Langmuir 2002, 18, 4465-4470
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Thermodynamic Effects of Alcohol Additives on the Cooperative Binding of Sodium Dodecyl Sulfate to a Cationic Polymer Hiroyuki Fukui, Iwao Satake, and Katumitu Hayakawa* Department of Chemistry and BioScience, Faculty of Science, Kagoshima University, Korimoto, Kagoshima 890-0065, Japan Received September 4, 2001. In Final Form: November 15, 2001 The binding isotherms for the interaction of sodium dodecyl sulfate (SDS) with poly(diallyldimethylammonium chloride) in aqueous solutions of 1,8-octanediol, 1,2-octanediol, or 1-butanol (BuOH) were determined by potentiometry, using a functional membrane responsive to anionic surfactants. Each alcohol caused the binding isotherm to shift to a lower SDS concentration. The magnitude of the effect was 1,2octanediol > 1,8-octanediol ∼ BuOH. The binding isotherms were analyzed by a linear lattice model for cooperative binding. The effect of the alcohols was analyzed by thermodynamic treatment of mixtures based on the regular mixing model, and the interchange energies between alcohol and water (4.06 kT for 1,8-octanediol, 4.82 kT for 1,2-octanediol, and 3.63 kT for BuOH) and between alcohol and SDS (-1.21 kT for 1,8-octanediol, -3.52 kT for 1,2-octanediol, and -1.65 kT for BuOH) were estimated. The pronounced effect of 1,2-octanediol on the binding isotherms was ascribed to its instability in the water phase and its strong hydrophobic interaction with the SDS anion in the polyion domain. The effect of 1,8-octanediol was comparable to that of BuOH, but 1,8-octanediol was more unstable in both the water phase and the polyion domain by about kT ln 2 compared to BuOH. This phenomenon can be attributed to the fact that binding of one 1,8-octanediol molecule structurally approximates the binding of two BuOH molecules.
Introduction Interactions between ionic polymers and surfactants are of interest from both academic and practical viewpoints. These interactions are relevant to biological systems and processes, and they have applications in detergents, paints and coatings, cosmetics, pharmaceuticals, tertiary oil recovery, and other industrial fields. The interaction of polyions with surfactants of opposite charge is characterized by highly cooperative surfactant binding arising from hydrophobic interactions between the bound surfactant ions.1-4 This cooperative binding has been found to depend on a variety of factors, such as the length of the surfactant ion carbon chain,5-9 the salt concentration,5,6,10-12 and the polyion charge density.7,13 The resulting surfactant ion clusters on the polyion chain are analogous to conventional surfactant micelles, in that (1) Robb, I. D. Polymer/surfactant interactions; Marcel Dekker: New York, 1981; p 109. (2) Goddard, E. D. Colloids Surf. 1986, 19, 301. (3) Hayakawa, K.; Kwak, J. C. T. In Cationic Surfactants: Physical Chemistry; Rubingh, D. N., Holland, P. M., Eds.; Marcel Dekker: New York, 1991; Vol. 37, p 189. (4) Goddard, E. D.; Ananthapadmanabhan, K. P. Interactions of Surfactants with Polymers and Proteins; CRC Press: Boca Raton, FL, 1993; p 427. (5) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Biophys. Chem. 1983, 17, 175. (6) Malovikova, A.; Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1984, 88, 1930. (7) Malovikova, A.; Hayakawa, K.; Kwak, J. C. T. In Structure/ performance relationships in surfactants; Rosen, M. J., Ed.; American Chemical Society: New York, 1984; p 225. (8) Satake, I.; Hayakawa, K.; Komaki, M.; Maeda, T. Bull. Chem. Soc. Jpn. 1984, 57, 2995. (9) Hayakawa, K.; Murata, H.; Satake, I. Colloid Polym. Sci. 1990, 268, 1044. (10) Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1982, 86, 3866. (11) Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1983, 87, 506. (12) Shirahama, K.; Yuasa, H.; Sugimoto, S. Bull. Chem. Soc. Jpn. 1981, 54, 375. (13) Satake, I.; Takahashi, T.; Hayakawa, K.; Maeda, T.; Aoyagi, M. Bull. Chem. Soc. Jpn. 1990, 63, 926.
they not only solubilize hydrophobic dyes but also serve as a hydrophobic medium for photophysical reactions in aqueous solutions.14-21 Most previous studies of surfactant ion/polyion systems have been confined to interactions occurring in aqueous solutions containing only trace amounts of additives. Shirahama et al. investigated the binding isotherms for the interaction of dodecylpyridinium chloride with poly(styrene sulfonate) in ethanol-water mixed solvents and found that both intrinsic binding and the cooperative effect were reduced by added ethanol.22 They interpreted these characteristic behaviors as being caused by the replacement of water molecules involved in hydration of the hydrophobic moieties of the polymer and surfactant by ethanol molecules. However, Hayakawa et al. found that a hydrophobic water-insoluble dye enhanced cooperative binding of surfactant ions.18 These findings prompted us to examine the effects of alcohols more hydrophobic than ethanol on cooperative surfactant/polyanion binding. Here, we report the effect of 1-butanol (BuOH), 1,8octanediol, and 1,2-octanediol on the cooperative binding of sodium dodecyl sulfate (SDS) to the cationic polymer poly(diallyldimethylammonium chloride) (PDAC) in aqueous solution. The results were analyzed with the aid of a (14) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Macromolecules 1983, 16, 1642. (15) Hayakawa, K.; Ohta, J.; Maeda, T.; Satake, I.; Kwak, J. C. T. Langmuir 1987, 3, 377. (16) Hayakawa, K.; Ohyama, T.; Maeda, T.; Satake, I.; Kwak, J. C. T. Langmuir 1988, 4, 481. (17) Hayakawa, K.; Satake, I.; Kwak, J. C. T.; Gao, Z. Colloids Surf. 1990, 50, 309. (18) Hayakawa, K.; Fukutome, T.; Satake, I. Langmuir 1990, 6, 1495. (19) Hayakawa, K.; Satake, I.; Kwak, J. C. T. Colloid Polym. Sci. 1994, 272, 876. (20) Hayakawa, K.; Shinohara, S.; Sasawaki, S.; Satake, I.; Kwak, J. C. T. Bull. Chem. Soc. Jpn. 1995, 68, 2179. (21) Hayakawa, K.; Nakano, T.; Satake, I.; Kwak, J. C. T. Langmuir 1996, 12, 269. (22) Shirahama, K.; Liu, J.; Aoyama, I.; Takisawa, N. Colloids Surf., A 1999, 147, 133.
10.1021/la0113926 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/26/2002
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thermodynamic treatment based on the regular mixing model, and the interchange energies between alcohol and water and between alcohol and SDS were estimated. Experimental Section Materials. SDS (99%, Nacalai Tesque, Osaka), PDAC (standard reagent for colloid titration, Wako, Osaka), 1,8-octanediol (TCI, Tokyo), 1,2-octanediol (TCI, Tokyo), and BuOH (GR, Cica, Tokyo) were used without further purification. Vinyl chloride (99.5%, Aldrich, Milwaukee, WI), 3-acrylamido-N,N-dimethylpropylamine (Wako, Osaka), and methyl iodide (GR, Wako, Osaka) were used without further purification for synthesis of modified poly(vinyl chloride) (PVC). All solutions were prepared using water obtained from a Milli-Q Millipore water filtration system. Measurements. The equilibrium concentration of SDS in aqueous solutions of PDAC was determined by potentiometry using the cell assembly shown below.
Figure 1. Plots of emf against the logarithm concentration of SDS selective electrode: (a) calibration (no polyion) and (b) 0.50 mM PDAC.
The functional membrane consisted of 40% (w/w) modified PVC and 60% (w/w) Elvaroy L-742.23 The modified PVC was synthesized by copolymerization of vinyl chloride and 3-acrylamido-N,N-dimethylpropylamine followed by quaternization with methyl iodide. In this concentration cell assembly, the electromotive force (emf, E) in the presence of polyelectrolyte is given by eq 1:
E)
RT CS ln ) E0 + S log CS nF C0
(1)
where R is the gas constant, T is degrees Kelvin, F is the Faraday constant, C0 and CS are the SDS concentrations in the reference and test solutions, respectively, and S is the slope of the emf versus log CS plot in the absence of polyion (calibration). The emf reading was detected with a stability within 0.1 mV. NaCl at 0.01 mol dm-3 was used as a buffer. In all experiments, the air temperature was thermostatically maintained at 25 °C, and test solutions were stirred using a magnetic stirrer during measurements.
Figure 2. The binding isotherms of SDS binding by PDAC in the presence of 1,8-octanediol. The concentration of 1,8octanediol: (a) 0.0, (b) 0.025, (c) 0.050, (d) 0.075, and (e) 0.100 M.
Results and Discussion Binding Isotherms. Semilogarithmic plots of E versus log CS for SDS solutions containing alcohol and 0.01 mol dm-3 NaCl had nearly Nernstian slopes between 2 µmol dm-3 and the critical micelle concentration of SDS in the absence of PDAC. This result indicated excellent Nernstian response of the functional membrane to the dodecyl sulfate anion. For example, the calibration plot (taken in the absence of PDAC) for 1,8-octanediol at 0.05 mol dm-3 and NaCl at 0.01 mol dm-3 yielded a slope of 59.2 mV, as shown in Figure 1. The deviation from the calibration curve observed in the presence of PDAC is due to binding of surfactant ion moieties by PDAC. Assuming that the activity coefficient for the free surfactant is constant, the degree of binding (z) can be calculated by comparing the binding curves observed in the presence and absence of PDAC: z ) Cb/CP ) (CS - Cf)/CP. Here, Cb and Cf are the concentration of bound and free surfactant ions, respectively, and CP is the PDAC (monomer) concentration. Binding isotherms were constructed by plotting z against Cf as shown in Figures 2-4. The binding isotherms observed in the presence of alcohols displayed characteristics typical of a cooperative process; binding started abruptly and saturated within a narrow range of surfactant concentration (Figures (23) Wan Yunus, W. M. Z.; Taylor, J.; Bloor, D. M.; Hall, D. G.; WynJones, E. J. Phys. Chem. 1992, 96, 8979.
Figure 3. The binding isotherms of SDS binding by PDAC in the presence of 1,2-octanediol. The concentration of 1,2octanediol: (a) 0.0, (b) 0.010, and (c) 0.025 M.
2-4).2,10,24 Cooperative binding of surfactant ions by polyions of opposite charge is a well-known phenomenon. The surfactant concentration at which binding begins to occur is called the critical aggregation concentration (cac), because surfactant clusters form on the polymer chain (24) Shirahama, K.; Ide, N. J. Colloid Interface Sci. 1976, 54, 450.
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Figure 4. The binding isotherms of SDS binding by PDAC in the presence of BuOH. The concentration of BuOH: (a) 0.0, (b) 0.015, (c) 0.025, (d) 0.055, and (e) 0.100 M.
via hydrophobic interactions among bound surfactant ions.25-30 For all the alcohols tested, the binding isotherms shifted to a lower equilibrium concentration of SDS as alcohol content increased. The largest shift was caused by 1,2-octanediol. At 0.025 mol dm-3, 1,8-octanediol caused a smaller shift than BuOH, but at higher concentrations, 1,8-octanediol caused a larger shift than BuOH. By treating the complex of polyion and surfactant as a pseudo-phase of regular mixing of the empty binding site (O) and the site occupied by the surfactant ion (S) on the polyion, surfactant binding can be expressed by eq 2,
(- -O- -) + S h (- - -S- -)
(2)
This treatment allows derivation of eq 3 (Appendix 1):
z)
[
1 12
1-s
]
x(1 - s)2 + 4s/u
s ) KCf
(3)
where K is the equilibrium constant and the parameter u indicates the cooperativity of surfactant binding and relates to the interchange energy between the empty site and the site occupied by surfactant.
u ) exp
{kT2 [E
OS
1 - (EOO + ESS) 2
]}
(4)
where k is the Boltzmann constant and Eij is the pairwise interaction energy between sites represented by i and j. The bracketed term is defined as the interchange energy. Satake and Yang derived the same equation based on a matrix model for the coil-to-helix transition of a biological polymer derived by Zimm and Bragg.31 The constant K is a measure of the binding affinity of the surfactant and is defined by eq 5.
-kT ln K ) µ0S,p - µ0O,p - µ/S 0 µi,p
Table 1 lists the values of binding affinity K and cooperativity parameter u which give the best fit of the calculated binding isotherms to the experimental data. The notation K0 is used for the system that does not contain alcohol. The simulated curves are represented by the solid lines in Figures 2-4. The binding affinity K increased as the alcohol content increased, but the cooperativity parameter u displayed no correlation with alcohol content. This finding suggests that the surfactant clusters formed in the polyion domain solubilize the alcohol, and the solubilized alcohol promotes surfactant binding by increasing the hydrophobicity of the clusters. Regular Mixing Model. The alcohol-induced shift in the binding isotherms of surfactant by polyion has been thought to arise from the solubilization of alcohol. The effect of alcohol content on the binding of SDS by PDAC was analyzed by a pseudo-phase separation model for mixed clusters of surfactant and alcohol in the polyion domain. In this model, the polyion pseudo-phase consists of two components: alcohol and surfactant. The mole fractions are represented as yA for alcohol and yS ()1 yA) for surfactant. Polyion is not counted as a component. On the other hand, the aqueous phase consists of three components: water, alcohol, and surfactants. The mole fractions are presented as xA for alcohol, xS for surfactant, and xW for water. This analysis leads to the following relationship (Appendix 2):
ln xA + βAW ) ln yA + βSA,p(1 - yA)2
Here, xA and yA represent the mole fractions of alcohol in the bulk phase and the polyion pseudo-phase, respectively, and βAW and βSA,p represent the interchange energy parameters between alcohol and water in the bulk phase and between surfactant and alcohol in the polyion pseudophase, respectively. βij is defined by eq 7,
βij )
{
m 1 E - (E + Ejj) kT ij 2 ii
and are the standard chemical potentials of the corresponding sites in the polyion pseudo-phase and of the surfactant in the bulk phase based on infinite dilution. (25) Abuin, E. B.; Scaiano, J. C. J. Am. Chem. Soc. 1984, 106, 6274. (26) Lissi, E. A.; Abuin, E. J. Colloid Interface Sci. 1985, 105, 1. (27) Almgren, M.; Hansson, P.; Mukhtar, E.; van Stam, J. Langmuir 1992, 8, 2405. (28) Cabane, B. J. Phys. Chem. 1977, 81, 1639. (29) Cabane, B.; Duplessix, R. J. Phys. (Paris) 1982, 43, 1529. (30) Cabane, B. In Solution Behavior of Surfactants; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; Vol. 1, p 661. (31) Satake, I.; Yang, J. T. Biopolymers 1976, 15, 2263.
}
(7)
where m is the number of nearest neighbors. Equation 6 indicates that the mole fraction of alcohol (yA) in the polyion pseudo-phase depends only on the alcohol content (xA) in the bulk phase. The total alcohol content is much greater than the bound surfactant content that is responsible for the solubilization of alcohol in the polyion pseudo-phase, and the bulk phase alcohol content (xA) is nearly equal to the total alcohol content and can be presumed to be independent of the degree of surfactant binding by the polyion. Therefore, xA was calculated from the total alcohol content by CA/55.5 where the value 55.5 corresponds to the mole of 1 L water. This consideration means that yA is independent of the degree of surfactant binding by the polyion and the bound surfactant always accompanies the alcohol at a constant ratio r defined by eq 8.
(5)
µ/S
(6)
r)
yA 1 - yA
(8)
When we introduce an imaginary chemical species R ()S + rA), surfactant binding in the presence of alcohol can be considered as the binding of the imaginary species R by the polyion,
(- -O- -) + (S + rA)bulk h (- -R- -)
(9)
Equation 3 can be applied to the surfactant binding expressed by reaction 9, but the cooperativity parameter
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Table 1. The Binding Parameters of SDS by PDAC at 25 °C in the Presence of Alcohol alcohol
alcohol concn CA/M
binding constant K or K0
cooperative parameter u
ERR/kT
none 1,8-octanediol
0 0.025 0.050 0.075 0.100 0.010 0.025 0.015 0.025 0.050 0.100
166 ( 0.2 (K0) 176 ( 0.2 196 ( 0.3 215 ( 0.5 239 ( 0.4 223 ( 0.4 388 ( 0.8 173 ( 0.3 191 ( 0.3 193 ( 0.4 204 ( 0.3
2300 ( 260 (u0) 4900 ( 1,200 4120 ( 560 2430 ( 440 2960 ( 510 2400 ( 320 1180 ( 150 1790 ( 230 2360 ( 310 1960 ( 270 2510 ( 280
-7.74 (ESS/kT) -8.50 -8.32 -7.80 -7.99 -7.78 -7.07 -7.49 -7.77 -7.58 -7.83
1,2-octanediol BuOH
average -8.15 ( 0.32
average -7.67 ( 0.16
Table 2. Alcohol Content and Interchange Energy Parameters in PDAC/SDS Complexes mole fraction of alcohol alcohol 1,8-octanediol
1,2-octanediol BuOH
a
alcohol concn CA/M
in bulk phase xA
in P/S complexa yA
partition coefficient yA/xA
mole ratio in P/S complexa r ) yA/yS
0.025 0.050 0.075 0.100 0.010 0.025 0.015 0.025 0.050 0.100
0.000 45 0.000 90 0.001 35 0.001 80 0.000 18 0.000 45 0.000 27 0.000 45 0.000 90 0.001 80
0.054 0.134 0.19 0.25 0.175 0.34 0.038 0.11 0.12 0.156
120 150 140 140 970 760 140 240 130 87
0.06 0.15 0.24 0.33 0.21 0.52 0.04 0.12 0.14 0.18
exchange energy parameter βSA,p
βAW
-1.21 ( 0.10
4.06 ( 0.16
-3.52 ( 0.27
4.82 ( 0.28
-1.65 ( 0.21
3.63 ( 0.23
“P/S complex” corresponds to the polyion pseudo-phase.
u and the binding affinity K are expressed by eqs 10 and 11.
-kT ln K ) µ0R,p - µ0O,p - (µ/S + rµA) u ) exp
{kT1 (2E
OR
- EOO - ERR)
}
(10) (11)
Here, µA is the chemical potential of the bulk phase alcohol. Since R is the 1:r mixture of the surfactant and alcohol in the surfactant cluster, the standard chemical potential is expressed by eq 12:
µ0R,p ) µ0S,p + kT ln(1 - yA) + kTβSA,pyA2 + rµA,p (12) This equation leads to eq 13 (Appendix 3).
ln(K0/K) ) ln(1 - yA) + βSA,pyA2
(13)
K0 and K are the binding affinities of the polyion for surfactant in the absence and presence of alcohol, respectively. βSA,p and βAW are constant and independent of the alcohol content. Values of K0 and K at xA for this study were calculated and are shown in Table 1. Equations 6 and 13 make it possible to determine βSA,p, βAW, and yA (at xA) from the K values at two different alcohol concentrations. From eq 8, r at xA could be determined. Table 2 gives the resulting values of yA, r, βSA,p, and βAW. The Interactions between Alcohol and SDS and between Alcohol and Water. The data shown in Table 2 indicate that the alcohol content yA in the surfactant cluster increases with total alcohol concentration. The fact that yA is much larger than the alcohol content xA in the bulk phase indicates the strong solubilization capacity
of SDS clusters formed in the polyion domain. The ratio of yA to xA is an estimate of the partitioning of alcohol between bulk phase and surfactant clusters and is much larger for 1,2-octanediol than for 1,8-octanediol. Presumably, the hydrophobic dodecyl group of the surfactant fits better with 1,2-octanediol than with 1,8-octanediol. The average partition coefficients are estimated to be 140 ( 10 for 1,8-octanediol and 870 ( 150 for 1,2-octanediol. The Gibbs energies of transfer from bulk phase to the SDS/PDAC complex are -4.9 kT and -6.8 kT, respectively. The yA/xA ratio decreases as the BuOH concentration increases, because yA tends toward saturation at high BuOH concentrations. This indicates that excess BuOH in the bulk phase does not promote the solubilization of BuOH in SDS clusters in the polyion domain. The interchange energy parameters βAW for the bulk phase are all positive and greater than 2 where the mixture is immiscible and induces two phases, indicating that all alcohols have solubility limits. 1,2-Octanediol is the most unstable in water. On the other hand, the interchange energy parameters βSA,p are negative, indicating a favorable interaction between SDS and alcohols. These interchange energy parameters are compared in Figure 5. The lowest value, which was obtained for 1,2-octanediol, indicates the highly favorable lateral interactions between the hydrophobic alkyl groups of SDS and 1,2-octanediol. Both interchange energy parameters, βAW and βSA,p, of 1,8-octanediol are larger by 0.4 than those of BuOH. The entropy loss of k ln 2 (∼0.7 k) is expected for 1,8-octanediol compared to BuOH, because the structure of the former approximates that of two bonded BuOH molecules. This difference occurs in both phases (bulk and SDS/PDAC complex) and leads to the similar effect of both alcohols on the binding of SDS by PDAC. From the cooperativity parameter u, we can calculate the interchange energy parameter βOR,p by eq 11. Since the pairwise interaction energy for empty sites (EOO and EOR) is expected to be negligible, we can estimate the
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R ) x1 + 4z(1 - z)(u - 1)
(A3)
At equilibrium, for eq 2
µ/S + kT ln[S] + µO,p ) µS,p
(A4)
Since the binding constant K is expressed by eq 5, eq A5 is derived:
R - 1 + 2z ) kT ln K[S] ) kT ln s kT ln R + 1 - 2z
(A5)
Therefore, Figure 5. The interchange energy between alcohol and water (βAW) and between SDS and alcohol (βSA,p).
pairwise interaction energy between the bound surfactants or the bound imaginary species R in the presence of alcohol:
βOR ≈ -kTERR
(14)
The values of ERR are largest for 1,2-octanediol at 0.025 mol dm-3 among three SDS/alcohol complexes and SDS itself (Table 1); 1,2-octanediol interacts strongly with SDS to form a bulky complex (R) with high alcohol content (r ) 0.52), leading to unfavorable pairwise interactions between the bulky complexes. Conclusions Binding isotherms for the interaction of the surfactant SDS with the polycation PDAC were determined in the presence of various alcohols. A thermodynamic treatment based on the regular solution model allowed calculation of the interchange energies between alcohol and water in the bulk phase and between alcohol and surfactant in the polyion domain. The data revealed that the pronounced effect of 1,2-octanediol on binding can be ascribed to its instability in the water phase and its strong hydrophobic interaction with the SDS anion in the polyion domain. The effect of 1,8-octanediol was comparable to that of BuOH. Thermodynamic analysis indicated that the instability of 1,8-octanediol in both the water phase and the polyion domain is ca. kT ln 2 compared to that of BuOH. This difference can be attributed to the fact that the structure of 1,8-octanediol approximates that of two BuOH molecules. Appendix 1 When the polyion/surfactant complex is treated as a pseudo-phase of regular mixing of the empty binding site (O) and the site (fraction z) occupied by a surfactant ion (S) on the polyion, the chemical potentials are expressed by A1 and A2, respectively, using the lattice model with the first-order (Bethe) approximation.32,33
R + 1 - 2z (A1) µO,p ) µ0O,p + kT ln(1 - z) + kT ln (1 - z)(R + 1) R - 1 + 2z z(R + 1)
µS,p ) µ0S,p + kT ln z + kT ln
(A2)
where (32) Hildebrand, J. H. J. Am. Chem. Soc. 1929, 51, 66. (33) Hildebrand, J. H.; Scott, R. L. Regular Solutions; Prentice Hall: New York, 1962.
R)
1+s (1 - 2z) ) x1 + 4z(1 - z)(u - 1) (A6) 1-s
Solving eq A6 for z results in eq 3.
z)
{
1 1+ 2
1-s
x(1 - s)2 + 4s/u
}
(3)
Appendix 2 There are three components (water, surfactant, and alcohol) in the bulk phase and two components (surfactant and alcohol) in the polyion pseudo-phase. The polyion is not counted as a component and is considered as a matrix for the mixture of surfactant and alcohol. Applying the lattice model with the Bragg-Williams approximation for the bulk phase and the polyion pseudo-phase, the activity coefficients fi for surfactant (subscript S), alcohol (subscript A), and water (subscript W) are given as follows.34 In the polyion pseudo-phase,
ln fS,p ) βSA,pyA2
(A7)
ln fA,p ) βSA,pyS2 ) βSA,p(1 - yA)2
(A8)
where yi is the mole fraction of component i in the polyion pseudo-phase and βij,p is the interchange energy parameter in the polyion pseudo-phase defined by eq 7. In the bulk phase, the excess Gibbs energy terms for each component are given by
ln fS ) βSAxA2 + βSWxW2 + (βSA + βSW - βAW)xAxW (A9) ln fA ) βAWxW2 + βSAxS2 + (βSA + βAW - βSW)xSxW (A10) ln fW ) βSWxS2 + βAWxA2 + (βAW + βSW - βSA)xSxA (A11) where xi is the mole fraction of component i in the bulk phase and βij is the interchange energy parameter in the bulk phase. Since xS and xA are very small and xW ≈ 1,
ln fS ) βSW
ln fA ) βAW
(A12)
At equilibrium, the chemical potential of the alcohol in the bulk phase equals that of the alcohol in the polyion pseudo-phase.
ln xA + βAW ) ln yA + βSA,p(1 - yA)2
(7)
(34) Saito, N. Kobunnsi Butsurigaku, 6th ed.; Syokabo: Tokyo, 1973.
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Appendix 3 When an imaginary chemical species, (S + rA) ) R, is introduced, surfactant binding can be taken as the binding of the imaginary species R to the polyion in the presence of alcohol. Equation 3 can be applied to the surfactant binding expressed by eq 9, but the binding affinity K and the cooperativity parameter u are expressed by eqs 10 and 11. Since R is the mixture of surfactant and alcohol of 1:r in the surfactant cluster, the standard chemical potential is given by eq 12. At equilibrium, the chemical potential of the alcohol in the
Fukui et al.
bulk phase is equal to that in the polyion pseudophase:
µA ) µA,p
(A13)
In the absence of alcohol, the binding constant K0 is given by eq 5. Substituting eqs 12, 5, and A13 into eq 11,
ln(K0/K) ) ln(1 - yA) + βSA,pyA2 LA0113926
(13)
