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In this paper a general multi-phase-field model is presented which is anextension and modification of the model proposed by Folch and Plapp for threephase fields [R. Folch and M. Plapp, Phys. Rev. E 72 011602 (2005)] to thearbitrary number of phases. In the model a physical constraint requiring thatthe sum of all phase fields in the system is equal to one is resolved by themethod of Lagrange multipliers. Namely, the thermodynamic driving force isreduced to its projection on the plane of the constraint. The general modelfunctions in a $N$-dimensional phase field space were derived which justify therequirements for the stability of the total free energy functional on dualinterfaces and hence the absence of "ghost" phases. Furthermore, the case ofthe different interface energies and mobility parameters on the individualinterfaces is resolved in a comprehensive manner. It is shown that the staticequilibrium for three or four phases fulfils Young's law for contact angleswith high accuracy. Also the model is verified by the quantitative simulationof the solidification in an Al-Cu-Ni alloy in the case of the four-phasetransformation reaction. We found the way to control the character of new phasenucleation using additional terms in free energy functional.

General Phase-Field Model with Stability Requirements on Interfaces in $N$-Dimensional Phase-Field Space