# Solution Thermodynamics - Durham University

Mechanics

Oct 27, 2013 (4 years and 6 months ago)

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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

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

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

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

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
than p1

-
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