A new scaling theory for polyelectrolyte brushes

Polyelectrolyte (PE) brushes are charged polymers grafted on solid-liquid interfaces, which render special surface functionalities in many applications such as colloidal stabilization, wear protection in biological systems, and drug delivery. In this work, we introduce new scaling laws for the brush height H as a function of the grafting density σ, salt concentration c_s, and charge fraction φ, by employing a cell model to provide a detailed analysis of the electrostatic interactions between the charged monomers and mobile ions based on the Poisson-Boltzmann theory. Our predictions are consistent with the classical scaling theory for a charged “salted brush”, H~(σ/c_s)^1/3, within the Debye-Hückel limit and conditions of overlapping electrical double layers (EDLs) that only apply to weak PEs. Further, we obtain new scaling laws that are expected for both weak and strong PE brushes: 1) H~(σ/c_s)^0.11 beyond the Debye-Hückel limit which is applicable for low salt and high grafting density, and 2) H~(σ/c_s)^1/5 for non-overlapping EDLs, which is applicable for high salt and low grafting density. We also provide predictions and insights on the abrupt shrinkage of PE brushes in the presence of multivalent ions, which has been demonstrated in experiments.