A thermodynamic analysis of the hydrogen ion titration of micelles

Journal of Colloid and Interface Science 263 (2003) 277–287 www.elsevier.com/locate/jcis

A thermodynamic analysis of the hydrogen ion titration of micelles Hiroshi Maeda Department of Chemistry, Faculty of Science, Kyushu University, Fukuoka 812-8581, Japan Received 5 December 2002; accepted 27 February 2003

Abstract A thermodynamic analysis of hydrogen ion titration is presented for association colloids with particular emphasis on surfactant micelles. When a particular type of the micellar Gibbs–Duhem relation (MGD), αM dµI + (1 − αM ) dµN = 0 [αM : the degree of ionization of micelles; µI , µN : chemical potentials of ionized and nonionized species], holds, the free energy change accompanying the ionization of the micelle Gex can be evaluated from the titration data in the same manner as for covalently bonded colloids such as linear polyions. In the case where the regular solution approximation is valid for mixed micelles, the titration curve should be a straight line with a slope yielding the interaction 2 . For dodecyldimethylamine oxide micelles for which the MGD relation has been shown parameter, and Gex is given as a function of αM to hold, values of the calculated electrostatic free energy Gel were close to but significantly greater than experimental Gex values when the former were calculated on the basis of the Poisson–Boltzmann equation for either a sphere or a plate with smeared charges in a salt solution of infinite volume. When the critical micelle concentration (cmc) data are combined with the hydrogen ion titration data, we obtain a criterion to judge whether the above MGD relation holds or not. When the MGD relation holds, the monomer concentration C1 can be evaluated from the hydrogen ion titration. For most cases examined, the C1 /C1 (αM = 0) from the titration agrees well with cmc/cmc(αM = 0), suggesting cmc = C1 above the cmc. For tetradecyldimethylamine oxide, the MGD relation does not hold in the range of low ionic strength and even at 0.1 M NaCl it has been found that C1 /C1 (αM = 0) < cmc/cmc(αM = 0), due probably to enormous micelle growth.  2003 Elsevier Science (USA). All rights reserved. Keywords: Hydrogen ion titration; Mixed micelles; Composition of mixed micelles; The micellar Gibbs–Duhem relation; Regular solution approximation of mixed micelles; Electrostatic free energy

1. Introduction Hydrogen ion titrations of linear polyelectrolytes have been proved a useful method to evaluate the free energy change accompanying the ionization of polyions [1–3]. In many polyions of a random coil nature, the electrostatic free energy has been shown to be the most significant contribution to this free energy change, compared to the contributions of conformational entropy and/or nonideal mixing entropy of charged and uncharged sites on the polyions and others [4]. For those polymers undergoing discrete conformational changes, such as the α-helix–random coil conversion of poly(L-glutamic acid), the method can provide the free energy difference between the two conformations in their uncharged states [5–9]. The application of the method is extended to cover the association or aggregation systems, the subunit dissociation of a protein [10], strand separation E-mail address: [email protected].

of polynucleotides [11], and a poorly characterized aggregation of the β-sheet of polypeptides [12], or to the case of phase separation with precipitates of polymers [13–16]. The hydrogen ion titration of heterogeneous dispersions of fatty acids has been examined [17–22]. As to other colloidal systems, ionic exchangers [23,24] and inorganic minerals [25–27] have been examined. The hydrogen ion titrations of surfactant micelles have been also carried out on alkylamine oxides [28–34] and others [35–38]. In the titration of surfactant micelles, however, there are several characteristics to be considered, in contrast with the case of linear polyions. For example, the aggregation number, a counterpart of the degree of polymerization of polyelectrolytes, varies with ionization. Also, the equilibrium between the micelles and the monomeric species in the solution is always maintained during the titration. Hence, the concentration of the micelles also changes with pH since the monomer concentration changes with pH. The effort to take this particular situation into account in the analyses was made but it was not satisfactory [39]. In the present study,

0021-9797/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00237-6

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H. Maeda / Journal of Colloid and Interface Science 263 (2003) 277–287

a thermodynamic analysis of the hydrogen ion titration of micelles is presented to take these issues into account.

2. Thermodynamics of hydrogen ion titration We consider a proton dissociation reaction between a cationic (RH+ ) and a nonionic (R) surfactant species: RH+ = R + H+ .

(2.1)

The excess free energy Gex is defined as follows: 
Gex = mRT αM ln γRH + (1 − αM ) ln γR .

Here, we define the apparent dissociation constant Ka and the intrinsic dissociation constant on the surface of the colloid particle KM as follows:
 ln Ka ≡ ln aH − ln αM /(1 − αM ) ,
 pKa = pH + log αM /(1 − αM ) , (2.12) ∗ M −RT ln KM = µM R (αM = 0) + µH − µRH (αM → 0). (2.13)

For the monomeric species in the solution, the proton dissociation equilibrium is expressed as follows:

Combining Eqs. (2.6), (2.8), and (2.9), we have

µRH = µR + µH .

pKM − pKa = log(γRH /γR ).

(2.2)

Chemical potentials of the monomeric species are written as follows in terms of the degree of ionization α1 and the concentration C1 assuming ideal behavior:
 µR = µ∗R + RT ln C1 (1 − α1 ) , (2.3) µRH = µ∗RH

+ RT ln(C1 α1 ).

(2.4)

The proton dissociation constant of the monomer K1 is defined as follows: −RT

ln K1 = µ∗R

+ µ∗H

− µ∗RH ,


 pK1 = pH + log α1 /(1 − α1 ) .

(2.5)

When, on the other hand, these species are fixed by any means on the surface of a colloidal particle, the equilibrium condition is written as follows in terms of the chemical potentials of µM i : M µM RH = µR + µH .

Ct α = (Ct − C1 )αM + C1 α1 .

(2.7)

(2.14)

Experimentally, pKM can be obtained by extrapolating pKa to αM = 0: pKM = pKa (αM → 0).

(2.15)

From Eqs. (2.11) and (2.12),   αM (pKM − pKa ) = Gex mRT (ln 10) − log γR .

(2.16)

3. Permanent colloid particles and polyelectrolytes It is pertinent to examine the case of permanent colloidal particles as a reference system, where the number of sites m is kept constant during the titration, just as for a linear polyelectrolyte. The change of Gex with αM is related to the ratio of the activity coefficients as follows:    ex ∂G /∂αM m m = (∂Gex /∂nRH )nR − (∂Gex /∂nR )nRH = RT ln(γRH /γR ).

(2.6)

We denote the chemical potential of the micelle as G(m, n) with the aggregation number m and the number of charges n. Numbers of protonated and unprotonated sites, nRH and nR , are nRH = αM m and nR = (1 − αM )m, respectively. Degrees of ionization of the solution α and of the micelle αM = n/m and the total surfactant concentration Ct are related to C1 and α1 by

(2.11)

(3.1)

An area A on the titration curve, pKa vs αM , is defined as follows: αM αM A = (pKM − pKa ) dαM = log(γRH /γR ) dαM . (3.2) 0

0

Thus, the excess free energy is directly given by the area A by simply integrating Eq. (3.1) [1–3]: αM   A = (pKM − pKa ) dαM = Gex (αM ) mRT (ln 10) .

We choose the nonionic micelle as a reference state (an asymmetric convention) and the chemical potentials are written in terms of the activity coefficients γR and γRH as
 M µM R = µR (αM = 0) + RT ln γR (1 − αM ) ,

If the electrostatic contribution is dominant in tric potential ψ0 is given as

γR (αM = 0) = 1,

e0 ψ0 /kT = φ0 = (ln 10)pKa = (ln 10)(pKM − pKa ).

(2.8)

M µM RH = µRH (αM → 0) + RT ln(γRH αM ),

γRH (αM → 0) = 1.

(2.9)

The term ln(micelle concentration) in G is ignored throughout the present study. Then the concentration-independent part of G, G∗ , is given as
 M G∗ ∼ G = m αM µM (2.10) RH + (1 − αM )µR .

0

Gex ,

(3.3) the elec-

(3.4) Here e0 denotes the protonic charge. The activity coefficients are obtained as follows: − log γR = (pKM − pKa )αM − A = A > 0,

(3.5)

log γRH = A + (1 − αM )(pKM − pKa ) > 0.

(3.6)

The areas A and A , shown in Fig. 1a, correspond, respectively, to the two terms at the rhs of Eq. (2.16).

H. Maeda / Journal of Colloid and Interface Science 263 (2003) 277–287

279

ciation colloids. The hatched area in Fig. 1b, that is equal to αM (pKM − pKa ), is related to Gex by Eq. (2.16). The condition of equilibrium between micelles and the solution can be given as µM R = µR ,

µM RH = µRH .

(4.2)

Introducing Eqs. (2.3), (2.4), (2.8), and (2.9) into Eq. (4.2), we have

 log γR = log C1 /C1 (αM = 0) + log (1 − α1 )/(1 − αM ) , (4.3)
 log γRH = log C1 /C1 (αM = 1) + log(α1 /αM ) + log γRH (αM = 1).

(4.4)

If we know C1 as a function of pH, we can calculate the activity coefficients. This relation has been given by Rathman and Christian in a different expression [31]. Once log γR is known from C1 , we can evaluate Gex from Eq. (2.16):

Fig. 1. Schematic representations of the titration curves. (a) Covalently bonded colloids and associative colloids with the Gibbs–Duhem relation of Eq. (5.1). At a given αM , pKM − pKa corresponds to φ0 /(ln 10) if the electric work is predominant in Gex . The areas A and A are equal to Gex /{(ln 10)mRT } and (− log γR ) according to Eqs. (3.3) and (3.5), respectively. (b) Associative colloids in general. The hatched area, A + A , is equal to Gex /{(ln 10)mRT } − log γR according to Eq. (2.16). (c) Mixed micelles under the regular solution approximation where pKa is a linear function of αM . The interaction parameter β is directly evaluated from the slope k as β = −(k/2) ln 10 according to Eq. (6.6).

Gex /(mRT ) = (ln 10)αM (pKM − pKa ) 
+ ln C1 /C1 (αM = 0) 
+ ln (1 − α1 )/(1 − αM ) .

(4.5)

Also, 

A = αM log C1 (αM = 0)/C1 (αM = 1)  + log γRH (αM = 1) αM
  + log α1 (1 − αM ) αM (1 − α1 ) dαM .

(4.6)

0

4. Association colloids–micelles In the case of surfactant micelles, the association–dissociation equilibrium exists between micelles and the surrounding solution and hence the aggregation number m varies with αM ; number of micelles Nm and C1 also change with pH. In the thermodynamic approach of the present study, we may take all these effects into the activity coefficients without recourse to an explicit description of various changes accompanying the protonation reaction, as was done in the previous study [39]. The area A defined by Eq. (3.2) is written as follows and further reduction is not possible:

As αM approaches zero, the ratio (α1 /αM ) at a given pH remains finite,


log(α1 /αM )

 α=0

= (pK1 − pKM ),

(4.7)

while (α1 /αM ) approaches 1 as αM goes to 1. When both α1 and αM go to zero in Eq. (4.4),
 γRH (αM = 1) = C1 (αM = 1)/C1 (αM = 0) (K1 /KM ). (4.8) Rewriting Eq. (4.4) with the aid of Eq. (4.8), we have
 log γRH = (pKM − pK1 ) + log C1 /C1 (αM = 0)

αM A = (pKM − pKa ) dαM

+ log(α1 /αM ).

(4.9)

0


= G /(mRT )(ln 10) − log (1 − αM )γR + ex

αM αM dpH. 0

(4.1) Thus, in contrast to the case of permanent colloids, we see that the area A does not always give Gex in the case of asso-

Then, Eq. (4.6) reduces to A = αM (pK1 − pKM ) αM
  + log α1 (1 − αM ) αM (1 − α1 ) dαM . 0

(4.10)

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5. The micellar Gibbs–Duhem relation without counterions When the activity coefficients are related to each other, they can be separately evaluated by combining the relation with Eqs. (4.3) and (4.4). An example is the following version of the micellar Gibbs–Duhem (MGD) relation without counterions, which is an exact expression for the nonionic mixed micelles but is an approximate one for ionic/nonionic mixed micelles, as discussed in Section 9. It is written as follows: M M αM dµM RH + (1 − αM ) dµR = αM dµH + dµR = 0,

(5.1)

αM d ln γRH + (1 − αM ) d ln γR = 0, d ln γRH /d ln γR = −(1 − αM )/αM < 0.

(5.2)

Equation (5.2) is rewritten as follows:
 ln(γRH /γR ) dαM = d αM ln(γRH /γR ) + d ln γR .

(5.3)

does not hold. We can use this criterion, with the modification that C1 = cmc, to judge the validity of Eq. (5.1) for a given mixed micelle, as shown later in Table 4. By integration of Eq. (5.9),  
log C1 /C1 (αM = 0) = (αM − α1 ) dpH
 = A − (pKM − pKa )αM + log (1 − αM )/(1 − α1 ) . (5.10) This relation is rewritten as follows in terms of the average degree of ionization α and the total concentration Ct , 


log C1 /C1 (αM = 0) − C1 − C1 (αM = 0) (ln 10)Ct    = α − αM dpH. (5.11) Under the condition that the second term on the lhs can be ignored, α is practically independent of Ct . Equation (5.10) or (5.11) provides us with the monomer concentration C1 from the titration data when the MGD relation (5.1) holds, as shown by Rathman and Christian [31].

If Eq. (2.14) is taken into account, integration of Eq. (5.3) gives log γR = A − αM (pKM − pKa ).

(5.4)

The activity coefficient γRH is then given by Eqs. (2.14) and (5.4): log γRH = A + (pKM − pKa )(1 − αM ).

(5.5)

Combination of Eqs. (5.4) and (5.5) gives A = αM log γRH + (1 − αM ) log γR   = Gex (αM ) mRT (ln 10) .

(5.6)

Equations (5.4), (5.5), and (5.6) are the same as Eqs. (3.5), (3.6), and (3.3), respectively. Thus, we have arrived at an important conclusion that the same results as for permanent colloids are obtained in the case of the association colloids if Eq. (5.1) holds. Using the equilibrium condition Eq. (4.2), Eq. (5.1) can be written as follows: αM dµRH + (1 − αM ) dµR
 = RT d ln C1 + RT (αM − α1 )/α1 (1 − α1 ) dα1 = 0.

(5.7)

(5.8)

Combining Eqs. (5.7) and (5.8), we have RT d ln C1 + (αM − α1 ) dµH = 0 or d log C1 /dpH = αM − α1 .

It has been known that thermodynamic properties of many mixed micelles are well described by the regular solution theory [43–46]. The conditions for this approximation is valid are: (1) The mixing enthalpy Hmix is given in terms of a micellar mole fraction α of one component by βRT α(1 − α). (2) The mixing entropy can be approximated with the ideal behavior. In the present analysis, however, the following relation in terms of the interaction parameter β, Gmix = βRT α(1 − α) is sufficient as the condition for the regular solution approximation to be valid. When the activity coefficients are given explicitly, as in the case of the regular solution approximation, further reduction of the analysis of the titration curves is possible. In the regular solution theory where the symmetric convention is usually employed for the reference states of the chemical potentials, we have in place of Eq. (2.9)   S M µM RH = µRH (αM = 1) + RT ln γRH αM , S (αM = 1) = 1. γRH

Equation (5.7) has been derived by Motomura et al. [41] and Hoffmann and Pössnecker [42]. From Eq. (2.5), d log α1 = −(1 − α1 ) dpH.

6. The regular solution approximation

(5.9)

According to Eq. (5.9), the minimum or the maximum of the monomer concentration C1 should be observed at the pH where αM = α1 [40]. When this is not the case, Eq. (5.1)

(6.1)

Activity coefficients of the two conventions are related to each other: S γRH /γRH = γRH (αM = 1).

(6.2)

According to the regular solution theory, the activity coefficients are given as Eq. (6.3) in terms of the interaction parameter β [43–46]: 2 ln γR = βαM ,

S ln γRH = β(1 − αM )2 ,

(6.3)

ln γRH = β(1 − αM ) + ln γRH (αM = 1).

(6.4)

2

Equation (6.4) gives, in the limit as αM approaches zero, ln γRH (αM = 1) = −β.

(6.5)

H. Maeda / Journal of Colloid and Interface Science 263 (2003) 277–287

281

Table 1 Approximate expression of the titration curves of alkylamine oxide 2 + a α 3 (α
0.8) micelles:a pKa = pKM + a1 αM + a2 αM M 3 M

C10DMAO C12DMAO C14DMAO C12DHEAO

C14DHEAO

pKM

a1

a2

a3

CS /M NaCl

pK1

Ref.

5.61 5.89 6.00 6.30 6.50 5.56 5.60 5.74 5.88 5.49

−1.85 −1.54 −0.75 −5.12 −2.89 −4.56 −3.17 −2.58 −2.61 −3.65

−2.97 −2.70 −4.25 3.55 −0.25 4.79 1.75 1.01 1.20 2.14

2.21 2.25 3.16 −1.46 0.31 −2.24 −0.35 −0.16 −0.21 −0.21

0.10 0.10 0.20 0.01b 0.10c 0.10 0.20 0.50 1.0 0.10

4.90 ± 0.1 4.90 ± 0.2 4.95

[34] [29] [29] [33] [33] [34] [34] [34] [34] [34]

4.90

4.90

a Surfactant concentration C was 20 mM expcept for C10DMAO (C = t t

60 mM). b Counterion concentration C = 0.03 M. g c Counterion concentration C = 0.13 M. g Fig. 2. The titration curve of dodecyldimethylamine oxide in 0.2 M NaCl solutions at 25 ◦ C. Open and closed circles refer to pKa and pK1 , respectively. The straight line is pKa = 6.03 − 2.11αM .

We have summarized some titration data on alkylamine oxides in the form

and the results are shown in Table 1. Most data in the range of αM greater than about 0.85 were not reliable and the polynomials were obtained by fitting the data for αM < 0.8–0.85. When the contributions of the last two terms on the rhs of Eq. (7.1) are negligible, the regular solution approximation becomes valid and a1 = 2β/(ln 10). One of the characteristics of the titration of micelles can be seen in Table 1: pKM > pK1 . The introduction of pKM in addition to pK1 was once criticized [31]. We have shown, however, that the criticism is not pertinent and the use of pKM and hence pKa is reasonable as well as useful [40]. Moreover, as shown in Fig. 2, we can estimate the interaction parameter β from the titration curve without recourse to the cmc measurements if we introduce pKa . Mille has compiled several examples exhibiting pKM = pK1 [30]. The relation pKM > pK1 indicates the presence of a noncoulombic interaction favoring protonation on the surface of the micelle over that in the bulk solution. This is the opposite of the effect expected from the image charge effect [54]. As a mechanism for this short-range interaction, hydrogen bonding between the cationic head group and one of its neighboring nonionic head groups has been proposed [30,39,55–59]. Recent spectroscopic studies are in favor of this hydrogen bond [60,61]. It is reasonable to conclude that the observed difference is contributed from these two opposing mechanisms, −G∗ = RT ln(KM /K1 ) = Ghb + G(image charge), where Ghb denotes the free energy of hydrogen bond formation (Ghb < 0). To evaluate the image charge contribution approximately, we now calculate the work of charging up a protonic charge e0 at a position z distant from the flat interface between the two media of different dielectric constants: ε1 for the aqueous solution and ε2 for the internal hydrocarbon core of the micelles (ε1 > ε2 ). The electric potential ψ due to the image charge is given as [54]   ψ = 1/(4πε0) e0 f/(2zε1 ),

2 3 + a3 αM pKa = pKM + a1 αM + a2 αM

f = (ε1 − ε2 )/(ε1 + ε2 ).

When the activity coefficients are introduced into Eq. (2.14), we have   pKM = pKa − 2β/(ln 10) αM . (6.6) In this situation, linear titration curves as shown in Fig. 1c are expected and the parameter β is evaluated from the slope of the plot pKa vs αM . An example on dodecyldimethylamine oxide in 0.2 M NaCl solution is shown in Fig. 2. A slightly sigmoidal titration curve is approximated by a straight line with a slope of −2.1, which gives β = −2.4. From the initial slope (αM < 0.25) of −1.9, we have β = −2.2 which is in good agreement with the β value obtained from the cmc analysis, −2.08 [39,40]. From the linear dependence of pKa on αM , Gex is now given as a function of 2 : αM 2 . Gex (αM )/mRT = (ln 10)A = −βαM

(6.7)

2 was discussed The quadratic dependence of Gex on αM previously in relation to the regular solution approximation [47]. The result can be applied to ionic/nonionic mixed micelles in general. The dependence of Gex on the composition of ionic/nonionic mixed micelles has been extensively examined by several researchers [48–53]. When the regular solution approximation is valid, Eq. (5.10) reduces to 
log C1 /C1 (αM = 0)   2
 = β/(ln 10) αM (6.8) + log (1 − αM )/(1 − α1 ) .

7. The intrinsic proton dissociation constants of micelles KM and the monomer K1

(7.1)

(7.2)

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Thus, 1 G(image charge) =


dλ ψ(λ)e0

0

     = f/(4πε0 ) e02 (4zε1 ) = f/(zε1 ) 5.85 × 10−29 J m. (7.3) For z = 0.4 nm, G(image charge) = (f/ε1 )1.46×10−19 J. For ε1 = 78.25 and ε2 = 4, f = 0.902 and G(image charge) = 1.69 × 10−21 J or 1.02 kJ/mol. Hence, G(image charge) = 0.411RT (T = 298 K) [RT = 2.48 kJ/mol for T = 298 K]. The observed difference (pKM − pK1 ) increased in magnitude as the ionic strength increased in the case of dodecyldimethylamine oxide (C12DMAO). Values of (pKM − pK1 ) = pK were 1.06, 1.01, 0.94, 0.91, 0.85 (and corresponding pK1 values were 4.95, 4.89, 4.88, 4.85, and 4.78) for 0.2, 0.1, 0.05, 0.01, and 0 M NaCl at 25 ◦ C [29]. Assuming the same pK1 values for C14DMAO, values of (pKM − pK1 ) were 1.4 and 1.8 for 0.1 and 1 M NaCl, respectively [33]. Effects of alkylchain length has been examined for CnDMAO at 25 ◦ C in 0.1 M NaCl [34]. Values of pKM were 6.2 (C16 at 45 ◦ C), 6.3 (C14) > 5.9 (C12) > 5.6 (C10). Corresponding values of G∗ = 2.303RT (pKM − pK1 ) in kJ/mol were 7.92 (C16) ∼ 8.00 (C14) > 5.71 (C12) > 4.00 (C10). The difference is well interpreted in terms of the effect of curvature of the micelle surface on the stability of the hydrogen bond. On the other hand, values of (pKM − pK1 ) in 0.1 M NaCl were 0.7 and 0.6 for dodecyl- and tetradecyldihydroxyethylamine oxide (C12and C14DHEAO), respectively [34]. The relatively small pK values of CnDHEAO compared with CnDMAO are well interpreted in the same way as the chain length effect described above since CnDHEAO micelles are spherical and have greater curvatures than C12- or C14DMAO due to their bulky head group. Correspondingly, the cmc showed a very shallow minimum at αM = 0.2–0.3 [34]. If we assume the contribution of G(image charge) is 0.41RT (0.77 kJ/mol) independent of CS , −Ghb = G∗ + G(image charge) amounts to 8.69, 9.02, 6.73, 5.02 kJ/mol for C16-, C14-, C12-, and C10DMAO, respectively. As to the ionic strength dependence in the case of C12DMAO, −Ghb = 7.07, 6.79, 6.39, 6.22, and 5.87 kJ/mol for 0.2, 0.1, 0.05, 0.01, and 0 M NaCl at 25 ◦ C. These values are reasonable for the hydrogen 1/2 bond and the stability increases almost linearly with CS as 5.89 + 2.66(CS /M)1/2 . A difference between pKM and pK1 has been observed for other micelles [30]: pK = pKM − pK1 = 0.80 for N, N, N -trimethylaminododecanimide (C12 IM) [36], 1.25 for acylcarnitines [37], and ∼0.3 for alkyldimethylammoniohexanoate [38]. In these three examples, pK is positive, suggesting some stabilization of protons on the surface of micelles. Hydrogen bonds can be formed in these examples: –NH· · ·O=C– for C12 IM and –CO2 H· · ·− O2 C– for the latter two. On the other hand, pK1 and pKM values of the carboxyl groups were similar for both N -

dodecyl-β-aminopropionic acid (DAPA) and N -dodecyl-βiminopropionic acid (DIPA) [35]. Some remarks on the titration of rodlike micelles are in order. Generally, αM of the end cap region is expected to be greater than αM of the cylindrical part, αM (cap) > αM (cyl). In the case of amine oxides, the stability of the hydrogen bond is greater for the cylindrical part than for the end cap due to different curvatures that determine the –NO· · ·HON+ – distance. Consequently, ionization of the rodlike micelle of C14DMAO is likely to proceed in the cylindrical part in the range of low αM . In parallel with the ionization (protonation) of amine oxides, the micelle growth takes place more or less, depending on the ionic strength, resulting in the increased fraction of the cylindrical part. 8. The electrostatic contribution Gel to the excess free energy Gex In Section 5, it was shown that the titration data of micelles can be analyzed in the same way as covalently bonded colloids in the case where Eq. (5.1) holds. Under this condition, the excess free energy Gex accompanying the ionization of micelles can be evaluated from Eq. (5.6). It is interesting to calculate the electrostatic free energy Gel based on the Poisson–Boltzmann equation and compare it with Gex . Prior to calculating Gel , we need to model the micelles feasible to the calculation, as shown below: (1) The hydrocarbon core volume V is calculated from the aggregation number m and the number of carbon atoms nC in the hydrocarbon chain according to the formula [62]   VC /nm3 = 27.4 + 26.9(nC − 1) m × 10−3 . (8.1) (2) The length of the hydrocarbon chain lC is evaluated as lC = 0.75lmax   = 0.75 × 0.154 + 0.1265(nC − 1) nm.

(8.2)

(3) We assume a prolate boundary for the hydrocarbon core with a minor axis b that is equal to lC or lmax . Thus, the major axis c is determined by VC = 4π(b )2 c /3. (4) Electric charges are assumed to be distributed continuously on the surface of the prolate boundary with major and the minor axes c and b, where c = c + l0 and b = b +l0 in terms of l0 , the distance from the hydrocarbon core surface. Thus, the surface area S of the prolate is given as 1/2    
arccos(b/c) . (8.3) S = 2π b 2 + c2 b c2 − b2 (5) For the micelles of charges mαM e0 , the surface charge density σ is given as σ/e0 = mαM /S

or σ = e0 /a0 ,

(8.4)

where a0 denotes the area per charge and hence per monomer in the case of ionic micelles.

H. Maeda / Journal of Colloid and Interface Science 263 (2003) 277–287

(6) Since the Poisson–Boltzmann equation for a prolate boundary is not easy to handle, we calculate instead the electric surface potential ψ0 and the free energy Gel for an infinite plate or a sphere with the same σ . In the latter case, its radius rS is given as 4πrS2 = S. We first compare Gex and Gel for dodecyldimethylamine oxide (C12DMAO) in 0.1 M and 0.2 M NaCl in the case of αM = 1, in which the condition for the micellar Gibbs– Duhem relation Eq. (5.1) is expected to be satisfied [40]. For C12DMAO, VC = 0.3233m, nC = 12, lC = 1.156 nm, lmax = 1.542 nm, and l0 = 0.188 nm. The aggregation numbers m are 70 and 80 for 0.1 and 0.2 M NaCl, respectively [63]. Relevant parameters are given in Table 2. The ionic strength parameter κ and a reduced charge parameter s are defined as follows in terms of the Bjerrum length lB and the ion concentration ni /m−3 : κ 2 = 4πlB (8.5) ni and s = 4πlB σ/(e0 κ). The electrostatic free energy Gel is calculated for a single plate or a sphere immersed in a salt solution of infinite volume. Under the approximation, Gel is given by the charging up as follows. The Poissson–Boltzmann equation is written as follows in terms of the reduced potential φ = e0 ψ/kT and the reduced distance x = κr. We take φ = 0 at infinite x. For a plate, r denotes the distance from the micelle surface and d 2 φ/dx 2 = sinh φ.

(8.6)

For a sphere, r denotes the distance from the center and d 2 φ/dx 2 + (2/x) dφ/dx = sinh φ.

(8.7)

The boundary conditions are as follows. At the micelle surface (x = 0 for a plate and x = κrS for a sphere): dφ/dx = −s.

(8.8)

283

At x → ∞: dφ/dx = 0.

(8.9)

For a plate, s = 2 sinh(φ0 /2), and Gel and Gel /m are given as follows [64]: Gel (unit area) 1 s/2   = ψ0 (λ)σ dλ = 4σ kT /(se0 ) sinh−1 (x) dx 0

0


 1/2  = 4σ kT /(se0 ) (s/2) ln (s/2) + (s/2)2 + 1 1/2   +1 , (8.10) − (s/2)2 + 1 




Gel /mkT = (e0 /σ )Gel (unit area)/kT .

(8.11)

U el

The electrostatic energy is given as
 1/2  −1 . U el /mkT = (s/2) (s/2)2 + 1

(8.12)

For a sphere of radius rS and the surface charge density σ , the Poissson–Boltzmann equation was solved numerically. The charging up of micelles was carried out by simple integration after φ0 (s) was expressed as a polynomial of s: s G /mkT = (1/s) el

φ0 (s ) ds .

(8.13)

0

In the calculations both for a plate and a sphere, the employed smeared charge model overestimates φ0 and Gel [65]. On the other hand, we ignored the finite size of counterions, which underestimates φ0 and Gel . The results are summarized in Table 3. The analytic expressions proposed by Hayter [66] gave consistent results, as shown in Table 3. In the plate model, we can see the contribution from U el to Gel is rather small. We look for the model that gives the best agreement with the observed φ0 values, which are equal to

Table 2 Parameters of C12DMAOH+ micelles in NaCl solutions CS /M m VC /nm3 b

c /nm c /b

b c c/b S/nm2a σ/C m−2 a0 /nm2b a0−1 /nm−2 rS /nmc sd a b c d

0.1 70 22.63 lC /nm = 1.156 4.050 3.50 1.343 4.238 3.16 58.40 0.1920 0.8343 1.199 2.156 10.34

0.2 80 25.86 lmax /nm = 1.542 2.272 1.47 1.730 2.460 1.42 48.55 0.2310 0.6936 1.442 1.966 12.44

S denotes the area where charges are distributed, Eq. (8.3). a0 denotes the area per monomer in nm2 , a0 = (e0 /σ ) × 1018 . rS denotes the radius of an equivalent sphere. The charge parameter s is defined by Eq. (8.5).

lC /nm = 1.156 4.629 4.00 1.343 4.817 3.59 65.85 0.1946 0.8232 1.215 2.289 7.41

lmax /nm = 1.542 2.597 1.68 1.730 2.785 1.61 53.59 0.2392 0.6699 1.493 2.065 9.11

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Table 3 The reduced electric surface potential φ0 and the electrostatic free energy per monomer Gel /m for C12DMAOH+ micelles in NaCl solutions CS /M 0.1 s xD a

b = lmax

b = lC

b = lmax

10.34 2.24

12.44 2.04

7.41 3.65

9.11 3.04

4.041 3.792 3.782

4.442 4.200 4.192

φ0 plate sphereb spherec Expd

4.690 4.385 4.375

Gel /mkT plate sphereb spherec Expe

3.040 2.657 2.618

U el /mkT plate a b c d e

0.2

b = lC

5.054 4.764 4.756 4.63

4.38 3.350 2.88 2.909

2.509 2.238 2.215

2.56 0.825

9. The Gibbs–Duhem relation for ionic–nonionic mixed micelles 2.833 2.558 2.516

2.33 0.852

0.766

from the calculation above might suggest that the hydrogen bond was preserved even between neighboring cationic head groups, probably through some intervening water molecules. It has been shown recently that the dependence of the degree of counterion binding on the composition of nonionic– ionic mixed micelles is well explained by the plate model, while the sphere model fails to reproduce the observed dependence except for low s values [67]. However, there seems to be little difference between the two models in the case of hydrogen ion titration.

0.804

xD = κrS in the case of the spherical model. Numerical solutions. According to Hayter’s formula, Ref. [66]. φ0 = 2.303(pKM − pKa ). Gex /mkT . Gex is given by Eq. (5.6).

pKa = (ln 10)(pKM − pKa ), rather than with Gex , since Gel and Gex need not coincide with each other. From Table 3, the sphere model with b = lmax and the plate model with b = lC gave results in fair agreement with the observed values and the former looks better than the latter. It is seen from Table 3 that there is about a 10% difference between the results obtained by b = lC and b = lmax . When relevant b values are not known, we should expect a range of uncertainty of about 10% concerning calculations of this kind. Thus, we find that both models gave Gel values that are greater than Gex . There are differences between the integrations leading to Gel and Gex . One of them is as follows: during the integration A defined by Eq. (3.2) in the case of micelles, changes in the size and shape of micelles are allowed to occur, while the integration in either Eq. (8.10) or Eq. (8.13) is carried out for a fixed size and shape of micelles. It is obvious that σ varies linearly with αM in the latter case. For actual micelles, on the other hand, this point is generally not clear due to possible size and shape changes. However, in the case of C12DMAO, we have found that σ linearly changes with αM [59]. We can therefore conclude that Gex is significantly smaller than Gel though the difference is small. This suggests that the micelles change the shape and/or the size to counteract the increase of Gel during the charging process, unless the present calculation overestimates Gel . On the other hand, if the hydrogen bond that accounts for pKM < pK1 becomes unstable at high charge densities (αM > 0.5), a corresponding positive contribution to Gex is expected and hence a relation Gex > Gel should be observed. That the trend Gex > Gel does not show up

It is pertinent to examine here the conditions for the MGD relation of Eq. (5.1) to hold. For ionic (I)–nonionic (N) mixed micelles in a salt solution containing a common counterion species (C) and coions, the following GD relation has been derived when the bound amount of coions, which is likely to be negative, is negligible in comparison with that of counterions [68,69]:   xI dµI + (1 − xI ) dµN + xI β dµC = 1/(Ct − C1 ) dΠP . (9.1) Here xI , β, and ΠP denote the mole fraction of the ionic species, the degree of counterion binding, and the colloid (micelle or polymer) contribution to the Donnan osmotic pressure, respectively. When the rhs of Eq. (9.1) can be ignored, we have at a constant Ct xI dµI + (1 − xI ) dµN + xI β dµC = 0.

(9.2)

This relation has been confirmed experimentally [70]. Under the condition of excess salt, dµC in Eq. (9.2) can be approximated to vanish and Eq. (9.2) reduces to Eq. (5.1). On the other hand, we cannot expect that Eq. (5.1) holds at low ionic strengths. The micelle composition αM at the cmc is evaluated from Eq. (5.7) if Eq. (5.1) is valid. Hence, the validity of Eq. (5.1) can be examined by comparing the two sets of αM values: one from Eq. (5.7) and the other from Eq. (2.7). As shown by Eq. (5.9), a simpler procedure is given by comparing the αM,min value where the cmc exhibits the minimum or max∗ , where pK = pK (α = α ∗ ) or α ∗ = α ∗ . imum with αM 1 a M M M 1 ∗ are close to each other As shown in Table 4, αM,min and αM and the validity of Eq. (5.1) is confirmed for salt concentrations higher than about 0.1 M. In the case of C14DMAO, however, the validity of Eq. (5.1) is questionable. At the counterion concentration Cg = 30 mM, the dependence of the cmc on pH exhibited either a monotonic or a shallow minimum that could be in ∗ the range αM,min = 0.1–0.2, that is clearly different from αM (= 0.38–0.40) [33], suggesting the breakdown of Eq. (5.1). At this low ionic strengths, the relation C1 < cmc is likely if Ct  cmc and the evaluation of αM based on Eq. (2.7) under the assumption C1 = cmc will be subject to some error. It is

H. Maeda / Journal of Colloid and Interface Science 263 (2003) 277–287

285

Table 4 Examination of the validity of Eq. (5.1)

∗ αM αM,min a Validity of Eq. (5.1) C1 /C1 (0) = cmc/cmc(0)b

C10DMAO 0.1 M NaCl

C12DMAO 0.1 M NaCl

C12DMAO 0.2 M NaCl

C14DMAO 30 mM Cg

C14DMAO 0.1 M NaCl

C12DHEAO 0.1 M NaCl

∼0.40 0.4–0.5 Yes Yes

0.45 0.45 Yes Yes

0.50 0.50 Yes Yes

∼0.35 0.1–0.2 No

∼0.55 ∼0.50

0.22–0.25 0.2–0.3 Yes Yes

No

Surfactant concentration Ct was 20 mM expcept for C10DMAO (Ct = 60 mM). a The degree of ionization of micelles that gives the minimum cmc value. b This row describes whether approximate equivalence holds (Yes) or not (No) between C /C (0) values evaluated by Eq. (5.10) from the hydrogen ion 1 1 titration and cmc/cmc(0) values determined from the methods of the surface ternsion or others.

increase of ΠP with ionization at a constant Ct , we may point out an increased intermicellar repulsion due to micelle growth. The suggested enormous micelle growth with increasing αM has been observed by viscoelastic properties and cryoTEM pictures [71]. If we accept this mechanism, we can evaluate the difference of the colloid osmotic pressure from the nonionic micelle as follows, assuming cmc = C1 :
 ΠP /RT = ΠP − ΠP (αM = 0) RT

  = (ln 10) log cmc/cmc(αM = 0)   − (αM − α1 ) dpH . (9.3)

Fig. 3. Comparison of the cmc and C1 of tetradecyldimethylamine oxide (C14DMAO) in 0.1 M NaCl at different degrees of ionization αM . Filled circles refer to cmc values determined from the surface tension and pH variation method and open circles refer to the C1 values evaluated from Eq. (5.10) with the assumption that cmc(αM = 0) = C1 (αM = 0). Data are taken from Ref. [33].

hoped, however, that the error about the difference between ∗ is not significant. Hence, the conclusion that αM,min and αM Eq. (5.1) does not hold is unchanged after this issue is taken into account. In 0.1 M NaCl, although the relation αM,min ∼ ∗ appears to be satisfied approximately (Table 4), a signifαM icant difference has been observed between C1 /C1 (αM = 0) predicted from Eq. (5.10) and cmc/cmc(αM = 0). Data taken from Ref. [33] are shown in Fig. 3 under a reasonable approximation, C1 (αM = 0) = cmc(αM = 0) [33]. When Eq. (5.1) holds, we can evaluate the monomer concentration ratio C1 /C1 (αM = 0) from Eq. (5.10). The obtained C1 /C1 (αM = 0) values have been in good agreement with cmc/cmc(αM = 0) for C10DMAO [34], C12DMAO [40], and C12DHEAO [34]. Since C1 , which is a rough estimate of the activity, cannot decrease with increasing the total concentration C1 , and since the counterion activity is expected not to change much with Ct , we are obliged to conclude that Eq. (5.10) and hence Eq. (5.1) does not hold even in 0.1 M NaCl solutions. If the dΠP term in Eq. (9.1) is positive and plays a significant role, then this contribution should be added to the rhs of Eq. (5.10) and it makes the value of C1 /C1 (αM = 0) larger. As a mechanism of this expected

From the data shown in Fig. 3, values of ΠP /RT are (in mM) 2.3, 6.2, 10.6, 14.4, 16.8, 17.4, 16.6, 14.9, 13.3, and 12.8 for αM of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0, respectively. It is to be noted that when C1 differs from the cmc, the procedure proposed by Rathman and Christian [31] to evaluate the cmc from the titration data cannot be applied.

10. NMR titration The hydrogen-ion titration can be done using the NMR spectra, where appropriate absorption bands show different chemical shifts for a protonated and an unprotonated species. Two issues, among others, should be examined with respect to the difference between the NMR-based and the usual titration methods. The first point refers to different ways of evaluating the degree of ionization α in the two methods. For this point, the NMR method has been believed to be superior to the usual one, since α is determined directly from the actual amounts of protonated and unprotonated species. In the usual method, on the other hand, α is evaluated by the reacted amount of an acid (or alkali), which is calculated as a difference between the added and the unreacted amounts of the acid (or alkali). A popular approximation employed to evaluate the latter is to assume the activity coefficient of hydrogen ions to be unaffected by the presence of charged colloid ions. With this approximation, the unreacted amount can be estimated from so-called blank titrations on those solutions which contain no colloid ions but otherwise are

286

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the same as the actual solutions. It is to be noted, however, that linearity between α and the chemical shift change is a prerequisite for the NMR method to be superior. We have carried out the hydrogen-ion titration of dodecyldimethylamine oxide (C12DMAO) based on NMR spectra [72] and we have obtained a titration curve that is not much different from that obtained in the usual way [29], exhibiting a generally observed trend dpKa /dα < 0. On the other hand, a significantly different NMR-titration curve on C12DMAO was reported where this general trend was not well presented [73]. We have found that the chemical shift of the N -methyl group of C12DMAO varies with α in a slightly but significantly nonlinear way when compared with α evaluated by approximating it with the degree of neutralization, a reasonable approximation at a high surfactant concentration (0.3 M) in 0.1 M NaCl solutions [72]. This result suggests that the chemical shifts are determined not only by the relative populations of the protonated and unprotonated species but also by various interactions among head groups. It is therefore important in the NMR-titration to examine the linearity between α and the chemical shift in question. The other difference between the NMR and the usual methods arises from the different media employed, H2 O and D2 O. It has been proposed that to convert the apparent pH readings in D2 O media to pD, we need to add 0.40–0.41 [74, 75]: pD = pH (pH meter reading in D2 O) + 0.41. Another point is the different proton dissociation activities of a given group in the two media. Brønsted acids become weaker in D2 O: pK(D2 O) − pK(H2O) = 0.5 and 0.7 for carboxyl and amino groups, respectively [75–77].

11. Conclusions A thermodynamic analysis of the hydrogen ion titration is presented for association colloids with a particular emphasis on surfactant micelles. (1) When the micellar Gibbs–Duhem relation (MGD), αM dµI + (1 α− αM ) dµN = 0, Eq. (5.1), holds, the integral A = 0 M (pKM − pKa ) dαM is related to Gex , the free energy change accompanying the ionization of micelle, as A = Gel /{mRT (ln 10)}, in the same way as for covalently bonded colloids like linear polyions. (2) In the case where the regular solution approximation is valid for mixed micelles, the titration curve, pKa vs αM , should be a straight line and Gex is given as a 2 in terms of the interaction parameter β: function of αM 2 . This result can be ex G (αM )/mRT = (ln 10)A = −βαM applied to ionic/nonionic mixed micelles in general. (3) We define the αM,min value as the micelle composi∗ as tion where the cmc exhibits the minimum and αM ∗ ∗ ∗ that where pK1 = pKa (αM = αM ) or α1 = αM . We obtain a criterion to judge whether the MGD holds or not: ∗ holds when Eq. (5.1) holds. αM,min = αM

(4) For dodecyldimethylamine oxide micelles (C12DMAO) ∗ has been shown to hold, values of to which αM,min = αM the calculated electrostatic free energy Gel were close to but significantly greater than experimental Gex values. (5) For tetradecyldimethylamine oxide (C14DMAO), Eq. (5.1) does not hold at a low ionic strength. (6) When Eq. (5.1) holds, the monomer concentration C1 can be evaluated from the hydrogen-ion titration according to Eq. (5.10). For most cases examined, the C1 /C1 (0) from the titration agree fairly well with cmc/ cmc(0), indicating constant C1 above the cmc. An exceptional behavior, C1 /C1 (0) < cmc/cmc(0), has been found for C14DMAO at 0.1 M NaCl and this inconsistency could be ascribed to intermicellar repulsion due to enormous micelle growth.

Acknowledgments The author thanks Dr. Kakehashi for discussion and H. Uchiyama for some calculations and preparing the manuscript. This work was supported, in part, by a Grant-in-Aid for Scientific Research (B) (No. 12440200) from The Ministry of Education, Culture, Sports, Science and Technology, Japan.

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