Cette thèse de doctorat est consacrée à l’étude analytique et numérique de systèmes de polymères et de fibres à deux dimensions. Des systèmes de polymères confinés en films ultra-minces présentent un très grand intérêt technologique et expérimentale et posent de nombreux défis théoriques en raison de leur fort comportement non-champ moyen qui se manifeste par divers exposants critiques non triviaux. Nous nous concentrons sur la limite strictement 2D où le croisement des chaînes est interdit et nous étudions, en fonction de la densité et de la rigidité des chaînes, les propriétés élastiques et conformationnelles de trois classes de systèmes: polymères flexibles et semi flexibles à température finie et polymères macroscopiques athermiques (fibres) à courbure spontanée imposée. Pour les polymères flexibles, il est démontré que bien que les polymères auto-évitant denses adoptent des configurations compactes avec un exposant de Flory ν = 1/2, ils ne se comportent pas comme des chaines gaussiennes. En particulier un exposant de contact non-nul θ2 = 3/4 implique une dimension fractale de périmètre dp = 5/4. Par conséquence, en accord avec la loi généralisée de Porod, le facteur de structure intramoléculaire F(q) révèle un comportement non-gaussien et la température de démixion des mélanges de polymères 2D devrait être réduite. Nous étudions également les effets de la rigidité des chaînes sur les systèmes de polymères à 2D et constatons que le comportement universel n’est pas modifié lorsque la longueur de persistance est beaucoup plus petite que la longueur de confinement. La nature de la transition de phase nématique à haute rigidité, qui est dans le cas 2D l’objet d’un débat de longue date, est également explorée. Des résultats préliminaires semblent indiquer une transition du premier ordre. Enfin, motivés par un travail théorique récent sur les modules élastiques de faisceaux de fibres, nous étudions les effets de la courbure spontanée sur l’élasticité d’ensembles de fibres. Nous montrons que en jouant sur le désordre des amplitudes des modes de Fourier de l’état fondamental il est possible de régler le module de compression, en accord qualitatif avec la théorie. / This PhD thesis is devoted to a theoretical study of polymer and ’polymer like’ systems in strictly two dimensions. Polymer systems in reduced dimensions are of high experimental and technological interest and present theoretical challenges due to their strong non-mean-field-like behavior manifested by various non-trivial universal power law exponents. We focus on the strictly 2D limit where chain crossing is forbidden and study as function of density and of chain rigidity conformational and elastic properties of three system classes: flexible and semiflexible polymers at finite temperature and macroscopic athermal polymers (fibers) with imposed quenched curvature. For flexible polymers it is shown that although dense self-avoiding polymers are segregated with Flory exponent ν = 1/2 , they do not behave as Gaussian chains. In particular a non-zero contact exponent θ2 = 3/4 implies a fractal perimeter dimension of dp = 5/4. As a consequence and in agreement with the generalized Porod law, the intramolecular structure factor F(q) reveals a non-Gaussian behavior and the demixing temperature of 2D polymer blends is expected to be reduced. We also investigate the effects of chain rigidity on 2D polymer systems and found that universal behavior is not changed when the persistence length is not too large compared to the semidilute blob size. The nature of the nematic phase transition at higher rigidities, which is in the 2D case the subject of a long standing debate, is also briefly explored. Preliminary results seem to indicate a first order transition. Finally, motivated by recent theoretical work on elastic moduli of fiber bundles, we study the effects of spontaneous curvature at zero temperature. We show that by playing on the disorder of the Fourier mode amplitudes of the ground state, it is possible to tune the compression modulus, in qualitative agreement with theory.

This PhD thesis is devoted to a theoretical study of polymer and 'polymer like' systems in strictly two dimensions. Polymer systems in reduced dimensions are of high experimental and technological interest and present theoretical challenges due to their strong non-mean-field-like behavior manifested by various non-trivial universal power law exponents. We focus on the strictly 2D limit where chain crossing is forbidden and study as function of density and of chain rigidity conformational and elastic properties of three system classes: flexible and semiflexible polymers at finite temperature and macroscopic athermal polymers (fibers) with imposed quenched curvature. For flexible polymers it is shown that although dense self-avoiding polymers are segregated with Flory exponent ν = 1/2 , they do not behave as Gaussian chains. In particular a non-zero contact exponent θ2 = 3/4 implies a fractal perimeter dimension of dp = 5/4. As a consequence and in agreement with the generalized Porod law, the intramolecular structure factor F(q) reveals a non-Gaussian behavior and the demixing temperature of 2D polymer blends is expected to be reduced. We also investigate the effects of chain rigidity on 2D polymer systems and found that universal behavior is not changed when the persistence length is not too large compared to the semidilute blob size. The nature of the nematic phase transition at higher rigidities, which is in the 2D case the subject of a long standing debate, is also briefly explored. Preliminary results seem to indicate a first order transition. Finally, motivated by recent theoretical work on elastic moduli of fiber bundles, we study the effects of spontaneous curvature at zero temperature. We show that by playing on the disorder of the Fourier mode amplitudes of the ground state, it is possible to tune the compression modulus, in qualitative agreement with theory.