Chemical Engineering 160/260
Polymer Science and Engineering
Lecture 2 - Polymer Chain
Configuration and Conformation
January 12, 2001
Sperling, Ch 2
Outline
■
■
Polymer Isomerism for All-carbon Backbones
◆ Constitutional
◆ Configurational
◆ Conformational
Spatial Parameters for Polymer Chains
◆ End-to-end distance
◆ Radius of gyration
◆ Freely-jointed model chain
Spatial Parameters for All-carbon Backbones
1.54 Å
109.5°
2.52 Å
Contour length of the fully extended chain = nl
n = number of repeat units
l = length of repeat unit
Constitutional Isomerism
Head-to-head defects
-CH2-C-CH2-C-CH2
Configuration of Pseudoasymmetric Carbon
Stereoconfiguration
!"
#$
#%&'
# !"
#$
#%'
Pseudoasymmetric Carbons:
Meso Dyad
&
(
&
(
+
)#
*
#
+ $ +,
Pseudoasymmetric Carbons:
Racemic Dyad
& ( &
+
)#
*
#
+ $ +,
Configurational Isomerism
%&'%&'%&'%&'%&'%&'%&'%&'
%&'%&'%&'%&'%&'%&'%&'%&'
%&'%&'%&'%&'%&'%&'%&'%&'-* Conformational Isomerism
. #)
%+'
-$+
+*
$ # $#$$$!$$
,
/$!$$
0
1,
)&),
23
,
Conformational Isomerism
4"*5
$,
6! )
$
+,
7
8**
7
7
7
Conformational Isomerism in Butane
9 +: # ,,7+9,;,9 *
<<*=>,
Meso (dd, ll) Rotational Dyad
) &
)
)&
)&)&
)
)&)
))&
))
Racemic (dl, ld) Rotational Dyad
)
) &
)&
)&)&
)&)
)
))
))&
Outline
■
■
Polymer Isomerism for All-carbon Backbones
◆ Constitutional
◆ Configurational
◆ Conformational
Spatial Parameters for Polymer Chains
◆ End-to-end distance
◆ Radius of gyration
◆ Freely-jointed model chain
End-to-end Distance of a Random Coil
?) "" -*
@#
A),
〈r 〉
2 1/ 2
+;: 9 +
θ
πθ
πθ
θ
φ
>
.
#$ ) -
,
4
*
A $) %πθ' ) %φ'+*+
+,
+;: 9 +
:
@#/
n r
r
r = ∑ li
r
2
i =1
r
r
r r
r r
r r
= r • r = ∑l • l = ∑l + 2 ∑l • l
2
i
j
i
i
0 <i < j ≤ n
i
i, j
j
r r
〈 r 〉 = ∑ 〈li 〉 + 2 ∑ 〈li • l j 〉
2
2
Average over
configurations.
i
r r
2
〈 r 〉 = nl + 2∑ 〈li • l j 〉 = nl
2
Vector
2
i< j
0 <i < j ≤ n
There is no
correlation.
Freely Jointed Model Polymer Chain:
Mean-squared End-to-end Distance
Relationship to molecular weight
〈 r 〉 = nl ≈ M
2
2
Characteristic ratio
〈r 〉 o
Cn =
2
nl
2
The subscript refers to the
unperturbed reference state in
which there are no external forces
or solvent effects.
Cn = 1 for a freely rotating model chain.
Equivalent Chain
One may account for fixed bond angles and hindered
rotation by letting several real bonds be represented by
a longer equivalent bond.
Real chain: n, l
Equivalent chain: n’, l’
n’ < n, l’ > l
〈 r2 〉 = nl 2 = n' (l' ) 2
n' (l' ) 2
C=
nl 2
n' ∝ M
1/ 2
Mean-squared Radius of Gyration
Ao
A1
s1
s0
sn
An
n
s =
2
2
m
s
∑ ii
0
n
∑m
i
0
A n-2
sn-2
Center of mass
sn-1
A n-1
n
m∑ si 2
n
1
2
0
s
=
=
∑
m(n + 1) n + 1 0 i
2 1/ 2
s
〈 〉 ≡ Rg
Mean-squared Radius of Gyration
The mean-squared radius of gyration is related to the
mean-squared end-to-end distance by
2
nl
 n + 2
2
〈 s 〉o =
6  n + 1
Details of this calculation may be found in Statistical
Mechanics of Chain Molecules, P.J. Flory, 1969, pp 16-17.
In the limit as
n→∞
2
〈
〉o
r
2
〈s 〉 =
= Rg 2
6
Rg may be determined from light, x-ray, or neutron
scattering.