# Capillary pressure

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Figure 1 Origin of surface tension at liquid-vapor interface.
Figure 2 Arbitrarily-curved surface with two radii of curvature RI and RII.

Since the distance between the molecules in the vapor phase is much greater than that in the liquid phase, the intermolecular force between the molecules in the vapor phase is very weak. The intermolecular attractive force in the liquid phase holds the molecules in the liquid close to each other. For the molecules within the liquid phase, the intermolecular forces from all directions are balanced. Although the forces acting on the molecules at the liquid-vapor interface are balanced along the tangential direction, the attractive force from the molecules in the liquid phase, Fi (in normal direction), tends to pull the molecules at the liquid-vapor interface toward the liquid phase because the attractive force from the vapor phase, Fo, is much weaker. The net inward force FiFo causes movement of the liquid molecules until the maximum number of molecules is in the interior, which leads to an interface of minimum area (Fig. 1).

It is generally necessary to specify two radii of curvature to describe an arbitrarily-curved surface, RI and RII, as shown in Fig. 2. The surface section is taken to be small enough that RI and RII are approximately constant. If the surface is now displaced outward by a small distance, the change in area is

 $\Delta A=\left( x+dx \right)\left( y+dy \right)-xy$ (1)

If $dxdy\approx 0$, then

 $\Delta A=y\,dx+x\,dy$ (2)

The work required to displace the surface is obtained from:

 $\delta W=\sigma (x\,dy+y\,dx)$ (3)

Displacement acting on the area xy over the distance dz also creates a pressure difference Δp across the surface – capillary pressure (pcap). The work attributed to generating this pressure difference is

 $\delta W=\Delta p\,xy\,dz={{p}_{cap}}xy\,dz$ (4)

From the geometry of Fig. 2, it follows that

 $\frac{x+dx}{{{R}_{I}}+dz}=\frac{x}{{{R}_{I}}}$ (5)

or

 $dx=\frac{x\,dz}{{{R}_{I}}}$ (6)

Similarly,

 $dy=\frac{y\,dz}{{{R}_{II}}}$ (7)

For the surface to be in equilibrium across this differential change, the two expressions for the work must be equal:

 $\sigma (x\,dy+y\,dx)=\Delta p\,xy\,dz$ (8)

i.e.,

 $\sigma \left( \frac{xy\,dz}{{{R}_{I}}}+\frac{xy\,dz}{{{R}_{II}}} \right)=\Delta p\,xy\,dz$ (9)

The pressure difference between two phases becomes

 ${{p}_{cap}}=\Delta p=\sigma \left( \frac{1}{{{R}_{I}}}+\frac{1}{{{R}_{II}}} \right)=\sigma ({{K}_{1}}+{{K}_{2}})$ (10)

where K1 and K2 are curvatures of the surface. This expression is called the Young-Laplace equation, and it is the fundamental equation for capillary pressure. It can be seen that when the two curvature radii are equal, in which case the curved surface is spherical, eq. (10) can be reduced to:

 ${{p}_{cap}}=\Delta p= \frac{2\sigma}{{{R}}}$ (11)

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.