Statistics of growth

  • Lorenz Ratke
  • Peter W. Voorhees
Part of the Engineering Materials book series (ENG.MAT.)


In this chapter we will study the evolution of a number of spherical particles described by a particle size density distribution. We assume that the particles do not interact, meaning for example if they grow by heat or mass diffusion at constant supercooling or supersaturation the diffusion fields do never overlap. The particles never touch each other and grow without any limit. Thus, the description of a particle ensemble does not take into account for example mass conservation applied to the entire sample. This simplification allows us to treat the statistics in a simple and analytical way. However, the situation described is not purely theoretical: E.g. when cooling a metal one can have for a limited period of time following nucleation just growth of these particles — very much as described here. As the particles grow the matrix concentration decreases and competitive growth — coarsening — sets in. This situation will be treated in chapters 7,8. The general statistical concepts presented below are useful in these chapters.


Cauchy Problem Continuity Equation Average Radius Convective Diffusion Scaling Solution 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lorenz Ratke
    • 1
  • Peter W. Voorhees
    • 2
  1. 1.Institute for Space SimulationGerman Aerospace Center DLRCologneGermany
  2. 2.Dept. of Materials Science and EngineeringNorthwestern UniversityEvanstonUSA

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