# Core-softened water-alcohol mixtures: the solute-size effects

Murilo Sodré Marques<EMAIL_ADDRESS>Centro das Ciências Exatas e
das Tecnologias, Universidade Federal do Oeste da Bahia
Rua Bertioga, 892, Morada Nobre, CEP 47810-059, Barreiras-BA, Brazil
Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento
Gonçalves 9500, Caixa Postal 15051, CEP 91501-970, Porto Alegre - RS, Brazil
Vinícius Fonseca Hernandes  Departamento de Física, Instituto de Física e
Matemática, Universidade Federal de Pelotas. Caixa Postal 354, 96001-970,
Pelotas-RS, Brazil.  José Rafael Bordin  Departamento de Física, Instituto de
Física e Matemática, Universidade Federal de Pelotas. Caixa Postal 354,
96001-970, Pelotas-RS, Brazil.

###### Abstract

Water is the most anomalous material on Earth, with a long list of
thermodynamic, dynamic and structural behavior that deviate from the expected.
Recent studies indicate that the anomalies can be related to a competition
between two liquids, which means that water has a potential liquid-liquid
phase transition (LLPT) that ends in a liquid-liquid critical point (LLCP). In
a recent work [J. Mol. Liq. 320 (2020) 114420], using molecular dynamics
simulations and a core-softened potential approach, we have shown that adding
a simple solute as methanol can ”kill” the anomalous behavior as the LLCP is
suppressed by the spontaneous crystallization in a hexagonal closed packing
(HCP) crystal near the LLPT. Now, we extend this work to see how alcohols with
longer chains will affect the complex behavior of water mixtures in the
supercooled regime. Besides CS methanol, ethanol and 1-propanol were added to
CS water. We observed that the anomaly vanishes as the alcohol chain
increases. Curiously, the LLCP did not vanishes together with the anomalous
behavior. The reason is that the mechanism for ethanol and 1-propanol is
distinct to the one observed previously for methanol: here, the longer chains
affect the competition between the scales in the CS potential. Also, the chain
size affects the solid phases, favoring the HCP solid and the amorphous solid
phase over the body-centered cubic (BCC) crystal. This findings helps to
elucidate the behavior of water solutions in the supercooled regime, and
indicates that the LLCP can be observed in systems without anomalous behavior.

††preprint: APS/123-QED

##  I Introduction

Life, as we known, started and evolved in water solutions. Then, we can say
that elucidate the demeanor of complex molecules in water, at both micro- and
macro- levels, is of paramount importance in modern science [1, 2]. Although
the behavior of complex biological proteins in water is a huge and complex
problem, with many hydrophobic and hydrophilic sites, we can draw some
information from simpler systems. In this way, a special class of aqueous
solutions are those containing short-chain alcohols (i.e., alcohols with a
small number of carbon atoms in the chain, like methanol, ethanol, or
1-propanol), most of which are miscible in water over the full range of
concentrations [3, 4]. They have attracted a great attention of scientific
community for decades for a number of reasons: (i) they are ubiquitous in the
medical [5], food industries [6], transportation [7] and personal care [8],
among others, and thus have attracted much theoretical and experimental
attention; (ii) comparatively to water, the molecular structure of alcohols
shows the presence of an organic radical in place of one of the hydrogen
atoms; as a consequence, alcohols do not form a completely developed hydrogen-
bonded network, as in the case of water; (iii) On the other hand, the presence
of both the hydroxyl group and the organic radical, usually non-polar
(amphiphilic character), allows for the interaction with a huge number of
organic and inorganic compounds, making alcohols good solvents, since the
solute-solvent interaction can have the same order of magnitude as the solute-
solute and solvent-solvent interactions [9]. In addition, (iv) although the
interactions with molecules of a dual nature, such as alcohols, involve not
only the hydrophobic hydration of the non-polar moiety of the molecule but
also the hydrophilic interactions between the polar groups and the water
molecules, still they constitute a model for the investigation of the
hydrophobic effect [10, 11].

With water as a solvent, both the complexity of analysis and the richness of
phenomena observed for such solutions are highlighted. Water is the most
anomalous material, with more than 70 known anomalies [12]. Probably, the most
well known is the density anomaly. While most of the materials increases the
density upon cooling, liquid water density decreases when cooled from 4oC to
0oC at atmospheric pressure [13]. Recent findings indicated a relation between
the anomalies and another unique feature of water: the liquid polymorphism and
the phase transition between two distinct liquids [14]. In addition to the
usual liquid-gas critical point (whose near-critical properties are so
drastically different from those of liquid water [15]), since the 1990s [16]
the existence of a second critical point - the liquid-liquid critical point
(LLCP) has been hypothesized by simulations and subject of extensive debate
[17, 18, 19, 20, 21]. It has not yet been reported from experiments once is
located in the so-called no-man’s land: due the spontaneous crystallization it
is (almost) impossible to reach this region by experiments - however, some
recent experiments show strong evidence of the existence of the LLCP [22, 23,
24]. The LLCP locates at the end of a first order liquid-liquid phase
transition (LLPT) line between low-density liquid (LDL) and high-density
liquid (HDL) at low temperatures [16, 25, 26, 27, 28, 29, 30, 31, 14, 32].
Unlike the liquid–gas phase transition (LGPT), which always has a positive
slope of the first-order phase transition line because $\Delta S>0$ and
$\Delta V>0$ (Clausius–Clapeyron relation), the LLPT line can indeed be either
negative or positive, depending on the pair atomic interactions of the model
[33].

The critical behavior of mixtures has been widely analyzed from both an
experimental and theoretical point of view [34, 35, 36, 37, 38, 39, 40, 41,
42, 43]. Studies on the effects of solute size on the thermodynamic and
structural properties of the aqueous solutions have been done [44, 45, 46, 47,
48] and, according to the authors’ knowledge, there are still no analysis of
alcohol chain size influence on the critical properties and on polymorphism of
these mixtures. In this way, some questions that arise are: how longer chains
of alcohol will affect the behavior of CS water in the supercooled regime? How
possible changes in the critical behavior can be related to the anomalous
behavior? Also, there is some effect in the spontaneous crystallization
observed near the LLCP?

To answer this question, we perform extensive simulations in order to analyze
the phase diagram of water and water-alcohols mixtures, where both water and
hydroxyls are modeled by core-softened potentials, which already have a long
history of describing water anomalies both in bulk and confined environments
[49, 50, 51, 52, 53, 54]. In our very recent work [55] we have explored the
supercooled regime of pure water, pure methanol and their mixtures using a
core-softened (CS) potential model [2]. In this scheme, hydroxyl groups of
different molecules interact by a continuous shouldered well (CSW) potential,
while all other interactions are reproduced by LJ-like contributions, and our
principal focus in that publication was on the relations between density
anomaly, liquid-liquid phase transition and spontaneous crystallization by
means of Molecular Dynamics simulations. In the present work, in a similar way
to that performed by Urbic et al [56], we extend this scheme to ethanol
($CH_{3}-CH_{2}-OH$) and 1-propanol ($CH_{3}-CH_{2}-CH_{2}-OH$), by modeling
such alcohols as linear chains constituted by three (trimers) and four
(tetramers) partially fused spheres, respectively, and the primary objective
is to complement our previous study by exploring the influence of the size of
the solute on the thermodynamic properties of mixtures of core-softened water
and alcohols: methanol, ethanol and 1-propanol.

The remaining of the paper is organized as follows. In Section II we present
our interaction models for water and alcohols molecules, and summarize the
details of the simulations. Next, in section III our most significant results
for our CS mixture model are introduced. In particular, we will focus on the
concentration dependence of the LLCP and influence on the excess entropy of
mixture in comparison with pure CS water. The paper is closed with a brief
summary of our main conclusions and perspectives.

##  II Model and simulation details

Our water-like solvent here will be modeled by a 1-site core-softened fluid in
which particles interact with the potential model proposed by Franzese [57].
Water-like particles $W_{CS}$ are represented by spheres with a hard-core of
diameter $a$ and a soft-shell with radius $2a$, whose interaction potential is
given by

$\displaystyle U^{CS}(r)$ $\displaystyle=$
$\displaystyle\frac{U_{R}}{1+exp\left[\Delta(r-R_{R})\right]}$ (1)
$\displaystyle-
U_{A}exp\left(-\frac{(r-R_{A})^{2}}{2\delta_{A}^{2}}\right)+U_{A}\left(\frac{a}{r}\right)^{24}.$

With the parameters $U_{R}/U_{A}=2$, $R_{R}/a=1.6$, $R_{A}/a=2$,
$(\delta_{A}/a)^{2}=0.1$, and $\Delta=15$ this potential displays an
attractive well for $r\sim 2a$ and a repulsive shoulder at $r\sim a$, as can
be seen in figure 1(a) (red curve). The competition between these two length
scales leads to water-like anomalies, as the density, diffusion anomalies, and
to the existence of a liquid liquid critical point [57, 58, 59, 60].

As a direct extension and building on our previous work on mixtures of water
and methanol, where a single methanol molecule was constituted by two tangent
spheres, in this work we have followed the extension made by Urbic et al
towards ethanol ($CH_{3}-CH_{2}-OH$) and 1-propanol
($CH_{3}-CH_{2}-CH_{2}-OH$) being modeled as linear trimers and tetramers,
respectively (Fig. 1). Trimers are constructed as linear rigid molecules
consisting of three partially fused spheres, where two adjacent spheres are
placed at fixed distance $L_{ij}=0.60a$, with $a$ as the unit length.
Analogously, tetramers are modeled as linear rigid molecules consisting of
four partially fused spheres, where two adjacent spheres are placed at fixed
distance $L_{ij}=0.78a$. In all models, hydroxyl groups interact through a
soft-core potential (eq 1), while $CH_{2}$ and $CH_{3}$ groups are nonpolar
and interact through LJ-like potential,

$U^{LJ}=\frac{4}{3}2^{2/3}\epsilon\left[\left(\frac{\sigma}{r}\right)^{24}-\left(\frac{\sigma}{r}\right)^{6}\right],$
(2)

$OH-CH_{2}$ and $OH–CH_{3}$ interactions are of $LJ$ type as well. $LJ$
parameters are reported in Table 1. As in last work, quantities are reported
in reduced dimensionless units relative to the hydroxyl group diameter and the
depth of its attractive well.

| $L_{ij}$ | $\epsilon_{1n}$ | $\epsilon_{nn}$ | $\sigma_{1n}$ | $\sigma_{nn}$
---|---|---|---|---|---
Dimers | 1.000 | $0.316$ | $0.100$ | $1.000$ | $1.000$
Trimers | 0.600 | $0.400$ | $0.400$ | $1.115$ | $1.230$
Tetramers | 0.780 | $0.500$ | $0.500$ | $1.115$ | $1.230$
Table 1: Potential parameters for dimers ($n=2$), trimers ($n=2,3$), and
tetramers ($n=2,3,4$); the $OH$ group is labeled with $1$. As for the bond
length, $j=i+1$ [56].

Figure 1: In (a), (b) and (c), our model for methanol, ethanol and 1-propanol
is outlined, while in (d) we see the interaction between water and hydroxyl’s
is described by the CSW potential, while other interactions behave like a 24-6
Lennard-Jones potential.

The simulations were performed in the $NPT$ ensemble with a fixed number of
molecules ($N_{tot}=1000$). $N_{alc}=x_{alc}N_{tot}$ is the number of alcohol
(methanol, ethanol or propanol) molecules and $N_{w}=N_{tot}-N_{alc}$ that of
water molecules, where $x_{alc}$ is the alcohol mole fraction, which has been
varied from 0.0 (pure water), 0.01, 0.05 and 0.1 (we’ve focused in low-
concentration range). The temperature and pressure were controlled using the
optimized constant pressure stochastic dynamics proposed by Kolb and Dünweg
[61] as implemented in the ESPResSo package [62, 63]. This barostat
implementation allows for the use of a large time step. This was set to
$\delta t^{*}=0.01$, and the equations of motion were integrated using the
velocity Verlet algorithm. The Langevin thermostat [64], that keeps the
temperature fixed, has a coupling parameter $\gamma_{0}=1.0$. The piston
parameters for the barostat are $\gamma_{p}=0.0002$ and mass $m_{p}=0.001$.
The particles were randomly placed in a cubic box, and then dynamics was run
for $5\times 10^{6}$ time steps in the $NVT$ ensemble to thermalize the
system. This was followed by $1\times 10^{6}$ time steps in the $NPT$ ensemble
to equilibrate the system’s pressure and $1\times 10^{7}$ time steps further
for the production of the results, with averages and snapshots being taken at
every $1\times 10^{5}$ steps. To ensure that the system temperature and
pressure were well controlled we averaged this quantities during the
simulations. As well, to monitor the equilibration the evolution of the
potential energy along the simulation was followed. Here, the molecule density
$\rho$ is defined as $N_{m}/<V_{m}>$ with $<V_{m}>$ being the mean volume at a
given pressure and temperature. Isotherms were evaluated from $T^{*}=0.20$ up
to $T^{*}=0.70$ with changing intervals - a finer grid was used in the
vicinity of the critical points. In the same sense, the pressure was varied
from $P=0.005$ up to $P=0.30$ with distinct intervals.

Also in a similar way to the previous work, we evaluated the temperature of
maximum density (TMD) and the locus of the maximum of response functions close
to the critical point at the fluid phase (the isothermal compressibility
$\kappa_{T}$, the isobaric expansion coefficient $\alpha_{P}$ and the specific
heat at constant pressure $C_{P}$):

$\kappa_{T}=\frac{1}{\rho}\left(\frac{\partial\rho}{\partial
P}\right)_{T},\alpha_{P}=-\frac{1}{\rho}\left(\frac{\partial\rho}{\partial
T}\right)_{P},C_{P}=\frac{1}{N_{tot}}\left(\frac{\partial H}{\partial
T}\right)_{P},$ (3)

where $H=U+PV$ is the system enthalpy, with $V$ the mean volume obtained from
the $NPT$ simulations. The quantities shown in the Electronic Supplementary
Material (ESI)† were obtained by numerical differentiation. As consistency
check, we have obtained the same maxima locations when using statistical
fluctuations: the compressibility is a measure of volume fluctuations, the
isobaric heat capacity is proportional to the entropy fluctuations experienced
by N molecules at fixed pressure, and the thermal expansion coefficient
reflects the correlations between entropy and volume fluctuations [64, 65].

In order to describe the connection between structure and thermodynamics, we
have analyzed the radial distribution function (RDF) $g(r^{*})$, which was
subsequently used to compute the excess entropy. $s_{ex}$ can be obtained by
counting all accessible configurations for a real fluid and comparing with the
ideal gas entropy [66]. Consequently, the excess entropy is a negative
quantity since the liquid is more ordered than the ideal gas. Note that
$s_{ex}$ increases with temperature just like the full entropy $S$ does; in
fact $s_{ex}\rightarrow 0$ as temperature goes to infinity at fixed density
because the system approaches to an ideal gas [67, 68]. Analytically, the
excess entropy may be computed if the equation of state is known [69]. A
systematic expansion of $s_{ex}$ exists in terms of two-particle, three-
particle contributions, etc.,

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

The two-particle contribution is calculated from the radial distribution
function $g(r)$ as follows:

$s_{2}=-2\pi\rho\int_{0}^{\infty}\left[g(r)lng(r)-g(r)+1)\right]r^{2}dr,$ (5)

since $s_{2}$ is the dominant contribution to excess entropy [70, 71] and it
is proved to be between $85\%$ and $95\%$ of the total excess entropy in
Lennard-Jones systems [72]. Also, the translational order parameter $\tau$ was
evaluates. It is defined as [73]

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

where $\xi=r\rho^{1/2}$ is the interparticle distance $r$ scaled with the
average separation between pairs of particles $\rho^{1/2}$. $\xi_{c}$ is a
cutoff distance, defined as $\xi_{c}=L\rho^{1/2}/2$, where $L$ is the
simulation box size. For an ideal gas (completely uncorrelated fluid),
$g(\xi)=1$ and $\tau$ vanishes. For crystal 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.

Another structural quantity evaluated was the orientational order parameter
(OOP), that gives insight on the local order [74, 73, 75, 76]. 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 vector from particle $i$ to its neighbour $j$. The OOP for a whole system
is obtained taking the average over the parameter value for each particle $i$,
$q_{l}=\langle q_{l}(i)\rangle_{i}$. In this work we evaluated the OOP for
$l=6$, using the freud python library [77], and the number of neighbors for
each particle was found computing Voronoi diagrams using voro++ [78].

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)

For alcohol molecules we have considered the center of mass displacement. 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 [79, 80]. Ovito was also employed to
visualize the phases and take the system snapshots.

##  III Results and discussion

###  III.1 Pure CS Water phase diagram

Figure 2: (a) $PT$ phase diagrams for pure CS water showing the solid phases I
(BCC solid), II (HCP solid) and III (amorphous solid) and the low and high
density liquid phases. The points in the phase separations indicate distinct
discontinuities or maxima in the evaluated response functions. The Widom Line
corresponds to maxima in $\kappa_{T}$. The solid-liquid coexistence lines were
draw based in the discontinuities in the pair excess entropy, the structure
factor (not shown here for simplicity) and in the diffusion constant, as
indicated in the figure (b). Also, the pair excess entropy (not shown here for
simplicity), the structure factor and $D^{*}$ have discontinuities in the LDL-
HDL transition, as we show in the figure (c) for the subcritical isobar
$P^{*}=0.13$, and a fragile to strong transition for the supercritical isobar
$P^{=}0.15$ as it crosses the Widom Line.

Water is fascinating and unique. From its numerous anomalous behaviors the
most well known is, probably, the density anomaly. It be characterized by the
Temperature of Maximum Density (TMD) line, that corresponds to the maxima in
the $\rho\times T$ isobar. Recent findings indicates that the water anomalies
are related to the Liquid-Liquid Critical Point (LLCP) [16, 81]. Although
hypothetical and not observed (yet) in experiments, there is strong and many
evidences that the water liquid polymorphism in the supercooled regime ends in
the LLCP [14, 82]. We can estimate the critical point location using the
isothermal density derivatives of the pressure [83]

$\left(\frac{\partial
P}{\partial\rho}\right)_{T}=\left(\frac{\partial^{2}P}{\partial\rho^{2}}\right)_{T}=0\;.$
(11)

Coming from the supercritical region, the LLCP lays at the end of the Widom
Line (WL) - a line in the $P\times T$ phase diagram that can be obtained by
the maxima in the response function $\kappa_{T}$ and corresponds to the
separation between LDL-like and HDL-like behavior in the supercritical regime.
In the figure 2 (a) we show the CS water phase diagram obtained from our
simulations. The WL, indicated by the dotted purple line and the purple
squares, ends at the LLCP. Below the LLCP we have the transition between the
liquid phases, indicated by the discontinuity in the thermodynamic property
$\kappa_{T}$ – shown in the ESI† for all isotherms – and in the structural and
dynamic properties. For instance, the upper panel in the figure 2(b) shows the
structure factor $\tau$ for the subcritical pressure $P^{*}=0.13$ and for the
supercritical pressure $P^{*}=0.15$ as function of the inverse of temperature.
As $T$ decreases we can see a discontinuity in the subcritical isobar,
indicating an abrupt change in the fluid structure. On the other hand, the
supercritical isobar has a monotonic increase in $\tau$ as $T$ decreases,
indicating an increase in the particles order. Similarly, the dependence of
the diffusion coefficient $D$ with $T^{*-1}$ is discontinuous in the
supercritical isobar. For the supercritical pressure $P^{*}=0.15$ we see a
change in the diffusion inclination with temperature when it crosses the Widom
Line, indicating the HDL-dominated to LDL-dominated regime. These results are
in agreement with our recent work obtained by the heating of the system [55]
and previous works employing this potential [57, 58, 59, 60].

Figure 3: (a) CS water-water radial distribution function (RDF) along the
isotherm $T^{*}=0.26$. Black lines are the pressures in the BCC phase, red
lines in the HCP phase and cyan in the amorphous phase. System snapshots in
the (b) BCC phase, HCP phase with (c)straight or (d) rippled planes and in (e)
the amorphous solid phase.

However, the solid phases weren’t explored for this system. In fact, the HCP
phase was observed in our work [55] and in the study by Hus and Urbic using
the methanol model [84, 85]. Here, exploring a larger region in the phase
diagram, we characterized three distinct solid phases. The solid phase I
corresponds to a body-centered cubic (BCC) crystal at lower pressures.
Increasing $P$ it changes to the solid phase II, with a hexagonal closed
packed (HCP) structure, and at even higher compression we observe the
amorphous solid, named phase III. The transition between the solid phases, and
from LDL to HCP, are well defined by the discontinuous behavior in
$\kappa_{T}$, shown in the SI. The transition from solid phase I to LDL and
from solid phase II to HDL have discontinuities in the response functions
$\alpha_{p}$ and $C_{p}$, shown in the SI. Also, the structure (here
characterized by the pair excess entropy) and the dynamic behavior (given by
the diffusion coefficient) are discontinuous for these solid-fluid transition.
This can be observed in the figure 2(c). Here, the pressure $P^{*}=0.05$ is a
isobar that cross the BCC-LDL transition, $P^{*}=0.11$, 0.14 and 0.20 the HCP-
HDL transition and $P^{*}=0.28$ the amorphous-HDL transition. As we can see,
the structure and dynamics change smoothly for this last transition, as the
magenta line for $P^{*}=0.28$ indicates in the figure 2 (c). Also, the
response functions $C_{p}$ and $\alpha_{p}$ are not discontinuous in this
transition, but have a maximum – indicating that this is a second order,
smooth transition. The distinct phases can also be observed when we analyze
the water-water radial distribution function (RDF) $g_{ww}(r)$ along one
isotherm. For instance, we show in the figure 3(a) the $g_{ww}(r)$ for
pressures ranging from $P^{*}=0.01$ to $P^{*}=0.30$ along the isotherm
$T^{*}=0.26$. We can see clear changes in the structure as $P$ varies. At
lower pressures the particles are separated mainly at the second length scale,
characterizing the BCC phase – a snapshot at $P^{*}=0.01$ and $T=0.26$ is
shown in the figure 3(b). Increasing the pressure the system changes for the
HCP phase, where the occupation in the first length scale dominates the
structure. In fact, the HCP planes are separated by a distance equal to the
second length scale, while the distance between particles in the same plane is
the first length scale. It becomes clear when we use the Ovito [80] feature
”create bonds” if the distance is equal to the first scale. While for the BCC
snapshot we did not see any bond, for the HCP snapshot at $P^{*}=0.14$ in the
figure 3(c) we can see the bonds between particles in the same plane. As $P$
grows, the HCP planes get rippled, as we can see in figure 3(d) for
$P^{*}=0.24$ and in the behavior of the RDF red lines. Finally, it changes to
an amorphous structure at high pressure, as we show for $P^{*}=0.30$ in figure
3(e).

Now, with the phase behavior of the CS water model depicted, we can see how
the presence of short alcohol affects the observed phases and the density
anomaly.

###  III.2 Water-short alcohol mixtures

Figure 4: (a) TMD behaviour of all CS alcohols used in this work: for
$\chi_{alc}=0.05$, 1-propanol didn’t show TMD and for $\chi_{alc}=0.10$, only
methanol shows TMD. (b) CSW-CSW radial distribution function $g_{ww}(r)$ along
the isobar $P^{*}=0.08$ for water-ethanol at $\chi_{alc}=0.10$ case with
temperatures ranging from $T=0.50$ (black solid line) to $T=0.68$ (magenta
solid line). The intermediate temperatures are shown with red dashed lines.
The arrows indicate the competition between the scales as $T$ increases. (c)
is for the case with ethanol concentration at $\chi_{alc}=0.05$, were the
competition between the scales and the TMD are still observable, while for (d)
$\chi_{alc}=0.10$ both competition and TMD vanish.

Small concentrations of short chain alcohols such as methanol, ethanol and
1-propanol create a very interesting effect in the TMD line [86, 87]. They act
as ”structure maker”, promoting the low density ice-like water structure and
increasing the TMD. This is usually observed for alcohol concentrations
$\chi_{alc}$ smaller than 0.01 - and is not our goal here. We want to analyze
the TMD vanishing and what happens in the phase diagram as it vanishes. This
can be observed as $x$ increases and methanol acts as ”structure breaker”.
Using the CS model for water-methanol mixtures [55] we found that the TMD
persists up to high methanol concentrations, as $\chi<70$% – much higher than
in experiments. This is a consequence of the model: the same potential is
employed for water-water, water-OH and OH-OH. On the other hand, the energy
for the water-water h-bonds is equal to the water-OH h-bonds. Nevertheless, we
can increase the structure breaker effect by increasing the solute size. In
fact, for ethanol the TMD line vanishes at $\chi_{alc}=0.10$, while for
1-propanol the TMD is only observed at $\chi_{alc}=0.01$ \- the TMD lines are
shown in the figure 4(a).

Figure 5: $PT$ phase diagrams for aqueous solutions of (a) methanol, (b)
ethanol and (c) propanol for all concentrations analyzed in this work.

The water anomalous behavior is related to the competition between two liquids
that coexist [14, 88, 89]. This competition can be observed using the
$g_{ww}(r)$. Here we show the RDFs the isobar $P^{*}=0.080$ between the
temperatures $T^{*}=0.50$ and $T^{*}=0.68$ for a fraction of ethanol of
$\chi_{alc}=0.01$ in the figure 4(b), for $\chi_{alc}=0.05$ in the figure 4(c)
and for $\chi_{alc}=0.10$ in figure 4(d). As we can see, for the fractions
$\chi_{alc}=0.01$ and $\chi_{alc}=0.05$ we observe the competition between the
scales: the water particles migrates from the second length scale to the first
length scale as $T$ increases, as indicated by the arrows. On the other hand,
for $\chi_{alc}=0.10$ there is practically no increase in the occupation of
the first length scale as the occupation in the second length scale decreases.
Once there is no competition, we do not observe the density anomaly. This
indicates that adding higher concentration of alcohol changes the competition
between the scales in the CSW water model and that the CS alcohol chain length
also affects the competition - in our previous work for methanol, we only
observe this at $\chi_{alc}=0.70$ [55].

Figure 6: Diffusion coefficient versus pressure for all solutions analyzed in
this work. From bottom to top in each diagram, we have the isotherms
$T^{*}=0.34,0.38,0.42,...0.66$. Each row represents a concentration of solute:
$x_{alc}=0.01,0.05\;and\;0.1$

Curiously, even without the density anomaly, all the water-alcohol mixtures
have liquid-liquid phase transition. In the figure 5 we show the $PT$ phase
diagrams for all the fractions and types of alcohols. After the liquid-liquid
critical point, The Widom Line (WL) separates water with more HDL-like local
structures at high temperatures from water with more LDL-like local structures
at low temperatures [90]. Looking at the diffusion coefficient $D^{*}$
isotherms we can see the distinct transitions. At lower temperatures, as
$T^{*}=0.34$ it melts from the solid phase to the HDL phase at high pressures,
as we show in the figure 6. Increasing $T^{*}$, we can see the system going
from the LDL phase (with $D^{*}>0$) to the HCP phase, with diffusion near
zero, to the HDL phase, were $D^{*}$ increases again. In the isotherms that
cross the LDL-HDL transition we can see the discontinuity in the curve,
indicating the phase transition. Above the LLCP we can see a change in the
$D^{*}\times P^{*}$ curve behavior as it crosses the WL.

Figure 7: Local bond orientation order parameter $q_{6}$ as function of
pressure for the isotherms (a) $T^{*}=0.46$, (b) $T^{*}=0.56$ and (c)
$T^{*}=0.66$.

The same discontinuity can be observed in the structural behavior. Besides
$\tau$ and $s_{2}$, we also evaluated the $q_{6}$ to analyze the structure.
For simplicity, we show here only the case of pure water and water-ethanol
mixtures. Along the isotherm $T^{*}=0.46$, that crosses the phases LDL, HCP
and HDL, we can see in the figure 7(a) clearly the discontinuities
correspondents to this transitions for all fractions of ethanol. Also, it is
noticeable how the ethanol is structure breaker: the value of $q_{6}$ in the
LDL and HCP phases are smaller as $\chi_{alc}$ increases. Along the isotherm
$T^{*}=0.56$, shown in the figure 7(b), we can see the same structure breaker
effect in the LDL regime, and a discontinuous transition to the HDL phase. In
contrast, above the LLCP there is no discontinuity. The isotherm $T^{*}=0.66$,
shown in the figure 7(c), have a smooth decay in $q_{6}$ as it crosses the WL
and goes from LDL-like to HDL-like. This is interesting, once it indicates
that a system can have a LLCP without have density anomaly.

Figure 8:  CSW-CSW radial distribution function $g_{ww}(r)$ along the isotherm
$T^{*}=0.26$ for water-alcohol mixtures with $\chi_{alc}=0.10$ for (a)
methanol, (b) ethanol and (c) propanol. Snapshots for the correspondents
mixtures: $\chi_{alc}=0.10$ of (d) methanol, (e) ethanol and (f) propanol. (g)
Local bond orientation order parameter $q_{6}$ as function of pressure for the
isotherm $T^{*}=0.26$.

As in the pure CSW water case, three solid phases were observed. However, an
interesting finding is how the carbon chain length affects the solid phases.
It is clear that longer apolar chains affects the extension of the region
occupied by each solid phase: the BCC crystal (region I) loses space, moving
to lowers pressures. Likewise, the HCP crystal (region II) moves to lower
pressures as the alcohol chain increase. However, the area occupied in the
$PT$ phase diagram remains - it just shifts to lower pressures as the size and
fraction of alcohol increases. For this two solid phases, the temperature
range in the $PT$ phase diagram seems to be independent of the fraction
$\chi_{alc}$. On the other hand, the amorphous solid phase (region III) is
favored as the it expands its extension to lower pressures and higher
temperatures as the apolar chain grows. To understand why the HCP phase
remains occupying a large area in the phase diagram while the BCC area shrinks
we show in the figure 8 the $g_{ww}(r)$ for the three alcohols with fraction
$\chi_{alc}=0.10$ along the isotherm $T^{*}=0.26$ from $P^{*}=0.01$ to
$P^{*}=0.30$. Comparing the three cases with the pure CSW case, figure 3(a),
is clear that increasing the size of the apolar tail in the alcohol favor the
occupation in the second length scale. While for methanol and ethanol we see a
low occupation in the first length scale at the lower pressures, for propanol
this occupancy is high even for $P^{*}=0.01$. This is consequence of a alcohol
bubble formation: for the CS methanol, once the molecule size is comparable to
the second length scale and the OH is modeled as a CSW particle, the molecules
merge in the BCC structure, as we show in the figure 8(d). However, the longer
alcohols creates bubbles, as we can see in figure 8(e) and (f) for ethanol and
propanol, respectively. It alters the water structure near the bubbles,
favoring the first length scale. It reflects in the local orientation. In the
figure 8(g) we show the $q_{6}$ for this isotherm for the case of pure CS
water ($\chi_{alc}=0.00$) and the three fractions of ethanol. We can see that
for both crystal phases, BCC and HCP, the local order is affected by the
alcohol once $q_{6}$ is small for higher fractions.

##  IV Conclusions

In this paper we have explored the supercooled regime of pure water and
mixtures of water and short chain alcohols: methanol, ethanol and propanol
using a two-length scale core-softened potential approach. Our aim has been to
understand the influence of chain size on density anomaly, the liquid-liquid
phase transition and on the polymorphism which are generally observed in these
models. There’s a pronounced influence of the apolar chain size on solid
polymorphism. The BCC phase loses space in the phase diagram once longer
alcohols favor the occupancy in the first length scale. Once the HCP phase has
a higher occupancy in this length scale it is shifted to smaller pressures,
while the amorphous solid phase grows favored by the disorder induced by
alcohol.

The density anomaly vanishes as the competition between the scales is
suppressed in the LDL phase. However, the LDL-HDL phase transition persists
for all cases. This indicates that the competition between two liquids is
connected with waterlike anomalies, but the system will no necessarily have
the anomalies if it has a liquid-liquid phase transition and liquid-liquid
critical point. This results helps to understand the complex behavior of water
and mixtures with amphiphilic solutes in the supercooled regime.

###### Acknowledgements.

MSM thanks the Brazilian Agencies Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) for the PhD Scholarship and Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for the support to the
collaborative period in the Instituto de Química Fisica Rocasolano. VFH thanks
the CAPES, Finance Code 001, for the MSc Scholarship. JRB acknowledge the
Brazilian agencies CNPq and Fundação de Apoio a Pesquisa do Rio Grande do Sul
(FAPERGS) for financial support. All simulations were performed in the SATOLEP
Cluster from the Group of Theory and Simulation in Complex Systems from UFPel.

##  V References

## References

  * Rowlinson and Swinton [1982] J. Rowlinson and F. Swinton,  _Liquids and Liquid Mixtures_, Butterworths monographs in chemistry and chemical engineering (Butterworth Scientific, 1982).
  * Franzese and Rubi [2008] G. Franzese and M. Rubi,  _Aspects of Physical Biology: Biological Water, Protein Solutions, Transport and Replication_, Lecture Notes in Physics (Springer Berlin Heidelberg, 2008).
  * Koga [2007] Y. Koga,  _Solution Thermodynamics and its Application to Aqueous Solutions: A Differential Approach_ (Elsevier Science, 2007).
  * Ruckenstein and Shulgin [2009] E. Ruckenstein and I. Shulgin,  _Thermodynamics of Solutions: From Gases to Pharmaceutics to Proteins_, SpringerLink: Springer e-Books (Springer New York, 2009).
  * Hermkens  _et al._ [2018] N. K. Hermkens, R. L. Aspers, M. C. Feiters, F. P. Rutjes, and M. Tessari, Trace analysis in water-alcohol mixtures by continuous p-h2 hyperpolarization at high magnetic field, Magnetic Resonance in Chemistry 56, 633 (2018).
  * Nguyen  _et al._ [2020] T. V. N. Nguyen, L. Paugam, P. Rabiller, and M. Rabiller-Baudry, Study of transfer of alcohol (methanol, ethanol, isopropanol) during nanofiltration in water/alcohol mixtures, Journal of Membrane Science 601, 117907 (2020).
  * Prslja  _et al._ [2019] P. Prslja, E. Lomba, P. Gómez-Álvarez, T. Urbic, and E. G. Noya, Adsorption of water, methanol, and their mixtures in slit graphite pores, The Journal of Chemical Physics 150, 024705 (2019), https://doi.org/10.1063/1.5078603 .
  * Vane [2019] L. M. Vane, Review: membrane materials for the removal of water from industrial solvents by pervaporation and vapor permeation, Journal of Chemical Technology & Biotechnology 94, 343 (2019).
  * Franks  _et al._ [2000] F. Franks, R. S. of Chemistry (Great Britain), and R. S. of Chemistry (Great Britain).,  _Water: A Matrix of Life_, RSC paperbacks (Royal Society of Chemistry, 2000).
  * Southall  _et al._ [2002] N. T. Southall, K. A. Dill, and A. D. J. Haymet, A view of the hydrophobic effect, The Journal of Physical Chemistry B 106, 521 (2002), https://doi.org/10.1021/jp015514e .
  * Xi and Patel [2016] E. Xi and A. J. Patel, The hydrophobic effect, and fluctuations: The long and the short of it, Proceedings of the National Academy of Sciences 113, 4549 (2016), https://www.pnas.org/content/113/17/4549.full.pdf .
  * Chaplin [2020] M. Chaplin, Anomalous properties of water, http://www.lsbu.ac.uk/water/anmlies.html (2020).
  * Kellu [1975] G. S. Kellu, Density, thermal expansivity, and compressibility of liquid water from 0.deg. to 150.deg.. correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale., J. Chem. Eng. Data 20, 97 (1975).
  * Gallo  _et al._ [2016] P. Gallo, K. Amann-Winkel, C. A. Angell, M. A. Anisimov, F. Caupin, C. Chakravarty, E. Lascaris, T. Loerting, A. Z. Panagiotopoulos, J. Russo, J. A. Sellberg, H. E. Stanley, H. Tanaka, C. Vega, L. Xu, and L. G. M. Pettersson, Water: A tale of two liquids, Chemical Reviews 116, 7463 (2016), pMID: 27380438, https://doi.org/10.1021/acs.chemrev.5b00750 .
  * Anisimov  _et al._ [2004] M. A. Anisimov, J. V. Sengers, and J. M. Levelt Sengers, Chapter 2 - near-critical behavior of aqueous systems, in  _Aqueous Systems at Elevated Temperatures and Pressures_, edited by D. A. Palmer, R. Fernández-Prini, and A. H. Harvey (Academic Press, London, 2004) pp. 29–71.
  * Poole  _et al._ [1992] P. Poole, F. Sciortino, U. Essmann, and H. Stanley, Phase-behavior of metastable water, Nature 360, 324 (1992).
  * Limmer and Chandler [2011] D. T. Limmer and D. Chandler, The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water, The Journal of Chemical Physics 135, 134503 (2011), https://doi.org/10.1063/1.3643333 .
  * Limmer and Chandler [2013] D. T. Limmer and D. Chandler, The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water. ii, The Journal of Chemical Physics 138, 214504 (2013), https://doi.org/10.1063/1.4807479 .
  * Poole  _et al._ [2013] P. H. Poole, R. K. Bowles, I. Saika-Voivod, and F. Sciortino, Free energy surface of st2 water near the liquid-liquid phase transition, The Journal of Chemical Physics 138, 034505 (2013), https://doi.org/10.1063/1.4775738 .
  * Palmer  _et al._ [2013] J. C. Palmer, R. Car, and P. G. Debenedetti, The liquid–liquid transition in supercooled st2 water: a comparison between umbrella sampling and well-tempered metadynamics, Faraday Discuss. 167, 77 (2013).
  * Palmer  _et al._ [2018] J. C. Palmer, A. Haji-Akbari, R. S. Singh, F. Martelli, R. Car, A. Z. Panagiotopoulos, and P. G. Debenedetti, Comment on “the putative liquid-liquid transition is a liquid-solid transition in atomistic models of water” [i and ii: J. chem. phys. 135, 134503 (2011); j. chem. phys. 138, 214504 (2013)], The Journal of Chemical Physics 148, 137101 (2018), https://doi.org/10.1063/1.5029463 .
  * Caupin [2015] F. Caupin, Escaping the no man’s land: Recent experiments on metastable liquid water, Journal of Non-Crystalline Solids 407, 441 (2015), 7th IDMRCS: Relaxation in Complex Systems.
  * Taschin  _et al._ [2013] A. Taschin, P. Bartolini, R. Eramo, R. Righini, and R. Torre, Evidence of two distinct local structures of water from ambient to supercooled conditions, Nat. Comm. 4, 2401 (2013).
  * Hestand and Skinner [2018] N. J. Hestand and J. L. Skinner, Perspective: Crossing the widom line in no man’s land: Experiments, simulations, and the location of the liquid-liquid critical point in supercooled water, The Journal of Chemical Physics 149, 140901 (2018), https://doi.org/10.1063/1.5046687 .
  * Stanley  _et al._ [1997] H. Stanley, L. Cruz, S. Harrington, P. Poole, S. Sastry, F. Sciortino, F. Starr, and R. Zhang, Cooperative molecular motions in water: The liquid-liquid critical point hypothesis, Physica A: Statistical Mechanics and its Applications 236, 19 (1997), proceedings of the Workshop on Current Problems in Complex Fluids.
  * Mishima and Stanley [1998] O. Mishima and H. Stanley, The relationship between liquid, supercooled and glassy water, Nature 396 (1998).
  * Stanley  _et al._ [2000] H. Stanley, S. Buldyrev, O. Mishima, M. Sadr-Lahijany, A. Scala, and F. Starr, Unsolved mysteries of water in its liquid and glassy phases, Journal of Physics: Condensed Matter 12, A403 (2000).
  * Sciortino  _et al._ [2003] F. Sciortino, E. la Nave, and P. Tartaglia, Physics of the liquid-liquid critical point, Physical review letters 91, 155701 (2003).
  * Debenedetti [2003] P. G. Debenedetti, Supercooled and glassy water, Journal of Physics: Condensed Matter 15, R1669 (2003).
  * Debenedetti and Stanley [2003] P. Debenedetti and H. Stanley, Supercooled and glassy water, Physics Today 56, 40 (2003).
  * Eugene Stanley [2005] H. Eugene Stanley, Liquid and glassy water: Two materials of interdisciplinary interest, in  _Handbook of Materials Modeling: Methods_, edited by S. Yip (Springer Netherlands, Dordrecht, 2005) pp. 2917–2922.
  * Handle  _et al._ [2017] P. H. Handle, T. Loerting, and F. Sciortino, Supercooled and glassy water: Metastable liquid(s), amorphous solid(s), and a no-man’s land, Proceedings of the National Academy of Sciences 114, 13336 (2017), https://www.pnas.org/content/114/51/13336.full.pdf .
  * Gibson and Wilding [2006] H. M. Gibson and N. B. Wilding, Metastable liquid-liquid coexistence and density anomalies in a core-softened fluid, Phys. Rev. E 73, 061507 (2006).
  * Forman and Thodos [1962] J. C. Forman and G. Thodos, Experimental determination of critical temperatures and pressures of mixtures: The methane-ethane-n-butane system, AIChE Journal 8, 209 (1962).
  * Stillinger and Helfand [1964] F. H. Stillinger and E. Helfand, Critical solution behavior in a binary mixture of gaussian molecules, The Journal of Chemical Physics 41, 2495 (1964), https://doi.org/10.1063/1.1726293 .
  * Levelt Sengers  _et al._ [1983] J. Levelt Sengers, G. Morrison, and R. Chang, Critical behavior in fluids and fluid mixtures, Fluid Phase Equilibria 14, 19 (1983).
  * Chang and Sengers [1986] R. F. Chang and J. M. H. L. Sengers, Behavior of dilute mixtures near the solvent’s critical point, The Journal of Physical Chemistry 90, 5921 (1986), https://doi.org/10.1021/j100280a093 .
  * Ludmer  _et al._ [1987] Z. Ludmer, R. Shinnar, and V. Yakhot, Solubility in binary mixtures at the immiscibility critical point, AIChE Journal 33, 1776 (1987).
  * Jiang and Prausnitz [2000] J. Jiang and J. M. Prausnitz, Critical temperatures and pressures for hydrocarbon mixtures from an equation of state with renormalization-group theory corrections, Fluid Phase Equilibria 169, 127 (2000).
  * Pérez-Pellitero  _et al._ [2006] J. Pérez-Pellitero, P. Ungerer, G. Orkoulas, and A. D. Mackie, Critical point estimation of the lennard-jones pure fluid and binary mixtures, The Journal of Chemical Physics 125, 054515 (2006), https://doi.org/10.1063/1.2227027 .
  * Artemenko  _et al._ [2010] S. Artemenko, T. Lozovsky, and V. Mazur, Critical lines in binary mixtures of components with multiple critical points, in  _Metastable Systems under Pressure_ , edited by S. Rzoska, A. Drozd-Rzoska, and V. Mazur (Springer Netherlands, Dordrecht, 2010) pp. 217–232.
  * Yamamoto and Matsumoto [2011] T. Yamamoto and M. Matsumoto, Solute effects on supercritical fluid, Molecular Simulation 37, 1091 (2011), https://doi.org/10.1080/08927022.2011.582104 .
  * Bell and Jäger [2017] I. H. Bell and A. Jäger, Calculation of critical points from helmholtz-energy-explicit mixture models, Fluid Phase Equilibria 433, 159 (2017).
  * Harris and Newitt [1998] K. R. Harris and P. J. Newitt, Diffusion and structure in dilute aqueous alcohol solutions: Evidence for the effects of large apolar solutes on water, The Journal of Physical Chemistry B 102, 8874 (1998).
  * Ashbaugh and Paulaitis [2001] H. S. Ashbaugh and M. E. Paulaitis, Effect of solute size and solute-water attractive interactions on hydration water structure around hydrophobic solutes, Journal of the American Chemical Society 123, 10721 (2001), pMID: 11674005.
  * Chowdhuri and Chandra [2006] S. Chowdhuri and A. Chandra, Solute size effects on the solvation structure and diffusion of ions in liquid methanol under normal and cold conditions, The Journal of Chemical Physics 124, 084507 (2006), https://doi.org/10.1063/1.2172598 .
  * Okamoto and Onuki [2018] R. Okamoto and A. Onuki, Theory of nonionic hydrophobic solutes in mixture solvent: Solvent-mediated interaction and solute-induced phase separation, The Journal of Chemical Physics 149, 014501 (2018), https://doi.org/10.1063/1.5037673 .
  * Gao  _et al._ [2020] A. Gao, R. C. Remsing, and J. D. Weeks, Short solvent model for ion correlations and hydrophobic association, Proceedings of the National Academy of Sciences 117, 1293 (2020), https://www.pnas.org/content/117/3/1293.full.pdf .
  * Jagla [1998] E. A. Jagla, Phase behavior of a system of particles with core collapse, Phys. Rev. E 58, 1478 (1998).
  * Jagla [1999] E. A. Jagla, Core-softened potentials and the anomalous properties of water, The Journal of Chemical Physics 111, 8980 (1999), https://doi.org/10.1063/1.480241 .
  * Oliveira  _et al._ [2008] A. Oliveira, P. Netz, and M. Barbosa, Which mechanism underlies the water-like anomalies in core-softened potentials?, The European Physical Journal B 64, 481 (2008).
  * Meyer and Stanley [1999] M. Meyer and H. E. Stanley, Liquid–liquid phase transition in confined water: A monte carlo study, The Journal of Physical Chemistry B 103, 9728 (1999), https://doi.org/10.1021/jp984142f .
  * Krott  _et al._ [2015] L. B. Krott, J. R. Bordin, N. M. Barraz, and M. C. Barbosa, Effects of confinement on anomalies and phase transitions of core-softened fluids, The Journal of Chemical Physics 142, 134502 (2015), https://doi.org/10.1063/1.4916563 .
  * Bordin and Barbosa [2016] J. Bordin and M. Barbosa, Flow and structure of fluids in functionalized nanopores, Physica A: Statistical Mechanics and its Applications 467 (2016).
  * Marques  _et al._ [2020] M. S. Marques, V. F. Hernandes, E. Lomba, and J. R. Bordin, Competing interactions near the liquid-liquid phase transition of core-softened water/methanol mixtures, Journal of Molecular Liquids 320, 114420 (2020).
  * Munaò and Urbic [2015] G. Munaò and T. Urbic, Structure and thermodynamics of core-softened models for alcohols, The Journal of Chemical Physics 142, 214508 (2015), https://doi.org/10.1063/1.4922164 .
  * Franzese [2007] G. Franzese, Differences between discontinuous and continuous soft-core attractive potentials: The appearance of density anomaly, Journal of Molecular Liquids 136, 267 (2007), eMLG/JMLG 2006.
  * de Oliveira  _et al._ [2008] A. B. de Oliveira, G. Franzese, P. A. Netz, and M. C. Barbosa, Waterlike hierarchy of anomalies in a continuous spherical shouldered potential, The Journal of Chemical Physics 128, 064901 (2008), https://doi.org/10.1063/1.2830706 .
  * Vilaseca and Franzese [2010] P. Vilaseca and G. Franzese, Softness dependence of the anomalies for the continuous shouldered well potential, The Journal of Chemical Physics 133, 084507 (2010), https://doi.org/10.1063/1.3463424 .
  * Vilaseca and Franzese [2011] P. Vilaseca and G. Franzese, Isotropic soft-core potentials with two characteristic length scales and anomalous behaviour, Journal of Non-Crystalline Solids 357, 419 (2011).
  * Kolb and Dünweg [1999] A. Kolb and B. Dünweg, Optimized constant pressure stochastic dynamics, The Journal of Chemical Physics 111, 4453 (1999), https://doi.org/10.1063/1.479208 .
  * Limbach  _et al._ [2006] H. Limbach, A. Arnold, B. Mann, and C. Holm, Espresso—an extensible simulation package for research on soft matter systems, Computer Physics Communications 174, 704 (2006).
  * Arnold  _et al._ [2013] A. Arnold, O. Lenz, S. Kesselheim, R. Weeber, F. Fahrenberger, D. Roehm, P. Košovan, and C. Holm, Espresso 3.1: Molecular dynamics software for coarse-grained models, in  _Meshfree Methods for Partial Differential Equations VI_ , edited by M. Griebel and M. A. Schweitzer (Springer Berlin Heidelberg, Berlin, Heidelberg, 2013) pp. 1–23.
  * Allen  _et al._ [2017] M. Allen, D. Tildesley, and D. Tildesley,  _Computer Simulation of Liquids_ (Oxford University Press, 2017).
  * Tuckerman [2010] M. Tuckerman,  _Statistical Mechanics: Theory and Molecular Simulation_, Oxford graduate texts (Oxford University Press, 2010).
  * Dzugutov [1996] M. Dzugutov, Universal scaling law for atomic diffusion in condensed matter, Nature 381, 137 (1996).
  * Dyre [2018] J. C. Dyre, Perspective: Excess-entropy scaling, The Journal of Chemical Physics 149, 210901 (2018), https://doi.org/10.1063/1.5055064 .
  * Bell  _et al._ [2020] I. Bell, J. Dyre, and T. Ingebrigtsen, Excess-entropy scaling in supercooled binary mixtures, Nature Communications 2020, 015012 (2020).
  * Galliero  _et al._ [2011] G. Galliero, C. Boned, and J. Fernández, Scaling of the viscosity of the lennard-jones chain fluid model, argon, and some normal alkanes, The Journal of Chemical Physics 134, 064505 (2011), https://doi.org/10.1063/1.3553262 .
  * Raveché [1971] H. J. Raveché, Entropy and molecular correlation functions in open systems. i. derivation, The Journal of Chemical Physics 55, 2242 (1971), https://doi.org/10.1063/1.1676399 .
  * Baranyai and Evans [1989] A. Baranyai and D. J. Evans, Direct entropy calculation from computer simulation of liquids, Phys. Rev. A 40, 3817 (1989).
  * Sharma  _et al._ [2006] R. Sharma, S. N. Chakraborty, and C. Chakravarty, Entropy, diffusivity, and structural order in liquids with waterlike anomalies, The Journal of Chemical Physics 125, 204501 (2006), https://doi.org/10.1063/1.2390710 .
  * Errington and Debenedetti [2001] J. R. Errington and P. D. Debenedetti, Relationship between structural order and the anomalies of liquid water, Nature (London) 409, 318 (2001).
  * Steinhardt  _et al._ [1983] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Bond-orientational order in liquids and glasses, Phys. Rev. B 28, 784 (1983).
  * de Oliveira  _et al._ [2006] A. B. de Oliveira, P. A. Netz, T. Colla, and M. C. Barbosa, Structural anomalies for a three dimensional isotropic core-softened potential, The Journal of Chemical Physics 125, 124503 (2006).
  * Yan  _et al._ [2005] Z. Yan, S. V. Buldyrev, N. Giovambattista, and H. E. Stanley, Structural order for one-scale and two-scale potentials, Phys. Rev. Lett. 95, 130604 (2005).
  * Ramasubramani  _et al._ [2020] V. Ramasubramani, B. D. Dice, E. S. Harper, M. P. Spellings, J. A. Anderson, and S. C. Glotzer, freud: A software suite for high throughput analysis of particle simulation data, Computer Physics Communications 254, 107275 (2020).
  * Rycroft [2009] C. H. Rycroft, Voro++: A three-dimensional voronoi cell library in c++, Chaos: An Interdisciplinary Journal of Nonlinear Science 19, 041111 (2009), https://doi.org/10.1063/1.3215722 .
  * Larsen  _et al._ [2016] P. M. Larsen, S. Schmidt, and J. Schiøtz, Robust structural identification via polyhedral template matching, Modelling and Simulation in Materials Science and Engineering 24, 055007 (2016).
  * Stukowski [2010] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO - the Open Visualization Tool, Modelling and Simulation in Materials Science and Engineering 18, 015012 (2010).
  * Stanley, H. E.  _et al._ [2008] Stanley, H. E., Kumar, P., Franzese, G., Xu, L., Yan, Z., Mazza, M. G., Buldyrev, S. V., Chen, S.-H., and Mallamace, F., Liquid polyamorphism: Possible relation to the anomalous behaviour of water, Eur. Phys. J. Special Topics 161, 1 (2008).
  * Mallamace  _et al._ [2020] F. Mallamace, G. Mensitieri, D. Mallamace, M. Salzano de Luna, and S.-H. Chen, Some aspects of the liquid water thermodynamic behavior: From the stable to the deep supercooled regime, International Journal of Molecular Sciences 21, 7269 (2020).
  * Corradini  _et al._ [2010] D. Corradini, S. V. Buldyrev, P. Gallo, and H. E. Stanley, Effect of hydrophobic solutes on the liquid-liquid critical point, Phys. Rev. E 81, 061504 (2010).
  * Huš and Urbic [2013] M. Huš and T. Urbic, Core-softened fluids as a model for water and the hydrophobic effect, The Journal of Chemical Physics 139, 114504 (2013), https://doi.org/10.1063/1.4821226 .
  * Hus and Urbic [2014] M. Hus and T. Urbic, Existence of a liquid-liquid phase transition in methanol, Phys. Rev. E 90, 062306 (2014).
  * Wada and Umeda [1962] G. Wada and S. Umeda, Effects of nonelectrolytes on the temperature of the maximum density of water. i. alcohols, Bulletin of the Chemical Society of Japan 35, 646 (1962), https://doi.org/10.1246/bcsj.35.646 .
  * González-Salgado  _et al._ [2020] D. González-Salgado, J. Troncoso, and E. Lomba, The temperature of maximum density for amino acid aqueous solutions. an experimental and molecular dynamics study, Fluid Phase Equilibria , 112703 (2020).
  * Salcedo  _et al._ [2013] E. Salcedo, N. M. Barraz, and M. C. Barbosa, Relation between occupation in the first coordination shells and widom line in core-softened potentials, The Journal of Chemical Physics 138, 164502 (2013), https://doi.org/10.1063/1.4802006 .
  * S. Marques  _et al._ [2020] M. S. Marques, T. P. O. Nogueira, R. F. Dillenburg, M. C. Barbosa, and J. R. Bordin, Waterlike anomalies in hard core–soft shell nanoparticles using an effective potential approach: Pinned vs adsorbed polymers, Journal of Applied Physics 127, 054701 (2020), https://doi.org/10.1063/1.5128938 .
  * Mallamace [2009] F. Mallamace, The liquid water polymorphism, Proceedings of the National Academy of Sciences 106, 15097 (2009), https://www.pnas.org/content/106/36/15097.full.pdf .


# Neutron-proton differential transverse flow in 132Sn + 124Sn collisions at
270 MeV/nucleon

Xin Huang  School of Physics and Electronic Science, Guizhou Normal
University, Guiyang 550025, China  Gao-Feng Wei<EMAIL_ADDRESS>School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550025, China  Guizhou Provincial Key Laboratory of Radio Astronomy and Data
Processing, Guizhou Normal University, Guiyang 550025, China  Qi-Jun Zhi
School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550025, China  Guizhou Provincial Key Laboratory of Radio Astronomy and Data
Processing, Guizhou Normal University, Guiyang 550025, China  You-Chang Yang
Guizhou University of Engineering Science, Bijie 551700, China  Zunyi Normal
University, Zunyi 563006, China  Zheng-Wen Long  College of Physics, Guizhou
University, Guiyang 550025, China

###### Abstract

Within a transport model, we study the neutron-proton differential transverse
flow and its excitation function in central 132Sn + 124Sn collisions at 270
MeV/nucleon. To more accurately evaluate effects of the high-density behavior
of symmetry energy $E_{sym}(\rho)$ on this observable, we also consider the
uncertainties of $E_{sym}(\rho)$ around the saturation density $\rho_{0}$. It
is shown that the neutron-proton differential transverse flow and its
excitation function are mainly sensitive to the slope $L$ of $E_{sym}(\rho)$
at $\rho_{0}$. However, the effects of low-density behavior of $E_{sym}(\rho)$
on this observable should also be considered. Therefore, it is suggested that
measurements of the neutron-proton differential transverse flow and its
excitation function may provide useful complements to the constraints on $L$
extracted from the spectral pion ratio in S$\pi$RIT experiments.

##  I introduction

Heavy-ion collisions (HICs) can directly generate high density nuclear matter,
and thus provide the opportunity to explore the properties of strong
interacting matter at extreme conditions. As an important input in simulations
of HICs, the isovector component of nuclear mean field, i.e.,
symmetry/isovector potential, is rather uncertain because of the extreme
challenge of relatively direct detection of isovector potential in
experiments. Using the nucleon-nucleus scattering and ($p$,$n$) charge-
exchange reactions Hoff72 ; Pat76 ; Kwi78 ; Rap79 ; Jeu91 ; Kon03 , one can
only extract limited information of isovector potential at $\rho_{0}$. As a
result, the determination of the $E_{sym}(\rho)$ term of the equation of state
(EoS) of asymmetric nuclear matter (ANM) is still unsatisfactory compared to
the relatively good determination of the isospin-independent part of the EoS
of ANM Dan02 ; Oert17 . Presently, the best knowledge of $E_{sym}(\rho)$ is
around $2\rho_{0}/3$, for which its value is determined to be 25.5$\pm 1$ MeV
from nuclear masses and isobaric analog states Wang13 ; Brown13 ; Dan14 . At
densities greater than $2\rho_{0}/3$, the uncertainties in $E_{sym}(\rho)$
grow monotonically. For example, the $E_{sym}(\rho)$ at $\rho_{0}$, that is
commonly used as one of the criterion in fitting the parameters of the
isovector potential, still has greater uncertainties than that at
$2\rho_{0}/3$, e.g., 32$\pm 2$ MeV Cozma18  and 32.5$\pm 3.2$ MeV Wang18 .
Also, the uncertainties for $E_{sym}(\rho)$ at suprasaturation densities are
rather larger, such as the reported value $L=106\pm 37$ MeV Bren21  by a
calculation of $L$ correlated to the improved $R_{\rm skin}^{{}^{208}{\rm
Pb}}$ in the PREX-II experiment PREX-II .

Very recently, the S$\pi$RIT collaboration reported the results of pion
production in Sn + Sn collisions at 270 MeV/nucleon Estee21 ; Jhang21 .
Moreover, through comparing the spectral pion ratio with the simulation from a
dcQMD model Cozma21 , they deduced that the value of $L$ is within the range
from 42 to 117 MeV Estee21 . Obviously, this value is consistent with that
deduced from the correlated calculation Bren21 . However, the uncertainty for
$L$ is still rather larger and thus need to be further constrained. We note
that the motions of energetic nucleons are directly influenced by the
$E_{sym}(\rho)$ and its $L$ value, and thus might provide a more direct detect
of the $E_{sym}(\rho)$ at suprasaturation density. This is because these
energetic participants can originate in the regions that are compressed during
the violent early stages of HICs and be accelerated by the symmetry potential,
resulting in their momenta reflecting the $E_{sym}(\rho)$ and its $L$ value.
Naturally, studies on the observables relavant to these nucleons, as an
important complement to the pion spectra Tsang17  and/or spectral pion ratio
Estee21 , may shed more lights on the $E_{sym}(\rho)$ at suprasaturation
densities. Actually, as one of this kind of observable, elliptic folw is more
suitable to probe the $E_{sym}(\rho)$ at suprasaturation density, and the
corresponding studies in Au + Au collisions have already placed constraints on
the $E_{sym}(\rho)$ and its $L$ value Cozma18 ; FOPI . As another candidate of
this kind of observable, the free neutron-proton differential transverse flow
has also been found to be more sensitive to the high-density behavior of
$E_{sym}(\rho)$ LiBA00 ; LiBA02 ; Yong05 ; Yong06 . Therefore, we attempt to
predict the sensitivities of this observable to the symmetry energy, and thus
to benefit for the upcoming or ongoing measurements in S$\pi$RIT experiments.

##  II The Model

This study is carried out within an isospin- and momentum-dependent Boltzmann-
Uehling-Uhlenbeck (IBUU) transport model that has incorporated constraints
Wei21b  on the momentum dependence of isovector potential from the recently
reported pion data in S$\pi$RIT experiments Jhang21 . Specifically, the
nuclear interaction is an improved momentum-dependent interaction (IMDI)
expressed as

$\displaystyle U(\rho,\delta,\vec{p},\tau)$ $\displaystyle=$ $\displaystyle
A_{u}\frac{\rho_{-\tau}}{\rho_{0}}+A_{l}\frac{\rho_{\tau}}{\rho_{0}}+\frac{B}{2}{\big{(}}\frac{2\rho_{\tau}}{\rho_{0}}{\big{)}}^{\sigma}(1-x)$
(1) $\displaystyle+$
$\displaystyle\frac{2B}{\sigma+1}{\big{(}}\frac{\rho}{\rho_{0}}{\big{)}}^{\sigma}(1+x)\frac{\rho_{-\tau}}{\rho}{\big{[}}1+(\sigma-1)\frac{\rho_{\tau}}{\rho}{\big{]}}$
$\displaystyle+$ $\displaystyle\frac{2C_{l}}{\rho_{0}}\int
d^{3}p^{\prime}\frac{f_{\tau}(\vec{p}^{\prime})}{1+(\vec{p}-\vec{p}^{\prime})^{2}/\Lambda^{2}}$
$\displaystyle+$ $\displaystyle\frac{2C_{u}}{\rho_{0}}\int
d^{3}p^{\prime}\frac{f_{-\tau}(\vec{p}^{\prime})}{1+(\vec{p}-\vec{p}^{\prime})^{2}/\Lambda^{2}},$

where $\tau=1$ for neutrons and $-1$ for protons, and $A_{u}$, $A_{l}$,
$C_{u}(\equiv C_{\tau,-\tau})$ and $C_{l}(\equiv C_{\tau,\tau})$ are expressed
as

$\displaystyle A_{l}$ $\displaystyle=$ $\displaystyle
A_{l0}+U_{sym}^{\infty}(\rho_{0})-\frac{2B}{\sigma+1}$ (2)
$\displaystyle\times$
$\displaystyle\Big{[}\frac{(1-x)}{4}\sigma(\sigma+1)-\frac{1+x}{2}\Big{]},$
$\displaystyle A_{u}$ $\displaystyle=$ $\displaystyle
A_{u0}-U_{sym}^{\infty}(\rho_{0})+\frac{2B}{\sigma+1}$ (3)
$\displaystyle\times$
$\displaystyle\Big{[}\frac{(1-x)}{4}\sigma(\sigma+1)-\frac{1+x}{2}\Big{]},$
$\displaystyle C_{l}$ $\displaystyle=$ $\displaystyle
C_{l0}-2\big{(}U_{sym}^{\infty}(\rho_{0})-2z\big{)}\frac{p_{f0}^{2}}{\Lambda^{2}\ln\big{[}(4p_{f0}^{2}+\Lambda^{2})/\Lambda^{2}\big{]}},$
(4) $\displaystyle C_{u}$ $\displaystyle=$ $\displaystyle
C_{u0}+2\big{(}U_{sym}^{\infty}(\rho_{0})-2z\big{)}\frac{p_{f0}^{2}}{\Lambda^{2}\ln\big{[}(4p_{f0}^{2}+\Lambda^{2})/\Lambda^{2}\big{]}}.$
(5)

In the framework, the present IBUU model originates from the IBUU04 Das03 ;
IBUU  and/or IBUU11 CLnote  models. However, the present version has been
greatly improved to more accurate simulations of HICs as briefly discussed in
the following.

First, a separate density-dependent scenario Xu10 ; Chen14  for in-medium
nucleon-nucleon interaction has been adopted for a more delicate treatment of
the in-medium many-body force effects Chen14 , which also affects
significantly the pion production in HICs Wei20 . Therefore, the $B$-terms in
Eq. (1) as well in the expressions of $A_{u}$ and $A_{l}$ are different from
those in Refs. Xu15 ; Xu17 .

Second, a quantity $U_{sym}^{\infty}(\rho_{0})$ proposed in Ref. CLnote ,
i.e., the value of symmetry potential at infinitely large nucleon momentum, is
used to characterize the momentum dependence of the symmetry potential at
$\rho_{0}$ as in Refs. Xu15 ; Xu17 . It should be mentioned that this quantity
is treated as a free one in Refs. Xu15 ; Xu17 . However, considering that the
symmetry potential with different $U_{sym}^{\infty}(\rho_{0})$ even with the
identical $E_{sym}(\rho)$ can also lead to different pion yields in HICs, we
have carried out a study to constrain the $U_{sym}^{\infty}(\rho_{0})$ through
comparing the pion observables of theoretical simulations for reactions 108Sn
+ 112Sn and 132Sn + 124Sn with the data in S$\pi$RIT experiments Jhang21 , the
central value of $U_{sym}^{\infty}(\rho_{0})$ is constrained approximately to
be $-160$ MeV, see, Ref. Wei21b  for the details.

Table 1:  The parameters $x$ and $z$ as well the corresponding $L$ values for Case-I and Case-II. Parameters | Case-I | Case-II | L (MeV)
---|---|---|---
$x$, $z$ |  $0.4$, $-2.149$ |  $0.491$, $0$ | $30.3$
$x$, $z$ |  $0$, $0.767$ |  $-0.032$, $0$ | $62.0$
$x$, $z$ |  $-0.4$, $3.712$ |  $-0.557$, $0$ | $93.8$
$x$, $z$ |  $-0.8$, $6.656$ |  $-1.081$, $0$ | $125.6$
Figure 1: (Color online) Density dependence of the $E_{sym}(\rho)$ for Case I
and Case II. Figure 2: (Color online) Upper: Kinetic-energy dependent
isoscalar potential (a) at $\rho_{0}$ in comparison with the Schrödinger-
equivalent one obtained by Hama et al. Lower: Kinetic-energy dependent
isovector potential (b) at $\rho_{0}$ for Case I and Case II.

Third, considering that the $E_{sym}(\rho)$ at $\rho_{0}$ still has greater
uncertainties than that at $2\rho_{0}/3$, we therefore introduce a parameter
$z$ for $C$ terms in the expressions of $C_{u}$ and $C_{l}$ as in Refs. Xu15 ;
Xu17  to adjust the $E_{sym}(\rho)$ at $\rho_{0}$. As the first case, i.e.,
Case I, we take the value for the $z$ parameter to ensure
$E_{sym}(2\rho_{0}/3)$ is $25.5$ MeV. For comparison, we also set the value of
$z$ as zero and take the value for $E_{sym}(\rho_{0})$ of 32.5 MeV, as
commonly used; this case is denoted as Case II. The parameters embedded in
IMDI interactions are determined by fitting identical experimental/empirical
constraints on properties of nuclear matter at $\rho_{0}$ for both Case I and
Case II, except the value of 25.5 MeV for $E_{sym}(2\rho_{0}/3)$ used in Case
I and 32.5 MeV for $E_{sym}(\rho_{0})$ used in Case II. Specifically, the
values of these parameters are $A_{l0}=A_{u0}=-66.963$ MeV, $B=141.963$ MeV,
$C_{l0}=-60.486$ MeV, $C_{u0}=-99.702$ MeV, $\sigma=1.2652$,
$\Lambda=2.424p_{f0}$ and $U_{sym}^{\infty}(\rho_{0})=-160$ MeV, where
$p_{f0}$ is the nucleon Fermi momentum in symmetry nuclear matter (SNM) at
$\rho_{0}$. Moreover, to study the effects of high-density behavior of
$E_{sym}(\rho)$, we adjust the $x$ parameter to obtain four different $L$
values for both Case I and Case II. The parameters $x$ and $z$ as well the
corresponding $L$ values are listed in Table 1. It is seen that except for the
value of 6.656 for the $z$ parameter being slightly larger than the
uncertainties of $E_{sym}(\rho)$ at $\rho_{0}$, the values are all basically
within the allowed uncertain range of $E_{sym}(\rho)$ at $\rho_{0}$ Cozma18 ;
Wang18 . This feature can also be seen in the density dependent
$E_{sym}(\rho)$ as shown in Fig. 1. It should be mentioned that except the
different symmetry energy criterion the parameters for Case I and Case II can
lead to identical properties for nuclear matter at $\rho_{0}=0.16$ fm-3, i.e.,
the binding energy $-16$ MeV, the incompressibility $K_{0}=230$ MeV for SNM,
the isoscalar effective mass $m^{*}_{s}=0.7m$, the isoscalar potential at
infinitely large nucleon momentum $U^{\infty}_{0}(\rho_{0})=75$ MeV as well
$U^{\infty}_{sym}(\rho_{0})=-160$ MeV. Shown in Fig. 2 are the isoscalar and
isovector potentials at $\rho_{0}$ for Case I and Case II. Since the
parameters $x$ and $z$ only affect the isovector properties of ANM, the
isoscalar potentials for Case I and Case II are the same and also compatible
with the results of Hama et al Hama90 ; Buss12 . Moreover, since the $x$
parameter only affects the isovector properties of ANM at nonsaturation
densities for both Case I and Case II, as well the identical $z$ values used
for Case II, we can observe that the isovector potentials at $\rho_{0}$ even
with different $L$ values for Case II are also identical to each other. In
contrast, since the $z$ parameter is used to ensure $E_{sym}(2\rho_{0}/3)$ to
be $25.5$ MeV for Case I by adjusting $E_{sym}(\rho_{0})$, it naturally
affects isovector potentials at $\rho_{0}$. Therefore, we can see that the
isovector potentials at $\rho_{0}$ with different $L$ for Case I are slightly
different due to the different $z$ values used for different $L$ in Case I.
However, the differences of isovector potentials in Case I from those in Case
II are very tiny.

Fourth, to more accurately simulate HICs, we also give detailed consideration
of the $\Delta$ and pion potentials as well as electromagnetic effects in
HICs, see, Refs. Wei18a ; Wei18b ; Wei21 ; Wei21b  for more details.

##  III Results and Discussions

Figure 3: (Color online) Differential transverse flows of free neutrons
$p_{x}^{n}$ and protons $p_{x}^{p}$ as a function of the center-of-mass
rapidity $y_{c.m.}$.

Now, we present the results for the 132Sn + 124Sn reaction at 270 MeV/nucleon
with an impact parameter of $b=3$ fm. The free neutron-proton differential
transverse flow is measured by LiBA00 ; LiBA02 ; Yong05 ; Yong06

$\displaystyle p_{x}^{np}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{N(y)}\sum_{i=1}^{N(y)}p_{x_{i}}\tau_{i}$ (6)
$\displaystyle=$
$\displaystyle\frac{N_{n}(y)}{N(y)}<p_{x}^{n}(y)>-\frac{N_{p}(y)}{N(y)}<p_{x}^{p}(y)>,$

where $N_{n}(y)$, $N_{p}(y)$ and $N(y)$ denote, respectively, the number of
free neutrons, protons, and total nucleons with local densities less than
$\rho_{0}/8$ at rapidities $y$, and $\tau_{i}$ is $1$ for neutrons and $-1$
for protons. From this formula, we can see that the isospin fractionation
effects are incorporated into the collective flow Dan85  of both neutrons and
protons through $N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$. To understand this
effect, we show first in Fig. 3 the collective flow of free neutrons and
protons as a function of the center-of-mass rapidity. First, it is seen that,
consistent with the previous result in Ref. LiBA00 , the proton flow is less
sensitive to $L$ but higher than the neutron flow due to the Coulomb repulsive
effects. Second, with a certain $L$, we can observe that the transverse flows
either for neutrons or protons are not changed essentially in the left and
righ panels of Fig. 3. This means that the transverse flow is relatively
insensitive to the actual value for $E_{sym}(\rho)$ at any given density, but
is mainly sensitive to the slope of $E_{sym}(\rho)$ , which governs the
pressure that the symmetry energy provides in a neutron-rich system as well
dense astrophysical environments such as neutron stars. It should be mentioned
that this finding is similar to that observed in the ratios/differences of
neutron vs proton elliptic flows in Refs. Cozma18 ; Cozma11  as well that
observed much earlier in Ref. Dan02 . Nevertheless, the sensitivities of
transverse flows to $L$ are not obvious either for neutrons or protons. Shown
in Fig. 4 are $N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$ as a function of the center-
of-mass rapidity, respectively. First, since the mass between the target and
projectile has a little difference, we thus observe that the shapes of both
$N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$ are a little asymmetric between the target
and projectile rapidities. Second, it is seen that varying $L$ from $30.3$ to
$125.6$ MeV causes more (less) free neutrons (protons) on the whole for both
Case I and Case II. Certainly, we can also find that effects of $L$ on either
$N_{n}(y)/N(y)$ or $N_{p}(y)/N(y)$ are reduced somewhat in mid-rapidities for
Case II compared to those for Case I. Also, the effects even reverse at the
target and/or projectile rapidities. To understand these observations, we show
in Fig. 5 the isovector potential of Case I and Case II at both $0.5\rho_{0}$
and $2\rho_{0}$. First, it is seen that $L$ affects the isovector potentials
mainly at high densities, and thus the high-density behavior of
$E_{sym}(\rho)$ dominates $N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$. Moreover,
varying $L$ from $30.3$ to $125.6$ MeV, the neutrons feel stronger repulisve
effects while the protons feel stronger attractive effects on the whole for
both Case I and Case II. Naturally, we can observe more free neutrons but less
free protons with larger values for $L$. Second, we observe that the
differences of isovector potentials at $2\rho_{0}$ between Case I and Case II
are negligible, while the differences of isovector potentials at $0.5\rho_{0}$
with different $L$ are smaller in Case I than those in Case II. This is
exactly due to the more accurate criterion, i.e., $E_{sym}(2\rho_{0}/3)=25.5$
MeV, used in Case I that reduces the effects of $L$ on the isovector
potentials and $E_{sym}(\rho)$ at subsaturation densities. Moreover, since the
effects of $L$ on the isovector potentials at low densities are opposite to
those at high densities, the effects of high-density symmetry energy/potential
on observables at compress stages are likely to be smeared out by low density
symmetry energy/potential at the expansion stages. In particular, nucleons
with target and/or projectile rapidities experience longer time in the
expansion stages and emit later compared to those at midrapidities, the
effects of low density symmetry energy/potential on observables are naturally
considerable. This is the reason we observe in Case II that the isospin
fractionation effects are reduced slightly at midrapidities but reversed at
target and/or projectile rapidities.

Figure 4: (Color online) $N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$ with $L=30.3$ MeV
and $125.6$ MeV as a function of the center-of-mass rapidity $y_{c.m.}$.
Figure 5: (Color online) Kinetic-energy dependent isovector potential at
$\rho=0.5\rho_{0}$ and $\rho=2\rho_{0}$ for Case I and Case II. Figure 6:
(Color online) Neutron-proton differential transverse flows as a function of
the center-of-mass rapidity $y_{c.m.}$. Figure 7: Excitation function of the
neutron-proton differential transverse flows at the mid-rapidity
$|y_{c.m.}|\leq{0.1}$ as a function of $L$. The lines are drawn to guide the
eye.

Now, we turn to the free neutron-proton differential transverse flows in the
same reaction. Shown in Fig. 6 are the free neutron-proton differential
transverse flows as a function of the center-of-mass rapidity. First, it is
observed that the free neutron-proton differential transverse flow inherits
the asymmetry in shapes from $N_{n}(y)/N(y)$ and $N_{p}(y)/N(y)$. Second, it
is seen that this observable indeed combines constructively the in-plane
transverse momenta generated by the isovector potentials while reducing
significantly influences of the isoscalar potentials of both neutrons and
protons as indicated in Ref. LiBA02 , and thus it is more sensitive to $L$.
Moreover, comparing observations between Case I and Case II, we can find that
the sensitivities of this observable to $L$ are indeed reduced in Case II.
Therefore, to show clearly this observation and facilitate the experimental
measurement, we calculate the excitation function of this observable, which is
defined as LiBA02b

$F={\Big{(}}\frac{d<p_{x}^{np}>}{dy_{c.m.}}{\Big{)}}_{y_{c.m.}=0}.$ (7)

Shown in Fig. 7 is the excitation function of neutron-proton differential
transverse flows as a function of $L$. It is seen that effects of $L$ on the
excitation function are rather obvious for both Case I or Case II, and thus
can be used to further verify the extracted $L$ from the spectral pion ratio
in S$\pi$RIT experiments Estee21 . Nevertheless, we can also find that the
curve of excitation function in Case I indeed is more steeper compared to that
in Case II due to fewer effects of low-density $E_{sym}(\rho)$. This feature
indicates that the uncertainties of $E_{sym}(\rho)$ around the $\rho_{0}$
should also be considered when using this observable to probe the high-density
behavior of $E_{sym}(\rho)$. Actually, the authors of Ref. Stone17  have
already discussed systematically the effects of uncertainties of low density
$E_{sym}(\rho)$ on the determination of high-density $E_{sym}(\rho)$.

Before ending this part, we give two useful remarks. First, the value of
$-160$ MeV used for $U_{sym}^{\infty}(\rho_{0})$ in this study is more
negative than that used in other models, e.g., $-115$ MeV, used in Refs. Xu15
; Xu17 . This large and negative $U_{sym}^{\infty}(\rho_{0})$ appears to have
the feature that interactions between two protons or two neutrons with
relatively large velocity will be far more attractive than those between a
neutron and proton moving at the same relative velocity. The reasons might be
due to different isospin states in the former and latter that may originates
from the different constituent quarks between neutrons and protons. This issue
also deserves serious consideration. Second, we predict the emission of
nucleons as free particles and do not consider the clustering effects of
nucleons that may change the quantitative results of the present study.
Therefore, it will be interesting to see how clusters change the quantitative
results of the present study.

##  IV Summary

In conclusion, we have studied the free neutron-proton differential transverse
flow and its excitation function in 132Sn + 124Sn collisions at 270
MeV/nucleon within a transport model. It is found that the sensitivities of
free neutron-proton differential transverse flow and its excitation function
to $L$ are rather obvious, and thus can be used to further verify the
extracted $L$ from the spectral pion ratio in S$\pi$RIT experiments.
Therefore, we conclude that measurements of the neutron-proton differential
transverse flow and its excitation function may provide useful complements to
the constraints on $L$ extracted from measurements of the spectral pion ratio
in S$\pi$RIT experiments. Moreover, after examining the effects of
$E_{sym}(\rho)$ around $\rho_{0}$ on this observable, it is also suggested
that the uncertainties of $E_{sym}(\rho)$ around $\rho_{0}$ should also be
considered when using the free neutron-proton differential transverse flow to
probe the high-density behavior of $E_{sym}(\rho)$.

###### Acknowledgements.

G.-F. Wei would like to thank Profs. Wei Zuo and Bo-Chao Liu for helpful
discussions. This work is supported by the National Natural Science Foundation
of China under grant Nos.11965008, 11405128, 11865019 and Guizhou Provincial
Science and Technology Foundation under Grant No.[2020]1Y034, and the PhD-
funded project of Guizhou Normal university (Grant No.GZNUD[2018]11).

## References

  * (1) G. W. Hoffmann, and W. R. Coker, Phys. Rev. Lett. 29, 227 (1972).
  * (2) D. M. Patterson et al., Nucl. Phys. A 263, 261 (1976).
  * (3) K. Kwiatkowski et al., Nucl. Phys. A 301, 349 (1978).
  * (4) J. Rapaport et al., Nucl. Phys. A 330, 15 (1979).
  * (5) J. P. Jeukenne, C. Mahaux, and R. Sartor, Phys. Rev. C 43, 2211 (1991).
  * (6) A. J. Koning et al., Nucl. Phys. A 713, 231 (2003).
  * (7) P. Danielewicz, R. Lacey, and W. G. Lynch, Science 298, 1592 (2002).
  * (8) M. Oertel, M. Hempel, T. KIähn, and S. Typel, Rev. Mod. Phys. 89, 015007 (2017).
  * (9) N. Wang, L. Ou, and M. Liu, Phys. Rev. C 87, 034327 (2013).
  * (10) B. A. Brown, Phys. Rev. Lett. 111, 232502 (2013).
  * (11) P. Danielewicz, J. Lee, Nucl. Phys. A 922, 1 (2014).
  * (12) M. D. Cozma, Eur. Phys. J. A 54, 40 (2018).
  * (13) R. Wang, L. W. Chen, and Y. Zhou, Phys. Rev. C 98, 054618 (2018).
  * (14) B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Phys. Rev. Lett. 126, 172503 (2021).
  * (15) D. Adhikari et al. (PREX Collaboration), Phys. Rev. Lett. 126, 172502 (2021).
  * (16) G. Jhang, J. Estee, J. Barney, G. Cerizza, M. Kaneko, J. W. Lee, W. G. Lynch, T. Isobe et al. (S$\pi$RIT Collaboration), M. Colonna, D. Cozma, P. Danielewicz, H. Elfner, N. Ikeno, C. M. Ko, J. Mohs, D. Oliinychenko et al. (TMEP Collaboration), Phys. Lett. B 813, 136016 (2021).
  * (17) J. Estee, W. G. Lynch, C. Y. Tsang, J. Barney, G. Jhang, M. B. Tsang, R. Wang, M. Kaneko et al. (S$\pi$RIT Collaboration), and M. D. Cozma, Phys. Rev. Lett. 126, 162701 (2021).
  * (18) M. D. Cozma, and M. B. Tsang, Eur. Phys. J. A 57, 309 (2021).
  * (19) M. B. Tsang, J. Estee, H. Setiawan, W. G. Lynch, J. Barnery, M. B. Chen, G. Cerizza, P. Danielewicz et al., Phys. Rev. C 95, 044614 (2017).
  * (20) W. Reisdorf et al. (FOPI Collaboration), Nucl. Phys. A 781, 459 2007; Nucl. Phys. A 848, 366 (2010); Nucl. Phys. A 876, 1 (2012).
  * (21) B. A. Li, Phys. Rev. Lett. 85, 4221 (2000).
  * (22) B. A. Li, Phys. Rev. Lett. 88, 192701 (2002).
  * (23) G. C. Yong, B. A. Li, and W. Zuo, Chin. Phys. Lett. 22, 2226 (2005).
  * (24) G. C. Yong, B. A. Li, and L. W. Chen, Phys. Rev. C 74, 064617 (2006).
  * (25) G. F. Wei, X. Huang, Q. J. Zhi, A. J. Dong, C. G. Peng, and Z. W. Long, arXiv: 2112.13518.
  * (26) C. B. Das, S. Das Gupta, C. Gale, and B. A. Li, Phys. Rev. C 67, 034611 (2003).
  * (27) B. A. Li, C. B. Das, S. Das Gupta, and C. Gale, Phys. Rev. C 69, 011603(R) (2004).
  * (28) L. W. Chen, B. A. Li, A note of an improved MDI interaction for transport model simulations of heavy ion collisions , Texas A&M University-Commerce, 2010 (Unpublished).
  * (29) C. Xu, B. A. Li, Phys. Rev. C 81, 044603 (2010).
  * (30) L. W. Chen, C. M. Ko, B. A. Li, C. Xu, and J. Xu, Eur. Phys. J. A 50, 29 (2014).
  * (31) G. F. Wei, C. Xu, W. Xie, Q. J. Zhi, S. G. Chen, and Z. W. Long, Phys. Rev. C 102, 024614 (2020).
  * (32) J. Xu, L. W. Chen, and B. A. Li, Phys. Rev. C 91, 014611 (2015).
  * (33) H. Y. Kong, J. Xu, L. W. Chen, B. A. Li, and Y. G. Ma, Phys. Rev. C 95, 034324 (2017).
  * (34) S. Hama, B. C. Clark, E. D. Cooper, H. S. Sherif, and R. L. Mercer, Phys. Rev. C 41, 2737 (1990).
  * (35) O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich, A. B. Larionoov, T. Leitner, J. Weil, and U. Mosel, Phys. Rep. 512, 1 (2012).
  * (36) G. F. Wei, C. Liu, X. W. Cao, Q. J. Zhi, W. J. Xiao, C. Y. Long, and Z. W. Long, Phys. Rev. C 103, 054607 (2021).
  * (37) G. F. Wei, B. A. Li, G. C. Yong, L. Ou, X. W. Cao, and X. Y. Liu, Phys. Rev. C 97, 034620 (2018).
  * (38) G. F. Wei, G. C. Yong, L. Ou, Q. J. Zhi, Z. W. Long, and X. H. Zhou, Phys. Rev. C 98, 024618 (2018).
  * (39) P. Danielewicz, and G. Odyniec, Phys. Lett. B 157, 146 (1985).
  * (40) M. D. Cozma, Phys. Lett. B 700, 139 (2011).
  * (41) B. A. Li, Nucl. Phys. A 708, 365 (2002).
  * (42) J. R. Stone, P. Danielewicz, and Y. Iwata, Phys. Rev. C 96, 014612 (2017).