The angular weighting coefficient is key to accurate particle size distribution (PSD) measurement using multiangle dynamic light scattering (MDLS). However, determining the weighting coefficient is affected by the noise in the measured MDLS data. In this paper, a novel successive updating of the angular weighting (AWSU) method is proposed. By using information character weighting, the angular weighting coefficient and the character weighting matrix are updated once with each additional angle. The effect of autocorrelation function (ACF) denoising and information extraction is improved by using information character weighting step-by-step. The results for broad unimodal and closely-spaced bimodal PSDs demonstrate the effectiveness of this method. Compared with another routinely used inversion method, it is found that the increase in PSD information retrieved from MDLS depends not only on increasing the number of angles, but also on the calculation of the angular weightings. An accurate weighting algorithm can reduce the number of scattering angles required. Using the AWSU method for PSD recovery with 4 scattering angles can give better results than the usual method with 6 scattering angles. It is also found that the AWSU method not only improves the accuracy of the weightings, but it also contributes directly to the PSD recovery by reconstructing the ACF with the PSD obtained by the weighted inversion. The combination of these two contributions not only reduces the number of scattering angles needed, but also keeps the PSD recovery results from getting worse because of excessive scattering angles.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Dynamic light scattering (DLS) is a widely applied technique for measuring submicron and nanoparticle size distribution (PSD) in science and industry [1–4]. In this technique, the PSDs are acquired by inverting the autocorrelation function (ACF) of the scattered light intensity. The recovery of the PSDs involves solving a Fredholm integral equation of the first kind, which is a highly ill-posed problem [5–8]. Due to the ill-posed nature of the equation, accurate inversion for broad unimodal and bimodal PSDs has not been well solved. This is especially so for closely-spaced bimodal PSDs. Multiangle dynamic light scattering (MDLS), which uses data from a number of scattering angles and analyses the data simultaneously to provide more information and reduce the ill-posedness of the equation, has improved PSD recovery [9–12].
Cummins and Staples  proposed MDLS in 1987 and applied the nonnegative least squares algorithm to recover the PSD from two or three scattering angles. They used the ratio of scattered light intensity from different angles obtained by Mie scattering theory as the angular weighting coefficient. For a 250/520 nm bimodal particle sample, a clear bimodal PSD was obtained by measuring at two scattering angles, while only a unimodal distribution was obtained from a single-angle measurement. Bryant and Thomas  combined MDLS with static light scattering (SLS), and used the SLS measurement data as the angular weighting coefficient, which improved the accuracy of the PSD recovery. However, the acquisition of SLS data requires a high precision measurement system and a long measuring time. To overcome these issues and make the MDLS technique more useful, Bryant et al  proposed and demonstrated that MDLS could be done without directly measuring SLS data. They recognized that the SLS angular dependence is inherently contained in the ACF data so that the SLS data could be recovered iteratively during the analysis to provide the angular weighting for subsequent iterations during the PSD recovery. Vega et al.  proposed a recursive least squares method to obtain the angular weighting coefficient and Liu et al.  proposed an iterative version of this method. In 2018, Xu et al.  extended Zhu’s  idea of single-angle weighting into MDLS. Using the particle size information distribution (PSID) in the ACF, an information-weighted constrained regularization (IWCR) method  and a character-weighted constrained regularization (CW-CR) method  was proposed. In IWCR method, the ACF data as the base and the PSID as the exponent were used to construct the weighting coefficient to enhance the weight of initial delay section in the ACF data. In the CW-CR method, the weight of this section in the ACF data was further enhanced by using the PSID and an adjusting parameter to construct the weighting coefficient. These methods not only reduce the noise influence in long delay time section of the ACF, but also improve the utilization of particle size information in the ACF. However, these methods only benefit the extraction of particle size information in the ACF from a particular scattering angle, as the angular weighting method is still the same as used by Liu et al . None of this has helped to obtain more accurate angular weighting coefficients, a critical parameter to MDLS measurement.
MDLS can provide more particle size information  than single-angle DLS, but the advantage of MDLS is strictly restricted by the number of angles  and the accuracy of the angular weighting calculation which is the prerequisite of obtaining more PSD information than single-angle DLS. The angular weightings can be calculated using Mie theory or the measured ACF data. The former is a theory-based method, and usually seldom used because of the errors between the theoretical values and the measured data. The latter is a measurement-based method and affected by noise [20,21] in the ACF data. In the MDLS measurement, the angular weighting for each scattering angle is calculated once, and the angular weighting are calculated sequentially in the PSD inversion, which mean that only one de-noising process was made and the weighting coefficient of the first angle could not be revised, and its error will affect the subsequent weighting calculation. In this paper, a successive updating of the angular weighting (AWSU) method is proposed, in which the information character weighting is used to calculate the weighting coefficient of each angle, and the initial angular weighting can be revised by re-calculating the weighting of all angles based on the reconstructed ACF with successive updates. By successive denoising and information weighting during the angular weighting calculation, more accurate angular weighting coefficients can be obtained, and then the PSD results are significantly improved.
2. MDLS theory and information character weighting
In MDLS, the intensity ACF and the electric field ACF are related by the Siegert relationship  so that15] is23,24], f(di) is the discrete particle size distribution, is an a priori unknown weighting coefficient at the scattering angle, and is defined by
In vector form, Eq. (2) can be written as:17]25,26].
For measurements with broad or closely-spaced bimodal PSDs, MDLS is superior to single-angle DLS. However, it is critical to determine the angular weighting accurately. Noise in the MDLS ACF data limits the accuracy with which the weighting factors can be determined and this, in turn, affects the accuracy of the PSD results. In calculating the weighting coefficients usually no special attention is paid to the effects of the noise. However, in MDLS measurements, the accuracy of the recovered PSD depends largely on the accuracy of the angular weighting coefficients which is affected by noise in the ACF data.
3. The AWSU method
To effectively reduce data noise and make further use of the PSD information, in the AWSU method the angular weighting coefficients are calculated and updated step-by-step in the order the angular data was measured. In the updating process the initial PSD obtained by single-angle DLS inversion with information character weighting is used to calculate the angular weighting coefficients for two angles. Then, using this weighting, the information character weighting matrix is reconstructed, and the two-angle DLS inversion is carried out to obtain the updated PSD. The angular weighting coefficients for three angles are then obtained and a new weighting matrix is constructed based on these weighting coefficients. This is used to carry out a three-angle DLS inversion to obtain the updated PSD and the process is repeated over all the scattering angles until all the angular weighting coefficients are obtained. The update process for the AWSU method is as follows:
- (1) Calculate the average particle size D by the method of cumulants and reconstruct the electric field ACF from the average particle size D. Then construct the character weighting matrix from the difference between the actual electric field ACF and the reconstructed electric field ACF. The expression obtained by weighting the ACF is, and f1 is obtained by inverting this equation.
- (2) f1 is taken into Eq. (3) to obtain angular weighting coefficients and and the information character weighting matrices, and are reconstructed. The expression obtained by weighting the ACF is ,f2 is obtained by weighted inversion of the ACF from scattering angles and .
- (3) Similarly, the angle weighting coefficients , andare obtained by bringing f3 into Eq. (3), and the character weighting matrices ,and are constructed to weight the ACF, as follows , f3 is obtained by inversion at scattering angles ,and.
- (4) The rest is done in the same manner, using the obtained fr-1 to calculate ,…, finally get the required number of angle weighting coefficients, and then construct ,, …, . The expression obtained by weighting the ACF is , and fr is obtained by inverting this equation.
During the weighted inversion, Tikhonov regularization was used and the regularization parameter was determined by L-curve criterion.
4. Simulation and results
The method was tested using four samples of simulated data consisting of one unimodal and three bimodal PSDs. The AWSU method was used to recover PSDs from data containing 3 × 10−4, 3 × 10−3 and 3 × 10−2 noise levels. The log-normal distribution  was used to simulate unimodal PSDs and a combination of two log-normal PSDs was used to form a bimodal PSD. The log-normal PSD is
The intensity ACF without noise can be obtained using Eqs. (1) and (4). Gaussian random noise is added to make the simulated ACF data more realistic so that,
The MDLS data were simulated with λ0 = 632.8 nm, T = 298.15 K, nm = 1.3316, kB = 1.3807 × 10−23 J/K, η = 0.89 cP, B = 1, β = 0.7, and dmin = 0.01 nm, dmax = 900.01 nm, dmin, dmax are the smallest and largest particle sizes. Scattering angles in the range = 30° to 140° in increments of 10° were used. Five sets of PSD and their corresponding ACF data were simulated with the parameters shown in Table 1.
In Table 1, PP is the peak position; RPP is the ratio of the peak positions; and D is the mean particle size obtained by the method of cumulants. In addition, to characterize the accuracy of PSD inversion, we introduce three indices: the peak height ratio for bimodal PSDs (RPH), the relative error of peak position (EPP), and the PSD recovery error (V). These three indices are defined as follows:
The recovery results for a 425nm unimodal broad PSD, a 400/700 nm bimodal PSD, a 350/575 nm bimodal PSD, and 500/700 nm bimodal PSD are shown in Figs. 1-4. The corresponding performance indices are shown in Tables 2-5. In Figs. 1-4, “true” represents the true PSD, “CW-CR-4” is the PSD obtained through the CW-CR method using four scattering angles (50°, 70°, 90° and 110°); and “AWSU-4” is the PSD obtained through the AWSU method using the same four scattering angles. “CW-CR-6”is the PSD obtained through the CW-CR method using six scattering angles (30°, 50°,…, 130°); and “AWSU-6” is the PSD obtained through the AWSU method using the same six scattering angles. The figure labels (a), (b) and (c) respectively indicate that the noise level is 3 × 10−4, 3 × 10−3, and 3 × 10−2.
For the 425nm broad unimodal PSD, it is clear that the AWSU method gives a better PSD estimation than the CW-CR method. According to Table 2, at the same noise level, the PSD obtained using the AWSU method is closer to the true PSD. When the noise level is 3 × 10−4, the relative error in peak position decreases from 0.040 to zero and the PSD recovery error (V) also decreases from 0.064 to 0.017 with four scattering angles, and the results with the AWSU-6 method are even closer to the true PSD than with CW-CR-6 method. When the noise level is 3 × 10−3, the value of V drops from 0.128 to 0.035 with four scattering angles and the value of V drops from 0.037 to 0.013 with six scattering angles. When the noise level is 3 × 10−2, using the AWSU method, the peak position is 424nm, and the relative error of the peak position is reduced to 0.002.
For the simulated 400nm and 700nm bimodal PSD, the recovered PSDs are shown in Fig. 2. Table 3 shows the performance parameter values of the true and recovered PSDs. Cearly the recovered PSDs from the AWSU method are closer to the true PSD than the recovery with the CW-CR method. At the same noise level, the inversion results of the AWSU method are better than the inversion results of the CW-CR method. At low noise level (3 × 10−4), the peak position errors are similar for the two methods but as the noise level increases the AWSU method has considerably lower errors than the CW-CR method. This is particularly so for four-angle measurements. As the noise level increases, a better estimation of the relative peak heights of 400nm and 700nm components is also obtained with the AWSU method.
For the simulated closely-spaced bimodal PSDs, where the ratio of the peak positions was less than 2:1 (350:575nm and 500:700nm bimodal PSDs), the recovered PSDs are shown in Fig. 3 and Fig. 4. These results clearly show that the proposed ASWU method has better repeatabilty and accuracy than the CW-CR method. As shown in Table 4 and Table 5, when the noise level is the same, the peak position obtained with the AWSU method is more accurate than with the CW-CR method. Using four scattering angles and a noise level of 3 × 10−2, the peak positions obtained with the ASWU-4 method are 345/581nm and 504/712nm, respectively.
5. Experimental results
Real experimental data were obtained from a light scattering system consisting of a vertically polarized He-Ne laser with a wavelength of 632.8 nm, a stepper-motor controlled goniometer (model BI-200SM, Brookhaven Instruments Inc.), and a 64-channel photon correlator (model BI-2030AT, Brookhaven Instruments Inc.). The sample was made by mixing 300 nm ± 3 nm and 502 nm ± 4 nm standard polystyrene latex spheres in 1mM NaCl. Using the manufacturer’s nominal concentration, the spheres were mixed in a number ratio of 5:1. The sample temperature was 298.15 K and MDLS measurements at four angles (50°, 70°, 90° and 110°) and six angles (30°, 50°, 70°, 90°, 110° and 130°), respectively. The recovery of the 300/502 nm bimodal PSD is shown in Fig. 6 and the corresponding performance indices are shown in Table 7.
As shown in Fig. 5 and Table 6, the results obtained using the AWSU method are better than CW-CR method at the same number of scattering angles. The peak position with the AWSU method is closer to the true value, and the peak position is 300/510 nm and the peak height ratio is 6.05:1 using four-angle analysis. Moreover, according to Table 6, the recovery of the AWSU method with four scattering angles is better than the results with the CW-CR method at six scattering angles, and the peak positions and relative peak heights are closer to the true values.
From the results with simulated and measured data, it can be seen that both CW-CR method and AWSU method show similar behaviour: with an increase in the number of measurement angles, the accuracy of the results gradually improves. It appears that the results obtained by the AWSU method using four measuring angles are better than those obtained by the CW-CR method using six measuring angles. To compare the difference in the results with the increase in the number of scattering angles, at 3 × 10−3 noise level, the PSD recovery error, V, and the relative peak position error, EPP, with the increase of scattering angle are calculated using CW-CR method and AWSU method to recover the PSD for a 400/700 nm bimodal particle mixture as a function of the number of scattering angles used. These results are shown in Figs. 6, 7 and 8. The PSD recovery error results using the CW-CR method is shown in Fig. 6(a), and the results using the AWSU method are shown in Fig. 6(b). From Fig. 6(a) and Fig. 7, it can be seen that the PSD recovery error and the relative peak position error of the two peaks decrease with an increase in the number of scattering angles. The PSD recovery error and the relative peak position error reach their minimum at six scattering angles, where V is 0.058 and the EPP are 0.010/0.006. When the number of scattering angles six, the PSD recovery error and the relative error of peak position of the two peaks start to increase again. This phenomenon is completely consistent with our previous observation  that the accuracy of MDLS PSD recovery results does not improve continuously with increasing number of scattering angles, but there is an optimal number of angles - usually six scattering angles. When the number of angles exceeds six, the results become worse. The reason is that more angle errors and measurement noise are introduced as the number of scattering angles is increased but relatively less information about the angular dependence of the light scattering (i.e. Mie scattering) is gained at larger scattering angles. Figures 6(b) and 8 show that the PSD recovery error and the relative error of peak position for the AWSU results show a decrease as do the CW-CR results as the number of scattering angles begins to increase. However, for the AWSU method these errors reach a minimum of around four angles compared with six for the CW-CR method. For the AWSU method these minima are V is 0.043 and EPP are 0.008/0.006. The AWSU method successive updates the angular weighting according to the number of angles. Every update of angular weighting corresponds to one ACF denoising, and the every angular weighting will be recalculated according to the results of subsequent denoising and weighting coefficient. In addition, the information character weighting matrix is reconstructed according to the updated angle weighting coefficient, which not only reduces the influence of noise on the angle weighting coefficient in ACF, but also improves the utilization of PSD information in the inversion process.
In order to compare the information utilization of the two methods, also take 400/700 nm bimodal particle are taken as an example, the utilized particle size information distribution (UPSID) of CW-CR method and AWSU method at 50°, 70°, 90° and 110° scattering angles is calculated, as shown in Figs. 9(I), 9(II), 9(III) and 9(IV). UPSID is the difference between PSID and unutilized particle information distribution (the difference between the true and the reconstructed ACF). From Figs. 9(I), 9(II), 9(III) and 9(IV), it is clear that in the ACF data the PSD information utilized by AWSU method is more than that by CW-CR method from all angles.
Compared with routine angular weighting methods such as CW-CR, the AWSU method not only reduces the number of scattering angles needed, but also avoids an excessive number of scattering angles worsening fitting errors and peak position errors for bimodal PSDs. It is generally believed that increasing the number of scattering angles can provide more measurement information, but it can also introduce measurement noise and angle error, which results in an excessive number of scattering angles leading to worse measurement results . However, Figs. 6 and 7 show that the negative effect of increasing the number of scattering angles can be offset by the effect of the angular weighting calculation algorithm. It seems unlikely that this can be explained solely by improving the accuracy of the weightings. In fact, from the above analysis, we can see that the effect of successive updating of the angular weighting on MDLS is not limited to the accuracy of the weighting calculation, but also effects signal denoising and PSD information utilization by successive reconstruction of the information character weighting matrix. As the number of signal denoising and information weighting increases with the number of scattering angles, the side effects caused by the increase in the number of angles are significantly weakened.
In MDLS, the increase in the number of scattering angles increases the particle size information, so as to improve the measurement accuracy. However, an excessive number of scattering angles may introduce more measurement noise and angular errors, thereby reducing or even offsetting the benefits from the increased number of scattering angles. The number of angles used depends on the balance of benefits and losses from the increased number of angles. Usually, the best measurement results require six scattering angles. This work shows that the increase of PSD information in MDLS depends not only on the increase of the number of angles, but also on the calculation of the angular weightings. An accurate weighting algorithm can reduce the number of scattering angles used. In MDLS measurement, using the AWSU method for PSD recovery with 4 scattering angles can give better results than the usual method with 6 scattering angles. This means that in practical application, the measuring device or the measurement procedure can be simplified. Moreover, the algorithm for calculating angular weightings is not restricted to the contribution to the accuracy of weightings. It also contributes directly to the particle size inversion by reconstructing the ACF with the PSD obtained by weighted inversion. The combination of these two contributions both reduces the number of scattering angles needed, and keeps the PSD inversion results from getting worse because of an excessive number of scattering angles.
The Natural Science Foundation of Shandong Province (ZR2018MF032, ZR2018PF014, ZR2017MF009, ZR2017LF026).
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