A phase-field model for bainitic transformation T.T. Arif, R.S. Qin ⇑ Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ, UK article info Article history: Received 6 January 2013 Received in revised form 16 March 2013 Accepted 18 April 2013 Keywords: Displacive phase transitions Phase-field model Bainite transformation Cubic crystal Autocatalysis abstract A phase-field model for the computation of microstructure evolution for the bainite transformation has been developed. The model has a classical phase-field foundation, incorporates the phenomenological displacive transformation theory and the symmetric analysis of cubic crystals, and is able to reproduce realistic grain morphology and crystal orientation after adequate calibration. Using the free energy expression for the shape change of displacive transformations along with the free energy formula for the chemical free energy change of the two phases derived from established regular solution models, the current model is able to deal with autocatalysis. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The phase-field (PF) method is used in the domain of materials science for simulating microstructure evolution on the mesoscale. The locations, sizes and shapes of the grains involved are repre- sented by the PF variables. The evolution of the PF variables is then governed by a set of partial differential equations. They have nearly constant values within the grains but vary continuously over the interfaces between them. This means that the interface has a width and is not infinitesimally thin. Thus the method is said to have a diffuse-interface description. Unlike sharp interface methods, the diffuse-interface nature enables the computation of complex microstructures without explicitly tracking the interfaces. The method has been reviewed extensively, such as that in Refs. [1,2]. A PF model involves the formulation of a free energy functional of the PF variables and their gradients. - A form of the free energy functional includes the PF variable / as a means of distinguishing coexisting phases along with the composition field C and temper- ature T as [3] G ¼ Z V g 0 ð/; C; T Þþ 1 2 e 2 C ð ~ rCÞ 2 þ 1 2 e 2 / ð ~ r/Þ 2   d ~ r ð1Þ where e C and e / are the gradient energy coefficients and g 0 (/, C, T) is the free energy density over the volume V. The governing equation for / is derived from G in a thermodynamically consistent manner adhering to the second law of thermodynamics. Eq. (1) can be cou- pled with a thermodynamic database [4–7] and used for the simu- lation of solidification [8], grain growth [9], solute drag [10] and many other processes. The method has thus gained momentum in microstructure formation and evolution. In solid state phase trans- formations there have been PF models that consider displacive transformations [11] and those that consider diffusive transforma- tions [9,12]. Considered within the present work is a PF model able to utilise both diffusive and displacive mechanisms in its treatment of microstructure evolution. The microstructure focussed upon is bai- nite. Bainite is a microstructure resulting from the decomposition of austenite usually found to occur at a temperature between the pearlite reaction and the martensite start temperature (Ms). Ini- tially detected as a unique microstructure in the early 1900s [13,14], interest in this multi-phase product of austenite grew once its benefits were realised. Bainitic steels boast improved strength without the expense of weldability and toughness and have appli- cations in the railway, automotive industries and structural engi- neering [15–19]. The time-consuming process of producing bainitic steels prompted the desire to understand and formulate models for the kinetics and formation of bainite. Following the kinetic model of Bhadeshia [20] the supersatu- rated ferrite sub-units form in austenite via a displacive mecha- nism. The ferrite sub-units form martensitically without the partitioning of alloying elements. Due to the higher temperatures when compared to martensite, the partitioning of the interstitial carbon from the supersaturated ferrite into the residual austenite follows soon after. Upon carbide precipitation in the austenite, the upper bainite microstructure forms. As the temperature is re- duced, this diffusion process is slowed down which results in carbon precipitation within the bainitic ferrite giving lower bainite 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.044 ⇑ Corresponding author. Tel.: +44 207 5946803; fax: +44 207 5946757. E-mail address: r.qin@imperial.ac.uk (R.S. Qin). Computational Materials Science 77 (2013) 230–235 Contents lists available at SciVerse ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci