## Thinning of the Water Film Attachment

Induction time-critical thickness Li, Fitzpatrick and Slattery have developed a mathematical model from which the attachment between the bubble and the particle can be characterized. The model relates to an initially spherical bubble approaching a stationary spherical particle. As the bubble-particle separation reduces, the water film separating them will thin and possibly reach a critical thickness at which it will rupture.

The induction time is the time between the initial contact, defined by the formation of a dimple on the bubble surface, and the final rupture of the separating film. Restating the previous sentence, a condition for attachment is for the induction time to be shorter than the contact time.

The model is defined in axisymmetric cylindrical coordinates with the z-axis coinciding with the line of approach of the bubble and particle centres. The particle is assumed to be spherical of radius Rp and the bubble away from the neighbourhood of close approach is also to be assumed spherical with a radius Rb. The origin of the system is taken as the centre of the spherical particle, the frame of reference being stationary with reference to the particle.

Figure 10 shows the bubble-film interface z = h1(r, t) and the surface of the solid z = h2(r). The surface of the bubble is dimpled on the nearest point of approach to the particle on the line of centres, the rim of the dimple is located at a radius R(t) from the line of centres.

It has been assumed that the initial time of the bubble-particle contact t = 0 occurs when the thinning rate in the film becomes dependent on radial position.

The film thickness between the two surfaces hx(r, t) is given by the difference:

hx(r, t) = hi(r, t) - hair) h0 = hx(r,0) and R0 = R(0) For the spherical solid particle:

The thinning of the film is due to the pressure in the film P(r, t) being greater than the hydraulic pressure ph remote from the line of centres; at this point r"Rh. The pressure inside the bubble p0 is linked to the hydraulic pressure ph by:

Components of the thinning pressure in the film

The local total film pressure P(r, t) has three components: (a) the London-van der Waals interactive pressure nv, (b) the electrostatic nd pressure due to the charges on the solid and the bubble surfaces, and (c) a pressure recap caused by the deformation of the bubble surface from that of a sphere to z = ht(r, t), in the neighbourhood of the solid particle.

For the van der Waals component, nv, for a film of thickness hx the following is considered:

When the film thickness was greater than 400 A, m = 4 and B = 10"19 erg cm, but when the film thickness was less than 120 A the corresponding values were taken as m = 3 and B = 10~14 erg.

These values are consistent with those quoted earlier for the Hamaker constant A. When B is positive the London-van der Waals forces between liquid-gas and the solid-liquid interfaces are mutually attractive.

The electrostatic interaction potential nei between the particle-water and the water-bubble surfaces is given by:

where:

K2"

8nnz}e2 DkT

A = 64nkT tanh

tanh zie^w-g

Figure 10 Thin liquid film formed as a small bubble approaches a small solid particle. Reproduced with permission from Li, Fitzpatrick and Slattery (1990) Rate of collection of particles by flotation. Industrial Engineering Chemistry Research 29, 955-957.

where and are the electrostatic potentials at the bounding interfaces. When the interfaces have the same sign, A is positive and the surfaces are mutually repulsive and when the signs are different, A is negative and the surfaces are mutually attractive.

As deformation of the bubble surface occurs, the capillary pressure in the film ftcap(r, t) is a function of the mean curvature H1 and the bubble pressure:

where y is the surface tension of the distorted film:

A lateral pressure gradient will develop in the water film normal to the separation distance hx. Under this pressure gradient water will move to the bulk liquid, causing thinning:

dr[hm

R2vr

Rr p Rz2

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