Mathematical modeling of the crystallization kinetics of cocoa butter

The crystallization of fats has been extensively investigated, probably due to its importance in everyday life. An understanding of the fat crystallization process plays a critical role in determining overall product quality, e.g. in chocolate and confectionary coatings. The kinetics of fat crystallization (i.e. the amount of crystallization as a function of time / temperature), being dependent on the composition and on the processing conditions, is important for controlling operations in the food industry to produce the desired product characteristics. The most generally used approach for the description of the phase transformation kinetics is the Avrami model developed in the 1940s. The fit of this model to the experimental data is however not satisfactory. Sometimes, also the Gompertz model, mainly used to describe the growth kinetics of bacteria, is used.
It is the aim of this project to develop a model which is able to better describe the crystallization kinetics of fats. As a main substrate cocoa butter was chosen, but other substrates like milk fat (fractions) and chocolate are studied too. The kinetics of the crystallization is monitored by means of differential scanning calorimetry (DSC) (which measures the crystallization heat released by the sample as a function of time / temperature) and by pulse nuclear magnetic resonance (pNMR) which immediately measures the solid fat content of a sample. For simplicity reasons, the research started with isothermal crystallization kinetics.
First, the DSC method to follow the crystallization was developed. The influence of sample preparation and melting protocol was studied. A calculation algorithm was developed to integrate the DSC peaks in an objective way.
We were able to develop a better model, which has the additional advantage that it is written in the form of a differential equation. This makes it easier to adapt it for non-isothermal measurements, to interpret the equation mechanistically and to make changes to the model on the basis of acquired knowledge. Our model describes the crystallization process as a first order forward reaction and a reverse reaction of order n. Expressed mathematically this leads to:

in which h = (a-f)/a, with f being the amount of crystallization at time t and a being the ultimate amount of crystallization (for t approaching infinity). The variable h is thus the remaining crystallizable fat. K1 and K2 are two rate constants, which can be assumed equal to simplify the model. The quality of this model was compared to that of existing models and it turned out to perform better for most of the samples. Currently the model is used to study the influence of chemical composition and process conditions (e.g. isothermal crystallization temperture) and the isothermal crystallization kinetics. In the future we are also planning to adapt the model to describe two-step processes (first a crystallization in one polymorphic form, followed by a recrystallization to and further crystallization in another polymorphic form) and to describe non-isothermal processes (by adding secondary models describing the temperature dependency of the parameters).

Project duration: 1999-10-01 - 2003-09-30

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Imogen Foubert
Department of Applied Mathematics, Biometrics and Process Control
Coupure links 653,
9000 Gent
Tel: +32(9)2646168
Fax: +32(9)2646218

Last update: 01 december 2008,

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