1. Last Lecture: The root-mean-squared end-to-end distance, 1/2, of a freely-jointed polymer molecule is N 1/2 a, 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:

2. 3SM Polymers in Solvent; Rubber Elasticity 19 March, 2009 Lecture 9 See Jones’ Soft Condensed Matter, Chapt. 5 and 9

3. 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 1/2 is given by aN (instead of aN 1/2 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.

4. 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 R 3. 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,  : where c is a constant

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

6. 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.

7. 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: 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.

8. Total Free Energy of an Expanded Coil The total free energy change is obtained from the sum of the two contributions: F exc + F el At equilibrium, the polymer coil will adopt an R that minimises F tot. At the minimum, dF tot /dR = 0: F el F tot R F exc F tot

9. Characterising the Self-Avoiding Walk So,  The volume of a repeat unit, b, can be approximated as a 3.  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?

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

11. Polymer/Solvent Interaction Energy So 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: w pp There is similarly an interaction energy between the solvent molecules (w ss ). Finally, when the polymer is dissolved in the solvent, a new interaction energy between the polymer units and solvent (w ps ) is introduced. w ss w ps

12. Polymer/Solvent  -Parameter When 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: 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 ,  U int is more negative and F is reduced. We note that N / R 3 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:

13. 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  U int. (But also - elastic effects, in which F el ~ R 2, are also still active!) As the form of the expressions for F exc and  U int 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.

14. Types of Solvent When  = 1/2, the two effects cancel: F exc +  U int = 0. The coil size is determined by elastic (entropic) effects only, so it adopts a random-coil conformation. When  0. The solvent is called a “ theta-solvent ”. as shown previously (considering the balance with the elastic energy). The molecule is said to be swollen in a “ good solvent ”.

15. Types of Solvent When  > 1/2, the term goes negative, and the polymer/solvent interactions dominate in determining the coil size. F exc +  U int < 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.

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

17. 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

18. 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.

19. 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

20. 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

21. Radius of Gyration of a Polymer Coil R For a hard, solid sphere of radius, R, the radius of gyration, R g, is: R A polymer coil is less dense than a hard, solid sphere. Thus, its R g 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.

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

23. 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: Bulk: l Strand: lolo

24. x z y z y x R 2 = x 2 +y 2 +z 2 Transformation with Affine Deformation z y x Bulk: RoRo Single Strand R o = x o + y o + z o R R = x x o + y y o + z z o If non-compressible: x y z =1

25. 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(R o ) = S( x x o, y y o, z z o ) - S(x o, y o, z o ) Finding  S: Initially:

26. Entropy Change in Polymer Deformation But, if the conformation of the coil is initially random, then = =, so: For a random coil, =Na 2, and also R 2 = x 2 +y 2 +z 2 = 3x 2, so we see: Substituting: This simplifies to:

27.  F for Bulk Deformation If the rubber is incompressible (volume is constant), then x y z =1. For a one-dimensional stretch in the x-direction, we can say that x =. Incompressibility then implies Thus, for a one-dimensional deformation of x = : The corresponding change in free energy: (F = U - ST) will be If there are n strands per unit volume, then  S per unit volume for bulk deformation:

28. Force for Rubber Deformation At the macro-scale, if the initial length is L o, then  = L/L o. Substituting in L/L o =  + 1: Realising that  F bulk 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. In Lecture 3, we saw that  T = Y . The strain, , for a 1-D tensile deformation is

29. Young’s and Shear Modulus for Rubber 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.

30. Experiments on Rubber Elasticity Treloar, Physics of Rubber Elasticity (1975) Rubbers are elastic over a large range of ! Strain hardening region: Chain segments are fully stretched!

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

32. 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