Lifshitz and Slezov

According to Lifshitz and Slezov [58,59], the value of Ac is a function of the difference between the radius of particles, r, and the average radius, r ... 

Using this, Lifshitz and Slezov showed that... 

Closely similar expressions for particle (grain) growth in dispersed systems of widely-separated particles by solution of the smaller particles in the matrix and precipitation on the larger particles have been derived by Greenwood ( °), Wagner ( ), Lifshitz and Slezov ( ), and Li and Oriani ( ). All these authors assumed the particles to be spherical, so that the variation of solubility with particle size was given by the Thomson-Freundlich equation,... 

Fig. 1. Time-independent normalized size-distribution function given by Lifshitz and Slezov(i2). Here r is the particle radius, f is the mean particle radius.

Lifshitz and Slezov on the other hand showed that in a dispersion in which grain growth was occurring according to Eq. (4) a time-independent normalized size distribution function would be approached in which the radius of the largest particles would be only 1.5 r (See Fig. 1.) From this steady-state distribution function, however, they derived an equation that predicts growth rates of the same order as Greenwood s equation, viz.. 

Lifshitz and Slezov deduced that in a dispersion of spherical particles in which grain growth is occurring by diffusion-controlled solution precipitation a time-independent size-distribution function is approached asymptot-... 

Therefore, the only process that may produce coarsening of nano-emulsions is Ostwald ripening. It is described by the LSW theory, formulated by Lifshitz and Slezov [68] and independently by Wagner [69], Several authors have indicated that this theory can be applied to macroemulsions with reasonable accuracy [79,80], It has also been reported that the presence of microemulsion droplets in the continuous phase accelerates the Ostwald ripening rate by increasing the diffusion coefficient [80,81], However, this effect is relatively small because microemulsion droplets have much smaller diffusion coefficients than molecules. 

The course of the process at a later stage, where the second assumption is not satisfied, was studied by I. M. Lifshitz and V. V. Slezov.1 The kinetics of phase transitions of the first kind near absolute zero, where fluctuations have a quantum character, were described by I. M. Lifshitz and Yu. M. Kagan2 and by S. V. Iordanskii and A. M. Finkelshtein.3 In these works the ideas of Ya.B. s paper also play an important role. 

Unfortunately the ensemble of nanoparticles cannot grow with the same rate as it was suggested for Fig. 2. The Lifshitz-Slezov instability leads to the formation of one large nanoparticle and a cloud of smaller ones [3]. Such pictures are really observed in many experiments [4]. 

Abstract The Lifshitz - Slezov theory is applied to study the metastable states of the matrix damage clusters, MA, and the copper enriched clusters, CEC, in neutron irradiated steels. It was found that under irradiation conditions the CE Cs are at the Ostwald stage for a neutron fluence of about 0.0002 dpa. The time dependence of number density, MDn, is determined by summarizing all differential equations of the master equation for MA with neglecting of dimmers concentration in comparison with concentration of the single vacancies and subtraction of the number CEC that replace the MA, namely vacancy clusters, due to the diffusivity of copper and other impurity atoms to them. For binary Fe-0.3wt%Cu under neutron irradiation with dose 0.026, 0.051, 0.10 and 0.19 dpa the volume content of the precipitates from the SANS experiment is found to be about 0.229, 0.280, 0.237 and 0.300 vol% respectively. The volume fraction of CEC, in these samples is 0.195 vol% and the calculated volume fraction ofMA is 0.034, 0.085, 0.042 and 0.105 vol% for doses 0.026, 0.051, 0.10 and 0.19 dpa respectively. 

Exact solution to the problem of change of the average particle size and of particle size distribution with time was given by E.M. Lifshitz, V.P. Slezov and C. Wagner [58-61]. We will present their treatment with some simplifications. 

H. J, Oel Max Planck Ins tit ut fur Silikatforschung, Wurzburg, Germany) Have distributions of the Lifshitz-Slezov type shown in your Fig. 1 been determined experimentally Attempts to prove the existence of distributions of this type in precipitates, where they would be expected to occur, have not been too successful, and I think that your Figure may be the first experimental evidence for this type of distribution. 

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