Film in Electrical Field
From Multiflow Wiki
<usebib>Eleiht</usebib>
We have a liquid film of thickness h(x) = const in a capacitor of gap width d, voltage U and electrical field 
in the film and 
in air. x is parallel to the condensor plates and z vertical.
The liquid has the dielectric constant εr, viscosity μ and surface tension γ; and the permittivity of vacuum is ε0.
For a flat film the electrical field is everywhere orthogonal to the plates 
.
At the film surface 
where 
is the normal vector of the surface, i.e. for a flat film we have εrEf = Ea.
The electrical potential is given by 
, i.e.
i.e.
The energy of the electrical field at a point in x is
![W_c=\int_0^d\frac{\epsilon}{2}\,E^2 dz\,=\frac{1}{2}\left[\epsilon_0\epsilon_r E_f^2 h + \epsilon_0 \epsilon_r^2 E_f^2 (d-h)\right]](/mwiki/images/math/5/3/e/53efc663f63aaf191ec0b962b4085825.png)
i.e.
is the energy the capacitor stores per area for a given voltage U and layer thickness h.
However, it is not the energy of the overall circuit because when changing the capacity of the capacitor (by changing the thickness of the dielectric) to keep the voltage constant charges have to be moved from the battery to the capacitor.
To move a charge Q against a voltage V changes the energy of the battery by
Wb = − QU.
The necessary charge is Q = CU where C is the capacity that can be read of from
Wc above using the relation 
.
The overall energy is W = Wc + Wb = − Wc that corresponds to the local free energy f(h) that enters the film thickness equation:
Writing the film thickness equation as
![\partial_t h = \partial_x \left[Q\,\partial_x\,\frac{\delta F}{\delta h}
\right]](/mwiki/images/math/1/7/b/17b715e19aff76dcc1b8c786cfc78545.png)
with
being the energy of the liquid film
gives
![\partial_t h = -\partial_x \left[Q\,\partial_x\,[\gamma \partial_{xx} h - \partial_h f(h)]\right].](/mwiki/images/math/a/3/9/a39c4e957f46ac2cc2aff6fa23c22e6c.png)
This is consistent with equations used in Lin01,Lin02,MPBT05,ThKn06 that all have the same sign for the 'electrical pressure'.
Note that the argument is different for a capacitor separated from the battery, i.e. with constant charge Q.
