March 06, 2008.
Statistical Thermodynamics
Theory of Powders. S.F. Edwards and R.B.S. Oakeshott
Journal Review Report
Introduction
The present paper presents a derivation of basic Statistical Mechanics principles to study powder systems. The approach was followed given the nature of powders, which are formed by a large number of particles (ranging from 100 to 1016 /ml) and still can be completely defined by a small number of parameters and constructed in a reproducible way( by macroscopic means).
Development
The authors start setting up the analogies between the descriptive variables of powders and Stat Mech. in order to be able to use the Stat Mech. ideas. Given that powders are static systems, where each of the particles is at rest and in contact with the other particles, the energy due to thermal temperature is neglected (there is no variation of the particles speed due to changes in T), and it is the volume who plays the role of the energy.
Stat Mech. Principles: In the regular systems, the easiest approach is the microcanonical ensemble (constant N, V and E) where the system takes all possible configurations with equal probability subject to the Hamiltonian function giving constant E. In this case the distribution function as studied in the course is given by:
However, experimentally it is easier to use the canonical ensemble, given that E is not easy to measure. In this ensemble the distribution function is given by the expression below, where F refers to the Helmholtz free energy.
Analogy to powders: As mentioned before, to describe powders, the volume occupied by it is the most useful variable. Therefore the function W is introduced, which similarly to the Hamiltonian gives the volume of the system based on the coordinates of the particles. Besides W, the function Q is introduced as well, which picks out valid configurations of the particle system (particles can’t overlap, and need to be in a stable state).
Given the assumption that for a given volume all the configurations are equally probable, a table is constructed comparing regular systems with powders.
Statistical mechanics / PowdersE / V
H / W
Kb /
F=E-TS / Y=V-XS
As we can see, the new variable X is the analogue of the Temperature, and measures the ‘fluffiness” of the powder, where X=0 represents the most compact powder and X= the least. On the other hand Y is similar to the free Energy, but in this case, the free volume or “effective volume”.
Overall we can see that the new system for powders has less independent variables:
Stat mech / PowdersE=E(S,V, N) / V=V(S,N)
F=F(T,V,N) / Y=Y(X,N)
After defining the analogies between stat mech and the powders theory, the authors study the volume of a simple particle powder via two methods:
In the fist one they study a 2 or 3 dimension system by looking at a 1-D problem, where particles of length “a” can be separated by a maximum distance b. They define the volume for the system as well as the phase space available:
and
And calculate the entropy of the system to obtain both the free energy and the volume as:
The second approach consists on using a smaller set of variables. In this case they consider the compactivity of uniform grains, and assume that their contribution to the overall volume will depend on the coordination number, where vc is the volume of a particle coordinated with c neighbors, and vcc’ is a correction factor for how different coordinated particles contribute to the volume. Then, the total volume will be given by:
Or, naming each grain “i”,
Finally, analyzing a simple example where there are only 2 types of coordination C0 and C1 the distribution function becomes:
From this formulas it can be calculated the maximum and minimum volume corresponding to X=0 and X=.
As we can see both approaches are simplified models, but let us characterize a powder system using Statistical Mechanics. The paper argues that it is needed more detailed information of the local structure for a more realistic theory of powder, but gives us the overall picture and connection to Stat Mech.
Alfredo Guariguata