Open Matlab File in the Editor MTEX

Misorientations

Misorientation describe the relative orientation of two grains with respect to each other. Important concepts are are twinnings and CSL (coincidence site lattice),

On this page ...
Misorientations between grains
Coinsident lattice planes
Twinning misorientations
Highlight twinning boundaries
Phase transitions

Misorientations between grains

Let us import some EBSD data set, compute grains and plot the colorized according to their meanorientation and lets highlight grain 70 and grain 80

mtexdata twins
grains = calcGrains(ebsd('indexed'))
CS = grains.CS; % extract crystal symmetry

plot(grains,grains.meanOrientation)
hold on
plot(grains([70,80]).boundary,'edgecolor','w','linewidth',2)
 
grains = grain2d  
 
 Phase  Grains  Pixels    Mineral  Symmetry  Crystal reference frame
     1     121   22833  Magnesium     6/mmm        X||a*, Y||b, Z||c
 
 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)

The misorientation between those two grains can be computed from the meanorientations of the grains. Remember that an orientation always maps crystal coordinates into specimen coordinates. Hence, the product of an inverse orientation with another orientation transfers crystal coordinates from one crystal reference frame into crystal coordinates with respect to another crystal reference frame.

mori = inv(grains(70).meanOrientation) * grains(80).meanOrientation
 
mori = misorientation  
  size: 1 x 1
  crystal symmetry : Magnesium (6/mmm, X||a*, Y||b, Z||c)
  crystal symmetry : Magnesium (6/mmm, X||a*, Y||b, Z||c)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2    Inv.
  149.583 94.2717 150.165       0
 
 

In the present case the misorientation describes the coordinate transform from the reference frame of grain 80 into the reference frame of crystal 70. Take as an expample the plane {11-20} with respect to the grain 80. Then the plane in grain 70 which aligned parallel to this plane can be computed by

round(mori * Miller(1,1,-2,0,CS))
 
ans = Miller  
 size: 1 x 1
 mineral: Magnesium (6/mmm, X||a*, Y||b, Z||c)
  h  2
  k -1
  i -1
  l  0

Conversely, the inverse of mori is the coordinate transform from crystal 70 to grain 80.

round(inv(mori) * Miller(2,-1,-1,0,CS))
 
ans = Miller  
 size: 1 x 1
 mineral: Magnesium (6/mmm, X||a*, Y||b, Z||c)
  h  1
  k  1
  i -2
  l  0

Coinsident lattice planes

The coincidence between major lattice planes may suggest that the misorientation is a twinning misorientation. Lets analyse whether there are some more alignments between major lattice planes.

m = Miller({1,0,-1,0},{1,1,-2,0},{1,0,-1,1},{1,1,-2,1},{1,1,-2,2},{0,0,0,1},CS);

close all
% cycle through all major lattice planes
for im = 1:length(m)

  % plot the lattice planes of grains 80 with respect to the
  % reference frame of grain 70
  plot(mori * symmetrise(m(im)),'symmetrised','MarkerSize',10,...
    'DisplayName',char(m(im)),'fundamentalRegion','figSize','normal')
  hold all
end
hold off
annotate(m,'labeled')

% show legend
legend({},'location','NorthWest','FontSize',13);

we observe an almost perfect math between the {11-20} lattice planes and the {10-11} lattice planes and good coincidences for the lattice plane {10-10} to {0001} and {11-22}; and for the lattice plane {10-10} to {11-22}. Lets compute the angles explicitly

angle(mori * Miller(1,1,-2,0,CS),Miller(1,1,-2,0,CS)) / degree
angle(mori * Miller(-1,0,1,1,CS),Miller(1,0,-1,1,CS)) / degree
angle(mori * Miller(0,0,0,1,CS) ,Miller(1,0,-1,0,CS)) / degree
angle(mori * Miller(1,1,-2,2,CS),Miller(1,0,-1,0,CS)) / degree
angle(mori * Miller(1,0,-1,0,CS),Miller(1,1,-2,2,CS)) / degree
ans =
    0.4592
ans =
    0.1766
ans =
    4.2919
ans =
    2.6341
ans =
    2.5686

Twinning misorientations

Lets define a misorientation that makes a perfect fit between the {11-20} lattice planes and between the {10-11} lattice planes

mori = orientation('map',Miller(1,1,-2,0,CS),Miller(2,-1,-1,0,CS),...
  Miller(-1,0,1,1,CS),Miller(1,0,-1,1,CS))

% the rotational axis
round(mori.axis)

% the rotational angle
mori.angle / degree
 
mori = misorientation  
  size: 1 x 1
  crystal symmetry : Magnesium (6/mmm, X||a*, Y||b, Z||c)
  crystal symmetry : Magnesium (6/mmm, X||a*, Y||b, Z||c)
 
  Bunge Euler angles in degree
  phi1     Phi    phi2    Inv.
   330 93.7008     330       0
 
 
 
ans = Miller  
 size: 1 x 1
 mineral: Magnesium (6/mmm, X||a*, Y||b, Z||c)
  h  1
  k  1
  i -2
  l  0
ans =
   86.2992

Lets plot the same figure as before with the exact twinning misorientation.

for im = 1:length(m)
  plot(mori * symmetrise(m(im)),'symmetrised','MarkerSize',10,...
    'DisplayName',char(m(im)),'fundamentalRegion')
  hold all
end
hold off
annotate(m,'labeled')

% show legend
legend({},'location','NorthWest','FontSize',13);

Highlight twinning boundaries

It turns out that in the previous EBSD map many grain boudaries have a misorientation close to the twinning misorientation we just defined. Lets Lets higlight those twinning boundaries

% consider only Magnesium to Magnesium grain boundaries
gB = grains.boundary('Mag','Mag');
% check for small deviation from the twinning misorientation
isTwinning = angle(gB.misorientation,mori) < 5*degree;

% plot the grains and highlight the twinning boundaries
plot(grains,grains.meanOrientation)
hold on
plot(gB(isTwinning),'edgecolor','w','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)

Phase transitions

Misorientations may not only be defined between crystal frames of the same phase. Lets consider the phases Magnetite and Hematite.

CS_Mag = loadCIF('Magnetite')
CS_Hem = loadCIF('Hematite')
 
CS_Mag = crystalSymmetry  
 
  mineral : Magnetite    
  symmetry: m-3m         
  a, b, c : 8.4, 8.4, 8.4
 
 
CS_Hem = crystalSymmetry  
 
  mineral        : Hematite         
  symmetry       : -3m1             
  a, b, c        : 5, 5, 14         
  reference frame: X||a*, Y||b, Z||c
 

The phase transition from Magnetite to Hematite is described in literature by {111}_m parallel {0001}_h and {-101}_m parallel {10-10}_h The corresponding misorientation is defined in MTEX by

Mag2Hem = orientation('map',...
  Miller(1,1,1,CS_Mag),Miller(0,0,0,1,CS_Hem),...
  Miller(-1,0,1,CS_Mag),Miller(1,0,-1,0,CS_Hem))
 
Mag2Hem = misorientation  
  size: 1 x 1
  crystal symmetry : Magnetite (m-3m)
  crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c)
 
  Bunge Euler angles in degree
  phi1     Phi    phi2    Inv.
   120 54.7356      45       0
 
 

Assume a Magnetite grain with orientation

ori_Mag = orientation('Euler',0,0,0,CS_Mag)
 
ori_Mag = orientation  
  size: 1 x 1
  crystal symmetry : Magnetite (m-3m)
  specimen symmetry: 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0    0    0    0
 
 

Then we can compute all variants of the phase transition by

symmetrise(ori_Mag) * inv(Mag2Hem)
 
ans = orientation  
  size: 48 x 1
  crystal symmetry : Hematite (-3m1, X||a*, Y||b, Z||c)
  specimen symmetry: 1
 

and the corresponding pole figures by

plotPDF(symmetrise(ori_Mag) * inv(Mag2Hem),...
  Miller({1,0,-1,0},{1,1,-2,0},{0,0,0,1},CS_Hem))