A Generalized Lattice-Point Method for Reconstructing Heterogeneous Materials from Lower-Order Correlation Functions

Review Article

Ann J Materials Sci Eng. 2014;1(1): 11.

A Generalized Lattice-Point Method for Reconstructing Heterogeneous Materials from Lower-Order Correlation Functions

Jiao Y*

Department of Materials Science and Engineering, Arizona State University, USA

*Corresponding author: Jiao Y, Department of Material Science and Engineering, Arizona State University, Tempe, 85287, USA

Received: April 10, 2014; Accepted: May 05, 2014; Published: May 22, 2014


The versatile physical properties of heterogeneous materials are intimately related to their complex microstructures, which can be statistically characterized and modeled using various spatial correlation functions containing key structural features of the material’s phases. An important related problem is to inversely reconstruct the material microstructure from limited morphological information contained in the correlation functions. Here, we present in details a generalizedlattice-point (GLP) method based on the lattice-gas model of heterogeneous materials that efficiently computes a specific correlation function by updating the corresponding function associated with a slightly different microstructure. This allows one to incorporate the widest class of lower-order correlation functions utilized to date into the Yeong-Torquato stochastic reconstruction procedure, and thus enables one to obtain much more accurate renditions of virtual material microstructure, to determine the information content of various correlation functions and to select the most sensitive micro structural descriptors for the material of interest. The utility of our GLP method is illustrated by modeling and reconstructing a wide spectrum of random heterogeneous materials, including “clustered” RSA disks, a metal-ceramic composite, a two-dimensional slice of Fontainebleau sandstone and a binary laser-speckle pattern, among other examples.


Heterogeneous materials (or random media) are those composed of domains of different materials or phases or the same material in different states. Such materials are ubiquitous in nature and in man-made situations; examples include sandstones, concrete, animal and plant tissue, gels and foams and distribution of galaxies [1-9]. Their versatile macroscopic (e.g., transport, mechanical and electromagnetic) properties which are of great interest in various engineering applications are intimately related to the complex material microstructure [1-3,10-12]. Accordingly, a larger number of statistical morphological descriptors have been devised to quantify the key structure features of different material systems [13- 18]. One family of such descriptors includes the standard n-point correlation functions Sn(x1; . . . ; xn) [1]. In particular, Sn gives the probability of simultaneously finding n points with positions x1; . . . ; xn, respectively, in one of the phases of the media. Of particular interest are the lower-order Sn (such as S1, S2 and S3), which have been computed for various models of heterogeneous materials and, as a result, excellent estimations of the effective properties of these mediahave been obtained under certain situations [1]. Recently, lower order correlation functions have been also been employed in computational material design schemes [19-21].

In the study of heterogeneous materials, an intriguing and important inverse problem is the reconstruction of these media from a knowledge of limited microstructural information (a set of lowerorder correlation functions) [22-35]. An effective reconstruction procedure enables one to generate accurate digitized representations (images) of the microstructure from lower-order correlation functions obtainable in experiments or from theoretical considerations, and subsequent analysis can be performed on the images to obtain macroscopic properties of the material without damaging the sample. Reconstruction of a three-dimensional medium using information extracted from two-dimensional plane cuts through the material is another application of great practical value, especially in petroleum engineering, biology and medicine, because in many cases only twodimensional information such as a micrograph is available. One can also determine how much information is contained in the correlation functions by comparing the original and reconstructed media. Construction often refers to generating realizations of heterogeneous materials from a set of hypothetical correlation functions, which enables one to test the realizability of various types of hypothetical functions, which is an outstanding theoretical question [36,37]. Recently, the (re)construction techniques have been employed to identify and categorize heterogeneous materials based on their correlation functions [26,27] and to model a wide spectrum of engineering materials, including sandstone [38,39], porous metal/ ceramics composite [40], alloys [41-44], and textile composites [45,46].

A significant number of reconstruction studies focus on the standard two-point correlation function S2(r), which gives the probability of finding two points separated by a displacement vector r in the phase of interest. This statistical descriptor can be obtained in small-angle X-ray scattering experiments [47]. As pointed out in Reference [48], S2 alone does not provide sufficient information to uniquely determine the microstructures in general; and it is not clear at all that including higher order n-point correlation functions such as S3, S4 etc. would lead to better reconstructions, since these quantities only introduce local information about n-point polygons (polyhedra). As noted in Reference [1], one can never reconstruct the target microstructure perfectly using limited information, i.e., such a reconstruction is generally non-unique. Thus, the objective here is notthe same as that of data decompression algorithms which efficiently restore complete information, but rather to generate realizations of random microstructure with the key morphological features depicted by the correlation functions. Instead of the aforementioned natural and obvious extension to higher-order versions of S2, one could look at additional lower-order correlation functions other than the standard S2 for a better signature of the microstructure.

In Reference [48], we introduced a novel reconstruction procedure called the Generalized Lattice-Point (GLP) method, which enables one to incorporate the widest class of lower-order correlation functions examined to date. The GLP method allows one to generate the accurate renditions of the media of interest using various combinations of the correlation functions and to determine the most sensitive statistical descriptors for the materials of interest. Moreover, we showed through several illustrative examples in [48] that the two-point cluster function C2(r), which gives the probability of finding two points separated by a displacement vector r in the same cluster [49] of the phase of interest, is a superior statistical descriptor to a variety of “two-point” quantities besides S2, including surface correlation functions Fss and Fsv, the pore-size function F, linealpath function L and the chord-length density function p [1] (all of which are defined precisely in Sec. 2). However, the details of the GLP method were not provided in Reference [48]. In this paper, we present the algorithmic details of the generalized lattice-point method. In particular, the discretized heterogeneous material is considered as a lattice-gas system [27], in which pixels with different local states are “molecules” of different “gas” species, or a point process on a lattice. The correlation functions of interest can be obtained by binning the separation distances between the selected pairs of molecules from particular species. For simplicity we only provide the formalism for binary random media here. The generalization of the methodology to multi-phase microstructures is straightforward. The GLP method is combined with the Yeong-Torquato stochastic reconstruction technique [22,23] to evolve a trial microstructure to match the specific target correlation functions as accurately as possible. The GLP method is necessary to efficiently update the correlation functions of the system during the reconstruction process to make it computationally feasible to incorporate those functions into the reconstruction: direct re-sampling is too computationally expensive to implement in practice.

To demonstrate its utility, we apply the GLP method to reconstruct a wide spectrum of random systems from a wide range of correlation function, including “clustered” RSA disks, a metalceramic composite, a two-dimensional slice of a Fontainebleau sandstone and a binary laser-speckle pattern, among other examples. To quantitatively ascertain the accuracy of a reconstruction, correlation functions other than the targeted ones are measured and compared to those of the original medium and the lineal-path function L(r) is used here. Except for the laser-speckle pattern, which processes a multi-scale structure with percolating phases, reconstructions incorporating C2 always produce the most accurate renditions of the target microstructures. This is consistent with our conclusion in Reference [48] that incorporation of C2 significantly reduces the number of compatible microstructures, even superior to certain higher order n-point correlation functions. Statistical descriptors that could be used to characterize multi-scale structures are also suggested.

The rest of the paper is organized as follows: In Sec. 2, we define and discuss various correlation functions used in the reconstructions. In Sec. 3 and 4, we provide the details of the GLP method and how to incorporate it into the general stochastic reconstruction procedure. In Sec. 5, we apply the methodology to reconstruct a variety of random media. In Sec. 6, we make concluding remarks.

Definition of Correlation Functions

Consider a d-dimensional two-phase (binary) microstructure in which phase i has volume fraction φi (i = 1; 2) and is characterized by the indicator function I(i)(x) defined as

I ( i ) (x)={ 1  x in phase i 0     otherwise                      ( 1 ) [email protected]@[email protected]@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaCaaaleqabaWaaeWaaeaacaWGPbaacaGLOaGaayzkaaaaaOGaaiikaiaahIhacaGGPaGaeyypa0ZaaiqaaeaafaqabeGabaaabaGaaGymaiaabccacaqGGaGaaCiEaiaabccacaqGPbGaaeOBaiaabccacaqGWbGaaeiAaiaabggacaqGZbGaaeyzaiaabccacaWGPbaabaGaaeimaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae4BaiaabshacaqGObGaaeyzaiaabkhacaqG3bGaaeyAaiaabohacaqGLbaaaaGaay5EaaGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacck[email protected][email protected]

The two-point correlation function is defined as

S2(i)(x1 ,x2 ) =< I(i)(x1)I(i)(x2)>

where <> denotes ensemble average. This correlation function is the probability of finding two points x1 and x2 both in phase i. hence forth, we will drop the superscript “i” and only consider the correlation functions for the phase of interest. For statistically homogeneous and isotropic microstructures, which is the focus of the rest of the paper, two-point correlation functions will only depend on the distance r = | x1 - x2| between the points and hence S2(x1; x2) = S2(r). In the absence of long-range order, which is the most common occurrence, S2 rapidly decays to Á2, i.e., the probability of finding two points independently in the phase of interest.

The surface-void and surface-surface correlation functions are respectively defined as

Fsv(r) = <M(x1)I(x2)> ; Fss(r) = <M(x1)M(x2)> ;

where M(x) =|∇I (x) | is the two-phase interface indicator function. By associating a finite thickness with the interface, Fsv and Fss can be interpreted, respectively, as the probability of finding x1 in the “dilated” interface region and x2 in the void phase and the probability of finding both x1 and x2 in the “dilated” interface region but in the limit that the thickness tends to zero [1].s

The lineal-path function L(r) is the probability that an entire line of length r lies in the phase of interest, and thus contains a coarse level of connectedness information, albeit only along a lineal path [1,50]. The chord-length density function p(r) is the probability density function associated with finding a “chord” [51] of length r in the phase of interest and is directly proportional to the second derivative of L(r) [52]. The pore-size function F(r) is related to the probability that a sphere of radius r centered at a random point can lie entirely in the phase of interest [1], and therefore is the “spherical” version of the lineal measure L.

The two-point cluster function embodies a greater level of connectedness information than either L or F. In particular, C2(r) is defined to be the probability of finding two points separated by a distance r in the same cluster of the phase of interest [49,53], as schematically shown in Figure 1. When the phase is not percolating, C2 is short ranged and decays to zero rapidly. As the size of the clusters in the systems increases, C2 becomes a progressive longer-ranged function such that its volume integral diverges at the percolation threshold [1]. Thus C2 is extremely sensitive to topologically connectedness information and it takes into account all possible connecting paths, not only the “lineal” and “spherical” ones.