Zivkovic et al _text + figures

© V. Zivkovic, M. J. Biggs, D.H. Glass, P. Pagliai, A. Buts, 2007

Particle dynamics in a dense vibrated fluidized bed as revealed

by diffusing wave spectroscopy

V. Zivkovic, M. J. Biggs,* D.H. Glass, P. Pagliai, A. Buts

Institute for Materials and Processes, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, United

Kingdom, EH9 3JL.

Abstract

We report granular temperature data and long-time dynamics of mono-

disperse glass particles in a three-dimensional dense bed subject to vertical

sinusoidal vibrations over a wide range of conditions. The granular

temperature of the particles was found to scale with the square of the peak

vibrational velocity. The mean time of flight between the collisions was

found to scale with inverse of the square of the peak vibrational velocity,

whilst the mean free path of the particles was observed to scale with the

inverse of this velocity. The movement of the particles throughout the bed,

which was observed to be sub-diffusive over macroscopic timescales for all

conditions considered here, appears to be governed by collective motion of

particles between cavities defined by their neighbours.

* Author for correspondence. Phone: +44-131-650-5891. Fax: +44-131-650-6551. Email: [email protected]

1 © V. Zivkovic, M. J. Biggs, D.H. Glass, P. Pagliai, A. Buts, 2007

1. Introduction

Vibrated granular materials have received much attention over many years because they are a

simple example of dissipative non-equilibrium systems that demonstrate rich and complex

behaviour [1, 2]. They are, however, also of wide interest across industry. For example, flow of

bulk solids from hoppers and silos is often controlled through the application of vibration [3], whilst

more recent work has shown that vibration can also improve the fluidization of fine powders [4].

Vibration has long been known to cause segregation in mixtures of different sized particles [5] as

well as compaction, whether desired as in the manufacture of sintered products [6] or otherwise [7].

Squires [8] has recently highlighted how vibrated granular systems may lead to improved operation

of reactors, combustors and heat exchangers. Finally, vibration of granular materials is important in

the construction industry including, for example, in the manufacture of building materials [9], soil

improvement [10] and soil response to seismic events [11].

A key quantity in granular systems is the ‘granular temperature’, which is defined as the mean

of the square of the velocity fluctuations about the mean velocity [12, 13]. The granular temperature

underpins the kinetic theory of granular flows [14, 15], which has been widely used to model

various non-equilibrium granular systems [16-19], as well as theories for heat transfer in granular

materials [20], erosion in fluidized beds [21] and granulation [22]. The validation of these theories

demands the experimental elucidation of the particle dynamics in general and determination of

granular temperature in particular.

Much understanding of the dynamics of particles in vertically vibrated fluidized beds (vibro-

FBs) and associated granular temperature data has been accumulated over the past decade or more

using high-speed video imaging [23-31], nuclear magnetic resonance [32, 33], and positron

emission particle tracking (PEPT) [34-37]. Various constraints inherent to these methods – multiple

light scattering for video methods, small bed-to-particle diameter ratios in NMR, and limited spatial

and temporal resolutions – mean they have all been restricted to one particle thick [23-28],

relatively shallow [29-31], confined [32, 33] or dilute three-dimensional (3D) [34-37] beds.


A few methods have been (or in principle can be) applied to dense systems not accessible to

video-based methods, NMR and PEPT including solids pressure data inversion [38], mechanical

spectroscopy [39, 40], diffusing wave spectroscopy (DWS) [41], and speckle visibility spectroscopy

(SVS) [42]. The pressure-data based approach is, however, limited to the wall region, whilst

mechanical spectroscopy is an intrusive method that appears to give results that depend on the

nature of the probe used [40]. Light scattering methods such as DWS and SVS, on the other hand,

are neither intrusive nor restricted to the wall region, yet they are able to probe length and

timescales well below those accessible to NMR, their nearest rival in this regard.

The single DWS-based study to date of dense 3D vibro-FBs [41] was very limited in scope,

focussing on a single frequency and short time dynamics of the particles. The SVS study [42] was

similarly limited, with it focusing on detailing a new method rather than phenomena. We, therefore,

undertook a far more detailed study of the dynamics of particles in a dense 3D vibro-FB using DWS

in which we considered both the short and long-time behaviour of the particles across a wide range

of conditions and at various point in the bed – this study is reported here. We first outline the

experimental details, including those pertaining to the apparatus, the particulate material and the

experimental procedures used. This is followed by presentation of the results obtained, which

include the particle dynamic regimes observed and the variation of the granular temperature with

process conditions and height above the base of the bed, and their discussion.

2. Experimental details

2.1. Experimental setup

The experimental apparatus is illustrated in Figure 1. The granular material was held in a

rigidly fixed rectangular column constructed from two 500 mm high and 196 mm wide borosilicate

glass plates separated by two aluminium edges 14.5 mm thick. The granular material was subject to

vertical vibrations at the base of the column by a piston fixed to an air-cooled electromagnetic

driven shaker (V721, LDS Ltd., Hertfordshire, UK) controlled by a Dactron COMET USB

controller (LDS Ltd.) with feedback from two integrated circuit piezoelectric accelerometers (model


353B03, PCB Piezotronics Inc., NY, US) attached to the piston. The shaker was capable of

delivering a range of different vibrational modes at frequencies in the range of 5-4000 Hz,

accelerations up to 70g, and amplitudes as large as 12.7 mm.

The dynamics of the particles in the vibro-FB were studied using DWS in transmission mode

[43-47]. This method involves illuminating one side of the bed at the point of interest with an ~2

mm diameter laser beam and collecting the scattered light from the opposite side of the bed over

time, t, with a single mode optical fibre (OZ Optics Ltd., Ottawa, Canada). A 400 mW diode

pumped solid state linearly polarized laser (Torus 532, Laser Quantum Ltd., Cheshire, UK)

operating at a wavelength of λ = 532 nm in single longitudinal mode was used. The collected light

signal was bifurcated and fed into two matched photomultiplier tubes (PMTs) to reduce spurious

correlations due to possible after-pulsing effects of the detector. The outputs from the PMTs were

amplified and fed to a multi-tau digital correlator (Flex 05, Correlator.com, US), which performed a

pseudo cross-correlation analysis in real time to give the intensity autocorrelation function (IACF),

g2(t), that was stored on a PC for further offline analysis as detailed below. Both the laser and the

fibre optic cable were mounted on linear stages so that different points above the base of the vibro-

FB could be investigated with ease.

2.2. Particulate material

The particulate material, which was selected to ensure that all experiments were conducted in

the DWS regime, consisted of semi-transparent, strongly light-absorbing, spherical glass particles of

diameter dp = 0.95 ± 0.05 mm. In all cases the vibro-FB was filled to a depth of 75 mm with the

material and subject to at least 10 minutes of vibration at 50 Hz and 4.17g before any data was

collected so as to ensure, as far as was practically possible, that the bed was in a ‘stationary state’.

2.3. Experimental procedure

2.1.3. Study details

The vibro-FB was subject to sinusoidal vibrations in all cases reported here. The vertical

position of the piston at time, t, for such vibrational motion is governed by


( )sinpy A tω= (1)

where A and 2f ω π= are the amplitude and frequency respectively. The associated peak

vibrational velocity and acceleration of the piston are pv Aω= and 2pa Aω= respectively; the

latter is presented here in the non-dimensional form, pa gΓ = , where g is the acceleration due to

gravity.

The IACF was determined for the frequency/acceleration combinations indicated in Table 1

at three points in the vibro-FB whose coordinates were x = 49 mm and y = 25, 40 and 55 mm, where

x is the horizontal distance from the centreline of the bed, and y is the distance from the vibrating

base of the bed (see Figure 1). As this table shows, the amplitude of the forcing varied from just

under half a particle diameter at f = 50 Hz and Γ = 4.170, to just 1.5% of the particle size at f = 500

Hz and Γ = 1.555. Further inspection of this table also shows that the frequency/acceleration

combinations were selected so as to allow systematic study of the effect of peak vibrational velocity

and amplitude over wide ranges.

2.2.3. Elucidating the particle dynamics and determining the granular temperature

Intensity autocorrelation functions (IACF) were obtained by collecting and correlating four

blocks of data of 60 s long each. Each IACF was then subject to further analysis as follows. The

normalized electric-field autocorrelation function (FACF), g1(t), was obtained from the intensity

autocorrelation function , g2(t), using the Siegert relationship [43]

2

2 1 12

(0) ( )( ) 1 ( )

I I tg t g t

Iβ≡ = + (2)

where β 1 is a phenomenological parameter determined from the intercept of the IACF; this

phenomenological parameter was always found to be β1 ≈ 0.5, as expected. The mean square

displacement (MSD) of the particles, 2 20( ) 6r t t kτ∆ = , was determined by inverting the FACF

using [43]


* * * *0 0

*0

1* * * *

* *

/ 4 / 3 6 3 2 6 3 6 3sinh cosh

/ 2 / 3 * 3 *( )

4 6 4 6 3 4 6 3 6 31 sinh cosh

9 3 3

a a a

a a a a

L l z t l t l z t l

z l l l l l lg t

t l L t l t l L t l

l l l l l l

τ τ τ

τ τ τ τ

+ + + + + + =

+ ⋅ + + + + +

(3)

where L is the sample thickness (14.5 mm here), l* is the transport mean free path, la is the

absorption path length, *0z lγ= is the distance over which the incident light is randomized,

21/ ok Dτ = , D is the photon diffusion coefficient, and 0 2k π λ= . The scaling factor, γ, was set to

unity in line with common practise [43, 46]. Note that this equation is for the case where light

absorption is not insignificant.

The mean square particle velocity fluctuation about the mean velocity, 2vδ , was derived

from the short-time region of the MSD where ballistic motion occurs using [45-47]

2 2 2lim( )

0r t v t

tδ∆ =

→ (4)

2.3.3. Determination of the transport mean free path and absorption path length

Equation (3) requires knowledge of the transport mean free path, l*, and absorption path

length, la, at the positions and conditions considered. Because the experimental set-up meant the

incident light intensity could not be measured with ease and only the L = 14.5 mm bed described

above could be vibrated, a two-stage method based on the approach of Leutz and Rička [48] was

used to obtain the transport mean free path and absorption path length data – this method is

described here.

The transport mean free path of a 0.1 vol % aqueous suspension of latex spheres (0.6 µm

polystyrene, G. Kisker GbR., Germany) was first determined by obtaining the electric-field

autocorrelation function of the suspension, and then inverting the following equation, which is valid

in the weak absorption limit [43]


*

1

* *

4 6

3( )

8 6 4 6 61 sinh cosh

3 3

L t

lg t

t L t t L t

l l

τ

τ τ τ τ

+ =

+ +

(5)

A value of * 2.30 0.02Sl = ± mm was obtained, which is very close to the theoretical value of 2.27

mm predicted by Mie scattering [49, 50].

A small amount of absorbing dye (Rhodamine B) was then added to the latex solution so as to

bring its degree of absorption into line with that of the strongly light-absorbing packed beds used

here (see below). The absorption path length of this dyed latex solution was then determined by

fitting the equation [48]

( ) ( )*0( ) , , expa aI L g l l I L l= − (6)

to the transmitted light intensity data, I(L), measured for several different thickness, L = 15.5, 20.6

and 24.2 mm, of the dyed solution. A value of laS = 5.5 ± 0.2 mm was obtained.

The absorption path length of a well tapped packed bed of the particulate material was then

determined by fitting equation (6) to the transmitted intensity data for beds of thickness L = 15.5,

20.6 and 24.2 mm. A value of laP = 4.0 ± 0.2 was obtained. The transport mean free path of the

packed bed was then determined using [48]

( )( )

( )( )* * 2 1R Ra aR

R

I L I Ll l

l l I L I Lβ

= + −

(7)

where the latex suspension data was identified with laR, *Rl and ( )RI L , and the packed bed data

with la, l* and ( )I L . A value of * =1.89 0.2Pl ± mm was obtained.

Our previous work on air fluidized beds [46, 47] suggests that the absorption path length in

such beds is not particularly sensitive to spatial position or solids density even for large degrees of


bed expansion. On this basis and the fact that the degree of expansion experienced in the vibro-FB

considered here was very small, we assumed here that the value obtained for the packed bed was

also valid for all points and conditions in the vibro-FB; i.e. laV = 4.0 ± 0.2. Using this assumption,

the transport mean free path of the vibro-FB was determined using equation (7) for the points and

conditions of interest, where the packed bed data was identified with laR, *Rl and ( )RI L , and the

vibro-FB data with la, l* and ( )I L .

As Figure 2(a) shows, the ratio of the transmitted intensities of the vibro-FB and packed bed,

V PI I , were found to correlate very well with the amplitude, A, of the vibrations. Thus, following

equation (7), the transport mean free path of the vibro-FB, *Vl , was correlated with A as indicated in

Figure 2(b). This figures shows that whilst *Vl increases with the amplitude of the vibrations and, to

a lesser extent, the height above the base of the vibro-FB, its variation over the range of conditions

and positions investigated here is small at ~10-15%.

3. Results and discussion

3.1. Overview of the particle dynamics observed

Figure 3 shows an intensity autocorrelation function (IACF), g2(t), typical of mid-range

frequencies along with the electric field autocorrelation function (FACF), g1(t), and mean square

displacement (MSD), 2r∆ , derived from the analysis outlined in §2.2.3. The IACF decays from

g2 ≈ 1.5 over the timescale of 10−6 ∼ 10−5 s to an intermediate plateau of g2 ≈ 1.01 where it remains

before once again decaying over the timescale of 10−1 ∼ 101 s, this time towards unity. Tests [47]

were undertaken to check that the double decay and intervening plateau observed here were not an

experimental artefact [51] but, rather, a true reflection of the physics involved – we shall return to

the origin of this double decay shortly.

Figure 3(b) shows that the double decay seen in the IACF is clearly passed into the FACF, as

one would expect given the Siegert relationship, equation (2). As the height of the intermediate


plateau in the FACF is significantly enhanced relative to that of the IACF, it is now possible to see

that whilst the second decay may possibly go to g1 ≈ 0 (i.e. g2 ≈ 1), poor statistics at long times

mean it is not possible to rule out further plateaus and decays. A better picture of this long time

behaviour would be obtained using alternative light scattering approaches [42, 52].

The MSD shown in Figure 3(c) is characterised by three well defined regions. The first, which

terminates at t ≈ 50 µs for this case, is associated with the ballistic motion of the particles, with the

curvature beyond t ≈ 6 µs indicating a spread of fluctuating velocities about the mean, which is

estimated here as 1 22 3.17vδ = mm s−1 using the construction shown in the figure. The associated

time between collisions and the mean free path are 16ct ≈ µs and 39cλ ≈ nm respectively;

these values are very much in line with those obtained by others for dense granular systems [44-47],

and are indicative of the level of resolution that is possible with DWS. Beyond the ballistic region,

the MSD remains unchanged up to t ≈ 0.3 s, indicating that the particles are in effect moving about

a point in space during this period. The third region of the MSD, which is located in this case over

the timescale of t = 0.3 ∼ 6 s, appears to be a sub-diffusive where 2 ~r tα∆ with α < 1. Inadequate

statistics at long times means it is not possible to say if a diffusive region – i.e. where α = 1, which

is indicated in Figure 3(c) by the dashed line at long time – exists for this or any other condition

considered here.

On the basis of video images of a shallow air fluidized bed and MSDs derived for these

images, Abate and Durian [30] identified four characteristic regimes: ballistic motion at short times,

caged motion at intermediate times where the MSD is largely unchanging as the particles rattle

around in cages defined by neighbouring particles, sub-diffusive motion at longer times where

particles break free of their cages only to become trapped once again in new cages nearby, and

finally at even longer times, diffusive motion. Given the similarities between their MSDs and ours

here, it is tempting to draw parallels and ascribe the decays and intervening plateau seen in the


ACFs accordingly. However, care needs to be exercised, as DWS is not particularly suited to the

study of timescales that are comparable to the forcing dynamics [42], and equation (3) is based on

certain approximations that may break down to a greater or lesser extent at longer times [53]. As the

first decay in the ACFs occurs at a time approximately 103 times smaller than the forcing period, it

is safe to ascribe it to the ballistic motion of the particles, Figure 4(a). In order to attribute the

intermediate plateau and second decay to caged motion and cage breaking, we appeal to the results

of others [52, 54], who have obtained double decays similar to those observed here, albeit for

colloidal gels, using light scattering methods that are more appropriate for longer time scales. Both

these groups attribute the second decay by implication [52] or explicitly [54] to collective re-

arrangement of the colloidal particles following a period of virtually no change at all. On this basis,

we believe that despite the potential uncertainties associated with the long time character of the

MSDs obtained here, it is reasonable to ascribe the second decay in the ACFs of Figure 3 and the

preceding plateau to particles moving in a collective manner between cages in which they remain

trapped for some time, Figure 4(b) and (c).

Figure 5 shows an FACF typical of the higher frequencies considered here. The FACF

exhibits an initial decay at short times followed by a finite number of decaying oscillations (or

‘echoes’ [55]) whose period is identical to that of the forcing (see insert of this figure) and, finally, a

gradual decrease in the FACF towards zero. If the amplitude of the echoes were constant, the bed

would in effect be oscillating as a single body in tune with the base of the bed [55]. Decaying

oscillations, on the other hand, indicate that only a fraction of the particles are oscillating, with the

remainder undergoing aperiodic motion [55]. Kim et al. [56], who also observed this behaviour in a

vibro-FB, likened the behaviour to that of supercooled liquids near the glass transition point. The

dynamics of the particles that are undergoing non-periodic motion may, therefore, be the same as

that depicted in Figure 4 [57].

Kim et al. [56] observed oscillatory behaviour similar to that of Figure 5 at 50 Hz for

accelerations slightly above 1g. Our results suggest, however, that the upper limit where oscillations


are observed is frequency dependent [58] – both acceleration and frequency are, therefore,

important. Figure 6 shows that the height of the first echo, ( )1 1g tf = , which Kim et al. [56] used

as a proxy for the presence or otherwise of oscillations, is well correlated by the peak vibrational

velocity, whilst no correlation was evident between ( )1 1g tf = and the amplitude of forcing – this

suggests that energetic rather than geometric factors dictate the behaviour of that fraction of the

particles that are undergoing non-periodic motion. Figure 6 also shows that the fraction of particles

that are undergoing non-periodic motion, which is inversely related to the height of the echoes [55],

at a fixed peak vibrational velocity decreases with the distance above the base of the bed – the

degree of fluidization in effect increases with height in the bed.

The FACFs at the lowest frequencies considered here, examples of which are shown in

Figure 7, also include decaying echoes with a period equal to that of the forcing. These echoes,

which are superimposed on the intermediate plateau or the second decay depending on the

acceleration and frequency of the forcing, are much weaker than those observed at the upper end of

the frequency range considered here (compare with Figure 5). The strength of the oscillations also

appear to increase as the acceleration of the forcing increases and its frequency decreases (i.e. in

effect, they tend to increase with the amplitude and peak vibrational velocity), contrary to the high

frequency behaviour. Both this trend and the presence of echoes at 50 Hz for accelerations as high

as 4.17g are also in stark contrast to the results of Kim et al. [56], who observed no echoes at 50 Hz

for accelerations beyond 1.12g. It is not clear precisely why the behaviour seen here at low

frequencies differs from that of Kim et al. [56] in these two respects as well as the higher frequency

behaviour observed here, but the differences in apparatus of the two studies suggests two possible

causes: (1) the difference in the way the bed is vibrated – Kim and co-workers shake the bed as a

whole whilst we ‘pump’ a piston a the base of a column – and the presence of air in our system,

which was removed in the experiments of Kim et al. [56], could lead to greater oscillatory driving

of the particles at the lower frequencies where amplitudes and peak velocities are significant, and


(2) the much smaller amplitude of forcing relative to the particle size here, could lead to less

particles being able to take on aperiodic motion for the same energy input.

3.2. Granular temperature, mean time of free flight, and mean free path

Figure 8 shows that the granular temperature is correlated very well with the peak vibrational

velocity, vp, whilst there is no correlation with the vibrational acceleration. The data above vp = 18

mm s−1 appears to be well described by 1 22 ~ 0.11 pv vδ , a scaling that appears both qualitatively

and quantitatively to be very much in line with that obtained by Yang and co-workers [32, 33] using

NMR in a relatively confined dense vibro-FB bed. The data below vp = 18 mm s−1 is also described

reasonably well by a linear scaling, 1 22 ~ 0.018 pv vδ . This crossover point corresponds very well

to the point where echoes appear (see the data for y = 40 mm in Figure 6), suggesting that it is near

the jamming point [30, 55, 56].

The data here indicates that the granular temperature scales with the square of the peak

vibrational velocity

2 2~ pv vδ (8)

This scaling is in line with that obtained experimentally by others for a thin bed [26], a shallow bed

[29], a confined bed [32, 33], and a bed similar to ours [41]. This scaling is also in line with a

number of theoretical models, including those of Kumaran [16] and Warr et al. [24], who both

obtain a scaling exponent of 2, and Martin et al. [59] who obtained a scaling exponent of 1.85 ±0.11

in the absence of significant wall friction. These models are, however, restricted to relatively dilute

systems, a point we shall return to further below.

Figure 9 shows that both the mean time between collisions and the mean free path of the

particles decrease in a non-linear manner as the peak vibrational velocity increases. The insert in

Figure 9(a) indicates a scaling of


2

1~c

p

tv

(9)

which, when combined with the scaling for the mean velocity fluctuations, gives

1

~cpv

λ (10)

Figure 10, which shows the mean velocity fluctuations about the mean as a function of the

height above the bed base for three representative frequencies, suggests that the granular

temperature varies little with height for the conditions investigated here. This lack of variation is at

odds with the experimental data of You and Pak [41], who also studied a dense three-dimensional

vibro-FB, where an exponential increase in the granular temperature with height was observed. The

reason for this difference is not precisely clear, but the system of You and Pak [41] differed from

that used here in a number of potentially important respects including the amplitudes – which were

at least 5dp compared to approximately 0.5dp here – and the absence of gas in their bed, which You

and Pak noted increased the granular temperature. The data of You and Pak [41] also appears to be

at odds with that of other studies, albeit for geometries [28] or densities [32-36] that differed from

those considered here. In particular, whilst these studies revealed granular temperature can change

with height under some conditions, the changes were all far weaker than those observed by You and

Pak [41] and were essentially linked to a corresponding variation in the solids density in every case,

a variation which is barely seen in You and Pak [41] (~0.02% change across the bed height) or,

indeed, here.

4. Conclusions

Using diffusing wave spectroscopy, we have elucidated the dynamics of particles in a dense

three-dimensional fluidized bed in which the density variations are small, and have determined the

granular temperature of particles as a function of the vibrational conditions and height above the

base of the bed.


The granular temperature was found to scale very well with the square of the peak vibrational

velocity; no correlation was observed between granular temperature and other vibrational variables

such as the acceleration or amplitude. The granular temperature was found to vary little with height

in the bed, although it was not possible to probe near the top and bottom of the bed where variations

may be expected, due to limitations with the experimental apparatus and data analysis procedures

adopted. The mean time of flight between collisions was found to scale inversely with the square of

the peak forcing velocity, whilst the mean free path of the particles scaled inversely with this

velocity.

The longer time dynamics of the particles, which is depicted in Figure 4, indicates that in

systems such as that investigated here, macroscopic particle motion in the bed is governed by the

collective motion of particles between cavities in the bed, possibly by acquiring fluctuating

velocities that are substantially greater than the local mean of the velocity fluctuations. This

suggests that any model that seeks to describe dense vibrated granular systems in which the density

fluctuations are small must account for the distribution of fluctuating velocities, not just the mean,

and must include spatial correlations. If this is the case, the standard kinetic theory of granular

systems [14, 15] may not be appropriate for dense vibro-FBs such as that investigated here without

significant development.

ACKNOWLEDGEMENTS

We thank to the EPSRC (EP/C546849) for support of this research.

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58. Oscillations were observed at accelerations up to 4.17g, 3g, 2.16g and 1.555g for 500 Hz, 359.84 Hz, 258.97 Hz and 186.38 Hz respectively.

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Tables

f (Hz) Γ

50.00 69.48 96.54 134.14 186.38 258.97 359.84 500.00

1.555 0.1540 48.56

0.0798 34.95

0.0413 25.15

0.0214 18.10

0.0111 13.03

0.0057 9.37

0.0030 6.75

0.0015 4.86

2.160 0.2145 67.45

0.1112 48.54

0.0576 34.93

0.0298 25.14

0.0155 18.09

0.0080 13.02

0.0041 9.37

0.0021 6.74

3.000 0.2980 93.68

0.1544 67.42

0.0800 48.52

0.0414 34.92

0.0215 25.13

0.0111 18.09

0.0058 13.02

0.0030 9.37

4.170 0.4145 130.21

0.2147 93.71

0.1112 67.44

0.0576 48.54

0.0298 34.93

0.0155 25.14

0.0080 18.09

0.0041 13.02

Table 1. Sinusoidal forcing frequency-acceleration combinations considered in this study along

with the associated (top) amplitude, A, in mm, and (bottom) peak forcing velocity, vp, in mm s−1.

The acceleration and frequency were controlled to a resolution of 0.005g and 0.001 Hz respectively.


Figure captions

Figure 1. Schematic of experimental setup.

linear stage

laser

borosilicate glass

PMT

transmittedlight

optical fibre

electromagnetic shaker

correlator

laserbeam

linear stage

moving piston PMT

beam splitter

controller

accelerometersy

linear stage

laser

borosilicate glass

PMT

transmittedlight

optical fibre

electromagnetic shaker

correlator

laserbeam

linear stage

moving piston PMT

beam splitter

controller

accelerometersy


Figure 2. (a) Variation of the transmitted light intensity of a vibro-FB compared to that of a packed

bed, V PI I , with vibration amplitude, A, at x = 49 mm and y = 20 mm for a range of accelerations

(2.16g – green, 3.00g – blue, 5.79g – black) and frequencies (50 Hz – circle, 70 Hz – asterisk, 100

Hz – diamond, 134 Hz – triangle, 186 Hz – cross, 260 Hz – star, 360 Hz – square, 500 Hz –

inverted triangle). (b) Variation of transport mean free path of the vibro-FB, *Vl , with the vibration

amplitude, A, at x = 49 mm for y = 25 mm (circles), y = 40 mm (squares) and y = 55 mm

(diamonds). Lines are a guide for the eye only.

0 0.05 0.1 0.15 0.2 0.25 0.31.90

1.94

1.98

2.02

2.06

2.10

10-3

10-2

10-1

100

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

(mm)

(a)

(b)

A (mm)

I V/I P

0 0.05 0.1 0.15 0.2 0.25 0.31.90

1.94

1.98

2.02

2.06

2.10

10-3

10-2

10-1

100

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

(mm)

(a)

(b)

A (mm)

I V/I P


Figure 3. (a) The intensity autocorrelation function, 2g , for the point x = 49 mm and y = 40 mm at

Γ = 3g and f = 96.535 Hz (corresponding to Vp = 48.52 mm s−1 and A = 0.08 mm). (b) The

normalised electric field autocorrelation function, 1g , obtained from 2g using the Siegert

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

1

1.1

1.2

1.3

1.4

1.5

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-18

10-17

10-16

10-15

10-14

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0.0

0.2

0.4

0.6

0.8

1.0

(b)

g 1

(a)

g 2

(c)

t (s)

1 22 39 nmcλ ≈

16 µsct ≈

1 22 13.17 mmsvδ −=

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

1

1.1

1.2

1.3

1.4

1.5

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-18

10-17

10-16

10-15

10-14

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0.0

0.2

0.4

0.6

0.8

1.0

(b)

g 1

(a)

g 2

(c)

t (s)

1 22 39 nmcλ ≈

16 µsct ≈

1 22 13.17 mmsvδ −=


relationship, equation (2). (c) The mean square displacement obtained from 1g by inverting

equation (3); the mean free path of the particles, cλ , the average time between successive

collisions, ct , and the mean fluctuation velocity, 1 22vδ , are indicated.


Figure 4. A schematic showing the three regimes experienced by the particles in the vibro-FB

studied here: (a) ballistic dynamics at very short times in which the distance travelled between

collisions (i.e. free path) is λc; (b) ‘rattling’ around in cages defined by neighbouring particles at

intermediate times; and (c) particles moving between cages in a cooperative manner at long times.

λc

λc

(a) (b) (c)

λc

λc

λc

λc

(a) (b) (c)


Figure 5. The electric field autocorrelation function for the point x = 49 mm and y = 40 mm at

Γ = 2.16g and f = 500 Hz (corresponding to vp = 6.74 mm s−1 and A = 0.0021 mm). The insert

shows a fraction of the oscillatory data on a linear time axis to emphasis that the period of the

oscillations is the same as the forcing. Note that the time axes have been nondimensionalized by the

forcing frequency to emphasis that the period of the oscillations is the same as that of the forcing.

10-4

10-2

100

102

104

0

0.2

0.4

0.6

0.8

1.0

×

g 1

tf

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5×

00

0.2

0.4

0.6

0.8

1.0

10-4

10-2

100

102

104

0

0.2

0.4

0.6

0.8

1.0

×

g 1

tf

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5×

00

0.2

0.4

0.6

0.8

1.0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5×

00

0.2

0.4

0.6

0.8

1.0


Figure 6. Variation of the height of the first echo in the FACF, 1( 1)g tf = , with peak forcing

velocity, vp, at points x = 49 mm and y = 25 mm (blue), y = 40 mm (black) and y = 55 mm (red), in

the bed for accelerations 1.555g (circles), 2.16g (squares) and 3g (diamonds). Lines are shown as a

guide for the eye only.

4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

G1

(tf=

1)

vp (mm s−1)

4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

G1

(tf=

1)

vp (mm s−1)


Figure 7. The electric field autocorrelation function at the point x = 49 mm and y = 40 mm for: (a)

Γ = 2.16g and f = 50 Hz (corresponding to vp = 67.45 mm s−1 and A = 0.2145 mm), (b) Γ = 4.17g

and f = 50 Hz (corresponding to vp = 130.21 mm s−1 and A = 0.4145 mm), (c) Γ = 2.16g and

f = 69.48 Hz (corresponding to vp = 48.54 mm s−1 and A = 0.1112 mm), (d) Γ = 4.17g and f = 69.48

Hz (corresponding to vp = 93.71 mm s−1 and A = 0.2147 mm). Note that the time has been

nondimensionalized by the forcing frequency to emphasis that the period of the oscillations seen

here is the same as that of the forcing.

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

×

g 1g 1

tftf

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

(c)

(a)

(d)

(b)

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

×

g 1g 1

tftf

10-6

10-4

10-2

100

102

0

0.2

0.4

0.6

0.8

1

(c)

(a)

(d)

(b)


Figure 8. Variation of the mean velocity fluctuations about the mean, 1 22vδ , with peak forcing

velocity, vp, at the point x = 49 mm and y = 40 mm for accelerations Γ = 1.555g (circles),

Γ = 2.160g (squares), Γ = 3.000g (diamonds) and Γ = 4.170g (triangle). Lines are linear fits to the

data and errors are no larger than the symbols.

0 20 40 60 80 100 120 1400

2

4

6

8

10

12

14

18 mm s−1

1 22 ~ 0.11 pv vδ

1 22 ~ 0.018 pv vδ

vp (mm s−1)

0 20 40 60 80 100 120 1400

2

4

6

8

10

12

14

18 mm s−1

1 22 ~ 0.11 pv vδ

1 22 ~ 0.018 pv vδ

vp (mm s−1)


Figure 9. Variation of (a) the mean time between collisions, ct , and (b) particle mean free path,

cλ , with peak forcing velocity, vp, at the point x = 49 mm and y = 40 mm for accelerations of

Γ = 1.555g (circles), Γ = 2.160g (squares), Γ = 3.000g (diamonds) and Γ = 4.170g (triangle).

0 20 40 60 80 100 120 14035

40

45

50

55

60

65

70

75

vp (mm s−1)

0 20 40 60 80 100 120 1400

50

100

150

200

250

100

101

102

103

10-6

10-5

10-4

10-3

slope -2

0 20 40 60 80 100 120 14035

40

45

50

55

60

65

70

75

vp (mm s−1)

0 20 40 60 80 100 120 14035

40

45

50

55

60

65

70

75

vp (mm s−1)

0 20 40 60 80 100 120 1400

50

100

150

200

250

100

101

102

103

10-6

10-5

10-4

10-3

slope -2

0 20 40 60 80 100 120 1400

50

100

150

200

250

100

101

102

103

10-6

10-5

10-4

10-3

slope -2


Figure 10. Variation of the mean velocity fluctuations about the mean, 1 22vδ , with height above

the vibrating base at f = 50 Hz (right triangle), f = 70 Hz (left triangle), f = 500 Hz (triangle). The

errors are no larger than the symbols.

20 30 40 50 600

2

4

6

8

10

12

h (mm)

20 30 40 50 600

2

4

6

8

10

12

h (mm)

Documents

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