Multiscale prediction of microstructure length scales in metallic alloycasting

B. Bellon1,2 , A.K. Boukellal1 , T. Isensee1,2 , O.M. Wellborn3 , K.P. Trumble3 , M.J.M. Krane3 , M.S. Titus3 ,D. Tourret1 , J. LLorca1,2∗

1 IMDEA Materials Institute, C/ Eric Kandel 2, 28906, Getafe, Madrid, Spain.

2 Department of Materials Science, Polytechnic University of Madrid/Universidad Politecnica de Madrid, E.T.S. deIngenieros de Caminos. 28040 - Madrid, Spain.

3Purdue Center for Metal Casting Research, School of Materials Engineering, Purdue University, West Lafayette, Indiana,USA.

Abstract

Microstructural length scales, such as dendritic spacings in cast metallic alloys, play an essential role inthe properties of structural components. Therefore, quantitative prediction of such length scales throughsimulations is important to design novel alloys and optimize processing conditions through integrated com-putational materials engineering (ICME). Thus far, quantitative comparisons between experiments andsimulations of primary dendrite arms spacings (PDAS) selection in metallic alloys have been mainly limitedto directional solidification of thin samples and quantitative phase-field simulations of dilute alloys. In thisarticle, we combine casting experiments and quantitative simulations to present a novel multiscale modelingapproach to predict local primary dendritic spacings in metallic alloys solidified in conditions relevant toindustrial casting processes. To this end, primary dendritic spacings were measured in instrumented castingexperiments in Al-Cu alloys containing 1 wt% and 4 wt% of Cu, and they were compared to spacing stabil-ity ranges and average spacings in dendritic arrays simulated using phase-field (PF) and dendritic needlenetwork (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 cannotachieve quantitative predictions for the Al-4 wt%Cu alloy because they are too computationally demandingdue 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 goodagreement with our experimental results as well as with those of an extensive literature review. Simulationsconsistently suggest a widening of the PDAS stability range with a decrease of the temperature gradient asthe microstructure goes from cellular-dendrites to well-developed hierarchical dendrites.

Keywords: Solidification, Casting, Metallic alloy, Microstructure, Multiscale modeling.

1. Introduction

Solidification is the initial processing step formost metallic alloys. Microstructural features thatdevelop during solidification play a key role in theproperties, performance and lifetime of cast metalparts. Even when the microstructure later evolvesthrough additional complex thermomechanical pro-

∗Corresponding author; Email address:javier.llorca@imdea.org

cessing steps, the starting point is the microstruc-ture that emerges from the liquid state. Dendrites,which are branched structures with primary, sec-ondary and higher-order branches, are among themost common microstructural features in cast al-loys [1–3]. Dendritic microstructures exhibit sev-eral important characteristic length scales, such asdendrite tip radius or dendrite arm spacings. Thesespacings directly affect the mechanical properties ofindividual grains [1, 2], in particular ultimate ten-sile strength and yield strength (see, e.g., [4–6]).

Preprint submitted to Acta Materialia, accepted for publication January 20, 2021

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They also set the scale for solute (micro)segregation[1, 2, 7], which has a strong effect on the potentialappearance of defects (e.g. freckles) or secondaryphases (e.g. precipitates or eutectics), as well as onelectrochemical properties, such as corrosion resis-tance (see, e.g., [6, 8, 9]). Primary dendrite armspacings (PDAS) also play a key role in the perme-ability of semi-solid microstructures [10–14], whichhas major influence on the appearance of criticaldefects, such as hot tears.

During growth of primary dendrites, the selec-tion of dendrite tip radius ρ is unique for a given setof solidification conditions (namely, solute concen-tration c∞, solidification velocity V , and thermalgradient G) and follows the microscopic solvabilitytheory [15–19].

The selection of PDAS, on the other hand, is notunique. A given set of parameters may yield a widedistribution of spacings [20–26], even though a sim-ilar processing history tends to lead to narrowlydistributed spacings [26, 27]. Over the years, manystudies have focused on the link between the so-lidification conditions and the morphological char-acteristics of dendrites (see e.g. [22, 26, 28, 29]among others). Most models for the PDAS relyon phenomenological relations linking the primaryspacing λ to processing parameters in the form ofpower laws such as [28]:

λ = KG−aV −b (1)

where the exponent a is often close to 0.5, b typi-cally varies between 0.25 and 0.5, and K is a pref-actor that depends on alloy parameters, such as itscomposition and phase diagram features (e.g. so-lute partition coefficient) [22, 26, 29, 30]. However,experimental evidence indicates that a wide rangeof PDAS can be obtained under similar solidifica-tion conditions. The lower limit of the PDAS stabil-ity range, λmin, is linked to an elimination instabil-ity [26, 31–34], while the upper stability limit, λmax,stems from a branching instability. The determina-tion of this range is not, however, trivial consideringall the mechanisms involved, including: tip elimi-nation through thermal and solutal interaction, tipsplitting, secondary and tertiary arm development,lateral migration, thermo-solutal convection, anddefect interaction with the growth front.

Directional solidification is the standard experi-mental approach to establish the link between so-lidification conditions and microstructure in metal-lic alloys. Particularly, dendrite growth in the Al-Cu system has been studied for many solidification

conditions (G and V ), different Cu concentrationsand different experimental set-ups. They includestandard Bridgman furnaces in which the solidifi-cation velocity is controlled [35–38], or small vol-umes in which solidification takes place under agiven thermal gradient [39–43], while other stud-ies used larger molds in which the temperature wasmeasured along the solidification direction to obtainthe thermal gradient and the solidification velocity[44–47]. The differences in solidification conditionshas led to a significant scatter in the PDAS for sim-ilar nominal values of the solidification process, asgiven by the cooling rate R = GV , but the overalltrend is clearly indicative of the power law behav-ior from phenomenological scaling laws (see Fig. Iof the Supplementary Material).

Further analysis of the parameters controlling mi-crostructural development during solidification canbe obtained through modeling and simulation.

Conventional models for macroscopic solidifica-tion processes (e.g. casting) rely on conservationequations averaged over multi-phase domains, suchthat a field corresponding to the average fractionof phase over representative volume elements ap-pears in the equations [48–54]. The fraction ofphases may be tabulated as a function of tempera-ture using classical solidification paths such as leverrule or Gulliver-Scheil model [29]. Alternatively,microstructural length scales (e.g. dendritic spac-ings) may be introduced in the equations to modelthe kinetics of solute microsegregation between den-dritic branches a bit more accurately [50–56]. Theselength scales are then typically estimated from phe-nomenological power laws such as Eq. (1).

Models capable of estimating microstructurallength scales require a relatively more precise spa-tial description, consistent with the scale of the fea-tures one aims at predicting. This makes simula-tions computationally challenging since they haveto combine phenomena across a wide range ofscales, from microscopic capillarity at dendritic tipsto macroscopic transport of heat and solute in themelt. Simulations of dendritic spacings have beenperformed with a range of models based on cellu-lar automata (CA) [57–59], grain envelopes [60, 61],dendritic needle networks (DNN) [62–65] and phasefields (PF) [66–68]. Approximate — and compu-tationally efficient — models (e.g. CA) rely onstringent assumptions and phenomenological laws,for instance directly relating the growth kinetics ofdendrite tips to some measure of the local drivingforce for solidification (e.g. undercooling or solute

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supersaturation measured at a given distance fromthe solid-liquid interface). The most accurate mod-els (e.g. PF) rely solely on fundamental conceptsof diffuse interface thermodynamics [69, 70] andallow quantitative calculations directly from ther-mophysical alloy parameters and processing condi-tions. However, they remain limited in length andtime scales by their relatively high computationalcost. Hence, a multiscale multi-model bottom-upstrategy appears to be a promising route to addressthe limitations of each models.

Here we specifically focus on two modeling meth-ods, namely PF and DNN, and we show that quan-titative PF simulations of dendritic growth in di-lute alloys can be quantitatively upscaled and ex-panded to non-dilute compositions using the multi-scale DNN approach.

The PF method is now recognized as the out-standing tool to predict microstructural patternsresulting from nonequilibirum solidification [69–71].The method allows modeling the evolution of solid-liquid interfaces and complex microstuctural pat-terns, such as dendrites. It has been applied to thesolidification of Al-Cu alloys, at first qualitatively,e.g. to reproduce scaling laws for secondary den-drite arm spacing in cast samples [72]. Then, thedevelopment of quantitative models for dilute bi-nary alloys [73–76] has allowed direct quantitativecomparisons with directional solidification experi-ments of dilute Al-Cu thin-samples imaged in situwith X-ray radiography [68]. However, quantitativePF predictions are still constrained by the require-ment of an accurate description of the local solid-liquid interface curvature. This limits the size of thegrid elements required in the vicinity of each den-dritic tip for primary, secondary, and higher-orderside-branches. This limitation may, to some extent,be addressed through parallelization [77, 78] andadvanced numerics (e.g. adaptive meshing [79–81]or implicit time stepping [82, 83]). However, quan-titative PF simulations become challenging, if feasi-ble at all, beyond the dilute alloy limit, as the sepa-ration of scale between interfacial capillary length,dendrite tip radius, and solute transport in the bulkbecomes extreme.

In order to overcome this limitation, the dendriticneedle network (DNN) method was developed tobridge the scale of the microscopic dendrite tip ra-dius, ρ, and that of solute transport in the liquidphase, e.g. the diffusion length lD = D/V with Dthe solute diffusivity. The DNN model represents adendritic crystal as a network of parabolic-shaped

needles, that approximate the primary stems andhigher order branches. The DNN model combinesthe microscopic solvability theory established at thescale of the tip radius [15–18] with a solute massbalance established at the “mesoscale” between ρand lD, to calculate the instantaneous growth ve-locities of individual dendrite tips [62, 63, 84, 85].The numerical spatial discretization can be taken atthe same order as the tip radius, such that the do-main size can be considerably larger than in equiv-alent PF simulation, in particular for concentratedalloys with ρ � lD. DNN simulations have beenverified to reproduce analytical and phase-field pre-dictions for steady-state and transient growth ki-netics [62, 63, 86]. First DNN applications to theprediction of primary dendritic spacings in binaryAl alloys, compared to well-controlled thin-sampledirectional solidification experiments [63, 87, 88],have shown promising results.

This investigation is aimed at bridging lengthscales in the simulation of alloy solidification. Onecentral objective is to demonstrate that microstruc-tural length scales, such as PDAS, can be computa-tionally predicted for a bulk specimen, under con-ditions distinctive from a Bridgman type experi-ment, namely nonuniform G, V and non-negligiblefluid flow, making it closer and thus more repre-sentative of solidification conditions for industrialcasting processing. To do so, we performed castingexperiments of binary Al-Cu alloys, and combinedsimulations at distinct length scales. First, we showthat both PF and DNN can provide reliable predic-tions of stability range of PDAS for an Al-1 wt%Cualloy. Then, we use the computationally-efficientDNN model to calculate the PDAS stability rangefor a more concentrated Al-4 wt%Cu alloy. We dis-cuss and compare our results with a broad range ofexperimental literature data for Al-Cu alloys. Over-all, the article shows how multiscale modeling canprovide quantitative predictions of important mi-crostructural features (e.g. PDAS) during solidi-fication of non-dilute alloys at experimentally rel-evant length and time scales using physics-basedcomputational models.

2. Methods

Here, we provide a short description of our exper-imental and computational methods. A more thor-ough description with full details of equipments,procedures, equations, and algorithms is providedin the joint Supplementary Material document.

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2.1. Experiments and characterization

2.1.1. Casting experiments

We performed casting experiments in the facili-ties of the School of Materials Engineering at Pur-due University. Two different Al-Cu alloys with 1and 4 wt%Cu were prepared by melting Al gran-ules and Cu rods in a clay-graphite crucible usingan air induction furnace. The alloys were pouredinto a stainless steel mold, of dimensions shown inFig. 1a. The mold assembly was wrapped with aheating tape to reach approximately 500◦C beforepouring. Prior to pouring through the riser, theheating tape was disconnected. The exterior of themold was covered with several layers of an insulat-ing blanket to minimize the heat losses through themold walls.

The heat was extracted from the side of the moldby a water-cooled chill plate (Fig. 1b), promotingpredominantly horizontal directional solidification.For each alloy, we explored different cooling rates byperforming two sets of experiments: either (1) witha 0.2 mm thick copper shim between the chill plateand the mold, thus ensuring good thermal contact,or (2) with an additional 5 mm thick alumina platebetween the chill plate and the copper shim, thusreducing the heat extraction rate. The mold was

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instrumented with 7 K-type thermocouples, shownin Fig. 1b (also see Table I of the SupplementaryMaterial), mechanically joined and protected fromdissolution by high-temperature cement.

2.1.2. Microstructure characterization

We sectioned the cast ingot in between the loca-tions of thermocouples and cut out cross-sections ofabout 30× 30 mm2. Their surface was polished us-ing SiC paper up to 4000 grit, followed by diamondpastes down to 0.25 µm particle size, and finishingwith a solution of colloidal silica (OP-S).

The polished samples were etched with Kellersreagent and observed under two different opti-cal microscopes with magnification from 6.3× to1000×. We used the software ImageJ to enhancethe image contrast and manually located the ap-parent centers of dendrites, measuring at least 400primary dendrites in each specimen. The averagePDAS was calculated from these locations using astandard Delaunay triangulation Python code.

We performed electron backscatter diffraction(EBSD) on the cross-sections in order to assess themisorientation between the typical [100] growth di-rection of Al dendrites and the cutting/observationplane. Several EBSD scans were combined intograin orientation maps in areas of about 16×4 mm2

and analyzed using the MTEX toolbox [89].

2.1.3. Estimation of local solidification conditions

In this article we focus primarily on the colum-nar dendritic structures observed between the loca-tions of thermocouples TC1 and TC2, i.e. in theearly stage of the experiment when the solidifica-tion direction is still expected to be aligned withthe (horizontal) main heat extraction direction.

We estimated the velocity of isotherms betweenTC1 and TC2 by dividing the distance between thethermocouples by the times t1 and t2 at which thetemperatures from TC1 and TC2 reach a given tem-perature T−L chosen up to 20 K below the alloy liq-uidus temperature TL (see Section 3.1 and Fig. 3).

The local temperature gradient was estimatedas an average cooling rate R of the temperaturesrecorded by TC1 and TC2 between times t1 and t2.Practically, we performed a linear fit of TC1 andTC2 in that time range to extract their respectivecooling rates R1 and R2, and then considered thatR ≈ (R1 +R2)/2 to calculate G = R/V .

Since the estimation of the local velocity ofisotherms V was more accurate than that of the

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temperature gradient G (see section 3.1), we de-cided to use V as a fixed input value in both PFand DNN simulations, and to explore different val-ues of G, i.e. different cooling rates R = GV .

2.2. Modeling and simulations

We used two models already described and usedin several previous publications. Therefore, we onlygive a brief description of the models and the mainsimulations procedures. Nevertheless, a completedescription of model equations, algorithms, and anexhaustive list numerical and alloy parameters, to-gether with a brief discussion of the latter, are pro-vided in the joint Supplementary Material.

2.2.1. Phase-field

We used a quantitative three-dimensional (3D)PF model for the directional solidification of a di-lute binary alloy in a temperature gradient [73–76] to calculate, in conditions representative of ourexperiments: (1) the limits of the PDAS stabilityrange λmin and λmax, (2) the dynamically selectedspacing in a spatially extended array with severalprimary dendrites, and (3) the dendrite tip selec-tion parameter σ. All PF simulations were per-formed for the Al-1 wt%Cu experiment using onegrowth velocity V = 160 µm/s, estimated in theexperiment with the insulating alumina plate (seeSection 3.1), and exploring different temperaturegradients G = 102, 103, and 104 K/m

First, we calculated λmin and λmax in a regu-lar quincunx (fcc-like) dendritic array, illustrated

Figure 2: Simulation domains for a quincunx (fcc-like) den-dritic pattern of primary spacing λ, as used in the phase-field(green and red squares) and DNN (blue square) simulations.For clarity only the tip region is shown here (up to ≈ 165 µmbehind the tip) and not the region at lower temperature, inwhich side-branching may occur.

in Fig. 2. Thus, we simulated one quarter of a den-drite (green square in Fig. 2) with mirror symme-try imposed along all boundaries. Using a proce-dure already used in several previous publications[33, 90, 91] (see Supplementary Material), we pro-gressively increased the size of the domain along yand z until a branching event occurs, thus mark-ing the upper limit of spacing stability λmax. Inorder to identify the lower stability limit λmin, weperformed simulations with four quarters of den-drites in opposite corners of the simulation box (redsquare in Fig. 2), and reduce the domain size untilone or more primary dendrite is eliminated, thusidentifying λmin. Using a moving frame followingthe tip location in the x direction, the size of the do-main in x was taken long enough for no-flux bound-ary conditions to have a negligible effect on the tipgrowth dynamics (see Supplementary Material).

Then, we performed one simulation of a spatiallyextended array of dendrites in a thermal gradientG = 103 K/m, using a simulation domain of cross-section Ly × Lz = 512× 512 µm2 and length Lx =1185 µm, starting from a planar interface located atthe liquidus temperature. The simulated time was200 s, which was found to be enough to approachan apparent steady-state microstructure.

We also used the PF results to estimate the den-drite tip selection parameter σ, which is an in-put parameter of the DNN model. To do so, weextracted primary tip radii in simulations of onequarter of dendrite, using least-square fitting of thesolid-liquid interface in (010) and (001) planes toa second order parabola (purple lines on the left-side dendrite of Fig. 2) using the method presentedin Ref. [68] (see Supplementary Material). We cal-culated σ for a handful of steady-state dendrites atdifferent spacings in order to verify its independenceto the PDAS in the simulations of one quarter ofdendrite at G = 102 K/m. We selected the lowestof the explored temperature gradients as it is wellwithin the well-developed dendritic regime, i.e. farfrom the cell-to-dendrite transition regime.

2.2.2. Dendritic Needle Network

We used a multiscale 3D DNN model for direc-tional solidification already presented elsewhere [62,63, 84] in order to calculate, for both Al-1 wt%Cuand Al-4 wt%Cu alloys: (1) the PDAS stability lim-its λmin and λmax and (2) dynamically selectedspacings in a spatially extended dendritic array.

For each alloy, we used one growth velocity es-timated between thermocouples TC1 and TC2 in

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the experiments with an insulating alumina plate,namely V = 160 µm/s (Al-1 wt%Cu) and V =200 µm/s (Al-4 wt%Cu). We scanned a range oftemperature gradient G = 102, 103, and 104 K/mfor c∞ = 1 wt%Cu and G = 5 × 102, 5 × 103, and5 × 104 K/m for c∞ = 4 wt%Cu. The value of thetip selection parameter, σ = 0.063, was estimatedfrom phase-field simulation results.

To estimate both λmin and λmax, DNN simula-tions were initialized with two primary dendritesin opposite corners of the simulation domain (bluesquare in Fig. 2). Simulations were performed fordifferent domain sizes Ly = Lz = λ/

√2 in order to

identify spacings leading to the elimination of oneof the two primary dendrite (when λ < λmin) or tothe emergence of a tertiary branch as a new primarydendrite (when λ > λmax). Similarly as in PF sim-ulations, we use a moving frame in the x directionand the length of the domain in x was chosen longenough for boundary conditions to have a negligibleeffect (see Supplementary Material).

Additionally, we performed two simulations ofspatially extended dendritic arrays to determine adynamically selected average PDAS 〈λ〉. For theAl-1 wt%Cu alloy, we used G = 103 K/m andV = 160 µm/s in a domain of size Ly = Lz =1138 µm, and for the Al-4 wt%Cu alloy, we usedG = 5 × 102 K/m and V = 200 µm/s in a do-main of size Ly = Lz = 692 µm, both with a lengthLx = 1280 µm. The domain was initialized with re-spectively 1849 and 576 uniformly distributed pri-mary needles evenly spaced in y and z, in order toapproximate a planar interface located at the al-loy liquidus temperature. The simulated time were200 s and 100 s, respectively, which was found suffi-cient to approach a steady microstructure with nofurther primary dendrite elimination.

3. Results

3.1. Casting

The temperature-time curves recorded by the 7thermocouples are plotted in Fig. 3 for the four ex-periments. All curves exhibit a plateau or a no-table inflection below the liquidus temperature, in-dicative of the release of latent heat (recalescence)during solidification.

Readings from thermocouples close to the cool-ing surface (i.e. TC1, TC2, and TC3) are nearlyparallel to one another for a notable range of tem-perature below liquidus temperature TL. Thus, the

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velocity of isotherms at T = T−L . TL (dash-dotlines in Fig. 3) — calculated as V ≈ d12/(t2 − t1),with d12 the distance between TC1 and TC2, andt1 and t2 the times at which TC1 and TC2 respec-tively cross the temperature T−L — provides a rea-sonably accurate estimation of the average solidifi-cation front velocity between TC1 and TC2.

The local temperature gradient is less straightfor-ward to obtain, as it depends significantly on themethod used to calculate it. As explained in Sec-tion 2.1.3, it is calculated from the time-averagedslopes of TC1 and TC2 between t1 and t2.

Resulting solidification conditions between TC1and TC2 are summarized in Table 1. In the sim-ulations, we specifically focus on the two experi-ments performed with an insulating alumina plate,i.e. cases (a) and (c) in Fig. 3 and Table 1.

Case V (µm/s) G (K/mm) R (K/s)

(a) 0.16± 0.01 0.91± 0.29 0.15± 0.04(b) 0.23± 0.02 0.50± 0.01 0.12± 0.03(c) 0.19± 0.01 0.37± 0.12 0.07± 0.02(d) 0.28± 0.02 0.50± 0.20 0.14± 0.04

Table 1: Solidification velocity V , temperature gradient G,and cooling rate R between thermocouples TC1 and TC2:Al-1 wt%Cu alloy (a) with and (b) without alumina platebetween mold and chill plate; Al-4 wt%Cu alloy (c) with and(d) without alumina plate between mold and chill plate.

3.2. Microstructure characterization

Cross-section micrographs between TC1 andTC2 appear in Fig. 4 for the two alloys solidified us-ing the alumina plate between mold and chill plate.

Figure 5 shows the [100] pole figures from EBSDanalysis of the dendrite orientation with respect tothe cutting plane in 16×4 mm2 cross-section ar-eas. We assumed that dendrites grow along their[100] crystalline direction, since they are in a well-developed dendritic regime, far from a cellular-to-dendritic transition regime in which a devia-tion may exist between [100] and growth direction[92, 93].

For the Al-1 wt%Cu alloy, between TC3 and TC4the cutting plane was nearly normal to the [100]direction (Fig. 5b) but closer to the chill plate,e.g. between TC1 and TC2, the misorientationwas more notable (Fig. 5a). In the Al-4 wt%Cu al-loy (Fig. 4b), most cross-sections presented severalgrains of different orientations with no remarkablepreferential orientation (Fig. 5c).

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Figure 4: Micrographs of the cross-section of the (a) Al-1 wt% Cu and (b) Al-4 wt% Cu alloy solidified using thealumina plate between the mold and the chill plate at themidpoint between thermocouples 1 and 2. Orange squaresshow the regions compared to simulation results in Fig. 11.

We used the Euler angle Φ to estimate the de-viation between the apparent spacing in the cross-sections λm and the actual spacing normal to thegrowth direction λ ≈ cos(Φ)λm (Fig. 6). The an-gle Φ, negligible for most grains between TC3 andTC4 in the Al-1 wt%Cu alloy, reached up to 40◦

between TC1 and TC2 and in the the Al-4 wt%Cualloy, thus leading to λ up to 25% lower than λm.Hence, for all samples, we estimated λ ≈ 0.9λm asan average approximate angle correction.

The average measured and corrected PDAS areplotted as a function of the cooling rate R in Fig. 7(Al-1 wt%Cu) and Fig. 8 (Al-4 wt%Cu), togetherwith literature data for relevant composition ranges

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Figure 5: EBSD [100] pole figures for the Al-1 wt% Cu alloy(a) between TC1 and TC2 and (b) between TC3 and TC4 aswell as (c) for the Al-4 wt% Cu between TC3 and TC4. Thescatter in a and c shows the lack of overall alignment of the[100] dendrite growth direction with the observation plane,in contrast to panel b showing a higher density of grainsgrowing along the principal temperature gradient direction.

Figure 6: Schematic for the correction of the dendrite spac-ing from the dendrite orientation, where λm is the PDASmeasured on the cross-section and λ is the actual PDAS.

(see graph legend), and a dotted line illustrating aR−0.5 slope.

Even though PDAS were slightly lower when themold was in direct contact with the chill plate,this effect was relatively small because the twoachieved cooling rates remained of the same or-der (see Fig. 3), which was not sufficient to yielda significant change microstructural length scale.Therefore, in Figs 7 and 8, we did not distinguishbetween experiments with and without the insu-lating alumina plate, since the resulting scatter isnegligible compared to the overall cloud of points.

Table 2: Fitted tip radii and selection parameters from thePF simulations of the Al-1 wt%Cu alloy at V = 160 µm/sand G = 102 K/m.

λ (µm) ρ (µm) σ

336 8.94 0.0605387 8.93 0.0606438 8.83 0.0619490 9.09 0.0585505 9.06 0.0589521 9.02 0.0593

3.3. Simulations

The first quantitative results extracted from thePF simulation are the dendrite tip radii and thecorresponding tip selection parameter σ. Resultsfrom simulations of the Al-1 wt%Cu alloy at V =160 µm/s and G = 102 K/m for different spac-ings λ appear in Table 2. The average radius re-mains within ρ = 8.96 ± 0.13 µm, with less than4% difference between radii measured in the y andz directions. The resulting tip selection parame-ter, accounting for the tip temperature, i.e. for thetemperature-dependent capillarity length, is withinσ = 0.06 ± 0.002. This compares well with lin-ear solvability theory, which gives σ ≈ 0.06 for aone-sided model (see, e.g., Fig. 2 of Ref. [94] withα = 15ε4 = 0.18 and Eq. (4.3) therein with µ = 0).In the DNN simulations we used σ = 0.063, whichwas extracted from preliminary PF simulations per-formed before we obtained all PF results. However,we expect that such small difference had little in-fluence on any of our results and conclusions.

For both PF and DNN models, results of single-spacing simulations appear in Figs 9 (Al-1 wt%Cu)and 10 (Al-4 wt%Cu), showing (a) primary tip un-dercooling versus spacing within the stable PDASrange, as well as (b-c) typical primary dendriteswithin the stable range and their surrounding solutefield in a (010) plane. We also report the calculatedvalues of λmin, λmax, as well as the ratio λmax/λmin

for all simulations in Table 3 (Al-1 wt%Cu) and Ta-ble 4 (Al-4 wt%Cu).

Final microstructures in the spatially extendedsimulations are illustrated and compared to exper-imental micrographs in Fig. 11 for the Al-1 wt%Cu(a-c) and the Al-4 wt%Cu (d,e) alloys, superim-posed with a Delaunay triangulation similar to theone used to calculate average primary spacings. Av-erage spacings 〈λ〉 from simulations are reported inFigs 7 and 8 as thick crosses (6), together with

8

10 2 10 1 100 101

R (K/s)

102

103

PDAS

(m

)

Grange et al., 1995, Al-0.73 wt%CuEskin et al., 2005, Al-0.98 wt%Cu

Okamoto and Kishitake, 1975, Al-1.0 wt%CuClarke et al., 2017, Al-1.5 wt%Cu

Geying and Lixin, 1987, Al-2.0 wt%CuEskin et al., 2005, Al-2.12 wt%Cu

Young and Kirkwood, 1975, Al-2.4 wt%Cu

This Work

PF [ min, max]PF

DNN [ min, max]

DNN

Figure 7: Comparison of PDAS as a function of cooling ratefor the Al-1 wt%Cu alloy in our experiments (both coolingconditions), together with literature results, as well as PF(red) and DNN (blue) predictions of dendritic stability range(error bars) and average spacing in spatially extended den-dritic arrays (6). PF and DNN simulations are performed atthe same (G,V ) and hence R, but PF data points in the plotare slightly shifted to the right, for the sake of readability.

Table 3: Predicted spacing stability limits and ratioλmax/λmin from PF and DNN simulations of Al-1 wt%Cusolidification at V = 160 µm/s.

G (K/m) 102 103 104

λmin (µm) PF 161.5 107.8 46.2DNN 183.6 108.8 61.5

λmax (µm) PF 959.0 318.1 128.3DNN 940.7 320.2 143.5

λmax/λmin PF 5.94 2.95 2.78DNN 5.12 2.94 2.33

predicted stability ranges as vertical error bars.

In terms of computational performance (see briefdiscussion in the Supplementary Material), a repre-sentative simulation in the vicinity of λ ≈ λmin atG = 103 K/m for the Al-1 wt%Cu alloy, performedon the same hardware (one NVIDIA GeForcer

GTX 1080Ti and a single core of an Intelr i7-6850K) was performed about six times faster with

10 2 10 1 100 101

R (K/s)

102

103

PDAS

(m

)

Gündüz and Çard rl , 2002, Al-3.0 wt%CuSpinelli et al., 2004, Al-3.0 wt%CuEskin et al., 2005, Al-3.24 wt%Cu

Badon-Clerc and Durand, 1974, Al-4.0 wt%CuGrange et al., 1995, Al-4.1 wt%Cu

Eskin et al., 2005, Al-4.3 wt%CuYoung and Kirkwood, 1975, Al-4.4 wt%Cu

This Work

DNN [ min, max]DNN

Figure 8: Comparison of PDAS as a function of cooling ratefor the Al-4 wt%Cu alloy in our experiments (both coolingconditions), together with literature results, and DNN (blue)predictions of dendritic stability range (error bars) and av-erage spacing in a spatially extended dendritic array (6).

Table 4: Predicted spacing stability limits and ratioλmax/λmin from DNN simulations of the Al-4 wt%Cu al-loy at V = 200 µm/s.

G (K/m) 5× 102 5× 103 5× 104

λmin (µm) 134.8 77.2 43.5λmax (µm) 619.2 224.4 86.9λmax/λmin 4.59 2.88 1.99

DNN than PF. However, in this regime where bothPF and DNN simulations are achievable, DNN sim-ulations are far outside of their optimal perfor-mance, which can only be obtained when ρ� D/Vand PF simulations become unfeasible.

4. Discussion

4.1. Experiments

The thermocouples farthest from the chill plate,namely TC5, TC6, and TC7 provide an estimationof the vertical thermal gradient. The temperatures

9

b

c

100 μm

1.0

1.2

1.4

1.6

1.8

2.0wt%Cu

a

102 103

PDAS ∏ (µm)

0.06

0.07

0.08

0.09

0.10

Und

erco

olin

g¢

G = 102 K m°1: DNNPF

G = 103 K m°1: DNNPF

G = 104 K m°1: DNNPF

Figure 9: Simulation results for PDAS stability rangefor the Al-1 wt%Cu alloy at V = 160 µm/s: (a) Steady-state primary tip undercooling ∆ versus λ, showing only thestable spacings λmin < λ < λmax predicted by PF (fullsymbols) and DNN (open symbols) models for G = 102

(red), 103 (green), and 104 K/m (blue), and dendrite mor-phologies simulated with (b) PF and (c) DNN models forstable spacings λ ≈ 110, 180, 250, and 320 µm (left toright) for G = 103 K/m. Panels (b) and (c) appear atthe same scale, showing the solid-liquid interface and sim-ilar iso-concentration lines (wt%Cu) in a (010) plane. Inpanel (a), the leftmost and rightmost data points show thelowest (λmin) and highest (λmax) stable spacings identifiedfor each set of parameters, even though within the stabilityrange only a small subset of the simulations are shown assymbols for the sake of readability.

b

5.4wt%Cu

a

200 μm

5.25.04.84.64.44.24.0

102 103

PDAS ∏ (µm)

0.0425

0.0450

0.0475

0.0500

0.0525

0.0550

0.0575

Und

erco

olin

g¢

G = 5£ 102 K m°1

G = 5£ 103 K m°1

G = 5£ 104 K m°1

Figure 10: Simulation results for PDAS stability range forthe Al-4 wt%Cu alloy at V = 200 µm/s: (a) Steady-stateprimary tip undercooling ∆ versus PDAS, showing only thestable spacings λmin < λ < λmax predicted by the DNNsimulations for G = 5 × 102 (red), 5 × 103 (green), and 5 ×104 K/m (blue), and (b) view of the dendrite morphologiesfor stable spacings λ ≈ 137, 296, 454, and 603 µm (left toright) for G = 5×102 K/m, showing the solid-liquid interfaceand iso-concentration lines (wt%Cu) in a (010) plane. Inpanel (a), the leftmost and rightmost data points show thelowest (λmin) and highest (λmax) stable spacings identifiedfor each set of parameters, even though within the stabilityrange only a small subset of the simulations are shown assymbols for the sake of readability.

of TC6 and TC7 mostly overlap, except in the Al-4 wt%Cu case with alumina plate (Fig. 3a), whereTC7 is at a slightly higher temperature. This islikely due to a second solidification front movingfrom the opposite end of the mold and promotingearlier solidification of that region. Something sim-ilar occurs in the Al-4 wt%Cu alloy when the moldis in direct contact with the chill plate (Fig. 3d):TC5 to TC7 exhibit a faster initial cooling ratesthan TC4 until the temperatures converge, suggest-ing that solidification occurs around the same timein this region. This uncertainty in the temperaturedistribution and history makes it more challengingto establish a clear processing-microstructure rela-tionship in this central region of the mold. There-fore, we chose to focus primarily on the solidifica-tion behavior between TC1 and TC2, where the so-lidification conditions can be estimated with greaterconfidence.

Dendritic arrays have a more regular arrange-

10

b

250 μm

a

500 μm

c

100 μm

e

150 μm

d

500 μm

Al – 1wt%Cu Al – 4wt%Cu

Expe

rimen

tsDe

ndrit

ic N

eedl

e Ne

twor

kPh

ase-

Fiel

d

Figure 11: Experimental micrographs (a,d) and steady statemicrostructures observed in the (x−) direction simulated us-ing the DNN (b,e) and PF (c) methods for the Al-1 wt%Cu(a-c) and the Al-4 wt%Cu (d,e) alloy processed in the pres-ence of an insulating alumina plate between the mold andthe chill face, i.e. for V = 160 µm/s and G = 103 K/m forAl-1 wt%Cu (a-c) and V = 200 µm/s and G = 5 × 102 K/mfor Al-4 wt%Cu (d,e).

ment and a better defined microstructure in themicrographs of the central region of the mold (i.e.between TC2 and TC3, or TC3 and TC4) than nearthe chill plate (i.e. between TC1 and TC2) in theAl-1 wt%Cu alloy (see Fig. 5). This is likely a signof transient thermal conditions at the beginning ofsolidification. Moreover, the insulation of the moldis poorer near the chill face, leading to possible heatfluxes in different directions, such that the dendritesgrowth direction may deviate from the main ther-mal gradient direction. Further away from the chillplate, the solidification front is stabilized and thesedisturbances vanish for the most part. In spite ofthese transient conditions, microstructures betweenTC1 and TC2 are sufficiently defined to extract areasonable measure of the PDAS.

The expected power law dependence of the PDASwith the cooling rate R appears in Figs 7 and 8,as the data follows the illustrated R−0.5 slope sug-gested in earlier studies [36–47, 95]. While ourexperiments did not permit to scan an extensiverange of cooling rates, they still suggest a decreasein PDAS for higher cooling rate, consistently withthis λ ∼ R−0.5 power law.

For the Al-4 wt%Cu alloy, our measured PDAScoincide well with the results of [40, 42, 95] for sim-ilar cooling rates. However, our measurements forboth alloys appear on the higher end of the mea-sured spacing distribution from the literature forcomparable compositions. They also appear on thehigh end of the calculated stable ranges. This maybe attributed to several factors.

First, the representation of λ as a sole function ofR is a simplification. For a given alloy, the averagePDAS is a function of both G and R, such that, fora given cooling rate R = G × V , different sets ofpoints might represent substantially different con-ditions and even different growth regimes. A betterrepresentation would be a three-dimensional graphwith axes λ, G, and V (or even 4D including c∞).

Second, experimental setups and configurationscompared here differ substantially from one an-other. Most previous data comes from thin-sampleor Bridgman directional solidification experiments.Even among these, whether the solidification is hor-izontal, vertical upwards, or vertical downwards,has important consequences, particularly in termsof thermo-solutal convection [96–102]. For exam-ple, in the experiments from Ref. [68] (full orangecircles in Fig. 7), performed vertically upwards, itwas shown that, in spite of the low sample thickness(200 µm), the extent of convection was sufficientfor these experiments to occur within a cellular-to-dendritic transition regime, explaining the inversetrend with ∂λ/∂R > 0 in Fig. 7 [68], in place of theexpected λ ∼ R−0.5 in the fully dendritic regime.

Third, we relied on limited local thermocouplemeasurements, which arguably leaves room for er-ror in the estimation of the local (G,V ) conditionsat exact times and locations within the mold. Amore accurate estimation of the local temperatureprofile in the sample could be achieved, for in-stance extending our multiscale modeling approachto thermal and solutal transport simulations at themacroscopic scale, which is the focus of an ongoingfollow-up study.

Finally, we based our estimation of the PDASon two-dimensional cross-sections. Even though we

11

attempted to correct for the misorientation of thedendrite growth direction, we expect that underly-ing assumptions also lead to some amount of errorin the PDAS measurements. Such issues could betackled in further studies using three-dimensionalvisualization methods, such as X-ray tomography[103–107], which would provide an unambiguousrepresentation of dendritic arrays and spacings.

4.2. Simulations

For the Al-1 wt%Cu alloy, the calculated valuesfor λmin and λmax exhibit a striking agreement be-tween PF and DNN methods, in spite of the lowerundercooling predicted by DNN compared to PF(Fig. 9a), which was already observed and reportedin an ongoing benchmark study [108]. Some dis-crepancy between DNN and PF appears at higherundercooling (i.e. for G = 104 K/m in Fig. 9a),which we attribute to the fact that a higher G pro-motes a transition toward an intermediate regimebetween cellular and dendritic. Consequently, theagreement is best at low G, when microstructuresare well-developed dendrites made of thin hierarchi-cal branches, for which the DNN model was specif-ically designed. For both compositions and bothsimulation methods, the dependence of both λmin

and λmax fall close to the phenomenological R−0.5

power law (dotted lines in Figs 7 and 8).

Hence, the ratio λmax/λmin (Tables 3 and 4) alsopresents a striking agreement between the two sim-ulation methods. This ratio is as high as 5 or 6 forthe lowest values of G, and decreases consistentlywith increasing G, as microstructures approach anintermediate cellular-dendritic regime. The factthat λmax/λmin is much higher than 2 is consis-tent with recent studies [68, 90, 91, 109] and incontrast with earlier theories, which suggested thatλmax/λmin ≈ 2 [23, 24], but mostly by lack of aproper simulation method to study branching in-stability at λ ≈ λmax. The decrease in λmax/λmin

withG and the fact that it may tend toward 2 as thegradient increases are new observations that war-rant further investigation.

Overall, this comparison between simulationsand experiments, in conditions relevant to indus-trial casting, provides a compelling demonstrationthat physics-based models can be coupled acrossscales to yield quantitative predictions of key mi-crostructural length scales — here specifically il-lustrated for the PDAS. However, our simulations,while in good agreement with literature data, seem

to underestimate the spacings measured in our ex-periments. This is due to error sources in bothsimulations and experiments, such that some limi-tations in the current calculations and assumptionsdeserve to be discussed, as they point toward direc-tions for improvement and further research.

First, it was previously shown that a microstruc-tural steady state, with a fixed PDAS, may requirea substantially longer time than that required toreach a steady undercooling and growth velocity ofthe solidification front [68]. Hence, even thoughmicrostructures may appear at steady state, a slowinteraction and elimination process might still beoccurring, possibly leading to an increase of aver-age spacing at longer time scale.

Then, in the single-spacing simulations, we madethe assumption of a quincux (fcc-like) spatial orderof the dendritic arrays (Fig. 2). As such, these sim-ulations are only representative of λmin and λmax

in a perfectly ordered array with this specific ar-rangement. This ordering can be identified in termsof nearest neighbors as an irregular hexagonal pat-tern (dotted line in Fig. 2), with two longer edgesalong the directions corresponding to side-branchesgrowth. It was shown to be predominant in di-rectionally solidified single crystals (see, e.g., thehigh density of such irregular hexagons in the bot-tom left region of Fig. 2b in Ref. [110]) — in con-trast to predominantly regular hexagonal arrays incellular growth due to the axisymmetry of cells[33, 111]. While we chose to leave this aspect outof the scope of the current study, it would be inter-esting, among future directions, to investigate theinfluence of the spatial order on the resulting spac-ing stability range, as well as the possible transitionfrom regular to irregular hexagonal arrays with theappearance of side-branches between cellular anddendritic growth regimes.

In both experiments and spatially extended sim-ulations, spatial order of the array is never perfect.In the experiments, array disorder is primarily pro-moted by the inhomogeneous processing conditions,with gradients in G, V , and c∞ due to a large ex-tent to convection, and the fact that microstruc-tures are polycrystalline, thus leading to sustainedelimination and branching events [90, 112].

Due to this disorder, a more advanced algorithmthan our Delaunay triangulation could have led toa more accurate PDAS estimation [110, 113–115].However, we expect that such methods would haveonly filtered a handful of nearest neighbor outliers,such that the resulting correction would have been

12

minimal once reported within the nearly two or-ders of magnitude of PDAS variation discussed here(Figs 7-8).

In all simulations, we made the assumptions thatG and V are constant, that heat transport is in-finitely fast compared to solute transport, and thatsolute transport occurs primarily by diffusion in theliquid phase. While these are all reasonable as-sumptions for the solidification of a metallic systemconsidered at a scale lower than the characteristichydrodynamic length, we expect that such assump-tions introduce some amount of discrepancy by de-viating from exact experimental conditions.

Finally, one intrinsic limitation of the DNNmodel is that primary dendrite trunks cannot driftlaterally, such that spacings can only be adjustedthrough elimination and branching events. Thisleads to a distribution of spacing less homoge-neous than in both experiments and PF simulations(Fig. 11), as already noted for thin-sample solidifi-cation of Al-Si alloys [109]. The DNN model mightbe adapted to allow such a drifting, but this wouldrequire (1) a new mathematical formulation of theFIF that would incorporate the directionality of theincoming solute flux, (2) establishing laws for den-drite drifting velocities in inhomogeneous arrays,and (3) a notably more complex implementation.

In spite of these limitations, the present compari-son between experiments and simulations highlightsa promising pathway toward quantitative predic-tions of microstructural length scales (e.g. PDAS)in casting processes. The striking agreement be-tween PF and DNN predictions is a strong indi-cation that discrepancies between simulated andexperimental data stem from the fundamental as-sumptions behind the models rather than from theupscaling from PF to DNN simulations.

5. Summary and perspectives

In this article, we demonstrated that primarydendritic spacings in metallic alloys cast in in-dustrially relevant conditions can be calculated bydirectly combining physics-based models — withbarely any adjustable parameters. On the onehand, we performed casting experiments of Al-Cu alloys and related the cooling conditions mea-sured by thermocouples to the PDAS measuredfrom cross-sections micrographs. On the otherhand, we calculated PDAS stability ranges andsimulated entire dendritic arrays using two differ-ent methods, namely phase-field (PF) and den-

dritic needle network (DNN) models. For the al-loy with lower copper content (Al-1 wt%Cu), bothtypes of simulations are feasible, and they lead tovery similar results. For the more concentrated al-loy (Al-4 wt%Cu), the wide separation of scale be-tween dendrite tip radius ρ and diffusion lengthlD makes PF simulations prohibitively challenging,while DNN simulations still yield a good agreementwith experiments.

We compared our results to the extensive bodyof literature on PDAS selection in Al-Cu alloys (seeFig. I in the Supplementary Material), and foundan overall good agreement between our results andpublished data. It is, to our knowledge, the firsttime that this broad data set was compiled anddirectly compared to simulation predictions. Theagreement between simulations and experimentaldata provides a conclusive validation of the mod-els’ capability to reliably predict primary spacings.Moreover, in spite of a slight underestimation ofundercooling by the DNN method as compared toPF (already identified elsewhere [108]), the excel-lent agreement in terms of stable spacing ranges fordifferent sets of parameters provides a further vali-dation of the models. In summary, we have shownthat quantitative PF simulations, computationallylimited to dilute alloys, could be readily extended tomore concentrated alloys using the DNN approach.This opens the way to further direct comparisonsof experiments and simulations at experimentallyrelevant length and time scales.

Perspectives from this work are manyfold. Froma fundamental standpoint, the relationship betweenthe width of the PDAS stability range (i.e. the ra-tio λmax/λmin) and the temperature gradient re-mains to be explored in further details. From anICME perspective, ongoing works include the ex-tension of this multiscale modeling strategy towardboth the macroscopic scale — e.g. coupling tothermomechanical simulations of the process withpossible integration of convection into DNN sim-ulations [84, 85] — and toward the nanoscale —e.g. calculating phase diagrams from first principles[116, 117] and solid-liquid interface properties frommolecular dynamics [118, 119]. Such computationalframeworks could also be complemented by similarmulti-scale strategies for later processing steps, e.g.including the formation of secondary phases suchas precipitates [120, 121].

13

Acknowledgements

This research was funded by the European Re-search Council Advanced Grant VIRMETAL underthe European Union’s Horizon 2020 research andinnovation programme (Grant Agreement 669141).D.T. acknowledges support from the EuropeanUnion’s Horizon 2020 research and innovation pro-gramme through a Marie Sk lodowska-Curie Indi-vidual Fellowship (Grant Agreement 842795). Ad-ditional support from the HexaGB project of theSpanish Ministry of Science (reference RTI2018-098245) is also gratefully acknowledged. B.B. ac-knowledges support by the Spanish Ministry of Ed-ucation through the Fellowship FPU15/00403 andD.T. and J.LL. also gratefully acknowledge supportby the NVIDIA Corporation with the donation oftwo Titan Xp GPUs. O.M.W. and M.S.T. wouldlike to acknowledge Purdue University for support-ing this work through start-up funds. The valuablehelp from undergraduate students Jesus MartınezJusto and Luis Montes Mota (Polytechnic Univer-sity of Madrid) for the metallography sample prepa-ration is also gratefully acknowledged.

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