Three dimensional modeling of complex heterogeneous materials via statistical microstructural descriptors

In general, the microstructure of a heterogeneous material can be uniquely determined by specifying the indicator functions associated with all of the individual phases of the material [3], i.e.

where i = 1, …, q and q is the total number of phases. In practice, a digitized version of the indicator function, i.e., a 3D array whose entries specifying the phases that the voxels belong to, is used to fully characterize a material microstructure. There exist several approximation schemes for I⁽ⁱ⁾(x) (i..e, the material microstructure), including the Gaussian random field (GRF) models [30] and reduced-order representation [31]. The key idea involved in these approximation schemes is to reduce the large number of degrees of freedom required to specify unnecessary structural details to a small set of parameters or data points that statistically characterize the material microstructure for purpose of modeling and visualization. However, the physical significance associated with these parameters is usually not obvious. On the other hand, as we will show in the ensuing sections, the parameters involved in the correlation functions such as the “correlation length” often have clear physical meaning and directly correspond to a salient structural feature of the material.

We use the Yeong-Torquato (YT) reconstruction procedure [25, 26] to generate virtual 3D microstructures from a specific set of correlation functions discussed in the previous Statistical microstructural descriptors. In the YT procedure, the reconstruction problem is formulated as an “energy” minimization problem, with the energy functional E defined as follows

where f α ¯ r is a target correlation function of type α and f^α(r) is the corresponding function associated with a trial microstructure. The simulated annealing method [45] is usually employed to solve the aforementioned minimization problem. Specifically, starting from an initial trial microstructure (i.e., old microstructure) which contains a fixed number of voxels for each phase consistent with the volume fraction of that phase, two randomly selected voxels associated with different phases are exchanged to generate a new trial microstructure. Relevant correlation functions are sampled from the new trial microstructure and the associated energy is evaluated, which determines whether the new trial microstructure should be accepted or not via the probability:

where T is a virtual temperature that is chosen to be initially high and slowly decreases according to a cooling schedule [23]. Initially, T is chosen to be high in order to allow a sufficiently large number of “up-hill” (energy increasing) trial microstructures. As T gradually decreases, the “up-hill” moves become less favorable. This significantly decreases the chances that the final microstructure gets stuck in a shallow local energy minimum, as illustrated in Figure 3. In practice, it is usually extremely difficult for the system to converge to the global minimum. Thus, the annealing process is considered complete if E is smaller than a prescribed tolerance, which we choose to be 10^-10 here.

The probabilistic interpretations of the correlation functions discussed in Statistical microstructural descriptors enable us to develop a general sampling method for reconstruction of statistically homogeneous and isotropic materials based on the “lattice-gas” formalism [22‐24]. In this formalism, pixels with different values (occupying the lattice sites) correspond to distinct local states and pixels with the same value are considered to be “molecules” of the same “gas” species. The correlation functions of interest can be obtained by binning the separation distances between the selected pairs of molecules from particular species.

In the case of S₂, all molecules are of the same species. We denote the number of lattice-site separation distances of length r by N_S(r) and the number of molecule-pair separation distances of length r by N_P(r). Thus, the fraction of pair distances with both ends occupied by the phase of interest, i.e., the two-point correlation function, is given by S₂(r) = N_P(r)/N_S(r). To obtain C₂, one needs to partition the molecules into different subsets Г_i (species) such that any two molecules of the same species are connected by a path composed of the same kind of molecules, i.e., molecules that form a cluster, which is identified using the “burning” algorithm [24]. The number of pair distances of length r between the molecules within the same subset Г_i is denoted by Nⁱ_P(r). The two-point cluster function is then given by C₂(r) = i Nⁱ_P(r)/N_S(r). The calculation of F_SS and F_SV requires partitioning the molecules into two subsets: the surface set K_S containing only the molecules on the surfaces of the clusters and the volume set K_V containing the rest. In a digitized medium, the interface necessarily has a small but finite thickness determined by the pixel size. Thus, the surface–surface and surface–void correlation functions can be regarded as probabilities that are given by F_SS = N_SS(r)/N_S(r) and F_SV = N_SV(r)/N_S(r), respectively; where N_SS(r) gives the number of distances between two surface molecules with length r and N_SV is the counterpart for pairs with one molecule on the surface and the other inside the cluster.

The lineal path function L can be obtained by computing the lengths of all digitized line segments (chords) composed of pixels of the phase of interest, and for each chord incrementing the counters associated with the distances equal to and less than that chord length [24]. The chord-length density function p can then be easily obtained by binning the chord lengths that are used to compute L. The pore-size function F can be computed by finding the minimal separation distances of pixels within the phase of interest to those at the two-phase interfaces. The minimal distances are then binned to obtain a probability distribution function, the complementary cumulative distribution function of which is the pore-size function F[24].

Generally, several hundred thousand trials need to be made to achieve such a small tolerance. Therefore, efficient sampling methods [22‐24] are used that enable one to rapidly obtain the prescribed correlation functions of a new microstructure by updating the corresponding functions associated with the old microstructure, instead of completely re-computing the functions. Specifically, a distance matrix D that stores the separation distances of all “molecule” pairs is established when the system is initialized and the molecules are partitioned into different “species” depending on their positions, as discussed in the main article. The quantities N_P, Nⁱ_P, N_SS and N_SV (all defined in the previous subsection) can be obtained by binning the separation distances of selected pairs of molecules from particular species. Recall that the molecules are pixels associated with different values indicating the species, i.e., the clusters or the surface/volume set they belong to.

In the reconstruction procedure, a trial microstructure is generated by moving a randomly selected pixel of the phase of interest to an unoccupied site. This results in changes of the separation distances between the moved the pixel and all of the other pixels, and causes two kinds of possible species events. The first kind is a “cluster” event, which involves breaking and combining clusters. For example, if the selected pixel happens to be a “bridge” connecting several sub-clusters, removing the bridge will make the original single cluster break into smaller pieces, i.e., new species (clusters) are generated. Similarly, the reverse of the above process can occur, i.e., a randomly selected pixel moved to a position where it connects several small clusters to form a larger cluster, which leads to combination of clusters and annihilation of species. The other kind of species event is the transition of pixels between surface and volume sets. If a pixel originally on the surface is removed, certain volume pixels (i.e., those inside the phase of interest) will now constitute the new surface and vice versa. The selected pixel itself could also undergo such a transition, depending on its original and new positions, e.g., a volume pixel originally inside the phase of interest could be moved to the interface to become a surface pixel.

The contributions of the number of pair distances to N_P, Nⁱ_P, N_SS and N_SV from the pixels in old microstructure involved in the species events are computed and subtracted accordingly. The new contributions can be obtained by binning the separation distances of pixel pairs in the new microstructure involved in the species event, which are then added to the corresponding quantities N_P, Nⁱ_P, N_SS and N_SV. This method only requires operations on a small number of pixels, including retrieving and binning their separation distances and updating the species sets (i.e., the clusters and surface/volume set). The use of the distance matrix D speeds up the operations involving distances. However, for very large systems (e.g., those including millions of pixels), storing D requires very a large amount of computer memory. An alternative is to re-compute the separation distances of the pixel pairs involved in the species events for every trial microstructure, instead of explicitly storing all the distances in D. This may slightly slow down the reconstruction process but make it easy to handle very large systems. Correlation functions of the new microstructure can then be obtained from the updated N_P, Nⁱ_P, N_SS and N_SV (i.e., dividing those quantities by N_S). Importantly, the complexity of the algorithm is linear in the total number of pixels (molecules) within the system.

Aluminum (Al) alloys have been widely used in many engineering structures especially in automotive and aircraft applications due to their unique high strength-to-weight ratio and corrosion resistance. An aluminum alloy almost always has secondary phase inclusions and particles with impurities that are present in the microstructure. Due to rolling of the alloy, the inclusions usually possess an anisotropic distribution, with the dimension along the rolling direction being significantly elongated. This in turn results in an overall anisotropic alloy microstructure.

In order to model an alloy microstructure with anisotropic Fe-rich and Si-rich inclusions in Al matrix, we will use correlation functions along three orthogonal directions, i.e., the longitudinal or rolling (L), transverse (T) and short transverse (S) directions [46]. Specifically, we denote the directional two-point correlation function, two-point cluster function and lineal-path function of type α along direction β respectively by S 2 α , β r, C 2 α , β r and L^α,β(r) where α = {Fe, Si, Fe-Si} and β = {L, T, S}. Accordingly, in order to obtain these functions from a digitized microstructure, only the “molecules” along specific directions are considered (see Results).

The directional two-point correlation function of the inclusion phase α can be approximated with the following function:

where ϕ_α is the volume fraction, A β α, B β α and D ¯ β α are parameters to be determined. The approximation (9) includes two “basis” functions: the Debye random medium function exp(Br) [22, 47] and V_int(r; D) [3], which is the scaled intersection volume of two isotropic inclusions with linear size D separated by r:

and Θ(x) is the Heaviside function, i.e.,

In other words, one can consider the alloy microstructure as a mixture of “clusters of all shape sizes” (features of a Debye random medium) and well defined inclusions (characterized by V_int). For the anisotropic inclusions, we consider that each direction possesses distinct combination parameter A β α, correlation parameter B β α and effective size D ¯ β α, which is the effective linear size or “length” of the inclusions along that direction. The values of these parameters are determined such that the approximated functions best match the actual autocorrelation function (i.e., the squared difference between two functions is minimized). In Tables 1 and 2, we provide the values of the parameters for the Fe-rich and Si-rich inclusions in the alloy, respectively. Note that we do not model the cross-correlation function S 2 Fe − Si since in the low ϕ limit its values are several orders of magnitude smaller than the autocorrelation functions, which again suggests that there are no significant inter-particle spatial correlations.

By definition, the two-point cluster function C₂ measures the contributions from the point pairs in the same cluster (inclusion) in S₂; see Eq. (5). Thus, we approximate C₂ of the inclusion phase α as follows:

where the parameters A β α, B β α and D ¯ β α are given in Tables 1 and 2 respectively for the Fe-rich and Si-rich inclusions.

The YT stochastic reconstruction procedure is employed to generate virtual alloy microstructures from the associated S₂ [Eq. (9)] and C₂ [Eq. (12)]; see Figure 4. By quantitatively comparing the experimentally obtained structure and the reconstructions via the lineal-path functions of the inclusion phases, it can be clearly seen that the combination of S₂ and C₂ is sufficient to statistically characterize the microstructure, i.e., the complex alloy microstructure can be modeled by six directional correlation functions with closed analytical forms, which only depend on the parameters given in Tables 1 and 2. In other words, the large 3D microstructural array specifying the local state of each individual voxels can be reduced to a handful of scalar parameters that statistically model the alloy microstructure.

Tin has been widely used as solider materials in electronic packaging. Due to heterogeneous cooling and the existence of intermetallic particles, a tin solder joint usually processes a polycrystalline microstructure. The mechanical properties of the joint are largely determined by the grain size, orientation and the degree of mis-orientation between the grains. Such structural information can be obtained via electron back scattered diffraction (EBSD) experiments (see Figure 5A). However, the EBSD image technique only allows one to probe the surface of the material sample. Therefore, serial sectioning is required to acquire successive 2D images at different depth of the polycrystalline structure. A final reconstruction can be obtained by carefully stacking these 2D images [20, 21].

Our modeling procedure provides an alternative approach to generate virtual 3D polycrystalline structure models from single or a few EBSD images of the material surface. Specifically, the colored EBSD image, in which different colors represent different grain orientations, is first converted to a gray-scale image. Then the gray-scale image is thresholded to generate a multiphase material structure such that each phase corresponds to a narrow range of gray values (i.e., a narrow distribution of orientation). The morphology of each phase includes compact regions that are associated the grains of different orientations (i.e., colors in the original EBSD image). The two-point correlation functions S₂ associated with the individual phases are computed (see Figure 5B). Under the assumption that the structure is statistically homogeneous and isotropic, the sampled S₂ from 2D images are representative of the full 3D microstructure and thus, can be used to generate 3D reconstructions. Similarly, the surface-surface correlation functions F_SS(r) sampled from the 2D images incorporate 3D structural information for statistically homogeneous and isotropic systems, and thus, can also be employed to render virtual 3D microstructures.

Figure 5C and D respectively shows the reconstructed 3D polycrystalline structures using S₂ alone and using the combination of S₂ and F_SS(r). Both reconstructions statistically reproduce the key structural features including the size and shape of the grains and the orientation correlation between neighboring grains, as can be seen by comparing the surfaces of the 3D virtual microstructures to the target 2D image. Quantitative comparison between the reconstructions (not shown here) indicates that incorporating the surface function does not lead to additional improvement of the accuracy of the reconstruction. This suggests that S₂ alone would be sufficient to model this polycrystalline system. We emphasize that a quantitative comparison between the reconstructions and experimentally obtained 3D structure is necessary to definitely ascertain the accuracy of the reconstructions. However, the latter is currently not available.

Binary lead/tin alloys have been widely used as solders [48]. In particular, a eutectic alloy of 63% tin (Sn), 37% lead (Pb) has been used as interconnect due to its unique low melting point (183°C), good wettability, and excellent mechanical properties. The eutectic alloy microstructure contains a Pb-rich phase and a Sn-rich phase, which can possess both laminar and globular morphologies. The salient microstructural features such as the width and extent of the laminar phases as well as the size and spatial distribution of the globular phases can significantly affect the overall mechanical properties of the alloy. At temperatures below the eutectic melting point, the enhanced diffusion of Pb and Sn atoms can lead to significant coarsening in the alloy, which lowers the total interfacial energy [49]. A heat treatment (e.g, annealing) can then be employed to “tune” the eutectic microstructure to achieve desirable material performance. The rate of coarsening increases with temperature and it is of particular importance to quantitatively model and predict the degree of coarsening in the design of materials for high temperature applications.

Figure 6 demonstrates our modeling of the structural evolution in a binary alloy (Pb37Sn63) annealed at 175°C up to 216 h [29], which can be accurately characterized by the associated time-dependent scaled two-point correlation functions f(r), i.e.,

Using phase-field modeling, we have found that a growing length scale λ characterizing the coarsening process can be defined from the correlation length (i.e., a scalar parameter) in f(r) associated with microstructures annealed for different amounts of time. 2D optical images of polished surface of the sample were taken at different annealing times (see the upper panels of Figure 6) and careful measurements of λ from these images have shown that λ satisfies the scaling relation λ(t) ~ t^1/3, which is consistent with the phase-field modeling results and expected for systems that are diffusion-controlled. This allows us to construct a universal functional form of f(r) parameterized by λ(t) to statistically characterize the entire spectrum of microstructures undergone coarsening, i.e.,

where a = 3.5 and b = 0.6 are parameters depending on the initial microstructure but not the annealing time. Snapshots of evolving alloy microstructure at specific annealing times reconstructed from the correlation function f [λ(t)] are shown in the middle panels of Figure 6. The development of coarsening in the microstructure is clearly seen. The lower panels of Figure 6 show the universal function on to which the scaled f associated with experimentally obtained microstructures annealed at different times collapse. The quantitative relation between the material microstructure as represented by the correlation function f[λ(t)] and the processing-condition parameter t lies the mathematical foundation for material design and optimization. In particular, this procedure of 4D material microstructure modeling allows one to directly and quantitatively tie processing parameters (e.g., the annealing time t in this case) to the microstructure. Subsequent analysis and simulations on the physical properties and performance can be carried out to further establish the processing-structure-performance relations.

Finally, we employ the procedure to examine the information content of different correlation functions associated with a hard-sphere packing, i.e., a dispersion of hard spheres in a matrix [24]. This structure has been widely used in models for particle-reinforced composites, colloids, and granular materials. The packing shown in Figure 7A is generated using the standard Metropolis Monte Carlo technique for a canonical ensemble of hard spheres in a cubical box under periodic boundary conditions. The three reconstructions shown in Figure 7B-D are generated respectively using S₂ alone, the combination of S₂ and the pore-size function F, and the combination of S₂ and the cluster function C₂. A visual comparison of the reconstruction involving C₂ reveals that it accurately yields a dispersion of well-defined spherical inclusions of the same size, in contrast to the S₂ reconstruction, which grossly overestimates clustering of the “sphere” phase. In addition, although the reconstruction incorporating F provides improvement over the rendition of the S₂-alone reconstruction, it is still inferior to the S₂–C₂ reconstruction in reproducing both the size and shape of the “sphere” phase. The accuracy of the reconstructions can also be quantitatively ascertained by comparing the lineal-path function associated with the sphere phase in the rendered microstructures, as shown in Figure 7E. This example clearly illustrates the utility of our procedure in examining the information content of various statistical descriptors for structural characterization. In the sphere-packing system, the combination of S₂ and C₂ provide a sensitive descriptor of the structure, suggesting that similar material systems (e.g., particle reinforced composites) can be accurately modeled by these quantities.