Part III. Polymer Dynamics – molecular...
Transcript of Part III. Polymer Dynamics – molecular...
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Part III. Polymer Dynamics – molecular models
I. Unentangled polymer dynamics I.1 Diffusion of a small colloidal particle
Source: Polymer physics – Rubinstein,Colby
I.2 Diffusion of an unentangled polymer chain
II. Entangled polymer dynamics II.1. Introduction to Tube models II.2. « Equilibrium state » in a polymer melt or in a concentrated solution:
II.3. Relaxation processes in linear chains
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I. Unentangled polymer dynamics I.1 Diffusion of a small colloidal particle:
Diffusion coefficient
Simple diffusive motion: Brownian motion
Friction coefficient:
A constant force applied to the particle leads to a constant velocity:
Einstein relationship: Friction coefficient
Stokes law: (Sphere of radius R in a Newtonian liquid with a viscosity η)
Stokes-Einstein relation: Determination of
random walk
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I.2.1. Rouse model:
N beads of length b - Each bead has its own friction ζ - Total friction:
- Rouse time:
t < τRouse: viscoelastic modes t > τRouse: diffusive motion
- Kuhn monomer relaxation time:
- Stokes law:
I.2 Diffusion of an unentangled polymer chain:
R
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Rouse relaxation modes:
The longest relaxation time:
For relaxing a chain section of N/p monomers:
(p = 1,2,3,…,N)
N relaxation modes
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- At time t = τp, number of unrelaxed modes = p.
- Each unrelaxed mode contributes energy of order kT to the stress relaxation modulus:
(density of sections with N/p monomers:
- Mode index at time t = τp:
, for τ0 < t < τR
Volume fraction
Approximation: , for t > τ0
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, for τ0 < t < τR
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Viscosity of the Rouse model:
Or:
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Exact solution of the Rouse model:
with
Each mode relaxes as a Maxwell element
The Rouse relaxation time is the half of the end-to-end correlation time.
Or: The Rouse relaxation time is the end-to-end correlation time.
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Free energy U of a Gaussian chain (see Part I):
Forces equilibrium after a displacement of ΔL:
After the disorientation “spring” force from the free energy
Rouse time determined from the free energy:
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Example: Frequency sweep test:
For 1/τR < ω < 1/τ0
For ω < 1/τR
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Rouse model: also valid for unentangled chains in polymer melts:
For M < Mc : τrel ∝ M2
For M > Mc: τrel ∝ M3.2 3.4
Relaxation times
M
τrel
Mc
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Viscous resistance from the solvent: the particle must drag some of the surrounding solvent with it.
Hydrodynamic interaction: long-range force acting on solvent (and other particles) that arises from motion of one particle.
When one bead moves: interaction force on the other beads (Rouse: only through the springs)
The chain moves as a solid object:
Zimm time: (In a theta solvent)
I.2.2. Zimm model:
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- τZimm has a weaker M-dependence than τRouse.
- τZimm < τRouse in dilute solution
- real case: often a combination of both
Rouse time versus Zimm time:
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Slope: 5/9 to 2/3 (theta condition)
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Rheological Regimes: (Graessley, 1980)
Dilute solutions: Zimm model – H.I. – no entanglement
Unentangled chains (Non diluted): Rouse model – no H.I. – no entanglement
Entangled chains: Reptation model – no H.I. – entanglement
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entanglement
The test chain is entangled with its neighbouring
polymers
Molecules cannot cross each other
II. Entangled polymer dynamics
II.1. Introduction to Tube models
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Doi & Edwards (1967), de Gennes (1971)
The tube model
entanglement
Me: molecular weight between two entanglements
The molecular environment of a chain is represented by a tube in which the chain is confined
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Tube model and entanglements:
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G(t) Glassy region
Pseudo equilibrium state
t
Relaxation processes
Concentrated solutions or polymer melts:
Pseudo equilibrium state:
The chains are not in their stable shape. The other chains prevent their relaxation. The tube picture becomes active.
Me: Mw of the longest sub-chain relaxed by a Rouse process
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II.2. « Equilibrium state » in a polymer melt or in a concentrated solution:
l
Khun segment
Leq.
1
Ne
2Ne
3Ne
4ne
a = tube diam.
tube
Z = segments number =4 Segment between two entanglements
Each segment between two entanglements is relaxed by the Rouse process
The primitive path of the chain is defined by its segments between the entanglements. Its length = Leq. Coarse grained model
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Primitive path
- Simpler description of the chain
- Coarse grained model: subchains of mass Me are considered as a segment of length l
- relaxation time of subchain Me: Rouse relaxation
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l
Leq.
1
ne
a
tube
Leq = length of the primitive path
Z = number of segments between two ent.
l = length of a segment between two ent.
a = tube diameter
b = length of a Khun segment
Ne = number of Khun segments between two entanglements
N = total number of Khun segments in the chain
Equilibrium state in a polymer melt: parameters
Gaussian chain in solution: the end-to-end distance R02 = Nb2
Gaussian chain in a melt or concentrated solution:
l2 ≈Neb2
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Constitutive equation of a polymer melt For a Rouse chain (unentangled chain):
(Gaussian)
(free energy)
Entropic material:
S = k lnP (free energy)
In a fixed direction:
U = -TS
R0
U(R0)
0
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Constitutive equation of a polymer melt
At equilibrium (U min.): Nb2 ≈ a.Leq
U(Lchain)
0 Lchain Leq
For a polymer melt:
Additional boundary conditions: ψ(R) = 0 on the tube surface
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l
Leq.
1
Ne
a
tube
Leq = length of the primitive path
Z = number of segments between two ent.
l = length of a segment between two ent.
a = tube diameter
b = length of a Khun segment
Ne = number of Khun segments between two entanglements
N = total number of Khun segments in the chain
Equilibrium state in a polymer melt: parameters
Gaussian chain in solution: R02 = Nb2
Gaussian chain in a melt or concentrated solution: l2 ≈Neb2 Nb2 ≈ a.Leq
l ≈ a
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Material parameters in the tube model
(x 4/5)
The plateau modulus:
Defined at the “equilibrium state” of the polymer
The molecular weight between two entanglements: Me
The relaxation time of a segment between two entanglements:
Very simplified coarse-grained model
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Not anymore memory of the initial deformation
Unrelaxed fraction of the polymer melt
Plateau modulus
Relaxation of the polymer (as Reptation)
Φ = 0 Φ = 1
G(t) Glassy region Rouse process
t
Plateau modulus : tube effect
II.3. Relaxation processes in linear chains
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II.3.1. Main relaxation process in linear chain: Reptation
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As the primitive chain diffuses out of the tube, a new tube is continuously being formed, starting from the ends.
This new tube portion is randomly oriented even if the starting tube is not.
Part of the initial tube
1.
2.
3.
4.
1-D curvilinear diffusion
+ Movie
Reptation
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Reptation time:
Monomeric friction
Total friction of the chain
Curvilinear diffusion coefficient:
De: along Leq
<R02>
D
In solution:
Reptation
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Reptation PBD, 130 kg/mol.
M (Kuhn segment): 105 g/mol N=1240 Kuhn segments
τ0 = 0.3 ns. τe = 0.1 µs. Me=1900g/mol. Ne=18. M/Me= 68 entanglements
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Reptation: Doi and Edwards model
- First passage problem
- diffusion of a chain along the tube axis
p(s,t): survival probability of a initial tube segment localized at a distance s.
0 Leq
s
Variables separability
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x
p(x,t)
time
x x
0 1 0
Doi and Edwards model
Using the usual coordinate system
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Doi and Edwards model
Relaxation modulus:
Related viscosity:
for M < Mc for M > Mc in the real case
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Discrepancy between theory and experiments: inclusion of the contour length fluctuations process
Zero-shear viscosity vs. log (M)
Slope of 3.4
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Initial D&E model (with CR)
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Equilibrium length Leq
Entropical force: Topological force:
Equilibrium length of an arm:
Leq = the most probable length
real length
This part of the initial tube is lost
II.3.2. Contour length fluctuations
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x=1
No Reptation Only fluctuations (see the next lesson)
x=0
Fixed point Symmetric star:
II.3.2. Contour length fluctuations
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+ movie
II.3.2. Contour length fluctuations
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Linear chains: Doi and Edwards model including the contour length fluctuations (CLF):
- the environment of the molecule is considered as fixed.
Rouse-like motion of the chain ends:
For t < τR
Displacement along the curvilinear tube:
Displacement of a monomer in the 3D space:
Rem/ Before t = τRouse, the end-chain does not know that it belongs to the whole molecule
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Leq,0
L(t)
Partial relaxation of the stress:
For τe < t < τR At time t = τR:
With µ close to 1
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In the same way:
Rescale of the reptation time in order to account for the fluctuations process
Log(M)
Slope: 3.4
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D&E model with contour length fluctuations (with CR)
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Linear chains: model of des Cloizeaux:
De
Based on a time-dependent coefficient diffusion:
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Des Cloizeaux model (with CR)