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102
Fortunately, the end-to-end distance of random coils is related to R
G
(no proof
given):
2
G
1 R
r
6
=
Moreover, R
G
is (of course) the root-mean-square (r.m.s.) average just like
<r>:
.
We normally just write (and say) R
G
, but for flexible chains this actually means
the r.m.s. radius given above.
All in all, by combining equations above one obtains for the freely jointed
chain (vector model) R
G
∝
√
n
∝
√
M
R
G
∝
M
0.5
log
R
G
=
0.5log
M
+
const
.
(
n
→ ∞
)
Although the vector model is a dramatic simplification of real flexible chains,
we can see that it holds for pullulan. In the example above (R
G
vs. M) the log-
log plot gives an exponent quite close to 0.5.
2.2.6. Real
chains
In contrast to the vector model, the shape and extension of real chains are
also influenced by:
•
fixed angles between monomers
•
restricted rotation around bonds
R
G
=
R
G
2