Scaling, phase transitions

Barber, M.N. (1983). Finite-size Scaling. Phase Transitions and Critical Phenomena, vol. 8, p. 146-268, (Domb, C. and Lebowitz, J. L.). London Academic Press. 

Reiss H and Hammerich ADS 1986 Hard spheres scaled particle theory and exact relations on the existence and structure of the fluid/solid phase transition J. Phys. Chem. 90 6252... 

Binder K and Landau D P 1984 Finite size scaling at Ist-order phase transitions Phys. Rev. B 30 1477-85... 

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

Fig. 3. PF diagram for a pure fluid (not to scale) point c is the gas—liquid critical state, is the constant pressure at which phase transition occurs at...

In modern materials science topics of high interest are surface structures on small (nanometer-length) scales and phase transitions in adsorbed surface layers. Many interesting effects appear at low temperatures, where quantum effects are important, which have to be taken into account in theoretical analyses. In this review a progress report is given on the state of the art of (quantum) simulations of adsorbed molecular layers. 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

Phase transition occurs at a state of thermodynamic equilibrium, inducing a change in the microstructure of atoms. However, corrosion is a typical nonequilibrium phenomenon accompanied by diffusion and reaction processes. We can also observe that this phenomenon is characterized by much larger scales of length than an atomic order (i.e., masses of a lot of atoms), which is obvious if we can see the morphological change in the pitted surface. 

Applying MD to systems of biochemical interest, such as proteins or DNA in solution, one has to deal with several thousands of atoms. Models for systems with long spatial correlations, such as liquid crystals, micelles, or any system near a phase transition or critical point, also must involve a large number of atoms. Some of these systems, including synthetic polymers, obey certain scaling laws that allow the estimation of the behaviour of a large system by extrapolation. Unfortunately, proteins are very precise structures that evade such simplifications. So let us take 10,000 atoms as a reasonable size for a realistic complex system. 

Hence, close to the critical point thermodynamic quantities at comparatively distant spatial locations become correlated. Especially in the case of liquid micro flows close to a phase transition, these considerations suggest that the correlation length and not the molecular diameter is the length scale determining the onset of deviations from macroscopic behavior. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

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