Crystalline derivation, expressions

The time dependence of the overall (primary) crystallization is described by the Avrami equation, which was originally derived for the crystallization of metals. The crystallinity is expressed as the volume fraction of the crystalline material in the total sample. For the derivation, it is assumed that each nucleus leads to an entity (e.g., a rod, disk, or sphere). After an infinitely long time the whole sample is filled with these shapes. The crystallinity of the sample is then 4>oo. This, however, is the crystallinity of a single entity which is assumed not to change during crystallization. At the time r, the fraction of the sample volume filled by these entities is /randomly distributed nuclei the probability p that a point does not lie in any one entity is proportional to this fraction, i.e.,... 

One of the problems with using test schemes based on a simple Arrhenius relation is that it does not readily account for the influence of different imposed stresses on the reaction rate. This is important, since data measured at one stress level will not necessarily provide a good estimate of reaction rate under another condition. Zhurkov investigated the time to failure of polymeric, metallic, and nonmetaflic crystalline materials in uniaxial tension at a number of elevated temperatures and different stresses [9]. His experimental results fit the empirically derived expression shown in Equation 16B.2 that he called the thermofluctuational theory. ... 

Volume and mass-based expressions for the degree of crystallinity are easily derived from the experimentally measured density (p) of a semi-crystalline polymer. The method is based on an ideal crystalline and liquid-like two-phase model and assumes additivity of the volume corresponding to each phase... 

Brenner and Garrison introduced a potential which was derived by rewriting a valence force expression so that proper dissociation behavior is attained . Because the equations were extended from a set of terms which provided an excellent fit to the vibrational properties of silicon, this potential is well suited for studying processes which depend on dynamic properties of crystalline silicon. For example, Agrawal et al. have studied energy transfer from adsorbed hydrogen atoms into the surface using this potential . 

In order to derive the determining physical equation for the stress-deformation state of a two-component mixture, let us consider the expression for the change in the elastic potential energy during a continious transition from a liquid to a solid phase. Let this transition occur at time t = to and let the quantity of material that undergoes the transition be equal to the increase in the degree of crystallinity Act. The specific elastic potential characterizing the new state of the material up to the time of a new transition can be written as... 

Some crystallins are (as far as we know) ubiquitous in vertebrates. Of these aA-crystallin is the most abundant and has been used in large-scale phylogenetic analyses, using classic techniques of isolation and sequencing.6 However, it has recently been appreciated that other major crystallins show remarkably taxon-specific patterns of expression (Fig. 1). Furthermore, these taxon-specific crystallins are all identical to enzymes, or else rather recently derived from enzymes (Table I). Some modification in a single functional gene led to the acquisition by the protein product of a dual function as both enzyme and structural lens protein. In this model gene recruitment comes first duplication may or may not follow.7... 

Timmermans (1913) and Mathews (1916) introduced the concept of zero point density based on the extrapolation of densities of crystalline and liquid substances to 0 K. Sugden (1927) and Biltz (1934) developed an additive system for deriving values of V°(0) from chemical constitution. The zero point volume is closely related to the Van der Waals volume. According to Bondi (1968a) a good approximation is given by the following expression ... 

In the ordered state, the suspension can have a substantial modulus (see Fig. 6-31). Buscall et al. (1982b, 1991) have derived an expression for the high-frequency modulus from the interparticle potential by calculating the work required for particles to move affinely with the flow. The result of Buscall et al. follows from the Zwanzig-Mountain expression, Eq. (6-20), where g r ) can be replaced by a delta function since the suspension is crystalline. Neglecting the osmotic term, the result is (Evans and Lips 1990 Buscall 1991)... 

The present paper is concerned with mixtures composed of a highly nonideal solute and a multicomponent ideal solvent. A model-free methodology, based on the Kirkwood—Buff (KB) theory of solutions, was employed. The quaternary mixture was considered as an example, and the full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived on the basis of the KB theory of solutions. Further, the expressions for the derivatives of the activity coefficients were applied to quaternary mixtures composed of a solute and an ideal ternary solvent. It was shown that the activity coefBcient of a solute at infinite dilution in an ideal ternary solvent can be predicted in terms of the activity coefBcients of the solute at infinite dilution in subsystems (solute + the individual three solvents, or solute + two binaries among the solvent species). The methodology could be extended to a system formed of a solute + a multicomponent ideal mixed solvent. The obtained equations were used to predict the gas solubilities and the solubilities of crystalline nonelectrolytes in multicomponent ideal mixed solvents. Good agreement between the predicted and experimental solubilities was obtained. 

A more comprehensive theory for the thermodynamics of semi crystalline diblocks has been developed using self-consistent mean field theory applied to diblocks with one amorphous block and one crystallizable block [20].The amorphous regions were modelled as flexible chains, and the crystalline regions as folded chains. Both monolayers and bilayers of once-folded chains were considered. Expressions were derived for the thickness of the amorphous and crystalline region and the number of folds. The central result is the domain spacing scaling [Eq. (1)]. 

Zeolites are fascinating materials whose unusual chemical and physical properties derive from their porous yet crystalline structure (1-6). Since the reader is probably unfamiliar with these materials, it will be useful to express the attributes of zeolites that are pertinent to the photochemistry of organic molecules adsorbed on their surface in terms of qualitative concepts that will allow an appreciation of the strategies available to the experimentalist. Such concepts will allow both a clear understanding of experimental results in terms of zeolite attributes and a means of extending the.approaches to other areas of chemistry. 

From the latter analytical expression, it is easily shown that the expression cannot be derived from the interaction of particles. On the other hand, the second term is discontinuous at densities corresponding to filled shells. As stated above, the potential of all the electrons in a crystalline solid changes discontinuously when another electron is added. This case is not predicted by the LDA method but it is predicted in the orbital-dependent forms. For an orbital functional, the kinetic energy is evaluated first ... 

The oligomeric structures listed in Table I are all derived from purified recombinant proteins, typically expressed in E. coli. However, in many organisms multiple sHsps are found in the same cellular compartment and coassembly of sHsps into heterooligomers is often observed. A primary case in point is the a-crystallins a A- and aB-crystallin are complexed together in the eye lens at an a A B ratio of 3 1, forming polydisperse aggregates with an average molecular mass of 800 kDa... 

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