Phase transitions finite size effects

In Sec. II we briefly review the experimental situation in surface adsorption phenomena with particular emphasis on quantum effects. In Section III models for the computation of interaction potentials and examples are considered. In Section IV we summarize the basic formulae for path integral Monte Carlo and finite size scahng for critical phenomena. In Section V we consider in detail examples for phase transitions and quantum effects in adsorbed layers. In Section VI we summarize. 

E. V. Albano. Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface Scahng approach and Monte Carlo investigation. Phys Rev B 42 R10818-R10821, 1990. 

The study of how fluids interact with porous solids is itself an important area of research [6], The introduction of wall forces and the competition between fluid-fluid and fluid-wall forces, leads to interesting surface-driven phase changes, and the departure of the physical behavior of a fluid from the normal equation of state is often profound [6-9]. Studies of gas-liquid phase equilibria in restricted geometries provide information on finite-size effects and surface forces, as well as the thermodynamic behavior of constrained fluids (i.e., shifts in phase coexistence curves). Furthermore, improved understanding of changes in phase transitions and associated critical points in confined systems allow for material science studies of pore structure variables, such as pore size, surface area/chemistry and connectivity [6, 23-25],... 

We now briefly consider finite size effects at first-order phase transitions. The easiest case is transitions driven by the field conjugate to the order parameter in systems at T < T. Then Eq. (40) is easily generalized by introducing Boltzmann weight factors for the two states < according to their Zeemann energies... 

If a unimodal pore network of arbitrary size is considered then, if the spatial distribution of pore sizes is non-random, the desorption percolation transition would be apparently smeared out (in addition to any finite size effect). It is possible that particular pores occupied by liquid-like phase might gain access to the vapour phase before would be expected to be the case for a purely random system because the actual layout of the pores might provide a convenient access route that would not have existed at that bond occupation level in a random system. The simulations of the nitrogen sorption... 

C. Borgs and R. Kotecky (1990) A rigorous theory of finite-size scaling at Ist-order phase-transitions. J. Stat. Phys. 61, pp. 79-119 ibid. (1992) Finite-size effects at asymmetric Ist-order phase-transitions. Phys. Rev. Lett. 68, pp. 1734-1737... 

Now, unfortunately, there are several difficulties that severely hamper Monte Carlo approaches to polymer blends [88, 91, 92, 107] (i) To see the unmixing transition occur by diffusive motion of the polymers, which ultimately form A-rich and B-rich macroscopic domains would require enormous expenditure of computer time and huge system sizes. System sizes accessible to simulations in practice are so small that such domains would never fit in. (ii) Even in the one phase region, the simulation is very difficult - the correlation length being very large, one must watch out carefully for finite size effects, (iii) Since... 

Stable ferroelectric nematic phases have been found in high density simulations of DHS [134,135], DSS [101-103] and Stockmayer particles [127,136]. So far the transition densities at fixed X or temperatures (at fixed density) have been estimated only quahtatively by the value at which the polarization order parameter (rotmded by finite size effects) Pi = fi-i d)/N) pa o.5... 

The temperature dependence of the magnetic moment in small systems is also of importance in understanding finite-size effects on magnetic phase transitions. Experimental data reported in [185] show that for Co and Ni, as expected, the ferromagnetic transition is smeared over several hundred degrees in small particles of 10 -10 atoms and that the Curie temperature is lower (although only slightly) than the bulk value. 

Finite-size scaling has become one of the most powerful tools for analyzing computer simulation data of phase transitions. Instead of treating finite-size effects as errors to be avoided, one can simulate systems of varying size and test whether or not homogeneity relations such as Eq. [8] are fulfilled. Fits of the simulation data to the finite-size scaling forms of the observables then yield values for the critical exponents. We will discuss examples of this method later in the chapter. 

BF Variano, NE Schlotter, DM Hwang, CJ Sandroff. Investigation of finite size effects in a 1st order phase-transition—High-pressure Raman-study of CdS microcrystallites. J Chem Phys 88 2848-2850, 1988. 

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