2D foams above the jamming transition: Deformation matters
Jens Winkelmann, Friedrich F. Dunne, Vicent J. Langlois, Matthias E. Möbius, Denis Weaire, Stefan Hutzler
22D foams above the jamming transition: Deformation matters
J. Winkelmann a, ∗ , F.F. Dunne a , V.J. Langlois b , M.E. Möbius a , D. Weaire a , S. Hutzler a a School of Physics, Trinity College Dublin, The University of Dublin. Ireland. b Laboratoire de Géologie de Lyon, Terre, Planétes, Environnement, Department of Earth Science, Université Claude Bernard Lyon 1, France
Abstract
Jammed soft matter systems are often modelled as dense packings of overlapping soft spheres, thus ignoringparticle deformation. For 2D (and 3D) soft disks packings, close to the critical packing fraction φ c , this resultsin an increase of the average contact number Z with a square root in φ − φ c . Using the program PLAT, we findthat in the case of idealised two-dimensional foams, close to the wet limit, Z increases linearly with φ − φ c ,where φ is the gas fraction. This result is consistent with the different distributions of separations for softdisks and foams at the critical packing fraction. Thus, 2D foams close to the wet limit are not well describedas random packings of soft disks, since bubbles in a foam are deformable and adjust their shape. This is notcaptured by overlapping circular disks. https://doi.org/10.1016/j.colsurfa.2017.03.058
1. Introduction (a) (b) (c)Dry Wet(d) p b pressure p i p j ( x n , y n ) (e) Figure 1:
Sample simulations as obtained with PLAT((a)–(d)) using periodic boundary conditions of atwo-dimensional foam with 60 bubbles at gas fraction φ = 0 . (a), 0.896 (b) and 0.841 (c). The bubbles insuch a foam are deformed even close to the wet limit, asseen in the example of (d) for φ = 0 . . In contrast (e)shows an example of overlaps in a soft disk simulation atthe same value for φ . The vertex positions ( x n , y n ) arethe coordinates of the point where a Plateau border endsand connects smoothly to a film, separating two bubbles. ∗ Email of corresponding author: [email protected]
In the wet limit a disordered two-dimensional (2D)foam (Fig. 1 (a) – (c)), as represented by the usualmodel (incompressible gas and liquid) [1, 2], assumesthe form of a packing of circular disks, as shown inFig. 1 (c). Simple arguments, often included in de-scriptions of jamming of frictionless granular mate-rials, lead to the result that, while local stability re-quires at least three neighbours for each disk, overallstability requires four as an average in 2D [3, 4, 5].But how does the average contact number approach this limiting value, as the wet limit is approached?Here we address this question, using the simu-lation program PLAT [6, 7] as described below. Itprovides a direct and accurate representation of themodel (Fig. 1).Various experiments for quasi-2D foams [8] and2D elastic disks [9], and simulations with the moreapproximate soft disk model [10] have been in agree-ment in finding the limiting form for the average con-tact number Z , Z − Z c ∝ ( φ − φ c ) / , (1)where φ is the packing fraction (or gas fraction, inthe case of foams) and φ c is its critical value; in thelimit of an infinite system the critical contact numberis Z c = 4 .Surprisingly, the result for an ideal 2D foam, sim-ulated with the program by PLAT [6, 7, 11], which Preprint submitted to Elsevier November 13, 2018 a r X i v : . [ phy s i c s . c o m p - ph ] A ug rovides a direct and accurate representation of a 2Dfoam, is different. It exhibits a linear increase in thewet limit, Z − Z c ∝ φ − φ c .This result is consistent with the distribution ofseparations [12] f ( w ) for the 2D foam, which is con-nected to Z − Z c via an integration [5, 10]. Thisseparation w is defined as the shortest distance be-tween two bubbles/disk edges (see Fig. 2). While f ( w ) for the soft disks exhibits a square root diver-gence, it reaches a finite limiting value for the foamin the limit of w → . (a) w (b) w Figure 2:
An Illustration of the separation w betweentwo bubbles (a) and soft disks (b). The separation isdefined as the shortest distance between two bubblearcs/disk edges. Its distribution f ( w ) is connected to Z − Z c via an integration [5, 10] (see also eqn. (5)).
2. Computer simulation of 2D foams
The results for the average contact number Z ( φ ) presented below were produced by the PLAT simula-tion code from [11] as described in [6, 7, 13].It is a software for the simulation of random 2Dfoam [6, 7, 13, 11] which is not based on an en-ergy minimisation routine, but instead directly imple-ments Plateau’s laws for a 2D foam by modelling thefilms and liquid-gas interfaces as circular arcs, con-strained to meet smoothly at vertices, see Fig. 1 (d).The radius of curvature r of each arc is determinedby the Laplace law.For a film this law is p i − p j = 2 γ/r , where p i und p j are the pressures in the two adjacent bubblesand γ is the surface tension. For a liquid-gas interface p i − p b = γ/r , where p b is the pressure in the Plateauborder, set equal in all Plateau borders.The samples were generated as (nearly) dry foamsby standard procedures [6, 7, 14]: A random De-launey tessellation is used to compute a Voronoi net-work. This is then converted to a (as yet unequili-brated) dry foam by decorating its vertices with small three-sided Plateau borders. The equilibration pro-cess of the decorated Voronoi network consists of ad-justing cell pressure and the vertex positions ( x n , y n ) under the constraints of smoothly meeting arcs andarea conservation for each bubble. Equilibrium isreached when the change in vertex positions is small.A progressive decrease in steps of ∆ φ = 0 . ingas fraction was imposed and the system was equi-librated at each step. Decreases in gas fraction areperformed by proportionally reducing bubble areas.The bubble radius distribution of the sample, whichis calculated from bubble cell areas, follows a lognor-mal distribution with a standard deviation ∆ R/ (cid:104) R (cid:105) ≈ . . More details of the protocol for sample prepa-ration are given in [14].Note that PLAT is currently the only simulationthat can simulate a wet foam with zero contact anglebetween two liquid interfaces. The Surface Evolver[15], the standard software to simulate 2D and 3Dfoams, requires finite contact angles with consequencesthat are currently being examined [16].As in its earlier application [17], PLAT was foundto be susceptible to a lack of convergence close to φ c ,which has not yet been eliminated. In the presentcase, this was mitigated by using a fairly small sys-tem (with periodic boundary conditions), consistingof bubbles, as in Fig. 1 (a) to (c). Results from
600 000 independent simulations were combined tocompute the variation of Z ( φ ) . Finite size effectswere taken into account when estimating the criti-cal packing fraction φ c , as detailed below. We believethis procedure to be reliable for present purposes, al-though there is a slight possibility of undesirable biasin the surviving runs close to the wet limit.As a standard procedure [8, 10], rattlers, whichare bubbles with less than three contacts, were ex-cluded in our analysis. These do not contribute tothe connected network and are mechanically unsta-ble bubbles, which can be removed without changingthe packing. (In the wet limit, less than of allbubbles were rattlers.)For a comparison with the soft disk model, ran-dom packings with similar conditions (same polydis-persity, same sample preparation protocoll) as in PLATwere created using conjugate gradient energy min-imisation [ ? ]. The average for Z ( φ ) excluding rat-tlers were taken over
20 000 independent simulations.In analysing our results we need to take into ac-count a small finite-size correction. In an infinite dis-ordered packing of disks the critical packing fraction2 c is associated with a contact number Z = 4 , ac-cording to arguments based on counting constraints[3, 4, 19]. In the case of our finite system with peri-odic boundaries the critical value of the contact num-ber is given by Z c = 4(1 − / N ) , where N is the num-ber of bubbles; N = 60 in our case thus results in Z c = 3 . . This relation is obtained from match-ing the number of degrees of freedom, N for a two-dimensional packing, with the number of constraints,due to the ZN/ contacts. However, in a periodic sys-tem we can fix one bubble without loss of generality,leaving only N − bubbles free to undergo transla-tional motion.
3. The variation of Z ( φ ) for 2D foams .
85 0 .
86 0 .
87 0 .
88 0 .
89 0 . φ . . . . . . A v e r ag ec o n t a c t nu m b e r Z Z lin ( φ ) = Z c + 17 . φ − . Z sqrt ( φ ) = Z c + 3 . √ φ − . − − φ − φ c − Z − Z c
2D foam Z − Z c ∝ ( φ − φ c ) . Z c Figure 3:
For 2D foams close to the critical gas fractionthe average number of contacts Z without rattlers wasfound to vary linearly with φ − φ c (red data points). Theaverage was taken over
600 000 independent simulationswith bubbles. A linear fit (solid red line) in thedisplayed range gave a slope of k f = 17 . ± . and acritical gas fraction of φ c = 0 . ± . . In the wet limit(at φ c ), Z c is given by Z c = 4(1 − / N ) due to finite sizeeffects. This results in Z c = 3 . for N = 60 bubbles. Forcomparison, Z ( φ ) is also plotted for a soft disk systems( N = 60 with
20 000 realisations), which shows thementioned square-root scaling. Inset: Double-logarithmicscale for Z − Z c vs. gas/packing fractions φ − φ c up to φ = 1 . By fitting a linear function (solid line), the φ c which gives the best linear relationship is obtained as φ c = 0 . ± . . In order to investigate the variation of Z ( φ ) closeto φ c , and the value of φ c itself, we plotted log( Z ( φ ) − Z c ) vs. log( φ − φ c ) , varying φ c to obtain the valuewhich gives the best linear relationship between these quantities (see also inset plot of Fig. 3). In this way,the critical gas fraction was found to be φ c = 0 . ± . , and the slope was . ± . in the logarith-mic plot.The conclusion is therefore that Z approaches Z c linearly , i.e. ( Z − Z c ) ∼ ( φ − φ c ) as plotted in Fig. 3.Appropriately, fitting Z = Z c + k f ( φ − φ c ) , (2)with Z c = 4 − / gives k f = 17 . ± . and a criticalgas fraction of φ c = 0 . ± . . In a different ap-proach, by looking at the excess energy, we obtained φ c = 0 . ± . [14] for the same system.The value of φ c is consistent with previous ex-perimental and numerical results, obtained for ex-ample from measurements of packings of bidispersehard disks[20], bidisperse elastic disks[9], polydis-perse hard disks[20], experimental data for (quasi)two-dimensional foams[8], and computer simulationsof polydisperse soft disk packings [21]. In the drylimit at φ = 1 , the PLAT simulation leads to Z = 6 ,which is the expected average contact number [1].This is not the case for the soft disk model.Our findings are also consistent with cruder esti-mates from previous PLAT simulations [13, 22] andsimulations using a hybrid lattice gas model [23].
4. Discussion of previous results for Z ( φ ) The linear increase of the average contact numberwith gas fraction, close to the wet limit, eqn. (2), isunexpected, since it is at odds with many previousfindings from computation, theory, and experiment.As an illustration we plot in figure 3 also results fromsoft disk systems with the same radius polydispersityas our 2D foam.Thus, before presenting further results supportingour results, we want to discuss the contradiction withprevious results and how to resolve it.At first there might seem to exist an incontrovert-ible weight of evidence for the square-root scaling,eqn. (1), but this is not the case for the 2D foam. Wediscuss the two strands of contrary evidence in turn.These are, firstly, results from the soft-disk model,and secondly, experimental data for bidisperse 2Dfoams.The discovery of the square root scaling for Z ( φ ) appears to date back to the work of Durian using theso-called Bubble Model [21]. Durian developed this3odel primarily to investigate the rheological prop-erties of foams, of which it indeed provides a goodoverall description [24]. Two-dimensional bubblesare approximated as disks, subject to repulsive forceswhen they overlap.The same square-root scaling for Z ( φ ) was alsofound in computer simulations of packings of three-dimensional soft spheres [25], a system which hassince been called the “‘Ising model’ for jamming” [5].If one describes foams in the wet limit as packingsof disks (or spheres), then it is tempting to extendthis analogy also to the functional relationship for Z ( φ ) and thus expect the same square-root relation-ship in lowest order. However, Surface Evolver simu-lations have shown, while the energy is harmonic in2D, the bubble-bubble interactions are not pairwise-additive [26]. That is, the model of interaction thatlies at the heart of the soft disk model does not repre-sent realistic bubble-bubble interactions. One shouldtherefore treat this prediction with some caution.Experimental evidence of the square-root scaling,as found from measurements of two-dimensional pho-toelastic disks under compression [9], is in agree-ment with the prediction of the bubble-model, whichone might expect to be applicable in this case, at leastfor qualitative purposes.Let us now turn to the second strand of contraryevidence by examining further experimental resultswhich bear directly on 2D foams.Katgert and van Hecke [8] performed experimentswith disordered rafts of bidisperse bubbles beneath aglass plate. The distance between plate and liquidsurface was varied to obtain foams at different valuesof gas fractions. The concept of a gas fraction is notwell defined for such quasi -2D bubble rafts, in partic-ular in the wet limit where the gap between coveringplate and liquid interface is similar to the bubble ex-tension parallel to the plate. For this reason Katgertand van Hecke [8] proceeded by imaging their raftsfrom the top to obtain an area gas fraction. Basedon their analysis Katgert and van Hecke established Z − Z c ∝ ( φ − φ c ) α , with exponent α (cid:39) . , Z c closeto 4, and φ c close to . [8]. Due to the problemin defining a gas fraction for such a quasi-2D exper-iment, and in identifying contacting bubbles, we donot think that these experimental results can be takento contradict our PLAT findings, even though Katgertand van Hecke describe their wet foams as consistingof “soft frictionless disks”.For 3D foams our results suggest also a deviation from the square root scaling in Z ( φ ) , since we con-jecture the reason for the deviation in the 2D caseto be the model of interaction. However, the scal-ing does not have to be linear. Apart from the non-pairwise interaction, the energy for the 3D bubble-bubble interaction is also not harmonic. It scales withthe form f ln(1 /f ) , first predicted by Morse and Wit-ten, where f is the force exerted between droplets[27, 26, 28].However, similar to the 2D case, evidence for thesquare root scaling seems to be indisputable at firstglance. Experiments from Jorjadze et al. [29] withdroplet emulsion in 3D show a good agreement withthe square root increase in Z ( φ ) . But, as in the ex-periments of Katgert and van Hecke the identifica-tion of contacting bubbles and the definition of a gasfraction is not straight forward. Jorjadze et al. re-constructed the droplets as overlapping spheres anddefined contacts as overlaps. The gas fraction is thenthe spherical volume reduced by the overlaps. Thus,it cannot be ruled out that this procedure contains abias towards the square root scaling of Z ( φ ) as in thesoft disk model.The distribution of contacts in a packing can bepredicted via the granocentric model [30] which hasrecently been extended to 2D cellular structures [31]and 2D packings of discs [32]. However, this modelcannot predict the variation of Z with φ in packingsas it only applies to the wet limit (or jamming point).
5. Link between Z ( φ ) and the radial density func-tion g ( r ) For soft disk packings it has been argued that thesquare root scaling of Z as seen in (1) is connectedwith the variation of the radial density function g ( r ) via an integration [5, 10, 33], although the validityof this argument is still under discussion [34].The radial distribution function R ( r ) is defined asthe probability to find a particle a given distance r away from another particle. In 2D the radial density function is given by g ( r ) = πr R ( r ) . From simula-tions of 3D monodisperse soft spheres with diame-ter D = 1 close to the jamming transition the be-haviour of g ( r ) is found to be divergent, according tothe power law g ( r ) = c d √ r − , (3)where c d is a constant [10]. A similar divergence canbe found in 2D polydisperse systems, when the radial4ensity function is rescaled to g ( ξ ) with the rescaledinterparticle distance ξ = r / ( R i + R j ) , where R i and R j are the radii of two disks with distance r apart[5]. Using an affine Ansatz (see below), integrating g ( r ) over r then results in the square root scaling for Z ( φ ) of eqn. (1) [5, 10, 33].
6. Distribution of separation f ( w ) for 2D foamsand soft disk systems For 2D foams such an argument involving g ( ξ ) isnot straightforward to develop, since bubbles are de-formable and only have well-defined centres in thewet limit (at φ c ) where they are circular. For this rea-son we will in the following consider a different ap-proach, which involves a distribution of separations f ( w ) between bubbles (or disks), as in the work ofSiemens and van Hecke [12]. Here, the separation w is the shortest distance between two bubble arcs/diskedges (see Fig 2). For the soft disk system, this sepa-ration is then related to their distance by their radii, r = w + R i + R j . For the soft disk system f ( w ) is iden-tical to g ( ξ ) close to the divergence, when shifted bythe average disk diameter D , thus g ( ξ − D ) = f ( w ) .Fig. 4 shows the distribution f ( w ) for both foamsand packings of soft disks with the same system size( N = 60 ) and area polydispersity. The difference be-tween our results for simulated 2D foams and 2D diskpackings is striking. Whereas in the case of disks, f ( w ) diverges in the limit w/D → as f ( w ) = c d (cid:112) w/D (4)as expected from the divergence of g ( ξ ) with c d =0 . ± . , for the 2D foams a finite limiting value c f = 2 . ± . is reached in this limit. Only at valuesof w/D (cid:38) − , f ( w ) is the same for both foams andsoft disks; see Fig. 4.Let us now consider the compression of a two-dimensional, polydisperse foam/disk sample of ini-tial gas/ packing fraction φ c to a final value of φ > φ c .The fractional compression ∆ (cid:15) is given by ∆ (cid:15) = ( φ − φ c ) / (2 φ c ) , where ∆ (cid:15) is considered to be small.We can estimate Z ( φ ) for the case of an affinecompression from f ( w ) . In this case the deformationof the sample will lead to an increase in contact num-ber due to bubbles coming together that initially, i.e.in the wet limit (at φ c ), were closest to each other.For an affine deformation the fractional compressioncan be expressed as ∆ (cid:15) ≈ ∆ w/D . Thus, the average − − − − Separation w/D D i s tr i bu t i o n o f s e p a r a t i o n f ( w ) soft disks f ( w ) = . √ w/D
2D foam c f = 2 . finite limitingvalue for foamsdivergencefor softdisks Figure 4:
Distribution of separation f ( w ) for 2D foam(red circles) and 2D disk packing (blue crosses) at asimilar average contact number Z SD = 4 . ± . for softdisks and Z foam = 4 . ± . for the 2D foam ( D :average bubble/disk diameter). The data shown presentsaverages obtained from 1379 packings, each containing60 bubbles or disks. In the case of foams, the finite valueat f ( w ) in the limit of w/D → is consistent with theobserved linear increase of the average contact number Z , according to the approximate argument, given in thetext. The decay of f ( w ) ∝ ( w/D ) − / in the same limit inthe case of the disk packings is consistent with the squareroot increase of the average contact number Z . number of contacts in 2D can be estimated by inte-grating ρf ( w ) over a radial shell up to D ∆ (cid:15) , where ρ = φ c πD is the particle number density, Z ( φ ) − Z c = 2 πρ (cid:90) D ∆ (cid:15) d wf ( w )( D + w ) (5)When inserting the power law expression fromeqn. (4) into (5), we obtain for the soft disk simu-lation Z ( φ ) − Z c = (cid:112) φ c c d (cid:112) φ − φ c + O (cid:16)(cid:112) φ − φ c (cid:17) ≈ (2 . ± . (cid:112) φ − φ c , (6)where we neglected terms of higher order in φ − φ c .For φ c the value . ± . was used [21].Fitting the soft disk data for N = 60 to a squareroot function, Z − Z c = k d √ φ − φ c for all Z < gives k d = 3 . ± . and φ c = 0 . ± . .For the 2D foam simulation, the finite limitingvalue c f can be inserted for f ( w ) in the limit w/D → in eqn. (5). By integrating we then obtain for Z ( φ ) Z ( φ ) − Z c =4 c f ( φ − φ c ) + O (cid:0) ( φ − φ c ) (cid:1) ≈ (11 . ± . φ − φ c ) . (7)Again, we neglected terms of higher order in φ − φ c .Qualitatively both estimations are in accord withexpectations, although the apparent numerical dis-crepancy in the prefactor remains to be resolved. Inboth cases the prefactors are underestimated whenobtained from our data for soft disk/bubble separa-tions. lim w → f ( w ) Z ( φ ) − Z c Computed via f ( w ) Direct calculation2D foam: . ± . ± φ − φ c ) (18 . ± . φ − φ c ) soft disks: . ± . √ w/D (2 . ± . √ φ − φ c (3 . ± . √ φ − φ c Table 1:
A summary of our results for 2D foams and softdisks. The functional form for Z ( φ ) can be obtained fromthe distribution of separation f ( w ) for both 2D foams andsoft disks, the numerical prefactor is underestimated forboth by a factor of / . Table 1 summarises all results that we found todiffer in 2D foams and soft disks. It demonstratesthat the linear variation of Z close to φ c is consis-tent with the distribution of separation found in wetfoams. However, this is still short of a full explana-tion of the asymptotic properties of the wet limit.
7. Conclusions
The variation of Z as a function of gas fractionwas one of the first problems that were tentativelyaddressed with the PLAT software, as soon as it wasdeveloped in the early 1990s. The very limited datasets available at the time ( φ ≥ . , 100 cells [13],530 cells [22]) showed that a linear extrapolation ofthe data leads to Z = 4 at φ c (cid:39) . [20]. However,later simulations using a lattice gas model for foamsalso showed a linear variation of Z very close to φ c ,but this data was based on an even smaller sample ofonly 30 bubbles [23].The success of Durian’s bubble model [21, 35]in reproducing the Herschel–Bulkley type rheologythat is associated with emulsions and foams [24],and its ease in simulating packings of 10000 or morebubbles, led to it being treated as the most practi-cal model for simulations of 2D foams in general.Its square-root variation of Z with gas fraction away from φ c was thus expected to also hold for 2D foams.Here we have shown, based on a large amount ofnew data, that this is not the case. For 2D foams wefind that the average contact number varies linearlyin this limit.The reason for this differing behaviour must ul-timately lie in the different contributions that diskor bubble contacts make to the total energy of thepacking. In a foam the energy per bubble per contact increases with the number of contacts [26]. Energyminimisation might thus lead to the reduction in thenumber of contacts in the wet limit compared to diskpackings.In summary, we showed that the disordered struc-ture of a polydisperse 2D foam is significantly differ-ent compared to a soft disk packing with the samepolydispersity as evidenced by the different Z ( φ ) andcorresponding distribution of separations. This is dueto the deformation of the bubbles, which is absentin the soft disk model, and the lack of pairwise in-teractions. While this study only focussed on a 2Dfoam system, similar deviations are likely for other2D jammed systems with soft, deformable particles.The relevance to 3D packings of soft particles, suchas emulsions, biological cells [36, 37] and microgelparticles [38] remains to be examined. Acknowledgments
We would like to thank F. Bolton for updatingthe PLAT software and D. McDermott for carrying outsome of the initial numerical analysis of the distribu-tion of near contacts. Research supported in part by aresearch grant from Science Foundation Ireland (SFI)under grant number 13/IA/1926 and from an IrishResearch Council Postgraduate Scholarship (projectID GOIPG/2015/1998). We also acknowledge thesupport of the MPNS COST Actions MP1106 ‘Smartand green interfaces’ and MP1305 ‘Flowing matter’and the European Space Agency ESA MAP Metalfoam(AO-99-075) and Soft Matter Dynamics (contract:4000115113).
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