Forces and fluctuations in planar, spherical and tubular membranes

Lipid membranes constitute very particular materials: on the one hand, they break very easily under microscopical stretching; on the other hand, they are extremely flexible, presenting deformations even at small scales. Consequently, a piece of membrane has an area excess relative to its optically resolvable area, also called projected area. From a mechanical point of view, we can thus identify three tensions associated to lipid membranes: the mechanical effective tension $\tau$, associated to an increase in the projected area and to the flattening of the fluctuations; the tension $\sigma$, associated to the microscopical area of the membrane and thus non measurable, but commonly used in theoretical predictions; and its macroscopical counterpart measured through the fluctuation spectrum, $r$. Up to now, the equality between these quantities was taken for granted when analyzing experimental data. In this dissertation, we have studied, using the projected stress tensor, whether and under which conditions it is justified to assume $\tau = \sigma$. We studied three geometries (planar, spherical and cylindrical) and obtained the relation $\tau \approx \sigma - \sigma_0$, where $\sigma_0$ is a constant depending only on the membrane's high frequency cutoff and on the temperature. Accordingly, we conclude that neglecting the difference between $\tau$ and $\sigma$ is justifiable only to membranes under large tensions: in the case of small tensions, corrections must be taken into account. We have studied the implications of this result to the interpretation of experiments involving membrane nanotubes. Regarding $r$, we have questioned a former demonstration concerning its equality with $\tau$. Finally, the force fluctuation for planar membranes and membrane nanotubes was quantified for the first time.