Analysis of Marching-modulus Rubber Rheograms
Rheometry is widely used throughout the rubber industry to characterise the cure behaviour of rubber mixes. Industrial rheometers produce continuous traces or rheograms in which torque is plotted as a function of time as a rubber progressively cures. Rheograms in which a distinct maximum or steady-state value is obtained within a reasonable time present few problems when the main objective is to determine optimum vulcanisation times. Nevertheless, some rubber mixes display cure profiles that do not exhibit steady-state values within acceptable time scales; taking extremely long periods of time to do so. Running rheometers over prolonged periods in order to attain flat or steady state cures is neither productive nor economical. In such situations, it is desirable to develop a methodology whereby steady-state values can be estimated from short running times. Assuming that cure advances by way of first-order kinetics a graphical technique is introduced enabling the steady-state torque value to be estimated from short running times. The steady-state torque value obtained from the graphical procedure was compared with experimental values; poor agreement was found being within about 11%. Notwithstanding the poor agreement between the steady-state torque value observed experimentally and that predicted by the graphical method we nevertheless continue to recommend the use of the graphical technique. Our justification for the use of the graphical method is that the procedure provides reliable rate constants for the incipient stages of vulcanisation without having to determine full extents of cure. The rate constant determined by the graphical procedure was in good agreement with typical values reported in the literature7 thus affording credibility to our graphical methodology. As further confirmation as to the reliability of our technique, an analytical method introduced by Rigby2 was employed to verify the authenticity of our results. Comparison of our results with those ascertained from the analytical method showed reasonable agreement. In order to account for the significantly large difference between the graphically-estimated and experimental steady-state torque value, the ideal and simplistic assumption that curing proceeds by first order kinetics throughout the entire curing period was abandoned. Instead, it was assumed that the crosslinking process comprises two concomitant first-order curing reactions; notably normal and slow crosslinking reactions. The kinetic expression describing the two simultaneous vulcanisation reactions in terms of rheometry parameters is: T t − T a = ( T ∞ , c − T ) ( 1 − e − k c ( t − t i ) ) + ( T ∞ , s c − T a ) ( 1 − e − k s c ( t − t i ) ) where Tt is the torque at time t; Ta is the minimum torque; ti is the induction time; T¥,c and T¥,sc are the steady-state torque values for the normal and slow crosslinking processes respectively; and, kc and ksc represent the first-order rate constants for the normal and slow crosslinking processes respectively. Excellent agreement was found between the experimental steady-state torque value and that predicted by the composite kinetic expression.