Polyamorphism
J.L. Yarger and C. A. Angell
Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ
85287.
Contents
I. Introduction, Definitions and Classification
(a) Concepts, terminology and definitions
(b) Polyamorphic Classifications
(c) Special role of Computer Simulations
(d) Organization of the Review
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II. Brief historical account of the subject
(a) Developments before 1990.
(b) Developments since 1990
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III. Elementary level interpretation of polyamorphism
IV. Laboratory and Simulation Studies
(a) Equilibrium Liquid Polyamorphism (level 1)
(i) Laboratory studies
(ii) Computer simulations
(b) Metastable Liquid Polyamorphism (level 2)
(i) laboratory studies
(ii) computer simulations
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(c) Amorphous Solid Polyamorphism (level 3)
(d) Amorphous-Amorphous Transitions (level 4)
(e) Liquid State Critical Transitions (level 5)
IV. Prognosis and Future Studies
V. Concluding Remarks
VI. Acknowledgments
VII. References
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I. Introduction, Definitions and Classification
Polyamorphism, Vitreous Polymorphs and Amorphous Polymorphs are terms that
have been used to describe the existence of two or more distinct amorphous phases of the
same substance, phases that differ only in density. They may be elemental, stoichiometric
compound or mixed compound in character, but the compositions of the two phases must
1
be identical to fall within the scope of this review. To date, these terms have been used to
describe a range of structural changes in amorphous solids, glasses, supercooled liquids,
and equilibrium liquids. In this article we will overview the existing experimental,
computational, and theoretical studies of these phases, their transformations, and related
phenomena within isochemical systems. As an initial part of this work, we will attempt
to clear up some of the ambiguity in definitions and terms used to describe these
phenomena in the recent and current literature. With clear definitions in place, we will
then classify the existing literature results into several categories that pertain to the type
of phase transition behavior exhibited, and to the equilibrium state of the system. It
should be mentioned at the outset that the vast subject of liquid crystals is excluded from
our consideration by the fact that these are not isotropic liquids.
Over the past several years, several concentrated sources of information on our
subject have appeared. There is a proceedings volume1 from the first conference on this
subject area (“Unusual phase transitions...) and there have been both focused reviews
and overview
articles on the basic phenomenon of polyamorphism2-5 liquid-liquid6
transitions (LLT), and amorphous-amorphous transitions5,7 (AAT). However, to date,
there has been no comprehensive review of the literature nor any systematic classification
of the various types of systems that are said to show polyamorphic behavior. This article
is intended to fill that need and to establish a sound framework for the future
development of this growing field.
I(a). Concepts, Terminology, Definitions
Although it is only recently that the idea of polyamorphic forms has been given much
attention (some 400 papers since 1990), the concept of distinct forms of the glassy state
for specific substances is actually quite old. The term amorphous polymorphs was used
by Kreidl and co-workers8 in 1966 to distinguish between the structure of normal SiO2
glass and the structure which SiO2 glass adopts under long exposure to radiation. Shortly
afterwards (1970), a much stricter use of the equivalent term “vitreous polymorphs”
was made by Angell and Sare.9 Although it was only expressed in a footnote to the
discussion of the apparently different structures that could be adopted by water in the
pure state and in the solution state (now considered to be closely related to the high
2
density amorphous HDA structure10), we reproduce their concept here because it
corresponds closely with the definition we would like to recommend in this article. They
wrote, “The question raised here is whether of not it is proper to think in terms of vitreous
polymorphs of a substance, i.e. whether long-range disordered substances can generate
sufficient differences in their short-range order, when prepared under different
conditions, to have distinct and different thermodynamics properties, which are
maintained above their glass transition temperatures (when the amorphous phase is in
equilibrium) for a finite temperature interval”. This can serve as a definition for true
(unambiguous) polyamorphism if we add the caveat that such distinct phases can co-exist
in metastable, or stable equilibrium. Indeed, it turns out that some authority for this idea
had already been given by Landau in his 1969 treatise11 where direct reference was made
to the possibility of two liquid phases existing for the same compound.
The term “polyamorphism” itself was first used by Palatnik12 in 1980 who wrote
“by analogy with the phenomenon of polymorphism, well known for crystalline
materials, we will refer to the phenomenon of the existence of several such varieties of
the amorphous state of the same substance, as polyamorphism.” This term was later reintroduced by G.H. Wolf in 199213, and has now been used repeatedly to deal with the
many new amorphous “transitions” discovered over the past 2 decades.
Here we will define (true) polyamorphism as “that phenomenon in which two
distinct phases of a substance, both isotropic, internally equilibrated, and of the
same composition,
equilibrium.”
can
co-exist
in
stable or
metastable thermodynamic
“Polyamorphs”, then, are the phases that participate in the
polyamorphic equilibrium.
If we were to restrict this article to cases in which the above definition applied
faithfully, then the article could be short indeed. Most of the cases for which the term has
been used in the past, are excluded by it. However, we believe it is important to have a
thermodynamically unambiguous case as a reference for informed discussion of the
remainder. The key ingredient in this definition is the possibility of co-existence in
internal equilibrium, as is true for the familiar crystalline state phenomenon.
By way of contrast, it is important to point out phenomena that are truly excluded
from the discussion. For instance, when sufficient regions of P-T space are explored, it is
3
clear that all amorphous materials will exhibit great structural changes. As an extreme
case, many gases undergo transitions between different structures, the most commonly
cited (textbook) case being that of nitrogen dioxide and its dimer, dinitrogen tetroxide.
However, this change in structure, which occurs continuously over a broad range of P-T
conditions, is chemical equilibrium, not polyamorphism. On the other hand, if NO2 were
to form a liquid phase that could coexist, at particular P, T combinations, with a different
liquid whose structural units were N2O4 molecules, then this would indeed satisfy the
definition. What has been confusing is the circumstance that most cases so far discussed
are somewhere intermediate between these extremes. Liquids and glassformers can
greatly change their structures in response to changes in pressure and temperature, but
these changes are usually also continuous. It is so far only in a very few cases, mostly
glasses, that the structural change can become abrupt like the first-order transition
between crystalline polyamorphs.
The more-or-less abrupt change observed in glass cases may be referred to as the
amorphous-amorphous transition, AAT, a term popularized by Brazhkin7,14 to describe
the continuous transition from different local or intermediate range structural topologies
in amorphous solid materials. Solid polyamorphs13 (or "vitreous polymorphs"9), if
defined so as to be consistent with the strict criteria applied to the liquid state
phenomenon, present us with a problem. As with crystalline polymorphs, alternative
polyamorphic solid forms should, strictly, be capable of co-existence in thermodynamic
equilibrium. However, in the case of amorphous solid systems this is impossible because
the glasses are themselves not in equilibrium, by definition. Nevertheless,
transformations occur between alternative structures in the vitreous state, and they are
receiving a lot of attention. Therefore we need to introduce a classification for the
behavior that has been observed. We base it on how closely the behavior observed in a
particular case approaches that of a thermodynamically ideal polyamorphic transition.
Whether or not many cases of stable state isocompositional liquid-liquid equilibrium will
ever be demonstrated, is for the future to determine.
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I(b). Classification
In the following listing, we provide a basis for separating the diverse phenomena, to
date been ‘lumped together’ under the rubric of polyamorphism, into a hierarchy based
on closeness of approach to the ideal polymorphic behavior. In descending order, we
recognize the following levels in the subject. (The examples mentioned will be fully
discussed in the appropriate parts of Section III).
1. Stable liquids in coexistence (examples: elemental phosphorus [laboratory] and
model liquid systems [computer simulation]).
2. Metastable liquids in co-existence or distinguished by a clearly defined first-order
phase transition (Example: Si, in computer simulations].
3. Metastable liquids that undergo a transition from liquid to glass (only one phase
in equilibrium. (Example: triphenyl phosphate [laboratory, no simulations
available].
4. Amorphous solid phases not formed from the liquid but separated by well-defined
separating surfaces (Example: Amorphous H2O).
5. Amorphous solids that change structure in a more or less abrupt manner, and
show hysteresis without, so far, showing any separating surface (Example: SiO2,
GeO2 and BeF2).
6.
Liquid state anomalies of the “smeared first-order type” that are well
characterized and reversible, but are not polyamorphic (These highly cooperative
processes are included because of the possibility that they could evolve into first
order transitions by application of the right thermodynamic stress on the system.
They have the special advantage of manifesting their cooperative transitions in the
thermodynamically stable liquid state).
I(c). The special role of computer simulation experiments in exploring
polyamorphism.
Unfortunately, the number of examples of laboratory materials that can be classified
as exhibiting level 1 (equilibrium) polyamorphism is, at this time, exceedingly small,
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namely, one, the case of phosphorus. Nature does not encourage this phenomenon, for
reasons that can be understood and which will be discussed in more detail below. Level
2 (metastable liquid) polyamorphism is more common, but is fraught with problems
stemming from the propensity of the low temperature polyamorphs to crystallize,
seemingly a rather general phenomenon.
For each of these “top levels” of polyamorphism, computer simulation studies have
special advantages. At level 1, they are free from restriction to reality, which means that
models of non-existent substances that exhibit equilibrium liquid-liquid coexistence can
be created and characterized. In this way, computer simulations permit the physical
reality of the phenomenon to be demonstrated, as we will review in section II1(a)(ii)
below. At level 2 in the above classification, simulations of real substances can be
conducted on time scales of nanoseconds, and the first-order character of the transition
between liquid phases can be established unambiguously before any crystallization of the
low temperature phase can occur. In larger systems, it should be possible to observe a
true co-existence of the two liquid phases, though this has yet to be done. At the lower
levels of the above hierarchy, the advantages swing increasingly in favor of the
experimental studies because of the larger samples and longer timescales of the
experiments involved.
I(d). Organization of the review.
After (i) a brief account of the way the subject has developed, empirically, since
early times, and (ii) an elementary level account of how the phenomenon can be related
back to deviations from ideal gas behavior in terms of specific attractive repulsive
interactions between the systems’ particles,
we will (iii) review the accumulated
knowledge in the field according to the order of phenomenological rigor listed in the
previous section, giving a separate section to each level. In each of these sections, we will
review first the laboratory experiments that define the level in question, and will then
describe the corresponding computer simulation studies that help to establish the
appropriateness of the level in question.
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II. Brief historical account.
(a) Developments before 1990.
To our knowledge, the possible existence of equilibrium between liquid phases of the
same composition was first recognized in 1967 through the behavior of a simple
theoretical model of the “two-species” liquid variety, proposed by Rapoport15. Two
species models themselves go back to Roentgen’s early attempt to explain the properties
of water16, but in Roentgen’s model and all others of its type17, the different “species”
were assumed to mix together in an ideal (random) fashion. Rapoport15,18 introduced a
non-ideal form of mixing into the model in his attempt to account for the existence of
melting point maxima in certain substances. Rapoport’s contribution was to assume that
the species mixed according to the regular solution model of binary solutions (which is
well known to have a critical point below which two phases of different composition coexist). The model gave a quite reasonable account of the melting point behavior but its
further implication that, at some extreme, an isocompositional liquid-liquid phase
transition would occur, was regarded as an embarrassment, and discounted.
While the “species” of Rapoport's model, like those of most other two-species
models, have not been identified, the cooperative character of the model, and the phase
transition predicted by its simple equations, have recently gained considerable attention.6
Different versions of it have appeared, usually without awareness of Rapoport’s
contribution. In 1979, for instance, Aptekar19 published a paper predicting liquid-liquid
equilibria in silicon and germanium, based on a model with the same mathematical form,
in which the species were silicon (or germanium) atoms which somehow could have
either metallic or semiconducting character and could nevertheless mix together, even if
non-ideally, in different proportions above and below the phase transition. An equally
unphysical "two liquids" version of this model, in which two distinct liquids are assumed
to occupy the same physical space in different volume fractions depending on T and P,
was later (in the 90’s) to be used by Ponyatovsky and co-workers20,21 to predict liquidliquid transitions in a variety of systems, again without recognition of any of the earlier
versions. More physical versions of this type of model will be discussed later. Separately,
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Mitus et al, in 198122,23, discussed liquid-liquid transitions on the basis of the properties
of a Potts model.
Probably the earliest demonstration in the laboratory of the solid state generation of
an amorphous material quite different in properties from the known liquid phase, was
made for the case of germanium.
In 1970, DeNeufville and Turnbull24 observed
spherical droplets of a non-crystalline semiconducting germanium phase to separate out
from a reheated sample of a reduced form of germania glass in which the O:Si ratio was
only 1.9:1. This was not a polyamorphic phase change because the new phase, pure Ge,
was quite different in composition from the phase from which it formed. The point of
importance to the present review is that the phase of germanium that formed was a
tetrahedral semiconducting phase rather the higher coordination number, weak metal
phase that is the liquid state of germanium.25
The temperature at which this semiconducting Ge amorph formed was just above the
glass transition temperature of the reduced glass.
The droplets were observed by
scanning electron microscopy. A consequence of the glassy Ge precipitation was the
disappearance of the glass temperature endotherm on second scanning.24 This is not too
surprising since each of the newly formed amorphous phases (Ge and stoichiometric
GeO2) is well below its glass temperature at the Tg of the original glass. GeO2 has a Tg of
???ºC, and amorphous Ge only recrystallizes at 450ºC. The amorphous Ge indeed
crystallized….??…
DeNeufville and Turnbull were aware of the confusing thermodynamics of
germanium liquid and amorphous solid, and it seems they expected that a liquid-toamorphous transition would occur in the pure germanium phase if supercooling of the
high temperature phase could occur without crystallization or, conversely, if the low
temperature phase could be heated without crystallization. This transition was later
formally predicted by Spaepen and Turnbull26, simultaneously with Bagley and Chen27
using a Gibbs energy construction. The temperature of the transition was predicted to be
969K, somewhat below the value subsequently assigned by experiment, but above the
value obtained by simulation studies to be discussed. In fact, it was close to the value,
880K, predicted by Aptekar19. The phase change was thought by Turnbull et al.28 to be
"melting" of the amorphous semiconducting solid, though it has recently been pointed
8
out29 that even the crystalline form of silicon has a viscous liquid-like value of the
diffusivity at the putative phase transition temperature.
In other words, the silicon
transition would be seen to be a liquid-liquid transition (as Aptekar had suggested) if it
could be observed on normal time scales.. More recently the efforts to observe the phase
transition have been made from the liquid state by various supercooling techniques,
particularly that of levitation melting.30,31 It has also become a subject for intense
computer simulation efforts, which will be described further below.
The field received a major impetus in 1984 when Mishima et al.32 demonstrated that
ice collapsed to a dense amorphous phase under cold (196 K) compression. The
amorphization pressure was at 1.6 GPa. Of special interest to our subject was the
additional observation that, on decompression at an appropriate temperature, the dense
phase suddenly expanded to a less dense but still amorphous phase – apparently a direct
glass-glass transition.
This started a wave of studies on amorphization, and much
speculation on glass-glass transitions, the most notable effort being that of Mishima33 in
which the water was cycled back and forth many times between the two states.
(b) Key developments since 1990
A particular reason for the current level of excitement about the phenomenon derives
from the paper by Poole et al, in 1992.34 Using molecular dynamics simulations of water
in the Rahman-Stillinger ST2 potential, these authors demonstrated that, when the long
range part of the interaction was computed using the "reaction field" method, that the
glass-glass transition observed by Mishima et al probably had a liquid state equivalent,
though it could not be observed directly by the simulation. What was observed was
evidence of a van der Waals type P vs V relation with a loop in the supercooled liquid
regime. Such a loop should become a liquid-liquid equilibrium domain in a larger system.
The loop developed as the temperature fell below a critical temperature at moderately
high pressures. The notion that there could be a second critical point in an isotropic
liquid phase was novel because Aptekar's work on silicon was unknown. Furthermore,
water is intrinsically a more interesting case.
“Liquid-liquid” transitions were then directly observed in two laboratory systems
during cooling of the high temperature liquid phase. The first case35 involved a mixed
oxide system related to garnet, Y3Al5O12, and the second case concerned a pure
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molecular liquid, tri-phenyl phosphite36-38 In each case there has been much controversy
due to the presence of crystalline material in the low temperature phase and, later, of the
realization that the low temperature phase was actually a glass (i.e. was below its Tg) as it
was formed. The ambiguity created in the field by the latter observations has fortunately
been dispelled by the observation of liquid-liquid equilibria in the stable fluid state (the
case of elemental phosphorus39), and by the development of simple theoretical models in
which the coexistence of two stable liquid phases can be demonstrated, and characterized
in detail40,41. Details in each case will be given in the appropriate sections below.
The rate at which interest in this field has increased can be gauged from Fig. 1, which
is a histogram of papers addressing the phenomenon over the past two decades.
Figure 1. Time evolution of interest in the polyamorphism field.
III.
Phase diagrams, and an elementary level interpretation of
polyamorphism .
Before we proceed to discuss the details of these papers according to the scheme
given in section Ic, it is appropriate to give brief attention to the phase relations that can
exist. In Fig. 1(a) and (b) we reproduce the first such phase diagram, given in the work of
Aptekar19 and the generalized version used by Poole et al.42 In the case of silicon, the
Clapeyron slope dT/dP for the liquid-liquid equilibrium is negative because of the open
network character of the low temperature phase, while in the second case the slope is
shown positive. Instances of each type have now been identified in laboratory studies, as
will be discussed in section III below. `
10
Figure 1. (a) Phase diagram for the element silicon showing …. (from ref. )
11
(b) generalized phase diagram for the non-crystalline phases of a single
component system, showing both a gas-liquid critical point and the second
(liquid-liquid) critical point. (reproduced from ref. X, with permission of the
AAAS.)
Both cases shown in Fig.1 exhibit a critical point that terminates the liquid-liquid
coexistence line. In the case of silicon it is at negative pressure and impossible to verify.
In the more interesting case of water, which is expected to have the same general form,
such a critical point has been suggested to exist at positive pressure on the basis of both
simulation43 and laboratory44 studies. The positive Clapeyron slope, envisaged in Fig. 2,
is the type of behavior shown by the model potential of Ref. X, see section III.a.(ii), and
possibly manifested in practice by TPP (see section III.b. (i). The possible existence of
such novel critical points has provided one of the major stimuli to the field, and is
responsible for much of it’s present level of activity.
To understand briefly how such behavior can arise in the condensed states of a
system of simple particles, we may generalize the ideas behind the Poole model for
polyamorphism in water. This is a modification of the van der Waals model for the gasto-liquid transition. Instead of invoking just the short range repulsion (on contact) and the
long range attraction of the Van der Waals model, the Poole model introduces a second
length scale for attractions which lowers the system energy maximally at a density lying
between the close packed density and the gas density.
In the case of water, this intermediate density arises because of hydrogen bonds
between water molecules which cause water molecules to find their lowest energies when
in four coordination, i.e. at a much lower density than the 10-12 coordination associated
with close packing. Of course there are many other types of bonding that can lead to
energy minima at densities lower than close packing, and any of these could provide a
mechanism to drive a polyamorphic equilibrium. All of these open structure-promoting
interactions can be overcome by the application of pressure so that eventually at
increased pressures the system will be most stable in close packing. At intermediate
pressures, the two types of packing may be able to coexist. Whether such coexistence will
actually be manifested, i.e. whether a polyamorphic equilibrium will be seen in practice,
depends on cooperative effects that prevent the simple mixing of low density and high
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density structural elements. The interactions which cause the initial deviation from ideal
gas behavior and lead to the two phase liquid solid coexistence are illustrated in Fig. 2 (a)
and the manner in which the simple two phase coexistence range of the normal liquid is
split into two separate coexistence domains by the presence of the low coordination
number driving potential, is illustrated in Fig 2(b). The mathematical formulation of these
concepts and the resulting phase diagrams for different parameterizations of the model,
are given in ref. X (Poole et al).
The case of water has the low density phase a homogeneous hydrogen-bonded
network, which is somewhat unusual. A much more common way of arranging for an
intermediate density phase would for it to be a molecular liquid, while the high density
phase would be a, continuously bonded polymeric phase, obtained by rearranging the
electrons around the constituent atoms. In the polymeric phase the free volume of the
molecular liquid has been eliminated automatically increasing the density. This is
essentially the condition met by the only case of polyamorphism to satisfy our strictest
criterion, namely the elemental phosphorus case, in which both gas and low density
liquid consist of P4 molecules, while the high density phase, generated by sufficient
increase of pressure, is polymeric (of so-far undetermined structure).
It might be expected that there should be many cases in which the rearrangement
of electrons between atoms should permit this type of polyamorphism, and it is a little
surprising that more cases have not yet been found. Some indications of where the
problem lies may be seen in the failure, some years ago, of a conscious attempt to
produce a high density phase by rebonding of the atoms in the molecular substance NSF.
It is perhaps reassuring to find that NSF is extremely stable under pressure, withstanding
in our case some 25 GPA without change in structure. Chemically, (as opposed to
comically) it means that the PV work of compression to, say, 50% of the molar volume of
NSF, a volume change of 20 ml, does not produce an energy change sufficient to
compensate for the rearrangement of the bonding electrons in NSF to any other
configuration. But there will surely be other cases in which it is sufficient, and then
polyamorphism will follow unless crystallization intercedes
13
For further thermodynamic background material on conditions for, and the origin
of, liquid-liquid phase equilibria, the reader is referred to refs. X,Y and Z.
(a)
(b)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Figure 2. Illustration of the relation between normal van der Waals interpretation
of the coexistence of liquid and gas phases, and the modification necessary to
understand the existence of a second critical point. (based on lectures of Peter Poole,
private comm.)
Jeff: I need to put into this Figure, in the T-V plots, a line for the glass
transition temperature and a comment on the effect of pressure below the
glass temperature in promoting a glass/glass conversion.
III. Laboratory and Simulation Studies
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In the body of this review, we will deal with the literature on polyamorphism in
separate sections, according to the hierarchy in ambiguity described above in section Ib,
and starting with the least ambiguous. In each section, we will first discuss the laboratory
phenomena that have been reported and then discuss the parallel studies of the same types
of system that have been carried out by computer simulation experiments.
III(a). Equilibrium Liquid Polyamorphism (Level 1)
In this level of the classification, there is but a single unambiguous laboratory
example, (liquid crystals having been excluded from consideration because either one, or
both, phases are anisotropic). However there are several rather detailed theoretical
models to consider, along with their computer-simulated manifestations. Because of the
desirability of tying this subject to real world phenomena, we will consider the laboratory
case first, and then will describe the more complete accounts of isocompositional liquids
in equilibrium that have recently been provided by model system computer simulation
studies.
(i) Laboratory Systems.
Phosphorus
(ii) Computer Simulations and Theoretical Modeling
In this section we consider
(a) simulations of the laboratory case of phosphorus, which can only be executed
using ab initio computational methods and which in consequence are extremely
expensive, in computation time, to simulate. As a result, information is only available on
small samples studied for short times.
(b) simulations of model systems with simple potentials which are easy to study
in large samples and which therefore are now well elaborated. The systems so far
investigated in detail are systems with soft repulsions which effectively provide the
second length scale necessary for the liquid-liquid co-existence to be manifested. Because
of the greater detail obtainable in the latter cases, we find it most fruitful to discuss these
first.
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Model systems
Systems with step, multistep and ramp potentials
Reflecting on the “structure-modified van der Waals model” introduced in section 1, we
need to have some mechanism by which the volume change on passing from the gas to
close-packed system can be interrupted at a structurally stable intermediate density. The
simplest model approach to obtaining this result is to have repulsion component of the
interaction potential contain a step, such that a second shorter distance repulsion replaces
the initial repulsion above a given pressure. This “softened core” idea was used in
theoretical studies by Stell some time ago, though not with polyamorphism in mind. Jagla
reintroduced it and showed that it generated water-like features in the behavior. Then it
was realized that a two step repulsion would yield the same result and so, by adding more
steps, the idea of a repulsive ramp modle arose. This simple idea is all that is . ..
.
which can only be studied in very small samples, with limited output have been
executed by several groups (XYZ) but demand ab initio treatments also those of
model systems which are simple to simulate, and which therefore have yielded
Phosphorus
Several first-principles molecular dynamics simulations on liquid phosphorus at high
temperature and pressure have been carried out in order to investigate the observed
liquid-liquid phase transition52-55. These computer simulations all show that the
transition is caused by the breakup of P4 tetrahedral molecules to form a polymeric liquid.
The major electronic rearrangements in phosphorus involved in passing from the
tetrahedral molecular fluid phase to the rebounded polymeric phase, requires this system
be simulated by ab initio methods. Currently, these simulations are very computer-timeintensive and no more than 100 atoms can be simulated for physically useful times. This
size of simulation is typically considered too small to reveal accurate phase transition
behavior clearly. However, it has been used very successfully to show the different
molecular and network motifs adopted be the various liquid phosphorus phases.
16