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Abstract

We study the nucleation and growth of flame fronts in slow combustion. This is modeled by a set of reaction-diffusion equations for the temperature field, coupled to a background of reactants and augmented by a term describing random temperature fluctuations for ignition. We establish connections between this model and the classical theories of nucleation and growth of droplets from a metastable phase. Our results are in good argeement with theoretical predictions.PACS numbers: 64.60. My,05.40.+j,82.40.Py,68.10.Gw The kinetic process by which first-order phase transitions take place is an important subject of longstanding experimental and theoretical interest [1]. Nucleation is the most common of first-order transitions, and remains of a great deal of interest [2][3][4][5][6]. There are two fundamentally different cases, homogeneous and heterogeneous nucleation. Homogeneous nucleation is an intrinsic process where embryos of a stable phase emerge from a matrix of a metastable parent phase due to spontaneous thermodynamic fluctuations. Droplets larger than a critical size will grow while smaller ones decay back to the metastable phase [7,8]. More commonplace in nature is the process of heterogeneous nucleation. There, impurities or inhomogeneities catalyze a transition by making growth energetically favorable.Here we show that the concepts of nucleation and growth can be usefully applied to understand some aspects of slow combustion. We use a phase-field model of two coupled reaction-diffusion equations to study the nucleation and growth of combustion centers in twodimensional systems. Such continuum reaction-diffusion equations have been used extensively in physics, chemistry, biology and engineering to describe a wide range of phenomena from pattern formation to combustion. However, the connection of reaction-diffusion equations to nucleation and interface growth has received little attention.In a recent study of slow combustion in disordered media, Provatas et al. [9,10] showed that flame fronts exhibit a percolation transition, consistent with mean field theory, and that the kinetic roughening of the reaction front in slow combustion is consistent with the KardarParisi-Zhang (KPZ) [11] universality class. In this paper we make a further connection between slow combustion started by spontaneous fluctuations, and the classical theory of the nucleation and growth of droplets from a metastable phase.We generalize the model of Provatas et al. by including an uncorrelated noise source η(x, t), as a function of position x and time t. The model then consists of equations of motion for the temperature field T (x, t) and the local concentration if reactants C(x, t). The temperature satisfiesThe first term on the right-hand-side accounts for thermal diffusion, with diffusion constant D, the second term gives Newtonian cooling due to coupling with a heat bath of background temperature T 0 , with rate constant Γ, and the third term R(T, C) is the exothermic reaction rate as a function of temperature and concentration o...

Kinetics of propagating phase transformation in compressed bismuth

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