Statistical mechanics of vibration-induced compaction of powders

Abstract

We propose a theory which describes the density relaxation of loosely packed, cohesionless granular material under mechanical tapping. Using the compactivity concept we develop a formalism of statistical mechanics which allows us to calculate the density of a powder as a function of time and compactivity. A simple fluctuation-dissipation relation which relates compactivity to the amplitude and frequency of a tapping is proposed. Experimental data of E.R.Nowak et al. [Powder Technology 94, 79 (1997) ] show how density of initially deposited in a fluffy state powder evolves under carefully controlled tapping towards a random close packing (RCP) density. Ramping the vibration amplitude repeatedly up and back down again reveals the existence of reversible and irreversible branches in the response. In the framework of our approach the reversible branch (along which the RCP density is obtained) corresponds to the steady state solution of the Fokker-Planck equation whereas the irreversible one is represented by a superposition of "excited states" eigenfunctions. These two regimes of response are analyzed theoretically and a qualitative explanation of the hysteresis curve is offered.PACS numbers 81.05.R,05.40.,03.20.

Patterns and collective behavior in granular media: Theoretical concepts

“…Due to the clustering long-term temperature evolution is T ∼ t −2/3 which is different from Haff's law T ∼ t −2 derived for the spatially-homogeneous cooling. Efrati et al (2005) argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute granular gases consistent with the Burgers equation (19) description of one-dimensional gas of "sticky particles" suggested by Nie et al (2002).…”

“…Meerson and Puglisi (2005) conducted molecular dynamics simulations of the clustering instability of a freely cooling dilute inelastic gas in a quasi-one-dimensional setting. This problem was also examined in the framework of granular hydrodynamics by Efrati et al (2005). It was observed that, as the gas cools, stresses become negligibly small, and the gas flows only by inertia.…”

“…scaling exponents for temperature) are only assessed within molecular dynamics simulation because the hydrodynamic description often breaks in dense cold clusters. One recent theoretical approach to the description of the late stages of clustering instability consists in introducing additional regularization into the hydrodynamic description due to the finite size of particles (Efrati et al, 2005;Nie et al, 2002). Nie et al (2002) argued that cluster formation and coalescence in freely cooling granular gases can be heuristically described by the Burgers equation for hydrodynamic velocity v with random initial conditions:…”

Hysteresis in vibrated granular media

“…If the above is true, the compactivity will be an increasing function of the vibration intensity, because the density over the "reversible" curve decreases with the tapping strength. In fact, this kind of behavior has been found in a simple model of a granular system described in terms of a Fokker-Planck equation [6].…”

“…This is, for instance, the behavior found in the simple two-volume model considered in Ref. [6]. In general, this property must be checked once the relationship between Γ and X, X = f (Γ), has been derived for each specific model.…”