Interfacial Instability of Charged End-Group Polymer Brushes

We consider a polymer brush grafted to a surface (acting as an electrode) and bearing a charged group at its free end. Using a second distant electrode, the brush is subject to a constant electric field. Based on a coarse-grained continuum model, we calculate the average brush height and find that the brush can stretch or compress depending on the applied field and charge end-group. We further look at an undulation mode of the flat polymer brush and find that the electrostatic energy scales linearly with the undulation wavenumber, q. Competition with surface tension, scaling as q, tends to stabilize a lateral q-mode of the polymer brush with a well-defined wavelength. This wavelength depends on the brush height, surface separation, and several system parameters. Introduction. – There are different ways to bind polymers to surfaces. Either by adsorption from solution or grafting them onto the surface with a terminal group or having an adhering block in case of block copolymers. Such coated surfaces have many important applications in colloidal and interfacial science. The polymer layer can change the hydrophobicity of the surface, prevent absorption of other molecules from solution and, in general, plays an important role in colloidal suspensions by preventing flocculation and aggregation of coated colloidal particles [1, 2]. A densely grafted polymer layer is called a polymer brush. The layer is grafted irreversibly on a solid surface by an end–group. Both neutral and charged polymer brushes have been studied extensively in the last few decades [3–11]. If there is no strong interaction between the monomers and the surface, the brush properties are mainly determined by the chain entropy. Neutral brushes, to a large extent, are characterized by their height that scales linearly with N , the polymerization index [4–7]. Charged polymer brushes depend in addition on the charge density of the chain as well as the solution ionic strength [8–13]. In this Letter we aim at understanding another variant of polymer brushes having a terminal charge group, Ze, at their free end, where e is the electronic charge and Z the valency (see Fig. 1). The main advantage of having a charged end-group is that we can control the layer height and other properties by applying an external electric field and varying it continuously. This field stretches (or compresses) the chains and is in direct competition with their elastic energy and entropy. Even in the absence of any external field, we expect the brush to be affected by the charge end-groups, because of their repulsive interactions. Indeed, the height profile depends on the charged group and an instability of the flat brush towards an undulating one may occur. Flat end-charged polymer brush. – Let us briefly recall the equilibrium properties of a neutral grafted layer. The condition of highly dense layers (the brush regime) is l ≪ Rg, where Rg is the chain radius of gyration, and l is the average distance between chains (Fig. 1a). The brush height, defined as the average distance of chain ends from the substrate, is denoted by h. In the 1970s, a simple freeenergy was proposed by Alexander and de Gennes [4,5] to determine the layer equilibrium height. In the Alexander – de Gennes model, the brush height is taken to be the same for all chains; namely, the height distribution is step-wise. Later, in more refined theories, it was found that the freeend distribution is parabolic [6,7]. In the present work we remain within the step-wise distribution approximation,