Advances in Nanoparticle Sizing in Suspensions: Dynamic Light Scattering and Ultrasonic Attenuation Spectroscopy

Abstract

Size characterization of nanoparticles has gained wide concerns in the past decades, but it remains a challenge for measurement in suspensions up to now. The extremely small scales of particle size result in great difficulty for traditional static light scattering method and optical imaging. In addition to the electron microscopy techniques, the dynamic light scattering (DLS) method is another widely used technique for laboratory analysis of samples. Moreover, the ultrasonic attenuation spectroscopy (UAS) technique is also being developed rapidly to provide an alternative method for nanoparticle sizing. This paper focuses on the latest development in the above two technologies for nanoparticle size characterization. As for the former, advances about the image-based DLS technology in recent years are reviewed, including three different kinds of data processing methods and corresponding measuring experiments using standard polystyrene particles. Methodology principles, models and experimental setup were also reviewed for the latter UAS technology. Samples of the same nanoscale silver particles were tested by the above two methods, as well as by transmission electron microscopy. A sample of Antimony Tin Oxide (ATO) nanoparticles has also been adopted for measurements and comparisons. Relatively consistent results can be found by comparing the particle sizes or distributions with various methods. The dramatically reduced measurement time in image-based DLS indicates the potential for real-time and in-situ nanoparticle sizing. UAS also provides a suitable way for nanoparticle size characterization at high concentrations.

1. Introductions

Nanoparticles are generally referred to as particles with diameter less than 100 nm, which are also called as ultrafine particles. Ultrafine particles often exhibit some unique effects such as quantum size effect, surface effect, etc., which bring particles many special properties. However, there is no reason to think that here is much difference between a 90 nm and a 110 nm particle in ability and there is little evidence for the cutoff diameter for the ultrafine effect (Donaldson K. et al., 2001). There are broad application prospects for them in catalysis, light filter, light absorption, medicine, magnetic media, new material development, and so on. Detailed studies are essential on methods of nanoparticle characterization. At present, popular methods used for ultrafine particle sizing include electron microscopy, laser particle sizing (combined with large-angle scattered light reception technology), high-speed centrifugal sedimentation, dynamic light scattering, ultrasonic method, etc. In many cases, ultrafine particles are suspended in a liquid medium to form a suspension, which is the basic state for measurement using dynamic light scattering and ultrasonic method.

Dynamic light scattering (DLS) has also been called as photon correlation spectroscopy, quasi-elastic light scattering, or intensity fluctuation spectroscopy in the past. It was first used for the particle sizing in the early 1970s (Pecora R., 1972; Silbey R. and Deutch J.M., 1972) and then got a booming development in the applications. Tscharnuter W. (2006) has also made a remarkable review on the developments of this technology before the 21^st century for particle sizing, including histories, principles, sampling, instrument and measurement.

In conventional DLS method, a photomultiplier tube is used to receive the scattered light from nanoparticles, and a digital correlator is utilized to process the one-dimensional time-series signal to obtain the autocorrelation function of the intensity fluctuation. Features of the autocorrelation function (ACF) curve reflect the Brownian motion characteristics of nanoparticles and thus the information of particle size. Commercial instrument based on DLS has become popular in off-line laboratory analysis. However, in order to obtain accurate ACF results, a large number of data is required. The measuring time usually lasts tens of seconds to hundreds of seconds. It is difficult to realize a real-time and online nanoparticle size measurement in unstable or rapidly changing processes.

Study of DLS method using a CCD (Charge Coupled Device) or CMOS (Complementary Metal Oxide Semiconductor) camera as an area detector was first proposed in the nineties of the last century. Wong A.P.Y. and Wiltzius P. (1993) utilized the scattered light at the same scattering angles on the same ring in a CCD sensor. The space-averaged information was used instead of time averaging, which greatly reduced the measurement time. Kirsch S. et al. (1996) collected the scattered light by a CCD camera at nearly identical scattering vectors. The CCD sensor was equivalent to an array of conventional photon sensors. The measurement accuracy was improved since sequential results of different pixels were averaged based on the conventional time average. However, limited by the frame rate of cameras, the above experiments (Kirsch S. et al., 1996; Wong A.P.Y. and Wiltzius P., 1993) were carried out in pure glycerol as dispersion medium, in which Brownian motion of the nanoparticles was very slow.

In addition, Dynamic Light Scattering Microscopy was proposed by Dzakpasu R. and Axelrod D. (2004). A progressive scan CCD was used and exposure through a column of the CCD array was adopted. The rapid access to time- and space- sequence signal of dynamic light scattering was realized. That caused an equivalent effect as increasing of frame rate. In this work, the method was used for nanoparticle sizing in water. Zakharov P. et al. (2006) carried out theoretical and experimental researches in order to suppress complex scattering problems in DLS measurements with the CCD camera.

As the rapid development of computer technology and imaging technology in recent years, the high-speed camera with tens of thousands of frames per second (fps) is available. On the other hand, the developed AOI (area of interesting) function in general industrial camera can greatly improve the frame rate by reducing image size. It provides the possibility of nanoparticle sizing in water. In this paper, we are mainly focused on the development of DLS method using image sensors. Three different data processing algorithms were reviewed here for image-based DLS (IDLS) method. The first one is based on the traditional time-based autocorrelation function. The second one is also based on autocorrelation function but with two-dimensional image cross-correlation. The third one is based on one cross-correlation coefficient between two images in a certain time interval, without the autocorrelation functions. A high-speed camera was used in experiments and the scattered light signal in two-dimensional space, as well as in time series were acquired. Images were manipulated in comparison of the three different data processing methods.

In addition to DLS, the ultrasonic attenuation spectroscopy (UAS) method is also considered to be very promising as a rapid on-line measurement technology for nanoparticle suspension system, which has advantages of simple structure, non-invasive and on-line measurement and being available to relatively dense suspensions.

In the field of theoretical modelling and experimental validation involved, Challis R.E. et al. (2005), Povey M.J.W. (2013) and Mcclements D.J. and Coupland J.N. (1996) conducted intensive studies from numerical calculation problems of the ECAH model based on scattering theory to the simplification for long wave limit, as well as experimental validations. Hipp A.K. et al. (2002) derived the Core-shell model which was suitable for high concentration conditions. Starting from the elastic ultrasound scattering of particles and by introducing Lambert-Beer principle based on an analogy to the Mie theory in light scattering, Riebel U. (1992) developed a model for large solid particles to depict the particle-wave interactions, including entrainment, scattering, and resonances in the short-wavelength regime. He also concluded theoretically the proper order of importance for high concentration systems as (i) steric, (ii) dependent scattering, and (iii) multiple scattering. Dukhin A.S. and Goetz P.J. (1996) and Dukhin A.S. et al. (1996) presented the coupled-phase model, in which, the Happel cell model was exploited to calculate the drag coefficient and a friction coefficient proportional to the volume fraction was introduced to describe the relationship between the gradient of pressure and the relative speed between the particle and the medium. The resultant DT Series commercial instruments were then released by combining acoustic and electroacoustic spectroscopy, whose greatest feature is being suitable for measuring highly concentrated colloidal particles which have been able to extend the volume fraction to 30 % or even more. Applications to a wide variety of materials were introduced too. Evans J.M. and Attenborough K. (2002) extended the coupled-phase model by way of including irreversible heat transfer and visco-inertial forces, based on the volume-averaged balance equations. They compared the predictions of the core-shell model and the extended coupled-phase model with measurements of ultrasonic velocity and attenuation in two different emulsions for volume fractions up to 70 %, which showed both models were in good agreements with the experimental results. Recently, Challis R.E. and Pinfield V.J. (2014) proposed a novel way to describe the ultrasonic wave propagation in concentrated slurries, in which a simple effective medium modification to the ECAH/LB was put forward to reflect the hydrodynamic effects involved, while it did not yet cause expected coincident results with those of the core-shell models.

In the characterization of ultrafine particles with UAS, Mougin P. et al. (2001) conducted an in-situ examination of solution phase crystallization processes on two organic compounds: urea and (L)-glutamic acid, in which ultrasonic attenuation spectra were measured by a commercial ultrasonic spectrometer (Ultrasizer by Malvern Instruments Ltd.) with two pairs of transducers operating over a frequency range from 1 MHz to 150 MHz. The experiment was carried out during a cooling process with temperatures from 50 °C to 15 °C (−0.4 °C/min), in which the particle sizes (median diameter 0.023–130.8 μm) and concentrations (0.1–0.81 vol %) were monitored continuously. Hou H. et al. (2010) measured an ultrafine ZnO-H₂O suspension with a mean size of approximately 200 nm, in which a variable-path method and the short-time Fourier transform have been adopted to obtain and extract ultrasonic attenuation spectrum and phase velocity synchronously. Wang X.Z. et al. (2009) carried out the on-line characterization of nanoparticle suspensions using dynamic light scattering, ultrasound spectroscopy and process tomography simultaneously. A silicon nanoparticle (70–100 μm in diameter) suspension with initial volume concentration of 24 % was diluted to prepare a series of samples with eight different concentrations as low as 0.8 vol %, thus the measurement of particle size distribution was implemented at different concentrations, and analysis was highlighted on nonlinear effects of the concentration on the final results. Furthermore, a modification (Liu L., 2009) was presented that in ultrasound particle sizing, factor analysis as a method of dealing with multivariate data can be applied to ultrasound spectra to extract the information of linear dependence of attenuation on concentration and the contribution of frequency to attenuation. On the basis of this work a closed-loop control system was further built to measure and control process parameters (Boonkhao B. and Wang X.Z., 2012).

In this paper, researches of UAS method for nanoparticle sizing are particularly focused on, including the following issues: theories and models for ultrasonic attenuation prediction; inversion method for solving particle size; ultrasonic signal transmission and detection technology. In order to clearly distinguish the ultrafine particles, high-frequency broad-band ultrasonic signals are critical to be utilized, which causes higher requirements from the theoretical model calculation to ultrasound generating and receiving technology.

The rest of this paper is arranged as follows. First, detailed reviews of IDLS and UAS for particle sizing are executed, respectively, including measurement principles, data processing, measuring apparatus, and validations experiments using polystyrene particles with various sizes. Then we focus on the characterizations of nanoscale silver and Antimony Tin Oxide (ATO) particles using IDLS and UAS, comparisons and analyses are then executed to come to the final conclusions.

2. Dynamic light scattering

2.1 Measurement principles and data processing

The principle of particle size measurement by DLS technique is based on the Brownian motion of nanoparticles (King T.A., 1989; Thomas A.J.C., 1991; Tscharnuter W., 2006). For spherical nanoparticles dispersed in solutions, the relationship between diffusion coefficient D_T of the Brownian motion and particle diameter D can be given by Stokes-Einstein equation:   

$$D_T=\frac{K_{\text{B}}T}{3\pi \mu D}$$(1)

where, K_B is the Boltzmann constant, T is the absolute temperature and μ is the dynamic viscosity of the dispersion medium. D_T is determined using DLS technology by analysing the scattering signal from nanoparticles, which is illuminated using coherent light beam. The scattering light fluctuate randomly in relation with Brownian motion. Smaller particles move faster and the fluctuation of scattered light intensity is also faster. For monodisperse system, the following autocorrelation function (Tscharnuter W., 2006) can characterize such a steady fluctuation.   

$$g^{(2)}(\tau )=1+\beta \text{exp}(-2\mathit{\Gamma }\tau )$$(2)

where, τ is the time delay between signals, β is an instrumental constant between 0 and 1. The decay constant Γ is related to the diffusion coefficient of the particles D_T by   

$$\mathit{\Gamma }=D_T{q}^2$$(3)

where, q is the norm of scattering vector, which indicates the difference between wave vector of the scattered light and the incident light. It is related to the refractive index of the dispersion medium n, the scattering angle θ and the wavelength λ of light by

  

$$q=\frac{4\pi n}{\lambda }\text{sin}\left(\frac{\theta }{2}\right)$$(4)

2.1.1. IDLS with conventional autocorrelation function

To exploit the information of decay constant Γ and diffusion coefficient D_T from one-dimensional fluctuation information of scattered light I(t), the intensity autocorrelation function will be fitted according to Eqn. 5 as conventional DLS data processing method.

  

$$G(\tau )=⟨I(t)I(t+\tau )⟩=\frac{1}{N}\sum _{i=1}^{N}I(t_i)I(t_i+\tau )$$(5)

In order to obtain reliable and accurate result, averaging was adopted on a large number of time-series data, since the fluctuation goes in a random way. It needs a long time for both data acquisition and data processing. The typical measurement time for the current conventional DLS particle sizer is generally 1–2 minutes, which is difficult to meet the requirements of real-time and online measurement for rapidly changing process. Since the Brownian motion of particles is sensitive to the temperature, a temperature control device is necessary in conventional DLS. It can be generally applicable to off-line analysis in laboratory.

IDLS is proposed to capture the information of Brownian motion using an image sensor (Wang Z.Y. et al., 2014; Xu C.Z. et al., 2015). One pixel or several pixels of the sensor is equivalent to one photomultiplier tube. There are generally hundreds of thousands to millions of pixels in one image sensor. As shown in Fig. 1, the image is divided into i × j grids, which is equivalent to i × j photomultiplier tubes. i × j time-series of light intensity signals will be obtained simultaneously. Eqn. 5 will be used to calculate the averaged autocorrelation functions and particle size D, or averaging can also be taken on the results D_i,j from each autocorrelation function. This takes the place of conventional averaging on autocorrelation functions from a long time-series signal, so the data acquisition time can be reduced dramatically.

Fig. 1

Images division for data processing (Xu C.Z. et al., 2015).

But from the perspective of data processing, the present IDLS still needs a sufficient amount of data and a long processing time to ensure a high accuracy of statistical results. Moreover, as for solutions with low viscosity or small particles, higher requirements are expected on the frequency of image acquisition, which is in other words the frame rate of the camera.

2.1.2. IDLS with ACF from image cross-correlation

Autocorrelation function of discrete-time signals characterizes the similarity between interval signals. It is obtained by correlation processing between one-dimensional signals in a certain time interval. In IDLS method, a camera can capture two-dimensional spatial distribution of light intensity as images, whose similarity can be obtained by image cross-correlation as Eqn. 6.

  

$$\begin{array}{l}G(\tau )=R(A(m,n),B(m,n))\\ =\frac{{\sum }_m{\sum }_n\left(A_{m,n}-\overline{A}\right)\left(B_{m,n}-\overline{B}\right)}{\sqrt{\left({{\sum }_m{\sum }_n\left(A_{m,n}-\overline{A}\right)}^2\right)\left({\sum }_m{\sum }_n{\left(B_{m,n}-\overline{B}\right)}^2\right)}}\end{array}$$(6)

Where, τ is the time delay between two images A and B, both of which have m × n pixels. That is to say, for a time-series of images, the 2D ACF values at different time delays can be determined by cross-correlation processing between images at the corresponding time delays, such as between the 1^st image and the 2^nd one, the 1^st and the 3^rd … and the 1^st and the last one (Liu L.L. et al., 2015; Zhou W. et al., 2015).

Considering three different sizes (79/482/948 nm) of standard polystyrene particles, the theoretical time-dependent autocorrelation functions with β = 1 can be calculated from Eqn. 1–4, as illustrated in Fig. 2. It shows that the autocorrelation function values decrease from 1 to nearly 0 in approximately 2 ms, 20 ms and 40 ms, respectively, for the three sizes of particles. Values close to 0 after the respective times have small effects on the curve fitting. As long as the frame rate of the camera is high enough to characterize the light intensity fluctuation, the data acquisition time will be shortened to be about 2 ms, 20 ms and 40 ms, respectively. If the images are captured in the frame rate of 10000 fps, the number of images required to cover the typical time-scales in auto-correlation function curves should be about 20, 200 and 400 frames. The time intervals between the 1^st image and any other image are the corresponding time delays in the 2D ACF. Conventional DLS acquires one value at a time, requiring a longer time-series data to get an averaged result. IDLS utilizes the information on two-dimensional space and the cross-correlation coefficients between images are space-averaged values.

Fig. 2

Theoretical ACF for nanoparticles of different sizes in water (Liu L.L. et al., 2015).

2.1.3. IDLS by one cross-correlation coefficient without ACF

As shown in Fig. 2, under the same time delay, the 2D ACF is a monotonically decreasing function of particle size. Lager particles lead to greater correlation coefficient. So the particle size may be determined by one correlation coefficient, which can be obtained by cross-correlation of two-dimensional light intensity signals in a time interval. Fig. 3 illustrates the variation of correlation coefficient, in another word theoretical ACF value, as particle size under different time delays (from 1 μs to 10 ms), with the same conditions as Fig. 2. It indicates that the data acquisition time can be shortened to the acquisition time of two images, and the data processing time is shortened to the time needed for cross-correlation of these two images.

Fig. 3

Variation of theoretical ACF value as particle size at different delayed times (Zhou W. et al., 2015).

It should be noticed that the measuring error can be diminished by selecting a proper time interval to maintain a moderate correlation coefficient, since higher accuracy can be achieved at larger slopes of the curves. It shows from Fig. 3 that more accurate results can be acquired from correlation coefficient in the range 0.15–0.85. Take 200 nm particles for example, the time interval should be in the range from 100 μs to 2 ms. There are also other parameters, such as scattering angle and laser wave length, which can be adjusted to achieve moderate correlation coefficient.

2.2 Experimental system

Experimental setup was developed to validate and compare the above mentioned methods. Schematic drawing of the experimental setup is shown in Fig. 4. The measurement system included light source, lenses, sample cell, diaphragm iris, high-speed camera and computer. A green laser diode at 532 nm with 50 mW continuous wave output was used as the light source. The laser beam was focused in the sample cell, just as other commercial DSL nanoparticle sizer has done. The scattered light from the nanoparticles was collected by a lens with focal length of 60 mm. An adjustable diaphragm was applied to define the measured region, with an opening diameter of 0.8 mm in the experiments. Limited scattered light was recorded by a high-speed camera (MotionProX3), instead of a photomultiplier tube in conventional DLS. The camera has an 8-bit CMOS sensor with total effective area of 15.4 × 12.3 mm² and one pixel area of 12 × 12 μm². The camera was fixed at a 45° angle with the incident light during the experiments.

Fig. 4

Schematic drawing of IDLS experimental setup.

To avoid the influence of multiple scattering, the solutions were diluted sufficiently to the particle concentration of about 0.1 vol %. The tests were all carried out at room temperature, which was measured during experiments. To meet the requirement of repeatability and measurement accuracy, all of the measurements were repeated five times.

2.3 Validation of IDLS using standard nanoparticles

Standard polystyrene particles (manufactured by Beijing Haianhongmeng Reference Material Technology Co., Ltd.) with three different sizes (79/482/948 nm) were used for validation of the proposed data processing methods. Accuracies were analysed respectively and the measurement times were compared.

2.3.1. Results using conventional ACF

It was noticed from the captured images that the scattering speckle size was about equal to the size of 4 × 4 pixels in our experiments. The speckle size is also called as one “coherence area” or one spot. So the images were divided into grids of the same size of 4 × 4 pixels. One grid is equal to one photomultiplier tube and one mean size value can be determined from the corresponding time-series intensity fluctuation. Averaged value on all the grids was taken as the final result. First the samples were tested in distilled water using a frame rate of 8290 fps at 200 × 128 resolution. That is the maximum frame rate of the camera employed here at that resolution. 2000 images were captured continuously in 0.24 s at a time. As shown in Fig. 2, 0.24 s is long enough to cover the typical time scale of the ACF for these three samples. The final results are illustrated in Table 1. It shows that even the minimum measurement error can reach 4.4 % at 8290 fps in distilled water. The reason is that not enough efficient signals were captured in the typical time scale, especially for small particles. Camera with higher frame rate should be used to improve the accuracy.

Table 1 Measured results of standard nanoparticles in different dispersion mediums at different frame rate (Wang Z.Y. et al., 2014)

Dispersant Frame rate	Distilled water		55 % glycerol water solution
	8290 fps		8290 fps		4000 fps		2000 fps
DN (nm)	DM (nm)	Error (%)	DM (nm)	Error (%)	DM (nm)	Error (%)	DM (nm)	Error (%)
79	74.3	5.9	80.3	1.6	82.3	4.2	84.6	7.1
482	441.7	8.4	485.6	0.8	465.8	3.4	462.5	4.1
948	989.8	4.4	—	—	957.1	0.1	926.5	2.3

However, according to Eqn. 1, the diffusion coefficient of nanoparticles was dependent with the viscosity of dispersion medium. Larger viscosity leads to smaller diffusion coefficient, which can increase the timescale of the autocorrelation function and improve measurement accuracy. Glycerin is a Newtonian fluid with high viscosity, and it is easy to be mixed with water at any ratio. In this work, the glycerol water solution of 55 % mass concentration was adopted as another dispersion medium. The viscosity increased from 0.87 × 10⁻³ Pa·s in water to 6.15 × 10⁻³ Pa·s in glycerol-water solution. The measurement results were largely improved at the same frame rate as shown in Table 1. No result was obtained for 948 nm particles in glycerol since the Brownian motion was too slow to be detected. In order to investigate the influence of frame rate, tests were also carried out at 4000 fps and 2000 fps in 55 % glycerol-water solution. It shows that the accuracy decreases as the frame rate reduces. In conclusion, the measurement accuracy may be improved by increasing the viscosity of dispersion medium but IDLS using conventional ACF method is greatly affected by the frame rate of the camera.

2.3.2. Results using ACF from image cross-correlation

According to the method described in section 2.1.2, the cross-correlation coefficients between images in a time series were calculated to compose the ACF curves, as shown in Fig. 5. Particles were tested in distilled water at 8290 fps. Fig. 5(a) shows fitted ACF curves using 400 images for all the samples. Results in Table 2 show a higher accuracy as compared with the results in section 2.3.1 at the same conditions, but using less image data. It was caused by grid division in the previous method. The divided grid size was an estimated value, which should be actually different in different locations of the images. The latter method does not require grid division.

Fig. 5

Cross-correlation coefficient and corresponding fitted ACF curves. (a) by 400 images each; (b) by 20, 160 and 300 images, respectively, for 79, 482, 948 nm particles.

Table 2 Measured mean sizes from ACF by image cross-correlation (Liu L.L. et al., 2015)

DN (nm)	DM (nm)	Error (%)
79	77.3	2.15
482	452.6	6.10
948	982.4	3.63

Fig. 5(b) shows ACF results using 20, 160, and 300 images, respectively, for 79, 482, 948 nm particles. There is no change in the shape of ACF curves. It indicates that a certain result can be determined with images captured in the typical time scale duration, which dramatically reduced the acquisition time. In our experimental conditions, the acquisition time were shortened to about 2, 20 and 40 ms, respectively.

2.3.3. Results using one cross-correlation coefficient without ACF

Using the same series of images captured in section 2.3.2, nanoparticle size can be determined from one cross-correlation coefficient between two images in a certain time interval. Two time intervals, 0.25 ms and 0.5 ms, were investigated in our experiments for three different samples, 79/482/948 nm, respectively. The mean particle size can be calculated directly from corresponding cross-correlation coefficient with Eqn. 1–4. Theoretical ACF curves at the above two time intervals and corresponding coefficients are all illustrated in Fig. 6.

Fig. 6

Mean particle size determination from one cross-correlation coefficient.

The measured results are shown in Table 3. It shows that the measured mean particle size D_M agrees well with corresponding nominal size D_N. The respective relative errors are generally small. It can also be noticed that, with other conditions remaining the same, coefficients at larger slope on the ACF curves lead to higher sensitivity. Take the time interval of 0.5 ms for example, it costs 1 ms to acquire a couple of images and less than 10 ms to process them using a notebook with i7-3612QM quad-core processor. It shows that the acquisition and processing time are both significantly reduced, compared with hundreds seconds in traditional DLS.

Table 3 Measured results using one cross-correlation coefficient

τ = 0.25 ms			τ = 0.5 ms
DN (nm)	DM (nm)	Error (%)	DN (nm)	DM (nm)	Error (%)
79	81.5	3	79	80.8	2.3
482	495.6	2.8	482	490.3	1.7
948	970.2	2.3	948	965.7	1.9

3. Ultrasonic attenuation spectroscopy

3.1 Measurement principles

The interaction between ultrasonic beams and particulate two-phase flow is a quite complicated process, in which the sound scattering phenomena must be taken into account in the first place. By assuming small particles as elastic objects, it can be analogous to the light scattering to reach some similar deduction, and certainly the shape of particles, the types of ultrasonic beams and incident angles, as well as the ratio between sound wavelength and particle size could be significant for the description of sound scattering. The analytic solution can usually be obtained for those regular-shaped particles like a sphere, while for more irregular particles the analytic solution is rarely available up to now. Ye Z. et al. (1997) has performed quite a few numerical investigations on acoustic scattering for non-spherical objects from gas, to weak fluid and rigid spheroids, as well as the multiple scattering problems from an ensemble of scatterers (Alvarez A. et al., 1999). However for more cases, a numerical solving method must be relied on. For example, the boundary element method (BEM) can be employed for some special cases like the scattering sound pressure calculation for one or several irregular particles, which could bring us a helpful estimation for the shape effects on sound scattering directive properties and scattering cross section. Except scattering, there are still several important interaction behaviors, relevant with the viscous and thermal dissipations. A further observation of the aforementioned phenomena can help us find obvious intrinsic differences in that the sound scattering merely causes the redirection of acoustic energy away from the incident beam, while the dissipations essentially give rise to the conversion from acoustic energy to heat. Dukhin A.S. and Goetz P.J. (2001) has already pointed out that in fact the scattering and dissipations can be divided into two separated components and analyzed independently, which actually facilitated the understanding of acoustic loss and simplified the mathematical modelling. Typically, a complete mathematical model is still quite necessary in describing all these physical phenomena and establishing the rigorous relationship between the particle size and the ultrasonic attenuation spectra. Fortunately, much work has been done by quite a few researchers as mentioned in the previous section, among which, the thinking of Epstein P.S. and Carhart R.R. (1953) will be cited in particular to demonstrate a typical procedure of particle sizing.

3.1.1. Ultrasonic attenuation model and calculation

When modelling the interaction between particles and the surrounding sound waves, Epstein P.S. and Carhart R.R. (1953) focused on acoustic scattering and attenuation by droplets suspended in the air. Allegra J.R. and Hawley S.A. (1972) further extended EC’s model to solid particles and unified the formulization of ultrasound scattering for particle sizing by fulfilling the ECAH model. As a typical linear scattering theory, their theoretical model fully accounted for the influence on acoustic waves of mixture stickiness and heat dissipation, which coupled to the scattering in the form of viscous waves and heat waves, respectively. By solving the complex wavenumber, the macroscopic phenomena like ultrasonic attenuation and phase shift are judiciously connected with the microscopic process like the interaction between a single particle and surrounding waves. Some numerical validations and predictions were introduced in the previous papers (Su M. and Cai X., 2002; Tebbutt J.S. and Challis R.E., 1996), and an example was stressed to demonstrate the application of the theoretical model for the submicron droplet characterization by using ultrasonic attenuation and phase velocity spectra simultaneously (Su M. et al., 2009).

For a poly-dispersed particle system, the ECAH derived expression for the attenuation coefficient α_s can be given as:   

$${\alpha }_{\text{s}}=\frac{3\varphi }{16k^2}\sum _{j=1}^N\frac{q_j}{D_j^3}\sum _{n=0}^{\infty }(2n+1)Re(A_n(D_j,f))$$(7)

where ϕ denotes the volume fraction of particles. k is the wavenumber of incident compression wave. D is the diameter of the particle. q_j is the frequency distribution of particles with radii between D_j and D_j+1. The summation of series of the compression wave-scattering coefficients A_n (a function of particle diameter D and ultrasonic frequency f) plays the key role for the calculation of the frequency-dependent attenuation coefficients. The solution of A_n finally reduces to a six-order complex linear equations, whose coefficient matrix elements is composed of spherical Bessel and Hankel functions with complex variables as arguments. With the complex argument, a linear increase of the imaginary part of the input parameter leads an exponential variation of these functions. The matrix condition number increases accordingly and the solution of linear equations has numerical instabilities. When function overflow appears for large argument, numerical matrix solving will fail frequently. The existence of above-mentioned problems seriously confines the applicable range of the ECAH model in numerical implementation in aspects of frequencies and particle size. Pinfield V.J. et al. (2005) once proposed a recurrence relation of Bessel functions involved to express the incident wave in a simulation of acoustic propagation in dispersions. A modified algorithm has also been proposed in detail to fully overcome such a numerical instability problem (Su M. and Cai X., 2002). Avoiding the direct use of Hankel function and constructing new related variables, recurrence relations of both Bessel and Hankel functions are as following:   

$$F_n(z)=\frac{2n-1}{z}{F}_{n-1}(z)-F_{n-2}(z)$$(8)

  

$$F_n^{\prime }(z)=F_{n-1}(z)-\frac{n}{z}{F}_n(z)$$(9)

where z is the argument for Hankel function and n is the order. Several related variables and their derived recurrence formula are introduced.   

$$L_n(z)=\frac{F_n^{\prime }(z)}{F_n(z)}=\frac{F_{n-1}(z)}{F_n(z)}-\frac{n}{z}=\frac{1}{M_n(z)}-\frac{n}{z}$$(10)

  

$$Q_n(z)=\frac{F_n^{″}(z)}{F_n(z)}=\frac{n-1}{z}\frac{1}{M_n(z)}-\frac{n+1}{z}{L}_n(z)+(\frac{n+1}{z^2}-1)$$(11)

where variable M_n(z) is   

$$M_n(z)=\frac{2n-1}{z}-\frac{1}{M_{n-1}(z)}$$(12)

And

  

$$\{\begin{array}{ll}{M}_0(z)=\text{TAN}(z)\hfill & \text{for\hspace{0.17em}Bessel\hspace{0.17em}function}\hfill \\ M_0(z)=-\text{i}\hfill & \text{for\hspace{0.17em}Hankel\hspace{0.17em}function}\hfill \end{array}$$(13)

The dependence of these variables on the increase of the variable z is demonstrated in Fig. 7, which shows that as complex variables increase, the spherical Hankel function and its derivative rapidly decrease toward zero, while the newly constructed variables L_n(z) and Q_n(z) tend to 1 in the end. That is to say, they can be used safely to replace the Hankel function to reduce the condition number and avoid the potential numerical overflow, which is quite crucial for the increase of dimensionless parameter (i.e., a very high frequency). Fig. 8 illustrates the numerical results using the improved method. It manifests that calculation range extends to the ultrasonic frequency 1~100 MHz with the particle size 1 nm~1000 μm, which indeed covers the major particle size range under potential investigations, although particles outside this range have also been successfully measured. At the same time, it can be observed that the attenuations grow gradually with the increasing frequency. For a fairly small frequency like 1 MHz, a sensitive peak can be found for particles in the submicron range (i.e. 100 nm~1000 nm) due to dissipation mechanism, which indicates the ultrasonic signal would vary with the particle size seriously. With the increasing of frequency, the peak moves towards the smaller particles direction. Thus a broader frequency range can undoubtedly contain more information about the particle size, accordingly in favor of the improvement of resolution of particle sizing.

Fig. 7

Variation of different variants with arguments (n = 1, z = w(1 + i)).

Fig. 8

Predicted ultrasonic attenuation coefficients against particle diameter (0.5 vol % silver aqueous suspension).

3.1.2. Data inversion for ultrasonic attenuation

Generally, the inversion algorithms to determine particle size distribution have been classified into two types: dependent and independent modes. The dependent mode algorithm assumes that the particle system to be measured conforms to one or two given size distribution functions, e.g. lognormal or Rosin-Rammler (Allen T., 2003), which are usually characterized by two (unimodal distribution assumption) or five (bimodal distribution assumption) parameters to be determined by optimization. So far as we know, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) and DFP (Davidon-Fletcher-Powell) are two significant algorithms to perform the optimization process determining the particle size distribution, although they both possess the drawback that optimization might be trapped into the local optimized solutions instead that the global one.

Unlike the competitor, the independent mode algorithm does not assume a priori particle size distribution. The particle size solution is obtained directly by solving the Fredholm integral equations discretized into matrix form. By rewriting Eqn. 7 according to its discrete form, the linear equations can be generated in the following form:   

$$\underset{{\mathbf{A}}_{i,j}}{\underbrace{\frac{3\varphi }{16}·{\int }_{\mathrm{\Delta }D}\frac{1}{k_c^2{D_j}^3}\sum _{n=0}^{\infty }(2n+1)Re(A_n(D_j,f_i))·\text{d}D}}·\underset{{\mathbf{q}}_{\mathit{i}}}{\underbrace{\sum q(\mathrm{\Delta }{D}_i)}}=\underset{{\mathbf{p}}_i}{\underbrace{{\alpha }_s(f_i)}}$$(14)

where A is the coefficient matrix and q is the discrete particle size frequency distribution, the solution vector. The array p is the experimental spectrum. It must be noted that the matrix is invariably ill-posed. In addition, solution to such an equation group must obey the non-negative constraint. It requires a regularization method (also known as Twomey algorithm) in order to give stable and real solution q to ill-posed problems (Tikhonov A.N. and Arsenin V.Y., 1977; Twomey S., 1977).   

$$\mathit{q}=({\mathit{A}}^T\mathit{A}+\gamma \mathit{H}){\mathit{A}}^T\mathit{p}$$(15)

where, H is the smooth matrix. Furthermore, an Optimum Regularization Technique (ORT) (Brandolin A. et al., 1991) has been put forward by selecting an optimum regularization factor γ_opt and appropriately introducing a set of basic cubic splines to reconstruct the distribution in each size bin, which can contribute to reducing the condition number of the coefficient matrix and be beneficial to the smoothness of the solution vector, without altering the linearity characteristic of equations. In fact, the ORT method can be further developed by choosing of an optimized regularization factor γ_opt which serves as a balance between the smoothness of the PSD solution and the fitting of the reconstructed spectrum. This has been confirmed in particle sizing using light extinction measurement (Su M. et al., 2007) by introducing the L-curve method (Hanson P.C., 1992), and particle sizing in dense two-phase droplet systems using ultrasonic attenuation and velocity spectra by introducing the Generalized Cross-Validation (GCV) criterion (Su M. et al., 2009).

3.2 Experimental Measurement System

For the purpose of measuring the ultrasonic spectrum based on the insert-substitution method with variable-path mode (ISO20998-1:2006), an experimental set-up has been developed for acquisition of broad-band ultrasonic signals. As shown in Fig. 9, signals were generated and detected by a PR-5800 ultrasonic pulse-echo transmitter/receiver (Parametric, Inc) working with two pairs of broadband transducers, whose center frequencies were 10 MHz (V312-SU) and 25 MHz (V324-SU), respectively. According to the insert substitution method, the samples under study were injected into two cubic sample cells with thickness of l₁ = 10 mm and l₂ = 20 mm respectively. Afterward, they encountered successively ultrasonic waves generated from the transducer and the attenuated ultrasonic waves were received by the other transducer on the other side. Temperature of the samples was stabilized at room temperature during the testing to eliminate possible interference on ultrasonic waves.

Fig. 9

Sketch map of measurement system.

Two typical ultrasonic time-domain signal are showed in Fig. 10, which indicates evidently the differences in amplitude and phase between pure water (for calibration) and the sample suspension. An amplitude spectrum was obtained after processing the raw signal through fast Fourier transform (FFT). Correspondingly, the attenuation could be calculated by path-variation measurement:

Fig. 10

Ultrasonic pulse signals of water and the sample suspension.

  

$${\alpha }_l=\frac{\text{ln}(A_1/A_2)}{l_2-l_1}+{\alpha }_{\text{w}}$$(16)

where A₁ and A₂ are the amplitude spectra of ultrasonic signals at the sound path length l₁ and l₂, respectively. α_w is the sound absorption of water.

4. Measurement of polydisperse nanoscale particles

4.1 Nanoscale silver particles

Practical nanoscale silver particles were measured by IDLS using one correlation coefficient and UAS combined with the ORT algorithm, respectively. The nominal mean size was claimed as 20 ± 2 nm by the manufacturer (Shanghai HuZheng Nanotechnology Co., Ltd.).

During IDLS measurement, samples were dispersed in the 75 % glycerol water solution, since the frame rate of the camera used in our work cannot meet the requirement when using water. The viscosity of 75 % glycerol-water at 298 K is 0.03162 Pa·s, which is two orders of magnitude higher than that of water. The refractive index of 75 % glycerol-water is 1.42. Other experimental parameters were set the same as section 2.2. The glycerol-water solution makes it feasible to measure small particles using a lower frame rate. Take the time interval of 1 ms in the measurement, the calculated cross-correlation coefficient is 0.821. Theoretical ACF curve at the delay time of 1 ms is displayed in Fig. 11. The particle size corresponding to coefficient of 0.821 is 23.15 nm, which is the measured mean size for the sample of nanoscale silver particles.

Fig. 11

Measured mean size of nanoscale silver particles by IDLS using one correlation coefficient.

The experimental setup and measurement conditions in UAS were introduced in section 3.2, where high frequency ultrasound was adopted. Before testing, the aqueous suspension with 0.5 vol % silver particles experienced a 2-minutes sonication process (FS-300, Shanghai Sonxi Ultrasonic Instrument Company). The PSD measured by UAS is illustrated in Fig. 12. Furthermore, the volume-based PSD of the same nanoscale silver particles is also represented in Fig. 12, which was extracted by analysing their TEM image indicated in Fig. 13. It is obvious that the UAS yields a relatively broader size distribution than TEM. This is probably caused by the smoothness operation during the inversion of ultrasonic attenuation spectra to overcome the ill-posed problems inside. Looking back to the size of IDLS, it appears a slightly bigger value compared with the 19.2 nm (Dv₅₀) of UAS and 18.0 nm of TEM analysis. The authors infer that this is due to their rough surfaces slowing the velocity of Brownian motion, which leads to an overestimated result here.

Fig. 12

PSD of the same nanoscale silver particles by UAS and TEM.

Fig. 13

TEM image of nanoscale silver particles.

The recovered spectrum and the measured spectrum for UAS are plotted in Fig. 14. A good consistence in tendency of variation can be found, while obvious deviations occur when the frequency exceeds 25 MHz, which can yield a relative deviation ( $\sum \frac{|{\alpha }_{\text{M}}-{\alpha }_{\text{R}}|}{{\alpha }_{\text{M}}}/N$)) of 18.2 % between the measured and recovered spectra. Authors inferred that it was likely due to the non-linear loss effect caused by the particle-particle interaction. Thus we calculated the mean distance h between two adjacent particles according to the Woodcock equation, which yielded a value of 71.0 nm (corresponding to the median size of 19.2 nm). Meanwhile the thermal and viscous skin depths (Mcclements D.J. and Coupland J.N., 1996) at 25 MHz were also provided as δ_T = 42.7 nm and δ_V = 107.4 nm for water. The comparison (δ_V > h) shows that the viscous waves are very likely to exert an effect on the surrounding particles before vanishing, which makes the attenuation spectrum deviate from the prediction of the linear theory (ECAH model), especially over the range of higher frequency. Since the overall trend of recovered spectrum should satisfy the assumption of theoretical model which is merely sensitive to the overall trend, it does not cause significant effect on the inversed results even if the two attenuation values were lost. Thus to some extent such a deviation is acceptable.

Fig. 14

Recovered and measured ultrasonic spectrum for suspensions of nanoscale silver particles.

4.2 Nanoscale ATO particles

Antimony Tin Oxide (ATO) nanoparticles, as one kind of the advanced materials, has been employed in many industry areas for its attractive properties, such as admirable thermal insulation, unparalleled dispersion, the active resistance, and high performance in wear resistance and safety.

The nanoparticle ATO sample in an aqueous suspension (2 vol %) adopted in our work is also provided by Shanghai HuZheng Nanotechnology Co., Ltd. The particle size distribution of ATO samples measured by ultrasonic attenuation spectrum is ranging from 14.3 nm to 40.6 nm (D_v01–D_v99), as shown in Fig. 15. And the volume median size D_v50 is around 24.8 nm.

Fig. 15

PSD of nanoscale ATO particles by UAS.

Using the same conditions described in section 4.1, ATO samples are also prepared and measured by IDLS method. To reveal the reliabilities and stabilities of the measurement, 500 couples of images were captured and processed to get 500 data in a very short time. The results are shown in Fig. 16, which has the averaged value of 23.9 nm and the standard error of 3.2 %.

Fig. 16

Mean particle size of nanoscale ATO particles by IDLS.

According to the values of particle size, the measuring results of ATO samples by IDLS and UAS methods can be considered as in good agreement. However, it can be noticed that the small deviation exists with the results of nanoscale silver particles. The authors would like to clarify that in addition to errors caused by indirect solving algorithms like the mathematic inversion, various definitions and principles among methods could not be ignored when considering the aforementioned deviations. For example, the equivalent projection area diameter (Heywood diameter) is usually used for TEM. The UAS relies on the assumption of spherical sphere to fully interpret the sound scattering behaviour theoretically, which is similar with the IDLS, whereas the latter actually gets the diameter of a ball with the same characteristics of Brownian motion. Strictly speaking, all the principles are aimed at spherical particles, so the measuring results are unavoidably influenced by the sphericity, the surface roughness and other factors, which often exert different influences for different particle characteristics. For example the influences of sphericity on Brownian motion may be crucial compared to that on ultrasonic attenuation. Consequently, the results presented here also reveal that the size information of nanoscale particles being adequately retrieved by different principles.

5. Final remarks

Two methods (DLS and UAS) with potential for development for nanoscale particles characterization were reviewed. The former is mainly focused on the researches using CCD cameras, which we called as image-based IDS. Three different image processing algorithms were reviewed in detail. At the same time, UAS was addressed to the problems of theoretical modelling (including numerical calculations involved), inversion and experimental signal acquiring and analysis for the broad-band spectrum. Then a sample of nanoscale silver particles with nominal mean size 20 ± 2 nm and ATO was measured by using IDLS, UAS and TEM. The results were reasonable and agreed well with each other. The IDLS method based on one correlation coefficient can avoid acquiring and processing a large number of data in conventional DLS and represent an advantage of ultrafast and stable testing. At the same time, the broad-band ultrasound contains available information on particle size, which can ensure a reasonable and stable result by combining the ORT algorithm. Thus, a potential ultrasound-based technique will be available to permit a fast and invasive measurement for concentrated nanoparticle suspensions without any dilution, where the non-linear effect must be considered carefully.

Acknowledgements

The authors gratefully acknowledge supports from the National Natural Science Foundation of China (51176128, 51576130, 51573093), the Shanghai Science and Technology Commission of China (13DZ2260900) and the Shanghai Municipal Education Commission (14CG46).

Nomenclature

ACF

Autocorrelation Function

DLS

Dynamic Light Scattering

IDLS

Imaged-based Dynamic Light Scattering

TEM

Transmission Electronic Microscopy

UAS

Ultrasonic Attenuation Spectroscopy

A_n

compression wave-scattering coefficients

D

particle size (nm)

D_N

nominal particle size (nm)

D_M

measured particle size (nm)

D_T

diffusion coefficient (m² s⁻¹)

F

Bessel/Hankel function

f

ultrasonic frequency

g

normalized autocorrelation function

G

autocorrelation function

h

mean distance between two adjacent particles (nm)

I

light intensity

k

incident compression wave wavenumber (m⁻¹)

K_B

Boltzmann constant (J K⁻¹)

l

sound path length (m)

L_n

complex variables related with the ration of the Bessel/Hankel functions and their first derivatives

M_n

complex variables related with the ration of the Bessel/Hankel functions and their second derivative

N

number of frequency in a discrete ultrasonic attenuation spectrum

n

refractive index (−)

column number of image pixels (−)

order in Hankel function

p

experimental spectrum, with element α_M

q

norm of the scattering vector (m⁻¹)

frequency distribution

Q_n

complex variables in the recurrence formula of Bessel/Hankel functions

R

cross-correlation operator

t

time (s)

T

absolute temperature (K)

z

argument for Bessel/Hankel function

α_s

ultrasonic attenuation coefficient(Np m⁻¹)

α_M

measured ultrasonic attenuation (Np m⁻¹)

α_R

recovered ultrasonic attenuation (Np m⁻¹)

β

instrumental constant (−)

Γ

decay constant (s⁻¹)

δ_T

thermal skin depth (nm)

δ_V

viscous skin depth (nm)

θ

scattering angle (°)

λ

wavelength (nm)

μ

dynamic viscosity (Pa s)

τ

time delay (s)

ϕ

volume fraction of particles (−)

Author’s short biography

Wu Zhou

Wu Zhou received her degree from Nanjing University of Science and Technology in 2006 and a Ph.D. in Engineering from Southeast University in 2011. She is now an associate professor in the School of Energy and Power Engineering at the University of Shanghai for Science and Technology. Her research interests include particles and two-phase flow measurement, combustion measurement, mainly by image-based methods. She also interests in mathematical modelling of the above processes. She is a member of Youth Council of the Chinese Society of Particuology.

Mingxu Su

Prof. Mingxu Su received his degree in Power Engineering from Nanjing University of Aeronautics and Astronautics in 1996 and his PhD in Thermal Engineering from the University of Shanghai for Science and Technology, USST in 2002. In the same year, he got a postdoctoral position at UMR 6614 - CORIA in Rouen, France. After that, he went back to the USST being a teacher till now. His main research interests are methods of particle size characterization based on ultrasonic spectroscopy and light scattering theories, as well as the industrial applications on particulate two-phase flow measurement.

Xiaoshu Cai

Prof. Xiaoshu Cai graduated from the Department of Engineering Thermalphysics, Zhejiang University in 1982, and obtained PhD in Department of Power Engineering, Shanghai Institute of Mechanical Engineering in 1991. He became a professor in University of Shanghai for Science and Technology in 1996. Since 1988 he has been engaged in particle sizing and two-phase flow measurement techniques. His current research interests include particles and two-phase flow measurement, combustion measurement, emission and environmental monitoring, as well as power engineering and turbo machinery. He is vice president of Chinese Society of Particuology and president of Shanghai Society of Particuology.

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