Investigation of droplet oscillation on a vibrating elastic plate

P. H. Tsai1, C. H. Wang

1, A. B. Wang

1, A. Korobkin

2, R. Purvis

2, & T. Khabakhpasheva

2

1Institute of Applied Mechanics, National Taiwan University, 10617, Taipei, Taiwan

2School of Mathematics, University of East Anglia, NR4 7TJ, Norwich, United Kingdom

Abstract

Behaviour of a sessile droplet on a vibrating elastic plate is investigated. We obtain exper-

imentally the characteristics of droplet oscillation modes by two high speed cameras recording

from side view and top view synchronously. Theoretical shapes and frequencies are compared

with those measured in the experiments, and different regimes of the droplet oscillations are ex-

plained. Moreover, the circular patterns of the elastic plate oscillations, so-called “Chladni pat-

terns”, were visualized. The radius variations of the nodal lines in the plate with the frequency

of the external forcing are well predicted by our theoretical model. We further focus on the i n-

fluence of the non-uniform amplitude of the plate oscillation on the droplet oscillatory behav-

iour. It is important to notice that a drop placed at a nodal line oscillates due to the pitch motion

of the plate at this line but not due to the plate vertical vibration which is negligible at the nodal

line. A droplet located either outside or inside the nodal lines behaves very differently, resulting

in potential depinning of the contact line due to the gradient of the plate deflection.

1. Introduction

Over the past decade, increasing attention has been drawn to the dynamics of sessile drops

due to a wide range of application including drop manipulation in microfluidics [1], inkjet print-

ing [2], and mixing of fluids [3]. The diversification of drop shapes on a plate driven at a specif-

ic frequency has not only triggered the scientific curiosity but also inspired the ideas of applica-

tion in practice. With the knowledge behind the phenomena, we can control the motion of drops,

even forcing it to move upward [4-6] and preventing it from moving (as discussed in this study).

Research has shown that the shape of drop oscillation can be determined by the external forces

which would be supplied by different techniques. Trinh and Wang [7] have facilitated acoustic

levitation of a droplet to study the resonant frequencies, the damping of oscillations, and drop

fission. Tan et al. [8] observed particular jetting phenomena of a drop caused by large accelera-

tions associated with surface acoustic waves (SAW). Sharp [9] investigated the resonant proper-

ties of sessile drops perturbed by a puff of air and found the contact angle dependence. Recently,

Chang et al. [10-11] investigated the resonant behaviours of sessile water drops driven by verti-

cal vibration of a substrate. A rich collection of resonant modes were observed experimentally

and compared with theoretical prediction of the linear, inviscid, irrotational theory. However, to

the best of our knowledge, there is no literature focused on the influence of the non-uniform vi-

bration amplitude of a plate (i.e., the effect of the position of the droplet on the plate). Here, we

attempt to combine the drop dynamics and non-uniform vibration of a plate to investigate the

drop-plate interaction that display a significant difference in the behaviours of sessile drops de-

posited on different positions of the plate.

P. H. Tsai, C. H. Wang, A. B. Wang, A. Korobkin, R. Purvis & T. Khabakhpasheva, 2015

2. Method and materials

A schematic diagram of the experimental apparatus is shown in Fig. 1. Distilled water was

used as the drop liquid. The density and viscosity were kept constant as 998 kg/m3 and 1cP, re-

spectively. The volumes of drops were set to 20 μl, 50 μl , 70 μl, and 100 μl in the experiments.

The circular plate was made of Poly(methyl 2-methylpropenoate) (PMMA) which is transparent

thermoplastic, mounted with an electrodynamic shaker (PD4801, Brüel & Kjæ r). The contact

angle of the drop and PMMA-plate was measured to be 65 ± 5 degrees. The vertical vibration of

the plate was operated by a frequency-controlled sinusoidal wave with controllable voltages

supplied from the generator WAVETEK (model 395) and the power amplifier Crown D-45.

Beneath the PMMA-plate surface, meshes with 500 μm weave were made by CO2 laser engrav-

er (Mercury II, GCC LaserPRO). While the white light emitted by the LED lighter passed

through the drop deposited on the plate with meshes, the surface waves on the drop surface can

be clearly visualized due to the distortion of the mesh patterns. The dynamic characteristics of

sessile drops were then captured by two synchronous high speed cameras (NAC Memrecam

GX-1 and X-Stream XS-5 for the top- and side-view recordings with frame rates setting of 5000

and 1000 frames per second, respectively).

Droplet

PMMA circular plate

Mini-shaker

High-speed camera

Mirror

Amplifier

Function generatorz

r

( )sin 2z A r ft

LED lighter

Fig. 1 Experimental apparatus.

3. Results and discussions

3.1 Drop dynamics on a vibrating plate

The motion of a freely oscillating drop in air or in the medium, the viscosity of which is negli-

gible, was theoretically described by Lord Rayleigh [12]. The resonant frequency of the drop can be

expressed as

2 3

0

( 1)( 2), 0,1, 2, ......

4n

n n nf n

R

(1)

where R0 is the undisturbed radius of the drop and n denotes n-th drop oscillation mode, σ and ρ are

surface tension and density of drop liquid, respectively. It should be noted that only axisymmetric

shapes (zonal modes) are considered in the theoretical analysis by Rayleigh. The shape of a drop

that involves the non-axisymmetric oscillations can be written as [13]

* ( , ) 1 (cos )cos( ),

0 , 0 2 ,

l

kR P l

(2)

where R* is the scaled distance from the center of mass to the surface of a drop, ε is the deformation

amplitude, θ and ψ are the polar and azimuthal angles in spherical coordinates, respectively. The

function Pkl is the associated Legendre polynomial where the integer indices k and l are referred to

as the degree and order, respectively. When l is zero, this function represents the axisymmetric

mode corresponding to the Rayleigh’s model and hence, k is equal to n in equation (1). Equation (2)

provides non-zonal solutions to the problem of sessile drop. In this way, equation (2) with the fre-

quency given by equation (1) can be used to determine the dynamics of drop for each mode includ-

ing zonal and non-zonal modes.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140 160 180 200

Am

pli

tud

e (

mm

)

Frequency (Hz)

Plate thickness = 1 mm

Triangle

Jetting

Concentric Circle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160 180 200

Am

pli

tud

e (

mm

)

Frequency(Hz)

Square

Plate thickness = 5 mm

a

b

Fig. 2 Experimental observation of the different shapes of a water drop deposited on the acrylic plate of (a) 5 mm, (b) 1 mm in thickness against driving frequencies.

In the performed experiments, as can be seen in fig. 2, a spherical drop was initially deposited

on the PMMA-plate without position control. The drops turned into a characteristic oscillation

mode when a specific frequency and amplitude were supplied to the system. In order to gain further

insight into the influential factors associated with the drop oscillation, we firstly put emphasis on

the investigation of drop shapes at the prescribed position of the droplet on the plate. Fig. 3a shows

the drop morphologies for different driving forces. For instance, elliptic shape of the drop can be

found at the driving frequency of 32 Hz, sequential images of which can be observed in fig. 3c. Tri-

angular shape of the drop can then occur at the range of driving frequencies from 38 ~ 41 Hz as

shown in figs. 3f and 3g. When the driving frequency is increased up to 50 Hz, the jetting mode

with the small satellite droplets emitted from the primary drop at its centre can be also noticed.

Generally, one characteristic frequency corresponds to one shape mode associated with the drop

oscillation in air; however, the results shown in fig. 3a suggest that the drop motion requires suffi-

cient acceleration to arrive at the characteristic shape mode; otherwise, the drop shape remains a

spherical cap within the whole period. In other words, one can observe the drop oscillation with

spherical shape when the applied driving force is not large enough as shown in figs 3d and 3e.

However, one can observe the drop oscillation with multi-lobed shape once the applied driving

force increases across the threshold. It is caused by the fact that the sessile drops are constrained by

the plate, the shape and mode of drop oscillation are affected by the contact angle and mobility of

the contact line, which is so-called contact angle hysteresis. Moreover, the observed frequencies (f)

of the oscillation modes have a significant deviation from the predicted frequencies of Rayleigh (fn).

The values of f/fn can be seen in figs. 3c-3h. It points out that the Rayleigh spectrum is insufficient

to describe this kind of oscillatory behaviour of sessile drops.

Fig. 3b shows that the larger the liquid volume, the lower the characteristic frequency of a cer-

tain oscillation mode. This trend is consistent with Rayleigh theory. An elliptic shape of the drop is

obtained with a liquid volume of 20 μl at the driving frequency of 49 Hz, while at 32 Hz for drop

liquid volume of 50 μl. Likewise, as far as the triangular shape of drop are concerned, the onset fre-

quencies are observed at 55 Hz, 41 Hz, and 35 Hz for drop liquid volume of 20 μl, 50 μl, and 70 μl,

respectively. Furthermore, it was noticed in fig. 3b that there exists the square shape of drop oscilla-

tion for the liquid volume of 70 μl but this mode is missing for smaller volumes. The absence of the

multi-lobed modes with degree less than 4 (k = 4) for smaller drops in the experiment implies that

the interplay between the inertia and the capillary forces might be of strong importance in the mo-

bility of contact lines. The inertia of larger drops dominates, and has ability to overcome the energy

barrier of multi-lobed modes.

c d

e

f

g

h

50

100

150

200

250

300

25 30 35 40 45 50 55

Vo

lta

ge

(m

V)

Frequency (Hz)

50μl @ r = 30 mm

Spherical Elliptic Triangular

Jetting Irregular

0

10

20

30

40

50

60

70

80

90

100

25 30 35 40 45 50 55 60 65 70 75 80 85

Liq

uid

vo

lum

e (μ

l)

Frequency (Hz)

@ r = 30 mm, V = 140 mV

Elliptic Triangular Jetting

Spherical Square

a b

32Hz

f/f2=0.94

t/T 0 0.10 0.20 0.30 0.40 0.50 0.60 0.80 0.90 1.00

38Hz

f/f2=1.12

t/T 0 0.10 0.20 0.30 0.51 0.61 0.71 0.81 0.91 1.00

41Hz

f/f2=1.21

t/T 0 0.21 0.32 0.42 0.53 0.58 0.63 0.68 0.79 1.00

41Hz

f/f3=0.62

t/T 0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

41Hz

f/f3=0.62

t/T 0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

49Hz

f/f4=0.57

t/T 0 0.10 0.20 0.30 0.50 0.60 0.70 0.80 0.90 1.00

c

d

e

f

g

h

Fig. 3 (a) Drop morphology under different driving frequencies. Liquid volume is 50 μl. (b) The frequency dependency of the liquid volume. All drop positions are at the distance 30 mm from the centre of the PMMA-plate (r = 30 mm). The supplied voltage to the system remains constant (V = 140 mV). (c-h) A sequence of images corresponding to various shape modes in one oscillation period (T). The size of mesh is 500 μm.

3.2 Theoretical description of a vibrating plate

The classical differential equation of motion for the transverse displacement w of the plate is

given by 4 0,

ttD w w (3)

where D is the flexural rigidity defined by 3

2,

12(1 )

EhD

(4)

where E is Young’s modulus, h is the plate thickness, ν is the Poisson’s ratio, ρ is the mass density

per unit area of the plate, t is time and ∇4=∇

2∇2, where ∇2

is the Laplacian operator.

In the problems with forced vibrations of the plate, the deflection of the plate is expressed as

, cos(2 ),w r t W r ft (5)

where f is the frequency in Hz and W(r) is an amplitude of vibration, which depends only on the ra-

dial coordinate r.

We consider axisymmetric vibrations of the circular plate. In our experiments,

20.055m, 0.005 , 0.052 / , 0.3, 4.62R h m kg R E GPa (6)

It should be noted that the plate is clamped around the centre in the area with Rb = 3.5 mm (i.e., the

radius of the cap of the screw in the experiment). The Laplacian operator expressed in polar coordi-

nate is 2 2

2

2 2 2

1 1,

r r r r

(7)

and in axisymmetric case 2

2

2

1.

r r r

(8)

General solution of the equation (3) in the polar coordinates for circular plate vibration is

24

2 3 4( ) ( ) ( ) ( ) ( ), 2 / ,

1 0 0 0 0W r C J kr C Y kr C I kr C K kr k f D (9)

where Jn(r) and Yn(r) are the Bessel functions of the first and second kinds, respectively, and In(r)

and Kn(r) are the modified Bessel functions of the first and second kinds, respectively. Coefficients

Cj, j = 1, 2 ,3, 4 are determined from the boundary conditions at the outer edge of the plate r = R,

and at the inner edge r = Rb.

We have free-free edges conditions at r = R:

20, 0 ( )rr r

W W W r Rr r

(10)

At the inner radius r = Rb, the following boundary conditions are imposed:

( ) 1, ( ) 0,b r b

W R W R (11)

The boundary conditions (11) imply that the plate is clamped to the vibrator and that the amplitude

of the vibration is set to be one. The resulting problem (3), (10), (11) is linear.

The conditions (10) and (11) provide the following linear system for coefficients Cj in (9):

2 3 1 4 1( ) ( ) ( ) 2 1 ( ) ( ) 2 1 ( ) 0,

1 0 0 0 0C kaJ ka C kaY ka C kaI ka I ka C kaK ka K ka

(12)

1 2 1 3 1 4 1

0 2 0 3 0 4 0

1 2 1 3 1 4 1

( ) ( ) ( ) ( ) 0,

( ) ( ) ( ) ( ) 1,

( ) ( ) ( ) ( ) 0,

1

1

1

C J ka C Y ka C I ka C K ka

C J kb C Y kb C I kb C K kb

C J kb C Y kb C I kb C K kb

where a = R is outer radius and b = Rb is the inner radius of the plate.

Results of the calculations of the non-dimensional shape function W(r) = A(r)/A0 where A(r)

denotes the amplitude at certain radial position, and A0 denotes the amplitude of the vibrator, with

Rb = 3.5 mm are shown in figs. 4a and 4b. For small frequencies as illustrated in fig. 4a, amplitudes

of the vibration of the plate grow with the radial position. For higher frequencies as illustrated in fig.

4b, the amplitudes of vibration decrease initially to zero and then their absolute values grow again.

The radial position where the amplitude is zero is referred to as the nodal radius (Fig. 4c). The inset

of fig. 4c shows that tiny particles placed on the vibrating plate in the experiment tend to minimize

their energy by approaching the nodal radius and consequently the circular pattern, which is known

as the Chladni figure [14-15], is observed in the circular PMMA-plate. Due to the fact that the vi-

bration amplitude of the plate is different at different distances from the plate centre, it is reasonable

that the position of drop on the plate would have an influence on the drop dynamics. Fig 4c shows

that the theoretical predictions of the nodal radii agree very well with the nodal radii measured in

the experiments. This result is important because the vibration amplitude of the drop placed at a

given position can be obtained from the theoretical calculations.

a b

0

5

10

15

20

25

30

35

40

600 800 1000 1200 1400 1600

No

da

l ra

diu

s (

mm

)

Frequency (Hz)

0

5

10

15

20

25

30

35

40

0 500 1000 1500 2000

No

dal ra

diu

s (

mm

)

Frequency (Hz)

Theoretical predictions

300mV

400mV

500mV

Theory

Experiment

Nodal radius

c

Fig. 4 (a-b) Variations of the dimensionless amplitude along the radial position for different driving frequencies. (c) Comparison between the nodal radius from the experimental results and theoretical predictions. The thickness of the plate here is 5 mm. The inset in (c) is for f = 850 Hz.

3.3 Position-effect on drop shape of oscillation modes

Fig. 5 shows the oscillatory characteristics of the drop deposited on several different positions

of the plate. Figs. 5a-5d illustrate the behaviour of the drops deposited inside, on, outside, and far

outside the nodal line. From the results of experimental observation, it is apparent that the drop lo-

cated on the nodal lines has an insignificant response compared to the other cases. More interesting-

ly, the other three cases exhibit various phenomena of drop oscillation. To specifically classify the

drop oscillation mode, indices (k, l) defined by spherical harmonics are useful to represent the char-

acteristics of drop oscillation. The physical meanings of indices (k, l) are referred to as polar and

azimuthal wave number, respectively. The drop oscillation modes belong to zonal mode for k = l,

whereas non-zonal mode correspond to k ≠ l. It is worth noting that the observed frequency is the

same as the driving frequency for zonal modes. Alternatively, the observed frequency is half of the

driving frequency for non-zonal modes. According to the definition of the spherical harmonics, it

was found that the cases from (a) to (c) correspond to the pairs (k, l) = (3, 3), (1, 0), (6, 0). The case

(d) corresponds to the superposition of the two modes (6, 0) and (2, 3). The eigen frequencies of the

modes (1, 0) and (6, 0) are equal to 160 Hz which is the driving frequency. The observed frequen-

cies of (3, 3) and (6, 0) + (2, 3) are only 80 Hz that is half driving frequency. Hence, not only has it

been proved that the drop dynamics are indeed affected by the position of drop on the plate, but also

it has been demonstrated that even under the same parametric conditions (i.e., driving frequency,

drop size, and liquid properties etc.), the oscillation mode of the drop strongly depends on the posi-

tion of the droplet on the plate.

Nodal line

z

r

a b c d

a

b

c

d

r/R= 0.27

t/T 0 0.10 0.19 0.29 0.39 0.58 0.68 0.77 0.87 1.00

r/R= 0.45

(Nodal radius)

t/T 0 0.10 0.19 0.39 0.48 0.58 0.68 0.77 0.87 1.00

r/R= 0.64

t/T 0 0.10 0.19 0.29 0.39 0.48 0.58 0.77 0.87 1.00

r/R= 0.82

t/T 0 0.10 0.19 0.29 0.48 0.57 0.67 0.76 0.86 1.00

Fig. 5 Comparison of oscillatory behaviour of the drop deposited on different positions of the plate. The driving fre-quency is 160 Hz and the drop liquid volume is 100 μl. The case (a) r = 15 mm, (b) r = 25 mm, (c) r = 35 mm, and (d) r = 45 mm are located inside, on, outside, far outside the nodal line, respectively. The scale can be obtained from the size of mesh pattern that is 500 μm.

4 Concluding remarks

In this study, we investigated experimentally the characteristics of drop oscillations and Chladni figures for a PMMA-plate. Based on the developed linear theory, the prediction of the nodal radius of the plate is closely consistent with the experimental results. The dynamics of drops are shown to have a significant variation as the positions of the sessile drops are varied. Under the same driving frequency, sessile drops on the nodal line of the plate may keep its ini-tial shape with slight pitch motion due to the zero amplitude of nodal line. The oscillation modes associated with other positions of the drop can differ significantly. We obtained different oscillation modes at the same time with drops placed at certain distances from the plate centre. Our experimental observations give rise to an intriguing possible application in bio-reactors. Thinking of droplets formed after two miscible liquids have been merged together, we can con-sider these droplets as small reactors. By making these droplets vibrate with different frequen-cy levels, we expect that the mixing efficiency inside the droplets could be also different. Com-pared to the conventional mixers, the prescribed performance of mixing efficiency can be achieved with an advantage of providing a qualitative, reagent-saving technique based on the droplet-based microfluidics.

Acknowledgements

This study was carried out within the project “Splashing on flexible substrates - a basic study” supported by The Royal Society of London, UK and the National Science Council of Taiwan (NSC-103-2911-I-002-533). Also, the authors thank the National Science Council of Taiwan for financial support under Grant No. 102-2221-E-002-081-MY3.

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