Last Lecture:

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 diblock.• 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:

.++=)( constTNakR

RF 2

2

23

32315

2//)(= N

kTa

d

3SM

Polymers in Solvent; Rubber Elasticity

19 March, 2009

Lecture 9

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

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

But, when polymers are dissolved in solvents (e.g. water or acetone), they are often expanded to sizes greater than a random coil.

Such expanded conformations are described by a “self-avoiding walk” in which <R2>1/2 is given by aN (instead of aN1/2 as for a coil described by a random walk).

What is the value of ?

The conformation of polymer molecules in a polymer glass and in a melted polymer can be adequately described by random walk statistics.

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 microstates, , for a macrostate: 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 number 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 unit is excluded from the volume occupied by the other N units, each with a volume, b:

)ln(=))(

ln(= cbN

cRk

NNbRc

kSni

33

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

3

3

311

RbN

NcR

kRbN

NcR

kSni

)]ln[(+= 31RbN

kSS idealni

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

kbNSS idealni

RNth unit

Unit vol. = b

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

3RbN

kTFF 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

RbN

kTRFexc =)(

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

23NaR

kTRFel +=)(

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

.++=)( constTNakR

TR

kbNRFtot 2

2

3

2

23

The total free energy change 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=

NakRT

RkbTN

dRdFtot

Characterising the Self-Avoiding Walk

24

2 33NakRT

RkbTN

=

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:

But when are excluded volume effects important?

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 (polymer + solvent).

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). Finally, 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 (ps) contacts are made, while contacts between like molecules (pp + ss) 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. For a +ve , 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:

).)(--2(int unitsnowwwU SSPPPS 3

2

3 2)(2-R

bNkTRNNbkT

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

3

2

RbN

kTRFexc =)(

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

221

RN

kTbUFexc )(=+ 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

221

RN

kTbUFexc )(=+ int

• When < 1/2, the term is positive, and the excluded volume/energetic effects contribute to determining the coil size: Fexc + Uint > 0.

The solvent is called a “theta-solvent”.

5321

2 aNR =

as shown previously (considering the balance with the elastic energy). The molecule is said to be swollen in a “good solvent”.

Types of Solvent

3

221

RN

kTbUFexc )(=+ int

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

Both terms lower F (which is favourable) as R decreases. 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.

TNakR

RN

FFRF eltot 2

2

3

2

int 23

+~+=)(

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 CoilingNano-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.

Polymer Particles Adsorbed on a Positively-Charged Surface

Particles can contain small molecules such as a drug or a flavouring agent. Thus, they are a “nano-capsule”.

1 m 100 nm

Comparison of Particle Response in Solution and at an Interface

Light scattering from solution

Ellipsometry of adsorbed particles

Good solvent: particle is open

Bad solvent: particle is closed

Radius of Gyration of a Polymer Coil

RFor 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 radius of gyration is the root-mean square distance of an objects' parts from its center of gravity.

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 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

x

R2 = x2+y2+z2

Transformation with Affine Deformation

z

y

x

Bulk:

Ro

Single Strand

Ro = xo+ yo+ zo

R

R = xxo + yyo + zzo

If non-compressible: xyz =1

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

23

23

ozoyox zyxNa

kconst

NakR

RS

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

23

ozoyoxo zyxNa

kRSRS

Finding S:

)++(~)( 22222

3oooo zyx

Nak

RSInitially:

Entropy Change in Polymer Deformation])(+)(+)[(~ 222222

2 1112

3ozoyox zyx

Nak

S

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

2>=<yo2>=<zo

2>, so:

)](+)(+)[(~ 11123 222

2

2

zyxo

Nakx

S

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

Nak

S 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: (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), 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, 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.

Experiments on Rubber Elasticity

])1+(

1)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

AMN

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

mstrands 3

3

Looking at the units makes this equation easier to understand:

kTMN

nkTGx

A==Substituting for n:

strand

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