INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) R167–R202
doi:10.1088/0953-4075/39/9/R01
TOPICAL REVIEW
Insights into phase transitions from phase changes of
clusters
A Proykova1 and R S Berry2
1
2
University of Sofia, 5 James Bourchier Blvd. Sofia-1126, Bulgaria
University of Chicago, Chicago, IL 60637, USA
E-mail: [email protected] and [email protected]
Received 11 November 2005, in final form 20 March 2006
Published 18 April 2006
Online at stacks.iop.org/JPhysB/39/R167
Abstract
The phases and phase transitions of bulk matter differ in several important
ways from the phases or phase-like forms of small systems, notably atomic
and molecular clusters. However, understanding those differences gives
insights into the nature of bulk transitions, as well as into understanding
the behaviour of the small systems. Small systems exhibit dynamic phase
equilibria, large fluctuations and size-dependent behaviour in ways one cannot
see with macroscopic systems. The Gibbs phase rule loses its applicability,
the distinction between first-order and second-order transitions reveals a size
dependence, and one can see phases in equilibrium (as minority populations)
that are never the most stable thermodynamically.
1. Introduction: phases of clusters and of macroscopic systems
This review addresses the relations between phases and phase changes exhibited by bulk,
macroscopic systems and their counterparts in small systems, atomic and molecular clusters
and nanoscale particles. The differences between the phase behaviour of bulk matter and of
clusters can, for the most part, be attributed to two factors. The more important is the enormous
free energy differences between phases of bulk matter, which make only the most favoured
phase observable, compared with the small free energy differences for clusters, which make
less favoured phases nearly as observable as more favoured or most favoured phases [1]. The
other is the relative contribution of the surface in a macroscopic and a small system. In a
d-dimensional system containing N atoms, approximately N (d−1)/d of atoms will be on the
surface; for large N the relative fraction N −1/d is very small and the surface effects on the
bulk properties of the system are commonly neglected. In contrast, a high proportion of atoms
or molecules are in surface layers of clusters which makes the surface as important for many
cluster properties as the interior part is. In the spherical approximation used above to estimate
the surface contribution one finds that about one half of the particles are on the surface of a
cluster containing 500 particles. Because the ratio of surface area to bulk is large for small
0953-4075/06/090167+36$30.00 © 2006 IOP Publishing Ltd Printed in the UK
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particles, a generalization of the nucleation theorem must be performed in order to correctly
predict appearance of new phases [2, 3]. Even more interesting is the geometrical effect of the
surface that breaks the translational invariance and isotropy. Together, these two factors—the
surface and the free energy—lead to striking differences between the behaviour of phases of
clusters and those of their bulk counterparts. This review examines those differences and shows
how one can get new insights into phase transitions of bulk matter through understanding their
small-system counterparts.
Phases are forms of matter that we normally encounter and recognize in our surroundings.
Gases, liquids and solids are what are most likely to come to mind, but of course solids may
exist in different forms that we classify as different phases, and even liquids may exhibit
more than one kind of phase. There are other forms of matter that we would also classify
as phases, forms such as plasmas, nuclear matter, Bose–Einstein and fermionic condensates,
liquid crystals, superfluids and supersolids and the paramagnetic and ferromagnetic phases of
magnetic materials. Quantum aspects of large finite systems have been the subject of a book
[4] and have been recently reviewed [5].
We have developed an elaborate part of thermodynamics, and another of kinetics, to
describe the equilibrium behaviour of bulk phases and the ways that matter undergoes its
changes from one phase to another. In this review, we first briefly examine those bulk phases,
and then turn to the corresponding—but certainly different—behaviour of small systems.
Throughout, we work in the context of Gibbsian ensembles as the objects that reveal the
thermodynamic behaviour of the systems we study.
1.1. Phases: what they are for bulk systems
A ‘phase’ is a set of states of a macroscopic physical system that have relatively uniform
chemical composition and physical properties, notably density, crystal structure, index of
refraction, diffusion coefficient, heat capacity and others that we associate with a homogeneous,
reproducible system that can exist in thermodynamic equilibrium. It is useful to recognize
at the outset that there are truly stable phases that correspond to global thermodynamic
equilibrium states for a given set of conditions, and also metastable states that correspond only
to local free energy minima. It will be useful to use these two concepts when we turn to the
phase behaviour of small systems.
Phases are emergent phenomena produced by the self-organization of a macroscopic
number of particles. Emergence is a term usually used in complex systems, to indicate a
process in which some new structures, patterns and properties appear, which are not obviously
inherent for the entities that comprise the complex system. An emergent property occurs when
a number of these entities interact to develop some behaviour as a collective system that is
more complex than any exhibited by the constituent elements. That is, the emergent system
has properties beyond those one can attribute to the individual components, such as structural
order.
‘Phase’ is a simple concept, but it may have several definitions that are equivalent for bulk
matter but may not be equivalent when applied in the context of small particles. One definition
of a phase of a system is a region in the parameter space of the system’s thermodynamic
variables in which the free energy is analytic. Equivalently, two states of a system are in the
same phase if they can be transformed into each other without abrupt changes in any of their
thermodynamic properties. By the same token, two states are in different phases if there is a
discontinuity in the internal energy and entropy, or in a first or higher derivative of these.
The free energy F and its derivatives define all the thermodynamic properties of a system—
the entropy S, heat capacity, magnetization, compressibility. As long as the free energy
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remains analytic, all the thermodynamic properties are well defined. The free energy may
change continuously as a system goes from one phase to another. Nonetheless, the passage
between two phases (a ‘phase transition’, in bulk systems) is characterized by what we classify
as a non-analytic change of the free energy. This non-analyticity reveals itself in that the free
energies on the two sides of the transition are two different functions. Thus one or more
properties change abruptly at the transition, and the transition appears sharp. The property
most commonly examined in this context is the heat capacity. At the point of a transition,
the heat capacity may become infinite, jump abruptly to a different value, or exhibit a ‘kink’
or discontinuity in its derivative. The transition can be viewed as a result of failure of the
stability criteria: dS = 0, d2 S < 0 (maximum) or d2 S > 0 (minimum) in the fundamental
relation governing the system.
Examples of phase transitions are melting (solid to liquid), freezing (liquid to solid),
boiling (liquid to gas), condensation (gas to liquid) and sublimation (solid to gas). Another
example is polymorphism: a substance may exist in a variety of solid phases, each with a
unique crystal structure.
Typical bulk samples of matter contain about 1023 particles, on the order of Avogadro’s
number. In systems that are much smaller—even, say, a thousand atoms—the transition
between phases is not sharp but round-off [6]. The reason is that the appearance of nonanalyticity in the free energy requires that the system be composed of a huge, formally
infinite, number of particles. We shall see in the following discussion that, strictly, no finite
system, no matter how large, has a rigorously non-analytic free energy at a phase transition,
but that for all meaningful, observable purposes, we are completely justified in using the ideal,
infinite-system limit to describe phases and phase changes of bulk systems.
Now the question is, ‘why do real systems exhibit phases although they contain countable
numbers of particles?’
The answer is due to thermodynamic fluctuations in real systems. Far from a phase
transition, these fluctuations are unimportant, but as the system approaches it, the fluctuations
begin to grow in size (i.e. spatial extent). At the ideal transition point, their size would be
infinite, but before that can happen the fluctuations will have become as large as the system
itself. In this regime, ‘finite-size’ effects come into play, and we are unable to predict accurately
the behaviour of the system. Thus, phases in a real system are well-defined away from phase
transitions, and how far away it depends on the size of the system.
It is useful to distinguish transient macrostates from transient microstates. The former
are the succession of conditions, specified by local values of macroscopic quantities such as
temperature and density, through which a macroscopic system passes as it moves towards
equilibrium. The latter are the states specified by microscopic variables that include the shortlived states involved in fluctuations around equilibrium, as well as those involved in passage
towards equilibrium. Having said that, we can define and observe transient states whose
properties may be well-defined and observable for small systems and under some conditions.
These transient states may occur in dynamic equilibrium with one another with the passage
among them simply by normal fluctuations of a system in equilibrium. This behaviour is
quite analogous to dynamic isomerization of molecules that can interconvert readily among
structures.
For a macrosystem the Gibbs phase rule gives
f = c − p + 2,
(1)
where f is the number of degrees of freedom, c is the number of components in the system
and p is the number of phases. A component is a chemically distinct constituent of a system
and the number c of components is equal to the number of independent chemical constituents,
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minus the number of chemical reactions between them, minus the number of any constraints
(such as charge neutrality or balance of molar quantities).
The rule implies that f increases with the number of components. The simplest case is,
of course, a system with only a single component and only one phase, such as liquid water.
For this, we have f = 2. Usually the two degrees of freedom are taken as temperature T and
pressure P, but may be others such as volume V or energy E. If however vapour is present in
equilibrium with the liquid, then p = 2 so f = 1, and it is not possible to fix arbitrarily both
T and P. For a single component with three coexisting phases f = 0 and the system exhibits
a triple point; i.e. the system of one component and three phases in equilibrium exists only at
an isolated point.
1.2. Phase transitions in bulk
Phase transitions have a wide variety of important implications including the formation of
topological defects—domain walls, monopoles and textures. Recently, a series of articles
have discussed the origin of phase transition phenomena, [7] and the references therein.
Important necessity theorems have been proved. These theorems imply that for a wide class of
potentials, a first- or a second-order phase transition can only occur as a result of a topological
change of the submanifolds of configuration space. A formula relates the microcanonical
entropy with quantities of topological meaning of the configuration space submanifolds. After
the Morse theory, the topology changes of manifolds can be put in one-to-one correspondence
with the existence of critical points qc of the potential VN (q1 , q2 , . . .): ∇VN (q)q=qc = 0.
Passing a critical value VN (qc ), the manifolds Mv = VN−1 ((−∞, v]) change the topology. If
the potential VN is bounded below and with a non-degenerate Hessian (that is, no eigenvalue
vanishes), [7], the theorem can be proved. These requirements are satisfied in most physical
cases. Coulomb interactions should be considered with a care by explicitly taking into account
that they are effective only at a finite distance due to Debye shielding. The only relevant source
of degeneracy is a continuous symmetry that is easily removed by adding a small variation to
the potential. Due to the free surface of small systems, the symmetry is effectively broken at
the surface ensuring that the requirements are satisfied.
Among the properties of bulk phases that have interesting counterparts in small systems
is that of the order of a phase transition. A first-order transition is the one in which there is
a change in the internal energy—a ‘latent heat’ L—when the system passes from one phase
to another. A second-order transition does not involve a latent heat and properties such as the
internal energy change continuously but with discontinuous derivatives [8, 9]. The distinction
between the transitions is not so clearly maintained with small systems, yet it has a great
usefulness there [10–15].
Most of the features of the phase transitions can be retrieved from mean-field
considerations as in the van der Waals theory of the liquid–gas transition [17], or the Landau–
Ginsburg theory of superconductivity [18, 19], or the Weiss theory of ferromagnetism [20].
The common concept in these theories is the order parameter q, which determines the order in
the system. It should be stressed that there is no unique order parameter—it could be a scalar
or a multicomponent vector.
In most cases, the free energy F can be expanded in terms of q:
γ
σ
β
(2)
F (q, T ) = F (0, T ) + αq + q 2 + q 3 + q 4 ,
2
3
4
a form known as the Landau–Ginzburg expansion, which is used to distinguish between firstand the second-order transitions in bulk. If α = γ = 0 and β = b(T − Tc ), the transition
is of second order and the minimum of F moves continuously from qm = 0 for T > Tc to
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F
T>Tc
F
T1
T2
T=Tc
T<Tc
T3
T4
q
T5
q
T6
(a)
(b)
T
T
Tc
Tcoex
q
(c)
q
(d)
Figure 1. The free energy and the phase diagrams for the first-order (a), (c) and the second-order
(b), (d) phase transitions [16]. These are typical for bulk with one exception—the curve T2 in
part (b) has a shifted shallow minimum to stress the attention that for small sizes it might happen
that distinguishing the second- and the first-order transition is hampered by surface effects and the
difference between the two minima can vary which is not the case in bulk.
qm = 0 for T < Tc , see figure 1(b). We choose the scale so that imposing γ = 0 preserves
the maximum symmetries of the system. If γ = 0, then the free energy is bimodal as it is in
figure 1(a) and the phase transition is discontinuous. For temperatures higher than the transition
temperature, the free energy has a minimum at qm = 0; at T = T2 a second local minimum
appears. The two minima have the same depth at the coexisting temperature T = Tcoex ; at
T = T5 the minimum located at q = 0 disappears. The phase diagrams in figures 1(c) and (d)
show the local minimum (full line) and the local maximum (dashed line) of the free energy
for the first- and the second-order transition, respectively. In bulk, the phase transition occurs
at a single-valued T = Tcoex along the line of phase coexistence (the thin dashed line in
figure 1(c)) where the two phases have the same free energy. We shall see that the finite
systems can coexist in finite-temperature intervals T with widths that depend on the system
size [21–24].
It should be stressed that there is no unique order parameter—it could be a scalar or
a multicomponent vector. Precisely because the order parameter is not unique, one should
search for a suitable order parameter which would give a clear picture of the processes in the
system [11, 25]. One strategy of search is to inspect the loss of symmetry (if any) during the
phase transition [26]: the number of components of the order parameter equals the number
of the symmetry elements lost in the transition. In other words, if the phase transition is
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associated with symmetry breaking G → H → · · ·, then each step corresponds to the number
of generating elements of the groups that are lost in the step; those numbers are equal to the
number of order parameter components required to describe the symmetry breaking.
The analogue of water may again prove useful in understanding the concept of symmetry
and symmetry breaking. The liquid phase of water is rotationally symmetric, that is, it looks
the same around each point regardless of the direction in which we look. We could represent
this large three-dimensional symmetry by the group G (actually the rotation group in three
dimensions, SO(3)). The solid form of frozen water, however, is not uniform in all directions;
the ice crystal has preferential lattice directions along which the water molecules align. The
group describing these different discrete directions H, say, will be smaller than G. Through
the process of freezing, therefore, the original continuous symmetry G is broken down to the
discrete symmetry of H.
Another example of a symmetry that can be broken by a phase transition is ‘up-down
symmetry’ meaning symmetry under the reversal of the direction of electric currents and
magnetic field lines. This symmetry is broken during the transition to a ferromagnetic phase,
due to the formation of magnetic domains in which individual magnetic moments are aligned
with one another. Within each domain, the magnetic field points in a fixed direction chosen
during the phase transition.
Let us consider the liquid-to-glass transition with no symmetry broken. While the liquidto-crystal transition is thermodynamically driven, i.e. the crystal is the most favourable state,
having a lower free energy than the liquid at temperatures below the melting point, the glass
transition is driven by kinetics. While the glassy state is clearly locally stable, and typically
has a free energy below that of the liquid at the temperatures at which it is observed, the glassy
state is normally not the state of lowest free energy and hence, for a bulk system, would not
be observable in a system at thermodynamic equilibrium. Its local stability means that the
disordered glassy state does not have enough kinetic energy to overcome the potential energy
barriers required for movement of the molecules past one another to reach the thermodynamic
minimum. The molecules of the glass take on a fixed but disordered arrangement as is
shown in figure 15. Glasses and supercooled liquids are both metastable phases rather than
true thermodynamic phases like crystalline solids. In principle, a glass could undergo a
spontaneous transition to a crystalline solid at any time. There is one situation, however, in
which the most stable solid does not have a long-range order; this is the simple, monatomic
solid under constant pressure high enough that the dominant interparticle forces are entirely
repulsive [27].
The molecular arrangement is sometimes used to classify substances:
• crystalline solids: molecules are ordered in a regular lattice;
• quasi-crystals: quasi-periodic arrangement of the molecules;
• fluids: molecules have a short-range order but not a long-range order and are not rigidly
bound;
• glasses: molecules are disordered but are rigidly bound, usually with some degree of
short-range order;
• amorphous materials, disordered with not even short-range order.
The glass has a structure with properties distinctly different from either liquids or (regular)
solids; therefore, we say that it is neither a liquid nor a solid. When we talk about solids and
liquids, we refer to the arrangement of molecules, which may be dynamical as well as static,
in addition to considerations of their macroscopic material properties.
The main achievement of mean-field theory of phase transitions in bulk is that the nature
of the transition in principle can be determined by studying the symmetry of the problem. If we
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can construct a cubic invariant, the system has a first-order transition and can be distinguished
from systems having a second-order transition (no cubic invariants). In a real microscopic
system a second-order phase transition may occur only if the system does not undergo a
first-order transition before reaching the fixed point. A remark is due: in mean-field theory,
one usually distinguishes between an unstable fixed point in the order parameter space (which
can exceed the dimension of the real space) and a thermodynamic instability, which means
an existence of states of lower free energy, if the parameter space has higher dimensionality
than the real space of the system. For systems that do not have stable fixed points, one detects
‘slush’ instead of coexisting phases, as we discuss in the next section.
The classification of the transitions based on the free-energy expansion is very useful;
however there are notable exceptions in which fluctuation-induced transitions occur [28, 29].
Systems expected to transform continuously sometimes exhibit discontinuous transitions and
conversely [30].
1.3. Phase changes in finite systems
The concept of ‘phase’ is, in some ways, strictly applicable only for macroscopic systems.
However, even though the concept loses much of its precision in the context of small systems,
it remains useful. One important reason is that the phase behaviour of small systems gives
us new insights into bulk phases. It also opens possibilities for kinds of behaviour that are
forbidden for bulk systems but are quite allowable for small systems. Hence we continue to use
‘phase’ for small systems; we just must do it carefully. In any case, we use the same language
to describe phases of small systems that has long been used for macroscopic systems, but
we sometimes find that the meaning of common terms has to be somewhat different from that
in the context of bulk matter. The essential designation of a phase of a small system, e.g. as a
liquid, a glass or a regular solid, is entirely useful for small systems that spend time intervals
at equilibrium in forms that have the same kinds of structural and dynamic properties that we
associate with the bulk counterparts. The atoms of an atomic liquid cluster have mean square
displacements in time that allow us to identify their motions with the diffusive motion of atoms
in a bulk liquid; the atoms of a solid cluster remain in the vicinity of fixed, specific sites for
much longer times, just as the atoms of a bulk solid do. However one immediate difference
to recognize is that small clusters that we certainly call solids may have structures that are
not possible for bulk materials. The prime example is the icosahedral structure exhibited by
many kinds of clusters of a few hundred atoms or fewer. Of course, one might have a bulk
solid composed of an assembly of polyhedra, just as one can have a Penrose structure that
does not conform to a lattice, but for some clusters, polyhedral structures are definitely their
lowest-energy forms [31].
Clusters undergo changes of phase that are related to the phase transitions of bulk matter,
but this is one way in which clusters and bulk matter differ most. We shall go into this topic
at length, below. They of course exhibit solid–liquid, solid–vapour, liquid–vapour and solid–
solid transitions. They also may exhibit metal–insulator transitions [32], magnetic transitions
[33], even normal-superfluid transitions [34]. The revived interest in very small magnetic
nanoparticles is mainly caused by recent advances in synthesis techniques of nanometersize magnetic particles and their application in magneto-optics [35]. Atomically engineered
magnetic nanostructures such as superlattices, wires and dots have driven the magnetic
renaissance through the intriguing physical phenomena such as spin-reorientation transition
(SRT), giant magnetoresistance (GMR) and exchange bias. For specific sizes, clusters of
transition metal atoms have non-collinear arrangement of spins [36–38] and transition to a
paramagnetic state occurs at a very high magnetic field—the smaller cluster the stronger
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field. The complex non-collinear nature of the low temperature magnetic configurations
whose stability increases with disorder, the unusual scaling behaviour of the magnetic energy
as a function of the particle concentration, the possibility of long-range magnetic order due
to dipolar interactions, or the complexity of the hysteresis loops have been found in [39].
Change of the optical properties is observed for very small metallic clusters due to the electron
confinement [40, 41]. Because some phases may exist for clusters in equilibrium that are not
observable for bulk matter, there may be phases and transitions that have no counterparts in
bulk, e.g. polyhedral to lattice structures [1, 42–44].
At this point, it is sufficient to say just a bit about how clusters change phase. Evidence
from simulations is very strong that some clusters, that is, some sizes of some kinds of clusters,
exhibit clear coexistence of phases in equilibrium, in the sense just given [21, 42, 43]. Under
conditions that allow coexistence, they remain in one phase for intervals long enough to
establish properties we associate with a given phase, and then transform, spontaneously, to
a different phase, where they again remain for times long enough to take on well-defined,
observable properties of that new phase, as shown in several figures discussed subsequently.
The distribution of the density of states of a system with coexisting phases is bimodal,
figure 17.
However this description does not apply to all clusters. There are apparently some
kinds and sizes of clusters that, under conditions that would allow phases to coexist, the
passage between phases is so frequent and rapid that such systems cannot establish equilibrium
properties of any specific phase [15, 25]. Under these conditions, one would expect to observe
only ‘slush’, the average over the properties of the transient residences in the regions of phase
and configuration space corresponding to the phases being visited. The contrast between these
two kinds of behaviour appears with two very similar-seeming clusters of argon, Ar17 and
Ar19 [45–47]. Within their domains of solid–liquid coexistence, the former never remains
in one phase for an interval long enough to establish solid-like or liquid-like characteristic
equilibrium properties; the latter does this very effectively.
The transition rate can be estimated on the basis of the Rice–Ramsberger–Kramers–
Marcus or RRKM theory [48], whose applicability to such systems has been justified by
simulations [49]. For a microcanonical ensemble this rate is:
Etot
−Ets
g
ρ(E † ),
Krate =
hρ(E ∗ ) †
E =0
where ρ(E ∗ ) is the density of states at E ∗ = Etot − Emin , Emin is the local minimum, ρ(E † )
is the density of states at E † : 0 E † Etot − Ets , and Ets is the transition state or a saddle;
g accounts for equivalent configurations which can be distinguished only if the molecules are
labelled (ionization or other means). The transition states are equally important for both the
dynamics of the system and its thermodynamics as each transition is associated with a flat
saddle region of the potential energy surface that connects two minima.
One definition of a phase in a finite system can be inferred from the concept of behaviour
as a function of some suitable order parameter that defines a space in which the system may
exist, not necessarily in equilibrium. A phase, from that perspective, is a state in which the
free energy is a local minimum with respect to that order parameter, with given values of the
thermodynamic conditions in which the system resides. For example, we have definition (2)
of a first-order phase transition as a form of matter whose existence is a consequence of a
convex intruder in the generalized (thermodynamic) potential. This leads to an anomaly (a
minimum) in the relevant probability distribution along the transformation from one state to
another. The relevant coordinate can then be considered an order parameter characterizing a
symmetry or any other observable. Phases are then defined by regions of the maximal values
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of the probability distribution with respect to the order parameter. There are two local minima
in the free energy in a traditional first-order transition, and the bulk system may exist with
the two phases in equilibrium under conditions that make those free energy minima equal.
Away from those conditions, the local minimum of higher free energy, the thermodynamically
unfavoured state, may be metastable, e.g. supercooled water.
The direct use of equation (2) to distinguish the order of phase transition in a small system
meets the obstacle that the surface breaks the symmetry and the cubic term appears even if
the transition should be continuous. That is why additional investigations and diagnostics are
needed to determine the order of the transition.
A rich aspect of phase changes of clusters is the relation between those changes and the
traditional orders of bulk transitions—first order, ‘weak first order’, second order and possible
higher-order transitions. A number of different definitions of first- and second-order phase
transitions have been proposed which can be applied also for a finite number of constituents:
1. The first is an anomalous sign of the curvature determinant of the entropy as a function
of the conserved quantities S(E,N,V,L, . . . ) [50–52].
Usually the entropy is defined as
1
,
(3)
p(Ai )loga
S=
p(Ai )
i
which is usually a concave function of the total energy E; S is maximized if all p(Ai ) are
the same, as it is in a microcanonical ensemble. The base a = e in physics and a = 2 in
information theory where S is measured in bits. An important observation is that, in general,
the entropy depends on the observer because it is expressed in terms of probabilities. Thus,
if one of the probabilities is equal to 1, then all the others are 0 and the entropy is 0. An
equivalent expression of the entropy is
S(E) = lim
1
N→∞ N
ln (E),
(4)
where (E) denotes the density of microstates of the system with total energy E and many
particles N. If S(E) from equation (4) contains one or more nonconcave dips, the first derivative
of S(E) is not a monotonic function of E and we get an anomalous sign of the curvature.
A dip in the caloric curve may appear if the limiting melting temperature Tm is higher
than the limiting freezing temperature [53]. Computationally bimodality (multimodality)
in the entropy appears if phase space regions corresponding to intermediate forms—solidlike/liquid-like—have been neglected [54].
2. Yang and Lee presented the grand-canonical partition function as a function of its
zeros in the complex plane [55], which for simple systems (hard-core interactions and the
Ising model) lie on a unit circle. Later, it has been shown by Fisher and by Grossmann
that the localization of the zeros of the canonical partition sum Z in the complex plane
β = 1/kB T → β + iτ distinguishes between the types of phase transitions in bulk [56]. The
nonanalytic behaviour of the canonical free energy
1
ln ZN (β)
N
is related to the distribution of complex zeros of the partition function
ZN (β) = N (E/N ) exp(−βE) d(E/N ),
F (β) = lim −
N→∞
(5)
(6)
where N (E/N ) is the density of microstates with mean energy E/N, in the limit N → ∞.
In the case of first-order phase transitions the zeros are located around a positive βc where
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F (βc ) is not differentiable. The loci of the zeros is parallel to the imaginary axis. It has been
proved in [57] that such a distribution of the zeros is induced by the bimodal distribution of
N (E/N ).
Borrmann classified [58] the phase transitions in finite systems by analogy with the
macroscopic approach in [59]. The difference in the two approaches reflects the fact that
in a finite system the transition temperature is not the same as the transition temperature in
the bulk. In [60] it was pointed out that the curvature of the microcanonical entropy is not
sufficient to distinguish the order of the transitions.
3. Binder’s cumulant UBinder = 1 − M 4 /3M 2 2 has been introduced for magnetic
systems characterized with a power-law divergence of the magnetization M considered as an
order parameter, [61]. If the minimum of UBinder scaled with the linear size L of the system is
equal to 2/3 then the transition is continuous (II order); if (UBinder )min is less than 2/3, then it
is discontinuous. However, the conclusion drawn only from this parameter is not sufficient,
especially for systems whose linear size is less than the interaction range.
4. The scaling ansatz states [56]: let p(q, L, ξ ) be the probability density function of
the fluctuating order parameter q in a finite system of size L and order-parameter correlation
length ξ . The scaling function P(x, y) is defined with
p(q, L, ξ ) = Ld(dq −d
∗
)/(d−d ∗ )
P(qLd(dq −d
∗
)/(d−d ∗ )
(7)
, L/ξd ∗ ),
∗
where dq is the anomalous or scaling dimension of the order parameter, d is the anomalous
dimension of the vacuum and ξd ∗ is the thermodynamic length defined by Binder in [62]. If
hyperscaling holds, then d ∗ = 0, the thermodynamic length becomes the correlation length,
ξ0 = ξ and the exponent in equation (7) reduces to the form dq = β/ν, where β is the
order parameter exponent, ν is the correlation length exponent [63]. The hyperscaling in
the temperature-driven phase transition reads: νd = 2 − α, where α is the exponent for the
c
→ 0. The scaling function P(x, y)
singular part of the heat capacity C ∼ t −α for t = T −T
Tc
is expected to be universal up to the choice of boundary conditions [61]. The actual values of
the exponents determine the universality class to which a given system belongs. The scaling
ansatz means that in the phase transition the local interactions have lost their importance. We
will apply the finite-size scaling in a finite system in section 4.2.1.
2. Relevant observables and their size dependence
When one deals with small systems, it is very important to specify what kind of ensemble one
is describing. In macroscopic systems the observables determined from different ensembles
are easily related—for example, canonical and microcanonical ensembles predict the same
averages in bulk provided that the steepest descent method, or saddle-point approximation
(see the appendix) is valid. This means that the canonical partition function
Z(t) = exp[−βU (x) − tφ(x)] dx,
(8)
where the variable t is conjugated to the extensive phase function φ of the set of variables x
and the microcanonical partition function
Z = exp(−βU )δ(KN − φ) dx,
(9)
where KN = φ(x) can be related as follows. Using the integral representation of the
delta-function:
i∞
−1
exp[t (KN − φ)] dt
(10)
δ(KN − φ) = (2π i)
−i∞
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we express Z in terms of the canonical partition function
i∞
−1
exp[N Kt + ln Z(t)] dt.
Z = (2π i)
(11)
−i∞
The last equation shows that the averages computed in both ensembles are the same.
However, the fluctuations of the observables are not the same in the different ensembles.
Having in mind that the fluctuations dominate the behaviour of small systems, the relation of
the formulae, derived for the different ensembles is not straightforward. Near-equivalences for
different ensembles of macroscopic systems are often far from equivalences for corresponding
ensembles of small systems.
There is another important characteristic that distinguishes and complicates the study of
small systems. A small system is not truly extensive because if we divide the small system
into pieces, the sum of entropies is not equal to the entropy of the whole system. The
large fraction of constituent particles on the surface, and the consequent large change in that
fraction accompanying breakup of the particles, make apparent how this problem manifests
itself. Hence, as an illustration, the total energy of the system U is considered as a function of
the atomic positions xj (t) and a control parameter X(t):
∂U ∂U
dxj +
dU =
dx = dQ + dW.
(12)
∂xj
∂X xj
j
The first term corresponds to the energy change due to configurational changes, i.e. the heat Q,
while the second term yields the contribution of the control parameter, i.e. the work term W .
(The vibrational contribution is not considered explicitly here; it may be unimportant if it
undergoes very little change in the process, but it may be important if, for example, the surface
vibrations are very different from those of interior atoms and the relative numbers of these
changes in the process.) If the control parameter varies between 0 and Xf , then the total work
performed over the system is:
Xf
Xf
W =
dW =
F dx,
(13)
0
0
where
∂U
=F
∂X xj
(14)
is the force needed to change the control parameter. The heat Q exchangeable in the nonequilibrium process is (U − W ) where U is the energy change. Random fluctuations
dominate the system behaviour and cause different fluctuations of U along different
trajectories. This means that the heat exchanged with the environment (bath) and the work
also fluctuate by amplitude and even by sign.
We shall see this in several ways. Our discussion here will focus primarily on canonical
and microcanonical ensembles of clusters of a single size. We should note that grand canonical
ensembles and ensembles of distributions of various sizes, even if the distributions are fixed, can
be relevant but have not been explored for clusters. One nonequilibrium ensemble especially
relevant for experimental work is the ‘evaporative ensemble’, an ensemble prepared by
allowing clusters to cool by evaporation to produce a long-lived but unequilibrated distribution
of sizes and internal energies [64].
The internal energy has been central to simulations and theory and, more recently, to
experiment as well. Although some were done with algorithms to approximate constant
temperature [65], most of the early molecular dynamics simulations were done for conditions
of constant energy, sometimes identifying the temperature with the kinetic energy [66], but
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more for constant energy [67]. Monte Carlo simulations, on the other hand, were frequently
done for constant temperature [68–70]. Both Monte Carlo and molecular dynamics simulations
of clusters of rare gas atoms showed solid behaviour at low temperatures, liquid-like behaviour
at higher temperatures and, at still higher temperatures, evaporation. This was not unexpected.
What was a striking new finding was that the solid and liquid forms could apparently coexist
in an ensemble over a range of energies or temperatures, at a fixed external pressure (usually
zero).
The heat capacity of a cluster can be experimentally measured [71] and compared to
the computations. For simulations performed at a constant temperature T the heat capacity
is C = ∂E/∂T in kB units. If, however, the simulations have been held at a constant total
energy, then there is an arbitrariness to the choice of definition of the effective temperature.
One is, of course, the mean kinetic energy per degree of freedom. Another is the derivative of
the internal energy with respect to entropy. The latter, it must be recognized, must be based
on the microcanonical entropy, not the usual canonical entropy. These two are not equivalent
for microcanonical ensembles, and the difference becomes especially apparent for ensembles
of small systems such as clusters. Nonetheless, with the definition based on kinetic energy,
the use of the fluctuation–dissipation theorem, developed for a microcanonical ensemble in
[72], is justified. The original formula was obtained for systems away from a phase transition.
To account for the larger fluctuations in the vicinity of phase transformation and small N, we
have modified the Lebowitz formula as follows [25]:
−1
3 (N − 1)2 δT 2 3
1−
.
(15)
C/kB =
2
2
N
T 2
A second class of observable properties are the functions that characterize the local
structure of a fluid and distinguish fluid from solid. Among the most notable among these is
the so-called radial or pair distribution function g(r):
2V
−2
δ(ri )δ(rj − r) =
δ(r − rij ) .
(16)
g(r) = ρ
N (N − 1) i j =i
i j =i
The radial distribution function is of interest because both neutron and x-ray scattering
experiments yield information about it. In a simulation, it is straightforward to compute
g(r) which is the ratio between the average number density (r) at a distance r from any given
atom and the density at a distance r from an atom in the reference state (it might be liquid or
gas depending on the temperature evolution). In a gas state, g(r) = 1 by construction, while
in a crystalline state g(r) contains well-resolved peaks (figure 3).
If we are only interested in the temperature of the fluid–solid transition, we use the
modified Lindemann index δlin [47, 74, 75]: The Lindemann index is based on displacements
from equilibrium sites; the modification uses interparticle separations or bond lengths.
N
rij2 − rij 2
2
,
(17)
δlin =
N (N − 1) i,j (>i)=1
rij where |rij | = |ri (t)−rj (t)| and . denotes time averaging. In bulk δlin measures the deviations
from the fixed lattice sites and if it is greater than 10% then melting occurs. For clusters δlin
is the ratio of the second and the first momenta of the position distribution and is obviously a
size-dependent quantity that can be smaller than 0.1 for melting. Equation (17) distinguishes
between two configurations of the same cluster, figure 2, and to decide if the transition is to
a fluid phase or to another solid configuration we must simultaneously compute the radial
distribution. All results—the Lindemann index in figure 2, the radial distribution in figure 3
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0.050
Lindemann criterion
0.040
0.030
0.020
0.010
0.000
0
10
20
30
40
50
60
70
80
90
100
110
120
Temperature [K]
Figure 2. The Lindemann index for a (TeF6 )89 cluster [25]: the curve’s slope changes around
60 K indicating a continuous transformation, in contrast to the jump at ≈90 K, which shows a
configurational reconstruction.
10
Genetic Algorithm
Conjugate Gradient Optimization
9
8
7
g(r)
6
5
4
3
2
1
0
5
10
15
20
25
Radial distance r [¯]
Figure 3. The radial distributions of the same (TeF6 )89 cluster optimized at ≈90 K with the genetic
algorithm [73] and the conjugate-gradient method indicates a solid phase.
and the shape of the caloric curve in the region of 80 K, figure 10, point at a discontinuous
solid–solid transition.
2.1. Fluctuations
The fluctuations of small systems are of course large. As a result, attention has sometimes been
focused on their behaviour in the course of those fluctuations. One aspect of that approach has
been to ascribe to the fluctuations apparent violations of well-established laws. An example
is the claim that the entropy production in a small system can be negative, which would imply
that the second law of thermodynamics is seemingly violated. Of course the second law is not,
in fact, violated, in the sense that the entropy change of a system passing from one equilibrium
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state to another is zero or positive. The second law certainly allows for fluctuations to transient
‘states’ for which the effective entropy is lower than the average of the equilibrium state from
which the fluctuation occurred. But there are also necessarily fluctuations to transient states
to which we would attribute entropy values higher than that of the time-averaged equilibrium
state. Several fluctuation theorems have been proposed to clarify the situation for deterministic
[76, 77] and stochastic systems [78]. The field is currently very lively, and the results are
sometimes contradictory as the authors use different definitions of quantities such as work and
entropy. A study of polymer stretching in different ensembles shows that the corresponding
distributions of the work performed differ [79]. This can be related to the fluctuations. An
interesting conclusion is that the fluctuation theorem proved for stochastic systems is valid for
the forward process in linear systems such as a chain polymer although the original treatment
relates forward and time-reversed processes.
2.2. Topographies of potential energy surfaces
One important way to think about and analyse clusters and their behaviour is through an
understanding of the topography of their energy landscapes. The system of N particles has
3n − 6 internal degrees of freedom, and hence can be described through the dependence of
its internal energy E on the values of those 3N − 6 independent variables. That function,
in a (3N − 5)-dimensional space, can be thought of as a landscape [42, 80, 81]. Such
landscapes are very complex indeed; the number of geometrically-distinct minima increases
at least exponentially with N, and the number of permutational isomers of each of these
is approximately N!, so the total number of minima, i.e. of locally stable configurations,
increases roughly as N ! eN . This may seem a fantastically rapid rate of increase, but the
number of saddles grows even faster with N. Consequently, in order to use information about
the topographies of energy landscapes to indicate how systems behave, we must use selected,
reduced sets of data for all but the smallest clusters. For example, one can use a master
equation to describe the kinetics of relaxation or of phase change, but it is only feasible to
use a statistical sample of the surface to construct a corresponding statistical-sample equation,
for clusters of about 20 or more particles. How this can be done is a subject of study
now [82, 83]. It has proven possible to make a useful qualitative classification of clusters
according to whether they (and peptides and proteins as well) can, on cooling or relaxing,
find their way to select, low-energy structures such as the rocksalt structures of alkali halide
clusters or the native structures of proteins, or, on the other hand, can only ‘get stuck’ in
random, amorphous structures. The key to whether a system is a ‘structure-seeker’ or a ‘glassformer’ is revealed in the shape of sequences of energies of geometrically-linked stationary
points, sequences of minimum-saddle-minimum- . . . , arranged with the energies of the minima
forming a monotonic sequence. If the monotonic sequence, drawn as a zigzag pattern, looks
like a sawtooth, the system will be a glass-former; if, on the other hand, it looks like a rough
staircase, it is a structure-seeker [84].
2.3. Coexistence, caloric curves and heat capacities
A property that has been central to simulations and theory and, more recently, to experiment
as well is the internal energy. Although some were done with algorithms to approximate
constant temperature [65], most of the early molecular dynamics simulations were done
for conditions of constant energy, sometimes identifying the temperature with the kinetic
energy [66], but more for constant energy [67]. Monte Carlo simulations, on the other hand,
were frequently done for constant temperature [68–70]. Both Monte Carlo and molecular
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Figure 4. The caloric curve of Ar13 based on constant-energy molecular dynamics simulations,
with effective temperature defined by the mean kinetic energy [74]: both the separate branches
and the single average curve.
dynamics simulations of clusters of rare gas atoms showed solid behaviour at low temperatures,
liquid-like behaviour at higher temperatures and, at still higher temperatures, evaporation. This
was not unexpected. What was a striking new finding was that the solid and liquid forms could
apparently coexist in an ensemble over a range of energies or temperatures, at a fixed external
pressure (usually zero).
One diagnostic tool used to describe behaviour specific to small systems is the shape
of the caloric curve, the variation of mean internal energy with temperature for a canonical
ensemble or the variation of the effective temperature for a microcanonical ensemble. The
caloric curve for a canonical ensemble or a single system at constant temperatures is of course
a monotonically increasing function, whether E(T ) or T (E), for a pure solid or pure liquid
cluster. However because of the dynamic coexistence of solid and liquid forms, one can
either examine the average behaviour or divide the time intervals into those in which the
system is solid and those in which it is liquid. Hence one may construct a single, overall
average curve, or a two-branched curve. Figure 4 shows a caloric curve for Ar13 derived
from molecular dynamics simulation at constant energy. In the energy range of coexisting
phases, one may construct separate branches for each phase, here shown as black triangles, or
as a single curve based on the average over the entire history, indicated by the black circles.
Alternatively, one may carry out simulations at constant temperature. A very simple, clear
example is that of figure 5. This everywhere-positive slope is typical of all caloric curves
from canonical ensembles, for which the slope, i.e. the heat capacity, is a direct measure
of the mean energy fluctuations and hence is an essentially positive quantity. It is also the
form of many caloric curves from microcanonical, constant-energy simulations. However
there are clear exceptions, as figure 6 shows. Both Ar55 and Ar147 have regions in which
the microcanonical caloric curves must have negative slopes, which is equivalent to having
negative heat capacities in those regions. The physical basis of this apparent anomaly is
easy to understand; the effective microcanonical ‘temperature’ is defined here as the mean
kinetic energy per degree of freedom. The regions of the potential surfaces of these systems
corresponding to the solid forms are deep, so, at constant energy, the kinetic energies of the
clusters are high there and their effective temperatures are therefore also high. In the liquid
regions, the potential energies are high, so the kinetic energies must be low and therefore the
liquid clusters are cold. With these particular clusters, the density of states in the liquid region
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Figure 5. The caloric curve of Ar13 based on constant-temperature (triangles) and constant-energy
(circles) molecular dynamics simulations.
Ar55
Ar147
(b)
Temperature (K)
Temperature (K)
(a)
Energy (a.u.)
Energy (a.u.)
Figure 6. The caloric curves of Ar55 and Ar147 based on constant-energy molecular dynamics
simulations, with effective temperature defined by the mean kinetic energy, [86]. Both of these
show regions of downward slope in their regions of solid–liquid coexistence.
increases rapidly, in fact so much more rapidly with energy than the density of states in the
solid region, that a small increase in total energy puts many more clusters in the region of the
cold liquid. The result is that the mean kinetic energy and corresponding effective temperature
drop when the total energy increases [85].
How can two (or more) phases be observable over a range of temperatures and pressures,
as the simulations of clusters clearly show? This is understandable if we turn to basic
thermodynamics and compare the behaviour of ensembles of large and small systems. Consider
a system with just two phases, solid and liquid. In an ensemble of such systems, the ratio
of the amounts in each phase is fixed by the equilibrium constant at each temperature,
K(T ) = [liquid]/[solid]. This quantity is directly related to the difference in free energies of
the two forms:
K(T ) = exp(−F /kB T ) = exp(−N µ/kB T ).
20
(18)
If N is of order 10 , then even if the system differs from the point of classical equilibrium,
where µ/kB T is only of order 10−10 , positive or negative, then the equilibrium constant is
so large or so small that the unfavoured form is completely unobservable. However if N is
of order 103 or smaller, then the equilibrium constant remains close enough to unity over an
observable range of temperature that the unfavoured phase becomes quite observable. This
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Figure 7. The first four Mackay icosahedra, containing (from upper left to lower right) 147, 13,
309 and 55 atoms, respectively.
has, in fact, been done experimentally with charged clusters of sodium of selected single sizes
[50, 71, 87].
3. Atomic clusters
Atomic clusters here will include not only those composed of a single species. We also discuss
simple binary ionic clusters, specifically alkali halides, whose properties are very much like
those of true atomic clusters.
One of the most obvious properties that distinguishes atomic clusters of many kinds from
their bulk counterparts is the structure of their states of lowest energy. The rare gas clusters,
and many metallic clusters as well, have structures based on polyhedra, rather than on lattices.
Most common is the icosahedron. The rare gases form clusters built on this geometry. The
structures with closed icosahedral shells are particularly stable, and are said to correspond
to ‘magic numbers’. The first four of these are 13, 55, 147 and 309, and form the first
four Mackay icosahedra, as shown in figure 7. It is likely that icosahedral structure tends to
dominate the low-energy structures of rare gas clusters up to sizes in the thousands of atoms.
However there are some exceptions, such as the 38-atom and 75-atom clusters, whose global
minima have close-packed lattice structures [81].
Typically, clusters with one more atom (or, for larger clusters, a few more atoms) than
a magic number can lose an atom or atoms easily. These are shells whose stability results
from geometric structure; another kind of closed-shell stability is that associated with shells
of electrons [88]. The former stability appears with a variety of kinds of clusters, including
the weak rare-gas clusters bound by van der Waals forces and many kinds of clusters of metal
atoms; electronic shell structure is of course intimately linked with the ability of valence
electrons to move throughout the entire cluster and hence is strictly characteristic of metal
clusters, and more specifically of those with very free conduction electrons [89]. Atomic
clusters with structural magic numbers of atoms tend strongly to have melting temperatures
significantly higher than those with fewer or more atoms.
The dynamic equilibrium between solid and liquid phases is very apparent in molecular
dynamics simulations. Many clusters of tens of atoms or more exhibit this behaviour by
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Figure 8. The time history of the potential energy of a cluster of 13 argon atoms simulated by
molecular dynamics at a temperature at which there is a dynamic equilibrium between solid (low
potential) and liquid (high potential) [90].
Figure 9. The time history of the short-time mean total energy of a cluster of 55 argon atoms
simulated by molecular dynamics at a temperature at which there is a dynamic equilibrium between
three stable phases [42].
remaining in one phase, then very rapidly moving to another phase where the system again
stays long enough to establish clear properties associated with that phase. An example is
shown in figure 8, for an isothermal simulation of Ar13 ; the potential energy, averaged over
1800 steps of 3 fs, is plotted as a function of time, so one can see the passage between the
low-potential solid and the higher-potential liquid, in clear steps [90].
The behaviour of somewhat larger clusters is more complex, and reveals another way that
small systems differ from bulk matter—again as a consequence of the differences resulting
from small versus very large numbers. Figure 9 shows a time history for a constant-temperature
simulation of Ar55 , done by molecular dynamics, in a particular stripe of the coexistence range
of temperature where one can see three regions of stability, corresponding to three phases.
These three phases are solid, with the lowest total energy, the liquid, with the highest total mean
energy, and a form called ‘surface-melted’ when it was first observed [70]. The diagnostics one
usually uses made it appear that in this phase the inside shell is quite firm and solid-like, but
the outer layer has a random, loose structure. This concept was only refined when animations
revealed that in the ‘surface-melted’ state, while the motions of most of the outer-shell atoms
are anharmonic and have large amplitudes, the motions of these atoms are clearly collective
oscillations around a loose polyhedral structure [86]. However a few of the surface atoms,
roughly one in 30 for clusters of various sizes, are promoted to ‘float’ around the outside of the
cluster, while remaining bound to it. The ‘floaters’ are responsible for the apparent very loose
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motion shown in numerical diagnostics such as mean square displacement. The surface atoms
do exchange with the floaters every few thousand oscillations, so full permutational symmetry
is eventually established among all the atoms of the surface. Yes, one could consider it a kind
of surface melting, but not what one might naively expect. This example illustrates that clusters
may exhibit more than two phases in equilibrium over a range of temperatures and pressures.
Clusters violate the phase rule because of the small number of particles that they contain.
There is nothing to prevent unfavoured, minority phases of clusters from being present in
observable amounts in thermodynamic equilibrium with other, more favoured phases. Hence
it is possible to observe phases of clusters, in equilibrium, that are never observable for bulk
materials, because they are never the forms of lowest free energy.
3.1. Phase diagrams
Because clusters have ranges of coexistence rather than sharp dividing curves, and because the
relative amounts of the different phases vary throughout those ranges, one cannot represent
phase equilibrium of clusters with conventional, two-dimensional phase diagrams. One needs
another dimension. Two novel kinds of phase diagrams for clusters have been introduced
[42]. One uses as a third variable, in addition, e.g., to temperature T and pressure p, a
distribution D, which is just a transformation of the equilibrium constant K to change the
range from 0 to infinity to a more convenient range from −1 to +1. The transformation is
simply D = [K −1]/[K +1], so if the system is all solid, D = −1 and if it is all liquid, D = +1.
At any given pressure, there is a minimum temperature Tf , called the ‘freezing limit’, below
which the free energy has only one minimum with respect to the relevant order parameter,
corresponding to a stable solid. There is another temperature Tm , called the ‘melting limit’,
above which there is only a single minimum in the free energy corresponding to the single
stable liquid phase. Between Tf and Tm , there are two local minima (and there may be more,
as with Ar55 ) that represent the locally stable phases.
A second kind of phase diagram displays the locus of stationary points of the free energy,
as a function of temperature and one or, more usually, two order parameters, e.g. the density
of vacancies on the surface and the corresponding density of vacancies in the core [42]. The
most convenient form actually uses the inverse temperature at the upper end of the axis, rather
than the temperature itself. A region in which the locus slopes downward from the lowest
temperature corresponds to a stable form, and an upward-sloping region corresponds to an
unstable local maximum in the free energy.
4. Molecular clusters
Molecular substances exhibit a rich diversity of structures and dynamics at zero and finite
temperatures because of the anisotropy of their intermolecular potentials. Structural phase
transitions (SPT) occur between high-symmetry, disordered and low-symmetry or ordered
phases. SPT have been observed in small clusters of AF6 molecules (A = S, Se, Te, U, W)
produced in supersonic nucleation by means of electron and neutron diffraction [91–95].
Experimental techniques such as surface-enhanced Raman scattering and infrared
spectroscopy provide realistic information about how the properties of the systems change in
the transition region. Among the static properties of interest are the temperature dependence
of the specific heat and the static dielectric and elastic constants.
If the molecules are somewhat globular (AF6 or CH4 ), and generally in the case of ionic
molecules, it is common to observe orientationally disordered phases at ambient temperatures
known as plastic crystals. Such substances undergo either second-order or weak first-order
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transitions to more orientationally ordered phases as the temperature is lowered [96]. The
behaviour of these systems is rather similar to a paramagnetic material as they belong to
the same universality class [97]. ‘More ordered’ means that molecules tend to align their
molecular axes of symmetry [98–100], the free rotation is hindered and only libration
eventually occurs, while the lattice structure may be unchanged or slightly distorted but
much less than in a purely displacive transition. As a result of transformation the clusters pack
in different structures [94].
The reason for the orientational order–disorder transition is the anisotropy of the
intermolecular interactions [101] which keeps the molecules in specific wells of the potential
energy surface at low temperatures [25]. Their nonspherical shapes assure that the molecules
have favourable mutual orientations of their molecular axes of symmetry. Examples are given
in figure 14, which shows a 59-molecule TeF6 cluster at T = 20 K and in figure 15, showing
methane at 30 K. The octahedral molecules are more aligned than the tetrahedral molecules
that arrange themselves in spherical shells with no alignment down to 5 K, indicating a fluid
structure even for such a low temperature. The question of whether methane clusters are liquid
or solid below the temperature of the phase transitions of bulk methane (20.4 K) is still not
completely answered [102] and needs further investigation. Methane clusters might exhibit a
glass transition, if this reasoning holds.
4.1. Model potential for molecular interactions in clusters
Within the Born–Oppenheimer approximation [103], it is possible to express the classical
Hamiltonian H (p, q) of the system as a function of the nuclear coordinates and momenta,
with the rapid motion of the
averagedout. Then we write: H = K + Epot , with
electrons
m
0.5 mv2i + I ω 2i , where nm is the number of the molecules
K the kinetic energy K = ni=1
in a cluster, I is the tensor of inertia of each molecule with linear velocity v and angular
velocity ω . For rigid octahedra and tetrahedra, I is diagonal in a system fixed at the molecular
centre of mass. The anisotropy of the interaction potential has been accounted for by a
double summation over pair-wise atom–atom interactions. The early works only considered
the Lennard–Jones interactions in AF6 systems [101, 104]. However, both octahedral AF6 and
tetrahedral CH4 molecules exhibit a slight polarization of the electron distribution: in a AF6
molecule the fluorine atoms pull the electron cloud, while in CH4 , electrons move towards the
carbon atom. To account for the polarization a small Coulomb term is included in the pairwise
potential Upw (i, j ) [11, 99, 101, 102, 105]:



12 6 
na
q
q
σ
σ
iα
jβ
αβ
αβ
4αβ 
+

−
(19)
Upw (i, j ) =
αβ
αβ
αβ
r
r
4π
r
0
ij
ij
ij
α,β=1
Upot =
Upw (i, j ),
(20)
i,j =1(i<j )
where na is the number of atoms in a molecule ( na = 7 in AF6 , na = 5 in CH4 ), rij is the
distance between the ith and the j th atom; α and β indicate the type of atom. The following
values for bond length b, σαβ in Å and αβ in eV reproduce the experimental diffraction patterns
in octahedral clusters [93] and in methane clusters [103]:
for TeF6 ,
b = 1.815, σTe,Te = 4.294, σTe,F = 3.553, σF,F = 2.940,
T,T = 0.013 76, T,F = 0.005 31, F,F = 0.002 049;
for SF6 ,
b = 1.561, σS,S = 3.405, σS,F = 3.165, σF,F = 2.943,
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cooling
heating
second cooling
-10
total energy in eV
-12
-14
-16
-18
-20
20
40
60
80
temperature n K
100
120
140
Figure 10. The total energy of a 89 TeF6 cluster as a function of the mean temperature in
isoenergetic MD calculations.
S,S = 0.012 60, S,F = 0.002 72, F,F = 0.002 18;
for CH4 ,
b = 1.094, σC,C = 3.35, σH,H = 2.81, σC,H = 3.08,
CC = 4.257, HH = 0.715, CH = 1.7447.
The effective electronegativity of the fluorine atoms in AF6 and the carbon atoms in CH4 ,
e.g. the charge q < e has been computed by means of the linear combination of atomic orbitals
(LCAO) method and its value depends on the basis used. For example, a plane-wave basis
gives for the charge at a fluorine atom in TeF6 qF = −0.1e, while a Gaussian basis gives
qF = −0.25e. For a CH4 molecule we use qiα = −0.572e for a C-atom [102]. Although the
Coulomb term is a small contribution to the total interaction, it has a well-resolved role in the
cluster phase changes, namely it increases the cluster melting temperature [102, 106]. Being
a long-range interaction it contributes to the complex behaviour of the clusters which may
undergo both first- and second-order phase transitions when the temperature changes.
4.2. Temperature-driven phase transitions
The potential, equation (20), has been used in molecular dynamics and Monte Carlo [107]
simulations of clusters, initially arranged as spherical as possible, figure 1 in [100], at a
constant energy [11, 25, 99] and at a constant temperature [24, 108]. The aims of simulations
have been to:
• compare the heating and the cooling of the clusters in order to find out if the clusters
memorize their initial state (figure 10);
• find the structure of the solid phases;
• monitor the shift of the transition temperature as a function of the cluster size;
• determine the type of transition(s) below freezing.
4.2.1. Molecular dynamics simulations. The details of simulations and various results can
be found in the cited papers and the references therein. Here we would like to comment on
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Figure 11. The heat capacity computed from equation (15).
some results which give insight into the phase transitions. In isoenergetic simulations we
compute the mean temperature from the mean kinetic energy:
2Ekin
T =
,
(21)
gkB
where g is the number of degrees of freedom. In the high-temperature region, just below the
freezing, g = 6(N − 1) as the molecules can freely rotate and vibrate with respect to their
equilibrium configuration (figure 16), while the centre of mass of the molecular cluster is
fixed. A further cooling of the cluster causes orientational phase transitions and some of the
molecular degrees of freedom are constrained. In this case the temperature for the vibrational
and rotational degrees of freedom should be separately computed. Of course the internal
modes of the system are in equilibrium if Trot = Tvib = T .
We have studied the internal energy which is an observable that can be directly measured.
The energy as a function of the mean kinetic energy (mean temperature) of a 89-molecule
TeF6 cluster below its freezing point indicates clearly a phase transition in the region around
80 K and a tiny change of the slope at about 30 K (figure 10). The heat capacity computed in
the vicinity of 30 K from equation (15) has a λ-type shape typical for continuous transitions
(figure 11). If the transition is continuous we could find the critical exponents implementing
the scaling ansatz. As the system size is finite, the correlation length of the interactions is
limited and we obtain ξ(t = 0, H = 0; L) ∼ L instead of ξ(t, H ; ∞) ≈ |t|−ν . Replacing
the infinite correlation length in the singular part of the heat capacity determined for bulk
Cs (t, 0; ∞) ≈ |t|−α with L, we get
Cs (t = 0, H = 0; L) ∼ Lα/ν ,
(22)
where t = (T − Tc )/Tc , and T is close to Tc . We use the phase trajectories of clusters of
different sizes (27, 89, 137, 307) to determine the singular part of the heat capacity and the
plot ln(Cs ) as a function of ln L gives α/ν, which combined with the hyperscaling νd =
2 − α determines α = 0.043 ± 0.028; ν = 0.652 ± 0.009. The exponents published in [109]
for the β-brass are (α = 0.05 ± 0.06, ν = 0.65 ± 0.02) and for Ni (α = 0.04 ± 0.12, ν =
0.64 ± 0.1). The comparison of the three sets of values of the critical exponents indicates that
the molecular clusters at very low temperatures belong to the same class of universality as
paramagnetic materials.
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Figure 12. Total energy versus the mean temperature of TeF6 clusters.
To find a fundamental background of the claim that a continuous (second-order) change
occurs at ∼30 K, we have analysed the symmetry change and ro-vibrational coupling by means
of the point group techniques. Section 4.3 provides details.
The transition at about 80 K has the characteristics of a discontinuous transition: a jump
in the caloric curve and different slopes of cooling and heating branches (hysteresis). Having
in mind that the fluctuations in a continuous transition are finite, we increase the cluster size,
expecting that the jump would evolve into a gap for big clusters. Indeed, the plot of the
total energy per molecule as a function of the mean temperature of selected sizes shows that
(figure 12): (a) the temperature, at which the energy jumps, shifts towards higher temperatures
for larger clusters; (b) the small jump for the 59-molecule cluster becomes a gap for the
307-molecule cluster.
The last but not least confirmation of a discontinuous transition is the phase coexistence.
In atomic clusters, more than two phases may coexist [42]. Molecular dynamic simulations
at constant temperatures show phase coexistence if the thermostat acts as a proper bath.
Figure 13, taken from [24], is a plot of the Lennard–Jones potential versus the cluster
temperature.
Below 30 keV the TeF6 clusters of all sizes adopt orientationally ordered structures
(figure 14). The octahedral molecules are aligned in a contrast to the tetrahedral molecules
which are orientationally disordered down to 5 K as they are at 30 K (figure 15). Whether this
is because of the tetrahedral shape or because of the difference between fluorine and hydrogen
atoms is an open question. There are no comparable studies yet of, for example, CF4 clusters.
Comparing clusters made of molecules with different symmetries and electron charge
distributions, we understand from molecular dynamics simulations, both at constant T and
at constant E, that: (i) if the electrons are attracted by the outer atoms, as it is in the TeF6
molecules, and the ratio of the interaction range and the molecular size is small, then clusters
of small numbers of molecules adopt well-aligned structures at low temperatures; (ii) if the
electrons are more localized around the central atom, as it is in a methane molecule, clusters
of sizes up to several hundred do not form a fcc structure observed in bulk methane [102].
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Figure 13. Coexisting phases in a 89 TeF6 cluster in a bath at T = 88 K.
Figure 14. A 59 TeF6 cluster is orientationally ordered at T = 20 K.
Figure 15. A 59 CH4 cluster forms spherical shells at T = 30 K.
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Figure 16. Below freezing a 59-molecule TeF6 cluster adopts a bcc structure with no orientational
order of the molecular axes.
One of the most important results in our study of structural transformations is induction
of a Coriollis force in clusters transformed orientationally [11]. The reason is the rotation of
the cluster, which results from the conservation of the angular momentum of the system as a
whole: above the phase transition temperature the molecules
are randomly oriented and the
total angular momentum is zero: 0 = L = Lcluster + j Lj . Below the transition
temperature,
the molecules have a specific mutual orientation of their axes of symmetry: j Lj = 0. The
total momentum of the cluster is: Lcluster = Icluster , where Icluster is the tensor of inertia, and
is the angular velocity of the induced rotation. After the structural transition, the clusters
rotate meaning that all quantities are influenced by the Coriollis force including the cluster
shape that deforms after the transformation. The degree of deformation can be measured
within the framework of Raman spectroscopy.
4.2.2. Monte Carlo simulations. Monte Carlo simulations have been used for years to study
various properties of physical models, including phase transitions.
There is a great problem to be overcome in these simulations: in the vicinity of phase
transitions extremely slow relaxation of the system due to low-frequency modes makes it very
difficult to meet conditions of ergodicity and limits the accuracy of the predictions [6, 110].
The slow relaxation due to conservation (hydrodynamic slowing [111]) is observed in both
continuous (critical slowing down) and discontinuous (metastable states) transitions, and
also in supercooling—when the temperature is below a specific energy barrier in the system
(glasses).
The problem of slow relaxation in MC is as difficult as the bottleneck problem in MD.
Due to the bottleneck, some trajectories of the system are restricted to a subdomain of the
phase space [112, 113]. Both MD and MC communities have developed special strategies to
overcome the respective obstacles.
The conventional Metropolis algorithm is not appropriate to sample a rugged configuration
space. Usually techniques called extended (generalized) ensemble Monte Carlo algorithms
are used: jump-walking MC [114], exchange Monte Carlo [115–117], Simulated tempering
[118], parallel tempering, multicanonical Monte Carlo [119] and taboo search [120]. These
techniques not only overcome the slow relaxation times; they can also enable us to compute
multivariance integrals or search for rare events. They can be considered as extended versions
of the old umbrella sampling method invented in [121] for computing the free energy. Each
technique has specific advantages and disadvantages for studying phase transitions [16].
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0.03
105 K
120 K
0.025
2
76.5 K
0.02
P(V)
P(V)
3
78 K
74 K
112 K
0.015
0.01
1
0.005
-11
-11.5
-10.5
-10
0
-9.5
-12
-12.5
-11.5
V [eV]
V [eV]
(a)
(b)
Figure 17. The distributions of the potential energy indicate phase coexistence. The plots are
results of parallel tempering Monte Carlo simulations of TeF6 clusters: (a) liquid- and solidphases coexist at ≈112 K in a 59-molecule cluster; (b) two different solid structures coexist in a
59-molecule cluster at 76.5 K.
74.0 K
72.0 K
0.006
96 K
92 K
90 K
88 K
86 K
84 K
80 K
P(V)
P(V)
0.005
0.008
80.0 K
78.0 K
77.0 K
76.5 K
76.0 K
0.006 75.0 K
0.004
0.004
0.003
0.002
0.002
0.001
0
-12.5
-12
-11
-11.5
-19.5
-19
-18.5
V [eV]
V [eV]
(a)
(b)
-18
-17.5
-17
Figure 18. The distributions obtained with the parallel Monte Carlo simulations are fitted with the
function given with equation (23): (a) 59-molecule cluster; (b) 89-molecule cluster.
The simulated temperature-driven phase transitions of 59-molecule clusters are presented
in figure 17: in the region of 112 K clusters of this size transform from a liquid-like to a
solid-like state; at 76.5 ± 1.5 K another transition occurs which has been identified as a solid–
solid orientational transition, [100]. The bimodal distributions have been successfully fitted
with
y(x) = A0 e−A1 (x−x1 ) + A2 e−A3 (x−x1 ) .
2
2
(23)
Figure 18 shows the distributions for a 59 cluster (a), and an 89 cluster (b), approximated
with equation (23). The coefficients Aj , j = 0, 1, 2, 3, have been determined in a
ξ 2 -procedure.
The important comment here is that the Metropolis Monte Carlo does not reveal the
double-peak distribution of the potential energy. That is why a continuous transition would
be found if this technique would be used.
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Figure 19. A special plane cuts the parent configuration to preserve the packing symmetry.
4.2.3. Genetic algorithm to search for low-energy configurations. We have mentioned that
the lowest-lying state might be buried in numerous metastable phases that are accessible
to transforming clusters. Various strategies have been developed for finding the ground
state of atomic clusters—hill climbing, taboo search, simulated annealing, basin hopping
[81, 122, 123], adiabatic switching [124], [125] and genetic algorithms [126].
For molecular clusters a development of the genetic algorithm proposed for atomic
clusters [126] proved to work well, as we have shown in [73] if the underlying structure
of the clusters is properly taken into account (figure 19). Our algorithm starts with a set of
configurations generated with molecular dynamics or Monte Carlo simulations. Each molecule
x = {x, y, z} is the Cartesian coordinate of
is defined with a pair of coordinates {x, q} = X;
the molecular centre of mass, and q = {q0 , q1 , q2 , q3 } is the quaternion representation of the
molecular orientation. A cluster configuration with N molecules is represented with
2, . . . , X
N }.
1, X
G = {X
The genetic algorithm uses a constant population of n structures {G}. A mapping operator
P : P (G, G ) → G performs the following action upon two parent geometries G and
G to produce a child G : (1) selects parents from the population using the distribution
equation (24); (2) generates special planes cutting the parents through the centre of mass in a
way to account for the packing symmetry (figure 19).
The choice of cutting planes is crucial for the proper working of the algorithm. In other
words, it is very important to find out the packing symmetry of all clusters in the populations.
Different planes have been used: (a) a plane parallel to the already-organized planes that
contain the molecular centres of mass [127]; (b) a plane, which contains a ‘specific’ direction
[73]. Neither method imposes special requirements for the orientational alignment of the
molecules.
After cutting the parents, a child G is assembled from the molecules of G which lie
above the plane and the molecules of G which lie below the plane. If the child generated in
this manner does not contain the correct number of molecules, the plane is translated until the
number of molecules in the child configuration G is correct. Relaxation to the nearest local
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0.30
Conjugate Gradient Method
Genetic algorithm
0.25
Density [a.u.]
0.20
0.15
0.10
0.05
0.00
-1.0
-0.5
0.0
Cos(θ)
0.5
1.0
Figure 20. In 89-molecule clusters the GA finds a ‘new’ orientation configuration (broken line)
with an energy lower than the energy found with a conjugate method (full line).
minimum can be performed with any technique [122]. The selection of parents from {G} is
ruled by the Boltzmann distribution:
p(G) ∝ exp[−E(G)/Kb Tm ],
(24)
where E(G) is the energy of the candidate G, Kb is the Boltzmann constant and Tm is the
mating ‘temperature’, chosen to be roughly equal to the range of energies in {G}. For better
performance, mutations can be applied to some members (µ) of the population.
The orientational order in a cluster can be seen from the mutual orientation of the
molecules: p(θ, ψ), where θ is the polar angle between any two bonds of two different
molecules. For the case of rigid octahedral molecules it is sufficient to monitor one of the
bonds.
The comparison of orientational distributions of molecules in clusters of various sizes
allows us to conclude that: (a) the GA is more efficient in finding the global minimum for
orientationally ordered configurations; (b) there is an important size effect which is found only
with GA up to now: only ‘magic’ numbers of molecules can orient as it is in the full-line
distribution in figure 20: 89, 137, 307; clusters of 27 or 59 molecules do not—see figure 21.
It is possible that the ‘new’ orientational structure of 89- and 137 is a mixture of surface
and volume orders, which are different for specific sizes, while for other sizes the surface and
the volume adopt the same orientational structure. Further research is needed to find out the
cases of surface-induced order for small clusters.
4.3. Continuous transitions in clusters
The second transition at ≈30 in figure 10 has been found to be from a partially-ordered
structure, shown in figure 22, to a completely ordered orientation, as in figure 23 [128, 129].
The λ-type shape of the heat capacity seen in figure 11 indicates a continuous transition
which should be confirmed by other means as well. Let us recall that the onset of orientation
order means a lowering of symmetry. Hence a proper order parameter is needed to restore the
symmetry elements to that of the Hamiltonian which is rotationally invariant with the potential
energy given in equation (20). The symmetry analysis of the cluster’s sites and the molecules
themselves results in the following choice of the order parameter: rotator functions [130, 131]
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0.25
Genetic Method
Conjugate Gradient Method
Density [a.u.]
0.20
0.15
0.10
0.05
0.00
-1.0
-0.5
0.0
0.5
1.0
Cos(θ)
Figure 21. In 59-molecule clusters the GA and the conjugate method find the same configuration
in the ground state.
Figure 22. The orientation distribution of molecules in a 89 TeF6 cluster at 71 K [129].
Figure 23. The orientation distribution at 18 K [129].
defined by the symmetry properties of the molecule αlkλ and of the site αlmτ , [26]:
τl λ (ω) =
l
l
mτ mk
αl Dl (ω)αlkλ ,
k=−l m=−l
(25)
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where Dlmk (ω) are the Wigner matrices, and ω is an arbitrary rotation of a molecule with
respect to the lab system.
¯ τl λ (ω) is zero in the disordered phase and non-zero
The rotator function’s average value in the orientationally ordered phases. This property makes it suitable to be chosen as an order
parameter for the solid–solid transformations as has been suggested by James and Keenan for
solid CD4 (l = 3) [132].
Now we can find the transition temperature under the assumption of a continuous transition
¯ τl λ (ω)
by expanding the intermolecular arrangement in terms of the order parameter, following the Landau approach. We consider various contributions to the total intermolecular
interaction described in our simulations with a sum over pair-wise classical potentials,
equation (20).
The configuration of N molecules in the cluster is given by τl λ (ω(n)) = τl λ (n) where
n = 1, 2, . . . , N , labels each molecule’s centre at its site position rn . The νth atom in the nth
molecule at site rn is labelled with (n, ν) and its position in the lab system is given by
R(n, d ν ) = rn + rnν + u(n)
with u(n) being the displacement of the nth molecule from its equilibrium site position rn ; the
vector rnν [d ν , ν (n)] is determined by the length d ν and the orientation ν (n) in the space
system. For a rigid molecule d ν equals the bond length.
We have found in the simulations that the bcc structure with an orientational disorder
(figure 16) transforms to a monoclinic structure of lower symmetry (figure 2) with only partial
order of the molecular axes (figure 22). The lattice reconstruction induces a translation–
rotation coupling (TR), whose exact form has been derived in [26]. Two other interactions
contribute to the total interaction Upot : translation–translation (TT) and rotation–rotation
(RR):
R
TR
TT
RR
Upot = Upot
+ Upot
+ Upot
+ Upot
.
(26)
RR
with its translation–rotation
We compare the rotation–rotation contribution Upot
TR
counterparts Upot for the interaction of a TeF6 molecule with its nearest neighbours located
at cubic Oh and at monoclinic sites C2h . In a cubic symmetry environment the matrix is
quasi-diagonal, while in C2h surroundings, the matrix is diagonal (only one-dimensional
representations).
We correlate Oh and C2h in two steps: first, we pass from Oh to D4h and then use the
inferred table for D4h to go on to C2h . However, another two-step correlation is also possible:
Oh to D3d to C2h . The different possibilities have been realized in different structures observed
in the experiments with tellurium clusters at low temperatures [93].
In the approximation of only nearest-neighbour interactions, we determine the energy per
molecule: in Oh , the rotation–rotation interaction energy is 4 meV and the translation–rotation
is 0.3 meV. In C2h , these values are 0.18 meV and 0.002 meV, respectively. The conclusion
is that the translation–rotation interaction can be neglected in the ordering of molecules on
monoclinic sites, so that the lower-temperature transition is entirely driven by rotational
ordering and that transition is continuous even in small systems. However on cubic sites,
motions of the molecular centres of mass are large enough and the transition is orientationdisplacive first-order-like, with two local minima in the free energy, at least in small systems.
The highly degenerate state of an Oh molecule in an Oh environment is resolved by a distortion
of the cluster structure if the model requires rigid molecules. Thus we can think of the process
as a Jahn–Teller effect that distorts the cluster from its local Oh symmetry to C2h via D3d or
D4h . The large value of the rotation–rotation interaction leads to a distinct partial ordering of
the molecules. The transition from Oh to C2h is resolved by invoking the representation Eg
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Figure 24. Temperature dependence of the single-molecule susceptibility (multiplied by the
temperature), x(T ). The temperature is in K.
in D4h (or in D3d ), equivalent to splitting of a degenerate mode. Having the interactions and
the total field, we can calculate the free energy F of each phase as a function of the rotator
functions considered as order parameters [131]:
1̂χ0−1 + FT[Jˆ] δµ (q)δµ (−q),
(27)
F = 0.5
q
where FT(Jˆ) and δµ (q) are the Fourier images of the rotator matrix Jˆ and µ (ω), respectively;
1̂ is the 3 × 3 unit matrix; χ0 ≡ xT −1 is the single-molecule orientational susceptibility [133]:
A 2
x = Z −1 g4 1g
(28)
exp(−V R (ω)/T )(µ (ω))2 dω
with Z = exp(−V R /T ) dω, the partition function. The expectation value of x does not
depend on the components of the rotator function .
In the limit of N → ∞, a phase transition occurs at Teq , which is the point where an
eigenvalue of [1̂T +x(T )FT[Jˆ] vanishes. The temperature dependence of x is very weak which
means that the Curie–Weiss law χ0 = x(T )/(T − Tc ) is valid for negative diagonal elements
of Jˆ. The transition point Teq occurs at the largest value of the matrix for the representations
allowed by the symmetry of the system, i.e. Teq = max[−x Jˆ].
The rotation–rotation interaction in the total potential for the orientational ordering in C2h
gives the graphical solution for Teq = max[−x Jˆ] = 27 K for the lower-temperature transition
A
(figure 24). We have implemented the cubic rotator functions l 1g in calculating the crystal
field. This prediction of the transition temperature is in a good agreement with the results
(∼30 K) from the canonical Monte Carlo simulations [134].
To summarize, the near-neighbour intermolecular interactions of a cluster can be expanded
in terms of multicomponent order parameter (rotator) functions in a manner that accounts for
the multi-step phase changes that these clusters exhibit. In particular, we have shown how
translation–rotation and rotation–rotation interactions enter into the Oh –D4h –[D3d ] transition
of TeF6 to yield a phase change with two local free-energy minima for the small system, but
only rotation–rotation interactions enter into the lower-temperature phase change from partial
to complete orientational ordering of the molecules on a monoclinic lattice, a change that,
according to all indications, involves only a single local free-energy minimum.
What have we learned from cluster orientation transitions?
One important result is that if the molecules are rigid then the cluster distorts its shape
when it is cooled from a temperature just above the freezing point (in fact it can be from liquid
or from a point in the band of temperatures of solid–liquid coexistence) down to the transition
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temperature at which the cluster exhibits an orientation transition. The distortion is analogous
to the Jahn–Teller effect in molecules.
5. Summary
Small systems display forms that have properties very similar to those we associate with
specific phases of bulk matter, so much so that we refer to ‘solid’ or ‘liquid’ or ‘glassy’ phases
of atomic and molecular clusters. However there are distinct, important and even dramatic
differences between the behaviour of bulk matter and small systems, with respect to their
phases and especially phase changes. The sheer size of bulk systems gives rise to essentially
discontinuous changes when a bulk system undergoes a first-order phase change, one associated
with two different local minima in the free energy. The two minima have different internal
energies and entropies, and whenever the chemical potentials of the two forms are unequal, the
unfavoured phase is unobservable in a system at equilibrium. When the two phases have equal
chemical potentials, they can exist in what appears to be a static equilibrium. The conditions
for phase equilibrium are described by the Gibbs phase rule, connecting the number of degrees
of freedom, f , with the number of independent components, c and the number of phases in
equilibrium, p: f = c − p + 2 where perhaps the only term in the equation that is not obvious
is the 2.
Small systems are quite different. The phases of a cluster of atoms or molecules exist in
a dynamic equilibrium, like chemical isomers. Away from the conditions of equal chemical
potential, the less favoured phase may exist as a minority species, observable in an equilibrium
system as a minority species. In fact, more than two phases may coexist in equilibrium, in
observable amounts. The only condition for such coexistence is the existence of a local
minimum in the free energy for each of the phase-like forms. When conditions are changed
enough to make a local minimum disappear, then the corresponding phase has no local stability
and must vanish.
This description applies to the small-system counterparts of first-order transitions. There
are also small-system counterparts of second-order transitions, in which there is only a single
minimum in the free energy, a minimum that changes with the conditions. Some transitions
appear to be first-order-like in small systems but become second order as the system grows
and the two minima converge. Other phase changes of small systems behave as second order
for all sizes. Whether there are transitions that appear to be second-order for small systems
and become first-order in large systems is not known.
Because two or more phases may coexist over a range of conditions, representation of the
phase behaviour of clusters and nanoscale particles require more information to be exhibited
in their phase diagrams. Hence extensions of traditional phase diagrams have been introduced
to provide such descriptions.
Another apparent paradox in the phase behaviour of clusters appears in the heat capacities
of some systems, specifically under conditions of constant energy, rather than of constant
temperature. Some clusters exhibit negative heat capacities under the conditions of coexistence
of two phases at constant energy. If the effective temperature of such a system is specified
by the mean kinetic energy per degree of freedom, and one phase has a high density of
states but at a high potential energy, then increasing the total energy can drive population into
the high-potential, low-kinetic energy region and hence into a condition of lower effective
temperature.
The effective order of phase changes of small systems can be related directly to what kind
of symmetry change occurs at the phase change. If the change simply involves dropping from
the symmetry of a larger group to that of a smaller subgroup of that larger group, then the
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phase change can be second-order-like; if the symmetries of the two relevant phases are not
so related, then the transition must be first-order-like.
There are many open questions concerning the phases of small systems, such as, ‘How do
the minima converge with increasing system size, as a transition moves from first order towards
second order?’; ‘What apparent order do magnetic transitions of small clusters exhibit?’;
‘How do the transitions occur from the icosahedral structures exhibited by small Lennard–
Jones clusters to the close-packed structures of their bulk counterparts?’; ‘Can we relate the
relaxation times of systems with different sizes if both surface and volume phase transitions
occur?’. The challenge of such questions will keep this subject alive and interesting for some
time, we are confident.
Acknowledgments
The authors would like to acknowledge the hospitality of the Aspen Center of Physics. The
authors wish to thank R Radev for the parallel tempering Monte Carlo simulations, I Daykov
for the molecular dynamics computations at constant temperature, S Pisov for the genetic
algorithm optimization of cluster structures.
Appendix
The idea of the steepest descent method which is explored here is: let xo be the global
maximum of f (x), i.e. f (xo ) = 0. Then if M is large, we perform
b
2
Mf (x)
Mf (xo )
e−M|f (xo )|(x−xo ) /2 dx,
e
dx ≈ e
(A.1)
a
where the integral is taken in the neighbourhood of xo as f (xo ) is larger than any f (x). The
usefulness of the steepest descent method (known as the Laplace method as well) is to compute
the permutational entropy N ! of a system by
∞
e−x x N dx.
(A.2)
N ! = (N + 1) =
0
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