Holes theory

Using the modified hole theory and the appropriate mathematical manipulations, the set of equations shown in Table II resulted. 

The diffusion coefficients at infinite dilution (D]0, D 0, and Dr0) for the fuzzy cylinder reduce to those for the wormlike cylinder, which can be calculated as explained in Appendix B. On the other hand, these diffusion coefficients, D, Dx, and Dr, for the fuzzy cylinder at finite concentrations can be formulated by use of the mean-field Green function method and the hole theory, as detailed below. 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

On the basis of the hole theory of Hirai-Eiring32,33 as developed by Smith34, the volume of the whole system may be given as... 

These phenomenological theories are more complicated than the Flory-type theory, though they have certain advantages over the latter. In the quasichemical treatment, the molecular interaction responsible for the transition, which is hidden behind the parameter % in the Flory theory, appears with clearer physical meaning. In the hole theory of gels, some properties of gels which are... 

The Hole Theory of the Glass Transition and its Relation to Calorimetric Studies.256... 

In terms of the hole theory, the opposite phenomenon, namely, the slow absorption of heat (negative temperature drifts of the calorimeter) have to be explained in the following way Assuming that the negative... 

The hole theory was perceived as a Active mathematical construction and was initially rejected by prominent contemporary physicists such as Pauli and Bohr. The physical reality of antiparticles was not taken seriously even by Dirac himself. In 1931 he wrote about his anti-electron we should not expect to find it in Nature [2]. Surprisingly, the first anti-electrons were discovered already in 1932 by Anderson, who studied cosmic rays in Caltech s magnet cloud chamber. Anderson noticed abnormally bending trajectories indicating the presence of light positively charged particles and, as related by Fowler [3], "could not resist the devastating conclusion that they are caused by positive electrons The first piece of antimatter, a positron, made its physical appearance. 

This principle serves as the basis for a number of models of fused salt systems. Perhaps the best known of these is the Temkin model, which uses the properties of an ordered lattice to predict thermodynamic quantities for the liquid state [79]. However, certain other models that have been less successful in making quantitative predictions for fused salts may be of interest for their conceptual value in understanding room temperature ionic liquids. The interested reader can find a discussion of the early application of these models in a review by Bloom and Bockris [71], though we caution that with the exception of hole theory (discussed in Section II.C) these models are not currently in widespread use. The development of a general theoretical model accurately describing the full range of phenomena associated with molten salts remains a challenge for the field. 

Sidestepping the question of liquid structure with an empirical approach, Abbott has had considerable success in modeling IL viscosity [178] and conductivity [179] using hole theory, as described in Section II.A. While the model is a useful tool, as noted previously, it is based on the macroscopic surface tension and does not provide direct insight on many of the underlying properties of the liquid. Nevertheless, any framework that is developed to interpret the behavior of ILs must also explain the success of hole theory. 

The conductivities of Type I ionic liquids based on anhydrous zinc and iron salts tend to be lower than those of the corresponding aluminum ionic liquids. This is due largely to the higher viscosity of the former, primarily because of the large size of the ions and the availability of suitably sized holes in the ionic liquids for the ions to move into. This has been quantified by the application of hole theory as is explained in Section 2.3.4. In general imidazolium-based liquids have lower viscosities and higher conductivities than the corresponding pyridinium or quaternary ammonium eutectics formed under the same conditions. 

We have fitted the viscosity of ionic liquids using hole theory [123]. The theory was developed for molten salts but has been shown to be very useful for ionic liquids. It was shown that the value of En is related to the size of the ions and the size of the voids present in the liquid [103]. The viscosity of ionic liquids is... 

The second group of theories is based on a general approximation and consists of the use of a molten salt model to obtain a partition function from the molecular motion. This group includes the following theories the hole theory, the theory of significant structures and other structural models. The theories of the first group are mathematically more difficult but lead to good results for the molten salt structure. 

Table 3. Parameters of the hole theory for some linear polymers and networks...

On the basis of the hole theory of liquids, Wunderlich (1960) concluded that the difference Cp — Cp at the glass transition temperature should be constant per structural bead in the polymer. A structural bead in this sense is defined as the smallest section of the molecule that can move as unit in internal rotation. 

Smith (1970) derived an equation of state for liquid polymers based on the hole theory of liquids. For higher temperatures, this equation can be reduced to a form equivalent to that of Eq. (7.26). 

Tn recent years, developing interests in surface energetics and adhesion of liquid-like polymers, or polymer liquids, have prompted both theoretical and experimental work on surface tension. Unlike low molecular weight liquids, polymer liquids have not been extensively studied. Bondi and Simkin (1) mentioned surface tension in their study on high molecular weight liquids. Roe (28) applied both the cell theory of polymer liquids and the hole theory of surface tension of simple liquids to develop an approximate theory of surface tensions of polymer liquids. His approach has met some degree of success. Notably, both Bondi s and Roes work are somewhat related to the cell theory introduced by Prigogine and... 

The GC-EOS based on a series expansion of the nonrandom lattice-hole theory is written by[l,6],... 

The first anti-particle discovered was the anti-electron, the so-called positron, in 1933 by Anderson [3] in the cloud chamber due to cosmic radiation. The existence of the anti-electron (positron) was described by Dirac s hole theory in 1930 [4], The result of positron—electron annihilation was detected in the form of electromagnetic radiation [5]. The number and event of radiation photons is governed by the electrodynamics [6, 7]. The most common annihilation is via two- and three-photon annihilation, which do not require a third body to initiate the process. These are two of the commonly detected types of radiation from positron annihilation in condensed matter. The cross section of three-photon annihilation is much smaller than that of two-photon annihilation, by a factor on the order of the fine structure constant, a [8], The annihilation cross section for two and three photons is greater for the lower energy of the positron—electron pair it varies with the reciprocal of their relative velocity (v). In condensed matter, the positron—electron pair lives for only the order of a few tenths to a few nanoseconds against the annihilation process. 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Comparison of Calculated Isothermal Compressibilities with the Experimental Values (Hole Theory)... 

Mathematical treatment of molten salts that supercool was first carried out by Cohen and Turnbull. The principal idea of the hole theory—that diffusion involves ions that wait for a void to turn up before jumping into it—is maintained. However, Cohen and Turnbull introduced into their model a property called thefree volume, Vf. What is meant by this free volume It is the amount of space in addition to that, Vq, filled by matter in a closely packed liquid. Cohen and Turnbull proposed that the free volume is linearly related to temperature... 

R. Fiirth, On the Theory of the Liquid State, n. The Hole Theory of the Viscous Flow of Liquids, Proc. Cambridge Phil. Soc. 37 281 (1941). 

Thus, according to the hole theory, the random walk of holes between adjacent layers results in momentum transfer and therefore viscous drag in a moving fused salt (Fig. 5.45). On the basis of this model, the expression for the viscosity of an ionic iiquid is... 

This is the mean lifetime of a hole. This expression is consistent with the idea that the hole theory represents a Swiss-cheese sort of model of a liquid with holes of different sizes [for is the mean radius for holes varying in size according to Eq. (5.44)]. The holes keep on opening and shutting, and the mean time they are open is given by Eq. (5.93). 

Before one leaves expression (5.93), it is well to note the innocent acceptance with which A has been treated. It is the heat term associated with getting a hole unmade, with collapsing the hole, the negative of the work of forming the hole. However, it has not yet been said how this will be calculated, and what terms go into this. Such a calculation will be one test that will be made of the hole theory in Section 5.7.6. 

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