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# Simulation and analysis of small angle scattering (SAS) patterns of Ni-based superalloy microstructures generated by a phase-field model

Stéphane Berbenni, Rémi Dingreville, Thomas Schenk

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Stéphane Berbenni, Rémi Dingreville, Thomas Schenk. Simulation and analysis of small angle scattering (SAS) patterns of Ni-based superalloy microstructures generated by a phase-field model. Computational Materials Science, 2025, 258, pp.114086. 10.1016/j.commatsci.2025.114086. hal-05175812

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Full length article

# Simulation and analysis of small angle scattering (SAS) patterns of Ni-based superalloy microstructures generated by a phase-field model

Stéphane Berbenni<sup>a,b,●</sup>,*, Rémi Dingreville<sup>c,●</sup>, Thomas Schenk<sup>b,d,●</sup>

## ARTICLE INFO

### Keywords:

Microstructure

Small angle scattering

Phase-field

Simulations

Patterns

## ABSTRACT

This paper investigates the relationship between microstructural features and small-angle scattering (SAS) patterns in Ni-based superalloys using a combined phase-field and SAS simulation approach coupled with microstructure analyses. The simulated SAS patterns accurately capture key experimental observations previously reported in the literature, including the time-dependent transition from circular to square-shaped precipitates and the development of anisotropic SAS patterns. Importantly, our analysis reveals the correlations between characteristic length scales extracted from SAS profiles and microstructural descriptors, such as precipitate size and inter-precipitate distance. These findings provide a comprehensive understanding of the link between SAS profiles and microstructure evolution in Ni-based superalloys, offering valuable insights for materials characterization and design.<fnref n="1" />

## 1. Introduction

High-performing metallic alloys designed to operate at high temperatures, such as nickel-based superalloys for instance, exhibit excellent thermo-mechanical properties due to precipitation strengthening. In these alloys, the second-phase particles are generally precipitated in the solid-solution matrix to promote microstructural stability and endow the material with superior properties at high temperatures. The physical properties of these alloys are strongly influenced by the crystallographic character, the volume fraction, the shape, and the spatial and size distributions of the precipitates, all of which stem from the kinetics of precipitation [1,2]. This kinetics is influenced by both the precipitate/matrix interfacial energy and the elastic interaction energy between precipitates [3–7].

Many experimental and modeling techniques have been developed to study precipitation kinetics in metallic alloys, see for instance [8] for a comprehensive review. Of particular interest, Small-Angle Scattering (SAS) is a non-destructive experimental technique, allowing for in-situ and real-time measurements of those kinetics without the need of sample preparation, simplifying the experimental process. This technique is based on measuring fluctuations of the scattering factor of a sample. Both structural and spatial information can be obtained from the scattering intensity in the reciprocal space. SAS has been

used for a long time for the characterization of inhomogeneous materials (metals and alloys, polymers, ceramics) at the submicron and nanometer scale [9–11]. It is used quite often in nanoparticle research, see for instance [12], but also to characterize precipitation processes in metallic systems [13,14]. For metals, SAS is used either with a X-ray (SAXS) [15] or neutron (SANS) [16] source to evaluate the time evolution of the characteristic sizes of the microstructure from the angle- or q-dependence ($q = 4\pi/\lambda \sin \theta$ for a scattering angle $2\theta$ and incident-radiation wavelength $\lambda$) of the scattering intensity $I(q, t)$, where $q$ is the scattering vector with magnitude $q$ in the reciprocal space.

Notably, this technique has been employed to characterize precipitation kinetics in nickel-based superalloys and the time evolution of $\gamma'$ precipitate morphologies [16–18]. For example, Fährman and coworkers [17] and Paris and coworkers [18] used SAXS combined with transmission electron microscopy (TEM) to study the early stages of precipitate rafting in a model NiAlMo alloy. From the two-dimensional (2D) SAXS spectra, they extracted important structural information on the aspect ratio of the rafts and the waviness of their interfaces along with statistical information on the temporal evolution of the precipitate morphology. SANS has also been utilized to characterize microstructural features such as precipitate sizes, shapes and distribution, for

<footnote><sup>a</sup> Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France</footnote>

<footnote><sup>b</sup> Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures (LabEx DAMAS), Université de Lorraine, Metz, France</footnote>

<footnote><sup>c</sup> Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, NM 87185, USA</footnote>

<footnote><sup>d</sup> Université de Lorraine, CNRS, Institute Jean Lamour, F-54000 Nancy, France</footnote>

<footnote>* Corresponding author at: Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France.
E-mail address: stephane.berbenni@univ-lorraine.fr (S. Berbenni).</footnote>

<footnote n="1">¹ This paper is dedicated to the memory of Dr. Alain Jacques who passed away on Dec. 30, 2021.</footnote>
instance in single crystalline superalloys (AM1) [16]. For non-spherical particles like γ' precipitates in nickel-based superalloys, SAS patterns were found to be strongly anisotropic [9,15,16]. For instance, Bellet and coworkers [16] reported anisotropic SANS patterns with a fourfold symmetry due to both the shape and the spatial arrangement of γ' precipitates. Interestingly, they studied the time evolution of correlation peaks visible in one-dimensional (1D) SANS intensity profiles during the coarsening of precipitates. From the analysis of the *q*-dependence of the correlation peak positions at different annealing times, Bellet and coworkers concluded that the average spacing between the centers of mass of precipitates increases with annealing time. In another example, Véron and Bastie [19] studied the evolution of SANS intensity contours and profiles for rafting during ageing of pre-strained nickel-based single crystal AM1 superalloys. The observed rafts were different from those obtained after creep test. Interestingly, these authors showed that correlation peaks shift and that they are strongly dependent on the crystallographic direction. Despite great insights from experimental observations illustrated in the examples above, SAS analyses often rely on simplified models and approximations to interpret the complex and high dimensional patterns; these models are used to extract low-order microstructural measures such as interparticle interference, especially for high volume fraction precipitates [20].

Various models, often simple, are routinely used to extract the characteristic lengths of microstructures (e.g. the average precipitate size or the microstructure wavelength) from the scattering intensity $I(q, t)$. One such model is based on Porod's law [21] for the analysis of 1D SAS spectra. Porod's law describes the asymptotic behavior of the scattering intensity $I(q)$ for large scattering wave numbers $q$ and is used to characterize surface properties, i.e. sharp vs. diffuse interfaces [22,23], roughness or curvature [18]. For instance, for a sharp interface, Porod's law predicts that the scattering intensity scales as $I(q) \propto S_{\text{Porod}}q^{-4}$ for large $q$, where $S_{\text{Porod}}$ is the average surface area of the precipitates. Conversely, for rough surfaces, if the roughness can be described by a fractal dimension $d$ between 2 and 3, then Porod's law predicts that the scattering intensity scales as $I(q) \propto S_{\text{Porod}}q^{-6-d}$ [24]. Similarly, another commonly used method for analyzing SAS patterns is the Guinier model [9]. This model is used to evaluate the mean radius of gyration of the precipitates by fitting the scattering intensity to an equation of the form: $\ln[I(q,t)] = \ln(I_0) - (R_g^2/3)q^2$, where $R_g$ is the radius of gyration. Such analysis is typically done only for the low-$q$ portion of the 1D SAS spectrum. Linear (Guinier) scaling suggests that the system is essentially monodispersed. Finally, so-called Kratky plots, which are generally used for studying macromolecules and polymers, less so for metals, plot the scattering intensity as $I(q) \times q^2$ vs. $q$ (instead of simply $I(q)$ vs. $q$). This representation divides out the decay of the scattering intensity. By analyzing the shape and features of the Kratky plot, researchers can infer important information on the microstructure. For instance, for aluminum (Al) alloys, it was used to identify the volume fraction of precipitates [13,14,25].

These models and their interpretations are based on various assumptions, e.g. uniformly distributed precipitates or the absence of long-range correlations, but they are often used outside the domain of validity of such approximations [11,15,26]. Additionally, when exercised, these models make use of simplifications and idealization of the microstructure as made up of elementary-shaped precipitates (e.g. dilute spheres, perfect rods or platelets) to extract microstructural characteristic lengths. To remedy such approximation, various strategies combining simulated diffraction with mesoscale simulations have been proposed to compare directly the simulated scattering intensity to a microstructure analysis [14,15,27,28]. For example, Gille [27] presented a linear simulation model for interpreting SAS experiments of random two-phase systems, focusing on the distribution laws of random distances between phase interfaces. Duchêne et al. [29] studied multi-scale aggregates of nanoparticles (as in alumina catalyst supports) with 1D SAS spectra. Scattered intensities were analyzed and characterized through form and structure factors or by means of multi-scale

Boolean models of spheroids. Welborn and Detsi [30] characterized the morphological features of nanoporous materials, such as the pore size distribution, pore curvature, and specific surface area, using different mathematical, idealized models. Schmidt-Rohr [28] presented an efficient numerical approach based on the Fast Fourier Transform (FFT) algorithm, which was implemented to compute 1D SAS spectra of 2D or three-dimensional (3D) idealized microstructures. He applied this method to systems of long parallel particles at high densities, which are simple to treat by this method but not by competing approaches. Schmidt-Rohr and Chen [31] have later on simulated SAS curves for spherical-cluster material models, polymer and network structures, and analyzed the origin of correlation peaks on SAS curves for these specific microstructures. For two- or three-phase materials simply modeled with Gaussian Random Fields (GRFs), length scale parameters and volume fractions were predicted with accuracy from 1D SAS simulated with a fast FFT-GPU method [32] using a machine-learning-based framework. For more complicated microstructures, some of the examples are using mesoscale methods such as the phase-field method [33] to generate statistics on the microstructural features and compare those statistics with experimental in-situ SAXS measurements [34,35].

To complement these studies, phase-field simulations [36–41] are commonly used to generate realistic shapes and spatial distributions of γ' precipitates in the γ matrix during isothermal conditions and study the evolution of precipitate size, shape (aspect ratios) and deviation from cuboidal shape of γ' precipitates [37,38,42]. However, such studies are rarely combined with SAS analysis of those microstructures, rather they are analyzed with statistical descriptors of the microstructure. Nguyen et al. [35] used phase-field simulation to study the evolution of a synthetic γ/γ' microstructure under a tensile load and analyzed 2D and 3D moment invariants of these microstructures. Sudbrack et al. [43] focused their study on the spacing between the precipitates and pair correlation functions. Similarly, 2-point correlation functions with reduce-order models like PCA (Principal Component Analysis) were extracted by Yabansu et al. [44] from phase-field simulations. For Ni-based superalloys, the linkages between process variables (aging time and aging temperature) and higher-order representations of the microstructure were provided [45].

Our work here is intended to link SAS analysis to phase-field simulations for a more comprehensive analysis of the microstructure and extend these types of studies to more complex microstructures seen during precipitation kinetics in Ni-based superalloys. The phase-field model utilized is based on the Kim-Kim-Suzuki (KKS) [46] interface model with four order parameters related to antiphase boundaries and anisotropic (cubic) and heterogeneous elasticity. We performed both 2D and 3D simulations of precipitation kinetics by varying interfacial energy and elastic properties to generate various times series of coarsened and rafted microstructures with different precipitates characteristics in terms of sizes and morphologies. To keep the analysis tractable, we limited our phase-field model to elastic rafted microstructures (with negative misfit strain and heterogeneous anisotropic elasticity) as reported for example in [47] and we did not consider creep loading. From the generated microstructures, we simulated 1D and 2D SAS patterns with Schmidt-Rohr FFT-based approach [28] to study the time evolutive features of the generated Ni-based superalloy microstructure and their relation to the evolution of the SAS patterns. The results on time-dependent correlation peaks were analyzed in light of spatial microstructure statistics and metrics [48,49]. We studied both the morphology and spatial arrangements effects in these alloys and the linkages between SAS profiles and different microstructure features, like precipitate size distributions or sharp vs. diffuse interface. We also performed PCA to gain further insights into the fundamental microstructural fingerprints encoded in SAS patterns.
## 2. Methods

### 2.1. Phase-field method for microstructure evolution

We used the Kim-Kim-Suzuki (KKS) [46] phase-field model to simulate the coarsening kinetics of $\gamma'$ precipitates in the $\gamma$ matrix in a binary Ni-Al alloy. This model follows the evolution of the microstructure through the evolution of a conservative $\gamma'$ volume fraction field $f(x, t)$ and four non-conservative ordered parameters $\phi_{l(l=1,...,4)}(x, t)$, which describe the four antiphase domains inside the $\gamma'$ $L_{12}$ ordered phase [50]. The fraction field $f(x, t)$ is related to the concentration field $c(x, t)$ through:

$$f(x, t) = \frac{c(x, t) - c_{\gamma}^{0}}{c_{\gamma'}^{0} - c_{\gamma}^{0}}, \qquad (1)$$

where $c_{\gamma}^{0}$ and $c_{\gamma'}^{0}$ are the equilibrium composition of $\gamma$ and $\gamma'$ phase. The order parameters $\phi_{l}(x, t)$ vary from $\phi_{l}(x, t) = 0$ in the matrix phase to $\phi_{l}(x, t) = 1$ in the precipitate phase.

The time evolution of the fraction field (and therefore the concentration field) obeys the classical Cahn-Hilliard's equation [51]:

$$\frac{\partial f(x, t)}{\partial t} = M \nabla^2 \frac{\delta \Psi}{\delta f}, \qquad (2)$$

where $M$ is the diffusion mobility, which is assumed here to be constant. The antiphase order parameters obey the Allen-Cahn's equation [52]:

$$\frac{\partial \phi_{l(l=1,...,4)}(x, t)}{\partial t} = -K \frac{\delta \Psi}{\delta \phi_{l}}, \qquad (3)$$

where $K$ is the kinetic relaxation coefficient, which is assumed here to be constant. In general, for diffusion-controlled microstructure evolution, it is assumed that $K$ is larger than $M$ [37,39,42]. In both of these equations, $\Psi$ represents the total free energy functional of the system, with chemical, interfacial, and elastic contributions such that:

$$\Psi = \int_{x} [\psi^{\text{chem}} + \psi^{\text{el}} + \psi^{\text{int}}] d\mathbf{x}, \qquad (4)$$

where $\psi^{\text{chem}}$, $\psi^{\text{el}}$, $\psi^{\text{int}}$ represent respectively the chemical, elastic and interfacial energy densities.

The chemical free energy density $\psi^{\text{chem}}$ is defined as [42]:

$$\psi^{\text{chem}} = h(\phi_i) \psi_{\gamma'}^{\text{chem}}(f^{\gamma'}) + (1 - h(\phi_i)) \psi_{\gamma}^{\text{chem}}(f^{\gamma}) + \omega g(\phi_i), \qquad (5)$$

where $h(\phi_i)$ is a continuous function which values varies from 0 to 1, $\psi^{\text{chem}}$ and $\psi_{\gamma'}^{\text{chem}}$ are the chemical free energy densities of $\gamma$ and $\gamma'$ respectively, $g(\phi_i)$ is the double well potential, and $\omega$ is the double-well potential height [42]. In our study, $\psi_{\gamma}^{\text{chem}}$ and $\psi_{\gamma'}^{\text{chem}}$ are assumed to be parabolic functions of the $\gamma'$ volume fraction (i.e. $f$):

$$\psi_{\gamma}^{\text{chem}} = \frac{1}{2} A_{\gamma} f^2 \quad \text{and} \quad \psi_{\gamma'}^{\text{chem}} = \frac{1}{2} A_{\gamma'} (1 - f)^2, \qquad (6)$$

where both chemical free energy coefficients (curvatures) $A_{\gamma}$ and $A_{\gamma'}$ can be determined using thermodynamic data for both phases. The functions $h(\phi_i)$ and $g(\phi_i)$ in Eq. (5) are given by [42]:

$$h(\phi_i) = \phi_i^3 (10 - 15\phi_i + 6\phi_i^2) \quad \text{and} \quad g(\phi_i) = \sum_{i=1}^{4} [\phi_i^2(1 - \phi_i^2)] + \alpha \sum_{i=1}^{4} \sum_{j \neq i} \phi_i^2 \phi_j^2. \qquad (7)$$

Here, the diffuse interface approach is described as a mixture of both $\gamma$ and $\gamma'$ phase with different $\gamma'$ volume fraction, with equal chemical potentials following the Kim-Kim-Suzuki (KKS) framework [46]:

$$f = [1 - h(\phi_i)] f^{\gamma} + h(\phi_i) f^{\gamma'} \quad \text{and} \quad \left. \frac{\partial \Psi}{\partial f} \right|_{f=f^{\gamma}} = \left. \frac{\partial \Psi}{\partial f} \right|_{f=f^{\gamma'}} \qquad (8)$$

The interfacial energy per unit volume is classically defined as a penalizing terms stemming from the composition and structural inhomogeneities at the interface [33] such that,

$$\psi^{\text{int}} = \frac{\kappa_{\phi}}{2} \sum_{i=1}^{4} (\nabla \phi)^2, \qquad (9)$$

where both parameters $\omega$ and $\kappa_{\phi}$ in Eq. (5) and Eq. (9) are related to the interfacial energy per unit area $\Gamma_s$ and the diffuse interface thickness $2\lambda$ [46] via

$$\Gamma_s = \frac{\sqrt{\omega \kappa_{\phi}}}{3\sqrt{2}} \quad \text{and} \quad 2\lambda = \alpha \sqrt{\frac{2\kappa_{\phi}}{\omega}}. \qquad (10)$$

The elastic part, i.e. the mechanical part of the free energy function, is calculated from the strain energy density $\psi^{\text{el}}$ such that,

$$\psi^{\text{el}} = \frac{1}{2} \sigma_{ij}(\mathbf{x}) \varepsilon_{ij}^{e}(\mathbf{x}) = \frac{1}{2} \mathcal{C}_{ijkl}(\mathbf{x}) \varepsilon_{ij}^{e}(\mathbf{x}) \varepsilon_{kl}^{e}(\mathbf{x}), \qquad (11)$$

where $\sigma_{ij}(\mathbf{x})$ is the local Cauchy stress tensor, $\mathcal{C}_{ijkl}(\mathbf{x})$ is the local elastic stiffness tensor, and $\varepsilon_{ij}^{e}(\mathbf{x})$ is the local elastic strain tensor, such that the generalized Hooke's law holds at each material point $x$:

$$\sigma_{ij}(\mathbf{x}) = \mathcal{C}_{ijkl}(\mathbf{x}) \varepsilon_{kl}^{e}(\mathbf{x}) \quad \text{with} \quad \varepsilon_{ij}^{e}(\mathbf{x}) = \varepsilon_{ij}(\mathbf{x}) - \varepsilon_{ij}^{*}(\mathbf{x}). \qquad (12)$$

$\varepsilon_{ij}(\mathbf{x})$ is the total strain tensor and $\varepsilon_{ij}^{*}(\mathbf{x})$ is the eigenstrain tensor [53] due to the temperature dependent lattice mismatch $\varepsilon_0$ between $\gamma$ and $\gamma'$. This eigenstrain is given by:

$$\varepsilon_{ij}^{*}(\mathbf{x}) = \varepsilon_{0} \delta_{ij} h(\phi_{i}), \qquad (13)$$

where $\delta_{ij}$ is the Kronecker delta ($\delta_{ij} = 1$ if $i = j$ and $= 0$ otherwise) and $\varepsilon_0$ is the magnitude of the misfit strain evaluated as a function of both lattice parameters $a'_\gamma$ and $a_\gamma$ for $\gamma'$ and $\gamma$ phases respectively. The misfit strain is given by,

$$\varepsilon_0 = 2 \frac{a_\gamma - a'_\gamma}{a_\gamma + a'_\gamma}. \qquad (14)$$

For the elastic constants, considering a Ni-based superalloy as a two-phase material with diffuse interface, with stiffness $\mathcal{C}_{ijkl}^{\gamma}(\mathbf{x})$ for the $\gamma$ matrix phase, and stiffness $\mathcal{C}_{ijkl}^{\gamma'}(\mathbf{x})$ for the $\gamma'$ precipitates, the elastic stiffness tensor $C_{ijkl}(\mathbf{x})$ in Eqs. (11) and (12) reads:

$$\mathcal{C}_{ijkl}(\mathbf{x}) = (1 - h(\phi_i)) \mathcal{C}_{ijkl}^{\gamma}(\mathbf{x}) + h(\phi_i) \mathcal{C}_{ijkl}^{\gamma'}. \qquad (15)$$

To study the effects of external loading, we only considered strain-controlled boundary conditions as in [36,39,54] such that the stress equilibrium was evaluated as,

$$\frac{\partial \sigma_{ij}(\mathbf{x})}{\partial x_j} = \frac{\partial}{\partial x_j} \left( \mathcal{C}_{ijkl}(\mathbf{x}) \left[ \bar{\varepsilon}_{kl} + \delta \varepsilon_{kl}(\mathbf{x}) - \varepsilon_{kl}^{*}(\mathbf{x}) \right] \right) = 0, \qquad (16)$$

with $\bar{\varepsilon}_{ij}$ representing an applied homogeneous strain tensor and $\delta\varepsilon_{ij}(\mathbf{x}) = 1/2(u_{i,j}(\mathbf{x}) + u_{j,i}(\mathbf{x}))$ being the heterogeneous strain obtained from the strain/displacement relationship (i.e it satisfies Saint-Venant strain compatibility equations).

The precipitation kinetics model described above was implemented in Sandia's in-house multiphysics phase-field framework MESoscale Multi-physics PHase field Simulator (MEMPHIS) [55,56]. The kinetics equations, i.e. Eqs. (2) and (3) were discretized using finite differences with a second-order central finite difference stencil with periodic boundary conditions. Solution of the displacement vector $u_j$ used to calculate the strain $\delta\varepsilon_{ij}$, and the corresponding elastic part of the free energy (see Eq. (11)) was based on a discretization and numerical solver using a second-order finite difference scheme and an iterative Gauss-Seidel method until stress equilibrium was satisfied to within a user defined tolerance.

### 2.2. FFT-based algorithm of small-angle scattering (SAS)

To simulate SAS patterns in 1D and 2D from the phase-field simulations, we adopted the FFT-based algorithm developed by Schmid-Rohr [28]. In this algorithm, the 2D scattering intensity $I(q, t)$ at time $t$ is calculated from the Fourier transform ($\mathcal{F}[\cdot]$) of the autocorrelation (or Patterson) function of the scattering factor density $\rho_{\text{phasefield}}(\mathbf{x}, t)$ [14],

$$I(q, t) = |\mathcal{F}(\rho_{\text{phasefield}}(\mathbf{x}, t))|^2, \qquad (17)$$
where **q** is the scattering vector (of length **q** = |**q**|). In the equation above, ρ<sub>phasefield</sub>(**x**, **t**), can be interpreted as the number of scatterers per unit volume multiplied by their scattering factor. Practically speaking, ρ<sub>phasefield</sub>(**x**, **t**), corresponds to the fraction field f(**x**, **t**), from our phase-field simulations. As such the scattering density is defined on a lattice of N<sup>d</sup> points (d being the dimensionality of the microstructure: N<sup>2</sup> = 512 × 512 in 2D and N<sup>3</sup> = 256 × 256 × 256 in 3D). The period in **q**-space is 2π/l<sub>0</sub>, where l<sub>0</sub> corresponds to the grid spacing (i.e. 1 nm). The scattering intensity defined in Eq. (17) produces **q** values ranging between q<sub>min</sub> = 2π/(Nl<sub>0</sub>) and q<sub>max</sub> = π/l<sub>0</sub>, respectively.

Because the algorithm makes use of the discrete Fourier trans-
form, the scattering factor density is defined only at lattice points
(kl<sub>0</sub>, ml<sub>0</sub>, nl<sub>0</sub>), where k, m, n are integers ranging from 1 to N. However,
density distribution in SAS are generally considered as continuous. As
such, the set of discrete scattering factor density peaks is converted
into a continuous density distribution via a smoothing algorithm. This
smoothing correction is performed via a convolution of the discrete
scattering intensity with an elementary cube density of width l<sub>0</sub>. Thus,
the continuous scattering intensity is defined as,

$$
I(\mathbf{q}, t) = |\mathcal{F}(\rho_{\text{phasefield}}(\mathbf{x}, t))|^2 \left[ \prod_{m=1}^{d} \frac{\sin(q_m l_0 / 2)}{q_m l_0 / 2} \right]^2 . \quad (18)
$$

Eq. (18) can be considered as the product of the structure factor of the discrete microstructure and the form factor of the elementary cube. Eq. (18) gives the 2D scattering intensity in **q**-space. The 1D scattering intensity *I*(*q*, *t*) is obtained by scanning the **q**-space grid to calculate *q* = |**q**| and perform an orientational average of all central cross sections through *I*(*q*, *t*). Details of the general algorithm to compute *I*(*q*, *t*) and *I*(*q*, *t*) are provided in [28].

2.3. Microstructure statistics and metrics

To quantify the microstructure features generated from our phase-
field simulations, we used statistical descriptors and 2-point spatial
correlations [49,57,58]. To compute information on precipitate sizes
and inter-precipitate distance, we calculated the autocorrelation func-
tion of the spatially dependent fraction field f(x,t) by binarizing the
microstructure using an indicator function I<sub>BW</sub>(x,t), to identify the
dominant phase at a location x within a microstructure, such that

$$
I_{BW}(\mathbf{x}, t) = \begin{cases} 1 & \text{if } c(\mathbf{x}, t) > 0.5 \\ 0 & \text{otherwise.} \end{cases} \tag{19}
$$

The 2-point autocorrelation function S<sub>2</sub>(x,t) is defined as the expec-
tation of the product I(x<sub>1</sub>,t)I(x<sub>2</sub>,t). The autocorrelation function can
be interpreted as the conditional probability density that two points
at positions x<sub>1</sub> and x<sub>2</sub> are found to be the same phase. We refer the
reader to [57] for a detailed discussion on microstructural correlation
functions. The 2-point statistics resembles a convolution operator and
can be efficiently computed using FFT [49]. By implementing the
FFT computational procedure, we take advantage of the fact that the
microstructure is periodic. The radial average of the autocorrelation
function, S<sub>2</sub>(r), represents the correlation between microstructural fea-
tures as a function of their separation distance. The first minima of
the radial autocorrelation function, R<sub>min</sub>, corresponds to the precipitate
size (i.e. likelihood of finding the precipitate is lowest), while the first
maxima, R<sub>max</sub>, corresponds to the inter-particle distance (i.e. likelihood
of finding another precipitate is highest again).

To calculate the precipitate sizes and size distribution in the mi-
crostructure, we used the Hoshen–Kopelman algorithm [59] adapted
for periodic boundary conditions [60]. The Hoshen–Kopelman algo-
rithm is used to count and label precipitates in the microstructure. For
each precipitate, the precipitate size is calculated as R = √A/π where
A is the surface area of a given precipitate. Having calculated the size of
each individual precipitate, we calculated the distribution of precipitate
size.

For the clustering analysis, we used PCA as a linear dimensionality
reduction technique. Starting from the 2D SAS patterns I(q, t), PCA uses
Singular Value Decomposition (SVD) of the data to project it to a basis
of vectors, such that the truncated orthonormal representation of I(q, t),
denoted as I<sub>pca</sub>, is given by

$$
I(\mathbf{q}, t) \sim I_{\text{pca}} = \hat{I} + \sum_{i=1}^{Q} \alpha_i \phi_i, \quad (20)
$$

where αᵢ are the principal component (PC) scores, φᵢ are the PC basis
vectors, Q is the number of PC directions retained, and ̂I represents the
mean of the 2D SAS pattern. Iₚca is a low-dimension representation of
I(q, t) that only requires Q entries to describe the scattering intensity,
where Q < N² (with N² being the dimension of scattering intensity).

3. Results

3.1. Phase-field simulation parameters and conditions

We simulated the precipitation kinetics in Ni-based superalloys at 1273 K both in 2D and 3D using the KKS model described in the Method section. In 2D, the phase-field model was solved on a 512 × 512 grid with a grid spacing l₀ of 1 nm. In 3D, we used a 256 × 256 × 256 grid with a grid spacing l₀ of 1 nm. We used periodic boundary conditions in all directions. Default parameters used for these simulations are presented in Table 1. The parameters A<sub>γ</sub>, A<sub>γ'</sub>, C'<sub>11</sub>, C''<sub>12</sub>, C''<sub>44</sub>, C'''<sub>11</sub>, C'''<sub>12</sub>, C'''<sub>44</sub>, ε₀ correspond to those used by Tsukada et al. [39] for a practical Ni-Al based superalloy (CMSX-4). The initial volume fraction of γ' phase was set to 0.66. Parameters ω and κ<sub>φ</sub> were obtained from Eq. (10) with Γ<sub>s</sub>=0.0142 J.m<sup>-2</sup> [61], and 2λ=5 nm, with α=2.2 [46]. Hence, in these simulations, the diffuse interface was resolved over 5 grid elements. Additionally, the parameters were converted to dimensionless quantities by dividing them with the following scaling factors: RT for energy, l₀ for length, l<sup>2</sup><sub>0</sub>/(MRT) for time, where R is the perfect gas constant and T is the absolute temperature (T=1273 K) [39]. The dimensionless diffusion mobility, denoted M*, was set to 2 and the dimensionless relaxation coefficient, denoted K*, was set to 5, for diffusion-controlled microstructure evolution [39]. Here, the dimensionless time step was set to Δt*=0.02.

To generate a broad variety of microstructure types, we considered five cases. In the numerical examples presented in this work, Case 1 corresponds to the default parameters reported in Table 1 for the CMSX-4 case by Tsukada et al. [39]. Case 2 represents the situation where the misfit strain ε₀ is set to zero compared to Case 1, so that there is no elastic energy in the model. Case 3 corresponds to the case where a higher misfit strain is considered compared to Case 1, namely ε₀ = -0.003. Case 4 corresponds to a higher elastic shear modulus C<sub>44</sub> for both phases, i.e. we set it a much higher value than the default parameter. We chose to set the elastic shear modulus to be 17 times higher than the default modulus used in Case 1 to exacerbate elasticity effects. Lastly, Case 5 is the case where elastic rafting is simulated considering an applied non-zero homogeneous strain: ε̄₂₂ = 0.02 and starting from an initial configuration which is the final state of Case 4. For simulations without external loading (Cases 1–4), the simulations started from a supersaturated solid solution with Langevin noise terms in the fields f and φ<sub>i</sub>(i=1,...,4) added to Eqs. (2) and (3). We used Gaussian distributions, where normal deviates follow the Box–Muller method [62]. For simulations with external strain (Case 5), the initial condition was taken from the cuboidal microstructure obtained from the final state of Case 4 without noise terms.

3.2. Microstructure evolution during aging treatment for five different cases: 2D simulation results

Fig. 1 shows the time evolution of the precipitation kinetics for the
volume fraction field f(x, t) predicted by phase field for the five differ-
ent cases listed in the previous section. The four first cases illustrate the
<table><caption>**Table 1**

**List of parameters.** Default model parameters used in the phase-field model applied to the present CMSX-4 Ni-based superalloy (Case 1). Reference from which the values are extracted is provided.</caption><thead><tr><th>Parameters</th><th>Values</th><th>Ref.</th></tr></thead><tbody><tr><td>Temperature (K)</td><td>1273</td><td>[39]</td></tr><tr><td>Dimensionless diffusion mobility <i>M</i><sup>*</sup></td><td>2</td><td>[39]</td></tr><tr><td>Dimensionless relaxation coefficient <i>K</i><sup>*</sup></td><td>5</td><td>[39]</td></tr><tr><td>Chemical energy constants <i>A</i> (J.m<sup>-3</sup>)</td><td><i>A</i><sub>1</sub> = 2.58 × 10<sup>8</sup>, <i>A</i><sub>2</sub> = 3.11 × 10<sup>8</sup><br><i>C</i><sub>11</sub><sup>*</sup> = 204.9, <i>C</i><sub>12</sub><sup>*</sup> = 150.8, <i>C</i><sub>44</sub><sup>*</sup> = 94.0</td><td>[39]</td></tr><tr><td>Elastic stiffness constants <i>C</i><sub>ij</sub> (GPa)</td><td><i>C</i><sub>11</sub><sup>*</sup> = 251.6, <i>C</i><sub>12</sub><sup>*</sup> = 194.5, <i>C</i><sub>44</sub><sup>*</sup> = 95.0</td><td>[39]</td></tr><tr><td>Misfit strain ε<sub>0</sub></td><td>-0.0023</td><td>[39]</td></tr><tr><td>Double-well potential height ω (J.m<sup>-3</sup>)</td><td>3.75 × 10<sup>7</sup></td><td></td></tr><tr><td>Gradient energy coefficient κ<sub>φ</sub> (J.m<sup>-1</sup>)</td><td>9.68 × 10<sup>-11</sup></td><td></td></tr><tr><td>γ' volume fraction</td><td>0.66</td><td></td></tr></tbody></table>

effect of the dimensionless length-scale parameter $L = \frac{C_{44}^{\gamma} \bar{\epsilon} \epsilon_0^2}{\Gamma}$ (with $\bar{\epsilon}$ is an equivalent radius for precipitates) on the morphology changes with time. The effect of this parameter has been largely discussed in the literature, see e.g. [17,63–65]. The parameter $L$ reflects the relative effect of the elastic strain energy to the interfacial energy. On one hand, the surface energy, which is isotropic, favors circular (2D) or spherical (3D) $\gamma'$ precipitates, and on the other hand, the elastic energy, which is anisotropic, favors precipitates with large interfaces perpendicular to the elastically soft directions. This competition leads to a transition from circle-to-square (2D) or sphere-to-cube (3D) microstructures during precipitate growth. In the phase-field results illustrated in Fig. 1, the microstructure evolutions for Cases 1–4 during aging treatment are reported at four times. These results confirm that increasing $L$ through an increase of the magnitude of $\epsilon_0$ (Case 3) or an increase of the elastic shear modulus (Case 4) accelerates the transition from circle-to-square microstructures compared to Case 1. For Case 2 which corresponds to $L = 0$, the microstructure does not exhibit cuboidal precipitates but instead spherical-like precipitates. Case 5 depicts the development of a rafted structure with strong morphological changes. For this case, we observe that the rafts are parallel to the $\langle 10 \rangle$ direction while the applied strain is parallel to the $\langle 01 \rangle$ direction. Under this mechanical loading, the rafted structures can form in the direction either parallel (P-type) or normal (N-type) to the loading direction [54,66,67]. Defining the energetic parameter $S = \Delta C \cdot \epsilon_0 \cdot \bar{\epsilon}$ [36], where $\Delta C = \frac{C_{11}' - C_{11}'}{C_{44}'}$, P-type and N-type can be classified according to: if $S > 0$, rafts are parallel to the applied strain (P-type). Conversely, if $S < 0$, rafts are formed normal to the stress axis (N-type). In the present study, $S < 0$ since $\Delta C > 0$, $\epsilon_0 < 0$, and $\bar{\epsilon} > 0$, then a N-type rafted structure is found. We therefore observe the development of N-type elastic rafting due to the elastic inhomogeneity present within the microstructure (Note that homogeneous elasticity would not lead to driving forces for such raft development).

Results in Fig. 1 can be put in parallel with the evolution of the four antiphase boundaries (APBs) characterized in our phase-field simulations via the field variables $\phi_{i}(i=1,...,4)(x,t)$, highlighting their role on precipitate coarsening. We note from these evolutions that L-shaped precipitates observed in Fig. 1 are due to the spatial distributions of APBs of the same type (i.e. for a given $\phi_i(x, t)$ and the interactions and co-evolution of antiphases of different types (i.e. $\phi_i(x, t)$ vs. $\phi_j(x, t)$ when $i \neq j$). This observation is consistent with the previous conclusion from Wang et al. [50] who showed that APBs are mandatory to find the correct morphological evolutions as compared to experimental observations.

The evolution of the average precipitate size $R$ as a function of time, $t^*$, is presented in Fig. 3(a) for Cases 1–4. The average precipitate size $R$ is defined as the sum of equivalent circular radii $R_i = \sqrt{\frac{A_i}{\pi}}$ (with $A_i$ the area of the $i$th precipitate), divided by the total number of precipitates in the simulation box. As indicated by the linear fits on the phase-field data, see Fig. 3(a), the plots of $R^3$ versus $t^*$ clearly show that $R^3 \propto t^*$, which indicates that during this heat treatment the precipitates coarsen following the Lifshitz-Slyosov-Wagner’s theory [68,

[69], a trend consistent with experimental data on single crystalline Ni-based superalloys [16,17]. However, according to our simulations, more subtle differences occur between the different cases. Indeed, it is seen (see Fig. 3(a) and (b)) that the overall coarsening rate for Cases 3 and 4, which exhibit larger elastic interaction energy between precipitates, is lower than the coarsening rate for Case 2 (no elastic interaction energy), while it is intermediate for Case 1 (moderate elastic interaction energy). Additionally, looking at the final size distribution of precipitates in Fig. 3(b), we observe that Case 2 has a broader distribution than Cases 1, 3 and 4. Furthermore, Case 4 shows a more uniform distribution of precipitate sizes essentially due to the cuboidal form of those precipitates.

### 3.3. Small-angle scattering (SAS) 2D patterns and 1D spectra: precipitate morphology and size effects

The comparison of the final microstructures obtained by the 2D phase-field simulation results for the five different cases presented in the previous section (see the last columns in Fig. 1 and plotted in Fig. 4(a)) illustrates the fact that these microstructures present variation in their morphologies. This variation is further quantified when plotting the 2-point spatial autocorrelation function of these microstructures in Fig. 4(b). We note that the 2-point statistics features for Cases 1, 3, 4 are quite similar and described well the microstructure anisotropy. Similar quantitative 2-point statistics feature predictions of the aged state of Ni-based superalloy microstructures have been reported elsewhere [70]. Conversely, Case 2 presents more rounded 2-point statistics features which is a signature of an equiaxed microstructure.

Fig. 4(c) shows the corresponding 2D scattering intensity, $I(q)$, with the scattering vector $q$ defined in the $(q_{(10)}, q_{(01)})$ plane. As expected, we immediately observe that these 2D SAS patterns qualitatively reflect the variation in the microstructure morphology. The SAS pattern for Case 2 is clearly isotropic, displaying concentric rings, while the SAS patterns for Cases 1, 3, 4 are instead anisotropic, displaying a fourfold symmetry pattern which corroborates with the symmetry observed in the corresponding 2-point autocorrelation function in Fig. 4(b). For Case 5, which corresponds to the case where the microstructure was subjected to an applied uniaxial deformation ($\bar{\epsilon}_{22} = 0.02$) along the $\langle 01 \rangle$ direction leading to rafting of the $\gamma'$ precipitates along the $\langle 10 \rangle$ direction, we note that SAS pattern displays a twofold symmetry along the same rafting direction. Hence, these results show that 2D SAS patterns effectively capture the effect of morphology.

Fig. 5(a) displays the 1D scattering intensity $I(q)$ curve normalized by the peak intensity (denoted $I_{max}$) for the five different cases at the final simulation time. Porod’s law, i.e. $I(q) \propto q^{-4}$, and Guinier’s regime, i.e. $I(q) \propto q^2$, are represented as the slopes for the high- and low-q portion of the scattering intensity $I(q)$ respectively. These results show that Cases 1–4 exhibit nearly the same profiles with both well-identified regimes and a sharp transition with a clear peak intensity. For Cases 1–4, the sharp and well-defined interference peak observed in Fig. 5(a) suggests a highly uniform periodic distance between precipitates. In contrast, Case 5 exhibits a smooth transition between both regimes with
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Precipitation kinetics predicted by two-dimensional phase-field simulations for five cases: Case 1 corresponds to the default parameters reported in [38], Case 2 corresponds to the case there is no misfit strain, i.e. $\epsilon_0 = 0$, Case 3 corresponds to the case where $\epsilon_0 = -0.003$, Case 4 corresponds to the case where the shear modulus is exacerbated, and Case 5 corresponds to elastic rafting under tensile load.</figcaption></figure>

a maximum intensity along a plateau instead of a peak. Following this brief plateau at low $q$ values, the scattered intensity decreases, resulting in a pronounced interference peak. This smooth transition implies a distribution of inter-precipitate distances across various length scales. The attribute of this transition is due to the important precipitate morphology changes due to the N-type rafted structure (at larger times compared to the other cases), and the corresponding elongation of $\gamma'$ along the horizontal direction due to an applied strain in the transversal direction. This observation is consistent with the rafted microstructure morphology evidenced in Fig. 1.

The morphology effect is more thoroughly characterized for the five different cases in Fig. 5(b) using the curvature parameter $S_{SAS} = 1 - I(\Theta)/I(0)$ as used for instance in [17], where $I(\Theta)$ and $I(0)$ are the integrated intensities measured in the specific crystallographic directions $\Theta$ and $\langle 01 \rangle$ respectively. In this definition of $S_{SAS}$, the angle $\Theta$ is measured counterclockwise from the $\langle 10 \rangle$ axis. As such, an angle of $\Theta = 0^\circ$ corresponds to the $\langle 10 \rangle$ direction and an angle $\Theta = 90^\circ$ corresponds to the $\langle 01 \rangle$ direction. When the value of $S_{SAS}$ is close to zero, it indicates there is almost no angular variation of the SAS intensity, which means that precipitates are equiaxed, i.e. they are close

to being circular-shaped. This is confirmed in Fig. 5(b) for Case 2 for which the value of $S_{SAS}$ is low regardless of the angle $\Theta$. In contrast, when the value of $S_{SAS}$ is high and tends to unity, the precipitates have more oblong shapes. This configuration is indeed the case for Case 1 and Cases 3–5 in the range of $\Theta$ values in between $30^\circ$ and $60^\circ$, for which the precipitates become more square-shaped (and also L-shaped). These results illustrate that morphology effects at these final steps can be ranked from less to more pronounced squared shapes using the $S_{SAS}$ parameter such that: $S_{SAS}(\text{Case 2}) < S_{SAS}(\text{Case 1}) < S_{SAS}(\text{Case 3}) < S_{SAS}(\text{Case 4}) < S_{SAS}(\text{Case 5})$.

The main effect of heat treatment at 1273K is precipitate coarsening. During coarsening, the time-dependent transition from circular to square shaped precipitate is also operating. Such transition is illustrated with the time evolution of the 2D SAS patterns for Case 1 showed in Fig. 6(a). Indeed, as time increases, we observe a more pronounced fourfold symmetry of the SAS pattern. Such evolution is further illustrated in the 1D scattering intensity $I(q)/I_{max}$ for Case 1 showed in Fig. 6(b). This result is in qualitative agreement with the experimental results provided by Bellet and coworkers [16] and Fährman and coworkers [17] for the time evolution of $\gamma'$ precipitates in Ni-based superalloys. In Fig. 6(b),
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Evolution of the four antiphase boundaries (APBs) for the five cases studied. Each color represented in the microstructure corresponds to a given antiphase domain, $\phi_{i(=1,...,4)}(\mathbf{x},t)$, inside the $\gamma'$ L1$_2$ ordered phase.</figcaption></figure>

we observe a clear precipitate size effect regarding the time-evolution of the SAS curve with a decrease of the values of correlation peaks as a function of time. The values obtained at peak positions along the *q*-axis are denoted *q*\*. Overall, high- and low-*q* portion of the scattering intensity curve are translated to lower *q*-values with increasing time, which is the signature of increasing both precipitate size and inter-particle distance with time [16]. In contrast, the slopes identified on the time-dependent SAS curves for Guinier's regime and Porod's law (assuming sharp interfaces) are quasi time-independent and are similar to those reported on Fig. 5(a). Fig. 6(c) reports the time evolution of the $S_{SAS}$ curvature parameter. The evolution of $S_{SAS}$ shows a slow transition from circular (low $S_{SAS}$ values) to square shaped precipitates (high $S_{SAS}$ values), as this parameter always remains below a value of 0.8 for any $\Theta$.

Similarly, in Fig. 7 we plot the time evolution of the 2D SAS patterns and 1D scattering intensity $I(q)/I_{max}$ for Case 4 followed by Case 5. For those cases, we observe from the 2D patterns that the development of the fourfold (cubic) symmetry for Case 4 originates more rapidly than in Case 1. This fast development can be related to the increased value of *L* parameter as introduced in Section 3.2 to differentiate Cases 1 and 4. For Case 5, due to the N-type rafting process (Fig. 1), the

fourfold symmetry is lost and a twofold stabilized symmetry appears at long time under applied strain. This observation is consistent with the experimental results of Paris and coworkers [18] (see the SAXS patterns in Fig. 2 therein), Fratzl [11] (see the SAXS patterns in Fig. 3 therein), and Véron and Bastie [19] (see the time evolution of SANS patterns in Fig. 2 therein with the loss of fourfold symmetry in the course of rafting leading to twofold symmetry). For Case 4, a precipitate size effect is also observed in Fig. 7(b). Indeed, looking at the time evolution of the SAS curve, we observe a decrease with time of the values of correlation peaks at $q = q^*$, while Case 5 demonstrates a stabilized SAS curve with no change of correlation peak positions at larger times (i.e. from $t^* = 20 \times 10^4$ to $t^* = 54 \times 10^4$). A similar observation can be inferred from Fig. 7(c). In this figure, we note a rapid transition from circular to squared shape precipitates as indicated by the change in the curvature parameter $S_{SAS}$ for Case 4, while the curvature parameter for Case 5 appears to be relatively unchanged as a function of time (with a large plateau between $\Theta = 30^\circ$ and $\Theta = 60^\circ$) with maximal values exceeding 0.8 indicating cuboids precipitates. This stable curvature parameter for Case 5 corresponds to the rafted microstructure which stabilizes between $t^* = 20 \times 10^4$ and $t^* = 54 \times 10^4$, see Figs. 1 and 2.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Precipitate size kinetics and distributions for the four investigated cases: Cases 1-4. (a) Evolution of the average precipitate size *R* as a function of time *t*'. Linear fits correspond to a precipitate growth of the type: *R*³(*t*') = *R*₀³ + *A* × *t*'. (b) Distributions of precipitate sizes at final time step with a bin width size of 6 nm. Solid lines represent the Kernel Density Estimate (KDE) for the various precipitate size histograms. KDE is an estimate of the probability density function of a continuous random variable.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Effect of morphology on 2D SAS patterns. (a) Microstructures obtained by phase-field simulations for the five different cases. (b) Corresponding autocorrelation functions for each case. (c) 2D scattering intensity *I*(*q*) for each case plotted in arbitrary units.</figcaption></figure>

4. Discussion

4.1. *Correlations between microstructural descriptors and 1D SAS profiles*

The modeling results presented above are in good agreement with
many of the previously reported experimental analyses of precipitation

kinetics in nickel-based superalloys [11,15–17,19], further demonstrat-
ing that SAS analysis is sensitive to both the morphology and size of
precipitates. The challenge experimentally lies in analyzing 1D SAS pro-
files and 2D patterns to extract meaningful microstructure descriptors
from them.
<figure><img src="image_8.png" /><figcaption>Fig. 7. Time evolution of SAS patterns for Cases 4 and 5. (a) Evolution of the 2D scattering intensity $I(q, t)$ for Case 4 ($t^* = 0.5 \times 10^4$ and $8 \times 10^4$) and Case 5 ($t^* = 20 \times 10^4$ and $54 \times 10^4$), plotted in arbitrary units. (b) Evolution of the 1D scattering intensity $I(q, t)$ for Cases 4 and 5. Vertical dashed lines indicate position of the correlation peaks, $q^*$. $q^*$ at $t^* = 20 \times 10^4$ and $t^* = 54 \times 10^4$ coincide. (c) Evolution of the curvature parameter $S_{SAS} = 1 - I(\Theta)/I(0)$ for Cases 4 and 5.</figcaption></figure>

provided in Supplementary Note 2 of the Supplementary Information. The correlations between the correlation length $R_{I(q^*)} = 2\pi/q^*$ and the characteristic length scales $R_{S_2}^{\min}$ and $R_{S_2}^{\max}$ extracted from $S_2(r)$ or $R_{\text{Hoshen Kopelman}}$ extracted from the Hoshen-Kopelman analysis (see Fig. 3) are presented in Fig. 8(c). These correlation plots show that $R_{S_2}^{\min}$, $R_{S_2}^{\max}$, and $R_{\text{Hoshen Kopelman}}$ are all linearly correlated to $R_{I(q^*)}$, with a stronger correlation between $R_{I(q^*)}$ and $R_{S_2}^{\max}$. The strong correlation between $R_{I(q^*)}$ and $R_{S_2}^{\max}$ indicates that the maximum correlation peaks, $q^*$, is representative of the inter-precipitate distance. Furthermore, an analysis of the width of the scattering intensity $I(q)$ profile gives insight into the distribution of sizes. A broader peak indicates a wider size distribution (more polydispersity), while a sharper peak suggests a more uniform size distribution. Fig. 9 shows the time evolution of the full width half max ($\Delta q_{\text{FWHM}}$) of the 1D scattering intensity $I(q)$ profile for Cases 1–4. As previously illustrated in the time evolution of the precipitation kinetics in Fig. 1, the decrease in $\Delta q_{\text{FWHM}}$ is indicative of the microstructures having a more uniform size distribution of precipitate as the microstructure coarsens. Additionally, a comparison of the different cases provides additional information on the precipitation kinetics. Indeed, we can observe that for Case 2 (no elastic energy) the precipitation size distribution transitions much more rapidly to a uniform distribution as compared to Cases 1, 3 and 4 (with misfit strain and elastic anisotropy). This trend can be explained by the fact that Case 2 tends to form more circular precipitates as compared to Cases 1, 3, and 4 which exhibit a precipitation kinetics from circular square-shaped precipitates. The role of elasticity is also illustrated in this figure. The comparison of the time evolution of $\Delta q_{\text{FWHM}}$ for Cases 3 and 4 with that of Cases 1 and 2 shows that, when elastic effects are stronger, it takes more time during precipitation kinetics to achieve a more uniform size distribution.

While the above results show correlations between the scattering features of the SAS patterns with some microstructural features in

terms of precipitate morphology and spatial correlations, the analyzed features are not necessarily unique nor do they correspond to only one specific structure. For instance, in the case of mesoporous materials, Gommes [71] showed that different stochastic models of microstructures can actually fit the same SAS experimental data, because a matching correlation function does not unambiguously define a unique microstructure. It was also shown that different multiscale Boolean models of spheroids can share very similar SAS curves [29], as well as different poly-dispersed spherical or ellipsoidal precipitates [72]. Additionally, the 2D SAS data exhibits pronounced anisotropy (see e.g. Fig. 4) as evident from the morphological analysis in Figs. 5–7. This anisotropic behavior significantly complicates the interpretation of 1D scattering intensity profiles $I(q)$, which fail to fully capture the complex information encoded in the correlation peaks or unique microstructural features beyond simple correlation lengths. The anisotropy also affects the FWHM measurements of the 1D scattering intensities in Fig. 8(b), resulting in broad SAS profiles. To accurately account for these anisotropic effects, an orientation-dependent analysis of $I(q)$ would be necessary to reconcile these effects. While the impact of anisotropy would be minimal for Cases 1 through 4 (since they all present a uniform and isotropic distribution of precipitates), it becomes particularly pronounced in Case 5 (due to the formation of rafted precipitates), highlighting the importance of considering anisotropy in diffraction data analysis for complex microstructures.

## 4.2. Principal component analysis (PCA) of 2D SAS patterns

The simulated SAS patterns for the different cases studied in the previous section accurately captured key experimental observations previously reported in the literature in terms of precipitate morphology and anisotropy in the overall topological arrangement of those same precipitates. Such results indicate that SAS patterns contain intrinsic
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Correlations between microstructure internal length scales and characteristic scales from SAS profiles. (a) Radial autocorrelation functions, $S_2(r)$, for Cases 1–4 for time $r^* = 4 \times 10^4$. Average feature size, $R_{S_1}^{\min}$, is indicated by a red circle, inter-precipitate distance, $R_{S_2}^{\max}$, is indicated by a blue square. Vertical dashed lines indicate the location of $R_{S_1}^{\min}$ and $R_{S_2}^{\max}$. (b) Corresponding 1D scattering intensities for Cases 1–4 for time $r^* = 4 \times 10^4$. Location of the maximum peak, $q^*$, is indicated by a red star. The horizontal red line indicates full width half max ($4q_{FWHM}$). (c) Correlations between characteristic length scales extracted from 1D scattering intensities $R_{I(q^*)}$ (y-axis) and $R_{S_2}^{\min}$ (x-axis, left panel), $R_{S_2}^{\max}$ (x-axis, middle panel), $R_{Hoshen}$ (x-axis, right panel). Dashed diagonal line indicates linear correlation. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.).</figcaption></figure>

fingerprints of the microstructure features. We performed PCA on the 2D scattering intensities for all the microstructures simulated in Cases 1–4 (64 microstructures in total) to further reveal those fingerprints. Case 5 is excluded for clarity, see Supplementary Note 3 for the evolution of $I_{PCA}$ for all five Cases. As shown in Supplementary Note 3, only eight principal components are necessary to capture 95% of the explained variance across all the microstructures in our dataset. Figs. 10 (a) and (b) show the evolution of $I_{PCA}$ against the first three PC scores for Cases 1–4. We immediately observe that the SAS patterns are clearly separated for each case, indicating that those patterns contain distinct information about the precipitation kinetics depending on the set of assumption determining each of those cases (e.g. no misfit strain vs. misfit strain). Such clear separation is not necessarily obvious from the examination of the microstructures shown in Fig. 1 or from the 1D scattering intensity profiles. We also note the clear distinction between $I_{PCA}$ for Case 2 and $I_{PCA}$ for Cases 1, 3, and 4. Indeed, Case 2 corresponds to a microstructure with a distribution of spherical-like precipitates while Cases 1, 3, and 4 exhibit distributions of precipitates which are more cuboidal. Similarly, and as expected based on the set of parameters chosen, $I_{PCA}$ for Cases 3 and 4 are the most similar, yet different. Such similarities can be visually confirmed by comparing the 2D scattering intensities in Fig. 4.

Fig. 10(c) depicts the first four basis PC bases, $\phi_i$, for $i = 1...4$. These bases represent the first four fundamental 2D SAS patterns that can be used to reconstruct any SAS patterns in our dataset based on Eq. (20). Note the various symmetries captured by each basis. The first basis, $\phi_1$, which accounts for ~40% of the explained variance

across all the microstructures analyzed, displays a fourfold symmetry along the (10) and (01) direction while the second basis, $\phi_2$, which accounts for an additional ~25% of the explained variance, has an eightfold symmetry along the (10) and (01) and the diagonal directions. Interestingly, the third basis, $\phi_3$, displays a concentric radial feature along with a fourfold symmetry and slightly shifted direction compared to $\phi_1$. While a much larger dataset of simulated SAS patterns would be necessarily to unequivocally determine more general fundamental SAS patterns such as those illustrated by $\phi_1-\phi_4$, these results highlight the various features that can be further extracted from SAS patterns to gain deeper insights on the characterization of those microstructures and partially address a point made earlier. 2D SAS data is able to reveal nuanced anisotropic effects and intricate details of the microstructure topology, offering a more comprehensive characterization than 1D scattering intensity profiles, which are limited in their ability to capture such complex features and higher-order structural information.

### 4.3. Extension to a three-dimensional case

In what follows, we discuss the extension of the present analysis to a three-dimensional (3D) case by using the full 3D phase-field model to study precipitate size effects and the time-dependent sphere-to-cube transition. Materials parameters are the same as those used for Case 4 in 2D. We remind that parameters of Case 4 correspond to default parameters reported in Table 1, but with $C_{44}$ being 17 times higher than the default parameters for both phases to accelerate the sphere-to-cube transition (due to higher value of length-scale parameter $L$).
<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Time evolution of the width of the scattering intensity profile for Cases 1–4. For all cases the full width half max ($\Delta q_{FWHM}$) decreases with increasing time and as the precipitates get bigger. The decrease in $\Delta q_{FWHM}$ is indicative of the microstructures having a more uniform size distribution of precipitate as time increases.</figcaption></figure>

compared to Case 1, see Fig. 1). Fig. 11 reports the time evolution of the $\gamma'$ precipitates in terms of size and morphology at six different times during precipitate coarsening. For the same heat treatment as before (1273K), we observe a gradual stabilization of the microstructure in terms of precipitate size and morphology from non-dimensional times $t^* = 1.2 \times 10^4$ to $t^* = 2.4 \times 10^4$.

Fig. 12(a) shows snapshots of the corresponding evolution of the 2D SAS patterns at three different times. These patterns defined in the $(q_{(100)}, q_{(010)})$ plane were obtained by summing the contributions of the 3D SAS spectra along the $q_{(001)}$-direction in this plane. Similar to the results presented for the 2D simulations (see Fig. 7(a)), we observe a fourfold symmetry of the SAS pattern which is getting increasingly more pronounced with time. This extended result for a 3D case is also in agreement with the experimental results of Bellet et al. [16] (see Figure 5 in [16]) and Fratzl [11] (see Figure 3 in [11]). Such evolution is further analyzed using the 1D scattering intensity $I(q)/I_{max}$ shown in Fig. 12(b). In this figure, the Guinier's regime and Porod's law are also recovered and are quasi time-independent (as in Fig. 7(b)). Again, we observe a precipitate size effect in the time evolution of the 1D SAS curve with a decrease of the values of correlation peaks as a function of time. Such shift in the correlation peak is the signature of annealing time-dependent precipitate size and inter-precipitate distance. This result is also in agreement with Bellet et al. [16] (see Figure 6 in [16]). Finally, the time evolution of the curvature parameter $S_{SAS}$ as a function of the angle $\Theta$ is analyzed in the light of Fig. 12(c). Here again, the same conclusion holds true as for the 2D result in Fig. 7(c) (see Section 3.3 for Case 4). Indeed, an increase of $S_{SAS}$ is observed with a maximum close to 0.8 for $\Theta$ between 30° and 60°.

### 4.4. Effect of sharp vs. diffuse interfaces in phase field model on SAS analysis

Until now, only sharp interfaces were considered, as described for example in Figs. 1 and 2. Hence, a Porod's law was obtained from the slope of the scattering intensity $I(q)$ at high $q$ in Fig. 5. Considering 2D phase-field simulations, Fig. 13 illustrates the effect of interface

thickness on the scattering curve between sharp interface, i.e. zero thickness as previously studied, and diffuse interface, where interface thickness is parameterized by the phase field model, see Section 3.1. These results are reported at the final time step for Case 1. Here, Fig. 13(a) clearly shows a negative deviation of the Porod's law on the curve $I(q)$ vs. $q$ in the high $q$-range. This negative deviation, which is better characterized in the plot: $I(q) \times q^4$ (in logarithmic scale) vs. $q^2$, see Fig. 13(b), is due to the presence of a diffuse interface causing a depletion of high-angle scattering [22–24,73]. Fig. 13(c) shows the effect of the diffuse interface of the phase-field model on 2D scattering patterns. The depletion of high-angle scattering is also observed in all the directions of the $(q_{(10)}, q_{(01)})$-plane. Finally, Fig. 13(d) shows the angular dependence of the curvature parameter $S_{SAS}$ with the angle $\Theta$ accounting for the diffuse interface. While clear differences are seen in the 1D and 2D scattering intensity for high $q$, we observe that the angular dependence is almost unchanged when comparing the sharp vs. diffuse interface cases, with a maximum approaching 0.8 at $\Theta \approx 50^\circ$. This result leads to the conclusion that using sharp or diffuse interface has very little influence on studying the precipitate morphology using $S_{SAS}$.

### 4.5. Connection with SAXS/SANS experiments

Fig. 14 shows experimental 2D-SAXS spectra and TEM images from Paris et al. [18] and Fratzl [11] for Ni-based alloy systems for both cuboid and rafted microstructures. A visual comparison between these experimental SAXS patterns and the simulated patterns for the same types of microstructures displayed in Fig. 4 shows a strong qualitative agreement between the two. For instance, change of the aspect ratio and loss of the cubic symmetry for 2D patterns for the rafted microstructures is correctly captured by the simulated SAXS patterns. However, due to several experimental factors, some characteristics of the experimental SAXS/SANS patterns differ from the simulated ones. The finite $q$-range of the experimental instrument and its resolution limits due to instrument configuration (resolution smearing), beamstop, limited size and irreducible pixel size of detectors, and the presence of incoherent scattering and background noise all convolve into the SAS data to produce complex scattering profiles that are not directly comparable to the idealized, simulated SAXS patterns. The simulation-based data analysis approaches presented here could be improved to address these real-world experimental challenges by incorporating realistic instrumental effects into the simulation workflow. This adaptation could involve integrating resolution functions and background corrections to modify the idealized simulated SAS patterns, effectively mimicking experimental conditions.

## 5. Conclusions

This paper examines the usage of SAS patterns for the quantification of a comprehensive list of microstructural features in Ni-based superalloys using a combined phase-field, SAS simulation approach, and microstructure analysis. The simulated SAS patterns obtained from the phase-field model accurately captured key experimental observations reported in the literature, including the time-dependent transition from circular to square-shaped precipitates and the development of anisotropic SAS patterns with a specific symmetries, reflecting the microstructure's symmetry. Our analysis revealed a strong correlation between characteristic length scales extracted from SAS profiles and microstructural descriptors, such as precipitate size and inter-precipitate distance, obtained from the autocorrelation function analysis and the Hoshen–Kopelman algorithm. PCA analysis on the 2D scattering intensities showed the ability of SAS patterns to uniquely differentiate precipitation kinetics. This study also demonstrated the sensitivity of SAS analysis to the presence of diffuse interfaces, highlighting the need to account for this factor in interpreting SAS data. These findings provide a comprehensive understanding of the link between SAS profiles
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Principal component analysis of 2D SAS patterns. (a) First vs. second principal scores. (b) First vs. third principal scores. Each case is distinct in the principal component space. (c) Principal component bases $\phi_i$. These bases represent fundamental 2D SAS patterns that can be used to reconstruct any SAS patterns based on Eq. (20). Eight principal components are necessary in this case to capture 95% of the explained variance in the present dataset (Cases 1–4), see Supplementary Note 3 in the Supplementary Information.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Precipitation kinetics from a three-dimensional phase-field simulation (Case 4).</figcaption></figure>

and microstructure evolution in Ni-based superalloys, offering valuable insights for materials characterization and design.

The use of simulated experimental results, as demonstrated in this work for instance, provides a powerful tool for better interpretation of SAS data and more generally diffraction data as studied in [74–80].

By directly comparing simulated SAS patterns and curves with experimental measurements, researchers can validate their interpretations and gain deeper insights into the microstructural features responsible for the observed scattering behavior. This approach is particularly useful for complex microstructures, such as those found in Ni-based
<figure><img src="image_13.png" /><figcaption>Fig. 12. Time evolution of SAS patterns from three-dimensional phase-field simulation. (a) Evolution of the 2D scattering intensity $I(q,t)$ for Case 4, plotted in arbitrary units. (b) Evolution of the corresponding 1D scattering intensity $I(q,t)$. (c) Evolution of the corresponding curvature parameter $S_{SAS}$.</figcaption></figure>

superalloys, where traditional analytical models based on simplified assumptions may not be sufficient. For example, the simulated SAS patterns can be used to identify the presence of specific microstructural features, such as precipitate rafting or the formation of clusters, which can then be further investigated using complementary experimental characterization techniques like TEM. Furthermore, by systematically varying microstructural parameters in the simulations, it becomes possible to establish quantitative relationships between microstructural features and diffraction features, enabling more accurate and reliable characterization of advanced materials.

### CRediT authorship contribution statement

**Stéphane Berbenni:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Rémi Dingreville:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Thomas Schenk:** Writing – review & editing, Validation, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors dedicate this manuscript to the memory of Dr. Alain Jacques who initially inspired this research. The authors thank Andreas

Robertson from Sandia National Laboratories for providing comments and suggestions on the initial version of this paper. The authors also thank Pierre Bastie, Frederic De Geuser, and Alexis Deschamps from the University of Grenoble for fruitful discussions on SAS. This work was supported by the French State through the program "Investment in the future" (LabEx "DAMAS" referenced as ANR-11-LABX-0008-01) at the University of Lorraine. High Performance Computing resources have been provided by the EXPLOR center hosted by the University of Lorraine (Project 2018M4XXX0708). Phase-field and microstructure analysis capabilities are supported in part by the Center for Integrated Nanotechnologies, an Office of Science user facility operated for the U.S. Department of Energy. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title, and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doi-public-access-plan.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.commatsci.2025.114086.

### Data availability

Data will be made available on request.
<figure><img src="image_14.png" /><figcaption>**Fig. 13.** Sharp vs. diffuse interface. (a) 1D scattering intensity $I(q,t)$ for the 2D phase-field results in Case 1 (at final step: $t^* = 8 \times 10^4$) with sharp (Sharp int.) vs. diffuse (Diffuse int.) interface. (b) $I(q,t) \times q^4$ vs. $q^2$, highlighting the deviation from Porod's law in the case of a diffuse interface (c) 2D scattering patterns with sharp (Sharp int.) vs. diffuse (Diffuse int.) interface. (d) Difference in the curvature parameter $S_{SAS} = 1 - I(\Theta)/I(0)$ for sharp vs. diffuse interface case.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Experimental 2D-SAXS spectra and dark field TEM images. Images from Paris et al. [18] (left) were reproduced with permission from Paris et al. "Early stages of precipitate rafting in a single crystal Ni-Al-Mo model alloy investigated by small-angle X-ray scattering and TEM", *Acta Materialia*, 45(3), 1997, 1085-1097, Copyright 2025, Elsevier, while the images from Fratzl [11] (right) were reproduced with permission of the International Union of Crystallography. The experimental 2D scattering intensities for cuboid and rafted microstructures share similar features as the simulated SAXS patterns presented in Figs. 4 and 12. Difference can be attributed to several experimental factors including *q*-range, resolution, smear effects of the instrument and background level.</figcaption></figure>
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