Discontinuous Transforms

FIGURE 5.7 If we take the difference between the incident and reflected rays k and lc0 from (a) to form the diffraction vector s, then, by Bragg s law, it is clear that s must be identical to the reciprocal lattice vector h for any family of planes in diffracting position. This must be true no matter what the actual distribution of atoms, or scattering material, around the planes in the family may be. 

For CuKa radiation of 1.54 A, the minimum interplanar spacing capable of diffracting X rays is 0.77 A. Families of planes with d 0.77 A in this case are of no interest. Because of this limit to d spacing, the reciprocal lattice is also not infinite but bounded by some wavelength-dependent resolution limit. 

Extended to three dimensions, the diffraction pattern of a crystal is the Fourier transform of the molecules in the unit cells of the crystal, but visible only at discrete reciprocal lattice points permitted by the underlying crystal lattice. This is the price we pay to simultaneously observe the cooperative diffraction from all of the unit cells in the crystal, to obtain enough intensity to measure. While a single unit cell would have given a continuous transform, it would not yield experimentally measurable intensity. A crystal yields sufficient intensity, but only grudgingly, at discrete points. 

DIFFRACTION FROM POINTS, PLANES, MOLECULES, AND CRYSTALS 

FIGURE 5.9 In (a) an assembly of six scattering points has been convoluted with an orthorhombic point lattice, and in (c) the same arrangement with a monoclinic lattice. The corresponding diffraction patterns are, respectively, orthorhombic and monoclinic as well. The diffraction patterns, however, are very different in their intensities because the point lattices sample the continuous transform of the six scattering points, the asymmetric unit of these two-dimensional crystals, at different places. 

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Stage II Discontinuous transformation in the neck, of the spherulitic into the fibrillar structure ... 

The remainder of the book treats discontinuous transformations. Nucleation, which is necessary for the production of a new phase, is treated in Chapter 19. The growth of new phases under diffusion- and interface-limited conditions is treated in Chapter 20. Concurrent nucleation and growth is treated in Chapter 21. Specific examples of discontinuous transformations are discussed in detail these include solidification (Chapter 22), precipitation from solid solution (Chapter 23), and martensite formation (Chapter 24). 

CLASSIFICATION OF PHASE TRANSFORMATIONS CONTINUOUS VERSUS DISCONTINUOUS TRANSFORMATIONS... 

This leads to the general classification of continuous and discontinuous transformations. 

Discontinuous transformations In this type of transformation, there is a free-energy barrier to infinitesimal variations and the system is initially metastable. However, a sufficiently large variation can cause the free energy to decrease. The transformation therefore can be initiated at a finite rate only by a variation that is large in degree but small in extent (i.e., nucleation is required). Examples include the formation of B-rich precipitates from a supersaturated A-B solution. 

Discontinuous transformations will generally occur in the series of stages illustrated in Fig. 19.1. 

Figure 19.1 Number of particles formed during a discontinuous transformation, N, as a function of time at constant temperature.

A discontinuous transformation generally occurs by the concurrent nucleation and growth of the new phase (i.e., by the nucleation of new particles and the growth of previously nucleated ones). In this chapter we present an analysis of the resulting overall rate of transformation. Time-temperature-transformation diagrams, which display the degree of overall transformation as a function of time and temperature, are introduced and interpreted in terms of a nucleation and growth model. 

In previous chapters we have developed models for discontinuous transformations that treat nucleation and growth processes independently. However, when... 

Simulations of octahedral molecular clusters at constant temperature show two kinds of structural phase changes, a high-temperature discontinuous transformation analogous to a first-order bulk phase transition, and a lower-temperature continuous transformation, analogous to a second-order bulk phase transition. The former shows a band of temperatures within which the two phases coexist and hysteresis is likely to appear in cooling and heating cycles Fig. 10 the latter shows no evidence of coexistence of two phases. The width of the coexistence band depends on cluster size an empirical relation for that dependence has been inferred from the simulations. 

Ultimately the Schottky maximum becomes severly deformed and when a critical value for the aggregation, Aw rj, is exceeded, the maximum becomes discontinuous on its high-temperature side. The diffuse transformation is then changed into a discontinuous transformation (Fig. 2.8). The reason for this change to a first order transition is the loss of stability of the system if... 

The critical value Awe i od the temperature T at which the discontinuous transformation Incomes posible are given by the highest, but perhaps only fictitious, value Do = 2 for the concentration of the (D 0) pairs. According to Eq. (2.24) this... 

Tte transition tempa-ature is thus indepraident of Aw and depends solely on w -w, L, q, and Au. The discontinuous transformation does, however, only occur at T, if additionally the critical value obtained by inserting — 1/2 into Eq. (2.27) is exceeded, i.e. ... 

It then follows from Eqs. (2.28) and (2.29) that the transition temperature has also a lower finite limit (T, ). Only crystals with chain molecules whose length exceeds a given minimum can undergo discontinuous transformation. The value of depends on q, Au and the interaction parameters. The transition temperature increases with drain length, but is subject to the following restricticai ... 

Figure 4.35 illustrates the different temperature—time zones of age-hardening of the Pb—0.11 wt% Ca—0.57 wt% Sn alloy. Zone A covers the discontinuous transformations zone B is the zone of continuous precipitation of (Pbi xSnx)3Ca and zone C is the zone of discontinuous precipitation of lamellar (Pbi xSnx)3Ca phase [75]. 

Medronho B, Shafaei S, Szopko R, Miguel MG, Olsson U, Schmidt C (2008) Shear-induced transitions between a planar lamellar phase and multilamellar vesicles continuous versus discontinuous transformation. Langmuir 24 6480-6486... 

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