Phase transitions higher order

The inclusion of both three and four-particle correlations in nuclear matter allows not only to describe the abundances oft, h, a but also their influence on the equation of state and phase transitions. In contrast to the mean-field treatment of the superfluid phase, also higher-order correlations will arise in the quantum condensate. 

The Ehrenfest17 classification of phase transitions (first-order, second-order, and lambda point) assumes that at a first-order phase transition temperature there are finite changes AV 0, Aft 0, AS VO, and ACp VO, but hi,lower t = hi,higher t and changes in slope of the chemical potential /i, with respect to temperature (in other words (d ijdT)lowerT V ((9/i,7i9T)higherT). At a second-order phase transition AV = 0, Aft = 0, AS = 0, and ACp = 0, but there are discontinuous slopes in (dV/dT), (dH/<)T), (OS / <)T), a saddle point in and a discontinuity in Cp. A lambda point exhibits a delta-function discontinuity in Cp. 

Upon further compression of the monolayer, a pronounced break, or discontinuity in the Isotherm marks a region of co-existence of the LE and the LC states. In most research papers it is stated that x(A) still increases upon compression beyond the beginning of the LE-LC transition region. Breaks may be indicative of higher-order phase transitions. The order of such a transition depends on the extent of co-operatlvity between the aliphatic chains. However, the presence of such transitions is not always well established. Pallas and Pethica found that... 

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

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]. 

At higher temperatures, other degrees of freedom than the radius R must also be considered in the fluctuation. However, this becomes critical only near the critical point where the system goes through a phase transition of second order. The nucleation arrangement described here is for heterogeneous or two-dimensional nucleation on a flat surface. In the bulk, there is also the formation of a three-dimensional nucleation, but its rate is smaller ... 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

The dependence on the temperature of the specific resistance (Q/cm) of the pure MEPBr and MEMBr complexes, and a 1 1 mixture there of, as obtained in Ref. [73], is listed in Table 4. It is remarkable that within the complex phases consisting of Br2 and either pure MEP or MEM the change of specific resistance at the liquid —> solid phase transition amounts to about one order of magnitude, where as the value is only doubled in the 1 1 mixture. The table also indicates that MEMBr complexes possess higher melting temperatures. 

The boundary layers, or interphases as they are also called, form the mesophase with properties different from those of the bulk matrix and result from the long-range effects of the solid phase on the ambient matrix regions. Even for low-molecular liquids the effects of this kind spread to liquid layers as thick as tens or hundreds or Angstrom [57, 58], As a result the liquid layers at interphases acquire properties different from properties in the bulk, e.g., higher shear strength, modified thermophysical characteristics, etc. [58, 59], The transition from the properties prevalent in the boundary layers to those in the bulk may be sharp enough and very similar in a way to the first-order phase transition [59]. 

First of all the term stress-induced crystallization includes crystallization occuring at any extensions or deformations both large and small (in the latter case, ECC are not formed and an ordinary oriented sample is obtained). In contrast, orientational crystallization is a crystallization that occurs at melt extensions corresponding to fi > when chains are considerably extended prior to crystallization and the formation of an intermediate oriented phase is followed by crystallization from the preoriented state. Hence, orientational crystallization proceeds in two steps the first step is the transition of the isotropic melt into the nematic phase (first-order transition of the order-disorder type) and the second involves crystallization with the formation of ECC from the nematic phase (second- or higher-order transition not related to the change in the symmetry elements of the system). 

Experience indicates that the Third Law of Thermodynamics not only predicts that So — 0, but produces a potential to drive a substance to zero entropy at 0 Kelvin. Cooling a gas causes it to successively become more ordered. Phase changes to liquid and solid increase the order. Cooling through equilibrium solid phase transitions invariably results in evolution of heat and a decrease in entropy. A number of solids are disordered at higher temperatures, but the disorder decreases with cooling until perfect order is obtained. Exceptions are... 

It is concluded that the cooperative effect observed is of long-range nature and therefore of elastic rather than of electronic origin. Recently, the additional suggestion has been made [138] that, due to intermolecular interactions in the crystal environment of [Fe(ptz)g](BF4)2, domains of iron(II) complexes interconvert together. The observed kinetics would then correspond to a first- or higher-order phase transition rather than to the kinetics which are characteristic for the conversion of isolated molecules. 

---