Solid-state Kinetic Models Research Articles

Solid‐State Kinetic Models: Basics and Mathematical Fundamentals

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Combined Kinetic Analysis of Solid-State Reactions: A Powerful Tool for the Simultaneous Determination of Kinetic Parameters and the Kinetic Model without Previous Assumptions on the Reaction Mechanism

The combined kinetic analysis implies a simultaneous analysis of experimental data representative of the forward solid-state reaction obtained under any experimental conditions. The analysis is based on the fact that when a solid-state reaction is described by a single activation energy, preexponetial factor and kinetic model, every experimental T-alpha-dalpha/dt triplet should fit the general differential equation independently of the experimental conditions used for recording such a triplet. Thus, only the correct kinetic model would fit all of the experimental data yielding a unique activation energy and preexponential factor. Nevertheless, a limitation of the method should be considered; thus, the proposed solid-state kinetic models have been derived by supposing ideal conditions, such as unique particle size and morphology. In real systems, deviations from such ideal conditions are expected, and therefore, experimental data might deviate from ideal equations. In this paper, we propose a modification in the combined kinetic analysis by using an empirical equation that fits every f(alpha) of the ideal kinetic models most extensively used in the literature and even their deviations produced by particle size distributions or heterogeneities in particle morphologies. The procedure here proposed allows the combined kinetic analysis of data obtained under any experimental conditions without any previous assumption about the kinetic model followed by the reaction. The procedure has been verified with simulated and experimental data.

Solid-State Kinetic Models: Basics and Mathematical Fundamentals

Many solid-state kinetic models have been developed in the past century. Some models were based on mechanistic grounds while others lacked theoretical justification and some were theoretically incorrect. Models currently used in solid-state kinetic studies are classified according to their mechanistic basis as nucleation, geometrical contraction, diffusion, and reaction order. This work summarizes commonly employed models and presents their mathematical development.

Effect of tablet geometrical structure on the dehydration of creatine monohydrate tablets, and their pharmaceutical properties

The effects of compression and pulverization on the dehydration kinetics and hardness of creatine monohydrate tablets were studied using a variety of kinetic equations and physical models. The dehydration behavior of unpulverized and pulverized tablets was investigated by using differential scanning calorimetry (DSC) and powder X-ray diffraction (PXRD). The hardness of both unpulverized and pulverized monohydrate tablets was significantly decreased after dehydration. The relationship between the degree of dehydration and the tablet hardness of both unpulverized and pulverized monohydrate tablets formed a straight line. The results suggest that the reduction in tablet hardness is dependent on the dehydration of crystal water, and the values of the slopes indicate that the bonding energy of the unpulverized sample was stronger than that of the pulverized sample. The dehydration kinetics of the unpulverized and pulverized monohydrate tablets were evaluated by analyzing the fit of the isothermal DSC data using a variety of solid-state kinetic models. The dehydration of the unpulverized tablets at various levels of compression pressure followed the 3-dimensional growth of nuclei mechanism. In contrast, although the dehydration kinetics of pulverized monohydrate tablets compressed at 500 and 750 kg/cm2 followed the 3-dimensional diffusion mechanism, those compressed at 1000 kg/cm2 followed the 3-dimensional growth of nuclei mechanism. The PXRD analysis indicated that the diffraction intensity of the pulverized monohydrate powder was significantly lower than that of the unpulverized powder. The diffraction peaks of the (h00) planes and the micropore structure of the unpulverized monohydrate tablets were affected by pulverization and compression force, respectively.

Effect of pulverization on hydration kinetic behaviors of creatine anhydrate powders

The crystal orientation of creatine monohydrate varies significantly with tableting performance and pulverizing mechanism. Furthermore, the X-ray diffraction patterns of anhydrous forms of untreated creatine monohydrate and of pulverized creatine monohydrate exhibit different crystal orientations. However, hygroscopic forms of unpulverized creatine anhydrate and pulverized creatine anhydrate was exhibit the same diffraction peak pattern. The hygroscopicity of unpulverized and pulverized creatine anhydrate has been investigated by hydration kinetic methods using isothermal differential scanning calorimetry data. Testing of the hygroscopicity of unpulverized and pulverized creatine anhydrate at various levels of relative humidity (RH) at 25 °C revealed that the anhydrate was stable at less than 33% RH, but was transformed into the monohydrate at more than 52% RH. Hydration data of unpulverized and pulverized creatine anhydrate at 60% and 75% RH were calculated to determine hydration kinetics using various solid-state kinetic models. The hydration type of unpulverized and pulverized creatine anhydrate powder follows the zero-order mechanism (Polany–Winger equation) R1. The transition rate constant of pulverized creatine anhydrate, calculated from the slope of the straight line, was about 1.34–1.36 times higher than that of unpulverized creatine anhydrate.

Use of in situ FT-Raman spectroscopy to study the kinetics of the transformation of carbamazepine polymorphs

The solid-state transformation of carbamazepine from form III to form I was examined by Fourier Transform Raman spectroscopy. Using a novel environmental chamber, the isothermal conversion was monitored in situ at 130 °C, 138 °C, 140 °C and 150 °C. The rate of transformation was monitored by taking the relative intensities of peaks arising from two C H bending modes; this approach minimised errors due to thermal artefacts and variations in power intensities or scattering efficiencies from the samples in which crystal habit changed from a characteristic prism morphology (form III) to whiskers (form I). The solid-state transformation at the different temperatures was fitted to various solid-state kinetic models of which four gave good fits, thus indicating the complexity of the process which is known to occur via a solid–gas–solid mechanism. Arrhenius plots from the kinetic models yielded activation energies from 344 kJ mol −1 to 368 kJ mol −1 for the transformation. The study demonstrates the value of a rapid in situ analysis of drug polymorphic type which can be of value for at-line in-process control.

Determination of Heat of Hydration and Hydration Kinetics of Theophylline by Thermal Analysis.

Heat of hydration of theophylline anhydrate to the monohydrate at 298 K was evaluated by three different methods. The values determined by heat of solution with a microcalorimeter and heat of dehydration with a differential scanning calorimeter were -9.34±0.65 and -9.82±0.48 kJ/mol, respectively. The values obtained directly with a heat conduction microcalorimeter under 97.3, 95.4 and 93.7% relative humidities were -9.48±0.49, -9.28±0.63 and -9.37±0.74 kJ/mol, respectively. From thermograms obtained with a microcalorimeter at the isothermal transition of theophylline anhydrate to monohydrate, hydration rates were analyzed by a deconvolution method. Hydration rates were in good agreement with those obtained by the Karl Fischer method. Hydration mechanisms were also analyzed on the basis of solid-state kinetic models and the transition followed a mechanism of random nucleation and two-dimensional growth of nuclei.

Generalization of the kissinger equation for several kinetic models

In this work, a generalization of the Kissinger equation was extended to several solid-state kinetic models, and the expressions for the correction term were also deduced from the algebraic equations of the differential ( f( α)) and integral ( g( α)) functions.

The dehydration kinetics of theophylline monohydrate powder and tablet.

The dehydration of theophylline monohydrate in powder and tablet forms was studied by using an infrared water-content measuring instrument. Intact theophylline powder was recrystallized from distilled water, and ground powder was obtained by grinding in a ceramic mortar with a pestle. The samples were dehydrated as follows. The intact and ground powders were placed on a plate 8cm in diameter. A powder bed was formed on a plate 3cm in diameter as a model of a loosely packed tablet, and a tablet 2.0cm in diameter was obtained by compression at 1.0t/cm2. Dehydration kinetic mechanisms of various dosage forms of theophylline monohydrate were clarified by using various kinds of solid-state kinetic models. The dehydrations of intact and ground powders followed two-dimensional growth of nuclei and three-dimensional growth of nuclei equations, and the activation energies (Ea) were 20.1 and 23.0 kcal/mol, respectively. The dehydrations of powder bed and tablet followed two-dimensional phase boundary and threedimensional phase boundary equations, and the Ea values were 15.4 and 11.5 kcal/mol, respectively. It seems that the dehydration of intact powder starts randomly at each particle and proceeds two dimensionally, but that of ground crystals proceeds three-dimensionally from the cracks in particles. In the case of the powder bed the dehydration proceeds two-dimensionally from the surface particles of the powder bed into the inner particles. The dehydration of a tablet proceeds three-dimensionally from the outside of the tablet into its inside.

Kinetics of Solid‐State Reactions of Particulate Ensembles with Size Distributions

Rate equations for solid‐state reactions of monosized spheres cannot be applied directly to particulate ensembles with a distribution of sizes, such as are frequently encountered in practice. A method using experimental data for the overall extent of reaction as a function of time is proposed for identification of the reaction mechanism and computation of the rate constant and activation energy when the particle‐size spectrum of the starting material is known. The method, which is based on comparison of the characteristic paths of the assumed reaction mechanisms with experimental data, may be applied to a wide range of solid‐state kinetic models.