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.

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

• Copolymers can be random, statistical, alternating or block.

• 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 must be reached.

.++=)( constTNa

kRRF 2

2

2

3

Free Energy Minimisation

dNa

Na

dkTFtot

3

2

2 +

2

3

220d

Na

Na

dkT

dddFtot

=)(

2

3

22d

Na

Na

dkT

=

32315

2//)(= N

kTa

d Chains are NOT fully stretched -

but nor are they randomly coiled!

kTaN

d2

523

=

Two different dependencies on d!

Poly(styrene) and poly(methyl

methacrylate) copolymer

The thickness, d, of lamellae created by diblock copolymers is proportional to N2/3. Thus, the molecules are not fully-stretched (d ~ N1) but nor are they randomly coiled (d ~ N1/2).

Polymers in Solvent and Rubber Elasticity

3SCMP

16 March, 2006

Lecture 9

See Jones’ Soft Condensed Matter, Chapt. 5 and 9

Radius of Gyration of a Polymer Coil

R

For a hard, solid sphere of radius, R, the radius of gyration, Rg, is:

RRRRg 6320510

52

.===

21212

661

Na

RRg ==

R

A polymer coil is less dense than a hard, solid sphere. Thus, its Rg is significantly less than the rms-R:

The Self-Avoiding Walk

In describing the polymer coil as a random walk, it was tacitly assumed that the chain could “cross itself”.

In reality, of course, it cannot. Consequently, when polymers are dissolved in solvents, they are often expanded to sizes greater than a random coil.

Such expanded conformations are described instead by a “self-avoiding walk” in which <R2>1/2 is given by aN (instead of N1/2 as for a random coil).

What is the value of ?

Excluded Volume

Paul Flory developed an argument in which a polymer in a solvent is described as N repeat units confined to a volume of R3.

From the Boltzmann equation, we know that entropy, S, can be calculated from the number of ways of arranging a system, : S = k ln .

Each repeat unit prevents other units from occupying the same volume. The entropy associated with the chain conformation (“coil disorder”) is decreased by the presence of the other units. There is an excluded volume!

In an ideal polymer coil with no excluded volume, is inversely related to the density of units,:

NcR

RN

cc 3

3

~~~

where c is a constant

Entropy with Excluded Volume Hence, the entropy for each repeat unit in an ideal polymer coil is

)ln(=ln=N

cRkkSideal

3

In the non-ideal case, however, each segment is excluded from the volume occupied by the other N segments, each with a volume, b:

)ln(=))(

ln(= cbN

cRk

NNbRc

kSni

33

)]ln(+)[ln(=)](ln[= 3

3

3

3

11R

bNN

cRk

R

bNN

cRkSni

)]ln[(+= 31R

bNkSS idealni

But if x is small, then ln(1-x) -x, so:3R

kbNSS idealni

RNth unit

Unit vol. = b

Excluded Volume Contribution to F

For each unit, the entropy decrease from the excluded volume will lead to an increase in the free energy, as F = U - TS:

3R

bNkTFF idealni +=

Of course, a polymer molecule consists of N repeat units, and so the increase in F for a molecule, as a result of the excluded volume, is

3

2

R

bNkTRFexc =)(

Larger R values reduce the free energy. Hence, expansion is favoured by excluded volume effects.

In last week’s lecture, however, we saw that the coiling of molecules increased the entropy of a polymer molecule. This additional entropy contributes an elastic contribution to F:

2

2

2

3

Na

RkTRFel +=)(

Elastic Contributions to F

Coiling up of the molecules is therefore favoured by elastic contributions.

Reducing the R by coiling will decrease the free energy.

Total Free Energy of an Expanded Coil

.++=)( constTNa

kRT

R

kbNRFtot 2

2

3

2

2

3

The total free energy is obtained from the sum of the two contributions: Fexc + Fel

At equilibrium, the polymer coil will adopt an R that minimises Ftot. At the minimum, dFtot/dR = 0:

Fel

Ftot

RFexc

Ftot

24

2 3+

3=0=

Na

kRT

R

kbTNdR

dFtot

Characterising the Self-Avoiding Walk

24

2 33

Na

kRT

R

kbTN=

325 bNaR =So,

53 /= aNaNR

The volume of a repeat unit, b, can be approximated as a3.

355 NaR

This result agrees with a more exact value of obtained via a computational method: 0.588

Measurements of polymer coil sizes in solvent also support the theoretical (scaling) result.

Re-arranging:

Visualisation of the Self-Avoiding Walk2-D Random walks

21212 //= aNR

2-D Self-avoiding walks

53212 //= aNR

Polymer/Solvent Interaction EnergySo far, we have neglected the interaction energies between the components of a polymer solution.

Units in a polymer molecule have an interaction energy with other nearby (non-bonded) units: wpp

There is similarly an interaction energy between the solvent molecules (wss), and when the polymer is dissolved in the solvent, a new interaction energy between the polymer units and solvent (wps) is introduced.

wss

wps

Polymer/Solvent -ParameterWhen a polymer is dissolved in solvent, new polymer-solvent contacts are made, while contacts between like molecules are lost.

Following arguments similar to our approach for liquid miscibility, we can derive a -parameter for polymer units in solvent:

( )SSPPPS wwwkTz

= 22

where z is the number of neighbour contacts per unit or solvent molecule.

Observe that smaller coils reduce the number of P-S contacts because more P-P contacts are created. Uint is more negative and F is reduced.

We note that N/R3 represents the concentration of the repeat units in the “occupied volume”, and the volume of the polymer molecule is Nb.

When a polymer is added to a solvent, the change in potential energy, (from the change in w) will cause a change in internal energy, U:

3int )(=).)(2(=R

NNbkTunitsnowwwU SSPPPS 2

Significance of the -ParameterWe recall that excluded volume effects favour coil swelling:

3

2

R

bNkTRFexc =)(

Opposing the swelling, will be the polymer/solvent interactions, as described by Uint. (But also - elastic effects, in which Fel ~ R2, are also still active!)

3

2

21R

NkTbUFexc )(=+ int

As the form of the expressions for Fexc and Uint are the same, they can be combined into a single equation:

The value of then tells us whether the excluded volume effects are significant or whether they are counter-acted by polymer/solvent interactions.

Types of Solvent

• When = 1/2, the two effects cancel: Fexc + Uint = 0.

The coil size is determined by elastic (entropic) effects only, so it adopts a random-coil conformation.

2121

2 aNR =

3

2

21R

NkTbUFexc )(=+ int

• When < 1/2, then the excluded volume effects contribute to determining the coil size: Fexc + Uint > 0.

The solvent is called a “theta-solvent”.

5321

2 aNR =

The molecule is swollen in a “good solvent”.

Types of Solvent

3

2

21R

NkTbUFexc )(=+ int

When > 1/2, then the polymer/solvent interactions dominate in determining the coil size. Fexc + Uint < 0.

The molecule forms a globule in a “bad solvent”.

Energy is reduced by coiling up the molecule (i.e. by reducing its R).

Elastic (entropic) contributions likewise favour coiling.

TNa

kR

R

NFFRF eltot 2

2

3

2

int 2

3+~+=)(

Determination of Polymer Conformation

Good solvent: I q1/(3/5)

Scattering Intensity, I q-1/ or I-1 q1/

Theta solvent: I q1/(1/2)

Applications of Polymer Coiling

Nano-valves

Bad solvent: “Valve open”

Good solvent: “Valve closed”

Switching of colloidal stability

Good solvent: Sterically stabilisedBad solvent: Unstabilised

A Nano-Motor?

• The transition from an expanded coil to a globule can be initiated by changing .

A possible “nano-motor”!

> 1/2 < 1/2

Changes in temperature or pH can be used to make the polymer coil expand and contract.

Rubber ElasticityA rubber (or 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.

Chemical bonds between polymer molecules are called “crosslinks”. Sulphur can crosslink natural rubber.

Affine Deformation

With 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

Single Strand

Ro = xo+ yo+ zo

R

R = xxo + yyo + zzo

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:

)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

22

nkSbulk

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

The corresponding change in free energy will be

)+(+~ 32

22

nkTFbulk

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

Force for Rubber DeformationIf the initial length is Lo, then = L/Lo.

)+)((+~ 32

22

LL

LLnkT

F o

obulk

Substituting: ))+(

+)+((+~ 31

21

22

nkTFbulk

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

])1+(

2)1+(2[

2== 2

nkTddF

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

Young’s and Shear Modulus for Rubber

])1+(

1)1+[(= 2

nkTT

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, 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.

Experiments on Rubber Elasticity

])1+(

1)1+[(= 2

nkTT

Treloar, Physics of Rubber Elasticity (1975)

Rubbers are elastic over a large range of !

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