Nucleation dimensionless time

Figure 7 The fraction polymerized material fas a function of the dimensionless time Ty according the kinetic Landau model discussed in the main text, with h the nucleation rate. Shown are results valid in the limit where the nucleation reaction is rate limiting, for a quench to X/Xp = 2 where in equilibrium f— 0.5. Depolymerization is much faster than polymerization.

We choose to define the dimensionless time, 0, accotding to relation [10.2] (choosing Iq = xq when an interface reaction is the rate-determining step and /q = 1 if this is diffusion) and in the same manner we define the dimensionless date of nucleation by ... 

Introducing the dimensionless variables and the model parameter, we calculate starting from [10.21] the fraction of grains having nucleated at time ... 

At each pressure to calculate the reactivity of growth and the specific frequency of nucleation y, we determine the parameter of model A and the ratio 6 It of dimensionless time on real time, starting from the characteristics of the point of inflection given in Table 18.19. For that, we use Table A.7.2 (see Appendix 7) and carry out linear interpolations between two consecutive values of a. ... 

To calculate the reactivity of growth, if), and the specific frequency of nucleation, Y, corresponding to each quadruplet (ii, A, 6 ), we use the relation of definition (see section 10.5.5) of dimensionless time therefore. 

Table 18.38. Ratio of the dimensionless time to real times for the model of nucleation...

Both Eqs. (10.28) and (10.31) predict a current density which first rises as the perimeters of the clusters grow, and then decreases rapidly as the clusters begin to overlap. They can be cast into a convenient dimensionless form by introducing the maximum current density jmax and the time fmax at which it is attained. A straightforward calculation gives for instantaneous nucleation and progressive nucleation, respectivly,... 

Deposition of mercury at boron-doped diamond (BDD) and platinum electrodes has also been studied [33]. Deposition and oxidation of mercury was performed by cyclic voltammetry from the solution of 1 mM Hg2 ( 104)2 in 1 M Na l04. In order to learn more about this deposition, it was carried out also under chronoamperometric conditions. The results obtained are shown in Fig. 2 in the form of dimensionless current-time transients. Experimental curves obtained at two different overpotentials were compared with the theoretical curves calculated for instantaneous and progressive nucleation. A good agreement of experimental plots with the instantaneous nucleation mechanism was... 

Fig. 2 Comparison of the experimental dimensionless current-time transients for electrodeposition of mercury onto boron-doped diamond electrode with the theoretical transients for instantaneous (upper curve) and progressive (lower curve) nucleation overpotentials (x) 0.862 V and ( ) 0.903 V (from Ref 33).

Fig. 29. Electrodeposition of Ag from 0.017 M AgCN + 0.92 M KCN + 0.11 M K2CO3 solution dimensionless analysis of experimental potentiostatic current transients (/, and tm are the current and time corresponding to the maximum on the current transient curve, respectively). Upper curve calculated for the instantaneous nucleation mechanism lower curve, for the progressive nucleation mechanism. Different symbols/experimental points relating to different potentials [136], Reproduced by permission of The Electrochemical Society, Inc.

Regimes of nucleation may be defined (Fig. 21-107) with the help of dimensionless drop penetration time and spray flux / , or... 

Fig. lb shows a series of Te deposition transients at constant potentials. The transients are characterised by an initial increase in current followed by a drop at longer time. These features are consistent with 3D nucleation also followed by the diffusion limited growth. The nucleation mechanism of Te deposition on n-Si was determined from the analysis of current-time transients. For this purpose the transients were plotted in the dimensionless form by normalizing two variables i and r with respect to the maximum current imax and the time zmax at which the maximum current is observed [2]. The theoretical plots for progressive and instantaneous nucleation and experimental plot for Te(IV) reduction at E = -0.375 V are given in the inset of Fig. lb. The corresponding experimental deposition transient is in a... 

For the sake of simplicity and clarity, the kinetics of aggregation processes are usually examined for monodisperse particles in the absence of nucleation, dissolution, and aggregate breakage. In that case, clusters of dimensionless mass m (i.e. the number of particles in the aggregate) result from the aggregation of two sub-clusters with masses i and j = m - i. At the same time, clusters of m join with other clusters. Hence, the mass balance of aggregates m takes the form ... 

The dimensionless spray flux parameter can be used both as a scale-up parameter and as a parameter to estimate nuclei starting sizes for population balance modelling (see Section 13.3.3). When combined with the drop penetration time, ha forms part of a nucleation regime map (see Figure 13.4) (Hapgood... 

Figure 5.21 Dependence of the incubation time of phase A2B1 nucleation at the interface A-Ai 02 on the thickness h of the solid solution layer [d is the lattice constant). The incubation time is taken in dimensionless...

For drop-controlled nucleation, the drop penetration time must be small compared to the bed circulation time before that section of powder passes again through the spray zone, i.e., the dimensionless penetration time should be small ... 

Fig.2 Reduced location of crystallization front versus dimensionless contact times (Fourier numbers) for the wall temperatures indicated near the curves with unchanged bulk temperature of 180 C for an industrial polypropylene, according to ref.7. The short pieces of lines near the origin belong to = 100 C (O), 95°C ( D ) and 90 C (6 ). A farther growth was inhibited by diffuse nucleation in the bulk of the melt. Sample thickness - 0,7 mm.

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

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