Physical Picture for Diffusion of Polymers• Low Molecular Weight (M < Me) chains shown moving past one another.

Rouse chains,unentangled

• High Molecular weight (M > Me)• Entanglements in a polymer melt (a short portion of one chain is outlined in bold).• Lateral chain motion is severely restricted by the presence of neighboring chains.

Reptating chains,entangled

• Portion of an effective constraining tube defined by entanglements ⊗ about a

given chain

X

Y

Z

Figure by MIT OCW.

Figure by MIT OCW.

Figure by MIT OCW.

Rouse Chain

• A flexible string comprised of connected Brownian particles that interact with a featureless viscous medium.

• Number of repeat units is less than the entanglement limit, small N, where N < Ne

• Viscosity depends on monomeric friction factor ξM and x2η ~ ξ ~ ξM x2 ~ M

• Diffusivity depends on friction factor via Einstein equation: D = kT/ξ ~ kT/M

(Self) Diffusivity Regimes

• Small N or M – Rouse Regime – (follows directly from Einstein definition of D)

D ~ N-1

• Large N or M – Reptation Regime – (see derivation in next section)

D ~ N-2

(Two more scaling laws !)

Reptating Chains• Key idea: Consider a Rouse chain moving inside a tube

defined by the surrounding topological constraints.

• The number of repeat units in the chain is much larger than the entanglement limit, i.e. large N, where N > Ne

η0 ~ M3.4

• Basics<x2> = D t Fundamental equation of Brownian notion

Einstein relation for diffusion of Brownian particle with a friction factor ξ in host medium

F = ξ v Frictional force for particle moving with velocity v in host medium

D =kTξ

In this regime,zero shear rate viscosity

Reptation (P. de Gennes, S. Edwards, 1971)

1) Identify set of topological constraints to chain motion.

2) Construct a “tube” of diameter atube about the contour of the “test chain” which meanders through these constraints. (Edwards, 1967).

Ne is the average number of segments between entanglements. There are Z pieces of the tube such that the tube contour length L is the same as that of the diffusing chain Nb = L

Za2 = Nb2 (tube itself is a random walk; treat entangled chain as physicist’s universal chain)

3) Test chain moves by continuous creation/destruction of its tube. As the chain moves, one end comes out of the tube and a new part of the tube is created. The end of the tube from which the chain has withdrawn is considered destroyed since it no longer has any influence on the mobility of the test chain.

reptating stringinside tube

Figure by MIT OCW.

Reptation cont’d4) The diffusion coefficient of the Rouse-like chain within the tube (not entangled)

is given by:

where N is the number of statistical segments of the chain and ξM is the friction factor per statistical segment.

5) As a result of the local wiggling of the chain, the tube migrates. The time τ for the chain to completely escape its “old” tube of total length L is approximately:

where L is the length of the tube which is also the contour length of the chain, L = N b. τ is called the “longest relaxation time”, since it characterizes the motion of the entire chain. The relaxation time for the chain to exit its old tube thus scales with N3 since Nb = L.

τ ~ N3

Recall viscosity scales with flow time, so we predict that for an entangled melt η ~ M3

Dtube =kT

NξM

τ =L2

Dtube

Reptation cont’d6) Diffusion coefficient for entangled chains: Center of mass motion

of the chain defines a reptation diffusion coefficient, Drept. The distance diffused for a time t is given by

<x2> = Drept t

For t = τ, the chain center of mass can be estimated to have moved a distance of approximately the radius of gyration.

Rg2 = Drept τ

Therefore Drept scales with chain length as

Note Drept = DselfExperimentally, for well entangled polymer melts, the self-diffusion

coefficient is found to scale as 1/(molecular weight)2 in accord with the reptation model.

Drept ≈Nτ

≈1

N 2

Polymer-Polymer DiffusionEntropic Effects - consider diffusion of chains with different

molecular weights: 2 scenarios:1. “Fixed obstacles”: diffusion of short chain in

matrix of large, entangled chains

2. “Constraint release”: diffusion of long chain in matrix of short chains

Enthalpic Effects

Self diffusion (same species)

χ = 0 No effect

Mutual diffusion (different species)

χ > 0 Slows or stops inter-diffusion

χ < 0 Thermodynamic acceleration of inter-diffusion of species in addition to Brownian motion

Ass’t Prof ELTDeGennes’Prediction:PVC/PCL

Enthalpic Influence on Diffusion cont’d

(Mutual Diffusion)

• Recall F-H free energy of mixing per lattice site:ΔGM

NokT=

φ1

x1

lnφ1 +φ2

x2

lnφ2 + φ1φ2χ12

• In the case of the mutual interdiffusion of two (miscible)high MW polymers with x1 = x2 >> 1, the free energy of mixing for a negative χ is dominated by the enthalpic term.

For chains comprised of N monomers,

ΔGM ~ Nχ12 φ1φ2

• This enthalpic effect influences the scaling dependence of the diffusion coefficients by increasing the diffusion scaling lawby a factor of N for compatible, negative chi blends.

(self) Diffusion and Viscosity Scaling Laws

Property N < Ne N > Ne

ηD

NN-1

N3

N-2

Where Ne is the number of statistical segments between entanglements. Note Me = NeMo, where Mo is the molecular weight per monomer.

Rubber ElasticityAssumptions:1. Gaussian subchains between permanent crosslinks2. Temperature is above Tg3. Flexible chains with relatively low backbone bond

rotation potentials4. Deformation occurs by conformational changes only5. No relaxation (permanent network is fixed on time scale

of experiment)6. Affine deformation

(microscopic deformation : macroscopic deformation)7. No crystallization at large strains8. No change in volume (density) with deformation

Ideal Rubber Elasticity cont’d

crosslink

Gaussian subchain (n segments, segment step size l)

3D Gaussian Subchain Network

Mx = AverageMolecular weightbetween Xlinks

Relaxed subchains

Relaxed vs Extended Subchain Conformations

Relaxed

Extended 2/12r

2/120r

FF

Entropic Elastic Force

State 1

l0

l0

l0

ly

lz

lx

FF

State 2

F = −T ∂S

∂l

⎛ ⎝ ⎜

⎞ ⎠ ⎟

T,P

State 1: RelaxedF = 0Ω1

x,y,z

2/120r

State 2: ExtendedF > 0Ω2

x’,y’,z’

2/12r

Extension Ratio, αi

xx xα=' y'= α y y zz zα='

zyx ααα ,,

Assumption #8: the rubber is incompressibleduring deformation

⇒

are the 3 orthogonal extension ratios which depend on the type of loading geometry.

ΔV = 01=zyx ααα

0llx

x =α0l

lyy =α

0llz

z =α

Assumption #6: Affine Deformation

Macrocoordinates

Local coordinates

Entropic Elastic Force cont’d

The entropic elastic force is derived by considering the changein the conformational entropy of the extended vs. the relaxed state

⎟⎟⎠

⎞⎜⎜⎝

⎛ −α+−α+−α−=

ΩΩ

=Δ 2

2z

22y

22x

2

1

2

nl2)]1(z)1(y)1(x[3

explnklnkS

1

212 ln

ΩΩ

Δ kSSS ==−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅= 2

20

1 23expnl

rconstΩ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++−⋅= 2

222222

2 2

)(3exp

nl

zyxconst zyx ααα

Ω

r2

It’s all about entropy…again

20

222 rzyx =++

Entropic Elastic Force cont’d

20

222 rzyx =++

20

2 rnl =

and (for the relaxed subchain)

∴ ΔS(α i ) = − k2

α x2 +α y

2 +α z2 − 3( ) per sub-chain

(general relationship)

depends on details of initialsample shape and applied loads and this in turn controls

)( iS αΔ

3

20222

rzyx ===

ΔFi = −T ∂ΔS(α i )∂ li

⎛

⎝ ⎜

⎞

⎠ ⎟

T,P

n steps of size l in subchain

ΔFi

Example: Uniaxial Deformation

xz

y

Poisson contraction in lateral directions since αxαyαz = 1x

zyxx ααααα 1, ===

With Axi-symmetric sample cross-section

xll →000

,ldld

ll x

xx

x == αα→xDeform along ,

Rewriting explicitly in terms of )( iS αΔ xα

⎟⎟⎠

⎞⎜⎜⎝

⎛−

α+

α+α−=αΔ 311

2k)(S

xx

2xi

Fx = −T ∂∂lx

ΔS(α x )( ) T ,P = kT2

∂∂lx

α x

2 + 2α x

− 3⎛

⎝ ⎜

⎞

⎠ ⎟

Uniaxial Deformation cont’d

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎟⎠

⎞⎜⎜⎝

⎛−= 2

02

0

1222 x

xx

xx lkT

lkTF

αα

αα

At small extensions, the stress behavior is Hookean (Fx ~ const αx)

⎟⎟⎠

⎞⎜⎜⎝

⎛

α−α

ρ=ασ 2

xx

x

Axxx

1M

kTN)(

Stress-Extension Relationship

xlinksNVz

== eunit volumper subchains of #VlA =00

for z subchains in volume V⎟⎟⎠

⎞⎜⎜⎝

⎛−== 2

000

1)( totalx

xxx

xx lAzkT

AF

ααασ

Mx =ρ

Nx /NA

Usually the entropic stress of an elastomeric network is written in terms of Mxwhere Mx = avg. molecular weight of subchain between x-links

uniaxial

mass /volmolesofcrosslinks /volume

=

Young’s Modulus of a Rubber

Notice that Young’s Modulus of a rubber is :1. Directly proportional to temperature2. Indirectly proportional to Mx

• Can measure modulus of crosslinked rubber to derive Mx

• In an analogous fashion, the entanglement network of a melt, givesrise to entropic restoring elastic force provided the time scale of the measurement is sufficiently short so the chains do not slip out of their entanglements. In this case, we can measure modulus of non-crosslinked rubber melt to derive Me !

⎟⎟⎠

⎞⎜⎜⎝

⎛+=≡

→ 31

21limxxx

xx

MRT

ddE

x αρ

ασ

α

E = 3ρRT/Mx

Stress-Uniaxial Extension Ratio Behavior of Elastomers

Uniaxialtensile

σ(α)

Image removed due to copyright restrictions.

Please see Fig. 5 in Treloar, L. R. G. “Stress-Strain Data for Vulcanised Rubber Under Various Types of Deformation.” Transactions of the Faraday Society 40 (1944): 59-70

compressive

Small deformations

Theoretical

5?

4

-2St

ress

/ M

N m

3

2

1

01 2 3 4 5 6 7 8

σ(α)

High tensile elongationsFigures by MIT OCW.