Nucleation diffusion-controlled growth

Reactions of the general type A + B -> AB may proceed by a nucleation and diffusion-controlled growth process. Welch [111] discusses one possible mechanism whereby A is accepted as solid solution into crystalline B and reacts to precipitate AB product preferentially in the vicinity of the interface with A, since the concentration is expected to be greatest here. There may be an initial induction period during solid solution formation prior to the onset of product phase precipitation. Nuclei of AB are subsequently produced at surfaces of particles of B and growth may occur with or without maintained nucleation. 

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... 

As a rule, short nucleation times are the prerequisite for monodisperse particle formation. A recent mechanistic study showed that when Pt(acac)2 is reduced by alkylalu-minium, virtually all the Pt cluster nuclei appear at the same time and have the same size [86]. The nucleation process quickly consumes enough of the metal atoms formed initially to decrease their concentration below the critical threshold. No new metal cluster nuclei are created in the subsequent diffusion-controlled growth stage. 

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... 

Diagnostic Relationships Between Current, Maximum Current, and Time. Scharifker and Hills (26) developed a theory that deals with the potentiostatic current transients for 3D nucleation with diffusion-controlled growth. According to this theory, the theoretical diagnostic relationship in a nondimensional form is given by... 

Electrodeposition on transparent material such as indium tin oxide (ITO) can be used for electrochromic applications [328]. Pb deposition on indium-tin oxide electrode occurs by three-dimensional nucle-ation with a diffusion-controlled growth step for instantaneous nucleation [329], and the electrode process has also been studied using electrochemical impedance spectroscopy [328]. 

Fransaer J, Penner RM (1999) Brownian dynamics simulation of the growth of metal nanocrystal ensembles on electrode surfaces from solution. I. Instantaneous nucleation and diffusion-controlled growth. Phys Chem B 103 7643... 

Leubner, I. H. Crystal formation (nucleation) under kinetically controlled and diffusion-controlled growth conditions. J. Phys. Chem. 91,6069-6073 (1987). 

In the case of progressive nucleation- and diffusion-controlled growth, the initial part of the current transient follows a dependence and the nucleation rate can be... 

The first electrodeposition of aluminum from an ionic liquid was reported in 1994 by Carlin etal. [157], Two years later, Zhao et al. [158] smdied the aluminum deposition processes on tungsten electrodes in trimethylphenylanunonium chlo-ride/aluminum chloride with mole ratio 1 2. It was shown that the deposition of aluminum was instantaneous as a result of three-dimensional nucleation with hemispherical diffusion-controlled growth, underpotential deposition of aluminum, corresponding to several monolayers. Liao et al. investigated the constant current electrodeposition of bulk aluminum on copper substrates was in 1-methyl-... 

Jiang et al. studied the electrodeposition and surface morphology of aluminum on tungsten (W) and aluminum (Al) electrodes from 1 2 M ratio of [Emim]CI/AlCl3 ionic liquids [165,166]. They found that the deposition process of aluminum on W substrates was controlled by instantaneous nucleation with diffusion-controlled growth. It was shown that the electrodeposits obtained on both W and Al electrodes between -0.10 and -0.40 V (vs. AI(III)/A1) are dense, continuous, and well adherent. Dense aluminum deposits were also obtained on Al substrates using constant current deposition between 10 and 70 mA/cm. The current efficiency was found to be dependent on the current density varying from 85% to 100%. Liu et al. showed in similar work that the 20-pm-thick dense smooth aluminum deposition was obtained with current density 200 A/m for 2 h electrolysis [167],... 

M ratio of triethylamine hydrochloride to aluminum chloride by the constant potential electrolysis by Gao et al. [173]. They found that the deposition process of aluminum on Al substrates was controlled by instantaneous nucleation with diffusion-controlled growth and the electrodeposits obtained on Al electrodes were dense, continuous, and well adherent, and the current efhciency was 73% at -2.4 V... 

Figure 4 shows this plot for the transformation when the ferrous and ferric hydroxides were precipitated separately by the quick addition of the sum of 2E ammonia and mixed together subsequently. The plot consists of two straight lines of slopes 1 and 2.5. When 2E ammonia was added to the mixture of ferrous and ferric solutions quickly the slopes were 1 followed by 1.85 when added slowly, the slopes were 1 and 1.82. Slope 1 indicates grain boundary nucleation after saturation. Slopes between 1.5 and 2.5 are indicative of diffusion controlled growth from small dimensions (Hi). Note that the curve for 2E Q seems to begin with a larger slope. 

The transformation of the jointly precipitated (Step I) mixture of ferrous and ferric hydroxides to magnetite (Step II) was studied by the continuous measurement of the magnetic moment. It was found that at large excess ammonia, added in Step I, magnetite forms at a fast rate and the transformation is complete. At small excess ammonia the transformation is slow and incomplete. The application of Avrami s theory to the data indicated grain boundary nucleation in the former case and grain boundary nucleation after saturation followed by diffusion controlled growth in the latter case. Mechanisms are proposed for both cases. 

The mechanism of nucleation and growth was determined by analysis of deposition current transients as a function of potential. Figure 2 shows a series of current transients for copper deposition on TiN from 50 mM Cu(II) solution for potential steps from the open-circuit potential to deposition potentials in the range from -0.9 V to —1.5 V plotted on a semi-log plot. The nucleation and growth process is characterized by a current peak where the deposition current first increases due to the nucleation of copper clusters and three-dimensional diffusion-controlled growth, and then decreases as the diffusion zones overlap resulting in one-dimensional diffusion-controlled growth to a planar surface [3-... 

Experimental current transients have been first obtained for the optimal conditions of CdSe epitaxy onto InP and GaAs. The best fitting is achieved assuming, according to the Scharifker model (6), a three dimensionnal nucleation on a finite number No of active sites, followed by the diffusion controlled growth (6), according to the equation (1) ... 

This mechanism for the formation of primary particles can be described using Nielsen s (5) expressions for homogeneous nucleation with diffusion-controlled growth in precipitation. In his discussion, the nucleation rate, J(c), is expressed as a power-law function of supersaturation, c,... 

A dispersed-element model for kinetic-diffusion controlled growth. Assuming that a total number ns of spherical crystals are nucleated per unit volume at a supercooling of A Tsc =Tm-T(, then these crystals can grow to final grain radius of Rc... 

Ga(III) leads first to Ga(I), then upon further reduction the elemental Ga forms from Ga(I). On glassy carbon the electrodeposition involves instantaneous three-dimensional nucleation with diffusion-controlled growth of the nuclei. No alloying with A1 was reported if deposition of Ga was performed in the Ga(I) diffusion regime. Reproducible electrodeposition of Ga is a promising route to binary and ternary compound semiconductors. A controlled electrodeposition of GaX quantum dots (X = P, As, Sb) would be very attractive for nanotechnology. 

Spectra were recorded at 100 ps intervals and were displayed in 1 ms intervals. Bands caused by the formed radical developed as a function of time. It precipitated onto the electrode as a film by association with supporting electrolyte anions according to HV +Br . Band intensities correlated well with coulometricaUy determined charge consumption and they were proportional to for f < 12 ms and to r for r > 12 ms. These time dependencies imply a film formation via initial nucleation and subsequent diffusion-controlled growth. 

Fig. 13 Current transients i(t) for Au (111), miscut < 0.5°, in 0.05 M H2SO4 obtained after a singie potentiai step from 1 = 0.75 V (region II) to various final potentials in region iii. The experimentai traces are given as individual data points, the solid lines represent theoretical curves calculated with the parameters of the numerical fit to a model combining (a) an adsorption process (Eq. 7) and (b) one-step nucleation according to an exponential law with surface diffusion-controlled growth (Eq. 34), (reprinted from Ref. [299]. Copyright 1997 by VCH Verlagsgesellschaft mbH Weinheim).

Fig. 23 Current transients ofthe dissolution of a Cu UPD MLon Pt(lll) in 1 mM Cu + +0.1 M H2SO4, obtained when stepping the potential from 1 = 0.50 V to various final potentials as indicated in the figure. The transients with final potentials lower than 0.67 V could be modeled by assuming two successive hole nucleation processes according to an exponential law coupled with surface diffusion-controlled growth (cf Eq. (30) and Eq. (33)). The inset shows, as an example, the fit (----) for the experimental transient (-----) 1 = 0.50 V —> 2 = 0.65 V [407].

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