Phase Transitions and Critical Behavior

The purpose of this chapter is twofold In the following two sections. Phase Transitions and Critical Behavior and Quantum vs. Classical Phase Transitions, we give a concise introduction into the theory of quantum phase transitions, emphasizing similarities with and differences from classical thermal transitions. After that, we point out the computational challenges posed by quantum phase transitions, and we discuss a number of successful computational approaches together with prototypical examples. However, this chapter is not meant to be comprehensive in scope. We rather want to help scientists who are taking their first steps in this field to get off on the right foot. Moreover, we want to provide experimentalists and traditional theorists with an idea of what simulations can achieve in this area (and what they cannot do,. .. yet). Those readers who want to learn more details about quantum phase transitions and their applications should consult one of the recent review articles or the excellent textbook on quantum phase transitions by Sachdev.  

In this section, we briefly collect the basic concepts of the modern theory of phase transitions and critical phenomena to the extent necessary for the purpose of this chapter. A detailed exposition can be found in, e.g., the textbook by Goldenfeld.  

Most modem theories of phase transitions are based on Landau theory. Landau introduced the concept of an order parameter, a thermodynamic quantity that vanishes in one phase (the disordered phase) and is nonzero and generally nonunique in the other phase (the ordered phase). For the ferromagnetic 

Close to the phase transition, the coefficients r,w,u vary slowly with respect to the external parameters such as temperature, pressure, and electric or magnetic field. For a given system, the coefficients can be determined either by a first-principle calculation starting from a microscopic model, or, phenomenologically, by comparison with experimental data. The correct equilibrium value of the order parameter m for each set of external parameter values is found by minimizing with respect to m. 

The failure of Landau theory to describe the critical behavior correctly was the central puzzle in phase transition theory over many decades. It was 

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K. Binder. Critical behavior at surfaces. In C. Domb, ed. Phase Transitions and Critical Phenomena. London, Academic Press, 1989, Vol. 8, pp. 2-144. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

Recently, polyphilic compounds have been reported, where an Rp-chain and a carbosilane chain were attached at opposite sides of a terphenyl or oligo (p-phenylene ethinylene) core [42], Compound 188 with the longest oligo (p-phenylene ethinylene) core shows two hexagonal columnar phases separated by a thermoreversible continuous (second order) phase transition with critical behavior upon approaching the transition temperature. Based on XRD data it was... 

Let us briefly review the well-studied subject of phase transitions and critical phenomena [39], Examples of critical points include a magnet at the onset of ordering, a liquid-vapor system at the critical temperature and pressure, and a binary liquid system that is about to phase-separate. The key point is that the fluctuations in a system at its critical point occur at all scales, and the system is exquisitely sensitive to tiny perturbations. Even though sharp phase transitions can occur only in infinitely large systems, behavior akin to that at a phase transition is observed for systems of finite size as well. Indeed, for a system near a critical point, the largest scale over which fluctuations occur is determined either by how far away one is from the critical point or by the finite size of the system. 

G. Forgacs, R. Lipowsky, and T. Nieuwenhuizen, The behavior of interfaces in ordered and disordered systems. In eds., C. Domb and J. L. Lebowitz, Phase Transitions and Critical Phenomena. Vol. 14. London Academic Press, 135-363 (1991). ISBN 0-12-220314-3... 

Understanding the mechanism of adsorption is timely and important from a fimdamental scientiflc perspective. Adsorption is defined as a change in concentration of a given substance at the interface with respect to its concentration in the bulk part of the system. Such a perturbation in the local concentration is the most characteristic feature of nommiform fluids. Adsorption is one of the fascinating phenomena connected with the behavior of fluids in a force field extorted by the solid surface. This process has a great influence on the structure of thin films and it affects phase transitions and critical phenomena near the surface. Briefly, adsorption dictates the thermodynamical properties of nonuniform fluids. 

The understanding of continuous phase transitions and critical phenomena has been one of the important breakthrough in condensed matter physics in the early seventies. The concepts of scaling behavior and universality introduced by Kadanoff and Wi-dom and the calculation of non-gaussian exponents by Wilson and Fisher are undeniably brilliant successes of statistical physics in the study of low temperature phase transitions (normal to superconductor, normal to superfluid helium) and liquid-gas critical points. 

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],... 

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

Figure 21. The characteristic relaxation times for Process B. On the figure are marked the three phase transitions and the critical crossover temperature, T — 354 K, between Ahrrenius and VFT behaviors. The circles are the experimental data and the lines are the fitting functions Ahrrenius, VFT, and Saddle-like [78,179]. (Reproduced with permission from Ref. 257. Copyright 2005, Elsevier Science B.V.)...

We have emphasized the proper modeling of thermodynamic nonideality both with regard to molecular diffusion and interphase mass transfer. The benefits of adopting the irreversible thermodynamic approach are particularly apparent here it would not be possible otherwise to explain the peculiar behavior of the Fick diffusivities. The practical implications of this behavior in the design of separation equipment operating close to the phase transition or critical point (e.g., crystallization, supercritical extraction, and zone refining) are yet to be explored. In any case, the theoretical tools are available to us. 

Where and LH are the corresponding activation energy and enthalpy of phase transition and the coefficient defines the maximum probability that molecules will cross the interface between the liquid and SCF (vapor) phases. This simple relationship can explain the behavior of the mass transfer coefficient in Figure 15 when it is dominated by the interfacial resistance. Indeed, increases with temperature T according to Eq. (49) also, both parameters E and A// should decrease with increase of pressure, since the structure and composition of the liquid and vapor phases become very similar to each other around the mixture critical point. The decrease of A/f with pressure for the ethanol-C02 system has been confirmed by interferometric studies of jet mixing described in Section 3.2 and also by calorimetric measurements described by Cordray et al. (68). According to Eq. (43) the diffusion mass transfer coefficient may also increase in parallel with ki as a result of more intensive convection within the diffusion boundary layer. 

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... 

The total relaxation rate being the sum of the normal non-critical behavior (subscript nc) and critical behavior /Tr = ( /Tr)ac + ( /Tr)c, the critical behavior is manifested by a pseudodivergence at the transition (Fig. 7b). It is important to keep in mind this possibility when extracting activation enthalpies from the slope of ln(l/rr) as a function of 1 jT. As shown in Fig. 7b, the critical rate contains an almost linear contribution that cannot be easily distinguished from the true linear contribution of the non-critical rate, leading to a false value of the activation enthalpy. Therefore, a special attention is necessary to treat the data near a phase transition when critical fluctuations are suspected. 

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