Solution Thermodynamics

1. Solution Thermodynamics Richard Thompson Department of Chemistry University of Durham r.l.thompson@dur.ac.uk

2. Overview Part 1 • Statistical thermodynamics of a polymer chain • How much space does a polymer chain occupy? Part 2 • Chemical thermodynamics of polymer solutions • What determines solubility of a polymer? Examine • (i) Models of polymer chain structure in solution • (ii) Interactions between polymers and solvents

3. l . . . 1 2 3 n The freely jointed chain • Simplest measure of a chain is the length along the backbone • For n monomers each of length l, the contour length is nl

4. A more useful measure is the end-to-end distance r • For an isolated polymer in a solvent the end-to-end distance will change continuously due to molecular motion • But many conformation give rise to the same value of r, and some values of r are more likely than others e.g., • Only one conformation with r = nl - a fully extended chain • Many conformation have r = 0, (cyclic polymers) • Define the root mean square end-to-end distance

5. See handout notes for derivation Key result for a freely jointed chain …

6. Bond angles and steric effects • Real chains are not freely jointed • Links between monomers subject to bond angle restrictions • Rotation hindered by steric effects • E.g., n-butane • Each bond angle q= 109.5° • Different conformations arise from rotation of 1 and 2 about 3-4 bond • Steric interactions between methyl groups  not all angles of rotation have the same energy

7. Valence angle model • Simplest modification to the freely jointed chain model • Introduce bond angle restrictions • Allow free rotation about bonds • Neglecting steric effects (for now) • If all bond angles are equal to q, indicates that the result is for the valence angle model • E.g. for polyethylene q = 109.5° and cos q ~ -1/3, hence,

8. Rotational isomeric state theory • Steric effects lead to … • f is defined by f = 0 as the planar trans orientation • <cosf > is the average of cosf , based on the probability of each angle f , determined by its associated energy and the Boltzmann relation • Generally | f | < 90º are the most energetically favourable angles • Steric effects cause chains to be more stretched • What about temperature effects????

9. Steric parameter and the characteristic ratio • In general • where s is the steric parameter, which is usually determined for each polymer experimentally • A measure of the stiffness of a chain is given by the characteristic ratio • C typically ranges from 5 - 12

10. An equivalent freely jointed chain … • A real polymer chain may be represented by an equivalent freely-jointed chain • Comprised of N monomersof length b such that the chains have the same contour length, i.e., Nb = nl • Normally has fewer, longer ‘joints’

11. Excluded volume • Freely jointed chain, valence angle and rotational isomeric states models all ignore • long range intramolecular interactions (e.g. ionic polymers) • polymer-solvent interactions • Such interactions will affect • Define where is the expansion parameter

12. The expansion parameter • ar depends on balance between i) polymer-solvent and ii) polymer-polymer interactions • If (ii) are more favourable than (i) • ar < 1 • Chains contract • Solvent is poor • If (ii) are less favourable than (i) • ar > 1 • Chains expand • Solvent is good • If these interactions are equivalent, we have theta condition • ar = 1 • Same as in amorphous melt

13. The theta temperature • For most polymer solutions ar depends on temperature, and increases with increasing temperature • At temperatures above some theta temperature, the solvent is good, whereas below the solvent is poor, i.e., What determines whether or not a polymer is soluble? Often polymers will precipitate out of solution, rather than contracting

14. Flory Huggins Theory • Dissolution of polymer increases conformational entropy of system • Molar entropy of mixing normally written as …where fi is the volume and volume fraction of each component (solvent = 1 and polymer = 2), ri is approximately the degree of polymerisation of each component (r1 ~ 1, r2 ~ N) • Note that increasing the r2 decreases the magnitude of DSmix

15. Flory Huggins Theory 2 • Enthalpy of mixing DHMix = kT cf2N1 …where c is the dimensionless Flory Huggins parameter. For dilute solution of high molecular weight polymers, N~N1 DHMix = RT cf2 Remember condition for thermodynamically stable solution DGMix = DHMix - TDSMix < 0

16. Practical Use of Polymer TDsFractionation • Consider solution in poor solvent of two polymers, p1 and p2. • Flory-Huggins tells us that if p2 has higher molecular weight it should precipitate more readily than p1 • add non-solvent until solution becomes turbid • heat, cool slowly and separate precipitate • finite drop in temperature always renders finite range of molecular weight insoluble • some p2 will also remain soluble! 1 phase clear solution T p2 2 phase cloudy p1 f2 volume fraction polymer

17. Summary A little knowledge goes a long way! • Simple models enable us to predict the size of polymer chains in solution • Critical to dynamic properties of solutions (next lecture) • Solubility of polymers generally decreases with increasing molecular weight. • Can exploit this in fractionation procedures to purify polymers • There are practical limits to how well fractionation can work