Critical cluster composition from homogeneous nucleation data: application to water in carbon dioxide–nitrogen carrier gases

Knowledge on critical cluster composition is important for improving the nucleation theory. Thus, homogeneous water nucleation experiments previously carried out in nitrogen and 0%, 5%, 15% and 25% of carbon dioxide ( Campagna et al. 2020a, 2021) are analyzed. The tests were conducted at 240 K and 0.1 MPa, 1 MPa and 2 MPa. The observed nucleation rates are strongly dependent on supersaturation, pressure, temperature and mixture composition. These experimentally found dependencies can be used to derive the composition of critical clusters by means of the nucleation theorem. In this way, a macroscopic quantity, nucleation rate, reveals properties of critical clusters consisting of a few tens of molecules. Two novel methods are presented for the detailed application of the nucleation theorem. The first method extends to mixtures of $$\,\,\,\,\,\,\,N>2\,\,\,\,\,\,$$ N > 2 components the approach used in literature for two components. The second method not only applies to $$N>2$$ N > 2 mixtures in a more straightforward manner, but it can also be used for unary as well as for binary and multi-component nucleation cases. To the best of our knowledge, for the first time the critical cluster composition is computed for high pressure nucleation data of a vapor (here water) in mixtures of two carrier gases (here carbon dioxide–nitrogen). After a proper parameterization of the nucleation rate data, both methods consistently lead to the same critical nuclei compositions within the experimental uncertainty. Increasing pressure and carbon dioxide molar fraction at fixed supersaturation leads to a decrease in the water content of the critical cluster, while the adsorbed number of nitrogen and carbon dioxide molecules increases. As a consequence, the surface tension decreases. This outcome explains the observed increase in the nucleation rate with increasing pressure and carbon dioxide molar fraction at constant supersaturation. Graphic abstract

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated that is valid not only for one-component but generally for multi-component systems. In this analysis, one basic assumption of classical nucleation theory is utilized. In addition, commonly employed in application to crystallization, it is supposed that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)], also relying widely on the standard classical approach in the description of crystal nucleation, holds for multi-component systems as well. The general form of the nucleation theorem derived by us is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, expressions recently developed by us for the curvature dependence of the surface tension, respectively, its dependence on pressure and/or temperature are employed. The basic assumption of classical nucleation theory mentioned above is, however, in general, not true. The bulk and surface properties of the critical crystal clusters may differ considerably from the properties of the evolving macroscopic phases. Such effects can be incorporated into the theoretical description by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem, as derived based on classical nucleation theory, holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs' approach in the thermodynamic description of heterogeneous systems and assuming that the basic assumptions of classical nucleation theory commonly employed in application to crystallization hold, a general form of the nucleation theorem can be formulated valid not only for one-component but generally for multi-component systems. This result is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, recently developed by us expressions for the curvature dependence of the surface tension, respectively, the dependence of the surface tension on pressure and/or temperature are employed. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)] holds also for multi-component systems as far as mentioned above assumptions are fulfilled. In the application of classical nucleation theory to crystallization processes it is assumed as one of its basic ingredients that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. This assumption is, however, in general, it is not true. This limitation of the theoretical description can be overcome by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem as derived based on classical nucleation theory holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

Critical Size of Secondary Nuclei Determined via Nucleation Theorem Reveals Selective Nucleation in Three-Component Co-Crystals

The critical size of the secondary nuclei plays an important role in determining the crystal growth rate. In the past, the Nucleation Theorem has been applied to determine the number of molecules in the critical nuclei of a single-component crystal via variation of the crystal growth rate with dilution by the non-crystallizable component. In this work, we extend the method to the three-component co-crystal poly (ethylene oxide)/urea/thiourea inclusion compound. The theoretical crystal growth kinetics were deduced and the dependence of the radial growth rate of the inclusion compound spherulites on the mass fraction of urea in urea/thiourea was measured. The results reveal that the secondary nuclei of the poly (ethylene oxide)/urea/thiourea inclusion compound consist mainly of ethylene oxide repeating units and urea molecules. We propose that only urea molecules and ethylene oxide repeating units are selected to form the secondary nuclei while co-crystallization of the three components happens at the lateral spreading stage. As a result, the composition of the critical secondary nuclei is different from that of the bulk inclusion compound crystals. The work is expected to deepen our understanding of the nucleation of multi-component co-crystals.

Influence of aerosol lifetime on the interpretation of nucleation experiments with respect to the first nucleation theorem

The SAWNUC (Sulphuric Acid Water NUCleation) microphysical aerosol nucleation model is used to study the effect of reactor walls on the interpretation of nucleation experiments with respect to nucleation theory. This work shows that loss processes, such as wall losses, influence the interpretation of nucleation experiments, especially at low growth rates and short lifetimes of freshly nucleated particles. In these cases the power dependency of the formation rates, determined at a certain particle size, with respect to H2SO4 does not correspond to the approximate number of H2SO4 molecules in the critical cluster as expected by the first nucleation theorem. Observed ∂log(J)/∂log([H2SO4]) therefore can vary widely for identical nucleation conditions but different sink terms.

Influence of aerosol lifetime on the interpretation of nucleation experiments with respect to the first nucleation theorem

The SAWNUC microphysical aerosol nucleation model is used to study the effect of reactor walls on the interpretation of nucleation experiments with respect to nucleation theory. This work shows that loss processes, such as wall losses, influence the interpretation of nucleation experiments, especially at low growth rates and short lifetime of freshly nucleated particles. In these cases the power dependency of the formation rates, determined at a certain particle size, with respect to H2SO4 does not correspond to the approximate number of H2SO4 molecules in the critical cluster as expected by the first nucleation theorem. Observed ∂log(J)/∂log([H2SO4]) therefore can vary widely for identical nucleation conditions but different sink terms.