This work explores the equilibrium and non-equilibrium statistical mechanics
of small charged particles (counterions) at oppositely charged polymers and
cylindrical surfaces.
Processes involving charged polymers and
their neutralizing counterions are ubiquitous in soft-matter and biological systems,
where electrostatic interactions result in an impressive variety of phenomena.
The interplay between electrostatic interactions, that attract counterions
towards charged polymers, and the entropy gained by counterions
upon dissolution leads to a critical counterion-condensation transition,
which is the central theme of this thesis.
The universal and critical features of this transition are investigated in equilibrium
conditions using both analytical approaches and a novel Monte-Carlo
simulation method. The critical exponents as well as the singular behavior associated with thermodynamic
quantities are determined and demonstrated
to be universal and in accord with mean-field theory in two and three spatial dimensions.
The statistical correlation between counterions
comes into play below the critical temperature,
where counterions
are strongly bound to the oppositely charged surface of the
polymer (condensation phase).
It is shown using asymptotic analysis that in the strong-coupling limit,
which is realized by high-valency counterions
or highly charged surfaces, electrostatic correlations dominate and
result in an effective electrostatic attraction between two like-charged cylinders.
Such attractive pair interactions are in striking contrast with
the standard, purely repulsive mean-field interactions, and can trigger aggregation and
phase instability in solutions of highly charged macroions.
Another relevant system, in which counterions play a decisive role and will be
subject of theoretical investigation in the present work, are
charged polymer brushes, that consist of densely end-grafted polymer chains onto a surface.
It is shown that the coupling between osmotic pressure of counterions trapped inside
the brush and the polymer length variation due to the chain elasticity leads to
a weak grafting-density dependence for the brush layer thickness. This behavior goes beyond
the standard scaling theories. It has been observed
in recent experiments and simulations, which are compared with
the present theoretical results.
Finally, to investigate the non-equilibrium dynamics
of counterions at charged polymers, Brownian Dynamics simulation techniques are
employed both in the presence and absence of hydrodynamic interactions
between constituent particles. In particular, the influence of counterion condensation
on the electrophoretic mobility of a charged polymer and its counterions is studied under
the action of small and large external electric fields.
It is shown that hydrodynamic interactions enhance the polymer mobility but substantially
reduce the mobility of counterions. In fact, counterions located in the immediate vicinity of
the charged polymer are found to be dragged along
with the polymer. It is shown using different charge pattern models that the local structural details
of the polymer chain, such as the charge spacing, can drastically affect the mobility of
counterions and the charged polymer itself