# Solution Thermodynamics Richard Thompson Department of Chemistry University of Durham

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

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

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 123 n... l

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 A more useful measure is the end-to-end distance r

A freely jointed chain in 1D … Each link (monomer) can step to either left or right with equal probability End-to-end distance vector of each monomer =  l l

End-to-End Distance If i = j then r i. r j = l 2 If i ≠ j then r i. r j = +l.+ l = l 2 or- l.- l = l 2 or- l.+ l = -l 2 or+ l.- l = -l 2 all with equal probability Hence i ≠ j = 0

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

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  = 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

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 , indicates that the result is for the valence angle model E.g. for polyethylene  = 109.5° and cos  ~ -1/3, hence,

Rotational isomeric state theory Steric effects lead to … –  is defined by  = 0 as the planar trans orientation – is the average of cos , based on the probability of each angle , determined by its associated energy and the Boltzmann relation –Generally |  º are the most energetically favourable angles –Steric effects cause chains to be more stretched –What about temperature effects????

In general –where  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 Steric parameter and the characteristic ratio

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

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 Excluded volume

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

The theta temperature For most polymer solutions  r 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? T >  r > 1 T =  r = 1 T <  r < 1 Often polymers will precipitate out of solution, rather than contracting

R.M.S. Radius of Gyration 1/2 Another way of characterising size –Defined as the average distance of chain segments from the centre of the chain –For linear polymers, –Particularly useful for branched/cyclic polymers Cannot meaningfully define an end-to-end distance R.M.S. radius of gyration is uniquely defined and a useful measure of size (or volume occupied)

Flory Huggins Theory Dissolution of polymer increases conformational entropy of system Molar entropy of mixing normally written as …where  i is the volume and volume fraction of each component (solvent = 1 and polymer = 2), r i is approximately the degree of polymerisation of each component (r 1 ~ 1, r 2 ~ N) Note that increasing the r 2 decreases the magnitude of  S mix

Flory Huggins Theory 2 Enthalpy of mixing  H Mix = kT  2 N 1 …where  is the dimensionless Flory Huggins parameter. For dilute solution of high molecular weight polymers, N~N 1  H Mix = RT  2 Remember condition for thermodynamically stable solution  G Mix =  H Mix - T  S Mix < 0

Practical Use of Polymer TDs Fractionation 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! T  2 volume fraction polymer p1 p2 2 phase cloudy 1 phase clear solution

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

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