MARILYN F. EMERSON AND ALFREDHOLTZER
1898
On the Ionic Strength Dependence of Micelle Number. 1 1 ’ ~ ~
by Marilyn F. Emerson and Alfred Holtzer Department of Chemistry, Washington University, St. Louis, Missouri
(Received November $8, 1966)
Measurements have been made of the cmc’s and (light-scattering) molecular weights of sodium dodecyl sulfate (SDS) and of n-dodecyltrimethylammonium chloride, bromide, and nitrate in aqueous solvent containing varying concentration of NaC1, NaC1, NaBr, and KaN03, respectively. These experimental values are used with the theory developed earlier to calculate the hydrocarbon contribution to the standard free energy change for addition of one detergent molecule to a micelle of the most probable size (AGmHc). For the cationic detergents, resulting values of A G m H C are independent of ionic strength and roughly independent (-10% spread) of supporting electrolyte. For SDS, a small, systematic dependence on ionic strength is observed. The value for SDS differs from that for the cationic detergents by -20%. Possible explanations of the observed differences are discussed.
Introduction One of the most striking features of solutions of detergents in water is the existence of a critical micelle concentration (cmc). Since, furthermore, the cmc is a readily measurable property, many efforts have been directed toward obtaining a clear understanding of its physical significance. In an earlier paper3 it was shown that the quantity RT In (cmc) is equal to the standard free energy change (AGmfi)for the addition of one more detergent monomer to a micelle which already contains that number of monomers most probable at the cmc (I?). This result is independent of assumptions about phase separation and does not require, for its derivation, a detailed statistical-mechanical analysis of micellar solutions. Such an analysis is, however, consistent with this concl~sion.~ The standard free energy change in question can be divided, conceptually, into a hydrocarbon contribution and an electrostatic contribution. The latter can be expressed in terms of the magnitude of the electrostatic potential at the micellar surface Thus
RT In (cmc)
+ Noe$,(b, a,
= AGmHC
K,
N)
(1)
where No is Avogadro’s number and 8 the magnitude of the protonic charge and it is emphasized that depends on the micellar radius ( b ) , the distance of the closest approach of small ions to the micelle (a), the ionic strength (through the Debye K ) , and the most probable micelle number (I?). T h e Journal of Physical Chemistry
The quantity AG-Hc, being a direct measure of the stability of the hydrophobic bond, is not only of interest in the theory of micelle formation, but may have implications for the theory of the stability of protein secondary, tertiary, and quaternary structure. As eq 1 makes clear, an experimental determination of the cmc, combined with an estimate of $*, can be used to calculate this hydrocarbon part of the free energy change (AGmHC). The quantity K is, of course, readily computed from the known ionic composition of the solution. The micellar radius can be estimated using a combination of the few available results of low-angle X-ray studies and some reasonable assumptions based upon the known dimensions of the monomer molecule^.^ The distance of closest approach can be calculated from b and existing tables of ionic radii3 The micelle number can be evaluated from results of any of the usual kinds of measurement of macromolecular weights, usually light scattering in the case of detergents. Thus, values (1) This investigation was supported by Research Grant RG-5488 from the Division of General Medical Sciences, Public Health Service. (2) Support for some of the computation was provided by National Science Foundation Grant G-22296 to the Washington University Computer Center. (3) M. F. Emerson and A. Holtzer, J . Phy8. Chem., 6 9 , 3718 (1965). This reference is paper number I. (4) R. H. Aranow, ibid., 67, 556 (1963). The authors are grateful t o Dr. Aranow for pointing out to us the relevance of this reference in the present context.
ON THE IONIC STRENGTH DEPENDENCE OF MICELLENUMBER
are assigned to all the relevant parameters before the fact; none remains adjustable. Given the values of these variables, the micellar surface potential can be estimated a~ the appropriate solution to the Poisson-Boltzmann equation. Because the surface potentials are quite high (& 5kT), the nonlinearized form of the Poisson-Boltzmann equation is required, necessitating numerical solution by electronic computation. When this procedure was applied to data then extant for sodium dodecyl sulfate (SDS) in NaCl solutions and n-dodecyltrimethylammonium bromide (DTAB) in NaBr solutions, it was found that A G m ~is c roughly independent of detergent and ionic strength, although a small apparent difference exists between the values of A G m Hobtained ~ for the two detergenha While these results seemed promising, more detailed experimental evidence bearing on the validity of this approach was evidently needed. In this paper are presented results of measurements of the cmc (by light scattering and conductivity) and of i$ (by light scattering) for n-dodecyltrimethylammonium chloride (DTAC) in NaCl solutions and n-dodecyltrimethylammonium nitrate (DTAN) in NaN03 solutions, as well as for DTAB in NaBr solutions and SDS in NaCl solutions, from which AG"Hc for each of these systems is calcuc changing the lated. Thus, the effects on A G r n ~ of ionic strength, the supporting electrolyte, and the charged detergent head group are assessed.
-
Experimental Measurements and Treatment of Data A. Materials. 1. General Reagents. All water used was first distilled in a commercial still, then passed through a charcoal filter (Culligan Water Conditioning Go.), and finally through two successive mixed-bed ion-exchange resin columns (Culligan Water Conditioning Co.). The water thus produced has a specific resistance greater than 1 megohm cm (0.1 ppm as NaC1). The preparation of dust-free water for light scattering has been described previou~ly.~ Unless otherwise specified, all chemicals were reagent grade and were used without further purification. 2. Sodium Dodecyl Sulfate (SDS). SDS was prepared by the reaction of 1-dodecanol (99.6% Clz and 0.4% Clo; obtained from the Givaudan-Delewanna Co., N. Y.) with chlorosulfonic acid, followed by neutralization with NaOH.6-8 The SDS was then extracted into hot 95% ethanol, recrystallized several times from 95y0 ethanol, and finally washed with spectranalyzed n-hexane (Fisher)
1899
for 12 hr in a Soxhlet extractor to remove trace amounts of l-dodecan~l.~The final product was dried over a Molecular Sieve (Fisher, Type 4A) in a vacuum desiccator. The product was characterized by its conductance; values of the critical micelle concentration (cmc), the equivalent conductance at the cmc (Aomo), and the equivalent conductance at infinite dilution (Am) of the sample in water at 25" are presented in Table I, along with the corresponding literature values. The agreement is satisfactory. Table I: Characterization of SDS and DTAB Samples by Conductance &mcv
SDS SDSa DTAB DTABb (1
See ref 10-12.
L
P
Cmc,
cm*/ohm
moles/l.
equiv
om*/ohm equiv
0.00800 0.00813 0.0148 0.0146
67.0 66.8 93.0 91.0
74.3 71.7 101 102
See ref 13.
3. n-Dodecyltrimethylammonium Bromide (DTAB). DTAB was prepared from n-dodecyl bromide (Eastman) by a method described elsewhere.'*J4 The salt was recrystallized several times from acetoneether.* Values of the cmc, Acme and A, of the product in water at 25" are presented in Table I along with the corresponding literature values. l 8 Again, the agreement is satisfactory. 4. n-Dodecyltrimethylammonium Chloride (DTAC). DTAC was prepared by extensive dialysis of DTAB, first against NaCl solution (3 M ) until no Br- could be detected in the dialyzate (by oxidation with chlorine water and shaking with carbon tetrachloride), then against water.8 At the high concentration of added (5) T. M. Schuster, Ph.D. Thesis, Washington University, St. Louis, Mo., 1963. (6) M. J. Schick, private communication. (7) J. L. Kurz, private communication. (8) M. F. Emerson, Ph.D. Thesis, Washington University, St. Louis, Mo., 1966. (9) J. L. Kurz, private communication. (10) R. J. Williams, J. N. Phillips, and K. J. Mysele. Trane. Faraday Soe., 51, 728 (1955).
(11) A. Wilson, M. Epstein, and J. Ross, J. Colloid Sei., 12, 346 (1957). (12) P. Mukerjee, K. J. Mysels, and C. I. Dulin, J . Phy.3. Chem., 62, 1390 (1958). (13) A. B. Scott and H. V. Tarter, J. Am. Chem. SOC.,65,692 (1943). (14) W. Bruning and A. Holteer, ibid., 83, 4865 (1961).
Volume 71, Number 6 May I067
1900
electrolyte present during most of the dialysis, the detergent monomer concentration is very low, so that, although monomers can pass through the dialysis membrane,15 they do so much more slowly than does the added electrolyte. Thus, small-ion equilibrium can be reached without significant over-all loss of detergent. After dialysis, the water was removed from the sample by lyophilization. This detergent, unlike DTAB, is highly hygroscopic and was dried for several hours over P205in a vacuum desiccator before use. The sample was then transferred in a drybox from the desiccator to weighed vials, which were then stoppered, removed from the drybox, and weighed again. The total amount of chloride in a weighed sample of the detergent preparation was determined gravimetrically by precipitation of AgCl with AgN03 solution. Assuming that the major impurity is NaCl, the sample was found to be 99.38 wt DTAC. 5 . n-Dodecyltrimethylammonium Nitrate (DTAN). DTAN was prepared by reaction of a solution of DTAB with a stoichiometrically equivalent AgN03 solution.8 After the precipitation of AgBr was complete, the mixture was heated to just below the boiling point (to coagulate the AgBr precipitate), cooled, and passed through a medium porosity, glass sinter (pore size 10-15 p ) . The almost-clear filtrate was centrifuged at 20,000 rpm for 1 hr, the top two-thirds of the solution decanted, and one small portion of it tested for the presence of Ag+ (with NaC1) and another for Br(with AgN03). Both tests were negative for all solutions of DTAN prepared in this manner. The DTAN was recovered from the solutions by lyophilization. B. Preparation of Solutions and Expression of Concentrations. The solutions used in this study normally contained the detergent solute and a solvent that was itself an aqueous solution (e.g., a typical solvent would be 0.2 M NaC1). A large batch of this solvent was prepared in advance and stock detergent solutions were made up by dissolving a weighed detergent sample in the desired aqueous solvent in a volumetric flask. Subsequent volumetric dilutions of these stock solutions with the same solvent thus provided a series of Of known Of detergent and 'Onstant (and known) molality of additive (e.g., NaC1). The molarity of additive, of course, varies somewhat in such a series. Detergent concentrations are therefore given as grams of detergent per milliliter of solution or formula weights of detergent per liter of solution. Concentrations of additive are expressed as molalities, although the solvent solutions were originally made UP VolUThe Journal of Physical Chemistry
MARILYN F. EMERSON AND ALFREDHOLTZER
metrically. In all the cases involved, the difference between molarity and molality is small. C . Refractometry. Specific refractive index increments were measured using a modified16 Brice-Phoenix" differential refractometer (Phoenix Precision Instruments, Philadelphia, Pa.). The instrument was calibrated using the data of Kruisl8 for KC1 solutions, as interpolated by Stamm.ls This calibration procedure has been described earlierm5 The procedure for calculating reliable molecular weights of charged detergent micelles in aqueous electrolyte solutions from the light-scattering data r e quires the values of the excess scattering and of the excess refractive index of a solution at detergent concentration c (grams per milliliter) over that of a solution at the cmc (co) but at constant molality of added salt (m3).'Jo,21 Thus, the specific refractive index increment that must be used is (n, - n,,)/(c - co) = (An/Ac)m,,where n, is the refractive index of a solution at concentration c. However, for systems in which the refractive index is a linear function of concentration above the cmc, (AnlAc),,,, = (ac2- n,,)/(cs cl), where c1 and c2 are both greater than co. Thus, differences in refractive index between pairs of solutions at different detergent concentrations above the cmc, but at constant molality of added electrolyte, were measured. In this manner, (An/Ac)*, was determined for several different values of Ac in the concentration range used for light scattering and was, indeed, found to be constant within experimental error. Since the concentration of detergent monomers remains approximately constant in the concentration region of interest, i.e., above the cmc,20,22the contribution of monomers to the refractive increment will cancel out and (An/Ac)m,is that for the colloidal particle. For cases involving mixed systems (DTAB in NaX), it was invariably true that [X-] >> [Br-] so the solutions were considered to be mixtures of DTAX, NaX, and NaBr; the measured refractive index increments were corrected for differences in refractive index associated with differences in the concentrations of NaX and NaBr in the pair of solutions m e a s ~ r e d . ~ ~ ~ ~ (15)K. Mysels, p. Mukerjee, and M. Abu-Hamdiyyah, J . p h y s . Chem., 67, 1943 (1963). (16) A. Holtzer, R. Clark, and S. Lowey, Biochemistry, 4, 2401 (1965). (17) B. A. Brice and M. Halwer. J . Opt. Soc. Am., 41, 1033 (1951). (18) A. Kruis, z.Physik. Chem. (Frankfurt), ~34,i3(1936). (19) R. F. Stamm, J . Opt. SOC.Am., 40,788 (1950). (20) p. J. Debye, Ann. N . Y.A d . 8ci.V 51,575 (1949). (21) W.Prim and J. J. Hermans, KoninkZ. Ned. A M . Wetenachap. B59, 162 (1956). (22) P. J. Debye, J . Phys. cozzoid Chem., 53,i (1949).
ON THE IONIC STRENGTH DEPENDENCE OF MICELLENUMBER
At some of the intermediate salt concentrations, the quantity (AnlAc),, was not measured directly, but was obtained from the empirical equation of Gladstone and Palez4 (AnlAc),, = &(nz - nd
(2)
where n2 is the refractive index of the colloidal particle, n1 the refractive index of the solvent medium, and ijz the partial specific volume of the colloidal particle. If 2?2, n1, and (AnlAc),, for the solution at one concentration of additive are known, n2 can be obtained from eq 2. Then, assuming that n2 and & remain constant, (An/Ac),, can be calculated, to a good approximation, at other concentrations of additive. D. Light Scattering. 1. Experimental. a. Instrumentation. A commercial (Phoenix Precision Instrument Co., 1979-5 series) light-scattering photometer, built according to the design of Brice, Halwer, and S p e i ~ e r *and ~ modified to improve the experimental pre~ision,~ was used for all light-scattering measurements. A description of the erlenmeyer cells used and of the instrument and cell calibration has been given by S c h ~ s t e r . ~ b. Optical Clurification of Detergent Solutions. Detergent solutions were cleaned for light scattering by vacuum filtration through a clean, ultrafine (pore size, 0.9-1.4 p ) , sintered-glass filter funnel directly into a clean, light-scattering cell. The filtration was carried out at a pressure differential sufficiently small to prevent the formation of bubbles in the filter stem and in the cell. The procedure for cleaning the cells and the filters and a detailed description of the filtration apparatus is given e1sewhe1-e.~ The cleanliness of the solutions was checked by placing the filled cell in the unfiltered beam of the lightscattering instrument and viewing it with a small hand mirror at the lowest possible angle (-5") to the forward direction of the incident beam. In a completely darkened room, dust particles appear as small, bright spots moving about against a black background. Absolutely clean (by visual examination) solutions could be obtained in this manner, although some dust can be tolerated for measurement of the scattering at 90" to the incident beam. The scattering from solutions of detergents in several aqueous electrolyte solvents was measured as a function of detergent concentration. All solutions were clarified individually because it was not possible to obtain clean solutions by dilution of a clean stock solution in the cell. The Rayleigh ratio of the scattering at 90" (R,) to the incident beam was measured for all solutions and, for several of the solutions, the Rayleigh ratio at 45 (Rd6)and 135" (R135) to the incident beam
1901
was also determined. From these, the corresponding quantity for the same system at the cmc was subtracted to obtain the excess Rayleigh ratio ARo (see next section). All solutions for which the Rayleigh ratio at 45 and 135" to the incident beam was measured had dissymmetries (f&/AR135) which were within 1% of unity. The values of AR%, AR135, and ARw agreed to within f1.5%. 1. Treatment of Data. To obtain the micellar molecular weight from light-scattering data it is necessary to measure the excess scattering of a series of solutions (all at concentrations above the cmc) over that of a solution at the cmc and to extrapolate a suitable plot of these data to the cmc.20J6 The appropriate form for the extrapolation and the necessary relationship between the results and the molecular weight can be obtained from the discussion of Prins and her man^,^^^^^ who have applied the general theory of the scattering of light by a multicomponent system to the case of charged detergent micelles dissolved in solvents that may contain added salts. If the Prins-Hermans expression (including nonideality) for the scattering at the cmc is subtracted from that at some higher concentration, the result isz8 K(c - CO)ARw
where AR, is the Rayleigh ratio, at 90" to the incident beam, of a solution of concentration c (grams per milliliter) minus that of a solution of concentration co; C1 is the molar concentration of detergent monomer and C3 the molar concentration of added salt ions having the same charge as a monomer ion; p is the absolute value of the number of fundamental charges per micelle; Q is defined by Q = (CI C3)2/(C12dl 2C1C3dz Cs2d3), with dl = 1 - a a2/4 a/4N, dz = 1 - 4 2 - f a / 2 fa2/4 fa/4N, and d3 = 1 fa fza2/4 f2a/4N where a ( = p / N ) is the effective degree of dissociation of the micelles, N the number of monomers per micelle, and f = (dn/dm3>J(dn/
+
+
+
+
+
+ +
+
+
(23) E. W. Anacker and H. M. Ghoee, J . Phys. Chem., 67, 1713 (1963). (24) P. Outer, C. I. Carr, and B. H. Zimm, J . Chen. Phvs., 18, 830 (1950). (25) B. A. Brice, M. Halwer, and R. Speiser, J. Opt. SOC.Am., 40, 768 (1950). (26) L. H. Princen and K. J. Mysels, J . Colloid Sci., 12, 594 (1957). (27) W. Prina and J. J. Hermans, Koninkl. Ned. Akad. Wetenschap. Proc., B59, 298 (1956). (28) See eq 30 of ref 21.
Volume 71, Number 6 May 1967
MARILYN F. EMERSON AND ALFREDHOLTZER
1902
dml),,, with m f the molality of the ith component; 8v, is the excluded volume per micelle; and K = 279. no2(dn/dc),,2/NoX4.Terms higher than the second in the virial expansion are assumed small. According to eq 3, the intercept ( I ) and the limiting slope (S) of a plot of the experimental quantity K(c (co/ARw us. (c - co) are both functions of both the number of charges per micelle and the number of monomers per micelle. They are given explicitly by the expressions
I
=
(4)
q/M
and
fining an "ideal part" (S') of the slope (S),eq 5 can be written p2 +
-
")
E S - 8vrnNd2 (8)
and S' can be calculated from the observed slope and intercept of the light-scattering plot and the excluded volume per micelle; the latter quantity is estimated from
8v,
=
32ir -r3 3
(9)
where r is the micelle radius. We have assumed a micelle radius of 24 A, the experimental result for qp 16vrnN0I (5) SDS micelles in pure water, including a layer of bound counter ion^.^^ The resulting S' is then substituted directly into eq 6 and p is calculated. Since the reThus, the value of the micellar molecular weight obsulting value of p is rather insensitive to the value of tained from eq 4 is q times that obtained simply from S' used, this approximate way of obtaining S' is the reciprocal of the intercept. The numerical value adequate. Once p is known, eq 7 can be used to obof q is usually sufficiently close to unityz7 so that the tain the value of N . molecular weights obtained by the two methods are All the experimental data were thus plotted as K ( c not very different. co)/ARsous. (c - co). When N and p are determined Equations 3 and 4 are not in a very useful form, since they do not give p and N explicitly in terms of the from eq 6 and 7 .using the observed intercept, I, and experimental quantities I and S. To utilize these a value for S' calculated as described above, the values simultaneous relationships, therefore, Anacker, et U Z . , ~ ~ of N determined by this more complex method do not solved them, but only for the "ideal" case (v, = 0). differ significantly from those obtained directly from In the notation used here, their expressions for p and N the reciprocal of the intercept (i.e., q is, indeed, near unity). For this reason, the reciprocal of the intercept are is essentially within experimental error of, and therefore may just as well be taken as, the molecular weight. Two examples of the results of the more elaborate computation are given in Table 11. E . Determinations of the Critical Micelle Concentration (Cmc). 1 . Conductivity. The cmc's of the cationic detergents in dilute (GO.05 M ) electrolyte solutions are sufficiently high so that they can be determined with good precision by the conductivity method. Accordingly, these cmc's were measured using a where S' is the slope if v, = 0; E = (C1 jC,)/(Cl Kohlrausch-type bridge equipped with a dipping elecCa); and M o is the formula weight of the detergent trode, calibrated using the data of Jones and Bradmonomer. shaw for KC1 solution.31 Unfortunately, the approximation urn = 0 is not alConductance techniques used were conventional and ways a good one. Nevertheless, these equations can a complete description is given elsewhere.s Values still be used, because it is possible to estimate the of A were plotted aganst C'l'. A sharp break in the contribution to the slope due to the excluded volume curve is observed at the cmc. per micelle ( i e . , to estimate 16qv,No/2M2) and then 2. Light Scattering. The cmc's of SDS in 0.05 to calculate the value that the observed slope would have if the solution were ideal by subtracting the esti(29) E.W.Anacker, R. M. Rush, and J. S. Johnson, J. Phya. Chem., mate from the observed slope. This value of S' can 68, 81 (1964). then be used in eq 6 and 7 to obtain p and N . The pro(30) F. Reiss-Husson and V. Lussati, ibid., 68, 3604 (1964). cedure is as follows. (31) G. Jones and B. C. Bradshaw, J. Am. Chem. SOC.,55, 1780 The numerical value of q is close to unitylZ7so, de(1933).
+
+
The Journal of Physical Chemietry
+
ON THE IONICSTRENGTH DEPENDENCE OF MICELLE NUMBER
Table II: LighbScattering Results Molality of added
SDS in NaCl solution DTAB in NaBr soh tion DTAC in NaCl solution DTAN in NaNOt solution
Table HI: Specific Refractive Index Increments
salt
1061. mole/g
1045, mole ml/g:
0.050 0.201 0.506 0.050 0.100 0.508 0.050 0.201 0.506 0.050 0.101 0,253 0.509
4.10 3.23 2.75 4.50 4.47 3.70 6.71 6.10 5.60 5.48 4.74 4.32 4.02
32.470 10.48 -4.09 16.58b 6.26 1.51 12.60 3.87 -1.12 13.90 11.01 1.25 0.72
System
1903
Molality
1 MOT
84
107 126 72 73 88 57 62 68 63 73 80 86
System
SDS in NaCl solution
DTAB in NaBr solution
Use of eq 6-9 gives 10%'' = 27.97, a = 0.157, and N = 89.0. b Use of eq 6-9 gives 10'8' = 11.14, a = 0.130,and N = 0
76.
and 0.5 M NaCl and of all the cationic detergents in aqueous electrolyte solutions ( b 0.05 M ) were determined by light scattering. Measurements of RW for solutions at several different detergent concentrations above and below the cmc, but at constant molality of added electrolyte, were made and the results plotted against the detergent concentration. At the cmc, there is a sharp break in the curve, above which the solutions scatter much more strongly per unit weight concentration than below.
Experimental Results I n order to use eq 3 to obtain the micellar molecular weight, it is necessary to know the specific refractive index increment, the critical micelle concentration, and the scattering, the latter as a function of concentration of detergent at constant molality of added electrolyte. The results of each of these measurements will be discussed in turn. A . Specific Refractive Index Increment. Values of the specific refractive index increments (AnlAc),,,, for the various detergents in aqueous electrolyte solutions are given in Table 111. The measured values of (AnlAc),, at different Ac are constant within experimental error (*0.5%). Values of (An/Ac)t,, calculated from the Gladstone-Dale equation (eq 2) are also listed in the table and agree quite well with the corresponding measured values. At some of the intermediate concentrations of additive, the calculated values were used. For mixed systems (e.g., those containing DTAB dissolved in NaX) a small correction must be and was applied.802s These corrected values
DTAC in NaCl solution DTAB in NaCl solution DTAN in NaNOs solution
DTAB in NaNOl solution
of added salt
108~1, dml
lO*Ac, g/ml
0.050 0.050 0.201 0.201 0.506 0.506 0.506 0.050 0.050 0.050 0.100 0.100 0.100 0.100 0.508 0.508 0.508 0.508 0.050 0.201 0.506 0.506 0.506 0.050 0.101 0.253 0.253 0.509 0.509 0.509 0.509
3.014 3.014 5.582 2.232 5.824 3.494 2.118 5.550 2.024 2.220 5.388 2.155 2.307
6.530 9.738 5.582 8.928 5.824 8.154 8.474 5.550 8.096 8.880 5.388 8.621 9.229
4.810 1.924 1.973
4.810 7.696 7.891
3.582
8.358
...
...
...
...
,..
...
1.8OOc
7.198C
... ...
...
...
3.105
7.245
3.090 ... 3.060
7.210
1.807'
7.228'
...
...
... ...
... ... 7.130
...
(An/ Ac)mas mI/g
0.1199 0.1195 0.1166 0.1158 0.1114 0.1115 0.1115 0.1504 0.1513 0.1515 0.1519 0.1504 0.1520 0.15080 0.1459 0.1451 0.1459 0.14560 0.1588 0.1574b 0.1536* 0.1707 0.1545d 0.1408 0.1403' 0.1381 0.1390' 0.1363 0.1367' 0.1524 0.13429
0 Calculated using the GladstoneDale equation with nt calculated from the measured value of (An/AC),, of 0.05 M NaBr with Cis = 0.95610and 111 for NaBr solutions computed from data of Anacker and Ghose.28 b Calculated using the GladstoneDale equation with n2 calculated from the measured value of ( An/Ac)ma of 0.05 M NaCl, with OS = 0.96 and nl for NaCl solutions computed from the data of Kruis,'* as interpolated by Stamm.lg Grams of DTAC/ml. calculated using data of Anacker and Ghoae'l for NaBr solutions and data of Kruis,'* as interpolated by Stamm,'@for NaCl solutions. * Calculated using the Gladstone-Dale equation with n2 calculated from the measured value of (AnlAc),, of 0.05 M NaNOa with an assumed value of = 0.96 and n1 for NaNOa solutions computed from data of Anacker and Ghose.28 Grams of DTAN/ml. 9 Calculated using data of Anacker and Ghosesa for NaBr and NaNO, solutions.
of (Anlac),, for the mixed systems are also listed in the table. B. Critical Micelle Concentration (Cmc). Since the values of the cmc's of detergents in aqueous salt solutions depend somewhat on the method of measurement,8J0 the ones used in the light-scattering calculaVolume 71, Nurnber 8 May 1887
MARILYN F. EMERSON AND ALFREDHOLTZER
1904
tions were those either directly measured by light scattering or measured by conductivity-a method which has been shown to agree with light scattering.8 Because of its greater convenience and speed, conductivity is the method of choice wherever it can be employed without sacrifice in precision. Practically speaking, the method is not routinely useful for electrolyte concentrations 5 0 . 1 M because of the high solvent corrections that become necessary. Fortunately, the light-scattering method is more precise at higher concentrations of electrolyte because the micelle numbers are greater and there is a concomitantly larger increase in scattering upon micelle formation. The two methods are thus complementary. Of course, other methods have also been An example of a light-scattering determination of the crnc is given in Figure 1. The results are displayed (Figure 2) as -log Co us. -log (Co C3) because it is well known that such plots are linear. It is apparent from Figure 2 that conductivity and light-scattering results fall on the same line and that our data for SDS are in satisfactory agreement with those of Williams, et al.'O These results were fit to straight lines by the method of least squares. The resulting equations are
11
-
10
-
12
0
J
=
98-
76-
I
1
1
1
1
1
1
1
1
0.5
1
1
1
1.0
IO'dg/ml)
Figure 1. Light-scattering determination of the cmc of DTAB in 0.500 M NaBr.
+
+
log (Co) = -0.666 log (Co C3) 3.491 for SDS in NaCl
+
log (Co) = -0.621 log (Co C,) 3.021 for DTAB in NaBr
+
log (Co) = -0.631 log (Co C3) (10) 2.794 for DTAC in NaCl
+
log (Co) = -0.547 log (Co C3) 3.018 for DTAN in NaNOa 0.5
The cmc values used in the light-scattering calculations were obtained from these relationships. C. Light Scattering. The light-scattering data were plotted as K(c - CO)/ARW us. (c - C O ) ; some typical plots are shown in Figure 3. I n all cases the data fall on straight lines within 5 1 % . Values of the intercepts ( I ) and slopes (S) of these lines along with values of N calculated from the simple intercept (N = l/MJ) are given in Table 11. I n the Table I1 footnotes are also given, for two of the solutions, values of S' calculated from eq 8 and 9, N from eq 7, and a ( = p / " ) from the value of p obtained from eq 6. As noted above, the differences between the results of the exact and approximate (q = 1) calculations of N are inappreciable. The precision of the measurements of RW is *l% The Journal of Phwicai Chemistry
1.0
- log(C0 t C3)
1.5
+
Figure 2. Plots of -log COvs. -log (CO C,) for detergents in aqueous salt solutions. SDS in NaCl: 0,light-scattering values; . , conductivity data of Williams, et aZ.10 DTAN in NaNOa: V, light scattering; V, conductivity. DTAB in NaBr: 0,light scattering; 0, conductivity. DTAC in NaCl: A, light scattering; A, conductivity.
(probable error) ; however, the necessity of extrapolating the data to (c - co) = 0 produces an uncertainty in the intercept which is sometimes as large as f2.5%. The over-all precision of the light-scattering molecular weights is somewhat less than this, because of the dependence on other parameters. The measure(32) M. L. Corrin and W. D. Harkins, J. Chem. Phys., 14, 640 (1948).
ON THE IONIC STRENGTH DEPENDENCE OF MICELLE NTJMBER
4-
-I
n
Y
2
b
4
10Vc
- co)(dml)l
n
I)
ui
10
12
Figure 3. Lightmattering results for SDS in NaCI: 0,0.050 m NaCl; Al 0.201 m NaCI; Dl 0.506 m NaCI.
ment of (An/Ac),,,n, has been made to a precision of f0.575, but since it enters the light-scattering equation as the square, it contributes an additional probable error of f1% to the molecular weight. The value of c is, of course, known quite accurately, but the imprecision of f1% in co produces a probable error which is sometimes as large as *0.5% in (c - co). Thus, the micellar molecular weights and therefore the micelle numbers determined by light scattering are sometimes precise only to about f4%. D . Compar.ison of Light-Scattering Results with Literature Values. The results obtained here for the micellar molecular weights may be compared with those of other investigations on SDS in NaCl solution@ and on DTAB in NaBr s0lutions.~9 There exist no comparable studies of DTAC or DTAN. The values of the refractive index increments and the cmc’s obtained here agree quite well with those obtained by Mysels and P r i n ~ e on n ~SDS. ~ On the other hand, the observed intercepts of the light-scattering plots differ by as much as 15%. However, most of their measurements were apparently made on solutions which produced scatterings less than five times that of solvent, so that the probable error involved in their determinations of A& would have to be as large as *3% at low concentrations and the over-all error in the resulting molecular weights must be as great as f8%. In addition, these earlier workers discarded “a very large number of measurements” for reasons such as “high dissymmetry, visible dust, or presence of impurities revealed upon further dilution to the cmc” as well as “a few stray values.” No such dif-
1905
ficulties were encountered here, possibly because of preliminary removal of dodecyl alcohol by extraction. At first glance, it appears as though the results for DTAB in NaBr presented in Table I1 agree closely with those obtained by Anacker, et al.,29directly from the intercepts of their light-scattering plots. However, this is to some extent owing to a fortuitous cancellation of discrepancies, since the values of the refractive index increments, the cmc’s, and therefore the scattered intensities obtained by them differ from those found here. Although many of their measurements were made on solutions which produced scatterings greater than five times that of background, their data points scatter about a smooth curve, falling as far as *5% from it. This considerably higher probable error in the values A& must produce a correspondingly greater error in the molecular weights found by them. For these reasons, but, more importantly, for the sake of internal consistency, we have used our own results in making comparisonswith theory in all cases.
Theoretical Results Values of AG”m were determined with the use of eq 1. The values of fi determined by light-scattering experiments (Table 11) on SDS, DTAB, DTAC, and DTAN solutions at various ionic strengths were used in the computations along with an assumed micelle radius of 19.7 A. This radius is an experimental result only for SDS micelles in pure water30 and it is assumed that the same value is appropriate for all four detergents under all conditions considered. The use of this value for the radius has been discussed earlier.* The distance of closest approach (a) was than calculated as 19.7 roounterion; namely 21.8 A for SDS and 21.2 A for DTAB, DTAC, and DTAN. The values for the counterion radii used were those for the hydrated ions in aqueous solutions, as obtained from experimental data using the Debye-Hiickel limiting law.34 These two quantities, fi and a, along with the ionic strength of the solution, were then used to calculate the boundary condition (i.e., the electric field at the micelle surface) that applies and the appropriate computer solution was obtained from the computer tables by interpolation, as beforeS3 Insertion of &,fi and the experimental cmc (mole fraction basis36) into eq 1 allows computation of AG-Hc (infinitely dilute refer-
+
(33) K. J. Mysels and L. H. Princen, J. Phys. Chem., 63, 1696 (1959). (34) I. M. Klotz, “Chemical Thermodynamics,” W. A. Benjamin, Inc., New York, N. Y.,1964,p 417. (36) In an earlier paper (see ref 3), a molality basis was used. The mole fraction basis would seem, however, to be a better choice in that it is a more direct reflection of the cratic contribution and also ia in more common use in the micelle literature.
Volume 71, Number 6 May 1967
MARILYNF. EMERSON AND ALFREDHOLTZER
1906
ence state, mole fraction basis) as before.a Results of this computation for the four detergents a t several ionic strengths are presented in Table IV.
Table IV: Computation of - L I G ~ H C
System
SDS in NaCl solution DTAB in NaBr solution DTAC in N:tC1 solution DTAN in NaNOl solution
Molality 10aC0, of added cmc in saltD moles/l.
0.050
0.201 0.506 0,050 0.100
0.508 0 050 0.201 0.506 0,050 0.101 0.253 0 509
2.30 0.94 0.51 5.71 3.88 1.46 9.50 4.36 2.48 4.70 3.32 2.04 1.40
106x0, cmc in mole e&&)/ fraction kT
4.13 1.69 0.91 10.30 6.98 2.61 17.10 7.83 4.43 8.45 5.97 3.66 2.50
5.33 4.48 3.89 5.03 4.47 3.43 4.46 3.52 2.93 4.72 4.46 3.85 3.39
-AG=HC, cal
10,900 11,400 11,900 9,510 9,460 9,610 8,640 8,630 8,710 9,310 9,510 9,540 9,590
0 The micelles in these solutions exist in the presence of monomeric detergent ions of concentration equal to the cmc. Thus, the ionic strength used was the added electrolyte concentration plus Co. This has no appreciable effect on the results.
As is clear from the table, AGmHcfor the cationic surfactants is roughly independent of the nature of the supporting electrolyte (-10% spread in all values) and essentially completely independent of ionic strength (-3% spread). For SDS, on the other hand, a small, but systematic dependence of A G m x on ionic strength is observed (-10% increase in going from 0.05 to 0.5 M NaCI). As reported earlier, an apparent difference (-20%) does indeed exist between the values of AG-HCfor SDS and those for the cationic detergents.
Discussion First, it should be pointed out that eq 1 bears some resemblance to the empirical, linear log (Co) us. log (C, C3) relationship (eq 10). Both equations are of the form log cmc = constant f(ionic strength). Thus, the constant term of the empirical equation is related to A G r n ~and c the potential at the surface of the micelle must be approximately linear in log (CO C3). The magnitude of A G r nobtained ~~ agrees quite well with two other experimental estimates of this quantity; namely Wishnia’s36 (from measurements of solubilities of gaseous hydrocarbons in detergent solutions), which gives between -7100 and - 10,800 cal for a 12-carbon chain, and Corkill and co-w~rkers’~’ (from cmc meas-
+
+
+
The
JOUTnal
of Physical Chemistry
urements on nonionic detergents3*), which is -8600 cal for the same length hydrocarbon chain. Agreement is also manifest with the values previously calculated (from then extant data on SDS and DTAB), which fall between -9900 and - 12,700 ~ a l . ~ The results of Table IV show, as b e f ~ r e a, ~slight increase of A G m ~with c increasing ionic strength. This change may be a result of increased “salting out” of the hydrocarbon chains or may be a result of a more fundamental variation of A G r n ~with c #, as suggested on theoretical grounds by R e i ~ and h ~ 0~0 s h i k a . ~ ~However, the variation is small, even in the SDS case, and could merely be a result of the assumption that the micellar radius is independent of N . The largest micelles of SDS, for example, might require some increase in radius because of crowding of hydrocarbon chains in its interior. An increase in radius of, say, 10% over the complete range of ionic strength would explain the apparent change in AG”Hc as computed here. The differences in AG”Hc for the different detergents, especially those between SDS and the cationic detergents, may also be a result of small differences in radii of the micelles of the various detergents or it may reflect small, specific differences in the supporting electrolytes. It cannot, however, be explained by the difference in salting-out strengths of the various supporting electrolytes. Thus, NaC1 salts out nonpolar substances more strongly than does NaBr or Z\iaT\’03,41 but the tendency for the DTAC hydrocarbon tails to enter the micelle from a medium containing NaCl is not as great as that for the DTAB and DTAN hydrocarbon chains from NaBr and NaN03, respectively. Although small variations are encountered, this extension of the measurements to a wider range of detergents and supporting electrolytes has not altered the earlier suggestion3that the simple approach adopted here seems to be adequate as an approximate explanation of the ionic strength dependence of the micelle number; that is, that the micelle number is limited by electrostatic repulsions that may be estimated by numerical solution of the Poisson-Boltzmann equation. The approach appears to be particularly satis(36) A. Wishnia, J . Phys. Chem., 6 7 , 2079 (1963). (37) J. M.Corkill, J. F. Goodman, and S. P . Harrold, Trans. Faraday SOC.,6 0 , 202 (1964). (38) To eliminate the effect of the polar head groups the data of Corkill, et al., for various sizes of head group have been extrapolated to zero head group. (39) I. Reich, J . Phys. Chem., 60,257 (1956). (40) Y.Ooshika, J . Colloid Sci., 9,254 (1954). (41) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” Reinhold Publishing Corp., New York, N.Y.,1958,pp 531-534.
THEORY OF OPTICALLY ACTIVECOMPOUNDS OF HIGHLATENT SYMMETRY
factory for cationic detergents. The crucial remaining question of the constancy of micellar radius can only be resolved by direct measurement of the quantity (by low-angle X-ray scattering) for solutions of the various detergents containing added salt. It would also be of interest, of course, to measure AG-Hc as a function and ASrn~c of temperature in order to determine &Hrn~c under the various experimental conditions. From a theoretical point of view, the next step would be to compare the experimental AG"xc with a value predicted from, in essence, first principles. We do not believe that the present state of the statistical theory of aqueous solutions warrants such a comparison. Recent attempts in this direction are not promising.42 Indeed, with the current range of adjustability of most theories, it is not unlikely that an embarrassment of riches would result, Le., that theories based on quite different physical models could all be made to yield an answer in agreement with experiment.
1907
As an example, we might cite the very simple approach of Aranow and W i t t e ~ ~based ; ~ ~ on the increase in rotational freedom of a hydrocarbon chain inside, compared with outside, the micelle, this theory ignores all attendant alterations in the structure of water that most people feel is the very crux of the matter. Nevertheless, the Aranow and Witten theory provides a value of AGm~c ( ~ - . l O , O O O cal) that is in rather good agreement with expenment. Achowledgment. The authors wish to thank Dr. Martin Schick of Lever Brothers for expert advice on the synthesis of SDS samples and Professor Joseph Kurs of Washington University for stimulating and informative discussions. (42) D. C. Poland and H. A. Saheraga, J. Colloid Interface Sei., 21, 273 (1966). (43) R. H. Aranow and L. Witten, J. Phy8. C h . ,64, 1643 (1960)
Theory of Optically Active Compounds of High Latent Symmetry
by Dennis J. Caldwell Department of Chemistry, The Univereity of Utah, Salt Lake City, Utah 8411.8
(Received November $9, 1066)
The behavior of magnetic dipole transitions in different types of dissymmetric fields is discussed along with limitations of one-center models for optical rotation. The general properties of the angular wave functions are used in a second-order perturbation treatment to explain the qualitative features of the ORD of certain transition metal complexes.
I. Introduction In a given dissymmetric field the sign of rotation for an optically active transition in a chromophore is determined by the general shape of the ground and excited orbitals. In most cases a detailed knowledge of the electron cloud is only necemary for quantitative work. Since the basic mechanisms responsible for the phenomenon are still in contention, it is often advantageous to examine things from the simplest possible
viewpoint. For example, possible intramolecular interactions leading to optical activity are coupled oscillator effectsl1V2 charge transfer,a and incomplete ~creening.~The appropriate theories of these separate (1) J. G. Kirkwood, J. Chem. Phyu., 5,479 (1937). (2) L. L. Jones, Ph.D. Thesis, University of Utah, Salt Lake City, Utah, 1961. (3) S.F. Mason, Nature, 199, 139 (1963). (4) W. J. Kauzmann, J. E. Walter, and H. Eyring, Chem. Rev., 26, 339 (1940).
Volume 71, Number 6 May 1967
