# Melting upon cooling and freezing upon heating: Fluid-solid phase diagram for Švejk-Hašek model of dimerizing hard spheres

###### Abstract

A simple model of dimerizing hard spheres with highly nontrivial fluid-solid phase behaviour is proposed. The model is studied using the recently proposed resummed thermodynamic perturbation theory for central force (RTPT-CF) associating potentials. The phase diagram has the fluid branch of the fluid-solid coexistence curve located at a temperatures lower than those of the solid branch. This unusual behaviour is related to the strong dependence of the system excluded volume on the temperature, which for the model at hand decreases with increasing temperature. This effect can be also seen for a wide family of fluid models with an effective interaction that combines short range attraction and repulsion at a larger distance. We expect that for sufficiently high repulsive barrier, such systems may show similar phase behaviour.

Keywords: fluid-solid phase diagram, inverse melting, association, dimerization, thermodynamic perturbation theory, phase coexistence

Considerable efforts during the last decade have focused on the investigation of
fluid models with intermolecular pair potentials that are characterized by
short-range attraction and long-range repulsion (SALR) (see ^{1, 2}
and references therein). Much of the interest in studying these systems
of this kind is due to their unusual and rich phase behaviour,
i.e. because of the competition between attraction and repulsion, the possibility exists for so-called
’modulated’ phases of different type to appear. Effective interactions of the SALR type can be
observed in a number of different soft matter fluid systems, including protein solutions,
star polymers and weakly charged colloidal systems ^{2, 3}.
In addition, due to the substantial progress
obtained recently in the experimental techniques there is a possiblity to synthesize colloidal
particles with predefined character of their interaction. A number of different versions of the
SALR models with different parameters for the shape, width and strength of the attractive and
repulsive portions of the potential have been developed and studied recently ^{1, 2}.
These also include the shielded attractive shell (SAS) model, developed quite some time ago
^{4}. Although the SAS model was used as a simple model of chemical reaction
of dimerization, it clearly belongs to the family of SALR models.
An important feature of these models (as also of the corresponding real systems) is that due to
short range attractive interaction the particles form clusters. While this feature is important
, for fluids with repulsive interaction located at a distance larger than the attractive
interaction, the formation of the clusters changes the excluded volume of the system. This effect is especially
important for the models with strong repulsion. For the low temperatures, when the degree of clusterization is
substantial, excluded volume is small. With increasing temperature, the degree of clusterization
becomes less and excluded volume can be substantially increased.
This feature can have a crucial effect on the phase behaviour of the system.

In this Letter we propose a simple SALR-type model that exhibits so-called inverse melting, i.e. the fluid branch of the fluid-solid phase diagram is located at lower temperatures than the solid branch. The driving mechanism of this phase transition is related to the temperature dependence of the excluded volume of the system.

Our model is represented by an equimolar mixture of the hard spheres of the same size with the number densities of the particles of the species and . There is an additional square-well potential acting between the particles of different species, so that the potentials are given by

(1) |

where and denote the particle species and take the values and ,

(2) |

(3) |

and are the square-well depth
and width, respectively, and . We will be focused here on the version of the model
with very narrow square-well potential () and consider the limiting
case of .
The former of these conditions allows us to follow
earlier studies^{4} and approximate the square-well associating potential
by the sticky potential, defined through the Boltzmann factor relationship

(4) |

where is Baxter’s stickiness parameter ^{5}.
The relation between stickiness parameter
and temperature can be established by equating the second virial
coefficient of the limiting sticky case (Eq. 4) with that of the original model defined in Eqs. 1-3.
In particular, we will consider here the version of the model with the square-well placed in a center
of the sphere, i.e. . Thus upon association the
model can form only dimers with one hard sphere completely buried inside the other
hard sphere. We will refer to this model as Švejk-Hašek
(ŠH) model ^{6} of dimerizing hard spheres.

The theoretical description of the model at hand can be carried out using the recently proposed
extension of the resummed thermodynamic perturbation theory for central force associating
potentials (RTPT-CF) ^{7}. The important feature of the model is
that at any temperature the system can be treated as a one-component hard-sphere fluid
with effective density

(5) |

where is the total density of nonbonded particles. This feature enables us to calculate the fluid-solid phase coexistence diagram of the ŠH model using the phase diagram of the conventional hard-sphere model. We shall denote the coexisting densities of such phase diagram as and , where the lower indices and denote the fluid and solid phases, respectively.

According to RTPT-CF approach ^{7} the density of nonbonded
particles satisfies the following equation:

(6) |

where

(7) |

and is the cavity distribution function
of the hard-sphere fluid at the overlapping distance and
with the particle number density .
This cavity distribution function can be calculated using relation due to Hoover and Poirier
^{8}. For the model at hand we have:

(8) |

Taking into account the expression for (5) equation (6) can be used to calculate the effective density of the system as a function of the temperature and density . For equal to either or equation (6) couples the temperature and density along either fluid or solid branches of the fluid-solid phase diagram of our ŠH dimerizing model. In this case we have

(9) |

where

(10) |

and the subscript takes the value of either or .
The excess chemical potential , which is needed to calculate
liquid branch of the phase diagram, can be obtained using the Carnahan-Starling expression
^{9}.
At equilibrium, the chemical potential in both phases is equal. Hence,
the excess chemical potential in the solid phase
can be calculated using its value in the fluid phase,
i.e.

(11) |

Using the relation between and (9), along with
with computer simulation values of the hard-sphere fluid-solid coexisting densities
(i.e., and ^{10}),
the fluid-solid phase diagram for the ŠH model of dimerizing hard spheres was computed.
The corresponding phase diagram in vs coordinate frame is presented
in figure 1. In contrast to conventional fluid-solid phase diagram one can see that the
fluid branch of the diagram is located at lower temperatures than the solid branch. Thus at constant
density, with increasing temperature the system undergoes a fluid to solid phase transition
(and vice versa). At sufficiently low temperature all the particles will be dimerized and
the effective density of such system will be . If in this case
then the system will be in a fluid state. With increasing temperature, the number of dimers decreases and the effective density increases. As soon as
becomes equal , the system begins to freeze and upon reaching
the value of the system will be in a solid state. According to Figure 1 this
may happen if the density of the system is in the range .
A system with density and sufficiently high temperature
will be in a state of fluid-solid equilibrium with the densities of the coexisting phases
and . Similarly, for density in the range
and sufficiently low temperature the system will
split into the coexisting fluid and solid phases with the densities
and , respectively.

In summary, we have studied a simple model for dimerization with highly nontrivial fluid-solid phase behaviour. The phase diagram, which was build using recently proposed RTPT-CF, has the liquid branch of the coexisting curve located at a temperatures lower than those of the solid branch. This unusual behaviour is related to strong dependence of the system excluded volume on the temperature, which for the model at hand decreases with increasing temperature. This effect can be observed also for most of the SALR type of the models and we expect that systems with sufficiently high repulsive barrier may exhibit similar phase behaviour.

## References

- 1 Archer, A.J.; Wilding, N.B. Phase Behavior of a Fluid with Competing Attractive and Repulsive Interactions. Phys. Rev. E 2007, 76, 031501.
- 2 Cigala, G.; Costa, D.; Bomont, J.-M.; Caccamo, C. Aggregate Formation in a Model Fluid with Microscopic Piecewise-Continuous Competing Interactions. Mol. Phys. 2015, 113, 2583â2592.
- 3 Kovalchuk, N.; Starov, V.; Langston, P.; Hilal, N. Stable Clusters in Colloidal Suspensions. Adv. Colloid Interface Sci. 2009, 147-48, 144-154.
- 4 Cummings, P.T.; Stell, G. Statistical Mechanical Models of Chemical Reactions. Analytic Solution of Models of in the Percus-Yevick Approximation. Mol. Phys. 1984 51, 253-287.
- 5 Baxter, R. J. PercusâYevick Equation for Hard Spheres with Surface Adhesion. J. Chem. Phys. 1968 49, 2770-2774.
- 6 Hašek, J. The Good Soldier Švejk. Ch. 4. Švejk Thrown Out of the Lunatic Asylum. ”… And I also met a certain number of professors there. One of them used to follow me about all the time and expatiate on how the cradle of the gipsy race was in the Krkonoše, and the other explain to me that inside the globe there was another globe much bigger than the outer one. …” Penguin Classics 2005
- 7 Reščič, J.; Kalyuzhnyi, Y.V.; Cummings, P.T. Shielded Attractive Shell Model Again: Resummed Thermodynamic Perturbation Theory for Central Force Potential. J. Phys.: Cond. Matt. 2016, (in press).
- 8 Hoover, W.G,; Poirier, J.G. Determination of the Virial Coefficients from the Potential of Mean Forces. J. Chem. Phys. 1962, 37, 1041-1042.
- 9 Carnahan, N.F.; Starling, K.E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969, 51, 635-636.
- 10 Hoover, W.G.; Ree, F.H. Melting Transition and Communal Entropy for Hard Spheres. J. Chem. Phys. 1968, 49, 3609-3617.