Fluids and Rocks - Introduction to Carbon Capture and Sequestration - page 403

Box 9.4.1 ( Continued )

p g p l

R

2 R

Let us now change the radius of our meniscus, R , by an infi nitesimal amount. As we
do not change the temperature, the change in the Helmholtz free energy is given by:

( )

dA = p − p dV + γ dA ,

gas liquid LG

with:

14 1

3 2

V = π R + V and A = 4 π R ,

rest

23 2

where V rest is the part of the volume that does not depend on R . This gives:

2

dV =π 2 R dR and dA =π 4 RdR

At equilibrium, the Helmholtz free energy takes a minimum value:

dA

( )

2

dR =− π

2 Rp − p + π γ 4 R = 0,

gas liquid LG

or:

2

pp − = γ ,

gas liquid LG

R

which is the Young-Laplace equation. In the derivation we have assumed a zero con-
tact angle, which only holds for a fl uid that completely wets the substrate. If we
observe a non-zero contact angle, the surface has a slightly different shape and the
Young-Laplace equation becomes:

2

pp − = γ cos θ,

gas liquid LG

R

where θ is the contact angle. Note that R ′ = R /cos θ is now the radius of curvature of

the fi lm.

p g

p l

θ

2 R