Membranes: A Variety of Energy Landscapes for Many Transfer Opportunities

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 in 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. The ability of the model to describe the physics of the filtration is discussed in detail. This model is solved in 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 (such as 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. Asymmetric interactions thus lead to direction-dependent transfer properties and the membrane exhibits diode behavior. These new transfer opportunities are discussed.


S1 -Development and main assumptions in the model
The two-fluid formalism is based on the application of the momentum and the mass balances on the dispersed phase and on the fluid phase (Jackson, 1997). This formalism allows to account for the different transport phenomena (and then forces) acting on the "particles" (that can be solute or colloids) and on the fluid. In a recent paper (Bacchin, 2017), the presence of a membrane has been accounted for in this model through the solving of these equations on an energy landscape. The parameter describing the energy landscape is the interfacial pressure, ( , , ) representing the colloid/membrane interactions. If neglecting the gravity forces and if considering isotropic pressures, the momentum and the mass balances for the dispersed phase (dispersed colloids or solubilized molecules), the fluid phase (water molecules) and by addition for the mixture are: Momentum balance On the dispersed phase On the fluid On the mixture The different contributions in these equations can be dissipative or elastic in nature (Bacchin, 2017) : The dissipative contributions are : • the drag force, = − ( ) , represents the forces due to the friction induced by the relative velocity between the phases (colloid/fluid friction) • the viscous dissipation, − 2 , due to the viscosity mixture in the Eqs. 2 and 3 (fluid/fluid friction induced by the shear) • the viscous dissipation in the fluid due to the interface that can be link to a porous media permeability, − in the Eqs. 2 and 3 (fluid/membrane friction) The elastic storage (non dissipative contributions) are : • the thermodynamic (reversible) colloid pressure gradient , ℎ , that corresponds to the water activity difference (colloid/colloid interaction) • the mechanical (non-reversible) colloid pressure, , due the shear and the associated shear induced diffusion (colloid/shear interaction) or due the compressibility of an arrested deposit if any (aggregated colloid/fluid interaction) • the interfacial pressure, , in the Eq. 1 and 3 (colloid/membrane interaction) • the pressure drop, − , representing the energy dissipated in the system (fluid/fluid interaction) If considering a process where only the reversible part of the osmotic pressure is present, = 0, (i.e. no effect of shear induced diffusion and no consolidated deposit on the membrane), the momentum balance can be written : − ( ) − − = 0 (S1-7) The first equation means that the velocity of the colloids (or more exactly the force needed to have a colloid velocity larger than the mixture velocity) is induced by the gradient of osmotic pressure and is modified by colloid/membrane interactions. The second equation indicates that the mixture velocity is induced by the gradient of fluid pressure and is modified by colloid/membrane interactions.

S2 -Energy map model and Boltzmann exclusion
In the absence of advection, u c =u m =0, the equation 6 can be simplified: If considering the ideal case, where the osmotic pressure for colloids follows the Van't Hoff law, = , the differential equation can be written : When this expression is integrated through the exclusion barrier where the interfacial pressure is varying from 0 to , the partition coefficient representing the volume fraction on the exclusion boundary in the membrane side over the volume fraction on the exclusion boundary in the bulk side can be written : The exclusion number, Ex, is then directly linked to the partition coefficient, K, through the Boltzmann exclusion induced by the energy map.

S3 -Integration along the membrane composite layers
The concentration profiles can be determined through the integration of the eq. 2. When considering the continuity of the mass flux through the dimension perpendicular to the membrane, x, a constant mass flux has to be considered. This mass flux is also the one coming out through the membrane: The equation 2 can then be written: The gradient in osmotic pressure is related to a diffusive mass flux if defining the diffusion coefficient with the Stokes-Einstein generalized law: If considering that the ideal case where the osmotic pressure follows the Van't Hoff law, = and leads to a constant diffusion coefficient, the equation S3-2 can be written as a classical convectionadvection balance with an additional mass flux due to the interaction with the membrane.
The mass flux due to the interaction with the membrane (already introduced in model accounting for membrane-colloid interaction, (Bacchin et al., 2011(Bacchin et al., , 1995) can be written as an advective flow with a velocity, , due to the interface/colloid interaction : This last velocity is constant is considering the specific case where the interfacial pressure is varying linearly with x (Fig. 2). In these conditions, the equation 6 can be simplified to: This equation can be integrated through all the layers representing the membrane in order to determine the volume fraction profile (subscript are defined in Fig.1) :

Polarisation layer
The integration of the transport equation, where is equal to the volume fraction of the permeate .

S4 -Expression of the transmission
The equations given in S3 can be combined in order to determine the solute transmission : In the case of Pé clet exclusion tending to 0 (for an infinitively thin membrane defined with a partition coefficient), the equation can be simplified: This last equation is the one given in (Bacchin et al., 1996;Opong and Zydney, 1991) for convective and diffusive transfer through membrane or for electrokinetic salt rejection (Jacazio et al., 1972).
If the polarization effect are negligible, → 0 : From the model presented in the paper, the counter pressure can be expressed from different set of parameters. This section inventories the main relationships and gives the limits of these relationship for the conditions of thin exclusion layers.
Eq. 5 allows to determine the permeability of the membrane. When integrated along the layers of the membrane (Fig. 1), the counter pressure is expressed through the integration of the interfacial pressure,

∫
. The gradient is only existing inside the exclusion layers. The integral can then be defined as ∫ + . The integral can be linked to the osmotic pressure difference and the drag force acting on the exclusion layer: The filtration law can then be expressed as a function of the osmotic pressure difference and the drag forces inside the exclusion layers: The drag force can be expressed as a function of the volume fraction inside the layers: The out-of-equilibrium counter pressure is then be expressed as the product of the drag force on immobile particles 0 and the excess number of arrested colloids in the membrane ( − )/ .
The concentration profile being determined (S3-9), this term can be defined for each layers according: The counter pressure can then be expressed as a function of the osmotic pressure at the boundary of the exclusion layers, , , , : The counter pressure can also be expressed as a function of the osmotic pressure at the extremities of the system membrane+ exclusion layers This last equation (that can also be written in a slightly different way accounting for the transmission as given in Eq. 6) allows to link the counter pressure to the operating conditions that are the different Pé clet numbers and the exclusion number.
Counter pressure limit for an infinitively thin membrane If considering an infinitively thin exclusion layer, → 0, these equation are tending toward : The counter pressure relationship can also be simplified when considering, → 0 If considering an ideal case for the osmotic pressure, = = Π Π , the counter pressure can be expressed as a function of the transmembrane osmotic pressure : -22) It can be noted that this last writing is equivalent to the relationship including a Staverman coefficient equal to (1-K). The model is then coherent with the model classically used. It allows to give a new explanation for the counter osmotic pressure and a new framework for the dynamic description of the osmosis flow.
If the transmission through a membrane depends on the direction of the flow, the transmission may have different values for permeation fluxes (or Peclet numbers) having the same amplitude but different directions (positive or negative). The application of a pulsed flow through the membrane should make it possible to work with a series of filtration steps with a different transmission across the membrane as outlined in Figure S6-1. With this mode of filtration, a gradual increase in concentration in the compartment from which the transmission is lower should be expected. After a series of pulses, separation should then occur between the compartments without net overall filtration of the fluid through the membrane. It is interesting to note here that a pulsed flow (or shaking) could lead to a separation; when shaking and separation can be considered a priori as an antagonist.
Simple calculations taking into account a succession of stationary states can show how separation can occur. These calculations are based on the mass balances on the two compartments separated by the membrane. Fig. S6-2 shows the concentration in the left compartment and the right compartment during a flow pulsation. These calculations are performed for different conditions of asymmetry. When there is no asymmetry (left part in Fig. S6-2), the pulsed flow leads to the mixing and the concentration becomes identical. For weak asymmetry (determined from the experimental data presented in Fig. 7), the separation occurs after a series of square pulsed flows (median part of Fig. S6-2). This separation can be even faster in the case of a perfect diode with perfect transmission in one way and full retention in the opposite direction (right part in Figure S6-2).