Solution Thermodynamics

1
Solution Thermodynamics

• Richard Thompson
• Department of Chemistry
• University of Durham
• r.l.thompson_at_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
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
A freely jointed chain in 1D

• Each link (monomer) can step to either left or
  right with equal probability
• End-to-end distance

l
vector of each monomer ? l
6
End-to-End Distance

• If i j then ri . rj l2
• If i ? j then ri . rj l. l l2
• or - l.- l l2
• or - l. l -l2
• or l.- l -l2
• all with equal probability
• Hence lt ri . rj gt i ? j 0

7
See handout notes for derivation Key result for
a freely jointed chain
8
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

9
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,

10
Rotational isomeric state theory

• Steric effects lead to
• f is defined by f 0 as the planar trans
  orientation
• ltcosf gt is the average of cosf , based on the
  probability of each angle f , determined by its
  associated energy and the Boltzmann relation
• Generally f lt 90º are the most energetically
  favourable angles
• Steric effects cause chains to be more stretched
• What about temperature effects????

11
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

12
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

13
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

14
The expansion parameter

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

15
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?

T gt q ar gt 1
T q ar 1
T lt q ar lt 1
Often polymers will precipitate out of solution,
rather than contracting
16
R.M.S. Radius of Gyration lts2gt1/2

• 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)

17
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

18
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, NN1
• DHMix RT cf2
• Remember condition for thermodynamically stable
  solution
• DGMix DHMix - TDSMix lt 0

19
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
20
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