Statistics of large contact forces in granular matter

Granular materials such as sand have both liquid-like and solid-like properties similar to both liquids and solids. Dry sand in an hour-glass can flow just like water, while sand in a sand castle closely resembles a solid. Because of these interesting properties granular matter has received much attention from numerous physicists. Part of the research on these materials focuses on the statistics of contact forces between individual particles and how these statistics can be used to understand and predict material properties. Contact forces in granular materials are organized in so-called forces networks. A common quantity to characterize these force variations is the probability distribution of the contact force, P(f). A long-standing issue is the asymptotic behavior of this distribution. In particular, one discusses whether the tail of P(f) is exponential, Gaussian, or has a different form. Furthermore, its relation with material and system properties is unclear. In view of the technical difficulty to measure contact forces, especially in the bulk of the material, we used computer simulations in the so-called force network ensemble of Snoeijer et al. (Phys. Rev. Lett., 2004, 92, 054302). Unfortunately, the estimation of P(f) for large contact forces f is inefficient. The reason is that by far the largest fraction of generated force networks contains only small forces. For unambiguous conclusions on the asymptotic behaviour of P(f), extremely long (of the order of years or even centuries) computer calculations are needed. To obtain better statistics for large contact forces, we developed an umbrella sampling method for the force network ensemble. In Chapter 1, an overview is given of the different methods to study the statistics of force networks. We also explained the umbrella sampling method that we developed to obtain excellent statistics for large forces. In Chapter 2, we applied this method to study the tail of the force distribution P(f) for different systems. The average number of contacts of a particle and the packing configuration are shown not to be important for the asymptotic behavior of P(f). Only the dimensionality of the system has a significant influence: P(f)~exp[-c f a] with a?2.0 for two-dimensional systems, a?1.7 for three-dimensional systems and a?1.4 for four-dimensional systems. In Chapter 3, a possible explanation is presented for the Gaussian decay of large contact forces in two-dimensional systems. It was found that mechanical balance on each particle is essential for the tail of the contact force distribution, which throws serious doubts on the statement that exponential statistics are a generic property of static granular materials. In Chapter 4, we focused on several details of the contact forces and their distribution. We also investigated how well the force network ensemble describes systems with “real” interactions