Solute Diffusion Field Research Articles

Pinning/depinning dynamics of trijunction lines during faceted/nonfaceted eutectic growth

We present an in situ experimental study of decoupled- and coupled-growth patterns in a binary faceted/ nonfaceted (f/nf) eutectic alloy. Real-time optical observation is carried out at low growth velocity V (<0.3μms−1) during directional solidification of the transparent AminoMethylPropaneDiol-Succinonitrile (AMPD-SCN) eutectic in thin samples. On a large scale, the two-phase growth front mostly exhibits irregularly spaced decoupled-growth patterns involving faceted fiber-like AMPD crystals that protrude in the liquid ahead of, and in weak diffusive interaction with the nonfaceted SCN solid. The morphological stability of low-velocity, steady-state decoupled-growth patterns is evidenced. In addition, three types of (f/nf) coupled-growth patterns with fiber-like, tubular (or “hollow”) and C-shaped morphologies, respectively, are identified. New light is cast on the pinning/depinning dynamics of trijunctions at which the two solid phases and the liquid meet in equilibrium. A discussion is initiated on the interplay between the interfacial kinetics of the more or less mobile facets and the solute diffusion field, and on the stability of triple junctions with strongly anisotropic interfaces.

Nucleation bursts of primary intermetallic crystals in a liquid Al alloy studied using in situ synchrotron X-ray radiography

The nucleation of primary Pt-rich intermetallic compound (IMC) crystals was studied in a model Al-Pt-Er alloy as an analogue to Al13Fe4 that forms in commercial Al alloys. The Pt-rich IMCs provided strong X-ray absorption contrast that allowed earlier detection of formation events and the resolution of solute diffusion fields. Crystal formation behaviour was investigated during directional and isothermal solidification when the alloy was inoculated with TiB2 particles at cooling rates of 0.1 Ks−1 to 8 Ks−1 using in situ synchrotron X-ray radiography. Different from previous studies that concerned disordered, solid solution phases such as α-Al dendrites, for the first time nucleation bursts of ordered compound crystals were observed, where the nucleation of the IMC crystals proceeded in distinct cascades (in the time domain) or waves (in the spatial domain), the magnitude of which increased with increasing cooling rate and particularly with decreasing thermal gradient in the melt. Comparing the crystal number density with the estimated number density of TiB2 particles suggested that the IMC crystals nucleated only on the most potent TiB2 particles, accounting for ∼0.5% of the total TiB2 number density in the melt. The effect of the thermal gradient and cooling rate on the magnitude of the nucleation waves and the IMC number density was revealed in terms of the available undercooling in the liquid in front of an individual IMC and the solute depleted liquid fraction that arose from the interaction of an ensemble of IMC crystals.

A 2D model for prediction of nanoparticle distribution and microstructure evolution during solidification of metal matrix nanocomposites

A 2D numerical model to predict the microstructure evolution in binary alloy matrix nanocomposites is presented. The main novelty of the model is that it captures the effect of nanoparticles on binary alloy dendrite growth and microstructure evolution. Diffusion based transport of nanoparticles is solved using a nanoparticle concentration defined in a similar way to an alloying element. Engulfment or rejection of nanoparticles at the solidification interface is implemented by defining a partition coefficient. The diffusivity of nanoparticles is calculated based on nanoparticle movement due to Brownian motion. The governing equations for binary alloy solidification are formulated using the volume averaged enthalpy method with a probabilistic nucleation model for grain nucleation. The effect of nanoparticles on solidification is incorporated through modifications of the solute diffusion field and interfacial energy. The developed model is used for predicting the effect of nanoparticle concentration and nanoparticle radius on dendrite growth and microstructure evolution. From single dendrite study, it is observed that the dendrite size decreases with increase in nanoparticle concentration while it increases with increase in nanoparticle radius. During microstructure evolution, increased non-homogeneity in nanoparticle distribution is observed with increase in nanoparticle concentration and more uniform distribution is observed with increase in nanoparticle radius.

A cellular automaton approach for the prediction of grain size in grain refined alloys

Grain refinement by inoculation is widely practised in metal casting, in particular for aluminium alloys. During the last decades several modelling techniques have emerged to quantitatively predict solidification microstructures. The prediction of grain size remains however a challenge due to the complex competition between nucleation and growth, and physical phenomena taking place at very different length scales. Models often address the thermal recalescence and sometimes the suppression of nucleation by the solute boundary layers. The possible clustering tendency of the nucleant particles is however usually not considered, although it can play an important influence on the final grain size. This effect is particularly difficult to quantify because it requires a small-scale approach to describe the clusters, while a large computational domain is needed for grain size statistics. A cellular automaton (CA) model tracking the grain envelopes was developed to address this challenge. The spatial distribution of the nucleant particles and the solute diffusion field in the intergranular liquid were represented directly on the CA grid. Predicted grain sizes were first compared successfully with published experimental and modelling data. An investigation of the role of clustering of nucleant particles was then carried out, which demonstrated the potential of the model to address this topic.

Effect of cooling rate on the nucleation and growth of large TiC particles in Ti-Mo steel

The mechanism and kinetics of large TiC precipitates in Ti-Mo steel were investigated in situ with a high-temperature confocal laser scanning microscope and analytically evaluated through thermodynamic calculations in Thermo-Calc. The size of the austenite grains and TiC gradually decreased with increasing undercooling. During the solidification process, strong solute diffusion fields were formed via undercooling, which inhibited the nucleation of TiC; hence, the second phase precipitation temperature was lower than the equilibrium precipitation temperature. Moreover, there was a critical cooling rate between 0.2 and 1.5 K/s that rendered the capacity of γ formation equal to that of TiC formation. The nucleation location of TiC can be changed from grain boundary to grain interior by increasing undercooling. The growth pattern of TiC conformed to the Avrami equation; at higher undercooling, the time exponent was small, and the relationship between the growth length and time was approximately linear. During actual cooling, the driving force exceeded a critical value under a specific degree of undercooling, and the γ/TiC interface became morphologically unstable, leading to a cellular/dendritic pattern.

Three-dimensional dendritic needle network model for alloy solidification

We present a three-dimensional (3D) formulation of the multiscale Dendritic Needle Network (DNN) model for dendritic microstructure growth. This approach is aimed at simulating quantitatively the solidification dynamics of complex hierarchical networks in spatially extended dendritic arrays, hence bridging the scale gap between phase-field simulations at the scale of a few dendrites and coarse-grained simulations on the larger scale of entire polycrystalline structures. In the DNN model, the dendritic network is represented by a network of branches that interact through the solutal diffusion field. The tip velocity V(t) and tip radius ρ(t) of each needle is determined by combining a standard solvability condition that fixes the product ρ2V and a solute flux conservation condition that fixes the product ρV2 in 2D and ρV in 3D as a function of a solutal flux intensity factor F(t). The latter measures the intensity of the solute flux in the dendrite tip region and can be calculated by contour (2D) or surface (3D) integration around the tip of each needle. We first present an extended formulation of the 2D DNN model where needles have a finite thickness and parabolic tips. This formulation remains valid for a larger range of tip Péclet number than the original thin needle formulation and is readily extended to 3D needles with paraboloidal tips. The 3D DNN model based on this thick-needle formulation is developed for both isothermal and directional solidification. Model predictions are validated by comparisons with known analytical solutions that describe the early transient and steady-state growth regimes. We exploit the power of the DNN model to characterize the competitive growth of well-developed secondary branches in 3D on the scale of the diffusion length. The results show that the length of active secondary branches increases as a power law of distance behind the tip with an exponent in good quantitative agreement with experimental measurements. Finally, we apply the model to simulate the three-dimensional directional solidification of an Al-7wt% Si alloy, which we directly compare to observed microstructures from microgravity experiments onboard the International Space Station. The predictions of selected microstructural features, such as dendrite arm spacings, show a good agreement with experiments. The computationally-efficient DNN model opens new avenues for investigating the dynamics of large dendritic arrays at length and time scales relevant to solidification experiments and processes.

Effect of nucleation undercooling on the kinetics and mechanism of the peritectic phase transition in steel

The mechanism and kinetics of the peritectic phase transition in steel were studied by using a concentric solidification technique in a high-temperature laser-scanning confocal microscope. Strong solute diffusion fields are established during solidification of the primary delta phase, which leads to suppression of nucleation of the austenite phase, and hence the extent to which a steel can be cooled to a temperature below the equilibrium peritectic temperature before the peritectic reaction will occur. The magnitude of these diffusion fields was also determined as a function of carbon content, alloying additions, cooling rate and the fraction of primary delta phase that solidified before the peritectic reaction occurred. Depending on the extent of undercooling required to nucleate the first austenite platelet on the liquid/delta-ferrite interface, three different modes of the solid-state peritectic transformation of delta to austenite have been observed: a diffusion-controlled phase transformation that occurred by the progression of a planar interface; a diffusion-controlled transformation with cellular/dendritic morphology; and a massive-type of transformation. Under certain circumstances, a changeover between these modes has also been observed. The underpinning thermodynamic considerations that lead to these different modes of transformation are discussed and a new explanation is provided for the occurrence of a massive transformation of delta-ferrite to austenite. The relevance to and implications of these new findings to the continuous casting of steel of near-peritectic steel composition are also discussed.

Diffusional constrained crystal nucleation during peritectic phase transitions

The casting behaviour, microstructure formation and resulting mechanical properties of materials that undergo a peritectic phase transition during solidification are largely determined by the kinetics of this high-temperature phase transformation during the production process. However, current nucleation models do not accurately predict the nucleation behaviour and phase transition kinetics in many polycrystalline materials. Utilizing a newly developed experimental technique, we have performed in situ observations to study the nucleation behaviour of a newly forming intermediate phase by using the peritectic phase transition in Fe–C and Fe–Ni alloys as examples. In our experiments as well as by using thermodynamic and kinetic arguments we demonstrate that nucleation of a new intermediate (peritectic) phase can be constrained in the presence of solute diffusion fields that form during primary solidification due to an increase in the Gibbs free energy barrier for nucleation. We found a strong correlation between the magnitude of these diffusion fields and the resulting nucleation undercooling required for the formation of a new phase. Our study casts new light on, and clarifies for the first time, the much-debated underpinning reason for the occurrence of massive phase transformations occurring during solidification processing at large nucleation undercoolings. These new insights contribute to the improvement of nucleation theory and allow more accurate predictions on nucleation events and, in turn, physical properties of materials that undergo phase transitions in the course of materials production processes.

An analytical model for secondary phase dissolution kinetics

An analytical model for dissolution kinetics of secondary phase particles upon isothermal annealing has been proposed. Considering the interactions of solute diffusion fields in front of the secondary phase/matrix interface upon dissolution, a Johnson–Mehl–Avrami type equation, subjected to necessary modification, was derived, in combination with a classic dissolution model for single-particle system. Compared with the semiempirical dissolution models, which are used to fit the experimental results and phase-field method simulation, the current model follows an analogous form, but with the time-dependent kinetic parameters. Distinct from the model fitting work published recently, the current model is derived from the diffusion-controlled transformation theory, while the modeling quality is guaranteed by the physically realistic model parameters. On this basis, the current model calculation leads to a clear relationship between the secondary phase volume fraction and the time. Accordingly, model predictions for isothermal θ′ dissolution in Al–3.0wt%–Cu alloy and silicon dissolution in Al–0.8wt%–Si alloy were performed; good agreement with the published experimental data has been achieved.

Multiscale dendritic needle network model of alloy solidification

We present a novel dendritic needle network (DNN) model for simulating quantitatively the solidification of dendritic alloys. This approach is intended to reliably bridge the gap between phase-field simulations on the scale of dendrite tip radius ρ and cellular-automaton simulations on the several orders of magnitude larger scale of an entire dendritic grain. In the DNN model, the dendritic network of primary, secondary and higher other branches is represented by a network of sharp needles that interact through the solutal diffusion field. The tip velocity V of each needle is determined by combining a standard solvability condition that fixes the product ρ2V and an additional solutal flux balance condition that fixes the product ρV2∼F2, where F measures the intensity of the solutal flux in the dendrite tip region. This solutal flux intensity factor F can be accurately computed by contour integral methods commonly used to compute stress intensity factors in fracture mechanics. This formulation provides an asymptotically exact description of the dendritic network dynamics in the limit of small Péclet number ρV/D≪1, where D is the solute diffusivity. The DNN model is developed and implemented for both isothermal and directional solidification in two dimensions and is validated by comparison with analytical solutions for both early-stage and steady-state equiaxed growth as well as phase-field simulations. The latter comparison shows that DNN simulations are roughly four orders of magnitude faster than phase-field simulations while remaining reasonably accurate. DNN simulations of directional solidification demonstrate that the approach can be used to efficiently investigate the stability and dynamics of spatially extended dendritic arrays. This is illustrated for an Al–7wt.% Si alloy by computing the stable range of primary array spacing and the history-dependent dynamic selection of this spacing following an abrupt change of solidification rate or of sample cross-section. We also compare DNN model predictions to microgravity experiments for the same alloy.

INVESTIGATION OF EFFECT OF INTERFACE ENERGY ANISOTROPY ON DENDRITIC GROWTH IN UNIDIRECTIONAL SOLIDIFICATION BY FRONT TRACKING SIMULATION

The dendritic growth with the different solid/liquid (S/L) interface energy anisotropies in the unidirectional solidification has been investigated using the self-consistent front tracking model. It is found that, for a given solidification condition, there were two kind of interface shape solutions with the different spacing Peclect number ranges. The interface shape with the small spacing Peclect number range was similar with cellular tip, and that with the large spacing Peclect number range referred to dendritic tip. The higher S/L interface energy anisotropy was in favor of the widening of the dendritic growth solution range. There was a certain power exponential relationship between the dendritic tip marginal stability parameter σ ∗ and the S/L interface energy anisotropic parameter E4. A modified Fisher dendritic tip solution, which considered the effect of S/L interface energy anisotropy, was obtained as follows: RIMS =2 .5646( ΓD L Vk 0ΔT0 ) 0.5 E −0.1905 4 , ΔT0 = mC0(k0 − 1)/k0. The undercooling in front of the S/L interface decreased with increasing the anisotropic parameter. The primary dendritic spacing mainly depended on the interaction of solute diffusion field between the adjacent dendrite, and the S/L interface energy had little influence on

Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography

This paper addresses dendritic equiaxed growth during grain structure formation in alloy solidification, and focuses on the growth and interaction of primary dendrite arms, which drive and limit grain development, respectively. Dedicated solidification experiments on Al–10wt.% Cu alloy were carried out at the European Synchrotron Radiation Facility, enabling live observation by in situ and real-time synchrotron X-ray radiography. Equiaxed solidification was achieved by cooling nearly isothermal samples, and the growth of the grains was continuously monitored from the very early stages of solidification up to a state where changes in grain structure were no longer visible to the observer’s naked eye. The time evolution of individual equiaxed grains is analysed through measurements of the lengths and tip velocities of the primary dendrite arms driving grain growth. It is found that equiaxed dendrite propagation is characterized by two growth regimes. The first regime, in which a gradual increase in the tip growth rate is observed, corresponds to free dendritic growth, when the relative distance between grains is large enough to consider a grain as isolated. During this regime, the far-field liquid concentration ahead of the advancing dendrite tip remains constant, equal to the initial alloy composition, and the driving force for growth is the undercooling applied by cooling of the sample, in agreement with three-dimensional models of free dendritic growth. The second regime shows a decrease in the tip velocity towards zero and is attributed to grain interaction when the solute diffusion field in front of the advancing dendrite tip under consideration overlaps with the solute field associated with the growth of the grain(s) facing it. Solute concentration profiles are measured in the extra dendritic liquid ahead of dendrite tips interacting with neighbour grains, from which the far-field solute concentration and neighbour–interaction distance are deduced. These measurements confirm that impingement of the solute boundary layers is the main cause of dendrite growth being slowed down and stopped.

Local non-equilibrium diffusion model for solute trapping during rapid solidification

A local non-equilibrium diffusion model (LNDM) for rapid solidification of binary alloys has been briefly reviewed and used to modify a number of solute trapping models with different solid–liquid interface kinetics. The LNDM takes into account deviation from local equilibrium of a solute diffusion field in bulk liquid on the basis that the exact solutions to hyperbolic diffusion equations govern the solute concentration and solute flux in bulk liquid under local non-equilibrium conditions. The LNDM leads to a velocity-dependent effective diffusion coefficient in bulk liquid ahead of the solid–liquid interface DbLNDM(V), which goes to zero when the interface velocity V→VDb, where VDb is the bulk liquid diffusion speed. The results show an abrupt transition from diffusion-limited to purely thermally controlled solidification, with the diffusion coefficient in bulk liquid DbLNDM(V)=0 and complete solute trapping KLNDM(V)=1 at a finite interface velocity V=VDb for any type of solid–liquid interface kinetics. The bulk liquid diffusion speed VDb is a critical parameter for the transition. The velocity dependence of partition coefficients KLNDM(V) has been calculated for different types of solid–liquid interface kinetics, with allowance for local non-equilibrium diffusion effects. The calculation shows that the local non-equilibrium partition coefficients KLNDM(V) reduce to the standard K(V) at low interface velocity (V≪VDb) and differ substantially at high interface velocity (V∼VDb).

A quantitative study of solute diffusion field effects on heterogeneous nucleation and the grain size of alloys

The nucleation ability of inoculating particles inside the solute diffusion zone around growing grains during alloy solidification is studied using a spherical, equiaxed dendritic grain model coupled with a new modified free growth model to predict the final grain size of cast aluminium alloys with improved accuracy. We show that the nucleation potency of inoculating particles is reduced by the solute field that develops close to existing, growing equiaxed grains under near isothermal conditions. Solute suppressed nucleation leads to much lower nucleated grain densities, higher nucleation undercoolings and longer times to recalescence when further nucleation events are halted. Under solute suppressed conditions, nucleation events occur in two stages: an initial transient nucleation before significant solute build-up and then continuous nucleation. The significance of the transient nucleation regime depends upon the size of the transient solute diffusion zone, and has been explored using the model. Model predictions suggest that the grain refinement of alloys of high solute content is controlled primarily by solute suppressed nucleation conditions.

Stochastic modeling of columnar dendritic grain growth in weld pool of Al‐Cu alloy

AbstractA multi‐scale model is used to simulate columnar dendritic growth in TIG (tungsten inert‐gas) weld molten pool of Al‐Cu alloy. The grain morphologies at the edge of the weld pool are studied. The simulated results indicate that the average primary dendrite spacing changes during the solidification process in the weld pool because of the complicated thermal field, solute diffusion field and competitive growth. And it is shown that the secondary dendrite arms grow insufficiently in the space between dendrite trunks if the primary dendrite spacing is small. And the phenomenon has been explained by analyzing the influence of the solute accumulation on the constitutional undercooling and undercooling gradient when there are two different opposite solute diffusion fields. (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the strengthening response of aluminum alloys containing shear-resistant plate-shaped precipitates

The precipitate strengthening increment arising from a reasonably monotonic distribution of plate-shaped shear-resistant precipitates in an Al matrix is considered theoretically in terms of the average stress required to push dislocations between the particles. The radius of the solute diffusion field surrounding the tips of the plates is shown to be a key microstructural parameter in controlling the distance of closest approach between precipitates and therefore the maximum strengthening increment. Three new strategies for enhancing the strengthening increment are proposed based on minimizing the size of the solute diffusion fields surrounding plates. These approaches are contrasted with those previously advocated. A model Al–Cu alloy containing plate-shaped, shear-resistant θ′ (Al 2Cu) precipitates is used to experimentally test the predictions of the proposed strengthening description. Quantitative agreement between experiment and theory is observed under isothermal annealing conditions at 473 K. Furthermore, one of the strategies advocated for increasing the strengthening response is the use of a decreasing transformation temperature. This is tested experimentally at three temperatures and the peak strengthening increment is shown to be in excellent agreement with the theoretical predictions. This work suggests a reinterpretation of the origins of the strengthening observed in current high-strength Al alloys containing plate-shaped precipitates is required.

Simulation of Nucleation of Proeutectoid Ferrite at Austenite Grain Boundaries during Continuous Cooling

The nucleation kinetics of proeutectoid ferrite during continuous cooling in three Fe-C-Mn-Si steels, measured in-situ by three-dimensional X-ray diffraction microscope, are compared with numerical simulation that takes into account differences in the activation energy of nucleation among grain boundary faces, edges, and corners. The essential feature of ferrite nucleation in the 0.21 pct C steel, i.e., nucleation occurred just below Ae3 and ceased at a small undercooling, is reproduced taking into account the site consumption, primarily at grain corners and overlap of solute diffusion fields in the grain boundary region or the matrix and assuming a very small or almost null activation energy of nucleation. In the 0.35 and 0.45 pct C steels, small activation energy, as reported by Offerman et al., was not unequivocally obtained because ferrite nucleation occurred at considerably large undercoolings, even below the paraequilibrium Ae3 in these steels. The increasing rate of the observed particle number with decreasing temperature is considerably smaller than calculation.

Diffusionless crystal growth in a eutectic system during rapid solidification

Experiments on nonequilibrium rapid eutectic growth are surveyed. The applicability limits of the modern theoretical models describing rapid solidification of binary systems are assessed. A problem of rapid eutectic growth when the local equilibrium is violated in the solute diffusion field (in the bulk liquid and at the solid-liquid interface) is formulated. An analytical solution to the problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that the diffusion-limited growth of a eutectic pattern ceases as soon as a chemically homogeneous crystalline phase begins to grow when the critical point V=VD is achieved (V is the solid-liquid interface velocity and VD is the solute diffusion speed in the bulk liquid). At V ≥ VD, eutectic decomposition is suppressed and the nascent homogeneous crystalline phase has the initial (nominal) chemical composition of the binary system.

Diffusionless Crystal Growth in Rapidly Solidifying Eutectic Systems

Using a local nonequilibrium model of solidification, experiments on rapid eutectic growth are analyzed. An analytical solution of a problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that solute diffusion-limited growth of a eutectic pattern is completely finished, and diffusionless growth of the chemically homogeneous crystalline phase begins to proceed at a critical point V = V(D), where V is the solid-liquid interface velocity and V(D) is the solute diffusion speed in bulk liquid. A suppression of eutectic decomposition occurs in the range V > or = V(D) that results in a growth of homogeneous crystal phase with the initial (nominal) chemical composition of the binary system.

Linear morphological stability analysis of the solid-liquid interface in rapid solidification of a binary system.

The interface stability against small perturbations of the planar solid-liquid interface is considered analytically in linear approximation. Following the analytical procedure of Trivedi and Kurz [Acta Metall. 34, 1663 (1986)]], which is advancing the original treatment of morphological stability by Mullins and Sekerka [J. Appl. Phys. 35, 444 (1964)]] to the case of rapid solidification, we extend the model by introducing the local nonequilibrium in the solute diffusion field around the interface. A solution to the heat- and mass-transport problem around the perturbed interface is given in the presence of the local nonequilibrium solute diffusion. Using the developing local nonequilibrium model of solidification, the self-consistent analysis of linear morphological stability is presented with the attribution to the marginal (neutral) and absolute morphological stability of a rapidly moving interface. Special consideration of the interface stability for the cases of solidification in negative and positive thermal gradients is given. A quantitative comparison of the model predictions for the absolute morphological stability is presented with regard to experimental results of Hoglund and Aziz [D. E. Hoglund and M. J. Aziz, in Kinetics of Phase Transformations, edited by M.O. Thompson, M. J. Aziz, and G. B. Stephenson, MRS Symposia Proceedings No. 205 (Materials Research Society, Pittsburgh, 1991), p. 325] on critical solute concentration for the interface breakdown during rapid solidification of Si-Sn alloys.