MICRESS Forum

Hi janin,

Can you explain how delta1 and delta2 work in 3D simulation as you showed in 2D upstairs?
Thanks!

Kind regards
Rody

The 3D formulation is similar to the 2D, but stretches the classic cubic formula in both y and z directions.
The user specifies as before the value of delta1 and additionally the values of deltaXY and deltaXZ in the input file.
The standard cubic formulation in 3D is
a = a0 * (1-4*delta1 * ( n_x^4+n_y^4+n_z^4 -0.75)
where n is the interfacial normal vector.
This value is then multiplied by the elongation factor:
elongation_factor = 1 + n_y^2 * (deltaXY - 1) + n_z^2 * (deltaXZ - 1).
Regards,
Janin

Hi Janin,

I have some problems about standard cubic formulation and Elongation_factor in 2D.

In the standard function of Delta1, does cos(n*theta) mean n×theta or the same as described in 3D. And what does cos(theta)**2 mean？ What does "**" mean?

Kind regards
Lilly

Hi Lilly,
in the driving file we can only use ascii code. Therfore we use the star symbol instead of a dot for multiplication:
a*b means a multiplied by b.
Moreover we use two stars for exponentation:
a**b denotes raised to the power of b.

Anyway, I would recommend to have a look at the Micress manual, volume 2, section 4.8.3 on page 81.
Here you can find a table with all anisotropy equations in proper notation.

Janin

Hi janin,

Thanks very much! I want to simulate morphology of primary silicon, can I use the facet model? And Can you give me some literatures for derivation of facet model?

Thanks!

Lilly

Yes,
the facet model is the best choice to simulate primary silicon.
You can find the description of the input parameters in the Micress manual, volume 2, section 4.5.2, page 70
and the respective equations in section 4.8.3, pages 82/83.

Moreover, I recommed a recent paper about Micress simulations of primary silicon growth in AlSi-alloy by
K. Wang, M. Wei, L. Zhang, and Y. Du:
Morphologies of Primary Silicon in Hypereutectic Al-Si Alloys: Phase-Field Simulation Supported by Key Experiments
Metallurgical and Materials Transactions A 47A (2016)1510 DOI: 10.1007/s11661-016-3358-1

In this paper you can also find further references.

Regards,
Janin

Hi Brend,

I didn't know whether this question is appropriate in this topic or the topic on Gibbs-Thompson effect. I want to be able to calculate the Gibbs-Thompson coefficient from input the surface energy in the driving file.

From the MICRESS Manual (Vol.0 - Page 8 ; Version 6.3), I found the following equation. 1)In the expression, (circled by blue ), I think Tm should be in numerator according to dimensional analysis. Is that the case ?

2)In the expression, (circled by red), Shouldn't we have a plus sign i.e sigma+sigma'' instead of minus sign ?

Regards,
Kamal

Hi Kamal,

you are perfectly right. These equations are wrong and I'm afraid these are not the only ones, because Volume0 of the manual has once been written by a student and unfortunately we never had time to review it.

The original Gibbs-Thomson equation is written in terms of temperature:
velocity = mobility_T * (deltaT - Gamma * curvature)
where deltaT is the thermal undercooling and Gamma the Gibbs-Thomson-coeffecient.

Micress corresponds to the more general sharp-interface solution in terms of energies:
velocity = mobility_G * ( dG - sigma * curvature)
where dG is the thermodynamic driving force, or more precisely the difference in chemical potentials.

Using the simplifying approximation: dG = deltaS * deltaT (where deltaS is is the entropy of transition),
the original Gibbs-Thomson equation can be recovered:
velocity = mobility_G * ( deltaS * deltaT - sigma * curvature)
= mobility_G * deltaS * ( deltaT - sigma/deltaS * curvature)
= mobility_T * (deltaT - Gamma * curvature)

i.e. mobility_T = mobility_G * deltaS and Gamma = sigma/deltaS
with the additional approximation deltaS = L/T_m we further get:
mobility_T = mobility_G * L/T_m and Gamma = sigma*T_m/L

which means, T_m should actually be in the numerator!

(Note that the aforementioned simplifying approximations should not generally be applied to non-dilute alloys!)

In the ansisotropic equation the sharp-interface equation becomes :
veleocity = mobility * ( dG - stiffness * curvature)

In 2D the stiffness can be replaced by sigma+sigma''
(2D:) velocity = mobility * ( dG - (sigma+sigma'') * curvature)
(so you are right again, this should be a positive sign!)

Eventually, I still like to mention that the anisotropic sharp-interface equation is more complex in 3D where the stiffness is a matrix and the curvature a vector).

Regards,
Janin