In a recent communication,1 we showed that by properly incorporating the curvature dependence of the surface tension of water into the framework of classical nucleation theory (CNT), it was possible to correct the wrong temperature dependence of CNT and significantly improve the agreement with experimental results.
In the preceding comment,2 Barrett expands on two points that were only briefly discussed in our communication. We would like to thank Barrett for his positive remarks about our work and for elaborating our findings and take this opportunity to further clarify these points.
In his first comment,2 Barrett shows that including the liquid-phase compressibility in the CNT-framework changes the nucleation rates by approximately a constant factor which we shall call A. Barrett further estimates the magnitude of A for water by considering an expansion which includes both the liquid-phase compressibility and the curvature dependence of the surface tension (Eq. (1) in Ref. 2). A more accurate estimate of A can be obtained by combining the curvature expansion of the surface tension with the p-form of CNT presented in 2004 by Obeidat et al.3 This procedure accounts more accurately for non-ideal and compressibility effects since the full equation of state is used instead of just an expansion. Following this procedure increases A by roughly a factor of 2 compared to Barrett’s estimate. Prior to publication of our communication, we implemented this more accurate procedure to rigorously support the following statement in our communication:1
“the actual values of the nucleation rates might change slightly if one takes into account non-idealities, the compressibility of the liquid phase or uses a different model for the kinetic prefactor. However, all these modifications typically shift the rates by an approximately constant factor.”3,4
which to a large extent summarizes Barrett’s first comment. However, as mentioned in the previous paragraph, the liquid-phase compressibility is by no means exclusive in influencing the magnitude of the nucleation rates. Many corrections to CNT have been proposed in the literature, including the 1/S5 or nonequilibrium effects6,7 that would shift the predicted rates in the opposite direction of the liquid-phase compressibility. However, discussing the particular influence of all these corrections was beyond the scope of our 4-page communication. As we stated in the communication:1
“The really important outcome of this work is thus that including the curvature dependence of the surface tension corrects the wrong temperature dependence given by the classical theory, at least for water.”
The second comment by Barrett concerns the saturation dependence of the nucleation rates from CNT. As Barrett correctly states, and which was also mentioned in our communication, the nucleation rates J from some of the individual experimental data increase less rapidly with the supersaturation S, than CNT. With a negative Tolman length, the curvature corrected theory presented in our communication (c-CNT)1 increases the disagreement. This has traditionally been interpreted, for instance, by Holten et al.8 and most recently by Bennett and Barrett9 as evidence of a positive Tolman length for water. We shall next show that drawing such conclusions from individual data-sets is not a recommendable procedure.
In Fig. 1, we have plotted the experimental data at 220 K and 230 K where the S-dependence of the nucleation rates deviates most from our theoretical predictions. In the figure, we have included predictions from CNT, c-CNT, and a new case following the suggested procedure where the Tolman length has been fitted to best match the saturation dependence of the experimental nucleation rates by Wölk and Strey (blue diamonds).4
Using the Tolman length as a “fitting parameter,” a small positive value (0.03–0.04 nm) is required to reproduce the S-dependence of the experimental data, in agreement with previous assessments.8,9 By construction, the new case is superior to both CNT and c-CNT at low saturations, as illustrated by the blue dashed dotted lines. For the data at high saturations however, the model with a positive Tolman length gives results which deviate orders of magnitude from the experiments, e.g., for the data by Wyslouzil et al. (green triangles) and the data by Manka et al. (yellow squares), and performs actually worse than the classical theory. c-CNT with a small negative Tolman length as presented in our communication1 represents the whole range of experimental data best, showing only modest disagreement in the S-dependence even at the worst temperatures (220 K and 230 K). Figure 1 clearly illustrates that inferring the Tolman length on the basis of individual experimental data-sets is not a recommendable procedure, since including additional experimental data obtained from other techniques or in a different range of supersaturations can fundamentally change the conclusions.
Given the challenging nature and diverse experimental techniques used in the study of water-condensation, one may as Barrett in his comment2 attribute the deviations either to inaccuracies of the experimental data or to a missing additional supersaturation-dependent term in the work of formation. A recent work by Mullick et al.12 nicely discusses the challenges faced with nucleation experiments and the possibility that the deviations in the S-dependence of the rates may reflect an “experimental bias that is extremely difficult to quantify.” Their discussion ends with an insightful sentence that perfectly summarizes our point:12
“We note that it is, therefore, equally important to keep the big picture in mind and ensure that matching local measurements is not considered more important to advancing theory than is matching rate measurements that are widely separated in terms of supersaturation and temperature.”
To get further insight into these issues, we recommend that additional water condensation experiments are carried out at higher temperatures and supersaturations.