Interactions between sodium polyacrylate and mixed surfactants of polyoxyethylene tert-octyl phenyl ether and dodecyltrimethylammonium bromide

Abstract

The interactions between sodium polyacrylate (PANa) and mixed surfactants of polyoxyethylene tert-octyl phenyl ether (TX100) and dodecyltrimethylammonium bromide (DTAB) in 40 mM NaBr aqueous solutions were studied using isothermal titration calorimetry (ITC). It was found that whether DTAB was titrated into PANa/TX100 or TX100/DTAB was titrated into PANa, three endothermic peaks were detected; representing three processes: (1) binding of DTAB monomers to PANa chains through electrostatic interactions, (2) polymer-induced micellization, and (3) cross-linking of polymer chains. The interaction mechanism was interpreted using a thermodynamic model, and it was found that when the molar ratio of bound DTAB to the carboxylate group of the polymer (C^poly_DTAB/C_poly) reached about 0.5, the polymer-induced micellization occurred; while when C^poly_DTAB/C_poly reached about 1, indicating complete neutralization of the electrostatic charges on the polymer chains, the cross-linking of the polymer chains started and precipitation was observed; finally, as C^poly_DTAB/C_poly reached 1.3, the precipitate was redissolved slowly due to reversion of the charge ratio of the polymer chains.

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1. Introduction

A large number of researchers have devoted their attention to polymer–surfactant systems in recent decades^1–10 because of their importance to scientific interest and practical applications in biochemical and pharmaceutical products, paint, cosmetics, and other industry fields. Interactions of surfactants with oppositely charged linear polyions may result in a cooperative process above its aggregation concentration (cac) that is 1–3 orders of magnitude lower than the critical micelle concentration (cmc). This aggregation occurs through the binding of the surfactant onto the polymer chains and the binding behavior is usually affected by the polyelectrolyte structure,^11–14 the surfactant properties,^8,15,16 the solvent medium,^9,17 etc. It is generally accepted that the main interaction between the polyelectrolyte and oppositely charged surfactant is predominantly governed by the electrostatic forces; however it may be reinforced by hydrophobic forces between bound surfactant molecules.

Interactions between the polyelectrolyte and mixed surfactants have also been studied,^18–36 most of which involved the system of poly(diallyldimethylammonium chloride) (PDDAC) and mixed surfactants of polyoxyethylene tert-octyl phenyl ether (TX100) and sodium dodecyl sulfate (SDS).^{18–20,22–24,26–29,32,33} It was found in those studies that the polyelectrolyte/surfactant interactions and the phase behavior were mainly determined by the ionic strength and the composition of the mixed surfactants, not by the concentration of polymer or total surfactant.

Many different techniques have been used to study the interactions and phase behavior of polymer/mixed surfactants systems, such as ultrafiltration,²⁰ turbidimetric,^{18–22,24,26,28,31–34} light scattering (QELS),^{19,20,23–28,31–33,35} potentiomeric titration,²² calorimetry,^29,36 deuterium NMR spectroscopy,³⁰ electrophoresis,^31–34 fluorescence,³⁴ electron spin resonance (ESR),³⁵ and electron microscopy.³⁶ Isothermal titration calorimetry (ITC) is a sensitive technique and has an advantage that not only the interactions between polymer and surfactant but also the phase behavior of the polymer/surfactant systems can be investigated simultaneously. Calorimetric studies can also give the corresponding thermodynamic parameters such as various critical concentrations, and the changes of the enthalpy (ΔH), entropy (ΔS) and Gibbs energy (ΔG) in various phase transitions which are crucial to understanding the interactions of the polymer/surfactant. However, to the best of our knowledge, study of the interactions between polyelectrolytes and mixed surfactants using ITC has very seldom occurred,³⁶ and thus more detailed ITC studies are helpful to fully disclose the interactions between polyelectrolytes and mixed micelles.

Poly(acrylic acid) (PAA) is a weak polyelectrolyte, and has surprisingly strong affinity for cationic surfactants. The system of PAA/cationic surfactant has potential applications such as control of drug delivery and chemical reactivity, and nonspecific binding of DNA with basic proteins, and may be used as a simplified model for elucidating the behavior of biological systems.^37,38 We studied the interactions between sodium polyacrylate (PANa) and anion–cation mixed surfactants of sodium bis(2-ethylhexyl)sulfosuccinate (AOT) and dodecyltrimethylammonium bromide (DTAB) in 40 mM NaBr solution previously.³⁶ It was found that the interaction mechanism varied with the titration order. When DTAB was titrated into PANa/AOT, DTAB micelles dissociated into monomers first, then the monomers bound to the PANa/AOT complex; whereas AOT/DTAB mixed micelles bind to PANa when AOT/DTAB is titrated into PANa. This variation of the interaction mechanism with the titration order was attributed to formation of DTAB-rich AOT/DTAB mixed micelles in the low surfactant concentration region for the titration of AOT/DTAB into PANa.³⁶ It is interesting to investigate the polymer/mixed surfactant interactions for a similar system with AOT being replaced by non ionic surfactant TX100, which has smaller cmc value than AOT, hence possibly forms TX100-rich TX100/DTAB micelles.

In this paper, we study the interactions between PANa and the nonionic/cationic mixed surfactants TX100/DTAB using ITC at low ionic strength. Two types of titration experiments are performed in order to investigate the effect of the mixing order on the interaction mechanism. Detailed information on the enthalpy changes is further analyzed using a thermodynamic model to investigate the mechanism of adsorption of the surfactants on the polyelectrolyte chains.

2. Experimental section

2.1. Materials

The polyelectrolyte used in this study is poly(acrylic acid) (PAA, 25% solution from Alfa Aesar Chemical Co.), which has an approximate average molecular weight of 2.58 × 10⁵ determined by static light scattering measurements in our laboratory. The cationic surfactant and nonionic surfactant are dodecyltrimethylammonium bromide (DTAB, from J&K chemical Ltd, ≥99% mass fraction) and polyoxyethylene tert-octyl phenyl ether (TX100, Fluka, n_D²⁰ = 1.490–1.494), respectively. Sodium bromide (NaBr, ≥99.0% mass fraction) and sodium hydroxide (NaOH, ≥96% mass fraction) were purchased from Sitong Chemical Company (Tianjin, China) and Tianjin Chemical Company (Tianjin, China), respectively. All materials were used without further purification. Twice distilled water was used for preparation of the samples.

2.2. Preparation of the polyelectrolyte solutions

Sodium polyacrylate (PANa) was obtained by adding an appropriate amount of NaOH to a poly(acrylic acid) aqueous solution. Then NaBr was added, and the solution was diluted with water and stirred rigorously. Finally, more NaOH was added to adjust pH of the PANa/NaBr solution to a value larger than 9 (about 9.5) to ensure almost complete ionization of PANa.^39,40 The aim of the addition of NaBr was to weaken the interactions between PANa and DTAB to obtain more detailed interaction information. The concentrations of the carboxylate groups and NaBr in the solutions were 6.93 mM and 40 mM, respectively.

2.3. Isothermal titration microcalorimetry (ITC)

The isothermal titration data were collected using the TAM 2277-201 microcalorimetric system (Thermometric AB, Järfäfla, Sweden), which has 4 mL sample and reference cells. In the study of the interactions between the polyelectrolyte and mixed surfactants, the titrations were carried out in two ways. In “type I” titrations, a DTAB aqueous solution was added to 2.2 mL of NaBr solution containing different TX100 concentrations with or without PANa. A “type II” titration corresponds to the addition of TX100/DTAB mixed micelles with certain molar ratios of TX100 to DTAB into 2.2 mL of NaBr solution with or without PANa. The stirring speed in the sample cell was set at 60 rpm, and the experiment temperature was 25.00 ± 0.02 °C. The values of the observed differential enthalpy (ΔH_obs) for various concentrations of surfactants were obtained using the integrals of the areas under the calorimetric peaks and normalized using the small amounts of injected surfactants. The uncertainty in the measurement of ΔH_obs for an individual titration process was about 0.05 kJ mol⁻¹. Each of the titration experiments was carried out twice; the reproducibility is reported in Section 3.2. The enthalpy curves without PANa are named as the background curves in this paper, while the ones with PANa are named as the binding curves.

3. Results and discussion

3.1. Interactions between PANa and TX100

The plots of the observed differential enthalpy ΔH_obs vs. the concentration of surfactant for titrating 14 mM TX100 into 40 mM NaBr aqueous solutions with or without PANa are shown in Fig. 1. The background curve denoted by blank circles has a sigmoid shape with an abrupt change which corresponds to micelle formation. The critical micelle concentration (cmc) was identified using the intersection of extrapolated lines of the initial portion and the rapidly increasing portion of the curves. Meanwhile, the enthalpy of the micellization (ΔH_mic) was determined from the difference in the ΔH_obs between the two linear segments of the plot at the cmc.^41,42 It was found that the values of cmc and ΔH_mic were 0.19 mM and 8.02 kJ mol⁻¹, respectively; which are both smaller than the 2.3 mM determined by Majhi⁴³ and the 12.2 kJ mol⁻¹ determined by Sharma⁴⁴ in TX100 aqueous solutions. These differences may result from the addition of NaBr in our experiment. The electrolyte NaBr disrupts hydrogen bonds between ethylene oxide (EO) groups and H₂O in the solution, leading to dehydration and contraction of the TX100 monomer chains, which favors micelle formation and hence decreases the values of cmc and ΔH_mic. The cmc of TX100 is nearly a thirtieth of that for DTAB (5.91 mM).³⁶ This may be attributed to that the hydrophobicity of TX100 is stronger than DTAB, and no electrostatic repulsion exists on the TX100 micelle surface, thus TX100 forms micelles much easier than DTAB.

Fig. 1 Observed differential enthalpy for titrating 14 mM TX100 into NaBr and PANa/NaBr aqueous solutions: (○) NaBr solution, (●) PANa in NaBr aqueous solution.

When TX100 was added into the PANa solution (indicated by filled circles in Fig. 1), it was found that the binding enthalpy curve coincides with the TX100 background curve, and the cmc value of TX100 is not affected by PANa; indicating that the interactions of TX100 and PANa in the system are negligible. Our result is consistent with ref. 21, where it was found that poly(acrylic acid) could form complexes with TX100 by H-bonding only for pH values below 5, while the H-bonding was suppressed as the pH increased; and no interactions between TX100 and PANa were detected for pH values of about 9.5 in this system.

3.2. Interactions between PANa and TX00/DTAB mixed surfactants

The studies of the interactions between PANa and mixed surfactants were performed in two ways. In the type I titrations, 70 mM DTAB was added to the TX100 solutions or PANa/TX100 solutions with 40 mM NaBr, where the initial concentrations of TX100 (C⁰_TX100) in the ampoule were 5.6 mM, 14 mM, and 28 mM, respectively. The plots of ΔH_obs vs. DTAB concentration (C_DTAB) are shown in Fig. 2a–c. Fig. 2d shows the phase behavior of the polymer/surfactant solutions at various DTAB concentrations denoted by letters corresponding to those in Fig. 2b. As an example, in Fig. 2a, we present the reproducibility of two measurements, where it can be seen that the two curves from two independent titrations are consistent with each other and the three peaks are reproducible. In the “type II” titrations, PANa solutions were titrated using TX100/DTAB mixed micelles, in which the concentration of DTAB was 70 mM, while the concentration of TX100 was 5.6 mM, 14 mM and 28 mM for each of the three individual titrations, respectively. The plots of ΔH_obs vs. the total concentration of surfactant (C_st) are shown in Fig. 3a–c. Fig. 3d shows the phase behavior of the polymer/surfactant solutions at various surfactant concentrations denoted by letters corresponding to those in Fig. 3b. Although peaks A and B in Fig. 3 are somewhat smaller compared with those in Fig. 2, the heights of them are larger than 8 times the estimated uncertainties.

Fig. 2 Observed differential enthalpy for titrating 70 mM DTAB into TX100/NaBr and PANa/TX100/NaBr aqueous solutions with different TX100 concentrations. (a) C⁰_TX100 = 5.6 mM; (□) background curve, (■) binding curve; first measurement (black), second measurement (red). (b) C⁰_TX100 = 14 mM; (○) background curve, (●) binding curve; the locations denoted by letters A, B, C, D and E are discussed in the text. (c) C⁰_TX100 = 28 mM; (△) background curve, (▲) binding curve. (d) The phase behavior of the polymer/surfactant solutions denoted by letters A, B, C, D and E which correspond to those in (b).

Fig. 3 Observed differential enthalpy for titrating TX100/DTAB mixed micelles into NaBr and PANa/NaBr aqueous solutions. The mixed micelles have the following molar ratios of C_TX00 : C_DTAB: (a) 5.6 : 70, (□) background curve, (■) binding curve; (b) 14 : 70, (○) background curve, (●) binding curve; the locations denoted by letters A, B, C, D and E are discussed in the text; (c) 28 : 70, (△) background curve, (▲) binding curve. Insets show magnified views of the background curves. (d) The phase behavior of the polymer/surfactant solutions denoted by letters A, B, C, D and E which correspond to those in (b).

3.2.1. Background curves. As shown in Fig. 2, in the “type I” titrations, the three background curves denoted by the blank symbols show that ΔH_obs generally decreases with the increase of the DTAB concentration. The concentrations of TX100 in the sample cells for the three titrations are all above its cmc value in the 40 mM NaBr aqueous solutions, thus TX100 micelles and TX100 monomers coexist in the solutions. When DTAB micelles are added into the solutions, the DTAB micelles disassociate into monomers, and then a part of the monomers enter the TX100 micelles and form TX100-rich mixed micelles.

As shown in the insets of Fig. 3, in the “type II” titrations, the shape of the left part of each of the three background curves is similar to that of pure TX100; while the right part of the enthalpy curve is similar to that of pure DTAB, indicating that TX100-rich mixed micelles or pure TX100 micelles form first, then transform into DTAB-rich mixed micelles.

It may be observed from the insets of Fig. 3a and b that the rates of increase of ΔH_obs with C_surf abruptly change in the initial parts of the two titration curves with the molar ratios of TX100 to DTAB of 5.6 : 70 and 14 : 70. The points corresponding to the abrupt changes are determined as the first critical micelle concentration (cmc₁), denoted in the insets, indicating the formation of the TX100-rich mixed micelles. However, for the sample with the molar ratio of TX100 to DTAB of 28 : 70, no such change of the increase rate was observed (see the inset of Fig. 3c), which possibly results from that the mixed micelles are formed at very low concentration i.e. cmc₁ is even less than the concentration of 0.42 mM at the first titration point. The enthalpies for the three systems decrease after a certain total surfactant concentration, which is defined as the second critical micelle concentration (cmc₂) and indicates the transformation to DTAB-rich mixed micelles. The values of cmc₁ and cmc₂ are listed in Table 1. Our results are consistent with those of Cui et al.,⁴⁵ who studied ionic/nonionic mixed systems (12-2-12/TX100, 14-2-14/TX100, and SDS/TX100) and ionic/ionic mixed systems (12-2-12/TTAB, 14-2-14/TTAB, and SDS/CTAB) in heavy water solutions using ¹H NMR spectroscopy. They found that the component with the lower cmc in the mixed solutions aggregated first; then the other one entered the aggregates as the total surfactant concentration increased, resulting in mixed micelles.

Table 1 Values of cmc for TX100/DTAB mixed surfactant systems at 298.15 K

CTX100 (mM) : CDTAB (mM)	Cm (mM)	cmc1 (mM)	cmc2 (mM)
5.6 : 70	1.82	1.17	4.20
14 : 70	0.98	0.87	2.93
28 : 70	0.61	<0.42	1.67

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The dependence of the critical concentration C_m of an ideal mixture of two surfactants on the individual critical concentrations C_m1 and C_m2 of the two pure surfactants can be expressed in terms of Clint’s equation⁴⁶

where α is the mole fraction of TX100 in the solution. The values of C_m for the mixed surfactants of TX100/DTAB were calculated using eqn (1) and are compared with the measured values of cmc₁ in Table 1. As shown in Table 1, all the three values of cmc₁ are smaller than the values of C_m calculated using Clint’s equation, indicating attractive interactions between the two surfactants and negative deviations from the ideal mixing behavior in the mixed micelles. The values of cmc₁ and cmc₂ decrease with the increasing molar ratio of TX100 to DTAB, which may be attributed to the decrease of the electrostatic repulsion between the surfactant headgroups by mixing the charged surfactant with the non-ionic surfactant and hence stabilization of the micelles,⁴⁷ since the EO groups of TX100 have a negligible contribution to the interactions between cationic surfactant DTAB and nonionic surfactant TX100 in the surfactant mixtures.⁴⁸

3.2.2. Binding curves. From Fig. 2 and 3, whether for the “type I” or “type II” titration, it may be found that each of the three binding curves exhibits three endothermic peaks A, B, and C (denoted only in Fig. 2a and 3a) and five critical points C₁, C′, C′′, C_max and C₂ (denoted only in Fig. 2c and 3c). It is known from the above results that the interactions of TX100 and PANa can be ignored. At low surfactant concentration, each of the binding curves coincides with the background one, indicating that the added DTAB also does not interact with the polymer; thus the endothermic titration heat effect results only from dissociation of the micelles, dilution of the dissociated surfactant monomers, and the interactions between surfactants. As the surfactant concentration increases to C₁, the solution is still transparent (see bottle A in Fig. 2d or 3d), but the binding curve starts to deviate from the background one, which indicates that DTAB monomers bind to the anionic sites of the polymer chains mainly by electrostatic interactions to yield an additional endothermic heat effect and consequently form peak A. The solution then becomes slightly milk-white (see bottle B in Fig. 2 or 3). The value of ΔH_obs increases rapidly again when the concentration reaches C′ to form peak B; in this region, the neighbouring bound DTAB molecules start to aggregate through hydrophobic interactions and form the PANa/DTAB micelles,⁴⁰ and the solution becomes milk-white (see bottle C in Fig. 2 or 3). Although the bulk concentration of DTAB at C′ is below the cmc of DTAB in the solution without polymer, the local concentration of DTAB on the polymer chains is high enough to induce the micellization. When the surfactant concentration reaches C′′, the enthalpy increases again to form peak C, where the TX100/DTAB mixed micelles are possibly linked by different polymer chains⁴⁹ to induce polymer cross-linking; some small precipitates appear at the bottom of the bottle (see bottle D in Fig. 2 or 3). With further increasing of the surfactant, the enthalpy curve levels off; in this stage, the opaque solution becomes clear, however precipitation is observed. For DTAB titrated into PANa/TX100, some PANa/TX100/DTAB complexes aggregate to form flocculations suspended in the solution, and some of them appear at the bottom of the bottle (see bottle E in Fig. 2d); while for TX100/DTAB titrated into PANa, the polymer solution becomes very viscous and solid residues adhere to the bottom of the sample bottle (see bottle E in Fig. 3d). These different phenomena may be explained as follows. The relative amount of TX100 is much less in the “type II” titration than that in “type I”; thus less DTAB forms free mixed micelles with TX100 in the solution phase and more DTAB interacts with the polymer chains to neutralize the opposite charge density on the polymer chains and to more significantly induce polymer cross-linking in “type II” than in “type I”. This results in the higher viscosity in the “type II” titration. With further addition of surfactant, the binding curve merges with the background one at C₂, which represents the condition where the polymer chains are completely saturated with surfactant. The binding mechanism and the structural changes of the polymer/surfactant complex are illustrated in Fig. 4. All the critical concentrations determined by the above two titration methods, together with the results from titration of DTAB into PANa without TX100 (ref. 36), are listed in Tables 2 and 3, respectively.

Fig. 4 Illustration of the interaction mechanism and the structural changes of the polymer/surfactant complex during the titrations.

Table 2 Critical concentrations for the titration of DTAB into PANa/TX100 solutions with various TX100 concentrations

C^0TX100 (mM)	C1 (mM DTAB)	C′ (mM DTAB)	C′′ (mM DTAB)	Cmax (mM DTAB)	C2 (mM DTAB)
0	0.63	4.72	7.16	7.62	9.32
5.6	0.94	4.17	8.80	9.56	10.95
14.0	1.58	5.27	10.71	12.05	14.54
28.0	2.76	7.54	12.52	13.53	16.09

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Table 3 Critical concentrations for the titration of TX100/DTAB mixed micelles into PANa solutions

CTX100 : CDTAB	C1 (mM surf)	C′ (mM surf)	C′′ (mM surf)	Cmax (mM surf)	C2 (mM surf)
5.6 : 70	0.67	2.32	7.95	8.50	9.56
14 : 70	0.75	3.98	10.62	10.91	12.32
28 : 70	0.84	5.22	12.21	13.39	15.60

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As indicated in Tables 2 and 3, all the critical concentrations increase with the concentration of TX100, since more DTAB monomers are needed to participate in the free mixed micelles with TX100 before the DTAB monomers bind to negatively charged carboxylate sites along the polymer chains, form the micelles and induce the polymer cross-linking; thus more surfactants are required to reach the corresponding critical conditions.

Peak A indicating DTAB monomers binding to PANa chains through electrostatic interactions and peak B representing the micellization of the surfactants induced by the polymer are endothermic, which is consistent with that of the PANa/AOT/DTAB system;³⁶ however, peak C of DTAB titrated into PANa/TX100 and TX100/DTAB titrated into PANa is also endothermic, different from that of the PANa/AOT/DTAB system which showed an exothermic peak,³⁶ indicating that the enthalpy caused by cross-linking is system dependent.

It is interesting to compare the “type II” titration results of PANa/TX100/DTAB in this work with that of PANa/AOT/DTAB reported in the previous publication.³⁶ It was found that when the mixed micelles TX100/DTAB were titrated into the polymer solution, the micelles dissociated first, then individual DTAB monomers bound to the polymer; while when the mixed micelles AOT/DTAB were titrated into the polymer solution, the mixed micelles were capable of binding to the polymer. This may be explained as follows. Because TX100 has a much lower cmc value compared with AOT, under our experimental conditions and in low surfactant concentration regions, DTAB-rich AOT/DTAB mixed micelles and TX100-rich TX100/DTAB mixed micelles respectively exist in the corresponding systems. The electrostatic interactions cause DTAB-rich AOT/DTAB mixed micelles to bind onto the PANa chains; moreover, the high hydrophobicity of AOT possibly also reinforces this binding through the hydrophobic interactions. On the other hand, the relative amount of DTAB in the TX100-rich TX100/DTAB mixed micelles is small; hence the charge density of the micelle is too low to be attracted to the polymer chains.

3.2.3. Thermodynamic characterization of the polymer/surfactant interactions. A pseudo phase model³⁶ was used to analyze the experimental results obtained from the “type I” titrations to further understand how the concentration of TX100 affects the interactions in the PANa/TX100/DTAB system. Before the concentration of DTAB reaches the point C₁, there exists an equilibrium of DTAB between the aqueous solution phase and the TX100/DTAB mixed micelle phase, and this equilibrium can be thermodynamically characterized by

μ^aq_DTAB = μ^mic_DTAB (2)

where μ^aq_DTAB and μ^mic_DTAB are the chemical potentials of DTAB in the bulk aqueous solution and in the TX100/DTAB mixed micelles, respectively. These two chemical potentials are expressed bywhere μ^aq*_DTAB and μ^mic*_DTAB are the chemical potentials at which the concentration variables C^aq_DTAB/S and C^mic_DTAB/C^mic_TX100 are equal to 1 and the DTAB in the bulk solution and in the surfactant mixture is assumed to have the behavior of an ideal dilute solution; C^aq_DTAB, C^mic_DTAB and C^mic_TX100 are the concentrations of DTAB in the aqueous solution phase and the micelle phase, and TX100 in the micelle phase, respectively; S = 1 mM is used to normalize the concentration C^aq_DTAB. As shown in Fig. 1, no interactions between TX100 and the polymer were detected, thus TX100 only exists in the aqueous solution and micelle phases. Because the cmc of the mixed micelles is very low, we neglected the existence of the TX100 monomers in the water phase. Therefore the concentration of TX100 in the micelle phase C^mic_TX100 can be substituted by the total concentration of TX100 (C_TX100) in the system. Combining eqn (2)–(4) giveswhere K₁ is the equilibrium constant for the equilibrium of DTAB between the aqueous phase and TX100/DTAB mixed micelle phase, thus C^mic_DTAB is expressed by

C^mic_DTAB = K₁C_TX100C^aq_DTAB/S (6)

The total concentration of DTAB (C_DTAB) in the system can be written as

C_DTAB = C^aq_DTAB + C^mic_DTAB = C^aq_DTAB + K₁C_TX100C^aq_DTAB/S (7)

assuming that as the concentration of DTAB in the aqueous solution phase reaches a certain value at point C₁, binding of the DTAB monomers to the polymer immediately occurs. When the total concentration of DTAB is above C₁, there will be an amount of DTAB bound to the polymer; thus an additional equilibrium exists in the system, and the phase equilibrium can be characterized by

μ^poly_DTAB = μ^aq_DTAB = μ^mic_DTAB (8)

withwhere μ^poly_DTAB is the chemical potential of DTAB bound on the polymer; μ^poly*_DTAB is the chemical potential when the concentration variable C^poly_DTAB/C_poly is equal to 1 and the DTAB on the polymer surface has the behavior of an ideal dilute solution; C^poly_DTAB and C_poly are the concentrations of DTAB bound on the polymer and of the carboxylate groups of the polymer in the system, respectively. Substituting eqn (4) and (9) into eqn (8) giveswith K₂ being the equilibrium constant for the equilibrium of DTAB between the TX100/DTAB mixed micelle phase and the polymer phase, and thus

C^mic_DTAB = K₂C_TX100C^poly_DTAB/C_poly (11)

The total concentration of DTAB in the system can be written as

C_DTAB = C^aq_DTAB + C^poly_DTAB + C^mic_DTAB = C^aq_DTAB + C^poly_DTAB + K₂C_TX100C^poly_DTAB/C_poly (12)

According to eqn (7), at point C₁, a plot of C_DTAB vs. C_TX100 yields a straight line, as shown by line a in Fig. 5. A linear least-square fitting gives eqn (13)

C_DTAB = 0.08C_TX100 + 0.55 (13)

and then K₁ was calculated to be 0.14 using the slope and intercept of eqn (13). According to eqn (12), at point C′, a plot of C_DTAB vs. C_TX100 yields a straight line, as shown by line b in Fig. 5, and a linear least-squares fitting gives eqn (14)

C_DTAB = 0.13C_TX100 + 4.06 (14)

The slope K₂C^poly_DTAB/C_poly and the intercept (C^aq_DTAB + C^poly_DTAB) of eqn (14) were calculated to be 0.13 and 4.16, respectively. Combining eqn (6) and (11) gives C^aq_DTAB = 0.13/K₁; then, according to eqn (12), we have C^poly_DTAB = 4.16 − C^aq_DTAB and K₂ = 0.13C_poly/C^poly_DTAB. This allowed us to calculate C^poly_DTAB/C_poly and K₂ at C′, which were 0.49 and 0.26, respectively. The former represents that when the ratio of the concentration of DTAB bound on the polymer to the concentration of carboxylate groups reaches 0.49, the formation of polymer induced micelles occurs. Also according to eqn (12), at points C′′, C_max and C₂, plots of C_DTAB vs. C_TX100 yield three straight lines, as shown by lines c, d and e in Fig. 5, and linear least-squares fitting gives eqn (15)–(17)

C′′: C_DTAB = 0.23C_TX100 + 7.54 (15)

C_max: C_DTAB = 0.26C_TX100 + 8.16 (16)

C₂: C_DTAB = 0.32C_TX100 + 9.71 (17)

With the slopes and intercepts of eqn (15)–(17) and the calculation procedure used for the calculation at point C′, the values of C^poly_DTAB/C_poly and K₂ at the three critical points were also obtained. The calculation results for C^aq_DTAB, C^poly_DTAB, C^poly_DTAB/C_poly, K₁, and K₂ at the different critical points are listed in Table 4. Finally, according to eqn (12), the values of C^mic_DTAB at all the critical points for different TX100 concentrations were calculated and are summarized in Table 5.

Fig. 5 Plots of C_DTAB vs. C_TX100 at the critical points C₁, C′, C′′, C_max, and C₂. The symbols show the experimental results, and the lines represent the least-squares fittings.

Table 4 Values of C^aq_DTAB, C^poly_DTAB, C^poly_DTAB/C_poly, K₁, and K₂ at the critical points C₁, C′, C′′, C_max and C₂, calculated using the pseudophase model for DTAB titrated into PANa/TX100 aqueous solutions

Critical points	C^aqDTAB (mM)	C^polyDTAB (mM)	C^polyDTAB/Cpoly	K1	K2
C1	0.55			0.14
C′	0.89	3.17	0.49		0.26
C′′	1.57	5.98	0.99		0.23
Cmax	1.79	6.37	1.08		0.24
C2	2.23	7.48	1.31		0.25

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Table 5 Values of C^mic_DTAB (mM) at the critical points C₁, C′, C′′, C_max and C₂ for DTAB titrated into PANa/TX100 aqueous solutions with various TX100 concentrations (C⁰_TX100)

Critical points	C^micDTAB (mM)
	C^0TX100 5.6	C^0TX100 14	C^0TX100 28
C1	0.39	1.00	2.19
C′	0.11	1.21	3.56
C′′	1.30	3.17	4.93
Cmax	1.40	3.89	5.37
C2	1.24	4.83	6.38

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As shown in Table 4, the ratios of C^poly_DTAB/C_poly increase during the titration process; while K₂ is confirmed to be a constant independent of the concentrations in the system and the amount of DTAB bound to the polymer. The ratio of C^poly_DTAB/C_poly characterizes the extent of the neutralization of the opposite charges on the polymer. We calculated the value of C^poly_DTAB/C_poly at C′′, which was 0.99; however, the ratios of the total DTAB concentration to the carboxylate group concentration were found to be 1.59, 2.09, and 2.41 for the systems containing TX100 concentrations of 5.6 mM, 14 mM and 28 mM, respectively. This confirms that the charge neutralization of the polymer rather than of the whole system decides the occurrence of cross-linking of the polymer chains.³⁶ At C₂ the polymer chains are completely saturated with surfactant, and C^poly_DTAB/C_poly is equal to 1.3, indicating that the charge ratio reverses. At this point redissolution of the precipitate was observed although this process is very slow. From Table 5, it was found that for the same critical point, the value of C^mic_DTAB increases with increasing TX100 concentration, indicating that more DTAB participates in the free mixed micelles with TX100 in the solution phase of the system.

4. Conclusions

Interactions between PANa and TX100/DTAB mixed surfactants in 40 mM NaBr solutions were studied using isothermal titration calorimetry (ITC). It was found that when TX100/DTAB was titrated into the NaBr solution, TX100-rich mixed micelles formed first, then this changed to DTAB-rich mixed micelles. The values of the critical concentration cmc₁ of the TX100/DTAB mixed micelles were all smaller than that calculated using Clint’s equation, suggesting synergistic effects between TX100 and DTAB.

At low ionic strength (40 mM NaBr), whether DTAB was titrated into PANa/TX100 or TX100/DTAB was titrated into PANa, three endothermic peaks were detected; which represent the binding of DTAB monomers to PANa chains through electrostatic interactions, polymer-induced micellization, and cross-linking of the polymer chains, respectively. The sign of the observed differential enthalpy of peak C was found to be opposite to that of the system PANa/AOT/DTAB, which reflected that the enthalpy effect of cross-linking was system dependent. The interaction mechanism was interpreted using a thermodynamic model, through which it was found that when the molar ratio of the bound-DTAB to the carboxylate groups of the polymer (C^poly_DTAB/C_poly) reached about 0.5, the polymer-induced micellization occurred; when C^poly_DTAB/C_poly reached about 1, the cross-linking of the polymer chains started and precipitation was observed, indicating that the charge neutralization of the polymer rather than of the whole system decides the occurrence of cross-linking of the polymer chains; finally as C^poly_DTAB/C_poly reached 1.3, the polymer chains were completely saturated with surfactant, and the precipitate was redissolved slowly. This study may shed new light on understanding the interaction mechanism of the polyelectrolyte/mixed surfactant system, which would help in designing rational polymer/surfactant systems for various applications, however more investigations are required to further validate the proposed mechanism and thermodynamic model.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (projects 20973061, 21173080, 21373085 and 21403098) and the Fundamental Research Funds for the Central Universities (lzujbky-2014-181).

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