The Adsorption Behavior of Ionic Surfactants and Their Mixtures with

Article pubs.acs.org/JPCB

The Adsorption Behavior of Ionic Surfactants and Their Mixtures with Nonionic Polymers and with Polyelectrolytes of Opposite Charge at the Air−Water Interface Alireza Bahramian,† Robert K. Thomas,*,‡ and Jeffrey Penfold§ †

Institute of Petroleum Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom § Rutherford-Appleton Laboratory, STFC, Chilton, Didcot, Oxfordshire, OX11 0RA, United Kingdom ‡

S Supporting Information *

ABSTRACT: The surface phase approach of Butler has been used to derive a model of the surface tension (ST) of surfactant solutions in terms of the ST of the surfactant in the absence of water, an area parameter corresponding approximately to the limiting area per molecule, and the critical micelle concentration (CMC). This isotherm is then used to account for the ST behavior of aqueous solutions of weakly interacting polymer−surfactant (P− S) and strongly interacting polyelectrolyte−surfactant (PE−S) mixtures. For P−S systems, no additional parameters are required other than the critical aggregation concentration (CAC) and the onset of the ST plateau at micellization (T3). The model accounts for experimental isotherms for sodium dodecyl sulfate (SDS) with poly(ethylene oxide) or poly(vinylpyrrolidone). For PE−S systems, the initial CAC has no effect on the ST and is well below the decrease in ST that leads to the first ST plateau at T1. This decrease is modeled approximately using a Langmuir isotherm. The remaining ST behavior is analyzed with the model surfactant isotherm and includes modeling the ST when there is separation into two phases. The behavior in the phase separation region depends on the dissociability of the PE−S complex. Loss of surface activity accompanied by a peak in the ST may occur when there is part formation of a nondissociable complex (neutral with segment/surfactant = 1). The model successfully explains the ST of several experimental systems with and without ST peaks, including poly(dimethyldiallylammonium chloride)−SDS and poly(sodium styrenesulfonate)−alkyltrimethylammonium bromide (CnTAB) with n = 12, 14, and 16.

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INTRODUCTION In industrial formulations containing both surfactant and polymer, the surfactant is mainly present to reduce interfacial tension and the polymer to modify the rheological properties of the system. The combination of surfactant and polymer often drastically changes the individual properties of each component in ways that are not well understood. Thus, the attachment of polymer, especially oppositely charged polyelectrolyte, to surfactant aggregates can reduce the headgroup repulsion and induce micellization and surface activity at much lower concentrations than normal (critical aggregate concentration, CAC), a phenomenon first analyzed by Jones.1 Jones found that, in mixtures of the anionic surfactant sodium dodecyl sulfate (SDS) and the nonionic polymer poly(ethylene oxide) (PEO), the plot of ST versus the logarithm of surfactant concentration (σ − ln c) at constant polymer concentration shows two transitions, one below the usual CMC of the surfactant and one above. At the first transition, denoted T1, the slope changes abruptly to a plateau or approximate plateau. For weakly interacting systems, this is the initial point of P−S complex formation, i.e., the CAC. Further addition of surfactant only increases the number of surfactant aggregates in the P−S © 2014 American Chemical Society

complex, and the resulting approximately constant concentration of surfactant monomers leads to a constant ST. When the polymer is saturated with micelles (at T2), the monomer concentration increases and the ST drops to another abrupt transition and approximate plateau at T3, where free surfactant micelles form. Surface behavior of this type is typical of uncharged polymers, and its explanation is supported by conductivity measurements,1 direct surface concentration measurements using neutron reflection (NR),2,3 and measurements of the bulk solution structure using small angle neutron scattering (SANS).4 There are some deviations in detail, but they do not affect the basic explanation that the ST behavior of weakly interacting systems is dominated by the bulk solution phase.5,6 The ST curve for strongly interacting polyelectrolyte− oppositely charged surfactant systems (PE−S) generally also has at least two sharp changes, and it is convenient to use the same T1, T2, and T3 designations without necessarily assigning Received: January 20, 2014 Revised: February 18, 2014 Published: February 19, 2014 2769

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SINGLE SURFACTANT SOLUTION An interface of a surfactant solution can either be treated as a monolayer composed of solvent and adsorbed surfactants,29 or it can be treated as an insoluble film of surfactant molecules. The latter is more widely used and has been developed by Nikas et al.30 to a stage where the ST of a single surfactant solution can be predicted from a single data point at any concentration. The adsorbed layer is treated as a twodimensional gas film, and this requires parameters such as the 2D second virial coefficient and headgroup area, which have to be separately calculated and the number of parameters rapidly increases for mixed systems. In the alternative Butler approach, the chemical potential of the surfactant at the interface is related to its composition and activity coefficient at the interface. Equality of the chemical potentials at the interface and in the bulk phase leads to an equation relating the interfacial tension to the bulk and interfacial composition. This method, although rigorous, has not so far been successful in predicting the ST of surfactant solutions, but once the two or three required parameters are determined, it can easily be extended to mixed surfactant solutions without the need for any other empirical parameters.31 The ST isotherm for a single surfactant solution can be derived following Lucassen-Reynders.32 The chemical potentials for the bulk phase (α) and the interface (αβ) are29

the same significance to them. Addition of PE leads to ST lowering at much lower surfactant concentration than in the weakly interacting case. Above T1, some systems may exhibit a more or less sharp peak in the ST, e.g., poly(dimethyldiallylammonium chloride) (PDMDAAC)−SDS7 or no peak, e.g., poly(sodium styrenesulfonate) (PSSS)−dodecyltrimethylammonium bromide (C12TAB)8−10 superimposed on an approximate plateau up to T3. The origin of the ST peak is not yet agreed, e.g., refs 11 and 12. Direct measurements of the surface structure using NR have shown that it is usually different in the two cases.13 For systems with a peak, the maximum adsorption observed between T1 and T3 is a monolayer, whereas a more complex structure is usually observed over part of the plateau for systems without a peak. This structure uniformly covers the surface and consists of a monolayer to which is attached a thin layer of polyelectrolyte and another surfactant layer with average characteristics of a loosely packed bilayer, which we will refer to as a “trilayer” structure. NR also often observes multilayer structures, but these occur over narrower ranges of concentration, do not usually cover the whole surface, and sometimes exhibit nonequilibrium characteristics.14 T1 is approximately invariant with polymer concentration for some systems, e.g., PDMDAAC−SDS, but proportional to polymer concentration for others, e.g., PSSS−CnTAB. Finally, nonequilibrium effects, e.g., ref 15, and time dependence, e.g., refs 16 and 17, may complicate the behavior of PE−S systems. The strong interaction in PE−S systems leads to a variety of structures, of which soluble aggregates, precipitates, phase separation, and formation of colloidal particles have all been observed (see ref 18 for a recent review). This phase behavior is important for many applications of PE−S formulations but may be less so for the surface properties. Surface applications usually use dilute solutions, for which the phase separation involves the coexistence of a concentrated (Pconc) and a dilute phase (Pdil). This affects the ST through the activity of the adsorbed components and the Gibbs equation, which makes the compositions of the phases important but their structures less so. This should make it feasible to devise thermodynamically based models of the ST with relatively simple assumptions about the two condensed phases. There are several models that describe P−S interaction in bulk solution, e.g., refs 19−22, but the only attempt at a quantitative model for the ST of PE−S systems is that of Bell et al.,23,24 which is an extension of the model of Gilanyi and Wolfram.19 This makes use of the Langmuir−Szyszkowski isotherm together with qualitative hypotheses based on NR data for strongly interacting systems. Although the Bell model generates both types of ST profiles described above, it overemphasizes the sharpness of the transitions, a consequence of the use of mass action, and it neglects the phase separation. However, the Bell model provides a means for the quantitative characterization of the behavior of PE−S systems (see, e.g., ref 25) and it illustrates the dependence of the ST on concentration rather than structure. Nevertheless, there is a need for a model that takes a more explicit account of the phase behavior. In this paper, we derive an ST isotherm for a surfactant solution which is an extension of a model of Bahramian and Danesh for surface and interfacial tension.26−28 We then apply this model and the parameters derived for the surfactants to the P−S and PE−S systems.

μiα = μi0, α (T , P) + RT ln(xiαγiα)

(1)

μiαβ = μi0, αβ (T , P) + RT ln(xiαβγiαβ) − σAi

(2)

μ0,α i

μ0,αβ i

where and are the standard chemical potentials, xi is the mole fraction, γi is the activity coefficient, Ai is the partial molar surface area, and σ is the interfacial tension. Making use of the equality of chemical potentials between bulk and interface, eq 1 and eq 2 give σ = σi +

⎡ αβ αβ ⎤ RT ⎢ xi γi ⎥ ln Ai ⎢⎣ xiαγiα ⎥⎦

(3)

where σi refers to the ST of the pure component i. Rearranging eq 3, using ∑xαβ i = 1, and restricting ourselves to the air/water interface, we obtain ⎡ γα ⎤⎤ ⎡A ∑ ⎢xiα iαβ exp⎢⎣ i (σ − σi)⎥⎦⎥ = 1 RT ⎥⎦ ⎢ γi i=1 ⎣ c

(4)

where c is the number of components. Equation 4 can be simplified in two ways. First, the dividing surface is moved from its customary position of zero excess for water further into the solution until the surface excesses of water and surfactant match. Because of the small size of the water molecule and the monolayer character of the surfactant adsorption, this should only make a small change to the value of Ai for the surfactant and this is what is found (see below and the discussion in Bahramian28). This ensures that the Ai are all nonzero and positive, which is necessary for the full application of the Butler approach.32 Setting all the partial molar areas to A and taking a monovalent 1:1 ionic surfactant, we obtain γwα x wα αβ γw =1 2770

α 2 ⎡ A ⎤ ⎛⎜ α γ± ⎞⎟ ⎤ ⎡ A exp⎢ (σ − σw)⎥ + ⎜xs αβ ⎟ exp⎢ (σ − σs)⎥ ⎣ RT ⎦ ⎦ ⎣ RT γ ⎝ ± ⎠

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where γ± is the mean activity of the ionic surfactant. The second simplification is based on a regular solution model which relates the activity coefficient of a nonsurfactant component at a liquid−vapor interface to its activity coefficient in the bulk by27 γiαβ =

γiα

(6)

and, since aw ≈ 1 for dilute aqueous solution, eq 5 reduces to α 2 ⎡ A ⎤ ⎛⎜ α γ± ⎞⎟ ⎡ A ⎤ exp⎢ (σ − σw)⎥ + ⎜xs αβ ⎟ exp⎢ (σ − σs)⎥ = 1 ⎣ RT ⎦ ⎣ ⎦ RT ⎝ γ± ⎠

(7)

The ST of most pure ionic surfactants cannot be experimentally measured. However, if the two activity coefficients are known, σs can be determined from the experimental ST curve of the aquueous solution. First, it is reasonable to assume that the surface activity coefficient of the surfactant is approximately unity because the surface is mostly occupied by surfactant, even in dilute regions. Determinations of the bulk solution activity coefficients of surfactants are, however, sparse and are restricted to Henry’s law activity coefficients, e.g., refs 35 and 36, but the activity coefficients of eq 7 are Raoult’s law activity coefficients and at infinite dilution they become constant and equal to γ∞ ± . The two types of activity coefficients can be interchanged using the Henry’s law constant and the vapor pressure of surfactant, but there are few values in the literature.37 However, the available data and models indicate that activity coefficients of surfactants are approximately constant below the CMC35 and this result can be verified as follows. Rearranging eq 7 gives ⎛ ⎡ A ⎤⎞ Aσ ln⎜1 − exp⎢ (σ − σw )⎥⎟ − ⎣ ⎦⎠ RT ⎝ RT ⎛ γα ⎞ Aσs ± = 2 ln xsα + 2 ln⎜⎜ αβ ⎟⎟ − RT ⎝ γ± ⎠

Figure 1. (a) The linearity of the function in eq 8 for SDS in the absence of salt (C) and two different plots for SDS in the presence of 0.1 M NaCl, one using eq 8 with a slope of 2 (B) and one using a modification of eq 8 (given in the Supporting Information) with a slope of 1 (A). The fitted parameters σ/mN m−1 and A/Å2 for the plots are 25.5 mN m−1 and 43.0 Å2 for plot C, 21.4 and 40.9 for plot A, and 29.2 and 91 for plot B. The data are from from Elworthy and Mysels33,34 and Staples et al.7 (b) The fit of eq 11 to SDS data extended to well above the CMC and using a mass action model to calculate the mean activities through and above the CMC.34 The parameters used are the same as those for plot C in part a and a degree of ionization of 0.24 and micelle aggregation number of 64 for the mass action calculation. The plot is in reduced units (c/cmc).

(8)

A linear plot is obtained by applying eq 8 to SDS and is shown as plot C in Figure 1a. Since σs refers to the pure surfactant, it has a fixed value and the linearity of this plot therefore establishes that the ratio γα±/γαβ ± is constant up to the CMC. It then follows from the assumption that the surface activity coefficient is unity over this range that γα± for the bulk phase is also constant up to the CMC. Since the Raoult activity coefficient in the micelle at the CMC (γmicelle ) is unity, γα± can ± then be obtained using

The linear plot of eq 8 can also be used to determine σs and A, and the value of A = 43 Å2 per molecule obtained from Figure 1a is in reasonable agreement with the observed value of A = 42 Å2 per molecule obtained by NR.34 In the presence of an electrolyte with a common ion, eq 8 has to be modified and this modification is given in the Supporting Information. It leads to a change of slope from 2 to 1 and this is applied to data for SDS in 0.1 M NaCl in plot A in Figure 1a, which is also linear. Although it is not strictly correct to use eq 8 in the presence of electrolyte, the approximation is a good one, but gives approximately the sum of the areas per ion, and this is shown as plot B in Figure 1a. In the Supporting Information, we derive the corresponding equations for a nonionic surfactant and demonstrate the linearity of eq 8 for four nonionic surfactants. In all four cases, the A values are in close agreement with other determinations of the limiting area per molecule. Elworthy and Mysels measured the ST of SDS above the CMC and explained the ST on either side of the CMC in terms of a varying mean activity determined by the degree of

x+αx−α(γ±α)2 = (x±α)2 (γ±α)2 = x+micellex−micelle(γ±micelle)2 at

x+α = x−α = cmc

(9)

which becomes γ±α =

1 2cmc

(10)

Using eq 10, eq 7 can be reduced to ⎤ ⎡ ⎛ x±α ⎞2 ⎤ ⎡ A RT ⎢ ln 1 + ⎜ (σw − σs)⎥⎥ σw − σ = π = ⎟ exp⎢ ⎦⎥ ⎣ RT A ⎢⎣ ⎝ 2cmc ⎠ ⎦ (11) 2771

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where Φ is the volume fraction of the complex, xt is the total concentration, and at is the corresponding surfactant activity. The assumption that γα± is constant below the CMC (eq 10) is not necessarily the case in the presence of polymer. It is possible to derive simple expressions for the variation, but the fits to the data indicate that it is not significant for P−S complexes. Similarly, fits to the data indicate that the value of γc± for the activity coefficient in the P−S complex is close to unity. In Figure 2, we compare the ST calculated using eqs 13

ionization of the micelle and mass action.33,34 A further test of eq 11 is then to combine it with these mass action mean activities (the mass action calculation is made by Xu et al.34). The excellent fit over the whole range of data up to about 10 × cmc, shown in Figure 1b, illustrates the effectiveness of the surface phase approach of Butler. Equation 11 reduces to the Szyskowski equation for a nonionic surfactant (see the Supporting Information) but with defined rather than empirical constants, and it also leads to a direct relation between the mean activity and the surface excess (see the Supporting Information), which will be useful when we examine strongly interacting systems.

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WEAKLY INTERACTING POLYMER−IONIC SURFACTANT SYSTEMS Introducing polymer into the system changes eq 7 to ⎡ A ⎤ ⎡ A ⎤ exp⎢ (σ − σw)⎥ + (xsαγ±α)2 exp⎢ (σ − σs)⎥ ⎣ RT ⎦ ⎣ RT ⎦ ⎡ A ⎤ + (xpα γpα ) exp⎢ (σ − σp)⎥ = 1 ⎣ RT ⎦

(12)

where the ST of pure polymer is denoted by σp and the last term disappears if the polymer is not surface active. In cases where the polymer is weakly surface active, this can be accounted for approximately by replacing σw by σwp, the ST of the aqueous polymer solution, giving σwp − σ = π =

⎡ A ⎤⎤ RT ⎡ ln⎢1 + (xsαγ±α)2 exp⎢ (σwp − σs)⎥⎥ ⎣ ⎦⎦ RT A ⎣ (13)

The activity coefficient of the surfactant at the CAC can be calculated by equating the activities of surfactant monomers in the bulk and of surfactant in the P−S complex, which would give an activity coefficient the same as in eq 10 with cac replacing cmc. However, although the mole fraction of polymer in the saturated complex contains a sufficient number of surfactants and counterions that its mole fraction is effectively zero, at the point where it just starts to associate, i.e., one polymer, one surfactant, and one counterion ion, its mole fraction is 1/3. Equation 9 then gives γ±α =

γ±c 3 × cac

at

xsα = cac

Figure 2. Comparison of observed and calculated ST using eqs 13 and 15 for SDS with two nonionic polymers: (a) PEO and (b) PVP. (a) 0.1% vol PEO−SDS (PEO was Mn = 24k, Mw/Mn = 1.02) solution at 35 °C, experimental data from Cooke et al.,2 with cac = 4.3 mM and T3 = 15 mM (σwp = 61.5 mN m−1 from Glass38). (b) PVP−SDS, data from Chari and Lenhart39 with cac = 1.6 mM and T3 = 25 mM (σwp = 62 mN m−1 from Glass40). The fixed surfactant parameters were σs = 26.5 mN m−1 and A = 43 Å2 (the CMC is 8.1 mM). Note that the calculation is done in mol fraction units but has been plotted in the more usual molal units.

(14)

At T3, the activity of the P−S complex is approximately the same as that of micelles at the CMC; i.e., the activity coefficient is given by eq 10. Since between the CAC and T3 we can assume that the surfactant molecules either form premicelles as a P−S complex and the rest are free monomer, the mean activity of the surfactant over the range of concentration from CAC to T3 can be obtained by averaging the chemical potentials of the two states, which gives a surfactant activity between the CAC and T3 given by

and 15 for SDS solutions containing 0.1 wt %/vol PEO at 35 °C with data measured by Cooke et al.2 and for SDS−PVP measured by Chari and Lenhart.39 Both sets of results are well fitted with just the two parameters, the CAC and T3, in addition to the two fixed surfactant parameters, σs and A. The ST reaches a simple plateau at T3 because of the assumption of a fully dissociated micelle. This part of the plot is not our main concern, although it would be possible to fit it using a similar approach as in Figure 1b. The model has the unusual feature of predicting the absolute value of the ST, and we also note that the saturation point T2 is not needed as a parameter.

ast = xsαγ±α

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⎛ γ±c ⎞Φ⎛ 1 ⎞(1 −Φ) =⎜ ⎟ ⎜ ⎟ ⎝ 3 ⎠ ⎝2⎠

STRONGLY INTERACTING POLYELECTROLYTE−SURFACTANT SYSTEMS Phase Separation. At or above T1, PE−S systems generally phase separate into dilute and concentrated phases, Pdil and Pconc, where Pconc is usually the denser phase and Pdil contains

t

⎛ γ±c ⎞(x − T3)/(cac − T3)⎛ 1 ⎞(x t − cac)/(T3− cac) ⎜ ⎟ =⎜ ⎟ ⎝2⎠ ⎝3⎠

(15) 2772

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little PE. This is exemplified in the phase diagram determined analytically for PSSS(Mw = 90k)−C12TAB by Hansson and Almgren41 and redrawn with some modification in Figure 3.

has increased substantially by the concentration at which precipitation becomes observable, where the phase diagram suggests a stoichiometry of more than 4:1 (S:PE) and an increase of water to 80%, leading to a relative volume fraction of Pconc of around 0.4% (at 100 ppm PE). Surface Tension in Two-Phase Systems. We start with the simple case of two largely immiscible liquids, of which the denser one also has the lower ST, and for simplicity, we neglect the vapor. An example is the cyclohexane−methanol system, which has the advantage of having been well studied.45 The interest in such systems is the evolution of wetting of the surface near the upper consolute temperature, but we are here interested only in the simpler behavior. Wetting is governed by Antonow’s rule, which is that the interfacial tensions of the three potential interfaces in the two liquid phase + vapor system obey the relation σαγ = σαβ + σβγ

(16)

where γ denotes air and the αγ interface is taken to have the highest interfacial tension and α is the upper bulk liquid phase. The rule requires that at equilibrium the αγ surface is coated by a wetting film of the lower surface energy β phase,46 but if it is not obeyed, a drop of β phase forms only a lens on the α phase. Methanol has the lower ST and is more dense than cyclohexane, and therefore, a wetting film of methanol intervenes between cyclohexane and vapor/air. This film is not restricted to a monolayer and will be thick enough to realize the maximum lowering of the ST consistent with the increase of gravitational potential in moving methanol from the lower phase to the surface. Typical experimental conditions lead to a significant thickness of the surface layer.45 The Butler approach must lead to the same result. It requires the Raoult activities of methanol in the surface phase and the other two phases to be the same. Methanol has an activity close to unity in the surface phase because it is almost pure methanol and a similar activity in the methanol rich phase. It has a much lower concentration in the cyclohexane rich phase, but when the deviation from Raoult ideality is sufficiently positive to lead to phase separation, the activity coefficient will be large and will offset the low mole fraction so that the activity is equal to that of methanol in the methanol rich phase, as required. The reason for the large activity coefficient is that on average each methanol molecule in the dilute phase is surrounded by cyclohexane molecules, leading to a large repulsive interaction relative to the methanol rich phase where each methanol is surrounded by methanol molecules. The surface energy makes a major contribution to the activities of the components in a thin film of the surface phase. Since methanol forms a more stable interface with air than cyclohexane, the overall activity in a methanol film is lowered by a relatively large amount, which stabilizes the film (the interface with the cyclohexane will contribute to the chemical potential in the opposite direction, though to a much smaller extent). The activity based approach thus leads to the same result as from the intuitively easier wetting approach. The two phases in direct contact with the surface phase, i.e., the vapor and cyclohexane rich phases, have compositions that, at a simple level, are apparently unrelated to the surface phase. When the aqueous PE−S system separates into Pconc and Pdil phases, there is often strong lowering of the ST caused by adsorption of the surface active PE−S complex. Since the complex is known to be mostly in Pconc, it is intuitively obvious that Pconc on its own should have a low ST at its air interface

Figure 3. The pseudoternary phase diagram of PSSS−C12TAB−H2O adapted from Hansson and Almgren.41 The two-phase region enclosed by green continuous and gray dashed lines was estimated from four direct measurements, the tie lines for two of which are shown as blue lines and blue points (open squares are the overall composition, closed circles are the measured phase compositions). The green line can be regarded as an accurate estimate, and the dashed gray line is less certain. The equivalence point (charge neutralization) is a thin black line. The two lines in red are estimated tie lines for the onset of turbidity and the onset of precipitation, but extended to the axes, found by Abraham et al.42 The overall composition of the starting sample for these two compositions is not shown but lies close to the lower left-hand corner.

Two experimentally determined tie lines are shown in Figure 3, and red lines mark features of the separate turbidity observations of Abraham et al.42 The boundary of the phase coexistence region has been reproduced as a continuous line in green and a dashed line in gray. The green line is established by the measured tie lines. However, the only tie line measured in the dashed line region was not consistent with the schematic diagram and is not shown. It is possible that this is a result of the colloidal persistence in the redissolution region observed in different systems by Mezei et al.43 and Naderi et al.,44 which we will return to in the Discussion. Over most of the phase diagram, Pconc contains an excess of surfactant relative to the number of segments. The composition of the dilute solutions used for the ST measurements is always close to the left-hand axis of Figure 3, and the tie lines therefore essentially extend outward from this axis, giving a large excess of Pdil. Turbidity measurements17 and visual observations7 indicate no phase separation until about 2 × T1 and no precipitation until about 4 × T1; i.e., there is an initial region where the phase separation determined analytically is “invisible”. This may be because the initial separation is into phases that are not very different in refractive index and/or because the total amount of complex is small (of the order of 0.02% for a 100 ppm PE solution). The onset of turbidity occurs just before Pconc reaches its most concentrated point (about 40% complex, 60% water), but the relative amount of Pconc is still only around 0.1%. At this point, the stoichiometry is approximately two S to one PE segment, and at higher surfactant concentration, the stoichiometry and water content of Pconc continue to increase. Thus, the volume fraction of Pconc 2773

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PDMDAAC−SDS appears to be different in that the measurements of Staples et al.7 and Campbell et al.12 indicate that T1 in this system is neither the onset of precipitation nor the equivalence point, although this was established by turbidity rather than analysis. The different concentration dependence of T1 for the two systems is shown in Table 1. In the

whereas Pdil might not. This would depend on whether Pdil contained more than an insignificant amount of PE. We would therefore expect that if Pconc is more dense the Pdil/air interface would be wetted by Pconc and the ST of the two-phase system would be the low value of Pconc. We emphasize that wetting allows as thick a layer as necessary to bring about ST lowering consistent with gravitational effects, which are negligible in the situations normally studied experimentally. We have previously proposed that the wetting mechanism is important in determining the ST of PE−S mixtures,47 but the two approaches via wetting or via activities are thermodynamically equivalent and we now examine the Butler approach, which is more amenable to quantitative modeling. The reason that the PE−S complex is surface active at such low concentrations is that S interacts strongly and attractively with the PE, which leads to an enhanced effective repulsive interaction of S with water; i.e., the PE−S interaction greatly increases the activity of S in aqueous solution. This offsets the entropic penalty of adsorption, and thus, adsorption of surfactant (in the form of a complex) is increased. Since the activity of S in Pconc and Pdil must be identical, a knowledge of the activity of surfactant in only one of the phases is sufficient to determine the surface properties. This means that there must be enough PE present in Pdil to increase the activity of S to the same value as in Pconc. Direct experiment shows this to be less than 10 ppm, and it is probably substantially smaller. The Significance of T 1 . Surfactant electrode and fluorescence methods show that small aggregates attach to polyelectrolyte at concentrations 2−3 orders of magnitude below the CMC and 1−2 orders of magnitude below the T1 observed in ST measurements; i.e., critical aggregation starts well below T1. Thus, Monteux et al. found a value of 2 × 10−5 M for 500 ppm PSSS−C12TAB, whereas T1 was at about 1.6 × 10−3 M.9 Three measurements of the onset of phase separation, which coincides with T1, by Hansson and Almgren at 8.5 × 10−5, 4.5 × 10−4, and 4 × 10−3 M have substantially lower corresponding CACs of 4 × 10−6, 1.2 × 10−5, and 7.8 × 10−5 M, respectively.41 For PDMDAAC−SDS, Lee and Moroi obtained a value of the CAC of about 2.5 × 10−6 M,48 which is substantially lower than the value of 2 × 10−4 M for T1 found by Zhang et al.25 Nizri et al. found a CAC for PDMDAAC− SDS at 3 × 10−5 M using conductivity and a high concentration plateau at about 10−4 M using an electrode.49 Surfactant electrodes appear to become unreliable at concentrations in the vicinity of T1, possibly because they become too extensively coated with polyelectrolyte, and hence, they are no help in the interpretation of what is happening at T1. However, all of the above observations are consistent with the conclusion of Monteux et al. that non-surface-active PE−S complexes form at concentrations well below T1, and that adsorption at the air/ water interface only occurs when the complexes become more fully extended. They designated the two transitions as CACbulk for the low concentration one, i.e., what we have called the CAC, and CACsurface for the one that causes a sharp decrease in the ST, i.e., our T1. Hansson and Almgren showed that the onset of phase separation for PSSS−C12TAB occurs at the equivalence point. Three measurements were made at 20, 85, and 950 ppm, the first two of which agree with T1 as measured by Taylor et al.8 (Taylor et al. did not measure the third.) Monteux et al. also observed an unusually high ellipticity of the surface just above their T1 together with aggregates in corresponding foam films, which they took as indicating the onset of phase separation at T1.

Table 1. Variation of T1 and the Onset of the ST Peak in PSSS−C12TAB and PDMDAAC−SDS−NaCl (0.1 M) with Polyelectrolyte Concentrationa polyelectrolyte (ppm)

equivalence pt (mM)

T1 (mM)

peak onset (mM)

PSSS/20 PSSS/50 PSSS/100 PSSS/140 PSSS/500 PDMDAAC/10 PDMDAAC/20 PDMDAAC/50 PDMDAAC/80

0.097 0.24 0.49 0.69 2.43 0.06 0.12 0.31 0.50

0.16 0.28 0.49 0.68 ≈2 0.06 0.06 0.06 0.06

0.08 0.14 0.25 0.41

a

The equivalence point is the concentration of charged surfactant that exactly matches the total charge on the polyelectrolyte. T1 is taken to be where the ST reaches its plateau (Monteux et al. defined their T1 as the high ST point just before the drop to the plateau). Data are taken from Taylor et al.,8 Monteux et al.,9 and Staples et al.7

PDMDAAC−SDS system, it depends only on surfactant concentration, which shows that the complex formed at T1 is between an isolated PE and surfactant. In the PSSS−C12TAB system, T1 is, however, proportional to PE concentration and must therefore be associated with the formation of a complex containing a number of individual PE−S fragments. This larger association is also suggestive of a phase transition. The equivalence point for PSSS−C12TAB is close to T1, whereas for PDMDAAC−SDS it coincides approximately with the onset of the ST peak. To produce a plateau in the ST requires either a pseudophase change, e.g., a change of the micelles to rods, or a full phase change, e.g., precipitation. That it is a phase change for PSSS−C12TAB is already established.41 Given that this phase change occurs at the equivalence point, it seems probable that, for PDMDAAC−SDS, the phase change coincides with the onset of the ST peak and that T1 corresponds to a pseudophase change of the surfactant on isolated polyelectrolyte molecules. Quantitative Modeling. The arguments concerning the effects of phase separation on the ST only apply at fixed stoichiometry, whereas in most ST measurements on PE−S systems the stoichiometry is varied by addition of surfactant and, since the properties of the complex may change arbitrarily with stoichiometry, precise modeling would require the determination of the key parameters, σs and A, at each composition. However, we seek to describe the whole range of concentration behavior with the single pair of values of σs and A for the pure surfactant and it is therefore necessary to make some assumptions. Although the PE−S complex is highly surface active, there are two special compositions which may not be. The first is if small clusters of surfactant with their head groups oriented toward the aqueous solution form at points along the PE chain, so that the PE chain retains much of its hydrophilicity and the clusters of surfactant reduce the surfactant hydrophobicity. Complexes of this type form at the CAC, well below T1, and 2774

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Other properties of PE−S systems are consistent with their dissociability. The plateau of ST over the large composition range from T1 to T3 is consistent with λ nearly always being nonzero. The strong dependence of T1 on PE concentration for PSSS−C12TAB must be associated with the self-assembly of dissociable PE−S aggregates rather than simple precipitation; i.e., the phase separation at T1 must be of the liquid crystalline type, even though T1 coincides approximately with the equivalence point. Formation of the NSC in PSSS− C14/16TAB does not occur until well above the phase separation at T1, as indicated by the position of the ST peak. This shows that the NSC does not form in Pconc until the PE is saturated (close to T2) and that this requires quite a high surfactant/ segment ratio. The lack of dependence of T1 on PE concentration for the PDMDAAC−SDS system, on the other hand, indicates that the initial surface active PE−S complexes formed below the equivalence point are dissociable but further surfactant converts them to a part NSC, which precipitates close to the equivalence point. At this point, λ drops well below 1 to give a sharp rise in ST before it is reversed by changing stoichiometry. The dissociability at T1 also has an effect on the ST below T1. Under conditions where the concentration of PE and its counterion are constant, the Gibbs equation is

they are not at all surface active. The second nonamphiphilic form of the complex is when the surfactant ion replaces every counterion on the polyelectrolyte to give the neutral stoichiometric complex (NSC). The hydrophilicity of the NSC may be reduced so that it becomes insoluble, and its charge may be reduced so that the backbone loses any tendency to adopt extended conformations. It is in more extended conformations that the amphiphilicity is strong.9 The NSC is therefore expected to be insoluble and surface inactive. This is the basis of the commonly presented simple picture of the PE− S complexes, in which initial addition of surfactant incompletely neutralizes the PE, giving a hydrophilic and soluble complex, neutralization then leads to precipitation, and as further surfactant is taken up the hydrophilicity of the aggregate complex increases and redissolution occurs. This picture fits naturally with the idea of nonequilibrium states of the system, especially in the redissolution region. The presence of electrolyte allows a range of compositions and structures for a given PE:S stoichiometry. Thus, for p segments per PE, the complex can have the general formula (PE)Sp−x+mMmXx, where x and m are, respectively, the number of X (PE counterions) attached to charged sites on the PE and S/counterion pairs attached at other points, e.g., to an S ion already attached to a charge on the PE. For example, m = x defines a segment/surfactant ratio of 1, but the NSC has the additional constraint that m = x = 0. The NSC will be surface inactive, but other complexes containing X counterions can have stretches of hydrophilic backbone interspersed with surfactant aggregates well separated along the PE chain. The important feature of these mixed complexes is that the unevenness of the attachments not only separates hydrophobicity and hydrophilicity but allows the polyelectrolyte to be more extended, both of which lead to high amphiphilicity. At the surface, there is the additional freedom that the stoichiometry of the complex is constrained only by overall neutrality of the surface; i.e., the segment/surfactant ratio does not have to be the same as the bulk. The combination of flexibility of distribution of surfactant on the backbone and of the composition of the surface with the thermodynamic requirement to reduce the free energy of the surface should mean that adsorption of complex is the normal behavior even when the bulk complex is mainly in the NSC form. The patchiness of the hydrophobicity and hydrophilicity is also likely to assist complexes to self-assemble to form aggregates in addition to the basic self-assembly to form a single PE−S. Phase separation can therefore be of two types, one giving a Pconc consisting of a precipitate of fully neutralized complex and one where Pconc is more of a liquid crystalline phase. The difference between the NSC and the other complexes is one of dissociability. The insoluble NSC is not expected to dissociate and its activity will be unity, just as for a typical uncharged species. Although uncharged species do not have to be surface inactive, the arguments above suggest that the NSC is likely to be surface inactive, whereas the dissociability in the mixed complex leads to surface activity and the ability to selfassemble. An expression for the activity that combines these features is based on averaging the chemical potential of the two types of complex to give a = adλan1 − λ = adλ

−

⎡ p + 2m − x ⎤ dσ =⎢ ⎥Γsd ln a± RT p ⎣ ⎦

(18)

where Γs is the surface excess and a± is the mean activity of the surfactant. For the NSC, x = m = 0 and the slope of the Gibbs plot is then Γs, which will be similar to the result for the ST curve of the pure surfactant in constant electrolyte containing the surfactant counterion. Buckingham et al. used this argument to explain the slopes of ST curves of PE−S systems.50 The argument only holds when the complex formed is of the nondissociable type; otherwise, eq 18 gives a higher slope because 2m is likely to be greater than x. A higher slope is generally observed, and Noskov et al. have drawn attention to the failure of the model of Buckingham et al. in the case of PSSS−C12TAB.10 The addition of electrolyte with an ion in common with the surfactant should have no effect on the slope in eq 18 for the m = x = 0 complex, but it will decrease the slope for a dissociable complex, as observed for both PSSS− C12TAB and PDMDAAC−SDS (see Figure 6). For both PDMDAAC−SDS and PSSS−C12TAB, the strong surface activity shows that a dissociable complex is formed at T1. Using the same starting point as for the P−S systems, the activity coefficient at T1 is then expected to be similar to eq 14 for P−S complexes but with T1 replacing cac and possibly a change in the factor of 1/3. This factor will depend on the composition of the PE−S aggregate and its state of dissociation. For the NSC, the factor will be unity because the mole fraction of PE is effectively zero and there are no counterions in the complex. However, for a dissociable complex, it should be between 1/2 if only the surfactant counterion is involved and 1/3 if both counterions are involved. Fits to the data suggest that the factor lies between these two limits for both PDMDAAC−SDS and PSSS−C12TAB, consistent with significant involvement of the counterions in the T1 complex. We therefore assume that both complexes are fully dissociable at T1 and that a reasonable approximation is to take an average of 1/ 2 (surfactant ion pair involvement) and 1/3 (surfactant ion pair and polyelectrolyte counterion), i.e.,

(17)

where d and n indicate dissociability and nondissociability and the degree of dissociation λ depends on composition. The further equality in the equation results because an is unity. 2775

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=

Article

5γ±c 12T1

associated with the formation of the NSC. The discussion above concerning the presence of counterions at the surface, the strongly varying composition of the complex, and the fact that in PSSS−C12TAB the complex is always surface active all suggest that loss of surface activity will occur over only a limited range of concentration from a lower limit, LT, to an upper limit UT. To treat the ST peak quantitatively, the simplest assumption is then to assume that the nondissociable fraction does not contribute to adsorption and that the dissociable fraction has an activity with a concentration dependence similar to that at T3 and above, i.e., characteristic of free surfactant. We further assume that the surfactant activity changes linearly between LT and UT so that the activity is given by

(19)

γc±

where as before represents residual small perturbations of the activity coefficient due to interactions within the complex. Equation 11 cannot be used to describe the ST below T1 because the low CAC changes the lower limit in a way that is not easily defined, especially with respect to the counterions. As an alternative, we use a Langmuir isotherm to describe the binding of S to PE, which gives the mole fraction of S in the complex below T1 as ⎡ ⎤ kx t xsc = ⎢ t t ⎥ ⎣ (T1 − x + kx ) ⎦

(20)

⎛ xα ⎞⎛ x t − LT ⎞ asp = ast + (1 − λ)⎜ s − ast ⎟⎜ ⎟ ⎝ 2cmc ⎠⎝ UT − LT ⎠

xcs

where k is the binding constant, is the mole fraction of S in the complex, and xt is total S. For simplicity, the total concentration xt is used in eq 20 rather than the actual monomer concentration. This approximation is largely lost in the approximation of using a Langmuir isotherm. However, the aim is only to give a semiquantitative description of the behavior below T1 and experimental isotherms resemble the approach to a plateau in the range where the ST starts to drop,49 which supports the idea of using a Langmuir isotherm. The limiting coverage is normalized to the activity from eq 19 at T1. The region above T3 can be modeled similarly to the P−S systems except that there will be significant general interactions between S and PE, which can be treated as follows. In terms of the osmotic coefficient, ϕ, the activity of water is ln(x wαγw ) = ϕ ln(x wα)

for

(21)

1 d[xsα(1 − ϕ)] xsα

(22)

xαs

where is the surfactant monomer concentration. For a strong electrolyte in a sufficiently dilute solution, it is found experimentally that51 1 − ϕ = B(xsα)1/2

(23)

with B constant and independent of concentration. Integration of eq 22 gives γ±T3 = exp[− 3BT31/2]

(24)

⎛ x t − UT ⎞ asp = ast + Δ − Δ⎜ ⎟ ⎝ T3 − UT ⎠

where the mean activity coefficient at infinite dilution, γ∞ ± , has been taken to be unity. Using these results, the plateau region between T1 and T3 can be modeled using the same approach as for eq 15 to give an overall mean activity of t

ast

for

UT ≤ x t ≤ T3 (27)

where is again determined from eq 25 and Δ is as defined above; i.e., the ST returns to the plateau value at T3. For some of the systems shown in the figures, electrolyte is present and the additional equations including electrolyte effects are given in the Supporting Information. Other than changes to σs, A, and the CMC, which are obviously different in the presence of electrolyte, the effect of electrolyte on eqs 25, 26, and 27 is small, but it does have an effect below T1. An alternative model of the ST from UT to T3 assumes the equality of the surfactant activity in Pdil and Pconc and calculates the ST of Pdil. The obvious way of modeling the activity in Pdil is to assume that Pdil initially contains no PE and can therefore be modeled using eq 11 and the free surfactant concentration xsα. Given that the best estimate of the free surfactant ats

(x t − T1)/(T3− T1)

⎛ 5γ±c ⎞(x − T3)/(T1− T3)⎛ γ±T3 ⎞ ⎜ ⎟ ⎟ =⎜ ⎜ 2 ⎟ ⎝ 12 ⎠ ⎝ ⎠

(26)

where xαs is the mole fraction of free surfactant and ats is obtained using eq 25. Note that λ is not identical to λ in eq 20 because, in the absence of any model to describe the variation of λ, we have used the simpler linear approximation. The activity coefficient γc± is set to 1, partly because it is close to unity at concentrations much less than T3 and partly because it is less than the uncertainty in xαs . The minimum value of xαs is the CAC, which is found experimentally to be of the order of T1/10 or less (see the section on T1). Since binding curves generally indicate that binding is at least 80% at T1, the maximum value of xαs is probably about T1/5. The plateau in ST between T1 and LT and the observed continuing buildup of precipitate beyond UT suggest that xαs does not start to increase again until UT. We therefore assume that xαs = cac below UT. When λ = 0, the ST tends to a value appropriate to a solution of surfactant at the CAC, which is close to that of pure water, and when λ = 1, the ST has the plateau value of eq 25. Equation 26 changes the activity by Δ (λ(cac/2cmc −aUT s ) from the plateau value of eq 25 at UT. Above UT, additional surfactant reconverts NSC to dissociable material and the additional contribution Δ starts to decline back down to zero. This is not necessarily the same as redissolution, since it can occur within the phase separation region. However, redissolution can also contribute to the process. For simplicity, we combine these into a single linear change between UT and T3 where the activity is given by

Using the Gibbs−Duhem equation, we can write d ln(γ±c ) = −

LT ≤ x t ≤ UT

(25)

Equation 25 makes no allowance for phase separation and, in combination with eqs 20 and 24, leads to an ST curve qualitatively similar to those for P−S systems, i.e., with two approximate plateaus following the intitial drop in ST. The PDMDAAC−SDS system, however, has a sharp ST peak on the plateau between T1 and T3, and this must be associated with a region where λ in eq 17 deviates from unity and, given the proximity to the equivalence point, it is likely that this is 2776

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concentration is that xαs = cac at UT and that xαs must be equal to the total surfactant concentration at T3 if redissolution is complete at this point, this gives

■

⎛ x t − UT ⎞ xsα = cac + ⎜ s ⎟(T3 − cac) ⎝ T3 − UT ⎠

concentrations. Since the ST should be the same whether it is calculated from Pdil or Pconc, these discrepancies show that enough PE must be present in Pdil to bring the ST down to the observed value and that the surface tension of Pconc is determined by the PE−S complex for a longer concentration range than estimated from eq 28. The value of λ drops to about 0.2, which indicates that the PDMDAAC−SDS system is close to the simple picture of an insoluble precipitate outlined earlier. The variation of the surface excess, as directly measured by NR for the 10 ppm sample, is also well accounted for by the model, as shown in Figure 4b, whereas it is not well explained by the free monomer model. This agreement of model and experiment resolves an observation by Staples et al.7 Merta and Stenius had earlier interpreted ST peaks for mixtures of anionic surfactants and cationic starches of different charge density in terms of phase separation into a concentrated “gel” and a dilute phase, i.e., Pconc and Pdil.52 They attributed the ST peak to the ST of Pdil; i.e., there was little adsorption because Pdil was too dilute. Staples et al. noted that the significant adsorption at UT is not consistent with the simple picture of precipitation. Taking this and the noncoincidence of UT with maximum precipitation, Staples et al. stated, “although precipitation may well be a factor in determining the complex ST behavior, it cannot fully explain the SDS−PDMDAAC data”. This is confirmed by the present model, although there may be additional nonequilibrium factors determining the center of gravity of maximum precipitation. The formation of the nondissociable complex reaches its maximum at UT, and this suggests that this can be identified with T2,7 the point of PE saturation. However, the discussion above indicates that the situation may be more complicated than simple saturation. The limits of LT and UT are determined by the observed breaks in the ST, but the variation of σ between the two limits is artificially imposed by the assumptions in eq 27. Because LT and UT are not very different in the PDMDAAC−SDS system, the effect of any variation is swamped by the narrowness of the transition. Nevertheless, even when this is sharp, the description “cliff edge” introduced by Campbell et al.17 is not appropriate. The original measurements of Staples et al. show a finite width for the upturn, and fitted values using the present model give an average value of 0.02 mM for UT − LT, which is about half the value of T1; i.e., it is only the log scale that creates the illusion of an abrupt step. The further proposal by Campbell et al. that the change is linked with total precipitation of the PE would require the ST to rise to the values shown by the blue curves in Figure 4a. It might be argued that these are based on the assumption that the concentration of free surfactant at UT has the low value of the CAC (in all the calculations, it has been constrained to a value of T1/10, although it is usually lower). However, to reduce the ST at UT to its observed value would require the free surfactant concentration to be as high as T1 itself. Given that the known levels of binding at T1 are usually 80% or more, such a high monomer concentration is improbable. Experimental evidence that the free surfactant concentration at UT is not high enough to cause significant adsorption of the monomer also comes from the dynamic experiments of Campbell et al.16 which show that there is no significant adsorption of SDS monomer at this concentration in the dynamic experiment. Figure 5 compares observed and calculated results for the three CnTABs with PSSS. The curve for C12TAB can be fitted by the Langmuir isotherm below T1 and by eq 25 above T1, and it qualitatively resembles the curves for the P−S systems.

(28)

RESULTS Calculations are shown for two concentrations of PDMDAAC with SDS and 0.1 M in Figure 4a. These are based on eqs

Figure 4. Comparison of experimental (points) and calculated (lines) for (a) the ST of PDDAAC−SDS−NaCl (0.1 M) at two PDMDAAC concentrations and (b) the surface excess for 10 ppm PDMDAAC− SDS−NaCl (0.1 M) (points measured by NR). The ST was calculated using eqs 25−27. In both cases, the calculation was also done on the basis of the free monomer concentration (blue lines) using eq 28. The parameters used for the calculations are given in Table 2, and the data are from Staples et al.7

25−27 except for modifications for the presence of excess electrolyte (see the Supporting Information). The ST of Pdil is also calculated (as blue lines) on the assumption that it is determined by the concentration of free surfactant given by eq 28 and that it contains no polyelectrolyte. The fitting parameters are given in Table 2. The absolute value of the ST at the T1 plateau is reasonably accurately predicted using the simple model, which is fixed by the independently determined parameters σs and A and is only adjustable through γc±, which is found to be unity. The variation of ST between UT and T3 is only approximately described by the model and this can almost certainly be attributed to the overly simple linear approximation used in eq 27, which effectively overemphasizes the free monomer in Pconc. The ST curve based on Pdil containing only free monomer does not account for the ST in the region of UT and just above but is better at higher 2777

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Table 2. The Fitting Parameters Used in Figures 4−6a system

T1 (mM)

C12TAB/20 (a) C12TAB/100 (b) C12TAB/100 C14TAB/50 C16TAB/50 C12TAB/140 C12TAB−NaBr/140

0.16 0.4 0.4 0.14 0.18 0.6 0.036

2.4 0.14 0.18

1 1.95 0.16

SDS−NaCl/10 SDS−NaCl/50 SDS/10

0.036 0.036 0.24

0.076 0.24 0.32

PDMDAAC 0.025 0.02 0.14

LT (mM)

UT−LT (mM)

λ

γc±

γT±3 (mN m−1)

0.1 0.1 0.1 0.1 0.3 0.1 2

0.85 0.65 0.54

0.93 0.93 0.98 0.95 1.03 0.84 0.52

−2.5 −2.5 −2.5 −1.6 0.4 −1.6 −0.4

0.5 0.5 0.25

0.18 0.18 0.11

1.03 1.1 1.4

−0.5 −0.5 −0.8

T3 (mM)

k (mM)

16 16 16 3.6 1.0 17 2.5

PSSS

1.38 1.38 8.3

The two surfactant parameters were determined independently from the ST of aqueous solutions of the surfactants. The number in the first column is the polyelectrolyte concentration in ppm. The surfactant parameters, σ and A, were determined from ST data for the aqueous solution of pure surfactant. These parameters were 26 (21.5) mN m−1 and 42 (41) Å2 for SDS (SDS + 0.1 M NaCl) and 32 (29), 28, and 29 mN m−1 and 69 (59), 50, and 51 Å2 for C12TAB (C12TAB + 0.1 M NaBr), C14TAB, and C16TAB, respectively. The calculations are not sensitive to the CAC, and this has been constrained to T1/10. a

However, there are three quantitative differences between PE− S and P−S systems. First, the initial drop to T1 is much sharper, second the plateau above T3 is slightly lower than the pure surfactant, and finally the plateau from T1 to T3 is not a true plateau because of variations in λ, which will affect the ST. The result gives an adequate description of the PSSS−C12TAB system, and there is no indication of significant loss of surface activity at any concentration above T1, which suggests that there is always a sufficient fraction of dissociable complex at the surface to keep the ST low. Turbidimetry, visual observation, and the phase diagram all indicate that maximum precipitation of Pconc occurs well along the T1 to T3 plateau. There is also a weak bump in the ST in this region, and it is therefore interesting to apply the model with an ST peak. However, the difference from the PDMDAAC is that there is a large gap between LT and UT. The phase diagram requires LT to coincide with T1, and the other observations indicate that UT is not far below T3. As shown in Figure 5a, the model with a peak fits the small bump as well as the one with no peak, although λ only decreases to about 0.85. For completeness, the ST that would be associated with Pdil depleted of all its polyelectrolyte has also been calculated and is shown in Figure 5a. The NR experiment shows that in the region of the bump in the ST the surface is covered by the trilayer described earlier. This is consistent with the PE−S complex remaining largely dissociable through this region. The replacement of a monolayer by a trilayer could have a small effect on the ST and be partly responsible for the weak fluctuations, but additional bilayers are not thought to have a strong effect on the ST. Ellipsometry also shows that the surface undergoes structural changes across the T1−T3 plateau.9 The results for the two longer chain CnTABs have ST peaks and have been modeled assuming that LT coincides with T1, just as for C12TAB. This gives a broad transition in line with the observations with λ dropping to values of about 0.5. From the earlier discussion, this suggests that NSC is not formed until well above the nominal equivalence point, which is located closer to T1. This again suggests that formation of this complex is associated with near saturation of the PE; i.e., it is close to T2. An important part of the earlier discussion concerned the difference between T1 and the much lower CAC. That the CAC is largely driven by electrostatic interactions is illustrated by the

Figure 5. Predicted (lines) and experimental (points) ST versus surfactant concentration for (a) PSSS−C12TAB (10 ppm PSSS and 50 ppm PSSS) and (b) PSSS−C14TAB and PSSS−C16TAB (both 50 ppm PSSS). Three calculations have been done for PSSS−C12TAB (50 ppm PSSS), one with no ST peak (in black), one with an ST peak defined by LT = T1, as required by the known phase diagram (see text), and a best fit UT (in blue), and one for the ST associated with the free monomer above UT (dark pink) and using the same assumptions as for the corresponding calculation in Figure 4. The parameters used for all the calculations are in Table 2, and the data are from Taylor et al.8,14

2778

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fact that it always increases on the addition of electrolyte;54 i.e., electrolyte screening reduces the strength of the interaction. There are not as many systematic observations of T1, but such as there are show a decrease in T1 on addition of electrolyte. This is consistent with it being driven largely by hydrophobic interactions, since the addition of electrolyte generally promotes surfactant aggregation by screening the repulsive headgroup interactions. Calculated ST curves are shown for PDMDAAC−SDS and PSSS−C12TAB in Figure 6, in both of

about 5k in PSSS−C12TAB before the equilibrium properties diverge,9,55,56 suggesting that an Mw of 10k is optimum. Taylor et al. already found difficulty in making equilibrium surface measurements on a monodisperse 18k sample of PSSS and typically required 12 h for equilibration for NR experiments. On the basis of Buckingham et al.’s results, a 1 M PSSS sample, as used recently by Abraham et al.,42 would require 1 month to reach the same state. The Mw dependence makes measurements on polymers particularly prone to depletion and polydispersity. At low molar concentrations, a significant fraction of material is lost to the various surfaces of an experiment and the bulk concentration is depleted below its nominal value. The effect is particularly acute in ST experiments because surface active species adsorb strongly and measurements are made at low concentrations.57 It is already common in ST experiments at concentrations of about 10−6 M and becomes serious at 10−7 and lower concentrations. Depletion and polydispersity often combine to produce apparent equilibrium effects of some complexity. Thus, An et al. explained the generally reproducible but unexpected steepness of the low concentration part of the ST curves of nonionic surface active polymers in terms of polydispersity.58 At low concentrations, low Mw species dominate the adsorption because their molar concentrations make them less vulnerable to depletion and their low area per molecule gives rise to a steep ST slope. Since adsorption energy varies with Mw, these are gradually replaced by larger species as the concentration increases to a level where the larger species are no longer lost by depletion, and this causes the ST slope to become more shallow. At the typical concentration of 100 ppm polyelectrolyte, the molar concentrations of PSSS of 18k and 1 M are, respectively, 5 × 10−6 and 10−7 M. The latter is in the range where depletion effects start to become significant and even the former may not be completely free of them, especially given the high surface activity of these systems. Abraham et al. used a 1 M PSSS at 100 ppm (10−7 M) for their recent experiments on PSSS−C12TAB,42 and the difference in ST between the two plateau regions in their experiment, which was found to be about 10 mN m−1, compared with about 3 mN m−1 obtained for much smaller values of Mw of 18k,8 43k,9 and 70k,10 suggests that depletion or polydispersity is playing a significant role in the very high Mw system. A possible polydispersity effect of the kind explained by An et al. may account for the observed systematic deviation from proportionality to PDMDAAC concentration in the PDMDAAC−SDS system. Thus, the ST peak position in the PDMDAAC−SDS system at its lowest concentration of 10 ppm is well above the equivalence point, unlike the higher concentrations (see Table 1). Since this concentration is in a range where depletion is likely, lower Mw species would be expected to be more dominant. If these species also require greater amounts of surfactant for precipitation, which is plausible, this would cause the observed discrepancy. Support for this comes from the dynamic experiments of Campbell et al.16 Campbell et al.16 used the overflowing cylinder to study the surface of PDMDAAC−SDS−NaCl on a 1 s time scale. They found a constant PE−S ratio in the adsorbed layer over the whole concentration range up to the equivalence point and a broad dip down to zero adsorption at the equivalence point. The latter results from the expected slow diffusion of large aggregates. The former is, however, surprising because the equilibrium PE−S binding in bulk solution increases with surfactant concentration49 and the equilibrium NR measure-

Figure 6. Predicted (lines) and experimental (points) ST versus surfactant concentration for (a) PDMDAAC−SDS (10 ppm) with and without 0.1 M NaCl and (b) PSSS−C12TAB (140 ppm PSSS) with and without 0.1 M NaBr. The parameters for the calculation are given in Table 2. The data are from (a) Taylor et al.14,53 and (b) Warren et al.7 and Zhang et al.25

which addition of electrolyte decreases T1, in the opposite direction to the effect of electrolyte on the CACs. As discussed above, the curves below T1 without electrolyte are both steeper than predicted by the Buckingham assumption50 and decrease substantially on addition of electrolyte containing a common ion. Both observations are consistent with the complexes being of the fully dissociable type at T1. The modifications of eq 20 used for the calculation are given in the Supporting Information.

■

DISCUSSION Dynamic Processes and Nonequilibration in PE−S Systems. The dynamics of polymer systems are sensitive to molecular weight, Mw. Buckingham et al. found that the rate of equilibration increases approximately linearly with Mw, although the equilibrium surface properties are more or less independent of Mw.50 It is therefore desirable to use a low Mw for equilibrium studies and monodisperse samples for dynamic studies. Monteux et al. have shown that there is a lower limit of 2779

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ments of Staples et al. and Campbell et al.12 both show a steady increase in the SDS:PDMDAAC ratio with surfactant over the same concentration range and a higher starting ratio than observed in the dynamic experiment. Low Mw species must dominate the dynamic experiment for the polydisperse PDMDAAC used. The probable explanation of the low and constant SDS:PDSMDAAC ratio observed in the dynamic experiment may therefore be that the low Mw species have a weaker interaction with surfactant than high Mw species, possibly because the cooperative interaction is reduced. This would be consistent with the more subtle effect of polydispersity in the previous paragraph. Polydispersity/ depletion effects will be dominant in any dynamic situation, and this is also the probable explanation of the “light mechanical stress” experiments done by Campbell et al. on the PDMDAAC−SDS−NaC,17 which we discuss at the end of the next paragraph. The model presented in this paper shows that the ST peak can be explained in terms of equilibrium, but this does not prove that it is a genuine equilibrium. Campbell et al. have explored aspects of the equilibration in the vicinity of the maximum precipitation region.12 The steps in their experimental protocol are (i) equilibration for a suitably long period, (ii) careful removal of the supernatant, and (iii) measurement of the ST of the supernatant. The resulting ST is then assumed to be the ST of the original mixture. The measurement is therefore of the ST of each of a series of different Pdil formed at different overall concentrations. If the concentration of PE in Pdil is not too low, the procedure will give the correct ST for the two-phase system, even though Pconc is absent. However, at overall concentrations where precipitation is high, the PE in Pdil is very likely to drop below the depletion threshold. When this occurs, the procedure will give a different result from the simpler measurement on the two-phase system, where there is a reservoir of PE−S, and may indicate that there is no PE at all in Pdil; i.e., it is likely to show an apparent peak in the ST in the vicinity of maximum precipitation. However, this would simply be an artifact of the protocol. Abraham et al. do indeed observe an ST peak in the PSSS (1 M)−C12TAB system. They estimated their PSSS concentration in Pdil at the peak to be 2 ± 3%, i.e., about 2 ppm, which corresponds to a molar concentration of 2 × 10−9 M for their PSSS. At this concentration, depletion would be expected to be so large as to make a true ST unmeasurable. It would be further exacerbated by the high surface to volume ratio of their method of measurement, the pendant drop method (e g, see Figure 5 in ref 59). In the three conventional measurements by Taylor et al., Monteux et al., and Noskov et al., where Pconc is present as a reservoir of PE, no ST peak was observed. The ST peak of Abraham et al. therefore appears to be an artifact. Campbell et al. have used the same protocol to study the PDMDAAC−SDS−NaCl system, and although these results generally confirm the earlier measurements of the phase diagram and the presence of an ST peak made by Staples et al.,12,17 the Campbell protocol produces a truly vertical “cliff edge” to the ST peak, which differs from the finite width obtained by Staples et al. using the whole sample (see the values of LT and UT in Table 2). If the PE concentration in Pdil drops to a depletion level at LT, the ST measured by the Campbell protocol will effectively be that of free surfactant, i.e., there will be a “cliff edge”, but this is not representative of the system when both phases are present. Given the earlier evidence that polydispersity and hence depletion play a role in

this system, the vertical transition found by Campbell et al. is probably an artifact, although in this case the presence of the peak is genuine. Campbell et al. also found that the ST of the decanted Pdil is sensitive to small mechanical disturbances applied to the mixture just before withdrawal of Pdil. The discussion in the previous paragraph suggests that such a disturbance probably leads to a quick release of the smaller Mw species, which then dominate the subsequent ST measurement; i.e., this observation suggests limitations in the definition of the sample rather than failure to reach equilibrium. Much attention has been given to nonequilibration of particulate PE−S fragments, and the persistence or generation of colloidal particles in regions outside the phase coexistence region has been clearly demonstrated by using different mixing protocols. 60,43 These indicate that the deviation from equilibrium is small on the low surfactant concentration side of precipitation but is significant on the high surfactant side.44 The surfactant concentration is always likely to be high above precipitation, but our model indicates that only a small amount of PE needs to be present in Pdil to generate the equilibrium surface, and therefore, bulk equilibration does not have to progress far to ensure an equilibrium surface. Thus, provided the system is not totally locked in a nonequilibrium state, there should be enough free PE to give the equilibrium ST in and above the precipitation region. Mezei et al. have indeed shown that, although preparation protocols significantly affect the bulk properties of the poly(vinylamine)−SDS system, they do not seem to affect the surface properties.43 The corollary of these observations is that ST measurements can give no reliable information about this aspect of equilibration.

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COMPARISON WITH OTHER INTERPRETATIONS Although there are several models in the literature for P−S systems, the only previous attempt at a quantitative model of the ST of PE−S systems is by Bell et al.23,24 In this model, the formation of the surface complex causes a sharp decrease in ST to a plateau at T1. If the formation of the bulk complex is at a concentration not too far above T1, an ST peak occurs because the activity of the surfactant is sharply reduced by formation of the bulk complex. The use of mass action makes the two transitions much sharper than in reality. The assumption that the bulk complex is in solution has been criticized, but from the point of view of the ST, this is not an important issue in that the bulk complex is just a phenomenological device that removes surfactant monomer. Comparison with the present model shows that the formation of the bulk complex in the Bell model corresponds to the onset of the phase coexistence region but with a negligible separation between LT and UT; i.e., there is one equilibrium constant for complex formation (the “surface complex”, at T1) and there is another for formation of the concentrated phase (the “bulk complex” at LT). This equivalence means that, for such a simple model, the Bell model remains remarkably useful for analyzing the ST behavior of PE−S systems. Comparison with Surface Structure. NR studies on a wide range of PE−S systems have established that systems with an ST peak usually only have surfactant monolayer adsorption, whereas systems without a peak usually show the trilayer adsorption described in the Introduction; i.e., the trilayer acts to reduce the ST and could be thought of as a wetting layer of Pconc. The wetting explanation is consistent with its occurrence over only a limited extent of the plateau, typically in the last third or so before T3. The mean composition of the trilayer has 2780

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whose structure (and surface activity) may vary across the plateau. Above T3, the two systems are similar in that for both systems it is now the free surfactant that lowers the ST, but for both there may be a significant depression of the ST by general interaction with the polymer. It is in this region and below T3 where dissolution of the precipitate in PE−S systems may be incomplete for equilibrium or nonequilibrium reasons, but the ST is relatively insensitive to these details. The most striking overall difference in the two similar ST curves for P−S and PE−S is that the P−S behavior is dominated by the bulk solution to the extent that measurement of the ST is a good route to the bulk phase diagram, whereas the PE−S behavior is dominated by the surface properties of the strongly surface active complex and is relatively uninformative about the bulk solution (it is often not even possible to use the ST to identify the phase separation!), as was originally suggested by Goddard.61 In some systems, the changing composition of the PE−S complex may lead to a loss in surface activity and this is almost certainly associated with the formation of a significant proportion of the surface inactive and nondissociable NSC. If this process is separate from T1, it will lead to an increase in the ST, which may have a sharp or a gradual onset. The nature of these PE−S complexes is that loss of surface activity occurs over only a narrow range of composition and is reversed at some higher surfactant concentration. The reversal of the ST is caused by a change in the complex and by release of a small amount of PE into the dilute phase. There is no easy way to predict the onset of a loss of surface activity, and it is probable that a wide range of patterns of this loss occur, just as there will be a wide range of redissolution patterns. This is the probable cause of the different shapes observed for the ST peak in different systems. All of the above has been explained in an equilibrium framework, but kinetic effects always play a significant role in polymer systems, and it would be surprising if equilibrium were always established. However, the ST is not always sensitive to nonequilibrium effects in the bulk and one of the further results from the Butler approach is that it has been possible to rationalize some hitherto unexplained features of the dynamics and equilibration. The interpretation of the ST has been shown to be consistent with more direct observations of the surface, in particular ellipsometry and neutron reflectometry studies of the interfacial structure.

been established as about 2−3 surfactants per PE segment. The composition of the concentrated phase does not reach a S:PE ratio of 2 until well above the initial phase separation, and this may be the reason for the delay in the appearance of the trilayer structure along the ST plateau. The sharpness of the interference effects seen in the signal from the trilayer and its intensity show that it is uniform over the whole surface. Although NR is not directly sensitive to lateral heterogeneity below a certain length scale, it would be sensitive to the kind of heterogeneity that would result from attachment and gradual disintegration of colloidal particles (multilayer fragments are seen in many systems, and unlike the trilayer, these are generally consistent with only a fragmentary coverage of the surface). The composition of the bilayer part of the trilayer is less dense than a conventional bilayer, and both its thickness and its density are more consistent with it having a structure of micellar rods oriented parallel to the surface. The Pconc phase of several PE−S systems is known to have a component of hexagonal liquid crystalline structure, and these would have a clear structural relationship with the trilayer. In the PSSS−C12TAB system, the trilayer structure exists mainly in the precipitation region, which correlates with zone 3 of Monteux et al.’s ellipsometry results. The ellipsometry of their zone 3 is totally different from zones 2 and 4 on either side, and although they did not attempt to interpret the ellipsometric data quantitatively, they concluded that “the surface ellipticity remains well above that of a pure surfactant layer”.9

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CONCLUSIONS The main purpose of this work was to obtain a better understanding of the surface behavior of oppositely charged polyelectrolyte−surfactant (PE−S) systems. The basis for the analysis has been the use of the Butler concept of a surface phase to derive a quantitative, easily modeled σ − ln c curve, for a simple surfactant solution. This characterizes the ST of a surfactant in terms of two easily derived parameters, each with an accessible physical significance. Application of this to two weakly interacting polymer−surfactant systems proves straightforward and accounts quantitatively for the shape and absolute value of the σ − ln c curve with no new parameters other than the easily identified CAC and final CMC. The difficulty of modeling the ST of PE−S systems arises mainly from the difficulty of handling surface equilibria for phase separating, reactive mixtures. We have used the Butler approach to understand the ST behavior between the two main fixed points, T1 and T3, in the surface and bulk behavior. T1 marks the onset of the initial plateau in the ST, and is confirmed as either a full phase transition or a pseudophase transition from an aggregate phase to a surface active phase, possibly micelles to rods on the polyelectrolyte, as originally proposed by Monteux et al.9 Redissolution starts on the plateau and is assumed to be complete at T3. If the PE−S complex remains surface active at all compositions, T1 and T3 will be connected by a more or less flat plateau in the ST, even when there is precipitation. Although such a curve superficially resembles that, for weakly interacting P−S systems, all three branches of the ST curve are associated with different underlying physical processes. Below T1, there is no complex in the P−S system, whereas in PE−S systems there is extensive complex formation. From T1 to T3, micelles form on the polymer in P−S systems and reduce the ability of the surfactant to reduce the ST, but in PE−S systems, ST lowering is maintained by adsorption of a strong surface active complex

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ASSOCIATED CONTENT

S Supporting Information *

The surfactant isotherm for a nonionic surfactant is derived and shown to be of the same form as the Szyskowski equation. The equation is applied to experimental data for four nonionic surfactants to determine σ and A. The latter values are shown to be close to independently determined values. The modifications necessary to take into account added electrolyte are given for the surfactant alone, surfactant and weakly interacting polymer, and strongly interacting polyelectrolyte− surfactant systems. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 2781

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Notes

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The authors declare no competing financial interest.

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