Dynamic-Mechanical Analysis of Materials (Polymers)

Big Assist: Ioan I. Negulescu

Viscoelasticity

According to rheology (the science of flow), viscous flow and elasticity are only two extreme forms of rheology. Other cases: entropic-elastic (or rubber-elastic), viscoelastic; crystalline plastic.

SINGLE MAXWELL ELEMENT(viscoelastic = “visco.”)

All real polymeric materials have viscoelasticity, viscosity and elasticity in varying amounts. When visco. is measured dynamically, there is a phase shift () between the force applied (stress) and the deformation (strain) in response.

The tensile stress and the deformation (strain) for a Maxwellian material:

dt

d

dt

d

E MM

1

Generally, measurements for visco. materials are represented as a complex modulus E* to capture both viscous and elastic behavior:

E* = E’ + iE”

* = 0 exp(i (t + )) ; * = 0 exp(it)

E*2 = E’2 + E”2

It’s solved in complex domain, but only the real parts are used.

In dynamic mechanical analysis (DMA, aka oscillatory shear or viscometry), a sinusoidal or applied.

For visco. materials, lags behind . E.G., solution for a single Maxwell element:

0 = EM 0 / [1 + 22]

E’ = EM 2 2 / [1 + 22] = 0 cos/0

E” = EM / [1 + 22] = 0 sin/0

= M/EM = Maxwellian relax. t

Schematic of stress as a function of t with dynamic (sinusoidal) loading (strain).

COMPLEX MODULUS:

I E' I = I E* I cos I E" I = I E* I sin

LOSS MODULUSSTORAGE ( Elastic) MODULUS

I E* I = Peak Stress / Peak Strain

E*=E’ + iE”

t

0

o

o

2 /

/

STRESS STRAIN

Parallel-plate geometry for shearing of viscous materials (DSR instrument).

The “E”s (Young’s moduli) can all be replaced with “G”s (rigidity or shear moduli), when appropriate. Therefore:

G* = G’ + iG"where the shearing stress is and the deformation (strain) is . Theory SAME.

Definition of elastic and viscous materials under shear.

In analyzing polymeric materials:

G* = (0)/(0), ~ total stiffness.

In-phase component of IG*I = shear storage modulus G‘ ~ elastic portion of input energy

= G*cos

The out-of-phase component, G" represents the viscous component of G*, the loss of useful mechanical energy as heat

= G*sin = loss modulus

The complex dynamic shear viscosity * is G*/, while the dynamic viscosity is

= G"/ or = G"/2f

For purely elastic materials, the phase angle = 0, for purely viscous materials, 90.

The tan() is an important parameter for describing the viscoelastic properties; it is the ratio of the loss to storage moduli:

tan = G"/ G',

A transition T is detected by a spike in G” or tan(). The trans. T shifts as changes. This phenomenon is based on the time-temperature superposition principle, as in the WLF eq. (aT).

The trans. T as (characteristic t ↓)

E.G., for single Maxwell element:

tan = ( )-1 and W for a full period (2/) is:

W = 02 E” = work

Dynamic mechanical analysis of a viscous polymer solution (Lyocell). Dependence of tan on - due to complex formation.

• DMA very sensitive to T.

• Secondary transitions, observed with difficulty by DSC or DTA, are clear in DMA.

• Any thermal transition in polymers will generate a peak for tan, E“, G“

• But the peak maxima for G" (or E") and tan do not occur at the same T, and the simple Maxwellian formulas seldom followed.

DMA of recyclable HDPE. Dependence of tan on . The transition is at 62C, the transition at -117C.

Dependence of G", G' and tan on for HDPE at 180C. More elastic at high !

Data obtained at 2C/min showing Tg ~ -40C (max. tan) and a false transition at 15.5C due to the nonlinear

increase of T vs. t.

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.2

0.4

0.6

0.8

1.0

Continental CarbonSample A-97058

15.5oC

tan

E"E'

Te

mp

era

ture

, o C

80

40

0.0

-40

-80

tan E' E" Temperature

tan

Time, min

DMA of low cryst. poly(lactic acid): Dependence of tan upon T and for 1st heating run

30 40 50 60 70 80 90 100

0.0

0.2

0.4

0.6

0.8

Crystallization

Tg

Crystallization

Tg

PLALC

75oC

69oC

66oC

62oC

tan

Temperature, oC

1.0 Hz 50 Hz

DMA of Low Cryst. Poly(lactic acid). Dependence of E’ on thermal history. Bottom line – high info. content, little work.

40 50 60 70 80 90 1000

1G

2G

(Stiffening)Crystallization

10 HzPLALC

2nd heating

1st heating

TCR

GlassTransitionT

g

E' @ 10 Hz (1st h)

E' @ 10 Hz (2nd h)

Sto

rag

e M

od

ulu

s (

Pa

)

Temperature, oC