Information-weighted constrained regularization for particle size distribution recovery in multiangle dynamic light scattering

In particle size measurement with dynamic light scattering (DLS), it is difficult to get an accurate recovery of a bimodal particle size distribution (PSD) with a peak position ratio less than ~2:1, especially when large particles (>350nm) are present. This is due to the inherent noise in the autocorrelation function (ACF) data and the scarce utilization of PSD information during the inversion process. In this paper, the PSD information distribution in the ACF data is investigated. It was found that the initial decay section of the ACF contains more information, especially for a bimodal PSD. Based on this, an information-weighted constrained regularization (IWCR) method is proposed in this paper and applied in multiangle DLS analysis for bimodal PSD recovery. By using larger (or smaller) coefficients for weighting the ACF data, more (or less) weight can then be given to the initial part of the ACF. In this way, the IWCR method can enhance utilization of the PSD information in the ACF data, and effectively weaken the effect of noise at large delay time on PSD recovery. Using this method, bimodal PSDs (with nominal diameters of 400:608 nm, 448:608 nm, 500:600 nm) were recovered successfully from simulated data and it appears that the IWCR method can improve the recovery resolution for closely spaced bimodal particles. Results of the PSD recovery from experimental DLS data confirm the performance of this method. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (290.3200) Inverse scattering; (290.5820) Scattering measurements; (290.5850) Scattering, particles; (290.1990) Diffusion; (290.3700) Linewidth. References and links 1. A. Jillavenkatesa, S. J. Dapkunas, and L. S. H. Lum, Particle Size Characterization (Helsingin Yliopisto, 2001). 2. I. D. Morrison, E. F. Grabowski, and C. A. 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Introduction
The particle size distribution (PSD) is an important characteristic of many colloidal materials [1]. Dynamic light scattering (DLS) has been widely used in science and industry for measuring the PSD of submicrometer particles [2]. In this technique the autocorrelation function (ACF) of scattered light intensity is measured using the photon correlation spectroscopy (PCS) method [3,4] and the PSD is determined by analyzing the ACF data. However, inverting the ACF to recover the PSD is a highly ill-posed mathematical problem, and the results can be variable due to the limited PSD information using the common single angle DLS technique. Noise in the ACF data also reduces the robustness of the PSD recovery.
To improve the accuracy of the PSD recovery, multiangle dynamic light scattering (MDLS), which combines ACF data sets detected at different scattering angles and can provide extra information than single-angle analysis, has been developed [5][6][7][8]. Numerous approaches have been proposed to solve the inverse problem based on MDLS. These include the regularization method [9,10], the CONTIN constrained regularization method [11,12], Bayesian strategies method [13,14], and the neural network method [15], for example. These methods generally work well for unimodal PSDs, but less well for bimodal PSDs with a peak position ratio less than ~2:1, although bimodal PSDs with a peak position ratio larger than ~3:1 can be recovered accurately with the CONTIN method [16,17]. Zhu et al. [18] proposed a weighted constrained regularization (WCR) method, and applied it to single-angle DLS measurements, in which a bimodal PSD with peaks at 200nm and 350nm was successfully recovered giving resolution for a peak position ratio of ~1.7:1, and in spite of considering particle sizes <350nm.
There are two factors that limit acceptable PSD recoveries in DLS: i) the low information content on the PSD in the ACF data; and ii) the unavoidable presence of noise in the ACF data [19]. To overcome these problems, a novel information-weighted constrained regularization (IWCR) method is proposed in this paper, using the distribution of PSD information in the ACF. In this method, the noise and the PSD information in the ACF are considered simultaneously and applied in MDLS analysis (M = 1, 3, 6). The results show that there are distinct and significant advantages using this method in MDLS analysis for recovering bimodal PSDs. It appears that this method can effectively reduce the effect of the noise in the ACF data, enhancing the ability to extract PSD information as well as improving the capacity to discriminate the peaks of bimodal distributions. Results of PSD recovery from experimental data support these observations and provide the experimental basis for the feasibility of this method.

Basic DLS theory
When a laser beam is focused into a dilute suspension of colloidal particles, light is scattered in all directions. If the scattered light is detected over a small solid angle [20] at a particular angle, it is observed to undergo light intensity fluctuations, which are due to the Brownian motion of the particles. The timescale of the fluctuations relates to the diffusion coefficient of the particles in the medium, which in turn is related to their sizes. The fluctuations are characterized by measuring the ACF of the scattered light intensity [20], as follows: by the Siegert relationship [20,21] Vol. 26, No. 1 | 8 Jan 2018 | OPTICS EXPRESS 17 where B is the measured baseline of ( ) ( ) 2 j G θ τ , β(θ) (≤1) is the instrumental coherence constant, and j (1≤ j ≤M) is the channel number. For a polydisperse system of colloidal particles, the normalized electric field ACF has the form where θ, d i , k B , T, λ 0 , n m (λ 0 ), and η are the scattering angle, the diameter of spherical particle, the Boltzmann constant, the absolute temperature of the medium, the wavelength of the incident light, the refractive index of the non-absorbing suspending medium, and the viscosity of the dispersing medium, respectively. The decay constant, Γ, of ( ) ( )

Noise in ACF data
Generally, the scattered light signals detected in DLS are weak (usually less than ~10 −15 W) and may be extremely weak when data are recorded for short times or low count rates. Such situations arise in evolving systems or online applications where measurement time may be limited. Although the PCS technique can reduce the noise with time averaging to improve the statistics [23], the noise mixed in the scattered light signals can still make the intensity ACF inversion uncertain. So an understanding of the noise in ACF data is of importance for improving the performance of the DLS.
In 1983, Schatzel analyzed the sources of noise in ACF data [24]. In his classification, there are two types of noise, intensity noise and photon noise [24,25]. Other types of noises (e.g., those inherent to the light source) are typically negligible particularly for well-stabilized laser sources. The intensity noise, which can be characterized as Gaussian under certain assumptions, is related to the scattered light signal, while the photon noise, is Poissonian and inherent to the photon counting process. The intensity noise is strongly correlated and may dominate the photon noise after auto-correlation calculation [26,27] and give rise to an intensity ACF baseline error which causes the estimated electric field ACF data to deviate from the theoretical data, especially at large lag time. In DLS measurement, it is impossible to avoid this distortion although we can get a less biased estimation [28].
Taking the experimental data of sample AE (details in Section 7) as an example, the normalized light intensity and electric field ACF data at 90°, G(τ) and g(τ) are shown in the Figs. 1(a) and 1(b) respectively. It can be seen that the effect of noise on fitted data is worse in G(τ) and g(τ), especially at large delay time.
To further study the noise in ACF data, wavelet analysis, a common method to extract the noise mixed in noisy data, was used to extract the noise distribution n I (τ) and n f (τ) from the noisy ACF data G(τ) and g(τ) (Figs. 1(c) and 1(d)). The db3 wavelet, one of Daubichies wavelets, was used to compute the detail coefficients vector (cDg), via a single-level wavelet decomposition of G(τ) (or g(τ)) based on a one-dimensional wavelet transform (i.e. dwt in MATLAB software). The detail coefficients, substantially the noise distribution n I (τ) (or n f (τ)), were reconstructed with cDg using the db3 wavelet and a direct reconstruction function (i.e. upcoef in MATLAB software). As shown as Figs. 1(c) and 1(d), it is obvious that the noise in the light intensity ACF is not significantly different at different delay time. However, for the electric field ACF, larger noise occurs at long delay times, as a consequence of square root transformation required to calculate the electric field ACF ( ) ( ) Actually, the ACF data at small delay times contain the measurement information used for recovery of the PSD, while very little or no PSD information is contained in the ACF data at large delay times [19,29,30]. Additionally, at the large delay times, much more noise is mixed in the ACF data than at the small delay times and this seriously affects the PSD recovery. The WCR method, which uses the ACF weighted with the noise distribution, can reduce the effect of noise in ACF data [18], but the lack of the information on closely spaced components in bimodal samples, needed in the PSD recovery, remains a shortcoming. Fig. 1. Normalized light intensity ACF data (a), G(τ), obtained from experimental sample AE at 90°scattering angle; Normalized electric field ACF data (b), g(τ), calculated from G(τ); The noise in G(τ) (c) and g(τ) (d), n I (τ) and n f (τ), extracted by wavelet analysis.

PSD information in ACF data
The ACF data contains different amounts of the PSD information at the different delay times, and more PSD information content usually exists at the initial damping section. Generally, the fact of neglecting the PSD information distribution in the common constrained regularization (CR) method or the WCR method, can limit the PSD recovering efficiency.
Here, five simulated DLS data sets with different mean diameter (95nm, 188nm, 350nm, 500nm, 800nm) are used to investigate the PSD information distribution in the ACF data and, the relationship between the PSD information and the recovered PSD. For the simulation, unimodal and bimodal PSDs are used to generate the corresponding ACF data, and the properties of each data set are shown in the Table 1.
All simulations assumed that the particles were dispersed in water at 298 K (refractive index n = 1.33, viscosity η = 0.89 cP). The light source was vertically polarized with wavelength λ 0 = 633nm. The ACF data were simulated by Eq. (4) at a scattering angle of θ = 90°, adding Gaussian random noise to make the simulated ACF data more realistic.
The noisy light intensity ACF data were simulated by where n I (τ j ) is the zero-mean Gaussian random noise, and δ denotes the noise standard deviation.
Taking the PSD with a mean diameter of 95nm as an example, a unimodal PSD and three bimodal PSDs (25/300nm 50/200nm 75/125nm), with peak position ratio of 12:1, 4:1 and 1.67:1 respectively, were used to simulate the ACF data. Figure 2(a) shows the simulated light intensity ACF data for the respective PSD. It can be seen that the light intensity ACF data from the 95nm unimodal PSD and the 75/125nm bimodal PSD are quite similar, which makes it highly difficult to discriminate these two sets of light intensity ACF data.
To make the difference between the ACFs more clear, we define the particle size information distribution for bimodal PSDs of Fig. 2(b) as the absolute value of the difference between each datum point of the bimodal ACFs and the ACF corresponding to the equivalent monodisperse PSD (e.g. the 95 nm unimodal ACF in this case). Here, the equivalent monodisperse PSD was obtained by the cumulants method [31,32]. It can be seen that, as the peak position ratio decreases, the difference between the ACFs decreases and less particle size information is provided as the bimodal PSDs approach a unimodal PSD. So it is increasingly difficult to recover this kind of bimodal PSD.
To illustrate the limitation of the common CR method, the PSD estimations were made by this method and the results are shown in Table 1 and Fig. 2(c). It is easy to see that an accurate recovery of bimodal PSDs was reached when the peak position ratio is > 2:1, such as the 25/300nm and 50/200nm bimodal PSDs, but only one peak was found for the 75/125nm bimodal PSD, with the peak position ratio of 1.67:1.
Simulations were also made for PSDs with different mean diameters (188nm, 350nm, 500nm, 800nm), and the results are shown in Figs. 3-6. In all cases, the less PSD information in the ACF data, the worse the PSD recovery. This effect was particularly evident in the bimodal PSDs with a peak position ratio < 2:1. Clearly, it is critical to make the best use of the PSD information in the ACF data during the inversion to improve the recovery of bimodal PSDs with close peaks.    . Simulated light intensity ACF data for the PSDs with 500nm mean diameter (a); PSD information distribution in the simulated ACF data for the bimodal PSDs (b); PSDs estimated from the simulated ACF data by using the common CR method (c).
Simulated ACF PSD information PSD Fig. 6. Simulated light intensity ACF data for the PSDs with 800nm mean diameter (a); PSD information distribution in the simulated ACF data for the bimodal PSDs (b); PSDs estimated from the simulated ACF data by using the common CR method (c).

IWCR method
To increase the ability to extract the PSD information, the here-proposed IWCR method is applied in MDLS. This method basically consists on assigning a different weight to each data in the ACF, according to the corresponding information content on the PSD. For single-angle DLS (SDLS), the IWCR method can be expressed as a weighted constrained optimization problem, as follows: where M α is the objective functional to be minimized, f the estimated PSD, α the regularization parameter, L the regularization matrix, · the Euclidean norm, and Lf the penalty functional factor. These parameters guarantee the optimal stable and smooth solution.
The weighting matrix W θ in Eq. (10), is described by W θ = diag(w θ ). Here diag is the diagonal operator and w θ is a vector of weighting coefficients, w θ_j , that can be calculated by: where ( ) 1 _ j g θ is the j th element of ( ) 1 θ g , and PID θ_j is the j th element of the PSD information distribution in the ACF data. This information distribution at scattering angle θ, PID θ , substantially is the absolute value of difference between the ACF corresponding to the equivalent monodisperse PSD and actual ACF. Here, the equivalent monodisperse PSD was obtained by the cumulants method [31,32].
In the IWCR method, a larger (smaller) value of w θ is generally used to weight the ACF data which has more (less) PSD information. In this way, it is possible to provide more precise PSD information contained in the ACF to improve the PSD recovery. The effect of noise at large delay times is also weakened by using smaller weighting coefficients at those delay times.
In MDLS analysis, the estimated ACF data obtained at different scattering angles is treated as one set. Correspondingly, with IWCR method Eq. (7) is expressed as, where ( ) is a vector set containing the weighted electric field ACF data at scattering angles θ 1 , θ 2 , …, θ m , as described by the second member of Eq. (12).
In what follows, the simulated and experimental PSDs are recovered by the proposed IWCR method (W2) with single-angle or multi-angle ACF data and compared with PSDs recovered using the standard CR method (W0) proposed by Tikhonov et al. [33] and the WCR method (W1) proposed by Zhu et al. [18]. The following four indices are used to measure the relative performance of the methods: the peak position (PP), the relative error of peak position (E PP ), the peak height ratio of bimodal PSD (R PH ), and the rms PSD recovery error (V). The index PP here is the particle diameter at the PSD maximum. The other indices are defined as follows: where HIP stands for the value of the PSD at PP; and K is the number of modes in the PSD.

Simulation and results
Using a combination of two PSDs from Johnson's S B distribution [34], a realistic bimodal PSD was simulated as where t = (d-d min )/(d max -d min ), and μ 1 , σ 1 , μ 2 , σ 2 , a, and b are distribution parameters used to get different PSDs. Parameter d is the particle diameter, d min and d max are the maximum and minimum particle diameter respectively, and t is the normalized particle diameter.
The intensity ACF data without noise can be obtained using Eqs.
(3) and (4). Three sets of PSD and their corresponding ACF data were simulated with the parameters shown in Tables 2 and 3. Parameters d 1 and d 2 in Table 3 are the diameters at the maximum of each peak. And Gaussian random noise was added to the simulated data to make it more realistic, using the form in Eq. (9). The levels of noise are shown in Table 3. All ACF data were simulated with k B = 1.3807x10 −23 J/K, T = 298.15 K, η = 0.89 cP, n m = 1.3316, λ 0 = 632.8 nm, θ r = 30°, 40°, 50°, …., 140°; B = 1, and β = 0.7; d min = 0.01nm and d max = 800.01nm for Sample AS and BS, d min = 300.01nm and d max = 800.01nm for Sample CS. The number of points was 100.
Using M-angle DLS (M = 1, 3, 6), recovery of the simulated PSDs was done by analyzing the noisy simulated ACF data with different constrained regularization methods (W0, W1 and W2). SDLS was simulated at 90°; 3-angle DLS was simulated at 70°, 90° and 110°; 6-angle DLS was simulated at 30°, 50°, …, 130°. Recovered PSDs are shown in Figs. 7-9, and the values of the different performance indices are shown in Tables 4-6. For the simulated bimodal PSD containing 400:608 nm spheres with 1:1.046 peak height ratio (sample AS), the recovered PSDs are shown in Fig. 7 separately using single-angle, 3angle and 6-angle analysis. Table 4 shows the performance indices of the true and recovered PSDs.
According to Fig. 7, a better estimation of the 400nm and 608nm peaks was obtained with the W2 method. Also, the recovered PSDs were closer to the true PSDs with increasing number of scattering angles. The ability to recognize the bimodal PSD of the W0 method was very poor for either SDLS or MDLS, although the latter reach more accurate peak position. Using multi-angle methods, particularly 3-angle analysis, the recovered peak height ratios were closer to the true value (1:1.046) with different inversion methods (particularly, W2), which was 1:1.105. The recovered peak position was also more accurate for both the 400 nm and 608 nm peaks. In fact, their relative errors were 0.023 and 0.006 respectively, when analyzing the 3-angle DLS data with the W2 method.
For the simulated bimodal PSD containing 448:608 nm spheres with a 1:0.845 peak height ratio (sample BS) the recovered PSDs are shown in Fig. 8. Table 5 shows the performance indices of the true and recovered PSDs.
A better estimation of the 448 nm and 608 nm peaks was extracted with 3-angle analysis, especially when using the W2 inversion method. With SDLS, this PSD could not be recovered well when compared with MDLS, particularly using the W0 method. In addition, small, spurious 200 nm and 170 nm peaks were obtained by the W0 or W1 methods in 6angle DLS. In contrast, the W2 inversion method did not produce these spurious peaks. According to Table 5, better results are obtained when using 6-angle DLS data with the W2 method. Here the relative peak position errors of the 448 nm and 608 nm peaks were 0.026 and 0.006 respectively, the recovered peak height ratio was 1:1.083, and the value of V was 0.067.     Figure 9 and Table 6 show the results for the simulated bimodal PSD containing 500:600 nm spheres with a 1:0.921 peak height ratio (sample CS). The PSDs recovered are shown in Fig. 9. In the single angle analysis, there was no significant deference between the recovered PSD with the W1 and W2 method. Obviously the recovered PSDs from these two methods are more close to true PSD than the recovery with W0 method. As increase of the scattering angles, the W2 method could give better recovered PSD, not only than W0 method, but also than W1 method.
The most accurate peak positions were obtained using 3-angle analysis and the W2 method where the relative peak position errors were 0.00 and 0.008 for the 500 nm and 600 nm peaks respectively, and the recovered peak height ratio was 1:0.609, close to the true value 1:0.921.

Experimental section
Real experimental data corresponding to three bimodal samples (AE, BE, CE) were used to evaluate the proposed method. Samples AE and BE were both obtained by mixing 306 nm ± 8 nm and 974 nm ± 10 nm standard polystyrene latex spheres (Polyscience Inc.), all of which were diluted into the distilled and deionized water. For sample AE, the number concentrations of the small and large particles were 98.95% and 1.05% respectively (which correspond to 50% in weight for each mode). For sample BE, the number concentrations of the small and large particles were 97.90% and 2.10% respectively (which correspond to 34% and 66% in weight for each mode). Sample CE was obtained by mixing 300 nm ± 3 nm and 502 nm ± 4 nm standard polystyrene latex spheres (Duke Scientific) in 1mM NaCl in a number ratio of 5:1, based on the manufacturer's nominal concentration. For all bimodal samples, the regulated sample temperature was 298.15 K, and the dispersion medium refractive index (n m ) was 1.33.
MDLS measurements were made with a general purpose laser light scattering photometer (Brookhaven Instruments, Inc.), fitted with a 20 mW vertically polarized He-Ne laser (λ 0 = 632.8 nm), a photomultiplier tube as a detector, a stepper-motor controlled goniometer (model BI-200SM, Brookhaven Instruments Inc.), and a digital correlator (Brookhaven Instruments Inc. model BI-2000 AT for sample AE and BE, model BI-2030AT for sample CE). The intensity ACF data, from sample AE and BE, were recorded at twelve detection angles (R = 12): [30°, 40°, 50°, 60°, 70°, 80°, 90°, 100°, 110°, 120°, 130° and 140°]. For sample CE, the MDLS was carried out in the detection angle range 30° to 120° at intervals of 10°. The angular alignment of the goniometer was checked prior to all measurements. Total measurement times ranged from 200 s to 500 s and a single-mode fiber-optic probe was used to collect the scattered light, which can ensure high-coherence ACF data [35].
For the experimental bimodal PSD containing 306:974 nm spheres with a 1:1 peak height ratio (sample AE), the recovered PSDs are shown in Fig. 10. The different performance indices of the true and recovered PSDs are shown in Table 7. With these values, we can compare the recovery results obtained by different analysis methods.
For the 360 nm and 974 nm peaks, each of the MDLS analyses successfully obtained a good estimation of the PSD with the 6-angle DLS solution being more reliable, especially for the W2 method. Furthermore, the height ratio of the peaks recovered with the W2 method is much closer to the true ratio than with either the W0 or W1 methods. Also the peak positions were more accurate, with relative errors of 0.003 and 0.006 for the 360 nm and 974 nm peaks respectively, when we used 3-angle DLS and the W2 method. The recovered peak height ratio in this case was 1:1.06, very close to the true value of 1:1.
For the bimodal PSD containing 306:974 nm spheres with a 1:2 peak height ratio (sample BE), the recovered PSDs are shown in Fig. 11. The different performance indices are shown in Table 8.
In all cases, a bimodal PSD is extracted but they are all quite different. The recovered peak positions with the W2 method are much closer to the true positions than obtained with the W0 and W1 methods. The PSDs recovered from single-angle analysis resolve the large size peak poorly. In Fig. 11(b) (3-angle MDLS) the recovered PSDs are also poor, but the recovered peak height ratio is much closer to the true value than with SDLS. We can see also that a small, spurious 500 nm peak in the PSD is recovered by the W0 and W1 methods for 3angle MDLS. Furthermore, 3-angle and 6-angle analysis respectively obtained a closer recovered peak height ratio and a small relative error of peak position, when using the W2 method.
The PSDs, shown in Fig. 12, are the recovery results of the bimodal PSD containing 300:502 nm spheres with a 5:1 peak height ratio (sample CE). The performance indices are shown in Table 9. Clearly, bimodal PSDs are obtained in all cases, but they are all quite different. The recovered peak positions and peak height ratios with the W2 method are all much closer to the true value than obtained with the W0 and W1 methods, especially in the MDLS measurement. The results suggest that the 6-angle DLS solution with the W2 method is more reliable, with smaller relative errors of 0.000 and 0.016 for the 300 nm and 502 nm peaks respectively and a closer peak height ratio of 4.914:1 to the true value of 5:1.

Conclusions
Under some conditions, particularly with bimodal samples, the resolution of the DLS particle sizing measurement is limited by the noise mixed in the ACF data and the inherently lower information content of the data, as well as low PSD information utilization during the inversion process. Generally, the effect of noise can be reduced by optimizing the DLS measurement system and more robust PSD information can be provided by MDLS. A method to effectively extract and make the best use of the PSD information hiding in the noisy data is critical for PSD recovering methods.
MDLS can provide more PSD information than SDLS and, hence improve the measurement of bimodal PSDs. However, this improvement is restricted by the data inversion routine. Incorporating weighting by the noise distribution, the WCR method can enhance the ability to restrain the noise in ACF data and further improve the measurement. However, the improvement is still limited by the scarce utilization of PSD information during the inversion process.
This paper shows that there are distinct and significant advantages for recovering bimodal PSDs with IWCR data analysis combined with MDLS. The IWCR method can enhance utilization of the PSD information in the ACF data, and effectively weaken the noise in the ACF data at large delay time. Consequently, the IWCR method can provide considerably more accurate PSDs than other common CR methods, and is more able to resolve closely spaced components. By using this method, an accurate PSD recovery is possible for a peak position ratio of 1.2:1, containing large particles (≥ 350nm).

Funding
The Natural Science Foundation of Shandong Province (ZR2014FL027, ZR2016EL16); Key Research and Development Program of Shandong Province(2017GGX10125); The Shandong University of Technology and Zibo City Integration Development Project (2016ZBXC097).