Materials Science and Engineering, A 148 ( 1991 ) 141 - 143 141 Primary dendrite spacing in constrained solidification Lasse Makkonen Building MateriaLs" Laboratory, Technical Research Centre of Finland, VTT, 02150 Espoo (Finland) (Received March 19, 1991 ; in revised form May 6, 1991 ) Abstract A theory of primary dendrite spacing 21 in directional solidification of alloys is developed based on thermodynamic and geometric arguments. The result for a cubic material is ~t = (:rA TRIG) ~~, where A T is the unit thermal undercooling, R is the tip radius and G is the temperature gradient. This predic- tion is in quantitative agreement with experimental data. 1. Introduction sample is balanced by the specific heat related to Solidification of alloys at relatively high growth a change in the sample temperature T. rates occurs in the form of a regular array of den- (df~/ (dT) drites. This results in microsegregation charac- L -C = 0 (1) terized by primary and secondary dendrite ~dt] spacings. Microsegregation affects the mechani- cal properties of cast or welded materials and where t is time and C and L are the volumetric should be carefully controlled when producing specific heat and latent heat of fusion, respec- advanced engineering components. It is therefore tively. important to be able to predict dendrite spacing For a unit volume following the solid-liquid in directional solidification, interface, the local derivatives vanish and eqn. ( 1 ) Three theoretical models have been proposed becomes a balance of the advection terms in the literature for estimating primary dendrite spacing as a function of growth rate, thermal gra- Of OT (2) dient, and alloy composition [1-3]. However, LV~xx = Cv 0~ none of these models represents quantitatively the real behaviour [2, 4]. It is shown in this paper where v is the specimen velocity in relation to the that during chemically controlled dendritic temperature field. growth, at a given temperature gradient, the pri- The change in the solid fraction within the tip mary dendrite spacing 21 is controlled by the tip region can be derived from the geometry of the radius only. A simple equation for 2~ is derived tip (Fig. 1). The tip shape of a dendrite of a rea- based on the thermal balance. sonably isotropic material is that of a paraboloid of revolution [7J. Consequently, the cross- 2. Theory section is given by y2=_ 2Rx where R is the radius of curvature, and the volume of the tip by Directional solidification experiments are typi- + cally performed by driving a solid-liquid inter- :ry2( -x)/2 = :rRx-. Consider a unit cell in a cubic array of dendrites (Fig. 1). The total volume of face across a temperature gradient G at a velocity such a unit is X212, where 2j is the primary den- v [3, 5, 6]. Under such conditions there is no external heat extraction other than that caused by drite spacing. Thus temperature advection when the sample moves in the temperature field. Consequently, the latent Of, 0 (TtRx21 :rR (3) heat released in solidifying a fraction f, of the Oxx-0-xk x ~ - ~ 2 ] = 2, 2 0921-5093/91/$3.50 © Elsevier Sequoia/Printed in The Netherlands