Nucleation, instantaneous

The overall current-time relationships for the simultaneous nucleation and growth of nuclei are of the form iwhere f3 is a variable depending primarily on the model of nuclei (2D, 3D) and the type of nucleation (instantaneous, progressive). 

If a particular particle is located in proximity to other growing particles (within 10 particle radii, rj, it can be deprived of reactant relative to particles that are located in relative isolation on the surface. This means that even when nanoparticles nucleate instantaneously, a distribution of growth rates can exist for individual particles on the surface. This deleterious phenomena, which we have termed interparticle diffusional coupling or IDC, does not occur for the growth of colloid particle suspensions because particles are constantly moving during growth and they typically do not persist in proximity to other particles. 

Calculation of the total number of spherulites requires integration over the entire crystallization time Nt = It is rather obvious that in the case of two samples with the same number of spherulites, the first nucleated instantaneously and the second spontaneously, the crystallization in the first one required a shorter time than in the second one. 

General Equation (7.19a) and Equation (7.19b) describing nonisothermal crystallization can be derived based on the probabilistic approach in a simpler way [17,18]. It is obvious that the point A remains at time t outside of all growing spherulites nucleated instantaneously at zero time, if no nuclei appeared in a sphere around the point A, having a radius equal to r(0, t) and a volume of E(0, t). According to Equation (7.15), the probability of this event is ... 

Equations (4.24) and (4.29) are equivalent, except that the former assumes instantaneous nucleation at N sites per unit area while the latter assumes a nucleation rate of N per unit area per unit time. It is the presence of this latter rate which requires the power of t to be increased from 2 to 3 in this case. 

Fig. 8.7. The displacive f.c.c. —> b.c.c. transformation in iron. B.c.c. lenses nucleate at f.c.c. groin boundaries and grow almost instantaneously. The lenses stop growing when they hit the next grain boundary. Note that, when a new phase in any material is produced by o displacive transformation it is always referred to os "martensite". Displacive transformations ore often called "martensitic" transformations os o result.

Instantaneous boiling takes place only if the temperature of a liquid is higher than its supeiheat-limit temperature (also called the homogeneous-nucleation temperature), in which case, boiling occurs throughout the bulk of the liquid. This temperature is only weakly dependent on the initial pressure of the liquid and the pressure to which it depressurizes. As stated in Section 6.1., T has a value of about 0.89T,., where is the (absolute) critical temperature of the fluid. 

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,... 

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... 

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.)... 

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. 

The maintenance of product formation, after loss of direct contact between reactants by the interposition of a layer of product, requires the mobility of at least one component and rates are often controlled by diffusion of one or more reactant across the barrier constituted by the product layer. Reaction rates of such processes are characteristically strongly deceleratory since nucleation is effectively instantaneous and the rate of product formation is determined by bulk diffusion from one interface to another across a product zone of progressively increasing thickness. Rate measurements can be simplified by preparation of the reactant in a controlled geometric shape, such as pressing together flat discs at a common planar surface that then constitutes the initial reaction interface. Control by diffusion in one dimension results in obedience to the... 

As discussed earlier the whole process is a redox reaction. Selenium is reduced using sodium borohydride to give selenide ions. In the above reaction, the metal ion reacts with the polymer (PVP or PVA) solution to form the polymer-metal ion solution. Addition of the selenide ion solution to the polymer-metal ion solutions resulted in instantaneous change in the colour of the solutions from colourless to orange (PVA) and orange red (PVP). This indicates the formation of CdSe nanoparticles. The addition of the selenide solution to the polymer - metal ion solution resulted in gradual release of selenide ion (Se -) upon hydrolytic decomposition in alkaline media (equation 4). The released selenide ions then react with metal ion to form seed particles (nucleation). 

The simplest case occurs when a number of nuclei is formed at the beginning of the process and does not change with time instantaneous nucleation). This is the case when the electrolysis is carried out by the double-pulse potentiostatic method (Fig. 5.18B), where the crystallization nuclei are formed in the first high, short pulse and the electrode reaction then occurs only at these nuclei during the second, lower pulse. A second situation in which instantaneous nucleation can occur is when the nuclei occupy all the active sites on the electrode at the beginning of the electrolysis. 

The calculation for the important case of two-dimensional nuclei growing only in the plane of the substrate will be based on the assumption that these are circular and that the electrode reaction occurs only at their edges, i.e. on the surface, 2nrhy where r is the nucleus radius and h is its height (i.e. the crystallographic diameter of the metal atom). The same procedure as that employed for a three-dimensional nucleus yields the following relationship for instantaneous nucleation ... 

Figure 2.23 Instantaneous representation of nucleate boiling surface showing distribution of heat transfer mechanisms (a) plan view (b) profile view. (From Hsu and Graham, 1976. Copyright 1976 by Hemisphere Publishing Corp., New York. Reprinted with permission.)...

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