Phase-field modeling of θ′ precipitation kinetics in 319 aluminum alloys

Computational Materials Science 151 (2018) 84–94

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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Phase-field modeling of θ′ precipitation kinetics in 319 aluminum alloys a,⁎

b

b

Yanzhou Ji , Bita Ghaffari , Mei Li , Long-Qing Chen a b

T

a

Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA Ford Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48124, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Phase-field model θ′ (Al2Cu) Precipitation kinetics Isothermal aging Al-Cu-based alloys

Understanding the morphological evolution of precipitates is critical for evaluating their hardening effects and therefore improving the yield strength of an alloy during aging. Here we present a three-dimensional phase-field model for capturing both the nucleation and the growth kinetics of the precipitates and apply it to modeling θ′ precipitates in 319 aluminum alloys. The model incorporates the relevant thermodynamic data, diffusion coefficients, and the anisotropic misfit strain from literature, together with the anisotropic interfacial energy from first-principles calculations. The modified classical nucleation theory is implemented to capture the nucleation kinetics. The model parameters are optimized by comparing the simulation results to the experimentally measured peak number density, average diameters, average thicknesses and volume fractions of precipitates during isothermal aging at 463 K (190 °C), 503 K (230 °C) and 533 K (260 °C). Further model improvements in terms of prediction accuracy of the precipitate kinetics in 319 alloys and the remaining challenges are discussed.

1. Introduction

this purpose, meso-scale simulations could be a powerful tool, and several of such numerical models have been developed for θ′ precipitates [7,12–15]. Especially, the physics-based phase-field method [16], which introduces the diffuse-interface concept to avoid the explicit tracking of interfaces, stands out with the significant potential to accurately predict the precipitate morphology and kinetics. For example, Vaithyanathan et al. [7] combined atomic-scale calculations and phase-field simulations to predict the θ′ morphology, using atomistic calculations to provide the key phase-field input parameters including thermodynamic free energies, lattice parameters, as well as the anisotropic elastic constants and interfacial energies. Hu et al. [13] introduced the Kim-Kim-Suzuki (KKS) model [17] for the more accurate prediction of θ′ precipitation kinetics at larger size scales. More recently, Liu et al. [18] considered the more detailed lattice deformation pathway from the lattice of α-Al to that of θ′, and investigated the effect of dislocations on the nucleation and growth behavior of θ′ precipitates. Kim et al. [19] improved the description of interfacial energy anisotropy by introducing a 4th rank gradient tensor and using first-principles calculations to evaluate its coefficients. However, these existing investigations either exclusively focused on the growth behavior of θ′ particles without quantitative evaluation of nucleation or lacked comprehensive experimental validations in terms of θ′ volume fraction, size and morphology, all of which are necessary prerequisites for a reliable and predictive computation model of θ′ precipitate kinetics. In this study, we aim to develop a three-dimensional (3D) phasefield framework for morphology and kinetics predictions of θ′

Precipitation hardening is one of the major strengthening mechanisms in Al-Cu-based alloys. It leads to high strength-to-weight ratio and therefore enables wide application of these alloys in automotive industries for improving fuel efficiency. The metastable θ′ (Al2Cu) precipitates, which usually appear at the peak aging point, are recognized as the key strengthening precipitate in Al-Cu alloys [1]. Since its discovery, both experimental and theoretical investigations have been devoted to the crystal structures [2], thermodynamics [3], interface properties [4–7], precipitate morphologies [8,9], and growth kinetics [6,10,11] of θ′ precipitates. Based on these investigations, it is well established that θ′ precipitates, due to the anisotropic interfacial energies and lattice misfits with the Al matrix, typically form a plate-like morphology, which can effectively impede the motion of dislocations in the Al matrix and therefore improve the yield strength of the alloy. These studies have established a solid knowledge-basis for designing the practical heat treatment conditions of commercial Al-Cu-based, precipitate-hardened alloys. To optimize the heat treatment strategy and accelerate the alloy design, predicting precipitate hardening effects via theoretical analysis or computer simulations is a promising complement to trial-and-errorbased experiments, reducing the alloy development time and cost. The ability to determine the precipitate hardening effects significantly relies on our ability to predict the evolution of precipitate morphology, size, volume fraction and spatial distribution during heat treatment [1]. For ⁎

Corresponding author. E-mail address: [email protected] (Y. Ji).

https://doi.org/10.1016/j.commatsci.2018.04.051 Received 19 January 2018; Received in revised form 19 April 2018; Accepted 20 April 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.

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Y. Ji et al.

interface thickness 2λ (θi ) :

precipitates in a commercial 319 Al-alloy (Al-3.5wt.%Cu-6.0wt.%Si) under different isothermal aging temperatures, based on our preliminary work [1]. To obtain more quantitative simulation results, we incorporate nucleation and more accurate diffusion-controlled kinetics to the phase-field models developed by Vaithyanathan et al. [7,20] and Hu et al. [13]. All input parameters are taken from validated experimental or theoretical investigations. The simulation results, including the diameter, thickness and volume fraction of θ′ precipitates are compared with those measured from experiments, so as to best estimate phase-field parameters within their range, with the ultimate goal to predict the precipitate kinetics in other relevant alloy systems and under other aging conditions, and using the predicted precipitate kinetics for yield strength predictions.

κ (θi )2 =

3 γ (θi )·2λ (θi ) 2.2

(4a)

w = 13.2

γ (θi ) 2λ (θi )

(4b)

To describe the plate-shaped θ′ morphology, the anisotropic interfacial energy γ (θi ) is considered to be angle-dependent [13]:

⎧1 + ⎪

γ (θi ) =

2. Phase-field model 2.1. Free energy formulation

⎝

(1)

⎠

i=1

The first term flocal (XCu ,{ηi },T ) describes the bulk free energy density:

flocal (XCu ,{ηi },T ) =

1 α α ′ θ′ (f (XCu ,T )·(1−h ({ηi })) + f θ (XCu ,T )·h ({ηi })) Vm + w·g ({ηi }) θ′

(2)

θ′ (XCu ,T )

α ,T ) and f are molar Gibbs free energies of α-Al where f α (XCu and θ′, respectively, taken from existing thermodynamic optimizations 3 [3,13], h ({ηi }) = ∑i = 1 (3ηi2−2ηi3) is an interpolation function, g ({ηi }) is a double-well type function, w is the double-well height, and Vm is the molar volume. Similar to [13], the KKS model is employed to remove the extra interface potentials, which requires the solution to the following equations: ′

θ α XCu = (1−h ({ηi }))·XCu + h ({ηi })·XCu

eel =

sinϕ0

π

π

sinθi, − 2 ⩽ θi ⩽ − 2 + ϕ0 π

π

1+

α cosϕ α π − sinϕ 0 sinθi, 2 −ϕ0 sinϕ0 0 ∂ηi / ∂x

(∂ηi / ∂x )2 + (∂ηi / ∂y )2 + (∂ηi / ∂z )2

⩽ θi ⩽ π

π 2

)− 2 , ϕ0 =

(5) π 10000

is a reg-

1 Cijkl (εij−εij0)(εkl−εkl0 ) 2

(6)

where Cijkl are elastic constants, εij0 is the stress-free transformation strain tensor between the Al matrix and the θ′ phase, taken from [20]. The total strain εij is solved from the stress equilibrium ∇ ·σ = 0 at each simulation time step. Here, we assume the system is elastically homogeneous, i.e., Cijkl are constants rather than a function of positions, and stress equilibrium is reached much faster than the diffusional phase transformation [1].

(3a)

′

∂f θ ∂f α = α θ′ ∂XCu ∂XCu

α cosϕ0

ularization angle to avoid numerical issues due to the cusps in the γγ plot, and γcoh = 1 +0 α = 0.24 J/m2 and γsemi = γ0 = 0.49 J/m2 are the interfacial energies of the coherent and semi-coherent α-Al/θ′ interfaces, respectively [1]. Density functional calculations, using the Vienna ab initio simulation package, are performed to calculate γcoh and γsemi, with a complete relaxation of the cell volume, cell vectors and cellinternal atomic positions. The interface supercells and calculation parameters are chosen similarly to previous calculations [6], but with larger supercells and a basis-set cutoff energy of 300 eV. The formation energy for each supercell consists of contributions from the bulk constituents, Al and θ′, the interfacial energy, and the coherency strain energy. To separate these contributions for each interface supercell, further calculations are conducted. One or the other of the Al and θ′ supercell constituents is replaced with vacuum, allowing comparison of the formation energies of epitaxially-strained and unstrained supercells, allowing the coherency strain and interfacial energies be estimated. The third term in Eq. (1) describes the elastic strain energy contribution due to lattice misfit, and is evaluated by Khachaturyan’s microelasticity theory [23]:

3

∫V ⎛⎜flocal (XCu,{ηi},T ) + 12 ∑ κ (θi)2|∇ηi |2 + eel ⎞⎟ dV .

+

1 + α cosθi, − 2 + ϕ0 ⩽ θi ⩽ 2 −ϕ0

where θi = arccos(

The current phase-field model is extended from Hu et al. [13]. Although the major alloying elements in 319 alloys include both Cu and Si, Cu is the most responsible for θ′ precipitation. Therefore, we simplify the alloy into an Al-1.5at.%Cu binary system [1]. To describe the (α-Al + θ′-Al2Cu) two-phase mixture, we use the conserved order parameter XCu to account for Cu composition and the non-conserved order parameter sets {ηi} (i = 1, 2, 3) for the three energetically equivalent and symmetrically related variants of θ′. The total free energy of the (α-Al + θ′-Al2Cu) two-phase system can be expressed as [1]:

F=

γ0 1+α⎨ ⎪ ⎩

α sinϕ0

(3b) α f α (XCu ,T )

θ′

2.2. Governing equations

θ′ (XCu ,T )

and f To improve the computation efficiency, are fitted into parabolic functions by Taylor expansions at the equilibrium compositions up to the 2nd order, so that Eqs. (3a) and (3b) can be solved analytically [21]. It should be noted that in the thermodynamic database, θ′ is treated as a stoichiometric compound, with the second derivative values unavailable. In reality, similar to other metastable precipitates [22], θ′ may have a rather narrow solubility range and a steep curvature (i.e., very large second derivative values with respect to composition) in the free energy function. Since this second derivative value would not affect the precipitate kinetics (as discussed in [13]), we assume the two phases have the same second derivative values at equilibrium compositions. The second term in Eq. (1) describes the part of interfacial energies due to the inhomogeneous distribution of order parameters. The gradient coefficients κ (θi ) and the double-well height w are typically evaluated based on one-dimensional analytical solution to the phasefield equations, in terms of both the interfacial energy γ (θi ) , and

The governing equations of the evolution of θ′ precipitates include the modified Cahn-Hilliard equation (for XCu) and the Allen-Cahn equation (for {ηi}) [1]:

∂XCu DCu δF ⎞ = ∇ ·⎛⎜ 2 ∇ ⎟ 2 ∂t δX / ∂ f ∂ X Cu ⎠ Cu ⎝ local

(7a)

∂ηi δF = −L (θi ) δηi ∂t

(7b)

where DCu is the impurity diffusion coefficient of Cu in Al taken from [24], L (θi ) is the angle-dependent kinetic coefficient for the anisotropic precipitate interface mobility; L (θi ) is assumed to follow the same angle-dependence as γ (θi ) , but with a different anisotropy factor β (c.f. α for γ (θi ) ) [1,13]: 85

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Y. Ji et al.

⎧1 + ⎪

L0 L (θi ) = 1+β⎨ ⎪ ⎩

β sinϕ0

+

β cosϕ0 sinϕ0

π

way to model heterogeneous nucleation is to explicitly evaluate the effect of possible heterogeneous nucleation sites such as dislocations, grain boundaries and pre-existing precipitates on nucleation driving forces and barriers. However, this treatment requires explicit information about the type and distribution of these heterogeneous nucleation sites, which is technically difficult to obtain.

π

sinθi,− 2 ⩽ θi ⩽ − 2 + ϕ0 π

π

1 + β cosθi,− 2 + ϕ0 ⩽ θi ⩽ 2 −ϕ0 1+

β cosϕ0

β − sinϕ sinϕ0 0

π

sinθi, 2 −ϕ0 ⩽ θi ⩽

π 2

(8)

The kinetic coefficient for the semi-coherent interface, L0, is estimated using the thin-interface analysis [13,25]:

L0 =

γ0 κ 02

3. Results

1 1 M0

κ 0·ζ DCu 2w

+

Here, M0 is the interface mobility coefficient and ζ = ′ f XθCu XCu

θ′e (XCu )·

dη , η (1 − η)

where

1 α f Vm XCu XCu

To calibrate and validate the phase-field simulation results, the kinetic characteristics of θ′ precipitates, i.e., the precipitate number density, volume fraction, mean diameter and thickness, are needed as a function of aging time. The precipitate statistics, measured from the transmission electron microscopy (TEM) images of a cast 319 alloy in previous work [29], are utilized. The alloy compositions, as well as procedures of casting, heat treatment and TEM characterizations, are detailed in [29]. Based on the experimental details in the foil preparation, a foil thickness of 100 nm is assumed, allowing the 3D precipitate number densities to be estimated. The precipitate volume fractions are then calculated from the precipitate diameter, thickness and number density data.

αe (XCu )·

h (η)(1 − h (η)) θ′e 2 1 αe XCu −XCu θ′ 0 ( 1 − h ( η ) )·f α αe θ′e XCu XCu ( XCu ) +h ( η )·f XCu XCu ( XCu ) θ′e αe XCu and XCu are the equilibrium Cu compositions in the

(

3.1. Experimental setup

(9)

) ∫

two

phases at a given temperature, and κ0 is the gradient coefficient for the semicoherent interface. Since the lengthening of θ′ is a diffusion-controlled process, M0 → ∞ is assumed; in addition, as explained above, the second deri′ vatives of the molar free energies of the phases, f XαCu XCu and f XθCu XCu , are assumed to be equal. On the other hand, the thickening kinetics is much slower than for lengthening, and diffusion is no longer the controlling factor. Therefore, rather than using Eq. (9), a large anisotropic factor β is used in Eq. (8) to estimate the interface kinetic coefficient for the coherent interface. Eqs. (7a) and (7b) are solved using the semi-implicit Fourier spectral method [26,27].

3.2. Parameterization Accurate quantitative evaluations of the phase-field modeling parameters are important prerequisites for accurate quantitative predictions of precipitation behaviors. In this section, we discuss in detail the parameterization needed before performing phase-field simulations. Especially, we attempt to determine the values of the model parameters in a physics-based manner. For the nucleation model, the parameter B is key to the nucleation kinetics, since it determines the total number of nuclei in the simulation volume and the rate of introducing nuclei into the simulation volume. A larger B value will lead to a slower nucleation rate and lower total nucleation number density, but the average precipitate size can be larger (see Fig. S2 in supplementary information for details). In addition, since B is in the exponential term, even a slight variation can lead to significant changes in nucleation rate and number densities. The θ′ number densities at different aging times and different aging temperatures are measured from experimental TEM images, as shown in Fig. 1. Because of the relatively large error bars, it is not possible to discern a robust trend vs. time that is physically meaningful and consistent for the three temperatures. Therefore, a uniform number density at each temperature is the most reasonable estimate, allowing the

2.3. Nucleation model A modified classical nucleation theory is implemented to predict the nucleation kinetics of θ′: ∗ ΔG het ⎞ j = ZN0 β ∗exp ⎛− k T B ⎠ ⎝ ⎜

⎟

(10)

where j is the nucleation rate, Z is the Zeldovich factor, N0 is the number of atoms per unit volume, β∗ is the atomic attachment rate, ∗ is the energy barrier for heterogeneous nucleation, T is temΔG het perature (in K) and kB is the Boltzmann constant [28]. In particular, we ∗ ∗ ∗ take ΔG het where ΔG hom is the nucleation barrier for the = B·ΔG hom homogeneous nucleation of a plate due to anisotropic interfacial energy; B is a phenomenological parameter representing the degree of heterogeneous nucleation [1]. The values of B for different aging temperatures are estimated based on experimentally measured precipitate number densities, which will be discussed in detail in Section 3.2. The parameters in Eq. (10) are evaluated for a plate-shaped θ′ nucleus, as shown in Table 1. Note that a more physical and accurate Table 1 Evaluation of parameters in the nucleation model [28]. Parameters

Explanations

Expressions

Z

Zeldovich factor

N0

Number of atoms per unit volume

N0 = NA/ Vm

β*

Atomic attachment rate

β∗ =

2πr ∗ (r ∗ + h∗) DCu 4 ( X αe − X θ′e )2 aAl Cu Cu

r*, h*

Critical nucleus radius (r*) and thickness (h*)

r∗ =

−2γsemi ∗ ,h ΔgV

∗ ΔG hom

Homogeneous nucleation barrier

ΔgV

Nucleation driving force

Z=

3 Δg 2 aAl V 16π 6kB T γsemi γcoh

∗ ΔGhom =

=

2 γ 8πγsemi coh ΔgV2

ΔgV (XCu ) =

1 ⎛ θ′ θ′e ⎛ α f (XCu )−⎜f (XCu ) Vm ⎜

⎝

(aAl = 4A˚ is the lattice parameter of α-Al solid solution).

86

−4γcoh ΔgV

⎝

+

⎞⎞ df α θ′e (XCu −XCu )⎟ ⎟ α dXCu XCu ⎠⎠

Computational Materials Science 151 (2018) 84–94

Y. Ji et al.

Fig. 1. The experimentally measured number density values at different aging temperatures: (a) 463 K, (b) 503 K, and (c) 533 K. The data points in the figure show the average number density at each aging time. The number beside the error bar indicates the number of measured TEM foils for the data point. Adapted from [1]

Table 2 Mean number density and fitted B values. Mean number density (/m3)

Temperature

21

463 K (190 °C) 503 K (230 °C) 533 K (260 °C)

2.04 × 10 1.21 × 1021 7.07 × 1020

Table 3 Parameters for phase-field simulations. B

Parameters

Explanations

Values

0.00644 0.00456 0.0033

Vm DCu

Molar volume Cu impurity diffusivity in Al Interfacial energies

1.06 × 10−5 m3/mol 8.88 × 10−5exp(−133900/R/T) m2/s [24] γsemi = 0.49 J/m2 γcoh = 0.24 J/m2 2 nm

γsemi, γcoh (2λ)semi

weighted average of the values at each temperature to be the experimental peak number density. The value of B is then obtained by phasefield simulations to match these experimental number density values. The average number densities at different aging temperatures, together with the corresponding fitted B values, are listed in Table 2. B is further fitted to a linear function [1]: B (T ) = 0.2724−4.5 × 10−5T (T in K). The high precipitation number densities and B values at lower temperatures are consistent with the expected nucleation behavior, namely larger nucleation driving force, lower nucleation barrier and higher heterogeneity (e.g., more defects such as dislocations) of nucleation behavior at lower temperatures [1]. With B values available, 3D phase-field simulations are performed at the three aging temperatures, 463 K (190 °C), 503 K (230 °C) and 533 K (260 °C), in a 160Δx × 160Δx × 160Δx (Δx = 1 nm) simulation volume. θ′ nuclei with the critical diameter and thickness evaluated from Table 1 are sequentially put into the simulation volume following Eq. (10), and the total numbers of nuclei put into the system are 9, 5, and 3 for the three aging temperatures, respectively. Every time a nucleus is introduced, we randomly select its spatial center and variant number. Due to the relatively high diameter-to-thickness ratio of the observed θ′ precipitates, a large anisotropy factor β = 1000 is used for L(θi) for all the three temperatures [1]. This high anisotropy in growth kinetics can be attributed to the different atomic-scale growth mechanisms for lengthening and thickening of θ′ [6]. Table 3 summarizes the model parameters used for phase-field simulations. Note we use a larger value for the diffusion coefficient DCu, which is twice the original value in [24]. The reason of this treatment will be given in detail in Section 4.3.2. In addition, we use the elastic constants of pure Al, which are taken from [30]. The modeling parameters are non-dimensionalized in the phasefield simulations using C44, DCu and l0 (l0 = Δx = 1 nm): κ2

C11, C12, C44

Interface thickness for the semi-coherent interface Elastic constants

εij0

Misfit strain tensor

αe XCu

Equilibrium Cu composition in Almatrix (calculated from [13]) Equilibrium Cu composition in θ′ 2nd derivatives of the Gibbs free energy at equilibrium Cu compositions (calculated from [13]) Interface kinetic coefficient for the semi-coherent interface (from Eq. (9)) Anisotropy factor for interfacial energy (α) and interface mobility (β)

θ′e XCu

f Xα

Cu XCu

L0

α, β

αe (XCu )

C11 = 108 GPa, C12 = 61 GPa, C44 = 29 GPa [30] ⎛ 0.00746 ⎞ 0.00746 ⎜ ⎟·h ({ηi }) [20] − 0.051⎠ ⎝ 0.00173 at T = 463 K 0.00314 at T = 503 K 0.00447 at T = 533 K 1/3 2.24 × 106 J/mol at T = 463 K 1.37 × 106 J/mol at T = 503 K 9.9 × 105 J/mol at T = 533 K

3.87 × 10−12 m3/(Js) at T = 463 K 1.01 × 10−10 m3/(Js) at T = 503 K 8.53 × 10−10 m3/(Js) at T = 533 K α = 1.04, β = 1000

3.3. Phase-field simulations: precipitate morphology and kinetics Fig. 2 illustrates the simulated θ′ precipitate morphology at different aging temperatures and times, together with TEM images at the same temperature and aging time. As discussed in existing papers on phasefield simulations, the considerations of anisotropies in interfacial energy, elastic strain energy and interface mobility are critical for an accurate prediction of the θ′ precipitate morphology and kinetics. With all these anisotropies considered and quantified, the plate- or diskshaped θ′ precipitates are well reconstructed using phase-field simulations. The average precipitate diameter, thickness and volume fraction are further evaluated from the phase-field simulation results and quantitatively compared with those measured from the TEM images of the 319 alloy. Figs. 3–5 show the best agreement between simulation and experiments, using the above-mentioned model parameters. The error

L·C44·l 2

∗ f local = flocal / C44 , (κ 2)∗ = C / l 02 , L∗ = D 0 , Δx ∗ = Δx / l 0 . Notably, to be 44 Cu directly comparable with the experimentally measured precipitation kinetics, the discretization time step size in the simulation (Δt ∗) is converted to real time (Δt ) through Δt = (Δx )2 / DCu ·Δt ∗ [1]. For each simulation case, we performed five simulations with different random seeds and averaged the results to reduce the statistical errors associated with the relatively small number of precipitates in each simulation volume.

87

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Fig. 2. Simulated θ′ morphologies and comparison with TEM observations. Phase-field simulations are performed in a 160 × 160 × 160 nm3 volume and are shown in (a) 463 K, after 10 h, (c) 503 K, after 3 h, and (e) 533 K, after 0.5 h; the corresponding TEM images with [0 0 1] zone axis at the same aging temperatures and times are shown in (b), (d) and (f), respectively. The different colors represent different variants of θ′. The white dashed regions in TEM images have the same size as the simulation volume. The x-, y-, and z-axis represent [1 0 0], [0 1 0] and [0 0 1] directions, respectively.

4. Discussions on model improvements

bars in the simulated average diameter and thickness are calculated from their standard deviations. The error bars in the simulated volume fraction are calculated according to the standard deviations in both the diameter and the thickness. The simulated precipitate diameters and volume fractions agree well with the experimental measurements at both 503 K (230 °C) and 533 K (260 °C); while for 463 K (190 °C), both predictions are higher than the experimental values. The precipitate thickness predictions also need further improvement, since the simulated thicknesses of θ′ stay almost constant for all aging temperatures. In addition, the phase-field simulations underestimate the precipitate kinetics at early aging stages. These deviations prompt our efforts for model improvements.

Based on the comparisons in Figs. 3–5, the deviations mainly lie in three aspects: (I) volume fraction; (II) thickness; and (III) early-stage kinetics. In this section, we discuss the possible sources for the deviations and possible strategies to improve the simulation results. 4.1. Precipitate volume fraction With the thermodynamic databases for Al-Cu and Al-Cu-Si systems available, the equilibrium volume fraction of θ′ at a given temperature can be estimated by the lever rule. The lever rule prediction represents 88

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Y. Ji et al.

Fig. 3. Simulated θ′ precipitation kinetics and comparison with experimental measurements at 463 K (190 °C). (a) Average diameter. (b) Average thickness. (c) Volume fraction.

an upper limit for the volume fraction of θ′ in the system since it neglects the interface and strain effects and assumes an infinitely long aging time. In Section 4.1.1 we use this prediction as a benchmark, to discuss the possible strategies for improving the thermodynamic database and experimental measurements. On the other hand, the phasefield simulations may be able to predict more accurately the precipitate evolution, and the strategies to improve the phase-field model parameters will be detailed in Section 4.1.2.

could be a cause for the difference in Table 4 between the experimental measurements and the lever rule predictions. Secondly, the experimental quantification of θ′ volume fractions needs further improvement. In this work, the TEM foil thicknesses were not measured for each foil, leading to uncertainty in the estimated precipitate number density and consequently volume fraction. This is reflected, in part, in the error bars in Fig. 1, since a foil thickness of 100 nm is assumed for images. However, a systematic error in this foil thickness is possible. Based on experimental considerations and previous measurements, the magnitude of such a systematic error is not sufficiently large to explain the difference from the lever rule prediction, and matching the lever rule would require the foil thickness to be ∼50 nm, a difficult result to achieve consistently without destroying many foils. Therefore, although direct foil thickness measurements would be ideal, the current discrepancy in Table 4 cannot be largely due to the foil thickness assumption. Also, though as shown by the numbers beside the error bars in Fig. 1, several TEM images have been taken for each data point, these images can only represent the precipitation characteristics of a few microscopic zones of the whole macroscopic sample. In practice, micro-segregation of solutes can lead to the Cu compositions deviating from the nominal overall composition in the selected microscopic zones, and causing the difference in Table 4. Ideally, many more TEM images should be taken for different selected microscopic zones in each sample, and compositional analysis should be evaluated for each microscopic zone to evaluate the extent of microsegregation. On the other hand, some aspects of the lever rule prediction can be responsible for the difference in Table 4. The existing thermodynamic descriptions of θ′, especially the predictions of θ′ solvus boundary, are still far from satisfactory. Since θ′ is a metastable phase and is usually gradually replaced by the stable precipitate θ during prolonged aging, it

4.1.1. Lever rule vs. experimental results The lever rule predictions of θ′ precipitate volume fraction from different thermodynamic databases for the three temperatures are listed in Table 4, along with the experimentally measured “stable” volume fractions of θ′ precipitates. In contrast to the lever rule predictions, the experimental values are lower and show much less variations as temperature changes. From the experimental standpoint, this difference can be attributed to the following reasons. Firstly, the approximation of the 319 alloy as a binary, or ternary alloy, may not be sufficient. For example, in the TEM image in Fig. 2(b), in addition to the disk-like θ′, seen as line segments of varying length and thickness, there are small black dots present, which are experimentally confirmed to be the metastable Q′ (or Q) phase containing Al, Cu, Si and Mg. Although the presence of Q (Q′) phase has been confirmed by a series of studies in 319, conclusive results on its stoichiometry and thermodynamic descriptions are yet to be established. According to a recent study [31], the Q (Q′) phase can consume some of the Cu in the matrix and co-exist with θ′ during isothermal aging. The number densities of Q (Q′) decrease with increasing aging temperatures, consistent with the observed trend in Table 4. Therefore, it is hypothesized that the presence of Q (Q′) in 319, which has not yet been captured by the existing Al-Cu and Al-Cu-Si thermodynamic databases,

Fig. 4. Simulated θ′ precipitation kinetics and comparison with experimental measurements at 503 K (230 °C). (a) Average diameter. (b) Average thickness. (c) Volume fraction. 89

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Fig. 5. Simulated θ′ precipitation kinetics and comparison with experimental measurements at 533 K (260 °C). (a) Average diameter. (b) Average thickness. (c) Volume fraction.

Moreover, the lever rule prediction, for which only the bulk free energies of the phases are considered, is usually an overestimation of the precipitate volume fraction during aging. In reality, the contribution from both the interfacial energy and misfit strain energy can shift the free energy curve of the precipitate phase upward, leading to a decrease in precipitate volume fractions, which is the trend observed for the experimental volume fractions in Table 4. A more careful discussion on this topic will be presented in the next section.

Table 4 Comparison of θ′ volume fractions from lever rule predictions and the current experimental measurements. Temperature

463 K 503 K 533 K

Lever rule predictions from different thermodynamic databases Hu [13]

Ravi [32]

Al-Cu-Si [33]

4% 3.6% 3.2%

3.4% 3.1% 2.6%

4.1% 3.8% 3.4%

Measured “stable” volume fraction

2.0% 2.1% 1.7%

4.1.2. Phase-field simulations vs. the lever rule As mentioned in the previous section, the drawback of using the lever rule to predict the precipitate volume fractions lies in neglecting the interfacial energy and misfit strain energies. Therefore, phase-field simulations that include these energetics can give more quantitative evaluations than the lever rule predictions. Since the interfacial energies and misfit strain energies are temperature-independent, it is expected that the deviations of the simulated stable θ′ precipitate volume fractions from the lever rule predictions would be similar for the three aging temperatures. However, comparing the simulation results in Figs. 3–5 and the lever rule predictions in Table 4, this deviation increases with aging temperature, and becomes rather significant at 533 K (260 °C) (∼50%). To uncover the reason, we perform a set of phasefield simulations to investigate the effect of the different energy contributions on θ′ volume fraction. As shown in Fig. 7(a), at T = 533 K (260 °C), different simulation cases are considered: (1) all anisotropies are included, which is identical to the simulations in Section 3; (2) no elastic energy is included, i.e., εij0 are assumed to be zero while angle dependence of γ (θi ) and L (θi ) are included; (3) only interfacial energy anisotropy is included, i.e., εij0 are assumed to be zero and L = L0 for all interfaces, while only the angle dependence of γ (θi ) is included; (4) only interface mobility anisotropy is included, i.e., εij0 are assumed to be zero and γ = γ0 for all interfaces, while only the angle dependence of L (θi ) is included; (5) all anisotropies are excluded, i.e., εij0 = 0 , L = L0 and γ = γ0 for all interfaces. When all anisotropies are excluded, the simulated stable volume fraction is significantly closer to the lever rule prediction. The effects of elastic strain energy and interface mobility anisotropy are relatively slight compared to the interfacial energy anisotropy, as shown from the comparison between the curves for case (1) and case (3). The reason that the interfacial energy anisotropy significantly affects the volume fraction prediction of θ′ using phase-field simulations may be a numerical pinning effect, since in Eqs. (4a) and (4b) assuming a constant barrier height w means the interface thickness is proportional to the interfacial energy, both of which are angle-dependent. By using an interface thickness of 2 nm for the semi-coherent interface and a grid size of 1 nm, the interface thickness of the coherent interface is about 1 nm, which is resolved by only one grid point. In the diffuseinterface context, this may cause the numerical pinning issue, i.e., the

Fig. 6. Summary of existing experimental investigations (adapted from Murray [3]) and thermodynamic databases (Hu [13], Ravi [32], Al-Cu-Si ternary [33]) for θ′ solvus boundaries in Al-Cu.

is difficult to experimentally identify the θ′ solvus boundary. The solvus boundary data points, however, are required for the CALPHAD-type thermodynamic modeling of the θ′ phase. In almost all the existing thermodynamic databases of Al alloys, the θ′ solvus boundary is fitted from the experimental results collected by Murray [3], as adapted in Fig. 6. The predicted θ′ solvus boundaries from all existing thermodynamic databases are also shown on this figure. Among them, the prediction by Ravi et al. [32] has the least square deviation from the experimental data points. However, for the temperature ranges of interest (463 K–533 K), the experimental solvus line, as decided by the data points labeled Exp. 1, is steeper, which agrees with the volume fraction trend from the current experimental measurement but is not well captured by the predicted solvus boundaries in Fig. 6. Due to the limited experimental solvus boundary data points within this temperature range and the limited information on the confidence of the experimental data points in Fig. 6, more experimental investigations of the θ′ solvus boundaries are needed. 90

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Fig. 7. Effect of interface thickness on phase-field simulation results. (a) θ′ volume fractions with (2λ)semi = 2 nm at T = 533 K; (b) θ′ volume fractions with (2λ)semi = 4 nm at T = 533 K; (c) θ′ volume fractions with (2λ)semi = 2 nm at T = 463 K; (d) θ′ mean diameters with (2λ)semi = 4 nm at T = 533 K. The red dotted straight lines indicate the lever rule predictions of equilibrium θ′ volume fractions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

algorithm [34,35], with finer meshes at the interface regions, can be helpful. Alternatively, rather than making w constant, the interface thickness (2λ) can be made constant for all interfaces by considering an angle-dependent w in Eqs. 4(a) and 4(b) [36,37].

coherent interface can be almost immobile during the simulations, reducing the volume fraction of the precipitates. Presence of numerical pinning at T = 533 K (260 °C) is confirmed by the constant precipitate thickness in Fig. 5(c). To further validate this issue, we perform the same set of simulations at T = 533 K (260 °C) using a larger interface thickness, (2λ)semi = 4 nm. As shown in Fig. 7(b), the volume fraction results show much less deviation from the lever rule prediction. In addition, the volume fraction predictions under the five different simulation cases are much closer to each other than those in Fig. 7(a), which indicates an alleviated numerical pinning effect. Simulations are also performed for T = 463 K (190 °C) with (2λ)semi = 2 nm for the same cases of different anisotropies as mentioned above, as shown in Fig. 7(c). The T = 463 K (190 °C) simulations of the different cases show much less variations than for T = 533 K (260 °C), indicating much less numerical pinning at T = 463 K (190 °C). Since the same anisotropies in energetics and kinetics apply at these two temperatures, the different variations among the cases are attributed to the different growth driving forces of precipitates. At T = 463 K (190 °C), the growth driving force is higher than that of T = 533 K (260 °C), which can overcome the numerical pinning force. Increasing the interface thickness for all temperatures, however, does not lead to universal improvement in predictions, as will be discussed below. Therefore, to minimize the numerical pinning effect during phase-field simulations, it is critical to perform a set of simulations to quantify the numerical pinning force and to select a proper interface thickness to resolve the thinnest interfaces of the precipitate. The adaptive mesh

4.2. Precipitate thicknesses In the above section, the disagreements in θ′ volume fractions among the phase-field simulations, lever rule predictions and experimental measurements, are discussed in detail. Notably, for T = 533 K (260 °C), by increasing the interface thickness (2λ)semi in phase-field simulations, the numerical pinning issue can be alleviated. However, as shown in Fig. 7(d), predictions of precipitate diameters worsen, with commensurate worsening in precipitate thickness predictions. Especially, due to the increased interface thickness, precipitate thickening is accelerated, which could not be adjusted by changing the interface kinetics anisotropy factor β for the interface kinetic coefficient L (θi ). Specifically, at (2λ)semi = 2 nm, the effect of β is negligible due to numerical pinning; at (2λ)semi = 4 nm, an increase in β would lead to faster lengthening and slower thickening of the precipitates without significantly changing the precipitate volume fractions. However, due to the limited number of simulation grids, a further increase of β (for β > 1000) could not further slow down the thickening kinetics to match he experimental values (see Fig. S1 in supplementary information for details). In this sense, the numerical pinning issue, although unfavorable for the volume fraction prediction, can assist in controlling 91

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4.3.2. Underestimating kinetic coefficients Quantitative evaluation of the kinetic coefficients is an important prerequisite for evaluation of precipitate kinetics. In this study, we assume the lengthening of θ′ is a diffusion-controlled process. Therefore, the Cu diffusivity in α-Al is an important quantity for the conversion between simulation time steps and the real time, as discussed in Section 3.1. There have been a few experimental investigations of Cu diffusivity in α-Al, as well as development of diffusion mobility databases for AlCu-based alloys [24,39–42]. The fitted Cu diffusivity values in α-Al are summarized in Fig. 9, together with the experimentally reported data points. The fitted values are generally consistent with the experimental data points; however, due to the limited data [40] within the temperature range of interest (463 K–533 K), the quality of existing fittings is difficult to be verified. In the current phase-field simulations, we used a value of DCu that is twice of that in [24], to better reproduce the experimental precipitate kinetics, especially for the early-stage kinetics. This treatment is supported by the fact that most of the original experimental data of DCu collected in [24], vast majority of which are outside the range shown in Fig. 9, span a region within a factor of 0.5 and 2 from the assessed diffusion coefficient, and the low temperature measurements (463 K–533 K) tend to be underestimated by the assessment. Moreover, by averaging all the diffusivity values in Fig. 9, we find that increasing the diffusivity value by a factor of 2 is a reasonable estimate. Of course, this discrepancy indicates the demand for more accurate experimental calibrations of Cu diffusivity in α-Al at low temperatures.

the precipitate thickening. In reality, as indicated by high-resolution TEM images [5,6], the semi-coherent Al/θ′ interface is more diffuse than the coherent Al/θ′ interface. The mobility of the coherent interface is therefore rather limited, acting as a real lattice pinning phenomenon for the coherent Al/θ′ interface. The numerical pinning effect in phase-field simulations can therefore represent the realistic lattice pinning effect of the coherent Al/θ′ interface to a certain extent. 4.3. Early-stage kinetics As shown in Figs. 3–5, the precipitate kinetics predictions at the early aging stages need further improvement. In this section, we discuss in detail the possible sources of the deviations and the possible solutions. 4.3.1. Underestimating the nucleation behavior The nucleation behavior of θ′ in 319 alloys can be quite heterogeneous. Firstly, during isothermal aging, defects such as matrix dislocations and grain boundaries can serve as heterogeneous nucleation sites for θ′. For example, experimental observations [9] have shown that at early aging stages, small θ′ precipitates can form zig-zag arrays, implying the heterogeneous nucleation of θ′ along matrix dislocation lines. As discussed in [18], the elastic interactions between the matrix dislocations and precipitates can provide additional nucleation driving force at certain positions of the dislocation lines, forming the zig-zag precipitate arrays. Secondly, other metastable precipitates, such as G.P. zones, θ” and Q′, may precede θ′ during aging, and serve as either heterogeneous nucleation sites or precursors for nucleation of θ′. As reported in a recent study [31], within the temperature range of interest (463 K–533 K), Q′ (or Q) can be an important precursor for nucleation of θ′. However, careful quantification of the effect of Q′ (Q) phase on θ′ nucleation would need to wait for an improved thermodynamic description of Q′ (Q), as well as its interface structures and misfit strains. Moreover, since 319 alloys contain a relatively large amount of Si (∼6 wt%), the effect of Si should be considered for the θ′ precipitate kinetics. Although Si atoms may not partition to inside of θ′ [38], they can segregate at Al/θ′ interfaces to reduce the interfacial energy of both the coherent and the semi-coherent Al/θ′ interfaces. As a result, the nucleation barrier of θ′ is lowered and the nucleation rate becomes faster. Based on the quantitative estimates of the interfacial energy decrease due to Si segregation at 463 K (190 °C) [4,38] and the expressions in Table 1, the nucleation barrier can be 1.4 times lower, while the critical nucleus diameter is only 87% of the case without Si segregation. Therefore, a larger number of finer θ′ nuclei is expected at early aging stages due to Si segregation. In addition, as observed in [6], the semi-coherent interface of θ′ is of a diffuse nature, consisting of a θ” front, especially at lower aging temperatures. The diffuse semi-coherent interface leads to a higher interfacial energy, ∼0.9 J/m2, as well as an accelerated thickening kinetics. Contrary to the Si segregation effect, the diffuse semi-coherent interface would result in higher nucleation barrier and larger critical nucleus. Therefore, fewer θ′ nuclei with larger initial diameter and diameter-to-thickness aspect ratio are expected. To evaluate the effect of these possible sources, we perform two sets of simulations at T = 463 K (190 °C): (i) site-saturated nucleation of θ′ with decreased interfacial energies due to Si segregation; (ii) nucleation with larger semi-coherent interfacial energy and critical radii, but without a change in nucleation rates. The simulated precipitate mean diameters, as shown in Fig. 8, are slightly improved at early aging stages. Given the persisting deviations from experimental measurements at early aging stages, more careful investigations of these heterogeneous nucleation factors should be performed in future investigations.

5. Conclusions In this study, by integrating a nucleation model, we extend the existing phase-field models of θ′ precipitates to quantitatively investigate the precipitate kinetics. By using model parameters from validated theoretical works or experimental calibrations, systematic phase-field simulations are performed, obtaining the best match with experimental measurements of mean diameter, thickness, and volume fraction of θ′ precipitates in 319 alloys at different aging temperatures. In view of the existing deviations between the simulation and experimental results, possible sources of the disagreements and the corresponding solutions, in terms of both experimental investigations and phase-field simulations, are thoroughly discussed. Specifically, (1) In terms of experimental efforts, more accurate investigations are needed to improve the existing thermodynamic databases of Al-Cubased alloys containing θ′, and to elucidate the microscopic mechanisms of θ′ precipitation kinetics. This includes more accurate measurements of θ′ precipitate kinetics at different aging stages, more efforts on identifying the θ′ solvus boundary and measuring the related kinetic coefficients, and more quantitative analysis of the microscopic factors affecting the precipitation of θ′, e.g., elucidating the role of Q′ (Q) phase on θ′ precipitation. (2) In terms of phase-field simulations, the phase-field model and numerical accuracy need to be further improved. This includes avoiding the numerical pinning issue to resolve both the lengthening and thickening kinetics, and considering more microscopic precipitation mechanisms to more accurately predict the precipitate kinetics, when the related experimental investigations become available. We believe the current study can provide practical guidance for applying the phase-field approach, in combination with the necessary experimental investigations, to predict the precipitate kinetics, which lays the foundation of predicting the mechanical properties of alloys during heat treatment processes.

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Fig. 8. Investigations on possible improvements for early-stage precipitate kinetics at 463 K: (a) site-saturated nucleation with decreased interfacial energy due to Si segregation; (b) increased interfacial energy and critical nuclei size due to the diffuse nature of the semi-coherent interface. [4] A. Biswas, D.J. Siegel, D.N. Seidman, Simultaneous segregation at coherent and semicoherent heterophase interfaces, Phys. Rev. Lett. 105 (2010) 076102. [5] L. Bourgeois, C. Dwyer, M. Weyland, J.-F. Nie, B.C. Muddle, Structure and energetics of the coherent interface between the θ′ precipitate phase and aluminium in Al–Cu, Acta Mater. 59 (2011) 7043–7050. [6] L. Bourgeois, N.V. Medhekar, A.E. Smith, M. Weyland, J.F. Nie, C. Dwyer, Efficient atomic-scale kinetics through a complex heterophase interface, Phys. Rev. Lett. 111 (2013) 046102. [7] V. Vaithyanathan, C. Wolverton, L.Q. Chen, Multiscale modeling of precipitate microstructure evolution, Phys. Rev. Lett. 88 (2002) 125503. [8] J.Y. Hwang, R. Banerjee, H.W. Doty, M.J. Kaufman, The effect of Mg on the structure and properties of Type 319 aluminum casting alloys, Acta Mater. 57 (2009) 1308–1317. [9] W.M. Stobbs, G.R. Purdy, The elastic accommodation of semi-coherent-θ' in Al-4wt. %Cu alloy, Acta Metallur. 26 (1978) 1069–1081. [10] A. Biswas, D. Sen, S.K. Sarkar, Sarita, Mazumder S, Seidman DN. Temporal evolution of coherent precipitates in an aluminum alloy W319: A correlative anisotropic small angle X-ray scattering, transmission electron microscopy and atom-probe tomography study, Acta Mater. 116 (2016) 219–230. [11] D. Mitlin, V. Radmilovic, J.W. Morris Jr., Catalyzed Precipitation in Al-Cu-Si, Metallur. Mater. Trans. A 31A (2000) 2697–2711. [12] S. Hu, M. Baskes, M. Stan, L. Chen, Atomistic calculations of interfacial energies, nucleus shape and size of θ′ precipitates in Al-Cu alloys, Acta Mater. 54 (2006) 4699–4707. [13] S.Y. Hu, J. Murray, H. Weiland, Z.K. Liu, L.Q. Chen, Thermodynamic description and growth kinetics of stoichiometric precipitates in the phase-field approach, Calphad. 31 (2007) 303–312. [14] R. Martinez, D. Larouche, G. Cailletaud, I. Guillot, D. Massinon, Simulation of the concomitant process of nucleation-growth-coarsening of Al2Cu particles in a 319 foundry aluminum alloy, Modelling Simulat. Mater. Sci. Eng. 23 (2015) 045012. [15] W. Wang, J.L. Murray, S.Y. Hu, L.Q. Chen, H. Weiland, Modeling of plate-like precipitates in aluminum alloys-comparison between phase field and cellular automaton methods, J. Phase Equilib. Diffus. 28 (2007) 258–264. [16] L.-Q. Chen, Phase-field models for microstructure evolution, Ann. Rev. Mater. Res. 32 (2002) 113–140. [17] S.G. Kim, W.T. Kim, T. Suzuki, Phase field model for binary alloys, Phys. Rev. E 60 (1999) 7186–7197. [18] H. Liu, B. Bellón, J. Llorca, Multiscale modelling of the morphology and spatial distribution of θ′ precipitates in Al-Cu alloys, Acta Mater. 132 (2017) 611–626. [19] K. Kim, A. Roy, M.P. Gururajan, C. Wolverton, P.W. Voorhees, First-principles/ phase-field modeling of θ ′ precipitation in Al-Cu alloys, Acta Mater. 140 (2017) 344–354. [20] V. Vaithyanathan, C. Wolverton, L.Q. Chen, Multiscale modeling of θ′ precipitation in Al–Cu binary alloys, Acta Mater. 52 (2004) 2973–2987. [21] Y. Ji, Y. Lou, M. Qu, J.D. Rowatt, F. Zhang, T.W. Simpson, et al., Predicting coherency loss of γ″ precipitates in IN718 superalloy, Metallur. Mater. Trans. A 47 (2016) 3235–3247. [22] Y.Z. Ji, A. Issa, T.W. Heo, J.E. Saal, C. Wolverton, L.Q. Chen, Predicting β′ precipitate morphology and evolution in Mg–RE alloys using a combination of firstprinciples calculations and phase-field modeling, Acta Mater. 76 (2014) 259–271. [23] A.G. Khachaturyan, Theory of structural transformations in solids, Wiley, New York, 1983. [24] Y. Du, Y.A. Chang, B. Huang, W. Gong, Z. Jin, H. Xu, et al., Diffusion coefficients of some solutes in fcc and liquid Al: critical evaluation and correlation, Mater. Sci. Eng.: A 363 (2003) 140–151. [25] S. Hu, Phase-field models of microstructure evolution in a system with elastic inhomoheneity and defects, The Pennsylvania State University, 2004. [26] L.Q. Chen, J. Shen, Applications of semi-implicit Fourier-spectral method to phase

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