Interface thermodynamics and driving forces

[nlucas]

Hi MICRESS team,

I'm modelling bainite sheaf growth in a simple Fe-C system using coupling with ThermoCalc. The simulations begin with finite radii (see images below). I have two questions relating to my outputs and the underlying phase-field approach. My driving file is attached for reference. You'll see that I have removed driving force averaging in the interface; I did this to compare interface equilibrium concentrations of carbon in austenite (c1pha1) with my calculated common-tangent amount from ThermoCalc, and compare maximum chemical driving forces with my calculated amount based on parallel tangent displacement at the bulk steel composition (294 J/cm3 at 0.4 wt% carbon). Since averaging is removed, these maximum driving forces are observed, in my simulations, at the outer edge of the interface, where carbon is the bulk amount (0.4 wt%). My two questions:

(1) Why are equilibrium carbon contents not always produced in the interface? I've found that this depends largely on C diffusivity in austenite and the capillarity (so interfacial energy and tip radius). I'm guessing this is due to the quasi-equilibrium constraint, which only imposes equal diffusion potentials and mass balance across the interface.

(2) Why does the chemical driving force as outputted depend on capillarity? The attached image shows two simulations that differ only in terms of initial ferrite radius (0.4 um and 0.2 um); the driving file is for the 0.2 um run. In the right panels, both the curvature and chemical driving force are higher (more blue cells). As a result, the smaller ferrite grows more quickly (since V = MG) and eventually propagates with a steady-state radius (see the other simulation snapshot). The larger ferrite slows to a stop before the steady-state tip forms; in this simple diffusional scenario, nothing should stop growth as long as there is austenite ahead of the tip with 0.4 wt% carbon.

Thanks in advance for your help ,
Nick

[nlucas]

Correction: the delta G value of -270 J/cm3 labelled in the top-right panel of one of the images is the chemical driving force, not curvature. Apologies for that.
Clarification: in the two images, the left panels show the 0.4 um radius scenario and the right for 0.2 um. The simulation outputs in each of the two images are at the same time-step.

Nick

[Bernd]

Dear Nick,

Welcome to the MICRESS forum. You are modelling a quite interesting and challenging system, and to make a successful simulation setup requires physical as wel as numerical assumptions.

Before discussing your specific model questions, I think, it is necessary first to address some numerical odds which cause numerical artifacts and thus cover the physical and model-inherent behaviour:

1.) You have specified global updates of the thermodynamic data (i.e. averaged for the whole domain). By specifying a much too large updating interval, you trigger an oscillating type of behaviour (because after the average interface moved by about its own thickness an automatic update is triggered). The solution is to use a much smaller updating interval (<1.E-4 s) to avoid such behaviour. Please also note that specifying the updating interval twice (in section "Database" as a global value for all interfaces and in section "Phase Interactions") makes no sense, because then it is done twice.

2.) If you activate "tablog" outputs you will get (among others) the .TabD tabular output of the diagonal diffusion coefficients. There you will see that your definition of the diffusion coefficient leads to a very small value for the fcc phase which is 6 orders of magnitude below that of the bcc phase. Perhaps this is just an error of units (cm/m). Apart from that, please note that you always should activate "tablog" outputs, because it gives you valuable information (e.g. the .TabL output file) without which you are "flying blind".

3.) As a general rule, in simulations with diffusion-limited growth, the diffusion pile-up should always be bigger or at least similar to the numerical interface thickness. Otherwise, artificial solute trapping creates wrong behaviour and a lot of problems. Depending on your choice of the diffusion coefficients in fcc, this may be not the case in your setup. For given diffusion data, the numerical interface thickness must be reduced accordingly. This can be achieved by increasing grid resolution and/or by reducing the numerical interface thickness. In your case, the numerical interface thickness is set to 8 cells which is extremely large. Please note that MICRESS is strongly optimized for a small interface thickness, so that it is sufficient to use 3-4 cells. Nevertheless, it probably will still be necessary to increase grid resolution as well.

4. ) I recommend to introduce the initial bcc-grain not by nucleation of a "large" seed (which is numerically very odd), but by setting it as a second grain within the initial microstructure. If you really want to set them by nucleation, the right way would be to nucleate it with an initial radius of 0 and to define a critical radius (by specifying "analytical_curvature") which is big enough to allow growth at the given conditions.

With these numerical adjustments, the behaviour of your system will completely altered, and maybe your questions will be answered (or modified...). However, a short answer to your questions is that under diffusion-limited steady-state growth conditions your local quasi-equilibrium only depends on curvature which has a strong effect on the local compositions (Gibbs-Thomson effect) and thus on the driving force.

Best wishes
Bernd

[nlucas]

Dear Bernd,

Thanks so much for your response. I am running further models to implement your suggestions. A few questions in the meantime. First, how does this oscillating behaviour manifest in the simulation and how significant is it? An updating interval of <1E-4 s feels very frequent for a simple Fe-C system and slows down the computation. Second, at 400 C the diffusivity of carbon is on the order of 1E-12 cm2/s and observed bainite tip velocities >1 um/s, which for diffusion-controlled growth result in sub-atomic diffusion length scales (presently ignoring the effects of elasticity). Is usage of atc and mob_corr corrections enough to capture diffusion-controlled growth where the grid resolution is larger than this unphysical diffusion length scale? I suppose this is not much different than the interface being mathematical as opposed to physical?

More questions to come... Thanks again.

Nick

[admin]

Dear Nick,

Maybe you are right, and 1E-12 cm2/s is a realistic order of magnitude for the diffusivity of fcc in austenite at such low temperature. But then I would expect also diffusion in ferrite to be much slower.

If you assume diffusion to be so small, you must assume not to be in diffusion control, so you should not use "mob_corr". Otherwise, the front would practically not move at all. Also, you would have a massive reaction where compositions have no influence on morphology.

Bernd

[nlucas]

Dear Bernd,

Thanks for your response. You are right to question the diffusivities I'm using -- these are not well constrained at 400 C. The austenite diffusivity may be an order of magnitude higher if an effective value is chosen to reflect the range of carbon contents between the interface and bulk steel. Equally, ferrite diffusivity is not well constrained at this temperature. A work in progress...

I recalculated the diffusion length scale using the Ivantsov solution for dendrites, which is more appropriate for the parabolic tip, using a radius of ~8 nm for Fe-C steel at the conditions of my investigation based on the modelling work of Leach et al. (2015). This length scale is ~24 nm. I ran a simulation with an extreme cell size of 1 nm (with a 3 cell interface), nucleating from a radius = 0, with 1e-4 s updating interval and "atc mob_corr" enabled. The concentration gradients are around this width (outside the interface), which is great. Regarding the growth, Leach et al. (2015) calculated a lengthening rate of ~10 micron/s at the conditions I'm investigating; my modelled ferrite initially grows at this velocity but slows to a halt and widens. In the model I also re-enabled driving force averaging in the interface, and now the values are far too low (<50 J/cm3) compared to the -294 J/cm3 I calculated independently from parallel tangent construction. It appears that interface averaging results in the incorrect growth behaviour I'm observing (i.e. growth should be unimpeded) -- am I understanding this correctly?

I will trial larger initial ferrite radii within an initial microstructure as you advised.

Best wishes,
Nick

[Bernd]

Dear Nick,

I am not sure whether I understand your idea of comparing the chemical driving force during growth (which is what you see in the .driv output) with the maximum driving force obtained from the parallel tangent construction for the reference composition of 0.4 wt% carbon. You would see this latter value of 294 J/cm3 only if you set the interface mobility to zero or diffusivity to infinity, so that the reference composition can be established throughout the whole interface region. However, as soon as the front moves, the observed chemical driving force decreases as the compositions at the interface change.

In the other extreme case, i.e. if we assume diffusion limited growth conditions and an "ideal" phase-field model with high resolution, kinetic undercooling would vanish completely, and you would see only the curvature contribution mirrored in the .driv output.

In the real case with MICRESS and a modest grid resolution, when using "atc mob_corr", you have 3 contributions to the chemical driving force (.driv output):

a) Curvature
b) An additional gradient in the driving force which is due to the composition gradient over the interface. You may correct this in first order by averaging of the driving force over the interface
c) An extra kinetic contribution which is introduced by "mob_corr" in order to achieve correct interface kinetics inspite of artificial trapping effects of the numerical interface.

Thus, I think, the relevant parameter to look at are not the chemical driving force but rather the stationary growth velocity and tip radius.


Bernd

[nlucas]

Dear Bernd,

Thank you for your informative response. All greatly appreciated. I think I misunderstood how driving force manifests in MICRESS. My understanding now is that it is the free energy integrated over the domain volume that is relevant and not the local driving force; i.e. the latter will in an "ideal" model be zero in the interface where equilibrium compositions are maintained. So in theory, if diffusion was very large (as you say) and the whole microstructure domain re-equilibrates, then free energy of the system approaches zero? And you are suggesting that I should not monitor the local driving force as a gauge of how accurate the kinetics are as there are multiple contributions to this, i.e. the three you list in your response.

Your answers appear to address my initial questions!

Thanks,
Nick