Viscoelasticity

Books

Most polymer texts cover this reasonably well in outline.

More specialist texts for reference:

JJ Aklonis and WJ Macknight Introduction to Polymer Viscoelasticity, Wiley 1983

IM Ward Mechanical Properties of Solid Polymers Wiley

Introduction

A viscoelastic material is, as the name suggests, one which shows a combination of viscous and elastic effects.

The viscous term leads to energy dissipation.

The elastic term to energy storage.

Rate effects are very important for these materials

For a viscous liquid with viscosity h, the constitutive equation relating stress s to strain e is

There is dissipation of energy – and irreversible shape changes – associated with the flow.

The viscosity can be related to the diffusion equation.

If an external force f on a particle/atom gives rise to a velocity u then

u = µ f where µ is the mobility

Einstein relation gives

µ = D/kT where D is the diffusion coefficient

Stokes Law says for a particle of radius a

f =6ph a u

In general then h and D are inversely related, and as D increases with temperature viscosity decreases.

In contrast most solids exhibit pure elasticity

Ideal elastic material

s = Ee E is Young's modulus

Energy is stored as elastic energy.

Material returns to original shape once stress removed.

Polymeric liquids, and various solids, have attributes of both and these are known as viscoelastic materials.

Creep

A constant load is applied and the resulting strain is measured.

e₁ = immediate elastic deformation

e₂ = delayed elastic deformation

e₃ = Newtonian flow (i.e. permanent deformation)

Define creep compliance

so there are 3 components of the creep compliance J₁ in general associated with the 3 components of strain.

One exception to this is a crosslinked rubber: its memory effect means that there is no permanent shape change so that e₃ = 0 and so J₃ also is zero.

Division into J₁ and J₂ (or equivalently e₁ and e₂) fairly arbitrary.

J₁ and J₂ sometimes knows as unrelaxed and relaxed responses.

Stress Relaxation

A fixed extension (strain) is applied

Define stress relaxation modulus

If no viscous flow occurs, stress drops to finite value at infinite times ® relaxed modulus.

If there is viscous flow, stress can drop to zero.

Models for Viscoelastic Response

1. Maxwell Model

Spring and Dashpot model

Spring = elastic component

+ Dashpot = viscous component in series

Define characteristic time t for response

t = h/E

Equation of motion

= +

total spring dashpot

strain rate

Stress relaxation experiment

de/dt = 0 in equation of motion

E(t) = E_oexp-t/t

At very short times, the Maxwell model behaves as a simple spring.

Takes longer for the viscous component to respond.

For t>>t stress drops to zero as only the response of the dashpot remains.

2. Kelvin or Voigt Model

Spring and dashpot in parallel

Total stress

In a creep experiment, s is a constant s_o so, dividing by h

Can solve for e with integrating factor expt/t

e(t) = s_o/E (1-exp-t/t)

This model cannot be used for stress relaxation experiments, since it would require infinite force to strain viscous elements instantaneously.

Both these models are too simple.

Next step is to combine them to produce a 'standard linear solid'.

This is an improvement, but there is still only one characteristic time associated with the model.

In general there will be a whole spectrum of these – for instance in a polydisperse polymer melt, different chain lengths respond differently.

Fast chains respond faster than long chains.

So to model a polymer melt properly we might imagine we need a whole system of standard linear solids each with its own t.

Standard Linear Solid (due to Zener)

Add to each side

Instantaneous response (e=0), finite de/dt) – no response from dashpot

\ modulus = E_r + E_m

Long time response, dashpot takes all the strain and E_m does not contribute

\ modulus = E_r relaxed modulus

Energy dissipated (in dashpot) a maximum at some intermediate time.

At early and late times, no movement in dashpot and hence no dissipation.

Boltzmann Superposition Principle

To describe the general response of a system, must allow for details of loading history.

This can be done using the Boltzmann superposition theory.

Boltzmann proposed:

·
Creep is a function of the whole sample loading history.

· Each loading step makes independent contribution to total loading history.

· Total final deformation is the sum of each contribution.

In general, for a creep experiment, increments of stress ds at times t_n

For stress relaxation, incremental additions of strain de at times t_n

For a steady state shear rate, this can be rewritten as

s(t) = h_ode/dt Newton's law of viscosity

where (by change of variable)

Note that this theory only works for small deformations – this is linear viscoelastic theory.

Complex Modulus and Dynamic Experiments

Two types of processes occurring – storage and dissipation of energy.

Looking at the analogous situation of LCR circuits, where V and I are out of phase for oscillating signals, can anticipate that for viscoelastic materials, stress and strain will be out of phase in dynamic experiments.

Furthermore the modulus must be described by a complex modulus.

If an alternation stress/strain is applied to a viscoelastic solid, stress and strain are out of phase.

Complex modulus G = G₁ + G₂

storage modulus = real part G₁

loss modulus = imaginary part G₂

This is a general description for all viscoelastic materials.

Let phase angle be d, apply sinusoidal strain

e = e_oexpiwt s = s_oexpi(wt+d)

= G =

In general ½G₂½<<½G₁½

G₁ represents stress in phase with strain – i.e. energy stored during deformation

G₂ is a measure of energy dissipated/cycle.

Consider energy loss/cycle

or rate of loss/cycle =

_\ DE = pe_o²G₂

Phase angle d related to G₁ and G₂ by

tand = G₂/G₁

Measurement of Complex Modulus

Measuring the in phase and out of phase components of the response of strain to an imposed stress (or vice versa) at different frequencies provides the two components of G to be determined.

The modulus may vary greatly with frequency/time scale of the experiment.

Different techniques are used for different frequencies (see Ward).

Example – torsion pendulum

Motion is damped SHO

For thin walled tube, angular strain = r (q/l)

Restoring force for rotation q = Gr(q/l) x area

Torque = r Gr (q/l) 2p rdr

Total torque = =

Equation of motion becomes

This is the equation for damped SHM with the term being iwq

With and

L is the logarithmic decrement.

As expected frequency is determined by G₁ and

damping by G₂

This apparatus works over frequency range 0.01-50Hz. At higher frequencies wavelength of the stress waves becomes comparable with the dimensions of the specimen.

Results of Measurements

Rubbery Visco- Glassy

elastic

These terms apply to polymer melts, but the phenomena are much more general.

Tand and G₂ are both large at intermediate frequencies in the viscoelastic regime.

This behaviour is the same for solids with any damping mechanism.

In the case of metals etc this is sometimes known as internal friction.

Example – Snoek damping

In bcc metals, the damping occurs due to movement of interstitials e.g. C or N in a iron.

Interstitials sit at the centres of the cube edges, and slightly distort the lattice.

When an external stress applied, the energy associated with the different interstitial sites is no longer degenerate.

Under oscillatory stress the interstitials will try to move to accommodate this.

At high frequencies this is impossible.
At low frequencies it will occur to completion.

In both these cases stress and strain will be in phase.

However at intermediate frequencies, around the natural frequency of interstitial jumping, there is significant damping and G₁ and G₂ will be out of phase.

Jumping will be thermally activated and so the frequency at which damping is maximum will be temperature dependent.

This is generally true.

Time-Temperature Superposition

Using polymer analogy again.

This represents the behaviour over the whole temperature range at a given w (or time t).

Alternatively can study at fixed temperature and range of frequencies w.

Similar shaped curve is found.

Experimentally observed that there is a correspondence between time and temperature.

Can shift curves for viscoelastic properties at different temperatures onto a single curve at a single temperature to create a master curve.

Then G(T₁, t) = G(T₂, t/a_T)

where a_T is the shift factor and given by

WLF equation

(Williams-Landel-Ferry)

and T_o is the reference temperature.

C₁ and C₂ are approximately universal constants

C₁ = 17.4 and C₂ = 51.6K

Note that a_T is not a function of time, only temperature.

This same equation can be used for any of the viscoelastic constants including viscosity.

In which case it recovers the Vogel-Fulcher Law.

as we saw before

Experimentally the WLF equation is very important because it enables the response of a system under a wide range of conditions to be described from limited experimental data.

Theoretically it implies that all the timescales in the problem scale in the same way with T.

Implies there is a single basis parameter which for polymers turns out to be the segment mobility.

Dynamics of Polymer Chains

The flow of polymers is dominated by long range motions.

However because of the complexity of chains, there are many internal motions possible.

These show up in the loss modulus (below T_g).

In the glassy state, local segmental notions not sufficient to relax the strains imposed by external stress, and G₁ is high.

Situation more complicated for crystalline polymers, where also have to pass through T_m before bulk flow occurs.

However remember that polymer chains are entangled – so how do they move at all?

These entanglements will affect viscosity, diffusion…..

At first it was thought that this was too difficult to deal with at all, but much progress has now been made.

De Gennes conceived of the idea of reptation – moving like a snake (1971).

Ideas developed further here by Doi and Edwards.

See the book by M Doi and SF Edwards Theory of Polymer Dynamics 1986, OUP.

Think of the motion of snakes in a nest – constrained laterally but can move along their length.

Where one chain interacts strongly with another chain (naively as a simple knot), identify an entanglement.

All the surrounding chains provide constraints for the movement of a test chain.

These surrounding chains can be averaged to provide a tube of diameter equal to the entanglement separation.

Consider a chain confined in such a tube.

The chain can slowly escape this tube as it undergoes Brownian motion, thereby creating a new tube.

Mobility µ of whole chain = monomeric mobility µ₁/N

where N is the number of monomers in the chain.

Einstein relation D=µkT implies

If t is tube relaxation time, ie time length L of old tube takes to be lost and new length L to be created, then by random walk

Now L is curvilinear length of tube/chain\ LµN

And t µ N³ (or equivalently M³)

And since hµt, reptation model implies hµN³

This result is in contrast to small molecules (ie ones for which entanglements and the tube concept do not apply) for which

tµ N

Schematically one might expect

Experimentally the dependence in the entangled regime is found to be

h µ N^3.4

Origin of discrepancy with simple theory is thought to lie in fluctuations.

If we apply a deformation to a polymer melt, the constraints are deformed.

The chain can gradually escape from its tube, to form a new undeformed tube.

The relaxation time can therefore be found from experiment – often known as the terminal time.

tµ N³ in simple reptation theory

MW₁<MW₂

Once the chain has completely escaped no further resistance to deformation, and hence G drops.

Alternatively can find the terminal time from the creep compliance curves.

As with rubbers, for which we identified a plateau modulus inversely related to M_x (the MW between crosslinks), for entangled polymers we can find an equivalent quantity – M_e, the MW between entanglements – from the value of the plateau modulus.

By analogy with the theory of rubber elasticity, the value of G at the plateau: