Resonant osmosis across active switchable membranes
RResonant osmosis across active switchable membranes
Sophie Marbach,
1, 2
Nikita Kavokine, and Lyd´eric Bocquet Courant Institute of Mathematical Sciences, New York University, New York, New York,USA Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e Paris-Diderot, Sorbonne Paris Cit´e, Paris,France (Dated: 10 February 2020)
To overcome the traditional paradigm of filtration, where separation is essentially performed upon steric siev-ing principles, we explore the concept of dynamic osmosis through active membranes. A partially permeablemembrane presents a time-tuneable feature that changes the effective pore interaction with the solute andthus actively changes permeability with time. In general we find that slow flickering frequencies effectivelydecrease the osmotic pressure, and large flickering frequencies do not change it. In the presence of an asym-metric membrane, we find a resonant frequency where pumping of solute is performed and can be analyzedin terms of ratchet transport. We discuss and highlight the properties of this resonant osmotic transport.Furthermore, we show that dynamic osmosis allows to pump solute at the nanoscale using less energy thanreverse osmosis. This opens new possibilities to build advanced filtration devices and design artificial ionicmachinery.
I. INTRODUCTION
Modern processes for filtration are based on passivesieving principles: a membrane with specific pore prop-erties allows to separate the permeating components fromthe retentate . The domain has been boosted over thelast two decades by the possibilities offered by nanoscalematerials, such as graphene based or advanced mem-branes. Selectivity requires small and properly dec-orated pores at the scale of the targeted molecules, andthis inevitably impedes the flux and transport, makingseparation processes costly in terms of energy. Further-more, standard membranes suffer from an intrinsic lim-itation: to increase permeability, one must typically in-crease the size of the pores at the expense of inevitablydiminishing selectivity. This is commonly referred to asthe selectivity-permeability trade-off. However, this classical paradigm only considers mem-branes with fixed properties and pore size, and there-fore the constraints of selectivity-permeability are de-fined with static systems. Interestingly, Nature encom-passes a number of highly selective and highly permeableporins that operate far from equilibrium, and involve ac-tive parts.
Pore shape agitation was identified insome cases to be tightly connected to selectivity prop-erties. Therefore it is natural to revisit the trade-offparadigm by investigating how it is possible to harnessnon-equilibrium dynamics and active membranes to sep-arate solutes across active nanopores, see e.g.
Fig. 1.There is accordingly an interesting analogy with activematter and the osmotic pressure generated by ac-tive fluids in the vicinity of passive semi-permeable mem-branes has also been explored.
However how mem-brane dynamics may affect osmotic pressure remains tobe investigated. In this context we explore the conceptof dynamic osmosis and the possibility of tuning the os-motic pressure via the membrane dynamics. This corre-sponds to a non-equilibrium situation, which could allow L a b U ( x ) xL L xL U ( x, t )
Membranes as potential barriers . (a) Porousmembrane seen as an energy barrier U ( x ); (b) The porousmembrane has some temporal dependence (for instance time-dependent porous aperture), and can be seen as a time de-pendent energy barrier U ( x, t ). to some extent to bypass the equilibrium constraints ofseparation.Considering a nanopore with some dynamic feature( e.g. a flickering aperture, a time-dependent surfacecharge...), we raise the following questions: how is theosmotic pressure expressed? How does the osmotic pres-sure depend on the typical timescale of the dynamic fea-ture? To adress these question, we will consider a simple,yet insightful kinetic model of membrane separation, inwhich the membrane pores are assimilated to a potentialenergy barrier U ( x ) across the membrane (see Fig. 1-a) . This energy profile is allowed to vary with atypical time scale, modeling the dynamic feature of themembrane (see Fig. 1-b). We show that the osmotic pres-sure response is a highly non-trivial function of the fre-quency of the pore oscillations. In specific regimes wherethe energy barrier is asymmetric, the osmotic pressureexhibits a resonance at a characteristic frequency. Inter-estingly, we will harness the know-how of transport andpumping through oscillating ratchet potentials to pre-dict the properties of dynamical osmosis. This allows in a r X i v : . [ c ond - m a t . s o f t ] F e b particular to identify the design rules for a minimal os-motic pump. The properties of such active membranesare therefore extremely broad and could be harvested foradvanced nanofiltration. Finally, we show that dynamicosmotic solute pumping is energetically less costly thanstandard reverse osmosis. II. ACTIVE NANOPOREA. Active membrane model and qualitative considerations
We consider a porous membrane separating two sub-volumes, containing a solvent and a solute. There isa solute concentration difference between the two sub-volumes, ∆ C = C − C . Following Ref. 21, we considera model in which the pores are replaced by a potentialbarrier (see Fig. 1). Specifically, we model the membraneas an external potential U ( x, t ) acting on the solute only ,and not on the solvent molecules. The membrane thusstill remains permeable to the solvent, with a permeance L hyd , relating the flux Q to the pressure drop ∆ p in theabsence of a concentration difference: Q = −L hyd ∆ p .The potential U ( x, t ) varies only along the x axis acrossthe membrane. We denote L the characteristic thick-ness of the membrane, so that U vanishes outside the lat-eral range L , see Fig. 1. The potential U represents anykind of interaction between the solute and the membrane.These could be steric interactions for e.g. colloids or largemolecules, or electrostatic interactions between chargedsolutes and a charged membrane, etc. Although here weconsider that the primary interaction is between the so-lute and the membrane, our model could be extendedfurther to account for specific interactions between thesolute and the solvent. To give rise to osmosis – which isour interest here –, the necessary condition is that soluteand solvent do not interact in the same way with themembrane , and therefore to simplify we only considerone interaction.If the potential U ( x, t ) is static in time, the above ki-netic framework allows to recover, for instance, the van’t Hoff law for osmosis . Here we are interested in thedynamical case, where a time-dependent pore permeabil-ity is modeled by an oscillating potential U ( x, t ) = U φ ( x ) (1 + (cid:15) cos ωt ) . (1)The resulting configuration is schematically depicted inFig. 2. As the energy profile goes down with time, theconcentration profile is accordingly modified, as diffu-sion brings solute into the membrane. When the en-ergy profile goes up again, solutes diffuse outwards, andthe concentration profile flows away accordingly. A typ-ical system representing such an active membrane couldconsist in electrically gated pores or in mechanicallydriven pores with some external excitation . Note aswell that nearly every biological nanochannel works insuch nonequilibrium conditions with e.g. electrical or mechanical gating .At this stage one can note that the ingredients enter-ing our system are very similar to those composing anoscillating potential ratchet. Therefore, we mayexpect the flux of solute particles to be strongly depen-dent on the frequency of forcing, as well as the height U of the energy barrier. Here we are especially interestedin the consequences on the osmotic pressure, for whichthere is little intuition and no analytic result.In the following subsections we give details on howto compute the concentration profile, the effective fluxand the osmotic pressure in this oscillating case. In thefollowing steps, we will perform an expansion in (cid:15) for anypotential shape in order to obtain general results for theosmotic pressure as a function of the frequency. Then inthe next sections, we will apply these results to specificshapes of the potential and obtain explicit results. U ( x ) C
Expected behavior with oscillating symmet-ric energy barrier. (a) The energy barrier is initially fullyexpressed, with the same solute on each side but in differentconcentrations; (b) when the barrier is decreased, solute candiffuse, eventually mixing up between both sides; (c) when thebarrier goes up again, solute is pushed back outwards, and so-lute effectively originating from the right hand side ends up onthe left hand side, inducing an effective flux of solute. The ef-fective flux is expected to depend on the spatial and temporalcharacteristics of the potential.
B. Expansion of the Smoluchowski equation
In the 1D geometry described above, the solute concen-tration c ( x, t ) obeys the time-dependent Smoluchowskiequation ∂ t c = − ∂ x ( − D∂ x c + λc ( − ∂ x U ) + vc ) , (2)where D is the diffusion coefficient and λ = D/k B T themobility, with k B and T being the Boltzmann constantand the temperature, respectively. We further assume alow P´eclet number limit, Pe = vL/D (cid:28)
1, such that theconvective term of Eq. (2) is negligible. This is valid forlow permeability (nanoporous) membranes – note thatconvective terms are also of higher order in the concen-tration profile as the velocity field v typically scales as c ( − ∂ x U ) from the Stokes for flow eq. 17.The boundary conditions for the concentration are c ( x , t ) = C c ( x , t ) = C (3)with x = − L × δ and x = + L × (1 − δ ) and δ a di-mensionless parameter (see e.g. Fig. 4). In the followingwe will also use δ = 1 − δ .Using now the expression of U ( x, t ) in Eq. (1), theSmoluchowski equation becomes ∂ t c = − D∂ x (cid:18) − ∂ x c + c U k B T (1 + (cid:15) cos ωt ) ( − ∂ x φ ) (cid:19) . (4)We expand the solution as c ( x, t ) = c ( x ) + (cid:15) δc ( x, t ) + (cid:15) δc ( x, t )with δc ( x, t ) = R e (cid:2) δc ( x ) e jωt (cid:3) and δc ( x, t ) = R e (cid:2) δc ( x ) + δc ( x ) e jωt (cid:3) (5)where R e stands for the real part. This expansion isthought as an expansion in (cid:15) . The term c × cos ωt inEq. (4) leads at second order to the expected modes in 0and 2 ω , yielding the two second order terms in Eq. (5). Inaverage over time, we expect the first order terms to van-ish. To compute relevant quantities such as the osmoticpressure and the average flux through the membrane, wethus need to perform the expansion up to second order. C. Concentration profile equations
In this section we present an analytic derivation for theconcentration profile and solute flux up to second order.For readability we nondimensionalize the equations using˜ x = x/L , ˜ t = t/τ , ˜ c = c/C (where C = ( C + C ) / ω = ω/ω , with ω − = τ = L /D , u = U k B T ).We then drop the tilde signs to simplify. We also write∆ c = ( C − C ) /C , and fluxes are nondimensionalizedby D C /L .
1. Zeroth order equation
The zeroth order solution c ( x ) is assumed to be sta-tionary and thus obeys0 = − ∂ x ( − ∂ x c + c u ( − ∂ x φ )) , (6)whose solution is (see Ref. 23 for details) c ( x ) = e − u φ ( x ) − (∆ c ) e − u φ ( x ) (cid:82) δ x dx (cid:48) exp[+ u φ ( x (cid:48) )] (cid:82) δ − δ dx (cid:48) exp[+ u φ ( x (cid:48) )] , (7) The corresponding flux to zeroth order is J = − (1 − σ ) × ∆ c (8)with the rejection coefficient σ defined as σ = 1 − (cid:82) δ − δ dx (cid:48) exp[+ u φ ( x (cid:48) )] . (9)
2. First order equation
The equation for the time-dependent concentration δc ( x, t ) at order 1 is ∂ t δc ( x, t ) = ∂ xx δc ( x, t ) − u ∂ x [ δc ( x )( − ∂ x φ )] − u cos( ωt ) ∂ x [ c ( x )( − ∂ x φ )] . (10)Accordingly, the first order complex amplitude, δc ( x, t ) = R e (cid:2) δc ( x ) e jωt (cid:3) (see Eq. (5)), obeys jωδc ( x ) = ∂ xx δc ( x ) − u ∂ x [ δc ( x )( − ∂ x φ )] − u ∂ x [ c ( x )( − ∂ x φ )] , (11)The boundary conditions are assumed to be δc ( x = − δ ) = δc ( x = δ ) = 0. The last term of Eq. (11) isa driving term.This equation can be solved for some specific formsof φ ( x ), and we come back to analytic solutions in thefollowing sections. In the end we will find δc ( x, t ) = δc ( x ) cos( ωt + ϕ ) where the phase ϕ depends on all pa-rameters.
3. Second order equation
As pointed out above, the second order is asum of zero frequency and 2 ω terms: δc ( x, t ) = R e (cid:2) δc ( x ) + δc ( x ) e jωt (cid:3) . We focus on the zero fre-quency term, δc ( x ), which is relevant for the flux andosmotic pressure, while the 2 ω term will not contributeand averages to zero.The second order static term obeys the equation0 = ∂ xx δc ( x ) − u ∂ x [ δc ( x )( − ∂ x φ )] (12) − u ∂ x [ | δc ( x ) | ( − ∂ x φ )] cos( ϕ ) (13)where the last term originates from the time average ofthe first order term over one period. One can also justsolve Eq. (13) in the complex domain and we do that inthe following. We assume the following boundary condi-tions: δc ( x = − δ ) = δc ( x = δ ) = 0.Eq. (13) can be easily solved. Defining the second orderflux as J = − ∂ x δc ( x )+ u ( − ∂ x φ ) δc ( x )+ u ∂ x [ δc ( x )( − ∂ x φ )] , (14)one has J = const. This yields J = 12 (cid:82) δ − δ dx δc ( x )( − ∂ x exp[ u φ ]) (cid:82) δ − δ dx exp[ u φ ] (15)and δc ( x ) = − J e − u φ ( x ) (cid:90) x − δ dx (cid:48) e u φ ( x (cid:48) ) + e − u φ ( x ) (cid:90) x − δ dx (cid:48) δc ( x ) ( − ∂ x exp[ u φ ( x (cid:48) )]) . (16) D. Dynamic osmotic pressure and flux1. Osmotic pressure
We now turn to the expression of the osmotic pres-sure. We write accordingly the force balance on the fluid(composed of the solvent and the solute). It is crucial toremark that the membrane will act on the fluid as an ex-ternal force, − ∂ x U , exerted on the solute molecules. Thisis due to solute and solvent being in a dense interactingphase, where the force acts on the whole fluid volume assolvent molecules are dragged along the solute. This isexpressed writing the force balance on the fluid, repre-sented by the Stokes equation along the x direction (herefully dimensionalized): ρ∂ t v = − ∂ x p + c ( x )( − ∂ x U ) + η ∇ v, (17)where p is the fluid pressure, v is the flow velocity ofthe fluid in the x direction, η is the fluid viscosity and ρ its density. The driving force inducing solvent flowis accordingly written in terms of an apparent pressuredrop, − ∂ x P = − ∂ x p + c ( x )( − ∂ x U ). The membrane, viaits potential U , will therefore create a pressure force onthe fluid, which writes per unit surface σ ∆Π = (cid:90) δ L − δ L dx c ( − ∂ x U ) . (18)∆Π is identified as the osmotic pressure which in the di-lute case takes the simple van ’t Hoff expression ∆Π = k B T ∆ C ; σ is a screening parameter that takes into ac-count the specificities of the membrane. Assuming thatthe time scale to establish the flow is much faster thanthe time scale of oscillation of the potential barrier, thefluid flux will therefore write Q = −L hyd (∆ p − σ ∆Π).At high forcing frequencies, this assumption should bereconsidered to account for inertial effects and may leadto enhanced or decreased behaviors.Here we are interested in the averaged effective forceover a period (cid:104) σ ∆Π (cid:105) . Following the previous formal ex-pansion c ( x, t ) = c ( x )+ (cid:15)δc ( x, t )+ (cid:15) δc ( x, t ), we expand the osmotic pressure contribution as∆Π app ≡ σ app ∆Π ≡ (cid:104) σ ∆Π (cid:105) = Π + Π + Π (19)corresponding to contributions of the zeroth, first andsecond order terms in the concentration; σ app is an ap-parent screening parameter. Note that both terms ∆Π and ∆Π are of order 2 in (cid:15) . We come back to nondimen-sionalized equations, where ∆Π is nondimensionalized by k B T C . a. To zeroth order The corresponding osmotic pres-sure contribution matches the stationary solution (seealso Ref. 23) and writesΠ = σ × ∆ c where σ is defined by Eq. (9). Note that the osmoticpressure contribution at zeroth order satisfies the follow-ing relation to the particle flux: Π = J + ∆ c (in dimen-sionless form). b. To first order We average the solution over a pe-riod to obtainΠ = (cid:15) (cid:90) δ − δ dx (cid:48) R e [ δc ( x (cid:48) )] × u ( − ∂ x φ )( x (cid:48) ) (20) c. To second order Only the zero frequency term δc ( x ) contributes to the osmotic pressure, so that:Π = (cid:15) (cid:90) δ − δ dx (cid:48) R e (cid:2) δc ( x (cid:48) ) (cid:3) × u ( − ∂ x φ )( x (cid:48) ) (21)
2. Relation to the particle flux
The (fully dimensionalized) solute flux is defined as J = − D∂ x c + Dk B T c ( − ∂ x U ) . (22)From Eq. (2) one then deduces that the time averagedflux (cid:104) J (cid:105) obeys ∂ x (cid:104) J (cid:105) = 0, so that (cid:104) J (cid:105) ( x ) = − D∂ x (cid:104) c ( x, t ) (cid:105) + Dk B T (cid:104) c ( x, t )( − ∂ x U )( x, t ) (cid:105) = const (23)Using ∆Π app = (cid:104) (cid:82) x x dx c ( − ∂ x U ) (cid:105) , one can integrate thisresult to obtain∆Π app = k B T [ C − C ] + k B T LD (cid:104) J (cid:105) (24)and in dimensionless form:∆Π app = ∆ c + (cid:104) J (cid:105) . (25)Therefore the osmotic contribution may be related to thesolute flux at any order, and also in out-of-equilibriumconditions. We stress that Eq. 24 is highly interestingbecause it allows, from the description of the solute flow,to quantify the osmotic pressure contribution. In generalit is difficult to compute the osmotic pressure contribu-tion directly, and such a symmetry relation is of greathelp to obtain the expression for the apparent osmoticpressure.The averaged flux can be calculated as (cid:104) J (cid:105) = J + (cid:15) J , with the first order term averaging to zero. UsingEqs. (8)-(15), one deduces∆Π app = σ ∆ c + (1 − σ ) (cid:15) × ... (cid:90) δ − δ dx (cid:48) R e [ δc ] ( x (cid:48) )( − ∂ x exp[ u φ ( x (cid:48) )]) (26)where the rejection coefficient σ is defined in Eq. (9).One can check that this expression matches the directcalculation of the osmotic pressure from the force, seeabove. Writing ∆Π app = σ app ∆Π with ∆Π = k B T [ C − C ] (now fully dimensionalized), σ app plays the role ofan apparent rejection coefficient . Note that σ app maydepend on the concentrations C and C and simplifiesto σ app [ ω, C , C ] = σ + (1 − σ ) (cid:15) ∆ c × ... (cid:90) δ − δ dx (cid:48) R e [ δc ] ( x (cid:48) )( − ∂ x exp[ u φ ( x (cid:48) )])(27) E. Explicit solution for the triangular potential
In the following we will apply these results to the spe-cific case of a triangular shape for the potential U ( x ).This allows to obtain explicit analytic expressions for theconcentration profile as a function of frequency. The ana-lytic expressions are however cumbersome and we reportthe derivation and expressions in Appendix A. In the fol-lowing we will focus on the implications of this analysis. III. SYMMETRIC BARRIER, TOWARDS OSMOSIS ONDEMAND
We investigate first the symmetric barrier case (typi-cally as in Fig. 3-c), using both the analytic results andstandard numerical simulations (see Appendix B for nu-merical simulation details). We explore a range of mod-ulation frequencies and modulation depths (cid:15) while keep-ing the height u of the energy barrier fixed. In Fig. 3-aand b we show the analytic and numerical results forthe apparent flux (cid:104) J (cid:105) and the apparent osmotic pressure∆Π app . The analytic expansion at small (cid:15) is in fairly goodagreement with the full numerical simulation as long as (cid:15) (cid:46) . -0.3-0.2-0.10 numericalanalytic C
Transport through an active symmetric bar-rier . (a) Average solute flux dependence on ω/ω when C = 1 . C and C = 0 . C and U = 5 k B T where C is an arbitrary concentration unit. Note that the solute fluxis negative corresponding from solute current going from rightto left as expected from standard relaxation in sketch (c). It isrenormalized by ∆ C . (b) Apparent osmotic pressure ∆Π app as a function of ω/ω for same parameters. The osmotic pres-sure is given in units of k B T ∆ C . (c) Sketch showing thepotential barrier oscillation between two solute reservoirs atdifferent concentrations. (d) Numerical results and predic-tion from Eq. (30) for the low frequency apparent osmoticpressure. pressure ∆Π app approaches the usual van ’t Hoff con-tribution k B T ∆ C , in other words the rejection coeffi-cient plateaus to a constant value independent of the fre-quency, as for static membranes. In this regime, the con-centration profile does not follow the temporal variationsof U ( x, t ) and thus effectively sees only its time-averagedvalue (cid:104)U ( x, t ) (cid:105) t = U φ ( x ). We thus expect: σ app ( ω → ∞ ) = σ (28)As expected, this result does not depend on (cid:15) . It is indi-cated in Fig. 3-b by a small horizontal arrow.For very low forcing frequencies, we expect the concen-tration profile at any time t to be in quasi-static equilib-rium with the potential, so that: σ app ( ω →
0) = (cid:42) − (cid:82) − δ − δ dx (cid:48) e u φ ( x (cid:48) )(1+ (cid:15) cos ωt ) (cid:43) t ;(29)the latter may be approximated at small (cid:15) and in thecase of a triangular symmetric potential ( δ = 1 /
2) onegets σ app ( ω → (cid:39) σ − (cid:15) − σ ) e u u e u (1 − e u ) (30)Thus, when (cid:15) increases, we expect a decrease of σ app .That is not necessarily obvious since the barrier effec-tively goes up and down in cycles. This demonstrates infact that for a given amount of energy, more solute fluxis gained by lowering the barrier by that amount than islost due to raising the barrier by that same amount. Weplot Eq. (30) as a function of (cid:15) in Fig. 3-d, and the valuesobtained with the numerical results. The approximationof Eq. (30) is very robust in reproducing the numericalresults.These results show that the osmotic pressure contribu-tion is strongly affected by the active component of themembrane. It is therefore possible to tune the osmoticpressure, and achieve “on demand” values. Such a richbehavior is achieved while only assuming a symmetricpotential profile U ( x, t ). In the following, we seek theosmotic pressure response with an asymmetric potentialprofile, which is expected to be even more varied, andexplore the consequences for filtration and separation. IV. ASYMMETRIC BARRIER: OSMOTIC RESONANCEA. Towards an osmotic pump and sink
In this part we turn to asymmetric potential profiles,and investigate their consequences on osmotic pressure.We are inspired by the classical results on potential ratch-ets . Under an oscillating asymmetric potential pro-file, one may expect non-trivial pumping of the soluteto occur for specific values of the frequency and poten-tial shape. The qualitative principle of this ratchet-typemechanism is sketched in Fig. (4) for various potentialasymmetries, highlighting that an oscillating potentialmay lead to pumping, or, conversely, accelerate solutediffusion (‘sink’ regime). Moreover, an oscillating bar-rier is know to induce the so-called stochastic resonancephenomenon. Therefore, because of the fundamentalrelation between the osmotic rejection coefficient and thesolute flux demonstrated in Eq. (24), this various effectson the solute flux should convert into a non-trivial result-ing osmotic pressure acting on the fluid. The stochasticresonance phenomenon observed on the flux is thereforeexpected to result in an “osmotic resonance”. This iswhat we clarify in the present section.
B. Characterization of the osmotic resonance, time scalesand amplitude1. Osmotic resonance
As a proof of principle, we compute the solute flux andapparent osmotic pressure in the case of an asymmetricpotential profile. We use both our analytic expansionand standard numerical simulations (see Appendix B).We show the results for the pumping geometry and thesink geometry and for different barrier strengths in Fig. 5.Note that the analytic expansion is quite robust but at C
Principle for osmotic pump and sink. (a) Ini-tial configuration of the asymmetric energy barrier U ( x, t = 0)with concentration imbalance; (b) when the barrier is de-creased, the solute diffuses inwards; (c) when the barrier in-creases back, solute that crossed the maximal point will beflushed towards the right. If the frequency is well adjusted,essentially only solute from the left hand side will have dif-fused past the barrier and will be flushed to the high con-centration reservoir, therefore acting as a pump. The processiterates back to (a). (d) Initial configuration of an effectivesink with initial configuration inverted as compared to (a); (e)when the barrier decreases, solute diffuses inwards; (f) whenthe barrier increases again at an appropriate time, the solutefrom the right has diffused beyond the maximal point, andis effectively flushed to the left, thus increasing the effectiveflux as compared to a symmetric barrier. The process iteratesback to (d). This increased diffusion is termed as a ’sink’. high energy barrier strengths U /k B T and at large (cid:15) itdeviates quantitatively from the simulations (though theobserved trends are rather similar). In the case of nu-merical simulations, ∆Π app and (cid:104) J (cid:105) are obtained inde-pendently and are in good agreement with the relationof Eq. (24).First, we clearly observe a resonance in both cases inthe solute flux and in the apparent osmotic pressure. Apumping regime can indeed be achieved (left panels with (cid:104) J (cid:105) > C > C greater than k B T ∆ C – or an apparent osmotic reflection coeffi-cient greater than 1. This excess osmotic pressure trans-lates into fluid flow. Therefore, if hydrostatic pressuredoes not equilibrate osmotic pressure, an increased flowof the fluid (including the solvent and solute) is observedin the active osmotic pump regime (in contrast to thestatic case).Second, we clearly observe strong variations of the ap-parent osmotic osmotic pressure, that eventually can leadto a vanishing or a negative osmotic pressure in the sinkgeometry in some frequency range; see Fig. 6-b or Fig. 9-f. One may therefore tune the sign of the osmotic pres-sure contribution. When the apparent osmotic pressureis negative, this leads to a flow of fluid against the con-centration gradient (towards the dilute side). This fluid -0.4-0.3-0.2-0.100.1 -0.6-0.5-0.4-0.3-0.2-0.10 h J i ( ! ) a bdc C
Osmotic pump and osmotic sink (a) Schemat-ics showing the geometry relevant for pumping, with a steepenergy barrier near the low concentration reservoir, C > C and δ < .
5. (b) Schematics showing the opposite geometryrelevant to a sink, with C > C but δ > .
5. (c) and (d)Simulated (solid lines) and analytic (dashed lines) results forthe effective normalized flux (cid:104) J (cid:105) ( ω ) and (e) and (f) osmoticpressure ∆Π app ( ω ). The results are plotted for several energybarrier strengths in different colors, and (cid:15) = 1 . U /k B T = 10 shows a very smallpositive flux around the resonance. In the pumping geometry, δ = 0 . δ = 0 .
9. For all data C = 1 . C and C = 0 . C . flow is accompanied by a flow of solute towards the diluteside. If the permeability of the system is important, onemay therefore expect a net pumping of the fluid (hencewater).To further illustrate the origin of this phenomenon, itis interesting to investigate a simple toy model with anON/OFF potential instead of a sinusoidal time depen-dence. This allows to obtain analytic expression for thefrequency dependent osmotic pressure. We report theseresults in Appendix C. While such results do not aim at aquantitative comparison, they highlight the phenomenonof osmotic resonance in both the pump and sink regime,see Fig.9.
2. Resonance frequency
We now investigate in more detail the resonance fre-quency ω c at which osmotic resonance occurs. It isstrongly dependent on the parameters of the system (seefor example Fig. 6-b), e.g. on the parameters determin-ing the membrane interactions with the solute (barrierstrength U /k B T and asymmetry parameter δ ).In the pump or the sink process, there are two timescales of interest: (i) a diffusive time scale that describesthe typical time that the solute takes to reach the maxi-mal barrier point (when the barrier is down) and (ii) anadvection time scale corresponding to the time it takesto “slide down” to the other side when the barrier is upagain.. Let us take the example of the sink process toevaluate these time scales. For the sink process the dif-fusive time scale writes τ diff = L δ D (31)as δ is the distance between the highly concentrated sideand the barrier peak. The advection process correspondsto sliding down the other side of the barrier. It thus takesplace with a velocity that is the mobility multiplied bythe force Dk B T ∂ x U = Dk B T U L (1 − δ ) . Therefore the advec-tion time scale writes τ adv = L (1 − δ ) D k B TU . (32)At the resonance, one expects the period of oscillation ofthe barrier to be equal to the maximal time scale for thepump or sink process, so that τ c = max ( τ diff , τ adv ), andtherefore the resonance frequency obeys ω sink c /ω ∼ min (cid:18) δ , U k B T − δ ) (cid:19) (33)and similarly ω pump c /ω ∼ min (cid:18) − δ ) , U k B T δ (cid:19) . (34)We plot the resonance frequency dependence with re-spect to U /k B T and δ in Figs. 6 and 7. In Fig. 6 thelinear dependence on U /k B T expected from Eq. (33)is clearly observed for intermediate values of U /k B T .For large values of U /k B T , we may observe the ex-pected saturation when U /k B T (cid:39) δ − (in particular for U /k B T (cid:38)
10 and δ = 0 .
2; larger values of U /k B T werenot accessible either numerically or with the analytic ex-pansion due to convergence issues.). For small values ofthe barrier strength U /k B T the process becomes veryweak and the scaling laws are no longer relevant.In Fig. 7 the inverse quadratic dependence on δ is ob-served in a narrow region, since it is expected for large U /k B T and large δ (visible still for δ (cid:38) .
05 and U /k B T = 10). For small values of δ , the dependence -0.200.20.40.60.811.2 -2 -1 -2 -1 numerics, sinkanalytics, sink U /k B T = 3 5 10
Resonance frequency of active osmosis asa function of the barrier strength U /k B T (a) Osmoticpump geometry and parameters; (b) Apparent rejection co-efficient σ app calculated from simulations with respect to theforcing frequency ω for different values of the asymmetry pa-rameter δ , U /k B T = 10 and (cid:15) = 1 .
0. (c) and (d) Resonancefrequency ω c with respect to the forcing strength U /k B T atdifferent δ (same color scale for both graphs) and (cid:15) = 0 .
5, forthe sink and the pump geometries. Analytical curves are ob-tained from the expansion discussed in the main text. A scal-ing law with slope ω c /ω ∝ U /k B T is indicated in gray. Val-ues for the concentrations are C = 1 . C and C = 0 . C in the pump configuration with δ < .
5, and vice-versa forthe sink configuration. of ω c on δ is expected to saturate from Eq. (33), andthis is clearly observable in Fig. 7. In the intermedi-ate regimes, more entangled dynamics are involved thatmay in particular require the introduction of other rele-vant time scales for the system. We leave investigationof these more complex dynamics for future work. -3 -2 -1 -3 -2 -1 a
Resonance frequency of the active osmoticbarrier with respect to its asymmetry δ : (a) pump-ing configuration with C < C (here C = 1 . C and C = 0 . C ); and (b) sink configuration with C > C (here C = 1 . C and C = 0 . C ). In both panels, the reso-nance frequency ω c is plotted with respect to the asymme-try parameter δ , at different forcing strengths u = U /k B T (same color scale for both graphs), for the sink and the pumpgeometries. Numerical and analytical data are for (cid:15) = 0 .
5. Ascaling law with slope ω c /ω ∝ δ − is indicated in gray. Eq. (33) provides a simplistic understanding of the dy-namics involved and demonstrates that active osmoticflow may be strongly impacted by the specificities of themembrane in terms of asymmetry and solute interactionstrength. Note that the amplitude of the resonance mayalso be tuned with the different parameters at hand. Asa rule of thumb, the greater the asymmetry (so for largevalues of the potential strength U /k B T or small valuesof δ ), the greater the resonance. V. ENERGETIC EFFICIENCY OF ACTIVE OSMOTICPUMPING
In the context of filtration it is of utmost relevance toquantify the efficiency of the active osmotic process, andeventually compare it to other more common filtrationprocesses. We consider the active osmosis (AO) con-figuration in a geometry similar to Fig. 8-a, where thelateral reservoirs are closed and therefore the fluid flow Q = 0. When the membrane is dynamically activated – e.g. when the barrier U ( x, t ) is oscillated – the averagepower spent writes (fully dimensionalized) P AO = 1 T (cid:90) T dt (cid:90) Lδ − Lδ Sdx c ( x, t ) ∂ U ( x, t ) ∂t (35)where T = 2 π/ω and S is the total accessible surfacewhere the potential is exerted on the solute. The usefulpower generated by active osmosis corresponds to thechemical potential change of solute driven from one sideto the other, which writes P AO u = (cid:104) J (cid:105) Sk B T ln C C . (36)Therefore the efficiency of the active osmotic process issimply η AO = P AO u P AO . (37)We show in Fig. 8-c the efficiency of the active osmoticprocess as a function of the oscillation frequency ω , fora set of parameters, varying only the membrane interac-tion strength U . We find that the efficiency reaches amaximum (here up to η AO (cid:39) .
8) for a given value of thefrequency, say ω η . Remarkably, ω η is significantly higherthan the resonance frequency ω c . In fact although theenergy recovered P AO u is indeed maximal for ω = ω c , theenergy expense P AO is monotonically decreasing with ω .This can be understood from the fact that at large fre-quencies solute has less time to diffuse around and there-fore the energy expense to drive solute from a point toanother is smaller. Furthermore, the maximal efficiency η AO ( ω η ) strongly depends on the parameters of the sys-tem ( δ , ∆ c , U ). Although we do not carry here an in-depth study of these dependencies, we simply note thattypically there is an optimal value for the membrane in-teraction strength U . When U (cid:28) k B T there is almostno pumping flux; conversely, when U (cid:29) k B T more en-ergy than needed is spent to drive the solute.We now compare the active osmotic process to a pro-totypical filtration process: reverse osmosis, depicted inFig. 8-b. The reverse osmosis process similarly consistsof two fluid reservoirs containing solvent and solute inconcentration C > C . The reservoirs are separatedby a membrane which is permeable to the solvent alone(equivalent to a very large static barrier U ( x, t ), with U (cid:38) k B T ). An operator applies a pressure in order toimpose a reverse osmosis flow rate Q . The useful powerextracted from the process corresponds to the reductionin mixing entropy of the system and writes P RO u = Q ( C − C ) k B T. (38)Note that this expression is not the same as for the AOprocess eq. 36, which only involves transport of soluteand no flow of solvent. To compute the thermodynamicefficiency we now need to estimate the power that is dissi-pated. Without yet considering any physical membrane,the system necessarily dissipates energy through the fric-tion of the solvent on the solute. Indeed, as solvent passesfrom the left reservoir to the right, it leaves behind thesolute it contains, which gives rise to a relative velocitybetween the solvent and the solute particles. If we denote L the characteristic thickness of the membrane and S itssurface area, then each solute particle generates on thesolvent a friction force equal to µQ/S , where µ = Dk B T isthe mobility of the solute. Since there are C LS immo-bile solute particles, and the solvent moves with speed QS , the power dissipated through friction is P RO f = C LµS Q (39)If we now assume that the solvent has to pass through n physical channels of circular cross-section area s = πr (we assume S = ns ), then we have to take into accountthe power dissipated through the hydrodynamic resis-tance of the channels, R h = 8 πηL/s , where η is theviscosity of the solvent (assuming a no-slip boundary con-dition at the walls). The dissipated power reads P RO h = nR h ( Q/n ) = 8 πηL Q ns . (40)We have in fact estimated the hydrodynamic permeabil-ity L hyd of the RO membrane: P RO f + P RO h ≡ L hyd Q S , (41)with L − = s πηL + C Lµ . (42)Although this result relies on a model of discrete pores, it yields an estimate which agrees very well with the valuesreported for state-of-the-art RO polymeric membranes ,when evaluated for nanometre-sized pores.We may now compute the thermodynamic efficiency ofthe reverse osmosis process as η RO = P RO u P RO u + P RO f + P RO h (43)and expanding η RO = 11 + LD C C − C QS + 8 πη ( C − C ) k B T Ls QS . (44)As expected, the efficiency equals 1 for vanishing flowrate Q ; however, it decreases at increasing flow rates.To compare the two processes, we require that theygenerate the same useful power. For a given AO current (cid:104) J (cid:105) , this sets the RO flow rate Q as S (cid:104) J (cid:105) ln( C /C ) / ( C − C ). Substituting in eq. (44) yields: η RO = 11 + L (cid:104) J (cid:105) Dc (cid:16) c ∆ C (cid:17) ln C C (cid:18) C c + 43 πar c (cid:19) (45)where we made use of Einstein’s relation D = k B T / πηa with a the molecular size of the solute. From Eq. (45), itis clear that RO becomes inefficient in the limit of verysmall pore sizes, where the hydrodynamic resistance issignificant. Interestingly, it also shows that the efficiencyis a decreasing function of (cid:104) J (cid:105) , while the efficiency of AOis maximal around the highest values of (cid:104) J (cid:105) . Therefore,we expect RO to be inefficient at the fluxes where AO isat its peak efficiency. This can be seen in particular inFig. 8-c where we show the efficiency of both processes.We compare in Fig. 8-d the efficiency of the reverse os-mosis η RO and the active osmosis η AO processes for theoptimal value of U at different forcing frequencies ω . Theresults indeed show that there exists a broad range of pa-rameters (for example nearly all concentrations c (cid:46) r = 1 nm) where the active osmotic process is moreefficient (and even up to 100 times more efficient) thanthe reverse osmosis process. This is extremely encour-aging for filtration applications with active membranes.Furthermore, from a more fundamental point of view, itis fascinating to see how it is possible to bypass the limi-tations of filtration across static membranes by injectingenergy at the scale of membrane pores (and not at amacroscopic scale as is the case with reverse osmosis).To some extent this echoes the “apparent second prin-ciple breaking” in active matter (with active particles,self-spinners and so on ), where energy is also beingconsumed at the very local scale. In this strongly out-of-equilibrium regime, the principles underlying osmosisand selectivity can bypass the simple ’trade-off’ pictureof separation and has therefore a great potential for newseparation methodologies.0 FIG. 8.
Efficiency of active osmosis versus reverse os-mosis. (a) Active osmosis with oscillating asymmetric barrierand fixed reservoir volumes. (b) Reverse osmosis counterpart,where a large external pressure is applied on one reservoir,driving solvent flow through pores impermeable to the solute.(c) Efficiency of both processes under the conditions wherethe thermodynamic collected energy is the same in both cases(solid lines, active osmotic pumping as defined by Eq. (37) andcalculated from simulations with δ = 0 . (cid:15) = 1 in the pumpgeometry with C = 1 . c and C = 0 . c ; dashed lines,reverse osmosis as defined by Eq. (45), with r = 10nm and c = 0 . U /k B T = 8 and molecular size a = 1˚A; for differentvalues of r and c (here translated in mol/L). The efficiencyzone corresponds to active osmotic pumping being more effi-cient than reverse osmosis. VI. CONCLUSION
To summarize, we draw here a first picture to un-derstand osmosis across active membranes, or out-of-equilibrium osmosis. We provide a robust model to de-scribe and account for the osmotic pressure as a functionof the typical oscillating frequency of the membrane dy-namics. Remarkably, this kinetic model shows that os-motic flow through the membrane is still described by the Kedem-Katchalsky transport equations as (cid:104) Q (cid:105) = −L hyd (∆ p − σ app k B T ∆ C ) , (46)where σ app is an apparent rejection coefficient that takesinto account the specifics of the membrane and its dy-namics. The solute flow (neglecting convection) may alsobe written (cid:104) J (cid:105) = − DL ω app ∆ C (47)where ω app still verifies the fundamental reciprocal rela-tion ω app = 1 − σ app . However all coefficients are nowcomplex functions of the frequency of the active mem-brane.Our model clarifies the underlying principles of activeosmosis. In particular, we have rationalized that at verylow frequencies a dynamic membrane ( e.g. pore openingand closing) behaves as an apparently more permeable membrane; whereas at very large frequencies a dynamicmembrane behaves as an apparently static membrane.In the intermediate regime, very interesting functionali-ties may be achieved, provided the membrane has someasymmetry: resonant pumping or sink, with a variety oftuneable features. Interestingly, active osmosis may beeasily connected to potential ratchets and intuition fromthis field may be translated to the description of activeosmosis. Finally, we demonstrate that in nanofiltrationprocesses active osmosis may outperform reverse osmosisin terms of energetic efficiency.The model considered here is simple and provides abasis to study a number of effects. For example we ex-pect (see Fig. 10 in Appendix C) that asymmetry notjust in space but also in time, e.g. how fast the bar-rier is activated up versus down, may lead to more in-teresting regimes. Going further, a number of details atthe nanoscale could be accounted for, so as to provide amore systematic and thorough description of nanofiltra-tion across membranes: this includes, for instance, elec-trostatic effects or surface interactions. The impact ofnoise (of the membrane interaction potential , or due tothe small number of solutes in the channel ) on os-motic pressure is expected to be relevant at these scalesand has to be explored. Such extensions will be the sub-ject of future work. However the main generic features ofactive osmosis are expected to be captured by the presentmodel.Overall, our model, even simplistic, provides a num-ber of rules of thumb to design active membrane, e.g. in terms of the asymmetry of the membrane or thetypical frequency range at play. In practice compositemembranes with tuneable sieving properties, for exam-ple gated by applied voltage, are a natural lead to ex-plore the fabrication of such active membranes. Activeosmosis through dynamic membranes has a considerablepotential to broaden the current paradigm of filtration,building the basis for advanced filtration devices and ar-tificial ionic machinery.1 ACKNOWLEDGEMENTS
L.B. acknowledges funding from the EU H2020 Frame-work Programme/ERC Advanced Grant agreement num-ber 785911-Shadoks.
APPENDIXAppendix A : Explicit solution of the triangular profilebarrierTriangular profile
We assume that the potential is piece-wise linear, i.e. φ ( x ) = 1 + xδ for − δ < x < φ ( x ) = 1 − xδ for 0 < x < δ such that the force γ = − ∂ x φ = − /δ (resp. +1 /δ ) for x < x > jωδc ( x ) = ∂ xx δc ( x ) − u γ∂ x δc ( x ) − γ∂ x f ( x ) , (48)where we introduced f ( x ) = u c ( x ). The average con-centration c ( x ) is easily computed as, for x < c ( x ) = (1 − ∆ c ) e − u (1+ x/δ ) + ∆ c (cid:20) δ e − u (1+ x/δ ) − e u − (cid:21) (49)and for x > c ( x ) = e − u (1 − x/δ ) − ∆ c (cid:20) δ e − u (1 − x/δ ) − e u − (cid:21) (50)Eq. (48)e and a full expansion at second order for c ( x )can be readily calculated. The osmotic pressure canbe deduced accordingly. On the left domain or x < ∂ xx δc ( x ) + u δ ∂ x δc ( x ) − jωδc ( x ) = − δ ∂ x f ( x ) , (51)and on the right domain: ∂ xx δc ( x ) − u δ ∂ x δc ( x ) − jωδc ( x ) = 1 δ ∂ x f ( x ) . (52) Expression of δc Let us introduce λ L ± = 12 − u δ ± (cid:115)(cid:18) u δ (cid:19) + 4 jω (53) λ R ± = 12 u δ ± (cid:115)(cid:18) u δ (cid:19) + 4 jω (54)The solution for Eq.(48) then writes, for x < δc ( x ) = α L e λ L − x + β L e λ L + x + e λ L − x (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) x dx (cid:48) e − λ L − x (cid:48) ( ∂ x f )( x (cid:48) ) − e λ L + x (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) x dx (cid:48) e − λ L + x (cid:48) ( ∂ x f )( x (cid:48) )(55)and for x > δc ( x ) = α R e λ R − x + β R e λ R + x − e λ R − x (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) x dx (cid:48) e − λ R − x (cid:48) ( ∂ x f )( x (cid:48) )+ e λ R + x (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) x dx (cid:48) e − λ R + x (cid:48) ( ∂ x f )( x (cid:48) ) . (56) Boundary conditions
The boundary conditions are δc ( x = − δ ) = δc ( x = δ ) = 0. This imposes0 = α L e − λ L − δ + β L e − λ L + δ + e − λ L − δ (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) − δ dx (cid:48) e − λ L − x (cid:48) ( ∂ x f )( x (cid:48) ) − e − λ L + δ (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) − δ dx (cid:48) e − λ L + x (cid:48) ( ∂ x f )( x (cid:48) ) (57)and0 = α R e λ R − δ + β R e λ R + δ − e λ R − δ (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) δ dx (cid:48) e − λ R − x (cid:48) ( ∂ x f )( x (cid:48) )+ e λ R + δ (cid:114)(cid:16) u δ (cid:17) + 4 jω (cid:90) δ dx (cid:48) e − λ R + x (cid:48) ( ∂ x f )( x (cid:48) ) (58)2with f ( x ) = + u c ( x ).Two more subtle conditions are continuity conditionsat x = 0. The continuity of the concentration imposes δc (0 − ) = δc (0 + ), so that α R + β R = α L + β L (59)The condition for the continuity of the (first order) fluxcan be obtained by integrating Eq. (48) between x = 0 − and x = 0 + , which imposes ∂ x δc (0 + ) − u δ δc (0 + ) − u δ c (0)= ∂ x δc (0 − ) + u δ δc (0 − ) + u δ c (0) (60)After calculating the terms ∂ x δc (0 − ) = λ L − α L + λ L + β L and ∂ x δc (0 + ) = λ R − α R + λ R + β R , one deduces the conti-nuity equation − λ L + α L + λ R + α R − λ L − β L + λ R − β R = − u (cid:20) δ δ (cid:21) c (0)(61)where we used the expressions of the λ ’s to simplifythings. Full Solution
The system of Eqs. (57), (58), (59), (61) can be solvedto obtain the explicit expressions for α ± ( ω ), β ± ( ω ) asa function of frequency and potential parameters. Wedon’t provide the full expressions here, since they arehighly cumbersome. We investigate the results in themain text on several limiting situations. Appendix B : Numerical simulation details
The Smoluchowski equations are solved with a finitedifference scheme over 6 orders of magnitude of ω/ω andvarious other parameters. To ensure global convergence,we perform a Crank-Nicholson scheme and are especiallycareful that advection only carries upstream solute. Thetime step and space discretization were chosen such thatany reduction of either one ( e.g. by a factor 2) leads to nosignificant numerical difference in the results. The initialconcentration profile corresponds to the static barrier for (cid:15) = 0. As we seek averages over the oscillating process,we look for the average of the osmotic pressure over fiveperiods. When the simulation of an extra period willnot change the osmotic pressure by a significant amount,the initial conditions are forgotten and the result is con-verged.In the simulations time is nondimensionalized by ω such that typical simulations will roughly take the sametime to run. Note that for very large frequencies, the re-laxation from the initial conditions is much slower as the allowed flux is much smaller, and therefore simulationswhere run for longer times in the that case.The critical frequency at which the process is resonantcorresponds to the frequency at which the osmotic reflec-tion coefficient is maximum or minimum. As the simu-lation provides the osmotic reflection coefficient at onlydiscrete values of the frequency ω we perform a fit ona very narrow region around the maximum (resp. min-imum; with a 4th order standard polynomial fit to ac-count for peak slight distortion) and obtain the criticalfrequency from this fit. For each fit the agreement withthe simulation data is thoroughly asserted such that thecritical frequency obtained is a reliable value. Appendix C : Toy model for the asymmetric potentialprofile
We consider a time-dependent triangular potential,with a spatial extension similar to the previous analy-sis, i.e. φ ( x ) = 1 + xδ for − δ < x < φ ( x ) = 1 − xδ for 0 < x < δ (62)(63)where x is the dimensionless coordinate (in units of themembrane width, say L ); δ , δ are in dimension of L ( δ + δ = 1).But we now consider a simplified time-dependence,where this triangular potential is periodically ON/OFFfor time-lapse with period T : U ( x, t ) = U × f ( t ) × φ ( x ) (64)with f ( t ) = 0 for t ∈ [ kT ; k ( T + τ )] and f ( t ) = 1 for t ∈ [ k ( T + τ ); ( k + 1) T ], with k = E ( t/T ), an integer.Note that t is here in units of τ = L /D the diffusiontime-scale.Boundary conditions for the concentration in the reser-voirs are: C for x < − δ and C for x > δ .We will make several simplifying assumptions to ob-tain work out the model and obtain tractable results.First we assume that at the ON period, with duration τ = T − τ is sufficiently long so that particles reach anequilibrium state in the potential. This “re-initializes”the problem after each period T . Second we wil assumethat the energy barrier U is very large, so that no parti-cle can cross when the potential is on. Also such a highpotential will basically confine particles for x < − δ and x > δ ; i.e. we neglect the extension of the equilibriumdensity profile in the region [ − δ ; δ ] when the potentialis ON.Under these simplified assumptions, some interestingpredictions can be obtained. We recall that the solutionfor free diffusion with initial condition c ( x, t = 0) = Θ( x )(Heaviside) and boundary conditions c ( x = 0 , t ) = 1,3 c ( x → ∞ , t ) = 0 is c ( x, t ) = ψ (cid:20) x √ t (cid:21) (65)with ψ ( x ) = (cid:20) −
12 (1 + Erf( x )) + 12 (1 + Erf( − x )) (cid:21) (66)Accordingly, once the potential is released (ON → OFFperiod), one may simplify the solution for the concentra-tion by superposing diffusion from the two reservoirs intothe membrane: C ( x, t ) = C × ψ (cid:20) x + δ √ t (cid:21) + C × ψ (cid:20) δ − x √ t (cid:21) (67)The flux is defined as the number of particles whichcross the barrier maximum at x = 0 in the OFF period.Indeed, once the potential is back to ON, the particle for x > x < (cid:104) J (cid:105) = 1 T × ( N R − N L ) (68)with N R = (cid:90) δ dx C × ψ (cid:20) x + δ √ τ (cid:21) N L = (cid:90) − δ dx C × ψ (cid:20) δ − x √ τ (cid:21) (69) i.e. the number of particles which have crossed x = 0(from left to right, or right to left) at the time τ .Let us introduce Ψ( x ) = (cid:82) x ψ ( x ) dx . One may calcu-late: Ψ( x ) = 1 − e − x √ π + x (1 − Erf( x )) (70)Then the flux (in units of D/L ) is (cid:104) J (cid:105) = 1 T × √ τ × (cid:20) C × (cid:18) Ψ (cid:18) √ τ (cid:19) − Ψ (cid:18) δ √ τ (cid:19)(cid:19) − C × (cid:18) Ψ (cid:18) √ τ (cid:19) − Ψ (cid:18) δ √ τ (cid:19)(cid:19)(cid:21) (71)The characteristic frequency is ω = 2 π/τ . Then T =2 π/ω and τ = α × π/ω .Now we rewrite the expression in terms of frequency, ω = 2 π/T . Writing τ = αT (with α the fraction of timewith OFF potential), one obtains the flux in units of L/D as (cid:104) J (cid:105) LD =2 (cid:114) α ωω × (cid:20) ... -2 -3-2-101 -2 -0.6-0.4-0.200.20.4 h J i ( ! ) a bdc C
Toy model of resonant osmosis (a) Schematicshowing the geometry relevant for pumping, with a steepenergy barrier near the low concentration, C > C and δ < .
5. (b) Schematic showing the opposite geometry C > C but δ > .
5. (c) and (d) Simulated (orange lines)and analytic (black lines) results for the effective normalizedflux (cid:104) J (cid:105) ( ω ) Eq. 72 and (e) and (f) apparent osmotic pres-sure ∆Π app ( ω ) Eq. 74. The results are plotted for δ = 0 . δ = 0 . C = 0 . C = 1 .
0; and in the numerical computation U = 10 k B T . The couple of red arrows indicates regimeswhere pumping is seen, and the ON-OFF times are kept equal α = 0 . C (cid:18) Ψ (cid:18) √ α (cid:114) ωω (cid:19) − Ψ (cid:18) δ √ α (cid:114) ωω (cid:19)(cid:19) − C (cid:18) Ψ (cid:18) √ α (cid:114) ωω (cid:19) − Ψ (cid:18) δ √ α (cid:114) ωω (cid:19)(cid:19)(cid:21) (72)The osmotic pressure is accordingly defined as∆Π = k B T [ C − C ] + k B TD × L × (cid:104) J (cid:105) (73)(with L = δ + δ ) and the apparent rejection coefficientis σ app [ ω, C , C ] = 1 + LD × (cid:104) J (cid:105) C − C (74)The frequency dependent flux and osmotic rejectioncoefficient are plotted in Fig. 9, with several interestingfeatures. First a resonance is clearly observed. What isremarkable is that (i) a pump behavior is observed (left4panels, : J > C > C ) and (ii) a change of sign isobserved for the osmotic rejection coefficient. The lattermeans that one can tune the sign of the osmotic pressureand it can even be made vanish for a given frequency!Finally, note that the tow model is a very good proxy tobuild insight on the effect of asymmetric barriers not justin space but also in time . In Fig. 10 we show how asym-metry in time dramatically impacts solute flux aroundthe resonance frequency. We observe that the longer thebarrier is OFF, the more solute is pumped (or is sunk).This makes sense considering that the longer the bar-rier is OFF, the more solute can actually go past the thebarrier peak. To improve our insight on these differentregimes further computations have to be done that weleave for future work. -2 -4-3-2-10 -2 h J i ( ! ) a bdc C
Asymmetric ON/OFF pumping (a)Schematic showing the geometry relevant for pumping, witha steep energy barrier near the low concentration, C > C and δ < .
5. (b) Schematic showing the opposite geometry C > C but δ > .
5. (c) and (d) Analytic results for theeffective normalized flux (cid:104) J (cid:105) ( ω ) Eq. 72 for different ratios ofthe ON-OFF respective times of the barrier. Note that α measures how long the barrier is OFF. The results are plot-ted for δ = 0 . δ = 0 . C = 0 . C = 1 .
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