. If the height of the selected column is equal to the height of one or both of its nearest neighbors and neither of the nearest neighbors has a lower height, then the height of the selected column is increased. If both of the neighboring columns have the same height, lower than that of the selected column, then one of these neighboring columns is selected at random and its height is incremented. In

P. Meakin, R. Jullien / Deposition with (non)-sticky particles 217

model C (which is similar to model A) the height of the selected column at position i is incremented, irrespective of its height, if the deposited particle is sticky or if the particle at the top of the ith column in the deposit is sticky. In model D (which is similar to model B) the height of the selected column (the ith column) is always incremented if the selected particle is sticky, but if the deposited particle is not sticky the height of the lowest column at positions i, i - 1 or i + 1 is incremented. Fig. 3 shows some of the results obtained using these two models. In this figure only the scaled data are shown. Again, it is apparent that the results can be represented quite well by eq. (8) with a scaling exponent (A) of about 4 if the sticky particles remain sticky in the deposit and about 2 if they do not.

1.0 , , , , , , , , , , ,

0.5

0.0

- - -0.5 ¢ -

-1.0

-1.5

-20

(a)

c =1 + 1 (MODELC)

I 1 I |

-, -6 -5 -4 -3 I I I I I I I i

-2 -1 0 1 2 3 4 5

in (z 4 t)

6

1.0

0.5

0.0

- -0.5 t ' -

-1.0 f -1.5

-2.0 -6

(b!

d = 1 + 1

I I

-5 -4

(MODEL D)

I I I I I I I I I

-3 -2 -1 0 1 2 3 4 5

In (z2t)

6

Fig. 3. Scaling plots of In(sOl -'t) vs In(z~t) obtained from model C (a) and model D (b) (1 + l )-dimensional simulations. For model C results for z = 0.1, 0.3, 0.5, 0.7 and 0.9 are shown. For model D data f rom simulations with z = 0.05, 0.075, 0. l , 0.2, 0.3, 0.5, 0.7 and 0.9 are shown.

218 P. Meakin, R. Jullien I Deposition with (non)-sticky particles

3. (2 + l)-dimensional simulations

Simulations were also carried out using ( 2 + 1)-dimensional versions of models A and B. In these simulation lattices (substrates) with a size of 1024 x 1024 lattice units were used. In these models sliding particles move with equal probability to any of the lower nearest neighbor sites irrespective of their relative heights. The results shown in figs. 4 and 5 indicate that the dependence of the surface width (~:) on t and z (the sliding probability) can be described quite well by the scaling form given in eq. (7) and the exponent A has the same value for corresponding (1 + 1)- and (2 + 1)-dimensional models (i .e, , A = 2 if only the deposited particles are sticky and A = 4 if both the sticky particles remain sticky after they have become part of the growing deposit.

¢

o

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

-3.5

-4.0 0

(a)

Z=0.1 -0.9

d = 2 + 1 (MODELA) I I I

1 2 3

0.9 I I I I I

4 5 6 7 8

In (t)

0 . 0 L i = ' I ! i = ~ i i

-0.5

-1.0

-I .5

-2.0 (b)

-2.5 d = 2

, I I I I I I I I I , I [

"30-6 -5 -4 -3 -2 -1 0 1 2 3 4 5

In (t z 4)

Fig. 4. Dependence of lhe surface width ~ on the time (I) and sliding probability (z) obtained from (2 + l)-dimensional model A simulations. (b) shows how the data in (a) can be represented by the ~aling form given in eq. (8).

"7. Z,,

r , .

In (t)

-I.0 -1.5 ~

-3.o (a) -3.5 Z = 0.025o 0.05, 0.1, 0.2, 0.3o 0.7, 0.9 ~ ' ~ " -4.0 d = 2 + 1 ( M O D E L B) 0 .9 "

. 4 . 5 0 i , , , i I , i 1 2 3 4 5 6 7 8 9

0.0

-0.5

-1.0

-3.0

-1.5 O , I

-2.o

c: -2.5

-3.5 (MODEL B)

(b)

d = 2 + 1 I I

P. Meakin, R. Jullien I Deposition with (non)-sticky particles 219

I i i i i . . . . . ! l i i i ! ! i

"4"0-1, -9 -8 - 7 - 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 "-8

In ( tz )

Fig. 5. Some results obtained from (2 + l)-dinmnsi )nal model B simulations (a) sho~vs the dependence of ln(t~ 2~ :) on In(t) and ,b) shows scatin; of the curves in (a).

4. Discuss ion

In the independent column model the growth of the individua, columns can be represented by a vertical random walk in a coordinate system moving with the mean growth velocity of the interface. The effect of restructuring is to bias the random walk towards the origin (in the moving coordinate system) with a bias velocity that is proportional to the sliding probability (proportional to z if only the deposited particles are sticky and proportional to z 2 if the sticky particle remains sticky in the deposit). Consequently, a crossover from t ~:2 growth of the correlation length to a slower growth can be expected when the random walk displacement (proportional to t *~/2) is of the same order of magnitude as "bias" displacement (proportional to zt* or zEt *) where t* is the characteristic crossover time. This implies that

220 P. Meakin, R. JuUien I Deposition with (non)-sticky particles

r* 1,2 ... t* (9a)

o r

r * , , 2 ..~ z 2 t . " (9b)

where eq. (9a) applies to models in which only the deposited particles are sticky and eq. (9b) applies to models in which sticky particles remain sticky in the deposit. It follows from eq. (9) that

t* --- z -2 (10a)

O r

t* -- z -4 (lOb)

in accord with the simulation results presented above. The work described above was motivated by the earlier work of Pellegrini

and Jullien [22, 23] in which the surface scaling properties associated with models for the deposition of sticky and non-sticky particles (sites) were explored. However, in this work the z = 0 (non-sliding) limit corresponded to the ballistic deposition model in which all three terms on the right-hand side of the KPZ equation (eq. (1)) are present. Consequently, the crossover (or transition) found by Pellegrini and Jullien is of a different nature than that studied here.

Family and Coleman [24, 25] have studied the crossover from independent column growth to ballistic deposition using a model that is quite closely related to those used in this work. In (! + l)-dimensional simulations they find that the growth of the surface width ~ crosses over from t ~ ~2 at short times (characteris- tic of the independent column growth model) to t ~3 growth (characteristic of ballistic depositions with sticking) at longer times. Most of their simulations were carried out using relatively small system sizes so, that a second crossover to a time independent surface width was found at still later times. In this model the deposited particle has the same probability of sticking at each step in its downward motion on the surface of the deposit. Consequently, this model is most closely related to the version of the models described above in which sticky particles remain sticky after they have been incorporated into the growing deposit. In any event, the work of Family and Coleman was concerned primarily with the second crossover from t t/~ growth of the surface width to saturation and a direct comparison of the results of their simulations with those des,:'ribed above is not possible.

P. Meakin, R. JuUien / Deposition with (non)-sticky particles 221

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L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986) p. 3. [6] B.B. Mandelbrot, The Fractai Geometry of Nature (Freeman, New York, 1982). [7] F. Family and T. Vicsek, J. Phys. A 18 (1985) L75. [8] S.F. Edwards and D.R. Wilkinson, Proc. R. Soc. A 381 (1982) 17. [9] M. Kardar, G. Parisi and Y. Zhang, Phys. Rev. Lett. 56 (1986) 889.

[10] P. Meakin, CRC Critical Reviews in Solid State and Material Science 13 (1987) 143. [11] J. Krug and H. Spohn, in: Solids Far From Equilibrium: Growth Morphology and Defects,

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in: Current Topics, vol. 6, E. Kaldis, ed. (North-Holland, Amsterdam, 1980) ch. 4. [18] M. Popescu, Thin Solid Films 121 (1984) 317. [19] P. Meakin and R. Jullien, J. Phys. (Paris) 48 (1987) 1651. [20] R. Jullien and P. Meakin, Europhys. Lett. 4 (1987) 1385. [21] F. Family, J. Phys. A 19 (1986) L441. [22] Y.P. Pellegrini and R. Jullien, Phys. Rev. Lett. 64 (1990) 1745. [23] Y.P. Pellegrini and R. Jullien, Phys. Rev. A (1991), to be published. [24] F. Family, private communication. [25] M. Coleman, Dynamic Scaling of Rough Surfaces, B. Sci. Thesis, Emory University (1989).