1. Dynamic-Mechanical Analysis of Materials (Polymers)
Big Assist: Ioan I. Negulescu

2. 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.”)

3. 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:

4. Generally, measurements for visco
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(it)
E*2 = E’ E”2
It’s solved in complex domain, but only the real parts are used.

5. 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 + 22]
E’ = EM 2 2 / [1 + 22] = 0 cos/0
E” = EM   / [1 + 22] = 0 sin/0
 = M/EM = Maxwellian relax. t

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

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

8. 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.

9. Definition of elastic and viscous materials under shear.

10. 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

11. The out-of-phase component, G" represents the viscous component of G
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"/2f

12. 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',

13. A transition T is detected by a spike in G” or tan(). The trans
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

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

15. 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.

16. DMA of recyclable HDPE. Dependence of tan  on 
DMA of recyclable HDPE. Dependence of tan  on . The  transition is at 62C, the  transition at -117C.

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

18. Data obtained at 2C/min showing Tg ~ -40C (max
Data obtained at 2C/min showing Tg ~ -40C (max. tan) and a false transition at 15.5C due to the nonlinear increase of T vs. t.

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

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