Some questions inspired by / about Jeremie’s experiment with active (diffusio-phoretic) colloids.
QUESTIONS
Why do they from a 2D layer? Answer: because they are heavy!
For now consider the simple case of a single inert particle in a (fixed) concentration gradient. Say it moves towards the source of the chemical. It will follow some mean trajectory, but with plenty of scatter. This is an out-of-equilibrium process.
Understand why it moves (diffusiophoresis): how fast and in what direction?
Consider an uncharged particle for now.
Reading: \citet{2010PhRvL.104m8302P}; \citet{1982JFM...117..107A}; \citet{onne_Ybert_Ajdari_Bocquet_2008}.
An osmotic pressure gradient the interfacial layer leads to drift velocity \(V_{\rm DP} = \mu\nabla c\). I have been able to recover something which resembles this sort of.
This is a complicated and difficult question – probably more productive if I just accept the established results and move on to other questions.
What are the statistics of this motion – how much variance is there?
Look at notes. Imagine density/concentration/probability instead of a single particle.
Which part of this setup is out of equilibrium?
What does the scatter in trajectories tell us about the system? Anything useful? Use linear-response theory (is this justified? Why?).
Now consider active particle in concentration gradient. Do we get a simple superposition of two behaviours?
Re-read \citet{2005PhRvL..94v0801G}. Read \citet{2009JPCM...21t4104G}.
DIFFUSIOPHORESIS For charged interactions, the mobility \(\mu\propto 1/c\). This leads to diffusio-phoretic velocity \(V = D\nabla\log c\). This arises from the balance of osmotic and viscous forces in the Debye layer. The diffusion coefficient \(D \sim T/\eta\ell_{\rm B}\), where \(\ell_{\rm B}\) is the Bjerrum length (ratio of electrostatic to thermal energy). Derivation of fluid velocity done in \citet{1989AnRFM..21...61A}, page 14ish.