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
1
2
3
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
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
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
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,
Rotational isomeric state theory
•
Steric effects lead to …
–
f
is defined by
f
= 0 as the planar trans orientation
–
<cos
f
> is the average of cos
f
, based on the probability of each
angle
f
, determined by its associated energy and the Boltzmann
relation
–
Generally 
f
 90
º are the most energetically favourable angles
–
Steric effects cause chains to be more stretched
–
What about temperature effects????
•
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
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
>
q
r
> 1
T
=
q
r
= 1
T
<
q
r
< 1
Often polymers will precipitate out of solution,
rather than contracting
Flory Huggins Theory
•
Dissolution of polymer increases conformational entropy of system
•
Molar entropy of mixing normally written as
…where
f
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
D
S
mix
Flory Huggins Theory 2
•
Enthalpy of mixing
D
H
Mix
= kT
cf
2
N
1
…where
c
is the dimensionless Flory Huggins parameter.
For dilute solution of high molecular weight polymers, N~N
1
D
H
Mix
= RT
cf
2
Remember condition for thermodynamically stable solution
D
G
Mix
=
D
H
Mix

T
D
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
f
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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