Understanding Fast and Robust Thermo-Osmotic Flows through

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Chemical and Dynamical Processes in Solution; Polymers, Glasses, and Soft Matter

Understanding Fast and Robust Thermo-Osmotic Flows through Carbon Nanotube Membranes: Thermodynamics Meets Hydrodynamics Li Fu, Samy Merabia, and Laurent Joly J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00703 • Publication Date (Web): 06 Apr 2018 Downloaded from http://pubs.acs.org on April 6, 2018

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The Journal of Physical Chemistry Letters

Understanding Fast and Robust Thermo-osmotic Flows through Carbon Nanotube Membranes: Thermodynamics Meets Hydrodynamics Li Fu, Samy Merabia, and Laurent Joly∗ Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France E-mail: [email protected]

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Abstract Following our recent theoretical prediction of the giant thermo-osmotic response of the water-graphene interface, we explore the practical implementation of waste heat harvesting with carbon-based membranes, focusing on model membranes of carbon nanotubes (CNT). To that aim we combine molecular dynamics simulations and an analytical model considering the details of hydrodynamics in the membrane and at the tube entrances. The analytical model and the simulation results match quantitatively, highlighting the need to take into account both thermodynamics and hydrodynamics to predict thermo-osmotic flows through membranes. We show that, despite viscous entrance effects and a thermal short-circuit mechanism, CNT membranes can generate very fast thermo-osmotic flows, which can overcome the osmotic pressure of seawater. We then show that in small tubes, confinement has a complex effect on the flow and can even reverse the flow direction. Beyond CNT membranes, our analytical model can guide the search for other membranes to generate fast and robust thermo-osmotic flows.

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Introduction The development of sustainable alternative energies is one of the greatest challenges faced by our society, and nanofluidic systems could contribute significantly in that field. 1–3 For instance, nanoporous membranes subject to a salinity gradient generate an electric current. 4 Recently developed membranes with very high power density could be used to harvest the so-called blue energy contained in salinity differences between sea and river water. 5–7 Similarly, membranes could be used to harvest energy from waste heat. 8–10 Vapor-gap membranes have been successfully used to generate pressure gradients under small temperature differences, 11 but waste heat harvesting with nanoporous membranes has been much less explored. On a fundamental side, molecular dynamics simulations have been used to investigate the response of nanofluidic systems to thermal gradients at the atomistic scale. 12–22 In particular, the molecular mechanisms of thermo-osmotic flows, generated at interfaces by thermal gradients, have been explored recently. 23,24 Notably, we have shown 25 that standard descriptions of thermo-osmosis based on irreversible thermodynamics 26–32 had to be complemented to take into account interfacial hydrodynamics (stagnant layer or liquid-solid slip). In that context, we have recently put forward the giant thermo-osmotic response of the water-graphene interface, 25 which opened interesting perspectives for efficient energy harvesting from thermal gradients based on carbon nanostructures. However, the practical implementation of carbon-based membranes poses a number of challenges. First, the giant thermo-osmotic response was not due to a giant thermodynamic force, but to the ultralow interfacial friction displayed by graphitic surfaces. 33–38 In a real membrane, viscous entrance effects 39–43 could strongly reduce the amplitude of the thermo-osmotic flow. Second, the large thermal conductivity of graphitic structures 44,45 could also limit the flow by a thermal short-circuit mechanism. 46 Finally, thermo-osmotic flows could be used to desalinate water, using membranes with pores sufficiently small to reject ions. This potential application calls for an investigation

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of the thermo-osmotic response in ultra-confined systems. In particular, one needs to know whether thermo-osmosis can overcome the osmotic pressure induced by the salt difference in a desalination setup. In order to assess the practical feasibility of waste heat harvesting with thermo-osmosis, taking into account the effects of entrances, thermal conductivity, and confinement, we consider in this letter a model membrane made of carbon nanotubes (CNT). We first develop an analytical model predicting the velocity of the flow through the membrane. We then assess the model using molecular dynamics simulations, and finally we discuss its implications for realistic systems.

Model The total flow through the nanotube can be decomposed as the superimposition of the bare thermo-osmotic flow and of a Poiseuille backflow, which is generated by entrance pressure drops and possibly by a difference in osmotic pressure between the reservoirs. The thermo-osmotic flow is generated by the thermodynamic force density fthermo in the tube: 25,30 fthermo = δheq (r; T (z))

∇T (z), T

(1)

with T (z) the local temperature (neglecting radial variations of T ), δheq (r; T ) = heq (r; T ) − hbulk (T ) the enthalpy excess density, heq (r; T ) the local equilibrium enthalpy density in the tube, and hbulk (T ) the bulk enthalpy density. In the following, we will consider the linear response regime, obtained for small temperature differences as compared to the average temperature. One can then simplify the expression of the thermodynamic force:

fthermo ≈ δheq (r; Tavg )

∇T (z) , Tavg

(2)

where Tavg is the average temperature of the system, corresponding to the equilibrium value when no temperature difference is imposed between the reservoirs. Within the framework of continuum hydrodynamics, one could then solve Stokes equa-

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tion to obtain the exact velocity profile in the tube. However, both experiments 33,34,38 and simulations 35–37 have shown that water-CNT interfaces exhibit very large slippage, with slip lengths b (the ratio between the slip velocity and the shear rate at the wall 47 ) up to several hundred nanometers. One can therefore assume that b is much larger than the thickness w of the interaction layer where the liquid enthalpy density differs from its bulk value. In that case, the thermo-osmotic velocity profile is almost constant over the extend of the interaction layer. The flow is consequently plug-like over the entire channel section, so that there is no shear in the liquid and the velocity jump is localized at the wall. 25 The thermo-osmotic velocity vosm then results from a simple force balance between the total thermodynamic force experienced by the liquid and the interfacial friction force. 25 One can integrate fthermo over the volume of the tube to get the total thermodynamic force Fthermo : Z

L

R

Z

dr 2πr δheq (r; Tavg )

dz

Fthermo =

0

0

∇T , Tavg

(3)

where L and R are the length and the radius of the tube, respectively. We define the average enthalpy excess density over the section of the tube: 1 δh = πR2

Z

R

dr 2πr δheq (r; Tavg ),

(4)

0

so that the expression of Fthermo simplifies:

Fthermo = πR2 δh

∆T . Tavg

(5)

One can note in particular that Fthermo is only controlled by the temperature difference ∆T between the reservoirs, and that the temperature profile inside the tube plays no role. Since the whole liquid moves with a homogeneous velocity vosm , the total friction force between the liquid and the tube is: Ffriction = 2πRLλvosm , where λ is the liquid-solid friction coefficient, quantifying the ratio between the friction force per unit area and the slip

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velocity. 47 By writing that Fthermo = Ffriction , one obtains the thermo-osmotic velocity:

vosm =

R δh ∆T × . 2λL Tavg

(6)

However, the focusing of the streamlines at the entrances of the tube generates viscous dissipation, which results in pressure jumps at each entrance, with a corresponding total pressure drop along the tube ∆pin proportional to the average velocity U in the nanotube: 39–43

∆pin =

πCη U, R

(7)

where η is the liquid viscosity (at the entrances), C is a numerical constant, which depends on the hydrodynamic boundary condition inside the tube and on atomic chamfering effects in nanometric tubes. 48 In a desalination setup, an osmotic pressure ∆Π will add to the entrance pressure drops. The total pressure difference ∆p = ∆pin + ∆Π induces a Poiseuille backflow with an average velocity vback . Assuming b R, one can express vback :

vback =

R(∆pin + ∆Π) . 2λL

(8)

The final velocity U = vosm − vback can be obtained by combining eqs 6, 7, and 8: Tavg ∆Π R δh ∆T U= × × 1− . 2λL + πCη Tavg ∆T δh

(9)

This expression differs from that of the bare thermo-osmotic velocity, eq 6, in two aspects: first, the 2λL term arising from friction on the tube walls is complemented with a πCη term arising from viscous entrance effects; second, the new parenthesis on the right describes the effect of osmotic pressure, and it provides a simple criterion on the maximum osmotic pressure ∆Πmax that can be overcome by the thermo-osmotic flow (i.e., such that U > 0):

∆Πmax = δh × ∆T /Tavg . 6


(10)

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This criterion, obtained from a molecular perspective, is reminiscent of the one derived by Dariel and Kedem based on irreversible thermodynamics. 27 Note however two differences. First, this criterion was obtained in the limit of large slip. As demonstrated in the supporting information (SI), when the slip length is not large as compared to the pore size (e.g., in the case of a no-slip boundary condition), both the definition of the average excess enthalpy, eq 4, and the expression of the flow velocity, eq 9, must be corrected to take into account hydrodynamics within the pore: 1 δh = πR2

Z

R

dr 2πr δh

0

r2 1− 2 R + 2bR

,

(11)

and U=

R2 1 +

2b R

× δh × (∆T /Tavg ) × 4b 4ηL + πCηR 1 + 2 R

! 1 + 4b T ∆Π avg R × 1− . 2 1 + 2b ∆T δh R

(12)

Second, we will later check that, as discussed in Refs. 23 and 24, it is the component along the flow direction of the (anisotropic) enthalpy excess that controls the flow. We will now test the predictions of the model using molecular dynamics simulations.

Figure 1: (a) Illustration of a simulation system. The two reservoirs are maintained at the same pressure using pistons and thermostatted at different temperatures. The nanotube radius is Rc = 7.0 ˚ A for the present figure, and the length is fixed at L = 100 ˚ A. (b) Corresponding temperature profiles of water and carbon atoms in the nanotube.

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Methods Here we introduce briefly the simulation setup; details can be found in the SI. A membrane consisting of two parallel graphene sheets and a nanotube was connected to two reservoirs at the same pressure p = 1 atm, imposed by using two pistons, see Figure 1. All simulations were performed with LAMMPS package. 49 We used the TIP4P/2005f force field for water 50 and the LCBOP force field for carbon atoms. 51 For water-carbon interactions, we used Lennard-Jones interaction parameters calibrated from high-level quantum calculations of water adsorption on graphene, 52 using the generic geometry of TIPnP water models for the relative O and H positions. These parameters were shown to reproduce accurately interaction energies between water and CNTs obtained from quantum chemistry calculations. 53 The system was firstly thermostatted at a given temperature Tavg = 323 K and we ensured that water had filled the nanotube before the next step. We then ran the system for another 10 ns to measure the local equilibrium enthalpy density heq (r; Tavg ) in the nanotube and the bulk enthalpy density hbulk (Tavg ) in the reservoirs. The enthalpy density was expressed zz as h(r) = (ui (r) + pzz i (r))ρ(r), where ui (r) was the energy per particle, pi (r) the atom-

based virial expression for pressure parallel to the nanotube axis, and ρ(r) the local density. Following the Onsager relation between the excess heat flux and the thermocapillary flow, 23,24 For comparison, we also the local pressure was defined along the flow direction as pzz i (r). yy zz = (pxx computed the enthalpy density using the average pressure pave i + pi + pi )/3. After i

thermalization of the reservoirs at different temperatures Thigh = 353 K and Tlow = 293 K, we measured the total flow velocity from the time evolution of the number of particles in the reservoirs in the steady-state, with production runs lasting between 30 ns for the largest tube and 100 ns for the smallest. The nanotube radius Rc , defined using the center of carbon atoms, was varied from 5.1 ˚ A to 11.7 ˚ A, and the length L was fixed at 100 ˚ A. For each nanotube radius, we performed 6 independent simulations, from which we computed the error bars on U with a level of confidence of 95 %.

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Results and discussion Figure 1b shows typical temperature profiles for water and carbon atoms along the tube axis. The temperature profile of water is not linear in the nanotube, with large gradients close to the entrances, and smaller gradients in the middle of the tube. This type of temperature profile has been observed in a similar system, 46 and explained by a thermal short-circuit mechanism due to the very large thermal conductivity of graphitic structures. 44,45 Nevertheless, our analytical model predicts that the flow velocity depends only on the temperature difference between the reservoirs, see eq 9, so that the temperature profile inside the tube should not affect the flow velocity. We now compare the measured flow velocities with the prediction of the model, eq 9, in a thermo-osmotic pumping configuration (∆Π = 0) and for different tube radii. In the theoretical formula, the hydrodynamic radius was estimated as R = Rc − σCO /2, where σCO is the interaction distance between carbon and oxygen atoms defined in the force field. 52 The friction coefficient was previously measured for the water-graphene interface with the same interaction parameters: λ ≈ 1.7 × 104 Pa s/m. 25 This value represents an upper bound for λ, since friction inside CNT has been shown to be lower than on graphene. 36,37 The viscosity term arises from entrance effects, so that we used the average viscosity of TIP4P/2005f water 54 at both entrances, η = (η[293 K] + η[353 K])/2 = 0.662 mPa s. Finally, the value of the numerical constant C was taken from Fig. 7 in Ref. 48, and takes values between 1.7 and 2.5 for the tube radii considered. With these parameters, πCη was roughly ten times larger than 2λL in eq 9, i.e., entrance effects dominated over friction on the tube wall. Figure 2, which compares the prediction of the model and the MD results for the evolution of the flow velocity with the tube radius, reveals several interesting features. First, the velocities measured in the simulations and predicted by eq 9 match perfectly, validating the model introduced earlier. The validity of continuum hydrodynamics at such small scales could appear surprising. 4,55 In the simulated systems however, viscous dissipation mostly occurs at the entrances, and continuum equations for entrance pressure drop have been shown to hold even for subnanometric pores. 48 It is also surprising that a linear-response

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1.0

Total flow velocity (m/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.5

0.0 Measured -0.5

Model 4

5

6

7

8

9

10 11 12 13

CNT radius (Å)

Figure 2: Measured and predicted total flow velocity as a function of the nanotube radius. model can describe quantitatively MD results obtained for large temperature differences. However, we verified that in the simulation conditions, the approximate expression of the local thermodynamic force induced by temperature gradients, eq 2, was on average along the tube a very good approximation of the exact expression, eq 1, explaining the validity of the model. The assumption that the temperature does not vary in the radial direction is also justified in the present system, due to the very low thermal conductance of water-graphitic interfaces, 56 which localizes temperature gradients at the interface, and to the giant slip ensuring that viscous heating in the tube is negligible. Note finally that in the model, it was crucial to compute δh based on the local pressure along the tube axis, pzz i ; when using the average pressure pave i , the model failed to reproduce the MD results. Second, despite viscous entrance effects and thermal short-circuit effects, the total flow velocity can reach ca. 0.7 m/s, which substantially exceeds typically measured thermoosmotic velocities, e.g. on the order of 1 to 10 µm/s for water on bare glass or pluroniccoated glass surfaces. 30 The measured velocities are consistent with previous MD results for a similar system: 17 the authors measured a flow of 0.89 molecules/pore/ns with a tube diameter of 8.1 ˚ A and ∆T = 15 K, corresponding to U ≈ 0.07 m/s, and to U ≈ 0.3 m/s for the temperature difference we used (60 K). In terms of water flux, we measured a maximum value of ca. 2.5 × 106 kg/h/m2 (for Rc = 6.3 ˚ A, with the simulation geometry corresponding 10


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to a pore density of ca. 1013 cm−2 ), more than 40 times the one measured with MD in zeolites for ∆T = 20 K. 20 Note that these giant velocities are specific to our simulation setup, with a very short tube and a large temperature difference; we will estimate flow velocities and water fluxes in realistic situations later, based on the analytical model. Third, for large tubes (Rc > 7 ˚ A), the velocity appears to reach a plateau, while for small tubes the velocity strongly varies; in particular, the flow direction changes for Rc = 5.5 ˚ A.

Figure 3: Enthalpy excess density profiles for nanotubes with different radii. When the radius becomes large enough, the local enthalpy in the center of the nanotube approaches the bulk value, and the local enthalpy excess reaches zero. The analytical model can now provide insight into the mechanisms underlying the evolution of the flow velocity with the tube radius. In eq 9 (with ∆Π = 0), the dependency to the tube radius comes from the R δh term. To understand its evolution with R, we plotted the radial profiles of excess enthalpy density for all the nanotubes considered in Figure 3, and the corresponding average enthalpy excess δh in Figure 4. In the largest tube, the enthalpy excess differs from zero close to the tube wall, in an interaction layer of thickness w ∼ 5 ˚ A. For even larger tubes, in the limit of a thin interaction

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layer w R and assuming that the enthalpy profile close to the wall is weakly affected by wall curvature, eq 4 predicts that δh varies as 1/R. Therefore, according to eq 9, the velocity U ∝ Rδh should not depend on the radius. Indeed, in the limit of weak confinement, the bare osmotic velocity in the tube vosm should reach the value generated by a planar graphene sheet, 25 which does not depend on the tube size; we show in the SI that this is the case when w R, regardless of the hydrodynamic boundary condition. Additionally, entrance effects do not add any dependency of the total flow velocity U to the radius; this, however, is only true when b R, as detailed in the SI. This result is compatible with the plateau of velocity appearing at large radii in Figure 2, and indicates that the giant velocities we observed for the simulated tubes should persist in larger tubes as long as b R. In smaller tubes, the interaction layers start to overlap and interfere, resulting in a complex variation of δh and hence of the velocity. In particular, the change of flow direction observed for Rc = 5.5 ˚ A indeed corresponds to a change of sign of δh. Finally, the interaction layer overlap results in a maximum absolute flow velocity of ca. 0.7 m/s for a special tube radius of Rc = 6.3 ˚ A. Now that the analytical model has been validated by MD results, one can use it to predict the performance of experimental systems. Indeed, the flow velocity is proportional to ∆T /Tavg . In our simulations, we used a large value of ∼ 20 %. Experimentally relevant values might be 10 to 20 times smaller. We have discussed earlier that tubes with various radii should generate similar velocities as long as b R. In contrast, using longer tubes might reduce the performance. Indeed, in eq 9, the effect of dissipation is described by the 2λL + πCη term. For short tubes, viscous entrance effects dominate, πCη 2λL, and the velocity does not depend on the tube length. However, above a critical length Lc = πCη/(2λ) = πCb/2 (indeed, b is related to the viscosity and the interfacial friction coefficient: 47 b = η/λ), friction on the tube wall will dominate, and the velocity will scale as 1/L. Using C ≈ 2, η = 0.662 mPa s, and λ = 1.7 × 104 Pa s/m, one finds Lc ≈ 120 nm. Therefore, for a tube of length L = 1 µm, the velocity should decrease by a factor L/Lc ∼ 8.

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Overall, the velocity in experimental systems could typically be 2 orders of magnitude smaller than in our simulations, U ∼ 5 mm/s, which remains very large as compared to previously measured thermo-osmotic velocities, up to ∼ 10 µm/s. With a density of pores that can reach ∼ 1013 cm−2 , 57 the corresponding water flux would be on the order of 2000 kg/h/m2 .

Figure 4: Average enthalpy excess density over the section of the tube δh as a function of tube radius. Inset: minimum (∆T /Tavg ) necessary to overcome the osmotic pressure of seawater (c.a. 2.8 MPa); Two data points with (∆T /Tavg )min out of reasonable range (for Rc = 5.1 and 5.9 ˚ A) are not shown in the inset. Finally, we focus on desalination applications of such membrane systems. In a desalination setup, the average driving force over the section, δh × ∆T /Tavg , should be greater than the osmotic pressure ∆Π. The latter can be approximately deduced by the Van’t Hoff equation and yields c.a. 2.8 MPa for seawater. The value of δh for different nanotubes were compared with this value in Figure 4. Except for two nanotubes with radii Rc = 5.1 and 5.9 ˚ A, δh largely overcomes the osmotic pressure of seawater, demonstrating the possibility of using CNT membranes for desalination. We calculated the minimum value of ∆T /Tavg necessary to overcome the osmotic pressure for each nanotube, see inset in Figure 4. For the best tube, (∆T /Tavg )min ∼ 7 %. We can now use the analytical model to predict the influence of the tube geometry on the capacity to overcome an osmotic pressure. Since the latter is directly controlled by δh, the tube length should have no effect. In contrast, we have shown earlier that in the limit of large radii, δh should vary as 1/R, so that the maximum pressure 13



that can be overcome will quickly decrease with the tube radius. In terms of overcoming a pressure difference, it is therefore critical to use nanometric pores, so that the size of the pore compares with the thickness of the interaction layer, and δh is maximal. In any case, for desalination applications the membrane should be selective to ions, which imposes a small pore size. As a test of our theoretical prediction, we performed an additional set of simulations for a tube radius Rc = 6.3 ˚ A, where we imposed a pressure difference ∆Π = 2.8 MPa against the flow direction with the help of the pistons. We observed that the thermo-osmotic flow was able to overcome the pressure difference, with a velocity U = 0.38 ± 0.18 m/s compatible with the prediction of eq 9, U = 0.44 ± 0.06 m/s. Note that we imposed pressures of 1 atm and 1 + 28 atm in the reservoirs, so that in the model we used the excess enthalpy measured at the average pressure, p = 14.5 atm. Similarly, previous MD simulations observed that thermal gradients could in practice be used to desalinate seawater. 17,20

Conclusions We investigated thermo-osmotic flows in CNT membranes. To that aim, we proposed an analytical model, which describes quantitatively the velocities measured with MD simulations. This model complement standard descriptions, relating the thermo-osmotic velocity to the excess of enthalpy density, by considering the effect of the hydrodynamic boundary condition at the wall and the pressure drop at the entrances. Despite viscous entrance effects and a thermal short-circuit mechanism, total flow velocities up to c.a. 0.5 m/s are observed in several of the nanotubes investigated, with a 60 K temperature difference and 323 K average temperature in the tubes. Using the model, we show that very large velocities can still be expected for experimentally relevant parameters, so that efficient pumping systems could be made based on these membranes. We also observed large variations of the flow velocity in the smallest tubes due to overlapping and interference of the interaction layers where the enthalpy density is affected by the wall. In particular, the direction of the flow changed for a special tube radius of 5.5 ˚ A.

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Interestingly, the model provides a criterion on the maximum pressure difference that can be overcome by the thermo-osmotic flow, based on the enthalpy excess in the tube and on its hydrodynamic properties. Using this criterion, our simulations identify a number of candidates with great potential for seawater desalinization. Meanwhile, in the limit that the slip length greatly exceeds the nanotube radius, increasing the radius will decrease the pressure difference that can be overcome, but the flow velocity will remain constant. In contrast, increasing the CNT length will decrease the flow velocity, but the pressure drop that can be overcome will not be affected. To predict the performance of such nanoscale heat harvesting systems, the consideration of hydrodynamic details is essential. Our model could then help interpret molecular dynamics simulations 17,20 and guide the search for other high-performance membranes, 58–61 either also based on carbon 62 (e.g. nanoporous graphene or graphene oxide 63 ) or on different materials 64 (e.g. polymers, zeolites 20 or new 2D materials 7 ). With that regard, our model based on macroscopic hydrodynamic equations could fail to describe subnanometric pores, 4 but it could be refined using extended hydrodynamic equations, 65 or based on direct characterization of the hydrodynamic properties of the pores via MD. Finally, the model could also be improved by considering the effect of radial temperature variations in the pore, due to thermal short-circuit by the solid and viscous heating.

Acknowledgement The authors thank A.-L. Biance, O. Bonhomme, C. Ybert and C. Cottin-Bizonne for fruitful discussions, and thank in particular C. Ybert for a critical reading of the manuscript. Correspondence with R. Qiao is also acknowledged. This work is supported by the ANR, project ANR-16-CE06-0004-01 NECtAR. LJ is supported by the Institut Universitaire de France.

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Supporting Information Available Calculation of the thermo-osmotic velocity for an arbitrary hydrodynamic boundary condition; limit of a thin interaction layer; simulation details. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Understanding Fast and Robust Thermo-Osmotic Flows through

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