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Estimation of micellization parameters of sodium dodecyl sulfate in water + 1-butanol using the mixed electrolyte model for molar conductance K. Gunaseelan and K. Ismail ∗ Department of Chemistry, North Eastern Hill University, Shillong 793022, India Received 26 November 2001; accepted 11 October 2002

Abstract The mixed electrolyte model of Shanks and Franses has been applied to estimate the critical micelle concentration, aggregation number, and counterion binding constant of sodium dodecyl sulfate in a water + 1-butanol medium from its measured conductivity data at 25 ◦ C. The surface potential of the ionic micelle in this mixed solvent medium was computed by solving the nonlinear Poisson–Boltzmann equation. The standard free energy terms of micellization were also calculated. The present study confirms further the observation made in the previous studies that ionic micelles do not contribute to the ionic strength of a surfactant solution, an inference originally made by McBain and coworkers. 2003 Elsevier Science (USA). All rights reserved. Keywords: Sodium dodecyl sulfate; Water + 1-butanol; Electrical conductivity; Critical micelle concentration; Counter-ion binding constant; Aggregation number; Surface potential; Free energy terms; Ionic strength

1. Introduction In a solution of an ionic surfactant changes in electrical conductivity with concentration are caused by ion–ion interactions, ion–solvent interactions, and interactions responsible for micellization. The first two types of interactions operate in solutions of normal electrolytes also. It is therefore necessary to account for the different interactions appropriately while analyzing the conductivity data of ionic surfactant solutions so that the values of the micellization parameters are controlled exclusively by the interactions responsible for micellization. A similar view was also presented by Binana-Limbele and Zana [1]. Therefore, the method of estimating the values of the micellization parameters of ionic surfactants from the electrical conductivity data of their solutions after properly accounting for the ion–ion and ion– solvent interactions is better than the conventional method wherein values of the critical micelle concentration or cmc (c0 ) and counterion binding constant (β) are determined directly from the plot of conductivity versus surfactant con* Corresponding author.

E-mail address: [email protected] (K. Ismail).

centration. Accounting for the ion–ion interactions becomes essential, particularly in analyzing the conductivity data of surfactant solutions in nonaqueous media [1]. Out of the various methods used by different workers [2– 6] to analyze conductivity data of ionic surfactant solutions, the one proposed by Shanks and Franses [6] is more recent and uses the Debye–Hückel–Onsager treatment to account for the ion–ion and ion–solvent interactions. Another new approach, suggested by Moroi and co-workers [7,8], to estimating the micellization parameters of ionic surfactants using electrical conductivity and emf data of ionic surfactant solutions is certainly superior than the other methods. The emf data obtained using ion-selective electrodes actually provide information about the concentrations of monomeric ions and counterions needed to use the mass-action model. However, the value of the micellization equilibrium constant for SDS reported by Moroi and co-workers [7,8] from this approach, K = 10230, seems to be unrealistic. The value for the standard free energy change of micellization of SDS obtainable from this value of K is much different from the generally accepted value of about −37 kJ mol−1 . Moreover, the high value of equivalent conductance of ionic micelles derived from this approach warrants a new conduction mechanism in micelles, as pointed out by Moroi

0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00065-6

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and Matsuoka [7]. Therefore, we preferred to use the Shanks and Franses [6] model, known as the mixed-electrolyte model, to estimate c0 , β, and aggregation number (n) of ionic surfactants in different media [9–11]. The Shanks and Franses model has, however, not been applied to solutions of ionic surfactants in mixed solvent media. The purpose of the present work is therefore to apply the mixed-electrolyte model of Shanks and Franses [6] to the determination of c0 , n, and β of sodium dodecyl sulfate (SDS) in a water + 1-butanol medium. Accordingly, the specific conductivity, κ, of SDS in water + 1-butanol was measured at 25 ◦ C as a function of SDS concentration and of butanol amount. Surface potentials, ψ, of the SDS micelles in this mixed solvent were also computed by solving the nonlinear Poisson–Boltzmann equation.

2. Experimental SDS (SISCO, molecular biology grade) and 1-butanol (SD, AR grade, 99.5% assay) were used as supplied. Double-distilled water was used to prepare the solutions. Electrical conductance measurements of solutions of SDS in water + 1-butanol as functions of SDS and 1-butanol amounts at 25 ◦ C and 1 kHz were made using a Wayne– Kerr B905 automatic precision bridge and a cell of cell constant 102.4 m−1 , as described elsewhere [9]. Viscosity of the water + 1-butanol mixed solvent was measured at 25 ◦ C using a Cannon–Ubbelohde viscometer having viscosity constant 0.3693 × 10−2 cSt/s.

3. Results and discussion The measured values of specific conductivity of SDS in water + 1-butanol mixed solvent at 25 ◦ C are presented in Fig. 1 as plots of κ versus SDS concentration. These plots have the usual shape, which manifests the occurrence of micellization at the cmc. As the alcohol content increases the difference in the slopes of the plots below and above the cmc becomes progressively smaller, which is illustrated in Fig. 2, this trend is similar to that reported [12] in solutions of CTAB in water + alcohol mixtures. Owing to this trend in the slopes of the plots of κ versus SDS concentration, the conductance method cannot be accurately employed for the determination of cmc above ∼0.95 mol kg−1 of butanol. It is interesting to note that this concentration of 1-butanol is closely comparable to the solubility limit of 1-butanol in water [13]. Therefore, in the present work, although we measured the conductivity of SDS in water + 1-butanol mixtures containing up to 2.052 mol kg−1 1-butanol, determination of micellization parameters from the κ data could be done for SDS solutions containing 1-butanol up to 0.875 mol kg−1 only. The method of using the mixed electrolyte model for obtaining the micellization parameters of SDS in water +

Fig. 1. Plots of specific conductivity versus concentration of SDS in water containing different amounts of 1-butanol at 25 ◦ C. indicates the upward shift in the ordinate scale.

Fig. 2. Variation of the slopes of the plots in Fig. 1 with concentration of 1-butanol. Upper line: slope < cmc; lower line: slope > cmc.

1-butanol mixture is the same as described in the case of SDS in acetamide melt [10]. According to the mixed electrolyte model the molar conductance, Λ, of SDS in the mixed solvent can be written as Λ = Λ1m Pmono + Λ1mic Pmic .

(1)

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In Eq. (1) Λ1m is the molar conductivity of the surfactant in the monomer phase when it is in the monomer form and Λ1mic is the molar conductivity of surfactant in the micellar phase when it is in the micellar form. Pmono and Pmic are fractions of the total surfactant molecules existing in the monomer and micellar forms, respectively. After accounting for the effect of ion–ion interactions on electrical conductivity using the Debye–Hückel–Onsager treatment, Eq. (1) takes the form Λ = Λ01 − A1 I 1/2 /(1 + B0 a1 ) (c0 /ct ) + Λ0n − An I 1/2 /(1 + B0 an ) ncn (1 − β)/ct . (2) Λ0i ’s

In Eq. (2) ai ’s and correspond to effective ionic sizes and limiting equivalent conductivities, respectively, of monomer (i = 1) and micelle (i = n). I is the ionic strength, ct is the total molar concentration of the surfactant, and cn is the molar concentration of micelle. B0 and Ai ’s are given by the following expressions: 1/2 1/2 B0 = 8πNA e02 /(103 εkB T ) (3) I , Ai = 2.801 × 106 |z+ z− |qΛ0i (εT )3/2 (1 + q 1/2) + 41.25 |z+ | + |z− | η(εT )1/2 . (4) In Eqs. (3) and (4) kB is the Boltzmann constant, T is the absolute temperature, ε is the dielectric constant of the medium, η is the viscosity of the medium, NA is the Avogadro number, and e0 is the electronic charge. The q term is given by another expression of the form q = |z+ z− |(λ0+ + λ0− ) |z+ | + |z− | |z+ λ0+ | + |z− λ0+ | , (5) where λ0+ and λ0− represent limiting ionic equivalent conductivities of cationic and anionic species of effective charges z+ and z− , respectively. To compute the values of c0 , n, and β using Eq. (2), the values of r1 (radius of the surfactant monomer), rn (radius of the micelle), rc (radius of the counterion), a1 , an , A1 , An , Λ01 , Λ0n , λ0+ , λ0− , and I must be determined first. r1 was evaluated in Å using the relation [14] 1/3 r1 = (3/4π)(27.4 + 26.9nc ) (6) , where nc is the number of carbon atoms in the hydrocarbon chain of the surfactant used. In Eq. (6) the volume of a monomer is estimated from geometric considerations and then this volume is equated to a spherical volume, thereby assigning an effective radius r1 for the monomer. Presuming the micelle to be spherical in shape, rn is computed from the relation rn = n1/3 r1 . The dielectric constant, ε, of the mixed solvent required for data analysis was estimated using the principle of additivity and the values of ε obtained thus are listed in Tables 1 and 2. The measured values of viscosity, η, of the mixed solvent are also given in Tables 1 and 2. The limiting ionic equivalent conductance of dodecyl sulfate ion (λ01 ) in the water + 1-butanol mixed solvent was computed

Table 1 Computed values of the micellization parameters of SDS in water + 1-butanol mixture at 25 ◦ C (concentration of 1-butanol = 0.037 mol kg−1 , η = 0.89 × 10−3 Pa s, ε = 78.3) Model for I a

Parameters (c0 × 103 ) ± 0.1 (mol kg−1 ) n (β × 10) ± 0.1 (Std. dev. in κ) × 104 (S m−1 ) λ0c × 104 (Sm2 eq−1 ) Λ01 × 104 (Sm2 eq−1 )

1b

2

3b

4b

7.7 42 ± 5 7.1 3.4

7.7 50 ± 5 7.1 3.2 51.8 73.0

7.7 41 ± 5 7.1 3.5

7.7 7±3 5.9 4.0

Λ0n × 104 (Sm2 eq−1 ) rc × 10 (nm) rn × 10 (nm) −(ψ × 10−1 ) ± 1 (mV) G0m (kJ mol−1 ) G0el (kJ mol−1 ) G0hy (kJ mol−1 )

135 1.8 16 10 −38 10 −48

a At c > c , I = c (model 1); I = c + 0.5n(1 − β)c (model 2); t n 0 0 0 I = c0 + 0.5n(1 − β)2 cn (model 3); I = c0 + 0.5n(1 − β)[1 + n(1 − β)]cn (model 4). In all the four models I = ct at ct < c0 . b Since model 2 for I is chosen for data fitting (see text), values of parameters other than, c0 , n, and β are not listed for models 1, 3, and 4 for I .

using the Stokes–Einstein relation λ0i = zi e0 F /(6πηri ),

(7)

where F is the Faraday constant. The limiting ionic equivalent conductivity of the counterion, λ0c , in the mixed solvent was then obtained by subtracting λ01 from the experimentally determined value of Λ01 (Tables 1 and 2). The radius of the counterion, rc , in the mixed solvent was determined by substituting the value of λ0c for λ0i in Eq. (7). The limiting ionic equivalent conductivity of the micelle, λ0n , was also calculated from Eq. (7) by substituting n(1 − β) for zi and rn for ri . Once λ0n was known, An and Λ0n were determined easily. a1 and an were evaluated by adding rc to r1 and rn , respectively. As emphasized in our earlier studies [9–11] as well as by others [15–17], the ionic strength of an ionic surfactant solution required for the data fitting cannot be determined simply by the expression I = 0.5 ci zi2 , where ci and zi are the concentration and charge of the ionic species i, respectively. In fact, Shanks and Franses [6] proposed four models for estimating the ionic strength of an ionic surfactant solution, which are given in Table 1. We tried all these four models for I while fitting the conductivity data to Eq. (2). It was observed that models 1, 2, and 3 for I fit the data almost equally well. This is illustrated in Table 1 for solutions of SDS in one particular mixture of water and 1-butanol containing 0.037 mol kg−1 of 1-butanol. Model 4 for I provided unreasonable values for the aggregation number. A similar observation was made earlier while analyzing the conductivity data of SDS in water also [6,9]. Although models 1, 2, and 3 for I are equally good for fitting the conductivity data to Eq. (2), aggregation numbers computed us-

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Table 2 Computed values of the micellization parameters of SDS in water + 1-butanol mixture at 25 ◦ C 1-butanol concentration (mol kg−1 )

Parameters

(c0 × 103 ) ± 0.1 (mol kg−1 ) n (β × 10) ± 0.1 (Std. dev. in κ) × 104 (S m−1 ) λ0c × 104 (Sm2 eq−1 ) Λ01 × 104 (Sm2 eq−1 ) Λ0n × 104 (Sm2 eq−1 ) rc × 10 (nm) rn × 10 (nm) −(ψ × 10−1 ) ± 1 (mV) G0m (kJ mol−1 ) G0el (kJ mol−1 ) G0hy (kJ mol−1 )

0.081 (η = 0.90 × 10−3 Pa s; ε = 78.2)

0.175 (η = 0.93 × 10−3 Pa s; ε = 78.1)

0.263 (η = 0.95 × 10−3 Pa s; ε = 78.0)

0.395 (η = 0.99 × 10−3 Pa s; ε = 77.9)

6.4 46 ± 4 6.9 5.3 54.4 75.3

4.5 41 ± 6 6.5 4.8 53.8 74.0

4.2 38 ± 10 6.2 8.9 54.2 73.9

3.6 34 ± 5 5.6 9.9 51.0 70.0

136 1.7 16 10 −38 10

139 1.6 15 12 −38 11

138 1.6 15 12 −38 11

139 1.6 14 13 −37 12

−48

−49

−49

−49

1-butanol concentration (mol kg−1 )

Parameters 0.525 (η = 1.01 × 10−3 Pa s; ε = 77.7) (c0 × 103 ) ± 0.1 (mol kg−1 ) n (β × 10) ± 0.1 (Std. dev. in κ) ×104 (S m−1 ) λ0c × 104 (Sm2 eq−1 ) Λ01 × 104 (Sm2 eq−1 )

Λ0n × 104 (Sm2 eq−1 ) rc × 10 (nm) rn × 10 (nm) −(ψ × 10−1 ) ± 1 (mV) G0m (kJ mol−1 ) G0el (kJ mol−1 ) G0hy (kJ mol−1 )

0.657 (η = 1.05 × 10−3 Pa s; ε = 77.6)

0.875a (η = 1.11 × 10−3 Pa s; ε = 77.1)

3.3 25 ± 5 5.1 8.2 52.5 71.0

3.6 30 ± 5 5.0 6.6 52.2 70.0

3.8 28 ± 10 5.0 16 52.2 69.0

130 1.5 13 13 −36 12

138 1.5 14 14 −36 13

130 1.4 13 13 −36 12

−48

−49

−48

a In this solution the uncertainties in the values of c × 103 and β × 10 are found to be to the extent of ±1.0 and ±0.2, respectively. 0

Fig. 3. Variation of aggregation number of SDS in water + 1-butanol with concentration of 1-butanol at 25 ◦ C. (•) Present work; (∗) Ref. [21].

ing model 2 for I are found to be more close to the reported values (Fig. 3). Same inferences were also made in the case of solutions of SDS in the other mixtures of water and 1-

butanol containing varying amount of 1-butanol. Therefore, in Table 2 only the results of the data fitting using model 2 for I are given. The values of cmc obtained from the mixed electrolyte model are comparable with the reported [18–20] values, as apparent from Fig. 4. The cmc of SDS decreases with increase in 1-butanol content and passes through a minimum at ∼0.5 mol kg−1 1-butanol, which is in accordance with the reported [21,22] trend. The observed trends in the variation of n and β with increase in 1-butanol content are similar to the trends reported [12,22–31] in solutions of ionic surfactants in water + alcohol mixtures. The value of β decreases with increases in the amount of 1-butanol [22–25]. The computed values of β up to 0.525 mol kg−1 of 1-butanol are comparable with the reported [31] values within ±0.05. The β values obtained in the present study for SDS solutions in 0.657 and 0.876 mol kg−1 1-butanol are, however, considerably higher than the reported [23] values. For example, the reported β values for SDS solution in water + 1-butanol mixture containing 0.69 and 0.86 mol dm−3 1-bu-

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Knowing c0 , n, and β, we calculated (i) the standard free energy change of ionic micelle formation per mole of monomer, G0m , using the equation G0m = RT (1 + β) × ln xcmc , where xcmc represents c0 in a mole fraction unit, and (ii) the surface potential, ψ, by solving numerically the nonlinear Poisson–Boltzmann (PB) equation in spherical symmetry. The PB equation is of the form d 2 y/dx 2 = (1/2x 4)(ey − e−y ),

(8)

where x = (B0 r)−1 and y = e0 ψr /kB T . The boundary conditions are y=0

as x → 0

(9)

and dy/dx = 4πρr e0 /(εB0 kB T x 2 )

Fig. 4. Variation of cmc of SDS with concentration of 1-butanol at 25 ◦ C. (•) Present work; () Ref. [17]; (◦) Ref. [18]; () Ref. [19].

tanol are nearly 0.37 and 0.27, respectively. The value of n also decreases with increase in 1-butanol amount [21,28–31] as shown in Fig. 3. The values of n obtained in this study above ∼0.3 mol kg−1 1-butanol are in good agreement with the experimental values reported by Almgren and Swarup [22,29]. 1-Butanol is known to distribute between the aqueous phase and the micellar phase [30,32,33]. The presence of 1-butanol in the micellar phase affects the surface area per surfactant monomer and hence the surface charge density, thereby affecting n and β [22]. From the above data analysis it is evident that the mixed electrolyte model satisfactorily describes the conductance behavior of an ionic surfactant solution in a mixed solvent medium, as in pure aqueous and molten media. Furthermore, the failure of model 4 for I to fit the conductivity data to Eq. (2) in the present study confirms once again the inference made from previous studies that in an ionic surfactant solution the ionic micelles do not contribute to the ionic strength of the surfactant solution. In fact, McBain and coworkers [34,35] had reported that in solutions of colloidal electrolytes there was no indication of high ionic strength. It has therefore become important either to redefine ionic strength term or to correct the Debye–Hückel approach for solutions containing large ionic species. The recent attempt made by Abbas et al. [36], in which they proposed a semiempirical framework of the screening mechanism on the basis of an extension of the Debye–Hückel theory, is a step forward in this direction. Abbas et al. [36] observed that accounting for ion size effects within a restrictive primitive model of an electrolyte led to a shift in the Debye screening length.

as r = rn .

(10)

ψr and ρr are the electrostatic potential and surface charge density at a distance r from the center of the spherical micelle, respectively. The micellar surface charge density is calculated from the expression ρr (at r = rn ) = e0 n(1 − β)/(4πrn2 ).

(11)

The computational method used for the numerical analysis of Eq. (8) is the same as described elsewhere [9,10]. Once ψ values were known, the values of G0m for SDS in a water + 1-butanol mixed solvent were split up into standard electrostatic free energy, G0el (= F ψ), and standard hydrophobic free energy, G0hy . The values of G0m , ψ, G0el , and G0hy are also listed in Tables 1 and 2. Although surface potential of SDS micelles decreases on initial addition of 1-butanol to water (for SDS in water ψ = −141 mV [37]), it approaches the value in a pure aqueous medium at higher amounts of 1-butanol. Surface potentials of SDS micelles in aqueous sodium chloride solutions containing 0.35, 0.50, and 1.00 mol dm−3 of 1-butanol were derived by Attwood et al. [38] from quasi-elastic light-scattering measurements. Presuming that [(ψ in water + x mol dm−3 1butanol)/(ψ in water + 0.1 mol dm−3 NaCl + x mol dm−3 1-butanol)] ≈ [(ψ in water/(ψ in water + 0.1 mol dm−3 NaCl)], an estimate of ψ of SDS micelles in water + 1butanol medium was made from the relevant data reported by Attwood et al. [38]. This estimate gave the values 124, 118, and 113 mV for ψ of SDS micelles in 0.35, 0.50, and 1.00 mol dm−3 aqueous 1-butanol solutions, respectively. The agreement between these estimated values of ψ and its corresponding computed values (Table 2) seems to be reasonably good in view of the underlying assumptions. Furthermore, the computed ψ values of the SDS + water + 1-butanol system are found to be comparable to the reported [39] experimental values of surface potential of SDS + water + pentanol system below alcohol concentration 0.08 mol dm−3 .

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Acknowledgments We acknowledge financial assistance received from the UGC, New Delhi, under the DRS program. One of the authors (K.G.) is thankful to the CSIR, New Delhi, for the award of a senior research fellowship.

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