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Langmuir 1992,8, 2680-2689
Surface Tensions of Aqueous Nonionic Surfactant Mixtures Y. J. Nikas, S. Puwada,t and D. Blankschtein' Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received May 13,1992. I n Final Form: August 21,1992 We report results of theoreticaland experimentalstudies of the adsorptionof surfactants at the solutionair interface from two aqueous binary nonionic surfactant mixtures: (1) n-dodecyl hexa(ethy1ene oxide)n-dodecyloda(ethy1eneoxide) (C12EgC12Ee), and (2) n-dodecylhexa(ethy1eneoxide)-n-decyl tetra(ethy1ene The theory assumes that the Surfactant molecules adsorbed at the interface form oxide) (C12E&1&). a mixed monolayer, which is modeled as a binary mixture of hard disks interactingthrough attractive van der Waals forces. The hard-disk areas are calculated using a Monte Carlo approach which incorporates the conformationalcharacteristics of the chainlike poly(ethy1ene oxide) surfactant head groups in the context of the rotational isomeric state model. The attractive van der Waals interactions between the surfactanthydrocarbon tails are treated as a perturbation to the hard-disk repulsions, truncated at second order in surfacedensity. The corresponding second-order (virial)coefficientsare calculatedusing detailed molecular informationabout the twosurfactant speciespresent in the monolayer. In so doing, the molecular nature of the mixed monolayer is accounted for explicitly. The behavior of the surfactant molecules in the bulk solution, including the formation of mixed micelles beyond the critical micelle concentration (cmc),is modeled usingarecently developed molecular-thermodynamictheoryof mixed surfactantsolutions. The theoretically predicted surface tension versus bulk surfactant concentration curves at various bulk surfactant compositions are found to be in very good agreement with the experimentallymeasured ones. The variation of the monolayer composition with total bulk Surfactantconcentrationand composition is also predicted and is found to exhibit an abrupt change at the mixture cmc. This predicted variation of the monolayer composition is rationalized in terms of changes in the distribution of the two surfactant species between the bulk solution and the monolayer due to the onset of mixed micelle formation at the cmc.
I. Introduction Aqueous solutions of mixed surfactants have received considerable attention because they can exhibit bulk and interfacial properties which are more desirable than those displayed by the respective single-surfactant solutions.' While many experimental and theoretical studies have focused on determining and predicting bulk thermodynamic properties of mixed-surfactant solutions,' much less effort has been devoted to developing a molecularlevel description, having predictive capabilities, of interfacial properties of such The adsorption of surfactants at fluid-fluid interfaces from aqueoussingle-surfactantsolutions has been studied in great detaiLs Most of the thermodynamic descriptions of single-surfactant adsorption at fluid-fluid interfaces are based on the assumption that the adsorbed surfactant molecules form a monolayer at the i n t e r f a ~ e . The ~ theoretical approaches can be divided into two broad categories: (1) "two-dimensional (20)solution approachesn,6J in which the solvent is considered explicitly in modeling the monolayer, and ( 2 ) "two-dimensional ( 2 0 )
* To whom correspondence should ba addressed.
+ Present
address: Center for Bio/Molecular Science and Engineering, Naval Research Laboratory, Washington, DC 20375-5000. (1) For a comprehensive experimentaland theoreticalsurvey of mixed micellar solutions sea Phenomena in Mixed Surfactant S y s t e m ; Scamehorn, J. F., Ed.;ACS SymposiumSeries 311; American Chemical Society: Washington, DC, 1986, and references cited therein. (2) Lucaseen-Reynders, E. H. J. Colloid Interface Sci. 1973,42,554, 563. (3) Rosen, M. J.; Hua, X. Y. J. Colloid Interface Sci. 1982, 86, 164. Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 90,212. (4) Holland,P. M.Colloids Surf. 1986,19, 171. (5) For a review of experimental and theoretical developmente of surfactant adsorption, see Lucaesen-Reynders, E. H. In Anionic Surfactants-Physical Chemistry of Surfactant Action; LucassenReynders, E. H., Ed.;Marcel Dekker: New York, 1981; p 1. (6) Butler, J. A. V. Proc. R. SOC. Ser. A 1932, 135, 348. (7) Defay, R.; Prigogine, I,; Bellemans, A. Surface Tension and Adsorption; Longmans: Bristol, 1966.
in which the solvent is treated as a gas continuum background in modeling the monolayer. In both approaches, the fundamental challenge involves developing a physically reasonable description of the adsorbed surfactant monolayer. The 2D-solution approach has been utilized to study the adsorption of short-chain alcoholss and fatty acids13 at solution-air interfaces. In addition, the equation of state, that is, the variation of surface pressure with available area per solute molecule at the interface, derived using this formulation has also been used to interpret the experimental data of insoluble lipid monolayers.14J6 The advantage of this approach is that it does not require explicit information about the molecular structures of the various components present in the monolayer and the interactions between them. Consequently, this formulation can be applied to a wide variety of systems.ls However, the phenomenological nature of this approach also entails a major disadvantage,that is, several (typically two to three) adjustable parameters are often required to obtain a reasonably good fit to the experimental data. Furthermore, the values of these parameters are sometimes inconsistent with the molecular characteristics of the species present in the monolayer.6 The OD-gas approach has also been used to describe interfacial phenomena in a variety of systems,17being most successful in the study of insoluble monolayers of sur(8)Davies, J. T. J.Colloid Sci., 1966,11, 377. Davies, J. T.; Rided, E . K . Interfacial Phenomena, 2nd ed.;Academic Press: New York, 1963. (9) Hedge, D. G. J. Colloid Sci. 1967, 12, 417. (10) Chattoraj, D. K.;Chatterjee,A. K. J. Colloid Interface Sei. 1966, 21, 159. (11) Smith, T.J. Colloid Interface Sci. 1967, 23, 27. (12) Birdi, K. S. Lipid and Biopolymer Monolayers at Liquid Interfaces; Plenum Press: New York, 1989. (13) Lucaesen-Reynders,E. H.;van der Temples,M. In Proc. Int. Cong. Surface Actiue Substances 1964,2, 779. (14) Gaines, G. L. J. Chem. Phys. 1978, 69, 924. (15) Smaby, J. M.;Brockman, H.L. Langmuir 1991, 7, 1031. (16) Lucassen-Reynders, E.H. Prog. Surf. Membr. Sci. 1976,10,253.
0743-7463/92~2408-2680$03.00/0 0 1992 American Chemical Society
Surface Tensions of Surfactant Mixtures factants and lipids.18 A major limitation of this approach is that, so far, in all the proposed&12equations of state for the monolayer, it has been necessary to introduce two to three adjustable parameters to obtain good agreement with the experimental data. Furthermore, these parameters cannot be deduced from the molecular characteristics of the components present in the monolayer. It is noteworthy that, to date, the 2D-gas approach has not been used to model mixed surfactant or lipid monolayers. Theoretical descriptions of insoluble mixed monolayers, as well as of the adsorption of surfactants at interfaces from solutions containing several surfactant species, have generally proceeded along the lines of the 2D-solution a p p r o a ~ h . ~Specifically, ~?~~ Lucassen-Reyfor singlenders generalized2 the theory of Defay et surfactant adsorption to the mixed-surfactant case, Clint derivedz1 an expression for the surface pressure as a function of bulk surfactant concentration and composition assuming ideal mixing in both the monolayer and the bulk, and Rosen and Hua3 and Holland4 applied the regularsolution theory, introduced by RubinghZ2to describe micelliition of binary surfactant mixtures, to model the nonidealities associated with mixing of the various surfactant species in both the bulk and the mixed monolayer. The phenomenological nature common to all these theories requires the introduction of at least one more empirical parameter per surfactant pair, in addition to those (typically two to three) associated with each surfactant species, in order to obtain agreement with the experimental data, thus reducing their predictive power. In this paper, we generalize the 2D-gas approach to describe mixed surfactant monolayers, and formulate a theory of surfactant adsorption at solution-air interfaces from aqueoussolutions containing binary mixtures of alkyl poly(ethy1ene oxide), CiEj, nonionic surfactants. The mixed surfactant monolayer is modeled as a two-dimensional mixture of hard disksz3 which interact through attractive forces arising from the van der Waals interactions between the surfactanthydrocarbon tails. The harddisk areas are calculated using a Monte Carlo approachz4 which incorporates the conformational characteristics of the chainlike poly(ethy1ene oxide) surfactant head groups in the context of the rotational isomeric state model. The magnitude of the van der Waals attraction is also calculated using a molecular approach. In so doing, the molecular nature of the mixed monolayer is accounted for explicitly. The behavior of the surfactant molecules in the bulk solution, including the formation of micelles beyond the critical micelle concentration (cmc), is modeled using a recently developed molecular-thermodynamictheory of surfactant solution^,^"^^ which has been quite successful in predicting a wide range of micellar and phase behavior (17) Chattoraj, D. K.; Birdi, K. S. Adsorption and the Gibbs Surface Excess; Plenum: New York, 1984. (18) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience Publishers: New York, 1966. (19) Smaby, J. M.;Brockman, H. L. Langmuir 1992,8, 563. (20) Huber, K. J. Colloid Interface Sei. 1991, 147, 321. (21) Clint, J. H. J. Chem. SOC.,Faraday Trans. 1 1975, 71, 1327. (22) Rubingh, D. N. In Solution Chemistry of9urfactants;Mittal, K. L.; Ed.; Plenum: New York, 1979; Vol. 1, p 337. (23) Reisa, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959,31, 369. (24) Sarmoria, C.; Blankschtein, D. J. Phys. Chem. 1992,96, 1978. (25) Blankschtein, D.;Thurston, G. M.;Benedek, G. B. J. Chem.Phys. 1986,85, 7268. (26) Puwada, S.; Blankschtein, D. J. Chem. Phys., 1990, 92, 3710. Blaukschtein,D.; Puwada, S. MRSSymp. Proc. 1990,177,129. Puwada,
S.; Blankschtein, D. In Surfactants in Solution; Mittal, K.L., Shah, D. O., Eds.; Plenum: New York, 1991; Vol. 11, p 95. (27) (a) Puwada, S.; Blankschtein, D. J.Phys. Chem. 1992,96,5567. (b) Puwada, S.; Blankschtein, D. J . Phys. Chem. 1992, 96,5579.
Langmuir, Vol. 8, No.11, 1992 2681 properties of aqueous solutions of ~ i n g l eand ~ ~mix# *~ surfactants, with particular emphasis on CiEj nonionic surfactants. The theoretical formulation, resulting from combining the descriptions of the mixed monolayer and the bulk solution, is used to predict the variation of surface tension with bulk surfactant concentration and composition. The theoretical predictions are then compared with experimentally measured surface tensions (using the Wilhelmy-plate method) of two aqueous binary CiEj surfactant mixtures: (1)n-dodecyl hexa(ethy1ene oxide)n-dodecyl octa(ethy1ene oxide) (c12E6-c12E8), and (2) n-dodecyl hexa(ethy1ene oxide)-n-decyl tetra(ethy1ene oxide) (c12&&1~84), and are found to be in very good agreement. In addition, the theory is utilized to predict the variation of the monolayer composition with bulk surfactant concentration and composition. It is important to emphasize that the theory presented in this paper requires only one adjustable parameter for each surfactant species, which can be readily determined from the measured value of the surface tension of an aqueous solution containing that particular species at a single surfactant concentration (which can be below, at, or above the cmc of that species). In contrast, all other theories require two to three adjustable parameters for each surfactant species which are obtained by fitting to the experimental data. Furthermore, another advantage of the theory presented here, as compared to existingones, is that it does not require any additional empirical parameters, or ideal-mixing assumptions, to predict the interfacial properties of the resulting binary surfactant mixtures. The remainder of the paper is organized as follows. In section I1 we present the general theoretical formulation of adsorption at the solution-air interface from a mixed surfactant solution. Section I11 describes the materials and experimental procedures utilized to measure surface tensions. In section IV we present the results of our calculationsfor the ClzEeclzEa and C12&&1& mixtures and compare them with the available experimental data. Concluding remarks are presented in section V. 11. Theory A. Equation of State of Mixed Surfactant Monolayer. As described in the Introduction, interactions between uncharged surfactant molecules present in the monolayer include repulsive excluded-area interactions and attractive van der Waals interactions between the surfactant hydrocarcarbon tails. The effect of these interactions can be captured by (1)modeling a surfactant molecule as a hard disk having an area given by either the cross-sectional area of the head group or the tail of the surfactant molecule, depending on which one is larger, and (2) utilizing an equation of state for the mixed surfactant monolayer which, in our case, consists of an equation of state for hard-disk mi~tures,2~ incorporating the effect of steric repulsions, perturbed by the attractive van der Waals interactions to second order in surface der~sity.~ In calculating the attractive component of the surface pressure of the mixed monolayer, we assume that the various adsorbed surfactant species mix randomly. In general, this assumptionwill not be valid if the interactions are sufficiently strong so as to induce long-range ordering or segregation in the monolayer. This is not expected in the case of nonionic surfactant mixtures, but may occur in
Nikm et al.
2682 Langmuir, Vol. 8, No. 11, 1992
some anionic-cationic surfactant mixtures due to the existence of strong electrostatic interactionseB By combiningthe contributions from the excluded-area effectz3 and the attractive interactions to the surface pressure, 11, one obtains the following equation of state for a mixed surfactant monolayer having total area A .
P-
NiuNjb
where Ni' is the number of surfactant molecules of type i in the monolayer, ri and ai = rriz are the hard-disk radius and hard-disk area associated with surfactant molecules of type i, respectively, Bij is the second-order (virial) coefficient (Bij = Bji) associated with attractive pairwise interactions between surfactant molecules of type i and j , Ci denotes summation over all the surfactant species, kg is the Boltzmann constant, and T is the absolute temperature. The first two terms in eq 1 reflect repulsive interactionsbetween surfactant moleculesdue to excludedarea effects, and the last term incorporates the effect of the attractive van der Waals interactions between the hydrocarbon tails of the surfactants. In eq 1, the attractive interactions are treated as a perturbation to the hard-disk repulsions, truncated at second order in the surface density, Nj'/A. The error introduced by this truncation is negligible in the lowsurface-density region but increases in magnitude as the surface coverageincreases. Therefore, in the high-surfacedensity region, such as that encountered in liquid-solid transitions occurring in spread monolayers at sufficiently high surface pressures, the errors can be significant. However, since CiEj nonionic Surfactants possess relatively large head groups, when they adsorb at fluid-fluid interfaces the resulting monolayer remains in the gaseous state even at adsorption saturation.29 This follows because the van der Waals attractions between the CiEj surfactant molecules are not sufficiently strong, due to the relatively large separations between the molecules, to induce a gasliquid phase transition. Consequently, in general, the virial-type expansion of the surface pressure truncated at second order in surface density constitutes a reasonable approximation for adsorbed monolayers of this type. B. Adsorption Isotherms. From the equation of state given in eq 1, one can obtain an expression for the chemical potential of a surfactant molecule in the monolayer using two equivalent approaches. The first involves the use of the thermodynamic relation between the free energy of the monolayer and the surface pressure,7 and the second involves the use of the Gibbs adsorption equation.* Below we present a derivation based on the first approach. At constant temperature, T,and pressure, P, the free energy of a mixed monolayer, F', having a total area, A , can be written as the s u m of an ideal-gas mixture and a contribution reflecting deviations contribution, Piid, from the ideal-gas state, that is
where ;i'" is the standard-state chemical potential of a surfactant molecule of type i a t a referencesurface pressure no = 1 dyn/cm and temperature T, and X i u Ni'/CJViuis the mole fraction of surfactant molecules of type i in the mixed monolayer. The last term in eq 3 arises from the ideal entropy of mixing of the mixed monolayer. Nota that the standard state is defined at a surface preeeure & = 1 dyn/cm, where the mixed monolayer is sufficiently dilute so that it can still be described by an ideal-gas mixture. The chemical potential of a surfactant molecule of type i in the mixed monolayer, piup can be obtained by differentiating the expression for Eb, resulting from the combinationof eqs 2 and 3, with respect toNi' at constant total area, temperature, pressure, and Nj' for j # i, that is, piu = ( a E b / a N j ' ) ~ , ~ p ~This . ~ ~ jyields ..
Substituting the expression for II given in eq 1 into eq 4 and performing the integration yields piu = p?'
b
+ kBT
ai + 2rriCjx;rj
a-cjx;aj
+
a - cjx;aj
arcs
+
where a = A/CJVi" is the average per surfactant molecule in the monolayer, and pi'"' = piu" + kBT(1 + In (k~T/lb)). At thermodynamicequilibrium, the chemical potential of a surfactant molecule of type i in the mixed monolayer, piu, should be equal to that of a surfactant molecule of type i in the bulk solution (either as a monomer or in a micelle), pib, namely, piu = pib, for all i. In order toevaluate the bulk chemical potentials, pib, we have adopted a recently developed molecular-thermodynamic theory which has been quite successful in predicting the micellar properties and phase behavior of aqueous solutions of single26*z6 and mixed27surfactants, with particular emphasis on CiEj surfactants. Specifically, in the mixedsurfactant case,z7the theoretical formulation combines a thermodymamic theory of mixed micellar so1utions,z7a which captures the salient features of these complex fluids at the macroscopic level, with a molecular model of mixed mi~ellization,2~~ which captures the essential physical factors responsible for mixed micelle formation. The molecular-thermodynamic theory can predictz7 a broad spectrum of micellar and phase behavior properties including the mixture cmc, the micellar size and composition distribution, and the two-phase coexistence curves. For simplicity, in what follows, we present a theoretical formulation applicable to solutions containing binary surfactant mixtures. The derivations and results can be readily generalized to solutions containing additional where Hid= kBTCJVi'/A is the surface pressure of an idealgas mixture. The free energy of an ideal-gasmixture, PJd, surfactant species. Consider a solution containing N , water molecules, N Asurfactant molecules of type A, and is given by N B surfactant molecules of type B. Generally, in such a solution there are NIAsurfactant monomers of type A, (28) Gu, B.; Roeen, M. J. J. Colloid Interface Sci. 1889,129, 637. NIBsurfactant monomers of type B, and a distribution (29) Lange, H.; Jeachke, P. In Nonionic Surfactants-Physical Chemistry; Schick, M. J., Ed.;Marcel Dekker: New York, 1987; p 1. (Nna,nsJ of mixed micelles composed of nA surfactant
Surface Tensions of Surfactant Mixtures
Langmuir, Vol. 8, No. 11, 1992 2683
molecules of type A and nB surfactant molecules of type B. Since we are interested in solutions having very low surfactant concentrations (below and slightly above the cmc), the contribution of intermicellar interactions to the free energy of the system is negligible. In that case, the bulk chemical potentials of surfactant monomers of type A and B, p~~ and pgb, respectively, can be written as2' *A
= pAb'O
+ kBT[h X ~ +AX -
(6)
XnA,J n A m
and 2 -(BBAxA'
a
where p ~and~p *~are ~~thet bulk ~ standard-state chemical potentials of surfactant molecules of type A and B in the infinitely dilute limit, respectively, X1A = N1d(NW+ NA + NB)and X1B = NIB/(N~ + N A+ NB)are the bulk mole fractions of surfactant monomers of type A and B, respectively, XnA,nB = NnA,n$(Nw + N A + NB)is the mole fraction of mixed micelles, and X = (NA+ N B ) / ( N , N A NB)is the bulk surfactant mole fraction. For any given bulk surfactant mole fraction, X, bulk surfactant composition, asoh= NA/(NA+ NB),T, and P, we first use the molecular-thermodynamic theory of mixed surfactant solutions to calculate Xi~(X,a~h,T,l'),XIB' (X,ad,TQ), and XnGB(X,amh,T,l'). A detailed description of the calculational procedures can be found in refs 26 and 27. Values of these quantities can then be inserted into eqs 6 and 7 to compute the bulk monomer chemical potentials pAb(X,asoh,TIP)and p ~ ~ ( X , a ~ h , T , lrespec'), tively. At given T and P, the conditions of diffusional equilibrium between the surface phase (monolayer) and the bulk solution require that, for each component, the monomer chemical potential in the monolayer, pi', should be equal to that in the bulk solution, pibe For a binary surfactant mixture (i = A and B) this yields
+
+
C(B~(~B',~,TIP) = c ( B ~ ( x , ~ ~ ~ ~ , T I P )(9)
and
(11)
where Ap.t,O = p~'pO - MAbpo, A~BO= pgUio- pgbi0,r A and U A = ?rrA2, and I-B and OB = T I - Bare ~ the hard-disk radii and hard-disk areas associated with surfactant molecules of type A and B, respectively, XA' + XB' = 1, and BAB= BBA. At any given T and P, we want to calculate (i) the variation of the monolayer composition, XA' ( X B ~= 1 XA'), and (ii) the variation of the average area available to an adsorbed surfactant molecule, a, with bulk surfactant concentration, X, and bulk surfactant composition, asoh. Once XA" and a are known as a function of X and amh,the surface pressure, n, or equivalently the surface tension, u = uo ll (where uois the surface tension of pure water), can be calculated as a function of X and a-h using eq 1 with i j = A,B. As will be shown in sections IIC and IID, for the alkyl poly(ethy1ene oxide), CiEj, surfactants considered in this paper, the values of ai and Bij(ij = A,B), appearing in eqs 10 and 11, can be calculated from their molecular characteristics. Consequently, the two unknowns, XA' and a, can be obtained by solving eqs 10 and 11simultaneouslyonce the differences in the standardO Apso = state chemical potentials, ACAO = PA'~O - P A ~ ~ and p B 4 - pgblo, are known. The quantity Apia = pi'" - pib*' (i = A and B) reflects the free-energy change associated with transferring a surfactant molecule of type i from the bulk solution to the monolayer and can also be referred to as the "free energy of adsorption". Values of Apia for various surface active substances have been estimated in the past by analyzing the surface tension versus bulk solute concentration curve over a finite concentration range in the dilute low-surfacecoverage region.w32 However, this method of estimating &io utilizes the measured surface tension v e r w surfactant concentration curve, and therefore does not provide an independent way of calculating the Apia values. Consequently, here we deduce Apia by fitting the predicted , surface pressure of an aqueous solution of pure component i to that measured experimentally at a single surfactant concentration, Xi*(mole fraction), which can be chosen arbitrarily. Here we have chosen Xi* to be lo* for C1& and C12E8 (see Figure 3) and for (see Figure 4). More specifically, using the measured surface pressure value of pure component i (A or B) at Xi*, II(Xi*) = uo - u(Xi*) (where we have used the measured value a0 = 73.1 dyn/cm for the surface tension of pure water), along with the calculated values of ai and Bii derived in sections IIC and IID, in eq 1, we first find the value of a(Xi*). This value of a(Xi*) is then substituted in eq 10 for i = A, and and A~BO,respectively. The in eq 11 for i = B, to find A ~ A O
-
and
where ZB' = 1 - XA'. Using eq 5 (with i = A) for and (with i = B) for PB", and eqs 6 and 7 for I.(A~ and p ~ respectively, we can rewrite eqs 8 and 9 as follows
+ BBBxB')
~
(30) Ward, A. F. H.; Tordai, L. Nature 1946,158,416; Trans.Faraday SOC.1946, 42, 408, 413. (31) Pmner, A. M.; Anderson, J. R.;Alexander, A. E. J. Colloid Sci. 1962, 7, 623. (32) Rosen, M. J. Surfactants andlnterfacial Phenomena;John Wiley & Sons: New York, 1989.
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2684 Langmuir, Vol. 8,No. 11, 1992
surface coverages (a values). In so doing, we effectively valuesof AMiOfor C1oE4, c12E6,and C12E~at25 OC obtained is this way are -10.48 kBT, -14.65 kBT, and -15.42 kBT, incorporate the influence of a on Oh in the region where it is most important, while simplifying considerably the respectively. The relative magnitudes of the ApiO'S so deduced are consistent with the expectation that the freemathematical complexities associated with the actual energy change associated with transferring a CiEj surfaccoupling of ah and a. There is some experimentalevidencewhich suggestsB~34 tant molecule from aqueous solution to the solution-air that the head-groupareas of CiEj Surfactants adsorbed at interface should become more negative as the number of solution-air interfaces scale with the number of ethylene methylene groups in the hydrocarbon tail increases (due oxide groups, j, as j1I2. This polymer scaling-law type tothe increased hydrophobicity), and it should also become slightly more negative as the number of ethylene oxide relation can be rationalized as follows. In an adsorbed CiEjsurfactant monolayer having a finitesurface coverage, units in the head group increases because poly(ethy1ene the PEO volume fraction near the solution-air interface, oxide) is also known to be somewhat surface active. defined as the total dry volume of the PEO chains divided C. Calculation of the Hard-Disk Areas of Polyby the volume of the interfacial layer containing the PEO (ethylene oxide) Head Groups. In order to utilize eq chains, can be quite high p0.5). Under such conditions, 1 to calculatethe surface pressure of an aqueoussurfactant the environment seen by the polymer chains resembles solution containing two surfactant species (A and B), we that in a polymer melt near an impenetrable wall (which must first determine (i) the hard-disk radii rA and m, or mimica the solution-air interface). For long polymer equivalently the hard-disk areas QA = ?rrA2and OB = um2, chains, which obey polymer scaling the average and (ii) the second virial coefficients BAA,BBB,and Bm height (or PEO layer thickness), lh, of a PEO chain in the = BBA.In this section we calculate the hard-disk areas melt scales36as n1I2,where n is the number of bonds in corresponding to the poly(ethy1ene oxide) (PEO) head the chain (note that for PEO, n = 3j). The average area groups of CiEj nonionic surfactants, and in section IID we that a PEO chain occupies a t the solution-air interface calculate the second virial coefficients. can be estimated as a h = u p m / l h , where UPEO is the dry In general,if the surfactant head group is small compared volume of the chain26*37 and is given by upm(j) = 63.5j (A3) to the cross-sectional area of the hydrocarbon tail, then n.36137Since UPEO n,the average head-grouparea a h the hard-disk area is given by the latter, which is the case scales as Upm/lh n/n'I2 n1I2 j112, as observed for fatty acids and alcohols." On the other hand, for CiEj e ~ p e r i m e n t a l l y .Although, ~ ~ ~ ~ ~ strictly speaking, the desurfactants, whose head groups are rather bulky, the hardscription above is applicable in the limit of long polymer disk area is given by the average cross-sectional area of chains, arecent computer simulation study24indicatesthat, the PEO head group, which can be estimated from the in the case of PEO, polymer scaling laws are obeyed with average area, a h , that the head group occupies when it is a 4% accuracy for fairly short chains (i 1 4)of the type adsorbed at the solution-air interface. In other words, ai in this paper. Accordingly, we assume that = Oh, fori = A and B. The early work of Schick s ~ g g e s t s ~ 1 ~considered ~ ahG) = Aj1I2, where A is a numerical p r e f a c t ~ r .Below, ~~ that, for an adsorbed CiEj surfactant molecule, the PEO we estimate ah(j) fori = 6 (correspondingto which chain extends into the solution and assumes a randomis a convenient intermediate value between j = 4 (correcoil configuration with one end (that connected to the Ci sponding to c184) and j = 8 (corresponding to C12Ea). chain) anchored at the surface and the rest of the chain Subsequently, we use the scaling-law relation, (lh(j) = restricted to lie in the half-space below the solution-air ~h(j=6)(j/6)'/~, with j = 4 and 8,to estimate the average interface. head-group areas of C10E4 and C12E8, respectively. In estimating the average PEO head-grouparea, a h , one As stated earlier, to estimate the average head-group is faced with a number of complicating features. These area, ah(n), of a PEO chain having n = 3 j bonds, we will include (i)an inevitableambiguityin assigning an effective utilize the relation,a&) upm(n)/lh(n),where the average area to a diffuse and flexible chainlike macromolecule such height, B(n), needs to be estimated. This can be done by as PEO, (ii) the need to incorporate the conformational analyzing the behavior of the bond-density distribution, characteristics of a PEO chain adsorbed at one end to the P n ( Z ) , normal to the interface, where z is the normal solution-air interface, and (iii) the possible dependence distance from the interface. To obtain P n ( Z ) , we performed of PEO chain conformationson surface coverage,reflecting Monte Carlo simulations of PEO chains of various lengths interactions between different PEO chains at high surface attached at one end to an impenetrable wall (whichmimics coverages, when the average distance between the adsorbed the solution-air interface). The conformationsof the PEO PEO coils becomes comparableto the average dimensions chains are generated using the rotational isomeric state of the PEO coils. Point iii, in particular, suggests that, in (RIS) model.39 The Monte Carlo-RIS approach has been general, an accurate modeling of the surfactantmonolayer described" elsewhere and will not be repeated here. We should treat the average PEO head-group area, a h , as a found that P n ( Z ) exhibits a maximum near the wall and function of surface coverage or, equivalently, of the decays slowly to zero at large z values. It is noteworthy available area per surfactant molecule, a. As will be discussed in section V,a calculation that effectively couples (34) van Vooret Vader, F. Tram. Faraday SOC.1960,56, 1078. a h and a is very involved and can only be performed (35) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell numerically. Our aim here is to provide a "simplified University Press: Ithaca, NY, 1975. working scheme" to estimate Oh. In view of the fact that (36) Mulley, B.A. In Nonionic Surfactants;Schick, M. J., Ed.;Marcel Dekker: New York, 1967; p 421. the effect of repulsive steric interactions, captured in a h , (37) Nagarajan, R.;Ruckenstein,E.J . Colloid Interface Sci. 1979,71, is most pronounced at relatively high surface coverages 580. (when a and a h are comparable), below we estimate a h in (38)Note that the same scaling-law relation is used to calculate the average PEO head-group areas of CiEj surfactantmolecules which form the high-surface-coverage region. Since the precise a h micelleain order to predict bulk micellarsolution propertiea. Interentingly, value is less important in the dilute low-surface-coverage the value of ahG'6) used to predict bulk properties (40 A*) is somewhat region, where the monolayer behaves ideally (a >> Oh), we larger than the one used to predict surface properties (36.3A*, see Table I). This appears reasonable since the larger curvature of the micellar assume that the a h value obtained in the high-surfacesurface, as compared to that of a planar monolayer, d o w s for a larger coverage region can be utilized over the entire range of lateral expansion of the PEO chains.
-
-- - -
-
(33) Schick, M. J. J . Colloid Sci. 1962, 17, 801.
(39) Flory, P. J. Statistical Mechanics of Chain Molecules; Hanser Publishers: New York, 1986.
Surface Temiom of Surfactant Mixtures
Langmuir, Vol. 8, No. 11, 1992 2606
that the pn(z) calculated using the Monte Carlo-RIS approach is consistent with an analytical expression derived4 by Heeselink for a terminally-attached ideal chain on a simple-cubiclattice. Since the calculated p,(z) curves did not exhibit a sharp decrease to zero beyond a threshold value of z, which, if present, could be identified as the average polymer height, lh(n), the following analysis was pursued. We calculated the average number of bonds found in the space between the wall and a plane parallel to the wall and located at a normal distance z from the wall, n(z) = Jo2pPn(z)dz. We found that n(z) is a monotonically increasing function of z, which for z = (R,2)1/2 = root-mean-square end-to-end distance of the chain, satisfies n(z)/n= 0.96 for polymer chains having n > 30 (j > 10). In other words, on average, most of the bonds are at a distance smaller than (Rn2)'/' from the wall. Therefore, as an estimate for these chains, one can approximate the average polymer height, lh(n), as being equal to (Rn2)'/2. For the shorter PEO chains of interest in this paper (having j Ilo), we will utilize the same criterion of n(lh)/n = 0.96 to identify the value of lh(n).41 Implementing the criterion described above for j = 6, we obtain Mj=6) = 10.5 A. Using this value of k, along with the value up~o(j=6)= 381 A3, we find that ah(j-6) = Up~oG=G)/lh(j=6)= 36.3 A'. Using this value of ah(j=6)in the scaling-lawrelation, ah(j) = ~hG=6)(j/6)'/~, we find that ah(j=4) = 29.6 A2 and ah(j=8) = 41.9 A2. Values of ah(j) for other CiEj surfactants having 4 < j' I 10 can also be computed using the same scaling-law relation.42 D. Calculation of the Second Virial Coefficients. In our theoretical formulation, the mixed surfactant monolayer is modeled as a binary mixture of hard disks interacting through attractive van der Waals forces. This implies that the interaction potential between a surfactant molecule of type i and one of type j (ij = A,B) at a separation r (distance between the centers of the disks) is infinitely repulsive for r Idij and attractive for r > dij, where dij = ri + rj, and ri and rj are the hard-disk radii associated with surfactant molecules of type i and j , respectively. Values of ri = (ai/X)'/' for c10E4,C&6, and C1zE8, obtained from the ai values calculated in section IIC, are listed in Table I. The attractive interactions are treated as a perturbation to the hard-disk mixture, and their contribution to the surfacepressure is evaluated using a virial-type expansion truncated at second order in the surface density. Denoting the attractive part of the interaction potential by uij(r),the second virial coefficient, Bij, is given by43 Bij = XJdi(l - e-uij(r)/kBT)ydr
(12)
The attractive interaction potential, uij(r),between two parallel and opposing hydrocarbon chains having equal lengths in the all-trans configuration has been calculated by Salem.4 For chains having lengths much greater than their separations, u(r),is given by the following simple expression4 ~~
~~~
(40)Heeselink, F. Th.J. Phys. Chem. 1969, 73, 3488. (41) Interestingly, we fmd that to satisfy this criterion for j -< 10, the value of b(n) is somewhat smder than (Rn2)l12. This may reflect the fact that the effect of the wall, which c a w the chains to stretch in the direction normal to the wall, is lees pronounced for the shorter chains. (42) For PEO chains having j > 10, a similar scalinglaw relation,a h t i ) = Aj112can be used,provided that the numerical prefactor A is calculated for a PEO chain of sufficient length > 10). (43) Fowler, R.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University London, 1965. (44)Salem, L.J. Chem. Phys. 1962,37, 2100.
u(r) = -1240nJr5 (kcal/mol) (13) where n, is the number of carbon atoms in the hydrocarbon chains and r (in units of A) is the separation between the centers of mass of the two chains. However, the hydrocarbon chains of surfactant molecules adsorbed at the interface are not always in the all-trans configuration or parallel to each other. Therefore, in principle, the interaction potential has to be averaged over all chain conformations. An approximate calculation can be done using a Monte Carlo approach, in which two hydrocarbon chains are anchored at one end to an impenetrable wall (which mimics the solution-air interface) at a given separation r, and their conformations are generated simultaneously using the RIS model for hydrocarbon chains.39 In evaluating the interaction potential between the two chains, we regard each CHZand CH3 group in the hydrocarbon chain as a pseudoatom and use a LennardJones potential to model the attractive interactions . between pseudoatoms, that is, u(r) = - 4 4 ~ / r ) ~The Lennard-Jones parameters used are those given by Jorgensen:46 UCH, = 3.39 A, U C H ~ = 3.47 A, u c H d H a = (ucH~ucH~)'/~, C C H ~= 0.19 kcal/mol, ECH, = 0.23 kcal/mol, / ~ . intrachain and interchain and ~ C H =~ ( ~HC H ~ E C I ~ J ' The repulsive excluded-volume interactions between pseudoatoms are taken into account by discarding those conformations in which the centers of any two pseudoatoms are located at a distance which is less than the LennardJones diameter associated with them. The total interaction potential between the two hydrocarbon chains is given by the sum of the interactions between all pseudoatoms, and this is then averaged over all possible chain conformations. We would like to point out that this approximate calculation of u(r) tends to underestimate the magnitude of the interaction potential because it assumes that the statistical-mechanicalBoltzmann factor, reflecting the effect of interchain interactions on the probability that a given chain conformation is realized, is unity for all the allowed chain conformations. Interestingly, the Monte Carlo results indicate that for two hydrocarbonchains having as few as 10 carbon atoms, the average interaction potential between them varies as (as in eq 13) over a range of separations from 4.61 A, which is the minimum distance that the two chains can approach each other, to 20 A, beyond which the van der Waals attraction becomes negligible. Moreover, the magnitude of the computed& average potential is consistently lower by about 510% than that corresponding to two parallel and opposing chains given in eq 13.& For comparison, if the hydrocarbon chains are in the all-trans configuration, but can rotate freely about an axis perpendicular to the interface, then the interaction potential is much smaller in magnitude. This is because in this case the distance of closest approachof two CHZgroups residin in different chains at any separation greater than 4.61 is determined by the actual separation, rather than by the Lennard-Jones diameter, U C H ~ . Consequently, since the Lennard-Jonespotential decays asr*, the totalinteraction between the two chains is much weaker in this case. This suggests that in the calculation done by Smith," in which he made the simplifying assumption that the two hydrocarbonchains are in the all-trans configuration to estimate the interaction energy between them, the attractive interaction potential was underestimated. Since the value of the interaction potential obtained from our Monte Carlo simulations differs by less than
i
(45) Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. SOC.,1984,106,6638. (46)N i b , Y .J.; Sarmoria, C.; Blankachtein, D. Unpublishedresulta.
' Nikaa et al.
2686 Langmuir, Vol. 8, No. 11, 1992 Table I. Values of €lard-Disk Radii, r h Hard-Disk Areas, a&I, Second Virial Coefficients, B* and Free Energies of Adsorption, Ad', of the CiEj Surfactants Used in the Present Study.
a1 =
~~
29.6 3.07 36.3 3.40 41.9 C12Ee 3.65 a The temperature is 25 "C. C d 4 c12E6
-10.48 -14.65 -15.42
-172.7 -124.7 -87.7
Table 11. Values of Second Virial Coefficients, Bu,of the CiEj Surfactant Mixtures Used in the Present Study* surfactant mixtures C12EgC12Es Cl2EgCl& a
(ken -103.4 -144.8
Bij
The temperature is 25 "C.
10% from that calculated using eq 13, here we will adopt eq 13 because it provides a convenient analytical expression to estimate the attractive contribution to the surface pressure. The interaction potential between two hydrocarbon chains having different chain lengths can also be calculated using a simple extension of eq 13. If the hydrocarbon chain lengths do not differ by more than a few carbon atoms, then the interaction energy between them is approximatelyequal to -1240ildP (kcal/mol),where Re = (n,l + n,2)/2 is the average chain length. If the chain lengths are very different, then this approximation is no longer valid because the longer chains can occupy the empty spaces above the shorter ones, and the calculations need to be performed numerically using the Monte-Carlo approach discussed earlier.46 In order to perform the integration and obtain the various second virial coefficients, we insert eq 13, with the appropriate n, (or ilc) values, into eq 12, with the appropriate dij = ri + rj values (see Table I). The Bij values obtained in this way are listed in Tables I and 11.
Surfactant Mole Fraction, X
Figure 1. Predicted mole fraction of C& in the monolayer, xUc1*, as a function of total bulk surfactant Concentration, X, for aqueous solutions of C12EgC12Es mixtures at the following bulk surfactant mixture compositions (mole fraction of C1& in the mixture): = 0.3 (-), 0.5 (- -), and 0.7 * -1. The temperature is 25 "C.
-
(e
\
111. Experiments
A. Materials. Homogeneous surfactants
c l & (lot 9004), (lot 90111, and C1& (lot 9054)were obtained from Nikko Chemicals, Tokyo, and used without further purification. 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. In addition, to ensure uniformity in the results, all the measurements were conducted using the same lot for each surfactant. All solutions were prepared using deionized water which had been fed through a Milli-Q ion-exchange system. B. Surface Tension Measurements. The surface tensions of mixed surfactant solutions of given compositions were measured as a function of the bulk surfactant mole fraction using a Wilhelmy-plate tensiometer (Kruss KlOT). The accuracy of the surface tension measurements is *O. 1dyn/cm. All measurements were carried out in a thermostated device maintained at a constant temperature of 25 "C. Before use, all the glassware was washed in a 1 N NaOH-ethanol bath, then in a nitric acid bath, followed by thorough rinsing with Milli-Q water and baking in an oven. The Wilhelmy platinum plate was washed using acetone, rinsed in Milli-Q water, and flamed until red hot before each measurement.
C12&
IV. Results and Discussions Using the theoretical formulation presented in sections IIA and IIB, with the values of the hard-disk areas, second virial Coefficients, and ApiO's listed in Tables I and I1 (see also sections IIC and IID), one can predict the mole fraction of C12E6, x ' c l l ~= N'clzE$(Nuclz~ + nu^), whereB denotes either C12E8 or C10E.4, adsorbed at the interface as a
1
Figure 2. Predicted mole fraction of Ct& in the monolayer, xUclrpe,as a function of total bulk surfactant concentration, X, for aqueous solutions of C12EgClOE4 mixtures at the following bulk surfactant mixture compositions (mole fraction Cl& in the mixture): amb = 0.3 (-), 0.5 (- - -1, and 0.7 (. .). The temperature is 25 "C.
-
function of the bulk surfactant concentration, X, at various bulk surfactant compositions, asoh. Note that the bulk surfactant composition, asoh,is defined as the bulk mole fraction of C12& in the appropriate mixture. Theoretical predictions of x'ccII~versus X are shown in Figure 1 (for the C12& cmc(a,h) to be reasonably good. For X C cmc(aWh),there are no micelles in the solution (x,,,,, = o), and X1A = aeOhX,X1B = (1- c~,h)x,where A refers to ClzE6 and B refers to either C12& or C10E4. This leads to very simple expressions for p~~and p.gb (see eqs 6 and 7 ) . The very good agreement between the predicted and measured u values for X C cmc(aeOln) indicatesthat the essentialfeatures of the mixed surfactant monolayer,including the calculated values of ai's and Bij'S, are captured fairly well in the theoretical formulation presented in section 11. It is also interesting to observe that, in the case of the ClZE6> Oh) and assumes that this a h value can be utilized over the entire range of surface coverages (a values). In so doing, one effectivelyincorporates the influence of a on Oh (and indirectly on Bij) in the region where it is most important, while simplifying considerably the mathematical complexities associated with the actual coupling of a h a d a. In the 2D-gas approach presented in this paper, the equation of state of the monolayer incorporates explicitly
+
(47) Kahlweit,
M.;Busse, G.;Jen, J. J. Phys. Chem. 1991,95, 6580.
Surface Tensions of Surfactant Mixtures the molecular characteristics of the various adsorbed species (through the values of ai and Bij, see eq 1). Accordingly, the theory developed in section I1 provides a convenient and useful tool to predict the interfacial properties of single- and mixed-surfactant solutions of various types. For example, the theory can be extended to treat double-chain surfactant mixtures and lipid mixtures, as well as fluorocarbon-hydrocarbon surfactant mixtures. In addition, the theory can also be extended to describe ionic-nonionic and anionic-cationic surfactant mixtures once the electrostatic interactions are incorporated into the formulation. Work along these lines is in progress. For surfactant systemsof a more complex nature, for example, mixtures of surfactants having very different or mixtures of branched hydrocarbon chain surfactants, an accurate modeling of the mixed surfactant monolayer, including a description of repulsive and attractive interactions between the various adsorbed
Langmuir, Vol. 8, No. 11, 1992 2689 species, represents a very challenging problem which deserves further consideration. Work along these lines is in progress.
Acknowledgment. This research was supported in part by the National Science Foundation (NSF) Presidential Yound Investigator (PYI)Award to Daniel Blankschtein, and an NSF Grant (No.DMR-84-18778)administered by the Center for Materials Science and Engineering at MIT. Daniel Blankschtein is grateful for the support of the Texaco-Mangelsdorf Career Development Professorship at MIT. He is also grateful to the following companies for providing PYI matching funds: BASF, Kodak,and Unilever. The authors are grateful to Claudia Sarmoriafor making availableto us her simulation programs and results, as well as for her helpful comments on conformational aspects of polymer chains.
