FRIT (M. Thatcher, 19/04/1983, as per N. Rivier) Aberystwyth (Wales, UK) - June 27th and 28th, 2005 Questions : 1)Liquid fraction in 2D : define ? measure ? 2)2D dissipation : internal vs wall friction ? 3)Flow past obstacle ? Define strain : in Newtonian liquid, Bingham, foam,... ? 3b) What good is the texture tensor ? 4)Define yielding and plasticity ? yield stress, yield strain, T1s, ... 5)Predict T1s ? 6b) Road towards a constitutive equation ? 6)Shear banding : its existence, properties, localisation seen as phase transition, 2D vs 3D... 7)Bubble size distribution : effect on shear modulus, yield point? Kraynik in 2D ? 8)Picnic on the beach tonight? Effect of liquid fraction: H2O, CH2COOH Abbreviations : PB = Plateau Border LA = liquid-air = bubble raft LG = liquid-glass GG = glass-glass = Hele-Shaw Liquid fraction in 2D : define ? measure ? We don't have a good definition. Suggestion : look at the main element (which length) which is important for the problem under consideration: size of the films, capillary pressure... Eg in bubble trains the section of the PB is more important than the fluid fraction itself Looking through a glass plate, you see the trace of the PB, but it does not tell you how it relates with eg the fluid fraction in theory or experiment. It is not only hard to measure but also to measure, at least in LA and LG systems: you don't know where your foam ends, what amount of water to include in what you call the system. The reason to define a fluid fraction is the need to compare the 2D simulations with the 3 types of experimental set-ups. In the GG system you know how much liquid you have in, but it is not the same as in simulations. Separate the technical problem (definition, measurement) with the physical problem (critical length). Distinguish fixed pressure and fixed volume cases. Important of course in drainage; it is linked to the pressure inside the bubbles, see 3D pressure. What quantity in 2D would define a local pressure, analogous to the 3D case? In LA, you only have the meniscus; in GG, you have the asymmetry between lower and upper PB; in LG, it is also complicated. Look at the curvature of PB itself and do not try to convert it into fluid fraction. Two mains roles of fluid fraction : triggering of T1s, and capillary pressure A few techniques used : It is important to quantify the fluid fraction; sqrt(total area / area of the gas). Critical length is important even in simulations : Evolver (T1s cutoff; phi = 0.242 lc^2/A), Potts (lattice, phi = 2/A) In flow : impose the flow rate of gas, measure the velocity and section of the total foam itself (assumes the gas and liquid have the same velocity) Area measured by thresholding GG : measure the weight of water Measure the radius of PB and divide by bubble size Schlieren : light the foam with parallel light and focus it (using an OHP) : see dark shadow (contrasted) where PBs are Light interference Understand the light patterns (caustics) produced by the PB to determine their shape (Conductivity : only in 3D) 2D dissipation : internal vs wall friction ? Solid wall always dominates the internal dissipation: LG and GG In a monodisperse foam, plug flow = dominated by walls In a heterogeneous flow, what is interesting is the deviation from the plug flow, even if the internal dissipation is very small it is still the most interesting term LA : only in-between the bubbles Langmuir Monolayer : dominated by 3D subphase in the case of liquid-gas (2D foam), dominated by the 2D monolayer in liquid-liquid (2D emulsion) Solid walls do not change the elastic stresses (they change only the dissipation). They decouple (unless there is plasticity?). Simulations are either external (vertex model, viscous froth) or internal (bubble model) Is Plexiglass different from glass ? Solid substrate in general (the nature of boundary condition is more important than the chemical nature of the surface) The solid wall friction can be included explicitly in the foam equation (constitutive equation). Flow past obstacle ? Define strain : in Newtonian liquid, Bingham, foam,... ? Macroscopic scale : no difficulty: Landau Lifschitz = displacement field, at large scale (forget about small fluctuations), derive spatially = get the strain difficulty at local scale. Doi-Ohta model as foam = collection of local films, see also PRL by Fortes. Stored deformation and release : what do you get back Total strain = elastic strain + plastic strain Non-linear phenomena : large-definition strain tensor (infinitesimal strain = only for small deformation, linear approximation) Fortes defines strain from the orientation of the connections of the strain Texture tensor = looks at the orientation of films in 2 directions = basic way to describe the distribution : 2nd moment of the distribution Total = time integral of velocity gradient Elastic = thermodynmic definition (conjugate of stress) or from statistics on microscopic positions (best fit, Goldenberg, texture tensor...) Stress = hard to measure at small scale ? Fluctuations at the bubble size? The problem is to make the system self-consistent : total stress = sum of different constituents. Only if stress is continuous can we use it to derive it. Define and choose the good scale to do it. Should we follow the bubble boundaries to define boxes for stress measurements? Compare different definitions in academic cases ? eg well-defined box ? What good is the texture tensor ? Texture tensor on edges or centers : both definitions coincide except for the constant prefactor which does not affect the results. On bubble centers, it is more robust : with respect to fluid fraction Vertices are more non-affine than edges. Texture tensor = descriptor of the elasticity stored in bubble shapes. Statistical elastic strain tensor could be a good approximation to strain valid in different cases. Emulsion studies show that the number of neighbours affect the elasticity. Relate it to the strain measured from the local equilibrium state; characterise it by the Hessian matrix. Define yielding and plasticity ? yield stress, yield strain, T1s, ... Yield stress : distinguish dynamic and static Static = largest stress a material can stand (in statics) before flowing; not related to the (linear) shear modulus Dynamic = stress when you flow and decrease the flow rate and tend to low velocity They do not coincide; is it related to the quasistatic flow (exists or not)? to coarsening? Yield stress = 1st T1 event? solid but plastic, not elastic Not the 1st T1; the stress vs time curve shows a 1st T1 much before the macroscopic yield visible on the curve Does it depend on the foam only, but also on the external object acting on the foam? Eg for a small obstacle, there is a lack of PBs to pull. Is it a question of scale? Local stresses of clusters can be much larger than the macroscopic yield stress. It is a problem also of force (=integrated stress). An inhomogeneous structure can have weak and strong points ("weak regions" picture by Liu et al). The macroscopic flow will result from an average over all these regions. Statistical definition. Not a simple average, but rather a summation of deformations (one large movement in a small region can make an earthquake). Look at the derivatives (jumps), more significant than the quantities themselves? Eg : the point where the stress changes from increasing to decreasing. Do the T1s (which occur before yielding) undo themselves after the yielding? Reversibility? The plastic flow modifies the foam structure (Kabla simulations, Kader experiments in LG): yield stress decreases the disorder of the foam (2nd moment of the side number distribution). Zero-frequency shear modulus (infinite time): if it is non-zero, is it equivalent to say that there is a yield stress ? Yes, it is equivalent for a liquid (!) material to say that it has a shear modulus or a yield stress. [NB : for a solid material, this is not equivalent]. Yield stress is a function of material (bubble size distribution, fluid fraction) but also certainly function of history of the sample ! It can vary by a factor of 2. Predict T1s ? T1s in plastic deformation, T1s in flow : is it different ? Qualitatively, quantitatively? There can be flow due to coarsening, to stress that induces T1s,... If you go very fast, T1s overlap and are not well identified. Question is about the number, but also the orientation of the T1s T1s function of (product of?) velocity gradient and deformation? T1s affect more than 4 bubbles - the function of orientation is complex (dipolar at long distance?); the propagator (Kabla) characterises the other bubbles' movement. Unless there are avalanches. Requires suitable descriptors for the orientation. Experiments show that 3 bubble distances are affected. Plasticity could be defined as where deformation and deformation rate become independent variables (the latter ceases to be the time derivative of the former). This defines a yield strain rather than yield stress, and is related to many T1s rather than to the first one. Role of the shorter edge (and of the orientation of the stress) to trigger the 1st T1. Road towards a constitutive equation ? In elastic regime : Kraynik has a mooney Rivlin model, with some approximations. Well understood. Flow regime is more difficult. Fluid + plastic + elastic = continuous approach using the texture tensor, to be tested experimentally, and derive predictions Stress, strain-rate and microscopical details : linear viscoelasticity (Sebastien) There are progresses on this road (despite the difficulties due to non-affinities) Fluid regime (higher shear rate or fluid fraction): smooth flow, driven by the friction between the bubbles External force can be treated separately (it is not a stress, but it contributes to the dissipation) - see eg the role of solvent dissipation in polymers Avalanches are incompatible with constitutive equations. Intuition is that avalanches do not affect the macroscopic scale ? Lab experiments look at details rather than at large number of bubbles. Polydisperse foams decrease the correlation length (to 2-3 bubble sizes) and favor the convergence Avalanches might also be sensitive to the dissipation mechanism, or even the geometry of the system. Link between avalanches and shear bands ? see next question Shear banding : its existence, properties, localisation seen as phase transition, 2D vs 3D... Shear banding could eliminate constitutive equations : stress cannot be a simple function of strain, you have to distinguish cases (like in a phase transition); it can be a non-monotonous constitutive equation Maybe the minimum of this relation is so shallow that it can easily be modified or suppressed Shear banding appears often in grains. In foams it is still controversial. Debregeas, Dennin (for experiments), Jiang, Cox, Kabla (for simulations) see it in different systems (planar, annular). Localisation might have been seen (O'Hern) in the Durian bubble model. Disagreements about the necessary parameters, or band width, or position of the band (outside, inside, or in the middle). Shear bands are more probable in monodisperse foams. A large dislocation movement can be called a fracture or shear band. The width of the shear band decreases (= less bubbles move) if the fluid fraction increases. Stress (and thus strain) are expected to be conserved, hence not localised. No localisation in 3D ? Denkov does not see it (due to polydispersity? to the additional degrees of freedom in 3D?). Coussot sees one similar to Dennin. Rouyer sees it at 90% fluid fraction in oscillatroy strain at high amplitude. Does the shear band get wetter, with feedback on the shear modulus and thus on the flow? Role of the geometry : In 3D Stokes experiment, the T1s decrease exponentially with the distance to the obstacle (=localisation?). In 2D, they decrease more slowly, without discontinuity (=no localisation?) If there is a critical strain rate, the geometry (eg annular) can affect the place where the critical strain occurs, and hence the T1 distribution. With tooth-shaped obstacle (cog-wheel), the slip occurs at the next bubble layer; try with a polydisperse foam? See Debregeas. Do these bands heal? If you stop the shear, then try again, do they appear at the same place? Difficult to define in quasistatic limit; maybe try to invert the rotation. If the flow has induced a density variation, it might not heal. Kabla : shear band is where the fluctuations are stronger. Even for the fluctuations of the texture tensor in experiments. Bubble size distribution : effect on shear modulus, yield point? Cox is performing simulations. See Kraynik's work in 3D: energy and shear modulus variations with foam dispersity. See his use of R32 as characterisation of many polydisperse foams: in 2D, it would be R21? Who will do the experiments? Picnic on the beach tonight? Effect of liquid fraction: H2O, CH2COOH (try effect of C02 for coarsening + proteins for stabilisation + malt for taste)