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La mécanique statistique des membranes biologiques confinées

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par Khalid EL HASNAOUI
Faculté des sciences Ben M'Sik Casablanca - Thèse de doctorat  2011
  

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IV. DYNAMIC CASIMIR FORCE

To study the dynamic phenomena, the main physical quantity to consider is the time height-field, h (r, t), where r = (x, y) E R2 denotes the position vector and t the time. The latter represents the time observation of the system before it reaches its final equilibrium state. We recall that the time height function h (r, t) solves a non-dissipative Langevin equation (with noise) [28]

?h(r,t) ?t

= - äH0 [h]

äh (r, t) + í (r, t) , (4.0)

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

where > 0 is a kinetic coefficient. The latter has the dimension : [] = L4 0T -1

0 , where L0 is some length

and T0 the time unit. Here, í (r, t) is a Gaussian random force with mean zero and variance

?í (r,t)í (r',t')? = 2ä2 (r - r')ä (t - t') , (4.0)

and H0 is the static Hamiltonian (divided by kBT), defined in Eq. (13).

The bare time correlation function, whose Fourier transform is the dynamic structure factor, is defined by the expectation mean-value over noise í

G(r - r',t - t') = ?h(r,t)h(r',t')?? - ?h(r,t)?? ?h(r',t')?? , t > t' . (4.0)

The dynamic equation (28) shows that the time height function h is a functional of noise í, and we write : h = h [í]. Instead of solving the Langevin equation for h [í] and then averaging over the noise distribution P [í], the correlation and response functions can be directly computed by means of a suitable field-theory, of action [28 - 31]

A [h, h] = J dt J d2r { h?th + ?häâh° h?h } , (4.0)

so that, for an arbitrary observable, O [æ], one has 111 JJJ

?

?O?? = [] O [ö [í]] P [í] =

f DhD?hOe-A[h,?h]

(4.0)

f DhD?he-A[h,h]

where h (r, t) is an auxiliary field, coupled to an external field h (r, t). The correlation and response functions can be computed replacing the static Hamiltonian H0 appearing in Eq. (13), by a new one : H0 [h, J] = H0 [h] - f d2rJh. Consequently, for a given observable O, we have

ä ?O?J äJ (r,t)

J=0

? )

= ?h (r, t) O . (4.0)

108

[ ]

The notation ? . ?J means the average taken with respect to the action A h, ?h, J associated with the

Hamiltonian H0 [h, J]. In view of the structure of equality (33), h is called response field. Now, if O = h, we get the response of the order parameter field to the external perturbation J

R (r - r',t - t') =

?

ä ?h(r',t')?J = (ti (r,t)h(r',t')). (4.0)

äJ (r, t) J=0

The causality implies that the response function vanishes for t < t'. In fact, this function can be related to the time-dependent (connected) correlation function using the fluctuation-dissipation theorem, according to which

? )

?h (r, t) h (r', t') = -è (t - t') ?t ?h (r, t) h (r', t')?c . (4.0)

The above important formula shows that the time correlation function C (r - r', t - t') = ?ö (r, t) ö (r', t')?c may be determined by the knowledge of the response function. In particular, we show that

(4.0)

t

L2? (t) = (h2 (r, t))c = -2 f dt' Ch (r, t') h (r, t')) .

8

The limit t ? -8 gives the natural value L2? (-8) = 0, since, as assumed, the initial state corresponds to a completely flat interface.

Consider now a membrane at temperature T that is initially in a flat state away from thermal equilibrium. At a later time t, the membrane possesses a certain roughness, L? (t). Of course, the latter is time-dependent, and we are interested in how it increases in time.

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

109

We point out that the thermal fluctuations give rise to some roughness that is characterized by the appearance of anisotropic humps. Therefore, a segment of linear size L effectuates excursions of size [32]

L? = BL? . (4.0)

Such a relation defines the roughness exponent æ. Notice that L is of the order of the in-plane correlation length, L?. From relation (20), we deduce the exponent æ and the amplitude B. Their respective values

are : æ = 1 and B es., (kBT/ê)1/2.

In order to determine the growth of roughness L? in time, the key is to consider the excess free energy (per unit area) due to the confinement, ?F. Such an excess is related to the fact that the confining membrane suffers a loss of entropy. Formula (27) tells us how ?F must decay with separation. The result reads [32]

?F es., kBT/L2max es., kBT (B/L?)2/? , (4.0)

where Lmax represents the wavelength above which all shape fluctuations are not accessible by the confined membrane. The repulsive fluctuation-induced interaction leads to the disjoining pressure

??F

_
Ð DL?

es., L-(1+2/?). (4.0)

?

In addition, a care analysis of the Langevin equation (28) shows that

?L???F ?tes., - ?L?

= × Ð es., L-(1+2/?) . (4.0)

?

(4.0)

2 + 2æ 4

We emphasize that this scaling form agrees with Monte Carlo predictions [32, 33]. Solving this first-order differential equation yields [34]

1

= .

L? (t) es., ?lt?l , è? = æ

This implies the following scaling form for the linear size

L (t) es., ?iit?ii , è? =

1= 1

2 + 2æ 4 . (4.0)

Let us comment about the obtained result (39).

Firstly, as it should be, the roughness increases with time (the exponent è? is positive definite). In addition, the exponent è? is universal, independently on the membrane bending rigidity constant ê. Secondly, we note that, in Eq. (39), we have ignored some non-universal amplitude that scales as ê-1/4. This means that the time roughness is significant only for those biomembranes of small bending rigidity constant.

Fourthly, this time roughness can be interpreted as the perpendicular size of holes and valleys at time t. Fifthly, the roughness increases until a fine time, ô. The latter can be interpreted as the time over which the system reaches its final equilibrium state. This characteristic time then scales as

ô es., -1L1/?l

? , (4.0)

where we have ignored some non-universal amplitude that scales as ê. Here, L? es., D is the final roughness. Explicitly, we have

ô es., -1D4 . (4.0)

As it should be, the final time increases with increasing film thickness D. On the other hand, we can rewrite the behavior (39) as

L? (t) L? (ô)

= I T I?l. (4.0)

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

This equality means that the roughness ratio, as a function of the reduced time, is universal.

Now, to compute the dynamic Casimir force, we start from a formula analog to that defined in Eq. (24), that is

11(t) 1

=

kBT Ó

? ln Z? 1 ?u ? ln Z?

?D = ?u , (4.0)

Ó ?D

with the new partition function

fZ? = DhD?he-A[h,?h] . (4.0)

11(t) 1

kBT 2

?u

A simple algebra taking into account the basic relation (35a) gives

?D L2 ? (t) , (4.0)

which is very similar to the static relation defined in Eq. (25), but with a time-dependent membrane roughness, L? (t).

Combining formulae (43) and (46) leads to the desired expression for the time Casimir force (per unit area)

11(t) 11(ô)

( t \?'

= ,

ô

(4.0)

where 11 (ô) is the final static Casimir force, relation (25). The force exponent, èf, is such that

æ 1

èf = 2è? =

(4.0)

1 + æ 2

= .

110

The induced force then grows with time as t1/2 until it reaches its final value 11 (ô). At fixed time and separation D, the force amplitude depends, of course, on ê, and decreases in this parameter according to ê-3/2. Also, we note that the above equality means that the force ratio as a function of the reduced time is universal.

In Fig. 2, we draw the reduced dynamic Casimir force, 11 (t) /11 (ô), upon the renormalized time t/ô.

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

FIG. 2. Reduced dynamic Casimir force, 11 (t) /11 (r), upon the renormalized time t/r.

Finally, consider again a membrane which is initially flat but is now coupled to overdamped surface waves. This real situation corresponds to a confined membrane subject to hydrodynamic interactions. The roughness now grows as [35]

?L? (t) t???

,

?è? = æ

1 + 2æ

1

= 3 . (4.0)

111

Therefore, the roughness increases with time more rapidly than that relative to biomembranes free from hydrodynamic interactions.

In this case, the dynamic Casimir force is such that

11h (t) 11(ôh)

( t )

= ôh

??f

,

(4.0)

where 11 (ôh) is the final static Casimir force, relation (25). The new force exponent is

- èf = 2-è? = 2æ

1 + 2æ

2

= 3 . (4.0)

There, ôh D3 accounts for the new time-scale over which the confined membrane reaches its final

equilibrium state. Therefore, the dynamic Casimir force decays with time as t2/3, that is more rapidly than that where the hydrodynamic interactions are ignored, which scales rather as t1/2. As we said before, this drastic change can be attributed to the overdamped surface waves that develop larger and larger humps.

We depict, in Fig. 3, the variation of the reduced dynamic force (with hydrodynamic interactions), 11h (t) /11 (ôh), upon the renormalized time t/ôh.

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

112

FIG. 3. Reduced dynamic Casimir force (with hydrodynamic interactions), 11h (t) /11 (r), upon the renormalized time t/rh.

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