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