arXiv:0708.1961v1 [cond-mat.soft] 14 Aug 2007 The jamming transition and new percolation universality classes in particulate systems with attraction Gregg Lois, Jerzy Blawzdziewicz, and Corey S. O’Hern Department of Mechanical Engineering, Department of Physics, Yale University, New Haven, Connecticut 06520-8284 We numerically study the jamming transition in particulate systems with attraction by investi- gating their mechanical response at zero temperature. We find three regimes of mechanical behavior separated by two critical transitions—connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels. The provocative conjecture that mechanical response at zero temperature is linked to slow dynamics at non- zero temperature in repulsive glassy systems has sparked tremendous interest in the jamming transition at “point J” [1]. At point J athermal, frictionless systems with finite-range repulsive interactions [2] undergo a single critical transition from an unjammed state with a van- ishing static shear modulus to a jammed state with a non-zero static shear modulus as density is increased. At the transition, it is observed that the average number of particles participating in the connected network jumps discontinuously from 0 to L d (where d is the spatial di- mension and L is the system size), which makes the tran- sition first-order in the network order parameter. Recent work has indeed shown that properties of the jammed state are closely related to the slow dynamics in highly compressed repulsive glasses [3]. How does the jamming transition change for partic- ulate systems with attraction? It is likely that attrac- tive interactions will qualitatively change the nature of the jamming transition. For example, thermal systems with attraction can form repulsive glasses, but they can also form gels and attractive glasses [4]. Moreover, at zero temperature the mechanical properties of attractive granular materials and powders are quite different than those for dry granular media with purely repulsive inter- actions [5]. While the jamming phase diagram of Ref. [1] needs revision to include attraction, the close correspon- dence between mechanical properties at zero tempera- ture and dynamics at non-zero temperature is robust. This has been demonstrated in experiments on attrac- tive colloidal suspensions [6] where, as expected from the jamming phase diagram, gelation occurs upon increasing density, decreasing thermalization, and decreasing stress. In this Letter, we explore the jamming transition in attractive particulate systems by studying their mechan- ical response at zero temperature. A central conclusion from our work is that repulsive jamming is fundamentally different than attractive jamming, even for an infinitesi- mal amount of attraction. Instead of a single first-order transition in the purely repulsive systems, we observe two second-order transitions in the attractive systems— connectivity and rigidity percolation. These two transi- tions separate three distinct types of mechanical response and exhibit critical exponents that differ from those mea- sured in the connectivity and rigidity percolation transi- tions without force balance constraints. The transitions we observe at zero temperature are also present at small but finite temperatures and resemble gelation. Simulation procedure We investigate the quasistatic compression of an attractive particulate system in two and three dimensions at zero temperature. The system consists of N particles interacting via a pairwise, spher- ically symmetric potential, with a finite repulsive core and finite-range attraction. Simulations begin with a dilute collection of N spherical particles of diameter σ i randomly placed in a cubic cell with periodic boundary conditions. In each simulation step the diameters of all particles are increased by a small factor and then the potential energy is minimized using a conjugate gradient method. The force F (r ij ) between a pair of particles i and j depends on their separation r ij relative to the sum of their radii σ ij =(σ i + σ j )/2, and is plotted in the inset of Fig. 1(a). Since the overall scale Y of the force is irrel- evant at zero temperature, there are three independent parameters: the packing fraction φ (calculated using the location of the minimum in the force law), the range of the attraction ℓ, and the minimum scaled force −C. Mechanical Response At zero temperature, whether a particulate system with attractions is jammed depends on two important factors: a) does the system contain a percolating cluster and b) if so, does the percolating cluster possess any floppy modes? A floppy mode [7] is an infinitesimal deformation of the material that does not increase its energy. The number of floppy modes is equal to the number of non-trivial zero eigenvalues of the dynamical matrix for a given set of particle positions and radii [8]. If there are no non-trivial [9] zero eigenvalues, the potential energy increases with any deformation and all elastic constants are non-zero. We refer to this state as “jammed”. If there is at least one non-trivial zero eigenvalue, the system is termed “unjammed”. Note that this is a very strict definition of jamming. It is impossible to have a jammed system that does not