Jamming V: Jamming in two-dimensional packings

We investigate the existence of random close and random loose packing limits in twodimensional packings of monodisperse hard disks. A statistical mechanics approach— based on several approximations to predict the probability distribution of volumes— suggests the existence of the limiting densities of the jammed packings according to their coordination number and compactivity. This result has implications for the understanding of disordered states in the disk packing problem as well as the existence of a putative glass transition in two dimensional systems. Introduction. – The concept of jamming is a common feature of out of equilibrium systems experiencing a dynamical arrest ranging from emulsions, colloids, glasses and spin glasses, as well as granular materials [1]. For granular matter, it is argued that a statistical mechanical description can be used, with volume replacing energy as the conservative quantity [2]. In this framework, a mesoscopic model has been presented [3], allowing the development of a thermodynamics for jamming in any dimension. Here we develop this theoretical approach to investigate the existence of disordered packings in two-dimensional systems composed of equal-sized hard disks [4]. The existence of amorphous packings in 2d is a problem of debate in the literature: two dimensional systems are found to crystallize very easily since disordered packings of disks are particularly unstable [5]. In two dimensional Euclidean space, the hexagonal packing arrangement of circles (honeycomb circle packing) has the highest density of all possible plane packings (ordered or disordered) with a volume fraction φhex = π √ 12 ≃ 0.9069 and each disk surrounded by 6 disks. Regarding amorphous packings, experiments find a maximum density of random close packing (RCP) of monodisperse spheres at φrcp ≈ 0.82 [4] while the lower limit (random loose packing, RLP) has been little investigated and its existence has not been treated so far. The tendency of 2d packings to easily crystallize has led to consider bidisperse systems which pack at a higher RCP volume fractions of φrcp ≈ 0.84 [6, 7]. In parallel to studies in the field of jamming— which consider the packing problem as a jamming transition approached from the solid phase [6–8]— other studies attempt to characterize jamming approaching the transition from the liquid phase [9, 10]. Here, amorphous jammed packings are seen as infinite pressure glassy states [9]. Therefore, the existence of disordered jammed structures (of frictionless particles) is related to the existence of a glass transition in 2d [9], a problem that has been debated recently [11]. Here we treat the disordered disk packing problem with the statistical mechanics of granular jammed matter [2]. The formal analogy with classical statistical mechanics is the following: the microcanonical ensemble, defined by all microstates with fixed energy, is replaced by the ensemble of all jammed microstates with fixed volume. Hence, the appropriate function for the description of the system is no longer the Hamiltonian, but the volume function, Wi, giving the volume available to each particle unit such that the total system volume is V = ∑ iWi [2]. The aim of the present work is to develop the model presented in [3] for the calculation of the volume function in the case of 2d packings. The validity of the hypothesis employed in [3] are discussed, and they are modified according to the properties of 2d packings. We use our results to study the nature of the RLP and RCP limit in 2d through an elementary construction of a statistical mechanics that allows the study of the existence of a maximum and minimum attainable density of disordered circle packings. We find that amorphous packings can pack between the density limits of ∼77.5% and ∼80.6% defining the RLP and RCP respectively, according to system coordination number and friction, opening such predictions to experimental and computational investigation. While these values should be considered as bounds to the real