key: cord-1053177-8g15d6lc
authors: Ritacco, Hernán A.
title: Complexity and self-organized criticality in liquid foams. A short review
date: 2020-10-06
journal: Adv Colloid Interface Sci
DOI: 10.1016/j.cis.2020.102282
sha: d57b6e64f70a93382765323667bb724e47a8f1a1
doc_id: 1053177
cord_uid: 8g15d6lc

This short review deals with the work done on liquid foams within the framework of the physics of complexity. It aims to stimulate new theoretical and experimental work on foam dynamics as complex dynamical systems. In particular, it examines these systems in relation to Self-Organized Criticality (SOC), for which foams could be used as an accessible experimental model system.

Before starting this review, written for an issue honoring Professor Ramón González Rubio, let me first tell you why I have chosen complexity in foams as the topic to write about. Ramón has published more than 200 papers in the area of liquid interfaces, surface rheology, surfactants and polymer-surfactant complexes in bulk and at interfaces, among others. We have co-authored several papers on these subjects [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] . Thus, when I was invited to write this article, I thought first to write about one of those subjects, in the line of our joint work. Then, I realized that the rest of the researchers participating in this special issue would probably do the same, and I had a second thought. I spent five years in Ramón"s lab as a postdoctoral fellow, from 2006 to almost 2011; during those years, I was encouraged by him and other staff members, to pursue my research interests freely. One of these interests was foam dynamics within the framework of complexity and Self-Organized Criticality (SOC) [16] , which I had started studying in a previous postdoctoral position and continued in Ramon"s group at the Universidad Complutense de Madrid. The opportunity to choose my own line of research following my own interests was very important for me at that time, and that is why I finally decided to write about complexity in liquid foams. All my gratitude to Ramón for his support during all those years.

This review deals with the dynamics of foams within the framework of complexity and Self-Organized Criticality (SOC), thus let me start by introducing the physics of liquid foams in this section.

Foams are ubiquitous systems in nature and everyday life [17] . The physics of foams has been reviewed from different points of view in many books and reviews [17] [18] [19] [20] [21] . Here, I will just provide a brief description for those who are not familiar with the field of foam physics.

Liquid foams are two-phase systems formed by gas dispersed in a liquid matrix [18, 20] . The gas is dispersed in the form of bubbles, portions of gas enclosed by liquid films, packed together to form a closed cell structure: the foam (see Fig. 1 ). Both the shape of the bubbles and the whole structure of the foam depend on the relative contents of gas and liquid [22] . This is represented by the liquid volume fraction (or gas volume fraction,  g ),  l =V l /V f , being V l and V f the volumes of liquid and foam, respectively ( g =1- l ). For large liquid fractions, the bubbles are spherical (minimum area for a given gas volume), but as  l is reduced, bubbles deform adopting shapes of polyhedra, they pack together and compress against each other as liquid drains and films get thinner (see Fig. 1 ). In the limit of dry foams,  l <0.05, the liquid films separating bubbles smoothly meet three at a time, at 120 degrees angles (first Plateau equilibrium rule [18] ). The intersection of these films forms liquid channels between adjacent bubbles; these J o u r n a l P r e -p r o o f channels are called Plateau borders. Plateau borders meet four at a time in a vertex or node, at angles of approximately 109 (second Plateau equilibrium rule [18] ). Plateau borders and nodes configure a network through which liquid can flow by gravity and capillarity (see Fig. 1 ). Fig. 1 : Foam in the gravity field. Liquid drains out of the foam though the Plateau borders and nodes. A vertical liquid profile develops: at the bottom, where the liquid fraction is high, bubbles are spherical; at the top, they are polyhedral. For dry foams, at the top, Plateau's local equilibrium rules are maintained. Due to pressure differences between neighboring bubbles (Laplace, P~1/r), they coarsen, and gas flows from the small into the large bubbles.

Photo courtesy of Professor Douglas Durian.

The structure just described is not at equilibrium; foams are out of equilibrium systems. To create foams we need to do some work [23] (e.g.to agitate them). This work is used to create the interfaces enclosing the bubbles. The energy needed to create the interfaces is given by the product of the surface tension, , and the total area created A. This excess of interfacial energy continuously drives the whole system to minimize its area and towards the complete phase separation, i.e foam destruction, which is the real (absolute minimum) thermodynamic equilibrium state. That is why additives, such as detergent, are needed for foaming. These additives are surface-active agents, generally surfactants, but proteins, polyelectrolytes, and nano-and microparticles as well as mixtures of them can be used. Surfactants [24] are molecules that spontaneously adsorb at liquid-air interfaces; they reduce the surface tension and thus the energy needed to create the bubble interfaces. Foams, when created, are trapped in one of many metastable states; the selected state depends on the system history, but for a given liquid fraction, the geometry of both the bubbles and the foam structure in such metastable state is determined by the minimization (local minimum) of the interfacial energy [22] (i.e. the interfacial area). However, surface tension alone cannot explain foam stability [20] . Surface active agents also confer certain properties to the interfaces, such as surface viscoelasticity [25] J o u r n a l P r e -p r o o f for instance, helping to stabilize the liquid film against rupture. The presence of surfactant molecules not only reduces the surface tension but also helps to kinetically stabilize the system by slowing down the three main dynamical processes that drive liquid foams to their final end, namely drainage, coarsening and coalescence [18] . Drainage refers to liquid flow through the Plateau border network driven by gravity and capillarity. Immediately after freshly forming, the liquid begins to drain out of the foam due to gravity; the top of the foam becomes dry while the bottom, in contact with the solution from which the foam was formed, remains wet ( Fig. 1) . A vertical profile of liquid content develops along the height of the foam in such a way that, in the metastable state, the force of gravity is balanced by the vertical pressure gradient [26] .

Drainage, for which theoretical models exist [21, 27] , is the most widely understood of the three mentioned dynamical phenomena.

Coarsening or disproportionation refers to the continuous change in bubble sizes as foam ages, due to gas diffusion among them. Large bubbles grow at the expense of smaller bubbles in contact with them, which shrink. The driving force for this process is the difference in pressures between bubbles of different sizes. The Young-Laplace equation [28] states that the internal bubble pressure varies as the inverse of its radius, thus the gas tends to diffuse from small bubbles to large ones. In order to diffuse, the gas has to traverse the liquid films separating the bubbles; to accomplish this, the gas needs to solubilize in the liquid; thus, for a perfectly insoluble gas, coarsening is not possible. Coarsening is also arrested if the bubbles are all exactly the same shape and size (monodisperse). In this case, all the bubbles have the same internal pressure, and no driving exist for gas diffusion (except for the bubble layer in contact with the atmosphere). The rate of disproportionation not only depends on the gas solubility and the distribution of bubble sizes but also on the kind of surface-active agent used. The gas has to go through the interface covered by surfactant molecules (or polymer, particles, etc.) that could act as barriers. Moreover, in the case of foams stabilized by solid particles, they can completely arrest the coarsening process [29] . Coarsening is also arrested when the surface compression modulus of the surfactant monolayer reaches a value equal to about half the surface tension [30] . There exist theoretical models for coarsening in 2D foams (foams sandwiched between two glass plates separated by distances smaller than the bubble size) [18, 21] , and they are quite well understood. For 3D foams, the situation is different [31] . In general terms, coarsening is less well understood than drainage.

The process of foam coalescence is the least understood of the three [32] . It refers to the rupture of the liquid films separating bubbles in a foam structure. The first model of rupture of a single isolated liquid film was proposed by Sheludko and extended by Vrij [33] , and it is based on the assumption that film ruptures because of thermal fluctuations in the film thickness. At a certain critical thickness, van der Waals forces act between both gas/liquid faces of the film producing J o u r n a l P r e -p r o o f an instability that cannot be damped and grows leading to film rupture, being the process controlled by surface tension and disjoining pressure [19] . Exerowa et al. [34] and de Gennes [35] proposed that a single film ruptures via thermal fluctuations of the surfactant concentration at the film interfaces. These fluctuations could produce holes (i.e. regions without surfactant molecules) at the interfaces. If the size of these regions is greater than a certain critical value (typically in the order of half the film thickness), the hole grows and the film breaks. This process is controlled by compression surface elasticity. Both models are developed for single isolated films. However, no clear correlation exists between the dynamics of a single isolated film rupture and the dynamics of the coalescence of those films in macroscopic 3D foams [36] .

For them, it was reported that coalescence occurs when the bubble radius reaches a critical value (by coarsening) [37] ; other authors reported that bubbles coalesce when the pressure difference between the gas and the liquid within Plateau borders reaches a critical value [38] . Foam destabilization mechanisms in which the dynamics of fast rearrangements of films triggered by coarsening are involved were also proposed [39, 40] . Additionally, a film rupture within a foam might trigger a cascade of ruptures, a phenomenon that makes it more difficult to correlate single film stability with macroscopic foam stability. Coalescence in foams is by far less well understood than drainage and coarsening.

After all the above description of foam dynamics, we conclude that, in fact, most liquid foams are not even metastable; they evolve continuously by drainage, coarsening and coalescence.

Each of these processes, when considered in isolation, acts on different timescales. The rupture of a single film occurs in fraction of seconds; the metastable vertical liquid profile is reached by drainage in minutes; finally, coarsening might last for hours. Despite this separation of timescales, in macroscopic foams, the three processes affect each other. For instance, the rupture of a film releases liquid that is collected by Plateau borders and films, locally increasing the liquid fraction and the film thicknesses, which in turn modifies the coarsening rates and the film stability against coalescence. Quite recently, a new numerical simulation approach, under certain simplifying assumptions, has been presented; it simultaneously takes into account the occurrence of the three processes in the modeling of foam dynamics [41] .

I have already mentioned the occurrence of bubble rearrangements or topological changes. They refer to any process that changes the number of neighbors of a bubble (i.e. the number of faces of that bubble). In 2D foams, all rearrangements can be expressed as the combination of two elementary topological changes (Fig. 2) . One of them is film switching (Fig. 2a) , known as T1

process. The second elementary topological change is the disappearance of a bubble or cell (Fig.   2a, 2b) , called T2 process. In 3D, the topological changes are a bit more complex, but the general idea is the same [21] .

These topological changes often occur in cascades. These avalanche-like dynamics that, as I have mentioned earlier, can also be seen in the rupture of films when foams coalescence are the main subject of these review. 

Some years ago, Weaire and Hustzler [42] published an article entitled "Foams as complex systems" in which they summarized very briefly the existing papers on 2D foams within the framework of complex systems; since then, the literature on the subject has been scarce. Before discussing the research on foams as complex systems, and having introduced the reader to foam physics in the previous section, let me now introduce complexity in physics.

The word complexity derives from the Latin word plectere, which means to weave and/or entwine. But what do we mean for complexity in physics? To answer the question, let me propose you to examine some particular examples [43] . First, consider the behavior of a small number of ants, say twenty. If you release them into a wood, they will go around endlessly doing nothing, just walking until they die of exhaustion. However, release one hundred thousand of them and you will find an "emergent collective behavior" that ensures the survival of the colony by collecting food, eliminating enemies and building shelters for breeding their larvae. Each of these ants is a blind and unintelligent animal that communicates by means of a few simple chemical signals. However, thousands of them behave as if they were an intelligent superorganism. The very simple rules that govern the behavior of a single ant produce an emergent structure and a collective behavior that is much more than the sum of the features of the interacting entities (ants). Let me give you a second example, think of our brain, a group of hundreds of thousands of neurons arranged in a network, which can be essentially in one of only J o u r n a l P r e -p r o o f two possible states: inactive or firing a signal to other neurons. A particular neuron sends a signal if it receives enough signals from other neurons. These simple rules produce signal patterns (electric and chemical) in the neuron network that give place to an emergent collective behavior, which in turn results in our intelligence, thoughts, learning capacity, feelings and consciousness. Again, the emergent result of simple interactions among individual entities in the system is, by far, much more than the sum or the average behavior of these individual entities.

The same can be said of the immune system, the stock market (in this case, the single entities are humans acting by self-interest) and the spread of a disease (quite relevant in the current COVID-19 crisis) [43] .

All the examples I have given so far involve some living entity; but, of course, complexity is not restricted to living systems. Complexity can be found in a pile of rice or sand, in forest fires, in an earthquake, in droplets on a glass window and in vortices in superconductors [44] , as well as in solar flares and in other astrophysical systems [45] , to name just a few. The emergent complexity of these inanimate systems arises from their statistical behavior (what we mean by this will be clarified later on). All the mentioned examples are many-body systems, interacting via some kind of signals or information that organizes themselves without any tuning from outside the system, resulting in an emergent collective behavior that cannot be described simply by averaging or summing. Thus, let us define a complex system in physics, one that exhibits complexity, as a many body system exhibiting nontrivial emergent and self-organized behavior [43] .

All the systems mentioned are dynamical (i.e. systems that change and evolve in time). In these dynamical systems "the whole is not the sum of its parts" which means that they are nonlinear dynamical systems. These nonlinearities could eventually lead to chaos. Chaotic systems are those that exhibit a sensitive dependence on initial conditions; that is, even small uncertainties in the measurements of initial conditions can result in huge errors in long-term predictions.

Chaos, bifurcation, period-doubling cascades, attractors, renormalization, critical phenomena, avalanches, fractals, scale invariants, power laws, self-organization and random networks are all terms and concepts related to complexity and complex systems [43] , and almost all of them can be found in relation with liquid foams. Among these concepts, we are particularly concerned here with SOC.

Self-organized criticality attracted a lot of attention the last two decades [44, 46] . This concept was introduced by Bak, Tang and Wiesenfeld in 1987 [47] , and it proposes that complex behavior can develop spontaneously in certain nonequilibrium systems with many-body interactions. These systems are complex in the sense that there is not a single time or spatial scale characterizing the system behavior, but the statistical properties are well described by simple power laws. The absence of the characteristic length and temporal scales is what is observed in the context of equilibrium thermodynamics at critical temperature in a continuous J o u r n a l P r e -p r o o f (critical) phase transition and the reason why the word "critical" is used in SOC. The selforganized part in the name SOC implies that the system reaches the critical state by itself without any tuning from outside (unlike to what happens in a critical phase transition).

When Bak, Tang and Wiesenfeld (BTW) introduced the concept of SOC [47] , they aimed to explain the origin (and mechanism) of self-similar fractal structures, in time and space, found in so many physical systems and phenomena. The following is the main idea [44] : in many-body systems, a signal can travel through if there is a connected path above a certain threshold for that signal to travel. The region above the threshold forms a dynamic random network that changes continuously by the combination of the internal relaxation and the continuous (slow)

driving by the external field. The signal stops when it cannot find a region above the threshold to continue travelling, and the system reaches a new metastable state. Then, by the action of the external field, some regions of the system are slowly driven above the threshold once more, and the internal relaxation is restarted. The dynamics is intermittent, with periods of activity and inactivity. The authors suggested that the dynamical network formed by the path followed by the signal has a percolating-like fractal geometry. The fractals could be of any size, as well as the time of the internal relaxation processes that lead the system to a new metastable state (i.e.

power law distribution functions and lack of spatial and temporal characteristic scales).

Systems exhibiting SOC behavior and SOC models are defined in terms of some dynamical variable (e.g. the stress on the earthquake fault [48] [49] [50] or the slope in a pile of rice [51] ). These dynamical variables evolve over time by the presence of a "field" (e.g. the slow movement of tectonic plates or the slow addition of a rice grain to the pile). The field slowly drives the system to undergo an event or, using the previous vocabulary, the signal (e.g. an earth movement, the displacement of a rice grain on the slope of the pile) when a certain threshold (stress or slope) is locally overcome. These individual events could produce avalanches of events of different sizes leading, for instance, to an earthquake. The statistical size distribution (energy realized in an earthquake or the number of rice grains involved in an avalanche) of these events follows power laws (scale-invariant, long-range spatiotemporal correlations). Thus, the key ingredients for the SOC behavior are: the power laws, the presence of thresholds and metastability (e.g. the friction force between the plates for earthquakes), and finally a slow external driving when compared to the internal relaxation times of the system (e.g. the movement of the tectonic plates that increases the stress lasts decades or centuries, but the internal relaxation -an earthquake-occurs in minutes).

All these ingredients are present in the dynamics of liquid foams. Foams are many-body metastable systems that are continuously driven to equilibrium by drainage and coarsening. This J o u r n a l P r e -p r o o f dynamics lasts from minutes to several days, but the local internal relaxation, bubble bursting or topological changes occur in the range of seconds or fractions of a second [52] .

In foams, the dynamical variables of the system could be the bubble radius or the pressure difference between the gas and the liquid in Plateau borders. For example, for foam collapse, as it was mentioned in §1, some researchers suggested that coalescence occurs when the bubble radius reaches a critical value (by coarsening) [37] ; others reported that bubbles coalesce when the pressure difference between the gas and the liquid within Plateau borders reaches a critical value [38] . In both cases, coarsening acts as "field" driving parts of the system above a local threshold that produces the first bubble rupture or rearrangement. If the foam is in a SOC state, this first event, rupture or topological change, could produce avalanches of events of any size, in which the macroscopic foam dynamics is now controlled and dominated by the collective dynamics regardless of the microscopic features of the liquid films and bubbles. If this is the case, the size and temporal probability densities of events should follow power laws. For the limit of infinite system sizes, an exponent <2 for the power law implies that the average of the distribution does not exist, and that for exponents <3, the standard deviation is infinite. In general, for finite size systems, the distribution of the avalanche sizes s (or duration, ) should be [46] ,

with a certain lower cutoff, s 0 (e.g. in bubble rupture, an event involves at least one bubble, then s s 0 =1). The function f(L) is a certain function that tends to 1 as the linear system size, L, tends to infinity [46] . For example, some systems exhibit a crossover from power law to exponential behavior as s increases above a certain value s 1 , in such a way that ( ) [ ⁄ ] for s>s 1 , where s 1 scales as , with >0.

It is within the framework of SOC dynamics that liquid foams could be thought as complex dynamical systems. The search of power laws distributions in the size and temporal distributions of bubble ruptures or topological changes in liquid foams and its relation with SOC is reviewed in the following sections.

Foams can be considered as complex systems in several ways, for instance focusing in the fractal structure of foams [53, 54] and foams flowing through porous media [55] and in Hele J o u r n a l P r e -p r o o f Shaw radial cells [56] (see §3, Fig. 8 ) or even the fractal-like patterns in chaotic light scattering by foams [57] . Ensembles of soap films were even used as synthetic systems exhibiting some characteristics of the statistics of human mortality [58] . Foam production can also serve as an experimental realization of the "period doubling route to chaos" (bifurcation) mimicking the behavior of the logistic map [59] , as observed in the production of bubbles in microfluidic devices [60] (see Fig. 3 and its caption for a brief explanation). In 1995, Brunet et al. [61] described the dynamics of breaking foams stabilized with SDS surfactant and sandwiched between two Plexiglas plates. They used a light lamp for gently heating these 2D foams to trigger film ruptures whose dynamics was followed by means of a CCD camera. The authors clearly identified a regime where the dynamics is controlled by a collective, cooperative behavior: cascades of film ruptures.

Probably the first article on the subject of foam collapse related to foams and SOC is the one by Vandewalle and Lentz [64] continued the work done in reference [63] , studying 2D and 3D

foams by imaging the foams with a CCD camera. The 3D foams were produced in a cylinder vessel and observed from above at the air/foam interface. The rupture of bubbles was detected by subtracting two successive images. This procedure also allowed observing topological changes such as edge and vertex movements. They found that these dynamics are temporally and spatially correlated (avalanches). The 2D foams were studied in a vertical Hele-Shaw cell that allowed observing the dynamics not only at the foam/air interface, as it was the case of 3D

foams, but also in the bulk. This allowed them to observe film ruptures and topological changes (side switching, T1, and vertex disappearance, T2; See Fig. 2 ) triggered by the film rupture. Figure 5 , reproduced from reference [64] , shows a film rupture followed by topological rearrangements. For these 2D foams, they also found that, as in 3D foams, film ruptures and topological changes are temporally and spatially correlated. They also observed that film ruptures and topological changes are independent of film size and curvature. This independence from the local (microscopic) characteristics and the existence of cooperativity and avalanches are the hallmark of SOC. However, the authors neither statistically analyzed these events (ruptures and topological changes) nor mentioned that possibility (SOC). In line with the work done by Park and Durian [56] , Okuzono and Kawasaki [67] conducted computer simulation experiments on the rheology under steady shear of 2D foams both in the absence of coarsening and in the dry limit. They used the so-called vertex model, which considers the viscous dissipation in the liquid. In their simulations, the authors found an intermittent dynamics with avalanches of topology changes. The probability density, P(s), of the avalanche sizes, s, followed power laws with an exponent 1.5. They compared their results with J o u r n a l P r e -p r o o f models exhibiting SOC behavior, in particular with stick-slip models for earthquakes [48, 50] , and suggested that foams indeed could self-organize in a critical state. However, Gopal and

Durian [68] , using Diffusing-Wave Spectroscopy (DWS), performed experiments on 3D foams made from shaving cream (wet foam,  l =8%) to study the dynamics of topological rearrangements in those foams. They conducted DWS experiments before, during and after an applied shear strain. They found that, although the dynamics of topological changes is a nonlinear stick-slip process similar to the avalanches observed in piles of granular media and in earthquakes, DWS showed that the events are temporal and spatially uncorrelated; thus, a characteristic time and spatial scale can be identified in the foam dynamics contradicting the SOC scenario. Despite this, the authors stressed at the end of the article that the resolution of contradictions between the results of DWS and those of computer simulations [67] should lead to a deeper understanding of the dynamics not only of foams but also of other disordered materials. In this line, Durian presented computer simulation results of the complex macroscopic rheological behavior of foams [69] by implementing the so-called bubble model [70] . He found that the distribution function of released energy is a power law for small events but exhibits an exponential cutoff independent of system size (see Fig. 9a ). This result contrasts those obtained by Okuzono and Kawasaki with the vertex model [67] for which, as we have already seen, the behavior is compatible with SOC. Kawasaki and Okuzono [71] extended their work by performing simulations with the vertex model exploring the effects of shear rate and system size. They again found results compatible with SOC (see Fig. 9b ).

J o u r n a l P r e -p r o o f They found that the histograms of the frequency of occurrence of the avalanches are well fitted with exponentials, contradicting the SOC scenario for which power laws are expected.

However, as the authors stated, their simulations are quasi-static and do not involve dynamical variables, which are essential ingredients for SOC.

So, do foams evolve into a SOC dynamical state? Before discussing this question let me first comment on SOC in other physical systems. I will specifically discuss two physical systems, piles of granular materials and earthquakes, trying to draw a parallel between them and the dynamics of bubble rearrangements and collapse in foams.

In their original paper, BTW [47] introduced SOC referring to avalanches in sandpiles. They did not perform a real experiment; they implemented instead a cellular automaton simulation mimicking a sandpile. In the simulations, they found power laws both for the size distribution of the avalanches P(s)~s -1. 35 and for the distribution of the avalanche lifetimes P(t)~t -0.9 . However, experiments on real sandpiles [74] [75] [76] showed that only the sizes of small avalanches are distributed according to power laws, exhibiting (pseudo) SOC behavior. This behavior is cut off and buried for large avalanches. The latter dominate the whole dynamics of the pile slope; for them, the size and duration of the avalanches are narrowly distributed [76] . It seems that this behavior has its origin in inertial effects: once the grains role down the pile slope, they gain Knopoff spring-block model 1 for earthquake faults [77] . The OFC algorithm is as follows: a 2D lattice is defined and a dynamical variable E i is randomly assigned to each lattice site i,, being E i a force (or an energy). Now, all sites are driven simultaneously at the same rate E i =E i +a, being a a constant. When in a simulation step the dynamical variable of a site i becomes larger than a certain critical value Ec, the site is relaxed according to the rule,

where E n,n refers to the neighboring sites of the lattice site i. The excess energy of the overcritical site i is distributed among its neighboring sites. Depending on the value of α, part of the energy is lost (dissipated), the model is nonconservative, except when α=1/q i being q i the coordination number of the site i. The OFC model exhibits SOC behavior, and the distribution of size events followed power laws P(E)~E -B . The B values were found to be dependant on α and a transition from power laws to exponential decay was found for a certain value of α (α ≤ 0.05). This crossover from power law to exponential is what was observed in bubble rafts as bulk viscosity changes [16] (see Fig. 6 ). Here viscosity plays the role of the parameter α In fact, the OFC model could be directly applied to the dynamics of bubble ruptures in bubble rafts. We only need to imagine that each site of the lattice is occupied by a bubble. When the dynamical variable of a particular bubble goes over the threshold by slow driving (E i =E i +a, E i >E c ), it breaks; the energy (the dynamical variable) of the site goes to zero and part of the released (interfacial) energy is transferred to neighboring bubbles (rules in eq. (2)). This is what we did in our cellular automaton in reference [16] . Figure 10 , reproduced from [16] , shows a sequence of images taken with a fast CCD camera that shows the effect of a bubble rupture on its neighbors on the bubble raft. After the bubble rupture, a cascade of ruptures could follow due to the mechanical perturbation produced on the neighbors (see video in [78] ). The relation between the phenomenon observed and the OFC model is obvious. Fig. 10 : Sequence of a bubble rupture on a raft of bubbles and its effect on its neighbors. The images were taken with a fast CCD camera, being the time between snapshots of 0.00225 s. The energy released by the rupturing bubble is transferred to the neighbors that in turn can rupture, triggering cascades of ruptures (see video in reference [78] ). This is exactly the mechanism proposed in the cellular automaton OFC model (see text). From [16] Now, the question with which we started this section can be answered: not all foams organize themselves in a critical state, but some of them can indeed be in SOC dynamical state for both topological rearrangements and macroscopic foam collapse. Foams could set in SOC dynamical state depending on the particular mechanisms of energy dissipation and thresholds of the system.

Fractal, chaos, unpredictability, SOC, power laws, cooperativity and avalanches are all terms and concepts associated to complex nonlinear dynamical systems that can also be found in relation to foam dynamics. The concept of SOC is of particular interest. Self-organized criticality applied to foams does not necessarily have to capture all the details of the phenomena nor does it need to explain everything about foam dynamics. It could be for example, relevant for small scales, such as small avalanches of bubble rearrangements that are power law distributed exhibiting SOC (or pseudo-SOC), but not for large scales for which "inertia" dominates. At this respect, the result of the bubble model [69] and the DWS experiments on shaving foams [68] are similar to the random OFC model for sandpiles [79] for which and exponential cutoff, independent of system size, exists. The same might be said of the collective dynamics of the collapse of macroscopic foams. For them, the bulk viscosity of the foaming solutions as well as the interfacial properties, such as surface viscoelasticity, could operate to set the foam in a critical state or out of it. If by modifying one of those properties, say the bulk J o u r n a l P r e -p r o o f viscosity as in [16] , we induce a transition from SOC to non-SOC behavior, or vice versa, without noticing it, any attempts to understand the effect of viscosity on foam stability will be difficult or impossible. This could be the origin of the problems to correlate the stability of single isolated liquid films and macroscopic foam stability in some experimental systems. If a foam is in critical state, its dynamics will be independent of the microscopic features of the films, at least in a certain range. However, if the modification of these microscopic properties induces a transition to a noncritical state, there might be a correlation between the properties and the macroscopic behavior of the foam.

From the point of view of SOC, foams might be a perfect experimental system to test ideas and models. We know how to change the threshold/dissipation by adjusting bulk viscosity or surface elasticity and viscosity by modifying the chemical systems used to stabilize them; in this way, the experimental counterpart of the parameter α (or E c ) in the OFC model is changed. Moreover, by changing and controlling the liquid volume fraction in foams, we can change the equivalent to the aspect ratio of grains in piles of granular materials [51] . The slow driving, which is mainly coarsening in foams, can also be changed by using , for example, more or less soluble gases.

In view of the small number of works that have appeared on the subject, it seems that the community of foam physics researchers has ruled out the possibility that the foams are SOC dynamical systems. Certainly, some foams are not in critical state but, as it happens in piles of granular materials, some others could be. Studying how the transition from SOC to non-SOC behavior occurs will help understand more deeply not only foam dynamics and its relation with microscopic features, but also the emergence of SOC behavior in nature.

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Foams: From nature to industry

The physics of foams

Foam and foam films: theory, experiment, application

Aqueous Foams: A Field of Investigation at the Frontier Between Chemistry and Physics

Structure and energy of liquid foams

The science of foaming

Surfactants and Interfacial Phenomena

Influence of interfacial rheology on foam and emulsion properties

Liquid dispersions under gravity: Volume fraction profile and osmotic pressure

Physical chemistry in foam drainage and coarsening

Physical Chemistry of Surfaces

On the origin of the remarkable stability of aqueous foams stabilised by nanoparticles: link with microscopic surface properties

The Collected Works of J. Willard Gibbs

On the influence of surfactant on the coarsening of aqueous foams

Bubble coalescence in pure liquids and in surfactant solutions

Possible mechanism for the spontaneous rupture of thin, free liquid films

Stability and permeability of amphiphile bilayers

Some remarks on coalescence in emulsions or foams

New analysis of foam coalescence: From isolated films to three-dimensional foams

Link between surface elasticity and foam stability

Critical capillary pressure for destruction of single foam films and foam: effect of foam film size

Coalescence in Draining Foams

How topological rearrangements and liquid fraction control liquid foam stability

Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams

Foam as a complex system

A guided tour. 1 st

Self-Organized Criticality: Emergent Complex Behaviour in Physical and Biological Systems

25 Years of Self-Organized Criticality: Solar and Astrophysics

Self-Organized Criticality: Theory, Models and Characterization

Self-organized criticality: An explanation of the 1/f noise

Properties of earthquakes generated by fault dynamics

Mechanical model of an earthquake fault

Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes

Avalanche dynamics in a pile of rice

Relaxation Time of the Topological T1 Process in a Two-Dimensional Foam

Relaxation of fractal foam

Soap, cells and statistics-random patterns in two dimensions

Quantitative Characterization of Foam Transient Structure in Porous Media and Analysis of Its Flow Behavior Based on Fractal Theory

Viscous and elastic fingering instabilities in foam

Hyperbolic kaleidoscopes and Chaos in foams and Hele-Shaw cell

Can soap films be used as models for mortality studies?

Simple mathematical models with very complicated dynamics

Nonlinear Dynamics of a Flow-Focusing Bubble Generator: An Inverted Dripping Faucet

Structure and dynamics of breaking foams

Avalanches in draining foams

Avalanches of Popping Bubbles in Collapsing Foams

Cascades of popping bubbles along air/foam interfaces

Scaling Laws in the dynamics of collapse of single bubbles and 2D foams

The effects of plateau borders in the two-dimensional soap froth iii. Further results

Intermittent flow behavior of random foams: A computer experiment on foam rheology

Nonlinear bubble dynamics in a slowly driven foam

Bubble-scale model of foam mechanics: Melting, nonlinear behavior, and avalanches

Foam mechanics at the bubble scale

Self-organized critical behavior of two-dimensional foams

Bubble sorting in a foam under forced drainage

Statistics and topological changes in 2D foam from the dry to the wet limit

Experimental study of critical-mass fluctuations in an evolving sandpile

Persistent self-organization of sandpiles

Relaxation at the angle of repose

Model and theoretical seismicity

Avalanche of bubble ruptures

Sandpile models with and without an underlying spatial structure

This work was supported by Agencia Nacional de Promoción Científica y Tecnológica 

Hernán A. Ritacco.

1. Fractal, chaos, self-organized criticality (SOC), power laws and avalanches are all terms and concepts associated to complex nonlinear dynamical systems that can also be found in relation to foam dynamics.

Liquids foams have all the ingredients needed for exhibiting SOC dynamical behavior. 3. The dynamics of liquid foams are frequently governed by avalanche-like phenomena involving bubble rearrangements and ruptures. 4. The distribution functions of the avalanche sizes in liquid foams are frequently well described by power laws as in SOC dynamical systems. 5. Not all foams set in a critical state but some of them could. Studying how the transition from SOC to non-SOC behavior occurs in foams will help understand more deeply not only their dynamics, but also the emergence of SOC behavior in nature J o u r n a l P r e -p r o o f