arxiv: 2605.02419 · v1 · submitted 2026-05-04 · ❄️ cond-mat.soft · physics.flu-dyn

Recognition: 3 theorem links

· Lean Theorem

Diffusio-osmotic transport in nanochannels

Lyd\'eric Bocquet

Pith reviewed 2026-05-08 18:30 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn

keywords diffusio-osmosisnanochannelsentropic transportosmosisforce balanceosmotic energy conversionsurface diffuse layersconcentration gradients

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The pith

Diffusio-osmosis shows that osmotic flows can occur in nanochannels without any semi-permeable membrane by treating both osmosis and diffusio-osmosis as the same entropic process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that diffusio-osmosis arises from entropic forces acting inside thin diffuse layers next to solid walls when a concentration difference is present. These forces drive fluid and solute motion in the same way classical osmosis does, so the two phenomena become two sides of one mechanism once placed inside an Onsager framework of local and global force balances. Specifying the picture to nanochannels removes the usual requirement for a selective barrier, which means entropically driven transport can appear in any channel whose walls create a diffuse layer. A reader would care because the same mechanism then explains enhanced diffusion, rectified flows, and energy extraction in systems that lack perfect membrane selectivity. The chapter works through concrete examples to show how these flows appear and how they can be used.

Core claim

Osmosis and diffusio-osmosis are two faces of the same phenomenon, naturally embedded in an Onsager framework and quantified by local and global force balances. When diffusio-osmosis is applied to nanochannels, osmotic drivings appear without the prerequisite of semi-permeability, thereby extending the domain of entropically driven transport to a broader class of channels and membranes.

What carries the argument

The diffusio-osmotic mechanism, arising from entropic driving forces inside diffuse layers at solid boundaries and quantified by local and global force balances in an Onsager framework.

If this is right

Osmotic drivings arise in channels and membranes without semi-permeability.
Diffusio-osmosis produces enhanced diffusion under concentration gradients.
The same mechanism yields mechano-sensitivity in transport through nanochannels.
Rectified osmotic flows appear when geometry or boundary conditions break symmetry.
Diffusio-osmosis can be used as a lever for osmotic energy conversion from single nanopores to larger membrane modules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Design of artificial osmotic devices could drop the requirement for perfect membrane selectivity and rely instead on controlled surface diffuse layers.
Symmetric nanochannels with tunable wall properties offer a clean experimental test bed for isolating the predicted velocity from concentration gradients alone.
Similar force-balance reasoning may connect diffusio-osmosis to other interface flows such as thermo-osmosis when multiple gradients are present simultaneously.
Scaling the single-pore results to modules suggests a route to osmotic power that tolerates imperfect selectivity at the cost of lower efficiency per pore.

Load-bearing premise

Entropic driving forces inside the diffuse layers next to solid boundaries can be mapped directly onto local and global force balances inside an Onsager framework.

What would settle it

A measurement of zero net fluid velocity inside a nanochannel subjected to a controlled concentration gradient, when surface charge and ion mobility are known and the predicted diffusio-osmotic velocity is nonzero, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.02419 by Lyd\'eric Bocquet.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2. Force balance on diffusio-osmosis. A gradient of view at source ↗

**Figure 3.** Figure 3: FIG. 3. Experimental demonstration of DO flow, from Ref. view at source ↗

**Figure 4.** Figure 4: FIG. 4. From Ref. [11]: Osmotic streaming current ver view at source ↗

**Figure 5.** Figure 5: FIG. 5. From Ref. [55]: enhanced diffusion across CNT mem view at source ↗

**Figure 6.** Figure 6: FIG. 6. From Ref. [41]: Sketch of the nanofluidic experi view at source ↗

**Figure 7.** Figure 7: FIG. 7. Flow rectification with composite membranes: view at source ↗

read the original abstract

In this chapter, I will enter into the roots of entropically-driven transport with a focus on diffusio-osmotic transport in nanochannels. Diffusio-osmosis is a subtle surface transport, originating in entropic driving forces occuring within the diffuse layers at solid boundaries. Specifying diffusio-osmosis to nanochannels may first look like a marginal refinement, yet it reveals that osmotic drivings can arise in channels and membranes without the prerequisite of semi-permeability, so that diffusio-osmosis extends the domain of existence of entropically driven transport. Osmosis and diffusio-osmosis are two faces of the same phenomenon, naturally embedded in an Onsager framework and quantified by local and global force balances. This perspective clarifies why nanochannels are privileged arenas where diffusio-osmosis and its consequence do flourish. Throughout the chapter, I discuss a set of conceptually relevant examples to show how diffusio-osmosis "pops up" in various situations: as enhanced diffusion, mechano-sensitivity, rectified osmotic flows and, ultimately, as a lever for osmotic energy conversion from single nanopores to membrane modules approaching industrial reality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Bocquet unifies osmosis and diffusio-osmosis conceptually in nanochannels under Onsager relations but introduces no new findings or data.

read the letter

The key point is that Bocquet presents diffusio-osmosis in nanochannels as a way to get entropically driven flows without needing semi-permeable membranes, all within the usual Onsager setup. Osmosis and diffusio-osmosis end up as two faces of the same thing once you look at the surface layers properly. It does well at spelling out the physical picture with local and global force balances and then showing how this plays out in examples ranging from enhanced diffusion to osmotic energy conversion in pores and modules. The emphasis on nanochannels as the right scale for these effects to matter is helpful and connects to practical questions in the field. The discussion stays grounded in standard thermodynamics without introducing fitting parameters or untested assumptions. The soft spot is that nothing here is new in terms of equations or observations. It re-frames existing thermodynamic relations and literature results into a unified story, which is fine for a perspective but means the chapter stands or falls on how clearly it makes the connections rather than on fresh evidence. If the goal was to derive something parameter-free or test a prediction, this doesn't do that. This is aimed at people already working on transport in confined fluids or looking at blue energy ideas. A reader who knows the basics of diffuse layers will follow easily and might pick up a few new angles on applications, but someone outside the area might find it too specialized. It deserves a serious referee because the synthesis is coherent and the topic is active. I'd send it for review rather than reject outright, with the expectation that it gets positioned as a perspective piece and perhaps gets suggestions for more recent examples.

Referee Report

0 major / 1 minor

Summary. This perspective chapter synthesizes concepts of entropically-driven transport with a focus on diffusio-osmosis in nanochannels. It claims that osmosis and diffusio-osmosis are two faces of the same phenomenon, naturally embedded in an Onsager framework and quantified by local and global force balances. Specifying the phenomenon to nanochannels reveals that osmotic drivings can arise without semi-permeability, thereby extending the domain of entropically driven transport. The chapter discusses examples including enhanced diffusion, mechano-sensitivity, rectified osmotic flows, and osmotic energy conversion from single nanopores to membrane modules approaching industrial scale.

Significance. If the framing holds, the manuscript offers a coherent conceptual unification of osmosis and diffusio-osmosis grounded in established thermodynamic relations. It explicitly credits the absence of free parameters, ad-hoc axioms, or invented entities, relying instead on standard Onsager relations from prior literature. This synthesis clarifies the privileged role of nanochannels for such transport and has implications for applications in osmotic energy conversion. The interpretive approach provides a parameter-free view that could guide understanding in nanofluidics without introducing new mathematical inconsistencies.

minor comments (1)

[Abstract] Abstract: 'occuring' is a typo and should be corrected to 'occurring'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our perspective chapter on diffusio-osmotic transport in nanochannels. The recommendation for minor revision is noted; we are prepared to incorporate any specific minor suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript is a perspective chapter that synthesizes established concepts of entropically driven transport, framing osmosis and diffusio-osmosis within standard Onsager relations and local/global force balances drawn from prior literature. No new mathematical derivations, fitted parameters, or falsifiable predictions are advanced that reduce by construction to inputs internal to the paper; the central interpretive claim extends known diffuse-layer physics to nanochannels without introducing self-definitional steps, fitted-input predictions, or load-bearing self-citation chains. The framework remains self-contained against external benchmarks of classical irreversible thermodynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a conceptual review chapter; it does not introduce new fitted parameters, new entities, or ad-hoc axioms beyond standard thermodynamic relations already established in the field.

axioms (1)

standard math Onsager reciprocal relations apply to the coupled transport phenomena described
Invoked to embed osmosis and diffusio-osmosis as two faces of the same framework.

pith-pipeline@v0.9.0 · 5501 in / 1210 out tokens · 42508 ms · 2026-05-08T18:30:32.520918+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (washburn_uniqueness_aczel) washburn_uniqueness_aczel unclear
a linear relation can be written between thermodynamic forces and fluxes ... the transport matrix is symetric and positive definite according to the Onsager principle

Reference graph

Works this paper leans on

134 extracted references · 4 canonical work pages · 1 internal anchor

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Apparent osmotic pressure Let us focus on a situation where a salt concentra- tion difference is applied across the channel,i.e.c R ext = cs + ∆cs,c L ext =c s (RandLstands for the right and left reservoirs). The concentration gradient in the reser- voirs yields a concentration difference inside the channel, but also an electric potential difference and a...
[2]

The integral relation As an alternative approach, we may consider the inte- gral approach from equation in Eq.(79), which expresses the apparent osmotic pressure in terms of the ionic flux js. In the present overlapping regime of the capillary pore model, the ionic fluxj s can be readily computed from from Eq.(110) and the Donnan expressions in Eqs.(112)-...
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rejection

Apparent osmotic pressure from the Onsager symmetry From the previous calculation, we obtained that the water flow under a salt concentration gradient rewrites Qw =L w ×(−σ(Du)c s)× −2k BT∆logc s (125) with the negative of the “rejection” coefficient given by −σ(Du) = √ 1 +Du 2 −1 andc s is the bulk reservoir concentration. This result can also be infered...
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Sinceλ≪h, the flow identifies with that on a flat surface which was discussed in Sec

Solvent DO velocity and ion fluxes Let us assume that there is a local concentration gradi- ent along the nanochannel,c 0(x), with the concentration c0(x) defined in the middle of the channel cross section (see below). Sinceλ≪h, the flow identifies with that on a flat surface which was discussed in Sec. III. Tak- ing a distancezperpendicular to the pore s...
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standard

Diffusio-osmotic ionic currents An interesting consequence of diffusio-osmosis is the emergence of diffusio-osmotic ionic currents induced by a gradient of salt concentration. Again, current in- duced by salinty gradients is usually expected for selec- tive nanochannels/membranes, such as cation- or anion- exchange membranes (CEM/AEM). However, diffusio- ...

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1D-PNP equations integrating the diffuse layer contribution In the thin diffuse layer regime, there is a separation of length scales between the Electric Double Layer and the characteristic scales over which the geometry varies laterally (e.g.pore length). One can therefore integrate over the EDL in order to separate explicity the scales and derive one-di...
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Non-convective contributions Let me first ignore the convection effects (terms pro- portional to the fluid velocity), which we will estimate hereafter. Using Eq.(94), and identifying thatc v =c s the value of the salt concentration outside the EDL, one get the cross-sectional integrals of the non-convective contri- butions to the solute flux and ionic den...
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These are slightly more subtle to describe

Convective effects in the thin diffuse layer regime Now, one should add the convective contribution, which stem from the ion transport by the solvent flow. These are slightly more subtle to describe. This follows from the analysis developped in the previous sections, as well as for the capillary pore model. Gathering the pre- vious results, the Stokes equ...
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Osmotic Transport across Surface Functionalized Carbon Nanotube Membrane

Combining all elements... Gathering results in Eqs.(143), (145) & (148) we have obtained the complete expression of the local transport matrix. Differences of pressure/concentration/voltage are applied at the two ends of the channel will in- duce pressure/concentration/voltage profiles inside the nanochannel, whose equations obeys the conservation laws ab...
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But observations in Ref

Simple estimates Fick diffusion is associated with a flux of salt pro- portional to the salt concentration gradient, asJ Fick s = −D 2∆cs L (remembering that the total ion concentration is twice the salt concentration). But observations in Ref
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show an enhanced salt diffusion at low concentration, see Fig.5. It is easy to reconcile this observation with the effect of diffusio-osmosis. According to the various de- scriptions above, a DO flow will be induced under the concentration gradient obeying vDO =D DO × −∆ logc s L (149) where the DO mobility has the dimension of a diffusion coefficient. Fo...
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More insights into the inner transport The above approach gives the main trends for the effect of DO on diffusive transport. However it misses some subtleties of the inner transport. This is what I briefly discuss here. The diffusio-osmotic flow is expected to flush the con- centration profiles, hence affecting the concentration gra- dient along the nanoc...

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Charge contrast as a source of mechano-sensitivity The origin of the phenomenon is the charge discon- tinuity at the entrance of the nanochannel, which is at the origin of a cascade of transport couplings. In a few words, the ion in the EDL close to the activated surface are flushed towards (or away from) the nanochannel but entering the nanochannels thes...
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The cherry on the cake: a huge diffusio-osmotic boost The approach above however misses an important in- gredient. Indeed the created salinity gradient inside the nanochannel under flow is expected to induce a diffusio- osmotic velocity,v DO =−K DO ×(c s(L)−c s(0)) with KDO =D DO/cs. In the presence of slippage with b≫λ D, this yieldsv DO =− kB T η λDb×(c...

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Osmotic energy, a quick recap Osmotic, or “blue”, energy denotes the free energy that is released when solutions of different salinity mix, for example where rivers discharge into the sea. This mix- ing process is associated with a substantial entropy gain, corresponding to an ideal extractable energy of about 0.8 kWh per cubic metre of freshwater mixed w...

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Diffusio-osmosis as an alternative lever for osmotic energy harvesting The concept of diffusio-osmosis has changed the per- spective for osmotic energy conversion by showing that it does not require strict molecular selectivity as previ- ously conceived and can instead be driven by interfacially controlled mechanisms such as diffusio-osmosis. This di- rec...
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nanofluidic-informed

Scaling up nanofluidics Exploiting such surface-driven transport in highly charged nanopores actually provides a route to circum- vent the conventional permeability–selectivity compro- mise and to boost osmotic power densities as compared with PRO and RED. Interestingly, while existing PRO and RED plant are highly optimized in terms of engi- neering, the ...
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