Аннотация
\par
For a highly charged polyelectrolyte,
counterions are known to condense onto the polyelectrolyte.
Moreover, under certain conditions,
the polyelectrolyte itself condenses.
In this study,
the relation between
the the counterion condensation
and the conditions for
the self-condensation of single polyelectrolytes
is studied
by performing molecular dynamics simulations of
a single flexible polyelectrolyte.
We consider a system consisting of a polyelectrolyte
with $N_m$ monomers,
which is represented by a bead-spring model,
and $N_c$ counterions.
The charge of each monomer is $-e<0$ and
that of each counterion is $z_ce>0$.
We choose $N_m=z_cN_c$.
The Manning ratio
$łambda_M=e^2/(ak_BT)$
describes
the strength of the electrostatic interaction
relative to the thermal energy.
Here, $\varepsilon$, $a$, $k_B$ and $T$
are the dielectric constant of the solvent,
the distance between two consecutive monomers,
the Boltzmann constant
and the temperature of the system,
respectively.
Note that
$łambda_M=0$
corresponds to the case of the neutral polymer.
The simulations are performed
for various values of $\varepsilon$ at a constant temperature $T$
by using the Langevin-type equations of motion,
where $N_m=24,48$ and $96$ and
$z_c=1,2,3$ and $4$.
\par
As $łambda_M$ is increased from zero,
the mean square average $R_e^2 \rangle$
of the end-to-end distance of the polyelectrolyte
first increases and then decreases after reaching a maximum value.
See Fig.\ 1.
The increase is due to the increase of the strength of
the repulsive electrostatic interaction between the monomers.
Screening of this repulsive interaction by
the counterions condensed onto the polyelectrolyte
causes the decrease in $R_e^2\rangle$.
For large values of $łambda_M$,
$R_e^2\rangle$ is smaller than
that for $łambda_M=0$,
which signals the self-condensation of the polyelectrolyte.
Note that the self-condensation can occur
independent of the counterion valence.
The self-condensation
implies the existence of an effective attractive interaction
among the monomers.
If the repulsive interaction between the monomers
is counterbalanced with the effective attractive interaction,
the polyelectrolyte is expected to behave as an ideal chain,
where the ratio of
$R_e^2\rangle$
to the
the mean square average $R_g^2\rangle$
of the radius of gyration becomes 6.
By determining
the onset of the self-condensation
by the condition
$R_e^2/R_g^2=6$,
it is found that
the self-condensation occurs when
about 90\% of the charge of the polyelectrolyte is neutralized
by the condensed counterions,
which agrees with the experimental fact.
It is also found that
$R_e^2\rangle$
takes its maximum value
when
about 13\% of the charge of the polyelectrolyte is
neutralized.
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