ELSEVIER
Physica
A 210 (1994)
301-316
A model for the packing of irregularly
C.C. Mounfield*,
Cavendish Laboratory,
S.F. Edwards
Madingley Road, Cambridge
Received
shaped grains
25 March
CB3 OHE, UK
1994
Abstract
In this paper we will attempt to address the problem of the packing properties of
granular materials composed of irregularly shaped grains (using configurational statistical
mechanics). In particular, we will develop a model for a system of irregular grains based
upon perturbing a packing of mono- or poly-disperse spheres. In the mono-disperse case
we will show that the system packs less densely than a packing of perfect spheres, except
when local correlations between configurations of grains are taken into account. The
opposite is found to be true for a perturbation
expansion based upon poly-disperse
spheres. Finally we will show that for a bi-disperse packing of spheres phase segregation
occurs for any size ratio and discuss whether this is to be expected.
1. Introduction
In other publications [1,2] we have attempted to apply the concepts of
configurational statistical mechanics to systems composed of spheres, elongated
ellipsoids or a mixture of the two. In all these cases the system’s constituents were
assumed to be perfectly regularly shaped objects. Real systems however,
presumably, will not be composed of such perfect constituents. Consequently, the
packing properties of systems of irregular grains is an important problem [3].
In this paper we will develop a model whereby we may study the properties of
such systems. This is achieved by perturbing a packing of perfectly spherical
grains from mono- or poly-dispersity by some amount dependent upon the
relative volume interactions between the grains. We will then evaluate the
macroscopic properties of these systems after having written down a suitable
microscopic description.
It is useful of course to be aware of what empirical results we must be able to
account for. In particular, it is well known that a loose packing of irregular grains
* To whom
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302
C.C.
Momfield,
S.F. Edwards
I Physica
A 210 (1994) 301-316
is less dense than loose packings of spheres (due to arching effects [4] and
excluded-volume
effects). However it is also true that gentle vibration or
compaction of such loose irregular packings can lead to denser dense packings
(since the particles can rearrange to fit one another [5]). Finally, a packing of
irregular grains of varying sizes packs more densely than a single species [6].
In studying this problem it becomes apparent that there are several possible
regimes of behaviour. On the one hand there is the case where the grains are
almost spherical. In the other extreme we have grains that are completely
irregular. In this latter case the system’s behaviour is certainly altered by the
granular irregularities. However, in the former case is the same also true? If the
perturbation was such that each grain had its radius altered by either plus or
minus some amount (i.e. an AB alloy model), then as that amount became larger
and larger it may be the case that the systems behaviour would be unaffected until
some critical strength of perturbation were reached and the system’s behaviour
were changed (for example the small spheres may not fall through the gaps left by
the larger ones and phase segregate until the difference in radii has exceeded
some threshold value). On the other hand, any perturbation from perfect spheres
may alter the system’s behaviour. This is an interesting question that we may
hope to attempt to answer.
Before being able to do this, however, it is necessary to obtain some
elementary results regarding the packings of perfect spheres, before being able to
develop a model for irregular grains. Firstly, though, we will briefly review the
important concepts of configurational statistical mechanics.
2. The application
of statistical
mechanics
to granular
materials
A granular material is a system with a large number of individual grains. If we
hypothesise that this granular system may be characterised by a small number of
parameters (analagous to temperature, pressure etc.) and that this system has
properties which are reproducible given the same set of extensive operations (i.e.
operations acting upon the system as a whole rather than upon individual grains)
then we may apply the ideas of statistical mechanics to these granular systems (it
is important to note however that the interactions between the grains in these
systems are dominated by frictional forces).
There have been a number of previous attempts to apply statistical mechanics
to granular materials [7-lo]. However, we shall differ from these approaches in
that we will argue that it is the volume of the system that plays an analogous role
to that of energy in conventional statistical mechanics.
However, instead of averaging over all available energy states of the system, we
now must average over all possible configurations of the grains in real space. We
are thus led to a configurational statistical mechanical theory describing the
random packings of grains.
C.C.
Mounfield,
S.F. Edwards
I Physica
A 210 (1994) 301-316
303
We should also note that we are ignoring the microscopic details of the contacts
between the grains. Consequently we are assuming that these systems may be
characterised as a population of rigid grains with different distributions of shapes
(i.e., configurations).
The crucial point to note is that it is the distribution of grains over the available
volume states (subject to external constraints such as stability under gravity and
that the grains are impenetrable) which gives rise to a volume of configuration.
Our formal analogy then is to replace energy E by volume V. Traditionally E is
specified by a Hamiltonian H which specifies the energy state of the system in
terms of the position q and momentum p states of its constituent particles i.e.
E = H(p, q). We require then a granular equivalent W which specifies the volume
state of the system in terms of its constituent grains i.e. V= W(q) (and consequently will be referred to as the volume function). q is taken to mean the
particular granular property over which we will average.
Having defined this microscopic description of the system it is now possible to
go through a process completely analogous to that in conventional statistical
mechanics in order to determine probability distribution functions etc. [ll]. In
particular the canonical ensemble (whereby we relax the constraint that V is fixed
and allow fluctuations of volume between subsystems) will prove to be the most
useful for calculational purposes. A consequence of this is that it is necessary to
define a parameter (analogous to temperature) which characterises the volume
state of the granular material. This parameter is the compactivity and is defined as
[12]:
1
-=-
x
X3
(1)
av
and the partition function is:
Z=e-
Y(X)/AX _
-
I
e -w(q)/Ax
dq
,
(2)
where we have defined a thermodynamic potential analogous to the Helmholtz
free energy. This is the effective volume Y(X) which is
Y(X) = V(X) - xs
and the volume energy of the system is determined
of this, i.e.
V(X) = Y(X) -
(3)
from the Legendre transform
xg
(this is the macroscopic volume the system occupies as a function of X).
These are the important analogies of the thermodynamic functions that we will
use for calculational purpose (a full list of the correspondences
may be found
elsewhere [1,13]).
304
C.C. Mounfield,
2.1.
Interpretation of the compactivity
S.F. Edwards
I Physica A 210 (1994) 301-316
The parameter X has been called the compactivity, and is a measure of the
packing of the grains. The definition of X in this manner introduces a systematic
way in which these systems may be characterised
(in the same way that
temperature may characterise many different systems) since we have assumed that
all configurations of a given volume are equally likely. Also no reference is made
to the detailed structure of the granular constituents.
From the definition of X as 8V/XS it is apparent that it may be interpreted as
being characteristic of the number of ways it is possible to arrange the grains in
the system into a volume AV such that the disorder is AS. Consequently the two
limits of X are 0 and a, corresponding to the most and least compact arrangements of grains respectively (i.e. maximum close packing and minimum loose
packing). X also describes the balance between the tendency of a system to
increase or decrease its volume and the tendency to increase or reduce its
entropy.
The compactivity, like temperature, may thus be interpreted as a measure of
the disorder i.e. the probability that the system is in the state specified by {q} is
f({q}) - emw(4)‘“x and when X+ 0 the system picks out one particular configuration as being most likely (the ground state) and when X+ 00the system picks out
all configurations as being equally likely.
3. The packing of spheres
The packing of spheres is a problem that has been of interest for a very long
time [14-161. For example, such a packing is a prototypical model for the
structure of simple liquids [17,18], and hard spheres form a simple system showing
a glass transition [19]. As a consequence there is a vast literature concerning
experimental, simulational and theoretical attempts to determine quantities such
as packing fractions, average number of nearest neighbours etc. It is not our
intention here to add to this literature a detailed attempt at calculating any of
these quantities, but rather to d.emonstrate how we may attempt to characterise
these very complex problems in a simple phenomenological manner in order to
show how we may use the statistical mechanical theory we have developed.
3.1.
A packing of mono-disperse
spheres
Previously it has been assumed that a suitable parameter that one could choose
over which one could perform the statistical averaging (quantities such as p and q
are not necessarily suitable) was the coordination number c of a grain (or to be
more precise the number of grains with coordination number c, n,). This was
because each grain will have neighbours touching it with a certain coordination
C.C.
Momfield,
S.F. Edwards
I Physica
A 210 (1994) 301-316
305
and angular direction. Thus the coordination number is a property of each grain,
and this property interacts with its neighbours in the form of compatibility
conditions (e.g. one cannot have a grain of coordination 4 next to a grain of
coordination 12). These conditions are difficult to write down and so we will not
attempt to do so here. The point is that the coordination number is a suitable
granular property over which we can perform our statistical averaging. A suitable
volume function is then of the form
where u, is the volume of a configuration containing a grain with c nearest
neighbours and u,,. is the volume of a configuration containing a grain with c
neighbours and one with c’ neighbours (there are y1,,, such configurations). In
principle this is a summation which continues until one is including all N grains
within the system. This, however, is not practically useful and so we truncate the
volume function after the first term (but remain aware of the approximation we
have made).
.
The canonical ensemble is then
z
=
I
e-w(“c)‘Ax
n
dn,
.
c
We have neglected the stability constraint on the grounds that its only effect will
be to reduce the number of possible configurations the system may occupy (i.e.
reduce the dimensionality of phase space). As a consequence of this we could
simply choose a smaller set of relevant parameters than the complete set (this
involves a Jacobian which will also be hard to calculate. So we will ignore it since
if it is a smooth function then it will not affect the physics anyway). Similarly the
difficult no-overlap constraint is avoided by assuming implicitly that u, is of a
steric nature.
If we make the approximation that the system may only occupy two possible
configurations, where each configuration corresponds to a grain with either c1 or
c2 nearest neighbours (the volumes of these configurations being u1 and u2
respectively) and if we assume that the system is equally divided between these
two configurations then we may quite simply evaluate (6) to find the effective
volume Y(X). The macroscopic volume the system occupies as a function of the
packing is determined from the Legendre transform of Y(X) and is
V(X) = i N(u, + u2) + i N(u, - u,) tanh
(7)
At minimum close packing X = 0 and (7) becomes V(0) = Nu,. This simply states
that in the ground state at X = 0 the system chooses one particular configuration
as being most probable. Correspondingly in the limit of loose packing X-t m and
(7) becomes V(X + ~0)= +N(u, + u2) which again merely states that in the state of
306
C.C.
Momfield,
S.F. Edwards
I Physica
A 210 (1994) 301-316
maximum disorder the system chooses all of its configurations as being equally
likely.
The approximation that the system may only occupy two possible configurations
(with coordinations c1 and c2) is rather an unrealistic one. The integral in (6) is of
course simple to evaluate, i.e.
N
Z=
e- [“~u~+(N-n,hl/hX
I
dn,
,
(8)
0
from which we find that
V(X) = + N(u, + u2) + AX - ; N(u, - ul) coth (&u,
(9)
-4)
and (9) agrees with the previous result in the limits of close and loose packing.
Consequently we conclude that although our two-coordination approximation is
unrealistic it does provide correct results in the limits of close and loose packings.
Since these are the only two packings that are of any practical interest to us
(generally speaking) we can justify .using this approximation again.
3.2. A packing of poly-disperse spheres
Given that we have been able to determine the properties of a mono-disperse
packing of spheres, can we generalise this to a packing of poly-disperse spheres?
In order to do this it is necessary to include an extra index on any parameters we
average over i.e. IZ,--+IZ,,, since this is now the number of configurations of grains
of radius r with c nearest neighbours.
We thus choose our volume function to be
(10)
where u,, is the volume of a configuration
neighbours. The partition function is
Z
(we
that
that
the
=
I
e-w(ncr)‘A*
n
dn,,
of a grain of radius r with c nearest
(11)
&
c,r
have to average over r to account for the extra configurations of the system
may arise due to this additional degree of freedom). If we once more assume
there are only two coordinations c, and c, and in addition to this assume that
grains are bi-disperse with radii a or b then (11) becomes
z = Ax
e(Q,+Q~W+~~g
2 cosh(Q,-Qb)/2
(12)
>
where we have defined
Qr=-
Nu,,
hX
-
10&k, -
%)
+
-%
2hX
U2r
-
ulr)
+
log 2 sinh (&
(u2, - %))
.
(13)
C.C. Mounfield,
S.F. Edwards
We can calculate the macroscopic
I Physica A 210 (1994) 301-316
307
volume of the system to find that
V(X)=hX+;hX2&(Q,+Q,)
(14)
At minimum close packing V(0) = NV,, , hence the system picks out one particular
configuration as being most likely, as expected (unfortunately the X+m limit
does not appear to behave as we would hope).
The fact that it has chosen a configuration with a grain of radius b should not be
interpreted as meaning that the two species of granular constituents have phase
separated, as this model lacks the sophistication to account for this possibility. ulb
is merely an artefact of our microscopic description and is related to the fact that
at minimum close packing the system will choose the configuration with the lowest
coordination.
4. The perturbation model
For a system composed of mono-disperse spheres alone then as we have seen
we may characterise this system’s microscopic behavior with a volume function
such as (5). However, what if the grains were only roughly spherical or even
completely irregular? How then would we characterise such a packing?
Clearly a grain i has a volume interaction with each of its c nearest neighbours
which is dependent upon the relative orientation and the relative separation of the
two grains. This interaction is also going to be different for each pair of grains i
and j. Let us define it to be fij(Oij, xii) where Oij is the relative orientation of the
grains and xii is their relative separation (it is also true, however, that in addition
to grains i and j interacting with one another, we will also have a three-body
interaction between grains i, j and k, and so on until all c + 1 grains are
interacting. This, however, is an added complication and so we will only restrict
ourselves to considering two-body interactions).
Hence we propose a volume function for this system to be one based upon
perturbing a packing of perfect mono-disperse spheres by an amount dependent
upon the grains’ relative interactionsf,,(.n,i, xij). It is important to note that due to
the fact that we are perturbing the grains in this system from being perfectly
spherical ( f, >fl,j must be equal to zero. Consequently the volume function is
where n, and u, are the quantities we have defined before and E
parameter representing how strong the interaction between the grains
expression is of course an approximation since we should really integrate
all Oij and xii to obtain the complete interaction between the grains
is some
is. This
fii over
i and j.
308
However,
C.C. Momfield,
S.F. Edwards
I Physica A 210 (1994) 301-316
if there are only a finite number
can approximate
this integral
of contacts
between
the grains then we
by the sum Cf=,
This volume function is capable of describing a system of many irregular grains.
That is to say fi2 (grain l’s interaction
with nearest neighbour
2) need not be the
same as, for example,
fi4 and similarly
also capable of accounting
granular irregularities.
It is interesting
for all the other
for the excluded-volume
grains in the system.
effects
to note that from the way we have defined
It is
that arise due to the
the interactionf,
it is
it as defining the shape of the grains i and j (that
it should be possible to treat
is, if we replace C,“=*+ J . . . dL! d_x). Consequently
f, as a statistical
parameter
over which we average (i.e. averaging
over all
possible shapes of grain). This is something we will return to later, where we shall
see it is possible to reformulate
this problem in analogy to a spin glass (since the
partition
function will now become a functional
of fi,).
apparent
we may also interpret
A final observation
we may make about the granular
interaction
J.j is one
concerning
the parameters
upon which it is dependent.
Whilst it is certainly the
case that the grains’ interaction
is dependent
upon .Rii, one would expect the
physical state that the system is in to also be important
in determining
i.j. That is,
when the system is close packed the grains will interact
in a manner
that is
different to when the grains are at minimum
loose packing. Consequently
i, is
also dependent
upon X (i.e. we are accounting
for the disorder present in the
system).
This is in some way equivalent
to replacing
xii with X since in the
problems
we are studying
the system’s constituents
are all assumed to be in
contact with sufficient of their neighbours
in order to be stable under gravity (such
constraints
being absent in the type of problems
studied previously
such as
Onsager’s model of a system of hard rods [20]). Physically we would expect the
interaction
between the grains to be a maximum at minimum
close packing and
vice-versa
i.e. A,(X) -X-l
. However,
at this level of approximation
is such an
added complication
necessary? The answer we suggest is not, since we have seen
many times already that producing models in analogy to thermal ones has served
us very well in the past. In what follows then we will assume that the exchange
interaction
has no compactivity
5. The macroscopic
The partition
dependence.
volume of a loose packing
function
for this microscopic
description
is
It is of course not difficult to evaluate the integral over II, (making our usual
assumption
that there are only two coordinations
c1 and c2), and we obtain
C.C. Momfield,
2 = AXn
J exp{[-N(u,
S.F. Edwards
309
I Physica A 210 (1994) 301-316
+ v,)(l + .a2fi)/2hX-
log[(u, - v,)(l+
a”ft)]
ij
+ log 2 sinh[N(u, - u,)( 1 + .s*fi) /2hX]} dQj dxij .
(17)
To now evaluate this partition function we might try an approach based upon the
one Onsager took, i.e. approximating the angular integral and making an
appropriate definition of the excluded volume between two grains i and j (this is
apparent if we evaluate the II, integral in the high X limit and suitably redefine
the steric interaction. The result is a partition function very similar to the one
Onsager used). However, this would be, we believe, inappropriate in the current
context and so for convenience we simply take the high X limit and can show that
V(X+
m) = +A@, + lJ*)( 1 + c’( f’))
where we have defined
(f 2> =
)
(18)
(f “) by
J f2 exp(-Cij e*f*/AX) Il, dQj d.x,,
] (- C,
E*f*/AX) Ilij dQj d_r,
(19)
’
Eq. (18) is the macroscopic volume the system occupies for any strength of
perturbation in the high X limit. We see that it agrees with the result stated earlier
that a loose packing of irregular packs less densely than a loose packing of
spheres.
However, the high X limit is not really of any great interest, since in this regime
the system is on the verge of becoming a suspension. Consequently, whether the
grains are spherical or not will presumably have less importance in determining
the properties of this system (since when there are, on average, fewer contacts
less
between the grains, their microscopic details become correspondingly
important, although of course the contacts between the grains are very important)
than at denser packings.
It is also possible to perform this calculation starting from the assumption that
we have near spheres. Then E << 1 and (16) becomes
Z=
I
e-wspheres’hX(1 - WperturbationlAX+ . . -) F dn, n
ij
d.R,i &,
.
(20)
After another long (and tedious) calculation we can show that we obtain (18) in
the X+ 00limit. This agrees with our assertion that in this limit the precise details
of the granular shape become unimportant.
From this analysis however it is not clear how the system would behave for
irregularly shaped grains in the low X limit, which is when the system’s behaviour
is most interesting. Consequently we will now try to determine this.
310
C.C. Mounfield,
6. The macroscopic
S.F. Edwards
I Physica A 210 (1994) 301-316
volume of a dense packing
We have obtained a prediction regarding the systems behaviour in the less
interesting high compactivity limit. It is possible to obtain a similar prediction in
the low X limit if we rewrite (17) as
2 = 2AX exp{ -N(u,
+ u,)/2hX
- log(v, - ul) + log 2 cosh[N(u, - u1)/2X]}
(21)
where we have defined
gijCx)
=
-
%
2AX
u1 + u,)e”fi
- log( 1 + e2fz.) + log 2 sinh z(uZ-ul))
(
+log[l+tanh(&(u,-u,)coth(s(u2-ul)].
(22)
We now take the low X limit (such that tanh[N(u, - u,)I2AX] - 1, and
coth[NE2fi(u2 - u,)/2AX] - 1). Upon doing this we can show that
V(X) = Ypheres(X) + $%,
+ u,)e’(f”)
9
(23)
where Vspheres(X) is given by (7). The average interaction (f’) is defined in the
same way as before (at least for near spheres when E << 1).
Consequently we see that as X+ ~0then (23) (which is supposedly for small X)
agrees with the high X result we obtained earlier, Eq. (18). We can conclude with
some degree of certainty then that (at least for a packing of near spheres) the
macroscopic volume of this system is given by (23) for all packings.
The volume the system occupies is thus perturbed from that of a packing of
perfect spheres by an amount that is dependent upon the strength of the
perturbation and the average interaction between the grains (i.e. the average
relative shape). (f’)
is always greater than zero, this means that V(X) >
Vspheres(X) (for all packings) which in turn means that the average granular shape
is such that the grains do not in general fit neatly into the gaps of their
neighbours. This is a result that is consistent with the mean-field approximation
we have made, if not with our expectations (as we shall now see).
7. The effect of local correlations
We have seen how, using our mean-field microscopic description, a packing of
irregular grains (based upon perturbing mono-disperse spheres) in general packs
less densely than a packing of perfectly mono-disperse spheres. Whilst this is
certainly the case in the large X limit (loose packings), is it also true for the small
X limit? This is indeed what (23) predicts. However, as noted earlier this is not
C.C. Momfield,
S.F. Edwards
311
I Physica A 210 (1994) 301-316
consistent with what is observed, where local correlations between the grains calz
produce denser packings than for spheres alone. Presumably such local effects will
only begin to manifest themselves as the system becomes more and more densely
packed i.e. in order for the system to become even more densely packed the
grains must be capable of rearranging themselves cooperatively on short scales.
In terms of our formalism we may account for such local (non-mean-field)
correlations by adding extra terms to the volume function allowing for the
alteration in the granular volume due to fluctuations in the allowed granular
configurations.
For mono-disperse
spheres this would amount to writing a
microscopic description such as (5) but this time retaining the terms representing
the higher-order correlations between the grains, and so by analogy we postulate
a perturbed volume function accounting for local correlations between configurations of grains to be
2 c (n,u, +
w=
i=l
nrc’U,,,
+ *.
‘)(l+
&‘,$i
f,,(aijii,xij)2
+
c,c’
**')
'
(24)
The partition function is of course the same as (16) except for the fact that we
now average over both n, and IZ,,.. Evaluating it (in the low X limit) it is now
possible to show that V(X+O) <Vspheres(X+O), which given the arguments
above is what we would expect to be the case.
8. A perturbation
model for poly-disperse
spheres
Finally we attempt to determine the behaviour of a system of irregular grains,
basing our perturbation expansion upon grains of varying radii. The quantities IZ,
and u, will now have an extra index r attached to them to account for this
poly-dispersity and the volume function becomes
W= 5 c ncru,, (1 + s2,$
i=l
f,(aijYxij>2
+.
(25)
. ')
C,r
and so the partition function is
Z =
e-(W,pheres+Wperturbation)/AX
I
n
c,r
dnc,
&
n
da,
&,
.
(26)
ij
If we now make the assumption that there are only two radii of sphere present
in the system which we perturb about (with radii a or b) then we may
approximate J . . , dr-+ CF_. It is now possible to evaluate the partition function
in a manner similar to that which we have done before. However, this calculation
is extremely tedious (but straightforward)
and so we will not present it here.
Rather we will simply state the result that at minimum close packing the volume
the system occupies is
312
C.C. Momfield,
S.F. Edwards
I Physica A 210 (1994) 301-316
(0) - Nw2(f2) >
V(O)= Kpheres
whereVspheres(0)
= Nf4, is
at minimum
effect
close packing.
the volume a poly-disperse
packing of spheres occupies
Thus if we assume that un, > 0 then we see that the
of the poly-dispersity
(compared
expect
to a packing
(since
is to increase
of poly-disperse
irrespective
ones).
9. A bi-radial
of spheres
the density
spheres).
of the granular
gaps left by the larger
packing
(27)
of the packing
at X = 0
This is of course what we would
irregularities
the small
grains
can fill
So far we have neglected the nature of our granular exchange interaction.
This
is because we have previously
approached
this problem in a manner similar to
that of an Onsager model i.e. we have an interaction
(which is steric) that is
dependent
upon the grains relative orientation
and separation
and in the partition
function we average over these properties.
Consequently
the exact form off, is of
little importance;
it is its parameter
dependence
and its steric nature that is
important.
However,
as noted earlier, fi, in essence defines the relative shape of grains i
and j (integrating
over all possible orientations
Qj). What then if we were to
approach
this problem
from an un-Onsager-like
approach
and performed
our
averaging over all possible granular shapes? The partition function would then be
a functional
If we indeed approach our problem from this perspective
then, as we shall see, it
is possible to use our model to make predictions
about a system whose phase
behaviour
is well known, namely the AB alloy model.
If the grains are bi-disperse
spheres (hence there is no angular dependence
of
the perturbation)
then we must have fi, = -+l (or some other number,
the
distinction
being unimportant).
Consequently
the grains in the system will have
radii perturbed
by an amount +s2.
Evaluating
the II, integral in the partition function (28) for two coordinations
(in the high X limit) we obtain
Z-
I
esf2df,
where we have defined 9 = N(u, - u,),5’/2AX (and dropped the subscripts
j). If we now also use the fact that we only have two possible choices for
(29) becomes
(29)
i and
f
then
CC.
MounJield,
S.F. Edwards
I Physica A 210 (1994) 301-316
313
+m
Z-
I
exp[ - 3~” + log 2 cosh(%@%)] dx
(30)
-_
(using a Gaussian transformation).
x= @
tanh(V%%)
,
Evaluating this by steepest descents we obtain
(31)
which obviously predicts the existence of a second-order phase transition at
X, # 0 (as we would expect for the AB alloy) [21].
What is interesting about (31) however is 9. When E = 0 then x = 0 is the only
solution of (31). When E # 0 there exist other solutions to (31). Consequently we
see that no matter how small the perturbation from mono-dispersity is, as long as
it is not zero then the spheres will phase segregate (which is not altogether
obvious).
This answers our question as to whether there exists some form of critical
strength of perturbation at which point the system’s behaviour is discontinuously
altered. The answer is no since (31) is perfectly well behaved for all perturbations
E. This is a result that is somewhat surprising. That is to say, if we were to
consider a box filled with golf balls and footballs and shook it, then it is not
difficult to envisage a situation whereby the two components had separated (golf
balls at the bottom). However, what if we had a mixture of ordinary golf balls and
golf balls which were infinitesimally smaller? Intuition is now not so clear. Eq.
(31) however is; the two types of ball will phase segregate at some critical
packing.
This is an important prediction but is it correct? What if we consider the other
extreme? That is to say, if the difference in grains radii were very large (and not
infinitesimal). In this case (despite what we have predicted) total phase segregation (even at X = 0) of the grains presumably need not occur. For example, the
small grains may become wedged in between the gaps of the large ones (this
would be very evident at minimum close packing).
Simulations of bi-disperse systems [22] subjected to vertical shaking have
suggested that phase segregation does occur regardless of the grains size ratio. We
have not attempted to address the problem of vibrating systems (our models lack
any dynamics, frictional interactions, consideration of boundary conditions or
stability under gravity) but we could envisage a situation after the shaking has
stopped and the components have settled (this particular configuration corresponding to a particular X). If eventually phase segregation were seen to occur
then our prediction would be valid. But is our model even able to account for
such a greatly reduced characterisation as this situation represents? This is what
we must attempt to answer.
Unfortunately these are not questions that we may answer within the context of
a simple mean-field model. We conclude then that our formulation of this problem
is as yet incomplete.
314
9.1.
C.C. Momfield,
Formulation
S.F. Edwards
of the perturbation
I Physica A 210 (1994) 301-316
model as a spin glass
If we write the volume function in the form
(34
then
Z
=
I
e-w[fl’Ax
JJ dn,Bf
c
(33)
(for simplicity we are only considering f(r) to be a continuous variable here. In
reality c is also a continuous field).
The volume function has become a functional of a random (i.e. unknown)
interaction between the grains, and we have averaged over all possible random
interactions. This is rather like a spin-glass model. That is, the set of &‘s for each
i and j are a random set of variables, whose distribution
determines the
configuration of the system. These fii’s are unknown but we assume they obey
certain statistical rules (such as (fi,) = 0). If there are only a finite set of possible
distributions then the partition function for a particular distribution of the fil’s
becomes
Zr = F 1 e-w’f”Ax v dn,
and the probability of obtaining the system in a configuration characterised
particular distribution of the f’s is
by this
(35)
The sample averaged effective volume (averaged over all possible distributions
i.e. configurations) is
where Yr = -AX log Zr.
This then is exactly similar in form to how one formulates a spin-glass model
(with the randomly distributed fij’s replacing the magnetic exchange interactions
J,). There do of course exist many ways of solving these spin-glass problems (for
example, the replica approach). However, there is an added complication. In
conventional spin glass theory one has Ising-like Hamiltonians which are defined
on a relevant lattice. In this problem our microscopic description is neither
Ising-like (although this is not necessarily a problem) nor defined on a lattice.
That is to say, we do not have the luxury of any form of spatial homogeneity. As
a consequence it is necessary in this problem to not only average over all the
possible distributions of the granular exchange interactions, but also to average
C.C. Mounfield,
S.F. Edwards
I Physica A 210 (1994) 301-316
315
over all the possible distributions of the spatial disorder. Once more this is an
interesting possibility that we may hope to return to in the near future.
10. Conclusions
In this paper we have presented a model for the packing properties of systems
of many irregular grains. We were able to determine that a packing of near
spheres (based upon perturbing mono-disperse spheres) will on average be less
dense than a packing of mono-disperse spheres. However, when local correlations
in the granular configurations were taken into account then be were able to show
in the close packed regime that this is not necessarily true, and depends upon the
relative granular interactions. We were also able to show that a packing based
upon perturbing poly-disperse spheres will tend to be more dense than a packing
of mono-disperse spheres (at minimum close packing). All of these results are
valid for any strength of perturbation.
Finally we showed for an AB alloy model that the system will phase segregate
for any size ratio.
Acknowledgements
One of the authors (CCM)
support of British Petroleum.
gratefully
acknowledges
the generous
financial
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