1 Introduction

Nanoparticles are widely used in industrial domains such as bio-materials, lubricating additives, cosmetics and so on, aiming to improve the performance of products. The appropriate applications of nanoparticles rely on in-situ and precise measurement of particle size [1, 2]. SEM/TEM might be intuitional for nanoparticle observation, but tremendous pretreatment are needed which introduces unpredictable change of the natural form of the measurand, especially for particles in liquid. Dynamic light scattering (DLS) is a well-known, non-invasive and in-situ method which can be applied in the above circumstance [3,4,5,6].

Though the applications of DLS method has been so hot, the precision as well as the consistency with other method (such as SEM and AFM) of this method is still a problem needed to be discussed. Due to the unstable hydrodynamic radius/diameter measurement results with large uncertainty and large dispersion by adopting different apparatus, it is still controversial to propose an international key comparison, only several interlaboratory comparisons being carried out since the year of 2005–2019 [7,8,9,10]. Most of the commercial DLS instruments being used by now are single-angle measurement, providing two different choices at 90° and a higher one. However, previous researches [11, 12] show that even for mono-dispersed nanoparticle, the apparent diffusion coefficient varies with scattering angle due to Mie scattering as well as long-distance interaction between particles, which gives rise to the angular dependence of particle size measurement results by using DLS method. In addition, particle size, concentration, type, refractive index and electrolyte will all play a role in size measurement results [13,14,15].

Researchers also tried to apply multi-angle dynamic light scattering in theoretically simulation and obtaining of the size distribution from several correlation functions in the case of a polydisperse suspension.[16]. The purpose of doing this is to add more constraint equations to the inverse matrix so that more information from the unknown particle samples could be revealed. However, most recent research showed that DLS results could significantly deviate from that obtained in theory at some specific scattering angles [17], which is undesired and has not been thoroughly explained and studied. Some of the present test rigs or instruments provided the function of multiple-angle measurement by rotating the angle of the receiver. In addition, multi-angle dynamic scattering could also be used as a tool to analyze particle shape by introduce depolarized light path [18], provided that autocorrelation functions obtained from different scattering angles are valid. If the accuracy of the light path in depolarized DLS cannot be guaranteed, measurement will be incorrect due to the undesired multiple scattering.

In the present work, a multiple-angle dynamic light scattering apparatus is proposed and established. A special pin-hole set and precise alignment design are applied into the light path to increase signal-to-noise ratio. Index-matching fluid and beam-stops are used not only at the emergent end of the laser light, but also at all scattering angle to avoid the undesired error light signal coming from the supplementary angle caused by glass-air surface reflection mixing into our measuring angle. Polystyrene latex (PSL) particle sized from 23 to 500 nm diluted with different concentrations are tested at different scattering angles. Methods of precisely determine the apparent hydrodynamic radius of PSL sized from nano to sub0-micro meters by using extrapolation method has been studied and discussed.

2 Experimental

2.1 Multiple-angle DLS apparatus

Figure 1 shows the schematic diagram of multiple-angle DLS system.

Fig. 1
figure 1

Schematic diagram of multiple-angle DLS

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In the present system, a 632.8 nm He–Ne laser (21 mW, Thorlabs) beam illuminates PSL sample after passing through a series of pinholes, a polarizer, a lens and a slit. Pinholes and the slit are used to minimize stray light so that the signal-to-noise ratio can be highly improved. Polarizer is optional when (a) the sample is spherical or (b) intensities of scattering light at different angles are not concerned. Here in this system, Glan Thompson prism is used to provide vertical polarization state of laser radiation. The laser is then focused by the lens to the center of the sample to limit the scale of scatterer and to increase radiance. As the scatterer is illuminated, eight photomultipliers (PMTs, Hamamatsu Photonics) arranged by the side of the sample start to receive scattering light from 30° to 150° and convert it into pulse signals that can be further processed into auto-correlation functions. The coherence of scattering light is guaranteed by connecting a lens and a monomode optical fiber in front of each PMT. A CCD is used to ensure all the lenses are towards the same scattering center, and precise scattering angle can be then calculated according to the captured photograph. Temperature is maintained at 20.00 ± 0.02 °C by using water cooling system and monitored by using a high resolution platinum resistance thermometer (PRT).

As shown in Fig. 2a, the incident laser beam passes through sample cuvette and illuminates the scatterer. At the interface of glass-air where laser exits the cuvette, about 4% of the light will be reflected back (see case A). This has been verified by detecting light intensity before and after the laser passing through an empty cuvette. The reflected laser light then illuminates the scatterer again oppositely from the incident direction so that an error scattering light signal will mix into the valid signal. Theoretically, reflection inside the cuvette will last several times before dissipation. Similarly, scattering light at the supplementary angle of the measured angle will also be reflected at the glass-air interface and then mixed into the valid signal (see case B). Finally, the autocorrelation function may contain scattering signals from two angles: the measured angle and its supplementary angle. This may lead to a size deviation from true DLS result (if cumulant algorithm is adopted) or an additional “peak” indicating some nonexistent particle size (if CONTIN algorithm is adopted). Only at 90° can the above influence be eliminated, because the supplementary angle of 90° is still 90°. In the present apparatus, decalin is used as the index matching liquid and beam-stops are arranged inside the decalin bath at the corresponding supplementary angles mentioned above, so that the error signal can be eliminated to a great extent. The diameters of the sample and immersion cuvette is 20 mm and 120 mm, respectively.

Fig. 2
figure 2

Undesired light signal due to a surface reflection and b its elimination by using beam stops

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2.2 Sample preparation and test procedures

Four different sized PSL samples were supplied by Thermo Scientific with diameters of about 23 nm, 50 nm, 100 nm, and 303 nm. Another 500 nm PSL sample was supplied by Tianjin Saierqun Technology Co. Ltd. All the samples were observed by using TEM and SEM again so that the diameters could be re-confirmed. The PSL suspensions were diluted with ultrapure water by different multiples and filtered if necessary. The PSL suspensions were stable, and no additional salts or surfactants were needed as stabilizers during or after dilution. Each test lasted for 120 s with the minimum sampling time set to be 1 μs. For much diluted sample or some special case where the scattering light intensity was very low, test duration could be extended as appropriate. Each test was repeated at least six times to ensure the repeatability and reliability of the results.

2.3 Calculation of the apparent hydrodynamic diameter

From the above experimental set-up, electric-field time correlation function can be obtained from the scattered light, and then analyzed by the cumulant method as described in the International Standard ISO 13,321 to evaluate the average decay rate [5]. Apparent hydrodynamic diameter from one single scattering angle can be calculated by the following equation:

$$ d = \frac{kT}{{3\pi \eta D}}, $$
(1)

where D is the diffusion coefficient which can be calculated from the decay rate of the autocorrelation function Γ:

$$ \begin{gathered} \Gamma = Dq^{2} \hfill \\ q = (4\pi n/\lambda )\sin (\theta /2), \hfill \\ \end{gathered} $$
(2)

and where q is the scattering vector, n is the refractive index of the solvent and λ is the wavelength of the incident light in vacuum. k, T, and η are the Boltzmann constant, absolute temperature, and solvent viscosity, respectively.

After measuring the apparent hydrodynamic diameter from different scattering angles and different sample concentrations, the final diameters were extrapolated to infinite dilution and to zero angle to which reflected the true diffusion coefficients.

3 Results and discussions

3.1 Correction on the measurement results after introducing beam-stops

Data shown in Fig. 3 compares the particle size measurement results of 100 nm and 500 nm PSL before and after using beam-stops. As can be seen from the results, for smaller particles, the differences of whether beam-stops are used or not can be neglected, since the differences of scattering intensity at different angle is not significant enough. However, when particle size increased, the influence of Mie scattering became stronger. The 4% reflection of error scattering light became a major problem which resulted in huge deviation of the theoretical scattering signals receiving from the target scattering angle.

Fig. 3
figure 3

Comparison between measurements with and without beam-stops. The red hollow spheres represent measurements results without beam-stops; the black hollow squares represent measurement results with beam-stops

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For PSL sized of 100 nm, the difference with and without beam-stops were not so significant. This might mainly due to the differences of Mie scattering of small nanoparticles at different scattering angles were small. As shown in Fig. 3b, however, results obtained with and without beam-stops showed very large deviation. Without beam-stops, smaller size was measured at smaller scattering angle while extremely large size could be measured at larger scattering angle. At 150°, the apparent hydrodynamic diameters reached 2700 nm which cannot be displayed here on the scale of this figure. Larger size measurement results also can be observed at some certain angles with very weak scattering intensity. This was mainly due to the influence of Mie scattering on the apparent diffusion coefficient D, and was exaggerated when mixed into error scattering signals. After using beam-stops, abnormal values were corrected. Results obtained with beam-stops revealed the nature of Mie scattering at different angles of large particles. Based on the measurement results of different sized PSL samples, it was found that for particle size smaller around or smaller than 100 nm, experiments carried out only with one beam stop at the end of the incident light path was acceptable; however, for larger particle size measurement, beam stop should be placed at each scattering angle in order to obtain precise result.

3.2 Multiple scattering at different scattering angle

Multiple scattering in the highly concentrated sample usually caused wrong measurement results [19, 20] and should be avoid in the present study where concentration dependence was considered as a very important part of the extrapolation. So before the experiment was conducted, proper sample concentration at each scattering angle should be determined. Scattering intensity was considered here as a main reference of determine if multiple scattering occurs.

Figure 4 shows two examples (50 nm and 303 nm PSL) of the scattering intensity at different scattering angle. The solid lines represented ideal scattering intensities without multiple scattering. Actually, the measured intensities gradually deviated from the solid lines as sample concentration increased, which indicated multiple scattering occurred. It was clear that the critical concentration at which the multiple scattering occur varied with different scattering angle. With the increasing of scattering angle, the critical no-multiple-scattering PSL concentration increased and then decreased, and 90° was the best testing angle to avoid multiple scattering. Thus, the following experiments were all based on this precondition in order to avoid the influence of multiple scattering.

Fig. 4
figure 4

Examples of multiple scattering at different scattering angles of PSL sized of 50 nm and 303 nm

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3.3 Angular and concentration dependence of DLS measurement results

Figure 5 shows the results of apparent hydrodynamic diameter of the tested sample. Each sample shows very unique pattern of angular dependence. For very small PSL nanoparticle with diameter of 23 nm, the apparent hydrodynamic diameter decreased linearly with increasing scattering angle. When particle size increased to 50 nm, the apparent hydrodynamic diameter started to slightly increase linearly with increasing scattering angle. When particle size increased to 100 nm, the increasing trend was almost linear but started to show quadratic behavior. The angular dependence of 300 nm and 500 nm PSL samples showed very clear cubic behavior, and became more winding with increasing particle size.

Fig. 5
figure 5

Apparent hydrodynamic diameter of different PSL sample. a 23 nm, concentration 1 × 10–5% w/w, fitting method: linear; b 50 nm, concentration 1 × 10–5% w/w, fitting method: linear; c 100 nm, concentration 5 × 10–5% w/w, fitting method: binomial; d 300 nm, concentration 5 × 10–5% w/w, fitting method: cubic; e 500 nm, concentration 1 × 10–5% w/w, fitting method: cubic. The red dash line indicates the fitting line of the apparent hydrodynamic diameter against scattering angle

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According to Mie scattering theory, when particle size smaller than 1/10 λ (λ is the wavelength of the incident laser) in a diluted system, each particle can be seen as a whole so that there is no significant scattering interference caused by intraparticle interference. So for smaller particle, long-distance interaction played a major role in the present colloid system [12]. In the present research, for PSL particles smaller than 50 nm, the above reason caused a nearly linear change of the apparent diffusion coefficient with the increasing scattering vector q, which resulted in the corresponding pattern of the hydrodynamic diameter.

With the increasing of particle size, intraparticle interference must be taken into consideration when evaluating their diffusion patterns. Intraparticle interference is caused by wavelets scattered from different segment of one single particle [11]. These wavelets can add constructively into the scattering light at each angle of the observation, especially at large values of the scattering vector q. Due to the above reason, at certain scattering angles and particle sizes, the scattering intensity tends to zero; while at others, tends to be very large. So for a particle system with finite size distribution width, size measurement result shifts with scattering angle. In addition, if the intraparticle interference cannot be neglected, the interparticle interference pattern of the whole scattering center will also be changed.

The combined effects of long-distant interaction as well as intraparticle interference on scattering light lead to the much more complex patterns of the apparent diffusion coefficient at each scattering angle. Equations (3) and (4) describes the scattering correlation function I(q) for the case of large particles [12]:

$$ I_{ij} (q) = (n_{i} )(n_{j} )^{2} < N > \alpha^{2} S(q), $$
(3)

where ni and nj represent scattering segments from different particle of which the effects can be neglected in infinite diluted suspensions; α is the molecular polarizability; S(q) is defined as the molecular form or structure factor, and can be calculated as

$$ S(q) = \frac{1}{{n^{2} }} < \sum\limits_{l,m} {\exp (iq \cdot (r_{l} - r_{m} ))} > . $$
(4)

From the above descriptions it can be estimated that, only when q is close to zero and in the infinite diluted suspensions, will the most accurate particle size can be solved from the autocorrelation function of the scattering field.

In the present test, the PSL sample used in this test had negative zeta potential and could be stabilized in water at any concentration, no additional surfactant was needed. For other types of particles with different charges or with different kinds of surfactants, this trend might be different. In addition, the influence of particle shape will also become more significant for larger particles. By using this multiple angle scattering system, behaviors and mechanisms of all the above mentioned particle system can be further studied.

Figure 6 shows the concentration dependence of different samples. All the above results were obtained at the scattering angle of 90°. As we could see, the apparent hydrodynamic diameter decreased with increasing sample concentration. Sample concentration significantly contribute to the effects of long-distance interactions between particles. For solutions in the limit of the infinite dilution, scattering patterns only reflected the nature of one single particle without any interferences with other particles. However, much diluted samples were not suggested to be used in DLS measurement. So that extrapolation was often adopted as an alternative. Note that in the cases of larger particle measurement, for example, samples sized of 300 nm and 500 nm, the angular dependence was more complex than smaller nanoparticles. So that at other scattering angles, the slope of concentration dependence may change accordingly.

Fig. 6
figure 6

Apparent hydrodynamic diameter of different PSL sample measured at 90°. a 23 nm; b 100 nm; c 300 nm; d 500 nm. The blue dash line indicates the fitting line of the apparent hydrodynamic diameter against concentration

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3.4 Particle size determination by using extrapolation method

As shown in Fig. 7, particle size was then calculated by using extrapolation from the data fitting results of different scattering angle and different sample concentration. By doing this, the effects of both long-distance interactions and intraparticle interference can be theoretically eliminated to the max extent. Table 1 gives the final size after using extrapolation method.

Fig. 7
figure 7

Extrapolation of the apparent hydrodynamic diameters of different PSL samples. a 23 nm; b 100 nm; c 300 nm; d 500 nm. The blue dash line indicates the fitting line of the apparent hydrodynamic diameter by using extrapolation method

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Table 1 Apparent hydrodynamic diameters of the particle samples after extrapolation method
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The calculation of uncertainty combines the uncertainty of the apparent hydrodynamic diameter of each sample as well as the uncertainty of the extrapolation. The principle was based on the international document “Guide to the Expression of Uncertainty in Measurement” (GUM) [21]. Particle size calculation from the autocorrelation function was described in the above as Eq. (1), and the uncertainty can be calculated as follows:

$$ \begin{gathered} u^{2} = \left(\frac{\partial f}{{\partial k}}\right)^{2} u_{k} (k)^{2} + \left(\frac{\partial f}{{\partial T}}\right)^{2} u_{T} (T)^{2} + \left(\frac{\partial f}{{\partial \eta }}\right)^{2} u_{\eta } (\eta )^{2} + \left(\frac{\partial f}{{\partial \Gamma }}\right)^{2} u_{\Gamma } (\Gamma )^{2} + \left(\frac{\partial f}{{\partial q}}\right)^{2} u_{q} (q)^{2} \hfill \\ \quad = f^{2} \left[\frac{{u_{k} (k)^{2} }}{{k^{2} }} + \frac{{u_{T} (T)^{2} }}{{T^{2} }} + \frac{{u_{\eta } (\eta )^{2} }}{{\eta^{2} }} + \frac{{u_{\Gamma } (\Gamma )^{2} }}{{\Gamma^{2} }} + \frac{{4u_{q} (q)^{2} }}{{q^{2} }}\right], \hfill \\ \end{gathered} $$
(5)

and where

$$ \begin{gathered} u_{q} (q)^{2} = \left(\frac{{\partial f_{q} }}{\partial n}\right)^{2} u_{n} (n)^{2} + \left(\frac{{\partial f_{q} }}{\partial \lambda }\right)^{2} u_{\lambda } (\lambda )^{2} + \left(\frac{{\partial f_{q} }}{\partial \theta }\right)^{2} u_{\theta } (\theta )^{2} \hfill \\ \quad \quad = f_{q}^{2} \left[\frac{{u_{n} (n)^{2} }}{{n^{2} }} + \frac{{u_{\lambda } (\lambda )^{2} }}{{\lambda^{2} }} + \frac{{u_{\theta } (\theta )^{2} }}{{4\tan^{2} (\theta /2)}}\right]. \hfill \\ \end{gathered} $$
(6)

When extrapolation method was applied, an additional component contributed to the uncertainty calculation also should be considered: that is the repeatability of the extrapolation method based on several repeated experiments.

From Table 1, it can be seen that, the apparent hydrodynamic diameter of the tested PSL samples obtained by MDLS extrapolation are basically 6–8% larger compared to the corresponding nominal value determined by using SEM. The corresponding uncertainties of the two methods can be quite close. And for sample sized smaller than 1/10 λ, the apparent hydrodynamic diameter measured from one single scattering angle can be similar to that obtained from extrapolation method. It should be noted here that the consistency described above could highly depend on the properties of the dispersion system, such as particle type, charges, concentration, etc. Surfactants might be necessary to stabilize particle dispersity in some certain cases. In such cases, the hydrodynamic diameter might be even larger compared to the nominal diameter obtained from SEM or TEM due to the presence of adsorbed molecular layer(s).

4 Conclusions

By using home-made multiple dynamic light scattering apparatus, PSL sized from nano to sub-micro meters were precisely measured. The calibration of each parameter of the apparatus, the validity of signal acquisition as well as the avoidance of multiple scattering are taking into consideration in order to assure measurement accuracy to the full extent. Extrapolation method was used to diminish the influences of inter-particle interactions and inner particle scattering interference in the cases of larger particles. For PSL system without any surfactants, hydrodynamic diameter obtained from extrapolation method can be 6–8% larger than that obtained by using SEM. This may provide a perspective for the measurement consistency by using different DLS apparatus and when testing sample with different dilution ratios. Particles sized from nano to sub-micron meters can be precisely measured by using the present method, which also may further contribute to the validity and precision when calculating wide particle size distribution by adopting multiple angle inversion algorithm.