Waterlike thermodynamic anomalies in a repulsive-step potential system
ABSTRACT We report a computer-simulation study of the equilibrium phase diagram of a three-dimensional system of particles with a repulsive step potential. The phase diagram is obtained using free-energy calculations. At low temperatures, we observe a number of distinct crystal phases. We show that at certain values of the potential parameters the system exhibits the water-like thermodynamic anomalies: density anomaly and diffusion anomaly. The anomalies disappear with increasing the repulsive step width: their locations move to the region inside the crystalline phase. Comment: 6 pages, 5 figures
- SourceAvailable from: arxiv.org[show abstract] [hide abstract]
ABSTRACT: The transitions in disordered substances are discussed briefly: liquid--liquid phase transitions, liquid--glass transition and the transformations of one amorphous form to another amorphous form of the same substances. A description of these transitions in terms of many--particle conditional distribution functions is proposed. The concept of a hidden long range order is proposed, which is connected with the broken symmetry of higher order distribution functions. The appearance of frustration in simple supercooled Lennard--Jones liquid is demonstrated.01/2005;
arXiv:0812.4922v1 [cond-mat.soft] 29 Dec 2008
Waterlike thermodynamic anomalies in a repulsive-step potential system
N. V. Gribova
Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142190,
Moscow Region, Russia, and Frankfurt Institute for Advanced Studies,
J.W. Goethe-Universit¨ at, Ruth-Moufang-Str. 1, D-60438, Frankfurt am Main, Germany
Yu.D. Fomin and V. N. Ryzhov
Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia
FOM Institute for Atomic and Molecular Physics, Amsterdam,
The Netherlands and Dept. of Chemistry, Univ. of Cambridge, Cambridge, UK
(Dated: December 31, 2008)
We report a computer-simulation study of the equilibrium phase diagram of a three-dimensional
system of particles with a repulsive step potential. The phase diagram is obtained using free-energy
calculations. At low temperatures, we observe a number of distinct crystal phases. We show that
at certain values of the potential parameters the system exhibits the water-like thermodynamic
anomalies: density anomaly and diffusion anomaly. The anomalies disappear with increasing the
repulsive step width: their locations move to the region inside the crystalline phase.
PACS numbers: 61.20.Gy, 61.20.Ne, 64.60.Kw
Some liquids (for example, water, silica, silicon, car-
bon, and phosphorus) show anomalous behavior in the
vicinity of their freezing lines [1, 2, 3, 4, 5, 6, 7]. The
water phase diagrams have regions where a thermal ex-
pansion coefficient is negative (density anomaly), a self-
diffusivity increases upon pressuring (diffusion anomaly),
and the structural order of the system decreases upon
compression (structural anomaly) [6, 7].
where these anomalies take place form nested domains in
the density-temperature  (or pressure-temperature )
planes: the density anomaly region is inside the diffusion
anomaly domain, and both of these anomalous regions
are inside the broader structurally anomalous region. In
the case of water these anomalies are usually related to
the anisotropy of the intermolecular potential. However,
isotropic potentials are also able to produce density and
diffusion anomalies. It is interesting that such potentials
may be purely repulsive and can be considered as the sim-
plest models for the water-type anomalies. It has been
shown that water-like structural, thermodynamic, and
dynamic anomalies can be generated in systems where
particles interact via isotropic potentials with two char-
acteristic length scales, with shorter range correspond-
ing to a hard-corelike steep repulsion and longer range
representing softer repulsion - potentials in which two
preferable interparticle distances compete depending on
the thermodynamic conditions of the system [8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In
these studies was found that there is an exception case
– the repulsive-step potential – in which no anomalies
were reported yet . In this sense, it is very interest-
ing to mention the recent study  of evolution of the
behavior of the water-like anomalies in the system with a
tunable potential ranging from a ramp potential, which
has all mentioned above anomalies to the repulsive-step
potential, where no anomalies were found so far . In
 it was shown that potentials in which two preferred
distances are present always exhibit water-like anoma-
lies, but sometimes they are in an inaccessible region, as
inside a crystal phase. This is the case for the repulsive-
step potential studied in Ref. .
However, recently it was shown that water-like anoma-
lies can exist in the systems of particles interacting
through the repulsive step potential  for some values
of the potential parameters.
This potential was introduced in the early work of
Hemmer and Stell [8, 9] in order to describe isostructural
phase transitions in materials such as Ce or Cs and is the
simplest example of a repulsive intermolecular potential
that has a region of negative curvature in the repulsive
part, a feature that is known to be present in the inter-
atomic potentials of some pure metallic systems, metal-
lic mixtures, electrolytes and colloidal systems. Systems
of particles interacting through such pair potentials can
possess a rich variety of phase transitions and thermo-
dynamic anomalies, including liquid-liquid phase transi-
tions [30, 31, 32], and isostructural transitions in the solid
region [33, 34, 35].
In this sense, the purpose of this paper is straightfor-
ward. We will show that the water-like anomalies do
exist for the repulsive step potential, but with increasing
the width of the repulsive step they move to the inacces-
sible region inside the crystal phase. The width of the
repulsive step of the potential considered in Refs. [27, 28]
corresponds exactly to this limiting case.
The repulsive step potential has the form:
∞, r ≤ d
ε, d < r ≤ σ
0, r > σ
where d is the diameter of the hard core, σ is the width
of the repulsive step, and ε its height.
temperature limit˜T ≡ kBT/ε << 1 the system reduces
to a hard-sphere systems with hard-sphere diameter σ,
whilst in the limit˜T >> 1 the system reduces to a hard-
sphere model with a hard-sphere diameter d. For this rea-
son, melting at high and low temperatures follows simply
from the hard-sphere melting curve P = cT/σ′3, where
c ≈ 12 and σ′is the relevant hard-sphere diameter (σ and
d, respectively). A changeover from the low-T to high-
T melting behavior should occur for˜T = O(1). The
precise form of the phase diagram depends on the ratio
s ≡ σ/d. For large enough values of s one should expect
to observe in the resulting melting curve a maximum that
should disappear as s → 1 . The phase behavior in
the crossover region may be very complex, as shown in
In our simulations we have used a smoothed version
of the repulsive step potential (Eq. (1)), which has the
In the low-
2ε(1 − tanh(k0(r − σs))) (2)
where n = 14,k0= 10. We have considered the follow-
ing values of σs: σs= 1.15,1.35,1.55,1.8. In Fig. 1 the
repulsive step potential is shown along with its smooth
version which was used in our Monte-Carlo (MC) and
molecular dynamics (MD) simulations.
In the remainder of this paper we use the dimensionless
quantities: ˜ r ≡ r/d,˜P ≡ Pd3/ε,˜V ≡ V/Nd3≡ 1/˜ ρ. As
we will only use these reduced variables, we omit the
In  the phase diagrams of the repulsive step poten-
tial system were reported for σs= 1.15,1.35,1.55. In the
present article we also calculate the phase diagram of the
system for σs= 1.8. To determine the phase diagram at
non-zero temperature, we performed constant-NVT MD
simulations combined with free-energy calculations. In
all cases, periodic boundary conditions were used. The
number of particles varied between 250, 500 and 864. No
system-size dependence of the results was observed. The
system was equilibrated for 5×106MD time steps. Data
were subsequently collected during 3 × 106δt where the
time step δt = 5 × 10−5.
In order to map out the phase diagram of the system,
we computed its Helmholtz free energy using the thermo-
dynamic integration: the free energy of the liquid phase
was computed via thermodynamic integration from the
dilute gas limit , and the free energy of the solid phase
was computed by thermodynamic integration to an Ein-
stein crystal [36, 37].In the MC simulations of solid
FIG. 1: A repulsive step potential consisting of a hard core
plus a finite shoulder (dashed line) (ε = 1,σ = 1.5) along
with the continuous version of the potential (2) used in the
simulations (ε = 1,σs = 1.55).
phases, data were collected during 5 × 104cycles after
equilibration. To improve the statistics (and to check for
internal consistency) the free energy of the solid was com-
puted at many dozens of different state-points and fitted
to multinomial function. The fitting function we used
is ap,qTpVq, where T and V = 1/ρ are the temperature
and specific volume and powers p and q are connected
through p + q = N. The value N we used for the most
of calculations is 5. For the low-density FCC phase N
was taken equal to 4, since we had less data points. The
transition points were determined by a double-tangent
The region where we have expected thermodynamic
anomalies is situated close to the glassy phase, that
means that proper sampling of the phase space can be
problematic. To overcome this problem we have used
the parallel tempering method . Instead of simulat-
ing one system we consider n systems, each running in
the NVT ensemble at a different temperature. Systems
at high temperatures go easily over potential barriers and
systems at low temperatures sample the local free energy
minima. The idea of parallel tempering is to put over
MD the MC scheme of accepting/rejecting a move, but
in our case it would be accepting or rejecting a swap of
temperatures between different configurations after each
full (equilibration together with sampling) MD run. If
the low and high temperatures are far apart, the prob-
ability to exchange the configurations is quite low, that
is why we use a range of ’intermediate’ temperatures be-
tween them with a small temperature step. So after run-
ning the whole parallel tempering scheme we get a row
of systems with subsequent temperatures and each of the
systems was sampled several times. For our problem we
usually used 8 temperatures and tried to swap them 40
times. This simulation took almost 24 hours running it
on 8 processors in parallel at the Joint Supercomputing
Center of Russian Academy of Sciences.
Fig. 2 shows the phase diagrams that we obtain from
the free-energy calculations for four different values of σs
(we included the phase diagrams for σs= 1.15,1.35,1.55
for completeness). Fig. 2(a) shows the phase diagram
of the system with σs = 1.15.
the system with σs = 1.15 there are no maxima in the
melting curve. In a soft-sphere system described by the
potential 1/r14a face-centered cubic crystal structure has
been reported .However, the addition of a small
repulsive step leads to the appearance of the FCC-BCC
transition shown in Figs. 2(a).
Fig. 2(b) shows the phase diagram of the system with
σs = 1.35 in the ρ − T plane. There is a clear maxi-
mum in the melting curve at low densities. The phase
diagram consists in two isostructural FCC parts corre-
sponding to close packing of the small and large spheres
separated by a sequence of structural phase transitions.
This phase diagram was discussed in detail in our pre-
vious publication . It is important to mention that
there is a region of the phase diagram where we have
not found any stable crystal phase. We think that no
crystal structure is stable in this density range because
of frustration as it was discussed in . In  it was
shown that the glass transition occurs in this region with
Tg = 0.079 at ρ = 0.53. The apparent glass-transition
temperature is above the melting point of the low-density
FCC and FCO phases (see Fig. 2(b)). This suggests that
the “glassy” phase that we observe is thermodynamically
stable.This is rather unusual for one-component liq-
uids. In simulations, glassy behavior is usually observed
in metastable mixtures, where crystal nucleation is ki-
netically suppressed. One could argue that, in the glassy
region, the present system behaves like a “quasi-binary”
mixture of spheres with diameters d and σsand that the
freezing-point depression is analogous to that expected in
a binary system with a eutectic point: there are some val-
ues of the diameter ratio such that crystalline structures
are strongly unfavorable and the glassy phase is stable
even for very low temperatures. The glassy behavior in
the reentrant liquid disappears at higher temperatures.
One can expect the frustration to be even more pro-
nounced if we increase the step size. In Fig. 2(c) we show
the phase diagram of the system with the potential (2)
for σs= 1.55. One can see that the system also demon-
strates low- and high density FCC phases separated by
FCC to BCC transitions and the amorphous gap which is
much more wider than for σs= 1.35. We did not find any
crystal structure between these isostuctural phases in our
study. The glass transition temperature is Tg= 0.11091
at ρ = 0.5. One can see that the glassification tempera-
ture becomes higher. Given the lack of crystal structure
between crystalline phases and the increase of the glass
transition temperature one can assume that the frustra-
tion effects become higher with the increase of the step
One can see that for
FIG. 2: Phase diagram of the system of particles interact-
ing through the potential (2) with σs = 1.15,1.35,1.55,1.8
in ρ − T plane. In Figs.2 (b-d) it is shown the behavior of
the diffusivity as a function density. In Figs.2 (b-c) we also
represent the locations of the minima on the isochores (see
FIG. 3: Diffusion anomaly for σ = 1.35,1.55,1.8.
The phase diagram of the system with σs = 1.8 is
shown in Fig. 2(d). One can see that inside the disordered
gap in the phase diagram there appears the crystalline
phase with diamond structure, however, this phase does
not extend over the whole disordered region in the phase
As it was mentioned above, one can expect the appear-
ance of thermodynamic anomalies in the vicinity of the
anomalous points on the phase diagrams of the repulsive-
step potential system. To check this point, we calculated
the isochores and diffusivity for different values of σs.
In this sense, it is not surprising that there are no ther-
modynamic anomalies for σs = 1.15. It is known, that
FIG. 4: Density anomaly for σ = 1.35,1.55,1.8.
for normal liquids the diffusivity decreases monotonically
with increasing density at constant temperature. In con-
trast, we have observed in our model, that for a certain
values of the potential parameters, for the densities in the
vicinity and above the maximum of the melting curve, the
diffusivity curve has a bend (see Figs. 3(b-c)). In Figs. 3
the behavior of the diffusivity is shown in more detail for
different values of σs. One can see that with increasing
the width of the repulsive step σsthe anomaly is becom-
ing less pronounced and disappears for σs = 1.8. It is
interesting to note that this value of σs corresponds to
width of the repulsive step considered in [27, 28] where
no anomalies were found for the repulsive step potential.
FIG. 5: Re-scaled part of the phase diagram with locations
of the minima on the isochores.
The region where the diffusivity anomaly exists almost
coincides with a region in which the isochore has a min-
imum instead of growing monotonically (see Figs. 2(b-
c), where the locations of the minima of isochores are
shown, and Figs. 4). Using the thermodynamic relation
(∂P/∂T)V= αP/KT, where αP is a thermal expansion
coefficient and KT is the isothermal compressibility and
taking into account that KT is always positive and finite
for systems in equilibrium not at a critical point, one
can conclude that there is a range of densities and tem-
peratures where the thermal expansion coefficient αP is
To elucidate the behavior of the anomalies with in-
creasing the width of the repulsive step of the potential,
we re-scaled the parts of the phase diagrams correspond-
ing to the first maximum on the melting curve (see Fig. 2)
by multiplying the density by the σ3
temperature by Tmax, where Tmax is the temperature
corresponding to the maximum. In accordance with the
qualitative picture depicted above the re-scaled parts of
the phase diagrams should coincide. As it is seen in Fig. 5
this is approximately the case for σs = 1.35,1.55,1.8.
The discrepancies between the curves appear to be be-
cause we consider the smoothed version of the repulsive
step potential. In Fig. 5 we also show the locations of
the minima of the isochores for σs= 1.35,1.55. One can
see that with increasing the width of the repulsive step
the line of the minima moves to the melting line and be-
comes invisible in the metastable region. It should be
noticed that this scenario is similar to the one depicted
in Ref. 28.
At low densities, we have effectively a liquid consisting
of spheres with diameter σs, at high densities, the liquid
consists of spheres with diameter d. In the “anomalous
region” inbetween, our system appears as a mixture of
both sorts of particles, and one can expect that in this
region structural order should decrease for intermediate
values of σs. In this case, the entropy of the system
should increase with increasing density, and, due to the
sand dividing the
one gets the anomalous behavior in this region. This fur-
ther demonstrates that our model shows a quasi-binary
In summary, we have performed the extensive com-
puter simulations of the phase behavior of systems de-
scribed by the soft, purely repulsive step potential (2) in
three dimensions. We find a surprisingly complex phase
behavior. We argue that the evolution of the phase di-
agram may be qualitatively understood by considering
this one-component system as a quasi-binary mixture of
large and small spheres.Interestingly, the phase dia-
gram includes two crystalline FCC domains separated
by a sequence of the structural phase transitions and a
reentrant liquid that becomes amorphous at low temper-
atures. The water-like anomalies (density anomaly and
diffusion anomaly) were found in the reentrant liquid for
σs = 1.35,1.55. The anomalies disappear with increas-
ing the repulsive step width: their locations move to the
region inside the crystalline phase in the vicinity of the
maximum on the melting line.
We thank S. M. Stishov and V. V. Brazhkin for stim-
ulating discussions. N.G. thanks A. Arnold for introduc-
tion to parallel computing and valuable remarks. N.G.
and Y.F. thank the Joint Supercomputing Center of Rus-
sian Academy of Sciences for computational power. The
work was supported in part by the Russian Foundation
for Basic Research (Grant No 08-02-00781) and the Fund
of the President of Russian Federation for Support of
Young Scientists (MK-2905.2007.2).
 P. Debenedetti, J. Phys.: Condens. Matter 15, R1669
 S. V. Buldyrev, G. Franzese, N. Giovambattista, G.
Malescio, M. R. Sadr-Lahijany, A. Scala, A. Skibinsky,
and H. E. Stanley, Physica A 304, 23 (2002).
 C. A. Angell, Annu. Rev. Phys. Chem. 55, 559 (2004).
 P. G. Debenedetti, Metastable Liquids: Concepts and
Principles (Princeton University Press, Princeton, 1998).
 V. V. Brazhkin. S. V. Buldyrev, V. N. Ryzhov, and H. E.
Stanley [eds], New Kinds of Phase Transitions: Trans-
formations in Disordered Substances [Proc. NATO Ad-
vanced Research Workshop, Volga River] (Kluwer, Dor-
 J. R. Errington and P. G. Debenedetti, Nature (London)
409, 18 (2001).
 P.A. Netz, F.V. Starr, H.E. Stanley, and M.C. Barbosa,
J. Chem. Phys. 115, 318 (2001).
 P. C. Hemmer and G. Stell, Phys. Rev. Lett. 24,
 G. Stell and P. C. Hemmer, J. Chem. Phys. 56, 4274
 G. Malescio, J. Phys.: Condens. Matter 19, 07310 (2007).
 E.Velasco, L. Mederos, G. Navascues, P. C. Hemmer, and
G. Stell, Phys. Rev. Lett. 85, 122 (2000).
 P. C. Hemmer, E.Velasko, L. Mederos, G. Navascues, and
G. Stell, J. Chem. Phys. 114, 2268 (2001).
 M. R. Sadr-Lahijany, A. Scala, S. V. Buldyrev and H. E.
Stanley, Phys. Rev. Lett. 81, 4895 (1998).
 M. R. Sadr-Lahijany, A. Scala, S. V. Buldyrev and H. E.
Stanley, Phys. Rev. E 60, 6714 (1999).
 P. Kumar, S. V. Buldyrev, F. Sciortino, E. Zaccarelli,
and H. E. Stanley, Phys. Rev. E 72, 021501 (2005).
 L. Xu, S. V. Buldyrev, C. A. Angell, and H. E. Stanley,
Phys. Rev. E 74, 031108 (2006).
 E. A. Jagla, J. Chem. Phys. 111, 8980 (1999); E. A.
Jagla, Phys. Rev. E 63, 061501 (2001).
 F. H. Stillinger and D. K. Stillinger, Physica (Amster-
dam) 244A, 358 (1997).
 A. B. de Oliveira, P. A. Netz, T. Colla, and M. C. Bar-
bosa, J. Chem. Phys. 124, 084505 (2006).
 A. B. de Oliveira, P. A. Netz, T. Colla, and M. C. Bar-
bosa, J. Chem. Phys. 125, 124503 (2006).
 A. B. de Oliveira, M. C. Barbosa, and P. A. Netz, Physica
A 386, 744 (2007).
 J. Mittal, J. R. Errington, and T. M. Truskett, J. Chem.
Phys. 125, 076102 (2006).
 H. M. Gibson and N. B. Wilding, Phys. Rev. E 73, 061507
 P. Camp, Phys. Rev. E 71, 031507 (2005).
 A. B. de Oliveira, G. Franzese, P. A. Netz, and M. C.
Barbosa, J. Chem. Phys. 128, 064901 (2008).
 L. Xu, S. Buldyrev, C. A. Angell, and H. E. Stanley,
Phys. Rev. E 74, 031108 (2006).
 A. B. de Oliveira, P. A. Netz, and M. C. Barbosa, Euro.
Phys. J. B 64, 481 (2008).
 A. B. de Oliveira, P. A. Netz, and M. C. Barbosa,
 Yu. D. Fomin, N. V. Gribova, V. N. Ryzhov, S. M.
Stishov, and Daan Frenkel, J. Chem. Phys. 129, 064512
 V. N. Ryzhov and S. M. Stishov, Zh. Eksp. Teor. Fiz.
122, 820 (2002)[JETP 95, 710 (2002)].
 V. N. Ryzhov and S. M. Stishov, Phys. Rev. E 67,
 Yu. D. Fomin, V. N. Ryzhov, and E. E. Tareyeva, Phys.
Rev. E 74, 041201 (2006).
 D. A. Young and B. J. Alder, Phys. Rev. Lett. 38, 1213
(1977); D. A. Young and B. J. Alder, J. Chem. Phys. 70,
 P. Bolhuis and D. Frenkel, J. Phys.: Condens. Matter 9,
 S. M. Stishov, Phil. Mag. B 82, 1287 (2002).
 Daan Frenkel and Berend Smit, Understanding molecu-
lar simulation (From Algorithms to Applications), 2nd
Edition (Academic Press), 2002.
 D. Frenkel and A. J. Ladd, J. Chem. Phys. 81, 3188
 R. Agrawal and D.A. Kofke, Phys. Rev. Lett. 74, 122
 R. M. Lynden-Bell and P. G. Debenedetti, J. Phys.
Chem. B 109, 6527 (2005).