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SOME NEW FEATURES OF INTERFACE ROUGHENING DYNAMICS IN PAPER WETTING, BURNING, AND RUPTURlNG
EXPERIMENTS ALEXANDER S. BALANKIN Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Ed. 5, 3er. Piso, Instituto Politécnico Nacional, México, D.F., México,07738 E-mail:
[email protected]. Tel.: (525) 729-6000+54589
DANIEL MORALESMATAMOROS InstitutoMexicanode Petróleo,México,D.F. E-mail:
[email protected] The dynamics of interfaces growing in paper wetting, fracturing and buming processesis studied with use the same kinds of papers for different experiments. We were able to study five different types of kinetic roughening. Some new observations conceming the spatial-temporal dynamics of rough interfaces are reported Specifically, we found that the types of kinetic roughening, as well as the scaling exponents, are dependent on the paper structure and the mechanism of interface formation. Moreover, we have observed that the local roughness exponent of moving wet front logarithmically increases from 0.5 at initial stage up to its stationary value, achieved before front saturation. We also found that the stress-strain fracture behaviour of some kinds of paper exhibits a statistical self-affine invariance with scaling exponent, which is equal to the local roughness exponent of rupture line. The same exponent govems the ch~nges in the stress-strain curve as the strain rate increases. This gives rise to the facture-energy -time-to-fracture uncertainty relation, similar to the time-energy uncertainty relation in quantum mechanics. The physical implications of these points are discussed.
Introduction Kinetic rougheningof interfacesin nonequilibrium conditions has attached much attention in recent years [1-8]. This is mainly due to the many important applications of the theory of interface dynamics including fluid flow in porous media, self-aftine crack mechanics, forest tire, and molecular-beamepitaxy among others. From a phenomenologicalpoint of view, there are many commonaspectsbetweeninterfacedynamics in these systems,despite their different natureand huge scaledifferences.Different growth processeshave very often beenshownto exhibit scaling properties that allow one to divide the growth models into universality classes characterised by different sets of critical exponents. In this sense, experiments with papercan be used to understandsomeessentialfeatures of kinetic roughening in systemsof different nature. Although fractal growth models have provided a useful insight into a vast array of problems conceming interface rougheningdynamics, many important questions are still open. In fact, it is particularly important to reach more complete and predictive theoretical understandingof interface
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growth in presenceof correlated spatial-temporalnoise. Clearly, such a problem can only be investigatedby a comprehensiveeffort that involves computersimulations, analyticaltools and suitablydesignedexperiments. The purpose of this work is to understandthe nature of the quenched and temporal noises and their effect on the scalingdynamicsof interfaces of different nature, growing in a given disordered medium. The main questionsto be answeredare: (1) What propertiesofthe systemdetermine the type of scaling behaviourand the values of scaling exponents?(2) Do the scaling exponents take only universal values or they change continuously with the arder parameterof system?(3) What is the reason for difference in the values of scaling exponents obtained in similar experiments in different works? (4) How doesthe spatial roughnessaffect the temporal behaviour of growing interface?To gain an insight to these problems we perform a detailed study of the spatial-temporalbehaviourof interfaces growing in papers with different structures by three different mechanisms:imbibition, combustion,and fracturing.
2 Experimentaldetails Paper is a porous composite material with anisotropic structure associated with an asymmetric orientation distribution of fibres. Despite the stochastic nature of the fibre distribution, the fibre and the pare structures of papers are not random but possess long-range correlations characterised by powerlaw behaviour of the space-density autocorrelation function [5-8]. Different kinds of paper have different structures, giving rise to different pare distributions, characterised by different scaling exponents of the autocorrelation function [8]. This allows us to study the effects of correlated quenched noise and medium anisotropy on spatial-temporal dynamics of interfaces formed by different processes, characterised by different types oftemporal (thermal) noise. In this work, we study kinetic roughening dynamics of interfaces growing in paper wetting, buming, and fracturing experiments. These experiments were performed on rectangular sheets of three different kinds of commercial paper: "Secante "(thickness = O.34:tO.O7mm, areal density = 210:t18 g/mm2), "Filtro" (thickness = O.25:tO.O4mm, areal density = 11O:tlO g/mm2), and "Toilet" (thickness = O.11:tO.O6mm, areal density = 36.8:tO.3 gimm2). These papers have well different structures characterised by different scaling exponents of space-density autocorrelation function (cI> = O.12:tO.O5,cI>= O.27:tO.O8,and cI>= O.43:tO.13 for "Secante", "Filtro",
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and "Toilet" papers,respectively).Detailed multifractal analysisof paper structureswill be published elsewhere. The papers have a well-defined structural anisotropy associatedwith the preferred orientation of fibres in the so-called machine direction. Accordingly, we studied interfacesformed along and acrossthe machine direction of a paper. The length of paper sheets(in the direction of the interface propagation)used in all experimentswas 250 mm, whereasthe sheet width was varied from Wo = 10 mm to WM = 100 mm, with the following relation W = A.Wofor scaling factor A.= 1, 1.5,2,3,4,5,8 and 10. At least 30 experimentsof eachtype were carried out for each sheet width. Statistical data were analysed with the help of @RISK 4.05 software [9] To study the spatial-temporaldynamics of the wet interface, a vertical sheetof paper is wetted with black Chinese-inksolution. The evolution of the interface between the wet (black) and the dry (white) regions is recorded with a digital camcorderat 24 frames/sec(see [7]). We allow the interface to rige until it stops and no change in either the height or the shapeof the interfaceis observed. To obtain the tlame front we take a sheetof paperand ignite it at one end by a linear electric heating wire. The sheets were bumed in the horizontal position and the combustionrate was rather slow (in the range from 2 mm/sec to 10 mm/sec for different kinds of paper).Unfortunately, we were able to satisfactory record the tire propagation only in experiments with "Toilet" paper, since the combustion of other papers used in this work is accompaniedby high heat radiationand smoke, owing to theirs larger thickness and high density. Accordingly, we have studied the temporal scaling of tire front only in "Toi/et" paper,whereasin other papers we have analysed only the spatial scaling properties of the "postmortem" buming and tlame fronts (see figure 1 a), formed when the tire has beenquenched,after it reachesa middle line markedo the specimen. Mechanical tests were carried out on a 4505 INSTRON testing machine under carefully controlled temperatureT = 20 :t 3°C and humidity 38 :t 6%. The deformationrate was controlled by grip displacementspeeds of 0.5, 1.0, 5.0, and 10 mm/min. The paper failure processwas recorded with a digital camcorderat 60 frames/sec. The failure of these papers occur as the culmination of progressive damage, involving complex interactions betweenmultiple growing microcracks (see figure 2 a). As a result, the descendingparts of stress-strain curves display a stochasticbehaviour (seefigure 2 b). Note the difference
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betweentheseexperimentsand the studiesin references[4-6], in which the single-notchedsheetswere tested. We note that "Secante" and "Filtro" papers have an elasto-plastic behaviour, whereasthe "Toilet" paperdisplaysthe linear elastic behaviour up to the tensile stressO"M (seefigure 2 b). 6
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Figure l. (a) Black-and-white image ofwetting, buming, and flame fronts and rupture line in "Secante" papel and (b) the corresponding graphsb(x).
Figure 2. (a) Rupture line forrnation by cumulative damageand (b) stress-straincurve for fue "Toilet" paper
The interface images obtained from video frames have relatively low resolution, limited by the maximum frame size 600x600pixels. Therefore, to fine study of the roughnessof "post-mortem" burning, flame, and wet fronts and rupture liDes,the testedspecimenswere scannedwith HP-61000
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Scan-Jetin black-and-white BMP-format with 600x600 dpi resolution. The profiles of studied interface from video frames, as well as from scanned images,were plotted using the Scion Image software [10] as single-valued functions h(x, t), in the XLS-format (see figure 1 b). Notice that in paper fracturing experiments, h(x) is a continuous function of x only after a paper sheetis divided in two parts. A rough interface is characterisedby the height fluctuations aroundthe mean position of interface h*(t) = (h(x, t»w, where (...)w denotes an averageayer x. Therefore,a basic quantity to look at is the global interface width w(t, W) = maxxew[h(x,t)] -minxew[h(x,t)]. The spatial correlations of rough interface are characterized by the height-height correlation function G(A,t) = ([h(x+A,t)-h(x,t)fIA' where ("')A denotes an average ayer x in widows of size A < W. Another useful characteristic function is the structure factor or power-spectrumof interface, defined as S(q,t) = ({,(q,t){,(-q,t»), where {,(q,t) is the Fourier transform of the interface height in a systemof sizeW. Numerous observationshave shown that at initial staget < ts, where ts is the saturation time, the mean plane of interface moves as h*(t) oc tO, where 5 is the diffusion exponent [1,2]. At the same time, the global interface width scales as w(t « ts,W) oc tP, where /3 is the growth exponent,while for saturatedinterfacesw(t ~ ts,W) ocWX, where a is the global roughnessexponent[1-8]. Accordingly, G1/2(A,ify » 1, or s(y) ocy8, if y «
1;
(1)
here g = 2as + 1, as is the spectral roughnessexponent, and a is the global roughnessexponent. Notice that if a :;t:as the power spectrum scaleswith the systemwidth as S(q,W) ocW29,wheree = a -as. In the absenceof any characteristiclength in the system,the interface growth are expected to show a power-law behaviour of the correlation function in spaceand time, and the Family-Vicsek dynamic scaling ansatz [1,2], w(t, W) = taJzf(W/~(t», ought to boldo The scaling function f(y) behavesas foc ya, ify« 1 (~» W), or foc constant, ify» 1. In sucha case, the local interface width also scales as Aw(A,t) oc G1/2(A,t) = tPg(LV~),where g(y) behavesas g occonstant, ify » 1, and g oc yl;, ify « 1, and the local roughnessexponentl; is equalto the global roughness exponent a, becauseof there is no characteristiclength scale besidesthe systemsize. Therefore,fue shorttime interfacedynamic exponentis equalto z = a/j3 [1,2]. The Family-Vicsek scaling ofthe structurefactor readsas [2]: S(q,t) ocq-ss(qf/Z), where g = 2a + 1 and s(y) occonstant, when y» 1, and s(y) ocyS, when y « 1, i.e. l; = a = as [13]. Generally, however, l;;~ a [12]. Kinetic rougheningwith l;;< a is called an anomalous roughening [1,2]. Different scaling forms can be treated as subclassesofthe generic scaling ansatz(1). Authors of[13] have identified four different dynamic scaling regimes, associatedwith the bounds of scaling exponents(see table 1). In this work, we have observed all these regimes,as well as a new unconventionalanomalousroughening(seetable 1). Specifically:
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1. Wet fronts in "Secante" and "Filtro " paperspossessa statisticalselfaffine invariance with l; = a = as = O.63:tO.O2 and l; = a = as = O.75:tO.O3, respectively. Rupture liDes along the machine direction in these papers also exhibit a statistical self-affine invariance.However, the crack roughness exponents differ from the wet front roughness exponents.Namely, l; = a = as = O.55:tO.O4 for "Secante" and l; = a = as = O.42:tO.O4 for "Filtro" papers. 2. The rupture liDes acrossthe machine direction in these papers exhibit an intrinsically anomalousroughnesswith the local exponentfound to be equal to the crack roughness exponent for rupture liDes in the perpendiculardirection and the global roughnessexponentequal to a = O.71:tO.O5for "Secante" and a = O.67:tO.O7for "Filtro" papers, respective1y.Rupture liDes acrossand along the machine direction in the "Toilet" paper are characterised by the same local roughness exponent l; = as = O.75:tO.O5,but different global exponents: a = 1.45:tO.15along and a = 1.85:tO.35acrossthe papermachinedirection. 3. The super-rougheningregime was observed in wetting and buming experimentswith the "Toilet" paper.We find that the wet front across the machine direction in this paper is characterisedby l; = O.92:tO.O4 and a = as = 1.5:tO.2,whereasthe buming front acrossand along the machine direction in this paperis characterisedby l; = 1.0:tO.1and a = as = 1.9:tO.9.4. The wet front along the machine direction in the "Toilet" paper displays a anomalous roughening with as = 1.5:tO..2> 1, l; = O.92:tO.O6, and a = 1.1:tO.2< as. 5. Unconventionalregime of anomalouskinetic roughening(seetable 1) of flame fronts was observed in slow combustion of "Secante" (see [14]) and "Filtro" paper experiments, whereas the buming front roughnessin thesepaperspossessesstatistical self-affine invariance. Unconventional anomalous roughening is characterised by as < 1. Furthermore, we find that the local roughnessexponentof flame front, as well as the crossoverlength, ~c, separatingthe local and the global scaling intervals, are scaledwith the papersheetwidth (seealgo [14]) as ~ oc WV and ~c oc WO>,
(2)
where the exponentv < o was found to be equalto the exponent28, which govems the structure factor scaling (1), while the exponentro was found to
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be equal to the global roughnessexponent,that is O)= a > as. For flame fronts in the "Secante" paper we find as = 0.5(S -1) = 0.53::1:0.02, a = 0.83::1:0.03 = 0), and 2e = v = -0.40::1:0.05< O [14]. Flame fronts in the "Filtro " paperare characterisedby as = 0.5::1:0.1, a = 0.72::1:0.06 = 0),and 2e = v = -0.37::1:0.07.This type of roughnessbehaviourmay be attributed to the effect of turbulent airflow accompanyingthe combustion of the paper. It is intriguing that in both papersthe global roughnessexponentof the flame front is found to be equal to the local roughnessexponent of the correspondingburning front. Table 1. Scaling exponents for different kinetic roughening regimes
Roughening
regime
as
l;
a
S
9
ansatz
O
O
O
roughening
> 1
~1
as
2a + 1
O
O
O
New class of anomalous
> 1
1
:t: as
2as + 1
a -as
?
O
as
2as + 1
