DK
D.J. Koeze
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This thesis considers the mechanical properties of amorphous solids such as foams, emulsions, and granular media. Each of these systems consists of “particles” (bubbles, droplets, grains) in a dense, disordered structure. As a system is compressed, it eventually forms enough contacts between particles that it can support a load, including shearing stresses. We say that it has jammed. A major theoretical challenge is to describe material properties in the vicinity of this non-equilibrium phase transition. Jamming has been widely studied in the context of a specific model, namely non- Brownian packings of soft frictionless disks or spheres. The particles repel when they overlap, and otherwise do not interact. They come in two distinct sizes to prevent crystallization; by convention the concentration and diameter ratio of the species are fixed to specific values. Little is known about the nature of the jammed solids that result when these restrictions are lifted. This is a significant knowledge gap, because bubbles, droplets, and grains routinely experience some degree of attraction to their neighbors (e.g. due to depletion interactions or capillary bridges), and their size distribution can vary considerably. Hence the goal of this thesis is to determine how the soft sphere model jams (i) when the degree of attraction between particles is varied, and (ii) when the size and number ratio of particles in a repulsive bidisperse packing is varied. Attraction.—First, we study how soft particles with an attractive shell become rigid. By analyzing the percolation of rigid clusters of particles, we present evidence for two distinct jamming scenarios. Strongly attractive systems undergo a continuous transition in which rigid clusters grow and ultimately diverge in size at a critical packing fraction. Purely repulsive and weakly attractive systems jam via a first order transition, with no growing cluster size. We further show that the weakly attractive scenario is a finite size effect, so that for any nonzero attraction strength, a sufficiently large system will fall in the strongly attractive universality class. We therefore expect attractive jamming to be generic in the laboratory and in nature. Second, we probe the elasticity of the strongly attractive solid. By treating the jamming point as a critical point, we exploit critical scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. We find that each observable differs not just quantitatively but also qualitatively from the purely repulsive case. Size and number ratio.—We systematically map out the jamming transition of 2D bidisperse mixtures of disks in the hard particle limit. The critical volume fraction and multiple structural and mechanical properties all show a rich variation with mixture composition and particle size ratio, and can therefore be tuned by choosing certain mixtures. We identify two local minima in the critical volume fraction, both of which have low structural order; one minimum is close to the widely studied 50:50 mixture of particles with a ratio of radii of 1:1.4. We also identify a region at low size ratios characterized by increased structural order, with a corresponding enhancement in the stiffness.
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This thesis considers the mechanical properties of amorphous solids such as foams, emulsions, and granular media. Each of these systems consists of “particles” (bubbles, droplets, grains) in a dense, disordered structure. As a system is compressed, it eventually forms enough contacts between particles that it can support a load, including shearing stresses. We say that it has jammed. A major theoretical challenge is to describe material properties in the vicinity of this non-equilibrium phase transition. Jamming has been widely studied in the context of a specific model, namely non- Brownian packings of soft frictionless disks or spheres. The particles repel when they overlap, and otherwise do not interact. They come in two distinct sizes to prevent crystallization; by convention the concentration and diameter ratio of the species are fixed to specific values. Little is known about the nature of the jammed solids that result when these restrictions are lifted. This is a significant knowledge gap, because bubbles, droplets, and grains routinely experience some degree of attraction to their neighbors (e.g. due to depletion interactions or capillary bridges), and their size distribution can vary considerably. Hence the goal of this thesis is to determine how the soft sphere model jams (i) when the degree of attraction between particles is varied, and (ii) when the size and number ratio of particles in a repulsive bidisperse packing is varied. Attraction.—First, we study how soft particles with an attractive shell become rigid. By analyzing the percolation of rigid clusters of particles, we present evidence for two distinct jamming scenarios. Strongly attractive systems undergo a continuous transition in which rigid clusters grow and ultimately diverge in size at a critical packing fraction. Purely repulsive and weakly attractive systems jam via a first order transition, with no growing cluster size. We further show that the weakly attractive scenario is a finite size effect, so that for any nonzero attraction strength, a sufficiently large system will fall in the strongly attractive universality class. We therefore expect attractive jamming to be generic in the laboratory and in nature. Second, we probe the elasticity of the strongly attractive solid. By treating the jamming point as a critical point, we exploit critical scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. We find that each observable differs not just quantitatively but also qualitatively from the purely repulsive case. Size and number ratio.—We systematically map out the jamming transition of 2D bidisperse mixtures of disks in the hard particle limit. The critical volume fraction and multiple structural and mechanical properties all show a rich variation with mixture composition and particle size ratio, and can therefore be tuned by choosing certain mixtures. We identify two local minima in the critical volume fraction, both of which have low structural order; one minimum is close to the widely studied 50:50 mixture of particles with a ratio of radii of 1:1.4. We also identify a region at low size ratios characterized by increased structural order, with a corresponding enhancement in the stiffness.
Numerous soft materials jam into an amorphous solid at a high packing fraction. This nonequilibrium phase transition is best understood in a model system where particles repel when they overlap. Recently, however, it was shown that introducing any finite amount of attraction between particles changes the universality class of the transition. The properties of this "sticky jamming"class remain almost entirely unexplored. We use molecular dynamics simulations and scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. Each observable differs not just quantitatively but also qualitatively from the purely repulsive case.
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Numerous soft materials jam into an amorphous solid at a high packing fraction. This nonequilibrium phase transition is best understood in a model system where particles repel when they overlap. Recently, however, it was shown that introducing any finite amount of attraction between particles changes the universality class of the transition. The properties of this "sticky jamming"class remain almost entirely unexplored. We use molecular dynamics simulations and scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. Each observable differs not just quantitatively but also qualitatively from the purely repulsive case.
Sticky Matters
Jamming and Rigid Cluster Statistics with Attractive Particle Interactions
While the large majority of theoretical and numerical studies of the jamming transition consider athermal packings of purely repulsive spheres, real complex fluids and soft solids generically display attraction between particles. By studying the statistics of rigid clusters in simulations of soft particles with an attractive shell, we present evidence for two distinct jamming scenarios. Strongly attractive systems undergo a continuous transition in which rigid clusters grow and ultimately diverge in size at a critical packing fraction. Purely repulsive and weakly attractive systems jam via a first-order transition, with no growing cluster size. We further show that the weakly attractive scenario is a finite size effect, so that for any nonzero attraction strength, a sufficiently large system will fall in the strongly attractive universality class. We therefore expect attractive jamming to be generic in the laboratory and in nature.
...
While the large majority of theoretical and numerical studies of the jamming transition consider athermal packings of purely repulsive spheres, real complex fluids and soft solids generically display attraction between particles. By studying the statistics of rigid clusters in simulations of soft particles with an attractive shell, we present evidence for two distinct jamming scenarios. Strongly attractive systems undergo a continuous transition in which rigid clusters grow and ultimately diverge in size at a critical packing fraction. Purely repulsive and weakly attractive systems jam via a first-order transition, with no growing cluster size. We further show that the weakly attractive scenario is a finite size effect, so that for any nonzero attraction strength, a sufficiently large system will fall in the strongly attractive universality class. We therefore expect attractive jamming to be generic in the laboratory and in nature.
Scaling of flow curves
Comparison between experiments and simulations
Yield-stress materials form an interesting class of materials that behave like solids at small stresses, but start to flow once a critical stress is exceeded. It has already been reported both in experimental and simulation work that flow curves of different yield-stress materials can be scaled with the distance to jamming or with the confining pressure. However, different scaling exponents are found between experiments and simulations. In this paper we identify sources of this discrepancy. We numerically relate the volume fraction with the confining pressure and discuss the similarities and differences between rotational and oscillatory measurements. Whereas simulations are performed in the elastic response regime close to the jamming transition and with very small amplitudes to calculate the scaling exponents, these conditions are hardly possible to achieve experimentally. Measurements are often performed far away from the critical volume fraction and at large amplitudes. We show that these differences are the underlying reason for the different exponents for rescaling flow curves.
...
Yield-stress materials form an interesting class of materials that behave like solids at small stresses, but start to flow once a critical stress is exceeded. It has already been reported both in experimental and simulation work that flow curves of different yield-stress materials can be scaled with the distance to jamming or with the confining pressure. However, different scaling exponents are found between experiments and simulations. In this paper we identify sources of this discrepancy. We numerically relate the volume fraction with the confining pressure and discuss the similarities and differences between rotational and oscillatory measurements. Whereas simulations are performed in the elastic response regime close to the jamming transition and with very small amplitudes to calculate the scaling exponents, these conditions are hardly possible to achieve experimentally. Measurements are often performed far away from the critical volume fraction and at large amplitudes. We show that these differences are the underlying reason for the different exponents for rescaling flow curves.
Many soft matter systems are confined in some but not all dimensions; examples include microfluidic channels and inclined plane flows. Hence it is important to characterize finite size effects not only as a function of volume, but also for varying aspect ratio. For soft sphere packings close to the jamming transition, finite size effects are well understood, but only in square and cubic systems. In these cases there is clear evidence for a critical volume that diverges at jamming, but it is equally clear that this picture must break down for extreme aspect ratios. We perform simulations of soft spheres near jamming in two and three dimensions for aspect ratios as large as 1024. In addition to the previously identified critical volume, we find evidence for a non-trivial length scale that diverges at the jamming point.
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Many soft matter systems are confined in some but not all dimensions; examples include microfluidic channels and inclined plane flows. Hence it is important to characterize finite size effects not only as a function of volume, but also for varying aspect ratio. For soft sphere packings close to the jamming transition, finite size effects are well understood, but only in square and cubic systems. In these cases there is clear evidence for a critical volume that diverges at jamming, but it is equally clear that this picture must break down for extreme aspect ratios. We perform simulations of soft spheres near jamming in two and three dimensions for aspect ratios as large as 1024. In addition to the previously identified critical volume, we find evidence for a non-trivial length scale that diverges at the jamming point.
While jamming is best understood in the context of purely repulsive soft spheres, emulsions and other experimental realizations of the soft sphere model commonly display weakly cohesive forces. We perform simulations of soft spheres with a finite-ranged attractive tail in the pair potential. The resulting attractive soft sphere packings can be stable at volume fractions below the purely repulsive jamming point. These new jammed states have counter-intuitive properties -- for example, while attraction introduces tensile forces, their presence leads to an increase in the compressive stress. We use critical scaling analysis to characterize the geometry and mechanics of attractive soft sphere packings as a function of both the volume fraction and the range of the attractive interaction.
...
While jamming is best understood in the context of purely repulsive soft spheres, emulsions and other experimental realizations of the soft sphere model commonly display weakly cohesive forces. We perform simulations of soft spheres with a finite-ranged attractive tail in the pair potential. The resulting attractive soft sphere packings can be stable at volume fractions below the purely repulsive jamming point. These new jammed states have counter-intuitive properties -- for example, while attraction introduces tensile forces, their presence leads to an increase in the compressive stress. We use critical scaling analysis to characterize the geometry and mechanics of attractive soft sphere packings as a function of both the volume fraction and the range of the attractive interaction.
We systematically map out the jamming transition of all 2D bidisperse mixtures of frictionless disks in the hard-particle limit. The critical volume fraction, mean coordination number, number of rattlers, structural order parameters, and bulk modulus each show a rich variation with mixture composition and particle size ratio, and can therefore be tuned by choosing certain mixtures. We identify two local minima in the critical volume fraction, both of which have low structural order; one minimum is close to the widely studied 50 : 50 mixture of particles with a ratio of radii of 1 : 1.4. We also identify a region at low size ratios characterized by increased structural order and high rattler fractions, with a corresponding enhancement in the stiffness.
...
We systematically map out the jamming transition of all 2D bidisperse mixtures of frictionless disks in the hard-particle limit. The critical volume fraction, mean coordination number, number of rattlers, structural order parameters, and bulk modulus each show a rich variation with mixture composition and particle size ratio, and can therefore be tuned by choosing certain mixtures. We identify two local minima in the critical volume fraction, both of which have low structural order; one minimum is close to the widely studied 50 : 50 mixture of particles with a ratio of radii of 1 : 1.4. We also identify a region at low size ratios characterized by increased structural order and high rattler fractions, with a corresponding enhancement in the stiffness.