STATISTICAL MECHANICS OF POWDER MIXTURES ... - Science Direct

Physica A 157 (1989) 1091-1100 North-Holland, Amsterdam

STATISTICAL

MECHANICS

Anita

and S.F. EDWARDS

MEHTA

Cavendish

Received

Laboratory,

29 December

Madingley

OF POWDER

Road,

Cambridge

MIXTURES

CB3 OHE, UK

1988

In this paper we use a new formulation of the statistical mechanics of powders to develop a theory for a mixture of grains of two different sizes. We map this problem onto the spin formulation of the eight-vertex model and reproduce the features of the phase separation diagram of the powder mixture that we would intuitively be led to expect. Finally, we discuss the insight afforded by this solution on the “thermodynamic” quantities of interest in the powder mixture.

1. Introduction There has been considerable recent interest in the behaviour of granular systems, both from an experimental [4, 51 and a theoretical [l, 61 viewpoint. Although much of the interest so far has been focused on their dynamical aspects, e.g. the stability of granular piles, or the characteristics of the avalanches generated therein [4-61, there has also been an attempt to develop a microscopic theory [l] of their configurational aspects along the lines of conventional statistical mechanics. It is this latter approach that we adopt in this paper where our aim is to provide at least a qualitative understanding of the packings of powder mixtures observed in nature. We lay out in section 2 the fundamentals of a new theory [l], which defines a “Hamiltonian” function representing the volume of a powder: next in section 3, we consider a binary mixture of grains and show that although an Ising-like Hamiltonian is in crude accord with intuitive expectations, it is necessary to improve upon this in order to generate a richer phase diagram of observed configurations - we describe in this context our motivation for choosing the eight-vertex model as our model of the binary powder. In section 4, we formulate our model, and discuss its implications for real granular systems. 037%4371/89/$03.50 0 (North-Holland Physics

Elsevier Science Publishers Publishing Division)

B.V.

A. Mehta and S. F. Edwards

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2. Definition

of fundamental

In this section

we draw

I Statistical

mechanics

of powder

mixtures

variables analogies

statistical mechanics and those that start with the “ergodic hypothesis”

between

the

are of interest

variables

of conventional

in a granular

system.

We

of powders

- all powders with the same specification of contents, formed by extensive manipulution (by which we mean processes of stirring, pouring, etc. which do not act on individual grains) and occupying the same volume, have the same macroscopic properties. Next, we decide to ignore the temperature of the powder system - a hot bucket of sand is much the same as a cold one! However, powders are of variable density - so that e.g. a powder of spherical granules has a well-known minimum and maximum of density. This immediately suggests an analogy between the volume of a powder and the energy of a statistical system - equally then, we need to define the equivalent of a Hamiltonian for a powder, which expresses its macroscopic volume in terms of microscopic coordinates. In this spirit, we therefore define V (analogous to E, the total energy in the Hamiltonian formulation of statistical mechanics), the total volume occupied by the granular system: W (analogous to the Hamiltonain H), the function which gives V in terms of the specification of the grains in the system: the effective volume Y (analogous to the free energy F): S, the entropy, which here has the dimensions of volume: and finally the compactivity X (the analogue of the temperature T) such that the following relations hold:

x= avias, Y=V-xs, e

-Y/AX =

PAX

(over all configurations)

,

where A is the analogue of Boltzmann’s constant. The name “compactivity” will be seen to be a rational choice, as X = 0 and X = ~0 correspond respectively to the most and least compact powders. It will be obvious that W is in general not simple, either to formulate or to evaluate once defined: in addition to the usual difficulties that bedevil the calculation of most Hamiltonians in statistical mechanics, we have here the additional problem that since a powder would be a pathologist’s choice for a non-lattice-based system, the normal definition of “coordinates” here is considerably more complex. There are many tentative ways of approaching this problem [l]; since we are here only considered with getting qualitative insights, we make the considerable simplification of ignoring the non-lattice-based aspects of this system. This may seem at first to be a dangerous oversimplification, but if we restrict ourselves to trying to find a model for favourable configurations for the powder granules, it will be seen not to be so.

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A. Mehta and S. F. Edwards I Statistical mechanics of powder mixtures

a

b

Fig. 1. (a) Powder mixture of grains of size R, and R,. The black square encircles a typical configuration. (b) The encircled configuration of (a) is represented by its equivalent configuration in our model: orientations are not preserved but the nature of the neighbours is.

For the moment, we argue in the following qualitative way powder mixture looks much more like fig. la, we concentrate only on the domains of separation or miscibility of the two thus represent the powder grains as spins on a lattice, distinguish between configurations such as those equated in orientationally, but not configurationally different.

- although a real in the following components. We and cannot then fig. lb which are

3. Mean field solution for a binary mixture We reproduce here the model calculation of Edwards [l] to illustrate the spirit of the above approach - the problem of the mixture of two grains is mapped onto the Bragg-Williams problem of the A-B alloy, and we seek a solution for the variation of the domains of A, B and A-B with respect to the compactivity X and the degree to which the grains aggregate preferentially among themselves. As in that formulation of the problem, we point out that


1094

the Jacobian

of the transformation

coordinates and configurations is in

between

this problem non-trivial, and is intimately however in the spirit of qualitative enquiry

related to the lack of a basis lattice: that is inherent to this paper, we set

the Jacobian problematic

equal to 1 after having reminded ourselves of its rather real and existence. Thus we model the W function so as to reproduce the on average if grains of possible assumption - less volume is “wasted”

crudest the same size cluster

c

w=

together

than

if grains

of different

sizes do:

+ &l$UBB + n~n;uAB}

{$yu””

)

(ii)

with n;(B) being 1 (0) (absent). This is mapped

c

W=-4

depending on whether an A (B) as usual onto an Ising model,

grain

is present

Jqa;,

(5)

(ll)

with J, the exchange,

J=

$I,,

given

- $I,, -

by

;u,,]

and uj = 21 depending on whether site i is occupied by an A or a B atom. mean field solution of Bragg and Williams gives [l] as usual (a)

= tanh{zJ(

The

a) /AX} ,

with z as the coordination number of the lattice. Thus for J/AX < 1, the two powders are totally miscible, but, as J/AX > 1, the powders tend to have unequal mixed domains until, at X+ 0, the material separates into domains of pure A and pure B. We see from the above that even this relatively crude modelling reproduces a qualitative feature of powder mixtures that one observes in nature: if one shakes a mixture of two powders enough, one would expect to find at least one of two extremal configurations of the types shown in figs. la and 2 respectively. (We represent the powder grains as spheres for ease of construction, but the same general argument would apply here regardless of shape.) We interpret J as a measure of the relative ease with which grains pack when they are of the same size as opposed to when they are of different sizes: e.g., in the case of spheres, we would expect on intuitive grounds that there should be a certain optimal ratio R,/R, of the radii below which the smaller spheres would simply slip through the holes created by the layered packing of the larger ones, so that when well shaken the mixture would adopt the configurations shown in fig. 2,


Fig. 2. Illustration

of the “ferromagnetic”

ordered

state of the binary

1095

mixture.

whereas above this ratio the smaller spheres would be able to adopt locally stable configurations on top of the larger ones, leading to the type of packing shown in fig. la. The other parameter that counts is the compactivity X - the more compact a powder mixture needs to be (i.e. the smaller X is), the more important space-saving considerations are, leading to a preference for “ordered”,

denser

packing

(fig. 2) rather

than the “fluffier”

packing

shown

in fig.

la. It would crossover-it

be very interesting would be relatively

to investigate the detailed nature of this simple to find, for instance, the “critical”

ratio RI/R,(in the case of spherical granules) that is implicit in the argument following (7) above for powder mixtures of a given compactivity X which leads to the crossover discussed. However, bearing in mind that we have a latticebased model of a powder, we choose to look for further qualitative insights rather than stretch out thinly such quantitative analogies as it might be possible to draw. We observe therefore that although we have already one “ordered” state of our system where the regions of A and B are completely separate (fig. 2), we do not have a representation for the “stacking” solution shown in fig. 3,

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A. Mehta and S. F. Edwards 1 Statistical mechanics of powder mixtures

Fig. 3. Illustration of the “stacking” solution; manner shown when compactivity requirements

grains of disparate become important.

sizes choose

to nestle

in the

i.e., one where a “layered” arrangement of the grains is preferred so that each A rests on a base of B and vice versa. Now for suitable values of RI/R,, the space-saving considerations mentioned above (for low X) will obviously favour i.e. we expect intuitively that the real phase this form of “close-packing”; state corresponding separation diagram of a powder must contain an “ordered” to this. This then motivates the next section, where we claim that the eight-vertex model when mapped on to the powder mixture correctly reproduces this and other features of the problem at hand.

4. Eight-vertex

approach

to powder mixtures

Although detailed expositions of the eight-vertex model exist in the literature [2, 3, 71, we mention here those aspects that are relevant for our purposes. The original model [7] on a square lattice assigns energies ei to the eight possible arrangements (cf. fig. 4) of arrows at every site (subject to the stipulation that we place an arrow on every edge of the lattice, and allow only configurations such that there are an even number of arrows into, and out of, each site),

so that the resulting

partition

function

is

where the sum is over all allowed configurations C of arrows on the lattice, rzi is the number of vertex arrangements of type j in configuration C, and the other symbols are as usual. With the usual assumptions [7] that leave the model unchanged under a reversal of all arrows, we have the so-called “zero-field” eight-vertex model where


1097

a++++-I&

E

E

1

E

4

3

2

-t+

8

I

6

5

E

&

E

&

b

+

f

f

t

f

#

f

f

+

f

+

f

c f t

Fig. 4. (a) The eight possible configurations of the eight-vertex model with their associated energies E,. (b) Illustration of the Kadanoff equivalence between arrows and spins in the eight-vertex model; the first state corresponds to ferromagnetic ordering, while the second corresponds to our “stacking” solution.

El = -52 3

This model

E3 =

E4

>

can be mapped

E5 =

E6

>

E,

=

E8

.

[2, 31 onto an Ising model

(9) in zero magnetic

field

with finite two- and four-spin interactions in the following way: imagine a spin placed at the interstitial points of the lattice as in fig. 4b. An arrow to the right (or upward) corresponds to the case in which the adjacent spins are parallel; a leftward or downward arrow makes the adjacent spins antiparallel. This leads to the equivalent Hamiltonian analogies of the section above powder model as

with

of Kadanoff can be written

[3] and Wu [2], which with the in terms of the W-function of our

A. Mehta and S. F. Edwards

1098

I Statistical

mechanics

of powder

E~=E~=-J-J’-J~,

Ed = E‘, = J + J’ - J4 ,

Em= Ed = J’ - J + J4 ,

E~=E~=-J’+J+J~,

mixtures

(10’)

where, as before, a; k = 21 depending on whether site (j,k) is occupied by an A or a B grain. The situation is illustrated in fig. 5: next-nearest-neighbour atoms are coupled by interaction constants J or J’ depending on the direction of the diagonal, whereas J4 couples all four atoms. One can view this [3] as two interpenetrating sublattices on a square lattice with nearest-neighbour interactions J and J’ - the two lattices are noninteracting except when J4, then the interaction coupling together atoms 1, 2, 3, and 4 (fig. 5) is non-zero. It is this feature that is crucial in representing at least qualitatively a “layered” system; J and J’ represent the intruZayer couplings, whereas J4 represents the interlayer coupling in this quasi-three-dimensional representation of our binary mixture of powders. Our choice of model, while enabling us to preserve the mathematical simplicity of two explicit dimensions, will also in its solution be seen to provide a ground state for the powder system that is in accord with the intuitive expectations discussed above in section 3. The solution of the zero-field eight-vertex model is known [7] for each of the four regions defined by c1