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ABSTRACT
This paper presents a computational model for the ammonothermal gallium nitride (GaN) crystal growth process, including fluid flow, heat transfer, dissolution and crystallization rates, GaN metastable phase transport, and crystal interface advancement. The presented article solves the Navier–Stokes equations along with the Brickman–Darcy–Forchheimer extensions for nutrient porous medium and Boussinesq approximation for free convection. Piecewise Linear Interface Calculation (PLIC) method is adopted to construct and advance the crystal interface. Simulations, in particular, were performed for a common research autoclave with a retrograde ammonothermal system. Special attention is given to the regions close to the crystal interface.
Nomenclature
| A | = | area |
| b | = | Forchheimer coefficient |
| B | = | porous media binary |
| C | = | concentration |
| cp | = | heat capacity |
| D | = | effective diffusivity |
| Da | = | Darcy number |
| = | particle diameter | |
| f | = | volume fraction of solid |
| g | = | gravitational acceleration |
| GaN | = | gallium nitride |
| Gr | = | Grashof number |
| h | = | mesh size for a square grid |
| K, K | = | conductivity |
| KNH2 | = | potassium amide |
| L | = | length of line segment in PLIC method |
| m | = | gradient of f |
| M | = | molar mass |
| n | = | normal |
| NH3 | = | ammonia |
| p | = | pressure |
| PLIC | = | piecewise linear interface calculation |
| Pr | = | Prandtl number |
| r | = | radial direction |
| R | = | reaction product |
| ri | = | inside diameter |
| S | = | GaN solubility |
| Sc | = | Schmidt number |
| t | = | time |
| T | = | temperature |
| v | = | kinematic viscosity |
| V | = | velocity |
| z | = | vertical direction |
| α | = | thermal diffusivity |
| β | = | expansion coefficient |
| μ | = | viscosity |
| κ | = | heterogeneous reaction rate coefficient |
| ρ | = | density |
| ε | = | porosity |
| λ | = | permeability |
| φ | = | source term |
| τo | = | tortuosity factor |
| σ,δ | = | computational parameters |
| η,ζ,Λ | = | mathematical transformations |
| θ | = | dimensionless temperature |
| Mathematical symbols | = | |
| ΔH | = | dissolution heat |
| ∀ | = | volume |
| * | = | dimensionless parameter |
| <> | = | volume average over volume ∀ |
| <>f | = | volume average over fluid volume ∀ |
| Subscripts | = | |
| c | = | concentration |
| ch | = | characteristic |
| E | = | energy |
| eff | = | effective |
| f | = | fluid |
| fs | = | fluid–solid interface |
| H | = | high |
| i,j | = | r and z direction indices |
| L | = | low |
| Max | = | maximum |
| s | = | solid |
| Tri | = | triangle |
| Tra | = | trapezoid |
Nomenclature
| A | = | area |
| b | = | Forchheimer coefficient |
| B | = | porous media binary |
| C | = | concentration |
| cp | = | heat capacity |
| D | = | effective diffusivity |
| Da | = | Darcy number |
| = | particle diameter | |
| f | = | volume fraction of solid |
| g | = | gravitational acceleration |
| GaN | = | gallium nitride |
| Gr | = | Grashof number |
| h | = | mesh size for a square grid |
| K, K | = | conductivity |
| KNH2 | = | potassium amide |
| L | = | length of line segment in PLIC method |
| m | = | gradient of f |
| M | = | molar mass |
| n | = | normal |
| NH3 | = | ammonia |
| p | = | pressure |
| PLIC | = | piecewise linear interface calculation |
| Pr | = | Prandtl number |
| r | = | radial direction |
| R | = | reaction product |
| ri | = | inside diameter |
| S | = | GaN solubility |
| Sc | = | Schmidt number |
| t | = | time |
| T | = | temperature |
| v | = | kinematic viscosity |
| V | = | velocity |
| z | = | vertical direction |
| α | = | thermal diffusivity |
| β | = | expansion coefficient |
| μ | = | viscosity |
| κ | = | heterogeneous reaction rate coefficient |
| ρ | = | density |
| ε | = | porosity |
| λ | = | permeability |
| φ | = | source term |
| τo | = | tortuosity factor |
| σ,δ | = | computational parameters |
| η,ζ,Λ | = | mathematical transformations |
| θ | = | dimensionless temperature |
| Mathematical symbols | = | |
| ΔH | = | dissolution heat |
| ∀ | = | volume |
| * | = | dimensionless parameter |
| <> | = | volume average over volume ∀ |
| <>f | = | volume average over fluid volume ∀ |
| Subscripts | = | |
| c | = | concentration |
| ch | = | characteristic |
| E | = | energy |
| eff | = | effective |
| f | = | fluid |
| fs | = | fluid–solid interface |
| H | = | high |
| i,j | = | r and z direction indices |
| L | = | low |
| Max | = | maximum |
| s | = | solid |
| Tri | = | triangle |
| Tra | = | trapezoid |