Mass Conservation

The matrix concentration is updated at each time step according to the law of mass conservation. If the alloy concentration, i.e. initial mole fraction of component $initial mole fraction of component$ in the matrix phase is $initial mole fraction of component i in the matrix phase$ , the new concentration can be obtained from the following mass balance shown in

$initial mole fraction of component i in the matrix phase$

where

$mole fraction of element i at the interface in the precipitate phase of particle size r_p at time t$ is the mole fraction of element at the interface in the precipitate phase of particle size at time . , and are the PSD function, growth rate, and molar volume of the precipitate phase , respectively.
is the time at time step .

Apparently, the new matrix concentration, and thus the updated supersaturation, nucleation rate, and interface velocity are all functions of the PSD function. This inevitably leads to the complex non-linear behavior of and great difficulty in solving the continuity equation.