V. CONCLUSIONS
In this work, we have reexamined the computation of the
Casimir force between two parallel walls delimitating a fluctuating fluid
membrane that is immersed in some liquid. This force is caused by the thermal
fluctuations of the membrane. We have studied the problem from both static and
dynamic point of view.
We were first interested in the time variation of the
roughening, L1 (t), starting with a membrane that is inially
in a flat state, at a certain temperature. Of course, this length grows with
time, and we found that : L1 (t) t°?
(91 = 1/4), provided that the hydrodynamic
interactions are ignored. For real systems, however, these interactions are
important, and we have shown that the roughness increases more rapidly
as : ?L1 (t)
t?°? (91 = 1/3). The
dynamic process is then stopped at a final r (or rh) that
represents the required time over which the biomembrane reaches its final
equilibrium state. The final time behaves as : r D4 (or
rh ? D3), with D the film thickness.
Now, assume that the system is explored at scales of the order
of the wavelength q-1, where q =
(4ð/À) sin (9/2) is the wave vector modulus, with
À the wavelength of the incident radiation and 9
the
scattering-angle. In these conditions, the relaxation rate,
r (q), scales with q as : r-1
(q) q1/°? = q4
( h (q) q1/?°? = q3)
or r-1 . Physically speaking, the relaxation
rate characterizes the local growth of the
height fluctuations.
Afterwards, the question was addressed to the computation of
the Casimir force, II. At equilibrium, using an appropriate field theory, we
found that this force decays with separation D as : II
D-3, with a known amplitude scaling as
#c-1, where #c is the membrane bending rigidity
constant. Such a force is then very small in comparison with the Coulombian
one. In addition, this force disappears when the
K. El Hasnaoui et al. African Journal Of Mathematical Physics
Volume 8(2010)101-114
temperature of the medium is sufficiently lowered.
The dynamic Casimir force, II (t), was computed using
a non-dissipative Langevin equation (with noise), solved by the time
height-field. We have shown that : II (t)
t°f
(èf = 2è1 =
1/2). When the hydrodynamic interactions effects are important, we
found that the dynamic force increases more rapidly as :
( )
IIh (t) '--
t?°f
?èf =
2?è1 = 2/3 .
Notice that we have ignored some details such as the role of
inclusions (proteins, cholesterol, glycolipids, other macromolecules) and
chemical mismatch on the force expression. It is well-established that these
details simply lead to an additive renormalization of the bending rigidity
constant. Indeed, we write êeffective =
ê + äê, where ê is the
bending rigidity constant of the membrane free from inclusions, and
äê is the contribution of the incorporated entities.
Generally, the shift äê is a function of the inclusion
concentration and compositions of species of different chemical nature (various
phospholipids forming the bilayer). Hence, to take into account the presence of
inclusions and chemical mismatch, it would be sufficient to replace ê
by êeffective, in the above
established relations.
As last word, we emphasize that the results derived in this
paper may be extended to bilayer surfactants, although the two systems are not
of the same structure and composition. One of the differences is the magnitude
order of the bending rigidity constant.
APPENDIX
To show formula (17), we start from the partition function that
we rewrite on the following form
|
J I j
-H [h]
Z = Dh exp =
kBT
|
D/2
f
-D/2
|
dz (z) . (5.0)
|
113
Also, it is easy to see that the membrane mean-roughness is given
by
L2 1 =
| 
