Generic Maximum-Valence Model for Fluid Polyamorphism Nikolay A. Shumovskyi1and Sergey V. Buldyrev1 2 1Department of Physics Boston University Boston MA 02215 USA

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Generic Maximum-Valence Model for Fluid Polyamorphism

Nikolay A. Shumovskyi1and Sergey V. Buldyrev1, 2

1)Department of Physics, Boston University, Boston, MA 02215, USA

2)Department of Physics, Yeshiva University, New York, NY 10033, USA

(*Electronic mail: buldyrev@yu.edu)

(Dated: 25 January 2023)

Recently, maximal valence model has been proposed to model liquid-liquid phase transition induced by polymerization

in sulfur. In this paper we present a simple generic model to describe liquid polyamorphism in single-component

ﬂuids using a maximum-valence approach for any arbitrary coordination number. The model contains three types of

interactions: i) atoms attract each other by van der Waals forces that generate a liquid-gas transition at low pressures,

ii) atoms may form covalent bonds that induce association, and iii) additional repulsive forces between atoms with

maximal valence and atoms with any valence. This additional repulsion generates liquid-liquid phase separation and

the region of negative heat expansion coefﬁcient (density anomaly) on a P-T phase diagram. We show the existence

of liquid-liquid phase transitions for dimerization, polymerization, gelation and network formation for corresponding

coordination numbers z=1,2,..6 and discuss the limits of this generic model for producing ﬂuid polyamorphism.

The existence of two alternative liquid phases in a single-

component substance is known as “liquid polyamorphism”1–3.

A substance may be found to be polyamorphic by experimen-

tally or computationally detecting a liquid-liquid phase transi-

tion (LLPT), which can be terminated at a liquid-liquid critical

point (LLCP)4,5. Liquid polyamorphism has been observed

in a variety of substances including: hydrogen6–8, helium9,10,

sulfur11, phosphorous12,13 and liquid carbon14, while being

proposed to exist in selenium and tellurium15,16. It has also

been hypothesized in metastable deeply supercooled water be-

low the temperature of spontaneous ice nucleation1–3,17–25.

The phenomenon of liquid polyamorphism can be un-

derstood through the interconversion of the two alterna-

tive molecular or supramolecular states via a reversible

reaction2,26,27. While for some polyamorphic systems, like

supercooled water, this approach is still being debated, there

are substances (such as hydrogen, sulfur, phosphorous, and

liquid carbon) where liquid-liquid phase separation is indeed

induced by a chemical reaction. For example, it was recently

discovered that high-density sulfur, well above the liquid-gas

critical pressure (in the range from 0.5 to 2.0GPa), exhibits

a LLPT indicated by a discontinuity in density from a low-

density-liquid (LDL) monomer-rich phase to a high-density-

liquid (HDL) polymer-rich phase11. This liquid-liquid transi-

tion is found in a polymerized state of sulfur (observed above

160°C at ambient pressure28–32). Another liquid-liquid tran-

sition accompanied by a reaction has been observed in hydro-

gen at extremely high-pressures (above 325 GPa at ambient

temperature8), in which liquid-molecular hydrogen (dimers)

dissociates into atomistic-metallic hydrogen6,7.

In this work, motivated by the recent discoveries of the

LLPT in hydrogen8, and continuing our previous work on

the maximal valence model for sulfur33, we propose a simple

generic model to describe liquid polyamorphism in a variety

of chemically-reacting ﬂuids. The model combines the ideas

of two-state thermodynamics2,22 with the maximum-valence

approach34–36, in which atoms may form covalent bonds via

a reversible reaction, changing their state according to their

bond number. By mimicking the valence structure by max-

imum bond number, z, our model predicts the LLPT in sys-

tems with dimerization (z=1), polymerization (z=2), and

gelation (z>2). We show that when the atoms with maxi-

mal valence repel atoms with any valence, phase separation is

coupled to dimerization (z=1), polymerization (z=2), and

gelation (z>2), thus generating the LLPT in polyamorphic

substances. The key difference of this paper and the previ-

ously published one33 is that here we investigate the case of

repulsion between atoms with the maximum valence z and any

other atoms which causes the segregation of the atoms with

the maximum valence into the low density phase; while in the

previous paper in which we have attempted to model sulfur

we investigate the case of attraction of the atoms of maximal

valence (z=2)to each other which causes segregation of the

polymerized atoms into the high density phase. Thus we have

two classes of maximal valence models - one with attraction

and another with repulsion, which drastically differ from each

other. The models with repulsion studied here do have a den-

sity anomaly region with negative heat expansion coefﬁcient

αP<0, while the models with attraction do not.

I. MAXIMUM-VALENCE MODEL

We model LLPT induced by molecular interconversion in

polyamorphic substances by characterizing each atom by its

coordination number k≤z, the number of bonds it has with

other atoms. Depending on the coordination number, each

atom is assigned to distinguished z+1 states: B0(with zero

bonds), B1(with one bond), Bk(with kbonds), and ﬁnally Bz

(with zbonds). Atoms cannot form more than zbonds and,

consequently, will associated into either dimers, for z=1, or

linear polymers, z=2, or some network structure z>2. All

of the atoms in the system may change their state by forming

or breaking a covalent bond via a reversible reaction. Fig. 1a

depicts all z(z+1)/2 types of reversible reactions that may oc-

cur in the system. In this work, we demonstrate that the min-

imum ingredients required to produce a LLPT are the follow-

ing: i) the van der Waals interactions between atoms, which

produce a LGPT; ii) covalent bonds between atoms, which in-

duce association; and iii), as we hypothesize, additional re-

arXiv:2210.01968v2 [cond-mat.soft] 24 Jan 2023

FIG. 1. Reactions and interactions in the generic maximum-valence model with repulsion. (a) z(z+1)/2 types of covalent bond-forming

reversible chemical reactions that may occur in the system. If two atoms without bonds (B0) collide with each other, they may form a bond

and become B1atoms. If a B0and B1atom collide, they may form a bond and become B1and B2atoms, respectively. If two B1atoms collide

with each other, they form an additional bond and become B2atoms. This continues until the atoms reach their maximum valency - state Bz.

(b-d) The three major interactions between atoms, in which each atom is composed of a core and shell, both with a diameter σand mass m.

U(r)is the pair potential energy and ris the distance from the centers of the particles. (b) The cores of each atom interact with an attractive

square well of depth εand width w. (c) The shells may react to form covalent bonds that consist of a narrow well with depth εband width wb.

(d) Phase segregation is coupled to dimerization, polymerization, gelation, etc., via the additional repulsive interactions between atoms in state

Bzand atoms in a state Bkwith k≤z, described by a square shoulder of height −εzand width wz.

pulsive interactions between atoms with maximum valence

(k=z) and atoms with any valence (k≤z), that are needed

to couple phase segregation to dimerization, polymerization

or gelation. These three ingredients are illustrated by square-

well potentials in Figs. 1(b-d).

In case of hydrogen, the pioneering quantum calculations

of Wigner and Huntington37 suggested that at high density

the energy of the metallic lattice is lower than the energy of

molecular lattice. In other words, at high densities H2-dimers

disassociate due to a steric effect, i.e. electron shells of H2-

dimers are getting larger than intermolecular distances. This

steric repulsion is expected to happen not only for the case of

hydrogen, but for other molecular liquids with higher valence.

Similar rule may be applied to the case of water. It was

found experimentally that at high densities when the ﬁfth

neighbors are being pushed into the ﬁrst coordination shell

of a molecule, hydrogen bonds bifurcate and the energy of

the bifurcated bonds are roughly half of the energy of the

straight bonds38. Essentially this implies that the shells of the

atoms with coordination number four become impenetrable

for a ﬁfth intruder at low temperatures.

To verify our hypothesis, we implement these three ingre-

dients of interactions via an event-driven MD technique39,40;

in particular, we use a discrete MD package (DMD) that only

includes particles interacting through spherically-symmetric

step-wise potentials, which may form bonds via reversible

reactions41. We simulate an NVT ensemble of N=1000

atoms in a cubic box with periodic boundaries at various con-

stant densities and temperatures. The temperature is con-

trolled by a Berendsen thermostat42. The van der Waals and

covalent-bonding interactions are implemented by separating

each atom into two overlapping hard spheres (a core and a

shell), with the same diameter σand mass m, see Figs. 1(b-

d). The connection between the core and its shell is repre-

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GenericMaximum-ValenceModelforFluidPolyamorphismNikolayA.Shumovskyi1andSergeyV.Buldyrev1,21)DepartmentofPhysics,BostonUniversity,Boston,MA02215,USA2)DepartmentofPhysics,YeshivaUniversity,NewYork,NY10033,USA(*Electronicmail:buldyrev@yu.edu)(Dated:25January2023)Recently,maximalvalencemodelhasbeenpropo...

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Generic Maximum-Valence Model for Fluid Polyamorphism Nikolay A. Shumovskyi1and Sergey V. Buldyrev1 2 1Department of Physics Boston University Boston MA 02215 USA

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