5738
J. Phys. Chem. 1988, 92, 5138-5143
remain agglomerates. The particles stick to the sintered platinum thin film, just as they stick to the foil surface; hence there is no agglomerate growth in zone IV. It is interesting to note that these crystals are very similar in size to the single-crystal silicon particles that form on a silicon substrate during deposition of silicon from the vapor phase.45 In zone v the particle density is low because (45) Wajda, E. S.; Kippenhan, B. W.; White, W. H. I B M J . 1960,22,288.
it is far from the reaction focus. The particles that do "land" there are mobile, because there is only silica underneath, and eventually "wash off" that region. Acknowledgment. We are grateful to the National Science Foundation (Grant NO. CBT-8616502) for providing financial support for this work. Registry No. Pt, 7440-06-4; H2,1333-74-0.
Temperature and Concentration Dependence of Properties of Sodium Dodecyl Sulfate Mlcelies Determlned from Small-Angle Neutron Scattering Experiments V. Yu. Bezzobotnov? S. BorbBly,+** L. Cser,*i%B. Farag6,s I. A. Gladkih,s Yu. M. Ostanevich: and Sz. Vassi Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525, PO Box 49, Budapest, Hungary, and Joint Institute for Nuclear Research, Dubna, PO Box 79, Moscow, USSR (Received: November 3, 1987; In Final Form: February 29, 1988)
Sodium dodecyl sulfate (SDS) micelle heavy water solutions were studied by using small-angle neutron scattering in a wide range of surfactant concentrations (0.03-0.6 mol/d1n3) and of temperatures (20-60 "C). A simple model is proposed to describe the concentration and temperature dependence of average aggregation number.
Introduction Amphiphilic molecules dissolved in water over a certain critical micellar concentration (cmc) can form various types of aggregates or micelles. Various theoretical approaches have been proposed to interpret data collected from different types of experiments. In one of these approaches the formation of micelles in a solution of ionic surfactants is assumed to be a stepwise kinetic process in which assemblies of aggregation number n are formed from those having aggregation number n f 1. Thus, in a way similar to chemical reactions, the process may be described as being in equilibrium in terms of thermodynamics.' The chemical potential involved in thermodynamic considerations has its origin in solute/solvent, solute/solute, and aggregate/aggregate interactions. The first two interactions can be expressed by the specific area of the micelle/solution contact surface belonging to one head group. The solute/solvent interaction arises mainly from the hydrophobic effect exerted by water molecules on the long aliphatic chain of surfactants, and its contribution is proportional to the specific head-group The solute/solute interaction in the case of ionic surfactants is of electrostatic nature: its contribution comes mainly from the Coulombic repulsion among the head groups and is somehow inversely proportional to the specific head-group area.2*5-9 The opposite tendency of these two contributions ("opposing forces", introduced by Debyes) versus specific head-group area defines a minimum energy contact surface and with packing constraints determines the micellar size and shape at least for small surfactant concentrations. According to recent theories,iO*lla t higher surfactant concentrations the aggregate/ aggregate interaction plays an 'increasingly dominant role and becomes the size- and shape-determining factor. Interpretation of the results obtained from thermodynamic analysis requires that thermodynamic quantities be expressed in terms of the molecular parameters of the assumed microstructures. The existence of these microstructures is proven and their properties are investigated by using different light-~cattering'*-'~ as To whom correspondence should be addressed. Joint Institute for Nuclear Research Dubna. :On leave from Kossuth Lajos University, H-4010, Debrecen, Hungary. 'Central Research Institute for Physics of the Hungarian Academy of Sciences. 0022-3654/88/2092-5738%01 S O I O
well as small-angle X-rayis and scattering techniques. The fact that the neutron wavelength can be compared with the intermolecular distances and scattering lengths for different isotopes results in good and variable contrast. Despite the complicated apparatus required, the small-angle neutron scattering (SANS) technique is the most promising of the above-mentioned experimental methods. (For detailed summaries of neutron scattering studics see ref 19 and 20.) In particular, this technique can in principle measure the micellar size and aggregation number and the extent of water penetration into the micellar core. The main difficulty in interpreting the scattering patterns arises from the presence of intermicellar-aggregate type interactions.
(1) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1973. (2) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (3) Hermann, R. B. J . Phys. Chem. 1972, 76, 2754. (4) Tanford, C . Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 1811. (5) Debye, P. Ann. N.Y. Acad. Sci. 1949, 91, 575. (6) Deriagin, B. V. Usp. Khim. 1979, 48, 675. (7) Gunnarson, G.; Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1980, 84, 3114. (8) Mukerjee, P. J . Phys. Chem. 1972, 76, 565. (9) Konstantinovich, A. B. K.; Albota, L. A. Kolloidn. Zh. 1983,45,68. (10) Ben-Shaul, A.; Gelbart, W. M. J . Phys. Chem. 1982,86, 316. (1 1) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. Phys. Reu. Lett. 1985, 54, 955. (12) Huisman, H. F. Proc. Ned. Akad. Wet. Ser. E Phys. Sci. 1964,67,
746.
(13) Mazer, N . A.; Benedek, G. B.; Carez, M. C. J . Phys. Chem. 1976, 80, 1075. (14) Corti, M.; Degiorgio, V. J . Phys. Chem. 1981, 85, 711. (15) Mattoon, R. W.; Steams, R.S.; Harkins, W. D. J . Chem. Phys. 1948, 16, 644. (16) Hayter, J. B.; Penfold, J. J . Chem. Soc., Faraday Tram. 1 1981, 77, 1851. (17) Bendedouch, D.; Chen, S.H.; Koehler, W. C. J . Phys. Chem. 1983, 87, 153. (18) Kotlarchyk, M.; Chen, S . H. J . Chem. Phys. 1983, 79, 2461. (19) Allen, G.; Higgins, J. S.Rep. Prog. Phys. 1973, 36, 1073. (20) Jacrot, B. Rep. Prog. Phys. 1976. 39, 911.
Q 1988 American Chemical Societv
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 5739
Properties of SDS Micelles The following theoretical approach was developed16 to elicit microscopic information from neutron scattering data. This method is based on solving the Ornstein-Zernike equation2’ h(r) = c(r)
+ pJh(r
- r’) c(r’) dr’
where c(r) = -u(r)/kT is the interparticle pair potential and T is the temperature. The function h(r) is connected with the pair correlation function g(r) by the relation h(r) = g(r) - 1. This method requires laborious computation to fit the parameters of eq 1 to the experimentally obtained SANS pattern. This method was applied to SDS micelle solutions in the work of Hayter and Penfold.I6 Evaluation of the expected experimental intensity pattern requires that one make assumptions on the aggregation number distribution and also on the shape of micelles. A model of monodisperse mean spheres27is usually used to avoid mathematical (and numerical) difficulties. In addition, the intermicellar interaction is usually described by means of the Deryagin-Landau-Verwey-Overbeek (DLVO) “screened” Coulomb potential. This choice again contains of some arbitrary elements. In the present work we carried out a reexamination of the concentration-dependence behavior of SDS micelles and extended the examination to the temperature properties of these micelles. To determine the tzmperature and concentration dependence of the average aggregation number in present work, we used the above-mentioned approach. However, due to our concern that the assumptions on which this method is based are not proved to be 100% correct, we also used a simpler method that in turn is again not correct but suffers from another type of error. Given that the results obtained can be compared with those of the mean spheres approximation, it would allow us to reveal what type of information is model dependent and what type of information is model independent. For a collection of uniform spherical particles having a number density Np,the scattered intensity distribution is given in general by
where P(q) represents the particle structure factor and S(q) is the apparent interparticle structure factor. (For nonspherical particles, the so-called effective structure factor is usually introduced.28 Supposing that the condition S(q) i= 1 is fulfilled at the q range over which the Guinier approximation for a single-particle structure factor can be applied, then the radius of gyration value can be determined. In fact, if the concentration of micelles is low, then their average distance is large and the value of the transferred momentum (q) at which S(q) has its maximum tends to zero. Oscillations of S(q) a t higher q values are expected to be smeared out because of fluctuation of the aggregation number, so there is a hope that the tail of the Guinier range overlaps with the q range where S(q) = 1 and consequently I(q)
- P(q)
exp(-l/3q2R,2)
(3)
is valid. Here R, stands for the radius of gyration. This above hope is supported by the fact that-as was recently establishedI7-the effective scattering length density of the sulfate (21) Ornstein, L. S.;Zernike, F. Proc. K.Ned. Akad. Wer. 1914,17,193. (22) GilHnyi, T.;Wolfram, E.; Stevgiopoulos, Ch. Colloid Polym. Sci. 1976,254, 1018. (23) Tamas, J., personal communication, 1982. personal communication, 1982. (24) Decsy, Z., (25)Vagov, V. A.; Kunchenko, A. B.; Ostanevjch, Yu. M.; Salamatin, I. M. JINR Report No. P14-83/898,Dubna, USSR, 1983;in Russian. (26) Kratky, 0.; Porod, G. J. Colloid Sci. 1949,4, 35/70. (27) Bendedouch, D.;Chen, S. H.; Koehler, W. C. International School of Physics ’Enrico Fermi” Varenna, July, 1983. (28) Hayter, J. B. International School of Physics “Enrico Fermi” Varenna, July, 1983.
head group is, within the limits of experimental error, equal to the scattering length density of the D 2 0 solvent. Thus in a D 2 0 solution only the hydrocarbon core is observable in a SANS experiment. As a consequence, the apparent size of the micelles gets smaller than the real one and the Guinier range becomes more extended. In this case the aggregation number is equal to the ratio of the volume of the hydrocarbon core and the volume of the hydrocarbon tail of one SDS molecule. In the case of elongated particles there exists a q range where the so-called Porod’s approximation I ( q ) i= q-’ exp(-1/2q2R,2)
(4)
is valid. Here, R, is the so called radius of gyration of cross section.29 The limits of concentration within which the above approximate formulas are valid need to be determined empirically. We found that for concentrations from 0.03 to 0.25 M over the range q = 0.01-0.03 A-2 the Z(q) function can well be approximated by formula 2 and over the range q = 0.02-0.04 A-2 by formula 3. These approximations were valid for the temperature range 20-60 OC. The consistency of results obtained by the use of both methods should serve as a feedback proof of the applicability of both methods. Materials and Sample Preparation Technical grade SDS (Merck) was purified by fractional recrystallization from solutions in a 1:l mixture of ethanol and benzene. The purified materials were tested by surface tension measurements,22by mass spectroscopic analysis of the equilibrium vapor phase of the heated solid SDS sample, and by the combined liquid chromatography-mass spectroscopic technique. Before purification, equilibrium vapor field analysis indicated that the only impurity was 1-dodecene: semiquantitative results showed that the purification had reduced its amount by about 2 orders of magnitude compared with that measured in the original material.23 The liquid chromatography-mass spectroscopic analysis showed less than 0.3% decyl, 0.2% tetradecyl, and 0.1% hexadecyl sulfate: no other analogues were found in the solution.24 SDS was dissolved in freshly distilled D 2 0 (99.8%),and the solutions were poured into quartz sample holders to provide the 1-2-mm sample thickness required by the concentration of the investigated solution. Experimental Results and Data Evaluation The SANS experiments were performed at the small-angle time-of-flight spectrometer situated at the IBR-2 fast impulse reactor of the Joint Institute for Nuclear Research, D ~ b n a . 2At ~ this time-of-flight spectrometer the value of transferred momentum q usually varied from 0.02 to 0.22 A-1 with a neutron flux of lo7 neutrons cm-2 s-I. In some cases the upper limit of the q range was enlarged to 0.3 A-’. The scattering patterns were obtained from samples of surfactant concentrations varying in the range 3 X 10-2-0.6 mol/dm3 a t various temperatures between 20 and 60 OC; all scattering patterns were corrected for resolution and background effects. The coherent scattering cross section was obtained by using a vanadium standard scatterer. Two methods were used to obtain the wavelength-dependent corrections for sample transmission and direct beam intensity. For weak scatterers (Le., dilute solutions) a vanadium target was placed between the sample and the detector at an appropriate distance from the detector, and the experiment was carried out periodically by repeating the measurement with and without the vanadium in the beam. For strongly scattering samples the transmission was measured by utilizing a neutron detector of low efficiency that was positioned in the central hole of the main neutron detector.25 Typical scattering patterns are shown in Figure 1. Their general shape is very similar to the spectra observed by Hayter2* and Hayter and Penf01d.~~ (29)Guinier, A,; Fournet, G. Small Angle Scattering of X-Rays; Wiley: New York, 1955.
Bezzobotnov et al.
5740 The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 1
TABLE I: Average Aggregation Number of SD6 Micelles at Two Ternmatures and at Different SIX3 Concentrations T = 25 “C concn, M T = 60 OC 0.03 84 65 0.07 0.10 0.12s 0.16 0.2s
Figure 1. Typical scattering intensity distribution versus SDS concentration: (X) 0.125 M; (0)0.25 M; (0)O S M; ( 0 )1.0 M; T = 20 OC.
100 113 122 129 142
0
0.1
74 92 91 100 1os
0.2 C.C.SDS(M1
Figure 3. Values of fourth degree of the averaged aggregation number versus SDS concentration at 25 (0)and 60 OC (+).
t”
Rqlnml
150
f I I
11)
‘OOi
I
0
0.1
42
C.C.(Y)
Figure 4. Concentration dependence of radius of gyration of cross section for different temperatures: (a) T = 25 OC; (b) T = 60 OC.
A
50-
, 005
,
1
I
1
/
05
/
/
I
/
c
10 /c-c.m.cifmo(arl
Figure 2. Mean aggregation number of micelles versus the one-fourth power of the concentration of aggregate-forming surfactant molecules (c cmc (molar)) obtained for SDS micellar solutions at 20 (0),30 (D), 40 (O), 50 (A),and 60 (+) OC.
To determine the average aggregation number (n),we applied both of the above mentioned methods using the corresponding least-squares fit. The mean spheres approximation results are given in Figure 2. It is clearly seen that at all temperatures the fourth power of average aggregation number depends linearly on the SDS concentration while by increasing temperature a monotonic decrease of the aggregation number is observed. With use of formulas 3 and 4 the R, and R, values were evaluated for the same temperatures and concentrations. Then the volume of the micelle can be estimated from the above parameters by supposing that micelles form ellipsoids of revolution with axes a and b that are related to the radius of gyration and to the radius of gyration of cross section as R: = 1/5a2 -+ 2/5b2
(5)
b2 = 2R:
(6)
Thus, the volume of an ellipsoid can be expressed as
V = 8/3rRq2(5Rg2 - 4R,Z)’/’
(7)
The aggregation number could then be determined as n = V/u
(8)
(30) Hayter, J. B.; Penfold, F. J . Colloid Polym. Sci. 1983, 261, 1022.
a1
0
-
Qp
oalYl
Figure 5. Concentrationdependence of the aggregation number at T = 25 OC (a) and T = 60 OC (c). Curve b connects Hayter’s data2*obtained at 40 OC.
where v is the volume of the hydrocarbon chain, which is equal to 360 A3.” The n values obtained are given in Table I and in Figure 3 . The n4 c relation for two extreme temperatures (20 and 60 “ C ) seems to be valid again. Note that the radius of gyration of cross section slightly increases with increasing SDS concentration (see Figure 4).
-
Discussion One of the important characteristics of micelles is the aggregation number. In reality one always deals with the distribution of this parameter so that it can be characterized by the average value of the aggregation number and by its dispersion. In this paper we restricted our consideration to these parameters, so in the following the term “aggregation number” will be used for its average value. Consistency of Data. From a comparison of the data given in Figure 2 and Table I it follows that n values obtained by the mean spheres approach are about 20% lower than those obtained by the use of the Guinier-Porod approximation. This difference evidently
Properties of SDS Micelles
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 5741
may originate from the overestimation of the value of the radius of gyration. At the same time, Figure 5 shows that Hayter’s dataB fit very well with our data derived from Guinier-Porod approximation. In that Hayter used the same mean spheres approximation as we did, it points to there being only one possible explanation of the observed discrepancy. Namely, the correction for the resolution function should be made very carefully since the shape of the scattering pattern is appreciably distorted by the convolution of the scattering cross section with the resolution function. This correction was not made for Hayter’s data, but it is important since the width of the wavelength distribution of the D11 facility at the Institute Laue-Langevin had a value of 10% during the course of the cited experiment. Our experience shows that this effect alone is not large enough to explain the discrepancy between our results and those of Hayter. But with a computer simulation it was shown that if the mean spheres approximation was employed for our uncorrected data for the instrumental resolution, it again gave rise to increased values of the aggregation number by about 10%. It was also observed that the value of the aggregation number was sensitive to the value of the upper limit of the q range. These observations indicate that our guess concerning the difference in the aggregation number values are at least qualitatively correct. Quantitative comparison was not possible because the resolution function of our time-of-flight small-angle scattering spectrometer appreciably differs from that of the constant-wavelength device used in Hayter’s experiment. The increased value of the aggregation number obtained from the Guinier-Porod approximation obviously comes from the effect that the S(q) function makes the scattering pattern always steeper over the q range that was considered as the Guinier range. Despite the relatively large discrepancy in the absolute values of the aggregation number, their concentration dependence remains similar for all cases, i.e., for all sets of data the relation n4
-
c
(9)
is valid. Concentration Dependence. Formula 9 was obtained on an empirical basis. In the following let us carry out a simple mathematical consideration to explain this empirical observation. When a small number of SDS molecules is added to a solution containing NM micelles with average aggregation number n, both NM and n will be slightly increased. There are several competing effects to be taken into account. First, amphiphiles are kept together by the hydrophobic effect. This can be supposed as having a constant value, and its contribution in energetic balance is q, per one SDS molecule. Second, the increasing n gives rise to an increasing repulsion force inside the micelles because of electrostatic interaction between charged sulfate groups. This contribution we characterize through an average value el. Third, the increased number of micelles in the same volume gives rise to a shorter distance between them, and, as a consequence, the repulsive electrostatic force between them should also be increased. It is evident that the equilibrium of these competing effects determines the concentration dependence of the average aggregation number. The electrostatic interaction term should have a screened Coulomb type shape, as their charge is partially compensated for by counterions. Now, the change of energy of the whole system due to increasing the aggregation number by value dn is AEl = N ~ t odn - NMeln dn The contribution from the creation of dNMnew micelles is equal to where e2(n,r)describes the distance-dependent repulsive interaction between micelles. From the equilibrium condition one obtains (12) Equation 12 may be simplified if one takes the terms containing AE1=
AE2
factor eo to be equal on both sides. In fact, these terms may be omitted as they express that for an added SDS molecule there is no difference from the viewpoint of short-range attractive interaction between situations involving getting attached to an existing micelle or taking part in the formation of a new micelle. Since NM
= NSDS/n
(13)
dNM = dNsDs/n - NsDs dn/n2
(14)
Thus from eq 12 If NsDSrepresents the number of SDS molecules per unit volume, then it can be replaced by concentration c. The resulting equation then has the shape c2(n,r)(nc’- c) = tln3 (16) With regard to the expected shape of effective potential e2(n,r) it is quite reasonable to take it as consisting of a hard core plus a double-layer repulsion screened Coulomb1’ form: e2
-
exp(-kr)/r
(17)
where 2 is the effective charge of a micelle, which dependsamong other factors-on the SDS concentration. A remarkable feature of the above potential is that at small k values and at short distances (which situation arises at high concentration) it tends to the r-l power law. It is an obvious simplification to rewrite 17 as &,r) = f(r)nm (18) converting the main charge dependence into a factor nm, where the exponent m will be determined later. If eq 18 is then inserted into eq 16
f ( r ) = (ne’- ~ ) q n - ~ - ~ (19) It now turns out that the only solution of eq 19 that satisfies both the empirical connection c = K2n4and the required asymptotic shapef(r) l / r corresponds to m = 0. Here, we took into account that r N1l3 (c/n)Il3,from which one obtains
-
- -
n = K,c~-’/~ In fact, the use of the relation n =~
~
(20) c
~
/
~(21)
and eq 20 give
f(r) = K3dm+l)
(22)
where K3 = C1/3K2-(m+2)(K1 /K2)(m+1)/3 and the correct asymptotic form forf(r) is obtained only if m = 0. On the other hand, if m = 0 andflr) 1/r then for the solution of (16) one obtains
-
n =~ ~ c ~ / ~(23) It can thus be seen that there is a self-consistency between our initial suppositions and the observed concentration dependence of the aggregation number (see Figures 1 and 3). During the course of the above consideration no suppositions were made specific to the SDS system. From this circumstance it follows that relation 23 should be valid for any similar system. This expectation is confirmed by the data of Bendedouch et a1.I’ on the LiDS system. In Figure 6, one can see that relation 23 fits well with their experimental data, suggesting that the validity of our consideration giving the fourth power law could be extended to concentrations of about 1 mol. This consistency, however, prescribes that the effective charge of the micelles ( Z ) should be independent of the aggregation number over the whole concentration range under investigation.
5142
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988
Bezzobotnov et al.
n4t
0
OB
1
o.c.tY/Il
Figure 6. Average aggregation number versus concentration plot for LiDS taken from Chen's paper*' ( T = 35 "C).
It means that the product of n and the degree of ionization will have a constant value. The data in ref 27 seem to confirm this conclusion for LiDS solutions, for which the product n multiplied by the fractional charge of micelle varies very slightly over a wide concentration range and even decreases when the concentration approaches 1 mol/L. This suggestion prescribes a concentration dependence of the degree of ionization proportional to c - I / ~ . This dependence was almost exactly observed in the case of LiSD,I7 where the value of the degree of ionization at c = 0.1 5 M was 0.4 and at c = 0.845 was 0.24 and the effective charge is equal to 25.2 and 22.3, respectively. Temperature Dependence. During the course of our experiment a slight temperature dependence of aggregation number was observed. To understand-although qualitatively-this observation, one can start again from a simple consideration. When the temperature is increased by a small value d T the total energy added to the system is CpdT, where C, is the heat capacity at constant pressure. This amount of energy will partially be spent on "evaporating" some of the amphiphiles previously attached to micelles. At high concentration these evaporated amphiphiles are unable to remain in solution, so it is a necessary consequence that they form new micelles consisting of a smaller number of aggregates. So, as the number of micelles increases, the distance between them decreases. The first process takes energy t1nNM dn, while the second one is equal to 1/2e2NMdNM. The term containing to was not taken into account by reason of similar arguments used in the previous section of the present paper. The full balance is then C, d T = tlnNMdn += 1/2t2NMdNM (24)
-
Taking into account the conditions t2 Z Z / r= K,z2(NM]r/3 and NM = c / n (where K , is an unknown coefficient), eq 24 can be expressed in terms of n: Cp dT/dn = t1c - 5 / 6 K , Z 2 ~ 7 ~ 3 n - ' 0 / 3
(25)
A more meticulously performed experiment would enable the constants K 1 ,22, and t, to be determined. However, in the absence of any knowledge of these parameters one can be satisfied with a simpler result. Inegrating expression 25, one obtains
+
T = An - BnT7i3 C
(26)
where A = c~c/C,
B = 35/18K1Z2~7/3Cp C = constant
With a suitable least-squares fitting procedure for the experimental points the values of A, B, and C can be determined. This was done for data belonging to the concentration 0.125 M (see curve 2 in Figure 7). It shows that formula 26 fairly well approximates the experimental data. The validity of the formula is supported
OJ
t
,
0
-777-
so 40 80 m o w Figure 7. Temperature dependence of mean aggregation number: (1) 0.25 M; (2) 0.125 M. 0
lo
20
by the fact that curve 1 was evaluated from constants A , B, and C obtained for curve 2 by applying the concentrational renormalization of these constants (Le., A(0.25M) = 2A(0.125M) and B(0.25M) = 27/3B(0.125M)). Deviations between the computed curves and experimental data, however, indicate the limitations of the validity of our model. In conclusion, we note that despite the above-mentioned discrepancies, qualitatively speaking, the main features of the temperature dependence of the aggregation number are well mirrored by our oversimplified model. Hydrocarbon Chain Package. It is generally accepted that ionic micelles consist of a hydrocarbon core inside which the hydrocarbon chains are densely packed and almost fully stretched out." As a consequence, the scattering density distribution consists of two significantly different homogeneous areas, Le., the densely packed hydrocarbon core and the surrounding shell consisting of sulfate head groups. In the framework of this model the growth of micelles will evolve from a spherical shape to become more and more elongated, forming an ellipsoid of revolution with a constant diameter. Our data seem to be in contradiction with this expectation as the value of the radius of gyration of cross section shows a slight but unquestionable change with changing concentration and temperature (Figure 4). There are two possible explanations for this observation. Either the Porod formula is invalid for small micelles, or the thickness of ellipsoidal micelles really alters because of their growth. To elucidate this question, we carried out a computer simulation, i.e., we computed the scattering pattern given in Porod coordinates (In [f(q)q] versus q2) and succeeded in proving that the slope of the Porod plot over the range q2 = 0.02-0.04 hi-* is in a good correlation with the radius of cross section down to the axis ratio of ellipsoids 1 :1 :1.2. Thus, it seems that one must accept that the observed change of R, values corresponds to a real change of the diameter of ellipsoidal micelles. This conclusion, however, contradicts the densely packed core model. In fact, if the hydrocarbon chains are fully stretched out, the cross section of a micelle consists of radially and isotropically spread chains. The distance between the outer parts of the hydrocarbon chains becomes large enough to allow either water molecules or SDS molecules to penetrate between them. From the viewpoint of hydrophobic interaction, the inclusion of some SDS molecules in the holes between concentric SDS molecules is preferable. Naturally, this penetration cannot be perfect. They get inserted only partially, as is schemathically shown in Figure 8a.
Properties of SDS Micelles
The Journal of Physical Chemistry, Vol. 92, No. 20, 1988 5743
h
b
a.
b.
c
d.
Figure 8. Schematic structure of SDS micelle: (a) densely packed model; (b) compact model; (c) scattering length density profile for model a; (d)
scattering length density profile for model b. At low concentration and high temperature, micelles get smaller, and the number of partially attached SDS molecules also decreases. After further decrease in the size of the micelles the open holes between the outer parts of SDS molecules may be filled by deformation of hydrocarbon chains (see Figure 8b). A schematic view of the corresponding scattering length density of these extreme distributions is given in the same figure. Variation of Aggregation Number. The width of the distribution in aggregation number can be estimated from the variation of n with the concentration c of micellized SDS through a general thermodynamic relation2 d / n * = d In ( n ) / d In ( c )
(27)
Supposing an algebraic relation between n and c n=cX (28) and substituting it into (27), one obtains that the relative variation is equal to x. This variation was measured by Hayter and Penf01d.~~In a 2% solution of SDS this polydispersity should amount to x = 0.1, while Cabane et aL3’ found that x = 0.16. These values are appreciably lower than x = 0.25. This latter value follows from our empirical formula 9, which fits very well the experimental data. The large discrepancy between the x values estimated by formulas 9 and 27 and the data obtained in Hayter’s paper supposedly comes either from the roughness of the mean spheres approximation or from the inadequacy of the thermodynamic approach used in deriving formula 27. In any case, the c ni
-
(31) Cabane, B.; Duplessix, R.; Zemb, T. J . Phys. (Les U i s , Fr.) 1985, 45, 2161.
relation seems to be in the best accordance with most of the observations. Here one must note that the relative variation given in formula 27 is valid only for high concentration because at low concentration the distribution of the aggregation number is not symmetrical due to the existence of micelles of minimal size. The aggregation number corresponding to this minimal size cuts off the distribution from the lower side, making it appreciably asymmetrical. General Considerations. Now, one can give a schematic scenario of micelle formation as follows. Below the critical concentration (cmc), i.e., below 8.3 X 10” M for SDS, individually dissolved SDS molecules are present in the solvent. Above the cmc a number of minimal size micelles are formed that have nearly spherical shape, and the hydrophobic core is filled by deformed hydrocarbon chains (similarly to the scheme of Figure 8b). This process is conditioned by the balance of competing forces of hydrophobic interaction, repulsive interaction of head groups, and the energy spent in the deformation of hydrocarbon chains. By an increase of the SDS concentration the number of micelles is increased too. The main competition arises between the repulsive interaction of the micelles and the forces governing the dimensions of an individual micelle. The first interaction strongly depends on the distance between the micelles, and so a larger concentration means a shorter distance between them. This interaction is thus the main driving force toward micelles of larger size but in decreased number. Contrary to this, the second interaction resists the formation of too large micelles. As a result, one observes a moderate increase of micelle size with increasing SDS concentration. There are two ways to form larger micelles: either the growth develops with the conservation of minimal cross section and the whole process ends in the formation of elongated ellipsoids that, after a while, deform to become cylinders with hemispherical ends, or the micelles first become swollen and only in a second step does the growth process develop to the formation of elongated ellipsoids. Our data prove that the second model seems to be more acceptable with the remark that the growth along the longer axis of ellipsoids at high SDS concentration is accompanied by a moderate additional swelling. If the temperature is elevated, the most loosely bound SDS molecules are evaporated. This process is mirrored by the observation of decreasing values of both radius of gyration and radius of gyration of cross section, resulting in a decrease of the aggregation number. This scheme of evolution of micelles seems to remove the apparent contradiction between models given by Chen2’ for LiDS and those given by Hayter28 for SDS solutions. The two-step contrast model is valid for smaller micelles (e.g., for high temperatures and low concentrations), whereas three-step contrast model is a fair approximation of loosely packed larger micelles with the tendency of continuous radius dependent contrast (see Figure 8c). Finally, in favor of our interpretation we should like to emphasize that it elucidates some common features both of LiDS and SDS micelles. Registry No. SDS, 151-21-3.
