J. Phys. Chem. 1083, 87, 1220-1223
1220
Composition of Mixed Micelles of Polydisperse Nonionic Surfactants 0. 0. Warr, F. Grleser,t and T. W. Healy’ ColloM and Surface Chemistry Group, Department of Physlcal Chemistry, University of Melbourne, Parkvile, Victoria, Australia (Received August 4, 1982; In Final Form: October 21, 1982)
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Micellar solutions of polydisperse nonylphenol ethoxylates have been studied experimentally and theoretically in order to determine the composition of the mixed micelles of these nonionic surfactants. The model, based on phase separation and ideal mixing, predicta that the micelles are enriched in the more hydrophobic components of the distribution and also accounts for the increase in total monomer concentration observed in ultrafiltration experiments.
Introduction Simple nonionic surfactants can be prepared in the laboratory or commercially by the alkoxylation of alkyl alcohols or phenols. A typical series of nonionic surfactants is obtained by ethoxylation of nonylphenol to produce a homologous series of nonylphenol polyethoxylates of varying average number (N) of ethylene oxide adducts by varying the ethylene oxidefnonylphenol ratio in the reaction mixture. This method of preparation generates a Poisson distribution of ethoxy chain lengths about the mean ethoxide chain length of each mixture. The present paper is part of a general study of the solution and interfacial chemistry of such “polydispersed nonionic surfactants” of the ethoxylated nonylphenol type. We consider first the aqueous solution chemistry, including micellization of these systems since their solution chemistry departs quite markedly from that of single species of the same compounds. In subsequent papers we shall consider the distribution of monomer components of each mixture between oil and water and the composition of the “mixed” adsorbed layer at the model solid-aqueous solution interface. The phase separation model of micellization is used to interpret experimental surface tension, dye solubilization, fluorescence studies, and total monomer activity data for a homologous series of polydispersed ethoxylated nonylphenols having N values of 4, 5 , 9 , 11, 13, 15, 20, and 30. These samples are designated N4, N5, N9, etc. In particular, the present paper is directed at determining the activity of each species present as a monomer in the mixed solution. Phase Separation Model of Micellization The formation of micelles in mixed surfactant solutions has been investigated by a number of Much of this work has been directed to determining the critical micelle concentrations (cmc) of binary or ternary mixtures as a function of composition. In the present generalized model of micellization of polydispersed nonionic surfactants the approach used by Clint4 for binary surfactant systems has been used, but with the specific aim of predicting the composition of the bulk solution containing monomeric surfactant and of the mixed micelles produced from any given N sample with its composition give by a Poisson distribution. The distribution about each N is treated as an ideal mixture at all concentrations. In deriving both the cmc of the mixed system (CM) and the monomer concentrations of each species i above the Queen Elizabeth I1 Fellow.
cmc, we have followed the method used by Clint4to obtain the well-known result 1/ CM =
i
(ai/ CMi)
(1)
where Cw is the cmc of i in a solution of pure i, and ai is the mole fraction of i in the mixed solute. From the phase separation model of micellization, the mole fractions, xi,of i in the micellar phase are related to the equilibrium monomer concentration Cimonby Cimon = XiCMi
(2)
The mole fraction of i in micelles is given by
(3) where CT is the total surfactant concentration (CT 1 CM). We now introduce the quantity C,, the cmc of the mixed system containing all species except i. This is obtained similarly to CM and is given by (1 - .i)/C,i
= 1/cM - (Yi/cMi
(4)
Similarly from eq 2 we obtain
C C’mon = (1 - xi)Cpi ,
dl J # l
(5)
or all j # i
CJmon =
c,i(l- c i m o n / c M i )
(6)
Substitution of this result into eq 2 and 3 yields a quadratic in Cimon,the solution of which is Cimon = ([(CT+ CMi - C,i)’ - 4aiCT(C,i - cMi)]’/2 CT - cMi + c,i)/{2[c,i/cMi- 11) (7) The monomer concentration of i in equilibrium with mixed micelles for CT I CM is defined by eq 7 in terms of the cmc values of pure surfactants and the composition of the mixture. Since ionic effects are absent in these ethoxylated surfactant systems, we shall treat the variation of cmc values of a homologous series of pure surfactants by analogy with the fragmental hydrophobic concept of Tanford5 and Thus (1) Nishikido, N.; Moroi, Y.; Matuura, R. Bull. Chem. SOC.Jpn. 1975, 48,1387. (2)Lange, H.; Beck, K.-H. Kolloid 2.2.Polym. 1973, 251, 424. (3)Goto, A.;Sakura, R.; Endo, F. J. Colloid Interface Sci. 1978, 67, 491. (4) Clint, J. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327. (5) Tanford, C, “The Hydrophobic Effect”, 2nd ed.; Wiley-Interscience: New York, 1980.
0022-3654/83/2087-1220$0 1.5010 0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 7, 1983
Mixed Micelles of Polydisperse Nonionic Surfactants
In CMi = a - bi
(8) where CMiis the cmc of a pure nonionic surfactant containing i ethoxy adducts. The coefficient -b is the negative hydrophobic (repulsive) component of the dimensionless Gibbs free energy of transfer of an ethoxy unit from bulk solution into the micelle. From eq 7 we can obtain the total and/or individual concentrations of monomer(s) in equilibrium with mixed micelles for any distribution of species. For a Poisson distribution of chain lengths, the mole fraction of a species with i ethoxy units in the hydrophilic chain is given by f f i - e-ND/i! i = 0, I, 2, ... (9)
TABLE I: Experimental Conditions for Ultrafiltration Experiments
filter XM50 XM50 YM10-A YM10-B UnlO5
pore surf- filtering diam? nominal actant press., nm ~ , c u t o f f b soln psi
3.0 3.0 1.6 1.6 1.2
50000 50000 10000 10000 500
230 89 813 B9 x9
15 15
15 15 45
retardation: %
-0
-0 15t 5 24 t 3 69t 3
a Average pore diameter quoted by Amicon. Nominal molecular weight cutoffs are determined by ultrafiltration of globular macrosolutes; quoted by Amicon. Values determined by filtering solutions below the cmc ; average of three measurements.
where N is the mean of the distribution of the number of ethoxy adducts in the chain. Substituting eq 8 and 9 into eq 1, we obtain an expression for the cmc of a Poisson-distributed surfactant mixture, viz. l/cM = Ee-N(D/i!)ebi-a Downloaded by TEXAS A&M INTL UNIV on August 24, 2015 | http://pubs.acs.org Publication Date: March 1, 1983 | doi: 10.1021/j100230a025
1221
I ,
1
i
= e-(a+mE{(Neb)'/
i!)
(10)
8
Since the power series representation can also be expressed as an exponential function, we may write
cM = ea+N(l-e*)
(11) Now for a value of lbl
