Dynamics of micron-sized particles in dilute and concentrated suspensions probed by dynamic ultrasound scattering techniques

Ultrasonics 65 (2016) 59–68

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Dynamics of micron-sized particles in dilute and concentrated suspensions probed by dynamic ultrasound scattering techniques Tomoyuki Konno, Tomohisa Norisuye ⇑, Kazuki Sugita, Hideyuki Nakanishi, Qui Tran-Cong-Miyata Department of Macromolecular Science and Engineering, Graduate School of Science & Technology, Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan

a r t i c l e

i n f o

Article history: Received 30 September 2015 Received in revised form 22 October 2015 Accepted 23 October 2015 Available online 31 October 2015 Keywords: Sedimentation Scattering Microparticle Size distribution

a b s t r a c t A novel ultrasound technique called Frequency-Domain Dynamic ultraSound Scattering (FD-DSS) was employed to determine sedimentation velocities and the diameters of microparticles in a highly turbid suspension. The paper describes the importance of the scattering vector q for dynamic scattering experiments using broadband ultrasound pulses because q (or frequency) corresponds to the spatial length scale whereas the pulses involve inevitable uncertainty in the time domain due to the frequency distribution of broadband pulse. The results obtained from Stokes velocity of monodispersed silica and polydivinylbenzene (PDVB) particles were compared to those obtained by a Field Emission Scanning Electron Microscope (FE-SEM). A novel method to extract the particle size distribution is also demonstrated based on an ultrasound scattering theory. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Dynamics of microparticles in fluid can be evaluated by dynamic light scattering (DLS) techniques via measurement of the decay rate of time-correlation function [1]. The concentration dependence of apparent diffusion constant provides information on the particle interactions and the particle diameter. Such information can be determined by extrapolating the diffusion coefficient to zero concentration, followed by calculation using the well-known Stokes–Einstein formula. On the other hand, as the particle size becomes larger, sedimentation becomes dominating the particle dynamics due to gravity. For this particular case, the particle diameter may be evaluated by the terminal velocity of a single particle [2]. Despite those advantages of the nondestructive techniques, DLS measurements are not available for most of the systems with micron-sized particles and concentrated suspensions because of multiple scattering and/or serious light attenuation. In order to overcome the problem, we have developed a highfrequency Dynamic ultraSound Scattering (DSS) technique [3–7], which is an acoustic analog of DLS. Since the DSS technique employs ultrasound pulse instead of visible light, it can be applied to optically turbid systems. The DSS method was first proposed by Page and coworkers to investigate a complex dynamics of particle in fluids, such as velocity fluctuations in fluidized bed [8,9] or ⇑ Corresponding author. E-mail address: [email protected] (T. Norisuye). http://dx.doi.org/10.1016/j.ultras.2015.10.022 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.

shared flow [10] using several megahertz ultrasound. On the other hand, the higher frequencies with the better spatial resolution were required in order to apply this technique to systems of micron- or nano-sized particles. In our previous papers, we have shown that the high-frequency dynamic ultrasound scattering techniques enabled us to evaluate the dynamics and structure of microparticles [3–7,11]. When a micron-sized particle settles down, the motion of surrounding liquid is perturbed by the presence of the particle, resulting in a multi-body problem of particle dynamics. It leads to noticeable fluctuations of settling particle velocity, and is known to involve long-ranged hydrodynamic interactions [12]. The hydrodynamic interactions are also responsible for the unique and surprisingly large (in the order of millimeters) cooperative structures where the origin of velocity fluctuations is now believed to be fluctuations in the number of particle taking part in a cooperative domain structure called blob [13,14]. Therefore there exist dynamic structures in the velocity field in spite of the concentration profile being statistically random in both time and space. Such dynamic structures were visualized by temporal extraction of ultrasound phase by so called the phase-mode DSS technique [5,6]. Besides the advantages of the DSS techniques for many particle systems intractable by conventional optical techniques, the accuracy of the particle velocity still remained in about 10% [4]. Although ‘‘broadband” is crucial for probing the acoustic properties in a wide range of frequencies, and the corresponding short pulse in the time-domain could resolve the objects better in nondestructive testing, this drawback would be probably due to the frequency

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T. Konno et al. / Ultrasonics 65 (2016) 59–68

distribution of the broadband pulse. Therefore, alternative DSS approaches were expected to overcome the problem. In this paper, we first address the importance of the scattering vector q for ultrasound scattering experiments using broadband ultrasound pulses because q is responsible for the spatial length scale in dynamic measurements while the pulses involve inevitable uncertainty in the time domain due to the frequency distribution of broadband pulse. We introduce a novel DSS technique, called FrequencyDomain Dynamic ultraSound Scattering technique (FD-DSS), which allows us to precisely determine the settling velocity and the corresponding particle diameter using broadband pulses having a wavelength distribution. As a result, the technique significantly improved the experimental accuracy compared to the previous studies. Second, the particle diameters converted from terminal velocities of sedimentation are quantitatively evaluated for mono-dispersed silica and polydivinylbenzene standard particles. Comparison of the results with those obtained by a scanning electron micrograph will be also made. Third, the effect of particle size distribution is addressed by probing two types of settling velocity: (1) the velocity of sedimentation front and (2) that of submerged particles in a suspension using the FD-DSS techniques. Finally, the dependence of the concentration profile of the sedimentation velocity on the Reynolds number, Re, will be compared to the data published by Richardson and Zaki (R–Z) [15], and Garside and AlDibouni [16].

2. Experimental procedure A series of standard polydivinylbenzene (PDVB) and silica microspheres were kindly provided by Sekisui Chemical Co. LTD. The silica or PDVB particles were dispersed in distilled water to obtain aqueous suspensions with desired concentrations in range 0.1 < C < 15% in weight, followed by a brief immersion in a low power ultrasonic bath prior to ultrasound scattering experiments in order to avoid aggregation. In the case of the PDVB suspensions, 0.2 wt.% sodium dodecyl sulfate (SDS) was added to disperse the hydrophobic particles in water. Water was purified twice using 0.2 lm membrane filter after distillation. Polyphenylene sulfide (PPS) rectangular vessels with the dimension 10
27.5
30 mm3

(width
depth
height) with the wall thickness of 1 mm were used as a sample cell. The effect of sample height (hindered settling) was carefully investigated prior to the experiments. The density of particle was determined to three places of decimals by a density matching method with calibrated aqueous solutions of sodium chloride (NaCl) or sodium polytungstate, SPT (3Na2WO49WO3H2O). The density of NaCl or SPT solutions was calibrated using a 25 mL Gay–Lussac pycnometer prior to the density matching experiments. FE-SEM images (JEOL JSM-7600F) were taken to calibrate the particle size and its distribution. Prior to observation, the samples were dried in vacuo followed by a gold deposition by magnetron sputtering to enhance the quality of images. The obtained bitmap images were recorded with 2560
1920 pixels containing about 10 particles in each picture, followed by calculation of the diameter for at least 300 particles. Since the thickness of gold layer was much thinner (typically 50 nm) than the particle diameter (several tens of micrometers), further correction was not made. Examples of the SEM images and the corresponding particle size distribution obtained for the silica and PDVB particles are shown in Fig. 1. In the SEM images, both standard particles seemed to be monodispersed; however, it was found that the PDVB particles have a broader size distribution compared to those of silica. The quantitative information of the particle is summarized in Table 1 where d is the nominal particle size provided by the supplier, dSEM is the average particle diameter calibrated by FE-SEM, CV is the coefficient of variation corresponding to the standard deviation of the particle size normalized to its average, q is the particle density, Re is the particle Reynolds number, Pe is the particle Peclet number, and N is the number of particle evaluated by the FE-SEM analysis. An ultrasound pulse was generated using a broadband pulser/ receiver (iSL BLP12R) connected to a longitudinal plane wave transducer. The energy of pulser was kept as low as possible to avoid unexpected flow induced by excess ultrasound energy [17]. Various water-immersion sensors such as, KGK B5K6I (diameter 6 mm, nominal frequency 5 MHz), B10K2I (2 mm, 10 MHz), B20K2I (2 mm, 20 MHz) and 25C6I (6 mm, 25 MHz), B30K1I (1 mm, 30 MHz) were employed to measure ultrasound signals with different frequency ranges. Although the FD-DSS technique described in this paper allows us to evaluate the frequency

Fig. 1. Examples of SEM image and the corresponding particle-size distribution obtained for the (a) silica and (b) PDVB particles.

61

T. Konno et al. / Ultrasonics 65 (2016) 59–68 Table 1 List of d, dSEM, CV, q, Re, Pe, and N for the silica and PDVB particles. Sample

d (lm)

dSEM (lm)

CV

q (g/cm3)

Re

Pe

N

5

Silica

3 5 10

2.91 ± 0.02 4.68 ± 0.04 9.57 ± 0.05

0.007 0.009 0.005

1.935 2.080 2.115

1.25
10 6.01
105 5.30
104

7.11 5.49
10 9.90
102

300 300 350

PDVB

3 5 10 25 40 50

2.87 ± 0.22 4.79 ± 0.25 9.48 ± 0.46 24.76 ± 1.01 40.50 ± 1.38 48.15 ± 2.65

0.077 0.052 0.049 0.041 0.034 0.055

1.176 1.175 1.174 1.172 1.175 1.176

2.27
106 1.05
105 8.11
105 1.42
103 5.40
103 1.08
102

1.28 9.84 1.50
102 6.91
103 4.05
104 1.01
105

310 496 312 457 420 560

Fig. 2. Schematic representation of the experimental setup and the data obtained by (A) in the time domain – and (B) in the frequency domain – DSS techniques.

dependence of the dynamics from a single transducer, a series of experiments using various transducers were carried out to ensure the discussion over a wider range of the frequency. The transducer and the cell container were carefully aligned using a custom-made stainless stage coupled with rotational (a, b and c) and translational (x, y, z) stages prior to the scattering experiments in order to avoid the signal loss originated from misalignment with respect to the cell wall where a, b and c were the axially tilting angles for the yz, zx, and xy planes, respectively. The signal was recorded by a 14 bit high-speed digitizer, GaGe CS-14200, with the sampling rate 200 Mega samples/sec. All the

digital devices were synchronized with a 10 MHz reference clock to avoid phase jittering. The sample was set in a homemade thermostat bath regulated at 20 ± 0.01 °C. 3. Data analysis 3.1. Time-domain correlation function approach When a spike pulse is applied to a transducer, an ultrasound pulse is generated and emitted to a sample vessel containing a suspension of micro-particles as shown in Fig. 2(A). The ultrasound

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T. Konno et al. / Ultrasonics 65 (2016) 59–68

wave field, w, contains various contributions such as the excitation spike, four reflected echoes from the cell walls, and the scattered signals from the particles located between the cell walls. The scattered signals are extracted as a function of the pulse propagation time, t, to statistically analyze the displacement of particles (Fig. 2(a)). Such a pulse is repetitively recorded during the observation time, T, to obtain the two-dimensional scattered field wðt; TÞ (Fig. 2(b)). As a result, the T dependence of the amplitude is obtained at given t (Fig. 2(c)), resulting in the time fluctuations of the scattered amplitude at a fixed location during the sedimentation process. As long as the process is in a steady state, the time correlation function of the field amplitude at a fixed t may give the average sedimentation velocity hVzi at the corresponding sample depth (Fig. 2(d)). The field correlation function, g ð1Þ ðt; sÞ is defined by [3,4,18]

g ð1Þ ðt; sÞ ¼

hwðt; TÞw ðt; T þ sÞiT ¼ hexp½iq  Drðt; sÞiT hwðt; TÞw ðt; TÞiT

ð1Þ

where asterisk indicates complex conjugate, Dr is the displacement pffiffiffiffiffiffiffi 1, and q is the

of the microspheres during the lag time, s, i ¼ magnitude of the scattering vector given by

4p h 4p f h sin : q¼ sin ¼ 2 c 2 k

ð2Þ

k, h, f, and c are respectively the wavelength of ultrasound in suspension, the scattered angle (which is 180° in our study), the ultrasound frequency, and the speed of sound in suspension. The bracket indicates the average over the observation time T. For settling particles, we have two types of formulae depending on the geometry, allowing us to evaluate the average velocity and its variance. For the z-direction, hVz(t)iT, and the standard deviation of velocity,

(4). In the time-domain analysis, the peak frequency in the broadband spectrum has been used as the representative value of the transducer. Fig. 3(a) shows an example of frequency spectra obtained for two different transducers; one is a narrowband ceramic transducer, 25C6I, and the other is a broadband composite transducer, B20K2I, manufactured by KGK, Japan. It could lead to an error if the selected (peak) frequency is not the proper one. In general, broadband pulses are commonly utilized in nondestructive testing and/or medical applications because the broadband measurements allow us to obtain information over a wide range of frequency at one time, and to detect a small defect thanks to the short pulse width in the time-domain. On the other hand, however, the selection of frequency could affect the accuracy of the velocity measurements, motivating us to carry out a novel analysis in the frequency-domain as shown in Fig. 2(B). In the FD-DSS analysis, wðt; TÞ is transformed into a frequency component wðf ; TÞ by a Fast Fourier Transformation (FFT) at a given T, followed by the reconstruction of the signal in a polar coordinate (Eq. (6)) to eliminate spurious signals such as multiply reflected noise within the cell.

  sin hðf ; TÞ  hsin hðf ÞiT wðf ; TÞ ¼ rðf ; TÞ exp i tan1 cos hðf ; TÞ  hcos hðf ÞiT

The magnitude of the scattered wave, jwðf ; TÞj, as a function of frequency is shown in Fig. 2(e). Such a FFT analysis was systematically performed to have a two-dimensional complex wave field as functions of ultrasound frequency, f, and the observation time, T, as demonstrated by an image of amplitude (Fig. 2(f)). In order to evaluate the characteristic time of wðf ; TÞ demonstrated in Fig. 2(g), a time-field correlation function in the frequency domain,

1

hdV 2z ðtÞi2T , as a measure of velocity fluctuations, can be obtained from

  1 g zð1Þ ðt; sÞ ¼ cos ðqhV z ðtÞiT sÞ exp  q2 hdV 2z ðtÞiT s2 2

ð3Þ

ð6Þ

g ð1Þ ðf ; sÞ ¼

hwðf ; TÞw ðf ; T þ sÞiT ; hwðf ; TÞw ðf ; TÞiT

ð7Þ

is employed (Fig. 2(h)). Similar to the TD-DSS technique, the corre1

is solely evaluated from

  1 g yð1Þ ðt; sÞ ¼ exp  q2 hdV 2y ðtÞiT s2 2

ð4Þ

since the average component for the y-direction is zero. This is called the Time-Domain Dynamic ultraSound Scattering (TD-DSS) technique, and has been employed to evaluate the sedimentation velocity and its fluctuations as a function of the distance from the cell wall [4]. If the dynamics is dominated by thermal fluctuations, the correlation function is expressed by

  g ð1Þ ðt; sÞ ¼ exp hDðtÞiT q2 s

  1 g ð1Þ ðf ; sÞ ¼ cos ðqhV z ðf ÞiT sÞ exp  q2 hdV 2z ðf ÞiT s2 2

40

4

1 0 0.010

and

3.2. Frequency-domain correlation function approach Besides the advantage of TD-DSS technique, such as the position sensitivity of particle motion in a suspension (Fig. 2(a)), we have noticed that there is inevitable uncertainty in evaluation of each particle velocity due to employment of a broadband pulse, of which the signal is smeared by the contributions from the different length scales. In the time-domain analysis, one requires a fre1

quency (or q) value prior to calculation of hVz(t)iT and hdV 2z ðtÞi2T from the s dependence of the relaxation behavior in Eqs. (3) and

3 2

0

(b)

0.009

(mm/s)

after, we simply denote hhV z ðtÞiT it , hhDðtÞiT it as hV z i, DV z , DV y , and hDi, respectively.

1

hhdV 2y ðtÞi2T it ,

20 10

where D is the mass diffusion coefficient. The statistical uncertainty may be minimized by taking average over different propagation time, t, (different sample position, z), if the data are statistically equivalent within a limited region of the propagation time. Here1

6

(a)

5

ð5Þ

hhdV 2z ðtÞi2T it ,

ð8Þ

30

A (f)

while the

1

hdV 2y ðtÞi2T

lation function gives hVz(f)iT, and hdV 2z ðf Þi2T , as follows:

0.008 0.007 0.006 0.005

15

20 25 frequency, f (MHz)

30

Fig. 3. (a) Typical frequency spectra of two different transducers, 25C6I (open circle) and B20K2I (open triangle). (b) hVzi evaluated by the time-domain (open symbols) and the frequency-domain DSS (solid symbols) techniques.

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T. Konno et al. / Ultrasonics 65 (2016) 59–68

It should be noted that unlike the case in the TD-DSS analysis (Eq. (3)), now the proper f (or q) values are known prior to evaluation of the velocity because the filed amplitudes are analyzed as a function of frequency (Fig. 2(f)). In the case of diffusive motion, g ð1Þ ðf ; sÞ is given by,

  g ð1Þ ðf ; sÞ ¼ exp hDðf ÞiT q2 s

ð9Þ

where D is the diffusion coefficient. In our previous work, the frequency-domain technique was utilized to elucidate complex dynamics of the particle having Peclet number close to unity where the dynamics involves both sedimentation and Brownian motion [11]. If the velocity and diffusion constant are irrespective of the frequency, further averaging over different frequencies gives

hV z i  hhV z ðf ÞiT if ,

DV z 

1

hhdV 2z ðf Þi2T if ,

DV y 

1

hhdV 2y ðf Þi2T if ,

hDi  hhDðf ÞiT if . The potential of the FD-DSS technique is demonstrated in Fig. 3 (b) where the solid and open symbols respectively denote the results of the FD-DSS and the TD-DSS, and the circle and triangle markers respectively represent the data obtained for the 25C6I (narrowband) and B20K2I (broadband) transducers. As seen from the figure, the FD-DSS method successfully gives constant and stable results irrespective of both the frequency values and the type of transducer having significantly different bandwidth and central frequency. On the contrary, the results of the TD-DSS are highly affected by the employed frequency values. As described below, the value of hVzi evaluated by the FD-DSS method showed perfect agreement with that calibrated by FE-SEM. It should be noted here that the employment of the peak frequency does not always provide the correct result in the TD-DSS analysis as shown by the dotted lines. It could be one of the main reasons why the TDDSS results have 10% of error to determine the particle size via Stokes velocity. 3.3. Phase statistics In both time- and frequency-domain DSS analyses, a certain time period is required to construct the time-correlation function depending on the relaxation time. Furthermore, only the magnitude of the velocity (without its sign) is evaluated through the analysis so that the direction of particle movement cannot be obtained by such methods. Since not only the amplitude but also the phase can be analyzed in pulsed ultrasound experiments, the phase difference was utilized to evaluate an instantaneous velocity of particle with its sign (e.g., either settling or floating) by the phase-mode DSS technique [5]. The information of phase, UðtÞ, contained in the waveform, w(t, T), may be extracted by a following formula as described in the previous papers [6,19]:



UðtÞ ¼ tan1 

SðtÞ  hSðtÞiT CðtÞ  hCðtÞiT



ð10Þ

where

SðtÞ ¼ I1 ½IfwðtÞ sinð2pf c tÞg
Hðf Þ CðtÞ ¼ I1 ½IfwðtÞ cosð2pf c tÞg
Hðf Þ

ð11Þ

and fc is the lock-in frequency (importance of fc is, again, explained later), H(f) is a filter function, and I and I1 respectively denote forward and inverse Fourier transformation. The extracted phase is further transformed into the instantaneous velocity expressed by [5,6]



V z ðt; TÞ ¼ 



Cðt; TÞ  hCðtÞiT  dCðt;TÞ Sðt; TÞ  hSðtÞiT dT i ð12Þ q fCðt; TÞ  hCðtÞiT g2 þ fSðt; TÞ  hSðtÞiT g2

dSðt;TÞ dT h

The advantage of the technique is that one can visualize the dynamic structure via the image of particle velocity. For example, the method was utilized to image the settling and floating suspensions [5], and to visualize cooperative particle structure accompanying extraordinary large domains due to long-ranged hydrodynamic interactions [6]. Once the instantaneous velocities are obtained, they can be statistically analyzed in a small area of the image using the following probability function P(V)

PðV z Þ ¼

1  3 2 2 z iÞ 2DV z 1 þ ðV z hV 2 DV

ð13Þ

z

to obtain the hV z i and, DV z , which is equivalent to that obtained by the correlation function approach in Eq. (8) although the evaluated value is unfortunately dependent on the selection of the lock-in frequency fc. Therefore, by taking into account both advantages of the FD-DSS and the phase imaging methods, the instantaneous sedimentation velocity and its image can be accurately obtained. For example, as shown in Fig. 3, the lock-in frequency is selected by searching a frequency where the TD-DSS (or phase) method matches the value evaluated by the FD-DSS (Fig. 3), followed by acquiring a velocity image by the phase-mode DSS technique with the ‘‘proper” frequency as fc. Once the lock-in frequency is found, further calibration is not necessary for every measurement. Hereafter we call the technique ‘‘FD-DSS assisted Phase-Mode DSS” and it will be utilized in the following discussion. 4. Results and discussions Fig. 4(a) displays a velocity image obtained for a silica suspension with d = 3 lm at / = 0.005 as functions of the pulsepropagation time, t (sample position, z ¼ ct=2) and the observation time, T, during sedimentation. The velocity was obtained by the phase analysis (Eq. (12)) with the proper lock-in frequency calibrated by the FD-DSS method. As the particles settle down, the boundary between clarified water (upper fraction) and the silica suspension (lower fraction) became noticeable, followed by a decrease in the height of the suspension. The location of the interface could be easily confirmed by reflected echoes of ultrasound pulse. In Fig. 4, the sedimentation velocities of the particles are observed at the lower fraction of suspension, while the upper fraction is filled with a noise pattern because of the absence of particles. Therefore, two types of the sedimentation velocity Vsed, i.e., (1) a velocity at the shallow surface of suspension (the average sedimentation velocity, hVzi, introduced in the data analysis section), and (2) the slope of Z vs. T image (the interface velocity, Vint) were evaluated. The former is a unique quantity offered by ultrasound scattering techniques, while the latter is equivalent to the conventional method to evaluate the sedimentation velocity except that ultrasound waves are utilized to precisely determine the location of interface. The time evolution of Vz(t,T) extracted along the line (A) in the image is shown in Fig. 4(b) where the solid line is a smoothed guide for the eyes. The process can be divided into 3 stages: (i) just after mixing, (ii) non-steady state and (ii) final steady state. In the following discussion, the data at the stage (iii) will be employed. The position dependence of the particle velocity extracted along the arrows (B) in the image is shown in Fig. 4(c) where the data are divided into three regions: (i) in the vicinity of the interface, (ii) stable suspension phase and, (iii) multiply scattered or less-accurate area due to low-amplitudes. Therefore, the data from region (ii) will be used for the following discussion. Fig. 5 shows the volume-fraction, /, dependence of the two types of Vsed; hVzi and Vint, obtained for the silica suspensions with

T. Konno et al. / Ultrasonics 65 (2016) 59–68

0

4 8

4 6 8

(B) 500

(b) 0.015 (i)

(A)

1000 1500 2000 Observation time, T (s) (ii)

0.004

0.0040

10

0

0.04

0.06

0.010

0.08

φ (b)

(A)

0

0.014

0.013

0.012

0.012

0

0.010

0.010

Silica 5 μm

-0.005 0

500 1000 1500 Observation time, T (s)

0

1

Depth (mm) 2

0.008

2000

0

3

4

(i)

0.004

(B)

0.003

(ii)

(iii)

1 2 3 4 Propagation time, t (μs)

0.052 0.050 0

5

(a) d = 3 lm, (b) 5 lm, (c) 10 lm. All the data were taken by B20K2I. The solid and dashed lines are the best-fit results with the empirical Richardson–Zaki (R–Z) equation,

ð14Þ

where V0 and n respectively are the terminal velocity and an exponent. The inset indicates the data at the low volume fraction (/ < 0.01) to obtain V0 by linear extrapolation. Once V0 is determined, one can obtain the particle diameter d using the following relation:

0.010

Silica 10 μm

0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18gV 0 d¼ Dq g

0.08

0.056

0.05

0.04

Fig. 4. (a) Image of sedimentation velocity obtained for the silica suspension with d = 3 lm at / = 0.005. (b) T dependence of Vz extracted from the image-(a) at the shallow surface (along the arrow A) where the stage (III) indicates the steady state. (c) t (depth) dependence of hVziT extracted at fixed Ts (along the arrow B).

V sed ¼ V 0 ð1  /Þn

0.06

0.054

1000 - 1400 s 1200 - 1600 s 1400 - 1800 s 1600 - 2000 s

0.002 0

0.04

(c)

0.06

0.006 0.005

0.02

φ

Vsed (mm/s)

T (mm/s)

0.02

2500

0.010

(c)

0

Silica 3 μm Vint

0.003

(iii)

0.005

V0 (SEM) 0.0045

Vsed (mm/s)

12 0

Vz(t,T ) (mm/s)

2

(a)

0.005

Position Z , (mm)

0.02 0.01 0 -0.01 -0.02

Vsed (mm/s)

0 Vz (mm/s)

(a)

Propagation time, t (μs)

64

0.02

0.04

0.06

0.08

φ Fig. 5. /-dependences of Vsed obtained for the silica particles with (a) d = 3, (b) 5, and (c) 10 lm. The insets show the data in the dilute regime (/ < 0.01).

Table 2 List of d, dDSS, dINT, dSEM, for the silica and PDVB particles. dDSS (lm)

dint (lm)

Silica

3 lm 5 lm 10 lm

2.94 (1.0%) 4.73 (1.1%) 9.52 (0.5%)

2.91 (0.0%) 4.66 (0.4%) 9.54 (0.3%)

PDVB

5 lm 10 lm 25 lm 40 lm 50 lm

4.86 9.42 24.36 40.31 48.55

4.76 (0.5%) 9.48 (6.1%) 22.85 (7.7%) 35.92 (12.8%) 44.23 (8.9%)

dSEM (lm) 2.91 4.68 9.57 4.79 9.48 24.76 40.50 48.15

ð15Þ

for sufficiently small Re, where g is the viscosity, Dq is the density difference between the particle and the surrounding liquid, and g is acceleration of gravity. The particle diameters evaluated from hVzi (FD-DSS) and Vint (interface) are respectively listed in Table 2 as dDSS and dint, which are averaged over three trials, and seem to be in very good agreement with the diameter obtained by the SEM analysis, dSEM within the relative error about 1%. In the case of the silica suspension having extremely narrow particle-size distribution, both hVzi and Vint at the zero concentration agreed well with the terminal velocity predicted from the SEM calibration as indicated by the dotted line in Fig. 5. Although both hVzi and Vint followed the relation of the R–Z equations, decrease in hVzi is more pronounced than that of Vint, i.e., the n value is larger for the former case. This point will be

discussed later. As long as the V0 is determined by linear regression at / < 0.01, one can obtain the particle size under a single scattering approximation. On the other hand, by combining Eqs. (14) and (15), one can obtain the following equation to evaluate the particle diameter from a Vsed observed at a finite concentration /:

d¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18gV sed Dqgð1  /Þn

ð16Þ

as long as the single scattering assumption holds (up to 20% is allowed in the case of silica suspension). Since the velocity and the particle size of the monodispersed silica particles could be characterized by the FD-DSS techniques, now the technique is applied to the PDVB particles having a broader size

T. Konno et al. / Ultrasonics 65 (2016) 59–68

distribution compared to that of the silica particles. Similar to the case of the silica particles, the particle velocity was extracted at around t = 2 ls after reaching the steady state of sedimentation. Fig. 6 shows the / dependence of Vsed obtained for the PDVB particles with (a) d = 5 lm, (b) 10 lm, (c) 25 lm. All the data were taken by a B20K2I transducer except for d = 25 lm (taken by B10K2I). Unlike the Vint for the silica particles, the spreading of sedimentation front is more pronounced due to the particle size distribution [20,21]. Therefore the interface is carefully evaluated to give the correct Vint by taking several levels of the scattered intensities. The /-dependence of Vsed is similar to that of the silica particles, i.e., hVzi is systematically smaller than Vint at the larger /. However, it should be noted that the intercept of Vint was not equal to V0 expected from the SEM calibration. Such a disagreement is also observed in other systems where some authors introduce a proportional constant k to the volume fraction dependence of Vsed as [22–24]

V sed ¼ kV 0 ð1  /Þn :

ð17Þ

The particle diameters of the PDVB particles evaluated by hVzi and Vint are respectively listed as dDSS and dint in Table 2. While dint was underestimated, the dDSS is fairly close to the value obtained by SEM calibration, suggesting the importance of the individual evaluation of the submerged particle in suspension. 0.0025

(a)

Vsed (mm/s)

0.003 0.0020

0.002

0.0015 0

PDVB 5 μm Vint

0.001 0

0.02

0.04

0.06

0.010

0.08

0.10

φ (b)

Vsed (mm/s)

0.012

0.009

0.010

0.008

0.008

0.007

0.006

0.010

PDVB 10 μm

0.004 0

0.02

0.04

Next, we attempted to extract the particle size distribution, N (d), from the data obtained by DSS. The basic idea is (1) to utilize the small gradient of the particle size along the sedimentation direction because the larger particles settle down faster and the smaller particles remain at the upper part in the suspension, and (2) to obtain the distribution function by analyzing the relation of the scattering intensity, hI(t)iT, vs. sedimentation velocity, hVz(t)iT, at a given sample position (propagation time, t). Here, hI(t)iT can be converted into the number of particle, N, using a known singe-particle scattering function. The particle size d is also simultaneously determined by hVz(t)iT (or Vsed) with Eq. (16). Fig. 7 shows an example for the analysis of the PDVB suspension with d = 5 lm at / = 0.0017. As shown in Fig. 7(a), the hVz(t)iT slightly increased with t (or z), indicating the presence of a size distribution or a concentration gradient. If we assume all the difference is attributed to the variation of concentration, the increase in the sedimentation velocity for the larger z corresponds to the decrease in the volume fraction according to Eq. (14). Since it is difficult to envisage, the gradient of velocity can be regarded as the size distribution within the limited t area. Note that such a clear increase was not observed for the silica particles because of the extremely narrow distribution of particle as shown in Fig. 4(c). The timeaverage of the scattered intensity, hIðtÞiT  hjwðt; TÞj2 iT , is simultaneously obtained by extraction of the envelope of the fieldamplitude by Hilbert transformation, followed by taking square and averaging over a certain period of the observation time to have the better statistics as shown in Fig. 7(b). Then, two relationships of the t-dependence are combined into hVz(t)iT vs. hI(t)iT by eliminating the variable t, followed by further conversion of hVz(t)iT into d using Eq. (16) to give the hI(d)iT vs. d curve as shown in Fig. 7(c). Beside the dynamic measurements utilizing ultrasound scattering, ultrasound spectroscopy, which enables us to evaluate the attenuation and the sound velocity, is also a powerful tool to understand the structure and the property of microparticles dispersed in liquid. In our previous work, the frequency dependence of the attenuation coefficient and sound velocity was investigated for a series of micron-sized rigid particles where the experimental results were successfully reproduced by a theory of acoustic scattering and a dispersion relation [25]. The scattering function is given by [26,27]

f ðh; dÞ ¼

0

0.06

0.08

0.10

φ (c)

Vsed (mm/s)

0.07 0.06

0.08

0.05

0.06 0

0.04

0.010

PDVB 25 μm

0

0.02

0.04

0.06

0.08

0.10

φ Fig. 6. /-dependence of Vsed obtained for the PDVB particles with (a) d = 5, (b) 10, and (c) 25 lm. The insets show the data in the dilute regime (/ < 0.01).

1 1 X ð2n þ 1ÞAn ðdÞPn ðcos hÞ ik0 n¼0

ð18Þ

where k0 is the wave number of liquid, An are the partial wave amplitude, and Pn(cos h) are Legendre polynomials of the nth order. In the present study, the scattered angle h is 180°. In the case of PDVB particles, the longitudinal velocity vL = 2.70 mm/ls, and shear velocity vS = 1.36 mm/ls, while these of the silica particles are 4.80 and 2.40 mm/ls, respectively. The value of N is then obtained by

NðdÞ ¼ 0.10

65

hIðdÞiT jf ðdÞj2

ð19Þ

provided that the inter-particle structure factor can be ignored at the low volume fraction. Since the data interval of particle diameter is not constant, the apparent N values are integrated first to obtain a cumulative distribution function (CDF), then it is further differentiated after properly smoothing by a cubic spline method as shown in Fig. 7(d). The final result of the particle size distribution is exhibited in Fig. 7(e) with the histogram obtained by SEM. Fig. 8 shows the size distribution obtained by SEM (in (a) and (c)) and by DSS (in (b) and (d)) techniques. The first two are the results of silica particles, while the last two are those of PDVB. It is clearly seen from the figure that the results obtained by the DSS technique are consistent with those obtained by SEM, and the difference between the silica and PDVB standard particles in

4

8

2

0.003

< I(t) >T x10 (V )

T. Konno et al. / Ultrasonics 65 (2016) 59–68

T (mm/s)

66

0.002 0.001 1 2 3 Field time, t (μs)

8

2

< I(d) >T x10 (V )

0

4

2 0 0

1 2 3 Field time, t (μs)

4

4 3 2 1 0 2

3

4 5 6 Diameter, d (μm)

7

2

3

4 5 6 Diameter, d (μm)

7

2

3

4 5 6 Diameter, d (μm)

7

CDF. x10

-4

1.5 1.0 0.5

N / NMAX

0

1.0 0.8 0.6 0.4 0.2 0

Fig. 7. Diagram of the evaluation of the size distribution obtained for the PDVB suspension with d = 5 lm; (a) t-dependence of the sedimentation velocity, (b) t-dependence of the scattered intensity, (c) intensity vs. d plot, (d) a cumulative distribution function (CDF), (e) the size distribution obtained by this way (solid line) and by FE-SEM (gray bars).

terms of the diameter distribution is also nicely evaluated by the present technique. Note that the shape of the distribution function did not change during the observation time, T, as long as the data were taken from the steady state, suggesting that the spatial position of the particles having different sizes has been fixed at the early stage. Therefore, determination of the sedimentation front should be satisfactory carried out in the analysis to evaluate Vint, and the concentration dependence seen in Figs. 5 and 6 is properly performed. Although the present method is potentially useful to the particle sizing, our technique also has limitation. For example, due to long-ranged hydrodynamic interactions, the particle dynamics is always affected by the velocity fluctuations, resulting in the broadening of the size distribution because of a velocity distribution of the individual particle. The more averaging of the signals, the effect of velocity fluctuation could be eliminated better, and more sophisticated algorithm to extract the plausible average velocity at a given sample position could give the better results. However, it is beyond the scope of this paper so that the further correction was not made here. Finally, let us discuss about the concentration dependence of the sedimentation velocity. Although both hVzi and Vint decreased with / for the silica and PDVB particles investigated in this study, hVzi exhibited noticeable reduction compared to Vint. In order to compare the results quantitatively, the R–Z exponent n is evaluated by ln hVzi and ln(1  /) plots as demonstrated in Fig. 9 where

the plots are vertically shifted for clarity. In addition to the particles employed in Figs. 5 and 6, the results obtained for the PDVB particles with d = 40 and 50 lm taken by B5K6I were shown here to discuss the behavior in the broader range of Re. Fig. 9 (a) and (b) respectively shows the result obtained by the DSS and interface analysis. While almost the same slope is obtained for the interface velocity in Fig. 9(b), that obtained by the DSS method in Fig. 9(a) becomes larger as the particle size decreases. The characteristic is irrelevant to the density of particle or the particle size distribution because the results of silica and PDVB are shown altogether here. Fig. 10 shows the values of n as a function of Re, where the curves of Richardson and Zaki [15], and Garside and Al-Dibouni [16], are also shown in the graph. The closed symbols indicate the data obtained by the DSS method while the open symbols correspond to the results obtained by the interface analysis. According to their papers, n is constant below 0.2, which is supported by our results of Vint (or a weak function of Re). Note that in our case, the effect of d/D (the ratio of the particle diameter d to the width of the cell D) on n is negligibly small. On the other hand, noticeable deviation of n at the low Re was observed for the data obtained by DSS. This behavior corresponds to the significant reduction of hVzi for the larger / as seen in Figs. 5 and 6. The inset shows the same data as a function of Peclet number, Pe. As the particle size becomes smaller, the thermal diffusivity plays an important role to the

67

1.0 0.8 0.6 0.4 0.2 0

10

N / NMAX

2

4 6 8 Diameter, d (μm)

10

Silica

6 4

12

8

(b)

1.0 0.8 0.6 0.4 0.2 0

8

n

10

2 10

PDVB

0

10

2

10

4

10

6

Pe

6

Silica (DSS)

Garside and Al-Dibouni

4 0

N / NMAX

12

Silica (SEM)

0

1.0 0.8 0.6 0.4 0.2 0

2

4 6 8 Diameter, d (μm)

10

12

DSS Interface

Richardson and Zaki

2

(c)

-6

-4

10

PDVB (SEM)

-2

10

10

0

10

2

10

Re Fig. 10. Re dependence of the exponent n.

0

N / NMAX

12

(a)

n

N / NMAX

T. Konno et al. / Ultrasonics 65 (2016) 59–68

2

4 6 8 Diameter, d (μm)

10

12

(d)

1.0 0.8 0.6 0.4 0.2 0

PDVB (DSS)

0

2

4 6 8 Diameter, d (μm)

10

12

Fig. 8. The particle size distribution of the silica particles ((a) and (b)), and PDVB particles ((c) and (d)) obtained by SEM ((a) and (c)) and by DSS ((b) and (d)).

dynamics. Therefore the individual sedimentation velocity of the smaller particles at the larger / might be affected by the effect. Recently we have demonstrated that the DSS technique not only enables us to evaluate the sedimentation velocity and diffusion coefficient, but also allows us to distinguish the type of particle motion via the time-exponent of the correlation function, i.e., ln g ð1Þ ðsÞ / s2 for sedimentation, and / s1 for diffusion process. The correlation functions observed in this study always show the s2 behavior (up to / = 0.1) so that it is concluded that the upturn in n is not directly attributed to the diffusive motion. However, the value of n for linear polystyrene particles with d = 2 lm

( a)

-1

-1

(b) μm

μm n ~5.4

ln + β

-3

-6

-7

n ~ 4.4

μm n ~ 5.9

μm μm

-5

μm

μm

n ~ 4.6

-4

μm

-2 n ~ 4.1

n ~ 4.6

-3

n ~ 4.0

μm

n ~ 7.9 n ~ 8.7

μm

n ~ 9.6

μm

n ~ 9.2

ln + β

-2

-4

-5

μm μm

n ~ 4.5 n ~ 4.9

-6

-7

DSS

-0.2

μm

n ~ 5.5

μm

n ~ 5.4 Interface Silica PDVB

Silica PDVB

-8

μm

n ~ 4.5

-8 -0.1

ln (1 - φ)

0

-0.2

-0.1

0

ln (1 - φ)

Fig. 9. Plots of ln hVzi and ln(1  /) obtained by the (a) DSS and (b) interface analyses. The open and closed symbols respectively denote the results of the silica and PDVB suspensions.

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T. Konno et al. / Ultrasonics 65 (2016) 59–68

approaches unity as the particle concentration increases (not shown here), suggesting the diffusivity could more or less contribute to decrease the settling velocity to give the larger n. 5. Conclusions We have demonstrated that the Dynamic ultraSound Scattering (DSS) technique is a promising probe for the dynamics of microparticles. As has been reported previously, detecting submicron particles by the DSS method also becomes possible, suggesting it could be a complementary tool of dynamic light scattering technique [11]. However, the accuracy of DSS involved 10% of error due to employment of broadband pulse. In order to solve the problem, we proposed a novel technique demonstrated in this paper. Thanks to the technique, the stable results could be obtained regardless of characteristics of transducer, such as the peak frequency and bandwidth of transducers. With the FD-DSS technique, other DSS techniques such as the TD-DSS or phase-imaging technique could be utilized again as a powerful tool to obtain the spatio-temporal information of the sedimentation particles. In this study, we employed the silica and polydivinylbenzene (PDVB) standard particles as model systems. Among them, the silica particles were employed as a monodisperse system. The sedimentation velocities were evaluated by the FD-DSS technique and now excellent agreement between the evaluated particle size and that obtained by SEM was confirmed. Although the particle size was evaluated by a linear extrapolation of the sedimentation velocities (i.e., via the terminal velocity of a single particle), it could be briefly obtained at a finite concentration if the R–Z formula holds and n is calibrated prior to the determination of particle size. Therefore, the theoretical prediction of the Re dependence of n (particularly for the DSS data at the low Re) is important to be elucidated, but it is beyond the scope of this study. On the contrary to the conventional observation of the sedimentation interface, the DSS technique could probe the individual settling particles in suspensions. The extraction of the particle size distribution was also achieved by fully utilizing the scattering functions as well as the position sensitivity of the ultrasound scattering. While the terminal velocity of the PDVB particles is underestimated by the conventional interface analysis (at the concentration range accessible by the DSS method), the DSS technique did successfully provide the information about particle size distribution. Acknowledgements This work was supported by KAKENHI (Grant-in-Aid for Scientific Research), No. 15K05627 from the Ministry of Education, Science, Sports, Culture, and Technology. The authors would like to thank Prof. Takeshi Shiono in Kyoto Institute of Technology for the assistance of FE-SEM measurements. References [1] B.J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology and Physics, Dover Publications Inc., Mineola, NY, 2000.

[2] G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech. 52 (1972) 245–268. [3] M. Kohyama, T. Norisuye, Q. Tran-Cong-Miyata, High frequency dynamic ultrasound scattering from microsphere suspensions, Polym. J. 40 (2008) 398– 399. [4] M. Kohyama, T. Norisuye, Q. Tran-Cong-Miyata, Dynamics of microsphere suspensions probed by high frequency dynamic ultrasound scattering, Macromolecules 42 (2009) 752–759. [5] A. Nagao, M. Kohyama, T. Norisuye, Q. Tran-Cong-Miyata, Simultaneous observation and analysis of sedimentation and floating motions of microspheres investigated by phase mode-dynamic ultrasound scattering, J. Appl. Phys. 105 (2009) 023526. [6] A. Nagao, T. Norisuye, M. Kohyama, T. Yawada, Q. Tran-Cong-Miyata, Collective motion of microspheres in suspensions observed by phase-mode dynamic ultrasound scattering technique, Ultrasonics 52 (2012) 628–635. [7] K. Sugita, T. Norisuye, H. Nakanishi, Q. Tran-Cong-Miyata, Effect of electrostatic interactions on the velocity fluctuations of settling microspheres, Phys. Fluids 27 (2015) 013304. [8] M.L. Cowan, I.P. Jones, J.H. Page, D.A. Weitz, Diffusing acoustic wave spectroscopy, Phys. Rev. E 65 (2002) 0666605. [9] M.L. Cowan, J.H. Page, D.A. Weitz, Velocity fluctuations in fluidized suspensions probed by ultrasonic correlation spectroscopy, Phys. Rev. Lett. 85 (2000) 453– 456. [10] A. Strybulevych, T. Norisuye, M. Hasselfield, J.H. Page, Dynamic sound scattering investigation of the dynamics of sheared particulate suspensions, AIP Conf. Proc. 982 (2008) 354–358. [11] K. Igarashi, T. Norisuye, K. Kobayashi, K. Sugita, H. Nakanishi, Q. Tran-CongMiyata, Dynamics of submicron microsphere suspensions observed by dynamic ultrasound scattering techniques in the frequency-domain, J. Appl. Phys. 115 (2014) 203506. [12] P.N. Segrè, E. Herbolzheimer, P.M. Chaikin, Long-range correlations in sedimentation, Phys. Rev. Lett. 79 (1997) 2574–2577. [13] E.J. Hinch, Sedimentation of small particles, in: E. Guyon, J.-P. Nadal, Y. Pomeau (Eds.), Sedimentation of Small Particles in Disorder and Mixing, N.A.T. O. A.S.I. E, Kluwer Academic, Dordrecht, 1988, pp. 153–162. [14] É. Guazzelli, J. Hinch, Fluctuations and instability in sedimentation, Annu. Rev. Fluid Mech. 43 (2011) 97–116. [15] J.F. Richardson, W.N. Zaki, Sedimentation and fluidisation: part I, Chem. Eng. Res. Des. 32 (1954) 35–53. [16] J. Garside, M.R. Al-Dibouni, Velocity–voidage relationships for fluidization and sedimentation in solid–liquid systems, Ind. Eng. Chem. Process Des. Dev. 16 (1977) 206–214. [17] T. Sawada, T. Norisuye, M. Kohyama, K. Sugita, H. Nakanishi, Q. Tran-CongMiyata, Effects of pulse repetition rate and incident beam energy on the dynamic ultrasound scattering data, Jpn. J. Appl. Phys. 53 (2014) 07KC10. [18] M.L. Cowan, J.H. Page, D.A. Weitz, Novel techniques in ultrasonic correlation spectroscopy: characterizing the dynamics of strongly scattering materials, Acoust. Imag. 26 (2002) 247–254. [19] M.L. Cowan, D. Anache-Menier, W.K. Hildebrand, J.H. Page, B.A. van Tiggelen, Mesoscopic phase statistics of diffuse ultrasound in dynamic matter, Phys. Rev. Lett. 99 (2007) 094301. [20] L. Bergougnoux, S. Ghicini, E. Guazzelli, J. Hinch, Spreading fronts and fluctuations in sedimentation, Phys. Fluids 15 (2003) 1875–1887. [21] R.H. Davis, M.A. Hassen, Spreading of the interface at the top of a slightly polydisperse sedimenting suspension, J. Fluid Mech. 196 (1988) 107–134. [22] X.L. Yin, D.L. Koch, Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers, Phys. Fluids 19 (2007). [23] R.D. Felice, R. Kehlenbeck, Sedimentation velocity of solids in finite size vessels, Chem. Eng. Technol. 23 (2000) 1123–1126. [24] R.D. Felice, E. Parodi, Wall effects on the sedimentation velocity of suspensions in viscous flow, AIChE J. 42 (1996) 927–931. [25] K. Kubo, T. Norisuye, T.N. Tran, D. Shibata, H. Nakanishi, Q. Tran-Cong-Miyata, Sound velocity and attenuation coefficient of hard and hollow microparticle suspensions observed by ultrasound spectroscopy, Ultrasonics 62 (2015) 186– 194. [26] J.J. Faran, Sound scattering by solid cylinders and spheres, J. Acoust. Soc. Am. 23 (1951) 405–418. [27] R.E. Challis, M.J.W. Povey, M.L. Mather, A.K. Holmes, Ultrasound techniques for characterizing colloidal dispersions, Rep. Prog. Phys. 68 (2005) 1541–1637.