SSecond econd IInternational nternational WWorkshop on orkshop on SSoft oft CCondensed ondensed MMatter atter PPhysics and hysics and BBiological iological SSystems ystems

28 - 30 April 2010, Fez, Morocco28 - 30 April 2010, Fez, Morocco

K. El Hasnaoui, M.Benhamou, H.Kaidi, M.ChahidK. El Hasnaoui, M.Benhamou, H.Kaidi, M.Chahid

Laboratoire de Physique des Polymères et Phénomènes CritiquesLaboratoire de Physique des Polymères et Phénomènes CritiquesBen M’sik Sciences Faculty, Casablanca, MoroccoBen M’sik Sciences Faculty, Casablanca, Morocco

Consider a single polymeric fractal of arbitrary Consider a single polymeric fractal of arbitrary topology :topology :(D:Dimension spectrale)(D:Dimension spectrale)

- - Linear polymers Linear polymers :

- Branched polymers: Branched polymers:

- Polymer networks, ...Polymer networks, ...

We assume that the considered polymer is trapped in We assume that the considered polymer is trapped in a good solvent. Its Flory radius scales as :a good solvent. Its Flory radius scales as :

The dimension fractal can be obtained from standar The dimension fractal can be obtained from standar Flory theory : Flory theory :

Fd

For linear polymers For linear polymers : :

Branched polymers (animals)Branched polymers (animals) :

3d

TheThe upper critical dimensionupper critical dimension

For linear polymers For linear polymers : :

Branched polymers Branched polymers :

For linear polymers For linear polymers : :

Branched polymers Branched polymers :

20 Fd

40 Fd

Confinement condition :

Extended Flory theory :Extended Flory theory :

The parallel extension depends on polymer and tubular The parallel extension depends on polymer and tubular vesicle characteristics, through M and parameters (vesicle characteristics, through M and parameters (··, p), p)..

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

The above behavior is valid as long as the tube diameter The above behavior is valid as long as the tube diameter is greater than the typical value : is greater than the typical value :

The confined polymer is one-dimensional. The confined polymer is one-dimensional.

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

Confinement condition : Confinement condition :

Backgrounds :Backgrounds :

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

Mean-separation behavior :Mean-separation behavior : Lipowsky and Lipowsky and Leibler.Leibler.

Here, ψ is a critical exponent whose value is :Here, ψ is a critical exponent whose value is :

Extended Flory theory :Extended Flory theory :

This behavior combines two critical phenomena : This behavior combines two critical phenomena : long mass limit, unbinding transition.long mass limit, unbinding transition.

The parallel radius becomes more and more smaller The parallel radius becomes more and more smaller as the unbinding transition is reached. as the unbinding transition is reached.

The confined polymer is two dimensional. The confined polymer is two dimensional.

Sépartranapps

tran TTR '

//

Two objectives :Two objectives :

Conformational study of a polymeric fractal inside a Conformational study of a polymeric fractal inside a tubular vesicle.tubular vesicle.

Conformational study between two parallel membranes Conformational study between two parallel membranes forming an equilibrium lamellar phase.forming an equilibrium lamellar phase.

Il sera interésant de completer cette etude par une Il sera interésant de completer cette etude par une Investigation de la dynamique des fractales Investigation de la dynamique des fractales polymériques confinées polymériques confinées

That’s all for today!

Thanks for your interest!

13

SSecond econd IInternational nternational WWorkshop on orkshop on SSoft oft CCondensed ondensed MMatter atter

PPhysics and hysics and BBiological iological SSystems ystems

28 - 30 April 2010, Fez, 28 - 30 April 2010, Fez, MoroccoMorocco

14

Hydration energy :Hydration energy :

J/m 2.0~ 2hhh PA With

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

nm 3.0h

nm 54~

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

The bilayer thickness

ll

WlV

ll

WlV

W

W

1

12) (

²) (

2

4

It originates from the membranes undulations :

kB : Boltzmann constant

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

CH : Helfrich constant CH ~0.23

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 Ph, λh, δ and ·.

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

ideal radius

:

:

:

2

//

//HR

R

Parallel extension of the polymer.

Excluded volume parameter (for good solvents).

Volume occupied by the fractal.

Minimizing the above free energies with respect to gives :

//R

4/1

4

)2(

// ~

H

aaMR D

D

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

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

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

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

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

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

The The upper critical dimensionupper critical dimension

1DLinear polymers :

Ideal branched polymers : 34D

Membranes : 2D ucd

4ucd

8ucd

D

Dduc

2

4

12111 1

RCRC

- Mean –Curvature

- Gaussian Curvature

212

1CCC

21CCK

moyen

:

:

:

:

:

:

:

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

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:

022222 20

20 CKCCCCCCP

0222 2 CCCP

0 ,1

2 KR

C

0

/12With 02

0

2

C

CteRCC

01

43

RP RH

PH

R

P2

42

1

4

31

3

dVPdAdAKdACCFVSSGS 2

0222

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)

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 ~

3FRH

TTH C ~

D

D

C* aMTT 5

)2(

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.

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

3/2

3

)2(

// ~

H

aaMR D

D

//R

H :is the equilibrium diameter

9/233

)2(

// ~

PaaMR D

D

3/14

2

PH

3FRH

With

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.