The Mechanism of Grain Boundary Serration and Fan-Type Structure Formation in Ni-Based Superalloys

Abstract

A model of discontinuous precipitation of γ′ phase at grain boundaries (GBs) in polycrystalline Ni-based superalloys during continuous cooling from supersolvus temperature was developed. The model calculates the size and γ′ phase fraction in GB precipitates as a function of cooling rate and GB mobility. The model is based on the classical mechanism of diffusion-controlled precipitation. It considers fast diffusion of γ′-forming elements along the GB and the motion of the GBs under driving force induced by alloy decomposition. The model predicts either discrete GB gamma-prime particles and GB serration or the growth of fan-type structures depending on the cooling rate and GB mobility. We conclude from the model predictions that the GB mobility is the key factor controlling the type and morphology of GB precipitates. High GB mobility results in formation of fan-type structures, while the lower GB mobility leads to GB serration. The predicted dependence of the size of GB precipitates on the cooling rate is in good agreement with the experimental observations.

Appendix A

Let us assume the grain boundary to be an infinite flat layer of thickness δ_GB and the grain boundary precipitate to be a cylindrical particle of radius R and of height δ_GB located inside the layer (Figure A1). Grain boundary thickness is usually supposed to be equal to several lattice parameters.

Fig. A1

Grain boundary with a localized absorbing particle of radius R

The γ′-forming elements diffuse along the grain boundary and through the grain bulk with diffusivities D_GB and D_G, correspondingly. The initial concentration of γ′-forming elements (initial condition) as well as concentration at infinity (boundary condition) is c₀ (nominal concentration). The concentration of γ′-forming elements at the boundary of the particle is cγ. The solution of diffusion equation in that geometry with the described above initial and boundary conditions is presented in detail.[29] The final equation for the Laplace transform of γ′ forming element flux to the particle is:

$$ J_{\gamma '} \left( s \right) = \frac{{2\pi \delta D_{\text{GB}} }}{s}\frac{{1 + F_{1} \left( s \right)}}{{F_{2} \left( s \right)}}, $$

(A1)

where

$$ F_{1} \left( s \right) = \int\limits_{0}^{\infty } {\frac{{x^{2} J_{0} \left( x \right)J_{1} \left( x \right){\text{d}}x}}{{x^{2} + \beta_{\text{GB}}^{2} + z\sqrt {x^{2} + \beta_{\text{G}}^{2} } }}} , $$

(A2)

$$ F_{2} \left( s \right) = \int\limits_{0}^{\infty } {\frac{{xJ_{0}^{2} \left( x \right){\text{d}}x}}{{x^{2} + \beta_{\text{GB}}^{2} + z\sqrt {x^{2} + \beta_{\text{G}}^{2} } }}} . $$

(A3)

Here $z = \frac{{D_{\text{G}} R}}{{D_{\text{GB}} \delta }}$, $\beta_{\text{GB}}^{2} = \frac{{R^{2} }}{{D_{\text{GB}} }}s$ and $\beta_{\text{G}}^{2} = \frac{{R^{2} }}{{D_{\text{G}} }}s$. In the long-time asymptotic (small s), when $t \gg \frac{{R^{2} }}{{D_{\text{G}} }}\sim 10^{ - 2} s$, the integrals (A2) and (A3) can be simplified as:

$$ F_{1} \approx \int\limits_{0}^{\infty } {\frac{{xJ_{0} \left( x \right)J_{1} \left( x \right){\text{d}}x}}{x + z}} , $$

(A4)

$$ F_{2} \approx \int\limits_{0}^{\infty } {\frac{{J_{0}^{2} \left( x \right){\text{d}}x}}{x + z}.} $$

(A5)

Taking advantage of $z < 1$, we obtain that $F_{1} \approx {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}$ and $F_{2} \approx \ln \left( {z^{ - 1} } \right)$. Substituting these equations for F₁ and F₂ into Eq. (A1), we obtain the final equation:

$$ J_{\gamma '} \left( s \right) = \frac{{3\pi \delta D_{\text{GB}} }}{s}\frac{1}{{\ln \left( {z^{ - 1} } \right)}}. $$

(A6)

After the inverse Laplace transform, we obtain the flux to the particle in the t-domain:

$$ J_{{\gamma^{\prime}}} = \frac{{3\pi D_{\text{GB}} \delta_{\text{GB}} (c_{0} - c_{\gamma } )}}{{\ln \left( {\frac{{D_{\text{GB}} \delta_{\text{GB}} }}{{D_{\text{G}} R}}} \right)}}. $$

(A7)

The long-time asymptotic, $t \gg 0.01 s$, is independent of time, which is the feature of the 3D diffusion-controlled reaction rate, although the γ′-forming elements are transported to the absorbing particle by 2D diffusion through the grain boundary.