Elastic Properties of Liquid Surfaces Coated with Colloidal Particles

The physical mechanism of elasticity of liquid surfaces coated with colloidal particles is proposed. It is suggested that particles are separated by water clearings and the capillary interaction between them is negligible. The case is treated when the colloidal layer is deformed normally to its surface. The elasticity arises as an interfacial effect. The effective Young modulus of a surface depends on the interfacial tension, equilibrium contact angle, radius of colloidal particles and their surface density. For the nanometrically scaled particles the line tension becomes essential and has an influence on the effective Young modulus.

Solid particles attached to liquid surface initiate a broad diversity of physical effects including capillarity-induced self-assembly and particle-assisted wetting [21][22][23]. One of the most interesting phenomena related to the ensembles of colloidal particles located at the liquid/air interfaces is the quasi-elastic behavior of such systems [24][25][26]. Vella et al. proposed the physical mechanism of elasticity demonstrated by the layer of colloidal particles covering a liquid surface. In our paper we introduce an alternative mechanism of elasticity of these layers.

Results and discussion
It was demonstrated recently that liquid surfaces coated with colloidal particles behave as two dimensional elastic solids (and not liquids) when deformed [24][25][26].
For example, liquid marbles do not coalesce when pressed one to another, as shown in Figure 1. When liquid marbles collide they do not coalesce, but demonstrate quasielastic collision depicted in Figure 2. It was also demonstrated that the layers built of monodisperse colloidal particles can support anisotropic stresses and strains [26]. The observed pseudo-elastic properties of liquid surfaces coated with colloidal particles call for explanation.
Vella et al. treated the collective behavior of a close packed monolayer of non-Brownian colloidal particles placed at a fluid-liquid interface [26]. In this simplest case, however, the close-packed monolayers may be characterized using an effective Young's modulus and Poisson ratio [26]. These authors proposed an expression for the effective Young modulus E of the "interfacial particle raft" in the form 26 where γ is the surface tension of liquid, d is the diameter of solid particles, is the Poisson ratio of solid particles, and is the solid fraction of the interface. They concluded that the elastic properties of such an interface are not dependent on the details of capillary interaction between particles [26]. We propose an alternative mechanism of pseudo-elasticity of liquid surface covered by solid particles, which are not close-packed, explaining the "transversal" elasticity of surfaces, depicted in Fig. 4. As it will be seen below, the elasticity in this case is caused by the change in the liquid area under deformation, as it also occurs in Ref. 26.
Consider two media (one of which may be vapor) in contact (Fig. 5). The plane boundary between them is characterized by the specific surface energy (or the surface tension) 1,2 . At this surface floats a spherical body of the radius R modeling a colloidal particle. Note that this flotation is due to surface forces but not to gravity since the size of a colloidal particle is much lower than the capillarity length √ 1,2 / | 1 − 2 | ( 1 and 2 are the corresponding densities). Besides 1,2 , these forces are characterized by the surface tensions 1 and 2 at the corresponding solidfluid interfaces (see Fig. 5).
The total surface energy U is given by: The first and second terms represent the surface energies of the solid-fluid interfaces, while the third one describes the energy of the "disappearing" area due to the solid body. Also the so-called line tension, Γ, of the triple line, neighboring two fluid and one solid media, is included in Eq.
(2) since it may be important for very small particles [29]. As is known, the flotation of strongly hydrophilic nanoparticles may be explained by considering the line tension only.
The equilibrium depth of immersion h 0 can be found by differentiation of (2) In the widespread case when the result for equilibrium depth of floating is: Here we also use notation for the Young contact angle To provide a correct asymptotic behavior of expression (5), one should require /sin ≪ 1. In order to study the oscillatory response of the surface covered by the colloidal particles, we obtain for the force component acting on the solid body in the case of small deviations from the equilibrium depth ℎ 0 which is of a Hooke law form.
As it follows from (7), our model system is a harmonic oscillator with the eigen-frequency (m is the mass of the floating body; Exp. (8) is obviously applicable when /sin 3 < 1). Note that elastic properties don't depend on the surface tensions 1 and 2 which determine only the equilibrium of the system in (5), (6). Exp. (8) may be rewritten in the form revealing an explicit dependence of the eigen-frequency on the radius of a particle R: , ρ p is the density of a colloidal particle. Now let n be the surface density of colloidal particles on the area S of the boundary (the surface is supposed to be flat). Under the deviation ℎ − ℎ 0 , the total elastic force on the area S is ℎ . According to the Hooke law ℎ = ℎ − ℎ 0 |ℎ 0 | and for the effective Young modulus E it follows on account of (7) = 2 |ℎ 0 | 1,2 (1 − sin 3 ) ≈ 2 1,2 |cos − cot 3 | Note, that here the deformation is connected with the interface change but not with the deformation of particles. The whole approach is applicable for dilute colloids or colloids with a weak interparticle interaction. This assumption is justified for colloidal particles with the characteristic dimensions of particles μm 5  R , as shown in Ref. 21. Such particles do not deform the water/vapor interface; this leads to a negligible capillary interaction [21]. Exp. 9 will predict the effective elastic modulus for both of flat and curved surfaces coated by colloidal particles, in the case when the characteristic dimension of the deformed area L is much larger than that of a particle R (see Fig. 4). It should be stressed that Eq. 9 supplies the upper limit of the effective elastic modulus, because it assumes the simultaneous contact of a plate with all of colloidal particles.
An example of numerical estimation of (9) may be given using Fig. 3A. The radius of particles is of order R~1 μm; thus, taking into account the most reasonable estimation of line tension Γ~10 −10 N [30] and water surface tension 1,2~1 0 −1 N/m, we see that according to (4) ~10 −3 , and line tension is negligible. From Fig. 3A, ~10 9 m -2 , for lycopodium ~120°, and from (9) a realistic estimation follows E~100 Pa that is two orders of magnitude lower than that following from (1)  For nanometric particles, the line tension should be taken into account in Exps. (8) and (9). For example, for hydrophobic colloidal particles of a size R=10 nm on water/air interface with above-mentioned parameters we have ~0.1 , and, putting the average distance between particles equal to their size R, the concentration n~10 16 m -2 . Then from (9) it follows E~10 MPa, that is much more than in the case of microscaled particles. This is not surprising; Exp.1 yields for nanoparticles similar values of the elastic modulus. It is also obvious from (9) that in the case of materials with lower values of the contact angle , the value of the effective Young modulus will be much lower.
It is noteworthy that the proposed model of transversal elasticity of surfaces coated with colloidal particles implies zero contact angle hysteresis [31][32][33].
Considering the contact angle hysteresis will call for much more complicated mathematical treatment.
It should be stressed that the proposed mechanism of elasticity will work only at the initial stage of the deformation of droplets coated with colloidal particles (see . 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 drop. Thus, the effective mechanical scheme of a liquid surface coated with solid colloidal particles looks like as it is depicted in Figure 6 (η is the viscosity of the liquid).

Conclusions
We introduce the mechanisms of elasticity of liquid surfaces coated with colloidal particles in the case when the gravity and capillary interactions are negligible. We treat the situation when solid colloidal particles do not form close packaging and are separated by water clearings. The deformation normal to the surface of a colloidal layer is discussed. When a colloidal particle is displaced from its equilibrium position the pseudo-elastic force following the Hooke-like law arises.
This purely interfacial force gives rise to the pseudo-elasticity of the colloidal raft coating a liquid surface.
The effective Young modulus of a surface depends on the interfacial tension, equilibrium contact angle, radius of colloidal particles and their surface density.
Eigen-frequency of oscillations of colloidal particles is calculated. For the nanometrically scaled particles the line tension may dominate over interfacial tension effects. The proposed mechanism of elasticity will work only at the initial stage of the deformation, on "short" time spans of deformation.