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Thermodynamic Study of the Protonation of Dimethyldodecylamine N-Oxide Micelles in Aqueous Solution at 298 K. Establishment of a Theoretical Relationship Linking Critical Micelle Concentrations and pH Virginie Lair,† Sabbah Bouguerra,‡ Mireille Turmine,*,† and Pierre Letellier† Laboratoire d’Electrochimie et Chimie Analytique, UMR 7575, Energe´ tique et Re´ activite´ aux Interfaces, Universite´ Pierre et Marie Curie, case 39, 4 place Jussieu, 75252 Paris cedex 05, France, and Laboratoire Eau et Technologies membranaires, INRST, BP 95-2050 Hammam-lif, Tunis, Tunisie Received April 13, 2004. In Final Form: June 14, 2004 Dodecyldimethylamine N-oxide (DDAO) is a zwitterionic surfactant with acid-base properties. The proton dissociation constant of this surfactant was determined by a novel potentiometric method at “controlled chemical potential” of the proton using a classical pH-glass electrode. When the DDAO was in its monomeric form, the pKa was about 5, consistent with the value commonly reported in the literature. However, a unique proton dissociation constant specific to the micellar form of this surfactant could not be obtained. We found that the acid-base behavior of the DDAO micelles depended on their environment. Indeed, we were able to establish thermodynamic relations linking the critical micellar concentration to the degree of protonation of the micelles. The experimental values were in good accordance with this model.
I. Introduction Dodecyldimethylamine N-oxide (DDAO) is a zwitterionic surfactant which forms neutral micelles at pH above 7 or cationic micelles at low pH. It also has the ability of being protonated in water, in an equilibrated way, to give the cationic species, DDAOH+. There have been numerous studies reported describing the acid-base properties of DDAO and its homologues which have alkyl chains of different lengths, both as monomers and aggregate forms and giving a variation of critical micelle concentration (cmc) according to the pH.1-18 In all these works, it is assumed that the acid and basic monomers are linked in * Email:
[email protected]. † Laboratoire d’Electrochimie et Chimie Analytique, UMR 7575, Energe´tique et Re´activite´ aux Interfaces, Universite´ Pierre et Marie Curie. ‡ Laboratoire Eau et Technologies membranaires, INRST. (1) Kolp, D. G.; Laughlin, R. G.; Krause, F. P.; Zimmerer, R. E. J. Phys. Chem. 1963, 67, 51. (2) Herrmann, K. W. J. Phys. Chem. 1964, 68, 1540. (3) Tokiwa, F.; Ohki, K. J. Phys. Chem. 1966, 70, 3437. (4) Maeda, H.; Tsunoda, M.; Ikeda, S. J. Phys. Chem. 1974, 78, 1086. (5) Rathman, J. F.; Christian, S. D. Langmuir 1990, 6, 391. (6) Zhang, H.; Dubin, P. L.; Kaplan, J. I. Langmuir 1991, 7, 2103. (7) Maeda, H.; Muroi, S.; Ishii, M.; Kakehashi, R.; Kaimoto, H.; Nakahara, T.; Motomura, K. J. Colloid Interface Sci. 1995, 175, 497. (8) Maeda, H. Colloids Surf., A 1996, 109, 263. (9) Imaishi, Y.; Kakehashi, R.; Nezu, T.; Maeda,H. J. Colloid Interface Sci. 1998, 197, 309. (10) Katsuura, H.; Takisawa, N.; Manabe, M.; Maeda, H. Colloid Polym. Sci. 1999, 277, 261. (11) Geramus, V.; Kameyama, K.; Kakehashi, R.; Maeda, H. Colloid Polym. Sci. 1999, 277, 868. (12) Maeda, H.; Kakehashi, R. Adv. Colloid Interface Sci. 2000, 88, 275. (13) Maeda, H.; Kanakubo, Y.; Miyahara, M.; Kakehashi, R. J. Phys. Chem. B 2000, 104, 6174. (14) Kakehashi, R.; Yamamura, S.; Tokai, N.; Takeda, T.; Kaneda, K.; Yoshinaga, K.; Maeda, H. J. Colloid Interface Sci. 2001, 243, 233. (15) Kawasaki, H.; Maeda, H. Langmuir 2001, 17, 2278. (16) Pettersson, A.; Rosenholm, J. B. Langmuir 2002, 18, 8436. (17) Pettersson, A.; Rosenholm, J. B. Langmuir 2002, 18, 8447. (18) Maeda, H. J. Colloid Interface Sci. 2003, 263, 277.
a classical proton exchange equilibrium
DDAOH+ ) DDAO + H+ characterized by a proton dissociation constant of the monomer, Ka, equal to 4.95 in water at 298 K.3 However, the properties of the aggregate forms are not always described in the same way. Maeda14 analyzed the protonation curve of DDAO at concentrations above the cmc and concluded that the micellized DDAOH+ behaves as weak monoacid in equilibrium with its neutral conjugate base, DDAO, also in an aggregated form, with a pKa of 5.95. In our opinion, there are several reasons why the notion of a pKa is not appropriate for this surfactant in its micellar form. The first is that the protonation of an aggregate of neutral surfactant leads to the continuous creation of mixed micelles of variable charge and of different stabilities according to the acidity of the medium. The values of standard chemical potentials of species constituting the aggregates, DDAO and DDAOH+, depend on the degree of ionization of the micelle and, therefore, on the acidity of the medium. Consequently, the acidbase equilibrium of aggregated species cannot be described by a single value of pKa independent of the pH of the medium. Rathman et al.5 made this point and showed that there is no need to introduce an additional equilibrium constant for the protonation of the micellar surfactant because the surfactant in the micelle is in equilibrium with the amphiphilic monomer. The second reason is linked to the application of thermodynamic rules. Values of cmc of a surfactant susceptible to protonation depend directly on the degree of protonation of the micelle (see below). All published measurements of the cmc of DDAO aggregates5,14,19 show a minimum for pH values close to 5. This phenomenon is not compatible with there being a single constant equi(19) Peyre, V.; Baillet, S.; Letellier, P. Anal. Chem. 2000, 72, 2377.
10.1021/la049067g CCC: $27.50 © 2004 American Chemical Society Published on Web 08/26/2004
Protonation of a Zwitterionic Surfactant
Figure 1. Calibration curves of DDAO at various pH values and 298 K. Table 1. cmc Values of DDAO According to pH (from Ref 19) pH 2 3 3.5 4 4.5 5.5 7.12 7.4 7.93 9.3 cmc/10-3 2.40 1.91 1.26 1.0 0.708 1.0 1.41 1.74 1.91 1.91 mol L-1
librium for the protonation of the micellar system. Indeed, the model and parameters proposed by Maeda imply a continuous increase of cmc values with the pH. We investigated how the behavior of DDAO is affected by the pH and determined accurately the degree of protonation of micelles according to the acidity of the medium. We used a novel technique involving potentiometry at “controlled chemical potential” of protons such that the cmc value for DDAO could be linked to the degree of protonation of the micelles. II. Materials, Products, and Measurement Techniques II.1. Materials and Products. II.1.1. Materials. DDAO-Sensitive Electrode. A DDAO-sensitive electrode was prepared as described by Peyre et al.19 This electrode was made of a liquid membrane (PVC as the polymer and dinonylphthalate as the placticizer), with DTABΦ4 as the carrier of the surfactant. The DDAOselective electrode was combined with a KCl-saturated calomel reference protected from the amphiphile diffusion by an agar-agar salt bridge with 2 mol L-1 KCl. Electromotive force (emf) values of this electrochemical cell, measured at 298 K with a TACUSSEL LPH530T millivoltmeter, were stable within (0.2 mV and reproducible to (0.5 mV. As is shown in Figure 1, the cmc values in aqueous solution of 0.1 mol L-1 NaCl, at different pH values, were accurately determined from the calibration curve. Values of the cmc of DDAO as determined by this method are given in Table 1. pH Electrode. A radiometer, model XG 100, pH-glass electrode was used as the proton-sensitive electrode. The electrode was calibrated with two buffer solutions (pH ) 4.065 and 7.00). Its accuracy and reproducibility were about 0.01 pH unit. Electrochemical Chain. pH-sensitive and DDAOsensitive electrodes were combined with a KCl-saturated calomel reference electrode protected from amphiphile diffusion by an agar-agar salt bridge containing 2 mol L-1 KCl. The temperature of this electrochemical cell was maintained at 298 ( 0.1 K with a circulating water bath. The electromotive force values were measured with a TACUSSEL LPH530T millivoltmeter. II.1.2. Products. Dodecyldimethylamine N-oxide (DDAO) from Fluka was used as received. All solutions
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were prepared in ultrapure water (all water used was distilled and then filtered with an ELGA UHQ II system, κ ) 18 MΩ). The hydrochloric acid (HCl) solutions were normadose solutions from Prolabo. II.2. Determination of Degree of Protonation at “Constant Chemical Potential” of the Proton. The technique used was directly inspired by that previously used to study interactions between cationic surfactants and various supports20 and the behavior of mixed micelles.21,22 It consists of using a pH-sensitive glass electrode to measure the number of moles of protons bound per mole of neutral DDAO, both in and not in micellar forms, at a given pH. The raw data can be analyzed with models of stoichiometric equilibria (by means of equilibrium constants) or with nonstoichiometric models (isotherms). First, we considered a volume, V°, of an aqueous solution of 0.1 mol L-1 NaCl, at given pH fixed by either HCl or a “buffer solution” of DDAOH+/DDAO obtained by adding a hydrochloric acid solution to a DDAO solution. With the buffer, the experiment can be started below or above the cmc as required. The pH of the initial solution can be set between pH 3 and 6. Activities of each species in solution are equal to their concentrations since the surfactant solutions are diluted and the ionic strength of the medium is fixed by the electrolyte NaCl. All acid and base solutions were prepared in 0.1 mol L-1 NaCl. Initially, in the solution of proton concentration, CH+, the number of protons is
nH+ ) CH+V° ) 10-pHV°
(1)
To this solution, a quantity of surfactant, δnDDAO, is added in a volume, VTA, of a concentrated DDAO solution, CTA, such that
δnDDAO ) CTAVTA
(2)
Subsequent addition of base (DDAO) to the solution increases the pH. The exact quantity of proton needed to restore the initial pH as monitored by the glass electrode is then added. This quantity δnH+ is added in a volume, VA, of a concentrated solution CA, such that
δnH+ ) CAVA
(3)
The quantity of protons, nH+f, fixed by the added surfactant at the given pH can be calculated by mass balance: some of the added protons bind to the added surfactant and some serve to compensate for the dilution resulting from the additions
CH+(V° + VA + VTA) ) CH+V° - nH+f + CAVA (4) The number of protons fixed by the quantity of base added, δnDDAO, is thus
nH+f ) CAVA - 10-pH(VA + VTA)
(5)
DDAO being a base, binding of protons presumably formed DDAOH+. Thus, the apparent degree of protonation, τapp, (20) Gloton, M. P.; Mayaffre, A.; Turmine, M.; Letellier P.; Suquet,H. J. Colloid Interface Sci. 1995, 172, 56. (21) Palous, J. L., Turmine, M.; Letellier, P. J. Phys. Chem. B 1998, 102, 5886. (22) Turmine, M.; Mace, C.; Millot, F.; Letellier, P. Anal. Chem. 1999, 71, 196.
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can be defined as the ratio of number of moles of DDAO, nDDAO, to the number of moles of DDAOH+, nDDAOH+
(6)
After the additions, the solution has the same pH as initially. The same operation was repeated several times to check the reproducibility of the results and to verify that the number of protons bound by the base did not depend on the total mass of surfactant added to the medium. An apparent pK (named pKapp) was thereby defined
(7)
Two possible cases can be distinguished: First Case: DDAO and DDAOH+ Are Both in Monomer Forms. In this case, the pKa of the pair can be determined by calculating the quantity
pKapp ) pKa ) pH - log (τapp)
Ka
)
-1
10-pH
CDDAOm ) CDDAOt - cmc - CDDAOH+m
H
pKapp ) pH - log τapp
(
CDDAOH+m ) CH+f - cmc 1 +
f
nDDAO δnDDAO - nH+ τapp ) ) nDDAOH+ n +f
so
(8)
Repeating the same experiment at several pH values should always give the same value of pKa. Second Case: DDAO and DDAOH+ Are in Micellar Forms. In this case, the DDAO is differently protonated in the micelle and in the solution. The system can be totally described because the cmc value is known. Above the cmc, four forms of the DDAO surfactant are in equilibrium. By convention, we will attribute the exponent b to the monomer species and the exponent m to the aggregated species. For each surfactant addition to the solution, the total quantity of DDAO in solution, nDDAOt, is known
(14) (14′)
The degree of protonation of DDAO in the micelle, τm, is calculated as follows
τm )
CDDAOm CDDAOH+m
nDDAOm
)
nDDAOH+m
(15)
the degree of protonation of DDAO in its monomeric form, τb, being
τb )
CDDAOb
) b
CDDAOH+
nDDAOb nDDAOH+b
(16)
Since the degree of protonation of DDAO in the micelle can be calculated, a proton dissociation constant Kam at the chosen pH for the solution can be defined
pKam ) pH - log (τm)
(17)
This same series of operations was repeated at various pH values, to assess whether the proton dissociation constant was fixed and independent of the pH or whether it varied according to the pH of the solution. Thus, the degree of protonation of the micelle can be determined experimentally if the cmc values at the considered pH are available. Moreover, cmc values and the degree of protonation are directly linked by thermodynamics relations as we show below.
nDDAOt ) nDDAOb + nDDAOH+b + nDDAOm + nDDAOH+m (9)
III. Theoretical Part: Thermodynamic Relations between cmc Values and the Degree of Protonation of DDAO at Various pH Values
This equality is conserved in the concentration scale by referring to the total volume of the solution
Consider a system consisting of nw moles of water, nDDAOt moles of DDAO, and nH+ moles of strong acid, HCl. nMCl moles of a chloride salt is also included to fix the ionic strength of the medium. The derivative of the Gibbs free energy can be written
CDDAOt ) CDDAOb + CDDAOH+b + CDDAOm + CDDAOH+m (10) Monomer concentrations are linked to the cmc
cmc ) CDDAOb + CDDAOH+b
(11)
(12)
where µDDAO, µHCl, µw, and µMCl are the chemical potentials of DDAO, HCl, water, and supporting salt, respectively. In this case, water and the supporting salt are considered as constituting the solvent S. The concentrations of M+ and Cl- can be considered to be constant because the supporting electrolyte is concentrated. Consequently, eq 18 can be formally simplified to
(12′)
dG ) V dP - S dT + µDDAO dnDDAOt + µH+ dnH+ + µS dnS (19)
If the pKa value of the conjugate DDAOH+/DDAO pair is known, concentrations of free DDAO and DDAOH can be calculated at a given pH
(
CDDAOb ) cmc 1 +
(
)
10-pH Ka
CDDAOH+b ) cmc 1 +
Ka
-1
)
-1
10-pH
Properties of cross-differentiation relations lead to
Moreover, the total quantity of protons bound by all the bases of the medium is known. By referring to the total volume of the solution, V, one can write
CH+f )
nH+f ) CDDAOH+b + CDDAOH+m V
dG ) V dP - S dT + µDDAO dnDDAOt + µHCl dnHCl + µw dnw + µMCl dnMCl (18)
(13)
( ) ∂µDDAO ∂µH+
P,T,nDDAOt,nS
(
)-
∂nH+
)
∂nDDAOt
(20)
T,P,µH+,ns
The left-hand term of this equality describes the variations of the chemical potential of DDAO when the chemical
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potential of H+ changes (due to the addition of more H+ for example) at constant total number of moles of DDAO and solvent. These are modifications associated with titration. The right-hand term expresses the fact that when the quantity of DDAO in the solution varies, the quantity of H+ must also change if the chemical potential of the protons (i.e., the pH of the medium) is to remain constant. These two magnitudes are linked
µH+ ) µH+∞ + RT ln(aH+) ) µH+∞ - 2.3RTpH
(21)
Protons added to the solution presumably interact with the DDAO to give DDAOH+. The total number of moles of DDAO is the sum of acidic (DDAOH+) and basic (DDAO) forms of the surfactant
nDDAOt
) nDDAO + nDDAOH+
(22)
So, at the considered pH, the cross-differentiation relationship leads to
( ) ∂µDDAO ∂µH+
(
)P,T,nDDAOt,ns
)
∂nDDAOH+ ∂nDDAOt
1 )τ app + 1 T,P,µH+,ns (23)
by introducing the notion of apparent degree of DDAO protonation defined previously in eq 6 into the equation. Now consider the left-hand term. Whatever the forms (monomers or aggregates) adopted by the surfactant, its chemical potential can be written by referring to the infinitely dilute solution of the monomer in the solvent. If DDAO in monomeric form is at concentration CDDAOb, then
µDDAO ) µDDAO∞ + RT ln{CDDAOb}
(24)
In which {CDDAOb} corresponds to the value of the concentration of free DDAO. Suppose that the surfactant concentration is above the cmc. The chemical potential of DDAO can be expressed from eq 12 as a function of the cmc value and the dissociation constant (Ka) of the conjugate DDAOH+/DDAO pair in their monomeric forms.
(
µDDAO ) µDDAO∞ + RT ln{cmc} - RT ln 1 +
)
{CH} Ka (25)
Thus
(
)
∂µDDAO ∂µH+
(
) RT
P,T,nDDAOt,ns
(
)
∂ ln{cmc} ∂µH+
(
∂ ln 1 +
RT
))
∂µH+
(26)
P,T,nDDAOt,ns
The chemical potential of the proton can be derived
dµH+ ) RT
d{CH} {CH}
) -2.3RT dpH
(
)
∂µDDAO ∂µH+
(
)-
P,T,nDDAOt,ns
)
∂ log{cmc} ∂pH
-
P,T,nDDAOt,ns
{CH} Ka + {CH}
(28)
For monomer species, the equilibrium condition is easily expressed as a function of the degree of protonation τb defined in relation 16
Ka ) τb{CH}
(29)
This condition, introduced into eq 28, leads to
(
)
∂µDDAO ∂µH+
)-
P,T,nDDAOt,ns
(
)
∂ log{cmc} ∂pH
-
P,T,nDDAOt,ns
1 1 )(30) τb + 1 τapp + 1 Thus, there is a direct relation between the cmc and the pH. This relation involves the degrees of protonation of monomer DDAO and of DDAO in all its forms.
d log{cmc} 1 1 ) dpH τapp + 1 τb + 1
(31)
Values of τapp and τb were determined at fixed pH as described above. This relation is interesting as it can be used to interpret various results reported in the literature concerning the changes in cmc according to the pH. In acidic and basic media, the derivative vanishes because the two degrees of protonation have the same value (τapp ) τb ) 0 or 1). If we suppose that the acid-base equilibrium concerning the micellized species is characterized by a unique dissociation constant, Kam (see eq 17), different from Ka, then τapp will always be lower or higher than τb at given pH, according to the value of Kam. Were this the case, the variation in cmc with pH would be monotonic and take the shape of sigmoid curve. The value of 5.95, adopted by Maeda for the pKam, requires that τapp always be smaller than τb leading to a continuous increase of the cmc with pH. However, the experimental finding is of a “bell-shaped curve” with a minimum, implying that at a pH value close to 5, degrees of ionization τapp and τb are equal and therefore pKam ) pKa. At a pH lower than 5, the experimental finding is that the cmc decreases with pH. This implies that τapp > τb, i.e., in this domain pKam < pKa. Obviously, the inverse is observed at pH higher than 5 with pKam > pKa. Measurement of degrees of protonation of monomer and micellized DDAO can be used to confirm this analysis. IV. Experimental Results, Determination of Degree of Protonation
-
P,T,nDDAOt,ns
{CH} Ka
and by replacing in eq 25 and simplifying
(27)
IV.1. Case of Monomers. First we validated this method of determination of degree of DDAO protonation at a given pH: we determined the pKa of the conjugate DDAOH+/DDAO pair for the monomer species at 298 K in 0.1 mol L-1 NaCl aqueous solution at two pH values: (i) pH 4.30 fixed by inclusion of hydrochloric acid and (ii) pH 5.15 fixed by inclusion of a DDAOH+/DDAO buffer solution, at a concentration below the cmc. The pKa values obtained with this method were coherent for all additions and for the two pH values studied (Table 2). The mean value of the pKa was about 5 ( 0.1, which
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Table 2. Experimental Results for DDAO Monomersa pH 4.3
pH 5.15
VTA (mL)
VA (mL)
tapp
pKa
20 30 50 100 200 20 30 30 50 50 100
35 50 85 170 340 19 25 25 40 40 90
0.152 0.186 0.186 0.222 0.204 1,11 1.28 1.32 1.39 1.42 1.35
5.1 5.0 5.0 5.0 5.0 5.1 5.0 5.0 5.0 5.0 5.0
a V TA and VA are the volumes added of the surfactant and the HCl, respectively. The initial volume in the beaker was 10 mL, the concentrations of the added solutions were CTA ) 2 × 10-2 mol L-1 for the surfactant and CA ) 10-2 mol L-1 for the HCl. pKa was calculated using relation 8.
Table 3. Selected Findings with the “Controlled Chemical Potential” Method for Determining the Apparent Dissociation Constant (pKapp) at Various pH Valuesa VTA VA (mL) (mL) pH 4.3 τb ) 0.2 cmc ) 0.95 × 10-3 mol L-1 V° ) 10 mL
pH 5.15 τb ) 1.41 cmc ) 0.76 × 10-3 mol L-1 V° ) 10.11 mL
pH 5.75 τb ) 5.62 cmc ) 0.89 × 10-3 mol L-1 V° ) 10.19 mL
20 30 50 100 200 300 500 1000 1000 1000 2000 2000 2000 20 30 30 50 50 100 200 300 500 2500 2500 5000 20 30 30 40 50 70 100 210 305
35 50 85 160 340 490 680 1230 1280 1280 2550 2550 2570 19 25 25 40 40 90 170 250 400 2150 2150 4500 15 25 25 35 75 100 160 330 520
τapp
pKapp
0,152 0,186 0,186 0,220 0,204 0,217 0.315 0.446 0.484 0.505 0.529 0.542 0.548 1.11 1.28 1.32 1.39 1.42 1.35 1.35 1.37 1.42 1.39 1.38 1.33 5.8 5.38 5.28 5.12 3.96 3.45 2.99 2.67 2.39
5.1 5.0 5.0 4.9 5.0 4.8 4.7 4.6 4.6 4.6 4.6 4.6 4.6 5.1 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.2 5.2 5.3 5.3 5.4
τm
pKam
0.466 0.608 0.620 0.625 0.632 0.636 0.635
4.6 4.5 4.5 4.5 4.5 4.5 4.5
1.01 1.32 1.43 1.39 1.38 1.33
5.1 5.0 5.0 5.0 5.0 5.0
1.07 1.64 1.92 1.92
5.7 5.5 5.5 5.5
a The initial volume in the beaker was V° (value given in the table, the concentrations of the added solutions were CTA ) 2 × 10-2 mol L-1 (5.1 × 10-2 mol L-1 at pH 5.75) for the surfactant and CA ) 10-2 mol L-1 for the HCl solution. The dissociation constant of micellized DDAO was calculated from eq 17.
is in accordance with that reported by Maeda14 from the analysis of titration curves. IV.2. Case of Micellar Solutions. Properties of Micelles. We used the same technique of addition at constant pH to determine the degree of protonation above the critical micelle concentration at pH values 4.3, 5.15, and 5.75 (Table 3). Values of the degree of protonation τapp and τm were not very different for surfactant concentrations higher than the cmc. Thus, values of apparent and micellar pK are
Figure 2. pKam plotted against pH. Table 4. pKam Values at Various pH Values pH pKam
4.23 4.4
4.3 4.5
4.5 4.6
5 5
5.15 5
5.56 5.1
5.75 5.5
very close. All the values of pKam obtained in this work are reported in Table 4. The pKam of micellized species was not constant, but changed according to the acidity of the medium. This finding is consistent with the theory developed above. pKam is plotted against the pH in Figure 2. The result is a straight line of slope 0.65 and of Y-intercept 1.65. Consequently
pKam ) 1.65 + 0.65pH
(32)
log τm ) 0.35pH - 1.65
(32′)
or
The pKa of conjugate pairs involving monomer and micellized species are equal for pH about 5.2 which corresponds to the value of the minimum of the curve: cmc ) f(pH). This behavior is consistent with the idea, as suggested by Maeda, that there are strong hydrogen interactions between the protonated and the nonprotonated polar heads in the micelle structure. Thus, micelle stability depends on the degree of ionization. This leads to the standard Gibbs free energy of micellized DDAOH+ also being dependent on the pH. Note that at pH 5.1, the degrees of ionization of monomer and micelle forms are equal and close to 1. This suggests that the micellized dimer (DDAO)2H+ should play a particular role in the conditions of micellar equilibrium as mentioned in a recent paper.23 Thus, simple laws of variation for the degree of ionization of micellized DDAO according to pH can be established; they completely describe the system and can be used to calculate the cmc values according to the pH. V. Analysis of Results: Variation of cmc with the pH Equation 31 can be explained according to pH. The degree of protonation of the monomer species τb is linked to pH by
τb ) e-2.3pKae2.3pH
(33)
Moreover, at concentrations higher than the cmc, the apparent pK, pKapp, as measured experimentally, is identical to that of the micelle, pKam. pKapp varies with the (23) Majhi, P. R.; Dubin, P. L.; Feng, X.; Guo, X.; Leermakers, F. A. M.; Tribet, C. J. Phys. Chem. B 2004, 108, 5980.
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Thus, the results obtained from thermodynamic relations for degrees of DDAO protonation are coherent with the experimental values of DDAO cmc. VI. Conclusion
Figure 3. cmc of DDAO plotted against pH. The curve is calculated from eq 39. The points correspond to the values determined experimentally (see Table 1).
pH according to the following experimental rule
pKapp ) a + bpH ) -log τapp + pH
(34)
Thus, the apparent degree of DDAO protonation can be expressed according to pH as
τapp ) e-2.3ae2.3(1-b)pH
(35)
The expression to integrate is then
d log cmc )
dpH -2.3a 2.3(1-b)pH
e
e
+1
e
dpH e
-2.3pKa 2.3pH
+1 (36)
with pH as a variable. This relationship is easy to integrate, as
1 ln(a + bepx) ∫ a +dxbepx ) ax - ap
(37)
The potentiometric method we describe here to determine the degree of DDAO protonation with the pH is very general. It can be used to investigate the acid-base behavior of systems that, on protonation, do not give a compound of simple or defined stoichiometry. The principle of the determination is based on a method of equilibrium which takes into account the only partial retention of protons by the studied compound (aggregate) without requiring any assumptions about the fate of the protons. In the example illustrated here, DDAO is a base, and therefore we assume that the proton is fixed in the acidic form, DDAOH+, but we have not excluded the possibility that this species can be linked to other DDAO molecules by hydrogen bonds. Thus, this first hypothesis would not be necessary, if adsorption isotherms for the protons onto DDAO micelles should be established. cmc variations with the pH obtained by this way would be the same but, nevertheless, with slightly different formalism. One of the strengths of this experimental technique is of the exploitability of the raw data: it can be used to determine the number of moles of H+ fixed by a quantity of substance. For DDAO systems, this technique reveals that the acid-base properties of micellized DDAOH+/DDAO cannot be characterized by a unique equilibrium constant independent of pH. This is probably due, as suggested by Maeda, to the formation of a micellized dimer (DDAO)2H+ that is particularly stable at pH 5.1. This dimer acts as an amphoteric compound. At pH lower than 5.1, the equilibrium
2DDAOH+mic ) (DDAO)2H+mic + H+
(40)
leading to
(
1 pH ln(1 + log{cmc} ) 1 2.3(1 - b) 1 pH e-2.3ae2.3(1-b)pH) ln(1 + e-2.3pKae2.3pH) + 1 2.3 constant (38)
) (
)
Thus
log{cmc} )
1 ln(1 + e-2.3pKae2.3pH) 2.3
1 ln(1 + e-2.3ae2.3(1-b)pH) + constant (39) 2.3(1 - b) The value of the constant term is calculated for a cmc of 1.91 × 10-3 mol L-1 at pH 3. This relation was exploited by introducing the laws of degrees of protonation variation established in the experimental part (a ) 1.65, b ) 0.65). The cmc values given in Table 1 and the curve obtained from the last equation are compared in Figure 3. The fit is extremely good.
is presumably shifted toward the formation of basic dimer, and DDAOH+ is then a stronger acid, and the consequence is a pKam lower than pKa. For pH values higher than 5.1, the equilibrium
(DDAO)2H+mic ) 2DDAOmic + H+
(41)
is shifted toward the acidic dimer. The apparent acidity of DDAOH+ therefore decreases leading to pKam being higher than pKa. There is another point which seems to us relevant concerning the study of whether variables are physicochemically linked. Changes in protonation of the micelle and of the cmc with the pH are two aspects of the same problem, these two magnitudes being entirely linked by cross-differentiations. Any effects on one has implications for the other: we show this unambiguously for this DDAO system, and believe that the same is true for other functional amphiphiles involved in complex formation or redox reactions. LA049067G
