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