Martensite Variants

This page applies the variant concept from Parent and Child Variants to measured EBSD data. The goal is to fit an austenite-to-ferrite orientation relationship and classify each child grain by variant, packet, and Bain group.

The example is a plessite microstructure from the Emsland iron meteorite. Plessite comes from the Greek plythos, meaning filling iron. It is an intimate intergrowth of parent taenite (austenitic fcc) and child kamacite (bcc).

Plessite develops at low temperature from retained taenite. It fills the spaces in a Widmanstaetten pattern between volumes already transformed to kamacite, with very thin taenite ribbons around them.

Both child bcc and retained parent fcc phases occur in this map. The fcc orientations record the former parent grains in the planetary body. Those parent grains can easily reach dimensions of metres.

plottingConvention.default('y↑→x');

% Import the EBSD data.
mtexdata emsland

% Extract the crystal symmetries.
csBcc = ebsd('Fe').CS;
csAus = ebsd('Aus').CS;

% Segment and smooth the grains.
[grains,ebsd] = calcGrains(ebsd,'angle',5*degree,'minPixel',2);
grains = smoothBoundary(grains,4);

ebsd = EBSDsquare (y↑→x, row↑→col)
 
 Phase  Orientations       Mineral         Color  Symmetry  Crystal reference frame
     0  18393 (6.8%)    notIndexed          none                                   
     1  215769 (80%)  Ferrite, bcc  LightSkyBlue      m-3m                         
     2   35838 (13%)     Austenite  DarkSeaGreen      m-3m                         
 
 Properties: bands, bc, bs, error, mad, oldId
 Scan unit : um
 X × Y × Z : [0 → 604] × [0 → 453] × [0 → 0]
 Normal vector: (0,0,1)
 Square grid  :450 × 600

Inspect the retained parent phase

The map colours bcc measurements by the crystal direction parallel to the selected specimen direction. This is inverse pole figure colouring. Retained austenite grains are blue, and the grey lines are grain boundaries.

plot(ebsd('Fe'),ebsd('Fe').orientations)
hold on
plot(grains.boundary,'lineWidth',2,'lineColor','gray')
plot(grains('Aus'),'FaceColor','blue','DisplayName','Austenite')
hold off

Notice the small amount and size of the remaining fcc phase. Increasing nickel content stabilizes this high-temperature phase during cooling. The low-temperature bcc phase can dissolve at most 6% nickel, so the fcc phase must assimilate the excess.

The amount and size of retained fcc are therefore indicators of the overall nickel content. An axis-angle plot tests whether the separate fcc regions have one common orientation.

plot(ebsd('Aus').orientations,'axisAngle')

plot 2000 random orientations out of 34651 given orientations

The tight cluster shows that all fcc grains have nearly the same orientation. The small deviations are assumed to record deformation from high-speed collisions in the asteroid belt.

The mean estimates the common parent orientation. The std reports its angular spread in degrees.

parentOri = mean(ebsd('Aus').orientations)
parentFit = std(ebsd('Aus').orientations) ./ degree

parentOri = orientation (Austenite → y↑→x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  266.294 163.623 245.508
 
parentFit =
    1.4885

The measured spread is 1.49 degrees. This small value supports treating the separate retained regions as samples of one former parent grain.

Compare parent and child poles

The next figure plots mean bcc grain orientations as blue points. Red points mark symmetrically equivalent poles of the retained parent orientation.

childOri = grains('Fe').meanOrientation;

hBcc = Miller({1,0,0},{1,1,0},{1,1,1},csBcc);
hFcc = Miller({1,0,0},{1,1,0},{1,1,1},csAus);

plotPDF(childOri,hBcc,'MarkerSize',5,'MarkerFaceAlpha',0.05,...
  'MarkerEdgeAlpha',0.1,'points',500);

nextAxis(1)
hold on
plot(parentOri * hFcc(1).symmetrise,'MarkerFaceColor','r')
xlabel('\((100)\)','Color','red','Interpreter','latex')

nextAxis(2)
plot(parentOri * hFcc(3).symmetrise,'MarkerFaceColor','r')
xlabel('\((111)\)','Color','red','Interpreter','latex')

nextAxis(3)
plot(parentOri * hFcc(2).symmetrise,'MarkerFaceColor','r')
xlabel('\((110)\)','Color','red','Interpreter','latex')
hold off
drawNow(gcm)

Several red and blue poles nearly coincide. This pattern suggests a crystallographic OR between the two phases.

The Kurdjumov-Sachs (KS) OR maps one parent {111}-fcc plane to one child {110}-bcc plane. Within those planes, one parent \(\langle110\rangle\)-fcc direction is parallel to one child \(\langle111\rangle\)-bcc direction.

In a cubic crystal, a plane normal \((hkl)\) is parallel to the direction \([hkl]\) with the same indices. The pole figures can therefore be read as plane-normal or direction alignments.

Define and test the Kurdjumov-Sachs relationship

MTEX provides orientation.KurdjumovSachs(csAus,csBcc). Here the two parallelisms define the OR explicitly, which makes its meaning visible.

KS = orientation.map(Miller(1,1,1,csAus),Miller(0,1,1,csBcc),...
  Miller(-1,0,1,csAus),Miller(-1,-1,1,csBcc))

plotPDF(variants(KS,parentOri),'add2all',...
  'MarkerFaceColor','none','MarkerEdgeColor','k','linewidth',2)

KS = misorientation (Austenite → Ferrite, bcc)
 
 (111) || (011)   [101̅] || [111̅]

The black rings are the ideal KS child variants predicted from the retained parent. Their offsets from the blue measurements show that the mapped material does not follow the ideal relationship exactly.

A parent-to-child misorientation expresses the relative rotation between one measured child and the parent. The mean angular distance from these rotations to KS quantifies the mismatch.

mori = inv(childOri) * parentOri;
fitKS = mean(angle(mori,KS)) ./ degree

fitKS =
    3.9349

The ideal KS relationship has a mean deviation of 3.93 degrees from the measured rotations.

Fit the orientation relationship from a known parent

Because the parent orientation is known here, a robust mean of all measured parent-to-child misorientations is a direct candidate for a better OR.

p2cMean = mean(mori,'robust')

plotPDF(childOri,hBcc,'MarkerSize',5,'MarkerFaceAlpha',0.05,...
  'MarkerEdgeAlpha',0.1,'points',500);
hold on
plotPDF(variants(p2cMean,parentOri),'add2all',...
  'MarkerFaceColor','none','MarkerEdgeColor','k','linewidth',2)
hold off

fitMean = mean(angle(mori,p2cMean)) ./ degree

p2cMean = misorientation (Austenite → Ferrite, bcc)
 
  Bunge Euler angles in degree
    phi1     Phi    phi2
  109.08 8.19426 206.902
 
fitMean =
    2.4910

The rings now pass through the measured pole clusters more closely than the ideal KS rings. The mean deviation falls to 2.49 degrees, which confirms the visual improvement.

Fit the relationship without a known parent

If no parent orientation remains, calcParent2Child can estimate the OR solely from child-to-child misorientations. The method iteratively assigns symmetry operators and refines the OR.

The iterative method needs an initial OR not too far from the actual OR when used as a local fit. The current default also scans the fundamental region for promising starting points. We supply the Nishiyama-Wassermann (NW) OR as the initial candidate.

NW = orientation.NishiyamaWassermann(csAus,csBcc)

% Extract neighbouring pairs of child grains and their orientations.
grainPairs = neighbors(grains('Fe'));
oriPairs = grains(grainPairs).meanOrientation;

% Estimate the parent-to-child orientation relationship.
p2cIter = calcParent2Child(oriPairs,NW)

% Compare it with the measured parent-to-child misorientations.
fitIter = mean(angle(mori,p2cIter)) ./ degree

NW = misorientation (Austenite → Ferrite, bcc)
 
 (111) || (011)   [11̅0] || [1̅00]
 
 
p2cIter = misorientation (Austenite → Ferrite, bcc)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  177.287 98.0148 43.6465
 
fitIter =
    2.4926

The OR computed only from child-to-child misorientations fits the measured parent-to-child rotations with a mean deviation of 2.49 degrees. It fits about as well as the robust mean.

This agreement is the important check before the OR is used for classification.

Assign variant, packet, and Bain group IDs

A variant is one crystallographically equivalent child orientation predicted from a single parent orientation by a known OR. A variant ID identifies one member of that ordered set.

calcVariantId compares every measured child orientation with all predictions. It returns the closest variant ID and the associated packet and Bain group IDs.

[variantId,packetId,bainId] = ...
  calcVariantId(parentOri,childOri,p2cIter);

Classify individual variants

Ordered colours give nearby variant IDs related colours. In the pole figures, overlap makes the individual coloured points difficult to distinguish.

variantColor = ind2color(variantId,'ordered');
plotPDF(childOri,variantColor,hBcc,'MarkerSize',5);

I'm plotting 208 random orientations out of 5225 given orientations
  You can specify the number of points by the option "points".
  The option "all" ensures that all data are plotted

The axis-angle plot separates the clusters. Each compact colour cluster is a group of measured child grains assigned to one variant ID.

plot(childOri,variantColor,'axisAngle')

plot 2000 random orientations out of 5225 given orientations

Classify packets

A packet is a coarse grouping of variants that share the same habit plane. For KS-type martensite, it records which parent {111} plane aligns with the child lattice.

The habit plane is the interface along which atomic rearrangement occurs during the phase transition. Variants within one packet are related by specific symmetries, and the packet ID identifies that group.

packetColor = ind2color(packetId);
plotPDF(childOri,packetColor,hBcc,'MarkerSize',5,'points',1000);

nextAxis(1)
hold on
opt = {'MarkerFaceColor','none','MarkerEdgeColor','k','linewidth',3};
plot(parentOri * hFcc(1).symmetrise,opt{:})
xlabel('\((100)\)','Color','red','Interpreter','latex')

nextAxis(2)
plot(parentOri * hFcc(3).symmetrise,opt{:})
xlabel('\((111)\)','Color','red','Interpreter','latex')

nextAxis(3)
plot(parentOri * hFcc(2).symmetrise,opt{:})
xlabel('\((110)\)','Color','red','Interpreter','latex')
hold off
drawNow(gcm)

Blue, orange, yellow, and green mark the four packet IDs. In the {110}-bcc panel, each colour is selected by which equivalent (111)-austenite axis aligns with the (110)-ferrite axis.

Plotting the same colours on the grain map reveals the spatial extent of each child packet.

plot(grains('Fe'),packetColor)

Classify Bain groups

A Bain group is a coarse grouping by Bain correspondence. It records which parent {001} cube-axis plane aligns with the child lattice.

Bain notation concisely represents a transformation path and the geometric correspondence between the parent and child crystal structures. Each Bain group ID identifies one such correspondence.

Packet and Bain group are independent classifications of the same variant. They are not two levels of one hierarchy.

bainColor = ind2color(bainId);
plotPDF(childOri,bainColor,hBcc,'MarkerSize',5,'points',1000);

nextAxis(1)
hold on
opt = {'MarkerFaceColor','none','MarkerEdgeColor','k','linewidth',3};
plot(parentOri * hFcc(1).symmetrise,opt{:})
xlabel('\((100)\)','Color','red','Interpreter','latex')

nextAxis(2)
plot(parentOri * hFcc(3).symmetrise,opt{:})
xlabel('\((111)\)','Color','red','Interpreter','latex')

nextAxis(3)
plot(parentOri * hFcc(2).symmetrise,opt{:})
xlabel('\((110)\)','Color','red','Interpreter','latex')
hold off
drawNow(gcm)

Blue, orange, and yellow mark the three Bain group IDs. The colours are distinguished by which equivalent (100)-austenite axis aligns with the (100)-ferrite axis.

The map shows how these child Bain groups are distributed in the microstructure.

plot(grains('Fe'),bainColor)

References

Next

Continue with Parent Beta Reconstruction to use a fitted orientation relationship and variant consistency to recover a parent-grain map when only the child phase remains.