PHYSICAL REVIEW E 93, 032613 (2016)

Bubble-raft collapse and the nonequilibrium dynamics of two-state elastica

Chin-Chang Kuo,1,* Devin Kachan,2 Alex J. Levine,2,3,4 and Michael Dennin1,5

1Department of Physics and Astronomy, University of California, Irvine, California 92697, USA2Department of Physics & Astronomy, University of California, Los Angeles, California 90095, USA

3Department of Chemistry & Biochemistry, University of California, Los Angeles, California 90095, USA4Department of Biomathematics, University of California, Los Angeles, California 90095, USA5Institute for Complex Adaptive Matter, University of California, Irvine, California 92697, USA

(Received 10 July 2015; published 30 March 2016)

We report on the collapse of bubble rafts under compression in a closed rectangular geometry. A bubble raftis a single layer of bubbles at the air-water interface. A collapse event occurs when bubbles submerge beneaththe neighboring bubbles under compression, causing the structure of the bubble raft to go from single-layerto multilayer. We studied the collapse dynamics as a function of compression velocity. At higher compressionvelocity we observe a more uniform distribution of collapse events, whereas at lower compression velocities thecollapse events accumulate at the system boundaries. We propose that this system can be understood in terms of alinear elastic sheet coupled to a local internal (Ising) degree of freedom. The two internal states, which representone bubble layer versus two, couple to the elasticity of the sheet by locally changing the reference state of thematerial. By exploring the collapse dynamics of the bubble raft, one may address the basic nonlinear mechanicsof a number of complex systems in which elastic stress is coupled to local internal variables.

DOI: 10.1103/PhysRevE.93.032613

I. INTRODUCTION

There are a number of soft-matter systems exhibiting highlynonlinear elasticity, at least in part due to the complex interplayof elasticity and internal state variables. Some examplesinclude colloidal crystals and wormlike micellar solutions,which exhibit banding [1,2] under applied shear stress,monolayers and membranes that form localized folds [3] undercompression, and biopolymers such as DNA [4–8] or α-helicalpolypeptides [9,10] that can kink (and locally melt) undertorque. There are significantly more complex many-bodysystems that generate nonlinear responses to applied stressvia the adaptation of the internal dynamics of the constituentelements in response to applied stress. In many cases theseinternal dynamics are poorly understood, but for the purposesof elucidating their collective mechanical response, theseinternal states may be reduced to one of a small numberof states. For example, in studies of tissue dynamics, therate of cell division in tissues appears to depend on appliedstress [11,12], and this mechanical nonlinearity has beensuggested to play a key role in the pattern formation [13]in tissues, the development of flowers [14], and the invasionof tumors into surrounding healthy tissue [12,15].

A minimal description of these complex systems can beformed in terms of an elastic manifold coupled to a discreteinternal variable that determines elastic properties in at leastone of two distinct ways. In the first case, the state ofelastic stress causes local transitions in the material’s elasticconstants; for example, in shear banding and biopolymerkinking the material’s elastic constants locally transition froma stiff state to a less stiff one. To be more precise in the caseof polypeptides, bending can disrupt local hydrogen bondingin the helix, and in DNA may induce local melting, whichcorresponds to the breakdown of the base-pairing interaction.

*chinchak@uci.edu

These changes in internal molecular configurations affect thelocal bending modulus of the biopolymer. For example, localDNA melting decreases the bending modulus by a factor ofabout fifty. Because the transition is to a softer state of theelastic body at the expense of increasing the local free energyof the chain, the system trades elastic strain energy for locallyincreasing its internal free energy. This leads to a nonlinearstrain softening of the elastic body driven by the creationof localized defects of the softer, but thermodynamicallydisfavored, internal configuration. Such effects have beenexplored using a simple model—the helix-coil wormlikechain [9,10]; we refer to this nonlinear coupling betweenelasticity and internal state variables as a type I coupling.

In contrast, the underlying elastic reference state, i.e., thestate of zero strain, can itself evolve by, e.g., cell division in atissue. When that local rate of cell division itself depends onthe state of stress, even a tissue in the linear elastic regime canexhibit complex, nonlinear responses to applied stress. There isalso an inorganic example to be found in CeO2 thin films withnanoscale pores, developed for use as supercapacitors [16]. Insuch materials, charge storage occurs through the absorption ofLi ions, which lead to a local expansion of the pores, changingthe elastic reference state of the material. We refer to thesesystems as having a type II coupling, and they will be thefocus of the current studies using bubble rafts.

There is a long history in condensed matter physics of usingbubble rafts as models of atomic solids. By taking advantageof the ∼107 increase in scale between bubbles and atoms,one may more easily visualize features such as the motion oftopological defects in crystals [17,18], stress relaxation [19],and shear banding in disordered materials [20–23]. In thisarticle we propose that bubble rafts may also be used toexplore the interplay of elasticity and local internal variablesthat control the elastic reference state. This class of problemsis of particular interest with regard to modern studies oftissue growth and morphology. We perform experiments onthe compression of a disordered elastic solid composed of

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KUO, KACHAN, LEVINE, AND DENNIN PHYSICAL REVIEW E 93, 032613 (2016)

a monolayer of bubbles at the air-water interface. The totalnumber of bubbles is essentially fixed over the time of atypical experiment. However, under sufficient compression,this system stochastically forms local regions of double bubblelayers, the second layer being submerged under the one atthe surface. The doubling of the layer can be treated asan elementary example of a discrete (two-state or Ising)variable coupled to stress in the effectively two-dimensionalelastic manifold of the interfacial bubble layer. Because thesubmerged bubble layer does not form a continuous, stress-bearing network, the local transition between the single anddouble layer can be represented as a local discrete changein the elastic reference state—the material jumps between alower-density (single-layer) and higher-density (double-layer)state. Thus, the bubble layer is a simple model of a two-dimensional elastic manifold with type II interactions, usingthe terminology introduced above.

In this case, the simple advantage of scale is less pronouncedbecause the bubbles are typically only 100 times larger thancells, but there are other advantages stemming from thesimplicity of the bubble raft. By eliminating the biologicalcomplexity of the tissue, such as chemical signaling and theactive generation of internal stresses, the bubble-raft systemallows one to develop and test fundamental models of type IInonlinear elasticity. The bubble raft also provides experimentaladvantages in terms of time scale. The collapse dynamics westudy occurs on time scales of 1–1000 s, and is thus slowenough to be amenable to video tracking but fast enoughto run multiple experiments rapidly. We believe there aredirect applications of these results to more complex systemsincluding tissue growth and membrane or monolayer folding.

We report on experiments exploring the collapse dynamicsin a disordered bubble raft in a closed rectangular geometry,focusing specifically on determining the spatial probabilitydistribution for double-layer formation as a function ofcompression speed. The rate dependence of the single-to double-layer transition demonstrates the fundamentallynonequilibrium nature of the dynamics. From a simple analysisof the compression modulus of the bubble raft and itsviscous flow over the aqueous subphase, it is clear thatthe elastic compression mechanically equilibrates rapidlyon experimental time scales. The single-layer–double-layervariable, however, is slow. It falls out of equilibrium, trappingthe system temporarily in higher-energy metastable states,which then slowly relax through the formation of double-layerdomains.

We also propose a general model to describe the elasticityof any linear elastic solid in which the strain-free state of thematerial can discontinuously switch between two states—atype II elastic nonlinearity. We explore an elementary, one-dimensional example of this model to study the nonequilibriumdynamics of double-layer domain growth in the compressedsystem, using Glauber spin dynamics to account for the single-to double-layer transitions. The predictions of the model for thespatial probability distribution of such transitions as a functionof compression speed are then compared to experiments

The remainder of the article is organized as follows. InSec. II we detail the bubble-raft collapse experiments anddemonstrate the stochastic double-layer formation kineticsunder compression in Sec. III. In Sec. IV, we develop the

minimal model of this type II nonlinear elasticity using an Isingvariable representing the single-layer–double-layer degree offreedom, coupled to the linearly elastic bubble layer. Wethen explore a one-dimensional example of this model. Inour discussion, we compare the model predictions to ourexperiments, discuss open questions, and propose future work.

II. EXPERIMENTAL DETAILS

The bubble raft was prepared by flowing compressednitrogen through a needle under the bubble solution. Thebubble solution contains 15% of glycerol, 5% Miracle Bubble,and 20% deionized water. The focus of this study is theamorphous system composed of bubbles with radii in therange of 0.3–0.5 mm. These polydisperse bubble rafts werecreated by oscillating the needle during bubble formation,generating small variations in bubble size. The initial stateis a contiguous array of bubbles that fill the trough surface,forming a continuous elastic sheet. Before compression, thetypical gas-area packing fraction for our bubble raft (i.e., thefraction of the bubble-raft area taken up by the bubbles) is≈0.86. Details of the bubble-raft preparation can be found inour previous work [24].

The experimental setup is shown in Fig. 1. The initialbubble-raft size is described as L × W0 for its initial lengthand width, respectively. We considered two different initialgeometries for the bubble raft: 8 cm × 6 cm and 8 cm ×10 cm. This allowed for an initial test of system size and/oraspect ratio effects. No obvious impact of system size or aspect

FIG. 1. (a) A schematic of the experimental setup and thegeometry for the bubble raft. We applied the compressive strain bymoving the barriers towards to the center. (b) A typical image of thecompressed bubble raft. Darker areas show doubling of the bubblemonolayer and �X indicates distances measured from the center lineof the system.

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ratio was observed, and the results presented here all are fromthe 8 cm × 10 cm system.

The focus of the study was on the impact of the compres-sion velocity on the probability of double-layer formation.Compression of the bubble raft was achieved through twomovable barriers that were controlled by a single steppermotor. The bubble rafts were compressed at a constant velocity,which ranged from 1 × 10−4 mm/s to 1 × 102 mm/s. Thecompressive strain in our experiment is defined as L0−L

L0=

�L/L, where L0 is the initial raft length.Bubble-raft collapse images were recorded by a CCD

camera with a maximum frame rate of 30 s−1. Over theobservable time scales, the stepper motor reaches its steady-state velocity essentially instantaneously. The bubbles alsoreach their terminal velocity very rapidly on these time scales.One can make a simple estimate of this time from the ratioof the Stokes drag on the bubble to its mass. We estimate thebubble’s mass to be 4πa2tρ (where t is the thickness of thebubble, ≈10−4 cm, and ρ is the density of water), to arrive ata bubble mass of ∼10−6 g. We also estimate the Stokes dragon a bubble of radius a and velocity v to be ≈ηav, where η

is the dynamical viscosity of water. Thus, we find the time forthe bubble to accelerate to its terminal velocity is on the orderof 10−3 s, which is not observable at our current frame rates.Consequently, we do not observe the acceleration of bubblesor any dynamical transients associated with the accelerationof the barriers. In addition, bubble coarsening does not occurover the time scales of two hours in our experiment, consistentwith previous studies [25].

We used MATLAB to perform image processing and analysis.With appropriate lighting and thresholding of images, thesingle and double layers were distinguished by their differencein overall brightness. These intensity variations were used toidentify collapse events.

III. EXPERIMENTAL RESULTS

Figure 2 illustrates the qualitative difference between thecollapse dynamics of amorphous bubble rafts at slow [Fig. 2(a),left] and fast [Fig. 2(b), right] compression velocities. Thecompression velocity of Figs. 2(a), 2(b) are 0.0086 and8.6 mm/s, respectively. Both cases had the same initial sizeof 8 cm × 10 cm. To compare the collapse distribution atdifferent compression velocities, we consider equal the raft atequal values of compressive strain (�L/L0). The images inFigs. 2(a) and 2(b) were both made at a compressive strain of∼0.66. For the system with the higher compression velocity,the collapse events are uniformly distributed throughout theraft; whereas for the slower speed, nucleation of multiple layersnear the boundary dominate the dynamics. Also, the overalldensity of collapse events appears to be greater for the fasterspeeds.

We also examined monodisperse bubble rafts with poly-crystalline structure. Here we focus on the amorphous bubblerafts as this provides a simpler elastic system. The moreordered, monodisperse bubble rafts exhibited double-layergrowth that propagated along particular crystallographic axes.We will consider this behavior in detail in future work.

To study the effect of compression velocity on the nucle-ation dynamics, we focus on the formation of double layers.

FIG. 2. Typical images of amorphous bubble raft under com-pression at the same amount of the compressive strain (∼0.66)with different compression velocities. (a), (b) are images with thecompression velocity of 0.0086 mm/s and 8.6 mm/s. The initial sizeis 8 cm × 10 cm and the average bubble radius is 0.4 mm for both setsof experiments. The scale bars in images are 2 cm. A comparison ofthese images illustrates the increased nucleation rate of double layersin the middle of the raft when it is compressed at higher rates.

As discussed above, double layers are identified using thedifferent intensity levels for the single- versus double-layerregions. Using this analysis, we also observe multilayer regions(greater than two) near the boundaries. The penetration lengthof higher-order multilayers ranged from ≈ 3 mm to 1 cmover the course of compression. We avoid taking data onthis boundary regime by focusing on the central region ofthe trough, where only single and double layers are observed.

The formation of the double-layer regions is a stochasticevent so that the precise trajectory of double-layer formationis irreproducible. Given the macroscopic energies involvedin the transition from single to double layer, the underlyingsource of stochasticity is nonthermal. Multiple experiments,however, allow one to build up the probability density ofcollapse events (i.e., the transition from a single to a doublelayer) as a function of both space and time. We compute theprobability of a collapse event occurring a fixed distance awayfrom the compression barrier using the following procedure(that is shown schematically in Fig. 1). We assume thatthe probability of a collapse event is homogeneous alongthe direction parallel to the barriers. Thus, we may average thefraction of single layer in vertical slices taken parallel tothe barriers to obtain this probability. The position of thevertical slices is defined relative to the center of the trough,so that the zero position is fixed and positions left of centerare negative. It will prove convenient when discussing thecomparison between these experiments and our theory toconvert the measured double-layer probability Pi into spinvariable via si = 2Pi − 1. Thus, −1 < s < +1 correspondingto double-layer probabilities running from 0–1. In order tocompare the dynamics at different compression rates, weused the compressive strain �L/L0 as our measure of time.

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FIG. 3. The probability of double-layer formation expressed interms of −1 < s < 1 (red to yellow) as a function of normalizedoff-center distance at a fixed total compressive strain of 0.85. (a), (b),and (c) show results for compression velocities of 0.0086, 0.43, and8.6 mm/s respectively.

Finally, due to the decrease in system length, we normalizedthe off-center distance by half of the compressed raft length(2�x/L).

Typical probability distributions for the formation ofa multilayer are shown for low (0.0086 mm/s), medium(0.43 mm/s), and high (8.6 mm/s) compression velocities inFigs. 3(a)–3(c). The color code, which runs from s = −1 (red)to s = 1 (yellow), shows the statistical result for single layerand double layer. In general, a high probability of collapseevents accumulates at the boundary and propagates inwardfor both fast and slow compression velocities. The growthdynamics of collapsed raft probability near the center of

the trough, however, are quite distinct for slow and rapidcompression. For slow compression velocity, collapse prob-ability continuously and gradually propagates inward fromthe boundaries towards the center, as shown in Fig. 3(a). Incontrast, at fast compression velocity, the collapse probabilityinitially propagates continuously inward from the boundaries,but at a critical strain of ≈ 0.25, the collapse probability rapidlyjumps in the central part of the bubble raft. Upon reaching thecompressive strain of 0.55, the whole surface of the bubble raftis a double-layer (multilayer) structure. Thus, the stochasticdynamics of double-layer formation depends strongly on thecompression velocity.

IV. THEORY

A. Multilayer raft elasticity

To explore the collapse dynamics theoretically, we considera simple, one-dimensional model of the bubble layer.1 Wetreat the two-dimensional bubble layer as elastically equivalentto a one-dimensional array of Hookean springs with springconstant k. To account for the change in elastic referencestate upon local layer collapse, each spring selects one oftwo possible rest lengths; the longer one corresponds to thesingle-layer configuration and the shorter to the double-layerconfiguration. Thus, when compressed, elastic energy in thespring chain can be reduced by transitions of some (or all)springs from the longer to the shorter rest length.

We write the elastic energy of the spring chain

Hel = k

2

N−1∑i=0

[ui+1 − ui + �(σi)]2 (1)

in terms of the displacements of the spring ends (nondi-mensionalized by the bubble radius a) ui . The rest lengthsof the springs depend on a discrete, Ising-type variable σ ,which takes the values −1 and +1 when the bubble layeris in its expanded single layer or compressed double-layerconfiguration, respectively. In other words, we require that

�(σ ) ={�> σ = +1�< σ = −1 , (2)

where, as implied by the notation, �> > �<. Since thelength �> corresponds to two bubble radii 2a, and theshorter separation corresponds to only one, we expect that�> − �< = a, although one is not required to make thisassumption in the more general implementation of the model.We neglect the possibility of triple and higher-order layeringsof the bubble raft. While these appear near the walls in the

1We justify the reduction of the two-dimensional system to a one-dimensional analysis as follows. In the images of the compressedlayers, one observes patterns of single and double layers consistentwith a uniform probability distribution in the direction (y) transverseto the compression. Based on this observation and to improve thereproducibility of the data, experimental collapse data were averagedin thin strips whose long axis lies in the transverse direction. Thus,we have access to only P(x,t), which is given by the spatial averageof P over the y direction, and we believe that the full probabilitydistribution depends at most weakly on the transverse direction.

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experiment, they are not observed in the interior of the bubblelayer.

To complete the description of the energetics of a layerconfiguration, we include a simple Ising description of thelayer variables σi :

Hs = −J

N−1∑i=0

σi+1σi + h

N−1∑i=0

σi. (3)

The implications of this description are twofold. We assumea ferromagnetic coupling J > 0 for the layer variables, whichimplies that there is domain wall energy associated with thetransition between single and double layers. This results fromthe fact that bubbles on the boundary of the double layernecessarily have fewer contacting bubbles and thus suffer alarger surface energy cost due to their position. The secondterm, representing an applied magnetic field in the standardIsing description allows for a fixed energy cost per areaassociated with the formation of a double layer; for eachtransition: σi : −1 → +1, the energy of the configurationincreases by 2h. We attribute this term to the (very small)effects of buoyancy, although the energetic effects of differentpackings in the first and second bubble layers cannot be entirelydiscounted as contributing to this term as well. Finally, weexpect that h � J for the bubble raft, as explained in Sec. IV C.

The combination of Eqs. (1), (3) completes our descriptionof the energetics of the model. Of course, any predictions maderegarding the stochastic dynamics of double-layer creationrelies not only on the energetics of the system, but also on theassumptions regarding the source of noise in this inherentlynonequilibrium system. There are multiple nonthermal noisesources present including vibrations coming from the motorused to change the area of the bubble raft and other ambientsources of vibration in the laboratory. The simplest assumptionone can make is to treat this as Gaussian white noise actingon all of the degrees of freedom in an uncorrelated manner.This is equivalent to assuming an effective noise temperature.In what follows we make this assumption, but suppressall references to temperature to avoid any confusion withthe thermodynamic temperature of the system. Thus, the J

coupling and the symmetry-breaking field in h in Eq. (3) areinherently dimensionless quantities.

B. Raft collapse dynamics

To develop a dynamical theory for the system from theabove Hamiltonian, we write overdamped (model A [26])dynamics for the displacement field

ui = −�∂Hel

∂ui

, (4)

based on the assumption that any long-range hydrodynamicinteraction is sufficiently screened by the small depth of theaqueous subphase [27]. We necessarily introduce one newparameter, � with dimensions of a mobility. Using Eq. (1)and Eq. (4) we write a set of differential equations for thedisplacement variables as

ui = k�[ui+1 − 2ui + ui−1 + �(σi − σi−1)]. (5)

The right-hand side of the above equation represents theoverdamped (i.e., diffusive) relaxation of density modes in thebubble layer controlled by the constant k�, with dimensionsof frequency. We justify the use of overdamped dynamicsby appealing to the small mass and large drag forces on thebubbles; please see the discussion in Sec. II.

The second term on the same side of the above equationcouples spin variables to the elastic stresses in the bubble layer.To obtain this second term, we have written the

relation between the elastic reference state the local spinvariable as

�(σ ) = (�> + �<)

2+ �σ, (6)

where the constant � = (�> − �<)/2 measures the changein the reference state associated with a local transition betweenthe single and the double layer. The distinction between Eq. (6)and the original Eq. (2) is immaterial as long as σ takes discretevalues of ±1. We will, however, continue to use the linear formof the �(σ ) when we replace the discrete spin variable σ witha continuous one si = 〈σi〉 representing the ensemble averageof that spin variable.

We also introduce a simple Glauber-based first-order differ-ential equation to describe the dynamics of the time-dependentaverage of the spinlike layer number variable si = 〈σi〉:

τs si = −si + γ

2(si+1 + si−1) − tanh (heff(t))[1 − γ η]. (7)

This corresponds to selecting a particular nonequilibriumdynamics for the Ising system based on single stochasticspin flips that do not conserve the total magnetization, i.e.,the amount of the system in the single- or double-layerconfiguration. This sort of spin dynamics is well describedin the literature, see Refs. [28,29]. It represents a truncation ofa system of differential equations representing the dynamicsof the nth moment of the spin distribution 〈σ1 . . . σn〉 in termsof higher moments. The Glauber dynamics enforces a closurerelation at the second moment and is applicable in the limitof small applied field h. Fortunately, this limit is appropriatefor the bubble-raft system in which the energy differenceper bubble between the single and double layers is small.In what follows, we will see that this approximation maybreak down under large compressive loading, where the elasticstrain energy generates a large effective ordering field, whichdominates over the inherent one h. We return to this point inthe discussion of the results.

In the Glauber approximation, the effect of the localJ coupling that energetically penalizes transitions in thelayer number enters the dynamical equations in terms ofγ = tanh(2J ) and η = tanh(J ); γ = 2η/(1 + η2). Finally, thefunction

heff(t) = tanh{h + k�[ui+1(t) − ui(t)]} (8)

provides the effective local magnetic field to which the spinvariables respond. The first term accounts for the (small)energy difference between the single- and double-layer bubbleconfigurations that favors the single-layer geometry. Thesecond term is the elastic correction that, under compression,is negative and that drives the local spin variable from thesingle-layer (−) to the double-layer (+) structure. In the

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standard derivation of Glauber dynamics and in our modifiedone as well, both h and k�a have dimensions of energy. Due toour suppression of the thermal energy scale in this derivation,the corresponding energy denominator is missing here. Wecome back to this point in the next section.

Taken together the first two terms on the right-hand sideof Eq. (7) account for spin diffusion induced by that nearest-neighbor coupling and the local exponential relaxation to zeroof the average si of the microscopic layer variable σi . Ofcourse, the final term on the right-hand side shifts the time-stationary value of si = S to

S = −e2J tanh[heff]. (9)

This result is the usual mean-field solution of the Ising model,at least in the limit of small h, where the Glauber dynamicsassumption is applicable.

If one were to assume that the spin dynamics were fast,τs � heff/heff , the mean spin variables would simply trackthe effective magnetization. The spin dynamics representingthe formation of the double layer would be enslaved to thestate of raft compression and independent of compression rate.The other limit, in which we assume that the spin variablesare slow compared to the rate at which the elastic variablesrelax, however, frees the double-layer formation to follow itsown dynamics in a manner that depends on the compressionrate of the raft. This latter case is also reasonable since theformation of the double layer is slow, relying on rare stochasticevents. The elastic relaxation of the bubble raft, on the otherhand, is rapid. It occurs on time scales associated with thediffusive relaxation of compression in the elastic network ofbubbles coupled to the viscous subphase. Consequently, weexplore the nonequilibrium dynamics of the coupled elasticand spin system, where the transitions between spin up and spindown are slow in order to address the formation of particulartransient structures obtained in the experiments as a functionof compression rate.

To explore the dynamics of the coupled system of spins andelastic displacement variables, we numerically solve Eqs. (4)and (5), but replacing the stochastic spin variables σi by theirensemble average si to close the set of dynamical equations.These equations are solved subject to boundary conditionssuch that the bubble layer is contained to move with a givenvelocity at the boundaries

u1(t) = −uN (t) = vt, (10)

and that the system starts with a double layer at the boundary

s1 = sN = +1, (11)

due to the favorable bubble-wall interaction. To compareour results to experiment, we note that the probability ofobserving a double layer at i is related to the mean-field spinvariable by Pi = (si + 1)/2. Thus, our solutions for si may bedirectly compared to the measured probability distributions asa function of both compression speed v, see Eq. (10), and thestate of compression c = �L/L. A representative sample ofsuch solutions are shown in Fig. 4.

Figure 4 shows the si for various values of compressionspeed, increasing down the column. It is clear that the mainqualitative features of the experiment are reproduced here.

FIG. 4. Time evolution of the probability of double-layer ap-pearance measured in terms of 〈s〉 (where s = −1 (red),+1 (yellow)corresponds to single and double layers, respectively) as a functionof the distance from the walls. For slow compression V = 0.1 (top)the double layer, stabilized by the interaction with the walls smoothlyprogresses inward a constant rate. At intermediate compression ratesV = 1 (middle), however, the probability of double-layer appearanceacquires a distinct step, indicating the introduction of a quenched-indomain wall in the layer variable. At higher compression rates V = 10the number of these steps grows until they merge (bottom) and thedouble layer grows uniformly. In all cases velocities are given indimensionless units, see Eq. (14), and D = 104, h = 0.1, J = 1.0,� = 0.5, and k = 10. The time axis has been rescaled such that theraft has the same compression at each time point in all figures (Time =1, �L/L0 = 0.8). As time progresses (to the right) lengths (verticalaxis) are rescaled so that the remaining width of the raft is fixed to beunity.

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FIG. 5. Time evolution of the probability of double-layer ap-pearance as a function of the distance from the walls along thecompression direction for a significantly stiffer layer: D = 104,h = 10−2, � = 0.5, and k = 104 for V = 0.1. For these parametersthe system exhibits six distinct steps during the compression. As thevelocity is increased, fewer but larger steps occur; eventually, the stepsize reaches the system size so a single, uniform jump from single todouble layer occurs.

First, at sufficiently slow compression speeds, the double-layer regime begins at the walls (where double layers areheterogeneously nucleated) and a single front of single-to double-layer transition moves inward with compressionuntil the entire raft becomes a double layer. At very highcompression speeds, one observes a spatially uniform increasein the transition probability from single to double layer, asexperimentally observed, see Fig. 3. The most interesting resultof the theoretical analysis occurs at intermediate compressionspeeds where multiple steps in the probability distribution fordouble layers appear. One observes at least two such stepsin the numerical results presented in the middle panel ofFig. 4. There is some evidence of such steplike features inthe experimentally observed distribution of single and doublelayers, see Fig. 3(b). The theory also predicts the formation ofa hierarchy of many nested steps appearing in the double-layerprobability in sheets with much higher compressional modulias compared to the bubble raft. An example of this theoreticalresult is shown in Fig. 5, where one observes six such nestedsteps.

C. Time scales and comparison to experiment

In order to make a comparison between theory andexperiment, a discussion of time and length scales is in order.It is most simple to introduce nondimensional time and spacevariables: t = t/τs and u = u/a, respectively. In terms of thesevariables, the relaxation of compression is now given by

D = k�τs, (12)

and the dimensionless � appearing in Eq. (5) is replaced by� = �/a = 1/2. In the spin dynamics, given by Eq. (7), thespin relaxation time scale is now unity. The effective magnetic

field heff(t) now depends on the dimensionless time t via

heff(t) = tanh{h + k�a2[ui+1(t) − ui(t)]}. (13)

Finally, in these nondimensionalized units, the compressionspeed is given by

V = vτs/a. (14)

The dimensionless speeds corresponding to the experimentalfast and slow compressions are then 1.72 × 10−1 and 1.72 ×102, respectively.

The parameter D physically represents the separation oftime scales between the equilibration of in-plane stressesoccurring on the time scale of 1/(k�), see Eq. (5), and thetransition time for single- to double-layer exchange, given byτs . For our model to be relevant, we need the former to bemuch faster than the latter, or a large value of D. The value ofD in the experiment can be estimated from the lifetime of thedouble layers and the relaxation of compression waves acrossthe system. Given the observed lifetime of small double-layerregions, we take τs ∼ 10 s. From the observed rapid (on theorder of milliseconds) relaxation of compression across theraft as a whole, we estimate D ∼ 104.

We now turn to the estimate of the other parameters enteringthe model. We may express the bubble-spring constant in termsof the bubble surface tension Tsurf and radius a, using k ∼Tsurf/a ≈ 10 dyn/mm2. Thus, we find that k�a2 ∼ 2.5 erg.If we associate the magnitude of the symmetry-breaking fieldh with the work done against buoyancy ∼ρg(4π/3)a3 × a

in pushing a bubble down to the second layer in water ofdensity ρ ∼ 1g/cc3, we find that h ∼ 0.4 erg. Although thisresult justifies our assumption that h � k�a2, we still do notknow the magnitude of the noise temperature in the system.Consequently, we treat the overall scale of h,J as adjustableparameters, consistent with the fact that h � J ∼ k�a2.

V. DISCUSSION

Using these numerical estimates, we find that the ex-perimental data in the fast and slow compression regimesis consistent with the model prediction of an inwardlypropagating wave of double-layer formation in the slowcompression regime and a spatially homogeneous flip in to thedouble layer at high compression velocities. In all cases, thesystem accommodates the loss of surface area first by uniformcompressive strain in the single-layer raft (since D � 1, elasticstrain relaxes across the system rapidly), but eventually thiscompressive strain is relaxed by local transitions from single todouble layers. At slow compression rates, this transition beginsat the walls, due to heterogeneous double-layer nucleationthere, and propagates inward with two symmetrically placedtransition fronts, in order to minimize the domain-wall energyassociated with J . At sufficiently high compression rates,the elastic system is driven to higher compressive strainsas the single- to double-layer transition becomes trapped ina metastable state. Eventually, the energetically unfavorable,high-compression single layer jumps homogeneously into thefavored double-layer state.

Most interestingly, the model makes the nontrivial pre-diction of a stepwise double-layer probability distributionat intermediate compression rates. The dynamical crossover

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from the slow dynamics of a single propagating front tothe fast dynamics of homogeneous transition proceeds viathe formation of multiple propagating fronts of single- todouble-layer transition probability. In fact, for sufficientlylarge J and for sufficiently high k, one can follow thetransition from a single front to homogenous single- todouble-layer transition through a continuous reduction of thenumber of steps. One may consider the single front to bein actuality the maximal number of steps; the step size hasbeen reduced to the microscopic length cutoff (here the bubblesize) causing these steps to merge into a single smooth front.As the compression velocity increases, the number of stepsdecreases and consequently the typical step length grows.Eventually this step length approaches the system size. At thispoint the transition from single to double layer is effectivelyhomogeneous in space, and we recover the observed fastcompression dynamics.

As an example of transition region, one may observe inFig. 5 six distinct steps in the transition regime betweenthe (slow compression) front and the (fast compression)homogeneous jump. It is to be noted that in order to resolve thislarger number of steps, the spring constant had been increasedto 104. As the spring constant is reduced (and as J is reduced)the steps become less sharp so that the full hierarchy of stepscannot be resolved in the dynamics. Using the parametersconsistent with the current set of bubble-raft experiments, weexpect to resolve no more than one or two steps.

The experimentally observed difference between single-to double-layer dynamics associated with, respectively, slowand fast compression is clearly reproduced by the model. Inaddition, the experimental results, while noisy, support thebasic step-forming picture that emerges from the model in thatwe see secondary fronts at intermediate compression rates.The formation of these nested fronts is a nontrivial predictionof the model, and their experimental observation validates thebasic structure of the analysis. We cannot yet experimentallyobserve the much richer structure of a large number of steps.Based on our numerical solutions of the model, however, weconclude that the bubble raft is insufficiently stiff elastically(too small k) to support the full step hierarchy. With our current

bubble radius of a few tenths of a mm, the elastic constant ison the order of 10 dyn/mm2. Varying the bubble diameter by2–3 orders of magnitude in future experiments will allow usto explore a similar range of elastic constants.

Even if the stiffness of the raft were increased, observingthe birth of multiple fronts in experiment (e.g., as in thetheoretical prediction shown in Fig. 5) would still provedifficult. This difficulty stems from the fact that, as thenumber of fronts grows, the compression-rate range consistentwith their formation becomes quite narrow. Nevertheless, webelieve that finding experimental support (or refutation) of thismodel has broad implications for the nonlinear elasticity of avariety of biological and synthetic materials, as outlined inthe introduction. Moreover, this work suggests a number ofintriguing extensions, such as the correlation between the col-lapse and the applied force under compression, understandingthe mechanics of these systems under more complex inducedstrains (as opposed to uniform compression), and experimentsin which one directly drives the internal variable to induce newstresses into the elastic manifold via local and incompatiblechanges to the stress-free, reference state. Such experimentswill have important implications for the charging of superca-pacitors. Finally, we note that same analysis provided here foramorphous, high-symmetry elastic systems, can be extendedto lower-symmetry crystalline materials. These allow morecomplex changes to the local reference state beyond simplechanges in area. In addition, they allow the interaction ofchanges in the reference state and local topological defectsin the structure. These extensions can also be pursued usingbubble rafts as a model system providing a (perhaps unique)combination of access to collective mechanics and the trackingof the microscopic strain and internal state variables innonequilibrium systems.

ACKNOWLEDGMENTS

A.J.L. and D.K. acknowledge partial support from NSF-DMR-1309188. M.D. and C.-C.K. acknowledge the supportof NSF-DMR-1309402 and Research Corporation.

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