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Thesis (PhD)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Several experiments [1, 2, 3, 4] have shown that e ective attractive interactions exist between con ned like-charged macromolecules. Theoretical approaches have not reached consensus as to precisely what the mechanism for the attraction is, but it is agreed that comprehending the role of the counterion arrangement around macromolecules is crucial for understanding the e ective macromolecule interactions. It is generally assumed that attraction only occurs in the limit of strong electrostatic coupling and is driven by correlation e ects that are neglible in a mean- eld approach, which is valid in the weak-coupling limit. However, in some experimental situations attraction occurs even in the limit of weak-coupling. We consider a eld-theoretic approach that includes uctuations to study the Coulomb interactions of con ned counterions with a single exible charged spherical macromolecule that can expand or collapse uniformly by changing its radius. We show how the linearised eld-theory (valid in the weak-coupling limit) is mapped onto the square-well potential of Quantum Mechanics. The con nement leads to bound states being present in the spectrum at all times. Bound states are non-perturbative and we investigate the role they play in the physics of the system. Some of the e ects are rather counter-intuitive. Firstly, upon expanding the macromolecule in a xed con nement volume, the uctuation part of the free energy favours a decrease in the free energy. Secondly, upon increasing the temperature to high but nite values, the uctuation contribution does not dominate the free energy as would be expected. The mathematical origins of these e ects are dicussed in detail and as part of the analysis we introduce a novel regularisation scheme for computing the functional determinant arising in the model considered where the cut-o is speci ed unambiguously in terms of physical parameters. / AFRIKAANSE OPSOMMING: Verskeie eksperimente [1, 2, 3, 4] toon dat makro-ione met gelyksoortige ladings, in `n eindige volume, `n e ektiewe aantrekkende krag ondervind. Alhoewel daar nog geen konsensus oor die presiese meganisme vir die aantrekking bereik is nie, is dit duidelik dat die rol van \counter-ion" rangskikking rondom die makro-ione belangrik is om die e ektiewe wisselwerkings te verstaan. Dit word algemeen aanvaar dat die aantrekkende krag slegs in die limiet van sterk elektrostatiese koppeling plaasvind en dat dit `n gevolg van \counter-ion" korrelasies is wat weglaatbaar is in `n gemiddelde veld benadering, wat geldig is in die limiet van swak elektrostatiese koppeling. Daar bestaan egter eksperimentele situasies waar die aantrekking in die limiet van swak elektrostatiese koppeling waargeneem word. Ons bestudeer die Coulomb wisselwerking tussen \counter-ions" en `n enkele rekbare sferiese makro-ioon vanuit `n veld-teoretiese beskouing wat uktuasies in ag neem. Die sferiese makro-ioon kan vergroot of verklein deur sy radius uniform te verander. Ons toon aan dat die gelineariseerde veldeteorie (geldig in die limiet van swak elektrostatiese koppeling) op die eindige-diepte put Kwantummeganiese model afgebeeld kan word. Die eindige volume van die sisteem het tot gevolg dat daar altyd gebonde toestande in die spektrum voorkom. Gebonde toestande is `n suiwer nie-steuringsteoretiese e ek en ons ondersoek die rol wat dit speel in die sika van die sisteem. Die teenwoordigheid van die gebonde toestande in die spektrum het `n paar teen-intuitiewe e ekte tot gevolg. Eerstens word die vrye energie verlaag soos die makro-ioon in `n eindige volume vergroot. Tweedens oorheers die uktuasie bydrae nie die vrye energie met toenemende temperatuur soos verwag sou word nie. Ons bespreek die wiskundige oorsprong van hierdie e ekte. As deel van die analise ontwikkel ons `n nuwe regulariseringstegniek vir die berekening van funksionaalintegrale waar die regulariseringsparameter ondubbelsinnig in terme van siese hoeveelhede uitgedruk kan word.

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