Analysis of noisy multi-angle dynamic light scattering data Shanshan Gao,1 Jin Shen,1,* John C. Thomas,1,2 Yingying Xiao,1 Zuoming Yin,3 Wei Liu,1 Yajing Wang,1 and Xianming Sun1 1
School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255049, China 2
Group Scientific Pty Ltd., P.O. Box 190, Salisbury South, SA 5106, Australia
3
School of Civil and Environment Engineering, University of Science and Technology Beijing, Beijing 100083, China *Corresponding author: [email protected] Received 2 June 2014; revised 6 August 2014; accepted 8 August 2014; posted 11 August 2014 (Doc. ID 213172); published 8 September 2014
In multi-angle dynamic light scattering measurements, due to the inevitable presence of baseline measurement noise, the normalized intensity autocorrelation function (ACF) data deviates from the true value. This leads to incorrect angular weighting estimates, which affect the accuracy of inversion results and determination of particle size distributions (PSDs). We outline a method to calculate better angular weighting coefficients from the noisy intensity ACF data. The method involves first compensating for the baseline error in the ACF data and then determining the weighting coefficients. We demonstrate the method using simulated ACF data containing baseline error for unimodal and bimodal PSDs and also for experimental data for unimodal and bimodal samples. For the unimodal PSDs ACF data were simulated for 100–900 nm and 100–650 nm particle size ranges, and for bimodal PSDs 360–900 nm and 100–900 nm particle size ranges were used. The performance of our method was shown by comparing the results of weighting coefficient and PSD determination with and without baseline compensation to the known coefficient values and PSDs. With baseline compensation the relative error of the weighting coefficients decreased significantly. Furthermore, with baseline compensation, the PSD results for the four groups of simulated data were improved. The deviations between the known and recovered PSDs were decreased, the relative error of peak position obviously decreased, and the occurrence of false peaks was reduced. The PSD results from the experimental data further validates the conclusion that the method proposed apparently reduces the relative error of peak position, effectively eliminates the false peak, and improves the accuracy of the recovered PSD. © 2014 Optical Society of America OCIS codes: (290.3200) Inverse scattering; (290.4020) Mie theory; (290.5850) Scattering, particles; (290.5890) Scattering, stimulated. http://dx.doi.org/10.1364/AO.53.006001
1. Introduction
Dynamic light scattering (DLS) is a widely used technique for particle size and particle size distribution (PSD) determination in the submicrometer range [1,2]. The experimental setup is simple; laser light is focused into a dilute suspension of colloidal particles, and a photometer collects the light scattered at a particular angle. Brownian motion of the particles 1559-128X/14/266001-07$15.00/0 © 2014 Optical Society of America
induces temporal fluctuations in the scattered light, and a digital correlator calculates the autocorrelation function (ACF) of the scattered intensity to characterize the fluctuations. The PSD is then extracted from the measured intensity ACF by an inversion algorithm. Data analysis to recover the PSD is problematic due to the relatively low information content inherent in the measured signal. The multi-angle DLS (MDLS) technique can provide more accurate PSD information than singleangle DLS [3,4]. In the MDLS data analysis, light intensity ACFs for each angle are combined into a 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6001
single data analysis with proper angular weighting coefficients, which can be estimated by the light intensity mean value, baseline of the ACF, or other methods [5]. However, inaccurate estimation of the weighting coefficients will affect the accuracy of the recovered PSD significantly. Bryant and Thomas [3] combined static light scattering (SLS) and MDLS to get more information and improved the accuracy of the recovered PSD. However, SLS measurement requires a precision angular measurement system. Bryant et al. [4] put forward a method of using an iterative algorithm post measurement to reconstruct the SLS data and estimate MDLS weighting coefficients. This avoids measuring high-quality SLS data at different scattering angles. The inversion results for intensity ACF data with 0.01% noise measured at five scattering angles gave good PSD estimation. However, when the noise level is higher, 10 or more scattering angles are needed for reliable PSD determination. Subsequently, a neural network algorithm was applied to MDLS data analysis by Gugliotta et al. [6], and a Bayesian inversion algorithm was proposed by Clementi et al. [7]. Recently, a new method for determining angular intensity weighting in MDLS using an iterative recursion method in MDLS was developed by Liu et al. [8]. The method worked well but did not consider baseline error in the ACF. In MDLS, the precision of PSD determination depends on accurate estimation of the angular weighting coefficients. Baseline error in MDLS experiments affects PSD determination through two mechanisms. In a single-angle DLS experiment, baseline error distorts the ACF, and if this is not compensated, the resulting PSD will be distorted. In an MDLS experiment baseline error distorts the ACF, just as for DLS, but it also causes errors in the angular weights because the baseline is also used to estimate them. In this paper we propose an angular weighting coefficient estimation technique for MDLS which reduces the effect of ACF baseline errors. First, baseline error compensation [9] is applied to the ACFs at each scattering angle and the ACFs are normalized using the compensated baselines; then the average light intensity is calculated from the compensated baselines and used to estimate the angular weighting coefficients. 2. Theory of MDLS
In a single DLS experiment, measurements of the light intensity ACF are taken at one scattering angle. However, the particles in the sample cell scatter light in all directions, so to get more information, MDLS is proposed. At a given scattering angle θr of the MDLS measurement, the DLS data recorded is the ACF of the scattered light intensity, G2 θr τj . This is related to the normalized ACF of the electric field, g1 θr τj i through [10] 2 1 2 G2 θr τj G∞;θr 1 βjgθr τj j ;
r 1; 2; …; R 6002
and
j 1; 2; …; M:
(1)
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
Here τj is the delay time, G2 ∞;θr is the intensity ACF baseline, β≤1 is an instrumental coherence parameter, and M is the total number of correlator channels or points of the ACF measured at θr . If the PSD is described by a number distribution f Di of particle sizes, Di , the electric field ACF is g1 θr τj kθr
N X i1
exp−Ω0 θr τj ∕Di CI;θr Di f Di ; (2)
with Ω0 θr
16πkB T nm λ 2 2 θr : sin 2 3η λ0
(3)
Here kθr are a priori unknown weighting coefficients that adopt different values at different θr and are the inverse of the total intensity scattered at each angle. CI;θr Di is the fraction of light intensity scattered by a particle of diameter Di at θr and is calculated through Mie scattering theory [11]. λ0 is the in vacuo wavelength of the incident laser light, nm λ is the refractive index of the nonabsorbing suspending liquid, kB is the Boltzmann constant, T is the absolute temperature, and η is the medium viscosity at temperature T. We define the dimensionless weighting ratio kθr relative to a fixed reference angle θ1 30°, as [5]: kθr
2 1∕2 G∞;θ1 kθ r hI θ i 1 2 hI θr i kθ 1 G∞;θr
r 1; 2; …; R:
(4)
Equation (4) shows that the weighting ratios kθr can be determined from the autocorrelation baseline G2 ∞;θr or the mean light intensity hI θr i. However, this uses the experimental ACF values, which may contain sufficient noise to corrupt the determination of the weighting ratios. The theoretical intensity ACF baseline is B G2 ∞;θr . Normalization errors occur in the normalized intensity ACF when there is an error in the baseline ˆ hnti2, where nt is the number of estimator B photon counts at sample time t. Writing the baseline ˆ − B, the intensity ACF with baseline error as ΔB B error is [12] g2 θr τj
2 ΔBθr Gθr τj 1− ˆθ hntθr i2 B r G2 θr τj
2 ΔBθr Gθr τj − : ˆ θ hntθ i2 hntθr i2 B r r
(5)
2 ˆ Here the second term, g2 cθr τj ΔBθr ∕Bθr G∞;θr τj ∕ hntθr i2 , is the normalization error at scattering 2 angle g2 θr τ j gcθr τj and causes distortion of the recovered PSD. Zhu et al. [12] have shown that this value can be determined and used to compensate the normalized intensity ACF so that it becomes
2 g2 θr τj gcθr τ j , and the compensated mean light intensity is then
hI cθr i
M g2 τ g2 τ X j j θ cθ r
r
M
j1
× exp−0.5μ σ lnt∕1 − t2 :
hI cθ1 i : hI cθr i
(7)
The intensity ACFs at selected angles can be combined with the angular weighting coefficients to form the MDLS data sets, which can be analyzed to determine the underlying PSD. We have shown previously that a modified Chahine method [13–15] performs well in recovering both unimodal and bimodal PSDs from MDLS data and have used that method as the inversion algorithm in this paper. The Chahine method is an iterative method wherein the distribution function is given by PM j1 uj;i gmeas;j p−1 p fi P f ; (8) M u gp−1 i j1 j;i calc;j where f p is the component of the PSD due to the i discrete diameter Di after the pth iteration. gmeas is p−1 the measured ACF data, and gcalc is the calculated ACF data obtained from the p − 1th iteration, p−1 gcalc;j
N X i1
hj;i f ip−1 ;
(9)
p−1 p−1 are the elements of the vector gcalc , where gcalc;j and uj;i hj;i ∕ h exp−Ω0 θr τj ∕Di CI;θr Di , Pj;iN h . After each iteration the root-mean-square j;i i1 error between gp calc;j and gmeas;j is calculated as
RESMP
1∕2 M 1X p g − gcalc;j ; M j1 meas;j
(10)
and when it satisfies j1 − RMSEp ∕RMSEp−1 j ≤ 1 × 10−5 , the iteration is halted, and f f p is the resulting PSD function. 3. Simulation Results and Discussion
The simulated unimodal PSDs by number were described by Johnson’s SB function [16]: D t1 − t−1 f D p 2π Dmax − Dmin × exp−0.5μ σ lnt∕1 − t2 ;
D1 p t1 − t−1 2π Dmax − Dmin
(6)
This can then be used in Eq. (4) to calculate compensated angular weighting coefficients as kcθr
D f D p t1 − t−1 2π Dmax − Dmin
(11)
and the bimodal PSDs were a combination of two Johnson’s SB functions:
× exp−0.5μ1 σ 1 lnt∕1 − t2 ;
(12)
Here, t D − Dmin ∕Dmax − Dmin ; μ, σ, μ1 , and σ 1 are distribution parameters; t is the normalized particle size; and Dmax and Dmin are the maximum and minimum particle size, respectively. The unimodal and bimodal electric field ACF data were generated by substituting f D into Eq. (2). Then the noise-free intensity ACF data, g2 θr τ, were produced through Eq. (1), and noise was added to the noise-free correlation data to simulate the measurement data as follows [17]: 2 g2 N;θr τ gθr τ δε:
(13)
Here ε is a normally distributed number in the range 0–1, and δ denotes the noise level. The uncompensated weighting coefficient ratios were estimated from the uncompensated noisy ACF data and the reference angle through the third equality of Eq. (4), and the compensated weighting coefficient ratios were calculated from the ACF data after it had been compensated for the baseline error through Eq. (7) and the reference angle. For comparison, the true weightings were calculated through the second equality of Eq. (4), with the baseline values of the ACFs G2 ∞;θr calculated from the original PSD as [5]:
G2 ∞;θr
⌊
c
N X i1
⌋
CI;θr Di f Di
r 1; 2; …; R: (14)
Here c 10−6 . From the baselines and the reference angle, the true weighting coefficient ratios were calculated. The 100–900 nm unimodal ACF data were generated with shape distribution parameters μ 0.2, σ 3.1, Dmax 900 nm, Dmin 100 nm, and δ 10−5 . The 100–650 nm unimodal ACF data were generated with shape distribution parameters μ 0.1, σ 3.1, Dmax 1200 nm, Dmin 100 nm, and δ 10−6 . The 360–900 nm bimodal correlation data were generated with shape distribution parameters μ 3.5, σ 3.0, μ1 −3.2, σ 1 2.7, Dmax 900, Dmin 360, and δ 10−6 ; for the 100–900 nm bimodal correlation data the parameters were μ 0.5, σ 3.1, μ1 −3.2, σ 1 2.7, Dmax 900, Dmin 100, and δ 10−7 . In all cases 200 data points covering the size range were used to calculate the PSD. The ACF data were 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6003
Table 1.
Result Summary for the 490 nm Unimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° Diameter (nm) V
Table 2.
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 0.43 0.46 1.21 1.38 505/840 0.18
1 1.28 0.95 0.41 0.37 253/661 2.07
0 1.94 1.07 0.66 0.73
1 0.36 0.47 1.12 1.45 480 0.15
0 0.18 0.02 0.08 0.06
1.21 × 10−6 8.45 × 10−7 4.43 × 10−7
Result Summary for the 637 nm Unimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° Diameter (nm) V
k
k1
jk − k1 j∕k
k2
jk − k2j∕k
Baseline Compensation Value
1 0.97 1.97 1.17 1.01 678 0.26
1 1.23 0.51 0.93 1.06 301/648 0.99
0 0.26 0.74 0.21 0.05
1 0.88 1.63 1.03 1.20 666 0.12
0 0.10 0.17 0.12 0.18
7.10 × 10−6 2.60 × 10−6 2.09 × 10−6
calculated at 30°, 50°, 70°, 90°, and 110° scattering angles for the unimodal PSD. For the bimodal PSD ACFs were calculated at these angles as well as 130°. The simulated ACF data were combined using the angular weighting coefficients to give the MDLS data set. Three MDLS data sets were generated for each of the PSDs by using the true, uncompensated, and compensated weighting coefficient ratios. These ratios are labeled k, k1 and k2 , respectively. Each of the three MDLS data sets were analyzed with the modified Chahine method to recover the PSD. The following performance index was introduced to select the best PSD estimation fˆ Di , 1∕2 X N f Di − fˆ Di 2 ∕f Di 2 : (15) V i1
Results from analysis of the unimodal data sets are summarized in Tables 1 and 2 and the graphs in Figs. 1 and 2. For the unimodal PSDs it can be seen from Tables 1 and 2 that the relative error for the angular weighting coefficients determined from uncompensated ACF data compared with the true angular weighting coefficients is significant. This is most apparent at scattering angles of 50° and 70° for the unimodal simulation, where the error is 1.94 and 1.07, respectively, and at angles 50° and 70° for the 637 nm unimodal simulation, where the errors are 0.26 and 0.74, respectively. However, when the baseline-compensated ACF data are used, the estimate the angular weighting coefficients, the relative error values greatly decrease to 0.18, 0.02, and 0.10, Particle Size Distribution by Number
Particle Size Distribution by Number 0.08
0.03
True PSD k k1
0.07
True PSD k k1
0.025
k2
k2
0.06
0.02
f(D)
f(D)
0.05 0.04
0.015
0.03 0.01
0.02 0.005
0.01 0
0
200
400
600 Diameter (nm)
800
1000
1200
Fig. 1. 490 nm unimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients. 6004
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
0
0
200
400
600 Diameter (nm)
800
1000
1200
Fig. 2. 637 nm unimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients.
Result Summary for the 481/780 nm Bimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° 130° Diameter (nm) V
Table 4.
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 1.39 0.97 1.17 2.88 1.45 463/768 0.05
1 1.20 1.03 1.12 1.12 1.00 343/625/822 1.49
0 0.14 0.074 0.03 0.61 0.31
1 1.33 0.84 1.21 2.70 1.19 481/738 0.46
0 0.05 0.12 0.04 0.06 0.18
1.27 × 10−5 9.37 × 10−6 7.73 × 10−6 6.72 × 10−6 5.82 × 10−6 5.29 × 10−6
Result Summary for the 463/720 nm Bimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° 130° Diameter (nm) V
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 0.73 0.83 1.01 1.37 2.06 475/720 0.14
1 1.13 1.40 1.79 2.55 3.95 313/607/816 1.28
0 0.54 0.69 0.77 0.84 0.92
1 0.81 0.89 0.81 1.43 2.19 517/750 0.52
0 0.10 0.07 0.22 0.03 0.06
3.36 × 10−6 2.48 × 10−6 1.49 × 10−6 8.78 × 10−7 4.10 × 10−7 1.64 × 10−7
0.17, respectively. This is a decrease in the relative error of approximately an order of magnitude. The PSD results determined from inverting Eq. (2) with the true, uncompensated, and compensated angular weighting coefficients are shown in Figs. 1 and 2 along with the true PSDs for comparison. It can be seen that the PSDs recovered using the true (k) and uncompenstated (k1 ) weighting coefficients are not unimodal and contain spurious peaks as commonly found for results from noisy ACF data with baseline errors. These spurious peaks distort and often dominate the recovered PSD. With the weighting coefficients determined from compensated ACF data using our proposed method, the recovered PSDs are unimodal, and the spurious peaks are eliminated. The peak size position is also determined more accurately. There is overall better agreement between the true and recovered PSDs as can be seen by consideration of the performance index, which reduces from 2.07 and 0.99 to 0.15 and 0.12 for the 490 and 637 nm unimodal PSDs, respectively. For the bimodal PSDs it can be seen from Tables 3 and 4 that the relative errors of the angular weighting coefficients are large before compensation, just as for the unimodal distributions. This is most apparent at angles 110° and 130° where the errors are 0.61 and 0.31, and 0.84 and 0.92 for the 481/780 nm bimodal simulation and 463/720 nm bimodal simulation, respectively. The weighting coefficients determined from compensated ACF data have much smaller relative errors. The errors for these angles are reduced to 0.06 and 0.18, and 0.03 and 0.06 for the two samples. Figures 3 and 4 show the PSDs recovered using the various angular weighting coefficients, and it
can be seen that recovery of the bimodal PSD is more sensitive to noise than the unimodal case. The PSDs recovered using the weights determined from the uncompensated ACF data are trimodal, and the peaks do not coincide with those of the true PSD. With the weights determined from compensated ACF data, the recovered PSDs are essentially bimodal and the peak positions are close to those of the true PSD. The large false peak at small particle size has disappeared; however, a very small false peak has emerged at 900 nm. Furthermore, the relative error of the peak position has decreased, and the performance index for the 481/780 nm and 463/720 nm bimodal distributions reduced from 1.49 and 1.28 to 0.46, and 0.52, respectively.
Particle Size Distribution by Number
0.045
True PSD k k1
0.04 0.035
k2
0.03 0.025 f(D)
Table 3.
0.02 0.015 0.01 0.005 0
0
200
400
600 Diameter(nm)
800
1000
1200
Fig. 3. 481/780 nm bimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients. 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6005
Particle Size Distribution by Number
Particle Size Distribution by Number
0.03
0.7
True PSD k k1
0.025
k1 k2
0.6
Nominal PSD
k2 0.5
0.02
f(D)
f(D)
0.4
0.015
0.3
0.01 0.2
0.005
0
0.1
0
200
400
600 Diameter(nm)
800
1000
1200
0 0
100
200
300
400 500 Diameter (nm)
600
700
800
900
Fig. 4. 463/720 nm bimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients.
Fig. 5. Nominal 502 nm unimodal PSD and PSDs recovered with the uncompensated (k1 ) and compensated (k2 ) angular weighting coefficients.
4. Experimental Results and Discussion
deviation of the PSD based on the manufacturer’s data sheet is given for comparison purpose and is labeled “Nominal PSD.” To quantify how accurately the peak position is recovered in the PSDs we calculate the relative error in each peak position using
Experimental ACFs were obtained from unimodal and bimodal suspensions of polystyrene spheres. The unimodal sample data were recorded at five scattering angles: 30°, 50°, 70°, 90°, and 110°. The experimental setup has been described previously [3] and used a vertically polarized HeNe laser with a wavelength of 632.8 nm, a BI-200SM goniometer system, and a 64-channel BI-2030AT digital correlator (Brookhaven Instruments Inc., Holtsville, New York). The sample was made by suspending 502 4 nm latex spheres (Duke Scientific, California) in 1 mM NaCl. The bimodal sample was made by mixing 306 8 nm and 974 10 nm diameter latex spheres (Polysciences Inc.) in a number ratio of approximately 3∶1. For this sample, data were recorded at six scattering angles: 30°, 50°, 70°, 90°, 110°, and 130°. Measurements were made using a HeNe laser and a BI-2000AT digital correlator (Brookhaven Instruments Inc., Holtsville, New York) as described previously [15]. The sample temperature was 298.15 K, and the dispersion medium refractive index was nm 1.33 for both samples. The angular weighting coefficients were first calculated through the second equality of Eq. (4). Then the intensity ACF data were compensated for baseline error, and the weighting coefficients were recomputed. Finally, the compensated ACF data were analyzed and inverted to recover the PSD using the modified Chahine algorithm. A plot of a Gaussian distribution with the mean size and the standard Table 5. Result Summary for the Experimental Unimodal Sample: Diameter, and Relative Error of the Particle Size Peak Position (E p ) of PSDs Recovered with Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients
Diameter (nm) EP
6006
Nominal PSD (f 0 )
k1 (f 1 )
k2 (f 2 )
502 0
540 0.08
505 0.01
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
EP
jf meas − f true j : f true
(16)
Here f meas and f true are the measured and true peak positions, respectively. The unimodal results are summarized in Table 5 and Fig. 5, and the bimodal results are summarized in Table 6 and Fig. 6. Tables 5 and 6 summarize the results of the PSD determination using the uncompensated (k1 ) and compensated (k2 ) weighting coefficients. Figures 5 and 6 show the nominal PSD and the PSDs recovered using the uncompensated (k1 ) and compensated (k2 ) weighting coefficients. Comparing the results of PSD determination using the uncompensated and compensated angular weighting coefficients, it can be seen that with the compensated values the relative errors in the peak positions for the unimodal sample decreases from 0.08 to 0.01 and the error for bimodal sample decreases from 0.20/0.10 to 0.04/0.07. Besides, the peak intensity ratio of bimodal sample increases from 1.4∶1 to 2.8∶1, closer to the expected 3∶1. Also the Table 6. Result Summary for the Experimental Bimodal Sample: Diameter and Relative Error of the Particle Size Peak Position (E p ) and Peak Number Ratio of PSDs Recovered with Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients
Diameter (nm) EP Number ratio
Nominal PSD (f 0 )
k1 (f 1 )
k2 (f 2 )
306/974 0 3∶1
367/879 0.20/0.10 1.4∶1
318/903 0.04/0.07 2.8∶1
Particle Size Distribution by Number
0.7
estimation by compensating for baseline error in the ACF data and shown that this gives more reliable and more accurate particle size distribution information.
k1 k2
0.6
Nominal PSD
The authors are grateful for the financial support received from the National Nature Science Foundation of China (61205191) and the Natural Science Foundation of Shandong Province (ZR2012FL22, ZR2012EEM028). Special thanks are extended to Prof. Jorge R. Vega for supplying the bimodal experimental data.
0.5
f(D)
0.4
0.3
0.2
References
0.1
0
0
200
400
600 800 Diameter (nm)
1000
1200
1400
Fig. 6. Nominal 306/974 nm bimodal PSD and PSDs recovered with the uncompensated (k1 ) and compensated (k2 ) angular weighting coefficients.
width of the large particle peak decreases, becoming closer to that expected. 5. Conclusions
The MDLS technique compensates for the low information inherent in the ACF measured at only one scattering angle by combining ACFs at different scattering angles, so it provides a better estimate of the PSD. However, ACF baseline errors affect the accuracy with which angular weighting coefficients can be estimated. Incorrect weight coefficients corrupt the PSD estimation. We have developed a method for improving the accuracy of angular weighting coefficient estimation in MDLS by removing the effects of ACF baseline error. The method has been demonstrated and shown to work with simulated MDLS data for two sets of unimodal and two sets of bimodal PSDs. It was observed that our method decreases the relative error of the weighting coefficients significantly, by about an order of magnitude, and that the PSDs obtained are closer to the true distributions as measured by the performance index. The performance index values for the 490 and 637 nm unimodal distributions reduce from 2.07 and 0.99 to 0.15 and 0.12, respectively. For 481/780 nm and 463/720 nm bimodal distributions, the values reduce from 1.49 and 1.28 to 0.46 and 0.52, respectively. Furthermore, the method shows its advantage in eliminating false peaks, which commonly occur in the recovered PSD. By analyzing data from MDLS measurements on real unimodal and bimodal samples consisting of a suspension of polystyrene latex standard spheres, it was also shown that the proposed method improves the recovery of PSD information. In summary, we have demonstrated a method for improving MDLS angular weighting coefficient
1. B. J. Berne and R. Pecora, Dynamic Light Scattering (Courier Dover, 2000). 2. J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991). 3. G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995). 4. G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. refinements and applications,” Langmuir 12, 6224–6228 (1996). 5. J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003). 6. L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, V. D. Gonzalez, R. J. Minari, J. R. Leiza, and J. R. Vega, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009). 7. L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. R. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemometr. Intell. Lab. Syst. 107, 165–173 (2011). 8. X. Y. Liu, J. Shen, J. C. Thomas, S. J. Shi, X. M. Sun, and W. Liu, “Multiangle dynamic light scattering analysis using angular intensity weighting determined by iterative recursion,” Appl. Opt. 51, 846–854 (2012). 9. S. S. Gao, J. Shen, Y. J. Wang, and X. J. Zhu, “The baseline error compensation of multi-angle dynamic light scattering technique,” Acta Optica Sinica 33, s0112002 (2013). 10. K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987). 11. W. J. Wiscombe, “Improved Mie scattering algorithm,” Appl. Opt. 19, 1505–1509 (1980). 12. X. J. Zhu, J. Shen, and J. C. Thomas, “Analysis of noisy dynamic light scattering data using constrained regularization techniques,” Appl. Opt. 51, 7537–7548 (2012). 13. G. R. Markowski, “Improving Twomey’s algorithm for inversion of aerosol measurement data,” Aerosol Sci. Technol. 7, 127–141 (1987). 14. W. Winklmayr, H. C. Wang, and W. John, “Adaptation of the Twomey algorithm to the inversion of cascade impactor data,” Aerosol Sci. Technol. 13, 322–331 (1990). 15. X. Y. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. M. Sun, “Multiangle dynamic light scattering analysis using the modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012). 16. A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990). 17. Y. Sun and J. G. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6007
School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255049, China 2
Group Scientific Pty Ltd., P.O. Box 190, Salisbury South, SA 5106, Australia
3
School of Civil and Environment Engineering, University of Science and Technology Beijing, Beijing 100083, China *Corresponding author: [email protected] Received 2 June 2014; revised 6 August 2014; accepted 8 August 2014; posted 11 August 2014 (Doc. ID 213172); published 8 September 2014
In multi-angle dynamic light scattering measurements, due to the inevitable presence of baseline measurement noise, the normalized intensity autocorrelation function (ACF) data deviates from the true value. This leads to incorrect angular weighting estimates, which affect the accuracy of inversion results and determination of particle size distributions (PSDs). We outline a method to calculate better angular weighting coefficients from the noisy intensity ACF data. The method involves first compensating for the baseline error in the ACF data and then determining the weighting coefficients. We demonstrate the method using simulated ACF data containing baseline error for unimodal and bimodal PSDs and also for experimental data for unimodal and bimodal samples. For the unimodal PSDs ACF data were simulated for 100–900 nm and 100–650 nm particle size ranges, and for bimodal PSDs 360–900 nm and 100–900 nm particle size ranges were used. The performance of our method was shown by comparing the results of weighting coefficient and PSD determination with and without baseline compensation to the known coefficient values and PSDs. With baseline compensation the relative error of the weighting coefficients decreased significantly. Furthermore, with baseline compensation, the PSD results for the four groups of simulated data were improved. The deviations between the known and recovered PSDs were decreased, the relative error of peak position obviously decreased, and the occurrence of false peaks was reduced. The PSD results from the experimental data further validates the conclusion that the method proposed apparently reduces the relative error of peak position, effectively eliminates the false peak, and improves the accuracy of the recovered PSD. © 2014 Optical Society of America OCIS codes: (290.3200) Inverse scattering; (290.4020) Mie theory; (290.5850) Scattering, particles; (290.5890) Scattering, stimulated. http://dx.doi.org/10.1364/AO.53.006001
1. Introduction
Dynamic light scattering (DLS) is a widely used technique for particle size and particle size distribution (PSD) determination in the submicrometer range [1,2]. The experimental setup is simple; laser light is focused into a dilute suspension of colloidal particles, and a photometer collects the light scattered at a particular angle. Brownian motion of the particles 1559-128X/14/266001-07$15.00/0 © 2014 Optical Society of America
induces temporal fluctuations in the scattered light, and a digital correlator calculates the autocorrelation function (ACF) of the scattered intensity to characterize the fluctuations. The PSD is then extracted from the measured intensity ACF by an inversion algorithm. Data analysis to recover the PSD is problematic due to the relatively low information content inherent in the measured signal. The multi-angle DLS (MDLS) technique can provide more accurate PSD information than singleangle DLS [3,4]. In the MDLS data analysis, light intensity ACFs for each angle are combined into a 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6001
single data analysis with proper angular weighting coefficients, which can be estimated by the light intensity mean value, baseline of the ACF, or other methods [5]. However, inaccurate estimation of the weighting coefficients will affect the accuracy of the recovered PSD significantly. Bryant and Thomas [3] combined static light scattering (SLS) and MDLS to get more information and improved the accuracy of the recovered PSD. However, SLS measurement requires a precision angular measurement system. Bryant et al. [4] put forward a method of using an iterative algorithm post measurement to reconstruct the SLS data and estimate MDLS weighting coefficients. This avoids measuring high-quality SLS data at different scattering angles. The inversion results for intensity ACF data with 0.01% noise measured at five scattering angles gave good PSD estimation. However, when the noise level is higher, 10 or more scattering angles are needed for reliable PSD determination. Subsequently, a neural network algorithm was applied to MDLS data analysis by Gugliotta et al. [6], and a Bayesian inversion algorithm was proposed by Clementi et al. [7]. Recently, a new method for determining angular intensity weighting in MDLS using an iterative recursion method in MDLS was developed by Liu et al. [8]. The method worked well but did not consider baseline error in the ACF. In MDLS, the precision of PSD determination depends on accurate estimation of the angular weighting coefficients. Baseline error in MDLS experiments affects PSD determination through two mechanisms. In a single-angle DLS experiment, baseline error distorts the ACF, and if this is not compensated, the resulting PSD will be distorted. In an MDLS experiment baseline error distorts the ACF, just as for DLS, but it also causes errors in the angular weights because the baseline is also used to estimate them. In this paper we propose an angular weighting coefficient estimation technique for MDLS which reduces the effect of ACF baseline errors. First, baseline error compensation [9] is applied to the ACFs at each scattering angle and the ACFs are normalized using the compensated baselines; then the average light intensity is calculated from the compensated baselines and used to estimate the angular weighting coefficients. 2. Theory of MDLS
In a single DLS experiment, measurements of the light intensity ACF are taken at one scattering angle. However, the particles in the sample cell scatter light in all directions, so to get more information, MDLS is proposed. At a given scattering angle θr of the MDLS measurement, the DLS data recorded is the ACF of the scattered light intensity, G2 θr τj . This is related to the normalized ACF of the electric field, g1 θr τj i through [10] 2 1 2 G2 θr τj G∞;θr 1 βjgθr τj j ;
r 1; 2; …; R 6002
and
j 1; 2; …; M:
(1)
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
Here τj is the delay time, G2 ∞;θr is the intensity ACF baseline, β≤1 is an instrumental coherence parameter, and M is the total number of correlator channels or points of the ACF measured at θr . If the PSD is described by a number distribution f Di of particle sizes, Di , the electric field ACF is g1 θr τj kθr
N X i1
exp−Ω0 θr τj ∕Di CI;θr Di f Di ; (2)
with Ω0 θr
16πkB T nm λ 2 2 θr : sin 2 3η λ0
(3)
Here kθr are a priori unknown weighting coefficients that adopt different values at different θr and are the inverse of the total intensity scattered at each angle. CI;θr Di is the fraction of light intensity scattered by a particle of diameter Di at θr and is calculated through Mie scattering theory [11]. λ0 is the in vacuo wavelength of the incident laser light, nm λ is the refractive index of the nonabsorbing suspending liquid, kB is the Boltzmann constant, T is the absolute temperature, and η is the medium viscosity at temperature T. We define the dimensionless weighting ratio kθr relative to a fixed reference angle θ1 30°, as [5]: kθr
2 1∕2 G∞;θ1 kθ r hI θ i 1 2 hI θr i kθ 1 G∞;θr
r 1; 2; …; R:
(4)
Equation (4) shows that the weighting ratios kθr can be determined from the autocorrelation baseline G2 ∞;θr or the mean light intensity hI θr i. However, this uses the experimental ACF values, which may contain sufficient noise to corrupt the determination of the weighting ratios. The theoretical intensity ACF baseline is B G2 ∞;θr . Normalization errors occur in the normalized intensity ACF when there is an error in the baseline ˆ hnti2, where nt is the number of estimator B photon counts at sample time t. Writing the baseline ˆ − B, the intensity ACF with baseline error as ΔB B error is [12] g2 θr τj
2 ΔBθr Gθr τj 1− ˆθ hntθr i2 B r G2 θr τj
2 ΔBθr Gθr τj − : ˆ θ hntθ i2 hntθr i2 B r r
(5)
2 ˆ Here the second term, g2 cθr τj ΔBθr ∕Bθr G∞;θr τj ∕ hntθr i2 , is the normalization error at scattering 2 angle g2 θr τ j gcθr τj and causes distortion of the recovered PSD. Zhu et al. [12] have shown that this value can be determined and used to compensate the normalized intensity ACF so that it becomes
2 g2 θr τj gcθr τ j , and the compensated mean light intensity is then
hI cθr i
M g2 τ g2 τ X j j θ cθ r
r
M
j1
× exp−0.5μ σ lnt∕1 − t2 :
hI cθ1 i : hI cθr i
(7)
The intensity ACFs at selected angles can be combined with the angular weighting coefficients to form the MDLS data sets, which can be analyzed to determine the underlying PSD. We have shown previously that a modified Chahine method [13–15] performs well in recovering both unimodal and bimodal PSDs from MDLS data and have used that method as the inversion algorithm in this paper. The Chahine method is an iterative method wherein the distribution function is given by PM j1 uj;i gmeas;j p−1 p fi P f ; (8) M u gp−1 i j1 j;i calc;j where f p is the component of the PSD due to the i discrete diameter Di after the pth iteration. gmeas is p−1 the measured ACF data, and gcalc is the calculated ACF data obtained from the p − 1th iteration, p−1 gcalc;j
N X i1
hj;i f ip−1 ;
(9)
p−1 p−1 are the elements of the vector gcalc , where gcalc;j and uj;i hj;i ∕ h exp−Ω0 θr τj ∕Di CI;θr Di , Pj;iN h . After each iteration the root-mean-square j;i i1 error between gp calc;j and gmeas;j is calculated as
RESMP
1∕2 M 1X p g − gcalc;j ; M j1 meas;j
(10)
and when it satisfies j1 − RMSEp ∕RMSEp−1 j ≤ 1 × 10−5 , the iteration is halted, and f f p is the resulting PSD function. 3. Simulation Results and Discussion
The simulated unimodal PSDs by number were described by Johnson’s SB function [16]: D t1 − t−1 f D p 2π Dmax − Dmin × exp−0.5μ σ lnt∕1 − t2 ;
D1 p t1 − t−1 2π Dmax − Dmin
(6)
This can then be used in Eq. (4) to calculate compensated angular weighting coefficients as kcθr
D f D p t1 − t−1 2π Dmax − Dmin
(11)
and the bimodal PSDs were a combination of two Johnson’s SB functions:
× exp−0.5μ1 σ 1 lnt∕1 − t2 ;
(12)
Here, t D − Dmin ∕Dmax − Dmin ; μ, σ, μ1 , and σ 1 are distribution parameters; t is the normalized particle size; and Dmax and Dmin are the maximum and minimum particle size, respectively. The unimodal and bimodal electric field ACF data were generated by substituting f D into Eq. (2). Then the noise-free intensity ACF data, g2 θr τ, were produced through Eq. (1), and noise was added to the noise-free correlation data to simulate the measurement data as follows [17]: 2 g2 N;θr τ gθr τ δε:
(13)
Here ε is a normally distributed number in the range 0–1, and δ denotes the noise level. The uncompensated weighting coefficient ratios were estimated from the uncompensated noisy ACF data and the reference angle through the third equality of Eq. (4), and the compensated weighting coefficient ratios were calculated from the ACF data after it had been compensated for the baseline error through Eq. (7) and the reference angle. For comparison, the true weightings were calculated through the second equality of Eq. (4), with the baseline values of the ACFs G2 ∞;θr calculated from the original PSD as [5]:
G2 ∞;θr
⌊
c
N X i1
⌋
CI;θr Di f Di
r 1; 2; …; R: (14)
Here c 10−6 . From the baselines and the reference angle, the true weighting coefficient ratios were calculated. The 100–900 nm unimodal ACF data were generated with shape distribution parameters μ 0.2, σ 3.1, Dmax 900 nm, Dmin 100 nm, and δ 10−5 . The 100–650 nm unimodal ACF data were generated with shape distribution parameters μ 0.1, σ 3.1, Dmax 1200 nm, Dmin 100 nm, and δ 10−6 . The 360–900 nm bimodal correlation data were generated with shape distribution parameters μ 3.5, σ 3.0, μ1 −3.2, σ 1 2.7, Dmax 900, Dmin 360, and δ 10−6 ; for the 100–900 nm bimodal correlation data the parameters were μ 0.5, σ 3.1, μ1 −3.2, σ 1 2.7, Dmax 900, Dmin 100, and δ 10−7 . In all cases 200 data points covering the size range were used to calculate the PSD. The ACF data were 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6003
Table 1.
Result Summary for the 490 nm Unimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° Diameter (nm) V
Table 2.
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 0.43 0.46 1.21 1.38 505/840 0.18
1 1.28 0.95 0.41 0.37 253/661 2.07
0 1.94 1.07 0.66 0.73
1 0.36 0.47 1.12 1.45 480 0.15
0 0.18 0.02 0.08 0.06
1.21 × 10−6 8.45 × 10−7 4.43 × 10−7
Result Summary for the 637 nm Unimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° Diameter (nm) V
k
k1
jk − k1 j∕k
k2
jk − k2j∕k
Baseline Compensation Value
1 0.97 1.97 1.17 1.01 678 0.26
1 1.23 0.51 0.93 1.06 301/648 0.99
0 0.26 0.74 0.21 0.05
1 0.88 1.63 1.03 1.20 666 0.12
0 0.10 0.17 0.12 0.18
7.10 × 10−6 2.60 × 10−6 2.09 × 10−6
calculated at 30°, 50°, 70°, 90°, and 110° scattering angles for the unimodal PSD. For the bimodal PSD ACFs were calculated at these angles as well as 130°. The simulated ACF data were combined using the angular weighting coefficients to give the MDLS data set. Three MDLS data sets were generated for each of the PSDs by using the true, uncompensated, and compensated weighting coefficient ratios. These ratios are labeled k, k1 and k2 , respectively. Each of the three MDLS data sets were analyzed with the modified Chahine method to recover the PSD. The following performance index was introduced to select the best PSD estimation fˆ Di , 1∕2 X N f Di − fˆ Di 2 ∕f Di 2 : (15) V i1
Results from analysis of the unimodal data sets are summarized in Tables 1 and 2 and the graphs in Figs. 1 and 2. For the unimodal PSDs it can be seen from Tables 1 and 2 that the relative error for the angular weighting coefficients determined from uncompensated ACF data compared with the true angular weighting coefficients is significant. This is most apparent at scattering angles of 50° and 70° for the unimodal simulation, where the error is 1.94 and 1.07, respectively, and at angles 50° and 70° for the 637 nm unimodal simulation, where the errors are 0.26 and 0.74, respectively. However, when the baseline-compensated ACF data are used, the estimate the angular weighting coefficients, the relative error values greatly decrease to 0.18, 0.02, and 0.10, Particle Size Distribution by Number
Particle Size Distribution by Number 0.08
0.03
True PSD k k1
0.07
True PSD k k1
0.025
k2
k2
0.06
0.02
f(D)
f(D)
0.05 0.04
0.015
0.03 0.01
0.02 0.005
0.01 0
0
200
400
600 Diameter (nm)
800
1000
1200
Fig. 1. 490 nm unimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients. 6004
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
0
0
200
400
600 Diameter (nm)
800
1000
1200
Fig. 2. 637 nm unimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients.
Result Summary for the 481/780 nm Bimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° 130° Diameter (nm) V
Table 4.
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 1.39 0.97 1.17 2.88 1.45 463/768 0.05
1 1.20 1.03 1.12 1.12 1.00 343/625/822 1.49
0 0.14 0.074 0.03 0.61 0.31
1 1.33 0.84 1.21 2.70 1.19 481/738 0.46
0 0.05 0.12 0.04 0.06 0.18
1.27 × 10−5 9.37 × 10−6 7.73 × 10−6 6.72 × 10−6 5.82 × 10−6 5.29 × 10−6
Result Summary for the 463/720 nm Bimodal Simulation: True (k ), Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients, Recovered Diameter, and Performance Index (V )
Scattering Angle 30° 50° 70° 90° 110° 130° Diameter (nm) V
k
k1
jk − k1 j∕k
k2
jk − k2 j∕k
Baseline Compensation Value
1 0.73 0.83 1.01 1.37 2.06 475/720 0.14
1 1.13 1.40 1.79 2.55 3.95 313/607/816 1.28
0 0.54 0.69 0.77 0.84 0.92
1 0.81 0.89 0.81 1.43 2.19 517/750 0.52
0 0.10 0.07 0.22 0.03 0.06
3.36 × 10−6 2.48 × 10−6 1.49 × 10−6 8.78 × 10−7 4.10 × 10−7 1.64 × 10−7
0.17, respectively. This is a decrease in the relative error of approximately an order of magnitude. The PSD results determined from inverting Eq. (2) with the true, uncompensated, and compensated angular weighting coefficients are shown in Figs. 1 and 2 along with the true PSDs for comparison. It can be seen that the PSDs recovered using the true (k) and uncompenstated (k1 ) weighting coefficients are not unimodal and contain spurious peaks as commonly found for results from noisy ACF data with baseline errors. These spurious peaks distort and often dominate the recovered PSD. With the weighting coefficients determined from compensated ACF data using our proposed method, the recovered PSDs are unimodal, and the spurious peaks are eliminated. The peak size position is also determined more accurately. There is overall better agreement between the true and recovered PSDs as can be seen by consideration of the performance index, which reduces from 2.07 and 0.99 to 0.15 and 0.12 for the 490 and 637 nm unimodal PSDs, respectively. For the bimodal PSDs it can be seen from Tables 3 and 4 that the relative errors of the angular weighting coefficients are large before compensation, just as for the unimodal distributions. This is most apparent at angles 110° and 130° where the errors are 0.61 and 0.31, and 0.84 and 0.92 for the 481/780 nm bimodal simulation and 463/720 nm bimodal simulation, respectively. The weighting coefficients determined from compensated ACF data have much smaller relative errors. The errors for these angles are reduced to 0.06 and 0.18, and 0.03 and 0.06 for the two samples. Figures 3 and 4 show the PSDs recovered using the various angular weighting coefficients, and it
can be seen that recovery of the bimodal PSD is more sensitive to noise than the unimodal case. The PSDs recovered using the weights determined from the uncompensated ACF data are trimodal, and the peaks do not coincide with those of the true PSD. With the weights determined from compensated ACF data, the recovered PSDs are essentially bimodal and the peak positions are close to those of the true PSD. The large false peak at small particle size has disappeared; however, a very small false peak has emerged at 900 nm. Furthermore, the relative error of the peak position has decreased, and the performance index for the 481/780 nm and 463/720 nm bimodal distributions reduced from 1.49 and 1.28 to 0.46, and 0.52, respectively.
Particle Size Distribution by Number
0.045
True PSD k k1
0.04 0.035
k2
0.03 0.025 f(D)
Table 3.
0.02 0.015 0.01 0.005 0
0
200
400
600 Diameter(nm)
800
1000
1200
Fig. 3. 481/780 nm bimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients. 10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6005
Particle Size Distribution by Number
Particle Size Distribution by Number
0.03
0.7
True PSD k k1
0.025
k1 k2
0.6
Nominal PSD
k2 0.5
0.02
f(D)
f(D)
0.4
0.015
0.3
0.01 0.2
0.005
0
0.1
0
200
400
600 Diameter(nm)
800
1000
1200
0 0
100
200
300
400 500 Diameter (nm)
600
700
800
900
Fig. 4. 463/720 nm bimodal simulated PSD and PSDs recovered with the true (k), uncompensated (k1 ), and compensated (k2 ) angular weighting coefficients.
Fig. 5. Nominal 502 nm unimodal PSD and PSDs recovered with the uncompensated (k1 ) and compensated (k2 ) angular weighting coefficients.
4. Experimental Results and Discussion
deviation of the PSD based on the manufacturer’s data sheet is given for comparison purpose and is labeled “Nominal PSD.” To quantify how accurately the peak position is recovered in the PSDs we calculate the relative error in each peak position using
Experimental ACFs were obtained from unimodal and bimodal suspensions of polystyrene spheres. The unimodal sample data were recorded at five scattering angles: 30°, 50°, 70°, 90°, and 110°. The experimental setup has been described previously [3] and used a vertically polarized HeNe laser with a wavelength of 632.8 nm, a BI-200SM goniometer system, and a 64-channel BI-2030AT digital correlator (Brookhaven Instruments Inc., Holtsville, New York). The sample was made by suspending 502 4 nm latex spheres (Duke Scientific, California) in 1 mM NaCl. The bimodal sample was made by mixing 306 8 nm and 974 10 nm diameter latex spheres (Polysciences Inc.) in a number ratio of approximately 3∶1. For this sample, data were recorded at six scattering angles: 30°, 50°, 70°, 90°, 110°, and 130°. Measurements were made using a HeNe laser and a BI-2000AT digital correlator (Brookhaven Instruments Inc., Holtsville, New York) as described previously [15]. The sample temperature was 298.15 K, and the dispersion medium refractive index was nm 1.33 for both samples. The angular weighting coefficients were first calculated through the second equality of Eq. (4). Then the intensity ACF data were compensated for baseline error, and the weighting coefficients were recomputed. Finally, the compensated ACF data were analyzed and inverted to recover the PSD using the modified Chahine algorithm. A plot of a Gaussian distribution with the mean size and the standard Table 5. Result Summary for the Experimental Unimodal Sample: Diameter, and Relative Error of the Particle Size Peak Position (E p ) of PSDs Recovered with Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients
Diameter (nm) EP
6006
Nominal PSD (f 0 )
k1 (f 1 )
k2 (f 2 )
502 0
540 0.08
505 0.01
APPLIED OPTICS / Vol. 53, No. 26 / 10 September 2014
EP
jf meas − f true j : f true
(16)
Here f meas and f true are the measured and true peak positions, respectively. The unimodal results are summarized in Table 5 and Fig. 5, and the bimodal results are summarized in Table 6 and Fig. 6. Tables 5 and 6 summarize the results of the PSD determination using the uncompensated (k1 ) and compensated (k2 ) weighting coefficients. Figures 5 and 6 show the nominal PSD and the PSDs recovered using the uncompensated (k1 ) and compensated (k2 ) weighting coefficients. Comparing the results of PSD determination using the uncompensated and compensated angular weighting coefficients, it can be seen that with the compensated values the relative errors in the peak positions for the unimodal sample decreases from 0.08 to 0.01 and the error for bimodal sample decreases from 0.20/0.10 to 0.04/0.07. Besides, the peak intensity ratio of bimodal sample increases from 1.4∶1 to 2.8∶1, closer to the expected 3∶1. Also the Table 6. Result Summary for the Experimental Bimodal Sample: Diameter and Relative Error of the Particle Size Peak Position (E p ) and Peak Number Ratio of PSDs Recovered with Uncompensated (k 1 ), and Compensated (k 2 ) Weighting Coefficients
Diameter (nm) EP Number ratio
Nominal PSD (f 0 )
k1 (f 1 )
k2 (f 2 )
306/974 0 3∶1
367/879 0.20/0.10 1.4∶1
318/903 0.04/0.07 2.8∶1
Particle Size Distribution by Number
0.7
estimation by compensating for baseline error in the ACF data and shown that this gives more reliable and more accurate particle size distribution information.
k1 k2
0.6
Nominal PSD
The authors are grateful for the financial support received from the National Nature Science Foundation of China (61205191) and the Natural Science Foundation of Shandong Province (ZR2012FL22, ZR2012EEM028). Special thanks are extended to Prof. Jorge R. Vega for supplying the bimodal experimental data.
0.5
f(D)
0.4
0.3
0.2
References
0.1
0
0
200
400
600 800 Diameter (nm)
1000
1200
1400
Fig. 6. Nominal 306/974 nm bimodal PSD and PSDs recovered with the uncompensated (k1 ) and compensated (k2 ) angular weighting coefficients.
width of the large particle peak decreases, becoming closer to that expected. 5. Conclusions
The MDLS technique compensates for the low information inherent in the ACF measured at only one scattering angle by combining ACFs at different scattering angles, so it provides a better estimate of the PSD. However, ACF baseline errors affect the accuracy with which angular weighting coefficients can be estimated. Incorrect weight coefficients corrupt the PSD estimation. We have developed a method for improving the accuracy of angular weighting coefficient estimation in MDLS by removing the effects of ACF baseline error. The method has been demonstrated and shown to work with simulated MDLS data for two sets of unimodal and two sets of bimodal PSDs. It was observed that our method decreases the relative error of the weighting coefficients significantly, by about an order of magnitude, and that the PSDs obtained are closer to the true distributions as measured by the performance index. The performance index values for the 490 and 637 nm unimodal distributions reduce from 2.07 and 0.99 to 0.15 and 0.12, respectively. For 481/780 nm and 463/720 nm bimodal distributions, the values reduce from 1.49 and 1.28 to 0.46 and 0.52, respectively. Furthermore, the method shows its advantage in eliminating false peaks, which commonly occur in the recovered PSD. By analyzing data from MDLS measurements on real unimodal and bimodal samples consisting of a suspension of polystyrene latex standard spheres, it was also shown that the proposed method improves the recovery of PSD information. In summary, we have demonstrated a method for improving MDLS angular weighting coefficient
1. B. J. Berne and R. Pecora, Dynamic Light Scattering (Courier Dover, 2000). 2. J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991). 3. G. Bryant and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering,” Langmuir 11, 2480–2485 (1995). 4. G. Bryant, C. Abeynayake, and J. C. Thomas, “Improved particle size distribution measurements using multiangle dynamic light scattering. 2. refinements and applications,” Langmuir 12, 6224–6228 (1996). 5. J. R. Vega, L. M. Gugliotta, V. D. Gonzalez, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: novel data processing for multiangle measurements,” J. Colloid Interface Sci. 261, 74–81 (2003). 6. L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, V. D. Gonzalez, R. J. Minari, J. R. Leiza, and J. R. Vega, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009). 7. L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. R. Orlande, “A Bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,” Chemometr. Intell. Lab. Syst. 107, 165–173 (2011). 8. X. Y. Liu, J. Shen, J. C. Thomas, S. J. Shi, X. M. Sun, and W. Liu, “Multiangle dynamic light scattering analysis using angular intensity weighting determined by iterative recursion,” Appl. Opt. 51, 846–854 (2012). 9. S. S. Gao, J. Shen, Y. J. Wang, and X. J. Zhu, “The baseline error compensation of multi-angle dynamic light scattering technique,” Acta Optica Sinica 33, s0112002 (2013). 10. K. Schätzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987). 11. W. J. Wiscombe, “Improved Mie scattering algorithm,” Appl. Opt. 19, 1505–1509 (1980). 12. X. J. Zhu, J. Shen, and J. C. Thomas, “Analysis of noisy dynamic light scattering data using constrained regularization techniques,” Appl. Opt. 51, 7537–7548 (2012). 13. G. R. Markowski, “Improving Twomey’s algorithm for inversion of aerosol measurement data,” Aerosol Sci. Technol. 7, 127–141 (1987). 14. W. Winklmayr, H. C. Wang, and W. John, “Adaptation of the Twomey algorithm to the inversion of cascade impactor data,” Aerosol Sci. Technol. 13, 322–331 (1990). 15. X. Y. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. M. Sun, “Multiangle dynamic light scattering analysis using the modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012). 16. A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990). 17. Y. Sun and J. G. Walker, “Maximum likelihood data inversion for photon correlation spectroscopy,” Meas. Sci. Technol. 19, 115302 (2008).
10 September 2014 / Vol. 53, No. 26 / APPLIED OPTICS
6007
