“…Consequently, the brush attains the charge by ionization of a weakly acidic group HA, which ionizes as so that Here c i ( i = HA, A – , H + ) is the concentration in moles per liter at the location of the end charge (i.e., y = y e ) and K a is the ionization constant in moles per liter. Consequently, i.e., the concentrations are expressed as number densities (having units of 1/m 3 ) and K a ′ = N A 10 3 K a has a unit of 1/m 3 ( N A is the Avogadro number). Following the large volumes of literature relating the charge density (σ ch ) of an ionizable surface to the concentration of species in the reaction that produces this surface charging, − we can express where Γ 0 is the total surface site density of the end charges, and Γ HA and Γ A – are the corresponding values associated with HA and A – (see Behrens and Grier for details). For the present case, we are interested to obtain the pH-dependent charge density: therefore we would need σ A – , i.e., Of course, following Behrens and Grier (kindly see eq in Behrens and Grier and note that the condition expressed in eq below allows one to obtain eq in Behrens and Grier), the site density corresponding to the individual species can be related to their corresponding concentrations (or number densities) as Using eq , we can reduce eq to which is identical to eq in Behrens and Grier. Therefore, using eqs , , and , we can obtain where n H + is the hydrogen ion number density at the location of the end charge, i.e., n H + = n H + ( y e ).…”