Shifting the Shear Paradigm in the Crystallographic Models of Displacive Transformations in Metals and Alloys

Article Shifting the Shear Paradigm in the Crystallographic Models of Displacive Transformations in Metals and Alloys

Cyril Cayron

LMTM, EPFL (Ecole Polytechnique Fédérale de Lausanne), Neuchatel 2000, Switzerland; cyril.cayron@epﬂ.ch; Tel.: +41-21-695-44-56 Received: 2 March 2018; Accepted: 16 April 2018; Published: 23 April 2018

Abstract: Deformation twinning and martensitic transformations are characterized by the collective displacements of atoms, an orientation relationship, and specific morphologies. The current crystallographic models are based on the 150-year-old concept of shear. Simple shear is a deformation mode at constant volume, relevant for deformation twinning. For martensitic transformations, a generalized version called invariant plane strain is used; it is associated with one or two simple shears in the phenomenological theory of martensitic crystallography. As simple shears would involve unrealistic stresses, dislocation/disconnection-mediated versions of the usual models have been developed over the last decades. However, a fundamental question remains unsolved: how do the atoms move? The aim of this paper is to return to a crystallographic approach introduced a few years ago; the approach is based on a hard-sphere assumption and linear algebra. The atomic trajectories, lattice distortion, and shuffling (if required) are expressed as analytical functions of a unique angular parameter; the habit planes are calculated with the simple “untilted plane” criterion; non-Schmid behaviors associated with some twinning modes are also predicted. Examples of steel and magnesium alloys are taken from recent publications. The possibilities offered in mechanics and thermodynamics are briefly discussed.

Keywords: shear; martensitic transformation; deformation twinning; hard sphere

1. The Origin of the Concept of Simple Shear Mechanical twinning and martensitic transformations are known to form very rapidly, sometimes at the velocities close to the speed of sound; the atoms move collectively, and the product phase (martensite or twins) appear as plates, laths, or lenticles. These transformations are called “displacive” in metallurgy (please note that the meaning of this term is different from the one used in physics, where “displacive” implies only small displacements of atoms without breaking the atomic bonds). In his extensive review, Cahn [1] wrote: “Cooperative atom movements in phase transformations may extend over quite large volumes of crystals. When this happens, the transformation is termed martensitic (from Martens, who discovered the ﬁrst transformation of this type in carbon steel). If the atom movements in a crystal results in a new crystal of different orientation, but identical structure, the process is termed mechanical twinning.” Christian and Mahajan [2] also insisted on the great similarities between deformation twinning and martensite transformations: “all deformation twinning should strictly be regarded as a special case type of stress-induced martensitic transformation”. Simple shears are the cornerstone of the theories of deformation twinning and martensitic transformation developed during the last century [1–3]. The notion of simple shear can even be traced back to older times, as recalled by Hardouin Duparc [4]. One hundred and ﬁfty years ago, in 1867, William Thomson and Peter Guthrie Tait’s “Treatise on Natural Philosophy” [5] (from [4])

Crystals 2018, 8, 181; doi:10.3390/cryst8040181 www.mdpi.com/journal/crystals Crystals 2018, 8, x FOR PEER REVIEW 2 of 54 Crystals 2018, 8, 181 2 of 55 1867, William Thomson and Peter Guthrie Tait’s “Treatise on Natural Philosophy” [5] (from [4]) defined that “‘a simple shear’ is the property that two kinds of planes (two different sets of parallel deﬁned that “‘a simple shear’ is the property that two kinds of planes (two different sets of parallel planes) remain unaltered, each in itself”. The mathematical formalism, with the nomenclature K1, η1, planes) remain unaltered, each in itself”. The mathematical formalism, with the nomenclature K1, K2, and η2, was introduced by Otto Mügge in 1889 [6] (from [4]). At that time, crystallographers and η1,K2, and η2, was introduced by Otto Mügge in 1889 [6] (from [4]). At that time, crystallographers mineralogists already knew the periodic structure of minerals, thanks to Haüy’s works at the and mineralogists already knew the periodic structure of minerals, thanks to Haüy’s works at the beginning of the 19th century [7]. The crystal periodicity was expressed by the lattice, but the beginning of the 19th century [7]. The crystal periodicity was expressed by the lattice, but the concept concept of the unit cell was yet not fully clarified; it was just a “molecule intégrante”, like a “brick”. of the unit cell was yet not fully clariﬁed; it was just a “molecule intégrante”, like a “brick”. Therefore, Therefore, it was natural at that time to consider that the “bricks” could glide on themselves in order it was natural at that time to consider that the “bricks” could glide on themselves in order to rearrange to rearrange and form a new twinned lattice, as illustrated in Figure 1. and form a new twinned lattice, as illustrated in Figure1.

Figure 1. Crystal and deformation twinning. (a) Crystal imagined as a stacking of bricks called Figure 1. Crystal and deformation twinning. (a) Crystal imagined as a stacking of bricks called “molecules intégrantes”, as published in 1801 by Haüy [7], and nowadays called a “unit cell” or “motif”; “molecules intégrantes”, as published in 1801 by Haüy [7], and nowadays called a “unit cell” or (b) The way to envision deformation twinning at the end of the 19th century was probably as follows: “motif”; (b) The way to envision deformation twinning at the end of the 19th century was probably twinning can be obtained by bricks gliding on themselves in such a way that the lattice is restored after as follows: twinning can be obtained by bricks gliding on themselves in such a way that the lattice is the glide. The basis vectors are represented by the red arrows, they are modified by the shear, but the restored after the glide. The basis vectors are represented by the red arrows, they are modified by the newly formed red twinned lattice is equivalent to the initial one by a reflection or by a 180◦ rotation. shear, but the newly formed red twinned lattice is equivalent to the initial one by a reflection or by a 180° rotation. At the early beginning of the 20th century, these notions, originally from mineralogists, were acquiredAt the by early the metallurgists,beginning of the without 20th century any possibility, these notions at that, timeorigina tolly further from clarifymineralogists the nature, were of acquiredthe “bricks”. by the Remember metallurgists, that without atoms were any possibility still a speculative at that time idea. to Democritusfurther clarify and the ancient nature of Greek the “bricks”.philosophers Remember supposed that their atoms existence, were butstill the a speculative first evidence idea. was Democrit given inus the and early ancient 1800s byGreek the philosopherschemist John Dalton,supposed who their noticed existence, that the but chemical the first reactions evidence always was givenimply inratios the of early elements 1800s inby small the chemistintegers. John Dalton Dalton also, imagined who noticed the atoms that the as solidchemical spheres. reactions The idea always of discontinuous imply ratios matter, of elements however, in smallwas not integers. unanimously Dalton admitted.also imagined In 1827, the botanistatoms as Robertsolid spheres. Brown observedThe idea theof discontinuous erratic movements matter of, however,dust grains was floating not unanimously at the water surface, admitted. and In the 1827, Bownian botanist motion Robert was Brown explained observed in 1905 the by erraAlberttic movementsEinstein using of dust statistical grains physics. floating The at the mass water and surface dimensions, and ofthe the Bownian atoms weremotion soon was experimentally explained in 1905determined by Albert by theEinstein physicist using Jean statistical Perrin. Aphysics. new area The then mass opened and dimensions to specify the of properties the atoms of were the atoms. soon experimentallyVery quickly, it appeared determined with by the the discoveries physicist ofJean the Perrin. electron A by new Thomson area then and opened protons toby specifyRutherford, the propertiesalong with of quantum the atoms. physics Very bornquickly, with it Bohr,appeared that atomswith the were discoveries far more complexof the electron than Dalton’s by Thomson solid andspheres. protons However, by Rutherford, in many along metals, with the quantum hard-sphere physics assumption born with holds Bohr, quite that atoms satisfactorily, were far at more least complexto describe than the Dalton’s atomic soli packings.d spheres. William However, Barlow in many in 1883 metals, [8] noticed the hard that-sphere the symmetries assumption of holds some quitemetals satisfactorily, and alloys can at beleast represented to describe by the packings atomic ofpackings. hard spheres, William with Barlow the body-centered in 1883 [8] noticed cubic (bcc),that thethe symmetries face-centered of cubic some (fcc), metals and and the hexagonalalloys can close-packedbe represented (hcp) by packings structures of among hard spheres, them. The with ratio the of bodylattice-centered parameters cubic c/a (bcc), in magnesium, the face-centered cobalt, zirconium, cubic (fcc) titanium,, and the rhenium, hexagonal or scandium close-packed differs (hcp) from q structures among them. 8 The ratio of lattice parameters c/a in magnesium, cobalt, zirconium, the ideal packing ratio 3 by less than 3%. The precise determination of the lattice parameters, thanks titanium,to X-ray or rhenium electron, or diffraction, scandium revealed differs from that in the many ideal bimetallic packing solidratio solutions, by less the than lattice 3%. parameterThe precise of the mixture of two elements is the average of the lattice parameter of each pure element, in agreement determinationwith a hard-sphere of the packinglattice parameters model. During, thanks fcc-bcc to X-ray martensitic or electron phase diffraction transitions,, revealed the that difference in many of bimetalliclattice parameters solid solutions, expected the by lattice a hard-sphere parameter model of th ise mixture only 4%. of The two critical elements shear is requiredthe average to activate of the lattice parameter of each pure element, in agreement with a hard-sphere packing model. During Crystals 2018, 8, 181 3 of 55 dislocations was studied in 1947 by Bragg and Nye [9] with sub-millimeter bubble rafts, and the hard sphere model is still used nowadays in classrooms to explain the ABCABC and ABABAB packings of the fcc and hcp phases, respectively. The hard-sphere model is also an important approximation made in molecular dynamics. However, atoms, even with their simple hard-sphere image, are not explicitly present in the crystallographic theories of twinning and martensitic transformations, which are all based on the lattices and their transformation by shears, as in Mügge’s initial model. The atoms are the “big losers” of these theories.

2. Shears Used in the Theories of Deformation Twinning Cahn wrote about deformation twinning: “That part of the parent crystal which is thus transformed undergoes a macroscopic change of shape which can be described exactly as a simple shear” [1]. A shear is defined by its shear plane K1, its shear direction η1, and its amplitude s1. The shear leaves a second plane (K2) undistorted (but rotated). The direction η2 belongs to K2, and is perpendicular to the intersection between K1 and K2. The plane K2 is rotated by an angle φ, and the shear amplitude is given by s1 = 2 tan(φ/2). It is classic to define two types of twins: type I, where the plane K1 and the direction η2 are rational; and type II, where the plane K2 and the direction η1 are rational. One of these components is a symmetry element that is common to both parent and twin crystal; K1 is a common mirror plane for type I twins, and η1 is a common two-fold axis for type II twins. For any mode I twin, defined by K1 along η1, one can associate a conjugate mode II twin, defined by substituting K2 in place of K1 and η2 in place of η2 [6]. The four components K1,K2, η1, and η2, as well as their geometric representation with the shearing ellipsoid, as represented by Hall in 1954 [10] (Figure2a) and still used in classical textbooks, date from Thomson and Guthrie Tait, who clearly noted that “the planes of no distortion in a simple shear are clearly the [two] circular sections of the strain ellipsoid” [5] (from Ref. [4]). In the 1950s, the twinning theory was mathematically developed and refined, mainly by Kihô in 1954 [11], Jaswon and Dove in 1956 [12], and later by Bilby et al. [13–15], following some of the notions and crystallographic tools used by Bowles and Makenzie in the phenomenological theory of martensite transformation (PTMC), which will be detailed in the next section. The master equation of the crystallographic theory of twinning built by Bilby et al. can be summarized by C = R.S, where C is the correspondence matrix, R is a rotation, and S is a simple shear. These three matrices have a determinant equal to ±1. The correspondence matrix gives the coordinates in the twin basis of the parent basis vectors, once distorted by the shear. As these vectors should be vectors of the twin lattice, the coordinate should be integers, or rational if the cell of the Bravais lattice contains more than one atom (as it is the case for hcp metals) or if a supercell is chosen for the calculations. For these cases, the atoms inside the cell that are not positioned at the nodes of the cell do not follow the same shear trajectory as those positioned at the nodes; one says that they “shuffle” (we will come later on this term). An additional mathematical restriction comes from the fact that the rotation matrix should check R.RT = I, with I being the identity matrix, which permits the establishment of a list of correspondence matrices. Among the numerous (but finite) possible correspondence and shear matrices given by the shear-based twinning theory, only those with the lowest shear amplitude and minimum shuffles are considered to be realistic. For example, fifteen deformation twinning modes could be listed in titanium [15], some of which are reproduced in Figure2b. The theory is still used today (see for example [ 16]). Despite the theory’s mathematical sophistication and rigor, one must admit that its fundaments do not differ from the initial 150-year-old view. Despite the progress, the theory does not answer the simple question: how do atoms move during twinning? In the case of a cell made of one atom, it is assumed that the atoms follow the same shear as the nodes of the lattice, but for multiple-atom (super)cells, the shuffling of the atoms is not satisfactorily treated. Some possible modes were proposed by Bilby and Crocker [13]; they often consist in finding the different ways of translating the positions obtained by the shear, in order to reach those obtained by the reflections or the 180◦ rotation symmetries. The shuffles in these models are very Crystals 2018,, 8,, 181x FOR PEER REVIEW 4 of 54 55 shuffles in these models are very complex, as shown by the arrows of Figure 2b. Shuffling was not complex, as shown by the arrows of Figure2b. Shuffling was not examined anymore in the last version examined anymore in the last version of the theory proposed by Bevis and Crocker [14]. of the theory proposed by Bevis and Crocker [14].

Figure 2. 2.Deformation Deformation twinning. twinning. (a) Lattice (a) deformationLattice deformation and associated and (K associated1, η1,K2, η 2 (K) crystallographic1, η1, K2, η2) parameters,crystallographic represented parameters, by Hall represented in 1954. Reprinted by Hall in from 1954. [10 ]Reprinted with permission from [10 from] with Elsevier. permission There from is no evolutionElsevier. Thereof the fundamental is no evolution concept of thein comparison fundamental with concept Mügge’s in work comparison published with in 1889; Mügge’s (b) complex work deformationpublished in twinning1889; (b) incomplex titanium, deformation involving supercellstwinning in and titanium shufﬂes,, involving according supercells to Crocker and and shuffles Bevis in, 1970.according Reprinted to Crocker from [and15] withBevis permission in 1970. Reprinted from Elsevier. from [15] with permission from Elsevier.

Actually, even if not always acknowledged, it seems unrealistic that the atoms can simply move Actually, even if not always acknowledged, it seems unrealistic that the atoms can simply move along straight lines, such as drawn by the arrows of Figure 2b; even in the case of one atom per cell. along straight lines, such as drawn by the arrows of Figure2b; even in the case of one atom per cell. Indeed, for steric reasons the atoms cannot follow simple translations dictated by a simple shear. For Indeed, for steric reasons the atoms cannot follow simple translations dictated by a simple shear. a long time, the theories were focused on the way the lattices are deformed, omitting to confront to For a long time, the theories were focused on the way the lattices are deformed, omitting to confront the question of the atomic trajectories. Therefore, the crystallographic theory of twinning is a to the question of the atomic trajectories. Therefore, the crystallographic theory of twinning is a phenomenological theory. Assuming that twinning results from a simple shear implies that Schmid’s phenomenological theory. Assuming that twinning results from a simple shear implies that Schmid’s law [17] should be verified. Deformation twinning should occur when the shear stress resolved law [17] should be verified. Deformation twinning should occur when the shear stress resolved along along the slip direction on the slip plane reaches a critical value, called the critical resolved shear the slip direction on the slip plane reaches a critical value, called the critical resolved shear stress. stress. However, this law is difficult to confirm experimentally (see, for example, “Orientation However, this law is difficult to confirm experimentally (see, for example, “Orientation dependence: dependence: is there a CRSS for twinning?” Section 5.1 of [2]). In addition, it was experimentally is there a CRSS for twinning?” Section 5.1 of [2]). In addition, it was experimentally noticed that some noticed that some twinning modes in magnesium have “abnormal”, i.e., “non-Schmid” behavior twinning modes in magnesium have “abnormal”, i.e., “non-Schmid” behavior [18]. The reason was [18]. The reason was attributed to the fact that the local stresses differ from the global ones [19], but attributed to the fact that the local stresses differ from the global ones [19], but to our knowledge, there to our knowledge, there are no convincing quantitative experimental results that can confirm this are no convincing quantitative experimental results that can confirm this explanation. Non-Schmid explanation. Non-Schmid behavior was also observed at least for slip deformation in bcc metals. In behavior was also observed at least for slip deformation in bcc metals. In 1928, Taylor wrote “in 1928, Taylor wrote “in β-brass resistance to slipping in one direction on a given plane of slip is not β-brass resistance to slipping in one direction on a given plane of slip is not the same as resistance the same as resistance offered to slipping in the opposite direction” [20]. This effect, called the offered to slipping in the opposite direction” [20]. This effect, called the twinning-antitwinning effect, twinning-antitwinning effect, was attributed to the core structure of the dislocations in the bcc was attributed to the core structure of the dislocations in the bcc metals [21,22], but the fundamental metals [21,22], but the fundamental reason for the dissymmetry is that Schmid’s law is a continuum reason for the dissymmetry is that Schmid’s law is a continuum mechanics law that does not take into mechanics law that does not take into account the atomic structure of the metals, or more precisely, account the atomic structure of the metals, or more precisely, the configuration of the atoms in the the configuration of the atoms in the stacking of the glide planes or twinning planes. The normal to stacking of the glide planes or twinning planes. The normal to the {112} bcc twinning plane is not a the {112} bcc twinning plane is not a two-fold rotation axis, which means that a shear along a <111> two-fold rotation axis, which means that a shear along a <111> direction is not equal to a shear in its direction is not equal to a shear in its opposite direction. The same situation exists for deformation opposite direction. The same situation exists for deformation twinning in fcc metals: as the direction twinning in fcc metals: as the direction normal to the {111} plane is a three-fold axis, not a two-fold normal to the {111} plane is a three-fold axis, not a two-fold axis, the shears in the <110> directions axis, the shears in the <110> directions are not equal to their opposites. The use of simple or pure are not equal to their opposites. The use of simple or pure shear (both are deformation at constant shear (both are deformation at constant volume) was proved to be highly efficient in continuum volume) was proved to be highly efficient in continuum mechanics for describing the plastic behavior mechanics for describing the plastic behavior of materials, but its direct application to describe the of materials, but its direct application to describe the lattice transformation of crystals (packing of lattice transformation of crystals (packing of discontinuous atoms) raises some important questions discontinuous atoms) raises some important questions that should not be discarded. that should not be discarded.

Crystals 2018, 8, 181 5 of 55 Crystals 2018, 8, x FOR PEER REVIEW 5 of 54

3.3. Shears UsedUsed inin thethe TheoriesTheories ofof MartensiticMartensitic TransformationsTransformations

3.1.3.1. Early ModelsModels ofof MartensiticMartensitic TransformationsTransformations ForFor martensiticmartensitic transformations,transformations, the notion of shear was also so pregnant that one could read,read, “shear“shear transformationstransformations areare synonymoussynonymous withwith martensiticmartensitic transformations”transformations” [[2323].]. However,However, thethe firstfirst crystallographiccrystallographic model model of of fcc fcc-bcc-bcc distortion distortion proposed proposed by by Bain Bain in 1924 in 1924 [24] [ does24] does not imply not imply shear. shear. This Thisdiscrepancy discrepancy was wasused used to discard to discard Bain’s Bain’s proposal proposal in inthe the first first models models of of martensitic martensitic transformation. transformation. Indeed,Indeed, aa fewfew yearsyears afterafter Bain’sBain’s proposal,proposal, thethe orientationorientation relationshiprelationship (OR)(OR) betweenbetween thethe parentparent fccfcc andand thethe bccbcc martensite martensite was was experimentally experimentally determined, determined using, using X-ray X-ray diffraction, diffraction by, Youngby Young in 1926 in 1926 [25], Kurdjumov[25], Kurdjumov andSachs and Sachs in 1930 in [26 1930], Wassermann [26], Wassermann in 1933 in [27 1933], and [27 Nishiyama], and Nishiyama in 1934 in [28 1934]. Young [28]. workedYoung worked on Fe–Ni on meteorites, Fe–Ni meteorites, Kurdjumov Kurdjumov and Sachs and on Sachs Fe–1.4C on steel,Fe–1.4C and steel, Nishiyama and Nishiyama on a Fe–30Ni on a alloy.Fe–30Ni The alloy. OR discoveredThe OR discovered by Young by wasYoung very was close very to close that discoveredto that discovered by Kurdjumov by Kurdjumov and Sachs, and butSachs, history but history of metallurgy of metallurgy only kept only the kept names the ofnames the two of the latter two researchers. latter researchers. The now-called The now KS-called OR (for KS Kurdjumov-Sachs)OR (for Kurdjumov and-Sachs) NW ORand (for NW Nishiyama-Wassermann) OR (for Nishiyama-Wassermann) are at 5◦ from are eachat 5° other,from andeach at other, 10◦ from and theat 10° OR from expected the OR by aexpected direct Bain by distortion. a direct Bain The distortion. difference isThe non-negligible, difference is whichnon-negligible is why Kurdjumov,, which is Sachs,why Kurdjumov, and Nishiyama Sachs proposed,, and Nishiyama in their respective proposed papers,, in their a model respective of fcc-bcc papers transformation, a model of fcc that-bcc is “doestransformation not agree” that with is Bain. “does Kurdjumov not agree” and with Sachs Bain. imagined Kurdjumov a transformation and Sachs imagined made by a two transformation consecutive made by two consecutive shears: (111)[112] followed by (112)[111]. Nishiyama proposed a shears: (111)γ 112 followed by 112 111 . Nishiyama proposed a slightly different sequence, γ α α slightly different sequence( , in) which the first shear (111)[112] is followed by a stretch. These in which the first shear 111 γ 112 γ is followed by a stretch. These models are quite close and summarizedmodels are quite by Nishiyama close and summarized in his 1934 paper by Nishiyama [28] in Figure in his3. Therefore,1934 paper in [28 those] in models,Figure 3. the Therefore, fcc-bcc transformationin those models, is the not fcc a Bain-bcc distortion,transformation but neither is not a is Bain it a simpledistortion, shear—it but neither is a combination is it a simple of shearsshear— orit shearsis a combination and stretch. of shears or shears and stretch.

FigureFigure 3. 3.Kurdjumov-Sachs-Nishiyama Kurdjumov-Sachs-Nishiyama (KSN) (KSN model) model of face-centered of face-centered cubic cubic (fcc)–body-centered (fcc)–body-centered cubic (bcc)cubic phase (bcc) transformationphase transformation proposed proposed by Kurdjumov by Kurdjumov and Sachs and (1930) Sachs and (1930) Nishiyama and Nishiyama (1934). Reprinted (1934). fromReprinted [28] with from permission [28] with permission from Elsevier. from Elsevier.

The important point of the KSN model is the distortion of the dense plane {111}γ into a dense The important point of the KSN model is the distortion of the dense plane {111}γ into a dense plane {110}α. Such planar distortion was also noticed and largely discussed by Young, who wrote plane {110}α. Such planar distortion was also noticed and largely discussed by Young, who wrote “On “On account of the marked resemblance of the (111) plane of the solid solution [taenite, fcc] and the account of the marked resemblance of the (111) plane of the solid solution [taenite, fcc] and the (110) (110) plane in kamacite [bcc], it is possible to form the solid solution by simply shearing rows of plane in kamacite [bcc], it is possible to form the solid solution by simply shearing rows of atoms and atoms and rearranging the atoms in adjacent planes, as already described, a crystal of kamacite which is only a few planes in thickness but of considerable area.” [25] Crystals 2018, 8, 181 6 of 55 rearrangingCrystals 2018, 8, the x FOR atoms PEER in REVIEW adjacent planes, as already described, a crystal of kamacite which is only6 of 54 a few planes in thickness but of considerable area” [25]. InIn 1934, 1934, Burgers Burgers determined determined the the OR OR in zirconium, in zirconium between, between the high the high temperature temperature parent parent bcc phase bcc andphase the and low-temperature the low-temperature martensite martensite hcp phase, by hcp X-ray phase diffraction, by X-ray [29]; diffractio as Young,n Kurdjumov, [29]; as Young, Sachs, andKurdjumov, Nishimaya Sachs did, forand fcc-bcc Nishimaya transformations, did for Burgers fcc-bcc proposed transformations, a crystallographic Burgers modelproposed of the a bcc-hcpcrystallographic transformation. model of Actually, the bcc he-hcp treated transformation. three models, Actually, with one he oftreated them three implying models, an intermediate with one of fccthem phase—but implying onlyan intermediate the first is now fcc phase widely— accepted.but only the This first model is now is reproduced widely accepted. in Figure This4. model Burgers is explainsreproduced it as in the Figure combination 4. Burgers of explains “a shear it parallel as the combination to a {112} plane of “ ina shear the [111] parallel direction to a {112} lying plane in this in plane,the [111] followed direction by lying a definite in this displacement plane, followed of alternate by a definite atomic displacement layers [shuffle] of alternate and a homogeneous atomic layers contraction[shuffle] and (eventual a homogeneous dilatation) contraction parallel to (eventual definite dilatation) crystallographic parallel directions.” to definite [ 29crystallographic] Burgers also calculateddirections. the” [29 associated] Burgers shear also value calculated and found thes = associated 0.22. His description shear value is not and exactly found that s of= his0.22. figure, His asdescription two shears is arenot actually exactly appliedthat of his on twofigure, different as two {112} shears planes. are actually In order applied to overcome on two this different problem, {112} it is nowplanes. usual In to order replace to overcome the shears bythis a problem,diagonal distortion it is now of usual the orthorhombic to replace the cell, shears marked by by a diagonal the bold linesdistortion in part of B the of Figureorthorhombic4, as described cell, marked (for example) by the bold in Kelly lines and in Groves’part B of book Figure [ 30 4]., Thisas described distortion (for is analogousexample) in to Kelly the Bain anddistortion Groves’ book in the [30 fcc-bcc]. This transformation,distortion is analogous but the differenceto the Bain is distortion that a shuffle in the is requiredfcc-bcc transformation, to put half of the but atoms the differenc in their goode is that positions a shuffle and is obtain required the to hcp put phase, half of as the shown atoms in thein their part notedgood positions by the letter and D obtain in Figure the 4hcp. phase, as shown in the part noted by the letter D in Figure 4.

FigureFigure 4.4. Burgers’Burgers’ modelmodel ofof bcc-hexagonalbcc-hexagonal close-packedclose-packed (hcp)(hcp) transformation.transformation. Reprinted from [[2929]] withwith permissionpermission fromfrom Elsevier.Elsevier. TheThe twotwo simultaneoussimultaneous shearsshears areare markedmarked byby thethe redred ellipses. ellipses.

3.2. The Phenomenological Theory of Martensite CrystallographyCrystallography (PTMC)(PTMC) TheThe KSN KSN model model of of the the martensitic martensitic fcc-bcc fcc- transformationbcc transformation was criticized was criticized by Greninger by Greninger and Troaino and inTroaino their 1949in their paper 1949 [ 31paper], because [31], because the KSN the modelKSN model could could not explain not explain the observedthe observed habit habit planes planes in Fe–22Ni–0.8C,in Fe–22Ni–0.8C and, and because because of the of “relatively the “relatively large movementslarge movements and readjustments” and readjustments” needed needed to exactly to obtainexactly the obtain fcc structure.the fcc structure. They proposed They proposed that the that “martensite the “martensite crystal iscrystal formed is formed from austenite from austenite crystal almostcrystal entirelyalmost entirely by means by ofmeans two homogeneousof two homogeneous shears” shears [31]. The” [31]. details The weredetails then were clariﬁed then clarified by PTMC. by OnePTMC would. One nowwould say now that say the tha ﬁrstt the shear first explains shear explains the shape, the i.e.,shape, the i.e., habit the plane habit of plane martensite, of martensite, which iswhich the invariant is the invariant plane strainplane strain (IPS), (IPS) while, while the second the second shear shear deforms deforms the lattice the lattice without without changing changing the shape,the shape because, because of an “invisible”of an “invisible” compensating compensating lattice invariant lattice invariant shear (LIS). shear The (LIS idea). of The composing idea of twocomposing homogeneous two homogeneous shears to obtain shears an to inhomogeneous obtain an inhomogeneous structure with structure an invariant with an planeinvariant was plane then soonwas then associated soon associated with the notion with the of the notion correspondence of the correspondence matrix, previously matrix, introducedpreviously byintroduced Jaswon and by WheelerJaswon and in 1948 Wheeler [32]. Thisin 1948 matrix [32]. establishes This matrix a correspondence establishes a correspondence between the directions between of the the directions fcc and bcc of the fcc and bcc lattices after a Bain distortion. The result forms a mathematical and unified theory that “predicts” both the habit planes and the orientation relationships. This theory was proposed by Weschler et al. [33] and by Bowles and Mackenzie [34,35] in 1953–54. This theory, now called the Crystals 2018, 8, 181 7 of 55

Crystalslattices 2018 after, 8, ax BainFOR PEER distortion. REVIEW The result forms a mathematical and uniﬁed theory that “predicts”7 bothof 54 the habit planes and the orientation relationships. This theory was proposed by Weschler et al. [33] “pandhenomenological by Bowles and t Mackenzieheory of martensite [34,35] in c 1953–54.rystallography This theory,” (PTMC), now has called been theadopted “phenomenological by most of the metallurgists,theory of martensite thanks crystallography” to exhaustive review (PTMC), papers has and been books, adopted such by as most those of written the metallurgists, by Christian thanks [36] andto exhaustive Nishiyama review [37], and papers thanks and to books, the very such didactic as those books written written by Christian by Bhadeshia [36] and [38 Nishiyama,39]. The theory [37], ofand deformation thanks to thetwinn verying didactic(see previous books section) written reintroduces by Bhadeshia most [38 of,39 the]. Themathematical theory of tools deformation used in thetwinning PTMC. (see For previous the classic section) fcc-bcc reintroduces martensitic most transformations of the mathematical in steels, tools the used master in the equation PTMC. Forfor the PTMCclassic fcc-bccis RB = martensiticP1P2, where transformations RB is the product in steels,of the lattice the master deformation equation compri for thesed PTMC of the is RBsymmetric= P1P2, whereBain stretchRB is thematrix product B by ofan the additional lattice deformation rotation matrix comprised R, P1 is of an the IPS symmetric, and P2 is Bain a simple stretch shear. matrix PTMCB by −1 assumesan additional that the rotation simple matrix shearR P,2P is1 exactlyis an IPS, compensated and P2 is a simplefor by a shear.n LIS PTMC(P2) produced assumes by that twinning the simple or −1 dislocationshear P2 is exactly gliding, compensated such that the for martensite by an LIS shape (P2) isproduced only given by twinning by P1. The or IPS dislocation P1 appears gliding, as a generalizedsuch that the notion martensite of simple shape shearis only that given takes by P into1. The account IPS P1 theappears volume as a changegeneralized of the notion phase of transformation,simple shear that i.e., takes the into dilatation account or the contraction volume change component of the phase in the transformation, direction normal i.e., the to thedilatation shear porlane. contraction The IPS componentgives the shape in the of directionthe martensite normal product to the (lath, shear plate plane., or The lenticle); IPS gives the shear the shape plane of of the P1 ismartensite the habit product(interface) (lath, plane. plate, Both or parts lenticle); of the the equations, shear plane RB ofandP1 Pis1P the2, are habit invariant (interface) line strains plane. (ILS), Both partsand R of isthe theequations, rotation addedRB and to renderP1P2, are unro invarianttated the line line strains undistorted (ILS), and by theR is Bain the rotationstretch B added. This tois usuallyrender unrotatedgeometrically the lineillustrated undistorted in traditional by the Bain textbooks stretch, byB. Thisshowing is usually how a geometrically sphere is deformed illustrated into anin traditional ellipsoid and textbooks, by finding by showing the intersection how a sphere points is deformed between into the an sphere ellipsoid and and the by ellipsoid. ﬁnding the A summaintersectionrizing points scheme between of the the IPS sphere and and strain the ellipsoid.matrices usedA summarizing in the PTMC scheme is given of the IPSin Figure and strain 5b, accordingmatrices used to [39 in]. the PTMC is given in Figure5b, according to [39].

Figure 5. The use of shears in the phenomenological theory of martensitic transformations ( (PTMC).PTMC). (a) SchematicSchematic view of an invariant plane strain (IPS) (IPS),, with the shear value s and the dilatation part δ

perpendicularly toto the the shear shear plane; plane (b; )(b combination) combination of theof IPSthe PIPS1 (also P1 (also called called shape shape strain) strain) with a simplewith a simpleshear P 2 shearthat generates P2 that generates the correct the and correct well-oriented and well lattice-orienteRB. Thed lattice simple RB shear. The P 2 simpleis compensated shear P2 byis compensateda lattice invariant by shear a lattice to get invariant the correct shear shape to P get1. Reprinted the correct from shape [39] Pwith1. Reprinted permission from from [39 Elsevier.] with permission from Elsevier. The PTMC solves the initial issue between the non-shear Bain strain and the specific plate/lath morphologiesThe PTMC of martensitesolves the initial that are issue believed between to be the the non results-she ofar shears. Bain strain The theory and the is veryspecific subtle, plate/lath and its morphologiesmain inventors of introduced martensite importantthat are believed modern to concepts be the results in crystallography, of shears. The such theory as theis very coordinate subtle, andtransformation its main inventors matrix, the introduced transformations important from modern direct to concepts reciprocal in spaces, crystallography, clear references such as to the coordinatebases used for transform the calculations,ation matrix, correspondence the transformations matrices, equations from direct that tolink reciprocal the shear matrices spaces, clear with referencesthe correspondence to the bases and used coordinate for the transformationcalculations, correspondence matrices, etc. It matrices, is true that equations PTMC predictedthat link the shearexistence matrices of twins with inside the correspondence the bcc or body and centered coordinate tetragonal transformation (bct) martensite matrices formed, etc. It in is steels, true that and PTMC predicted the existence of twins inside the bcc or body centered tetragonal (bct) martensite formed in steels, and that these twins were not visible by optical microscopy. Their discovery by Nishiyama and Shimizu in the early era of transmission electron microscopy (TEM) in 1956 [40] probably made Nishiyama give up his two-step shear-stretch model (Figure 3) to fully adopt the Crystals 2018, 8, 181 8 of 55 that these twins were not visible by optical microscopy. Their discovery by Nishiyama and Shimizu in the early era of transmission electron microscopy (TEM) in 1956 [40] probably made Nishiyama give up his two-step shear-stretch model (Figure3) to fully adopt the PTMC. That was not a complete change of mind, according to Shimizu [41]: “he [Nishiyama] foresightedly expected the existence of lattice invariant deformation in martensite a few years before the phenomenological crystallographic theory of martensitic transformation was proposed”. It is plausible that Nishiyama’s rallying to PTMC had a huge impact on the rest of the scientific community, and finally convinced the most recalcitrant metallurgists of that time about the importance of this theory. However, the predictive character of PTMC is often oversold. Some researchers claim that PTMC “predicts all” the martensite features, but this affirmation should be considered thoughtfully. For example, one can be impressed by the apparent prediction of the ORs, but it should be remembered that the PTMC starts from the Bain OR, which is at 10◦ from the experimentally observed KS or NW ORs, and then it imposes the existence of an invariant line, which makes it closer to KS. If one considers the strong internal misorientations (up to 10◦) inside the martensitic products, “finding” an OR close to one belonging to the continuous list of experimentally observed ORs should not be considered as a “prediction”. One should also have a cautious look at the habit planes. The {259} habit planes were explained in 1951 by Machlin and Cohen [42], following Greninger and Troiano’s initial idea of composing ( ) shears. The {225} habit planes were explained by Bowles and Mackenzie [34] with a 225 γ 112 γ IPS ( ) followed by a 112 α 111 α shear, but with the help of a dilation parameter that was later subject of controversies. The PTMC did not predict these planes, because these planes were already observed before the establishment of the theory; PTMC “only” tried to explain them. The theory did not make predictions but postdictions, which is completely different when the reliability of a theory is considered. Numerous papers were published on the {225} martensite, trying to suppress the dilatation parameter by increasing the level complexity with extra shears (see for examples [43,44]). This “saga” of the {225} habit planes was summarized in 1990 by Wayman [45], and more recently by Dunne [46] as well as Zhang and Kelly [47]. Surprisingly, the fact that the {225} habit planes could be explained by using different methods and different parameters did not raise questions about the real relevance of the PTMC approach. We note here that most of these PTMC studies ignore Jaswon and Wheeler’s paper [32] for the explanation of the {225} habit planes; sometimes they refer to it, but only in the introduction for the use of Bain correspondence. It is worth recalling that Jaswon and Wheeler’s model was discarded by Bowles and Barrett in 1952 [48], because “Jaswon and Wheeler’s picture of the transformation as a simple homogeneous distortion of the lattice is not consistent with the observed relief effects”. We will come back later on this “rejection”, which we considered as an unfortunate missed opportunity. Apart from the {225}, the {557} habit planes have also been the object of much research for more than 60 years, without consensus; most of the studies include additional shears (see for example [49,50]). The last development of the PTMC has mainly consisted in adding or varying the shears and their combinations, i.e., by adding complexity, without being more effective than the initial Bowles-Mackenzie and Weschler-Liebermann-Read models. The “predictions” are sometimes written with six- or eight-digit numbers and compared with experimental results, where accuracy rarely exceeds one digit. The PTMC has been applied to other transformations, such as bcc-orthorhombic and bcc-hcp transformations [51], and here again the same criticisms can be raised. The theory does not predict orientation relationships, because it starts from Bain-type distortions, which already agree with experimental orientations. In the case of the bcc-hcp transformation, for example, PTMC starts with the ortho-hexagonal cell that was already defined by the Burger’s OR, shown in Figure4. The habit planes are not predicted, but “explained”, by choosing the LIS twinning systems among those reported by experimental observations, and by adjusting the dilatation parameter and the twinning amplitude in order to fit the calculated habit planes with those experimentally reported, as in [52]. The success of the PTMC for shape memory alloys, in which the transformation strains are lower than in steels, is probably more convincing. However, as admitted by its creators and by most of its promoters, Crystals 2018, 8, 181 9 of 55 the PTMC is and remains phenomenological. For example, Bhadeshia [38] clearly wrote that “the theory is phenomenological and is concerned only with the initial and final states. It follows that Crystals 2018, 8, x FOR PEER REVIEW 9 of 54 nothing can be deduced about the actual paths taken by the atoms during transformation: only a structuresdescription can of the be correspondence obtained.” Therefore, in position it is between the same the problem atoms in as the for two deformation structures can twinning; be obtained.” the theoryTherefore, does it not is the answer same the problem simple as question: for deformation how do twinning;the atoms themove theory during does the not transformation? answer the simple question: how do the atoms move during the transformation? 3.3. Bogers and Burgers’ Model 3.3. Bogers and Burgers’ Model In 1964, Bogers and Burgers developed a hard-sphere model to respond to this essential In 1964, Bogers and Burgers developed a hard-sphere model to respond to this essential question question for the fcc-bcc martensitic transformations [53], as illustrated in Figure 6. They noticed that for the fcc-bcc martensitic transformations [53], as illustrated in Figure6. They noticed that if a shear if a shear on a (111)γ plane is applied to a fcc crystal and stopped midway between the initial fcc on a (111)γ plane is applied to a fcc crystal and stopped midway between the initial fcc structure and structure and its twin, the operation distorts the two adjacent {111}γ planes into {110}α planes, as is its twin, the operation distorts the two adjacent {111}γ planes into {110}α planes, as is the case for a the case for a Bain distortion. This idea was actually not new, as Cottrell in his book [54] reported Bain distortion. This idea was actually not new, as Cottrell in his book [54] reported that Zener [55] that Zener [55] thought that “The face-centred cubic metals, for example, pass through twinning and thought that “The face-centred cubic metals, for example, pass through twinning and body-centered body-centered cubic configurations when sheared on their slip planes”. However, Bogers and cubic configurations when sheared on their slip planes”. However, Bogers and Burgers noticed that Burgers noticed that actually, the midway structure is not exactly bcc, and another shear on another actually, the midway structure is not exactly bcc, and another shear on another (111)γ plane is required (111)γ plane is required to obtain the final and correct bcc structure. to obtain the final and correct bcc structure.

FigureFigure 6. HardHard-sphere-sphere model model proposed proposed by by Bogers Bogers and and Burgers Burgers for for the the fcc fcc-bcc-bcc transformation transformation and and fccfcc-fcc-fcc deformation deformation twinning. twinning. The The initial initial fcc fcc structure structure ( (aa)) becomes becomes,, after after distortion distortion,, a a ne neww twinned twinned fcc fcc structurestructure ( (cc).). The The bcc bcc structure structure is is at at the the midway midway between between the the two two structures structures ( (bb));; however, however, additional additional shearsshears are are required required to to place place the the atoms atoms at at their their correct correct positions. positions. Reprinted Reprinted from from [ [5353]] with with per permissionmission fromfrom Elsevier. Elsevier.

BogersBogers and Burgers’ Burgers’ workwork was was later later corrected corrected and and refined refined by byOlson Olson and and Cohen Cohen in 1972 in 1972 and 1976, and 1976with, thewith introduction the introduction of two of shears two [shears56,57]. [ It56 can,57]. be It summarizedcan be summarized in Figure in7 asFigure follows: 7 as the follows: first shear the firston a shear {111} γonplane a {111} isγ achieved plane is achieved by 1/6 <112> by 1/6γ partial<112>γ dislocations,partial dislocations averaging, averaging one over one every over second every second(111)γ slip(111) plane,γ slip andplane, the and second the second shear along shear another along another {111}γ plane {111} isγ plane achieved is achieved by 1/6 <112>by 1/6γ <112>partialγ partialdislocations dislocations averaging averaging one over one every over thirdevery (111) thirdγ (111)slip plane.γ slip plane. The former The former is noted is noted T/2 and T/2 the and latter the latterT/3. This T/3. approach This approach is inqualitative is in qualitative agreement agreement with the with observations the observations of the martensite of the martensite formation formationat the intersection at the intersection of hcp plates of hcp or plates stacking or stacking faulted bands faulted on ban twods (111)on twoγ planes. (111)γ planes. The model The model has an hasinteresting an interesting physical physical basis, butbasis its, intrinsicbut its intrinsic T/2 and T/2 T/3 and asymmetry T/3 asymmetry between between the {111} theγ {111}planesγ planes seems seemsto be tooto be strict too tostrict be obtainedto be obtained in a real in a material.real material. The refinedThe refined model model is quite is quite complex, complex and, and does does not notanswer answer anymore anymor thee question the question about about the atomic the atomic trajectories. trajectories. It also raisesIt also questions raises questions about the about origin the of originthe partial of the dislocations, partial dislocations, but we will but come we will back come on thisback point on this in thepoint next in section.the next section. Crystals 2018, 8, 181 10 of 55 Crystals 2018, 8, x FOR PEER REVIEW 10 of 54

Figure 7. Olson and Cohen’s refinement of Bogers and Burgers’ model. The α martensite is formed at Figure 7. Olson and Cohen’s reﬁnement of Bogers and Burgers’ model. The α martensite is formed the intersection of the two planar faults T/2 and T/3. Reprinted from [56] with permission from at the intersection of the two planar faults T/2 and T/3. Reprinted from [56] with permission Elsevier. from Elsevier.

Another hard-sphere model of fcc-bcc transformation was proposed by Le Lann and Dubertret [58]. AnotherThis model hard-sphere contains model some ofessential fcc-bcc transformationingredients, such was as proposed the factby that Le Lannone of and dense Dubertret directions [58]. This model contains some essential ingredients, such as the fact that one of dense directions remains remains invariant <110>γ = <111>α, and the authors qualitatively envisioned the transformation as a invariantwave propagating <110>γ = perpendicularly<111>α, and the to authors this direction. qualitatively In addition, envisioned the the authors transformation proposed as atomistic a wave propagating perpendicularly to this direction. In addition, the authors proposed atomistic structures structures of the well-known {225}γ and {3,10,15}γ habit planes. However, the model seems to be ofwidely the well-known ignored, probably {225}γ and because {3,10,15} it isγ habitquite planes.difficult However, to understand the model, as it seems implies to bethe widely distortion ignored, of a probablyregular octahedron because it ismade quite of difﬁcult 19 atoms to, understand, and is mainly as itgeometric implies the; it does distortion not explain of a regular how octahedronto calculate madethe atom of 19 trajectories atoms, and oris the mainly distortion geometric; matrix. it does not explain how to calculate the atom trajectories or theMore distortion than 150matrix. years after Mügge’s model of deformation twinning, and more than 60 years afterMore the birth than of150 PTMC, years after the exact Mügge’s continuous model of paths deformation related twinning, to the collective and more displacements than 60 years of after the theatoms birth during of PTMC, the lattice the exact deformation continuous remain paths beyond related the to possi the collectivebilities of displacements the classic crystallographic of the atoms duringtheories the of lattice displacive deformation transformations remain beyond in metals. the possibilities We think that of the the classic absence crystallographic of progress should theories be ofattributed displacive to transformationsthe main paradigm, in metals. i.e., the We assumption think that that the absencethe lattice of distortion progress should should be be attributed a shear or to a thecomposition main paradigm, of shears. i.e., the assumption that the lattice distortion should be a shear or a composition of shears. 4. Simple Shear Saved by the Dislocations/Disconnections? 4. Simple Shear Saved by the Dislocations/Disconnections? Simple shear, or its derivative lattice invariant strain, is perfectly adapted to model the Simple shear, or its derivative lattice invariant strain, is perfectly adapted to model the collective collective displacements of the atoms that move all together at the same time. However, simple shear displacements of the atoms that move all together at the same time. However, simple shear has raised has raised important issues for a long time. Atoms are not bricks that can glide onto themselves, as important issues for a long time. Atoms are not bricks that can glide onto themselves, as represented represented in Figure 1. A collective shear displacement of the atoms would imply a magnitude of in Figure1. A collective shear displacement of the atoms would imply a magnitude of shear stress to shear stress to the same order as that of the Young modulus. It is worth recalling the usual the same order as that of the Young modulus. It is worth recalling the usual “demonstration” that was “demonstration” that was given by Frenkel [59] in 1926 and reported in different books [60,61]. For given by Frenkel [59] in 1926 and reported in different books [60,61]. For example, in [60], it is explained example, in [60], it is explained that “The shearing force required to move a plane of atoms over the that “The shearing force required to move a plane of atoms over the plane below will be periodic, since plane below will be periodic, since for displacements x < b/2, where b is the spacing of atoms in the for displacements x < b/2, where b is the spacing of atoms in the shear direction, the lattice resists the shear direction, the lattice resists the applied stress but for x > b/2 the lattice forces assist the applied applied stress but for x > b/2 the lattice forces assist the applied stress. The simplest function these stress. The simplest function these properties is a sinusoidal relation of the form = ( ) ≈ properties is a sinusoidal relation of the form τ = τ sin 2πx ≈ τ 2πx , where τ is the maximum m b m b m , where is the maximum shear stress at the displacement = b/4. For small displacements shear stress at the displacement = b/4. For small displacements the elastic shear strain given by x/a τ µ b isthe equal elastic to µ shearfrom strain the Hooke’s given by law, x/a where is equalµ is tothe shear from modulus, the Hooke’s so that lawτm ,= where2π a and is since the b shear≈ a, themodulus theoretical, so that strength = of a perfect and sincecrystal b is≈ ofa, thethe order theoretical of µ/10 strength.” The sinusoidal of a perfect form crystal was also is of used the byorder Peierls of in/10 his.”famous The sinusoidal paper introducing form was thealso friction used by force Peierls on a in dislocation his famous [62 paper]. However, introducing Frenkel’s the friction force on a dislocation [62]. However, Frenkel’s demonstration is actually misleading. Indeed, there is no reason to believe that the periodicity along the x-axis could explain the stress value at x = Crystals 2018, 8, 181 11 of 55 Crystals 2018, 8, x FOR PEER REVIEW 11 of 54 demonstration is actually misleading. Indeed, there is no reason to believe that the periodicity along 0. What could justify that the value ( = 0) is correlated to the maximum value ( = )? the x-axis could explain the stress value at x = 0. What could justify that the value τ(x = 0) is correlated Actually, the “trick” of the demonstration is hidden in the sinusoidal function. This function seems to the maximum value τ x = b ? Actually, the “trick” of the demonstration is hidden in the sinusoidal to be harmless, but it already4 contains the answer, because it is such that its derivative is function.proportional This to function its maximum. seems to Other be harmless, periodic but functions it already lead contains to completely the answer, different because results. it is such Let that us itsconsider derivative, for example is proportional. The soft to and its maximum. rigid functions Other defined periodic as functionsfollows: lead to completely different results. Let us consider, for example. The soft and rigid functions deﬁned as follows: (Soft) = ( ) (Soft) τ = τ sin3( 2πx ) m b ( √ = 2πx > 0 (1) τ y i f y = sin > 0 (1) m b (Rigid(Rigid)) τ = = √ 2πx −−τm −y i f y == sin <<00 b The sinusoidalsinusoidal function,function, andand thethe softsoft andand rigidrigid functionsfunctions areare shownshown inin FigureFigure8 .8.

Figure 8. Periodic functions ττ with different slopesslopes µμ atat thethe pointspoints ττ == 0. ( a) Sinusoidal function, µμ = 1 1;; ((bb)) softsoft function,function, µμ == 0;0; (c)) inﬁnitelyinfinitely rigidrigid function,function, µμ == ∞..

The soft case gives = =dτ = = 0, and the rigid case gives= d τ = = = ∞. In these two The soft case gives µ dx x=0 0, and the rigid case gives µ dx x=0 ∞ . In these two extreme µ cases,extreme there cases, is no there correlation is no correlation between thebetween shear modulusthe shear modulusand the maximumμ and the maximum stress τm. Onestress could . alsoOne findcould different also find functions different for functions which the for shear which modulus the shear is constant modulus but is theconstant maximum but the shear maximum stresses areshear very stresses different. are very Since different. the result Since is given the result withthe is given premise, with Frenkel’s the premise demonstration, Frenkel’s demonstration is biased; any argumentis biased; based any argument on the periodicity based on of the the structureperiodicity cannot of the be structure correct. Cottrell cannot mentions be correct. in the Cottrell first chaptermentions of hisin theexcellent first book chapter [54 ] of that his Orowan excellent told book him in [54 a] private that Orowan communication told him that in “the a private critical shearcommunication strain should that be “the less critical than this shear when strain a realistic should lawbe less is taken than forthis the when force a realistic between law the is atoms”. taken Cottrellfor the force then between made a rough the atoms”. estimation Cottrell using then a central made a law rough force, estimation and came using to estimate a central this law strain force to, beand about cameµ /30. to estimate He also this gave strain the adviceto be about that “the μ/30. sinusoidal He also functiongave the [..] advice should that be “the replaced sinusoidal by a [centralfunction force] [..] should relation”. be replaced However, by he a did [central not specify force] thatrelation”. the periodicity However, argument he did not should specify be definitely that the discarded.periodicity Despite argument Frenkel’s should error, be definitelyand as shown discarded. by Cottrell, Despite it is correct Frenkel’s to assume error, thatand the as maximumshown by shearCottrell, stress it is associated correct to withassume a simple that the shear maximum is a tenth shear the orderstress ofassociated magnitude with of thea simple shear shear modulus. is a Nabarrotenth the [ 63order] also of took magnitude a more appropriate of the shear function, modulus. as Nabarro suggested [63 by] also Cottrell, took and a more showed appropriate that the orderfunction, of magnitude as suggested of the by friction Cottrell, force and estimated showed that by Peierls the order is correct. of magnitude The fundamental of the friction reason force is notestimated the periodic by Peierls nature is correct. of the lattice, The fundamental but the fact thatreason the is interactions not the periodic between nature atoms of the located lattice in, thebut neighboredthe fact that unit the cells interactions create a betweenlattice friction. atoms This located can in be the understood neighbored by consideringunit cells create the case a lattice of a one-atomfriction. This unit can cell be of Figure understood9a, for by which considering the atom the interaction case of isa oneassumed-atom to unit be fullycell of elastic. Figure 9a, for whichIf the the atom trajectory interaction of the is atoms assumed could to be continuously fully elastic. follow a simple shear strain, they would interpenetrateIf the trajectory so much of that the the atoms repulsive could force continuously at the interface follow would a simple reach shear very strain, high values. they would In the 2Dinterpenetrate example√ of Figureso mu9cha, that by noting the repulsive the diameter force (atd) the of theinterface atoms, would the rate reach of interpenetration very high values. would In the be h 3 12D− exampled = 1 − of2 Figure≈ 13.4% 9a,, i.e., by farnoting above the the diameter usual elastic (d) of limit the atoms, (around the 0.1–1%). rate of Ininterpenetration the three-dimensional would q (3D)be 1 case− of= an 1 − ABCABC√3 ≈ 13 stacking,.4%, i.e., the far layers above are separated the usual by elastic a distance limit of (aroundh = 1 3 0.1d,– and1%). a simple In the 2 2 8 shearthree- displacementdimensional ( of3D the) case atoms of a ofn ABCABC the upper stacking, layer relative the layers to the twoare separated atoms of theby lowera distance layer of would ℎ = q 2 reduce the distance between them down to d0 = 1 h2 + d , which induces an interpenetration of , and a simple shear displacement of the atoms2 of the4 upper layer relative to the two atoms of q 1 − d0 = 1 − 11 = 8.3%. Here again, this value is too high to be compatible with the elastic limits d 12 the lower layer would reduce the distance between them down to ′ = ℎ + , which induces of the metals. Thus, even if incorrectly demonstrated by Frenkel (at least the sinusoidal function an interpenetration of 1 − = 1 − = 8.3%. Here again, this value is too high to be compatible Crystals 2018, 8, x FOR PEER REVIEW 12 of 54

Crystals 2018, 8, 181 12 of 55 with the elastic limits of the metals. Thus, even if incorrectly demonstrated by Frenkel (at least the sinusoidal function should be replaced by a more realistic function, as suggested by Cottrell), it is notshould possible be replaced, with by reasonable a more realistic stress, function,to induce as a suggested collective by movement Cottrell), it of is not the possible, atoms on with a crystallographicreasonable stress, plane. to induce This a result collective was movementconfirmed ofwith the the atoms experiments on a crystallographic made by Bragg plane. and This Lomer result withwas conﬁrmed sub-millimeter with the soap experiments bubble rafts made [64 by]. Please Bragg and note Lomer that the with real sub-millimeter argument is soap the bubble atom size/raftsinteratomic [64]. Please noteenergy, that and the real not argument the lattice is the periodi atomcity. size/interatomic The hard-sphere energy, assumption and not the lattice is a rudimentaryperiodicity. The expression hard-sphere of assumption the interatomic is a rudimentary energy. Similar expression calculations of the interatomic can also be energy. applied Similar to metalliccalculations glasses, can despite also be appliedtheir quasi to metallic-amorphous glasses, state. despite their quasi-amorphous state.

Figure 9. Schematic two-dimensional (2D) representation of a collective displacement of atoms during deformation twinning, according to (a) simple shear (with internal friction) or (b) angular distortion Figure 9. Schematic two-dimensional (2D) representation of a collective displacement of atoms (in a free crystal). during deformation twinning, according to (a) simple shear (with internal friction) or (b) angular distortion (in a free crystal). Anyway, the conclusion seems unescapable: the stresses that would be required to shear a perfect crystalAnyway, are far too the high. conclusion Dislocations seems are unescapable: thus required the for stresses any shear that deformation, would be required whether to by shear gliding, a perfecttwinning, crystal or martensite are far too transformation.high. Dislocations Dislocation are thus required theory was for any emerging, shear deformation, and was soon whether successful by gliding,in 1940–1950 twinning in explaining, or martensite the plasticity transformation. of metals, Dislocation recrystallization, theory was and emerging, many other and phenomena was soon successfulin materials in science. 1940–1950 Brilliant in explain confirmationsing the plasticity of the existence of metals, of dislocationsrecrystallization were, and also many obtained other by phenomenaTEM in the early in materials 1950s [65 science.]. Consequently, Brilliant confirmations most of the researchers of the existence came to of assume dislocations that dislocations were also obtainedwere the by cause TEM and in the the early most 1950 fundamentals [65]. Consequentl part of deformationy, most of the twinning. researchers According came to assume to Hardoin that dislocationsDuparc [4], thewere term the “twin cause dislocation” and the most was fundamental coined by Seitz part and of Readdeformation in 1941 [twinning.66], and this According concept hasto Hardoinbeen pursued Duparc till [4 now], the by term Vladimirskii “twin dislocation” [67], Frank was and coined van der by MerweSeitz and [68 Read], Sleeswyk in 1941 [[6966],], Christianand this conceptand Mahajan has been [2,70 pursued], and many till now other by great Vladimirskii scientists. [67 Let], Frank us cite and Cottrell van der [54 ]Merwe when he[68 explains], Sleeswyk the [arguments69], Christian and and the Mahajan reasoning [2 at,70 the], and origin many of theother concept great ofscientists. twinning Let dislocations. us cite Cottrell “Ithas [54] often when been he explainssuggested the that arguments mechanical and the twinning reasoning takes at place the origin by the ofcontinuous the concept growth of twinning on an dislocations. atomic scale “It of hastwinned often materials, been suggested and the that general mechanical arguments twinning for a takesdislocation place mechanism by the continuous are the growthsame as on in thean atomiccase of scale slip; first,of twinned it is scarcely materi believableals, and the that general the atom arguments should allfor move a dislocation simultaneously mechanism and, are second, the sametwinning as occursin the at case stresses of slip; comparable first, it with is those scarcely for slip, believable i.e., far below that the the theoreticalatom should strength all move of the simultaneouslyperfect lattice.” andPlease, second note that, twinning it is supposed occurs that at stresses all the atoms comparable move withsimultaneously, those for slip i.e.,, alli.e., at far the belowsame time the—but theoretical we will strength come backof the later perfect on this lattice.” point. Please Cottrell note continues that it is directly,supposed by that proposing all the atoms a first moveversion simultaneously, of what will become i.e., all the at “polethe same mechanism”: time—but we will come back later on this point. Cottrell continues directly, by proposing a first version of what will become the “pole mechanism”: Since a new configuration is produced by twinning, the dislocations that cause it must be imperfect. While it is usually not difficult to discover a suitable imperfect dislocation to cause

the required shear of neighbouring planes as it passes between them, the problem is to explain how twinning develops homogeneously through successive planes. The homogeneous shear Crystals 2018, 8, x FOR PEER REVIEW 13 of 54

Since a new configuration is produced by twinning, the dislocations that cause it must be Crystalsimperfect.2018, 8, 181 While it is usually not difficult to discover a suitable imperfect dislocation to13 cause of 55 the required shear of neighbouring planes as it passes between them, the problem is to explain requiredhow twinning a twinning develops dislocation homogeneously on every throughplane without successive exception, planes. which The homogeneousseems unlikely, shear orrequired the motion a twinning of a single dislocation dislocation on fromevery plane plane to without plane in exception a regular, manner.which seems Cottrell unlikely and , or Bilbythe motio haven recentlyof a single suggested dislocation a mechanism, from plane based to plane on that in ofa regular Frank andmanner. Read, Cottrell whereby and the Bilby latterhave recentlyprocess can suggested occur in a certain mechanism crystals, based containing on that dislocations. of Frank and Figure Read, 51 whereby [reported the in latter Figureprocess 10 canb] illustratedoccur in certain the mechanism. crystals containing Here OA, dislocations. OB, and OC, Figure represent 51 [reporte three dislocationd in Figure 10b] linesillustrated and CDE the mechanism. is a slip or twinning Here OA plane., OB, Theand dislocationOC, represent OC three (“the dislocation sweeping” lines dislocation) and CDE is anda slip its or Burgers twinning vector plane. both Thelie in dislocation this plane (the OC “sweeping” (“the sweeping plane)” dislocation) and the dislocation and its can Burgers rotatevector inboth the lie plane in this about plane the point(the “ O.sweeping If it is a” unitplane) dislocation and the dislocation and remains can in therotate plane in the as it plane rotatesabout the a slip point band O. isIf formed.it is a unit The dislocation requirements and forremains twinning in the (or plane a shear as transformation)it rotates a slip band to is occurformed. are The as follows: requirements 1. The sweepingfor twinning dislocation (or a shear must transformation) be imperfect and to produceoccur are the as correct follows: 1. shearThe sweeping displacement disloca ontion the must sweeping be imperfect plane; 2. and Successive produce sweeping the correct planes shear must displacement be joined to on the formsweeping a helical plane surface.; 2. Successive [54] sweeping planes must be joined to form a helical surface [54].

grainFrank boundaries and Read’s [ RE model,, mentioned mentioned by Cottrell by Cottrell,, was was explained explained a little a little earlier earlier in in his his book; book; it it is a formerformer modelmodel ofof whatwhat isis nownow knownknown asas “Frank–Read”“Frank–Read” sourcesource ofof dislocations.dislocations. In thisthis model,model, FrankFrank and ReadRead imagined a dislocation rotating around a point,point, producing a new slip for each revolution, as explainedexplained by CottrellCottrell inin thethe FigureFigure 4949 ofof hishis bookbook [[5454]] reportedreported inin FigureFigure 1010aa,b.,b. CottrellCottrell made aa parallelparallel withwith thethe spiralsspirals formedformed at the surfacesurface of a growinggrowing crystal,crystal, eveneven ifif itit waswas clearlyclearly statedstated thatthat thisthis spiralspiral dislocationdislocation resultsresults fromfrom growthgrowth andand notnot fromfrom deformation.deformation. TheThe Frank-ReadFrank-Read modelmodel usedused by CottrellCottrell toto explainexplain twinningtwinning waswas thethe ancestorancestor ofof thethe “pole“pole mechanism”mechanism” model.model. ItIt waswas followedfollowed by Cottrell and and Bilby’s Bilby’s model [71],], reﬁnedrefined by Sleeswyk [69,72] and again later by Venables [ [73]] (Figure(Figure 10 10cc).). Sleeswyk Sleeswyk [ [6969]] also also imagined how the twinning twinning dislocations dislocations at the the interface interface could could dissociate at the interface and move away, andand hehe introducedintroduced thethe conceptconcept of “emissary“emissary dislocations”.dislocations”. General considerationsconsiderations onon twinningtwinning mechanisms mechanisms and and twinning twinning dislocations dislocations in in metals metals can can be be found found in Ref.in Ref. [2, 3[2].,3 An]. An updated updated review review was was recently recently published published by by Mahato Mahato et al.,et al., in Sectionin Section 4 of 4 [of74 ].[74].

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FigureFigure 10. 10.Schematic Schematic representation representation of twinningof twinning dislocations dislocations and poleand mechanism.pole mechanism. (a) An ( earlya) An model early mademodel by made Frank by and Frank Read and explains Read explains the dislocation the dislocation “multiplication” “multiplication” in a crystal, in froma crystal, Cottrell’s from bookCottrell’s [54]; (bookb) this [54 model]; (b) was this used model by Cottrell was used to explain by Cottrell how dislocationsto explain how can build dislocations twins; ( c can) Scheme build of twins the pole; (c) mechanismScheme of the proposed pole mechanism by Venables, proposed adapted by from Venables, [73]. adapted from [73].

TEM observations confirmed the existence of dislocations, dislocation pile-ups, dislocation TEM observations conﬁrmed the existence of dislocations, dislocation pile-ups, dislocation dissociations, climb, etc. However, to our best knowledge, the spiraling dislocations that should be dissociations, climb, etc. However, to our best knowledge, the spiraling dislocations that should be expected from a pole mechanism have never been proven in metals. Spiraling/helical dislocations CrystalsCrystals2018 2018, ,8 8,, 181 x FOR PEER REVIEW 1414 ofof 5554

expected from a pole mechanism have never been proven in metals. Spiraling/helical dislocations werewere observedobserved by by TEM TEM in in cast cas Al–Cut Al–Cu alloys alloys [75 ],[75 but], but they they come come from from a vacancy a vacancy collapse collapse and not and from not stresses.from stresses. What areWhat often are presented often presented as TEM as images TEM ofimages “twinning of “twinning dislocations” dislocations” are dislocation are dislocation pile-ups inpile front-ups of in micro-twins front of micro [74,76-twins–79], such[74,76 as–79 those], su shownch as those in Figure shown 11. inThey Figure are generally 11. They interpreted are generally in terminterpreted of dissociations in term of of dissociations full dislocations of full associated dislocations with complexassociated pole with mechanisms; complex pole however, mechanisms to the; besthowever of our, to knowledge, the best of theour initialknowledge, spiraling the dislocation initial spiraling source dislocation has never source been shown. has never been shown.

FigureFigure 11. 11. TEM image of dislocation pileups in in front front of of micro-twins micro-twins presented presented as as “twinning “twinning dislocations”,dislocations”, ( a(a)) in in an an Fe–Mn–Si–Al Fe–Mn–Si–Al Transformation Transformation Induced Induced Plasticity Plasticity (TRIP) (TRIP steel,) steel, from from [74 ];[74 and]; and (b) in(b a) in tetragonal a tetragonal Ni–Mn–Ga Ni–Mn– alloy.Ga alloy. Reprinted Reprinted from from [79] with[79] with permission permission from from Elsevier. Elsevier.

The notion of “twinning dislocations” has been progressively enlarged to displacive phase The notion of “twinning dislocations” has been progressively enlarged to displacive phase transformations. For fcc-hcp and hcp-fcc transformations, the classical models imply the coordinate transformations. For fcc-hcp and hcp-fcc transformations, the classical models imply the coordinate displacement of partial Shockley dislocations, which are supposed to be created by a pole or similar displacement of partial Shockley dislocations, which are supposed to be created by a pole or complex mechanism [80–83]. For fcc-bcc transformations, we have seen in Figure 7 that Olson and similar complex mechanism [80–83]. For fcc-bcc transformations, we have seen in Figure7 that Cohen [56,57] used partial dislocations in order to correct the initial hard-sphere Bogers and Burgers Olson and Cohen [56,57] used partial dislocations in order to correct the initial hard-sphere Bogers model [53]. The use of dislocations as a fundamental part of the transformation models was and Burgers model [53]. The use of dislocations as a fundamental part of the transformation generalized with the introduction of the concept of “disconnection”, which became the core of the models was generalized with the introduction of the concept of “disconnection”, which became “topological model” (TM) developed by Hirth, Pond, and co-workers [84–88]. A disconnection is a the core of the “topological model” (TM) developed by Hirth, Pond, and co-workers [84–88]. kind a dislocation located at the interface of two misoriented crystals, which can be of same phase A disconnection is a kind a dislocation located at the interface of two misoriented crystals, which can (for twinning) or different phases (for phase transformation)—it is the assembly of a classical be of same phase (for twinning) or different phases (for phase transformation)—it is the assembly dislocation and a geometrical “step”, both required to assure the compatibly at the interface. The of a classical dislocation and a geometrical “step”, both required to assure the compatibly at the dislocations/disconnections are assumed to be glissile, because twinning and martensitic interface. The dislocations/disconnections are assumed to be glissile, because twinning and martensitic transformations are envisioned as the consequence of the propagation of the interface, and transformations are envisioned as the consequence of the propagation of the interface, and researchers researchers came to introduce the concept of “glissile interfaces” [70] by extrapolating the concept of came to introduce the concept of “glissile interfaces” [70] by extrapolating the concept of glissile glissile dislocations. Despite its efficiency in describing locally, at a nanometer scale, the structure of dislocations. Despite its efficiency in describing locally, at a nanometer scale, the structure of the the interface between the parent matrix and the daughter crystal, the TM does not answer yet the interface between the parent matrix and the daughter crystal, the TM does not answer yet the most most important questions: where the dislocations come from? How are they created? How can they important questions: where the dislocations come from? How are they created? How can they move collectively and in coordinate way to build a new phase? How can they move at the sound move collectively and in coordinate way to build a new phase? How can they move at the sound velocity? Why is twinning activated when the temperature decreases? Why do hcp metals exhibit so velocity? Why is twinning activated when the temperature decreases? Why do hcp metals exhibit many twinning modes with so few dislocation modes? Atomistic simulations [89,90] show that some so many twinning modes with so few dislocation modes? Atomistic simulations [89,90] show that dislocation loops can be created “ex-nihilo” (helped by the external stress field and dislocation some dislocation loops can be created “ex-nihilo” (helped by the external stress field and dislocation pile-ups) and that these loops can be the very first nucleation step of twinning. These studies pile-ups) and that these loops can be the very first nucleation step of twinning. These studies propose propose partial responses to the first two questions, but the others remain open. partial responses to the first two questions, but the others remain open.

5.5. AngularAngular DistortiveDistortive TransformationsTransformations

5.1.5.1. TwinningTwinning DislocationsDislocations ReplacedReplaced bybyTransformation Transformation Waves Waves and and Dislocations Dislocations Induced Induced by by Twinning Twinning LetLet us us now now explain explain the way the we way envision we deformation envision deformation twinning and martensitictwinning and transformations. martensitic Thetransfo TEMrmations. images The of the TEM dislocation images of pile-upsthe dislocation in front pile the-ups micro-twins, in front the such micro as-twins those, ofsuch Figure as those 11, doof notFigure prove 11, thatdo not the prove dislocations that the are dislocations the cause of are twinning; the cause they of twinning; may be as they well may a consequence be as wellof a twinning.consequence Despite of twinning. the fact Despite that for the lastfact 60that years, for the nearly last all60 theyears theoretical, nearly modelsall the theoretical and experimental models and experimental observations have been interpreted in terms of twinning dislocations or Crystals 2018, 8, 181 15 of 55 observations have been interpreted in terms of twinning dislocations or disconnections, it is difficult to believe that dislocations could be created in a periodic sequential way by a pole mechanism, and that they could propagate at the speed of sound and produce a new crystallographic structure (twin or martensite). The high speed of twin/martensite propagation, and the fact they are favored by low temperatures and high deformation rates, should constitute important arguments against theories based on dislocations. Cottrell is right when he says that “it is scarcely believable that the atom should all move simultaneously” [54], because that would imply that all the bonds break instantaneously, which would require very high energy for an instantaneous moment. Nevertheless, an important point should be considered: in non-quantum physics, the word “instantaneously” has no meaning. Information cannot travel faster than a limited speed, i.e., the speed of light for electromagnetism, or the speed of sound for waves in materials (even if supersonic dislocations are reported for extreme conditions). Let us explain this general idea with the classical Ising model. Reversing a spin has an influence on the neighboring spins, and when the temperature is close to the transition (Curie) temperature, i.e., when the local spin–spin interaction energy is not masked by the high temperature Brownian motion, nor counterbalanced by the low temperature global magnetic field, this influence becomes predominant on the system, and the magnetic susceptibility and the correlation length become infinite [91]. The effect of flipping a spin on the other spins is not instantaneous, but travels at the speed of light. The spin-up and spin-down domains are thus not formed instantaneously; they appear under the effect of a solitary phase transition wave, i.e., a soliton. It can be imagined as a domino cascade or as a seismic wave. This idea is not new, but it has appeared only sporadically in metallurgy literature, with long periods of oversight. Machlin and Cohen [42] envisioned martensitic transformation in Fe–30Ni alloys as a strain wave; Nishiyama wrote in his book that “martensite transformation is like Shôgidaoshi (a Japansese word meaning ‘falling one after another in succession’) rather than like military notion” [37]. Meyers [92] proposed an equation of martensite growth that takes into account the velocities of the longitudinal and shear elastic waves. Le Lann and Dubertret [58] developed their crystallographic model of fcc-bcc transformation by maintaining the direction <110>γ = <111>α invariant, because they considered it to be the wave vector of the transformation. With this method of envisioning displacive phase transformations, there is no need of dislocations, at least not in a free single crystal. In 1984, Barsch and Krumansl introduced the concept of a soliton creating boundaries without interface dislocation for ferroelastic transitions [93,94]. Solitary waves were also introduced by Flack in a one-dimensional shear model [95]. Unfortunately, there are few works in metallurgy based on this physical idea. A Russian team lead by Kashchenko and Chashchina [96] are developing the concept of dynamics and wave propagation of martensite in steel, but the physical details (l-waves, s-waves) are not easy to understand for a non-physicist, and would need experimental confirmations. We consider here that during deformation twinning or martensitic transformation in a free crystal, the atoms move following a solitary “phase transformation wave”, i.e., a phase soliton, as geometrically shown in Figures9a and 12. Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

Burgers: [111]α = [100]ε and� (110)α�//(001)ε (16)

TheThe displacementdisplacement ofof eacheach atomatom hashas anan influenceinfluence ofof thethe neighboringneighboring atoms,atoms, thethe wayway thatthat aa spinspin flipflip hashas anan influenceinfluence of the neighboring neighboring spins spins in in an an Ising Ising model. model. If Ifthe the parent parent single single crystal crystal is fr isee, free, the thelattice lattice distortion distortion becomes becomes a macroscopic a macroscopic shape shape distortion. distortion. This This idea idea is quite is quite new new to tous us,, and and should should be bedeveloped; developed; however, however, it itseems seems possible possible that that the the parent/martensite parent/martensite accommodation accommodation area, area, in inorange orange in inFigure Figure 12 12, is, is spread spread over over a alarge large area area,, such such that that the the distances distances between between the neighboring atomsatoms remainremain lowerlower thanthan thethe limitlimit imposedimposed byby thethe criticalcritical yieldyield strain,strain, andand thethe interfaceinterface isis accommodatedaccommodated purelypurely elastically.elastically. SuchSuch aa modelmodel willwill bebe detaileddetailed inin thethe lastlast section.section. ThereThere isis nono reasonreason toto believebelieve thatthat thethe dislocation-freedislocation-free model model proposed proposed by Barsch by Barsch and Krumansl and Krumansl is limited to is ferroelastics. limited to For ferroelastics. polycrystalline For materialspolycrystalline or non-free materials crystals, or non one-free must crystals, distinguish one must two cases,distinguish depending two cases on the, depending martensite on size. the Whenmartensite the size size. of theWhen martensite the size domain of the martensite and its elastic domain accommodation and its elastic zone accommodation are significantly zone lower are thansignificantly the grain lower size, the than distortion the grain can size, still the be distortion elastically can accommodated still be elastically inside accommodated the grain, at least inside during the high-speedgrain, at least transitory during stageshigh-speed of the transitory transformation stages process.of the transformation The elastic zone process. is then The quickly elastic relaxed zone is bythen the quickly formation relaxed of dislocation by the formation arrays in of the dislocation surrounding arrays matrix, in the and surrounding by interfacial matrix dislocations, and by (disconnections).interfacial dislocations When (disconnections). the martensite grows When and the martensite the size of grows the elastic and accommodationthe size of the elastic zone becomesaccommodation comparable zone with becomes the grain comparable size, the incompatibilities with the grain size, must the then incompatibilities be plastically accommodated, must then be asplastically schematized accommodated, in Figure 13. The as schematized volume change in associatedFigure 13. with The the volume phase change transformation associated is averaged with the inphase all the transformation grains, and distributed is averaged in in a allwhole the sample grains, and (it can distributed be measured in a bywhole dilatometry), sample (it but can the be measured by dilatometry), but the deviatoric parts of the lattice distortion are “blocked” by the grain boundaries. The accommodation zone cannot be spread any further, and dislocations become necessary. Crystals 2018, 8, 181 17 of 55 deviatoric parts of the lattice distortion are “blocked” by the grain boundaries. The accommodation Crystalszone cannot 2018, 8, bex FOR spread PEER anyREVIEW further, and dislocations become necessary. 17 of 54

Figure 13. ElasticElastic and and plastic plastic accommodation accommodation zone zone around around the martensite the martensite product product in a parent in a parent grain ofgrain a polycrystalline of a polycrystalline material. material. (a) During (a) During the transitory the transitory stages stages of the of martensite the martensite propagation, propagation, the parent/martensitethe parent/martensite incompatibilities incompatibilities can can be bespread spread to to be be elastically elastically accommodated, accommodated, and and martensite can bebe formedformed asas an an acoustic acoustic wave; wave (;b ()b when) when the the martensite martensite size size becomes becomes comparable comparable to the to grainthe grain size, size,the incompatibilities the incompatibilities must must be plastically be plastically accommodated. accommodated.

5.2. The Concept of Angular Distortion Freed of of the the limitations limitations imposed imposed by the by shear the paradigm shear paradigm and the associated and the twinning associated dislocations, twinning dislocations,one can now imagineone can how now the imagine atoms howmove the during atoms displacive move during transformations. displacive Of transformations. course, atoms are Of course,not hard atoms spheres, are butnot in hard some spheres, metals (fcc,but inbcc, some and metals hcp with (fcc, the bcc, ideal and packing hcp with ratio), the the ideal hard-sphere packing ratio),assumption the hard is a-sphere good starting assumption point is [8 a]. good Simple starting (or pure) point shears [8]. Simple are deformation (or pure) shear modess are at deformationconstant volume, modes which at constant is well-adapted volume, which for plasticity is well-adapted by dislocation, for plasticity but is notby dislocation, suited for displacive but is not suitedtransformations for displacive in these transformations metals. Indeed, in it isthese known metals. from Indeed, Kepler in it 1611 is known (Kepler’s from conjecture Kepler inbecame 1611 (Kepler’sa theorem conjecture in 1998, thanks became to Hales’ a theorem demonstration) in 1998, thatthanks the two to Hales’ densely-packed demonstration) structures that are the fcc twoand densehcp, whichly-packed means structures that any are intermediate fcc and hcp, state which between means twothat fccany crystals, intermediate two hcpstate crystals, between or two an fcc crystals,and a hcp two crystal hcp crystals, should have or an a fcc higher and volumea hcp crystal than should that of anhave fcc a or higher hcp crystal. volume Simple than that shear of an is notfcc orcompatible hcp crystal with. Simple the hard-sphere shear is not assumptioncompatible with if one the is hard looking-sphere for continuous assumption deformation if one is looking models. for continuousLet us look again deformation at Figure models.9a. When Let a ussimple look shear again is at applied Figure continuously 9a. When a to simple a crystal, shear the is distance applied continuouslyh between the to atomic a crystal, layers theis distance supposed h between to be constant, the atomic as if layers the crystal is supposed were constrained to be constant, between as if thetwo crystal infinitely-rigid were constrained horizontal between walls. Now,two in withfinitely a hard-sphere-rigid horizontal model, walls. we consider Now, with the a case hard in-sphere which model,the crystal we consider is free to the move case in in the which direction the crystal normal is free to the to move atomic in layers,the direction as illustrated normal into Figuresthe atom9bic layers,and 14 a. as Of illustrated course, the in atomsFigure cans 9b slightly and 14a interpenetrate. Of course, theeach atoms other canaccording slightly to interpenet a reasonablerate elastic each otherlimit (thataccording should to bea reasonable defined), butelastic on firstlimit approximation (that should be itdefined), is assumed but thaton first the approximation sphere size is only it is assumedslightly elastically that the sphere relaxed, size such is only that theslightly trajectories elastically of the relaxed atoms, such are the that same the trajectories as those of hardof the spheres atoms (infinitelyare the same rigid) as those with aof slightly hard sphe lowerres diameter(infinitely than rigid) the with initial a slightly one, as lower shown diameter in Figure than 14b. the initial one, as shown in Figure 14b. CrystalsCrystals 20182018, ,88, ,x 181 FOR PEER REVIEW 1818 of of 54 55

FigureFigure 14. 14. AngularAngular distortive distortive model, model, ( (aa) )with with hard hard-spheres;-spheres; ( (bb)) with with elastic elastic-hard-hard spheres. spheres.

Thus,Thus, on on first first approximation, approximation, the the atomic atomic displacements displacements are are not not essentially essentially modified modified by by the the elasticityelasticity of of the the spheres; spheres; the the trajectories trajectories remain remain close close to to that that of of “rolling “rolling balls”. balls”. Now, Now, if if the the lat latticetice notednoted by by the the vectors vectors ( (aa, ,bb)) in in Figure Figure 14a14a is is considered, considered, the the distortion distortion is is no no longer longer a a simple simple shear shear alongalong aa; ;it it is is a a disto distortionrtion in in which which the the length length of of the the vector vector bb remainsremains constant constant (it (it is is and and remains remains the the atomatom diameter) diameter),, and and the the angle angle between between the the vectors vectors aa andand bb, ,noted noted here here as as ββ, ,continuously continuously changes. changes. ThisThis angle angle becomes the uniqueunique parameterparameter of of the the lattice lattice distortion; distortion; in in physics physic wes we would would say say that that it is itthe is theorder order parameter parameterof the of transition.the transition. The The trajectories trajectories of the of atoms the atoms naturally naturally become become arcs of arcs a circle, of aand circle, the andlattice the distortion lattice distortion is not a shear is not strain a shear anymore, strain but anymore, an angular but distortion. an angular With distortion. such a simple With approach, such a simplethe shape approach, of the crystal the shape is given of bythe the crystal same isdistortion given by as the that same of the distortion atomic displacements. as that of the Of atomic course, displacements.in polycrystalline Of course materials,, in polycrystalline a grain is not free materials, to arbitrarily a grain deform,is not free due to toarbitrarily the surrounding deform, due grains, to theand surroundingcomplex accommodation grains, and modes complex by variant accommodation coupling and modes by dislocations by variant are required. coupling and by dislocations are required. 5.3. Accommodation Dislocations and Disclinations 5.3. Accommodation Dislocations and Disclinations The accommodation dislocations are not randomly distributed; they are such that their strain fieldThe compensates accommodation for the deviatoricdislocations parts are ofnot the randomly angular distortion,distributed; and they constitute are such the that plastic their trace strain of fieldthe mechanism. compensates In for our the opinion, deviatoric these parts dislocations of the angular are those distortion shown, and in theconstitute TEM images the plastic of Figure trace of11 . theThe mechanism. elastic field associatedIn our opinion, with thethese dislocation dislocations arrays are shouldthose shown be equal in tothe the TEM back-strains images of between Figure 11 the. Themartensite elastic productfield associated and the parentwith the crystal. dislocation These dislocationarrays should arrays be areequal very to probably the back at-strains the origin between of the thecontinuous martensite features product observed and the inthe paren Electront crystal. Back These Scatter dislocation Diffraction arrays (EBSD) are pole very figures probably of martensitic at the originsteels (see of the next continuous section). It features can be expected observed that in for the softElectron parent/hard Back Scatter daughter Diffraction phases, the (EBSD distortion) pole figuresis accommodated of martensitic inside steels the (see parent next phase;section). for It hardcan be parent/soft expected that daughter for soft phases, parent/ thehard distortion daughter is phases,accommodated the distortion inside the is accommodated daughter phase, inside and one the can parent expect phase an equal; for proportionhard parent/ forsoft deformation daughter phases,twinning. the According distortion is to accommodated this model, the inside mesoscopic the daughter strain field phase, should and onebe dictated can expect by the an equphaseal proportiontransformation for deformation distortion, andtwinning. the exact According and detailed to this knowledge model, the ofmesoscopic the structure strain of thefield dislocations should be dictatedadopted by by the the phase material transformation to produce this distortion, strain field and is the not exact required and to detailed explain knowledge it. As the of lattice the structuredistortion of is the now dislocations modelled by adopted an angular by the distortion, material the to dislocationproduce this arrays strain should field beis not formed required in order to explainto compensate it. As the for latti thece angulardistortion deficit is now (or modelled excess) implied by an angular by the distortion.distortion, the The dislocation most appropriate arrays shouldplastic be mode formed for suchin order types to compensate of accommodation for the angular implies deficit the concept (or excess) of “disclination”. implied by the This distortion. concept Thewas most first introduced appropriate in plastic 1907 by mode Volterra for [such97], who types considered of accommodation two types implies of defects the in concept a periodic of “disclination”.solid: the rotational This concept dislocations was first (disclinations) introduced and in 1907 the translationalby Volterra [ dislocations97], who considered (simply referredtwo types as ofdislocations defects in nowadays). a periodic The solid: disclination the rotational strength dislocations is given by an (disclinations) axial vector w , and called the the translation Frank vector,al dislocationsencoding the (simply rotation referred needed to as close dislocations the system nowadays). in a similar The way disclination that the dislocation strength strength is given is by given an axialby its vector Burgers w, called vector theb, encodingFrank vector, the translationencoding the needed rotation to closeneeded the to Burgers close the circuit. system If dislocationsin a similar wayconstitute that the a dislocation fundamental strength part of is metallurgy,given by its Bu disclinationsrgers vector are b, encoding less known. the translation The theory needed has been to close the Burgers circuit. If dislocations constitute a fundamental part of metallurgy, disclinations are less known. The theory has been developed by Romanov [98], and by Kleman and Friedel [99], Crystals 2018, 8, 181 19 of 55 Crystals 2018, 8, x FOR PEER REVIEW 19 of 54 developedbut the applications by Romanov are [98 mainly], and by limited Kleman to highly and Friedel deformed [99], but metals the applications [100]. To our are knowledge, mainly limited only toMüllner highly deformed and co-workers metals [ 100 use]. To discli ournations knowledge, to calculate only Müllner the and strain co-workers field and use strain disclinations energy to of calculatehierarchically the strain twinned field and structures strain energy formed of hierarchically by deformation twinned twinning structures [101 formed] and by by deformation martensitic twinningtransformation [101] and in Ni by–Mn martensitic–Ga Heusler transformation alloys [102]. in Their Ni–Mn–Ga models Heuslerare based alloys on the [102 classical]. Their concept models of areshear, based but on it the could classical of interest concept to of see shear, if the but classic it could concept of interest could to be see substituted if the classic by concept that of could angular be substituteddistortion. by The that disclinations of angular distortion. shown in Figure The disclinations 15 would beshown the naturalin Figure a ccommodation15 would be the modes natural of accommodationpositive or negative modes angular of positive misfits. or negative angular misfits.

FigureFigure 15. 15. VolterraVolterra cut cut for for a a negative negative wedge wedge disclination disclination,, where where material material is inserted is inserted (a–c (),a – andc), andalternative alternative Volterra Volterra cut cutfor a for positive a positive wedge wedge disclination disclination,, where where material material is removed is removed (d–f). Fig (d–uref). Figureand caption and caption reprinted reprinted from Müllner from Müllner and King and King[102] [with102] permission with permission from fromElsevier. Elsevier.

In this approach, martensitic transformations and deformation twinning are not the result of a In this approach, martensitic transformations and deformation twinning are not the result of cause that would be the sweep of “twinning dislocations”, or of the propagation of a “glissile a cause that would be the sweep of “twinning dislocations”, or of the propagation of a “glissile interface”; instead, they directly result from a lattice distortion, whose driving force is (a) for interface”; instead, they directly result from a lattice distortion, whose driving force is (a) for martensite, martensite, the difference of chemical energy between a parent phase and its daughter phase; and (b) the difference of chemical energy between a parent phase and its daughter phase; and (b) for for twinning, the difference of mechanical energy given by the work of the applied stress along the twinning, the difference of mechanical energy given by the work of the applied stress along the distortion path. The interface and accommodating dislocations in its surroundings then appear as distortion path. The interface and accommodating dislocations in its surroundings then appear as the consequences of the distortion. Large volumes of parent phase are suddenly transformed into the consequences of the distortion. Large volumes of parent phase are suddenly transformed into daughter phase (at the acoustic wave velocity), and the parent–daughter interface created by the daughter phase (at the acoustic wave velocity), and the parent–daughter interface created by the distortion is made of glissile and sessile dislocations/disconnections. At mesoscale, these dislocations distortion is made of glissile and sessile dislocations/disconnections. At mesoscale, these dislocations induce rotation gradients (disclinations), such as those schematically represented in orange in the induce rotation gradients (disclinations), such as those schematically represented in orange in the hexagon–square transformation of Figure 12. This way of envisioning deformation twinning and hexagon–square transformation of Figure 12. This way of envisioning deformation twinning and martensite transformation completely reverses the usual paradigm, which gives a central driving martensite transformation completely reverses the usual paradigm, which gives a central driving role role to the dislocations in the transformation mechanism. There is no need here to imagine a to the dislocations in the transformation mechanism. There is no need here to imagine a hypothetical hypothetical complex pole mechanism, because the dislocations are now created by twinning or by complex pole mechanism, because the dislocations are now created by twinning or by martensitic martensitic transformation; similarly, there no need to impose glissile properties to the interface. It is transformation; similarly, there no need to impose glissile properties to the interface. It is true that true that plasticity is required to accommodate the lattice distortion in polycrystalline metals; plasticity is required to accommodate the lattice distortion in polycrystalline metals; however, in no however, in no way does that mean that dislocations/disconnections are the cause or can explain way does that mean that dislocations/disconnections are the cause or can explain the distortion. the distortion. 6. Angular Distortion versus the Phenomenological Theory of Martensite Crystallography, Illustrated6. Angular with Distortion a Simple versus Example the Phenomenological Theory of Martensite Crystallography, Illustrated with a Simple Example The concept of angular distortion is so simple and natural that it can be directly applied to martensiticThe concept transformations, of angular without distortion requiring is so simple the four and (or natural more) matrices that it can of the be PTMC.directly Let applied us use to againmartensitic the example transfor ofmations, a 2D hexagonal–square without requiring phase the transformationfour (or more) ofmatrices Figure of12 .the The PTMC. classical Let PTMC us use decomposesagain the example this transformation of a 2D hexagonal into– twosquare distinct phase paths. transformation The ﬁrst pathof Figure is the 12 matrix. The classical product PTMCR.U, whichdecomposes says that this ﬁrst transformation a (Bain) stretch intoU is two applied, distinct and paths. then a rotationThe firstR pathmust is adjust the ma thetrix stretch product in order R.U, which says that first a (Bain) stretch U is applied, and then a rotation R must adjust the stretch in order to maintain an invariant line (here the x-axis). The notation U is now preferred to B (Bain) to Crystals 2018, 8, 181 20 of 55

Crystals 2018, 8, x FOR PEER REVIEW 20 of 54 to maintain an invariant line (here the x-axis). The notation U is now preferred to B (Bain) to avoid confusionavoid confusion with our with notation our notation of the bases. of the The bases. second The path second is P 1path.P2, whereis P1.P2P, 1whereis an IPSP1 is and an PIPS2 a and simple P2 a shear.simple Here, shear.P2 Here,is reduced P2 is reduced to the identity to the identity because because only P1 onlyis required P1 is required to get the to ﬁnalget the square final lattice. square However,lattice. However, even by keepingeven by onlykeepingP1, oneonly must P1, one admit must that admit the casethat isthe not case direct, is not because direct, there because are twothere parametersare two parameters in P1—i.e., thein P simple1—i.e., shearthe simplePs and the shear dilatation Ps and P theδ (see dilatation Figure5 a).Pδ The(see directions Figure 5a of). the The vectorsdirectionss and ofδ theare vectors known, s but and nothing δ are isknown, said on but how nothing the norms is said of these on how two the vectors norms evolve of these during two thevectors transformation. evolve during The PTMC the transformation. treats the two paths The PTMCR.U and treatsPsPδ theas two two independent paths R.U and steps, Ps asPδ shownas two inindependent Figure 16. One steps, can as understandshown in Figure with 16 this. One simple can understand example why with PTMC this simple is still example phenomenological why PTMC andis stillcontinues phenomenological to be mute on atomicand continues displacements to be during mute the on transformation, atomic displacements even seventy during years the aftertransformation, its birth. Do the even atoms seventy follow years the after trajectory its birth.R.U or Do the the trajectory atoms followPsPδ? Onlythe trajectory an atom withR.U aor gift the oftrajectory ubiquity P orsPδ behaving? Only an asatom a quantum with a gift particle of ubiquity could or follow behaving both as paths a quantum at the same particle time. could The follow only wayboth to paths tackle at this the issue same is to time impose. The rules only between way to the tackle parameters this issue involved is to impose in R, U rules, Ps, and betweenPδ, such the thatparameters an equality involved of the twoin R, paths U, Ps, is and obtained Pδ, suchduring that anthe equality transformation of the two for paths all the is intermediate obtained during states. the Thesetransformation calculations for will all bethe given intermediate a little later states. in theThese text. calculations will be given a little later in the text.

FigureFigure 16. 16.Hexagonal–square Hexagonal–square transformation transformation with with PTMC. PTMC.

NowNow it it is is easier easier to to directly directly propose propose our our alternative alternative to to the the PTMC. PTMC. The The trajectories trajectories of of the the atoms atoms cancan be be determined determined by by using using an an angular angular distortion distortion matrix. matrix. The The only only thing thing to to do do is is to to write write the the way way thatthat the the lattice lattice is is distorted distorted by by the the atomic atomic displacements. displacements. In In that that aim, aim, we we use use the the crystallographic crystallographic basis = (, ) and the orthonormal basis = (, ). During the distortion, the vector a basis Bc = (a, b) and the orthonormal basis B0 = (x, y). During the distortion, the vector a remains remains invariant, and the vector b is rotated. Its image is noted b’. The angle betweenβ a and b’ is β. invariant, and the vector b is rotated. Its image is noted b’. The angle between a and b’ is . The angle The angle β decreases during the transformation;2π it is = in the starting state, andπ it is = β decreases during the transformation; it is βs = in the starting state, and it is β f = in the ﬁnal 3 2 in the final state, as shown in Figure 17. state, as shown in Figure 17. Crystals 2018, 8, 181 21 of 55 Crystals 2018, 8, x FOR PEER REVIEW 21 of 54

FigureFigure 17. 17. HexagonalHexagonal–square–square transformation transformation with with angular angular distortion. distortion.

The vectors (, ′) define a crystallographic basis () expressed in by The vectors (a, b0) deﬁne a crystallographic basis Bc(β) expressed in B0 by 1 Cos() () = [ ⟶ ()] = ! (2) 10 CosSin((β)) Bc(β) = [B −→ Bc(β)] = (2) 0 0 Sin(β) The matrix of lattice distortion () between the starting basis = () and any distorted basis (), expressed in the crystallographic basis , is simply () = [ ⟶ The matrix of lattice distortion D (β) between the starting basis B = B (β ) and any distorted ][ ⟶ ()] = (). This matrixc can now be expressed in thec orthonormalc s basis by basis B (β), expressed in the crystallographic basis B , is simply D (β) = [B −→ B ][B −→ B (β)] using thec usual coordinate change equation, i.e., c() = [ ⟶c ]. (c). [ 0⟶ 0 ]. It thenc = B −1B (β). This matrix can now be expressed in the orthonormal basis B by using the usual becomesc c 0 coordinate change equation, i.e., D0(β) = [B0 −→ Bc].Dc(β).[Bc −→ B0]. It then becomes () = () (3) D (β) = B (β) B −1 (3) This equation is very general and can0 be appliedc c to any displacive transformation. In the hexagonThis–square equation example, is very combined general andwith canEquation be applied (2), it directly to any displacivegives transformation. In the hexagon–square example, combined with Equation1 (2),+ 2 itCos directly() gives 1 ⎛ ⎞  + √3 ( )  (β) = ⎜ 1 1 2Cos√ β ⎟ (4) 2Sin3() D0(β) = 0 ( )  (4) 0 2Sin√√3β ⎝ 3 ⎠

The matrix of complete distortion is obtained for = : The matrix of complete distortion D0 is obtained for β = β f : 1 ! 11 √1 ⎛ √3⎞3 D0 ==D0 β == (5(5)) f ⎜ 0 2√2⎟ 0 3 ⎝ √3⎠ The volume (here, surface) change during the distortion is given by the determinant of the distortionThe volume matrix: (here, surface) change during the distortion is given by the determinant of the distortion matrix: V 0 2Sin(β) β f = Det D0(β) = √ (6) V 2Sin(3) = () = (6) We take the opportunity given by Formula (6) to mention√3 an important point. It was assumed by Weschsler, Liebermann, and Read in their seminal paper [33], which gave birth to PTMC, that the We take the opportunity given by Formula (6) to mention an important point. It was assumed average distortion matrix D resulting from two distortion matrices D and D of variants in proportions by Weschsler, Liebermann, and Read in their seminal paper [33], which1 gave2 birth to PTMC, that the λ and 1 − λ, with 0 ≤ λ ≤ 1 is average distortion matrix D resulting from two distortion matrices D1 and D2 of variants in D = λD + (1 − λ)D (7) proportions and 1 − , with 0 ≤ λ ≤ 1 is 1 2 This formula only works in the special condition called “kinematical compatibility”; in the general = + (1 − ) (7) case, the volume is not conserved, i.e., det(D) 6= λ.det(D1) + (1 − λ).det(D2). We note that, to the bestThis of our formula knowledge, only Bowles works in and the Mackenzie special condition never used called this formula,“kinematical and theycompatibility”; always composed in the generalmatrices case, by multiplication.the volume is not That conserved, is why we i.e., are det not( convinced) ≠ . det( when) + it(1 is− said). det that( the). WechslerWe note that, et al. toand the Bowles best of and our Mackenzie knowledge, forms Bowles of the andPTMC Mackenzie are similar; never they used differ this at least formula in the, wayand the they distortion always composed matrices by multiplication. That is why we are not convinced when it is said that the Wechsler et al. and Bowles and Mackenzie forms of the PTMC are similar; they differ at least in the Crystals 2018, 8, 181 22 of 55 matrices are averaged. This is probably one of the reasons why both PTMC versions have continued their own developments without intermixing, nearly ignoring each other. The approach based on hard spheres is very effective in comparison with the PTMC, because it gives in one simple step the continuous analytical expression of the lattice distortion that is compatible with the atom size, and is thus realistic from an energetic point of view. The PTMC can determine the distortion matrix only once the transformation is finished, i.e., Equation (5). Of course, one could try developing a complex continuous infinitesimal version of the PTMC, as in [103], but the method is artificial and needlessly heavy. For example, let us come back to 0, and try to find the expressions R, U, Ps, and Pδ such that the two paths are maintained equal during the transformation. We note O, A, B, and C as the nodes of the lattice, with the vectors OA = [1,0], OB(β) = [Cos(β), Sin(β)], and OC(β) = OA + OB(β). The stretch component UL in a local orthonormal basis positioned along the diagonal OC and AB is

 k ( )k   β  OC β 0 2Cos 0 kOC(βs)k 2 UL(β) =  kAB(β)k  =   (8) 0 0 √2 Sin β kAB(βs)k 3 2

2π This local basis is obtained from the reference basis B0 by a rotation of angle 6 . Thus, the stretch distortion matrix in the basis B0 is

2π −2π U0(β) = R 6 .UL(β).R 6  √ √  Cos β + 3Sin β 3Cos β − Sin β (9) 1 2 2 2 2 = 2  √  3Cos β − Sin β 3Cos β + √1 Sin β 2 2 2 3 2

1 2π The compensating rotation matrix R is a rotation of angle 2 β − 3

 β √ β √ β β  1 Cos 2 + 3Sin 2 3Cos 2 − Sin 2 R(β) =  √ √  (10) 2 β β β β − 3Cos 2 + Sin 2 Cos 2 + 3Sin 2

The other path of the PTMC is made of the shear displacement along the x-axis ( ) = ( ) − 2π s β Cos β Cos 3 of the√ point B, located at a distance h from the shear axis. For a simple shear, h is constant and equal to 3/2. Thus

( ) ! 1+2Cos(β) ! 1 s β 1 √ Ps(β) = h = 3 (11) 0 1 0 1

The second part of the second path is an extension along the y-axis that dilates the distance h and 2π makes it become h(β) = h + Sin(β) − Sin 3 . The dilatation matrix is thus ! ! 1 0 1 0 Pδ(β) = = ( ) (12) h(β) 0 2Sin√ β 0 h 3

The ﬁrst path is continuous, and its analytical expression is given by the product R(β) U0(β) with the matrices given in Equations (9) and (10). The second path is continuous, and its analytical expression is given by the product Pδ(β)Ps(β) with the matrices given in Equations (11) and (12). The atoms do not need ubiquity anymore, because the reader can check that the two paths are actually equal: the path corresponding to the angular distortion given by Equation (4). We let the reader make his own opinion on the physical relevance and simplicity of Figure 17 and Equation (4), and that of Figure 16 and the set of four Equations (9)–(12). Crystals 2018, 8, 181 23 of 55 Crystals 2018, 8, x FOR PEER REVIEW 23 of 54

7.7. The Angular DistortiveDistortive ModelModel forfor Face-CenteredFace-Centered Cubic–Body-CenteredCubic–Body-Centered CubicCubic Martensitic MartensiticTransformati Transformationon

7.1.7.1. The IntriguingIntriguing ContinuitiesContinuities inin thethe PolePole FiguresFigures WhenWhen fullyfully transformed,transformed, manymany steelssteels andand titaniumtitanium oror zirconiumzirconium alloysalloys areare onlyonly constitutedconstituted ofof thethe daughter daughter phase, phase, without without a sufﬁcientsufficient amount amount of of retained retained parent parent phase phase to to know know the the sizes sizes and and orientationsorientations ofof the the prior prior parent parent grains grains that had that existed had at existedhigh temperatures at high temperatures before the transformation. before the Thistransformation. apparentlylost This information apparently is, lost however, information crucial is to, however get a better, crucial understanding to get a better of the fatigue,understanding impact, andof the corrosion fatigue, properties, impact, and because corrosion the prior properties parent, grainbecause boundaries the prior are parent preferential grain location boundaries sites are of impuritypreferential segregation. location sites A method of impurity to reconstruct segregation. the prior A method parent to beta reconstruct grains in the titanium prior alloysparent from beta EBSDgrains maps in titanium was proposed alloys byfrom Gey EBSD and Humbertmaps was [104 proposed], but at that by Gey time and it was Humbert based on [104 misorientations], but at that betweentime it was grains based and on was misorientations not applicable to between steel, due grains to the and highest was number not applicable of symmetries to steel and, due variants. to the Newhighest constraints number had of symmetries to be found and to reduce variants. the inﬂuence New constrain of thets tolerance had to angle.be found These to constraints reduce the wereinfluence found of in the the tolerance algebraic angle. structure These of constrain variants.t Thes were orientational found in the variants algebraic and structure their misorientations of variants. formThe orientational a groupoid [ 105 variants], and the and theoretical their misorientations groupoid composition form a groupoid table can [105 be used], and to the automatically theoretical reconstructgroupoid composition the parent grainstable can from be EBSDused to data automatically [106,107]. Anreconstruct example the of reconstructionparent grains from is given EBSD in Figuredata [106 18.,107]. An example of reconstruction is given in Figure 18.

Figure 18. Reconstruction of the parent grains from EBSD maps. (a) Initial EBSD map of fully Figure 18. Reconstruction of the parent grains from EBSD maps. (a) Initial EBSD map of fully martensitic Fe–0.1C-9Cr–1Mo–0.4Si steel; (b) prior parent austenitic grains, reconstructed with the martensitic Fe–0.1C-9Cr–1Mo–0.4Si steel; (b) prior parent austenitic grains, reconstructed with the software ARPGE. The orientations of the austenitic grains and those of the martensitic grains they software ARPGE. The orientations of the austenitic grains and those of the martensitic grains they contain are plotted in the pole figures, with the calculated <111>γ directions in red and the contain are plotted in the pole ﬁgures, with the calculated <111>γ directions in red and the experimental experimental <110>α directions in blue. <110>α directions in blue.

The reconstruction method was applied to many alloys, and was something striking: for The reconstruction method was applied to many alloys, and was something striking: low-alloyed steels, continuous features were observed in the pole figures of the martensitic grains for low-alloyed steels, continuous features were observed in the pole ﬁgures of the martensitic grains contained in the prior parent grains. That was surprising because only a discrete distribution of contained in the prior parent grains. That was surprising because only a discrete distribution of orientations was expected from the 24 KS variants. As these features were observed in many orientations was expected from the 24 KS variants. As these features were observed in many different different steels, and were also reported by X-ray diffraction and EBSD in Fe–Ni meteorites steels, and were also reported by X-ray diffraction and EBSD in Fe–Ni meteorites [108,109], we made [108,109], we made the hypothesis that they were the plastic trace of the distortion mechanism the hypothesis that they were the plastic trace of the distortion mechanism itself. The features could be itself. The features could be simulated phenomenologically by applying two continuous rotations A simulated phenomenologically by applying two continuous rotations A and B to the 24 KS variants, and B to the 24 KS variants, one around the normal to the common dense plane, and one around the common dense direction, with rotation angles continuously varying in the range [0°, 10°]. Crystals 2018, 8, 181 24 of 55 one around the normal to the common dense plane, and one around the common dense direction, with rotation angles continuously varying in the range [0◦, 10◦].

7.2. A Two-Step Model Developed to Explain the Continuous Features What is the physical meaning of these rotations? The PTMC tells nothing about them, and we checked that they were not correlated to the usual plastic deformation modes of the bcc or fcc structure. We found that the rotations A and B could be associated with the fcc-hcp and hcp-bcc transformations, respectively; a two-step model was then established, implying the existence of an intermediate fleeting hcp phase, even in alloys without a retained ε phase [110]. It was later realized that this two-step model has some similarities with the initial KSN model shown in Figure3, the first shear in the KSN model being replaced by the fcc-hcp step made by sequential and coordinate movements of partial Shockley dislocations, supposed at that time to be originated by a pole mechanism [111]. However, further investigations could not confirm the two-step model. No trace of the intermediate hcp phase could be found in the microstructures of the martensitic steels; and ultrafast in-situ X-ray diffraction experiments in two synchrotrons (ESRF and Soleil) could not put into evidence the hypothetical fleeting hcp phase [112]. In addition, the pole mechanism appeared to us more and more doubtful (see previous sections), and many questions remained unanswered. All these “negative” results prompted us to consider other models for the fcc-bcc transformation.

7.3. One-Step Hard-Sphere Model with Pitsch OR Would it be possible to establish a mechanistic model, as simple as possible, without combining shears and without dislocation in its core—a model in which the atoms would move collectively in one step? The important thing is to start the orientation relationship between the fcc and bcc phases. It was noticed that the KS, NW, and Pitsch ORs were located in the continuous features observed in the EBSD pole figures, and we made the assumption that all these ORs actually result from a unique OR, a “natural” spontaneous OR of the transformation for stress-free samples. What could this “natural” OR be? The Bain OR obtained by lattice stretching maintains the highest number of common symmetries between the parent fcc and the daughter bcc, and thus could be a good candidate if one imagines that all the chemical bonds “break” instantaneously at the same time. However, a criterion that predicts the OR only from symmetry considerations would not agree with experiments, and would probably not be relevant. Indeed, the transformation does not occur instantaneously; we envision it as a wave. The existence of common dense directions or planes obtained for special ORs, as it is the case for the KS OR, is probably favorable to wave propagation, whereas stretch distortions in general do not maintain the parallelism of dense directions. Besides any theoretical considerations, the experimentally determined ORs are more than 10◦ far away from the Bain OR. Could the Pitsch OR be a good candidate? This OR is less well-known than KS or NW; it was observed by Pitsch in 1959 after cooling a TEM lamella of iron nitrogen alloy [113], which means that it was formed without the surrounding stresses that exist in bulk samples. Maybe Pitsch OR was the missing link to build a theory. The Pitsch OR is at the midway between two low-misoriented KS variants, in the same way that the NW OR is at the midway between the two other low-misoriented KS variants. The simulations of the pole figures with 24 KS variants and the rotations A and B with angles limited to 0–5◦ are shown in Figure 19a. The distortion matrix associated with the Pitsch distortion was calculated, and the rotations A and B were qualitatively explained by the Pitsch distortion, even if the simulations were based on KS OR and not the Pitsch OR. For the calculations, the lattice parameters of the fcc and bcc phases had to be chosen, and those of hard-sphere packing with a constant atomic radius for the iron atoms were taken as a first approximation. Even if a hard-sphere model is not perfect, because the size of Fe atoms are not exactly the same in the fcc and bcc crystals (4% of difference), it is a good starting point to build an atomistic model. The Pitsch distortion matrix could be diagonalized, and the strains were surprisingly lower than those of Bain—but the basis of diagonalization is not orthonormal [114]. One year later, we developed a method to show in the EBSD maps the regions oriented according Crystals 2018, 8, 181 25 of 55 to Pitsch, KS and NW ORs with a red, green, and blue (RGB) color coding [115]. It was applied to variousCrystals 2018 steels, 8, x andFOR confirmedPEER REVIEW that each bcc martensitic grain exhibits internal gradients between25 of the 54 Pitsch–KS–NW orientations, as shown in Figure 19b. However, the gradient NW/KS/Pitsch/KS/NW, expectedNW/KS/Pitsch/KS/NW by the Pitsch, distortionexpected model, by the could Pitsch not distortion be clearly model identified,, could which not be cast clearly doubts identified, on the Pitschwhich distortioncast doubts model. on the Pitsch distortion model.

Figure 19. Continuous gradients of orientations inside bcc martensitic grains. (a) Simulations of the Figure 19. Continuous gradients of orientations inside bcc martensitic grains. (a) Simulations of the pole figures made with 24 KS variants and two rotations A and B, with angular range 0–5° (left pole figures made with 24 KS variants and two rotations A and B, with angular range 0–5◦ (left column), column), compared to the experiments (right column), from [114]; (b) RGB coloring of the Pitsch, KS, compared to the experiments (right column), from [114]; (b) RGB coloring of the Pitsch, KS, and NW and NW ORs inside the martensitic grains of the EMT10 steel shown in 0. Reprinted from [115] with ORs inside the martensitic grains of the EMT10 steel shown in 0. Reprinted from [115] with permission permission from Elsevier. from Elsevier. Despite the absence of unambiguous experimental confirmation, we had a feeling that the fundamentsDespite of the the absence model (one of unambiguous-step and hard experimental-spheres) were confirmation, correct. However, we had could a feeling KS, in thatplace the of fundamentsPitsch, actually of the be modelthe “natural” (one-step OR? and The hard-spheres) KS OR was were also correct. reported However, in a quenched could KS, TEM in place lamella of Pitsch,[116]. The actually KS OR be also the“natural” gives the OR?best Thesimulations KS OR was of the also continuous reported in features a quenched in the TEM pole lamella figures [,116 and]. Theis the KS only OR alsoOR givesthat a thellow bests the simulations parallelism of of the the continuous dense directions features inand the the pole parallelism figures, and of isthe the dense only ORplanes that of allows the fcc the and parallelism bcc phases of the at dense the same directions time. and What the if,parallelism contrary of to the what dense weplanes thought of the at thefcc andbeginning, bcc phases nobody at the had same tried time. to build What a if,one contrary-step model to what of transformation we thought at the based beginning, on the KS nobody OR, i.e., had a triedcontinuous to build version a one-step of the model KSN ofmodel transformation shown in Figure based on3? the KS OR, i.e., a continuous version of the KSN model shown in Figure3? 7.4. One-Step Hard-Sphere Model with KS OR 7.4. One-Step Hard-Sphere Model with KS OR A one-step continuous model of the fcc-bcc lattice distortion leading to a KS OR was built A one-step continuous model of the fcc-bcc lattice distortion leading to a KS OR was built similar similar to the one made with the Pitsch OR in [114]. This model also uses the hard-sphere to the one made with the Pitsch OR in [114]. This model also uses the hard-sphere assumption to assumption to explicitly calculate the trajectories of the atoms during the transformation, i.e., to explicitlyobtain a calculatecontinuous the atomistic trajectories model of the of atoms the duringfcc-bcc the transformation transformation, [117 i.e.,]. The to obtain main a idea continuous can be atomistic model of the fcc-bcc transformation [117]. The main idea can be briefly explained as follows. briefly explained as follows. During the Bain distortion the fcc lattice is contracted along a <100>γ During the Bain distortion the fcc lattice is contracted along a <100> direction; the two other <100> direction; the two other <100> directions are expanded; when the transformationγ is complete, half a bcc crystal is formed, as shown in Figure 20. Crystals 2018, 8, 181 26 of 55

directions are expanded; when the transformation is complete, half a bcc crystal is formed, as shown

Crystalsin Figure 2018 20, 8., x FOR PEER REVIEW 26 of 54 Crystals 2018, 8, x FOR PEER REVIEW 26 of 54

Figure 20. Bain distortion with hard hard-spheres.-spheres. ( (aa)) Initial Initial fcc fcc lattice lattice;; ( (bb)) s sameame crystal crystal with with hard hard spheres. spheres. Figure 20. Bain distortion with hard-spheres. (a) Initial fcc lattice; (b) same crystal with hard spheres. The direction of the contraction axisaxis alongalong aa <100><100> direction is marked by the white arrows.arrows. ( c) Final The direction of the contraction axis along a <100> direction is marked by the white arrows. (c) Final distorted crystal, which is half a bcc crystal cut on a (110) plane;plane; ( d) Initial fcc crystal and final final bcc crystaldistorted after crystal, Bain distortion.which is half Movie a bcc at crystalhttp://lmtm.epfl.ch/researchhttp://lmtm.epfl.ch/research cut on a (110) plane; . (d. ) Initial fcc crystal and final bcc crystal after Bain distortion. Movie at http://lmtm.epfl.ch/research. For the distortion that leads to the KS OR, simply called KS distortion, the trajectories of the For the distortion that leads to the KS OR, simply called KS distortion, the trajectories of the atomsFor are the slightly distortion different that from leads those to the obtained KS OR, with simply the calledBain OR. KS Letdistortion, us consider the Figuretrajectories 21. The of theKS atoms are slightly different from thosethose obtainedobtained with the Bain OR. LetLet us consider Figure 2121.. The KS OR implies the parallelisms of a dense plane (POK) = {111}γ//{110}α and a dense direction PO = ½1 OR implies the parallelisms of a dense plane (POK) = {111}γ//{110}α and a dense direction PO = OR implies the parallelisms of a dense plane (POK) = {111}γ//{110}α and a dense direction PO = ½2 <110>γ = ½1 <111>α. The KS distortion is thus obtained as follows. The atom P is fixed. The angle β <110>γ = <111>α. The KS distortion is thus obtained as follows. The atom P is fixed. The angle β made<110> γby = ½the2 <111> denseα. Thedirections KS distortion (PO, PK is) opensthus obtained from 60° as to follows.70.5° while The maintaining atom P is fixed. the direction The angle PO β made by the dense directions (PO, PK) opens from 60◦ to 70.5◦ while maintaining the direction PO invariant;made by the the dense atoms directions O and K lose (PO contact,, PK) opens and thefrom atom 60° M to in70.5° the whileupper maintaininglayer, located the above direction O and PO K, invariant; the atoms O and K lose contact, and the atom M in the upper layer, located above O and K, moveinvariant;s (“roll thes ”)atoms over Othem and toK loseget closercontact, to andthe theatom atom P. When M in the the upper transformation layer, located is complete above O, andhalf K a, moves (“rolls”) over them to get closer to the atom P. When the transformation is complete, half a bcc bccmove crys stal(“roll is sformed”) over (themfor Bain to getdistortion closer to), but the nowatom directly P. When in theKS OR.transformation is complete, half a crystalbcc crystal is formed is formed (for ( Bainfor Bain distortion), distortion but), nowbut now directly directly in KS in OR. KS OR.

Figure 21. KS distortion explained with hard-spheres. The continuity is given by a unique parameter, Figure 21. KS distortion explained with hard-spheres. The continuity is given by a unique parameter, theFigure angle, 21. βKS = ( distortionPO, PK) that explained changes with from hard-spheres. 60° (bcc) to 70.5° The (fcc). continuity Movie isat givenhttp://lmtm.epfl.ch/research by a unique parameter,. the angle, β = (PO, PK) that changes from 60° (bcc) to 70.5° (fcc). Movie at http://lmtm.epfl.ch/research. the angle, β = (PO, PK) that changes from 60◦ (bcc) to 70.5◦ (fcc). Movie at http://lmtm.epﬂ.ch/ Forresearch the. KS OR expressed by (111)γ//(110)α & [110]γ = [111]α, the continuous form of the distortionFor the matrix KS is OR expressed by (111)γ//(110)α & [110]γ = [111]α, the continuous form of the distortion matrix is For( the) KS OR expressed by (111)γ//(110)α & [110]γ = [111]α, the continuous form of the () 2 distortion matrix1 1 X is 1 1 X 1 X 1 1 X 2 X  2 X  X 1 X  2 X  2 X  2 X  2 X     2   ⎛ 16 11 X 16 1 X 13X 16 1 X ⎞ DKS(β)  2 X  2 X  X 1 X   2 X  2 X    2 X  2 X  ⎜⎛ 6 1√X √ 6 1 X √ √ q3 6 1 X √ ⎟⎞√  q − 1 q1 X− − 2 q −  √1 1 X 2X + 2 X + X 11− XX− √1 1 X 2 X2X2 +X2 X 12 X − √1 1 X 2X + 2 X = ⎜6 11+X1  X 6 1 1X+X 1  X 3 1 61  X1+X ⎟ (13)  √ 2 X √2 X  X 16 1 X √ √  q   2 X√ 2 X  √  q −   1 X  q −  2 X  2 X  2− 2 q −  (13) =  −⎜√1 161 X11 X + + − + √1 1 1X X + 1−3X 1 X 1+6 √11 X1 X + ⎟  (13) =  1+X 2X2 X 2 2 XX  XX 1 X 6 1+X 2X 2 X  3 1+X2 X 22XX 2 X   ⎜ 6   6 6  ⎟  ⎜ 6q 1  X√ √ q √ √ q3 2 6 1 qX √ ⎟√ √2 1−X √2 1−X 12−X √2 1−X 2 1+X1 X 2X − X − 2 11+XX 2X − X 12 X 3 2− 1 X 1+X 2X − X ⎜ 6 2 X  X  6 2 X  X 2  6 2 X  X ⎟     2   ⎝ 26 1 X 26 11 X 13X 26 1 X ⎠  2 X  X    2 X  X  2   2 X  X  ⎝ 6 1 X 6 1  X 3 6 1 X ⎠ where x = cos(β). It can be checked that this distortion matrix is the identity matrix for the starting statewhere β =x 60°= cos( (X β=). 1/2); It can when be checkedthe transformation that this distortion is complete, matrix β = is70.5° the (identityX = 1/3), matrixit becomes: for the starting state β = 60° (X = 1/2); when the transformation is complete, β = 70.5° (X = 1/3), it becomes: 2 √6 √6 1 √6 1 2 + √6 √6+ 1 √6− 1 ⎛3 + 18 18 + 3 6 − 3⎞ ⎜⎛3 √186 18 √63 16 √63⎟⎞ = ⎜ − 1 + − ⎟ (14) ⎜ 18√6 18√6 31 √66⎟ = ⎜ − 1 + − ⎟ (14) ⎜√6 181 1 √186 13 √66⎟ ⎝√6 − 1 1 − √6 1 + √6⎠ 9 − 3 3 − 9 3 + 3 ⎝ 9 3 3 9 3 3 ⎠ Crystals 2018, 8, 181 27 of 55 where x = cos(β). It can be checked that this distortion matrix is the identity matrix for the starting state β = 60◦ (X = 1/2); when the transformation is complete, β = 70.5◦ (X = 1/3), it becomes: √ √ √  2 + 6 6 + 1 6 − 1  3 √18 18 √3 6 √3 DKS =  − 6 + 6 1 − 6  (14)  √ 18 1 √18 3 √6  6 1 1 6 1 6 9 − 3 3 − 9 3 + 3

If one prefers using the equivalent KS OR, deﬁned by (111)γ//(110)α &[110]γ = [111]α, the matrix of complete distortion is obtained by a change of reference frame √ √ √  2 + 6 6 + 1 6 − 1  3 √18 18 √3 6 √3 DKS =  − 6 + 6 1 − 6  (15)  √ 18 1 √18 3 √6  6 1 1 6 1 6 9 − 3 3 − 9 3 + 3

Even if the trajectories of the atoms are different, the calculations prove that the KS distortion matrix DKS is linked to the Bain distortion B by a polar decomposition DKS = RB. All the calculations used here are analytical; they result from simple geometrical considerations, and not from density functional theory (DTF) or molecular dynamics (MD). MD simulations help one understand displacive transformations, but one can doubt the capacity of brute force MD to simulate the coordinated “wave-like” displacements of atoms, even if it was proved that the phonon dispersion curves can be obtained by MD simulations [118]. Simulating a “phase transition wave” seems more complex than an acoustic wave, and it is probable that a better understanding of the transformations is required to add new crystallographic constrains to MD. The first MD simulations proposed by different groups in the last decade are going in the right direction [119–122]. For example, in 2008 Sinclair and Hoagland could simulate by MD the formation of bcc martensite in Pitsch OR, at the intersection of stacking faults [119]. Sandoval et al. showed in 2009 [120] that a pure Bain path would necessitate compressive stresses five times higher than with an NW path (see KSN model described earlier). Wang et al. could reproduce in 2014 the effects of the carbon content and cooling rates on the martensite start (Ms) temperature [121]. It is certain that MD will be an indispensable tool in the future to compare the “realisms” of the different crystallographic models, so efforts will be made in the future to combine the angular distortion with MD simulations. The PTMC has the habit planes of martensite for output, but as already discussed, it makes post-dictions, not pre-dictions. Can a simple criterion be introduced to explain only the habit planes from the unique value of the distortion matrix? As there is no common plane between the fcc and bcc structures (even if this has never been formally proven), the IPS condition cannot be used. The PTMC made the choice to use it anyway, by combining it with other shears. The other possibility is finding a condition that is less restrictive than IPS. A criterion in which the habit plane is only untilted is physically relevant, because it avoids accumulating defects on long distances. The habit plane should be an eigenvector of the distortion matrix calculated in the reciprocal space, i.e., of the inverse of the transpose of DKS. The two only untilted planes determined with the matrix (15) are (111) and √ γ ◦ 11 6 ; the latter plane is only at 0.5 away from the expected (225)γ habit plane. The calculation γ is direct, without any fitting parameter, and without knowing the details of the accommodation mechanisms in this plane. While writing the paper, [117], we realized that Jaswon and Wheeler [32] had already postdicted the (225)γ habit planes by calculating the distortion matrix related to KS OR, and by using an “untilted plane” criterion; therefore, it is true that part of our work [117] is actually an independent rediscovery of Jaswon and Wheeler’s. One should keep in mind, however, that Jaswon and Wheeler’s study is indeed rarely cited, and when it is, it is for the use of the Bain correspondence matrix and for introducing matrix algebra, not for the calculation of the KS distortion or for the “unrotated plane” criterion. As explained previously, Bowles and Barrett discarded Jaswon and Wheeler’s model very early in 1951 [48], that is why it is rarely mentioned in books on the Crystals 2018, 8, 181 28 of 55

PTMC; for example, it is not cited by Bhadeshia in his excellent and didactic book on martensitic transformations [38]. Bowles, Dunne, and many other metallurgists did not try to come back to Jaswon and Wheeler’s model, despite the huge difficulties they encountered in explaining the (225)γ habit planes, as recalled by Dunne [46]: “In contrast [to Mackenzie] John Bowles retained his intense concentration on martensite crystallography until he retired from the University of New South Wales in 1983. Bowles’ focus, moreover, was predominantly on the seemingly intractable {225}F problem and six of his nine PhD students works on aspects of this problem: Peter McDougall, Allan Morton, Druce Dunne, Don Dautovich, Peter Krauklis and Barry Muddle”. Recently, an anonymous reviewer claimed that the model of the KS distortion proposed in [117] was an “outright copy of Jaswon and Wheeler’s 1948 approach”. This is unfair if one considers the way we followed to come to the result, and if one compares the papers. Experienced scientists like to repeat Satayana’s aphorism “Those who cannot remember the past are condemned to repeat it”, but practically, how is it possible to remember or even know more than 5000 papers devoted to martensitic transformations in steels? More importantly, rediscovering also allows for exploring new directions. Only the matrix of a complete transformation was given by Jaswon and Wheeler, whereas the continuous KS distortion and analytical expression of the paths of the atoms could be calculated in [117]. Now, if one compares the model of [117] with the PTMC, the former gives directly and continuously the correct bcc structure, without adjusting parameters and without choosing additional shears, whereas the latter gives only the final structure and needs other parameters, such as the lattice parameters and a choice of LIS systems. All the steps of the transformation are now given analytically by a unique continuous parameter: the angle β made by the dense directions (PO, PK). This angle is the natural order parameter that is usually so difficult to define for first-order transitions. The distortion matrix has singular properties. Contrary to the Bain and Pitsch distortion matrices, it is not diagonalizable, and it has only two eigenvalues: 1 and 1.088 (which is the volume change due to the hard-sphere assumption). The eigenvector associated with unity is the parallel dense direction PO, and the eigenvector associated with 1.088 is a direction that lies in the untilted dense plane (POK), which means that contrary to an IPS, where the volume changes occurs perpendicularly to the invariant plane, here the volume change occurs inside the untilted (but distorted) plane. We will come back later to this property. This result is new, and implies that martensitic transformations should not be considered as a shear transformation. The notion of angular distortive transformation was thus introduced in order to replace the notion of shear. With a large step back, the hard-sphere displacive model of fcc-bcc transformation appears as a convergent model of nearly all the models reported in the past. It substitutes a one-step angular distortion for the two shears of the 1930–34 KSN model, or for the two-step fcc-hcp → hcp-bcc model [110]; it is a continuous version of the underestimated 1948 Jaswon and Wheeler model of the KS distortion [32], and it replaces the four matrices of the 1952–53 PTMC models [33,34] by only one matrix. The model uses hard spheres as in the 1969 Bogers and Burgers model [53], but now it opens the correct angle on the correct dense plane. Indeed, in the Bogers and Burgers model shown in Figure6, it is the angle between the untilted (111) dense plane (noted VQP) and another {111} plane (noted VSP) that was changed. The 60◦ angle between the dense directions in the plane VQP is not modified, i.e., the plane VQP is fully invariant before the second shear that allows the forming of the bcc structure. In the model shown in Figure 21, the untilted (111) plane is noted POK, and it is the angle between two of the three the dense directions of this plane (PO and PK) that is changed to directly obtain the bcc structure. In our model, there is no need for complementary shears as those proposed by Olson and Cohen in 1972–76 [56,57]. In addition, the important suspected crystallographic similarity with deformation twinning can be detailed and quantified. For example, fcc-bcc martensitic transformation and fcc-fcc deformation twinning can be geometrically represented in Figure 22, and quantitatively compared [123]. Crystals 2018, 8, 181 29 of 55 Crystals 2018, 8, x FOR PEER REVIEW 29 of 54

Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

It is probable that, in many cases, mainly for the initial stages and for the midrib formation, the habit plane is rendered invariant by the co-formation of a pair of variants (as in Figure 24); however, that condition does not seem to be necessary for the growth of individual martensite, and cannot be used as a fundamental part of a mechanistic theory of martensitic transformation. In brief, discarding the assumption of shear permitted the establishment of simple and coherent models of the {225}γ and {557}γ martensite, which is encouraging if one considers the very long history and the high number of publications on this subject. Besides, it was stated from the first quantitative metallographies that the habit planes of martensite are irrational, even if noted with high index rational numbers. Irrational values can now be proposed: according to our model the {225}γ observed in high-carbon steels and the {557}γ habit planes in low-carbon steels are {11 6}γ and {11 2}γ planes, respectively. The former are fully accommodated by coupling the twin-related √ variants that share the same dense direction, and the latter both by coupling the low-misoriented variants√ that share the same dense plane and by additional dislocations. The difference between these two modes might come from the difference of Ms temperatures; in the former case, the low Ms temperatures do not allow dislocation plasticity, but promote variant-pairing IPS accommodation, whereas in the latter case, dislocation plasticity allows for conditions that are less FigureFigure 22. 22. HardHard-sphere-sphererestrictive model model than of of ( apure)) fcc fcc-fcc -IPS.fcc deformation twinning and (b) fcc-bccfcc-bcc martensitic martensitic transformation.transformation. The The distance distance h ish is the the distance distance between between the the planes planes,, initially initially POK POK = (111) =γ (111). Thisγ. Thisdistance distance changes changes during8. during Generalization the thetransformation transformation of the process. process.Angular Both Both Distortive fcc fcc-fcc-fcc twinn twinning Modeling and andto Facefcc fcc-bcc-bcc-Centered martensitic Cubic –Body-Centered transformationstransformations are are modelled Cubic modelled–Hexagonal as functions as functions Close of a unique of-Packed a unique angle Martensitic of angle distortion, of Transformations distortion, noted η and notedβ, respectively. η, and and to β Face, -Centered Cubic– Inrespectively. both transformations, In bothFace transformations, the-Centered dense plane Cubic the POK dense Deformation and the plane dense POK directionTwinning and the PO dense remain direction untilted. PO Reprinted remain fromuntilted. [123 Reprinted] with permission from [123 from] with Elsevier. permission from Elsevier. The hard-sphere model allows description with the same formalism the mechanisms of both deformation twinning and martensite transformation. The model permits a natural introduction of They shows that twinning results from a simple atomic displacement on each {111}γ layer, as in They shows that twinning results from a simple atomic displacement on each {111}γ layer, as in the the usual representation,an but order the parametercontinuous in atomicto the transitiontrajectories; the do ordernot follow parameter a simple is simply shear strai the nangle of distortion (and usual representation, but the continuous atomic trajectories do not follow a simple shear strain because because the atom M mustnot goan overill-def theined atoms multidimensional O and K (and strainthe angle state). β remains The work locked done at for 60°). the Thisfcc-bcc transformation was the atom M must go over the atoms O and K (and the angle β remains locked at 60◦). This trajectory is trajectory is familiar to thusany studentquickly ingeneralized metallurgy to who other has displacive already usedtransformations ball stacking between in ABCABC the fcc, bcc, and hcp phases familiar to any studentobserved in metallurgy in other who metal has alreadyalloys [123 used]. By ball noting stacking that in fcc ABCABC = γ, bcc = sequences α, and hcp in = ε, as it is usually done practicalsequences classes. in practical The fcc-bcc classes. transformation The fcc- resultsbcc transformation from the same reatomicsults displacements, from the same except atomic that displacements, except thatfor steels,now the the movement classical ORs of thebetween atom theM isthree combined phases with are thean KSincrease OR for of fccthe bcc, the Burgers OR now the movement of the atom M is combined with an increase of the distance between the atoms O distance between the atoms[29] for O andbcc K (the hcp, angle and Shojiβ is unlocked-Nishiyama and (SN) changes [37] for from fcc 60° hcpto 70.5°) transformations:, which and K (the angle β is unlocked and changes from 60◦ to 70.5◦), which explains why the “jump” h of⟶ explains why the “jump” h of the atom M (the distance between the atom M and its projection on the atom M (the distance between the⟶ atom M and its projectionKS: [110] onγ = the[111] POKα and plane) (111)⟶ isγ// lower(110)α than for the POK plane) is lower than for twinning. For fcc-fcc deformation twinning, as the surface of the twinning. For fcc-fcc deformation twinning, as the surfaceBurgers: of the[111] planeα = [100] POKε and is constant,� (110)α�//(001) the changeε (16) plane POK is constant, the change of the distance h implies a volume change during the distortion, of the distance h implies a volume change during the distortion, even if, when twinning is complete, even if, when twinning is complete, the volume comes backSN: [110]to itsγ initial = [100] value.ε and (This111) resultγ//(001) isε actually the volume comes back to its initial value. This result is actually expected from a hard-sphere� model, expected from a hard-sphere model, because for fcc-fcc twinning the intermediate states are because for fcc-fcc twinningThese the intermediate respect the states parallelism are necessarily of the close less-packed dense thandirections� fcc (Kepler–Hales, and form a close set of ORs that is necessarily less dense than fcc (Kepler–Hales theorem). The curve of Figure 22b shows that the theorem). The curve of Figureof interest 22b showsto build that a unifying the distance theory,h also as changesalready noticed during theby fcc-bccBurgers distortion. in 1934 and illustrated in Figure distance h also changes during the fcc-bcc distortion. When the distortion is complete, this distance When the distortion is26a complete,. The three this phases distance projected comes back along to the its initialnormal value, to their as common it does for dense fcc-fcc plane is shown in Figure comes back to its initial value, as it does for fcc-fcc twinning, despite the fact the bcc phase has a twinning, despite the fact26b the. The bcc phaseuse of has angular a volume distortion higher thanmatrices for the offers fcc phase.the possibility The volume of treat changeing these transformations volume higher than for the fcc phase. The volume change between the initial and final states is between the initial and ﬁnalsimilarly states in is completelya unique framework, due to the change with ofthe the same surface paradigm. of the plane The POKfcc hcp and bcc hcp completely due to the change of the surface of the plane POK ⟶ POK’ because of the angular POK’ because of the angulartransformations distortion in involve which β shuffling= (PO, PK, because) increases the fromparent 60 ◦andto 70.5 daughter◦, as shown phases in do not have the same distortion in which β = (PO, PK) increases from 60° to 70.5°, as shown in Figure 23. ⟶ ⟶ Figure 23. number of atoms in their Bravais cell (the bcc and fcc primitive cells contain one atom, whereas the hcp primitive cell contains two atoms). The distortion matrices and shuffle trajectories (when required for the hcp phase) were analytically determined as functions of an angular parameter that depends on the transformation. The choice of an angular parameter and the details of the calculations are given in [123]. It is important to note that both the lattice distortion and shuffling are the two sides of the same coin; there is no reason to disconnect them. Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

Crystals 2018, 8, 181 30 of 55 8. Generalization of the AngularCrystals Distortive 2018, 8, x ModelFOR PEER to REVIEW Face-Centered Cubic–Body-Centered 30 of 54 Cubic–Hexagonal Close-PackedCrystals Martensitic 2018, 8, x TransformationsFOR PEER REVIEW , and to Face-Centered Cubic– 30 of 54 Face-Centered Cubic Deformation Twinning The hard-sphere model allows description with the same formalism the mechanisms of both deformation twinning and martensite transformation. The model permits a natural introduction of an order parameter into the transition; the order parameter is simply the angle of distortion (and not an ill-defined multidimensional strain state). The work done for the fcc-bcc transformation was thus quickly generalized to other displacive transformations between the fcc, bcc, and hcp phases observed in other metal alloys [123]. By noting that fcc = γ, bcc = α, and hcp = ε, as it is usually done for steels, the classical ORs between the three phases are the KS OR for fcc bcc, the Burgers OR [29] for bcc hcp, and Shoji-Nishiyama (SN) [37] for fcc hcp transformations: ⟶ ⟶ KS: [110]γ = [111]α and (111)⟶γ//(110)α

Burgers: [111]α = [100]ε and� (110)α�//(001)ε (16)

26a. The three phases projected along the normal to their common dense plane is shown in Figure Figure 23. Representation of the fcc-bcc distortion with the KS OR. The volume change is entirely due 26b. The use of angular distortion matricesFigureFigure 23.23. offers RepresentationRepresentation the possibility ofof thethe fcc-bccfcc of- bcctreat distortiondistortioning these withwith transformations thethe KSKS OR.OR. TheThe volume volume changechange isis entirelyentirely duedue to the angular distortion in the plane POK ⟶ POK’, for which β β= (PO, PK) increases for 60° to◦ 70.5°. ◦ similarly in a unique framework, totowith thethe angularangularthe same distortiondistortion paradigm. inin thethe planeplanThee POKfcc ⟶ POK’,hcp for and which bcc β == ( PO (POhcp, ,PK PK) )increases increases for for 60° 60 toto 70.5°. 70.5 . transformations involve shuffling, because the parent and daughter phases do not have the same In addition, it is qualitatively understood⟶ that the two ⟶components of the KS distortion, one number of atoms in their Bravais cellInIn (the addition, addition, bcc and it it is fcc is qualitatively qualitativelyprimitive cells understood understood contain thatone that theatom, twothe whereas two components components the of the of KS the distortion, KS distortion, one given one given by the rotation of the direction PK around the normal to the dense plane POK, and the other hcp primitive cell contains twobygiven theatoms). rotationby the The rotation of distortion the direction of the matrices directionPK around and PK theshufflearound normal thetrajectories to normal the dense to(when the plane dense POK, plane and POK, the other and by the the other tilt by the tilt of the plane POM around the direction OK, produce the continuous rotations A and B required for the hcp phase) wereofby theanalytical the plane tilt of POMly thedetermined aroundplane POM the as direction functionsaround OKthe of ,andirection produce angular theOK parameter continuous, produce that the rotations continuousA and rotationsB observed A inand the B observed in the pole figures Figure 19a. In this approach, the special arrangements of dislocations depends on the transformation.poleobserved The figures choice in Figurethe ofpole 19an a.figures angular In this Figure approach, parameter 19a. In the andthis special approach,the arrangementsdetails the of special the of dislocations arrangements are of generated dislocations by are generated by the rotational parts of the distortion: they form the disclinations at the origin of the calculations are given in [123]. theItare is rotationalgenerated important parts by to the note of rotational the that distortion: both parts the they oflattice the form distortion:distortion the disclinations theandy shufflingform at the the disclinations origin of the at rotations the originA and of theB, rotations A and B, retained in the material as plastic traces of the mechanism. We are presently are the two sides of the same coin;retainedrotations there inis A theno and reason material B, retained to asdisconnect plastic in the traces them. material of the mechanism. as plastic traces We are of presently the mechanism. working We to quantitatively are presently working to quantitatively simulate the continuous special features in the pole figures only from the simulateworking theto quantitatively continuous special simulate features the continuous in the pole special figures features only from in the the pole analytical figures expressions only from the of analytical expressions of the distortion. theanalytical distortion. expressions of the distortion. Even if, in first approximation, the mechanism of the fcc-bcc transformation can be modelled EvenEven if,if, inin firstfirst approximation,approximation, thethe mechanismmechanism ofof thethe fcc-bccfcc-bcc transformationtransformation cancan bebe modelledmodelled without knowing the exact nature of the accommodation processes, one can ask whether or not the withoutwithout knowingknowing thethe exactexact naturenature ofof thethe accommodationaccommodation processes,processes, oneone cancan askask whetherwhether oror notnot thethe “untilted plane” condition can be transformed into an IPS condition. The PTMC often changes the “untilted“untilted plane”plane” conditioncondition cancan bebe transformed transformed intointo anan IPS IPS condition. condition. TheThe PTMCPTMC oftenoften changeschanges thethe lattice distortion RB into an IPS by using an LIS on a (112)α plane, explaining that it corresponds to latticelattice distortiondistortion RBRBinto into anan IPS IPS by by using using an an LIS LIS on on a a (112) (112)α plane,plane, explainingexplaining thatthat itit correspondscorresponds toto a mechanical twinning of the bcc phase, and forgetting thatα bcc metals generally twin only at very aa mechanicalmechanical twinningtwinning ofof thethe bccbcc phase,phase, andand forgettingforgetting thatthat bccbcc metalsmetals generallygenerally twintwin onlyonly atat veryvery low temperatures (far below the usual Ms temperatures). The hypothesis made in [114] is that the lowlow temperaturestemperatures (far(far belowbelow thethe usualusual MsMs temperatures).temperatures). TheThe hypothesishypothesis mademade inin [[114114]] isis thatthat thethe “mechanical twins” are in fact KS twin-related variants. We discovered that the “untilted” (225)γ “mechanical“mechanical twins” twins” are are in factin fact KS KStwin-related twin-related variants. variants. We discovered We discovered that the that “untilted” the “untilted” (225) plane (225)γ plane could be transformed into an IPS by combining two twin-related KS variantsγ [124] couldplane be could transformed be transformed into an IPS into by combining an IPS by two combining twin-related two KS variants twin-related [124] misoriented KS variants by [124 60◦] misoriented by 60° around the common dense direction PO = <110>γ//<111>α. A movie of the aroundmisoriented the common by 60° dense around direction the commonPO = <110> dense//<111> direction. APO movie = <110> of theγ//<111> continuousα. A movie formation of the of continuous formation of a (225) martensite composedγ of two KSα variants could be simulated (Figure 24). acontinuous (225) martensite formation composed of a (225) of martensite two KS variants composed could of betwo simulated KS variants (Figure could 24 be). simulated (Figure 24).

Figure 24. Continuous distortion model of the fcc-bcc transformation on a (225) habit plane. This FigureFigure 24. 24.Continuous Continuous distortion distortion model model of the of the fcc-bcc fcc- transformationbcc transformation on a (225) on a habit (225) plane. habit plane.This plane, This plane, which is only untilted during the formation of one variant, becomes an IPS when two twin-related whichplane, iswhich only is untilted only untilted during during the formation the formation of one of one variant, variant, becomes becomes an an IPS IPS when when two two twin-related twin-related KS variants are formed together. The atoms of the fcc crystal are in green, and those of the two final KSKS variantsvariants areare formedformed together.together. TheThe atomsatoms ofof thethe fccfcc crystalcrystal areare inin green,green, andand thosethose ofof thethe twotwo ﬁnalfinal twin-related bcc variants are in red and blue. Movie available at https://www.nature.com/ twin-relatedtwin-related bcc bcc variants variants are are in red in and red blue. and Movie blue. available Movie available at https://www.nature.com/article- at https://www.nature.com/ article-assets/npg/srep/2017/170120/srep40938/extref/srep40938-s1.avi. From [124]. assets/npg/srep/2017/170120/srep40938/extref/srep40938-s1.aviarticle-assets/npg/srep/2017/170120/srep40938/extref/srep40938-s1.avi. From From [ [124124].]. These simulations are good agreement with the TEM observations of the midrib structure These simulations are good agreement with the TEM observations of the midrib structure observed in the center of lenticular martensite products [125,126]. However, one should keep in observed in the center of lenticular martensite products [125,126]. However, one should keep in mind that far from the midrib, martensite is only constituted of one variant, which means that mind that far from the midrib, martensite is only constituted of one variant, which means that Crystals 2018, 8, 181 31 of 55

These simulations are good agreement with the TEM observations of the midrib structure observed in the center of lenticular martensite products [125,126]. However, one should keep in mind that far from the midrib, martensite is only constituted of one variant, which means that during martensite growth the transformation obeys an angular distortion accommodated by dislocations, and no longer an IPS. A model of the {557}γ martensite in low-carbon steels was also established [127]. The natural transformation is still the KS distortion for the {225}γ martensite in high-carbon steels, but the average is now made between the two low-misorientated variants that share the same dense plane; these two variants formCrystals anassembly 2018, 8, x FOR PEER called REVIEW a “block”. The average distortion is exactly the distortion31 of 54 associated with the NWduring OR, and martensite an eigenvector growth the of transformation the NW distortion obeys an matrix angular expressed distortion accommodated in the reciprocal by space (i.e., √ an untilted plane)dislocations, is the and no11 longer2 anplane, IPS. which is only 0.3◦ far away from the expected (557) plane. A model of the {557}γγ martensite in low-carbon steels was also established [127]. The natural transformation is still the KS distortion for the {225}γ martensite in high-carbon steels, but the 7.5. Habit Planeaverage and is Surface now made Relief between the two low-misorientated variants that share the same dense plane; these two variants form an assembly called a “block”. The average distortion is exactly the Assumedistortion that the associated habit with plane the NW is OR an, and invariant an eigenvector shear of the plane NW distortion at a mesoscopic matrix expressed scale in (made of (11 2) the martensitethe reciprocal product space associated (i.e., an untilted with plane) twins, is the twin-related √ plane, variants, which is only or 0.3° periodic far away arrangementsfrom of the expected (557) plane. dislocations) is the classical paradigm of PTMC. We have seen that Bowles and Barrett in 1952 [48] discarded Jaswon7.5. Habit and Plane Wheeler’s and Surface Relief model of (225) martensite because of this paradigm. Two years later, Bowles and MackenzieAssume that [34 ] the detailed habit plane the is reasons an invariant for shear which, plane according at a mesoscopic to them, scale (made the habit of the plane should be invariant.martensite First, they product stated associated that, with“the twins, amount twin- ofrelated plastic variants, deformation or periodic would arrangements be[unreasonable] of dislocations) is the classical paradigm of PTMC. We have seen that Bowles and Barrett in 1952 [48] extensive evendiscarded for small Jaswon rotations and Wheeler’s of the model habit of plane (225) martensite [ ... ] It because can therefore of this para bedigm. concluded Two years that the habit plane is not rotated”later, Bowles (as and also Mackenzie assumed [34] detailed by Jaswon the reasons and Wheeler).for which, according Then, to they them, stated the habit that plane “no line in the [habit] planeshould can be be rotated invariant. [ ... First], Athey direct stated test that, of “the the accuracy amount of of plastic this deformation conclusion would is furnished be by the [unreasonable] extensive even for small rotations of the habit plane […] It can therefore be observation thatconcluded a martensite that the habit plate plane and is not the rotated” neighbouring (as also assumed matrix by Jaswon can kept and Wheeler). in focus Then, under they a microscope while traversingstated the thatwhole “no line length in the [habit] of plate.” plane This can be sentence rotated […] is A essential direct testbecause of the accuracy it isused of this to afﬁrm that the habit plane,conclusion in addition is furnished to beingby the observation untilted, that is alsoa martensite undistorted, plate and i.e.,the neighbouring fully invariant. matrix can Unfortunately, kept in focus under a microscope while traversing the whole length of plate.” This sentence is the argumentessential is not clear,because and it is we used do to notaffirm understand that the habit why plane, compatibility in addition to between being untilted the, martensiteis also and its parent austeniteundistorted, necessarily i.e., fully implies invariant. that Unfortunately, the interface the plane argument is undistorted. is not clear, and Peet we and do Bhadeshia not [128] could observe,understand by atomic why compatibility force microscopy, between the the martensite surface and relief its parent produced austenite bynecessarily the formation implies of bainite ◦ that the interface plane is undistorted. Peet and Bhadeshia [128] could observe, by atomic force at 200 C. Themicroscopy surface,, the shownsurface relief in Figureproduced 25 bya, the is formation interpreted of bainite by at these200 °C. authorsThe surface, as shown the in resultant of a homogeneousFigure shear 25a (Figure, is interpreted 25b). by It these is in authors agreement as the resultant with Bowles of a homogeneous and Mackenzie; shear (Figure however, 25b). It such a relief can be explainedis in agreement as well with by an Bowles angular and Mackenzie; distortion, however, as illustrated such a relief in can Figure be explained 25c,d. as well by an angular distortion, as illustrated in Figure 25c,d.

Figure 25. Interpretation of the surface relief formed by an fcc-bcc transformation. (a) Image made by Figure 25. Interpretationatomic force microscopy of the surface of bainite relief plates formed, and (b) by their an interpretation fcc-bcc transformation. by a shear mechanism. (a) Image made by atomic force microscopyReprinted from of[128 bainite] with per plates,mission and from ( bSpringer) their Nature. interpretation (c,d) Alternative by a shearexplanation mechanism. with an Reprinted from [128] withangular permission distortive mechanism. from Springer Surface (c) Nature. before transformation, (c,d) Alternative and (d) after explanation transformation with; the an angular tent-shape is compatible with an intra-planar distortion of the habit plane (without rotation). distortive mechanism. Surface (c) before transformation, and (d) after transformation; the tent-shape is compatible with an intra-planar distortion of the habit plane (without rotation). Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

It is probable that, in many cases, mainly for the initial stages and for the midrib formation, the habit plane is rendered invariant by the co-formation of a pair of variants (as in Figure 24); however, that condition does not seem to be necessary for the growth of individual martensite, and cannot be used as a fundamental part of a mechanistic theory of martensitic transformation. In brief, discarding the assumption of shear permitted the establishment of simple and coherent models of the {225}γ and {557}γ martensite, which is encouraging if one considers the very long history and the high number of publications on this subject. Besides, it was stated from the first quantitative metallographies that the habit planes of martensite are irrational, even if noted with high index rational numbers. Irrational values can now be proposed: according to our model the {225}γ observed in high-carbon steels and the {557}γ habit planes in low-carbon steels are {11 6}γ Crystals 2018, 8, 181 32 of 55 and {11 2}γ planes, respectively. The former are fully accommodated by coupling the twin-related √ variants that share the same dense direction, and the latter both by coupling the low-misoriented variants√ that shareIt is the probable same that,dense in plane many and cases, by mainlyadditional for dislocations. the initial stages The anddifference for the between midrib formation, these two modesthe habit might plane come is rendered from the invariant difference by the of co-formationMs temperatures of a pair; in ofthe variants former (as case, in Figurethe low 24 ); however, Ms temperaturesthat condition do not does allow not seem dislocation to be necessary plasticity for the, but growth promote of individual variant martensite,-pairing IPS and cannot be accommodation,used as whereas a fundamental in the latter part ofcase, a mechanistic dislocation theoryplasticity of martensitic allows for transformation.conditions that are less restrictive than pureIn brief, IPS. discarding the assumption of shear permitted the establishment of simple and coherent models of the {225}γ and {557}γ martensite, which is encouraging if one considers the very long history 8. Generalizationand the of high the numberAngular of Distortive publications Model on this to Face subject.-Centered Besides, Cubic it was–Body stated-Centered from the first quantitative Cubic–Hexagonalmetallographies Close-Packed that the Martensitic habit planes Transformations of martensite are, and irrational, to Face even-Centered if noted Cubic with high– index rational Face-Centerednumbers. Cubic IrrationalDeformation values Twinning can now be proposed: according to our model the {225}γ observed in √ o √ o The hardhigh-carbon-sphere model steels andallows the description {557}γ habit with planes the in same low-carbon formalism steels the are mechanisms {11 6 γ and of {11 both2 γ planes, deformationrespectively. twinning and The martensite former are transformation. fully accommodated The model by coupling permits thea natural twin-related introduc variantstion of that share an order parameterthe same denseinto the direction, transition and; the the order latter bothparameter by coupling is simply the low-misorientedthe angle of distortion variants (and that share the not an ill-defsameined dense multidimensional plane and by strain additional state). dislocations. The work done The for difference the fcc-bcc between transformation these two was modes might thus quicklycome generalized from the to difference other displacive of Ms temperatures; transformations in the between former the case, fcc, the bcc low, and Ms hcp temperatures phases do not observed inallow other dislocation metal alloys plasticity, [123]. By but noting promote that fcc variant-pairing = γ, bcc = α, and IPS hcp accommodation, = ε, as it is usually whereas done in the latter for steels, thecase, classical dislocation ORs plasticitybetween the allows three for phases conditions are the that KS are OR less for restrictive fcc bcc, than the pure Burgers IPS. OR [29] for bcc hcp, and Shoji-Nishiyama (SN) [37] for fcc hcp transformations: 8. Generalization of the Angular Distortive Model to Face-Centered⟶ Cubic–Body-Centered Cubic–Hexagonal⟶ Close-PackedKS: [110]γ = Martensitic[111]α and (1 Transformations,11)⟶γ//(110)α and to Face-Centered Cubic–Face-Centered Cubic Deformation Twinning Burgers: [111]α = [100]ε and� (110)α�//(001)ε (16) The hard-sphere model allows description with the same formalism the mechanisms of both SN: [110]γ = [100]ε and (111)γ//(001)ε deformation twinning and martensite transformation.� The model permits a natural introduction of These anrespect order the parameter parallelism into of the the transition; close-packed the order directions� parameter, and form is simply a close the set angle of ORs of distortion that is (and not of interest toan build ill-defined a unifying multidimensional theory, as already strain noticed state). Theby Burgers work done in 1934 for theand fcc-bcc illustrated transformation in Figure was thus 26a. The threequickly phases generalized projected to along other the displacive normal transformations to their common between dense theplane fcc, is bcc, shown and hcpin Figure phases observed 26b. The usein otherof angular metal distortion alloys [123 matrices]. By noting offers that the fcc possibility = γ, bcc = αof, andtreat hcping =theseε, as transformations it is usually done for steels, similarly inthe a classicalunique ORsframework, between thewith three the phasessame areparadigm. the KS OR The for fcc fcc bcc,hcp the and Burgers bcc OR hcp [29] for bcc transformationshcp, and involve Shoji-Nishiyama shuffling, because (SN) [37 the] for parent fcc andhcp daughter transformations: phases do not have the same ⟶ ⟶ number of atoms in their Bravais cell (the bcc and fcc primitive cells contain one atom, whereas the hcp primitive cell contains two atoms).KS The : [110 distortion]γ =[111 ]αmatricesand 111 andγ// shuffle110 αtrajectories (when required for the hcp phase) were analyticalBurgersly determined : [111]α =[100 as ]functionsε and 110 of αan// angular(001)ε parameter that (16) depends on the transformation. The SNchoice : [110 of] γan=[ angular100]ε and parameter111 γ// (and001) εthe details of the calculations are given in [123]. It is important to note that both the lattice distortion and shuffling These respect the parallelism of the close-packed directions, and form a close set of ORs that is of are the two sides of the same coin; there is no reason to disconnect them. interest to build a unifying theory, as already noticed by Burgers in 1934 and illustrated in Figure 26a. The three phases projected along the normal to their common dense plane is shown in Figure 26b. The use of angular distortion matrices offers the possibility of treating these transformations similarly in a unique framework, with the same paradigm. The fcc hcp and bcc hcp transformations involve shuffling, because the parent and daughter phases do not have the same number of atoms in their Bravais cell (the bcc and fcc primitive cells contain one atom, whereas the hcp primitive cell contains two atoms). The distortion matrices and shuffle trajectories (when required for the hcp phase) were analytically determined as functions of an angular parameter that depends on the transformation. The choice of an angular parameter and the details of the calculations are given in [123]. It is important to note that both the lattice distortion and shuffling are the two sides of the same coin; there is no reason to disconnect them. Crystals 2018,, 8,, 181x FOR PEER REVIEW 33 of 54 55

Figure 26.26. PhasePhase transformationstransformations in in the the fcc-hcp-bcc fcc-hcp-bcc system. system. (a )( Asa) As represented represented by Burgersby Burgers in 1934 in 1934 [29]; ([b29)] planar; (b) p representationlanar representation of the offcc, the bcc, fcc, and bcc hcp, and structures. hcp structures The positions. The positionsl = 1 and l l== 1 2 andrepresent l = 2 therepresent level of the the atoms level in of their the stacking, atoms in perpendicular their stacking to the, perpe densendicular planes (1to11) γthe//( 1 dense10)α//(001) planesε. Reprinted(111)γ//(110) fromα//(001) [123ε]. withReprinted permission from [ from123] with Elsevier. permission from Elsevier.

Some movies showing the atomic displacements were simulated, with some snapshotssnapshots shown in Figure 2727.. They illustrate the atomicatomic displacementsdisplacements during the transformation process by using an initial single crystal with a cubic shape. The exact morphology of the martensite product and the details of the accommodation mechanisms werewere notnot yetyet consideredconsidered inin thesethese simulations.simulations. The habit planes planes are are calculated calculated according according to to the the “untilted “untilted plane” plane” criterion. criterion. For bcc-fccbcc-fcc n √ √ o (7 + 2 6, 2 + 2 6, 5) transformations and and for for bcc-hcpbcc-hcp transformations, transformations, they they are are 7 + 2 6,√ 2 + 2 √6, 5 and α n(√√6, √6√, 2) o, respectively. They are less than 15° away from the habit planes reported in literature. 6, 6, 2 , respectively. They are less than 15◦ away from the habit planes reported in literature. This agreement αis thus quite good, if one considers that there is no free parameter at all in the theory. This agreement is thus quite good, if one considers that there is no free parameter at all in the theory. Of course, the interface plane of the martensite in these alloys is more complex, in that a simply Of course, the interface plane of the martensite in these alloys is more complex, in that a simply untilted untilted plane, along with the dislocations, twins, and twin-related variants that structure the plane, along with the dislocations, twins, and twin-related variants that structure the martensite martensite interface are not taken into account by the model for the moment, except for the (225) interface are not taken into account by the model for the moment, except for the (225) martensite martensite (Figure 24). In the same spirit, one-step crystallographic models of martensitic (Figure 24). In the same spirit, one-step crystallographic models of martensitic transformation using transformation using the hard-sphere assumption were proposed recently by Sowa [129]; the main difference with our approach is that the models are based on group-subgroup chains of intermediate structures. Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

Burgers: [111]α = [100]ε and� (110)α�//(001)ε (16)

SN: [110]γ = [100]ε and (111)� γ//(001)ε These respect the parallelism of the close-packed directions� , and form a close set of ORs that is of interest to build a unifying theory, as already noticed by Burgers in 1934 and illustrated in Figure 26a. The three phases projected along the normal to their common dense plane is shown in Figure 26b. The use of angular distortion matrices offers the possibility of treating these transformations Figure 27. Snapshots of 3D simulation movies of (a) an fcc ⟶ bcc martensitic transformation. (b) fcc similarly in a uniqueFigure framework, 27. Snapshots with of the 3D simulationsame paradigm. movies of The (a) an fcc fcc bcchcp martensitic and bcctransformation. hcp (b) fcc ⟶ fcc mechanical twinning. (c) fcc ⟶ hcp martensitic transformation, with, in blue, the initial parent transformations involvefcc shuffling mechanical, because twinning. the ( cparent) fcc andhcp martensiticdaughter transformation,phases do not with,have in the blue, same the initial parent fcc cube with its {100}γ facets; in red, the resulting transformed⟶ daughter crystals⟶; and in yellow, the number of atoms in theirfcc cube Bravais with cell its {100} (theγ bccfacets; and in fcc red, primitive the resulting cells transformedcontain one daughteratom, whereas crystals; the and in yellow, intermediate states stopped at midway (half of the maximum distortion angle). The black arrow hcp primitive cell containsthe intermediate two atoms). states stoppedThe distortion at midway matrices (half of theand maximum shuffle distortiontrajectories angle). (when The black arrow represents the invariant line PO, and the white arrow the direction PK (also invariant for the fcc ⟶ required for the hcp phase)represents were the analytical invariant linely determinedPO, and the as white functions arrow the of directionan angularPK parameter(also invariant that for the fcc hcp and fcc ⟶ fcc transformations). Reprinted from [123] with permission from Elsevier. depends on the transformation.hcp and fcc fccThe transformations). choice of an Reprintedangular fromparameter [123] with and permission the details from Elsevier.of the calculations are 9.given Angular in [123 Distortive]. It is important Model for to Deformationnote that both Twinning the lattice in distortion Hexagonal and Close shuffling-Packed Metals are the two sides9. of Angular the same Distortive coin; there Model is no reason for Deformation to disconnect Twinning them. in Hexagonal Close-Packed Metals 9.1. Application to Extension Twinning The cases cases of of deformation deformation twinning twinning in bcc in bcc and andhcp metals hcp metals were notwere investigated not investigated in [123 ], in because [123], becausethey are theymore are complex more than complex for fcc-fcc than twinning. for fcc-fcc The twinning. investigations The investigations on deformation on deformation twinning in twinning in magnesium were only started three years ago. Among hcp metals, magnesium was magnesium were only started three years ago. Among hcp metals, magnesium was chosen√ because chosenthe γ = becausec/a ratio the≈ 1.625, = / which ratio is ≈ close1.625, to which the ideal is close hard-sphere to the ideal packing hard-sphere value ofpacking8/3 value≈ 1.633. of For8/ extension3 ≈ 1.633. twinning, For extension the displacements twinning, the of the displacements Mg atoms were of quite the Mg easy atoms to determine were quite [130]. easy A brief to determineexplanation [130 is given]. A brief here. explanation The twin–parent is given misorientation here. The twin is– aparent rotation misorientation of 86◦ around is thea rotation common of 86°a-axis. around Instead the ofcommon working a- directlyaxis. Instead with thisof working misorientation, directly it with is simpler this misorientation, working on a it “prototype” is simpler workingBain-like stretch on a “prototype” twin with a 90 Bain◦ misorientation,-like stretch twin and do with the a ﬁrst 90° calculations misorientatio inn, the and orthogonal do the basis first calculationsxyz shown in in Figure the orthogonal 28a. The lattice basis stretch xyz shown results in from Figure an exchange 28a. The between lattice stretch the basal results and prismatic from an planes,exchange as between shown in the Figure basal 28 andb. This prismatic exchange planes, can be as visualizedshown in Figure by considering 28b. This the exchange XYZ supercell can be visualizedcontaining fourby considering atoms, and isthe a consequence XYZ supercell of acontaining simultaneous four rotation atoms, of and angle is aη consequenceof the three atoms of a simultaneousinside the cell. rotation The lattice of angle stretch η of induces the three a slight atoms rotation inside the of the cell. 0 The112 latticeplane, stretch called induces obliquity. a Theslight obliquity rotation angleof the ξ(0shown112) plane in Figure, called 28 cobliquity. can now The be calculated obliquity angle and continuously ξ shown in Figure compensated 28c can now be calculated and continuously compensated for, in order to leave the (0112) plane untilted during the twinning distortion, such that the final twin–parent misorientation is (86°, a). The analytical equations of trajectories of all the magnesium atoms are established as functions of a unique angular parameter (as for martensitic transformations). It was shown that the (0112) habit plane is not fully invariant during the twinning process; it is just untilted, and restored when the Crystals 2018, 8, 181 35 of 55 for, in order to leave the 0112 plane untilted during the twinning distortion, such that the ﬁnal twin–parent misorientation is (86◦, a). The analytical equations of trajectories of all the magnesium atomsCrystals are2018 established, 8, x FOR PEER as REVIEW functions of a unique angular parameter (as for martensitic transformations).35 of 54 It was shown that the 0112 habit plane is not fully invariant during the twinning process; it is just untilted,distortion and is complete. restored whenIndeed, the the distortion distance isOV complete. in this plane Indeed, is not the constant distance (Figure OV in 28d this). planeIn addition, is not constantthere is a (Figure volume 28 changed). In addition, of 3% during there isthe a distortion volume change (Figure of 3%28e), during as expected the distortion from Kepler (Figure–Hales’ 28e), astheorem expected for from a Kepler–Hales’ transition between theorem two for a hcp transition states. between This volume two hcp change states. This cannot volume be changetotally cannotaccommodated be totally elastically accommodated, which elastically, means that which plasticity means is that required plasticity to isaccommodate required to accommodate the twinning thedistortion twinning in polycrystalline distortion in polycrystalline alloys. alloys.

FigureFigure 28. 28.Hard-sphere Hard-sphere displacive displacive model model of extension of extension twinning twinning in magnesium. in magnesium. (a) Unit (a) cellUnit with cell labels with Olabels, X, M O, N, ,XZ, ,M and, NT, givenZ, and to T the given magnesium to the magnesiumatoms. (b) Schematic atoms. (b view) Schematic of the atomic view displacements of the atomic associateddisplacements with associated a stretch distortion; with a stretch (c) obliquity distortionξ of; ( thec) obliquity {0112} plane ξ of thatthe {0 should112} plane be compensated that should for; be (compensatedd) evolution of for the; ( distanced) evolution OV belonging of the distance to the {0OV112} belonging plane; and to (thee) change {0112} ofpl theane unit; and volume (e) change during of thethe distortionunit volume process, during as the functions distortion of κ process,, the cosine as functions of the angular of κ, parameterthe cosine ofη. Adaptedthe angular from parameter [130]. η. Adapted from [130]. This work conciliates inside a unique model the (90◦, a) twins observed in nano-pillars [131] This work conciliates inside a unique model the (90°, a) twins observed in nano-pillars [131] and the (86◦, a) usual twins observed in bulk samples. Both twins result from the same atomic and the (86°, a) usual twins observed in bulk samples. Both twins result from the same atomic displacements and lattice distortion, and differ only by their obliquity. At this step, the model does displacements and lattice distortion, and differ only by their obliquity. At this step, the model does not imply imagining twinning dislocations, disconnections, or any complex mechanism involving not imply imagining twinning dislocations, disconnections, or any complex mechanism involving dislocations. Some elastic strains are induced, and some dislocations are emitted by the distortion, dislocations. Some elastic strains are induced, and some dislocations are emitted by the distortion, and that could be interesting to determine how they can be predicted from the distortion matrix and and that could be interesting to determine how they can be predicted from the distortion matrix from the usual slip systems in magnesium, but that is beyond the model in its current form. and from the usual slip systems in magnesium, but that is beyond the model in its current form. As the distortion is not a simple shear, a new criterion had to be introduced to substitute Schmid’s law. It is often reported that twins can form in hcp metals despite low or even negative Schmid factors [132–135]. This “non-Schmid” behavior is usually explained by invocating that the local stress fields in the grains differ from the applied stresses, but another explanation is worth being considered. The intermediate state, at the maximum volume change, implies the largest strains perpendicularly to the {1012} habit plane; therefore, it is conceivable that this state Crystals 2018, 8, 181 36 of 55

As the distortion is not a simple shear, a new criterion had to be introduced to substitute Schmid’s law. It is often reported that twins can form in hcp metals despite low or even negative Schmid factors [132–135]. This “non-Schmid” behavior is usually explained by invocating that the local stress fields in the grains differ from the applied stresses, but another explanation is worth being considered. Crystals 2018, 8, x FOR PEER REVIEW The intermediate state, at the maximum volume change,36 impliesof 54 the largest strains perpendicularly to the 1012 habit plane; therefore, it is conceivable that this state constitutes an energetic gap for the constitutes an energetic gap for the twintwin formation. formation. Twinning Twinning occurs occurs when when the work the work performed performed by by the external stresses accompanying the external stresses accompanying the twinthe twin formation formation is higher is higher than than this thisenergetic energetic gap, gap, or is or at isleast at least positive. The general formula of the positive. The general formula of the workwork W doneW done during during a la attice lattice distortion distortion =E = D −, Iwhere, where DD isis the distortion matrix and I the identity the distortion matrix and I the identity matrixmatrix occurring occurringCrystals in in a2018 a fixed fixed, 8, x FOR stressstress PEER REVIEW field field isis 36 of 54 𝓔𝓔 𝐃𝐃 − 𝐈𝐈 W = . constitutes an energetic gap for the twin formation. Twinning occurs when the work performed by 𝚪𝚪 W = .E (17) (17) the external stresses accompanying the twin formationij isij higher than this energetic gap, or is at least Whereas the usual shear model with Schmid’s𝚪𝚪𝑖𝑖𝑖𝑖 law𝓔𝓔positive𝑖𝑖𝑖𝑖 predicts. The general the formulaformation of the workof extension W done during twins a lattice distortion = − , where D is during uniaxial tensile tests with parent crystalsWhereas tiltedthe in distortion the usual range matrix shear [ and model43° I the, 47° identity with], as matrix Schmid’sshown occurring in lawFigure in a predictsfixed stress field the formation is of extension twins ◦ ◦ W = . [− ] 29a, the hard-sphere displacive modelduring associated uniaxial with tensile the testswork with criterion parent crystalspredicts tilted a range in the of range 43 , 47 , as shown(17) in Figure 29a, − ◦ ◦ [ 59°, 59°], as shown in Figure 29b. theExperimental hard-sphere studies displaciveWhereas would the modelusual beshear associatedneeded model with to withSchmid’s confirm the law work predictsthis criterion the formation predicts of extension a range twins of [−59 , 59 ], during uniaxial tensile tests with parent crystals tilted in the range [−43°, 47°], as shown in Figure prediction. It should be noted that foras shownfcc-fcc indeformation Figure 29b. Experimentaltwinning, the studies difference would between be needed to confirm this prediction. It should − 29a, the hard-sphere displacive model associated with the work criterion predicts a range of Schmid’s law and the work criterion at bemaximum noted that volume for[−59° fcc-fcc, 59° change], as deformation shown is hardly in Figure twinning,perc 29beptible,. Experimental the as difference shown studies would between be needed Schmid’s to confirm law this and the work by Figure 29d–f (unpublished result);criterion which could at maximumprediction. explain volume It why should non change be noted-Schmid is that hardly forbehavior fcc- perceptible,fcc deformation of the as twinning, shown the by difference Figure between29d–f (unpublished result); whichSchmid’s could explainlaw and the why work non-Schmid criterion at maximum behavior volume of change the mechanicallyis hardly perceptible, twins as shown in fcc metals has mechanically twins in fcc metals has never been observed.by Figure 29d–f (unpublished result); which could explain why non-Schmid behavior of the never been observed.mechanically twins in fcc metals has never been observed.

Figure 29. InteractionFigure 29. Interaction work during work during a tensile a tensile stress stress along along the the z-axisz -axisof a parent of a crystal parent, tilted crystal, by an tilted by an angle Φ aroundangle the Φx around-axis and the x rotated-axis and rotatedby an by angle an angleθ around θ around the the nn-axis-axis (normal (normal to the to twinning the twinning plane), with (a–c) for extensionplane), with (a twinning–c) for extension in hcptwinning metals, in hcp andmetals, the andx the-axis x-axis being being the [100] [100] axis axis.. (d–f) F (ord– f) For fcc-fcc fcc-fcc deformation twinning, the x-axis is the [110] axis. (a,d) Interaction work Wf calculated with the deformation twinning,complete distortion the x-axis (simple is the shear) [110] matrix. axis. W (fa is,d ) proportional Interaction to thework usualW f Schmidcalculated factor. with(b,e) the complete Interaction work Wi calculated with the intermediate distortion matrix corresponding to the distortion (simple shear) matrix. Wf is proportional to the usual Schmid factor. (b,e) Interaction work maximum volume change. (c,f) Schematic view of the orientation of the parent crystal. The distortion W calculated with the intermediate distortion matrix corresponding to the maximum volume change. i matrix of extension twinning in hcp metal is given in [130]. The distortion matrix for fcc-fcc (c,f) Schematicdeformation view of the twinning orientation is given in of [123 the]. parent crystal. The distortion matrix of extension twinning in hcp metal is given in [130]. The distortion matrix for fcc-fcc deformation twinning is given in [123]. Figure 29. Interaction work during a tensile stress along9.2. Comparisonthe z-axis with of thea parent Pure-Shuffle crystal Model, tilted by an angle Φ around the x-axis and rotated by an angle θ around the n-axis (normal to the twinning Another model of extension twinning without twinning dislocations exists in literature. In plane), with (a–c) for extension twinning9.2. in Comparisonhcp metals,2009, and with Lithe the and x Pure-Shufﬂe- Maaxis “observed” being the Model the [100] results axis of. ( MDd– f simulations) For and proposed an “atomic shuffling fcc-fcc deformation twinning, the x-axis is theAnother [110] axis model. (a,d) Interaction of extension work twinning Wf calculated without with twinning the dislocations exists in literature. In 2009, complete distortion (simple shear) matrix. Wf is proportional to the usual Schmid factor. (b,e) Li and� Ma “observed” the results of MD simulations and proposed an “atomic shufﬂing dominated Interaction work Wi calculated with the intermediate distortion matrix corresponding to the maximum volume change. (c,f) Schematic view of the orientation of the parent crystal. The distortion matrix of extension twinning in hcp metal is given in [130]. The distortion matrix for fcc-fcc deformation twinning is given in [123].

9.2. Comparison with the Pure-Shuffle Model Another model of extension twinning without twinning dislocations exists in literature. In 2009, Li and Ma “observed” the results of MD simulations and proposed an “atomic shuffling Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

8. Generalization of the Angular Distortive Model to Face-Centered Cubic–Body-Centered Cubic–Hexagonal Close-Packed Martensitic Transformations, and to Face-Centered Cubic– Face-Centered Cubic Deformation Twinning Crystals 2018, 8, 181 37 of 55 The hard-sphere model allows description with the same formalism the mechanisms of both deformation twinning and martensite transformation. The model permits a natural introduction of an ordermechanism” parameter [136 into]. Inthe 2013, transition Wang; et the al. order also noticed parameter in the is MDsimply simulations the angle a nucleationof distortion step (and obeying not ana ill “pure-defined shuffle multidimensional mechanism”, thestrain growth state). being The work accomplished done for the “via fcc the-bcc conventional transformation glide-shuffle was thus quicklymechanism” generalized [137]. to These other authorsdisplacive also transformations used the expressions between the “unit fcc, cellbcc, reconstruction”and hcp phases [138], observedand in “twinning other metal with alloys zero [ twinning123]. By noting shear” that [139 fcc], which= γ, bcc can = α make, and hcp one = think ε, as it that is usually the mechanism done is for steels,diffusive the classical across short ORs distances. between the Li and three colleagues phases are define the KS shuffling OR for as fcc “inhomogeneous bcc, the Burgers displacements OR [29] forof bcc an ensemble hcp, and of Shoji atoms-Nishiyama in the layers (SN) immediately [37] for fcc adjacent hcp transformations: to the TB [twin boundary]” [136], and twinning growth is imagined as a boundary migration where “the TBs⟶ can migrate freely and fully consume⟶ the parent grains”KS: [139 [110]]. Thisγ = [111] definitionα and (1 can11)⟶γ// be(1 quite10)α confusing. Shuffling is historically

invocated when some atomsBurgers: in the [111] unitα cell= [100] doε not and follow� (110)α the�//(001) sameε trajectories as those located(16) at the nodes of the lattice. As written by Christian et al., “a shuffle only rearranges the atom positions within SN: [110]γ = [100]ε and (111)γ//(001)ε a unit cell” [140]. For example, as the hcp unit cell contains� two atoms, the martensitic transformations Thesebetween respect fcc and the hcp,parallelism or between of the bcc close and-packed hcp, require directions� an additional, and form shufflinga close set of of half ORs of that the is atoms, of interestas shown to build previously a unifying in theory, Figure as26 .already The shuffling noticed by equations Burgers arein 1934 analytically and illustrated determined in Figure for these 26a. Thetransformations three phases in projected [123]. Transformations along the normal between to their a simple common phase dense with plane a small is unit shown cell, in and Figure a complex 26b. Thephase use with of angular a large unitdistortion cell containing matrices offers a high the number possibility of atoms, of treat implying largethese andtransformations complex shuffles, similarlyas it in happens a unique inmany framework, shape memorywith the alloys—for same paradigm. example, The the fcc B2 B19’hcp transformationand bcc hcp in NiTi transformationsalloys [141 ].involve shuffling, because the parent and daughter phases do not have the same ⟶ ⟶ number ofIt atoms is not in surprising their Bravais that shufflingcell (the bcc and and distortion fcc primitive are correlated cells contain in metals, one asatom, experimentally whereas the found hcp primitiveby Wang etcell al. contains for the bcc-hcp two atoms). transformation The distortion in titanium matrices [142], and because shuffle the latticetrajectories distortion (when and the requireddisplacement for the hcp of phase) the atoms were in analytical the latticely are determined intrinsically as correlated.functions of Similarly, an angular deformation parameter twinning that in dependshcp crystalson the imaginedtransformation. as an hcp-hcpThe choice transformation of an angular involves parameter shuffling. and In thethe exampledetails ofof extensionthe calculationstwinning are in given magnesium, in [123]. aIt quarteris important of the to atoms note that (those both at thethe nodeslattice ofdistortion the XYZ and lattice) shuffling follow the are thetrajectories two sides given of the by same the distortioncoin; there matrix, is no reason and three to disconnect quarters of them. the atoms shuffle inside their local XYZ cell. Consequently, in its traditional meaning, shuffling cannot exist for itself; if it exists, it is always concomitant with a lattice distortion. In Christian et al.’s words: “shuffles can only be avoided if both structures have primitive unit cells containing only one atom” [140]. This meaning has digressed over time, with the introduction of the concept of “shuffle transformations”. Christian et al. define them as follows: “A pure shuffle transformation requires that some unit cell of one lattice is almost identical with a cell of the other lattice” [140]. Delanay uses a more precise definition: “A shuffle is a coordinated movement of atoms that produces, in itself, no lattice distortive deformations but alters only the symmetry or structure of the crystal; a sphere before the transformation remains the same sphere after the transformation” [143]. He adds later: “Shuffle transformations are not necessarily pure; small distortive deformations may additionally occur. They therefore also include those transformations involving dilatational displacements, in addition to the pure shuffle displacements, provided that they are small enough not to alter significantly the kinetics and morphology of the transformation” [143]. Therefore, a pure shuffle transformation is a transformation that should not significantly change the lattice; this is not the case for extension twinning, and that is why the idea of “pure shuffle” is unclear. It is difficult to imagine that the atoms can move independently of the lattice in which they are contained. This would be possible with a big lattice containing many, largely-spaced atoms of small size, but is completely impossible with a hard-sphere model, in which the atoms are in contact, as proved by the calculations of [130]. When the atoms move, the lattice is distorted, and vice versa. There is no possibility of elastically accommodating atomic displacement in the unit cell while maintaining the unit cell dimension. Liu et al. are more cautious than Li and Zhang in their use of vocabulary; they clearly specify that “UCR [unit cell reconstruction] produces tetragonal deformation instead of simple shear” [138]. It should be interesting to clarify the role of the external stresses in the “pure shuffle” model. Despite the possible confusion raised by the term “shuffle”, we can notice that this model is probably not so different from the hard-sphere model we propose. Indeed, the trajectories of the atoms shown in Figure 3 of [137] and reported in Figure 30a seem to be the same as those shown in Figure 28a. The trajectories reported in Figure 4 of [131], shown in Figure 30b, are different, but the rotation and mirror symmetries between the figures makes Crystals 2018, 8, 181 38 of 55

Crystals 2018, 8, x FOR PEER REVIEW 38 of 54 an exact comparison difﬁcult. The conclusions reached by both models are also convergent: “the {10–12}during twinning” plane and a cannotmechanism remain “without invariant the during need of twinning” twinning dislocations” and a mechanism is possible “without [139 the]. It needis also of interesting twinning dislocations” to note the very is possible good agreement [139]. It is alsobetween interesting the DTF to notecalculations the very made good agreementby Ishii et betweenal. [144] and the DTFthe analytical calculations calculations made by presented Ishii et al. in [ 144Section] and 9.1 the. For analytical example, calculations the energy presentedlandscape ingiven Section by DFT9.1. For presents example, an energy the energy barrier landscape due to shuffling,given by DFT as if presents “something an energy internal barrier ‘got stuck’” due to shufﬂing,[144], and as this if “something barrier was internal found ‘got at the stuck’” midway [144], point and this of the barrier transformation was found at process, the midway as with point the of themaximum transformation volume process, change as calculated with the maximum analytically volume (Figure change 28e calculated). The DTF analytically results can (Figure thus 28 bee). Theinterpreted DTF results by the can fact thus th beat interpreted the atoms in by the the unitfact that cell the get atoms too close in the to unit each cell other get (the too close spheres to eachinterpenetrate other (thed spheresone another) interpenetrated when a simple one shear another) is applied, when a i.e., simple when shear no isdilatation applied, perpendicular i.e., when no dilatationto the twin perpendicular plane is allowed to the. T twinhis type plane of isstrain allowed. does Thisnot let type the of cell strain “breath” does not and let reach the cell the “breath”volume andchange reach required the volume by the change intermediate required state by the (+3%). intermediate If the analytical state (+3%). approach If the permitsanalytical obtaining approach a permitsqualitative obtaining understanding a qualitative of the understanding mechanism, ofDFT the remains mechanism, necessary DFT remainsto quantify necessary the energies to quantify and thecompare energies them and with compare the experiments. them with the experiments.

FigureFigure 30.30. ShufﬂeShuffle modelsmodels ofof extensionextension twinning in hcp metals proposed by (a) Wang etet al.,al., andand byby ((bb)) Liu Liu etet al. al. Reprinted from [137] and from [ [131],], respectively, respectively, with with permission permission from from Nature Nature PublishingPublishing Group.Group.

WeWe believebelieve thatthat thethe couplingcoupling betweenbetween shufflingshuffling andand latticelattice distortiondistortion observedobserved inin thethe extensionextension twinstwins is is a a general general phenomenon, phenomenon, and and we we do do not not think think that that “[the“[the extension extension twinning] twinning] mechanism mechanism distinctivelydistinctively differs differs from from any any other oth twinninger twinning modes” modes [139”]. [ We139 have]. We shown have that shown deformation that deformation twinning cantwinning be modelled can be bymodelled a distortive by a mechanism, distortive mechanism in which all, in atoms which can all move atoms collectively can move atcollectively the speed ofat sound,the speed orat of lower sound, speeds or at if lower time isspeeds needed if totime reorganize is needed the t dislocationso reorganize emitted the dislocations by the distortion emitted and by relaxthe distortion the back-stresses and relax in the the back surrounding-stresses in matrix. the surrounding Therefore, matrix. to our point Therefore, of view, to theour mechanismpoint of view, of extensionthe mechanism twinning of in extension magnesium twinning does not in differmagnesium from the does other not deformation differ from twinning the other modes. deformation All the modestwinning should modes. imply All anthe angular modes distortion,should imply and an many angular of them distortion, (but not and all many of them) of them become (but a simplenot all shearof them) when become the distortion a simple isshear complete. when the distortion is complete. TheThe advantages advantages of of the the distortive distortive model model of ofdeformation deformation twinning twinning is that is that it can it be can combined be combined with mechanicalwith mechanical calculations, calculati suchons, as such Equation as Equation (17) in order(17) in to order explain to whyexplain twins why are twins formed are and formed for which and orientationsfor which orientations they are formed. they are Contrary formed. to Contrary the shuffle to model,the shuffle the distortive model, the model distortive makes model macroscopic makes predictions.macroscopic On predictions. the other hand,On the it isother true hand, that the it is distortive true that model the distortive does not model yet explain does thenot microscopic yet explain characteristicsthe microscopic at characteristics the twin interface, at the suchtwin interface, as the fact such that as the the interface fact that planethe interface is often plane not straightis often not straight but made of basal-prismatic segments. Partisans of “pure-shuffle” insist on these but made of basal-prismatic segments. Partisans of “pure-shuffle” insist on these features as if they werefeatures a specific as if they property were a of specific extension pro twinning,perty of extension but the existence twinning, of but a segmented the existence interface of a segmented is a quite generalinterface property is a quite of generaldisplacive property transformations. of displacive The transformations. formation of terraces The formation and ledges of are terraces observed and forledges the deformationare observed twinsfor the in deformation the Twin Induced twins in Plasticity the Twin (TWIP) Induced steels, Plasticity and for (TWIP many) steels, diffusive and and for many diffusive and displacive transformations, as proved by the HRTEM observations made by displacive transformations, as proved by the HRTEM observations made by Ogawa and Kajiwara in Ogawa and Kajiwara in martensitic iron alloys [145]. At a mesoscale, the lenticular shape of extension twins in magnesium can also be observed for other deformation twin modes and in other Crystals 2018, 8, 181 39 of 55

Crystalsmartensitic 2018, 8 iron, x FOR alloys PEER [REVIEW145]. At a mesoscale, the lenticular shape of extension twins in magnesium39 of 54 can also be observed for other deformation twin modes and in other metals and alloys, as in TWIP metalssteels (Figure and alloys, 31b). as Thus, in TWIP neither steels the ( basal-prismaticFigure 31b). Thus, segments neithernor the thebasal lenticular-prismatic shape segments are proof nor ofthe a pure-shufﬂelenticular shape mechanism; are proof they of a canpure be-shuffle formed mechanism; displacively they as well.can be formed displacively as well.

Figure 31. 31. EBSDEBSD maps maps of of lenticular lenticular deformation deformation twins, twins, ( (aa)) formed formed in in recrystallized pure pure (hcp) magnesium;magnesium; andand ( b ()b in) ina (fcc) a (fcc) TWIP TWIP steel (kindly steel (kindly given by given K. Zhu, by ArcelorMittal). K. Zhu, ArcelorMittal). The twin boundaries The twin are marked in red; they are identiﬁed by the misorientations (87◦ ± 5◦, [001] ) and (60◦ ± 5◦, <111> ) boundaries are marked in red; they are identified by the misorientations (87°hex ± 5°, [001]hex) and (60°fcc ± in (a,b), respectively. 5°, <111>fcc) in (a,b), respectively.

In brief, it seemsseems thatthat thethe “pure-shuffle”“pure-shuffle” andand thethe “distortive”“distortive” visions could be mixed,mixed, because they are not antagonistic.antagonistic. Both approaches have have strong strong argument argumentss [146]; they agree to say that the dislocation twinning twinning and and the the complexity complexity of the of topologicalthe topological model model are actually are actually unnecessary unnecessary to explain to explainthe twinning the twinning mechanism. mechanism. In addition, In addition, both discard both the discard classical the assumption classical assumption that extension that twinning extension is twinninga simple shearis a simple mechanism. shear mechanism. However, Li However, and Ma think Li and that Ma the think non-shear that the character non-shear is limited character to the is {10limited12} twinning to the {10 mechanism,12} twinning and mechanism, that “this should and that not “ bethis deemed shouldas not the be failure deemed of theas the classical failure theory”, of the whereasclassical wetheory believe”, whereas that there we is believe nothing that special there in the is nothing mechanism special of extension in the mechanism twinning in of magnesium, extension twinningand thatall in the magnesium deformation, and twinning that all modes the deformation and all the martensitic twinning modes transformations and all the obey martensitic the same transformationscrystallographic rules.obey the The same angular crystallographic distortive paradigm rules. The proposes angular a way distortive to get thisparadigm unification. proposes a way to get this unification. 9.3. Application of the Distortive Model to Other Twinning Modes 9.3. ApplicationThere is another of the Distortive important Model difference to Other between Twinning the Modes pure-shuffle and the distortive approaches. The formerThere is relies another on MDimportant computer difference simulations between to explore the pure new-shuffle twinning and the modes, distortive and extractapproaches. some Thehints former and rules relies to on build MD crystallographic computer simulations models, to whereas explore innew the twinning latter, the modes computer, and is e justxtract used some to hintshelp theand analytical rules to build calculations crystallographic that, at least models, in theory, whereas could in be the done latter by hand., the computer The displacive is just model used to is purelyhelp the geometric; analytical this calculations simplicity that, is a at force least to in understand theory, could and be explain done by the hand. various The twinning displacive modes mod inel ismetals. purely For geometric; example, this the hard-spheresimplicity is displacivea force to understand model was alsoand recentlyexplain the applied various to {10 twinning11} contraction modes intwinning metals. in For hcp example, metals [147 the], as hard shown-sphere in Figure displacive 32, leading model to was similar also conclusions, recently applied i.e., the to volume {1011} contractionof the unit cell twinning was not in constant hcp metals and [147 the], habit as shown plane in was Figure not invariant 32, leading during to similar the process; conclusions, it was justi.e., theuntilted volume and restoredof the unit when cell the was process not constant was complete, and the and habit the distortion plane was matrix not theninvariant became during a simple the process;shear matrix. it was The just result untilted was and a shear restored of amplitude when the s =process 0.358 on was the complete, plane (01 1)and along the distortion the axis [18, matrix5, 5], whichthen became was not a predictedsimple shear by Bevis matrix. and The Crocker’s result theory.was a It shear was recentlyof amplitude realized, s = by0.358 writing on the the plane shear (direction011) along with the four axis indices, [18, 5 i.e.,, 5],− [ which41, 28, was 13, 5], not that predicted this shear by mode Bevis is andexactly Crocker’s the same theory. as that It given was recentlyby Serra etrealized al. in their, by writing list of three the shear possible direction {1011} twinningwith four systemsindices, giveni.e., −[ in41 Table, 28, 13, 1 of 5], [148 that], calculatedthis shear modefrom theis exactly translation the same vectors as that between given the by parentSerra et and al. twinin their crystals list of (i.e., three vectors possible of the{101 dichromatic1} twinning systemsstructure) given [149]. in This Table confirms 1 of [148 the], validity calculated of the from calculations the translation [147], andvectors also between proves that the an parent approach and twindifferent crystals from (ai.e., disconnection vectors of model the dichromatic is possible. structure) [149]. This confirms the validity of the calculations [147], and also proves that an approach different from a disconnection model is possible. Crystals 2018, 8, 181 40 of 55 Crystals 2018, 8, x FOR PEER REVIEW 40 of 54

Figure 32. Snapshots of three states during the formation of a (56°, a) contraction twin on the (0111) Figure 32. Snapshots of three states during the formation of a (56◦, a) contraction twin on the (0111) plane in magnesium,magnesium, simulated with the displacive hard-sphere hard-sphere model. ( (aa)) Initial Initial hchcpp crystalcrystal;; ( (b)) intermediate statestate atat midway;midway; ( c()c) newly newly formed formed hcp hcp crystal. crystal. This This ﬁgure figure is madeis made to helpto help understand understand the 1 crystallographicthe crystallographic exchange exchange between between the basalthe basal plane plane and theand (01the1 1)(01 plane.1) plane. Reprinted Reprinted from from [147] [ with147] permissionwith permission from from the International the International Union Union of Crystallography. of Crystallography.

There is no technical obstacle to apply the same approach to other twinning modes in There is no technical obstacle to apply the same approach to other twinning modes in magnesium, magnesium, or to bcc-bcc twinning modes, and a general theory of deformation twinning or to bcc-bcc twinning modes, and a general theory of deformation twinning compatible with a compatible with a hard-sphere assumption can be developed in the near future. hard-sphere assumption can be developed in the near future. It is important to note that the habit plane is not an invariant plane, but just a plane restored It is important to note that the habit plane is not an invariant plane, but just a plane restored from the intermediate state when the transformation is complete. The possibility that the plane is from the intermediate state when the transformation is complete. The possibility that the plane transformed into a new plane different from the initial one is offered by the new paradigm. We is transformed into a new plane different from the initial one is offered by the new paradigm. recently found, by EBSD, a new and unconventional twin in a magnesium single crystal [150] that is We recently found, by EBSD, a new and unconventional twin in a magnesium single crystal [150] that incompatible with simple shear but corresponds exactly to this case. This twin is a cousin of the is incompatible with simple shear but corresponds exactly to this case. This twin is a cousin of the usual (1122) and (1126) twins; actually, the three twins derive from the same prototype “Bain” usual (1122) and (1126) twins; actually, the three twins derive from the same prototype “Bain” stretch stretch detailed in [151], and differ only by their obliquity correction. detailed in [151], and differ only by their obliquity correction. 9.4. Comparison with Earlier Atomistic Models of Hexagonal Close-Packed Twinning 9.4. Comparison with Earlier Atomistic Models of Hexagonal Close-Packed Twinning In 1980, Dubertret and Le Lann proposed models of twinning in hcp metals [152,153] that were In 1980, Dubertret and Le Lann proposed models of twinning in hcp metals [152,153] that were cited by Wang et al. [137] as references for the concept of shuffling. These models were nearly cited by Wang et al. [137] as references for the concept of shuffling. These models were nearly forgotten, forgotten, despite or because of the huge amount of literature on deformation twinning. They are despite or because of the huge amount of literature on deformation twinning. They are also based on also based on hard-spheres, but a direct comparison with our models is difficult because the atomic hard-spheres, but a direct comparison with our models is difficult because the atomic displacements displacements are expressed from tetrahedrons associated with the odd condition√ where = √3. c are expressed from tetrahedrons associated with the odd condition where a = 3. The main initial Theidea mainis the sameinitial as id thatea is used the in same the pure-shuffle as that used model, in the and pure the-shuffle c/a ratio model, was and probably the c/a chosen ratio to was let probablythe atoms chosen move andto let exchange the atoms their move positions and exchange in the lattice, their positions without changingin the lattice the, latticewithout dimensions changing thein order lattice to dimensions be in a pure-shuffle in order to case. be in Despite a pure- thisshuffle choice, case. and Despite the absencethis choi ofce correlation, and the absence between of correlationthe atomic displacements between the atomic and the displacements lattice distortion, and Dubertret the lattice and distortion, Le Lann Dubertret clearly understood and Le Lann that {10clearly12} extension understood twinning that {10 and12} {10extension11} contraction twinning twinning and {101 are1} obtainedcontraction by twinning atomic displacement are obtained that by producesatomic displacement a crystallographic that produce exchanges a crystallographic (correspondence) exchange between (correspondence) the basal and a prismatic between planesthe basal in andthe formera prismatic case, andplanes between in the the former basal case, plane and and between a {1101} the plane basal in theplane latter and case. a {1 They101} plane called in these the latterexchanged case. planesThey called the “corrugated these exchanged planes”. planes Going the deeper “corrugated into historical planes”. considerations, Going deeper it into is interesting historical considerations,to note that the initialit is interesting 1968 Kronberg to note model that the [154 initial] on which1968 Kronberg Dubertret model and Le [154 Lann’s] on which works Dubertret are based andalso Le uses Lann’s a hard-sphere works are representation based also uses of a the hard atoms,-sphere and representation creates a correlation of the atoms between, and the creates atomic a corrtrajectorieselation between and the lattice the atomic distortion, trajectories as shown and inthe Figure lattice 33 distortion,. as shown in Figure 33. Crystals 2018, 8, 181 41 of 55 Crystals 2018, 8, x FOR PEER REVIEW 41 of 54

Figure 33. 33. Kronberg’sKronberg’s model of extension twinning in hcp hcp metals. metals. ( (aa)) Initial Initial crystal with the “hard-sphere” atoms projected along the [010]hex direction, (b) corresponding lattice, (c) lattice “hard-sphere” atoms projected along the [010]hex direction, (b) corresponding lattice, (c) lattice distortion during twinning,twinning, and (d) lattice distortion and atomic displacements during twinning. Reprinted from [154154]] withwith permissionpermission fromfrom Elsevier.Elsevier.

Our model in [130] is a kind of “rediscovery” of Kronberg’s work [154], but now it includes the Our model in [130] is a kind of “rediscovery” of Kronberg’s work [154], but now it includes the calculation of the distortion, correspondence, and orientation matrices, and analytical equations of calculation of the distortion, correspondence, and orientation matrices, and analytical equations of the the atomic trajectories. atomic trajectories.

10. Future Works and Perspectives

10.1. The Distortion Angle As a Natural Order Parameter Thermodynamic models of phase transitions are based on a parameter called the “order parameter”. In In phase phase field field models models (see for example [155,156]), the order parameter is simply the proportion of phases, i.e., a real number between 0 (parent phase) and 11 (daughter(daughter phase), with the interface encoded encoded by by a a re realal value value between between 0 0and and 1. 1.Some Some models models try tryto find to find a physical a physical parameter parameter that thatcontrols controls the orderthe order parameter, parameter, but but the the choice choice depends depends on on the the type type of of transition. transition. It It is is usual to differentiate second-ordersecond-order and first-orderfirst-order phase transitionstransitions (note that the meaning ofof term “order” is different from from that that of “order of “order parameter”). parameter”). Second-order Second transitions-order transitions imply small implyatomic displacements, small atomic sodisplacements that the parent, so andthat daughter the parent phases and daughter are very close.phases The are daughter very close phase. The isdaughter formed directlyphase is from formed the totalitydirectly of from the parent the totality phase of without the parent coexistence phase of witho the twout coexistence phases. First-order of the two transitions phases. imply First-order large displacementstransitions imply of atomslarge anddisplacements an important of atoms lattice distortion;and an important in addition, lattice the distortion transition; in is accompaniedaddition, the bytransition a latent is heat. accompanied The energy by required a latent heat to accommodate. The energy required the daughter to accommodate phase inside the the daughter parent phase phase is suchinside that the theparent two phase phases is generallysuch that the coexist, two phases and an generally important coexist, undercooling and an important is required undercooling to reach the is required to reach the complete transformation. The order parameter that defines the daughter phase continuously changes from 0 to 1 in second-order transitions, and is abruptly and Crystals 2018, 8, 181 42 of 55 complete transformation. The order parameter that defines the daughter phase continuously changes from 0 to 1 in second-order transitions, and is abruptly and discontinuously modified from 0 to a non-nullCrystals value 2018, 8, inx FOR first-order PEER REVIEW transitions. The distinction between second- and first-order42 transitions of 54 is quite arbitrary. All the structural transitions imply an accommodation process in the surrounding discontinuously modified from 0 to a non-null value in first-order transitions. The distinction matrix, and thus a dissipation of energy. Of course, if the distortion is small, as in the “displacive” between second- and first-order transitions is quite arbitrary. All the structural transitions imply an transitionsaccommodation (with the process physical in the meaning, surroundi i.e.,ng matrix small, and atomic thus displacements),a dissipation of energy. the accommodation Of course, if is mainlythe elastic,distortion the is transition small, as in is the reversible, “displacive” and transitions the hysteresis (with is the small; physical whereas meaning, if the i.e., transition small is “reconstructive”atomic displacements), (with the physical the accommodation meaning, i.e., is mainly breaking elastic, of the the atomic transition bounds is reversible and formation, and the of new ones),hysteresis the distortion is small; is large whereas and if implies the transition plasticity. is “reconstructive” It is often assumed (with that the the physical former meaning, are second-order i.e., andbreaking the latter of are the first-order, atomic bounds but the and distinction formation is of more new quantitativeones), the distortion than qualitative. is large and It implis alsoies usual to assumeplasticity. that It is when often aassumed group–subgroup that the former relation are second exists,-order the and transition the latter is are second-order, first-order, but and the when thisdistinction relation does is more not exist, quantitative the transition than qualitative. is first order; It is alsohowever, usual one to assume can imagine that when a cubic-tetragonal a group– subgroup relation exists, the transition is second-order, and when this relation does not exist, the distortion with very high distortion amplitudes that would be irreversible (and thus would be of the transition is first order; however, one can imagine a cubic-tetragonal distortion with very high first order). Mnyukh [157] came to the extreme conclusion that the distinction between the first and distortion amplitudes that would be irreversible (and thus would be of the first order). Mnyukh second[157 order] came transitions to the extreme is artificial, conclusion and that theall the distinction transitions between are first-order, the first and only second differing order by the amplitudetransitions of the is artificial, discontinuity. and that all the transitions are first-order, only differing by the amplitude of theSecond-order discontinuity. transitions are generally modelled by the Landau theory [158]. The order parameter is arbitrarilySecond chosen-order depending transitions on are the generally physical property modelled that by is the judged Landau of interest; theory [ it158 can]. The be the order variation of theparameter local density is arbitrarily for liquid-crystal chosen depending transitions, on the as physical in Landau’s property paper that [is158 judged], the of atomic interest; displacement it can or polarizationbe the variation for ferroelectrics, of the local density the variation for liquid of-crystal the probability transitions, of as site in occupancyLandau’s paper for order–disorder[158], the transitions,atomic displacementetc. The free energy or polarization is expressed for ferroelectrics, by a Taylor decomposition the variation of into the a probabilitypolynomial of form site of the occupancy for order–disorder transitions, etc. The free energy is expressed by a Taylor order parameter, and the polynomial coefficients depend on the temperature and pressure. The order decomposition into a polynomial form of the order parameter, and the polynomial coefficients parameter is the solution of the null derivative of this polynomial; it is zero at high temperature and depend on the temperature and pressure. The order parameter is the solution of the null derivative non-nullof this below polynomial; the critical it is transition zero at high temperature temperature Tc, and it non continuously-null below variesthe criti ascal a function transition of the temperaturetemperature (Figure Tc, and 34 it a,b). continuously varies as a function of the temperature (Figure 34 a,b).

Figure 34. The free energy F function of the order parameter ξ and temperature T, and the order Figure 34. The free energy F function of the order parameter ξ and temperature T, and the order parameter function of the temperature T, in the case of (a,b) second order transitions, or of (c,d) first parameter function of the temperature T, in the case of (a,b) second order transitions, or of (c,d) first order transitions. order transitions. The Landau theory [158] is phenomenological because of the ad-hoc choices of the polynomial exponentsThe Landau and theory coefficients, [158] is but phenomenological it gives an important because mathematical of the ad-hoc framework choices for of the the usepolynomial of exponentssymmetries and coefficients, in thermodynamics. but it gives Landau an important stated that mathematical the order parameter framework is for a multidimensional the use of symmetries in thermodynamics.parameter, which has Landau as many stated degrees that of freedom the order as the parameter number of is dimensions a multidimensional of the irreducible parameter, space chosen for the irreducible representation of the group of the parent phase. This concept is now which has as many degrees of freedom as the number of dimensions of the irreducible space chosen didactically explained in many modern textbooks. In the example of cubic-tetragonal ferroelectric for thetransitions, irreducible one irreducible representation space ofis the one group-dimensional of the parent space phase.z (the direct Thision concept that will is becomes now didactically the Crystals 2018, 8, 181 43 of 55 explained in many modern textbooks. In the example of cubic-tetragonal ferroelectric transitions, one irreducible space is the one-dimensional space z (the direction that will becomes the c-axis of the tetragonal phase), and the other irreducible space is the two-dimensional space (x, y). The use of the group representation’s theory in physics was quite natural, as it was already introduced few years earlier in fundamental physics, mainly by Wigner [159] and Weyl [160]. Group representation was and still is considered to be a major part of group theory; it has acquired important success in materials science, mainly in spectroscopy with the calculations of the Raman frequencies. The martensitic transformations and deformation twinning are first-order structural transitions. It is agreed that the order parameter is a discontinuous function of temperature (Figure 34c,d). What is the order parameter for these transformations? Defining it is an old and difficult problem. Clapp, for example, wrote: Certainly one of the stumbling blocks in defining a martensitic transformation is that there is no obvious order parameter" associated with it, such as one has with ferromagnetic transformations (magnetic moment), order-disorder (long range order parameter), etc. Of course, one always has the possibility of using the volume fraction transformed as such an order parameter, analogous to the use of the fractional density of superconducting electrons for superconducting transformations, but this is a much more erratic quantity (markedly dependent on sample history) in the case of interest here. [161] Different order parameters have been proposed beside the “volume fraction” mentioned by Clapp. For example, Flack [95], and more recently Clayton and Knap [162], proposed to use the shear strain(s). Beside the problem of defining the order parameter, Landau’s theory is based on the idea that the transition is a loss of symmetry, and thus cannot be applied directly to first-order transitions. Indeed, in metallurgy, many transformations are in between very distinct phases without group–subgroup relations, for example fcc-bcc transformations. Some symmetry elements are lost, but others are created. These transitions are also called in physics “reconstructive” (the meaning here is different from that used in metallurgy, where it is synonymous of “diffusive”). An interesting and powerful generalization of Landau’s theory of reconstructive transitions was proposed by Tolédano and Dmitriev [163]. The effective order parameter is a variation of a density function that is a sum of wave functions. The periodic form of this parameter permits one to decompose the free energy as a truncated Fourier series of the order parameter (and no longer a Taylor series). Many examples of applications from crystal to quasicrystals are studied and described in [163]. The density-wave description is an important element of generalization that establishes a first link with the wave propagation mode. However, the systematic use of a latent lattice, relevant for order–disorder transitions, would require more physical justifications for martensitic transformations. Besides, Tolédano and Dmitriev’s book [163] aims at physicists, but it does not use the classical vocabulary and concepts of metallurgy, and it does not respond to some basic questions raised by metallurgists—for example, about orientation relationships and the habit planes. Important work on mutual understanding remains to be done, to build a real bridge between the developments made in physics and those made in metallurgy. The angular-distortive models presented in the previous sections show that a one-dimensional order parameter (simply the angle of distortion) can be chosen for many martensitic transformations and deformation twinning modes. This parameter is physical, controlling both the atomic trajectories and the lattice distortion. It is, however, not yet clear to us how this parameter could enter into the framework of Landau’s theory or into its advanced reconstructive version. Indeed, there is a fundamental difference in the way the symmetries are mathematically treated. Let us explain it with the simple questions: how many variants (domains) and how many types of domain boundaries are created by a phase transition? We were confronted to this problem when we had to find a method to reconstruct the prior parent grains from EBSD data (Figure 18). Instead of using the theory of group representation (based on the action of the group by conjugation), as in Landau’s and Tolédano and Dmitriev’s theories, the action (distortion) was treated by a left multiplication. The distinct orientations of the domains are called Crystals 2018, 8, 181 44 of 55 Crystals 2018, 8, x FOR PEER REVIEW 44 of 54 daughterorientational phases have variants. some symmetry For some operations special parent/daughter in common; these orientation symmetries relationships, form a subgroup the parent of and the pointdaughter group of phases the parent have phase some, symmetrycalled an intersection operations group. in common; The orientational these symmetries variants form are athe subgroup left cosetsof based the point on this group subg ofroup. the parent More phase,explicitly, called we ancall intersection Gγ and Gα the group. point The groups orientational of the parent variants are and daughterthe left phases, cosets respectively; based on this there subgroup. is no need More to explicitly, imagine them we call as Gabstractγ and Ggroupsα the pointthat should groups of the be decomposedparent and into daughter irreducible phases, representations respectively;. thereA point is no group need here to imagine is simply them the as abstractset of 3 groups× 3 that matricesshould of symmetries be decomposed tabulated into in irreducible the International representations. Tables for A pointCrystallography. group here isLet simply us also the call set of 3 × 3 γ→α γ→α T0 thematrices coordinate of symmetries transformation tabulated matrix in the International deduced from Tables the for orientation Crystallography. relationship. Let us also The call T0 the coordinate transformation matrix deduced from the orientation relationship. Theγ→α symmetries symmetries of the daughter crystal, expressed in the parent basis, are given by the matrices →T Gα → of the daughter crystal, expressed in the parent basis, are given by the matrices Tγ 0αGα (Tγ α)−1. → → →α 0 0 γ α −1 γ γ γγ γγ→α α γ →α −−1 1 γ γ γ ( T0 ) The. The intersection intersection group group is is thus thusH H = =G G TT00 G ((TT00 )) . The. The set set H H is ais subgroup a subgroup of of G . γ γ Gγ. TheThe distinct distinct orientational orientational variants variants ααi iareare defined defined byby thethe left left cosets cosetsαi = g =i H ,H andγ, and their their orientations γ→αi γ⌒ αi γ γ→α γ relative to the parent basis are T = [ Bγ→→αi B ] = g T with g ∈ αi.𝛾𝛾 One must understand orientations relative to the parent basis0 are 0 =0 i 0 = i𝑖𝑖 with . One γ γ T0 𝛼𝛼 𝑔𝑔𝑖𝑖 γ “gi ∈ αi”, as a matrix gi arbitrarily chosen in the cosetγ of matricesα𝑖𝑖 αγi.γ By→α convention,γ g1 is the identity must understand “ ”, as a matrix arbitrarilyα chosen in the coset of matricesγ 𝑖𝑖 . By matrix. The number of orientational variants N �is𝐁𝐁 the0 → number𝐁𝐁0 � 𝑔𝑔 of𝑖𝑖 𝐓𝐓 cosets0 on H𝑔𝑔;𝑖𝑖 it∈ is𝛼𝛼 the cardinal of γ γ convention,γ γ is the identity matrix. The number of orientationalα γ variantsγ Nα is the number of cosets G /H , and is𝑖𝑖 given𝑖𝑖 by the Lagrange𝑖𝑖 formula: N = |G |/|H |. More details are given𝑖𝑖 in [105]. γ 𝑔𝑔 ∈ 𝛼𝛼 𝑔𝑔 𝛼𝛼 on Hγ; it is the cardinal of Gγ/Hγ, and is given by the Lagrange formula: Nα = Gγ / Hγ . More details are 𝑔𝑔Now,1 we consider the different distortion matrices at the origin of the orientational variants. given in [105]. A distortional variant is defined by its effect on the initial shape of a crystal.∣ ∣ ∣ An∣ initial parent crystal of Now, we consider the different distortion matricesγ at the origin of the orientational variants.γ→α A a shape whose symmetries form a group G becomes, after complete distortion, D0 , a daughter γ→α γ→α distortionalcrystal variant whose is defined shape has by symmetriesits effect on the given initial by theshape matrices of a crystal.D AnGγ initial(D parent)−1. The crystal “distortional” of 0 0 → γ γ α a shape intersectionwhose symmetries group is form formed a group by symmetries G becomes that, after are common complete to distortion the crystal, beforeD0 , anda daughter after distortion, γ→α γ→α γ γ γ −1 γ γ→α γ→α i.e., K = G D0 G (D0 ) , with K as a subgroupγ of G . The−1 distortional variants di are crystal whose shape has symmetries givenγ by the matrices D0 G ( D0 ) . The “distortional” defined by the left cosets d = g Kγ. The number of distortional variants Mα is the number of cosets intersection group is formed by i symmetriesi that are common to the crystal before and after on Kγ; it is the cardinal of Gγ/Kγ and is given by the Lagrange formula: Mα = |Gγ|/|Kγ|. Generally, γ→α γ→α γ γ γ γ γ −1 γ γ distortion,K i.e.,is a K subgroup = G ofD0H G, which ( D0 means) , with that K the as numbera subgroup of distortionalof G . The distortional variants is variants higher than the α α di are definednumber by ofthe orientational left cosets variants: = K Mγ. The≥ numberN . More of detailsdistortional are given variants in [123 Mα]. is the number of ⌒ cosets on KγLet; it usis comethe cardinal back to theof G orientationalγ/γKγ and is variants.given by The the special Lagrange misorientation formula: M betweenα = Gγ those/ Kγ . variants, 𝑑𝑑𝑖𝑖 𝑔𝑔𝑖𝑖 Generally,simply Kγ is called a subgroupoperators of, H areγ, which defined means by the that double the number cosets H ofγ \distortionalGγ/Hγ. The variants number is ofhigher operators is ∣ ∣ ∣ ∣ than thegiven number by of Burnside’s orientational formula. variants: In addition, Mα ≥ Nα the. More algebraic detailsstructure are given of in the [123 variants]. and their operators Letis us a groupoid,come back and to the groupoidorientational composition variants. tableThe special (imagined misorientation for the purpose) between can those be used as a variantscrystallographic, simply called signatureoperators,of are the defined phase transition by the [double105]. These cosets ideas Hγ are\G notγ/Hγ completely. The number new; of important operatorsresults is given about by cosetBurnside’s and double-coset formula. In decompositions addition, the algebraic were already structure discovered of the variants by Janovec and [164 ,165] their operatorsin ferroelectrics, is a groupoid, and theand intersection the groupoid group composition was found table by (imagined Kalonji and for Cahn the purpose) for transitions can be without used as group–subgroupa crystallographic relation signature in theirof the theoretical phase transition study of[105 grain]. These boundaries ideas are [166 not]. completely The mathematical new; importantnotion of results a groupoid about unifies coset theseand notionsdouble-coset and allows decomp theirositions generalization. were already A groupoid discovered structure by for the Janovec distortional[164,165] in variants, ferroelectrics, similar and to that the of intersection the orientational group variants, was found seems by conceivable. Kalonji and Since Cahn the for distortion transitionsmatrices without simply group depend–subgroup on a one-dimensionalrelation in their theoretical order parameter study (theof grain angle boundaries of distortion), [166 these]. The matrices, mathematicalor their notion associated of a groupoid angular unifies order parameter,these notions could and beallows used their to build generalization. an Mα-dimensional A groupoid space for structurethe for polynomial the distortional form variants, of the free similar energy. to Inthat a broadof the orientational way, one would variants, replace seems the group conceivable. representation Since thetheory distortion used inmatrices Landau’s simply theory depend by a groupoid on a one theory,-dimensional based onorder cosets parameter and double-cosets. (the angle Oneof could distortion),hope these that matrices to find a, polynomialor their associated form of theangular free energyorder parameter whose solutions, could wouldbe used constitute to build aa groupoid.n Mα-dimensionWe recallal space that groupoidsfor the polynomial were initially form of introduced the free energy. by Brandt In a broad in 1926 way, [167 one,168 would] for quadratic replace forms. the groupThe representation main idea here theory would used be toin findLandau’s a groupoid theory on by the a groupoid set of solutions theory of, based the polynomial on cosets and form of the double-cosets.free energy One thatcould would hope be that an isomorphto find a polynomial to the groupoid form of of distortional the free energy variants. whose One solutions way to show the would constituteisomorphism a groupoid. could be We to compare recall that the groupoids composition were tables initially of both introduced groupoid by structures. Brandt in The 1926 idea is still [167,168vague,] for quadratic and such forms. research Thewould main idea require here the would help be of mathematicians.to find a groupoid on the set of solutions of the polynomial form of the free energy that would be an isomorph to the groupoid of distortional variants10.2.. One Dynamics way to show of Phase the Transformation isomorphism and could Accommodation be to compare Phenomena the composition tables of both groupoid structures.During a martensiticThe idea is transformation, still vague, and all thesuch atoms research move would collectively; require however, the help as previouslyof mathematicians.discussed, this does not mean that all the atoms move together at the same time. Imagining that the transformation propagates in the material as a wave permits escaping Frenkel and Cottrell’s 10.2. Dynamics of Phase Transformation and Accommodation Phenomena Crystals 2018, 8, 181 45 of 55 issue about the unrealistic critical stress required for an instantaneous and homogenous shear. It is Crystals 2018, 8, x FOR PEER REVIEW 45 of 54 indeed possible to use the continuous analytical form of the angular distortion matrix of a structural transformationtransformationD(β propagates), where β inis the material distortion as a angle, wave topermits generate escaping the martensite Frenkel and phase Cottrell’s continuously issue but heterogeneouslyabout the unrealistic in space critical and stress time. required One wayfor an could instantaneous be to find and an homogenous analytical equation shear. It is for indeed the order parameterpossibleβ that to use depends the continuous both on analytical the position formr ofand the the angular instant distortiont. For example, matrix of let a structural us introduce (i) thetransformation position of the D(β parent), where interface β is the distortionL(t), which angle, moves to generate at a wavethe martensite velocity phasec; and continuous (ii) a lengthly of accommodationbut heterogeneously (LA), without in space and detailing time. One for way the momentcould be to whether find an analytical the accommodation equation for the is order elastic or plastic.parameter An interesting β that depends form of both the on order the parameterposition r andβ( rthe, t) instantis given t. byFor a example, linear ramp let us function introduce (i) the position of the parent interface L(t), which moves at a wave velocity c; and (ii) a length of accommodation (LA), without detailing for the moment whether the accommodation is elastic or I f r ≤ L(t) : β = β plastic. An interesting form of the order parameter (, ) is fgiven by a linear ramp function ( ) = (r−L(t)) (L(t)+LA−r) β r, t ⎧ I f L(t) <≤ r(<): L(t) + LA : β = =LA βs + LA β f (18) ⎪ − () (() + − ) (, ) = I f( r)> (): = where β f is the finished distortion angle, i.e., D(β f ) is the matrix of complete distortion, βs is the start where is the finished distortion angle, i.e., D() is the matrix of complete distortion, is the distortion angle, i.e., D(βs) = Identity, and L(t) = c.t. The regular letters r and L mean the norm of the start distortion angle, i.e., D() = Identity, and L(t) = c.t. The regular letters r and L mean the norm of vectorsther vectorsand L. r This and rampL. This function ramp function is shown is shown in Figure in Figure 35. 35.

FigureFigure 35. Ramp 35. Ramp function function for for the the angular angular parameter parameter β ofof the the lattice lattice distortion distortion transformation transformation D(β)D. (β).

One can imagine that L(t) is a function of the position vector r; for example, the length of L(t) L t L t Onecan becan inversely imagine that proportional( ) is a function to the amplitude of the position of strain vector at ther; for position example, r, such the length that. of(, ()) = can be inversely proportional. to the amplitude of strain at the position r, such that. L(r, t) = c.t n, , where = , and ε is an additional constant added to avoid infinityk( D issues.(β f )− IThe).nk+ ε . ε = r ε wheremodel,r , applied and is to anthe additional2D hexagon constant–square distortion added topresented avoid inﬁnity in Figure issues. 12, allows The the model, simulation applied of to the 2Dsimple hexagon–square dynamic and crystallograp distortion presentedhic atomistic in Figuremovies; 12 some, allows snapshots the simulation are given in of Figure simple 36. dynamic The and crystallographicmovie shows a hexagonal atomistic movies;lattice of some hard-disk snapshots atoms are constituting given in a Figure round 36 crystal. The, moviewhich showsis a hexagonalcontinuously lattice and of heterogeneously hard-disk atoms distorted constituting to be transformed a round crystal, into a square which lattice is continuously of hard disks and heterogeneouslyforming an elliptic distorted crystal to at be completion. transformed The intoLA was a square chosen latticeto be equal of hard to the disks size of forming the crystal an, ellipticin crystalorder at completion. to minimize Thethe change LA was of chosen distance to d be between equal tothe the neighboring size of the atoms. crystal, Actually, in order the to LA minimize could the changebe of arbitrarily distance chosend between such the that neighboring d does not atoms. deviate Actually, from the the initial LA could interatomic be arbitrarily distance chosen (atom such diameter) by more than a fixed and low value that agrees with elasticity. In addition, it seems that d does not deviate from the initial interatomic distance (atom diameter) by more than a ﬁxed and possible to replace the linear ramp by a more complex form (for example, the one indicated by the low value that agrees with elasticity. In addition, it seems possible to replace the linear ramp by a more dashed curve in Figure 35) that could minimize the maximum interatomic distance (di-d) on the set complexof the form atoms (for i. example,This approach the one could indicated be also by investigated the dashed to check curve whether in Figure or 35 not) that the could existence minimize of the maximumparallel dense interatomic directions an distanced planes ( dwouldi-d) on favor the the set wave of the propagation, atoms i. which This approachcould then be could used beto also investigateddetermine to a check new criterion whether to or predict not the the existence “natural” of parent parallel–daughter dense directionsOR. This also and starts planes establishing would favor the wavea link propagation, with the Ginzburg which–Landau could thentheory be [169 used], in to which determine the gradient a new of criterion the order to parameter predict the is taken “natural” parent–daughterinto consideration OR. Thisin thealso analytical starts form establishing of the free a energy, link with here the with Ginzburg–Landau the advantage of including theory [the169 ], in whichatoms, the gradient the lattices of the, orderand many parameter crystallographic is taken into tools consideration that are in of the interest analytical for form martensitic of the free energy,transformations here with the (distortion, advantage correspondence, of including theand atoms, orientation). the lattices, and many crystallographic tools that are of interest for martensitic transformations (distortion, correspondence, and orientation). Crystals 2018, 8, x FOR PEER REVIEW 32 of 54

Burgers: [111]α = [100]ε and� (110)α�//(001)ε (16)

Crystals 2018, 8, x; doi: FOR PEER REVIEW www.mdpi.com/journal/crystals Crystals 2018, 8, 181 47 of 55

11. Conclusions The concept of simple shear has been the corner stone of the classical theories of the crystallography of martensitic transformations and deformation twinning for 150 years. The early models of the fcc-bcc transformations proposed by Young, Kurdjumov, Sachs, and Nishiyama, those of the bcc-hcp transformations made by Burgers, the phenomenological theory of martensitic transformations (PTMC) initiated by Greninger and Troiano and developed by Weschler, Read and Liebermann, and Bowles and Mackenzie, and lastly the theory of deformation twinning developed by Bilby, Bevis, and Crockers, among others, are all based on compositions of simple shears, or imply an invariant plane strain (IPS), which is a generalized form of shear that accounts for the volume change induced by the transformation. The mathematical treatment is irreproachable, although complex due to the initial premises. The predictive character of these theories is more limited than what is often claimed, which may explain why the theories have evolved over time by the addition of new shears (double- or triple-shear, double-shear versions of twinning), i.e., by adding new layers of complexity. An important problem of the shear paradigm was raised by Frenkel’s calculations (corrected later by Orowan and Cottrell); it shows that an instantaneous simple shear is impossible under realistic stresses, or without the help of dislocations. Cottrell [54] and Bilby [13], followed by Sleeswyk [69], Venables [73], and Christian [2], among others, thus came to imagine additional mechanisms by which the deformation twins are created by sequential and coordinated movements of partial dislocations, called “twinning dislocations”. Sleeswyk also proposed that the twinning dislocations dissociate into “emissary dislocations”, before moving away from the interface to reduce the stresses [69]. Often, TEM images of partial Shockley dislocations in front of twins or martensite are shown as a proof of the existence of “twinning dislocations”. The concept of a dislocation-mediated mechanism of twinning was generalized to displacive transformations by Hirth, Pond, and co-workers [84]; here, the twinning dislocations were replaced by “disconnections”, specifically designed to model the ledges at the parent–martensite interfaces. In this framework, the mechanism of the martensitic transformation is envisioned as the result of the interface displacement, and the martensite growth depends on the glissile character of the disconnections at the interface. Important questions are not addressed by these models. The pole mechanism and its derivatives do not convincingly explain how the dislocations/disconnections are created, and how they propagate in a sequential and coordinated way at speeds close to the acoustic velocities. In addition, the fundamental question of “how do the atoms move?” remains unsolved. In order to respond to this question, in the case of transformation between fcc, bcc, and hcp phases in metals, the assumption was made that the atoms move as hard spheres. The concept of simple shear has thus to be replaced by that of angular distortion, because the volume change of the intermediate states is higher than that allowed by simple elasticity. The sole input data is the final orientation relationship between the parent and the twin/martensite daughter phases. There is no other free parameter. A unique one-dimensional angular parameter, specific to each transition, allows the analytical calculation of the trajectories of all the atoms. The lattice distortion and shuffling (when required) appear as both sides of the coin. Obviously, the distortion can be decomposed into infinitesimal combinations of stretches and rotations, or LIS and IPS, as in the PTMC, but this decomposition would be artificial and of low interest. In the framework of this new paradigm, it is not required that all the atoms move instantaneously; martensite may actually be formed as a “lattice distortion wave” that propagates at high velocity and around which an accommodation zone can be spread across large distances, in order to maintain the interatomic distances in the material below the elastic limit. It is conceivable that the accommodation is elastic in the first stages of nucleation and growth, and becomes plastic when the martensite size becomes comparable to the grain size. In this way, contrary to the usual dislocation/disconnection–mediated models, the dislocations are not the cause but appear as a consequence of the lattice distortion. The partial dislocations shown by TEM in front of the mechanical twins may be created and emitted by the twin/martensite itself, and not by hypothetical pole mechanisms. When the lattice transformation is achieved, the angular distortive Crystals 2018, 8, 181 48 of 55 paradigm allows finding the same results as in shear-based theories in some classical cases, as for the {225} martensite in high-carbon steels, the deformation twins in fcc metals, and the extension twins in hcp metals. The angular distortive paradigm is, however, more general than the shear one; it allows the calculation of the atomic trajectories, and also opens new possibilities. The habit planes can be determined with a criterion that is less restrictive than with the shear paradigm; it is assumed that the habit planes are those untilted by the distortion (they are not necessarily fully invariant). Different accommodation mechanisms are possible: variant pairing, special dislocation arrays that close the rotational gaps (disclinations), etc. In our opinion, these mechanisms can be studied in a second stage; there is no need to imply them directly in the mechanism of transformation, and no need to impose gliding conditions on the interface in order to understand the propagation of a martensitic transformation. Undercooling or external stresses are important to obtain sufficient chemical or mechanical energy, in order to create the elastic or plastic accommodation zone required by the lattice distortion. In athermal martensite, the stress field accumulated around the product phase blocks the transformation. An additional undercooling is thus required to continue to form martensite in these deformed zones. In isothermal martensite, dislocation climbing and recovery phenomena may occur, which reduces the blocking strength in the accommodation zone and allows the transformation to continue with time. Even if quite new, the angular-distortive paradigm is already permitted to obtain significant results in various transformations that are classic in metallurgy. It allows the calculations of the irrational {225} and {557} habit planes in martensitic steels, without any adjustment; it naturally introduces a unique “order parameter” (the distortion angle), and establishes quantitative correlations between deformation twinning and martensitic transformations, also explaining the unconventional twinning modes recently discovered in magnesium. The change of paradigm also allowed us to predict a volume change during the twinning process, and propose a solution to the non-Schmid formation of twins in magnesium. Dedicated experiments are conceivable to test these predictions. Interesting links between thermodynamics and crystallography are now offered, even if mathematical tools different from those currently used in the physics of phase transitions are required. As the angular-distortive paradigm is only geometric, it can be blamed for its over-simplicity, and the few initial assumptions can be criticized: (a) the hard-sphere hypothesis ignores the electronic structure and real interatomic potential; (b) the “arbitrary” choice of the orientation relationship is based on the assumption of the existence of a “natural” distortion, without clear absolute criterion to define it; (c) the formation of martensite or twins conceived as “phase transformation waves” is not yet physically and mathematically detailed; and (d) the accommodation mechanisms are not yet correlated to the plasticity modes of the parent and daughter phases. Consequently, the approach will evolve in the future by replacing the hard spheres with elastic spheres, by incorporating more fundamental physics with DFT calculations, and with more mechanics for crystalline plasticity simulations. Investigations will be continued to establish a mathematical formalism for the wave propagation and deduce a criterion that defines the “natural” orientation relationship. The features observed in the EBSD pole figures are qualitatively explained as the plastic traces of the lattice distortion (as if the transformation waves were “frozen” by plasticity), and we hope to quantitatively simulate them uniquely with the distortion matrices; this would bring a response to the initial question that triggered this research ten years ago.

Acknowledgments: I would like to show my gratitude to Roland Logé, director of LMTM, and our student Annick Baur for our discussions and common work on martensite. The PX group is also acknowledged for the laboratory funding. The reviewers are also sincerely thanked for their positive and constructive comments. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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