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

1ArielUniversity, Physics Department, P.O.B. 3, 40700, Ariel, Israel

2ArielUniversity, Chemical Engineering and Biotechnology Department, P.O.B. 3, 40700, Ariel, Israel

3Department of Chemistry, Bar-IlanUniversity, 52900 Ramat-Gan, Israel

4Faculty of Mechanical Engineering, Technion – IsraelInstitute of Technology, Haifa32000, Israel.

Abstract

Liquid marbles are non-stick droplets coated with colloidal particles. Liquid marbles do not coalesce when pressed one to another or whencolliding. 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 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, Fujiiet 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 Matsukumaet 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 iscovered 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 wasabout 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 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 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 βand 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 collisionτ(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:. 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 dependencehas been predicted and observed [45-49].

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.

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 massU before the impact. The total kinetic energy of a rolling marble is thus expressed as:. Part of this kinetic energy 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:

(1)

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 modulusG of a liquid possessing the surface tension γ and coated with monodisperse particles with the diameter of dwas estimated in Ref. 43 as

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 particlesencapsulating 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:

(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 . Finally, for elastic energy one obtains:

(3)

here 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 thekinetic 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 massm = = (9/10)Mand effective stiffnessk = πGh.The frequency of such oscillator is written as:

(4)

One can assume that the collision time corresponds to a half-period of oscillation. Then, for the time of collision, one obtains:

(5)

We assume; ; (see Ref. 42), and obtainτ ≈ 3 · 10–2s, 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 thatτ~~, in complete agreement with the results presented for lycopodium-coated marbles in 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 Teflon-coated marbles were the same. It is noteworthy that Richard et al. obtained in Ref. 50asimilar scaling law for the characteristic time of water droplets bouncing solid substrates, i. e: (compare with Exp. (4)). This is not surprising, because in the case of bouncing droplets the role of the effective spring stiffness kis 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.

Fig. 7. The ln-ln plot of the collision (contact) time τ vs. the volume of a

lycopodium-coated marble V.

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:

, (6)

where η is the viscosity of the liquid (water in our case), V is the volume of the marble is the gradient of velocity of liquid filling the marble [48]. For the sake of a very crude estimation we assume:, 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:

. (7)

And finally we derive for the estimation of viscous dissipation:

. (8)

It is easily seen from Exp.(8)that the viscous dissipation is much lower than the total kinetic energy of the cue marble, indeed:

. (9)

Here we assume. 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); is adopted for the sake of the rough estimation). This is expressed by:

. (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 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 studiedthe 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 given by (γ is the surface tension [51,52]), the approach developed by Quere et al. predicted the following scaling law:

. (11)

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 shellsand 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 energyΔEfrom a sphere tangent to the plane can be written dimensionally as [1]:

. (12)

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)

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 ψmay be introduced:

(14)

Substituting into Exp. (14) , which is the effective surface tension of a marble8 and aforementioned values of G and h, results in.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 parameterψ is close to unity. The effective surface tension of marbles is not a well-defined 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 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.

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