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Modeling of primary spacing selection in dendrite arrays during directional solidification

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Abstract

By analyzing the competitive growth in a dendrite array, an analytical model is proposed to describe the characteristic features in dendrite spacing selection. Nominal spacing, according to scale law at steady state, is defined as the baseline for spacing variation. Variation in dendrite spacing is attributed to the creation of new dendrites or to the elimination of existing dendrites. Newly formed spacing is equated to current nominal spacing factored by a kinetic factor. Critical velocities are proposed for the initiation of dendrite creation/elimination during increasing/decreasing velocity, thus yielding the length of incubation periods during which the spacing remains unchanged. The evolution of spacing distribution between its upper and lower limits, and thus weighted average spacing, can be calculated in the process of dendrite creation/elimination. Quantitative calculations have been performed for a model alloy succinonitrile-2.5 wt pct ethanol and compared with the experimental data reported in the literature. The results show that this model provides a reasonable description of the characteristic features of dendrite spacing selection, such as the wide range in spacing distribution, the delayed response in spacing variation, and its history dependence.

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Abbreviations

G :

temperature gradient

V :

growth velocity

D :

diffusion coefficient in liquid

k :

distribution coefficient

Γ:

Gibbs- Thomson parameter

ΔT 0 :

freezing range

C 0 :

initial concentration

C*:

concentration at the solid/liquid interface

δ c :

solute boundary layer thickness

r :

radius in radial coordinate

R :

dendrite tip radius

λ :

primary dendrite spacing

λ N :

nominal primary spacing

λ i :

initial side arm spacing

λ p :

distance between the tip and the first side-branch perturbation

λ q :

distance between the tip and the newly formed dendrite trunk

x :

dimensionless spacing (x=λ/λ m 0)

P β (x):

beta distribution of spacing

V 0 :

constant growth velocity at stable stage

λ 0N :

nominal spacing at V 0

λ 0m , λ 0a , λ 0M :

minimum, average, and maximum spacing at V 0

B 0 :

constant in beta distribution

f iv :

kinetic factor for dendrite creation at increasing velocity

V iv1 :

critical velocity at the start of the first period of spacing variation

V iv2 :

critical velocity at the start of the second period of spacing variation

λ iv m , λ iv a , λ iv M :

minimum, average, and maximum spacing

x iv a :

dimensionless average spacing (x iv a =λ iv a /λ 0 m )

B iv1 , B iv2 :

constant in beta distribution

f dv :

kinetic factor for dendrite elimination at increasing velocity

V dv1 :

critical velocity at the start of the first period of spacing variation

V dv2 :

critical velocity at the start of the second period of spacing variation

λ dv m , λ dv a , λ dv M :

minimum, average, and maximum spacing

x dv a :

dimensionless average spacing x dv a =λ dv a /λ 0 m

B dv1 , B dv2 :

constant in beta distribution

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  1. ACCESS Institute, RWTH, 52056, Aachen, Germany

    Dexin Ma (Research Scientist)

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Ma, D. Modeling of primary spacing selection in dendrite arrays during directional solidification. Metall Mater Trans B 33, 223–233 (2002). https://doi.org/10.1007/s11663-002-0007-4

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  • DOI: https://doi.org/10.1007/s11663-002-0007-4

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