Melting upon cooling and freezing upon heating: Fluid-solid phase diagram for Švejk-Hašek model of dimerizing hard spheres
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
where and denote the particle species and take the values and ,
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 studies4 and approximate the square-well associating potential by the sticky potential, defined through the Boltzmann factor relationship
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
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:
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:
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
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.
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.
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