1. SSecondecond IInternationalnternational WWorkshop onorkshop on SSoftoft CCondensedondensed MMatteratter PPhysics andhysics and
BBiologicaliological SSystemsystems
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 Critiques
Ben M’sik Sciences Faculty, Casablanca, MoroccoBen M’sik Sciences Faculty, Casablanca, Morocco
2. Consider a single polymeric fractal of arbitraryConsider a single polymeric fractal of arbitrary
topology :topology :(D:Dimension spectrale)(D:Dimension spectrale)
-- Linear polymersLinear polymers :
- Branched polymers:Branched polymers:
- Polymer networks, ...Polymer networks, ...
3. We assume that the considered polymer is trapped inWe 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 standarThe dimension fractal can be obtained from standar
Flory theory :Flory theory :
Fd
For linear polymersFor linear polymers ::
Branched polymers (animals)Branched polymers (animals) :
3=d
4. TheThe upper critical dimensionupper critical dimension
For linear polymersFor linear polymers ::
Branched polymersBranched polymers :
For linear polymersFor linear polymers ::
Branched polymersBranched polymers :
20
=Fd
40
=Fd
6. Extended Flory theory :Extended Flory theory :
The parallel extension depends on polymer and tubularThe 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 importantAt 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 diameterThe 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.
7. The aim is the conformation study of a polymer ofThe aim is the conformation study of a polymer of
arbitrary topology confined to two parallel fluctuatingarbitrary topology confined to two parallel fluctuating
fluid membranes :fluid membranes :
Confinement condition :Confinement condition :
8. Backgrounds :Backgrounds :
Consider a lamellar phase formed by two parallelConsider a lamellar phase formed by two parallel
bilayer membranes, their total interaction energybilayer membranes, their total interaction energy
(per unit area) is the following sum :(per unit area) is the following sum :
9. Mean-separation behavior :Mean-separation behavior : Lipowsky andLipowsky and
Leibler.Leibler.
Here, ψ is a critical exponent whose value is :Here, ψ is a critical exponent whose value is :
10. 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 smallerThe 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.
Sépartran
apps
tran TTR −
↓
→⇔ '
//
11. Two objectives :Two objectives :
Conformational study of a polymeric fractal inside aConformational study of a polymeric fractal inside a
tubular vesicle.tubular vesicle.
Conformational study between two parallel membranesConformational study between two parallel membranes
forming an equilibrium lamellar phase.forming an equilibrium lamellar phase.
Il sera interésant de completer cette etude par uneIl sera interésant de completer cette etude par une
Investigation de la dynamique des fractalesInvestigation de la dynamique des fractales
polymériques confinéespolymériques confinées
14. 14
Hydration energy :Hydration energy :
J/m2.0~ 2
hhh PA λ=With
: is the hydration length.hλ
: is the hydration pressure.hP ( )Pa4.10Pa4.10 97
<< hP
nm3.0≅hλ
15. nm54~ −δ
The Hamaker constant is in the range W ~10-22
- 10-21
J
The bilayer thickness
δ
π
δ
δ
<<−≈
>>≈
l
l
W
lV
l
l
W
lV
W
W
1
12
)(
²
)(
2
4
16. 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
17. 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 ·.
18. Standard Flory de Gennes theory based on the following free
energy :
ideal radius
:
:
:
2
//
//
HR
R
υ
Parallel extension of the polymer.
Excluded volume parameter (for good solvents).
Volume occupied by the fractal.
19. Minimizing the above free energies with respect to
gives :
//R
4/1
4
)2(
// ~
+
H
a
aMR D
D
20. 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
d
Dd à
F
//R
21.
22. 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.
Fd
F aMR
1
~
The mean square distance between two
monomers i and j is twice as large as Rg
23. d
F
F
B R
N
R
R
Tk
F 2
2
0
2
υ+=
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 .
2
0
2
R
R
Tk
F F
B
el
=
The first term is an elastic Hookean spring contribution
d
FB R
N
Tk
F 2
int
υ=
24. The dimension fractal gets himself while minimizing the
free energy of Flory with report to , we arrive to:
Fd
2
2
+
+
=
D
d
DdF
FR
2
5
)3(
+
=
D
D
dF
For dimension 3,we have
For linear polymers :
Ideal branched ones (animals) :
3
5)3( =Fd
2)3( =Fd
25. D
D
dF
−
=
2
20
20
=Fd1=DLinear polymers :
Ideal branched polymers : 3
4=D 40
=Fd
Membranes : 2=D ∞=0
Fd
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:
0
1
0 ~ Fd
aMR
0
Fd
26. 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
)/(2
2
0
0
<<→
→
−−
−−
F
F
ddd
ddd
d
F
Na
Na
R
N
υ
υυ
0
2 Fdd ≥
30. :
:
:
:
:
:
:
0C
p
V
dA
G
γ
κ
κ
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
VSS
G
S ∫∫∫∫ ++++= γγκ
κ 2
022
2
Curvature : la courbure
31. With the surface Laplace Bertlami operator :
( )ij
ij
gg
g
det
:
=
is the metric tensor on the surface
∂
∂
∂
∂
=∇ j
ij
i
u
gg
ug
12
( )( ) ( ) 022222 2
0
2
0 =∇+−+++− CKCCCCCCP κκγ
The general shape equation has been derived via variational
calculus to be:
32. ( )( ) ( ) 022222 2
0
2
0 =∇+−+++− CKCCCCCCP κκγ
( )( ) 0222 2
=+− CCCP κγ
0,
1
2 =−= K
R
C
( )
0
/12With02
0
2
=
=−==∇
C
CteRCCκ
0
1
4 3
=
−+
R
P κ ( )RH
P
H
R
P
2
4
2
1
4
3
1
3
=
=⇔
=
κ
κ
( ) dVPdAdAKdACCF
VSS
G
S ∫∫∫∫ ++++= γγκ
κ 2
022
2
33.
34. 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 =−= K
R
C
For very long tubes, the uniform solution to equation (a) is:
3/1
4
2
=
p
H
κ
where H is the equilibrium diameter.
(b)
35. 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 <<
( )ψ
TTH C −~
ψD
D
C
*
aMTT 5
)2( +−
−=
36. The standard Flory- de Gennes theory based on the
following free energy
2
//
2
2
0
2
//
HR
M
R
R
Tk
F
B
υ+=
ideal radius0
1
0 ~ Fd
aMR
:
:
:
2
//
//
HR
R
υ
the polymer parallel extension to the tube axis
is the excluded volume parameter (for good solvents)
represents the volume occupied by the fractal.
37. Minimizing the above free energies with respect to
yields the desired results :
3/2
3
)2(
// ~
+
H
a
aMR D
D
//R
H :is the equilibrium diameter
9/2
33
)2(
// ~
−
−
+
P
aaMR D
D
κ
3/1
4
2
=
P
H
κ
3FRH <<
With
38. Standard Flory de Gennes theory based on the following free
energy :
HR
M
R
R
Tk
F
B
2
//
2
2
0
2
//
υ+=
ideal radius
0
1
0 ~ Fd
aMR
:
:
:
2
//
//
HR
R
υ
Parallel extension of the polymer.
Excluded volume parameter (for good solvents).
Volume occupied by the fractal.
