Of phase transition

Yeomans Y M 1992 Statistical Mechanics of Phase Transitions (Oxford Oxford University Press)... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

Islands occur particularly with adsorbates that aggregate into two-dimensional assemblies on a substrate, leaving bare substrate patches exposed between these islands. Diffraction spots, especially fractional-order spots if the adsorbate fonns a superlattice within these islands, acquire a width that depends inversely on tire average island diameter. If the islands are systematically anisotropic in size, with a long dimension primarily in one surface direction, the diffraction spots are also anisotropic, with a small width in that direction. Knowing the island size and shape gives valuable infonnation regarding the mechanisms of phase transitions, which in turn pemiit one to leam about the adsorbate-adsorbate interactions. 

Flere we discuss the exploration of phase diagrams, and the location of phase transitions. See also [128. 129. 130. 131] and [22, chapters 8-14]. Very roughly we classify phase transitions into two types first-order and continuous. The fact that we are dealing with a finite-sized system must be borne in mind, in either case. 

Ddnweg B 1996 Simulation of phase transitions critical phenomena Monte Carlo and Molecular Dynamics of Condensed Matter Systems vol 49, ed K Binder and G Ciccotti (Bologna Italian Physical Society) pp 215-54... 

Lovett R 1995 Can a solid be turned into a gas without passing through a first order phase transition Observation, Prediction and Simuiation of Phase Transitions in Compiex Fiuids vol 460 NATO ASi Series O ed M Baus, L F Rull and J-P Ryckaert (Dordrecht Kluwer) pp 641-54... 

This method is used to locate phase transitions via measurements of the endothennic enthalpy of phase transition. Details of the teclmique are provided elsewhere [25, 58]. Typically, the enthalpy change associated with transitions between liquid crystal phases or from a liquid crystal phase to the isotropic phase is much smaller than the melting enthalpy. Nevertheless, it is possible to locate such transitions with a commercial DSC, since typical enthalpies are... 

Frenkel D 1992 Computer simulations of phase transitions in liquid crystals Phase Transitions in Liquid Crystals ed S Martellucci and A N Chester (New York Plenum)... 

Lekkerkerker FI N W, Buining P, Buitenhuis J, Vroege G J and Stroobants A 1995 Liquid orystal phase transitions in dispersions of rodlike oolloidal partioles Observation, Prediction and Simulation of Phase Transitions in Complex Fluids ed M Baus, L F Rull and J P Flansen (Dordreoht Kluwer) pp 53-112... 

Type-n structures are formally the out-of-phase transition states between two type-I structures, even if there is no measurable banier. 

It was estabhshed ia 1945 that monolayers of saturated fatty acids have quite compHcated phase diagrams (13). However, the observation of the different phases has become possible only much more recendy owiag to improvements ia experimental optical techniques such as duorescence, polarized duorescence, and Brewster angle microscopies, and x-ray methods usiag synchrotron radiation, etc. Thus, it has become well accepted that Hpid monolayer stmctures are not merely soHd, Hquid expanded, Hquid condensed, etc, but that a faidy large number of phases and mesophases exist, as a variety of phase transitions between them (14,15). 

The entropy change AS/ - and the volume change AV/ - are the changes which occur when a unit amount of a pure chemical species is transferred from phase I to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition ... 

Carter, W.J., Shock Wave Techniques for the Examination of Phase Transitions, in Phase Transitions—1973, Proc. Conf. on Phase Transitions and Their Applications on Materials Science (edited by Cross, L.E.), Pergamon Press, New York,... 

Ross, M., A Review of Some Recent Theoretical Calculations of Phase Transitions and Comparisions with Experimental Results, in Shock Waves in Condensed Matter—1983 (edited by Asay, J.R., Graham, R.A., and Straub, G.K.), North-Holland Physics, Amsterdam, 1984, pp. 19-26. 

Crystallography is a very broad science, stretching from crystal-structure determination to crystal physics (especially the systematic study and mathematical analysis of anisotropy), crystal chemistry and the geometrical study of phase transitions in the solid state, and stretching to the prediction of crystal structures from first principles this last is very active nowadays and is entirely dependent on recent advances in the electron theory of solids. There is also a flourishing field of applied crystallography, encompassing such skills as the determination of preferred orientations, alias textures, in polycrystalline assemblies. It would be fair to say that... 

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

In 1970 Widom and Rowlinson (WR) introduced an ingeniously simple model for the study of phase transitions in fluids [185]. It consists of two species of particles, A and B, in which the only interaction is a hard core between particles of unlike species i.e., the pair potential v jsir) is inflnite if a P and r < and is zero otherwise. WR assumed an A-B demixing phase transition to occur in dimensions D >2 when the fugacity... 

Another example of phase transitions in two-dimensional systems with purely repulsive interaction is a system of hard discs (of diameter d) with particles of type A and particles of type B in volume V and interaction potential U U ri2) = oo for < 4,51 and zero otherwise, is the distance of two particles, j l, A, B] are their species and = d B = d, AB = d A- A/2). The total number of particles N = N A- Nb and the total volume V is fixed and thus the average density p = p d = Nd /V. Due to the additional repulsion between A and B type particles one can expect a phase separation into an -rich and a 5-rich fluid phase for large values of A > Ac. In a Gibbs ensemble Monte Carlo (GEMC) [192] simulation a system is simulated in two boxes with periodic boundary conditions, particles can be exchanged between the boxes and the volume of both boxes can... 

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... 

Phase transitions in adsorbed layers often take place at low temperatures where quantum effects are important. A method suitable for the study of phase transitions in such systems is PIMC (see Sec. IV D). Next we study the gas-liquid transition of a model fluid with internal quantum states. The model [193,293-300] is intended to mimic an adsorbate in the limit of strong binding and small corrugation. No attempt is made to model any real adsorbate realistically. Despite the crudeness of the model, it has been shown by various previous investigations [193,297-300] that it captures the essential features also observed in real adsorbates. For example, the quite complex phase diagram of the model is in qualitative agreement with that of real substances. The Hamiltonian is given by... 

Freezing transitions have been examined in recent years by density functional methods [306-313]. Here we review the results [298] of a modification of the Ramakrishnan-Yussouff theory to the model fluid with Hamiltonian (Eq. (25)) a related study of phase transitions in a system of hard discs in two dimensions with Ising internal states which couple anti-ferromagnetically to their neighbors is shown in Ref. 304. First, a combined... 

O. G. Mouritsen. Computer Studies of Phase Transitions and Critical Phenomena. Berlin Springer, 1984. 

From the time when Thorny and Duval presented the results of their early experiments (late 1960s) the field has grown enormously. Hundreds of papers and several monographs have been published and many eonferenees have been held to present new results of experimental and theoretieal studies and to exehange ideas as well as to stimulate further developments. A vast majority of all that aetivity has been direeted towards the understanding of the fundamental problems of phase transitions on uniform surfaees, whereas problems of the surfaee heterogeneity efleets have been mueh less intensively studied [11,57,122-126],... 

In this seetion we begin the diseussion with some basie thermodynamie eonsiderations whieh, as we hope, will show elearly why the efleets of surfaee heterogeneity make the study of surfaee phase transitions so diflieult. Then we shall eoneentrate mainly on the results of the reeent eomputer simulation studies of phase transitions in model systems exhibiting various forms of disorder. 

There have been other promising lines along which the theory of quenched-annealed systems has progressed recently. One of them, worth discussing in more detail, is the adsorption of fluids in inhomogeneous, i.e. geometrically restricted, quenched media [31,32]. In this area one encounters severe methodological and technical difficulties. At the moment, a set of results has been obtained at the level of a hard sphere type model adsorbed in sht-like pores with quenched distribution of hard sphere obstacles [33]. However, the problem of phase transitions has remained out of the question so far. 

It is of special interest for many applications to consider adsorption of fiuids in matrices in the framework of models which include electrostatic forces. These systems are relevant, for example, to colloidal chemistry. On the other hand, electrodes made of specially treated carbon particles and impregnated by electrolyte solutions are very promising devices for practical applications. Only a few attempts have been undertaken to solve models with electrostatic forces, those have been restricted, moreover, to ionic fiuids with Coulomb interactions. We would hke to mention in advance that it is clear, at present, how to obtain the structural properties of ionic fiuids adsorbed in disordered charged matrices. Other systems with higher-order multipole interactions have not been studied so far. Thermodynamics of these systems, and, in particular, peculiarities of phase transitions, is the issue which is practically unsolved, in spite of its great importance. This part of our chapter is based on recent works from our laboratory [37,38]. 

Another interesting class of phase transitions is that of internal transitions within amphiphilic monolayers or bilayers. In particular, monolayers of amphiphiles at the air/water interface (Langmuir monolayers) have been intensively studied in the past as experimentally fairly accessible model systems [16,17]. A schematic phase diagram for long chain fatty acids, alcohols, or lipids is shown in Fig. 4. On increasing the area per molecule, one observes two distinct coexistence regions between fluid phases a transition from a highly diluted, gas -like phase into a more condensed liquid expanded phase, and a second transition into an even denser... 

J. Tiaden, B. Nestler, H. J. Diepers, I. Steinbach. Physica D 115 11, 1998 G. J. Schmitz, B. Nestler. Simulation of phase transitions in multiphase systems, peritectic solidification of YBaCuO-superconductors. Mater Sci Eng B 53 11, 1998. 

A.I. Kolesnikov, A.M. Balagurov, I.O. Bashkin, V.K. Fedotov, V.Yu. Malyshev, G.M. Mironova, E.G. Ponyatovsky, A Real-Time Neutron Diffraction Study of Phase Transitions in the Ti-D System after High Pressure Treatment, J. Phys. Condensed Matter 5 5045 (1993). 

Ti ansients grow exponentially fast near the transition (Ac f/2) in the study of phase transitions in statistical physics this is known as critical slowing down. 

Table 7.4 Oi dor of phase transition and threshold probabilities versus space dimension for rules R, ...Rn, as determined by mean-field theory and numerical calculation ([bidaux89a], [bidaux89b]).

Langton s tentative answer to the question above is therefore We expect that information processing can emerge spontaneously and come to dominate the dynamics of a physical system in the vicinity of a critical phase transition. Langton speculates that the dynamics of phase transitions is fundamentally equivalent to the dynamics of information processing. 

If a fundamental equivalence between the dynamics of phase transitions and information processing along the lines suggested by Langton really exists, it would also lend support to the Church-l iring hypothesis [lang92] ... 

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