Crystallinity theory

One possible reason suggested by Flory and Mandelkern (1956) for or contribution to the discrepancies indicated in Table 2 is the inability of the penetrometer method to detect the maximum temperature of melting because of its relative insensitivity and because of the upper "tail to the crystallinity-temperature curve as deduced from the copolymer crystallinity theory of Flory (1955). Dole and Wunderlich (1957, 1959) found, however, that two melting regions each of the quenched 80/20 and 60/40 copolymers could be observed in the calorimetric experiments. The specific heat curves of these two copolyesters are illustrated in... 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

At a sufficiently low temperature, the phase nucleated will be crystalline rather than liquid. The theory is reviewed in Refs. 1 and 7. It is similar to that for the nucleation... 

RHEED is a powerful tool for studying the surface structure of crystalline samples in vacuum. Information on the surface symmetry, atomic-row spacing, and evidence of surfece roughness are contained in the RHEED pattern. The appearance of the RHEED pattern can be understood qualitatively using simple kinematic scattering theory. When used in concert with MBE, a great deal of information on film growth can be obtained. 

Figure 3.6). This theory known as the fringed mieelle theory or fringed crystallite theory helped to explain many properties of crystalline polymers but it was difficult to explain the formation of certain larger structures such as spherulites which could possess a diameter as large as 0.1 mm. 

Figure 3.6. Two-dimensional representation of molecules in a crystalline polymer according to the fringed micelle theory showing ordered regions (crystallites) embedded in an amorphous matrix.

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take... 

Appropriately, this was called the Folded Chain Theory and is illustrated in Fig. A.ll. There are several proposals to account for the co-existence of crystalline and amorphous regions in the latter theory. In one case, the structure is considered to be a totally crystalline phase with defects. These defects which include such features as dislocations, loose chain ends, imperfect folds, chain entanglements etc, are regarded as the diffuse (amorphous) regions viewed in X-ray diffraction studies. As an alternative it has been suggested that crystalline... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

The singlet-level theories have also been applied to more sophisticated models of the fluid-solid interactions. In particular, the structure of associating fluids near partially permeable surfaces has been studied in Ref. 70. On the other hand, extensive studies of adsorption of associating fluids in a slit-like [71-74] and in spherical pores [75], as well as on the surface of spherical colloidal particles [29], have been undertaken. We proceed with the application of the theory to more sophisticated impermeable surfaces, such as those of crystalline solids. 

The singlet-level theory has also been used to describe the structure of associating fluids near crystalline surfaces [30,31,76,77]. The surface consists explicitly of atoms which are arranged on a lattice of a given symmetry. The fluid atom-surface atom potential can also involve an associative term, i.e., the chemical-type bonding of the adsorbate particles with the surface may be included into the model. However, we restrict ourselves to the case of a nonassociative crystalline surface first. 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

Density functional theory was originally developed by solid-state physicists for treating crystalline solids and almost all applications were in that field until the mid-1980s. It is a current hot topic in chemistry, with many papers appearing in the primary journals. 

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