Phase field study of the tip operating state of a freely growing dendrite against convection using a novel parallel multigrid approach

Alloy dendrite growth during solidification with coupled thermal-solute-convection fields has been studied by phase field modeling and simulation. The coupled transport equations were solved using a novel parallel-multigrid numerical approach with high computational efficiency that has enabled the investigation of dendrite growth with realistic alloy values of Lewis number ~ 10 4 and Prandtl number ~ 10 - 2 . The detailed dendrite tip shape and character were compared with widely recognized analytical approaches to show validity, and shown to be highly dependent on undercooling, solute concentration and Lewis number. In a relatively low flow velocity regime, variations in the ratio of growth selection parameter with and without convection agreed well with theory.

[1]  A. G. Quarrell.  Solidification , 1965, Nature.

[2]  Chih-Jen Shih,et al.  Phase field simulation of non-isothermal free dendritic growth of a binary alloy in a forced flow , 2004 .

[3]  Juan C. Ramirez,et al.  Examination of binary alloy free dendritic growth theories with a phase-field model , 2005 .

[4]  R. Sekerka,et al.  Stability of a Planar Interface During Solidification of a Dilute Binary Alloy , 1964 .

[5]  M. E. Glicksman,et al.  Equiaxed dendrite growth in alloys at small supercooling , 1987 .

[6]  J. Warren,et al.  A general mechanism of polycrystalline growth , 2004, Nature materials.

[7]  Saville,et al.  Growth of needle-shaped crystals in the presence of convection. , 1988, Physical review. A, General physics.

[8]  Christina Freytag,et al.  Using Mpi Portable Parallel Programming With The Message Passing Interface , 2016 .

[9]  Michel Rappaz,et al.  Orientation selection in dendritic evolution , 2006, Nature materials.

[10]  Jonathan A. Dantzig,et al.  Dendritic growth with fluid flow in pure materials , 2003 .

[11]  H. Müller-Krumbhaar,et al.  THEORY OF DENDRITIC GROWTH—I. ELEMENTS OF A STABILITY ANALYSIS* , 1988 .

[12]  Heike Emmerich,et al.  Phase-field investigation of microstructure evolution under the influence of convection , 2011 .

[13]  Anand Kumar,et al.  Isotropic finite-differences , 2004 .

[14]  A. Karma,et al.  Regular Article: Modeling Melt Convection in Phase-Field Simulations of Solidification , 1999 .

[15]  William N. Gill,et al.  THE EFFECT OF CONVECTION ON AXISYMMETRIC PARABOLIC DENDRITES , 1988 .

[16]  A. Karma,et al.  Phase-field model of dendritic sidebranching with thermal noise. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  D. Stefanescu.  Fundamentals of Solidification , 2004 .

[18]  S. Vanka.  Block-implicit multigrid solution of Navier-Stokes equations in primitive variables , 1986 .

[19]  A. Karma,et al.  Quantitative phase-field modeling of dendritic growth in two and three dimensions , 1996 .

[20]  Eshel Ben-Jacob,et al.  Boundary-layer model of pattern formation in solidification , 1984 .

[21]  Chia-Jung Hsu.  Numerical Heat Transfer and Fluid Flow , 1981 .

[22]  Eshel Ben-Jacob,et al.  Dynamics of Interfacial Pattern Formation , 1983 .

[23]  R. Trivedi,et al.  Solidification microstructures and solid-state parallels: Recent developments, future directions , 2009 .

[24]  A. Karma,et al.  Phase-field simulations of dendritic crystal growth in a forced flow. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Barbara Chapman,et al.  Using OpenMP: Portable Shared Memory Parallel Programming (Scientific and Engineering Computation) , 2007 .

[26]  A. Karma,et al.  Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Zhipeng Guo,et al.  An implicit parallel multigrid computing scheme to solve coupled thermal-solute phase-field equations for dendrite evolution , 2012, J. Comput. Phys..

[28]  A. Karma.  Phase-field formulation for quantitative modeling of alloy solidification. , 2001, Physical review letters.

[29]  Wilfried Kurz,et al.  Rapid dendrite growth in undercooled alloys , 1987 .

[30]  Bouissou,et al.  Effect of a forced flow on dendritic growth. , 1989, Physical review. A, General physics.

[31]  Richard C. Brower,et al.  Geometrical models of interface evolution , 1984 .

[32]  Juan C. Ramirez,et al.  Three-dimensional phase-field simulations of the effect of convection on free dendritic growth , 2005 .

[33]  Barbara Chapman,et al.  Using OpenMP - portable shared memory parallel programming , 2007, Scientific and engineering computation.

[34]  Kessler,et al.  Velocity selection in dendritic growth. , 1986, Physical review. B, Condensed matter.

[35]  H. Müller-Krumbhaar,et al.  Theory of dendritic growth—III. Effects of surface tension☆ , 1978 .

[36]  G. P. Ivantsov.  Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in a pre-cooled melt , 1985 .

[37]  E. Ben-Jacob,et al.  The formation of patterns in non-equilibrium growth , 1990, Nature.

[38]  Herbert Levine,et al.  Pattern selection in fingered growth phenomena , 1988 .