Foams : structure and dynamics

1 Uses of foams . . 1
1 The foams around us . . 1
1.1 Foams in mythology . . 1
1.2 On your plate and in your glass . . 1
1.3 Detergents and cosmetics . . 3
1.4 Spontaneous or undesirable foams . . 4
2 Foam identification . . 4
2.1 Physico-chemical constituents . . 4
2.2 Geometrical and physical properties . . 5
2.3 Mechanical properties . . 6
3 What are foams used for? . . 6
3.1 Desirable functions . . 6
3.2 Mineral flotation . . 8
4 Solid foams and other cellular systems . . 9
4.1 Solid foams . . 9
4.2 Other cellular structures . . 11
5 Experiments . . 13
5.1 Three ways to make a foam . . 13
5.2 Chocolate mousse . . 14
References . . 15

2 Foams at equilibrium . . 17
1 Description at all length-scales . . 17
1.1 At the scale of a gas/liquid interface . . 17
1.2 At the scale of a film . . 19
1.3 At the scale of a bubble . . 21
1.4 At the scale of a foam . . 22
2 Local equilibrium laws . . 23
2.1 Equilibrium of fluid interfaces . . 23
2.2 Plateau's laws . . 26
3 Dry foams . . 30
3.1 Number of neighbours: topology . . 31
3.2 Bubble geometry . . 35
3.3 Topology and geometry  . 38
4 Wet foams . . 45
4.1 Modification of the structure . . 46
4.2 Osmotic pressure . . 51
4.3 Role of gravity . . 54
5 2D and quasi-2D foams . . 55
5.1 3D structure of a monolayer of bubbles between two plates . . 57
5.2 A model for a dry 2D foam . . 58
5.3 Two-dimensional liquid fraction . . 60
5.4 2D foam flows . . 61
6 Experiments  . 63
6.1 Surface tension and surfactants . . 63
6.2 Creation and observation of 2D and quasi-2D foams . . 65
6.3 Giant soap films . . 66
6.4 Kelvin cell . . 68
7 Exercises . . 69
7.1 Interfacial area of a foam . . 69
7.2 Film tension and the Young-Laplace law . . 69
7.3 Plateau's laws in 2D . . 70
7.4 Euler's formula . . 71
7.5 Perimeter of a regular 2D bubble . . 71
7.6 Energy and pressure . . 72
References . . 72

3 Birth, life, and death . . 75
1 Foam evolution . . 75
1.1 The competition between different processes . . 75
1.2 Elementary topological processes . . 78
2 Birth of a foam . . 82
2.1 Foamability: introduction to the role of surfactants . . 82
2.2 Interfacial properties and foamability . . 82
2.3 Properties of liquid films and foamability . . 92
2.4 Summary of the microscopic origins of foamability . . 98
3 Coarsening . . 99
3.1 Growth rate of a bubble in a dry foam . . 99
3.2 Evolution of bubble distributions in a dry foam . . 104
3.3 Effects of different parameters . . 109
4 Drainage . . 113
4.1 What is drainage? . . 114
4.2 Free drainage . . 114
4.3 Forced drainage . . 115
4.4 Modelling flows in solid porous media . . 116
4.5 Modelling the permeability of a liquid foam . . 119
4.6 Drainage equations . . 127
4.7 Comparison of theoretical predictions with experiments . . 128
4.8 Summary and remarks . . 133
5 Rupture and coalescence . . 134
5.1 Rupture at the scale of a single film . . 134
5.2 Rupture at the scale of a foam . . 140
5.3 Defoamers and antifoams . . 140
6 Appendices . . 145
6.1 Stabilizing agents . . 145
6.2 Dissipation due to surfactant motion during the steady expansion of a film . . 151
7 Experiments . . 154
7.1 Flow in a soap film . . 154
7.2 Free drainage in a foam and the vertical motion of bubbles . . 156
7.3 Forced drainage in a foam: observation of the wetting front . . 157
7.4 Life and death of a foam measured by electrical conductivity . . 158
8 Exercises . . 161
8.1 Exponent in the scale-invariant regime . . 161
8.2 Frumkin equation of state . . 161
8.3 Foam drainage and equilibrium height . . 161
8.4 Drainage in the bulk and at the wall . . 162
8.5 Free drainage: characteristic times and liquid fraction profiles . . 162
8.6 The true 3D pressure and 2D surface pressure . . 162
References . . 162

4 Rheology . . 167
1 Introduction . . 167
2 Overview of the rheological behaviour of complex fluids . . 168
2.1 Constitutive laws . . 168
2.2 Shear tests . . 172
2.3 Small and large strains . . 173
2.4 Stress tensor in a complex fluid . . 174
3 Local origin of rheological properties . . 178
3.1 Elastic shear modulus of a dry monodisperse foam . . 178
3.2 The elastic limit of a dry foam . . 183
3.3 Dissipative processes . . 187
4 The multiscale character of foam rheology . . 193
4.1 Solid behaviour . . 194
4.2 Transition from solid to liquid behaviour . . 208
4.3 Foam flow . . 211
5 Appendix: Prom the discrete to the continuous . . 215
6 Experiments . . 217
6.1 Observation of T1s . . 217
6.2 Visualization of the yield stress . . 217
7 Exercises . . 218
7.1 The Young-Laplace law and the stress in a spherical bubble . . 218
7.2 Elasticity of a dry 2D foam . . 219
7.3 Poynting's law . . 219
7.4 Stress and strain in a square lattice . . 220
7.5 Elasticity and plasticity . . 220
7.6 Compressibility of a foam . . 221
References . . 221

5 Experimental and numerical methods . . 225
1 Experimental methods . . 225
1.1 Methods used to study interfaces and isolated films . . 225
1.2 Methods for studying foams . . 230
2 Numerical simulations . . 242
2.1 Predicting static structure . . 242
2.2 Predicting dynamics . . 244
3 Methods of image analysis . . 248
3.1 Image treatment . . 248
3.2 Image analysis . . 250
3.3 Image analysis, liquid fraction, and stress in 2D . . 254
Exercises . . 255
4.1 Measurement of the average liquid fraction of a foam . . 255
4.2 Pressure in the Potts model . . 255
References . . 256

Notation . . 259
Index . . 263