Linear nucleation

Growth habit Homogeneous nucleation Heterogeneous nucleation" Linear growth... 

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]. 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

Figure 11.7 shows how the mechanical properties of normalised carbon steels change with carbon content. Both the yield strength and tensile strength increase linearly with carbon content. This is what we would expect the FejC acts as a strengthening phase, and the proportion of FojC in the steel is linear in carbon concentration (Fig. 11.6a). The ductility, on the other hand, falls rapidly as the carbon content goes up (Fig. 11.7) because the a-FejC interfaces in pearlite are good at nucleating cracks.

In sulphur dioxide linear kinetics are generally observed due to control by phase boundary reactions, i.e. adsorption of SOj. RahmeF suggested that this is one of the conditions which favours simultaneous nucleation of sulphide and oxide at the gas/scale interface. The main reaction products are NiO, NijSj, Ni-S,j, and NiS04, depending on the temperature and gas pressure for example, according to the following reaction ... 

Many authors studying the formation of ECC from melts and solutions suggested that preliminary unfolding and extension of macromolecules occurs. Keller and Maehin25 have shown that in all known cases (including such extreme variants as the crystallization of natural rubber under extension and a polyethylene melt under flow) the same initial process of linear nucleation occurs and fibrillar structures is formed by the macromolecu-lar chains oriented parallel to the fibrillar axes27. ... 

In contrast, there is no nucleation barrier for rough surface growth at any supercooling. The growth rate is then simply proportional to v as given by Eq. (3.4), and hence is expected to be linear in AT for small undercoolings. 

Fig. 3.3. Growth rate versus supercooling for two different face orientations. T is above its roughening temperature and is approximately linear. 2 is below its roughening temperature and is nucleation controlled at low supercooling but the growth rapidly increases after kinetic roughening...

The exponent n = j3 + X, where j3 is the number of steps involved in nucleus formation (frequently j3 = 1 or 0, the latter corresponding to instantaneous nucleation) and X is the number of dimensions in which the nuclei grow (X = 3 for spheres or hemispheres, 2 for discs or cylinders and 1 for linear development). Most frequently, it is found that 2 < n < 4. Since n is a compound term, the value determined does not necessarily provide a unique measurement of both j3 and X. Ambiguity may arise where, for example, n = 3 could be a consequence of (j3 = 2, X = 1), (j3 = 1,... 

Linear plots of ln(l — a) against t were obtained [454] for the removal of water from CaC03 6 H20. The slope of the line gives the rate coefficient for the nucleation step. 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

The initiation of dehydration at the first-formed nuclei does not necessarily preclude the continued production of further nuclei elsewhere on unreacted surfaces. During dehydration of CuS04 5 H20, the number of nuclei was shown [426] to increase linearly with time, whereas during water removal from NiS04 7 H20 [50] the number of nuclei increased with the square of time, Nt = kN(t — t0)2. (The latter behaviour contrasts with the instantaneous nucleation of NiS04 6 H20 mentioned above.)... 

Kinetic data for the decompositions of several metal hydrides are summarized in Table 12 to which the following information can be added. The acceleratory period in the decomposition of BeH2 (a < 0.35) is ascribed [673] to the random formation of metal nuclei followed by linear growth. The increase in rate consequent upon exposure to X-irradia-tion is attributed to enhanced nucleation. Grinding similarly increased the... 

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