Multiscale prediction of microstructure length scales in metallic alloy casting

Abstract

Microstructural length scales, such as dendritic spacings in cast metallic alloys, play an essential role in the properties of structural components. Therefore, quantitative prediction of such length scales through simulations is important to design novel alloys and optimize processing conditions through integrated computational materials engineering (ICME). Thus far, quantitative comparisons between experiments and simulations of primary dendrite arms spacings (PDAS) selection in metallic alloys have been mainly limited to directional solidification of thin samples and quantitative phase-field simulations of dilute alloys. In this article, we combine casting experiments and quantitative simulations to present a novel multiscale modeling approach to predict local primary dendritic spacings in metallic alloys solidified in conditions relevant to industrial casting processes. To this end, primary dendritic spacings were measured in instrumented casting experiments in Al–Cu alloys containing 1 wt.% and 4 wt.% of Cu, and they were compared to spacing stability ranges and average spacings in dendritic arrays simulated using phase-field (PF) and dendritic needle network (DNN) models. It is first shown that PF and DNN lead to similar results for the Al-1 wt.%Cu alloy, using a dendrite tip selection constant calculated with PF in the DNN simulations. PF simulations cannot achieve quantitative predictions for the Al-4 wt.%Cu alloy because they are too computationally demanding due to the large separation of scale between tip radius and diffusion length, a characteristic feature of non-dilute alloys. Nevertheless, the results of DNN simulations for non-dilute Al–Cu alloys are in overall good agreement with our experimental results as well as with those of an extensive literature review. Simulations consistently suggest a widening of the PDAS stability range with a decrease of the temperature gradient as the microstructure goes from cellular-dendrites to well-developed hierarchical dendrites.

Introduction

Solidification is the initial processing step for most metallic alloys. Microstructural features that develop during solidification play a key role in the properties, performance and lifetime of cast metal parts. Even when the microstructure later evolves through additional complex thermomechanical processing steps, the starting point is the microstructure that emerges from the liquid state. Dendrites, which are branched structures with primary, secondary and higher-order branches, are among the most common microstructural features in cast alloys [1], [2], [3]. Dendritic microstructures exhibit several important characteristic length scales, such as dendrite tip radius or dendrite arm spacings. These spacings directly affect the mechanical properties of individual grains [1], [2], in particular ultimate tensile strength and yield strength (see, e.g., [4], [5], [6]). They also set the scale for solute (micro)segregation [1], [2], [7], which has a strong effect on the potential appearance of defects (e.g. freckles) or secondary phases (e.g. precipitates or eutectics), as well as on electrochemical properties, such as corrosion resistance (see, e.g., [6], [8], [9]). Primary dendrite arm spacings (PDAS) also play a key role in the permeability of semi-solid microstructures [10], [11], [12], [13], [14], which has major influence on the appearance of critical defects, such as hot tears.

During the growth of primary dendrites, the selection of dendrite tip radius $\rho$ is unique for a given set of solidification conditions (namely, solute concentration $c_{\infty },$ solidification velocity $V,$ and thermal gradient $G$) and follows the microscopic solvability theory [15], [16], [17], [18], [19].

The selection of PDAS, on the other hand, is not unique. A given set of parameters may yield a wide distribution of spacings [20], [21], [22], [23], [24], [25], [26], even though a similar processing history tends to lead to narrowly distributed spacings [26], [27]. Over the years, many studies have focused on the link between the solidification conditions and the morphological characteristics of dendrites (see e.g. [22], [26], [28], [29] among others). Most models for the PDAS rely on phenomenological relations linking the primary spacing $\lambda$ to processing parameters in the form of power laws such as [28]:

$$\lambda =K{G}^{-a}{V}^{-b}$$

where the exponent $a$ is often close to 0.5, $b$ typically varies between 0.25 and 0.5, and $K$ is a prefactor that depends on alloy parameters, such as its composition and phase diagram features (e.g. solute partition coefficient) [22], [26], [29], [30]. However, experimental evidence indicates that a wide range of PDAS can be obtained under similar solidification conditions. The lower limit of the PDAS stability range, ${\lambda }_{\min },$ is linked to an elimination instability [26], [31], while the upper stability limit, ${\lambda }_{\max },$ stems from a branching instability. The determination of this range is not, however, trivial considering all the mechanisms involved, including: tip elimination through thermal and solutal interaction, tip splitting, secondary and tertiary arm development, lateral migration, thermo-solutal convection, and defect interaction with the growth front.

Directional solidification is the standard experimental approach to establish the link between solidification conditions and microstructure in metallic alloys. Particularly, dendrite growth in the Al–Cu system has been studied for many solidification conditions ($G$ and $V$), different Cu concentrations and different experimental set-ups. They include standard Bridgman furnaces in which the solidification velocity is controlled [34], [35], [36], [37], or small volumes in which solidification takes place under a given thermal gradient [38], [39], [40], [41], [42], while other studies used larger molds in which the temperature was measured along the solidification direction to obtain the thermal gradient and the solidification velocity [43], [44], [45], [46]. The differences in solidification conditions has led to a significant scatter in the PDAS for similar nominal values of the solidification process, as given by the cooling rate $R$ = $GV,$ but the overall trend is clearly indicative of the power law behavior from phenomenological scaling laws (see Fig. I of the Supplementary Material).

Further analysis of the parameters controlling microstructural development during solidification can be obtained through modeling and simulation.

Conventional models for macroscopic solidification processes (e.g. casting) rely on conservation equations averaged over multi-phase domains, such that a field corresponding to the average fraction of phase over representative volume elements appears in the equations [47], [48], [49], [50], [51], [52], [53]. The fraction of phases may be tabulated as a function of temperature using classical solidification paths such as lever rule or Gulliver-Scheil model [29]. Alternatively, microstructural length scales (e.g. dendritic spacings) may be introduced in the equations to model the kinetics of solute microsegregation between dendritic branches a bit more accurately [49], [50], [51], [52], [53], [54], [55]. These length scales are then typically estimated from phenomenological power laws such as Eq.  (1).

Models capable of estimating microstructural length scales require a relatively more precise spatial description, consistent with the scale of the features one aims at predicting. This makes simulations computationally challenging since they have to combine phenomena across a wide range of scales, from microscopic capillarity at dendritic tips to macroscopic transport of heat and solute in the melt. Simulations of dendritic spacings have been performed with a range of models based on cellular automata (CA) [56], [57], [58], grain envelopes [59], [60], dendritic needle networks (DNN) [61], [62], [63], [64] and phase fields (PF) [65], [66], [67]. Approximate, and computationally efficient, models (e.g. CA) rely on stringent assumptions and phenomenological laws, for instance directly relating the growth kinetics of dendrite tips to some measure of the local driving force for solidification (e.g. undercooling or solute supersaturation measured at a given distance from the solid-liquid interface). The most accurate models (e.g. PF) rely solely on fundamental concepts of diffuse interface thermodynamics [32], [68], [69] and allow quantitative calculations directly from thermophysical alloy parameters and processing conditions. However, they remain limited in length and time scales by their relatively high computational cost. Hence, a multiscale multi-model bottom-up strategy appears to be a promising route to address the limitations of each models.

Here we specifically focus on two modeling methods, namely PF and DNN, and we show that quantitative PF simulations of dendritic growth in dilute alloys can be quantitatively upscaled and expanded to non-dilute compositions using the multiscale DNN approach.

The PF method is now recognized as the outstanding tool to predict microstructural patterns resulting from nonequilibirum solidification [32], [68], [69], [70]. The method allows modeling the evolution of solid-liquid interfaces and complex microstuctural patterns, such as dendrites. It has been applied to the solidification of Al–Cu alloys, at first qualitatively, e.g. to reproduce scaling laws for secondary dendrite arm spacing in cast samples [71]. Then, the development of quantitative models for dilute binary alloys [72], [73], [74], [75] has allowed direct quantitative comparisons with directional solidification experiments of dilute Al–Cu thin-samples imaged in situ with X-ray radiography [67]. However, quantitative PF predictions are still constrained by the requirement of an accurate description of the local solid-liquid interface curvature. This limits the size of the grid elements required in the vicinity of each dendritic tip for primary, secondary, and higher-order side-branches. This limitation may, to some extent, be addressed through parallelization [76], [77] and advanced numerics (e.g. adaptive meshing [78], [79], [80] or implicit time stepping [81], [82]). However, quantitative PF simulations become challenging, if feasible at all, beyond the dilute alloy limit, as the separation of scale between interfacial capillary length, dendrite tip radius, and solute transport in the bulk becomes extreme.

In order to overcome this limitation, the dendritic needle network (DNN) method was developed to bridge the scale of the microscopic dendrite tip radius, $\rho ,$ and that of solute transport in the liquid phase, e.g. the diffusion length $l_D=D/V$ with $D$ the solute diffusivity. The DNN model represents a dendritic crystal as a network of parabolic-shaped needles, that approximate the primary stems and higher order branches. The DNN model combines the microscopic solvability theory established at the scale of the tip radius [15], [16], [17], [18] with a solute mass balance established at the “mesoscale” between $\rho$ and $l_D,$ to calculate the instantaneous growth velocities of individual dendrite tips [61], [62], [83], [84]. The numerical spatial discretization can be taken at the same order as the tip radius, such that the domain size can be considerably larger than in equivalent PF simulation, in particular for concentrated alloys with $\rho \ll l_D$. DNN simulations have been verified to reproduce analytical and phase-field predictions for steady-state and transient growth kinetics [61], [62], [85]. First DNN applications to the prediction of primary dendritic spacings in binary Al alloys, compared to well-controlled thin-sample directional solidification experiments [62], [86], [87], have shown promising results.

This investigation is aimed at bridging length scales in the simulation of alloy solidification. One central objective is to demonstrate that microstructural length scales, such as PDAS, can be computationally predicted for a bulk specimen, under conditions distinctive from a Bridgman type experiment, namely nonuniform $G,$ $V$ and non-negligible fluid flow, making it closer and thus more representative of solidification conditions for industrial casting processing. To do so, we performed casting experiments of binary Al–Cu alloys, and combined simulations at distinct length scales. First, we show that both PF and DNN can provide reliable predictions of stability range of PDAS for an Al-1 wt.%Cu alloy. Then, we use the computationally-efficient DNN model to calculate the PDAS stability range for a more concentrated Al-4 wt.%Cu alloy. We discuss and compare our results with a broad range of experimental literature data for Al–Cu alloys. Overall, the article shows how multiscale modeling can provide quantitative predictions of important microstructural features (e.g. PDAS) during solidification of non-dilute alloys at experimentally relevant length and time scales using physics-based computational models.