Question on anisotropy of interfacial energy

[Feng]

Hello,
I have some questions on the anisotropy of interfacial energy when simulating the 2D dendrite growth.
In the manual book, Vol0_MICRESS_phenomenology, it is written “In order to include anisotropy into the model, both the interfacial energy  and the interface mobility  are assumed to be anisotropic. In 2 dimensions this can be accomplished by making these parameters dependant on the angle θ between the growth direction and the crystal orientation i.e. σ= σ(θ) and μ= μ(θ). For a simple cubic symmetry in 2D these functions could look like σ= σ0 (1-cos(4 θ)) and μ= μ0 (1-cos(4 θ)).”
In the example a dri file of 2D simulation of dendrite for Al-Cu alloy, it is shown as:
# Type of interfacial energy definition between 0 (LIQUID) and 1 (AL-FCC) ?
# Options: constant temp_dependent
constant
# Interfacial energy between 0 (LIQUID) and 1 (AL-FCC) ? [J/cm**2]
# [max. value for num. interface stabilisation [J/cm**2]]
1.6E-05 1.60000E-04
# Type of mobility definition between LIQUID and AL-FCC?
# Options: constant temp_dependent dg_dependent [fixed_minimum]
constant
# Kinetic coefficient mu between LIQUID and AL-FCC [cm**4/(Js)] ?
10.
# Is interaction isotropic?
# Options: isotropic
# anisotropic [junction_force] [harmonic_expansion]
anisotropic
# Anisotropy of interfacial stiffness? (cubic)
# 1 - delta * cos(4*phi), (delta =delta_stiffness =15*delta_energy)
# Coefficient delta (<1.) ?
0.4500000000000
# Anisotropy of interfacial mobility? (cubic)
# 1 + delta * cos(4*phi)
# Coefficient delta (<1.) ?
6.0000000000000E-02

The first question is the if “σ= σ0 (1-cos(4 θ)) and μ= μ0 (1-cos(4 θ)) ” should be written as “σ= σ0 (1-γcos(4 θ)) and μ= μ0 (1-γcos(4 θ)) ”?
The second question is that is the “-” should be “+” in above equation, such as σ= σ0 (1-γcos(4 θ)) and μ= μ0 (1-γcos(4 θ)) should be σ= σ0 (1+γcos(4 θ)) and μ= μ0 (1+γcos(4 θ))? because I found these equations are different from some published papers, which makes me a little confused. or is there any special definition in Micress? I did not find any further information in the manual books of Micress.
In addition, is the interfacial energy, i.e. 1.6E-05, in the dri file the σ0 in σ= σ0 (1+γcos(4 θ))?
Many thanks.

[Feng]

Sorry, the first question is the if “σ= σ0 (1-cos(4 θ)) and μ= μ0 (1-cos(4 θ)) ” should be written as “σ= σ0 (1+γcos(4 θ)) and μ= μ0 (1+γcos(4 θ)) ”?

[Bernd]

Dear Feng,

Please excuse my late reply, it seems that the notification function is not working correctly at our side.

What you refer to is from the old Manual which has not been updated since MICRESS version 6.4. You are right that it is not correct what has been written there.

Please note that the current MICRESS Manual is an online source which you can access via the MICRESS webpage under Dokumentation. Then you find the correct functions under MICRESS/MICRESS/Topics/Phase Interactions in Table 2, or by using the search function. For cubic symmetry and 2D-equivalent the anisotropy functions are:

σ=σ₀*(1-δ_σcos(4Θ)
μ=μ₀*(1+δ_μcos(4Θ)

Sorry for the confusion!

Bernd

[Feng]

Thanks a lot for your kind reply. As you have shown, for cubic symmetry and 2D-equivalent, the anisotropy stiffness function is σ=σ0*(1-δσcos(4Θ)). However, from some references, it is written σ=σ0*(1+δσcos(4Θ)), where Θ is the angle between n and a fixed reference axis. I also checked the book, titled A Phase-field model for technical alloy solidification, by Janin Eiken, where the anisotropy is also written as a(2D)=1+δσcos(4Θ). If the stiffness function is σ=σ0*(1-δσcos(4Θ)), is that meant the Θ here has a different reference axis from that mention above? what is the definition of Θ in the Micress for a cubic symmetry?
In addition, is the interfacial energy set in dri file is the σ0 in above equation?

[janin]

Hi Feng,
It is important to distinguish between the interfacial ENERGY σ and the interfacial STIFFNESS σ*.
- The cubic 2D-anisotropy of the interfacial energy is σ = σ0 (1 + δ cos(4Θ)).
- The cubic 2D-anisotropy of the interfacial stiffness is σ*= σ0* (1- δ* cos(4Θ)).
From σ*= σ + σ'' we get σ0* = σ0 and δ* = 15 δ.

If you have selected the standard anisotropy model, σ0*=σ0 corresponds to the interfacial energy you have already specified in your driving file (i.e. 1.6E-05) and the anisotropy coefficient δ* corresponds to the anisotropy of the stiffness, which is 15 times the value of the anisotropy of the interfacial energy, usually given in the literature.

The different models are described in the manual:
https://docs.micress.rwth-aachen.de/7.2 ... mobilities.
The pragmatic use of the stiffness is described in "A Phase-field model for technical alloy solidification" on page 120.



With best regards,
Janin