J. Phys. Chem. 1983, 87, 2621-2828
2621
Determination of Interparticle Structure Factors in Ionic Micellar Solutions by Small Angle Neutron Scattering Dallla Bendedouch, Sow-Hdn Chen; Nuclaer Englnmrlng Depclrtment, Massachusetts Insmuie of Technology, Cambrhlge, Messachusetts 02 139
and Wallace C. Koehler National Center for Small Angle Scetterlng Research, Oak RMge National Laboratory, Oak Rklge, Tennessee 37830 (Received: October 20, 7982; I n Flnal Fwm: December 74. 7982)
A series of small-angle neutron-scattering (SANS) intensity spectra measurementa were carried out on aqueous micellar solutions of ionic detergent lithium dodecyl sulfate (LDS). A systematic study of the system was conducted as a function of both detergent concentration, from 0.008 to 1.107 M, and salt (LiCl) concentration, from 0.0 to 1.0 M, at around 37 O C . The data show that LDS micelles stay "small" within this range of solution conditions,and therefore analysis can be made regarding the micelles as approximatelymonodisperse macroions suspended in a solution of given ionic strength (I). The SANS spectra exhibit a pronounced interaction peak due to strong intermicellar Coulombic repulsions, especially at low I. Analysis is made by using a mean spherical approximation ( M A ) for a model charged hard-sphere system (as developed by Hayter and Penfold'2) to compute the interparticle structure factor S(Q),and a uniform prolate spheroidal structure for the micellar core to model the intraparticle structure factor P(Q). Excellent agreement is obtained between our data and this model using only two free parameters: the aggregation number ii, and the degree of ionization of a micelle a. rt is found to increase with detergent concentration and I. The effective micellar diameter (r varies from 44 to 52 A and a ranges from 0.4 at low concentrations to 0.12 at the highest.
1. Introduction
Traditionally, methods of small-angle X-ray scattering (SAXS)and neutron scattering (SANS) were used mostly to study the internal structure, or what we will call the intraparticle structure factor P(Q),of macromolecules such as proteins or supramolecular aggregates, such as micelles in s ~ l u t i o n . ~These * ~ methods implicitly assume the solution to be ideal in the limit of low concentration, so that the interparticle interactions can safely be neglected. The scattering crow section in this case is proportional to P(Q) of a single particle for a nearly monodispersed system. In the case of ionic micellar solutions, these conditions are difficult to satisfy. The fact that micelles are charged means there are long-range Coulombic repulsions which, even for a solution containing only the minimal critical concentration (cmc) for micelle formation and in the absence of electrolyte, can produce a significant structural effect. The method of extrapolation to zero concentration (or more exactly to cmc) is valid only for a narrow range of concentrations because of micellar growth as a function of c~ncentration.~ The investigation of an ionic micellar structure at any arbitrary concentration is difficult for the same reasons, unless a large amount of electrolyte, which changes the micellar state, is added. Furthermore, many intemsting properties such as the sphere-to-rod transition6-' and more recently discovered critical phenomenaatg in (1) Hayter, J. B.; Penfold, Mol. Phys. 1981,42, 1, 109. (2) Haneen, J.-P.; Hayter, J. B. Mol. Phys. 1982,46, 651. H. B.; Miller, A. Appl. Crystallogr. 1978,11, 325. (3)&uh", (4)ilacrot, B. Rep. Prog. Phys. 1976,39,911. (5) kfuisman, H. F. Proc. K . Ned. Akad. Wet., Ser. B 1964,67, 367, 376, 388,407. (6)Mazer, N. A.; Benedek, G. B.; Carey, M. C. J.Phys. Chem. 1976, 80, 1075. (71gdieSe1, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. d. Phys. Chem. 1980,84,1044. ( 8 ) Corti, M.; Degiorgio, V. Phys. Rev. Lett. 1980,45,13, 1045. 0022-385418312087-2621$01.50/0
micellar solutions occur at high detergent concentrations. It has been known in the literature that at high concentration and low ionic strength (I), charged colloids in solution tend to develop an ordered structureloJ1due to strong electrostatic repulsion between particles. A quantitative theory of the interparticle structure factor taking into account a realistic interaction potential has been lacking. Recently a pioneer study by Hayter and Penfold12 showed that a pronounced interaction peak could be observed in sodium dodecyl sulfate (SDS)micellar solutions containing low salt and moderate detergent concentrations. They were able to account for the sharpening interaction peak in terms of building up of the interparticle structure as a result of Coulombic repulsions between micelles at high concentrations. They developed a mean spherical model of the interparticle structure S(Q)based on an interaction potential between micelles consisting of a hard sphere plus screened Coulomb potential. In a later publication: the accuracy of the mean spherical approximation (MSA) for this class of potentials was further verified even at low volume fractions. Strictly speaking, the scattering cross section for a system of interacting particles can be calculated from P(Q) and S(Q) only for a system consisting of monodisperse spherical particles. It is questionable in view of the results of light-scattering experiments' whether SDS system can be regarded as monodispersed in the full range of temperature, salt, and detergent concentrations of interest. In fitting the SANS spectra, Hayter and Penfold12adopt a ~~
~
~
~
(9)Corti, M.; Degiorgio, V. J.Phys. Chem. 1981,85, 1442. (10) (a) Riley, D. P.; Oster, G. Discuss.Faraday. SOC.1951,1I,107. (b)Ise, N.; Okubo, T. Acc. Chem. Res. 1980,13, 303. (11) (a) Magid, L. J.; Triolo, R.; Johnson, J. S.; Koehler, W. C. J.Phys. Chem. 1982,86,164.(b)C a b , C.; Delord, P.; Martin, J. C. J.Phys. Lett. 1979,40,16, L-407. (12) Hayter, J. B.; Penfold, J. J. Chem. Soc., Faraday Trans. 1 1981, 77. 1851.
0 1983 American Chemical Society
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Bendedouch et at.
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983
three-layer spherical model of micelles (P(Q))which was not independently verified. Even with this reservation, it was remarkable that they were able to fit their SANS spectra with good accuracy using only two independent parameters: ii and a. In an attempt to improve on the results of Hayter and Penfold, we undertook an extensive study on a related system. The LDS micellar system was chosen because a light-scattering study7 shows it is not subject to large variations of size and shape under experimental conditions which would trigger a large increase in size and polydispersity in SDS system. We could, hence, safely assume the system to be reasonably monodispersed for our purposes. Independently, a systematic study of P(Q) was undertaken and the results were reported elsewhere.13 We use it as a basis for our analysis. In addition, the experimental temperature was chosen to be >30 "C so that no significant growth was e ~ p e c t e d . ~ In a previous publi~ation,'~ we proposed a method for extracting both intra- and interparticle structure factors for macroions in solution. The method is based on measurement of P(Q)for a low detergent and high salt concentration solution for which S(Q) 1 and we use it to deduce S(Q)for concentrated and electrolyte free solutions, assuming that P(Q)at the corresponding condition does not change. This procedure is perhaps better suited for globular protein solutions where the conformation of the particle does not change with concentration and ionic strength. But it is less accurate for a micellar system because P(Q)changes with concentration even when no salt is added to the solution. In this case, it is necessary that we establish a realistic model for P(Q)based on an independent experiment at some experimental condition, and then perform a self-consistent analysis using a concentration dependent P(Q)and HaytepPenfold procedure' to calculate S(Q). Our previous P(Q)model13 can be shown to depend only on one parameter ii, assuming a prolate spheroidal shape for the hydrocarbon core of LDS micelles. The Hayter-Penfold S(Q) depends only on the effective hard-core diameter of the micelle Q, which can further be related to A, and an effective a. In this way we were able to obtain an excellent fit to all our data using only two parameters, ii and a. ii obtained in this fashion agrees with our previous determination by an independent method at the same ~0ndition.l~ This SANS technique is essentially an extension of the Debye plot15 method to finite momentum transfer Q and also to finite concentration of colloidal particles. In section 2 we describe the experimental procedure, followed by a presentation of the models for P(Q)and S(Q) in section 3. The results are discussed in section 4. We conclude with a remark that this method can be viewed as a refinement of the previously proposed method for extracting an experimental S(Q) for an interacting and approximately spherical and monodispersed micellar system. l4
-
The solutions were prepared by dissolving known amounts of detergent in aqueous solvent made of D20 and 0.0, 0.2, 0.4, or 1.0 M LiCl. The five most concentrated solutions in pure D20 had their final LDS concentrations determined by weight. A known volume from each solution was dryed to obtain the weight of the solute. The cmc in salt-free DzO solutions was checked by integrated neutron intensity measurements and was found to agree with tabulated values for LDS solutions in H20.16 The cmc for LDS in various LiCl concentrations in H 2 0 is reported in ref 17 to follow an empirical law which we used to compute the various cmc's. The sample cells were quartz cells from NSG Precision Cells, Inc., and had windows of thickness 1 mm and path length of 5 and 2 mm. In each measurement, the path length of the cell was chosen such that the transmission was in the range of 65-85%. Sample cells were mounted on an aluminum container whose temperature was maintained to f 0 . l . V . Samples were freshly prepared before the experiments and kept frozen when not in use to avoid any deterioration (hydrolysis) of the material. Data Collection. The experiments were conducted at the National Center for Small Angle Scattering Research (NCSASR) in Oak Ridge National Laboratory (ORNL) with a neutron wavelength X = 4.75 A and a sample-todetector distance of 3.10 and 1.60 m. This spans a Q range (Q = (4.n/X) sin 8/2, B is the scattering angle) from 0.02 to 0.22 A-1. The fmite experimental resolution in Q space is calculated from the spectrometer geometry to be AQ/Q 3%. Measurement times ranged anywhere from 20 min to 2 h, depending on the concentration. Sample transmission was taken before and after each measurement to detect possible alteration of material (solvent evaporation, bubble formation, etc.) and to correct for self-absorption of the sample. An absolute intensity calibration was performed by using a 1-mm H20 sample18and an aluminum standard supplied by NCSASR. The calibration was estimated to be good to 5 % . Raw scattering data were manipulated in a standard fashion to subtract background and solvent contributions and to correct for detector sensitivity. The final data were presented in a radially averaged form which directly gives (dZ/dQ)(Q) (coherent scattering probability per unit solid angle per unit sample length) denoted by I(Q) in the text. Data Analysis. The single particle form factor F(Q) is defined by
-
F(Q) = lexp(iQ.r)(p(r)
- P,)
dr
(1)
where p ( r ) and ps are the scattering length densities of the particle and of the solvent, respectively. The integral is over the volume of the particle. For an assembly of interacting particles, the scattering cross section takes the form
2. Experimental Procedure
Samples and Cells. LDS was obtained from Sigma Chemical Co. with a specified purity better than 99%. The uniformity of the hydrocarbon chain length was checked by the manufacturer by gas and thin layer chromatography. DzO used was 99.8 atom % D pure (Merck and Co., Inc.). All other chemicals were of reagent grade. (13) Bendedouch, D.; Chen, 5.-H.; Koehler, W. C. J. Phys. Chem. 1983, 87.153. (14) Bendedouch, D.;Chen, S.-H.; Koehler, W. C.; Lin, J. S. J . Chem. PhYS. 1982, 76, 10, 5022. (15) Berne, B. J.; Pecora, R. (1976) "Dynamic Light Scattering"; Wiles New York, 1976; p 173.
with c n=-N
io3
1 n
-
A -
(3)
ii is the number density of micelles, and c is the molar concentration of LDS monomers in the micellar state. It (16) Mukerjee, P.; Mysels, K. J. Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. 1971, No.36. (17) Mukejee, P.;Mysels, K. J.; Kapauan, P. J.Phys. Chem. 1976, 71, 13, 4166. (18) Jacrot, B.; Zaccai, G. Biopolymers 1981, 20,2413.
The Journal of Physical Chemlstty, Vol. 87, No. 14, 1983
Interpartlcle Structure Factor Determination by SANS
2623
TABLE I: Results of SANS Data Analysis for LDS Micellar System" [LDS],
[LiCl],
M
t, "C
Q
k
0.008 0.018
0.2 0.2
0.037
0.0 0.2 0.3 0.4
37 37 50 37 37 35 37 37 37 37 37 37 37 37 37 22 37 37 37 37 35 35 35 35 35
0.003 0.008 0.008 0.015 0.017 0.017 0.017 0.03 0.03 0.03 0.03 0.07 0.07 0.07 0.06 0.14 0.14 0.14 0.13
6.9 7.0 6.8 1.7 7.2 8.9 10.3 2.1 7.3 10.5 17.0 2.8 7.5 10.6 17.4 4.1 3.9 7.9 10.9 16.5 4.0 5.3 5.5 5.0 4.4
M
0.074
0.0
0.294
0.2 0.4 1.0 0.0 0.2 0.4 1.o 0.0
0.325 0.598 0.845 0.985 1.107
0.2 0.4 1.0 0.0 0.0 0.0 0.0 0.0
0.147
0.0
0.10 0.15 0.27 0.39 0.44 0.45
Gek
0 ,A
alb
3.0 2.9 2.8 19.2 3.8 2.9 2.3 18.4 3.9 2.4 1.1 16.0 3.4 1.9 1.5 14.9 13.1 3.5 1.9 2.3 12.1 8.2 4.7 2.9 1.8
47.5 47.9 47.5 44.3 49.2 49.5 49.9 45.1 49.3 51.0 52.5 46.2 49.3 51.0 53.3 48.4 47.7 49.8 51.3 50.0 49.0 50.1 51.1 51.3 50.0
1.3 1.3 1.3 1.o 1.5 1.5 1.6 1.1 1.5 1.7 1.9 1.2 1.5 1.7 2.0 1.4 1.3 1.6 1.7 2.0 1.5 1.6 1.7 1.7 1.8
-
n
70 72 70 53 80 82 85 57 81 92 103 63 81 92 109 75 71 84 94 111 79 86 93 94 96
CY
0.29 0.28 0.28 0.39 0.30 0.30 0.30 0.40 0.30 0.29 0.26 0.40 0.29 0.26 0.30 0.41 0.40
0.30 0.26
0.34 0.35 0.33 0.24 0.17 0.12
K 0.78 0.82 0.85 0.85 0.95 1.01 0.90 0.87 0.93 1.05 1.04 0.92 0.92 0.99 1.06 1.13 1.03 1.02 1.06 1.01 1.01 1.03 1.06 1.09 1.09
a q is the volume fraction. Gek is the dimensionless contact potential y exp(- k ) . K is the ratio of the theoretical to the experimental intensity factor (see eq 5). The cmc values used for 0.0, 0.2, 0.3, 0.4,and 1.0 M LiCl are respectively" 8.9 X 1.1 X 0.9 X 0.6 X and 0.0 M LDS.
is usually taken to be the total detergent concentration minus the cmc. N Ais Avogadro's number. In the case of F(Q12 monodisperse and spherically symmetric micelles, = F(Q)= P(Q) and eq 2 becomes
I(&) = nP(Q)S(Q)
(4)
The assumed structure of the micelle, on which P(Q) depends directly, will be described in the next section. S(Q) is computed with a Fortran package written by Hayter and Penf01d.l~ It depends essentially on three parameters, ii, a,and 6. This dependence will be made explicit in the following section. The general procedure for data analysis can be summarized as follows. The data are normalized at a point, usually the maximum of the interaction peak, to the theoretically calculated I(&) according to eq 4. The normalization factor thus obtained
K = Y(xj)/Yj
(5)
should be close to one for the best cases. Inspection of Table I shows that it is 80 for most cases. In the other cases where K is appreciably far from unity (e.g., K = 0.78) we speculate that the discrepancy can be attributed partly to the experimental error on LDS concentration and partly to the error on the value of the cmc. A t the lowest concentrations, the relative error on concentration is larger. To obtain the absolute experimental cross-sections from the graphs, one must multiple the data by K. The parameters ii and a are varied until satisfactory agreement is achieved by visually monitoring the fit. The goodness of fit given by the average percentage deviation per data point as
(19)Hayter, J. B.; Penfold, J. A Fortran Package to Calculate SCQ)for Mamion Solutione Report No. 80HA 075, Institut hue-Langevin,1980.
was around 5% for most spectra, and never exceeded 13% for the few worst cases. N is the total number of data points; yi and y ( x J are the experimental and theoretical data points, respectively.
3. Models P(Q)Model. According to our previous in~estigation,'~ the LDS micelle in DzO environment can be well represented by a prolate spheroid of uniform density equal to the density of liquid paraffin whose specific volume is 1.30 f 0.08 cm3/g. As far as neutron scattering is concerned, the micelle can be modeled accurately to consist of two regions: an inner unwetted hydrocarbon core of volume VI containing all the carbons, and an outer layer containing the polar groups (sulfate groups plus "bound" hydrated counterions). This latter is heavily impregnated with water and has a total volume VO. The "dry" volume of the micelle, as obtained from our previous e~periment'~ in the case of 0.037 M LDS and 0.3 M LiCl in D20 solution, is made of two parts: VI and V , which corresponds to the volume occupied by the polar groups as defined above. The ratio VI/ (VI + V,) was found to be equal to 0.8. ii in this case turned out to be 78, giving an. average volume per polar group, not impregnated by bulk solvent, which corresponds to a sphere of 5.56 A in diameter. The neutron-scattering density of the wetted outer layer, po, is very close to that of the solvent. As a result, the micelle appears as an oil drop with a spheroidal shape with semiaxis a,b,b. In order to arrive at the model we use subsequently, we fix b at 16.7 A which is the length of a fully extended dodecyl chain. a is computed by assuming a close-packed core and 4* VI = flul = -ab2 3 where uI = 364 f 22 A3 is the volume per dodecyl chainI3 consistent with the specific volume of liquid paraffin. The scattering length density of the core is PI = bl/Ul (7)
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Bendedouch et al.
The Journal of Physical Chemistty, Vol. 87, No. 14, 1983
defined by the ionic strength I of the solution
I = cmc + [LiCl] + Yzac
I is determined by the concentration of LDS monomers, taken to be equal to cmc, of added LiCl and of ionized Li+ from the micelles. E is the dielectric constant of the solvent medium, e is the electronic charge, and y exp(-k) = ~z2e2/7rttOu(2 + k)2 (16)
I
t
I
Figure 1. Schematic micellar model for P ( Q ) (a) and S ( Q )(b). The relationship between the two is explained in detail in the text. The ellipsoid and the sphere have an equal inner hydrocarbon core volume and are surrounded by the spherical polar head groups with an average diameter equal to 5.56 A. The scattering length density profile corresponding to P ( Q ) model is also shown.
where bl is the sum of the scattering length of all atoms in a dodecyl chain. For a prolate ellipsoid20 P(Q) =
with
Re =
[a2p2 p
+ b2(1 - p2)]1/2
(9)
= cos 8
Re is the radius of an equivalent sphere and tl is the angle between the directions of a and the scattering vector Q. The integral represents the average over all possible orientations of the particle. From the above presentation, one can see that P(Q)depends essentially on one single parameter, ii (see eq 6 and 9). The micellar model for P(Q) is pictured in Figure la. S(Q) Model. In the model proposed by Hayter and Penfold,l S(Q) is calculated by assuming the micelle to be a rigid charged sphere of diameter u, interacting through a dimensionless screened Coulomb potential pU(x) = y exp(-kx)/x x >1
PU(x) =
x
0)
0.2). However, the argument given later by Hansen and Hayter2 demonstrated that it can be made accurate by performing a so-called rescaling procedure. This operation is incorporated in the computer program,l9 and is performed automatically whenever q < 0.2. Aside from numerical accuracy of the solution of the OrnsteinZemicke equation by MSA, one can argue about the basic validity of the interaction potential used. It has been shown by Verwey and Overbeek21from the solution of the Poisson-Boltzmann equation for the charged double layer that the screened Coulomb potential of eq 10 for the charged micelles is accurate provided k < 6. It is generally believed that, when the contact potential is much larger than kBT,the interacting particles have a rare chance of approaching close enough to sample the inaccuracy of the potential at short distances. Thus, when the condition y exp(-k) >> 1 is met, one can also justify the use of a simplified hard-core instead of a more realistic soft-core potential. To calculate S(Q) according to this model one needs three input parameters: k = K U ; z = iia (the effective charge on a micelle to compute y); the volume fraction q = (7r/6)nd. u can be specified by using a constraint that the effective radius of the hydrocarbon core is defined by
R13= ab2
(20) Chen, 9.-H.; Hole, M.; Tartaglia, P. Appl. Opt. 1977, 16, 187. (21) Verwey, E. J. W.; Overbeek, J. Th. G. "Theory of the Stability of
(18)
and u = 2(RI
+ 5.56) 8,
(19)
Figure 1 illustrates how the micellar model for P(Q)(Figure l a ) translates into the one used for S(Q) (Figure lb). In this scheme, the fitting of the intensity spectra is reduced to specification of two independent parameters, r i and a. Remarks on Model Fitting. It is important at this point to discuss sensitivity of the data fitting to variations of model parameters. From the model of P(Q) given above, eq 8, one can deduce the half-width of P(Q)for a given ii as
-
112 A-1
(111.6+ 0.019fi2) Hence, the variation of It directly affects the steepest part (or half-width) of P(Q), Q1/z
Lyophobic Colloids"; Elvesier: New York, 1948.
(15)
(22) Lebowitz, J. L.; Percue, J. K. Phys. Rev. 1966, 144, 251.
Interparticle Structure Factor Determination by SANS 34
-
00 I 00
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983 2625
-2c
I
I
Q [i-']
too
00
00
025
00
0 2
Flgure 2. I@), P(Q), and S ( 0 )for a nearly ideal LDS micellar solution in DO , at 37 "C. The conditions of low detergent (0.008M) and moderate electrolyte (0.2M LiCI) concentrations are necessary to 1 for most of the experimental 0 range. I n this maintain S ( 0 ) governs the shape of the scattering curve. Since the data case, P(0) are in an absolute scale our model for the structure of a micelle as having a prolate hydrocarbon core containing all the chains (Figure la), can be critically tested and the average micellar size can also be deduced. The solid line is the theoretical fff as calculated with eq 4, and the open circles represent the data points normalizedto the fit as explained in the text.
-
The dependence of S(Q) on r i is more complex, but, roughly speaking, the position of the first diffraction peak is a rather unique function of ri, when the Coulomb force is weak (k > 6). It can be seen, for example, from Figure 4 in ref 1 that for k > 6, Qm,u 5, for q = 0.05 and 5.5 as q increases. Q, gradually increases to Qm,su is the position of the peak. Since both 7 and u are functions of only, it follows that Q, depends mainly on when k > 6. In this case the role of a is merely to broaden or sharpen the peak, i.e., when a is large the peak is sharper. However, when the Coulombic interaction is strong (k < 6), the repulsion keeps the particles further separated and Q, is smaller. The value of Q , is in this case affected by a,too. Since the interaction peak is a result of the product of P(Q) by S(Q),the above consideration leads to the following conclusion. If the repulsion is screened (k > 6) the position and height of the interaction peak is most directly related to It, and the magnitude of a merely affect its broadening. However, when there is no salt (k < 6) present, the position and height of the interaction peak are sensitive functions of both It and a. As we already stated, the form of the screened Coulomb potential (eq 10) becomes inaccurate for k > 6. But fortunately, from the above argument, the position and height of the interaction peak is a rather unique function of ri, and thus it could also be extracted reliably. However, the parameter a becomes a fitting parameter only, and looses its significance as a fractional charge of a micelle.
Figure 3. Scattering intensity spectrum for 0.037 M LDS in DO , with 0.3M LiCl at 35 "C. The open circles are the data points and the solid line, the theoretical I ( 0 ) curve. S ( 0 )for this case is also presented. The aggregation number In this case in 17 = 82,a value in very good agreement with our previous independent determination.13 60
/?
L #:.I4
[cm-'1
--
4. Results and Discussion
The experiment has covered a detergent concentration range from 0.008 to 1.107 M which is equivalent to q = 0.003-0.45, with LiCl concentrations varying from 0.0 to 1.0 M. Although most of the experiments were performed at 37 "C, there are few cases done at 22,35, and 50 "C. The results of the analysis are summarized in Table I. For the lowest LDS concentration, 0.008 M with 0.2 M LiC1, where analysis gives Iz = 70 and a = 0.29, the intensity spectrum is dominated by P(Q). S(Q) is almost unity throughout the Q range. This situation is illustrated in Figure 2. This case with k = 6.9 is, according to our criteria discussed previously, on the border line of validity
I
0.C
'i
-
0.0
Q [i-']
0.
Figure 4. Scattering for 4 LDS concentrations: 0.037(0),0.074(X), 0.147(O), 0.294 M (0)in pure DO , at 37 "C. The corresponding volume fractions q are indicated. The lines are theoretical fits and the symbols are data points. The building up of structure is clearly seen to follow the increased micellar concentration. The higher the concentration, the sharper the peak. The peak shifts to larger Q values when the LDS concentration increases, reflecting a decrease in the mean distance which results in a stronger interaction between particles in solution.
of the repulsive potential function. We have used this intensity spectrum as a reference P(Q)in one of our previous p u b l i c a t i o n ~ . ~ ~ Figure 3 presents a case where an excellent fit is attained for 0.037 M LDS in 0.3 M LiCl and D20 at 35 "C. The values of It and CY which fit best the data agree well with our previous determination by an independent method.13 The enhancement of interparticle structure is best illustrated by Figure 4. As 7 is changed from 0.015 to 0.14 corresponding to 0.037 to 0.294 M LDS solutions, when no salt is present, the interaction peak shifts to higher Q's continuously and the peak height increases steadily. The different symbols are data points and the solid lines indicate the theoretical fits to the data. One sees the agreement is excellent and the extracted S(Q)and P ( Q ) are shown in Figure 5. P(Q=O) incremes like It2(eq 8) with ri increasing from 53 to 71. The particle shape changes from a sphere ( a / b 1) to a prolate ellipsoid ( a / b 1.3) to accomodate larger It. The lowest It (53) obtained for 0.037 M LDS in pure D 2 0 is consistent with the radious of a spherical hydrocarbon core of 16.7 A and a tail volume of 364 A3.13 Thus we see that in an electrolyte-free aqueous solution the micelles still assume their smallest spherical size for a concentration of up to 0.037 M LDS.
-
-
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The Journal of Physical Chemistry, Vol. 87, No. 14, 7983
s(Ql I 5
oo
7 ' 45 1 1
39 00
I
ooo
0
I
7' 4 5
4 4 4'1
1 ~
I (Q) [cm-lli
OOL0 00
Figwe 6. Same as Figure 4 but for higher concentratlons: 0.325 (0), 0.598 (X), 0.845 (O), 0.985 (O),and 1.107 M (I). The temperature is 35 O C . The largest concentration corresponds to a volume fraction of 0.45 where liquid crystalline structure may set in.
For this series k is below 6, thus the S(Q)theory should be valid. The buildup of the first diffraction peak is clearly seen as a function of increased concentration. It should be noted that S(Q-0) is between 0.1 and 0.2. Even for 0.037 M LDS, the solution is highly nonideal. The degree of ionization a stays nearly constant at 0.4, a value consistent with the literature." Figure 6 presents I(Q) vs. Q for the high density series of LDS micelles in pure DzO. The volume fraction ranges from 9 = 0.15 to 0.45. The model fitting to the data are uniformly good with k < 2. As 9 rises, Q increases and approaches a limiting value,Q = 0.133 at the highest 9 = 0.45. For this micellar density, the mean interparticle distance, assuming a simple cubic arrangement, is 52.6 A, a value close to the effective diameter of a micelle which is u = 50 A. We find it difficult to make samples with detergent concentrations above 1.107 M because the so-
oooooooop~~~
0 0 ooB 0Q C
0
0
0
Do I I I I
%py$
I 1
I
I
O[i-lI
"~
I
00;
0 00 I 0_0 _ . i i "@qPp'I,
Flgure 5. Theoretical S ( Q )(lower graph) and P ( Q ) (upper graph) for the various LDS concentrations in D,O at 37 O C corresponding to Figure 4. The change In P ( Q ) is due to the change in a l b ratio which is only dependent on I?, according to our model (see text). S(Q) in these low ionic strength condltlons show a marked depression at low Q, and a sharper and higher peak as the concentration increases.
I
15
,
I
1
I
0 2
Flgure 7. Corresponding theoretical S ( Q ) (lower graph) and P ( Q ) (upper graph) to Figure 8, but for only three of the LDS Concentrations. From Table I, we see that i ) does not Increase signlfkxntly from 0.845 to 1.107 M LDS, and this is r e M d in P ( 0 ) . However, S(Q) is more structured due to the increased number density of micelles.
lution becomes extremely viscous. It has been knownz3 that viscosity of globular protein solutions approaches infinity at q N 0.45, signifying that colloidal system has assumed a nearly close-packed structure. The P(Q)and S(Q) resulting from the data fitting are shown in Figure 7. It is somewhat surprising that the theory was still able to account for the general features of the interaction peak at such a high micellar density and yields reasonable value for the parameter ii = 96 corresponding to u = 51.6 A. We found that the computer programl9 does not converge for this case, and therefore we reduced u to 50 A. The resulting fit is shown in Figure 6. This may be evidence that the S(Q)theory cannot account for the density regime where -the liquid becomes close-packed, and an ordered structure begins to emerge. Generally, a decreases as LDS concentration increases for the high-density series. For 1.107 M LDS it is 0.12. When the distance between micelles becomes comparable to the micellar size, it is reasonable to expect that the counterions have no other alternative but to stay in the vicinity of the micelles and neutralize more effectively the charge on the micelles. It is also conceivable that in order to reduce the repulsion between micelles and between charged heads on the micellar surface, in this highly charged and dense system of colloidal particles, the counterions are more tightly bound. The results on salt dependence at various LDS concentrations are shown in Table I as well as in Figure 8-11. We notice from the table that as soon as one exceeds 0.2 M LiC1, k becomes larer than 6, and the screened Coulomb potential becomes inaccurate. However, from the previous argument we know that precisely in this region the position and height of the interaction peak are determined mainly by ri. Thus fi extracted from the experiment for this series are expected to be reliable. But a is only a fitting parameter, although, we note that its value stays rather (23) Menjivar, J. A. Sc.D. Thesis, MI",Department of Nutrition and Food Science, 1981.
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983 2827
Interparticle Structure Factor Determination by SANS 00 M LiCl
I W I
-"I
c:
I
0 0 M LiCf
0
02
X
04
0
10
0
I
00 00
1
1
Q
I
I
I
Flgure 8 . Ionic strength dependence of the Scattering intenslty for 0.037 M LDS at 37 OC. The solid lines are theoretically calculated flt and the symbols are data points as indicated on the figure. I t is interesting to note that, even at such a relatively low detergent concentration, lt is necessary to add a large amount of electrolyte to meen out the effect of CoUbmMc repulskn between c h a r w mice#es. 0.0MLiCs 02
02
Q [ i-']
Flgure 10. Effect of electrolyte concentration on scattering intensity for 0.147 M LDS at 37 OC. As the LCl concentration is increasedthe peak disappears gradually unit at the highest salt concentration it becomes more like a shoulder. C O M L ~ C I ~ 0 2 I O
,
0.05,
1
I
0000
0
[;-I]
i
0
oooo
S(Q1
t
001 00
Flgure 9. S(Q) (lower graph) and P ( Q ) (upper graph) for the case presented in Flgure 8, 0.037 M LDS at 37 OC In pure D,O (0),with 0.2 M LiCl (X), and with 0.4 UCI (0).Note the nonideality of the sdutlon as reflected In S(Q-0) below unity even at the highest salt concentration.
constant around 0.3. The striking feature is the weakening of structure as salt concentration increases beyond 0.2 M. Nevertheless, S(Q-4) stays significantly below 1even at 1.0 M LiCl. This result render the usual argument made by quasi-elastic light-scattering experimentalists,6 that when sufficient salt is added (>0.5 M) the interaction can effectively be screened out, questionable. Figure 12a shows 1(Q),S(Q),and P(Q) for 0.294 M LDS in DzO.One can clearly see how the interaction peak is generated from the joint contribution of P(Q)and S(Q). The position of the peak is in this case largely determined by S(Q). We also present in Figure 12b g(r) for this case. The peak position occurs at r,, = 74 8,which is the mean distance between nearest neighbors. The number of nearest neighbors surrounding a central micelle is estimated to be 10.6 from integration of the first peak of 4 ~ 9 n g ( r ) . It is to be noted that r,, is not equal to 2*/Q,, in general for solutions. It is thus not justified to identify
-
1
1
.L---
~-
Q[i-']
0 2
Flgure 11. S(Q) (lower graph) and P ( Q ) (upper graph) corresponding to Figure 10 for 0.147 M LDS at 37 OC, in pure D,O (0),with 0.2 M LiCl (X), and with 1.0 M LCl (0).I t is clear here that even the highest salt concentration 1 M LiCl is not sufficient to screen completely the electrostatic repulsion between micelles.
from a sharp peak in S(Q)as measured by SAXS an ordered structure parameter in colloidal solutions.1° The tabulated F? (Table I) are plotted in Figure 13 as a function of LiCl concentrations and for various LDS concentrations. For the lowest LDS concentration (0.037 M) there exists a similar curve for SDS obtained by Huisman5 which resembles our curve. In general, fi increases with LDS concentration at a given salt concentration and with salt at a given LDS concentration. Within our model, the prolate and oblate ellipsoid or capped cylinder of equal volumes will give essentially the same results because of ratio a l b stays 12,and it is thus not possible to differentiate between these various shapes. At low LDS (
