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
2
Consider a single polymeric fractal of arbitrary topology :
- Linear polymers :
- Branched polymers :
- Polymer networks, ...
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 upper critical dimension
1DLinear polymers :
Ideal branched polymers : 34D
Membranes : 2D ucd
4ucd
8ucd
D
Ddd Fuc
2
42 0
Consider a biomembrane of arbitrary topology. A point of this membrane can be de scribed by two local coordinates (u1, u2)
12111 1RCRC
- 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:
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)
The polymer is confined if only if its three dimensional Flory radius RF3 is must larger than the diameter H,
3FRH
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 :
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
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
)()()()( lVlVlVlV vdWéleh
Consider a lamellar phase formed by two parallel bilayer membranes, the total interaction energy per unit area is the following sum
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
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
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
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
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
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(
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.
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
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
.
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