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PHYSICAL REVIEW E 67, 056308 ~2003!

Anomalous roughening in experiments of interfaces in Hele-Shaw flowswith strong quenched disorder

Jordi Soriano,* Jordi Ortın, and A. Herna´ndez-MachadoDepartament d’Estructura i Constituents de la Mate`ria, Universitat de Barcelona, Avenida Diagonal, 647, E-08028 Barcelona, Spa

~Received 2 August 2002; revised manuscript received 21 February 2003; published 21 May 2003!

We report experimental evidence of anomalous kinetic roughening in the stable displacement of an oil-airinterface in a Hele-Shaw cell with strong quenched disorder. The disorder consists of a random modulation ofthe gap spacing that is transverse to the growth direction~tracks!. Experiments were performed by varying theaverage interface velocity and the gap spacing, and measuring the scaling exponents. The following values ofthe scaling exponents were obtained;b.0.50,b* .0.25,a.1.0, a loc.0.5, andz.2. When there is no fluidinjection, the interface is driven solely by capillary forces, and a higher value ofb of approximatelyb50.65 is measured. The presence of multiscaling and the particular morphology of the interfaces, characterizedby large height differences that follow a Le´vy distribution, confirms the existence of anomalous scaling. Froma detailed study of the motion of the oil-air interface, we show that the anomaly is a consequence of differentlocal velocities on the tracks plus the coupling in the motion between neighboring tracks. The anomalydisappears at high interface velocities, weak capillary forces, or when the disorder is not sufficiently persistentin the growth direction. We have also observed the absence of scaling when the disorder is very strong or whena regular modulation of the gap spacing is introduced.

DOI: 10.1103/PhysRevE.67.056308 PACS number~s!: 47.55.Mh, 68.35.Ct, 05.40.2a

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I. INTRODUCTION

The study of the morphology and dynamics of rough sfaces and interfaces in disordered media has been a suof much interest in recent years@1#, and is an active field ofresearch with relevance in technological applicationsmaterials characterization. These surfaces or interfacesusually self-affine, i.e., statistically invariant under an anitropic scale transformation. In many cases, the interfafluctuations follow the dynamic scaling of Family-Vicse~FV! @2#, which allows a description of the scaling propertiof the interfacial fluctuations in terms of a growth exponeb, a roughness exponenta, and a dynamic exponentz.However, many growth models@3–8#, experiments of frac-ture in brittle materials, such as rock@9# or wood@12,10,11#,as well as experiments on sputtering@13#, molecular beamepitaxy@14#, and electrodeposition@15# have shown that theFV scaling is limited, and a new ansatz, known asanomalousscaling @7,16,17#, must be introduced. The particularity othe anomalous scaling is that local and global interfacfluctuations scale in a different way, thus, in general,global growth and roughness exponentsb and a, respec-tively, differ from local exponentsb* anda loc .

An interesting observation from the experiments on frture in wood, where tangential and radial fractures were sied @12#, is that the morphology of the interfaces for thformer were characterized by large slopes, in contrast tosmoother appearance of the latter. As pointed out by Lo´pezet al. @9#, the origin of anomalous scaling in these expements seems to be related to these large slopes in thephology of the interfaces.

In a previous work@18#, we presented an experiment

*Electronic address: [email protected]

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study of forced fluid invasion in a Hele-Shaw cell witquenched disorder. We focused on the limit of weak capillforces and studied different types of disorder configuratioat different interface velocities and gap spacings. Wetainedb.0.50, which was almost independent of the expemental parameters. However, the measured roughness enents were dependent on the experimental parameters,two regimes were characterized,a1 at short length scales ana2 at long length scales. We also observed that in conditiof totally persistent disorder in the direction of growth, afor sufficiently strong capillary forces, a new regimemerged. This new regime must be described withinframework of anomalous scaling and was discussed in acent paper@19#. The scaling exponents were obtained forspecific set of experimental parameters, and the range olidity of the anomaly was explored. We also introducedphenomenological model that reproduces the interfacenamics observed experimentally and the scaling expone

The objective of the present paper is to analyze theperimental evidence of anomalous scaling in a systemway, introducing additional techniques for the detectionanomalous scaling, such as multiscaling and the statisdistribution of height differences. The experimental rangeanomalous scaling is studied in depth, and we investigateeffects of average interface velocity, gap spacing, and dider configuration. We also consider the situation in whichinterface is solely driven by capillary forces, which allowthe behavior of the interface to be better understood atdisorder scale and its relationship with the scaling exponeat particular limits of the experiments.

The outline of the paper is as follows. In Sec. II we rview the anomalous scaling ansatzs. In Sec. III, we introdthe experimental setup and describe the methodology usedata analysis. The main experimental results are preseand analyzed in Sec. IV. The discussion and final conclusiare given in Sec. V.

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SORIANO, ORTI´N, AND HERNANDEZ-MACHADO PHYSICAL REVIEW E 67, 056308 ~2003!

II. SCALING ANSATZS

The statistical properties of a one-dimensional interfadefined by a functionh(x,t) ~interface height at positionxand timet) are usually described in terms of the fluctuatioof h. If we consider an interface of lateral sizeL, the rmswidth calculated over a windowl ,L is defined asw( l ,t)5^@h(x,t)2h(t)#2&x

1/2, whereh(t)5^h(x,t)&x , and^&x is aspatial average in thex direction. While theglobal widthw(L,t) follows the FV dynamic scaling@2#, in the anoma-lous scaling assumption thelocal width w( l ,t) follows adifferent scaling given by

w~ l ,t !;H tb if t! l z

l a loctb* if l z!t!Lz

l a locLa2a loc if Lz!t.

~1!

Here, b and a are the global growth and roughness expnents, respectively.z is the dynamic exponent witha5zb.b* 5(a2a loc)/z is the local anomalous growth exponeand a loc the local roughness exponent. The local timegiven byt l; l z and the saturation time byt3;Lz. Therefore,in the case of anomalous scaling, five scaling expone~three of them independent! are needed to characterize thscaling behavior of the surface. It is common to defineu5a2a loc as theanomalous exponent, which measures thedegree of anomaly@16,17#. Whenu50 andb* 50 local andglobal scales behave in the same way, and we recoverusual FV scaling, i.e.,w( l ,t);tb for t! l z and w( l ,t); l a

for t@ l z.It is common to define a scaling function for the behav

of w( l ,t) in the form w( l ,t)/ l a5g( l /t1/z). From Eq. ~1!,g(u) is expected to scale as

g~u!;H u2(a2a loc) if u!1

u2a if u@1.~2!

One of the implications of anomalous scaling is that the lowidth saturates when the system size also saturates, i.etimes t3 , and not at the local timet l as occurs in the FVscaling. It is also interesting to note that anomalous scacan be identified in an experiment by plotting the local widw( l ,t) as a function of time for different window sizes. According to Eq.~1! the different curves would show a verticshift proportional tol a loc.

The roughness exponents are usually determinedstudying the power spectrum of the interfacial fluctuatioS(k,t)5^H(k,t)H(2k,t)&, where H(k,t)5(x@h(x,t)2h(t)#exp(ikx). While in the FV assumption, the powespectrum scales as

S~k,t !5k2(2a11)s~kt1/z!, ~3!

wheres is the scaling function

s~u!;H u2a11 if u!1

const if u@1,~4!

in the anomalous scalings(u) is given by

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For times t@t3 the power spectrum scales asS(k,t);k2(2a loc11)t2u/z and not simply ask2(2a11) of the ordi-nary scaling.

In general, one differentiates between two typesanomalous scaling:super-roughnessandintrinsic anomalousscaling. Super-roughness is characterized bya loc51 anda.1, although the structure factor follows the ordinary scing given by Eqs.~3! and~4!. Thus, for super-roughness, thpower spectrum provides information about the globroughness exponenta. The intrinsic anomalous scalinstrictly follows the scaling given by Eq.~5!. This implies thatthe decay of the power spectrum withk only gives a measureof a loc , not of a. Moreover, the particular temporal depedence of the scaling of the power spectrum allows idencation of the existence of intrinsic anomalous scaling bypresence of a vertical shift of the spectral curves at differtimes. One has to be cautious with this particularity becathe observation of a vertical shift, however, does not necsarily imply the existence of intrinsic anomalous scaling. Tdifferent forms of anomalous scaling are included in a gneric dynamic scaling ansatz recently introduced by Rmascoet al. @20#.

III. EXPERIMENTAL PROCEDURE AND DATA ANALYSIS

A. Experimental setup

The experimental setup used here has been describedviously in Refs.@18,19#. The setup consists of a horizontHele-Shaw cell that contains a pattern of copper traplaced on a fiber glass substrate. The tracks ared50.0660.01 mm high and are randomly distributed along the leral directionx without overlap, and fill 35% of the substratThe lateral size of the unit track is 1.5060.04 mm, but widertracks are obtained during design when two or more utracks are placed adjacently together. The separation betwthe fiber glass substrate and the top glass defines thespacing b, which was varied in the range 0.16<b<0.75 mm. A silicone oil~Rhodorsil 47V) is injected on oneside of the cell at constant volumetric injection rateQ anddisplaces the air initially present. A typical example of tmorphology of an interface and its temporal evolutionshown in Fig. 1@21#.

Although the disorder made by tracks~which will becalledT1.50, the numerical value indicating the width of thunit track in millimeters! has been the main disorder patteused in the experiments, other disorder configurations hbeen employed. These configurations are characterizedrandom distribution of squares with a lateral size of 1.500.40 mm, and will be called SQ1.50 and SQ0.40, resptively. A full description and characterization of these disoders, plus all additional details about the experimental seand procedure can be found in Refs.@18,19#.

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ANOMALOUS ROUGHENING IN EXPERIMENTS OF . . . PHYSICAL REVIEW E 67, 056308 ~2003!

B. Characterization of scaling

As explained in Sec. II, we have studied the roughenprocess of the interfaces by means of the interfacialwidth w( l ,t) and the power spectrumS(k,t). The scaling ofw( l ,t) with time provides the global~for l 5L) and local~forl !L) growth exponentsb and b* . The exponentb* hasalso been measured using an alternative scheme, proposLopez @22#, in which the scaling of the mean local slopwith time is analyzed.

The local roughness exponentsa loc were obtained fromthe scaling of the power spectrumS(k,t) with k. The resultsobtained were checked using other methods, such as thpendency of the height-height correlation functionC( l ,t) orW( l ,t) on thewindow size l, in both cases scaling asl a loc.Similar results to those fromS(k,t) were obtained withinerror bars.

FIG. 1. Closeup views of the oil-air interface with disordT1.50. The first three pictures are early stages, while the last pcorresponds to saturation. The position of the copper trackshown in the lower box. The dotted line shows the position ashape of the interface profile att50 s. The experimental parameters arev50.08 mm/s,b50.36 mm.

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Finally, within the framework of intrinsic anomalous scaing, the only direct way to measure the global roughnexponentsa, and in additionz5a/b, is to study the scalingof C(L,t) or w(L,t) with the systemsizeL. Both quantitiesscale asLa. Because it is not practical to perform expements by varying the system size, we have not measuglobal roughness exponents directly. Instead, they have bobtained from the collapse ofS(k,t) or w( l ,t). Note thatinformation about the value ofa is contained in both thevertical displacement of the spectral curves with time@Eqs.~3! and ~5!# and the vertical shift of thew( l ,t) curves fordifferent window sizes@Eq. ~1!#. The error bars that we givein the final numerical value of a particular scaling exponetake into account the various methods of analysis anderror in the different collapses.

C. Side wall effects

The profileh(x,t), obtained from the analysis of the images, contains 515 points corresponding to a lateral siz190 mm. Because the oil tends to advance along thewalls of the cell, we have a distortion of the front that peetrates a distancep from each side wall. This penetratiolength depends on the average interface velocity and is ecially sensitive to variations in gap spacing. An examplepresented in Fig. 2~a!, which shows the effect of the distortion on an interface with experimental parametersv50.08 mm/s,b50.75 mm and disorder SQ1.50.

To evaluate the effect of the distortion, we have measuthe penetration lengthp at different gap spacingsb for twodisorder configurations SQ1.50 andT1.50. The penetrationlength is measured at short times, when the interface is

toisd

FIG. 2. ~a! Example of an interface with strong side wall distotions. The dashed lines represent the regions cut at the end ointerface, and the solid line represents the interface finally studThe experimental parameters arev50.08 mm/s,b50.75 mm, anddisorder SQ1.50.~b! Penetration lengthp as a function of the gapspacingb for SQ1.50~dots! andT1.50 ~triangles!.

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bally flat except at the side walls. Fitting a straight line to tcentral region of the interface enables us to determinep asthe distance between the side wall and the point whereinterface deviates from the straight line. The results fortwo disorder configurations analyzed are presented in2~b!, and show thatp increases quickly with gap spacing. Fb50.75 mm, the largest value studied,p is about 8 mm.Notice also that the presence of columnar disorder signcantly reduces the effect of the distortion for the smallestspacings. To minimize the effect of the distortion, we hacut 8 mm from both ends of all interfaces, as shown in F2~a!, reducing them to a final lateral size of 174 mm containg 470 pixels, which gives a data spacing ofDx50.37 mm.

D. Data treatment

In order to defineh(x,t) at the edges of the copper trackwhere~due to the finite resolution! the interface may oscil-late between two neighboring pixel columns, we have tathe lowest value of the left column as first point and thighest value of the right column as second point, in the cof increasingh, and the highest left and lowest right in thcase of decreasingh. This gives a finite slope to the interfacat the edges of the tracks.

For data analysis convenience~particularly in the compu-tation of the power spectrum!, we have extended the numbof points toN5512 using linear interpolation, which givesfinal data spacing ofDx50.34 mm. Although taking equallyspaced points introduces an artificial cutoff on the heigradients that depends onDx, our spatial resolution is highenough for this not to be a problem. To prove it, we haartificially increased the data spacingDx ~by taking onepixel every n pixels! and performed again the differenanalysis. As the numbern tends to 1~andDx approaches theactual value used in our experiments!, the results becomeinsensitive to the spatial resolution.

Because the initial stages of roughening are so fast,interface profile att50 s is not a flat interface, but a profilwith an initial roughnessw(0), as can beobserved in Fig. 1~dotted line!. This roughness, on one hand, does not enaobservation of the growth of the small scales at short timon the other hand,w(0) changes from one disorder configration to another, which makes the early stages stronglypendent on the particular disorder realization. To eliminthese effects, we will always plot thesubtracted widthW( l ,t), defined asW( l ,t)5^w2( l ,t)2w2( l ,0)&1/2, where thebrackets indicate average over disorder realizations. Tcorrection is a standard procedure in the field of growinterfaces, and is reported in both experiments and numesimulations@1,13,23#.

The final treatment we use is to enforce periodic boundconditions by subtracting the straight line that connectstwo ends of the interface. This procedure, well documenin the literature on kinetic roughening@24#, eliminates anartificial overall slope22 in the power spectrum due to thdiscontinuity at the two ends@25#.

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IV. EXPERIMENTAL RESULTS AND ANALYSIS

A. Evidence of anomalous scaling

The first evidence of anomalous scaling in these expments was presented in our previous work@19# and corre-sponded to the experimental parametersv50.08 mm/s andb50.36 mm. The global and local growth exponents, msured fromW( l ,t), were b50.5260.02 ~corresponding tol 5L) andb* 50.2660.03 ~for l 5L/256). The fact thatb*is clearly different from zero is robust evidence of anomlous scaling. We also studied the evolution of the powspectrum, which gave two clear power law dependencesshort times with slope23.460.1, and at saturation withslope22.160.1. The former was a transient regime, whoorigin is the strong dominance of the capillary forces at veshort times. The latter provided the local roughness expona loc50.5560.10.

Here, we consider an alternative analysis that complethose presented in Ref.@19# and provide alternative ways tconfirm the existence of anomalous scaling.

To check the reliability of the exponentb* , we have stud-ied the scaling of the mean local slopes with time,r(t)5@^(¹h)2&x#

1/2, where the brackets denote a spatial averaalongx and the overbar denotes average over disorder cfigurations. Usuallyr(t) scales with time asr(t);tk, withk,0 for the ordinary scaling andk.0 for anomalous scal-ing. In this last case,k is identified with the local growthexponentb* . The result, presented in Fig. 3, isb* 50.3060.05, in good agreement with the result ofb* usingW( l5L/256,t).

The values ofa and z were obtained in Ref.@19# byplotting S(k,t)k2a11 as a function ofkt1/z. This collapserefined the numerical values of the experimental resultsgave

b50.5060.04, b* 50.2560.03,

a51.060.1, a loc50.560.1, z52.060.2. ~6!

Here, the exponentsu anda have also been determined bmeans of the scaling function defined in Eq.~2!. This col-lapse, shown in Fig. 4, givesu5a2a loc50.5 anda51.The presence of anomalous scaling can also be identifie

FIG. 3. Analysis of the mean local slopes as a function of timThe experimental parameters arev50.08 mm/s,b50.36 mm, anddisorderT1.50.

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checking for multiscaling@4#. For this purpose, we havstudied the scaling of the generalized correlations of ordeq,of the form Cq( l ,t)5$^@h(x1 l ,t)2h(x,t)#q&x%

1/q; l aq,where aq are local roughness exponents. The resultstained are shown in Fig. 5, which gives the values ofaq atdifferent ordersq for interfaces at saturation. We obtainprogressive decrease of the measured exponent, varyinga2.0.58 toa6.0.25, which confirms the existence of mutiscaling in the experiment and, therefore, of anomalous sing.

B. Experimental range of anomalous scaling

Once the existence of anomalous scaling is well chaterized for a specific set of experimental parameters, wenow interested in analyzing the range of validity of tanomaly. We have performed a series of experimentsvaryingv and fixing the gap spacing tob50.36 mm togetherwith a series of experiments varyingb and fixing the averageinterface velocity tov50.08 mm/s. As we can identify thexistence of anomalous scaling by looking at the behavio

FIG. 4. Collapse of theW( l ,t) plots using the scaling functionof Eq. ~2!. The experimental parameters arev50.08 mm/s, b50.36 mm, and disorderT1.50.

FIG. 5. Multiscaling analysis through the computation of tq-order height-height correlation function for interfaces at satution. The experimental parameters arev50.08 mm/s, b50.36 mm, and disorderT1.50.

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the smallest scales, i.e., the existence of ab* Þ0, we presentthe results by plotting the local growth exponentb* for thedifferent experiments.

1. Interface velocity

The results ofb* at differentv are presented in Fig. 6For a wide range of interface velocities, fromV to 10V,whereV50.04 mm/s is the minimum velocity investigateand will be used as a reference velocity, the measured enentb* is not very sensitive to changes in the velocity andvaries from 0.3560.04 for velocityV to 0.2860.04 for ve-locity 10V. For velocitiesv*15V, however, the measurevalue ofb* quickly decays to a value close to zero, indicaing that above a threshold velocity~about 13V for b50.36 mm) the anomaly disappears.

In terms of the power spectrum, when the velocity is vied from V to 15V, we always observe a vertical shift of thspectral curves and the power spectrum at different timesbe collapsed to obtain the set of scaling exponents. Abthe threshold velocity, the power spectrum follows ordinascaling, i.e., without vertical displacement.

2. Gap spacing

The results ofb* at differentb are presented in Fig. 7. Iis clear from the different plots that the exponentb* is rathersensitive to variations of the gap spacing. Three differbehaviors can be observed:

~i! For small values ofb (b50.16 mm), where capillaryforces are dominant, we obtainb* 50.5260.04, which isidentical to the corresponding value ofb. This similitude isdiscussed in Sec. IV E.

~ii ! For intermediatevalues ofb ~in the range 0.23<b<0.56 mm), we obtain values ofb* that vary from 0.3560.05 forb50.23 mm to 0.2560.04 forb50.56 mm. Thisrange of gap spacings is characterized by strong~but notdominant! capillary forces, and is the region where thanomalous scaling can be fully characterized.

~iii ! For large values ofb (b*0.60 mm), we obtainb*50.1060.04 forb50.75 mm, andb* nearly zero for larger

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FIG. 6. W( l ,t) curves for a window sizel 5L/256 and differentinterface velocities. The gap spacing isb50.36 mm and the disor-der T1.50 in all cases.V50.04 mm/s. The curves are verticallshifted for clarity.

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gap spacings. In this regime capillary forces are not suciently strong, and the anomalous scaling disappears.analysis ofb* through the scaling of the mean local slophas given identical results within error bars, althoughsensitivity of this analysis to short-time transients makeless reliable.

As regards the power spectrum, in the range 0.23<b<0.56, the vertical shift of the spectral curves is present,a regime characterized by anomalous scaling can bescribed. Forb50.16 mm, although the spectral curves avertically shifted, a scaling regime cannot be identified. Tpower spectra can be collapsed, but reliablea, a loc , andzexponents cannot be obtained. Forb*0.60 mm, a verticalshift in the power spectra is not observed, and the scalinthe interfacial fluctuations is best described within the framwork of FV ordinary scaling.

3. Disorder properties

Other modifications carried out to study the range oflidity of the anomalous scaling are the track width andpersistence of the disorder along the growth direction. Fig8 shows the variation of the local interfacial fluctuationsfour different disorder configurations. The gap spacing avelocity have been fixed tob50.36 mm andv52V, respec-tively. For T0.40, we obtain a similar result to that obtainwith T1.50, andb* 50.2960.05 was measured. Howevethere is evidence that the anomaly decreases as the widthe basic track is reduced, and it disappears when the wof the basic track is&0.20 mm due to the fact that the inplane curvature of the copper track becomes similar tocurvature of the meniscus in thez direction.

Now different disorder configurations with progressivesmaller persistence in the growth direction are examined.SQ1.50, although the saturation time for the full systemt3.1000 s, we observe that the local interfacial fluctutions saturate at a much shorter time, which is more chateristic of ordinary scaling. However, we still observe tracof anomalous scaling at earlier times, for whichb* 50.2860.04. This is really not very surprising if we take in

FIG. 7. W( l ,t) curves for a window sizel 5L/256 and differentgap spacings. The interface velocity isv52V50.08 mm/s and thedisorderT1.50 in all cases. The curves are vertically shiftedclarity.

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account the fact that, for this disorder, the first arraysquares is equivalent toT1.50, but is only 1.50 mm longThus, at very short times, the growth follows the samehavior as tracks. With a driving velocity 0.08 mm/s, the iterface takes about 10220 s to cross the first array osquares, which is the same time interval for which tanomalous scaling is observed. For SQ0.40, witht3

.300 s, we measureb* 50.0560.04, indicative of the ab-sence of anomalous scaling.

4. Characterization by the statistical distributionof height differences

We have also considered an alternative analysis to cfirm the results of anomalous scaling. In a recent work, Akainen et al. @26# showed that the presence of anomaloscaling can be associated with the presence of multiscaon one hand, and to high jumps in the morphology of tinterfaces on the other. According to these authors, thetribution function of the height differences,P(Dh), withDh[uh(xi 11)2h(xi)u, scales following a Le´vy distribution,P(Dh);Dh2g, where 0,g,2; while for ordinary scalingP(Dh) follows a Gaussian distribution. We have checkthis prediction with our experimental results. Figure 9~a!compares the shape of three different interfaces with difent degrees of anomalous scaling. In all cases,v52V andb50.36 mm. All the interfaces are in the saturation regimFor T1.50 ~the most representative case of anomalous sing! the interface presents high jumps at the edges ofcopper tracks. ForT0.40 some regions present high jumpbut, in general, the interface looks smoother. This case shanomalous scaling, but the degree of the anomaly is mweaker than the previous case. For SQ1.50, the interfacvery smooth and the interfacial fluctuations are bestscribed with the FV scaling, with only traces of anomaloscaling at very short times. The interfaces in these thcases are similar to those reported in fracture of woodEngo”y et al. in Fig. 1 of Ref. @12#. In particular, the inter-faces forT1.50 orT0.40 look like interfaces obtained fromtangential fracture in wood; both are characterized by h

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FIG. 8. W( l ,t) curves for a window sizel 5L/256 and differentdisorder configurations with progressively shorter persistence ingrowth direction.b50.36 mm andv52V50.08 mm/s in all casesThe curves are vertically shifted for clarity.

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jumps and anomalous scaling. Along the same lines, theterfaces for SQ1.50 are similar to those obtained from rafracture. These similarities indicate that the role of the wofibres is probably equivalent to the role of the copper train our system as the origin of the anomalous scaling.

The distribution functionsP(Dh) for the different inter-face morphologies are shown in Figs. 9~b! and 9~c!. ForT1.50, we have two different regimes. Up toDh.2 mm~dashed line in the plot!, we have the contribution of theheight differences corresponding to the structure wittracks. ForDh.2 mm ~solid line!, the contribution to thedistribution function arises from the edges of the tracks. Tlatter contribution is responsible for the anomalous scaliand scales withDh with an exponent 0.660.1. A similarresult is observed forT0.40. In this case, if we do not consider the contribution of the structure inside tracks~up to1 mm), we obtain a power law dependency with expon2.060.2. The fact that the anomaly is weaker forT0.40compared withT1.50 is illustrated in this case by the largvalue of the slope. If we compare the distribution functioof T1.50 and SQ1.50, we observe that, in the latter cathere is only a contribution from small height differences.this case,P(Dh) would scale with an exponent 3.6, far frothe values expected for anomalous scaling.

FIG. 9. ~a! Samples of interfaces for different disorder configrations and identical experimental parametersv52V50.08 mm/sand b50.36 mm. ~b! Distribution function P(Dh), with Dh[uh(xi 11)2h(xi)u, for local height differences, comparingT1.50and SQ1.50.~c! The same forT0.40. See the text for details of thpower law fits.

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C. Experiments at QÄ0

The previous results indicate that a more detailed studythe role of capillary forces at the disorder scale has tocarried out in order to understand the origin of the anomlous scaling. For this reason, we have performed experimat Q50, where interfacial growth is driven solely by capilary forces.

The experiments atQ50 were performed by initiallydriving the interface at high velocity up to the center of tcell. The injection is then stopped and the experiment starWe studied five gap spacings, in the range 0.16<b<0.75 mm, and characterized the interface velocity on cper tracks and fiber glass channels. For small gap spacup tob.0.36 mm, the morphology of the interface adoptsfingerlike configuration, which ultimately causes the inteface to pinch-off at long-time intervals. For this reason,have reduced the duration of the experiments to 1500 sall gap spacings.

Figure 10 shows a comparison of the interfaces at5500 s with different gap spacings. The horizontal line reresents the average position of the interface att50. For b50.16 mm, the points of the interface on the different tracmove almost independently, with weak coupling betweneighboring tracks: the oil on copper tracks always advanwhile the oil on fiber glass channels, except for some pticular locations, always recedes, driven by the different cillary forces associated with the different curvature of tmenisci on copper tracks and the fiber glass substrate. Asincrease the gap spacing, the coupling in the motion ofinterface tips on copper tracks with neighboring tracksprogressively more intense. Forb50.36 mm, the advanceon a specific copper track is coupled with its neighbors. Toil advances or recedes on a given track or channel deping on the particular disorder realization. Only the widecopper tracks or fiber glass channels have an indepenmotion. For b50.56 mm, the coupling reaches larger rgions of the cell, and forb50.75 the capillary forces are nlonger sufficient to allow a large deformation of the inteface.

It is also noticeable that the average position of the intface does not remain at rest. Indeed, it advances in the sdirection as the oil on the copper tracks. This is a conquence of the three-dimensional nature of the experimThe total mass of fluid in the cell is conserved, but not tarea measured on the images. However, the measured vity of the advance of the interface is very low. Forb50.36 mm, for example, the velocity is aboutV/10.

The first analysis that we have performed with the expements atQ50 is to study the growth of interfacial fluctuations with time. Figure 11~a! shows theW(L,t) curves forfour different gap spacings. We observe power laws withgrowth exponentb that decreases from 0.6560.04 for b50.16 mm to 0.5060.04 for b50.75 mm. Clearly, whenthe capillary forces are sufficiently strong to allow for bothe advance of the oil on the copper tracks and the recesof the oil on the fiber glass channels, we obtain a larger vaof the growth exponent than in all other cases. A simibehavior is observed forb* , which varies from 0.6560.04

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for b50.16 mm to 0.1560.05 forb50.75 mm.As regards the behavior of the power spectrum, show

Fig. 11~b!, the fact that the experiments do not reach a saration regime makes the analysis of the power spectrumcomplete. Qualitatively, we have observed a vertical shiftthe spectral curves as time increases for gap spacings ub50.56 mm. For b50.75 mm the vertical shift is nopresent, confirming the absence of anomalous scalingvalues of b>0.60 mm for any interface velocity. At longtimes, we have measured local roughness exponentsvary from a loc50.660.1 for b50.36 mm to a loc50.960.1 for b50.75 mm. Forb50.16 mm, no clear scalingcan be identified.

The second analysis that we have performed has bedetailed study of the motion of the interface on copper traor fiber glass channels separately. In Fig. 12 we repre^h&T , the mean height for those points of the interface t

FIG. 10. Example of interfaces atQ50 with different gap spac-ings and disorderT1.50. All the interfaces correspond tot5500 s. The horizontal dotted line indicates the position ofinterface att50 s.

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are advancing on copper tracks, as a function of time.gap spacings in the range 0.16<b<0.36 mm, ^h&T is apower law of time with an exponent 0.5060.03. For largergap spacings, the exponent is progressively smaller andpower law regime is only identifiable at short times. A simlar analysis can be carried out for the fiber glass channelsthis case, we obtain the same behavior in the range 0<b<0.36 mm, with an exponent 0.5060.03. For b>0.56 mm, no clear scaling can be identified.

Finally, we have also studied the motion on individutracks in order to find out the role of track widthD. Theresults are shown in Fig. 13. A track isuncoupledwhen themotion of the interface tip on the track grows ash;t0.5

independently of its neighbors’ configuration. Conversethe interface motion over acoupled track depends on theparticular location of this track in the disorder realizatioand the behaviorh;t0.5 is not observable or cannot be chaacterized at all times. The behavior of coupled tracks islustrated in Figs. 14~a! and 14~b!, which show two examplesfor D51.5 mm andD53.0 mm, respectively. In both caseb50.36 mm. These figures show that two tracks of the sawidth present different growth depending on their neighbo

FIG. 11. Experimental results forQ50 at four different gapspacings and disorderT1.50. ~a! Interfacial width W(L,t). Thestraight line with slope 0.65 corresponds to a fit of the experimewith b50.16 mm and 0.36 mm. The straight line with slope 0.50a fit to the experiment withb50.75 mm. Each curve is an averagover two disorder realizations, with two runs per disorder realition. ~b! Power spectraS(k,t) at t51000 s. The curves have beeshifted in the vertical direction for clarity. The vertical line gives thvalue ofk associated with the spatial scale of the disorder.

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configuration. While the uncoupled track grows ash;t 0.5,the coupled track follows the same behavior only at shtimes and afterwards reaches a stationary height. A compson of Figs. 14~a! and 14~b! also shows that the wider thtrack, the weaker the coupling. The widest tracks in a givdisorder realization and forb<0.36 mm remain uncoupledduring all the experiment. In addition, for small gap spaings,b<0.16 mm, all tracks are uncoupled.

From these studies, we conclude that it is possible toscribe the interface motion on a single copper trackv1

i , orfiber glass channelv2

i by the simple law

v6i ~ t !56A6

i ~b,D6i !t21/2 for t>1 s, ~7!

whereA6i (b,D6

i ) is a positive prefactor that depends on tgap spacingb and the individual copper track or fiber glachannel widthDi . In general,Ai increases withDi and de-creases withb. Averaging out the prefactorA6

i over differenttrack or channel widths, we obtain a characteristic maximvelocity vM6 for a given gap spacing. Equation~7! nowbecomes

FIG. 12. Log-log plot of the average height as a function of timfor the interface points advancing on copper tracks, for differentspacings and disorderT1.50. Q50 in all cases. The curves arvertically shifted for clarity.

FIG. 13. Log-log plot of the average height as a function of timfor the interface points advancing on uncoupled copper tracksspecific width D. The experimental parameters areQ50, b50.36 mm, and disorderT1.50. The curves are vertically shiftefor clarity.

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v6~ t !56vM6~b!t21/2 for t>1 s. ~8!

Our experimental observations show that Eq.~8! is valid forgap spacings in the range 0.16<b<0.36 mm. For b>0.56 mm, Eq.~8! only holds for copper tracks and at shotimes. The temporal range in which this behavior is obsergoes to zero as the capillary forces become weaker.

In general,vM1.vM2

due to the different gap spacing icopper tracks and in fiber glass channels. This differenhowever, is progressively smaller asb increases. For gapspacings aroundb50.36 mm, it is reasonable to considvM1

.vM2[vM . The timet51 s, determined from system

atic experimental observations, is a typical time in which tvelocity jumps from the average interface velocity out of tdisorder tovM inside the disorder.

D. From QÄ0 to low injection rates

Equation~8! gives the velocity on copper tracks or fibeglass channels whenQ50 ~no injection!. In the presence ofinjection, when the interface is driven at an average interfvelocity vÞ0, the behavior observed experimentallymodified in the form

v6~ t !5v6~vM2v !t21/2, ~9!

wherevM5vM(b), andv1 and v2 are the velocities averaged over copper tracks and fiber glass channels, restively. Thus, when the interface makes contact with the dorder, the velocity on copper tracks jumps to a maximumvMin a characteristic time of 1 s and then relaxes ast21/2 to thenominal velocity. In fiber glass channels, the velocity is zeor even negative at short times, and as time goes on,velocity increases until it reaches the nominal velocitylong times. Figure 1 illustrates this behavior. At short timet.10 s, all the oil on copper tracks advances, while thein the widest fiber glass channel recedes. Att.50 s the oil infiber glass channels has stopped receding. At later timesvelocity at different points of the interface tends asymptocally to the nominal velocity, reached at saturation.

p

a

FIG. 14. Log-log plot of the average height as a function of timfor the interface points advancing on a copper track of a specwidth D. ~a! D51.5 mm. ~b! D53.0 mm. In the two cases,Q50 andb50.36 mm.

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At low injection rates, the external velocityv can betuned appropriately to have an initial regime in which theadvances on copper tracks and recedes on fiber glass cnels, followed by a regime in which the oil advances atpoints of the interface. This leads to two scaling regimesW(L,t), at short times with a growth exponentb.0.65,which corresponds to a regime dominated by strong capilforces, and at longer times with a growth exponentb.0.50, which corresponds to a regime where capillary aviscous forces are better balanced. We have illustratedbehavior by performing experiments at lowv. In particular,we have studied velocitiesV/2 andV/10 using a gap spacinb50.36 mm and disorderT1.50. The results are presentedFig. 15. The difficulty of these experiments is that the intface pinches-off on the longest fingers before it reaches sration, so we can only find out the growth regime. ForV/2,we observe the two power law regimes described above.V/10, only the first regime can be observed before the inface breaks. These results are qualitatively in agreementthe analysis of Herna´ndez-Machadoet al. @27#, who pre-dicted two scaling regimes,b155/6 at short times~surfacetension in the plane dominant! and b251/2 at long times~viscous pressure dominant!. However, the presence oanomalous scaling and the complicated role of capillforces in our experiments makes a direct comparison wRef. @27# difficult.

Another result than can be obtained from Eq.~9! is aprediction for the saturation timet3 as a function of theaverage interface velocityv. We can estimatet3 for a givenv and gap spacingb by considering that saturation takeplace when all tracks reach the nominal velocityv. This isnot strictly exact due to the coupling between neighbortracks, but it gives a good estimate of the dependencyt3(v),as shown in Fig. 16. For a given experiment with velocv,vM the saturation takes place when the functionvMt21/2

crosses the nominal velocity. Thus, as shown in the insethe figure,t3 is expected to scale ast3;v22. This result isconfirmed by our experiments with disorderT1.50 andb50.36 mm~open triangles in the inset!.

FIG. 15. Log-log plot ofW(L,t) for experiments at low velocitywith disorderT1.50. The end of the experiments is the momenwhich the interfaces pinch-off. The gap spacing isb50.36 mm inboth cases, andV50.04 mm/s. Each curve is an average over tdisorder configurations, with one run per disorder configuration

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E. Experiments with a regular modulation of the gap spacing

In order to analyze the importance of the coupling btween neighboring tracks, we have performed experimewith a regular modulation of the gap spacing, consistingcopper tracks 6 mm wide separated by fiber glass channemm wide. Interface velocity and gap spacing are set tov50.08 mm/s andb50.36 mm, respectively. The regulamodulation and the temporal evolution of the resulting intfaces are presented in Fig. 17~a!. Due to the symmetry of theregular pattern there is no growing correlation length alothe direction of the interface. For this reason, a dynamscaling of the interfacial fluctuations cannot even be definThe growth of the interfacial width is due to the independeequivalent growth of interface fingers on copper tracwhose motion follow thet21/2 behavior@Eq. ~9!#. The sys-tem saturates when the fingers reach the nominal veloThis is shown in Fig. 17~b!, which shows the evolution oW( l ,t) for the global (l 5L) and local (l 5L/256) scales.The figure reveals that both scales grow in the same wThe absence of dynamic scaling is also confirmed bypower spectrum, shown in Fig. 17~c!. We obtain a nearly flatspectrum, with a large peak at the value ofk that correspondsto the spatial periodicity of the underlying regular patteWe also observe a vertical shift of the spectra at differtimes, caused by the progressive elongation of the oil fingon copper tracks during growth.

These experiments with regular modulation of the gspacing illustrate the need for a growing correlation lenalong the interface to reach a dynamic scaling regime. Isimilar way to these experiments, the lateral growth ofcorrelation length can also be inhibited even if the modution of the gap spacing is random, provided that the expmental parameters are such that the interface motion ontracks is not coupled and that Eq.~9! holds for each track.Thus, for very small gap spacings (b<0.16 mm) andv>V, we observe a growth of the interfacial width with e

t

FIG. 16. Prediction of the dependency oft3 with v. Main plot:for a given average velocityv, the system saturates when the aveage velocity on copper tracks decays tov. Inset: a log-log plot ofthe t3 obtained by the previous method~filled dots! as a function ofv gives a straight line with slope22. The triangles are the expermental results forT1.50 andb50.36 mm.

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ponent.0.5 for both global and local scales. Similarly,the experiments atQ50 with b50.16 mm the same exponent.0.65 is measured for both global and local scales. Tpower spectrum of this last case also demonstrates theof dynamic scaling, as can be seen in the top plot of F11~b!. When the modulation of the gap spacing is regular,transition to saturation is sharp due to the fact that all trahave the same width, and thus all the oil fingers reachnominal velocity at the same time. For a random patterntransition to saturation is smooth, and does not take pstrictly until the widest copper tracks~those that have thehighest value ofA1

i ) reach the nominal velocity. The plot fob50.16 mm of Fig. 7 illustrates this behavior.

V. DISCUSSION AND CONCLUSIONS

Our experimental results show that two ingredientsnecessary to observe anomalous scaling in a disordered HShaw cell: ~i! strong destabilizing capillary forces, sufficiently persistent in the direction of growth and~ii ! couplingin the interface motion between neighboring tracks. Th

FIG. 17. Experiments with a regular modulation of the gap sping, using tracks 6 mm wide. The experimental parameters av52V50.08 mm/s andb50.36 mm.~a! Temporal evolution of theinterfaces.~b! W( l ,t) plots for global (l 5L) and local (l 5L/256)scales.~c! Power spectrum att5300 s. The vertical line indicatethe value ofk corresponding to the spatial periodicity of the regupattern.

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two ingredients give rise to large slopes at the track edtogether with a growing correlation length along the inteface. This scenario is represented in the phase diagramFig. 18, which shows the different regions of scaling eplored experimentally. Each point of the diagram is an avage of six experiments~three disorder configurations and twruns per disorder configuration!. The dark gray area represents the region where anomalous scaling has been idfied. We have indicated the value of 2u/z, which is the valueneeded to vertically collapse the spectral curves and, thfore, indicates the degree of anomaly. For 0.16,b,0.60 mm, and for driving velocities belowvM(b), the cap-illary forces are strong enough to allow an initial acceleratand subsequent deceleration of the oil fingers on coptracks, but are sufficiently weak to allow coupling betweneighboring tracks. Forv.vM , the viscous forces becomdominant, and the anomalous scaling is unobservable.b>0.60 mm, the capillary forces are not sufficiently stroand we have ordinary scaling for any velocity. Finally, fb<0.16 mm~light gray area in Fig. 18!, the capillary forcesare dominant, and the oil fingers on copper tracks and figlass channels have independent motion. In this situatiodynamic scaling scenario is not adequate because theregrowing correlation length along the interface.W( l ,t) growsas a power law of time, but this is a consequence ofindependent behavior for each track given by Eq.~9!. Al-though it is not possible to determine the scaling exponeit is still possible to perform a vertical collapse of the spetral curves because only thedifferencebetweena and a loc~i.e., u) is needed. This provides the values of 2u/z for b50.16 mm.

In conclusion, experiments are reported of anomalousnetic roughening in Hele-Shaw flows with quenched disder. The displacement of an oil-air interface for differestrengths of the capillary forces, flow velocity, and disordconfigurations has been studied. It has been observedscaling of the interfacial fluctuations follows the intrinsanomalous scaling ansatz in conditions of sufficiently strocapillary forces, low velocities, and a persistent disorderthe growth direction. A random distribution of tracks hbeen used as representative of a persistent disorder. A d

-

FIG. 18. Phase diagram for the anomalous scaling. The symindicate the parameters explored experimentally. The numbersto the symbols in the anomalous scaling region give the value2u/z. The solid line represents the curvev5vM(b).

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ent scaling for the global and local interfacial fluctuatiohas been measured, with exponentsb.0.50, b* .0.25, a.1.0, a loc.0.5, andz.2. The presence of anomalous scing has been confirmed by the existence of multiscaliWhen capillary forces become dominant or when a regutrack pattern is used, however, there is no growing corrtion length along the interface, and therefore no dynamscaling. A detailed study of the interface dynamics atscale of the disorder shows that the anomaly is a conquence of the different interface velocities on copper traand fiber glass channels plus the coupled motion betwneighboring tracks. The resulting dynamics gives interfaprofiles characterized by large height differencesDh at the

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edges of the copper tracks. TheseDh follow an anomalousLevy distribution characteristic of systems which displanomalous scaling.

ACKNOWLEDGMENTS

We are grateful to M. A. Rodrı´guez, J. Ramasco, and RCuerno for fruitful discussions. The research has receifinancial support from the Direccioń General de Investigacion ~MCT, Spain!, project BFM2000-0628-C03-01. J.O. aknowledges the Generalitat de Catalunya for additionalnancial support.

at

d-

@1# A.-L. Barabasi and H.E. Stanley,Fractal Concepts in SurfaceGrowth ~Cambridge University Press, Cambridge, 1995!.

@2# F. Family and T. Vicsek, J. Phys. A18, L75 ~1985!.@3# M. Schroeder, M. Siegert, D.E. Wolf, J.D. Shore, and M. Pl

chke, Europhys. Lett.24, 563 ~1993!.@4# J. Krug, Phys. Rev. Lett.72, 2907~1994!.@5# S. Das Sarma, C.J. Lanczycki, R. Kotlyar, and S.V. Ghais

Phys. Rev. E53, 359 ~1996!.@6# C. Dasgupta, S. Das Sarma, and J.M. Kim, Phys. Rev. E54,

R4552~1996!.@7# J.M. Lopez and M.A. Rodrı´guez, Phys. Rev. E54, R2189

~1996!.@8# M. Castro, R. Cuerno, A. Sańchez, and F. Domıńguez-Adame,

Phys. Rev. E57, R2491~1998!.@9# J.M. Lopez and J. Schmittbuhl, Phys. Rev. E57, 6405~1998!.

@10# S. Morel, J. Schmittbuhl, J.M. Lo´pez, and G. Valentin, PhysRev. E58, 6999~1998!.

@11# S. Morel, J. Schmittbuhl, E. Bouchaud, and G. Valentin, PhRev. Lett.85, 1678~2000!.

@12# T. Engo”y, K.J. Malo”y, A. Hansen, and S. Roux, Phys. ReLett. 73, 834 ~1994!.

@13# J.H. Jeffries, J.-K. Zuo, and M.M. Craig, Phys. Rev. Lett.76,4931 ~1996!.

@14# H.-N. Yang, G.-C. Wang, and T.-M. Lu, Phys. Rev. Lett.73,2348 ~1994!.

-

s,

.

@15# S. Huo and W. Schwarzacher, Phys. Rev. Lett.86, 256~2001!.@16# J.M. Lopez, M.A. Rodrı´guez, and R. Cuerno, Phys. Rev. E56,

3993 ~1997!.@17# J.M. Lopez, M.A. Rodrı´guez, and R. Cuerno, Physica A246,

329 ~1997!.@18# J. Soriano, J. Ortıń, and A. Hernańdez-Machado, Phys. Rev. E

66, 031603~2002!.@19# J. Soriano, J.J. Ramasco, M.A. Rodrı´guez, A. Hernańdez-

Machado, and J. Ortıń, Phys. Rev. Lett.89, 026102~2002!.@20# J.J. Ramasco, J.M. Lo´pez, and M.A. Rodrı´guez, Phys. Rev.

Lett. 84, 2199~2000!.@21# Real-life motion pictures of the experiments are available

http://debian01.ecm.ub.es@22# J.M. Lopez, Phys. Rev. Lett.83, 4594~1999!.@23# G. Tripathy and W. van Saarloos, Phys. Rev. Lett.85, 3556

~2000!.@24# I. Simonsen, A. Hansen, and O.M. Nes, Phys. Rev. E58, 2779

~1998!.@25# J. Schmittbuhl, J.-P. Vilotte, and S. Roux, Phys. Rev. E51, 131

~1995!.@26# J. Asikainen, S. Majaniemi, M. Dube´, and T. Ala-Nissila, Phys.

Rev. E65, 052104~2002!.@27# A. Hernandez-Machado, J. Soriano, A.M. Lacasta, M.A. Ro

rıguez, L. Ramı´rez-Piscina, and J. Ortıń, Europhys. Lett.55,194 ~2001!.

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anomalous roughening in experiments of interfaces in hele ...fv scaling is limited, and a new...

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