Phase-field modelling of diffusive dynamics of slowly miscible liquids

The mixing of miscible liquids is essential for numerous processes in nature and industry. The rate of mixing is ultimately determined by the slow interfacial diffusion process that is initiated by the contact of two miscible liquids. The hydrodynamic flows near interfacial boundaries may strongly affect the diffusion process, sometimes resulting in deformation or even disintegration/disappearance of interfaces.The mixing dynamics of miscible liquids remains poorly understood. The diffusion flux is traditionally defined through the classical Fick’s law (i.e. with the diffusive flux being proportional to the gradient of concentration), which is only applicable to the cases of small concentration gradients. At least, at the moment of an initial contact of two liquids the concentration gradient across the interface boundary is strong, which renders the classical Fick’s inapplicable for accurate description of this system.To prove this statement, we have fulfilled the numerical studies of there cent experiment, in which the diffusive mixing of two miscible liquids was studied. The liquids were saturating a capillary tube. A visible liquid/liquid boundary was observed for prolonged time periods, and the time evolution of the boundary shape and its propagation dynamics were documented. Through the set of 1D, 2D, and 3D numerical simulations, we proved that the experimental observations cannot be reproduced on the basis of the Fick’s law. Neither the shape of the liquid/liquid boundary nor its time dynamics have been correctly reproduced. We have added to the model the effects of hydrodynamic motion and surface tension (the Korteweg’s term), and still the simulation results have remained different from the experimental observations.Further, we have performed the simulations on the basis of the phase-field model. This time the diffusion process has been defined by the extended Fick’s law, i.e. through the gradient of the chemical potential. In addition, the model included the capillary effects associated with the interface. We found that such an approach is capable of producing a realistic shape of the liquid/liquid boundary. However, the numerical predictions for the movement of the boundary have remained different from the experimental observations. With a hope to make the numerical results more aligned with the experimental observations, the hydrodynamic effects were added. Although, the flows induced in a capillary were too weak, and, in particular, the experimental dependence on the tube’s diameter have been not described by the flows.Finally, we have tried to use even more sophisticated mathematical models, e.g. the models that were previously developed for explanation of diffusive dynamics in polymer systems, where the non-Fickian behaviour is also frequently reported. However, the attempted modelling (namely, the inclusion of viscoelastic effects) still has failed to provide the experimentally observed dependences.To conclude we would like to state that the diffusive dynamics of miscible liquid/liquid interfaces cannot be explained by the classical Fick’s approach. The phase-field approach can be used to provide the accurate shape of the miscible interfaces. Nevertheless, the diffusive dynamics of miscible liquid/liquid interfaces have remained poorly described and thus the further research work is suggested