Effects of detector sensitivity and resolution on induction time reading

Abstract

The effects of sensitivity and resolution of a nucleation detector on induction time were mathematically analysed. The induction time was defined here as the time at which the number density of crystals nucleated under isothermal conditions had reached the minimum detectable number density of crystals (N/M)_det. The value of (N/M)_det depends on the detector used, and therefore can be called the detector sensitivity. Meanwhile, the detector resolution is defined as the minimum detectable crystal size L_d. The equation for induction time t_ind was derived for a simplified case wherein secondary nucleation and concentration reduction were both neglected. For a more general case where such a simplification cannot be made, the induction time t_ind was calculated numerically. The sensitivity and resolution had effects on induction time. The induction time increased as the sensitivity decreased (i.e., the value of (N/M)_det increased). It also increased with a decrease in the resolution (i.e., an increase in the minimum detectable size L_d). The effect of nucleus size L₀ on induction time t_ind was also analysed. The analysis suggested that the induction time in an actual experiment, where the minimum detectable size L_d would be much larger than the nucleus size L₀, was independent of nucleus size itself but was affected by the resolution L_d. The interfacial energy of a crystal nucleus in solution that is deduced from the widely used plot of ln t_indvs. 1/(ln S)² (S: supersaturation ratio) was pointed out to be questionable. The effect of agitation rate (or stirrer speed) on induction time was also discussed.