Modelling & Practice

 

1. Theory

The model used was a simple FCC lattice structure. To enable a simplified model of the lattice, two spheres were used which were in contact with each other. The spheres were assumed to be non-porous and rigid. This system was subjected to a perfect liquid vapour with the liquid condensing on the surface of the spheres. To begin with, the liquid coats just the surface of the spheres (chemisorbed). As the vapour condenses on the first layer (physisorption), a liquid bridge is formed at the place where the spheres are in contact.

This bridge continues to form until such a point where the bridge is no longer sustainable and the liquid flows.

From the study of this bridge, such factors as the area of the sphere covered by liquid and volume of the bridge can be obtained. It is convenient to consider these in terms of fraction area of the sphere surface wet (FAW), fraction area of the sphere covered by the liquid (FAL) and fractional volume of the liquid bridge (FVL).

2. A mathematical and computational analysis

S-S

S = ½ distance between spherical particles. Typically though, this is normally expressed as if S was the full distance (i.e. 2S) between the spheres. There is a limiting condition (Smax) after which a bridge is not formed. For a sphere with a radius of 1000Å and a vapour pressure (Vp equivalent of p/po) of 400, the maximum separation is around 74Å.

The system is considered to be inside of a capillary. This capillary size can be thought of as effectively being the vapour saturation.

A variant of the Kelvin equation is used to calculate the vapour pressure inside the capillary.

As a concequence of surface tension, there is a pressure difference across a curved interface, quantified by the Young - LaPlace equation (5). This is derived in appendix 3a.

equation 5

where Δp is an applied pressure, γ the surface tension and r the capillary radius.

It can be proven thermodynamically that when a liquid suffers an applied pressure, it's vapour pressure (pg) will be increased from the normal value (p*g) found in the absence of the applied pressure. This effect is quantified in (6) and derived in appendix 3b.

equation 6

6

By substituting Δp in (6) with -2γ/r gives the Kelvin equation (7).

equation 7

7

equation

The proof of (7) is that for an equilibrium to exist between the liquid and vapour of a pure substance that Gvap = Gliquid. If the vapour pressure is reduced by Δp, then Gvap must also reduce and to keep the equilibrium, Gliquid must reduce by the same amount. The overall reduction can be said to be

equation 7.1

7.1

The system used here is that the liquid is at one bar ( Δpliquid = 0), this transforms (7.1) into (7.2) and expressed using molar quantities.

equation 7.2

7.2

Subtraction of (7.2) from (7.1) gives (7.3)

equation 7.3

7.3

By re-arrangement of (7.3), Δpliquid is obtained (7.4)

equation 7.4

7.4

If (7.4) is combined with the equation of Young and Laplace (7.5), (7) is obtained.

equation 7.5

7.5

(7.5) is obtained by the following theory.

The cavities in a liquid are at equilibrium when the tendency for the surface to area to reduce is equal to the rise in internal pressure (which would occur). If the pressure inside the cavity is said to be pin and the cavity radius is denoted as r, then the outside force, pout, will equal the pressure x area = 4πr2pin. This external force is the sum of pin and the surface tension, γ.

Should the radius of the sphere change from r to r+ δr then the surface area, σ, changes to

equation 7.5.1

7.5.1

The work done to cause this increase in surface area is

equation 7.5.2

7.5.2

As force x distance is the work, the opposing force to the stretch through a distance of Γr (with corresponding radius of r) is

equation 7.5.3

7.5.3

When the outward force equals the inward force (i.e. at equilibrium)

equation 7.5.4

7.5.4

(7.5.4) can be re-arranged to give (7.5).

It is quite simple to see that the pressure the liquid will exert on a sphere will be different to that on a flat surface (by a factor of Δp = 2γ/-r).

The contact angle (β) (diagram 3) is the angle at which the liquid vapour comes into contact with the sphere. Water comes into contact with β = 0° (therefore sin β » 0). The rule though for a solid/liquid is that 0 < θ < θ max. (or expressed as sin, 0 < θ c < 1) the surface is covered with liquid.

equation

where θ, θc and θmax are the contact angles.

For the liquid bridge simulation, it has been assumed that the surface tension is zero (which to a very good approximation, it is), that sin β = 0 or another value up to sin β = 1 (i.e. obeying the solid/liquid rule) and have a fixed p/po maximum (the maximum capillary radius size). The bridge condition (S << Smax) is also observed.

Contact angles - Diagram 3

The simulation produces by use of elliptic integrals (of the first and second type) the fraction area of the liquid, the fraction of area wetted and the fractional volume of the liquid bridge between the spheres.

Elliptic integrals are by their nature very complex, but the mathematical theory is stated in appendix 3c. They are used in preference to a simpler method as it allows the study of the entire bridge (one calculation, but using a more complex variation of the Young & LaPlace and Kelvin equations) instead of having to calculate the two spherical cap interfaces and internal volumes (three series of calculations). This greatly increases the speed of data production.