Kinetic Studies of Complex Reactions. Part 2: Description of Diffusion Control

Introduction

Cross-linking reactions of «hot curing systems» are complex reactions. Depending on the reaction conditions, an initially chemically controlled reaction can change and become diffusion controlled. The reaction rate thereby decreases rapidly. The reaction almost stops. The reason for this is chemically induced vitrification as a result of which the material changes from a liquid to a glassy state.

In the first part of this study [1], the curing reaction of a two-component epoxy resin consisting of the diglycidylether of bisphenol A (DGEBA) and diaminodiphenylmethane (DDM) as hardener or curing agent was investigated by DSC and evaluated using the Model Free Kinetics (MFK) software.

Here it was shown that curing at heating rates greater than 1 K/min is chemically controlled over the entire reaction range. By means of MFK, it is possible to correctly predict the course of an isothermal reaction right up to vitrification. If the glass transition temperature is known as a function of conversion, MFK can be used to estimate the time at which the material vitrifies in an isothermal reaction.

MFK must be extended in order to describe the complete course of a curing reaction that includes the transition from chemically to diffusion-controlled kinetics [2-4]. In this article, MFK is enhanced sufficiently for it to take the effect of diffusion control on the reaction kinetics into account.

Prediction of Chemical Behavior 

The Conversion Curve 

If MFK is applied to DSC curves that were measured at sufficiently high heating rates, one obtains activation energy curves as a function of conversion that describe a purely chemically controlled reaction (curve a in Fig. 2 of Part 1 [1]). The activation energy curve can then be used to calculate the isothermal course of the reaction. A comparison of the conversion curve calculated using MFK and the measured conversion curve for T_react = 100 °C is displayed in Figure 1.

Up until about 40 min, the measured curve agrees well with the MFK prediction. In this range the cross-linking reaction is chemically controlled. Afterward, the influence of diffusion control increases and the measured reaction proceeds more slowly. A comparison of the two curves allows the influence of diffusion control on the reaction kinetics to be estimated. The reaction rate is described by the equation.

where dα/dt is the reaction rate (derivative with respect to time of the measured conversion curve), [dα/dt]chem is the reaction rate of the chemically controlled reaction (obtained using MFK) and ƒd(α) is the diffusion control function that describes the influence of diffusion control on the reaction kinetics. ƒd can be determined according to eq (1) by division of the measured reaction rate and the reaction rate calculated by MFK (see Fig. 1).

Conclusions 

MFK adequately describes the reaction kinetics if the DSC measurements are made at sufficiently high heating rates so that the reaction remains chemically controlled. If the material vitrifies during the reaction, the kinetics changes from being chemically controlled to diffusion controlled. In this case, a better description of the kinetics can be achieved through the introduction of a diffusion-control function. This function enhances MFK in such a way that it also describes the transition to diffusion control. It is then possible to calculate the maximum conversion that can be reached.

To predict the kinetics of the curing reaction with MFK requires at least three dynamic DSC measurements performed at different heating rates. Furthermore, at least four (better six) postcuring measurements to determine the conversion dependence of the glass transition temperature are needed. The glass transition temperature of a fully cured sample should be determined in an additional experiment. In comparison to the time required for isothermal measurements, the use of MFK presents a considerable saving of time.

_{Kinetic Studies of Complex Reactions Part 2: Description of Diffusion Control | Thermal Analysis Application No. UC 192 | Application published in METTLER TOLEDO Thermal Analysis UserCom 19}