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Improved Particle Size Distribution Measurements Using Multiangle Dynamic Light Scattering Gary Bryant and John C. Thomas* School of Applied Physics, University of South Australia, The Levels, S A 5095, Australia Received December 12, 1994. I n Final Form: April 24, 1995@ Dynamic light scattering is a widely used technique for the determination of particle size in colloidal systems. However, the resolution of the technique is limited by the inherently low information content of the data and the nonunique nature of the data analysis. In this paper, the extra constraints provided by the simultaneous analysis of multiangle dynamic and static light scattering data are used to improve the resolution, and reduce the run-to-run variability, of the resulting particle size distributions. It is shown that the multiangle approach provides considerably more robust particle size determinations than any single angle experiment of similar total duration. In addition, the multiangle analysis is better able to resolve closely spaced components in multimodal samples.
Introduction Dynamic light scattering (DLS) (also known as photon correlation spectroscopy (PCS))is one of the most widely used techniques for the study of colloidal systems. It is a fast, convenient and relatively simple technique, enabling absolute estimates of particle size for a wide range of colloidal suspensions. The technique does however have some limitations, the most critical being the relatively low information content inherent in the measured signal. In this technique a laser is focused into a small volume of solution which contains colloidal particles, and the scattered light is collected over a small solid angle. The phase and polarization of the light scattered by any individual molecule is determined by its size, shape, and composition. The random Brownian motion of the scatterers causes the total intensity at the detector to fluctuate with time, with the time scale of the fluctuation being determined by the time scale of the Brownian motion of the particles. In a DLS measurement, a n autocorrelation function is generated from the intensity fluctuations, and this function is then inverted to obtain the distribution of diffusion coefficients or particle sizes. The details of the technique are described in a number of books and reviews. 1-6 The major difficulty with the technique lies in the inversion of the data. This involves the numerical inversion of a Laplace transform, a problem which is mathematically ill-conditioned, possessing no unique solution. Over the last decade or so a range of approaches have been taken to tackle the mathematical problem. The most widely used techniques include the method of c~mulants,~ CONTIN,8 nonnegative least squares (NNLS),gJOsingular value analysis,'l and maximum Abstract published in Advance ACS Abstracts, June 1, 1995. (1)Berne, B. J.; Pecora, R. Dynamic Light Scattering; WileyInterscience: New York, 1976. (2) Dahneke, B. E., Ed. Measurement ofsuspended Particles by QuasiElastic Light Scattering; Wiley-Interscience: New York, 1983. (3)Chu, B. Laser Light Scattering, 2nd ed.; Academic Press: New York, 1991. (4)Thomas, J. C. Photon Correlation Spectroscopy: Multicomponent Systems. Proc. SPIE, 1991,1430, 2-18. ( 5 ) Brown, W., Ed. DynamicLight Scattering: The Method and Some Applications. Clarendon Press: Oxford, England, 1993. (6)Finsy, R.Adu. Colloid Interface Sci. 1994,52,79-143. (7)Koppel, D. E. J . Chem. Phys. 1972,57,4814-4820. ( 8 ) Provencher, S.W. Comput. Phys. Commun. 1982,27,213-227. (9)Grabowski, E. F.; Morrison, I. D. In ref 2. (10)Morrison, I. D.; Grabowski, E. F.; Herb, C. A.Langmuir 1985, 4,496-501.
entropy.12-15 These techniques attempt to extract the wanted information (particle size) from the measured autocorrelation function, using a range of constraints which limit the possible solutions to the problem. The advantages and disadvantages of each technique and comparisons between them under a range of conditions have been addressed by a number of authors.6J6-18 In general these techniques work well for monomodal samples, or for bimodal samples where the ratio of the diameters of the two peaks is a t least 2: 1,and the intensity contributions from the two peaks are comparable. However, none of these approaches solve the experimental problem that the autocorrelation function has a n inherently low information content. One way of doing this is to conduct DLS as a function of scattering angle. The angular variation of the total intensity of the scattered light is described (for spheres) by the Mie scattering function. The shape and magnitude of the scattering function is determined by the size of the particles (assuming the particles have the same refractive index), and for large spheres (greater than about 400 nm), the Mie scattering function contains a large amount of structure. This is the information which is measured in static light scattering (SLS). In a bimodal mixture it is often possible to choose one angle where the intensity due to the first species will dominat,eand another angle where the intensity due to the second species will dominate. In principle, if one performed DLS measurements a t these two angles, more information about the system can be obtained than from a single angle measurement. This thinking lies behind recent attempts to develop techniques for the analysis of multiangle DLS. A few groups have addressed this p r ~ b l e m , ~with ~ - ~slightly ~ different emphases, and have shown that the technique
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(11)Finsy, R.; de Groen, P.; Deriemaeker, L.; Van Laethem, M. J . Chem. Phys. 1989,91,7374-7383. (12)Livesey, A. K.;Licinio, P.; Delaye, M. J . Chem. Phys. 1986,84, 5102-5107. (13)Nyeo, S.-L.; Chu, B. Macromolecules 1989,22,3998-4009. (14)Bryan, R.Eur. Biophys. J . 1990,18, 165-174. (15)Langowski,J.; Bryan, R. Macromolecules 1991,24,6346-6348. Sieberer, J.; Schnablegger, H. Part. Part. Syst. (16)Glatter, 0.; Charact. 1901,8, 274-281. (17)Van der Meeren, P.; Van Laethem, M.; Vanderdeelen, J.; Baert, L. J . Liposome Res. 1992,2,23-42. (18)Finsey, R.; Deriemaeker, L.; De Jaeger, N.; Sneyers, R.; Vanderdeelen, J.; Van der Meeren, P.; Demeyere, H.; Stone-Masui, J.; Haestier, A.; Clauwaert, J.; De Wispelaere, W.; Gillioen, P.; Steyfkens, S.; Gelade, E. Par. Part. Syst. Charact. 1993,10, 118-128.
0 1995 American Chemical Society
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is in principle a viable one. In this paper we present a simple method of analyzing multiangle DLS data using NNLS. The utility of the technique is examined using a range of simulations, with particular emphasis on determining the effects of different levels of noise on the behaviour of the multiangle inversions and the effect of run-to-run variations on the reliability and robustness of the resulting particle size distributions. Finally, the technique is used to study a range of experimental systems, and the results are compared with both SLS and single angle DLS analyses. Theoretical Section Dynamic Light Scattering. A DLS experiment measures the normalized intensity (photon count) autocorrelation function (ACF) of the scattered light:
where B is a n instrumental constant (order 1). g‘l)(z) is the electric field ACF which is described, for a dilute, monodisperse suspension of noninteracting spheres, by
lg(l)(r)l= exp(-rr) where r is the decay constant, D is the particle diffusion constant, and K is the magnitude of the scattering vector
K=
4nn sin(W2)
(3)
*O
where n is the refractive index of the suspending liquid, 8 is the scattering angle, and A0 is the wavelength of the laser in vacuum. For spheres the Stokes-Einstein relationship relates the diffusion constant D to the particle radius r (4)
where Tis the absolute temperature, k g is the Boltzmann constant, and ~7 is the viscosity. For polydisperse samples each particle size contributes its own exponential decay to the ACF, which becomes
Ig(’)(r)l= LmG(r)exp(-rr) dT
(5)
where G(T) is the intensity scattered by particles with a decay constant between r and r d r . Extracting the desired information from eq 5 (i.e.,the distribution of decay constants G(r)) requires a Laplace inversion. Nonnegative Least-Squares Method. NNLS has been chosen a s the inversion method in this study for a number of reasons: it is the simplest of the useful techniques to implement; it requires little computer time; it provides good results for broad or multimodal distributions; and the only parameter is the range over which the fit is to take place. Note that this can easily be estimated automatically-one method is to perform a cumulants analysis prior to the fit and use the maximum slope to determine a reasonable size range.
+
(19)Bott, S.E. In Particle Size Analysis; Lloyd, P. J., Ed.; Wiley: London, 1988;p 77-88. (20)Cummins, P. G.; Staples, E. J . Langmuir 1987,3,1109-1113. (21)Finsy, R.;De Groen, P.; Deriemaeker, L.; Gelade, E.; Joosten, J. Part. Part. Syst. Charact. 1992,9,237-251. (22)Wu, C.: Unterforsthuber. K.: Like, D.: Luddecke, E.: Horn, D. Part. Part. Syst. Charact. 1994,11, 146-149.
The nonnegative least-squares method is based on the standard NNLS routine of Lawson and H a n ~ o ndevel,~~ oped for the study of DLS data by Morrison and Grabow~ki.~-l~ The integral equation, (51,is first rewritten in the discrete form g = Kz, where x is the distribution of decay constants a n d g is the ACF. K is a matrix which contains the kernel of the transform-each of the elements of the kernel matrix K is a n exponential of the form exp[-r,rjl, where the assumed values of the exponential decay constant distribution Ti are incremented across the rows of the matrix, and the exponential delay times rJ are incremented down the columns. A number of decay constants (or equivalently particle sizes) are chosen (order 10-25), and the algorithm finds the distribution of x which minimizes llKx - gll subject to the constraint that all elements of x be nonnegative. As any set of assumed particle sizes is equally valid, the procedure can be repeated with several different (but equivalent) sets of assumed sizes over the fit range,24and they can be averaged to give a smoother “regularized” solution. The implementation used here is based on that of Morrison and Grabowski, converted from FORTRAN to QuickBasic with some modifications, and averages five equivalent sets of assumed decay constants (or particle sizes), for a total of 55 basis decay constants. Static Light Scattering. The angular dependence of the time averaged intensity of scattered light is described (for spheres) by the Mie scattering function, which is dependent on the refractive indices of the particle and the medium and the size of the particles present. The Mie calculations used here are implemented in QuickBasic, based on the FORTRAN algorithm “BHMIE” of Bohren and H ~ f f m a n . ~ ~ Static light scattering (also known as classical light scattering) measures the total intensity of scattered light as a function of angle, and this information is used to determine particle size distributions.26-28 The inverse scattering problem for static light scattering is similar (though better conditioned) to that for DLS, and can be solved using similar techniques. A routine based on NNLS can be used to solve the problem. However, in order to compare the results ofmultiangle DLS with an established SLS analysis, a more complex routine based on CONTIN and developed by Prof. R. Finsy is used for the comparison. Multiangle Dynamic Light Scattering. For multiangle DLS, the techniques of DLS and SLS are combined. The ACF and absolute intensity are measured at each angle, and both sets of information are incorporated into the inversion routine. The formalism used here follows that of Bott,lgwith the difference that Bott used CONTIN rather than NNLS. The extension of the technique to a multiangle analysis is in principle straightforward. If experiments are carried out a t a number of different angles, there will be an equation g = Kx for each angle. To combine data from several angles in a single analysis, these equations are combined using the Mie scattering function as the weighting factor. For n angles the equation becomes (23)Lawson, C.L.; Hanson, R. J. Solving Least Squares Problems. Prentice-Hall: Englewood Cliffs, NJ, 1974. (24)McWhirter, J. G.; Pike, E. R. J.Phys. A: Math. Gen. 1978,11, 1729-1745. - .- . .-.
(25)Bohren, C. F.;Huffman, D. R.Absorptionand ScatteringofLight by Small Particles. Wiley: New York, 1983. (26)Glatter, 0.;Hofer, M. J. Colloid Interface Sei. 1988,122,496mi (27)Finsy, R.; Deriemaeker, L.; Gelade, E.; Joosten, J . J. Colloid. Interface Sci. 1992,153,337-354. (28)Schnablegger, H.; Glatter, 0.J.ColZoidInterface Sci. 1993,158, 228-242.
2482 Langmuir, Vol. 11, No. 7, 1995
Bryant and Thomas
where the S(&) are m x m diagonal matrices whose diagonal components are the number to intensity conversion factors (Miefactors) for particles with decay constants rl,..., rm-l, Tm. In the original work by Bott,lgthe static light scattering information was also introduced by adding the standard SLS matrix to eq 6 by row augmentation. In this study however we found that this augmentation does not improve the results. The computing requirements for the multiangle analysis are minimal. Running on a n IBM compatible 33 Mhz 386SX with numeric coprocessor, the 5- and 10-angle analyses require approximately 20 and 45 s, respectively (in nonoptimized QuickBasic code).
Simulations Simulations were carried out using an in-house program coded in QuickBasic. The inputs for the program are the diameter and standard deviation for up to three peaks, a diameter calculation range, and the angular range. The number distributions for each peak are generated assuming either normal or log-normal distributions, and using 400 points to characterize the distribution. The program then computes the combined, Mie-corrected intensity distribution for the particle mixture a t each angle and generates the corresponding autocorrelation function. Autocorrelation functions were simulated at 10 angles between 30 and 120" for each particle size distribution. The effects ofrun-to-run variability were studied by adding the same quantity of random (Poisson) noise a number of times, using different seed values for the random number generator. The resulting RMS noise (relative to the baseline) is calculated by subtracting the noisy ACF from the original ACF. The simulations were analysed using combinations of 1,3,5,or 10 angles. For the 3- and 5-angle analyses, several combinations of angles were used to investigate how the choice of angles affects the results. Experimental Section Experiments were carried out using a BI-2OOSM stepper-motor controlled goniometer system (Brookhaven Instruments). A 5 mW He-Ne laser beam (632.8 nm) was focused down to -100 pm diameter in the center of the scattering cell, and the scattered light was detected by a single mode fiber optic probe mounted on the goniometer arm,29 before being focused onto a photomultiplier tube (EM19863B/350)using a short focal length lens. The samples were placed in an index matching bath (toluene)at room temperature (293C! 2 K). Dilute suspensions of polystyrene standard spheres (Duke Scientific, Palo Alto, CA) were used for the experimental studies. Spheres with certified diameters of 200,300, and 500 nm were used in various combinations. Mixtures were made on a number ratio basis, using the manufacturer's stated concentration values. The spheres were suspended in 1mM NaCl solution. Solutions were made from water purified using a Milli-RO/Milli-Q filtration system and finally filtered through a Millex-GS 0.22 um filter (Millipore, Bedford, MA). The samples were sealed into 25 mm diameter cylindrical quartz cells for the measurements. DLS measurements were carried out at each specified angle, with each measurements being of 10-min duration. The static data were collected independently using an in-house program, and consisted of either three or five measurements of 2-min duration at each angle. In principle, with our detector optics,28the static (29) Suparno; Deurloo, K.; Stamatelopolous, P.; Srivastva, R.; Thomas, J. C. Appl. Opt. 1994,33,7200-7206.
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Figure 1. Particle size distributions (relative number vs diameter) determined from single angle analyses of the 300: 500 nm (1O:l number ratio) simulated mixture at (a)30, (b) 60, (c) 90, and (d) 120". The intensity ratios at each angle are approximately 0.7, 4, 13, and 1.6 respectively. Each frame shows the results from several independent noisy simulations. Noise -0.1%. data can be collected simultaneously with the dynamic, but the current software does not allow this option.
Results Simulations. Figure 1shows single angle analyses of a simulated mixture containing 300:500 nm spheres with a 1O:l number ratio. Each run has a simulated noise of 0.1%. The distributions shown are the number distributions, which are extremely sensitive to changes in the fitted data and are the most difficult distributions to determine. Each of the single angle analyses extract the 300 nm peak successfully, but, with the exception of one run at 30", show no evidence of the 500 nm peak. The results for 60 and 120" are relatively consistent between runs, whereas those for 30 and 90" show more variation (300 nm peak position varies by up to 20%). Figure 2 shows multiangle analyses of the same data. In all cases both peaks are clearly resolved. For the fiveangle analyses there are some differences between the three different angle choices,and for some ofthe individual runs there are a number of spurious peaks (usually only one bin wide) in the lower size range. The 10-angle result is considerably better, with almost complete reproducibility between runs, both peaks well resolved, and only a small spurious peak on one of the runs. (Note that the existence of these peaks is common in NNLS analysed as there is no "smoothness" constraint.) In general it can be seen that the multiangle results considerably improve the resolution, even for such relatively large noise levels. To investigate whether these benefits remain at lower
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Langmuir, Vol. 11, No. 7, 1995 2483
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400 500 600 700 Diameter (nm) Figure 2. Particle size distributions (relative number vs diameter)determined from multiangle analyses of the 300:500 nm (10:1number ratio) simulated mixture using (a) the lower 5 angles (30, 40, 50, 60,and 70"),(b) the upper 5 angles (80, 90, 100, 110, and 120°),( c ) 5 evenly spaced angles (30,50, 70, 90,and l l O O ) , and (d) all 10 angles (30-120"). Noise -0.1%.
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noise levels, the same simulations were analysed with 0.01% noise. Figure 3 shows data for the same simulated mixture as in Figures 1and 2, but with 0.01% noise. The single angle results (a, b) are not dramatically improved-all angles resolve only one peak, and some spurious peaks remain for some runs. The multiangle results (c, d) on the other hand are dramatically improved-the results for the worst (c)and best (d)cases are virtually indistinguishable, with almost 100%reproducibility between runs, and two wellresolved peaks. The multiangle analysis is clearly beneficial here. Note that the simulated distributions are not completely reproduced-the number of molecules in the 500 nm peak is lower than predicted by approximately a factor of 2, and the position of the 500 nm peak has been shifted down to around 460 nm. However, the results are still very good, considering that the peaks are separated by less than a ratio of 2:l in size, considered to be the practical limit for traditional dynamic light scattering.19 Simulations were also carried out using the same model sizes, but number ratios of 5:l and 2:l (data not shown), which showed similar trends. In all cases only single peaks were extracted for single angle analyses, and two peaks for multiangle. The effects ofnoise and the general quality of the fits obtained were similar to the results shown in Figures 1-3. Results for mixtures of 300 and 600 nm spheres (data not shown) also exhibited similar behavior. These simulations were carried out in the range where the Mie scattering function exhibits substantial structure-the 500 nm sample has its first minimum a t about 75". This raises the question, how does the technique perform a t sizes below which substantial structure is
300 400 500 600 700 Diameter (nm) Figure 3. Particle size distributions (relative number vs. diameter) for the 300:500 nm (1O:l number ratio) simulated mixture. Representative single angle analyses are shown in parts a (30")and b (60"). Multiangle analyses are shown in parts c (5 angles in the range 30-70") and d (10angles in the range 30"-120"). Noise -0.01%. 100
200
observed? In order to test this, a mixture of 100:200 nm spheres (20:l number ratio) was simulated, and the results are shown in Figure 4. The single angle results shown in parts a and b are typical of the single angle analyses. Some of the runs yield two peaks, though the peaks are invariably in the wrong positions. Reproducibility a t any ofthe 10 angles studied is extremely poor. The multiangle analyses are shown in parts c and d. Each run yields a clear peak at 100 nm and a small peak a t 230 nm (the 230 nm peaks are shown on a n expanded vertical scale). The position of the second peak is higher than it should be, and its number fraction is lower. However, the runs are highly consistent and reproducible. Experiment. Single angle and multiangle analyses of two experimental samples are shown in Figure 5, together with the SLS analyses using the same 10-angle data set. Figure 5a,b shows the single angle and multiangle analyses for a 300:500 nm (51)mixture. The single angle analyses (a)produce a broad peak between 400 and 500 nm (600 nm for 120'1, do not resolve the two peaks, and sometimes show spurious peaks. The multiangle analyses (b) each yield two distinct peaks centered a t approximately the correct values (about 270 and 470 nm), although the relative height of the 500 nm peak is overestimated. The SLS analysis (at the same 10 angles) yields a single broad peak centered a t approximately 400 nm. Figure 5c,d shows the analyses for a 200:500 nm (50:l) mixture. Again the single angle analyses are inconsistent and do not in general yield two peaks. The multiangle analyses are consistent and yield peaks a t approximately
2484 Langmuir, Vol. 11, No. 7, 1995 1
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diameter) for the 100:200 nm (20:l number ratio) simulated mixture. Representative single angle analyses are shown in parts a (90") and b (120"). Multiangle analyses are shown in parts c (5 angles in the range 30-120") and d (10 angles in the range 30-120"). The intensity ratio varies from 0.3 (30")to 0.7 (120"). The second peaks in parts c and d are shown on an expanded scale. Noise -0.01%. the correct position and with approximately the correct number ratio. The SLS results are considerably better here than for the previous sample, accurately reproducing both peak positions, though overestimating the relative fraction in the 500 nm peak.
Discussion For any single angle measurement on a n unknown sample, one does not know a priori if there are any components whose Mie scattering function is near a minimum a t that angle and which therefore will be under represented in the DLS measurement. Because of this, the particle size distribution determined a t any single angle can a t best be only a measure of those sizes which contribute significantly to the signal a t that angle. This is a physical limit and applies regardless of the inversion technique used on the data. One can combat this by performing a series of single angle measurements a t several angles, and if they all give similar answers, then the answer is probably representative. However, for multimodal (or broad unimodal) samples the results a t different angles will be significantly different, with the contribution of each component varying relative to the others. The obvious solution to these difficulties is to analyze the data a t a range of angles simultaneously. The results presented above demonstrate that there are clear advantages to this approach. First, the simultaneous analysis of multiangle measurements provides far more reproducible and accurate results than any single angle measurement with similar noise, and is significantly
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better than a single angle measurement of similar total duration. Second, the technique inherently avoids problems associated with the relative contributions of different components a t certain angles. Finally, the technique is far more successful at resolving closely spaced components. The technique is not however perfect-the major limitation is a problem which is common to all light scattering techniques, that all components must scatter a n appreciable amount of light a t some angle in order to be detected: a 100 nm sample mixed with a 500 nm sample must have a huge number ratio in order to be detected above the noise. Consequently, large numbers of particles (which contribute small intensities and have small volumes) can be hidden from all light scattering techniques. The second limitation is that the technique is not useful for very small particle sizes-particles of 100 nm or lower have inherently featureless Mie scattering patterns, so the static light scattering data contains much less information. In this regime single angle DLS is equally effective. These problems aside, the technique is a very useful one. It should be pointed out that the analysis used in this paper is a very simple one. It is based on the NNLS algorithm, and provides very good results. However, other inversion techniques could also be used. BottlS used CONTIN, and Cummins and Staplesz0used a variant of NNLS, though each of these authors used slightly different
Dynamic Light Scattering formalisms to the one used here and carried out their analyses on only two or three angles. Recently Finsy and co-workersZ1used CONTIN and SVR methods to analyse multiangle data, and Wu and coworkers22used a n analysis with an adjustable smoothness regulariser (similar in principle to CONTIN). Wu and co-workers used up to 11angles in the range 40-90" for their study, and found that the multiangle analysis offered significant improvements. Finsy and co-workers found that both CONTIN and SVR gave good results, though not always markedly better than single angle results. They conducted their multiangle analyses in the range 35-60" and analyzed the data a t two, three, or six angles. They found that there were no significant differences between these results, that is, that a two-angle analysis was sufficient to extract any extra information. This is a t odds with the results reported here. As can be seen in Figure 2, the 10-angle result is clearly better than the five-angle ones. In addition, the five-angle analysis which used angles in the 30-120" range is better than high or low angle sets. Analyses using three-angle sets (data not shown) showed considerably more variability than the five-angle sets. The likely explanation of the differences between our results and those of Finsy and co-workers is that their results were taken over a small angular range of only 25" (35-60'). Thus their results show little benefit of the angular dependence-indeed, for the samples they studied there is little structure in the Mie scattering functions in this range. Thus their results are not as strongly influenced by the intensity constraint of the SLS data as the results presented here, for which measurements were conducted over a much wider angular range. The differences between the various five-angle sets studied here suggest that the number of angles studied is not the only relevant parameter and that the best results are obtained by analysing data from as wide a range of angles as possible. The total number of angles needed to give a reliable result depends on the noise-five angles is sufficient for 0.01% noise, but insufficient for 0.1% noise. Finsy and co-workers21also carried out a comparative study of multiangle DLS with standard SLS and concluded that SLS provides considerably better results in similar experimental times. However, the SLS experiments were carried out a t 61 angles in the range 30-EO", and the multiangle (three-angle) DLS study was carried out over a range of 35-60'. The results presented here suggest that the improved resolution observed for the SLS data is a direct consequence of the larger range and number of angles used, and not necessarily due to any inherent superiority over multiangle DLS. A multiangle DLS experiment carried out over a similar angular range could produce considerably better results. The results of our SLS analyses (at the same ten angles used for the multiangle fit) indicated that SLS provided much poorer results for the 300:500 nm mixture (Figure 5b). For the 200:500 nm mixture, for which there is a greater difference in the Mie scattering functions, the results were comparable. Thus for the same accumulation time and number of angles the multiangle DLS provided results a t least as good a s and sometimes better than the SLS analysis. Indeed it has been claimed22that multiangle DLS is less
Langmuir, Vol. 11, No. 7, 1995 2485 sensitive to errors in the refractive index of the particles. Our studies also suggest that multiangle DLS is also less sensitive to uncertainties in the scattering angle. Further investigations are needed before the question of the relative merits of the two techniques can be resolved. There are also some situations where multiangle DLS may be a more practical technique than SLS. One example is in the development of small, compact, relatively inexpensive light scattering systems which have no moving parts and use optical fibers to collect light at several fixed angles. Physically, the number of angles is limited to around 20 or fewer, and the angular separation is a t least several degrees. High precision SLS methods which require angular measurements every degree or so cannot be carried out here. However, the system can be easily used to conduct multiangle DLS experiments-as few as five angles can provide useful information. The development of such systems in the future may greatly benefit from multiangle DLS analysis techniques. Finally, it should be noted that the analysis carried out here is a relatively simple one. A range of relatively simple modifications can in principle be made to the data analysis and data acquisition to improve the data and the inversion. The dynamic range could be improved by the use of multiple sample times. The noise characteristics could be improved by averaging several short duration measurements a t each angle, instead of a single long measurement. Accumulation times a t each angle could be optimized so that each angle generates an ACF with similar noise (which is the major difference between the simulated and experimental data studied here). Alternatively, the amount of noise on the ACF could be used as a weighting factor in the inversion routine. The inversion routine could benefit from the use of a more sophisticated method for baseline correction and the handling of negative data points (in the current analysis data beyond the first negative point is discarded). Other more esoteric modifications are also possible, though the incremental gains are likely to be slight.
Conclusions The results show that there are distinct and significant advantages in using a multiangle simultaneous DLS and SLS analysis of light scattering data. Multiangle analyses provide considerably more robust, reproducible, and accurate particle size distributions than any single angle analysis with a similar experimental duration and are better able to resolve closely spaced components. The major advantage of a multiangle analysis over conventional DLS is that it avoids problems associated with a n inappropriate choice of scattering angle for a n unknown sample. It supplements the standard methods (SLS and DLS) and can provide superior results to both under some conditions. Acknowledgment. We are indebted to Dr. Ian Morrison (XeroxWebster, NY)for providing a copy ofhis NNLS program, and Prof. R. Finsy (Vrije Universiteit Brussel) for providing a copy of his SLS analysis program. This work was carried out with the support of ARC Grant No. AE9332546. LA940976N
