Hard order parameter transitions

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

In the case of a localized 1/n adsorption, which is observed in many Me UPD systems at relatively high AE or low F (formation of expanded Meads superlattice structures, cf. Section 3.4), the adsorption process can be described by the so-called hard-core lattice gas models using different analytical approximations or Monte Carlo simulations [3.214, 3.262-3.264]. Monte Carlo simulation for 1/2 adsorption on a square lattice is dealt in Section 8.4. Adsorption isotherms become asymmetrical with respect to AE and are affected by the structures of the Meads overlayer and S even in the absence of lateral Meads interactions [3.214, 3.262-3.264]. Furthermore, the critical interaction parameter for a first order phase transition, coc, which is related to the critical temperature, Tc, increases in comparison to the 1/1 adsorption. 

The conventional van der Waals approach where model parameters d and a are the constants cannot describe more than one first order phase transition and one critical point. Therefore a key question is a formulation of temperature -density dependency for EoS parameters generating more than one critical point in the mono-component matter. There are several approaches of the effective hard sphere determination from spherical interaction potential models that have a region of negative curvature in their repulsive core (the so-called core softened potentials). To avoid the sophistication of EoS and study a qualitative picture of phase behavior we adopt an approach Skibinsky et al. ° for one-dimensional system of particles interacting via pair potential... 

It is known that water-like anomalies and liquid-liquid transition can be observed for liquid systems of spherically symmetric potentials [80-82]. So the link between this type of models and our model based on bond orientational order is interesting to study. We argue that the constraint of packing and the resulting selection of symmetry with hard-core repulsion leads to the link between this type of potential and bond orientational order even for particle interacting with spherical potentials. This problem is related to an even more fundamental question about what is the relevant order parameter to describe liquid-liquid transition. Thus, further careful studies are highly desirable. The link between liquid-liquid transition and kinetics of water is also an interesting issue [83]. 

The behaviour of the adsorbed film in this approximation depends on only two parameters, /x the adsorption affinity and w or more generally Wij, the potential of mean force of two adsorbate moeties, i,j. The first quantity is obtained from the smooth wall approximation discussed in the previous section. The potential of mean force w determines the behaviour of the adsorbed film If it is attractive, then first order phase transitions may occur. If it is repulsive then second order phase transitions can take place, such as order-disorder rearrangements. If the interaction is strongly repulsive not allowing first nearest neighbors on the triangular lattice then this becomes equivalent to the hard hexagon problem solved exactly by Baxter [59]... 

Figure 4. The dependence of the order parameter and the discontinuity in density at the N-I phase transition. The liquid is composed of hard rectangular parallelepipeds with dimensions a = l,b and c=5 (after Gelbart and Barboy [42]).

Figure 6. Phase diagram of a fluid of hard sphero-cylinders in the (axial ratio/order parameter) plane. The circles are the simulation results for the smectic A transition [45]. The N-SmA transition obtained in [45] is denoted by squares N and triangles SmA (after Poniwierski and Sluckin [69]).

At extremely high fields the isotropic phase should be indistinguishable from the nematic one, even well above the zero field transition temperature, since the uniaxial order induced by a magnetic or ac electric field in the isotropic phase will be comparable with the nematic orientational order. However, such fields are hardly accessible, even with the pulse technique. Much stronger changes in order parameter may be achieved with ferroelectric transitions (see below). 

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