Polymer Viscoelasticity & Rheology

Minshiya P. Assistant Professor

M E S Keveeyam College Valanchery

• Incompressible – the density is constant

• Irrotational – the flow is smooth, no turbulence

• Nonviscous – fluid has no internal friction ( η = 0)

• Steady flow – the velocity of the fluid at each point is constant in time.

Ideal or Newtonian fluids

• Under fixed and constant stress – strian or

deformation increases continuosly

• Strain is non recoverable (zero yield value)

• Strain, ε = f(σ, t)

Ideal or Newtonian fluids

Ideal or Newtonian fluids

• σs (T) α dv/dr • σs (T) = ηdv/dr = (1/φ) dv/dr ------- (1)

• η = 1/φ = σs/dv/dr --------(2)

η : coeff of viscosity or viscosity or internal friction of the liquidφ : fluidityeqn (1) & (2) : Newton’s law

Ideal or Newtonian fluids

• d ε /dt = dv/dr ----- (3) d ε /dt :Strain rate

• Strain rate : rate of change in strain (deformation) of a material with respect to time

• (1) σs = η d ε /dt = (1/φ) d ε /dt ------ (4)

• Shear stress α rate of shear strain

Non-Newtonian fluid

• A non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids.

• Viscosity dependent on shear rate or shear rate history

• Eg:- Many salt solutions, molten polymers & many commonly found substances such as ketchup, custard, toothpaste, starch suspensions, paint, blood, and shampoo

Newtonian V/s Non –Newtonian

Newtonian fluid

Relation between shear stress and shear rate is linear, passing through the origin

The constant of proportionality being the coefficient of viscosity

Non -Newtonian fluid

Relation between the shear stress and the shear rate is different

Can even be time-dependent.

Coefficient of viscosity cannot be defined.

The concept of viscosity is used to characterize the shear properties of a fluid

Inadequate to describe non-Newtonian fluids

Types of non-Newtonian behaviour

τy

Ideal or Elastic solids

Force needed to extend or compress a spring by some amount is proportional to that amount

is a constant characteristic of the spring, stiffness.

Hooke's law

Ideal or Elastic solids

• Under stress, deformed instantaneously and experiences an internal force that oppose the deformation and restore it to its original state if the external force (stress) is no longer applied

• Strain α Applied stress • F/A = k ∆L/ L0

• σ = Eε E : Youngs modulus

• Under a constant stress, an ideal elastic solid deform immediately to a fixed & constant level in eqm with applied stress and will not deform or strain further with time.

Ideal or Elastic solids

                                                                                      

Stress–strain curve

1. Ultimate strength2. Yield strength – corresponds to yield point3. Rupture4. Strain hardening region5. Necking regionA: Engineering stress (F/A0)B: True stress (F/A)

Also termed as tensile strength (TS) or ultimate strength : maximum stress that a material can withstand while being stretched or pulled before failing or breaking.

1. Ultimate tensile strength (UTS)

2. Yield (engineering)

• Yield strength or yield point : The stress at which a material begins to deform plastically

• Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed.

• Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible.

3. Fracture

* Separation of an object or material into two, or more, pieces under the action of stress

* The fracture of a solid almost always occurs due to the development of certain displacement discontinuity surfaces within the solid.

* Displacement develops in this case perpendicular to the surface of displacement, it is called a normal tensile crack or simply a crack

* Displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band, or dislocation.

4. Strain hardening

Also known as Work hardening (cold working)

Strengthening of a metal by plastic deformation.

This occurs because of dislocation movements and dislocation generation within the crystal structure of the material.

Most non-brittle metals (with high melting point) & several polymers can be strengthened

5. Necking

• Necking: a mode of tensile deformation • Relatively large amounts of strain localize

disproportionately in a small region of the material

• The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck"

A PE sample with a stable neck

Viscoelasticity & Rheology

• Pure Newtonian viscosity & pure elastic behaviour are ideal

• All motions of real bodies have flow and elastisity• Elastoviscous liquids or viscoelastic solids• Polymers are viscoelastic

Polymer viscoelasticity

• Inter-relationship among elasticity, flow & molecular motion

• Time dependent flow behaviour

• Creep & Stress relaxation

OVERVIEW

• POLYMERS TREATED AS SOLIDS Strength Stiffness Toughness

• POLYMERS TREATED AS FLUIDS Viscosity of polymer melts Elastic properties of polymer melts

• VISCOELASTIC PROPERTIES Creep Stress relaxation

• Stress Relaxation and Creep

– Chemical versus Physical Processes– Analysis with Springs and Dashpots– Relaxation and Retardation Times– Physical Aging of Glassy Polymers

Creep

3 kg

3 kg3 kg

3 kg

Time

Deformation under a constant load as a function of time

Creep or Creep strain may be either fully recoverable with time or involve permanant deformation with partial recovery

Stress Relaxation

Constant deformationexperiment

Stress decreases with time

Slowly drops to zero as the material undergo permanant set to the stretched length

3 kg4 kg5 kg6 kg

Time

Reasons for Stress Relaxation and Creep

1. Chain Scission:• Oxidation, Hydrolysis.

2. Bond Interchange:• Polyester, Polysiloxane• Polyamides, Polyethers

Reasons for Stress Relaxation and Creep (Contd.)

3. Segmental Relaxations:• Thirion Relaxation

Reasons for Stress Relaxation and Creep (Contd.)

• Viscous Flow: slippage of chains past one another, fast reptation

All physical reasons for stress relaxation and

creep are related to molecular motion (strain) induced by stress, therefore, must be related to conformational changes

Mechanical models of viscoelsatic materials

• Maxwell & Kelvin (Voigt) Element

• Using Springs (purely elastic) and Dashpots (purely viscous)

• For a spring σ = E ε (Hooke’s Law)

• For a dashpot σ = η dε/dt (Newton’s Law)

Springs Stretched instantaneously under stress & holding that stress indefinitely

Store energy and respond instantaneously.

Spring Dashpot

Under stress piston moves through the fluid

Rate α stress

No recovery on removing

Dissipate energy in the form of heat.

Maxwell & Kelvin (Voigt) Element

• Maxwell Model

Spring & Dashpot are in series Developed to explain the behavior of pitch

and tar Maxwell assumed that such materials can

undergo viscous flow and also respond elastically

Combine Hooke’s and Newton’s laws

E

η

Creep Experiment

Strains are additive , ie; ε = εelast + εvisc

ε(t) = σ/G + σt/η 1st term -- instantaneous elastic response 2nd term -- the viscous retarded response.

Both elements feel the same stress Residual stress at time t, σ = σ0 exp ( -t/λ)

λ : relaxation time , σ0 : initial stress

E

η

ε =

σ /E

Creep Experiment - Maxwell Model (contd)

• Kelvin (Voigt) Element

Spring & dashpot are in parallel

Combine Hooke’s and Newton’s laws

Both elements feel the same strain

Stress is additive , ie; σ = σelast + σvisc

σ = ε Ε + η dε/dt

Creep Experiment

Both spring & dashpot undergo concerted motion

Dashpot slowly responds to the stress & viscous flow decreases

asymptotically

Strain rate at Dashpot, d ε /dt = σ/ η

Spring bears all the stress – spring & dashpot stop deforming together-

creep stops

Overall stress is constant• ε(t) = σ0 [1 - exp(-t /η)]• This is a Creep Experiment.• The strain reaches a limiting value at long times.

Creep Experiment - Kelvin (Voigt) Element

Stress Relaxation Experiments

• Maxwell Element Apply an instantaneous deformation, ε0 and

keep it constant, record the decaying stress: • ε = εelast + εvisc

• dε/dt = (1/G) dσ/dt + σ/η

E

η

• Since the shear strain is kept constant• dσ/dt = -Gσ/η• σ(t) = σ0 exp(-Gt/η)

• Stress relaxation modulus G(t) = σ(t) / ε0

• This is a Stress Relaxation Experiment.• The quantity τ1 = η/G has units of time and is

called the Relaxation Time

Stress Relaxation Experiments (contd.)

Stress Relaxation

Stress-Strain Behaviors

Maxwel Model Kelvin-Voigt Models

Stress Relaxation and Creep The Four Element Model

• Combination of Maxwell and Kelvin elements in series

• Used to describe the viscoelastic deformation of polymers in simple terms

The Four Element Model• OA: Instantaneous extension (Maxwell element,

E1)• AB: Creep or retarded deformation (Kelvin element

)

• BC1: Instantaneous & partial recovery (Maxwell element E1)

• C1D:Time dependant recovery (Creep recovery) – t1 to t2 (Kelvin element)

• DE: Permanant deformation• Note that the recovery is not complete.

A

B

C1

D

----------------------

--------

----------------C2

C3E

t1 t2

F

Relaxation time

Maxwell Model

Spring & Dashpot are in series

ε = εelast + εvisc

dε/dt = 1/E dσ /dt + σ /η

dσ /dt = E dε/dt - E σ /η

= E dε/dt - σ /λ : λ = η/E – Relaxation time

• Relaxation time: Order of magnitude of time required

for a certain proportion of polymer chain to relax (ie;

to responds to the external stress )

• Time required for a chemical reaction to takes place

• Bond interchange, degradation, hydrolysis &

oxidation

Retardation time• Kelvin (Voigt) Element

Spring & dashpot are in parallel

σ = σelast + σvisc

σ = ε Ε + η dε/dt

ε= σ/E - η/E dε/dt

Under cont stress the eqn integrated to

ε= σ/E (1-e - (E/ η)t )

ε= σ/E (1-e- t/τ) : τ = η/E Retardation time

• For four element model under cont stress• ε = ε1+ ε2 + ε3

• ε= σ/E1 + σ/E2 (1-e- t/τ) + (σ /η3) t

elastic term Viscoelastic effect Viscous effect

• Retardation time: time required for E2 & η2 in kelvin element to deform 1-1/e or 63.21 % of the total expected creep

Time-Temperature Superposition Principle

• – Method of Superposition• – WLF Equation and Application

Time-Temperature Superposition

The Stress relaxation modulus depends on temperature and time.

• First: Choose an arbitrary reference temperature T0.

• Second: Shift, along the time axis, the stress relaxation modulus curve recorded at T just above or below T0, so that these two curves

superpose partially.

• Third: Repeat the procedure until all curves have been

shifted to be partially superposed with the previous ones.

• Four: Keep track of aT, the amount a curve recorded at T is shifted. No shift for curve recorded at T0.