Simulating Surfactant Spreading: Impact of a Physically Motivated Equation of State

TL;DR

This study introduces an empirically-motivated equation of state for simulating surfactant spreading on thin liquid films, improving agreement with experiments and highlighting the importance of the EoS choice on dynamics.

Contribution

It proposes a new, empirically-based EoS for surfactant spreading models, addressing discrepancies with experimental data and exploring the influence of initial conditions and non-dimensional parameters.

Findings

Improved match between simulations and experiments in surfactant distribution.

The choice of EoS significantly affects spreading dynamics.

The gravitational vs. capillary force ratio impacts the spreading behavior more than other parameters.

Abstract

For more than two decades, a single model for the spreading of a surfactant-driven thin liquid film has dominated the applied mathematics literature on the subject. Recently, through the use of fluorescently-tagged lipids, it has become possible to make direct, quantitative comparisons between experiments and models. These comparisons have revealed two important discrepancies between simulations and experiments: the spatial distribution of the surfactant layer, and the timescale over which spreading occurs. In this paper, we present numerical simulations that demonstrate the impact of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically-motivated equation of state; this provides a route to resolving some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and impacts the dynamics of the simulations. The relevant feature of the EoS, the gradient, has distinct regions for empirically motivated choices, which suggests that future work will need to consider more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational vs. capillary forces has a larger impact on the dynamics than the other non-dimensional groups. Finally, we observe that the experimental approach of using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the meniscus itself does not significantly affect the dynamics.