《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_Defect modelling of martensitic interfaces in plate martensite

MATERIALS HIENGE& ENGIEERING ELSEVIER Materials Science and Engineering A 481-482(2008)404-408 www.elsevier.com/locate/msea Defect modelling of martensitic interfaces in plate martensite Institute of Materials Science and Technology, School of Mechanical Engineering, South China University of Technology guangzhou 510640, China Department of Engineering, University of Liverpool. Brownlow Hill, Liverpool, 169 3BX, UK Received 17 May 2006: received in revised form 13 December 2006: accepted 22 December 2006 A physical model of martensitic transformations describing the structure of the parent-martensite interface and the transformation mechanism is presented. This approach is used here to model martensitic transformations in lath and plate martensite. The habit plane comprises coherent (1 1 1)y/(o l I)a terraces where the coherency strains are accommodated by a network of near screw dislocations, originating in the martensite phase, and disconnections(transformation dislocations), also in near screw orientation. The lateral motions of the disconnections effecting the transformation shear; moreover, the transformation process is explicitly shown to be diffusionless Experimental observations from the literature of the dislocation and disconnection arrays, habit plane and orientation relationship are in good agreement with the model 2007 Elsevier B V. All rights reserved Keywords: Disconnection; Interfacial defects; Interface structure; Martensite crystallography 1. ntroduction formations in lath and plate martensite. The principles of the TM are set out in the next section, and subsequently applied The phenomenological theory of martensite crystallography to ferrous alloys. Finally, experimental observations of transfor- (PTMC) was developed more than 50 years ago [1, 2] and is the mations in ferrous systems previously reported in the literature cornerstone of our present understanding of martensitic crys- are compared with the present modelling tallography. This theory is a quantitative algorithm for finding the invariant habit plane of a shape-transformation 3, 4], and embodies the kinematic compatibility criterion for interfaces 2. Topological model of martensitic interfaces between two semi-infinite continua having no long-range dis- placement fields [5]. while a considerable body of experimental A fundamental step in the TM of a transformation is to observations is consistent with the PTMc 3-5], other data are tify a candidate interface between the phases that exhibits at variance with its predictions, for example see Ref. [6]for a coherency; this is referred to as a terrace plane. Feasible terrace review of ferrous alloys. An alternative approach has been devel- planes in stiff engineering materials are expected to have rela- oped recently in terms of interfacial defects [7, 8]and is refered tively modest coherency strains. Once a terrace plane has been to here as the"topological model"(TM). The TM is a descrip-(b, 0)and glissile disconnections(b, h)that can arise therein tion of the structure of the parent-martensite interface and the line-defects therein; the transformation proceeds by movement can be determined using the theory of interfacial defects [13] of transformation dislocations, or disconnections [9]as they are The coherency strains arising at a terrace plane must be relieved known, across the interface. This defect motion produces the by arrays of interfacial defects. An array of appropriately ori transformation shear and can be shown explicitly to be diffu- ented and spaced glissile disconnections can be one of these sets, ionless [10]. To date, the TM has been applied successfully to and synchronous motion of this set can thereby provide the dual transformations in TiMo alloy [7], ZrO2[11] and PuGa[l2]: the function of effecting the transformation and partially relieving objective of the present work is to report progress with trans- the coherency strain. The second set of defects necessary for complete misfit relief need not be glissile; however, these will intersect the disconnection array, and these intersections must Corresponding author. Tel: +86 20 2223 6396: fax: +86 20 8711 2762. not impede the glissile motion of the former. In other word E-mailaddress:maxiao@@scut.edu.cn(X.Ma) the intersections must themselves be glissile[14, 15 ]: it has been 0921-5093/S-see front matter O 2007 Elsevier B v. All rights reserved doi:10.1016/msea.2006.2.196

Materials Science and Engineering A 481–482 (2008) 404–408 Defect modelling of martensitic interfaces in plate martensite X. Ma a,∗, R.C. Pond b a Institute of Materials Science and Technology, School of Mechanical Engineering, South China University of Technology, Guangzhou 510640, China b Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool, L69 3BX, UK Received 17 May 2006; received in revised form 13 December 2006; accepted 22 December 2006 Abstract A physical model of martensitic transformations describing the structure of the parent–martensite interface and the transformation mechanism is presented. This approach is used here to model martensitic transformations in lath and plate martensite. The habit plane comprises coherent (1 1 1)//(0 1 1) terraces where the coherency strains are accommodated by a network of near screw dislocations, originating in the martensite phase, and disconnections (transformation dislocations), also in near screw orientation. The lateral motions of the disconnections effecting the transformation shear; moreover, the transformation process is explicitly shown to be diffusionless. Experimental observations from the literature of the dislocation and disconnection arrays, habit plane and orientation relationship are in good agreement with the model. © 2007 Elsevier B.V. All rights reserved. Keywords: Disconnection; Interfacial defects; Interface structure; Martensite crystallography 1. Introduction The phenomenological theory of martensite crystallography (PTMC) was developed more than 50 years ago [1,2] and is the cornerstone of our present understanding of martensitic crys￾tallography. This theory is a quantitative algorithm for finding the invariant habit plane of a shape-transformation [3,4], and embodies the kinematic compatibility criterion for interfaces between two semi-infinite continua having no long-range dis￾placement fields[5]. While a considerable body of experimental observations is consistent with the PTMC [3–5], other data are at variance with its predictions, for example see Ref. [6] for a review of ferrous alloys. An alternative approach has been devel￾oped recently in terms of interfacial defects [7,8] and is refered to here as the “topological model” (TM). The TM is a descrip￾tion of the structure of the parent–martensite interface and the line-defects therein; the transformation proceeds by movement of transformation dislocations, or disconnections [9] as they are known, across the interface. This defect motion produces the transformation shear and can be shown explicitly to be diffu￾sionless [10]. To date, the TM has been applied successfully to transformations in TiMo alloy [7], ZrO2 [11] and PuGa [12]; the objective of the present work is to report progress with trans- ∗ Corresponding author. Tel.: +86 20 2223 6396; fax: +86 20 8711 2762. E-mail address: maxiao@scut.edu.cn (X. Ma). formations in lath and plate martensite. The principles of the TM are set out in the next section, and subsequently applied to ferrous alloys. Finally, experimental observations of transfor￾mations in ferrous systems previously reported in the literature are compared with the present modelling. 2. Topological model of martensitic interfaces A fundamental step in the TM of a transformation is to identify a candidate interface between the phases that exhibits coherency; this is referred to as a terrace plane. Feasible terrace planes in stiff engineering materials are expected to have rela￾tively modest coherency strains. Once a terrace plane has been identified, the set of lattice invariant deformation (LID) defects (b, 0) and glissile disconnections (b, h) that can arise therein can be determined using the theory of interfacial defects [13]. The coherency strains arising at a terrace plane must be relieved by arrays of interfacial defects. An array of appropriately ori￾ented and spaced glissile disconnections can be one of these sets, and synchronous motion of this set can thereby provide the dual function of effecting the transformation and partially relieving the coherency strain. The second set of defects necessary for complete misfit relief need not be glissile; however, these will intersect the disconnection array, and these intersections must not impede the glissile motion of the former. In other words, the intersections must themselves be glissile [14,15]; it has been 0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.12.196

X Ma, R.C. Pond/ Materials Science and Engineering A 481-482(2008)404-408 23. Stage 3: interface structure and transformation crystallography b,0 The first step in this final stage is to determine the line direc- tions and spacings of the arrays of disconnections and LID that accommodate the coherency strain. These are related to terrace plane the coherency strains by the Frank-Bilby equation [16]; if the strains defined in Eq. (1)are elastic and are relieved plastically by defect arrays, we have LiD glide or twinning plane -=nEcK Fig. 1. Schematic illustration of a parent-martensite interface showing the ter- race segments and defect arays Coherently strained terraces are reticulated by where B is the total Burgers vector crossed by a probe vector v rrays of disconnections (b, h)and crystal slip or twinning dislocations(b, 0) lying in the interface from the(lower)martensite crystal. 3. Interface structure in plate martensite shown that, in general, this only arises if the second set is com- prised of LId able to reach the interface by gliding through in the 3. 1. Stage 1: reference structures e interface is envis aged as coherent terraces reticulated by arrays of disconnections A convenient choice of natural reference structure for y-FCC and Lid(slip or twinning), as depicted schematically in Fig. 1. and a-BCC crystals is the Nishyama-Wassermann(NW)OR, as Because of the step character of the disconnections, the overall depicted schematically in Fig. 2. In this OR the closest-packed interface plane, or habit plane, deviates from the terrace plane. planes of the two phases are parallel, i.e. (11 1)//(01 1)a; these At equilibrium, the spacing of the disconnections, d, and LID, are potential terrace planes and the axes of the terrace coordi &The three stages for determining a misfit-relieved glissile aa =0.2870nm, leading to misfit parallel to x, yad onma cty d, and their line directions, ED and 5, must be adjusted until nate frame are indicated in the figure. The lattice para the misfit is fully relieved along the habit plane bit plane are briefly described below The misfit parallel tox and y is now removed by the coherency strain, thereby forming the coherent reference, and the corre- 2.1. Stage 1; reference states sponding dichromatic pattern is depicted in Fig 3. The strain Initially, the two phases are juxtaposed with a chosen ori- entation relationship(OR) and exhibiting their natural lattice -0.125400 parameters-the natural reference state. Next, the parent and 00.07720 martensite crystals are strained into coherence on the terrace plane-the coherent reference state; the necessary deformations of the parent and martensite crystals are represented by the matri- ces ePn and Mn, respectively. The total strain in the terrace plane uo loofa is then defined by nEc=(cPn-cM) Alternative natural reference states can be chosen where the two crystals are, respectively rotated by the angles to/2 about the normal to the terrace plane, represented by the matrices r 口■△ and R, giving E8={(R+cPa)--(R-Mn)- 口 2.2. Stage 2; interfacial defects 口△口△○口 The set of lid and disconnections that can arise in the coher- By layers +1, -2 △ y layers+2,-1 ent reference state are determined using the topological theory ●- O layer0 ■ layers±l of interfacial defects [13]. From this set, the subset of disconnec tions(b, h)that are glissile in the terrace plane can be determin and also the slip or twinning LID systems that do not form sessile and y crystals exhibiting the NW OR viewed perpendicular to the lattice sites are depicted in white, y, and black, a, forming a dichromatic intersections with the disconnections

X. Ma, R.C. Pond / Materials Science and Engineering A 481–482 (2008) 404–408 405 Fig. 1. Schematic illustration of a parent–martensite interface showing the ter￾race segments and defect arrays. Coherently strained terraces are reticulated by arrays of disconnections (b, h) and crystal slip or twinning dislocations (b, 0) from the (lower) martensite crystal. shown that, in general, this only arises if the second set is com￾prised of LID able to reach the interface by gliding through in the martensite phase. Thus, the parent–martensite interface is envis￾aged as coherent terraces reticulated by arrays of disconnections and LID (slip or twinning), as depicted schematically in Fig. 1. Because of the step character of the disconnections, the overall interface plane, or habit plane, deviates from the terrace plane. At equilibrium, the spacing of the disconnections, dD, and LID, dL, and their line directions, D and L, must be adjusted until the misfit is fully relieved along the habit plane. The three stages for determining a misfit-relieved glissile habit plane are briefly described below. 2.1. Stage 1; reference states Initially, the two phases are juxtaposed with a chosen ori￾entation relationship (OR) and exhibiting their natural lattice parameters—the natural reference state. Next, the parent and martensite crystals are strained into coherence on the terrace plane—the coherent reference state; the necessary deformations of the parent and martensite crystals are represented by the matri￾ces cPn and cMn, respectively. The total strain in the terrace plane is then defined by nEc = (cP−1 n − cM−1 n ). (1) Alternative natural reference states can be chosen where the two crystals are, respectively rotated by the angles ±ω/2 about the normal to the terrace plane, represented by the matrices R+ and R−, giving nER c = {(R+cPn) −1 − (R−cMn) −1 }. (2) 2.2. Stage 2; interfacial defects The set of LID and disconnections that can arise in the coher￾ent reference state are determined using the topological theory of interfacial defects[13]. From this set, the subset of disconnec￾tions (b, h) that are glissile in the terrace plane can be determined, and also the slip or twinning LID systems that do not form sessile intersections with the disconnections. 2.3. Stage 3; interface structure and transformation crystallography The first step in this final stage is to determine the line direc￾tions and spacings of the arrays of disconnections and LID that accommodate the coherency strain. These are related to the coherency strains by the Frank–Bilby equation [16]; if the strains defined in Eq. (1) are elastic and are relieved plastically by defect arrays, we have −B = nEcv, (3) where B is the total Burgers vector crossed by a probe vector v lying in the interface. 3. Interface structure in plate martensite 3.1. Stage 1: reference structures A convenient choice of natural reference structure for -FCC and -BCC crystals is the Nishyama–Wassermann (NW) OR, as depicted schematically in Fig. 2. In this OR the closest-packed planes of the two phases are parallel, i.e. (1 1 1)//(0 1 1); these are potential terrace planes and the axes of the terrace coordi￾nate frame are indicated in the figure. The lattice parameters of the cubic crystals used in this work are a = 0.3580 nm and a = 0.2870 nm, leading to misfit parallel to x, y and z. The misfit parallel to x and y is now removed by the coherency strain, thereby forming the coherent reference, and the corre￾sponding dichromatic pattern is depicted in Fig. 3. The strain is given by nEc = ⎛ ⎜⎝ −0.1254 0 0 0 0.0772 0 0 00 ⎞ ⎟⎠ . (4) Fig. 2. Schematic illustration showing the natural reference state formed by and crystals exhibiting the NW OR viewed perpendicular to (1 1 1)/(0 1 1); the lattice sites are depicted in white, , and black, , forming a dichromatic pattern.

X Ma, R.C. Pond Materials Science and Engineering A 481-482(2008)404-408 l0/100l4 口■△。口■ [101l 口■△C 口■△D口■ l2h[01]a t(a) t(?) Schematic illustration of the coherent dichromatic pattern; the b of ca ions, joining sites of opposite colour, and LID, joining sites of ne colour are also shown 3. 2. Stage 2; interfacial defects The elastic strain defined above is relieved plastically by incorporation into the interface of admissible interfacial t(a)t(r) defects. Their Burgers vectors and step heights are conveniently illustrated in the coherent dichromatic pattern, Fig 3. Slip dislo- cations of the martensite have b corresponding to black-to-black translation vectors, t(a), for example b= 1/2[11 l ]a; this dis- location do not exhibit step character when embedded in the terrace plane Disconnections have b corresponding to black- Fig 4 Schematic illustration of a(a)b=l-I and(b)b=23/-2disconnections in to-white vectors, t(y)-t(o), in a dichromatic pattern, and two the coherent reference interface examples are depicted in Fig 3 and their parameters are listed in Table 1. One is designated bD-i to indicate that its step goes For this stage, the small components b: of the disconnections are downwards into the(lower)martensite crystal by one(111)y suppressed temporarily Solutions can then be found using the and one(01 1)a plane, and the other, having a step twice this Frank-Bilby equation either as in expression (3) for the NW Or, height is b a schematic illustration of these discon- or alternative natural reference states with finite values of o. The nections, viewed along [1o1Jy/ 1]a, is shown in Fig. 4. spacing of the disconnections, d, and LID, dl,and their line their /-l and b=2/-2 are glissile in the terrace directions, sD and EL, are treated as variable quantities in order plane despite their components bz. Intersections of disconnec tions b=l/- and b=2)-2 with slip dislocations are glissile if the two arrays are present it is possible to choose a probe vector b of the latter is not perpendicular to the terrace [15] as is the vo parallel (or anti-parallel)to sd that intersects only LID,and case for the crystal dislocation denoted b-in Fig 3 and Table 1. similarly a vector VL parallel (or anti-parallel)to 5, cutting only disconnections. Thus in the former case. the burgers vector cut 3.3. Stage 3: misfit relief per unit length, B, can be expressed as The first step is to find a network comprising one array of BL- bl sin(OD-8L) LID, and a second array of disconnections, i.e. defects b-1/-I and in the latter as or b-2/-2, that accommodates the misfit on the terrace plane Table 1 Topological parameters of interfacial defects where AD and el are the angles subtended by ED and 5lfrom br (nm) b,(nm) b(nm) h t() t(a) the positive x-axis. Eq. ( 3) can now [1I gniol, Julii v=-nE- Bl, 0135-0070-00821[2 11y [0111 and similarly for vl. Thus, both the directions and mag a The"overlap"step height, h, is defined as the smaller of the terrace plane of vD and v- can be determined, hence giving the unit vectors E spacings, i.e. (0 11)a in the present case. and E(from which od and e- are found) and the separations d

406 X. Ma, R.C. Pond / Materials Science and Engineering A 481–482 (2008) 404–408 Fig. 3. Schematic illustration of the coherent dichromatic pattern; the b of can￾didate disconnections, joining sites of opposite colour, and LID, joining sites of the same colour, are also shown. 3.2. Stage 2; interfacial defects The elastic strain defined above is relieved plastically by incorporation into the interface of admissible interfacial defects. Their Burgers vectors and step heights are conveniently illustrated in the coherent dichromatic pattern, Fig. 3. Slip dislo￾cations of the martensite have b corresponding to black-to-black translation vectors, t(), for example bL = 1/2[1 1 1] ¯ ; this dis￾location do not exhibit step character when embedded in the terrace plane. Disconnections have b corresponding to black￾to-white vectors, t() − t(), in a dichromatic pattern, and two examples are depicted in Fig. 3 and their parameters are listed in Table 1. One is designated bD −1/−1 to indicate that its step goes downwards into the (lower) martensite crystal by one (1 1 1) and one (0 1 1) plane, and the other, having a step twice this height, is bD −2/−2. A schematic illustration of these discon￾nections, viewed along [1 0 1] ¯ /[1¯ 1 1] ¯ , is shown in Fig. 4. Disconnections bD −1/−1 and bD −2/−2 are glissile in the terrace plane despite their components bz. Intersections of disconnec￾tions bD −1/−1 and bD −2/−2 with slip dislocations are glissile if the b of the latter is not perpendicular to the terrace [15], as is the case for the crystal dislocation denoted bL in Fig. 3 and Table 1. 3.3. Stage 3; misfit relief The first step is to find a network comprising one array of LID, and a second array of disconnections, i.e. defects bD −1/−1 or bD −2/−2, that accommodates the misfit on the terrace plane. Table 1 Topological parameters of interfacial defects bx (nm) by (nm) bz (nm) ha t() t() bL 0.135 −0.211 0 0 – 1 2 [1 1 1] ¯ bD −1/−1 −0.135 −0.141 −0.004 −1 1 2 [1¯ 1 0] ¯ 1 2 [1 1¯ 1]¯ bD −2/−2 −0.135 −0.070 −0.008 −2 1 2 [2¯ 1¯ 1]¯ [0 1¯ 1]¯ a The “overlap” step height, h, is defined as the smaller of the terrace plane spacings, i.e. (0 1 1) in the present case. Fig. 4. Schematic illustration of a (a) bD −1/−1 and (b) bD −2/−2 disconnections in the coherent reference interface. For this stage, the small components bz of the disconnections are suppressed temporarily. Solutions can then be found using the Frank–Bilby equation either as in expression (3) for the NW OR, or alternative natural reference states with finite values of ω. The spacing of the disconnections, dD, and LID, dL, and their line directions, D and L, are treated as variable quantities in order to find solutions. A practicable procedure is as follows; since two arrays are present it is possible to choose a probe vector vD parallel (or anti-parallel) to D that intersects only LID, and similarly a vector vL parallel (or anti-parallel) to L, cutting only disconnections. Thus, in the former case, the Burgers vector cut per unit length, BL, can be expressed as BL = bL sin(θD − θL) dL (5a) and in the latter as BD = bp/q sin(θD − θL) dD , (5b) where θD and θL are the angles subtended by D and L from the positive x-axis. Eq. (3) can now be expressed as vD = −nE−1 c BL, (6) and similarly for vL. Thus, both the directions and magnitudes of vD and vL can be determined, hence giving the unit vectors D and L (from which θD and θL are found) and the separations dD

X Ma, R.C. Pond/ Materials Science and Engineering A 481-482(2008)404-408 Misfit-relieving network parameters for b- LID and b2,_,disconnections 6(°) d l(nm) p(°) dD(nm) 0.5 1.25 35.42 3.37 1179 27.71 4.24 119.71 121.2 12264 5.96 123.89 [lll]a 4.0 12503 0.86 [01]y would require the bz components of the disconnections to be re-instated and the defect content on the habit plane determined using probe vectors in that plane. At equilibrium, the misfit along e=1225 this plane must be accommodated by the defect network, and fur- ther adjustments of defect line directions and separations may be needed. Defect content with resultant component of B per- pendicular to the final habit plane does not affect misfit relief; it 70.39 acts as a low-angle tilt boundary thereby introducing an ancil lary change in the OR. This refinement procedure for ferrous 3771 alloys will be reported fully in a separate paper [17] llll a 4. Discussion ig. 5. Schematic illustration of misfit-relieving networks ofbL LID andbD For a transformation mechanism to be feasible. the free deviates from this by a=2.5 towards the Ks Ol energy of the system containing an interface with struc- ture as described above must be favourable. In the case of the ferrous alloys studied here, the provisional solutions for and d. a schematic illustration of a misfit-relieving network of LID/disconnection networks outlined show that a range of possi- bl LID and b=1/-1 disconnections for the NW Or is shown in bilities corresponding to different ORs satisfies the geometrical Fig5(a), and a second example where the OR deviates from this constraints of the Frank-Bilby equation. Finding an optimal by a=2.5 towards the Kurdjumov-Sachs(KS)OR is depicted solution for a given material would therefore require consid in Fig. 5(b) Network parameters, 0L, d, eD and d for a rang eration of the networks energies which must be added to the of deviations o are listed in table 2 for b and b =1/-ldefects, energy of the coherent terraces [18]. Tables 2 and 3 and Fig.5 and in Table 3 for b- and b,, defects show that the LID separations, dl, increase whereas the dis- he first stage of refinement for the solutions described above connection separations, d, decrease with deviation o from the is to introduce the step character of the disconnections and hence NW towards the KS OR. Also, the line-directions of the defects define the provisional habit plane. The normal to this plane is change over this range of OR. All of these factors influence the determined by rotating the normal to the terrace plane by the free-energy through the elastic strain fields of the defects and angle y=tan-(h/dd)about an axis parallel to s and the values their core energies [19] so obtained are included in Tables 2 and 3. Further refinement Although the habit plane structures reported here are not fully refined, the provisional structures show good agreement with experimental observations in the literature. For example, Sand- Misfit-relieving network parameters for b- LID and b=1/-1 disconnections vik and Wayman [20] studied lath martensite in an Fe-Ni-Mn 60) d(nm) 8o) d (nm) wco) alloy using transmission electron microscopy(TEM).They observed an array of 1/2[11 1]a Lid dislocations with dlin 0 59.62 111.49 6.87 the range 2.6-6.3 nm, sL varying between 10 and 150from 61.04 251 113.92 screw orientation in a habit plane with y=9.45, and o ranging 8.19 between 0.16 and 3. 16.. This observation resembles closely 8.64 the array of b and b=1/-I defects predicted here(Table 2)for a=2.5, namely: d=3.77 nm, EL oriented 12.990from orientation and y=9.10. We note that in the present work 5L 10.52 was treated as a variable, whereas, in our earlier work on TiMo [7], it was taken as being the intersection of the active slip or

X. Ma, R.C. Pond / Materials Science and Engineering A 481–482 (2008) 404–408 407 Fig. 5. Schematic illustration of misfit-relieving networks of bL LID and bD −1/−1 disconnections (a) for the NW OR and (b) a second example where the OR deviates from this by ω = 2.5◦ towards the KS OR. and dL. A schematic illustration of a misfit-relieving network of bL LID and bD −1/−1 disconnections for the NW OR is shown in Fig. 5(a), and a second example where the OR deviates from this by ω = 2.5◦ towards the Kurdjumov–Sachs (KS) OR is depicted in Fig. 5(b). Network parameters, θL, dL, θD and dD for a range of deviations ω are listed in Table 2 for bL and bD −1/−1 defects, and in Table 3 for bL and bD −2/−2 defects. The first stage of refinement for the solutions described above is to introduce the step character of the disconnections and hence define the provisional habit plane. The normal to this plane is determined by rotating the normal to the terrace plane by the angle ψ = tan−1(h/dD) about an axis parallel to D and the values so obtained are included in Tables 2 and 3. Further refinement Table 2 Misfit-relieving network parameters for bL LID and bD −1/−1 disconnections ω ( ◦) θL ( ◦) dL (nm) θD ( ◦) dD (nm) ψ ( ◦) 0 59.62 2.31 111.49 1.66 6.87 0.5 61.04 2.51 113.92 1.56 7.30 1.0 62.73 2.74 116.08 1.47 7.74 1.5 64.77 3.02 117.99 1.39 8.19 2.0 67.27 3.35 119.71 1.32 8.64 2.5 70.39 3.77 121.25 1.25 9.10 3.0 74.37 4.28 122.64 1.19 9.57 3.5 79.55 4.90 123.89 1.13 10.05 4.0 86.45 5.64 125.03 1.08 10.52 Table 3 Misfit-relieving network parameters for bL LID and bD −2/−2 disconnections ω ( ◦) θL ( ◦) dL (nm) θD ( ◦) dD (nm) ψ ( ◦) 0 40.46 2.77 111.49 1.33 16.77 0.5 38.18 3.05 113.92 1.25 17.77 1.0 35.42 3.37 116.08 1.18 18.77 1.5 32.01 3.76 117.99 1.11 19.80 2.0 27.71 4.24 119.71 1.05 20.82 2.5 22.23 4.80 121.25 1.00 21.84 3.0 15.14 5.48 122.64 0.95 22.88 3.5 5.96 6.23 123.89 0.90 23.89 4.0 −5.67 6.91 125.03 0.86 24.89 would require the bz components of the disconnections to be re-instated and the defect content on the habit plane determined using probe vectors in that plane. At equilibrium, the misfit along this plane must be accommodated by the defect network, and fur￾ther adjustments of defect line directions and separations may be needed. Defect content with resultant component of B per￾pendicular to the final habit plane does not affect misfit relief; it acts as a low-angle tilt boundary thereby introducing an ancil￾lary change in the OR. This refinement procedure for ferrous alloys will be reported fully in a separate paper [17]. 4. Discussion For a transformation mechanism to be feasible, the free energy of the system containing an interface with struc￾ture as described above must be favourable. In the case of the ferrous alloys studied here, the provisional solutions for LID/disconnection networks outlined show that a range of possi￾bilities corresponding to different ORs satisfies the geometrical constraints of the Frank–Bilby equation. Finding an optimal solution for a given material would therefore require consid￾eration of the networks’ energies, which must be added to the energy of the coherent terraces [18]. Tables 2 and 3 and Fig. 5 show that the LID separations, dL, increase whereas the dis￾connection separations, dD, decrease with deviation ω from the NW towards the KS OR. Also, the line-directions of the defects change over this range of OR. All of these factors influence the free-energy through the elastic strain fields of the defects and their core energies [19]. Although the habit plane structures reported here are not fully refined, the provisional structures show good agreement with experimental observations in the literature. For example, Sand￾vik and Wayman [20] studied lath martensite in an Fe–Ni–Mn alloy using transmission electron microscopy (TEM). They observed an array of 1/2[1 1 1] ¯ LID dislocations with dL in the range 2.6–6.3 nm, L varying between 10◦ and 15◦ from screw orientation in a habit plane with ψ = 9.45◦, and ω ranging between 0.16◦ and 3.16◦. This observation resembles closely the array of bL and bD −1/−1 defects predicted here (Table 2) for ω = 2.5◦, namely: dL = 3.77 nm, L oriented 12.99◦ from screw orientation and ψ = 9.10◦. We note that in the present work L was treated as a variable, whereas, in our earlier work on TiMo [7], it was taken as being the intersection of the active slip or

X Ma, R.C. Pond Materials Science and Engineering A 481-482(2008)404-408 twinning plane with the habit plane as depicted in Fig. 1. An [2] J.S. Bowles, J.K. Mac Kenzie, Acta Metall. 2(1954)129, 138, important feature of ferrous alloys is that 1/21 1 l]a disloca- tions are glissile in the terrace plane and hence may be able to [31Jw Christian. The Theory of Transformations in Metals and Alloys, Perg- adjust their line directions after reaching the interface. Moritani [4] C M. Wayman, Introduction to the Crystallography of Martensite Trans- in an Fe-Ni-Mn alloy, the spacing of these dislocations was [5 K. Bains, Macmillan, New York, NY,1964 et al.[21]also observed an array of 1/2[11 1]a LId dislocations d=4.8 nm with 5 close to pure-screw orientation in a habit Inc, New York, NY. 2003. plane with y=1947 and o=1.56. This observation reser [6] P.G. McDougall, C M. Wayman, in: G.B. Olson, w.s. Owen (Eds ) bles the array of bland b=2/-2 defects predicted in Table 3 for The Crystallography and Morphology of Ferrous Martensites, Martensite, American Society for Metals Intemational, USA, 1992 a=1.5, namely: dl=3.76 nm and y=19.80, 5L inclined at [7]RC Pond, S. Celotto, J.P. Hirth, Acta Mater. 51(2003)5385 25.39 to screw. In addition, they were able to image the discon- [8]RC Pond, x. Ma, Y.w. Chai, J.P. Hirth, in: F.R. N. Nabarro, J.P. Hirth nection array using high-resolution TEM. Images obtained with Eds ), Dislocations in Solids, vol. 13, North-Holland, Amsterdam, 2007. the beam direction parallel [1 0 1ly closely resemble the defects [91 j.P. Hirth, J. Phys. Chem. Solids 55(1994)985 in Fig. 4, bearing in mind the screw component of their Burgers [10)R C Pond, S. Celotto, Int. Mater. Rev. 48(2003)225 vectors are not evident in such images. Moreover, the average [11] R.C. Pond, X Ma, J.P. Hirth, Mater. Sci. Eng. A438-440(2006)109 spacing d observed experimentally is in good agreement with [12] J.P. Hirth, J.N. Mitchell, D.S. Schwartz, T.E. Mitchell, Acta Mater. 54 the calculated value of 1.11 nm for b,a defects, Mahon et 20061917 al. [22] and Ogawa and Kajiwara[23] also published images of 3] R.C. Pond, in: ER N Nabarro (Ed ), Dislocations in Solids, vol8, North. disconnection arrays in ferrous alloys; these too are consistent [14)RC Pond, X Ma, Z. Metallkd 96(2005)1124 with the defects illustrated in Fig 4. [15] R.C. Pond, A Serra, D.J. Bacon, Acta Mater. 47(1999)1441 Finally, we note that the LID/disconnection networks pre- [16] B.A. Bilby, R. Bullough, E. Smith, Proc. R. Soc. A 231A(1955) dicted in the present work accommodate misfit on the habit plane of martensite in a manner reminiscent of screw dislocation (18)TNagano, M Enomoto, Metall. Mater. Trans. A 37(2006 networks in grain boundaries between orthorhombic crystals as [191 J.P. Hirth, J Lothe, Theory of Dislocations, second ed,McGraw-Hill,New discussed by Matthews [24]. The present result can be contrasted York. 1982 with misfit-relieving configurations involving edge disconnec- [20] B PJ Sandvik, C M. Wayman, Metall. Trans. A 14(1983)835 tions, as discussed for example by rigsbee and Aaronson[25] 21T. Moritani, N. Miyajima, T. Furuhara, T Maki, Scripta Mater. 47(2002) and Moritani et al. [21]. [22] G.. Mahon, J.M. Howe, S. Mahajan, Philos. Mag. Lett. 59(1989) References [23] K. Ogawa, S Kajiwara, Philos Mag. 84(2004)2919 124 J.w. Matthews, Philos Mag. 29(1974)797. [1] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197(1953) [25] J.M. Rigsbee. H.L. Aaronson, Acta Metall. 27(1979)36

408 X. Ma, R.C. Pond / Materials Science and Engineering A 481–482 (2008) 404–408 twinning plane with the habit plane as depicted in Fig. 1. An important feature of ferrous alloys is that 1/2[1 1 1] ¯ disloca￾tions are glissile in the terrace plane and hence may be able to adjust their line directions after reaching the interface. Moritani et al. [21] also observed an array of 1/2[1 1 1] ¯ LID dislocations in an Fe–Ni–Mn alloy; the spacing of these dislocations was dL = 4.8 nm with L close to pure-screw orientation in a habit plane with ψ = 19.47◦ and ω = 1.56◦. This observation resem￾bles the array of bL and bD −2/−2 defects predicted in Table 3 for ω = 1.5◦, namely: dL = 3.76 nm and ψ = 19.80◦, L inclined at 25.39◦ to screw. In addition, they were able to image the discon￾nection array using high-resolution TEM. Images obtained with the beam direction parallel [1 0 1] ¯ closely resemble the defects in Fig. 4, bearing in mind the screw component of their Burgers vectors are not evident in such images. Moreover, the average spacing dD observed experimentally is in good agreement with the calculated value of 1.11 nm for bD −2/−2 defects. Mahon et al. [22] and Ogawa and Kajiwara [23] also published images of disconnection arrays in ferrous alloys; these too are consistent with the defects illustrated in Fig. 4. Finally, we note that the LID/disconnection networks pre￾dicted in the present work accommodate misfit on the habit plane of martensite in a manner reminiscent of screw dislocation networks in grain boundaries between orthorhombic crystals as discussed by Matthews[24]. The present result can be contrasted with misfit-relieving configurations involving edge disconnec￾tions, as discussed for example by Rigsbee and Aaronson [25] and Moritani et al. [21]. References [1] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197 (1953) 1503. [2] J.S. Bowles, J.K. MacKenzie, Acta Metall. 2 (1954) 129, 138, 224. [3] J.W. Christian, The Theory of Transformations in Metals and Alloys, Perg￾amon Press, Oxford, UK, 2002. [4] C.M. Wayman, Introduction to the Crystallography of Martensite Trans￾formations, Macmillan, New York, NY, 1964. [5] K. Bhattacharya, Microstructure of Martensite, Oxford University Press, Inc., New York, NY, 2003. [6] P.G. McDougall, C.M. Wayman, in: G.B. Olson, W.S. Owen (Eds.), The Crystallography and Morphology of Ferrous Martensites, Martensite, American Society for Metals International, USA, 1992. [7] R.C. Pond, S. Celotto, J.P. Hirth, Acta Mater. 51 (2003) 5385. [8] R.C. Pond, X. Ma, Y.W. Chai, J.P. Hirth, in: F.R.N. Nabarro, J.P. Hirth (Eds.), Dislocations in Solids, vol. 13, North-Holland, Amsterdam, 2007, p. 225. [9] J.P. Hirth, J. Phys. Chem. Solids 55 (1994) 985. [10] R.C. Pond, S. Celotto, Int. Mater. Rev. 48 (2003) 225. [11] R.C. Pond, X. Ma, J.P. Hirth, Mater. Sci. Eng. A 438–440 (2006) 109. [12] J.P. Hirth, J.N. Mitchell, D.S. Schwartz, T.E. Mitchell, Acta Mater. 54 (2006) 1917. [13] R.C. Pond, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, vol. 8, North￾Holland, Amsterdam, 1989, p. 1. [14] R.C. Pond, X. Ma, Z. Metallkd. 96 (2005) 1124. [15] R.C. Pond, A. Serra, D.J. Bacon, Acta Mater. 47 (1999) 1441. [16] B.A. Bilby, R. Bullough, E. Smith, Proc. R. Soc. A 231A (1955) 263. [17] X. Ma, R.C. Pond, J. Nucl. Mater. 361 (2007) 313. [18] T. Nagano, M. Enomoto, Metall. Mater. Trans. A 37 (2006) 929. [19] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., McGraw-Hill, New York, 1982. [20] B.P.J. Sandvik, C.M. Wayman, Metall. Trans. A 14 (1983) 835. [21] T. Moritani, N. Miyajima, T. Furuhara, T. Maki, Scripta Mater. 47 (2002) 193. [22] G.J. Mahon, J.M. Howe, S. Mahajan, Philos. Mag. Lett. 59 (1989) 273. [23] K. Ogawa, S. Kajiwara, Philos. Mag. 84 (2004) 2919. [24] J.W. Matthews, Philos. Mag. 29 (1974) 797. [25] J.M. Rigsbee, H.I. Aaronson, Acta Metall. 27 (1979) 365.