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Simulating mesoscopic order

01 Jan 1994-Computational Materials Science (Elsevier)-Vol. 2, Iss: 1, pp 127-130

Abstract: Simulations of ordering transitions in mesoscopic systems are, at present, mainly of fundamental interest. It is argued that such simulations are potentially important for the design of novel materials.
Topics: Mesoscopic physics (59%)

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ELSEVIER
Computational Materials Science 2 (1994) 127-130
COMPUTATIONAL
MATERIALS
SCIENCE
Simulating mesoscopic order
D. Frenkel
FOM Institute for Atomic and Molecular Physics, PO Box 41883, 1009 DB Amsterdam, The Netherlands
(Received 21 June 1993)
Abstract
Simulations of ordering transitions in mesoscopic systems are, at present, mainly of fundamental interest. It is
argued that such simulations are potentially important for the design of novel materials.
I. Introduction
One of the attractive features of "nano-tech-
nology" is that it is now becoming possible to
build materials atom-by-atom. Still, nobody in his
or her right mind would try to build a silicon
crystal (say) in this way. The reason is, of course,
that nature already does this for us. In fact,
crystal formation is such a common phenomenon
that it is easy to forget that it is also something of
a miracle. In order to form a crystal, the crys-
talline phase must be thermodynamically stable
but, in addition, the kinetics of crystal formation
must be such that the stable phase forms on a
reasonable time scale. In fact, this is certainly not
always the case: many mixtures will vitrify before
they crystallize and this is also true for many pure
substances, in particular those consisting of large
molecules. Yet, many materials, even proteins
and colloids, do form crystals and this has consid-
erable relevance for materials science, as we all
know that ordered phases have electrical, optical
and mechanical properties that are completely
Elsevier Science
B.V.
SSDI 0927-0256(93)E0040-5
different from those of the corresponding disor-
dered phase.
By contrast, if we use bricks to build a wall, we
should not wait for these bricks spontaneously to
assemble: we put them in place by hand. How-
ever, also when we build smaller ordered struc-
tures (e.g., memory chips), we
impose
the order.
The lure of "nano-technology" is that it extends
this form of manual control to the nanometer
scale. However, as the example of crystal forma-
tion shows, some structures will form without our
direct intervention, as long as we provide the
correct building blocks and create the conditions
such that ordering is not kinetically inhibited. It is
for this reason that there is a great need to
understand the physical factors that determine
the spontaneous ordering of mesoscopic systems.
In fact, in several instances, present-day technol-
ogy exploits the spontaneous ordering of meso-
scopic systems, for instance in the fabrication of
high-strength fibers that are spun from liquid-
crystalline solution in which the polymeric build-
ing blocks of the fibre are already partially

128
D. Frenkel / Computational Materials Science 2 (1994) 127-130
aligned. However, compared with the way in
which nature exploits spontaneous ordering pro-
cesses in biological systems, our efforts in this
direction are still in their infancy. When it comes
to preparing ordered patterns in electronic de-
vices, we still rely on lithography, rather than on
self-assembly. I believe that this will change but,
in order for these changes to happen, we must be
able to predict what building blocks to choose in
order to achieve a specific type of mesoscopic
order. Moreover, it is not enough to know that a
particular mesoscopic structure is thermodynami-
cally stable: the formation must also be kineti-
cally allowed. In fact, kinetic factors are much
more important in ordering transitions of meso-
scopic systems than in microscopic ("atomic")
systems, because in the former systems vitrifica-
tion (gelation) is never far away.
Computer simulations can be used to predict
the structure and rate of formation of mesoscopi-
cally ordered systems. I shall not attempt to re-
view the extensive literature in this field. Let me
just indicate the three focal points of the simula-
tion of mesoscopic systems, viz., (a) the computa-
tion of structure and phase stability, (b) the simu-
lation of transport properties (including the study
of gelation) and (c) the determination of free-en-
ergy barriers for activated processes (e.g., crystal
nucleation). One feature that is common to all
mesoscopic systems and that is very different
from the corresponding atomic or simple molecu-
lar systems, is the important, often dominant, role
that entropy plays in determining the physical
properties. I shall illustrate this by discussing the
spontaneous formation of
AB13 (!)
crystals in
mixtures of essentially hard, repulsive colloidal
particles. However, in order to explain why the
formation of such crystalline superstructures is
surprising, I must first briefly explain how our
intuitive interpretation of entropy may conflict
with the statistical mechanical definition.
2. Order through disorder
The second law of thermodynamics states that
the entropy of an isolated system increases until
it reaches a maximum. Statistical mechanics has
given a "microscopic" interpretation to this law:
in suitable units, entropy is simply equal to the
number of (quantum) states accessible to the
system. If we constrain a system to be in a single
state, its entropy will be zero. If we then release
this constraint and allow the system to explore all
other states with the same energy, the entropy
will increase until the system is equally likely to
be in any one of its accessible states. At that
point the entropy attains its maximum value and
the system has reached equilibrium. Of course, in
the process, we have lost information about the
system: initially, we knew exactly what state it was
in, but after equilibration the system can be in
any one of a large number of states. We have
gone from an
ordered
initial situation to a more
disordered
final situation. In this sense, entropy is
a measure for the amount of disorder in a system.
Can we
see
order or disorder? Strictly speaking,
the answer is no, because we cannot see the
number of accessible states of a system. However,
we have another, more intuitive notion of order
and disorder based on things that we
can
see. We
call a crystal ordered, because its constituent
particles are constrained to be near specific points
in space (lattice sites). A fluid, where no such
constraints apply, is called disordered. Our intu-
itive notion about order and disorder suggests
that a system with a given density and energy
should have a higher entropy in the fluid phase
than in the crystalline phase, and that freezing
would result in a decrease of entropy. The second
law of thermodynamics tells us that this can only
happen if this decrease in entropy is offset by a
larger increase in entropy of some other part of
the universe. This can happen if the
energy
of the
crystalline phase is lower than that of the liquid
phase. In that case, the system releases heat on
freezing, and thereby increases the entropy of the
rest of the universe. This unsophisticated descrip-
tion of the thermodynamics of freezing explains
why, for a long time, it was commonly thought
that attractive forces between molecules are es-
sential for crystallization: a crystal can form be-
cause the lowering of the potential energy of the
system upon solidification pays the price for the
decrease in entropy.
It was therefore a great surprise when, in the

D. Frenkel / Computational Materials Science 2 (1994) 127-130
129
1950s, computer simulations by Alder and Wain-
wright [1] and Wood and Jacobson [2] indicated
that a fluid of hard spheres could freeze. In this
case, the potential energy of the system is always
zero. Hence, if crystallisation occurs, this can only
mean that the entropy of the crystalline solid is
higher than that of the fluid. Although, at pre-
sent, the hard-sphere freezing transition is well-
established and non-controversial, this was cer-
tainly not the case when it was first reported. In
fact, during a round-table discussion on this topic
held in 1957, the chairman (G.E. Uhlenbeck)
asked the experts on the panel (including two
Nobel laureates) to c, ote on the existence of the
hard-sphere freezing transition, and the vote was
even .... Uhlenbeck closed the discussion with
the following words: "... I am quite sure that the
transition goes a little bit against intuition: that is
why so many people have difficulty with it, and
surely I am one of those. But this transition - it
still might be true, you know - and I don't think
one can decide by general arguments" (the pro-
ceedings were published six (!) years later [3]).
Yet, in retrospect, we can understand the hard-
sphere freezing as follows: a naive picture of a
solid is a cell model in which all molecules are
confined to cells centered around lattice sites.
Confining the molecules to cells costs entropy
(we call this contribution to the entropy configu-
rational). However, we also gain entropy, be-
cause a molecule has more free volume to move
in this cell than it had in the fluid; in other words,
there is more jamming of molecules in a dense
fluid than in a solid of the same density. At
sufficiently high densities, the gain in entropy due
to the increase in free volume outweighs the loss
in configurational entropy. When this happens,
entropy will favour crystallization: an increase in
macroscopic order is driven by an increase of
microscopic disorder. In fact, it is now known
that numerous ordering transitions can be driven
by entropy alone. In particular, not only crystals,
but also many kinds of liquid crystal can form in
systems of hard repulsive particles [4].
The complexity of the order that can be in-
duced by entropy has become clear only recently
when Ottewill et al. [5] reported experiments on
the crystallisation of a colloidal suspension of a
mixture "large" and "small" polymethylmethacry-
late (PMMA) particles with a size ratio of 0.58.
These mesoscopic particles have steep repulsive
interaction, not unlike hard spheres. Bartlett et
al. observed that this mixed colloid would form a
crystal with a so-called ABe3 structure. This
structure is familiar in metal alloys (e.g., NaZn~3)
and consists of a simple cubic lattice of A parti-
cles. In the middle of each A-cube, we find a
B-particle surrounded by an icosahedral cluster
of other Bs. The full unit cell consists of eight
such sub-cells with neighbouring icosahedra alter-
nating in orientation by 7/2.
These findings immediately raised two ques-
tions: (1) is the observed ABI3 phase stabilized by
entropy alone or is the fact that PMMA particles
are not hard spheres crucial? and (2) can we
understand why this phase should form? A
definitive answer to the first question can only
come from computer simulations. In fact, the
results of an extensive numerical study by E1-
dridge et al. [6] indicate that entropy alone can
indeed account for the stability of the AB13 struc-
ture. In a recent publication, Xu and Baus [7]
consider both questions from a different point of
view. In their paper, Xu and Baus use classical
density-functional theory, i.e., the best analytical
theory of freezing to date, to estimate the stabil-
ity of the ABI3 phase in a mixture of large and
small hard spheres. Their calculations confirm
that, for a size ratio of 0.58, there is a density
range where the AB13 structure is more stable
than the fluid mixture, the pure A and B solids or
the AB 2 compound. Why is AB~ the stable solid
structure? Surprisingly, it turns out that, in this
case, it is not the free volume that is responsible.
In fact, both the pure A and B phases and the
AB 2 structure can fill space more efficiently, and
hence the free volume would favour those phases
over ABe3. The ABe3 phase is stable because it
has a higher configurational entropy. In a mix-
ture, this configurational entropy is closely re-
lated to the entropy of mixing. In the words of Xu
and Baus "...the larger entropy of mixing of the
ABI3 structure relative to that of the competing
structures is responsible for its stability". I guess
that George Uhlenbeck would have found this
observation more than a little counter-intuitit~e.

130 D. Frenkel / Computational Materials Science 2 (1994) 127-130
3. Discussion
I have included this discussion of the
AB13
crystal to illustrate that quite complex colloidal
structures are thermodynamically allowed. How-
ever, this is only part of the story, because I have
not discussed the kinetics of crystal nucleation
and growth. In fact, the experimental evidence
strongly suggests that kinetic factors play an im-
portant role in determining which phase actually
forms. In fact, we do have the techniques to
compute both the free-energy barrier for crystal
nucleation [8] and the rate at which this barrier is
crossed (for a clear discussion, see ref. [9]). The
latter calculations are non-trivial because the sim-
ulation of the dynamics of dense colloidal suspen-
sions requires special techniques [10] that I will
not go into here.
In summary, simulations of ordering in meso-
scopic systems are, at present, still very much a
technique to increase our fundamental under-
standing of the statistical mechanics of order-dis-
order transitions. However, in the near future,
such simulations may play a crucial role in design-
ing materials with taylor-made mesoscopic order.
the 'Nederlandse Organisatie voor Wetenschap-
pelijk Onderzoek' (NWO).
5. References
[1] B.J. Alder and T.E. Wainwright, J. Chem. Phys. 27 (1957)
1208.
[2] W.W. Wood and J.D. Jacobson, J. Chem. Phys. 27 (1957)
1207.
[3] G.E. Uhlenbeck et al., in: The Many-Body Problem, ed.
J.K. Percus (Interscience, New York, 1963) p. 498.
[4] D. Frenkel, in: Proc. NATO ASI on Phase Transitions in
Liquid Crystals, eds. S. Martellucci and A.N. Chester
(Plenum, New York, 1992) p. 67.
[5] R.H. Ottewill, P. Bartlett and P.N. Pusey, Phys. Rev.
Lett. 68 (1992) 3801.
[6] M.D. Eldridge, P.A. Madden and D. Frenkel, Molec.
Phys., 79 (1993) 105.
[7] H. Xu and M. Baus, J. Phys.: Condens. Matter 4 (1993)
L663.
[8] J.S. van Duijneveldt and D. Frenkel, J. Chem. Phys. 96
(1992) 4655.
[9] D. Chandler, Introduction to Modern Statistical Mechan-
ics (Oxford University, New York, 1987).
[10] A.J.C. Ladd, in: Computer Simulation in Materials Sci-
ence, eds. M. Meyer and V. Pontikis (Kluwer, Dordrecht,
1991) p. 481.
4. Acknowledgement
The work of the FOM Institute is part of the
research program of FOM and is supported by
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