Nucleation diffusion-controlled

Dimension Geometry Instantaneous Nucleation Sporadic Nucleation Sporadic Nucleation, Diffusion Controlled... 

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

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... 

Fig. 13. Plot of variations of activation energy ( /kJ mole"1) with water vapour pressure (PHjO/Torr) for dehydration of calcium sulphate. Data from Ball et al. [281,590, 591] who discuss the significance of these kinetic parameters. Dehydrations of CaS04 2 H2O, nucleation ( ), boundary (o) and diffusion (e) control Q-CaSC>4 5 H2O, diffusion control, below (X) and above (+) 415 K j3-CaS04 5 H20, diffusion control ( ).

An unusual variation in kinetics and mechanisms of decomposition with temperature of the compound dioxygencarbonyl chloro-bis(triphenyl-phosphine) iridium(I) has been reported by Ball [1287]. In the lowest temperature range, 379—397 K, a nucleation and growth process was described by the Avrami—Erofe ev equation [eqn. (6), n = 2]. Between 405 and 425 K, data fitted the contracting area expression [eqn. (7), n = 2], indicative of phase boundary control. At higher temperatures, 426— 443 K, diffusion control was indicated by obedience to eqn. (13). The... 

While it is possible that surface defects may be preferentially involved in initial product formation, this has not been experimentally verified for most systems of interest. Such zones of preferred reactivity would, however, be of limited significance as they would soon be covered with the coherent product layer developed by reaction proceeding at all reactant surfaces. The higher temperatures usually employed in kinetic studies of diffusion-controlled reactions do not usually permit the measurements of rates of the initial adsorption and nucleation steps. 

After polarization to more anodic potentials than E the subsequent polymeric oxidation is not yet controlled by the conformational relaxa-tion-nucleation, and a uniform and flat oxidation front, under diffusion control, advances from the polymer/solution interface to the polymer/metal interface by polarization at potentials more anodic than o-A polarization to any more cathodic potential than Es promotes a closing and compaction of the polymeric structure in such a magnitude that extra energy is now required to open the structure (AHe is the energy needed to relax 1 mol of segments), before the oxidation can be completed by penetration of counter-ions from the solution the electrochemical reaction starts under conformational relaxation control. So AHC is the energy required to compact 1 mol of the polymeric structure by cathodic polarization. Taking... 

Polymer formation icont.) diffusion components and, 421 diffusion control of oxidation in, 389 electrochemical responses, 400 influence of concentration, 397 and kinetic equations, 381 nucleation and, 379 oxidized area, 387... 

By electrodeposition of CuInSe2 thin films on glassy carbon disk substrates in acidic (pH 2) baths of cupric ions and sodium citrate, under potentiostatic conditions [176], it was established that the formation of tetragonal chalcopyrite CIS is entirely prevalent in the deposition potential interval -0.7 to -0.9 V vs. SCE. Through analysis of potentiostatic current transients, it was concluded that electrocrystallization of the compound proceeds according to a 3D progressive nucleation-growth model with diffusion control. 

In Table 4-2, we show both phase-boundaiy controlled and diffusion-controlled nucleation reactions as a function of both constant and zero rate of nucleation. 

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. 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

Westwater (W4, W5) has written a detailed review of boiling in liquids with emphasis on nucleation at surfaces. Although written in 1956, this is still very useful and it provides a detailed description of the factors affecting nucleation. In a more recent review, Leppert and Pitts (L2) have described the important factors in nucleate boiling and bubble growth, and Bankoff (B2) has reviewed the field of diffusion-controlled bubble growth in nonflowing batch systems. 

These newer PET formulations utilize not only a nucleating agent, but also a plasticizing agent [3-9], The crystallization rate of polymeric materials can be broken down into two different regions, i.e. nucleation-controlled and diffusion-controlled (Figure 15.3). In the injection molding process, hot polymer is injected... 

The dynamics of upd reactions have also been examined by STM. The formation of the ordered copper/sulfate layer [354] and copper chloride layer [355] on Au(lll) was examined in a dilute solution of Cu where the reaction was under diffusion control so that growth proceeded on a time scale compatible with STM measurements [354]. In another study, the importance of step density on nucleation was examined and the voltammetric and chronoamperometric response for Cu upd on vicinal Au(lll) was shown to be a sensitive function of the crystal miscut, as... 

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... 

Summary of the n Values Found in the Diffusion-Controlled Nuclei Growth Model for Different Growth Geometries and Nucleation Rates. 

The exponent n is Unked to the munber of steps in the formation of a nucleus (this is a zone in the soUd matrix at which the reaction occurs), ft, and the number of dimensions in which the nuclei grow, X. It can be difficult to distinguish ft and X without independent evidence, and ft can fall to zero following the consumption of external nuclei sites. Hulbert has analysed the possible values of the exponent, n, for a variety of conditions of instantaneous (/3 = 0), constant (ft = 1) and deceleratory (0 < /I < 1) nucleation and for growth in one, two and three dimensions (X = 1 - 3) [ 17]. He also considered the effects of a diffusion contribution to the reaction rate. This reduces the importance of the acceleratory process and reduces the value of n. For diffusion controlled processes, n = ft + Xjl, whereas for a phase boimdary controlled process n = ft + X. Possible values of n are summarised in Table 1. Interpretation of these values can be difficult, and a given value does not unequivocally allow the determination of the reaction mechanism. 

In both the above cases, we have 2D processes. Following nucleation, the reaction may be either phase boimdary controlled (i.e. the rate is limited by the rate at which the interlayer space expands to accommodate the guest) or diffusion controlled (i.e. the reaction rate is controlled by the rate at which the guests diffuse between the layers - the interlayer spacing expands instantly as the guests move). 

In addition to the above, there are further possibihties. When the rate of guest diffusion between individual layers is very large compared with the rate of nucleation at the edge of the crystal, there exists a situation in which the individual layers appear to fill instantly. In this case, when Avrami kinetics are applied to the system, the diffusion process being observed is not the diffusion of guest species between the layers, but the diffusion of filled layers parallel to the c-axis. Such ID processes will consist of nucleation followed by diffusion control in the vast majority of cases, although phase boundary control is also possible if the rate of advancement of the phase boundary is also very rapid with respect to nucleation. In this case, instantaneous nucleation is not a possibility [18]. 

Finally, there is the possibility of a reaction mechanism that is entirely diffusion controlled, where the rate of nucleation does not play a role in determining the rate of reaction, hi this case, Avrami-Erofe ev kinetics do not... 

The values of n for LiBr and Ii2S04 he between 1 and 2, implying a two-dimensional diffusion-controlled mechanism with deceleratory nucleation. However, the liNOs process has n = 0.5, indicating that this process is completely nucleation controlled. liOH has n = 2.2, consistent with a phase boundary controlled process in two dimensions, with deceleratory nucleation again. 

The temperature dependence of the reaction was studied, and the activation energy of the reaction was calculated to be approximately 100 kj mol The exponent n was found to lie in the range 1-2, which is consistent with a 2D diffusion controlled reaction mechanism with deceleratory nucleation. The rate of reaction increases markedly with the amount of water added to the LDH with very small amounts of water added, the deintercalation process does not go to completion. This effect is a result of the LiCl being leached into solution. An equilibrium exists between the LDH and gibbsite/LiCl in solution. The greater [LiCl], the further to the LDH side this lies. 

The intercalation of these species has been studied using time-resolved EDXRD. For intercalation into the LiAl - Cl system, a kinetic analysis of the data for naproxen (Nx), diclofenac (Df) and 4-biphenylacetic acid (4-Bpaa) suggests that the reactions are 2D diffusion controlled processes following instantaneous nucleation. In a number of cases, the importance of nucleation decreases at higher temperatures (T > 60 °C), with a corresponding reduction in the value of n from 1 to 0.5. This latter value corresponds to a situation where nucleation plays no part in controlling the reaction rate. The data in Fig. 22 relate to the intercalation of Nx. 

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... 

Geometric effects coupled with diffusion and nucleation usually control the rates of all solids deposition phenomena. Such effects can be put to good use in the production of special products such as cellulose yarn (rayon), by the precipitation of cellulose in filament form as it emerges as sodium cellulose xanthate liquid from the spinnerets into a bath containing sulphuric acid, which extracts the sodium as sodium sulphate, and the carbon disulphide. In a similar manner, the fabrication of aromatic polyimide fibres is performed by dissolving the polymer in concentrated sulphuric acid and forcing the solution through spinnerets into water. 

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