The Charge of Glass and Silica Surfaces

Sven H. Behrens and David G. Grier Department of Physics, James Franck Institute, and Institute for Biophysical Dynamics, The University of Chicago, Chicago, IL 60637 (Dated: S. H. Behrens & D. G. Grier, J. Chem. Phys. 115, 6716-6721 (2001)) We present a method of calculating the electric charge density of glass and silica surfaces in contact with aqueous electrolytes for two cases of practical relevance that are not amenable to standard techniques: surfaces of low speciﬁc area at low ionic strength and surfaces interacting strongly with a second anionic surface.

INTRODUCTION colloidal charge is large. A series of recent experiments has spurred interest in Ionization processes in aqueous solutions have long the charge on glass and silica surfaces of low specific area been a point of central interest to physical chemistry. in pure water, i.e. systems for which the usual picture Much progress has been made in recent years in under- of the “no salt” regime does not apply. For example, standing the charging properties of such different entities interaction measurements using digital video microscopy as small molecules, polyelectrolytes and various kinds of and optical trapping suggest that highly charged latex interfaces [1]. spheres may experience an anomalous long-ranged at- In this context, silica and silicate glass surfaces im- traction when confined by charged glass walls [9–12], con- mersed in water are known to acquire a negative surface trary to the predictions of Poisson-Boltzmann theory [13– charge density, primarily through the dissociation of ter- 15]. In one particular case [12], the attraction appears minal silanol groups. The degree of dissociation and thus to result from a hydrodynamic interaction driven by the the surface charge density results from an equilibrium spheres’ electrostatic repulsion from a nearby wall [16]. between counterions at the glass surface and free ions in This explanation hinges on the heretofore untested as- the bulk electrolyte. Experimentally, this type of equilib- sumption that the glass wall carries an effective charge 2 rium and its dependence on the solution conditions can density of −2000 ± 200 e/µm , where e is the elementary be studied by potentiometric acid-base titrations on col- charge. Such hydrodynamic coupling does not seem to loidal dispersions of non-porous silica particles [2]. This explain the like-charge attractions measured for spheres technique actually measures the volume concentration of confined between two charged glass walls [9–11], since protons transferred between the surfaces and the solu- the proposed effect is strongly suppressed by the second tion. In order for the surfaces to accommodate a suffi- wall. How the walls’ charge influence colloidal electro- cient amount of charge, the electrostatic interaction be- static interactions is not yet resolved, in part because of tween the surface sites must be screened at least partially open questions regarding the charging state of the glass. by added salt ions, and/or the available surface area must We recently reported that the pair interaction of silica be large. spheres remains monotonically repulsive even in the presence of a single glass wall [17]. The spheres’ effective These constraints can be relaxed to some degree by re- surface charge density of −700 ± 150 e/µm2 extracted sorting to alternative techniques like microelectrophore- from these measurements is considerably smaller than sis [3, 4], streaming potential measurements [5], con- the value posited in Ref. [16] for a compact glass surface. ductometry [6] and electroacoustic methods [7, 8]. All of these methods, however, rely heavily on approximate Because a silica surface’s charge density depends on models for electrostatic or hydrodynamic processes in the the local chemical environment, it necessarily varies with interfacial region, introducing uncertainties that are diffi- proximity to other charge-carrying surfaces. The in- cult to estimate. We are not aware of any way to measure terpretation of typical particle deposition experiments directly the surface charge of silica in a solution of very [18] and force measurements by atomic force microscopy low ionic strength. (AFM) [19, 20] or total internal reflection microscopy Theoretical studies of low ionic strength solutions typ- (TIRM) [21] for example, is complicated by the fact that ically deal with dense colloidal and macroionic systems the charge densities of the substrate and the probe are and consider the regime of “no salt”, “low salt”, or “coun- a function of their separation, a phenomenon known as terions only”. Here the defining assumption is that the “charge regulation” [22–24]. Since local properties of the overall ionic strength is due predominantly to the par- enclosed solution rather than bulk properties determine ticles or macroions and the compensating counterions in the charging state, a naive use of charging data from bulk solution, whereas the concentration of any additional ions measurements can lead to errors. is negligible. For colloidal dispersions this assumption is In the following section we discuss how the experimen- again legitimate if the specific surface area carrying the tally supported 1-pK Basic Stern Model [25, 26] for silica 2 surfaces may be used to calculate the elusive charge of Within the Basic Stern model the charge of silica is glass plates and strongly diluted silica particles in deion- regarded as localized entirely on the surface and arising ized water. In the remainder of this paper we take ad- from a concentration ΓSiO− of dissociated head groups vantage of a recently proposed theoretical treatment of [27], giving rise to the surface charge density charge regulation [27, 28] to discuss the the charge of a silica-like surface in close proximity to a second anionic σ = −eΓSiO− . (2) surface, which will be chosen, in view of the most com- mon applications, as either of the same type or of con- Under normal conditions, only a fraction of the total constant charge (like sulfate latex) or of carboxylic nature centration, (like carboxyl latex and many biological surfaces). Γ = ΓSiO− + ΓSiOH, (3)

EFFECTIVE CHARGE OF GLASS AND SILICA of chargeable sites dissociate. The relevant mass action IN DEIONIZED SOLUTIONS law for the deprotonation reaction, Eq. (1),

+ [H ] Γ − The principal mechanism by which glass and silica sur- 0 SiO = 10−pK Mol/l, (4) faces acquire a charge in contact with water is the disso- ΓSiOH ciation of silanol groups [2] is characterized by the logarithmic dissociation constant, SiOH )* SiO− + H+. (1) pK, and accounts for the influence of the surface’s electrostatic potential, ψ0, through the surface activity of Further protonation of the uncharged group is expected protons, only under extremely acidic conditions [26, 29] and will + + be disregarded. Similarly, we will not take into account H 0 = H b exp (−βeψ0) . (5) the protonation of doubly coordinated Si2 − O groups as + −pH these are generally considered inert [26]. Here, [H ]b = 10 Mol/l is the bulk activity of pro- −1 In addition to the hydronium or other counterions dis- tons, and β = kBT denotes the thermal energy. sociated from the surface, the bulk solution includes ions The dissociation constant is an inherent property of the due to the autodissociation of water. The latter can con- silicate-water interface and is estimated to be pK = 7.5 tribute more strongly to the overall concentration of mo- on the basis of a surface complexation model [26]. bile ions than the surface-dissociated counterions if the As counterions dissociate from the surface, they form ratio of surface area to solution volume is exceedingly a diffuse cloud of charge within the electrolyte. The small. In practice, even the purest water also contains Basic Stern model treats the counterions as being sepa- some residual electrolyte. Therefore the charging state rated from the surface by a thin Stern layer across which of low specific area surfaces is controlled by the ionic the electrostatic potential drops linearly from is surface strength and pH of the solution bulk, just as in the gen- value, ψ0, to a value ψd called the diffuse layer potential eral case of high salt concentrations. We propose to take [25, 27]. This potential drop is characterized by the Stern advantage of this similarity by applying insights gained layer’s phenomenological capacity, from studies at high electrolyte concentrations to calcu- σ late the charge density at silica-water interfaces with low C = (6) surface area and with no added salt. ψ0 − ψd Whether the specific surface area is indeed small This capacity, C, reflects the structure of the silicate- enough to warrant a “high salt” treatment, depends water interface and should vary little with changes in largely on the geometry of the considered experimen- surface geometry or electrolyte concentration. Titra- tal setup and has to be checked on a case-by-case ba- tion data on colloidal silica [32] are consistent with C = sis. For some of the aforementioned interaction measure- 2.9 F/m2 [26]. ments [12, 17] this approach is appropriate, in other cases Eqs. (2–6) can be solved for the diffuse layer potential it may provide a very rough upper limit for the surface as a function of the charge density on the interface: charge. If, on the other hand, the counterions due to the charged surfaces give a non-negligible contribution 1 −σ ln 10 σ ψ (σ) = ln − (pH − pK) − . (7) to the overall ionic strength, they have to be considered d βe eΓ + σ βe C explicitly, for instance within a cell model [30, 31]. Here, we concentrate on the former case of “high salt” and This relation reflects the chemical nature of the interface adopt the Basic Stern model [25], which has been shown and its charging process. to accurately describe titration data [32] obtained in the Another functional dependence follows from the dis- this regime for nominally nonporous, fully hydrated silica tribution of mobile charges in the solution. If the latter particles [26, 29]. is described by the Poisson-Boltzmann equation (PB), 3 then the charge of an isolated, flat surface satisfies the u(h), between two charged spheres of radii a1 and a2 as a Grahame equation function of their surface-to-surface separation h. In this approximation [35], 2εε0κ βeψd σ(ψd) = sinh . (8) 4π σ a2 σ a2 exp(−κh) βe 2 u(h) = 1 1 2 2 , (13) εε0 1 + κa1 1 + κa2 a1 + a2 + h −1 Here, εε0 is the permittivity of the solution and κ the 2 2 where σ1 and σ2 should be understood to be effective sur- Debye screening length given by κ = βe n/εε0, where n is the total concentration of small ions, all of which are face charge densities obtained from Eq. (11) rather than assumed to be monovalent. The generalization of Eq. (8) the bare charge densities from Eqs. (7) and (9). Using to account for a curvature of radius a, the effective surface charge densities implicitly accounts for overexponential decay of the electrostatic potential 2εε κ βeψ 2 βeψ near the surfaces that follows from the nonlinearity of the σ(ψ ) = 0 sinh d + tanh d , d βe 2 κa 4 Poisson-Boltzmann equation. The interaction between a (9) sphere and a planar wall is obtained by taking the limit is known to give the surface charge density to within 5% of one infinite radius in Eq. (13). for κa ≥ 0.5 and any surface potential [33]. Fig. 1 shows computed values for the bare and effective Combining Eq. (7) with Eq. (8) or (9) yields self- charges of a planar silica surface and a 1 µm-diameter consistent values for the surface charge density, σ, and silica sphere for pH values between 7 and the lowest the diffuse layer potential, ψd. These values character- pH compatible with an ionic strength of 1, 5, and 10 ize the equilibrium of bound and mobile charges in the µMol/l. These are reasonable values for deionized water interfacial region, but are not necessarily accessible ex- under usual experimental conditions. In addition to using perimentally, given the requirement of large surface areas C = 2.9 F/m2 and pK = 7.5, we have further assumed a for potentiometric titrations and the interpretive ambi- total site density of Γ = 8 nm−2, a commonly cited lit- guities inherent to other techniques. erature value for nonporous, fully hydrated silica [2]. Al- Most measurements of interfacial interactions probe though Γ could vary widely depending on surface prepa- the electrostatic potential ψ at distances for which eψ ≤ ration, the degree of protonation is determined mostly by kBT . Under these conditions ψ is described accurately the electrostatic interactions among the small fraction of by the linearized Poisson-Boltzmann equation, whose so- charged surface sites, rather than the large number of lution for a single flat surface has the form ψ(x) = neutral sites that Γ accounts for, and so our results are ψeff exp(−κx), where x is the distance from the surface. quite insensitive to this parameter. This robustness val- The effective surface potential ψeff in this experimentally idates our assumption that details in the structure of accessible regime is related to the actual diffuse layer po- nonporous surfaces do not matter in the present context. tential through [33] Indeed, the top graph of Fig. 1 also should represent the charging properties of a polished glass surface. Note how- βeψd ever that our arguments do not apply to some types of βeψ = 4 tanh . (10) eff 4 silica that are believed to be very porous and contain a much higher charge [2], the largest part of which seems Again, there is an approximate generalization for located in the porous volume [36] rather than on the sur- curved surfaces [34]: face. Fig. 1 demonstrates that the effective charge densi- βeψd 8 tanh 4 ties in deionized solutions do not depend as sensitively βeψ = . (11) eff h i1/2 on pH as their “bare” counterparts, but that a signifi- 1+2κa 2 βeψd 1 + 1 − (1+κa)2 tanh 4 cant variation with ionic strength persists. The top part of the figure indicates that the value of 2000 effective The associated effective charge density can be obtained charges per µm2 (= −0.32 mC/m2) assumed by Squires from and Brenner [16] for a glass plate in contact with deion- 1 ized solution of κ−1 = 0.275 µm (i.e. an ionic strength σ = εε κψ 1 + , (12) −6 eff 0 eff κa of 1.2 × 10 M) is very reasonable. The confirmation of this previously very uncertain value provides vital sup- which is just the linearization of Eq. (9). This effective port for their recent electro-hydrodynamic explanation of charge density characterizes essentially all of the recent apparent attractions between like-charged particles near measurements of electrostatic interactions between well- a single glass wall. separated charged surfaces. In a study of the equilibrium interaction between few The effective charge’s relevance to experimental obser- 1.58 µm silica sphere at the bottom of a large glass vations is based in the popularity of the linear superposi- container filled with deionized water, we found an ionic tion approximation for estimating the interaction energy, strength between 8.5 × 10−7 and 1.1 × 10−6 M and a 4

CHARGE REGULATION OF ANIONIC SURFACES

Many experimental techniques to measure colloidal forces (AFM, TIRM, and the surface force apparatus, for instance) provide good resolution at very short distances. Often the surface separations h of interest are comparable to or smaller than the screening length and much smaller than the radii of curvature, a1 and a2, of either surface. In this situation, the surfaces may be regarded as locally ﬂat, and the Derjaguin approximation a a f(h) = 2π 1 2 W (h). (14) a1 + a2 accurately expresses the magnitude f(h) of the acting force in terms of the interaction energy per unit area W (h) for two parallel (thick) plates of the same separation. This reduces the interaction problem to ﬁnding the energy W (h) or the dividing pressure Π(h) = −dW/dh resulting from a one-dimensional distribution of mediat- ing small ions. On the other hand, the superposition of the noninter- acting surfaces’ electrostatic potential is no longer war- ranted; nor can reliable results be expected from a solution of the linearized Poisson-Boltzmann equation, as it only captures the case of weak potentials (atypical for strongly interacting surfaces) or, with renormalized charges, the asymptotic behavior for large separations. Instead, we consider the nonlinear PB equation, which in one dimension and for an excess of monovalent electrolyte ions reads

d2Ψ (x) = κ2 sinh Ψ(x), (15) dx2 where Ψ = βeψ is the dimensionless electrostatic potential and x the coordinate normal to the surfaces. Apply- FIG. 1: The bare (full lines) and effective (dashed curves) ing this mean field formalism to more general electrolytes charge densities of a planar glass wall and a 1 micron silica shall not be discussed here, since neglected ion correla- sphere, assuming a density Γ = 8 nm−2 of chargeable sites, a pK value of 7.5 for the silanol dissociation, and a Stern tions and ion-specific interactions with the surfaces tend capacity of 2.9 F/m2. to complicate the case of polyvalent ions. ¿From Eq. (15) it is clear that Ψ(x) is a convex function for Ψ ≤ 0. The charge density on either surface (1 or 2) is given, according to Gauß’ law, by

εε0 dΨ σ1,2 = − , (16) βe dx 1,2 where the derivative is taken at the surface with respect charge density between 550 and 830 e/µm2 from a ﬁt of to its outward normal. It follows that between two nega- measured interaction energies to Eq. (13) [17]. Compar- tively charged surfaces, Ψ(x) has a maximum Ψm, which ison with Fig. 1 shows that these charge densities are a for identical surfaces lies exactly at the midplane. Choos- little below our expectation for isolated spheres, but have ing generally the position of this maximum as the origin the right order of magnitude. The remaining diﬀerence of our coordinate system (i.e. Ψ(0) = Ψm), we can ex- can be explained by the spheres’ close proximity to the press the solution of Eq. (15) as [22, 28] bottom wall of the glass container, as we describe in the next section. Ψ(x) = Ψm + 2 ln cd(u|m), (17) 5 with κx u = exp (−Ψ /2) (18) 2 m and

m = exp (2Ψm) , (19) where cd(u|m) is a Jacobian elliptic function of argument u and parameter m [37]. The derivative is

dΨ sn(u|m) = m3/4 − m−1/4 κ , (20) dx cn(u|m) dn(u|m) where sn(u|m), cn(u|m), and dn = cn / cd are again Ja- cobian elliptic functions of the argument u and parameter m given above. Eﬃcient numeric implementations of these functions are readily available from mathematical libraries [38].

FIG. 2: The charge density of surfaces with silanol, carboxyl, Equal Surfaces or sulfate head groups as a function of pH. Parameters for the silica-like surface are as in Fig. 1. For both the sulfate- We shall measure the separation h between the sur- and the carboxyl bearing surface we have assumed a density Γ = 0.25 nm−2 of sites, all of which are constantly charged faces by the distance between the head ends of the dif- in the sulfate case; further parameters of the carboxyl surface fuse layer, i.e., Ψ(h/2) = βeψd. Evaluating Eqs. (16- are a large (infinite) Stern capacity and a dissociation pK of 20) at x = ±h/2 and combining them with the chemical 4.9. boundary condition, Eq. (7), provides an expression for the actual surface charge density σ. The second boundary condition, dΨ/dx = 0 at x = 0, is already implied in Dissimilar Surfaces the solution for Ψ(x), Eq. (17). Note that for σ(ψd, h) defined by Eqs. (16) through (20), the long distance limit The procedure described before may be applied to limh→∞ σ(ψd, h) is given by Eq. (8), which we have pre- negatively charged surfaces other than glass or silica as viously used for isolated surfaces and weakly interacting long as the chemically imposed charge-potential relation surfaces in the superposition approximation. In general, σ(ψd) is modified to account for the surface properties of σ is a function of ψd as well as of the surface separa- the considered material. Carboxylated latex for instance tion and cannot be expressed analytically. As before, the can be described in the same framework as silica, with a electrostatic definition of σ, Eqs. (16) and (20), depends pK value of 4.9 for the dissociation of the carboxyl sur- parametrically on the Debye length, while the chemical face groups (COOH )* COO− + H+) and a large Stern definition, Eq. (7), depends only on pH and the surface capacity C (any value C 10 amounting to a negligible chemical parameters, Γ, pK, and C. potential drop |ψ0 − ψd| across the Stern layer) [39]. Sul- Technically, the combination of Eqs. (7) and (16-20) fate latex, on the other hand, may be considered as hav- results in a single transcendental equation for the mid- ing a constant charge density (σ(ψd) = −eΓ = const.), plane potential Ψm, whose numerical solution provides because the strongly acidic sulfate groups are fully dis- a very convenient alternative to numerical integration of sociated in all relevant solution conditions. Fig. 2 shows the differential equation (15) with nonlinear boundary the predicted (and experimentally confirmed [26, 39, 40]) conditions. charging behavior of the aforementioned materials. The −2 Solving for Ψm has the further advantage of immedi- site density of Γ = 0.25 nm chosen for both the sul- ately yielding the electrostatic force per unit area fate and the carboxyl surface lies in the typical range for commercially available latex spheres and has also been Π = nkBT (cosh Ψm − 1) , (21) cited as the density of carboxyl groups on the membrane of blood cells [22]. i.e. the excess osmotic pressure of small ions at the mid- A way to evaluate the interaction between two dissim- plane where the electric field and the associated Maxwell ilar surfaces starts by applying the described method for stress are zero. equal surfaces separately to both materials. For each 6 of these symmetric systems, one obtains the midplane equation for different systems are identical if they corre- potential Ψm and thus via Eq. (17) the full potential spond to the same Ψm (i.e. the same pressure), the only function Ψ(x) associated with any given separation be- difference being the surface separation h for which they tween equal plates. Since Ψ(x) is already fully deter- occur in the two systems. A solution Ψ(x) associated mined by the value of Ψm = Ψ(0) and the requirement with a separation h1 in one system and with separation dΨ/dx|x=0 = 0, solutions Ψ(x) of the Poisson-Boltzmann h2 in the second system clearly serves as a solution in a mixed system with a surface of type 1 at x = −h1/2 and a surface of type 2 at x = +h2/2. Moreover, the separation h = (h1 + h2)/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 h1(Ψm) and h2(Ψm). Our strategy therefore consists of computing Ψm for all separations of interest in the symmetric systems 1 and 2, finding the separations h1(Ψm), h2(Ψm) by inversion, and finally inverting their arithmetic mean h(Ψm) = [h1(Ψm) + h2(Ψm)]/2 to obtain Ψm(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 de- viate 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.

CONCLUSIONS

We have seen that the apparent charge on glass and silica surfaces of low specific area in pure water can be understood in terms of a simple model that was originally developed and tested for high electrolyte concentrations. Model predictions for effective charge densities compare favorably with interaction experiments on highly diluted FIG. 3: The charge density of the silica surface and the force silica spheres in deionized water [17]. They also sup- per unit area it experiences when interacting with any of the port a new kinematic explanation of spurious long range three presented types of surfaces at pH 6 and an ionic strength attractions between like-charged particles near a single of 1 mM (κ−1 = 9.6 nm). Surfaces parameters as in the glass wall [16]. previous figures. The regulated charge of silica and glass surfaces near 7 contact with a second anionic surface, as well as the Particle Deposition and Aggregation: Measurement, strength of the interaction, has been calculated from an Modeling and Simulation (Butterworth-Heinemann, Ox- exact solution of the nonlinear Poisson-Boltzmann equa- ford, 1995). tion. In commonly encountered solution conditions, the [19] H.-J. Butt, Biophys. J. 60, 1438 (1991). [20] W. A. Ducker, T. J. Senden, and R. M. Pashley, Lang- charge regulation of silica was found to be effective at muir 8, 1831 (1992). separations well beyond a Debye length. It also proves [21] D. C. Prieve, Adv. Colloid Interface Sci. 82, 93 (1999). very sensitive to the chemical nature of the opposing sur- [22] B. W. Ninham and V. A. Parsegian, J. Theor. Biol. 31, face. Although the additional presence of van der Waals 405 (1971). forces makes a quantitative measurement of these effects [23] D. Y. C. Chan, J. W. Perram, L. R. White, and T. 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