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Where, volume fraction of polymer i=A,B including segments
Flory Huggins interaction parameter. It is related to solubility
parameters as:
where, volume of one polymer segment
solubility parameter
Flory Huggins parameter depend upon considering interactions between pairs of
molecules/monomers/units/groups
Multiscale modeling of macromolecules
Thermodynamical Approach: Flory-Huggins Theory
The Flory Huggins theory is useful to predict:
the stability of the systems polymer/solvent at low temperature,
the interaction between polymers,
the general features of the phase behavior and swelling of networks
Dissipative Particle Dynamics (DPD): a promising approach
Extensions
Extensions
UniversitÚ de Reims
CHAMPAGNE ARDENNE
INRA UMR FARE
Fractionnement des Agro Ressources
et Emballages
CPCB Moulin de la Housse, BP 1039,
F 51687 REIMS cedex 2
Mesoscopic modelling
(I/II)
Examples
Examples
The system contains discrete beads.
Each bead represents the collective
degrees of freedom of many (o(10)
o(1000)) atoms. For polymers, beads are
linked by harmonic springs.
The beads interact through soft, short
ranged pair potentials, which allows for a
long dynamic time step and rapid
simulation.
According to a random matrix frame, Flory Huggins theory expresses the free energy of mixing
for the mixture of macromolecule systems including flexible chains of different sizes.
ln
ln
m
A
B
A
B
FH
A
B
A
B
G
RT
N
N
=
+
+
m
G
i
i
N
FH
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
B
B
A
A
B
B
B
B
B
B
A
A
A
A
A
A
B
B
(
)
2
seg
FH
A
B
V
RT
=
-
i
seg
V
Flory
Flory
-
-
Huggins theory
Huggins theory
Intrinsic interface width
for two immiscible
polymers is given in terms of both, statistical
segment length and Flory Huggins
interaction parameter, as .
6
FH
a
w
=
w
a
Schematic representation of
dilute (1), semi-
dilute (2) and concentrated (3) polymer
solutions.
(1)
(2)
(3)
Simulation of the coordination
number: Many clusters are
generated to determine
how many
solvent molecules that can be
packed around a monomer.
Simulation of the interaction
energy: avoiding close contacts
between cyclohexanal
and
styrene head and tail and
calculating the interaction
energy of each pair.
Morphology of
copolymer blends
with
varying ratios of head to tail group
size.
Examples
Examples
Bridging the gap between scales
Bridging the gap between scales
( )
;
R
ij
R
ij
ij
ij
F
r
e
DPD
DPD describe the
interactions between N
particles according to 3
forces (
conservative=C,
dissipative=D and
random=R
):
C
D
R
ij
ij
ij
ij
F
F
F
F
=
+
+
;
ij
C
ij
i
F
r
=-
( )
;
D
ij
D
ij
ij
ij
ij
F
r
e v
e
-
Torsion constraints
Disparate bead sizes
Non spherical shapes
Extended ensembles:
Grand Canonical, Gibbs
Non homogeneous densities
local properties
local properties
depend on chemistry
depend on chemistry
global properties
global properties
universal
universal
properties
properties
persistence length
gyration radius
End to end distance
intermingling chains
intermingling chains
e.g. Semi crystalline
oriented PP
(1.5 Ám)
Globally kT dominates attraction:
Thermal energy is sufficient to
overcome a net long range
attraction between chain segments.
Bond angles and rotations
Persistence is unchanged with
temperature
Locally attraction
overcomes kT (thermal
fluctuation)
Bond length~1 ┼
Persistence
length~5 ┼
Bond Stretch potential = Stiff spring with rest length
replace with rigid rod constraint
.
removes the highest frequency components from the dynamics.
Multiple Scale Dynamics:
slow
and
fast
variables or
slow
and
fast
forces
Voter dynamics: transition state theory, reaction paths,
kinetic Monte Carlo, Onsager Machlup action
vector position,
distance between i particle and j particle,
gradient of a two particle interaction potential,
unit vector between particles i and j,
relative velocity of two particles,
Gaussian random number,
and weight functions vanishing
( dependent),
ij
i
r
ij
r
ij
e
ij
v
ij
D
R
i
r