Nucleation time-independent

Analogously to 3D nucleation processes, a supersaturated state of an expanded structure is formed first which is characterized by actual Fep () and 9ep(0 values kinetically controlled by Me oiy bulk diffusion and charge transfer. 2D nucleation and growth start from a supersaturated expanded 2D Meads phase and lead to a condensed 2D Meads phase which is characterized, in a first approximation, by time-independent equilibrium values of Tcd(AEf) and cdfAEf). 

If the nucleation time is much larger then the propagation time, each nucleus has sufficient time to spread over the surface before the next nucleus is formed. Under these conditions, each layer is formed by one nucleus only. The current i corresponding to the development and decay of the peripheral edge of each layer is not stable, and fluctuates with the nucleation and spreading of the layers. Fig. 5.9. The mean current density is given by the nucleation fi-equency l/ nuc = /A and is independent of the propagation rate v ... 

Yet, the largest future challenge goes beyond time-independent descriptions, to irreversible thermodynamics, or kinetics. We know very little about a kinetic mechanism founded on hydrate measurements. Due to the stochastic nature of nucleation, experimentalists have dealt with the deterministic growth process. 

At constant temperature and supersaturation the monomer attachment and detachment frequencies are time-independent and nucleation can proceed in the stationary regime. The process is then characterized by the stationary nucleation rate J, which is the frequency of transformation of the nuclei (the j sized cluster) into the supernuclei (the j + 1 sized cluster). Therefore, if Z(j) is the time-independent stationary concentration of j sized clusters, J is the difference between the rate f(j )Z(j ) of all j -I-1 transitions per unit volume area of the system and the rate g(j + l)Z(j + 1) of all j + Iones ... 

Remark.- If super-saturation, that is, if temperature and partial pressures are maintained constant, the reactivity of condensation, and thus of nucleation, is independent of time, and the rate (or the frequency of germination) is separable. 

We assume that the reactivity of growth and the specific frequency of nucleation are independent of time (pseudo-steady state modes at constant tenperature and partial pressures). We will thus refer to relations [10.16] and [10.18], but in this case, a nucleus corresponds to a grain we can thus reveal in these expressions the space function of growth of a grain. 

With the pseudo-steady state mode, when experiments are performed at constant temperatures and partial pressures of various gases, the areal rate of nucleation is independent of time and only the variations in surface area Sl (see relation [14.52]) will vary with time due to nucleation, and also growth. 

In addition to adsorption, dissociation or ionization of the oxygen (i.e. charge transfer from zinc to oxygen) it is also possible for phase formation to be rate-determining (nucleation, early stages of layer formation) [489]. Whatever the decisive elementary step is, its rate will after a very short transient, to a good approximation, not be explicitly dependent on the layer thickness. If the surface reaction is stationary then L is time independent and we obtain a linear growth law... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

X= 2) or (P = 0, X = 3) and the distinction between these possibilities is most satisfactorily based upon independent evidence, such as microscopic observations. The growth of compact nuclei inevitably results in the consumption of surfaces and when these outer faces, the sites of nucleation, have been eliminated, j3 necessarily is zero this may result in a diminution of n. The continued inward advance of the reaction interface at high a results in a situation comparable with the contracting volume reaction (discussed below) reference to this similarity was also made in consideration of the Mampel approach discussed above. Shapes of the deceleratory region of a time curves for nucleation and growth reactions and the contracting volume rate process are closely similar [409]. 

Reid et al. [ 1.12] described the effect of 1 % addition certain polymers on the heterogeneous nucleation rate at-18 °C the rate was 30 times greater than in distilled, microfiltered water and at -15 °C, the factor was still 10 fold hogher. All added polymers (1 %) influenced the nucleation rate in a more or less temperature-dependent manner. However, the authors could not identify a connection between the polymer structure and nucleation rate. None the less it became clear that the growth of dendritic ice crystals depended on to factors (i) the concentration of the solution (5 % to 30 % sucrose) and (ii) the rate at which the phase boundary water - ice crystals moved. However, the growth was found to be independent of the freezing rate. (Note of the author the freezing rate influences the boundary rate). 

Mandelkow et alP provided direct kinetic studies of nucleation by using 1 A synchrotron radiation to obtain time-resolved scattering data during cycles of assembly and disassembly after temperature shifts between 4 and 36°C. Small-angle scattering theory requires independent scattering from all particles in the solution, and the theory relates the intensity of scattering to other parameters as follows ... 

We have mentioned above the tendency of atoms to preserve their coordination in solid state processes. This suggests that the diffusionless transformation tries to preserve close-packed planes and close-packed directions in both the parent and the martensite structure. For the example of the Bain-transformation this then means that 111) -> 011). (J = martensite) and <111> -. Obviously, the main question in this context is how to conduct the transformation (= advancement of the p/P boundary) and ensure that on a macroscopic scale the growth (habit) plane is undistorted (invariant). In addition, once nucleation has occurred, the observed high transformation velocity (nearly sound velocity) has to be explained. Isothermal martensitic transformations may well need a long time before significant volume fractions of P are transformed into / . This does not contradict the high interface velocity, but merely stresses the sluggish nucleation kinetics. The interface velocity is essentially temperature-independent since no thermal activation is necessary. 

Consider recrystallization and grain growth in an infinite thin sheet. Assume that the nucleation rate of recrystallized grains is a linear function of temperature above a critical temperature, Tc, and the nucleation rate is zero for T grain-growth rate, R, is constant and independent of temperature. Suppose that at time t = 0 the sheet is heated at the constant rate T(t) = Tc/2 + /3t. Using Poisson statistics, the probability that exactly zero events occur in a time t is p0 = exp(— Nc))-... 

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