31
UNIVERSITE HASSAN II UNIVERSITE HASSAN II MOHAMMEDIA - CASABLANCA MOHAMMEDIA - CASABLANCA - FACULTE DES SCIENCES BEN M’SIK FACULTE DES SCIENCES BEN M’SIK Laboratoire de Physique des Polymères et Phénomènes Laboratoire de Physique des Polymères et Phénomènes Critiques Critiques K .El hasnaoui,H.Kaidi, M.Benhamou and M.Chahid K .El hasnaoui,H.Kaidi, M.Benhamou and M.Chahid Second International Workshop on Soft Condensed Matter Physics and Biological Systems 28-30 April 2010

Communication fès nno modifié

Embed Size (px)

Citation preview

Page 1: Communication fès nno modifié

UNIVERSITE HASSAN IIUNIVERSITE HASSAN IIMOHAMMEDIA - CASABLANCAMOHAMMEDIA - CASABLANCA

--

FACULTE DES SCIENCES BEN M’SIKFACULTE DES SCIENCES BEN M’SIK

Laboratoire de Physique des Polymères et Phénomènes Laboratoire de Physique des Polymères et Phénomènes CritiquesCritiques

K .El hasnaoui,H.Kaidi, M.Benhamou and M.ChahidK .El hasnaoui,H.Kaidi, M.Benhamou and M.Chahid

Second International Workshop on Soft Condensed Matter Physics and Biological Systems

28-30 April 2010

Page 2: Communication fès nno modifié

2

Page 3: Communication fès nno modifié

Consider a single polymeric fractal of arbitrary topology :

- Linear polymers :

- Branched polymers :

- Polymer networks, ...

Page 4: Communication fès nno modifié

We assume that the considered polymer is trapped in a good solvent . We denote by

its gyration (or Flory) radius.

Hausdor fractal dimension.ff

:

:

:

a

M

dF

Molecular weight (total mass) of the considered polymer.

Monomer size.

FdF aMR

1

~

The mean square distance between two monomers i and j is twice as large as Rg

Page 5: Communication fès nno modifié

dF

F

B R

N

R

R

Tk

F 2

20

2

For a polymer of radius R, Flory wrote the free energy in the form:

The second terms is a middle interaction energy.

0R is the ideal radius .

20

2

R

R

Tk

F F

B

el

The first term is an elastic Hookean spring contribution

dFB R

N

Tk

F 2int

Page 6: Communication fès nno modifié

The dimension fractal gets himself while minimizing the free energy of Flory with report to , we arrive to:

Fd

2

2

D

dDdF

FR

2

5)3(

D

DdF

For dimension 3,we have

For linear polymers :

Ideal branched ones (animals) :

35)3( Fd

2)3( Fd

Page 7: Communication fès nno modifié

D

Dd F

2

20

20 Fd1DLinear polymers :

Ideal branched polymers : 34D 40 Fd

Membranes : 2D 0Fd

When the system is ideal(Without excuded volume forces),its radius is such that , stands for Gaussian fractal dimension, it is related to the spectral dimension D by:

01

0 ~ FdaMR0Fd

Page 8: Communication fès nno modifié

The upper critical dimension is obtained by using Ginzburg criterion, this criteria consists in considering the part interaction of the energy free of Flory, in which we replace

0RRF

1ideal )/(2

)/(22

0

0

F

F

ddd

ddddF

Na

NaR

N

02 Fdd

Page 9: Communication fès nno modifié

The upper critical dimension

1DLinear polymers :

Ideal branched polymers : 34D

Membranes : 2D ucd

4ucd

8ucd

D

Ddd Fuc

2

42 0

Page 10: Communication fès nno modifié

Consider a biomembrane of arbitrary topology. A point of this membrane can be de scribed by two local coordinates (u1, u2)

Page 11: Communication fès nno modifié

12111 1RCRC

Page 12: Communication fès nno modifié

- Mean –Curvature

- Gaussian Curvature

212

1CCC

21CCK

moyen

Page 13: Communication fès nno modifié

:

:

:

:

:

:

:

0C

p

V

dA

G

Area element Volume enclosed within the lipid bilayer

Bending rigidity constant

Gaussian curvature

Surface tension Pressure difference between the outer and inner sides of the vesicles

Spontaneous curvature

Vesicles also have constraints on surface and volume. According to Helfrich’s theory, the free energy

of a vesicle is written as : dVPdAdAKdACCF

VSSGS 2

0222

Curvature : la courbure

Page 14: Communication fès nno modifié

With the surface Laplace Bertlami operator :

ij

ij

gg

g

det

:

is the metric tensor on the surface

j

iji u

ggug

12

022222 20

20 CKCCCCCCP

The general shape equation has been derived via variational calculus to be:

Page 15: Communication fès nno modifié
Page 16: Communication fès nno modifié

For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have :

R is the radius of the cylinder

0 ,1

2 KR

C

For very long tubes, the uniform solution to equation (a) is:

3/14

2

pH

where H is the equilibrium diameter.

(b)

Page 17: Communication fès nno modifié

The polymer is confined if only if its three dimensional Flory radius RF3 is must larger than the diameter H,

3FRH

Page 18: Communication fès nno modifié

The standard Flory- de Gennes theory based on the following free energy

2//

2

20

2//

HR

M

R

R

Tk

F

B

ideal radius 01

0 ~ FdaMR

:

:

:

2//

//

HR

R

the polymer parallel extension to the tube axis

is the excluded volume parameter (for good solvents)

represents the volume occupied by the fractal.

Page 19: Communication fès nno modifié

Minimizing the above free energies with respect to yields the desired results :

4/1

3

)2(

// ~

H

aaMR D

D

//R

H :is the equilibrium diameter

9/233

)2(

// ~

PaaMR D

D

3/14

2

PH

3FRH

With

Page 20: Communication fès nno modifié

Firstly, notice that, in any case, naturally depends on polymer and tubular vesicle characteristics, through M and parameters (·, p), respectively.

Secondly, at fixed polymer mass M, the parallel extension is important for those tubular vesicles of small bending modulus ·..

Finally, the above behavior is valid as long as the parallel extension remains below the maximal extension of the polymer, that is we must have (maximal extension). This gives a minimal tube diameter

DaMR1

//

//R

à travers

//R

DD

aMH 2)1(

min ~

The confinement of the polymeric fractal implies that the tube diameter is in the interval

3min FRHH

Page 21: Communication fès nno modifié

)()()()( lVlVlVlV vdWéleh

Consider a lamellar phase formed by two parallel bilayer membranes, the total interaction energy per unit area is the following sum

Page 22: Communication fès nno modifié

22

J/m 2.0~ 2hhh PA

h

l

hh eAlV

)(

With

: is the hydration length. h : is the hydration pressure. hP Pa4.10 Pa4.10 97 hP

nm 3.0h

Page 23: Communication fès nno modifié

nm 54~

The Hamaker constant is in the range H ~10-22 - 10-21J

The bilayer thickness

)²2(

1

)²(

2

²

1

12)(

lll

HlUVdW

Page 24: Communication fès nno modifié

It originates from the membranes undulations :

2

2

)(

l

TkClV B

Hs

kB : Boltzmann constant

T : Absolute temperature·: Effective bending rigidity constant of the two membranes.

21

21

CH : Helfrich constant CH ~0.23

Page 25: Communication fès nno modifié

When the critical amplitude is approached from above, the mean separation between the two membranes diverges according to :

Here, ψ is a critical exponent whose value is :

The critical value Wc depends on the parameter of the problem, which are temperature T, and parameters Ah, λh, δ and ·.

)TTor W(W ,~~ -CCCC TTWWH

0.031.00 ψ

Page 26: Communication fès nno modifié

The aim is the conformation study of a polymer of arbitrary topology confined to two parallel fluctuating fluid membranes.

The necessary condition to have the confinement is such that :

3FRH

, when a fractal polymeric object is unconfined H

Page 27: Communication fès nno modifié

This condition implies that the polymer confinement is possible only when the temperature T is below some typical value :

We note that the polymer is confined only when its three dimensional gyration :

is much greater than the mean separation :

D

D

F aMR 5

)2(

3 ~

TTH C ~

3FRH

D

D

C* aMTT 5

)2(

Page 28: Communication fès nno modifié

Standard Flory de Gennes theory based on the following free energy :

HR

M

R

R

Tk

F

B2//

2

20

2//

ideal radius 0

1

0 ~ FdaMR

:

:

:

2

//

//HR

R

Parallel extension of the polymer.

Excluded volume parameter (for good solvents).

Volume occupied by the fractal.

Page 29: Communication fès nno modifié

Minimizing the above free energies with respect to gives :

//R

4/1

4

)2(

// ~

H

aaMR D

D

)T(T ,~~ *44

)2(

44

)2(

//

TTaMWWaMR CD

D

CD

D

)TTor W(W ,~~ -CCCC TTWWH

Page 30: Communication fès nno modifié

Firstly, the expression of the parallel extension combines two critical phenomena : long mass limit of the polymeric fractal, vicinity of the unbinding transition of the membranes.

Secondly, in this formula, naturally appears the fractal dimension (D + 2) /4D of a two dimensional polymeric fractal

Finally, the parallel radius becomes more and more smaller as the unbinding transition is reached. In other word, this radius is important only when the two adjacent membranes are strongly bound.

2

4

2

2 2d

D

D

D

dDd à

F

//R

Page 31: Communication fès nno modifié

.