Channel Flow Configuration for Studying the Kinetics of

Laboratory of Physical Chemistry and Electrochemistry, Helsinki University of Technology, P.O. Box 6100,. Fin-02015 HUT, Helsinki, Finland, and Conden...
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Anal. Chem. 2005, 77, 6895-6901

Channel Flow Configuration for Studying the Kinetics of Surfactant-Polyelectrolyte Binding Sanna Carlsson,† Peter Liljeroth,‡ and Kyo 1 sti Kontturi*,†

Laboratory of Physical Chemistry and Electrochemistry, Helsinki University of Technology, P.O. Box 6100, Fin-02015 HUT, Helsinki, Finland, and Condensed Matter and Interfaces, Debye Institute, University of Utrecht, P.O. Box 80000, 3508 TA. Utrecht, The Netherlands

A novel channel flow configuration was developed and utilized for studying the poly(styrenesulfonate)-cetylpyridinium ion interaction kinetics. The surfactant solution was continuously injected into a flow of polyelectrolyte solution, and the extent of the association reaction was probed at an ion-selective detector electrode. The system was modeled within an analytical approximation, which was tested by a finite-element simulation of the full convective mass transport problem including the homogeneous complexation reaction. The results show that association kinetics can be resolved and that the initial steps of the reaction are not influenced by intermolecular interactions between the bound surfactants. The presented methodology is general, and further development should enable the study of complex cooperative kinetics of surfactant-polyelectrolyte systems. The interaction between a polymer and ionic surfactants has been the subject of intense study due to its fundamental relevance for numerous applications (e.g., cellular delivery of DNA1). In the case of a polyelectrolyte, the binding of the oppositely charged surfactant occurs at a concentration of 2-3 orders of magnitude below critical micellar concentration (cmc) of the same surfactant. This is due to the strong electrostatic and hydrophobic interaction forces between the polyelectrolyte and the ionic surfactants. The interaction thermodynamics of such systems has been widely studied.2-5 However, the kinetics of the interaction is poorly understood, due mainly to the rapid rate of the association reaction. Some published results exist on the polymer (e.g., poly(vinylpyrrolidone) or poly(ethylene oxide)) surfactant (alkanesulfates) systems. The kinetics have been studied using relaxation techniques including ultrasonic,6-10 temperature jump,11 and * Corresponding author. Tel: +358 9 451 2570. Fax: +358 9 451 2580. E-mail: [email protected]. † Helsinki University of Technology. ‡ University of Utrecht. (1) Felgner, P. L.; Ringold, G. M. Nature 1989, 337, 387-388. (2) Sˇ kerjanc, J.; Kogej, K.; Vesnaver, G. J. Phys. Chem. 1988, 92, 6382-6385. (3) Wang, C.; Tam, K. C. J. Phys. Chem. B 2005, 109, 5156-5161. (4) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Macromolecules 1983, 16, 1642-1645. (5) Hakkarainen, S.; Gilbert, S. L.; Kontturi, A.-K.; Kontturi, K. J. Colloid Interface Sci. 2004, 272, 404-410. (6) Gettins, J.; Gould, C.; Hall, D. G.; Jobling, P. L.; Rassing, J. E.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 2 1980, 76, 1535-1542. (7) Wan-Badhi, W. A.; Wan-Yunus, W. M. Z.; Bloor, D. M.; Hall, D. G.; WynJones, E. J. Chem. Soc., Faraday Trans. 1993, 89, 2737-2742. 10.1021/ac050947p CCC: $30.25 Published on Web 10/07/2005

© 2005 American Chemical Society

pressure jump12 methods. These methods are capable of measuring the rather slow dissociation kinetics of the polymer-surfactant aggregate but are not applicable for studying the association kinetics in the absence of equilibrium data. Channel flow and other hydrodynamic electrodes provide a convenient means to study electron-transfer kinetics and associated homogeneous chemical reaction steps (e.g., EC and ECE mechanisms).13,14 However, the conventional channel flow electrode is limited to the study of redox-active species. This limitation has been overcome by the use of an immobilized liquid-liquid interface as a detector electrode whereby ion transfer across the interface between two immiscible electrolyte solutions (ITIES) is monitored.15 In addition, the channel flow geometry offers a possibility of controllably mixing two solutions, whose interaction time can be manipulated by changing the flow rates. The careful combining of two solutions is critical, so that the laminar velocity profile is maintained. Different cell configurations have been introduced for this purpose, such as the flow injection channel and the confluence reactor.16-19 In this work, the association kinetics of the polyelectrolyte and the ionic surfactant is studied in a novel channel flow system, in which the sodium poly(styrenesulfonate) (PSS) and n-cetylpyridinium chloride (CP) containing flows are combined in a controlled manner and the concentration of the unassociated surfactant is measured at a liquid-liquid detector electrode. The rate of surfactant ion transfer across ITIES (ion-selective electrode) is controlled by mass transfer in the channel and the possible (8) Bloor, D. M.; Wan-Yunus, W. M. Z.; Wan-Badhi, W. A.; Li, Y., Holzwarth, J. F.; Wyn-Jones, E. Langmuir 1995, 11, 3395-3400. (9) D′Aprano, A.; La Mesa, C.; Persi, L. Langmuir 1997, 13, 5876-5880. (10) La Mesa, C. Colloids Surf. A 1999, 160, 37-46. (11) Tondre, C. J. Phys. Chem. 1985, 89, 5101-5105. (12) Painter, D. M.; Bloor, D. M.; Takisawa, N.; Hall, D. G.; Wyn-Jones, E. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2087-2097. (13) Aixill, W. J.; Alden, J. A.; Prieto, F.; Waller, G. A.; Compton, R. G.; Rueda, M. J. Phys. Chem. B 1998, 102, 1515-1521. (14) Morland, P. D.; Compton, R. G. J. Phys. Chem. B 1999, 103, 8951-8959. (15) Liljeroth, P.; Johans, C.; Kontturi, K.; Manzanares, J. A. J. Electroanal. Chem. 2000, 483, 37-46. (16) Gooding, J. J.; Coles, B. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 175-181. (17) Gooding, J. J.; Coles, B. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 182-188. (18) Fulian, Q.; Stevens, N. P. C.; Fisher, A. C. J. Phys. Chem. B 1998, 102, 3779-3783. (19) Stevens, N. P. C.; Gooch, K. A.; Fisher, A. C. J. Phys. Chem. B 2000, 104, 1241-1248.

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Figure 1. Schematic of the channel flow cell used in this study. CE1, CE2, RE1, and RE2 denote the aqueous and organic counter and reference electrodes, respectively.

homogeneous association reaction. The laminar flow in a channel flow cell can be modeled, and the dependence of the limiting steady-state current on the flow rate and bulk ion concentration is obtained from the solution of the convective diffusion equation.20 The problem including the second-order homogeneous complexation reaction is considered using two different approaches. On a simple level, the flow rates only determine the residence time of the surfactant in the cell and, hence, the allowed time for the complexation reaction. The convective mass transport is taken into account by a mass-transfer coefficient, which is obtained from experiments in the absence of polyelectrolyte. The kinetic parameters derived from this elementary approach are tested by a finite-element simulation of the full convective diffusion equation including the homogeneous chemical reaction. EXPERIMENTAL SECTION Chemicals. The organic supporting electrolyte tetraphenyl arsonium tetrakis-4-(chloro)phenylborate (TPAsTPBCl4) was synthesized from tetraphenylarsonium chloride (TPAsCl, Aldrich, 97%) and potassium tetrakis-4-(chloro)phenylborate (Aldrich) as described earlier.21 The aqueous supporting electrolyte was sodium chloride (Merck, p.a.). The model ion, by which the channel flow cell was tested, was tetraethylammonium ion (TEA+), the complexing agent was CP ion and the polyanion was PSS. The cations were added to the aqueous phase as chloride salts: tetraethylammonium chloride (TEACl, Sigma, 98%) and n-cetylpyridinium chloride (Merck, 99%). The polyanion was added to the aqueous phase as sodium poly(styrenesulfonate) (Polysciences, Mn ) 4600, Mw/Mn ) 1.13). All the aqueous solutions were made in ion-changed Milli-Q water, and all the chemicals were used as received. Channel Flow Cell. The channel flow cell is shown schematically in Figure 1. The organic phase was 2-nitrophenyl octyl ether (NPOE, Fluka, Selectophore), which was immobilized by poly(vinyl chloride) (PVC, Sigma, very high molecular weight).22 (20) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962; p 112. (21) Cunnane, V.; Schiffrin, D. J.; Beltran, C.; Geblewicz, G.; Solomon, T. J. Electroanal. Chem. 1998, 247, 203-214. (22) Lee, H. J.; Beattie, P. D.; Seddon, B. J.; Osborne, M. D.; Girault, H. H. J. Electroanal. Chem. 1997, 440, 73-82.

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Immobilization was done by drop-casting a hot (120 °C) mixture of NPOE with the organic base electrolyte and 5 wt % PVC into the top part of the channel flow cell on glass plate in order to attain a smooth interface. The following electrochemical cell configuration was used:

The cell was made of poly(tetrafluoroethylene) apart from the metal cover and bottom plates. The dimensions of the channel were length l ) 47 mm, width w ) 10 mm, and height h ) 0.52 mm and those of the injection slit were width ws ) 7.5 mm and length ls ) 1.5 mm. The dimensions of the interface were width wσ ) 2 mm and length lσ ) 2 mm. The distance between the interface and the injection slit, d was 5 mm (center-to-center distance). Both the aqueous and the organic counter electrodes (CE1 and CE2 in Figure 1) were made of silver. Ag/AgCl electrodes were used as reference electrodes (RE1 and RE2 in Figure 1). The aqueous reference electrode (RE1) was placed in the tubing after the outlet slit. The potential was controlled by a four-electrode potentiostat (Autolab PGSTAT100, The Netherlands). The current across the liquid-liquid interface was measured under mass transport limited conditions, which were achieved by applying a suitable interfacial potential difference (the difference between the applied interfacial potential and the standard potential of transfer of the ion was sufficiently large to ensure that the surface concentration in aqueous phase tends to zero). Consequently, the current is only limited by the convective diffusion of surfactant in the aqueous phase. The flow rate of the main flow (V1) was controlled by a peristaltic pump (Ismatec, IPN-12), and the flow rate of the injection flow (V2) was controlled by a syringe pump (Sage Instruments model 341B). The total flow rate (VT) was typically kept between 5 × 10-4 and 5 × 10-3 cm3 s-1, which corresponds to a residence time (see eq 6) of 5-50 s. The flow rate of the injection slit (when in use) was 2.8 × 10-4 cm3 s-1.

THEORY Despite its relevance, the kinetics of surfactant binding to a polyelectrolyte has only received limited attention.23 Traditionally, the measurement of binding kinetics between a polymer and a surfactant involves the use of a pressure or temperature jump or ultrasonic techniques that perturb the complexation equilibrium enabling the dissociation kinetics to be monitored. The amount of the dissociated compound has been followed by measuring the emf of a surfactant-selective electrode. This only yields reliable kinetic parameters if mass transfer to the electrode is considerably faster than the rate of the reaction. In most published studies, the dissociation reaction has been described as a first-order reaction consisting of one or more subsequent steps.8,9,23 In principle, the binding of a surfactant to a polymer or to a polyelectrolyte is a multistep process with numerous forward and backward rate constants. However, if one assumes a noncooperative mechanism (no interaction between the bound surfactants), the reaction can then be considered to occur between the monomers of the polyelectrolyte and surfactant ions. This assumption is justified in our case, since the concentration of the surfactant was kept sufficiently low (relative to the concentration of the polyelectrolyte) such that the degree of binding did not exceed ∼0.15. Thus, the reaction can be described as a singlestep, reversible second-order reaction. The binding kinetics and the expected current response of the channel flow cell can be estimated using a simple model, in which the mass transfer and kinetics are considered separately and the surfactant ion transfer is taken into account by a mass-transfer coefficient. The rate law for a second-order reaction with forward, kf, and backward, kb, rate constants can be written as

dcx ) kf(cm0 - cx)(cs0 - cx) - kbcx dt

(1)

where cx is the concentration of the bound surfactant and a superscript 0 denotes the initial concentrations of the monomers, cm, and surfactant, cs. Those can be calculated from the volumetric flow rates.

cs0 )

V2 c V1 + V2 s,2

(2)

cm0 )

V1 c V1 + V2 m,1

(3)

where cm,1 and cs,2 are the monomer and surfactant concentrations in the main and injection flows, respectively. The rate law is easily solved, and the result is

A ) x1 + 2(cm0 + cs0)K + (cm0 - cs0)2K2

(5)

with K ) kf/kb. The average residence time can be written in terms of the volumetric rates and the channel dimensions.

t ) hwd/(V1 + V2)

(6)

The measured steady-state current is proportional to unbound surfactant, cs.

I ) mFcs ) mF(cs0 - cx)

(7)

where F is Faraday’s constant. The mass-transfer coefficient, m, which includes the effects of the size of the interface and the diffusion coefficient on the current, can be obtained from the measurements in the absence of PSS (see below). The dependence of steady-state current on the volumetric total flow rate with different rate constants is illustrated in Figure 2. As the residence time is inversely proportional to the total flow rate, the conversion is high (low current) and the reaction is close to equilibrium at slow flow rates. With low kb values, the current response is mainly determined by kf almost independently of the flow rate. High kb values affect the magnitude and the flow rate dependence of the current response in the slow flow rate range. In this channel flow configuration, the initial concentrations of associating compounds depend on the flow rates in the main and injection channels. Because of this, the behavior of conversion is more complicated than in traditional kinetic studies. After reaching the equilibrium-controlled region, the conversion begins to decrease again as the residence time increases, which is shown in Figure S-1 (Supporting Information). This occurs as the conversion depends not only on the rate constants but also on the concentration of the associating compounds, which in turn are set by the flow rates. The behavior of the flow cell was also modeled by solving the full convective diffusion equation with appropriate boundary conditions using the finite-element method (finite-element package FEMLAB, Comsol Ab, Sweden). These types of simulation have been extensively described in the literature.16,17 For the surfactant, s (and analogously for the monomer, m and the complex, c) the convective diffusion equation at steady state including a reversible chemical reaction is

(

Ds

∂2cs ∂x

2

+

)

∂2cs ∂y

2

- vx

∂cs - kfcscm + kbcc ) 0 ∂x

(8)

where vx is the parabolic velocity profile in the main channel, which is assumed to be unperturbed by the injection flow24

cx ) 2cm0cs0K

where

ekbAt - 1 - 1 - (cm0 + cs0)K + A + ekbAt(1 + A + (cm0 + cs0)K) (4)

(23) Maulik, S.; Chattoraj, D. K.; Moulik, S. P. Colloids Surf. B 1998, 11, 5765.

(

vx ) v m 1 -

)

(h - 2y)2 h2

(9)

where vm is the maximum velocity in the channel, which is related to the total flow rate Vt ) 2/3 hwvm. The injection is treated as a Analytical Chemistry, Vol. 77, No. 21, November 1, 2005

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Figure 2. Current as a function of total flow rate modeled by the eqs 4-7. (A) kb is kept constant (0.27 s-1), and kf is varied. kf ) 105, 1.5 × 104, 5000, 1000, 400, and 60 dm3 mol-1s-1from bottom to top. (B) kf is kept constant(1.5 × 104) and kb is varied. kb ) 0.001, 0.1, 0.5, 1, 2, 5, and 20 s-1 from bottom to top.

constant flux boundary condition for the surfactant across the injection slit, and this flux as a function of the total flow rate is obtained by fitting the current in the absence of the reaction (see below). The main flow entrance was taken to contain the given concentration of the polyelectrolyte. The current is obtained from the simulations by integrating the flux of surfactant over the detector interface. As the applied interfacial potential difference was sufficiently large (wrt the standard potential of transfer of the ion), the surface concentration in aqueous phase tends to zero and the current is only limited by the convective diffusion of surfactant in the aqueous phase

∫

I ) -FDswσ

lσ

0

|

∂cs dx ∂y y)0

(10)

The finite element mesh was refined adaptively to attain convergence of the calculated current with a reasonable number of elements, typically of the order of 104. In all the simulations, the diffusion coefficients of all the species (CP, PSS and CP-PSS complexes) were taken to be equal to 5.0 × 10-6 cm2 s-1. Simulated concentration profiles are plotted in Figure 3. The injection of the surfactant generates a locally high surfactant concentration, and this results in depletion of the polyelectrolyte due to the association reaction. The convective flow in the channel confines these changes close to the channel wall, and a diffusion layer is formed. RESULTS AND DISCUSSION The channel flow cell was tested with tetraethylammonium ion, which is a prototypical probe ion in liquid-liquid electrochemistry. Examples of the cell performance with respect to the sweep and flow rates are shown in Figure S-3 (Supporting Information). The limiting current was a linear function of the cubic root of total flow rate, and the slope was in accordance with the expected value given the dimensions of the channel and the diffusion coefficient (24) In addition, we have solved the Navier-Stokes equations simultaneously with the convective diffusion equation to study the solvent flow in the injection region of the cell. Some results are shown in Figure S-2 (Supporting Information). The main conclusions of these calculations are that the flow remains laminar, no eddies are formed, and the flow profile reaches the normal parabolic shape very soon after the injection.

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Figure 3. Simulated concentration profiles in the channel with typical values of the parameters: Vt ) 0.003 cm3 s-1, K ) 6 × 103 dm3 mol-1, kf ) 6 × 102 dm3mol-1s-1, and kb ) 0.1 s-1. From top to bottom, the concentrations for the surfactant, polyelectrolyte, and the complex, respectively. Red denotes high and blue low concentrations. The positions of the injection slit and the detector electrode are indicated by thick black lines.

of TEA+.15,20 In the actual measurements, the bulk of the data was collected by applying a constant interfacial potential difference and monitoring the resulting current. The actual applied potential difference is less important in this case as CP transfers from the aqueous to the organic phase throughout the available potential window; see Figure S-4 (Supporting Information).5 The injection mechanism was tested with CP as the probe ion in the absence of PSS. The current at the detector electrode as a function of the volumetric flow rate in the main channel is shown in Figure 4A. These experiments were used as a calibration to obtain the masstransfer coefficient by using the eq 7 with zero reaction rate. The results as a function of the flow rate are shown in Figure 4B.

Figure 4. Results of calibration experiments carried out with CP as the probe ion in the absence of polyelectrolyte. (A) The current as a function of total flow rate. (B) Filled circles: Mass-transfer coefficient, m, as a function of total flow rate. Empty circles: The injection flux, J as a function of total flow rate.

Surprisingly, the mass-transfer coefficient is linearly dependent on the total flow rate and can be fitted by the following equation.

m ) 0.11Vt - 1.95 × 10-5 cm3/s

(11)

where total flow rate Vt is given in cm3 s-1. These calibration experiments with CP were also used to extract the injection flux used in the solution of the full convective mass transport problem, and these results are also shown in Figure 4B. In the simulation, the diffusion coefficient of CP was taken to be 5 × 10-6 cm2 s-1.5 It can be seen that, with sufficiently high total flow rate (>0.002 cm3 s-1), the injection flux is essentially constant. This indicates that the injection stream does not disturb the flow profile in the main channel. As expected, the kinetics of the binding of CP to PSS was very fast. However, as will be demonstrated below, the available flow rate range was sufficient to guarantee that the CP-PSS system was not in equilibrium. In the experiment, the total flow rate was varied between 5 × 10-4 and 5 × 10-3 cm3 s-1 while the flow rate of the injection slit (when in use) was kept fixed at 2.8 × 10-4 cm3 s-1, corresponding to residence times between 5 and 50 s. The experimental runs were done with three different PSS main flow concentrations: 5 (PSS5), 10 (PSS10), and 97 µM (PSS97). The concentration of CP in the injection flow was 130 µM, which is below cmc of the surfactant. The best fits to the analytical model are given in Figure 5, and the corresponding rate constants are listed in Table 1. As can be seen from Figure 5, the correspondence between theory and experiment is good. This is not the case if the reaction is assumed to be irreversible, shown for comparison in Figure S-5 (Supporting Information). The results for PSS5 proved to be most suitable for evaluation of the kinetic parameters. The concentration ratios of CP and PSS for the cases of PSS10 and PSS97 were so low that the reaction was essentially at equilibrium throughout the studied flow rate range and the current response was not sensitive to kb. Nevertheless, these results could be used to establish the equilibrium constant (PSS10) or a lower bound for it (PSS97). The observed trends can be understood in the context of the Aniansson and Wall model, which was developed for micelle formation kinetics. In this model, two different steps can be distinguished: A fast, initial step that

Figure 5. Experimental results and fits to the analytical model. Empty triangles, filled circles, empty circles, and filled triangles refer to PSS0, PSS5, PSS10, and PSS97, respectively. Solid lines refer to the reversible model discussed in the text.

describes the binding of an individual (noninteracting) surfactant ion is followed by a slower stage when the micelle is close to equilibrium (cooperative effects).25 The downward trend of the forward rate constant with decreasing concentration of PSS can be rationalized by a higher rate of the association of the first surfactants. It should be borne in mind that, at highest concentration of the polyelectrolyte, the degree of binding is actually the lowest. This is also explains the fact that a model where the association step is assumed to be irreversible fits the experimental data reasonably only at the highest polyelectrolyte concentration; see Figure S-5 (Supporting Information). The further association of CP might induce some minor changes in the conformation of the surfactant-polyelectrolyte complex, slowing down the rate of further association, which is seen in the values for kf of the lowest polyelectrolyte concentration, PSS5. Although the cooperative effects are not included in the present model, they are reflected in the values of forward rate constant and the equilibrium constant. The equilibrium constant increases with increasing polyelectrolyte concentration (decreasing the degree of binding) and (25) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905922.

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Table 1. Kinetic Parameters from the Analytical and Numerical Models analytical model

PSS5 PSS10 PSS97

numerical model

kf (dm3 mol-1 s-1)

kb (s-1)

K (dm3 mol-1)

kf (dm3 mol-1 s-1)

kb (s-1)

K (dm3 mol-1)

1.0 × 103 >4 × 103 >1 × 104

0.2 >0.5 >0.5

4.4 × 103 7.9 × 103 >2 × 104

4.8 × 102 >1 × 104 >8 × 104

0.08 >0.4 >0.4

6 × 103 3 × 104 >1 × 105

Figure 6. Comparison between the experiment (PSS5, filled circles) and the numerical solution of the convective diffusion problem. (A) Effect of K: kb ) 0.08 s-1 and K ) 5 × 103 (dotted line), 6 × 103 (solid line), and 7 × 103 dm3 mol-1 s-1 (dashed line). (B) Effect of kb: K ) 6 × 103 dm3 mol-1 s-1 and kb ) 0.04 (dotted line), 0.08 (solid line), and 0.16 s-1 (dashed line).

approaches the intrinsic equilibrium constant of the cooperative theory.5 This is understandable, since at a low degree of binding, the reaction between the surfactant cation and the polyelectrolyte occurs through a non cooperative pathway. To check the validity of the simple model, the experimental results were also compared with the numerical solution of the full diffusion problem. The sensitivity of the theoretical response to the kinetic parameters is illustrated in Figure 6, where experimental data from PSS5 is compared with theoretical predictions. As can be seen from Figure 6, the value of the equilibrium constant has an approximately constant effect throughout the studied flow range. On the other hand, the actual values of rate constants can be obtained from the shape of the current as a function of the flow rate. We are still able to extract kinetic data; i.e., the reaction is not at equilibrium. The best fits for all PSS concentrations are shown in Figure 7. The obtained kinetic parameters are shown in Table 1. There is remarkable agreement between the simple analytical model and the more sophisticated numerical calculations. We can observe the same trends in the rate and equilibrium constants obtained from the two models. The lowest concentration of the polyelectrolyte (PSS5) again gives the most precise information, and the following discussion on the differences between the two modeling approaches concentrates on this set of data. There is a factor of 2 difference between the kinetic parameters derived from the different models. This overestimation is most probably due to the approximations made in deriving the residence time in the analytical model. This is given simply by the channel dimensions and the total flow rate, thus resulting in an average residence time of the surfactant. In reality, the velocity profile in the channel is parabolic and the velocity is very small close to the channel walls. This would result in an underestimation of the residence time, 6900 Analytical Chemistry, Vol. 77, No. 21, November 1, 2005

Figure 7. Experimental data (symbols) and the best fits to the numerical model (solid lines), from top to bottom PSS0, PSS5, PSS10, and PSS97. The parameters are given in Table 1.

which consequently would lead to an overestimation of the kinetic parameters. Also, the dimensions of the detector electrode and the injection slit are comparable to the distance between them, which contributes to further uncertainty in the residence time. These problems have been overcome in the numerical model, in which the velocity profiles and channel dimensions are given precisely. The equilibrium constants were same order of magnitude in both models. Considering the simplicity of the analytical model, it has been shown to give surprisingly accurate results on the association kinetics of the system under study. CONCLUSIONS The kinetics of the binding of CP ion to PSS was studied with the novel channel flow cell configuration. The experimental results

were modeled with the reversible second-order reaction, which was solved analytically with an approximate description of the mass transport. The predictions of this simple model were shown to be remarkably good by solving the full convective diffusion problem numerically using the finite-element method. The analytical model overestimates the kinetic parameters slightly due to the approximations made in deriving the residence time of the reactants. The surfactant-polyelectrolyte association is a complex series of interaction phenomena. The suitability of the simple reversible second-order model in explaining the results can be explained by the low concentration range used. In this case, only a few surfactants associate with the empty polyelectrolyte binding sites and the cooperativity, which is a typical property of polyelectrolyte-surfactant systems, can be assumed to have only a minor effect on the association reaction. This is corroborated by the measured values of the equilibrium constants, which were similar to the intrinsic equilibrium constant that describes the equilibrium in a noncooperative system. Further development of

this type of channel flow systems should enable the study of complex cooperative kinetics of surfactant-polyelectrolyte systems. ACKNOWLEDGMENT The Academy of Finland is thanked for financial support. SUPPORTING INFORMATION AVAILABLE The conversion as a function of the residence time, examples of the simultaneous solution of the Navier-Stokes and the convective diffusion equations, examples of the voltammetric response of the channel flow cell in the presence of TEA+ or CP, and fits of the measured current to an analytical model with irreversible association reaction. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review May 30, 2005. Accepted September 8, 2005. AC050947P

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