The Journal of
Physical Chemistry
0 Copyright, 1983, by the American Chemical Society
VOLUME 87, NUMBER 10
MAY 12, 1983
LETTERS Study of Intermicellar Interaction and Structure by Small Angle Neutron Scattering Dallla Bendedouch and Sow-Hsln Chen" Nuclear Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139 (Receive& January 1, 1983; I n Final Form: March 14, 1983)
We have carried out an extensive series of small angle neutron scattering measurements on both dilute and concentrated micellar solutions of the ionic surfactant lithium dodecyl sulfate over a wide range of salt (LiC1) concentrations. In a previous report we have demonstrated that it is possible to extract the intermicellar structure factor from the intensity distribution function. In this report we show that, in the dilute regime (volume fraction 7 < 0.15), and at high salt concentrations, the extracted structure factors can be interpreted in terms of an equivalent hard-sphere interaction. In the concentrated regime (7> 0.15) and with no salt added, the extracted structure factors can be Fourier transformed to yield the pair correlation functions. There is evidence that the number of first neighbors around a given central micelle decreases in the sequence 10 to 8 to 6 as 7 increases from 0.15 to 0.39 to 0.45. A possible structural ordering of the most concentrated micellar solution is discussed.
1. Introduction
It is now fairly well established from small angle neutron scattering (SANS) and quasi-elastic light scattering (QELS) experiment~l-~ that the ionic detergent lithium dodecyl sulfate (LDS) forms small micelles in solution over an extended range of surfactant and salt (LiC1) concentrations, at and above room temperatures. Owing to a substantial repulsive Coulombic interaction between the charged micelles, pronounced interaction peaks have been observed in SANS experiments. The interaction peak for a monodispersed system of spherical particles arises from the product of the intraparticle (1) Bendedouch, D.; Chen, S.-H.; Koehler, W. C.; Lin, J. S. J. Chem. Phys., 1982, 76, 10,5022. (2) Bendedouch,D.; Chen., S.-H.; Koehler, W. C. J.Phys. Chem. 1983, 87, 153. (3) Bendedouch, D.; Chen, S.-H.; Koehler, W. C. J. Phys. Chem., accepted for publication. (4) Missel, P. J. Ph.D. Thesis, Massachusetts Institute of Technology, 1981.
structure factor P(Q)and the interparticle structure factor S(Q). Q is the scattering vector equal to (47r/X) sin 6/2, X is the neutron wavelength, and 6 the scattering angle. In fact the measured intensity distribution is
I(&) = n P ( d S ( Q )
(1)
where n is the number density of the micelles. It would be desirable, theoretically and experimentally, to be able to determine separately P(Q)and S(Q)from the measured I ( Q ) at each solution concentration and ionic strength. This turned out to be possible due to the following fortunate circumstances: (1)In the case of ionic micellar systems, where the interactions between the micelles are largely repulsive, the functional form of P(Q) and S(Q) are rather orthogonal to each other in the small Q region. (2) In a previous study we demonstrated experimentally2 that the small micelle has a two-layer structure consisting of a purely hydrocarbon (water free) inner core surrounded
0022-3654/83/2087-1653$01.50/00 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87,No. 10, 1983
by a polar-head layer heavily permeated by the aqueous solvent molecules. The neutron scattering density of the outer layer is so close to that of the solvent that, as far as the neutron scattering is concerned, the micelles can be modeled as an ellipsoidal hydrocarbon core with an equivalent liquid density. As a result, P(Q)can be computed by assigning merely a mean aggregation number rt to the micelles. (3) In a more recent study3 we proposed a method for extracting S(Q)which made use of a model for charged hard spheres in a dielectric medium with the electrostatic screening determined by the ionic strength I of the solution. This model, as developed by Hayter and P e n f ~ l d , ~ is in principle applicable only to low I([LiCl] < 0.2 M) colloidal solutions, where the interparticle potential function is describable as a sum of a hard sphere plus a screened Coulomb potential. In this case the model yields an analytical structure factor capable of fitting the data uniquely by specifying one effective parameter a, when combined with an appropriate model for P(Q). a is the degree of ionization of a micelle defined as q = art, where q is the net surface charge of a micelle. However, in a solution with large ionic strength I , the interparticle Coulomb repulsion is no longer accurately describable by the screened Coulomb potential. Instead, according to Verwey and Overbeek6 it is given by a different logarithmic form. Nevertheless, we found in a previous study3 that we were still able to fit our intensity data satisfactorily with Hayter-Penfold S(Q)(SHp(Q)), with a reasonable choice of a even for solutions with high I and at all densities. This result is somewhat puzzling in view of the fact that the interparticle potential functions for high and low I are quite different. In this Letter, we shall give a physical basis for this agreement at least for the case of low volume fraction 7 < 0.15. We show that for the large I case S(Q)can be satisfactorily represented by an effective uncharged hard-sphere model with an effective diameter a*, becuase in this case the large screening reduces the interparticle potential to a relatively short-ranged potential. Replacement by an equivalent hard-sphere model can be justified for the low density case where the second virial coefficient is sufficient to specify the osmotic equation of state. u* can then be uniquely computed from the pair potential function which includes the hard-core diameter (a) and the appropriate electrostatic repulsion valid for the high I case. Using this procedure we achieve an excellent fit between the experimental data and the effective hardsphere model. We are thus able to extract the true degree of ionization a. When a is used to calculate I of a given solution, it is seen that the mean aggregation number f i is approximately linearly related to In I in the dilute concentration range. We also show that it is always possible to choose an effective charge q* = a*rt in such a way that the screened Coulomb potential can be made to closely approximate the true logarithmic potential appropriate for the high I case. In the second part of this Letter, we demonstrate that the extracted structure factor for the high concentration solutions without salt, for which Hayter-Penfolds theory is valid, can be Fourier transformed to obtain the pair correlation functions. We then derive an interesting variation of the number of the first neighbors as the micellar concentration increases to the highest value. A (5) (a) Hayter, J. B.; Penfold, J. Mol. Phys. 1981,42, 109. (b) Hansen, J.-P.; Hayter, J. B. Ibid. 1982, 46, 561. (6) Verwey, E. J. W.; Overbeek, J. Th. G. "Theory of the Stability of Lyophobic Colloids"; Elvesier: New York, 1948.
Letters
possible ordering of these charged hard spheres in the latter case can be inferred from analysis of the first and second neighbor shell distances. As for the detailed description of the experiments and the data analysis we refer the reader to ref 1-3 and 7 . 2. Theoretical Background In the case of a system of nonspherical but monodispersed particles the scattered intensity distribution function I(Q) as given in eq 1 can be generalized to
I(Q)= nP(Q)S(Q)
(1')
where
P(Q)= (lF(Q)l2)
S(Q)= 1 + P(S(Q)- 1) P = (F(Q))2/(IF(Q)12) F(Q) is the form factor for the particle and the brackets signify the orientational average of the form factor. For a prolate spheroid with an axial ratio less than 1.5 the factor P is nearly unity within the Q range of interest in our work, as a consequence of this eq 1' reduces to eq 1 in all cases we a n a l y ~ e d .Further ~ for these small values of axial ratios, the form factor P(Q)computed by assuming an equivalent sphere of the same volume is nearly identical with that of the orientationally averaged spheroid. The repulsive interaction between two charged colloidal particles is usually represented by a sum of a hard-core (VHS) and an electrostatic (VR) repulsion. Depending on the thickness ( 1 / ~of) the ionic atmosphere surrounding the particles (diffuse double layer) relative to the diameter of the hard core a, two approximate analytical expressions have been given by Verwey and Overbeek.6 a. Thick Double-Layer Case: K a = k < 6. The potential energy of interaction is J.02taexp(-k(x - 1))
vR(x) = 4
X
(2)
with a surface potential 4qe
lciO = 4 2
+ k)
x = r/a K2
=
(3) (4)
87NAe2I (tkBn(103)
(5)
q is the surface charge; t, the dielectric constant of water; NA, Avogadro's number; kg, Boltzmann's constant; T, the absolute temperature; I , the ionic strength of the solution in equiv2/(L mol); e, the electronic charge. b. Thin Double-Layer Case: k > 6. In this case J.o2ta
VR(x) = -In [l + exp(-k(x - l ) ) ]
(6)
fi0= (2kBT/e) sinh-' (2qe2/kuckB7'l
(7)
4
In case a an analytical expression for S ( Q )can be obtained in the mean spherical approximation (MSA) by the Hayter-Penfold p r ~ c e d u r e .Whenever ~ comparisons between the theory and experiments were possible, the agreement has been excellent3i8 In the case of LDS mi(7) Bendedouch, D. Ph.D. Thesis, Massachusetts Institute of Technology, 1983.
The Journal of Physical Chemistry, Vol. 87, No. 70, 7983
Letters
1655
TABLE I : Results of the Data Analysis for Various LDS and LiCl Concentrations from the Two Different Procedures As Described in the Text
[LDS], M
[LiCl], M
t, "C
0.008 0.018
0.2 0.2 0.2 0.2 0.3 0.4 0.2 0.4 1.0 0.2 0.4 1.o 0.2 0.4 1.o
37 37 50 37 35 37 37 37 37 37 37 37 37 37 37
0.037 0.074 0.147 0.294
-
n
70 72 70 80 82 85 81 92 103 81 92 109 84 94 111
0 ,
A
47.5 47.9 47.5 49.2 49.5 49.9 49.3 51.0 52.5 49.3 51.0 53.3 49.8 51.3 50.0
71
71*
Q
Q*
X*
flVR(X*)
0.003 0.008 0.008 0.017 0.017 0.017 0.03 0.03 0.03 0.07 0.07 0.06 0.14 0.13 0.10
0.007 0.016 0.017 0.035 0.028 0.026 0.067 0.051 0.038 0.12 0.09 0.08 0.21 0.17 0.13
0.23 0.23 0.23 0.23 0.24 0.26 0.22 0.25 0.26 0.20 0.21 0.27 0.1 7 0.19 0.27
0.29 0.28 0.28 0.30 0.30 0.30 0.30 0.29 0.29 0.29 0.26 0.30 0.30 0.26 0.34
1.263 1.257 1.263 1.280 1.192 1.162 1.258 1.157 1.057 1.217 1.118 1.064 1.165 1.092
0.429 0.428 0.434 0.399 0.4 17 0.411 0.4 11 0.409 0.400 0.441 0.433 0.4 20 0.449 0.419 0.4 1 2
1.080
The procedure for finding the equivalent hard-sphere diameter is as follows: The second virial coefficient is given by 2B2 = 'a3(CHs 6
+ ci,+)
(9)
where CHS = 8, p' = kBT,and
Ci, = 2 4 s m [ 1- exp(-PVR(x))]x2dx 1
(10)
We introduce
6 = J m [ l- exp(-PVR(x))]x2dx
(11)
and rewrite eq 9 into the form
which defines an effective hard-sphere diameter Flgure 1. Potential energy of repulsion between two spherical particles of diameter u. x is the distance between the center of the particles in units of u. k = KU, where K is the inverse screening length, indicates the thickness in units of u, of the electric double layer surrounding the particles. k = 2 represents the thick double-layer case (eq 2) and k = 8 the thin double-layer case (eq 6).
cellar solutions, the range of applicability of this intermicellar potential is limited to a salt-free solution and solutions with salt concentration < 0.2 M. In case b which includes solutions with significant amount of salt, the analytical expression for S(Q) cannot be obtained even in the MSA although, in this case, numerical solution via computer molecular dynamics simulation is still possible. If we limit ourselves to sufficiently low micellar concentrations, say 17 < 0.15, then one can demonstrate that an equivalent hard-sphere system exists which reproduces an exact value of the second virial coefficient, B2.9 Since
[I +, ,
S(Q=O) = 1/ 1
-
u* = u(1
+ 36)1/3
u*
by
(13)
The hard-sphere structure factor S,(Q) is computed from the Percus-Yevick equation.'O It is given by"
SHS(Q) = (1 - nc(Q))-'
(14)
where 4m3 C(Q) = -[cu@(sin K - K cos K)
K6
+
O,$P(2K sin K - (P- 2) cos K - 2) + 7[(4K324K) sin K - (K4 - 1 2 P + 24) cos K + 2411 (15) (Yo
= (1
Po = -617(1
+ 27)2/(1 - 7 ) 4 + 0.57)'/(1
(16)
- 7)4
(17)
7 = 0.57~~0
(18)
K = Qu
(19)
we may argue that S(Q)at finite Q can also be reproduced by SHs(Q) of an effective hard sphere.
The data analysis for solutions with high I and low density proceeds as follows: The intensity distribution function I(Q) is normalized to an absolute unit of probability of scattering per unit solid angle per unit sample path length. P(Q)is computed by assigning an aggregation
(8) Hayter, J. B.; Penfold, J. J. Chem. Soc., Faraday Trans. 1 1981, 77, 1851. (9) Van den Broeck, C.; Lostak, F.; Lekkerkerker, H. N. W. J. Chem. Phys. 1981, 74, 2006.
(10)Percus, J. K.; Yevick, G. J. Phys. Reu. 1958, 110, 1. (11) Ashcroft, N. W.; Leckner, J. Phys. Reu. 1966, 145, 83. (12) Hayter, J. B.; Penfold, J. Report No. 80HA 075, Institut LaueLangevin, Grenoble, France, 1980.
~
~
~~~
~~~
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Tne Journal of Physical Chemistry, Vol. 87, No. 10, 1983
number ii to the micelle and knowing the scattering length density of the micelle.2 A detailed description of P(Q) model can be found in ref 2 and 3. S(Q)is modeled as SH,(Q) (eq 14) with an equivalent hard-sphere diameter u*. With these two parameters ii and u*, eq 1 is used to fit the experimental data. It is found that for 7 < 0.15 and with salt concentration I 0.2 M, we always obtain excellent fits to the data with a unique set of these two parameters, within experimental accuracy (see Table I for the results). Equation 13 is then used to determine the parameter 6 knowing u. Since micelles are prolate ellipsoids with axial ratio < 2, u is defined as the diameter of an equivalent sphere, having the same hydrocarbon core volume as the micelle, plus twice the thickness of the head-group layer. We then vary q until the value of 6 is recovered by computing it numerically through eq 11. Thus a, the correct degree of ionization, is recovered from this procedure. An alternative way of data fitting is also investigated. We model the structure factor by SHp(Q) computed using the Hayter-Penfold program.12 This requires a choice of an effective charge q* = a*fi. We found interestingly that we can fit the data consistently with the same fi but with a* > a. The extracted structure factors from the two procedures turned out to be numerically nearly identical in most cases. For concentrated solutions (7 > 0.15) without salt added, k is less than 6 and the screened Coulomb potential (case a) is a good approximation and therefore the HayterPenfold procedure is appropriate. The extracted S ( Q ) , although available only in the limited experimental Q range, can be analytically continued into an extended Q range by the theory. We can therefore Fourier transform it to obtain the pair correlation function according to
Letters
X
Figure 2. Potential energy of repulsion in units of k,Tvs. x = r / u for LDS micelles in solution containing 0.037 M LDS and 0.3 M LiCI. The solid line is computed by using a = 0.24 and eq 6. The dashed line is computed by using a’ = 0.30 and eq 2. These potential functions have been used to calculate SHs(Q)and SHp(Q). respectively.
!
I-
The number of micelles surrounding a given micelle at the origin is given by
N ( x ) dx = 247x2g(x) dx
(21)
From N ( x ) we can obtain the first and second nearest neighbor distances, x1 and x 2 , respectively. The number of first nearest neighbors, N1,is computed by integrating N ( x ) dx around N(x,).
I OL--,-
0.0
3. Results and Discussion For a given surface potential +o, the intermicellar interaction depends strongly on the ionic strength of the solution through K or k. Depending on the value of k, two different approximate forms of the potential energy of interaction are adequate: for k < 6, eq 2 is valid; for k > 6, eq 6 is applicable. The two different forms of the potential are shown in Figure l for k = 2 and 8. It should be noted that the logarithmic form of the potential (eq 6) is short ranged compared to the screened Coulomb potential (eq 2). In the former case, because of the shortrange nature of the potential, it is possible to calculate S(Q) in terms of an equivalent hard-sphere system in dilute solution, as explained previously. In the latter case this approximation is poor because of the long-range of the potential, and we have fortunately the Hayter-Penfold theory to calculate S(Q)a t all densities. We have previously mentioned in section 2 that a t low densities and high salt concentrations, both the equivalent hard-sphere and Hayter-Penfold model give an identical result for S(Q)and fi provided a fictitious a* is chosen for the latter case. The fitting procedure indicates consistently that a* > a obtained from the first procedure. We can
~
P I
oi%
-
_-_
_ _
L
