Thank you Bernd,
I am sorry, I still do not understand how you correlate potential gradient (K term which is in [1/cm^2]) to curvature. Could you please describe me how you find a normal vector for each node? Is there any relationship between order parameter and normal vector?
Particle pinning
Re: Particle pinning
Dear mtoloui,
the K term in our phase-field formulation is not a curvature as it has the dimension 1/cm2. It has some relation to curvature if we compare the PF equation to the Gibbs-Thomson relation by dividing both sides of the equation by the norm of the gradient of the phase-field parameter. But even K/|grad phi| is not exactly the curvature because it still contains a stabilisation term which acts to keep the correct interface thickness.
For this reason, we independently calculate the curvature in the following way:
1.) Calculate the normal vector for each interface grid point:
Only the next neighbours are used for calculating the gradient of the phase-field variable in each direction. For dual interfaces, we write this in a symmetrically weigthed form:
2.) Calculate the curvature as the divergence of the normal vector:
Again, only the next neighbours are used. This curvature now has the dimension 1/cm and can be used for the particle pinning model.
Bernd
the K term in our phase-field formulation is not a curvature as it has the dimension 1/cm2. It has some relation to curvature if we compare the PF equation to the Gibbs-Thomson relation by dividing both sides of the equation by the norm of the gradient of the phase-field parameter. But even K/|grad phi| is not exactly the curvature because it still contains a stabilisation term which acts to keep the correct interface thickness.
For this reason, we independently calculate the curvature in the following way:
1.) Calculate the normal vector for each interface grid point:
- Normalenvektor.gif (243 Bytes) Viewed 4521 times
- Normalenvektor_gewichtet.gif (573 Bytes) Viewed 4521 times
- Krümmung.gif (149 Bytes) Viewed 4521 times
Bernd
Re: Particle pinning
Using above formulations, it looks to me that we get different values of curvature and different normal vectors at different nodes across the boundary nodes. This will lead to different values of mobility across the boundary which may destabilize the boundary. So how do you prevent unstable migration of the boundaries?
Re: Particle pinning
Dear mtoloui,
It is true that due to a variation of the evaluated curvature across the interface you may get an only partial pinning of the interface in a certain position. But this is not "unstable" because there is no positive feed-back - to the contrary there is a negative feed-back, because local "partial" pinning will lead to an increase in the local curvature which will remove the pinning. So, effectively, the effect of "partial pinning" introduces a smoother transition between pinned and unpinned interfaces.
The only measure against this smoothening effect would be the averaging of curvature across the interface like it is done for the chemical driving force in MICRESS. But - similar as in case of the chemical driving force - I would expect unwanted side effects on anisotropy...
Bernd
It is true that due to a variation of the evaluated curvature across the interface you may get an only partial pinning of the interface in a certain position. But this is not "unstable" because there is no positive feed-back - to the contrary there is a negative feed-back, because local "partial" pinning will lead to an increase in the local curvature which will remove the pinning. So, effectively, the effect of "partial pinning" introduces a smoother transition between pinned and unpinned interfaces.
The only measure against this smoothening effect would be the averaging of curvature across the interface like it is done for the chemical driving force in MICRESS. But - similar as in case of the chemical driving force - I would expect unwanted side effects on anisotropy...
Bernd
Re: Particle pinning
Thank you Bernd,
For a boundary with the width of for example 6 nodes, this formulation yields 6 different values of k (curvature) and vectors of n (normal) corresponding to the 6 boundary nodes across the interface. This will cause a huge mobility difference i.e. in some parts of the boundary width, pinning and in the rest unpinning occurs. As unpinned mobility is quite larger than pinned mobility, grain boundary gets unstable.
For a boundary with the width of for example 6 nodes, this formulation yields 6 different values of k (curvature) and vectors of n (normal) corresponding to the 6 boundary nodes across the interface. This will cause a huge mobility difference i.e. in some parts of the boundary width, pinning and in the rest unpinning occurs. As unpinned mobility is quite larger than pinned mobility, grain boundary gets unstable.
Re: Particle pinning
Do you have an example for such an instability?
Bernd
Bernd