J E Nie, B C Muddle/ Materials Science and Engineering A 438-440(2006)343-345 Fig. 1.(a) Transmission electron micrograph showing martensite plates of the monoclinic phase in the matrix of the tetragonal phase in a CeO2-stabilised ZrO2 loy, and (b)corresponding SAED pattern from the martensite plates labelled 1 and 2 and surrounding matrix. Electron beam is parallel to [0101//[001].The labelled reflections of the monoclinic phase in(b)refer to those from plate 1 in(a). The habit plane normal of plate 1 is indicated by the dash line in(b). Careful inspection of monoclinic plates reveals the exis- [001]t, if any, is accommodated plastically by dislocation loops, ce of another orientation relationship that can be described then the tetragonal to monoclinic transformation can be simpli by(002)m//(020)t, [0 1 0]m//00 1] and [100)m//[100J [6, 1 ]. fied into a two-dimensional case, as illustrated in Fig.2.For This second orientation relationship differs from the first one by this structural transformation, the(020) plane is sheared into a rotation of approximately 8.5 about the [01)m//[001] axis. (002)m, and the(200)t plane is expanded into(200)m.There The habit plane of monoclinic plates that are associated with the fore, s=tan A=tan(Pm-900), nx=d(oo2)m/(d(o20), sin Pm) second orientation relationship is about 5 from(020)t towards cm/at, ny= d(200m/d(200)=(am/at)sin Pm, where s is the (200)t magnitude of shear strain, nx is the dilatation of(02 O) in[01 OJt ny is the dilatation of (200)t in [100]t, and at, am, Cm and 3. Theoretical analysis are the lattice parameter of the tetragonal and monoclinic lat- tices, respectively. Substituting these parameters into following Inspection of the tetragonal and monoclinic lattices sug- equations [8] gests that the monoclinic lattice can be generated from the tetragonal lattice by shearing in the [0 10]t direction on(200) (1+n y)(nx+ny)+sn yV /52ny-(n2-1)(02-D and dilatation in the [0 101,[100]t and [001]t directions, COsp- (nx +nv)2+52n2 respectively(Fig. 2(a). Assuming that the misfit strain alon y t[1001t y si+(nx-1)-+ 2n(1-cos p+ ssin and assuming that at=0.5132 nm, Ct=0.5228 nm, am=0.5193nm,bm=0.5204nm,cm=0.5362nm,and [010 Bm=98.8%[1], the rotation angle, between the two lat- tices is predicted to be 0 and thus the monoclinic lattice is (200)t (200)m x expected to develop a rational orientation relationship with out of plane normal (001lt ut of plane normal [010)m he tetragonal lattice, i.e. (200hm/(200)t,[010Jm/00lt and [00 1]m/n[o i OJt. The theory also predicts that the habit y4(100 ▲00m (02 0)t and (002)m planes, is 16.15 clockwise from(200) i.e. B=(90-a)=16. 15, where a is the angle between the habit plane and(02 O)(Fig 3). These predictions are thus in excellent agreement with experimental observations(Fig. 1). It (200)m is to be emphasised that the predicted orientation relationship and the interface orientation in particular, are a function of the ratio of lattice parameters of the monoclinic and tetragonal lattices. The angle =0 only when n,=(am/at)sin Bm=1 out of plane normal [010]m Since(200)m and (200) planes are precisely parallel to each grams showing two possible structural transformat other and the macroscopic interface is inclined to them, the tetragonal to monoclinic lattices. The viewing direction is[0 10lm//1o macroscopic interface may adopt a form of regularly spaced344 J.F. Nie, B.C. Muddle / Materials Science and Engineering A 438–440 (2006) 343–345 Fig. 1. (a) Transmission electron micrograph showing martensite plates of the monoclinic phase in the matrix of the tetragonal phase in a CeO2-stabilised ZrO2 alloy, and (b) corresponding SAED pattern from the martensite plates labelled 1 and 2 and surrounding matrix. Electron beam is parallel to [0 1 0] ¯ m//[0 0 1]t. The labelled reflections of the monoclinic phase in (b) refer to those from plate 1 in (a). The habit plane normal of plate 1 is indicated by the dash line in (b). Careful inspection of monoclinic plates reveals the exis￾tence of another orientation relationship that can be described by (0 0 2)m//(0 2 0)t, [0 1 0] ¯ m//[0 0 1]t and [1 0 0]m//[1 0 0]t [6,1]. This second orientation relationship differs from the first one by a rotation of approximately 8.5◦ about the [0 1 0] ¯ m//[0 0 1]t axis. The habit plane of monoclinic plates that are associated with the second orientation relationship is about 5◦ from (0 2 0)t towards (2 0 0)t. 3. Theoretical analysis Inspection of the tetragonal and monoclinic lattices sug￾gests that the monoclinic lattice can be generated from the tetragonal lattice by shearing in the [0 1 0] ¯ t direction on (2 0 0)t and dilatation in the [0 1 0] ¯ t, [1 0 0]t and [0 0 1]t directions, respectively (Fig. 2(a)). Assuming that the misfit strain along Fig. 2. Schematic diagrams showing two possible structural transformations from tetragonal to monoclinic lattices. The viewing direction is [0 1 0] ¯ m//[0 0 1]t. [0 0 1]t, if any, is accommodated plastically by dislocation loops, then the tetragonal to monoclinic transformation can be simpli- fied into a two-dimensional case, as illustrated in Fig. 2. For this structural transformation, the (0 2 0)t plane is sheared into (0 0 2)m, and the (2 0 0)t plane is expanded into (2 0 0)m. There￾fore, s = tan θ = tan(βm − 90◦), ηx = d(0 0 2)m /(d(0 2 0)t sin βm) = cm/at, ηy = d(2 0 0)m /d(2 0 0)t = (am/at) sin βm, where s is the magnitude of shear strain, ηx is the dilatation of (0 2 0)t in [0 1 0] ¯ t, ηy is the dilatation of (2 0 0)t in [1 0 0]t, and at, am, cm and βm are the lattice parameter of the tetragonal and monoclinic lat￾tices, respectively. Substituting these parameters into following equations [8]: cos ϕ = (1 + ηxηy)(ηx + ηy) + sηy s2η2 y − (η2 x − 1)(η2 y − 1) (ηx + ηy) 2 + s2η2 y , (1) sin α = s cos ϕ + sin ϕ s2 + (ηx − 1)2 + 2ηx(1 − cos ϕ + ssin ϕ) , (2) and assuming that at = 0.5132 nm, ct = 0.5228 nm, am = 0.5193 nm, bm = 0.5204 nm, cm = 0.5362 nm, and βm = 98.8◦ [1], the rotation angle, ϕ, between the two lat￾tices is predicted to be 0◦ and thus the monoclinic lattice is expected to develop a rational orientation relationship with the tetragonal lattice, i.e. (2 0 0)m//(2 0 0)t, [0 1 0] ¯ m//[0 0 1]t and [0 0 1]¯ m//[0 1 0] ¯ t. The theory also predicts that the habit plane, the Moire plane defined by the intersection of the sets of ´ (0 2 0)t and (0 0 2)m planes, is ∼16.15◦ clockwise from (2 0 0)t, i.e. β = (90◦ − α) = 16.15◦, where α is the angle between the habit plane and (0 2 0)t (Fig. 3). These predictions are thus in excellent agreement with experimental observations (Fig. 1). It is to be emphasised that the predicted orientation relationship, and the interface orientation in particular, are a function of the ratio of lattice parameters of the monoclinic and tetragonal lattices. The angle ϕ = 0 only when ηy = (am/at)sin βm = 1. Since (2 0 0)m and (2 0 0)t planes are precisely parallel to each other and the macroscopic interface is inclined to them, the macroscopic interface may adopt a form of regularly spaced