Surfactant Systems with

Mar 30, 2004 - The molecular reorientation model of Fainerman et al. is conceptually adapted to explain the dynamic surface tension behavior in polyel...
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Langmuir 2004, 20, 3648-3656

Dynamic Surface Tension of Polyelectrolyte/Surfactant Systems with Opposite Charges: Two States for the Surfactant at the Interface Herna´n A. Ritacco* and Jorge Busch Laboratorio de Sistemas Lı´quidos, Facultad de Ingenierı´a, Universidad de Buenos Aires, Buenos Aires, Argentina Received November 7, 2003. In Final Form: February 19, 2004

The molecular reorientation model of Fainerman et al. is conceptually adapted to explain the dynamic surface tension behavior in polyelectrolyte/surfactant systems with opposite charges. The equilibrium surface tension curves and the adsorption dynamics may be explained by assuming that there are two different states for surfactant molecules at the interface. One of these states corresponds to the adsorption of the surfactant as monomers, and the other to the formation of a mixed complex at the surface. The model also explains the plateaus that appear in the dynamic surface tension curves and gives a picture of the adsorption process.

Introduction During the past few years, mixtures of polymers and surfactants have been actively studied. The surfactants are used in practical formulations to control the surface properties, and the polymers are added to control the rheology of the systems. Because of the necessity of controlling both properties, polymer/surfactant systems are nowadays highly present in industry. Among their technology applications, they are found in pharmaceutical formulation, paints and coatings, industrial detergents, oil recovery, personal care, and so forth.1 On the other hand, the knowledge of basic properties is challenged by the complexities of the interactions involved. To understand these systems, a great number of experimental techniques have been used: viscosity, conductance, thermochemical and volumetric parameters, surfactant binding isotherms, phase diagrams, and so forth.1,2 The modelization at a microscopic level of polymer-surfactant aqueous mixtures has been improved using more sophisticated experimental techniques as NMR or fluorescence.1 The behavior of a polyelectrolyte in a solution is ruled by the balance of hydrophobic and hydrophilic interactions of the polymeric segments either between themselves or with the solvent. In a similar way, surfactant aggregates in solution are ruled by hydrophilic, hydrophobic, or ionic interactions. Hence, aqueous solutions which contain polymer/surfactant mixtures present a great diversity of behaviors. This becomes richer when the surfactant and the polymer bear opposite electrical charges. As regards surface tension measurements, it has been found that many polymer/surfactant systems act in a cooperative way in the interface air/solution. As a result, a decrease in the surface tension, the value of which is greater than that expected for the polymer or the surf* To whom correspondence should be addressed. E-mail: [email protected]. (1) Polymer Surfactant Systems; Kwak, J. C. T., Ed.; Surfactant Science Series, Vol. 77; Marcel Dekker: New York, 1998. (2) Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1993.

actant in its aqueous solution, is observed. This “synergic” effect is particularly interesting, and it has been widely studied in equilibrium conditions.3-5 In most industrial applications, the processes take place out of equilibrium. The involved interfaces have very short effective lives, down from some seconds to 1 ms. Thus, from a technological point of view, knowledge about the adsorption dynamics is more important than that corresponding to the equilibrium.6-8 In this paper, we present complementary investigations of dynamic surface properties on the PAMPS/ DTAB systems previously reported.9,10 As far as we know, there are no theories for dynamic surface tension in these systems. We propose the use of a model to explain the adsorption process, which is based on the model for one surfactant with the possibility to adopt two different orientations at the interface air/solution.11 Materials and Methods Materials. PAMPS is an anionic statistical copolymer of neutral acrylamide monomers and charged monomers of acrylamido methyl propane sulfonate (PAMPS). In this work, we use polymers with a degree of charge of 25%, 20%, and 10% (ratio of number of sulfonated monomers to the total number of monomers), which we call PAMPS25%, PAMPS20%, and PAMPS10%, respectively. The polymers were synthesized by SNF Floerger, dissolved in deionized water, and purified with an ultrafiltration unit with a 20 000 Da cutoff membrane. The (3) Atef Asnacios, M. Melanges polyelectrolytes-tensiactifs en solution aqueuse: complexation a l’interface eau/air et structure des films de mouse. The`se, Universite´ de Paris VI, Paris, France, 1997. (4) Asnacios, A.; Langevin, D.; Argillier, J. F. Macromolecules 1996, 29, 7412. (5) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 1550. (6) Kretzschmar, G.; Miller, R. Adv. Colloid Interface Sci. 1991, 36, 65. (7) Malysa, K. Adv. Colloid Interface Sci. 1992, 40, 37. (8) Lucassen-Reynders, E. H.; Kuijpers, H. A. Colloids Surf. 1992, 65, 175. (9) Ritacco, H.; Kurlat, D.; Langevin, D. J. Phys. Chem. B 2003, 107, 9146. (10) Ritacco, H.; Kurlat, D. Colloids Surf., A 2003, 218, 27-45. (11) Fainerman, V. B.; Miller, R.; Wu¨stneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 7669.

10.1021/la036097v CCC: $27.50 © 2004 American Chemical Society Published on Web 03/30/2004

Dynamic Surface Tension of Surfactant Systems

Langmuir, Vol. 20, No. 9, 2004 3649

polymer molecular weight, the mean monomer molecular weight, and the polymerization degree are as follows:

agreement with known values acquired using the MBP technique, even for time slightly less than 1 s.

PAMPS25%: Mw ∼ 4 × 105 g/mol; MMonomer ∼ 121; N ∼ 3400

Theoretical Background

PAMPS20%: Mw ∼ 4 × 105 g/mol; MMonomer ∼ 114; N ∼ 3600 PAMPS10%: Mw ∼ 4 × 105 g/mol; MMonomer ∼ 100; N ∼ 4100 The surfactant is dodecyl trimethyl ammonium bromide (DTAB), which was supplied by SIGMA and used without further purification (>99% purity). Deionized water was obtained by DEIONEX MS 160 equipment (water resistivity, 12 MΩ). All the measurements were performed at T ) 298 K ((0.2). The solutions were prepared from concentrated stock solutions of polymer and surfactant, which were diluted and mixed. The experiments with mixed solutions were done 7 days after the preparation. All solutions were kept in a refrigerator between measurements and were discarded after 1 month. Methods. The equilibrium surface tension was measured with the Wilhelmy plate technique,4 and the dynamic surface tension, that is, the change in the surface tension (with time) until it reaches equilibrium, was measured with the maximum bubble pressure methods described below. Maximum Bubble Pressure Method. The dynamic surface tension was measured using homemade equipment.9 The maximum bubble pressure (MBP) technique involves measuring the maximum pressure necessary to detach a bubble in a liquid from the tip of a capillary. The surface tension can thus be determined from the Young-Laplace equation:

(

∆P ) γ

)

1 1 + R1 R2

(1)

where γ is the surface tension, ∆P is the pressure difference between the gas in the bubble and the liquid, and R1 and R2 are the two principal radii of curvature. For a spherical interface, we have

γ)

Adsorption Isotherms for a Surfactant with Two Different Orientations at the Interface. Fainerman et al. reported a model that explains the isotherms of surface tension for surfactants that may adopt two different orientations at the interface.11 These two orientations are characterized by different values for the partial molar area, wc (state 1) and wm (state 2), and for the parameters bc and bm, related to the surface activities of each orientation state (the reason for the subscripts c and m will be explained later). The authors used Buttler’s equation14 for surfactant mixtures (ideal system), considering the different surfactant orientations at the interface as pseudocomponents. Thus, they obtained the following relations for their model: The surfactant concentration in the solution, c, is related to the surface concentration of each orientation state, Γc and Γm, by

c)

(12) Miller, R.; Kretzschmar, G. Adv. Colloid Interface Sci. 1991, 37, 91. (13) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1993, 160, 435.

bm(1 - Γws)wm/ws

)

( )

wm R Γcws (2) wc b (1 - Γw )wm/ws m s

where the following relation is assumed to hold between the surface activities of each state and their partial molar areas:11

( )

b m ) bc

wm wc

R

(3)

R ) 0 corresponds to a system where the surface activity is not dependent on the partial molar area, while R > 0 means that the state with greater w has a greater surface activity. In eq 2, ws denotes the mean partial molar area:15

ws )

∆PR 2

where R is the radius of the bubble. The equipment has a glass capillary with an internal radius of 0.213 mm. The capillary is plunged into the solution with the tip situated 3 mm away from the surface. The pressure is generated by a peristaltic pump connected to a 250 cm3 air reservoir in order to damp the pressure fluctuation due to the pumping (and to increase the total volume of the system). This reservoir is connected to a valve which controls the air flow to the capillary. The pressure difference between the inner and outer sides of the bubble is measured with a pressure transducer (maximum pressure, 25 cm H2O; accuracy, 1% of the maximum pressure) connected to a computer. This setup allows the measurement of the dynamic surface tension of surfaces whose ages are greater than or equal to 0.05 s. The solution is located in a 40 mL closed glass container, placed in a chamber that is temperature-controlled via water circulation ((0.2 °C). In this technique, the surface grows continuously. So, the surface age is less than the time elapsed between the bubble formation and the detachment.12-13 The instrument was tested with TRITON X-100 aqueous solutions. The surface tension variation with time is in good

Γmws

wcΓc + wmΓm Γc + Γm

(4)

where the contribution of each state c and m is weighted by its surface concentration Γc and Γm. Assuming that w0 ) ws11 and locating the Gibbs separation surface in such form that the total adsorption of all the components equals the saturation adsorption value,16,17

Γ0 + Γc + Γm ) 1/ws

(5)

where the subscript 0 refers to the solvent. Thus, the surface state equation becomes

Π)-

RT RT ln(1 - Γcwc - Γmwm) ) ln(1 - Γws) ws ws (6)

where Γ ) Γc + Γm and Π ) γ0 - γ (definition of surface pressure). Here R denotes the gas constant and T the temperature. (14) Defay, R.; Prigogine, I. In Tension Superficielle et Adsorption; Desoer Ed.: Lie`ge, 1951; Chapter XI, p 144. (15) Fang, J. P.; Joos, P. Colloids Surf., A 1994, 83, 63. (16) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (17) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1972, 91, 156; 1982, 85, 178.

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The relationship between the orientation states is18

( ) [

Γc wc ) Γm wm

R

exp

]

Π(wm - wc) RT

c)

Πws 1 - exp RT Πwc Πwm wc R exp + exp wm RT RT

[( ) (

bm

)

)

φor ) km-cΓm - kc-mΓc

(7)

where the partition of the surfactant in the two states is ruled by their partial molar areas and R. The adsorption isotherm is then given by

(

The orientation flux is then

)]

(

We can give an expression for kc-m and km-c in terms of the isotherm parameters, using the equilibrium isotherm relations. The orientation flux is then

φor ) (8)

[( )

( ) ( ( ) (

ws )

wc wm

wc 1+ wm

R

R

exp -

)

Π(wc - wm) RT

)

Π(wc - wm) exp RT

Then eqs 10 may be rewritten as

dΓc dΓ ) χ + φor dt dt dΓm dΓ ) (1 - χ) - φor dt dt (9)

These last two equations define the adsorption isotherm. If we have the experimental curves, we may fit them by adjusting the values for the parameters using eqs 8 and 9. Dynamic Surface Tension for One Surfactant with Two Possible Orientations at the Interface. A complete description of the adsorption process should address the main subprocesses involved in its dynamics, that is, (1) transport and diffusion of surface-active agents from the bulk to the interface, (2) the adsorption-desorption kinetic process between the surface and the bulk, and (3) the reorientation process of the adsorbed molecules at the interface. When the three processes are taken into account, only a numerical solution is possible. Under this model, the randomly oriented molecules adsorb at the interface in one of the two states c and m, with probability χ and (1 - χ), respectively. The adsorption induces a flux from the bulk to the interface, while the adsorbed molecules reorient until they reach equilibrium. Thus, the evolution in time of the adsorption in both states may be described by19

dΓc ) φac - φdc + φor dt dΓm ) φam - φdm - φor dt

(10)

where φa and φd denote the adsorption and desorption fluxes, and φor denotes the contribution of reorientation to the flux. From now on, we will assume that the characteristic time of the adsorption-desorption process is very small when compared with those of the other processes (see above). The distribution in both states is analyzed as a kinetic process kc-m

]

wc R (1 - wsΓ)(wc-wm)/wsΓm - Γc kc-m (12) wm

Thus, eq 4 becomes

wm + wc

(11)

If we use Fick’s equation for the diffusion from the bulk to the interface, with the boundary conditions imposed by the mass balance at the surface, we obtain the whole description of the adsorption process by the well-known Ward and Tordai equation:20

Γ(t) ) 2

1/2

[ct1/2 -

∫0t cs(t - τ) dxτ]

(14)

( )

Γmws wm R Γcws + χ cs ) (1 - χ) wc b (1 - Γw )wm/ws bm(1 - Γws)wm/ws m s (15) Equations 15, 14, 13, 12, and 4 describe the adsorption dynamics resulting from the reorientation process and the kinetics of interchange between the bulk and the interface. The solution is obtained numerically as is to be shown in the following section. Numerical Solution of the Equations. We will follow the approach given in ref 21, with some modifications that we shall outline here. After the standard normalizations,22 and denoting with a hat the corresponding normalized variables (i.e., Γˆ for Γ, Γˆ c for Γc, ˆt for t, etc.), we obtain from eqs 14 and 13 the coupled equations

Γˆ (tˆ) )

(

2 xˆt xπ

∫0ˆt

cˆ s(tˆ - τ) 2xτ

dτ

)

d k (χΓˆ m - (1 - χ)Γˆ c) ) Γˆ c - cqΓˆ m dtˆ

Γc 98 Γm 98 Γc

(18) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1991, 145, 291. (19) Ravera, F.; Liggieri, L.; Miller, R. Colloids Surf., A 2000, 175, 51.

(Dπ )

where D is the diffusion coefficient, involving eventually the energetic barriers, cs is the subsurface concentration, and c is the bulk concentration. At this point, the problem has been reduced to find an expression for cs. In eq 1, the two terms are equal only at equilibrium. As the subsurface concentration is the sum of the concentrations of both adsorption states, linked to the probabilities χ and (1 - χ), we may state

km-c

where kc-m and km-c are the velocity constants for the transitions from state c to state m and vice versa.

(13)

(16) (17)

where (20) Ward, A. F. H.; Tordai, L. J. Phys. Chem. 1946, 14, 543. (21) Aksenenko, E. V.; Makievski, A. V.; Miller, R.; Fainerman, V. B. Colloid Surf., A 1998, 143, 311. (22) Miller, R.; Aksenenko, E. V.; Liggieri, L.; Ravera, F.; Ferrari, M.; Fainerman, V. B. Langmuir 1999, 15, 1328.

Dynamic Surface Tension of Surfactant Systems

(

cq ) β exp -

Π (wc - wm) RTΓ0

)

β)

( ) wc wm

k)

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R

( )( ) D

kc-m

c Γ0

2

Following ref 21, we will approximate the values cˆ s(tˆn), Γˆ c(tˆn), and Γˆ m(tˆn) at times ˆtn ) n2δ2. Assuming that we have obtained these approximations up to n steps, from eq 16 we obtain the following estimate for Γˆ (tˆn+1) in terms of Γˆ (tˆn) and cˆ s(tˆn), cˆ s(tˆn+1),

Γˆ (tˆn+1) )

n Γˆ (tˆn) + n + 1 + x2n + 1 2 - cˆ s(tˆn+1) - cˆ s(tˆn) 2 x2n + 1 δ (18) 2 xπ

After obtaining by finite differences a rough first estimate of cˆ s(tˆn+1) and cq(tˆn+1), a first estimation of Γˆ (tˆn+1) is calculated using eq 18. With these first estimates, we can start the iterative procedure: First, solving eq 17 by finite differences, an estimate of χΓˆ m(tˆn+1) - (1 - χ)Γˆ c(tˆn+1) is calculated. Then, Γˆ c(tˆn+1) and Γˆ m(tˆn+1) are obtained using the equation Γˆ c + Γˆ m ) Γˆ and the previous estimate of Γˆ . With these Γˆ c and Γˆ m values and using the normalized version of eq 15, a new estimate for cˆ s(tˆn+1) and cq(tˆn+1) is obtained. Finally a new estimate of Γˆ (tˆn+1) is calculated using eq 18. We have now new estimates for cq(tˆn+1) and Γˆ (tˆn+1), which will be the starting point at the new step of iteration. The iterative procedure will continue until two successive approximations of Γˆ c and Γˆ m are very close. Experimental Results Equilibrium Surface Tension. In Figure 1, we show the surface tension isotherms of all the systems addressed in this work. The surface tension for free surfactant solutions of polyelectrolytes is not different from that of the pure solvent, at least at the polymer concentrations concerned (50-500 ppm), Figure 1a. Analogously, in Figure 1b there is no surface activity in polyelectrolyte free surfactant solutions, up to a concentration of ∼0.2 mM. But in the polyelectrolytesurfactant systems, we see a steep descent of surface tension at concentrations where DTAB and PAMPS solutions show no surface activity. This behavior has been widely studied.3-5,9,10 The curves for mixed systems are, within the mentioned range, rather independent of the polyelectrolyte concentration, as already reported.3-5 The plateaus observed in the surface tension isotherms of systems that contain PAMPS25% and PAMPS20% are associated to the critical aggregation concentration (cac). This concentration is related to the moment when many polymer-surfactant complexes appear in bulk, and it is coincident with the beginning of the plateau in the γ(c) curve. The cac is usually much lower than the critical micelle concentration (cmc) for the polymer free surfactant solutions (for DTAB, cmc ∼ 15 mM). The cac for the PAMPS25%/DTAB system is close to 0.8 mM, and for the PAMPS20%/DTAB system, cac ∼ 1 mM. For PAMPS10%/DTAB, the cac is not determined because the γ(c) curve shows no plateau, Figure 1b. Dynamic Surface Tension. In Figure 2, a typical dynamic surface tension curve for mixed systems below the cac is illustrated.

Figure 1. (a) Equilibrium surface tension as a function of polymer concentration. PAMPS(10%) (squares); PAMPS(20%) (circles); PAMPS(25%) (up triangles). (b) Surface tension as a function of surfactant concentration. DTAB solutions (squares); PAMPS(10%) 175 ppm/DTAB systems (up triangles); PAMPS(20%) 65 ppm/DTAB (down triangles); PAMPS(25%) 65 ppm /DTAB (diamonds).

Figure 2. Typical dynamic surface tension curve. System: PAMPS(25%) 65 ppm/DTAB 0.2 mM.

In Figure 3, the corresponding curves with the same DTAB concentration, polyelectrolyte free, are compared with those for the mixed system, at DTAB concentrations close to or above the cac. In these curves, plateaus are observed. The first plateau coincides with the static value for the polyelectrolyte free surfactant solution (∼equilibrium). After some time, the surface tension continues its descent, falling finally to the equilibrium values for the mixed system (which is not noticeable due to the limited MBP time window).

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Figure 4. Schematic of the state change caused by the polymer arrival at the interface.

Figure 3. Dynamic surface tension. (a) PAMPS(25%) 65 ppm + DTAB 0.7 mM (circles); +DTAB 1 mM (up triangles); +DTAB 2 mM (squares). DTAB solutions of concentrations 0.66 mM (continuous line); 1 mM (dashed line); 2 mM (dashed-dotted line). (b) PAMPS(20%) 65 ppm + DTAB 1 mM (circles); +DTAB 4 mM (squares).

For concentrations above the cac, in systems with PAMPS25% or PAMPS20%, oscillations and steep falls of the dynamic surface tension are observed, Figure 3a,b. This behavior is associated with the appearance of the mixed complexes in bulk (cac) and their adsorption.9,23 This produces a block incursion of surfactant in the interface, causing the steep descents and oscillations in γ(t).23 This behavior is not present in the PAMPS10% systems at any surfactant concentration below 10 mM. In the following, we will focus on results with concentrations under the cac. Discussion Equilibrium Isotherms. We shall assume that surfactant molecules may adsorb in two different states. In our case, these states are not associated with the orientation of the molecules at the interface, as in the original model. The first state corresponds to surfactant adsorption in the absence of polyelectrolyte (subscript m for “monomer”), with parameters wm, bm, and Γm. In the other state, the presence of the polyelectrolyte at the interface forces the surfactant to change its values to wc, bc, and Γc (subscript c for polymer/surfactant “complex”). Figure 4 illustrates these ideas. Let us assume that the DTAB molecules approaching the interface have essentially the macro-ions as counter(23) Ritacco, H. Polielectrolitos y surfactantes de carga opuesta en solucio´n acuosa. Un estudio de efecto Kerr y tensio´n superficial dina´mica. Tesis Facultad de Ingenierı´a, Universidad de Buenos Aires-Laboratoire de Physique de Solides, Universite´ de Paris XI, 2003.

Figure 5. Experimental points (squares); Model (line) (a) PAMPS10%/DTAB, model parameters: wc ) 5.45 × 10-5 m2 mol-1, wc ) 2.7 × 105 m2 mol-1, R ) 7.8, bm ) 0.5 m3 mol-1. (b) PAMPS25%/DTAB, model parameters: wc) 5.5 × 10-5 m2 mol-1, wm ) 2.6 × 105 m2 mol-1, R ) 8.8, bm ) 0.42 m3 mol-1.

ions.23 Once there, they adsorb (in state m), forcing the polyelectrolyte with opposite charge to come into the interface, causing the reorganization of the surfactant molecules (state c). With this picture in mind, we can use eqs 8 and 9 to fit the experimental curves. Figure 5 illustrates the fitting results for PAMPS10% and PAMPS25%. The isotherms are represented by Π(c) instead of γ(c) to be consistent with the form in which these equations are presented. The results of these fittings are shown in Table 1. The obtained data for polyelectrolyte free DTAB solutions are also included. In this last case, eqs 8 and 9 result in Szyskowski’s equation24 (wc ) wm ) w). The values obtained from this fit for the surfactant solutions are w ) (2.6 ( 0.1) × 105 m2 mol-1 and b ) (0.15 ( 0.02) m3 mol-1. These last values are in good agreement with the values obtained for the same surfactant.25 (24) von Szyszkowski, B. Z. Phys. Chem. (Leipzig) 1908, 64, 385. (25) Fainerman, V. B.; Lucassen_Reynders, E. H. Adv. Colloid Interface Sci. 2002, 96, 295.

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Figure 6. Monolayer conformation in PAMPS/DTAB systems. Table 1. Model Parameters wc [m2 mol-1]

solutions DTAB PAMPS(10%)/DTAB PAMPS(20%)/DTAB PAMPS(25%)/DTAB

(5.45 ( 0.3) × 105 (5.5 ( 0.2) × 105 (5.5 ( 0.2) × 105

wm [m2 mol-1] w ) (2.6 ( 0.1) × (2.7 ( 0.2) × 105 (2.6 ( 0.1) × 105 (2.6 ( 0.2) × 105

On the other hand, the value obtained for w is close to the expected one. This parameter can be estimated by the limit slope of the γ(ln(c)) curve:26

w)-

nRT 1 ) (dγ/d ln(c))cfcmc Γ∞

(19)

For DTAB, the w value obtained corresponds to a surface area occupied by a surfactant polar head of ∼43 Å2, very close to the value determined previously by neutron reflectivity.27,28 All the fits for the systems with PAMPS give the same values for the parameters wc and wm, which are estimated to be 5.5 × 105 m2 mol-1 and 2.6 × 105 m2 mol-1, respectively. Substituting these values in eqs 4, 6, and 7, we obtain a surfactant polar head area of ∼90 Å2, a little greater than the value obtained using a model derived from the Gibbs adsorption equation.3-5,9,10 Notice the coincidence in the values obtained for wm in all the systems, also coincident with the value found for the DTAB/water system. It is thus obvious that the state with the smallest partial molar area in the mixed systems corresponds to the adsorption of DTAB molecules as monomers, which is exactly our original hypothesis. The value of wc corresponds to the area occupied by the surfactant, but this time forced by the presence of polyelectrolyte molecules into the interfacial zone (Figure 4). At the surface, the polymer acts like spacer between DTAB molecules. Owing to the relative errors in b and R, we shall not derive any conclusion about these values in an absolute sense (b and R are related to the relative surface activities of both states and, in the state c, it is influenced by the polymer presence). (26) Fainerman, V. B.; Miller, R. J. Phys. Chem. B 2001, 105, 46, 11432. (27) Simister, E. A.; Thomas, R. K.; Penfold, J; Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A. J. Phys. Chem. 1992, 96, 1383. (28) Lyttle, D. J.; Lu, J. R.; Su, T. J.; Thomas, R. K. Langmuir 1995, 11, 1001.

105

bm [m3 mol]

R

b ) 0.15 ( 0.02 0.5 ( 0.4 0.45 ( 0.3 0.42 ( 0.4

independent 7.8 ( 3 8.7 ( 2 8.8 ( 3

On the other hand, for any values of R and b in a particular fitting, the ratio between the surface concentrations given by eq 7 decreases continuously when the surfactant concentration increases. This means that the surface concentration of state m (surfactant monomer) increases with respect to that of state c (mixed complex), which is the qualitative behavior indicated in Figure 1b. When the surfactant concentration increases, the surfactant replaces completely the mixed complex at the interface, driving the isotherms of the mixed system to coalesce with the surfactant ones. For instance, for the PAMPS25%/DTAB system, in the range 0 < c < 1 mM, the ratio of surface concentrations varies from Γc/Γm ∼ 700 to Γc/Γm ∼ 34. It is also clear that at equilibrium, state c (polymer/surfactant complex) is dominant for bulk concentrations lower than the cac. This result is in agreement with the synergistic decay of the surface tension due to the formation of mixed complexes at the interface. Comments on the Mixed Monolayer. The previous results show that the surface area occupied by one surfactant polar head, ∼90 Å2, is independent of the charge degree of the polymer (PAMPS10%, 20%, and 25%). We can estimate the distance between adjacent surfactant molecules at the interface by the square root of this area, that is, ∼9 Å. But the average distance between adjacent charged groups in the polymer chain is different for each one. These distances obtained from the bond angles and lengths are ∼7.5 Å for PAMPS25% and 24 Å for PAMPS10%.23 So, at the interface, if each charge on the polymer is neutralized by a surfactant molecule (at equilibrium), the mixed monolayer must be denser (δ) and rougher (R) for PAMPS10% than for PAMPS25%. These ideas are summarized in Figure 6. On the other hand, from the molecular geometry, we can estimate the monolayer thickness to be ∼24 Å (see Figure 6). All these predictions are in good agreement with results from X-ray reflectivity measurements.29,30 (29) Stubenrauch, C.; Albouy, P. A.; Klitzing, R.; Langevin, D. Langmuir 2000, 16, 3206.

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Dynamic Surface Tension. The adsorption dynamics has a complex behavior involving several subprocesses with different characteristic times. For the analysis of this dynamics, it is useful to estimate the time for each involved subprocess. For the diffusion characteristic time τD from the bulk to the subsurface, the following expression was proposed:31

τD )

( )

1 Γ∞ D c

2

(20)

where D is the diffusion coefficient, and the ratio Γ∞/c (equilibrium surface excess/bulk concentration) represents the surface activity properties of the system. For DTAB at a concentration c ) 1 mM, with D ) 4 × 10-10 m2 s-1 and Γ∞ ) 9 × 10-7 mol m-2, τD is ∼0.002 s, while for c ) 0.1 mM, τD is ∼0.2 s.10,23 In the case of PAMPS and using D ∼ 1 × 10-12 m2 s-1 and c ) 100 ppm ) 2.44 × 10-4 mol m-3, the polymer diffusion characteristic time is τD ∼ 100 s.9,23 As a conclusion in the first stage, for time close to 1 s, the adsorption is mainly of the surfactant monomers, whose dynamics is retarded by the presence of polyelectrolyte. This is confirmed by the superposition of the γ values at equilibrium for polymer free surfactant solutions and the plateau that appears in mixed systems at t ∼ 1 or 2 s (see Figures 2 and 3). The second stage is dominated by the arrival of the polyelectrolyte molecules and by the reorganization of surfactant molecules to accommodate them. The equilibrium isotherm parameters determined by the fittings can be now used to fit the dynamic curves γ(t) (or Π(t)). To obtain the dynamic surface pressure, Π(t), we shall use the scheme previously described. In that, the adsorption-desorption kinetics is neglected; it is assumed that its characteristic time is very small when compared with those of the other subprocesses. We can estimate the characteristic time for this process, τk, in the absence of any potential barrier19 as

1

τk ) wsc

x

RT 2πm

(21)

where m is the molar mass and c is the bulk concentration. For the surfactant, we estimated τk ∼ 10-8 s, and for the polyelectrolyte τk ∼ 10-6 s. It is much smaller than τD and, we will see, than τor. The computer program allows us to give alternative values to the model parameters, wc, wm, bm, R, D, kc-m, and χ. Of these parameters, we have obtained the first four by fitting the equilibrium isotherms. The diffusion coefficient was obtained from the dynamic surface tension curves of the DTAB/water system and the asymptotic solutions for the Ward and Tordai equation, D ∼ 2 × 10-10 m2 s-1.9,10,23 Thus, to fit the experimental Π(t) curves, we shall adjust the parameters kc-m, which describes the kinetics of interchange between both states, and χ, the probability with which the molecules adsorb in state c. The parameter χ should be interpreted as the probability of the surfactant molecule to adsorb directly onto an adsorbed macromolecule. It is not clear how we could estimate this value, so we shall take it as a free parameter. The numerical (30) Ritacco, H.; Albouy, P. A.; Bhattacharyya, A.; Langevin, D. Phys. Chem. Chem. Phys. 2000, 2, 5243. (31) Fang, J. P.; Joos, P. Colloid Surf. 1992, 65, 113.

Figure 7. PAMPS(10%) 175 ppm + DTAB 0.2 mM: (a) Π(t) experimental (points) and calculated (line); (b) surface concentrations of both states as a function of time, Γc(t) (continuous line) and Γm(t) (dashed-dotted line).

experiments show that the dependence of the results on χ is rather weak. Finally, the fitting procedure consists of finding the value of kc-m that better fits the experimental results and varying very slightly the other parameters until the best fitted curve is obtained. The curves that we show are obtained by manual adjustment of the parameters, instructed by the experience from the previous adjustments. We show a typical result for PAMPS10%/DTAB systems in Figure 7a. The parameters are wc ) 5.45 × 105; wm ) 2.6 × 105, R ) 6.5; χ ) 0.3; c ) 0.2; kc-m ) 7 × 10-4; bm ) 0.5; D ) 2 × 10-10; Γ∞ ) 1.69 × 10-6. The equilibrium surface concentration, Γ∞, may be obtained from the equilibrium value for the surface pressure, Π∞ ∼ 12.5 mN, using eqs 6 and 9. The fitting suits well the observed kinetics, in the MBP time window (t < 200 s). In Figure 7b, we show the numerical results for the surface concentration for both states as a function of time. The curve in points corresponds to state m, and the continuous one to state c. Both behaviors are illustrative. For state m, we have a maximum at t ∼ 2 s (the same time for the first plateau in γ(t), Figures 2 and 3) and then it falls continuously down to a value close to zero. This happens while Γc increases at the expense of Γm because of the state change. In relation with this last remark, let us make some comments about the value of kc-m.

Dynamic Surface Tension of Surfactant Systems

Langmuir, Vol. 20, No. 9, 2004 3655

Figure 8. PAMPS(25%) 65 ppm + DTAB 0.3 mM: (a) Π(t) experimental (points) and calculated (line); (b) calculated surface concentrations for both states as a function of time, Γc(t) (continuous line) and Γm(t) (dashed-dotted line). Model parameters: wc ) 5.55 × 105; wm ) 2.6 × 105; R ) 8.2; χ ) 0.3; c ) 0.3; kc-m ) 0.00025; bm ) 0.42; D ) 1.6 × 10-10; Γ∞ ) 1.83 × 10-6.

To estimate the characteristic time of state change, we have19

τor )

1

[ ( )]

kc-m 1 +

Γc Γm

(22)

∞

where τor depends on the kinetic constant kc-m of the state change process and on the ratio between the surface excesses of the two states at equilibrium. Using eq 7 to obtain this last ratio and substituting in eq 22, the reorganization process has a characteristic time of τor ∼ 50 s, consistent with the diffusion characteristic time for the polyelectrolyte (∼100 s) but accelerated by the presence of the oppositely charged surfactant molecules at the interface. In the same way, the surfactant molecules are retarded by the presence of the polymer. Remember that at these concentrations, the polyelectrolyte has no surface activity without surfactant (see Figure 1b). The resulting picture for the process is now clear. Due to its greater adsorption rate, the surfactant molecules arrive first at the interface. They adsorb until they reach the equilibrium γ (or Π) value for pure surfactant solution at the same bulk concentration. Once there, the surfactant molecules induce the polyelectrolyte adsorption. This forces the change of the surfactant molecules from state m to state c. Γc increases at the expense of Γm until almost all the surfactant molecules are in state c (complex). From

Figure 9. PAMPS(20%) 65 ppm + DTAB 1 mM: (a) Π(t) experimental (points) and calculated (line); (b) calculated surface concentrations for both states as a function of time, Γc(t) (continuous line) and Γm(t) (dashed-dotted line). Model parameters: wc ) 5.55 × 105; wm ) 2.6 × 105; R ) 8.7; χ ) 0.15; c ) 1; kc-m ) 0.0002; bm ) 0.45; D ) 1.7 × 10-10; Γ∞ ) 1.9 × 10-6.

then on, it is the polyelectrolyte molecules that attract more DTAB molecules to the interface, because not all the polymer charges are neutralized. In this last stage, the surfactant molecules adsorb preferably in state c, that is, associated with the macromolecules as surface complex. In Figure 8, we show a typical result for PAMPS25%/ DTAB systems. In this case, kc-m is ∼2.5 × 10-4 s-1 and τor ∼ 35 s. In Figure 8a, we see a suggestion of a second plateau in the γ(t) curves at times on the order of 100-1000 s, but with our MBP we access a restricted time window and could not follow the complete adsorption dynamics till the equilibrium. The second plateau is also suggested for the PAMPS10%/ DTAB and PAMPS20%/DTAB systems, Figures 7a and 9a. Conclusions We have described the surface adsorption in surfactant/ polyelectrolyte systems with opposite charges, assuming that there exist two different states for the surfactant at the interface. The first one corresponds to the surfactant adsorption as a monomer. Then, the arrival of the polymer to the surface causes a second state of adsorption for the surfactant, increasing its surface concentration and its surface activity. These results give a complex picture for the process, which involves several superposed subprocesses. The picture can well describe the observed adsorption dynamics. At the initial stage, the surfactant adsorption prevails

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(first plateau). It is retarded because the surfactant molecules are dragging the macromolecules toward the interfacial zone. At the second stage, the adsorbed surfactant forces the incursion of the polymer molecules at the interface, provoking a charge reversal, which in the third stage will induce the adsorption of new surfactant molecules. All these phenomena have their own characteristic times. The kinetics of state change is ruled by the polymer diffusion time because the surfactant state change begins with the arrival of the polymer molecules to the interfacial zone. Due to the limited MBP time window, we cannot confirm the existence of the second plateau. Although the twostate model suits well in the observed region, t < 200 s, it is clear that more work is needed to clarify the whole adsorption process. For example, to prove the charge reversal at the interface it would be very useful to measure the surface potential variations. Obviously, it will be necessary to complete the dynamic surface tension results with other techniques, such as “pendant drop”, to follow the complete adsorption dynamics to the equilibrium.

Ritacco and Busch

The observed oscillations9 in the behavior of the surface tension at high surfactant concentrations (>cac) could be explained by a quick adsorption of the surfactant in the first stage. It causes a quick and steep decrease of surface tension. Then, the arrival of the macromolecules produces a change of state in the adsorbed surfactant molecules and possibly their desorption. As a consequence, a transient increase in the surface tension occurs. The characteristic time for the surfactant state change process gives also a plausible explanation for the adsorption barriers found in the simple diffusion model.9,10 Acknowledgment. We thank Dominique Langevin for her helpful comments and critique on our work and also J. F. Argillier from IFP for the polymers. Supporting Information Available: An outline of a mathematical deduction for eq 17. This material is available free of charge via the Internet at http://pubs.acs.org. LA036097V