Statistical Mechanics of Vibration-Induced Compaction of Powders

TL;DR

This paper develops a statistical mechanics framework using the concept of compactivity to model how powders densify under mechanical tapping, explaining experimental observations of reversible and irreversible compaction behaviors.

Contribution

It introduces a formalism linking compactivity to tapping parameters and provides a theoretical explanation for hysteresis in powder compaction.

Findings

Density approaches RCP under tapping

Reversible and irreversible compaction branches identified

Hysteresis explained via eigenfunction superposition

Abstract

We propose a theory which describes the density relaxation of loosely packed, cohesionless granular material under mechanical tapping. Using the compactivity concept we develope 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. [{\it 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.