Effective Charge of Glass and Silica in Deionized Solutions

The principal mechanism by which glass and silica surfaces acquire a charge in contact with water is the dissociation of silanol groups [2]

$${\rm SiOH \rightleftharpoons SiO^- + H^+ . }$$ (1)

Further protonation of the uncharged group is expected only under extremely acidic conditions [26,29] and will be disregarded. Similarly, we will not take into account the protonation of doubly coordinated ${\rm Si}_2-{\rm O}$ groups as these are generally considered inert [26].

In addition to the hydronium or other counterions dissociated from the surface, the bulk solution includes ions due to the autodissociation of water. The latter can contribute more strongly to the overall concentration of mobile ions than the surface-dissociated counterions if the ratio of surface area to solution volume is exceedingly small. In practice, even the purest water also contains some residual electrolyte. Therefore the charging state of low specific area surfaces is controlled by the ionic strength and pH of the solution bulk, just as in the general case of high salt concentrations. We propose to take advantage of this similarity by applying insights gained from studies at high electrolyte concentrations to calculate the charge density at silica-water interfaces with low surface area and with no added salt.

Whether the specific surface area is indeed small enough to warrant a ``high salt'' treatment, depends largely on the geometry of the considered experimental setup and has to be checked on a case-by-case basis. For some of the aforementioned interaction measurements [12,17] this approach is appropriate, in other cases it may provide a very rough upper limit for the surface charge. If, on the other hand, the counterions due to the charged surfaces give a non-negligible contribution to the overall ionic strength, they have to be considered explicitly, for instance within a cell model [30,31]. Here, we concentrate on the former case of ``high salt'' and adopt the Basic Stern model [25], which has been shown to accurately describe titration data [32] obtained in the this regime for nominally nonporous, fully hydrated silica particles [26,29].

Within the Basic Stern model the charge of silica is regarded as localized entirely on the surface and arising from a concentration $\Gamma_{\mathrm{SiO^-}}$ of dissociated head groups [27], giving rise to the surface charge density

$$\sigma = - e \Gamma_{\mathrm{SiO^-}}.$$ (2)

Under normal conditions, only a fraction of the total concentration,

$$\Gamma = \Gamma_{\mathrm{SiO^-}}+ \Gamma_{\mathrm{SiOH}},$$ (3)

of chargeable sites dissociate. The relevant mass action law for the deprotonation reaction, Eq. (1),

$$\frac{\left[\mathrm{H}^+\right]_0\Gamma_{\mathrm{SiO^-}}}{\Gamma_{\mathrm{SiOH}}} = 10^{-\mathrm{pK}}\, \mathrm{Mol/l},$$ (4)

is characterized by the logarithmic dissociation constant, pK, and accounts for the influence of the surface's electrostatic potential, $\psi_0$, through the surface activity of protons,

$$\left[\mathrm{H}^+\right]_0= \left[\mathrm{H}^+\right]_{\mathrm b} \exp\left(-\beta e\psi_0\right).$$ (5)

Here, $\left[\mathrm{H}^+\right]_{\mathrm b} = 10^{-\mathrm{pH}}\, \mathrm{Mol/l}$ is the bulk activity of protons, and $\beta^{-1}=k_BT$ denotes the thermal energy. The dissociation constant is an inherent property of the silicate-water interface and is estimated to be $\mathrm{pK}= 7.5$ on the basis of a surface complexation model [26].

As counterions dissociate from the surface, they form a diffuse cloud of charge within the electrolyte. The Basic Stern model treats the counterions as being separated from the surface by a thin Stern layer across which the electrostatic potential drops linearly from is surface value, $\psi_0$, to a value $\psi_d$ called the diffuse layer potential [25,27]. This potential drop is characterized by the Stern layer's phenomenological capacity,

$$C = \frac{\sigma}{\psi_0 - \psi_d}$$ (6)

This capacity, $C$, reflects the structure of the silicate-water interface and should vary little with changes in surface geometry or electrolyte concentration. Titration data on colloidal silica [32] are consistent with $C = 2.9~\mathrm{F/m^2}$ [26].

Eqs. (2-6) can be solved for the diffuse layer potential as a function of the charge density on the interface:

$$\psi_d (\sigma) = \frac{1}{\beta e} \, \ln\frac{-\sigma}{e\Gamma ... ...igma} - (\mathrm{pH}- \mathrm{pK}) \frac{\ln 10}{\beta e} - \frac{\sigma }{C} .$$ (7)

This relation reflects the chemical nature of the interface and its charging process.

Another functional dependence follows from the distribution of mobile charges in the solution. If the latter is described by the Poisson-Boltzmann equation (PB), then the charge of an isolated, flat surface satisfies the Grahame equation

$$\sigma(\psi_d) = \frac{2\varepsilon \varepsilon _0\kappa}{\beta e} \, \sinh\left(\frac{\beta e \psi_d}{2}\right).$$ (8)

Here, $\varepsilon \varepsilon _0$ is the permittivity of the solution and $\kappa^{-1}$ the Debye screening length given by $\kappa^2 = \beta e^2 n /\varepsilon \varepsilon _0$, where $n$ is the total concentration of small ions, all of which are assumed to be monovalent. The generalization of Eq. (8) to account for a curvature of radius $a$,

$$\sigma(\psi_d) = \frac{2\varepsilon \varepsilon _0\kappa}{\beta e... ...ght) + \frac{2}{\kappa a} \, \tanh\left(\frac{\beta e\psi_d}{4}\right) \right],$$ (9)

is known to give the surface charge density to within 5% for $\kappa a \ge 0.5$ and any surface potential [33].

Combining Eq. (7) with Eq. (8) or (9) yields self-consistent values for the surface charge density, $\sigma$, and the diffuse layer potential, $\psi_d$. These values characterize the equilibrium of bound and mobile charges in the interfacial region, but are not necessarily accessible experimentally, given the requirement of large surface areas for potentiometric titrations and the interpretive ambiguities inherent to other techniques.

Most measurements of interfacial interactions probe the electrostatic potential $\psi$ at distances for which $e\psi \le k_BT$. Under these conditions $\psi$ is described accurately by the linearized Poisson-Boltzmann equation, whose solution for a single flat surface has the form $\psi(x)=\psi_{\rm eff} \exp(-\kappa x)$, where $x$ is the distance from the surface. The effective surface potential $\psi_{\rm eff}$ in this experimentally accessible regime is related to the actual diffuse layer potential through [33]

$$\beta e \psi_{\rm eff} = 4 \tanh\left(\frac{\beta e \psi_d }{4}\right).$$ (10)

Again, there is an approximate generalization for curved surfaces [34]:

$$\beta e \psi_{\rm eff} = \frac{8 \tanh\left(\frac{\beta e\psi_d}{... ...a}{(1+\kappa a)^2} \tanh^2\left(\frac{\beta e \psi_d}{4}\right) \right]^{1/2}}.$$ (11)

The associated effective charge density can be obtained from

$$\sigma_{\rm eff} = \varepsilon \varepsilon _0\kappa\psi_{\rm eff} \left[1+\frac{1}{\kappa a} \right],$$ (12)

which is just the linearization of Eq. (9). This effective charge density characterizes essentially all of the recent measurements of electrostatic interactions between well-separated charged surfaces.

The effective charge's relevance to experimental observations is based in the popularity of the linear superposition approximation for estimating the interaction energy, $u(h)$, between two charged spheres of radii $a_1$ and $a_2$ as a function of their surface-to-surface separation $h$. In this approximation [35],

$$u(h) = \frac{4 \pi}{\varepsilon \varepsilon _0} \left( \frac{\sig... ... \frac{\sigma_2 a_2^2}{1+\kappa a_2} \right) \frac{\exp(-\kappa h)}{a_1+a_2+h},$$ (13)

where $\sigma_1$ and $\sigma_2$ should be understood to be effective surface charge densities obtained from Eq. (11) rather than the bare charge densities from Eqs. (7) and (9). Using the effective surface charge densities implicitly accounts for overexponential decay of the electrostatic potential near the surfaces that follows from the nonlinearity of the Poisson-Boltzmann equation. The interaction between a sphere and a planar wall is obtained by taking the limit of one infinite radius in Eq. (13).

Fig. 1 shows computed values for the bare and effective charges of a planar silica surface and a 1 $\mu$m-diameter silica sphere for pH values between 7 and the lowest pH compatible with an ionic strength of 1, 5, and 10 $\mu$Mol/l. These are reasonable values for deionized water under usual experimental conditions. In addition to using $C = 2.9~\mathrm{F/m^2}$ and $\mathrm{pK}= 7.5$, we have further assumed a total site density of $\Gamma = 8~\mathrm{nm}^{-2}$, a commonly cited literature value for nonporous, fully hydrated silica [2]. Although $\Gamma$ could vary widely depending on surface preparation, the degree of protonation is determined mostly by the electrostatic interactions among the small fraction of charged surface sites, rather than the large number of neutral sites that $\Gamma$ accounts for, and so our results are quite insensitive to this parameter. This robustness validates our assumption that details in the structure of nonporous surfaces do not matter in the present context. Indeed, the top graph of Fig. 1 also should represent the charging properties of a polished glass surface. Note however that our arguments do not apply to some types of silica that are believed to be very porous and contain a much higher charge [2], the largest part of which seems located in the porous volume [36] rather than on the surface.

Figure 1: The bare (full lines) and effective (dashed curves) charge densities of a planar glass wall and a 1 micron silica sphere, assuming a density $\Gamma =8$ ${\rm nm}^{-2}$ of chargeable sites, a pK value of 7.5 for the silanol dissociation, and a Stern capacity of 2.9 ${\rm F}/{\rm m}^2$.

Fig. 1 demonstrates that the effective charge densities in deionized solutions do not depend as sensitively on pH as their ``bare'' counterparts, but that a significant variation with ionic strength persists. The top part of the figure indicates that the value of 2000 effective charges per $\mu {\rm m}^2 \xspace$ ( $=-0.32~\mathrm{mC/m^2}$) assumed by Squires and Brenner [16] for a glass plate in contact with deionized solution of $\kappa^{-1}=0.275$ $\mu$m (i.e. an ionic strength of $1.2 \times 10^{-6}$ M) is very reasonable. The confirmation of this previously very uncertain value provides vital support for their recent electro-hydrodynamic explanation of apparent attractions between like-charged particles near a single glass wall.

In a study of the equilibrium interaction between few 1.58 $\mu$m silica sphere at the bottom of a large glass container filled with deionized water, we found an ionic strength between $8.5 \times 10^{-7}$ and $1.1 \times 10^{-6}$ M and a charge density between 550 and 830 $e/\mu {\rm m}^2 \xspace$ from a fit of measured interaction energies to Eq. (13) [17]. Comparison with Fig. 1 shows that these charge densities are a little below our expectation for isolated spheres, but have the right order of magnitude. The remaining difference can be explained by the spheres' close proximity to the bottom wall of the glass container, as we describe in the next section.