Collective motion of microspheres in suspensions observed by phase-mode dynamic ultrasound scattering technique

Ultrasonics 52 (2012) 628–635

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Collective motion of microspheres in suspensions observed by phase-mode dynamic ultrasound scattering technique Ayumi Nagao, Tomohisa Norisuye ⇑, Teppei Yawada, Mariko Kohyama, 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 27 April 2011 Received in revised form 6 January 2012 Accepted 6 January 2012 Available online 17 January 2012 Keywords: Velocity fluctuation Scattering Microsphere

a b s t r a c t Compared with a nano-sized particle, dynamics of a micron-sized particle in a liquid is often associated with sedimentation (or floating) due to its relatively large mass. The motion of more than two particles is dominated by the hydrodynamic interactions, which are known to persist over a fairly long range, e.g., several millimeters, in suspensions. The particle size may be obtained from the dynamic ultrasound scattering (DSS) technique by the analysis of velocity fluctuations, whose origin is believed to take root in the particle-number fluctuations among temporally formed domains involving collective motion of particles with a certain cut-off length. In this study, such collective particle motion in highly turbid solutions was visualized by means of the phase-mode DSS technique with a single element transducer. Quantitative agreement between the velocity fluctuations obtained by the phase- and conventional amplitude-mode analyses was confirmed, followed by examination of the concentration and the particle size dependences on the dynamic structures induced by the long-ranged interactions. It was found that the phase modeDSS was a promising method to evaluate the time-dependent structures of the micro-particles in highly turbid suspensions. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Characterization of suspended micro-particles without dilution nor drying of the samples has been a strong demand both from the academic and industrial point of views [1]. As for an industrial application, particulate filler is often embedded in thermoset plastics in order to improve the mechanical and/or thermal stability. Since the thermal and mechanical properties such as thermal expansion, glass transition temperature and elastic modulus are considered to be highly affected by the particle size, filler–matrix adhesion or the time-dependent structure [2–4], the direct observation of the particle dynamics and structures during curing of resin might provide useful information for fabricating versatile composites. While dynamic light scattering (DLS) [5] technique and similar optical techniques are promising technologies for these purposes, it is unfortunately difficult to characterize opaque samples such as concentrated suspensions or slurry by conventional optical techniques due to the serious attenuation of the incident beam and/or complexity of the analysis involving multiply scattered signals. Therefore, in general, the sample concentration should be moderately dilute or the optical path should be sufficiently short to allow transmission of the light source. Recently, a novel approach based on a combination of interferometry and ⇑ Corresponding author. E-mail address: [email protected] (T. Norisuye). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2012.01.002

confocal optics was proposed in the literature to extract the scattering contribution on the short pass [6,7]. The sophisticated hybrid techniques are very unique and have some success in the field of nano-particle sizing at high concentrations. On the other hand, for the micron-sized particles which may have some costadvantages in industrial applications, the hydrodynamic interaction exerted over a surprisingly long-ranged obeys the entire dynamics. It is well known that there is a significant difference in the particle dynamics between the sample at different locations, e.g., near the container wall and the center of the sample caused by the shear stress from the walls [8]. Therefore establishment of the analysis method for turbid systems with micron-sized particles is required to elucidate the particle dynamics as a function of the sample depth. In this view point, ultrasound-based methods, the dynamic ultrasound scattering (DSS) technique, as an example, would be a good candidate for investigation of micron-sized samples. DSS was originally proposed by Page and co-workers [9] and was utilized to investigate the complex fluid media, such as fluidized bed [10] and sheared particle dynamics [11–13]. Since the source wavelength was too large for our particles, we have developed this technique by using a back scattering geometry associated with high frequency ultrasound transducers to analyze highly opaque suspensions with the micron depth-resolution [14]. Ultrasound Doppler velocimetry [15,16] has essentially the same basis with dynamic ultrasound scattering, in terms of utilization of the time-dependent phase evolution. However, most of the

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e, T

(a)

Evolution tim

A Field time, t or Position, z

(b)

A1A2

A3 A4

Field time, t or Position, y

A Evolution time, T

B s) T(

(c)

Amplitude Field amplitude,

studies were carried out at relatively low frequencies, and were not intended to apply for studying of micro-particles. As for the higher frequency study, one can find literatures based on a cross-correlation technique for ultrasound field data acquired by oriented beam setups [17–19]. This is similar to the auto-correlation function approach, but the cross-correlation technique requires determination of an appropriate area to be analyzed prior to the data analysis. Furthermore, averaging over several wave numbers was required to obtain the satisfactory result. In contrast, the method demonstrated in this study provides instantaneous velocities not only with the evolution time but also with each field-time containing the scattering path information. In the previous works [20], DSS equipped with a high frequency (20 MHz) transducer was utilized to investigate the settling dynamics of microsphere with the particle size ranging from several to several tens of micro-meters. In the case of the back scattering geometry, where the direction of the scattering vector coincides with that of the beam emission, the beam setup parallel (z-direction) and perpendicular (y-direction) to the direction of sedimentation provides versatile information on the particle dynamics. For example, an instantaneous ultrasound pulse emitted along z-direction finds the particles undergoing sedimentation so that the average velocity can be obtained from the time evolution of the scattering amplitude at a fixed field time as illustrated in Fig. 1a. If the average velocity is found, the particle diameter can be evaluated by the balance equation of buoyant force against gravity. Here it should be noted that the velocity field is not uniform because of the long-ranged hydrodynamic interaction [21,22], leading to the emergence of noticeable velocity fluctuations accompanying a large velocity–vortex even at the low-Reynolds number. Since the average velocity is zero for the horizontal component, the velocity fluctuations could be solely evaluated from the horizontal DSS setup as illustrated in Fig. 1b. The auto-correlation function which is constructed from the waveform amplitude as a function of the evolution-time T at a fixed field time t, is suitable to evaluate the relaxation of the amplitude associated with the concentration fluctuations of the micro-particles. In addition to evaluation of the average velocity, the horizontal components could be utilized to understand the concentration and the particle size dependences on the particle dynamics accompanying the long-ranged hydrodynamic interactions. More recently, we have shown that the time-evolution of the instantaneous velocity with its sign as a function of the sample position can be obtained by phase information of ultrasound pulse based on the phase-extraction technique, enabling one to visualize the particle motions, involving settling, floating or those mixed motions, as a function of the sample position [23]. This leads to an idea for visualizing the particle structure formed by the long-ranged hydrodynamic interactions. The velocity fluctuations described above are still the complex matter of fluid dynamics. Among the wide variety of the literatures on theory [24–26], simulation [27– 29] and experiments [21,30,31] of the velocity fluctuations, the most accepted model may be a collective motion of the settling particles driven by formation of swirls, where the velocity fluctuations are ascribed to the fluctuations of particle number among the instantaneous blobs. As realized from the literatures dealing with the non-Poisson statistics [32–34], the number fluctuations are still the matter of interests in the field of particle suspensions accompanying the instantaneously formed particle structures. Such dynamic structures do not exist in nano-particles showing random Brownian motion, but do exist in micro-particles. Therefore, the aim of this paper is to visualize such blobs and to quantitatively confirm the validity of the velocity data obtained by the DSS techniques. Since the functional form of the phase distribution was unclear in the previous paper [23] (Gaussian as a rough approximation), only

τ

ΔT

Phase

Evolution time, T (s)

Fig. 1. Schematic illustrations of the DSS geometry for (a) z- and (b) y-component measurements. (c) The particle velocity is evaluated from the fluctuation time of the scattering signals.

the average velocity was calculated so far. Now this paper enables us to calculate the precise velocity variance from the phase distribution function in order to evaluate the second order fluctuation variable with short acquisition time, leading to further analysis of the long-ranged interactions from the 2D image of blob structures. Evaluation of the average settling velocity is time-consuming as the particle size becomes smaller, thereby the present method could be a promising technique to briefly evaluate the dynamics of microsphere suspensions. Our goal is to establish this method as a complementary tool to characterize variety of the micron-sized particles. 2. Theory As an ultrasound pulse propagates through a cell containing a suspension of microspheres, four reflected echoes A1, A2, A3, A4 from the cell walls are observed as shown in Fig. 1b. If there is noticeable scattering contribution from the microspheres, the complicated scattering patterns could be observed between A2 and A3 as well. The pulse wave w for the scattering component may be written as,

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wðtÞ ¼ AðtÞ cos½2pfc t þ UðtÞ

ð1Þ

where t is the field-time, fc is the central frequency, A and U are respectively the amplitude and phase of the temporal pulse. In the case of the backscattering geometry, t contains the spatial information of the scatterers along the beam direction enabling us to obtain the information about the location of the particles as indicated by the line A in Fig. 1b [20]. Since the time difference between A2 and A3 is fairly short, c.a. 13.5 ls for the round-trip of cell size L = 10 mm, compared to the relaxation time of the particle dynamics (s) for the corresponding wavelength, the time-evolution of the pulse can be visualized as an image of the sound field by successive recording of pulses at an interval DT(ms). As the results, we obtain the time fluctuations of the scattered signals along the axis of the evolution-time, T, as indicated by the line arrow B. The characteristic time of the fluctuations may be evaluated by the time correlation function g(1)(s) defined by,

g ð1Þ ðsÞ ¼

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

ð2Þ

where Dr is the displacement of the microspheres during the lag time s, q is the magnitude of the scattering vector, asterisk indicates complex conjugate. The bracket indicates the average over the evolution 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, hVzi and the standard D E12 deviation of velocity dV 2z can be obtained from,

  1 g zð1Þ ðsÞ ¼ cosðqhV z isÞ exp  q2 hdV 2z is2 2 while the y-component of the standard deviation evaluated from,

The detail of the extraction method is described elsewhere [23]. The advantages of this technique are (1) one can obtain the velocity within the short time as short as the pulse repetition time, (2) the velocity V has the sign so that one can recognize the direction of the particle motion unlike the correlation function approach giving only the average of the absolute velocity, and (3) the velocity can be evaluated as a function of the sample position. If necessary, averaged velocity hVi and its variance hdV2i may be estimated from the interested region of the velocity map by statistical average. In the following section, we will briefly describe the statistics of phase to interpret the distribution of the instantaneous velocity obtained by phase-mode dynamic ultrasound scattering. In order to focus on the velocity fluctuations, the subsequent discussion deals with a system involving no average component for the displacement. First, we assume arbitrary two points of a correlated multi-variable Gaussian distribution of the amplitude A and A0 and the phase U and U0 [37,38]. The joint distribution function of the phase difference, DU  U0  U, is obtained by integration over the amplitudes, followed by transformation of the variables (U0 , U) to the phase difference (DU, 1). After the nominal mathematical procedure [39], we obtain,

0

b ¼ g ð1Þ cos½DU if g

PðDUÞ ¼ is solely

ð4Þ

ð8Þ

is the real number. Eq. (7) has an asymptotic form,

ð3Þ

  1 g yð1Þ ðsÞ ¼ exp  q2 hdV 2y is2 2

q2 hDr 2 i

ðDrÞ

2

E

¼

D

dV 2y

E

after taking short time limit by Taylor expansion. Eq. (9) has already derived before [40,41] and was utilized to investigate the phase statistics of optics and acoustics [35,36,42,43]. By substituting Eq. (5) into Eq. (9), we obtain,

PðDUÞ ¼

s

2

Vðt; TÞ ¼

1 DUðt; TÞ q DT

2 1=2

2qhdV i

ð5Þ

Those quantities were successfully evaluated for each geometry as described in the previous paper [20] Although fitting of the correlation function to these functions is an effective approach to evaluate the quantitative velocities, it requires an averaging time to construct the reliable correlation function. Therefore the conventional approach may be insufficient to evaluate the velocity within a short window of the experimental time. As can be realized from line A in Fig. 1a, one needs to define the appropriate time region of interests for sedimentation, i.e., quasistationary region of T at a fixed field time t, prior to construction of the correlation function. Therefore it motivates us to propose a novel technique allowing the evaluation of instantaneous velocities from the phase difference containing the information about the particle positions [23]. Although the phase is the matter of interests, the Fourier transformation is not a suitable operation because the resultant phase would not be a function of time t but frequency f, resulting in disappearance of the information on the sample position. Therefore the lock-in approach [35,36], which was essentially equivalent to the Hilbert transformation, was employed. The extracted phase difference DU(t, T) contained in the waveform (Eq. (1)) is further transformed into the instantaneous velocity,

ð6Þ

ð9Þ

3

2ðq2 hDr 2 i þ ðDUÞ2 Þ2

since the average component for the y-direction is zero and,

D

ð7Þ

for the phase difference distribution function where b is given by,

(1)

1 hdV 2y i2

1

1 1  jg ð1Þ j2 B b cos1 ðbÞC PðDUÞ ¼ @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2p ð1  b Þ 1  b2



1

DT 1 þ

1 hdV 2 i



1 DU q DT

2 32

ð10Þ

Beside the divergence of P(DU) at the short time limit DT ? 0, we notice that further transformation of the variable d(DU) = (qDT)dV allows us to find a finite value for the particle velocity. In the case of observation from the horizontal direction, 1 the velocity corresponds to hdV 2y i2 , therefore with Eq. (6),

PðVÞ ¼

1  32 1 V2 2hdV 2y i2 1 þ hdV 2 i

ð11Þ

y

is finally obtained. Note that above equation indicates that there is no fitting parameter other than hdV 2y i without any proportional constant. If there is noticeable contribution from the average displacement, the parameter V in RHS of Eq. (11) should read (V  hVi). 3. Experimental Mono-disperse polystyrene microspheres with different particle diameter d were purchased from Sekisui Chemical Co. Ltd. The particles were dispersed in an aqueous solution containing 0.2% sodium dodecyl sulfate (SDS) to obtain a suspension with desired volume fraction in range 1 < / < 30%, followed by a brief immersion in a low power ultrasonic bath prior to DSS experiments to avoid aggregation. SEM micrographs (Hitachi S-3000N) were also taken to verify the particle size and its distribution.

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is obtained for the settling particles observed from the y-direction 1 so that the standard deviation hdV 2y i2 are expected to be solely evaluated. While capturing of the scattering signals were carried out for sufficiently long pulse transit-time and observation time, figures (a–c) were extracted from the source matrix as shown in the insets for the sake of demonstration for the readers. The position z or y was converted from the field time by taking account of the round-trip time and the sound velocity as described later. Fig. 2d–f shows the histograms constructed from the corresponding velocity images. The distribution of (d) the settling particles and (e) the floating particles respectively exhibited the positive and negative mean velocity, while the distribution function obtained from (f) the y-direction has the zero mean. The solid lines indicate the theoretical prediction of the velocity distribution from Eq. (11). The excellent agreement between the experiments and theory was achieved, suggesting us to apply this technique to the complex system where the particles form a blob moving in different directions. When ultrasound beam incident in a suspension containing microspheres surrounded by fluid, the propagation of sound is fairly rapid compared with the heat diffusion, enabling us to assume the propagation is adiabatic. If the energy of sound is sufficiently low, one can expect the linear response of displacement from the pressure field. However, if we apply too much sound energy into the suspension, the displacement would become non-linear with respect to the applied pressure, resulting in the emergence of harmonic and DC frequency contributions. The latter cannot be cancelled out even if the time averaging is taken, leading to the extra-energy to invoke unexpected particle motion, e.g., upflow against gravity brought by sound [45]. Fig. 3 shows the velocity field of the settling particles acquired by a focused-beam ultrasound transducer (6/ – 20 MHz). As clearly seen from the negative velocity in the image, the particles exhibited floating against gravity, while the regular sedimentation

Gay-Lussac pycnometer was used to measure the densities to three places of decimals. Polystyrene rectangular vessels with the dimension 10  10  40 mm3 were used as the sample cells. The wall thickness was 1 mm. All the experiments were performed at 20.0 ± 0.05 °C. The signal recording was initiated after approximately 90 s for all the particles except for the largest particles with d = 32 lm (20 s) in order to avoid the effects from the turbulent flow due to initial mixing of the particles. A negative impulse emitted from a pulser/receiver (Olympus, model 5800PR) was transferred to a 10 or 20 MHz – longitudinal plane wave transducer (Olympus) immersed in a water bath to generate broadband ultrasound pulse. Note that the pulse repetition time should not be too high to avoid reverberation of the sound on the recording field and/or unexpected particle flow due to excess energy while most of the conventional pulser/receiver could be employed for the measurements. The transducer and the cell container were carefully aligned by using a custom-made stainless stage coupled with rotational and translational slides prior to the scattering experiments. The obtained signals were then amplified by the receiver, followed by successive recording with a 14 bit high-speed digitizer (GaGe, Compuscope CS14200) at the sampling rate 200 MS/s. The detail information of the experiments is given in the previous papers [20,23,44].

4. Results and discussions Fig. 2a–c shows the velocity field obtained for the settling particles dispersed in water (a) and (c) and for the floating particles in deuterium water (b). For the images (a) and (b), experiments were carried out from the z-direction to obtain the average velocity hVzi 1 and the standard deviation of velocity hdV 2z i2 , while the image (c) 10 8 6 4 2

10 8 6 4 2 120

0 6.0

5.5 5.0 4.5

0.04 0 -0.04

10

20

30

(b)

5.5

5.0

4.5

0.04 0 -0.04

(d)

30

P(Vz )

15 10 5

10

20

30

5.0 4.5

0

40

0.04 0 -0.04

20

20 15

Vz

10

10

20

30

40

Evolution time, T (s)

(e)

(f)

15

Vy

10 5

5

0

120

5.5

25

20

80

(c)

Evolution time, T (s)

Evolution time, T (s)

40

4.0 0

40

0

6.0

P(Vy)

0

P(Vz )

120

4.0

4.0

25

80

Vz (mm/s)

Position, z (mm)

(a)

40

Vy (mm/s)

80

Vz (mm/s)

Position, z (mm)

40

Position, y (mm)

0

6.0

10 8 6 4 2

0

-0.15 -0.10 -0.05 0

0.05 0.10 0.15

Vz (mm/s)

-0.15 -0.10 -0.05 0

0.05 0.10 0.15

Vz (mm/s)

-0.15 -0.10 -0.05 0

0.05 0.10 0.15

Vy (mm/s)

Fig. 2. Images of the particle velocity showing (a) sedimentation and (b) floating observed from the z-direction, and (c) y-component velocity involving sedimentation. The corresponding velocity distribution functions are given in figure (d–f) where the solid lines indicate the theoretical prediction with Eq. (11). The figures (a–c) are extracted from the raw data capturing the whole process of settling or floating as shown on top of the figures (a–c).

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A. Nagao et al. / Ultrasonics 52 (2012) 628–635

diameters obtained by the horizontal setup. The correlation functions monotonically decreased from unity to zero, which was the characteristic feature of the dynamics in the absence of the average component of the velocity. The standard deviation of velocity 1 hdV 2y i2 can be systematically calculated by fitting the data to Eqs. (4) and (11). As seen from the figure, the curve fitting is satisfactory so that we can proceed to the next stage to quantitatively understand the velocity fluctuations as a function of the particle diameter. As described in the introduction, the velocity fluctuations of settling particles have been investigated by many researchers. Caflisch and Luke [24], and Hinch [25] derived the following formula,

V (mm/s)

V (mm/s)

-0.05

Position, z (mm)

8 6

0

0.02

8

0 -0.02

6

-0.04

4

4

2

2

0

0

0

5

10

15

20

Evolution time, T (s)

hdV 2y i1=2 ¼ C y V 0

FTD Fig. 3. Image of particle velocity showing upflow motion against gravity induced by focused ultrasound energy.

was observed at the out-of-focus region. The distance of the detector to the center of sample was assigned to be the focal distance of the transducer, i.e., 38 mm, so that the maximum power was applied to the middle of the cell. Without the focused beam, i.e., using the plane beam with the same frequency, the velocities are all positive indicating that the particles exhibit regular sedimentation as described before. In the next section, we will discuss the dependence of the par1 ticle size on the standard deviation of velocity hdV 2y i2 obtained from the horizontal beam setup. Fig. 4 shows the correlation functions and the velocity distribution functions with different particle

rffiffiffiffiffiffiffiffiffi 1 1 1 2/L 22 C y Dqg/2 L2 32 ¼ d d 18g0

ð12Þ

for the settling micro-particles where Cy is a constant related to the structure factor, V0 is the terminal velocity determined by the viscosity g0, gravitational acceleration g, density difference Dq between the particle and liquid and so on. It is now well-known that divergence of the velocity fluctuations with respect to L (see Eq. (12)) is unphysical and there must be a cut-off length for the blobs. This idea was suggested by Segre, resulting in modification 1

1

of the scaling relation with respect to / ð/2 ! /3 Þ [21]. However, Brenner argues that Eq. (12) is still valid if the screening of the hydrodynamics interaction is taken into account [28]. Since the volume fraction dependence of hdV 2y i1=2 for our suspensions followed 1

/2 dependence, as predicted by the model, the diameter depen3

dence hdV 2y i1=2 / d2 was subsequently examined. As shown in 3

Fig. 5, hdV 2y i1=2 increased with d and roughly followed the d2 depen-

1.0 80

0.8 0.6

60

3.45μm

0.4

40

0.2

20

0

0

1.0

80

0.8

60

0.6

4.87μm

40

0.4

20

P(Vy)

0

(1)

gy ( )

0.2

1.0

0 40

0.8

30

0.6

10μm

0.4

20

0.2

10

0

0

1.0

20

0.8

15

0.6

18μm

0.4

10 5

0.2 0 0.1

1

(s) ð1Þ

10

0 -0.10

-0.05

0

0.05

0.10

Vy (mm/s)

Fig. 4. Particle size dependences of the correlation functions g y ðsÞ and velocity distribution functions P(Vy). The solid lines indicate the result of fitting to Eqs. (4) and (11).

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A. Nagao et al. / Ultrasonics 52 (2012) 628–635

y = ct/2. When we probe the sedimentation process at a fixed height from the bottom, particles and its blob come into the observation window, followed by escape from the detectable region at an average rate hVzi. Without scanning along the vertical direction, the blob size of the z-direction may be obtained from fixed location measurements if the instantaneous velocities as a function of T are converted by using the relation z ¼ hV z iT. Here we assume that the blobs are not broken during the short time period to construct the 2D image. This hypothesis is valid as long as the small number fluctuations between blobs are the matter of interests. It should be noted here that the quantitative analysis can only be performed along the z-axis at fixed y location since the transducer has a finite beam area faced to the xz plane and the y(or t) axis has the high resolution provided by the maximum sampling rate of the digitizer. Fig. 6 shows the concentration dependence of the velocity image. The scattered field data was processed by the method described above to give the velocity images. As indicated by the images, the blob size n becomes smaller with increasing the concentration, in agreement with the relation proposed by Segre [21].

0.45 0.35 0.30 0.25 0.20

-1

0

10

20

30

4

d (μm)

2 1/2

< Vy> /

100

Phase Amplitude d dependent Cy Cy=0.28

10 1

10

d (μm) Fig. 5. The diameter dependence of the velocity fluctuations evaluated by the phase (solid circle) and amplitude modes (open circle) of the DSS techniques. The proportional constant Cy appeared in Eq. (12) is shown in the inset of the figure.

1

n / d/3

V (mm/s)

1%

0.010

V (mm/s)

dence as indicated by the dotted line where hdV 2y i1=2 was normalized by Dq in order to account for the difference on the densities. While Cy = 0.28 was proposed in the literature [27], Cy seemed to be dependent on the particle size if it was assigned as an adjustable parameter [20]. The solid line is reproduced from Eq. (12) with an empirical form Cy = 0.24 + 0.31 exp (0.17d). Although there is no convincing evidence nor plausible model dealing with Cy at this stage, we found a good agreement between the velocity fluctuations calculated by the conventional-amplitude (the open circle) and phase mode (the solid circle) DSS. So far, we have focused on the evaluation of the velocity fluctuations to extract the information on the size and concentration of the micron-sized particles in turbid suspensions. Formation of swirls and resultant collective motion of the particles induced by the long-ranged hydrodynamic interaction play an important role for understanding the velocity fluctuations. Therefore, establishment of the DSS technique utilizing scattering of high frequency ultrasound would be crucial to visualization and elucidation of such instantaneous structures. As demonstrated in Fig. 1b, the backscattered echoes probe the dynamics of particles at a specific location along the y-axis. The position y may be obtained from the velocity of sound c and the pulse transit-time t for a round-trip,

2.0

3%

0 -0.010

ð13Þ

The proportional constant were so far experimentally determined and depended on the geometry. According to this relation, one may expect the blob size would increase with increasing the particle size. Although it is not shown here, such a behavior is also confirmed for the size dependence of the velocity images. In order to quantitatively evaluate the blob size, Fig. 6 was further analyzed by a spatial-correlation function defined by hdVy(z)dVy(z + Dz)i. If the correlation function can be expressed   by a single exponential function, i.e., exp  Dnz , the harmonic average of n may be obtained by,

 n ¼  lim

Dz!0

1 d lnhdV y ðzÞdV y ðz þ DzÞi dDz

0.02 0.01 0 -0.01 -0.02

7.5 %

0.04 0.02 0 -0.02 -0.04

30 %

0.02 0 -0.02

Position, z (mm)

1.5

1.0

0.5

0 0

2 4

6

8

0

2

4

6

8

ð14Þ

Since appropriate direction for the analysis of the matrix of dVy was the z-direction, the correlation function was first constructed along the direction and subsequently averaged over different y positions. Fig. 7 shows the volume fraction dependence of the correlation length n evaluated by Eq. (14) with different particle diameters. As seen from the figure, n increased with increasing the particle diameter and decreased with the volume fraction as expected from the model described above (see Eq. (13)). The lines 1 were calculated from n ¼ 7:5d/3 [46]. Although n evaluated from at least 10 different runs is more or less scattered in the figure, it is

V (mm/s)

-1

(mm s g )

Cy

0.40

V (mm/s)

1000

0

2

4

6

8

0

2 4

6

8

Position, y (mm) Fig. 6. The concentration dependence of the velocity images showing collective motion of the particles.

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A. Nagao et al. / Ultrasonics 52 (2012) 628–635

Correlation length, (mm)

2

3.6 μm

1

5.5 μm

8 6

10 μm

4 2

0.1

8 6 4 8 9

2

3

4

5

6 7 8 9

1

2

3

4

10

Volume fraction,

(%)

Fig. 7. The correlation length evaluated by Eq. (14) as a function of the volume fraction / with different particle diameters.

concluded that above discussion seems to be consistent within our experimental findings. In order to improve further the quality of the data deviating from the power-law line, the image matrix should be accumulated over several tens of images because of the extraordinary large correlation length, which sometimes exceeds the size of cell, so that one requires the longer sampling range or enormous number of average to obtain the stable results. The effect of cell size to the velocity fluctuations is also an important issue, which will be addressed in the forthcoming article. 5. Conclusions In this study, we have demonstrated that the phase-mode dynamic ultrasound scattering (DSS) technique is an effective tool to investigate the dynamics of micron-sized particles accompanying sedimentation with the long-ranged hydrodynamic interaction. Since the technique employs ultrasound, one can apply this technique to highly turbid systems or to investigate the real time dynamics for micron-sized particles. Comparing to the conventional correlation function approach, one can evaluate (1) instantaneous velocity, (2) its sign, and (3) the velocity as a function of the spatial location of the scatterers. These advantages allowed us to investigate the long-ranged interaction of the complex fluid systems. Visualization of the time-dependent structure and subsequent analysis of the correlation length were carried out, and the results were in agreements with the effects of particle size and concentration on the velocity fluctuations and the correlation length. On the contrary to the analysis of nano-particles accompanying Brownian motion, sizing of micron-sized particles is rather challenging because of the complicated hydrodynamic interactions. Unlike the particle size and volume fraction dependences as described in this article, the cell size dependence (effects of spatial confinement) still remains unclear. In this meaning, developing this technique and understanding the dynamic structure are crucial for practical applications using micro-particles. Acknowledgements This work was supported by KAKENHI (Grant-in-Aid for Scientific Research), No. 22750205 and KAKENHI (Grant-in-Aid for Scientific Research) on Priority Area, ‘‘Soft Matter Physics’’ (No. 463/ 19031018) from the Ministry of Education, Science, Sports, Culture, and Technology. References [1] T. Provder, J. Texter, Particle Sizing and Characterization, American Chemical Society, Washington, DC, 2004.

[2] T. Adachi, M. Osaki, W. Araki, S.-C. Kwon, Fracture toughness of nano- and micro-spherical silica-particle-filled epoxy composites, Acta Mater. 56 (2008) 2101–2109. [3] C.G. Robertson, Influence of particle size and polymer-filler coupling on viscoelastic glass transition of particle-reinforced polymers, Macromolecules 41 (2008) 2727–2731. [4] Y. Rao, T.N. Blanton, Polymer nanocomposites with a low thermal expansion coefficient, Macromolecules 41 (2008) 935–941. [5] B.J. Berne, R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology and Physics, Dover Publications, Inc., Mineola, NY, 2000. [6] K. Ishii, R. Yoshida, T. Iwai, Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer, Opt. Lett. 30 (2005) 555–557. [7] H. Xia, K. Ishii, T. Iwai, Hydrodynamic radius sizing of nanoparticles in dense polydisperse media by low-coherence dynamic light scattering, Jpn. J. Appl. Phys. 44 (2005) 6261–6264. [8] P. Peysson, An experimental investigation of the intrinsic convection in a sedimenting suspension, Phys. Fluids 10 (1998) 44–54. [9] M.L. Cowan, J.H. Page, D.A. Weitz, Novel techniques in ultrasonic correlation spectroscopy: characterizing the dynamics of strongly scattering materials, Acoust. Imaging 26 (2002) 247–254. [10] 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. [11] A. Strybulevych, D.M. Leary, J.H. Page, Shear-induced structure and velocity fluctuations in particulate suspensions probed by ultrasonic correlation spectroscopy and rheology, AIP Conf. Proc. 708 (2004) 444–445. [12] 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. [13] A. Strybulevych, J.H. Page, Shear-induced structure of concentrated suspensions of non-Brownian particles in a microgravity environment, Microgravity Sci. Technol. 16 (2005) 249–252. [14] M. Kohyama, T. Norisuye, Q. Tran-Cong-Miyata, High frequency dynamic ultrasound scattering from microsphere suspensions, Polym. J. 40 (2008) 398– 399. [15] U. Lemmin, T. Rolland, Acoustic velocity profiler for laboratory and field studies, J. Hydr. Eng. 123 (1997) 1089–1098. [16] L. Zedel, A.E. Hay, R. Cabrera, A. Lohrmann, Performance of a single-beam pulseto-pulse coherent doppler profiler, IEEE J. Oceanic Eng. 21 (1996) 290–297. [17] S. Manneville, L. Bécu, A. Colin, High-frequency ultrasonic speckle velocimetry in sheared complex fluids, Eur. Phys. J. Appl. Phys. 28 (2004) 361–373. [18] S. Manneville, L. Becu, P. Grondin, A. Colin, High-frequency ultrasonic imaging. A spatio-temporal approach of rheology, Colloids Surf., A: Physicochem. Eng. Aspects 270–271 (2005) 195–204. [19] Y. Takeda, Velocity profile measurement by ultrasonic doppler method, Exp. Therm. Fluid Sci. 10 (1995) 444–453. [20] M. Kohyama, T. Norisuye, Q. Tran-Cong-Miyata, Dynamics of microsphere suspensions probed by high frequency dynamic ultrasound scattering, Macromolecules 42 (2009) 752–759. [21] P.N. Segre, Long-range correlations in sedimentation, Phys. Rev. Lett. 79 (1997) 2574–2577. [22] É. Guazzelli, Sedimentation of small particles: how can such a simple problem be so difficult?, C R. Mec. 334 (2006) 539–544. [23] 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. [24] R.E. Caflisch, J.H.C. Luke, Variance in the sedimentation speed of a suspension, Phys. Fluids 28 (1985) 759–760. [25] E.J. Hinch, Sedimentation of small particles, in: E. Guyon, J.-P. Nadal, Y. Pomeau (Eds.), Sedimentation of Small Particles in Disorder and Mixing, NATO ASIE, Kluwer Academic, Dordrecht, 1988, pp. 153–161. [26] G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech. 52 (1972) 245–268. [27] P.J. Mucha, S.Y. Tee, D.A. Weitz, B.I. Shraiman, M.P. Brenner, A model for velocity fluctuations in sedimentation, J. Fluid Mech. 501 (2004) 71–104. [28] M.P. Brenner, Screening mechanisms in sedimentation, Phys. Fluids 11 (1999) 754–772. [29] F.R. Cunha, G.C. Abade, A.J. Sousa, E.J. Hinch, Modeling and direct simulation of velocity fluctuations and particle velocity correlations in sedimentation, J. Fluids Eng. 124 (2002) 957–968. [30] P.N. Segre, F. Liu, P. Umbanhowar, D.A. Weitz, An effective gravitational temperature for sedimentation, Nature 409 (2001) 594–597. [31] P. Snabre, B. Pouligny, C. Metayer, F. Nadal, Size segregation and particle velocity fluctuations in settling concentrated suspensions, Rheol. Acta 48 (2009) 855–870. [32] B.J. Ackerson, X.L. Lei, P. Tong, Subtle order in settling suspensions, Pure Appl. Chem. 73 (2001) 1679–1688. [33] L. Bergougnoux, E. Guazzelli, Non-poisson statistics of settling spheres, Phys. Fluids 21 (2009) 091701. [34] X. Lei, B.J. Ackerson, P. Tong, Settling statistics of hard sphere particles, Phys. Rev. Lett. 86 (2001) 3300–3303. [35] J.H. Page, M.L. Cowan, D.A. Weitz, B.A. Tiggelen, Diffusing Acoustic Wave Spectroscopy: Field Fluctuation Spectroscopy With Multiply Scattered Ultrasonic Waves, Kluwer Academic, Amsterdam, 2003.

A. Nagao et al. / Ultrasonics 52 (2012) 628–635 [36] 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. [37] J.W. Goodman, Speckle Phenomena in Optics, Roberts and Company Publishers, South Yosemite, 2007. [38] E. Jakeman, K.D. Ridley, Fluctuations in Scattered Light, optica.unican.es, pp. 1–8. [39] D.K.C. Macdonald, Some statistical properties of random noise, Proc. Cambridge Phil. Soc. 45 (1949) 368–372. [40] O.S. Rice, Statistical properties of a sine wave plus random noise, Bell Syst. Tech. J. 27 (1948) 109–157. [41] E. Jakeman, S.M. Watson, K.D. Ridley, Intensity-weighted phase-derivative statistics, J. Opt. Soc. Am. A 18 (2001) 2121–2131. [42] A.Z. Genack, P. Sebbah, M. Stoytchev, B.A. van Tiggelen, Statistics of wave dynamics in random media, Phys. Rev. Lett. 82 (1999) 715–718.

635

[43] B.A. van Tiggelen, P. Sebbah, M. Stoytchev, A.Z. Genack, Delay-time statistics for diffuse waves, Phys. Rev. E 59 (1999) 7166–7172. [44] T. Norisuye, S. Sasa, K. Takeda, M. Kohyama, Q. Tran-Cong-Miyata, Simultaneous evaluation of ultrasound velocity, attenuation and density of polymer solutions observed by multi-echo ultrasound spectroscopy, Ultrasonics 51 (2011) 215–222. [45] C.J. Hartley, Characteristics of acoustic streaming created and measured by pulsed Doppler ultrasound, IEEE Trans. Ultrason., Ferroelec., Freq. Contr. 44 (1997) 1278–1285. [46] S.Y. Tee, P.J. Mucha, L. Cipelletti, S. Manley, M.P. Brenner, P.N. Segre, D.A. Weitz, Nonuniversal velocity fluctuations of sedimenting particles, Phys. Rev. Lett. 89 (2002) 054501.