Let us throw a tiny amount of a polyelectrolyte salt, such as deoxyribonucleic acid sodium salt or sodium polystyrene sulfonate, into an aqueous solution containing a known amount of a simple electrolyte such as potassium chloride. In due course, the polymer salt dissolves into a homogeneous solution consisting of the charged polymer molecules, their dissociated counterions, and the cations and anions of the simple electrolyte. As in the cases of rigid charged particles in electrolyte solutions, an ion cloud would surround a polyelectrolyte chain, due to an optimization between the attractive interaction among opposite charges and the loss of translational freedom of free ions. A generic picture of a polyelectrolyte is drawn in Figure 4.1a. For one particular conformation of a polymer molecule, some of the counterions hover around the chain backbone. However, unlike rigid bodies, the polymer has the intrinsic capacity to assume enormous number of conformations due to the chain flexibility. The polymer conformations are influenced by the electric forces arising from the charges of counterions and salt ions, and the charges on the chain backbone itself. The polymer conformations, in turn, influence the spatial distribution of the small ions. After an elapse of a certain short time, the polymer conformations in equilibrium would have changed (Figure 4.1b). Now the ion cloud surrounding the polymer skeleton will contain different configurations of the counterions, with some of the original counterions replaced by new ones at new locations. Thus, on an average, we imagine a counterion worm around a polymer chain. The counterion worm is dynamic, with the counterions continuously binding and unbinding with the polymer backbone at random locations, but maintaining an average number inside the worm. As a result, the effective charge of the polyelectrolyte is not the same as its chemical charge that could be estimated by assuming that all ionizable groups of the polymer fully dissociate. The effective charge is unique to the average polymer conformation, and it self-regulates with changes in polymer conformations accompanying changes in experimental conditions. The repulsion among the effective charges on the polymer backbone is manifest as electrostatic swelling for flexible polymers, or equivalently as chain stiffening for semiflexible polymers. The primary focus of this chapter is to combine the key concepts of electrostatic interactions (Chapter 3) and the various polymer models (Chapter 2), toward a description of the equilibrium properties of polyelectrolyte molecules in dilute solutions. After

presenting major concepts and some key experimental and simulation results, we shall describe simple treatments of electrostatic effects endowed on polymer conformations.