Young–Laplace equation

In physics, the Young–Laplace equation () is an equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension acting against the curvature of the surface (although use of the latter is only applicable if assuming that the wall is very thin).

The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness):

Δ

p

=

−

γ

∇

⋅

n

^

=

−

2

γ

H

f

=

−

γ

(

1

R

1

+

1

R

2

)

{\displaystyle {\begin{aligned}\Delta p&=-\gamma \nabla \cdot {\hat {n}}\\&=-2\gamma H_{f}\\&=-\gamma \left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)\end{aligned}}}

where

Δ

p

{\displaystyle \Delta p}

is the Laplace pressure, the pressure difference across the fluid interface (the exterior pressure minus the interior pressure),

γ

{\displaystyle \gamma }

is the surface tension (or wall tension),

n

^

{\displaystyle {\hat {n}}}

is the unit normal pointing out of the surface,

H

f

{\displaystyle H_{f}}

is the mean curvature, and

R

1

{\displaystyle R_{1}}

and

R

2

{\displaystyle R_{2}}

are the principal radii of curvature. Note that only normal stress is considered, because a static interface is possible only in the absence of tangential stress.

The equation is named after Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year. It is sometimes also called the Young–Laplace–Gauss equation, as Carl Friedrich Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions using Johann Bernoulli's virtual work principles.

Soap films

If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface.

Emulsions

The equation also explains the energy required to create an emulsion. To form the small, highly curved droplets of an emulsion, extra energy is required to overcome the large pressure that results from their small radius.

The Laplace pressure, which is greater for smaller droplets, causes the diffusion of molecules out of the smallest droplets in an emulsion and drives emulsion coarsening via Ostwald ripening.