Mathematical Modelling of Dendritic Growth in Metals

Dendrites can be imagined as Pine Trees with vertical growth with branches growing perpendicular to the primary directions. There is secondary growth as well as tertiary and many higher-level branches. The growth is in the crystallographic directions. “Dendrite” is drawn from the Greek word “Dendron” meaning Trees.

Crystal growth in naturally occurring substances especially metals resembles a dendritic growth. The best-known example of this phenomenon is the ide flakes. Scientists have studied ice-flake growth comprehensively, and a good level of understanding has been drawn on such Dendritic Growth.

?Metallurgists have been long fascinated by the phenomena of Dendritic Growth. The nature of this growth defines the way an intergranular crack occurs besides the segregation zones of the alloys. Another interesting study is the Dendritic Growth in pure alloys essential to design energy storage devices – a need for Green Energy Devices!

Understanding the mechanism and modeling the formation and growth of Dendrites requires an in-depth understanding of the physical phenomena involved. Jaafar et al in 2017 state that it is important to understand that a phase-change material will see dendritic growth under suitable conditions. The degree of supercooling combined with a nucleating site are the key parameters to study and fit in a mathematical model.

The parameters are considered when formulating the Gibbs-Thomson Equation. This Model includes the heat equation in both solid and liquid phases and the heat conservation equation separating the two phases. To solve this complex non-linear problem, several numerical methods have been used.

Salhoumi & Galenko et al in 2016 state that the model is described by equations for the hyperbolic transport and fast interface dynamics, which are reduced to a sole equation of the phase field with the driving force given by deviations of temperature and concentration from their equilibrium values within the diffuse interface. It is shown that the obtained?interface condition?presents the acceleration- and velocity-dependent Gibbs–Thomson interfacial condition.?

The modeling approach has undergone progressive evolution as follows:

-????????? The Length Scale Approach

-????????? The Phase Field Approach

-????????? Atomistic Simulation Methods

A combination of the Phase Field Approach with the Atomistic Simulation Method was proved by Hoyt et al in 2003 to yield a parameter-free prediction of the dendrite growth velocity as a function of undercooling for pure Ni melts.

The study of Dendritic Growth provides a clear understanding of the mechanical integrity of cast ingots, such as solute segregation, grain size, and porosity, all depend critically on the morphologies and velocities of individual or arrays of growing dendrites. However, insights into dendritic formation and growth is not captured even in the latest versions of solidification simulation software.

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References:

Hoyt, J. (2003). Atomistic and continuum modeling of dendritic solidification. Materials Science and Engineering: R: Reports, 41(6), 121–163. https://doi.org/10.1016/S0927-796X(03)00036-6

Jaafar, M. A., Rousse, D. R., Gibout, S., & Bédécarrats, J.-P. (2017). A review of dendritic growth during solidification: Mathematical modeling and numerical simulations. Renewable and Sustainable Energy Reviews, 74, 1064–1079. https://doi.org/10.1016/j.rser.2017.02.050

Salhoumi, A., & Galenko, P. K. (2016). Gibbs–Thomson condition for the rapidly moving interface in a binary system. Physica A: Statistical Mechanics and Its Applications, 447, 161–171. https://doi.org/10.1016/j.physa.2015.12.042