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Langmuir 1992, 8, 1082-1085
Micellization Process with Emphasis on Premicelles Marjorie J. Voldt Rancho Bernard0 Convalescent Hospital, 16532 Pomerado Road, Poway, California 92064 Received September 30, 1991 A study of the packing of n-octyl sulfate chains to reduce their external (hydrophobic) area has lead to a greater understanding of the micellization process. The distribution of concentrations of all species of aggregates of amphiphiles in water, including monomers, being controlled by their formation constants, is such that a critical micellization concentration exists where a small fraction of the material is present in a distribution of premicellesof low aggregationnumber in equilibrium with the micelles. The key words are 'distribution of premicelles", replacing 'monomer" in earlier interpretations. For sodium n-octyl sulfate the optimum arrangement of the monomers in premicelles is for the chains (in all-transconformations) to lie parallel with the sulfate heads far apart. The outer chains have almost half their hydrocarbon surface removed from contact with water. For interior chains this loss is over 80%. A t its experimental critical micelle concentration (cmc) in water at 25 "C of 0.130 m, the concentrations in monomers is 0.49 mm, dimers 0.76 mm, trimers 0.054 mm, and tetramers 0.002 mm for a total concentration in premicelles of 1.306 mm, 1%of the amphiphile present. Premicelles from monomers with a gauche link are less stable because each gauche link costs about 500 cal/mol and interior monomers lose a smaller fraction of their contact area with water.
Introduction Micelles of amphiphiles in water and in brine and other polar solvents have a long and honorable history. A skeleton includes discovery by J. W. McBain in 1911,Hartley's spherical micelles and the concept of counterion binding and critical micelle concentration (cmc) in 1936, micellar weight by Debye based on incoherent light scattering, charge from effects of some low molecular weight electrolytes on the cmc values of sodium n-dodecyl sulfate (K. J. Mysels), and distribution of micellar aggregation numbers near the cmc by several researchers, mostly in 1972, P. Mukerjee, B. W. Ninham e t al., and Charles Tanford among them. The author asks forgiveness for omissions and for any mistakes in assigning priority. The first paper of this series' dealt with the process of forming dimers of sodium n-octyl sulfate and its dodecyl analogue, using the algorithms postulated by Abe, Jernike, and Flory and their parameters (derived to fit butyl and isobutyl spectra).2 This paper uses a new approach. Theory There are 43 conformations of an n-octyl chain and 683 of an n-dodecyl chain, described by 5- and 9-letter codes, respectively. For both of the chains the all-trans conformations are the only ones whose potential energies need to be evaluated. For each the hydrophobic contribution to the potential energy is taken as the loss in interfacial chain area with water with aggregation multiplied into a so-called hydrophobic constant of 25 cal mol-' A-* a t 298.3 K. The chain atoms are made of spheres whose normal van der Waals radii (1.7 A for carbon, 1.2 A for hydrogen) are augmented by 1.5 A to allow for change in the water structure.
Confining potential energy calculations for premicelles to the all-trans conformation of monomers is dictated by the energy cost of 500 cal per gauche link. In the absence +Deceased November 4, 1991. (1) Vold, M. J. J. Colloid Interface Sci. 1990, 135, 520. (2) Abe, A,; Jernike, R. I.; Flory, P.J. J . A m . Chem. SOC.1966,88,633.
of aggregation probability considerations lead to an average number of gauche links of 1.68 for n-octyl and 2.77 for ndodecyl amphiphiles.' Just above the cmc other considerations lead to an estimate of three gauche links per mole for sodium dodecyl sulfate, confirmed by small angle neutron ~ c a t t e r i n g . ~ Monomers are then assembled into dimers, trimers, etc. For dimers the E,,f energy (named for Abe, Jernike, and Flory) is minimized by arranging the two monomers with molecular axes parallel to the 2 coordinate axis but running in directions from the first to eighth carbon atoms and vice versa to maximize head separations (and thus reduce head group electrostatic repulsion). If one of the monomers is shifted along its 2 axis, charge separation is increased, thus reducing the repulsion between head groups. However, the chain overlap giving rise to the hydrophobic effect (attractive) is also reduced. For greatest stability a balance is struck. The new feature over ref 1is that the hydrophobic effect uses a fraction of each monomer surface instead of the change in absolute surface area. Averaging potential energies over a selection of conformations is abandoned. As is well known, there is a region of concentration for each amphiphile-solvent system, its critical micelle concentration, a t which the percent of amphiphile in nearly monodisperse full micelles approaches 100. Tanford4uses 5 % , a bit higher than most other hypotheses. (That is, if the stoichiometric concentration of amphiphile at the cmc is 0.01 m, the molal concentration in monomeric form is 0.5 mm.) This work reports the behavior of n-octyl sulfate in water and leads to a new interpretation of the cmc.
Computational Procedure Calculations were performed on a series of IBM personal computers except that some surface areas were evaluated on a Sun workstation at the University of Arizona, Tucson, accessed by modem. The hierarchy of programs to create a premicelle starts with one generating the x , y, and 2 , Cartesian coordinates of the 17atom chain -(CH&CH3 in its all-trans conformation with its molecular axis (line C 1 C8) parallel to the 2 coordinate axis.
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(3) Gruen, D. W. R. Prog. Colloid Polym. Sci. 1985, 79, 6. (4) Tanford, C . J . Phys. Chem. 1974, 78, 2469.
0 1992 American Chemical Society
Langmuir, Vol. 8, No. 4, 1992 1083
Micellization Process with Emphasis on Premicelles
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MOLECULAR A i IS
MOLECULAR AXIS
Figure 1. A trans n-octyl sulfate chain as viewed down the Y axis (top) and along the X axis (lower). The Z axis of the Cartesian system runs horizontally. The scale block is 2 A on edge: squares, carbon atoms; diamonds,hydrogen atoms;stars, sulfate. Note the zigzag of the chain and the aptness of the designation 'molecular axis". The sulfate group (four oxygen atoms forming a regular tetrahedron with the sulfur atom at ita center) is then attached to the hydrocarbon chain through an oxygen atom with a tetrahedral LSOC, angle. The length, girth and sulfate and hydrocarbon interfacial areas are then obtained for later comparisonwith corresponding properties of aggregates. Two (or more) such monomers treated as rigid bodies are then positioned near each other with the aid of a workhorse program that can shift a monomer's center of mass, rotate it through any angle about any axis, or reflect it in any of the three Cartesian planes (For XU,z -z for all 30 atoms, etc.). Their final position must avoid excessive Born repulsion of the electrons of atoms on different chains, measured here using the pseudo-Lennard-Jones function and parameters of Abe, Jernike, and F10ry.~No pair of atoms of two hydrocarbon chains are permitted closer than the inflection point between ita high maximum and shallow minimum. We use 0.46 A for carbon-carbon pairs, 0.74 A for carbon-hydrogen pairs, and 0.97 A for hydrogen-hydrogen pairs. Additional restrictions are imposed on the sulfur atoms and oxygen atoms to ensure that these hydrophilicelectricallycharged sulfate groups remain in contact with water and as far apart as possible. The separation of every atom pair involving either sulfur or oxygen must be larger than the sum of their radii. Fitting is considered complete when a closer approach by shifting any monomer coordinate by 0.1 A leads to constraint violation. Electrostatic repulsion is taken as
Figure 2. Dimer of n-octyl sulfate as viewed down the Y axis (Le., projected onto the ZX plane). The scale block is 2 A on edge: squares, carbon atoms; diamonds, hydrogen atoms; stars, sulfate. Note how the chains are nested.
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1 -Q2NAv (2) &Dss(i) The sodium sulfate is taken as fully ionized so the lines of force, running through the water, lead to a coulombic repulsion, E, where L = 1 for a dimer, 3 for a trimer, and 6 for a tetramer. D is the dielectric constant of water and ss(i) the sulfur to sulfur separation. Q is the electronicchargeand N*,Avogadro's number; suitable conversion factors expressE in units of calories per mole of premicelle.
E=
. .-
MOLECULAR AXIS
Figure 3. Trimer of n-octylsulfate projected onto the ZX plane. The scale block is 2 A on edge. The atoms have been given different symbols to help distinguish crosses in circles, Maltese crosses, and stars for the sulfate atoms of each monomer; diamonds are hydrogen; squares represent carbon. With trimers not all three heads can be widely separated. I
L
Results and Discussion Visualization of premicelles is facilitated by viewing them along each of the coordinate axes, i.e., projections on the ZX,ZY,or XY planes. Figure 1 illustrates this strategy for the n-octyl sulfate monomer. The top monomer is viewed down the Y axis and shows the zigzag nature of the chain. The lower one is its appearance looking down the X axis onto the ZY plane. The view down the Z axis of the XY plane (not shown) has so many of the atoms so close together that it is not useful. Figures 2-4 display some of these features for the dimer, trimer, and tetramer. Figure 5, the tetramer of Figure 4 with each of the chain atoms ringed by a circle of radius equal to its van der Waals radius, shows how the chainlike monomers develop into knobby ovoids as they aggregate.
MOLECULAR AXIS
Figure 4. Tetramer of n-octyl sulfate projected onto the ZX plane. The scale block is 2 A on edge. Carbon and hydrogen atoms of a chain are given the same symbol. It looks as if the top monomer could be positioned closer to the others but the heads interfere. The final positioningfor all premicelles is done algebraically. Numerical results include electrical interaction of the sulfate heads (two electronic charges separated by distances equal to those between sulfur atoms, hydrophobic interaction between hydrocarbon chains, the net potential energy (Table I), and the percent loss in interfacial area for each monomer (Table 11) (nos. 1and 4 are the monomers a t the top and bottom of Figure 2 with nos. 2 and 3 between them). The net potential energy is the lowest found for any 0.1-A change in relative position of the monomers. Successive addition leads to a "sheet" rather than a "bundle". The 500 cal/mol per gauche link argues intuitively against premicelles involving other conformations. The prediction for one gauche link (chain gtttt replacing
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1084 Langmuir, Vol. 8, No.4, 1992
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O .
Table 111. Calculated Properties of the Micellar Solution at Its Cmc monomer concentrations O.OO0 488 m 1.59 X 1od mole fraction all specid 0.001 30 molarity of amphiphileaa premicellee percent amphiphile aa premicelles 1.00 a Assumed (4.87729 X lo-' m ) . b Except Na+, believed to be ca. 0.04 m.
MOLECULAR AXIS
Figure 5. hemicello of Figure 4 with atoms ringed by circles of their own normalvan der Waals radii. It wrvea to suggestthat the premicellm have a knobby oval shape. Table I. Characteristics of Premicelles dimer trimer dimer (g)" trimer (g)" tetramer electricalpart &mol premicl 436.7 339.3 1335.4 1091.8 2696.3 hydrophobic part cal/molpremicl -7132 -8334 -13164 -8931 -19001 net potl energy cal/molpremicl -6746 -6995 -11829 -6339 -16105 RT at 25 O C -11.39 -11.81 -19.97 -10.70 -27.19 a Monomer chains gtttt. Table 11. Loss of ExChain Area uDon Micellization low of chaiaarea for each monomer. % no. 1 no. 2 no.3 no.4 49 dimer 50 53 dimer @)a 53 52 trimer 45 83 56 trimer (g)a 41 19 84 48 tetramer 45 83 a Monomer chains gtttt.
0
1
2
3
4
5
6
AGGREGATION NUMBER
Figure 7. Molarity of n-octyl sulfate in premicellea at ita cmc (0.13m). The area under the curve is 1%of the total.
dimer formation, is 8.6RT on going from there to trimer, 7.2RTto tetramer.' The counterion (Na+) concentration near the cmc suggests that higher premicelles will not be completely ionized sa that the electrical part is more complicated. Accordingly, pentamera and higher were not modeled directly. Instead the net potential energies were used to obtain formation constants for .reactions of the type
I
The K(q) and [A(@] are in mole fraction unite; oonmntrations in moles per liter are obtained by multiplication by 55.55 (mol/L of water at 25 "0. K(q) is aseumed to have the form I
K(d = + & em(-@& (5) T h e data of Table I are just sufficient to solve for the three constants (81 = 3352.037; 82 = -3644358; /33 = 3.7685). Finally the molarity of each species of aggregate
Q
MOLECULAF AXIS
Figure 6. l b o dimers projeded onto the ZY plane, showing only the carbon atoms and sulfate groups. The upper dimer hae one gauche lipk (code gtttt);the lower is the standard ttttt. The two have nearly the same potential energy. With trimers the pmt type is much leas stable.
chainttttt) isnot borne out for the dimer but is conspicuous for the trimer. These results are included in Tablee I and II. The largest destabilizing influence is Qe ,@weredloss of surface area for the interior monomer. Figwe. 6, comparing views down the X axis (projectians on the ZY plane) of trimers based on chains gtttt VB those based on chains ttttt, show that only a small displacement along the molecular axis suffices to destabilizethe trimer by the 9RT of Table I(19.97-10.70). The premicellesevolvein a pattern with the twooutside monomers loeing about half their interfacial interaction with water, internal ones upward from 80%. The net potential energy change per added monomer, 11.4RTfor
is calculated using e q ' 4 and an assumed, monomer concentration. (C(1) is chosen so that the total concentration of amphiphileequalsthe experimentalcmc of 0.130 m.9 The procedure is similar to that of Mukerjeee but with different assumptions. Reaults are given in Table 111and displayed in Figure 7. Note that the amount in dimers is higher than in monomers, but thereafter decremes rapidly, becoming negligible above an aggregation number of 4. The area under this curve is the molarity of amphiphile in premicelles, as presented in Table II. The critical micelle concentration is the concentration range of rapid change in many experimental properties. (5) Vold, R.D.; Volct,M.J. Colloid a d l n t e r f a c e Chembtry;AddiaonWmley, Reading, MA,1985, p 816. Anhason, E. A. 0.; Wall, S. N.; AlmSren,M.;Hotfmann,H.;Kielmann,J.;Ulbricht,W;Zana,R;~, J.; Tondre, C. J. Phys. Chem. 1976,80,905. (8) Mukerjee, P . J. Phys. Chem. 1972,76, 565. (7) Mukejeo, P.;Mys~le,K. J. Critical Micelle Concentrationr of Aqueous Surfactant Solutions; National Bureau of Standardr: Washington DC, 1970.
Micellization Process with Emphasis on Premicelles A valuable analysis by Mukerjee and Mysels precedes their evaluation of hundreds of reported results. It has been variously interpreted4%**as a concentration below which only monomers are found and above which the amphiphile exists almost entirely ~ E Inearly monodisperse micelles a t the low end up t o that a t which half the amphiphile present is momeric, half-micellized. We here propose a new interpretation as fully compatible with the multiple equilibria nature of the micellization process: The distribution of concentrations of all species of aggregate of (8)Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (9) Debye, P. J. Phys. Colloid Chem. 1949, 53, 1. (10) Philips, J. N.; Mysels, K. J. J. Phys. Chem. 1955, 59, 325. (11) Kahlweit, M.; Teubner, M. Adu.Colloid Interface Sci. 1980,3,1. (12) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci.1983,93, 500.
Langmuir, Vol. 8, No. 4, 1992 1085 amphiphiles i n water, including monomers, being controlled by their formation comtants,is such that a critical micelle concentration exists where a small fraction of the material is present in a distribution of premicelles of low aggregation number in equilibrium with the micelles. The key words are "distribution of premicelles", replacing "monomer" in earlier interpretations.
Acknowledgment. Thanks are due to Research Corp., Tucson, AZ, for financial assistance, to Professor Bruno H. Zimm, Department of Chemistry, University of California a t San Diego, Coprincipal Investigator, and Professor Willis E. Lamb, Jr., Optical Sciences Research Center, University of Arizona, Tucson, for access to a Sun workstation and for programming assistance. Registry No. Sodium n-octyl sulfate, 142-31-4.
