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The rheology of foam
Current Opinion in Colloid & Interface Science 13 (2008) 171 – 176 www.elsevier.com/locate/cocis
The rheology of foam Denis Weaire ⁎ Université Paris-Est, Laboratoire de Physique des Matériaux Divisés et des Interfaces, UMR CNRS 8108, 5 Bd Descartes, 77454 Marne-la-Vallée cedex 2, France Received 1 October 2007; accepted 31 October 2007 Available online 12 November 2007
Abstract Foam rheology has arrived at an interesting stage of active debate, some of it focused on two dimensional systems. Theorists are ambitious to formulate comprehensive models and experimentalists want to look more critically and constructively at the techniques of rheometry. Are there grounds for optimism, that the difficulties of this topic may be overcome? Whatever the eventual extent of success, we argue that liquid foam should stand as the appropriate prototype for that wide class of substances that exhibit elasticity and a yield stress, above which they flow. We focus mainly on the constitutive law for steady shear and the recent debate over shear localization in two dimensions. © 2007 Elsevier Ltd. All rights reserved. Keywords: Foam; Rheology; Shear; Localisation
1. Introduction Foam is a solid under low stress but it flows like a liquid under high stress. It therefore has a yield stress, and belongs to that category of complex materials that may be termed “Bingham fluids”, attributing to them (perhaps rashly) a particular model representation of rheological properties. The importance of those properties in applications is immense, and so they have attracted plenty of attention from industrial researchers. After many years of such research, a general, systematic and practical theory of foam rheology is still lacking. At present there is a tentative sense of convergence towards this challenging goal. Is it justified? We shall argue this case, in reviewing various contributions. We will still be left with more questions than answers. The author is not a rheologist, but comes to the subject from a wide background of experience in modelling foam properties. Rheology is perhaps the next big problem to be faced in that field, following twenty years of substantial accomplishment by an international community in framing and testing satisfactory models for most static or quasistatic properties. ⁎ Permanent address: School of Physics, Trinity College, Dublin, Ireland. Tel.: +353 1 8961675; fax: +353 1 6711759. E-mail address: [email protected]. 1359-0294/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2007.11.004
When foam flows, many more physical effects come into play, and we face a more complex challenge. Books by Barnes et al. [1] and Hutzler and Weaire [2], and a review by Kraynik [3•] reflect various earlier stages of the subject, and may be contrasted with the sophistication of some of the studies that are under way today. We give only exemplary references here, and they may be used to find a way into the already extensive literature. Fig. 1 represents two ways of visualizing the rheological properties in question: either in terms of a detailed structure of cells which are deformed and rearranged by local forces when stress is applied, or a continuum described in terms of average quantities. We shall ask how they are related. 2. Foam as an elastic solid Ordinary foam is polydisperse and isotropic. Its elastic properties are characterized by its bulk modulus and its shear modulus. The bulk modulus is dominated by that of the enclosed gas and is so large as to render the foam effectively incompressible in many practical contexts. We are left with the shear modulus due entirely to surface tension. It is generally a small fraction of (surface tension/mean bubble radius), as might be estimated from simple arguments. There is a range of strain over which the foam is elastic, that is, deformation is recoverable. There is some scope for
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yield stress. Upon further reduction of the stress, so that it eventually the opposite sign, its absolute value can again reach the yield stress. Shear flow again occurs, in the opposite direction to that before. This is the hysteresis to which we have referred. 4. Simple or complicated?
Fig. 1. Microscopic and macroscopic viewpoints. (a) A foam (here in 2D, in equilibrium) is made up of local elements with specific dynamic properties. (b) Its continuum rheological properties may be represented by the dependence of stress on strain and strain rate. How are these two pictures related?
argument about this, strictly speaking, but let us set that aside. The effective elastic range is large, reaching to strains that are a significant fraction of unity (contrasting with most solids, and making nonlinear elasticity interesting). Beyond the elastic range of strain, the foam is plastic. 3. Plasticity and flow The underlying mechanism of plasticity is simple and welldefined for low shear rates (the quasistatic regime); it consists of sudden topological events that rearrange bubbles, so that their local configurations remain stable. These events, which interrupt the otherwise smooth evolution under stress, are easily described in relatively dry foams, for which Plateau's rules apply: they require such changes whenever it happens that more that six films meet at a vertex in 3D, or three lines in 2D, in a dry (that is, low liquid fraction) foam. These are elementary “T1” or neighbour-switching rearrangements, and are a vital ingredient of most theories that seek to link microscopic and macroscopic mechanics. As strain increases yet further the yield stress is reached, beyond which the plastic events continue indefinitely under constant stress, and the foam flows. When stress increases above the yield stress, the flow rate increases. Note that the schematic curve shown is for monotonically increasing strain only. There is strong hysteresis in this system, when the strain rate changes sign. Upon reduction of the stress, the foam reverts to an elastic solid when the stress falls below the
The above scenario seems, on the face of it, rather simple. It is however a mere narrative of the local behaviour of a foam under increasing stress, in crude qualitative terms. A closer look reveals a multitude of possible complications and difficulties. It is often difficult to decide which of these are important. Which should be conscientiously confronted and which are mere niceties, that is, small effects for later consideration? How can we steer a prudent course between Scylla and the Charybdis: naïve idealised models that throw away too much physics and elaborate multi-parameter formalisms that promise much and deliver little? Whatever the complications envisaged, it as well to recall the essential simplicity of a foam in equilibrium, as regards its welldefined local structure and its evolution, and our ability to simulate this in detail. However hard it may be to adequately and generally characterize its rheology, this gives us some hope of eventual theories that are meaningful, transparent and effective. They can set out from the quasistatic case and move progressively towards high shear rates. That is the programme that is under way. 5. Some measurements One particular set of earlier rheological measurements may be indicated as belonging among the best and representative of many: those of Khan et al. [4]. Let us state one regret: it is a great pity that in this and many other cases, a wider range of data was not acquired, to lay down the basis for proper tests of later theories. Instead the authors embarked on lengthy theoretical analysis of their own, based on the inadequate models of the time. Their conclusions were consistent with our sketch in Fig. 1(b), with stress varying with strain rate, to a power which was roughly one-half. For an example of more recent data, roughly consistent with this; see Denkov et al. [5]. 6. The Herschel–Bulkley relation Such data as that of Khan et al. is commonly interpreted in terms of the Herschel–Bulkley relation: r ¼ ry þ gHB ðdg=dt Þn
ð1Þ
(σ is stress, σy is yield stress, γ is strain). For n = 1 this is called the Bingham equation, a common favourite in toy models, and ηHB may be called the plastic or Bingham viscosity. While it may be dangerous at this stage to take any of the exact published values of the parameter n seriously, they do indicate that it is probably not equal to unity. The obvious next question is: what determines n? Vaguely, the answer must lie in the micromechanics of the system, and this shifts the question on to difficult ground, since local
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dynamic effects remain mysterious, as regards the extent and nature of dynamic surface effects associated with surfactants. Recent work of Besson and Debregeas [6•] provides an example of an attempt to clarify these effects. We would prefer to narrow the question to one which has not been recognised in the past, and should now be answerable with the aid of simulations: for a given microscopic model (however simplified or realistic), is there a definite relation between its microscopic parameters and the macroscopic ones of a Herschel– Bulkley model (Eq. (1))? The question is rather general; foams offer a context in which appropriate simple models seem to be closest to reality. This seems to lead inevitably to detailed consideration of the role of T1s (or other elementary processes) in non-affine microscopic deformation. The peculiarity of the mechanics of disordered (non-crystalline) materials is that the deformations (elastic or plastic) are not strictly affine on any scale. See for example the recent analysis of Goldenberg [7•], in search of a formalism that can cope with the subtleties of this kind of response, and another exercise of this kind by Takeshi and Sekimoto [8]. Many other questions are raised by the work of Khan et al., and others. One is: what is the significance of the peak in the curve (of the order of ten percent above the yield stress in their case, and found to a varying degree in many published simulations. And what are its implications? Also, what modifications are required at high strain rates? There are plenty of clues (for example ref [9]) that deformation is quite different above a critical strain rate which is the inverse of the relaxation rate for T1s. Even at low strain rates, individual T1 events are not independent. As the liquid fraction increases, they are no longer separate, but form cascades or avalanches [10]. 7. Modelling at the cellular level An example of a static simulation of a 2D foam was shown in Fig. 1(a). This kind of calculation has been used many times to establish elastic properties, simply by varying the (periodic) boundary conditions. Potentially, much insight can be gained from such detailed cellular models, which have been extended to 3D by Surface Evolver, but they remain poorly developed for dynamic effects. They are still being explored in the simpler two dimensional case. 8. A tactical retreat to two dimensions—or just a diversion? Ever since the days of Plateau and Kelvin, foam physicists have always found simplicity and inspiration in the study of 2D foams. Again and again, these have been shown to have properties that are similar to those of their 3D counterparts, and they are often more tractable in theory, in experiment and in visualization. This strategy presumably motivated the group of Debregeas to construct a rheometer for 2D foam [11••]. It is analogous to the classic Couette rheometer of two concentric cylinders. Two walls (the inner one being rotatable) surround a 2D foam between two glass plates. The results obtained from this have provoked quite a debate, fuelled by subsequent experiments by
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the group of Dennin [12•,13], using both Couette and simple shear geometry (two parallel straight boundaries). The salient feature of the results of Debregeas is strong shear localization or shear banding, located at the boundary which is moving (the inner circle in the original experiment) [12]. It was argued that the localisation effect was not attributable to Couette geometry, and indeed it was found to persist for the straight geometry of simple shear, so we will not take time here to explore this delicate point. For our purposes, the effect of cylindrical geometry is largely irrelevant. Such localisation can be also be seen in various other 2D experiments, especially when flow around obstacles is concerned [14]. Two explanations have been offered for it. Firstly, the cellular simulations of Kabla [15•], with simple shear, showed similar localization, and were then analysed in terms of an elaborate description of local topological changes. This model is quasistatic, and represents the detailed cellular structure of the foam, rather as in Fig. 1(a). Such a dynamic application of a quasistatic model always raises several questions [16]. The program proceeds from one equilibrium to another by topological changes which must be, to some extent, dependent on the details of the algorithm, rather than the unique consequences of dynamical laws. It may seem natural to take the last equilibrium structure as the first approximation to the next, as the boundary shear increases, but at least one alternative has been applied by others—that of affine increments. Also, in a quasistatic picture, there can be no distinction between moving the inner boundary and moving the outer one. In experiment, localisation always occurs at the moving boundary but a quasistatic picture does not distinguish between the boundaries, since we can change to a moving reference frame, maintaining the same equations. Only relative motion is significant. Let us therefore ask: what is it that distinguishes the moving boundary in reality, and will stop us using any frame of reference with the same equations for the internal motions of the bubbles? One answer lies in external drag force associated with the motion of the foam relative to the two confining plates (this being the nature of the 2D foam used by the Debregeas group: others are slightly different, but the same argument applies). The inclusion of the drag force is essential to the alternative picture that follows. The second explanation of localisation is based on a dynamic continuum model [16,17••,18] and it does include the drag force. Indeed it is the key ingredient of the model, for the localization length is found to be infinite when the magnitude of the drag tends to zero. The rest of its formulation is essentially based on the Bingham model of a yield–stress material (Eq. (1) above, n = 1). The experiments by the Dennin group confirmed and extended the Debregeas results (using simple shear). However when a Bragg raft was used to create a 2D foam system with no confining plates and hence little or no drag force, localisation was not observed. The quasistatic model does not distinguish this from the original case, and its status is further questioned by this experiment. The two models are not variants of a common viewpoint, and appear at first to be sharp alternatives. Or are they in some sense
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complementary, two parts of a puzzle that need to be combined? We incline to the latter view. The observed localization length may contain distinct contributions to be found in the two models, respectively. How could that be? We might argue that the better primary model is the continuum, as it is free of logical inconsistencies, and it is normal to deal with “mean field” theory first and next consider the effect of fluctuations/inhomogeneities. Suppose then that the continuum model gives a localisation length which is of the order of the bubble size (as seems likely, but we are unsure of the quantities involved). Then we might expect a further contribution to the (de)localization of stress coming from the random nature of foam. Is this what is represented by the finite localisation length found in the quasistatic simulations? Could the two localisation lengths (or more likely their squares, drawing inspiration from convolution theorems) simply be additive, in a full picture? We await more quantitative experimental tests and cellular simulations (incorporating what we might call internal viscosity and external viscosity, or wall drag, already in progress), to see if such a reconciliation emerges. In this, drag localises shear and polydispersity delocalises it to some extent. For the time being, an important message to be taken from this is surely that we must be very careful in using 2D foams as a guide to 3D foams, in the case of dynamic properties. The 2D foam is subject to an external body force, the surface drag, whenever it is in motion, and this has no counterpart at all in three dimensions. It may be of great interest, and indeed it has been intriguing and instructive (“diversion” having the extra meaning of “entertainment” in the Irish version of the English language), but it is an unreliable guide to 3D rheology. According to these arguments we should, in particular, not expect to see such localization in the 3D case. However, by coincidence or otherwise, there have recently been strong suggestions of localization in 3D [19••], and a further debate surrounds these. 9. The constitutive relation: who needs it? We return to the historic goal of representing the rheology of the 3D foam in some general way. The traditional choice is that of a constitutive relation, such as Eq. (1), expressing the general dependence of local stress on local strain and strain rate, with parameters that should depend on other local quantities. This can then be applied locally to arbitrary geometries (itself an interesting technical challenge). This must certainly be realisable for some limited cases, but any attempt at realism and generality in such an empirical formalism will confront many difficulties. They include the treatment of hysteresis (for which Jiang et al. and others have suggested practical formulations [20•]), but also further physical effects such as: 9.1. Effects at higher strain rates In addition to the expected failure of the power law that obtains for low flow rates, high flow rates are likely to entail
instabilities, especially that which causes break-up of bubbles (the mechanism of moany common types of foam formation). 9.2. Segregation due to shear Under continuous shear, polydisperse foams show some tendency to separate into regions of smaller and larger bubbles. This is an obvious possible source of localisation in 3D. See for example the experiments of Herzhaft [21]. 9.3. Drainage It would be desirable to be able to model rheological properties in the presence of drainage. Since the latter is fairly well understood [2], a combined model seems possible, including both a constitutive law (with dependence on liquid fraction) and a drainage law. 9.4. Dilatancy This celebrated effect is, in terms of foam physics, a coupling between drainage and shear. The osmotic pressure (or each normal component of stress, more generally) depends on shear [22•] and also on shear rate, neither effect having been yet explored adequately. 9.5. Coarsening It should be possible to combine coarsening theories[2] with a rheological model, to take account of the evolution of the bubble sizes due to gas diffusion. See for example ref [23]. 9.6. Particles Particle-laden foams are of current interest. Models that incorporate particles (of all sizes) will be required. Faced with all this the establishment of a really general constitutive relation begins to look like an impossible task. 10. Is there any alternative? We begin to see that formidable challenges face us on many fronts, in the inherent complexity of any general constitutive relation to be applied to a general geometric configuration, and its supplementary equations for drainage, coarsening, break-up of bubbles, particle concentration… It is likely to contain an overwhelming number of empirical parameters that we cannot easily deduce from first principles or experiments at a deeper level. To try to simply fit them to experiment is unlikely to be fruitful. To this author the situation is analogous to that in which the theory of solid properties such as cohesive energy found itself in the mid-20th century. Crude nearest-neighbour interactions with a few parameters were qualitatively successful. For some people the road ahead seemed very clear: just add more distant and complex interatomic interactions with more and more parameters, in pursuit of ever greater accuracy/generality. However
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the number of such parameters that deserve more or less equal standing soon outstripped the ability to fit the model meaningfully to any conceivable data. That approach came to a dead end in a “labour of Sisyphus”, and today is replaced by quantum mechanical (“ab initio”) calculations with little or no adjustable parameters. Is this a doctrine of despair for traditional rheology in our case? Hardly—we would argue that the framing of simple constitutive relations and their application to various geometries should continue to be highly informative. Qualitative understanding is valuable: “toy models” are much more than toys. But rather than seeking to elaborate and tune these methods quantitatively, some alternative should be found. This may possibly be found in simulation. Such simulations will not necessarily be of the obvious kind (Fig. 1(a)) that simulates detailed bubble structure very accurately, although these will always be interesting. More pragmatic approaches are possible, using local elements, in the same spirit as the “discrete element” methods, or DEM, in fluid dynamics, along the lines of the simulations of Durian [24•] These local elements, moving according to prescribed laws, can perhaps “solve the problem” for us, while requiring relatively few parameters to capture the physics. For example, drainage and size segregation may proceed automatically, and such awkward difficulties as those posed by free surfaces may become insignificant. Such an approach has been advocated by Gardiner and Tordesillas [25]. 11. What we have set aside In order to concentrate the mind on a reasonable number of questions, we have omitted several important recent developments that bear on other aspects of foam rheology for relatively dry foams. In addition to the more general models of the Graner group, mentioned above, there has also been progress in understanding the response for small strains as a function of frequency and also for large amplitudes, and the related problem of creep. These and other topics from this subject have been reviewed in detail by Höhler and Cohen-Addad [26•]. We have also refrained from delving into many recent investigations of local dynamic properties of bubbles, films, Plateau borders… Such work will provide essential ingredients for an eventual synthesis. The wet limit has become an area of specialised research, of great theoretical interest but severe experimental difficulty, often under the heading of “jamming” (or “melting” or “rigidity loss”) [27•], Again we have set that aside. 12. Conclusion The recent excursion into two dimensions and shear localisation has been somewhat of a distraction, but nevertheless highly worthwhile. In the end, it may have little direct relevance to three dimensions. Whatever the dimension, we would make a plea for more, and more complete, experiments. Again and again, attention has been diverted into theory and heroic efforts to fit limited data
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and claim agreement. It needs to be said that the most significant experiments are often those in which theory and experiment do not agree. We have also pointed to the possibility of a new departure in simulation that steers a practical middle course between simplicity and realism. Are there new directions to be commended in experiment as well? Certainly tomographic techniques (here taken to refer to any method that can generate 3d information from the interior) are promising. We have referred to the impact of MRI techniques, and others have also pursued this line as well. X-ray tomography can already be performed at table-top scale, and may yet be made quick enough for rheometry. Given such tomography, simpler geometry may have its virtues, especially that of pipe rheometry [28]. In some cases, microfluidics may also be useful, for example in assessing wall slip effects. The prospects for progress with new extensions of rheometry are now excellent, and there will be parallel theoretical progress, if undisciplined empiricism can be avoided. Acknowledgements This review drew much inspiration from the Dourdan GDR Meeting organised by Michele Adler. Discussions with R Hoehler, F Graner, S Cox and S Hutzler have been essential, and M Dennin and D Durian are thanked for helpful communications. The research upon which this was based was supported by SFI and ESA, and by CNRS through a visiting appointment. References and recommended reading [1] Barnes HA, Hutton JF, Walters K. An introduction to rheology. Amsterdam: Elsevier; 1989. [2] Hutzler S, Weaire D. The pysics of foams. Oxford: Oxford University Press; 1999. [3] Kraynik A. Foam flows. Ann Rev Fluid Mech 1988;20:325. A standard • reference for early ideas (eg those of Princen) and Kraynik's contributions to rheology of 2D foams. At this stage, the influence of disorder was not fully appreciated: models that use an ordered honeycomb structure are misleading, as regards plastic properties and the nature of the wet foam limit. [4] Khan SA, Schnepper CA, Armstrong RC. Foam rheology, III. Measurement of shear flow properties. J Rheol 1988;32:69. [5] Denkov ND, Subranian V, Gourovich D, Lips A. Wall slip and viscous dissipation in sheared foams: effect of surface mobility, Colloids and Surfaces A. Physicochem Eng Aspects 2005;263:129. [6] Besson S., Debregeas G., in press. Statics and dynamics of adhesion between two soap bubbles. Eur. Phys. J. E. An important contribution to clarifying the forces that act at the level of films and Plateau borders, which are still poorly understood. No attempt to discuss them is made in the present review. [7] Goldenberg C. Particle displacements in the elastic deformation of • amorphous materials: local fluctuations vs non-affine field. Europhys Lett 2007;80:16003. This paper should contain important clues to the nature of the mechanical response of disordered systems, for which there is still no established theory, despite a substantial literature of formal developments.
• ••
of special interest. of outstanding interest.
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[8] Takeshi O, Sekimoto K. Internal stress in a model elastoplastic fluid. Phys Rev Lett 2005;95:108301. [9] Rouyer F, Cohen-Addad S, Vignes-Adler M, Hoehler R. Dynamics of yielding in a three-dimensional aqueous foam. Phys Rev E 2003;67:021405. [10] Jiang Y, Swart PJ, Saxena A, Asipauskas M, Glazier JA. Hysteresis and avalanches in two dimensional rheology simulations. Phys Rev E 1999;59: 5819. [11] Debregeas G, Tabuteau H, di Meglio JM. Deformation and flow of a two•• dimensional foam under continuous shear. Phys Rev Lett 2001;87:17–8305. This paper launched the (still not concluded) debate on the source of localisation in 2D and the possibility of its occurrence in 3D. [12] Lauridsen J, Chanan G, Dennin M. Velocity profiles in slowly sheared • bubble rafts. Phy Rev Lett 2004;93:18303. Important work which complements that of ref 11. See also ref 13. [13] Wang Y, Krishan K, Dennin M. Impact of boundaries on velocity profiles in bubble rafts. Phys Rev E 2006;73:031401. [14] Cantat I, Poloni C, Delannay R. Experimental evidence of flow destabilization in a two-dimensional bidisperse foam. Phys Rev E 2006;73: 011505. [15] Kabla A, Debregeas G. Local stress relaxation and shear banding in a dry • foam under shear. Phy Rev Lett 2003;95:218301. The first interpretation of ref 12, this uses quasistatic simulations: the main conclusions have been confirmed by others. Despite qualitative agreement with experiment, this approach remains under active debate, especially in comparison with ref 16. See for example, Wynn A, Devies IT, Cox SJ, submitted to Eur Phys J. E 2007. [16] Janiaud E, Weaire D, Hutzler S. A simple continuum model for the dynamics of a quasi-two dimensional foam. Colloids and Surfaces A. Physicochem Eng Aspects 2007;309:125. [17] Janiaud E, Weaire D, Hutzler S. Two dimensional foam rheology with •• viscous drag. Phys Rev Lett 2006;97:038302. This introduces a continuum model for 2D rheology with wall drag, and obtains exponential localisation, as well as transient effects. Currently attempts are being made to reconcile it with ref 15.
[18] Clancy R, Janiaud E, Weaire D, Hutzler S. The response of 2D foams to continuous applied shear in a Couette rheometer. Europhys Lett 2006;21:123. [19] Coussot P, Raynaud JS, Bertrand F, Moucheron P, Guilbaud JP, Huyth HT, •• et al. Coexistence of liquid and solid phases in soft-glassy materials. Phys Rev Lett 2002;88:218301. This paper raises the issue of 3D localization: further experimental investigations are necessary to test the generality of the conclusions. [20] Marmottant P, Graner F. An elastic, plastic, viscous model for slow shear of • a liquid foam. Eur Phys J E 2007;23:337. This group has led the way in framing model constitutive relations which can be applied to foams under general boundary conditions and imposed shears. [21] Herzhaft B. Correlation between transient shear experiments and structure evolution of aqueous foam. J Colloid Interface Sci 2002;247:412. [22] Weaire D, Hutzler S. Dilatancy in liquid foams. Phil Mag 2002;83:2747. • Introduces the concept of dilatancy (but only within elastic theory) into foam physics. While dilatanccy is frequently mentioned in the mechanics of granular materials, it is yet to be explored in the analogous case of a foam. [23] Vincent-Bonnieu S, H¨ohler R, Cohen-Addad S. Slow viscoelastic relaxation and aging in aqueous foam. Europhys Lett 2006;74:533. [24] Durian DJ. Foam mechanics at the bubble scale. Phys Rev Lett • 1995;75:4780. An early example of the use of “soft spheres” to model bubble mechanics is a simple way, using linear springs on contact. Still a useful approach for qualitative purposes. [25] Gardiner BS, Tordesillas A. The link between discrete and continuous modeling of liquid foam at the level of a single bubble. J Rheology 2005;49:819. [26] Höhler R, Cohen-Addad S. Rheology of liquid foam. J Phys Condens • Matter 2005;17:R1041. An excellent source of detailed information on recent experiments and theories for foam rheology, rigorously analysed. [27] Liu AJ, Nagel SR. Jamming and rheology: constrained dynamics on • microscopic and macroscopic scales. New York: Taylor and Francis; 2001. A useful review of rheology of granular materials and other systems, concentrated on the jamming transition. [28] Yoon WB, McCarthy K. Rheology of yogurt during pipe flow as characterized by magnetic resonance imaging. J Texture Studies 2002;33:431.
The rheology of foam Denis Weaire ⁎ Université Paris-Est, Laboratoire de Physique des Matériaux Divisés et des Interfaces, UMR CNRS 8108, 5 Bd Descartes, 77454 Marne-la-Vallée cedex 2, France Received 1 October 2007; accepted 31 October 2007 Available online 12 November 2007
Abstract Foam rheology has arrived at an interesting stage of active debate, some of it focused on two dimensional systems. Theorists are ambitious to formulate comprehensive models and experimentalists want to look more critically and constructively at the techniques of rheometry. Are there grounds for optimism, that the difficulties of this topic may be overcome? Whatever the eventual extent of success, we argue that liquid foam should stand as the appropriate prototype for that wide class of substances that exhibit elasticity and a yield stress, above which they flow. We focus mainly on the constitutive law for steady shear and the recent debate over shear localization in two dimensions. © 2007 Elsevier Ltd. All rights reserved. Keywords: Foam; Rheology; Shear; Localisation
1. Introduction Foam is a solid under low stress but it flows like a liquid under high stress. It therefore has a yield stress, and belongs to that category of complex materials that may be termed “Bingham fluids”, attributing to them (perhaps rashly) a particular model representation of rheological properties. The importance of those properties in applications is immense, and so they have attracted plenty of attention from industrial researchers. After many years of such research, a general, systematic and practical theory of foam rheology is still lacking. At present there is a tentative sense of convergence towards this challenging goal. Is it justified? We shall argue this case, in reviewing various contributions. We will still be left with more questions than answers. The author is not a rheologist, but comes to the subject from a wide background of experience in modelling foam properties. Rheology is perhaps the next big problem to be faced in that field, following twenty years of substantial accomplishment by an international community in framing and testing satisfactory models for most static or quasistatic properties. ⁎ Permanent address: School of Physics, Trinity College, Dublin, Ireland. Tel.: +353 1 8961675; fax: +353 1 6711759. E-mail address: [email protected]. 1359-0294/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2007.11.004
When foam flows, many more physical effects come into play, and we face a more complex challenge. Books by Barnes et al. [1] and Hutzler and Weaire [2], and a review by Kraynik [3•] reflect various earlier stages of the subject, and may be contrasted with the sophistication of some of the studies that are under way today. We give only exemplary references here, and they may be used to find a way into the already extensive literature. Fig. 1 represents two ways of visualizing the rheological properties in question: either in terms of a detailed structure of cells which are deformed and rearranged by local forces when stress is applied, or a continuum described in terms of average quantities. We shall ask how they are related. 2. Foam as an elastic solid Ordinary foam is polydisperse and isotropic. Its elastic properties are characterized by its bulk modulus and its shear modulus. The bulk modulus is dominated by that of the enclosed gas and is so large as to render the foam effectively incompressible in many practical contexts. We are left with the shear modulus due entirely to surface tension. It is generally a small fraction of (surface tension/mean bubble radius), as might be estimated from simple arguments. There is a range of strain over which the foam is elastic, that is, deformation is recoverable. There is some scope for
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yield stress. Upon further reduction of the stress, so that it eventually the opposite sign, its absolute value can again reach the yield stress. Shear flow again occurs, in the opposite direction to that before. This is the hysteresis to which we have referred. 4. Simple or complicated?
Fig. 1. Microscopic and macroscopic viewpoints. (a) A foam (here in 2D, in equilibrium) is made up of local elements with specific dynamic properties. (b) Its continuum rheological properties may be represented by the dependence of stress on strain and strain rate. How are these two pictures related?
argument about this, strictly speaking, but let us set that aside. The effective elastic range is large, reaching to strains that are a significant fraction of unity (contrasting with most solids, and making nonlinear elasticity interesting). Beyond the elastic range of strain, the foam is plastic. 3. Plasticity and flow The underlying mechanism of plasticity is simple and welldefined for low shear rates (the quasistatic regime); it consists of sudden topological events that rearrange bubbles, so that their local configurations remain stable. These events, which interrupt the otherwise smooth evolution under stress, are easily described in relatively dry foams, for which Plateau's rules apply: they require such changes whenever it happens that more that six films meet at a vertex in 3D, or three lines in 2D, in a dry (that is, low liquid fraction) foam. These are elementary “T1” or neighbour-switching rearrangements, and are a vital ingredient of most theories that seek to link microscopic and macroscopic mechanics. As strain increases yet further the yield stress is reached, beyond which the plastic events continue indefinitely under constant stress, and the foam flows. When stress increases above the yield stress, the flow rate increases. Note that the schematic curve shown is for monotonically increasing strain only. There is strong hysteresis in this system, when the strain rate changes sign. Upon reduction of the stress, the foam reverts to an elastic solid when the stress falls below the
The above scenario seems, on the face of it, rather simple. It is however a mere narrative of the local behaviour of a foam under increasing stress, in crude qualitative terms. A closer look reveals a multitude of possible complications and difficulties. It is often difficult to decide which of these are important. Which should be conscientiously confronted and which are mere niceties, that is, small effects for later consideration? How can we steer a prudent course between Scylla and the Charybdis: naïve idealised models that throw away too much physics and elaborate multi-parameter formalisms that promise much and deliver little? Whatever the complications envisaged, it as well to recall the essential simplicity of a foam in equilibrium, as regards its welldefined local structure and its evolution, and our ability to simulate this in detail. However hard it may be to adequately and generally characterize its rheology, this gives us some hope of eventual theories that are meaningful, transparent and effective. They can set out from the quasistatic case and move progressively towards high shear rates. That is the programme that is under way. 5. Some measurements One particular set of earlier rheological measurements may be indicated as belonging among the best and representative of many: those of Khan et al. [4]. Let us state one regret: it is a great pity that in this and many other cases, a wider range of data was not acquired, to lay down the basis for proper tests of later theories. Instead the authors embarked on lengthy theoretical analysis of their own, based on the inadequate models of the time. Their conclusions were consistent with our sketch in Fig. 1(b), with stress varying with strain rate, to a power which was roughly one-half. For an example of more recent data, roughly consistent with this; see Denkov et al. [5]. 6. The Herschel–Bulkley relation Such data as that of Khan et al. is commonly interpreted in terms of the Herschel–Bulkley relation: r ¼ ry þ gHB ðdg=dt Þn
ð1Þ
(σ is stress, σy is yield stress, γ is strain). For n = 1 this is called the Bingham equation, a common favourite in toy models, and ηHB may be called the plastic or Bingham viscosity. While it may be dangerous at this stage to take any of the exact published values of the parameter n seriously, they do indicate that it is probably not equal to unity. The obvious next question is: what determines n? Vaguely, the answer must lie in the micromechanics of the system, and this shifts the question on to difficult ground, since local
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dynamic effects remain mysterious, as regards the extent and nature of dynamic surface effects associated with surfactants. Recent work of Besson and Debregeas [6•] provides an example of an attempt to clarify these effects. We would prefer to narrow the question to one which has not been recognised in the past, and should now be answerable with the aid of simulations: for a given microscopic model (however simplified or realistic), is there a definite relation between its microscopic parameters and the macroscopic ones of a Herschel– Bulkley model (Eq. (1))? The question is rather general; foams offer a context in which appropriate simple models seem to be closest to reality. This seems to lead inevitably to detailed consideration of the role of T1s (or other elementary processes) in non-affine microscopic deformation. The peculiarity of the mechanics of disordered (non-crystalline) materials is that the deformations (elastic or plastic) are not strictly affine on any scale. See for example the recent analysis of Goldenberg [7•], in search of a formalism that can cope with the subtleties of this kind of response, and another exercise of this kind by Takeshi and Sekimoto [8]. Many other questions are raised by the work of Khan et al., and others. One is: what is the significance of the peak in the curve (of the order of ten percent above the yield stress in their case, and found to a varying degree in many published simulations. And what are its implications? Also, what modifications are required at high strain rates? There are plenty of clues (for example ref [9]) that deformation is quite different above a critical strain rate which is the inverse of the relaxation rate for T1s. Even at low strain rates, individual T1 events are not independent. As the liquid fraction increases, they are no longer separate, but form cascades or avalanches [10]. 7. Modelling at the cellular level An example of a static simulation of a 2D foam was shown in Fig. 1(a). This kind of calculation has been used many times to establish elastic properties, simply by varying the (periodic) boundary conditions. Potentially, much insight can be gained from such detailed cellular models, which have been extended to 3D by Surface Evolver, but they remain poorly developed for dynamic effects. They are still being explored in the simpler two dimensional case. 8. A tactical retreat to two dimensions—or just a diversion? Ever since the days of Plateau and Kelvin, foam physicists have always found simplicity and inspiration in the study of 2D foams. Again and again, these have been shown to have properties that are similar to those of their 3D counterparts, and they are often more tractable in theory, in experiment and in visualization. This strategy presumably motivated the group of Debregeas to construct a rheometer for 2D foam [11••]. It is analogous to the classic Couette rheometer of two concentric cylinders. Two walls (the inner one being rotatable) surround a 2D foam between two glass plates. The results obtained from this have provoked quite a debate, fuelled by subsequent experiments by
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the group of Dennin [12•,13], using both Couette and simple shear geometry (two parallel straight boundaries). The salient feature of the results of Debregeas is strong shear localization or shear banding, located at the boundary which is moving (the inner circle in the original experiment) [12]. It was argued that the localisation effect was not attributable to Couette geometry, and indeed it was found to persist for the straight geometry of simple shear, so we will not take time here to explore this delicate point. For our purposes, the effect of cylindrical geometry is largely irrelevant. Such localisation can be also be seen in various other 2D experiments, especially when flow around obstacles is concerned [14]. Two explanations have been offered for it. Firstly, the cellular simulations of Kabla [15•], with simple shear, showed similar localization, and were then analysed in terms of an elaborate description of local topological changes. This model is quasistatic, and represents the detailed cellular structure of the foam, rather as in Fig. 1(a). Such a dynamic application of a quasistatic model always raises several questions [16]. The program proceeds from one equilibrium to another by topological changes which must be, to some extent, dependent on the details of the algorithm, rather than the unique consequences of dynamical laws. It may seem natural to take the last equilibrium structure as the first approximation to the next, as the boundary shear increases, but at least one alternative has been applied by others—that of affine increments. Also, in a quasistatic picture, there can be no distinction between moving the inner boundary and moving the outer one. In experiment, localisation always occurs at the moving boundary but a quasistatic picture does not distinguish between the boundaries, since we can change to a moving reference frame, maintaining the same equations. Only relative motion is significant. Let us therefore ask: what is it that distinguishes the moving boundary in reality, and will stop us using any frame of reference with the same equations for the internal motions of the bubbles? One answer lies in external drag force associated with the motion of the foam relative to the two confining plates (this being the nature of the 2D foam used by the Debregeas group: others are slightly different, but the same argument applies). The inclusion of the drag force is essential to the alternative picture that follows. The second explanation of localisation is based on a dynamic continuum model [16,17••,18] and it does include the drag force. Indeed it is the key ingredient of the model, for the localization length is found to be infinite when the magnitude of the drag tends to zero. The rest of its formulation is essentially based on the Bingham model of a yield–stress material (Eq. (1) above, n = 1). The experiments by the Dennin group confirmed and extended the Debregeas results (using simple shear). However when a Bragg raft was used to create a 2D foam system with no confining plates and hence little or no drag force, localisation was not observed. The quasistatic model does not distinguish this from the original case, and its status is further questioned by this experiment. The two models are not variants of a common viewpoint, and appear at first to be sharp alternatives. Or are they in some sense
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complementary, two parts of a puzzle that need to be combined? We incline to the latter view. The observed localization length may contain distinct contributions to be found in the two models, respectively. How could that be? We might argue that the better primary model is the continuum, as it is free of logical inconsistencies, and it is normal to deal with “mean field” theory first and next consider the effect of fluctuations/inhomogeneities. Suppose then that the continuum model gives a localisation length which is of the order of the bubble size (as seems likely, but we are unsure of the quantities involved). Then we might expect a further contribution to the (de)localization of stress coming from the random nature of foam. Is this what is represented by the finite localisation length found in the quasistatic simulations? Could the two localisation lengths (or more likely their squares, drawing inspiration from convolution theorems) simply be additive, in a full picture? We await more quantitative experimental tests and cellular simulations (incorporating what we might call internal viscosity and external viscosity, or wall drag, already in progress), to see if such a reconciliation emerges. In this, drag localises shear and polydispersity delocalises it to some extent. For the time being, an important message to be taken from this is surely that we must be very careful in using 2D foams as a guide to 3D foams, in the case of dynamic properties. The 2D foam is subject to an external body force, the surface drag, whenever it is in motion, and this has no counterpart at all in three dimensions. It may be of great interest, and indeed it has been intriguing and instructive (“diversion” having the extra meaning of “entertainment” in the Irish version of the English language), but it is an unreliable guide to 3D rheology. According to these arguments we should, in particular, not expect to see such localization in the 3D case. However, by coincidence or otherwise, there have recently been strong suggestions of localization in 3D [19••], and a further debate surrounds these. 9. The constitutive relation: who needs it? We return to the historic goal of representing the rheology of the 3D foam in some general way. The traditional choice is that of a constitutive relation, such as Eq. (1), expressing the general dependence of local stress on local strain and strain rate, with parameters that should depend on other local quantities. This can then be applied locally to arbitrary geometries (itself an interesting technical challenge). This must certainly be realisable for some limited cases, but any attempt at realism and generality in such an empirical formalism will confront many difficulties. They include the treatment of hysteresis (for which Jiang et al. and others have suggested practical formulations [20•]), but also further physical effects such as: 9.1. Effects at higher strain rates In addition to the expected failure of the power law that obtains for low flow rates, high flow rates are likely to entail
instabilities, especially that which causes break-up of bubbles (the mechanism of moany common types of foam formation). 9.2. Segregation due to shear Under continuous shear, polydisperse foams show some tendency to separate into regions of smaller and larger bubbles. This is an obvious possible source of localisation in 3D. See for example the experiments of Herzhaft [21]. 9.3. Drainage It would be desirable to be able to model rheological properties in the presence of drainage. Since the latter is fairly well understood [2], a combined model seems possible, including both a constitutive law (with dependence on liquid fraction) and a drainage law. 9.4. Dilatancy This celebrated effect is, in terms of foam physics, a coupling between drainage and shear. The osmotic pressure (or each normal component of stress, more generally) depends on shear [22•] and also on shear rate, neither effect having been yet explored adequately. 9.5. Coarsening It should be possible to combine coarsening theories[2] with a rheological model, to take account of the evolution of the bubble sizes due to gas diffusion. See for example ref [23]. 9.6. Particles Particle-laden foams are of current interest. Models that incorporate particles (of all sizes) will be required. Faced with all this the establishment of a really general constitutive relation begins to look like an impossible task. 10. Is there any alternative? We begin to see that formidable challenges face us on many fronts, in the inherent complexity of any general constitutive relation to be applied to a general geometric configuration, and its supplementary equations for drainage, coarsening, break-up of bubbles, particle concentration… It is likely to contain an overwhelming number of empirical parameters that we cannot easily deduce from first principles or experiments at a deeper level. To try to simply fit them to experiment is unlikely to be fruitful. To this author the situation is analogous to that in which the theory of solid properties such as cohesive energy found itself in the mid-20th century. Crude nearest-neighbour interactions with a few parameters were qualitatively successful. For some people the road ahead seemed very clear: just add more distant and complex interatomic interactions with more and more parameters, in pursuit of ever greater accuracy/generality. However
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the number of such parameters that deserve more or less equal standing soon outstripped the ability to fit the model meaningfully to any conceivable data. That approach came to a dead end in a “labour of Sisyphus”, and today is replaced by quantum mechanical (“ab initio”) calculations with little or no adjustable parameters. Is this a doctrine of despair for traditional rheology in our case? Hardly—we would argue that the framing of simple constitutive relations and their application to various geometries should continue to be highly informative. Qualitative understanding is valuable: “toy models” are much more than toys. But rather than seeking to elaborate and tune these methods quantitatively, some alternative should be found. This may possibly be found in simulation. Such simulations will not necessarily be of the obvious kind (Fig. 1(a)) that simulates detailed bubble structure very accurately, although these will always be interesting. More pragmatic approaches are possible, using local elements, in the same spirit as the “discrete element” methods, or DEM, in fluid dynamics, along the lines of the simulations of Durian [24•] These local elements, moving according to prescribed laws, can perhaps “solve the problem” for us, while requiring relatively few parameters to capture the physics. For example, drainage and size segregation may proceed automatically, and such awkward difficulties as those posed by free surfaces may become insignificant. Such an approach has been advocated by Gardiner and Tordesillas [25]. 11. What we have set aside In order to concentrate the mind on a reasonable number of questions, we have omitted several important recent developments that bear on other aspects of foam rheology for relatively dry foams. In addition to the more general models of the Graner group, mentioned above, there has also been progress in understanding the response for small strains as a function of frequency and also for large amplitudes, and the related problem of creep. These and other topics from this subject have been reviewed in detail by Höhler and Cohen-Addad [26•]. We have also refrained from delving into many recent investigations of local dynamic properties of bubbles, films, Plateau borders… Such work will provide essential ingredients for an eventual synthesis. The wet limit has become an area of specialised research, of great theoretical interest but severe experimental difficulty, often under the heading of “jamming” (or “melting” or “rigidity loss”) [27•], Again we have set that aside. 12. Conclusion The recent excursion into two dimensions and shear localisation has been somewhat of a distraction, but nevertheless highly worthwhile. In the end, it may have little direct relevance to three dimensions. Whatever the dimension, we would make a plea for more, and more complete, experiments. Again and again, attention has been diverted into theory and heroic efforts to fit limited data
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and claim agreement. It needs to be said that the most significant experiments are often those in which theory and experiment do not agree. We have also pointed to the possibility of a new departure in simulation that steers a practical middle course between simplicity and realism. Are there new directions to be commended in experiment as well? Certainly tomographic techniques (here taken to refer to any method that can generate 3d information from the interior) are promising. We have referred to the impact of MRI techniques, and others have also pursued this line as well. X-ray tomography can already be performed at table-top scale, and may yet be made quick enough for rheometry. Given such tomography, simpler geometry may have its virtues, especially that of pipe rheometry [28]. In some cases, microfluidics may also be useful, for example in assessing wall slip effects. The prospects for progress with new extensions of rheometry are now excellent, and there will be parallel theoretical progress, if undisciplined empiricism can be avoided. Acknowledgements This review drew much inspiration from the Dourdan GDR Meeting organised by Michele Adler. Discussions with R Hoehler, F Graner, S Cox and S Hutzler have been essential, and M Dennin and D Durian are thanked for helpful communications. The research upon which this was based was supported by SFI and ESA, and by CNRS through a visiting appointment. References and recommended reading [1] Barnes HA, Hutton JF, Walters K. An introduction to rheology. Amsterdam: Elsevier; 1989. [2] Hutzler S, Weaire D. The pysics of foams. Oxford: Oxford University Press; 1999. [3] Kraynik A. Foam flows. Ann Rev Fluid Mech 1988;20:325. A standard • reference for early ideas (eg those of Princen) and Kraynik's contributions to rheology of 2D foams. At this stage, the influence of disorder was not fully appreciated: models that use an ordered honeycomb structure are misleading, as regards plastic properties and the nature of the wet foam limit. [4] Khan SA, Schnepper CA, Armstrong RC. Foam rheology, III. Measurement of shear flow properties. J Rheol 1988;32:69. [5] Denkov ND, Subranian V, Gourovich D, Lips A. Wall slip and viscous dissipation in sheared foams: effect of surface mobility, Colloids and Surfaces A. Physicochem Eng Aspects 2005;263:129. [6] Besson S., Debregeas G., in press. Statics and dynamics of adhesion between two soap bubbles. Eur. Phys. J. E. An important contribution to clarifying the forces that act at the level of films and Plateau borders, which are still poorly understood. No attempt to discuss them is made in the present review. [7] Goldenberg C. Particle displacements in the elastic deformation of • amorphous materials: local fluctuations vs non-affine field. Europhys Lett 2007;80:16003. This paper should contain important clues to the nature of the mechanical response of disordered systems, for which there is still no established theory, despite a substantial literature of formal developments.
• ••
of special interest. of outstanding interest.
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[8] Takeshi O, Sekimoto K. Internal stress in a model elastoplastic fluid. Phys Rev Lett 2005;95:108301. [9] Rouyer F, Cohen-Addad S, Vignes-Adler M, Hoehler R. Dynamics of yielding in a three-dimensional aqueous foam. Phys Rev E 2003;67:021405. [10] Jiang Y, Swart PJ, Saxena A, Asipauskas M, Glazier JA. Hysteresis and avalanches in two dimensional rheology simulations. Phys Rev E 1999;59: 5819. [11] Debregeas G, Tabuteau H, di Meglio JM. Deformation and flow of a two•• dimensional foam under continuous shear. Phys Rev Lett 2001;87:17–8305. This paper launched the (still not concluded) debate on the source of localisation in 2D and the possibility of its occurrence in 3D. [12] Lauridsen J, Chanan G, Dennin M. Velocity profiles in slowly sheared • bubble rafts. Phy Rev Lett 2004;93:18303. Important work which complements that of ref 11. See also ref 13. [13] Wang Y, Krishan K, Dennin M. Impact of boundaries on velocity profiles in bubble rafts. Phys Rev E 2006;73:031401. [14] Cantat I, Poloni C, Delannay R. Experimental evidence of flow destabilization in a two-dimensional bidisperse foam. Phys Rev E 2006;73: 011505. [15] Kabla A, Debregeas G. Local stress relaxation and shear banding in a dry • foam under shear. Phy Rev Lett 2003;95:218301. The first interpretation of ref 12, this uses quasistatic simulations: the main conclusions have been confirmed by others. Despite qualitative agreement with experiment, this approach remains under active debate, especially in comparison with ref 16. See for example, Wynn A, Devies IT, Cox SJ, submitted to Eur Phys J. E 2007. [16] Janiaud E, Weaire D, Hutzler S. A simple continuum model for the dynamics of a quasi-two dimensional foam. Colloids and Surfaces A. Physicochem Eng Aspects 2007;309:125. [17] Janiaud E, Weaire D, Hutzler S. Two dimensional foam rheology with •• viscous drag. Phys Rev Lett 2006;97:038302. This introduces a continuum model for 2D rheology with wall drag, and obtains exponential localisation, as well as transient effects. Currently attempts are being made to reconcile it with ref 15.
[18] Clancy R, Janiaud E, Weaire D, Hutzler S. The response of 2D foams to continuous applied shear in a Couette rheometer. Europhys Lett 2006;21:123. [19] Coussot P, Raynaud JS, Bertrand F, Moucheron P, Guilbaud JP, Huyth HT, •• et al. Coexistence of liquid and solid phases in soft-glassy materials. Phys Rev Lett 2002;88:218301. This paper raises the issue of 3D localization: further experimental investigations are necessary to test the generality of the conclusions. [20] Marmottant P, Graner F. An elastic, plastic, viscous model for slow shear of • a liquid foam. Eur Phys J E 2007;23:337. This group has led the way in framing model constitutive relations which can be applied to foams under general boundary conditions and imposed shears. [21] Herzhaft B. Correlation between transient shear experiments and structure evolution of aqueous foam. J Colloid Interface Sci 2002;247:412. [22] Weaire D, Hutzler S. Dilatancy in liquid foams. Phil Mag 2002;83:2747. • Introduces the concept of dilatancy (but only within elastic theory) into foam physics. While dilatanccy is frequently mentioned in the mechanics of granular materials, it is yet to be explored in the analogous case of a foam. [23] Vincent-Bonnieu S, H¨ohler R, Cohen-Addad S. Slow viscoelastic relaxation and aging in aqueous foam. Europhys Lett 2006;74:533. [24] Durian DJ. Foam mechanics at the bubble scale. Phys Rev Lett • 1995;75:4780. An early example of the use of “soft spheres” to model bubble mechanics is a simple way, using linear springs on contact. Still a useful approach for qualitative purposes. [25] Gardiner BS, Tordesillas A. The link between discrete and continuous modeling of liquid foam at the level of a single bubble. J Rheology 2005;49:819. [26] Höhler R, Cohen-Addad S. Rheology of liquid foam. J Phys Condens • Matter 2005;17:R1041. An excellent source of detailed information on recent experiments and theories for foam rheology, rigorously analysed. [27] Liu AJ, Nagel SR. Jamming and rheology: constrained dynamics on • microscopic and macroscopic scales. New York: Taylor and Francis; 2001. A useful review of rheology of granular materials and other systems, concentrated on the jamming transition. [28] Yoon WB, McCarthy K. Rheology of yogurt during pipe flow as characterized by magnetic resonance imaging. J Texture Studies 2002;33:431.