Counterions at Charged Polymers

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.