CSL Boundaries

Most misorientations bring the lattices on the two sides of a boundary into no particular relation. At a coincidence site lattice (CSL) relationship, some sites of the two lattices coincide.

The number \(\Sigma\) is the reciprocal density of those sites. Thus, an exact \(\Sigma 3\) relationship has one coincidence site for every three lattice sites. The \(\Sigma 3\) relationship is the misorientation of a coherent annealing twin in a cubic metal.

This does not make every measured \(\Sigma 3\) segment a coherent or low-energy boundary. A CSL relationship fixes the three misorientation degrees of freedom, but not the two degrees that specify the boundary plane. The plane, chemistry, and deviation from the exact relationship also affect boundary energy and properties. Low \(\Sigma\) is therefore a useful geometric classification, not a monotonic measure of specialness.

Some low \(\Sigma\) populations, especially coherent twins, resist intergranular corrosion and cracking. Increasing their fraction and breaking up the connected network of susceptible boundaries is the aim of grain boundary engineering. This page shows how to find and analyse them in an EBSD map.

The workflow assumes grain reconstruction, boundary misorientations, and misorientation theory. The density section also uses kernel density estimation.

Reconstruct the boundary network

The example is a single-phase cubic iron map. MTEX's CSL generator is for cubic symmetry. The plotting convention below matches the specimen frame stored with this data set.

plottingConvention.default('y↓→x');
mtexdata csl silent

% grain segmentation
[grains,ebsd] = calcGrains(ebsd);

% grain smoothing
grains = smoothBoundary(grains,5);

% plot the reconstructed grains by their mean orientations
plot(grains,grains.meanOrientation,'ipfDirection',zvector)

What the orientation map shows

Thin bands of similar colours cross many of the larger grains. They are the first visual clue that this recrystallised material contains many annealing twins.

Compare the reconstruction with image quality

Diffraction-pattern image quality often drops at a boundary. Plotting it beneath translucent orientation colours checks the reconstruction against a signal that was not used to classify the boundary.

plot(ebsd,log(ebsd.prop.iq),'figSize','large')
mtexColorMap black2white
setColorRange([.5,5])

% make the orientation layer translucent
hold on
plot(grains,grains.meanOrientation,'ipfDirection',zvector,...
  'FaceAlpha',0.4,'linewidth',3)
hold off

What the image-quality overlay shows

Many dark image-quality bands follow the reconstructed interfaces. The narrow coloured lamellae remain visible through the translucent layer.

Detect the Sigma 3 boundaries

angle measures the smallest distance between a boundary misorientation and the symmetrically equivalent ideal relationships. Here the fixed tolerance is 3 degrees.

% restrict the analysis to iron--iron boundaries
gB = grains.boundary('iron','iron')

% construct the ideal Sigma 3 misorientation
csl3 = CSL(3,ebsd.CS);

% select boundary segments within 3 degrees of Sigma 3
gB3 = gB(angle(gB.misorientation,csl3) < 3*degree);

% report their segment fraction
sigma3SegmentPercent = 100 * length(gB3) ./ length(gB)

% overlay the Sigma 3 segments on the existing plot
hold on
plot(gB3,'lineColor','gold','linewidth',3,'DisplayName','CSL 3')
hold off

gB = grainBoundary (y↓→x)
 
 Segments    length  mineral 1  mineral 2
    17569  16291 µm       iron       iron
sigma3SegmentPercent =
   42.6831

How much of the network is Sigma 3

The summary reports 17,569 iron--iron segments. Of these, 7,499, or 42.7 percent, lie within 3 degrees of \(\Sigma 3\). The gold segments are not scattered randomly: many continue along complete runs from one triple point to the next.

The tolerance is a choice. The Brandon criterion uses \(15^\circ / \sqrt{\Sigma}\), which is 8.7 degrees for \(\Sigma 3\) and narrows as \(\Sigma\) increases. The 3 degree tolerance above is stricter. Neither tolerance determines the unmeasured boundary-plane inclination.

Triple points where Sigma 3 boundaries meet

A triple point is where exactly three boundary segments meet and separate three real grains. Its boundaryId property gives the three incident segments. The condition below selects points where at least two of them match the \(\Sigma 3\) relationship.

isTwinning is the convenience form of the angular comparison used above. It also checks the phases on the two sides of each segment.

% logical list of Sigma 3 boundary segments
isCSL3 = grains.boundary.isTwinning(csl3,3*degree);

% logical list of triple points with at least two Sigma 3 segments
tPid = sum(isCSL3(grains.triplePoints.boundaryId),2) >= 2;

% report and plot the selected triple points
numberOfSelectedTriplePoints = nnz(tPid)

hold on
plot(grains.triplePoints(tPid),'color','red','linewidth',2,...
  'MarkerSize',8)
hold off

numberOfSelectedTriplePoints =
    83

What the selected triple points mean

The map contains 83 such points. Many mark where a twin lamella ends against another boundary. Their number is one measure of how strongly twinned a material is, although it does not describe network connectivity by itself.

Merge across the twins

A twin belongs to the grain in which it grew. Passing the selected segments to merge dissolves those interfaces and groups the grains on their two sides. See Merging Grains for the bookkeeping after this operation.

% merge grains that share a selected Sigma 3 segment
[mergedGrains,parentIds] = merge(grains,gB3);

% report the number of grains before and after merging
numberOfGrainsBeforeAndAfter = [length(grains),length(mergedGrains)]

% overlay the merged-grain boundaries on the previous plot
hold on
plot(mergedGrains.boundary,'linecolor','w','linewidth',3)
hold off

numberOfGrainsBeforeAndAfter =
   885   415

What merging changes

The merge reduces 885 reconstructed grains to 415 groups. Many white outlines enclose several coloured grains; those regions were one grain before it twinned.

Merging records which grains belong together, but it does not identify which child was the original grain. That distinction needs an additional rule, as explained on the merging page.

Compare other low Sigma relationships

Other low \(\Sigma\) boundaries can be selected in the same way. This comparison deliberately uses the wider fixed tolerance of 5 degrees for every relationship.

delta = 5*degree;
gB5 = gB(gB.isTwinning(CSL(5,ebsd.CS),delta));
gB7 = gB(gB.isTwinning(CSL(7,ebsd.CS),delta));
gB9 = gB(gB.isTwinning(CSL(9,ebsd.CS),delta));
gB11 = gB(gB.isTwinning(CSL(11,ebsd.CS),delta));

lowSigmaSegmentCounts = [length(gB5),length(gB7),...
  length(gB9),length(gB11)]

hold on
plot(gB5,'lineColor','b','linewidth',2,'DisplayName','CSL 5')
plot(gB7,'lineColor','g','linewidth',2,'DisplayName','CSL 7')
plot(gB9,'lineColor','m','linewidth',2,'DisplayName','CSL 9')
plot(gB11,'lineColor','c','linewidth',2,'DisplayName','CSL 11')
hold off

lowSigmaSegmentCounts =
    26    41   504   187

What the other relationships contribute

In the order \(\Sigma 5\), \(\Sigma 7\), \(\Sigma 9\), and \(\Sigma 11\), the output gives 26, 41, 504, and 187 segments. Thus, \(\Sigma 9\) and \(\Sigma 11\) account for 2.9 and 1.1 percent of the network, while \(\Sigma 5\) and \(\Sigma 7\) together contribute fewer than 70 segments.

These colours occur mainly in short pieces rather than along complete boundaries. The prominence of \(\Sigma 9\) is not accidental: when two different \(\Sigma 3\) twin variants meet, their composition can be a \(\Sigma 9\) relationship.

The misorientations in their fundamental region

The preceding tests compare each segment with one ideal relationship. A complementary view plots all boundary misorientations in the symmetry-reduced fundamental region. Grain-exchange symmetry identifies a misorientation with its inverse because a boundary has no preferred side.

% compute and plot the boundary of the fundamental region
oR = fundamentalRegion(ebsd.CS,ebsd.CS,'antipodal');
close all
plot(oR)

% plot a reproducible sample of 500 boundary misorientations
rng default;
mori = discreteSample(gB.misorientation,500);
hold on
plot(mori.project2FundamentalRegion)

% mark the ideal Sigma 3 misorientation
plot(csl3.project2FundamentalRegion('antipodal'),...
  'MarkerColor','r','DisplayName','CSL 3','MarkerSize',20)
hold off

What the fundamental-region plot shows

The cloud is not uniform. It forms a dense clump at a corner of the region, and the red \(\Sigma 3\) marker lies inside that clump.

Estimate the boundary misorientation distribution

A density estimated from the segment misorientations makes the same observation quantitative. The halfwidth is the angular smoothing scale, while bandwidth sets the harmonic truncation. The displayed summary confirms that the MDF retains grain-exchange symmetry.

This is a boundary, or correlated, MDF. It contains one sample per segment, so long or finely sampled boundaries contribute more than short ones. Misorientation Distribution Function compares this population with the uncorrelated MDF implied by texture alone.

mdf = calcDensity(gB.misorientation,'halfwidth',5*degree,...
  'bandwidth',48)

mdf = SO3FunHarmonic (iron → iron)
  antipodal: true
  bandwidth: 48
  weight: 1

Plot axis--angle sections

Sections at constant misorientation angle show where the density lies. The low \(\Sigma\) relationships are annotated for comparison.

plot(mdf,'axisAngle',(25:5:60)*degree,'colorRange',[0 15])

annotate(CSL(3,ebsd.CS),'label','\(CSL_3\)','backgroundcolor','w')
annotate(CSL(5,ebsd.CS),'label','\(CSL_5\)','backgroundcolor','w')
annotate(CSL(7,ebsd.CS),'label','\(CSL_7\)','backgroundcolor','w')
annotate(CSL(9,ebsd.CS),'label','\(CSL_9\)','backgroundcolor','w')

drawNow(gcm)

Locate the density maximum

The density is concentrated in the 60 degree section at the \(\Sigma 3\) label. The two strongest local maxima provide a numerical check.

[peakMRD,peakMori] = max(mdf,'numLocal',2);

peakMRD

peakAngles = angle(peakMori) ./ degree

peakAxes = round(axis(peakMori))

peakMRD =
   54.1994
    1.6775
peakAngles =
   59.9957
   37.6160
 
peakAxes = Miller (iron)
 size: 2 × 1
  h k l
  1 1 1
  1 0 1

Read the strongest maximum

The first maximum is 54.2 multiples of a random distribution (mrd), at 60.0 degrees about a member of the \(\langle 111 \rangle\) family. It is the \(\Sigma 3\) twin relationship.

Count segments near an ideal relationship directly

volume gives the fraction of sampled segment misorientations within a chosen radius. It does not require an MDF.

sigma3WithinTwoDegrees = 100 * ...
  volume(gB.misorientation,csl3,2*degree)

sigma9WithinTwoDegrees = 100 * ...
  volume(gB.misorientation,CSL(9,ebsd.CS),2*degree)

sigma3WithinTwoDegrees =
   40.7650
sigma9WithinTwoDegrees =
    2.0718

Compare the direct segment fractions

Within 2 degrees, 40.76 percent of the segments are \(\Sigma 3\), compared with 2.07 percent for \(\Sigma 9\). These are segment fractions, not equal votes from neighbouring grain pairs.

Evaluate the MDF along low-index axes

The density can also be evaluated along paths through misorientation space. The three paths below are rotations about low-index axes. The \(\Sigma 3\) relationship is a 60 degree rotation about \(\langle 111 \rangle\).

omega = linspace(0,60*degree);
fibre100 = orientation.byAxisAngle(xvector,omega,mdf.CS,mdf.SS);
fibre111 = orientation.byAxisAngle(vector3d(1,1,1),omega,mdf.CS,mdf.SS);
fibre101 = orientation.byAxisAngle(vector3d(1,0,1),omega,mdf.CS,mdf.SS);

close all
plot(omega ./ degree,mdf.eval(fibre100),'LineWidth',2)
hold on
plot(omega ./ degree,mdf.eval(fibre111),'LineWidth',2)
plot(omega ./ degree,mdf.eval(fibre101),'LineWidth',2)
hold off
legend('[100]','[111]','[101]')
xlabel('misorientation angle');
ylabel('mrd');

Read the low-index-axis profiles

The [111] curve rises to a sharp peak of 54.2 mrd at 60 degrees. The [101] curve stays below 3.4 mrd, and the [100] curve stays below 1 mrd. One misorientation dominates this material, and it is the twin.

Evaluate the MDF at one misorientation

Finally, the MDF can be evaluated at a single misorientation. This asks how common that particular relationship is in this boundary network.

testMori = orientation.byEuler(15*degree,28*degree,14*degree,...
  mdf.CS,mdf.CS);

testMisorientationMRD = mdf.eval(testMori)

sigma3MRD = mdf.eval(csl3)

testMisorientationMRD =
    1.5507
sigma3MRD =
   54.2051

Compare the two density values

The chosen misorientation has density 1.55 mrd, close to the random baseline. The \(\Sigma 3\) relationship has density 54.2 mrd, about 54 times the random baseline.

References

Next

Continue with Twinning Analysis to infer an unknown twin relationship from measured boundaries. The chapter next turns from crystallographic character to geometry in Boundary Curvature. Use Merging Grains when the merged parent--child bookkeeping matters.