# Small particles moving large droplets

The moving contact line problem is one of the main unsolved fundamental problems in fluid mechanics, with relevant physical phenomena spanning multiple scales, from the molecular to the macroscopic scale.

At the mesoscale, a unified derivation for single and binary fluid diffuse interface models is presented, consolidating two models present in the literature. Results from an asymptotic analysis of the sharp interface limit of the binary fluid diffuse interface model are compared with numerical computations of the inner region in the vicinity of a moving contact line.

We want to understand the nanoscale structure of the density profile in the vicinity of the contact line. This is a hard problem, because of its small scale, large fluctuations and inhomogeneities. Typically, nanoscale properties are studied using molecular dynamics (MD) simulations. MD resolves single particles, but needs to impose high contact line velocities, therefore requiring some other simplifying assumptions.

Here, we follow an alternative approach by using hydrodynamic dynamic density functional theory (HDDFT). It assumes that locally, the chemical $\mu$ potential is a functional of the fluid density profile,

where $\mathcal{F}[n]$ is the free energy functional of the number density $n$ and $V_{\textnormal{ext}}$ is the external potential. The external potential models the force acting from the wall on the fluid particles. This formulation can then be employed in a momentum equation

This framework allows to study the interactions of particle interactions with hydrodynamic shear and compression forces $\boldsymbol \tau$. This allows to see how a shear layer emerges close to the wall or where compression emerges -- depending on the mean fluid density profile in the immediate vicinity of the contact line.

# Code

Because at the nanoscale, the fluid behavior is described by an integral-equation, we developed our own numerical scheme based on Pseudospectral methods. The code is available on Github and was used to compute results published in the following papers:

- Benjamin D. Goddard, Beth Gooding, Grigoris A. Pavliotis, Hannah Short ,2020, "Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions" ArXiv
- B. D. Goddard, R. D. Mills-Williams, Grigorios Pavliotis, 2020, "Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory" ArXiv
- Goddard, Mills-Williams, Sun, 2020, "The singular hydrodynamic interactions between two spheres in Stokes flow", Phys. Fluids, 32, 062001 PoF
- Nold, A., MacDowell, L. G., Sibley, D. N., Goddard, B. D., Kalliadasis, S., 2018, “The vicinity of an equilibrium three-phase contact line using density-functional theory: density profiles normal to the fluid interface”, Mol. Phys. (5 pgs.) Links: core.ac.uk, Mol Phys
- Nold, A., Goddard, B. D., Yatsyshin, P., Savva, N., & Kalliadasis, S., 2017 ”Pseudospectral methods for density functional theory in bounded and unbounded domains." J. Comp. Phys. 334 639-664. Links: ArXiv, J Comp Phys
- Goddard, B. D., Nold, A., & Kalliadasis, S., 2016 “Dynamical density functional theory with hydrodynamic interactions in confined geometries” J. Chem. Phys. 145 Art. No. 214106 (19 pgs.). PubMed
- Nold, A., Sibley, D.N., Goddard, B.D. & Kalliadasis, S., 2015 “Nanoscale fluid structure of liquid-solid-vapour contact lines for a wide range of contact angles,” Math. Model. Nat. Phenom. 10 111–125. Links: ArXiv, MMNP

# Selected publications

- Nold, A., MacDowell, L. G., Sibley, D. N., Goddard, B. D., Kalliadasis, S., 2018, “The vicinity of an equilibrium three-phase contact line using density-functional theory: density profiles normal to the fluid interface”, Mol. Phys. (5 pgs.) Links: core.ac.uk, Mol Phys
- Nold, A., Goddard, B. D., Yatsyshin, P., Savva, N., & Kalliadasis, S., 2017 ”Pseudospectral methods for density functional theory in bounded and unbounded domains." J. Comp. Phys. 334 639-664. Links: ArXiv, J Comp Phys
- Nold, A. 2016 "From the Nano- to the Macroscale – Bridging Scales for the Moving Contact Line Problem", PhD Thesis ResearchGate
- Nold, A., Sibley, D.N., Goddard, B.D. & Kalliadasis, S., 2015 “Nanoscale fluid structure of liquid-solid-vapour contact lines for a wide range of contact angles,” Math. Model. Nat. Phenom. 10 111–125. Links: ArXiv, MMNP
- Sibley, D.N., Nold, A. & Kalliadasis, S. 2014 “The asymptotics of the moving contact line: Cracking an old nut,” J. Fluid Mech. 764 445-462. Links: lboro.ac.uk
- Nold, A., Sibley, D.N., Goddard, B.D. & Kalliadasis, S. 2014, “Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory,” Phys. Fluids 26 Art. No. 072001 (16 pgs) (Featured in the “Research highlights from Physics of Fluids”). Links: Loughborough repository, PoF
- Goddard, B.D., Nold, A., Savva, N., Yatsyshin, P. & Kalliadasis, S. 2013 “Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments,” J. Phys.: Condens. 25 Art. No. 035101 (14 pgs). PubMed
- Goddard, B.D., Nold, A., Savva, N., Pavliotis, G.A., Kalliadasis, S. 2012 “General dynamical density functional theory for classical fluids,” Phys. Rev. Lett. 109 Art. No. 120603 (5 pgs). ArXiv
- Nold, A., Malijevský, A. & Kalliadasis, S. 2011 “Wetting on a spherical wall: influence of liquid- gas interfacial properties,” Phys. Rev. E 84 Art. No. 021603 (17 pgs). ArXiv