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A gravity droplet crossing a liquid-liquid interface is covered on the forefront with a film of the leaving liquid phase. The film thickness is not homogeneous over the droplet surface, and it reduces as the droplet penetrates the interface, particularly in the stretched area where it then ruptures. An expression for the film thickness in the stretched region is deduced from a force balance. The film rupture is expected to occur at a droplet position when the normal stress in the stretched film reaches the tensile strength of the liquid. By using some experimental data from literature the expression delivers 26 nm for the film thickness at rupture, while Burrill and Woods [1] obtained experimentally values between 30 nm and 50 nm.

Dynamics of thin fluid films governs kinetics of many processes like coalescence and encapsulation of particles. Prior to coalescence the fluid separating the particles must be squeezed out and the remaining film ruptured. The flow in the film is mostly treated on the basis of the lubrication model, see e.g. Davis et al. [

In a recent paper Oldenziel et al. [

In addition to the film shape, of certain importance is also the question of the thickness of the film at the instant of its rupture, both in coalescence and encapsulating processes. The present paper provides a model for estimation of the thickness of this film at the rupture. The model is based on a static force balance of the film formed on a droplet crossing a horizontal liquid-liquid interface. The droplet is assumed to be an oblate spheroid formed by revolution of an ellipse about its minor (vertical) axis.

The physical system consists of two sufficiently thick horizontal liquid layers of different densities,

Many experiments dealing with such penetration phenomena have shown that the droplet shape depends on the system properties and the velocity the droplet approaches the interface, see e.g. Chan et al. [

Referring to

where R is an equivalent radius of the droplet taken as a sphere.

Prior to interaction start with the upper liquid phase and deformation of the interface, the whole droplet volume is involved and causes this force. When the droplet is in interaction with the upper phase and is deformed as illustrated in

With this notion and the assumption of an ideal (rotation-symmetric) ellipsoid with respect to its vertical axis, described parametrically,

the volume V causing the buoyancy is

The Archimedes force, , may thus be expressed as

For m = b the ellipsoid is surrounded by the heavier (lower) liquid and its total volume is involved in the buoyancy force.

The geometrical parameters of the ellipsoidal droplet depend on the process parameters. The movement of the droplet is assumed to be very slow and the dynamical effects are neglected. The shape of the droplet is then caused only by the force F_{AE} which generates certain curvature of the droplet surface; the maximum curvature is expected in the horizontal symmetry plane, at y = 0. The local curvature of the ellipse is

which for y = 0, that is for t = 0, gives the curvature of the ellipsoid surface in the vertical plane at y = 0,

The curvature of the ellipsoid surface in the horizontal plane y = 0 is, and the average curvature is

This curvature is used to obtain the ratio of the main ellipsoid axes. Neglecting the capillary pressure at x = 0, y = b, the force balance at x = a, y = 0,

delivers

, , (9)

where the Bond number B_{0} is defined with the axis b of the ellipsoid as the characteristic length. Obtaining b from experiments, Equation (9) determines the other axis, a; for,.

The movement of the droplet penetrating the interface is a complex process whose treatment requires a detailed analysis of fluid dynamics with overlapped interfacial effects. The so called lubrication model is mostly used for treatment of such or similar problems, see e.g. [2-5,9] for details. However, the dynamical effects are getting weaker as the film thins and may be neglected immediately prior to film rupture in comparison to static effects; the film rupture is viewed as follows.

As the droplet moves upwards, the liquid film covering the droplet becomes stretched particularly in its outer (thinnest) region, and the thickness d decreases,

Using the average

for the capillary pressure one obtains

This pressure also retards the film drainage. It is larger than the capillary pressure in the film near the vertical axis () which may support formation a concave-convex interface in that region.

To obtain an expression for the thickness d at the instant of the film rupture we adopt the following model assumptions:

a) The liquid film is a fluid sheet able of flowing and stretching.

b) The force F_{AE} given in Equation (4) presses the liquid film above the droplet against the lighter upper phase, and a certain amount of the film liquid flows out across the gap (thickness d) at the inflection line, Figures 1(b) and 2. Due to the upwards droplet movement the liquid in this gap is continuously stretched and exposed to the action of a steadily increasing capillary pressure, decreasing m in Equation (11). The film stretching arises both verti cally and azimuthally (increasing c in

c) The liquid film is held together by molecular interaction in the film cross-section. At a large film thickness the number of molecules in the film cross section is large and the resulting force arising from the molecular interactions prevents the film rupture (the origin of the molecular interaction is not the subject of this paper).

d) As the film thins, the number of molecules occupying the cross-section decreases; at certain film thickness the stretching force overcomes the molecular interaction, and the film ruptures.

Taking the sketched process sufficiently slow to neglect dynamical effects and disregarding capillary effects, one obtains the following force balance for the film cross-section along the inflection line. The projection of the buoyancy force F_{AE} on the tangent at the inflection line (

where, is balanced by the internal, resulting molecular force in the film that may be stated in terms of the tensile strength of the liquid. Denoting the tensile strength by Ã, the force balance may be written as

hence

Using Equation (9), Equation (16) can be written as

The shape function F contains only the geometrical parameters of the ellipsoid, but it depends implicitly on the physical properties via the ratio according to Equation (9). For m = 0 it is and

which is valid for a particular position of the droplets,

_{0}.

_{0}.