  We study water adsorption-induced deformation of a monolithic, mesoporous
silicon membrane traversed by independent channels of $\sim$8 nm diameter. We
focus on the elastic constant associated with the Laplace pressure-induced
deformation of the membrane upon capillary condensation, i.e. the pore-load
modulus. We perform finite-element method (FEM) simulations of the
adsorption-induced deformation of hexagonal and square lattices of cylindrical
pores representing the membrane. We find that the pore-load modulus weakly
depends on the geometrical arrangement of pores, and can be expressed as a
function of porosity. We propose an analytical model which relates the
pore-load modulus to the porosity and to the elastic properties of bulk silicon
(Young's modulus and Poisson's ratio), and provides an excellent agreement with
FEM results. We find good agreement between our experimental data and the
predictions of the analytical model, with the Young's modulus of the pore walls
slightly lower than the bulk value. This model is applicable to a large class
of materials with morphologies similar to mesoporous silicon. Moreover, our
findings suggest that liquid condensation experiments allow one to elegantly
access the elastic constants of a mesoporous medium.
