Structural Ledges Research Articles

Atomic structure of interphase boundary of an a precipitate plate in a β Ti[sbnd]Cr alloy

Abstract The atomic structure of the interphase boundary enclosing an α (hcp) precipitate formed from the β (bcc) matrix in a Ti-Cr alloy was examined. All the interfaces of an α lath exhibit continuity of atomic planes between the matrix and the precipitate and thus are considered to be coherent or semicoherent. Such observations imply that there is an atomic site correspondence during transformation. The shear component of the transformation strains arising from the passage of (a/12) [111]β transformation dislocations (structural ledges) on (112)β∥[1100)α planes is accommodated by a set of glissile dislocation loops existing on every sixth (1100)α plane at the side facet and the edge of an α lath. The volumetric component of the transformation strain is accommodated by a set of sessile misfit dislocations (b = (a/2)[110]β = (c/2) [0001]α J on the risers of the growth ledges at the side facet. The presence of sessile dislocations implies that the migration of α-β interfaces is a non-conservative process ...

Atomic structure of interphase boundary enclosing bcc precipitate formed in fcc matrix in a Ni-Cr alloy

The atomic structure of the interphase boundaries enclosing body-centered cubic (bcc) lath-shape precipitates formed in the face-centered cubic (fcc) matrix of a Ni-45 mass pct Cr alloy was examined by means of conventional and high-resolution transmission electron microscopy (HRTEM). Growth ledges were observed on the broad faces of the laths. The growth ledge terrace (with the macroscopic habit plane\( \sim (112)_{fcc} //(23\bar 1)_{bcc} \)) contains a regular array of structural ledges whose terrace is formed by the (111)fcc//(110)bcc planes. A structural ledge has an effective Burgers vector corresponding to an\(a/12[1\bar 21]_{fcc} \) transformation dislocation in the fcc → bcc transformation. The side facet (and presumably the growth ledge riser) of the bcc lath contains two distinct types of lattice dislocation accommodating transformation strains. One type is glissile dislocations, which exist on every six layers of parallel close-packed planes. These perfectly accommodate the shear strain caused by the stacking sequence change from fcc to bcc. The second set is sessile misfit dislocations (∼10 nm apart) whose Burgers vector isa/3[111]fcc =a/2[110]bcc. These perfectly accommodate the dilatational strain along the direction normal to the parallel close-packed planes. These results demonstrate that the interphase boundaries enclosing the laths are all semicoherent. Nucleation and migration of growth ledges, which are controlled by diffusion of substitutional solute atoms, result in the virtual displacement of transformation dislocations accompanying the climb of sessile misfit dislocations and the glide of glissile dislocations simultaneously. Such a growth mode assures the retention of atomic site correspondence across the growing interface.

The role of structural ledges as misfit- compensating defects: fcc-bcc interphase boundaries

An energetic justification is given for structural ledges as interfacial misfit compensating defects, based on elastic considerations. We detail the NW-x configuration in which the orientation is imposed by close matching along the (21 l)fcc and (110)bcc directions and (111) fcc and (110) bcc planes, respectively. Special prominence is given to the terrace between structural ledges, where it is shown that an elastic relaxation obtains at specific terrace lengths based on atomic matching. It is this relaxation that makes a stepped interface more stable than a planar one at all meaningful values of misfit.

Intrinsic ledges at interphase boundaries and the crystallography of precipitate plates

The structure of intrinsic ledges at interphase boundaries has been interpreted with extended O-lattice/DSC-lattice approaches. The distribution of structural ledges can be predicted if the spacing difference between parallel matrix and product planes is treated as a measure of the relaxed coincidence condition. A small rotation away from the low-index planar parallelism introduces a series of interfacial dislocations that cancels the spacing difference, resulting in a lattice invariant line. Misfit-compensating ledges at bcc: hcp interfaces are produced as a ledged interface intersects additional O-points that are recognized with the incorporation of previously omitted bcc atom positions into the O-lattice construction. Energetic consideration suggests that structural interfacial energy may decrease when a flat interface becomes ledged with misfit-compensating ledges. Burgers vectors associated with structural ledges and misfit-compensating ledges are displacement shift complete (DSC) lattice vectors. Precipitate and martensite crystallography may both include a lattice invariant line, but they are involved in different interphase boundary characteristics. Assumptions and implications in precipitate and martensite crystallography are discussed in the framework of the O-lattice theory and phenomenological theory of martensite crystallography.

Ledges and dislocations in phase transformations

The coupling of dislocations and ledges in phase transformations is considered in terms of the symmetry of phases meeting at an interface. Growth and structural ledges and a variety of dislocation arrays are treated. Symmetry and the absence of long-range strain fields alone are shown to produce few restrictions on possible interface structures.

O-lattice modeling of ledged, partially coherent b.c.c.:h.c.p. boundaries

The structure of ledged, partially coherent b.c.c.:h.c.p. ( β: ζ) interfaces has been modeled by extending the Bollmann O-lattice theory to permit the matching of all atom positions in the two crystal structures and the matching between relaxed atom positions, each of which creates a set of additional O-points. Misfit-compensating ledges and structural ledges are predicted when interphase boundaries made up of parallel steps and terraces intersect additional O-points. Misfit-compensating ledges with Burgers ( 1 100) ζ(2 11) β and Pitsch-Schrader (2 110) ζ( 100) β terraces have Burgers vectors and interledge spacings both one half those associated with misfit dislocations on the corresponding flat interfaces. Biatomic structural ledges with Burgers (1 100) ζ(2 11) β and Potter (1 101) ζ(10 1) β terraces are shown to have Burgers vectors and interledge spacings both one sixth of their misfit dislocation counterparts.

The role of structural ledges at phase boundaries—I. Interfaces with rectangular atomic nets

Misfitting crystal interfaces are examined as to the mechanisms of minimizing interfacial energy. A comparison between the energetics of planar and stepped interphase boundaries is made. Structural ledge interfaces with terrace patches of (dimensions l x × l y ) are examined by employing, with increased sophistication, the rigidlike or purely geometrical energetics, followed by the relaxed energetic properties. For the stepped interface the mismatch which builds up along each terrace pair patch is compensated for by a relative displacement of the atomic patterns of the two terraces (pattern advance) at the next step. In this manner the average misfit parallel to the interface is minimized and the necessity for the introduction of misfit dislocations to accommodate this misfit eliminated. Since the atomic plane spacings parallel to the terraces are different on the two sides of the interface, misfit also exists in a direction normal to the terraces. This misfit is accommodated by a tilt type misfit dislocation in every superperiod. The present paper proposes models for calculating the energies associated with (i) the interaction between opposing terrace patches (including elastic relaxation), (ii) the line energy of misfitting opposing risers at the steps and (iii) the energy of the tilt misfit dislocation in a superperiod. This energy is compared with the energy of a planar interface containing conventional misfit dislocations. The calculations predict the regimes of crystal parameters for which misfit accommodation by either planar or stepped interfaces are energetically favored. In general it is shown that steps become more favorable as the misfit decreases. The calculated resolved and normal stresses associated with steps are a significant fraction of the shear modulus and may contribute to the plasticity of metals.

The role of structural ledges at phase boundaries—III. F.C.C.-B.C.C. interfaces in Kurdjumov-Sachs orientation

The ledge mode of misfit accommodation is extended to {111} f.c.c.∥{110} b.c.c. interfaces with the Kurdjumov-Sachs (KS) orientation relationship. As with the Nishiyama-Wassermann (NW) relationship the geometric quantities are uniquely related by the misfit ratio r of atomic diameters. A rigid lattice analysis shows that the driving forces for a KS orientation relationship are significantly larger than those for the NW- x and even more so than those for the NW- y relationships. The rigid analysis also predicts that the terrace width which satisfies the periodicity conditions for a stepped interface are well within the terrace widths which allow significant energy gains. The models developed for the NW analysis are modified for KS configurations and employed to sum energies for the terrace patches, tilt misfit dislocations and the riser interface energy. Comparing this energy with that of a planar interphase boundary comprised of only misfit compensating dislocations shows that a stepped interphase boundary is energetically favored at r values near and greater than r KS but less than r NW- y .

The role of structural ledges at phase boundaries—II. F.C.C.-B.C.C. interfaces in Nishiyama-Wasserman orientation

The ledge mode of misfit accommodation was first postulated for boundaries between b.c.c. and f.c.c. metal phases; the interfaces being {111} f.c.c.-{110} b.c.c. planes and relative orientations varying from Nishiyama-Wassermann (NW) to Kurdjumov-Sachs (KS). Here the geometrical quantities are uniquely related by the misfit ratio r of atomic diameters. We consider the so-called NW- x configuration in which the orientation is imposed by close matching along the 〈 211〉 ; f.c.c. and 〈 110〉 b.c.c. (taken as x-) directions. From the fact that no net shear pattern displacements are present with x-ledges (ledges normal to the x-direction) it is concluded that they are energetically preferable to y-ledges and justifies the approach of an energetic comparison between stepped interfaces with x-ledges and a planar interface containing conventional misfit dislocations (MDs). The NW- x stepped configuration is at first subjected to a rigid model analysis, i.e. a model with rigid crystals and periodic (truncated Fourier representation) interfacial interaction. This analysis provides (i) energetic justification for a relation between terrace periodicity and misfit cancellation, (ii) values of upper and lower average energy bounds, (iii) a method for estimating interfacial shear moduli, and (iv) a motivation for the suggestion that a relative rigid translation of the crystals is needed for energy minimization. The average energy per atom of a stepped interface ( ϵ s ) provides for misfit at the terrace patch and riser interface, as well as for misfit normal to the interface. The planar interface energy ϵ p accounts for interfacial misfit only. From a display of the excess energy Δ ϵ = ϵ s − ϵ p plotted against r it is concluded that stepped interfaces are more stable than planar interfaces for all meaningful values of the misfit ratio r, i.e. within the range where the NW- x orientation is viable.

Interface structures of epitaxial β-SiC on α-SiC substrates

Epitaxial β-SiC (3C) films were grown on (0001) 6H-SiC and 15R-SiC substrates by chemical vapor deposition (CVD). Transmission electron microscopy (TEM) characterization revealed that both films exhibited large areas of atomically flat, coherent interfaces. These areas were sometimes separated by structural ledges that were integral numbers of α-SiC unit cells in height. When 3C-SiC films were grown on 6H substrates, the primary defects were double position boundaries (DPBs), and islands of 6H were occasionally embedded in the predominantly 3C film. Films of 3C-SiC grown on 15R substrates exhibited relatively few grain boundaries, although islands of 15R were sometimes observed. The effect of substrate surface terraces on the observed interfacial microstructures and on defect formation mechanisms is discussed.

An EAM simulation of the alpha - gamma iron interface

The embedded atom method was used for the design of an atomic potential model for the simulation of the alpha - gamma iron interface. The model takes the form of three cubic expressions covering different ranges, and is normalized to give equilibrium to the two-body potential as well as to the embedding function at the perfect lattice configurations of both the alpha and gamma phases. The model yields good results for surface energies and the vacancy formation energy. Simulations were carried out for alpha - gamma interfaces with the Nishiyama-Wasserman orientation relationship. Interface energies were calculated for interfaces without structural ledges, with monoatomic ledges and with triatomic ledges with results close to the experimental values. The relaxation behaviour of the lattices from their initial configurations was also studied, and it was found that most of the relaxation occurs along the atomic habit planes instead of perpendicular to them. No major effect of different boundary conditions on the lattice relaxation and the interface energy was observed.

Structural ledges at precipitate/matrix interfaces in a (Ni, Co) superalloy: atomic structures and elastic fields

The broad faces of plate-like DO 19 precipitates of a (Ni, Co) base superalloy are decorated with triangular networks of linear singularities (LS's) accomodating the 2.5% isotropic length misfit in the atomically dense planes (111) fcc∥(0001) DO19. High-resolution electron microscopy images depict the detailed atomic cores of two of these LS's and show that they are associated with ledges of two (111) spacings in height. The elastic fields surrounding the LS's are analyzed by using both multislice computations and the Somigliana dislocation model in anisotropic elasticity theory. As a result, the Burgers-vector content of the observed ledges is found to be 1 6 〈112〉 vectors.

Interphase boundary structures of intragranular proeutectoid α plates in a hypoeutectoid TiCr alloy

Interphase boundary structure on the board faces of proeutectoid α plates in a retained β matrix has been studied in a Ti-6.62at. %Cr alloy by means of conventional and high resolution transmission electron microscopy. Misfit compensating ledges with b = c /2[0001] α are observed on the broad faces of α plates. Growth ledges, irregular in path, spacing and alignment, are also observed. The apparent habit plane of the broad faces of α plates and the riser directions of the misfit compensating ledges, determined by trace analysis, imply the presence of the structural ledges predicted in the companion modeling study. High-resolution TEM observation has revealed that biatomic structural ledges with ( 1 100) α//(2 11) β terraces do indeed exist on the broad faces of α plates, with exactly the characteristics predicted. This result provides the first experimental evidence for structural ledges in any b.c.c.:h.c.p. system.

Computer modeling of partially coherent B.C.C.:H.C.P. boundaries

A modeling study of partially coherent b.c.c.:h.c.p. coherent boundaries, using a computer-aided graphical technique, has been made for parallel pairs of low index planes in the presence of specified lattice orientation relationships. Three well-known lattice orientation relationships between the b.c.c. and h.c.p. phases, each of which was also slightly perturbed in various ways, were utilized. All of the planar interfaces thus formed which were based upon parallel pairs of low index planes can be described by two arrays of parallel misfit dislocations. The possibility of replacing one array of misfit dislocations with an array of structural ledges was then analyzed. In the presence of near-Burgers orientation relationships, the most probable structural ledges were found to have (1 100) h.c.p. //(2 11) b.c.c. terraces with risers 2, 4 or 6 atomic layers high. This type of structural ledge has a Burgers vector of 1/12[111] b.c.c., which lies in the terrace plane, associated with its riser. Thus it can replace a set of a-type misfit dislocations; hence only a single set of c-type misfit dislocations is now necessary on the terrace of structural ledges. This types of structural ledge was also found to step down along the lattice invariant line in order to accommodate simultaneously the misfit normal to the terrace plane. Further, the possibility of structural ledges with (1101) h.c.p.//(110) b.c.c. terrace was also discerned in the presence of near-Potter orientation relationships.

Structural ledges in interphase boundaries

This article addresses the properties of stepped misfitting interfaces and their energetic preference to planar misfitting interfaces. It highlights: (a) the purely geometrical or rigidlike, (b) the rigid (unrelaxed) energetic, and (c) the relaxed energetic properties of stepped interfaces. In (a), we address (1) the accommodation of misfit by the step or ledge mode through the cancellation of the mismatch, that builds up along a terrace, by the forwardpattern advance effected by a step,i.e., the relative displacement of atomic patterns on either side of the interface as observed in crossing a structural ledge along the interface, (2) the sideways (shear) pattern advance which seems to be energetically undesirable, (3) the need for tilt-type misfit dislocations to accommodate the misfit normal to the interface, and (4) the fact that at {III}fcc(face-centered cubic)/{110}bcc(body-centered cubic) interfaces with rhombic symmetries, the misfits, as well as the pattern advances, are interrelated through the ratior = b/a of nearest-neighbor distances in the crystals. In (b), we exploit the rigid model approach that (1) yields ideality criteria for minimum energy and provides energetic justification for the step mode of misfit accommodation, (2) confirms that the average terrace widthl[inx defined by this mode also meets the condition for positive energy gain, and (3) defines the upper and lower energy bounds to provide a perspective of the system energetics. In (c), the foregoing considerations are refined by a transition to the harmonic (elastic) model to yield (1) the dependence of the mean energy per atom of a stepped interface on interfacial misfit and pattern advance, as well as the dependence of the mean energy per atom of a planar interface on misfit, (2) expressions for the stresses related to the atomic interaction between opposing terraces, (3) atomic displacements that might be probed by modern analytical techniques, and (4) resolved shear stresses and normal stresses that may facilitate the formation of glide dislocations in the presence of applied stresses. The boundary in a two-dimensional space—spanned by misfit and pattern advance—between regions where stepped interfaces are more stable than planar ones has been determined, suggesting that a critical misfit exists above which only planar interfaces are stable. Whereas the resolved shear stress related to the formation of structural ledges may facilitate the formation of dislocations in the presence of a subcritical applied stress, the corresponding displacements (bending) of atomic planes are probably observable only with strain contrast electron microscopy techniques.

The role of ledges in the proeutectoid ferrite and proeutectoid cementite reactions in steel

The influence of interphase boundary ledges on the growth and morphology of proeutectoid ferrite and proeutectoid cementite precipitates in steel is examined. After reviewing current theoretical treatments of growth by the ledge mechanism, investigations that clearly document the presence and motion of ledges with thermionic emission electron microscopy (THEEM) and transmission electron microscopy (TEM) are reviewed. A fundamental distinction is made between two types of ledges: (1) mobile growth ledges whose lateral migration displaces the inter-phase boundary and (2) misfit-compensating structural ledges. Both types of ledges strongly affect the apparent habit plane and aspect ratio of precipitate plates. Agreement between measured growth rates of proeutectoid ferrite and cementite (plates and allotriomorphs) and predicted growth kinetics assuming volume diffusion-controlled migration of ledge-free disordered boundaries is shown to be consistently poor. Physically realistic growth models should incorporate the ledge mechanism. More accurate comparisons of the growth models with experimental data will need to account for observed ledge heights, interledge spacings, and ledge velocities. In this vein, the sluggish growth kinetics of cementite allotriomorphs observed in an Fe-C alloy are shown to be quantitatively consistent with a strong increase in interledge spacing with time.

Growth of epitaxial CoSi2 on (100)Si

A bilayer CoSi2/TiN has been grown on (100)Si, starting from a (100) p-type Si wafer deposited with thin layers of Ti followed by Co metal, through a two-stage annealing in a nitrogen environment and an intervening etch. In this process, Co and Ti switch places to form CoSi2 covered with TiN on Si. Cross-section transmission electron microscopy, and Rutherford backscattering/channeling spectrometry were used to characterize the bilayer sample. The CoSi2 was found to be single crystal, fully coherent, and epitaxial with (100)Si, whereas the TiN at the top of CoSi2 was polycrystalline. The stress in the CoSi2/TiN layer was found to be 1.9×1010 dynes/cm2. The planar (100) CoSi2/Si interface was interrupted with {111} ledges which are believed to be structural ledges present to maintain the coherency at the interface.

Descriptions of low energy misfit dislocation structures using the parabolic interaction potential

The need for and development of the parabolic models as a power series approximation of the sinusoidal Frenkel-Kontorowa model, used to study low energy misfit dislocation (MD) structures in epicrystals, is briefly discussed. Its application to monolayers on thick substrates, thickening epilayers, superlattices and structural ledges is outlined with special emphasis on critical misfit and thickness in coherent to incoherent transitions. The “nucleation” of an MD by climb from the free surface is calculated. The critical misfits obtained from (a) the continuum (approximation) and exact solutions, using the parabolic model, and (b) the continuum solutions of the sinusoidal and parabolic models are compared and the differences between the various approaches assessed.

Surface relief and the mechanism of a phase transformation

Surface relief of the invariant plane strain type is invariably observed during martensitic transformations. However, some diffusion-controlled transformations also exhibit this effect. The observation of a surface relief effect is therefore no proof for a martensitic transformation mechanism. The same reasoning applies to the formation of groups of variants that accomodate each other's strain. Self-accommodating groups of four variants of martensite plates have been observed in several martensites but they are also found in precipitation reactions. While these observations cannot be used as proof for a martensitic mechanism, neither does the presence of structural ledges in a reaction interface necessitate diffusion. Both these arguments have been used by different sides in a recent controversy (1,2), and the present paper is an attempt to clarify the situation. Starting with the simple example of AlAg, the mechanism of surface relief in a diffusional fcc-hcp transformation and its relation to ledge growth will be examined. Applying the same mechanism to increasingly complex cases the relation to structural ledges, fcc-bcc transformations and finally a general martensite interface will be developed.

A computer modeling study of partially coherent f.c.c.:b.c.c. boundaries

A computer modeling study has been made of the structure of f.c.c.:b.c.c. boundaries which are expected to be partially coherent. These boundaries are defined by {111} f. c. c ∥{110} b. c. c and a rotation about an axis normal to these conjugate habit planes leading to orientation relationships ranging from Nishiyama-Wasserman to beyond Kurdjumov-Sachs. A wide range of ratios of the f.c.c. to the b.c.c. lattice parameter was employed. Regions of good atomic matching exist in the {111} f. c. c ∥{110} b. c. c interface for all orientation relationships and lattice parameter ratios investigated. Extending the study of Hall et al., incorporation of structural ledges in the boundary allows the switching of matching layers and results in an increase in the interfacial coherency. The interledge spacing is a function of orientation and lattice parameter ratio and varies from ca. 3 to 30 Å. The structural ledges result in a deviation of the apparent habit plane from {111} γ although on an atomic scale this plane is always retained. The amount of deviation of the apparent habit plane from {111} γ is a function of orientation and lattice parameter ratio. Interfacial misfit dislocations lie midway between coherent regions on the broad faces of the structural ledges. The misfit dislocations consist of a single array whose spacing is a function of the orientation and lattice parameter ratio and ranges from ca. 8 to 40 Å. At least one partially coherent interface, normally incorporating both structural ledges and misfit dislocations, is predicted for each orientation and lattice parameter ratio investigated. In every case, the misfit dislocations are sessile with respect to slip out of the atomic conjugate habit planes, thus making a martensitic f.c.c. → b.c.c. transformation mechanistically impossible for these interfaces.