pgnd = 2ϑ/ub (pgnd= GND density, ϑ = misorientation angle, u = unit length, b = magnitude of burger's vector)
Several papers reference this formula as a way to calculate dislocation density (see links), using the ϑ misorientation angle measured by KAM maps. How is this possible when ϑ only varies a few degrees, but dislocation density can vary from 1013 to 1016? Also, how are u and b calculated?
I already have made KAM maps with optimized step size, my final goal is to measure dislocation density in these samples, but I'm having trouble understanding how this formula works in practice. What steps are required to make my KAM map into a pGND map?
Thank you in advance.
Sincerely,
Kameron Hansen
http://www.sciencedirect.com/science/article/pii/S0921509310000195
http://www.sciencedirect.com/science/article/pii/S1359646202003354?np=y
There is a simple and well known formula, which gives the dislocation density in the low angle grain boundaries (dislocation tilt boundaries). D = b/ Theta where D is distance of separation of edge dislocation and Theta is angle of misorientation, and b is Burgers vector. This formula is used to calculate low angle grain boundary energy, which is given by E = Eedge /D = Eedge Theta /b = Eo Theta [A- Ln Theta]. where
Eedge = b ThoNot /2 where ThoNot= G/ 2Pi( 1- v) , G is shear modulus, v Poisson's ratio.
Note: 1/D gives the number of dislocation lying down per cm2 area of tilt boundary. 1/D is Surface density of dislocations and the unit is [ 1/cm].
SEE: Dislocations in Crystals by W. T. Read.
Dear Cameron!
It is a rather long talk needful. So I would address you to the 2 discussions concerning GND meaning at my page of RG. See"questions" subsection.
Dr. Rohan is right in his remark. If you have complete polygonization that mean all the dislocations which were in opposite sign in annealed out after gliding and climbing and the rest were lying out along the twist or twin boundaries then you can calculate their global density using above formula in connection with the mean distance of separation between low angle dislocation walls, which may be denoted by U by the following connection. N= (UD)-1 #/ cm2 . This figure doesn't give you the bulk dislocation density after the deformation but the dislocation density remained in the body after the polygonization annealing treatment. especially high stacking fault materials such as Aluminum.
The above remarks are quite correct; you will only obtain the GND densities from the dislocation walls that give rise to significant misorientations. The total dislocation density may be similar to the GND density for hot deformed or annealed high SFE materials like Al but may be several times greater for low SFE materials. Also you should be aware of the errors induced by using standard EBSD for low angle boundary misorientations - see our paper "HR-EBSD and its Materials Applications"
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