Liquid Bridges

3a. The pressure across a curved interface.

 

Surface tension (γ) is defined as the work required to create a unit area of new interface, hence w = γδα. It quantifies a thermodynamic driving force to minimise surface area.

A curved interface between two phases (α and β) will be at equilibrium when the three forces due to the surface tension and to the pressures in the two phases just balance. Let α (diagram a) be the phase on the convex side of the interface.

If the interface is displaced towards the concave, left hand side, then the surface area decreases. Thus the force of surface tension acts together with p_α and is counterbalanced by p_β. Consequently, the pressure on the concave side of a curved interface is always greater than that on the convex side p_&beta > p_α.

The three forces may be quantified (as pressure = force/area). If the radius of curvature changes by δr, the surface area changes by 4π(r + δr)² - 4πr² &8776; 8πrδr. This involves an amount of work, 8πrγδr (since w = γδα). As work = force/distance, the force opposing the change dr is 8πrγ. It follows that 8πrγ + 4πr²p_α = 4πr²p_β. Rearrangement of this gives (a3a-1)

a3a-1

This is the Young - LaPlace equation for a "spherical" interface.

 

Effect of applied pressure on the vapour pressure of a liquid ^[11]

 

One of the fundamental equations of thermodynamics is (a3b-1)

a3b-1

Therefore for an isothermal change (δ T=0) (m signifies molar quantities)

a3b-2

If a pure liquid (l) under a total pressure (P) is in equilibrium with its vapour (g) at a pressure (P_g) then the chemical potentials must be the same in each phase (for a pure substance, μ = G_m). If these pressures are altered, yet maintaining equilibrium, then the changes, δG_m, must be the same in each phase. Therefore

a3b-3

Approximating the vapour to an ideal gas gives V_m,g = RT/P_g. Therefore

a3b-4

For moderate pressures, V_m,l can be taken to be constant. By integration of (a3b-4) gives (a3b-5a)

a3b-5a

a3b-5b

Setting P₁ as the normal vapour pressure p*_g (no applied pressure), p_g = p₁ = p*_g. It follows then that

a3b-6

Δp is the additional applied pressure, p_g is the new vapour pressure.

 

Elliptic integral theory and equations.

 

The integral (for the real part) can be expressed as

a3c-1

where

a3c-2

a3c-3

Other terms within equations (a3c-1 to a3c-3) which need to be defined are

a3c-4a

a3c-4b

a3c-4c

a3c-4d

a3c-4e

The limiting factor is that b < a. l (q) is the arc length of the bridge between the spheres for a given value of (typically that of the hyperbola from q = 0 to q). e is known as the eccentricity.

The calculation for the complete elliptic integrals of the first and second kind (F and E in (a3c-1) respectively) are of the calling form F (x,y,z). x and y have to be non-negative and z must be positive. It is possible to simplify matters by excluding the z parameter as this normally is set as 1.

The two integrals are determined using the following formulae

a3c-5a

a3c-5b

where R_F and R_D are the two functions called to do the calculations. These are both available freely in the SLATEC ^[12] libraries as source code. The code can be amended to negate the error routines which make the overall source code a great deal larger than it needs to be.

R_F is given by Carlson^[13] as

a3c-6a

R_D^[14] can also be assigned by

a3c-6b

t in both cases is the varied position of the point of integration.

Given that these two equations only differ by , it is fair to say that R_F deals with the negative (say, left hand sphere) slope while R_D deals with the other side.

(a3c-6a) and (a3c-6b) can be converted into a simpler form by the assignment of the special variables, s nu, d nu, c nu and t nu. (a3c-7a and a3c-7b) show this.

a3c-7a

a3c-7b

This can be transferred into (a3c-6a) thus

a3c-8

The transformation of (a3c-6b) into this style requires a far more complex set of transformations. These are given below (a3c-9a - h)

a3c-9a

a3c-9b

a3c-9c

a3c-9d

a3c-9e

a3c-9f

a3c-9g

a3c-9h

These can then be written into (a3c-5a) and (a3c-7b) thus

a3c-10a

a3c-10b

What are these s nu, d nu, c nu and t nu terms?

u is simple. It is another way of expressing the elliptic integral of the first type, F (a3c-10a). n stands for amplitude with s, c and t meaning sine, cos and tan. This leaves d. d should actually be written as δ as it stands for delta. Therefore, s nu is the sine amplitude of the elliptic integral of the first kind.

The evaluation of R_D and R_F is performed by a loop with the governing condition the x << x_max. On entry, the x, y and z parts of the equations for R_F and R_D are provided by the values c n²u, d n²u and 1. Each have the conditions that none are negative with x and y being non zero.

Due to the nature of programming, it is necessary to initialise a number of extra variables. These need not be discussed, suffice to say, their only real purpose is to make the program simpler to understand.

Both sets of loops follow the conditions

a3c-11

where

(the same treatment for defining root_x and root_y as well as xndev and yndev can be performed. x, y and zndev are the deviations from the initial values of xn, yn or zn).

At the end of each loop, xn, yn and zn are incremented by (xn + γ x 0.25) (where xn can be either yn or zn). The loop then returns to the initial definition of μ. When the condition ε < the error limit (defined in the program as around 1 x 10^-25), the loop is exited (the integration has reach the computers mathematical calculation error tolerance, this can be counted as the approx. value). Count is incremented by taking the current value of count and multiplying 0.25

In standard notation, we can express the range of values for xndev, yndev or zndev as below :

>

By defining a number of smaller variables, the final calculations for R_D (a3c-12a) and R_F (a3c-12b) become far simpler to express.

a3c-12a

For R_F, the variables used for R_D are not used, but simpler ones are used.

a3c-12b

While all this maths is all well and good, it is of very little use unless it can be outputted and displayed in a form which can be used.

By running the program via a loop in the main calling routine (going from 0.5 to RC_MAX) and storing each successive value of FAW, FAL, FVL and SM (bridge distance max.) with the counter as a reference and outputting this as a simple to use spreadsheet file (a CSV file is accepted by all spreadsheet or data plotting packages), it is possible to see how the isotherms are produced for the varied RC_MAX. A typical output will look like this.

Graph of fraction liquid vs. radius of capillary.