Topic 2: Viscoelasticity and Diffusion

“Viscosity” world “Elasticity” world

Strain rate Strain

i

k

k

iik

x

u

x

u

i

k

k

iik

x

v

x

v

Stress sik = {force}i per unit {area}┴k

Elastic stress ikikaa aaik GB

s 2

13

Young modulus E

Shear modulus

Bulk modulus

Poisson ratio

)1(2

EG

)21(3

EB

You’ll remember from last year 1B:

etc.;3211 sss E

BE

621

||

Viscous shear stress & pressure

ikikik p s 2

Incompressibilty

0div kk kkv

Only shear deformations allowed

32

33

Non-linear elasticity and non-Newtonian fluids

Newtonian Hookean

Shear stress s

Shea

r m

od

ulu

s G

(s)

Creep, plastic flow

Stress

stiffening

Strain rate d/dt

Vis

cosi

ty

Shear

thinning

Shear

stiffening

Yield

stress

For a Hookean solid or a Newtonian fluid, the elastic modulus or the viscosity are strictly a constant. For a higher deformation or a greater shear rate – the system just breaks down…

However many complex systems do not obey this simple linearity.

Examples of different types of complex solids and fluids:

There is a great richness of behaviour depending on the internal structure, at higher deformations / rates of flow.

An additional complexity occurs in the time-domain: some materials behave like solids at high strain rates, but liquids at low. Such materials are known as viscoelastic.

34

A typical elastic, viscous and viscoelastic response

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

• Polymeric fluids and some disordered solids are examples.

• The elastic aspect leads to energy storage. Its contribution to a shape change will be lost once the stress is removed.

• The viscous aspect leads to energy dissipation, and irreversible shape changes associated with the flow.

• Rate effects are very important for these materials.

Creep experiment

A constant load is applied and the

resulting strain is measured.

1 = immediate elastic deformation 2 = delayed elastic deformation 3 = plastic flow (permanent deformation)

J(t) (t)

so

time

Applied stress (load)

time

Resulting strain (deformation)

Viscoelastic creep

1

1

2

3

35

A typical elastic, viscous and viscoelastic response

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

• Polymeric fluids and some solids are examples.

• The elastic aspect leads to energy storage. Its contribution to a shape change will be lost once the stress is removed.

• The viscous aspect leads to energy dissipation, and irreversible shape changes associated with the flow.

• Rate effects are very important for these materials.

Stress relaxation experiment

A fixed extension (strain) is applied

and the resulting stress is measured.

s1 = instant stress (glassy response) seq = equilibrium stress (rubbery response) Ds(t) = stress relaxation

o

ttG

s )()(

time

Applied strain (deformation)

time

Resulting stress

Viscoelastic relaxation

s1

seq Ds

36

Storage and dissipation of energy

So both the creep compliance J(t) and the elastic modulus G(t) vary with time

Elastic response: G =Geq. =1/Jeq.

Viscous response: G =∙(t) J =(1/∙t

Approach to equilibrium = relaxation = viscous flow = energy dissipation

Hence the “effective viscosity” is also a function of time – or a function of frequency in an oscillating test

o

ttG

s )()(

time from onset

Modulus G(t)

Energy dissipation Log[time]

Modulus G(t)

37

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 linear superposition theory.

Proposals:

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

2. Each loading step makes an independent contribution to total loading history.

3. Total final deformation is the sum of each contribution.

Input

Response

Time

Str

ess

Str

ain

Ds1

Ds2

Ds3

e (t)1

e (t)2

e (t)3

1 2 3

t

dudu

udutJt

)()()(

s

And for a general creep experiment with

increments of stress ds at times n

In the same way, for stress relaxation, with incremental steps of strain d

taking place at times t’< t, we have:

t

dd

dtdt

tdttGtdttGt '

'

)'()'()'()'(lim)(

0

s

38

Pure elasticity and pure viscosity

The “retarded” linear response – the sum over the whole history of

applying deformation and relaxation:

Pure elastic system: no relaxation, no dissipation – no retardation:

G(t-t’) = G0

Pure viscous system:

t

dtdt

tdttGt '

'

)'()'()(

s

)(''

)'()( 00 tGdt

dt

tdGt

t

s

time

Resulting stress

)(''

)'()'()( tdt

dt

tdttt

t

s

)'()'( ttttG

39

Viscoelasticity in oscillating regime

Consider applying a sinusoidal deformation:

Then using this in:

gives:

where we have defined the complex modulus G*

(think of the analogy with complex impedance Z*(w) in V(w)-I(w)

Pure elastic:

Pure viscous:

t

dtdt

tdttGt '

'

)'()'()(

s

tiet w 0)(

tit

ti eGdtettGit ww wws 0

'

0 )(*')'()(

)(FT)()(*0

tGideGiG i

www w

'tt

00

0)(* GdeGiG i

ww w

www w ideiG i

0)()(*

''' GiG

40

Example in question sheet:

Torsional rod rheometer

Apply sinusoidal strain:

= oexp(iwt)

s = soexp(i(wt+))

Complex Modulus

If there is an oscillatory driving force, then

real part G‘ = storage modulus

(This gives the part of the strain in phase with the driving force)

imaginary part G ‘‘ = loss modulus

(This represents the out of phase component of linear response).

This is a generic description for all systems with a retarded linear response:

The real and imaginary parts of a complex linear-response modulus are related by the Kramers-Kronig relations.

s

w ieiGGG

0

0''')(*

one end

cylindrical (disk) sample radius r, height h

rod rigidly clamped at

one end

moment of inertia I

Set in oscillatory motion

ti

ti

ti

eIZV

eHM

eEP

w

w

w

ww

ww

ww

0

0

00

)(*)(

)(*)(

)(*)(

M

41

Maxwell fluid

)/exp()( 0 tGtG

Simplest models of viscoelasticity

0G

How is related to G0 and ?

Kelvin-Voigt solid

s /

eq.

0 1)( teG

t

0G

Glassy

Rubbery

2

1

42

Examples of viscoelastic behaviour

Polyisobutylene - overview

Theory: crosslinked rubber

Silicone – loss mechanisms

Plastic flow

Yield point

Examples of viscoelastic behaviour

Metallic glass / chocolate / sintered granular system

Initial elastic response

(high modulus)

High shear rate

Low shear rate

44

Einstein in 1905 (5 key papers)

In 1905 the atomic hypothesis was not fully accepted.

Despite Brownian motion having been known about for 75 years, its significance was not appreciated (nor understood).

The kinetic theory of gases was thought of as a 'mechanical analogue', but implied reversibility.

The 2nd law of thermodynamics required irreversibility.

Einstein understood that taking a statistical approach, and assuming atoms existed, reconciled the paradox.

His paper on Brownian motion, the second of the 5, was written in April 1905.

A macroscopic particle – such as the

pollen particle of Brown – would be

buffeted by the atoms/molecules in the

surrounding water.

The particle would undergo diffusion and

measuring the diffusion constant (or

equivalently the displacement) should

show an increase with t (not linear).

Perrin's subsequent experiments on

sedimentation showed how all this

hung together.

Microscopic origin of fluctuations

The origin of fluctuations of all thermodynamic quantities is, of

course, the random thermal motion of particles in the medium

(the energy of which we called “heat”).

Fluctuations are an essential element of Nature, since they are the

mechanism by which the system explores its available phase space; the

statistical summation of quantities such as partition function requires this to

achieve equilibrium.

On the other hand, thermal fluctuations make the value of any system

parameter, or the dynamical trajectory of a particle, uncertain. No individual

observation has any value, only the probabilities and the averages do.

We now study the Brownian motion, which is the simplest and

most transparent effect that allows us to explore microscopic

origins of thermal fluctuations. The classical experiments of

Robert Brown tracked the motion of small (1mm) particles in

water. 45

Brownian motion

The free particles should move according to the dynamical

equation v

dt

dvm

where the friction drag coefficient for a

sphere of radius a in a fluid of viscosity

has the Stokes’ form: 6pa

Movement in a viscous fluid is dissipative, so if a particle

starts with a velocity v0, that will decay as

Yet Robert Brown has recorded particles moving continuously,

following a random-walk trajectory:

tmevv )/(

0

This type of motion is called stochastic.

It clearly requires an additional force to

act on the particle, which we shall call

the stochastic force x(t) (sometimes it

is called the random noise)

46

Stochastic force

The dynamical equation of free particle changes into

)(tvdt

dvm x

where the stochastic force may have a typical

time-dependence (in a 1D projection):

If we zoom in, there is a characteristic

time scale between collisions of our

particle with molecules of water, which

are in thermal motion. t

x(t)

x(t) in water at room-T:

s10~ 13

T

Let us finish by defining the

statistical characteristics of

such a stochastic force:

It is pretty clear that: const)(but0)( 2 tt xx

0)()()()(also 2121 tttt xxxx This is the definition of “white noise”,

complete lack of correlation between

different pulses! 47

Langevin equation

The stochastic equation has a generic form: start with the

dynamical equation (generic Newton’s) and add the stochastic

force to all other forces present. Even for a free particle we

must include friction, to provide the sink for the energy

delivered by the collisions.

)(tvdt

dvm x

t

ttmtm dttm

eevtv0

]')[/()/(

0 ')'(1

)( x

Either work out the solution from the Green function of

the corresponding homogeneous equation – or just

check it by direct differentiation.

The general solution:

As with the stochastic force itself, there is no meaning in the

value of the (also stochastic) particle velocity v(t). Only the

averages make sense. Clearly 0)( tv

48

Mean square velocity, #1

t

ttmtm dttm

eevtv0

]')[/()/(

0 ')'(1

)( x

The general solution:

t

ttmttm

tt

tmtmtm dtdtttm

eedttm

eevevv0

21212

])[/(])[/(

00

')/()/(2

0

)/(22

0

2 )()(1

')'(1

2 21 xxx

Taking the

average:

t

ttmttm

t

tm dtdtttm

eeevv0

21212

])[/(])[/(

0

)/(22

0

2 )()(1

21 xx

)( 21 tt

t

ttmtm dtem

evv0

1

])[/(2

2

)/(22

0

2 1

tmtm em

evv )/(2)/(22

0

2 12

Decay of the

initial condition mv

2

2

Maxwell distribution of the velocity gives:

m

Tkv B2

TkB2

49

Mean square velocity, #2 The mean kinetic energy:

2

0

][)/(2

22

x dm

)(e

mvmt

tm

As before, the only object that needs “averaging” is x(t) under the integral:

2121

][)/(

0

][)/(

0

21

2

1xx

dd)()(ee

m

tmt

tmt

421

0

][)/(2 1

de

m

tm

Long times: steady-

state mean K.E. TkB

2

1

Equipartition The important result we just found says that the

mean-square intensity of the random force is

proportional to temperature kBT (that’s expected)

but also the dissipative friction constant !

This is one of many forms of Fluctuation-Dissipation Theorem

50

Fluctuation-Dissipation relationship

Intensity of the random

noise (thermal motion of

molecules): the energy supplied to the particle

2x

TkB2 Dissipation of energy of

the particle due to the

friction drag against the same medium

ap 6

The Langevin equation has the underlying relaxation dynamics,

with the characteristic time of velocity decay

)(tvdt

dvm x x(t)

/mV

The equilibrium (Maxwell) distribution of velocities in the ensemble of such particles is established after this time scale.

v(t)

52

Velocity correlation function

Correlation functions in general measure the extent to which there is any correlation between a quantity at two separate times.

It can be applied to position, force, velocity etc, and can be defined in the same way as for the velocity correlation function.

The larger the value of the function is, the greater the extent of correlation.

For random motion, the velocity correlation function equals <v2(0)> for t~0, and = 0 at large t, when all correlation is lost.

)/exp()0()()0( 2 mtvtvv <v2(0)>

t

<v2(0)>e-t/m

<v(0

)v(t

)>

Velocity correlation function

For timescales shorter than V the velocity retains correlation, but not for time scales much longer than this.

For a sphere of radius 10nm (typical size for a large polymer) in water,

this leads to V = m/ ~10-10 s. 1mm particles in water: m/ ~ 2·10-8 s

DTk

dtem

Tkdttvv

B

mtB

0

/

0

)()0(

Overdamped limit If our “observation window” is much wider than the velocity

relaxation time V, then the particle effectively has no

acceleration, and we are left with the balance of forces,

viscous vs. stochastic:

)(1

or0 tdt

dx(t)v x

x Which we can easily solve

to produce the expression

x

d)(tx

t

0

1)(The stochastic expression for the

particle position x(t) is useless!

We can, however, build other quantities which can be evaluated

and studied. For example, the mean square displacement:

2121

0 0

2

2 )()(1

)( xx

ddtx

t t

53

Mean square displacement )()()( 2121 tttt xx

tDtdtx

t

21

)(21

0

2

2

This represents diffusion, and defines the diffusion constant D

As before, averaging under the integral:

The F.D. relationship:

Gives the Einstein relation between the

diffusion constant of a particle, and its

kinetic friction:

TkB2

/BTkD

a

TkD

p6

B

For a spherical particle with the

Stokes form of the friction drag

54

55

Typical values for D in water

Molecule T(oC) MW

(g / mol)

D

(m2 s-1)

Oxygen 25 32 2 x 10-9

Sucrose 25 342 5 x 10-10

Myosin 20 493,000 1 x 10-11

DNA 20 6,000,000 1.3 x 10-12

Tobacco mosaic

virus

20 50,000,000 3 x 10-12

Random walk

Let us consider the simple example of free diffusive motion

along one axis, the variable x(t), as a sequence of random

steps: to the left and to the right:

Steps of length a occur at time intervals

Dt. The total of N steps is made: t=N·Dt

)!(!

!

2

1),(

NNN

NNNP

N

If we count the number of steps to the right, N+ , and to the

left, N=NN+ , then we can find the probability P(x) as that

of making N+ steps out of the total of N – in any order!

The total displacement after N steps is x(t),

but the total length traveled is L=N·a

Use the Stirling formula,

as always: NNNeN ln!

taD

D

2

2

tD

x

Na

x

ee

42

2

2

2Where we defined

the constant:

x(t)

Random walk Diffusion

The sequence of ±steps along one axis is an example of

diffusive motion!

Steps of length a occur at time intervals Dt.

tD

dxe

dxexdxxPxx

tDx

tDx

2)(

4

42

222

2

The average displacement <x(t)> =0, of course. But the

mean square displacement is determined by the Gaussian

probability P(x) as the variance:

x(t)

taD

D

2

2

Compare the Einstein relation for the diffusion constant,

with our “counting states” result: dividing the elementary

step length by the time to make this step

/BTkD

57

Free diffusion equation

The sequence of ±steps along one axis.

),1(),1(),1(),1()1,( NkPkkwNkPkkwNkP

The transition probability of making a step from the position k

to (k+1) is w(k,k+1)=1/2. Obviously, w(k,k-1)=1/2 too. Now, the

probability to end up in a position k after N+1 steps is made of

two parts:

x(t)=k·a

Subtract P(k,N) from both sides, and use the fact that w(k-1,k)+w(k+1,k)=1

),(][),1(),1(),()1,( NkPwwNkPwNkPwNkPNkP

t

tkP

t

NkPNkP

D

),(),()1,(

2

22

2

2 ),(

2

),(

2

1),1(),(2),1(

2

1

x

txP

t

a

k

tkP

tNkPNkPNkP

t

D

D

D

Diffusion equation: 2

22 ),(

2

),(

x

txP

t

a

t

txP

D

58

This, as the random walk, is completely

generic to the stochastic system that has

equal ±½ probabilities of a step.

If you recall that P(x,t) represents the concentration c(x,t) (or density) of

particles, then we realize that empirically this relation has been known long

ago: Fick (1885) empirically described mass diffusion by assuming that the

flux of particles is proportional to the gradient of concentration:

),( txcDJ

2

22 ),(

2

),(

x

txP

t

a

t

txP

D

Conservation of mass implies the continuity relation: to

increase c(x,t) you need “influx” ),(),( txJtxc

dt

d

Also, notice a remarkable analogy between the free diffusion and the

Schrödinger equation for a free quantum particle:

2

22

2 xmti

Not surprisingly, the early quantum theory was

considering ideas of “imaginary time”, or an

effective imaginary diffusion constant: miD 2/

Free diffusion equation

60

Diffusion Control

Diffusion may limit:

Growth of 2nd phase particles

Supply of nutrients to organisms

Colloidal aggregation

Consider spherically symmetric growth, so work in spherical coordinates.

Assume steady state so dc/dt=0.

Diffusion equation becomes

As usual the solution depends on boundary conditions.

Example 1: Diffusion of molecules which react at the surface.

If these are transformed/lost during the reaction, then c(r = a) = 0. If the concentration well away from particle is c , the solution is:

For such a concentration profile, the flux is

given by

01 2

2

2

dr

dcr

dr

d

rc

r

acrc 1)(

2)(

r

aDc

dr

dcDrJ

a

61

Hence the number of collisions at the surface per unit time is:

The rate seen here sets radius a(t)

growing at a ~ t1/2, which is the fastest possible rate at which any diffusion-limited process can occur

If the reaction at the surface itself is rate limiting, the process will be slower.

If the diffusing molecule is a nutrient to an organism, then we also have to think about its consumption and this will be proportional to the volume.

Thus if e.g. a bacterium were too large, it would not get sufficient supply of the nutrients by diffusion, and so this will set an upper limit on size.

Example 2: Growth of particles at late stages post-nucleation

After the initial spinodal decomposition, in many systems (with conserved order parameter) there is an evaporation/condensation mechanism.

The smaller particles are unstable with respect to the larger ones, and material diffuses to the surface of larger particles, causing growth.

Process known as Ostwald ripening or coarsening.

The growth of the average cluster size follows

This is known as the Lifshitz-Slyozov law, and applies to a wide range of coarsening systems (droplets, polycrystals, phase decomposition in mixtures).

acDaaJaN pp 44)()( 2

3/1tr

Diffusion Control

Confined Brownian motion

Previously the free particle “started” with a

velocity v0 and no reference position in space.

After a characteristic time =(m/) the initial

velocity is “forgotten” and the Maxwell distr. P(v)

is established. But there is still no reference

position in space, and the diffusion continuously

spreads the particle “envelope”:

Let us now look how the thermal noise affects the system with

a potential energy, using the examples of a particle on a spring

Dtx 22

)(txdt

dxx

When the particle is confined by the potential U=½ x2 around

the equilibrium at x=0, the situation changes.

The overdamped Langevin eq.

The solution is obvious, from the analogy

with the earlier free Langevin eq. for the

velocity!

t

ttt dtteextx0

]')[/()/(

0 ')'(1

)( x

x TkB22

with

62

Particle in a potential well First of all, there is now a new time scale in the problem, x/, which

determines how long does it take for the particle to “forget” its initial position

and adopt the equilibrium distribution.

t

ttt dtteextx0

]')[/()/(

0 ')'(1

)( x

Finding the m.s. <x2> follows the same steps

as we made in the evaluation of the m.s. <v2>

We have seen this very recently!

t

tttt

t

t dtdttteeexx0

21212

])[/(])[/(

0

)/(22

0

2 )()(1

21 xx

tt eexx )/(2)/(22

0

2 12

After the time t >> x

Tkx B

2

2

The m.s. deviation of x from zero is the m.s. fluctuation

about the equilibrium. <x2> then is given by our general

relation:

eq

2

Y

XTkX B

D Here the force conjugate to the displacement

x is just the spring force f=ax 63

Diffusion and drift Consider our earlier scheme of deriving the free

diffusion equation from analysing the probabilities

of ±steps, with transition probabilities w±=½.

Let us now consider a constant bias:

w(k,k+1)=1/2 and w(k,k-1)=1/2.

x(t)=k·a

subtracting P(k,N) from both sides, and forming the discrete derivatives:

),(][),1(),1(),()1,( NkPwwNkPwNkPwNkPNkP

k

tkP

tk

tkP

tt

tkP

t

NkPNkP

D

D

D

),(2),(

2

1),(),()1,(2

2

Diffusion with a

constant drift: x

txP

t

a

x

txP

t

a

t

txP

D

D

),(2),(

2

),(2

22

),1(),1(),1(),1()1,( NkPkkwNkPkkwNkP

Starting from the balance:

64

Diffusion in external potentials With a constant drift “force” the diffusion

equation adopts a modified form. There is a

general solution of this, but let us look at the

steady state - the equilibrium distribution.

x

txPC

x

txPD

t

txP

),(),(),(2

2

)()(

:)()(

0 eq

eqeq

2

eq

2

xPD

C

x

xP

x

xPC

x

xPD

xDCePP /

0eq

Example: Sedimentation by gravity: V=mgz, force f=mg

which will follow if we take C=mg/

Tkmgz

BePP

0eq

This suggests that a general diffusion equation of motion in an arbitrary

external potential V(x) would take the form

),(

1),(),(

force),(),(txP

x

V

x

txPD

xtxP

x

txPD

xt

txP

In different sources this modified diffusion equation is often called the

Fokker-Planck equation. However, strictly speaking, the F-P equation is

for the joint probability P(v,x,t) – whereas this reduced form has the name

of Smoluchowski equation (1906)

Flux of probability We now see how the equilibrium distribution

arises as the limiting (t→) steady state of

generalised diffusion:

P

x

V

x

PD

xt

txP

1),(

dxx

V

DP

P

1eq

Tk

xV

BePP)(

0eq

The probability distribution P(x,t) is a continuous field with a fixed

normalisation. As all such fields, it must satisfy the continuity relation:

),(div),(

txJt

txP

P(x,t)

),(),(f),( )()(

txPex

eDtxPx

txPDJ

xVxV

Here the flux of probability J(x,t)

has to be defined as:

For the force-free diffusion: ),( txcDJ

66

Escape over potential barrier, #1 Consider a system in a metastable state,

separated by a potential barrier D from

reaching its thermodynamical equilibrium.

The generic question is:

How long would it have to “wait” for a sufficiently

large thermal fluctuation to take it over the barrier?

)()()(

xPex

eDJxVxV

The classical Kramers problem (1940): steady-state current of particles

(or states) from A to B. The constant flux is given by

V(x)

D

A B x0

Integrate both sides between A and B:

BA

B

A

xPeDdseJxVsV

)()()(

If the potential well at B is deep

)()(

A

B

A

xPDdseJsV

Assuming the basic shape of potential barrier: V(x)=D ½kC (x-x0)2 ,

the integral gives the approximate steepest-descent result:

)(42

021 )(

A

C

BTkxsTkxPDeJdseeJ

TkBCB

D

D

k

pk

We have obtained the steady-state flux to

carry our system from A to B (over the

energy barrier towards its true equilibrium):

V(x)

D

A B x0

The rate of this escape (i.e. the inverse life-time at A) is by definition the

ratio of the flux to the {number of particles present at A}, which in turn is

given by

A

BA

x

AAAA

TkxPdxexPNdxexPdN AAV

k

pk 4)()(:)(

2

21

)(4

A

B

CTkxPeDJ

Tk

B

D

p

k

The rate of escape over the barrier D:

D

D

TkCATk

CA

BA

BB eeD

N

J

Tk p

kkkk

p 44

1

This is a very important (and famous) result explaining the so-called

Arrhenius law, empirically relating the life time of any metastable state to

the Boltzmann exponential with the energy barrier (found anywhere from

biology/chemistry to mechanical engineering).

Escape over potential barrier, #2

69

Observing Brownian Motion

0 100 200 300 400 500 600

0

50

100

150

200

250

300

350

400

450

15 20 25 30 35 40

30

35

40

45

50

70

Using particle tracking for “microrheology”

The mean square displacement as a function of time can be followed eg. during some process.

(d is dimensionality, a is particle radius, is lag time)

We have seen that viscoelasticity can be treated as a complex viscosity.

p

a

Tkdr B

62)(2

<r2>

)(6

2)(2 srsa

TkdsG B

p

“generalised Stokes Einstein equation”

What happens with Brownian motion in a complex fluid (viscoelastic medium)?

A generalised version of the Stokes-Einstein equation allows the complex moduli

to be determined by tracking the motion of the particles.

(r2(s) is the Laplace transform of r2 (t),

s is a Laplace frequency,

G(s) is related to G*(w) with s=iw )

71

Motion of beads within cells

Fluorescence image of an individual

human dermal fibroblast cell showing

clustering of the beads in the perinuclear

region, with a few beads in the periphery of

the cell.

Two dimensional mean square

displacements of beads in this cell. A

selection of particles have been

highlighted in red, for clarity. The slopes

exhibit significant scatter, which we

attribute to local variations in the rheology

according to cytoskeletal arrangements.

Note: resolution of position ! 10nm is possible optically