## diffusion problem

### diffusion problem

Hi, Bernd.

when solving the diffusion equation in the interface, how does MICRESS calculate the gradient of the concentration at the edge of the interface?

Because if the point is at the edge, we have to use the points out of the interface to calculate the gradient . For phase field equation, the value of is 0 out of the interface. But for the concentration, if the value of is 0, what's the concentration ?

By the way, the diffusion equation can be formulated like this?:

+

where is the diffusivity of phase .

when solving the diffusion equation in the interface, how does MICRESS calculate the gradient of the concentration at the edge of the interface?

Because if the point is at the edge, we have to use the points out of the interface to calculate the gradient . For phase field equation, the value of is 0 out of the interface. But for the concentration, if the value of is 0, what's the concentration ?

By the way, the diffusion equation can be formulated like this?:

+

where is the diffusivity of phase .

### Re: diffusion problem

Hi zhubq,

Diffusion as implemented in MICRESS can be described for a single component like you did, your formulation can be obtained from the standard formulation

using the product rule of differentiation and assuming a constant value of .

For discretisation, the fundamental question is how to obtain and because they are defined between the grid points like . Normal averaging between the left and right grid cell like

+

would lead to the problem that, if e.g. in an interface one of the phases had no solubility at all for the component under consideration, there would be still a flux into this stoichiometric phase, even if for the other phase is zero. Therefore, we instead apply the modified averaging scheme

+

which avoids such problems

By this way, it also doesn't matter that we do not know , if in one grid point is zero.

Diffusion as implemented in MICRESS can be described for a single component like you did, your formulation can be obtained from the standard formulation

using the product rule of differentiation and assuming a constant value of .

For discretisation, the fundamental question is how to obtain and because they are defined between the grid points like . Normal averaging between the left and right grid cell like

+

would lead to the problem that, if e.g. in an interface one of the phases had no solubility at all for the component under consideration, there would be still a flux into this stoichiometric phase, even if for the other phase is zero. Therefore, we instead apply the modified averaging scheme

+

which avoids such problems

By this way, it also doesn't matter that we do not know , if in one grid point is zero.

### Re: diffusion problem

Hi, Bernd.

I am not sure whether my understanding is right.

First of all, you get the values of e.g. (Y(i-0.5) and Y(i+0.5)) between each two grid points e.g. X(i-1), X(i) and X(i), X(i+1). Then you use the two values ((Y(i-0.5) and Y(i+0.5))to get at point i.

I am not sure whether my understanding is right.

First of all, you get the values of e.g. (Y(i-0.5) and Y(i+0.5)) between each two grid points e.g. X(i-1), X(i) and X(i), X(i+1). Then you use the two values ((Y(i-0.5) and Y(i+0.5))to get at point i.

### Re: diffusion problem

if and , how to calculate in between?

### Re: diffusion problem

If X(i-1)=0, like in your case, then Y(i-0.5) must be 0, whatever the value of is, i.e. you don't need to know it! The total composition change is then only Y(i+0,5).

The trick is that the total composition change in grid cell i is calculated as the sum of all pair-wise fluxes with the neighbour grid cells. The fluxes of all pairs which have a fraction of 0 in at least one cell are 0, and you need not calculate for them!

The trick is that the total composition change in grid cell i is calculated as the sum of all pair-wise fluxes with the neighbour grid cells. The fluxes of all pairs which have a fraction of 0 in at least one cell are 0, and you need not calculate for them!

### Re: diffusion problem

Hi, Bernd.

Thank you. Now I know it. A numerical trick has to be used to enforce the pair-wise flux to be 0 if the neighbor has a zero value of order parameter.

Thank you. Now I know it. A numerical trick has to be used to enforce the pair-wise flux to be 0 if the neighbor has a zero value of order parameter.

### Re: diffusion problem

Well - I would not call it a trick, because when it comes to discretize the diffusion equation, one has to make a choice for the averaging scheme applied, and the one we use is not better or worse than another one like the arithmetic or geometric average!

We have chosen this scheme not only to remove the problem of the unknown gradient in the boundary cells of interfaces, but - what is much more important - to remove unphysical fluxes into stoichiometric phases without solubility and to allow defining a stabilty criterion for time-stepping for the explicit diffusion solver, which would not be possible with arithmetic averaging!

We have chosen this scheme not only to remove the problem of the unknown gradient in the boundary cells of interfaces, but - what is much more important - to remove unphysical fluxes into stoichiometric phases without solubility and to allow defining a stabilty criterion for time-stepping for the explicit diffusion solver, which would not be possible with arithmetic averaging!