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Last Lecture:

Last Lecture:. The root-mean-squared end-to-end distance, < R 2 > 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

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Last Lecture:

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  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 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”. The conformation of polymer molecules in a polymer glass and in a melted polymer can be adequately described by random walk statistics. 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 aNn(instead of aN1/2 as for a coil described by a random walk). What is the value of n?

  4. Excluded Volume In an ideal polymer coil with no excluded volume, is inversely related to the number density of units, r : where c is a constant 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 microstates, , for a macrostate:S = k ln.

  5. Entropy with Excluded Volume Unit vol. = b R Nth unit 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: Hence, the entropy for each repeat unit in an ideal polymer coil is

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

  7. 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.

  8. Ftot Ftot Fel Fexc R Total Free Energy of an Expanded Coil 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:

  9. So, The volume of a repeat unit, b, can be approximated as a3.   Characterising the Self-Avoiding Walk Re-arranging: This result agrees with a more exact value of n obtained via a computational method: 0.588 Measurements of polymer coil sizes in solvent also support the theoretical (scaling) result. But when are excluded volume effects important?

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

  11. Polymer/Solvent Interaction Energy wss wps 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: 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.

  12. Polymer/Solvent -Parameter 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. 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: When a polymer is dissolved in solvent, new polymer-solvent (ps) contacts are made, while contacts between like molecules (pp + ss) are lost. 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. Observe that smaller coils reduce the number of P-S contacts because more P-P contacts are created. For a +ve c, DUint is more negative and F is reduced.

  13. As the form of the expressions for Fexc and DUint are the same, they can be combined into a single equation: 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!) The value ofcthen tells us whether the excluded volume effects are significant or whether they are counter-acted by polymer/solvent interactions.

  14. Types of Solvent The coil size is determined by elastic (entropic) effects only, so it adopts a random-coil conformation. •When c = 1/2, the two effects cancel: Fexc + DUint = 0. The solvent is called a “theta-solvent”. • Whenc< 1/2, the term is positive, and the excluded volume/energetic effects contribute to determining the coil size: Fexc + DUint > 0. 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 Whenc> 1/2, the term goes negative, and 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. Both terms lower F (which is favourable) as R decreases. The molecule forms a globule in a “bad solvent”.

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

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

  18. A possible “nano-motor”! c> 1/2 c< 1/2 A Nano-Motor? • The transition from an expanded coil to a globule can be initiated by changing c. 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 100 nm 1 mm Particles can contain small molecules such as a drug or a flavouring agent. Thus, they are a “nano-capsule”.

  20. Comparison of Particle Response in Solution and at an Interface Ellipsometry of adsorbed particles Bad solvent: particle is closed Light scattering from solution Good solvent: particle is open

  21. A polymer coil is less dense than a hard, solid sphere. Thus, its Rg is significantly less than the rms-R: R Radius of Gyration of a Polymer Coil The radius of gyration is the root-mean square distance of an objects' parts from its center of gravity. For a hard, solid sphere of radius, R, the radius of gyration, Rg, is: R

  22. 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.

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

  24. z z R = lxxo + lyyo + lzzo Ro = xo+ yo+ zo Single Strand Ro R y y x x Transformation with Affine Deformation z Bulk: y R2 = x2+y2+z2 x If non-compressible: lxlylz =1

  25. Initially: Finding DS: 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:

  26. 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: Entropy Change in Polymer Deformation

  27. 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: (F = U - ST) will be DF for Bulk Deformation If there are nstrands per unit volume, then DS per unit volume for bulk deformation: If the rubber is incompressible (volume is constant), then lxlylz =1.

  28. In Lecture 3, we saw that sT = Ye. The strain, e, for a 1-D tensile deformation is Substituting inL/Lo = e + 1: Force for Rubber Deformation At the macro-scale, if the initial length is Lo, thenl = L/Lo. 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.

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

  31. Alternative Equation for a Rubber’s G  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: strand Looking at the units makes this equation easier to understand: Substituting for n: We have shown that G = nkT, where n is the number of strands per unit volume.

  32. Network formed by H-bonding of small molecules Blue = ditopic (able to associate with two others) Red = tritopic (able to associate with three others) H-bonds can re-form when surfaces are brought into contact. For a video, see: http://news.bbc.co.uk/1/hi/sci/tech/7254939.stm P. Cordier et al., Nature (2008) 451, 977

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