1. 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.

2. Free Energy Minimisation
Poly(styrene) and poly(methyl methacrylate) copolymer
Two different dependencies on d!  
Chains are NOT fully stretched - but nor are they randomly coiled!
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).

3. Polymers in Solvent and Rubber Elasticity 3SCMP
16 March, 2006
Lecture 9
See Jones’ Soft Condensed Matter, Chapt. 5 and 9

4. Radius of Gyration of a Polymer Coil
For a hard, solid sphere of radius, R, the radius of gyration, Rg, is: R R
A polymer coil is less dense than a hard, solid sphere. Thus, its Rg is significantly less than the rms-R:

5. 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 aNn (instead of N1/2 as for a random coil).
What is the value of n?

6. 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.
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!
From the Boltzmann equation, we know that entropy, S, can be calculated from the number of ways of arranging a system, : S = k ln .
In an ideal polymer coil with no excluded volume,  is inversely related to the density of units,r :
where c is a constant

7. Entropy with Excluded Volume
Hence, the entropy for each repeat unit in an ideal polymer coil is R
Unit vol. = b
Nth unit
In the non-ideal case, however, each segment is excluded from the volume occupied by the other N segments, each with a volume, b:
But if x is small, then ln(1-x)  -x, so: 

8. 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:
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
Larger R values reduce the free energy. Hence, expansion is favoured by excluded volume effects.

9. Elastic Contributions to F
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:
Reducing the R by coiling will decrease the free energy.
Coiling up of the molecules is therefore favoured by elastic contributions.

10. Total Free Energy of an Expanded Coil
The total free energy is obtained from the sum of the two contributions: Fexc + Fel
Ftot R
Fexc
Ftot
Fel
At equilibrium, the polymer coil will adopt an R that minimises Ftot. At the minimum, dFtot/dR = 0:

11. Characterising the Self-Avoiding Walk
Re-arranging:
So,
The volume of a repeat unit, b, can be approximated as a3.  
This result agrees with a more exact value of n obtained via a computational method:
Measurements of polymer coil sizes in solvent also support the theoretical (scaling) result.

12. Visualisation of the Self-Avoiding Walk
2-D Self-avoiding walks
2-D Random walks

13. Polymer/Solvent Interaction Energy
So 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
wss
wps
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.

14. Polymer/Solvent -Parameter
When 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 c-parameter for polymer units in solvent:
where z is the number of neighbour contacts per unit or solvent molecule.
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, DU:
Observe that smaller coils reduce the number of P-S contacts because more P-P contacts are created. DUint is more negative and F is reduced.

15. Significance of the -Parameter
We recall that excluded volume effects favour coil swelling:
Opposing the swelling, will be the polymer/solvent interactions, as described by DUint. (But also - elastic effects, in which Fel ~ R2, are also still active!)
As the form of the expressions for Fexc and DUint are the same, they can be combined into a single equation:
The value of c then tells us whether the excluded volume effects are significant or whether they are counter-acted by polymer/solvent interactions.

16. Types of Solvent
• When c = 1/2, the two effects cancel: Fexc + DUint = 0.
The coil size is determined by elastic (entropic) effects only, so it adopts a random-coil conformation.
The solvent is called a “theta-solvent”.
• When c < 1/2, then the excluded volume effects contribute to determining the coil size: Fexc + DUint > 0.
The molecule is swollen in a “good solvent”.

17. Types of Solvent
When c > 1/2, then the polymer/solvent interactions dominate in determining the coil size. Fexc + DUint < 0.
Energy is reduced by coiling up the molecule (i.e. by reducing its R).
Elastic (entropic) contributions likewise favour coiling.
The molecule forms a globule in a “bad solvent”.

18. Determination of Polymer Conformation
Scattering Intensity, I  q-1/n or I-1  q1/n
Good solvent: I  q1/(3/5)
Theta solvent: I  q1/(1/2)

19. Applications of Polymer Coiling
Nano-valves
Bad solvent: “Valve open”
Good solvent: “Valve closed”
Switching of colloidal stability
Good solvent: Sterically stabilised
Bad solvent: Unstabilised

20. A Nano-Motor?
• The transition from an expanded coil to a globule can be initiated by changing c.
c < 1/2
A possible “nano-motor”!
c > 1/2
Changes in temperature or pH can be used to make the polymer coil expand and contract.

21. Rubber Elasticity
A rubber (or elastomer) can be created by linking together linear polymer molecules into a 3-D network. 
Chemical bonds between polymer molecules are called “crosslinks”. Sulphur can crosslink natural rubber.
To observe “stretchiness”, the temperature should be > Tg for the polymer.

22. Affine Deformation
With an affine deformation, the macroscopic change in dimension is mirrored at the molecular level.
We define an extension ratio, l, as the dimension after a deformation divided by the initial dimension:
Bulk: l
Strand: lo

23. Transformation with Affine Deformationz
Bulk: y x z y x z y x R
R = lxxo + lyyo + lzzo
Ro = xo+ yo+ zo Ro
Single Strand
R2 = x2+y2+z2

24. Entropy Change in Deforming a Strand
The entropy change when a single strand is deformed, DS, can be calculated from the difference between the entropy of the deformed coil and the unperturbed coil:
DS = S(R) - S(Ro) = S(lxxo, lyyo, lzzo) - S(xo, yo, zo)
We recall our expression for the entropy of a polymer coil with end-to-end distance, R:
Initially:
Finding DS:

25. Entropy Change in Polymer Deformation
But, if the conformation of the coil is initially random, then <xo2>=<yo2>=<zo2>, so:
For a random coil, <R2>=Na2, and also R2 = x2+y2+z2 = 3x2, so we see:
Substituting:
This simplifies to:

26. DF for Bulk Deformation
If there are n strands per unit volume, then DS per unit volume for bulk deformation:
If the rubber is incompressible (volume is constant), then lxlylz=1.
For a one-dimensional stretch in the x-direction, we can say that lx = l. Incompressibility then implies
Thus, for a one-dimensional deformation of lx = l:
The corresponding change in free energy will be

27. Force for Rubber Deformation
If the initial length is Lo, then l = L/Lo.
In lecture 3, we saw that sT = Ye. The strain, e, for a 1-D tensile deformation is
Substituting:
Realising that DFbulk is an energy of deformation (per unit volume), then dF/deT is the force (per unit area) for the deformation, i.e. the tensile stress, sT.

28. Young’s and Shear Modulus for Rubber
In the limit of small strain, sT  3nkTe, 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.

29. Experiments on Rubber Elasticity
Rubbers are elastic over a large range of l!
Treloar, Physics of Rubber Elasticity (1975)

30. Alternative Equation for a Rubber’s G
We have shown that G = nkT, where n is the number of strands per unit volume. 
strand
For a rubber with a known density, r, in which the average molecular mass of a strand is Mx (m.m. between crosslinks), we can write:
Looking at the units makes this equation easier to understand:
Substituting for n: