# Solid-amorphous transition is related to the waterlike anomalies in a fluid
without liquid-liquid phase transition

José Rafael Bordin
Departamento de Física, Instituto de Física e Matemática,
Universidade Federal de Pelotas
Caixa Postal 354, CEP 96001-970, Pelotas, RS, Brazil
<EMAIL_ADDRESS>
Leandro B. Krott
Centro de Ciências, Tecnologias e Saúde, Campus Araranguá
Universidade Federal de Santa Catarina
Rua Pedro João Pereira, 150, CEP 88905-120, Araranguá, SC, Brazil
<EMAIL_ADDRESS>
###### Abstract

The most accepted origin for the water anomalous behavior is the phase
transition between two liquids (LLPT) in the supercooled regime connected to
the glassy first order phase transition at lower temperatures. Two length
scales potentials are an effective approach that have long being employed to
understand the properties of fluids with waterlike anomalies and, more
recently, the behavior of colloids and nanoparticles. These potentials can be
parameterized to have distinct shapes, as a pure repulsive ramp, such as the
model proposed by de Oliveira et al. [J. Chem. Phys. 124, 64901 (2006)]. This
model has waterlike anomalies despite the absence of LLPT. To unravel how the
waterlike anomalies are connected to the solid phases we employ Molecular
Dynamics simulations. We have analyzed the fluid-solid transition under
cooling, with two solid crystalline phases, BCC and HCP, and two amorphous
regions being observed. We show how the competition between the scales creates
an amorphous cluster in the BCC crystal that leads to the amorphization at low
temperatures. A similar mechanism is found in the fluid phase, with the system
changing from a BCC-like to an amorphous-like structure in the point where a
maxima in $k_{T}$ is observed. With this, we can relate the competition
between two fluid structures with the amorphous clusterization in the BCC
phase.Those findings help to understand the origins of waterlike behavior in
systems without liquid-liquid critical point.

##  1 Introduction

Every known fluid solidifies upon cooling. However, the cooling process can
lead to solid structures with distinct types of ordering. While crystalline
solid phases show a long-range, macroscopic order, in condensed amorphous
phases, as a glass or amorphous solid, no long-range order is observed
Debenedetti (1997); Petrenko and Whitworth (2002). In fact, understanding the
glass transition and the behavior of fluids in the vicinity is a huge task in
condensed matter physics; and the challenge grows for polyamorphic systems. A
one-component system has polyamorphism if it has the coexistence (often
metastable) of two amorphous phases Kivelson and Tarjus (2002); Brazhkin and
Lyapin (2003); Loerting et al. (2009); Angell. (2014); Anisimov et al. (2018);
Woutersen et al. (2018); Bachler et al. (2019); McMillan and Wilding (2021);
Hernandes et al. (2021).

Certainly, the most famous polyamorphous material is water. There are growing
evidences that, deep into the supercooled liquid region, water have a
liquid–liquid first-order phase transition that ends in a liquid-liquid
critical point (LLCP), which is connected to the first order phase transition
observed between two glassy phases at lower temperatures Mishima et al.
(1985); Poole et al. (1992); Stanley et al. (2009); Giovambattista et al.
(2012); Gallo et al. (2016); Kim et al. (2020); Shi and Tanaka (2020); Bachler
et al. (2021); Foffi and Sciortino (2021); Giovambattista and Poole (2021);
Caupin and Anisimov (2021); Lucas et al. (2022); Verde et al. (2022). It is
possible to investigate the location of a critical point through an analysis
of the thermodynamic response functions. For instance, the specific heat and
isothermal compressibility display a line of maxima in the P-T plane (Widom’s
line) that typically ends at the critical point where the maximum evolves into
a divergence in the thermodynamic limit Simeoni et al. (2010); Brazhkin et al.
(2011, 2018); Zerón et al. (2019); Losey and Sadus (2019); Bianco and Franzese
(2019). Water, in addition to the Widom line ending at the vapor-liquid
critical point, shows a second ending at the LLCP Xu et al. (2005); Franzese
and Stanley (2007); Stanley, H. E. et al. (2008); Kumar et al. (2008); Abascal
and Vega (2010); Luo et al. (2015); Gallo et al. (2016). Predicted in 1992 by
Poole and co-authors Poole et al. (1992), experimental studies to observe the
LLCP have been hampered by the spontaneous crystallization in the no-man’s
land for 25 years. Only recently the Widom line was detected in experiments
Kim et al. (2017, 2020); Amann-Winkel et al. (2023); Kringle et al. (2020).
Nevertheless, many other systems have evidences of polyamorphism, as silicon
Sastry and Austen Angell (2003); Beye et al. (2010); Goswami et al. (2021),
silica Saika-Voivod et al. (2004); Lascaris et al. (2014); Chen et al. (2017);
Trang et al. (2020); Yu et al. (2022), hydrogen Zaghoo et al. (2016); Fried et
al. (2022), sulfur Plašienka et al. (2015); Henry et al. (2020); Shumovskyi et
al. (2022), phosphorous Katayama et al. (2000, 2004); Yang et al. (2021),
carbon Glosli and Ree (1999); Sundqvist (2021), triphenyl phosphite Tanaka et
al. (2004), gallium Liu et al. (2021), molten elements Brazhkin et al. (1999),
metallic glasses Sheng et al. (2007); Greaves et al. (2008); Li et al. (2022),
germania Smith et al. (1995), tellurium Sun et al. (2022) and more Angell
(1995); Rizzatti et al. (2019); Tanaka (2020); Lucas (2021); Desgranges and
Delhommelle (2018).

The existence of LLCP is the mainly accepted scenario to explain the anomalous
behavior of water Gallo et al. (2021). In fact, water is the most anomalous
known material, with more than 70 thermodynamic, dynamic and structural
behaviors that strongly diverge from the observed in most fluids Chaplin
(2023); Podgornik (2011). For instance, it is long known that water density
increases as the temperature grows from 0oC to 4oC at 1 atm Kellu (1975),
whereas in most materials heating is associated naturally with the thermal
expansion. Once most of these anomalies, and the LLCP, occur in the
supercooled regime, and due to the difficulty to access this region
experimentaly Gallo et al. (2021), computer simulations have long been applied
to overcome the experimental barriers and understand the behavior of water and
others anomalous fluids Stanley et al. (2009); Gallo et al. (2016); Liu et al.
(2012); Gallo et al. (2021). In this sense, many models are able to reproduce
the water polyamorphism and the LLCP Giovambattista and Poole (2021); Palmer
et al. (2013, 2018); Chiu et al. (2013). It is interesting that models without
LLCP, such as the SPC/E, show a low density amorphous (LDA) - high density
amorphous (HDA) phase transition and waterlike anomalies Giovambattista et al.
(2012).

Water molecules interaction are known to be computationally expensive and
difficult to treat, especially in many body atomistic models Abella et al.
(2023). Is this way, many isotropic core-softened potentials with two
characteristic length scales have been employed to unveil the waterlike
behavior, searching for the origins of anomalies Vilaseca and Franzese (2011).
Since the early 1970’s, where Hemmer and Stell Hemmer and Stell (1970)
observed that we can induce polyamorphism by softening a hard core interaction
judiciously, obtaining a core-softened (CS) potential, passing by the seminal
works by Jagla in the late 1990’s Jagla (1998, 1999, 1999), many spherical
symmetric core-softened potentials have been employed to study polyamorphous
systems Xu et al. (2010); Gnan et al. (2014); Xu (2019); Dasgupta et al.
(2020); Fomin (2020) and to understand the relation between waterlike behavior
and the competition between distinct structures in the fluid Yan et al.
(2006); Gibson and Wilding (2006); Franzese (2007); Xu et al. (2011); Reisman
and Giovambattista (2013); Das et al. (2013); Krott et al. (2015a); Luo et al.
(2015); Bordin et al. (2015); Bordin (2016); Pinheiro et al. (2017); de Haro
et al. (2018); Higuchi et al. (2018); Ryzhov et al. (2020); Martín-Roca et al.
(2022); Bretonnet and Bomont (2022). In fact, these simple models can be
applied for a variety of systems Likos (2001). Experimental works have shown
that the effective interaction in solutions of pure or grafted PEG colloids
are well described by core-softened potentials Quesada-Perez et al. (2001);
Contreras-Aburto and amd R.C. Priego (2010); Haddadi et al. (2020), and
computational studies indicate the same type of effective interactions for
polymer-grafted nanoparticles S. Marques et al. (2020); Lafitte et al. (2014)
or star polymers Bos et al. (2019).

The existence of two length scales in the CS potential does not guarantee the
existence of waterlike anomalies Quigley and Probert (2005). The competition
between the length scales is a key ingredient to observe the anomalous
thermodynamic, dynamic and structural behavior in such fluids de Oliveira et
al. (2008); Franzese (2007); Stanley et al. (2009); da Silva et al. (2010);
Barraz et al. (2009); Marques et al. (2020, 2021), reproducing the competition
between two liquids in supercooled water Gallo et al. (2016, 2021). However,
for purely repulsive potentials, as the one proposed by de Oliveira and co-
authors, no LLCP was observed - despite the competition between the scales and
the waterlike anomalies de Oliveira et al. (2008, 2006); da Silva et al.
(2010). This is a similar scenario from the observed in SPC/E water
Giovambattista et al. (2012) \- however, the existence of solid
poly(a)morphism and their relation with the waterlike anomalies were not
investigated.

To unravel the solid phases and the relation with the waterlike anomalous
behavior, we perform large scale $NpT$ and $NVT$ Molecular Dynamics
simulations cooling a purely repulsive CS from the fluid to the solid phase.
To this end, we employ the model proposed by de Oliveira et al. de Oliveira et
al. (2006), reproducing the waterlike anomalies obtained in previous works de
Oliveira et al. (2006); Barros De Oliveira et al. (2006); De Oliveira et al.
(2006); de Oliveira et al. (2008); da Silva et al. (2010). We expand these
works by looking to the low temperature regime. Four distinct solids
structures were observed: two crystalline and two amorphous. Our results
indicate that the system undergoes a fluid-crystal solid first order phase
transition, and a second order fluid-amorphous transition. Also, we show how
the anomalies are connected to a crystal-amorphous phase transition, and found
a maxima in the isothermal comprehensibility that does not end in a LLCP but
it is connected to the amorphization of a crystal structure.

The remainder of the paper is organized as follows. In Section II we present
our anomalous fluid model and summarize the details of the simulations. Next,
in section III our most significant results. In particular, we will focus on
the solidification scenario for the distinct solid structures and in their
relations with waterlike anomalies. The paper is closed with a brief summary
of our main conclusions and perspectives.

##  2 Model and Simulation Details

###  2.1 The Model

Once it is an effective model, and for the sake of simplicity, all quantities
will be depicted in standard LJ units Scott et al. (1991). The fluid particles
interaction was modeled by the CS potential given by Barros De Oliveira et al.
(2006); De Oliveira et al. (2006)

$U_{(}r)=4\epsilon\bigg{[}\bigg{(}\frac{\sigma}{r}\bigg{)}^{12}-\bigg{(}\frac{\sigma}{r}\bigg{)}^{6}\bigg{]}+$
$u_{0}\exp\bigg{[}-\frac{1}{c_{0}^{2}}\bigg{(}\frac{r-r_{0}}{\sigma}\bigg{)}^{2}\bigg{]},$
(1)

which is composed of a short-range attractive Lennard-Jones potential and a
repulsive Gaussian potential centered at $r_{0}$, with depth $u_{0}$ and width
$c_{0}$. Using the parameters $u_{0}=5\epsilon$, $c_{0}^{2}=1.0$, and
$r_{0}/\sigma=0.7$ Barros De Oliveira et al. (2006); De Oliveira et al.
(2006); Bordin and Barbosa (2018) the potential (1) acquire the purely
repulsive ramp-like shape shown in Fig.1. The waterlike anomalous behavior in
this potential arises from the competition between two characteristic length
scales da Silva et al. (2010); Bordin and Barbosa (2018): a closer one located
at the shoulder in $r_{1}=1.2\sigma$, where the force has a local minimum (as
we can see in Fig. 1 graph inset), while the further length scale is located
at $r=2.0\sigma$, where the fraction of imaginary modes of the instantaneous
normal modes spectrum has a local minimumBarros De Oliveira et al. (2010). The
cutoff radius for the interaction is $r_{c}=3.5\sigma$.

Figure 1: Ramp with shoulder potential. Graph inset: the force ($F=-\partial
U/\partial r$) times the distance $r$. Molecule insets is the schematic
depiction of the hard core-soft shell particle.

###  2.2 Simulation Details

The phase diagram was obtained by simulations in the isothermal-isobaric
ensemble ($NpT$) with fixed number of particles $N=5000$, as implemented in
the ESPResSo package Limbach et al. (2006); Arnold et al. (2013). For the
$NpT$ simulations, isobars raging from $p=0.0001$ up to $p=4.00$ with distinct
intervals were simulated. Along each isobar we perform a cooling process,
starting at $T=0.600$, reducing the temperature up to $T=0.025$ with a
temperature step $\delta T=0.025$. A random system is created at the highest
temperature - at this temperature, the system is always in the fluid state
Barros De Oliveira et al. (2006). Then, $5\times 10^{6}$ time steps in the
$NVT$ ensemble thermalizes the system, followed by $1\times 10^{6}$ time steps
in the $NpT$ ensemble to equilibrate the system’s pressure and density.
$1\times 10^{7}$ time steps are then performed for the production of the
results, with averages and snapshots being taken at every $1\times 10^{5}$
steps. After that, we cool down the system to $T-\delta T$ along the isobar,
repeating the thermalization and the equilibration process in the $NVT$ and
$NpT$ ensembles, respectively, in order to relax the system at the new
temperature and achieve the proper new density before run the production
steps. In this way, the two equilibration steps allow the system to cool down
and relax at the new state. To ensure that the system is in the equilibrium we
averaged quantities as energy, entalpy, density, instantaneous pressure and
kinetic temperature along the simulations, including in the equilibration
processes. The Langevin thermostat Allen et al. (2017), that keeps the
temperature fixed in our ESPResSo implementation, has a coupling parameter
$\gamma_{0}=1.0$. The piston parameters for the Anderson barostat Allen et al.
(2017) are $\gamma_{p}=0.0002$ and mass $m_{p}=0.001$. Here, the molecule
density $\rho$ is defined as $N/V$ with $V$ being the mean volume at a given
pressure and temperature.

To confirm the low temperature phase transitions, simulations using the
isothermal ensemble ($NVT$) were performed in the range $0.020\leq T\leq
0.250$ and $0.048\leq\rho\leq 0.296$ as implemented in a fortran90 code
developed in the house Krott and Bordin (2013); Krott et al. (2015b). The
temperature was controlled using the Nosè-Hoover thermostat Frenkel and Smit
(1996) with coupling parameter $Q=2.0$. Two different initial configurations
of the systems were simulated: solid and liquid states. Using different
initial configurations allow us to identify precisely the final state of the
system, avoiding metastability. The equilibrium state was reached after
$4\times 10^{6}$ steps, followed by $2\times 10^{6}$ simulation run. We used a
time step $\delta t$ = 0.001. All the physical quantities were obtained from
50 uncorrelated samples. To check the stability of the systems, we verified
the energy as a function of time and the pressure, evaluated by the virial, as
a function of density.

The temperature of maximum density (TMD) along each isobar was used to
characterize the density anomaly. Also, the locus of the maximum of response
functions were obtained: the isothermal compressibility $\kappa_{T}$, the
isobaric expansion coefficient $\alpha_{P}$ and the specific heat at constant
pressure $C_{p}$ and at constant volume $C_{V}$,

$\kappa_{T}=\frac{1}{\rho}\left(\frac{\partial\rho}{\partial
p}\right)_{T}\;,\quad\alpha_{p}=-\frac{1}{\rho}\left(\frac{\partial\rho}{\partial
T}\right)_{p}\;,$ $C_{p}=\frac{1}{N}\left(\frac{\partial H}{\partial
T}\right)_{p}\;,\quad C_{V}=\frac{1}{N}\left(\frac{\partial U}{\partial
T}\right)_{V}\;.$ (2)

Here, $H=U+PV$ is the system entalpy. The quantities shown were obtained by
numerical differentiation and, as consistency check, using statistical
fluctuations: the compressibility as the measure of volume fluctuations, the
isobaric heat capacity is proportional to the entropy fluctuations experienced
by $N$ particles at fixed pressure, and the thermal expansion coefficient
reflects the correlations between entropy and volume fluctuations Allen et al.
(2017); Tuckerman (2010).

In order to describe the connection between structure and thermodynamics, we
have analyzed translational and orientational properties. The radial
distribution function (RDF) $g(r)$ was evaluated and subsequently used to
compute the excess entropy $s_{ex}$ and the translational order parameter
$\tau$. The first one can be obtained by counting all accessible
configurations for a real fluid and comparing it with the ideal gas entropy
Dzugutov (1996). Then, $s_{ex}$ is a negative quantity that increases with
temperature, as the full entropy $S$ does Dyre (2018); Bell et al. (2020).
Expanding $s_{ex}$ in terms of two-particle, three-particle contributions,
etc. Galliero et al. (2011), we have

$s_{ex}=s_{2}+s_{3}+s_{4}+...$ (3)

The two-particle contribution,

$s_{2}=-2\pi\rho\int_{0}^{\infty}\left[g(r)\ln g(r)-g(r)+1)\right]r^{2}dr.$
(4)

corresponds to $85\%$ and $95\%$ of the total excess entropy in Lennard-Jones
systems, being the dominant contribution to excess entropy Raveché (1971);
Baranyai and Evans (1989); Sharma et al. (2006).

In order to ascertain how the short and long range coordination shells
behaves, we evaluated the cumulative two-body entropy Krekelberg et al.
(2008); Cardoso et al. (2021)

$C_{s_{2}}(r)=-2\pi\rho\int_{0}^{r}[g(r^{\prime})\ln(g(r^{\prime}))-g(r^{\prime})+1]r^{\prime
2}dr^{\prime}\;.$ (5)

Here, $r$ is the upper integration limit. It is relevant to address that the
two-body excess entropy is not a thermodynamic quantity, but a structural
order metric that connects thermodynamics and structure. Then, $C_{s_{2}}$ is
a useful tool to analyze long-range structural characteristics of the core-
softened system.

The translational order parameter is defined as Errington and Debenedetti
(2001)

$\tau\equiv\int^{\xi_{c}}_{0}\mid g(\xi)-1\mid d\xi,$ (6)

where $\xi=r\rho^{1/3}$ is the interparticle distance $r$ scaled with the
average separation between pairs of particles $\rho^{1/3}$. $\xi_{c}$ is a
cutoff distance, defined as $\xi_{c}=(L\rho^{1/3})/3$, where $L$ is the
simulation box size. For an ideal gas (completely uncorrelated fluid),
$g(\xi)=1$ and $\tau$ vanishes. For crystals or fluids with long range
correlations $g(\xi)\neq 1$ over long distances, which leads to $\tau>0$. The
excess entropy and the translational order parameter $\tau$ are linked, once
both are dependent on the deviation of $g(r)$ from unity. The translational
structural anomaly can then be characterized by an anomalous decrease of
$\tau$ under compression.

Another structural quantity evaluated was the orientational order parameter
(OOP), that gives insight on the local orientational order Steinhardt et al.
(1983); Errington and Debenedetti (2001); De Oliveira et al. (2006); Yan et
al. (2005). The OOP for a specific particle $i$ with $N_{b}$ neighbors is
given by

$q_{l}(i)=\sqrt{\frac{4\pi}{2l+1}\sum_{m=-l}^{l}|q_{lm}|^{2}},$ (7)

with

$q_{lm}(i)=\sqrt{\frac{1}{N_{b}}\sum_{j=1}^{N_{b}}Y_{lm}(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij}))}.$
(8)

where $Y_{lm}$ are the spherical harmonics of order $l$ and $\vec{r}_{ij}$ is
the distance between two particles. The OOP for a whole system is obtained by
taking the average over the parameter value for each particle $i$,
$\tilde{q}_{l}=\langle q_{l}(i)\rangle_{i}$. In this work we evaluated the OOP
for $l=4$ and 6, using the Freud python library Ramasubramani et al. (2020) –
the number of neighbors for each particle was found computing Voronoi diagrams
using voro++ Rycroft (2009), as explained in details in our previous work
Hernandes et al. (2021). The anomalous behavior is characterized by a maxima
in $\tilde{q}_{6}$ and a minima in $\tilde{q}_{4}$

The dynamic behaviour was analyzed by the mean square displacement (MSD),
given by

$\langle[\vec{r}(t)-\vec{r}(t_{0})]^{2}\rangle=\langle\Delta\vec{r}(t)^{2}\rangle\;,$
(9)

where $\vec{r}(t_{0})=$ and $\vec{r}(t)$ denote the particle position at a
time $t_{0}$ and at a later time $t$, respectively. The MSD is then related to
the diffusion coefficient $D$ by the Einstein relation,

$D=\lim_{t\rightarrow\infty}\frac{\langle\Delta\vec{r}(t)^{2}\rangle}{6t}\;.$
(10)

The diffusion anomaly region was obtained using the $(D,p)$ isotherms. For
regular fluids $D$ decreases by compressing. However, for a range of
pressures, $D$ increases with $p$ in fluids with diffusion anomaly. With this,
the diffusion anomaly extrema (DE) may be defined using the minima and maxima
in these curves de Oliveira et al. (2008); Vilaseca and Franzese (2010); S.
Marques et al. (2020).

Finally, the onset of crystallization was monitored analyzing the local
structural environment of particles by means of the Polyhedral Template
Matching (PTM) method implemented in the Ovito software Larsen et al. (2016);
Stukowski (2010).

##  3 Results and Discussion

###  3.1 The Fluid-Solid Transitions

Figure 2: Low temperature $T\times\rho$ phase diagram obtained from the $NpT$
simulations. Isobars ranging from $p=0.050$ up to $p=4.200$ are shown in
dashed grey lines with circular points - the deviations from the mean value of
$\rho$ are smaller than the data points. The Extrema (E) – that can be a
maxima or a minima – and the Discontinuity (D) observed in the response
functions are indicated by the points: magenta filled right triangle for a
discontinuous behavior in the $c_{p}\times T$ and $\alpha_{p}\times T$
isobars, magenta open right triangle for a extrema in the $c_{p}\times T$ and
$\alpha_{p}\times T$ isobars, cyan filled left triangle for a discontinuous
behavior in the $c_{V}\times T$ isochores, cyan open left triangle for a
maxima in the $c_{V}\times T$ isochores, green filled up triangle for a
discontinuous behavior in the $k_{T}\times p$ isotherm, green open up triangle
for a maxima in the $k_{T}\times p$ isotherms. Figure 3: Dependence of $c_{p}$
as function of $T$ for two distinct isobars that crosses the (a) fluid-BCC
solid discontinuous phase transition, (b) fluid-amorphous solid 1 continuous
phase transition, (c) fluid-HCP solid discontinuous phase transition, and (d)
fluid-amorphous solid 2 continuous phase transition.

Although anomalous, fluid polymorphism was not reported for the purely
repulsive CS model employed in this work De Oliveira et al. (2006); Barros De
Oliveira et al. (2006); da Silva et al. (2010). One possibility is that the
fluid properties vary very smoothly, with the fluid near a crystalline phase
behaving distinctly from water near an amorphous phase. In the original works
by de Oliveira and co-authors De Oliveira et al. (2006); Barros De Oliveira et
al. (2006), and in their next articles Oliveira et al. (2008); da Silva et al.
(2010), the crystallization limit was not explored. Recently, Bordin and co-
authors Cardoso et al. (2021); Nogueira and Bordin (2022) explored the 2D
limit, and found three distinct crystalline solid phases separated by two
amorphous solid phases. To check if a poly(a)morphism will be observed in the
3D limit, we can first look to the specific heat behavior along an isobar (for
the $NpT$ simulations), and corroborate the findings evaluation the specific
heat at constant volume using the $NVT$ simulations. In this way, we show in
Fig. 2 the $T\times\rho$ phase diagram for temperatures below $T=0.250$. Here,
the gray circles connected by gray dashed lines indicate the behavior of the
isobar upon cooling and, to identify the fluid-solid phase transitions, the
response functions were evaluated. Interesting, both $c_{p}$ and $\alpha_{p}$
can show discontinuities ($c_{p}D$ and $\alpha_{p}D$) or extrema ($c_{p}E$ and
$\alpha_{p}E$) depending on the isobar - in a similar way, $c_{v}$ can also
have extrema or discontinuities along distinct isochores.

Figure 4: Dependence of $c_{V}$ as function of $T$ for two distinct isochores
that crosses the (a) fluid-BCC solid discontinuous phase transition, (b)
fluid-amorphous solid 1 continuous phase transition, (c) fluid-HCP solid
discontinuous phase transition, and (d) fluid-amorphous solid 2 continuous
phase transition.

In Fig. 3 we show the $c_{p}$ behavior. At low pressures, up to $p=0.840$, the
specific heat has a discontinuity (D) at low temperatures, indicating a first
order phase transition, as we can see in Fig. 3(a). Analysing isochores in the
same region, as illustrated in Fig. 4(a), we can see that $c_{V}$ also has a
discontinuity in a close temperature. On the other hand, for pressures ranging
from $p=0.860$ to 1.360, the $c_{p}$ has a maxima. This $\lambda-$like curve
is typical from second order phase transitions, and was observed for both
$c_{p}$, Fig. 3(b), and $c_{v}$, Fig. 4(b). Compressing the system, the
discontinuous behavior is recovered for both specific heats, see Fig. 3(c) and
Fig. 4(c) and, for the higher pressures and densities, it changes once again
to a $\lambda$ shape. Such behavior was also observed in another response
function, $\alpha_{p}$, shown in Fig. 5. Here is also possible to observe two
waterlike features of this model: the density anomaly and a fluid phase denser
than the solid phase (as the isobars in Fig. 2 shown).

Figure 5: Dependence of $\alpha_{p}$ as function of $T$ for two distinct
isobars that cross the (a) fluid-BCC solid discontinuous phase transition, (b)
fluid-amorphous solid 1 continuous phase transition, (c) fluid-HCP solid
discontinuous phase transition, and (d) fluid-amorphous solid 2 continuous
phase transition.

The discontinuous/continuous transitions can also be observed using structural
parameters. First, we analyze the behavior of $s_{2}$ and $\tau$ along two
pressures isobars. The first one, $p=0.500$, has a discontinuity in the
response functions $c_{p}$ and $\alpha_{p}$, and the second, $p=1.100$, a
continuous behavior. By looking to the $s_{2}$ and $\tau$ curves, shown in
Fig. 6, we can also see a discontinuous behavior along the fluid-solid
transition, evidenced by the abrupt change in the RDF behavior in the lower
panel of Fig. 6. Cooling down the system from $T=0.075$ to $T=0.050$ leads to
an abrupt change in the RDF, from a fluid to a solid with defined structure –
with the PTM method Larsen et al. (2016); Stukowski (2010) and $q_{6}$ and
$q_{4}$ analyzes Lechner and Dellago (2008) we identified as a body-centered
cubic (BCC) structure. On the other hand, along the isobar $p=1.100$ the
translational structural parameters $s_{2}$ and $\tau$ change continuously
once the structure at the amorphous phase is similar to the fluid structure,
as shown in the RDF of the lower panel of Fig. 6. This amorphous region will
be denominated as AS1.

Figure 6: Behavior of $s_{2}$ (upper panel) and $\tau$ (middle panel) as
function of temperature for the isobars $p=0.500$ and $p=1.100$. Lower panel:
at left we show the RDF for the system in the BCC phase in the point
$p=0.500,T=0.050$ and in the fluid phase at $p=0.500,T=0.075$, and in the
right the RDF for the system in the amorphous solid phase 1 in the point
$p=1.100,T=0.050$ and in the fluid phase at $p=1.100,T=0.075$.

Compressing the system, $s_{2}$ and $\tau$ show a discontinuous behavior along
the isobar $p=2.000$ and continuous at $p=3.200$ in Fig. 7, as the
thermodynamic response functions. The discontinuous behavior can be associated
to a fluid-solid transition, with a hexagonal closed packed (HCP) Lechner and
Dellago (2008); Larsen et al. (2016); Stukowski (2010) structure in the solid
crystalline phase, while the continuous behavior correspond to a fluid-
amorphous transition - the AS2 phase. As in the fluid-AS1 transition, the main
difference between the fluid and AS2 structures is the occupancy in the two
characteristic length scales, as the RDF in the lower panel of Fig. 7 show:
the amorphous phases have higher peaks at $r_{1}$ and $r_{2}$ and the
occupancy between the peaks vanishes, indicating the solidification.

Figure 7: Behavior of $s_{2}$ and $\tau$ ad function of temperature for the
isobars $p=2.000$ and $p=3.200$. Lower panel: at left we show the RDF for the
system in the HCP phase in the point $p=2.000,T=0.100$ and in the fluid phase
at $p=2.000,T=0.125$, and in the right the RDF for the system in the amorphous
solid phase 2 in the point $p=3.200,T=0.075$ and in the fluid phase at
$p=3.200,T=0.100$.

The existence of two crystalline solid phases and two amorphous phases can
also be inferred from the longer range translational ordering. For a ordered
solid, $|C_{s2}(r)|$ increases with $r$, while for disordered systems - as
fluid or amorphous solids – $|C_{s2}(r)|$ reaches a plateau as the
RDF$\rightarrow$ 1.0 at short distances. It can be seen in the cumulative two-
body entropy, Fig. 8: the BCC and HCP phases entropy continually increase
until large distances $r$, indicating a longer range ordering in the
coordination shells. On the other hand, the fluid and amorphous solid phases
reach a plateau after a short distance, characterizing a shorter range
ordering.

Figure 8: Cumulative two-body entropy in the vicinity of the phase
transitions. In (a) we show $|C_{s2}(r)|$ for the point $p=0.500,T=0.050$ in
the BCC phase and $p=0.500,T=0.075$ in the fluid phase, (b) for
$p=1.100,T=0.050$ in the AS1 phase and $p=1.100,T=0.075$ in fluid phase, (c)
HCP phase in the point $p=2.000,T=0.100$ and fluid phase at $p=2.000,T=0.125$,
and (d) AS2 phase at $p=3.200,T=0.075$ and fluid phase at $p=3.200,T=0.100$.

To understand the orientational ordering we can plot the behavior of it’s mean
value, $\tilde{q}_{6}$, for each particle as function of the mean value of
$q_{4}$ Hernandes et al. (2021); Steinhardt et al. (1983); Lechner and Dellago
(2008); Eslami et al. (2018); Geiger and Dellago (2013); Boattini et al.
(2018, 2019). In Fig. 9(a) each point corresponds to one particle in the
system in the isobar $p=0.500$ at $T=0.50$ (black points, BCC phase) and
$T=0.075$ (red squares, fluid phase). As we can see, the sixfold ordering
drastically decreases upon cooling. Similar behavior can be seen in the isobar
$p=2.00$,(c), where there is a drastic change in the OOP values from the solid
HCP phase at $T=0.100$ to the fluid phase at $T=0.125$. However, in the fluid-
amorphous transitions, (b) and (d), the orientational behavior is practically
the same, without an abrupt change in the OOP values.

Figure 9: $\tilde{q}_{6}\times\tilde{q}_{4}$ for (a) the point
$p=0.500,T=0.050$ in the BCC phase and $p=0.500,T=0.075$ in the fluid phase,
(b) for $p=1.100,T=0.050$ in the AS1 phase and $p=1.100,T=0.075$ in fluid
phase, (c) HCP phase in the point $p=2.000,T=0.100$ and fluid phase at
$p=2.000,T=0.125$, and (d) AS2 phase at $p=3.200,T=0.075$ and fluid phase at
$p=3.200,T=0.100$. Each point corresponds to one particle in the system.

Not only the thermodynamic and structural behaviors indicate a discontinuous
transition between fluid and crystalline solids and a fluid-amorphous
continuous transition, but also the dynamics leads to this conclusion. In Fig.
10 we show the MSD along distinct isobars for all simulated temperatures. It
is clear that the MSD – and, consequently, the diffusion coefficient $D$ –
have a discontinuity in the fluid-crystalline solid transition, while it has a
continuous behavior in the fluid-amorphous solid transition. With this, we can
see that the thermodynamic response functions, the translational order
parameter, the two-body excess entropy, the OOP and the dynamic of the system
indicate the existence of two crystalline solid and two amorphous solid for
the potential, Eq. (1), with discontinuous fluid-crystalline transition and
continuous fluid-amorphous transition. Next, we explore the solid-solid
transitions in the system.

Figure 10: MSD as function of time for pressures that cross (a) the fluid-BCC,
(b) fluid-AS1, (c) fluid-HCP and (d) fluid-AS2 transitions. The temperature
ranges from $T=0.025$ up to $T=0.600$.

###  3.2 The Solid-Solid transitions

Figure 11: Dependence of $k_{T}$ as function of $p$ for the lowest simulated
temperatures, indicating a continuous transition as the system goes from BCC
solid to amorphous solid 1, and discontinuous AS1-HCP and HCP-AS2 transitions.

We start this discussion by analyzing the isothermal compressibility, $k_{T}$,
looking for the solid-solid pressure transitions between the four phases: BCC,
AS1, HCP and AS2. The BCC and AS1 phases were observed for $T\leq 0.050$, and
the HCP and AS2 phases in the $T\leq 0.100$ region. The correspondent
$k_{T}\times p$ curves are shown in Fig. 11. As we can see, at the lowers
temperatures, $T=0.025$ and 0.050, the curve has tree peaks. For $T=0.025$,
they are located at $p=0.86$, the BCC-AS1 transition, at $p=1.38$ and
$p=3.00$, the AS1-HCP and HCP-AS2 transitions, respectively. Upon heating, at
$T=0.075$ and $T=0.100$, the peak at low temperatures is replaced by a small
bump in the $k_{T}$ curve in the regions near the BCC-AS1 transition observed
in the colder isotherms. This bump vanishies at higher temperatures. As shown
in the $\rho\times T$ phase diagram at low temperatures, Fig. 2, the extrema
$k_{T}$E line and discontinuous $k_{T}$D lines correspond to the borders
between the crystalline and amorphous solid phases - with the $k_{T}$E line
going to inside the fluid phase, as a prolongation of the BCC-AS1 transition.
But how the structural properties are related to this?

Figure 12: Left panel:$\tilde{q}_{6}\times\tilde{q}_{4}$ diagram for pressures
in the border of the phase transitions along the isotherm $T=0.025$. Here,
each symbol stands for the mean value of each fluid particle. Solid black
symbols stand for the lower temperature, and open red symbols for the higher
one. From top to bottom, we have the BCC-AS1 transition, AS1-HCP transition
and HCP-AS2 transition, respectively. Right panel: correspondent probabilities
distribution $\tilde{q}_{6}$, $P(\tilde{q}_{6})$. Black curves stands for the
lower pressure and red curves for the higher pressure

First, lets take a look at the orientational ordering, given by the OOPs. In
Fig. 12 we show in the left panel the $\tilde{q}_{6}\times\tilde{q}_{4}$
diagram for pressures in the border of the phase transitions along the
isotherm $T=0.025$. In the right panel are the correspondent probabilities of
a particle have a OOP $\tilde{q}_{6}$, $P(\tilde{q}_{6})$. As the system is
compressed from $p=0.840$ to $p=0.860$ the structure change from a BCC to the
AS1, as the $k_{T}$ curve in Fig. 11. Interestingly, at $p=0.840$ some
particles have an amorphous-like orientational ordering Lechner and Dellago
(2008), as indicated at Fig. 12(a). The correspondent $P(\tilde{q}_{6})$, in
the right panel, shows a overlap in the probability distributions for both
pressures. This indicates that AS1 clusters grow in the BCC phase, promoting
the amorphization. On the other hand, the AS1-HCP and HCP-AS2 transitions have
a distinct behavior. As the Fig. 12(b) and (c) show, there is no overlap in
the probabilities, the values of $\tilde{q}_{6}$ are distinct for each phase.
This indicates a discontinuous orientational transition, with all particles
changing from AS1 to HCP and, under compression, from HCP to AS2.

Figure 13: RDF for pressures between 0.600 and 1.00 at temperatures $T=0.025$
(upper panel) and 0.100 (lower panel). The black curves indicate the RDF where
$k_{T}$ has a maxima, cyan curves are the RDF for lower pressures and in red
curves we present the RDF for higher pressures.

From the radial distribution function $g(r)$ is possible to correlate the
maxima in $k_{T}$ at lower pressures with the competition between the two
characteristic length scales. The RDF for the isotherm $T=0.025$, shown in the
upper panel of Fig. 13, shows how as the BCC phase is compressed the particles
start to move from the second length scale, $r_{2}\approx 2.2$, to the first
length scale, $r_{1}=1.2$, reaching $g(r_{1})\approx 1.0$ at the pressure of
transition, $p=0.860$. This corroborates the idea that an amorphous cluster
grows in the BCC phase as the particles start to occupy the first length
scale, leading to the observed decrease in $\tilde{q}_{6}$ for some particles,
and the overlap in $P(\tilde{q}_{6})$ for this transition. At this point, it
is relevant to analyze the RDF behavior along the $T=0.100$ isotherm. At this
temperature and in the range of pressures $0.300<p<1.00$ the system is in the
fluid phase. However, $k_{T}$ has a maxima at $p=0.920$ due the small bump.
Interestingly, this is the pressure where $g(r_{1})\approx 1.0$, following the
observed in the BCC-AS1 transition. Although this purely repulsive model does
not have fluid-fluid phase transition, it is striking to find this maxima line
that converges to the BCC-AS1 transition. Sure, it can not be characterized as
the Widom line since it does not end in a LLCP.

Figure 14: Behavior of $s_{2}$ (upper panel) and $\tau$ (middle panel) as
function of pressure along the isotherm $T=0.025$. Lower panel: RDF at the
phase-transition borders: BCC($p=0.84$)-AS1($p=0.86$),
AS1($p=1.36$)-HCP($p=1.38$) and HCP($p=2.95$)-AS2($p=3.00$)

Then, as the occupancy in the first length scale grows under compression, the
amorphous cluster grows, leading to the order-disorder transition indicated by
the $s_{2}$ and $\tau$ at $T=0.025$ curves, Fig. 14 upper and lower panel,
respectively. A sharp change in the structure can also be seen at the AS1-HCP
transition. However, in the HCP-AS2 phase transition the translational
structure varies smoothly – unlike the OOPs, where the orientation changes
abruptly, as the Fig. 12 indicates. Looking to the RDFs at the lower panel of
Fig. 14, we can see that while the BCC-AS1 phase transition is characterized
by the particles moving from the first to the second scale, at the AS1-HCP the
first scale occupancy did not change, but the ordering for distances from the
second scale, $r_{2}\approx 2.2$, and beyond are affected. The last solid
transition, HCP-AS2, does not have a significant change in the occupancy in
the first two coordination shells, but the long range behavior is affected. It
can be easily seen in the correspondent cumulative pair entropy, shown in Fig.
15 for the three phase boundaries.

Figure 15: Cumulative two-body entropy in the vicinity of the phase
transitions along the isotherm $T=0.025$. In (a) we show $|C_{s2}(r)|$ BCC-AS1
transition, (b) AS2-HCP transition and (c) HCP-AS2 transition.

###  3.3 The waterlike anomalies

Figure 16: (a) $\rho\times T$ isobars from $p=0.500$ (bottom curve) to
$p=1.00$ (top curve). The TMD line is indicated by the red dashed curve. (b)
Response function $\alpha_{p}$ for isobars ranging from $p=0.300$ to
$p=0.700$. The minima in the fluid phase is indicated by the red line.

Our findings show that the BCC-AS1 transition is ruled by the competition
between the scales, with a sharp change in the translational ordering and the
nucleation of a amorphous cluster in the BCC crystal under compression - on
the other hand, in the HCP-AS2 transition we have not observed the
amorphization of the HCP crystal. The amorphization of the BCC phase seems to
affect the fluid phase near the fluid-BCC transition, where maxima in $k_{T}$
where observed. This indicates that, although this model does not have LLCP,
there are two structures, one more BCC-like and another one AS1-like,
competing in the fluid phase. Therefore, if there is competition then there
should exist waterlike anomalies as well de Oliveira et al. (2008); Stanley et
al. (2009); Gallo et al. (2016); Angell. (2014).

Among the thermodynamic anomalies observed in real water Chaplin (2023), here
we show the density anomaly, characterized by a maximum in $\rho$ as function
of $T$ at constant $p$, and the minimum in $\alpha_{p}$ \- which corresponds
to the region where the fluid expand over cooling. The isobars are shown in
black in Fig. 16(a) and (b), with the extrema connected by the red curve.

Figure 17: (a) Orientational and (b) translational order parameter as function
of pressure for isotherms in the fluid phase, from $T=0.025$ up to 0.600. The
red lines indicate the extrema in each parameter, delimiting the anomalous
region.

The orientational structural anomaly is usually characterized by a maxima in
$\tilde{q}_{6}$ under compression, as we observed – Fig. 17(a). Also, a minima
in $\tilde{q}_{4}$ was observed in a similar region of the phase diagram. The
translational structural anomaly can be observed by an anomalous decrease in
$\tau$ under compression - in normal fluids, the translational ordering
increases with $p$. Likewise, for regular fluids the pair entropy $s_{2}$
decreases under compression, but water has an anomalous region delimited by a
minima and a maxima in the $s_{2}\times p$ isotherm. The behavior of both
translational parameters are shown in Fig. 17(b).

Figure 18: $D\times p$ isotherms from $T=0.025$ (bottom curve) up to 0.600.
The dashed red lines indicate the minima and maxima in $D$.

Finally, the diffusion anomaly is characterized by the increase in the self-
diffusion coefficient $D$ under compression. As the $D\times p$ isotherms in
Fig. 18 show, there is a region in the phase diagram where the fluid diffuses
faster as the pressure increases. All these anomalies are indicated in the
$T\times\rho$ phase diagram, Fig. 19. As in the original works by de Oliveira
and co-workers De Oliveira et al. (2006); Barros De Oliveira et al. (2006), a
waterlike hierarchy of anomalies was obtained as the diffusion, $\tau$,
$s_{2}$, $\tilde{q}_{6}$ and $\tilde{q}_{4}$ extrema (DE, $\tau$E and
$s_{2}$E, $\tilde{q}_{6}$E and $\tilde{q}_{4}$E curves, respectively) show. At
last, it is interesting to notice how the diffusion and structural anomalous
region limit begin at the point were $k_{T}$ has a maxima, and $g(r_{1})=1.0$,
reinforcing the idea that the competition between two fluids leads to the
anomalous behavior.

Figure 19: $T\times\rho$ phase diagram. Isobars ranging from $p=0.050$ up to
$p=4.000$ and $T=0.025$ to 0.0600 are shown in dashed gray lines with circular
points. The Extrema (E) – that can be a maxima or a minima – and the
Discontinuity (D) observed in the response functions that separate the phases
are indicated by the points: magenta filled right triangle for a discontinuous
behavior in the $c_{p}\times T$ and $\alpha_{p}\times T$ isobars, magenta open
right triangle for a extrema in the $c_{p}\times T$ and $\alpha_{p}\times T$
isobars, cyan filled left triangle for a discontinuous behavior in the
$c_{V}\times T$ isochores, cyan open left triangle for a maxima in the
$c_{V}\times T$ isochores, green filled up triangle for a discontinuous
behavior in the $k_{T}\times p$ isotherm, green open up triangle for a maxima
in the $k_{T}\times p$ isotherms. The anomalous region are defined by: the TMD
line shown by the black filled circles connected by a solid line, the
$\alpha_{p}$ minima indicated by open circles, the diffusion extrema in red
squares connected by a solid line, the $\tau$ and $s_{2}$ extrema lines
indicated by filled and open violet diamonds and the $\tilde{q}_{4}$ and
$\tilde{q}_{6}$ extrema by filled and open violet triangles, respectively.

##  4 Conclusions and Perspectives

In this paper we have explored the phase diagram of the core-softened
potential proposed by de Oliveira and co-authors De Oliveira et al. (2006);
Barros De Oliveira et al. (2006). We depicted for the first time the solid
phases of this fluid, showing that it has solid poly(a)morphism. At low
densities, a BCC crystalline phase is observed, followed by an amorphous phase
as we compress the system. This ordered-disordered solid transition is a
consequence from the competition between the two length scales in the
particle’s interaction, with an amorphous cluster growing in the BCC crystal ,
promoting the amorphization. It reflects in the fluid near the BCC and AS1
phases, where we can correlate a maxima in $k_{T}$ to a transition from a BCC-
like fluid to a AS1-like fluid, where the first coordination shell reaches
$g(r)=1.0$. Compressing the first amorphous phase, the system changes to a HCP
crystal phase, with changes in the second coordination shell, and at high
pressures, a second amorphous region is observed – here the long range
ordering rules the ordered-disordered transition. However, no amorphous
cluster was observed in the HCP phase. Our findings show that even for fluid
without LLCP we can find a maxima in the $k_{T}$ connected to the BCC-AS1
phase transition and to the existence of amorphous clusters in the BCC
crystal. It is remarkable that the mechanism observed in the 3D system is
similar to what we have observed in the 2D case Cardoso et al. (2021);
Nogueira and Bordin (2022). However, in the 2D limit there were more than one
anomalous regions, corresponding to ordered-disordered transitions where an
amorphous cluster grows inside the crystal. Then, for confined anomalous
fluids, there should be a minimal pore size where we will observe a 2D like or
a 3D like behavior.

##  5 acknowledgments

Without public funding this research would be impossible. $NpT$ simulations
were performed in the Quindim/Camafeu Workstations from the Bordin Lab at
UFPel – proc. 403427/2021-5 from the Brazilian National Council for Scientific
and Technological Development (CNpq). JRB is also grateful to the CNPq, proc.
407818/2018-9, and Research Support Foundation of the State of Rio Grande do
Sul (FAPERGS), TO 21/2551-0002024-5, for the funding support. JRB thanks L. C.
de Matos for the illuminating insights and L. Pinheiro and T. P. O. Nogueira
for the critical reading of the manuscript.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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