Letter | Published:

Transport and dispersion across wiggling nanopores

Abstract

The transport of fluids at the nanoscale has achieved major breakthroughs over recent years1,2,3,4; however, artificial channels still cannot match the efficiency of biological porins in terms of fluxes or selectivity. Pore shape agitation—due to thermal fluctuations or in response to external stimuli—is believed to facilitate transport in biochannels5,6,7,8,9, but its impact on transport in artificial pores remains largely unexplored. Here we introduce a general theory for transport through thermally or actively fluctuating channels, which quantifies the impact of pore fluctuations on confined diffusion in terms of the spectral statistics of the channel fluctuations. Our findings demonstrate a complex interplay between transport and surface wiggling: agitation enhances diffusion via the induced fluid flow, but spatial variations in pore geometry can induce a slowing down via entropic trapping, in full agreement with molecular dynamics simulations and existing observations from the literature. Our results elucidate the impact of pore agitation in a broad range of artificial and biological porins, but also, at larger scales, in vascular motion in fungi, intestinal contractions and microfluidic surface waves. These results open up the possibility that transport across membranes can be actively tuned by external stimuli, with potential applications to nanoscale pumping, osmosis and dynamical ultrafiltration.

Main

“Everything that living things do can be understood in terms of the jigglings and wigglings of atoms”: this quotation by R. P. Feynman highlights the importance of fluctuations in living matter. Transport driven by fluctuations in biological nanopores is no exception to this rule, and has been widely investigated5,6,7,8,9. Similar observations have been made in (non-living) fluid phases, where bulk hydrodynamic fluctuations dramatically affect nanoscale dynamics10,11,12. More recently, numerical simulations have noted the impact of phonon modes in carbon nanotubes on the transport of particles confined within them13,14,15. At much larger scales, microfluidic surface waves have been exploited to sort or mix solute particles16, and peristalsis in fungal species or in the small intestine significantly modifies nutrient and bacterial transport17,18. While these observations—which span a considerable range of length scales—point to a quantitative impact of surface agitation on the transport in confined geometries, a general theory, predicting the dependence of transport properties on the surface fluctuations, is lacking. Even the simple question ‘do pore surface fluctuations enhance or diminish transport?’ is surprisingly difficult to answer. While surface fluctuations are naively expected to enhance diffusive transport via induced hydrodynamic flows, which underlies the canonical Taylor–Aris mechanism19,20, geometrical bulges may entrap particles, resulting in entropic slowing down, as captured by the Fick–Jacobs framework21,22, sketched in Fig. 1d. The more general situation of transport with temporally varying geometry remains open despite its obvious importance.

In this paper, we establish a general relationship between diffusive transport and the dynamical spectrum of surface fluctuations. Our framework applies to the case where structural fluctuations of the confining pore are induced by thermal noise, but crucially also to active, non-equilibrium fluctuations induced by external stimuli. We apply our formalism to several situations relevant to nanopore transport13,14, as well as to larger-scale configurations, for example active contractions in fungal species, which influence nutrient transport17,23,24; see Fig. 1a–c.

Our theory starts with the perturbative analysis of the diffusion of a particle confined between fluctuating surfaces. For simplicity, we consider the two-dimensional geometry presented in Fig. 1e, but the results and the formalism can readily be extended to three dimensions (Supplementary Information 1). The upper surface is located at z = H + h(x, t), where H is the mean height and h(x, t) represents fluctuations about this mean (x represents the coordinate along the channel and z the height coordinate). The tracer particle—for instance, a molecule of dye in the fluid—locally diffuses with a bare diffusion constant D0 and is advected by a hydrodynamic flow u. The probability density p(x, z, t) of the particle obeys the Fokker–Planck (or Smoluchowski) equation: ∂tp = −(up) + D0 Δp. At this stage, the convective degrees of freedom arising from the membrane fluctuations are not averaged out and explicitly enter the Fokker–Planck equation. The Fokker–Planck equation is valid as long as the timescales associated with the tracer position are larger than the microscopic relaxation timescales, and we can verify that this timescale separation assumption holds in practice down to molecular confinement25. In the case of thin channels, a standard expansion (Supplementary Information 1 and 2) yields a reduced advection diffusion equation for the marginal probability density $$p^ \ast (x,t)$$ = $${\int}_0^{H + h(x,t)} {\kern 1pt} p(x,z,t)\,{\rm d}z$$,

$$\partial _tp^ \ast = - \partial _x\left( {\overline u _xp^ \ast } \right) - D_0\partial _x\left( {\frac{{\partial _xh}}{{(H + h)}}p^ \ast } \right) + D_0\partial _{xx}p^ \ast$$
(1)

where $$\overline u _x$$ = $$\frac{1}{{H + h(x,t)}}{\int}_0^{H + h(x,t)} {\kern 1pt} u_x(x,z,t)dz$$ is the height averaged longitudinal hydrodynamic flow. Incompressibility of the flow also gives $$\partial _x\left( {(H + h)\overline u _x} \right)$$ = $$- \partial _th$$. Equation (1) contains two advection terms, which both depend explicitly on height fluctuations. The first is advection by the height-averaged hydrodynamic flow, expected to enhance diffusion. The second term is present even in the absence of any fluid in the channel and represents the effect of geometric fluctuations of the confining channel. When h does not depend on time, the advection term is zero and the second term represents an entropic potential, which slows down diffusion21.

We analyse equation (1) via a perturbation expansion in the amplitude of the fluctuations of h (with $$\left\langle h \right\rangle = 0$$, where $$\left\langle \cdot \right\rangle$$ is the average over the thermal fluctuations). Using a diagrammatic expansion, and to first non-trivial order26, we then demonstrate that the renormalized late-time diffusion constant is given in terms of the spectrum of the pore surface fluctuations S(k, ω), where k and ω are Fourier frequencies in space and time, respectively, as

$$D_{\rm e} = D_0\left( {1 - \frac{1}{{H^2}}{\int} \frac{{{\rm d}k{\kern 1pt} {\kern 1pt} {\rm d}\omega }}{{(2\uppi )^2}}\frac{{\left( {D_0k^2} \right)^2 - 3\omega ^2}}{{\left( {D_0k^2} \right)^2 + \omega ^2}}S(k,\omega )} \right)$$
(2)

where S(k, ω) is defined by $$\left\langle {\tilde h(k,\omega )\tilde h(k^\prime ,\omega^ \prime )} \right\rangle$$ = (2π)2δ(k + k′)δ(ω + ω′)S(k, ω). Equation (2) is the main result of this work. The full derivation of this key result is reported in Supplementary Information 3. This equation can be generalized to any dimension; it can also be expressed in terms of the response function, related to the spectrum via fluctuation dissipation, here evaluated at an imaginary frequency (Supplementary Information 4). Note also that a similar expression can be obtained for the wave-induced velocity drift (Supplementary Information 3).

At this stage, the formalism is completely general and can be used to compute the renormalized diffusion resulting from any pore or interface motion in terms of the pore or interface fluctuation spectrum. It may be interpreted as a generalization of the Fick–Jacobs formalism to dynamical confinement. It also applies to the case where fluctuations originate from thermal noise, as well as active (non-equilibrium) fluctuations driven by external stimuli.

Equation (2) highlights that the renormalization of the diffusion constant can be either positive or negative depending on the fluctuation spectrum: jiggling can both speed things up and slow things down! Indeed, when the height field is frozen, we have S(k, ω) = (2π)δ(ω)Sf(k), where Sf(k) is the spectral density of the spatial variations of the frozen height. Here we find De/D0 = 1 − $$\left\langle {h^2} \right\rangle {\mathrm{/}}H^2$$, in agreement with Fick–Jacobs21,22; diffusion is decreased via entropic trapping. When the height fluctuations are very rapid in time, but have a large spatial wavelength, we find De/D0 = 1 + $$3\left\langle {h^2} \right\rangle {\mathrm{/}}H^2$$ and diffusion is enhanced via advection coupling. We highlight that these limiting regimes are very general and can be recovered in the case of more complex surface spectra.

In the following we give two example applications of this theory. We first focus on thermal fluctuations of a thin layer. Thin-film dynamics can be analysed within the lubrification approximation27 and in the limit of low Reynolds numbers. The fluctuation interface spectrum can be obtained from the Hamiltonian $${\cal H}[h]$$ of the height fluctuations h. The energy functional depends on the Van der Waals interactions between the surfaces, possibly the discrete properties of the confined liquid28,29 and also contributions associated with constraints fixing the average gap size. The dynamics for small height fluctuations are given by27

$$\partial _th = - \partial _x\left( {(H + h)\overline u _x} \right) = \partial _x\left( {\lambda (H)\partial _x\frac{{\delta {\cal H}}}{{\delta h}} + \sqrt {\lambda (H)} {\cal N}} \right)$$
(3)

with λ(H) the permeance of the slit. The term $${\cal N}$$ is a Gaussian white noise with $$\left\langle {{\cal N}(x,t){\cal N}(x^\prime ,t^\prime )} \right\rangle$$ = 2(kBT/W)δ(x − x′)δ(t − t′), where T is the temperature and W the channel width, generated by hydrodynamic fluctuations within the liquid27. Focusing more specifically on a thin film confined by an elastic membrane, the Hamiltonian takes the Landau form

$${\cal H}[h] = \phi (H)WL + \frac{W}{2}{\int}_0^L {\kern 1pt} {\rm{d}}x\;\left( {\phi ^{\prime\prime} (H)h^2 + \kappa (H)\left( {\partial _{xx}h} \right)^2} \right)$$
(4)

where ϕ(H) is the free energy per unit area of a flat interface (verifying ϕ′(H) = 0 for the equilibrium height) and κ(H) can be interpreted as the surface’s bending rigidity; L is the channel length. In this case S(k, ω) = 2kB(H)k2/(ω2 + k4λ(H)2(ϕ″ + κk4)2) and, directly, one finds using equation (2)

$$D_{\rm e} = D_0\left( {1 + \frac{{\left\langle {h^2} \right\rangle }}{{H^2}}\left[ {\frac{4}{{(1 + D_0{\mathrm{/}}D(H))^{3/4}}} - 1} \right]} \right)$$
(5)

with $$\left\langle {h^2} \right\rangle$$ = $$k_{\rm B}T\ell ^3{\mathrm{/}}2\sqrt 2 W\kappa$$, where $$\ell$$ = (κ/ϕ″(H))1/4 is the characteristic wavelength of the surface fluctuations. The quantity D(H) = λ(H)ϕ″(H) is a diffusion coefficient scale associated with interface fluctuations. For no slip boundary, one has D(H) = $$\frac{{H^3}}{{12\eta }}\phi^{\prime\prime} (H)$$, where η is the viscosity of the liquid. Other fluctuating confinement cases, with different Hamiltonians, geometries (e.g. tubes) and boundary conditions (e.g. partial slip), can easily be analysed, and further examples are given in Supplementary Information 6, leading to qualitatively similar conclusions.

Equation (5) shows that if bulk diffusion is large (specifically, D0/D(H) ≥ $${\cal M}_{\mathrm{0}}$$ with $${\cal M}_{\mathrm{0}}$$ = 44/3 − 1), then effective diffusion will be reduced, i.e. De ≤ D0. Particles with high bare diffusivity see an effectively quenched surface profile, which generates an entropic trapping potential, thus slowing down diffusion. On the other hand, for slowly diffusing particles, D0/D(H) ≤ $${\cal M}_{\mathrm{0}}$$, the fluctuation-induced advection enhances diffusion, i.e. De ≥ D0. This results in a compensation effect, summarized in Fig. 2a, whereby fast particles are slowed down and slow particles are sped up. This difference in effective diffusion as a function of bare diffusion has been observed in simulations of gas particles in metal–organic frameworks30 and water transport in biological porins9. Equation (5) also describes typical low-inertia phonon modes on an interface (such as phonons on the surface of graphene or carbon nanotubes31; see also Supplementary Information 7). The impact of phonon modes of a carbon nanotube on confined water has indeed been measured and shown to depend substantially on phonon frequency13.

As a second example, we consider a situation where the fluctuations of the pore’s shape are driven out of equilibrium by an external stimulus. This situation is particularly interesting in the sense that the pore structure undergoes non-equilibrium fluctuations, therefore leading to out-of-equilibrium transport. In general, this situation can be accounted for by adding a force to equation (3) with a specific frequency ω0 and wavenumber k0. We consider here a propagating wave of amplitude a0 on the interface of a deforming tube with bare radius R as δR(x, t) = a0 cos(ω0t − k0x). In this context, the relevant driving spectrum is given by S(k, ω) = π2$$a_0^2$$[δ(ω0 + ω)δ(k0 − k) + δ(ω0 − ω)δ(k0 + k)], and using the result of equation (2) for a tubular geometry we find (see Supplementary Information 5 and 8 for the exhaustive derivation)

$$D_{\rm e} = D_0\left( {1 - \frac{{a_0^2}}{{2R^2}}\frac{{1 - 3\left( {D_ \approx {\mathrm{/}}D_0} \right)^2}}{{1 + \left( {D_ \approx {\mathrm{/}}D_0} \right)^2}}} \right)$$
(6)

where D = $$\omega _0{\mathrm{/}}k_0^2$$ can be interpreted as a diffusion coefficient associated with the surface wavepacket. Interestingly, one recovers, from equation (6), the two regimes highlighted previously, but now for the actively fluctuating surfaces: diffusion is enhanced or slowed down depending on whether the bare diffusion D0 is lower or higher than the wavepacket diffusion D. The results are summarized in Fig. 2b.

Altogether, for the in- and out-of-equilibrium cases respectively, one can define a dimensionless Péclet-like number to characterize the transition from decreased to enhanced mixing. For equilibrium fluctuations Pe = D(H)/D0 with D(H) the dispersion induced by surface-induced flow, while for active fluctuations Pe = D/D0, introducing active surface transport via D. One may accordingly expect that $$Pe\sim \ell ^2{\mathrm{/}}D_0\tau$$ where $$\ell$$ and τ are characteristic length and time scales of the fluctuations. An outcome of our theoretical framework is that this Péclet number enables us to quantify the impact of surface fluctuations on the transport. Typically, for high (low) Péclet number Pe ≥ 1 (Pe ≤ 1), the transport is expected to increase (decrease) under structural jiggling; see Fig. 3.

We show in Fig. 3 a variety of systems where the structural agitation of the confinement is expected to have an impact on particle transport. These fluctuating systems are classified according to their corresponding Péclet number (full details are given in Supplementary Information 10). (1) Biological channels, such as aquaporins, will undergo thermal fluctuations with typically Pe ~ 10−2–102 (Supplementary Table 1), indeed both behaviours were observed for water in a KcsA channel9, and diffusion is indeed enhanced where fluctuation timescales are fast. In the selectivity filter of the KcsA channel, fluctuations are typically 0.5–0.8 Å, so the enhancement factor is $$a_0^2{\mathrm{/}}R^2$$ 0.1–0.26,32. (2) In metal–organic frameworks the range is similar (Supplementary Table 2): small gas molecules such as H2 have Pe ≤ 1, and indeed such molecules are not accelerated by framework flexibility30; while large gas molecules have Pe ≥ 1, and indeed molecular diffusion is increased, well beyond perturbation theory, e.g. $$D_{\rm e}{\mathrm{/}}D_0 \gg 1$$. In this context, the spatial and temporal vibrations yield $$\left\langle {h^2} \right\rangle {\mathrm{/}}H^2$$ 0.333 and thus a significant enhancement of De. (3) Graphene sheets show thermal ripples that are expected to affect water transport, typically with Pe ~ 101–103 (Supplementary Table 3). We have fully confirmed the theoretical picture in this situation by carrying out our own molecular dynamics simulations of water confined between graphene sheets: the thermally fluctuating sheets do increase particle diffusion by 150% as compared to the flat rigid sheets, while diffusion between undulated but frozen graphene sheets is reduced to 80% with respect to the flat rigid sheets; see Supplementary Information 11. Putting in the numbers for the fluctuating graphene case, the typical enhancement factor for De in our simulations is actually found to be larger than the prediction $$1 + 3\left\langle {h^2} \right\rangle {\mathrm{/}}H^2$$ 110%, pointing to complementary molecular effects. (4) Finally, we find that non-dissipative phonon coupling is in the range Pe ~ 104–105 (Supplementary Table 4) and always enhances diffusion, in accordance with the observations of ref. 13.

Moving to actively driven dynamics, various examples include (1) nanotubes, in which longitudinal mechanical vibrations could typically yield Pe ~ 105 (Supplementary Table 5), echoing the numerical results of ref. 34, (2) the small intestine, which demonstrates contractile activity with typically Pe ~ 106 (Supplementary Table 6) (evidence of an increased dispersion has also been observed, though not quantified, in a microfluidic device mimicking the contractions of the intestine18; in the small bowel, one finds $$a_0^2{\mathrm{/}}R^2$$ 0.12–0.25 and thus the enhancement of De is expected to be large35), and (3) slime mould vascular networks, in which nutrient dispersion is characterized typically by Pe ~ 10−1–102, and increasing for larger organisms (Supplementary Table 7). The latter example of nutrient dispersal in contracting vascular networks of organisms is actually of particular interest, since it is essential for survival to generate alternative strategies for fast nutrient dispersal. For example the unicellular organism P. polycephalum, shown in Fig. 1c, is composed of a connected network of veins, containing cytoplasm that can deliver nutrients to the entire body: it exhibits non-stationary periodic shuttle flows36 driven by a peristaltic wave of contractions17 spanning the organism. For an individual such as the one in Fig. 3 with length L = 8 mm, k0 2π/L ~ 800 m−1 and ω0 ~ 0.06 s−136, we find D ~ 10−7 m2 s−1. This is two orders of magnitude larger than the bare diffusion of nutrient molecules, with at most D0 ~ 10−9 m2 s−1, as indeed observed in simulations in ref. 17. In this organism the hierarchical structure of veins leads to a0/R 0.9 (a0 being the typical difference between large and small veins), such that the enhancement of De is expected to be large.

This figure highlights the fact that a wide variety of situations, covering a huge range of scales, can be put in perspective under the framework of spectral mixing by confinement jiggling. Going further, it is possible to study the relative effect of coupled passive and active dynamics, for instance in biological membranes, with an active component37, or nanoscale systems such as nanotubes that are sensitive to thermal as well as activated vibrations38. Taking a step further, it would be interesting to extend the theory and investigate how coloured noise or nanoscale non-Markovian effects might influence transport. Moving to macroscopic scales, it would be interesting to connect this formalism with cases where the flow velocity presents some recirculation, as may occur in the intestine18 but also in ocean waves, which have been shown to influence mixing and dispersion39.

Our formalism allows us, in particular, to identify the key components to design active channels. The results here show that it is possible to actively tune the diffusion of species, therefore dynamically differentiating solutes as a function of their own Peclet number, and thus providing the conditions for active separation40. Moreover, the modification of osmosis processes in fluctuating channels remains to be explored, along with their applications to filtration and desalination. In a different context, to improve the sensitivity of sequencing via translocation through nanopores, a slowing down of some species may indeed be required. Our results also demonstrate that this fine tuning is possible by harnessing out-of-equilibrium dynamics of the pore, with considerable implications for DNA sequencing41,42.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors are indebted to B. Rotenberg for several fruitful discussions on molecular dynamics, and to K. Alim for bringing to the discussion biologically related examples. The authors also thank F.-X. Courdert, D.avid Lacoste and J.-F. Joanny for interesting discussions. S.M. acknowledges funding from a J.-P. Aguilar grant of the CFM foundation. D.S.D. acknowledges funding from the ANR grant FISICS. L.B. acknowledges support from ANR grant Neptune. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche.

Author information

Affiliations

1. Laboratoire de Physique Statistique, Ecole Normale Supérieure and CNRS, PSL University, Paris, France

• Sophie Marbach
•  & Lydéric Bocquet
2. Laboratoire Ondes et Matière d’Aquitaine (LOMA), CNRS, UMR 5798 Université de Bordeaux, Talence, France

• David S. Dean

Contributions

L.B. designed the research. S.M., D.S.D. and L.B. conducted research. S.M. carried out the molecular dynamics simulations. S.M., D.S.D. and L.B. wrote the paper.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Lydéric Bocquet.

Electronic supplementary material

1. Supplementary Information

7 Figures, 7 Tables, 42 References