Dynamics of surface roughening in disordered media

Physica A 200 (1993) 136-154 North-Holland

SDI: 037%4371(93)EOlll-Q

Dynamics of surface roughening in disordered media

Z. Csahbk”, K. HondaaTb, E. Somfai”, M. Vicsek” and T. VicsekaTd “Department of Atomic Physics, EijtvBs University, Puskin u 5-7, 1088 Budapest, Hungary bDepartment of Applied Physics, Faculty of Engineering, Nagoya University, Nagoya 464-01, Japan

‘Computer and Automation Institute of HAS, P.O.B. 63, I.518 Budapest, Hungary ‘Institute for Technical Physics, P.O.B. 76, 1325 Budapest, Hungary

In this paper the roughening of interfaces moving in inhomogeneous media is investigated by examining the corresponding stochastic differential equations using (i) numerical methods and (ii) dimensional analysis. We consider interface evolution equations where disorder is represented by quenched noise which can be both additive and multiplicative. Our main finding is that quenched noise leads to a new universality class as concerning the exponents a and p describing respectively the spatial and temporal scaling of the surface roughness. In particular, additive noise close to the pinning transition results in a behaviour with LY = 0.712 0.08 and p = 0.61 f 0.06 up to a crossover time. These estimates are in very good agreement with the theoretical prediction p = 3/5 and (Y = 3/4 that we derive from a dimensional analysis of the equation. Furthermore, we argue that multiplicative noise is the appropriate choice to describe experiments where the interface between two flowing phases is considered. By numerically integrating the proposed equation we have obtained (i) surfaces remarkably similar to those observed in the experiments and (ii) a scaling of the surface width as a function of time with an exponent p = 0.65 being in an excellent agreement with the experimental value. In addition to the exponents we discuss other relevant features of the surfaces, including the scaling of the average velocity of the surface u, close to pinning and the non-trivial, power law distribution of waiting times and noise along the interface in the stationary regime.

1. Introduction

If the conditions of a growth process are such that the development of the interface is not unstable and the fluctuations play a relevant role, the resulting structure typically has a rough surface which can be well described in terms of self-affine fractals [l-3]. In such phenomena the fluctuations (which usually die out quickly for a stable surface and grow exponentially for an unstable one) survive without drastic changes for a long time leading to surfaces with spatial scaling over many length scales.

0378-4371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

2. Csahdk et al. I Surface roughening in disordered media 137

There are two major types of this kind of growth: roughening (i) in the presence of temporally uncorrelated fluctuations and (ii) in disordered media, representing quenched noise. The first case is theoretically better understood due to much recent interest in the Kardar-Parisi-Zhang (KPZ) [4] and closely related equations. Less is known about the dynamics of interfaces whose motion is dominated by the pinning forces present in an inhomogeneous medium. Since this case is relevant to many experimental situations such as two phase fluid flows in porous media [5-81, the motion of domain walls in magnetically ordered systems [9] or the pinning of charge density waves [lO,ll], it is of great importance to understand the dynamics of such growing surfaces.

The main effect of above mentioned pinning forces is that they are able to dramatically slow down the motion of the interface. This slowing down may occur locally (the surfaces is pinned at a given position) or globally. In particular, at the so called pinning transition the pinning forces completely halt the moving interface.

Studies of kinetic roughening with quenched noise have been concentrating on the two exponents cx and p describing respectively the static and time dependent scaling of the surface width in the pinning dominated regime. The few very recent results for these exponents obtained from experiments, theory and simulations have not led to a consistent picture yet. In the most studied (1 + 1)-dimensional case the experiments on viscous flows gave estimates 0.63 < (Y < 0.81 [5,6] and p 2: 0.65 [6]. The theoretical value for the exponent p was suggested by Parisi [ll] and Nattermann et al. [12] to be 3/4. In a recent preprint Kaganovich [13] (using a power counting type approach of Hentschel and Family [14]) proposed that the exponents corresponding to the KPZ equation with quenched noise are p = 0.6 and (Y = 0.75. Finally, the simulation of various growth models and model equations resulted in diverse values as well. Two closely related growth models [15,16] which were interpreted in terms of directed percolation yielded (Y and p both close to 0.63. p = 0.75 was obtained from the model of Parisi [ll]. The numerical integration of an Edwards-Wilkinson [17] type continuum equation with additive quenched noise led to the conclusion that (Y crosses over from 1 to 0.5 as the velocity of the surface is increased [18].

Thus, the main questions to be answered are: (i) Does a well defined, distinct universality class exist for kinetic roughening in disordered media close to pinning? (ii) Do the surfaces exhibit any unusual behaviour related to the appearance of new exponents?

Here we present the results of our numerical and theoretical investigations of various equations relevant from the perspective of the above questions. This paper is partially based on our two related recent papers [19,20] and contains

138 Z. Csahdk et al. I Surface roughening in disordered media

further numerical findings uncovering additional interesting aspects of the motion of interfaces in the presence of pinning forces.

2. Dynamic scaling

The roughening of growing surfaces can successfully be described in terms of dynamic scaling based on the general concepts of scale invariance and fractals. It has become a standard tool in the study of growing surfaces and has been applied to the study of a variety of theoretical models of growing interfaces, and some recent experiments.

We consider the time evolution of a rough interface in a (1 + 1)-dimensional space starting from an initially flat surface of length L at time t = 0. Let us concentrate on a part of the surface having extended over a distance r along the substrate. The growing surface typically can be described by a single-valued function h(x, t) which gives the height (distance) of the interface at position x (along the substrate) at time t measured from the original (d = 1)-dimensional flat surface. During growth the interface heights fluctuate about their average value h(t) - t and the extent of these fluctuations characterizes the width or the thickness of the surface. The root-mean-square of the height fluctuations w(r, t)

is a quantitative measure of the surface width and is defined by

w(r, t) = [(h2(X, t)), - (h(x, t)y. (1)

Since we assume that the surface is a continuous function, its width has to saturate to some value after some relaxation time r. The only scale in the problem is the linear size of the substrate r (there is no intrinsic time scale), thus w depends only on some power of r and a ratio of the form t/r’, where z is an exponent describing how the relaxation time depends on the system size. Correspondingly, the dynamic scaling of the surface width has the form

w(r, t) = rmf(tlra’P) , (2)

where crlp = z is the dynamic scaling exponent. In the limit where the argument of the scaling function f(y) is small, y 4 1, the width depends only on t and this implies that for small y the scaling function f(y) has to be of the form f(y) - y’. Saturation at large times means that f(y) goes to a constant in that limit. From the above it follows that for long times and r < L

w(r, t+ 00) - ra , (3)

and for fixed r (e.g., r = L) and short times

2. Csahbk et al. I Surface roughening in disordered media 139

w(r, t) - tP . (4) The expressions (3) and (4) can be used to determine the exponents (Y and p

for the numerically or experimentally obtained surfaces.

3. Equations

One of the most common approaches to surface roughening is the investiga-

tion of the corresponding stochastic di$ferential equations. In particular, the Edwards-Wilkinson (EW) and the Kardan-Parisi-Zhang (KPZ) equations have been shown to be very useful in the description of interface dynamics in the presence of temporally uncorrelated (shot) noise. In this work we follow the same strategy and first consider the quenched versions of these equations in 1 + 1 dimensions,

ah dt = uV2h + +h(Vh)’ + u +7(x, h) ,

where v and A are constants. In (5) (~(x, h)) = 0 represents uncorrelated quenched noise with a Gaussian distribution of amplitudes and a correlator which in the continuum limit is formally given by

bh,, h,) v(xo + x, ho + h’)) = Da@‘) 6(x) , (6)

where D is a constant. As we shall see, in the simulations and the theoretical interpretation of the numerical results we have to consider that the discretiza- tion of eq. (1) corresponds to a somewhat modified form for the noise correlations.

For A = 0 eq. (5) is equivalent to the Edwards-Wilkinson equation with quenched disorder. This latter equation has been studied more intensively, in part, because it has been assumed that the nonlinear term of the KPZ type equation is automatically generated in the quenched EW version by the noise which depends on h in a highly nonlinear way. Since our dimensional analysis indicated that the two equations may exhibit different scaling behaviour, we decided to simulate both equations.

Next we describe an equation corresponding to physical situations in which the interface is formed between two phases flowing in a disordered medium. We propose that the development of the interface h(x, t) in the experiments on quasi (1 + 1)-dimensional viscous flows is described by the equation

$ = {V’h + u[l + (Vh)2]“2}(p’ + n) , (7)


where p > 0 is some constant, u is the normal velocity and the term n> 0 corresponds to quenched noise with no correlations as in (6). However, in this case we do not assume that the distribution of the noise amplitudes is Gaussian-like with a zero mean; it would be equivalent to supposing that flat parts of the interface would move backward at places with n < -p. Rather, we shall assume that 77 follows some other simple distribution, e.g., the Poisson or the uniform distribution. In this way, we can avoid (unlike in the case of Gaussian distribution) the occurrence of the unphysical values p + 7) < 0.

The main idea behind incorporating noise in a multiplicative form into (7) is that the motion of the flowing phase in an inhomogeneous medium is determined by the simultaneous effects of surface tension, capillary forces and local flow properties (permeabilities of the channels). The advancement of the interface at a given point is proportional to the local driving force and the permeability (Darcy’s law) (just as the electric current j is proportional to the conductivity cr and the electric field E, j = Ea). In the case of viscous flows the driving forces are the wetting or capillary forces which would produce velocity u for unit permeability and the forces due to the surface tension which are represented by the term V2h. Thus, eq. (7) is equivalent to u, = FE, where u, is the velocity of the surface in the vertical direction, E is the randomly changing local permeability and F denotes a general driving force. Although here we used the wetting experiment as an example to justify the necessity to take into account multiplicative noise, we think that in many other situations where transport of the phases takes place our approach should be adequate.

We would like to take a few remarks on the other aspects of the proposed equation. (i). The term u[l + (Vh)2]*‘2 is included in its full form (instead of its linearized version used in the KPZ equation), because in the actual experiments at the majority of the points along the interface the condition ]Vh] 4 1 is not satisfied. This statement becomes very relevant when pinning forces are present and the interface develops deep valleys with ]Vh( * 1 playing a determining role in the process of roughening. (ii) It has to be noted that even without including additive noise explicitly, the term u[l + A(Vh)2]“2n in (7) has a contribution which corresponds to additive noise. If both types of noises are present, one expects that in the limit of very large system sizes and long times the additive noise will dominate the growth since the various derivatives of the surface become very small on a coarse grained scale.

4. Dimensional analysis

Before describing our numerical results obtained for the scaling behaviour of the solutions of eqs. (5) and (7) we carry out a detailed dimensional analysis of

eq. (5).

Z. Csahbk et al. I Surface roughening in disordered media 141

The key point in a dimensional analysis is the appropriate choice for the form of the noise term. In other words, the results are dependent on the assumed dependence of the noise on h and X. If we intend to compare our theoretical predictions with the results of the experiments and numerical integration (simulation) of the above equations we need to consider a noise term with correlations corresponding to the conditions of the experiments and the numerical methods. When doing this, we first need to point out that in real materials and the simulations the correlations describing the disorder are not delta function like. The reason is that in the numerical studies the surface is discretized in the horizontal direction (along the substrate), but is assumed to move continuously in the vertical direction. Similarly, in the experiments there always exists a typical size of the inhomogeneities in the disordered medium (e.g., the diameter of the glass beads) which plays the role of the lattice constant.

As will be described, in the simulations the noise (disorder) can be represented by random numbers attributed to the sites of a grid. In this way the noise is discretized and is uncorrelated in both directions in the sense that its value at a given site does not depend on the noise values at the neighbouring sites. On the other hand, during the continuous advancement of the surface between two grid points the surface experiences a correlated effect of the local value of n in the growth direction (in the other direction no such effect takes place). Correspondingly, we are led to study the effects of noise having the correlator (see also ref. [12])

(rlh h,)v(ro + r, 4, + h’)) = 4lh’l) 6d(r) 9

where d is the dimension of the substrate. A(lh’l) is a monotonically decreasing function with a cutoff at a small characteristic size a (corresponding to the lattice constant) playing the role of the typical correlation length of the fluctuations of noise in the h direction. Since h is the relevant direction from the point of growth, for the other directions we can keep the original assumption (6) for the correlations.

By denoting the dimension of the corresponding quantities by [h], [t], [r][A],

bl, bl, b11 and PI we get the following equations from (5) and (8):

[h]l[t] = [v][h]l[r]* = [A][h]*/[r]* = [v] = [q] = [A]1’2[r]-d’2.

Expressing the dimension of h and r through the dimensions of A, A and t we

get

H=A- dl(4+d) A 2/(4+d) (4-d)l(4+d) t 3 R=h 21(4+d)A1/(4+d)t4/(4+d) 3 (10)


where H and R are such that [H] = [h] and [R] = [v]. Using these relations we can also express [u] and [v] through [A], [A] and [t]. The dimension of the squared surface width w*(L, t) = (h*) - (h)* over a region of linear extent L

is [HI*. Therefore, the dimensionless width w*(L, t)lH* has to be the function of the dimensionless variables L/R, V”l(Hlt) and vI(R*It), where v”= u - u, with u, being the critical driving force to be discussed below. Thus, we have

w2(L, t) A- 2dl(4+d)

A t 4/(4+d) 2(4-d)l(4+d) =f( A2’(4+d)dlf;‘+d)t4,‘4+d) 3

u” v

A- dl(4+d) A t

2/(4+d) -2d/(4+d) 7 . A

4/(4+d) A t

2/(4+d) (4-d)l(4+d) (11)

Writing f in the form f(a, b, c), next we eliminate the explicit t dependence from c. First we note that t-2d’(4+d) = ~Tlh-~‘(~+~)A*‘(~+~)b. Then we insert this expression for time into the complete form of c as it appears in (11). After simple algebra we get

c = (v2d~4-dA-dA-2~l/2db(d-4)/2d .

(12)

From here it follows that

w2(L, t) = A -2dl(4+d)

A t 4/(4+d) 2(4-d)l(4+d)

g

L U -4+dhdt2d ,,2dc4&d

X h A

2(4+d) 1/(4+d) 4/(4+d) 7 t A2 ’ A2Ad

(13)

Now we are in the position to extract the exponents (Y and /3 from the above expression and discuss briefly the possible crossovers.

Let us consider the case when v*~U”-~ /A2Ad is fixed and t is much smaller than t* = (c44+dAd/A2)-“2d_ We also introduce the size dependent crossover time t,(L) = L(4+d)‘4/(A2A)“4. There are two important limiting cases: (i) t + t,(L). In this case w* should be independent of L which on the basis of (13) means that it scales as a function of time according to w3 - c*(~-~)‘(~+~). (ii) For t 9 t,(L) the width becomes independent of time w* - L*“. This is realised by assuming that in the given limit g(a, b, c) - u(~-~)‘* since in this way the time dependence is cancelled out. The above are equivalent to

4-d 4-d a=- and P=4+d. 4

In particular, for the much studied d = 1 case the exponents are

(14)

cu=3/4=0.75 and @=3/5=0.6. (15)

Z. Csahdk et al. I Surface roughening in disordered media 143

We conclude this section by making the following comments: (i) According to (14) d, = 4 is the upper critical dimension of the problem in agreement with the previous theories (see, e.g., refs. [lo-121). (ii) Above the crossover time t* it is expected that the pinning forces do not effect the scaling of the surface width and a behaviour analogous to that of the standard (non-quenched) KPZ equation sets in. In other words, if the velocity u is large, even for short times a KPZ type behaviour is expected, since the correlation time for the applied noise (the time the surface spends between two grid points) becomes small.

5. Numerical method

The numerical integration of (5) for the (1 + 1)-dimensional case was carried out using a simple single step method (see, e.g. ref. [21]). The discretized

version of (5) is

h(x, t + At) = h(x, t) + At [h(~ - 1, t) - 2h(x, t) + h(x + 1, t)]

+ At{ +A[+ + 1, t) - h(x - 1, t)12 + u + V(X, [h(x, t)])} ,

(16)

where [e] denotes integer part, and n was chosen to be uniformly distributed on the interval (- $, +). We assumed that as the surface was moving between two grid points in the vertical direction, it experienced the effect of the fixed value of n associated with the closest grid point below the surface. Obviously, this does not correspond to a S(h’) kind of correlations; on the other hand, this is the only reasonable choice in the framework of the given method. It should be pointed out that in the present case the mesh size used in the course of discretizing the n dependence in eq. (5) and (7) has a physical meaning: it corresponds to the lower cutoff length of the fluctuations of the media (for example, it can be identified with the diameter of the glass beads in the experiments of refs. [5,6], since n is assigned to the grid points of a square lattice with mesh size Ax).

The control parameter in (5) is the driving force u; above a critical u, the interface keeps moving for arbitrary large times. However, if u <u, the interface becomes pinned after some characteristic time, i.e., due to the randomly distributed negative values of n it stops completely.

In order to produce simulations for long times we used At = 0.1, and At = 0.01. We checked our choice for At by making runs for smaller At and comparing the results. The appropriate time step depends on the actual values of u or V. We have carried out calculations for system sizes L = 1300 and L = 5000 lattices points along the x direction, and for A = 0 and A = 4.

The simplest algorithm to implement quenched randomness would be to


assign the random numbers to all of the grid points from the beginning. According to our tests the surface almost never grows backward passing more than one grid point in the additive noise case (for multiplicative noise it always moves forward by definition). Thus, the random numbers which are already behind the surface never are used again. This allows us to keep track of random numbers only in the surface layer making possible much larger scale simulations.

The following quantities were determined: (i) the total surface width w(L, t) = ((h2) - (h)2)1’2 as a function of time, (ii) the average position of the interface h(t) = (h(~, t)) and (iii) th e surface width W(T) as a function of r in the saturated regime (in which w(L, t) 2: const.).

Eq. (7) was treated in an analogous manner. In this case the results were obtained for system sizes L = 800 and L = 1500.

6. Results

In this section we present our results for various features of the growing surfaces obtained by numerically integrating eqs. (5) and (7).

6.1. Additive noise, KPZ case

Fig. 1 shows the visual appearance of the growing surface near pinning when additive quenched noise (eq. (5)) is present. It can be clearly seen that after “breaking away” at a locally almost pinned point the surface grows isotropical- ly in both directions.

For A = 4 we found that the pinning threshold was u: = 0.05. Just above this value we observed a scaling regime of w(t) - tP. The exponent p can be obtained by fitting a straight line on a log-log plot of w(t) for u = u,. From fig. 2 we conclude that the dynamics of surface roughening is described by

p = 0.61 & 0.06

in the case of the discretized, quenched version of the KPZ equation. This value is clearly different from 0.75 given for p in refs. [11,12].

A further important aspect of the growth is the geometry of the surface in the stationary regime. We found that as the system size becomes larger the behaviour of the surface approaches self-affine scaling with a static exponent close to (Y = 0.71* 0.06. This value is in agreement with our result cy = 314 predicted by dimensional analysis. Thus, our simulations and theoretical considerations suggest that (Y = 0.75 is not likely to be a virtual exponent due to crossover [18], but the result of a genuine scaling regime. Fig. 3 shows the


m m N N d 3

146 Z. Csahbk et al. I Surface roughening in disordered media

OC

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 i.8

log t

Fig. 2. The time dependence of the surface width w in the quenched version of the KPZ equation for L = 1300 and v = 0.04.

0 0.5 1 1.5 2 2.5 3 3.5

log T

Fig. 3. The behaviour of the surface width w for h = 4, v = 1, L = 5000 and u = 0.04 as a function of the length x over which its average value was calculated. The middle part of the plot indicates self-affine scaling with an exponent a = 0.71.

corresponding results for L = 5000. If, for example, L = 1300 the scaling part

of the plot is much shorter.

6.2. Additive noise, EW case

For A = 0 (EW equation) we found that close, but above the pinning

threshold (uC 2: 0.19) the dynamics of the interface roughening is described by

an effective exponent 0.62 as well. However, if we get closer to u,, for longer

times a continuously growing effective exponent can be observed. This explains


an earlier numerical estimate /3 = 0.75 [ll]. The reason for the crossover to higher (near to 0.9) effective exponents is not completely understood. We think that very close to complete pinning, when only a very limited part of the surface is not still, the effective exponent corresponding to a more or less regularly moving localized region is close to 1. At earlier stages of the growth process the surface does not reach this “almost pinned” state.

We have obtained detailed data for the scaling of the asymptotic surface velocity u, = lim,,, ah/at as a function of the distance from the critical velocity u, (fig. 4a). We find that u, vanishes at u = u, and assume that for u - u, > 0 4 1 it scales as u, - (u - u,)’ [11,12]. Aeppli and Bruinsma [lo] predicted that u - L -2. Fig. 4a shows the asymptotic velocity versus the driving force for v’= 1 and system sizes L = 1300,260O and 5000. It demonstrates that for the parameters we used u, is independent of L. Instead of using our preliminary methods ref. [20] (simple fit to a (u - u,)’ dependence overestimating 8 because of a crossover from the scaling close to u, to a scaling with an exponent 1 for large u) to estimate 8 we decided to extract 0 from a more sophisticated approach. In brief, we assumed that the combined scaling behaviour of u, in the two regimes (close and far from pinning) can be well represented by the expression

1

‘a = ’ - UC 1 + A(ulu, - 1)” ’ (17)

‘b 1.5

I -1’ 1 0.25 0.3 0.35 0.4 0.45 0.5 -3 -2 -1 0 1 2

21 21 - zIc

log ~ 21,

Fig. 4. (a) The dependence of the average velocity of the interface as a function of the driving velocity for various system sizes and Y = 1. (b) This plot was obtained for the EW case with L = 1300 and for various Y values. It shows the resealed data for the average velocity u, as a function of the distance of the driving velocity from its critical value. The dotted line is the theoretical prediction for Y = 0. The data for Y > 0 are scattered around the continuous line which is a fit with 0 = 0.49.


where A is some constant providing the best fit (together with the approprop- riately chosen u, and 0) to the data provided by the numerical integration. Since such an analysis takes into account crossover effects, it is expected to provide a considerably better estimate for 8 than a simple fit of a straight line to the data for relatively (but not sufficiently) small u - u,. Fig 4b shows the resealed data for various v and L = 1300. The dotted line is the theoretical prediction (see below) for v = 0. The data for v > 0 are scattered around the continuous line which is a fit with 8 = 0.49. On the basis of fig. 4b we conclude that 8 is close to 0.5. This value is definitely smaller than 2/3 proposed for the d = 4 case in ref. [12] and our preliminary estimate 0.64 [20]. On the other hand it is good agreement with the expectation that an extrapolation of the d = 4 case to lower dimensions should lead to 8 smaller than 2/3 [22].

The detailed dynamics of the growing surface can be characterized by the distribution time intervals or waiting times the interface spends between two grid points. Let us consider the development of the surfaces obtained for large times (in the stationary regime) and determine the number of places N(7) where the surface spends a given amount of time 7, where T is the called waiting time. This time strongly depends on the value of n as well as on the local geometry of the surface. The results for v = 1 and u = 0.202 > u, are presented in fig. 5. It is remarkable how the uniform (Gaussian type) input noise is transformed by the growth mechanism into a power law distribution of the waiting times. The slope -2.3 of the nearly perfectly scaling region is non-trivial.

In this respect our results for the exponents (Y and p are consistent with the predictions of Tang et al. [23], who showed that in models with a power law

100 ’ 10' 102 103 104

r/At

Fig. 5. The distribution of the times T the surface spends between two grid points during its advancement. This figure indicates a power law output distribution resulted from a uniform input

noise.


distribution of waiting times the statics and the dynamics of the interface is described by non-universal exponents.

The following scaling arguments provide support for the above findings. In the case v = 0 there are no correlations in the x direction, the columns are independent. Eq. (5) can be rewritten as

ah X=U +7(h). (18)

The pinning threshold is u, = min(n) since the motion stops in a column when u + q(h) < 0. For a Gaussian distribution this implies u, = M. In our simulations we used uniformly distributed random numbers, P,(n) on the interval (-0.5, OS), thus there is a finite pinning threshold with u, = 0.5.

We can easily calculate the noise distribution along the surface (the distribution P,(n) of 7(x, h(x)). The surface spends a time ~(77) = a(u + q)-l (a is the lattice spacing) at a place where the noise is 7. As far as the input noise has a uniform distribution, P,(n) - (u + 7)-l. By averaging (18) we get

I

1 X,, = u + 77pS(77) drl = ln[(u + OS)l(u - 0.5)] ’

-0.5

(19)

The asymptotic velocity is u, = lim,,, d(h),/&, in this case u, = llln[(u + OS/ (u - 0.5)]. It vanishes as - 1 lln(u - 0.5) if u + 0.5 from above, thus in this case 8 =o.

There is a simple connection between the waiting time r and n and, therefore, we can obtain the waiting time distribution Z’,(r) from PO,

thus

Pw(7)=Po(q) & =Po(++‘. I 1 (21)

Our measured exponent -2.3 for the Y > 0 case is close to this value, however, because of the almost perfect scaling regime we believe that the -2 is outside of our error bars. This question needs further investigation.

6.3. Multiplication noise

In the multiplicative noise case we have assumed that p + 7) is always larger than zero, thus, the surface never becomes completely pinned. On the other


hand, at places where p + 77 -=s 1, the motion of the interface slows down dramatically. These points can be considered as temporarily pinned. Like in all of the existing experiments on growth involving flow of matter, after some time the surface passes by these places or regions of low permeability and advances further without a complete stop.

In order to better understand what are the most relevant factors determining the behaviour of experimental surfaces, we have numerically studied eq. (7) for times and system sizes compatible to those which have been realized in refs. [5,6]. For simplicity we assumed that 7 was distributed uniformly on (0, 11. Fig. 6 shows the calculated surfaces for the following sets of parameters: L = 1500, p = 0.0001 and u = 0.5; (a) A = 1 and (b) A = 0. The simulated surfaces look very similar to the observed ones. Interestingly, the EW case (A = 0) resulted in interfaces having a closer resemblance to there is no particular reason to think that during the development of wetting fronts.

Next we investigated w(L, t) for A = 1, the the entire system. The results shown in fig. 7 exists a non-trivial scaling

W--t P

the experimental ones, although lateral growth can be neglected

time dependence of the width of indicate that at early times there

(22)

with p 2: 0.65 + 0.05 in a surprisingly good agreement with the only published experimental result [6]. This value is close to 3/5 obtained for the additive noise case and also agrees with the corresponding estimates obtained in refs. [15,16]. The crossover to a behaviour described by a smaller exponent p = 0.26 -+ 0.05 is well pronounced and according to our simulations the crossover time t, only weakly depends on L. Our preliminary calculations indicate that

Fig. 6. Subsequent interfaces obtained by numerically integrating eq. (7) for p = 0.0001 and u = 0.5 for a system of linear size L = 1500. (a) A = 1, (b) A = 0.


10

z- s 1

3

0.1

1000

100

u” 10

1

0.1 1

100 10' 102 lo:' 10’ 105 0.0001 0.001 0.01 0.1 1

t P

Fig. 7. Fig. 8.

Fig. 7. The time dependence of the surface width w for various values (ranging from 0.0001 to 1) of the parameter p related to the strength of the pinning forces (smaller p corresponds to stronger pinning). There is a well defined crossover at a time t,, depending onp, from a scaling according to an exponent p = 0.65 to a scaling with p about 0.26. These results were obtained for by averaging over 20 runs of 50000 time steps for system sizes L = 1500.

Fig. 8. The behaviour of the crossover time t, as a function of the parameter p. In a limited region the dependence can be approximated as t, -P-‘.~.

for larger system sizes and longer times the value 0.26 does not change significantly.

The crossover time has a dependence on p which in a limited region of the p

values can be interpreted in terms of power law scaling. This is demonstrated in fig. 8. Assuming that for the intermediate values t, -p-’ a linear fit to the data gives the estimate y = 0.7 + 0.1.

We have also calculated the spatial scaling of w. According to our results for L = 800 and 1500 there exist no well defined, extended straight parts in the log w versus log r plots for the sizes and times we could realize. However, for larger sizes (L = 5000) a scaling of w(r) as a function of r over a limited range of length scale could be observed (just as in the EW case of the quenched additive noise). The corresponding exponent (Y 2: 0.47 was close to 0.5.

The actual status of the growing surface can be characterized by the distribution P(n) of n values. We consider a large set of surfaces obtained for times t S t, and determine the number of lattice sites with a given 7. The time the interface spends in a particular grid point (with the corresponding n) strongly depends on the value of n as well as on the local derivatives of the surface. These factors lead to a P(n) which is rather different from the original uniform distribution for q. The results are presented in fig. 9. This figure suggests that in the limit of very small p there is a range of 17 in which the distribution of the random numbers along the interface scales with an exponent


100

10-l

s c lo-’

10-s

in-4 - __ I”-.5 10-4 10-z 10-X 10-l 100

7 Fig. 9. The distribution of the noise values (output noise) along the surface for t B t,. The solid line

is for p = 0.0001, the thin line is for p = 0.0003, the dotted line is for p = 0.001.

close to 0.5. Again, a trivial, Gaussian type input noise distribution is transformed by the growth mechanism into a power law distribution of the noise values along the interface. This observation is in analogy with the findings of Sneppen and Jensen [24] described in a very recent preprint.

An interesting special case of the noise is when n depends only on X. An existing aspect of the physics is reflected by this choice: the motion of the interface is determined by not only the conditions at the surface, but also by the permeability of regions already left behind (which may partially block the supply of additional fluid). A possible realization of this case includes a Hele-Shaw cell with parallel grooves of different depth engraved onto one of the glass plates.

Fig. 10 shows a typical series of surfaces for L = 1500, p = 0.0001 and u = 0.5. In this model there is well pronounced scaling both in time and space,

Fig. 10. Series of surfaces obtained as a function of time for the model with 17 depending on x only

(see the text).


with numerically determined exponents close to 1 (a = /3 = 0.96 + 0.06). A temporal scaling with an exponent 1 can be considered as a trivial consequence of the particular choice for the noise. On the other hand, the result (Y = 1 is far less trivial and indicates that the surfaces attain a limiting case when the interface is both a single valued self-similar (and self-affine) function.

7. Conclusions

There are several relevant conclusions which can be drawn on the basis of our numerical results and dimensional analysis.

As concerning the additive noise case, our studies have shown that (i) the quenched versions of the KPZ and EW equations represent a new universality class with a well defined scaling in time. The theoretical prediction p = 3/5 is in good agreement with our estimate obtained from the numerical integration and is close to most of the related previous results p -0.63 [15,16]. Our value is significantly different from 3/4 proposed on the basis of a different theoretical and simulational method (ref. [18]). (ii) A well defined spatial scaling has been found with (Y = 0.71 in contrast with the suggested crossover behaviour (ref. [ll]). (iii) Close to the pinning transition, the average velocity scales with u - u, according to an exponent 8 close to 0.5. (iv) In spite of a uniform input distribution of r], the distribution of times spent by the interface between two grid points is a power law.

Our stochastic partial differential equation with multiplicative noise has led to (i) surfaces remarkably similar to those observed in the experiments (ii) and a scaling behaviour of the surface width with an exponent p = 0.65 being in an excellent agreement with the measured value. (iii) In addition, the transforma- tion of the uniform input noise to a power law distribution of the 77 values along the surface could be observed.

The appearance of power law distributed waiting times and “output” noise values should raise interesting questions about the connections of real and simulated anomalous roughening with the Zhang model [25] in which a power law input noise is assumed. In this respect our results can be used to interpret the experimental findings of Horvath et al. [26] indicating the presence of a power law output noise in wetting fronts.

Acknowledgements

Useful discussions with H.J. Herrmann, V.K. Horvath and J. Kertesz are acknowledged. One of us (Z. Cs.) is grateful for the kind hospitality during his


visit to HLRZ (Supercomputing Centre) Jiilich. The present research was supported by the Hungarian Research Foundation Grant No. T4439.

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Dynamics of surface roughening in disordered media