Grain Boundary Diffusion Input for Carbide Simulation

[Ku shihyeon]

Hello! Thank you for all your previous help.

I am planning to simulate carbide nucleation (both bulk and grain boundary nucleation) within a polycrystalline matrix. This is a step to organize initial microstructure for a subsequent simulation that will be linked via a restart function.

In the subsequent simulation, I aim to calculate the dissolution (or decomposition), growth, and additional precipitation of these carbides within the microstructure under conditions reflecting the HAZ thermal cycle during welding. The depletion and redistribution of the carbide-forming elements due to carbide formation are also of interest. Consequently, I am investigating variables related to solute diffusion.
This change is expected to manifest differently depending on whether the secondary phase is located in the bulk or at the interface/boundary position.

In the process of searching for some information on this forum, I noticed an extra line in the diffusion section specifically for considering grain boundary diffusion. However, I have not previously considered any additional options in my MICRESS usage to reflect the difference between bulk diffusion and interfacial diffusion of the solute (more accurately, I assumed it was already being implicitly reflected without an extra option).

Up until now, I have only activated the diagonal distribution for each element and phase (e.g., # # diagonal_dilution z). Although I haven't run the calculation yet, I tentatively added a row 9 +b after the existing diagonal diffusion rows, and confirmed there was no syntax error.

In this context, my questions are as follows:
1. Specific Explanation and Usage of the Extra Line for Interfacial Diffusion
I would like to know the specific explanation and usage of this additional row for interfacial diffusion. The existing rows are entered in the sequence: (Element) (Phase) (Option). However, this interfacial diffusion input seems to allow (Element) +b. I input it this way because I thought it should only specify the element, as the option relates to the interface. However, I have seen other posts where it is entered as (Element) (Phase) +b. In this case, does it mean that the (Phase)/(Phase) interface for the specified (Element) is being designated?

2. Difference in Diffusion Calculation Without the Interfacial Option
If the calculation is performed without adding this interfacial option, is there no difference calculated between bulk diffusion and interfacial diffusion?

Thank you.

[Bernd]

Dear Ku shihyeon,

Grain boundary diffusion (GBD) often leads to confusion because it is a quite complex issue. When GBD is not explicitly addressed in MICRESS, diffusion is calculated inside each phase using the defined bulk diffusivity values. That means that within the diffuse interfaces, bulk diffusion is solved for each coexisting phase independently. This is the usual homogenisation approach for diffuse interfaces.

In our approach implemented in MICRESS, GBD is not considered as an interface-specific contribution to diffusion. Instead it is assumed that bulk diffusion can be enhanced within interfaces, i.e. it is always a phase-specific contribution. This means, at the example of a dual phase interface, that diffusion can be considered to be enhanced either for one or for the other phase of the interface (or for both). For grain boundaries with identical phases, instead, there is only one phase present which may have enhanced diffusion at interfaces.

In any case, the enhancement of bulk diffusion at interfaces is described by a reduction of the activation energy of diffusion. Defining a difference in activation energy is equivalent to factor on the diffusion coefficient which is T-dependent. This has the side effect that DBG can only be defined for phases which already have diffusion (and e.g. not for interfaces between stoichiometric phases). Furthermore, GBD scales with the interface thickness, so that the physical interface thickness is needed to account for the difference to the numerical interface thickness. Finally, the user must define in which type of interfaces (i.e. in the context of which interacting second phase) enhanced diffusion shall be applied. MICRESS cannot account for GBD automatically, because there is no respective data available in the mobility databases.

When using grain boundary diffusion in MICRESS, it is strongly recommended to use "terse" mode for inpout of the diffusion data. This means that the element and phase number (which defines the diffusion "term", i.e. the flux of one element in one phase, or a horizontal line of the diffusion matrix) are written at the beginning of each line. By this way, it is possible to use more than one line for definition of the same diffusion term, which is needed for many "extra line" options which are available in MICRESS. For BGD it would read like e.g.:

# How shall diffusion of component 5 in phase 1 be solved?
5 1 multi gg
5 1 +b nnbn
# Grain boundary diffusion settings for component
# 5 at boundary of phases 1 and 2:
Correction for activation energy? [J/mol]
1.8E3
Physical width of the interface? [nm]
2.5

In this example, diffusion of element 5 is considered in phase 1. Grain boundary diffusion can be added for all interfaces with those phases defined afterwards. The string "nnbn" has a character "b" for each phase number where phase 1 gets enhanced diffusion, and a character "n" where not. In this case, phase 1 will have enhanced diffusion only in interfaces with phase 2, while interphases with phases 0, 1, and 3 won't. The string must exactly have the length of the number of phases (including phase 0). The definition "(Element) +b" is not valid (although it perhaps is erroneously accepted).

In practice, it may be a bit difficult to get a feeling about the GBD-parameters, given the complex expoential effect of the correction of the activation energy and the additional rescaling to the numerical interface thickness. In case diffusion coefficient are read from a thermodynamic .GES5-file, the .diff output gives additional information on the maximum diffusion coefficients which are effectively reached inside the corresponding interfaces. This is very helpful to estimated how big the effects of GBD will be for the chosen parameter set.

Please tell me if some of your doubts have not been sufficiently answered.

Bernd

[Ku shihyeon]

Thank you for your response.

Regarding the part of your answer: "Defining a difference in activation energy is equivalent to factor on the diffusion coefficient which is T-dependent."
Can I understand this to mean that the diffusion enhancement in the interfacial region is considered to be the effect of a decrease in activation energy, which is equivalent to the enhancement of the diffusion coefficient within the bulk due to a rise in temperature?

Also, upon checking the .diff file from the previous calculation, I confirmed that the activation energy is outputted per phase over time. Should I consider this as the activation energy for bulk diffusion?
And should I input a smaller value compared to this bulk diffusion value in the following line: [Correction for activation energy? [J/mol] ]?
It seems likely that there is no explicit criterion for how much smaller this value should be. For now, I'll need to check if I can refer to papers that experimentally studied the desired alloy system under different process

Additionally, we aim to simulate carbide precipitation within the initial solid-state matrix under a specific thermal history, as well as the re-dissolution (re-solution/decomposition) of the precipitate into the matrix or liquiaction (local melting).
In our calculations so far, we have only experienced secondary phase precipitation from liquid phase.

The formation of a secondary phase within the solid phase appears similar to what we experienced in the prior calculation, but do re-solution of the precipitate phase or local liquiaction require a separate seed setting?
While I'm not certain, I recall seeing a post in a forum previously suggesting that to implement re-dissolution when applying a thermal input to re-melt a system after solidification, it was necessary to set up nucleation for the liquid phase.

Thank you.

[Bernd]

Dear Ku shihyeon,

Sorry for the confusion caused by my explanation. If the bulk diffusion coefficient is defined by

D = D₀ exp(-E_A/RT)

the user is requested to specify a correction (reduction) of the activation energy ΔE which corresponds to the increase of diffusivity at the interface, then the (physical) diffusivity at the interface will be

D^* = D₀ exp(-(E_A-ΔE)/RT)

This increase of the diffusivity at the interface is not a constant factor, because the temperature T is inside the exponential function. So, my comment was essentially that the effect of the user-defined grain or phase boundary diffusion is T-dependent.

Additionally, the user is requested to input a physical interface thickness η^*, so that the diffusivity can be correctly scaled to the numerical interface thickness η:

D^* = D₀ (η^*/η) exp(-(E_A-ΔE)/RT)


In the .diff-file this looks e.g. like that:

diff.PNG

In this case, the effect at the interface should be pronounced as the rescaled diffusion coefficient is about 2-3 orders of magnitude higher than the bulk diffusion coefficients. This maximum value is only reached at the center of the interface where both fraction are idential (φ_α=φ_β=0.5), it gets smaller towards both sides of the interface.


Regarding your second question, it is important to notice that in MICRESS new phases can only appear during the simulation time if nucleation is explicitly requested by the user. For melting this means that liquid must be nucleated, and carbides will also form only if a corresponding seed type is defined. To the contrary, any phase can vanish without specific definition (e.g. dissolution of carbides in the matrix phase).

Nucleation of liquid is a bit tricky, because usually the formation of different liquid "grains" is not intended (they would form grain boundaries when growing together!) This can be avoided by using "add_to_grain" with suboption "new set", which makes sure that all seeds from the given seed type definition will belong to the same grain number, or by adding them explicitly to the grain number 0 (suboption "grain_number". As a reference, you can have a look at seed types 4 and 5 in the application example 'A018_Al4Cu_Additive_Rosenthal.dri'. Please note that in this case the initial radius of the liquid seeds was chosen >0 (but still <Δx) to give the new seeds a slighly bigger initial phase fraction. The reason is that, due to the fact that all seeds belong to the same grain number, the specific "small grain" models (stabilisation or analytical curvature) cannot work correctly, and initial grains with radius r=0 could occasionally disappear before getting a chance to grow. Please also note that the critical nucleation "undercooling" in case of liquid should still be set as a positive value, although it of course means "overheating".

Bernd