Growth kinetics of solid-liquid Ga interfaces: Part II. Theoretical

The faceted (111) and (001) Ga interfaces grow at low supercoolings with either of the lateral growth mechanisms, two-dimensional nucleation growth (2DNG) or screw dislocation-assisted growth (SDG), depending on the perfection of the interface. The classical theories regarding the growth kinetics of smooth interfaces describe the results qualitatively but not quantitatively. The latter is due to the inadequacy of the assumptions made in the classical theories, which treat the interfacial atomic migration the same as the liquid bulk diffusion process and the step edge energy as independent of the supercooling. Beyond a threshold supercooling, the results show that the faceted interfaces gradually become kinetically rough as the supercooling increases. The step energy, treated as a function of the supercooling, is shown to diverge exponentially with the supercooling at the faceted/nonfaceted transition. At supercoolings exceeding the transition value, dislocations do not affect the growth rate. Furthermore, beyond the transition, the growth rates are linearly dependent on the supercooling, which implies that the growth mode changes from lateral to normal. A generalized lateral growth equation, which includes the interfacial diffusivity and supercooling-dependent step edge free energy, is given to describe the growth kinetics of both interfaces up to supercoolings marking the kinetic roughening transition. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

On the role of elastic interactions in the nucleation and growth of ledges

Elastic interactions among ledges on transformation interfaces have noticeable consequences when chemical and interfacial tension (capillary) forces are small, namely, near equilibrium. This occurs just at nucleation (unstable equilibrium) or during the slow coarsening regime. When the interface lies perpendicular to the misfit strain (as do the large faces of misfits in Al-Cu alloys), ledges of like sign repel one another, and nucleation of new ledges occurs as far as possible from existing ones. However, when the interface lies parallel to the misfit strain, ledges of like sign attract one another. We then expect the formation of superledges. Essentially, such an interface with ledges is elastically unstable. Expressions are derived for the kinetics of ledge amaleamation. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformation Committee of the Materials Science Division, ASM INTERNATIONAL.     

Thickening of grain-boundary α allotriomorphs in a Ti-Cr alloy by multiple sets of ledges

A computer model is developed to simulate the growth of grain-boundary allotriomorphs having more than one set of growth ledges at their interfaces. The growth is controlled by the volume diffusion of solute to or from the riser of a ledge. The time dependence of the growth rate of two orthogonal sets of ledges is found to be somewhat different from that of a single set of ledges. However, the operation of multiple sets of ledges is unlikely to alter significantly the growth kinetics of grain-boundary allotriomorphs from those predicted from the disordered growth theory, except at small ledge spacings or at short reaction times. Faster growth kinetics of proeutectoid α allotriomorphs than those of either planar or ellipsoidal disordered boundaries which have been reported in a Ti-6.6 at. pet Cr alloy are not likely to be accounted for with the heights and spacings of double sets of ledges actually observed on the interfaces of allotriomorphs. Hence, the grain-and interphase-boundary diffusion-assisted growth of precipitates, (rejector plate mechanism, RPM) appears to be operative during the growth of a allotriomorphs, as previously proposed on the basis of growth-rate measurements. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Nonequilibrium effects during the ledgewise growth of a solid-liquid interface

Directional solidification studies have been carried out in the napthalene-camphor system in which the interface advances through the formation and motion of ledges. Growth conditions have been varied so as to characterize both the planar interface growth and the condition for the instability of the planar interface. It is found that the mechanisms of planar interface instability and the subsequent morphological development of the interface depend significantly on the crystallographic orientation of the interface. Appreciable interface kinetics effects are present during the growth of a planar interface, and experimental studies have been designed to quantitatively evaluate the nonequilibrium conditions at the interface. These nonequilibrium effects for the napthalene-camphor system have been determined experimentally and have been characterized by two response functions that describe the interface temperature and the interface composition in the liquid under nonequilibrium conditions. The interface kinetic law is found to be exponential, indicating that the growth of the interface occurs through the process of nucleation of new layers. Formerly with Ames Laboratory. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Interfacial steps and growth mechanism in ferrous pearlites

The role of steps at the growth and interlamellar interfaces in ferrous pearlites is examined. The direction steps at the ferrite/cementite interlamellar interface (FCI) first characterized by Hackney and Shiflet (HS)^1-5 are quantified by relating the degree of macroscopic curvature to step height and spacing using lattice imaging techniques. These interlamellar steps associated with alloy curvature are demonstrated to result directly from pearlite growth ledges. Interphase boundary carbide precipitation in an Fe-C-V alloy is employed to further demonstrate that the pearlite growth mechanism occurs through the migration of steps laterally across the growth front and that the reported mechanism is not specific to the high Mn-containing alloy. Analysis is based within the context of the dynamics of the phase transformation, with the argument made that the static interpretation of interphase boundary structure can be misleading. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Atomic structure of the crystalline/amorphous interface in a directionally crystallized Pd80Si20 alloy

An amorphous ribbon of Pd₈₀Si₂₀ alloy was directionally crystallized under an imposed temperature gradient of 25 K/mm with a growth velocity of 0.0785 mm/s, and the structure of the crystalline/amorphous interface was investigated by conventional and high-resolution transmission electron microscopy (TEM). Under these conditions, the amorphous Pd₈₀Si₂₀ crystallizes into a broken-lamellar eutectic of the Pd₃Si and Pd₉Si₂ equilibrium phases. The Pd₃Si phase is faceted and grows along the [010] direction by nucleation and propagation of unit-cell ledges parallel to the (010) terrace plane. The Pd₉Si₂ phase is largely coherent with Pd₃Si and grows along a high-index crystallographic direction. Microscopic facets were not observed on the Pd₉Si₂ phase either by conventional or high-resolution TEM, indicating that its crystalline/ amorphous interface is comparatively rough. These observations are related to the crystallography and interphase boundary (IPB) energies of the phases and discussed in terms of mechanisms of lamellar growth. Formerly Graduate Research Assistant, Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Enhanced Morphological Stability in Sb-Doped Ge

The axial heat processing (AHP) crystal growth technique was used to investigate the morphological stability of faceted solid/liquid (s/l) interfaces. Six Sb-doped Ge single crystals containing 2.3 × 10⁻² to 2.3 × 10⁻¹ at. pct Sb were grown at pulling rates of 10 to 20 mm/h. These include two bicrystals specifically designed to investigate the effect of slight misorientation on stability. Faceted growth with a kinetic supercooling on the order of 0.15 K was achieved, and a characteristic two-dimensional W instability boundary, an inverted crater in three dimensions, was observed. The crystals exhibited enhanced morphological stability over the predictions of the constitutional supercooling (CS) criterion and the Mullins and Sekerka (MS) stability criterion, with the highest stability in the center of the W. These results are examined with current analytical stability theories accounting for convection and kinetics. An alternate model is proposed based on anisotropic kinetics and the competition between lateral spreading on a faceted interface and the amplification rate of an interfacial perturbation.

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The crystallography and atomic structure of line defects in twin boundaries in hexagonal-close-packed metals

The crystallographic analysis of line defects in interfaces is discussed and applied to the particular case of twinning dislocations in hexagonal-close-packed (hop) metals, which have been studied here by atomistic simulation. Two crystallographic approaches are used; first, the concept of bicrystal structure maps is developed for the case of interfaces between crystals having multiple-atom bases, and second, the topological theory of line defects based on symmetry theory is used. On the basis of the atomistic calculations, some general conclusions concerning the relative contribution to the total energy of dislocations made by their elastic fields and core structures are presented. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Misfit dislocations in epitaxy

This article on epitaxy highlights the following: the definition and some historical milestones; the introduction by Frenkel and Kontorowa (FK) of a truncated Fourier series to model the periodic interaction at crystalline interfaces; the invention by Frank and van der Merwe (FvdM)—using the FK model—of (interfacial) misfit dislocations as an important mechanism in accommodating misfit at epilayer-substrate interfaces; the generalization of the FvdM theory to multilayers; the application of the parabolic model by Jesser and van der Merwe to describe, for growing multilayers and superlattices, the impact of Fourier coefficients in the realization of epitaxial orientations and the stability of modes of misfit accommodation; the involvement of intralayer interaction in the latter—all features that impact on the attainment of perfection in crystallinity of thin films, a property that is so vital in the fabrication of useful uniformly thick epilayers (uniformity being another technological requirement), which also depends on misfit accommodation through the interfacial energy that function strongly in the criterion for growth modes, proposed by Bauer; and the ingenious application of the Volterra model by Matthews and others to describe misfit accommodation by dislocations in growing epilayers. This article is based on a presentation in the symposium “Interfacial Dislocations: Symposium in Honor of J.H. van der Merwe on the 50th Anniversary of His Discovery,” as part of the 2000 TMS Fall Meeting, October 11–12, 2000, in St. Louis, Missouri, sponsored under the auspices of ASM International, Materials Science Critical Technology Sector, Structures.     

Thermal considerations on the recalescence of alloy powders

The principles involved in the solidification of supercooled binary alloy droplets are discussed with particular emphasis on solute redistribution. The effects of alloying elements on the relevant parameters of the thermal history of recalescing aluminum droplets are studied with the aid of enthalpytemperature relationships. Thermal considerations indicate that the critical supercoolings to achieve partitionless solidification change rather modestly for the alloys investigated. In addition, the rate of recalescence after nucleation is likely to be slowed down by the addition of solute. A Newtonian model for solidification of nonideal binary alloys with morphologically stable interfaces is derived and used to study the thermal history and solute redistribution during recalescence. The effects of different solutes, alloy concentration, initial supercooling, and interfacial kinetics are discussed.     

Observations of the formation and kinetics of surface steps during evaporation and condensation

Surface step patterns produced by crystal growth or evaporation can be observed, e.g., on NaCl, AgBr, Ag, and Si, by means of the electron microscopic method of decoration. These observations give insight into the mechanisms (random two-dimensional (2-D) nucleation, formation of hills or pits by spirals or repeated preferential 2-D nucleation, kinematic step interaction, orientation dependence of step motion, light-influenced evaporation, and stage of coalescence of thin films) and molecular processes (surface and edge diffusion) of crystal and thin film growth. Combining the decoration and platinum-carbon replica techniques enables an interesting insight into the step kinetics during the process of faceting. The pinning of moving steps at impurities and their piling up are decisive particulars. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Kinetics of the peritectic phase transformation: In-situ measurements and phase field modeling

An experimental study has been conducted into the role of cooling rate on the kinetics of the peritectic phase transformation in a Fe-C alloy. The interfacial growth velocities of the peritectic phase transformation were measured in situ for cooling rates of 100, 50, and 10 K/min. In-situ observations were obtained using high-temperature laser scanning confocal microscopy (HTLSCM) in a concentric solidification configuration. The experimentally measured interface velocities of the liquid/austenite (L/γ) and austenite/delta-ferrite (γ/δ) interphase boundaries were observed to increase with higher cooling rates. A unique finding of this study was that as the cooling rate increased, there was a transition point where the L/γ interface propagated at a higher velocity than the γ/δ interface, contrary to the findings of previous researchers. Phase field modeling was conducted using a commercial multicomponent, multiphase package. Good correlation was obtained between model predictions and experimental observations in absolute values of interface velocities and the effect of cooling rate. Analysis of the simulated microsegregation in front of the L/γ and γ/δ interfaces as a function of cooling rate revealed the importance of solute pileup. This microsegregation plays a pivotal role in the propagation of interfaces; thus, earlier modeling work in which complete diffusion in the liquid phase was assumed cannot fully describe the rate of propagation of the L/γ and δ/γ interfaces during the course of the peritectic transformation.     

Kinetics of the peritectic phase transformation: In-situ measurements and phase field modeling

An experimental study has been conducted into the role of cooling rate on the kinetics of the peritectic phase transformation in a Fe−C alloy. The interfacial growth velocities of the peritectic phase transformation were measured in situ for cooling rates of 100, 50, and 10 K/min. In-situ observations were obtained using high-temperature laser scanning confocal microscopy (HTLSCM) in a concentric solidification configuration. The experimentally measured interface velocities of the liquid/austenite (L/γ) and austenite/delta-ferrite (γ/δ) interphase boundaries were observed to increase with higher cooling rates. A unique finding of this study was that as the cooling rate increased there was a transition point where the L/γ interface propagated at a higher velocity than the γ/δ interface, contrary to the findings of previous researchers. Phase field modeling was conducted using a commercial multicomponent, multiphase package. Good correlation was obtained between model predictions and experimental observations in absolute values of interface velocities and the effect of cooling rate. Analysis of the simulated microsegregation in front of the L/γ and γ/δ interfaces as a function of cooling rate revealed the importance of solute pileup. This microsegregation plays a pivotal role in the propagation of interfaces; thus, earlier modeling work in which complete diffusion in the liquid phase was assumed cannot fully describe the rate of propagation of the L/γ and δ/γ interfaces during the course of the peritectic transformation.     

Kinetics of the peritectic phase transformation: In-situ measurements and phase field modeling

An experimental study has been conducted into the role of cooling rate on the kinetics of the peritectic phase transformation in a Fe−C alloy. The interfacial growth velocities of the peritectic phase transformation were measuredin situ for cooling rates of 100, 50, and 10 K/min.In-situ observations were obtained using high-temperature laser scanning confocal microscopy (HTLSCM) in a concentric solidification configuration. The experimentally measured interface velocities of the liquid/austenite (L/γ) and austenite/delta-ferrite (γ/δ) interphase boundaries were observed to increase with higher cooling rates. A unique finding of this study was that as the cooling rate increased there was a transition point where the L/γ interface propagated at a higher velocity than the γ/δ interface, contrary to the findings of previous researchers. Phase field modeling was conducted using a commercial multicomponent, multiphase package. Good correlation was obtained between model predictions and experimental observations in absolute values of interface velocities and the effect of cooling rate. Analysis of the simulated microsegregation in front of the L/γ and γ/δ interfaces as a function of cooling rate revealed the importance of solute pileup. This microsegregation plays a pivotal role in the propagation of interfaces; thus, earlier modeling work in which complete diffusion in the liquid phase was assumed cannot fully describe the rate of propagation of the L/γ and δ/γ interfaces during the course of the peritectic transformation.     

Growth kinetics and mechanism of the massive transformation

During the composition invariant massive transformation, the propagation of a reaction interface is motivated by the free energy change due to a change in crystal structure and results in either single or duplex product morphologies. A common feature underlying the growth of massive product phases is a thermally activated control by boundary diffusion. Microstructural and kinetics information indicate that massive growth involves the migration of mobile, incoherent interfaces through a lateral growth process that may evolve into continuous growth at the highest velocity. The high driving free energy for massive growth is reflected by a relative insensitivity to microstructural obstacles and by a rapid transformation in the absence of local interfacial equilibrium. For the limiting conditions under which massive growth can prevail at the transition into lattice shear or solute partitioning, an incomplete dissipation of driving free energy is indicated by product structures of metastable supersaturated solid solutions. This paper is based upon a presentation made at a symposium on The Massive Transformation, held at the Pittsburgh meeting of The Metallurgical Society of AIME and the Materials Science Division of ASM, October 9, 1980, under the sponsorship of the MSD Phase Transformations Committee.     

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. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Diffusion-controlled kink motion

A simple model for the growth of kinks by volume diffusion is discussed, and singular perturbation methods, valid for supersaturations much less than one, are used to derive coupled integral equations for the motion of trains of (well-spaced) kinks. Numerical results are presented for the motion of two-and three-kink trains. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

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. Formerly with the Physics Department, University of Pretoria. Formerly Visiting Scientist, Physics Department, University of Pretoria, Pretoria, South Africa. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.     

Interface dislocations and ledges in oxidation and diffusional phase transformations

The origin of ledge concepts in growth from the vapor is reviewed. The ideas are extended to solid-state phase transformations with the added effects of strain and misorientation. Types of ledges and dislocations are classified. The concepts are illustrated for the example of oxidation of a metal. Further extensions to diffusional phase transformations are briefly discussed. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.