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Simulation of cellular-dendritic solidification structures of binary alloys in three-dimensional growth using a multiparticle diffusion-limited aggregation model

  • T. A. M. Haemers , D. G. Rickerby and E. J. Mittemeijer EMAIL logo
Published/Copyright: February 10, 2022
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International Journal of Materials Research
From the journal International Journal of Materials Research Volume 95 Issue 12

Abstract

Solidification in binary alloy melts was simulated with two different three-dimensional models, that differ in the way the role of the solid/liquid interface energy on the solidification is expressed: either surface rearrangement or a surface curvature-dependent attachment probability was applied. In the melt, the simultaneous diffusion of all diffusing particles (atoms) was taken into account. Cellular –dendritic growth modes, as observed in practice, could be well simulated. The surface rearrangement model is essential for exhibiting details (as facetting) and the effect of the next-nearest neighbour interaction.

Keywords: Solidification; Binary-alloy melt; Solid/liquid interface; Cellular – dendritic growth
Prof. E. J. Mittemeijer Max-Planck-Institut für Metallforschung Heisenbergstr. 3, D-70569 Stuttgart, Germany Tel.: +49 711 689 3311 Fax: +49 711 689 3312

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Appendix: Discretization of the Laplacian in an f. c. c. lattice

In one dimension, the discretization of the second derivative of ρi,t is (in the following, the subscripts i and t have been omitted.

∂x2ρx=(ρx−1+ρx+1−2ρx)/Δx2

where Δx is the distance between two lattice points.

In two dimensions this becomes

∇2ρx,y=∂x2ρx,y+∂y2ρx,y               =( ρx−1,y+ρx+1,y−2ρx,y+ρx,y+1                       +ρx,y−1−2ρx,y )/Δx2

and for three dimensions, it follows

(A-1) ∇2ρx,y,z=∂x2ρx,y,z+∂y2ρx,y,z+∂z2ρx,y,z                    =( ρx−1,y,z+ρx+1,y,z+ρx,y+1,z+ρx,y−1,z                             +ρx,y,z−1+ρx,y,z+1−6ρx,y,z )/Δx2

In order to arrive at an expression for Δ2ρx,y,z for the f. c. c. lattice where the individual ρ values in the discretized expression (as Eq. (A-1)) pertain to ρ values at nearest neighbour lattice sites, we proceed as follows.

First, recognizing that p nearest neighbours in the f. c. c. lattice are on a distance 122 times the lattice constant times Δx, an Equation analogous to Eq. (A-1) can be given

(A-2) ∇2ρx,y,z=( ρx−122,y,z+ρx+122,y,z+ρx,y+122,z+ρx,y−122,z                        +ρx,y,z−122 +ρx,y,z+122 −6ρx,y,z )/(122Δx)2                   =2( ρx−122,y,z+ρx+122,y,z+ρx,y+122,z                            +ρx,y−122,z+ρx,y,z−122 +ρx,y,z+122 −6ρx,y,z )/Δx2

where positions with coordinates (x, y, z) do not correspond with sites of the f. c. c. lattice. To get (partial) coincidence with sites of the f. c. c. lattice sites, Eq. (A-2) can be rotated over each of the three axes. Rotation of 45° over the x axis results in

(A-3) ∇2ρx,y,z=2( ρx−122,y,z+ρx+122,y,z+ρx,y+12,z+12+ρx,y−12,z+12 +ρx,y+12,z−12+ρx,y−12,z−12−6ρx,y,z )/Δx2

with only the terms (ρ values) on the x axis not on a site of the f. c. c. lattice.

Rotation of 45° over the z axis results in

(A-4) ∇2ρx,y,z=2( ρx+12,y+12,z+ρx+12,y−12,z+ρx−12,y+12,z+ρx−12,y+12,z                        +ρx,y,z−122 +ρx,y,z+122 −6ρx,y,z )/Δx2            

with only the terms (ρ values) on the z axis not on a site of the f. c. c. lattice.

Rotation of 45° over the y axis results in

(A-5) ∇2ρx,y,z=2( ρx+12,y,z+12 +ρx+12,y,z−12 +ρx,y+122,z+ρx,y−122,z                      +ρx−12,y,z+12 +ρx−12,y,z−12 −6ρx,y,z )/Δ2x

with only the terms (ρ values) on the y axis not on a site of the f. c. c. lattice.

Now, to finally get an Equation with all terms (ρ values) pertaining to nearest neighbour sites of the f. c. c. lattice, first, Eqs. (A-3), (A-4) and (A-5) are added, secondly, Eq. (A-2) is subtracted from this sum. Then, after dividing the left- and right-hand sides of the Equation by 2, the final result is

∇2ρx,y,z=( ρx+12,y+12,z+ρx−12,y+12,z+ρx+12,y−12,z+ρx−12,y−12,z                       +ρx+12,y,z−12 +ρx+12,y,z+12 +ρx−12,y,z+12 +ρx−12,y,z−12                       +ρx,y+12,z+12 +ρx,y−12,z+12 +ρx,y+12,z−12                       +ρx,y−12,z−12 −12ρx,y,z )/Δx2

which for an f. c. c. lattice represents the discretized Laplacian of ρi,tin terms of nearest neighbour values of ρi,t
Received: 2004-02-16
Accepted: 2004-09-17
Published Online: 2022-02-10

© 2004 Carl Hanser Verlag, München

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https://doi.org/10.1515/ijmr-2004-0205
Keywords for this article
Solidification; Binary-alloy melt; Solid/liquid interface; Cellular – dendritic growth

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Editorial
  4. Werner-Köster-Preis
  5. den Werner-Köster-Preis 2003
  6. Festabend zum 85. Geburtstag der DGM
  7. Festabend zum 85. Geburtstag der DGM
  8. Articles Applied
  9. NiAl – material for the application in gas turbines
  10. High-temperature creep behavior and hot-pressing consolidation of NiAl
  11. Direct Strip Casting of Magnesium
  12. Thermo-mechanical analysis of cast/mould interaction in casting processes
  13. Aluminum–Lithium alloy development for thixoforming
  14. On the influence of heavy warm reduction on the microstructure and mechanical properties of a medium-carbon ferritic –pearlitic steel
  15. Thermodynamic assessment of the Dy–Mg system
  16. Influence of pre-straining on mechanical properties of HSLA steel by using ball indentation technique
  17. Fabrication and application of co-continuous Al2O3/Al composite by reactive infiltration of molten Al into the preform of SiO2 with cordierite addition
  18. Articles Basic
  19. Simulation of cellular-dendritic solidification structures of binary alloys in three-dimensional growth using a multiparticle diffusion-limited aggregation model
  20. Microstructure and age-hardening effects of aluminium alloys with additions of scandium and zirconium
  21. Notifications / Mitteilungen
  22. Personal/Personelles
  23. Materials Week
  24. News/Aktuelles
  25. Books/Bücher
  26. Conferences / Konferenzen
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