Dynamic stability of detached solidification

Mazuruk, K.; Volz, M. P.

doi:10.1016/j.jcrysgro.2016.03.032

Cited by 3 publications

(1 citation statement)

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“…For the Czochralski process, Mika and Uelhoff [22] used free energy minimization along with the concepts of variational calculus to numerically determine instability conditions for the meniscus. Mazuruk and Volz [23,24] addressed the static stability problem for the Bridgman process using numerical simulations to calculate the sign of the second order variations for the governing free energy formualtion. Using the observation that the contact angle between the crystal and the melt should converge towards a constant value-11 • for the case of Silicon as investigated by Mazuruk et al [25]-Surek [26] developed a theory for shape stability in capillary shaped crystal growth systems based on deviations from the growth angle.…”

Section: Introductionmentioning

confidence: 99%

Weierstrass' variational theory for analysing meniscus stability in ribbon growth processes

Noronha,

Oliveros,

Ydstie

2020

Preprint

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We use the method of free energy minimization based on the first law of thermodynamics to derive static meniscus shapes for crystal ribbon growth systems. To account for the possibility of multivalued curves as solutions to the minimization problem, we choose a parametric representation of the meniscus geometry. Using Weierstrass' form of the Euler-Lagrange equation we derive analytical solutions that provide explicit knowledge on the behaviour of the meniscus shapes. Young's contact angle and Gibbs pinning conditions are also analyzed and are shown to be a consequence of the energy minimization problem with variable end-points. For a given ribbon growth configuration, we find that there can exist multiple static menisci that satisfy the boundary conditions. The stability of these solutions is analyzed using second order variations and are found to exhibit saddle node bifurcations. We show that the arc length is a natural representation of a meniscus geometry and provides the complete solution space, not accessible through the classical variational formulation. We provide a range of operating conditions for hydro-statically feasible menisci and illustrate the transition from a stable to spill-over configuration using a simple proof of concept experiment.

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