Nucleation Induced Nanostructures
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Nucleation Induced Nanostructures A. ten Bosch Laboratoire de Physique de la Matière Condensée, CNRS 6622, Parc Valrose, F-06108,Nice Cedex 2, France ABSTRACT Numerical simulations in films and aggregates have repeatedly shown the presence of vibrations during a phase transition and/or the appearance of periodic structures. A phase transition could be controlled and novel nanostructures created by astute manipulation of such phenomena. In order to study the occurrence and effect of wave-like phenomena, the dynamics of a first order phase transition is described using kinetic theory. At a first order phase transition, the initial phase is replaced in time by the new phase on propagation of a density front through the sample. The dynamic stability analysis studies the transition to the uniform phase by propagation of the front and provides the conditions for the formation of transient periodic structures by a local increase of density. The results apply also to spherical geometry and a discussion of cluster dynamics follows the planar case. INTRODUCTION In a first order phase transition, the emerging stable phase displaces the original phase by the dynamic process of nucleation and growth. Between the two phases, a contact forms which propagates as an interfacial profile, providing a practical method to produce materials in thin layers. Applications are well known in thin films and coatings of metals, semiconductors and polymer materials for use in electronics, optics and in the surface modification of materials to improve mechanical or chemical properties.[1] A phase transition could be controlled and novel nanostructures created by excitation of waves. Numerical simulations in films and aggregates have repeatedly shown the presence of vibrations during a phase transition and/or the existence of specific wave-like motions[26] which effect the shape, the morphology and the propagation of the interface. THEORY In order to study the occurrence and effect of wave-like phenomena, the dynamics of a first order phase transition is described using kinetic theory [7] for the evolution of the probability distribution of particles f(r, t) or the number density n(r,t)= nf(r,t), n being the total number of particle per volume. The function n(r, t) measures the probability for a particle to arrive at position r after a time t and fulfills the equation of continuity:
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v
∂n ( r , t ) ∂ r v = − v ⋅ j (r , t ) (1) ∂t ∂r The evolution of the flux j (r,t) = n(r,t) v(r,t) , v(r,t) being the average velocity of the particles of mass m, is described by the dynamic flux equation, r r 1 ∂j ∂ r v = − βj − n ( r , t ) r µ ( r , t ) (2) ∂t ∂r m The friction forces of the medium are included in β. We concentrate on the dynamics of a system which, having achieved uniform temperature, is dominated by particle flow. The driving force of the transition is the gradient of the local chemical potential µ(r,t). The chemical potential is enslaved by the density and in the effective interaction model [8]: κ v v µ (r , t ) = µ (n) − ∆n(r
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