8354
J . Phys. Chem. 1989, 93, 8354-8366
More interesting than the reaction of the aromatic compounds on the (0001)-Zn polar surface was their reaction on the (OOOl)-O polar surface. Unlike the C , through C3 oxygenates and the alkynes, which did not react on the (OOOT)-O surface, the aromatic oxygenates were found to react on this surface. Benzoic acid, benzaldehyde, and phenol decomposed on the (0001)-0 polar surface at high temperatures to produce CO, C02, H2, and H20. The XPS results clearly demonstrated that these aromatic compounds were stabilized on the (OOOi)-O surface via a direct interaction of the aromatic ring with the surface. This result serves
to illustrate further the effects of surface crystallographic and electronic structure in determining the reactivity of metal oxide surfaces and suggests that different surfgce structures can lead to the activation of different functional groups. Acknowledgment. We gratefully acknowledge the National Science Foundation for support of this research through Grant CBT-87 14416. Registry No. C6H5COOH,65-85-0; C6H5CH0,100-52-7; C6H5CHzOH, 100-51-6; C~HSOH, 108-95-2; ZnO, 1314-13-2.
Influence of Alkali-Metal Counterion Identity on the Sphere-to-Rod Transition in Alkyl Sulfate Micelles Paul J. Missel,*.+ Norman A. Mazer,$ Martin C. Carey,#and George B. Benedek Department of Physics, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 021 39, and Department of Medicine, Harvard Medical School, Division of Gastroenterology, Brigham and Women's Hospital, Boston, Massachusetts 0221 5 (Received: May 17, 1989)
Data are presented on the first extensive study of the influence of counterion identity on the size of dodecyl sulfate micelles as measured by quasielastic light scattering spectroscopy (QLS). The tendency for micelles to grow from spherical to long rodlike structures follows the pattern LiDS < NaDS < KDS < CsDS < RbDS. Deductions of a thermodynamic parameter K governing the sphere-to-rod transition reveal that the energetic advantage of forming cylindrical micellar regions appears to decrease approximately linearly with the increasing hydrated radius rHof the counterion, which accounts for the counterion dependence observed for micellar growth. To understand the counterion size effect we have refined our previous analysis of the electrostatic contributions to micellar growth using the Gouy-Chapman model of a curved electric double layer allowing for a finite distance of closest approach for the counterion. Binding in the Stern layer is neglected. Based on all of our experimental studies of micellar growth in this system, the functional dependence of K on temperature (T), added electrolyte concentration (CJ, alkyl chain length (nc), and hydrated counterion size ( r H )can be described as In K = no [ a / T (K) + A In (C,)+ rn, - PrH + p"], where a,p", r, A, and p are fitted parameters and no is the calculated number of surfactants in a spherical micelle of maximal size. On the basis of our present electrostatic models, the contributions to the various parameters governing K are discussed.
I. Introduction In previous studies'-8 we have investigated experimentally and theoretically the spherocylindrical growth of sodium alkyl sulfate micelles in aqueous solutions of NaCl. Quasielastic light scattering (QLS) spectroscopy has demonstrated that, for this system, micellar size is a very sensitive function of detergent and NaCl concentration, temperature, and alkyl chain length. We have developed a theory for this micellar growth. The fundamental parameter of our theory is K = exp[no(p,O - p c o ) / R T ] ,where is the number of molecules in a spherical micelle (the state of no micellar growth, -58 for the SDS micelles (see eq 6 of ref 8), R is the gas constant, T i s absolute temperature, and p,O and pco are the standard chemical potentials per monomer associated with spherical and cylindrical micellar regions, respectively. In the present study we have investigated the influence of the identity of the alkali-metal counterion on micellar growth for a variety of alkali dodecyl sulfates (CI2HZSOSO3-, X+; X+ = Li+, Na+, K+, Rb+, and Cs+). We have used QLS to measure the hydrodynamic radius (ah)of alkali-metal dodecyl sulfate micelles over a wide range of temperatures (10-80 "C), detergent concentrations (0.33-2 g/dL), and alkali-metal chloride concentrations (0.45-1.0 M X'Cl-). The identity of the alkali-metal counterion is found to have a strong influence on R h and by inference on the thermodynamic parameter K. Analysis of the *To whom correspondence should be addressed. Present address: Alcon Laboratories, 6201 South Freeway, Fort Worth, TX 76134. *Present address: Thera-Tech, Inc., Research Park, 410 Chipeta Way, Suite 219, Salt Lake City, UT 84108. $ Harvard Medical School.
0022-3654/89/2093-8354$01 SO10
dependence of K on counterion identity suggests that the hydrated radius of the counterion ( r H )plays an important role in determining the electrostatic contribution to the parameter K . We have also found that the identity of the counterion strongly influences the temperature of the micellar/crystal phase boundary (the critical micellar temperature, cmt) in alkali-metal dodecyl sulfate/alkali-metal chloride/water systems at fixed alkali-metal chloride concentration. 11. Materials and Methods A . Reagents. Several grams of homogeneous lithium, potas-
sium, rubidium, and cesium dodecyl sulfates (LiDS, KDS, RbDS, and CsDS, respectively) were synthesized according to the procedure outlined in Appendix I of our previous paper.8 We have employed >99.5% pure dodecyl alcohol (Humphrey Chemical Co., North Haven, CT), and reagent grade alkali-metal hydroxides (1) Mazer, N . A.; Benedek, G. B.; Carey, M. C. J . Phys. Chem. 1976,80, 1075. (2) Mazer, N . A.; Carey, M. C.; Benedek, G. B. Micellization, Solubilization, and Microemulsions; Mittal, K. L. Ed.; Plenum: New York, 1977; Vol. I, p 359. (3) Missel, P. J. S.B. Thesis, MIT, 1977. (4) Young, C. Y.; Missel. P. J.: Mazer. N . A.; Benedek. G. B.: Carey, M. C. J. Phys. chem. 1978.82, 1375. ( 5 ) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980,84, 1044. (6) Missel, P. J. Ph.D. Thesis, MIT, 1981. (7) Missel, P. J.; Mazer, N . A.; Benedek, G. B.; Carey, M. C. Solution Behavior of Surfactants; Fendler, E. J., and Mittal, K. L., Eds.; Plenum: New York, 1982; Vol. I, p 373. (8) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J . Phys. Chem. 1983, 87, 1264.
0 1989 American Chemical Societv
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8355
Influence of Counterion on Micellar Size and Shape TABLE I: Deductions of Crystal Phnse Boundary Temperatures, the cmt ("C) of Alkali-Metal Dodecyl Sulfates in Solutions of Alkali-Metal Chloride
concentration by using the following equation
cmt, OC CS,OM
0.45 0.6
0.8
LiDS in LiCl 12
N a D S in NaCl 24.5 26.9 29.2
CsDS in CsCl 38.5 39.5 41
RbDS in RbCl 45 46 47
KDSin KCI 51 52 54
Alkali-metal chloride concentration.
(Alpha Chemical Co., Danvers, MA). By this procedure, we ensured that the alkali-metal counterion purity for each detergent synthesized was >99+%. The lithium, potassium, rubidium, and cesium chlorides (LiCl, KCl, RbCl and CsCl, respectively) were all >99% pure (Alpha) and were mixed with appropriate stock solutions of the detergents. The water purification procedure was identical with that used previously.'" B. Deductions of the cmt. The cmt is defined as the temperature at which the solid phase is in equilibrium with the micellar phase.'V2 It is closely related to the Krafft temperature (see ref 2). Above the cmt, only the micellar phase exists. Detergent solubility falls rapidly below the cmt. The cmt values were determined by the method of Demarcq and Dervi~hian.~Frozen aliquots of the solutions were slowly heated from below the crystalline to micellar phase transition while periodically stirring them and recording the temperature at which the hydrated crystal disappeared. In this way it was possible to avoid the "coagel", a metastable transparent viscous gel which can be obtained upon slowly cooling from above to below the cmt.I0 We frequently encountered this phase in dodecyl sulfate systems containing the K+, Rb+, and Cs+ counterions, when solutions were stored at room temperature, which is below the cmt for these systems. The gel state eventually transformed into a milky-white hydrated crystal after remaining transparent for several hours (several days for CsDS in 0.45 M CsC1). Upon reheating above the cmt, we found it more difficult to redissolve the material into an isotropic solution if the gel, rather than the crystalline state, was present. It was for this reason that the cmt's were determined by heating the solutions from the frozen state, as the crystalline solid phase was easily obtained upon freezing. C. QLS Measurements and Data Analysis. The detergent solutions were mixed in IO-mL amounts and stored frozen in stoppered test tubes. Prior to QLS measurements, the samples were carefully heated; 0.5 mL of the solution was centrifuged above the cmt, exactly as described previouslyS (see Table I for the cmt's measured in this study). The same apparatus and method of cumulants analysis11*t2 of the autocorrelation function R ( T ) = ( 1 ( 0 ) 1 ( ~ )described ), in our earlier work,1*2~5s6*8 were employed. The method of cumulants analysis allows the deduction of the mean micellar diffusion coefficient D from the autocorrelation function, along with various moments of the micellar size distribution, such as the variance V, a measure of the degree of micellar polydispersity.t*2-SJ2The data presented in this study represent averages of at least five and up to 12 measurements of D on at least two identically prepared samples. The mean hydrodynamic radius of the micelles, R h , was deduced from the following formula, analogous to the Stokes-Einstein relationI3 kB T Rh = -
67~10
(1a)
where kBis the Boltzmann constant, ~i~ the absolute temperature (K), and r ) is the viscosity of the solvent at temperature T which was corrected for the influence of each alkali-metal chloride (9) Demarcq: M.; Dervichian, D. Bull. Soc. Chim. Fr. 1945, 12, 939. (10) Dervichian, D. Chim. Rev. 1943, 217, 299. (11) Koppel, D. E. J . Chem. Phys. 1972,57, 4814. (12) Mazer, N. A. S.B.Thesis, MIT, 1973. (13) Einstein, A. Investigation on the Theory of Brownian Movement; Dover: New York, 1956; p 58.
where vo( T ) is the temperature dependence of the viscosity of pure water and o(C,) is the viscosity of an aqueous solution containing a concentration C,of alkali-metal chloride at 20 OC. The refractive index of the solution n, an important parameter influencing the magnitude of the scattering vector (see eq 2, ref 2), was also corrected for the addition of alkali-metal chloride, but ignoring the temperature dependence of the refractive index, taking its value for 20 "C. The dependences of qo(T), v(Cs),and n(CJ were taken from the CRC Handb00k.l~ D. Deductions of K . We have previo~sly'+~-~ derived how R h varies as a function of the product K ( X - X B ) for a surfactant system which forms spherocylindrical micelles. Here X is the added detergent concentration and XB is approximately equal to the cmc, each in mole fraction units. (For a detailed derivation of the parameter XB,see ref 5. In this paper, for the most part we have set X B = X,,, which should agree to within from Figure 6 of ref 8 by ( X - XB)for each value of R h measured. For the most part, the values of K were deduced from a single detergent concentration. We have shown previously that the influence of surfactant concentration is as predicted for sodium dodecyl sulfate in 0.8 M NaCl (ref 5) and for sodium decyl sulfate in 2 M NaCl (ref 8). The deductions of K presented for the LiDS, KDS, RbDS, and CsDS were obtained from experiments at a single detergent concentration. 111. Results A . Phase Boundaries. In Table I we record our observations
of the cmt for the solutions examined in this study. The cmt varies quite strongly with counterion identity according to the following sequence: cmt(LiDS) C cmt(NaDS) C cmt(CsDS)
< cmt(RbDS) < cmt(KDS) (2)
In comparison, as shown in Table I, the cmt changes relatively slowly with added alkali-metal chloride concentration for fixed counterion identity. It is interesting that this sequence corresponds neither to the periodic ordering nor the ordering of the hydrated counterion sizes. B. Micellar Growth. In Figure la-c, we present measurements of Rh versus temperature fo; our series of aqueous solutions of alkali-metal dodecyl sulfate and matching alkali-metal chloride. The data for sodium dodecyl sulfate (NaDS) was taken from our previous Figure l a contains data taken from solutions with an alkali-metal chloride concentration of 0.45 M (except for the LiDS data, obtained from solutions of 1 M LiCl). Parts b and c of Figure 1 contain data obtained from solutions with alkali-metal chloride concentrations of 0.6 and 0.8 M, respectively. (14) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics, 58th ed.; CRC Press; Cleveland, OH,1977. (15) Mukerjee, P.; Mysels, K. J.; Kapauan, P. J . Phys. Chem. 1967, 71, 4166.
8356
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989
Missel et al.
TABLE 11: Experimental Deductions of R band K from Solutions of Various Alkali-Metal Dodecyl Sulfates at Various Added AW-Metal Chloride Concentrations (Cs),T,and X
c,, M
c,
a/dL
Xn”
Xa
T, ‘C
Rhr8,
K
b b b
cs, c,
M g/dL (a) LiDS 1.o 2 1.o 2
0.45 1.0 1.0
2 2 2
1.32 X lo-’ 1.32 X 10‘’ 1.32 X lo-’
1.24 X 6.36 X 10“ 6.36 X 10”
40 10 20
26.12 26.85 26.77
0.45 0.45 0.45 0.45 0.45 0.45 0.6 0.6 0.6 0.6 0.6
2 2 2 2 2 2 2 2 2 2 2
1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X
9.21 9.21 9.21 9.21 9.21 9.21 7.58 7.58 7.58 7.58 7.58
10” 10” 10” 10” 10” 10” 10” 10” 10” 10” 10”
10 15 20 30 40 50 18 20 25 30 35
56.12 46.67 39.17 30.96 26.69 25.54 100.76 90.26 74.97 63.99 52.53
3.52 X 1.75 X 8.31 X 2.29 X 6.51 X 3.79 x 2.52 X 1.76 X 9.53 x 5.57 x 2.76 X
(b) NaDS 10’ 0.6 lo7 0.6 lo6 0.8 lo6 0.8 lo5 0.8 105 0.8 lo8 0.8 lo8 0.8 107 0.8 107 0.8 lo7 0.8
0.45 0.45 0.45 0.45 0.45 0.6 0.6 0.6
0.5 0.5 0.5 0.5 0.5 0.507 0.507 0.507
2.96 X lo4 2.96 X lo4 2.96 X lo4 2.96 X lo4 2.96 X lo4 3.00 X lo4 3.00 X lo4 3.00 X lo4
7.49 X 10” 7.49 X 10” 7.49 X 10” 7.49 X 10” 7.49 X 10” 6.16 X 10” 6.16 X 10” 6.16 X 10”
45 50 55 60 65 50 55 60
124.1 86.7 60.9 39.1 30.1 184.7 145 107.6
2.15 X 6.62 X 2.02 X 3.54 x 8.11 X 7.99 x 3.55 x 1.32 x
0.45 0.45 0.45 0.45 0.45 0.45 0.6 0.6 0.6
0.5 0.5 0.5 0.5 0.5 0.5 0.585 0.585 0.585
2.56 X 2.56 X 2.56 X 2.56 X 2.56 X 2.56 X 3.00 X 3.00 X 3.00 X
lo4 lo4 lo4 lo4 lo4 lo4 lo4 lo4
5.67 X 10” 5.67 X 10” 5.67 X 10” 5.67 X 10” 5.67 X 10” 5.67 X 10” 4.68 X 10” 4.68 X 10” 4.68 X 10”
40 45 50 55 60 65 50 55 60
197 154 112 79.2 50.6 36.6 194.2 153.8 120.2
1.16 X 5.10 X 1.76 X 5.66 X 1.19 X 2.97 X 9.39 x 4.31 X 1.89 x
0.45 0.45 0.45 0.45 0.45 0.45 0.6 0.6 0.6
0.5 0.5 0.5 0.5 0.5 0.5 0.663 0.663 0.663
2.26 X 2.26 X 2.26 X 2.26 X 2.26 X 2.26 X 3.00 X 3.00 X 3.00 X
lo4 lo4 lo4 lo4 lo4 lo4 lo4 lo4 lo4
5.67 5.67 5.67 5.67 5.67 5.67 4.68 4.68 4.68
40 45 50 55 60 65 50 55 60
181.6 138 94.9 67.9 41.7 35.1 188.3 150.2 116.6
1.01 x 4.01 x 1.17 x 3.85 X 6.18 x 2.72 X 8.49 x 3.97 x 1.71 x
lo-’ lo-’ lo” lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-’
IO4
X
X X
X X X
X X X X X
X X X X X X X X X
10” 10” 10” 10” 10” 10” 10” 10” 10”
Xa 1.32 1.32
Xna
T,‘C
Rh,A
K b b
X X
lo-’ lo-’
6.36 6.36
X X
10“ 10”
30 40
26.79 26.79
2 2 2 2 2 2 2 2 2 2 2
1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X 1.25 X
lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-’
7.58 X 7.58 X 6.24 X 6.24 X 6.24 X 6.24 X 6.24 X 6.24 X 6.24 X 6.24 X 6.24 X
10” 10” 10” 10” 10” 10” 10” 10” 10“ 10“ 10”
40 50 17 20 25 30 40 50 55 60 70
47.28 34.86 263.59 217.6 176.31 144.92 91.15 61.6 53.76 36 33.2
1.85 x 4.65 X 6.04 x 3.26 x 1.62 x 8.38 X 1.81 X 4.89 x 3.01 x 5.52 X 3.54 x
107 lo6 109 109 109 lo8 lo8
(c) KDS lo9 0.6 lo8 0.6 los 0.8 107 0.8 lo6 0.8 109 0.8 109 0.8 109
0.507 0.507 0.253 0.253 0.253 0.253 0.253
3.00 X 3.00 X 1.50 X 1.50 X 1.50 X 1.50 X 1.50 X
lo4 lo4 lo4 lo4 lo4 lo4 lo4
6.16 X 10” 6.16 X 10” 5.08 X 10“ 5.08 X 10” 5.08 X 10” 5.08 X 10” 5.08 X 10”
65 70 60 65 70 75 80
89.0 71.8 169.9 133.5 107.4 83 70.3
7.07 X 3.48 X 1.22 x 5.46 x 2.66 x 1.14 x 6.58 x
lo8 los 10’0 109 109 109 109
(d) RbDS 1O1O 0.6 lo9 0.6 log 0.8 lo8 0.8 lo8 0.8 lo7 0.8 109 0.8 lo9 0.8 109
0.585 0.585 0.292 0.292 0.292 0.292 0.292 0.292
3.00 X 3.00 X 1.50 X 1.50 X 1.50 X 1.50 X 1.50 X 1.50 X
lo4 lo4 lo4 lo4 lo4 lo4 lo4 lo4
4.68 X 4.68 X 3.85 X 3.85 X 3.85 X 3.85 X 3.85 X 3.85 X
10” 10” 10“ 10” 10” 10” 10“ 10“
65 70 55 60 65 70 75 80
97.9 78 230.2 186 143.6 116.5 92.3 76.8
9.62 X 4.56 X 3.33 x 1.64 X 6.91 x 3.45 x 1.60 x 8.77 X
lo8 lo*
(e) CsDS ldld 0.6 109 0.6 109 0.8 IO8 0.8 107 0.8 lo’ 0.8 109 0.8 109 0.8
0.663 0.663 0.331 0.331 0.331 0.331 0.331 0.331
3.00 X 3.00 X 1.50 X 1.50 X 1.50 X 1.50 X 1.50 X 1.50 X
lo4 lo4 lo4 lo4 lo4 lo4 lo4 lo4
4.68 X 4.68 X 3.85 X 3.85 X 3.85 X 3.85 X 3.85 X 3.85 X
10” 10“ 10“ lod 10” 10” 10” 10”
65 70 55 60 65 70 75 80
93 75 223.4 178.1 139.6 113 88.3 72.6
8.15 4.01 3.01 1.42 6.28 3.11 1.39 7.26
lo-’
107 107
lo6 106
1010
10’” 109 109 109
lo8
lo8 lo8 lo1” 1O1O x 109 x 109 x 109 X lo8 X
X X X
109
Mole fraction. *See text for estimate.
In the figure captions we indicate the detergent concentrations for each alkali-metal dodecyl sulfate solution represented. We have chosen values of the detergent concentration for each data set to be such that Rh is brought into the region 50 8, < Rh < 250 A (if possible), to enable reliable deductions of the parameter K, free from possible complicating effects of micellar flexibility16J7 and entanglements as the concentration of micelles exceeds the semidilute limit18,19by analogy with polymer solutions.20v21 In Figure la, at an ionic strength of 0.45 M alkali-metal chloride, the RbDS, CsDS, and KDS micelles show a very marked temperature-dependent micellar growth at the relatively low detergent concentration of 0.5 /dL, with Rh increasing from 30-37 8, at 65 OC to over 150 i f at 40 O C . For NaDS in 0.45 M NaC1, despite the fact that the NaDS is 4 times higher in (16) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (17) (a) Safran, S. A.; Turkevich, L. A,; Pincus, P. A. J . Phys. (Paris) Left. 1984, 45, L-69. (b) Safran, S. A. Polym. Mater. Sci. Eng. 1987, 57, 949. (18) (a) Candau, S. J.; Hirsch, E.; Zana, R. J. Phys. (Paris) 1984, 45, 1263. (b) Candau, S. J.; Hirsch, E.; Zana, R. Polym. Mater. Sci. Eng. 1987, 57, 953. (19) Kato, T.; Anzai, S.; Seimiya, T. J . Phys. Chem. 1987, 91, 4655. (20) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (21) Cates, M. E. Polym. Mater. Sci. Eng. 1987, 57, 956.
concentration than the Rb, Cs, and KDS solutions, only a moderate growth to 50 A is observed for temperatures below 15 O C . LiDS micelles show virtually no growth (Rh < 27 A) at the relatively high detergent concentration of 2 g/dL, even upon increasing the ionic strength to 1 M LiCl. In Figure 1, b and c, a similar pattern is observed for the growth of the various alkali-metal dodecyl sulfates at the higher alkali-metal chloride concentrations. Thus, at a given ionic strength, detergent concentration, and temperature, the micellar sizes would be ordered as follows: &(RbDS) > &(CSDS) > &(KDS) >> &(NaDS) >> Rh(LiDS) (3) In contrast to the cmt values, the ordering corresponds exactly to the inverse of the hydrated counterion ~ i z e s . ~In~Table , ~ ~ I1 we present a complete tabulation of the Rh data for all the experiments represented in Figure la-c. The table is organized by counterion in five sections (a-e) in which & values are listed according to the experimental parameters of alkali-metal chloride concentration, detergent concentration, and temperature. Also shown are the values of X , appropriate for each experiment and (22) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Academic: New York, 1959; pp 463-465. (23) Owen, B. B. J. Chim. Phys. 1952, 49, C-72.
Influence of Counterion on Micellar Size and Shape
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8357 TABLE III: Values of a,@, and A Deduced from Applying Eq 4 to Data for In K for Various Counterions in Figure 2a-d alkalimetal counterion Li+
Na+ K+
Rb+ Cs+
rHlaA
2.35 1.85 1.318 1.257 1.268
B
a,K-'
218 324 321 329
f f f f
6 15 11 12
-0.32 -0.52 -0.51 -0.54
f 0.02 f 0.04 f 0.03 f 0.03
A
9
0.180 f 0.003 0.992 0.143 0.006 0.983 0.122 f 0.005 0.988 0.129 f 0.005 0.986
*
aEffective ion size (ref 22 and 23).
OA
Ib
o:
io
40
do
$0
i o do
Temperature ("C)
0.6M Chloride
b
the theoretical parameter K deduced from each value of Rh. The data for NaDS was taken from our previous study8 and are presented here again for comparison. (Data for NaDS at other NaCl concentrations as well as for other chain length sodium alkyl sulfates are available also from ref 8.) Deductions of the thermodynamic micellar growth parameter K are shown in Figure 2a-d. We plot In K versus 1 / T (K) for the various alkali-metal chloride concentrations for the Na+, K+, Rbf, and Cs+ counterions, respectively. In our previous study8 of sodium alkyl sulfate micelles, we found the following semiempirical formula to be useful in describing the dependence of In K on temperature and added alkali-metal chloride concentration for a given chain length: In K / n o = [ a / T (K)
+ A In (C,) + (31
(4)
where C, is the molar concentration of alkali-metal chloride, and a, (3, and A are fitted constants. The solid lines in Figure 2a-d represent results of a multivariate linear regression fit24to the data plotted as solid points for each of the counterions separately. The deduced values of a, A, and (3 from these fits are shown in Table 111.
-
0
Temperature ("C)
In our previous paper,8 we found that the term fl had an approximately linear dependence on alkyl chain length n, (the number of carbon atoms per alkyl chain) and thus expanded our semiempirical expression to also include a linear term to express the dependence of the bracketed quantity on chain length, by setting 0 = (3' + rn,. On the basis of the observation that micellar growth diminished with increasing hydrated counterion size, we have also extended this semiempirical relationship by means of an additional linear term in the brackets: In K / n o = [ a / T (K)
+ A In (C,) + rn, - prH + p"]
(5)
Here rHis the effective Stoke's radius of hydration in 8,as deduced from ionic mobility measurement^,^^,^^ and p is a constant of units A-'. We have used eq 5 to fit all the data for the sodium octyl through tetradecyl chain lengths from our previous together with the data plotted as solid points in Figure 2b-d using multilinear regression. One data point for LiDS at 40 O C was also included.26 The values for the constants deduced from such a fit appear in Table IV, together with theoretical estimates described in the Discussion section. The experimentally observed influence of counterion identity on these micellar systems can be more explicitly shown by plotting In K versus rH,as seen in Figure 3 for the case 0.45 M alkali-metal chloride, 40 O C . The dependence of In K on rH is quite strong,
Figure 1.
Rbversus absolute temperature for various alkali-metal dodecyl
sulfate micelles in solutions of alkali-metal chloride: (a) (+) 2 g/dL of LiDS (X = 1.32 X lo-)) in 1 M LiCI. The remaining data are from solutions of the appropriate alkali-metal chloride a t a concentration of 0.45 M: (0)2 g/dL of NaDS (X= 1.25 X IO-), from ref 5 and 8); (A) 0.5 g/dL of KDS (X= 2.96 X IO4); ( 0 )0.5 g/dL of CsDS (X= 2.26 X IO4); (0) 0.5 g/dL of RbDS (X= 2.56 X lo4). (b) Solutions of 0.6 M alkali-metal chloride, with total detergent concentration for each curve as follows: (0)X = 1.25 X NaDSs; (A)X = 3 X IO4 KDS; ( 0 ) X = 3 X lo4 CsDS; (0)X = 3 X IO4 RbDS. (c) Solutions of 0.8 M alkali-metal chloride, with total detergent concentration for each curve as follows: (0)X = 1.25 X 10-3NaDS;S (A)X = 1.5 X lo4 KDS; ( 0 ) X = 1.5 X lo4 CsDS; (0)X = 1.5 X lo4 RbDS.
(24) SAS Institute, SAS Circle, Box 8000, Carey, N.C. 2751 1. (25) The following data points were included from Table I1 of ref 8. Sodium nonyl, decyl, undecyl, and hexadecyl sulfate: all data points. Sodium dodecyl, tridecyl, and tetradecyl sulfate: all data points plotted as filled symbols in Figure 7d-f of ref 8 respectively. (26) For 2 g/dL of LiDS in 0.45 M LiCl at 40 OC,Rb = 26.1 & 0.8 A. If we assume the effective Rh includes a layer of associated Li+ counterions, and if these counterions remain fully hydrated, we may subtract the Li+ ion hydrated diameter of 4.7 A from this value to obtain the radius of the hydrocarbon core plus head roup. Using a radius of 21.4 f 0.8 A, from our previous theoretical curve$ calculated for dodecyl sulfate micelles ignoring counterion adsorption, we obtain an In K value of 3.9 f 4.5. Although the remainder of the In K values for the other counterions were calculated by ignoring counterion adsorption contributions to R,, the errors introduced are quite small because of the great amount of growth which has occurred. This single data point was weighted twice in the fit to eq 5.
Missel et al.
8358 The Journal of Physical Chemistry, Vol. 93, No. 26, 1989
a)
b)
Temperature ("C)
28
1
~
0 70
1
'
60
1
'
50
1
40
'
1
30
'
l
20
~
I
'
Temperature ('C)
26
l
10
I 80
'
I
'
I
70
'
I
'
50
60
40
O.6M
23
24 1 n
1 n
20
20
K
K
0
17
16 0.45M
0
0
Potassium
Sodium 14
12 2.8
3.0
3.4
3.2
c)
2
3.0
2
1000 / T(K)
IOOO / T(K)
4
Temperature ('C)
Tem parat u r r
("C)
0
1
14
17.
0
Cesium
Rubidium
2.8
3.0
14 2.8
3.2
3.0
I 3.2
1000 / T(K)
IOOO / T(K)
Figure 2. Deductions of In K, plotted versus 1000/T (K). (a) N a D S in NaCI, taken from ref 8. The remaining data are from this study: (b) KDS in KCI; (c) RbDS in RbCI; (d) CsDS in CsCI. Solid lines represent least-squares fit to eq 4 for each counterion (see Table I11 for parameter values resulting from these fits). TABLE I V Values of a. B".
r. A, and P Deduced from Applying Ea 5 to Data for In F pll -0.47 f 0.04
r
A
0.043 f 0.002
0.208 & 0.005
fit to data
a,K-' 241 f 10
total electric
Theory Assuming the Stern Layer Begins outside the Headgroups 141 f 7 -2.10 f 0.03 0.080 f 0.001 0.191 f 0.004
P,
A-'
0.232 f 0.007 9 = 0.938 0.494 f 0.005
9 = 0.980 double layer
26 f 4
-0.92 f 0.01
0.007 f 0.001
0.193 f 0.002
0.028 f 0.003
9 = 0.995 . . -1.19 f 0.03
0.467 f 0.007 9 = 0.984
0.074 f 0.001
Stern layer
115 f 9
total electric
Theory Assuming the Stern Layer Begins outside the Hydrocarbon Core 0.056 f 0.001 0.199 f 0.002 113 f 4 -1.74 f 0.02
0.392 f 0.003
9 = 0.992 double layer
27 f 4
-0.94 f 0.02
0.008 f 0.001
Stern layer
87 f 5
-0.80 f 0.02
0.048 f 0.001
0.199 f 0.002
0.033 f 0.003 9 = 0.995 0.359 f 0.004 9 = 0.988
"Data for a range of chain length sodium alkyl sulfate^^.^^ have been pooled together with data for other alkali-metal dodecyl sulfates in Figure 2a-d .26
decreasing almost 4-fold as rH increases from 1.2 to 2.4 A. The experimental deduction of p of 0.232, when multiplied by the is responsible for a change in In K of 16. A theoretical prefactor 4, explanation for the strong dependence of In K on counterion size
is presented in the Discussion section.
Discussion The specific influence of each counterion on the alkali-metal
Influence of Counterion on Micellar Size and Shape
0.4%
Chloride
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8359
No data could be found for the cmc of RbDS; thus, in this study, we have assumed that the cmc of RbDS is equal to that of CsDS for the same amount of added alkali-metal chloride. No data exists either for SI( T,CJ for the various alkali-metal dodecyl sulfates. If SIwere independent of counterion identity, the theoretical cmt’s (points of intersection between the cmc and SI)would increase in the same ordering as the cmc’s, which clearly disagrees with the data. To produce the observed ordering of cmt values requires that SIvaries as SI(LiDS) > Sl(NaDS) > S1(CsDS) > S1(KDS) L SI(RbDS) (6b)
1.2
1.4
1.6
1.8
2.0
2.2
2.4
STOKES ION RADIUS (A)
Figure 3. Values of In K plotted versus rH for 0.45 M alkali-metal chloride at 40 “C.The value for ?f was calculated from eq 4 by using the parameters in Table 111. The solid line was generated by using eq 5 and the experimentally deduced parameters in Table IV.
dodecyl sulfate/alkali-metal chloride/water system can arise from various sources. By virtue of their finite size, the various counterions will exert systematically varying influences on the electrostatic free energy of micellar formation and growth. Independent of this, specific chemical binding of counterions to surfactant aggregates can occur. These specific chemical affects need not necessarily follow the same pattern indicated by finite size effects, especially if water of hydration is lost upon binding. We present evidence that, for this system, counterion identity influences micellar growth largely through electrostatic effects. However, we first consider the effect of various counterions of the cmt, which appears to arise from a different origin. A. Influence of Counterion Identity on the cmt. We have previously demonstrated how2 the ideas set forth by Murray and Hartley2’ regarding detergent solubility offer an accurate prediction of the dependence of the cmt on counterion concentration for the NaCS/NaCl/water system. Stated briefly, this approach assumes that the cmt is determined by the intersection of the cmc with the phase boundary determining the solubility of the monomer in solution, denoted SI.Below the intersection point, the solubility of the monomer is below the cmc and thus is too low to support the formation of micelles. Above the intersection point, detergent solubility increases markedly since much more material can be accommodated in solution in the micellar state. The kinetics of dissolution are very rapid. As the temperature is raised from below to above the cmt, the solution clarifies in a matter of seconds with mild agitation. Both the cmc and SIare functions of temperature, counterion identity, and counterion concentration. For the moment, consider the case of fixed alkali-metal chloride concentration. The variation of the cmc for the alkali-metal dodecyl sulfates follows the same order as the hydrated counterion sizes:15~28-38 cmc(LiDS)
> cmc(NaDS) > cmc(KDS) > cmc(CsDS) (6a)
(27) Murray, R. C.; Hartley, G. S. Trans. Faraday SOC.1935, 31, 183. (28) Kodama, M. J . Sci. Hiroshima Univ. Ser. A 1973, 37, 53. (29) Goddard, E. D.; Harva, 0.;Jones, T. G. Trans. Faraday SOC.1953, 49, 980. (30) Williams, R. J.; Phillips, J. N . ; Mysels, K. J. Trans. Faraday SOC. 1955, 51, 728. (31) Meguro, K.; Kondo, T.; Yoda, 0.;Ohba, N . Nippon Kagaku Zasshi 1956, 77, 1236. (32) Goddard, E. D.; Benson, G. C. Can. J . Chem. 1957, 35, 986. (33) Mysels, K. J.; Princen, L. H. J . Phys. Chem. 1959, 63, 1698. (34) Schick, M. J. J . Phys. Chem. 1964, 68, 3585. (35) Moroi, Y.; Motomura, K.; Matuura, R. J. Colloid Interface Sci. 1974, 46, 1 1 1 . (36) Volkov, V. A.; Vins, V. G.; Rodionova, R. V. Izv. Vyssh. Uchebn. Zaoed., Khim. Khim. Teknol. 1975, 18, 983.
which is a different ordering from that of the bare or hydrated counterion radius. Such nonperiodic ordering could reflect chemical interaction between the headgroups and counterions in the hydrated crystal state. This suggests that the cmt is influenced by counterion identity in a very different manner than is micelle formation and growth. This is contrary to some ideas proposed earlier suggesting a possible connection between the cmt and the observation of micellar g r o ~ t h . ~ ” Other ~ ~ physicochemical studies for the NaDS/NaCl/water system have confirmed that these two phenomena are not casually related.43.M B. Potential Influences of Intermicellar Interactions. We have interpreted the diffusion coefficient as providing an actual measure of micellar size using eq 1, ignoring the potential complicating influences of intermicellar interactions and entanglements. The recent experimental data by Ikeda et al!5*46 is relevant here. They found the Debye plot to be highly nonlinear, with significant upward curvature at low detergent concentration (C0.02M SDS). This upward curvature is reproduced exactly by using our theory for micellar growth. Such a comparison is shown in Figure 9 of ref 5 for another set of Debye plot experiments. In fact, when excluded volume interactions are considered, we can predict the entire concentration dependence of Ikeda’s Debye plots without any consideration of long-range attractive interactions. Using the thermodynamic parameter K as a single adjustable parameter, we obtain K values from Ikeda’s data that agree to within less than 5% of ours at identical salt concentration and temperature!’ Thus, for low concentrations, we find no need to appeal to the attractive micellar interactions proposed by Corti and Degiorg i ~ . ~ We ” ~ have assumed that this is the case also for the other alkali-metal dodecyl sulfates as well. However, at higher detergent concentrations, it is quite likely that the diffusion coefficient will be altered as the mean distance between micelles approaches the mean micellar radius of gyration.l* The concentration at which this occurs is designated as X . We have calculated the ratio X / X (see Appendix) and find that it did not exceed -0.4 for any of the experiments reported here. We have made similar calculations for another system, sodium hexadecyl sulfate, which undergoes an extreme degree of micellar growth with added NaC1. A careful analysis of the concentration dependence of D for the condition 0.45 M NaCI, and for values of X ranging from - 5 X IO” to 6 X lo“, shows that deviations from the square root dependence become apparent (37) Kishimoto, H.; Sumida, K. Chem. Pharm. Bull. Jpn. 1976,24, 1235. (38) In the presence of added alcohol, the reverse order is observed, suggesting that hydration effects have changed. See: Singh, H. N.; Swarup, S.; Saleem, S.M. J. Colloid Interface Sci. 1979, 68, 128. (39) Corti, M.; Degiorgio, V. Ann. Phys. 1978, 3, 303. (40) Corti, M.; Degiorgio, V. Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum: New York, 1979; Vol. I, p 377. (41) Corti, M.; Degiorgio, V. Light Scattering in Liquids and Macromolecular Solutions; Degiorgio, V., Corti, M., Giglio, M., Eds.; Plenum: New York, 1980; p 109. (42) Corti, M.; Degiorgio, V. J . Phys. Chem. 1981, 85, 711. (43) Frames, E. I.; Davis, H. T.; Miller, W. G.; Scriven, L. E. J . Phys. Chem. 1980,84, 2413. (44) Staples, E. J.; Tiddy, J. T. J . Chem. Soc., Faraday Trans. 1 1978, 74, 2530. (45) Hayashi, S.; Ikeda, S. J . Phys. Chem. 1980,84, 744. (46) Ikeda, S.; Hayashi, S.; Imae, T. J . Phys. Chem. 1981,85, 106. (47) Unpublished results.
8360 The Journal of Physical Chemistry, Vol. 93, No. 26, 1989
-
Missel et al. CHART I spherical regions
= 2RTge,(S,)+ N A d ,
m=2
Figure 4. Flat cross-sectionalview of an idealized micellar region depicting the Stern layer. No counterion charge is allowed inside region I; thus, the Laplace equation (A-1) applies here. In region 11, the Poisson-Boltzmann equation (A-2) applies. For three-dimensional views, see Figure 8.
only when X / X * exceeds -0.45. Deductions of the thermodynamic parameter K appear to be independent of detergent concentration as long as X/x*5 0.45. Thus,the experiments reported here for the KDS, RbDS, and CsDS systems are conducted within the regime where K values can be deduced correctly independent of detergent concentration, and should not be strongly affected by entanglements. If entanglement effects were present, the true K values and micellar sizes would be even larger than those reported in Table 11. C. Influence of Counterion Identity on Micellar Growth. Our original theoretical treatment of the influence of added NaCl on SDS micellar growth2q5employed the Gouy-Chapman theory of the flat electric double This simple approach admittedly ignored the contributions of counterion binding,s2-59 surface curvature,6M2 detailed considerations of the structure of the micellar s ~ r f a c eand , ~deficiencies ~ ~ ~ ~ ~in the Poisson-Boltzmann (PB) equation due to the finite size of the counterion- and other f a ~ t o r s . ~ ~Porte - ~ l and Appell later demonstrated that a massaction model for counterion binding could also predict the influence of added salt concentrations2 and specific counterion effectss3 on the sphere-to-rod transition for micellar solutions of the cetylpyridinium and cetyltrimethylammonium salts.
(48) Gouy, G. J . Phys. 1910, 9,457; Ann. Phys. 1917, 7, 129. (49) Chapman, D. L. Philos. Mag. 1913, 25,475. (50) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability Lyophobic Colloids; Elsevier: New York, 1948. (51) Stigter, D. J . Colloid Interface Sei. 1974, 47, 473. (52) Porte, G.; Appell, J. J . Phys. Chem. 1981, 85, 2511. (53) Porte, G.; Appell, J. Surfactants in Solution; Mittal, K. L., Ed.; Plenum: New York, 1984; Vol. 11; pp 805-825. (54) Stigter, D. J . Phys. Chem. 1964, 68, 3603. (55) Stigter, D. Prog. Colloid Polym. Sci. 1978, 65, 45. (56) Beunen, J. A.; Ruckenstein, E. J . Colloid Interface Sei. 1983,96,469. (57) Gross, L. M.; Straws, U. P. Chemical Physics of Ionic Solutions; Conway, B. E., Barradas, R. G., Eds.;Wiley: New York, 1966; pp 361-389. (58) Mukerjee, P. J . Phys. Chem. 1962, 66, 943. (59) Rathman, J. F.; Scamehorn, J. F. J . Phys. Chem. 1984, 88, 5807. (60) Mitchell, D. J.; Ninham, B. W. J . Phys. Chem. 1983, 87, 2996. (61) Evans, D. F.; Ninham, B. W. J . Phys. Chem. 1983,87, 5025. (62) Evans, D. F.; Mitchell, D. J.; Ninham, B. W . J . Phys. Chem. 1984, 88, 6344. (63) Stigter, D. J . Phys. Chem. 1974, 78, 2480. (64) Stigter, D. J . Phys. Chem. 1975, 79, 1008. (65) Stigter, D. J . Phys. Chem. 1975, 79, 1015. (66) Stern, 0. Z . Electrochem. Angew. Phys. Chem. 1924, 30, 508. (67) Sparnaay, M. J. J . Electroanal. Chem. 1972, 37, 65. (68) Levine, S.; Bell, G. M. Trans. Faraday Soc. 1966, 42, 69. (69) Bell, G . M.; Levine, S. Reference 57, pp 409-454. (70) Jonsson, B.; Wennerstrom, H.; Halle, B. J . Phys. Chem. 1980, 84, 2179. (71) Gunnarsson, G.;Jonsson, B., Wennerstrom, H. J . Phys. Chem. 1980, 84. 31 14.
cylindrical regions
@,d= 2RTge,(S,) + NAw6,
(7a.b)
m=l
2 C,CkB T
It has been proposed that effects which have been phenomenologically attributed to counterion binding could simply be a consequence of purely electrostatic causes (see, for example ref 60-62 and 7 1). The further theoretical development presented here reflects this concept. We investigated the possibility that extensions of the Gouy-Chapman theory would alone be sufficient for predicting our observed counterion effects without invoking specific counterion binding. Therefore we have included two new elements to our previous theory for the electrostatic contribution to micellar growth, which was originally based upon the Gouy-Chapman flat electric double layer.48s49The first modification is to include the influence of surface curvature by solving the PB equation in spherical and cylindrical coordinates as appropriate. Ninham and co-workers have developed some useful analytical approximations for the influence of surface curvature on the nonlinear PB equation and have applied them in modeling the energetics of the formation of spherical SDS micelles.6M2 We have adapted their approximations appropriately for modeling the energetics of spherocylindrical micellar growth. A second and more significant addition to our electrostatic theory is the incorporation of a constraint of a finite distance of approach, 6, for the counterion. In Figure 4 we show a hypothetical cr0s.s-sectional view of a micelle (either hemispherical or cylindrical region; refer to Figure 8 for threedimensional views). The charge associated with the metal counterion is assumed to be excluded from a shell (region I) of thickness 6, taken to be equal to the hydrated counterion radius, surrounding the idealized negatively charged micellar surface of radius Rg.This is essentially a S t e r P layer where the potential for specific adsorption of the counterion has been neglected. In all other respects, the charge distributions in solution (region 11) and on the surface are taken to be smooth distributions. This provides a purely electrostatic interpretation of counterion effects, which exert their specific influences by virtue of their differing hydrated sizes. D. The Curved Electric Double Layer with Finite Distance of Closest Approach. In the Appendix, we derive the electrostatic free energy necessary to form a curved electric double layer assuming a finite distance of closest approach for the counterion. Calculated are the electrical chemical potentials per mole of anions in cylindrical micellar regions and hemispherical endcaps, M:I and ~.c;l, respectively. For convenience, the key results are reproduced in eq 7-1 1 of Chart I. Here no is the aggregation number of the maximum spherical micelle (aggregates larger than this must extend in at least one dimension). R8 is the radius of the surface at which the Stern layer begins. In view of the uncertainty associated with surface roughness, R6 can fall between the length of the hydrocarbon chain I,, or the full length of the detergent anion R,. For the alkali-metal dodecyl sulfates we set n, = 55.5, R, = 20.34 A and 1, = 16.68 A, consistent with our previous work (using eq 6 of ref 8). Further, e is the electronic charge (in esu),
Influence of Counterion on Micellar Size and Shape
The Journal of Physical Chemistry, Vol. 93, No. 26, 1989 8361
Molarity added NaCl I
I
Molarity Added NaCl I
0.45 0.6
0.3
I 0.8
e C
t r 1 -1 C
a
I
F r e
e
E -1 n e
r 9 Y
-2.4
-2.0
-1.5
-1.0
-0.5
-1
0.0
by various models for dodecyl sulfate micelles. Dashed curve: GouyChapman approach ignoring surface curvature.’V5 Solid lines include effects of surface curvature as derived in the Discussion section. Solid points are exact numerical calculations using the same set of boundary conditions.’* dielectric constant of water, kBis the Boltzmann constant, Tis absolute temperature (K), N A is Avogadro’s number, C,is the counterion concentration in ions/cm3, and K is as usual the inverse Debye length: c is the
Each of the chemical potentials p:l and p:’ are composed of two parts. The first part arises from solving the PB equation outside the Stern layer (region 11) and is represented by the function gel(S) of eq A-1 (taken from ref 60-62). The second arises from calculating the contribution, represented by d,,,, electrostatic energy in the unshielded Stern layer (region I). Both geland the w6 parameters depend upon the surface charge density cr, for the spherical region and u, for the cylindrical region. (gel depends upon surface charge density through the variable S.) We assume the micellar anions are fully charged; i.e., no specific counterion binding occurs. The values of us and ucare deduced from packing considerations developed in our previous paper.* The expression for gelis functionally equivalent to our previous formula for the flat electric double layer2q5with an extra term to account for the influence of surface curvature. Because of an added set of boundary conditions at the Stern layer, the variable S is now influenced by the surface curvature and the thickness of the Stern layer, unlike the previous case for the flat electric double layer. The contributions arise as the surface charge c is effectively redefined as that at the surface of a curved plane displaced a distance 6 outside the micellar surface. This imparts a weak dependence of gelon amphiphile chain length n, and on the thickness of the Stern layer 6 . gelis a strong function of counterion concentration C,. The second contribution to each of the chemical potentials represents the energy required to charge the unshielded Stem layer. As such, it is tacitly independent of counterion concentration. However, it depends strongly upon both the thickness of the Stern layer and upon surface curvature (and therefore chain length). ~ dCfor the spherical and We denote this contribution by w * and cylindrical micellar regions, respectively. Both of these terms vary approximately linearly with 6 , provided 6
