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Unusual exponents in interface roughening: the effects of pinning
Mogens Jensen, Itamar Procaccia
To cite this version:
Mogens Jensen, Itamar Procaccia. Unusual exponents in interface roughening: the effects of pinning.
Journal de Physique II, EDP Sciences, 1991, 1 (10), pp.1139-1146. �10.1051/jp2:1991124�. �jpa- 00247580�
J Phn. Ilfrance 1(1991) 1139-1146 CCIDBRE1991, PAGE 1139
Classificafion
PlgsksAbsmwts
05.40-47.55M-68.70
Unusual exponents in interface roughening:
the effects of pinning
Mogens H. Jensen and Itanmr Procaccia
NiRDrLABlegdanwvej 17, DK-21W Copenhagen 13, Denmark (Received31Ji~~y lwl, accepted l2Aupat lwl)
Absmct. Recent e~periments indicate that interfbce rougheningin fluid displacement in porous media is characterhed by exponents that differ &om the predictions of standard roughening models.
By introducing two simple models that take into account the effect of internee pinningby pores, We can easily account for the unusual exponents.
The growth of interfaces is studied in the context of a variety ofphysical systems, most notably
in fluid displacements in porous media [1,2], growth of solids from the vapor, stochastic ballistic
depositions[3-8] and even in the grown of bacterial colonies [9]. The theoretical interest in these
phenomena arises mainly due to the dynamical creation of self-afline interfaces in all these exam-
ples. Indeed, most of the theoretical effon was devoted to the characterization of the interface
roughening and to the understanding of the exponents appearing in the relation of the "tvidth" of tile interface to the linear size of the system.
There are ovo main theoretical approaches to the subject: in approach A one relies on a stochastic differential equation description of the interface growth, whereas in approach B one
employs various ballistic deposition models on a lattice substrate. The two most typical repre- sentatives of these two approaches are the Kardar-Parisi-Zhang ~KPZ) equation [10] and the
Kbn-Kosterlitz ~KK) model [8]. The former is an equation for the local deviation from the mean of the height of an interface, denoted byh, and it reads
where z E Ld and qis a gattssian white noise. The latter model selects randomly a site on a cubic d dimensional lattice and permits a growth in the height of the interface by one unit whenever the local gradient does not exceed a chosen threshold. It has been found by extensive numerical
simulationjthat the approaches A and B predict the same exponents for the width of the interface, defined by h(x, t = h(x, t < h(x, t) > and
w2jt)
= < iijx,1)12 > j2)
i140 JOURNALDEPHYSIQUEII N°10
where < > stands for an average over z. Fbr short times the width was found to scale like
1()
-~ 1))~ 13)
whereas for long times the width is saturated to an L dependent limit which satisfies
1()
-~ 1))~. 14)
The static exponent x also characterhes the structure function of the illterface, sy) = < jhjx + i,t) h(x,t)i >
-~ ix is)
for times long enough such that stationary growth has been achieved.
The values of X and fl found by simulations seems to agree approximately tvith the ansatz [8]
x(d) =
]
fl(d~ =
j.
(6)There is really no analytic theory to account for these values accept i11d = I where the exponent
x = 1/2 is dictated by standard random walk theories. The renormalization group analysis indi- cated that the long wavelength limit of equation (I) may be governed by a strong coupling fixed
point which may or may not account for the exponents found numerically [10].
These theoretical models seem however to miss something fundamental in the physics of fluid
displacement in porous media. Experiments in this field discovered that the values of x and fl
differ significantly from the those quoted above [1,2]. In these experiments a more viscous fluid
displaces a less viscous one, and the displacing fluid also wets the porous medium more effectively.
One way to account theoretically for the differences in exponents tvithin approach Ah to change
the properties of the random noise term in equation (1). I~1deed, such a tack was recently adopted by Zhang [iii and Parisi [12]. The former author inuoduced a non-gaussian, power law noise
pin)
-w
q-°-I which is uncortelated in space. The latter author introduced a quenched Gaussian noise which depends on h and z. Both authors found indeed noise-dependent exponents. Wis
however leaves open the question what is the physical reason for the different exponents. In fac~
some authors realized that the main physical source of difference must be related to the pinning of the interface on the wider pores in the medium [12.14]. The displacing, better wetting fluid flows
easily through the narrow necks, but can get stuck rather effectivelyin the vicinity of tvide pores.
The aim of this letter is to explore the effects of such a pinning phenomenon on the roughening
indices tvithin the approach B, with the help of a simple generalization of the KK model.
In constructing a sensAle model, one faces two severe unknowns. The first is what is the dis- tribution of tvider than usual pores in a porous medium which is artificially constructed in the
laboratory ~by a gravitational packing of small spheres). The second b how strongly would the interface pin to a wider than usual pore. Also there is the question of how to account for pinning
in the actual growth algorithm. The last question can be easfly solved. Simply, mea lattice point
x b assigned a counter Mix), and growth of the illterface by one unit is only allowed when the counter of an interface point exceeds the "pinningstrength" p(x), that is when Mix) > p(x). The distribution of p(x) values should be the answer to the first two questions.
Maybe the simplest model of interest is a model in which p(x) is distributed according to a
power law,
Prob(p(x) = a)
-J
a~~~~~~ (7)
N°10 UNSUALEXPONENTSINWWRFACEROUGHENmG 1141
We refer to this model as model I. We shall show below that thb model suffices for obtaining roughening eTponents that differ significantly from (6).
A model that is based more on the physics of gravitational packing ofspheres in two dimensions
can be constructed following some recent findings of Jullien and Meakin [ls~;they discovered that in a gravitationally packed 2-ditnensional medium of mono-dbpersed spheres the density of voids decreases as a power law with the height of the medium. The voids in thb case are roughly equal~
and therefore their effect on the interface should be roughly equally strong. Wis motivates our model II: Here p(z) = I or p(z)
= pm > I, and the density of sites with p = pm decreases as a power law with the height of the medium. We shall show that also this model leads to effective
scaling exponents that differ from (6).
In this letter we repon results for simulations in d
= I only. We consider a semi,infinite lattice of lateral size L, and denote the finite coordinate by z, and the infinite coordinate by y. Every point ix,y) is assigned a value p(z, y), for all y < yaw which is chosen tc be big enough. The interface growth is achieved as follows: at time t = 0, an interface function is defined as h(z) = I for every z, y = I. Each point on this interface has a counter Mix, h(z)). Now sites belonging to
h(z) are chosen randomly, and their counter is increased by one: Mix, h(z))
- Mix, h(z)) + I.
Whenever the counter exceeds p(z, h(z)), then h(z) is increased by one unit, provided tllat the difference (h(z) + I h(z + I) does not exceed I. Fresh interface sites are given an empty counter but all the other counters on the interface keep their score. The time of the experiment is taken in units of added particles.
in s(11
14
~=2
13
iz
ii
O 2 4 6
In L
fig. 1. A plot of the structure function, equation (5), on log-log scale for p
= 2 and L = 7000 after addition of2 x 10~ particles. We find X
= 0.85 £ 0.03. Note however that the scaling above £
~- 300 begins
to saturate probably due to tile finite size of the system and the finite growth time.
In model I the values of p(z,y) are chosen randomly according to the distribution (7). In practice one chooses for each point ix,y) a random number R in the the interval (0, 1), and assigrts
a value p(z, y) = R~ f. Figure Idisplays a typicalscaling of the structure function equation (5)
calculated after saturation. In figure 2 a corresponding dynamical scaling of the width of the
1142 JOURNAL DE PHYSIQUE II N°10
jaw
W~ 4~
~z2
z
O
+ ~
+
~
+ +
-2
~
~
-5 -2.5 o 2.5 5
in t
t~
Pg. ~ We dynanfical scaling ~Eq.(3)) on log-log smle for a system of size L = 2048. The surfing is
averaged over ~-iW independent runs. Note the initial growth for W < i after which the dynamica1scaling
takes over with an exponent # = 0.80 * 0.05.
interface, equation (3), is dhplayed. The dynamical scaring has two distinct regimes; for times t < to, W -w )°.~ Fort > to, W
-w ())°."*°.°~ The first regime obtains in the initial stage
o o
of the random thickening of the interface up to values W -w i. The second regime, which b the more interesting one, pertains to the dynamical roughening of the interface. The exponent fl of
equation (3) pertains to the second regime. In figure 3 we displayx and fl as a function of ~. It is interesting to notice that the usual sum rule of KK mode( Le x + ~ = 2, seems roughly to be
fl
fulfilled (within numerical errors) in this model as wefl~ One should notice that in our model, like in KK~ the width of the interface b bounded ~om above byL, and therefore x b bounded by I. In fact, not only the width itself but all the moments W9it) =< (I(z,t)(9 > are bounded by L9. Our model is free of some divergencies of higher moments which appear in the canter of approch A~
like in Zhang's model, in which W9 does no ewt for q > ~ [iii.
In model II the values ofp~z, y) take on two values: I (I.e. no pinning), or the constant pinning strength pm. The density pig) of sites vith pinning strength pm is chosen such that
Ply)
°'
~° ~ ~
~ ~ ~°
18)
Po Or y < yo
The actual sites at each yior which p(z,y)
= pm are chosen randomly tvith this density. The cutoff in the power law has been introduced in order to mimic the results of reference [15] (Cf.Fig.3).
One could have a power law density down to y = i, without major changes in the conclusions. In
principle the salve of K can be changed atwili~ but we use only the value suggested in reference
[15],I.e K = 2. The only parameter in model II that influences X b thus the pinning suength pm.
Figures 4aAc show typical results for the structure functions at various stages of the growth.
one sees that at shon and intermediate titnes there is a buildup of a nontriAal exponent for the structure function~ At long times the sca1illg crosses over to the KK behaviour; the density of
N° lo UNSUAL EXPONENTS IN WWREACE ROUGHENING 1143
x
~~
a
O.7
~i 2 5 4 5 6
~l
#
16)
off o
i
O.4
5 2 2 5 3 3.5 4
Rg. 3. A plot of the roughening exponents x (a) and #(b) as a function of the exponent p of the power law distribution of the strength of the pinning center~ Note the similarity slith the corresponding plot of Zhang, reference [i11.
pmnwg sites Mcomes too small to affect the scaling. There is always a well defined time, when the width of the interface b madmal, when the scaling law extends over a madtnal range of scales.
Itis the value ofX at thb time that we display for a range of pinning strength pm infigure5. Notice tha~ of course, for pm - i the KK value of x is regained.
Thedynanflcalbehavior isq~mfitativelysimilarto the one ofmodell. On shon times, for W < i, there b a different growth rate of W then on longer times, when W > I. It is less obvious however that flhas awell defined meaning here, since asymptotically we expeaa KK dynamics to take over.
the apparent values offl for the intermediate time scales do not fulfill the usual sum rule, but we
1144 JOURNAL DE PHYSIQUE II NO lo
In S(i)
(al
o 2 3 4 5
In I In s(t)
(b)
ii
O 2 3 4
In t
~~ ~~ ~~
(C)
9
o , z 3 4 5
In I
Fig. 4. Plots of the structure functions for model II Mith the following parameters: L = 1500, pm = so,
n = 2, go = 10, pa = 0.5. a)AIter addition 0f335000 particles; theslope for small distances is approximately
o.5 which crosses oNer to a slope ~- o.8. b) After addition of 485000 particles the slidth of the interftce is maximal and this is where we obtain goodscaling slith x
= 0.84 £ 0.04. c) AIter a growth of io~ particles
there isagain a cross-ever; at silali smies
we ham the x offigure b and at large scales we begin to see the KK result x = 0.5.
NO lo UNSUAL EXPONENTS IN WTEREACE ROUGHENWG 1145
x
o~
i
IO 20 3O 40 50 6O
Pm
Fig. 5. The value of x for model II as a function of the pinning strength pm; the other parameters as in figure4. the value of X is edracted where the width assumes its maximal value.
cannot insist on Wb issue due to the questionable nature of fl here.
The main conclusion of these results is that it is indeed sufficient to incorporate pinning effects in order to obtain scaling indices that fall close to the experimental values with either model I
or model II. In model I we obtain good scaling behavior that continues for times longer than the initial build up time to- In model II we get a cross over behaviour. However, we draw the atten- tion of the reader to the similarity in the scaling plots of model II, figure 4c, and the experimental
results displayed in reference [2], figure 3. Of course, this similarity in results cannot be taken as a proof for identity of mechanism before further experimental research addresses the pinning
mechanism explicitly. It should be noticed that the scaling indices depend sensitively on the nu- merical values of the parameters in the models, I.e. ~, n and pm respectively. Thus, if our models bear any relevance to the experiments in question, we can expect that the exponents measured in related experiments would not be univemL In panicular, changing the wetting properties of
the displacing fluid, or the nature of the porous medium might result in a significant change in the exponents. It is certainly premature to speculate on these issues. We shall feel that our point
has been well taken if this note would motivate further research into the nature of the pinning of interfaces for fluid flows in porous medh, and into the relation of this phenomenon to the scaling
indices that characterize the roughened interfaces.
After this manuscript was submitted we loomed about a communication by LH. fling, J.
Kert£sz and D. Wolf which addresses similar issues, using distn~utions of waiting times, finding
also unusual exponents.
Acknowledgements.
We are grateful to J. Gollub, M. Kosterfitn P Meakin, Y:C. Zhang and R. Zeitak for discussions and communications.
1146 JOURNAL DE PHYSIQUE II NO10
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