The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions

Optics & Laser Technology 43 (2011) 1128–1137

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The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions Xinjun Zhu, Jin Shen n, Yuanlei Wang, Jia Guan, Xianming Sun, Xianqiang Wang School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255049, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 May 2010 Received in revised form 15 February 2011 Accepted 22 February 2011 Available online 17 March 2011

In this paper, the reconstruction of particle size distributions (PSDs) using particle swarm optimization (PSO) techniques from dynamic light scattering (DLS) data was established. Three different objective functions containing non-smooth constrained objective function, smooth functional objective function of Tikhonov regularization and L objective function, were employed. Simulated results of unimodal, bimodal and bi-dispersed particles show that the PSO technique with non-smooth constrained objective function produces narrower PSDs focusing on peak position in the presence of random noise, the PSO technique with smooth functional of Tikhonov regularization creates relative smooth PSDs, which could be successfully applied to the broad particles inversion, and the PSO technique with L objective function yields smooth PSDs, which saves calculation amount. Experimental results as well as comparisons with CONTIN algorithm and Cumulants method demonstrate the performance of our algorithms. Therefore, the PSO techniques employing the three different objective functions, which only require objective function and need a few initial guesses, may be applied to the reconstruction of PSDs from DLS data. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Particle size distributions Particle swarm optimization Objective function

1. Introduction

angle. The discrete form of Eq. (1) is

Dynamic light scattering (DLS) is a powerful tool for measuring particle size distributions (PSDs) in solution or colloidal suspensions [1–5]. The fluctuation of scattered light of particles under Brownian motion contains information about particle sizing [1]. From the correlation function of scattered light intensity, the PSDs can be obtained. For the poly-dispersed solutions, the normalized autocorrelation function of electric field is

g ð1Þ ðtÞ ¼

g ð1Þ ðtÞ ¼

Z

N X

expðGi tÞai

with constraint conditions N X

ai ¼ 1

ð3Þ

i¼1

ai Z 0

1

GðGÞexpðGtÞdG

ð1Þ

0

where GðGÞ is normalized distribution function of the decay   constant G ¼ KB T=3pZd q2 , KB is Boltzmann constant, T is the absolute temperature, Z is viscosity coefficient of water, d is     diameter of particles, q ¼ 4pn=lJ sin y=2 is the scattering vector where l is the wavelength of incident beam in vacuum, n is the refractive index of suspending medium, y is scattering

ð2Þ

i¼1

ð4Þ

where ai is the amplitude scattered by a particle with decay constant Gi , which stands for the mean value of each interval ½Gi , Gi þ 1  from total intervals N. In experiment, the equality constraint Eq. (3) should be transformed into an inequality constraint equation N X

ai r 1

ð5Þ

i¼1

In fact, constraint condition of Eq. (5) can be further limited to a subset, which could be expressed as follows: n

Corresponding author. E-mail address: [email protected] (J. Shen).

0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.02.010

g1 Dg r

N X i¼1

ai rg1 þ Dg

ð6Þ

X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

where g1 is correlation function value of the first channel while Dg is a constant that can be set to be 0.1 approximately. The retrieved PSD is obtained by inverting ai from Eq. (2) under the constraint conditions of Eqs. (3) and (4) in theory or Eqs. (4) and (6) in experiment. However, as a well-known Fredholm integral equation of the first kind, the discrete equation Eq. (2) is very ill-conditioned in the sense that arbitrary noise in equation will yield large deviations in solutions. Therefore, special approaches should be employed to solve this problem. Numerous approaches have been proposed in the recovery of PSDs from DLS data, such as Cumulants method (actually they merely supply average diameter and polydispersity index) [6], Laplace Transform method [7,8], Exponential Sampling method [9], CONTIN [10], Non-Negative Least Squares method [11] and Bayesian strategies method [12]. However, these techniques are still in improvement [13–18] to recover more realistic PSDs. During the development of the traditional algorithms presented above, stochastic optimization techniques, more recently, have been adopted in inverse problems. The advantages are that they are agreeable to both linear and nonlinear problems, as well as not sensitive to the initial values. Compared with the gradientbased approaches, meanwhile, they only require objective function. In the case of PSD reconstruction, several stochastic optimal algorithms, such as Monte Carlo, Genetic Algorithms and PSO techniques, have been proposed. In static light scattering, Ligon et al. [19] firstly employed Monte Carlo method to retrieve the PSD from light scattering data, Ye et al. [20] applied a genetic algorithm to extract PSDs from static light scattering data. Qi et al. [21], as well as Liu et al. [22] adopted improved PSO techniques to estimate PSDs. In dynamic light scattering, Li [23] used Gray encoded genetic algorithm to recover the PSD. In this study, we employed the PSO technique to retrieve PSDs from DLS data. Besides, the least squares error between the calculated data and the measured data was chosen to be the objective function (non-smooth constrained objective function) in the above-mentioned stochastic optimization approaches, which may cause relative unstable solutions for the ill-posed problems [24]. Smooth functional of the Tikhonov regularization method, recently, has been used as the objective function of stochastic optimization techniques in inverse problems [25,26], since the Tikhonov regularization method produces stable solutions for ill-posed problems. Therefore, in this paper, we employed the PSO technique to characterize PSDs from DLS data with different objective functions involving non-smooth constrained objective function, smooth functional of Tikhonov regularization approach and another objective function (called as L objective function) owning the characteristics of the Tikhonov regularization method according to the L-curve rule. The simulated results with the three different objective functions as well as experimental results were discussed and analyzed.

2. Particle swarm optimization

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In this study, we employed the inertia weight particle PSO [28], which is also called Stand PSO. Stand PSO updates particle position and velocity according to the following the equations: vijt þ 1 ¼ wvtij þc1 r1 ðpbtij xtij Þ þ c2 r2 ðgbtj xtij Þ

ð7Þ

where w ¼ wmax tðwmax wmin Þ=tmax

ð8Þ

xtijþ 1 ¼ xtij þvtijþ 1

ð9Þ

In Eqs. (7) and (9), the inertia weight coefficient w and the constant values c1, c2 affect the ability of the PSO to find the optimum solution, r1 and r2 are uniformly distributed random numbers, pbtij is the history best position of i particle at j dimension, gbtj is global best position. x and v are the particle position and velocity, respectively.

3. Statement of reconstruction of PSDs with PSO technique 3.1. The PSO technique with non-smooth constrained objective function It is noted that many inverse problems in engineering can be reformulated as optimization problems [29]. Likewise, the PSD reconstruction using the PSO technique can be reformulated as the optimization problem with objective function as follows: F¼

M X

ðgðtj Þ

j¼1

N X

expðGi tj Þai Þ2

ð10Þ

i¼1

F is the least squares error between the first order correlation function value and the calculated autocorrelation function value. The PSD is retrieved by implementing the optimization of problem (10). We employed the commonly used penalty function method to implement constrained conditions in Eqs. (3) and (4), as they are essential for the PSO technique to produce reasonable solutions especially the non-negative constraint. 3.2. The PSO technique with smooth constrained objective function 3.2.1. The PSO technique with smooth functional of Tikhonov regularization For the operator equation Az ¼ u

ð11Þ

where A is the finite linear operator between Hilbert spaces Z and Hilbert U, u A U,z A Z. For the ill-posed problems, arbitrary disturbance in the right-hand side of Eq. (11) will lead to big deviation in solution. Tikhonov regularization method is a wellknown approach for the inverse problems, which seeks to provide a fair balance between the accuracy and stability of solution. Tikhonov solution as the approximation solution of problem (11) [30], is obtained by minimizing smooth functional 2

Ma ðz,uÞ ¼ :Azu: þ aOðzÞ Particle swarm optimization inspired by the behavior of birds and insects was introduced by Kennedy and Eberhart [27]. It is one of the latest evolutionary algorithms, which is sought to find the overall optimal solution of a function in a search space. Each particle in swarm has current position, velocity, fitness value determined by optimized objective function, as well as history best position and history best fitness value. Each particle’s position stands for a potential solution. In every iteration, each particle moves toward the two best values, the global value and the history best value. The iteration is performed until the PSO satisfies the stop criterion.

ð12Þ

where a is the regularization parameter controlling balance of accuracy and the stability of solution. OðzÞ is the penalty func2 tional to ensure the stability of solution, usually OðzÞ ¼ :z: . According to Eq. (12), we take the following equation as the objective function of the PSO technique: F¼

M X j¼1

ðgðtj Þ

N X

2

expðGi tj Þai Þ2 þ a:ai :

ð13Þ

i¼1

The PSD can be obtained from the optimization of Eq. (13) under the constraint conditions of Eqs. (3) and (4). In fact, the PSO

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X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

where t ¼ ðaamin Þ=ðamax amin Þ

ð17Þ

In Eq. (16), m and s are distribution parameters, t is the normalized particle size. amax and amin denote maximum and minimum particle size, respectively [28]. Consequently, the second order autocorrelation functions could be further simulated with the acquired PSDs according to Eq. (1). The unimodal PSD simulation was accomplished with the parameters of Johnson’s SB u¼3.8, s ¼3.0, amin ¼ 2nm, amax ¼100 nm, and the bimodal one was also established with sum of two Johnson’s SB functions of equal intensity quotients and their parameters were m1 ¼ 3:8, s1 ¼ 3:0, u2 ¼ 4:2, s2 ¼ 2:8, amin ¼ 2nm, amax ¼ 100 nm, respectively. To simulate the real conditions we added a specific amount of random Gaussian noise to the simulated second order correlation function, using the following equation: pffiffiffiffi g ð2Þ ðtÞ ¼ g ð2Þ ðtÞ þ D:g ð2Þ ðtÞ:e= N ð18Þ Fig. 1. Choosing Tikhonov regularization parameter with L-curve rule.

technique with non-smooth constrained objective function is a special case of the PSO technique with smooth functional objective function where regularization parameter is set to be zero. 3.2.2. The PSO technique with L objective function L-curve is called for that its plot of the norm :xa : of the regularization solution versus the corresponding residual norm :yAxa : in the log–log scale is like letter ‘‘L’’ in Fig. 1. In this way, the L-curve clearly displays the balance between minimization of the two parts. The vertical part of the L-curve corresponds to solutions where :xa : is sensitive to the changes in regularization parameter while the horizontal part corresponds to solutions where :yAxa : is sensitive to regularization parameter changes. The optimal regularization parameter is determined where the corresponding point is at the ‘‘vertex’’ of the curve ‘‘L’’ and it reflects the tradeoff between the accuracy and stability of solution. Engl [31] pointed out that in many cases the L-curve criterion can be implemented by minimizing the following functional with respect to the regularization parameter a: 2

2

FðaÞ ¼ :Axa y: :xa :

where e is random number, d denotes noise level and N is data length of correlation function data g ð2Þ ðtÞ. The parameters wmin and wmax of the PSO are set to be 1.0 and 0.4, respectively, and the size of particle swarm population is 200. The iteration numbers of the PSO technique to reconstruct unimodal particles and bimodal ones are set to be 5000 and 10,000, respectively, since the retrieval of PSDs from bimodal particles is more difficult than that from unimodal ones and large iteration number tends to get the global optimal solution. 4.1. Recovered PSDs using the PSO technique with non-smooth constrained objective function Figs. 2 and 3 show recovered PSDs from unimodal and bimodal particle size distributions presented above, respectively, by the PSO technique with non-smooth constrained objective function. Random noise was added to the free noise data with a noise level of 0.05, according to Eq. (18). In Fig. 2 and the succeeding figures, the white bar, the black bar and the black solid line indicate reconstructed PSDs at no noise 0, at noise level 0.05 and the exact PSDs, respectively. It should be stressed that the average of five running results should be used as the final result, since not only does the PSO technique belongs to a stochastic technique [20], but also the objective function adopted in this section may cause unstable solutions for the ill-posed nature of this problem.

ð14Þ

However, in order to obtain the point at ‘‘vertex’’ of the curve ‘‘L’’, or to get the optimizer a in Eq. (14), an array of points corresponding to a set of regularization parameters should be calculated and plotted, which yields a large calculation amount. For this case, we employ the functional 2

2

F ¼ :Axy: :x:

ð15Þ

and minimize it from the view of optimization theory, instead of Eq. (14). Thus, optimizing Eq. (15) yields only once calculation because it does not require the regularization parameter a.

4. Numerical simulation results Johnson’s SB function [32] was used to simulate PSDs, and it could be expressed as follows:

s

f ðaÞ ¼ pffiffiffiffiffiffi ðtð1tÞÞ1 2pðamax amin Þ expð0:5ðm þ s lnðt=ð1tÞÞÞ2 Þ

ð16Þ

Fig. 2. Recovered PSDs from unimodal particle size distribution covering particle size ranging from 2 to 100 nm with non-smooth constrained objective function.

X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

Fig. 3. Recovered PSDs from bimodal particle size distribution covering particle size ranging from 2 to 100 nm with non-smooth constrained objective function.

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As can be seen from Figs. 2 and 3, the PSO technique with nonsmooth constrained objective function yields recovered PSDs with accurate peak positions for the unimodal particle size distribution and for the first mode of bimodal particle size distribution, in the cases of noise level 0 and 0.05. Compared with the theoretical value, however, the proportion at peak position is higher in the case of noise level 0.05, as described in Figs. 2 and 3. This drawback decreases the resolution of unimodal and bimodal inversion results. Quantitatively, Table 1 reveals that big deviations in real solutions occur with inversion errors of 0.66 and 0.51 for unimodal and bimodal particle size distributions, respectively. However, as can be concluded from Fig. 4, the addition of noise results in higher proportion at peak position, which improves the resolution of extracted results from bi-dispersed particle size distribution. As listed in Table 1, the relative error was 0.74 at noise level 0, while it declined to 0.46 in the presence of noise level 0.05. In conclusion, the PSO technique with non-smooth constrained objective function produces narrower PSDs, which tend to focus on the peak position with random noise. It decreases the accuracy of unimodal and bimodal particles inversion results while it increases the accuracy of bi-dispersed particles inversion results.

4.2. Recovered PSDs with smooth functional objective function

Fig. 4. Recovered PSDs from 200 to 800 nm bi-dispersed particle size distribution with non-smooth constrained objective function.

The PSO technique with smooth constrained objective function was examined with the same simulation profiles as Section 4.1. Figs. 5–7 show the retrieval PSDs from unimodal, bimodal and bidispersed particle size distributions, respectively, in the cases of noise levels of 0 and 0.05. The Tikhonov regularization parameters determined with GCV rule [33] are 1.04  10  12, 2.43  10  2 for unimodal particle size distribution, 5.95  10  10, 4.14  10  3 for bimodal particle size distribution and 1.33  10  10, 2.28  10  4 for bi-dispersed particle size distribution, respectively. Figs. 5 and 6 depict that smooth PSDs are produced in the presence of noise level 0.05. It also improves the resolution of estimated results compared with the PSO technique with nonsmooth constrained objective function. As summarized in Tables 1 and 2, the PSO technique with non-smooth constrained objective function at noise level of 0.05 gives results with relative errors of 0.66 and 0.51 for unimodal and bimodal particle size distributions, respectively. In contrast, the PSO technique with smooth functional objective function yields relative errors of 0.29 and 0.50 for the above particles. Influence of the regularization

Table 1 Relative errors of inversion results by the PSO technique with non-smooth constrained objective function. Noise level

Unimodal particles

Bimodal particles

Bi-dispersed particles

0 0.05

0.089 0.66

0.34 0.51

0.74 0.46

Besides, the ability of the PSO technique to recover PSDs from bi-dispersed particle size distribution was also investigated. They are a special kind of bimodal particles for the infinitely narrow width. Fig. 4 displays the results estimated from the simulated 200 and 800 nm bi-dispersed particle size distribution with light intensity ratio 1:1 for its two peaks. Table 1 displays relative errors of the recovered results from the above three kinds of particles by using: e ¼ :xxtrue :=:xtrue :

ð19Þ

where x denotes the recovered PSD while xtrue represents the true PSD of Eq. (2).

Fig. 5. Recovered PSDs from unimodal particle size distribution covering particle size ranging from 2 to 100 nm with smooth functional objective function.

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X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

Fig. 6. Recovered PSDs from bimodal particle size distribution covering particle size ranging from 2 to 100 nm with smooth functional objective function. Fig. 8. Recovered PSDs from unimodal particle size distribution covering particle size ranging from 2 to 100 nm by the PSO technique with smooth functional objective function (the open circle) and Tikhonov regularization approach (the filled circle).

Fig. 7. Recovered PSDs from 200 to 800 nm bi-dispersed particle size distribution with smooth functional objective function.

Table 2 Relative errors of inversion results by the PSO technique with smooth functional objective function. Noise level

Unimodal particles

Bimodal particles

Bi-dispersed particles

0 0.05

0.11 0.29

0.41 0.50

0.72 0.83

parameter on the PSD has been discussed by Su et al. [34]. Small regularization parameter leads to a narrow PSD and vice versa. In the case of noise level of 0, the recovered PSDs depicted in Fig. 5 and 6 are not smooth for their small regularization parameters. Fig. 7 displays determined PSDs from 200 and 800 nm bidispersed particle size distribution. Compared with Fig. 4, the PSD characterized by the PSO technique with smooth functional objective function yielding inversion error of 0.83 is not as accurate as that retrieved by the PSO technique with non-smooth constrained objective function producing inversion error of 0.46,as listed in Table 2. The reason is that the stability and smoothness of Tikhonov regularization solution is under the

sacrifice of the singularity of real solution [24], but the real solution of bi-dispersed particles share singularity. For instance, the theoretical PSD of bi-dispersed is a sum of two Dirac delta functions and has narrow distribution width, but the smoothness of regularization impose a wide distribution on the recovered PSD. Hence, the PSO technique with smooth functional objective function disadvantages the results from bi-dispersed particles. Further, Tikhonov regularization approach and the PSO technique with smooth constrained objective were compared since they conform to the regularization theory. Fig. 8 shows recovered results from unimodal particle size distribution employing the Tikhonov regularization approach and the PSO technique with smooth functional objective function. Both these techniques used same regularization parameter 3.70  10  4, which was chosen by the GCV rule. Clearly the recovered PSD using the Tikhonov regularization approach has an artifact in the large particle range and produces wider distribution than that by the PSO technique with smooth functional objective function. In this case, the PSO technique with smooth functional objective function yielding inversion relative error of 0.11 display higher performance than Tikhonov regularization approach creating inversion relative error of 0.18; this is attributed to the additional implementation of constrained conditions of Eqs. (3) and (4) as well as the ability of the PSO technique to find global optimal solution. Difficulties of broad particles reconstruction exist in references and therein [7]. A simulated broad particles, covering particle size range from 250 to 600 nm, with shape parameters u ¼0, s ¼ 1:5, was achieved. Fig. 9 illustrates recovered results by the PSO technique with non-smooth constrained objective function (Fig. 9a) and smooth functional objective function (Fig. 9b) with regularization parameter 1.20  10  4. The former with inversion error 0.32 is inferior to the latter with inversion error 0.12. Therefore, the PSO technique with smooth functional objective function can be well applied to recover PSDs from broad particles.

4.3. Recovered PSDs with L objective function Figs. 10–12 show recovered PSDs from unimodal, bimodal and bi-dispersed particle size distributions, respectively, by the PSO technique with L objective function. The same simulated data as presented in Sections 4.1 and 4.2 was used in this inversion. We see

X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

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Fig. 9. Recovered PSDs from broad unimodal particle size distribution covering particle size ranging from 250 to 600 nm by the PSO technique with non-smooth constrained objective function (a) and smooth functional objective function (b).

Fig. 10. Recovered PSDs from unimodal particle size distribution covering particle size ranging from 2 to 100 nm with L objective function.

Fig. 11. Recovered PSDs from bimodal particle size distribution covering particle size ranging from 2 to 100 nm with L objective function.

that the PSO technique with L objective function yields smooth results at a noise level of 0.05, which exhibits the smoothness feature like regularization techniques without knowing the

Fig. 12. Recovered PSDs from 200 to 800 nm bi-dispersed particle size distribution with L objective function.

Tikhonov regularization parameter. But it could not resolve bimodal or bi-dispersed particles at noise level of 0.05 as shown in Figs. 11 and 12. Indeed, the PSO technique with L objective function could be successfully applied to these problems with noise level approximately less than 0.02 for both bimodal and bi-dispersed particle size distributions in this study. We tried to explain this reason but in vain. To test the ability of PSO techniques to recover PSDs in various conditions or harsh conditions, different distributions were involved: unimodal and bimodal particle size distributions with a larger particle size range between 100 and 1200 nm, 100 and 300 nm bi-dispersed particle size distribution with a smaller intensity for the small size (1:11) as well as 100 and 200 nm bidispersed particle distribution which has peaks with 2:1 difference in size. Figs. 13 and 14 show recovered results from unimodal and bimodal particle size distributions with a large size range (100–1200 nm), which come to the same conclusion as in small size range (2–100 nm). Fig. 15 indicates that the PSO techniques with non-smooth constrained objective function and smooth functional objective function could resolve bi-dispersed particle size distribution, which also yields the recovered intensity ratio of 0.062 and 0.054 relative to the true ratio of 0.091. However, we can see from Fig. 16, the PSO techniques with all three different objective functions fail to resolve 100 and 200 nm bi-dispersed particle size distribution. It is hoped that the

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X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

Fig. 13. Recovered PSDs from unimodal particle size distribution covering particle size ranging from 100 to 1200 nm by the PSO techniques with the three objective functions.

Fig. 15. Recovered PSDs from bi-disperse particle size distribution with intensity ratio 1:11 by the PSO techniques with the three objective functions.

Fig. 14. Recovered PSDs from bimodal particle size distribution covering particle size ranging from 100 to 1200 nm by the PSO techniques with the three objective functions.

Fig. 16. Recovered PSDs from bi-dispersed particle size distribution with diameter ratio 2:1 by the PSO techniques with the three objective functions.

question will be resolved with multi-angle dynamic light scattering technique [35–37]. Finally, in all recovered PSDs from bimodal and bi-dispersed particle size distributions, the distribution of the second peak is wider than that of the corresponding first peak. This phenomenon is also existed in other algorithms [11,12,15,17,20,22,23]. We may attribute it to the equally spaced diameter intervals ½Di ,Di þ 1  presented in Section 1. However, this speculation should be examined in future.

Laser

Cuvette Polarizer Lens Pin holes

Filter

PC

5. Experiments

Detector The experiment setup is shown in Fig. 17, which includes He–Ne laser (30 mW, 632.8 nm), photon detector (BI-DS1), Digital Autocorrelator (BI-9000AT, Brookhaven Instruments Corporation). All measurements were made at 29872 K and a scattering angle of 901. The standard polystyrene latex spheres with certified

Correlator

Fig. 17. Experimental setup.

X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

diameter 60 72.5 nm (3060 A),10273 nm (3100 A) and 1997 6 nm (3200 A) from Duke Scientific ( Palo Alto, CA) were investigated. They have a refractive of index of n ¼1.59 and a density of r ¼1.05 g/ml. The refractive index of scattering medium was distill water and filtered through a 0.45 mm membrane filter. It has refractive index and viscosity coefficient of 1.331, 0.89  10  3 Pa s, respectively. After that, both of these standards are supplied at 0.05% w/v concentration. In the following we considered samples with two different distributions: mono-dispersed and bi-dispersed particles. The first sample was 10273 nm particles while the latter was a mixture of 60 72.5 nm, 200 nm 19976 nm. The mixture sample was approximately made with a ratio of mass concentration of 4:1. For the former, the experiment time was 180 s, while for the latter, it exhibited a longer elapsed time of 900 s. The reason for this is that the analysis of poly-dispersed or bimodal particle data is a much more difficult problem and longer time measurement yields more accurate results. Figs. 18 and 19, respectively, show the normalized second order correlation functions for mono-dispersed (102 73 nm) and bi-dispersed polystyrene standards (60 72.5 and 19976 nm), and their inversion results under circumstances described above. Both the PSO techniques with the three different objective functions and CONTIN method were examined in this inversion.

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The regularization parameter chosen by GCV rule was 0.05. In this study, an emulation of CONTIN implemented with MATLAB was adopted. Fig. 18b shows that the PSO technique with non-smooth constrained objective function provides the narrowest PSD, CONTIN and the PSO technique with smooth functional objective function give the successive broader PSDs, and the PSO technique with L objective function creates the broadest PSD. For the case of average diameters, the PSO techniques with the three different objective functions offer three values presented in Table 3, which are 99.99, 100.5 and 102.5 nm, respectively. While other methods also give their values, that is,100.1 nm for CONTIN and 99.72 nm for Cumulants. For the bi-dispersed particles with diameter of 60 and 200 nm, we see from Table 4 that inversion results share the

Table 3 Average diameters of recovered PSDs by the PSO technique with non-smooth constrained objective function (PSO-N), the PSO technique with smooth functional objective function (PSO-R), the PSO technique with L objective function (PSO-L) and the CONTIN method for 100 nm mono-dispersed particles. Algorithm

PSO-N

PSO-R

PSO-L

CONTIN

Cumulants

Average diameter (nm)

99.99

100.5

102.5

100.1

99.72

Fig. 18. Correlation function data of 100 nm mono-dispersed particles (a) and recovered results (b) using the PSO technique with non-smooth constrained objective function (solid line), smooth functional objective function (solid line with open circles ), L objective function (solid line with filled circles) and using CONTIN method (solid line with diamonds).

Fig. 19. Correlation function data of 60 and 200 nm poly-dispersed particles (a) and inversion results (b) using the PSO technique with non-smooth constrained objective function (solid line), smooth functional objective function (solid line with open circles ), L objective function (solid line with filled circles) and using CONTIN method (solid line with diamonds).

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X. Zhu et al. / Optics & Laser Technology 43 (2011) 1128–1137

Table 4 Average diameters and relative errors of recovered PSDs by the PSO technique with non-smooth constrained objective function (PSO-N), the PSO technique with smooth functional objective function (PSO-R), the PSO technique with L objective function (PSO-L) and the CONTIN method for 60 and 200 nm bi-dispersed particles. Algorithms First mode

PSO-N PSO-R PSO-L CONTIN

Second mode

Average diameter

Relative error (%)

Average diameter

Relative error (%)

43.83 45.74 105.4 47.50

27 24 – 21

156.3 156.4 105.4 157.9

22 22 – 21

average diameters of two peaks of 43.83 and 156.3 nm for the PSO technique with non-smooth constrained objective function, 45.74 and 156.4 nm for the PSO technique with smooth functional objective function, the only one mode average diameter of 105.4 nm for the PSO technique with L objective function, and 47.50 and 157.9 nm for CONTIN, respectively. However, all these approaches result in large relative errors of average diameters for 60 and 200 nm bi-dispersed particle size distribution. From these results it is found that the PSO technique with non-smooth constrained objective function and smooth functional objective function could recover mono-dispersed and bi-dispersed particle size distributions and complement CONTIN algorithm in DLS.

6. Conclusions In this paper, we employed the PSO techniques with different objective functions to estimate PSDs from simulated and experimental data in DLS. The PSO technique with non-smooth constrained objective function provides narrower PSDs in the presence of random noise, which is a disadvantage to the unimodal and bimodal particle size distributions recovery but an advantage to the bi-dispersed particle size distribution inversion. The PSO technique with smooth functional objective function yields smooth PSDs for unimodal and bimodal particle size distributions at noise level of 0.05 and could be successfully applied to broad particles size distribution recovery; however, it is not beneficial for the bi-dispersed particle size distribution recovery. As to the reconstruction by the PSO technique with L objective function, it produces smooth solutions and yields a little calculation amount at low noise levels, whereas, it cannot resolve PSDs of bimodal and bi-dispersed particle size distributions at noise levels greater than 0.02. The experimental results of 100 nm mono-dispersed particles as well as 60 and 200 nm bi-dispersed particles indicate that the PSO techniques with non-smooth constrained objective function and smooth functional objective function could compliment Cumulants and CONTIN algorithms. However, they fail to resolve bimodal particle size distribution with peaks position ratio of 2:1, which is the main problem to be solved in future.

Acknowledgments The authors thank Dr. Guan from the Institute of Theoretical Chemistry, Jilin University for providing helpful discussion and Dr. Marino from University of Parma for the matlab source code of CONTIN algorithm. Special thanks are extended to the financial support received from the National Science Foundation of China (NSFC 60877050) and the National Science Foundation of Shandong Province (ZR2010FM005).

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