pubs.acs.org/Langmuir © 2009 American Chemical Society
Determination of the Zeta Potential of Porous Substrates by Droplet Deflection: II. Generation of Electrokinetic Flow in a Nonpolar Liquid Dominik P. J. Barz* Karlsruhe Institute of Technology, 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, Olin Hall 120, Ithaca, New York 14853 Received August 18, 2009. Revised Manuscript Received November 3, 2009 Here we study the nature and extent of free electrical charges in nonpolar liquids, using a recently introduced technique of observing droplet deflection generated by electrokinetic flow in a porous substrate. In the presence of dispersed water, surfactant molecules agglomerate and inverted micelles are generated which may act as charge carriers. In the present work, the conductivities of solutions of a nonpolar liquid with several concentrations of a dissolved surfactant are measured by electrical transients. The induced current densities are proportional to the applied voltage, indicating that the solutions represent an ohmic system. The conductivity does not scale simply with the surfactant concentration, though. It is inferred that different micellization mechanisms exist depending on the surfactant concentration, and a model is sketched. Further experiments reveal that flows of such solutions can be generated within saturated porous substrates when they are subjected to moderate electric fields. An investigation of the phenomena leads to the conclusion that these flows exist due to the presence of an electrical double layer; that is, they are of electrokinetic (electroosmotic) origin. Hence, the measured electrokinetic flow rates can be related to the zeta potential of the porous substrate saturated with the solution. Plotting the zeta potential against the logarithm of the ionic strength reveals a linear relationship.
I. Introduction The presence of electrical charges in nonpolar liquids is not well-understood. Several techniques, including electron injection from a tunnel cathode1 or from an electron beam2 and the photoinjection of electrons,3 enable the generation of unipolar charges. The voltages which are necessary to inject the charges are very high, and there is no linear relationship between applied voltage and induced current. Bipolar charges can exist in nonpolar liquids as well; the origin of the charges is based on the presence or the dissociation of (ionic) “impurities”. While the dissociations of free electrical charges in polar liquids are wellunderstood, the dissociation of electrically charged species in nonpolar liquids still raises fundamental questions.4 One main difference is the different impact of electrostatic forces. The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory describes the interaction of two dispersed charges.5,6 Here, the Bjerrum length gives the characteristic separation between two charges at which the electrostatic interaction is comparable in magnitude to the thermal energy scale. The much larger Bjerrum length in nonpolar liquids has important consequences for free charges.7 First, the concentration of such charges is extremely
small because the solvation energy of an ionic species scales exponentially with the negative ratio of Bjerrum length to ion radius. This practical absence of charged carriers in nonpolar liquids results in almost no screening of the electrostatic interactions which are, therefore, extremely long-ranged. Second, much higher surface potentials are generated in nonpolar liquids in comparison to polar liquids, even though the surface charges are considerably fewer. The presence of considerable free charge is also the basis for electrokinetic phenomena which are related to the existence of an electrical double layer. In polar liquids, numerous varieties of electrokinetic phenomena are known; good overviews are given by Lyklema8 and Delgado et al.9 In colloid science, it is wellknown that particle suspensions in nonpolar liquids can be stabilized by manipulation of surface charges.7,10 For this purpose, surfactants are added which enhance the zeta potential of the particles in order to prevent their agglomeration. In such nonpolar suspensions, the electrokinetic phenomena of electrophoresis, essential for advanced electronic display technology,11-13 can be induced. Research on electrophoresis in nonpolar liquids was performed by Kornbrekke et al. and by Morrison et al. who measured the electrophoretic mobility of
*To whom correspondence should be addressed. E-mail: dominik.barz@ kit.edu. (1) Silver, M.; Onn, D.; Smejtek, P. J. Appl. Phys. 1969, 40, 2222–2227. (2) Watson, P.; Schneider, J.; Till, H. Phys. Fluids 1970, 13, 1955–1962. (3) Brignell, J.; Buttle, A. J. Phys. D: Appl. Phys. 1971, 4, 1560–1566. (4) Morrison, I. Colloids Surf., A 1993, 71, 1–37. (5) Verwey, E.; Overbeek, J. Theory of the Stability of Lyophobic Colloids: The interactions of sol particle having an electrical double layer; Elsevier: Amsterdam, 1948. (6) Derjaguin, B.; Landau, L. Acta Physicochim. URSS 1941, 14, 194–224. (7) Roberts, G.; Sanchez, R.; Kemp, R.; Wood, T.; Bartlett, P. Langmuir 2008, 24, 6530–6541.
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(8) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: New York, 1995. (9) Delgado, A.; Gonzalez-Caballero, F.; Hunter, R.; Koopal, L.; Lyklema, J. J. Colloid Interface Sci. 2007, 309, 194–224. (10) Hsu, M.; Dufresne, E.; Weitz, D. Langmuir 2005, 21, 4881–4887. (11) Evans, P.; Lee, H.; Maltz, M.; Dailey, J. Color display device. U.S. Patent 3612758, 1971. (12) Novotny, V.; Hopper, M. J. Electrochem. Soc. 1979, 126, 2211–2215. (13) Hopper, M.; Novotny, V. IEEE Trans. Electron Devices 1979, 26, 1148– 1152.
Published on Web 11/23/2009
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carbon black particles in dodecane. Clean carbon black particles are practically uncharged; the addition of the surfactant OLOA 1200 imparts a negative charge to the carbon.14,15 Other studies of the electrical characteristics of nonpolar suspensions reveal that the addition of a surfactant not only influences the charge (electrophoretic mobility) of the particles but also dominates the conductivity of the nonpolar liquid. Novotny and Hopper studied the conductivity of dielectric liquids (xylene, dodecane) when small amounts of an ionic surfactant aerosol-OT (AOT) were added.16 The addition of the surfactant leads to an enhanced conductivity. Current-voltage experiments exhibit a non-ohmic behavior with the steady state currents saturating at higher voltages. Moreover, some measurements indicate a high sensitivity to the presence of water. The important role of water for the free charge generation is confirmed in a subsequent work on ternary solutions containing surfactant and water dissolved in a nonpolar liquid.17 Given a constant surfactant concentration, the conductivity is observed to change by 3 orders of magnitude when only small amounts of water are added. In comparison to the pure liquid, the electrical conductivity of the ternary system was 5 orders of magnitude higher. Such ternary solutions are known to contain surfactant agglomerations, so-called inverted micelles with a size of about 4 nm, detectable by quasielastic lightscattering measurements. The generation of inverted micelles in AOT-dodecane systems is also confirmed by the work of others. Above a critical micellar concentration (cmc) of about 1 mM in dodecane, AOT forms nanometer-sized micelles containing about 80 surfactant molecules.18,19 It is of interest whether electrokinetic phenomena other than electrophoresis can occur in nonpolar liquids. The capability to generate an electrokinetic (electroosmotic) flow would be especially promising and would give opportunities for a variety of applications in microfluidics. In terms of electrophoresis, an electrical double layer is arranged around the mobile phase, that is, the particle. To generate electrokinetic flow, however, an electrical double layer present at a stationary solid is necessary. Flows in nonpolar liquids under the influence of an electric field can also be induced by several electrohydrodynamic mechanisms not affiliated with the existence of an electrical double layer. In this work, we investigate electrical and electrokinetic characteristics of a nonpolar liquid when different amounts of surfactant are added. The system selected for this work consists of dodecane and OLOA 11000. In the next section, the experimental methods and materials are described. Then experiments regarding the conductivity of the mixture are introduced, and a simple model regarding the micellization and charge generation is proposed. In the following section, the results of the electrokinetic experiments are presented and discussed. The micellization model is used to determine the electrokinetic parameters of the system. Finally, we summarize this Article with some concluding remarks.
II. Experimental Methods and Materials In this work, the electrical and electrokinetic characteristics of a nonpolar liquid containing different concentrations of a surfactant are investigated. Solutions of dodecane (anhydrous, >99%, Aldrich) and OLOA 11000 (Chevron Oronite) are prepared. Dodecane is an alkane hydrocarbon with the chemical formula CH3(CH2)10CH3; a thick, oily liquid of the paraffin series. The (14) (15) (16) (17) (18) (19)
Kornbrekke, R.; Morrison, I.; Oja, T. Langmuir 1992, 8, 1211–1217. Morrison, I.; Thomas, A.; Tarnawskyj, C. Langmuir 1991, 7, 2847–2852. Novotny, V.; Hopper, M. J. Electrochem. Soc. 1979, 126, 925–929. Novotny, V. J. Electrochem. Soc. 1986, 133, 1629–1636. Mukherjee, K.; Moulik, S.; Mukherjee, D. Langmuir 1993, 9, 1727–1730. Mathews, M.; Hirschhorn, E. J. Colloid Sci. 1953, 8, 86–96.
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Article density and viscosity of dodecane are FD = 0.75 kg/L and μD = 1.336 10-3 Pa 3 s, respectively. The viscosity of dodecane depends strongly on the surfactant concentration cS, though. Supporting experiments result in the correlation μðcS Þ 0:1517 2 0:0074 ¼ cS þ 1:336 10 -3 cS þ Pa 3 s mol=L mol2 =L2
ð1Þ
for the viscosity of the solutions. It has been mentioned that water plays an important role as a source of charge (micellization) in nonpolar liquid-surfactant solutions. The quantity of water needed to stabilize the micelles is extremely small. Some experimental evidence indicates that a single water molecule is sufficient to trigger the formation of a micelle.20 Dodecane of anhydrous quality is used in this work, since the water content is known to be cD,W e 0.05 g/L. The nonionic surfactant OLOA 11000 is a polyethyleneimine substituted succinimide derivative of polyisobutylene (PIBS). OLOA 11000 is comparable to OLOA 1200 which is often used to stabilize particles in low dielectric constant media and in certain electroimaging compositions.15 We assume that the cmc of OLOA 11000 is comparable to that of OLOA 1200 which is 0.005 wt %.21 The density and average molecular weight of OLOA 11000 are FS = 0.937 kg/L and MS = 950 g/mol, respectively.22 The OLOA macromolecule has three principle parts. There is a polar polyisobutylene head and a nonpolar amine group tail which are connected by the succinimide functional group. The polar head features an approximate length of 1 nm, and the nonpolar tail is about 4 nm. All experiments in this work are performed using the electroosmotic droplet switch (EODS) device which was conceived and realized by Vogel et al.23 In the first Article of this series of two, it is shown that the EODS device can be used to measure the zeta potential of porous substrates in contact with aqueous electrolytes.24 The theory and experimental setup of the method are introduced in that Article in detail; hence, we give only a brief overview here. The idea of the EODS is based on switching between two stable equilibrium states which result when two pinned droplets are coupled by their internal pressure. Inside each droplet, there is a pressure owing to surface tension (capillarity). Equilibrium occurs when the interdroplet pressure difference vanishes, that is, when both droplets have the same curvature. If the combined volume of both droplets is greater than a sphere of diameter equal to the contact line diameter, the two stable symmetric equilibria feature a large and a small droplet that are two complementary pieces of a sphere. The pressure necessary to flip the droplets between their bistable states can be induced by electroosmosis arising at the solid-liquid interface of a porous substrate located between the droplets. In practice, the EODS device is made out of two chambers with a dividing middle plate. Chambers are connected by a porous substrate which is mounted in the middle plate. At each outer side of the chambers, gilded copper electrodes are attached. The gilded copper electrodes induce the electric field which generates the electrokinetic flow in the substrate. Each chamber has a connected nozzle from which protrudes a droplet pinned along the circular nozzle lip. The EODS device is incorporated into an electrical circuit including a DC voltage supply. A precise resistor is added to the circuit to facilitate the time-dependent measurement of the electric current. The data acquisition is realized by using a PC featuring a PCI data acquisition board. A high-speed CCD camera is used to image the droplet deflections which occur when the electrokinetic flow is generated in the porous substrate. Before every measurement, the (20) Eicke, H.-F. Top. Curr. Chem. 1980, 87, 85–145. (21) Strubbe, F.; Verschueren, A.; Schlangen, L.; Beunis, F.; Neyts, K. J. Colloid Interface Sci. 2006, 300, 396–403. (22) Korosec, P. MidContinental Chemical Co., Olathe, KS. Personal communication, 2008. (23) Vogel, M.; Ehrhard, P.; Steen, P. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 11974–11979. (24) Barz, D.; Vogel, M.; Steen, P. Langmuir 2009, 25, 1842–1850.
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Article parts of the EODS are cleaned with isopropanol, followed by an extensive DI water rinsing, and then dried in an oven. The dodecane-OLOA 11000 solutions are prepared immediately before the measurements and directly filled in the assembled EODS to prevent absorption of humidity from the atmosphere. These procedures aim to prevent the contamination of the solutions with impurities and to keep the water content in the solution constant. We did not explicitly measure the water content in the solutions, although this should be done in future research. The influence of the surfactant concentration on the conductivity of the solution is investigated first. The EODS is used for these measurements as well. However, the nozzles are sealed and no middle plate is used. Hence, the electrode spacing is very small relative to the electrode areas so that the electrical system is onedimensional, to good approximation. There are several possible ways to investigate the electrical conductivity of liquids. An important technique is the measurement of the electrical transients. We use the “sweepout transient” approach, whereby the voltage difference is suddenly applied across the sample in equilibrium and then maintained. Such measurements can be performed with blocked electrodes in order to distinguish between the liquid bulk phenomena and the (electrochemical) transport processes at the electrode. Usually, the current then decreases to zero due to the polarization of the electrodes. In the present work, however, nonblocking electrodes are used. A decrease of the current to a quasi steady-state value is expected, whereby the biggest kinetic resistance, either within the bulk or at the electrode, determines the value of the steady-state current. For every dodecane-OLOA solution, we perform measurements with a step-by-step increase of the applied voltage. Between the measurements, the electrodes are short-circuited to mitigate polarization effects. In order to perform the electrokinetic measurements, a porous substrate is incorporated within the middle plate of the EODS. A disk-shaped borosilicate frit (RH1000) by R&H Filter Company, Georgetown, DE is used as the porous substrate. The frit has a radius of rf = 25.2 mm and a thickness of lf = 4.6 mm. The average pore radius is rp = 1.9 μm. Porosity and tortuosity are ψ = 0.23 and χ = 1.2, respectively. Between the measurements of different surfactant concentrations, the frit is chemically cleaned according to a procedure described in ref 24. Then the EODS is filled with the respective dodecane-OLOA solution. Two droplets featuring different volumes are adjusted at the nozzles, and an electrical potential difference is applied across the electrodes. The electrokinetic flow which is generated within the porous substrate is measured by the deflection of the droplets. Measurements with different OLOA-dodecane solutions and for different voltages across the electrodes are performed.
III. Conductivity Experiments The aim of the conductivity measurements is to obtain the ionic strength of the solutions, a measure for the overall charged species content. Typically, we apply average electrical fields ranging from 1 to 10 kV/m which are comparable with the electric field strengths used in the works of Kim et al.25 and Prieve et al.26 When the EODS is filled with pure dodecane, no current is detectable, as anticipated owing to the very low background conductivity of dodecane (less than 10-3 pS/m10). The situation is different when the surfactant OLOA 11000 is dissolved in the dodecane. Figure 1 shows the transient electric current densities which are induced in dodecane containing cS = 15.8 mmol/L OLOA 11000 by applying several voltages across the electrodes. For instance, applying a voltage of 25 V, an electric current (25) Kim, J.; Anderson, J.; Garoff, S.; Schlangen, L. Langmuir 2005, 21, 8620– 8629. (26) Prieve, D.; Hoggard, J.; Fu, R.; Sides, P.; Bethea, R. Langmuir 2008, 24, 1120–1132.
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Figure 1. Transient current density measured in cS = 15.8 mmol/L OLOA 11000 in dodecane for various applied voltages.
density of about 2.5 nA/cm2 is observed. When a higher voltage is applied, the generated current is also higher. Surprisingly, there is no hint of electrode polarization with a time resolution of some measurements down to 1 ms. For every voltage investigated, a steady-state current is observed during the measurement time of 20 s, to good approximation. However, at lower OLOA concentrations (cS j 10 mmol/L), higher fluctuations of the currents are observed. Polarization, the accumulation of charges near the electrode, is a problem in polar and nonpolar liquids alike. The accumulation of charges diminishes the electric field so that the generated electric current decreases. Oxidation and reduction reactions are the mechanisms by which the electrode injects charge into the liquid to neutralize the accumulated charge. In practice, electrodes in aqueous (polar) solutions can only polarize a volt or so before some species (often water) is oxidized or reduced. Nonaqueous media do not usually contain sufficient quantities of oxidizable or reducible species (particularly water) to permit charge injection. Therefore, the electrodes can polarize up to hundreds of volts if the electric field is turned on long enough.14 In the measurements performed for this Article, no noticeable polarization is observed. There are several possible explanations. (i) The time scale of the polarization is shorter than the time resolution of the experimental setup. Measurements of OLOA 371 in dodecane by Kim et al.25 revealed polarization times on the order of 100 ms. However, the measurements were performed with blocking electrodes. (ii) The time scales of the respective electrode reactions are much shorter than the time scales of the other transport processes. Hence, the charges in the vicinity of the electrodes are immediately consumed and electrode polarization cannot occur. Here, the question arises as to what the oxidizing/ reducing species is. (iii) The concentration of charges is too low. The charges indeed accumulate in the vicinity of the electrodes, but their influence on the electric field is very small. In Figure 2, the current density is plotted against the applied voltage for dodecane solutions containing OLOA in a range of cS = 0.8-59.2 mmol/L. For clarity of presentation, not every measured OLOA-dodecane solution is plotted. (In general, conductivity can only be measured by applying a potential difference. It is possible that the potential difference induces the formation of more charged species. Such a deviation from Ohm’s law was first analyzed by Onsager.27) We find a linear relationship between current density and applied voltage for all measured (27) Onsager, L. J. Chem. Phys. 1934, 2, 599–615.
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Figure 2. Current densities plotted versus applied voltage for dodecane solutions containing different OLOA concentrations. Solid lines indicate the linear correlations obtained by least-square regressions.
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surfactant concentration of about cS ≈ 20 mmol/L, the trend changes again and the conductivity increases with the surfactant concentration and finally approaches a plateaulike region (range III). Clearly, the physicochemical processes which drive the charge generation depend on the concentration of the surfactant. Hence, if one is to determine the ionic strength of the different dodecane-OLOA solutions, models which reflect the observed behavior are needed. A. Modeling of the Charge Generation Processes. In view of the large Bjerrum length in nonpolar liquids, charged species remain dissociated in low dielectric media only if they are large or contained in large structures. Inverted micelles are one such structure, and large polymers are another. Micelles are aggregations of surfactant molecules above the critical micelle concentration (cmc), whereas the core of an inverted micelle consists of a polar medium (e.g., water). The quantity of water needed to stabilize inverted micelles is undetectably small.20 It is also assumed that electrical charges are dissolved in the polar interior of inverse micelles enabling opposite charges to be held sufficiently far apart (i.e., larger than the Bjerrum length).4 For the concentration range I of Figure 3, there is a linear relationship between conductivity and surfactant concentration. Here, we propose two steps which are necessary to generate charged inverted micelles. First, OLOA molecules attach to the dispersed water, forming micelles of type A according to ns, A S T A
Figure 3. Measured (circles) and modeled (line) conductivity as a function of OLOA concentration. Ranges of behavior I, II, and III are discussed in the text.
solutions. Hence, the OLOA-dodecane solutions represent an ohmic system; the amount of electrical charge is independent of the applied potential difference. The linear correlation obtained by a least-square regression of the experimental data is also plotted in Figure 2. A direct proportionality between current and applied voltage was also observed by Kim et al.25 and Prieve et al.26 Both works used OLOA 371 and blocking electrodes, and, in this case, the linearity is related to the initial current. The slope of the correlations gives the conductivity of the solution which is plotted in Figure 3 as a function of the OLOA concentration. The error bars are based on the standard error of the regression. No error bar indicates that the standard error at most corresponds to the size of the symbol. (Note that the solid line is a result of computations using a model proposed and discussed in subsection IIIA.) Figure 3 reveals that the conductivity of the solutions does not scale simply with the OLOA concentration. Rather, we find different intervals where, evidently, different charge generation mechanisms occur. In the concentration range between zero and approximately cS ≈ 9 mmol/L (range I), there is a linear relationship between conductivity and surfactant concentration, to good approximation. Then the situation changes, and the conductivity decreases as the surfactant concentration further increases (range II). At a Langmuir 2010, 26(5), 3126–3133
ð2Þ
where S is a single surfactant molecule and ns,A is the number of OLOA molecules in a micelle A. There are indications in the literature that the number of surfactant molecules forming a micelle does not change over a certain range of concentrations.10,25 Hence, we assume the number of surfactant molecules ns,A forming a micelle A as constant. In the second step, some of the neutral micelles are getting charged. Here, two principle mechanisms are conceivable. (i) The micelle dissociates into two oppositely charged fractions accordcase, the law of mass action would give ing to A T Dþ þ D-. In this√ a conductivity scaling with cS, though. (ii) The neutral micelles exchange charges by a collision mechanism according to 2A T Aþ þ A -
ð3Þ
which results in a linear conductivity-concentration relationship. Therefore, we choose reaction 3 to be used in our proposed model. Within this concentration range, the micelles are assumed to be of spherical shape. The size of the aqueous core of the micelle corresponds to the length of OLOA’s polar group, that is, rA,W ∼ 1 nm. Hence, the hydrodynamic radius is about rA ∼ 5 nm, that is, the sum of the aqueous core and the nonpolar tail of an OLOA molecule (4 nm). This size is in accordance with other values given in the literature. Pugh et al. assumed the hydrodynamic radius of PIBS micelles to be about 5 nm based on the length of the polymer chain.28 Kim et al.25 cite data by Kornbrekke who used a light-scattering method to measure the equivalent radius of PIBS micelles and also found a value of about 5 nm. A straightforward mass calculation using density and molecular weight of OLOA 11000 reveals that a micelle A consists of nS,A ∼ 350 surfactant molecules. The Einstein-Smoluchowski relationship correlates the hydrodynamic radius to the electrophoretic mobility, that is λA ðcS Þ ¼
F fA ðcS ÞNAG
ð4Þ
(28) Pugh, R.; Matsunaga, T.; Fowkes, F. Colloids Surf. 1983, 7, 183–207.
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Here, F is the Faradaic constant, NAG is the Avogadro number, and fA(cS) = 6πμ(cS)rA is the drag of a sphere with a hydrodynamic radius rA. Finally, the conductivity of the solution can be determined according to σ A ðcS Þ ¼ FcA, i ðcS Þ λA ðcS Þ
ð5Þ
Here, cA,i(cS) is the concentration of ionized micelles. This can be obtained by fitting the model to the experiments. We determine an ionization degree of δ = 0.014. Our estimates are consistent with the literature25 where spherical micelles in OLOA 371-dodecane solutions are proposed to have a hydrodynamic radius of about 10 nm, a number of 800 surfactant molecules per micelle, and an ionization degree of 0.025. These authors consider a spheroidal micelle shape as also possible. In that case, a micelle consists of 1300 surfactant molecules and the ionization degree is determined to be 0.016. Within range II of Figure 3, located between ∼9 and ∼20 mmol/L, the conductivity of the solution decreases while surfactant is added. If we again assume that the ionization degree is independent of the surfactant concentration, that is, δ = 0.014, there are two possible mechanisms for the conductivity drop: (i) the number of charge carriers decreases, which would necessarily mean a diminishment of the dispersed water; (ii) the concentration of dispersed water remains constant, but the size of the micelles grows, which results in lower electrophoretic mobilities. The number of surfactant molecules per micelle would consequently depend on the surfactant concentration of the solution. We focus on the content of dispersed water in the dodecane to evaluate these two mechanisms. Using the overall water concentration cD,W and the radius of an aqueous micelle core rA,W, we calculate the maximum concentration of (monodispersed) water droplets as ∼20 μmol/L. The OLOA concentration which is necessary to convert all dispersed droplets into micelles of type A is cS ≈ 7 mmol/L, which corresponds reasonably to the end of concentration range I. This suggests that the dispersed water is entirely consumed by the micellization process at the beginning of concentration range II, and we therefore choose mechanism (ii). The amount of surfactant which is then added is incorporated into the micelles, called B, and increases the hydrodynamic radius. This is contrary to the assumption that the number of surfactant molecules of a micelle is independent of the surfactant concentration as proposed in refs 10 and 25, though. In their work, however, different surfactants were used and the water concentrations were much higher so that a regime of surfactant excess might not have been accessed. When we use our model to estimate the ionic strength of our solutions, the outcome compares favorably to the results of Prieve et al.,26 supporting our assumption. We propose that the micelle is getting elongated due to consumption of the excess surfactant molecules. That is, the geometric shape of the micelles changes from a sphere to a prolate spheroid. On one hand, this is consistent with the experimental data. On the other hand, the micelle elongation will support the transition to range III behavior. We assume that the additional surfactant molecules which attach only increases the (polar) radius. For the calculation of the electrophoretic mobility of a prolate spheroid, the drag of a perimeter-equivalent sphere can be used, f ðcS Þ ¼ 6πμðcS Þ KðcS Þ rB ðcS Þ
ð6Þ
Here, K(cS) is the shape factor and rB(cS) is the radius of a sphere whose perimeter is the same as the projected perimeter of the spheroid, whereas both parameters depend on the direction of motion. We use the arithmetic mean of these parameters in the direction of the polar and equatorial radius. Clift et al. give a 3130 DOI: 10.1021/la903075w
shape factor for the polar and equatorial radius direction as Kp(cS) = 0.244 þ 1.035S(cS) - 0.712S(cS)2 þ 0.441S(cS)3 and Ke(cS) = 0.392 þ 0.621S(cS) - 0.040S(cS)2.29 Here, the parameter S(cS) stands for the surface area of the spheroid divided by the surface area of a perimeter-equivalent sphere. We further propose that the charge generation (collision) mechanism and the ionization degree are essentially the same as in range I. Applying this model, we estimate that a micelle B consists of about 780 OLOA molecules at the end of concentration range II (cS = 20 mmol/L). Finally, in range III of Figure 3, distinguished by concentrations between ∼20 and 60 mmol/L, the conductivity increases again while surfactant is added. The relationship is apparently nonlinear, though. Conductivity may rise due to an increase of the ionized micelle concentration or due to higher electrophoretic mobilities while the ionized micelle concentration is constant. Higher micelle concentrations might occur with the addition of water; however, we tried to prevent this during the sample preparation. Another possibility is that the overall amount of water remains constant but a greater dispersion is approached. Since the micelles are considerably elongated at the end of range II, a breakup of the micelles B is proposed. Such a breakup may be related to a bistability between spherical and spheroidal shapes or to the input of mechanical energy (stirring) during the sample preparation. Hence, if we assume that in range III the micelles B break into two equal parts, consisting of 390 surfactant molecules, each broken micelle is able to consume 390 OLOA molecules again. The conversion of neutral into charged micelles according to a mechanism comparable to reaction 3 with an ionization degree of δ = 0.014 is assumed in range III as well. According to the proposed heuristic model, the conductivity as a function of the surfactant concentration can be computed. The results of the computations are plotted in Figure 3 as a line. We find a good agreement between experimental and modeled conductivities. In range I, the model has been fitted to the experimental data to obtain the ionization degree. However, this parameter in combination with the further simple assumptions listed above allows for a good description of the other concentration ranges. Alternative computations assuming a viscosity independent of the surfactant concentration have been performed as well. It turns out that the nonlinear behavior in range III is solely related to the strong dependence of solution’s viscosity on OLOA concentration. The model also predicts the concentration of ionized micelles and hence allowsPfor the estimation of the dimensionless ionic strength I = 1/2 i zici/cΘ of the different OLOA-dodecane solutions, where cΘ is the standard molar concentration. A plot of the dimensionless ionic strength over the surfactant concentration is given in Figure 4. In range I, the ionic strength linearly rises from 0 to about 1.7 10-7. In range II, this value remains constant over the surfactant concentration since the number of charges does not change. In range III, the ionic strength rises linearly again until a value of about 5 10-7 is approached. Note that the ionic strengths of the solutions are the same order of magnitude as those in deionized water. When we compare our model results to the experimental results of Prieve et al.,26 we find some supporting features. Prieve et al. investigated the nature of OLOA 371-heptane solutions. They used heptane of HPLC grade; its water content is at least 4 times higher than that of our dodecane, but this is still a very small amount of water. The ionic strength found by Prieve et al. is in the lower concentration regime similar to the ionic strength that we have estimated; for example, for a (29) Clift, R.; Grace, J.; Weber, M. Bubbles, drops, and particles; Academic Press: New York, 1978.
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Figure 4. Dimensionless ionic strength I as a function of the OLOA concentration in dodecane.
PIBS concentration of c = 8 mmol/L, the dimensionless strengths are 1.3 10-7 and 1.4 10-7, respectively. At a PIBS concentration of about c = 10 mmol/L, we have a plateaulike trend of the ionic strength where we concluded that there is no dispersed water left for further micellization. In Prieve’s results, there is also a strong plateaulike behavior of the ionic strength obvious but located at higher PIBS concentrations of about c = 40 mmol/L, which is likewise higher by a factor of 4 compared to our results. Moreover, Prieve et al. reported that the hydrodynamic micelle radii change over the PIBS concentration. In principle, they found a similar qualitative behavior as proposed in our model. In a first concentration range, the hydrodynamic radius is constant (which we assumed in range I). Then the hydrodynamic radius increases with increasing PIBS concentration (as proposed in our range II). Finally, there is a decrease of the hydrodynamic radius with increasing PIBS concentration (which we also assumed in range III).
IV. Electrokinetic Experiments There are different types of flow phenomena possible when a liquid is subjected to an electric field. One distinguishes between conductive (electrolytic) and nonconductive (dielectric) liquids. In electrolytic liquids, based on polar solvents, the induced flow phenomenon is called electroosmosis or electrokinetic flow. In contrast, electrohydrodynamics (EHD) deals with the interaction between electric fields and dielectric (nonpolar) liquids. The Korteweg-Helmholtz body force due to an electric field EB acting on an isotropic liquid summarizes these various effects, " # 2 Dε 1 2 1 FBel ¼ qEB - EB rε þ r FEB 2 2 DF T
ð7Þ
Here, q is the volume density of free electric charges, ε is the permittivity, T is the temperature, and F is the mass density of the liquid. All three different components of the body force can arise for fluid flow. The first term accounts for the Coulomb force which acts on free electric charges in the presence of an electric field. The second term refers to the dielectric force which is induced when a permittivity gradient is present. The third term describes the electrostriction force which may be generated in compressible media only. Electrokinetic flows are related to the Coulomb term. A local excess of free charges, either positive or negative, is required, something typically found in electrical double layer (EDL) configurations. The formation of the EDL Langmuir 2010, 26(5), 3126–3133
is based on the existence of surface charges at solid boundaries. These surface charges attract oppositely charged free charges in the vicinity of the solid-liquid interface and hence form a layer featuring a net charge density. Outside of the EDL, the liquid remains electrically neutral. When an electric field is applied, Coulomb forces act on the liquid within the EDL and induce the electrokinetic flow. In terms of electrohydrodynamics, a flow in an isothermal, incompressible, and nonpolar liquid can only be induced if either free space charges are present or if a permittivity gradient exists. To date, four EHD mechanisms to generate flow are known: ion injection pumping, conduction pumping, Maxwell pressure gradient pumping, and induction pumping. While the first two mechanisms rely only on the presence of free charges in the liquid, the remaining two depend on liquid polarization. A good overview of these mechanisms is available.30 Raghavan et al. also list important attributes in order to identify the different mechanisms:31 (i) The induced flow velocity of the ion injection and the conduction mechanism scales linearly with the electric field. (ii) The flow velocity of induction and Maxwell pressure gradient mechanisms scale quadratically with the electric field. (iii) Ion injection and conduction mechanisms have threshold values greater than ∼100 kV/m. (iv) No threshold values are observed for induction and Maxwell pressure gradient mechanisms. Moreover, to pump a nonpolar liquid, all EHD mechanisms require complex electrode geometries and arrangements. For instance, the ion injection mechanism requires a needlelike emitter electrode and a hollow collector electrode. The conduction pumping mechanism results only in a net flow when the electrode configuration is asymmetric. We perform measurements in the EODS device with different concentrations of surfactant dissolved in dodecane and several electrical potential differences applied at the electrodes. The applied voltages induce electric fields within the porous substrate with strengths of EB = 2-40 kV/m. When no porous substrate is installed within the EODS, flow is not detectable. Likewise, when pure dodecane fills the EODS, flow is not observed. In contrast, when the EODS is filled with OLOA-dodecane solutions, the application of an electric field generates a flow through the porous substrate. Switching the polarity of the applied voltage results in a flip in the flow direction. Figure 5 shows the results of the electrokinetic flow measurements of dodecane containing cS = 7.9 mmol/L OLOA for different applied voltages. Measurements at a constant applied voltage were repeated several times. The boxes plotted in Figure 5 indicate the average value of these measurements. The standard deviation among the measurements is given by error bars. We find relative deviations always to be less than ∼15%. The reason behind the standard deviation is not known. The plotted line is the result of a linear regression. Figure 5 clearly reveals a linear relationship between the flow rate through the frit and the applied voltage at the electrodes. In the present case, we are not able to measure any flow below an applied voltage of ∼50 V which corresponds to an electric field of E ≈ 5 kV/m. The upper limit of the applied voltage is determined by the power source. Figure 6 illustrates the normalized flow rate as a function of the OLOA concentration of the solution. The normalization is done by dividing the absolute flow rate by the averaged electric field, present within the porous substrate, and by the inner surface area of the porous substrate. The error bars indicate the standard error of the linear regression of the flow rate against the applied (30) Laser, D.; Santiago, J. J. Micromech. Microeng. 2004, 14, R35–R64. (31) Raghavan, R.; Quin, J.; Yeo, L.; Friend, J.; Takemura, K.; Yokota, S.; Edamura, K. Sens. Actuators B: Chem. 2009, 140, 287–294.
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pore-scale electrokinetic flow (cf. ref 24), that is ψ εζ Δjf 2I1 ðΠÞ 1V_ f ≈ πrf 2 χ μ lf ΠI0 ðΠÞ
Figure 5. Electrokinetic flow rate through the porous substrate as a function of the applied voltage. Solid line is obtained by a leastsquare regression. The OLOA concentration is cS = 7.9 mmol/L.
Figure 6. Normalized flow rate through the porous substrate for different surfactant concentrations in dodecane.
voltage data (cf. Figure 5). We find a behavior which is comparable to the conductivity measurements. At first, the normalized flow rate increases when the surfactant concentration increases too. When a concentration of about cS ∼ 10 mmol/L is approached, the situation changes and the flow rate decreases with increasing OLOA concentration. Here, some features of the flow phenomena that are found in our measurements should be discussed. On one hand, the electric field strengths necessary to induce the flow are unambiguously below the threshold values of ion injection pumping and conduction pumping. Moreover, there are no complex electrode configurations in the experiments reported in this Article. The results of the experiments reveal that the flow rate scales linearly with the applied voltage (Figure 5) so that induction pumping and Maxwell pressure gradient pumping can also be excluded as possible mechanisms. On the other hand, since the flows are only induced if the porous substrate is incorporated within the EODS, a surface phenomenon must be involved. The presence of the porous substrate enlarges the wetted surface area by 2 orders of magnitude. In conjunction with the linearity of flow with applied voltage, one concludes that electrohydrodynamics can be excluded as responsible for the physical phenomenon and that an electrokinetic phenomenon is the origin of the flow. As a consequence, we can use the correlation derived in part I which relates the net flow through a porous substrate with the 3132 DOI: 10.1021/la903075w
ð8Þ
Here, ζ is the zeta potential, Δjf is the average potential drop over the frit, and I0 and I1 are the zero-order and first-order modified Bessel function of the first kind, respectively. The parameter Π is the ratio of pore radius rp to Debye length lD = (εRT/2F2IcΘ)1/2. The dimensional ionic strength inferred above allows for the calculation of the Debye length. We estimate the range of the Debye length to be lD ≈ 67-174 nm and therefore we obtain Π ≈ 11-28. Hence, the size of the EDL is always smaller than the average pore radius, that is, we do not have an overlapping EDL. However, the electrokinetic experiments are conducted with “finite electrical double layers” (cf. ref 32). The thickness of the EDL is only 1 order less than that of the pore radius, and this finite EDL effect can be estimated using the Bessel functions term of eq 8. For the case of the thickest Debye length, the value of the Bessel functions term is about 0.175. This can be interpreted as the loss due to the finite EDL regime. In other words, if we operated in a regime with an infinitely thin EDL, the flow rate would be about 17.5% higher. In the case of the thinnest EDL present here, the value of the term is 0.07. At least two different relationships for zeta potential and ionic strength are known in literature, depending on whether countercharged ions adsorb onto the solid surface (cf., e.g., ref 33). If there is no adsorption of the counterions, the surface potential (charge) remains constant. The ionic strength affects only the thickness of the diffusive electrical double layer. In the case of a symmetric electrolyte with |ζ|,|z|RT/F, the zeta√potential directly scales then with the Debye length ζ µ lD µ 1/ I. In the case of counterion adsorption, a zeta potential scaling linearly with the logarithm of the ionic strength ζ µ a log(I) þ b is typically found (cf., e.g., ref 34). Since the addition of OLOA 1200 imparts a negative charge to previously uncharged carbon black,14,15 we assume in the present case that the surface charges are also (partly) related to the addition of the surfactant. Hence, the zeta potentials of borosilicate in contact with different solutions are plotted against log(I) in Figure 7. We find typical values for the zeta potentials on order of ζ ∼ -10 mV. It can be clearly seen that the higher the value of the ionic strength, the lower the absolute value of the zeta potential. This corresponds well with the theory, since a higher ionic strength results in a smaller thickness of the EDL and therefore in a higher shielding of the surface charge. The zeta potential that corresponds to the lowest ionic strength does not obey this finding, though. The absolute value is smaller than all other zeta potentials which are found for higher ionic strengths. It is possible that, at this low concentration, the amount of ionized micelles is just too small to establish a distinctive electrical double layer. Another explanation could be that solid surface charging (adsorption) still occurs. Hence, we may not expect that the correlation between electrokinetic flow and zeta potential entirely obeys the Debye-H€uckel theory. The zeta potentials estimated at surfactant concentrations of 7.9 and 15.8 mmol, however, provide another piece of evidence to support the electrokinetic flow theory. According to the proposed charge-generation mechanism, which is the outcome of the conductivity experiments, the ionic strengths for both concentrations are similar even though the surfactant concentrations differ (32) Yao, S.; Santiago, J. J. Colloid Interface Sci. 2003, 268, 133–142. (33) Kirby, B.; Hasselbrink, E., Jr. Electrophoresis 2004, 25, 187–202. (34) Scales, P.; Grieser, F.; Healy, T. Langmuir 1992, 8, 965–974.
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Figure 7. Zeta potentials of borosilicate in contact with OLOAdodecane solutions featuring different ionic strengths.
considerably. In fact, the electrokinetic experiments, entirely independent of the conductivity measurements, reveal only a minor difference between the flow rates and therefore, in either case, zeta potential is about ζ ≈ -15 mV. We also add a best fit linear trend of the experimental data in Figure 7 obtained by neglecting the value at the lowest ionic strength. We find the correlation ζ/mV = 10.5 log(I) þ 56.8. Kirby and Hasselbrink33 claim that the counterion adsorption does not play a significant role when the intercept is about zero, which is an empirical observation for simple (monovalent) ions on silica surfaces. We find an intercept which differs considerably from zero which opens the possibility of adsorption, as assumed here. However, it remains unclear if these findings33 can be transferred to systems consisting of charged inverted micelles in nonpolar solvents.
V. Concluding Remarks The focus of the present work is the investigation of solutions of a surfactant dissolved in a nonpolar liquid. In the presence of dispersed water, the surfactant molecules agglomerate and inverted micelles are formed which may act as charge carriers. The conductivities of different mixtures of OLOA 11000 in dodecane are measured by electrical transients. The induced current densities are proportional to the applied voltage; that is, the OLOA-dodecane solutions represent an ohmic system. The conductivity does not scale simply with the surfactant concentration. It is proposed that different charge generation mechanisms exist depending on the surfactant concentration, and a model which gives a consistent picture of the conductivity as a function of the surfactant concentration is put forward. We estimate that the addition of OLOA to the dodecane results in ionic strengths which are comparable to those of deionized water. Further experiments reveal that flows of OLOA-dodecane solutions can be generated within porous substrates when an electric field is applied. Comparison to characteristics of electrohydrodynamic flows leads to the conclusion that an electrokinetic (electroosmotic) flow is generated. So far, electrokinetic flows have only been known in polar electrolytic solutions, since a certain amount of charges is necessary to establish the required electrical double layer. Our measured electrokinetic flow rates are then related to the zeta potential of the porous substrate saturated with the OLOA-dodecane solution. Plotting the zeta potential against the logarithm of the ionic strength shows a linear correlation.
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A number of open questions remain. The charge generation mechanisms merit a more thorough investigation. Other (more complex) charge generation mechanisms may well be relevant. It is conceivable that there are competing charge generation mechanisms, for example, dissociation and collision, which operate concurrently to each other. Depending on the surfactant concentration or on the ratio of water to surfactant, one mechanism may prevail over the others. The assumption of a monodisperse water distribution is a further simplification. There are indications that size and shape of an inverted micelle depend on the amount of water which forms the micelle core.35-37 Furthermore, we do not know how the surface charges originate. Borosilicate in aqueous solution forms silanol groups whose deprotonation is the major contribution to the surface charge. So far, it is unexplored whether there is relevant surface chemistry in surfactant nonpolar solutions taking place. The surface charge may (partly) result from impurities or just from defects in the molecular surface structure. Adsorption of surfactant or of entire micelles is certainly an issue which creates surface charges and maybe influences the conductivity of the solutions too. The deviation from ζ µ log(I) in the case of the lowest OLOA concentration raises the question: what is the minimum amount of charged micelles necessary to establish an electrical double layer? Moreover, the electrical double layer theory is derived for infinitely small ions. In our system, in contrast, the size of an inverted micelle is 2 orders of magnitude larger than a regular monovalent ion (e.g., Naþ) and steric effects can therefore be expected. These questions should be the subject of further investigations. Micelle generation could be investigated in more detail with analytical methods such as, for example, light scattering which helps to determine size and shape of the micelles. In this regard, the preparation of monodisperse solutions with an exactly adjusted water content would be crucial. Further understanding of the charge in the micelle can possibly be made by investigation of the oxidation and reduction potential by means of electrochemical methods. In summary, this series of two papers introduces the droplet deflection technique for inferring zeta potentials of saturated porous substrates which are the site of electroosmosis. In the first paper (part I), electroosmosis arising from borosilicate glass in contact with an aqueous electrolyte, a much-studied system, was found to have zeta potentials consistent with the literature. In this second paper (part II), a borosilicate glass in contact with a doped nonpolar liquid, less well-studied, is found to exhibit electroosmotic behavior sensitively dependent on the concentration of surfactant doping. To the extent that the droplet deflection technique has delivered results that contribute to the field of study, its value as an alternative technique for the determination of electrokinetic parameters is self-evident. Acknowledgment. The authors would like to thank Wolfgang Wenzel (INT, Karlsruhe Institute of Technology) for his useful suggestions regarding the molecular structure of the surfactant. We thank Ravi Sharma (Bausch & Lomb) for introducing us to this problem (while at Kodak) and Chevron Oronite Company for kindly providing the OLOA 11000. An anonymous referee is thanked for kindly bringing ref 26 to our attention. The authors acknowledge support from DARPA and NSF CBET0653831 for this work. D.P.J.B.’s stay at Cornell University was generously supported by the Nano- and Microsystems program of the Helmholtz Association of German Research Centres. (35) Muller, N. J. Colloid Interface Sci. 1978, 63, 383–393. (36) Petit, C.; Lixon, P.; Pileni, M. Langmuir 1991, 7, 2620–2625. (37) Zhu, D.-M.; Feng, K.; Schelly, Z. J. Phys. Chem. 1992, 96, 2381–2385.
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