Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives

Diffusiophoresis, the motion of a particle in response to an externally imposed concentration gradient of a solute species, is analysed from both the traditional coarse-grained macroscopic (i.e. continuum) perspective and from a fine-grained micromechanical level in which the particle and the solute are treated on the same footing as Brownian particles dispersed in a solvent. It is shown that although the two approaches agree when the solute is much smaller in size than the phoretic particle and is present at very dilute concentrations, the micromechanical colloidal perspective relaxes these restrictions and applies to any size ratio and any concentration of solute. The different descriptions also provide different mechanical analyses of phoretic motion. At the continuum level the macroscopic hydrodynamic stress and interactive force with the solute sum to give zero total force, a condition for phoretic motion. At the colloidal level, the particle’s motion is shown to have two contributions: (i) a ‘backflow’ contribution composed of the motion of the particle due to the solute chemical potential gradient force acting on it and a compensating fluid motion driven by the long-range hydrodynamic velocity disturbance caused by the chemical potential gradient force acting on all the solute particles and (ii) an indirect contribution arising from the mutual interparticle and Brownian forces on the solute and phoretic particle, that contribution being non-zero because the distribution of solute about the phoretic particle is driven out of equilibrium by the chemical potential gradient of the solute. At the colloidal level the forces acting on the phoretic particle – both the statistical or ‘thermodynamic’ chemical potential gradient and Brownian forces and the interparticle force – are balanced by the Stokes drag of the solvent to give the net phoretic velocity. For a particle undergoing self-phoresis or autonomous motion, as can result from chemical reactions occurring asymmetrically on a particle surface, e.g. catalytic nanomotors, there is no imposed chemical potential gradient and the backflow contribution is absent. Only the indirect Brownian and interparticle forces contribution is responsible for the motion. The velocity of the particle resulting from this contribution can be written in terms of a mobility times the integral of the local ‘solute pressure’ – the solute concentration times the thermal energy – over the surface of contact between the particle and the solute. This was the approach taken by Córdova-Figueroa & Brady (Phys. Rev. Lett., vol. 100, 2008, 158303) in their analysis of self-propulsion. It is shown that full hydrodynamic interactions can be incorporated into their analysis by a simple scale factor.