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The Gallium Arsenide Wafer Problem

The Gallium Arsenide Wafer Problem. Industrial Needs versus Mathematical Capabilities. Margarita Naldzhieva joint work with Wolfgang Dreyer, Barbara Niethammer. DFG Research Center M ATHEON Mathematics for key technologies.

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The Gallium Arsenide Wafer Problem

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  1. The Gallium Arsenide Wafer Problem Industrial Needs versus Mathematical Capabilities Margarita Naldzhieva joint work with Wolfgang Dreyer, Barbara Niethammer DFG Research Center MATHEONMathematics for key technologies BMS Days Berlin 18 / 02 / 2008

  2. Industrial problem The Becker-Döring model From Becker-Döring to Fokker-Planck Quasi-stationaryand longtime behaviour of solutions Outline BMS Days Berlin 18 / 02 / 2008

  3. Arsen concentration = 0.500082 The Gallium Arsenide Wafer Problem Single crystal gallium arsenide BMS Days Berlin 18 / 02 / 2008

  4. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  5. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  6. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  7. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  8. + 1 n n + 1 + n 1 n-1 The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  9. + 1 n n + 1 + n 1 n-1 Variables density ofn- cluster The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  10. + 1 n n + 1 + n 1 n-1 Variables Becker-Döring system density ofn- cluster with The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  11. + 1 n n + 1 + n 1 n-1 Variables Becker-Döring system density ofn- cluster with The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  12. Transition rates Lyapunov Function Becker-Döring system with Different Strategies, I: Ball Carr Penrose Equilibria BMS Days Berlin 18 / 02 / 2008

  13. Lyapunov Function Becker-Döring system with Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  14. Lyapunov Function n n Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  15. Transition rates Lyapunov Function Becker-Döring system with Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  16. Transition rates Lyapunov Function 2nd law of thermodynamics Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  17. Condensation rates 2nd law of thermodynamics Evaporation rates Comparison of Transition Rates BMS Days Berlin 18 / 02 / 2008

  18. Lyapunov Functions Lyapunov Functions with BMS Days Berlin 18 / 02 / 2008

  19. Water droplets in vapour An p0 T0 n NNNN

  20. Quasi stationary Flux An p0 T0 n J25(t) p0 nmax = 70 nmax = 50 nmax = 40 t NNNN

  21. Quasi stationary Flux monomer density constant Model system for calculation Skim of droplets with nmax + 1 atoms: zn + 1= 0 max J25(t) J25(t) p0 p0 nmax = 70 nmax = 50 nmax = 40 T0 NNNN

  22. Quasi stationary Flux monomer density constant Model system for calculation Skim of droplets with nmax + 1 atoms: zn + 1= 0 max J25(t) p0 p0 T0 t t J25(t) nmax = 70 Model system nmax = 50 nmax = 40 NNNN

  23. Equilibrium solutions : Jn=0 Conservation of mass number n-clusters total volume number n-clusters total number Becker-Döring system with Variables Fluxes BMS Days Berlin 18 / 02 / 2008

  24. Equilibrium solutions : Jn=0 Constraints Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  25. Equilibrium solutions : Jn=0 Convergence radii RBCP and RDD Constraints Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  26. Equilibrium solutions : Jn=0 Convergence radii RBCP and RDD Equilibrium solutions Constraints Equilibrium conditions BMS Days Berlin 18 / 02 / 2008

  27. Longtime behaviour of solutions I: Penrose et al. Equilibrium condition Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  28. Longtime behaviour of solutions I: Penrose et al. Equilibrium condition Equilibrium solutions Asymptotical behaviour BMS Days Berlin 18 / 02 / 2008

  29. Longtime behaviour of solutions I: Penrose et al. Asymptotical behaviour • existence of metastable states (Penrose 1989) • excess density described by the LSW equations (Penrose 1997, Niethammer 2003) • convergence rate to equilibrium e-ct1/3 (Niethammer, Jabin 2003) BMS Days Berlin 18 / 02 / 2008

  30. Simplified Dreyer/Duderstadt Model Current state of the art Mathematical results for a modified DD model! Modified Flux Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  31. Simplified Dreyer/Duderstadt Model New time scale Modified Becker-Döring system with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  32. Longtime behaviour of solutions II: mod. Dreyer Equilibrium condition Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  33. Longtime behaviour of solutions II: mod. Dreyer Equilibrium condition Equilibrium solutions depending on the Availability of the System: Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  34. Longtime behaviour of solutions: GaAs wafer Assumption n Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  35. Longtime behaviour of solutions: GaAs wafer Assumption Equilibrium solutions relevant? Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  36. Longtime behaviour of solutions: GaAs wafer Assumption J25(t) nmax = 70 nmax = 50 nmax = 40 t relevant? Distribution of liquid droplets J25(t) BMS Days Berlin 18 / 02 / 2008

  37. From Becker-Döring to Fokker-Planck Becker-Döring system with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  38. From Becker-Döring to Fokker-Planck Becker-Döring system with Continuous System (Duncan) Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  39. From Becker-Döring to Fokker-Planck Continuous System (Duncan) Dreyer/Duderstadt Continuous thermodynamics with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  40. From Becker-Döring to Fokker-Planck Continuous System (Duncan) Dreyer/Duderstadt Continuous thermodynamics Lyapunov Funktion, minimal at equilibria! Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  41. Mixed System Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  42. Mixed System Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  43. Mixed System Thermodynamics and mass conservation Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  44. Mixed System Boundary values Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  45. Becker-Döring model for homogeneous nucleation thermodynamically consistent nucleation rates equilibrium solutions and asymptotical behaviour existence of a metastable phase befor equilibrium? Duncan`s PDE approximation of the discrete System? Summary and outlook BMS Days Berlin 18 / 02 / 2008

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