EBSD-based continuum dislocation microscopy Calvin J. Gardner * , Brent L. Adams, John Basinger, David T. Fullwood Brigham Young University, USA article info Article history: Received 19 October 2009 Received in final revised form 17 May 2010 Available online 12 June 2010 Keywords: A. Dislocations A. Grain boundaries B. Polycrystalline material C. Electron microscopy C. Boundary integral equations abstract Recent advances in high-resolution electron backscatter diffraction (EBSD)-based micros- copy are applied to the characterization of elastic fields and incompatibility structures near the grain boundaries (GBs) in polycrystals. Two main recoveries are reported here: surface geometrically necessary dislocation (density) tensors, as described by Kröner, and the elas- tic fields near cracks (unconsolidated portions of interface) in loaded samples. Context for the application of these recoveries is described, using Green’s function solutions for com- bined heterogeneity and dislocation. Featured recoveries required the cross-correlation based determination of the elastic distortion tensor, aided by application of the simulated pattern method, and determination of the absolute pattern center utilizing the expected pattern properties in a spherical Kikuchi reference frame. High-resolution data obtained along an ultrasonically consolidated nickel boundary of varying amalgamation indicates that the imposed traction free boundary condition at free surfaces is well observed in the data structure. Further, high-resolution data acquired near a single grain boundary in well-annealed, low content steel suggests that it may be possible to measure the intrinsic elastic properties of GBs. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction This paper describes recent advances in scanning electron microscopy that relate to the continuum theory of dislocations, as applied to polycrystals. The simplest possible view is described here – that of single-phase polycrystals, where the main defect structures are dislocations, within the constituent crystallites, and in the grain boundaries that separate the primary grains of the material. That which is observable about these defects is the (elastic) self-stresses that preserve the compati- bility of the displacement field. These distortions of the crystal lattice can be measured in the electron backscatter diffraction (EBSD) patterns. This paper will focus exclusively upon EBSD-based microscopy. The notion that incompatible strains must be coupled with compensating elastic strains in order to maintain compatibil- ity, reaches back at least to the foundational work of Eshelby (1957), where these incompatible strains were referred to as stress-free transformation strains in treating ellipsoidal inclusion problems. Nye’s (1953) geometrical theory of dislocation was extended to include consideration of self-stresses in the masterwork of Kröner (1958). A significant retrospective of that work, written by Kröner himself, was published in 2001. Much of the focus of this paper will be with Kröner’s work in mind; and to the extent possible his notation will be maintained here. For those who might prefer a systematic, detailed presen- tation of the continuum theory of dislocations, the monographs of Teodosiu (1982) and Mura (1987) are recommended. The first observations of diffraction patterns, taken in the backscattering mode, were reported by Nishikawa and Kikuchi (1928). These images were recorded on film. The physics of backscattering was first detailed in the work paper of Alam et al. (1954). Venables and co-workers (1973, 1977) seem to have been the first investigators to employ a camera located in the 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.05.008 * Corresponding author. Address: Department of Mechanical Engineering, Brigham Young University, 435 CTB, Provo, UT 84602, USA. Tel.: +1 970 449 2889. E-mail address: [email protected](C.J. Gardner). International Journal of Plasticity 26 (2010) 1234–1247 Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas
Transcript
EBSD-based continuum dislocation microscopy
Calvin J. Gardner *, Brent L. Adams, John Basinger, David T. FullwoodBrigham Young University, USA
a r t i c l e i n f o
Article history:Received 19 October 2009Received in final revised form 17 May 2010Available online 12 June 2010
Keywords:A. DislocationsA. Grain boundariesB. Polycrystalline materialC. Electron microscopyC. Boundary integral equations
a b s t r a c t
Recent advances in high-resolution electron backscatter diffraction (EBSD)-based micros-copy are applied to the characterization of elastic fields and incompatibility structures nearthe grain boundaries (GBs) in polycrystals. Two main recoveries are reported here: surfacegeometrically necessary dislocation (density) tensors, as described by Kröner, and the elas-tic fields near cracks (unconsolidated portions of interface) in loaded samples. Context forthe application of these recoveries is described, using Green’s function solutions for com-bined heterogeneity and dislocation. Featured recoveries required the cross-correlationbased determination of the elastic distortion tensor, aided by application of the simulatedpattern method, and determination of the absolute pattern center utilizing the expectedpattern properties in a spherical Kikuchi reference frame. High-resolution data obtainedalong an ultrasonically consolidated nickel boundary of varying amalgamation indicatesthat the imposed traction free boundary condition at free surfaces is well observed inthe data structure. Further, high-resolution data acquired near a single grain boundary inwell-annealed, low content steel suggests that it may be possible to measure the intrinsicelastic properties of GBs.
! 2010 Elsevier Ltd. All rights reserved.
1. Introduction
This paper describes recent advances in scanning electron microscopy that relate to the continuum theory of dislocations,as applied to polycrystals. The simplest possible view is described here – that of single-phase polycrystals, where the maindefect structures are dislocations, within the constituent crystallites, and in the grain boundaries that separate the primarygrains of the material. That which is observable about these defects is the (elastic) self-stresses that preserve the compati-bility of the displacement field. These distortions of the crystal lattice can be measured in the electron backscatter diffraction(EBSD) patterns. This paper will focus exclusively upon EBSD-based microscopy.
The notion that incompatible strains must be coupled with compensating elastic strains in order to maintain compatibil-ity, reaches back at least to the foundational work of Eshelby (1957), where these incompatible strains were referred to asstress-free transformation strains in treating ellipsoidal inclusion problems. Nye’s (1953) geometrical theory of dislocationwas extended to include consideration of self-stresses in the masterwork of Kröner (1958). A significant retrospective of thatwork, written by Kröner himself, was published in 2001. Much of the focus of this paper will be with Kröner’s work in mind;and to the extent possible his notation will be maintained here. For those who might prefer a systematic, detailed presen-tation of the continuum theory of dislocations, the monographs of Teodosiu (1982) and Mura (1987) are recommended.
The first observations of diffraction patterns, taken in the backscattering mode, were reported by Nishikawa and Kikuchi(1928). These images were recorded on film. The physics of backscattering was first detailed in the work paper of Alam et al.(1954). Venables and co-workers (1973, 1977) seem to have been the first investigators to employ a camera located in the
0749-6419/$ - see front matter ! 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijplas.2010.05.008
* Corresponding author. Address: Department of Mechanical Engineering, Brigham Young University, 435 CTB, Provo, UT 84602, USA. Tel.: +1 970 449 2889.E-mail address: [email protected] (C.J. Gardner).
International Journal of Plasticity 26 (2010) 1234–1247
Contents lists available at ScienceDirect
International Journal of Plasticity
journal homepage: www.elsevier .com/locate / i jp las
chamber of the electron microscope for recording these EBSD patterns. Fully-automated scanning and real-time analysis ofEBSD-patterns to form images was first reported by Adams and co-workers (1993). The acronym OIM, which stands for ori-entation imaging microscopy, is used to describe 2-D images of microstructure, where the constituent features are discrim-inated by lattice orientation. Numerous refinements and applications of OIM and EBSD-related microscopy have beendetailed in the monographs edited by Schwartz and co-workers (2000). Various simulations have been developed that val-idate measurements made using OIM based methods (Petit et al., 2007; St-Pierre et al., 2008; Zhao et al., 2008).
Using conventional OIM capabilities, Sun et al. (2000) made the first measurements of Nye’s dislocation density tensor –in an Al bicrystal. The resolution of the conventional EBSD techniques, based upon the Hough transformation for indexinglattice orientation, has been addressed by El-Dasher et al. (2003), and Field et al. (2005), and more recently by Kacherand Adams (2009). The Hough-based methods achieve a limiting angular resolution of !0.5", which restricts their applica-tion to materials with high dislocation densities (>1015 m"2). Of particular interest here, however, are the more recenthigh-resolution techniques pioneered by Troost and co-workers (1993), and more particularly by Wilkinson et al. (2006).Important advances to Wilkinson’s approach, named the simulated pattern method (SPM), were recently introduced by Kacheret al. (2009). This paper describes the application of the Wilkinson method, and its SPM augmentation, to the characteriza-tion of dislocation tensor fields. It will not describe the higher-order spatial correlation functions of the dislocation tensorfield; if the reader is interested in these ideas, the recent work of Landon et al. (2008) can be accessed.
The plan for the paper is as follows: A brief review of the fundamental relationships of continuum dislocation theory ispresented in Section 2. This presentation will necessarily be limited to just a brief sampling of the many results that areavailable. Section 3 will present a short exposition of the high-resolution EBSD-based methods, with an emphasis placedupon the particular experimental challenges encountered, such as precision placement of the pattern center. Section 4 pre-sents some experimental recoveries of elastic stress fields, lattice curvature fields, and surface dislocation densities. In Sec-tion 5, a brief summary discussion and some conclusions are presented.
2. Basic relations of continuum dislocation theory
2.1. The dislocation density tensor
Our purpose for this section of the paper is to describe the context for the experimental measurements. Microstructurerepresentation, with its varied and multiplied possibilities, must be guided by physical theory to settle its context. Theauthors contend that localization relations present a compelling context for EBSD-based microscopy, and vice-versa. It isillustrated that the details of the local stress condition are dependent upon the local elastic polarization, and upon the fieldof dislocation, contained in both the grain interiors and the grain boundaries.
In this section we wish to connect the observable features (‘observable’ meaning that the information can be gleanedfrom EBSD patterns) of a given sample with the associated stress fields that are known to determine the response of thematerial to mechanical loading. The parameters that are immediately extracted from an EBSD pattern include the orientationof the local crystal lattice, g, and components of the local elastic strain tensor, eij (the next section details the methodology forrecovering these values). One particularly important structure metric that can be inferred from these measurements is thedislocation density tensor.
The dislocation density (encompassing only geometrically necessary dislocations) may be described by a second-ranktensor, with components aij, which gives the line density of dislocations with Burgers vectors in the direction of xj and linesextending in the direction of xi. Nye originally defined this tensor using only the lattice curvature components, jij, but laterKröner included the effect of the lattice strain in a form that can be expressed by the following representation (Kröner, 1981)
aki # jki " dkijpp $ 2kljeeij;l; %1&
where !ijk is the permutation tensor. This representation has been used previously to connect the measured parameters withthe dislocation structure (e.g. Landon et al., 2008).
This relation may be immediately derived from a consideration of the elastic distortion tensor, b, where bij # @uej =@xi are
spatial derivatives of the elastic displacement field. Fundamental to these developments is the idea that compatibility of thetotal displacement field requires that
a # curlb # "curlbp
i:e: ahi # 2hljbji;l%2&
where bp is the plastic distortion tensor. It is this fundamental curl-free condition that enables the high-resolution EBSD-based experimental measurements to be connected to the constructs of the continuum dislocation theory of Kröner(2001). This relationship is utilized implicitly in the results below as we employ a Green’s function approach to connectthe local stress field to the measurable quantities that we have just discussed.
Before considering the local field equations, we also introduce the surface dislocation tensor, defined at surface R betweenregions I and II in the material:
ahi # 2hljfbji%II& " bji%I&gnl; %3&
where n is the unit normal vector on the surface directed from domain I into domain II.
C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247 1235
2.2. Green’s function solutions for combined heterogeneity and dislocations
The Green’s function solutions to the underlying field equations have been solved for both the case of a (heterogeneous)polycrystal (Fullwood et al., 2010; Kröner, 1986; Torquato, 2002), and for the case of a field of dislocations (Mura, 1987). Tothe author’s knowledge, the combined case has not been considered, or published. Here these two contributions are consid-ered together, with their intrinsic coupling.
Let eij represent the total strain in the material, eij represents the elastic strain, and e'ij represents the ‘eigenstrain’ (theterminology of Mura (1987)) caused by the presence of dislocations in the sample. Note that the connection with the distor-tion tensor introduced above is given by eij # 1
2 %bij $ bji&. Then
eij # eij $ e'ij: %4&
And the compatibility relation may be written:
eij #12%ui;j $ uj;i& %5&
in terms of the displacement components ui and their spatial derivatives, as denoted by ui,j. We also have the elastic consti-tutive relations:
rij # Cijklekl # Cijkl%ekl " e'kl&: %6&
It suits our purposes here to write the local stiffness tensor in terms of a perturbation from a reference stiffness tensor:
Cijkl%x& # Crijkl $ C 0ijkl%x&: %7&
In the absence of external body forces, the equilibrium condition rij,j = 0 may be combined with the previous two equationsto give:
Crijkl $ C 0ijkl%x&
! "ekl%x& " e'kl%x&# $h i
0j# 0;
Crijklekl;j%x& $ C 0ijkl%x&ekl%x& " Cr
ijkl $ C 0ijkl%x&! "
e'kl%x&h i
0j# 0:
%8&
From the symmetry of Crijkl we may write Cr
ijklekl;j # Crijkluk;lj, and combining the second term into a fictitious body force:
Fi%x& # C 0ijkl%x&ekl%x& " Crijkl $ C 0ijkl%x&
! "e'kl%x&
h i
0j: %9&
Thereby we arrive at the fundamental differential equation that we wish to solve:
Crijkluk;lj $ Fi # 0: %10&
Define a Green’s function, G, such that:
CrijklGkm;lj%x" x0& $ d%x" x0&dim # 0: %11&
Now multiply Eq. (10) by G, and integrate. Using integration by parts twice on the first term to obtain second derivatives of G,and then using Eq. (11):
uk%x& #Z
VGki%x" x0&Fi%x0&dx0: %12&
Hence
ekl%x& #Z
V
12
Gki;l%x" x0& $ Gli;k%x" x0&# $
Fi%x0&dx0
#Z
V
12
Gki;l%x" x0& $ Gli;k%x" x0&# $
C 0ijmn%x0&emn%x0& " Cr
ijmn $ C 0ijmn%x0&
! "e'mn%x
0&h i
0jdx0: %13&
Using Eq. (4) this becomes:
ekl%x& #Z
V
12
Gki;l%x" x0& $ Gli;k%x" x0&# $
C 0ijmn%x0&emn%x0& " Cr
ijmne'mn%x0&
h i
0jdx0: %14&
To deal with the singularity in the Green’s function at x = x’ we integrate by parts between a sphere about the singularity andan infinite sphere (Torquato, 1997):
1236 C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247
ekl%x& #Z
S
12%Gki;l%x" x0& $ Gli;k%x" x0&&(C 0ijmn%x
0&emn%x0& " Crijmne'mn%x
0&)nj dS% & x0!1
sphere with
x0!xsphere with
"Z
V 0
12%Gki;lj%x" x0& $ Gli;kj%x" x0&& C 0ijmn%x
0&emn%x0& " Crijmne'mn%x
0&h i
dx0: %15&
The volume integral may be compared with Mura (1987, Eq. (6.1)) for the case that C is constant. Ignoring the eigenstrain surfaceintegrals for now; let them be combined in the tensor, S(x). Consider the value of the elastic surface integrals. For the integral onthe infinitesimal sphere x0 ? x one may assume that the local stiffness and strain are approximately constant, and can be movedoutside of the integral sign. Then for an isotropic reference tensor we may write (Kröner, 1986; Torquato, 2002),
Eijkl #Z
S:x'!x
12%Gki;l%x" x0& $ Gli;k%x" x0&&nj dS # 1
15lk$ lk$ 2ldijdkl "
3k$ 8lk$ 2l Iijkl
' (: %16&
The integral on the infinite sphere may be approximated as (Fullwood et al., 2009):
I1 * EC0e1; %17&
where e1 is the applied strain at infinity. Hence we may rewrite the elastic surface integrals as:
Z
S
12%Gki;l%x" x0& $ Gli;k%x" x0&&C 0ijmn%x
0&emn%x0&nj dS% & x0!1
sphere with
x0!xsphere with
* EC0e1 " EC0%x&e%x&: %18&
Clearly, if CR # !C (the over-bar signifying the volume average) then C0 # 0, and I1, as calculated from Eq. (17), tends to zero.Using the usual notation (Fullwood et al., 2009; Kröner, 1986):
The right hand side of Eq. (21) could be reached directly from the eigenstrain component of Eq. (14); hence we can deal withthe singularity relating to the eigenstrain term (temporarily referred to as S(x) above) by integrating Eq. (21) by parts:
Z
V2lnhGki;j%x" x0&Cr
ijmnahm%x0&dx0 #Z
S2lnhGki%x" x0&Cr
ijmnahm%x0&nj ds% &x0!1
sphere
x0!xsphere
"Z
V 02lnhGki%x" x0&Cr
ijmnahm;j%x0&dx0: %23&
We treat two surface integrals as for the elastic case (Eq. (18)), with an approximate value for the infinite sphere. Let
Sijk%x" x0& #Z
S:x0!xGki%x" x0&nj ds: %24&
Then
2lnhGki%x" x0&Crijmnahm%x0&nj ds
h ix0!1sphere
x0!xsphere * 2lnhSijkCr
ijmn%a1hm " ahm%x&&: %25&
Then
rpq%x& # Cpqkl%x&Eklmn%C 0mnrse1rs " C 0mnrs%x&ers%x&& " Cpqkl%x&2lnhSijkCr
ijmn%a1hm " ahm%x&&
$ Cpqkl%x&Z
V 02lnhGki%x" x0&Cr
ijmnahm;j%x0&dx0 " Cpqkl%x&Z
V 0UklijC
0ijmn%x
0&emn%x0&dx0: %26&
C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247 1237
Although quite complex in the form of its various terms, relation (26) defines the familiar dependence of local stress at posi-tion x upon the detailed spatial placement of lattice orientation, through terms such as the elastic stiffness tensor, or itspolarization. Less familiar is its dependence upon the placement of dislocations, throughout the body. Dislocation densityand orientation measurements taken using high resolution analysis of EBSD patterns may be inserted into this solution toinvestigate issues of stress localization, fatigue initiation, and other local phenomena that have been inaccessible to detailedanalysis in the past. However, care must be taken when incorporating the surface dislocation density into the volumeintegral.
3. High resolution cross-correlation analysis of EBSD patterns
3.1. Diffraction geometry
An EBSD system consists of a scanning electron microscope (SEM), a sample tilted at 70" from the horizontal, and a phos-phor screen with a CCD (charge coupled device) camera to view the diffraction patterns. For EBSD, a beam of electrons isdirected to a point of interest on a tilted crystalline sample (Fig. 1). The incident electrons are diffracted from planes withinthe crystalline sample according to Bragg’s Law. Diffracted electrons from a particular {h k l} plane form a pair of wide anglecones. When these cones intersect the fluorescent phosphor screen, the intense area between them is known as a Kikuchiband. The center-line of a Kikuchi band nominally corresponds to the intersection of the diffracting plane with the phosphorscreen. Therefore, changes in lattice structure from elastic strains in the material result in shifted Kikuchi bands.
Fig. 2 illustrates how an elastic strain produces a shift on the phosphor screen. In the figure, q is the measured shift, r isthe unit vector from the specimen origin to the center of a region on the phosphor screen, r0 is the shifted unit vector fromthe specimen origin, rpc is a unit vector normal to the phosphor screen from the sample origin, and z* is the perpendiculardistance from the screen to the sample origin. The displacement under deformation is shown by the vector Q, whereQ # br; b is the elastic distortion tensor, (b + I) = F, and F is the local deformation gradient tensor. With these definitionswe can write the following geometric relationship.
qk# Q " %Q + r0&r0 $ q + r0
kr0; %27&
where
k # z'
rpc + r: %28&
Eq. (27) contains three independent equations, one for each component of q. As is apparent in Fig. 2, r is easily calculatedfrom knowledge of microscope geometry and the appropriate coordinate frame. Subsequently, r0 can be evaluated usingthe measured shifts q (to be discussed in the next section) and, therefore, the elastic distortion tensor is the only remainingunknown.
Fig. 1. Schematic of SEM. The electron beam from the pole piece diffracts off the sample and forms a pattern on the screen of the EBSD detector. Figurecourtesy of TSL.
1238 C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247
3.2. Cross-correlation analysis of EBSD images, recovery of displacement gradient tensor
Troost et al. and Wilkinson et al. demonstrated how the use of cross-correlation analysis can be used to recover the dis-placement gradient tensor and to increase angular sensitivity by two orders of magnitude beyond the Hough/Radon trans-form (Troost et al., 1993; Wilkinson et al., 2006). The reduced error in misorientation axis determination is sufficient toenable an estimation of lattice curvature and to recover some components of Nye’s dislocation tensor (Nye, 1953). Moreover,increased accuracy facilitates the partial recovery of the elastic distortion tensor.
Wilkinson’s method compares two measured EBSD patterns utilizing the small shifts in the position of pattern features todetermine the difference in strain and orientation between the patterns. Therefore, in order to obtain absolute measurements,one of the two compared patterns must be a strain free reference pattern at (or near) the correct orientation. The patterns arecompared by selecting a number of regions of interest (ROIs) distributed over each pattern. The cross-correlation between ROIsin the reference and experimental patterns are then calculated using Fourier transforms as follows:
P # J"1fJffg ' conj%Jfgg&g; %29&
where Jfg is the Fourier transform, conj( ) is the complex conjugate, f and g are corresponding ROIs from the two patterns, *indicates the element wise multiplication of matrices, and P is the resulting image. The line emanating from the pattern cen-ter to the peak in each of the ROI cross-correlation images gives the shift vector q for that ROI. The shift is assumed to beequal to the average shift in the center of the ROI and is measured perpendicular to r. Combining equations for componentsof q results in two simultaneous equations:
Initially it appears that the two equations can be solved to obtain all nine components of b. However, in magnitude, q, rand consequently, r and r0 are not sufficiently discernible. Therefore, realizing that r0 * r, the two simultaneous equationsbecome:
In these equations, the differences between diagonal components of b are treated as single unknowns for a total of eightunknowns in two equations. By measuring the shift vectors of at least four ROIs, the components of b can be solved. Unfortu-nately, Wilkinson’s method cannot fully resolve the terms along the diagonal of b, only the difference between them isobtainable. Still, if knowledge of the crystal elastic constants is available, traction free boundary conditions may be applied, con-sistent with the presence of the free surface of the sample, and the final of the 9 degrees of freedom can be resolved. In addition tothe three independent equations from the components of the above relationship, the traction free boundary condition results inthree additional equations:
Fig. 2. Illustration of how elastic strain within a lattice produces a measurable shift upon the phosphor screen.
C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247 1239
0 # rij rpcj # Cijkleklrpc
j ; %34&
where r is the stress, C is the elastic stiffness, and e is the strain. Once the displacement gradient tensor is determined it is asimple step to find the strain tensor and the rotation tensor as follows:
eij #12%bij $ bji& and xij #
12%bij " bji&; %35&
where e, the infinitesimal strain tensor, is the symmetric part of b, and x, the infinitesimal rotation tensor, is the anti-sym-metric part.
Wilkinson’s method represents a substantial advance in the angular and strain resolution of EBSD analysis and opens thedoor for accurate analysis of lattice curvature, elastic strain, and geometrically necessary dislocation (GND) densities. How-ever, without a strain free pattern the Wilkinson method is limited to measuring gradients of elastic strain and lattice ori-entation rather than absolute values. This makes comprehensive application of Wilkinson’s method to a polycrystallinesample difficult, particularly for small grain sizes and plastically deformed samples. The simulated pattern method presentedin the recent work of Kacher et al. (2009) offers a potential solution to these difficulties.
3.3. Simulated pattern method
In order to avoid the difficulty of obtaining a strain free pattern, it is possible to use a simulated reference pattern. Becausehigh fidelity simulations are computationally expensive, the simulated pattern method uses a simple kinematical model(Bragg’s Law based) to generate a strain-free reference pattern. By iteratively generating these simple patterns at each cal-culated deformation state of a measured pattern, and then repeating the calculation with the new simulation, a high reso-lution result is rapidly found by convergence. The simulated pattern method augmentation of Wilkinson’s method ispresented in detail in Kacher et al. (2009). Here a brief presentation of some of the key ideas is repeated to facilitate conti-nuity in the presentation of this paper.
The deformation tensor F, determines how diffraction cones are oriented with respect to the phosphor frame and mayalso alter the inter-planar spacing dhkl. Combining the equation for a cone with the various parameters determining orien-tation and intensity results in the following equation for a simulated Kikuchi band:
B%~p; F;Rv!c;Rc!co; %hkl&& # S2hkl if %(Rc!coFRv!c~p)1&
2 $ %(Rc!coFRv!c~p)2&2 P %(Rc!coFRv!c~p)3&
2
tan%h&
! "2
0 otherwise
8<
:
9=
;; %36&
where B is the simulated band,~p represents a point in the phosphor screen reference frame v, Rv?c is the rotation tensor fromthe phosphor screen frame to the crystal frame c, Rc?co is the rotation tensor from the crystal frame to the cone referenceframe co, (h k l) is the chosen diffraction plane, h is the cone angle, and Shkl is the structure intensity factor. Summing thecontributions of each band and its symmetry variants generates the complete approximation of the EBSD image, thus:
I%~p; F& #X
i
X
j
B%~p; F; S%i&; %hkl&%j&&; %37&
where I is the composite image, S(i) are the elements of the symmetry subgroup, and (h k l)(j) are the elements of the set thatincludes all of the considered diffracting planes. The final composite pattern is then filtered using high and low pass filters tomore accurately reflect variation in the measured EBSD pattern background.
The simulated pattern analysis algorithm begins by measuring a local lattice orientation to within 0.5" using the Houghtransform method of standard OIM. A simulated pattern is then generated from the known crystal structure and the Houghtransform estimate of the orientation. Cross-correlation analysis as described above is used to compare the ROI shifts fromthe simulated pattern and the measured EBSD image. The deformation tensor is calculated using geometric relationshipsand the traction free boundary condition. The fit of the deformation tensor is evaluated by calculating the average error ordifference between measured and calculated shifts. Finally, a new pattern is simulated at a deformation state closer to the actualstate of the material. This process is repeated iteratively until the deformation converges as close as possible to that of the actuallattice structure.
At present, the resolution of the simulated pattern method does not reach the level of Wilkinson’s method. The simulatedpattern method can determine the strain and rotation tensors to a resolution of ±7 - 10"4 indicating a misorientation res-olution of ±0.04". However, though the Wilkinson’s method is more accurate, the simulated pattern method readily extendsthe high resolution advantages to polycrystalline and deformed samples where Wilkinson’s method becomes difficult.
3.4. Pattern center location
Knowledge of the absolute pattern center is required for greater accuracy in resolution of angular crystal orientation, anddeformation gradient determination. An incorrect pattern center on the phosphor screen means that the simulated patternwill be centered at some displacement from the electron source on the sample (see Fig. 3). This incorrect assumption addsstrain artifacts to the simulated patterns used to iteratively match collected EBSD patterns (Villert et al., 2009). Pattern cen-
1240 C.J. Gardner et al. / International Journal of Plasticity 26 (2010) 1234–1247
ter approximations can be achieved using ‘strain-free’ samples with a known orientation for calibration. However, there isalways some error due to present elastic strains, and potentially due to errors in the assumed orientation.
A more generic (and accurate) method can be developed by observing that when the electrons leave the sample, they cre-ate a pattern of bands (with parallel edges) that are centered upon great circles when projected onto any sphere that is cen-tered on the interaction point on the sample. When the electrons hit the phosphor screen, the pattern is distorted due to theflat surface. If the pattern center is correctly identified, and one maps the image back onto a sphere that has its center givenby this pattern center, then one again arrives at a pattern with parallel bands centered on great circles. If the pattern center isguessed, then mapping onto the sphere from the phosphor image will cause a distorted pattern on the sphere, with bandsthat lose their properties of parallelism and are no longer centered on great circles. The nature and magnitude of this dis-tortion is directly determined by the error in the pattern center approximation.
Development of a framework to accurately determine the pattern center based upon this approach has established thatvery slight shifts in pattern center (1/10 of a pixel), on the order of the sensitivity of HROIM, cause noticeable changes in theparallelism of bands when rotated into the spherical unit sphere reference frame (Kacher et al., 2010). Several methods fordetermining the distortion that occurs when mapping the phosphor image back onto a sphere are possible. They include useof kinematical simulation of diffraction bands in the spherical frame, and the use of a spherical Hough transform on bandedges. Here, we describe an approach utilizing the simulated pattern method and summation of intensities.
The parallelism method of pattern center calibration uses a band profile obtained by the integration of pixel intensitiesthat belong to the actual EBSD pattern and lie in between the simulated Kikuchi band edges. As illustrated in Fig. 4, the sim-ulated bands and the actual EBSD patterns are overlaid in a spherical Kikuchi reference frame (see Day (2008) for an over-view of spherical EBSD). A coordinate frame transformation based on a guessed pattern center rotates the real EBSD patternonto the unit sphere and the simulated bands are calculated based on the orientation and crystal parameters. EBSD pointsfrom a square grid mapped onto the sphere at a given pattern center have non-uniform spacing, density, and the grid will nolonger be square.
Next, crystal symmetry is applied to each reflecting plane, which re-orients the simulated bands along other great circleson the unit sphere. Repetitive or non-unique bands are disregarded for efficiency. For each band that intersects the actualEBSD pattern, the intensity profile is calculated within the simulated band edges by averaging the intensities down thelength of the band in small steps moving from one edge to the other. The simulated band edges are artificially widenedto encompass real EBSD band edges that might not be aligned with the simulated edges in an initial guess. Several indirectmeasures of parallelism from these band intensity profiles are possible, including trough intensity minimization and the dis-tance of these troughs/band edges from the great circle the simulated bands are centered on. Such measures of band edgeparallelism were applied in the pattern center recovery for strain-free single crystal germanium (Kacher et al., 2010). Theprocess is repeated for all {h k l} planes and symmetric bands being simulated.
A single parallelism score is generated from the band intensities for the particular pattern center and orientation simu-lated. An iterative optimization routine repeats the process and searches for the most parallel fit. This, then, is the absolutepattern center, where the simulated bands are most accurately aligned with captured EBSD bands. Generally, the optimiza-tion is done in a discrete way (genetic algorithm, simulated annealing, and so forth) due to variance in the fitness score atsmall changes in pattern center and orientation. Note that the right half of Fig. 4 indicates fairly large steps in pattern center,
Fig. 3. The pattern center is the x*, y* location on the phosphor screen at z* distance from the sample where the reflected beam intersects the screen at 90".For an EBSD pattern, incorrect pattern center gives misleading information about sample geometry.
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at which scale the search space appears smooth. Only two dimensions of the search space are shown, out of the six param-eters used (x*, y*, z*, u1, U, u2).
This is an initial approach to the recovery of absolute pattern center. Other methods mentioned above, such as the spher-ical Hough transform, are being developed to eliminate the simulated spherical band edge method’s reliance on accuratelyobtaining orientation and strain, which are additional dimensions that complicate the optimization problem.
4. Experimental recovery of micro-fields in polycrystals
Two examples of micro-field recovery by high-resolution EBSD-based methods are described in this section. In the firstexample precision recovery of the elastic strain field within the grains of a small section of polycrystal is presented. Theexample selected was prepared from !80 lm thick foils of commercial purity Ni 201, built up by ultrasonic consolidation(UC). Further details of UC lamination and material texture were presented in a previous paper (Adams et al., 2008). Thissample was placed under a static compressive stress, using a C-clamp, and held under load during microscopic evaluation.A region of this material with imperfect consolidation was deliberately selected in order to evaluate whether the recoveredhigh-resolution stress fields satisfy the boundary conditions naturally imposed by the presence of free surface (unconsoli-dated interface) within the material.
In the second example the lattice curvature field near a single grain boundary was considered at a very fine step size(20 nm). The material chosen was a minimum-content steel (nominally Fe), heavily annealed to achieve a large grain size(>1 mm) with columnar grain morphology. The recovery presented comprises all six components of the GB dislocation ten-sor. The underlying question raised is whether or not it is possible to measure the intrinsic state of dislocation at grainboundaries, using the high-resolution EBSD-based microscopy.
Each example was viewed in the FEI-Philips XL-30 SFEG scanning electron microscope, EBSD images were captured usinga DigiView CCD camera supplied by TSL-EDAX at full resolution (1000 - 1000 pixels). In both of these examples the inclina-tion of the interfaces of primary interest is known.
4.1. Elastic stress fields in UC laminated samples
The scan region for the UC Ni sample contained a partially consolidated boundary running horizontally across thescanned region. The non-consolidated regions can be seen on both sides of the sample in Fig. 5. Given the UC process geom-etry, the normal to the non-consolidated interfaces is known to be in the 1-direction. This sample was placed in elastic com-pression parallel to the consolidated boundary – nominally in the 2-direction. The magnitude of compression was notcontrolled, but it is believed that the stress level is high. Traction free conditions at the unconsolidated interfaces requirethat nr #~0, and therefore near these surfaces r11 = r12 = r13 = 0. These conditions are augmented by the traction free con-dition of the metallographic section plane, with normal n # e3, which requires that r31 = r32 = r33 = 0. From these two sets of
Fig. 4. Left: Germanium EBSD pattern with overlaid simulated bands for three reflecting planes. Bands that do not intersect the experimental pattern areomitted. Right: Example parallelism measurements (lower is better) using a simulated reference pattern to show convergence to the correct pattern centerat x* = 0.51494, z* = 0.73308 starting from a position predicted by OIM. Other dimensions of the optimization not show include y* and crystal orientationangles.
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boundary conditions, it follows that the unconsolidated interfaces can only support stress of the type r22. Correct recoveriesof the local elastic fields should demonstrate these characteristics. Since several grains abut the unconsolidated interfaces,
Fig. 5. Left images depict stress fields in an ultrasonically consolidated nickel sample with an obviously consolidated central region and two surroundingunconsolidated regions. The images on the right depict the average stress for each region running from the left of the scan to the right of the scan. This figuredemonstrates the validity of an imposed traction free boundary condition when considering a free surface.
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and each has a distinctive lattice orientation and differing elastic constants in the common frame, the considered boundaryconditions provide a strong test of the consistency of the recovered strain fields. Additionally, the comparison of an in-situtest and Green’s function predictions could be used to validate the remainder of the stress field. The scanned region is150 lm - 50 lm in area, and the step size was 0.5 lm. Selected maps of the resultant elastic stress field, obtained by exer-cising Hooke’s law on the measured elastic strain field, and accounting for local lattice orientation, are presented in Fig. 5.Note that the black overlaying lines represent the positions of grain boundaries on each side of the horizontal consolidatedboundary. The maps on the left show the plotted values of the stress fields and the maps on the right show the averagedvalues of the stress, as a function of horizontal location, taken from the three perceptible regions of the stress field maps.
It is apparent that in the regions where the boundary did not consolidate, in the upper and lower portions of the scan, theexpected traction free condition is measured in all of the participating grains. The r11, and r12 components, which are non-zero as result of load-shedding in the vicinity of the consolidated region, do approach zero near the traction free boundariesin the upper and lower unconsolidated regions. The r22 component, however, remains largely constant throughout thescanned area. The far-field view and wide colormap scale combined with the predominance of mesoscale mechanical stres-ses tends to washout the grain to grain stress variation in Fig. 5. Obviously, a narrower scale and near-field view could beused to examine grain to grain variations if that particular property were of interest.
One of the most difficult problems encountered with the study of stress/strain localization in polycrystals is the unknownelastic properties of grain boundaries. (Relation (26) quantifies these effects in the context of Green’s function solutions tothe stress equilibrium equation.) Although molecular dynamics simulations have given some insight into the nature of theelastic properties of GBs (Alber et al., 1992), only a very limited sampling of the space of GB character has been possible, withthe available examples typically possessing special character, such as a low-sigma coincidence site relationship. If it werepossible to directly measure these properties by high-resolution EBSD-based microscopy, this could be of great benefit tothe study of localization phenomena. To explore this possibility the authors examined a single GB in Fe. The results presentedhere do not constitute a comprehensive evaluation of requirements or possibilities – only a first attempt at recovering GBincompatibilities by high-resolution EBSD.
Minimum-content steel was heavily annealed to achieve near-columnar grain morphology, with grain size in excess of1 mm. A single sample was polished and scanned over a 1 lm - 1 lm region, containing one GB near the center of the fieldof view. In consideration of the elongated interaction volume, the grain boundary was aligned to minimize pattern overlap.Two thousand five hundred EBSD patterns were collected over a square grid of points at 20 nm step size. The full1000 - 1000 available array of pixilated data were stored for off-line analysis by the high-resolution EBSD methods de-scribed in Section 3. The plane normal associated with the featured GB was oriented nominally parallel to a plane definedby the e-beam direction and the sample normal. This is the 2-direction, as shown in Fig. 6. The consistency of image qualitythroughout the map indicates little to no grooving or pitting along the polished boundary.
Several characteristics of the featured GB are evident from Fig. 6. Two distinct lattice orientations are evident from theprimary red and green colors. The featured boundary fits the common classification as ‘random’ and ‘high angle’ (see thework of McCabe for greater analysis of twinned boundaries (McCabe et al., 2009)). Clearly evident from this false-color imageis the non-planarity of the interface. Also very evident is a region of disturbance of lattice orientation, of thickness ! ±40 nm,roughly centered on the GB. This disturbance may in fact, given the processing conditions applied to the material, be asso-ciated with the intrinsic incompatibility of the GB.
Fig. 6. False color image of the recovered lattice orientation near the studied iron grain boundary in a 1-lm square region with a 20-nm step size. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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At several points taken near the centroids of the two participating grains, reference EBSD patterns were taken. These refer-ence images were used for pattern center calibration, and to provide strain free reference patterns from which Fast FourierTransform (FFT) algorithms were applied to recover estimates of the local elastic distortion tensor. The equations used are thosegiven in (32)–(34) above. Initial estimates of lattice orientation were then corrected using the recovered off-diagonal compo-nents of the elastic distortion tensor. Pattern overlap was corrected for the patterns nearest the boundary using the techniquessuggested by Kacher et al. (2008). Forming derivatives in the 1- and 2-directions from the precision cross-correlation determi-nations between adjacent scan points, six available components of the lattice curvature tensor, j11, j21, j31, j12, j22, j32 weremeasured over the complete view field. Fig. 7 depicts the average of the magnitudes of the measured curvatures,j # 1
6
P3i#1P2
j#1jjijj, using a color scale, shown on the right of the image. The elastic disturbance of the grain boundary regionis evident in this image.
Fig. 7 depicts the average of the magnitudes of the six available components of the lattice curvature tensor. The very smallmagnitude of these curvatures is notable, as is the increased intensity near the grain boundary where average curvatures ofup to 0.14"/micron were recorded. Over a step size of 20 nm, these correspond to maximum angle differences of !0.0028".Clearly these tiny levels of curvature lie just at the threshold of measurability with the current high-resolution EBSD system.And yet, as evident in Fig. 7, there is a systematic measured presence of these effects near the GB, and over distances that arelarger than the spatial resolution limit of the system.
Following the expressions given in Section 2, the six available components of the surface dislocation tensor field are pre-sented in Fig. 8. It is presumed, in forming these values that the interface normal lies in the 2-direction, as defined in Fig. 6.Thus, in Eq. (3), n1 and n3 are zero and only as
11;as12;as
13;as31;as
32;as33 are accessible. Note that these components represent the
geometrically necessary dislocations to satisfy compatibility and not the total grain boundary dislocation density. As itwould be presumed from the very small magnitudes of the associated lattice curvatures, the measured magnitudes of thecomponents of the GB surface dislocation are also very small. Thus, the noisiness of the data in these plots is understandable.The data suggest that some components are positive, others negative. Differences of magnitude are also evident. We con-clude that a state of dislocation density or incompatibility, at this GB, may in fact be measurable, at least in a statistical sense.
5. Summary and conclusions
Elastic continuum theory, based upon solutions to the stress equilibrium equations and considerations of compatibility,clearly identify two sources of influence upon stress localization in single-phase polycrystals: the spatial heterogeneity of theelastic stiffness present within the grain structure, and the presence of dislocation within grain interiors and at grain bound-aries. Experimental capability for recovering the first of these effects is now routine, through EBSD-based microscopy of or-dinary resolution. The results presented in this paper suggest that proper application of cross-correlation based high-resolution analysis, in consort with the simulated pattern extension to Wilkinson’s method and a novel technique to deter-mine absolute pattern center location, can access the intrinsic dislocation structure associated with elastic disturbances neargrain boundaries.
Elastic strain fields measured near (internal) free surfaces in loaded samples were shown to be completely consistentwith the traction free mechanical requirement, across multiple grains that abut these surfaces. This is a strong indicationof the consistency and reliability of the elastic fields recovered by the high-resolution EBSD methods. It may be possible
Fig. 7. Average of the magnitudes of the six available components of the lattice curvature tensor, in units of degrees/micron.
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to utilize these conditions, in conjunction with samples placed under mechanical loads within the microscope, to ascertainthe presence of new/developing free surface associated with crack nucleation – even before a void is discernable.
Although the presented results are still preliminary, if these capabilities can be satisfactorily developed and exploited, itmay be possible to map the elastic (and yield) properties of individual grain boundaries back to their five degrees of
Fig. 8. The six available components of the surface dislocation tensor field calculated along the grain boundary in Fig. 6.
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crystallographic freedom, and to local loading conditions. Such a mapping could have profound significance in extendingtheory for microstructure design to the design of grain boundary character and distribution, for the purpose of mitigatingextreme value properties, such as fracture and fatigue, where it is widely held that properties/lifetimes depend upon thepeaks within the elastic stress/strain field.
However, extension of this technique to non-columnar polycrystalline samples would require three-dimensional grainstructure information and current focused ion beam scanning electron microscopy (FIBSEM) technology presents numerouschallenges. Stereologically, reliable accurate alignment of sectioned scan planes is both essential to measurement of the sur-face dislocation tensor and difficult to ensure. Furthermore, the limitations of step size between adjacent scanned planes,artifacts introduced by the FIB technique, and the constant presence of a traction free surface limit the techniques effective-ness. One possible solution to these issues which presents an opportunity for further research is to take a statistical approachto the problem based on scans of orthogonal faces.
Acknowledgements
Support from the Materials Division of the U.S. Army Research Office, under the direction of Dr. David Stepp, is gratefullyacknowledged. Additionally, support from the National Science Foundation for our work with Green’s functions is gratefullyacknowledged.
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