Langmuir 1987,3, 1109-1113
1109
Particle Size Distributions Determined by a "Multiangle" Analysis of Photon Correlation Spectroscopy Data P. G. Cummins* and E.J. Staples Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, United Kingdom Received December 26, 1986. In Final Form: April 13, 1987 A developmentof the photon correlation spectroscopy technique involving the inclusion, and simultaneous analysis, of data collected at two scattering vectors is described. The additional information provided by the change in scattering power of the individual components at the separate scattering vectors provides a powerful constraint on the derived solution. The method is illustrated by using precharacterized latices of 0.25 and 0.52 pm and the solution is demonstrated to be stable to the range, and number, of the fitting parameters (sizes).
Introduction/Theory The development of the analysis for particle size distributions from photon correlation spectroscopy (PCS) data has become an active area of research in the last few years.' From such an experiment the measured intensity autocorrelation function G ( 2 ) ( tis) related to the field autocorrelation f ~ n c t i o n by ~-~ Gm(t)= A ( l
+P~(')(T)~)
having a variance of
(1)
In this expression, A is the square of the time-averaged photon count and hence constitutes the base line and P is an instrumental constant dominated by the number of coherence areas sampled. The field autocorrelation function g(')(t)for a system having size polydispersity takes the form
where G ( r )represents the amplitude distribution for the normalized decay rates and
r=D@
bution function G ( r ) is described as an infinite series in power of the delay time (7)about the mean value of the distribution G ( r ) such that
(3)
where K (scattering vector) = 4IIn/X sin (8/2) and Dt is the translational diffusion coefficient. The amplitudes are related to the particle number distribution P ( r ) by application of the appropriate Mie coefficientss
G ( r ) = Mie(r)P(r) where Mie(r) is the Mie scattering coefficient associated with decay constant r. The validity of such a transform is dependent on the errors present in G ( r ) . However, a problem arises through the very nature of eq 2 since, as a representative of a class of ill-conditioned functions, no unique solution exists when noise is present on the data. A much-used approach that circumvented the mathematical problems associated with a direct inversion was the method of cumulants.6 In this approach the distri(1)Measurement of Suspended Particles by QELS; Dahneke, B. E., Ed.; Wiley: New York, 1983. (2) Dynamic Light Scattering; Pecora, R., Ed.; Plenum: New York,
The method of cumulanta can be considered very powerful if a large number of coefficients in T may be determined. This,however, is not the case, and this has perhaps led to the belief that data of unobtainable accuracy/precision were required not only for the direct Laplace transform of eq 2 but for any method that would yield significant detail of the size distribution. The reason for the instability inherent in eq 1was clearly s ~ o w I I ' ~by~ ~recourse * to an eigenvector equation representing the correlation function. This showed that at high frequency ((0 m); high resolution) the representative eigenfunctions were transmitted across the equation so weakly that they could not be distinguished from the noise. The information was effectively filtered out. This leads to the conclusion that no method of inversion can claim to produce a distribution to any more accuracy than is imposed naturally by the noise, and any attempt to obtain detail, no matter what constraints are placed on the solution, on a scale that would correspond to extracting data from within the noise band must produce physically meaningless results. From an experimental viewpoint where noise is always present, the number of equally correct solutions, which may differ from each other substantially in form, increases or decreases with the magnitude of the noise term. Hence the requirement for highly accurate data. With this in mind, two methods of tackling the problem arose: the truncated expansion of G ( r ) in the eigenfunctions of the Laplace transform operator mentioned above1J8and a method by which the derived curvature in G(r)was explicitly restricted to yield a smooth function?JO In the truncated expansion method the data are described by a set of linear equations assuming a fixed set
-
1985.
(3) Light Scattering in Liquids & Macromolecular Solution; Degiorgio, V., Corti, M., Giglio, M., Eds.; Plenum: New York, 1980. (4) Photon Correlation &Light Beating Spectroscopy; Cummins, H. Z., Pike, E. R., Eds.; Plenum: New York, 1974. (5) Kerker, M. The Scattering of Light; Academic: New York, 1969. (6) Koppel, D. E. J. Chem. Phys. 1972,57,4814.
0743-7463/87/2403-llO9$01.50/0
(7) McWhirter, J. C.; Pike, E. R. J. Phys. A: Math. Gen. 1978, 11, 1729. (8) Ostrowsky, N.; Sornette, D.; Parker, P.; Pike, E. R. Opt. Acta 1981, 28(8), 1059. (9) Provencher, S. W. Comput. Phys. Commun. 1982,27, 213. (10)Provencher, S. W. Macromol. Chem. 1979, 180, 201.
0 1987 American Chemical Society
1110 Langmuir, Vol. 3, No. 6, 1987
of decay constants and spanning the required range of sizes. Although any arbitrary set will not match the true set exactly, it is tacitly assumed to be sufficiently similar as to give approximately the true distribution. This choice of r values removes the transcendental nature of the minimization procedure such that only amplitude terms remain. The introduction of exponential sampling wherein the limit of resolution could be obtained by using a geometric progression in particle radii over the appropriate time scales was estab1ished.l~~ In order to keep physically realistic solutions (i.e., positive amplitudes) for the inversion, a limited number of sizes (-5-10) needed to be retained. This results in a loss in resolution which, however, may be improved by independently analyzing the same data set over the same range with differing sets of assumed particle sizes and then averaging all the individual solutions. This method of ”regularization”does not discriminate against multimodal distributions. Such a discrimination does occur in the application of the regularization procedure outlined for the restricted curvature meth~d.~JO In this approach to the inversion, the curvature in G ( r ) is explicitly restricted by the addition of an operator which adds a positive penalty to the x2 minimization for any solution requiring maxima and minima. This method thus yields the most “parsimonious” solution consistent with the data. Recently, Morrison et al.” have extended the above methods, and they argue that although the “Pike distribution” is only an approximation it is very often more important to determine the general shape of the distribution rather than the specific values of the particle size. In their work they introduce a nonnegative nonlinear least-squares (NNLS) program rather than inversion routines. This routine is guaranteed to converge in a finite number of iterations (removing any need to consider fitting termination criteria) and to generate a “unique” solution to the problem posed. Finally, they make the obvious though previously ignored comment that spurious correlations that propagate through a single correlation function and appear in the fitting as additional peaks even after large data collection times are more readily removed if many shorter experiments are independently fitted with the same I’, values. The solutions of their “displacive interpolation” can then be averaged with all the independently collected data sets.I2 In the multiangle analysis described below, we demonstrate that the additional information provided by the particle-scattering factor can impose a very severe constraint on the range of possible solutions. Resolutions can be achieved that are unobtainable by using a displacive interpolation approach, and recourse to result averaging is unnecessary. Multiangle Analysis. Following the assumptions that the scattering particles are isotropic and noninteracting and that number density fluctuations are insignificant, we can assume that the measured autocorrelation functions G@)(7)characterize processes that are purely diffusive. As a consequence, eq 1, 2, and 3 are valid. Further, normalizing G(2)(7)by using the calculated base line ((I)2) shows there is an implicit assumption that the optical field (11)Morrison, I. D.; Grabowksi, E. F.; Herb, C. A. Langmuir 1985, I ,
496. (12) Quote from ref 11: “The average of the constrained inverses of
individual auto correlation functions is a more probable representation of the real size distribution than the constrained inverse of the average of several auto-correlation functions”. (13) Data Reduction and Error Analysis for the Physical Sciences; Bevington, P. R., Ed.; McGraw-Hill: New York, 1969.
Cummins and Staples possesses Gaussian statistics. Such normalized correlation functions, however, collected at different scattering angles, will typically possess different intercept and “residual”base lines (the residual base line reflects the presence of some laser noise and a small contribution from number density fluctuations). We have, therefore, extended the list of fitting parameters used in the analyses to include not just the amplitude terms a@,) and u(Dl)[Il(8,)/Zl(~l)](see eq 4 and 5) but also separate intercept and noise terms for each correlation function. Normalization of the correlation function obviously removes any explicit information regarding the intensity variation with scattering angle; however, intensity information is still implicitly preserved within the correlation function since the relative contribution due to any given particle size can still vary with scattering angle. For example, by suitable selection of scattering angle one may arrange for different particle species to dominate the correlation function. These relative changes in contribution, and hence the result, can be normalized to any selected angle. To provide direct comparison between multiangle and single-angle analyses, we have displayed the result as the intensity distribution a(D,) associated with one of the angles at which data have been collected. We can, therefore, contrast the result of multiangle analyses with a result (the single-angle intensity analysis) that is not subject to any distortion associated with assumed Mie coefficients. We achieve the normalization to a single angle as follows: The electric field autocorrelation function associated with the displayed angle (8,) is given by g1(701)=
Ca(D,) ~ X P [ - D , K ~ ~ K ~ ~ , I (4) 1
whilst the electric field autocorrelation function associated with other angles (e,) is given by g1(70,) = Ca(D,)[Il(e,)/l,(el)l exp[-D,K,,%,]
(5)
1
where D, is a preselected set (geometric progression) of diffusion coefficients;74, 70, are the correlation decay times for data collected at angles el, On, K,, K, are the scattering vectors associated with particles of radius r, (at the display angle); and a, is the scattering intensity associated with particle radius r, (at the display angle) where rl = k t / 6117rD1. Note that C,u(D,)and Cla(D,)[ I 1 ( ~ , J / I l ( 6 are ,)] normalized to equal the difference between intercept and base line for the appropriate correlation function. 11(8,), I1(Ol) are scattering intensities for the ith component at the angle 8, evaluated by using Mie theory. In ascribing the scattering powers, we have assumed that the particle size distributions are broad with respect to the selected size interval and have therefore obtained a more representative estimate of the scattering power by integrating the seattering functions over the associated size interval. By preselection of the diffusion coefficients (D), the correlation functions (eq 2 and 3) can be restated as linear functions. However, as we need to restrict u(DJto positive values, we employ a Marquadt algorithm to minimize x2:
where glhb includes the correlograms collected at a l l angles and each correlelogram is the average of K spectra. u, represents the expected uncertainty in the data: j=k
- (g’j”’’2
g i = --C((glj)2
Nj=l
“Multiangle” Analysis of Photon Correlation Spectroscopy Table I. Dispersion Characterization relative scattering 1 2 m / I b 2 W = 0.62 power at 4505 number ratiob NZm/N52ch, = 11.6 mass ratiob m a ~ s ~ ~ / m a s s=~ 1.29 -
Langmuir, Vol. 3, No. 6,1987 1111 LATEX MIXTURE SINGLE ANGLE ANRLYSIS
(45O)
I
a Dispersion composition determined from scattering profile only. Inferred by using the Mie scattering coefficients associated with the “nominal” sizes and Z2sonm/1520nm = 0.62.
The standard deviation ui was experimentally determined by using several correlation functions. This reduced error term x2 is used as a criterion of success in the fitting procedure in that values less than 2 1 imply not an improvement of fit but reflect some fortuitous fitting to the experimental noise. From the measured intensity distribution the Mie scattering terms can be used to generate number and mass distributions and also the scattering angle dependence of the intensity. This intensity profile can be used for comparison with the measured angle dependence to validate the distribution/scattering function combination. It is also possible that the measured intensity profile information can be included as extra data directly within the multiangle analysis and thus further constrain the solution. In these cases we have adopted a purely pragmatic approach in the solution of the weighting term ui associated with the intensity data.
100
200
400
600
800 1000
2000
DIAMETER ( n M ) LATEX MIXTURE SINGLE RNGLE ANRLYSIS ( 7 0 O )
Experimental Section Two well-defined and independently characterizedpolystyrene latices made by a surfactant-free procedure were mixed in known proportions. Their sizes were 0.25 and 0.52 pm as determined by photon correlation spectroscopy. The correlation functions were recorded on a Malvern 7032 series multibit correlator (128 channels). The instrument was situated in a thermostated room and isolated from mechanical motion by an air-cushioned antivibration table (Newporth e a r c h Corp.). The samples were filtered through 1.2-pm millipore fiters and held at constant temperature (*0.1 “C) in the scattering volume. The data acquisition was invariably carried out overnight, and as many as 20 independent correlation functions were obtained at each angle. The correlation functions from different angles were transferred to HP desk-top computers (HP 9826/HP 9816) for data analysis. The data were fitted to a single angle or to two angles simultaneously. In the latter case, fitting parameters were ascribed to the intercepts ( 7 = 0) and base lines of the separate correlation function in addition to the (now normalized)gradient amplitudes (intensities),
Results and Discussion Figure 1shows the intensity distributions obtained for the single-angle analysis at 4 5 O , 70°, and 90°. The simple fitting procedure is unable to separate the two particle size contributions to the correlation function. Figure 2 shows the results of double-angle analyses for both intensity and mass using the data employed in Figure 1 but in the combinations 45O,7Oo and 45O,9Oo, respectively. It is evident that the separate Components can now be resolved. This improvement in resolution stems from the extra boundary conditions placed on the solution. From the areas of the resolved peaks we identify the relative scattering contributions of the two species present (Table 11, column 8). In Figure 4 we show, and contrast with Mie theory, the variation in scatter with angle of the separate, assumed monodisperse systems. Also plotted is the intensity variation obtained for the mixture of latices. It was found necessary to filter dispersions directly into the cell to minimize dust contamination, a process that inevitably results in a change in particle concentration.
h 100
400
200
600
B O O 1000
2000
DIAMETER ( n M ) LATEX MIXTUQE S I N G L E ANGLE
(9~2’)
n
I
I00
ANRLYSIS
2 00
400
DIAMETER
600
8 0 0 1000
2000
(nM)
Figure 1. Intensity versus size for the single-angle analysis a t 45”, 70°, and 90°, respectively.
The dispersion composition was therefore determined by comparing the scattering angle dependence of the mixture with that of the separate components by using a leastsquares fitting procedure. It is then possible to predict the relative scattering contributions of the two components at any angle and infer, subject to the validity of the Mie coefficients, the number and mass fractions (Table I). The success of the multiangle analysis in predicting both the latex mean sizes and relative scattering contributions is
1112 Langmuir, Vol. 3, No. 6, 1987 DCuBLE
PNG,E
Cummins and Staples
FNALYSIS
(4S0/
70°
DOCiELE
)
ANGLE
RNPLYSIS \ 4 5 ' /
90n 1
t r
100
200
4C3
DIRME-EP
600
830
3OC
20c3
I00
200
409
ZIPYE-ER
?M>
500
E 0 0 '30C
2c3c
I-M,
Figure 2. Intensity and mass versus size for the angle combinations 450,70° and 45",90°. Table 11. scattering angles, deg 45, 70 45, 70 45,90 45,90 45,90 45,90 70, 90 90, 70 45, 70
size range limit, nm lower upper 100 200 100 100 100 100 100 100 200
1300 600 1300 1300 1300 1300 1300 1300 10000
no. of sizes
no. of interpolations"
peak '1, nm
peak 2, nm
relative intensityb
12 12 12 20 40 72 40 40 62
6 6 6
270 266 268 263 261 260 262 256 278
513 505 516 508 501 499 502 496 555
0.67 0.6 0.71 0.65 0.66 0.63
1 1 1 1 1 1
~
0.75
aValues other than 1 involve displacive interpolation; see ref 11. [Peak 1 (area)]/[peak 2 (area)] 1250nm/Z520nm.
indicated in Table I1 (column 7). The utility of a multiangle analysis is obviously dependent on the amount of extra information available. Figure 5 indicates the variation of the ratio Ii(OJ/Ii(02)with particle size for three separate scattering angle combinations as derived by Mie scattering theory (nsolvent = 1.33, n particle = 1.6). Also indicated (vertical lines) are the mean sizes of the two latex systems used in the dispersion analyzed. It can be seen, for the scattering angle combinations chosen, that the detail of the additional information varies markedly; however, as indicated in Tables I1 and 111, this does not have a large effect on the result. The result of an analysis of multiexponential decays using a preselected set of gradients (ri)is generally dependent on the number (N) and range of the gradients selected. In some circumstances the value of N necessary
to provide the required resolution in G ( r ) can produce unphysical results. In such cases several separate sets of sizes (equal to number of interpolations) can be fitted in turn to each data combination and the results averaged.lJ~~ From Table I1 it can be seen that the extra information provided by a multiangle analysis imposes such a severe constraint on the possible solutions that the results of the analysis are little affected by the range and/or number of terms (sizes) employed in the analysis. It is possible to extend this procedure to several angles, and this may be justified where enhanced resolution is required or where more complex distributions are to be studied. For the dispersion and scattering angles used in this work the correlation functions above, in both double-angle and triple-angle analysis, provide a sufficient constraint
Langmuir, Vol. 3, No. 6,1987 1113
"Multiangle" Analysis of Photon Correlation Spectroscopy TRIPLE RNGLE RNRLYSIS
0.52 MICRON LRTEX 0.25 MICRON LRTEX
"xi'
,,
--- -\--. \
-
'
.
IC
"0"
MIXTURE
o
\
--.a
\
-.
\ \ X
\\
MIE
v \
30 IO0
200
400
600
E 0 0 1000
55
2000
DIRMETER ( n M )
e0
130
105
RNGLE
Figure 4. Mie theory analysis for 0.25-pm ( 0 )and 0.52-pm (X) latex and the measured intensity versus angle for the mixture.
TRIPLE RNGLE RNRLYSIS
L1 Lo
a
L
10-1 0 I00
2 00
400
DIRMETER
600
2000
B O O 1000
(nM)
Table 111. PCS Double-Angle Analysis (with Intensity Constraint) intensitv constraint mass ratio (calcd) ITQ/I46 = 0.27 1.22 = 0.66 = 0.18
1.47 1.39
Table IV. Triple-Angle Analysis (No Intensity Constraint) intensity ratio maSSzmnm/
measd calcd
2
3
4
2*PI*n*a/l a m b d a
Figure 3. Triple-angle (45°,700,900) analysis of intensity and mass versus size.
IW/I70 IW/I46
\/
1
I7QI145
IW/I70
Id145
0.27 0.26
0.60 0.66
0.18 0.17
mass520nm
1.29 1.25
to achieve resolution of the separate components. Also, results acquired in the presence and absence of the intensity profile information are essentially identical. Transformation of the intensity distribution to mass provides a more severe test of the results of the analysis.
Figure 5. Mie theory variation of Ii(81)/Ii(82)with particle size for three separate scattering angles: 45O, 70°, and 90' (n, = 1.33; np = 1.6).
Table I11 shows the relative mass ratios predicted for the three double-angle combinations in which the scatter angle dependence (Table 111, column 1) has been included as data within the analysis. The results are quite stable. In Table IV the intensity profile information measured directly is compared with that predicted by the triple-angle analysis in the absence of any intensity constraint. The agreement is very good and serves to confirm the validity of the approach and that the assumptions made with regard to the chosen Mie terms (i.e. refractive index, etc.) are adequate.
Acknowledgment. We wish to thank I. D. Morrison et al. for providing us with a listing of their "Caedmon" program. Registry No. Polystyrene, 9003-53-6.
