We consider the phenomenon of heterogeneous nucleation in electrolytic cells. Through the proposition of a stochastic model, we derive an equation to describe the migration of solute ions and their reaction near the electrode surface. Defining particular boundary conditions, the stochastic equation is solved and the time dependent probability density is related to the flux of ions towards the growing nucleus. The flux of ions is identified as the current transient expression, which is adequate for a comparison with the experimental results. The theory is powerful enough to reproduce results obtained by electrodepositing cobalt on n-type silicon and to explain deviations from predictions of the standard model for three-dimensional nucleation diffusion limited growth.

The formation of nanocrystalline and amorphous phases and unstable and stable solid solutions by mechanical alloying is analyzed by considering the mixture of the interfacial components of the elements as an ideal solution. We used the Gibbs free energy and equilibrium volume equations together with the results obtained for the excess Gibbs free energy for the metals, in nanometric form, to calculate activation energies associated with diffusive processes responsible for grain growth, grain boundary migration, atomic migration, and nucleation of new phases. Results obtained from this thermodynamic approximation seem to be confirmed experimentally by the formation of nanocrystalline Fe 67 Ge 33 and amorphous Co 67 Ge 33 alloys by mechanical alloying in our laboratory.

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We considered a Hamiltonian system that can be described by two generalized variables. One of them relaxes quickly when the system is in contact with a heat bath at fixed temperature, while the second one, the slow variable, mimics the interaction of the system with another heat bath at a lower temperature. The coupling between these variables leads to an energy flow between the heat baths. Allahverdyan and Nieuwenhuizen [Phys. Rev. E 62, 845 (2000)] proposed a formalism to deal with such problem and calculated the steady states of the system and some related properties as entropy production, energy dissipation, etc. In this work we applied the formalism to a coupled system of ideal gases and also to an ideal gas interacting with a harmonic oscillator. If the temperatures of the heat baths are not too close, the Onsager relations do not apply.

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