Theory Of Dendritic Growth Research Articles

Microstructural evolution during containerless rapid solidification of Ni–Mo eutectic alloys

Ni–45%Mo hypoeutectic, Ni–47.7%Mo eutectic and Ni–50%Mo hypereutectic alloys are rapidly solidified during containerless processing in drop tube. The microstructures of Ni–47.7%Mo eutectic alloy are composed of lamellar eutectic plus anomalous eutectic of Ni and NiMo phases. When the droplet size decreases, the volume fraction of anomalous eutectic becomes larger. The structural morphology transforms into Ni dendrite plus lamellar eutectic in very small droplets which are highly undercooled. The microstructures of Ni–45%Mo hypoeutectic alloy are characterized by primary Ni dendrite plus lamellar eutectic, whereas those of Ni–50%Mo hypereutectic alloy consist of NiMo dendrite plus lamellar eutectic. For both off-eutectic alloys, the experimental results show that the microstructure evolution depends mainly on droplet size. In the case of Ni–45%Mo hypoeutectic alloy, with the decrease of droplet size, the primary Ni phase transforms from dendrites to equiaxed grains. As for Ni–50%Mo hypereutectic alloy, when droplets become smaller and smaller the microstructural transition proceeds from primary NiMo dendrite plus lamellar eutectic to anomalous eutectic. The calculated highest undercoolings of the three alloys are 226, 182 and 135 K, respectively. By classical nucleation theory, Ni phase is the primary phase to nucleate for Ni–47.7%Mo eutectic alloy. The TMK eutectic growth and LKT/BCT dendritic growth theories are applied to analyze the rapid solidification process and investigate the microstructural transition mechanisms. The coupled zone of Ni–Mo eutectic alloy has also been calculated on the basis of TMK and LKT/BCT models, which covers a composition range from 45.7% to 57.1% Mo.

Microstructure evolution and solidification kinetics of undercooled Co–Ge eutectic alloys

The Co-29.7% Ge eutectic and Co-33% Ge hypereutectic alloys processed in drop tube attain large undercoolings and show “lamellar eutectic to anomalous eutectic” and “dendritic to equiaxed” morphology transitions respectively. The eutectic coupled zone is calculated on the basis of current eutectic and dendritic growth theories, which takes the shape of peanut and leans toward Co-rich side.

Growth of Solutal Ice Dendrites Studied by Optical Interferometry

Optical interferometry was used to study ice dendrite growth in a homologous series of sugar solutions, chosen in order to probe the effect of solute diffusion coefficient on solutal ice dendrite growth. Solute concentration fields, three-dimensional dendrite morphologies, and dendrite growth velocities were measured, and the results were compared with predictions from the modified Ivantsov and solvability theories of dendrite growth. The dendrites were not axially symmetrical but had a parabolic profile parallel to the basal plane and were surrounded by a paraboloidal solute concentration field, in accordance with the Ivantsov predictions combined with anisotropic growth kinetics. Solvability theory, for a system governed by solute diffusion, accounted for the relation between the dendrite tip morphology and the growth velocity, with a value of 0.006 ± 0.001 measured for the stability parameter, σ*, from the basal tip radius. Different values were measured when the dendrites were formed in an imposed temperature gradient, which was ascribed to the increased importance of nonequilibrium effects in these cases.

Containerless rapid solidification of highly undercooled Co-Si eutectic alloys

Droplets of Co–62.1%Si and Co–43.5%Si eutectic alloys with different sizes are rapidly solidified during containerless processing in drop tube. The microstructures of Co–62.1%Si eutectic alloy are mainly characterized by lamellar eutectic plus anomalous eutectic of CoSi2 and Si phases, whereas those of Co–43.5%Si eutectic alloy are mainly characterized by lamellar eutectic plus anomalous eutectic of CoSi and CoSi2 phases. For both alloys, the experimental results show that the microstructural evolution depends mainly on undercooling. In the case of Co–62.1%Si eutectic alloy, the smaller the droplet diameter, the larger the volume fraction of anomalous eutectic. When its diameter is very small, the droplet exhibits anomalous eutectic morphology entirely. As for Co–43.5%Si eutectic alloy, with the decrease of droplet size, the microstructural transition proceeds from lamellar eutectic to anomalous eutectic, even to dendrites. The nucleation rates of each phase have been calculated. The TMK eutectic growth and LKT/BCT dendritic growth theories are applied to analyze the rapid solidification process and investigate the microstructural transition mechanisms. The coupled zone around Co–43.5%Si eutectic alloy has also been calculated on the basis of TMK and LKT/BCT models, which covers a composition ranging from 40.8 to 43.8%Si. And calculated coupled zone of Co–62.1%Si covers a composition ranging from 60.2 to 81.6%Si.

Theory of plane front and dendritic growth in multicomponent alloys

Analytical solutions for the solute diffusion fields during plane front and dendritic growth in multicomponent alloys have been developed, taking into account the diffusive interaction between the species. It is found that the composition field for each of the n solutes is given by a sum of n expressions, each corresponding to the binary solution, but where the diffusion coefficients are replaced by the eigenvalues of the diffusion matrix. An extended constitutional undercooling criterion is deducted from the solution for plane front growth. A linear stability analysis of plane front growth is also presented. For dendritic growth, the diffusion field ahead of a growing paraboloid is calculated. Using the two latter solutions, growth of a dendrite at marginal stability is modelled. As an example, these models are applied to an hypothetical ternary system. From these results, some effects of diffusional interaction are shown and discussed.

Research of grain refinement in undercooled DD3 single crystal superalloy

The structure evolution of commercial DD3™ 1 Trademark of superalloy made in People's Republic of China. 1 single crystal superalloy has been systematically investigated within the achieved range of undercooling 0–210K. The grain structure of the alloy can be refined notably if it solidifies in a range of lower undercooling 30–70 K or above the critical undercooling (Δ T *=180 K). These two kinds of grain refinements arise in completely different ways. Based on current dendrite growth theory, a thermodynamic concept, dimensionless superheating, is adopted to evaluate the tendency of the dendrite remelting. With the increase of undercooling, the dimensionless superheating of the alloy increases first and then decreases, which suggests that the dendrite remelting driven by recalescence superheating should only be responsible for the grain refinement at lower undercooling range. Whereas the decrease of grain size above Δ T * is attributed to the high strain energy and lattice deformation energy that originate from the extremely rapid solidification, and lead to the dendrites distortion and disintegration. Subsequently, the recrystallization occurring in cooling could enable the grain size to decrease further. Dislocation morphology evolution in as-solidified structure is also provided by TEM technique to verify the recrystallization mechanism.

Rapid solidification processes of semiconductors from highly undercooled melts

Highly pure Si and Ge were undercooled by an electromagnetic levitator combined with a laser heating unit. Their crystal growth velocities were measured as a function of undercooling by means of two photodiodes and the appearance of the solid–liquid interface was observed by high-speed video camera. The result was compared with the predicted value based on the dendrite growth theory. The growth behaviors of Si and Ge were found to be classified into three categories of lateral, continuous, and successive nucleation at low, moderate, and high undercooling values, respectively in the measured range of undercooling. The transition temperatures for the classifications are 85 and 170 K for Ge. The microstructures of samples solidified from undercooled liquid were investigated and grain refinement showing the successive nucleation is observed.

Further examinations of dendritic growth theories

Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory have been proposed during the past decade, attempting to resolve the problem of selection of dendrite growth, and explain the essence of the pattern formation. This article attempts to clarify the differences and commonalities between these two theories and compare the predictions of these theories with some latest numerical evidence and experimental data. Since the MSC theory is most well-developed for the two-dimensional case, the comparisons of the theories with the numerical simulations are made mainly by using, but not restricted to, the two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been lately carried out by Wheeler et al. (Physica D 66 (1993) 243), Provatas et al. (Phys. Rev. Lett. 80 (15) (1998) 3308; 82 (22) (1999) 4496) and Karma et al. (Phys. Rev E 53 (1996) 3071; Phys. Rev. Lett. 77 (1996) 4050; J. Crystal Growth 174 (1997) 54) with the phase field model, and by Ihle and Müller-Krumbhaar with the free-boundary problem model (1994). It is seen that in a region where the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not self-sustaining. These results support the conclusions drawn by both the MSC and IFW theories. However, the numerical simulations also showed that there exists ‘a smallest value of the anisotropy parameter’, less than which ‘no steady needle solution was found’ (refer to Wheeler et al. Physica D 66 (1993) 243). This numerical evidence appears to be in agreement with the IFW theory and contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. in the microgravity of space and an excellent overall agreement is found.

Dendrite growth in undercooled DD3 single crystal superalloy

Crystal growth in the undercooled DD3 single crystal super-alloy is investigated in this paper. With the increase of undercooling, two kinds of dendrites are obtained. When ΔT < 30 K, dendrite growth is dominantly controlled by solutal diffusion, and the as-solidified morphologies are similar to the conventional as-cast highly branched dendrites. In the undercooling range of 78–150 K, the severe solute trapping that originates from high dendrite growth velocity weakens the effect of solutal diffusion on the dendrite growth. Thermal diffusion, controlled by heat gradient, instead of solutal diffusion, is found to become the key factor in determining the formation of largely developed fine dendritic morphologies. Furthermore, the measured dendrite arm spacing are explained by employing the current dendrite growth theories.

Influence of undercooling on solid/liquid interface morphology in semiconductors

Highly pure Si and Ge were undercooled by an electromagnetic levitator combined with a laser heating unit. Their crystal growth velocities were measured as a function of undercooling and the appearance of the solid/liquid interface was observed by means of a high-speed camera. The result was compared with the predicted value based on the dendrite growth theory. The growth behavior of Si was found to be classified into three categories of lateral growth, isolated dendrite growth and closer dendrite growth at low, moderate and high undercooling values, respectively. The first transition in the classification was caused by a change in the growth mechanism from stepwise growth in a single plane to that in multiple planes. The second transition was observed as a flattening of the macroscopic appearance of the interface. The transition undercoolings for the classifications were 100 and 210 K for Si, and 85 and 170 K for Ge.

Rapid dendritic growth investigated with artificial neural network method

Rapid dendritic growth of γ-(Ni, Fe) phase, β-CoSb intermetallic compound and α-Fe phase was realized by undercooling Ni-10%Fe single phase alloy, Co-60.5%Sb intermetallic alloy and Fe-40%Sn hypomonotectic alloy to a substantial extent. Their experimentally measured dendrite growth velocities were 79.5m/s, 12m/s and 0.705m/s, corresponding to undercooling levels of 303K(0.18TL), 168K(0.11 TL) and 219K(0.15 TL) respectively. Since the usual dendrite growth theory deviates significantly from reality at great undercoolings, an artificial neural network incorporated with stochastic fuzzy control was developed to explore rapid dendrite growth kinetics. It leads to the reasonable prediction that dendritic growth always exhibits a maximum velocity at a certain undercooling, beyond which dendrite growth slows down as undercooling increases still further. In the case of Fe-Sn monotectic alloys, α-Fe dendrite growth velocity was found to depend mainly on undercooling rather than alloy composition.

Growth kinetics of quasicrystalline and polytetrahedral phases of Al-Pd-Mn, Al-Co, and Al-Fe from the undercooled melt

Melts of the alloys Al-Pd-Mn, Al-Co, and Al-Fe are containerlessly undercooled and solidified using the technique of electromagnetic levitation. The (quasi)crystal growth velocity is measured as a function of undercooling for the various phases by a high-speed infrared photosensing device. Additionally, a video image technique is applied to measure growth velocities especially at low undercoolings where sluggish growth prevails. Scanning and transmission electron microscopy as well as x-ray diffraction is utilized to identify the phases primarily solidified from distinct levels of undercooling. The experimental results are analyzed within current theories of dendritic growth. The investigations give insight into topological and chemical short-range order effects in the growth kinetics of phases with polytetrahedral ordering in the solid state.

An integrated model for dendritic and planar interface growth and morphological transition in rapid solidification

Rapid solidification can be achieved by quenching a thin layer of molten metal on a cold substrate, such as in melt spinning and thermal spray deposition. An integrated model is developed to predict microstructure formation in rapidly solidified materials through melt substrate quenching. The model solves heat and mass diffusion equations together with a moving interface that may either be a real solid/liquid interface or an artificial dendrite tip/melt interface. For the latter case, a dendrite growth theory is introduced at the interface. The model can also predict the transition of solidification morphology, e.g., from dendritic to planar growth. Microstructure development of Al-Cu alloy splats quenched on a copper substrate is investigated using the model. Oscillatory planar solidification is predicted under a critical range of interfacial heat-transfer coefficient between the splat and the substrate. Such oscillatory planar solidification leads to a banded solute structure, which agrees with the linear stability analysis. Finally, a microstructure selection map is proposed for the melt quenching process based on the melt undercooling and thermal contact conditions between the splat and the substrate.

RAPID SOLIDIFICATION: FUNDAMENTALS AND MODELING

Rapid solidification can produce materials with superior, and sometimes astonishing, physical and mechanical properties. This has led to many advanced materials processing techniques such as melt atomization, thermal spray coatings, melt-spinning, laser melting and resolidification, and high-energy beam treatment of surfaces, to name a few. Because of large melt undercooling and very high rate of solidification, the rapid solidification processes inherently involve nonequilibrium kinetics of phase change and coupled transport of energy and species. This coupling ultimately determines the phase selection, microstructure formation and final properties of the rapidly solidified materials. The fundamentals of rapid solidification, and recent progress made in modeling of these processes are reviewed in this article. Both single component and alloy materials are considered, and the issues related to thermodynamics of rapid solidification, nucleation and growth kinetics of crystalline phase, planar interface stability, dendritic growth theory, and microsegregation are discussed. Thermal transport models incorporating the basic principles of rapid solidification and their integration with kinetics models are presented. Special features of theoretical results are discussed in detail, particularly, for phase selection and microstructure formation in atomized powders, pulsed laser processing, and melt-substrate quenching.

Single Crystal Solidification of Undercooled Single-phase Alloys

The correlation between dendritic microstructure and melt undercooling is investigated to present a new technique for rapid single crystal growing, or high undercooling directional solidification (HUDS). In a certain range of undercooling for stable dendrite growth in a negative temperature gradient, this new technique is employed to prepare single crystal Ni 50 Cu 50 bars of approximately 10 mm in diameter and 65 mm in length. Optical metallography and X-ray diffraction experiments indicate that the resulting single crystals in the directionally solidified bars are composed of dendrites approximately parallel to each other and growing mainly in the (100) directions. The formation process and growth velocity of single crystals in an undercooled liquid are analyzed according to the obtained results and calculated by a current dendrite growth theory, respectively. A comparison of the HUDS technique with the as-proposed autonomous directional solidification (ADS ) process is made also in the present paper.

Dendritic Growth tip velocities and radii of curvature in microgravity

Dendritic growth is the common mode of solidification encountered when metals and alloys freeze under low thermal gradients. The growth of dendrites in pure melts depends on the transport of latent heat from the moving crystal-melt interface and the influence of weaker effects like the interfacial energy. Experimental data for critical tests of dendritic growth theories remained limited because dendritic growth can be complicated by convection. The Isothermal Dendritic Growth Experiment (IDGE) was developed specifically to test dendritic growth theories by performing measurements with succinonitrile (SCN) in microgravity, thus eliminating buoyancy-induced convection. The first flight of the IDGE in 1994 operated for 9 days at a mean quasi-static acceleration of 0.7 × 10−6g0. The velocity and radius data show that at supercoolings above approximately 0.4 K, dendritic growth in SCN under microgravity conditions is diffusion limited. By contrast, under terrestrial conditions, dendritic growth of SCN is dominated by convection for supercoolings below 1.7 K. The theoretical and experimental Peclet numbers exhibit modest disagreement, indicating that transport theories of dendritic solidification required some modification. Finally, the kinetic selection role for dendritic growth, VR2=constant, where V is the velocity of the tip and R is the radius of curvature at the tip, appears to be independent of the gravity environment, with a slight dependence on the supercooling.

Comments on ‘Solidification modes and microstructure of Fe–Cr alloys solidified at different undercoolings’

The critical growth velocity for a planar solidification front in undercooled alloy melts is discussed on the basis of the absolute stability theory to reexamine the interpretation of a current analysis of the solidification modes in undercooled bulk Fe–Cr alloys contributed by Xuezhi Zhang. Theoretically, it is possible to produce a planar front in the solidification of undercooled bulk melts. But practically, it is imopssible for the undercooled bulk Fe–Cr melts to produce a planar front in the solidification. The dendritic growth theories as well as the calculations due to Zhang have been analyzed.

Dendrite growth processes of silicon and germanium from highly undercooled melts

The crystal growth behavior of a semiconductor from a very highly undercooled melt is expected to be different from that of a metal. In the present experiment, highly pure undoped Si and Ge were undercooled by an electromagnetic levitation method, and their crystal growth velocities (V) were measured as a function of undercooling (ΔT). The value of V increased with ΔT, and V=26 m/s was observed at ΔT=260 K for Si. This result corresponds well with the predicted value based on the dendrite growth theory. The growth behaviors of Si and Ge were found to be thermally controlled in the measured range of undercooling. The microstructures of samples solidified from undercooled liquid were investigated, and the amount of dendrites immediately after recalescence increased with undercooling. The dendrite growth was also observed by a high-speed camera.

The double-pyramid structure of dendritic ice growing from supercooled water

It is known that ice growing freely from supercooled water has a morphological transition at T=−2.7°C, from a flat dendritic structure at higher temperatures to a twelve-sided double-pyramid structure at lower temperatures. The double-pyramid structure, which can be described as two hollow six-sided pyramids joined at their apices, is built from dendrites growing in well-defined growth directions which are noncrystallographic in the planes normal to the basal plane while their projections on the basal plane retain the hexagonal symmetry. Similar structures have been reported in other hexagonal materials. In order to understand the growth mechanism better, we measured the temperature field in the water around the growing crystals by using the temperature dependence of its refractive index. Since this dependence happens to be zero at the freezing point for regular water (H 2O), we use heavy water (D 2O), and achieve considerably greater sensitivity. The free growth experiments performed with heavy ice reveal that their morphological behavior is similar to regular ice, as well as their velocities and the angle between the pyramids as a function of supercooling. The temperature measurements showed that the interaction between the two sides of the pyramid via the temperature field is weak. This leads to the conclusion that the solution for the growth mode of the dendrites should be found in the single dendrite level. Explanations of this phenomenon are discussed in the light of recent advances in dendritic growth theory – in particular the concept of microscopic solvability – combined with the behavior of the surface tension and the kinetic effect as a function of crystallographic orientation. It can be shown that growth in a low symmetry direction leads to an asymmetrically growing crystal and to asymmetry in the observed temperature field.

Microstructural characterization and solidification behavior of atomized Al–Fe powders

The effect of solidification history on the resultant microstructure in atomized Al–2.56 wt% Fe and Al–6.0 wt% Fe powders was studied, with particular emphasis on droplet size, undercooling, and phase stability. The atomized Al–Fe powders exhibited four microstructural features, i.e., Al3Fe phase (now known as Al13Fe4), Al + Al6Fe, α–Al dendrite, and a predendritic microstructure. The presence of these phases was noted to depend on alloy composition and a kinetic phase competitive growth mechanism due to the initial undercooling experienced by the powders. The occurrence of structures of the predendritic, cellular, and/or dendritic type was properly predicted by the theory of dendrite growth into undercooled alloy melts for the case of large undercoolings.