Anal. Chem. 2007, 79, 8114-8117
Shapes of Polyelectrolyte Titration Curves. 1. Well-Behaved Strong Polyelectrolytes Robert Pelton,*,† Bernard Cabane,‡ Yuguo Cui,† and Howard Ketelson§
Department of Chemical Engineering, JHE-136, McMaster University, Hamilton, Ontario, Canada, L8S 4L7, Alcon Research, Ltd., 6201 South Freeway, Fort Worth, Texas 76134, and CNRS UMR 7636, ESPCI, 75231, Paris Cedex 05, France
The charge content of aqueous polymers is measured routinely by polyelectrolyte titrations in which an unknown polymer is titrated with an oppositely charged standard polymer solution, usually poly(diallyldimethyl ammonium chloride) or potassium poly(vinyl sulfate). Polyelectrolyte titration end points are frequently determined with a streaming current detector (SCD). The shapes of polyelectrolyte titration curves from polyelectrolytes with fixed electrical charges were simulated by a diffuse electrical double layer model. Well-behaved titration curves obtained with fixed charged polyelectrolytes were fit by the modeling, giving support for the basic hypothesis that the net charge of material adsorbed on the SCD wall is a linear function of the volume of added titrant. The shapes of titration curves from deviant systems such as poly(vinyl alcohol)-borate cannot be predicted by the model. Figure 1. Schematic illustration of a streaming current detector cell.
Automated polyelectrolyte titration employing streaming current end point detection is widely used in industry, both in laboratories and as on-line sensors to measure the charge content of water-soluble polymers and dispersed colloids. For example, in papermaking technology, polyelectrolyte titration is used to control the addition of cationic polymeric coagulants, which are added to sequester indigenous anionic polymer material in aqueous streams.1 In Terayama’s2 original description of polyelectrolyte titrations, end points were detected with a dye indicator. Although automated detection of dye indicators has been described,3 most practitioners of polyelectrolyte titrations employ a streaming current detector (SCD) to determine the charge equivalence end point. First described in 1966,4 streaming current detectors are commercially available from a number of suppliers. Figure 1 shows a schematic diagram of a SCD.1,4 The heart of the device is a reciprocating fluorocarbon piston in a loosely fitting fluorocarbon cylinder. When the piston moves in the cylinder, the solution is forced to move in the annulus between the piston * To whom correspondence should be addressed. Phone: (905) 529 7070 ext 27045. E-mail:
[email protected]. † McMaster University. ‡ CNRS UMR 7636. § Alcon Research, Ltd. (1) Phipps, J. S. Tappi J. 1999, 82 (8), 157-165. (2) Terayama, H. J. Polym. Sci. 1952, 8 (2), 243-253. (3) Horn, D.; Heuck, C. C. J. Biol. Chem. 1983, 258 (3), 1665-1670. (4) Bockenhoff, K.; Fischer, W. R. Fresenius. J. Anal. Chem. 2001, 371 (5), 670-674.
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and the cylinder wall. Electrodes embedded in the cylinder wall measure the induced streaming current. Polyelectrolyte titration with streaming current detection relies on two primary assumptions: (a) The reaction between the polycations and polyanions is complete. In other words, uncomplexed cationic and anionic groups do not coexist. The cooperative nature of polyelectrolyte complex formation gives very high effective binding constants, making this a good assumption.5(b) The species of interest in solution are also adsorbed on the fluorocarbon surfaces of the SCD, and the titrant volume corresponding to the isoelectric point of the adsorbed layer, measured by streaming current, corresponds to the isoelectric point of the polyelectrolyte complex species in solution. Figure 2, taken from Kam and Gregory,6 shows typical polyelectrolyte titration results. The three data sets correspond to three cationic polymers with varying density of cationic charged groups. The solid lines and the terms β and σdl come from our modelsthese are described in the next section. The polymers were titrated with potassium poly(vinyl sulfate) (PVSK). The y-axis in Figure 2 shows the SCD output, and the x-axis is the volume of added PVSK. The titration end points are taken as the PVSK volume corresponding to a detector output of zero. (5) Horn, D. Prog. Colloid Polym. Sci. 1978, 65, 251-264. (6) Kam, S. K.; Gregory, J. Colloids Surf. A 1999, 159 (1), 165-179. 10.1021/ac071210y CCC: $37.00
© 2007 American Chemical Society Published on Web 09/29/2007
Figure 2. Comparison of published data with curves calculated with a diffuse layer model (eq 5). κ ) 3.3 × 107 m-1 and R ) 1 (eq 1).
Recent publications have discussed the polyelectrolyte-streaming current titration in terms of reproducibility,4 salt sensitivity,7 time effects,1 order of polyelectrolyte addition,7 and comparison with other methods for measuring titratable charge6 or ζ potential.8 A detailed hydrodynamic-electrokinetic model of the streaming current detector was published a decade ago.9 Approximate equations relating streaming current to the ζ potential of the detector wall were given. Polyelectrolyte titration with streaming current detection has become a standard technique in the applied polymer and colloid literature; most references employing the technique simply report the end point results. This is reasonable because the end of the titration is usually characterized by a steep decline (or rise) through the zero SCD signal line. In these cases, the titrant volume corresponding to zero SCD signal can be determined with high precision and the results are reproducible. Although some of the publications listed above present the total titration curves (such as those in Figure 2), there has been no analysis of shape of the titration curves. Predicting the shapes of the titration curves is important from two perspectives. First, a successful analysis confirms the fundamental assumptions. Second, understanding the shape should make it easier to identify deviant systems where zero SCD signal does not correspond to an obvious end point. Figure 3 shows an extreme example of deviant behavior displayed by the titration of poly(vinyl alcohol)borate complex with poly(diallyldimethyl ammonium chloride) (PDADMAC). Clearly there is no reason to believe that the PDADMAC volume giving zero SCD represents the content of charged borate groups on the poly(vinyl alcohol). A future publication will show that the deviant nature of Figure 3 reflects an increasing concentration of bound borates over the course of the titration. (7) Chen, J. H.; Heitmann, J. A.; Hubbe, M. A. Colloids Surf. A 2003, 223 (13), 215-230. (8) Barron, W.; Murray, B. S.; Scales, P. J.; Healy, T. W.; Dixon, D. R.; Pascoe, M. Colloids Surf. A 1994, 88 (2-3), 129-139. (9) Walker, C. A.; Kirby, J. T.; Dentel, S. K. J. Colloid Interface Sci. 1996, 182 (1), 71-81.
Figure 3. Polyelectrolyte titration of PVA-borate with PDADMAC. A total of 10 mL of PVA (95 mg/L) plus 0.095 M borax were titrated with PDADMAC (1 mequiv/L) using a PCD T3 titrator with a Mutek PCD 03 SCD.
We were intrigued by the shape of the curves in Figure 2. In particular, why is the SCD output nearly constant over most of the titration when the net charge density in solution must be changing linearly with the titrant volume? We considered effects such as counterion condensation on the polymer layer adsorbed on the SCD surface. However, the following analysis shows that simple diffuse electrical double layer theory is sufficient to predict the shape of the curves in Figure 2. Modeling the Titration Curve for Polymers with Fixed Charges. Our approach starts with the following assumptions: (1) The detector output, SCD, is proportional to the zeta potential, ζ, of the detector surface. Walker’s modeling suggests that this is valid, although the proportionality constant, R, is model dependent.9
SCD ) Rζ Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
(1) 8115
Figure 4. Influence of salt concentration on the polyelectrolyte titration from eq 5.
(2) The surface charge density on the SCD surface at the shear plane, σd, is the following linear function of the volume of added titrant, VT, where σdI is the initial shear plane charge density on the surface and β is the proportionality constant. This follows directly from assumption b given above. That is, if the same volume of titrant produces an isoelectric SCD surface and an isoelectric solution species, then the surface and solution charge densities must be linearly related on the approach to the isoelectric point.
σd ) σdl + βVT
(2)
The electrical double layer was modeled by a simplified Gouy-Chapman model in which the diffuse layer potential was assumed to be equal to the ζ potential. Therefore, the charge density at the shear plane, σd, is related to ζ by the following equation, where is the combined dielectric constant, k is Boltzmann’s constant, T is temperature, eo is the elementary charge, and κ is the Debye-Hu¨ckel screening parameter, which is a function of Avogadro’s number, NA, and the 1:1 electrolyte concentration, c.10
σd )
( )
eoζ - 2kTκ sinh eo 2kT
where
κ)
x
eo2NA 2c kT
(3)
Rearranging gives
( )
ζ ) sinh-1
eoσd 2kT 2kTκ eo
(4)
(10) Hunter, R., Zeta Potential in Colloid Science, Principles and Applications; Academic Press: London, 1981; p 386.
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Substituting eqs 1 and 2 into eq 4 gives
SCD ) sinh-1
(
)
eo(σdl + βVT) 2kT 2kTκ Reo
(5)
The only parameters in eq 5 changing in the course of a titration are the volume of polyelectrolyte titrant, VT, and the corresponding signal, SCD. The solid lines in Figure 2 compare Kam and Gregory’s data for the PVSK SCD titration of cationic poly(acrylamide-co-trimethylaminoethyl acrylate) with the predictions of the diffuse double layer model (eq 5). The values of the initial surface charge density of the cationic polyelectrolyte adsorbed on the detector wall, σdI (see eq 2), were chosen so that the initial calculated SCD values equaled the initial experiment readings. Each curve has a unique β value (eq 2) reflecting the different charge contents of the three polymers being titrated. The simulated curves showed good agreement with the data, suggesting that diffuse double layer theory predicts the shape of the polyelectrolyte titration curves. This result shows that the shape of the titration curve can be explained by the hyperbolic sinh function relating ζ potential to charge density (eq 3), a classical relationship in electrical double layer theory. The term Debye-Hu¨ckel κ in eq 5 accounts for the effects of electrolyte concentration. Figure 4 shows an example of calculated SCD versus volume curves for three salt concentrations. The curves flatten with increasing electrolyte. The calculations in Figure 4 look very much like the experimental curves recently published by Chen et al.,7 again suggesting that our model predicts the general features of polyelectrolyte titrations. In conclusion, the polyelectrolyte titration with automated SCD detection is an excellent technique for measuring the concentration of ionic groups on soluble polymers and dispersed particles. However, the shape of the titration curve for new systems should be examined to ensure that zero SCD output
corresponds to a meaningful end point. The good performance of the simple model presented herein supports the basic hypothesis that the isoelectric point of material adsorbed on the detector wall corresponds to the isoelectric point of polyelectrolyte complexes in solution. Furthermore, the model gives a creditable explanation for the hitherto unexplained shapes of the titration curves. On the other hand, in view of the many assumptions behind the model, the computed parameters β and σdl should be used with caution. Future work will involve
an analysis of some deviant curves such as the one shown in Figure 3. ACKNOWLEDGMENT The authors thank Professor Hiroo Tanaka for useful discussions. Alcon Laboratories, Fort Worth, are thanked for funding Y.C. Received for review June 8, 2007. Accepted August 16, 2007. AC071210Y
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