Effects of Substitutional Solute Accumulation at α/γ Boundaries on the Growth of Ferrite in Low Carbon Steels

Abstract

The growth of proeutectoid ferrite in Fe-C-X alloys containing ∼3 at. pct X, where X is Mn, Ni, Cr, and Si, is re-examined in terms of solute drag using the Hillert–Sundman theory. The differences of measured growth rates from those calculated under paraequilibrium (PE) reported previously were accounted for taking into account not only the binding energy of substitutional solute with the boundary, but also the transformation temperature of the alloy. The ferrite growth in quaternary Fe-C-Mn-Si alloys was modeled using the stationary-interface approximation for the matrix of finite grain size. The principal features of growth in these alloys, i.e., initial fast unpartitioned growth and subsequent slow partitioned growth with a high level of carbon supersaturation in austenite, were reproduced incorporating cosegregation of Mn and Si at the boundary. Thus, a strong Mn-Si interaction is likely to enhance accumulation of these elements at the boundary and yield the growth behavior that resembles the growth stasis in Fe-C-Mo alloys.

Appendix

Stationary interface approximation for the growth of ferrite in a finite austenite matrix

For the diffusion fields in ferrite and austenite (Figure 1), the solution is given by an infinite series as[43]

$$ \frac{{x_{i} - x{}_{i}^{0} }} {{x_{i} ^{\alpha } - x_{i} ^{0} }} = 1 - \frac{\pi } {4}{\sum\limits_{n = 0}^\infty {\frac{{{\left( { - 1} \right)}^{n} }} {{2n + 1}}{\text{ }}\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\alpha } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4S^{2} }}} \right]}{\text{ }}\cos {\text{ }}{\left[ {{\left( {n + \frac{1} {2}} \right)}\pi \frac{s} {S}} \right]}} } $$

(A1)

and

$$ \frac{{x_{i} - x{}_{i}^{0} }} {{x_{i} ^{\gamma } - x_{i} ^{0} }} = 1 - \frac{\pi } {4}{\sum\limits_{n = 0}^\infty {\frac{1} {{2n + 1}}{\text{ }}\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\gamma } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4{\left( {d - S} \right)}^{2} }}} \right]}{\text{ }}\sin {\text{ }}{\left[ {{\left( {n + \frac{1} {2}} \right)}\pi \frac{{s - S}} {{d - S}}} \right]}} } $$

(A2)

respectively, where x _i is the solute concentration (or mole fraction) in the matrix, and x ^α _i and x ^γ _i are the concentrations at the boundary in ferrite and austenite. The term x ⁰ _i is the bulk concentration; $D_{i} ^{\nu }$ (ν = α or γ) is the solute diffusivity; d is half the grain size; s is the distance; and S is the location of the boundary at the real time t, which is determined by the condition of mass balance in ferrite and austenite as

$$ {\int\limits_0^S {{\left( {x_{i} ^{0} - x_{i} } \right)}ds = {\int\limits_S^d {{\left( {x_{i} - x_{i} ^{0} } \right)}ds} }} } $$

(A3)

Performing integration with respect to s, Eq. [A3] becomes

$$ {\left( {1 - \Omega _{i} } \right)}S\phi _{\alpha } {\left( t \right)} = \Omega _{i} {\left( {d - S} \right)}\phi _{\gamma } {\left( t \right)} $$

(A4)

where Ω _i = (x ^γ _i – x ⁰ _i )/(x ^γ _i – x ^α _i ) is the supersaturation of solute i, and

$$ \phi _{\gamma } {\left( t \right)} = 1 - \frac{8} {{\pi ^{2} }}{\sum\limits_{n = 0}^\infty {\frac{1} {{{\left( {2n + 1} \right)}^{2} }}{\text{ }}\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\gamma } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4{\left( {d - S} \right)}^{2} }}} \right]}} } $$

(A5)

and

$$ \phi _{\alpha } {\left( t \right)} = 1 - \frac{8} {{\pi ^{2} }}{\sum\limits_{n = 0}^\infty {\frac{1} {{{\left( {2n + 1} \right)}^{2} }}{\text{ }}\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\alpha } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4S^{2} }}} \right]}} } $$

(A6)

Differentiating Eq. [A4] with respect to t,

$$ \begin{aligned}{} &\dot{S}{\left( {x_{i}^{0} - x_{i} ^{\alpha } } \right)}\phi _{\alpha} - S\dot{x}_{i} ^{\alpha } \phi _{\alpha } + S{\left( {x_{i} ^{0} -x{}_{i}^{\alpha } } \right)}\dot{\phi }_{\alpha } \\ & = -\dot{S}{\left( {x_{i} ^{\gamma } - x_{i} ^{0} } \right)}\phi_{\gamma } + {\left( {d - S} \right)}\dot{x}_{i} ^{\gamma } \phi_{\gamma } + {\left( {d - S} \right)}{\left( {x_{i} ^{\gamma } -x_{i} ^{0} } \right)}\dot{\phi }_{\gamma } \\\end{aligned}$$

(A7)

where

$$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\phi }_{\alpha } {\left( t \right)} = \frac{{2D_{i} ^{\alpha } }} {{S^{2} }}{\left( {1 - \frac{{2\ifmmode\expandafter\dot\else\expandafter\.\fi{S}t}} {S}} \right)}{\sum\limits_{n = 0}^\infty {\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\alpha } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4S^{2} }}} \right]}} } $$

(A8)

and

$$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\phi }_{\gamma } {\left( t \right)} = \frac{{2D_{i} ^{\gamma } }} {{{\left( {d - S} \right)}^{2} }}{\left( {1 + \frac{{2\ifmmode\expandafter\dot\else\expandafter\.\fi{S}t}} {{d - S}}} \right)}{\sum\limits_{n = 0}^\infty {\exp {\text{ }}{\left[ { - \frac{{D_{i} ^{\gamma } {\left( {2n + 1} \right)}^{2} \pi ^{2} t}} {{4{\left( {d - S} \right)}^{2} }}} \right]}} } $$

(A9)

As described in Section II–B, when diffusion in ferrite can be neglected, one can put

$$ \phi _{\alpha } \sim 1\quad {\text{and}}\quad {\dot{\phi}} _{\alpha } \sim 0 $$

in Eqs. [A4] and [A7].