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 was58 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