Time-Temperature Superposition WHICH VARIABLES ARE TEMPERATURE DEPENDENT in the ROUSE and REPTATION MODELS?

Time-Temperature Superposition WHICH VARIABLES ARE TEMPERATURE DEPENDENT IN THE ROUSE AND REPTATION MODELS?

Rouse Relaxation Modulus:

N ρ(T )RT X 2 G(t) = e−tp /λR M p=1 Rouse Relaxation Time:

a2N 2ζ(T ) λ = R 6π2kT Reptation Relaxation Modulus:

N 8 ρ(T )RT 1 2 X −p t/λd G(t) = 2 2 e π Me p p odd Reptation Relaxation Time:

a2N 2ζ(T ) M λd = 2 π kT Me Both models have the same temperature dependence. This is hardly surprising, since reptation is just Rouse motion conﬁned in a tube.

Modulus Scale G ∼ ρ(T )T

ζ(T ) Time Scale λ ∼ ∼ λ T N

Since the friction factor is related to the shortest Rouse mode λN . a2ζ(T ) λ = N 6π2kT

1 Time-Temperature Superposition METHOD OF REDUCED VARIABLES

All relaxation modes scale in the same way with temperature.

λi(T ) = aT λi(T0) (2-116)

aT is a time scale shift factor T0 is a reference temperature (aT ≡ 1 at T = T0).

Gi(T0)T ρ Gi(T ) = (2-117) T0ρ0 Recall the Generalized Maxwell Model

N X G(t) = Gi exp(−t/λi) (2-25) i=1 with full temperature dependences of the Rouse (and Reptation) Model

N T ρ X G(t, T ) = G (T ) exp{−t/[λ (T )a ]} (2-118) T ρ i 0 i 0 T 0 0 i=1 We simplify this by deﬁning reduced variables that are independent of temperature. T ρ G (t) ≡ G(t, T ) 0 0 (2-119) r T ρ t tr ≡ (2-120) aT

N X Gr(tr) = Gi(T0) exp[−tr/λi(T0)] (2-118) i=1

Plot of Gr as a function of tr is a universal curve independent of temperature.

2 Time-Temperature Superposition EMPIRICAL MASTER CURVE

Figure 1: Reduced Variables Generates a Master Curve from the Stress Re- laxation Modulus Data for a Polymer Melt.

Can now estimate the relaxation modulus at any temperature in the range 192K< T < 350K, covering EIGHTEEN ORDERS OF MAGNITUDE IN TIME!

3 Time-Temperature Superposition TEMPERATURE DEPENDENCE OF VISCOSITY (P. 1)

Recall Z ∞ η0 = G(t)dt (2-19) 0 T ρ G (t) ≡ G(t, T ) 0 0 (2-119) r T ρ Substitute for G(t). T ρ Z ∞ η0(T ) = Gr(t)dt T0ρ0 0

Change integration variable to tr ≡ t/aT , so dt = aT dtr. Z ∞ aT T ρ η0(T ) = Gr(tr)dtr (2-122) T0ρ0 0

Since Gr(tr) refers to the reference temperature T0 Z ∞ Z ∞ η0(T0) = G(t, T0)dt = Gr(tr)dtr (2-123) 0 0 we can rewrite (2-122)

aT T ρ η0(T ) = η0(T0) (2-125) T0ρ0

Solve for aT

η0(T )T0ρ0 aT = (2-126) η0(T0)T ρ Since viscosity is far more temperature dependent than T ρ

∼ η0(T ) aT = (2-127) η0(T0)

4 Time-Temperature Superposition TEMPERATURE DEPENDENCE OF VISCOSITY (P. 2) Arrhenius equation — Viscosity is thermally activated (ﬁrst applied to viscosity by Andrade)

E η ∼ exp a T > T + 100K 0 RT g

Ea is the activation energy for ﬂow η0(T ) Ea 1 1 = exp − T > Tg + 100K (2-128) η0(T0) R T To

Doolittle equation — Viscosity is determined by free volume vf B(v − vf ) η0 ∼ exp vf B(v − v ) ln η = ln A + f (10-11) vf

If one assumes free volume vf is proportional to temperature

vf = vf (Tg) + αf (T − Tg) (10-12) one gets the WLF (Williams-Landel-Ferry) EQUATION

(B/2.3fg)(T − Tg) log η0(T ) = log η0(Tg) − (10-13) (fg/αf ) + T − Tg

fg = vf (Tg)/ [vf (Tg) + vo(Tg)] is the fractional free volume at Tg

WLF EQUATION for an arbitrary reference temperature T0

0 −C1 (T − T0) log(aT ) = 0 (2-129) [C2 + T − T0] At high temperatures WLF approaches Arrhenius behavior with an activation energy

0 0 ∼ Ea = 2.303RC1 C2 = 4 kcal/mole

5 Time-Temperature Superposition SUMMARY

The modulus scale shift factor is really more complicated than T ρ G (t) ≡ G(t, T ) 0 0 (2-119) r T ρ because the entanglement molecular weight can depend weakly on temperature. Thus we generally write

G(t, T ) Gr(t) ≡ bT The Rouse Model has T ρ bT = T0ρ0

but most polymers do not obey this equation, so bT is treated as an empirical modulus scale shift (analogous to aT for the time scale shift, but bT is much smaller). t tr ≡ (2-120) aT t G(t, T ) = bT G ,T0 aT

0 0 G (ω, T ) = bT G (ωaT ,T0)

00 00 G (ω, T ) = bT G (ωaT ,T0)

t J a ,T0 J(t, T ) = T bT

6 Time-Temperature Superposition EXAMPLE

Figure 2: Time-Temperature Superposition for Storage Compliance of ◦ Poly(N-octyl methacrylate) with T0 = 100 C.

Time-Temperature Superposition COUNTER-EXAMPLE

Figure 3: Failure of Time-Temperature Superposition in the Glass Transition ◦ Zone of Polystyrene with T0 = 126.7 C.

Time-Temperature Superposition does not always work!!!