Grain Boundary Properties edit page

In this section we discus geometric properties that can be derived from grain boundaries. Lets start by importing some EBSD data and computing grain boundaries.

% load some example data
plottingConvention.default('y↑→x');
mtexdata twins silent
ebsd.prop = rmfield(ebsd.prop,{'error','bands'});

% detect grains
[grains,ebsd] = calcGrains(ebsd,'angle',10*degree,'minPixel',3);

% smooth them - this page reads ebsdId per segment, which only means something
% as long as every segment still runs between one pair of pixels, so the
% coarsening and resampling steps are switched off
grains = grains.smoothBoundary(5,'noSimplify','noRefine');

% visualize the grains
plot(grains,grains.meanOrientation)

% extract all grain boundaries
gB = grains.boundary;

hold on
plot(gB,'LineWidth',2)
hold off

Property overview

A variable of type grainBoundary contains the following properties

ebsdId

neighboring pixel ids

phaseId

neighboring phase ids

grainId

neighboring grain ids

F

vertices ids of the segments

segLength

length of each segment

direction

direction of each segment

midPoint

mid point of the segment

curvature

curvature of each segment

misorientation

between ebsdId(:,1) and ebsdId(:,2)

triplePoints

list of all triple points

componentId

connected component id

componentSize

connected component size

The first three properties refer to \(N \times 2\) matrices where \(N\) is the number of boundary segments. Each row of these matrices contains the information about the EBSD data, and grain data on both sides of the grain boundary. To illustrate this consider the grain boundary of one specific grain

gB4 = grains(4).boundary
gB4 = grainBoundary (y↑→x)
 
 Segments  length  mineral 1  mineral 2
        8  1.3 µm  Magnesium  Magnesium

This boundary consists of 8 segments and hence ebsdId forms a 8x2 matrix

gB4.ebsdId
ans =
        1103        1104
         966         967
         966         829
         965         828
         965         964
         965        1102
        1103        1102
        1103        1240

It is important to understand that the id is not necessarily the same as the index in the list. In order to index an variable of type EBSD by id and not by index the following syntax has to be used

ebsd('id',gB4.ebsdId)
ans = EBSDsquare (y↑→x, row←col↑)
 
 Phase  Orientations    Mineral         Color  Symmetry  Crystal reference frame
     1     16 (100%)  Magnesium  LightSkyBlue     6/mmm        X||a*, Y||b, Z||c
 
   Id   Phase             orientation    bc    bs   mad   oldId   grainId
 1103       1   (114.7°,15.7°,218.5°)   182   174   0.5    1011         4
  966       1   (113.7°,15.5°,219.3°)   164   158   0.5    1010         4
  966       1   (113.7°,15.5°,219.3°)   164   158   0.5    1010         4
  965       1   (115.3°,15.6°,218.2°)   170   176   0.7     843         4
  965       1   (115.3°,15.6°,218.2°)   170   176   0.7     843         4
  965       1   (115.3°,15.6°,218.2°)   170   176   0.7     843         4
 1103       1   (114.7°,15.7°,218.5°)   182   174   0.5    1011         4
 1103       1   (114.7°,15.7°,218.5°)   182   174   0.5    1011         4
 1104       1     (4.7°,80.8°,195.4°)   174   181   0.5    1178        42
  967       1     (4.5°,80.7°,195.2°)   168   171   0.3    1177        42
  829       1     (4.6°,80.7°,195.4°)   156   160   0.4    1009        42
  828       1     (4.9°,80.4°,195.2°)   167   170   0.5     842        42
  964       1     (4.9°,80.4°,195.2°)   176   196   0.5     676        42
 1102       1     (4.4°,80.5°,195.1°)   174   197   0.3     844        42
 1102       1     (4.4°,80.5°,195.1°)   174   197   0.3     844        42
 1240       1     (4.4°,80.5°,195.2°)   176   168   0.4    1012        42
 Scan unit : um
 X x Y x Z : [1.8 → 2.7] x [1.2 → 2.1] x [0 → 0]
 Normal vector: (0,0,1)
 Square grid  :8 x 2

Similarly

gB4.grainId
ans =
     4    42
     4    42
     4    42
     4    42
     4    42
     4    42
     4    42
     4    42

results in 8x2 matrix indicating that grain 4 is a tiny inclusion of grain 42.

plot(grains(4),'FaceColor','DarkBlue','micronbar','off')
hold on
plot(grains(42),'FaceColor','LightCoral')
hold off

Grain boundary misorientations

The grain boundary misorientation defined as the misorientation between the orientations corresponding to ids in first and second column of ebsdId, i.e. following two commands should give the same result

gB4(1).misorientation

inv(ebsd('id',gB4.ebsdId(1,2)).orientations) .* ebsd('id',gB4.ebsdId(1,1)).orientations
ans = misorientation (Magnesium → Magnesium)
  antipodal: true
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  329.875  86.401 150.169
 
 
ans = misorientation (Magnesium → Magnesium)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  329.875  86.401 150.169

Note that in the first result the antipodal flag is true while it is false in the second result.

Obviously, misorientations of a list of grain boundaries can only be extracted if all of them have the same type of phase transition. Let us consider only Magnesium to Magnesium grain boundaries, i.e., omit all grain boundaries to an not indexed region.

gB_Mg = gB('Magnesium','Magnesium')
gB_Mg = grainBoundary (y↑→x)
 
 Segments  length  mineral 1  mineral 2
     3161  722 µm  Magnesium  Magnesium

Then the misorientation angles can be plotted by

plot(gB_Mg,gB_Mg.misorientation.angle./degree,'linewidth',4,'micronbar','off')
mtexColorbar('title','misorientation angle (°)')

Geometric properties

The direction property of the boundary segments is useful when checking for tilt and twist boundaries, i.e., when we want to compare the misorientation axis with the interface between the grains

% compute misorientation axes in specimen coordinates
ori = ebsd('id',gB_Mg.ebsdId).orientations;
axes = axis(ori(:,1),ori(:,2),'antipodal')

% plot the angle between the misorientation axis and the boundary direction
plot(gB_Mg,angle(gB_Mg.direction,axes),'linewidth',4,'micronbar','off')
axes = vector3d (y↑→x)
 size: 3161 x 1
 antipodal: true

We observe that the angle is quite oscillatory. This is because of the stair casing effect when reconstructing grains from gridded EBSD data. The weaken this effect we may average the segment directions using the command calcMeanDirection 

% plot the angle between the misorientation axis and the boundary direction
plot(gB_Mg,angle(gB_Mg.calcMeanDirection(4),axes),'linewidth',4,'micronbar','off')

The midPoint property gives the position of each segment as a vector3d. It is what one needs whenever something has to be drawn at, or selected by, the location of a segment - the misorientation axes above, for instance, are best shown as arrows sitting on their segments

plot(grains,grains.meanOrientation,'faceAlpha',0.3,'micronbar','off')
hold on
quiver(gB_Mg(1:3:end),axes(1:3:end),'color','black','autoScaleFactor',0.6)
hold off

and it is equally the natural handle for a spatial selection

pos = gB_Mg.midPoint;
isTop = pos.y > 30;

plot(grains,'faceAlpha',0.3,'micronbar','off')
hold on
plot(gB_Mg(isTop),'linewidth',3,'lineColor','red')
plot(gB_Mg(~isTop),'linewidth',3,'lineColor','blue')
hold off

While the command length(gB_Mg) gives the total number of all Magnesium to Magnesium grain boundary segments the command segLength(gB_Mg) gives the length of each segment in µm. The total length of all Magnesium to Magnesium grain boundary segments is hence

sum(gB_Mg.segLength)
ans =
  722.2508

Connected components

When analyzing the topology of boundary networks connected components of certain subsets of boundaries are of interest. Using the commands|gB.componentId| and gB.componentSize we are able to separate the boundary network into groups of connected components and analyze them separately. We do so below at the example of twin boundaries which we first colorize according to length.

CS = ebsd.CS;
twinning = orientation.map(Miller(1,-1,0,1,CS),Miller(1,0,-1,-1,CS),...
  Miller(0,1,-1,1,CS,'uvw'),Miller(1,-1,0,1,CS,'uvw'))

gBTwin = gB(gB.isTwinning(twinning));

plot(grains,grains.meanOrientation,'faceAlpha',0.25,'micronbar','off')

hold on
plot(gBTwin,gBTwin.componentSize,'lineWidth',4)
hold off
mtexColorbar
twinning = misorientation (Magnesium → Magnesium)
 
 (101̅1̅) || (011̅1)   [011̅1] || [11̅01]

Next we compute how curvy each twin boundary component is, by dividing it spatial extension by its total length. This measure has proven to be useful to tell apart misindexing due to pseudosymmetries and true twin boundaries.

numComponents = max(gBTwin.componentId);
xmax = accumarray(gBTwin.componentId,gBTwin.midPoint.x,[numComponents,1],@max);
ymax = accumarray(gBTwin.componentId,gBTwin.midPoint.y,[numComponents,1],@max);
xmin = accumarray(gBTwin.componentId,gBTwin.midPoint.x,[numComponents,1],@min);
ymin = accumarray(gBTwin.componentId,gBTwin.midPoint.y,[numComponents,1],@min);

ext = sqrt((xmax-xmin).^2+(ymax-ymin).^2);
len = accumarray(gBTwin.componentId,gBTwin.segLength,[numComponents,1],@sum);
value = ext ./ len;

plot(grains,grains.meanOrientation,'faceAlpha',0.25)
hold on
plot(gBTwin,value(gBTwin.componentId),'lineWidth',4)
hold off
mtexColorbar
mtexColorMap blue2red

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/BoundaryProperties.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX 
@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.