  An integrated mechanical model for fiber-laden membranes is presented and
representative predictions of relevance to cellulose ordering and orientation
in the plant cell wall are presented. The model describes nematic liquid
crystalline self-assembly of rigid fibers on an arbitrarily curved fluid
membrane. The mechanics of the fluid membrane is described by the Helfrich
bending-torsion model, the fiber self-assembly is described by the 2D Landau-de
Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane
interactions (inspired by an extension of the 2D Maier-Saupe model to curved
surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic
(curvature-avoiding) effects. Analysis of the free energy reveals three fiber
orientation regimes: (a) along the major curvature, (b) along the minor
curvature, (c) away from the principal curvatures, according to the competing
curvo-philic and curvo-phobic interactions. The derived shape equation (normal
stress balance) now includes curvature-nematic ordering contributions, with
both bending and torsion renormalizations. Integration of the shape and nematic
order equations gives a complete model whose solution describes the coupled
membrane shape/fiber order state. Applications to cylindrical membranes,
relevant to the plant cell wall, shows how growth decreases the fiber order
parameter and moves the fiber's director from the axial direction towards the
azimuthal orientation, eventually leading to a state of stress predicted by
pure membranes. The ubiquitous 54.7o cellulose fibril orientation in a
cylindrical plant cell wall is shown to be predicted by the preset model when
the ratio of curvo-phobic and curvo-philic interactions is in the range of the
cylinder radius.
