Development of a level set methodology to simulate grain growth in the presence of real secondary phase particles and stored energy – Application to a nickel-base superalloy Andrea Agnoli a,b,⇑ , Nathalie Bozzolo a , Roland Logé a,c , Jean-Michel Franchet b , Johanne Laigo b , Marc Bernacki a a Mines ParisTech, CEMEF – Centre de Mise en Forme des Matériaux, CNRS UMR 7635, CS 10207 rue Claude Daunesse, 06904 Sophia Antipolis cedex, France b Snecma Gennevilliers, 171 boulevard Valmy, 92702 Colombes, France c Thermomechanical Metallurgy Laboratory – PX Group Chair, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rue de la Maladière 71B – CP 526, CH-2002 Neuchâtel, Switzerland article info Article history: Received 3 December 2013 Received in revised form 18 March 2014 Accepted 25 March 2014 Available online 18 April 2014 Keywords: Grain growth Finite element Zener pinning Simulation Microstructure Level set abstract The influence of secondary phase particles on grain growth (Zener pinning) is simulated in two dimensions using a level set method. Several simulations with circular particles are performed to study the influence of particle surface fraction and distribution. The limiting grain sizes are compared to previous numerical simulations from literature. It is shown that the method allows to take into account the morphology of real particles as measured by microscopy. Moreover, the model also allows to introduce stored energy driven grain boundary migration in Zener pinning simulations. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction In microstructures containing secondary phase particles Zener pinning [1] is the phenomenon that hinders grain growth, eventually leading to a limiting grain size. The understanding and modeling of Zener pinning is of great interest as it is practically exploited to control and limit grain size in several industrial alloys. During the last six decades, the phenomenon has been widely studied and many different analytical models have been proposed in the literature (see [2] for a comprehensive review). Despite the variety of all these studies, there is a global agreement that the limiting grain size can be predicted by this general relation: R ¼ K r f m ; ð1Þ where R is the average radius of grains, r and f are respectively the average radius and the volume fraction of the secondary-phase particles. K and m are two parameters that fluctuate according to the assumptions that are made to obtain the equation. In the literature the pinning effect of secondary phase particles on grain growth has already been modeled by Monte Carlo, phase- field and boundary-tracking methods. Monte Carlo models [3–5], which were the first to be developed, and phase-field models [6–8], which have gained more attention in the last years, are able to simulate both the 2D and 3D Zener pinning phenomenon. On the contrary, boundary-tracking models are limited to 2D simulations [9] or to 3D simulation of the interaction of a single grain boundary with particles [10,11]. Overall, while all methods can effectively simulate Zener pinning in the simple case of circular or spherical particles, there is not yet a model that can deal with particles whose shape is more complex than ellipsoidal. In addition, all models still assume that particles are incoherent with the matrix, that is the interface energy between grains and particles is isotro- pic. The present paper introduces the simulation of Zener pinning based on a level set description of interfaces in a finite element context. It is to note that the level set approach has been also developed in the context of uniform grids with a finite difference formulation [12,13]. The numerical approach presented in this work was already used to simulate both 2D and 3D primary recrystallization [14] and grain growth [15] in polycrystals in a finite element context where anisotropic meshing and adaptive remeshing techniques were used. In the conference paper [16], first http://dx.doi.org/10.1016/j.commatsci.2014.03.054 0927-0256/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author at: Mines ParisTech, CEMEF – Centre de Mise en Forme des Matériaux, CNRS UMR 7635, CS 10207 rue Claude Daunesse, 06904 Sophia Antipolis cedex, France. Tel.: +33 0493957415. E-mail address: andrea.agnoli@mines-paristech.fr (A. Agnoli). Computational Materials Science 89 (2014) 233–241 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci