Modelling phase transition kinetics of chenodeoxycholic acid with the Runge–Kutta method

The phase transition kinetics of two chenodeoxycholic acid polymorphic modifications—form I (stable at high temperature), form III (stable at low temperature) and the amorphous phase has been examined under various conditions of temperature and relative humidity. Form III conversion to form I was examined at high temperature conditions and was found to be non-spontaneous, requiring seed crystals for initiation. The formation kinetic model of form I was created incorporating the three-dimensional seed crystal growth, the phase transition rate proportion to the surface area of form I crystals, and the influence of the amorphous phase surface area changes with an empirical stage pointer q that contained the incomplete transition of the amorphous phase to form I   with a residue ωA∞ωA∞. The extent of transition and the phase transition rate constant depended on form I seed crystal amount in the raw mixture, and on the sample preparation. To describe phase transition kinetic curves, we employed the Runge–Kutta differential equation numeric solving method. By combining the Runge–Kutta method with the multi-point optimization method, the average quadratic deviation of the experimental results from one calculated series was under 2%.