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Abstract
In ultra-low concentration suspensions, particle number fluctuations in the scattering volume add a long delay component to the intensity autocorrelation function (ACF) in dynamic light scattering (DLS) measurements. This gives a strong artifact peak in the particle size distribution (PSD) recovered. To improve the accuracy of DLS at ultra-low concentrations, we analyzed the different decay characteristics of particle Brownian motion and particle number fluctuation in the ACF. By differentiating the ACF we were able to identify and separate the number fluctuation term and then analyze the ACF to recover the PSD. The results for simulated DLS data at 151nm and 690nm diameters with average particle numbers of 6, 12, 24 and 48 in the scattering volume at four noise levels show that, compared with the usual DLS data processing method, inversion of the ACF after the separation of the number fluctuation term effectively eliminates the strong artifact peaks, and the relative errors of peak positions and distribution errors are significantly reduced. This was further verified with experimental results from samples of standard polystyrene spheres.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Dynamic light scattering (DLS) is an effective method for measuring particle size and particle size distribution (PSD) in the submicrometer and nanometer range [1–5]. This technique uses the measured scattered light signal from randomly fluctuating particles to calculate the scattered intensity autocorrelation function (ACF) in real time, and then the electric field ACF is obtained from the Siegert relation, and the PSD is obtained by inversion of the electric field ACF [5]. However, DLS measurement is limited by sample concentration [6]. When the concentration is too high, particle interactions and multiple scattering distort the particle size measurement. When the concentration is too low, the measured intensity fluctuations arise not only from the Brownian motion of the particles, but also from the particle number changes in the scattering volume. This latter fluctuation term is often not accounted for and gets lumped in with the Brownian motion term, during the particle size inversion, which lead to the artifact peaks in the recovered PSDs.
A number of methods have been developed to improve DLS measurements at high sample concentration. These include the cross-correlation method [7,8], the diffusion spectrum method [9,10], the fiber optic method [11–13] and the backscattering method [14,15] have been proposed successively and proved to be able to improve the distortion of particle size measurement at high concentrations. However, no significant progress has been made in the ultra-low concentration regime where the number fluctuations lead to an additional incoherent decay in the intensity ACF that is not present at normal concentrations, and which distorts recovered PSDs unless properly accounted for. Willemse et al. [16–20] used a low-pass filter to separate the high and low frequency of the output signal of the photodetector, and then used the high frequency signal to recover the particle size. Although this method is theoretically feasible, it requires an additional device between the photodetector and the photon correlator in the DLS system. This may not only introduce additional noise, but is also incompatible with the classical DLS measurement mode, since the real-time mode for calculating ACF data is changed. Around 2015, Chaikov et al. [21], Kirichenko et al. [22], and Bunkin et al. [23], conducted ultra-low concentration measurements with commercial instruments. They found that the artifact peak area is inversely proportional to the particle concentration, and the appearance of artifact peaks causes the main peaks in the PSD to shift in the direction of smaller sizes, and this shift increases as the concentration decreases. We have conducted PSD recovery using the full ACF model, including the number fluctuation component, and an ACF model with the number fluctuation term removed by resetting ACF baseline at ultra-low concentration. However, accurate inversion results from measured data were not obtained in either case [24]. The former is due to the fact that the actual measured intensity ACF does not match the non-Gaussian intensity ACF model, and the latter is due to the accurate separation of Gaussian and non-Gaussian terms, which was significantly affected by noise.
Analyzing DLS data from very dilute samples using the same model as for normal concentrations gives rise to a strong false peak due to the additional slow decay ACF component resulting from number fluctuations. This slow decay component’s amplitude depends linearly on the number of particles in the scattering volume. When the particle number is greater than 100, this component will be relatively small and can be ignored [25]. A typical DLS measuring volume is about 10−6 cm3 [26] implying a lower detection limit of 108 particles/cm3. This measurement condition limits the application of DLS and is one of the reasons DLS is difficult to use for aerosol measurements. If the Brownian motion and number fluctuation decay components in the ACF could be separated, more reliable PSD results could be obtained. In this paper we show that differentiating the ACF provides a means of determining the boundary between the two decay components and separating them out in the ACF. When the measured intensity ACF is noisy it can be smoothed by function fitting. Using the simple Brownian motion model for the ACF once the number fluctuation term has been removed to perform particle size inversion on the measured data after the non-Gaussian term is separated, it can effectively eliminate artifact peaks and yield more accurate PSD results.
2. Theory
2.1 Basic principle of DLS
When a beam of laser light is focused into a suspension of particles, light is scattered in all directions. Under a specific scattering angle, the scattered light intensity fluctuations due to the Brownian motion of the particles can be observed. These fluctuations can be characterized by the ACF of the scattered light intensity. When the amplitude distribution of the scattered field is Gaussian, as is the case for light scattered from Brownian particles, the intensity ACF and the normalized electric field ACF satisfy the Siegert relation as follows:
Here ${G^{(2)}}(\tau )$ is the intensity ACF at delay time $\tau$, $B$ is the baseline, $\beta ({ \le 1} )$ is an instrumental coherence factor, and ${g^{(1)}}(\tau )$ is the normalized electric field ACF. For a monodisperse suspension of particles,
where $\Gamma = D{q^2}$ is the decay constant, $D = {{{K_B}T} / {3\pi \eta d}}$ is the diffusion coefficient of the particles, and $q = ({{{4\pi n} / \lambda }} )\sin ({{\theta / 2}} )$ is the magnitude of the scattering vector. Here, ${K_B}$, $T$, $\eta$, $d$, $n$, $\lambda$ and $\theta$ are the Boltzmann constant, the absolute temperature of the sample solution, the viscosity coefficient of the suspending liquid, the particle diameter, the refractive index of the suspending liquid, the wavelength of the incident beam in vacuum, and the scattering angle respectively. For a polydisperse suspension of particles, where $G(\Gamma )$ is the normalized intensity distribution function of the decay constant $\Gamma $. The discrete form of Eq. (3) is2.2 Particle number fluctuations in the scattering volume
When the average number of particles in the scattering volume is small, the amplitude distribution of the scattered field is no longer Gaussian [27]. In this case, particle number fluctuations in the scattering volume will also cause fluctuations in the scattered light intensity, resulting in an additional decay component in the intensity ACF. The intensity ACF and the electric field ACF no longer satisfy the Siegert relation, instead [28,29]
The correlation function in Eq. (5) decays on two different time scales. The second term on the right side of Eq. (5) is the fast decay part of the intensity ACF, which reflects the changes in phase difference of the scattered field caused by Brownian motion. This is why it is often called the coherent decay or Gaussian term. The third term is the slow decay part and reflects the contribution to the ACF from the fluctuations in the number of particles entering and leaving the scattering volume. It is also often called the incoherent decay or non-Gaussian term. The effect of the non-Gaussian term on the intensity ACF is not negligible when $\langle N\rangle < 100$. The characteristic time of the slow decay is [29]
where $\sigma$ is the radius of the scattering volume. ${\tau _N}$ is much larger than the characteristic time of the fast decay ${\tau _D}$,If only Brownian motion is used to model the ACF and number fluctuations are ignored, this latter term can be mistaken for a slow decay due to Brownian motion, resulting in artifact peaks in the large particle size range in the PSD obtained.
3. Intensity ACF decay characteristics and Tikhonov regularization
As discussed above, at low particle concentrations, the intensity ACF is composed of a coherent and an incoherent decay component. In an ideal situation, when the measurement time is long enough (>105s [30]), the number fluctuation characteristics will be captured, and the coherent and incoherent decays can be separated far enough in delay time to obtain good PSD inversion results using a full non-Gaussian ACF model. However, most measurements are quite short (often 60s∼120s) and the intensity ACF will not have decayed to its long-time baseline value.
To see how to separate the coherent and incoherent part of the intensity ACF, the decay change characteristics of the ACF are analyzed. Taking a system of 690nm unimodal particles as an example, we calculated its intensity ACF (Fig. 1(a)) and differentiated this to get the ACF characteristic curve (Fig. 1(b)). To highlight the characteristics of the ACF after differentiation, in Fig. 1(b) the Y-axis is logarithmic. It can be seen that the decay rate of the same ACF at different delay times is different, showing the characteristics of first increasing, then decreasing, then increasing and decreasing. The first increase and decrease corresponds to the coherent decay of ACF, and the second increase and decrease corresponds to the incoherent decay of ACF. The boundary point of the two increases and decreases is at the minimum value of the ACF characteristic curve.
Fig. 1. (a) The intensity ACF of a 690 nm unimodal particle system; (b) Decay characteristic curve of the ACF.
It can be also seen from Fig. 1(b) that the characteristic curves of the ACF completely overlap in the first increase/decrease segment at all ultra-low concentrations. Examination of Fig. 1(a) shows that for all concentrations the ACFs in the coherent decay segment have the same decay characteristics as the ACF at normal concentration, the only difference is that the ACFs increase in amplitude by a height of ${\gamma / {\langle N\rangle }}$ (the height of incoherent decay). In the second increase/decrease segment, the characteristic curve of ACF has the same changing behaviour and the amplitude increases as the concentration decreases, which indicates that the incoherent decay has a tendency to become faster as the concentration decreases. Since the critical point of these two increase/decrease segments is the minimal value point of the characteristic curve, the value of the vertical coordinate of this point, ${\gamma / {\langle N\rangle }}$, which is both the minimum of the Gaussian decay term and the maximum of the non-Gaussian decay term. This minimum is used to truncate the ACF, and the truncation position is the delay time ${\tau _\textrm{c}}$ corresponding to the minimum [see in Fig. 1(a)], so that the separation of Gaussian and non-Gaussian terms can be achieved.
The intensity ACFs after separating the non-Gaussian terms are shown in Fig. 2. These ACFs are now the same as Eq. (1). During the delay time from 10−5s to 4×10−2s, the changes of the non-Gaussian term in magnitude between 10−6 and 10−8, which can be ignored compared with the amplitude of the intensity ACF. Therefore, we took the amplitude of the non-Gaussian term for the above delay period as a new baseline. This baseline can been calculated from measured ACFs by non-linear least square fitting, and then Eq. (4) can be obtained from the Siegert relation, whose matrix form is
where ${\textbf{g}^{(1 )}}$ is a vector with elements of ${g^{(1 )}}({{\tau_j}} )$ and whose dimension is M×1, $\textbf{A}$ is a kernel matrix including elements of $A(j,i) = \exp ( - \frac{{16\pi {n^2}{K_B}T}}{{3\eta {\lambda _0}^2{d_i}}}{\sin ^2}\left( {\frac{\theta }{2}} \right){\tau _j})$ and whose dimension is M×K, $\textbf{f}$ is a vector containing $f({{d_i}} )$ and whose dimension is K×1.The solution of $\textbf{f}$ from Eq. (8) is an ill-posed problem, which usually resorts to regularization methods such as constrained Tikhonov regularization method to obtain reliable results. With this method, the solution of Eq. (8) can be transformed into a constrained function optimization problem, as follows
4. Simulation data analysis
To verify the effect of the above separation of Gaussian and non-Gaussian terms, simulated intensity ACF data for two unimodal PSDs at 151nm and 690nm particle size were inverted using the normal DLS method and the non-Gaussian term separation method, and the results were analyzed. The simulated PSDs were based on a log-normal distribution [34]
The PSD can be obtained by adjusting the log-normal distribution parameters, and the corresponding intensity ACF data without noise are obtained by combining Eqs. (4) and (5). Two groups of PSD and their corresponding ACF data were simulated with the parameters shown in Table 1. All ACF data were simulated with ${\lambda _0}$=532nm, ${w_0}$=54µm, $l$=200µm, $T$ = 298.15K, $n$=1.3316, $\theta$=90°, ${K_B}$=1.3807×10−23J/K, $\eta$=0.89×10−3cP, and $\beta$=0.7. The number of discrete points in the PSD was set to K=150, and four particle number concentration cases of < N>=6, 12, 24 and 48 were taken in the range of < N>≤50.
Table 1. Parameters of the Simulated PSDs
The simulated ACF data without noise were processed by the normal DLS data processing method and the non-Gaussian term separation method, respectively, and then Tikhonov regularized inversion was performed. The inversion results are shown in Figs. 3 and 4.
Fig. 3. Unimodal PSD results for number concentrations < N>=6, 12, 24 and 48. Analysis using the normal DLS method. (a) 151 nm; (b) 690 nm.
Fig. 4. Unimodal PSD results for number concentrations < N>=6, 12, 24 and 48. Analysis using the non-Gaussian term separation method. (a) 151 nm; (b) 690 nm.
It can be seen from Fig. 3 and Fig. 4 that for the single-peak PSD of 151nm and 690nm, in the ideal case of no noise, compared with the normal DLS data processing method, the use of the non-Gaussian term separation method to process the intensity ACF eliminates the artifact peaks at large sizes in the PSDs, thus making the recovered PSD more accurate.
Since measured intensity ACF data inevitably contain measurement noise, different levels of Gaussian random noise are added to make the simulated ACF data more realistic so that,
where $G_{noise}^{(2)}(\tau )$ is the noisy intensity ACF, ${G^{(2)}}(\tau )$ is the noiseless intensity ACF, $\delta$ is the noise level, and $n(\tau )$ denotes Gaussian random noise.Function differentiation is problematic. Any noise in the ACF will result in unpredictable errors to the derivative calculation. The decay characteristic curve of the ACF is shown in Fig. 5 for the derivative of the ACF with a very low level of noise, $\delta = {10^{ - 7}}$. It can be seen that, due to the influence of noise, the decay characteristic curve exhibits fluctuations in the long delay time region and there are multiple minima. This makes it difficult to accurately distinguish the Gaussian and non-Gaussian terms.
Fig. 5. Decay characteristic curves of intensity ACF for 690 nm particles at number concentrations of < N>=6, 12, 24 and 48, at noise level δ=10−7.
To overcome the problem of differentiating the noisy ACF data, the data can be smoothed by fitting it to Eq. (5) using nonlinear least squares. The noisy ACF and the fitted ACF are shown in Fig. 6(a), where the solid line represents the fitted ACF. The difference between the noisy ACF and the fitted ACF is shown in Fig. 6(b). It can be seen that Eq. (5) fits the noisy ACF well, and the fitting error gradually decreases with increasing < N>. Figure 6(c) is the ACF decay characteristic curve obtained by differentiating the fitted ACF.
Fig. 6. (a) Noisy and fitted intensity ACFs for 690 nm particles and concentrations of < N>=6, 12, 24 and 48 at noise level δ=10−7; (b) error between noisy ACF and fitted ACF; (c) the decay characteristic curve of the fitted ACF.
To characterize the accuracy of the PSD recovery, we introduce two performance evaluation indices: the relative error of the peak position (${E_P}$) and the PSD recovery error ($V$), which are defined as
Here ${f_{true}}(d )$ and ${f_{meas}}(d )$ are the true PSD and the PSD obtained by inversion respectively, ${P_{true}}$ and ${P_{meas}}$ are the corresponding peak particle sizes.
The simulated inversion results of the noisy intensity ACF using the two methods are shown in Figs. 7–10. Figures 7 and 9 are the inversion results obtained by the normal DLS method at different noise levels, and Figs. 8 and 10 are the inversion results obtained by the non-Gaussian term separation method at different noise levels. The performance indices of the corresponding inversion results are shown in Tables 2 and 3. In the tables, Method 1 denotes the normal DLS data processing method and Method 2 denotes the non-Gaussian term separation method.
Fig. 7. PSD recovery results for simulated ACF data for 151 nm particles using the normal DLS data processing method at particle concentrations of < N>=6, 12, 24 and 48, and different noise levels. (a) 10−4, (b) 10−3, (c) 10−2, and (d) 2×10−2 noise levels.
Fig. 8. PSD recovery results for simulated ACF data for 151 nm particles using the non-Gaussian term separation method at particle concentrations of < N>=6, 12, 24 and 48, and different noise levels. (a) 10−4, (b) 10−3, (c) 10−2, and (d) 2×10−2 noise levels.
Fig. 9. PSD recovery results for simulated ACF data for 690 nm particles using the normal DLS data processing method at particle concentrations of < N>=6, 12, 24 and 48, and different noise levels. (a) 10−4, (b) 10−3, (c) 10−2, and (d) 2×10−2 noise levels.
Fig. 10. PSD recovery results for simulated ACF data for 690 nm particles using the non-Gaussian term separation method at particle concentrations of < N>=6, 12, 24 and 48, and different noise levels. (a) 10−4, (b) 10−3, (c) 10−2, and (d) 2×10−2 noise levels.
Table 2. Performance Parameter Values for the Recovery of the 151nm Unimodal PSD by Using Normal DLS Data Processing Method (Method 1) and Non-Gaussian Term Separation Method (Method 2)
Table 3. Performance Parameter Values for the Recovery of the 690nm Unimodal PSD by Using Normal DLS Data Processing Method (Method 1) and Non-Gaussian Term Separation Method (Method 2)
Figure 7 shows that when the average number of particles in the scattering volume is as low as < N>=6, 12, 40 and 48, regardless of the noise level, the PSDs obtained by the normal DLS method contain strong artifact peaks at large particle size. It can be seen from Fig. 8 and Table 2 that, compared with Fig. 7, the PSD obtained using the non-Gaussian term separation method to process the intensity ACF can effectively eliminate the strong artifact peak, and the relative error of the peak position and the PSD recovery error are significantly reduced. At low noise levels, $\delta = {10^{ - 4}}$ and $\delta = {10^{ - 3}}$, the relative error of the peak position is as low as 0; while at noise levels $\delta = {10^{ - 2}}$ and $\delta = 2 \times {10^{ - 2}}$, the inversion result is a main peak and a very weak artifact peak. This weak artifact peak, which is caused by noise and tends to become stronger as the concentration decreases to < N>=6 and 12. In all cases this peak is negligible compared with the main peak.
As can be seen from Fig. 9, the PSD results obtained by the normal DLS method for the simulated ACF data for the 690nm unimodal PSD are very similar to those for the 151nm simulation results. The PSD also contains the strong artifact peak that makes the main peak deviate from the true peak. Figure 10 and Table 3 show that for the 690nm unimodal distribution, using the non-Gaussian term separation method improves the PSD results as it did for the 151nm unimodal distribution, but the degree of improvement is more significant. At noise level $\delta = {10^{ - 2}}$, the relative errors of the peak positions of the inversion results obtained by the non-Gaussian term separation method are as low as 0, 0, 0, and 0.01 at < N>=6, 12, 40 and 48, respectively. At noise level $\delta = 2 \times {10^{ - 2}}$, the peak errors are as low as 0.04, 0.01, 0.01, and 0. As the noise increases the distribution broadens and weak peaks appears in the PSD. In Fig. 10(d), the weak peak is slightly enhanced when < N>=6.
5. Experimental results
Experimental ACF data were obtained from an experimental setup consisting of a solid-state laser (MGL-III-532nm-15mW) with a wavelength of 532nm and a power of 15mW, a photon counter (model CH326, Hamamatsu Photonics) and a 512-channel digital correlator [35]. The particles used were standard polystyrene latex spheres of size 152 ± 5nm (Duke, 3150A) and 693 ± 10nm (GBW(E)120087). These particles were diluted with filtered distilled water to the desired sample concentrations. The volume fractions of the 152nm samples were 9.6×10−9, 1.9×10−8, 3.8×10−8 and 7.7×10−8, respectively, and the volume fractions of the 693nm samples were 9.1×10−7, 1.7×10−6, 3.6×10−6 and 7.3×10−6. The volume fractions of these two groups of samples corresponded to average particle numbers of 6, 12, 24 and 48 in the scattering volume, respectively. The sample temperature was 298.15K, the scattering angle was 90°, the measurement time was 120s, the focal length of the incident lens was 175mm, and the receiving aperture of the detector was 400um. Results of measurements from these standards at normal concentrations (NC) are shown in Fig. 11. The recovered PSD results from the measured data at ultra-low concentrations are shown in Fig. 12, and the corresponding performance indexes are shown in Table 4.
Fig. 11. The standard measurement results of standard polystyrene latex spheres at normal concentrations, (a) 152nm; (b) 693nm.
Fig. 12. PSD recovery results for standard polystyrene latex spheres at particle concentrations of < N>=6, 12, 24 and 48. 152nm samples using the normal DLS data processing method (a), and the non-Gaussian term separation method (b); 693nm samples using the normal DLS data processing method (c), the non-Gaussian separation method (d).
Table 4. Performance Parameter Values for the Recovery of the 152nm and 693nm Unimodal Standard Polystyrene Latex Particles by Using Normal DLS Data Processing Method (Method 1) and Non-Gaussian Term Separation Method (Method 2)
At normal concentrations, the peak values of recovered PSD for 152nm and 693nm particle samples were 158nm and 723nm, respectively. The corresponding peak position relative errors are 0.04 and 0.04 respectively.
As shown in Fig. 12 and Table 4, when the same measured data are processed using the non-Gaussian term separation method, the strong artifact peaks are removed and accurate peak positions that cannot be obtained by the normal DLS method are obtained. Table 4 also shows that the relative errors of the peak position recovery results using both methods tend to decrease with increasing particle number concentration, but this trend is more obvious for Method 2. At a particle concentration of < N>=48, these errors are as low as 0.03 for both the 152nm and 693nm particle samples. By comparing the recovery results of Method 2 with the standard measurement results at normal concentrations, it can be seen that for both the 152nm and 693nm samples, the recovered PSDs at < N>=48 are close to those obtained for the standard measurement at normal particle concentrations. Comparing the results from the measured and simulated data, it can be seen that the measured data results are closer to the recovery results from the simulated data at higher noise levels. This simply indicates that when measuring at ultra-low concentrations, the weaker scattering signal due to the low number of particles in the scattering volume, reduces the signal-to-noise ratio of the measured signal.
6. Conclusions
For ultra-low concentration particle suspensions, number fluctuations lead to a slow decay in the intensity ACF in DLS measurements, and this slow decay causes the recovered PSD to contain a strong artifact peak. To achieve more accurate measurement of PSDs at ultra-low concentrations, the different decay characteristics of particle Brownian motion and number fluctuations in the intensity ACF make it possible to separate the two terms. We propose separating the non-Gaussian term by differentiating the intensity ACF, and then using the intensity ACF containing only Gaussian term for the recovery of the PSD. The recovery results for simulated DLS data for 151nm and 690nm particles with < N>=6, 12, 24 and 48 in the scattering volume at four noise levels show that, compared with the normal DLS data processing method, the inversion of the ACF after the separation of non-Gaussian terms by the Tikhonov regularization method can effectively eliminate the strong artifact peaks and significantly improve the PSD results. This degree of improvement is more significant at lower noise levels. Under the same dilution conditions as the simulated data, the recovery results of the measured data for standard polystyrene latex particles verify this conclusion. The accuracy of the results gradually increases with < N>, and the recovery of the measured data at a concentration of < N>=48 gives recovered results close to those of particle measurements under normal measurement concentration conditions.
Funding
National Natural Science Foundation of China (61801272); Natural Science Foundation of Shandong Province (ZR2020MF124); Key Technology Research and Development Program of Shandong (2019GGX104017); Shandong University of Technology and Zibo City Integration Development Project (2019ZBXC011).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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