Jamming (physics)

Jamming is the physical process by which the viscosity of some mesoscopic materials, such as granular materials, glasses, foams, polymers, emulsions, and other complex fluids, increases with increasing particle density. The jamming transition has been proposed as a new type of phase transition, a strongly idealized hard-sphere glass transition, but very different from the formation of crystalline solids. While a glass transition occurs when the liquid state is cooled, the jamming transition happens when the density, or the packing fraction of the particles, is increased. This crowding of the constituent particles prevents them from flowing under an applied stress and from exploring phase space, thus making the aggregate material behave as a solid. The system may be able to unjam if volume fraction is decreased, or external stresses are applied such that they exceed the yield stress. This transition is interesting because it is nonlinear with respect to volume fraction. The jamming phase diagram relates the jamming transition to inverse density, stress and temperature. The density at which systems jam is determined by many factors, including the shape of their components, the deformability of the particles, frictional interparticle forces, and the degree of dispersity of the system. The overall shape of the jamming manifold may depend on the particular system. For example, a particularly interesting feature of the jamming transition is the difference between attractive and repulsive particle systems. Whether the jamming surface diverges for high enough densities or low temperatures is uncertain. Simulations of jammed systems study particle configurations leading to jamming in both static systems and systems under shear. Under shear stress, the average cluster size may diverge after a finite amount of strain, leading to a jammed state. A particle configuration may exist in a jammed state with a stress required to "break" the force chains causing the jam. The simplest realization of a static jammed system is a random sphere packing of frictionless soft spheres that are jammed together upon applying an external hydrostatic pressure to the packing. Right at the jamming transition, the applied pressure is zero and the shear modulus is also zero, which coincides with the loss of rigidity and the unjamming of the system. Also, at the jamming point the system is isostatic. Above the jamming point, the applied pressure causes an increase of volume fraction by squeezing the soft spheres closer together, and thus creates additional contacts between neighboring spheres. This leads to an increase of the average number of contacts z {\displaystyle z} . As shown in numerical simulations by Corey O'Hern and collaborators, the shear modulus G increases with increasing z {\displaystyle z} following the law: G ∼ ( z − 2 d ) {\displaystyle G\sim (z-2d)} , where d is the dimension of space. A first-principles microscopic theory of elasticity developed by Alessio Zaccone and E. Scossa-Romano quantitatively explains this law in terms of two contributions: the first term is a bonding-type contribution, thus proportional to z {\displaystyle z} , and related to particle displacements which exactly follow the applied shear deformation; the second (negative) term is due to internal relaxations needed to keep local mechanical equilibrium in a strained disordered environment, and thus proportional to the total number of degrees of freedom, hence the dependence on space dimension d. This model is relevant for compressed emulsions, where the friction between particles is negligible. Another example of static jammed system is a sand pile, which is jammed under the force of gravity and no energy is being dissipated. A high dimensional limit mean-field theory for jamming was obtained by Giorgio Parisi and Francesco Zamponi, where clustering density and jamming density ϕ {\displaystyle \phi } scales as: ϕ ∼ d log ⁡ ( d ) 2 d {\displaystyle \phi \sim {\frac {d\log(d)}{2^{d}}}} Near jamming from below ( ϕ → ϕ J − ) {\displaystyle \left(\phi \to \phi _{J}^{-}\right)} , the pair correlation function delta peak scales as:

g ( r ) ∼ 1 δ f ( r − 1 δ ) , where δ ≡ ϕ J − ϕ → 0 {\displaystyle g(r)\sim {\frac {1}{\delta }}f\left({\frac {r-1}{\delta }}\right){\text{, where }}\delta \equiv \phi _{J}-\phi \to 0}

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