Lecture handout including QS - Department of Materials Science ...

BH19 Course B: Materials for Devices BH19
How big is a polymer molecule
A polymer chain can be modelled as a series of shorter, rigid segments (each of length l ) that can
rotate freely where they join.
“Random walk”:
r 1
• If the chain consists of n segments, and it
were stretched out straight, its length, L, would
be nl.
• end-to-end vector,
• A useful measure of the size of the polymer molecule is given by the
2
root mean square of R n
: R n
€ €
r n-1
r 2
r 3 r 4
R n -1
R n
r n
€
also,
R n
=
R n−1
=
n
∑ and
• The average value € of the end to end vector,
R n
, is zero: R n
= 0
€
i=1
r i
n−1
∑r i
i=1
R n
= R
n−1
+ r
n
1 2
where γ is the angle between the vector
R 2 n
= ( R n−1
+ r n ) ⋅
Rn−1 + r n
€ 2
= R n−1
+ € 2R n−1
× l cosγ
( ) = R 2
n−1
( ) + l 2
The average value of cosγ must be zero, and therefore,
( ) + r n
+ 2 R n−1
⋅ r n
and the last segment,
R n
2
2
= R 2
n−1
€ = R 2
n−2
r n
+ l 2
€
+ l 2 + l 2 etc.
R n−1
And, by induction:
mean square end-to-end distance,
R n
2
= nl 2
root mean square,
R n
2
1 2
= n 1 2
l
We might assume that each segment of the chain, € l, corresponds to a single C-C bond. But such
segments can never be completely freely jointed (bond angles are limited, and adjacent segments can
simply get in the way of each other). € Hence real polymer chains are always stiffer than shown by this
model using a single C-C bond as a segment length, and their stiffness is dependent upon the nature
of the monomer units involved.
The model can be used more generally if we define l, the segment length, as the length scale below
which the chain is effectively straight and rigid – this is called the Kuhn length. The simple
polyethylene structure has a Kuhn length of about 3.5 times the C-C bond length, whilst polystyrene,
which has more bulky side-groups, has a Kuhn length of 5 × C-C (it’s stiffer).