approximation. If one includes local chain stiffness in the model, the swollen state
and the Theta state in Fig. 1 still persist, whereas the compact states are modified.
Bond vectors between neighboring monomers along the chain try to develop some
local nematic-type order, but the chain connectivity constraint and the tendency to
form compact structures are then in some conflict, which can, e.g., be resolved by
the formation of toroidal structures [4, 16]. However, an exhaustive study of the
phase diagram of a single chain as a function of stiffness, strength, and range of
attractive effective monomer–monomer interactions, and chain length is still a task
for the future.
1.2 Force Versus Extension Behavior in the Good Solvent
Regime
In this subsection, we focus on long chains under good solvent conditions, where
a rather universal behavior of the polymers (irrespective of the details of the
model that is studied) [15] can be expected. Hence, we focus on the simplistic
self-avoiding walk (SAW) model on the simple cubic lattice, but we include
the effect of chain stiffness (introducing an energy penalty E b whenever the SAW
makes a 90
kink). This extension of the model is crucial when one has in mind the
application to biopolymers (such as double-stranded DNA, which has a persistence
length ‘ p ¼ 50 nm but a chain diameter of only D ¼ 3 nm) [1, 2]. This model can
be studied very efficiently with the PERM algorithm (pruned-enriched Rosenbluth
method) [17, 18], which allows the study of rather long chains (e.g., up to
N ¼ 50,000). This algorithm directly estimates the partition function of the chain
(and hence its free energy) as a function of N, q B ¼ exp(ÀE b /k B T ) and the
Boltzmann factor due to the force [exp f ‘ b =k B T
ð
Þ , where the bond length‘ b is simply
the lattice spacing, taken as unit of length]. This model hence allows contact with
the Kratky–Porod [19] worm-like chain (WLC) model, which is used as a standard
model of semiflexible macromolecules [1, 2]. However, the Kratky–Porod model
neglects excluded volume completely and hence necessarily fails (under good
solvent conditions) for long chains.
Figure 3a shows a plot of 3 R
2
g
D E
=‘ p L, where L ¼ N‘ b ¼ N is the contour length
of the semiflexible chain, versus n p ¼ L=‘ p , the contour length in units of the
persistence length, for zero stretching force, f ¼ 0 [20]. This representation
is chosen such that the Kratky–Porod result reduces to a master curve, which
saturates at unity for large n p . It is seen that for n p 1 all data coincide
on a straight line, described by hR g
2
i ¼ L
2 /12 (or 3 R
2
g
D E.
‘ p L
À Á ¼ 0:25 L=‘ p
À
Á
,
respectively). This is the trivial result for rigid rods. For 1 < n p < 10, a gradual
crossover towards the behavior of SAWs occurs, hR g
2
i/L / L
2ν À 1 , if the chains are
rather flexible. If the chains are rather stiff, an intermediate Gaussian regime sneaks
in (in d ¼ 3 dimensions only), before a second crossover to SAW-like behavior
Mechanical Properties of Single Molecules and Polymer Aggregates
5
and the Theta state in Fig. 1 still persist, whereas the compact states are modified.
Bond vectors between neighboring monomers along the chain try to develop some
local nematic-type order, but the chain connectivity constraint and the tendency to
form compact structures are then in some conflict, which can, e.g., be resolved by
the formation of toroidal structures [4, 16]. However, an exhaustive study of the
phase diagram of a single chain as a function of stiffness, strength, and range of
attractive effective monomer–monomer interactions, and chain length is still a task
for the future.
1.2 Force Versus Extension Behavior in the Good Solvent
Regime
In this subsection, we focus on long chains under good solvent conditions, where
a rather universal behavior of the polymers (irrespective of the details of the
model that is studied) [15] can be expected. Hence, we focus on the simplistic
self-avoiding walk (SAW) model on the simple cubic lattice, but we include
the effect of chain stiffness (introducing an energy penalty E b whenever the SAW
makes a 90
kink). This extension of the model is crucial when one has in mind the
application to biopolymers (such as double-stranded DNA, which has a persistence
length ‘ p ¼ 50 nm but a chain diameter of only D ¼ 3 nm) [1, 2]. This model can
be studied very efficiently with the PERM algorithm (pruned-enriched Rosenbluth
method) [17, 18], which allows the study of rather long chains (e.g., up to
N ¼ 50,000). This algorithm directly estimates the partition function of the chain
(and hence its free energy) as a function of N, q B ¼ exp(ÀE b /k B T ) and the
Boltzmann factor due to the force [exp f ‘ b =k B T
ð
Þ , where the bond length‘ b is simply
the lattice spacing, taken as unit of length]. This model hence allows contact with
the Kratky–Porod [19] worm-like chain (WLC) model, which is used as a standard
model of semiflexible macromolecules [1, 2]. However, the Kratky–Porod model
neglects excluded volume completely and hence necessarily fails (under good
solvent conditions) for long chains.
Figure 3a shows a plot of 3 R
2
g
D E
=‘ p L, where L ¼ N‘ b ¼ N is the contour length
of the semiflexible chain, versus n p ¼ L=‘ p , the contour length in units of the
persistence length, for zero stretching force, f ¼ 0 [20]. This representation
is chosen such that the Kratky–Porod result reduces to a master curve, which
saturates at unity for large n p . It is seen that for n p 1 all data coincide
on a straight line, described by hR g
2
i ¼ L
2 /12 (or 3 R
2
g
D E.
‘ p L
À Á ¼ 0:25 L=‘ p
À
Á
,
respectively). This is the trivial result for rigid rods. For 1 < n p < 10, a gradual
crossover towards the behavior of SAWs occurs, hR g
2
i/L / L
2ν À 1 , if the chains are
rather flexible. If the chains are rather stiff, an intermediate Gaussian regime sneaks
in (in d ¼ 3 dimensions only), before a second crossover to SAW-like behavior
Mechanical Properties of Single Molecules and Polymer Aggregates
5