Experimental link of coarsening rate and volume distribution in dry foam

Transcription

1 August 2012 EPL, 99 ( doi:.1209/ /99/ Eperimental link of coarsening rate and olume distribution in dry foam J. Lambert 1, F. Graner 2,3, R. Delannay 1 and I. Cantat 1(a 1 IPR UMR 6251, Uniersité Rennes I - F Rennes Cede, France, EU 2 Polarité, Diision et Morphogenèse, Laboratoire de Génétique et Biologie du Déeloppement, Institut Curie 26 rue d Ulm, F Paris Cede 05, France, EU 3 Matière et Systèmes Complees, Uniersité Paris Diderot, CNRS UMR 7057 rue Alice Domon et Léonie Duquet, F Paris Cede 13, France, EU receied May 2012; accepted in final form 17 July 2012 published online 23 August 2012 PACS PACS Rr Aerosols and foams Iz Emulsions and foams Abstract In dry foam, we inestigate the link between coarsening rate and bubble distribution. Reisiting Mullins predictions (Mullins W. W., J. Appl. Phys., 59 ( , we predict, without any phenomenological assumption, that the olume distribution behaes as (2 D/D (where D is the dimension of space in the limit of small olumes. This is in agreement with recent eperimental results. In the self-similar growth regime, we alidate eperimentally a relation between the bubbles growth rates and their size distribution. In 2D, we discuss the relation between the areas distribution and the correlation between edge numbers and bubble area. Copyright c EPLA, 2012 Introduction. Coarsening in cellular systems like liquid foams and grains in crystals has been studied for long both eperimentally and theoretically. In liquid foams, for eample, the coarsening is due to the gas flow between neighboring bubbles induced by their pressure difference [1,2]. The equations goerning the dynamics of those systems has been inestigated by Mullins who focused on the late-stage dynamics assuming the eistence of a Self-Similar Growth Regime (SSGR of the olume distribution at different times. This assumption led Mullins to a now classical prediction on the aerage bubble size time eolution [3]: d dt = C α, (1 where is the aerage bubble olume in 3D and the aerage bubble area in 2D, C is a constant that depends on the physico-chemistry of the foam and α is a constant that depends on the dimension of the space D: α = (D 2/D, so α 3D = 1/3 and α 2D = 0. A direct consequence of eq. (1 is that R(t 2 R(0 2 t, with R 1/D the aerage bubble radius. Mullins also established that the growth rate of indiidual bubbles can be deduced from the bubble size distribution (the eact relation is gien below eq. (, which had (a isabelle.cantat@uni-rennes1.fr been successfully tested by Streitenberger et al. using 3D numerical simulations [4]. In this paper, we show a first eperimental proof of this relation for the case of lowliquid fraction 3D foams. In addition, we establish new consequences of the general theory deeloped by Mullins. We predict a power law behaior at small olumes for the 3D bubble size distribution and test this prediction eperimentally. In the last part of the paper, we focus on 2D foam. A linear relation between the number of edges aeraged oer the set of bubbles sharing the same area and the bubble area is often assumed in the literature, which implies an eponential decay of the area distribution in SSGR [5]. It has been recently proposed that this aerage number of edges aries as the bubble area to the power 1/2 [6], thus leading to another area distribution [4]. We compare both area distributions to eperimental data, and get a better agreement with the last one. Growth rate. In a dry foam, the bubbles shapes strongly depart from the spherical shape. In the limit of anishing liquid fraction, the bubbles are polyhedra with slightly cured faces and well-defined edges. The curature of the faces and the pressure in each bubble are related to each other through the Laplace law. The pressure difference between two neighboring bubbles induces a slow gas flu goerned by Fick s law, and the olume eolution p1

2 J. Lambert et al. is gien by [1,2] d dt = D eff Kds, (2 where K is the bubble mean curature; and ds are, respectiely, the bubble olume and an element of its surface in 3D, and the bubble area and an element of its perimeter in 2D; D eff is an effectie diffusion coefficient depending on the physico-chemistry of the foam and on the films thickness. This last parameter is constant in time for dry foams. The growth rate of a bubble is defined as G = α d/dt and can be epressed as K G = D eff ds. (3 α As K scales as 1/D, ds as (D 1/D and α as (D 2/D, it follows that the integral does not depend on, meaning that G is not modified by a homothetic transformation of the bubble and only depends on its shape [7]. In 2D, the on Neumann law states that G only depends on the number of neighbors F [8]: G 2D = D 2D eff (F 6. (4 The etension of this epression in 3D inoles more comple geometrical quantities, and not only the faces number [9]. Howeer, statistically, G increases with the number of faces (see [ 13] and the reiew [14]. As the bubble olume, compared to the aerage bubble olume in the foam, and the number of faces are strongly correlated, G also increases with statistically. Model: small olume limit of the olume distribution. The distribution of G determines the time eolution of the bubble size distribution as well as the eolution of the bubble number. We consider N(t bubbles coarsening in a closed olume V, and P (, t the olume distribution at time t, normalized to unity. Defining G as the growth rate aeraged oer the set of bubbles sharing the same olume, we get = lim G ( α P (, t. (5 0 This relation, established by Mullins (eq. (6 in ref. [3], see A in the appendi requires a few comments. First, this relation only makes sense if the foam sample is large enough so that its arious properties can hae a continuous representation. More precisely, the time scale of the obseration must be large enough to consider the bubble number as a continuous function of time, and the typical length scale of the description must be large enough for the limit of anishing olume to be defined. Second, een in a ery large foam, the smallest bubble olume may be limited by the finite olume of the ertices in which the bubbles eentually dissole, so we only discuss here the limit of ery dry foams. Finally, let us underline the fact that eq. (5 does not assume that the SSGR is reached. As the bubble size decreases, its number of faces decreases and, on aerage, its internal pressure increases to a much higher pressure than its neighbors. This eplains why G reaches a finite and strictly negatie alue in the limit of small olumes. So we define G 0 as the limit of G at anishing olumes and obtain = G 0 lim α P (, t. (6 0 For a foam in which the coarsening process is already established (but possibly still in its transient stage, the bubble number regularly decreases (/dt < 0. This imposes the asymptotic behaior of the olume distribution at small scale. In 3D, α = 1/3 and /dt 0 only if P (, t dierges in 0 as 1/3 : P (, t A 1/3 (7 with A = G0 1. This consequence of Mullins theory is in good agreement with our eperiments as will be shown hereafter. The theoretical prediction of G 0 would require the knowledge of the foam dynamics at small olumes, which is not yet described. Howeer, it may be assumed that tetrahedra dominate the small size bubble shape distribution. In this case, the high internal pressure enforces equality of the 4 faces curatures and thus a symmetrical shape for the tetrahedron. This symmetry is more and more precisely achieed as the olume decreases and the pressure increases. Under this assumption, the alue of G 0 can be calculated from eq. (3 if the prefactor D eff is known: G 0 = 3.97 D eff [14]. The same asymptotic bubble size distribution is found for bubbly liquids, goerned by Oswald ripening, for the interface limited case [15]. This comes from the fact that, also in that case, G reaches a finite limit at small olume. Howeer, the full distribution is significatiely different from what is found for dry foams. In 2D, α = 0, leading to the relation lim P (, t = 0 NG 0 dt (8 which simply states that P (, t reaches a finite alue at small olume. If G 0 is gien by the growth rate of 3-faces bubbles, then G 0 = 3D 2D eff. Note that using the bubble radius R instead of the olume as ariable leads to the same relation in all dimensions: P (R R at small R. Model: self-similar growth regime. In the SSGR, all scale-independent quantities become statistically time independent. The normalized distribution of reduced olume = / (t, with (t = V/N(t the aerage olume, is defined as ˆP ( = (tp (, t and does not depend on time. The distribution ˆP ( is likely to be uniersal [16] and can thus be considered as a p2

3 Dry foam coarsening well-defined continuous function, independent of the initial preparation of the foam. In the SSGR, the time eolution of P (, t is entirely goerned by the ariation of the bubble number /dt, or equialently, by the ariation of the aerage olume (t = V/N(t. This becomes eplicit by using the reduced distribution which is, by definition, inariant during the SSGR: P (, t = 1 (t ˆP (/ (t. (9 On the one hand, we can predict P (, t + dt from eq. (9 if we know P (, t, (t and (t + dt. On the other hand, by definition of the growth rate G, we can deduce P (, t + dt from P (, t and the distribution of G at any gien time. Both predictions of P (, t + dt must lead to the same alues, which simply implies that, in the SSGR, the bubble size distribution, the time eolution of the bubble number and the growth rates distribution are not independent. The eact relation between these three quantities has been established by Mullins in 1986 [3]. More precisely, Mullins gae the prediction of the growth rate aeraged oer the set of bubble of same olume, G, as a function of the bubble size distribution, P (, and of the bubble number ariation: G ( = 1 N dt P ( α P (d P (. ( This relation is the eq. (13 in [3], epressed as a function of the ariables G and P ( for which we recently proided 3D eperimental measurement for foams, respectiely, in [17] and [16] (the transformation is gien in B in the appendi. Comparison with 3D eperimental results in SSGR. Eperimental data hae been obtained with a dry dishwashing liquid foam, 190 min after its making (same data as in [16], fig. 3. At this time of its eolution, the relatie ariation of the bubble number is = s 1 and the liquid fraction is 2.0% ± 0.2. The olume distribution is plotted in fig. 1. Fiing the eponent at 1/3, we fit P ( as A 1/3 with A = 13.5 mm 2 1. Gien the alue of, this corresponds to G 0 = 11.1 µm 2 s 1, in good agreement with the alue found eperimentally (see fig. 2. At large olume, an eponential decay is found. A Gamma distribution Γ( = (/ 0 s 1 e /0 /( 0 Γ(s is classically used to fit grain olume distributions, with the shape eponent s as adjustable parameter [18]. The fact that the distribution must ary like 1/3 at small olumes imposes the alue of the shape eponent s = 2/3. The agreement with the bubble olume distribution is good, as shown in fig. 1. From the eperimental bubble size distribution shown in fig. 1, we compute G cal ( is plotted in fig. 2 and compared with the direct measurement of G ep ( using eq. (. G cal ( based on the bubble tracking between times 188 min and 194 min on the same foam. The image processing method has already been used for a slightly P( (mm (mm 3 Fig. 1: Normalized olume distribution in self-similar growth regime. The full line is the Γ distribution P ( = (/ Γ 1/3 e / Γ /(Γ(2/3 Γ with Γ = mm 3 ; the dotdashed line is the eponential law P ( = (1/ 0 e / 0, with 0 = mm 3 ; the dashed line is the power law P ( = /3. G (µm 2 s D (mm 3 Fig. 2: (Colour on-line Bubble growth rate in 3D. The indiidual growth rate in a 3D dry foam during the SSGR is aeraged oer a set of bubbles haing the same olume. : G mes directly measured from the olume ariation of bubbles followed between t = 188 min and t = 194 min. Each point represents an aerage oer 0 bubbles of similar olume. : G cal obtained from eq. ( with the distribution plotted in fig. 1 and = s 1. Full line: G ep gien by eq. (11, based on the assumption of an eponential olume distribution. wetter foam in [17] where the different steps of the analysis are discussed. The difference between G mes and G cal is within the statistical fluctuations, without any adjustable parameter. Equation ( is thus eperimentally erified. One may simply notice a small discrepancy at ery small olumes: G mes is larger than epected at small olumes. This is ery likely due to the non-negligible size of the meniscus for those olumes. This reduces the accessible area for gas transfer and leads to a smaller G mes in absolute alue [17]. A simple analytical prediction G epo is obtained for an eponential olume distribution: G epo = 3D. (11 1/ p3

4 J. Lambert et al. 1 2D 2D 12 0,1 P( F 0, ,001 (a (b Fig. 3: (Colour on-line Self-similar growth regime in 2D. (a : bubble area distribution (fig in ref. [19]. The full line is the eponential distribution and the dash-dotted line the distribution gien by eq. (18. (b + : F plotted as a function of the eperimental alue of = a F /ā (from fig. 3.17a in ref. [19]. The aerage area is adjusted. : F obtained from eq. (14, with the eperimental distribution (a and the adjustable prefactor C/Deff 2D = The full line and the dash-dotted line correspond, respectiely, to the relations between faces number and aerage area gien by eq. (15 and eq. (17. roughly fits the eperimental distribution, with an underestimation of the smallest olumes (see fig. 1. The agreement between G mes and G epo is satisfying, and eq. (11 is thus a ery conenient analytical prediction for the growth rate in SSGR (see fig. 2. G epo 2D case and correlation between area and edges number. In two dimensions, the eponent α is zero and the bubble area a replaces the olume. Equation ( becomes G 2D (a = P (ada ap (a P (a a (12 or, using the reduced ariable = a/ā and the constant C = dā/dt ( G ( = C ˆP (d. (13 ˆP ( Furthermore the growth rate in 2D is an eact function of F, the face number of the bubble (see eq. (4. Aeraging oer the set of bubbles haing the same area we get, from eq. (13 (see also [6], eq. (49 ( F a = 6 + C Deff 2D 1 ˆP ( ˆP d. (14 ( Eperimental data hae been obtained by Pignol [19] on 2D foam in SSGR. We plot the area distribution of a coarsening 2D foam in SSGR in fig. 3(a. The quantity that is usually plotted to characterize the foam structure is a F (see, e.g., [19]. Despite the fact that this aerage may differ from F a, we compare a F to the predictions obtained with eq. (14 in fig. 3(b. As neither the kinetic factors C/Deff 2D nor the aerage area are aailable, these quantities are taken as adjustable parameters. For an eponential area distribution, eq. (14 becomes [5] F a = 6 + C Deff 2D ( 1. (15 In the SSGR, there is thus an equialence between a linear behaior of the aerage edge number F a as a function the area and an eponential area distribution. Both laws are represented in fig. 3 using full lines and compared to eperimental data. Recently, the relation F a = 3(1 + a/ a (16 has been predicted theoretically [6], thus questioning the eponential behaior of the area distribution in coarsening foams. The general epression of the area distribution in a foam as a function of an arbitrary growth rate has been deried in [4]. We proide below the eplicit epression for the distribution corresponding to the growth rate of interest here, obtained by approimating eq. (16 by F a = 3(1 +. (17 Inerting eq. (14 with C/Deff 2D = 0.95 (see fig. 3(a, dashdotted line leads to the semi-analytical area distribution: ˆP ( = ˆP (0 ep[ arctan ( ] ( (18 This distribution proides a better agreement with eperimental data than an eponential decay for large bubbles. For small bubbles this law is less satisfying p4

5 Dry foam coarsening Neertheless, the eperimental distribution in SSGR shown in [20] (fig. 6 differs significantly from fig. 3 for small bubbles and is in qualitatie agreement with the prediction (eq. (18. Conclusion. In this paper we highlighted the strong relation between the bubble s growth rates and their size distribution in the self-similar growth regime. One can be deduced from the other and the other way around. This opens new perspecties both in the theoretical and eperimental fields. On the eperimental ground, P ( is far easier to measure than G since the latter quantity requires the correlation of two successie images. In contrast one theoretical challenge is to predict P ( in the SSGR. Most attempts to derie P ( rely on assumptions like maimum entropy [21]. Howeer, another way to proceed could be based on a local estimation of G. This way has been less inestigated despite many efforts to predict G [9 14] and might be reealed as productie. Appendi Hereafter, as in Mullins paper, f(,, t is the density of bubble of olume and olume ariation = d/dt, with f(,, tdd = 1 and f(,, td = P (; ρ(,, t = Nf(,, t; φ(, y is the reduced density (normalized to unity defined by φ(, y = (1+α f(,, t, with = / and y = / α, so that φ(, y does not depend on time in the SSGR; finally, φ( = φ(, ydy = P (. A Transformation of eq. (6 from [3] into the relation (5: 0 dt = lim d ρ( 0,, t 0 0 = N lim 0 0 P ( 0 =0 (A.1 (A.2 = lim 0 0 NP ( 0G ( 0 1/3 0. (A.3 B Transformation of eq. (13a from [3] into the relation (. Equation (13a is C d ( φ( d = φ(, yydy. d We get for the right-hand side φ(, yydy = f(,, t (1+α α d α = (1 α α f(,, t α d (A.4 (A.5 (A.6 = (1 α α P (G (A.7 and for the left-hand side, using eq. (1 C d ( ( φ( d = C φ( d d φ( (A.8 = d ( dt α P (d P (. (A.9 Finally (1 α α P (G = d dt α G = 1 N dt REFERENCES ( P (d P (, (A. P ( α P (d P (. (A.11 [1] Weaire D. and Hutzler S., The Physics of Foams (Oford Uniersity Press, Oford [2] Cantat I. et al., Les mousses. Structure et dynamique (Belin, Paris 20. [3] Mullins W. W., J. Appl. Phys., 59 ( [4] Streitenberger P. and Zöllner D., Scr. Mater., 55 ( [5] Flybjerg H., Phys. Re. E, 47 ( [6] Durand M. et al., Phys. Re. Lett., 7 ( [7] Glazier J., Phys. Re. Lett., 70 ( [8] on Neumann J., in Metal Interfaces (American Society for Metals, Cleeland 1952, pp [9] MacPherson R. and Sroloitz D., Nature, 446 ( [] Mullins W. W., Acta Metall., 37 ( [11] Hilgenfeldt S., Kraynik A. M., Koehler S. A. and Stone H. A., Phys. Re. Lett., 86 ( [12] Hilgenfeldt S., Kraynik A. M., Reinelt D. A. and Sullian J. M., Europhys. Lett., 67 ( [13] Wang H., Liu G. Q., Song X. Y. and Luan J. H., EPL, 96 ( [14] Jurine S., Co S. and Graner F., Colloids Surf. A, 263 ( [15] Markworth A. J., Metallography, 3 ( [16] Lambert J. et al., Phys. Re. Lett., 4 ( [17] Lambert J. et al., Phys. Re. Lett., 99 ( [18] Vaz M. F. and Fortes M., Scr. Metall., 22 ( [19] Pignol V., Éolution et caractérisation de structures cellulaires bidimensionnelles epérimentales, en particulier les mousses de saons, et simulées, PhD Thesis, INPL, 1996, [20] Duplat J., Bossa B. and Villermau E., J. Fluid Mech., 673 ( [21] Thomas G. L., de Almeida R. M. C. and Graner F., Phys. Re. E, 74 ( p5