  A membrane can be represented by an energy landscape that solutes or colloids
must cross. A model accounting for the momentum and the mass balances on the
membrane energy landscape establishes a new way of writing for the Darcy law.
The counter pressure in the Darcy law is no longer written as the result of an
osmotic pressure difference but rather as a function of colloid-membrane
interactions. Physically, the colloid-membrane interactions are slowing down
the colloid velocity thus inducing a relative fluid-colloid motion that in turn
leads to the counter pressure. The ability of the model to describe the physics
of the filtration is discussed in detail. This model is solved on a simplified
energy landscape to derive analytical relationships that describe the
selectivity and the counter pressure from ab-initio operating conditions. The
model shows that the stiffness of the energy landscape has an impact on the
process efficiency: a gradual increase in interactions (like with hourglass
pore shape) can reduce the separation energetic cost. It allows the
introduction of a new paradigm to increase membrane efficiency: the
accumulation that is inherent to the separation must be distributed across the
membrane.
