It was also found [69] that in two dimensions (relevant for strongly adsorbed
semiflexible polymers), excluded volume effects are much stronger than in three
dimensions. This is not just because the exponent ν in the relation hR
2
e,b i / N b
2n
,
ν ¼ 3/4 [68] differs more strongly from the Gaussian value ν ¼ 1/2 than the value
ν ¼ 0.59 in three dimensions. As mentioned above, for very stiff and thin polymers
in three dimensions, a regime of Gaussian behavior occurs for chains of intermediate
length, in between the rod regime (for short chains) and the self-avoiding walk regime
(for long chains). In two dimensions, however, there is never a Gaussian regime
intermediate between the rod regime (hR
2
e,b i ~ N b
2
) and the swollen coil
regime (hR
2
e,b i ~ N b
2ν
¼ N b
3/2
), rather the two regimes join smoothly for N b l b ¼ l p .
It is also interesting to note that for a given energy of local bond bending, the
persistence length l p depends distinctly on the conditions in which the polymer exists:
for an adsorbed stiff thin chain, l p is about 2.4 times larger than when the same chain
is not yet adsorbed. Also, the solvent conditions matter and this has been
demonstrated by a simulation [71] of a bead-spring model of a bottlebrush polymer,
comparing good solvent conditions (a temperature about 30% higher than the Theta
temperature) with Theta conditions. At the Theta temperature, the persistence length
is smaller, as expected, because the cross-sectional radius is also smaller.
3.3 Experiments on Individual Brushes
The main conformation of cylindrical brush polymers is governed by two opposing
forces: The steric repulsion of the densely grafted side chains leads to stretching of
the side chains and of the main chain, whereas the entropy elasticity causes both
main and side chains to adopt a coiled conformation. In the case of rigid side chains,
the arguments above apply to the main chain only. Rigid side chains were
investigated by analytical theory as well as by simulations, but experimental data
are rare (see below). An increase in main chain stiffness with the length of the side
chains was one major goal of experimental investigations. Usually, the
Kratky–Porod wormlike chain model is applied to fit experimental data such as
form factors, dimensions such as the radius of gyration (R g ) and the hydrodynamic
radius (R h ), and intrinsic viscosities. For a reliable application of the wormlike
chain model, the contour length, L, needs to be known. Typically L ¼ l m P, with
P being the (main chain) degree of polymerization and l m the length per repeat unit.
For vinyl main chains, l m ¼ 0.25 nm. There was considerable discussion about
whether l m ¼ 0.25 nm also applies to cylindrical brush polymers, because AFM
investigations resulted in smaller cylinder lengths than expected by the weightaverage degree of polymerization, P w , determined by static light scattering (SLS),
even if corrected for main chain polydispersity [72–74]. It was speculated that
within the cylindrical geometry the main chain was locally coiled, but persistent on
larger length scales. This picture seemed to be supported by simulations, which
showed a bimodal decay of the bond angle correlation function [66, 70, 75–77].
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
135
semiflexible polymers), excluded volume effects are much stronger than in three
dimensions. This is not just because the exponent ν in the relation hR
2
e,b i / N b
2n
,
ν ¼ 3/4 [68] differs more strongly from the Gaussian value ν ¼ 1/2 than the value
ν ¼ 0.59 in three dimensions. As mentioned above, for very stiff and thin polymers
in three dimensions, a regime of Gaussian behavior occurs for chains of intermediate
length, in between the rod regime (for short chains) and the self-avoiding walk regime
(for long chains). In two dimensions, however, there is never a Gaussian regime
intermediate between the rod regime (hR
2
e,b i ~ N b
2
) and the swollen coil
regime (hR
2
e,b i ~ N b
2ν
¼ N b
3/2
), rather the two regimes join smoothly for N b l b ¼ l p .
It is also interesting to note that for a given energy of local bond bending, the
persistence length l p depends distinctly on the conditions in which the polymer exists:
for an adsorbed stiff thin chain, l p is about 2.4 times larger than when the same chain
is not yet adsorbed. Also, the solvent conditions matter and this has been
demonstrated by a simulation [71] of a bead-spring model of a bottlebrush polymer,
comparing good solvent conditions (a temperature about 30% higher than the Theta
temperature) with Theta conditions. At the Theta temperature, the persistence length
is smaller, as expected, because the cross-sectional radius is also smaller.
3.3 Experiments on Individual Brushes
The main conformation of cylindrical brush polymers is governed by two opposing
forces: The steric repulsion of the densely grafted side chains leads to stretching of
the side chains and of the main chain, whereas the entropy elasticity causes both
main and side chains to adopt a coiled conformation. In the case of rigid side chains,
the arguments above apply to the main chain only. Rigid side chains were
investigated by analytical theory as well as by simulations, but experimental data
are rare (see below). An increase in main chain stiffness with the length of the side
chains was one major goal of experimental investigations. Usually, the
Kratky–Porod wormlike chain model is applied to fit experimental data such as
form factors, dimensions such as the radius of gyration (R g ) and the hydrodynamic
radius (R h ), and intrinsic viscosities. For a reliable application of the wormlike
chain model, the contour length, L, needs to be known. Typically L ¼ l m P, with
P being the (main chain) degree of polymerization and l m the length per repeat unit.
For vinyl main chains, l m ¼ 0.25 nm. There was considerable discussion about
whether l m ¼ 0.25 nm also applies to cylindrical brush polymers, because AFM
investigations resulted in smaller cylinder lengths than expected by the weightaverage degree of polymerization, P w , determined by static light scattering (SLS),
even if corrected for main chain polydispersity [72–74]. It was speculated that
within the cylindrical geometry the main chain was locally coiled, but persistent on
larger length scales. This picture seemed to be supported by simulations, which
showed a bimodal decay of the bond angle correlation function [66, 70, 75–77].
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
135