Determination of the Zeta Potential of Porous Substrates by Droplet

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Langmuir 2009, 25, 1842-1850

Determination of the Zeta Potential of Porous Substrates by Droplet Deflection. I. The Influence of Ionic Strength and pH Value of an Aqueous Electrolyte in Contact with a Borosilicate Surface Dominik P. J. Barz* Forschungszentrum Karlsruhe, IKET - Chemical Process Technology, P.O. Box 3640, D-76021 Karlsruhe, Germany

Michael J. Vogel and Paul H. Steen Cornell UniVersity, School of Chemical and Biomolecular Engineering, 120 Olin Hall, Ithaca, New York 14853 ReceiVed September 8, 2008. ReVised Manuscript ReceiVed NoVember 28, 2008 This paper presents a new method to determine the zeta potential of porous substrates in contact with a liquid. Electroosmosis, arising near the solid/liquid boundaries within a fully saturated porous substrate, pumps against the capillary pressure arising from the surface tension of a droplet placed in series with the pump. The method is based on measuring the liquid/gas interface deflection due to the imposed electric potential difference. The distinguishing features of our technique are accuracy, speed, and reliability, accomplished with a straightforward and cost-effective setup. In this particular setup, a bistable configuration of two opposing droplets is used. The energy barrier between the stable states defines the range of capillary resistance and can be tuned by the total droplet volume. The electroosmotic pump is placed between the droplets. The large surface area-to-volume ratio of the porous substrate enables the pumping strength to exceed the capillary resistance even for droplets small enough that their shapes are negligibly influenced by gravity. Using a relatively simple model for the flow within the porous substrate, the zeta potential resulting from the substrate-liquid combination is determined. Extensive measurements of a borosilicate substrate in contact with different aqueous electrolytes are made. The results of the measurements clarify the influence of the ionic strength and pH value on the zeta potential and yield an empirical relationship important to engineering approaches.

I. Introduction Research has been conducted on electrokinetic phenomena for almost 200 years. Interest in this work has recently experienced a renaissance due to the utilization of electrokinetic phenomena in innovative microfluidic concepts such as “Miniaturized Total Analysis Systems”1 and “Lab on a Chip”. These concepts include the arranging of microcomponents like pumps, valves, and detectors onto a plastic or glass chip and connecting those components by a system of microchannels. Typical microfluidic processes taking place are pumping, mixing of liquids, and separation of chemical and biological species. A special feature of small dimension devices is their high surface area-to-volume ratio which results in interfacial effects such as electrokinetic phenomena becoming prominent. Electroosmosis and electrophoresis are electrokinetic phenomena related to the presence of an electrical double layer (EDL) in the liquid phase. The zeta potential is the electric potential at the plane of shear and is commonly considered as the relevant parameter for electrokinetic phenomena. Electroosmosis is the motion of a liquid under the influence of an applied electric field relative to a charged solid surface. It is used to generate flows in microfluidic structures without moving (mechanical) parts, allowing for the realization of pumps and mixers. A review of electroosmotic pumps for microfluidic devices can be found in ref 2. Electrophoresis is the migration of dispersed charged particles under the influence of an applied electric field. * Corresponding author. E-mail: [email protected]. (1) Manz, A.; Graber, N.; Widmer, H. Sens. Actuators B 1990, 1, 244–248. (2) Laser, D.; Santiago, J. J. Micromech. Microeng. 2004, 14, R35–R64. (3) Bazant, M.; Thornton, K.; Ajdari, A. Phys. ReV. E 2004, 70, 021506-1– 021506-24.

Excellent reviews of the historical development of the field are available (e.g., ref 3). It is well-known that electrical charges in aqueous solution can influence the zeta potential and the thickness of the electrical double layer and, thereby, the strength of electroosmosis and electrophoretic mobility. There are a number of experimental methods for the indirect determination of the zeta potential. Common methods are based on electrophoresis, streaming current/potential, and electroosmosis. An extensive review of these methods is available.4 A variety of practical issues are faced. For any specific application, the suitability of each experimental method depends on the geometric form of the substrate (particulate or bulk solid) and how much substrate material and liquid is available. The typical commercially available “zeta machine” measures the electrophoretic mobility of a particle by electrophoretic light scattering (cf., e.g., ref 5) or by laser-Doppler electrophoresis (cf., e.g., ref 6). Analytical measurements of this kind typically require equipment of considerable cost. Moreover, the most straightforward electrophoretic zeta potential measurements require monodisperse and spherical particles. Otherwise, complex correlations between mobility and zeta potential must be invoked. Hence, this method is best suited to a large number of spherical particles of nearly the same size. For bulk materials, in contrast, extensive sample preparation is necessary. Here, the question arises as to whether or not the sample preparation influences the surface characteristics. In this article, we introduce a conceptual approach based on electroosmotic flow counterbalanced by capillarity for the (4) Delgado, A.; Gonzalez-Caballero, F.; Hunter, R.; Koopal, L.; Lyklema, J. J. Colloid Interface Sci. 2007, 309, 194–224.

10.1021/la802949z CCC: $40.75  2009 American Chemical Society Published on Web 01/07/2009

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Table 1. Properties of the Glass Frits, Obtained by Characterization Experiments frit

porosity ψ

tortuosity χ

Γ

1 2 3 4 5

0.39 0.38 0.31 0.28 0.31

1.17 1.31 1.19 1.16 1.17

0.48 0.55 0.54 0.56 0.53

determination of the zeta potential of porous substrates. With a large number of pores with sufficiently small average radius, volumetric flow rates can be induced even under small applied voltage. Furthermore, the pressure drops generated are sufficient to pump against liquid/gas interfaces with small radii-of-curvature. Measurements of the consequent deflection of the liquid/gas interfaces allow for the estimation of the zeta potential. The distinguishing feature of our approach is that imaging the liquid/ gas interface deflection is an accurate and relatively inexpensive means to observe liquid motion. Moreover, our concept requires only very small amounts of liquids. The conceptual approach of using capillarity to counterbalance electroosmosis can be used with bulk substrates, too. In a recent study, for example, the zeta potential of two parallel-plate channel walls is measured using capillary deflection. In this case, the capillary deflection drives a flow seeded by particles whose velocities are measured using microparticle imaging velocimetry.7 We proceed in the next section by giving a short overview of electroosmotic flow in a porous substrate. Next, we will introduce our method and describe the experimental setup and materials we used in this article. An extensive discussion of the experimental results follows, including comparison with data from the literature. A borosilicate-type glass substrate is chosen for study largely to validate the experimental approach. Finally, we summarize with concluding remarks. In the companion to this paper, our method is applied to verify electrokinetic flow in a doped nonpolar liquid.8

II. Flow Model An attractive feature of porous substrates is their large surface area-to-volume ratios. This makes them especially well-suited for measuring zeta potentials, both because of the flow rates that they can generate and for the good statistics that come from measuring averages from a large ensemble of small pores. The glass frit substrates used in the experiments reported below (cf. Table 1) fit into the palm of one’s hand (volume of 10 cm3) but possess a surface area of a snooker table (about 4 m2). A model is needed to relate macroscopic flow through the porous material to the flow in the pore and ultimately to the zeta potential driving force. Note that, in this section, we focus on electroosmotic flow and ignore contributions to the flow from capillarity. In this paper, we use a model first established by Mazur and Overbeek for porous diaphragm structures.9 Vallano and Remco have applied this model to packed beds of macroporous particles10 and Yao and Santiago to porous glass frits.11 Figure 1 shows a sketch of the modeled system. The porous substrate can be idealized as an arrangement of a large number of parallel single microchannels (pores). Every pore in the model has the same average pore radius rp and the same zeta potential. Moreover, the pores have an average tortuosity χ accounting for the nonstraight shape of the pores. The tortuosity (5) Hartford, S.; Flygare, W. Macromolecules 1975, 8, 80–84. (6) Miller, J.; Yalamanchili, M. Langmuir 1992, 8, 1464–1469. (7) Lin, C.-H.; Chaudhury, M. Langmuir, 2008, 24, 14276–14281. (8) Barz, D.; Vogel, M.; Steen, P. 2008, in preparation. (9) Mazur, P.; Overbeck, J. Rec. TraV. Chim. 1951, 70, 83–91. (10) Vallano, P.; Remcho, V. Anal. Chem. 2000, 72, 4255–4265. (11) Yao, S.; Santiago, J. J. Colloid Interface Sci. 2003, 268, 133–142.

Figure 1. Schematic of the porous substrate model.

is defined as the ratio of the average tortuous length of the pore to the thickness of the substrate χ ≡ lp/lf. Subscript f will stand for “frit” throughout this paper. The flow rate through the substrate is taken as the sum of the flow rates through the single pores, i.e.

V˙f ) NpV˙eof

(1)

The number of pores Np can be expressed in terms of the experimentally measured porosity ψ of the substrate according to

Np )

()

ψ rf χ rp

2

(2)

Rice and Whitehead12 related the flow within a cylindrical tube to the zeta potential, a result which we briefly review. They found an analytical solution for the potential distribution φi within a capillary by solving the Poisson-Boltzmann equation. For smaller values of the zeta potential, the so-called Debye-Hu¨ckel approximation (i.e., a linearization of the source term) can be applied, and the potential distribution of symmetric electrolytes (i.e., the absolute valencies |zi| are equal) becomes

φi(r) ) ζ

I0(r ⁄ lD) . I0(Π)

(3)

Here, r is the coordinate of the cylindrical cross section; I0 is the zero-order modified Bessel function of the first kind; and Π ) rp/lD is the ratio of the pore (capillary) radius to the Debye length. The Debye length also results from Debye-Hu¨ckel approximation and is related to the properties of the liquid only

lD )



jT εR 2F2IcΘ

(4)

j is the universal gas constant; T is Here, ε is the permittivity; R the temperature; F is the Faraday constant; and cΘ is the standard molar concentration. The dimensionless ionic strength I ) 1/2Σizici/cΘ reflects the influence of ionic species i of concentration ci and charge zi within the liquid. If no pressure gradient is applied, the electroosmotic flow rate within the pore corresponds to

[

2I1(Π) εζ V˙eof ) A0 ∇ φ 1 µ ΠI0(Π)

]

(5)

where A0 ) πrp2 is the pore cross-sectional area; µ is the dynamic viscosity; ∇φ is the local electric potential gradient; and I1 is the first-order modified Bessel function of the first kind. The zeta potentials of silica in contact with aqueous electrolytes are usually on the order of ζ ∼ -100 mV, which is significantly higher than appropriate for the Debye-Hu¨ckel linearization approximation. (For example, the Debye-Hu¨ckel limit for a monovalent j T/F electrolyte at standard temperature is approximately |ζ| ) R (12) Rice, C.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017–4024.

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≈ 25 mV.) However, comparison of the potential distribution within a pore with zeta potential of ζ ) -100 mV based on the Debye-Hu¨ckel approximation versus numerical solutions of the Poisson-Boltzmann equation still shows good agreement as long as Π > 100.11 Hence, the experiments we report in this paper use materials selected such that Π > 100, in which case eq 5 gives us the flow rate with sufficient accuracy. Note that the value of the bracketed term in eq 5 is then close to unity (between 0.98 and 0.99 for these experiments). Even so, it is retained in our calculations. Finally, the net flow through the porous frit can be related to the pore-scale electroosmotic flow by substituting eqs 2 and 5 into eq 1 to obtain

[

2I1(Π) ψ εζ ∆φf V˙f ≈ πr2f 1χ µ lf ΠI0(Π)

]

(6)

Here, the local electric potential gradient is replaced by an averaged gradient defined as ∆φf/(χlf). In summary, to obtain the zeta potential, one must measure the electroosmotic flow through the substrate as well as the geometric, physicochemical, and electrical parameters in eq 6 to back out the value of the zeta potential.

III. Experimental Methods and Materials We now illustrate the general strategy of pumping against a resistance due to the force of surface tension (capillarity) with a particular configuration where the liquid can be pumped back and forth between two droplets which act like liquid containers. Advantages of the general strategy of countering electroosmosis with capillarity are outlined in the next subsection after which the physics of the two-droplet configuration is described in more detail. The experimental setup and methods are explained thereafter. Finally, the substrate and the electrolytes used in the present experiments are discussed. A. Countering Electroosmosis with Capillarity. Small droplets have large pressures owing to surface tension γ. It has been shown that such capillary pressure can be counterbalanced effectively by electroosmotically generated pressure originating in a porous substrate.13 In particular, given any capillary surface of dimension R0, one needs to choose a pore size rp sufficiently small and/or a voltage drop ∆φf sufficiently large to make the dimensionless parameter (ε|ζ|∆φfR0)/(γr2p) greater than unity. Furthermore, as droplet size diminishes, the voltage required to pump electroosmotically scales as ∆φf ∝ rp2/R0. Accordingly, the voltage needed to pump against smaller higher-pressure droplets can actually decrease provided the pump pore size scales down with droplet size appropriately. Of course, deflection of any interface could be observed, in principle, to measure the electroosmotically generated pressure. But perhaps the simplest configuration to characterize is that of two connected droplets. Vogel et al.13 used such a configuration with a porous frit placed in between to demonstrate the above predictions in a practical context (similar to the present setup, shown below in Figure 3). The two droplets consist of the same liquid with different volumes V1 and V2. Each droplet is pinned on a circular contact line and has an internal pressure pi proportional to its curvature, according to the Young-Laplace law. If the droplets are smaller than about a millimeter (capillary length scale), gravity has a negligible influence, and droplet shapes are spherical caps, to good approximation. Equilibrium between the two droplets occurs when the interdroplet pressure difference ∆pi vanishes. The equilibrium states of the droplets can be described in terms of the volume difference V1 - V2 and the total volume V1 + V2. Figure 2A shows the bifurcation diagram of the bistable system. Below a total volume of one, a single stable state is present where the droplets are identical. At a total volume (13) Vogel, M.; Ehrhard, P.; Steen, P. Proc. Natl. Acad. Sci. 2005, 102, 11974– 11979.

Figure 2. (A) Bifurcation diagram of a bistable system of two connected droplets according to ref 13. The droplet volumes are nondimensionalized with the volume of a sphere of radius equal to the nozzle radius. (B) Normalized energy landscape (droplet surface areas) shows the energy barrier to switching at the unstable equilibrium II.

Figure 3. Schematic of the experimental setup, including the electroosmotic droplet switch (EODS).

of one, there is a bifurcation to three branches. The two stable symmetric equilibria (I, III) consist of complementary pieces of the same sphere, depicted by photos in the upper and lower insets. Located between the stable branches is an unstable equilibrium (II) which forms an energy barrier to overcome for switching between the stable equilibria. The barrier between the equilibria is illustrated in Figure 2B by means of the energy landscape. The height of the energy barrier is controlled by the total volume of the droplets and shrinks to zero as volume one is approached. Hence, the energy barrier can be readily tuned to the typical strength of the pump. Electroosmotic pumping can overcome the energy barrier and flip the droplets between their bistable equilibrium states much like

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Langmuir, Vol. 25, No. 3, 2009 1845

Figure 4. (A) Equivalent circuit diagram of the EODS. (B) Schematic of voltage and current vs time measurements to determine the overall ohmic resistance of the EODS.

a mechanical toggle switch can be flipped back and forth by the human hand. This mechanism is referred to as the electroosmotic droplet switch (EODS).13 The direction of pumping changes with the polarity of the electric potential difference. Switching time scales as R30/(r2p∆φf). With millimeter-sized water droplets and micrometersized pores, 5 V can yield switching times of 1 s. B. Experimental Setup and Methods. Figure 3 shows a schematic of the EODS device configuration and the experimental setup used in this study. The box is constructed of a number of polyoxymethylene (POM) plates to form two chambers with a dividing middle plate. The chambers are connected by the porous substrate which is positioned in the middle plate. Different diskshaped substrates can be readily investigated by changing the middle plate only. For our study, we use a glass frit as the porous substrate. At the outer side of each chamber, graphite electrodes are attached. All plates and the electrodes are sealed with o-rings. There are two ports connected to each chamber. One port leads to a nozzle which acts as the pinning line for the droplet. The other port is connected to a syringe (not shown), used to fill the chamber with liquid and to adjust the droplet volume. Power for the EODS comes from a DC voltage supply (HEATH ZENITH, model SP-2717) which generates a potential difference between the electrodes. A polarity switch controls the direction of flow. We further include a defined electrical resistance within the circuit allowing for the time-dependent measurement of the electrical current. The data are acquired by a PC with a data acquisition board (National Instruments, NI PCI6221). A high-speed CCD camera (Redlake, MotionPro X3) is used to simultaneously image the volumes of the two droplets. The porosity of the frit is measured separately by weighing the frit in a dry state and in a wet state by using a precise scale (AND, HR-200)

ψ≡

Vvoid mwet - mdry ) Vtotal F πr2l H2O

(7)

f f

The tortuosity of the frit and the potential drop over the frit (cf. eq 6) can be estimated using the regular experimental setup to measure the electrical resistance of the substrate. Hence, it is useful to first discuss the equivalent circuit diagram of the EODS, shown in Figure 4A. The EODS can be modeled as a system of two electrodes and several electrolytic resistances. A well-accepted model for electrodes in electrochemical systems is an ohmic resistance and capacitance in parallel, as shown in Figure 4A. Here, the capacitance Celectrode corresponds to the electrical (electrochemical) double layer, and the ohmic resistance Relectrode corresponds to the charge transfer resistance of the electrode.14 The electrolyte present in the left and right chambers can be considered as ohmic resistances (Rel, l and Rel, r, (14) Schmidt, V. Electrochemical Process Engineering; Wiley-VCH: Weinheim, 2005.

respectively). We perform resistance measurements with and without the frit in place. In either case, the resistance in the middle of the equivalent circuit diagram is of ohmic nature. When the saturated frit is in place, the resistance is Rf (as shown in Figure 4A), or in the absence of the frit, the resistance is that of the pure electrolyte Rel, m. This latter value can be inferred from a measured value of the specific conductivity of the electrolyte σel. In this work, the electrolyte conductivity is measured by a conductivity meter (VWR Scientific Products, model 2052). One characteristic difference between ohmic and capacitive electrical elements is their response time when an input signal is changed. The transient for capacitive charging is identified with the polarization time of an electrode. A step change in voltage of amplitude ∆φapp across the circuit shown in Figure 4A yields the output current shown in Figure 4B. The instantaneous contribution to the current Iin (Figure 4B) is related to the overall ohmic resistance of the EODS, Rohm ≡ Relectrode, l + Rel, l + Rf +... according to Iin ) ∆φapp/Rohm. The desired parameter Rf is obtained by subtraction of the resulting ohmic resistances (with frit versus without frit) according to Rf ) ∆Rohm + Rel, m ) ∆Rohm + lf/(πrf2σel). To obtain the tortuosity, the resistance of the frit is assumed to be due solely to the electrolyte present in the void volume. The surface conductivity of the substrate is neglected, which is justified as long as electrolytes are used. Hence, corresponding to the model established for the electroosmotic flow, the resistance of the entire frit is the sum of the resistances of all individual pores so that Rf ) Npχ2lf/(σelπrp2). In terms of measured quantities, the tortuosity can be estimated by

(

σelπr2f ∆Rohm lf

∆φf ) ∆φapp

Rf ≡ ∆φappΓ Rohm

χ)

ψ 1+

)

(8)

The measurement of the ohmic resistances with and without the frit also allows for an estimate of the potential drop across the frit. This potential gradient must be known to determine the zeta potential, and we choose to indirectly estimate its value based on the resistance measurements. Considering the equivalent circuit diagram, a balance based on Ohm’s law defines the voltage-drop fraction of the total potential applied at the electrodes, Γ, and results in

(9)

This expression yields the value of ∆φapp to be used in eq 6. With respect to the configuration of the EODS, one notes that a constant electroosmotic flow cannot be expected. Preliminary experiments are performed to estimate the polarization at the electrodes. We switch on the potential and track the decay of the current which is generated within the EODS. The current shows an exponential decay. A typical time scale of polarization is defined by constructing two tangents to the electrical current curve. One tangent is constructed where the voltage is applied and the other with respect to the asymptotic final value of the current. The intersection of both tangents gives us the time scale of the polarization. In the experiments, the electroosmotic flow within the EODS is indirectly measured by tracking the volume change of the two coupled droplets via an in-house Matlab code that analyzes the captured images. C. Materials. The porous substrate investigated is a borosilicate fritted disk obtained from R&H Filter Company, Georgetown, Delaware (RH1000), and commonly used as a filter. Borosilicate glass composition is typically about 73% silica SiO2, 10% boron oxide B2O3, 8% sodium oxide Na2O, 8% potassium oxide K2O, and 1% calcium oxide CaO. The thickness and the radius of all frits used in this study are lf ) 4.9 mm and rf ) 25.2 mm. The average pore radius rp ) 1.9 µm is determined by porosity measurements conducted by Porous Materials, Inc., Ithaca, NY. This value is consistent with the nominal value reported by the manufacturer and with in-house characterization techniques.13 The frits are subjected to a chemical treatment to clean the surfaces and ensure consistent initial conditions

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for every frit. First, the frits are immersed in isopropanol C3H8O for 10 min to remove soluble organic contaminants. Second, the frits are treated in a solution of sulfuric acid H2SO4 (98%vol) and hydrogen peroxide H2O2 (30%vol), mixed in a volume ratio of 4:1, for 10 min at a temperature of 100-110 oC in order to oxidize the nonsoluble organic contaminants. Third, to remove anorganic contaminants, the frits are treated in a hydrochloric acid HCl (36%vol), hydrogen peroxide (30%vol), and deionized (DI) water mixture, mixed in a volume ratio of 1:1:6, for 10 min at 75 oC. Between every treatment step, the frits are extensively rinsed with DI water. After the treatment, the frits are stored in DI water until used in the EODS. DI water is also the solvent of the electrolytes used presently. The primary experiments to determine the zeta potential of the substrates use electrolytes with different pH values and ionic strengths, created by the addition of buffer mixtures. To adjust the pH value of the electrolytes, we use a citric acid-disodium hydrogen phosphate buffer. Sodium hydrogen phosphate Na2HPO4 (Sigma, for electrophoresis, >99%) and citric acid monohydrate C6H8O7 (Fluka, purris. p.a., >99.5%) were used. We prepared stock solutions of 0.2 M disodium hydrogen phosphate and 0.1 M citric acid. Mixing the stock solutions in certain ratios creates several buffers for a wide range of pH levels from about 3 to 7. After the preparation of the electrolytes, the pH value is measured by a pH meter (Hannah Instruments, HI 98130). The pH value of the electrolyte is controlled by adding small amounts of the buffers. Since the EODS is a batch-type system, the buffer capacity is consumed during the switching process. By preliminary experiments, we determined the needed amount of buffer to be typically 1-2 mL per liter of electrolyte. This small amount allows for a sufficient number of switching events to obtain reliable results and also to maintain a constant pH value. The addition of the buffer determines the minimum of the ionic strength. Thus, the knowledge of the equilibrium composition of every buffer mixture is required. Regarding the citric acid, the dissociation and association reactions

Figure 5. Polarization time of the EODS at different ionic strengths (pH ) 6.1).

purris. p.a., >99.5%) is added. Salt concentrations for the experiments reported here range from cNaCl ≈ 0.02 gm/L to 3 gm/L, corresponding to ionic strengths of I ≈ 2.5 × 10-4 to 2.5 × 10-2.

IV. Experimental Results and Discussion

Here, reaction 13 tends entirely toward the direction of the product side until the solubility limit is approached. The corresponding pK’s for the phosphate oxidation set are pKPH, I ) 12.3, pKPH, II ) 7.2, and pKPH, III ) 2.15 Knowing the reaction scheme (10-16) and the corresponding equilibrium constants allows the calculation of the composition for every electrolyte mixture by means of the law of mass action and element balances. The resulting equilibrium concentrations of the ionic species determine the lowest ionic strengths of the electrolytes for the respective pH value. To obtain an ionic strength above the minimum, sodium chloride NaCl (Fluka,

The experimental results are given in two subsections. First, we discuss the results from the preliminary experiments which characterize the frit and the EODS. Then, the main experiments to determine the zeta potentials are given and compared to a number of literature results to validate the technique. A. Frit and EODS Characterization. The frit and electrical parameters necessary to determine the electroosmotic flow are investigated with the methods described in section IIIB. Five frits of the same size are prepared, each to be used for a different pH value. Table 1 is a compilation of the results. The porosity and the tortuosity of all frits are in the range of ψ ) 0.28 to 0.39 and χ ) 1.16 to 1.31. Yao et al.16 determined a similar value of tortuosity χ ) 1.20 (based on our definition of tortuosity) for a comparable borosilicate substrate by fitting the substrate model to their experimental electroosmotic pump flow rates. Also, the fraction of the applied potential which drops over the frit is calculated from eq 9 based on these measurements, and we find that fraction to be in the range of Γ ) 0.48 to 0.56. We also perform characterization experiments (without frit) to investigate the polarization time of the EODS. The working liquid within the EODS is an electrolyte buffered at pH ) 6.1. We investigate the influence of different ionic strengths (I ) 0.001 to 0.05) on the polarization. Figure 5 shows the polarization time, obtained by the procedure described in section IIIB. We find polarization times greater than two seconds for ionic strengths of I ≈ 10-3. When we increase the ionic strength, however, the time scale decreases to less than one second for I ) 0.05. These rather short times verify that we cannot expect a constant electroosmotic flow during the switching process of the droplets. Consequently, the electrodes are short circuited after each measurement to depolarize the electrodes. B. Zeta Potential. Table 2 shows the buffer compositions, the buffer concentration of the electrolytes, the respective pH values, and the resulting minimum ionic strength I0 for the basic electrolytes used to determine zeta potential. For example, basic electrolyte 1 has pH ) 3. The pH value is adjusted by the addition of 2 mL of buffer per liter of 80%vol 0.1 M citric acid and 20%vol

(15) Pearse, A. Histochemistry, Theoretical and Applied: PreparatiVe and Optical Technology, 4th ed.; Churchill Livingstone: Edinburgh, 1980; Vol. 1.

(16) Yao, S.; Hertzog, D.; Zeng, S.; Mikkelesen, J., Jr.; Santiago, J. J. Colloid Interface Sci. 2003, 268, 143–153.

KCA,I

+ C6H8O7 + H2O S C6H7O7 + H3O

(10)

KCA,II

2+ C6H7O7 + H2O S C6H6O7 + H3O

(11)

KCA,III

3+ C6H6O27 + H2O S C6H5O7 + H3O

(12)

take place. The corresponding pK’s of the equilibrium constants are pKCA, I ) 3.15, pKCA, II ) 4.77, and pKCA, III ) 6.4.15 The reaction scheme for the dissociation and association of the disodium hydrogen phosphate is

Na2HPO4 + H2O S 2Na+ + HPO24

(13)

KPH,I

3+ HPO24 + H2O S PO4 + H3O

(14)

KPH,II

2+ H2PO4 + H2O S HPO4 + H3O

(15)

KPH,III

+ H3PO4 + H2O S H2PO4 + H3O

(16)

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Figure 6. Droplet volume against time for frit 5 and electrolyte 5 (pH ) 7.1, I ) 5 × 10-3). The inset illustrates the volume distribution and the curvatures of the droplets before and at the end of the switching event (∆φapp ) 12.5 V). Table 2. Buffer Mixtures and Resulting Minimum Ionic Strength I0 of the Basic Electrolytes, With Corresponding Ratio Π0 electrolyte

0.1 M C6H8O7

0.2 M Na2HPO4

cbuffer (mL/L)

pH

I0

Π0

1 2 3 4 5

0.80 0.60 0.45 0.30 0.03

0.20 0.40 0.55 0.70 0.97

2 1 1 1 1

3.0 4.0 5.2 6.1 7.1

7.1 × 10-4 2.9 × 10-4 3.1 × 10-4 4.1 × 10-4 5.8 × 10-4

165 105 109 125 149

0.2 M disodium hydrogen phosphate. The minimum ionic strength of electrolyte 1 is calculated according to the reaction schemes (10-16) to be I0 ) 7.1 × 10-4. We also list the ratio Π0, calculated using the minimum ionic strength I0. We find that Π0 > 100 for every basic electrolyte value, so that using the Debye-Hu¨ckel approximation to estimate the flow rate within a single pore is justified. For every basic electrolyte, the corresponding frit from Table 1 is used, and measurements with different ionic strengths are performed. We proceed from low to high ionic strengths to minimize the influence of the previous electrolyte. Moreover, the frit is excessively rinsed with DI water after a series of measurements at a constant ionic strength. The upper ionic strength which can be used within the EODS is certainly limited by the polarization time. Furthermore, when the ionic strength, and therefore the conductivity, is too high, the electrode processes are pronounced. This leads to electrolysis and unwanted gas production at the electrodes which may influence the droplet dynamics. Hence, we limit the upper ionic strengths for all electrolytes to I ) 0.025. Figure 6 shows the droplet volumes against time. Here, the EODS is assembled with frit 5 and electrolyte 5 (pH ) 7.1) with an ionic strength of I ) 5 × 10-3. Initially, there is an equilibrium droplet distribution characterized by a large droplet (V1 ≈ 45 mm3) and a small droplet (V2 ≈ 15 mm3). When a voltage difference of ∆φapp ) 12.5 V is applied at the electrodes, the small droplet starts to grow while the large droplet shrinks. After about 4 s, a new (quasi-) equilibrium state is achieved. However, the inset shows that the volume distribution of the droplets is not a mirror-image of the initial distribution since the electric potential is still applied and the system is not in mechanical equilibrium, as verified by the different curvatures of the two droplets. With respect to Figure 2B, if we are switching from state I to state III, then keeping the potential on causes the system to move past III up the energy landscape. Alternatively, one can think of the

Figure 7. Electroosmotic flow as a function of the applied voltage for frit 5 and electrolyte 5 (pH ) 7.1, I ) 5 × 10-3).

energy landscape of the system as being shifted by the applied voltage. If the voltage is turned off, the system will return to mechanical equilibrium with a volume distribution opposite the initial state. When we take a closer look at the volume curve of a single droplet, two different behaviors are found. For the first ∼1 s of the switching time, the volume change vs time follows a linear relationship. Subsequently, the volumes tend to their final values with a decreasing slope. This nonlinear behavior is on one hand related to the electrode polarization. The polarization of the electrode decreases the potential drop within the frit and therefore decreases the electroosmotic flow. On the other hand, the capillary pressure of the growing droplet acts against the electroosmotic flow and diminishes the flow between the droplets as well. Due to the difficulties in separating these transient effects, we use only the initial flow rate to determine zeta potential since we know that the flow is purely due to electroosmosis at that moment. Using the measured tangent, we find the electroosmotic flow, averaged by considering both droplets, to be V˙f ) 19.32 mm3/s. To get accurate and reliable results, a number of measurements for each ionic strength are performed. However, instead of repeating measurements at a constant voltage, the voltage is incrementally increased. Figure 7 shows a series of measurements conducted for frit 5 and electrolyte 5 with an ionic strength of I ) 5 × 10-3. A linear behavior for the electroosmotic flow rate is seen over an applied voltage range of ∆φapp ) 2.5 V to 20 V, which is consistent with eq 6 which predicts a linear relationship between electroosmotic flow and potential difference for constant geometric and physicochemical parameters. Furthermore, the axis intercept of the linear function gives us an estimate of the voltage losses at the electrodes. If the applied voltage is entirely consumed by the electrode reactions, no electroosmotic flow would occur. In the present case, the electrode losses are about 1 V. However, electrode reactions and therefore the losses depend strongly on the electrode material in combination with several characteristics of the electrolyte. On the basis of the measured electroosmotic flow, the zeta potential for different ionic strengths is calculated according to eq 6. The influence of the ionic strength on the zeta potential, given constant pH value, temperature, and permittivity, gives rise to at least two effects:17 (i) The counter-charged ions either adsorb at the surface or at the Stern layer and thereby change the surface potential φS. Predictive models may result in zeta (17) Kirby, B.; Hasselbrink, E., Jr. Electrophoresis 2004, 25, 187–202.

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Figure 8. Zeta potential of frit 5 and electrolyte 5 at different ionic strengths. Additionally, values from comparable experiments from the literature are plotted.

potentials varying linearly with the logarithm of the ionic strength ζ ∝ a log(I) + b (cf., e.g., ref 18). (ii) If there is no adsorption of the counter-ions, neither at the surface nor at the Stern layer, the surface potential remains constant. The ionic strength affects only the thickness of the diffusive electrical double layer, changing the zeta potential directly. Two relations can be found, depending on whether the Debye-Hu¨ckel approximation can be applied or j T/F, the zeta not. For a symmetric electrolyte with |ζ| , |z|R potential scales with the Debye length ζ ∝ lD ∝ 1/I. For |ζ| . j T/F, the Debye-Hu¨ckel approximation should not be applied, |z|R and ζ ∝ log(lD) ∝ a log(I) + b is derived. Specific adsorption at silica surfaces can be expected by evaluation of retention times in ion exchange cromatography. Monovalent cations like potassium, sodium, and lithium ions tend not to adsorb.19 Figure 8 shows the zeta potential of frit 5 in contact with electrolyte 5 with pH ) 7.1 and ionic strengths of I ≈ 5 × 10-4 to 2.5 × 10-2. When we plot the ionic strength on a logarithm scale, a good agreement with the linear assumption is found. The zeta potential linearly drops from a value of ζ ≈ -100 mV at I ≈ 5 × 10-4 to ζ ≈ -44 mV at I ≈ 2.5 × 10-2. A linear regression gives ζ/mV ≈ 34.6 log(I) + 12.7. To validate the method described in this article, we plot results from comparable experiments found in the literature. Caslavska and Thormann measured the electroosmotic flow in fused-silica capillaries.20 They used a 100 mM N-(2-acetamido)-2-aminoethanesulfonic acid (ACES)/90 mM sodium hydroxide (NaOH) buffer with pH ) 7.85 and a basic ionic strength of I0 ) 90 × 10-3. The ionic strength is varied by dilution of the buffer. A linear regression of the results leads to ζ/mV ≈ 32.9 log(I) + 3.5. Another comparable data set was generated by Scales et al., who investigated the streaming potential at a flat fused-silica plate for pH ) 7 and different ionic strengths of a potassium chloride (KCl) solution.18 These authors found a deviation from linear behavior at an ionic strength of I ) 10-4. However, when we only consider the data above this ionic strength, the fitting results in ζ/mV ≈ 34.5 log(I) + 8.7. To summarize, there is good agreement between the literature data and the results measured using the EODS. Minor discrepancies can be attributed to the difference in pH values, the different electrolytes (counter-ions) used to adjust the ionic strength, and fundamental differences in the experimental techniques. Moreover, the substrates of the (18) Scales, P.; Grieser, F.; Healy, T. Langmuir 1992, 8, 965–974. (19) Melanson, J.; Baryla, N.; Lucy, C. TrAC 2001, 20, 356–374. (20) Caslavska, J.; Thormann, W. J. Microcolumn Sep. 2001, 13, 69–83.

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Figure 9. Zeta potential of borosilicate glass in contact with aqueous electrolytes at different pH values and ionic strengths. Table 3. Results of the Linear Regression of Data in Figure 9 electrolyte

a (mV)

b (mV)

1 2 3 4 5

12.8 20.0 27.9 30.6 34.6

6.0 7.6 6.9 6.5 12.7

compared experiments, fused-silica and borosilicate glass, are not identical. However, according to Kirby and Hasselbrinck Jr., the zeta potential does not vary significantly between silicate types.17 Considering all these differences, we find fairly good agreement. Figure 9 extends the above results to the zeta potential of borosilicate glass as a function of ionic strength for different aqueous electrolytes. For every set with a constant pH value, the zeta potential scales with the logarithm of the ionic strength. Also plotted in Figure 9 are linear regressions with respect to the logarithm of the ionic strength. It can be seen that, for a constant pH value, the magnitude of the zeta potential decreases with increasing ionic strength. At a constant ionic strength, the magnitude of the zeta potential decreases with increasing pH values, with the differences between the lower pH values being more pronounced than for the higher pH values. Table 3 shows the results of the regression according to ζ(pH ) const) ) a log(I) + b. We find an increasing gradient a with increasing pH values. Moreover, the intercept b for all electrolytes is in the same range of about 6-7 mV except for electrolyte 5 (pH ) 7.1) which is twice as large. To obtain a better insight into the relationship between zeta potential and pH values, the regression results listed in Table 3 are used to obtain interpolated zeta potential vs pH values at given ionic strengths as shown in Figure 10. We find a number of curves with the same basic trend. Between the pH values of 3 and 5, there is a linear drop of the zeta potential. When the pH value goes above 5, the slope decreases and a plateau at pH ≈ 7 is seen. The zeta potential approaches a limiting value. The curves are in good qualitative agreement with the results of Scales et al.18 who use potassium chloride electrolytes in contact with a fused-silica surface. They relate the occurrence of the plateau near a pH value of 8 to the entire deprotonation of the silanol surface sites so that a maximum of negatively charged sites is reached. Since borosilicate glass contains a considerable fraction of materials other than silica and should therefore have fewer silanol sites, a shift to lower pH values seems reasonable.

Zeta Potential of Porous Substrates

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of the ionic strength leads to a collapse of the data onto a single curve, to a good approximation. A second-order regression results in ζ(pH)/(-log(I)) ) (1.015 pH2 - 14.95 pH + 25.84) mV, which can be used to give good accuracy for engineering design. When we extrapolate the curve we obtain an isoelectric point between 1.7 and 2, depending on whether we linearly or quadratically extrapolate. This value differs from other observations, though, which found the isoelectric point to be about pH ) 2.6-2.8 for fused-silica.17,18 However, Ermakova et al. investigated the isoelectric point of sodium borosilicate glass membranes for different background electrolytes and found isoelectric points even below pH ) 1 for 0.001-0.1 M sodium chloride background electrolytes.21 They draw the conclusion that, in comparison to pure silica, these shifts to more acidic regions are due to the presence of aluminum and boron oxide sites. Figure 10. Zeta potential of borosilicate glass in contact with aqueous electrolytes as a function of pH. Note that these values are interpolated from Figure 9.

Figure 11. Zeta potential normalized with the negative logarithm of the ionic strength as a function of pH.

To summarize, the qualitative characteristics of a borosilicate glass surface in contact with a sodium chloride electrolyte and of a fused-silica surface in contact with a potassium chloride electrolyte are comparable. This is also proposed by Kirby and Hasselbrinck Jr.17 based on an extensive review of relevant literature. They claim that there is only a minor difference in the zeta potentials of different glasses. The differences in valency and size of the counter-ion are the crucial features influencing the zeta potential. Sodium and potassium ions have a valency of one and are of comparable size. Kirby and Hasselbrinck Jr. further suggested scaling the zeta potential with the negative logarithm of the counter-ion concentration and plotting against the pH value. If the counterion adsorption at the glass surfaces does not play an important role, which is consistent with the empirical observation that the intercept b ≈ 0, all zeta potential vs pH values collapse to a single curve. Since we have also found rather small intercepts b (cf. Table 3), we now make the same comparison. However, the concept of ionic strength perfectly expresses the influence of the counter-ion concentration and is well-established in the literature. Therefore, we chose to scale the zeta potential with the negative logarithm of the ionic strength, as shown in Figure 11. However, we do not follow Kirby and Hasselbrinck Jr. in setting the intercept b to zero since this would influence considerably the position of the isoelectric point. Nevertheless, the normalization of the zeta potential with the negative logarithm

V. Concluding Remarks In this article, we determine the zeta potential of porous substrates using an electroosmotic flow counterbalanced by a capillary resistance force. The flow rate is measured by imaging the time course of the interface deflection. The conceptual approach is realized by an electroosmotic droplet switch apparatus. With a porous substrate and electrodes arranged between a pair of droplets, an induced electroosmotic flow enables switching between the two stable equilibrium states, and the droplet volume change yields the electroosmotic flow rate through the porous substrate. The zeta potential is then estimated by using a relatively simple model for the flow within the porous material. This fast, reliable, and accurate approach of imaging liquid/gas interface deflection can be realized by a simple and cost-effective setup. With this technique, zeta potentials are determined for a borosilicate substrate in contact with aqueous electrolytes for a range of pH values and ionic strengths. It turns out that for a constant pH value there is a linear relationship between zeta potential and the logarithm of the ionic strength: the higher the ionic strength, the lower the absolute zeta potential. Also, the zeta potential dependence on pH is linear at small values of pH, but a plateau behavior is found at larger values of pH. This behavior is consistent across all ionic strengths investigated. Furthermore, when we normalize the zeta potential with the negative logarithm of the ionic strength, an empirical second-order correlation arises, which will likely be useful for engineering approaches. The results of the experiments are compared to fused-silica data obtained from literature with good qualitative agreement, which validates our method. There are minor discrepancies, mainly with respect to the position of the plateau and the isoelectric point, which are related to the different compositions of borosilicate and fusedsilica glass. A noteworthy observation of the present results is the linear change of volume over time at the beginning of switching, as seen in Figure 6. The implication here is that a high-speed camera, which is the most costly part of our experimental setup, is not necessary, and a much simpler camera would suffice, thus making this technique even more attractive. Additionally, using a constant current source instead of a constant voltage source would be beneficial. Then, the voltage drop across the frit is not affected by electrode polarization, at least for a certain time of operation. (21) Ermakova, L.; Sidorova, M.; Bogdanova, N. Colloid J. 2006, 68, 411– 416.

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Another variant, which could be realized in the EODS apparatus, measures the potential difference (and/or current) at zero net-volume-transport and thereby infers the zeta potential. Indeed, with regard to Figure 6, there is no volume transport between droplets after about 4.5 s. This represents a balance between the opposing mechanical pressure due to surface tension and the applied electroosmotic pressure. The voltage at which this electromechanical equilibrium is obtained depends also on the electrode polarization and the liquid/gas surface tension. Hence, in contrast to the variant employed in this paper (and in ref 7), additional measurements would be required. In

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some situations, this disadvantage might be outweighed by the advantage that different electromechanical states would be probed. Acknowledgment. The authors would like to thank David A. Anderson for his help and useful suggestions and Darpa for support. D.P.J.B.’s stay at Cornell University was generously supported by a scholarship of the Nano- and Microsystems program of the Helmholtz Association of German Research Centres. LA802949Z