ISSN 1061933X, Colloid Journal, 2010, Vol. 72, No. 6, pp. 788–798. © Pleiades Publishing, Ltd., 2010.

Physicochemistry of Mixed Micellization: Binary and Ternary Mixtures of cationic Surfactants in Aqueous Medium1 Chanchal Dasa, Tanushree Chakrabortyb, Soumen Ghoshb, and Bijan Dasa a

b

Department of Chemistry, North Bengal University, Darjeeling734013, W.B., India Centre for Surface Science, Department of Chemistry, Jadavpur University, Kolkata700 032, W.B., India email: [email protected] Received December 9, 2009

Abstract—The importance of studying mixed micellization lies in tuning the performance of an amphiphile to bend through variation of stoichiometry of the blend. In this study, the binary and ternary mixed systems of cetylpyridinium chloride (CPC), tetradecyltrimethylammonium bromide, and dodecylpyridinium chlo ride (DPC) have been studied at 30°C using tensiometry and conductometry. In most cases, the cmc observed from either method is in close proximity whereas in CPC/DPC mixtures, tensometric cmc precedes conduc tometric cmc which may arise from a lowering in degree of counterion binding on micellar interface in the mixed system with lower stoichiometric mole fraction of CPC. Various existing theories have been used and the results were compared with the experimental observations. DOI: 10.1134/S1061933X10060098 1

1. INTRODUCTION

Mixed surfactant systems exhibit better perfor mance compared to the individuals [1, 2] in terms of lowering surface tension of aqueous solution and hence, exhibit enhanced detergeney, dispersing water insoluble substances in aqueous solution, and are widely used in the area of suspension, wetting, emulsi fication, and different technological, biochemical, pharmaceutical directions [3]. The molecular struc ture of amphiphiles, their concentration, and compo sition along with the environmental conditions, such as temperature, pH, pressure, and presence of addi tives [3] significantly govern the activity of the surfac tant mixtures. Because of the amphiphilic chemical structure, surfactant has a preference towards interfacial adsorp tion at low concentration region; whereas above a crit ical concentration, it selfaggregates to form assem bled structure whose size, shape and average number of amphiphile per aggregated structure depend on the amphiphile concentration and other physicochemical parameters like temperature, presence of salt, etc. The critical amphiphile concentration required for the onset of formation of more or less spherical aggregated structure, called micelle, in aqueous medium is called critical micellar concentration (cmc). Micelles can be treated as a separate phase within a surfactant solution and various physicochemical properties change dra matically depending on surfactant concentration. Measurement of any of such properties with variation of surfactant concentration leads to a discontinuity in 1 The article is published in the original.

the profile leading to determination of cmc of the sur factant. Frequent measurements of surface tension [4–9], conductance [4–9], fluorescence intensity [10], heat capacity [11, 12], light scattering [13], etc., are used in determination of cmc. The change in these properties occurs over a narrow region of total surfac tant concentration. Moreover, concentration depen dent changes of properties of premicellar and postmi cellar regions can be joined by two straight lines and the point of intersection of these two lines is taken as the cmc of the surfactant mixture. The selfaggregation and associated thermody namics of different mixed surfactant combinations, such as, nonionic–nonionic [7, 8, 14, 15], cationic– nonionic [4, 7, 9, 16, 17], anionic–nonionic [8, 18, 19], cationic–anionic [20, 21], anionic–biosurfactant [22], cationic–cationic [4, 11, 23], anionic–anionic [24, 25], etc. have been studied for a long period of time. In this report, we have physicochemically stud ied three cationic surfactants namely, dodecyl pyridin ium chloride (DPC), cetylpyridinium chloride (CPC), and tetradecyltrimethylammonium bromide (TTAB) in pure and mixed states. The first two (DPC and CPC) have the same pyridinium head group and dif ferent tail lengths (12 and 16 C) whereas TTAB has a methyl substituted quaternary ammonium head group linked with a 14 C tail. Three permutations of binary combinations, CPC/TTAB, DPC/TTAB, and CPC/DPC and ternary mixtures of CPC/DPC/TTAB have been studied. Tensiometry and conductometry were used for determining the cmc of all the mixtures. Various parameters such as cmc, Gibbs surface excess (Γmax ) , minimum area of exclusion per surfactant

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monomer at the air/solution interface ( Amin ) , pC 20 (where C 20 is the surfactant concentration required to decrease the surface tension of solvent by 20 unit), degree of counterion binding (g) on micellar interface along with various thermodynamic parameters like 0 Gibbs adsorption energy ΔGad , Gibbs micellization

(

(

)

)

energy ΔG m0 , can be found out by using these two methods. Existing theories of Clint [26, 27], Rosen [27–29], Rubingh [27, 30], Motomura [27, 31–33], Blankschtein [27, 34–36], and Rubingh and Holland [37, 38] are applied in these systems to evaluate theo retical cmc, micellar and interfacial mole fractions and interaction parameters among the surfactants in micellar and interfacial monolayer, surface free energy, activity coefficient, etc. 2. EXPERIMENTAL 2.1. Materials The cationic surfactants DPC, CPC, and TTAB were purchased from Sigma (USA). All the products were used without further purification. All solutions were prepared in doubly distilled water and the exper iments were performed at 303 ± 0.1 K.

789

γ/mN m–1 70 1 2 3 4

65 60 55 50 45 40 35 –4.0

–3.6

–3.2 –2.8 –2.4 –2.0 log([surfactant]/mol dm–3)

Fig. 1. Tensiometric plot of binary mixtures (1) CPC/TTAB, αCPC = 0.25; (2) DPC/TTAB, αDPC = 0.4; (3) CPC/DPC, αCPC = 0.9; and ternary mixtures (4) DPC/CPC/TTAB (0.333/0.333/0.333) at 303 K.

3. RESULTS AND DISCUSSION

Beyond this concentration, the added surfactants can hardly affect the interfacial topology of the interfa cially adsorbed surfactant monolayer but prefer to self associate in the bulk solution to form micelles whereby the hydrophobic tails of individual surfactant mono mers are buried within a hydrophobic encapsulation provided by the polar head group of the surfactant structures [5, 19]. On initiation of the selfassociation, the counterions of ionic surfactants with significantly large ionic mobility start to adsorb onto the micelle– solution interface leading to a dramatic change in the electrical transport property of the solution as a result of decrease in number of effective charge carriers in solution and is reflected in a sharp break in the specific conductance of the solution (κ) vs. surfactant concen tration isotherm (Fig. 2). The conductometric cmc, therefore, corresponds to onset of selfassociation of ionic surfactant in bulk solution. If the interfacial sat uration and bulk association processes are coherent, the cmc values observed in either method corroborate each other and vice versa. For all the pure surfactants, cmc determined by either method are in close proxim ity (Table 1). The cmc values increase with decrease in tail length. Although, CPC and DPC have the same pyridinium head groups, their cmc’s differ ~19 times reflecting the hydrophobic interaction prevailing over the electrostatic interaction in micellization process.

3.1. Determination of Critical Micellar Concentration (cmc) Tensiometrically, cmc corresponds to the surfac tant concentration at which there is a discrete break in the air/solution interfacial tension vs. log[surfactant] isotherms (Fig. 1). Phenomenologically, this cmc cor responds to the saturation of interfacial adsorption.

Among the binary mixtures, for CPC/TTAB and DPC/TTAB systems, the cmc values obtained from either method closely resemble each other whereas for CPC/DPC mixed system, the tensometric cmc is much lower than that determined by conductometry signifying a concentration delay (lag) between interfa cial saturation and selfaggregation process. In all the mixtures, the cmc values are close to the component

2.2. Methods 2.2.1. Tensiometry. The tensiometric experiments were performed using a platinum ring by the ring detachment method in a calibrated K9 Tensiometer (Krüss, Germany). Detailed procedure has been reported earlier [4–10, 19]. Each experiment was repeated several times to achieve good reproducibility. The measured surface tension (γ) values were cor rected according to the procedure of Harkins and Jor don. The γ values were accurate within ±0.1 mN m–1. 2.2.2. Conductometry. The conductance measure ments were taken with a PyeUnicam PW9509 con ductivity meter at a frequency of 2000 Hz using a con ductivity cell of cell constant 1.0 cm–1. The same pro cedure of addition of surfactant as in tensiometry was followed. The accuracy of the measurements was within ±1%. The measurement details can be found elsewhere [4–10, 19].

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Table 1. Critical micellar concentrationa (cmc) of pure, bi nary and ternary mixtures of surfactants at 303 K αCPC (I) or Conducto Tensiometry αDPC (II) metry 0.00 0.10 0.25 0.50 0.75 0.90 1.00 0.00 0.10 0.25 0.40 0.50 0.60 0.75 0.90 1.00 0.00 0.10 0.25 0.40 0.50 0.60 0.75 0.90 1.00

Average cmc

cmcC

CPC/TTAB (103 cmc, mol dm–3) (I) 3.715 3.524 3.620 – 2.564 2.716 2.640 2.846 1.556 2.255 1.906 2.154 1.247 1.562 1.404 1.534 0.998 1.250 1.124 1.191 0.920 1.313 1.026 1.050 0.910 1.036 0.973 – 3 –3 DPC/TTAB (10 cmc, mol dm ) (II) 3.715 3.524 3.620 – 3.908 3.992 3.950 3.931 4.645 4.731 4.688 4.513 5.297 5.781 5.539 5.297 5.929 6.435 6.182 5.991 7.211 7.131 7.171 6.895 9.453 9.387 9.420 8.910 11.695 12.414 12.054 12.588 17.132 17.608 17.370 – 3 –3 CPC/DPC (10 cmc, mol dm ) (I) 17.132 17.608 17.370 – 4.291 8.631 6.461 6.468 3.254 7.764 5.509 3.332 2.218 5.190 3.704 2.244 1.527 3.950 2.739 1.843 1.465 2.775 2.120 1.563 1.445 2.469 1.957 1.273 1.096 2.296 1.696 1.074 0.910 1.036 0.973 –

a The average error in cmc is ±2%.

Conduc Average cmc α DPC α CPC αTTAB Tensiometry tometry (Clint cmc) 0.125/0.250/0.625 0.125/0.625/0.250 0.250/0.125/0.625 0.250/0.625/0.125 0.333/0.333/0.333 0.625/0.125/0.250 0.625/0.250/0.125

a The average error in cmc is ±3%.

3.2. Interfacial Adsorption Due to the surfaceactive properties of surfactants, they first accumulate at the air/water interface when added in water. On interfacial adsorption, they break the water structure at the interface and thus the surface tension of water decreases on increasing addition of surfactant. After completion of interfacial saturation by surfactant, they start to gather in the bulk phase in order to form the surfactant aggregates, called micelles. The process is, of course, mainly entropy driven whereby there is an expulsion of water mole cules solvating the monomer in bulk solution through hydrophobic hydration on micellization. In some cases, exothermic enthalpy change associated with the micellization process adds to the entropic effect to make the process thermodynamically spontaneous. The onset of interfacial saturation can be experienced from a sharp break in the tensiometric isotherm and the interfacial tension remains more or less invariant with surfactant concentration thereafter. The interfa cial tension at the point of cmc ( γ cmc ) is a measure of efficacy of the surfactant to lower the interfacial ten sion of the solution in the form of a monolayer. The same of a surfactant or the blend of surfactants toward interfacial adsorption is quantified in terms of Gibbs surface excess (Γ max ) at the air/water interface and can be calculated using the least square slope of γ vs. logC plot near the cmc [4–10] as,

Γ max = −

Table 2. Critical micellar concentrationa (cmc) of ternary mixtures of surfactants at 303 K

103 cmc, mol dm–3 1.517 1.862 1.153 1.367 2.138 2.322 1.183 1.388 1.462 1.628 2.667 3.981 1.941 2.213

having lower cmc (higher tail length) and increases with increasing stoichiometric mole fraction of the shorter tail component in the mixture. Later, a thor ough discussion in this context will be addressed in the theoretical section. The cmc values of ternary mixture, DPC/CPC/TTAB have been presented in Table 2 showing that the cmc determined by tensiometry is slightly lower than that by conductometry. The lowest average cmc is for the combination of 1 : 5 : 2 whereas the highest one is for the ratio of 5 : 1 : 2.

1.690 (2.289) 1.260 (1.392) 2.230 (3.169) 1.286 (1.447) 1.545 (2.206) 3.324 (4.282) 2.077 (3.054)

dγ 1 Lt , 2.303nRT C~cmc d log C

(1)

where γ is the interfacial tension, R is the universal gas constant, T is the absolute temperature, and n is the average number of ions present in solution per surfac tant monomer. Here, n is taken as unity for pure sur factants, their binary and ternary mixtures, i.e., the contribution of counterions of cationic surfactants has been neglected. Its estimation from the knowledge of counterion condensation at the micellar interface seems to be a crude approximation due to the large discrepancy in radius of curvature, which significantly affects the counterion condensation between the micellar interface and the air/solution interface. C is the total surfactant concentration in molarity scale [6]. COLLOID JOURNAL

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The area of exclusion ( Amin ) per surfactant mono mer at complete saturation of air/water interface near cmc can be calculated using the equation, 18 (2) Amin = 10 , N AΓ max where the 1018 factor arises out of conversion from m to nm and NA is the Avogadro’s number. Γ max and Amin are expressed in mol m–2 and nm2 molecule–1 units, respectively. Conductometrically, we can calculate the fraction of counterion binding (g) at the micellar surface [4, 7– 10, 19]. From the ratio of post and pre micellar slopes of specific conductance ( κ) vs. [conec.] plot, we obtain the fraction of counterion dissociation (f) and from which g can be calculated as g = 1 – f. C20 is the surfactant concentration [39, 40] required to decrease the surface tension of water by 20 mN m–1 and negative logarithm of C20 is called pC20. Higher the surface activity of a surfactant, smaller is the C20. The cmc/C20 ratio dictates the preference of a surfactant

791

κ/μS cm–1 800 700 1 2 3 4

600 500 400 300 200 100 0

0.002 0.004 0.006 0.008 0.010 0.012 0.014 [Surfactant]/mol dm–3

Fig. 2. Conductometric plot of binary mixtures (1) CPC/TTAB, αCPC = 0.1; (2) DPC/TTAB, αDPC = 0.4; (3) CPC/DPC, αCPC = 0.75; and ternary mixtures (4) DPC/CPC/TTAB (0.125/0.625/0.250) at 303 K.

Table 3. Interfacial parameters and counter ion binding of pure and binary surfactant combinations at 303 K αCPC (I) or αDPC (II)

103 Πcmc, J m–2

106 Γmax, mol m–2

0.00 0.10 0.25 0.50 0.75 0.90 1.00

37.9 36.1 35.7 34.0 31.5 29.2 27.7

4.52 4.63 4.89 4.97 5.19 5.50 5.99

0.00 0.10 0.25 0.40 0.50 0.60 0.75 0.90 1.00

37.9 37.4 37.6 37.2 36.6 35.7 34.6 33.7 32.6

4.52 4.29 4.48 5.13 4.25 3.47 3.30 3.12 2.6

0.00 0.10 0.25 0.40 0.50 0.60 0.75 0.90 1.00

32.6 31.6 30.7 29.8 29.6 28.7 28.1 27.6 27.7

2.60 2.27 2.51 2.75 3.74 3.96 4.14 4.51 5.99

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Amin, nm2 molecule–1

g

CPC/TTAB (I) 0.37 0.645 0.36 0.583 0.34 0.564 0.33 0.557 0.32 0.602 0.30 0.615 0.28 0.601 DPC/TTAB (II) 0.37 0.645 0.39 0.710 0.37 0.639 0.32 0.606 0.39 0.592 0.48 0.579 0.51 0.542 0.53 0.500 0.64 0.554 CPC/DPC (I) 0.64 0.554 0.73 0.338 0.66 0.371 0.60 0.37 0.44 0.355 0.42 0.521 0.40 0.509 0.37 0.503 0.28 0.601 2010

o o −ΔGm, kJ mol–1 −ΔGad, kJ mol–1 pC20 cmc/C20

39.94 39.70 40.50 41.52 43.62 44.34 44.17

48.33 47.50 47.81 48.36 49.69 49.65 48.79

3.09 3.19 3.36 3.38 3.36 3.34 3.28

4.41 4.09 4.38 3.38 2.59 2.23 1.84

39.94 41.14 38.73 37.27 36.51 35.62 33.73 31.88 31.59

48.33 49.86 47.11 44.52 45.12 45.92 44.21 42.67 44.14

3.09 3.07 3.01 2.86 2.91 2.93 2.80 2.67 2.48

4.45 4.62 4.85 3.97 5.07 6.14 5.94 5.66 5.29

31.59 29.56 30.65 33.19 33.85 38.98 38.98 39.36 44.17

44.14 43.48 42.88 44.03 41.76 46.23 45.77 45.48 48.79

2.48 3.20 3.24 3.28 3.24 3.23 3.25 3.24 3.28

5.29 10.24 9.57 7.13 4.73 3.61 3.47 2.97 1.84

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CHANCHAL DAS et al. –ΔG °m/kJ mol–1

44 42 40 38 36 34 32

1 2 3

30 28 0

0.2

0.4

0.6

0.8

1.0 α1

o

Fig. 3. Variation of ΔGm as a function of composition in mixed binary systems at 303 K: (1) CPC/TTAB, (2) DPC/TTAB, and (3) CPC/DPC.

blend towards micellization compared to interfacial adsorption [39, 40]. Among the pure surfactants, TTAB leads to the lowest γcmc value (34.1 mN m–1) followed by DPC (39.4 mN m–1) and CPC (44.3 mN m–1). Table 3 shows that the Γ max value increases with increasing tail length, and eventually, Amin decreases. The pC 20 values also follow the same trend as Γ max as expected if the tensometric isotherms are linear in premicellar region. The increasing Γ max and pC 20 values with increasing tail length is a consequence of increasing surface activ ity arising out of increasing hydrophobicity of the longer tail surfactant. The lowering in Amin signifies greater compactness of the surfactantsaturated monolayer arising from increased van der Waals attraction among the surfactant tails. The cmc/C20 value increases with decreasing chain length dictating the preference of the shortertailed surfactant toward micellization over interfacial adsorption. The degree of counterion condensation onto the micellar inter face is the largest for TTAB and it decreases from CPC to DPC. The lowering of g with decreasing tail length within a homologous series may originate from a change in micellar size, the higher ‘g’ value for TTAB, however, signifies a greater zeta potential at the TTAB micellar surface. Among the CPC/TTAB mixtures, γ cmc and Γ max increase with increasing mole fraction of CPC; Amin being following a reverse trend. There is a significant lowering in ‘g’ of pure TTAB micelle on mere presence of CPC and it reaches a minimum at α CPC = 0.5 and increases thereafter, pC20 is also maximized at α CPC = 0.5. Within the DPC/TTAB blend, γ cmc reaches min

ima at α DPC = 0.4 and increases thereafter. Γ max is again maximized at α DPC = 0.4. The pC20 values, how ever, follow no regular trend. The variation in ‘g’ is, however, regular and decreases with increasing sto ichiometric proportion of DPC. In case of ternary sys tems presented in Table 4, the value of Γ max is maximal for 1 : 1 : 1 composition and the corresponding area is minimum indicating higher compactness of surfac tantsaturated monolayer in the system. γ cmc of CPC/DPC system slowly decreases and Γ max increases with increasing α CPC. The most impor tant observation is a drastic decrease in ‘g’ value for pure DPC in presence of CPC up to α CPC = 0.5. This indicates some kinds of decrease in surface charge density at the surfactant head group probably through the stacking interaction among the π electron cloud operating only when there exists a disparity in the chain length of the surfactant. This decreases the extent of counterion condensation destabilizes the micellar structure and can account for the concentra tion lag between interfacial saturation and bulk micel lization in these compositions. 3.3. Energetics of Micellization and Interfacial Adsorption With the help of g value, the standard free energy of micellization per mole of monomer unit Δ G mo for pure cations and their mixtures can be obtained as,

(

)

ΔG mo = (1 + g ) RT ln X cmc,

(3)

where X cmc is the cmc in mole fraction scale and the (1 + g) factor accounts for the part of free energy con tribution arising out of counterion condensation on micellar surface. The standard free energy of interfacial adsorption o at the air/water interface is obtained from the Δ G ad relation,

(

)

(

o ΔG ad = Δ G mo − Π cmc

Γ max

),

(4)

where Π cmc is the surface pressure at cmc, defined as Π cmc = γ H2O − γ cmc,, where γ H2O is the surface tension value of water (72 mN m–1 at 30°C) and γ cmc is the sur face tension value of solution at cmc point. The free energy changes associated with the micel o lization and interfacial adsorption ΔG mo , ΔGad for binary and ternary mixtures are given in Tables 3 and 4, respectively. The g values for binary mixtures of CPC/TTAB and DPC/TTAB are higher compared to CPC/DPC system. For ternary mixtures, the values of g and pC20 are maximal for higher α CPC and minimal for higher α DPC. The values of Δ G mo for CPC/TTAB and CPC/DPC systems increase with increasing α CPC COLLOID JOURNAL

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Table 4. Interfacial and thermodynamica parameters of ternary combinations of DPC, CPC, and TTAB at 303 K

a

α DPC α CPC αTTAB

103 Πcmc, J m–2

106 Γmax, mol m–2

Amin, nm2 molecule–1

g

0.125/0.250/0.625 0.125/0.625/0.250 0.250/0.125/0.625 0.250/0.625/0.125 0.333/0.333/0.333 0.625/0.125/0.250 0.625/0.250/0.125

34.6 32.2 35.4 30.4 33.2 33.5 32.1

5.76 5.21 4.45 4.96 5.90 4.90 5.34

0.29 0.32 0.37 0.33 0.28 0.34 0.31

0.391 0.566 0.338 0.529 0.416 0.321 0.328

o −ΔGmo , kJ mol–1 −ΔGad, kJ mol–1 pC20 cmc/C20

–36.44 –42.19 –34.12 –41.11 –37.42 –32.36 –34.1

42.45 48.37 42.08 47.24 43.05 39.20 40.12

3.31 3.33 3.24 3.27 3.24 3.06 3.13

3.43 2.68 3.92 2.41 2.67 3.81 2.81

The average errors in each of ΔGm and ΔGad are ±3%. o

o

(Fig. 3), indicating that the micellization process becomes more and more spontaneous with CPC, but the reverse trend is observed in case of DPC/TTAB system. The values of Δ G mo for ternaries indicate micellization becoming more favorable for greater o mole fraction of CPC. The values of Δ G ad follow the same trend to that of Δ G mo for CPC/TTAB, DPC/TTAB, and ternaries, but for CPC/DPC, it does not follow any regular trend. Considering the experimentally obtained average cmc from tensometric and conductometric methods, the following theoretical section is processed.

4.1. Clint Model This [14, 26] is a tool for predicting cmc of mixed surfactant systems cmc C with known proportion from the knowledge of individual cmcs of the compo nents. The involved equation is,

( )

( X ) ln ⎡⎣cmc α C X ⎤⎦ =1 (1 − X ) ln ⎡⎣cmc (1 − α ) C (1 − X )⎤⎦ σ 2

mix

)

n

(1 cmc ) = ∑ (α c

i

cmc i ).

(5)

i=1

Here, αi denotes the stoichiometric mole fraction of C ith component in solution. The terms cmci and cmc are the critical micellar concentrations of ith compo nent and the mixture, respectively. Any negative devi C ation in experimental cmc from that of cmc reflects an overall synergistic (attractive) interaction whereas overall antagonistic (repulsive) interaction is inferred from a negative deviation. It is observed from Table 1 that for binary cationic mixed micelles, the Clint’s cmc’s are lower than the average experimental cmc’s for DPC/TTAB and CPC/DPC systems, but higher for CPC/TTAB com bination indicating the presence of nonideality due to mutual interaction of the surfactants in the micelle. COLLOID JOURNAL

4.2. Rosen Model This model focuses on the adsorbed Langmuirian mixed surfactant film at the air/solution interface [28, 29] and is basically an optimization algorithm. A closer resemblance with the experimental area of exclusion ( Amin ) is obtained from the Amin values of the respective pure components using computational iteration which leads to the mole fraction of the component X 1σ and in teraction parameter at the interface among the compo nents βσ as the optimization parameters. The two equa tions involved in the iteration procedure are,

( )

4. THEORETICAL SECTION

(

Clint cmc’s overcome the average experimental cmc’s in case of ternaries (Table 2), and do not follow any rule with surfactant compositions.

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σ 2

mix

σ

0 1

1

1

σ

0 2

(6)

and σ

β =

ln ⎡⎣cmcmix α1 C10X σ⎤⎦

(1 − X )

σ 2

,

(7)

where cmcmix, C10 and C 20 are the molar concentrations of the mixture, pure surfactants 1 and 2 respectively at a fixed γ value corresponding to tensiometric cmc of pure component 1, α1 is the stoichiometric mole frac tion of surfactant 1 in solution. The βσ and Xσ values of the binary mixtures are pre sented in Table 5. The negative value of βσ for the cat ionic/cationic combinations indicates synergistic interaction. The interaction parameter decreases reg ularly with increasing α1 for CPC/TTAB and DPC/TTAB mixtures, but for CPC/DPC mixture, the values are highly irregular. Higher Xσ compared to α1 for all cationic/cationic systems indicates propen sity of surfactant 1 to preferentially adsorb at the

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air/water interface as compared to surfactant 2. These Xσ values for CPC/DPC system is very much higher compared to other two binaries reflecting more surface activity of those cationic/cationic combinations.

αCPC whereas these are irregular with other systems. The f2 values of CPC/TTAB and CPC/DPC deviate highly from unity indicating nonideality of the systems whereas it tends to unity for DPC/TTAB system.

4.3. Rubingh Model Applying iterative method, the following Rubingh equation [30] is solved to evaluate the micellar mole fraction ( X R ) of a surfactant in the mixed system and with the aid of X R, the molecular interaction parame ter (βR) can be found out. The equations are,

4.4. Motomura Model Motomura [31, 32] has considered mixed micelles as a macroscopic bulk phase from thermodynamic point of view. In this model, excess thermodynamic quantities are used to evaluate various energetic parameters. The fundamental equation is,

( X R ) 2 ln [cmc mix α1 cmc1X R ] =1 (1 − X R ) 2 ln [cmc mix (1 − α1) cmc 2 (1 − X R )]

(8)

ln [cmc mix α1 cmc1X R ] , (1 − X R ) 2

(9)

where cmc1, cmc2, and cmcmix are the critical micellar concentrations of surfactants 1, 2, and their mixtures, respectively, at a mole fraction α1. The activity coefficients of surfactants 1 and 2 in the mixed micelle ( fR1 and f R2 ) can be evaluated from the equations, 2 R f1 = exp ⎡⎣β (1 − X R ) ⎤⎦

(12)

ν 2α 2 ˆ = (ν1α1 + ν 2α 2 ) cmc. and cmc ν1α1 + ν 2α 2 Subscripts 1 and 2 denote surfactants 1 and 2, respectively, α and ν represent stoichiometric mole fraction and the number of dissociated ions by a sur factant in solution. The micellar mole fractions ( X Mo ) of a surfactant in the binary mixtures evaluated by Motomura equation are shown in Table 5. The mean activity coefficient of surfactant 1 in micelle ( f1) according to this model [33] is presented by the equation where Xˆ 2 =

and

βR =

ˆ )(∂cmc ˆ ∂Xˆ 2 ) , X Mo = Xˆ 2 − ( Xˆ1Xˆ 2 cmc T, P

(10)

and 2 R (11) f 2 = exp ⎡⎣β ( X R ) ⎤⎦ . The values of XR, βR, f1, and f2 for the binary mix tures are tabulated in Table 5. It shows that for cat ionic/cationic mixed systems, the micellar mole frac tion (XR) increases with increasing mole fraction of α1, but for CPC/TTAB and CPC/DPC systems, they are very much higher than the corresponding αCPC, but lower than that in case of DPC/TTAB mixture. XR val ues are higher for CPC/TTAB mixtures compared to DPC/TTAB mixtures. For CPC/DPC mixtures, the order of XR values is irregular. On comparing Xσ and X R values (Table 5), it can be shown that DPC/TTAB mixtures have much greater Xσ values than X R, indicating greater surface activity of the component DPC rather micellization in the mixed state. On the other hand, for CPC/TTAB mixed micelles, Xσ values are little bit lower compared to X R values (ex cepting α CPC = 0.1) indicating equal or to some extent higher priority in surface saturation and micelle forming properties. The values of βR of all mixtures are negative (excepting α CPC = 0.25 and 0.40 for CPC/DPC sys tems) indicating synergistic interaction. This conclusion can also be drawn from the lower experimental cmc in most of the cases of those mixtures than the Clint cmc. The f1 values of CPC/TTAB increase with increasing

ˆ cmc10 = f1X 1(Mo) Xˆ 1 cmc

(13)

and the f1 values of all binary mixtures are shown in Table 5. For CPC/TTAB and CPC/DPC, X Mo values are much higher than the stoichiometric mole fraction of α CPC. On the other hand, for DPC/TTAB, all the X Mo values are very much lower than α DPC. Motomura model is independent of the nature of surfactant and their counterions and predicts the micellar composition only. 4.5. SPB Model Another thermodynamic theory developed by Blankschtein et al. (SPB) [34–36] also predicts quan titatively the cmc, micellar composition, shape, and phase behavior on the basis of hydrophobic, struc tural, and electrical interactions between the binary components. Thus, Clint equation is written in the form

α1 1 − α1 1 = + , cmc mix f1cmc1 f 2cmc 2

(14)

where the term f is the activity coefficient of the sur factant in the mixed micelle and is expressed by the relations given below,

⎡β (1 − α * ) 2⎤ f1 = exp ⎢ 12 ⎥ kT ⎣ ⎦ COLLOID JOURNAL

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Table 5. Molecular interaction parameters of binary mixtures in aqueous medium at 303 K Rosen Model

Rubingh Model

αCPC (I) or αDPC (II)

Xσ

βσ

XR

0.10

0.43

–3.49

0.34

0.25

0.52

–3.48

0.50

0.62

0.75 0.90

βR

Motomura Model f1/f2

XMo

f1

–0.46

0.82/0.95

0.30

0.91

0.55

–1.21

0.78/0.69

0.60

0.81

–2.71

0.73

–0.88

0.94/0.62

0.73

0.99

0.74

–2.16

0.84

–1.16

0.97/0.43

0.88

0.98

0.84

–1.94

0.92

–1.40

0.99/0.31

–

0.94

CPC/TTAB (I)

DPC/TTAB (II) 0.10

0.37

–7.82

0.05

–0.89

0.45/1.00

0.07

0.29

0.25

0.40

–6.17

0.06

–0.09

1.08/1.00

0.08

0.82

0.40

0.43

–4.93

0.14

–0.18

0.87/1.00

0.07

1.75

0.50

0.45

–4.38

0.20

–0.19

0.88/0.99

0.10

1.75

0.60

0.47

–3.54

0.23

–0.16

1.10/1.01

0.20

1.26

0.90

0.65

–1.58

0.64

–0.32

0.96/0.88

0.76

0.82

0.10

0.73

–2.18

0.60

–1.56

0.78/0.57

0.12

7.31

0.25

0.76

–2.96

0.89

+0.33

1.00/1.30

0.64

3.13

0.40

0.81

–3.16

0.97

+1.16

1.00/3.02

–

–

0.50

0.80

–4.04

0.86

–1.51

0.97/0.33

–

–

0.60

0.89

–1.83

0.97

–0.01

1.00/1.01

0.78

1.67

0.75

0.97

–

–

–

–

0.77

1.96

0.90

0.99

–

–

–

–

0.91

1.73

CPC/DPC (I)

and

⎡β (α *) 2⎤ (16) f 2 = exp ⎢ 12 ⎥, ⎣ kT ⎦ where β12 is the predicted interaction parameter between surfactants 1 and 2, α * is the optimal micellar composition (denoted by X SPB, where the free energy of mixed micellization reaches its minimum value). The following equation,

(

)

⎛ α1cmc 2 ⎞ β12 (1 − 2α *) + ln α * = ln ⎜ ⎟ kT 1−α* ⎝ (1 − α1) cmc1 ⎠

(17)

is solved iteratively to obtain α* and β12 and by using these values, f can be calculated according to the equations (15) and (16). It is observed from Table 6 that X SPB increases with increasing α1 for all the binaries and the values are much lower than α1 in DPC/TTAB, but much higher than α1 in case of other two systems. X SPB is comparable with X R, X Mo or Xσ. In the CPC/TTAB combination, βSPB < βR (in most cases) and negative values of βSPB indi cate attractive interaction. In case of DPC/TTAB (ex COLLOID JOURNAL

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cepting α DPC = 0.9) and CPC/DPC systems, positive βSPB value denotes repulsive interaction between the sur factants. In case of all binary systems, the activity coeffi cients of component 1 are nearly unity. The cmc values of CPC/TTAB (excepting α CPC = 0.1) and DPC/TTAB mixtures obtained by the SPB method are lower than those obtained experimentally and more or less compa rable with those obtained by Clint method. Again, in case of CPC/DPC combination, the cmc values follow the re verse order (SPB > obs > Clint). There are more devia tions of cmc value found in case of CPC/DPC mixture particularly at lower mole fraction of CPC denoting the presence of nonideality for mutual interaction of am phiphiles in the micelle. This deviation of cmc is the lim itation of SPB theory. 4.6. Rubingh–Holland Model Theoretical treatment of surfactant mixtures of more than two components is very limited [37, 38]. The theory of Rubingh and Holland (RH) [37] has been properly tested to determine the micellar composition, activity coefficient and cmc of ternary system. The activity coef

796

CHANCHAL DAS et al.

Table 6. Micellar compositions (XSPB), interaction parameter [βSPB (kT)], activity coefficient (f), and cmc’s of binary mix tures at different stoichiometric compositions (X) at 303 K αCPC (I) or αDPC (II)

βSPB (kT)

XSPB

0.10 0.25 0.50 0.75 0.90

0.292 0.553 0.788 0.918 0.971

–0.146 –0.608 –1.962 –8.605 –27.221

0.10 0.25 0.40 0.50 0.60 0.90

0.022 0.064 0.122 0.172 0.238 0.652

0.035 0.060 0.058 0.050 0.069 –0.358

0.10 0.25 0.40 0.50 0.60 0.75 0.90

0.665 0.856 0.923 0.947 0.964 0.982 0.994

2.567 – – – – – –

f1 CPC/TTAB (I) 0.93 0.89 0.92 0.94 0.98 DPC/TTAB (II) 1.034 1.054 1.046 1.035 1.041 0.958 CPC/DPC (I) 1.334 2.330 1.650 1.486 1.356 1.536 1.578

ficients of surfactants in a binary mixed micelle can be es timated from the relations (10) and (11). In a multi ncomponent mixture, the activity coefficients fi, fj, fk, … of mixed micelle forming amphiphilic species i, j, k, … can be expressed by the RH equation [37] n

ln f i =

∑

j =1 ( j ≠i )

n

2 βij x j

+

ij

+ βik − β jk)x j x k,

(18)

j =1 k =1 (i ≠ j ≠ k)

where β ij denotes the net (pairwise) interaction between components i and j, and xj is the mole fraction of the jth component in the micelles; βik, β jk, and xk have similar significance. The Eq. (19) is valid at cmc. x i = α i C j f j x j ( C iα j f i ) ,

(19)

where the new terms, Ci and Cj represent cmc’s of the ith and jth components, respectively, and αi and αj are the mole fractions of the ith and jth components in the solution.

( )

0.99 0.83 0.30 – –

2.640/2.760/2.846 1.906/1.853/2.154 1.404/0.972/1.534 1.124/0.010/1.191 0.973 /–/1.050

1.000 1.000 1.001 1.001 1.004 0.859

3.950/3.934/3.931 4.688/4.529/4.513 5.539/5.330/5.297 6.182/6.033/5.991 7.171/6.980/6.895 12.054/11.59/12.588

3.112 – – – – – –

6.461/10.67/6.468 5.509/9.070/3.332 3.704/4.013/2.244 2.739/2.892/1.843 2.120/2.199/1.563 1.957 /–/1.273 1.696/–/1.074

the activity coefficients of f1, f2, and f3 for the ternary mixtures of DPC/CPC/TTAB system using the comput er controlled “successive substitution” method. By put ting the f value in Eq. (19), the cmc of the mixed micelle was found out. The mole fraction of the individual com

(

)

(

)

ponents X RH , the activity coefficient f RH , and the

j −1

∑ ∑(β

103cmc (mol dm–3) Obs/SPB/Clint

f2

The average interaction parameter β Rav values of bi nary mixtures of CPC/TTAB, DPC/TTAB, and CPC/DPC combinations, obtained from the equation of Rubingh (Eq. (9)), were used in equation (18) to evaluate

RH cmcRH were presented in Table 7. It is found that X DPC values are much higher than the stoichiometric mole RH fraction α DPC (excepting 6th combination) and X CPC values are fairly lower than α CPC, whereas most of the RH values are lower than α TTAB (exception lies in 4th, X TTAB 6th, and 7th sets of combinations) indicating DPC mainly forms the mixed micelles. In literature survey, RH theory has been applied in various combinations, namely, ion ic/nonionic/nonionic [7–10], ionic/ionic/nonionic [4, 41], ionic/ionic/ionic [42]. The activity coefficient of RH RH CPC ( f CPC ) is close to unity, whereas f DPC is moderately RH high compared to f TTAB indicating that CPC and DPC control the activity of the third component. The cmc val ues obtained by RH method are higher than those ob tained experimentally (excepting the 2nd combination). But in all cases, cmcClint is higher than cmcRH and cmcobs indicating the synergistic nonideal nature of the ternary

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Table 7. Micellar composition (XRH), component activity coefficient (fRH), and cmcRH in ternary mixtures of DPC/CPC/TTAB by RH method at 303 K and at different stoichiometric compositions (α)

α DPC α CPC αTTAB

X DPC X

0.125/0.250/0.625 0.125/0.625/0.250 0.250/0.125/0.625 0.250/0.625/0.125 0.333/0.333/0.333 0.625/0.125/0.250 0.625/0.250/0.125

0.574/0.117/0.414 0.830/0.008/0.162 0.450/0.033/0.517 0.713/0.013/0.274 0.700/0.032/0.267 0.547/0.116/0.336 0.741/0.093/0.166

RH

RH CPC

RH

X TTAB

system. cmcRH deviates from cmcobs due to the limita tions of RH theory. 5. OVERVIEW OF THE PRESENT WORK This paper reports the mixed micellization behav ior of three cationic surfactants with different tail lengths using tensiometry and conductometry from a thermodynamic viewpoint. The increasing cmc value with decreasing tail length signifies the dominant effect of hydrophobic interaction in dictating self aggregation behavior. The coherence of cmc values obtained by tensiometry and conductometry for pure and CPC/TTAB and DPC/TTAB binary combina tions signifies concurrence of interfacial saturation by surfactant monomer and their selfaggregation in bulk solution. A finite and significant lag in cmc deter mined by those two methods for CPC/DPC and ter nary system may arise out of a decrease in charge den sity at the micellar interface which may arise out of π π interaction among the pyridinium head groups of surfactants with dissimilar tail lengths. Negative devi ation in experimental cmc for the binary combinations again infers the van der Waals attractive interaction among the similarly charged surfactant head groups. The Rosen model predicts the interfacially adsorbed layer and interaction parameters. Rubingh, Motomura and SPB models, on the other hand, deal with the micellar phase. For CPC/TTAB system, these models predict more or less similar composition of micellar phase. The interaction parameters in micellar phase predicted by Rubingh model are all negative signifying synergistic interaction. With increasing stoichiometric proportion of CPC, its pro portion in monolayer and micellar phase increases. The increase in composition of CPC in interfacial layer is, however, less prominent than that in micellar phase. In DPC/TTAB system again, the Rubingh, Motomura, and SPB models predict similar composi tion in micellar phase. The variation in composition with changing stoichiometric composition is again more pronounced in micellar phase compared to monolayer. For CPC/DPC system, however, there is a discrepancy in micellar composition as predicted by COLLOID JOURNAL

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RH

f DPC f

RH CPC

RH

f TTAB

0.835/1.093/0.707 0.972/1.010/0.489 0.749/1.082/0.798 0.923/1.063/0.587 0.921/1.610/0.590 0.857/1.072/0.682 0.958/1.017/0.524

103cmc (mol dm–3) RH/Obs/Clint 1.791/1.690/2.289 1.236/1.260/1.392 2.492/2.230/3.169 1.301/1.286/1.447 1.833/1.545/2.206 3.453/3.324/4.282 2.707/2.077/3.054

Rubingh, Motomura, and SPB models. For ternary system, due to the presence of dissimilarity in nonpo lar tail and ionic head groups of three amphiphiles, they are incompatible among them. For this reason, the RH theory gives unexpected results. ACKNOWLEDGMENTS T.C. thanks Centre for Surface Science, Jadavpur University, for giving an opportunity to perform exper imental works. The authors are grateful to the Univer sity Grant Commission, New Delhi, for financial assistance extended through DRS Projects. REFERENCES 1. Scamehorn, J.F., in Phenomena in Mixed Surfactant Systems, ACS Symp. Ser., 1989, vol. 311, p. 1. 2. Holland, P.M., in Mixed Surfactant Systems, ACS Symp. Ser., 1992, vol. 501, p. 31. 3. Hill, R.M., in Mixed Surfactant Systems, Surfactant Sci. Ser., 1993, vol. 46. 4. Chakraborty, T., Ghosh, S., and Moulik, S.P., J. Phys. Chem. B, 2005, vol. 109, p. 14813. 5. Ghosh, S. and Chakraborty, T., J. Phys. Chem. B, 2007, vol. 111, p. 8080. 6. Chakraborty, T. and Ghosh, S., Colloid Polym. Sci., 2007, vol. 285, p. 1665. 7. Ghosh, S., J. Colloid Interface Sci., 2001, vol. 244, p. 128. 8. Ghosh, S. and Moulik, S.P., J. Colloid Interface Sci., 1998, vol. 208, p. 357. 9. Moulik, S.P. and Ghosh, S., J. Mol. Liq., 1997, vol. 72, p. 145. 10. Das Burman, A., Dey, T., Mukherjee, B., and Das, A.R., Langmuir, 2000, vol. 16, p. 10020. 11. Basu Ray, G., Chakraborty, I., Ghosh, S., et al., Lang muir, 2005, vol. 21, p. 10958. 12. Basu Ray, G., Chakraborty, I., Ghosh, S., and Moulik, S.P., J. Colloid Interface Sci., 2007, vol. 307, p. 543. 13. Gracia, C.A., GomezBarreiro, S., GonzalezPerez, A., et al., J. Colloid Interface Sci., 2004, vol. 276, p. 408. 14. Clint, J.H., Surfactant Aggregation, New York: Blackie, Chapman and Hall, 1992.

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29. Rosen, M.J. and Dahanayake, M., in Industrial Utiliza tion of Surfactants. Principles and Practice, Champaign: AOCS, 2000, p. 28. 30. Rubingh, D.N., in Solution Chemistry of Surfactants, K. Mittal, Ed., New York: Plenum, 1979, vol. 1, p. 337. 31. Motomura, K., Yamanaka, M., and Aratono, M., Col loid Polym. Sci., 1984, vol. 262, p. 948. 32. Motomura, K., Aratono, M., Ogino, K., and Abe, M., in Mixed Surfactant Systems, K. Ogino and M. Abe, Eds., New York: Marcel Dekker, 1993, p. 99. 33. Aratono, M., Villeneuve, M., Takiue, T., et al., J. Col loid Interface Sci., 1998, vol. 200, p. 161. 34. Sarmoria, C., Puvvada, S., and Blankschtein, D., Langmuir, 1992, vol. 8, p. 2690. 35. Puvvada, S. and Blankschtein, D., in Mixed Surfactant Systems, ACS Symp. Ser., 1992, vol. 501, p. 96. 36. Shiloach, A. and Blankschtein, D., Langmuir, 1998, vol. 14, p. 4105. 37. Holland, P.M. and Rubingh, D.N., J. Phys. Chem., 1983, vol. 87, p. 1984. 38. Graciaa, A., Ben Ghoulam, M., Marion, G., and Lachaise, J., J. Phys. Chem., 1989, vol. 93, p. 4167. 39. Tsubone, K. and Ghosh, S., J. Surfactant Detergent, 2003, vol. 6, p. 225. 40. Tsubone, K. and Ghosh, S., J. Surfactant Detergent, 2004, vol. 7, p. 47. 41. Dar, A.A., Rather, G.M., Ghosh, S., and Das, A.R., J. Colloid Interface Sci., 2008, vol. 322, p. 572. 42. Basu Ray, G., Chakraborty, I., Ghosh, S., et al., J. Phys. Chem. B, 2007, vol. 111, p. 9828.

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