《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_Multi-scale phase field approach to martensitic transformations

MATERIALS 兴 HIENGE& ENGIEERING ELSEVIER Materials Science and Engineering A 438-440(2006)55-63 www.elsevier.com/locate/msea Multi-scale phase field approach to martensitic transformations Yunzhi Wang,, Armen G Khachatur a Department of Materials Science and Engineering. The Ohio State University, Columbus, OH 43210, USA b Department of Ceramic and Materials Engineering. Rutgers University, 607 Taylor Road, Piscataway, NJ08854, USA Received 18 August 2005: received in revised form 17 March 2006: accepted 17 April 2006 Because of its flexibility in treating complex geometries and topological changes, the phase field method has been used widely in modeling microstructural evolution during various phase transformations, grain growth and, most recently, plastic deformations. We review formulations and applications of the method in the context of martensitic transformations(MTs). Examples are chosen to illustrate the capabilities of the method at both mesoscopic and microscopic length scales. At the mesoscopic level we present simulation predictions on structural configurations of critical nuclei generated by homogeneous nucleation through Langevin thermal fluctuations under large undercooling, formation of herringbone structures by autocatalytic growth, and microstructural evolution and transformation hysteresis in a polycrystalline alloy under uniaxial stresses. At the microscopic level we discuss new developments in phase field model of dislocation dissociation and core structure and phase field model of mislocation-a new elementary defect introduced to describe the initiation and growth of martensite. o 2006 Elsevier B. v. All rights reserved Keywords: Martensitic transformation; Phase field; Nucleation; Dislocation; Multi-scale modelin 1. Introduction formation of a critical nucleus configuration in an arbitrary pr existing microstructure and its subsequent growth without any The essential features of martensitic transformations(MTs) a priori assumptions about its shape, orientation, spatial loca- are characterized by(a) thermoelastic two-phase equilibrium tion, and arrangement of different orientation variants within that violates the Gibbs phase rule;(b) non-ergodicity that it. The conventional treatment that considers an isolated parti leads to path-dependent equilibrium microstructure and trans- cle of a given shape without capturing the effect of pre-existing formation hysteresis;(c)unique morphological patterns; and microstructures may not be sufficient to quantitatively describe (d) stress-induced transformation and shape memory effect. the nucleation and growth processes leading to self-organization These features are associated with elastic strain accommoda- of multivariant and multiphase martensitic microstructures. tion of different orientation variants of the martensitic phase in The work by Cahn [2-5] on spinodal a parent phase matrix, which dominates the transformation ther- coherent fluctuations is, in fact, the earliest application of modynamics, kinetics and crystallography. The strain-induced the phase field method to coherent transformations in solids long-range elastic interactions among the orientation variants where the transformation-induced coherency strain was consid and between them and pre-existing strain-carrying defects such ered. The extension of the method to arbitrary microstructures as dislocations, precipitates, microcracks and grain boundaries produced by diffusional and displacive transformations with determine the activation pathways for nucleation and growt arbitrary transformation strains is made possible by using the and control the overall transformation kinetics. Since the elastic microelasticity theory of Khachaturyan and Shatalov(Ks the strain energy is in general a function of size, shape, orien- ory)[1, 6, 7] who developed a reciprocal-space formulation of the tation and spatial arrangement of precipitates [1], a rigorous strainenergy as an explicit functional of arbitrary continuous dis- treatment of microstructural evolution during MTs requires a tributions of structural and/or compositional non-uniformities kinetic approach that is able to describe self-consistently the The method has been applied to various coherent phase trans- formations including MTs and many complicated strain-induced morphological patterns have been predicte Corresponding author. Tel. +1 614 292 0682: fax: +1 614 292 1537. see[8-11)). Most recently, the KS theap. ed (for recent reviews vas successfully inte- E-mail address: wang.363@osu.edu(Y. Wang). grated with the phase field models of dislocation dynamics and 0921-5093/S-see front matter

Materials Science and Engineering A 438–440 (2006) 55–63 Multi-scale phase field approach to martensitic transformations Yunzhi Wang a,∗, Armen G. Khachaturyan b a Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA b Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA Received 18 August 2005; received in revised form 17 March 2006; accepted 17 April 2006 Abstract Because of its flexibility in treating complex geometries and topological changes, the phase field method has been used widely in modeling microstructural evolution during various phase transformations, grain growth and, most recently, plastic deformations. We review formulations and applications of the method in the context of martensitic transformations (MTs). Examples are chosen to illustrate the capabilities of the method at both mesoscopic and microscopic length scales. At the mesoscopic level we present simulation predictions on structural configurations of critical nuclei generated by homogeneous nucleation through Langevin thermal fluctuations under large undercooling, formation of herringbone structures by autocatalytic growth, and microstructural evolution and transformation hysteresis in a polycrystalline alloy under uniaxial stresses. At the microscopic level we discuss new developments in phase field model of dislocation dissociation and core structure and phase field model of mislocation—a new elementary defect introduced to describe the initiation and growth of martensite. © 2006 Elsevier B.V. All rights reserved. Keywords: Martensitic transformation; Phase field; Nucleation; Dislocation; Multi-scale modeling 1. Introduction The essential features of martensitic transformations (MTs) are characterized by (a) thermoelastic two-phase equilibrium that violates the Gibbs phase rule; (b) non-ergodicity that leads to path-dependent equilibrium microstructure and trans￾formation hysteresis; (c) unique morphological patterns; and (d) stress-induced transformation and shape memory effect. These features are associated with elastic strain accommoda￾tion of different orientation variants of the martensitic phase in a parent phase matrix, which dominates the transformation ther￾modynamics, kinetics and crystallography. The strain-induced long-range elastic interactions among the orientation variants and between them and pre-existing strain-carrying defects such as dislocations, precipitates, microcracks and grain boundaries determine the activation pathways for nucleation and growth and control the overall transformation kinetics. Since the elastic strain energy is in general a function of size, shape, orien￾tation and spatial arrangement of precipitates [1], a rigorous treatment of microstructural evolution during MTs requires a kinetic approach that is able to describe self-consistently the ∗ Corresponding author. Tel.: +1 614 292 0682; fax: +1 614 292 1537. E-mail address: wang.363@osu.edu (Y. Wang). formation of a critical nucleus configuration in an arbitrary pre￾existing microstructure and its subsequent growth without any a priori assumptions about its shape, orientation, spatial loca￾tion, and arrangement of different orientation variants within it. The conventional treatment that considers an isolated parti￾cle of a given shape without capturing the effect of pre-existing microstructures may not be sufficient to quantitatively describe the nucleation and growth processes leading to self-organization of multivariant and multiphase martensitic microstructures. The work by Cahn [2–5] on spinodal decomposition and coherent fluctuations is, in fact, the earliest application of the phase field method to coherent transformations in solids where the transformation-induced coherency strain was consid￾ered. The extension of the method to arbitrary microstructures produced by diffusional and displacive transformations with arbitrary transformation strains is made possible by using the microelasticity theory of Khachaturyan and Shatalov (KS the￾ory)[1,6,7] who developed a reciprocal-space formulation of the strain energy as an explicit functional of arbitrary continuous dis￾tributions of structural and/or compositional non-uniformities. The method has been applied to various coherent phase trans￾formations including MTs and many complicated strain-induced morphological patterns have been predicted (for recent reviews see [8–11]). Most recently, the KS theory was successfully inte￾grated with the phase field models of dislocation dynamics and 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.04.123

Y Wang, A G Khachaturyan/Materials Science and Engineering A 438-440(2006)55-63 microcrack propagation at mesoscales [12]and with phase field with a homogeneous strain that is responsible for the martensitic model of dislocation core structures at microscales [13]. These features of the transformation advances have paved the road for realistic nano-and meso-scale <3 A key step in developing a phase field model is to formulate phase field modeling of MTs involving simultaneous plastic the total free energy of a system as a function of the chosen deformation and fracture. The latter could have important appli- SOP fields. In the discussion of MTs, the total free energy is cations to transformation toughening in ceramics. usually separated into two competing terms, i. e, the chemical This article provides an overview of the applications of the free energy that provides the driving force for the transformation phase field methods to computer simulation of MTs at two dif- and the elastic strain energy that suppresses the transformation ferent length scales. Thus, at the mesoscopic level, the examples The chemical free energy of a homogeneous system is usually include fundamental characteristics of microstructural evolution approximated by a Landau polynomial expansion with respect to (stain-accommodating packing of martensitic variants, auto- the SOPs [31-33]. For example, the simplest fourth-order poly catalytic effect on nucleation and growth, and transformation nomial approximation of the Landau free energy used [34-38 hysteresis)during a generic cubic ->tetragonal MT in a sin- for a proper fcc- bcc MT is presented as a sum of the symmetry gle crystal [14] and a more complex cubic->trigonal MT in invariants a polycrystalline system [15]. At the microscopic leve scopic phase field model of dislocation core structures [13, 161 f(n1, n2, n3= Ar discussions focus on the most recent developments of micro- 1(m2+n+n)-32(m+n+n nd its application to the study of micromechanics of nucle ation and growth of martensite[17]. It should be pointed out +A3(m+n+32 (1) that extensive work has been done at the mesoscopic level in pursuing ferroelastic and phase field approaches to martensitic where np(p=1, 2, 3)are the SOPs that describe the tetragonal transformations [18-27] and the applications discussed in this distortion of the three orientation variants along the Bain path article are just a few examples from our own work, by no means andAi(i=1, 2, 3)are the expansion coefficients. For an improper complete MT the sixth-order polynomial forming independent symmet invariants of the appropriate order has been introduced [14] 2. Phase field method at meso-scale and its applications to martensitic transformations f(m1,n2,…,m)=(m+n+…+听 Using gradient thermodynamics of non-uniform systems [28-30], the theory of microelasticity [1,6,7), and continuum A fields of conserved and non-conserved parameters, the phase m+n+…+m field method describes spatial-temporal evolution of arbitrary microstructures consisting of various types of extended defects A3(+n+… such as homo- and hetero-phase interfaces, antiphase domain boundaries, ferromagnetic and ferroelectric domain walls, and, most recently, dislocations and cracks (for the most recent 6+听吗 reviews, see [12]). A typical example of the conserved fields is +…+n2-272-1m2 concentration that characterizes chemical non-uniformity and a typical example of the non-conserved fields is the long-range (⑦m+n2+…+m) order parameter that characterizes crystal symmetry change during an order-disorder transformation. The microstructures developed in a MT can be characterized fully by a set of non- +6+吃 onserved fields, the structural order parameter(SOP)fields Different from phase field models of solidification where the where np (p=l, 2,..., n) are the SOPs that characterize the non-conserved field is introduced as a numerical technique to amplitudes of the optimal displacement modes and Ai(i=1 avoid boundary tracking, the non-conserved fields employed in 2,..., 6) are the expansion coefficients. The free energies pre- phase field models of solid-state phase transformations have sented in Eqs. (1)and(2)are specific free energies of structurall well defined physical meanings. In a proper MT such as the homogeneous finite elements with uniform values(equilibrium face centered cubic to body centered cubic ( cc- bcc) lattice or non-equilibrium) of the SOP, np. The minima of the free rearrangement in iron, for example, the SOP distinguishing the energy correspond to the equilibrium free energies of the stress- martensite and austenite phases is the tetragonal Bain distor- free parent and product phases. The minima for the product tion(a homogeneous strain that transforms an fcc lattice to a(martensite) phase are degenerate, with each of them describing bcc lattice). In an improper MT such as the cubic>tetragonal a crystallographically equivalent orientation domain transformation in Zro2 and many MTs in ferroelastic materi- In a structurally non-uniform system such as a mixture of als, the SOP is the amplitude of a"soft "optical displacement martensite and austenite phases, the finite elements characte mode that results in a mutual displacement of atoms within a unit ized by the Landau free energy have different values of np cell of the parent phase(shuffle). This primary SOP is coupled According to gradient thermodynamics [28-30], the chemical

56 Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 microcrack propagation at mesoscalses[12] and with phase field model of dislocation core structures at microscales [13]. These advances have paved the road for realistic nano- and meso-scale phase field modeling of MTs involving simultaneous plastic deformation and fracture. The latter could have important appli￾cations to transformation toughening in ceramics. This article provides an overview of the applications of the phase field methods to computer simulation of MTs at two dif￾ferent length scales. Thus, at the mesoscopic level, the examples include fundamental characteristics of microstructural evolution (stain-accommodating packing of martensitic variants, auto￾catalytic effect on nucleation and growth, and transformation hysteresis) during a generic cubic→tetragonal MT in a sin￾gle crystal [14] and a more complex cubic→trigonal MT in a polycrystalline system [15]. At the microscopic level, the discussions focus on the most recent developments of micro￾scopic phase field model of dislocation core structures [13,16] and its application to the study of micromechanims of nucle￾ation and growth of martensite [17]. It should be pointed out that extensive work has been done at the messoscopic level in pursuing ferroelastic and phase field approaches to martensitic transformations [18–27] and the applications discussed in this article are just a few examples from our own work, by no means complete. 2. Phase field method at meso-scale and its applications to martensitic transformations Using gradient thermodynamics of non-uniform systems [28–30], the theory of microelasticity [1,6,7], and continuum fields of conserved and non-conserved parameters, the phase field method describes spatial-temporal evolution of arbitrary microstructures consisting of various types of extended defects such as homo- and hetero-phase interfaces, antiphase domain boundaries, ferromagnetic and ferroelectric domain walls, and, most recently, dislocations and cracks (for the most recent reviews, see [12]). A typical example of the conserved fields is concentration that characterizes chemical non-uniformity and a typical example of the non-conserved fields is the long-range order parameter that characterizes crystal symmetry change during an order-disorder transformation. The microstructures developed in a MT can be characterized fully by a set of non￾conserved fields, the structural order parameter (SOP) fields. Different from phase field models of solidification where the non-conserved field is introduced as a numerical technique to avoid boundary tracking, the non-conserved fields employed in phase field models of solid-state phase transformations have well defined physical meanings. In a proper MT such as the face centered cubic to body centered cubic (fcc→bcc) lattice rearrangement in iron, for example, the SOP distinguishing the martensite and austenite phases is the tetragonal Bain distor￾tion (a homogeneous strain that transforms an fcc lattice to a bcc lattice). In an improper MT such as the cubic→tetragonal transformation in ZrO2 and many MTs in ferroelastic materi￾als, the SOP is the amplitude of a “soft” optical displacement mode that results in a mutual displacement of atoms within a unit cell of the parent phase (shuffle). This primary SOP is coupled with a homogeneous strain that is responsible for the martensitic features of the transformation. A key step in developing a phase field model is to formulate the total free energy of a system as a function of the chosen SOP fields. In the discussion of MTs, the total free energy is usually separated into two competing terms, i.e., the chemical free energy that provides the driving force for the transformation and the elastic strain energy that suppresses the transformation. The chemical free energy of a homogeneous system is usually approximated by a Landau polynomial expansion with respect to the SOPs [31–33]. For example, the simplest fourth-order poly￾nomial approximation of the Landau free energy used [34–38] for a properfcc→bcc MT is presented as a sum of the symmetry invariants f (η1, η2, η3) = A1 2 (η2 1 + η2 2 + η2 3) − A2 3 (η3 1 + η3 2 + η3 3) + A3 4 (η2 1 + η2 2 + η2 3) 2 (1) where ηp (p = 1, 2, 3) are the SOPs that describe the tetragonal distortion of the three orientation variants along the Bain path and Ai (i = 1, 2, 3) are the expansion coefficients. For an improper MT the sixth-order polynomial forming independent symmetry invariants of the appropriate order has been introduced [14] f (η1, η2,...,ηn) = A1 2 (η2 1 + η2 2 +···+ η2 n) + A2 4 (η4 1 + η4 2 +···+ η4 n) + A3 4 (η2 1 + η2 2 +···η2 n) 2 + A4 6 (η2 1η2 2η2 3 + η2 1η2 2η2 4 +· · · + η2 n−2η2 n−1η2 n) + A5 6 (η6 1 + η6 2 +···+ η6 n) + A6 6 (η2 1 + η2 2 +···+ η2 n) 3 (2) where ηp (p = 1, 2, ..., n) are the SOPs that characterize the amplitudes of the optimal displacement modes and Ai (i = 1, 2, ..., 6) are the expansion coefficients. The free energies pre￾sented in Eqs.(1) and (2) are specific free energies of structurally homogeneous finite elements with uniform values (equilibrium or non-equilibrium) of the SOP, ηp. The minima of the free energy correspond to the equilibrium free energies of the stress￾free parent and product phases. The minima for the product (martensite) phase are degenerate, with each of them describing a crystallographically equivalent orientation domain. In a structurally non-uniform system such as a mixture of martensite and austenite phases, the finite elements character￾ized by the Landau free energy have different values of ηp. According to gradient thermodynamics [28–30], the chemical

Y Wang, A.G. Khachaturyan/ Materials Science and Engineering A 438-440 (2006)55-63 free energy of such a system is given by energies to experimental or calculated data [43, 44]. This process also defines the length scale of the numerical simulations If the MIron. m:.. 0) +2 2A, P)V m, mo d, lattice correspondence between the parent and product phases their lattice parameters and elastic constants are known, the elas- (3) tc strain energy(4)can be formulated in rather straightforward fashion. Detailed examples can be found in [11] re Bij are the gradient energy coefficients. The first term, The spatial-temporal evolution of the SOPs, which com- pecific free energy, depends only on local values of the pletely defines the microstructural evolution during a MT, field np(r)(where r is the spatial coordinate), while the sec- obtained by solving the Onsager-type semi-phenomenological ond term, the gradient energy, accounts for contributions from kinetic equations of motion, which assume a linear dependence spatial variation of np(r). The interplay between the two deter- of the rate of evolution, anp/at, on the driving force, 8 F/6np mines the equilibrium profiles of np and hence the widths and energies of interfaces between different orientation variants of the martensitic phase and between the martensite and parer 9=l ong(r, t +sp(r, t) To remove the structural incompatibility between adjacent finite elements of different values of SOP, one need to introduce where F=Fch +Eel is the total free energy of the system,with additional inhomogeneous displacements providing the continu- Fcn being the chemical free energy and e the elastic energy ity of the total displacement field( the crystal lattice coherency). Lpg are the kinetic coefficients and 5p is the Langevin random By definition, this is an elastic displacement field that provides force term that describes thermal fluctuations [31,45-471 the elastic strain energy. According to the Ks theory [1, 6,7] The phase field method formulated above offers a self- the elastic strain energy for arbitrary structural non-uniformities consistent treatment of the thermodynamics and kinetics of MTs characterized by np(r)in an elastically anisotropic crystal under involving arbitrary multivariant configuration and particle mor- the homogeneous modulus assumption is given by an explicit phology at mesoscopic length scales. The effect of long-range elastic interactions on the initiation and growth of martensite is taken into account automatically by including the elastic strain ∑+8m(7) (4 energy in the total free energy in Eq (6). The models have been (4) applied successfully to proper and improper MTs in single crys- where Inp(r))g is the Fourier transform of the function np(r) that tals [14,37], polycrystalline alloys [15, 48), multi-layer systems [49], and thin-films [50] and the numerical simulations have characterizes an arbitrary structural non-uniformity of the pth type(in principle, the structural non-uniformities could be asso- Shed light on the micromechanisms underlying microstructural ciated with any types of extended crystal defects such as coherent development during MTs One of the interesting observations from the phase field sim precipitate, dislocations, grain boundaries, microcracks, etc. ),g ulations is the formation of multivariant polytwinned embryos represents an arbitrary point in the reciprocal space, e=g/g, g through homogeneous nucleation under large undercooling con- indicates complex conjugate, m=l for a proper MT [37] and ditions. Fig. I shows an example of the phase field simulation of m=2 for an improper MT[14], and an improper cubic-tetragonal MT in an elastically Bpg(e)=CKIE (P)e&(q)-e00(P)2 (e)ogr(q)er (5) single crystal with the SFTS being a pure shear [14. The where Cijkl are the elastic constants, e!(P)is the stress-free trans- strain energy, GEt, and the chemical driving force, Af(D,i.e formation strain(SFTS)of type p martensite variant, o(p)= 5=GE0/Af(T), where G is the typical shear modulus, Eo is the CiReR (p), and 2*(e)=Cike; e is the inverse of the Green's typical SFTs, and a/D) is the ditterence of the specific chemical free energy between the martensite and parent phases. This ratio function in the reciprocal space. f in Eq (4)represents the prin- was chosen as 0.056 to mimic a large undercooling(high driving ciple value of the integral that excludes a small volume in the force)condition. The three orientation variants were character reciprocal space.(21)/, at g=0(V is the total volume of the ized by three SOP fields and fi(ul, m2, ..,m)=f(n, n2, n3)in Eq (2). To visualize clearly a homogeneous nucleation event and 3. For a given MT, the symmetry invariants of the SOPs in configurations of critical or operational nuclei, only two orienta Landau free energy can be derived by the symmetry of tion variants were considered. They are represented in Fig. I by the co-existing phases [31-33] and the expansion coefficients the bright and dark shades, respectively. The parent phase is rep- of a Landau polynomial can be determined by fitting the Lan- resented by a gray background. The nine small squares presente dau free energy to thermodynamic databases or first principles in each micrograph are the consecutive equally spaced (010) calculations [ 39, 40](for recent reviews see 35, 36]). Ther re are cross sections of a three-dimensional (3D) cubic computational increasing efforts in calculating the gradient energy coefficients cell with 643 mesh points and a mesh size of 1.0nm. The initial for various types of interfaces from first principles [41, 42]. If condition is an"as-quenched"homogeneous cubic phase that is such data are not available for a particular system, the gradi- in a metastable state. Since no other crystal defects were con- ent energy coefficients can be determined by fitting interfacial sidered, the transformation developed through a homogeneous

Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 57 free energy of such a system is given by Fch = V ⎡ ⎣f (η1, η2,...,ηn) + 1 2 n p=1 βij(p)∇iηp∇jηp ⎤ ⎦d3r (3) where βij are the gradient energy coefficients. The first term, the specific free energy, depends only on local values of the field ηp(r) (where r is the spatial coordinate), while the sec￾ond term, the gradient energy, accounts for contributions from spatial variation of ηp(r). The interplay between the two deter￾mines the equilibrium profiles of ηp and hence the widths and energies of interfaces between different orientation variants of the martensitic phase and between the martensite and parent phases. To remove the structural incompatibility between adjacent finite elements of different values of SOP, one need to introduce additional inhomogeneous displacements providing the continu￾ity of the total displacement field (the crystal lattice coherency). By definition, this is an elastic displacement field that provides the elastic strain energy. According to the KS theory [1,6,7], the elastic strain energy for arbitrary structural non-uniformities characterized by ηp(r) in an elastically anisotropic crystal under the homogeneous modulus assumption is given by an explicit close form (4) where {ηp(r)}g is the Fourier transform of the function ηp(r) that characterizes an arbitrary structural non-uniformity of the pth type (in principle, the structural non-uniformities could be asso￾ciated with any types of extended crystal defects such as coherent precipitate, dislocations, grain boundaries, microcracks, etc.), g represents an arbitrary point in the reciprocal space, e = g/g, “*” indicates complex conjugate, m = 1 for a proper MT [37] and m = 2 for an improper MT [14], and Bpq(e) ≡ Cijklε0 ij(p)ε0 kl(q) − eiσ0 ij(p)Ωjk(e)σ0 kl(q)el (5) whereCijkl are the elastic constants, ε0 ij(p) is the stress-free trans￾formation strain (SFTS) of type p martensite variant, σ0 ij(p) ≡ Cijklε0 kl(p), and Ω−1 jk (e) = Cijkleiel is the inverse of the Green’s function in the reciprocal space. in Eq. (4) represents the prin￾ciple value of the integral that excludes a small volume in the reciprocal space, (2π) 3/V, at g =0(V is the total volume of the system). For a given MT, the symmetry invariants of the SOPs in the Landau free energy can be derived by the symmetry of the co-existing phases [31–33] and the expansion coefficients of a Landau polynomial can be determined by fitting the Lan￾dau free energy to thermodynamic databases or first principles calculations [39,40] (for recent reviews see [35,36]). There are increasing efforts in calculating the gradient energy coefficients for various types of interfaces from first principles [41,42]. If such data are not available for a particular system, the gradi￾ent energy coefficients can be determined by fitting interfacial energies to experimental or calculated data [43,44]. This process also defines the length scale of the numerical simulations. If the lattice correspondence between the parent and product phases, their lattice parameters and elastic constants are known, the elas￾tic strain energy (4) can be formulated in a rather straightforward fashion. Detailed examples can be found in [11]. The spatial-temporal evolution of the SOPs, which com￾pletely defines the microstructural evolution during a MT, is obtained by solving the Onsager-type semi-phenomenological kinetic equations of motion, which assume a linear dependence of the rate of evolution, ∂ηp/∂t, on the driving force, F/ηp, ∂ηp(r, t) ∂t = − N q=1 Lpq δF δηq(r, t) + ξp(r, t) (6) where F = Fch + Eel is the total free energy of the system, with Fch being the chemical free energy and Eel the elastic energy, Lpq are the kinetic coefficients and ξp is the Langevin random force term that describes thermal fluctuations [31,45–47]. The phase field method formulated above offers a self￾consistent treatment of the thermodynamics and kinetics of MTs involving arbitrary multivariant configuration and particle mor￾phology at mesoscopic length scales. The effect of long-range elastic interactions on the initiation and growth of martensite is taken into account automatically by including the elastic strain energy in the total free energy in Eq. (6). The models have been applied successfully to proper and improper MTs in single crys￾tals [14,37], polycrystalline alloys [15,48], multi-layer systems [49], and thin-films [50] and the numerical simulations have shed light on the micromechanisms underlying microstructural development during MTs. One of the interesting observations from the phase field sim￾ulations is the formation of multivariant polytwinned embryos through homogeneous nucleation under large undercooling con￾ditions. Fig. 1 shows an example of the phase field simulation of an improper cubic→tetragonal MT in an elastically isotropic single crystal with the SFTS being a pure shear [14]. The “strength” of the MT is characterized by the ratio of the typical strain energy, Gε2 0, and the chemical driving force, f(T), i.e., ζ = Gε2 0/f (T ), where G is the typical shear modulus, ε0 is the typical SFTS, and f(T) is the difference of the specific chemical free energy between the martensite and parent phases. This ratio was chosen as 0.056 to mimic a large undercooling (high driving force) condition. The three orientation variants were character￾ized by three SOP fields and f(η1, η2, ..., ηn) = f(η1, η2, η3) in Eq.(2). To visualize clearly a homogeneous nucleation event and configurations of critical or operational nuclei, only two orienta￾tion variants were considered. They are represented in Fig. 1 by the bright and dark shades, respectively. The parent phase is rep￾resented by a gray background. The nine small squares presented in each micrograph are the consecutive equally spaced (0 1 0) cross sections of a three-dimensional (3D) cubic computational cell with 643 mesh points and a mesh size of 1.0 nm. The initial condition is an “as-quenched” homogeneous cubic phase that is in a metastable state. Since no other crystal defects were con￾sidered, the transformation developed through a homogeneous

Y Wang, A G Khachaturyan/Materials Science and Engineering A 438-440(2006)55-63 A N Fig. 1. Microstructural development during a cubic-tetragonal martensitic transformation through homogeneous nucleation simulated by the phase field method 14). The nine small squires in each micrograph are the consecutive 2D cross sections of a 3D cube along the [0 1 0] axis. (A-D)Correspond to reduced time =2, 6. nucleation process simulated by the Langevin random force term three orientation variants form initially an internally twinned given in Eq. (6). No a priori assumptions about possible crit- martensitic platelet( Fig. 2 at [=2), the third orientation variant ical nucleus configurations were made. The simulation result nucleates spontaneously at the interfaces between the matrix and presented in Fig. 1(A)shows that the stochastic random noises the martensitic platelet(Fig. 2 at [=5). The growth of the third produce critical nuclei consisting of two twin-related domains. variant induces the outgrowth of one of the two variants in the These compound nuclei were formed by either a collective pro- existing martensitic platelet, forming two new internally twined cess or a correlated process assisted by the autocatalytic effect platelets, one on each side of the original martensitic platelet discussed in [51]. Such an internally twinned structure of the(Fig. 2 at t=7). Recurring of this process results in the herring embryos reduces considerably the strain energy [l](see also bone structure consisting of adjacent internally twinned plates Fig. 7 in Section 3). This allows us to assume safely that het -(Fig. 2 from [=9 to T= 13). The herringbone structure was also erogeneous nucleation on lattice defects should also produce predicted for the cubic-trigonal proper MT[15]. These pre compound nuclei of polytwinned structures dictions agree well with experimental observations. The growth process of the polytwinned nuclei(Fig. 1(B One of the most complicated MTs that have been studied by and(C)is highly anisotropic, with the alternating twin-related the phase field method is the cubic- trigonal proper MT in a domains mostly expanding in directions parallel to their 110 polycrystalline Au-Cd alloy [15]. The trigonal lattice of the 5? habit planes. The growing martensitic particles have basically a martensite in Au-Cd can be visualized as a stretched cubic lattice lenticular shape before they finally joined to form two intersect- in one of the body diagonal (i.e, [11 1]) directions. Four lattice ing internally twinned thin plates of the invariant plane habits correspondence variants are associated with the transformation When all three variants were considered, a herringbone struc- the phase field model, the spatial distribution of the four vari- ture developed by self-assembly of the three variants(rep- ants is characterized by four SOPs and the chemical free energy resented by different shades of gray in Fig. 2)during the is approximated by the fourth-order Landau expansion polyno- cubic-tetragonal MT. It was found that autocatalysis plays an mial presented in Eq (1). Eight randomly oriented grains were important role in the self-assembly process during growth which considered in the polycrystalline system(Fig 3). To present the leads to the herringbone structure. For example, when two of the SFTS in a global coordinate system, a rotation operation was

58 Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 Fig. 1. Microstructural development during a cubic→tetragonal martensitic transformation through homogeneous nucleation simulated by the phase field method [14]. The nine small squires in each micrograph are the consecutive 2D cross sections of a 3D cube along the [0 1 0] axis. (A–D) Correspond to reduced time = 2, 6, 10 and 20. nucleation process simulated by the Langevin random force term given in Eq. (6). No a priori assumptions about possible crit￾ical nucleus configurations were made. The simulation result presented in Fig. 1(A) shows that the stochastic random noises produce critical nuclei consisting of two twin-related domains. These compound nuclei were formed by either a collective pro￾cess or a correlated process assisted by the autocatalytic effect discussed in [51]. Such an internally twinned structure of the embryos reduces considerably the strain energy [1] (see also Fig. 7 in Section 3). This allows us to assume safely that het￾erogeneous nucleation on lattice defects should also produce compound nuclei of polytwinned structures. The growth process of the polytwinned nuclei (Fig. 1(B) and (C)) is highly anisotropic, with the alternating twin-related domains mostly expanding in directions parallel to their {110} habit planes. The growing martensitic particles have basically a lenticular shape before they finally joined to form two intersect￾ing internally twinned thin plates of the invariant plane habits (Fig. 1(D)). When all three variants were considered, a herringbone struc￾ture developed by self-assembly of the three variants (rep￾resented by different shades of gray in Fig. 2) during the cubic→tetragonal MT. It was found that autocatalysis plays an important role in the self-assembly process during growth which leads to the herringbone structure. For example, when two of the three orientation variants form initially an internally twinned martensitic platelet (Fig. 2 at τ = 2), the third orientation variant nucleates spontaneously at the interfaces between the matrix and the martensitic platelet (Fig. 2 at τ = 5). The growth of the third variant induces the outgrowth of one of the two variants in the existing martensitic platelet, forming two new internally twined platelets, one on each side of the original martensitic platelet (Fig. 2 at τ = 7). Recurring of this process results in the herring￾bone structure consisting of adjacent internally twinned plates (Fig. 2 from τ = 9 to τ = 13). The herringbone structure was also predicted for the cubic→trigonal proper MT [15]. These pre￾dictions agree well with experimental observations. One of the most complicated MTs that have been studied by the phase field method is the cubic→trigonal proper MT in a polycrystalline Au–Cd alloy [15]. The trigonal lattice of the ζ 2 martensite in Au–Cd can be visualized as a stretched cubic lattice in one of the body diagonal (i.e., [1 1 1]) directions. Four lattice correspondence variants are associated with the transformation, which correspond to the four 111 directions of the cube. In the phase field model, the spatial distribution of the four vari￾ants is characterized by four SOPs and the chemical free energy is approximated by the fourth-order Landau expansion polyno￾mial presented in Eq. (1). Eight randomly oriented grains were considered in the polycrystalline system (Fig. 3). To present the SFTS in a global coordinate system, a rotation operation was

Y Wang, A.G. Khachaturyan/ Materials Science and Engineering A 438-440 (2006)55-63 T 5 IT 自=最自 Fig. 2. Formation of herringbone structure through autocatalytic growth during a cubic- tetragonal martensitic transformation. The nine small squires in each micrograph are the consecutive 2D cross sections of a 3D cube along the [0 1 0] axis. t is reduced time. applied to the SftS derived for each grain within each grain but differs from one grain to another. Fig. 4 shows the microstructural evolution obtained under external uni- ey (r)=Rik()rj(r)eu(r). (7) axial stresses in a system of 1283 mesh points and a mesh size where Ri (r) is a 3 x 3 matrix that defines the orientation of the of 0.5 pm. The strength parameter of the MT, s, is chosen to grain in the global coordinate system. It has a constant value be 5.0, which corresponds to a small undercooling. The four orientation variants are represented by four different shades of gray in the figure(color online). The multivariant microstru tures observed in the polycrystalline system is found to be quite different from the one observed in a single crystal. Because of the elastic coupling between neighboring randomly oriented par tially transformed grains, the MT in the polycrystalline system did not go to completion and the multi-domain structure is sta- ble against further growth, which is contrary to the simulation results obtained in a single crystal where the mt went to com- pletion under exactly the same conditions. This difference is caused by the geometrical constraints imposed on the mt in a polycrystalline system. The stress-strain hysteresis correspond ing to the microstructures under different uniaxial stresses also shown in Fig. 4. The nucleation of new variants and the domain boundary movement is clearly reproduced in the simu- lation. This example demonstrates well the potential of the phase field method in predicting very complex strain accommodating assemblages of multiple orientation domains produced by mts in polycrystalline materials under applied stresses. 3. Microscopic phase field model of dislocation core structures and its applications to martensitic transformations Polycrystal structure of the parent phase with eight randomly oriented While the mesoscopic phase field model of MT has achieved remarkable success in predicting stain-accommodating packing

Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 59 Fig. 2. Formation of herringbone structure through autocatalytic growth during a cubic→tetragonal martensitic transformation. The nine small squires in each micrograph are the consecutive 2D cross sections of a 3D cube along the [0 1 0] axis. τ is reduced time. applied to the SFTS derived for each grain, ε 0,g ij (r) = Rik(r)Rjl(r)ε0 ij(r), (7) where Rij(r) is a 3 × 3 matrix that defines the orientation of the grain in the global coordinate system. It has a constant value Fig. 3. Polycrystal structure of the parent phase with eight randomly oriented grains in the computational volume. within each grain but differs from one grain to another. Fig. 4 shows the microstructural evolution obtained under external uni￾axial stresses in a system of 1283 mesh points and a mesh size of 0.5m. The strength parameter of the MT, ζ, is chosen to be 5.0, which corresponds to a small undercooling. The four orientation variants are represented by four different shades of gray in the figure (color online). The multivariant microstruc￾tures observed in the polycrystalline system is found to be quite different from the one observed in a single crystal. Because of the elastic coupling between neighboring randomly oriented par￾tially transformed grains, the MT in the polycrystalline system did not go to completion and the multi-domain structure is sta￾ble against further growth, which is contrary to the simulation results obtained in a single crystal where the MT went to com￾pletion under exactly the same conditions. This difference is caused by the geometrical constraints imposed on the MT in a polycrystalline system. The stress–strain hysteresis correspond￾ing to the microstructures under different uniaxial stresses is also shown in Fig. 4. The nucleation of new variants and the domain boundary movement is clearly reproduced in the simu￾lation. This example demonstrates well the potential of the phase field method in predicting very complex strain accommodating assemblages of multiple orientation domains produced by MTs in polycrystalline materials under applied stresses. 3. Microscopic phase field model of dislocation core structures and its applications to martensitic transformations While the mesoscopic phase field model of MT has achieved remarkable success in predicting stain-accommodating packing