Important Points from Last Lecture:• The root-mean-squared end-to-end distance, <R2>1/2, of a freely-

jointed polymer molecule is N1/2a, when there are N repeat units, each of length a.

• The radius-of-gyration of a polymer, Rg, is 1/6 of its root-mean-square end-to-end distance <R2>1/2.

• Polymer coiling is favoured by entropy. The elastic free energy of a polymer coil is given as

• Thinner lamellar layers in a diblock copolymer will increase the interfacial energy and are not favourable. Thicker layers require chain stretch and likewise are not favourable! A compromise in the lamellar thickness, d, is reached as:

.++=)( constTNa

kRRF 2

2

2

3

32315

2//)(= N

kTa

d

Last Lecture:• Elastic (entropic) effects cause a polymer molecule to coil up.• Excluded volume effects cause polymer molecules to swell (in a

self-avoiding walk).• Polymer-solvent interactions, described by the -parameter, also

have an effect, depending on whether polymer/solvent interactions are more favourable than self interactions.

• Thus there is a competition between three effects!• When = 1/2, excluded volume effects are exactly balanced by

polymer/solvent interactions. Elastic effects (from an entropic spring) lead to a random coil: <R2>1/2 ~ aN1/2

• When < 1/2, excluded volume effects dominate over polymer/solvent interactions. They dominate over elastic effects and result in a swollen coil: <R2>1/2 ~ aN3/5

• When > 1/2, unfavourable polymer/solvent interactions are dominant over excluded volume effects. They lead to polymer coiling: a globule results.

PH3-SM (PHY3032)

HE3 Soft Matter Lecture 10

Polymer Elasticity, Reptation,

Viscosity and Diffusion

13 December, 2011

See Jones’ Soft Condensed Matter, Chapt. 5

Rubber ElasticityA rubber (or elastic polymer = elastomer) can be created by linking together linear polymer molecules into a 3-D network.

To observe “stretchiness”, the temperature should be > Tg for the polymer.

Covalent bonds between polymer molecules are called “crosslinks”. Sulphur can crosslink natural rubber (which is liquid-like) to create an elastomer.

Affine DeformationWith an affine deformation, the macroscopic change in dimension is mirrored at the molecular level.

We define an extension ratio, , as the dimension after a deformation divided by the initial dimension:

o

=

oo ll

==

o

Bulk:

l

Strand:

lo

y x

z

x

z

y yy

zz

xx

z

y

xR2 = x2+y2+z2

Transformation with Affine Deformation

z

y

x

Bulk:

Ro = xo+ yo+ zo

R

R = xxo + yyo + zzo

If non-compressible (volume conserved): xyz =1

Ro

Single Strand:

yoxo

zo

Entropy Change in Deforming a Strand

We recall our expression for the entropy of a polymer coil with end-to-end distance, R:

The entropy change when a single strand is deformed, S, can be calculated from the difference between the entropy of the deformed coil and the unperturbed coil:

S = S(R) - S(Ro) = S(xxo, yyo, zzo) - S(xo, yo, zo)

)++(~.+=)( 22222222

2

2

3

2

3ozoyox zyx

Na

kconst

Na

kRRS

])(+)(+)[(~)()( 2222222 111

2

3ozoyoxo zyx

Na

kRSRS

Finding S:

)++(~)( 22222

3oooo zyx

Na

kRSInitially:

Entropy Change in Polymer Deformation

])(+)(+)[(~ 2222222 111

2

3ozoyox zyx

Na

kS

But, if the conformation of the coil is initially random, then <xo

2>=<yo2>=<zo

2>, so:

)](+)(+)[(~ 1112

3 2222

2

zyxo

Na

kxS

For a random coil, <R2>=Na2, and also R2 = x2+y2+z2 = 3x2, so we see:

3

22 Na

xo >=<

)++)((~ 332

3 2222

2 zyxNa

Na

kS Substituting for xo

2:

)3++(2

~ 222zyx

kS

This simplifies to:

)++(~ 32

222zyxbulk

nkS

F for Bulk Deformation

If the rubber is incompressible (volume is constant), then xyz =1.

For a one-dimensional stretch in the x-direction, we can say that x = . Incompressibility then implies

1== zy

)32

(2

~ 2 --

nk

Sbulk

Thus, for a one-dimensional deformation of x = :

The corresponding change in free energy: (F = U - ST) will be

)+(+~ 32

22

nkTFbulk

If there are n strands per unit volume, then S per unit volume for bulk deformation:

Force for Rubber DeformationAt the macro-scale, if the initial length is Lo, then = L/Lo.

)+)((+~ 32

22

LL

LLnkT

F o

obulk

Substituting in L/Lo = + 1:

))+(

+)+((+~ 31

21

22

nkTFbulk

Realising that Fbulk is an energy of deformation (per unit volume: Nm/m3), then dF/d is the force, , per unit area, A (units: N/m2) for the deformation, i.e. the tensile stress, T.

In Lecture 3, we saw that T = Y. The strain, , for a 1-D tensile deformation is

1===oo

o

o LL

LLL

LL

])1(

2-)1(2[

2A 2

nkT

d

dFT

A

Young’s and Shear Modulus for Rubber

22

1]

)1(

1)1[(

nkTnkTT

In the limit of small strain, T 3nkT, and the Young’s modulus is thus Y = 3nkT.

The Young’s modulus can be related to the shear modulus, G, by a factor of 3 to find a very simple result: G = nkT

This result tells us something quite fundamental. The elasticity of a rubber does not depend on the chemical make-up of the polymer nor on how it is crosslinked.

G does depend on the crosslink density. To make a higher modulus, more crosslinks should be added so that the lengths of the segments become shorter.

This is an equation of state, relating together , L and T.

Experiments on Rubber Elasticity

]1

[2

-nkTT

Treloar, Physics of Rubber Elasticity (1975)

Rubbers are elastic over a large range of !

Strain hardening region: Chain segments are fully stretched!

Alternative Equation for a Rubber’s G

We have shown that G = nkT, where n is the number of strands per unit volume.

xMRT

=

x

A

MN

n

=

For a rubber with a known density, , in which the average molecular mass of a strand is Mx (m.m. between crosslinks), we can write:

)(

)#)((=

#

moleg

molestrands

mg

m

strands 3

3

Looking at the units makes this equation easier to understand:

kTMN

nkTGx

A==Substituting for n:

strand

This is a useful equation when constructing a network from “strands”.

P. Cordier et al., Nature (2008) 451, 977

H-bonds can re-form when surfaces are brought into contact.

Network formed by H-bonding of small molecules

Blue = ditopic (able to associate with two others)

Red = tritopic (able to associate with three others)

For a video, see:

http://news.bbc.co.uk/1/hi/sci/tech/7254939.stm

With a constant shear stress, s, the shear modulus G can change over time:

)(=)(

ttG

s

s

G(t) is also called the “stress relaxation modulus”.

Viscoelasticity of Soft Matter

t

G(t) can also be determined by applying a constant strain, s, and observing stress relaxation over time:

s

s ttG

)(

=)(

Example of Viscoelasticity

High molecular weight polymer dissolved in water. Elastic recovery under high strain rates, and viscous flow under lower strain rates.

Relaxation Modulus for Polymer Melts

Viscous flow

Gedde, Polymer Physics, p. 103

Elastic T = terminal relaxation time

Experimental Shear Relaxation Moduli

Poly(styrene)

GP

Low N

High N

~ 1/tG.Strobl, The Physics of Polymers, p. 223

Relaxation Modulus for Polymer Melts • At very short times, G is high. The polymer has a glassy

response.• The glassy response is determined by the intramolecular

bonding.• G then decreases until it reaches a “plateau modulus”, GP. The

value of GP is independent of N for a given polymer: GP ~ N0.

• After a time, known as the terminal relaxation time, T, viscous flow starts (G decreases with time).

• Experimentally, it is found that T is longer for polymers with a higher N. Specifically, T ~ N3.4

• Previously in Lecture 3, we said that in the Maxwell model, the relaxation time is related to ratio of to G at the transition

between elastic and viscous behaviour. That is: T~/GP

Viscosity of Polymer Melts

Poly(butylene terephthalate) at 285 ºCFor comparison: for water is 10-3 Pa s at room temperature.

Shear thinning behaviour

Extrapolation to low shear rates gives us a value of the “zero-shear-rate viscosity”, o.

o

From Gedde, Polymer Physics

Scaling of Viscosity: ~ N3.4

~ TGP

~ N3.4 N0 ~ N3.4

Universal behaviour for linear polymer melts!

Applies for higher N: N>NC

Why?

G.Strobl, The Physics of Polymers, p. 221

Data shifted for clarity!

Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: o

3.4

An Analogy!

There are obvious similarities between a collection of snakes and the entangled polymer chains in a melt.

The source of continual motion on the molecular level is thermal energy, of course.

Concept of “Chain” Entanglements

If the molecules are sufficiently long (N > ~100 - corresponding to the entanglement mol. wt., Me), they will “entangle” with each other.

Each molecule is confined within a dynamic “tube” created by its neighbours so that it must diffuse along its axis.

Tube G.Strobl, The Physics of Polymers, p. 283

Network of Entanglements

There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.

The physical entanglements can support stress (for short periods up to a time Tube), creating a “transient” network.

Plateau Modulus for Polymer Melts • Recall that the elastic shear modulus of a network depends on molecular weight between crosslinks, Mx. In a polymer melt, GP therefore depends on the molecular weight between entanglements, Me.

• That is, GP ~ N0 (where N is the number of repeat units in the molecule).

• Using an equation for the polymer melt that is analogous to a crosslinked network:

eP M

RTG

=

• It makes sense that Me is independent of N - consistent with experimental measurements of GP versus t for various values of M.

Entanglement Molecular Weights, Me, for Various Polymers

Poly(ethylene) 1,250

Poly(butadiene) 1,700

Poly(vinyl acetate) 6,900

Poly(dimethyl siloxane) 8,100

Poly(styrene) 19,000

Me (g/mole)

Me corresponds to the Nc that is seen in the viscosity data.

Reptation Theory: Molecular Level

• Polymer molecules “dis-entangle” after a time, Tube.

• Chain entanglements create restraints to other chains, defining a “tube” through which they must travel.

• The process by which a polymer chain moves through its tube formed by entanglements is called “reptation”.

• Reptation (from the Latin reptare: “to crawl”) is a snake-like diffusive motion that is driven by thermal motion.

• Models of reptation consider each repeat unit of the chain as diffusing through a tube with a drag coefficient, seg.

• The tube is considered to be a viscous medium surrounding each segment.

• For a polymer consisting of N units: pol = Nseg.

x

xx x

x

xx

x

xx

x x x

xx

xx

x

x

x

xx

xx

xx

x

x xx

x

x

xx

Experimental Evidence for Reptation

Chu et al., Science (1994) 264, p. 819.Chain follows the path of the frontInitial state Stretched

Fluorescently-stained DNA molecule

Development of Reptation Scaling Theory

Sir Sam Edwards (Cambridge) devised tube models and predictions of the shear relaxation modulus.

In 1991, de Gennes was awarded the Nobel Prize for Physics.

Pierre de Gennes (Paris) developed the concept of polymer reptation and derived scaling relationships.

Polymer Diffusion along a Tube

In our discussion of colloids, we defined an Einstein diffusion coefficient as:

kTD =

If we consider the drag on a polymer molecule, we can express D for the diffusion of the molecule in a tube created by an entangled network as:

segpoltube N

kTkTD

==

Hence, the rate of 1-D tube diffusion is inversely related to the length of the molecules.

Tube Relaxation Time, tube

The polymer terminal relaxation time, T, must be comparable to the time required for a polymer to diffuse out of its confining tube, tube.

The length of the tube must be comparable to the entire length of the polymer molecule (contour length): Na

By definition, a diffusion coefficient, D, is proportional to the square of the distance travelled (x2) divided by the time of travel, t.

For a polymer escaping its tube: tubetube

Nat

xD

22 )(~~

Comparing to our previous Einstein definition:tubeseg

aNN

kT

22

~

We thus can derive a scaling relationship for tube:

3Ntube ~

Scaling Prediction for Viscosity

Then: 330 ~~~ NNNG tubeP

But, recall that experiments find ~ N3.4. Agreement is not too bad!

3Ntube ~We see that which is comparable to experiments in which T ~ N3.4

We have also found that GP ~ N0

Recalling that ~ G

We can think of T as the average time required for chains to escape the confinement of their tube, tube.

Polymer Self-Diffusion

X

Time = 0 Time = t

Reptation theory can also describe the self-diffusion of polymers, which is the movement of the centre-of-mass of a molecule by a distance x in a matrix of the same type of molecules.

In a time tube, the molecule will diffuse the distance of its entire length. But, its centre-of-mass will move a distance on the order of its r.m.s. end-to-end distance, R.

In a polymer melt: <R2>1/2 ~ aN1/2R

Polymer Self-Diffusion Coefficient

X

tubetubeself

NaaNt

xD

22212

=)(

~~/

A self-diffusion coefficient, Dself, can then be defined as:

Larger molecules are predicted to diffuse much more slowly than smaller molecules.

But we have derived this scaling relationship: 3Ntube ~

Substituting, we find:2

3

2

~~ NN

NaDself

Testing of Scaling Relation: D ~N -2

M=Nmo

-2

Experimentally, D ~ N-2.3

Data for poly(butadiene)

Jones, Soft Condensed Matter, p. 92

• Reptation theory predicts ~ N3, but experimentally it varies as N3.4.

• Theory predicts Dself ~ N-2, but it is found to vary as N-2.3.

• One reason for this slight disagreement between theory and experiment is attributed to “constraint release”.

• The constraining tube around a molecule is made up of other entangled molecules that are moving. The tube has a finite lifetime.

• A second reason for disagreement is attributed to “contour length fluctuations” that are caused by Brownian motion of the molecule making its end-to-end distance change continuously over time.

• Improved theory is getting even better results!

“Failure” of Simple Reptation Theory

• DNA is a long chain molecule consisting of four different types of repeat units.

• DNA can be reacted with certain enzymes to break specific bonds along its “backbone”, creating segments of various sizes.

• Under an applied electric field, the segments will diffuse into a gel (crosslinked molecules in a solvent) in a process known as gel electrophoresis.

• Reptation theory predicts that shorter chains will diffuse faster than longer chains.

• Measuring the diffusion distances in a known time enables the determination of N for each segment and hence the position of the bonds sensitive to the enzyme.

Application of Theory: Electrophoresis

Application of Theory: Electrophoresis

From Giant MoleculesOne common technique: polyacrylamide gel electrophoresis (PAGE) (or SDS-PAGE)

Relevance of Polymer Self-DiffusionWhen welding two polymer surfaces together, such as in a manufacturing process, it is important to know the time and temperature dependence of D.

Good adhesion is obtained when the molecules travel a distance comparable to R, such that they entangle with other molecules.

R

Interfacial wetting: weak adhesion from van der Waals attraction

Chain extension across the interface: likely failure by chain “pull-out”

Chain entanglement across the interface: possible failure by chain scission (i.e. breaking)

Stages of Interdiffusion at Polymer/Polymer Interfaces

Example of Good Coalescence

J.L. Keddie et al., Macromolecules (1995) 28, 2673-82.

Immediate film formation upon drying!

Hydrated film Tg of polymer 5 °C;

Environmental SEM

• Particles can be deformed without being coalesced. (Coalescence means that the boundaries between particles no longer exist!)

Bar = 0.5 m

K.D. Kim et al, Macromolecules (1994) 27, 6841

Strength Development with Increasing Diffusion Distance

Full strength is achieved when the diffusion distance, d, is approximately the radius of gyration of the polymer, Rg.

Rg

d

Relaxation Modulus for Polymer Melts

Viscous flow

T

Gedde, Polymer Physics, p. 103

Problem Set 61. A polymer with a molecular weight of 5 x 104 g mole-1 is rubbery at a temperature of 420 K. At this temperature, it has a shear modulus of 200 kPa and a density of 1.06 x 10 3 kg m-3. What can you conclude about the polymer architecture? How would you predict the modulus to change if the molecular weight is (i) doubled or (ii) decreased by a factor 5?

2. Two batches of poly(styrene) with a narrow molecular weight distribution are prepared. If the viscosity in a melt of batch A is twice that in a melt of batch B, what is the predicted ratio of the self-diffusion coefficient of batch A over that of batch B? Assume that the reptation model is applicable.

3. The viscosity for a melt of poly(styrene) with a molecular weight of 2 x 104 g mole-1 is given as X. (This molecular weight is greater than the entanglement molecular weight for poly(styrene)). (i) According to the reptation theory, what is for poly(styrene) with a molecular weight of 2 x 105 g mole-1. (ii) Assuming that poly(styrene) molecules exist as ideal random coils, what is the ratio of the root-mean-square end-to-end distance for the two molecular weights?

4. The plateau shear modulus (GP) of an entangled polymer melt of poly(butadiene) is 1.15 MPa. The density of a poly(butadiene) melt is 900 kg m-3, and the molecular mass of its repeat unit is 54 g mole-1.(i) Calculate the molecular mass between physical entanglements.(ii) The viscosity (in Pa s) of the melt can be expressed as a function of the degree of polymerisation, N, and temperature, T (in degrees Kelvin), as:

Explain why has this functional form.(iii) Estimate the self-diffusion coefficient of poly(butadiene) in the melt at a temperature of 298 K when it has a molecular mass of 105 g mole-1.

433

1281404

10683 .]exp[.= NT

x