ELASTIC PROPERTIES OF LIQUID MARBLES
E. Bormashenko1,2, R. Pogreb1, R. Balter2,3, H. Aharoni2,
Y. Bormashenko1,2, R. Grynyov1, L. Mashkevych2, D. Aurbach3,
O. Gendelman4
1
Ariel University, Physics Department, P.O.B. 3, 40700, Ariel, Israel
2
Ariel University, Chemical Engineering and Biotechnology
Department, P.O.B. 3, 40700, Ariel, Israel
3
Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan,
Israel
4
Faculty of Mechanical Engineering, Technion – Israel Institute of
Technology, Haifa 32000, Israel.
edward@ariel.ac.il
Abstract
Liquid marbles are non-stick droplets coated with colloidal
particles. Liquid marbles do not coalesce when pressed one to another or
when colliding. The paper is devoted to the study of the quasi-elastic
properties of liquid marbles under collisions. It was established that the
contact time under collision is independent of the velocity of the cue
marble. The liquid marble was modeled by a droplet coated by an elastic
shell, representing the colloidal layer covering the marble. The elasticity
of the shell is due to the capillary interaction between colloidal particles.
The physical model of collisions is proposed. Pathways of viscous
dissipation are discussed. Scaling laws describing the collision are
derived. The proposed scaling laws governing the marbles’ collisions
were verified experimentally. The contact time of the collision scales as
the square root of the marbles’ volume as it occurs under bouncing of
droplets. Pathways of viscous dissipation are discussed.
Introduction
Liquid marbles, shown in Figure 1a and b, are non-stick droplets
encapsulated with micro- or nano-scaled solid particles [1-4]. Since
liquid marbles were introduced in the pioneering works of Quèrè et al.,
they have been exposed to intensive theoretical and experimental
research [5-11]. An interest in liquid marbles arises from both their very
unusual physical properties and their promising applications. Liquid
marbles present an alternate approach to superhydrophobicity, i.e.
creating a non-stick situation for a liquid/solid pair. Usually
45
superhydrophobicity is achieved by a surface modification of a solid
substrate. In the case of liquid marbles, the approach is opposite: the
surface of a liquid is coated by particles, which may be more or less
hydrophobic [12]. Marbles coated by graphite and carbon black, which
are not strongly hydrophobic, were also reported [13,14].
A variety of media, including organic and ionic liquids and liquid
metals, can be converted into liquid marbles [15-17]. Liquid marbles
were successfully exploited for: microfluidics [17-21], water pollution
detection [22], gas sensing [23], electrowetting [24], blood typing [25]
and optical probing [26]. Respirable liquid marbles for the cultivation of
microorganisms and Daniel cells based on liquid marbles were reported
recently by Shen et al [27,28]. Stimulus (pH, UV and IR) responsive
liquid marbles were reported by Dupin, Fujii et al [29-31]. The stability
of marbles is crucial for their microfluidics and sensing applications.
Marbles possessing increased mechanical and time stability were
prepared by Matsukuma et al [32]. Liquid marbles coated with
monodisperse micron-size particles of poly(methylsilsesquioxane),
forming a hexagonally close-packed (HCP) structure were reported
recently [33]. Plasmonic liquid marbles were reported recently [34]. It is
noteworthy that liquid marbles retain non-stick properties on a broad
diversity of solid and liquid supports [35-36]. Actually, liquid marbles
are separated from the support by air cushions in a way similar to
Leidenfrost droplets [37]. The state-of-the-art in the study of the
properties and applications of liquid marbles is covered in recent
reviews [38-40]. Actually, liquid marbles demonstrate not only viscous
but also elastic properties [41-43]. Our paper is devoted to the study of
non-coalescent collisions of liquid marbles, which are governed partially
by quasi-elastic phenomena.
Experimental
Polytetrafluoroethylene (PTFE) powder (100-200 nm) was
supplied by Aldrich. Lycopodium (the average diameter of particles was
about 30 µm) was supplied by Fluka. SEM image of the lycopodium
particle is supplied in Fig. 1b. The average diameter of particles,
specified above, was established with SEM imaging, carried out with
high resolution SEM (JSM-6510 LV). Distilled water (with the electric
conductance of 0.6 mS) was used for manufacturing the marbles. Liquid
marbles were prepared according to the protocol described in detail in
46
Ref. 19-22. Rapid camera (Viework model VC-2MC-C340EO),
equipped with DVR Express Core software (IOI Industries), was used
for the visualization of collisions. All experiments were repeated 10
times and averaged.
A
B
C
Fig. 1. A. 15 µl lycopodium-coated water marble. B. 20 µl Teflon-coated water
marble (colored for the purposes of visualization with Potassium
permanganate). C. SEM image of the lycopodium particle. Scale bar is
5 µm.
Results and discussion
Non-coalescent collisions of liquid marbles
Coalescence of liquid marbles was studied by Bhosale et al [44].
We focus on the opposite experimental situation, namely non-coalescent
47
quasi-elastic collisions of liquid marbles. Liquid surfaces coated with
colloidal particles behave as two dimensional elastic solids (and not
liquids) when compressed [41-43]. The stretching modulus and bending
stiffness of such surfaces were reported recently [42, 43].
Hence, when we place liquid marbles close to one another, and
even press them slightly, they do not coalesce, as shown in Fig. 2.
Fig. 2. 10 µl Teflon (white) and lycopodium (yellow)-coated marbles do not
coalesce even when pressed one to another.
We studied the collisions of liquid marbles with the simple device
presented schematically in Fig. 3 and introduced in Ref. 5. One of the
marbles (the object marble) deposited on the horizontal surface was in
rest, and the second (the cue marble) rolled down from the inclined
plane, as shown in Fig. 3. The slope of the inclined plane β and the
initial height of the rolling marbles allowed control of the velocity of the
center mass of the cue marble at the instant of collision in the range of
0.12–0.3 m/s. The time of the collision τ (i.e. the interval between the
first contact of marbles and their disconnection) was established
experimentally with the rapid camera. A sequence of images displaying
the typical collision is displayed in Fig. 4.
First of all, it was observed that within a broad range of velocities
of the cue marble, the marbles did not coalesce. The results obtained for
lycopodium and Teflon coated marbles were very close: the average
collision times were established as:  lyc  0.03 s;  TEFLON  0.04 s . Rather
surprisingly, the time of collision τ was independent on the velocity of
the cue marble, as demonstrated in Fig. 5. This definitely contradicts the
dependence inherent in the collision of elastic balls (the Hertz problem),
where the weak dependence   v 1 / 5 has been predicted and observed
[45-49].
48
cue marble
object marble
β
U
S
Fig. 3. Sketch of the experimental device used for the study of collisions of
liquid marbles. Marbles rolled down from a height of 10-17 mm, β =
= 15-300.
Fig. 4. Sequence of images illustrating the collision between lycopodium-coated
liquid marbles. Scale bar is 1 mm.
49
0,04
0,035
τ (s)
0,03
0,025
0,02
0,015
0,01
0,1 0,12 0,14 0,16 0,18 0,2 0,22 0,24 0,26 0,28 0,3
U (m/s)
Fig. 5. The contact time τ vs. velocity of the center of mass U of the cue
lycopodium-coated marble.
In order to describe the physical processes occurring under
collisions, we approximate liquid marbles by water droplets coated by a
thin elastic shell, shown in Fig. 6. Let us adopt that both cue and object
marbles have the same mass M. Then, the cue marble has the velocity of
the center of mass U before the impact. The total kinetic energy of a
rolling marble is thus expressed as: Ttot  (7 / 10) MU 2 . Part of this kinetic
energy Tc  MU 2 / 4 is related to the motion of the center of masses of
the system and is conserved in the course of collision due to momentum
conservation. We neglect the effects related to friction between the
marbles and the plane in the course of collision, due to the extremely
short time scale of the process – see below. The other part of the kinetic
energy is related to the relative motion of the marbles and is converted
into elastic energy and dissipated by viscous forces in the course of
collision. This kinetic energy of the relative motion is expressed as:
Trel  Ttot  Tc 
50
9
MU 2
20
(1)
model of a marble
marble
elastic shell
colloidal particles
R
R
h
Fig. 6. Mechanical model of a liquid marble: the colloidal layer is represented
by an elastic shell.
In order to estimate the elastic energy of collision, we recall that
the quasi-elastic shell represents colloidal particles coating a marble.
The effective Young modulus G of a liquid possessing the surface
tension γ and coated with monodisperse particles with the diameter of d
was estimated in Ref. 43 as
G
1  
1 d
where  is the Poisson ratio of solid particles, φ is the solid
fraction of the interface. It was concluded in Ref. 43, that the elastic
properties of such an interface are relatively insensitive to the details of
capillary interaction between particles. However, this is true for a model
situation in which the marble is coated with monodisperse particles,
forming a close-packed structure, as it occurs in Ref. 33. In our
experiments the solid coating is random, and complicated aggregates of
solid particles encapsulating a marble are separated by water clearances,
as it is seen from ESEM images displayed in Refs. 7, 35. Thus, the
nature of interaction between solid particles is rather intricate, and we
keep the phenomenological approach in which a solid coating is
represented by an elastic shell possessing the elastic modulus G. The
energy of the elastic shell is given by:
 
Eel  Gh   S
R
2
51
(2)
where R is the radius of the shell, which may be considered equal
to the radius of the marble, h is the thickness of the shell, which is on the
order of magnitude of a lycopodium particle, G is the elastic modulus of
the shell, and  is the radial displacement of the points of the deformed
shell [45]. S is the characteristic area of the contact. It follows from Fig.
4 that the contact area is rather large; then, we could estimate it as
S  2 R 2 . Finally, for elastic energy one obtains:
2
 Ghsrel
(3)
Eel  2 Gh 2 
2
here srel  2 is the relative displacement between the marbles in the
course of interaction (again, we suggest that the marbles are completely
symmetric and are deformed in the same way). Looking at Estimation
(1) for the kinetic energy of the relative motion and expression (3) for
the elastic energy, we recognize that these expressions are formally
equivalent to ones obtained for linear oscillator with effective mass m =
= (9/10)M and effective stiffness k = πGh. The frequency of such
oscillator is written as:

k
10 Gh


m
9M
Gh
 R3
(4)
One can assume that the collision time corresponds to a halfperiod of oscillation. Then, for the time of collision, one obtains:



 R3
(5)

Gh
5
m ; h  10 m ; G  103 Pa (see Ref. 42), and
We assume R  10 3
obtain τ ≈ 3 · 10–2 s, in a good agreement with the experimental findings
displayed in Fig. 5. Moreover, the model of linear oscillator nicely
explains the independence of the collision time on the velocity (Fig. 5).
Expression (5) predicts higher times of collisions for the Teflon-coated
marbles, which was actually observed experimentally. However, it
should be emphasized that h is not a well-known parameter, because
actually marbles are coated with agglomerates of colloidal particles and
not by isolated ones [7, 14, 35]. Hence it may be estimated only
qualitatively.
Besides, from (4) and (5) it follows that τ~ M ~ V , in complete
agreement with the results presented for lycopodium-coated marbles in
52
Fig. 7. The slope of the ln-ln plot varied from 0.47 to 0.51 for various
initial velocities of the cue marble. The results obtained for Tefloncoated marbles were the same. It is noteworthy that Richard et al.
obtained in Ref. 50 a similar scaling law for the characteristic time of
water droplets bouncing solid substrates, i. e:   R 3 /  (compare with
Exp. (4)). This is not surprising, because in the case of bouncing droplets
the role of the effective spring stiffness k is played by the surface tension
of a bouncing droplet γ [49]. Richard et al. in Ref. 50 also reported
independence of the contact time from the velocity of bouncing drops in
complete accordance with our results. Thus, the observed scaling laws
could be related both to the pseudo-elastic and surface tension-induced
effects, and the distinguishing between these is not simple, as it will be
shown below.
lnτ
y = 0.4707x + 5.0497
R2 = 0.9881
-3.45
-3.4
-3.35
-3.3
-3.25
-3.2
-3.15
-3.1
-3.05
-17.2
-17.4
-17.6
-17.8
-18
lnV
Fig. 7. The ln-ln plot of the collision (contact) time τ vs. the volume of a
lycopodium-coated marble V.
53
-18.2
From Fig. 4, it is obvious that a collision between two marbles is
by no means purely elastic. Still, the employed model of simple linear
oscillator does not take the damping into account in any way. It is then
instructive to discuss the pathways of the viscous dissipation under
collisions. The viscous dissipation may be estimated according to:
dEvisc
2
   u  dV ,
dt
V
(6)
where η is the viscosity of the liquid (water in our case), V is the volume
of the marble u is the gradient of velocity of liquid filling the marble
[48]. For the sake of a very crude estimation we assume: u  U / 2R ,
where U is the velocity of the center of mass of the cue marble (we
adopt that viscous dissipation takes place over the entire volume of the
marbles, as it is seen from Fig. 4); thus we obtain:
dE visc
U  4 3 
2
 
 R  U R .
dt
2
R
3
3


2
(7)
And finally we derive for the estimation of viscous dissipation:

(8)
U 2 R .
3
It is easily seen from Exp. (8) that the viscous dissipation is much
lower than the total kinetic energy of the cue marble, indeed:
E visc 
E visc


 10 2 .
2
Ttot
R
(9)
Here we assume   103 ;   102 s;   103 kg/m 3 . Moreover, the
energy of viscous dissipation is even less than the energy of “elastic”
deformation of the shell, formed by colloidal particles (see Expressions
(3) and (8); srel  2  2u is adopted for the sake of the rough
estimation). This is expressed by:
E visc R

 10 2 .
Eel
Gh
(10)
Estimations (9)-(10) lead to the conclusion that viscous
dissipation inside the marble is not the key factor in the energy
dissipation under the collisions of liquid marbles, and this is due to the
low viscosity of water. A similar situation occurs under rolling of liquid
54
marbles. As it was shown in Ref. 5, the friction of rolling water marbles
is mainly due to the disconnection of the contact line from the substrate.
It is plausible to suggest that in the case of collisions the main pathways
of the energy dissipation are the “non-elastic” part of the deformation of
the shell formed by the colloidal particles, and the disconnection of the
triple line from the substrate, accompanied by the viscous dissipation in
the colloidal layer coating a marble.
The energy of the “elastic” shell coating a marble and its
influence on scaling laws governing its shape
The shape of a liquid marble may be described by two main
geometric parameters, its radius R and the radius of the contact area a.
Quere et al. theoretically and experimentally studied the scaling law
relating a to R, and established that it is different for various dimensions
of marbles. When the radius of a marble is close to the capillary length
 (γ is the surface tension [51, 52]), the approach
g
developed by Quere et al. predicted the following scaling law:
given by lca 
(11)
a  lca R 2 .
The scaling law supplied by Exp. (11) has been successfully
verified in a number of studies [1, 14]. The truth of the scaling law given
by Exp. (11) is rather surprising, due the fact that Quere et al. neglected
the elastic component of the energy of a liquid marble in their treatment.
The apparent discrepancy is resolved if we consider that scaling laws
governing the energy of a deformed elastic shells and the surface energy
of droplets are the same (both are quadratic, see Exp. (3)). When the
center of mass of a marble is lowered by a quantity δ, the difference in
energy ΔE from a sphere tangent to the plane can be written
dimensionally as [1]:
(12)
E   2  gR 3 .
Minimization of Exp. (12) considering geometrical constraints
resulted in the scaling law given by Exp. (11). Considering the elastic
component of the energy yields (see Exps. (2)-(3)):
(13)
E  Gh 2  gR 3 .
55
Obviously, minimization of Exps. (12)-(13) will give rise to the
same scaling law, represented by Exp. (11). That is why the scaling law
expressed by Exp. (11) has been successfully verified experimentally [1,
14].
The dimensionless constant describing the interrelation of
“elastic” and surface energies ψ may be introduced:

(14)
  eff
Gh
Substituting into Exp. (14)  eff  50 mJ/m 2 , which is the effective
surface tension of a marble8 and aforementioned values of G and h,
results in   1 . This means that the “elastic” energy is not negligible.
Thus, the description of phenomena related to liquid marbles may
be carried out both in terms of the effective surface tension induced
effects and quasi-elastic phenomena. Accurate distinguishing between
these effects is problematic due to the fact that the parameter ψ is close
to unity. The effective surface tension of marbles  eff is not a welldefined physical value, due to its pronounced hysteretic behavior [8].
The hysteretic behavior is also expected for the elastic modulus of their
shell G; indeed, the capillary interaction between colloidal particles
forming the shell will be varied in the course of collisions. This makes a
qualitative description of effects related to the dynamics of liquid
marbles challenging.
Conclusions
Liquid marbles are fascinating non-stick droplets presenting an
alternative approach to the achievement of superhydrophobicity. Our
paper is devoted to quasi-elastic collisions of liquid marbles coated with
lycopodium and Teflon-coated marbles. We established that the time of
collisions is independent of the velocity of the cue marble. It was also
established that it increases as a square root of the marbles’ volume.
These observations are well explained theoretically when marbles are
approximated by water droplets encapsulated within a thin elastic shell
representing the colloidal layer. The elasticity of the shell is due to the
capillary interactions between the colloidal particles coating the marble.
Thus, a collision between two marbles may be considered with the help
of a simple model of linear oscillator, moved by a spring with the
56
stiffness, governed by the elasticity and thickness of the elastic shell.
This model explains the experimental observations both quantitatively
and qualitatively. Considering this “elastic” energy is essential for
understanding collisions of liquid marbles and scaling laws governing
their shape.
Acknowledgements
Acknowledgement is made to the donors of the American
Chemical Society Petroleum Research Fund for support of this research
(Grant 52043-UR5). We thank Mrs. N. Litvak for the SEM images.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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