Langmuir 1994,10, 109-121
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Salt Effects on Intramicellar Interactions and Micellization of Nonionic Surfactants in Aqueous Solutions Teresa R. Carale, Quynh T. Pham, and Daniel Blankschtein' Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received August 16, 1993. I n Final Form: October 14, 1 9 9 P The effects of adding LiC1, NaC1, KC1, KBr, and KI to aqueous solutions containing the alkyl poly(ethylene oxide), CiEj, nonionic Surfactants, C12E6, CIZEE,and CIOE~, have been investigated through a combined theoretical and experimental approach. The theoretical studies involve (1) the generalization of a recently developed molecular theory of micellization to characterize and quantify salt effects on intramicellar interactions and the utilization of this theory to predict and rationalize (i) the various contributions to the free energy of micellization, (ii) the critical micellar concentration (cmc),and (iii) the salt constants and (2)the development of a new description of the interactions between the poly(ethy1ene oxide), PEO, hydrophilic (Ej) chains which are "grafted" at one end to the micellar-core surface. The theoretical description in (2)accounts explicitlyfor the chainlike character of PEO and utilizes a rotational isomericstate Monte Carlo approachto generatethe average conformationalcharacteristicsof the "grafted" PEO chains. The experimental study involves surface tension measurements, conducted at 25 OC, which are utilized to determine cmc's as a function of salt type and concentration, as well as the salt constants of the five salts examined. The theoretically predicted cmc's and salt constants are in very good agreement with those determined experimentally. The theoretical and experimental results are consistent with the observation that the salts examined have a much more pronounced effect on the hydrophobic alkyl (CJ moieties than on the hydrophilic PEO (Ej) moieties, primarily by inducing a decrease in the solubility of the alkyl moieties when added to aqueous solutions. This, in turn, results in both a lowering of cmc's and surface tensions of the CiEj aqueous salt solutions examined.
I. Introduction The addition of salts to aqueous surfactant solutions may result in a modification of both intramicellar and intermicellar interactions. Consequently, solution properties, such as the critical micellar concentration, the surfactant concentration at which micelles first begin to form, as well as the phase behavior of the surfactant solution, may be modified significantly upon the addition of salts. Accordingly, a fundamental understanding of how salts affect the behavior of aqueous surfactant solutions may lead to a more effective utilization of these systems in various practical applications. In this paper, we examine how the addition of salts affects the behavior of aqueous solutions of nonionic surfactants belonging to the alkyl poly(ethy1ene oxide), CiEj, family. These surfactants are useful model systems because by varying the number of carbon atoms, i, in the hydrophobic alkyl tail, or the number of ethylene oxide units, j, in the hydrophilic PEO head, a systematic study of the effects of salts on these surfactants can be conducted. Specifically, experiments were performed using the surfactants Cl2E6, C12E8, and C&6. The salts were chosen in order to identify the effects of varying both the anion and the cation by using KC1, KBr, KI, and LiC1, NaC1, KC1, respectively. Experimental studies of salt effects on the critical micellar concentration of nonionic surfactants, primarily of the alkyl (Ci) or alkylphenoxy (CiPh) poly(ethy1ene oxide), Ej, type in aqueous solutions, have been conducted in the past.'-' These studies indicate that most salts lower
* To whom correspondence should be addressed. *Abstract published in Advance ACS Abstracts, December 1, 1993. (1)Becher, P. J. Colloid Sci. 1962, 17, 326. (2) Hsiao, L.; Dunning, H. N.; Lorenz, P. B. J. Phys. Chem. 1966,60, 657. (3) Schick, M. J. J. Colloid Sei. 1962,17,801. Schick, M. J.; Atlas, S. M.; Eirich, F. R. J. Phys. Chem. 1962,66, 1326.
the critical micellar concentration (cmc),and, in most cases, the salt effect on the cmc follows the relation In cmc = In cmc,,
- k,m,
(1) where k, is referred to as the salt constant and m, is the salt molality. The observed salt effects were correlated to the Hofmeister series, which is derived from studies on the salting-out of proteins.8 It was also found that the salt effects are virtually independent of the number of ethylene oxide units, j , in the hydrophilic head.6 Theoretically, salt effects on the cmc of nonionic surfactants have been investigated in the context of standard thermodynamic treatments, including the massaction model9of micellization, which assumes equilibrium between monomers and monodisperse micelles, and the phase-separation model10 of micellization, which considers micelles as another phase and thereby describes micellization as a phase-separation phenomenon. In both models, the activity coefficient of the surfactant monomer is expressed as the sum of two contributions, one resulting from the hydrophobic (tail) moiety and the other from the hydrophilic (head) moiety. The effect of the salts on the activity coefficient of the surfactant tail can then be estimated from solubilities of these alkanes in aqueous salt solutions. More specifically, the empirical solubility relation of the series methane to butane in aqueous NaCl solutionsll was used to estimate the solubility of higher alkanes, and thereby estimate the activity coefficient of a surfactant monomeric tail in aqueous NaCl solutions. The contribution from the head was then deduced from (4) Schott, H.; Suk Kyu Han J . Pharm. Sci. 1976,65,976. Jpn. 1977,60,1690. (5) Nishikido, N.; Matuura, R. Bull. Chem. SOC. (6) Ray, A.; Nemethy, G. J . Am. Chem. SOC.1971,93,6787. (7) Zourab, Sh. M.; Sabet, V. M.; Abo-El Dahab, H. J. Dispersion Sci. Technol. 1991, 12, 25. (8)Hofmeister Arch. Exp. Pathol. Pharmakol. 1888,24, 247. (9) Mukerjee, P. J. Phys. Chem. 1966,69,4038. (10) Gordon, J. E. J. Phys. Chem. 1970, 74, 3823. (11) Morrison, T. J.; Billett, F. J . Chem. S O C .1962, 3819.
0743-7463/94/2410-0109$04.50/00 1994 American Chemical Society
110 Langmuir, Vol. 10, No. 1, 1994
the experimentally measured variation of the cmc with salt concentration, and the independently estimated activity coefficient of the tail. The utilization of the massactions and the phase-separation1°models of micellization yielded conflicting results regarding the salt effects attributed to the heads. Specifically,using the mass-action model, the effect of increasing the concentration of NaCl on the activity coefficient of the poly(ethy1ene oxide), Ej, head was found to be negligible as compared to the effect on the tail? while the use of the phase-separation model showed a very significant contribution of the heads6 In this paper, a recently developed molecular theory of micellization12is implemented and generalized to predict salt effects on the micellization of nonionic surfactants belonging to the CiEj series. By more systematically accounting for the various molecular contributions to the micellization process, as well as working with free-energy changes rather than with activity coefficients, the molecular theory of micellization circumvents the existing controversybetween the mass-action and phase-separation treatmetns of salt effects on micellization. The McDevitLong theory,13utilized in earlier studies to rationalize salt effects on the activity coefficients of hydrocarbon tails,g is utilized in this paper in a slightly different context to quantitatively capture the effect of salts on the formation of the micellar hydrocarbon cores. Macroscopic determinations of salt effects on interfacial tensions are used to capture the corresponding effects at the micellar level, that is, on the micellar core-aqueous salt solution interfacial free energy. An additional generalization of the original theory12 consists of explicitly incorporating the chainlike character of the short PEO, Ej, heads. Indeed, instead of representing the PEO heads as effective hard disks residing at the micellar-core surface, as was done in the original theory,12wemodel that PEO heads as short polymer chains “grafted” at one end to the micellar-core surface. In the original theory,12 the contribution of the hydrophilic heads to the micellization free energy was described in terms of a steric repulsive term. In the theory presented in this paper, the contribution of the PEO heads is described in terms of a free-energy change associated with transferring the polymeric PEO heads from the aqueous salt solution to the micelle head layer, which is modeled as a solution of “grafted”PEO chains. The magnitude of the PEO head transfer free energy depends on several factors, including the number of EO units in the head, the shape of the micellar core, and the quality of the solvent, which, in turn, may be affected by the type and concentration of added salt, as well as by changes in the solution temperature. The theoretical predictions presented in this paper are compared to results from a systematic experimental cmc study at 25 “C, using the surface-tension method, on the effects of various salts (LiCl, NaC1, KC1, KBr, KI) on the cmc’s of the nonionic surfactants C&6, C12E8, and c1(& in aqueous solutions. The remainder of the paper is organized as follows. Section I1describes the modifications made to the recently developed molecular theory of micellization in order to account for the salt effects on the various contributions to the free energy of micellization, gmic, as well as on the cmc. Section I11describes the experimental determination of the cmc’s using the surface-tension method. Section IV presents, discusses, and compares the experimental (12) Puwada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. Blankschtein, D.; Puwada, S. MRS Symp. h o c . 1990,177, 129. (13) McDevit, W. F.; Long,F.A. J . Am. Chem. SOC.1952, 74,1773.
Carale et al. and theoretical results which include (i) surface tensions, (ii) the various contributions to gmio(iii) cmc’s, and (iv) salt constants, for the CiEjaqueous salt solutions examined. Finally, Section V presents some concluding remarks. 11. Theory The theoretical description presented below is based on a recently developed molecular-thermodynamic theory of micellization,12which was generalized to predict cmc’s of CiEj nonionic surfactants in aqueous solutions in the presence of various salts. This theoretical approach blends a molecular theory of micellization,12which incorporates the essential intramicellar interactions responsible for micelle formation, with a thermodynamic free-energy description14of the collective macroscopic phase behavior of the micellar solution. The molecular-thermodynamic theory of micellization has been successfullyutilized to predict micellar properties of aqueous solutions of single nonionic surfactants belonging to the CiEj and glucoside families,’2 as well as mixed (nonionic-nonionic, ionic-nonionic, and anioniccationic) surfactants16asa function of surfactant molecular architecture, surfactant concentration and composition, temperature, and urea concentration’6 (an additive that modifies the solvent quality). The predicted properties, including the cmc, the micellar shape, micellar size distribution and its characteristics, and the coexistence curve (including the critical surfactant concentration for phase separation) were found to compare favorably with available experimental data. The thermodynamic freeenergy description has also been successfully utilized to describe the phase behavior of aqueous solutions of zwitterionic surfactants, in the presence of added elect r o l y t e ~and ~ ~ urea,18 over a wide range of surfactant concentrations and temperatures. More recently, the theoretical framework has also bees generalized to predict quite accurately the crossover surfactant concentration (marking the transition from the dilute to the semidilute as well as to predict surface micellar solution regime~),’~ tensions,m of aqueous solutions of CiEj surfactants. The following discussions will describe the molecular theory of micellization, emphasizing each of the contributions to the free energy of micellization, and the salt effects on these contributions, Adiscussion of the method for estimating the cmc from the free energy of micellization, in the context of the molecular theory of micellization, will then follow. A. Molecular Theory of Micellization. Micellization in aqueous solutions involvesthe aggregation of monomers such that the hydrocarbon tails cluster together, with the heads oriented toward the aqueous solution forming a “polar”shell. This aggregation behavior limits the contact between the tails and the aqueous solution. There is no direct method to calculate the free-energy change involved in micellization. Therefore, models are (14) Blankschtein, D.; Thurston, G. M.; Benedek, G.B. Phys. Rev. Lett. 1985,54,955. Blankschtein,D.; Thurston, G. M.; Benedek, G. B. J . Chem. Phys. 1986,85,7268. (15) Puwada, S.P.; Blankschtein,D. J. Phys. Chem. 1992,96,5567. Puwada, S. P.;Blankschtein, D. J. Phys. Chem. 1992,96,5579. Puwada, S . P.; Blankschtein,D. ACS Symp. Ser. 1992, No. 501,96. Sarmoria, C.; Puwada, S. P.; Blankschtein, D. Langmuir 1992,8,2690. (16) Briganti, G.; Puwada, S.;Blankschtein, D. J. Phys. Chem. 1991, 95, 8989. (17) Huang, Y. X.;Thurston,G.M.; Blankschtein,D.; Benedek, G. B. J. Chen. Phys. 1990,92, 1956. (18) Carvalho, B. L.; Briganti, G.; Chen, S. H. J. Phys. Chem. 1989, 93, 4282. (19) Carale, T. R.; Blankschtein, D. J.Phys. Chem. 1992,96,459. (20) Nikaa, Y. J.; Puwada, S.; Blankschtein, D. Langmuir 1992, 8, 2680.
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typically devised to simulate micelle formation. The molecular theory of micellization,12 which generalizes earlier micellization descriptions,21922is based on the wellknown thermodynamic principle that the free energy is a state function, and, therefore, an overall reversible process, like micellization, can be replaced by a series of simpler, well-characterized, reversible processes connecting the initial state with the final state. Thus, the chosen initial state for the formation of an n-mer is n monomers dispersed in solution, while the final state is an n-mer in the same solution. As stated earlier, we have generalized the molecular theory of micellization12to incorporate the effects of salts on the free energy of micellization, gmic(n,lc,S).Note that gmic(n,lc,S) is the free-energy change per surfactant molecule associated with forming a micelle at a fixed position in the solution, having aggregation number, n,core-minor radius, lc, and shape, S, from n monomers dispersed in solution. In the context of this theory, a conceptual thought process was introduced, whereby the micellization of the surfactant molecules (CiEjin the present case) can be broken down into a series of steps, each reflecting an important physicochemical factor involved in micellization. Specifically, the following steps are involved: (1) breaking the surfactant head-tail bond, (2) transferring the tail from the aqueous salt solution (s) to bulk hydrocarbon (hc), characterized by a free-energy contribution, ga/hc,(3) forming a hydrocarbon core-aqueous salt solution interface, characterized by a free-energy contribution, g,, (4) packing the hydrocarbon chains within the micellar core with one end of each chain attached to the interface, characterized by a free-energycontribution, gpack, ( 5 ) re-forming the head-tail bond (where the free energy of forming the bond is assumed to be equal in magnitude and opposite in sign to that of breaking it, and therefore the free-energy changes in steps 1 and 5 cancel altogether), and (6) transferring the heads from the aqueous salt solution (hs) to the micelle head layer (hl), where they are subsequently "grafted" at one end to the micellar-core surface, characterized by a free-energy contribution,gi,+. Note that since the heads are nonionic, we have assumed that the electrostatic free-energy contribution, gelec,is vanishingly small, that is, we have set gelec= 0 for these surfactants. Accordingly, the free energy of micellization can be expressed as follows (2) gmic e gs/hc + gu + gpack + gha/hl The four free-energy contributions to gmic shown in eq 2 will be briefly discussed below, with the aim of characterizing and quantifying the modes by which the presence of salts in the solution would affect each of these contributions. a. Transfer of the Hydrocarbon Tails from the Aqueous Salt Solution to Bulk Hydrocarbon. The hydrophobic driving force for micellization can be viewed operationally as the free-energy change involved in the transfer of the hydrocarbon tails, the hydrophobic moieties of the surfactants, from the aqueous salt solution (s) to bulk hydrocarbon (hc). This transfer free-energy contribution, gs/hc, can be estimated from solubility data of hydrocarbons in aqueous salt solutions. Specifically
temperature. However, there is very little available solubility data of hydrocarbons in aqueous salt solutions, primarily because of their very low solubility as compared to that (which is already low) in pure water. In general, the low solubility of hydrocarbons in aqueous solutions can be rationalized in terms of the relatively large work contribution required to create a cavity in order to accommodate the nonpolar hydrocarbon moiety. This work reflects the strong attractive forces (particularly of the hydrogen-bond type) that exist between water molecules, which have to be weakened in order to create the cavity. The addition of salts to water increases this already large work contribution even more, because the polarizing power of the ions binds water molecules and ions more strongly to each other, thus resulting in stronger solventsolvent interactions, where the ions are considered part of this "hypersolvent". This low solubility of hydrocarbons in aqueous salt solutions makes solubility measurements difficult, hence the absence of available experimental solubility data. In view of this, the transfer of the surfactant tails from the aqueous salt solution ( 8 ) to bulk hydrocarbon (hc), a reversible thermodynamic process, was decomposed into two transfer steps: (1) the transfer of the tails from the aqueous salt solution (8) to water (w), with a corresponding free-energy change,g,/,, and (2) the subsequent transfer of the tails from water (w) to bulk hydrocarbon (hc),with a correspondingfree-energychange, gw/hc. Accordingly, &/hc can be expressed as gs/hc gs/w + gw/hc (4) The free-energy change corresponding to the second transfer step, gw/hc, can be estimated from available solubility data of hydrocarbons in pure water.23 Specifically (5) gw/hc = kT In sw where sw is the solubility of the hydrocarbon tails in pure water. Solubility data23were utilized to arrive at a relation betweengw/hcand the number of carbon atoms in the alkyl tail, i.12 Specifically, at T = 25 "C (298 K), one obtains
gw/hc(i)= -(2.02 + 1.49i)kT (6) The free-energy change involved in the first transfer step, galw,can be estimated using the McDevit-Long theory.13 Specifically lim
d@,/,)
--
- vhc(v80
-
where ss is the solubility of the hydrocarbon in the salt solution, k is the Boltzmann constant, and Tis the absolute
(7) dC, BO where c h c is the concentration of the hydrocarbon tails in the aqueous salt solution, C,is the molarity of the salt, v h c is the volume of a hydrocarbon tail, V, is the molecular volume of pure (liquid) salt, V," is the partial molecular volume of salt at infinite dilution, and BO is the compressi bility of water (=4.52 X lo4 b a r ' at 25 "C)." Note that eq 7 is valid in the limit of low solute concentrations and, therefore, is particularly appropriate for hydrocarbons in aqueous salt solutions. In deriving eq 7, it was assumed that the only role of the solute molecules (the hydrocarbon tails in this case) is to modify the ion-water interactions by occupying volume. Building on this simple concept, it was shown13 that the excess work done against the ionsolvent forces upon the introduction of the nonpolar solute volume, vhc, into an aqueous salt solution having low salt molarity, is proportional to the volume change which occurs
(21) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. (22) Nagarajan,R.; Ruckenstein, E. J . Colloid Interface Sci. 1977,60, 221; 1979, 71, 580.
(23) Abraham,M.H.J. Chem. SOC.,Faraday Trans. 1 1984,80,153. (24)Handbook of Chemistry and Physics; We&, R. C., Ed.; CRC Press: Boca Raton, FL, 1988.
(3)
ch~-.OG+
~~
Carale et al.
112 Langmuir, Vol. 10, No. 1, 1994 Table 1. Estimated Values of Liquid Salt Volumes salt
v. (A3)
vao (A3)
V,”- v,(A31
LiCl NaCl KC1 KBr KI
35.08 36.63 52.87 61.54 78.77
28.22 27.62 44.40 56.05 74.98
-6.86 -9.01 -8.47 -5.49 -3.79
upon mixing (liquid) salt and water. Equation 7 indicates that the contraction in the total solution volume upon mixing salt and water, which results in an increase in internal pressure, makes it more difficult to “insert” the volume of the nonpolar solute. Equation 7 can be expressed in terms of salt molalities, m,, by relating m, to C,. For this purpose, we examined experimentallymeasured%specificgravities of the aqueous salt solutions considered in this paper. We found that for the range of salt concentrations examined, m, varies linearly with C,, that is, m, = KC,,with K equal to 1.068 (for LiCl), 1.068 (for NaCl), 1.104 (for KCl), 1.124 (for KBr), and 1.180 (for KI). experimentally that the free energy It has been of transfer, g,/,, varies linearly with salt concentration over a wide range of salt concentrations. Utilizing this fact, an integration of eq 7, using the relation m, = KC,, yields
In order to calculate the free energy of transferring a hydrocarbon tail from the aqueous salt solution to the micellar core, it is necessary to define precisely which part of the surfactant molecule constitutes the tail and which part constitutes the head. In our considerations, the methylene (CH2) group nearest to the head in a hydrocarbon chain composed of i carbon atoms is considered part of the head, since it is in direct contact with the aqueous salt solution. Therefore, this methylene group is not explicitly “transferred“ from the aqueous salt solution to the micellar core. The tail is thus defined as the hydrocarbon chain containing (i - 2) CHZgroups and one terminal CH3 group, that is, a total of (i - 1)carbon atoms, while the head includes the last CH2 group, the j ethylene oxide units, and the terminal hydroxy (OH) group. The volume of the hydrocarbon tail, v h c , containing (i - 1) carbon atoms can be estimated using the empirical relation introduced2l by Tanford, Vh, = 27.4 + 26.9(i - 11, in A3. A compilation of data has been done26and is reported in the third column of Table 1. Note that values of V , cannot be measured (except at temperatures above the melting point of the salt) and, therefore, need to be estimated as discussed below in section IVBl (see also the second column in Table 1). b. Formationof a HydrocarbonCore-Aqueous Salt Solution Interface. The next step in the micellization thought process is the formation of the hydrocarbon coreaqueous salt solution interface, which represents a freeenergy cost to micellization. The resulting interfacial freeenergy change per monomer is estimated as follows12
vSo
g, = uo( 1 - T)(a (S- 116
-a,)
where uo is the interfacial tension between bulk hydro(25) Lange, N. A. Handbook of Chemistry, loth ed.; McGraw-Hill: New York, 1967; pp 1174-1179. (26) Millero, F. J. In Water and Aqueolls Solutions: Structure, Thermodynamics and Transport Processes; Home, R. A., Ed.;Wiley Interscience: New York, 1972; p 519.
Table 2. Variation in uo (Dodecane-Water) and .g with Salt Molality for Various Salts salt
LiCl NaCl KCl KBr KI
duddm. (dyn/(cm m)) 1.56 1.41 1.37 0.86 -0.07
CIZ& 0.092 0.083 0.081 0.051 -0.004
C1zE.g 0.103 0.093
CI& 0.091 0.082
0.090
0.080
0.066 -0.005
-0.004
0.050
carbon and the aqueous salt solution, 6 is the Tolman distance, a measure of the interfacial thickness [6 is estimated to be 2.25 A for CIZand 1.88 A for Clo and is assumed to be independent of salt concentration (see ref 12 for details)], a is the interfacial area per monomer [ = s v h J & , where, as stated earlier, S is the shape factor (3 for spheres, 2 for rods, and 1 for disks or bilayers), and 1, is the micellar core-minor radius], and a0 is the screened interfacial area per monomer which corresponds to the cross-sectional area of a fully-stretched hydrocarbon tail (a0 = VhJlmax = 21 A2,where211,= 1.54 1.265(i - 11, in A, is the fully extended length of a hydrocarbon tail having (i - 1) carbon atoms). The addition of most salts (except, for example, iodide) to pure water increases the (organic-aqueous) interfacial free energy, as reflected by the increase in the (organicaqueous) interfacial tension as the salt concentration is increased.z7 The low dielectric constant of the hydrocarbon medium drives the ions deeper into the aqueous phase where the ions can be surrounded by polarized water molecules. Therefore, the polarizing power of the ions makes it free energetically more favorable for them to be farther away from the interface, resulting in an apparent repulsion (desorption or negative adsorption) between the interface and the ions. This desorption of the ions from the organic-aqueous interface can be correlated to the change in the interfacial tension, UO, as a function of the salt activity, a, = Tsms,where ysis the activity coefficient of the salt in water, and m, is the salt molality. Indeed, it can be seen from the Gibbs adsorption equation28
+
that a negative relative salt adsorption onto the interface, I’JW),which is observed for most salts, except iodides, corresponds to apositiue gradient of the interfacial tension with salt activity, and, correspondingly, with salt concentration (molality). In general, the higher the ion surface charge density (the smaller the bare ion radius), the more polarizing it is and, consequently, the greater the desorption from the interface (more negative rll(w) values). Accordingly, as indicated by eq 10, the higher the ion surface charge density, the larger the magnitude of duo/ d(ln a,) duo/d(ln m,). The salt concentration dependence of the hydrocarbon-water interfacial tension, UO,at 25 “C has been determinedz7 for dodecane and decane and is reported for dodecane in the second column of Table 2. Indeed, Table 2 indicates that, for the chloride salts, duoldm, and, consequently, duo/d(ln m,), increase as the cation size decreases (or as the surface charge density increases), that is, in the order KCl < NaCl < LiC1. Similarly, for the potassium salts, daoldm, increases as the anion size decreases (or as the surface charge density increases), that is, in the order KI < KBr < KCl. Note
-
(27) Aveyard, R.;Saleem, S. M. J. Chem. Soc., Faraday Tram. 1 1976, 72,1609. (28)Hiemenz, P. C. Principles of Colloid and Surface Chemistry,2nd ed.; Marcel Dekker: New York, 1986; p 391.
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Langmuir, Vol. 10, No. 1, 1994 113
that, in the case of KI, there is even a weak adsorption a t the interface, such that duoldm, is slightly negative for this salt. Experimentally, it was observed27 that, for the salts examined in this paper, the interfacial tension is a linear function of salt concentration up to m, = 1 m. In the present studies, however, salt concentrations as high as 3 m were examined. It should be noted that the surface tensions of aqueous salt solutions have been found24to increase linearly with salt molality for concentrations of up to 4 m. Therefore, in the absence of available experimental data, we have assumed that the linearity in the variation of the hydrocarbon-aqueous salt solution interfacial tension with salt concentration observed in the studies reported in ref 27 extends to higher salt concentrations (up to 3 m), similar to the observed24linearity in the variation of the aqueous salt solution surface tension with salt concentration. With this in mind, the following relation between u, and me is obtained
An expression for u~(~,salt) as a function of the hydrocarbon chain length can be derived from available experimental uO(,,,dt) data at 25 0C.29 Specifically, for hydrocarbons composed of (i - 1)carbon atoms, a linear regression of the experimental data contained in ref 29 yields (in dyn/cm) uO(aosalt) = 48.673
+ 0.311(i - 1)
(12)
The values of uo determined from eq 11,using eq 12 and duo/dmllvalues from Table 2, can then be utilized in eq 9 to estimateg,. It should be noted that there is very little perceivable difference between the salt effects on uoas the alkane chain length is varied from CIOto C12.27 In view of this, we have assumed that the duoldm, values for dodecane (C12) reported in Table 2 can be utilized for both the C11 and Ce tails corresponding to the surfactants C12E6, C12E8, and C10E6 examined in this paper. c. Packing of the Hydrocarbon Tails in the Micellar Core. The free-energy contribution, &?pack,arises from the loss of conformational degrees of freedom of the tails upon anchoring one end of each tail at the micellar core-aqueous solution interface.12 As discussed in section IIAa, only (i - 1)carbon atoms are considered part of the tail, and consequently, only (i - 1) carbon atoms were included in the packing calculations. This step in the micellization process is independent of salt conditions, since it focuses exclusively on the environment within the micellar core. The corresponding repulsive contribution, gp&, is calculated by using a single-chain mean-field a p p r o ~ i m a t i o n .The ~ ~ calculation involves the use of the rotational isomeric state approximation to generate a large number of conformations of a single tail inside the micellar core. Typically, the evaluation of gp& needs to be performed numerically. Recently, however, the numerically generated values of &!pack were fitted to a secondorder polynomial of the form31 gpack = A2(lJlm-)2 + Al(lJl-1 + A0 (13) The values of the coefficients Ao, AI, and A2 have been (29) Aveyard, R.; Briscoe, B. J.; Chapman, J. J.Chem. SOC.,Faraday Tram. 1 1972, 6 8 , l O . (30) Ben-Shad, A.;Szleifer, I.; Gelbart, W. M. J.Chem. Phys. 1985, 83,3597. Szleifer,I.;Ben-Shad, A.; Gelbart, W. M. J.Chem.Phys. 1985, 83,3612. Gruen, D.W.R. J.Phys. Chem. 1985, 89, 146, 153. (31) Naor, A.; Puwada, S.;Blankschtein,D. J. Phys. Chem. 1992,96, 7830.
tabulatedgl for various micellar shapes and for hydrocarbon tails having i = 6 to 16 carbon atoms, d. Formation of the Grafted PEO Micelle Head Layer. The last contribution which may be affected by the presence of salts is the free-energy change, g b / N , due to the transfer of the PEO, Ej, heads from the aqueous salt solution (hs) to the micelle head layer (hl), and the subsequent “grafting” of the PEO head chains at one end to the micellar-core surface. In the original molecular theory of micellization,12this contribution was modeled (for the no salt case) using an ideal-localized monolayer, that is (14) where ah is the average cross-sectional area of the head and a is the interfacial area per monomer. In essence, in deriving eq 14, the head was modeled as an effective hard disk whose cross-sectional area, ah, was estimated from the conformational characteristics of a PEO chain, and was assumed to be independent of the shape of the micelle, S, and the micellar core minor radius, 1,. The effective hard-disk model was then utilized to computegb/Mthrough the use of a scaling relation, a h G ) = ah(6)G/6)J’,where the two parameters, ah(6) and y, were estimated using experimental data.12 I t is evident, however, that the effective hard-diskmodel, andconsequently eq 14,does not account explicitly for the chainlike character of the PEO heads, an issue which is being addressed in the calculation of gb/H presented below. The need to model explicitly the polymeric nature of the PEO, Ej, heads of CiEj surfactants prompted us to examine the feasibility of implementing some of the available theories of diblock copolymer micellization in the case of CiEj surfactants. Among these theories, polymer scaling descriptions (blob models)32and the selfconsistent field theory33 have been utilized extensively. “Blob” models, however, are only applicable in the case of sufficiently long polymer chains and are therefore not suitable to describe the relatively short PEO chains (E6 and Ea)present in the CiEjsurfactants under consideration. Although, in principle, the self-consistentfield theory could be utilized to model CiEjsurfactants, in practice this would require the availability of accurate experimental data on the variation of the binary interaction parameters associated with PEO, water, and salt as a function of temperature and salt type and concentration. Such experimental data are limited, thus making the implementation of this theoretical approach problematic in the present case. In view of the inherent limitations of the diblock copolymer approaches, we have adopted a somewhat different description. Specifically, we describe the PEO micelle head layer as a solution of PEO chains which are transferred to that layer from the aqueous salt solution, and are subsequently “grafted” at one end to the micellarcore surface. This description of the micelle head layer is conceptually similar, but mechanistically different, to one which was recently proposed in ref 34. While in ref 34 use was made of the well-known Flory-Huggins theory of polymer solutions to model the free energy of a solution of “grafted” PEO chains, we have instead examined the relative magnitudes of the enthalpic and entropic con(32) de Gennes, P.G.Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (33) See, for example, Schuetjens, J. M. H. M.; Fleer, G.J. J.Phys. Chem. 1979,83, 1619, and references therein. (34) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7 , 2934.
Carale et al.
114 Langmuir, Vol. 10, No. 1, 1994 tributions to the free-energy change, gh/M, associated with the formation of the PEO micelle head layer. We concluded that the enthalpic contribution to g h / H is the dominant one, and then proceeded to estimate g h / H by relating it to experimental PEO enthalpy of dilution data. A detailed description of the g h / H calculation along the lines outlined above is presented next. In the micellization process, a monomeric head, in particular a short PEO chain in the present case, is transferred from the aqueous salt solution (he) to the micelle head layer (hl), where it is subsequently “grafted” at one end to the micellar-core surface. The PEO head of a monomeric surfactant molecule in the aqueous salt solution is represented as an isolated PEO chain in an infinitely dilute solution, having a head volume fraction, 4h 0 (recall that the monomer concentration is of the order of the cmc, which is very low). On the other hand, the PEO chains present in the micelle head layer can be modeled as a solution of “grafted” PEO chains having a head volume fraction, 4 ~ The . free-energy change corresponding to the overall transfer and “grafting” process, gh/M, is therefore given by
-
= ghl(4hl)- g h s ( h )
(15) where gM(4M) is the free energy of a ”grafted” PEO head chain in the micelle head layer and gh(4h) is the free energy of a singly-dispersed PEO chain in the aqueous salt solution. Based on the thermodynamic relation, g, = h, - Ts,, where h, and sLare the enthalpy and entropy per molecule in phase x , respectively, eq 15 can be rewritten as &/M
Note that the entropy of a monomer in the aqueous salt solution, and, consequently, the entropy of the PEO head in the aqueous salt solution, sh(&J, should not be Instead, accounted for explicitly in the evaluation 0fgmiCaB6 in the evaluation of g,ic, it is assumed that the surfactant monomers are transferred into the micelle from a fixed position in the aqueous solution. Furthermore, since the head present in the micelle head layer is a short PEO chain “grafted” to the micellar-core surface, it has only very limited available degrees of freedom, both of the conformationaltype (sincethe number of EO units is small) as well as of the translational type (since the chain is “grafted”). Consequently, as a first approximation, it is reasonable to assume that the dominant contribution to the free energy, gh/M, arises from the enthalpy terms in eq 16 [minimal entropic contribution, namely, Ah >> TAsl. Making this approximation, eq 16 yields
[hM(dJhi) - hhB(4h)l= h b p (17) The enthalpy change, hk/H,in eq 17 originates from the increase in the number of PEO head-head contacts, as well as the corresponding increase in the solvent-solvent contacts and the decrease in the number of PEO headsolvent contacts, when a PEO chain is transferred from the aqueous salt solution to the micelle head layer. This enthalpy change, hhalhl, can be related to the enthalpies of dilution of PEO aqueous solutions. Dilution enthalpies of aqueous solutions of PEO of various molecular weights have been measured.3’j3* It was found36 that the dilution gh/M
=S
(35)See refs 12 and 14,wheregdcaccountsfor the free-energychange associated with forming a stationary micelle from free surfactant monomers and the entropic contribution is explicitly accounted for in the expression for the total micellar solution free energy. (36)Maron, S. H.; Filisko, F. E. J. Macromol. Sci.-Phys. 1972, B6, 79.
enthalpies per gram of PEO, a t constant temperature, vary approximately linearly with the difference between the PEO concentrations before and after dilution. This results in the relation
ha/, = hb - ha
h($b - 4,)
(18)
where hs/b is the enthalpy change associated with the transfer of one EO segment from phase a, having PEO concentration $a, to phase b, having PEO concentration $b, and h is a positiue enthalpy parameter. The positiue sign of h reflects the fact that in a typical dilution experiment, where $a > 4b,the dilution of PEO with water is found to be exothermic, that is, ha/b< 0.36137In the case of the transfer of a surfactant PEO head composed of j EO units from the dilute aqueous salt solution characterized by a concentration, $h,to the micelle head layer characterized by a concentration, &,I, the enthalpy of transfer, hh/hl, takes the form
hh,M0’) z jh(& - &)
j W M
where the reasonable approximation that made. Utilizing eq 19 in eq 17 yields
&/,0’)
=
I$H
(19)
>> & was (20)
The estimation of the enthalpy parameter, h, is discussed in section IVB1. e. Determination of the PEO Head Volume Fraction in the Micelle Head Layer, hl. As mentioned in section IIAd, the PEO chains present in the micelle head layer can be modeled as a solution of “grafted”PEO chains having a head volume fraction, 4 ~ As . a first approximation, 4~ will be assumed to be independent of the radial distance from the micellar-core surface, that is, 4~ is taken to be uniform in the micelle head layer. This assumption appears to be reasonable a t the level of one micelle and should be contrasted to situations where the detailed structure of the micelle head layer may be important, for example, in the estimation of the magnitude of intermicellar interactions resulting from the interpenetration of micelle head layers of two approaching micelles.39 Note that &I, can be estimated as the ratio of the volume of a PEO head divided by the volume of the micelle head layer per surfactant molecule, YH. As discussed in section IIAc, the last methylene group in the alkyl chain is considered as part of the surfactant head. Accordingly, each PEO head, Ej, having j ethylene oxide units, is comprised of 3j + 2 bonds, the first being a C-C bond, followed by j (C-0, 0-C, and C-C) bonds, and finally a C-0 bond corresponding to the terminal hydroxy (OH) group. In other words, each PEO head has effectively (3j + 2)/3 = j’ = (j + 2/3) EO units. The volume of the PEO head is then given by the product of the effective number of EO units in the head, j’, and the volume of an EO unit, VEO,where VEO= 63.5 A3 is the dry volume of a monomer of poly(ethy1ene oxide).@ Therefore
&
=S
?vEO/UM
(21)
Note that YM can be calculated by dividing the total volume of the micelle head layer, VH, which is a function of the average thickness of the micelle head layer, ZM, by the number of surfactant molecules per micelle, n (= V m j c e h C o d v h c ) . For spherical micelles (8= 3) one obtains (37)Daoust, H.; St-Cyr, D. Macromolecules 1984,17, 596. (38)Kagemoto,A,;Murakami, S.; Fujishiro, R. Makromol.Chem. 1967,
105, 154.
(39)Carale, T.R.;Blankschtein, D. Unpublished results. (40) Sarmoria, C.; Blankschtein, D. J. Phys. Chem. 1992, 96,1978.
Salt Effects on Zntramicellar Interactions
Langmuir, Vol. 10, No. 1, 1994 116
(22)
and for rodlike micelles ( S = 2) of length, L,one obtains
Utilizing eq 22 in eq 21, one obtains the following expression for &phcorresponding to spherical micelles
Similarly, combining eqs 23 and 21, one obtains the following expression for corresponding to rodlike micelles
Figure 1. Confinement of a PEO head chain within a cone, where 1, is the micellar core-minor radius and R,is the radius of the cone at a distance r from the micellar-core surface. using well-known geometrical argument^,^^ one finds that
in eq 20, respectively, yields Using eqs 24 and 25 for the following ghs/hl expressions for spherical micelles
and for rodlike micelles
Recall that in eqs 24-27, j’ = j + 2/3. f. Determination of the Micelle Head-LayerThickness, &I. The solution of eqs 24-27 requires the a priori determination of the micelle head-layer thickness, l ~ , which was defined earlier as the average distance from the micellar-core surface up to which a “grafted” PEO chain extends into the aqueous environment. In order to estimate IN, the PEO head chain-segment distribution was calculated as a function of the distance from the micellarcore surface, and the value of ZH was subsequently determined from it (see below). The determination of the average chain-segment distribution was done by generating possible PEO head chain conformations using a method based on the rotational isomeric state (RIS) approximation, combined with a Monte Carlo approach. This method has been utilized recently to determine the root-mean-square end-to-end distance of short chains of poly(ethy1ene oxide) attached O have generalized at one end to a planar inert ~ a l l . ~ We this method to incorporate curvature effects associated both with the shape of the micellar-core surface (spherical or rodlike), and the micellar core minor radius, 1,. In addition, we have also estimated the effect of other PEO chains present in the micelle head layer on the conformations of a given PEO chain. For this purpose, the PEO head is confined within a conical section (see Figure 11, whose radius, R,, varies with the distance from the micellarcore surface, r. It is noteworthy that both the value of this cone radius at the micellar-core surface, Rr=0, as well as its value, Rr, as a function of r, depend on the shape of the micellar-core surface. Note that R,,o is the radius corresponding to the area per surfactant molecule at the micellar-core surface, which can be estimated by assuming a constant density in the hydrocarbon core. Specifically,
where, as described earlier, S is the shape factor (S = 2 for rods and S = 3 for spheres), v h c is the volume of the hydrocarbon tail, and I, is the micellar core-minor radius. In a similar manner, the area per surfactant molecule can be calculated as one moves away from the micellar-core surface by a distance, r. The resultingradius, R,, for rodlike micelles is given by
and for spherical micelles, it is given by Rtph= R,=,
(y)
(30)
In addition to the constraints on the chain conformations discussed in ref 40, whereby a chain conformation is rejected if a bond penetrates the micellar core, we have imposed the additional constraint due to the presence of the other chains, as described above. Accordingly, while generating the chains, a particular conformation of a chain is rejected when a bond goes beyond the conical section defined by the radius given in eq 29 (for rodlike micelles) and in eq 30 (for spherical micelles). This additional restriction generates more extended conformations of the PEO head chains, as compared to those corresponding to a single chain attached to a planar wall. As S decreases from 3 (for a sphere) to 2 (for a rod), the radius, Rr, of the confining conical section decreases (compare eqs 30 and 291, which results in more extended PEO chain conformations for rodlike micelles. In other words, the smaller the curvature of the micellar-core surface, the more extended the chains are, since there is less available space for the chains to expand laterally. Applying the above-mentioned constraints, we then proceeded to determine the PEO bond-density distribution and, consequently, the average position of the chain segments farthest from the micellar-core surface, which would correspond to IN. The micelle head layer was divided into discrete “shells”, such that the number of bonds, n,, (41)Israelachvili,J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72, 1525.
Carale et al.
116 Langmuir, Vol. 10,No. 1,1994 c
3.88
+ 1.83j’
(in A)
(31)
and, for rodlike micelles (S = 21, we find that
ZHrd = 4.82 + 1.85j’ (in A) (32) g. Determination of the Optimal Micellar Shape, 9, and Optimal MicellarCore-MinorRadius, IC*.The
Figure 2. Division of the miceIle head layer into concentric shells, each having a thickness of 0.5 A, where m is the shell number.
16 i7
m
5 tl
1615 14131211 -
(33)
and
zl:: .-?
-1 6 2
X=CnXn
8-
76 5 2
free-energy of micellization, gmiocan be determined from its various contributions (see eq 2) as a function of the micellar core-minor radius, Z, for the two micellar shapes considered, spherical (S = 3) and rodlike (S = 2). The resulting gmicexpression was then minimized with respect to I, for each of these shapes in order to determine the optimal shape, S*,and associated optimal micellar coreminor radius, I,*. For details of the minimization procedure, see refs 12 and 31. This minimization procedure has been utilized in order to calculate the gmic values reported in section IVBP (see also Table VI). B. Prediction of the Critical Micellar Concentration. Application of the principle of multiple chemical equilibrium between micelles of different sizes42p43 makes possible the determination of the micellar size distribution through the simultaneous solution of the distribution and mass balance equations.12J4~42~43~u Specifically
(34)
n
3
I
Ihl
where X is the total surfactant mole fraction, XI is the monomer mole fraction, and Xn is the mole fraction of micelles having aggregation number, n. Note that, for the spherical (S = 3) and rodlike (S = 2) micelles considered in this paper, gmic(n)is approximated by the relation12
’ I
‘0
2
4
6
8
10
12
14
16
Distance from Micellar-Core Surface, r
(A)
16
20
Figure 3. Cumulative number of bonds between the micellarcore surface and a distance, r, from the micellar-coresurface, for spherical micelles 0’ = 6), with 1, = 15.41 A ( O ) , 14.64 8, (+), 13.90 8,( O ) ,13.21 8, (A),12.55 8, (X), and 11.92 8, (v).The dashed line corresponds to 3j’ = 3j + 2 = 20, for which r = l~ (see arrow). Note that the cumulative number of bonds is not a strong function of I,.
in the mth shell (see Figure 2) could be counted while each of the chain conformations is generated. We have chosen to set the thickness of each shell to be 0.5 A, since the thickness of the shells determines the precision of the estimated lhl value, that is, the segment which is farthest from the micellar-core surface will be anywhere within a shell of said thickness. The value of n, was taken as the average of 40 000accepted conformations. The cumulative number of bonds, En,, that is, the total number of bonds between the micellar-core surface and the mth shell, was then plotted as a function of r. The thickness, IN, was estimated as the value of r at which the limiting value of the total number of bonds 3j’ = 30’ + 2/3) = 3j + 2, was approached. This is seen in Figure 3, which presents plots of En, as a function of r for ClzEa spherical micelles (S = 3) and for various values of I,. As seen in Figure 3, the IN values determined numerically are not strong functions of I,. However, we found that they do depend significantly on S. We also found that we can fit the estimated values of IN of Eq, Eg, Ea, and El0 chains to a linear function of j’. Specifically, for spherical micelles (S = 3), we find that
where nsph* is the aggregation number of the “optimum” spherical micelle and the values of g&Ph and gmicrd are determined using the formalism presented in section IIAg. Equation 33 results from the conditions imposed by chemical equilibrium involving the micelles and the free monomers in solution, while eq 34 constitutes the massbalance relation. The cmc can be evaluated by plotting XI as a function of X, the estimated cmc being the concentration at which the plot exhibits a sharp break due to the onset of micelli~ation.~~ An approximate way of calculating the cmc makes use of a relation between the cmc and gmicwhich is valid for micelles having large aggregation numbers.12 Specifically cmc
-
exp( gmic - 1)
Combining eq 36 with eq 1, which can be rewritten as, -k, = d(ln cmc)/dm,, gives an approximate relation for the salt constant, k,, namely, (42) Corkill, J. M.; Goodman, J. F.; Walker, T.;Wyer, J.Pr0c.R. SOC. London, Ser A 1969,312,243. (43) Mukerjee, P. J . Phys. Chem. 1972, 76, 565. (44) For a more detaileddescription of the thermodynamicframework, see ref 14. (45) Puwada, S.;Blankschtein,D.Proceedings of thedthhternationul Symposium on Surfactants in Solution; Mittal, K. L., Shah, D. O., Eds.; Plenum: New York, 1991; Vol. 11, p 95.
Salt Effects on Intramicellar Interactions
Langmuir, Vol. 10, No. 1,1994
117
Table 3. Variation of Alkyl Tail Transfer Free Energy with Salt Molality, dg,/j,Jdm, = dg,/,/dm, (kT/m),for CsHls and CllHp Chains salt LiCl NaCl KC1 KBr KI
%Hie -0.5596 -0,7350 -0.6684 -0.4262 -0.2802
CiiHzs -0.6713 -0.8817 -0.8018 -0.5112 -0.3362
Note that the various free-energy contributions in eq 37 are in units of kT,and, therefore, the kT term does not appear explicitly. Equation 37 is a central result which indicates that it is possible to predict the salt constant, k,, in the context of the theoretical formulation developed in this paper. Indeed, values of dg Jdm,, dgddm,, and dgt+ddm, can be evaluated for eac type of salt and surfactant considered. Values of dg+, Jdm, = Q,/,/dm, (see eqs 3-5) were calculated utilizing eq 8 in conjunction with Table 1and are reported in Table 3. Similarly, values of dgu/dm,were determined using eq 9, in conjunction with eqs 11and 12 and Table 2, where we have assumed that, to leading order, the main effect of the salt is on 00. We have thus assumed that the Tolman distance, 6, is a weak function of m,, while the effect of the salts on the geometric factors, 1, and a, occurs through the minimization of 1,described in section IIAg. Calculated values of dgddm, are reported in Table 2. It can also be deduced from eq 20 that dghlddm, = j h dhldm,. We have therefore shown that by breaking down the salt constant, k,, into its qpmponent parts, it is possible to determine the relative magnitude of each contribution to k,. This will be discussed in more detail in section IVB4.
"h"
111. Experiments A. Materials and Sample Preparation. HomogeneousCiEj surfactants were obtained from Nikko Chemicals, Tokyo, and used without further purification. To ensure uniformity in the results, measurements were conducted using the same lot number for each surfactant: ClzEs (lot 90111, C I ~ E(lot E 90541, and CloEe (lot 1054). The high purity of the surfactants was confirmed by the absence of any detectable minimum in the measured surface tension versus surfactant concentration curves of aqueous solutions of each surfactant. Salts were of the analytical reagent grade from Mallinckrodt, and were further purified to remove any organics by ignition at 450 OC overnight. Salt solutions were prepared by weight, using deionized water which was purified using a Milli-Q ion-exchange system. A known weight of surfactant was then added to each of the salt solutions. The prepared surfactant solutions were utilized within the same day. Since the occurrence of evaporation was possible at room temperature, which, in turn, could change the surfactant concentration of the samples,proper precautions against evaporation were taken. These included placing samples in stoppered flasks and sealing the flasks while awaiting measurement. Before use, all glassware were immersed in a 1N NaOH-ethanol bath for at least 8 h and then in a 1N nitric acid bath for another 8 h, followed by thorough rinsing with Milli-Q water. The glassware were then dried in an oven. The Wilhelmy platinum plate, to be used in the surface tension measurements, was rinsed with distilled water, then with acetone, and again with distilled water and flamed until red hot before each surface tension measurement. B. Measurement of the Critical Micellar Concentration. The critical micellar concentrations of surfactant solutions with and without added salts were measured using the surface-tension method. It is well-known that as the surfactant concentration,
0
0.5
1
I.5
2
2.5
3
Sail Molality
Figure 4. Comparison of predicted cmc's of Cl& in aqueoue solutions of LiCl (..*), KC1 (-1, and NaCl (---) with experimentally measured cmc's in aqueous solutions of LiCl (A), KCl (D), and NaCl(0).
X,is increased, both the hydrophobicity of the surfactant tails and the high watel-air surface free energy promote the adsorption of the surfactant molecules onto the surface. The increase in the surface pressure due to surfactant surface adsorption leads to a lowering of the surface tension, u. Beyond a certain surfactant concentration, the cmc, it becomes more favorable, from a freeenergy point of view, for the surfactant molecules added to the solution to form micelles, rather than continue to adsorb at the surface. This is reflected in a negligible change in surface tension with increasing surfactant concentration beyond the cmc. The "break" in the u versus X curve, therefore, approximates the concentration at which micellization first takes place. In order to determine this "break", the surface tensions of at least 12 surfactant solutions were measured and plotted as a function of the logarithm of the surfactant mole fraction, X. Linear regression was utilized to determine the best fit line on either side of the break in the curve, the intersection of these two lines being taken as the experimental cmc value. The accuracy of the experimental cmc determination is bounded by the surfactant concentrations corresponding to the experimental points immediately preceding and following the experimental cmc value, adjusted to account for errors from solution preparation. The accuracy was found to be between 5 and 20% of the cmc values. Surface tension measurements were performed using the Wilhelmy plate tensiometer (Kruss KlOT). All measurements were carried out in a thermostated devicemaintained at a constant temperature of 25 O C . Each surfactant sample was equilibrated for 45-90 min, depending on the time needed to attain a constant surface-tensionvalue, maintained for at least 30 min, as monitored using a chart recorder. The concentrations (mole fractions) of the surfactant samples were varied within a 2 order of magnitude range. The cmc's of c&e in water, and in 1,2, and 3 m solutions of NaC1, LiCl, KCl, KBr, and KI were determined using the method outlined above. In addition, in order to validate the salt trends established from the Cl& experiments, cmc determinations of ClzEEand C&S in water and in 1and 2 m solutions of KC1, KBr, and KI were conducted. Note that in the latter experiments, only the anion effects were studied, since the initial cmc studies with C & 3indicated that the variation of the anion has a more significant effect on the cmc as compared to that of the cation. IV. Results and Discussions A. Experimental Results. 1. Critical Micellar Concentration. The experimentally determined cmc's of C12&, C12E8, and cl0E6in various aqueous salt solutions are shown in Figures 4-7. The resulting uncertainty in the measured cmc's is in the range 5-20 5% ,as determined using the method described in section IIIB and shown for illustrative purposes by the error bars in Figure 5 for C12E8 in KC1solutions. The range of uncertainty for each reading
Carale et al.
118 Langmuir, Vol. 10,No. 1, 1994
Table 4. Comparison of Predicted and Experimentally Deduced Salt Constants, k,,for CliEs in Various Aqueous Salt Solutions k, (predicted k, (experimentally
Cl& in LiCl solutions C12& in NaCl solutions Clz& in KC1 solutions Clz& in KBr solutions Cl2& in KI solutions
using eq 37) 0.58 0.80 0.72 0.46 0.34
deduced using eq 1) 0.63 0.81 0.69 0.48 0.30
585854-
0
1
0.5
1.5
2
2.5
-E "1
3
52
SaR Molality
Figure 5. Comparison of predicted cmc's of Cl2& in aqueous solutions of KI (. .),KBr (- - -), and KCl (-) with experimentally measured cmc's in aqueous solutions of KI (A),KBr (e), and KCl (m), Note that error bars are only shown for illustrative purposes in the KCl case.
-
2 a
-
E&-
9
44-
b
42-
40-
381.6,
I
36 34 32
-
m * "-8.0 " " "-7.6 " " "-7.2 "
0.2
1
,
,
,
I
0.5
1
1.5
2
Salt Molality
Figure 6. Comparison of predicted cmc's of ClzEe in aqueous with experisolutions of KI .), KBr (- - -), and KC1 (-) mentally measured cmc's in aqueous solutions of KI (A),KBr (e),and KC1 (D). (a
1.5
0.5
Salt Molality
Figure 7. Comparison of predicted cmc's of C1& in aqueous solutions of KI (. .), KBr (- - -), and KC1 (-) with experimentally measured cmc's in aqueous solutions of KI (A),KBr (e),and KC1 (m). is not shown in the other plots since the curves are in close proximity to each other. Figures 4-7 indicate that, for the three surfactants examined, the cmc decreases as the salt concentration increases for all the salts examined. The order of decreasingthe cmc is consistent with that found in previous studies? namely, C1- > B r > I- for the anion (see Figures 5-71) and Na+ > K+ > Li+ for the cation (see Figure 4).
.
-6.8
-6.4
6.0
-5.6
-5.2
Furthermore, comparing cmc values at the same salt molality for C12E6 (Figure 5) and CloE6 (Figure 7) reveals that the cmc's of the less hydrophobic Cl& (shorter alkane chain, and therefore more soluble in water), in the aqueous salt solutions examined, are an order of magnitude higher than those of C12E6. Increasing the length of the PEO head from an E6 to an Eg chain, which corresponds to making the PEO head more hydrophilic, gives rise to a slightly higher cmc for (see Figure 6) as compared to C12E6 (see Figure 5) at the same salt conditions. This reflects the fact that it is more difficult to transfer the longer, more hydrophilic, Eg chain to the micelle head layer, as compared to an E6 chain, thereby increasing the micellization free energy and, consequently, the cmc. 2. Salt Constant, k8. In agreement with previous experimental work on salt effects on the cmc of nonionic surfactant^,'-^ we have found that the logarithm of the measured cmc's varies linearly with salt concentration, according to eq 1. As an illustration, the salt constants, k,, for Cl&& in various aqueous salt solutions were determined from the linear regression of In cmc versus m,, and the results are reported in the third column of Table 4. 3. Surface Tensions of Aqueous CiEj Solutions. The surface tensions of the aqueous CiEj solutions examined reflect primarily the balance between the solubilities of the hydrophobic alkyl tails and the hydrophilic PEO heads. The addition of salts decreases the solubility of the surfactant molecules. This leads to an increased surfactant adsorption at the surface, and, therefore, decreases the surface tension of the solution, as seen in Figure 8, which plots the surface tension of Cl2Eg in water (0)) and in 2 m KC1 ( O ) , 2 m KBr (A),and 2 m KI (v)aqueous solutions at 25 OC. Furthermore, the salt effect on the surface-tension lowering varies according to the salting-out trend found for the hydrocarbon tails,which is in the order KC1 > KBr > KI. Recall that this order
Salt Effects on Intramicellar Interactions
is related to the volume contrytion effects discussed in section IIAa, expressed in the V," - V, values shown in the fourth column of Table 1. Thus, as seen in Figure 8,uKCl
< UKBr < am.
It is interesting to note that an examination of the surface tension plots in Figure 8 indicates a slight increase in the surface tension of the surfactant solution a t concentrations beyond the cmc upon the addition of KI (v),as compared to that in pure water (0). We speculate that the observed increase in u may be due to specific interactions between the iodide ions and the poly(ethy1ene oxide) heads.46147 This "complexation" adds to the PEO heads a pseudocharge such that the concentration of surfactants adsorbed at the surface will decrease due to the added repulsive electrostatic interactions between the "effectively charged" PEO heads of the adsorbed surfactant molecules. Note that at surfactant concentrations below the cmc of pure water, the decrease in the solubility of the alkyl tails upon the addition of KI dominates over the increased electrostatic repulsion between the surfactant heads adsorbed at the surface. The net result is a higher concentration of surfactant molecules adsorbed at the surface and a corresponding lower surface tension of ClzE8 in KI solutions as compared to that in pure water (see Figure 8). The existence of an electrostatic repulsion between the heads, however, limits the maximum concentration of surfactant molecules that can be adsorbed at the surface, making it smaller than the corresponding concentration for the pure water case. Consequently, C H ~ O< UKI beyond the cmc (see Figure 8). B. TheoreticalResults. 1. Parameter Estimation. The theoretical prediction of cmc's was described in detail in section 11. In that section, it was pointed out that two parameters could neither be determined experimentally nor taken from literature sources and, therefore, have to be estimated. These parameters are the molecular volume of pure (liquid) salt, V,, and the enthalpy parameter, h, whose estimation is described below. Liquid Salt Volume, V,. In order to determine the free-energy contribution associated with transferring a surfactant tail from the aqueous salt solution to bulk hydrocarbon, it is necessary to determine the free-energy change corresponding to the transfer of the surfactant tail from the aqueous salt solution to pure water, g,/w. This, in turn, can be determined through the use of eq 8 once the three molecular parameters vhc, V," ,and V, are known. The first two parameters can be determined, as described earlier. However, liquid salt volumes at atmospheric conditions cannot be determined experimentally, thus the necessityfor estimation. The estimation was done by using the theory embodied in eq 8,where the experimentally determined variation of gs/hc with salt concentration, m, (from solubility experiments), as well as inputs of vhc, V,",and 00, makes possible the determination of V,. Considerable experimental data on the solubility of benzene in various salt solutions is a~ailable.~314Accordingly, the results of these experiments were utilized to estimate V, values of LiC1, NaC1, KC1, KBr, and NaI. In these calculations, v h c of benzene was estimated from density measurements reported in ref 24. These values, together with 00 (=4.52 X 106 b a r 1 at 25 "C) from ref 24 and V," from Table 1, were substituted in eq 8 in order to estimate V,. No data for benzene solubility in KI (46)Lundberg, R.D.;Bailey,F. E.; Callard, R.W. J.Appl. Polym. Sci. A-1 1966,4, 1563.
(47)Binana-Limbele, W.; Van Os,N. M.; Rupert,L. A. M.; h a , R. J. Colloid Interface Sci. 1991, 144, 458. (48) Saylor, J. H.; Whitten, A. I.; Claibome, I.; Gross,P. M. J. Am. Chem. Soc. 1952, 74, 1778.
Langmuir, Vol. 10, No. 1, 1994 119 Table 5. Values of the Enthalpy of Dilution, A H ~ . ~ for n, PEO in Pure Water, Where (1 and A are the PEO Concentrations before and after Dilution (see Table 3 of ref 36), and 4 Is the Enthalpy Parameter Deduced from the Experimental Data Using Equation 18 0.0179 0.0346 0.0474 0.0474 0.0474 0.0619 0.0995 0.1133
0.0149 0.0289 0.0090 0.0406 0.0068 0.0530 0.0911 0.0283
0.00238 0.00513 0.03497 0.00565 0.03869 0.00744 0.00818 0.08779
0.79 0.90 0.91 0.83 0.95 0.84 0.97 1.03
solutions are available, however. Therefore, an estimate of the V, value for KI was made by using a relation that was found to be valid at high temperature^.^^ Specifically (38) IK' % K ' C1 - VNeCl -k v N d The molecular volume of KI predicted using eq 38 at 900 "C,a t which temperature these salts are in the molten state, was found to be accurate to within 2.4% of the measured ~ a l u e . 4A~summary of V,values deduced using the procedure outlined above is shown in the second column of Table 1. Enthalpy Parameter, h. The second parameter that needs to be determined is the enthalpy parameter, h, appearing in eq 20, which characterizes the transfer of an EO unit of a short PEO chain (E6and Ea) from an infinitely dilute solution (for which 4h~ 0) to the micelle head layer (for which, for example, 4~ = 0.3, as estimated for rodlike C12E6 micelles using eq 25). The linear relation between the dilution enthalpy and the difference between the PEO concentrations before and after dilution, shown in eq 18, was established based on experiments done on PEO solutions where the PEO concentrations before dilution ranged from 4 = 0.018 to 4 = 0.113 and the PEO concentrations after dilution ranged from 4 = 0.015 to 4 = 0.091.36 The concentration ranges examined in ref 36 are therefore quite different from those involved in the PEO, Ej, head transfer considered in this paper. Furthermore, the PEO chains examined in ref 36 had a higher molecular weight (6000) than that of the E6 or E8 heads (295 and 383, respectively). Experiments involving low molecular weight PEO,38 on the other hand, were conducted primarily on solutions having very high PEO concentrations (4 l),which are much greater than the PEO concentrations of interest in this paper. In view of the discussion above, there appears to be some ambiguity as to whether the available experimental PEO dilution enthalpy data are directly applicable to the treatment of the transfer of the PEO, Ej, heads considered in this work. Consequently, it appears necessary to find an alternative, possibly more reliable, way to estimate the value of h associated with the transfer of the Ej heads examined. To this end, we fitted the theoretically predicted cmc to the experimental cmc of C12E6 in pure water and deduced a value of h = 0.85kT/EO. Note that the h value so deduced is consistent with the values deduced from the application of eq 18to the PEO dilution enthalpy data reported in ref 36 (see the reported h values in Table 5). The value of h = 0.85kT/EO so determined was then utilized, without any further adjustments, to predict the cmc's of C I Z Eand ~ CloE6. Regarding salt effects on h, a comparison of the gmic contributions shown in Table 6 (to be discussed in section
-
-
(49)A Comprehensiue Treatise on Inorganic and Theoretical Chemistry; Mellor, J. W., Ed.; Longmans, Green and Co.: London, 1941;pp
533 and 599.
Carale et al.
120 Langmuir, Vol. 10, No. 1, 1994 Table 6. Various Contributions to the Free Energy of Micellization, gdc (in k!l“), at 25 OC, Calculated at the Optimal Micellar Core-Minor Radius, l,,, for Various Micellar Shapes gwck
ghn/hl
-18.46 2.84 1.42 -12.41 -19.13 2.92 1.42 -13.00 2.91 1.42 -19.34 -13.21 -19.26 2.91 1.42 -13.14 -18.97 2.88 1.42 -12.87 -18.80 2.83 1.42 -12.75 Optimal Shape: Spherical Micelles
1.80 1.80 1.80 1.80 1.80 1.80
3.78 -18.46 -19.26 3.88 -18.97 3.85 -18.80 3.78 Rodlike Micelles
1.50 1.50 1.50 1.50
1.15 1.15 1.15 1.15
-11.93 -18.46 2.99 1.35 -19.26 3.07 -12.65 1.35 -18.97 3.04 1.35 -12.39 -18.80 2.99 1.35 -12.27 Optimal Shape: Rodlike Micelles
2.19 2.19 2.19 2.19
gmic
&/he
ga
Optimal Shape: Rodlike Micelles CIZ& ._in
Hi0
1m LiCl 1m NaCl 1m KCl 1m KBr
lmKI
ClzEs in HzO 1m KCl 1m KBr lmKI ClzEs in HzO 1m KC1 1m KBr lmKI
cl~&in
HzO 1m KCl 1m KBr 1m KI
-12.02 -12.73 -12.47 -12.36
-9.75 -10.35 -10.13 -10.04
-15.47 -16.14 -15.90 -15.75
2.65 2.72 2.70 2.65
1.31 1.31 1.31 1.31
1.76 1.76 1.76 1.76
IVB2) for C12E6, C12E8, and CloE6 in pure water indicates that the dominant contribution to gd, is gs/hc, which is an order of magnitude larger than g b / H . Based on this observation, we have made the plausible assumption that, as a first approximation, a change ingmi,due to salt effects on gblH should be very small as compared to the overall magnitude ofg,i,. With this assumption in mind, we have chosen the value, h = 0.85kT/EO, for all the aqueous PEO salt solutions as well. This, in turn, implies that dhldm, = 0 and, consequently, that dgb/hl/dm, = 0 (see eq 20). Nevertheless, it should be emphasized that salt effects on gb/H, however small, may be significant when predicting micellar solution properties other than the cmc. For example, in predicting the micellar size distribution, the difference, gmic,rod- gmic,sph, constitutes the relevant freeenergy factor. In this case, gs/hc cancels out because this contribution is independent of micellar shape, thus making the other contributions togmi,more important (recall that in predicting cmc’s, &/hc is the dominant term). Accordingly, when predicting salt effects on properties which depend on the micellar size distribution, such as the critical surfactant concentration for phase separation, X,,salt effects on gb/H may need to be accounted for. Work in this direction is in progress. 2. Predicted Free Energy of Micellization, gmjo. A very useful feature of the theoretical approach presented in this paper is that the magnitude of the salt effects on the various contributions to the free energyof micellization, gmic, can be computed, rationalized, and compared. For example, values of the various contributions to gmicfor C12E6, C12E8, and C1OE6 are reported in Table 6 for the optimal micellar shape, S*,and the corresponding optimal micellar core minor radius, Zc*. Note that, for the purpose of comparison, values of these gmic contributions are also reported for rodlike C12Eamicellesat Z,*(rod) at conditions where spherical C12E8 micelles are the optimal shape. The ability to examine the various gmiccontributions in detail should be contrasted with previous work which has lumped all the salt effects in terms of changes in activity coefficients.
The transfer free energies,g,/,, for C11 (thetail for Cl2E6 and C12E8) and Cg (the tail for c186)chains for five types of salts were calculated using eq 8 (see section IIAa and Table 3). Values of gw/hc were calculated using eq 6 and subsequently added to the calculatedg,/, values to obtain &/he (see the third column of Table 6). The g, values calculated using eq 9, in conjunction with eqs 11 and 12 and Table 2, are reported in the fourth column of Table 6. The values of gpack were calculated as for the salt-free ~ a s e ~ using ~93~ eq 13 and are reported in the fifth column of Table 6. The gb/H values were calculated using eqs 26 and 27 and are reported in the last column of Table 6. For each micellar shape considered, spherical (S = 3) and rodlike (S = 21, the gmic contributions were incorporated into eq 2 to calculate the free energy of micellization, gmic, as a function of the micellar core-minor radius, 1,. As mentioned in section IIAg, gd, was then minimized with respect to I , for each of these shapes in order to determine the optimal value, lc*. The optimal shape, S*,was then determined from the minimumgd,(S) value. For example, in Table 6, gmic and its various components are reported for both spherical (S = 3)and rodlike (S = 2) ClzEamicelles. A comparison of the free energies of micellization for both shapes indicates thatgd,values for spherical C12& micelles are lower than those corresponding to rodlike C12E8 micelles. Accordingly,the theory predicts that the optimal ClzE8 micellar shape is spherical at the conditions considered. As seen in Table 6, the dominant contribution to the free energy of micellization, gd,, is the hydrocarbon transfer free energy, gs/hc. I t is not surprising, therefore, that the observed enhancement of micellization (cmc lowering) is in the order C1- > B r > I- for the anion and Na+ > K+ > Li+ for the cation, since this is the order of the salt effect on gs/hc. The dominance of the salt effect on the hydrocarbon tail transfer free-energy contribution, gs/hc, is also in agreement with earlier findings that the salt effects are virtually independent of the number of ethylene oxide units in the PEO, Ej, heads6 Salt effects on the formation of nonionic surfactant micelles are primarily a consequence of the polarizing power of the ions, which is a function of the ion-charge density. Specifically, smaller ions, such as C1- as compared to Brand I-, cause a greater electrostriction of water, that is, a larger contraction in total volume upon mixing water and salt. This is shown in Table 1,where the electrostriction of water, as reflected in the (Vso - V,) values, is greatest for C1-, which causes the greatest increase in the internal pressure, and thereby “squeezes outwmore effectively the hydrocarbon tails. The cation trend is not as clear, since the bare ion radius of Li+ is much smaller than that of Na+ and K+, but it has - Valvalue. This anomalous behavior of the smallest ITSo Li+ is also reflected in the lyotropic and the Hofmeister series. Li+ is very strongly hydrated, such that water molecules in the inner hydration shell are very strongly bound. It is speculated that for the case of Li+,therefore, the inner hydration shell becomes part of the “ion”, such that it is the bare ion radius plus the inner hydration shell that determines the electrostriction of the rest of the “unbound” solvent molecules. The actual mechanism is far from being understood, however. In any case, the results indicate that the higher the charge density of the ion (which in this case is equivalent to a smaller hydrated radius), the lower is the solubility of the hydrocarbon tails in the aqueous salt solution, which would enhance micellization (decrease gmic). These salt effects on the hydrophobic driving force are
Langmuir, Vol. 10, No. 1, 1994 121
Salt Effects on Intramicellar Interactions slightly balanced by the interfacial free-energy contribution, g,, to gmjc, whereby the more polarizing ions increase the interfacial free energy to a larger extent, and thereby inhibit micellization (increase gmic). The fourth column in Table 6 shows that there is very little perceivable difference between the effects of the cations in the various chlorides ong, (in the case of C12E6),while the anion effect ong, (shown inTable 6 for C12E6, C12E8, and CloE6) follows that for &/he, namely, KCl > KBr > KI. Table 6 also reveals trends in the values of gmic as a function of surfactant type. For example, shortening the tail, which is equivalent to making the surfactant less hydrophobic, decreases the free-energy advantage of forming micelles, as shown by the lower g,icvalues Of C12E6 as compared to those Of C1&6. Both C12E6 and Cl&6 tend to form rodlike micelles in aqueous salt solutions at 25 "C, that is,gmicsod
