Dear all,
Negative compositions is a nasty problem when simulating microstructure formation in complex alloy systems. In almost every case where it occurs it is a consequence of the linear extrapolation methods used for redistribution of the elements between two or more phases. In mild cases the problem may constitute itself only as optically disturbing and not easily explicable artefact, but there might be errors occurring as a consequence (e.g. if MICRESS tries nucleating in such a region), or in extreme cases even the total mass balance is harmed.
Given that many MICRESS users do not regularly check all the relevant output files, negative composition often remain undetected or manifest only by secondary errors. They are even not always visible in the concentration (.conc*) outputs, so that checking also the phase concentration outputs (.c*pha*) is required. A good practice is to set the concentration range close to 0 (e.g. -0.001 to 0.001), so that negative phase compositions can easily be seen, even if many results are opened simultaneously in DP_MICRESS. The focus should be on interconnected blue (negative) regions, while single isolated grid cells with negative value are often not critical. Most often, negative compositions occur in the liquid phase (solidification), or in the primary phase if further precipitate phases are involved. Typically, they appear if phases with high composition grow into a phase with very low composition of the same element (and the phase diagram is far from ideal solution).
To avoid (or at least minimize) the problem, there are different strategies which can be followed. Even if the reason always is linear extrapolation, the circumstances are different and need different solutions:
1.) The simple case:
The simplest case involves only 2 phases: A precipitate phase with high composition, e.g. a TiC-carbide, grows into a liquid phase with very low Ti-content. Because of the numerically broad interface region, there will be a gradient of Ti-composition in the liquid inside the interface, which can easily turn negative at the inner side. The reason is that linear extrapolation treats negative values as "normal", while physically the driving forces must get extreme, inhibiting the phase transformation the closer the composition comes to zero. The problem aggravated if updating of linearisation data is retarded or averaged over larger areas.
Solution 1: More frequent TC-updating
Theoretically, the occurrence of negative compositions should be solved with sufficiently frequent updating of the thermodynamic data which isused for redistribution. However, this will only work for sure if thermodynamic data are obtained for every single interface cell ("local" updating scope). Otherwise, the spatially averaged quasi-equilibrium data cannot account for all different locations, and partially negative values of some compositions can remain. Reducing the averaging area for each reference set of thermodynamic data (see .refR output) will help further, but this comes (like also more frequent updating) with a high computational cost.
Solution 2: Penalty on the driving force
To correct for the wrong interpretation of negative compositions when calculating the driving force, penalty terms can be used (option "penalty" in section Numerical Parameters). For a specific phase in a specific interface, compositions which are outside of the allowed range for the given phase are translated to a penalty driving force which increases exponentially the more the range is violated, and which acts in the direction which avoids the violation. In most cases, using a penalty term for each phase composition which gets negative (and the corresponding phase interaction) is quite effective resolving the problem and also improving the correctness of the results. However, it should be avoided to implement unnecessary penalties because they can sometimes get contradictory at triple junctions, eventually leading to major issues. Furthermore, the use of a cut-off value for the driving force ("dGMax" option) is mandatory because the penalty terms are not limited and may get huge.
2.) The wicked cases
Apart from the simple case treated above, there other reasons where negative compositions may occur and which cannot be solved by the methods above (alone). They can be divided into two subcategories (which also tend to occur simultaneously):
a) Precipitate phases which are treated in MICRESS as "stoichiometric":
While many intermetallic phases are really stoichiometric (i.e. without any solubility range in the database), in MICRESS it is sometimes necessary to treat them as such. Indeed, it makes some sense for the user to define intermetallics as completely stoichiometric by default. One reasons may be that the solubility range is very small. More relevant however are the cases where the matrix component is absent or has a very small concentration, or independent sublattices without vacancies exist. Then, calculation of slopes against the matrix component are not feasible. As a consequence, elements of these phases are treated as stoichiometric for redistribution, but can still vary strongly for different sets of quasi-equilibria. I.e. dissolved elements are very variable against each other, but cannot be exchanged against the matrix component!
b) If a precipitate phase gets overgrown by the primary phase, compositions of the matrix can change rapidly and with formation of gradients, while the interaction of the precipitate with the matrix after overgrowth is very slow. This common process, in combination with precipitate phases with segregation (either simple or wicked) can easily lead to negative compositions which cannot be handled with penalties or frequent relinearisation.
Solution 1:
In the default uses of "database" coupling, the averaging regions for the different phase interaction which belong to a triple ore multiple junction are typically inconsistent. This is because e.g. if complete interfaces are constituting the averaging region (scope), they are also used within triple junctions and thus overlap with the different scopes of the other interfaces. Thus, extrapolation inside the triple junctions, in wicked cases, can lead completely negative compositions there.
Using "database consistent" forces a consistent choice of the scopes, i.e. averaging regions cannot overlap. Then, within such a scope, compositions should be correct in average. However, due to composition gradients, there still may be partially negative compositions.
In conjunction with "database consistent", concentration gradients and the related partially negative compositions can be reduced by increasing diffusion at the interface between the matrix and the precipitate phase. In reality, diffusion is probably always increased at interfaces, and taking this effect into account may help to reduce the problem of negative compositions further.
Solution 2:
One characteristic of these wicked cases is that penalties don't work because of the low interface mobility of the matrix-precipitate interface. Instead of adjusting mobility (what we do not want because it is physically based or determined by the diffusion limited kinetics) we can also increase the cut-off value ("max") of the driving force. This allows the interface to move despite low mobility, and thus avoid negative concentrations.
After all, it may still be difficult to combine all the above options in order to remove or minimize the negative compositions in your simulation. But at least, you now know the tools...
Bernd