CSL Boundaries

Explains how to analyze CSL grain boundaries

On this page ...
Data import and grain detection
Detecting CSL Boundaries
Mark triple points
Merging grains with common CSL(3) boundary
Colorizing misorientations
Misorientations in the 3d fundamental zone
Analyzing the misorientation distribution function

Data import and grain detection

Lets import some iron data and segment grains within the data set.

mtexdata csl
plotx2east

% grain segementation
[grains,ebsd.grainId] = calcGrains(ebsd('indexed'))

% grain smoothing
grains = smooth(grains,2)

% plot the result
plot(grains,grains.meanOrientation)
 
grains = grain2d  
 
 Phase  Grains  Pixels  Mineral  Symmetry  Crystal reference frame
    -1     885  154107     iron      m-3m                         
 
 boundary segments: 21982
 triple points: 1451
 
 Properties: GOS, meanRotation
 
 
ebsd = EBSD  
 
 Phase   Orientations  Mineral       Color  Symmetry  Crystal reference frame
    -1  154107 (100%)     iron  light blue      m-3m                         
 
 Properties: ci, error, iq, x, y, grainId
 Scan unit : um
 
 
grains = grain2d  
 
 Phase  Grains  Pixels  Mineral  Symmetry  Crystal reference frame
    -1     885  154107     iron      m-3m                         
 
 boundary segments: 21982
 triple points: 1451
 
 Properties: GOS, meanRotation
 
  I'm going to colorize the orientation data with the 
  standard MTEX colorkey. To view the colorkey do:
 
  oM = ipdfHSVOrientationMapping(ori_variable_name)
  plot(oM)

Next we plot image quality as it makes the grain boundaries visible. and overlay it with the orientation map

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

% the option 'FaceAlpha',0.4 makes the plot a bit transluent
hold on
plot(grains,grains.meanOrientation,'FaceAlpha',0.4)
hold off
  I'm going to colorize the orientation data with the 
  standard MTEX colorkey. To view the colorkey do:
 
  oM = ipdfHSVOrientationMapping(ori_variable_name)
  plot(oM)

Detecting CSL Boundaries

In order to detect CSL boundaries within the data set we first restrict the grain boundaries to iron to iron phase transitions and check then the boundary misorientations to be a CSL(3) misorientation with threshold of 3 degree.

% restrict to iron to iron phase transition
gB = grains.boundary('iron','iron')

% select CSL(3) grain boundaries
gB3 = gB(angle(gB.misorientation,CSL(3,ebsd.CS)) < 3*degree);

% overlay CSL(3) grain boundaries with the existing plot
hold on
plot(gB3,'lineColor','g','linewidth',2,'DisplayName','CSL 3')
hold off
 
gB = grainBoundary  
 
 Segments  mineral 1  mineral 2
    20356       iron       iron

Mark triple points

Next we want to mark all triple points with at least 2 CSL boundaries

% logical list of CSL boundaries
isCSL3 = grains.boundary.isTwinning(CSL(3,ebsd.CS),3*degree);

% logical list of triple points with at least 2 CSL boundaries
tPid = sum(isCSL3(grains.triplePoints.boundaryId),2)>=2;

% plot these triple points
hold on
plot(grains.triplePoints(tPid),'color','r')
hold off

Merging grains with common CSL(3) boundary

Next we merge all grains together which have a common CSL(3) boundary. This is done with the command merge.

% this merges the grains
[mergedGrains,parentIds] = merge(grains,gB3);

% overlay the boundaries of the merged grains with the previous plot
hold on
plot(mergedGrains.boundary,'linecolor','w','linewidth',2)
hold off

Finaly, we check for all other types of CSL boundaries and overlay them with our plot.

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));

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

Colorizing misorientations

In the previous sections we have checked whether the boundary misorientations belong to certain specific classes of misorientations. In order to analyze the distribution of misorientations we may colorize the grain boundaries according to their misorientation. See S. Patala, J. K. Mason, and C. A. Schuh, 2012, for details. The coresponding orientation to color mapping is implemented into MTEX as

oM = patalaOrientationMapping(gB)
oM = 
  patalaOrientationMapping with properties:

          CS1: [24×2 crystalSymmetry]
          CS2: [24×2 crystalSymmetry]
    antipodal: 1

Colorizing the grain boundaries is now straight forward

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

% and overlay it with the orientation map
hold on
plot(grains,grains.meanOrientation,'FaceAlpha',0.4)

hold on
plot(gB,oM.orientation2color(gB.misorientation),'linewidth',2)
hold off
  I'm going to colorize the orientation data with the 
  standard MTEX colorkey. To view the colorkey do:
 
  oM = ipdfHSVOrientationMapping(ori_variable_name)
  plot(oM)

Lets examine the colormap. We plot it as axis angle sections and add 300 random boundary misorientations on top of it. Note that in this plot misorientations mori and inv(mori) are associated.

plot(oM,'axisAngle',(5:5:60)*degree)

hold on
plot(gB.misorientation,'points',300,...
  'MarkerFaceColor','none','MarkerEdgeColor','w')
hold off
  plotting 300 random orientations out of 20356 given orientations

Misorientations in the 3d fundamental zone

We can also look at the boundary misorienations in the 3 dimensional fundamental orientation zone.

% compute the boundary of the fundamental zone
oR = fundamentalRegion(oM.CS1,oM.CS2,'antipodal');
close all
plot(oR)

% plot 500 random misorientations in the 3d fundamenal zone
mori = discreteSample(gB.misorientation,500);
hold on
plot(mori.project2FundamentalRegion)
hold off


% mark the CSL(3) misorientation
hold on
csl3 = CSL(3,ebsd.CS);
plot(csl3.project2FundamentalRegion('antipodal') ,'MarkerColor','r','DisplayName','CSL 3','MarkerSize',20)
hold off

Analyzing the misorientation distribution function

In order to analyze more quantitatively the boundary misorientation distribution we can compute the so called misorientation distribution function. The option antipodal is applied since we want to identify mori and inv(mori).

mdf = calcMDF(gB.misorientation,'halfwidth',2.5*degree,'bandwidth',32)
 
mdf = MDF  
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
  antipodal:         true
 
  Harmonic portion:
    degree: 32
    weight: 1
 

Next we can visualize the misorientation distribution function in axis angle sections.

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)

The MDF can be now used to compute prefered misorientations

mori = mdf.calcModes(2)
 
mori = misorientation  
  size: 1 x 2
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2    Inv.
  206.659 48.4404 117.001       0
  35.5941  47.343 306.581       0
 
 

and their volumes in percent

100 * volume(gB.misorientation,CSL(3,ebsd.CS),2*degree)

100 * volume(gB.misorientation,CSL(9,ebsd.CS),2*degree)
ans =
   40.9904
ans =
    2.0338

or to plot the MDF along certain fibres

omega = linspace(0,55*degree);
fibre100 = orientation('axis',xvector,'angle',omega,mdf.CS,mdf.SS)
fibre111 = orientation('axis',vector3d(1,1,1),'angle',omega,mdf.CS,mdf.SS)
fibre101 = orientation('axis',vector3d(1,0,1),'angle',omega,mdf.CS,mdf.SS)

close all
plot(omega ./ degree,mdf.eval(fibre100))
hold on
plot(omega ./ degree,mdf.eval(fibre111))
plot(omega ./ degree,mdf.eval(fibre101))
hold off
legend('100','111','101')
 
fibre100 = misorientation  
  size: 1 x 100
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
 
 
fibre111 = misorientation  
  size: 1 x 100
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
 
 
fibre101 = misorientation  
  size: 1 x 100
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
 

or to evaluate it in an misorientation directly

mori = orientation('Euler',15*degree,28*degree,14*degree,mdf.CS,mdf.CS)

mdf.eval(mori)
 
mori = misorientation  
  size: 1 x 1
  crystal symmetry : iron (m-3m)
  crystal symmetry : iron (m-3m)
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    15   28   14    0
 
 
ans =
    5.6826