The interaction of a particle and a polymer brush coating a permeable surface

Coating of filtration membranes with a polymer brush holds great promise for efficiently preventing the deposition of fouling particles. The polymer chains are compressed by incoming particles, carried with the permeation flow towards the membrane, and consequently exert a repulsive force that acts to keep the particles away from the membrane surface. Here, we theoretically investigate the effect of a polymer brush coating on the permeation-induced hydrodynamic force, pulling a particle towards the membrane, and its balance with the steric repulsion exerted by the compressing brush, resisting the particle's approach. Lubrication theory yields an ordinary differential equation for the pressure, from which is calculated numerically. Further, an asymptotic analysis is performed for the limiting cases of a dilute or dense brush, providing analytic expressions that demonstrate how brush properties affect. Finally, the equilibrium position of a particle is evaluated by considering a balance between the opposing forces. The results provide an upper boundary for the brush properties, beyond which the brush is barely compressed under conditions typical of membrane filtration processes. Further increasing the brush density or thickness only decreases the total system permeance, resulting in increased energy consumption. The results shed light on the mechanisms by which a polymer brush affects the forces acting on a foulant particle, providing quantitative measures for assessing the potential efficacy of brush coatings.