Hi,

To question1: Relaxation approach

The MPF order parameters are linked by the unit sum constraint, i.e. each phase can only change its state in interaction with another phase. The given relaxation approach describes the evolution of the multiphase system towards a minimum of the free energy under comprehensive consideration of pairwise interactions with individually defined mobilities. A combined handling of the dependent derivatives based on a single Lagrange multiplier (used in other MPF models) is only valid under assumption of identical or locally averaged interface mobilities. In this simplified case, the individual dependent contributions can be extracted from the brackets and combined. This is however not suitable for realistic quantitative application. I attach some PPT-slides for more detailed information.

A comprehensive derivation of the MPF kinetic equations can be found in my thesis:

J.Eiken: A Phase-Field Model for Technical Alloy Solidification

https://www.researchgate.net/publicatio ... dification
To question 2: Multi-obstacle potential

The advantage of the multi-obstacle potential is that the interfaces have a finite and clearly defined thickness. Thus, the MPF equations have to be solved within restricted interface areas, only, and artificial ghost contributions of dormant faces are avoided. However, it is very difficult to mathematically formulate the model equations in the singular points and in fact, we never published this correctly, so far. The multi-obstacle potential combines an inner "multi-parabolic" potential ∑(Φ_α * Φ_β) with outer "obstacles". The inner multi-parabolic potential is defined with respect to those phases which are “locally interacting”, only. A phase is defined as “locally interacting” if the amount of the gradient of the respective order parameter differs from zero. Thus, a point with order parameter 0 or 1 may be either inside or outside the interface region. Inside, its finite derivatives are clearly defined by the multi-parabolic potential ∑(Φ_α * Φ_β), outside all associated interfacial mobilities and energies are defined as zero and therefore the theoretical ‘obstacle’ part of the free energy functional is irrelevant and never defined in practice.

The discretization of the (double-obstacle) PF-equation based on a dedicated Finite-difference scheme is outlined in:

https://www.researchgate.net/publicatio ... tion_error
With best regards,

Janin