## The role of the phase minimum (phMin)

### The role of the phase minimum (phMin)

The phase minimum (phMin, at the very end of the input file) is a purely numerical parameter which controls below which fraction (phase-field parameter) an interface cell is assumed to convert to a bulk cell. If the fraction of a phase is falling below this value the concentration of this small rest amount is frozen into the other phase(s) which is (are) present in this cell. So, a high value of the phase minimum increases artificial solute trapping.

Small values of phMin reduce this effect, but may require smaller phase-field time steps for not producing instabilities related to segregation, and thus slow down the simulation. This is automatically included into the automatic time step functionality.

On the other hand, if a new phase is appearing in a grid cell, it has at least the fraction corresponding to the value of the phase minimum. In cases where the new phase has a stoichiometric composition which is much higher than the actual composition in the matrix phase, there may be not enough solute of this component contained in the grid cell if the value of phMin is too high.

In most cases a value of 1.E-4 can be recommended. Lower values down to 1.E-7 can be used if stoichiometric components with extreme concentration differences are involved (like precipitation of carbides from a extremely low-carbon matrix).

---

original message from Bernd

Small values of phMin reduce this effect, but may require smaller phase-field time steps for not producing instabilities related to segregation, and thus slow down the simulation. This is automatically included into the automatic time step functionality.

On the other hand, if a new phase is appearing in a grid cell, it has at least the fraction corresponding to the value of the phase minimum. In cases where the new phase has a stoichiometric composition which is much higher than the actual composition in the matrix phase, there may be not enough solute of this component contained in the grid cell if the value of phMin is too high.

In most cases a value of 1.E-4 can be recommended. Lower values down to 1.E-7 can be used if stoichiometric components with extreme concentration differences are involved (like precipitation of carbides from a extremely low-carbon matrix).

---

original message from Bernd

### Re: The role of the phase minimum (phMin)

Hi, Jan

Based on the formula given in another thread (see below), the phMin is not involved in the time step.

Based on the formula given in another thread (see below), the phMin is not involved in the time step.

### Re: The role of the phase minimum (phMin)

Hi, zubq,

The post to which you are replying has been copied from the old forum to the new one in June 2008, that is why it carries the name jan, who is the guy who made the copying - essentially, the post was originally from me...

The content is not any more actual in two respects: First - in so far you are right - phMin is no longer influencing the time-stepping criterion, and, secondly, smaller values of phMin down to about 1E-12 are now possible because MICRESS is by default double-precision since version 5.5.

From the formula which you posted, and which in principle is used by MICRESS, the time stepping would be phMin dependent: Remember that the criterion is calculated for each interface cell, with different values of and , and the minimum value of all cells is used.

Now imagine the extreme case of a stoichiometric phase where one of the slopes is infinite. Then, the time step is directly proportional to the fraction of the non-stoichiometric phase, and the smaller phMin the smaller is the time step (assuming that such small fractions are appearing somewhere in the domain).

But experience has shown that fluctuations are more and more improbable the closer one gets to the edge of the interface. Therefore we restricted the fractions in the criterion:

That is the reason why time-stepping according to the segregation criterion essentially is not dependent from phMin!

Bernd

The post to which you are replying has been copied from the old forum to the new one in June 2008, that is why it carries the name jan, who is the guy who made the copying - essentially, the post was originally from me...

The content is not any more actual in two respects: First - in so far you are right - phMin is no longer influencing the time-stepping criterion, and, secondly, smaller values of phMin down to about 1E-12 are now possible because MICRESS is by default double-precision since version 5.5.

From the formula which you posted, and which in principle is used by MICRESS, the time stepping would be phMin dependent: Remember that the criterion is calculated for each interface cell, with different values of and , and the minimum value of all cells is used.

Now imagine the extreme case of a stoichiometric phase where one of the slopes is infinite. Then, the time step is directly proportional to the fraction of the non-stoichiometric phase, and the smaller phMin the smaller is the time step (assuming that such small fractions are appearing somewhere in the domain).

But experience has shown that fluctuations are more and more improbable the closer one gets to the edge of the interface. Therefore we restricted the fractions in the criterion:

That is the reason why time-stepping according to the segregation criterion essentially is not dependent from phMin!

Bernd

### Re: The role of the phase minimum (phMin)

Hi, Bernd.

From the equation above, if is stoichoi phase, is infinite,then . So there will be a lower bound of when .

By the way, do you mean now PhiMin does not affect the time step in MICRESS [new version?]? I am using MICRESS_408, time step decreases with smaller PhiMin.

From the equation above, if is stoichoi phase, is infinite,then . So there will be a lower bound of when .

By the way, do you mean now PhiMin does not affect the time step in MICRESS [new version?]? I am using MICRESS_408, time step decreases with smaller PhiMin.

### Re: The role of the phase minimum (phMin)

Hi zubq,

It is correct what you say! I mixed things up in my mind, the formula you gave us above is for the double-well potential - for double well (which is no longer supported in MICRESS) there is no such problem. But for double obstacle, has to be replaced by , and then things are different, and ...

Sorry for the wrong information, I hope I have it correct now!

Bernd

It is correct what you say! I mixed things up in my mind, the formula you gave us above is for the double-well potential - for double well (which is no longer supported in MICRESS) there is no such problem. But for double obstacle, has to be replaced by , and then things are different, and ...

Sorry for the wrong information, I hope I have it correct now!

Bernd

### Re: The role of the phase minimum (phMin)

HI, Bernd.

Thanks。

Is this right:

not ?

When I ran a simulation of cementite precipitation from austenite in 1D, I found that the driving force decreases across the interface from austenite side to cementite side. The carbon concentration in austenite decreases even below the equilibrium value which is not true (this happens when approach zero on cementite side). I tried to use a small value of PhiMin (1.0e-7), the time step is also quite small (1.0e-7), but from the .TabC file, I still found minum C-concentration below the equilibrium and maximum C-concentration extremely high. How could I avoid this kind of numerical error?

Thanks。

Is this right:

not ?

When I ran a simulation of cementite precipitation from austenite in 1D, I found that the driving force decreases across the interface from austenite side to cementite side. The carbon concentration in austenite decreases even below the equilibrium value which is not true (this happens when approach zero on cementite side). I tried to use a small value of PhiMin (1.0e-7), the time step is also quite small (1.0e-7), but from the .TabC file, I still found minum C-concentration below the equilibrium and maximum C-concentration extremely high. How could I avoid this kind of numerical error?

### Re: The role of the phase minimum (phMin)

Hi zhubq,

Again you are right, it must read

I think yesterday I was not at my best!

With respect to the numerical errors you mention, I would say that it is basically a consequence of the averaging of the driving force. I guess that you are using averaging to prevent the interface from spreading. This averaging, even if not complete (e.g "avg 0.5" in 'deltaG' options), leads to a violation, especially at the borders of the interface. You could use "avg 0" with high resolution (or try our new anti-trapping correction which will be available with the next release).

On the other hand, some unphysical values at the boundaries of the interface may be acceptable as long as they only occur in cells with very low fraction...

If the problem is also occuring without averaging of the driving force, then it could be due to a hidden instability and probably could be removed by optimising numerical parameters like interface mobility, resolution, etc.

Bernd

Again you are right, it must read

I think yesterday I was not at my best!

With respect to the numerical errors you mention, I would say that it is basically a consequence of the averaging of the driving force. I guess that you are using averaging to prevent the interface from spreading. This averaging, even if not complete (e.g "avg 0.5" in 'deltaG' options), leads to a violation, especially at the borders of the interface. You could use "avg 0" with high resolution (or try our new anti-trapping correction which will be available with the next release).

On the other hand, some unphysical values at the boundaries of the interface may be acceptable as long as they only occur in cells with very low fraction...

If the problem is also occuring without averaging of the driving force, then it could be due to a hidden instability and probably could be removed by optimising numerical parameters like interface mobility, resolution, etc.

Bernd