Gradient dynamics models for liquid films with soluble surfactant

In this paper we propose equations of motion for the dynamics of liquid films of surfactant suspensions that consist of a general gradient dynamics framework based on an underlying energy functional. This extends the gradient dynamics approach to dissipative non-equilibrium thin film systems with several variables, and casts their dynamic equations into a form that reproduces Onsager's reciprocity relations. We first discuss the general form of gradient dynamics models for an arbitrary number of fields and discuss simple well-known examples with one or two fields. Next, we develop the gradient dynamics (three field) model for a thin liquid film covered by soluble surfactant and discuss how it automatically results in consistent convective (driven by pressure gradients, Marangoni forces and Korteweg stresses), diffusive, adsorption/desorption, and evaporation fluxes. We then show that in the dilute limit, the model reduces to the well-known hydrodynamic form that includes Marangoni fluxes due to a linear equation of state. In this case the energy functional incorporates wetting energy, surface energy of the free interface (constant contribution plus an entropic term) and bulk mixing entropy. Subsequently, as an example, we show how various extensions of the energy functional result in consistent dynamical models that account for nonlinear equations of state, concentration-dependent wettability and surfactant and film bulk decomposition phase transitions. We conclude with a discussion of further possible extensions towards systems with micelles, surfactant adsorption at the solid substrate and bioactive behaviour

Gradient dynamics models for liquid films with soluble surfactant

In this paper we propose equations of motion for the dynamics of liquid films of surfactant suspensions that consist of a general gradient dynamics framework based on an underlying energy functional. This extends the gradient dynamics approach to dissipative non-equilibrium thin film systems with several variables, and casts their dynamic equations into a form that reproduces Onsager's reciprocity relations. We first discuss the general form of gradient dynamics models for an arbitrary number of fields and discuss simple well-known examples with one or two fields. Next, we develop the gradient dynamics (three field) model for a thin liquid film covered by soluble surfactant and discuss how it automatically results in consistent convective (driven by pressure gradients, Marangoni forces and Korteweg stresses), diffusive, adsorption/desorption, and evaporation fluxes. We then show that in the dilute limit, the model reduces to the well-known hydrodynamic form that includes Marangoni fluxes due to a linear equation of state. In this case the energy functional incorporates wetting energy, surface energy of the free interface (constant contribution plus an entropic term) and bulk mixing entropy. Subsequently, as an example, we show how various extensions of the energy functional result in consistent dynamical models that account for nonlinear equations of state, concentration-dependent wettability and surfactant and film bulk decomposition phase transitions. We conclude with a discussion of further possible extensions towards systems with micelles, surfactant adsorption at the solid substrate and bioactive behaviour

An introduction to inhomogeneous liquids, density functional theory, and the wetting transition

Classical density functional theory (DFT) is a statistical mechanical theory for calculating the density profiles of the molecules in a liquid. It is widely used, for example, to study the density distribution of the molecules near a confining wall, the interfacial tension, wetting behavior, and many other properties of nonuniform liquids. DFT can, however, be somewhat daunting to students entering the field because of the many connections to other areas of liquid-state science that are required and used to develop the theories. Here, we give an introduction to some of the key ideas, based on a lattice-gas (Ising) model fluid. This approach builds on knowledge covered in most undergraduate statistical mechanics and thermodynamics courses, so students can quickly get to the stage of calculating density profiles, etc., for themselves. We derive a simple DFT for the lattice gas and present some typical results that can readily be calculated using the theory

Influence of the fluid structure on the binding potential: comparing liquid drop profiles from density functional theory with results from mesoscopic theory

For a film of liquid on a solid surface, the binding potential $g(h)$ gives the free energy as a function of the film thickness $h$ and also the closely related structural disjoining pressure $\Pi = - \partial g / \partial h$. The wetting behaviour of the liquid is encoded in the binding potential and the equilibrium film thickness corresponds to the value at the minimum of $g(h)$. Here, the method we developed in [J. Chem. Phys. 142, 074702 (2015)], and applied with a simple discrete lattice-gas model, is used with continuum density functional theory (DFT) to calculate the binding potential for a Lennard-Jones fluid and other simple liquids. The DFT used is based on fundamental measure theory and so incorporates the influence of the layered packing of molecules at the surface and the corresponding oscillatory density profile. The binding potential is frequently input in mesoscale models from which liquid drop shapes and even dynamics can be calculated. Here we show that the equilibrium droplet profiles calculated using the mesoscale theory are in good agreement with the profiles calculated directly from the microscopic DFT. For liquids composed of particles where the range of the attraction is much less than the diameter of the particles, we find that at low temperatures $g(h)$ decays in an oscillatory fashion with increasing $h$, leading to highly structured terraced liquid droplets

Liquid drops on a surface: using density functional theory to calculate the binding potential and drop profiles and comparing with results from mesoscopic modelling

The contribution to the free energy for a film of liquid of thickness h on a solid surface due to the interactions between the solid-liquid and liquid-gas interfaces is given by the binding potential, g(h). The precise form of g(h) determines whether or not the liquid wets the surface. Note that differentiating g(h) gives the Derjaguin or disjoining pressure. We develop a microscopic density functional theory (DFT) based method for calculating g(h), allowing us to relate the form of g(h) to the nature of the molecular interactions in the system. We present results based on using a simple lattice gas model, to demonstrate the procedure. In order to describe the static and dynamic behaviour of non-uniform liquid films and drops on surfaces, a mesoscopic free energy based on g(h) is often used. We calculate such equilibrium film height profiles and also directly calculate using DFT the corresponding density profiles for liquid drops on surfaces. Comparing quantities such as the contact angle and also the shape of the drops, we find good agreement between the two methods. We also study in detail the effect on g(h) of truncating the range of the dispersion forces, both those between the fluid molecules and those between the fluid and wall. We find that truncating can have a significant effect on g(h) and the associated wetting behaviour of the fluid

Bifurcation analysis of the behavior of partially wetting liquids on a rotating cylinder

We discuss the behavior of partially wetting liquids on a rotating cylinder using a model that takes into account the effects of gravity, viscosity, rotation, surface tension and wettability. Such a system can be considered as a prototype for many other systems where the interplay of spatial heterogeneity and a lateral driving force in the proximity of a first- or second-order phase transition results in intricate behavior. So does a partially wetting drop on a rotating cylinder undergo a depinning transition as the rotation speed is increased, whereas for ideally wetting liquids the behavior only changes quantitatively. We analyze the bifurcations that occur when the rotation speed is increased for several values of the equilibrium contact angle of the partially wetting liquids. This allows us to discuss how the entire bifurcation structure and the flow behavior it encodes changes with changing wettability. We employ various numerical continuation techniques that allow us to track stable/unstable steady and time-periodic film and drop thickness profiles. We support our findings by time-dependent numerical simulations and asymptotic analyses of steady and time-periodic profiles for large rotation numbers

Films, layers and droplets: The effect of near-wall fluid structure on spreading dynamics

We present a study of the spreading of liquid droplets on a solid substrate at very small scales. We focus on the regime where effective wetting energy (binding potential) and surface tension effects significantly influence steady and spreading droplets. In particular, we focus on strong packing and layering effects in the liquid near the substrate due to underlying density oscillations in the fluid caused by attractive substrate-liquid interactions. We show that such phenomena can be described by a thin-film (or long-wave or lubrication) model including an oscillatory Derjaguin (or disjoining/conjoining) pressure, and explore the effects it has on steady droplet shapes and the spreading dynamics of droplets on both, an adsorption (or precursor) layer and completely dry substrates. At the molecular scale, commonly used two-term binding potentials with a single preferred minimum controlling the adsorption layer height are inadequate to capture the rich behaviour caused by the near-wall layered molecular packing. The adsorption layer is often sub-monolayer in thickness, i.e., the dynamics along the layer consists of single-particle hopping, leading to a diffusive dynamics, rather than the collective hydrodynamic motion implicit in standard thin-film models. We therefore modify the model in such a way that for thicker films the standard hydrodynamic theory is realised, but for very thin layers a diffusion equation is recovered

Films, layers and droplets: The effect of near-wall fluid structure on spreading dynamics

We present a study of the spreading of liquid droplets on a solid substrate at very small scales. We focus on the regime where effective wetting energy (binding potential) and surface tension effects significantly influence steady and spreading droplets. In particular, we focus on strong packing and layering effects in the liquid near the substrate due to underlying density oscillations in the fluid caused by attractive substrate-liquid interactions. We show that such phenomena can be described by a thin-film (or long-wave or lubrication) model including an oscillatory Derjaguin (or disjoining/conjoining) pressure, and explore the effects it has on steady droplet shapes and the spreading dynamics of droplets on both, an adsorption (or precursor) layer and completely dry substrates. At the molecular scale, commonly used two-term binding potentials with a single preferred minimum controlling the adsorption layer height are inadequate to capture the rich behaviour caused by the near-wall layered molecular packing. The adsorption layer is often sub-monolayer in thickness, i.e., the dynamics along the layer consists of single-particle hopping, leading to a diffusive dynamics, rather than the collective hydrodynamic motion implicit in standard thin-film models. We therefore modify the model in such a way that for thicker films the standard hydrodynamic theory is realised, but for very thin layers a diffusion equation is recovered

Dynamic unbinding transitions and deposition patterns in dragged meniscus problems

We sketch main results of our recent work on the transfer of a thin liquid film onto a flat plate that is extracted from a bath of pure non-volatile liquid. Employing a long-wave hydrodynamic model, that incorporates wettability via a Derjaguin (disjoining) pressure, we analyse steady-state meniscus profiles as the plate velocity is changed. We identify four qualitatively different dynamic transitions between microscopic and macroscopic coatings that are out-of-equilibrium equivalents of equilibrium unbinding transitions. The conclusion briefly discusses how the gradient dynamics formulation of the problem allows one to systematically extend the employed one-component model into thermodynamically consistent two-component models as used to describe, e.g., the formation of line patterns during the Langmuir-Blodgett transfer of a surfactant layer

Efficient calculation of phase coexistence and phase diagrams: Application to a binary phase-field crystal model

We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary Phase-Field-Crystal model for the crystallisation of a mixture of two types of particles. The resulting phase diagram is determined for one- and two-dimensional domains. In the former case it is compared to the diagram obtained from a one-mode approximation. The various observed liquid and crystalline phases and their stable and metastable coexistence are discussed as well as the temperature-dependence of the phase diagrams. This includes the (dis)appearance of critical points and triple points. We also relate bifurcation diagrams for finite-size systems to the thermodynamics of phase transitions in the infinite-size limit