How grain boundary diffusion works

technical aspects of .dri file generation (e.g. debug mode ) etc...
Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

How grain boundary diffusion works

Post by Bernd » Tue Jun 29, 2010 10:24 pm

Dear all,

Grain boundary diffusion is available in MICRESS since more than 2 years, but nearly nobody used it up to now. It also has never been documented in detail. (but in the new manuals we are working on, it will of course be included!)

Grain boundary diffusion in MICRESS is implemented as an increased diffusivity of a phase in the interface region to other phases. This increase is expressed as a reduction of the activation energy, leading to a temperature-dependent factor on the diffusion coefficients. The user has to specify this activation energy correction in J/mol as well as the physical interface thickness which is needed to scale to some real thickness. Thus, grain boundary diffusion is only possible for elements in phases which have also diffusion in the bulk and not e.g. for stoichiometric intermetallics without solubility range!

The input works only in the terse mode, i.e. if each diffusion term is defined with a proceeding element and phase number. First, bulk diffusion has to be defined for this contribution. Afterwards, for the same contribution, grain boundary diffusion can be added by using the keyword "+b", followed by a string with length <number of phases +1> consisting of "n" or "b". This string defines for each phase whether in grain boundaries to the actual phase enhanced diffusion is to be considered. For each "b" character, an actication energy difference and a physical interface thickness has to be entered in extra lines, e.g.(ternary alloy, 3 solid phases):

# How shall diffusion of component 1 in phase 1 be solved?
1 1 multi gg
# How shall diffusion of component 2 in phase 1 be solved?
2 1 multi gg
# How shall diffusion of component 1 in phase 1 be solved?
1 1 +b nnbn
# Grain boundary diffusion settings for component
# 1 at boundary of phases 1 and 2:
Correction for activation energy? [J/mol]
1.8E3
Physical width of the interface? [nm]
2.5
...
end_diffusion_data

Bernd

Hamid
Posts: 15
Joined: Fri Feb 27, 2009 12:47 am

Re: How grain boundary diffusion works

Post by Hamid » Wed Aug 18, 2010 11:56 pm

Dear Bernd,
I could not add '+' to the equations. Thus I had to make an image out of my question :(.

Thank you,
Hamid
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Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

Re: How grain boundary diffusion works

Post by Bernd » Tue Aug 24, 2010 2:21 pm

Dear Hamid,

sorry for the delay!

I have looked up what exactly is implemented in the MICRESS code. This has been done by Philippe in former times, and now is a good opportunity to recall how it is done!

As already said above, the physical model or idea is to assume a higher diffusivity in a given phase at the interface between identical phases or to other phases as the consequence of a reduced activation barrier for diffusion. This increased diffusivity shall be applied in a small channel or plate with a thickness of \eta_{phys} which is small compared to the numerical interface thickness \eta:

D^{channel}=D_0 \,e^{\frac{-(E_A-E_{GB})} {RT}}

The reduction of the activation energy E_{GB} can be specified by the user. For interfaces between different phases, seperate interface contributions to diffusion must be specified for both involved phases.

For the numerical implementation, the diffusion in the small surface "channel" has to be rescaled to the numerical interface thickness. For this purpose it is necessary to divide the total diffusivity in a numerical grid cell into the normal diffusion contribution and an extra grain boundary diffusion:

D^{tot}=D + D^{GB}

with

D=D_0 \,e^{\frac{-E_A} {RT}}

and

D^{GB}=D_0 \,e^{\frac{-E_A} {RT}}\,*\,f\frac{\eta}{\eta_{phys}}\phi_1\phi_2\,(e^{\frac{E_{GB}}{RT}}-1)


Edit 2018-03-01: \eta and \eta_phys have been mixed up in the last formula. The correct one is:

D^{GB}=D_0 \,e^{\frac{-E_A} {RT}}\,*\,f\frac{\eta_{phys}}{\eta}\phi_1\phi_2\,(e^{\frac{E_{GB}}{RT}}-1)


The factor f is for normalizing, in our implementation its value is 8.0.

Note that the resulting total diffusivity of this phase D^{tot} in case of an interphase \alpha/\beta is additionally weighted like


D=\phi_{\alpha}D_{\alpha} + \phi_{\beta}D_{\beta}


Bernd

Hamid
Posts: 15
Joined: Fri Feb 27, 2009 12:47 am

Re: How grain boundary diffusion works

Post by Hamid » Thu Aug 26, 2010 8:59 pm

Dear Bernd,

Thank you for the response. Honestly, I am looking to see how 'f' was calculated. How did you come up with f=8 as the normalizing factor?

Thank you
Hamid

Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

Re: How grain boundary diffusion works

Post by Bernd » Thu Aug 26, 2010 10:23 pm

Dear Hamid,

also honestly, I do not know, but it should be the reciprocal value of the integral of _{\phi_1 \times \phi_2} over the interface which has a sinus shape...

Bernd

Hamid
Posts: 15
Joined: Fri Feb 27, 2009 12:47 am

Re: How grain boundary diffusion works

Post by Hamid » Sat Sep 04, 2010 1:04 am

Dear Bernd,

Are you aware of anyone who can help me in more detail on that since I am going to report that in a paper.

Best regards,
Hamid

Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

Re: How grain boundary diffusion works

Post by Bernd » Tue Sep 07, 2010 6:52 pm

Hi Hamid,

we are trying to find that out and give you the correct information soon...

Bernd

Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

Re: How grain boundary diffusion works

Post by Bernd » Wed Sep 15, 2010 4:39 pm

Hi Hamid,

I discussed it with Markus, and it is as I told you, the integral over the interface is:

\int_0^\eta \, \phi_1(x) \phi_2(x) dx = \frac {\eta} {8}

Using a sinus function, which is the equilibrium interface profile for the double obstacle potential, the exact calculation is:

\int_{-0.5}^{0.5} \, (1-(1\oplus sin(\pi \frac x \eta)/2) ((1 \oplus sin(\pi \frac x \eta)/2)\, dx=\eta/8

(please excuse the use of the stupid \oplus instead of the plus character, which I still not manage to use in this Latex implementation :cry: )

The calculation was done by Markus using Matlab.

Apart from that, your formulation of grain boundary diffusion, which you originally got from Markus, is completely compatible to mine, just that DGB is defined differently!

Bernd

Hamid
Posts: 15
Joined: Fri Feb 27, 2009 12:47 am

Re: How grain boundary diffusion works

Post by Hamid » Tue Oct 05, 2010 5:39 am

Thank you Bernd,

Then, the approach that I presented above is also correct. The only difference is that I used Tanh function to present phi instead of sinus function.

Bernd
Posts: 1504
Joined: Mon Jun 23, 2008 9:29 pm

Re: How grain boundary diffusion works

Post by Bernd » Tue Oct 05, 2010 2:44 pm

Ah, ok!

I guess you used the "double_obstacle" potential for simulations in MICRESS, so the profile should be a sinus function (the "double_well" potential with its tanh profile is no longer available in the current MICRESS version).


Bernd

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