Phase Transitions Basic Concepts

The term hysteresis is used to describe the lagging of an effect behind its cause. Hysteresis is a common phenomenon that is observed in a great variety of physical, chemical, and biological systems. However, the magnetization response of a ferromagnet in the 

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For all kinds of transitions, the system tends to hesitate between order and disorder and is prone to exhibit thermodynamic fluctuations which reflect the search for a compromise between the simultaneous requirements for minimum energy and maximum entropy. As the conducting polymers are pseudo-one-dimensional/two-dimensional systems, the probability of thermodynamic fluctuation increases significantly, resulting in a decrease in the ordered phase. The basic concept is that all electrochemical reactions proceed by adsorption from solution. This amounts to the replacement of solvent molecules by substrate, a process which is simultaneously governed by solvent-electrode, solvent-solute and solute-electrode interactions. Water, which is the most common solvent, possesses a high dielectric constant and, as such, tends to reject at its bulk periphery all molecules with a low dielectric constant. 

This chapter introduces additional central concepts of thermodynamics and gives an overview of the formal methods that are used to describe single-component systems. The thermodynamic relationships between different phases of a single-component system are described and the basics of phase transitions and phase diagrams are discussed. Formal mathematical descriptions of the properties of ideal and real gases are given in the second part of the chapter, while the last part is devoted to the thermodynamic description of condensed phases. 

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

The QET is not the only theory in the field indeed, several apparently competitive statistical theories to describe the rate constant of a unimolecular reaction have been formulated. [10,14] Unfortunately, none of these theories has been able to quantitatively describe all reactions of a given ion. Nonetheless, QET is well established and even the simplified form allows sufficient insight into the behavior of isolated ions. Thus, we start out the chapter from the basic assumptions of QET. Following this trail will lead us from the neutral molecule to ions, and over transition states and reaction rates to fragmentation products and thus, through the basic concepts and definitions of gas phase ion chemistry. 

It is common to begin the discussion of second-order phase transitions, including their symmetry aspects, by a concept whose basic idea is a series expansion of the Gibbs energy in terms of the order parameter... 

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

The OOA was not designed for and does not apply to temperature dependencies of any kind in JT crystals. In particular, one cannot expect a reasonable estimate of the temperature of phase transitions in crystal lattice (structural), electron orbital, and/or spin system. This follows from the partitioning procedure that includes averaging over vibrational degrees of freedom. One can see the same reason from another perspective. The pseudo spin of a JT site, as the basic concept used in the OOA, operates in the basis of degenerate ground state wave functions. Excited vibronic states are beyond the pseudo spin setup. Therefore, in the OOA, by its very definition, temperature population of excited states does not make sense. 

In this review, we shall introduce some basic concepts about sohd-state NMR of half-integer quadrupolar nuclei and discuss the most useful and promising methods presently available to study them. These include older methods, such as Double Rotation (DOR) and Dynamic Angle Spinning (DAS), and novel techniques including MQMAS, Quadrupolar Phase Adjusted Spinning Sidebands QPASS, SateUite Transition (ST) MAS and Inverse-STMAS NMR, and Fast Amplitude Modulation (FAM). We also discuss several techniques based on dipolar interactions between quadrupolar and spin-1/2 nuclei, such as Cross-Polarization (CP) MQMAS, MQ Heteronuclear Correlation Spectroscopy... 

In Landau theory, the information about the change of physical quantities is gathered in the order parameter Qo = V f d rQ(r) which is a macroscopic quantity that neglects spatial and temporal fluctuations. The basic concept of the description of phase transitions is the introduction of the Landau free energy, iF = J d rf, which takes into account the symmetry of the system through a power series expansion in terms of the scalar invariants of the order parameter, whereas the equation of state of the system reads... 

The general concept of the excitation of coherent oscillations in biological systems, with limited reference to membranes, has been nicely reviewed by Kaiser. A discussion of Frohlich s model is presented in terms of how it possesses the requisite characteristics to undergo various types of phase transitions subsequent to external perturbations. Reference, however, to the specific involvement of biological membranes in such phase transitions is only briefly mentioned. The emphasis in Kaiser s analysis of Frohlich s basic hypothesis is on the idea that stable limit cycle behavior... 

Chapter 4 will introduce the reader to the basic concepts of the X-ray analysis of crystals and its applications to particular liquid crystal phases. It should be noted that in the present literature this problem is not adequately dealt with anywhere, and this chapter attempts to rectify this deficiency. Chapter 5 covers phase transitions, one of the key problems of the liquid crystal physics, and which has been widely discussed in other texts at very different levels. In this chapter I give only a detailed explanation of the basic crmcepts of the phase transitions between most important mesophases. 

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. ... 

To obtain the fundamental understanding of the above basic concepts one must consider first the paint components. Most paint formulations consist of disperse systems (solid in liquid dispersions). The disperse phase consists of primary pigment particles (organic or inorganic) which provide the opacity, color and other optical effects. These are usually in the submicron range. Other coarse particles (mostly inorganic) are used in the primer and undercoat to seal the substrate and enhance adhesion of the top coat. The continuous phase consist of a solution of polymer or resin which provides the basis of a continuous film that seals the surface and protects it firom the outside environment. Most modern paints contain latexes which are used as film formers. These latexes (with a glass transition temperature mostly below ambient temperature)... 

Chapters 1 and 2 introduce the main phases and basic properties of liquid crystals and other anisotropic fluids, such as soaps, foams, mono-layers, fluid membranes and fibers. These chapters do not include difficult mafliematical formulas and are probably suitable for imdergraduates or for other professionals, such as K-12 teachers. Chapter 3 describes the nature of phase transitions based on the phenomenological Landau-de Gennes theories, and on the self-consistent mean-field theories that use concepts in statistical physics. 

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