J. Phys. Chem. 1988, 92, 1533-1538 to dissociation into PbO when the latter is present in concentrations l l O d M. Despite the large uncertainties in AfHo(PbO)2 the maximum concentration in equilibrium with PbO at 9000 K is about 10-2.3(PbO). If such polymers formed at lower temperatures (-700 K) they would rapidly dissociate at 900 K or at the much higher temperatures involved in knocking conditions. Kinetic considerations also favor molecular species over clusters as the active antiknock agents. When PbO molecules associate to form dimers or smaller particles they lose kinetic activity because they lose potentially active surface area. In the case of dimers this is the area of contact between the monomers and comprises roughly half of the areas of the initial monomers. For small crystals, all of the interim molecules are buried and hence inactive while molecules in the faces will have lost about three fourths of their area. Molecules in edges will have covered about half and only molecules in corners will have more than half their surfaces available for collision. More significant than these geometrical constraints are the chemical constraints. For an atom in a surface to be chemically active it must be able to form chemical bonds. An 0 atom already bound to two adiacent Pb atoms is chemically saturated. It is unlikely to form a third bond to HOz or H202. By the same token a Pb atom bound to two adjacent 0 atoms in a surface has lost its most active valence sites. It can form at best a very weak bond to the 0 in H 0 2 and certainly cannot abstract the H atom. It is even less likely to attack H2O2. Only Pb or 0 atoms at corners or steps are likely to be able to form chemical bonds. Such considerations dictate that particulate PbO is likely to have collision efficiencies that are very small. Molecular PbO,
1533
PbOH, and Pb are all expected to be very much more active. There is no reason to believe that Pb is the only species capable of such debranching behavior. Much evidence points to other species as possible substitutes. Potassium has been described' as having antiknock properties as has Fe(CO),. Effect of Pb on Combustion Finally it is important to consider the effect of Pb species on normal combustion. Will they inhibit burning? The answer to this must depend on the temperature. In the region between 1100 and 1200 K an important competing process for atomic Pb is
Pb
-
+ 0, 6'
PbO
+ 0 - 29 kcal
Reaction 6' has a high A factor which is estimated at about 1010.5 L/(mol s) at 1100 K and an activation energy of about 29 kcal. At 1100 K it will act to initiate chains in competition with its chain debranching behavior in step 6. In similar fashion PbO will be able to attack hydrocarbon to initiate chains above 1100 K in competition with step 7: PbO
+ RH 7' PbOH + R'
We thus expect the debranching efficiency of Pb species to be significant only in the range from 800 to 1100 K. Acknowledgment. This work has been supported by grants from the National Science Foundation (CHE-86-46922) and the U.S. Army Research Office (DAAG29-85-K-0019). Registry No. (C2H5),Pb, 78-00-2.
Study of Schultz Distribution to Model Polydisperslty of Mlcroemulsion Droplets Michael KotlarchykJ Richard B. Stephens,$and J0hn.S.Huang*$ Exxon Research and Engineering Company, Route 22 East, Annandale, New Jersey 08801, and Department of Physics, Rochester Institute of Technology, One Lomb Memorial Drive, P.O. Box 9887, Rochester, New York 14623-0887 (Received: June 3, 1987; In Final Form: September 30, 1987)
We have employed a histogram analysis method to study the spectra obtained by small-angle neutron scattering (SANS) in order to investigate the size distribution of a model water-in-oil three-component microemulsion. It is found that the often used Schultz distribution function is indeed one of the reasonable choices to represent the moderately polydispersed droplet system. Analysis based on mean spherical approximationsof the SANS data and inverse Laplace transformation of quasi-elastic light scattering data, assuming a Schultz distribution, yield consistent results compared with the histogram method. The polydispersity index, M / R ,found in the model microemulsion is roughly constant at larger R, but increases at small R , consistent with a theory proposed by Safran.
1. Introduction Recently, small-angle neutron scattering (SANS) was used to study the structure of AOT/D20/decane (where AOT stands for sodium bis(2-ethylhexy1)sulfosuccinate) water-in-oil (w/o) microemulsions in the vicinity of a phase separation critical point.'s2 In doing so, it was found that, for a water-to-surfactant molar ratio of 40.8, the microemulsion droplets could be satisfactorily represented by a system of polydispersed spheres, according to the Schultz distribution function'
Here R is the mean sphere radius and z is a parameter related to the width of the di'stribution. In particular, one has a polydispersity index p = aR/R,where uR is the root-mean-square Rochester Institute of Technolow.
* Exxon Research and Engineerie Company.
deviation from the mean size, given by
At the above molar ratio, it was found that R and p remain approximately constant with respect to changes in temperature and volume fraction of the dispersed phase. The reason for the choice of the Schultz distribution is mainly that of the ease in analytical computation. There are two questions that arise out of these studies. First, is the Schultz distribution an appropriate function to represent the spread of particle sizes? And second, how does the polydispersity depend on the mean radius of the droplets as determined by the molar ratio X = [D20]/[AOT]? The last question is one (1) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S . Phys. Reu. A 1 8 3 , 28, 508. (2) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S.; Kim,M. W. Phys. Rev. A 1984. 29. 5054. (3) Zimm, B. H. J . Chem. Phys. 1948, 16, 1099.
0022-3654/88/2092-1533$01.50/0 0 1988 American Chemical Society
1534
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
Kotlarchyk et al.
of particular interest because the answer can be compared with theories of m i c r o e m ~ l s i o n ~size . ~ and shape fluctuations. In particular, a theory by Safran6 predicts that the polydispersity should depend on X and not on the overall volume fraction 4. In this paper, we describe our analysis of a series of S A N S and quasi-elastic light scattering (QELS) spectra from room temperature w/o microemulsions obtained for a constant volume fraction (kept at 5%) a t different values of X . The importance of studying the polydispersity of the microemulsion is clearly illustrated in a detailed statistical model of the micellar systems proposed by Stecker and Benedek.’
4 Radius
.-F
Water
A’OT
Exterior Medium
B
s Radius (A)
11. Experimental Section
The X values of our three-component microemulsions are 5.41, 14.41, 32.37, 40.61, and 44.35. By specifying the [micellar] volume fraction 4 to be 0.05, the composition of each microemulsion is completely determined. Here
4 = 4 ~ +~ ~ A0 O T
4 2R
(3)
where 4Dt0and dAoTare the respective volume fractions of the are 0.01, 0.02, heavy water and surfactant. The values of 4D20 0.03, 0.0325, and 0.0335. The values of d A oare ~ 0.05 - 4D0. The volume of AOT is calculated by using the density of dry surfactant (1.1 3 g/cm3). The chemical components and surfactant purification procedure are described in a previous paper.2 The SANS experiments were performed at the High-Flux Beam Reactor of the Brookhaven National Laboratory. The neutron wavelength was selected at X = 5.28 8,. A sample-to-detector distance of 173 cm was used, and the available range of the where Q = 4*/X wave-vector transfer was 0.01 d Q d 0.21 sin (0/2) is given by the incident wavelength X and the scattering angle 8. The sample temperature was held at 22 f 0.2 O C in 1-mm-path-length cylindrical quartz cells, providing a total transmission of about 52%, mainly due to the incoherent scattering of the solvent (decane). Dynamic light scattering was also employed to study the polydispersity in microemulsion systemss in the system with the large droplets (X= 40.61). Our droplet size is below the resolution of classical light scattering.
+
13A-
I
-
Radius (A)
Figure 1. The construction of the scattering amplitude function for an AOT/D20 micelle: (a) The intraparticle structure, H ( r ) , and (b) the interparticle correlation, G(r).
than the more general distribution of scattering scale lengths shown by the usual radial distribution function. The analysis procedure is very straightforward. The experimental SANS spectrum is assumed to be the sum of the scattering intensity of particles with a given structure over a range of sizes
I(Q)= C~o(r)lFr(Q)IZSr(Q) r
where the P,(Q) 0 IFr(Q)IZcontains all the Q dependent terms from the intraparticle scattering and the partial structure factor, Sr(Q),contains the terms from interparticle scattering. They are defined so that Pr(0) = S,(O) = 1. As a result, lo(?) contains the contrast of the scattering-length density between the particle and the solvent, AI, particle volume, V(r),and number density, N, Zo(r) = N , V ( ~ ) ~ A Z ~
111. Analysis of SANS Data
Each of the SANS spectra was reparametrized in terms of the Q = 0 contributions from populations of micelles of discrete radii Zo(r). The micellar form factors of the scattering law were calculated from the structure determined in earlier studiesG2We then examined the form of the size distribution histogram. We refer to this approach as a histogram analysis? It has the advantage over more usual techniques1° of analyzing for the size dispersion of the scatterers without presupposing an analytic form for the dispersity, and the program usually converges to a unique set of fitting parameters. This analysis is in effect an inverse Fourier transform from Z(Q) Zo(r)that transforms the variable of the scattering spectrum from reciprocal to real space. Since we have explicitly included the form of the scatterer, the resulting real-space spectrum shows the distribution of scatterers of the specific form considered rather
-
(4) For general review, see: Surfactants in Solution; Mittal, K., Lindman, B., Eds.; Plenum: New York, 1984. ( 5 ) For thermodynamic theory of microemulsion, see: Mitchell, D. J.; Ninham, B. W. J. Chem. Soc. Faraday Trans 2,1981,77 601. Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986,85, 7268. (6) Safran, S.A. J. Chem. Phys. 1983 78, 2073. Safran, S. A. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.;Plenum: New York, Vol. 3, p 1781. (7) Stecker; Mark M.; Benedek, George D. J. Phys. Chem. 1984,88,6519. (8) See: Huang, J. S.; Kim, M. W. Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems; Chen, S.H., Chu, B., Nossal, R., Eds.; Plenum: New York, 1981; p 809. (9) Stephens, R. B. J. Appl. Phys. 1987, 61, 1348. (10) Glatter, 0. Small Angle X-ray Scattering, Glatter, O., Kratky, O., Eds.; Academic: New York, 1982; p 148.
(4)
(5)
The Zo(r)’s are determined by minimizing the weighted mean square sum of the differences between the calculated values given by eq 4 and the S A N S s p e ~ t r u m . ~ J ~ F(Q)is calculated from the Fourier transform of the neutron scattering length density profile, H ( r ) , shown in Figure la. Previous work by Kotlarchyk et all2 has characterized the average structure of the AOT/water micelles in detail (Figure 4 of their paper). The AOT head groups plus their associated water molecules form a shell 5 8, thick around the core and have a scattering-length density estimated to be 0.9 that of the core. The radius of the core (including the AOT head groups), r,, is presumed to be variable, causing size dispersion amongst the micelles. The external medium, decane, has a scattering-length density almost equal to that of the AOT tails (they are assumed to be the same in this work), so the tails would make no contribution to the intraparticle scattering. At finite concentrations of micelles, one has to also include the effects of interparticle correlations. The tails define a minimum separation between micelles, which are found to be 13 8, (twice the length of the AOT tails less a 3-8, overlap). That effect is included through C ( r ) (Figure lb) where only hard-sphere interactions are assumed. For that case, the concentration of spheres is a constant outside a minimum center-to-center distance, rmin, ( 1 1) The transformation in this case is calculated by a least-squares fit between the experimental and calculated I(Q). The transformation functions have been worked out analytically for the case of a distribution of solid spheres. Bertero, S.M.; Pike, E. R.; Optica Acta 1983, 30, 1043. Bertero, M.; Boccacci, P.; Pike, E. R. Inverse Probl. 1985, 1 , 111. (12) Kotlarchyk, M.; Huang, J. S.; Chen, S.-H. J . Phys. Chem. 1985,89,
4382.
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1535 I
10.1 Relative Volume Fraction
-
*( *.c *\
-
I
1
1000
I
Generating Function Fit to Calculated I(Q)
' . \
-
0.0001
*
I
\*
*
I
I
I
I
I
1
1 1 1 1
1
a
-
,
Legend
-:p
ol o[\
b-..
o.ol 0.001
I
I
1
Intensity
-
10 0.01
I
Q (a-1)
O.'
b Relative Volume Fraction 20
c
0 0
a
.
'* \
Legend
-Expt.
ool,~-
C
O.'
1 0.01
r
I
I
i
-
-
i
A I
40
60
...I
I
I l l 1
I
80 I
0.01
I
0.1
I
I
I
I
- - -160 -100920 I
I
I
I
I
I
40
60
80
120 140 Sphere Radius (A)
160
100
180
histogram
I
distribution fitted to the histogram. (c) Part b with the ordinate expressed as the logarithm of the relative volume fraction.
and 0 inside that distance except for the 6 function at the center. The number density of micelles in the external medium is just proportional to the ratio of the water concentration to the average drop volume, U,so [ h 2 0-k
0.14dAOTI/P
G(r) = 6(r)
(r
( r < rmin)
QELS
MSA
x
6n.n
R,A
z
R,A
z
R, A
z
5.41 14.41 32.37 40.61 44.35
0.010 0.020 0.030 0.0325 0.0335
5.9 26.7 48.3 57.5 61.5
1.1 22.2 13.2 12 11
6.0 21.2 46.4 54.9 60.1
0.76 5.66 9.32 8.73 8.93
70
9.5
'
rmin)
(6)
figure shows (a) the experimental Z(Q) and its fit, (b) the calculated distribution, and the Schultz distribution fit to that, and (c) the calculated distribution and its fit, using a log ordinate. The other fits appear equally reasonable. Table I shows the parameters extracted from those fits for each of the microemulsions in the series. The self-consistencyof the extracted parameters can be checked in several ways. Both the total scattering volume and surfaceto-volume ratio of the microemulsions are completely determined by their known composition. In both cases one can form from the calculated distribution a sum which, aside from a constant, is proportional to the externally determined value. In the case of the total dispersed-phase volume one uses the Q = 0 limit of the scattered intensity (eq 5 ) . Then
where rmin= 2r,
60
50
Figures 3 and 4 show data for microemulsions with X = 32.37 (dDZ0 = 0.03) and X = 5.41 (dDIO= 0.01), respectively. Each
180
Figure 3. (a) I ( Q ) vs Q experimental and fit for a microemulsion system with X = 32.37 (#D20 = 0.03). (b) Relative droplet volume fraction vs size from the histogram analysis of part a. The heavy line is the Schultz
G(r) =
1
TABLE I: z$ from SANS, Using Two Independent Techniques and OEM
0.1 0.01 20
I
20 30 40 Sphere Radius (A)
with X = 5.41 (bDZ0= 0.01). (b) Relative droplet volume fraction vs size from the histogram analysis of part a. The heavy line is the Schultz distribution fitted to the histogram. (c) Part b with the ordinate expressed as the logarithm of the relative volume fraction.
1
1
I
I
10
Figure 4. (a) I ( Q ) vs Q experimental and fit for a microemulsion system
I
- 1
140
I
tI
0
I-
I
C
Volume Fraetian
I
.'.
i
10
I
Q (a-1)
Volume Fraction
0 20
1
C Z d r ) / V ( r ) = A P C N r W = AZ2CVtOt(r)a Vtot (7)
+ 13 A
The Fourier transform of G ( r ) corresponds to S(Q).Since < 0.04, this causes only a very small perturbation to Z(Q). The points in Figure 2 shows the distribution resulting from a fit to data calculated from the sum of two Schultz distributions of micelles. The solid line shows the function used to generate the data. One can see that the calculated p i n t s follow the rather complex generating function quite well.
r
r
r
whereAl, the scattering-length density of D20, is invariant with size, and V, is the total scattering volume of the microemulsion and should be proportional to the volume fraction of water plus AOT heads: dD9 O.14dAoT. The total surface area S, is given by a similar sum
+
x3Zo(r)/rV(r) = 3A12CNrV(r)/ r = AZ2CSt0,(r) a S,,, r
r
r
(8)
Kotlarchyk et al.
1536 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
2.5
-t
1
3
00
0.005
0.015
0.01
0.02
0.025 0.03 0.035
Water Content
Figure 5. [Cl,,(r)/$]/[#D20 + 0.14~#~,~~] vs water content. The sum is proportional to total scattering volume if the particle shape does not change.
251 20
Dmplet 15 Radiur
01 0.00
I
I
I
I
1.00 1.50 2.00 (Water % + 0.14’AOTYo) / (AOT 010)
0.50
2.50
Figure 6. R vs (9)/3(.2) from histogram data. One set of points were calculated from volume/C surface and the other from Schultz distribution fits to the histogram data. They are nearly indistinguishable. The solid lines are straight-line fits to the two sets of data.
S , should correspond to the surface area per AOT head, aH,times the number density of AOT molecules. The ratio of the calculated and externally determined values for Vtot, shown in Figure 5, should be constant. This ratio is somewhat lower (=20%) for the microemulsion containing 1% water than for the others. The difference in that point could be caused by a shape change for small micelle radii and will be discussed later. It was previously shown2 that the average surface-to-volume ratios of these micelles are VtOJStOt
= (r3)/3(?)
= FcI1
+ 2(uR/7c)21/3
a
where the particle factors P(Q) and @(Q)can be computed2*’’by assuming a Schultz distribution, and S(Q) is assumed to be the structure factor for a hard-sphere interaction potential.’* This approximation is known as the decoupling approximation and is valid for weakly interacting systems. Since the hard-core diameter u is constrained by u
(9)
A more accurate relationship, which does not presuppose a form of the distribution, is (r3)/3(?2)
where D is the sample thickness (0.1 cm), ko is the incident wave vector (2n/h), and A is the difference between the scatteringlength densities of oil and water. Using the largest values of R and 4D20from Table I, i.e. 61.5 A and 0.0335, respectively, we calculate a value of S(0)/ko2 0.04. In their paper, Schelten and Schmatz clearly show that the extrapolated Q = 0 intensity, the Guinier slope, and the Porod constant are all insignificantly affected by multiple scattering as long as S(0)/ko2is less than 0.1. There are still some details to consider. When the histogram data is plotted with a logarithmic ordinate, as in Figures 3c and 4c, one can see a shoulder on the large radius side of the Schultz distribution. This shoulder appears in every fit. In every case, the maximum value on this shoulder is about 1% of the major peak and for each case the radius at that point lies at the appropriate minimum interparticle separation for the particular microemulsion. This component is not big enough to substantially affect the fits obtained above, but it does illustrate the sensitivity of this technique to the real details of the microemulsion. It was found by Huang14 that the micelles exhibit a weak short-range attraction. As a result, there should exist in all of these microemulsions a somewhat enhanced number of touching micelles. Their number can be estimated from the ratio between the sticking time15and diffusion time between micelles.16 It comes out to =3%, in good agreement with the observed height of the shoulder. As a check, the SANS spectra were also analyzed, as in previous publication^,^^'^ by assuming that the scattered intensity is representable by
r
(z n
C IMr) / VWl / CI 3 W / r Vt 1 ( 10) r r where the slope of the relationship gives the effective AOT head area. The slope of the line determined by eq 10 gives aH = 62.2 A2, in excellent agreement with the value of 62.5 A2 determined earlier2. The points calculated from eq 9 and 10 are plotted in Figure 6. Note that they are virtually identical except for the microemulsion with the lowest water content. For that point, the value derived from eq 9 is some 20% lower than the value determined from eq 10. The problem is t h a t the distribution found for t h e
lowest water content (X= 5.41) is not a good Schultz distribution. One can see in Figure 3b that the distribution is cut off at about 9 A on the small R side. That corresponds to the minimum micellar size determined in the earlier papers. The effects of multiple scattering are negligible. This can be seen by computing what Schelten and SchmatzI3 call the total scattering probability, S ( 0 ) / k o 2for the worked out case of monodisperse, homogeneous spheres. Although our droplets are not monodisperse, the effects of multiple scattering should be comparable to that for spheres of radius R having a volume ~ ~ . 13 shows that fraction c $ ~ Reference (13) Schelten, J.; Schmatz, W. J. Appl. Crystallogr. 1980,13, 385-390.
(12)
where
CV(r)/CS(r)= r
= (64~/~5,)’/~
+ 1)2
-
.. = A -
there are only the two fitting parameters: R and z. The fitted values are listed in Table I along with the values derived from the histogram analysis.
IV. Analysis of Dynamic Light Scattering Data A similar histogram method was introduced by Gulari et al.19 to analyze the scattered nonexponential intensity autocorrelation function in terms of a distribution of decay times. A particlesize distribution was then inferred by using the Stokes-Einstein relations, assuming spherical particles. This method required high-precision data and accurate determination of the base line. Unique convergence was typically very hard to obtain. We have (14) Huang, J. S. J. Chem. Phys. 1985,82,480. (IS) Dozier, W. D.; Kim, M. W.; Chaiken, P., submitted for publication in J. Colloid Interfacial Sei. (16) Diffusion time T = P / D where I is the average intermicelle separation, and D = 1.2 X lod cmz/s is the micellar diffusion constant. (17) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983,79,2461. (18) Sharma, R. V.;Sharma, K. C. Phys. Lett. A 1976,56A,107. (19) Gulari, Esin; Gulari, Erdogan; Tsunashima, Y.; Chu, B. J. Chem. Phys. 1979,70,3965.
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1537
Polydispersity of Microemulsion Droplets h
'"'R
'
l
I
(
I
'
I
'
I
104
'
I
I
I1
Delay Channels
Figure 7. The intensity autocorrelation function measured by QELS as fitted by eq 13. The best fit values = 70 A and z = 9.5. I
I
80 -
50
s
'
30
10
1
1.5%
2.0% 2.5% Water Content (Vol%)
3.0%
3.5%
Figure 9. R vs vol % water. The line was calculated from Safran's theory by using r , = 60 A and K = 0.01 eV. The R were determined from histogram fits to data.
of approximately 2.9 A, which is a very reasonable result. One should note that the linearity of eq 14 is satisfied only if one includes the polydispersity factor a ( p ) . V. Comparison to Theory The calculated polydispersity in the microemulsions, p(R), can be compared to the model developed by Safram6q7 H e finds that the dominant spherical harmonics in these fluctuations are the am (breathing) and the azm(waist pinching) modes, where the mean fluctuation amplitudes are
60
'9
I
1
1.0%
I
I
0.6
0.0
I
I
;/"
0 0
-
1
I
10
I
I
20
40
30
50
X
Figure 8. a(p)R plotted against X = [H,O]/[AOT]. The straight-line fit corresponds to a constant packing area of surfactant head uH = 60
A=.
instead chosen to calculate the line-width function explicitly by assuming that the Schultz distribution (eq. 1 ) is appropriate and then fitting this function to the measured values to obtain z and W , the two parameters that characterize the size distribution. With that assumption, we can calculate the mean line width for the scattered light F(R,z,Q) to be
1
R6flR) exp [-( k ~ T / 6 a ~ # ) Q ~dRt ]
(e-rr) =
$R6f(R)
dR
were r0 = (keT@/6?rqi?)-' is the viscosity of the continuous phase (decane), R is the mean radius, and Kz+7is the modified Bessel function of the second king of the order ( z 7). R6 is the proper weighting of the intensity scattered from small spheres of radius R. The best fit values for i? = 70 8, (I? is obtained from the best fit value for the hydrodynamic radius RH: R = R H - 6 where 6 = 18 8, is the surfactant length as determined in ref 12). The best fit value for z = 9.5 (Figure 7) corresponds to a 31% polydispersity. It was previously shown in ref 2 that the microemulsion droplets must satisfy the relation a(p)R = 3uw/aHX + 3uH/aH (14)
+
where uw = specific volume of a D,o molecule, uH = volume of an AOT head group, aH = area per AOT head group, and the prefactor a ( p ) for a Schultz distribution is a(p)
=1
+ 2p2
(15)
A plot of a(p)R vs Xis shown in Figure 8. The result is aH = 60 A2 and uH = 103 A3, corresponding to a head-group radius
and T kBT/16aK,K is the surfactant surface bending energy, and r l and R are the natural and actual radius of curvature for the surfactant layer, respectively. According to this analysis, the amplitude of the fluctuations are inversely related to K , and the dominant type are aoofor R = r l , and azmfor R
