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**Polymer Viscoelasticity & Rheology**

Minshiya P. Assistant Professor M E S Keveeyam College Valanchery

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**Ideal or Newtonian fluids**

• 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.

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**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)

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**Ideal or Newtonian fluids**

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**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 liquid φ : fluidity eqn (1) & (2) : Newton’s law

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**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

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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

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**Newtonian V/s Non –Newtonian**

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. Newtonian fluid Relation between shear stress and shear rate is linear, passing through the origin The constant of proportionality being the coefficient of viscosity Inadequate to describe non-Newtonian fluids The concept of viscosity is used to characterize the shear properties of a fluid

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**Types of non-Newtonian behaviour**

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**Ideal or Elastic solids**

Hooke's law Force needed to extend or compress a spring by some amount is proportional to that amount is a constant characteristic of the spring, stiffness.

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**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

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**Ideal or Elastic solids**

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.

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Stress–strain curve 1. Ultimate strength 2. Yield strength – corresponds to yield point 3. Rupture 4. Strain hardening region 5. Necking region A: Engineering stress (F/A0) B: True stress (F/A)

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**1. Ultimate tensile strength (UTS)**

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.

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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.

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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.

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**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

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**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

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**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

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**Polymer viscoelasticity**

Inter-relationship among elasticity, flow & molecular motion Time dependent flow behaviour Creep & Stress relaxation

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**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

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

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**Creep Deformation under a constant load as a function of time**

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

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**Stress Relaxation Constant deformation experiment**

3 kg 4 kg 5 kg 6 kg Time Constant deformation experiment Stress decreases with time Slowly drops to zero as the material undergo permanant set to the stretched length

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**Reasons for Stress Relaxation and Creep**

1. Chain Scission: Oxidation, Hydrolysis.

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**Reasons for Stress Relaxation and Creep (Contd.)**

2. Bond Interchange: Polyester, Polysiloxane Polyamides, Polyethers

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**Reasons for Stress Relaxation and Creep (Contd.)**

3. Segmental Relaxations: Thirion Relaxation

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**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

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**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)

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**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. Springs Stretched instantaneously under stress & holding that stress indefinitely Store energy and respond instantaneously.

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**Maxwell & Kelvin (Voigt) Element**

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**Creep Experiment • Maxwell Model Spring & Dashpot are in series**

η • 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

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**Creep Experiment - Maxwell Model (contd)**

η 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 ε = σ /E

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**Creep Experiment 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

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**Creep Experiment - Kelvin (Voigt) Element**

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.

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**Stress Relaxation Experiments**

η 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

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**Stress Relaxation Experiments (contd.)**

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

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**Stress Relaxation Stress-Strain Behaviors Maxwel Model**

Kelvin-Voigt Models

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**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

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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 C3 E t1 t2 F

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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

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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

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**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

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**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 % of the total expected creep

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**Time-Temperature Superposition Principle**

– Method of Superposition – WLF Equation and Application

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**Time-Temperature Superposition**

The Stress relaxation modulus depends on temperature and time.

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**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.

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