aa r X i v : . [ c ond - m a t . s o f t ] S e p epl draft Statistical mechanics of two-dimensional foams
Marc Durand
Mati`ere et Syst`emes Complexes (MSC), UMR 7057 CNRS & Universit´e Paris Diderot, 10 rue Alice Domon et L´eonieDuquet, 75205 Paris Cedex 13, France, EU
PACS – Self-organized systems
PACS – Emulsions and foams
PACS – Pattern formation
PACS – General Statistical Methods
Abstract. - The methods of statistical mechanics are applied to two-dimensional foams undermacroscopic agitation. A new variable - the total cell curvature - is introduced, which playsthe role of energy in conventional statistical thermodynamics. The probability distribution ofthe number of sides for a cell of given area is derived. This expression allows to correlate thedistribution of sides (“topological disorder”) to the distribution of sizes (“geometrical disorder”)in a foam. The model predictions agree well with available experimental data.
Introduction. –
Foams and related physical systems(like emulsions, biological tissues, or polycrystals) areubiquitous, and serve as a paradigm for a wide rangeof physical phenomena and mathematical problems [1–4].One of them deals with the general topological and ge-ometrical properties of cellular materials [5–12]. In thisconnection, Quilliet et al. [5, 6] studied recently the topo-logical features of two-dimensional (2D) soap froths un-der slow oscillatory shear. Such macroscopic strain in-duces rearrangements within the foam, and the number ofsides of every cell evolves in time through local topologicalchanges ( T et al. reported the existence of an equilibrium state after fewcycles, characterized by a stationary probability distribu-tion of the number of sides per cell (topological disorder).They also showed that the width of this distribution ofsides is strongly correlated to the distribution of bubblesizes (geometrical disorder) within the foam. These resultssuggest that the macroscopic state of a homogeneouslysheared foam can be adequately described using the ideasand formalism of statistical thermodynamics. Indeed, thepioneering work of Edwards on granular matter [15] hasshown how the powerful arsenal of statistical physics canbe extended to athermal systems. This method has provenits applicability to other fields as well [16]. Because thereis no thermal averaging due to Brownian motion, this ap-proach requires the presence of a macroscopic agitation(analogue to an effective temperature) allowing the sys-tem to explore its entire phase space. Various attempts have been made in the past to de-scribe the geometrical and topological properties of 2Dfoams using the concepts of statistical thermodynamics[17–22]. However, these former theoretical approaches relyon strong assumptions: either they use an ad-hoc inter-action potential between bubbles [17, 18], involve (ratherthan deduce) empirical laws correlating size and side dis-tributions [19–21], or ignore some geometrical constraints(for instance, only the mean bubble area is specified, notthe individual bubble areas [19, 20, 22]). Some of thesemodels are based on the maximum entropy (informationtheory) formalism [19, 20], which has been subject to con-troversy [23]. Other models invoke minimisation of theenergy [21], or a combination of both principles [22] todescribe the state of a foam. However, it has been estab-lished [12, 24–27] that different arrangements (topologies)of a large number of bubbles do not really affect the energy(see discussion below). Furthermore, none of these modelscan account for the correlations between topological andgeometrical disorders reported by Quilliet et al. [5, 6].In this letter we set up a framework for describing theequilibrium state of a two-dimensional foam, basing ourdevelopment on analogies with conventional statistical me-chanics. As for other athermal systems [15, 16], we showthat the energy is not relevant to describe the macroscopicstate of a foam. Instead, a more appropriate state vari-able is introduced: the total cell curvature. We establishthe function of state which is minimized for a finite clusterof bubbles at equilibrium. This thermodynamic potentialp-1arc Durandfunction differs from entropy or energy used in previoustheories. The formalism developed here allows to derivean analytical expression for the distribution of the num-ber of sides. We show that the semi-empirical expressionconjectured by other authors [17, 18, 28, 29] is recovered,in a certain limit. Finally, the bubble size-topology corre-lations deduced from the present theory are investigated,and compared with experimental data. Physical, geometrical, and topological con-straints. –
Consider a given set of N B bubbles with pre-scribed areas { A i } (we focus on time scales much shorterthan those typical of bubble coarsening and coalescence,so the bubbles preserve their integrity and size ). A 2Dfoam is a partition of the plane without gaps or overlaps,and its structure must obey certain constraints [1] (oneconsiders the dry foam limit where liquid volume fractionis negligible). The physical constraints follow from themechanical equilibrium throughout the system: first, thebalance of film tensions at every vertex implies that theedges, or Plateau borders, are three-connected making an-gles of 120 ◦ with each other (Plateau’s laws). Then, thebalance of gas pressures in adjacent bubbles implies thatevery edge is an arc of circle, whose algebraic curvature κ ij is proportional to the pressure difference between thetwo adjacent bubbles i and j (Laplace’s law): κ ij = − κ ji = P j − P i γ , (1)where γ is the film tension, and P i and P j are the pressuresin bubble i and j , respectively (by convention, κ ij ≥ i , i.e.: when P j ≥ P i ). As a consequence, the algebraic curvatures ofthe three edges that meet at the same vertex must add tozero: κ ij + κ jk + κ ki = 0 . (2)The foam must also satisfy topological and geometri-cal requirements: apart from the constraint of prescribedareas, its structure must obey Euler’s rule, which relatesthe number of bubbles N B , with those of Plateau borders N P b , and vertices N v : N B + N v − N P b = c , where c isthe Euler-Poincar´e characteristic ( c = 0 for a torus, c = 2for an infinite plane). This rule, combined with Plateau’slaws, immediately gives: N P b = 3( N B − c ) . (3)Finally, the Gauss-Bonnet theorem applied to a n -sidedcell yields P nj =1 l ij κ ij = π ( n − /
3, where l ij denotes thelength of the edge common to bubbles i and j . Microcanonical ensemble ( N B , κ tot , N s ) . – In sta-tistical mechanics, the most fundamental entry is certainly Rigorously, the area and pressure of a cell vary from one config-uration to another; only the number of molecules of gas it containsremains unchanged. We consider that the area fluctuations are smallso that each bubble can be identified by its area. via the microcanonical ensemble. Suppose that the foamis agitated slowly as compared to the characteristic relax-ation time after a T accessible microstates .By analogy with the fundamental postulate of statisti-cal mechanics [30], we hypothesize that all the accessiblemicrostates of a given set of bubbles filling the 2D spacehave equal probability, under a slow macroscopic agita-tion. Obviously, a periodic or infinite foam fills the 2Dspace. It must be pointed out that the postulate appliesto a free cluster too, provided that the surrounding air isincluded as a supplementary bubble. Then, this “extrabubble” must also be taken into account in the countingof the number of accessible microstates. In the following,an infinite or periodic foam, or a free cluster plus the sur-rounding air shall be referred to as an unbounded foam .Conversely, a free finite cluster, a cluster within a largerfoam, or a foam enclosed in a container shall be referredto as a bounded foam . Figure 1 summarizes the differentkinds of boundaries that exist for a 2D foam. (a) (b)(c) (d) Fig. 1: Various situations for the boundary of a 2D foam. (a): infinite or periodic foam. (b): free cluster (figure taken from[25]). (c): cluster of bubbles within a larger foam. (d): foamenclosed in a container. Situation (a) is referred to as an un-bounded foam, while situations (c) and (d) are referred toas bounded foams. Situation (b) is either an unbounded orbounded foam, depending on whether or not the surroundingair is included as a supplementary bubble.
Rigorously, the accessible microstates do not all corre-spond to the same total surface energy. It is tempting, byanalogy with the statistical mechanics of a gas, to restrictp-2tatistical mechanics of two-dimensional foamsthe equiprobability hypothesis to the microstates of equalenergy. This refinement appears to be unnecessary: pre-vious studies have shown that different arrangements of alarge number of bubbles of given areas do not affect theenergy very much : in their computational studies of 2Dfoams under shear, Jiang et al. [27] reported energy fluctu-ations less than 2%. Graner et al. [25] obtained the sameresults, both numerically and experimentally: the energyvalues of the different metastable states of a 2D foam liewithin 2% of the “ground state” value. Kraynik et al. [12]conducted similar studies on 3D foams, and reported fluc-tuations below 4% for the surface energy of every bubble.These observations are also consistent with the so-called“equation of state” of a foam [25,31–33], which relates sur-face energy E , areas A i , and pressures P i within a clusterof bubbles: E = P i ( P i − P ) A i , where P is the exter-nal pressure. In the limit of fixed bubble pressures, thetotal surface energy of such a cluster would be conserved.Hence, to a first approximation, one can reasonably as-sume that the energy of a large cluster does not dependon its configuration, but is directly determined from thenumber of bubbles N B and the area distribution p ( A ).On a macroscopic scale, the state of the foam should bedescribed by a limited number of independent variables(besides the number of bubbles and the size distribution).These quantities must be “constants of motion” [30], i.e:they must keep constant values throughout the dynamicsof the system (here, a slowly agitated foam). An impor-tant observation is that one can define two independentquantities which are conserved for an unbounded foam.The first one is the total number of sides N s = P i n i ,where n i is the number of sides of bubble i . One has N s = 2 N P b for an unbounded foam ( N s , and not N P b , isan extensive and fluctuating variable for a finite cluster).Thus, according to Eq. (3), N s = 6( N B − c ). The secondone is the total cell curvature κ tot = P i κ i , where κ i isthe cell curvature of bubble i , defined as: κ i = X j ∈N ( i ) κ ij , (4) N ( i ) denoting the neighbouring cells of bubble i . Obvi-ously, κ tot is additive. Since κ ij = − κ ji , the terms cancelin pairs in the double sum, yielding κ tot = 0 for an un-bounded foam. The property κ ij = − κ ji correlates thetwo adjacent bubbles that share a common edge. It isalso important to note that – regardless of this mathe-matical property – the constraint κ tot = 0 is also imposedby the curvature sum rule (2), as it is illustrated on Fig.2. This rule, which correlates the three adjacent bub-bles that share a common vertex, has a physical origin(Laplace’s law). Although it is unclear whether N s and κ tot are the only constants of motion for an unboundedfoam, the constraints N s = 6( N B − c ) and κ tot = 0 mustbe taken into account in the statistical description of such Actually, there is also some uncertainty on the value of the en-ergy of an isolated volume of gas [30]. a system. Surprisingly enough, the constraint on the totalcurvature has always been ignored in previous theoreticalmodels [17–22].
Fig. 2: Illustration of the equivalence (for an unbounded foam)between the sum (over all vertices) P α S α and the sum (overall cells) κ tot = P i κ i , where S α denotes the sum – turningclockwise – of the algebraic curvatures κ ij + κ jk + κ ki betweenthe three bubbles i , j , k that share the same vertex α , while κ i denotes the cell curvature of bubble i (see Eq. (4)). Each term κ ij in the first sum is represented by an arrow pointing from i to j . The equivalence between the two sums is immediate:the cell curvature κ i of cell i is represented by the collectionof arrows coming out of that cell. For an unbounded foamat mechanical equilibrium, the curvature sum rule (2) yields S α = 0 at every vertex α , and thus κ tot = 0. In the light of this discussion, we may restate the pos-tulate as: all the accessible states of a given set of N B incompressible cells corresponding to the same values of N s and κ tot are equally probable, under macroscopic agita-tion. By analogy with thermal physics, the microcanonicalensemble ( N B , κ tot , N s ) refers to a large number of copiesof a foam (with given bubble size distribution p(A)) whoseparameters N B , κ tot , and N s are fixed. Thermodynamic limit. –
For a bounded foam, κ tot reduces to the sum of the side curvatures along its bound-ary. Both N s and κ tot are fluctuating variables, althoughtheir values are usually restricted within a certain range.For instance, for a free cluster, N s < N P b and κ tot < N s < N P b , but κ tot can be positive ornegative. For a foam enclosed in a container, κ tot = 0(the container walls can be regarded as zero curvaturesides), but N s fluctuates. One can argue whether ornot the fluctuations of N s and κ tot become negligible inthe thermodynamic limit ( N B → ∞ , total surface area A tot → ∞ , but h A i = A tot /N B remains finite). The av-erage number of sides of a large cluster ( N B ≫
1) scalesas h N s i ∼ N B , while the total curvature, equal to thesum of the side curvatures along the cluster boundary,scales as h κ tot i ∼ p N B / h A i . The relative standard devi-p-3arc Durandations of such quantities scale as N − / B [30]. Thus, thefluctuations of N s and κ tot become vanishingly small as N B → ∞ . Moreover, their thermodynamic limit valuesare known: h N s i /N B →
6, and h κ tot i /N B →
0. Notehowever that h N ∗ s i /N ∗ B ∼ N ∗ B converges much faster than h κ ∗ tot i /N ∗ B ∼ N ∗ B − / . Idealized foam. –
In order to obtain the probabilitydistribution of the number of sides for a cell of given size,we need to enumerate the microstates which correspond tothe same macroscopic state. Ideally, a microstate is spec-ified by the curvature, length, position and orientationof each Plateau border. However, this description is toocumbersome to handle. Moreover, even for a given foamtopology, the number of different accessible microstatesdepend in a non-trivial way on boundary conditions (e.g.:free, periodic, or enclosed cluster) and bubble size distri-bution [25, 35–38]. i jk i jk (a) (b)
Fig. 3: (a):
Real two-dimensional foam under shear. (b):
Idealization of the foam (mean-field approximation): adjacentbubbles i , j and k are now regular cells surrounded by an ef-fective foam and disconnected from each other. We shall use a simplified microscopic description of thefoam: we consider that for each particular bubble i , therest of the foam can be replaced on average by a “meanfield” of identical bubbles. Thus, the bubble i is a regularcell with identical curved sides joining at each vertex withangles of 120 ◦ , as sketched on Fig. 3b. The cell curvatureof a n -sided regular cell with area A is [22, 25]: κ ( A, n ) = π n ( n − e ( n ) √ A , (5)where e ( n ) = π | n − | √ n (cid:16) πn − π − sin( π/n − π/
6) sin( π/ π/n ) (cid:17) − / isthe elongation of the cell (ratio of perimeter to square-root of area). e ( n ) is a slowly decreasing function, lyingbetween e (3) ≃ .
74 and e ( ∞ ) ≃ .
71 [25]. It can benoticed that while the (surface) energy of such a bubble( ∼ e ( n ) √ A ) is almost independent of n , its cell curva-ture κ ( A, n ) increases rapidly with n , and is not upper-bounded.In this idealized foam description, each bubble is “dis-connected” from the others. By construction, all the phys-ical, geometrical and topological constraints are satisfied, except the curvature sum rule (2) and the Euler-Plateaurelation (3), which both involve adjacency of the bubbles.Consistent with the discussion above, the accessible mi-crostates of a foam tiling the entire plane are those whichsatisfy N s = 6( N B − c ) and κ tot = 0 (the number of bub-bles is large enough so N s and κ tot can be treated as con-tinuous variables). A microstate of the idealized foam isspecified by the numbers of sides { n i } of all the bubbles.Hence, the number of accessible microstates is obtained byenumerating the distributions of the N s = 6( N B − c ) sidesover the N B bubbles which satisfy the constraint κ tot = 0.This number is not easy to evaluate, and a grand-canonicaldescription shall be more appropriate. Grand-canonical ensemble (cid:0) N B , β − , µ (cid:1) . – Con-sider then a sample of N ∗ B bubbles in the unbounded foam(the asterisk denotes the variables of this grand-canonicalensemble). This sample can be a cluster of bubbles, or acollection of isolated bubbles. Let p ∗ ( A ) be the distribu-tion of bubble size within this sample. Although N ∗ B isfixed, this system can exchange sides and curvature withthe rest of the foam, through T N ∗ s and the total curvature κ ∗ tot arenow internal variables free to fluctuate for this system .We assume that possible other variables required to de-scribe the macroscopic state remain fixed. Surface energyin particular does not fluctuate, since it depends only on N ∗ B and the distribution p ∗ ( A ), and not on the specificconfiguration of the system. Suppose that the rest ofthe foam is large in comparison with the system, so thatit constitutes a reservoir of sides and curvature. Usingthe formalism of conventional statistical mechanics [30], itcomes that the probability for the system to be in the mi-crostate (cid:0) n , n , . . . , n N ∗ B (cid:1) is proportional to e − βκ ∗ tot + µN ∗ s ,with κ ∗ tot = P N ∗ B i =1 κ ( A i , n i ) and N ∗ s = P N ∗ B i =1 n i . β − and µ (rigorously, µβ − ) denote respectively the “temperature”of the reservoir of curvature, and the “chemical potential”of the reservoir of sides. A large number of copies of a foamwhose parameters (cid:0) N B , β − , µ (cid:1) are fixed shall be referredto as a grand-canonical ensemble . As noticed before, thecell curvature of a bubble is not upper-bounded as n → ∞ ,what ensures that the temperature β − is always positive[30]. Finally, the probability for a given cell of size A tohave n sides is: p A ( n ) = χ − ( A ) e − βκ ( A,n )+ µn , (6)where χ ( A ) = P n ≥ e − βκ ( A,n )+ µn denotes the parti-tion function of the cell. The average total cell cur-vature and average number of sides are, respectively, h κ ∗ tot i = − ∂ ln Ξ /∂β and h N ∗ s i = ∂ ln Ξ /∂µ , with ln Ξ = N ∗ B R ∞ p ∗ ( A ) ln χ ( A ) dA . Ξ is the partition function ofthe system. Using the formalism of conventional statisti-cal mechanics [30], one concludes that the thermodynamic N ∗ s and κ ∗ tot are independent variables: there are different waysof distributing N ∗ s sides over N ∗ B bubbles; each of these distributionscorresponds to a different value of κ ∗ tot . p-4tatistical mechanics of two-dimensional foams Table 1: Statistical description of an ideal gas and a 2D foam. ideal gas 2D foamsource ofergodicity
Brownianmotion /collisions macroscopicshear / T1events constants masses { m i } areas { A i } degrees offreedom positions andmomenta { r i , p i } side numbers { n i } fixedparameters(microcanoni-calensemble) energy E ,number ofmolecules N ,volume V total curvature κ tot , number ofsides N s ,number ofbubbles N B fixedparameters(grand-canonicalensemble) temperature T ,chemicalpotential µ ,volume V effectivetemperature β − , effectivechemicalpotential µ ,number ofbubbles N B equilibrium of a foam in contact with a reservoir of curva-ture and sides coincides with the minimum of the thermo-dynamic potential Φ = − β − ln Ξ. The analogy betweenthe statistical descriptions of an ideal gas and a 2D foamis summarized in Table 1.As noted by Graner and coworkers [25], the elongationof a regular cell is almost constant: e ( n ) ≃ e ≃ .
72. Withthis simplification, the total surface energy of the clusteris strictly conserved (i.e., it does not depend on the clusterconfiguration). Besides, the distribution (6) simplifies to p A ( n ) = χ ′− ( A ) e − β ′ ( n − + µ ′ ( n − , (7)with χ ′ = χe − µ , β ′ ( A ) = πβ/ (3 e √ A ) and µ ′ ( A ) = µ − πβ/ ( e √ A ). This is the exact distribution intuitedindependently by Schliecker and Klapp [17, 18] and Sher-rington and coworkers [28, 29], except that here β ′ and µ ′ explicitly depend on the bubble size A . Such a dependenceis necessary to reflect the correlations between bubble sizeand bubble shape which have been observed experimen-tally [5, 6]. It can be noted that the average number ofsides n ( A ) = P n > np A ( n ) increases with A . This re-sult is consistent with experimental observations [1, 39]:larger bubbles have more sides since they are surroundedby smaller bubbles.It must be pointed out that the present theory containsno free parameters in the thermodynamic limit: h N ∗ s i /N ∗ B and h κ ∗ tot i /N ∗ B have known values, and β and µ are di-rectly obtained by extremizing the grand-canonical en-tropy per bubble S ( β, µ ) = Z ∞ p ∗ ( A ) ln χ ( A )d A + β h κ ∗ tot i N ∗ B − µ h N ∗ s i N ∗ B , (8)with h N ∗ s i /N ∗ B = 6 and h κ ∗ tot i /N ∗ B = 0. Discussion. –
The expression (7) allows to de-duce the distribution of number of sides per cell p ( n )from the distribution of bubble size p ( A ) [ p ( n ) = R ∞ p ( A ) p A ( n )d A ], and thus to correlate topological andgeometrical disorders. Let us study the implicationsof the present theory with the simple case of an infi-nite/periodic monodisperse foam: p ( A ) = δ ( A − A ). Inthat case, the thermodynamic limit values h N ∗ s i /N ∗ B = 6and h κ ∗ tot i /N ∗ B = 0 give, respectively, h n − i = 0 and h ( n − i = 0. Thus, the distribution of side number tendsto the Kronecker delta distribution: p ( n ) = p A ( n ) = δ n, .As expected, all the bubbles of an unbounded monodis-perse foam have hexagonal shape. n p ( n ) ∆ A/ h A i ∆ n / h n i Fig. 4: Theoretical and experimental distributions p ( n ). Theexperimental distribution is taken from [5]. The theoreti-cal distribution is obtained by fitting the function p ( n ) = R ∞ p ( A ) p A ( n )d A to the data. Inset: topological disorder∆ n/ h n i vs geometrical disorder ∆ A/ h A i for four differentfoams. Black dots: obtained from theory; red curve: power-law fit a (∆ A/ h A i ) b , with a = 0 . ± .
03 and b = 0 . ± . a ∆ A/ h A i + b , with a = 0 . ± .
02 and b = 0 . ± . We also compare the model with the four experimen-tal distributions reported in the Quilliet et al. paper [5].Foam samples contain a few hundreds of bubbles. Thisnumber is large enough to consider that h N ∗ s i /N ∗ B ≃ h κ ∗ tot i p h A ∗ i /N ∗ B ≃ h κ ∗ tot i /N ∗ B is unknown, β and µ cannot be obtained by extremizing S ( β, µ ). Instead, theyare obtained by fitting the theoretical distribution of sides p ( n ) = R ∞ p ( A ) p A ( n )d A – in which p ( A ) is the experi-mental size distribution, and p A ( n ) is given by Eq. (7)– to the data. Regression is performed under the con-straint P n ≥ np ( n ) = 6, so there is only one adjustableparameter really. Fig. 4 compares a theoretical distribu-tion p ( n ) obtained this way with the corresponding exper-imental distribution taken from [5]. The theoretical curvereproduces the experimental data well. Comparison of themodel with the other distributions of [5] (not shown here)gives similar results. We also plotted the relative standarddeviation ∆ n/ h n i = p h n i − h n i / h n i of the theoreticalp-5arc Duranddistributions, as a function of the relative standard devia-tion ∆ A/ h A i of the size distributions for the four samples(inset of Fig. 4). A power-law fit of this plot is in goodagreement with the relationship found by Quillet et al. :∆ n/ h n i = 0 .
27 (∆ A/ h A i ) . . However, it can be noticedthat the power-law fit can be hardly distinguished from alinear fit.In summary, we developed a theoretical model to de-scribe the state of a two-dimensional foam under slow ag-itation, using a formulation closer to conventional statis-tical mechanics than information theory. We show thatthe total number of sides and the total cell curvature– rather than energy – are the relevant variables to de-scribe the macroscopic state of a foam. The distributionof sides of a cell of given size is derived. This result allowsto correlate the size and shape distributions. Theoreti-cal size-topology relations deduced from the theory arein very good agreement with existing experimental data.However, a free parameter has been added for the com-parison with experiments, due to the indeterminacy ofthe experimental values of h κ ∗ tot i /N ∗ B . Furthermore, thegrand-canonical description requires that 1 ≪ N ∗ B ≪ N B .This condition cannot be checked on the available data.Further experiments taking these considerations into ac-count should allow to confirm the validity of the present(zero-free-parameter) model.The formalism developed here can be extended to three-dimensional foams, and to coarsening (and coalescing)foams. Coarsening has consequences for the size-topologycorrelations particularly, since the average area of n -sidedcells increases more rapidly in that case [6, 39, 40]. Exten-sion of this formalism to other cellular systems (concen-trated emulsions, cell aggregates) is also under investiga-tion. ∗ ∗ ∗ I wish to thank J. B. Fournier, F. Graner and F. VanWijland for useful discussions and suggestions.
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