ReadShockley
The Read-Shockley equation describes energies of low-angle grain boundaries (GBs). We can define this function in fityk:
define ReadShockley(sigma0=1, a=1) = sigma0 * t * (a - ln(t)) where t=x*pi/180
Note that different books use different symbols in this equation.
The equation above is derived using isotropic dislocation theory (which is based on isotropic elastic approximation) with the assumption that dislocations are equally spaced. Additionally, small angle approximations are used to derive this equation. If we avoid the latter, we get a bit more complex equation [1,2]:
where η0 = πb/αd = (2π/α) sin(θ/2), α = r0/b, μ is shear modulus, ν is Poisson's ratio, D is the distance between dislocations in the GB, θ is the GB misorientation.
α is a factor accounting for the dislocation core energy, usually it has order of unity.
We define this function in fityk, using k=μb/(4π2(1-ν)):
define FullGBEw(k=1, alpha=1) = k * alpha * eta * (eta / tanh(eta) - ln (2*sinh(eta))) where eta = 2*pi/alpha * sin(theta/2), theta=x*pi/180
Then, with a couple of mouse clicks, we fit the equation to data:
In this case the data are energies of SiC symmetric tilt GBs, calculated from MD simulations with Tersoff potential. The details are given in: M. Wojdyr et al., Modelling Simul. Mater. Sci. Eng. 18 (2010) 075009.
[1] Hirth, J. P. and J. Lothe (1992). Theory of Dislocations, Krieger Publishing Company.
[2] Reed-Hill, R. E. and R. Abbaschian (1992). Physical Metallurgy Principles, International Thomson Publishing.