ABSTRACT

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.

1. Introduction

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. [2], Chesters [3], Klaseboer et al. [4], Bazhlekov et al. [5]. From this model, equations for calculating the film thickness as func­ion of time were derived, but a determination of the film thickness at the rupture instant is not possible. The situation is similar with encapsulation of a particle. The particle to be encapsulated is forced to cross a thin layer of encapsulating material whereby a thin film of this material envelops the particle, see e.g. Massoth et al. [6] and Abkarian et al. [7]. In contrast to coalescence, the film enveloping the particle should be stable and posses a homogeneous thickness.

In a recent paper Oldenziel et al. [8] present experimental results on the behaviour of liquid droplets at a two-liquid interface, mainly focusing on the time history of the film covering the upper part of the droplet. The movement of the droplet is provided by gravity; the soft interaction of the originally spherical droplet with the interface flattens the sphere to a rotational ellipsoid. The thickness of the film on the droplet is not homogeneous; it is maximal at the droplet vertex and decreases radially outwards. The film rupture was observed to occur in the outer region, and not at the droplet vertex. This information is important in context with the estimation of the probability that the droplet during the interaction time mix with the upper liquid layer, being miscible with the droplet liquid. The time history of the layer observed in the experiments shows a spatial asymmetry, caused by experimental reality. The reported results also provide important insights into encapsulation and collision process e.g. of two droplets of different sizes. They are in agreement with the theoretical models; several such models have been analysed in the review paper by Chan et al. [9], and the reader is referred to this source, also for further references.

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.

2. The Physical Model

The physical system consists of two sufficiently thick horizontal liquid layers of different densities, Figure 1(a). In the lower layer there is a liquid droplet of a lighter (upper) phase that moves upwards by gravity. As the droplet approaches the upper phase, it deforms the originally even interface, penetrates that phase, but remains at first separated from it by a thin film of the heavier (lower) phase. Since the droplet is not rigid, it also becomes deformed and changes its shape from nearly spherical to a flattened ellipsoid, Figure 1(b). As the droplet penetrates further the lighter phase, the liquid layer enveloping the droplet on its upper part thins down to a certain thickness when it then ruptures, and the droplet liquid becomes sucked by the lighter phase.

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. [9]. The thickness of the liquid layer is not uniform over the droplet surface, and at certain distance from the vertical symmetry axis, near the inflection line of the interface separating the continuous phases, the liquid film is extremely thin. This hinders the outflow of the upper portion of the film (bottle neck effect). The movement of the droplet results in a continuous thinning of the film in its outer region which subsequently ruptures.

2.1. The Buoyancy Force

Referring to Figure 1(a), the Archimedes force causing the droplet to ascend is

(1)

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 Figure 1(b), the situation changes and the buoyancy force depends on the position of the droplet relative to the undisturbed interface. It seems reasonable to consider the droplet volume below the plane placed a distance m above the horizontal symmetry plane of the ellipsoid to cause the Archimedes force. The position of this plane is determined by the position of the horizontal circular line connecting the minima of the film thickness.

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

, (2)

the volume V causing the buoyancy is

(3)

The Archimedes force, , may thus be expressed as

. (4)

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

2.2. The Geometry of the Droplet

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

, (5)

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

(6)

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

(7)

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,

(8)

delivers

, , (9)

where the Bond number B₀ 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,.

2.3. The Film Thickness at Rupture

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, Figure 1(b). This impedes the liquid above the droplet from outflow. In addition, the droplet penetration increases the capillary pressure in the region of the inflection line of the interface, Figure 2. At this line, the tangential curvature of the bell-shaped interface is zero,; its normal curvature may be obtained from the equation of the ellipse, for small d. Equation (5) then gives, for:

(10)

Using the average

for the capillary pressure one obtains

(11)

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 Figure 1(b)).

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 (Figure 2) in the direction of the film flow,

(12)

(13)

(14)

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

(15)

hence

(16)

(17)

(18)

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

(19)

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

(20)

which is valid for a particular position of the droplets, Figure 1(b) .

3. Illustration of Results

Figure 3 illustrates Equation (9), the shape of the droplet. The ratio of the ellipsoid axes increases almost linearly with B₀.

Figure 4 quantifies Equation (11) for chosen values of the axes ratio. The pressure enhances the thinning of the film in the region of the inflection line of the interface and resists the outflow of the liquid locked above the droplet. This effect is larger at larger B₀.