A thermodynamically consistent diffuse interface model for the wetting phenomenon of miscible and immiscible ternary fluids

Abstract

The wetting effect has attracted great scientific interest because of its natural significance as well as technical applications. Previous models mostly focus on one-component fluids or binary immiscible liquid mixtures. Modelling of the wetting phenomenon for multicomponent and multiphase fluids is a knotty issue. In this work, we present a thermodynamically consistent diffuse interface model to describe the wetting effect for ternary fluids, as an extension of Cahn's theory for binary fluids. In particular, we consider both immiscible and miscible ternary fluids. For miscible fluids, we validate the equilibrium contact angle and the thermodynamic pressure with Young's law and the Young–Laplace equation, respectively. Distinct flow patterns for dynamic wetting are presented when the surface tension and the viscous force dominate the wetting effect. For immiscible ternary fluids, we manipulate the wettability of two contact droplets deposited on a solid substrate according to three scenarios: (I) both droplets are hydrophilic; (II) a hydrophilic droplet in contact with a hydrophobic one; (III) both droplets are hydrophobic. The contact angles at each triple junction from the simulations are compared with Young's contact angle and Neumann's triangle rule. Simulations for the validation of our work are performed in two and three dimensions. In addition, we model the evaporation process of a ternary droplet and obtain the same power law as that of previous experiments. Our model allows one to relate the interfacial energies with surface composition, enabling the modelling of the coffee-ring phenomenon in further perspective.

Evolution dynamics of thin liquid structures investigated using a phase-field model

“…43 This dynamic boundary condition has already been validated in our previous work and applied to analyze the nonequilibrium contact line movements of droplets under condensation and evaporation. 75 To accurately model extremely low and high contact angles, we have addressed this issue in a separate paper by considering the presence of surface composition 76,77 based on Cahn's theory. The present work directly employs the contact angle correction method.…”

Wetting Effect Induced Depletion and Adsorption Layers: Diffuse Interface Perspective

“…(1) depicts the wall free energy resulting from the interaction between the liquid solution and the solid substrate S . To account for the repulsive and attractive interaction between the liquid and the solid wall, we adopt the following expression for the wall free energy density [1,20,32] $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr f_w /f_w^{\ast} = {1 \over 2}\gamma _2 \,\psi ^2 + \gamma _1 \psi + \gamma _0, \hfill\cr}}$$$ …”

“…To simulate the depletion/adsorption of the A−B binary system in this work, we use the composition of component A, namely φ to distinguish A from B inside the bulk region Ω, so that the composition of component B is $${1 - \phi }$$ , while applying ψ to represent the composition of component A on the substrate S . To establish the equilibrium depletion/adsorption layer, the spatiotemporal evolution of φ and ψ follows the Cahn‐Hilliard equation with the natural wetting boundary condition as [20,33] $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr {{\partial \phi } \over {\partial t}} = \nabla \cdot \left( {{\rm{{\cal M}}}\nabla \mu + {\bf{{\bf\rm\xi} }}} \right),\;{\rm{in}}\;\Omega, \hfill\cr}}$$$ $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr {{\partial \psi } \over {\partial t}} = \tau \,\left( {2\kappa \nabla \psi \cdot {\bf{n}} + f_w^{^\prime} } \right),\;{\rm{on}}\;S.\hfill\cr}}$$$ …”

“…[34]. The numerical convergence of the model has been comprehensively discussed in our previous study [20,22,28] . The simulations are performed on the parallel computer bwUniCluster of Baden‐Wuerttemberg equipped with Intel Xeon Gold CPUs in the environment of Red Hat Enterprise.…”

“…The thermodynamic equilibrium of the two immiscible bulk phases delineates the corresponding equilibrium bulk compositions within the respective fluid phases. Considering the thermodynamic equilibrium between the fluid phases and the solid substrate, the equilibrium composition in the fluid phase can differ from that results from liquid‐liquid equilibrium [1,19–21] . In essence, treating the solid substrate as an inert object, as in Young's law, disrupts the symmetry at the solid‐fluid interface, resulting in the accumulation or dispersion of species on the solid substrate.…”