10
Accurate modeling and reconstruction of three-dimensional percolating filamentary microstructures from two-dimensional micrographs via dilation-erosion method En-Yu Guo a, b , Nikhilesh Chawla b , Tao Jing a , Salvatore Torquato c , d, e , f , Yang Jiao b, a Key Laboratory for Advanced Materials Processing Technology, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China b Materials Science and Engineering, School for Engineering of Matter, Transport, and Energy, Arizona State University, Tempe, AZ 85287, USA c Department of Chemistry, Princeton University, Princeton, NJ 08544, USA d Department of Physics, Princeton University, Princeton, NJ 08544, USA e Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USA f Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA ARTICLE DATA ABSTRACT Article history: Received 30 October 2013 Received in revised form 21 December 2013 Accepted 23 December 2013 Heterogeneous materials are ubiquitous in nature and synthetic situations and have a wide range of important engineering applications. Accurate modeling and reconstructing three-dimensional (3D) microstructure of topologically complex materials from limited morphological information such as a two-dimensional (2D) micrograph is crucial to the assessment and prediction of effective material properties and performance under extreme conditions. Here, we extend a recently developed dilationerosion method and employ the YeongTorquato stochastic reconstruction procedure to model and generate 3D austeniticferritic cast duplex stainless steel microstructure containing percolating filamentary ferrite phase from 2D optical micrographs of the material sample. Specifically, the ferrite phase is dilated to produce a modified target 2D microstructure and the resulting 3D reconstruction is eroded to recover the percolating ferrite filaments. The dilationerosion reconstruction is compared with the actual 3D microstructure, obtained from serial sectioning (polishing), as well as the standard stochastic reconstructions incorporating topological connectedness information. The fact that the former can achieve the same level of accuracy as the latter suggests that the dilationerosion procedure is tantamount to incorporating appreciably more topological and geometrical information into the reconstruction while being much more computationally efficient. © 2013 Elsevier Inc. All rights reserved. Keywords: 3D microstructure reconstruction Dilationerosion Topologically complex microstructure Austeniticferritic stainless steel 1. Introduction Heterogeneous materials such as composites, alloys, granular materials and porous media abound in nature and in engineer- ing applications. Applications of heterogeneous materials in civil, industrial and aerospace engineering require accurate assessments and predictions of the effective material properties and their performance under extreme conditions, which in turn rely on the accurate knowledge of the complex material microstructures. In the past few decades, a quantitative under- MATERIALS CHARACTERIZATION 89 (2014) 33 42 Corresponding author. Tel.: +1 4809654560; fax: +1 4809659973. E-mail address: [email protected] (Y. Jiao). 1044-5803/$ see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.matchar.2013.12.011 Available online at www.sciencedirect.com ScienceDirect www.elsevier.com/locate/matchar

Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

Ava i l ab l e on l i ne a t www.sc i enced i r ec t . com

ScienceDirect

www.e l sev i e r . com/ loca te /matcha r

Accurate modeling and reconstruction of

three-dimensional percolating filamentarymicrostructures from two-dimensional micrographsvia dilation-erosion method

En-Yu Guoa,b, Nikhilesh Chawlab, Tao Jinga, Salvatore Torquatoc,d,e,f, Yang Jiaob,⁎aKey Laboratory for Advanced Materials Processing Technology, School of Materials Science and Engineering, Tsinghua University,Beijing 100084, ChinabMaterials Science and Engineering, School for Engineering of Matter, Transport, and Energy, Arizona State University, Tempe, AZ 85287, USAcDepartment of Chemistry, Princeton University, Princeton, NJ 08544, USAdDepartment of Physics, Princeton University, Princeton, NJ 08544, USAePrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USAfProgram in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

A R T I C L E D A T A

⁎ Corresponding author. Tel.: +1 4809654560;E-mail address: [email protected] (Y. Ji

1044-5803/$ – see front matter © 2013 Elseviehttp://dx.doi.org/10.1016/j.matchar.2013.12.0

A B S T R A C T

Article history:Received 30 October 2013Received in revised form21 December 2013Accepted 23 December 2013

Heterogeneous materials are ubiquitous in nature and synthetic situations and have a widerange of important engineering applications. Accurate modeling and reconstructingthree-dimensional (3D) microstructure of topologically complex materials from limitedmorphological information such as a two-dimensional (2D) micrograph is crucial to theassessment and prediction of effective material properties and performance under extremeconditions. Here, we extend a recently developed dilation–erosion method and employ theYeong–Torquato stochastic reconstruction procedure to model and generate 3D austenitic–ferritic cast duplex stainless steel microstructure containing percolating filamentary ferritephase from 2D optical micrographs of the material sample. Specifically, the ferrite phase isdilated to produce a modified target 2D microstructure and the resulting 3D reconstructionis eroded to recover the percolating ferrite filaments. The dilation–erosion reconstruction iscompared with the actual 3D microstructure, obtained from serial sectioning (polishing), aswell as the standard stochastic reconstructions incorporating topological connectednessinformation. The fact that the former can achieve the same level of accuracy as the lattersuggests that the dilation–erosion procedure is tantamount to incorporating appreciablymore topological and geometrical information into the reconstruction while being muchmore computationally efficient.

© 2013 Elsevier Inc. All rights reserved.

Keywords:3D microstructure reconstructionDilation–erosionTopologically complexmicrostructureAustenitic–ferritic stainless steel

1. Introduction

Heterogeneous materials such as composites, alloys, granularmaterials and porous media abound in nature and in engineer-ing applications. Applications of heterogeneous materials in

fax: +1 4809659973.ao).

r Inc. All rights reserved.11

civil, industrial and aerospace engineering require accurateassessments and predictions of the effectivematerial propertiesand their performance under extreme conditions, which inturn rely on the accurate knowledge of the complex materialmicrostructures. In the past few decades, a quantitative under-

Page 2: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

34 M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

standing of the microstructure and structure–property relationof heterogeneousmaterials has begun to emerge,mainly due tothe development of advanced experimental and computationalmaterial microstructure characterization techniques [1–3].Specifically, advanced imaging techniques such as X-raytomographic microscopy [4–6], which eliminates destructivecross-sectioning and allows for superior resolution andimage quality with minimal sample preparation [7,8], havebeen widely used to obtain high-resolution three-dimensional(3D) microstructure for a wide range of heterogeneousmaterials,including Sn-rich alloys [9], powder metallurgy steels [10], metalmatrix composites [11–15], and lightweight alloys [16–20]. On thetheory side, a zoologyof statisticalmicrostructuredescriptorshasbeen presented and derived from rigorous structure–propertyanalysis [1,2]. For example, the canonical n-point correlationfunction Hn [2], the integrals of which are involved in variousrigorous bounds [21] and contrast expansions [22,23] of effectivematerial properties, gives the probability of finding a specificn-point configuration in the phase of interest. A variety ofnon-canonical descriptors providing topological connectednessinformation [24–26] and interface information [27] have also beendevised and applied to study a wide spectrum of materials.

Despite of rapid advances of non-destructive 3D imagingtechniques, there are still material systems for which only 2Dimages are available. For example, in certain alloys the contrastbetween the absorption rates of probing rays associated withdifferent phases is too weak to resolve the individual phases. Insuch situations, usually the only available structural dataobtained from non-destructive means are 2D electron micro-graphs or optical images of the sample surfaces, which does notcontain topological connectedness information of the materialphases [2]. Although serial sectioning can be employed to obtainfull 3D microstructure, this procedure is very tedious and willcompletely destroy the material sample. Therefore, it is highlydesirable to render realistic virtual 3D microstructuresthat faithfully present the limited morphological informationcontained in the available 2D data sets, albeit with 3Dexperimental verification.

Over the past two decades, a variety of microstructurereconstruction methods from limited structural informationhave been developed, including the Gaussian random fieldmethod [28], stochastic reconstruction procedure [29,30], phaserecovery method [31], multi-point reconstruction method [32],and raster-path method [33]. The Gaussian random fieldmethod [28] was originally devised to reconstruct realizationsof statistically homogeneous and isotropic randommedia fromthe associated two-point correlation functions. Specifically, afield-field correlation function is constructed basedon the giventwo-point correlation function S2 (see definition in Section. 2). AGaussian random field is then generated using the field-fieldcorrelation function, whose level-cut results in a binarymicrostructure associated the target two-point function. Al-though a wide class of microstructures can be obtained usingthis method, the morphological information used for recon-struction is limited to the two-point correlation functions [28].The phase recovery method enables one to take into accountthe full vector information contained in the two-point statisticsassociated with the material, and thus allows the reconstruc-tions of complex anisotropicmicrostructure and polycrystallinematerials [31]. This method proceeds by iteratively solving for

the phase information that is lost when representing themicrostructure using the vector two-point correlation func-tions. Once such phase information is successfully recovered,the microstructure can be directly obtained via fast Fouriertransform.

The multi-point reconstruction method was originally devel-oped for the reconstruction of porous geomaterials from 2Dimages [32]. Instead of using two-point statistics associated withthe entire 2D microstructure, this method incorporates alln-point statistics within a smaller window containing a portionof the microstructure. The window size is determined by thecorrelation length of the material. A sequential addition methodis then employed to put down voxels in an initially emptyreconstructiondomainbasedon the n-point statistics to generatea 3D microstructure. Although more accurate reconstructionscan be obtained due to the additionalmorphological informationcontained in the multi-point statistics, it is difficult to efficientlystore, represent and retrieve the n-point statistics even for a verysmall window. Recently, a raster-path method is devised whichallows one to employ the multi-point statistics in a much moreefficient way [33]. Specifically, instead of extracting the statisticsfrom the 2D microstructure, a cross-correlation function isintroduced to directly compare a reconstructed portion of thematerial to the target 2D image. Only the reconstructed portionsthat sufficientlywellmatchaportion in the targetmicrostructurewill be accepted and inserted into an initially empty reconstruc-tion domain along a 1D raster path [33].

The stochastic reconstruction procedure [29,30] (also re-ferred to as Yeong–Torquato procedure in literature) enablesone to incorporate an arbitrary number of correlation functionsof any types into the reconstructions. Specifically, a randominitial microstructure is evolved using simulated annealingprocedure such that the set of correlation functions sampledfrom the reconstructed microstructure match the correspond-ing set of target statistics up to a prescribed small tolerance (seeSection. 3 for algorithmic details). This reconstruction proce-dure is very flexible and versatile, however, due to its stochasticnature a large number of intermediate microstructures need tobe generated and analyzed, which makes it computationallyintensive. Several different implementations of the Y–T proce-dure have been devised to improve efficiency [34–36], preserveisotropy [37–39] or handle anisotropic materials [40]. In partic-ular, a dilation–erosion (DE) method based on the Y–T proce-dure has recently been proposed by Zachary and Torquato toimprove the accuracy of the reconstructions of topologicallycomplex microstructures [41]. The DE method (discussed indetail in Section. 3) has been applied to successfully reconstruct2D model microstructures including multiply connected“donut” media and random distributions of micro-cracks [41].

In this paper, we will extend the dilation–erosion proceduredevised by Zachary and Torquato [41] for 2D random textures toaccurately model and reconstruct 3D filamentary microstruc-tures from limited morphological information (e.g., certainspatial correlation functions associated with the filamentaryphase) available in 2D images of the material. We will focus onaustenitic–ferritic cast duplex stainless steels for which the 2Doptical micrographs of the sample surfaces (see Fig. 1) will beused as the input for the reconstruction. As shown in Fig. 1, thefilamentary structures correspond to the ferrite phase whichpercolates in three dimensions. Such percolating filamentary

Page 3: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

Fig. 1 – Left panel: Optical micrograph of the surface of anaustenitic–ferritic cast duplex stainless steel. Right panel: Thesegmented binary image associated with the opticalmicrograph.

35M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

structures abound in a wide spectrum of material systemsincluding grain-boundary networks, binary composites, andgeological structures, to name but a few. And a variety ofquantitative morphological descriptors for such structures havebeen devised [42–45]. A thin 3D microstructure reconstructionwill be obtained via serial sectioning in order to ascertain theaccuracy of the stochastic reconstructions. The two-pointcorrelation function S2 sampled from 2D micrographs will beused in both the standard reconstruction (without dilation–erosion of the phases) and the dilation–erosion reconstruction,in which the filamentary phase is dilated when sampling S2 anderoded in the 3D reconstructions. For the austenitic–ferritic castduplex stainless steel under consideration, which is statisticallyhomogeneous and isotropic, S2 sampled from 2Dmicrographs isrepresentative of that obtained from full 3D material [2]. Inaddition, the two-point cluster function C2 of the un-dilatedferrite phase, which contains additional topological informationof the microstructure [26,36] (see definition below), will bedirectly obtained from the 3D experimentally reconstructedobtained microstructure and incorporated into the standardreconstructions without dilation/erosion (unlike in Ref. [41]). Weshow that the dilation–erosion method can lead to much moreaccurate reconstructions than the standard method using S2obtained from un-dilated 2D micrographs, which underes-timates the linear extent and connectivity of the filamenta-ry ferrite phase. Our results indicate that the dilation–erosion process considered in this work is tantamount tothe use of the three dimensional cluster function C2, andthus can better capture the topological connectednessinformation of the filamentary structures. The importantpoint to note, since only S2 of the dilated phase isincorporated into the reconstruction procedure, the compu-tational cost associated with the DE reconstruction issignificantly lower than that of the reconstruction incorpo-rating C2, which involves keeping track of the complexclusters during the evolution of the microstructure. Thissuggests that the dilation–erosion method could provide anefficient tool for the modeling and characterization ofcomplex filamentary microstructures.

The rest of the paper is organized as follows: In Section. 2,we provide definitions of the statistical descriptors used to

characterize the filamentarymicrostructures. In Section. 3, wedescribe the stochastic reconstruction procedure and thedilation–erosion method in detail. In Section. 4, we presentand analyze the reconstruction results. In Section. 5, we makeconcluding remarks.

2. Statistical Microstructure Descriptors

2.1. Two-Point Correlation Function

In general, themicrostructure of a heterogeneousmaterial can beuniquely determined by specifying the indicator functionsassociatedwith all of the individual phases of thematerial [2], i.e.,

I ið Þ xð Þ ¼ 1 x in phase i0 otherwise

�ð1Þ

where i = 1,…, q and q is the total number of phases. The volumefraction of phase i is then given by

φi ¼< I ið Þ xð Þ > ð2Þ

where <> denotes the ensemble average overmany independentmaterial samples or volume average over a single large sample ifit is spatially “ergodic” [2]. The two-point correlation functionS2(ij)(x1,x2) associated with phases i and j is defined as

S ijð Þ2 x1;x2ð Þ ¼< I ið Þ x1ð ÞI jð Þ x2ð Þ > ð3Þ

which also gives the probability that two randomly selectedpoints x1 and x2 fall into phase i and j respectively (see Fig. 2A). Fora material with q distinct phases, there are totally q2 different S2.However, it has been shown that only q − 1 of them areindependent [2,31] and the remaining q*(q − 1) + 1 functions canbe explicitly expressed in terms of the q independent ones. In ourcase, q = 2 (filamentary structures in a matrix) and thus, we canonly consider the two-point correlation function associated withthe filamentary structures. In the ensuing discussion, we willfocus the correlation function for the phase of interest and dropthe superscript in the function for convenience.

For a statistically homogeneous and isotropic material suchas the stainless steel considered here, there are no preferredcenter and direction in the microstructure. Therefore, theassociated S2 for the phase of interest depends only on therelative distance between the two points [2], i.e.,

S2 x1;x2ð Þ ¼ S2 x2−x1j jð Þ ¼ S2 rð Þ ð4Þ

where r = |r| = |x2 - x1|. At r = 0 the quantity S2(0) gives theprobability that a randomly selected point falls into the phaseof interest, i.e., the volume fraction of the associated phaseS2(r = 0) = φ. At large r values, the probabilities of findingthe two points in the phases of interest are independent ofone another, leading to S2(r → ∞) = φ2. Importantly, it hasbeen shown that for statistically homogeneous and isotropicmicrostructures, the associated S2 can be obtained from thelow-dimensional cuts of the microstructure (e.g., 2D slices of a3D material) [2]. In other words, the S2 of 2D slices containsmorphological information that is representative of the original

Page 4: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

Fig. 2 – Schematic illustrations of the events contributing to different correlation functions. The two-point correlation functionS2 gives the probably of finding two points in the phases of interest. In (A), we show events that contribute to S2 associatedwiththe blue phase. The lineal-path function L(r) gives the probability that a randomly chosen line segment of length r entirely fallsinto the phase of interest. In (B), we show events that contribute to L associated with the blue phase. The two-point clusterfunction C2(r) gives the probability of finding two points separated by r in the same cluster of the phase of interest. In (C), weshow events that contribute to C2 associated with the blue phase.

36 M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

3D microstructure. This makes it a very useful quantity forreconstructing 3Dmicrostructures from 2D images.

We note that the general n-point correlation function Snwhich gives the probability of finding a particular n-pointconfiguration in specific phases can be defined in a similarmanner as S2 [c.f. Eq. (2)], i.e.,

Sn x1;x2;…;xnð Þ ¼< I x1ð ÞI x2ð Þ⋯I xnð Þ > ð5Þ

It has been shown that the effective properties of aheterogeneous material can be explicitly expressed as seriesexpansions involving certain integrals of Sn. Interestedreaders are referred to Ref. [2] for detailed discussions of Snand their properties.

2.2. Lineal-Path Function

The lineal-path function L(i)(r) gives probability that a randomlyselected line segment of length r = |r| along the direction of vectorr entirely falls intophase i (see Fig. 2B) [2,24]. At r = 0, L(i)(0) reducesto the probability of finding a point in phase i and thus, L(i)(0) = φi.In materials that do not contain system-spanning clusters, thechances of finding a line segment with very large length entirelyfalling into any phases are vanishingly small. Accordingly,for large r values L(i) decays to zero rapidly in such materials,i.e., L(i)(r → ∞) = 0. The lineal-path function contains partialtopological connectedness information of the material's phases,i.e., that along a lineal-path. Generally, the lineal-path functionunderestimates the degree of clustering in the system (e.g., twopoints belonging to the samecluster butnot alonga specific linealpath will not contribute to L(i)).

2.3. Two-Point Cluster Function

The two-point cluster correlation function C2(i)(x1,x2) gives the

probability that two randomly selected points x1 and x2 fallinto the same cluster of phase i (see Fig. 2C) [2,26]. Forstatistically homogeneous and isotropic materials, C2 for thephase of interest depends only on the relative distancebetween the two points, i.e., C2(x1,x2) = C2(|x1 − x2|) = C2(r). Incontrast to the lineal-path function, C2 contains completeclustering information of the phases, which has been shown

to have dramatic effects on the material's physical proper-ties [2]. Moreover, unlike S2 and L, the cluster functionsgenerally cannot be obtained from lower-dimensional cuts(e.g., 2D slices) of a 3D microstructure, which may not containcorrect connectedness information of the actual 3D system.

It has been shown that C2 is related to S2 via the followingequation [26]

S2 rð Þ ¼ C2 rð Þ þ D2 rð Þ ð6Þ

whereD2(r)measures theprobability that twopoints separatedbyr fall into different clusters of the phase of interest. In otherwords, C2 is the connectedness contribution to the standardtwo-point correlation function S2. For microstructures withwell-defined inclusion, C2 (r) of the inclusions is a short-rangedfunction that rapidly decays to zero as r approaches the largestlinear size of the inclusions. We note that although C2 is a“two-point” quantity, it has been shown to embody higher-orderstructural information which makes it a highly sensitive statis-tical descriptor over and above S2 [36,41].

3. Stochastic Reconstruction Procedures

3.1. Yeong–Torquato Reconstruction Procedure

We use the Yeong–Torquato (YT) reconstruction procedure[29,30] to generate virtual 3D microstructures from a specificset of correlation functions discussed in the previous Section. 2.As discussed in Section. 1, there are many other differentmicrostructure reconstruction procedures, such as theGaussianrandom field method [28], phase recovery method [31],multi-point reconstruction method [32] and the recently devel-oped raster path method [33]. Importantly, the YT procedure isthe most flexible and versatile one that enables us to incorpo-rate an arbitrary number of correlation functions of any type.

In theYTprocedure, the reconstructionproblem is formulatedas an “energy”minimization problem,with the energy functionalE defined as follows

E ¼Xα

Xr

f α rð Þ− f̂ α rð Þh i2

ð7Þ

Page 5: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

37M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

where f̂αrð Þ is a target correlation functionof typeαand f α(r) is the

corresponding function associated with an intermediate micro-structure. In the case that the set of target functions only includesthe standard two-point correlation function, the “energy” formu-lation of the reconstruction problemhas been successfully solvedusing other schemes such as the gradient-basedmethod [46] anditerative procedures [31]. The simulated annealingmethod [47] isemployed here to solve the aforementioned minimizationproblem. Specifically, starting from an initial microstructure(i.e., oldmicrostructure) which contains a fixed number of voxelsfor each phase consistentwith the volume fraction of that phase,two randomly selected voxels associated with different phasesare exchanged to generate a new microstructure. Relevantcorrelation functions are sampled from the new microstructureand the associated energy is evaluated, which determineswhether the new microstructure should be accepted or not viathe probability:

pacc old→newð Þ ¼ min 1; expEold−Enew

T

� �� �ð8Þ

where T is a virtual temperature that is chosen to be initially highand slowly decreases according to a cooling schedule [29,30]. Theabove process is repeated until E is smaller than a prescribedtolerance, which we choose to be 10−10 here. Generally, severalhundred thousand trials need to bemade to achieve such a smalltolerance. Therefore, efficient samplingmethods [36–38] are usedthat enable one to rapidly obtain the prescribed correlationfunctions of a newmicrostructure by updating the correspondingfunctions associated with the old microstructure, instead ofcompletely re-computing the functions.

3.2. Dilation–Erosion Method

For a topologically complex microstructure such as percolatingthin filaments in amatrix, it is difficult to accurately capture thekey structural features using a limited number of associatedcorrelation functions [48,49]. Instead, we consider a modifiedmicrostructure that involves uniformly “dilating” or “eroding” areference phase in the direction normal to the two-phaseinterface in the original microstructure [41]. Specifically, thedilation can be achieved by placing a virtual sphere of radius δinto the non-reference phase and determining the space

Fig. 3 – Schematic illustration of the dilation–erosion procedure.microstructure. As shown in the middle panel, the reference (blatwo-phase interface. The resulting modified microstructure afterpanel. It can be seen that the dilation process maps the originalThe erosion process is just the inverse of the dilation procedure,un-dilated system.

available to the sphere as if the reference phase was impene-trable to the virtual sphere [1]. The space unavailable to thesphere is considered to be the dilated reference phase. Theerosion process simply reverses this procedure such that thereference phase is allowed to be penetrated by a distance δnormal to the two-phase interface. In our implementations, wechose δ = 1 pixel and the dilation (erosion) process simplycorresponds to converting the pixels of the non-reference(reference) phase bordering the reference (non-reference)phase into the reference (non-reference) phase; see Fig. 3 forillustration. We will elaborate on this selection of δ in thediscussion section.

To reconstruct a filamentarymicrostructure from 2D images,a dilationoperation is first applied to the original 2D images. Thisleads to a modified 2D microstructure with thick “stripe-like”structures. The two-point correlation functions associated withthe modified 2Dmicrostructure are then computed and used asthe input (i.e., target functions) for the Yeong–Torquato recon-struction procedure. As discussed in Section. 2.1, although theS2's are sampled from2D images, themorphological informationcontained in such correlation functions is representative of the3D microstructure. After a 3D reconstruction is obtained, anerosion operation is applied to the 3D microstructure to recoverthe thin filaments. The degree of dilation (erosion) is determinedby theparameter δ, which is the “thickness”of thedilated region.The choice of this parameter depends on the systems underconsideration. A general rule is that the selected δ should belarge enough to produce a percolated phase in the dilatedmicrostructure such as the two-point correlation function S2 ofthe dilated phase can lead to an accurate reconstruction.

4. Results

4.1. Experimental Results

The material studied in this work is a commercially availableaustenitic–ferritic cast duplex stainless steel (Z3CN20-09 M). Thesampleused to obtain themicrostructureswasmachined fromacentrifugally cast pipewhichwasheat treated at 1105 ± 10 °C for4.5 h and then water-quenched. More microstructural informa-tion of the material can also be found in Ref. [50].

The leftmost panel shows an original filamentaryck) phase is uniformly dilated in the direction normal to thethe dilation process is complete is shown in the rightmostfilamentary structures into percolating stripe-like structures.which should lead to a better rendition of the original

Page 6: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

Fig. 4 – Left panel: Experimentally reconstructed microstructure of the cast duplex stainless steel showing the austenite matrix(in green) and ferrite phase (in red) obtained via the serial sectioning technique. Right panel: Extracted 3D ferrite phasemorphology.

Fig. 5 – The correlation functions (defined in the text)associated with the percolating filamentary ferrite phase inthe stainless steel microstructure. The unit of length is theside length of a pixel and 1 pixel = 3.8 μm.

38 M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

In order to acquire the 3Dmicrostructure of thematerial, theserial-sectioning method [51] was utilized and the samplewas polished using a semi-automatic polisher (MiniMet@ 1000,Buehler company, Illinois, U.S). Initially, the sample waspolished with 0.3 μm Al2O3 solution and then, followed bypolishing with 0.05 μm colloidal silica solution to minimize thematerial removal rate and to restore a good surface quality. Aftereach polishing cycle, themicrostructurewas revealed by etchingusing a solution of 25 mlHCl, 25 mlHNO3 and 50 mlH2Oand theoptical micrographs of the microstructure were acquired usingan optical microscope (FS100, Mitutoyo, Japan). A removalmaterial thickness of around 1.7 μm for each slice, which wasmeasured to be much smaller than the microstructural phasesof interest, was obtained and great effort was made to maintainthe same removal rate for each slice. The micrographs werealigned using MATLAB software and then segmented into blackand white images using a commercial image analysis software(Image J, Bethesda. MD). Finally, all the 2D segmented imageswere imported into a 3D visualization software for reconstruc-tion and analysis (Mimics, Materiallise NV, Leuven, Belgium).

The experimentally reconstructed 3D microstructure com-posed of ferrite phase (red) in the austenite matrix (green) isshown in Fig. 4A. The extracted 3D ferrite phase morphology isshown in Fig. 4B, fromwhich the connectivity of the phase can beclearly seen. The volume fraction of the ferrite phase is φ = 0.26.We employ the two-point correlation function S2, the lineal-pathfunction L and the two-point cluster function C2 to quantitativelycharacterize the geometrical and topological features of theferrite phase. Since only a thin 3D microstructure is available,the correlation functions are only sampled in 2D slices perpen-dicular to the short direction of the sample. However, the 3Dconnectedness information is properly taken into account tocompute the cluster function C2. Specifically, a pair of pointsmaybelong to apparently distinct clusters in the 2D slice, which areactually connected in 3D. Such point pairs will contribute to C2.The computed correlation functions are shown in Fig. 5. It can beseen that the 3DC2 is long-ranged, indicating high connectivity ofthe ferrite phase in three dimensions.

4.2. Three-Dimensional Reconstructions

Three-dimensional reconstructions are obtained using boththe standard YT procedure and the dilation–erosion method.

The microstructures obtained from the standard reconstruc-tion using S2 sampled from 2D slices without dilation–erosion,the reconstruction using the combination of S2 and C2

sampled from the actual 3D microstructure, as well as fromthe dilation–erosion reconstruction using S2 sampled from2D slices, are respectively shown in Fig. 6A,B, and C. Thecorresponding reconstructed correlation functions are respec-tively shown in Fig. 7A,B, and C.

It can be clearly seen from Figs. 6A and 7A that the standardS2-alone reconstruction without dilation/erosion significantlyunderestimates the linear extent and connectivity of the ferritefilaments, although the reconstructed S2 is virtually indistin-guishable from the target S2. This is because in general S2 alone isnot sufficient to uniquely determine a microstructure, i.e., thereare an enormous number of distinct microstructures thatare compatible with a specific S2 [49]. At low volume fractions(i.e., φ = 0.26), the number of compatible microstructures con-taining short filaments and compact clusters ismuch larger thanthe number of compatible microstructures with percolatingfilaments. Due to the stochastic nature of YT procedure, theprobability of finding a microstructure with disconnected clusteris thus much higher. On the other hand, the standard recon-struction incorporating both S2 and C2 leads to a significantly

Page 7: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

Fig. 6 – 3Dmicrostructures obtained from the standard reconstruction using S2 obtained from 2Dmicrographswithout dilation/erosion (A), reconstruction incorporating S2–C2 sampled from 3D experimental obtained material microstructure (B), and thedilation–erosion reconstruction using S2 of the dilated phases obtained from 2D micrographs (C). The linear size of the systemis 400 μm. In (A), the linear sizes of the filaments are underestimated. The renditions of the filaments in (B) and (C) aresignificantly improved, due to the incorporation of additional topological connectedness information by directly using thecluster function C2 (B) or the dilation–erosion procedure (C).

39M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

improved 3Dmicrostructure,which canbe seenbothvisually (seeFig. 6B) and from the comparison of the lineal-path functionsassociated with the reconstruction and the original microstruc-ture (see Fig. 7B). The improved accuracy is due to theincorporation of additional topological connectedness informa-tion contained in the three dimensional C2, which is crucial tocapturing the salient features of percolating filamentary ferritephase.

Fig. 6C shows the 3D microstructure reconstructed fromS2 alone using the dilation–erosion method. It can be seen bothvisually and from the comparison of the associated C2 and L(seeFig. 7C) that theDE reconstructionalso significantly improvesupon the standard reconstructionusing S2 alone. Specifically, thecluster function C2 associated with the DE reconstruction islong-ranged, indicating the high connectivity of the reconstruct-ed filamentary ferrite phase. This is because by dilating theferrite phase in the target 2D image, the phase volume fractionincreases from0.26 to 0.43, leading to percolating thick stripe-like

Fig. 7 – Comparison of the target and reconstructed correlation fuphase. (A) Standard reconstruction using S2 obtained from 2D mincorporating S2–C2 sampled from 3D experimental obtained mareconstruction using S2 of the dilated phases obtained from 2D mand 1 pixel = 3.8 μm.

structures. At φ = 0.43, the majority of the microstructurescompatible with the given S2 are composed of percolatingstripe-like structures. Thus, the YT procedure has a highprobability to find a microstructure with high connectivity. Byapplying an erosion operation on the initial reconstruction, thepercolating filamentary ferrite phase is then recovered.

We note that although only S2 from 2D slices is used, the 3Dreconstruction obtained by the dilation–erosion method iscomparable in accuracy to the reconstruction incorporating 3DC2, which contains additional topological connectedness infor-mation. This implies that dilating or eroding the reference phaseis tantamount to including connectedness information of thephase. However, since the reconstruction using C2 requires oneto keep track of the dynamics (e.g., forming and breaking) of theclusters for each intermediate microstructure [36], it is muchmore computationally intensive than the DE reconstruction,which only incorporates S2; see Table 1 for the comparison of theCPU hours for different reconstructions.

nctions associated with the percolating filamentary ferriteicrographs without dilation/erosion (A), the reconstructionterial microstructure (B), and the dilation–erosionicrographs (C). The unit of length is the side length of a pixel

Page 8: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

Fig. 8 – Effect of the dilation thickness δ on the accuracy of thedilation–erosion reconstructions.

40 M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

5. Summary and Discussion

We have extended the dilation–erosion method [41] toreconstruct topologically complex 3D microstructures fromlimited morphological information contained in 2D images.Specifically, we have focused on an austenitic–ferritic castduplex stainless steel that contains percolating filamentaryferrite phase. The Yeong–Torquato reconstruction procedurewas employed to generate 3D microstructures from thetwo-point correlation function S2 sampled from 2D opticalmicrographs of the sample surface as well as from thetwo-point cluster function C2 obtained from experimentallyreconstructed 3D material microstructure. The key points aresummarized in the following bulleted list:

• The standard reconstruction based on S2 from 2D micro-graphs without dilation–erosion significantly underesti-mates the linear extent and connectivity of the ferritephase, leading to isolated elongated clusters.

• On the other hand, the reconstruction incorporating threedimensional C2, and thus additional topological connected-ness information, reproduces the salient features of thepercolating filamentary ferrite phase very well.

• The dilation–erosion reconstruction incorporating S2 obtain-ed from dilated 2D micrographs can lead to 3D microstruc-tures with the level of accuracy comparable to the S2-C2

reconstruction using 3D structural information directlyobtained from actual material microstructure. This suggeststhat dilating or eroding the reference phase is tantamount toincluding 3D connectedness information of the phase in thereconstruction.

• The computational cost of the DE reconstruction is signif-icantly lower than the reconstruction incorporating thecluster function C2.

A key parameter in the dilation–erosion reconstructionmethod is the dilation thickness δ which was chosen to be 1pixel length in our implementation. As we mentioned inSection. 3.2, a general rule to determine the value of δ is thatthe selected value should lead to a percolating phase in thedilated microstructure. Too small δ values result in isolatedphases which S2 is insufficient to accurately characterize. Onthe other hand, too large δ values will produce significantlydilated phases that are difficult to recover during the erosionprocess. To quantify the effects of δ on the accuracy of thereconstruction, we reconstruct the filamentary stainless steelstructure using different δ values, and compare the “errors” of

Table 1 – Comparison of the CPU hours for differentreconstruction methods including the standard Y–Tstochastic reconstruction using S2 alone withoutdilation–erosion, the standard reconstruction using both S2and C2, as well as the dilation–erosion (DE) reconstruction.The simulations were carried out on a standard Dell T5600workstation.

S2 alonereconstruction

S2–C2

reconstructionDE

reconstruction

CPU hours 1.6 29.5 1.8

the reconstructions, which are defined to be the squareddifference between the reconstructed and target two-pointcluster functions C2; see Fig. 8. At δ = 0, no dilation/erosionoperations are applied during the reconstruction and thus, theprocedure corresponds to the standard S2 reconstruction,whichis associated with an error E0. For δ > 0, the error E of thereconstructions is normalized by E0. It can be clearly seen thatδ = 1 is associated with the minimal E among all δ considered.This clearly indicates the optimality of our choice of δ.

The dilation–erosion method provides a computationallyefficient means for modeling and reconstructing realistic 3Dmicrostructure of topological complex materials from limitedmorphological information for subsequent analysis of theireffective properties and performance. This technique isreadily applied to characterize and model other complexfilamentary structures such as composites and alloys withmicrocracks. However, for materials composed of compactwell-isolated phases such as particle reinforced composites,directly incorporating the cluster function C2 into the recon-struction has been shown to be a superior method toaccurately characterize and model such structures [36].

Acknowledgments

This work was supported by the Division of MaterialsResearch at National Science Foundation under award No.DMR-1305119. E. G. and T. J. thank the financial support fromthe National Science and Technology Major Project of theMinistry of Science and Technology of China, under ProjectNos. 2011ZX04014-052 and 2012ZX04012011. E. G. is alsograteful to the Chinese Scholarship Council for financialsupport during his stay at Arizona State University (USA)and to the Doctoral Fund of Ministry of Education of China,under Grant No. 20130002110012.

R E F E R E N C E S

[1] Torquato S. Microstructure characterization and bulkproperties of disordered two-phase media. J Stat Phys1986;45:843–73.

Page 9: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

41M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

[2] Torquato S. Random heterogeneous materials: microstructureand macroscopic properties. New York: Springer; 2002.

[3] Sahimi M. Heterogeneous materials I: Linear transport andoptical properties. New York: Springer; 2003.

[4] Kak AC, Slaney M. Principles of computerized tomographicimaging. New York: IEEE Press; 1988.

[5] Brandon D, Kaplan WD. Microstructural characterization ofmaterials. New York: John Wiley & Sons; 1999.

[6] Herman RT. Fundamentals of computerized tomography:image reconstruction from projection. 2nd ed. New York:Springer; 2009.

[7] Baruchel J, Bleuet P, Bravin A, Coan P, Lima E, Madsen A, et al.Advance in synchrotron hard X-ray based imaging. C RPhysique 2008;9:624–41.

[8] Kinney JH, Nichols MC. X-ray tomographic microscopy (XTM)using synchrotron radiation. Annu Rev Mater Sci1992;22:121–52.

[9] Padilla E, Jakkali V, Jiang L, Chawla N. Quantifying the effectof porosity on the evolution of deformation and damage inSn-based solder joints by x-ray microtomography andmicrostructure-based finite element modeling. Acta Mater2012;60:4017–26.

[10] Chawla N,Williams JJ, Deng X,McClimonC. Three dimensional(3D) characterization and modeling of porosity in powdermetallurgy (P/M) steels. Int J Powder Metall 2009;45:19–27.

[11] Babout L, Maire E, Buffière JY, Fougères R. Characterization byX-ray computed tomography of decohesion, porosity growthand coalescence in model metal matrix composite. ActaMater 2001;49:2055–63.

[12] Borbély A, Csikor FF, Zabler S, Cloetens P, Biermann H.Three-dimensional characterization of the microstructure ofa metal-matrix composite by holotomography. Mater Sci Eng2004;A367:40–50.

[13] Kenesei P, Biermann H, Borbély A. Structure–propertyrelationship in particle reinforced metal-matrix compositesbased on holotomography. Scripta Mater 2005;53:787–91.

[14] Williams JJ, Flom Z, Amell AA, Chawla N, Xiao X, De Carlo F.Damage evolution in SiC particle reinforced Al alloy matrixcomposites by x-ray synchrotron tomography. Acta Mater2010;58:6194–205.

[15] Silva FA, Williams JJ, Mueller BR, Hentschel MP, Portella PD,Chawla N. 3Dmicrostructure visualization of inclusions andporosity in SiC particle reinforced Al matrix composites by x-raysynchrotron tomography. Metall Mater Trans 2010;A41:2121–8.

[16] Weck A, Wilkinson DS, Maire E, Toda H. Visualization byx-ray tomography of void growth and coalescence leading tofracture in model materials. Acta Mater 2008;56:2919–28.

[17] Toda H, Yamamoto S, Kobayashi M, Uesugi K, Zhang H. Directmeasurement procedure for three-dimensional local crackdriving force using synchrotron X-ray microtomography.Acta Mater 2008;56:6027–39.

[18] Wang MY, Williams JJ, Jiang L, De Carlo F, Jing T, Chawla N.Three dimensional (3D) experimental characterization byx-ray synchrotron tomography and phase-field simulations.Scripta Mater 2011;65:855–8.

[19] Wang MY, Williams JJ, Jiang L, De Carlo F, Jing T, Chawla N.Three dimensional (3D) microstructural characterization andquantitative analysis of solidified microstructures inmagnesium alloys by x-ray synchrotron tomography.Metallogr Microstruct Anal 2012;1:7–13.

[20] Williams JJ, Yazzie KE, Phillips NC, Chawla N, Xiao X, De CarloF. On the correlation between fatigue striation spacing andcrack growth rate: a 3D x-ray synchrotron tomography study.Metall Mater Trans 2011;A42:3845–8.

[21] Torquato S. Random heterogeneous media: microstructureand improved bounds on the effective properties. Appl MechRev 1991;44:37–76.

[22] Torquato S. Exact expression for the effective elastic tensor ofdisordered composites. Phys Rev Lett 1997;79:681–4.

[23] Pham DC, Torquato S. Strong-contrast expansions andapproximations for the effective conductivity of multiphasecomposites. J Appl Phys 2003;94:6591–602.

[24] Lu B, Torquato S. Lineal path function for randomheterogeneousmaterials. Phys Rev 1993;A45:922–9.

[25] Torquato S, Avellaneda M. Diffusion and reaction inheterogeneous media: pore size distribution, relaxationtimes, andmean survival time. J Chem Phys 1991;95:6477–89.

[26] Torquato S, Beasley JD, Chiew YC. Two-point cluster functionfor continuum percolation. J Chem Phys 1988;88:6540–7.

[27] Torquato S. Interfacial surface statistics arising in diffusionand flow problems in porous media. J Chem Phys1986;85:4622–8.

[28] Roberts AP. Statistical reconstruction of three-dimensionalporous media from two- dimensional images. Phys Rev 1997;E56:3203–12.

[29] Yeong CLY, Torquato S. Reconstructing random media. PhysRev 1998;E57:495–506.

[30] Yeong CLY, Torquato S. Reconstructing random media II.Three-dimensional media from two-dimensional cuts. PhysRev 1998;E58:224–33.

[31] Fullwood DT, Niezgoda SR, Kalidindi SR. Microstructurereconstructions from 2-point statistics using phase-recoveryalgorithms. Acta Mater 2008;56:942–8.

[32] Hajizadeh A, Safekordi A, Farhadpour FA. A multiple-pointstatistics algorithm for 3D pore space reconstruction from 2Dimages. Adv Water Resour 2011;34:1256–67.

[33] Tahmasebi P, Sahimi M. Cross-correlation function foraccurate reconstruction of heterogeneous media. Phys RevLett 2013;110:078002.

[34] Sheehan N, Torquato S. Generating microstructures withspecified correlation functions. J Appl Phys 2001;89:53–60.

[35] Rozman MG, Utz U. Uniqueness of reconstruction ofmultiphase morphologies from two-point correlationfunctions. Phys Rev lett 2002;89(13):135501.

[36] Jiao Y, Stillinger FH, Torquato S. A superior descriptor ofrandom textures and its predictive capacity. Proceedings ofthe National Academy of Sciences of the United States ofAmerica; 2009. p. 17634–9 106.

[37] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneousmaterials via two-point correlation functions: basicprinciples. Phys Rev 2007;E76:031110.

[38] Jiao Y, Stillinger FH, Torquato S. Modeling heterogeneousmaterials via two-point correlation functions: II.Algorithmic details and applications. Phys Rev 2008;E77:031135.

[39] Jiao Y, Pallia E, Chawla N. Modeling and predictingmicrostructure evolution in lead/tin alloy via correlationfunctions and stochastic material reconstruction. Acta Mater2013;61:3370–7.

[40] Singh S, Williams JJ, Jiao Y, Chawla N. Modeling anisotropicmultiphase heterogeneousmaterials via direction correlationfunctions: simulations and experimental verification. MetallMater Trans 2012;A43:4470–4.

[41] Zachary CE, Torquato S. Improved reconstructions of randommedia using dilation and erosion processes. Phys Rev 2011;E84:056102.

[42] Frary ME, Schuh CA. Correlation-space description of thepercolation transition in composite microstructures. PhysRev E 2007;76:041108.

[43] Wilding SE, Fullwood DT. Clustering metrics for two-phasecomposites. Comput Mater Sci 2011;50:2262–72.

[44] Basinger J, Homer ER, Fullwood DT, Adams BL. Susceptiblegrain boundary percolation 304 stainless steel. Scrip Mat2005;53:959.

[45] Gueguen Y, Dienes J. Transport properties of rocks fromstatistics and percolation. Math Geol 1989;21:1–13.

[46] Fullwood DT, Kalidindi SR, Niezgoda SR, Fast A, Hampson N.Gradient-based microstructure reconstructions from

Page 10: Accurate modeling and reconstruction of three-dimensional ...chemlabs.princeton.edu/.../12/files/publications/papers//paper-356.pdf · efficient tool for the modeling and characterization

42 M A T E R I A L S C H A R A C T E R I Z A T I O N 8 9 ( 2 0 1 4 ) 3 3 – 4 2

distributions using fast Fourier transforms. Mater Sci Eng A2008;494:68–72.

[47] Kirkpatrick S, Gelatt CD, Vecchi MP. Optimization bysimulated annealing. Science 1983;220:671–80.

[48] Jiao Y, Stillinger FH, Torquato S. Geometrical ambiguity of pairstatistics. II. Heterogeneous media. Phys Rev 2010;E82:011106.

[49] Gommes CJ, Jiao Y, Torquato S. Density of states for aspecified correlation function and the energy landscape. PhysRev Lett 2012;108:080601.

[50] Guo E-Y,WangM-Y, Jing T, ChawlaN. Temperature-dependentmechanical properties of an austenitic–ferritic stainlesssteel studied by in situ tensile loading in a scanningelectron microscope (SEM). Mater Sci Eng 2013;A580:159–68.

[51] Chawla N, Ganesh VV, Wunsch B. Three-dimensional (3D)microstructure visualization and finite element modeling ofthe mechanical behavior of SiC particle reinforced aluminumcomposites. Scripta Mater 2004;51:161–5.