Thermal motion is able to produce capillary waves at the molecular scale. At this scale, gravity and hydrodynamics can be neglected, and only the

surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...

contribution is relevant. Capillary wave theory (CWT) is a classic account of how thermal fluctuations distort an interface. It starts from some intrinsic surface

h(x,y,t)

that is distorted. Its energy will be proportional to its area: :

E_\mathrm= \sigma \int dx\, dy\, \left sqrt-1\right approx \frac \int dx\, dy\,  \left \left( \frac\right)^2+\left(\frac\right)^2 \right

where the first equality is the area in this ( de Monge) representation, and the second applies for small values of the derivatives (surfaces not too rough). The constant of proportionality,

\sigma

, is the

. By performing a

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...

treatment, normal modes are easily found. Each contributes an energy proportional to the square of its amplitude; therefore, according to classical statistical mechanics, equipartition holds, and the mean energy of each mode will be

kT / 2

. Surprisingly, this result leads to a divergent surface (the width of the interface is bound to diverge with its area). This divergence is nevertheless very mild: even for displacements on the order of meters the deviation of the surface is comparable to the size of the molecules. Moreover, the introduction of an external field removes the divergence: the action of gravity is sufficient to keep the width fluctuation on the order of one molecular diameter for areas larger than about 1 mm2 (Ref. 2).J.S. Rowlinson and B. Widom "Molecular theory of capillarity" 2002

References

See also