# Kinetics of Solid Phase Transitions

Detailedmechanismsformost physicaldegradationprocessesaffectingthe efficacy and safety of drug products have not been extensively studied because of their complexity. Unlike chemical degradation rates in solution, physical degradation rates usually cannot be predicted on the basis of kinetic parameters estimated from data obtained under accelerated conditions. However, prediction of some physical degradation pathways such as polymorphic changes has been attempted. Several reports dealing with the prediction of polymorphic transitions based on kinetic principles are summarized below.

The Hancock-Sharp equation610 is often used to describe the kinetics of polymorphic transitions:

where B is a constant. In this equation, a is the fraction of drug in the product state over the fraction in the starting state. By plotting the left-hand side of Eq. (3.2) against the logarithm of time, a linear relationship with a slope of m is obtained. The value of m is then used as an indicator of the transition mechanism. Because each of the mechanisms for polymorphic transitions shown

 m value in Eq. (3.2) Equation Mechanism 0.62 a2 = kt D1(a), one-dimensional diffusion 0.51 (1-a)In(l- a) + a = kt D2(a), two-dimensional diffusion 0.54 [1 - (1 - a)1'3]2 = kt D3(a), three-dimensional diffusion 0.57 1 - 2a/3 - (1 - a)2'3 = kt D4(a), three-dimensional diffusion 1. 0 -1n ( 1 - a) = kt Fi(a), first-order kinetics 1.11 1 - (1 - a)1/2 = kt R2(a), phase boundary (cylindrical) 1.07 -(1 - a)1'3 = kt R3(a), phase boundary (spherical) 1.24 a = kt Zero order, zero-order kinetics 2.00 [-In( 1 - a)] "2 = kt A2(a), two-dimensional growth of nuclei 3.00 [-In( 1 - a)] 1/3 = kt A 3(a), three-dimensional growth of nuclei

a Reference 610.

a Reference 610.

in Table 10 exhibits a characteristic value of m, determining m according to Eq. (3.2) makes it possible to select a suitable rate equation and to estimate a descriptor rate constant, k.

The polymorphic transition of carbamazepine from form I to form III and that of benoxaprofen from form I to form II exhibited m values of 2.23 and 2.24, respectively, indicating possible mechanisms involving two-dimensional growth of nuclei.61 1 578 When m is approximately equal to 2, the reaction conforms to the Avrami-Erofe'ev equation:

[-ln(1 - a)]1'2 = kt (3.3)

The data for carbamazeine and benoxaprofen plotted according to Eq. (3.3) are shown in Figs. 143 and 144, respectively; and the rate constants k obtained from the slopes gave linear Arrhenius plots, indicating that it might be possible to predict the polymorphic transition rates at other temperatures (Fig. 145).

Polymorphic transitions of bromovalerylurea from form I to form II and from form III to form I conformed to mechanisms involving one-dimensional diffusion and two-dimensional nuclei growth processes, respectively. Both transitions also exhibited good Arrhenius behavior in the temperature range studied, as shown in Fig. 146.579 Transitions of phenyl- Figure 143. Polymorphic transition kinetics of carbamazepine from form I to form III according to the Avrami-Erofe'ev equation (Eq. 3.3). (Reproduced from Ref. 611 with permission.)

Figure 143. Polymorphic transition kinetics of carbamazepine from form I to form III according to the Avrami-Erofe'ev equation (Eq. 3.3). (Reproduced from Ref. 611 with permission.) Figure 144. Polymorphic transition kinetics of benoxaprofen from form I to form II according to the Avrami-Erofe'ev equation (Eq. 3.3). (Reproduced from Ref. 578 with permission.)

Figure 144. Polymorphic transition kinetics of benoxaprofen from form I to form II according to the Avrami-Erofe'ev equation (Eq. 3.3). (Reproduced from Ref. 578 with permission.)

butazone polymorphs from form a to 8 and from form P to 8 conformed to two-dimensional patterns and first-order kinetics, respectively (Fig. 147).612

The transition of phenobarbital from forms C and E to an anhydrous form conformed to the Jander equation (Eq. 3.4), indicating a three-dimensional diffusion mechanism.587 Some results are shown in Fig. 148.

The transition of 5-(4-oxo-phenoxy-4#-quinolizine-3-carboxamide)-tetrazolate from a tetrahydrate to a monohydrate conformed to zero-order kinetics, and the depend-

[1 - (1 - a)1'3]2 = kt+ C Figure 145. Linear Arrhenius plots for the polymorphic transitions of carbamazepine (o,) and benoxaprofen (•,) The rate constants k were obtained according to the Avrami-Erofe'ev equation (time unit: min). (Reproduced from Refs. 578 and 611 with permission.) Figure 146. Arrhenius plots of the polymorphic transitions of bromovalerylurea. o, Transition from form I to form II; transition from form III to form I. The rate constant k is in units of min-1. (Reproduced from Ref. 579 with permission.)

ence of the rate constant on temperature and humidity could be described by Eq. (2.113) (Fig. 149).613

The transition of sulfaguanidine from a monohydrate form to an anhydrous form in the presence of differing water vapor pressures586 conformed to different rate equations among those listed in Table 10. Similarly, the dehydration kinetics of hydrated theophylline614 of different particle sizes was also described by different rate equations. Polymorphic transitions of anhydrous theophylline in tablets conformed to different rate equations depending on the tablet size and pore size.583 584

The rate equations shown in Table 10 have also been used to describe the kinetics of crystallization, that is, the conversion from the amorphous state to a crystalline state. Crystallization of amorphous furosemide dispersed in Eudragit conformed to the rate equation proposed for a three-dimensional diffusion process.615 Figure 147. Polymorphic transition kinetics of phenylbutazone according to the Avrami-Erofe'ev equation (a) and by first-order kinetics (b), as a function of exposure to varying relative humidities (60°C). ( 0% RH; A,a> 50% RH; □, ■, 70% RH; O, : 80% RH. (Reproduced from Ref. 612 with permission.)

Figure 147. Polymorphic transition kinetics of phenylbutazone according to the Avrami-Erofe'ev equation (a) and by first-order kinetics (b), as a function of exposure to varying relative humidities (60°C). ( 0% RH; A,a> 50% RH; □, ■, 70% RH; O, : 80% RH. (Reproduced from Ref. 612 with permission.) Figure 148. Transition kinetics of two hydrated forms of phenobarbital to an anhydrous species according to the Jander equation (Eq. 3.4). T= 45°C. (a) Transition from form C to an anhydrous state; (b) transition from form E to an anhydrous state. (Reproduced from Ref. 587 with permission.)

Some polymorphic transitions can be described by equations other than those listed in Table 10. The transition of nitrofurantoin from its anhydrous form to a monohydrate was described by the following equation585:

kt+C

The Arrhenius equation has been employed as a first approximation in an attempt to define the temperature dependence of physical degradation processes. However, the use of the WLF equation (Eq. 3.6), developed by Williams, Landel, and Ferry to describe the temperature dependence of the relaxation mechanisms of amorphous polymers, appears to have merit for physical degradation processes that are governed by viscosity. Figure 149. Temperature and humidity dependence of the tetrahydrate-to-monohydrate transition kinetics for 5-(4-oxo-phenoxy-4if-quinolizine-3-carboxamide)-tetrazolate. Here k refers to the apparent zero-order rate constant for the process (time unit: h) and k' = k/PS, where P is water vapor pressure. (Reproduced from Ref. 613 with permission.)

Figure 149. Temperature and humidity dependence of the tetrahydrate-to-monohydrate transition kinetics for 5-(4-oxo-phenoxy-4if-quinolizine-3-carboxamide)-tetrazolate. Here k refers to the apparent zero-order rate constant for the process (time unit: h) and k' = k/PS, where P is water vapor pressure. (Reproduced from Ref. 613 with permission.) -C,(T-Tg)/(C2+T-T8)

Figure 150. The crystallization rate of nifedipine plotted according to the WLF equation. A,, Relationship between crystallization rate and temperature; i relationship between crystallization rate and Tg. (Reproduced from Ref. 604 with permission.)

-C,(T-Tg)/(C2+T-T8)

Figure 150. The crystallization rate of nifedipine plotted according to the WLF equation. A,, Relationship between crystallization rate and temperature; i relationship between crystallization rate and Tg. (Reproduced from Ref. 604 with permission.)

In this equation, fa and k are the crystallization rates at temperatures T and Ts, respectively, and Ci and C2 are constants. The role of viscosity in the crystallization of amorphous sucrose was suggested by the observation that crystallization is enhanced by a lowered Ts resulting from moisture adsorption.574-577 Also, the crystallization rate of amorphous nifedipine exhibited a temperature dependence best represented by the WLF equation. The increase in the crystallization rate caused by the decrease in Ts under higher humidity conditions was also described by the WLF equation, as shown in Fig. 150.604