Dissimilar Surfaces

Figure 2: The charge density of surfaces with silanol, carboxyl, or sulfate head groups as a function of pH. Parameters for the silica-like surface are as in Fig. 1. For both the sulfate- and the carboxyl bearing surface we have assumed a density $\Gamma =0.25$ ${\rm nm}^{-2}$ of sites, all of which are constantly charged in the sulfate case; further parameters of the carboxyl surface are a large (infinite) Stern capacity and a dissociation pK of 4.9.

Figure 3: The charge density of the silica surface and the force per unit area it experiences when interacting with any of the three presented types of surfaces at pH 6 and an ionic strength of 1 mM ( $\kappa ^{-1}=9.6$ nm). Surfaces parameters as in the previous figures.

The procedure described before may be applied to negatively charged surfaces other than glass or silica as long as the chemically imposed charge-potential relation $\sigma(\psi_d)$ is modified to account for the surface properties of the considered material. Carboxylated latex for instance can be described in the same framework as silica, with a pK value of 4.9 for the dissociation of the carboxyl surface groups ( ${\rm COOH \rightleftharpoons COO^- + H^+}$) and a large Stern capacity $C$ (any value $C \gg 10$ amounting to a negligible potential drop $\vert\psi_0-\psi_d\vert$ across the Stern layer) [39]. Sulfate latex, on the other hand, may be considered as having a constant charge density ( $\sigma(\psi_d)=-e\Gamma={\rm const.}$), because the strongly acidic sulfate groups are fully dissociated in all relevant solution conditions. Fig. 2 shows the predicted (and experimentally confirmed [26,39,40]) charging behavior of the aforementioned materials. The site density of $\Gamma=0.25~\mathrm{nm^{-2}}$ chosen for both the sulfate and the carboxyl surface lies in the typical range for commercially available latex spheres and has also been cited as the density of carboxyl groups on the membrane of blood cells [22].

A way to evaluate the interaction between two dissimilar surfaces starts by applying the described method for equal surfaces separately to both materials. For each of these symmetric systems, one obtains the midplane potential $\Psi_{\rm m}$ and thus via Eq. (17) the full potential function $\Psi(x)$ associated with any given separation between equal plates. Since $\Psi(x)$ is already fully determined by the value of $\Psi_{\rm m}=\Psi(0)$ and the requirement $d\Psi/dx\vert _{x=0}=0$, solutions $\Psi(x)$ of the Poisson-Boltzmann equation for different systems are identical if they correspond to the same $\Psi_{\rm m}$ (i.e. the same pressure), the only difference being the surface separation $h$ for which they occur in the two systems. A solution $\Psi(x)$ associated with a separation $h_1$ in one system and with separation $h_2$ in the second system clearly serves as a solution in a mixed system with a surface of type 1 at $x=-h_1/2$ and a surface of type 2 at $x=+h_2/2$. Moreover, the separation $h=(h_1+h_2)/2$ at which this solution occurs in the mixed system is unique, because the pressure is a monotonic function of separation in the symmetric systems and can thus be inverted to give the two separation functions $h_1(\Psi_{\rm m} )$ and $h_2(\Psi_{\rm m})$. Our strategy therefore consists of computing $\Psi_{\rm m}$ for all separations of interest in the symmetric systems 1 and 2, finding the separations $h_1(\Psi_{\rm m} )$, $h_2(\Psi_{\rm m})$ by inversion, and finally inverting their arithmetic mean $h(\Psi_{\rm m})=[h_1(\Psi_{\rm m})+h_2(\Psi_{\rm m}) ]/2$ to obtain $\Psi_{\rm m}(h)$ and all the ensuing properties of interest in the mixed system.

Some results of this type are shown in Fig. 3, where we have plotted the charge density of a glass or silica surface and the electrostatic pressure as it interacts with either its own kind or with a surface of the carboxyl or the sulfate type. At the chosen ionic strength of 1 mM and pH 6, the charge of the silica surface is seen to deviate significantly from its value in isolation (horizontal line and Fig. 2) up to separations of several screening lengths. Moreover, the nature of the second surface also has a profound effect not only on the strength of the interaction, but also on the charging state of the silica. While all anionic surfaces will reduce the effective charge on silica upon approach, the rate at which they do so strongly depends on the amount and variability of their own charge. Neither of these dependencies are usually considered in the discussion of interaction measurements. A recent attempt to determine these ionization properties of silica experimentally with atomic force microscopy [41] has been limited to symmetric surfaces, and relies on model assumptions both for the charge regulation and for the strong van der Waals forces at short surface separations.