8048
Langmuir 2007, 23, 8048-8052
Electrical Charges in Nonaqueous Solutions I: Acetone-Water Mixtures as Model Polar Solvents Rong Fu and Dennis C. Prieve* Center for Complex Fluids Engineering and Department of Chemical Engineering, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213 ReceiVed January 22, 2007. In Final Form: April 24, 2007 To establish the mechanism of charging of solute entities in nonaqueous solvents, the number density of charges must be measured. Conductivity measurements alone are not always sufficient. Here we explore the use of total internal reflection microscopy (TIRM) to determine the Debye length and ionic strength by measuring the potential energy profile between a charged microscopic sphere and a charged plate separated by a solution of 0.1-2 mol‚m-3 KOH dissolved in a mixture of acetone with 10-20% v/v water as a model nonaqueous solvent. KOH behaves as a weak electrolyte in these solvents. The dissociation constant was determined to be 1.99 mol‚m-3 in the 15% v/v solution by both conductivity and TIRM experiments. Conductivity is much preferred for determining the concentration of charge in solutions like this in which the mechanism of charging is simple dissociation of ion pairs. But in nonpolar solvents, in which the mechanism might be proton transfer between two neutral micelles, TIRM can yield the number density of charges while conductivity alone cannot.
Introduction Unlike dispersions prepared in water, dispersions in hydrocarbon media such as dodecane are often assumed to be electrically neutral,1 and the media itself is assumed to be free of any ions. But Koelmans and Overbeek2 showed that carbon black particles dispersed in toluene were electrostatically stabilized. Their explanation was that the trace amount of water, which is soluble in toluene, accumulates at the hydrophilic surface of the carbon particles and dissociates ionizable groups there. Klinkenberg and van der Minne3 showed that trace compounds found in commercial oil products can form equal numbers of positive and negative ions within the medium and lead to significant charge on surfaces. This early work was motivated by the attempt to explain how the flow of oil can lead to electrical sparks, which can trigger an explosion. Other applications range from motor oil additives, which retard the deposition of soot on engine parts, to electronic displays. The pixels in one highly flexible display4 consist of microcapsules filled with a mixed dispersion of positive and negative particles in a hydrocarbon media. Depending on the polarization of a dc electric field, the microcapsules appear either black or white. The mechanism for the charging of molecules in hydrocarbon media and for the charging of solid-hydrocarbon interfaces is quite different than that in water and is much less studied. In his review, Morrison5 points out that the very low dielectric constant of nonpolar hydrocarbon media means that stable charge can be borne only by large entities such as micelles or polymers: small ions such as Na+ or Cl- never separate because electrostatic * Corresponding author. (1) Markovic´, I.; Ottewill, R. H.; Underwood, S. M.; Tadros, T. F.; Interactions in Concentrated Nonaqueous Polymer Latices. Langmuir 1986, 2, 625-630. (2) Koelmans, H.; Overbeek, J. T. G. Stability and Electrophoretic Deposition of Suspensions in Non-aqueous Media. Discuss. Faraday Soc. 1954, 18, 52-63. (3) Klinkenberg, A.; van der Minne, J. L. Electrostatics in the Petroleum Industry; Elsevier: New York, 1958. (4) Chen, Y.; Au, J.; Kazlas, P.; Ritenour, A.; Gates, H.; McCreary, M. Electronic Paper: Flexible Active-Matrix Electronic Ink Display. Nature 2003, 423 (8 May 2003), 136. (5) Morrison, I. D. Electrical Charges in Nonaqueous Media. Colloids Surf., A 1993, 71, 1-37.
attraction is too strong. Even when micelles or polymers are large enough to bear a stable charge without associating with an oppositely charged micelle or polymer, only a very small fraction of them have any charge. For example, in a recent study, Kim et al.6 concluded that the charge carriers in dodecane doped with poly(isobutylene succinimide) (PIBS) were PIBS micelles having a diameter of about 20 nm; only about 3% of the micelles were charged. Conductivity measurements depend not only on the concentration of charge carriers, but also on their charge and mobility (size). Moreover, for a solution to be electrically neutral, both positive and negative carriers must coexist, thus increasing the number of unknowns that affect conductivity. Consequently, conductivity measurements alone are not sufficient to establish the nature of electrical charges in nonaqueous media. Morrison5 mentions a number of techniques (such as light scattering, flow birefringence, and osmotic pressure) that have been used to understand the nature of charge in nonaqueous media. In this paper, we use total internal reflection microscopy (TIRM)7 to measure the interaction between a microscopic sphere and a flat plate to explore what such force measurements can tell us about charges in nonaqueous media. In particular, we determine the Debye length or ionic strength of the media, which depends on the concentration and charge of the charge carriers, but not on their size. Thus the Debye length contains information on charge carriers that is independent of that inferred from conductivity. Electrostatic repulsion between particles and/or planar interfaces has been measured previously in nonaqueous media using either a modified surface force apparatus8 or atomic force microscopy9,10 or has been inferred from the two-particle radial (6) Kim, J.; Anderson, J. L.; Garoff, S.; Schlagen, L. J. M. Ionic Conduction and Electrode Polarization in a Doped Nonpolar Liquid. Langmuir 2005, 21, 8620-8629. (7) Prieve, D. C. Measurement of Colloidal Forces with TIRM. AdV. Colloid Interface Sci. 1999, 82, 93-125. (8) Briscoe, W. H.; Horn, R. G. Direct Measurement of Surface Forces Due to Charging of Solids Immersed in a Nonpolar Liquid. Langmuir 2002, 18, 39453956. (9) Tulpar, A.; Subramanian, V.; Ducker, W. Decay Lengths of Double-Layer Forces in Solutions of Partly Associated Ions. Langmuir 2001, 17, 8451-8454.
10.1021/la070189t CCC: $37.00 © 2007 American Chemical Society Published on Web 06/15/2007
Acetone-Water Mixtures as Model Polar SolVents
Langmuir, Vol. 23, No. 15, 2007 8049
correlation function determined for semidilute concentrations of particles confined to two dimensions.11 One advantage of TIRM is that, not only does it measure two-body interactions directly, but, by being able to detect very weak interactions, TIRM is able to probe the asymptotic limit of very large separations in which the electrostatic interactions decay exponentially regardless of the mechanism for surface charging.12 One disadvantage of TIRM is that both surfaces need to be amply charged and of like sign so that a microscopic sphere can be levitated robustly above the interacting plate.7 This limitation initially prevented us from taking measurements in a nonpolar solvent such as dodecane. So, in this note we report some preliminary studies made in a polar organic solvent: acetone with some water added. The electrolyte used was KOH, which only partially dissociates in this nonaqueous solvent. In a subsequent paper, we will describe how we overcame this limitation to obtain results for a nonpolar solvent. Experimental Materials. Solutions were prepared using as-received semiconductor-grade KOH and acetone purchased from Fisher Scientific (Fairlawn, NJ) and deionized water. Because the solid KOH contained a significant but unknown amount of adsorbed water, we first prepared aqueous stock solutions of nominal concentration (either 2 or 25 mM), whose actual concentration was established by measuring its conductivity. These stock solutions were then diluted with acetone and/or water to obtain the solutions used below. Surfactant-free, sulfonated polystyrene latex particles, having a nominal particle diameter of 5.7 ( 0.5 µm, were obtained from Interfacial Dynamics Corp. (Portland, OR). Glass microscope slides, obtained from Fisher Scientific (Pittsburgh, PA), were cleaned using detergent solution, then thoroughly rinsed with ethanol, followed by deionized water, soaked for at least 4 h in a 10 mM KOH solution, and again rinsed with distilled water just before mounting in the TIRM flowcell. Soaking in a dilute basic solution is necessary to ensure a large negative charge density on the glass for near-neutral pH. Methods. The conductivities of the solutions were measured with an Accumet model AR60 meter from Fisher Scientific (Pittsburgh, PA), employing an Accumet conductivity probe (either model 13620-156 or model 13-621-155) having a cell constant of either 0.1 or 1 cm-1. The potential energy φ(h) of interaction between a latex particle and a glass slide was determined by repeatedly observing the elevation h of a levitated particle while it underwent Brownian motion in the force field. Elevations were determined to (1 nm using TIRM, which is described elsewhere.7 The potential energy (PE) relative to that at the most probable elevation was determined from N(I)I φ(h) ) ln kT N(Im)Im
(1)
where N(I) is the number of observations of intensity in a narrow interval of intensity centered at I, and Im is the most probable intensity. We used about 100,000 observations of intensity I, sampled at 10 ms intervals, to form the histogram denoted as N(I). Intensity was converted into elevation using (10) McNamee, C. E.; Tsujii, Y.; Matsumoto, M. Interaction Forces between Two Silica Surfaces in an Apolar Solvent Containing an Anionic Surfactant. Langmuir 2004, 20, 1791-1798. (11) Hsu, M. F.; Dufresne, E. R.; Weitz, D. A. Charge Stabilization in Nonpolar Solvents. Langmuir 2005, 21, 4881-4887. (12) Biggs, S.; Prieve, D. C.; Dagastine, R. R. Direct Comparison of Atomic Force Microscopic and Total Internal Reflection Microscopic Measurements in the Presence of Nonadsorbing Polyelectrolytes. Langmuir 2005, 21, 5421-5428. (13) Timmermans, J. The Physico-chemical Constants of Binary Systems in Concentrated Solutions; Interscience: New York, 1960; Vol. 4.
I(h) ) I0e-βh + Ib where I0 is the intensity of the particle when stuck, Ib is the background intensity measured in the absence of any particle, and β-1 is the penetration depth of the evanescent wave, which can be calculated from 4π (n sin θi)2 - n22 λ x 1
β)
where λ is the wavelength in vacuo of the laser used to generate the evanescent wave, n1 and n2 are the indices of refraction of the glass and fluid, respectively, and θi is the angle of incidence. We used 1.52 for the index of the glass and 68° for the angle of incidence. The index of the fluid depends on the relative amounts of water and acetone used and is given in Table 1.
Theory Dissociation Equilibria. In acetone-water mixtures, KOH behaves like a weak electrolyte and only partially dissociates:
KOH f K+ + OH-
(2)
Let the dissociation constant be defined by the mass-action law
K)
[K+][OH-] [KOH]
Denote the total molar concentration of KOH added to solution as C, and denote the degree of dissociation as R; then,
[K+] ) [OH-] ) RC and
[KOH] ) (1 - R)C Combining these equations, R(C) can be calculated as the root of the quadratic
K)
R2 C 1-R
(3)
A plot of (1 - R)/R2 versus C should be a straight line through the origin. From the slope, we can infer K. Conductivity of Weak Electrolyte Solutions. In Arrhenius’ theory of conductivity of weak electrolyte solutions, each ion has a mobility mi that is independent of the concentration of electrolyte. Then the specific conductance of a KOH solution is given by
L ) m+eF[K+] + m-eF[OH-] ) (m+ + m-)eFRC where e is the charge on one proton, and F is Faraday’s constant (charge per mole). The ion mobility can be expressed in terms of its hydrodynamic radius using Stokes’ law:
m( )
1 6πµa(
(4)
The equivalent conductance is defined as
Λ)
L ) (m+ + m-)eFR C
At infinite dilution (C f 0), eq 3 predicts complete dissociation of the electrolyte (R ) 1). Then the limiting conductance is
8050 Langmuir, Vol. 23, No. 15, 2007
Fu and PrieVe
Table 1. Physical Properties of Water-Acetone Solutionsa water concn (vol %)
r
n2
F (g/cm3)
10 15 20
24.6 30.3 33.4
1.3482 1.3454 1.3431
0.8524 0.8753 0.8939
a
Obtained from ref 13.
Λ0 ) (m+ + m-)eF
(5)
In terms of conductivity measurements L(C), the degree of dissociation is given by
R)
Λ L ) Λ0 Λ0C
(6)
Combining eqs 3 and 6,
L ) KΛ02
C - KΛ0 L
(7)
Thus a plot of L versus C/L should be linear. The parameters K and Λ0 can be deduced from the slope and intercept. Expected PE Profile. When the separation distance is several Debye lengths, we expect van der Waals attraction to be severely retarded and screened and double-layer repulsion to be well modeled using linear superposition and Derjaguin’s approximations. Then, for a 1:1 electrolyte, the total PE profile is expected to obey
φ(h) ) B exp(-κh) + Gh
(8)
where
() ( ) ( )
eψ1 eψ2 4 kT 2 tanh tanh B ) a π e 4kT 4kT
(9)
is the dielectric permittivity of the fluid, a is the radius of the sphere, e is the elemental charge, ψ1 and ψ2 are the Stern potentials of the sphere and the plate, respectively,
κ)
x2IeF kT
(10)
is the Debye parameter, I is the total ionic strength, and
4 G ) πa3(Fs - Ff)g 3
(11)
is the net weight of the sphere. Equation 8 has a single minimum at
κhm ) ln
κB G
(12)
Eliminating B between eqs 8 and 12, we obtain the PE relative to that at the most probable elevation hm:
φ(h) - φ(hm) G G ) {exp[-κ(h - hm)] - 1} + (h - hm) kT κkT kT (13) Experimentally determined PEs, calculated from eq 1, are compared to those predicted by eq 13. By fitting the experimental PE profile to eq 13, values for G, κ, and hm can be determined. If desired, a value for B can then be estimated from eq 12. Thus, from the PE profile, we can deduce the ionic strength of the solution from the Debye parameter κ using eq 10. In terms of Debye parameter measurements κ(C), the degree of dissociation
Figure 1. Effect of water content on PE profile measured in 0.5 mM KOH in acetone-water solutions. Solid curves are fits to eq 13.
is given by
R)
I kTκ2 ) C 2eFC
(14)
Results and Discussion r from TIRM: Water Content. KOH is completely dissociated in pure water and is insoluble in pure acetone. To find an water-acetone mixture in which KOH is only partially dissociated, we first examined a series of solutions with different water content at a KOH concentration convenient for TIRM measurements. Figure 1 shows the PE profiles for 0.5 mM KOH in several water-acetone solutions having different amounts of water. For each water concentration, the PE profile was obtained three times with the same sphere, but only one of the three profiles at each water content appears in Figure 1. PE profiles are fit to eq 13 using a weighting factor of exp[-2{(φ - φm)}/{kT}] applied to each contribution to the total variance. Notice the factor of 2 in the exponent: owing to variance being proportional to the square of the deviation, this weighting is equivalent to weighting each absolute error by the Boltzmann factor for that energy level. The fact that the PE profiles in Figure 1 are well fit by eq 13 suggests that other forces (e.g., van der Waals attraction) are not significant. This fitting procedure yields values for G, κ-1, and hm. Assuming KOH dissociates to form equal concentrations of K+ and OH- and that no other ions contribute significantly to ionic strength, the ionic strength I calculated from eq 10 can be equated with the concentration of KOH that actually dissociates in this nonaqueous system, and the degree of dissociation R can be calculated from eq 14. The apparent diameter d of the latex spheres is calculated as 2a deduced from G using eq 11. The charge parameter B is deduced from eq 12, and an apparent average surface potential ψ is then obtained using eq 9, taking ψ1 ) ψ2 ) ψ. The results are summarized in Table 2. The purpose of showing results from multiple runs in Table 2 is to gauge reproducibility. The ionic strength I is reproduced within (0.01 mM for each water content. The ionic strength deduced for 20% water is 0.51 mM, corresponding to a degree of dissociation exceeding 100% and thus outside the range of R’s admitted by the dissociation model given by eq 3. Although this degree of dissociation is unreasonable, an ionic strength of 0.51 mM is still within experimental error of 0.50 mM corresponding to 100% dissociation.
Acetone-Water Mixtures as Model Polar SolVents
Langmuir, Vol. 23, No. 15, 2007 8051
Table 2. Results from Curve Fitting of PE Profiles in Figure 1 for Which C is 0.5 mM water concn hm κ-1 I (% v/v) run (nm) (nm) (mM) 10
15
20
1 2 3 1 2 3 1 2 3
55.5 10.94 55.0 10.93 55.3 10.94 45.0 9.14 45.8 9.14 45.1 8.98 30.6 8.82 30.5 8.80 30.9 8.81
0.24 0.24 0.24 0.43 0.43 0.44 0.51 0.51 0.51
G (pN) 0.181 0.189 0.189 0.169 0.156 0.169 0.156 0.169 0.144
d R B |ψ| (µm) (%) (kT) (mV) 5.5 5.6 5.6 5.6 5.5 5.6 5.7 5.8 5.5
48 48 48 86 86 88 102 102 102
76.8 77.0 78.9 51.5 52.1 55.9 10.8 11.5 10.3
6.5 6.4 6.5 4.7 4.8 4.9 2.0 2.1 2.0
The particle diameters deduced from the regression vary from 5.5 to 5.8 µm. All latex particles came from the same vial marked by the manufacturer as 5.5 ( 0.3 µm. Thus the G values reported in Table 2 are quite reasonable: the small systematic decrease in G with water content arises from the small increase in density of the solution (see Table 1), which causes the particle to have a slight decrease in net buoyant weight. Although the three runs reported for 10% water were reproducible, with longer exposure of the polystyrene sphere to this solvent, the particle increased 1-2 µm in diameter, and the surface became sticky. No swelling was observed for larger water contents, even for quite long exposures. The most probable separation distance hm decreases with water content. This is primarily due to increasing the ionic strength (decreasing Debye length) as the degree of dissociation increases with greater water content. A secondary effect is the decrease in the surface potential with increasing water content (see Table 2), which also decreases the most probable separation distance hm according to eq 12. The charge parameter B and associated mean surface potential ψ also have reasonable, albeit low values. The observed decrease in surface potential with water content would be expected if the surface charge density was the same for each water content. We choose 15% for the remaining experiments since swelling was observed at 10% and complete dissociation was observed at 20%. r from TIRM: KOH Concentration. PE profiles were measured for KOH concentrations between 0.1 and 2 mM. A sampling of those profiles are shown in Figure 2. As with the analysis of Figure 1, the PE profiles of Figure 2 are fit to eq 13. This yields a value for the Debye length κ-1, which can be used to deduce the ionic strength I and the degree dissociation R. The
Figure 2. Effect of KOH concentration on the PE profiles measured in an acetone solution containing 15% water. Solid curves are fits to eq 13.
Table 3. C (mM)
κ-1 (nm)
I (mM)
R
0.1 0.2 0.5 0.8 1.0 2.0
19.0 13.4 9.1 7.8 7.3 5.4
0.10 0.20 0.43 0.60 0.67 1.24
1.00 1.00 0.86 0.76 0.67 0.62
results are summarized in Table 3. As expected for an increase in total KOH concentration, the ionic strength increases while the degree of dissociation decreases. r from Conductivity. Figure 3 summarizes the results of conductivity measurements over a range of KOH concentrations. At low concentrations, the specific conductance increases in direct proportion to the concentration, thus suggesting complete dissociation of KOH. At high concentrations, the specific conductance increases at a slower pace, owing to partial dissociation. Replotting the data as suggested by eq 7 results in the open symbols of Figure 3, which are well fit by the straight line shown. Using the slope and intercept of the regression line, we can infer values for the two parameters in eq 7: K ) 1.99 ( 0.06 mol‚m-3 and Λ0 ) 8.51 ( 0.20 mS‚m2‚mol-1. Having the value of the limiting equivalent conductance Λ0, we can infer a value for the average ion mobility m j ) (m+ + m-)/2 from eq 5 and the corresponding harmonic-mean ion radius from eq 4. Thus we deduce an average ion radius of 0.216 ( 0.005 nm in the acetone-water solution. This is considerably larger than the harmonic mean of the hydrodynamic radii of K+ and OH- in pure water: using tabulated values of the limiting ionic conductances of 74.5 and 192 S‚cm2‚mol-1 reported at 298 K in handbooks, we infer an average radius of 0.069 nm in pure water. The ratio of ion radii in the two solvents is roughly the same as the ratio of the Bjerrum lengths in the two solvents. Recall that the Bjerrum length is defined as
λBj )
e2 kT
which represents the distance between two point elemental charges at which the electrostatic energy equals the thermal energy kT.
Figure 3. Specific conductance L measured for different total concentrations C of KOH in a solvent consisting of 15% water in acetone (filled symbols). Open symbols are the same set of data replotted as L versus C/L as suggested by eq 7. The solid line is a regression of the open symbols.
8052 Langmuir, Vol. 23, No. 15, 2007
Figure 4. Comparison of the degree of dissociation determined by conductivity measurements with those determined from TIRM.
Generally, two oppositely charged ions (or an ion and a water dipole) have to come closer together than the Bjerrum length in order to stick. In pure water, the Bjerrum length is 0.7 nm, whereas, in a 15 vol % water-in-acetone solution, it is 1.9 nm. The lower dielectric constant for the acetone-water solutions means that electrostatic attraction is stronger than in water. Perhaps the hydration shell surrounding each ion is consequently thicker, thus leading to a larger radius for the hydrated ion. Determination of Equilibrium Constant. Knowing the limiting equivalent conductance Λ0, we can use the conductivity data in Figure 3 to deduce the degree of dissociation R according to eq 6. The result, plotted in Figure 4, shows that the degree of dissociation decreases monotonically, as expected, from unity to about 0.6 at 2 mol‚m-3 KOH. As suggested by eq 3, replotting the data as (1-R)/R2 versus C in Figure 4 yields a straight line through the origin. From the regressed value of the slope, we infer K ) 1.991 ( 0.012 mol‚m-3. The TIRM data for R versus C given in Table 3 are also plotted as (1-R)/R2 versus C in Figure 4 (the open circles). Error bars are based entirely on standard error estimates reported for the κ parameter by the nonlinear regression wizard in Sigmaplot version 9.01. Clearly there is greater random uncertainty in these TIRM results than that for the results deduced from conductivity. Nonetheless, all the points lie within experimental error of the straight line, except the point corresponding to 1 mol‚m-3. Faced with this one-point discrepancy, we sampled additional PE profiles at a number of KOH concentrations, yielding the open squares in Figure 4. This second set of data falls virtually on top of the regression line for the conductivity results. The much better agreement for set #2 is attributed to improved experimental technique (this second set of data was obtained several months after the first set). Regressing set #2 of the TIRM results yields K ) 1.99 ( 0.03 mol‚m-3. This is virtually identical to the value of the dissociation constant inferred from conductivity. Thus, an experienced user of TIRM can obtain the same concentration of dissociated ions as deduced from conductivity. Of course, in the case of polar
Fu and PrieVe
nonaqueous solutions such as KOH in acetone-water mixtures, conductivity is a much easier and more accurate technique compared to TIRM. We will now show why, in the case of nonpolar solvents, conductivity measurements alone are not sufficient to determine the concentration of charge carriers. Advantage of TIRM. To study charging mechanisms in nonpolar solvents, the number density or molar concentration of charge carriers in solution needs to be measured. Let us look at what is needed to determine that concentration. For strong electrolytes, which are completely dissociated, you only need to know the molecular weight of the solute, which is then divided into the mass concentration to obtain the molar concentration. Charge carriers in nonpolar solvents such as dodecane often are oligomers such as PIBS.6 Even if the molecular formula is known, the number of monomer units needs to be measured. Let us assume the molecular weight is known, either by knowing the molecular formula or by measuring colligative properties of the solution such as osmotic pressure. Charge carriers in nonpolar solvents are often micelles;6 thus, you also need to know the aggregation number, which might vary with the concentration of solute or other components. Even if you know the aggregation number, what fraction of the micelles are charged? These questions are analogous to finding the degree of dissociation of KOH in the present study. Equation 6 was used to infer an answer from the measured specific conductance L of the solution, which requires a value for the limiting equivalent conductance Λ0. The latter value is deduced from conductivity measurements at infinite dilution. In the limit C f 0, the massaction law (eq 3) associated with the dissociation reaction (eq 2) implies that R f 1. The limiting equivalent conductance Λ0 depends primarily on the ionic mobilities (see eq 5). Unfortunately, dissociation according to eq 2 is not the only mechanism by which solutes become charged. Another mechanism by which solute entities become charged in nonpolar solvents is proton exchange between two neutral micelles:5
2M a M+ + MThe mass-action law associated with this reaction leads to a degree of dissociation that is independent of the total concentration. Therefore, the ionic mobilities cannot be determined from conductivities measurements alone. Thus the concentration of charged micelles cannot be determined from conductivity, even if the molecular weight of the micelles is known. The micelle mobility or hydrodynamic radius is still needed. The main advantage of TIRM (relative to conductivity) is that the mobility of the ions is not needed to interpret the results. TIRM is used to determine the Debye parameter κ, which, in turn, is related to the ionic strength by eq 10. Since the charge number z is usually unity for nonpolar solvents, the ionic strength is usually equal to the molar concentration of charge. No additional properties of the charged solute are needed. LA070189T
