La mécanique statistique des membranes biologiques confinées

D/2

f dzz² (z)

. (5.0)

-D/2

D/2

_f

-D/2

dz (z)

The restricted partition function is

? { _j

_-H [h]

(z) = Dhä [z - h (x0, y0)] exp . (5.0)

kBT

Here, H [h] is the original Hamiltonian defined in Eq. (3). Of course, this definition is independent on the chosen point (x0, y0), because of the translation symmetry along the parallel directions to plates. Notice that the above function is not singular, whatever the value of the perpendicular distance.

Since we are interested in the confinement-regime, that is when the separation D is much smaller than (z h << L0

the membrane mean-roughness L0 ), we can replace the function par its value at z = 0,

1 1

denoted 0. In this limit, Eq. (A.2) gives the desired result.

This ends the proof of the expected formula.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the »First International Workshop On Soft-Condensed Matter Physics and Biological Systems», 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

K. El Hasnaoui et al. African Journal Of Mathematical Physics Volume 8(2010)101-114

114

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Dans cette deuxième contribution originale, nous considérons une solution colloïdale au contact d'une biomembrane, qui est confinée dans une fente. L'on suppose que l'épaisseur de cette fente est beaucoup plus petite que la rugosité en volume, afin d'assurer le confinement de la membrane. Le but étant l'étude de la dynamique Brownienne de ces particules, sous la variation d'un paramètre adéquat, tel que la température, par exemple. L'objet de base est la densité locale des particules.

Nous déterminons exactement cette densité, qui est fonctionde la distance et du temps. L'outil pour cela est l'équation de Smolokowski

Article N°2

Dynamique Brownienne de colloïdes

au contact d'une biomembrane confinée.

101

African Journal Of Mathematical Physics Volume 8(2010)101-114

Casimir force in confined biomembranes

K. El Hasnaoui, Y. Madmoune, H. Kaidi,
M. Chahid, and M. Benhamou

Laboratoire de Physique des Polymères et Phénomènes Critiques

Facultédes Sciences Ben M'sik, P.O. 7955, Casablanca, Morocco
[e-mail]

abstract

We reexamine the computation of the Casimir force between two parallel interacting plates delimitating a liquid with an immersed biomembrane. We denote by D their separation, which is assumed to be much smaller than the bulk roughness, in order to ensure the membrane confinement. This repulsive force originates from the thermal undulations of the membrane. To this end, we first introduce a field theory, where the field is noting else but the height-function. The field model depends on two parameters, namely the membrane bending rigidity constant, k, and some elastic constant, u - D^-4. We first compute the static Casimir force (per unit area), 11, and find that the latter decays with separation D as : 11 - D^-3, with a known amplitude scaling as k^-1. Therefore, the force has significant values only for those biomembranes of small enough k. Second, we consider a biomembrane (at temperature T) that is initially in a flat state away from thermal equilibrium, and we are interested in how the dynamic force, 11 (t), grows in time. To do calculations, use is made of a non-dissipative Langevin equation (with noise) that is solved by the time height-field. We first show that the membrane roughness, L? (t), increases with time as : L? (t) - t¹/⁴ (t < r), with the final time r - D⁴ (required time over which the final equilibrium state is reached). Also, we find that the force increases in time according to : 11 (t) - t¹/² (t < r). The discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the surrounding liquid. In this case, we show that : L? (t) - t¹/³ (t < rh) and 11h (t) - t²/³ (t < rh), with the new final time rh - D³. Consequently, the hydrodynamic interactions lead to substantial changes of the dynamic properties of the confined membrane, because both roughness and induced force grow more rapidly. Finally, the study may be extended, in a straightforward way, to bilayer surfactants confined to the same geometry.

Key words: Biomembranes - Confinement - Casimir force - Dynamics.