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==Thermal capillary waves== | ==Thermal capillary waves== | ||
Thermal '''capillary waves''' are oscillations of an [[interface]] which are thermal in origin. These take place at the molecular level, where only the | Thermal '''capillary waves''' are oscillations of an [[interface]] | ||
which are thermal in origin. These take place at the molecular level, where only the surface tension | |||
contribution is relevant. | |||
Capillary wave theory (CWT) is a classic account of how thermal fluctuations distort an interface (Ref. 1). It starts from some [[intrinsic surface]] that is distorted. In the Monge representation, the surface is given as <math>z=h(x,y)</math>. | |||
An increase in area of the surface causes a proportional increase of energy: | |||
:<math> | :<math> | ||
E_\mathrm{st}= \sigma \iint dx\, dy\ \sqrt{1+\left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2} -1 | E_\mathrm{st}= \sigma \iint dx\, dy\ \sqrt{1+\left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2} -1 | ||
</math> | </math> | ||
for small values of the derivatives (surfaces not too rough): | for small values of the derivatives (surfaces not too rough): | ||
:<math> | :<math> | ||
E_\mathrm{st} \approx \frac{\sigma}{2} \iint dx\, dy\ \left[ \left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2 \right]. | E_\mathrm{st} \approx \frac{\sigma}{2} \iint dx\, dy\ \left[ \left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2 \right]. | ||
</math> | </math> | ||
A [[Fourier analysis]] treatment begins by writing the intrinsic surface as an infinite sum of [[normal modes]]: | |||
A [[Fourier analysis]] treatment begins by writing the intrinsic surface as an infinite sum of normal modes: | |||
:<math>h(x,y)= \sum_\vec{q} a_\vec{q} e^{i\vec{q}\vec{r}}.</math> | :<math>h(x,y)= \sum_\vec{q} a_\vec{q} e^{i\vec{q}\vec{r}}.</math> | ||
Since normal modes are [[orthogonal]], the energy is easily expressable as a sum of terms <math>\propto q^2 |a_\vec{q}|^2</math>. Each term of the sum is quadratic in the amplitude; hence [[equipartition]] holds, according to standard [[classical statistical mechanics]], and the mean energy of each mode will be <math>k_B T/2</math>. Surprisingly, this result leads to a '''divergent''' surface (the width of the interface is bound to diverge with its area) (Ref 2). 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 | |||
Moreover, the introduction of an external field removes | 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 mm<sup>2</sup> (Ref. 2). | of one molecular diameter for areas larger than about 1 mm<sup>2</sup> (Ref. 2). | ||
The action of gravity is taken into account by integrating the potential energy density due to gravity, <math>\rho g z</math> from a reference height to the position of the surface, <math>z=h(x,y)</math>: | The action of gravity is taken into account by integrating the potential energy density due to gravity, <math>\rho g z</math> from a reference height to the position of the surface, <math>z=h(x,y)</math>: | ||
:<math>E_\mathrm{g}= \iint dx\, dy\, \int_0^h dz \rho g z = \frac{\rho g}{2} \int dx\, dy\, h^2.</math> | :<math>E_\mathrm{g}= \iint dx\, dy\, \int_0^h dz \rho g z = \frac{\rho g}{2} \int dx\, dy\, h^2.</math> | ||
(For simplicity, we are neglecting the density of the gas above, which is often acceptable; otherwise, | |||
(For simplicity, | instead of the density the difference in densities appears.) | ||
Recently, a procedure has been proposed to obtain a molecular intrinsic | Recently, a procedure has been proposed to obtain a molecular intrinsic | ||
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==Gravity-capillary waves== | ==Gravity-capillary waves== | ||
These are ordinary waves excited in an interface, such as ripples on | These are ordinary waves excited in an interface, such as ripples on | ||
a water surface. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth: | a water surface. | ||
Their dispersion relation reads, for waves on the interface between two fluids of infinite depth: | |||
:<math> | :<math> | ||
\omega^2=\frac{\rho-\rho'}{\rho+\rho'}gk+\frac{\sigma}{\rho+\rho'}k^3,</math> | \omega^2=\frac{\rho-\rho'}{\rho+\rho'}gk+\frac{\sigma}{\rho+\rho'}k^3,</math> | ||
where ''ω'' is the [[angular frequency]], ''g'' the acceleration due to [[standard gravity|gravity]], ''σ'' the [[surface tension]], ''ρ'' and ''ρ‘'' the [[mass density]] of the two fluids (''ρ > ρ‘'') and ''k'' the [[wavenumber]]. | |||
===Derivation=== | ===Derivation=== | ||
This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As Richard Feynman put it (Ref. 6): | This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As [[Richard Feynman]] put it (Ref. 6): | ||
<blockquote> | <blockquote> | ||
...[water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses... are the worst possible example... they have all the complications that waves can have | |||
</blockquote> | </blockquote> | ||
====Defining the problem==== | ====Defining the problem==== | ||
Three contributions to the energy are involved: the [[surface tension]], gravity, and | |||
Three contributions to the energy are involved: the [[surface tension]], gravity, and hydrodynamics. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above. | |||
The new contribution involves the [[kinetic energy]] of the fluid: | The new contribution involves the [[kinetic energy]] of the fluid: | ||
:<math>T= \frac{\rho}{2} \iint dx\, dy\, \int_{-\infty}^h dz v^2,</math> | :<math>T= \frac{\rho}{2} \iint dx\, dy\, \int_{-\infty}^h dz v^2,</math> | ||
where <math>v</math> is the module of the velocity field <math>\vec{v}</math>. | where <math>v</math> is the module of the velocity field <math>\vec{v}</math>. | ||
(Again, we are neglecting the flow of the gas above for simplicity.) | (Again, we are neglecting the flow of the gas above for simplicity.) | ||
====Wave solutions==== | ====Wave solutions==== | ||
Let us try separation of variables: | |||
:<math>h(x,y,t)=\eta(t)e^{i\vec{q}\vec{r}},</math> | :<math>h(x,y,t)=\eta(t)e^{i\vec{q}\vec{r}},</math> | ||
where <math>\vec{q}=(q_x,q_y)</math> is a two dimensional wave number vector, and <math>\vec{r}=(x,y)</math> the position. | |||
We may take <math>\vec{q}=(q_x,0)</math> without loss of generality. | |||
In this case, | |||
:<math>E_\mathrm{g} \propto \frac{\rho g}{2} \eta^2,</math> | |||
:<math>E_\mathrm{st} \propto \frac{\sigma}{2} q^2 \eta^2,</math> | |||
where a factor of <math>A</math> that will appear every | |||
<math>\int dx\, dy\ </math> integration is dropped for convenience. | |||
To tackle the kinetic energy, suppose the fluid is | |||
incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow|potential]]: <math>\vec{v}=\nabla\psi</math>, and <math>\psi</math> is a potential (scalar field) which | |||
must satisfy [[Laplace equation]] <math>\nabla^2\psi=0</math>. | |||
To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow|potential]]: <math>\vec{v}=\nabla\ | |||
If we try try separation of variables with the potential: | If we try try separation of variables with the potential: | ||
:<math>\psi(x,y,z,t)=\xi(t) f(z) e^{i\vec{q}\vec{r}},</math> | |||
:<math>\ | with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math> | ||
Laplace equation then requires on the later | |||
with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math> | <math>f''(z)=-q^2 f(z) .</math> | ||
Laplace | This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in | ||
oceanography). Therefore <math>f(z) = a \, \exp(|q| z) </math>, with some constant <math>a</math>. The less trivial | |||
condition is the matching between <math>\psi</math> and <math>h</math>: the potential field must correspond to a velocity | |||
field that is adjusted to the movement of the surface: <math>v_z (z=h) =\partial h/\partial t</math>. This means that | |||
This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in oceanography). Therefore | <math>\xi(t)=\eta(t)'</math>, and that <math>f'(z=h) = 1 </math>, so that <math>f(z) = \exp( -|q|(h-z))/|q| </math>. | ||
We may now find <math>v^2=|\nabla\psi|^2</math>, which is <math>2 \exp( -2 |q|(h-z)) (\partial h/\partial t)^2 </math>. Performing the <math>\int_{-\infty}^h</math> integration first we are left with | |||
:<math>T=\frac{\rho}{2|q|} \int dx\,dy\, \left( \frac{\partial h}{\partial t} \right)^2 \propto | |||
\frac{\rho}{2|q|} \left( \eta' \right)^2, | |||
with some constant <math>a</math>. The less trivial condition is the | |||
:<math>v_z (z=h) =\partial h/\partial t</math>. | |||
This | |||
so that | |||
We may now find | |||
<math>v^2=|\nabla\ | |||
:<math>T= \frac{\rho | |||
\frac{\rho}{q} \ | |||
</math> | </math> | ||
where we have dropped a factor of <math>A/4</math> in the last step. | |||
The problem is thus specified by just a potential energy involving the square of <math>\eta(t)</math> and a kinetic energy involving the square of its time derivative: a regular [[Harmonic spring approximation|harmonic oscillator]]. Its equation of motion will be | |||
:<math>\frac{\rho}{|q|} \eta'' + (\rho g+ \sigma q^2) \eta=0,</math> | |||
whose oscillatory solution is | |||
:<math>\omega^2=g k+\frac{\sigma}{\rho}k^3,</math> | |||
the same dispersion as above if <math>\rho'</math> is neglected. | the same dispersion as above if <math>\rho'</math> is neglected. | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)] | #[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)] | ||
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#[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)] | #[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)] | ||
#R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153 | #R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153 | ||
==External links== | ==External links== | ||
*[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in | *[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in wikipedia] | ||
*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in | *[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in wikipedia] | ||
[[Category: Classical thermodynamics ]] | [[Category: Classical thermodynamics ]] |