C 10 Lecture Notes
Freely jointed chain, a “phantom” chain n links of same length, ak
Joints permit completely free rotation

Mean squared end to end length = nkak2
Contour (stretched out) length
Lc=nkak

Persistence length aq=ak/2 What is the size of the blob?

For an ideal Kuhn chain
RG2 = RL2/6 (approx.)
RG: Characteristic radius of blob

3. Non-crystalline Polymer (Physical States of Matter)
3.1 Glass transition temperature
Specific
volume
a) Tg occur in all materials where crystallinity doesn’t get in the way
b) Because it is not an equilibrium phase, glass transition is not a thermodynamic transition

Melt/Rubber

Glass

Equilibrium line

T

0
Specific
volume

Specific volume 1. Cool slowly
2. Heat quickly

Tg depends on time scale of observation

Slow cool
Fast heat
Anneal

0

T

0

T

3.2 Polymeric states
Molecular
weight

Increase molecular weight to infinity
(chemically linked)
→ All RUBBER in this region

Tg
Glass

Rubber

Viscous melt

T

1 GPa
Glass

log E

Rubber
1 MPa

~ 10 MPa
Viscous creep

Temperature

Spring: purely elastic, σ = Eε
Dashpot: purely viscous, σ = ηε

stiff spring model elasticity of glass (Eg) dashpot controls short relaxation time processes
(frees upon reaching Tg)

Eg-Er

Er

η1(T)

η2(T)

weak spring model elasticity of glass (Er) dashpot for longest relaxation time processes (frees for rubber-melt transition)

η1 and η2 solid → glass: Eg η2 solid, η1 free → rubber: Er

viscosity, η = σ/ε

4.1 Behavior of spring/dashpot models

σ

ε

Maxwell element Time

tot

=

E

+

− CON. EQN

σ σ0 at time t = τ, σ = σ0/e

0

t

σ ε = σ0/E

t
Voigt element
Add stresses,

σtot = Eε + ηε − CON. EQN ③

Constant ε, ε = 0

σ = Eε

Constant σ, σ = 0

4.2 How realistic?
Assumptions:
(i) Viscosity Newtonian η ≠ f (ε)
Not very good assumption at high strain rate
(ii) Only two relaxation processes

τ1

τ2

LHS controls Tg
Relaxation of molecular chain segments
RHS longest τ
Unraveling of entangled chain

N (τ)
Relaxation Time Spectrum

τ
Note: N (τ): number of elements with τ between τ and τ + d τ, like ‘density of state”

5.1 Modulus of glass
Eg ~ E of van der Waals solid (held together by VdW forces), approx. 1 GPa
Einorganic glass ~ 65 GPa → combinations of ionic bonding and Si – Si
Emetal : steel ~ 210 GPa Al ~ 70 GPa
Epolymer chain pulled out ~ 250 GPa

5.2 Viscosity of η1 dashpot
In region of Tg,
(a) Time-temperature superposition (Fig 5.1)
(b) Superimposed curve, made by shifting data by log aT (Fig 5.2 b)
(c) Plot of shift factor aT vs. temperature (Fig 5.2 c)

Williams-Landel-Ferry (WLF) found empirical equation to describe (c)

Ts ~ reference temperature, works well if Ts = Tg
With

C1 = 17.4 , C2 = 51.6 K

Implication: log aT (i.e. η1) turns to ∞ when T = Tg – C2 → (Tg – 51.6) K
∴Creep of polymer glass turns to zero at temperature (Tg – 51.6) K or lower

Justification of WLF in terms of free volume
Specific
volume

Occupied volume

0

T

Second Input: Doolittle Equation

5.3 Elasticity of Rubber Spring

5.4 Viscosity of

dashpot

Rubber ⇄ viscous melt?
Viscous flow ~ relative motion of centers of masses of molecules

A. Reptation Model
Molecules effectively confined in tube by entangling neighbours
Constraints physical cross-linked points
Reptation - Molecule can only escape lengthwise
Parameters:

Need two other equations:

B
Above critical molecular weight for