Grain Boundary Normal Distribution

Grain Boundary Normal Distribution edit page

In this section we discuss a stereographic approach of determining the grain boundary normal distribution from two dimensional EBSD data following the publications

D.M. Saylor, G.S. Rohrer: Determining crystal habits from observations of planar sections in J. Am. Ceram. Soc., 85(11):2799–2804, 2002.

R. Hielscher, R. Kilian, K. Marquardt, E. Wünsche: Efficient computation of the grain boundary normal distribution from two dimensional EBSD data, not yet published.

The command calcGBND computes three closely related distributions, distinguished by what is passed to it

the specimen GBND - the distribution of the boundary normals in specimen coordinates. It answers whether the boundaries prefer a direction in the sample.
the crystal GBND - the distribution of the boundary normals in crystal coordinates. It answers which lattice planes the boundaries prefer, i.e. the habit planes.
the GBCD - the grain boundary character distribution, i.e. the crystal GBND restricted to those boundaries that have one fixed misorientation.

From two dimensional data only the crystal GBND and the GBCD are accessible, since a planar section reveals the boundary traces but not the inclination of the boundary planes. The specimen GBND requires three dimensional data and is discussed at the end of this section.

The crystal GBND from two dimensional EBSD data

We start by importing some Magnesium data and reconstructing the grain structure. Magnesium deforms by tension twinning on the \(\{10\bar{1}2\}\) plane, which makes it a good example here - the twin boundaries are coherent, i.e. they actually lie on that plane.

plottingConvention.default('y↑→x');
mtexdata twins

[grains,ebsd] = calcGrains(ebsd,'angle',5*degree,'minPixel',3);

% calcGBND looks the orientations on either side of a segment up through
% gB.ebsdId, which only means something as long as every segment still runs
% between one pair of pixels - so smooth the vertex positions but leave the
% segment list alone
grains = smoothBoundary(grains,10,'noSimplify','noRefine')

CS = grains.CS;

plot(ebsd,ebsd.orientations)
hold on
plot(grains.boundary,'lineWidth',2)
hold off

ebsd = EBSDsquare (y↑→x, row↑→col)
 
 Phase  Orientations     Mineral         Color  Symmetry  Crystal reference frame
     0     46 (0.2%)  notIndexed          none                                   
     1  22833 (100%)   Magnesium  LightSkyBlue     6/mmm        X||a*, Y||b, Z||c
 
 Properties: bands, bc, bs, error, mad, oldId
 Scan unit : um
 X x Y x Z : [0 → 50] x [0 → 41] x [0 → 0]
 Normal vector: (0,0,1)
 Square grid  :137 x 167
 
 
grains = grain2d (y↑→x)
 
 Phase  Grains  Pixels    Mineral  Symmetry         Color
     1      91   22794  Magnesium     6/mmm  LightSkyBlue
 
 boundary segments: 3762 (887 µm)
 inner boundary segments: 3 (0.67 µm)
 triple points: 122
 
 Properties: meanRotation, GOS

Misorientation angle at grain boundaries

Next we separate the grain boundaries according to the misorientation angle. The tension twins of Magnesium show up as a sharp peak at 86.3 degree, so we distinguish those grain boundaries with a misorientation angle larger then 80 degree and those with a smaller misorientation angle.

gB = grains.boundary('indexed');
cond = gB.misorientation.angle > 80 * degree;

plot(ebsd,ebsd.orientations)
hold on
plot(grains.boundary,'lineWidth',3)
plot(gB(cond),'lineWidth',2,'lineColor','w')
hold off

Passing the grain boundaries together with the EBSD data to calcGBND gives the crystal GBND, i.e. the distribution of the boundary normals in crystal coordinates.

Recovering a three dimensional distribution from planar sections is a deconvolution, and without any constraint the result rings, i.e. it oscillates around the true distribution and may even become negative. The option nonneg enforces a nonnegative result, which is what we want for a density.

gbnd1 = calcGBND(gB(cond),ebsd,'halfwidth',5*degree,'nonneg')
gbnd2 = calcGBND(gB(~cond),ebsd,'halfwidth',5*degree,'nonneg')

% the twinning plane. These are plots of the full sphere, so we mark all
% symmetrically equivalent variants - marking only (10-12) would single out
% one of the six maxima for no reason
tp = Miller(1,0,-1,2,CS,'hkil');

% both are plotted with the same color range, otherwise the almost uniform
% distribution on the right hand side would be stretched to look structured
contourf(gbnd1,'colorrange',[0.5 1.5])
mtexTitle('GBND for \(\omega > 80^{\circ}\)')
mtexColorMap parula
annotate(symmetrise(tp),'labeled','backgroundColor','w')
nextAxis
contourf(gbnd2,'colorrange',[0.5 1.5])
mtexTitle('GBND for \(\omega < 80^{\circ}\)')
mtexColorMap parula
mtexColorbar

gbnd1 = S2FunHarmonic (Magnesium)
  bandwidth: 62
  antipodal: true
 
 
gbnd2 = S2FunHarmonic (Magnesium)
  bandwidth: 62
  antipodal: true

We observe that for the twinning grain boundaries the boundary plane is mostly parallel to the \(\{10\bar{1}2\}\) twinning plane, while for all other grain boundaries no preferred boundary plane exists. Indeed, the maximum of the left hand side distribution is only a few degrees away from \(\{10\bar{1}2\}\)

[value,pos] = max(gbnd1);
[value, min(angle(pos,symmetrise(tp)))./degree]

ans =
    1.3905    5.5946

while the distribution of all remaining boundaries stays close to one everywhere

[min(gbnd2),max(gbnd2)]

ans =
    0.8626    1.0868

The grain boundary character distribution (GBCD)

Splitting the boundaries by misorientation angle as above is a crude selection. Passing a misorientation as a third argument restricts the computation to those boundaries whose misorientation is close to it and aligns them accordingly. The result is the GBCD for that misorientation.

% the ideal Magnesium tension twin
moriRef = orientation.byAxisAngle(Miller(1,1,-2,0,CS,'uvtw'),86.3*degree,CS,CS);

gbcd = calcGBND(gB,grains,moriRef,'halfwidth',5*degree,'nonneg')

plot(gbcd,'contourf')
mtexTitle('GBCD for the tension twin')
mtexColorMap parula
annotate(symmetrise(tp),'labeled','backgroundColor','w')
mtexColorbar

gbcd = S2FunHarmonic (Magnesium)
  bandwidth: 62
  antipodal: true

Two of the six \(\{10\bar{1}2\}\) variants carry the maxima - the specimen was deformed, so not every twin variant is equally active. Accordingly the maximum sits right on a twinning plane

[value,pos] = max(gbcd);
[value, min(angle(pos,symmetrise(tp)))./degree]

ans =
    2.1012    1.3070

Three dimensional data

For three dimensional grains the boundary normals are known directly, so no stereological reconstruction is required and all three distributions become accessible. We use the Dream3d data set that is also discussed in Three dimensional grains.

grains3 = grain3d.load(fullfile(mtexDataPath,'EBSD','SmallIN100_MeshStats.dream3d'));

% the faces at the surface of the measured volume belong to one grain only
gB3 = grains3.boundary;
gB3 = gB3(all(gB3.grainId > 0,2))

gB3 = grain3Boundary (y↑→x)
 
  faces  mineral 1  mineral 2  area
 637564    unknown    unknown  100%

Note that we have removed the faces at the surface of the specimen. Those are not grain boundaries at all, but the six sides of the measured box. They make up almost a quarter of the total face area and, being flat and axis aligned, they would dominate the specimen GBND computed below.

The specimen GBND

Called with the boundary alone, calcGBND gives the distribution of the boundary normals in specimen coordinates.

gbndSpecimen = calcGBND(gB3)

plot(gbndSpecimen,'contourf','upper')
mtexTitle('specimen GBND')
mtexColorMap parula
mtexColorbar

gbndSpecimen = S2FunHarmonic (y↑→x)
  bandwidth: 25
  antipodal: true

The crystal GBND

Passing the grains in addition rotates every normal into the coordinate system of its grain, which gives the crystal GBND.

gbndCrystal = calcGBND(gB3,grains3)

plot(gbndCrystal,'contourf')
mtexTitle('crystal GBND')
mtexColorMap parula
mtexColorbar

gbndCrystal = S2FunHarmonicSym (unknown)
  bandwidth: 25
  antipodal: true

For this data set the crystal GBND is almost uniform, i.e. taken over all grain boundaries there is no preferred habit plane

[min(gbndCrystal),max(gbndCrystal)]

ans =
    0.9588    1.0273

The GBCD in three dimensions

This changes as soon as we restrict to a single misorientation. About a quarter of the inner faces of this data set are \(\Sigma 3\) twin boundaries, and for those the boundary plane is a \(\{111\}\) plane.

cs3 = grains3.CSList(2);
sigma3 = orientation.byAxisAngle(Miller(1,1,1,cs3),60*degree,cs3,cs3);

gbcd3 = calcGBND(gB3,grains3,sigma3)

plot(gbcd3,'contourf')
mtexTitle('GBCD for \(\Sigma 3\)')
mtexColorMap parula
annotate(symmetrise(Miller(1,1,1,cs3)),'labeled','backgroundColor','w')
mtexColorbar

gbcd3 = S2FunHarmonic (unknown)
  bandwidth: 25
  antipodal: true

Of the four \(\{111\}\) variants only one is populated, namely the one that is the rotational axis of the \(\Sigma 3\) misorientation. These twins are coherent, and the maximum coincides with that plane

[value,pos] = max(gbcd3);
[value, min(angle(pos,symmetrise(Miller(1,1,1,cs3))))./degree]

ans =
    2.3322    0.0015