Nucleation, first-order

In Mampel s treatment [447] of nucleation and growth reactions, eqn. (7, n = 3) was found to be applicable to intermediate ranges of a, sometimes preceded by power law obedience and followed by a period of first-order behaviour. Transitions from obedience of one kinetic relation to another have been reported in the literature [409,458,459]. Equation (7, n = 3) is close to zero order in the early stages but becomes more strongly deceleratory when a > 0.5. 

Daw et al. [632] made a microscopic and crystallographic study of talc dehydroxylation. Nucleation, to yield enstatite, occurs inhomogeneously within the particles, perhaps at dislocations. Later, this product is topo-tactically orientated with respect to the reactant lattice, though with extensive faulting on the (010) plane owing to misfit, in addition to the attempt to preserve the oxygen lattice. In an isothermal study (1100— 1160 K) of the same reaction, Ward [633] found first-order obedience and the value of E determined (422 kJ mole-1) is close to that estimated for... 

Torkar et al. [702,706—708] identified nucleation as an autocatalytic process at the (hk0) planes of hexagonal platelets of NaN3. The decelera-tory reaction fitted the first-order equation [eqn. (15)]. Values of E tended to be irreproducible for the pure salt E was about 180 kJ mole 1 but this was reduced to about half by doping. This influence of an additive and the observed similarities in magnitudes of E for decomposition and for diffusion were interpreted as indicating that growth of nuclei was controlled by a diffusion process. 

The kinetic observations reported by Young [721] for the same reaction show points of difference, though the mechanistic implications of these are not developed. The initial limited ( 2%) deceleratory process, which fitted the first-order equation with E = 121 kJ mole-1, is (again) attributed to the breakdown of superficial impurities and this precedes, indeed defers, the onset of the main reaction. The subsequent acceleratory process is well described by the cubic law [eqn. (2), n = 3], with E = 233 kJ mole-1, attributed to the initial formation of a constant number of lead nuclei (i.e. instantaneous nucleation) followed by three-dimensional growth (P = 0, X = 3). Deviations from strict obedience to the power law (n = 3) are attributed to an increase in the effective number of nuclei with reaction temperature, so that the magnitude of E for the interface process was 209 kJ mole-1. 

Isothermal a—time curves were sigmoid [1024] for the anhydrous Ca and Ba salts and also for Sr formate, providing that nucleation during dehydration was prevented by refluxing in 100% formic acid. From the observed obedience to the Avrami—Erofe ev equation [eqn. (6), n = 4], the values of E calculated were 199, 228 and 270 kJ mole"1 for the Ca, Sr and Ba salts, respectively. The value for calcium formate is in good agreement with that obtained [292] for the decomposition of this solid dispersed in a pressed KBr disc. Under the latter conditions, concentrations of both reactant (HCOJ) and product (CO3") were determined by infrared measurements and their variation followed first-order kinetics. 

The most developed and widely used approach to electroporation and membrane rupture views pore formation as a result of large nonlinear fluctuations, rather than loss of stability for small (linear) fluctuations. This theory of electroporation has been intensively reviewed [68-70], and we will discuss it only briefly. The approach is similar to the theory of crystal defect formation or to the phenomenology of nucleation in first-order phase transitions. The idea of applying this approach to pore formation in bimolecular free films can be traced back to the work of Deryagin and Gutop [71]. 

First-order phase transitions exhibit hysteresis, i.e. the transition takes place some time after the temperature or pressure change giving rise to it. How fast the transformation proceeds also depends on the formation or presence of sites of nucleation. The phase transition can proceed at an extremely slow rate. For this reason many thermodynamically unstable modifications are well known and can be studied in conditions under which they should already have been transformed. 

Water soluble impurities and their effect can be easily included in equation (1-4), through which they are going to directly affect the particle nucleation rate, f(t). If one assumes a first order reaction of an active radical with a water soluble impurity (WSI) to give a stable non-reactive intermediate, then one simply has to add another term in the denominator of equation (1-4), of the form kwsr[WSI](t)-kv, and to account for the concentration of WSI with a differential equation as follows ... 

Structural changes on surfaces can often be treated as first-order phase transitions rather than as adsorption process. Nucleation and growth of the new phase are reflected in current transients as well as dynamic STM studies. Nucleation-and-growth leads to so-called rising transients whereas mere adsorption usually results in a monotonously falling transient. In Fig. 10 are shown the current responses to potential steps across all four current peaks in the cyclic voltammogram of Fig. 8a [44], With the exception of peak A, all structural transitions yield rising current transients sug-... 

The electrodeposited TiAl3 described above displays crystallographic features of distinctly different length scales. The deposit grain size is on the order of 0.1 to 0.5 pm while the Ll2 domain size is on the order of 5-10 nm. The domains appear to have grown through a first order nucleation and growth process that is independent... 

A real surface of a solid metal is inhomogeneous, and nucleation for the growing clusters is favored at certain active sites. To simplify the mathematics we consider an electrode with unit surface area. If there are Mo active sites, the number M(t) of growing nuclei is given by first-order kinetics ... 

The interphases between MDs should not be able to cause nucleation of the semicrystalline phase. If the interphase displays nucleation activity, first-order crystallization kinetics may be obtained however, the supercooling at which dynamic crystallization from the melt occurs will not be the maximum that could be possibly attained if homogeneous nucleation were in place. 

Many high-pressure reactions consist of a diffusion-controlled growth where also the nucleation rate must be taken into account. Assuming a diffusion-controlled growth of the product phase from randomly distributed nuclei within reactant phase A, various mathematical models have been developed and the dependence of the nucleation rate / on time formulated. Usually a first-order kinetic law I =fNoe fi is assumed for the nucleation from an active site, where N t) = is the number of active sites at time t. Different shapes of the... 

Eagland et al. ( 3) propose a different scheme. The helix formation is compared to a first order reaction concerning only individual chains. In the first step the helices are nucleated and stabilized by the solvent. Next, the chains slowly fold back and the helical sequences associate by hydrogen bonds. Van der Waals interactions or entanglements between the folded chains are responsible for the gel gormation (see Figure 1-b). 

The NMR results presented in Sect. 2 allow for D-RADP-x (with x = 0.20,0.25, 0.30), in fact for no other interpretation than a multitude of local first order phase transitions with a probabihty distribution of transition temperatures. We beheve therefore that we deal with a nucleation mechanism. To illustrate this possibihty we have to make some assumptions ... 

Under hydrothermal conditions (150-180 °C) maghemite transforms to hematite via solution probably by a dissolution/reprecipitation mechanism (Swaddle Olt-mann, 1980 Blesa Matijevic, 1989). In water, the small, cubic crystals of maghemite were replaced by much larger hematite rhombohedra (up to 0.3 Lim across). Large hematite plates up to 5 Lim across were produced in KOH. The reaction conditions influenced both the extent of nucleation and crystal morphology. The transformation curve was sigmoidal and the kinetic data in water and in KOH fitted a first order, random nucleation model (Avrami-Erofejev), i.e. 

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