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Faceted Crystals: Grown from Solution & Proven Stable by Stefan-Type Analysis

Explore the stability of crystal facets through a Stefan type problem, with a focus on interfacial energy. This study, conducted by Yoshikazu Giga in collaboration with Piotr Rybka, delves into the patterns of crystal growth to determine the stability of facets near equilibrium states. The research covers mathematical modeling, key results, and open problems in understanding crystal growth phenomena. Findings suggest that all facets of a cylindrical crystal are stable near equilibrium, presenting an intriguing puzzle for further exploration.

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Faceted Crystals: Grown from Solution & Proven Stable by Stefan-Type Analysis

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  1. Faceted Crystals Grown from Solution A Stefan Type Problem with a Singular Interfacial Energy Yoshikazu Giga University of Tokyo and Hokkaido University COE Joint work with Piotr Rybka December , 2005 Lyon

  2. A basic problem from pattern formation in the theory of crystal growth.In what situation a flat portion (a FACET) of crystal surface breaks or not ? Goal : We shall prove : ‘All facets are stable near equilibrium for a cylindrical crystal by analysizing a Stefan type problem’

  3. Contents 1 Model 2 Problem 3 Main mathematical results 4 Three ingredients － ODE analysis － Berg’s effect － Facet splitting criteria－ 5 Open problems

  4. 1 Model Crystals grown from vapor (snow crystal) from solution (NaCl crystal) <driving force : supersaturation> (density of atoms outside crystal is small)

  5. Stefan like Model

  6. unnormalized version :

  7. We shall consider (1)－(5) for given quasi-stationary One phase Stefan problem with Gibbs-Thomson + kinetic effect

  8. Solvability (smooth ) K. Deckelnik - C. Elliott ’99 ( Hele Shaw type ) No …Friedman –Hu ’92 Liu – Yuan ’94

  9. Others (No ) Kuroda-Irisawa-Ookawa ‘77 Stability of facets Experiment e.g. Gonda-Gomi ’85 (No ) : Fingering : Saffman-Taylor R.Almgrem ’95

  10. 2.Problem (specific to ours)

  11. 3. Main Math Results

  12. Th (Rybka-G ‘04) If is close to the Equilibrium then the solution solves the original problem (1),(2),(3), (4),(5),(6),(7) Near equilibrium Facet does not break.

  13. Reduction to ODE

  14. Near equilibrium : close to zero / bounded away from zero

  15. 5. Open problems • Existence of solution of the • Original • problem is widly open if • is not near • equilibrium • (Even if is given • M.-H. Giga – Y. Giga ’98 • graphs) • ( : constant • M.-H. Giga – Y. Giga ‘01 • level set approach : unique • existence of generalized sol • (2-D)) • Uniqueness of the solution of • the original problem • (Sol is unique for • Reduced problems)

  16. All my preprints are in Hokkaido University Preprint Series on Math. http:coe.math.sci.hokudai.ac.jp

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