Metallurgical and Materials Transactions A

, Volume 49, Issue 9, pp 4146–4157 | Cite as

Phase-Field Modeling of Precipitation Growth and Ripening During Industrial Heat Treatments in Ni-Base Superalloys

  • Michael Fleck
  • Felix Schleifer
  • Markus Holzinger
  • Uwe Glatzel
Topical Collection: Superalloys and Their Applications
Part of the following topical collections:
  1. Third European Symposium on Superalloys and their Applications


We develop a phase-field model for the simulation of chemical diffusion limited microstructure evolution, with a special focus on precipitation growth and ripening in multicomponent alloys. Further, the model accounts for elastic effects, which result from the lattice-misfit between the precipitate particles and the parent matrix phase. To be able to simulate particle growth and ripening in one dimension, we introduce an extra optional driving-force term, which mimics the effect of curved interfaces in one dimension. As a case study, we consider the one-dimensional (1D) \(\gamma \)′-precipitation growth and ripening under the influence of a realistic multistep heat treatment in the multicomponent Ni-based superalloy CMSX-4. The required temperature-dependent thermodynamic and kinetic input parameters are obtained from CALPHAD calculations using the commercial software-package ThermoCalc. The required temperature-dependent elastic parameters are measured in-house at the chair of Metals and Alloys, using resonance ultrasound spectroscopy and high-temperature X-ray defraction. Finally, the model is applied to calculate the equilibrium shape of a single \(\gamma \)′-particle with periodic boundary conditions. Relations to the shapes of \(\gamma \)′-particles in respect of heat-treated experimental microstructures are discussed.


\(\Omega \)

Grand potential energy

\(\omega \)

Grand potential energy density


The partial derivative with respect to the three possible spatial directions \(i=x,y,z\), i.e., \(\partial_{x}\equiv \partial /\partial x\), \(\partial_{y}\equiv \partial /\partial y\), and \(\partial_{z}\equiv \partial /\partial z\)


For the ith component \(Q_i\) of the n-component vector \(\mathbf {Q}\), the embedding of which into curly brackets means \(\{Q_{i}\}= (Q_{0},Q_{1},\ldots,Q_{n-1})\)

\(\omega_{{\rm int}}\)

Interfacial contribution to the grand potential density

\(\omega_{{\rm ch}}\)

Chemical contribution to the grand potential density

\(\omega_{{\rm bulk}}\)

Bulk contributions to the grand potential density

\(\omega_{{\rm el}}\)

Elastic contribution to the grand potential density

\(\varphi \)



Chemical diffusion potential field of ith component


Elastic displacement field for the spatial direction \(i = x,y,z\).

\(p(\varphi )\)

Double-well potential

\(\xi \)

Phase-field interfacial width

\(\Gamma \)

Interfacial energy density

\(h(\varphi )\)

Interpolation function


Phase-field kinetic coefficent

\(f_{{\rm art}}^{1{\rm D}}\)

Artifical Gibbs–Thompson driving force for the one-dimensional (1D) simulations

\(\kappa \)

Curvature of the interface

\(\Gamma_{{\rm art}}\)

Artifical interfacal energy density for the 1D simulations

\(g_{{\rm ch}}^{\alpha }\)

Gibbs free energy density of the phase \(\alpha \)

\(\alpha \)

Phase superscript takes the values \(\alpha = m\) for the matrix phase and \(\alpha = p\) for the precipitation- and particle-phase


Phase superscript p indicates that the respective precipitation- or particle-phase value has to be taken


Concentration field of the ith component

\(X_{ij}^{\alpha }\)

Thermodynamic factor of the components i and j within the phase \(\alpha \)

\({\mathcal{A}}_{i}^{\alpha }\)

Concentration of the ith. component, where the Gibbs free energy of phase \(\alpha \) is minimal

\({\mathcal{B}}^{\alpha }\)

Offset energy of the Gibbs free energy of phase \(\alpha \)



\(\chi_{ij}^{\alpha }\)

Generalized susceptibility matrix, which corresponds to the inverse of the thermodynamic factor matrix

\(\left( c_{{\rm eq}}^{\alpha }\right)_{i}\)

Phase equilibrium concentration of the ith component within the phase \(\alpha \)

\(\left( \mu_{{\rm eq}}\right)_{i}\)

Equilibrium diffusion potential of the ith component

\(g_{{\rm eq}}^{\alpha }\)

Equilibrium Gibbs free energy density of phase \(\alpha \)


Diffusion coefficent for the ith component in the presence of component j


Elastic strain field of the spatial directions i and j, \(i,j=x,y,z\)


Eigenstrain or inherent strain in the presence of no elastic stress, spatial directions i and j, \(i,j=x,y,z\)


Elastic stiffness coefficent between the stress component \(\sigma_{ij}\) and the elastic strain \(\varepsilon_{kl}\), \(i,j,k,l= x,y,z\)


Elastic stress field of the spatial directions i and j, \(i,j=x,y,z\)


Cooling rate

\(\gamma \)

Fcc-matrix phase in the CMSX-4 alloy

\(\gamma \)

Ordered fcc-L12 phase of the precipitate particles in the CMSX-4 alloy

\(\phi^{{\rm p}}\)

Phase fraction of the precipitation- or particle-phase


Elasic stiffness parameter \(C'=\left( C_{11}-C_{12}\right) /2\)


Dimensionless elasic anisotropy parameter (Zener Anisotropy) \(A=2C_{44}/\left( C_{11}-C_{12}\right) \)


Scaling constant for the energy density


Temperature-dependent scaling constant for the diffusion matrix


Scaling-superscript s indicates that the respective scaled value for the use inside the simulation has to be taken


Scaling constant for the lengths

\(V_{\rm m}\)

Molar volume

\(E_{0}^{{\rm el}}\)

Elastic energy scale constant \(E_{0}^{{\rm el}}=E_{0}/V_{\rm m}\)


Index eq indicates that the respective equilibrium value of the quantity has to be taken


Total volume of the system


Dimensionless characteristic length of precipitate particles, \({\mathcal{L}}=C_{44}^{\gamma '}\cdot \epsilon_{0}^{2}\cdot r/\Gamma \)


Radius of precipitate particles



The authors thank the Federal Ministry for Economics and Energy (BMWi) of the Federal Republic of Germany for the financial support under the running project COORETEC: ISar (Funding Code: 03ET7047D). Further, the authors thank the Federal Ministry of Education and Research (BMBF) for the financial support under the running project ParaPhase (Funding Code: 01IH15005B). Also, the financial support from the German Research Foundation (DFG) within the second phase of the priority program 1713 (FL 826/3-1|GL 181/53-1) is gratefully acknowledged. In addition, the authors are very thankful to their collaboration partners at the ACCESS, the MTU Aero Engines AG, and the Siemens AG for the fruitful discussions during their joint meetings for the research project. Finally, the authors thank their colleague Fabian Krieg for providing the elastic data on CMSX-4.

Supplementary material

11661_2018_4746_MOESM1_ESM.mpg (2.7 mb)
Electronic supplementary material 1 (MPG 2730 kb)


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Copyright information

© The Minerals, Metals & Materials Society and ASM International 2018

Authors and Affiliations

  • Michael Fleck
    • 1
  • Felix Schleifer
    • 1
  • Markus Holzinger
    • 1
  • Uwe Glatzel
    • 1
  1. 1.Metals and AlloysUniversity BayreuthBayreuthGermany

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