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Langmuir 1991, 7, 2060-2065
Photon Correlation Spectroscopic Studies of Bimodal Colloidal Dispersions G. A. Schumacher and T. G. M. van de Ven' Paprican and Department of Chemistry, Pulp and Paper Research Centre, McGill University, Montreal, Quebec, Canada H3A 2A7 Received September 1,1990. Zn Final Form: February 20,1991 Conventional light scattering theory predicts that large particles can act as pseudo local oscillators for small weak scatterers. We have verified this prediction by performing homodyne experiments on bimodal dispersions of spheres of radii ab and ag with ag >> aA and Ig >> ZA,Z being the intensity of the scattered light. Under these conditions a quasi-heterodyne autocorrelation function is obtained, which can be analyzed similar to a classical heterodyne autocorrelation function. A number of measurements were performed on bimodal systems to establish in which regime quasi-heterodyningis possible, to verify the underlying theory. Besides bimodal suspensions of spheres, suspensions containingspheres and doublets of spheres were studied. It was found that the percentage of doublets in the dispersion can be estimated from both static and dynamic light scattering experiments.
Introduction Our primary intention of making PCS measurements on bimodal colloidal dispersions was to perform heterodyne experiments. Usually heterodyne experiments are performed in one of the following two ways:' (i) a combination of beam splitters and mirrors is used to take a portion of the incident laser beam (local oscillator) and recombine it with the scattered light from the colloidal dispersion, or (ii)the scattered light from a motionless surface (sample cell, Teflon wedge, etc.) in the scattering volume is used as a local oscillator. Misalignment of the local oscillator beam will cause deviations from true heterodyne detection, therefore making difficulties with alignment of the optics a disadvantage of the first method. In contrast, the second method does not suffer from misalignment problems, since the local oscillator beam originates from the scattering volume. However the presence of a motionless surface (wall) would slow down the diffusion (Brownian motion) of those colloidal particles near its surface, thereby distorting the autocorrelation function.2 Adding large colloidal particles to a dispersion of relatively much smaller particles would be very similar to having a motionless surface in the scattering volume, since the large particles remain essentially motionless on the time scale of movement of the smaller ones. However the distortions of the autocorrelation function caused by the wall would not be present here, provided the dispersion is sufficiently dilute. We therefore thought that bimodal colloidal dispersions could be used as a relatively simple way of obtaining heterodyne autocorrelation functions. As a preliminary step, the effect of the size ratio aA/aB and the intensity ratio zB/zA of the component fractions A and B making up the bimodal dispersion on PCS measurements was studied. Although several experiments on bimodal dispersions are reported in the literature (cf. ref 3 and references therein), our experiments are designed to lay the ground work for the bimodal heterodyne experiments. An example of such a bimodal quasiheterodyne experiment is given, showing the feasibility of this method. Further, we have included experiments which study the effect of changing the scattering angle Bs. This (1) Fletcher, G. C.; Harnett, J. I. A u t . J . Phys. 1981,34, 575.
(2) Lan, K. H.; Ostrowski, N.; Sornette, D. Phys. Rev. Lett. 1986,57, 17. (3) Hallet, F. R.;Craig, T.; Marsh, J.; Nickel, B. Can. J . Spectrosc. 1989, 34, 63.
allows the estimation of the percentage of doubleta in the dispersion. A small amount of doublets is present in many colloidal dispersions, but their presence is often ignored.
Theory Monodisperse Colloidal Dispersions. The normalized heterodyne autocorrelation function obtained from PCSmeasurements on monodisperse colloidal particles is given by1 g(')(T) = e-'' (1) where I' = Dq2,D being the translational diffusion constant and q the magnitude of the scattering vector: q = (4?m/X) sin 19,/2;here n is the refractive index of the medium, XO is the wavelength of the light used (in vacuum), and 0, is the scattering angle. The theoretical relationship for the normalized homodyne autocorrelation function can be obtained by using the Siegert relationship* g'2'(7)
+ [g'"(7)]2
=1
(2)
to obtain g ( 2 ) ( T )=
1 + e-'"
(3) It is clear that both of these autocorrelation functions decay as single exponentials, with the homodyne autocorrelation function decaying twice as fast as the heterodyne autocorrelation function. The unnormalized versions of eqs 1 and 5 are' ~ ( 1 ) (= ~ )
le-''
(4)
and G ( 2 ) ( ~=) p[1 + e-2r']
(5)
where G denotes unnormalized and Z is the average amount of scattered light reaching the detector per unit time. Bimodal Colloidal Dispersions. Let the bimodal colloidal dispersion consist of two monodisperse fractions A and B of different sizes, where A is the smaller sized fraction. Then we can define (4) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley-Interscience: New York, 1976; p 40.
0143-7463/91/2401-2Q60$02.5Q/Q 0 1991 American Chemical Society
Langmuir, Vol. 7, No. 10, 1991 2061
Studies of Bimodal Colloidal Dispersions G f ) ( r ) = I A exp[-rAr]
(6)
and a similar expression for G!), as the unnormalized autocorrelation functions for the individual fractions, where = DAq2 and rB = DBq2. The diffusion constants are related to the sizes of the colloidal particles via the wellknown Stokes-Einstein relationship
D = kT/6uqa (7) where k T is the thermal energy, q is the viscosity of the medium, and a is the particle radius. The unnormalized and normalized heterodyne autocorrelation functions for a bimodal colloidal dispersion are then given by6 G ~ / ( T=)I A exp[-rAr]
+ I, exp[-rBz]
(8)
and g&)
IA
IB
*A + I B
A+IB
= -exp[-rAr] + 7 exp[-rBr]
(9)
Making use of the Siegert relationship (eq 2) leads to
and then G#(r) = (IA
+ IB)2g#(r)
(11)
Polydisperse Colloidal Dispersions. If we consider a polydisperse colloidal dispersion to be made up of N monodisperse fractions, then just as for eq 8, we obtain N
G E ~=~Eli ( ~ )exp[-rp]
(12)
ill
or in integral form GE,y(r) = eNpJmI(Oa,a)F(a) exp[-r(a)r] da (13)
where in this case, I(O,,a) is the intensity of light scattered from a particle of size a, F(a) is the normalized size distribution, and N p is the number of particles in the scattering volume. An optical constant c has been included, since the relevant intensity is that which impinges on the detector.
Experimental Section Determination of the Effect of Varying IBIIA. The experimental apparatus used to analyze the bimodal colloidal dispersions was a Brookhaven Instruments photon correlation spectrometer with a BI-2030, (64 + 8) channel, 6-bit autocorrelator. Only scattered light was allowed to impinge on the photomultiplier tube (homodyne detection). The colloidal particles used for this part of the experiment were all latex particles and their sizes are summarized in Table I. Stock dispersions for each latex were made with distilled, deionized, and filtered water as the dispersion medium (the filter had a pore size of 0.2 pm). Particle volume fractions were in the range of 10-6 to 104. All of the experiments in this part of the experimental work were carried out at a scattering angle of 150O. Each experiment consisted of fiist experimentally determining IA and IB for the two stock dispersions of a given size ratio by measuring the number of photon counts in a 10-5 duration. The average over 10 such repetitions was calculated. From these values were subtracted the average number of photon counts, if (5) Bargeron, C. B. A p p l . Phys. Lett. 1973,23, 379.
Table I. h a y Frequencies and Sizes of Colloidal Particles Used in the Bimodal Dispersions rIs-1 a diameterlpmb 1 7028 76 0.0454 0.005 2 2927 35 0.109 0.001 3 2044 & 37 0.156 0.003 4 1262 22 0.252 0.004 5 225 14 1.42 0.09 6 62+5 5.1 0.4 r measured at a scattering angle of 150"C. The aasociated error denotesthe reproducibility (95%confidencelevel). Calculated from r = Dqz and eq 7.
** *
* **
*
Table 11. Radius Ratios and Latexes Used in Experiments expt no. aA/ aB latexes used" 1 0.021 2and6 2 0.077 2and5 3 0.180 land4 4 0.29 1 land3 5 0.416 land2 6 0.617 3and4 7 0.698 2and3 0 Numbers refer to particle index of Table I. only water was used as the sample. These intensities, however, were very small compared to the particle intensities (ZA and ZB) and therefore had a negligible effect on the results. Secondly, r'A and rB were determined by experimentally measuring r for each stock dispersion. These experimental values for r A , re,ZA, and IB were used to construct theoretical autocorrelation functions (ggi versus 7 ) for values of I B I I A ranging from 0.01 to 100. Each of these theoretical autocorrelation functions (taking 72 data points and the same time scale as used when experimentally determining FA) was analyzed as a single _exponentialto obtain a force fitted r which we will refer to as l', the apparent decay frequency. A theoretical curve of I'A/P versus log (ZBIZA) was constructed (see below). The two stock dispersions were then mixed to give bimodal dispersions for various values of Z B I I A , where the ratio z B / z A was determined based on the assumption that both ZB and IA were directly proportional to their volume fractions or, in other words, that the effects due to multiple scattering were negligible. This assumption was experimentally checked and found to be valid. The experimental autocorrelation functions of the bimodal colloidal dispersions with given I B I I A were analyzed in the same way as the theoretical autocorrelation functions to obtain experimental f values (nondimensionalized to I'A/f). Thew values were then compared to the theoretical curves. This procedure was repeated for various values of aA/aB (see Table I1 for aA/aB ratios used in this Experimental Section). Some of these experiments were repeated to ensure reproducibility of the results. The procedure we chose for analyzing the PCS data from the bimodal dispersions was not the only method available to us. We could have used a multiple regression curve fitting routine to fit the data to the theoretical equations. This procedure would have been rather difficult, especially for bimodal dispersions with size ratios of order unity, due to the problem of separating exponential decays of similar decay frequency.e Measurement of Quasi-Heterodyne Autocorrelation Functions Using Bimodal Dispersions. In order to see whether bimodal colloidal dispersions could be used to perform heterodyne experiments, we used the same procedure for a bimodal dispersion ( a A , l a B = 0.009, latex 1 and 6) as the one described above except for two changes. The first change was that the exponential term involving only r B was not included in the construction of the theoretical autocorrelation functions. The second change involved the measurement and analysis of the experimental autocorrelation functions. The data channels were multiplexed' to include 7 X 72 (504)non-real-time channels. This (6) Wiscombe, W.J.; Evane, J. W.J. Comput. Phys. 1977,24,416. (7) D i g i t a l Correlutor Operator's M a n u a l ; Version 2; Brookhaven
Instrumenta Corp., p 5-11.
Schumacher and van de Ven
2062 Langmuir, Vol. 7, No. 10, 1991
x
I.
I
-'-I
k
3.35 LD
0 F
3.30v
sa L3
3.25-
4
a
3.20 0.0
J
1 .o
0.5
-r/ms 1 .o
0.8 ul
2 Ab
0.6
v
7u
0.4
n G J
I
-1
0
0.2
1
0.0 I
T/ms
C
Figure 2. Conversion of a (a) bimodal autocorrelation function into a (b) quasi-heterodyneautocorrelationfunction for a bimodal ~ The longer decaying colloidal dispersion with a ~ / =a 0.009. exponential due to r B is graphically separated to produce the quasi-heterodyneautocorrelationfunction, the natural logarithm of which is plotted in the inset.
-2
I
2.2,
0.9 -1
0
1
log(ls / [ A )
Figure 1. Plot of rA/i' versus log (ZB/ZA). The solid theoretical line corresponds to eq (10) and the experimental points are for g 0.021, (b) a ~ / a = g 0.291, and (c) a ~ / =a0.698. ~ The (a) a ~ / a = error bars denote reproducibility (standard deviation of ten measurements). introduced the possibility of measuring both the long time and short time decay, allowing the long time decay due to r B to be graphically separated. The remaining quasi-heterodyne autocorrelation function was force fitted to a single exponential, nondimensionalized, and compared to the theoretical curve. Determination of the Effect of Varying the Scattering Angle. Dilute dispersions of latex 1 and 2 (see Table I) were made such that the scattering from each dispersion at 0. = 30' was roughly equal. An equal portion of each was mixed to obtain a bimodal dispersion. PCS measurements were made to determine both the intensity of light scattered as well as the apparent diameter (from force fitting the autocorrelation function to a single exponential) from all these dispersions, as a function of the scattering angle e,, for values varying from 0. = 30' to 0, = 150'. The intensity was corrected for the dark counts and the dead time of the detector (photomultiplier tube). Corrections for the amount of light scattered by the water were not made, since this was found to be a negligible effect. The apparent diameter values obtained for the bimodal dispersion were compared to those predicted according to eq 10. Measurements of the apparent diameter of a partially coagulated gold sola as a function of scattering angle 0, were also performed, for 0. ranging again from 30 to 150'. These experiments were performed 3 times, correspondingto having filtered the gold sol through three different pore size filters (0.22, 0.1, 0.05 pm). (8)EnQetih,B.V.,Turkevich,J. J. J. Am. Chem. Soc. 1963,85,3317.
(L
\
L" N
0.8J -2.0
-1.0
0.0
1 .o
I
2.0
I O dB/l*)
Figure 3. Plot of 2rdi' as a function of log (ZB/ZA) for quasiheterodyne autocorrelation functions, where the experimental points are obtained from the graphically separated quasiheterodyne autocorrelations. Electron micrographs of latex 1 and 2 and the gold sol were taken with a Philips EM400T transmission electron microscope to ensure the monodispersenature of the particles. The particle diameters of 200 particles were measured from each electron micrograph and the corresponding diffusion constants were calculated from the Stokes-Einstein equation. Histograms were constructed for the diameter and diffusion constant values.
Results and Discussion Effect of I B / & on PCS Measurements of Bimodal Dispersions. Plots of the-dimensionless reciprocal apparent decay frequency r A / r as a function of t h e logarithm of the scattering ratio I B / I Aare shown in Figure 1,corresponding t o values of aA/aB found in Table I for experiments 1,4,and 7. Plots for the other experiments
Studies
Bimodal Colloidal Dispersions --
2063 70
I
a1
-
I
b
-._
60-
50-
-
40. 50
-
20.
10-
1
O r
c
60-
7
50. 40
-
-
5020
-
-
10. O r
7
Figure 4. Particle size distribution of (a) latex 1and (b) latex 2 (seeTable I), from electron microscopy and particle diffusion constant distribution for (c) latex 1 and (d) latex 2. where the diffusion constant has been calculated from the size by the Stokes-Einstein equation (eq 7).
are given elsewhere and show the same trend.g The apparent-decay frequency has been nondimensionalized to I'A/I' for a few reasons: (i) similar plots for mixed homodyne-heterodyne experimental0 have used the same convention, (ii) plotted in this way the final rA/r plateau is equal to the ratio of the larger to the smaller particles, and (iii) as the proportion of light-scattered from B increases, so does the value of rA/I', an aesthetically pleasing trend. It is clear that there is excellent agreement between experiment and theory for all of these experimenta. Equation 10 therefore applies to all ratios of QA/ OB and, because of the many different sizes of particles used, it does not depend on the absolute size of the particles. The shape of the curves indicates that small amounts of B have a relatively small effect on the measured decay frequency when the measurements are made on the time scale and the scattering angle is large (e, = 150'). This is even the case of experiment 1(Figure l),where the difference in the sizes of the particles used is quite large. This is not the case at smaller scattering angles, where the larger particles contribute relatively more to the scattering than at the larger angles. This dependence on scattering angle should be considered for samples with small amounts of larger sized impurities. Measurement of Quasi-Heterodyne Autocorrelation Functions Using Bimodal Dispersions. Equation 11 can be manipulated to give quasi-heterodyne autocorrelation functions. If the long time decay due to r B is fitted and then subtracted from the remaining curve (Figure Za), one is left with a quasi-heterodyne autocor(9) Schumacher, G. A. Ph.D. Thesis, McGill University, 1990. (10) Oliver,C.J. InPhoto CorrelationandLight BeatingSpectroscopy; NATO Advanced Study Institute Series B; Pike, E.R.,Cummins, H.Z., Eds.;Plenum: New York, 1974; p 151.
10,
1-r
=
-
1
- -
lotex 1
-
-
-
I
* ln ?
latex 2
1 '.
i 4-
toluene 0
0
0
0
0
0
0
0
0
relation function Q(l)(T)
= 2 I A I B exp(-rAT) exp(-rBT)
+ 12 exp(-2rA7) (14)
which for IB>> IAand OB >> OA reduces to
G ( ~ ) ( T= )WAIB exp(-I',.r)
(15)
when measured on the r A time scale. Equation 15 is of the same form as the equation for classical heterodyne detection (eq 61, where the condition IB>> IAis analogous
Schumacher and van de Ven
2064 Lungmuir, Vol. 7, No.10, 1991
501 48
1
latex 1
421 0
30
90
60
120
150
180
4 /" 120
b
' O 0I
P 70 -
4
d
E \
i
mixture
(U
55
8
i
__ 0
$ 1 30
60
90 65
120
a 150
180
/"
Figure 6. Angular dependence of the measured diameter for (a) latex 1, (b) latex 2, and (c) a bimodal mixture of latex 1 and 2 (A). The solid lines are predictions from eq 13with 0.4% doublets for latex 1and 5.3% doublets for latex 2. The dashed lines are for no doublets. The theoretically predicted values for the bimodal mixture are also shown ( 0 ) . Error bars denote reproducibility (95% confidence level).
to the condition ILO >> I, found in classical heterodyne experiments, where ILO and I, are the intensity of the local oscillator and the scattered light, respectively. Practically, ILO should be at least 30 times that of I,. In the quasiheterodyne case, the baseline is the first portion of a decaying exponential, whereas in the classical heterodyne case the baseline comes from the uncorrelated local oscillator and is, therefore, necessarily flat. Figure 2b shows a plot of I!%) obtained from Figure 2a, where aA/aB = 0.009 and IB/IA = 32. The natural logarithm of @ l ) ( ~ ) , which is plotted in the inset of Figure 2b, results in a slope of 0.49 times that of the homodyne case, as compared to 0.5 which is expected theoretically. The excellent agreement shows that quasi-heterodyne autocorrelation functions give the same results as expected from classical heterodyne experiments. The condition QB >> UA is necessary, so that the exponential involving r B in eq 14 is approximately equal to 1for the whole decay due to A, thereby minimizing the difference between the quasiheterodyne and the classical heterodyne methods. The 7 = 0 intercept for eq 11 is given by Ggj(0) = 2(1A + IB)~, whereas the 7 = 0 intercept for eq 15 is given by
C(l)(O)= 2IAIB. The values of the intercepts taken for Gg)(O)and 6(l)(O)from the data in Figure 2, when put into these two equations, give IB/IA = 31.8. This value is in direct agreement with the value determined from the amount used of each stock dispersion (see above). The nondimensionalized apparent decay frequencies of the subtracted autocorrelation functions (Figure 2b) are in Figure plotted as a function of the logarithm of IB/IA 3. The experimental points, which are the result of individual experiments, are in good agreement with the theoretically expected behavior. The largest scatter in the data occurs at large values of IB/IA. This should not be surprising since one is measuring the decay due to A, which is becoming an increasingly weaker signal as IB/IA increases. Suspensions Containing Singlets and Doublets. Many colloidal dispersions contain a small percentage of doublets (and possibly larger aggregates),which are either formed during the preparation stage of the dispersion or caused by aging. We can consider such dispersions as bimodal, one mode being the singlets and the second one the double&. It is possible to estimate the percentage of doublets from both dynamic and static light scattering data. According to eq 13,the autocorrelationfunction depends on I(O,,u). For Rayleigh scatterers I(O,,a) is proportional to a6 but independent of scattering angle. Therefore eq 13predicts that the measured size is independent of angle for a polydisperse particle size distribution. This is not the case for Rayleigh-Debye-Gans or Mie scatterers, for which I(O,,a) depends on the scattering angle. Latex 1and 2 were chosen for the angular experiments on a bimodal dispersion since they approximatedRayleighDebye-Gans scatterers, for which the theory is relatively simple. Size distributions and their corresponding diffusion constant distributions for latex 1and 2 are plotted in Figure 4. It is clear that the latexes are quite monodisperse,with latex 1and 2 having averagediameters (from electron microscopy) of d-39.7 nm and d=92.4 nm, respectively. The measured diameter from PCS should be somewhat higher (see Table I) due to the effect of the larger sized fractions scattering more light. Angular measurements of the intensity of the light scattered from latex 1and 2 and a bimodal mixture of the two are plotted in Figure 5. It should be noted that the incident light is perpendicularly polarized. The light scattered from toluene is also shown as a reference. Since toluene is a Rayleigh scatterer, we do not expect any angular dependence on the intensity of light scattered, as is indeed observed. Since the bimodal dispersion was made up of a 1:l mixture of latex 1and 2, we would expect the intensity values of the bimodal mixture to lie half way between that of its two components, as is clearly seen in Figure 5. The solid lines are calculated from RayleighDebye-Gans theory
I = kN,~"P(0,,a)a6F(a)da
(16)
where P(O,,a)is the form factor and k a constant depending on experimental conditions. The percentage of doublets is used as a variable parameter. The theory predicts the largest scattering at low angles, in agreement with the observations. The fraction of doublets has been expressed as the number of doublets divided by the total number of particles (the number of singlets plus doublets). The size distributions in Figure 4 are used for F(a). The dashed lines are for the case that no doublets are present. Expressions for the form factor P(O,,a)for spheres and for
Studies of Bimodal Colloidal Dispersions
sphere doublets are given elsewhere." The product kN, (eq 16) has been varied to match the magnitude of the experimental points. There is excellent agreement between experiment and theory for 0.4% doublets for latex 1 and 5.3% doublets for latex 2. The apparent diameters measured by PCS (Figure 6a,b) are higher than those obtained from electron microscopy, as expected, due to the larger sized fractions and the doublets (see above) scattering relatively much more than the smaller fractions. The solid lines are calculated by using eq 13, using the percentage of doublets obtained from the intensity results (see above). The expression for the diffusion constant of a doublet is given elsewhere.12 The agreement between experiment and theory is excellent. Not only is the shape of the theoretical curve in agreement with the data but so is its magnitude. The dashed lines are again for no doublets. In the case of latex 1, the dashed line (no doublets) gives a lower bound fit, whereasthe solid line (0.4% doublets) give an upper bound fit indicating the percentage of doublets is somewhere in between. The dashed line for latex 2 gives a considerably poor fit to the data. The consistency of the results of absolute (static) and dynamic light scattering indicates that the percentage of doublets in a dispersion can be accurately measured in this manner. The apparent diameters for the bimodal dispersions agree very well with those predicted from the intensity and diameter values for the component fractions (Figure 6 4 , calculated from eq 10. It is obvious that the lower angles again emphasize the larger particles and vice versa. We have also performed some experiments with gold sol particles, similar to those used in ref 13. As is the case for the latexes, the gold particles are very monodisperse. We observed large changes in apparent size with scattering angle for a gold sol filtered through 0.22 pm pore filter paper. This is due to the fact that the gold sol was partially coagulated, the coagulated particles behaving as large particles. Successive removal of the larger particles by filtration through smaller pore size filter paper diminishes the effect. The intensity of light scattered by gold sol is (11) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969; Chapters 3 and 8. (12) Vades. E. J. Colloid. Interface Sci. 1976.57. 308. (13) Schkacher, 0.A.; van de Ven, T. G. M. Faraday Discuss. Chem. SOC.1987,83, 75.
Langmuir, Vol. 7, No. 10, 1991 2065
qualitatively similar to that of latex 1and 2, even though they are not Rayleigh-Debye-Gans scatterers but Mie scatterer~.~~
Concluding Remarks We have shown that it is possible to perform quasiheterodyne scattering experiments by mixing large particles with small ones under the conditions IB >> IA and ag >> aA. Quasi-heterodyning is in many respects simpler than classical heterodyning since recombining beams requires high precision and is subject to vibrations. These studies could be extended e.g. to electrophoretic light scattering measurements, using quasi-heterodyne detection instead of the classical heterodyne approach. The theory underlying quasi-heterodyne detection has been verified by performing experiments on a variety of bimodal dispersions of spheres of different radii. Besides dispersions of spheres, dispersions containing a small amount of doublets were studied as well. It was shown that the percentage of doublets can be accurately obtained by performing both static and dynamic light scattering experiments. Several people have made attempts to obtain size distributions from PCS3J6l7 with varied success. Obtaining more experimental information, such as making measurements at various angles, could improve the ability to invert the integral (eq 13) and thereby solve for the size distribution. Experiments already performed on bimodal and trimodal latex dispersions, using data collected at two scattering angles, shownthat multiangle experiments help in determining the size distribution.'*
Acknowledgment. Special thanks are in order to Mr. Eric Zuck for writing the data acquisition program used for transferring the multiplexed autocorrelation functions from the BI-2030 to a PC and to Dr. J.-F.Revol for taking the electron micrographs. (14) Mie, G. Ann. Phys. 1908,25,25. (15) Chu, B.; Gulari, E.; Gulari, E. Phys. Scr. 1978,19,476. (16) Gulari, Esin; Gulari, Ergodar;Tsunashima, Y.; Chu, B. J. Chem. Phys. 1979, 70,3965. (17) Fletcher, G. C.; Ramsay, D. J. Opt. Acta 1983,30,1183. (18) Cummins, P. G.; Staples, E.J. Langmuir 1987,3,1109.
