Theory for growth of needle-shaped particles in multicomponent systems
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. INTRODUCTION
THERE is a variety of models dealing with the diffusioncontrolled growth of needlelike particles with the idealized shape of a paraboloid of revolution. Some of the original work in this area was done by Ivanstov[1] and Horvay and Cahn,[2] who presented exact solutions to the diffusion equation for an isoconcentrate boundary. Their model is expressed mathematically as ⍀ ⫽ p exp { p}E1{ p}
[1]
where p ⫽ v /2D is the Pe´clet number, v is the lengthening rate, is the radius of curvature of the tip of the paraboloid, D is the diffusion coefficient of the solute in the matrix phase, E1 is the exponential integral,[3] and ⍀ is the dimensionless supersaturation given by c ⫺ c␣ ⍀ ⫽ ␣ c ⫺ c␣
[2]
where c is the average solute concentration in the alloy, c␣ is the solute concentration of the matrix (␣ ) in equilibrium with the precipitate ( ), and c␣ is the corresponding concentration in the precipitate in equilibrium with the matrix. Trivedi[4,5] pointed out the nonisoconcentrate nature of the interface and incorporated capillarity and interface kinetics effects.
冋
v ⍀R1{ p} ⫹ c ⍀R2{ p} ⍀ ⫽ p exp { p}E1{ p} \1 ⫹ vc a \ \ b
册
c
[3] ␣
where vc ⫽ (c ⫺ c ) is the velocity of a flat interface during interface-controlled growth, i.e., when almost all the free energy is dissipated in the transfer of atoms across the boundary, causing the concentration difference in the matrix to vanish, and is the interface kinetics coefficient. For curved interfaces, the growth rate will become zero at a curvature, 1/c , via the Gibbs–Thomson effect.[6] The 1 1 functions R1 ⫽ N1{ p} ⫺ 1 and R2 ⫽ N { p} ⫺ 1 2p 4p 2 P.E.J. RIVERA-DI´AZ-DEL-CASTILLO, Ph.D. Student, and H.K.D.H. BHADESHIA, Professor of Physical Metallurgy, are with the Department of Materials Science and Metallurgy, Cambridge CB2 3QZ, United Kingdom. Contact e-mail: [email protected] Manuscript submitted July 12, 2001.
METALLURGICAL AND MATERIALS TRANSACTIONS A
have been evaluated numerically by Trivedi[4] for large supersaturations and have recently been extended for small values of ⍀ ;[7] such functions account for the concentration change along the parabolic surface. The term labeled a in Eq. [3] is the Ivanstov solution,[1] where interface kinetics and capillarity effects are neglected; terms b and c account, respectively, for those effects. Equation [3] does not provide a unique answer for the growth rate, v, which depends on the tip radius, . For solid-state transformations, it is common to adopt Zener’s assumption[7–10] that the radius of curvature is that which gives the maximum growth rate. This is obtained by differentiating Eq. [3] with respect to and setting ⭸v /⭸ ⫽ 0, which gives 0 ⫽ (g*{ p})2 ⫹
c p 1 2 R⬘1{ p} ⫺ R2{ p} ⫹ R⬘2{ p} q* p
冢
冣 [4]
g*{ p} ⫹ g*{ p} ⫺ 1 p
where g*{ p} ⫽ p exp { p}E1{p} and q* ⫽
(c ⫺ c␣) 2D/c
[5]
is a parameter that indicates the relative magnitudes of the interface kinetics and the diffusion effect. The values of the functions R⬘1 and R⬘2 are given in Reference 7; they can be used to s
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