Gibbs Adsorption Impact on a Nanodroplet Shape: Modification of

Article Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Gibbs Adsorption Impact on a Nanodroplet Shape: Modification of Young−Laplace Equation Mykola Isaiev,*,†,‡ Sergii Burian,‡ Leonid Bulavin,‡ William Chaze,† Michel Gradeck,† Guillaume Castanet,† Samy Merabia,§ Pawel Keblinski,∥ and Konstantinos Termentzidis*,†,⊥ †

LEMTA, CNRS-UMR7563, Université de Lorraine, Vandoeuvre les Nancy F-54500, France Faculty of Physics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Str., Kyiv, Ukraine 01601 § Université de Lyon 1, ILM, CNRS-UMR5306, 69621 Villeurbanne, France ∥ Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, United States ⊥ Univ Lyon, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, CETHIL UMR5008, F-69621 Villeurbanne, France ‡

ABSTRACT: We present an efficient technique for the evaluation of the Gibbs adsorption of a liquid on a solid substrate. The behavior of a water nanodroplet on a silicon surface is simulated with molecular dynamics. An external field with varying strength is applied on the system to tune the solid−liquid interfacial contact area. A linear dependence of droplet’s volume as a function of the contact area is observed. We introduce a modified Young−Laplace equation to explain the influence of the Gibbs adsorption on the nanodroplet volume contraction. Fitting of the molecular dynamics results with the analytical approach allows us to evaluate the number of atoms per unit area adsorbed on the substrate, which quantifies the Gibbs adsorption. Thus, a threshold of a droplet size is obtained, for which the impact of the adsorption is crucial. For instance, a water droplet with 5 nm radius has 3% of its molecules adsorbed on silicon substrate, while for droplets less than 1 nm this amount is more than 10%. The presented results could be beneficial for the evaluation of the adsorption impact on the physical−chemical properties of nanohybrid systems with large surface-to-volume ration.

1. INTRODUCTION The interfaces between two or more different phases dominate a variety of physical and chemical phenomena, especially at the nanometer scale, as the surface-to-volume ratio becomes very high. Interactions between different phases are crucial for the wettability, solubility, chemical reactions etc. The prediction of the interaction of different phases is a key challenge for a wide range of applications in chemical industry, catalysis, material science, microfluidics, pharmacology, medicine, etc. As an example, we can mention, the wetting properties of a liquid droplet on a solid substrate. For the case of nanoscale droplets additionally to the surface tension, the line tension should be considered as it has an important impact on the wetting angle. The free energy excesses between all possible pairs of different phases per unit length should be considered to achieve stability of deformable surfaces. The generalized Young equation taking into account line tension can be presented as follows:1−4 cos(θw) =

γvs − γls γvl

−

τ γvlr

the line tension, and r is the contact radius of the three-phase line. Barisic and Beskok5 have simulated spherical droplets with different volumes located on silicon substrate using molecular dynamics. They measured a wetting angle as a function of the curvature of three phases contact line (1/r). With the use of a linear dependence (Equation 1), they estimated the line tension and they defined several parameters of the interaction between water molecules and silicon atoms with the use of the experimental wetting angles of macrosize droplet as input. Later on, it was shown6 that the simulation of cylindrical droplets is more efficient than this of spherical droplets for the evaluation of the interaction parameters between a droplet and a substrate, due to the zero curvature of the three-phases contact line7,8 in the case of cylindrical droplets. Leroy et al.9 have shown recently that the evaluation of the force-fields parametrization using experimental wetting angles of droplet has limitations. They have stated that the dependence of water surface tension on the chosen water model should be taken into

(1) Received: December 15, 2017 Revised: February 9, 2018 Published: March 5, 2018

where θw is the wetting angle, γvs, γls, and γvl are the vapor/solid, liquid/solid, and vapor/liquid surface tensions respectively, τ is © XXXX American Chemical Society

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DOI: 10.1021/acs.jpcb.7b12358 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B account. Thus, additional markers for reliable description of water−solid interfacial properties should be found, such as “work of adhesion”.9 Numerous MD studies10−13 of solid− liquid interfaces refer to the density fluctuations of fluids near the solid free surfaces. These fluctuations appear as a result of molecules or atoms absorbed in or adsorbed on the solid substrate. The density of adsorbents strongly depends on how these atoms or molecules interact with the substrate. The density variation can even lead to experimentally perceptible contraction of the volume of the liquid containing nanoparticles.14,15 Moreover, the altered structure of confined water near the wall can lead to a drastic change of its physical properties.16−19 From all the above-mentioned works a question arises about the scale limitations to which droplet size the macroscopic approaches describes accurately the statistic of nanodroplet. Is there a threshold after which these approaches do not describe the physical phenomena or could these macroscopic approaches be corrected to achieve convergence of dimensions? In the current paper, the influence of the absorbed layer on the droplet shape and droplet volume is analyzed. Silicon first of all is a reference material and it is widely used in different microelectro-mechanical devices, while the absorbed humidity can affect functional regimes of such devices. Additionally, it should be noted that the surface of crystalline silicon is easy to treat and chemically functionalize. Therefore, the contraction of a nanoscale cylindrical water droplet on a silicon substrate with means of MD is studied. To be able to tune the contact area, an homogeneous external field with varying intensities is applied on the water droplet. We then correlate the shape of a nanoscale droplet modeled and simulated by MD with the one obtained by a macroscopic method of hydrostatic similarity of droplet shape in gravity field. The deviations between the results of the MD simulations and the analytical approach are interpreted. Furthermore, we propose a new model accounting the contraction of the droplet volume due to adsorption phenomena. The proposed model can be also relevant to experimentally observed volume contraction of nanocolloidal solutions14,15 or nanoporous materials filled with liquids.20,21 These systems are accessible to experimental measurements and they have high surface/volume ratio.

Figure 1. Initial (a, b) and after thermostatting (c, d) configurations of a cylindrical water droplet without substrate (a, c) and on a silicon substrate (b and d).

The interactions between water molecules are described by the extended simple point charge (SPC/E) water model,23 with parameters taken from Orsi.24 The SPC/E water model was chosen since it well represents the bulk density of the water25 and is reasonable concerning the CPU time. The 12−6 Lennard-Jones interatomic potential was used for the interactions between oxygen and silicon atoms, with εSi−O = 15.75 meV and σSi−O = 2.6305 Å6. In our study, the interactions between hydrogen and silicon atoms are neglected. The Stilling−Weber potential26 is used for the interactions between silicon atoms. Additionally, the harmonic force was applied to silicon atoms of the lower four atomic layers (z < 2a0) to tether them to their initial positions and thus prevent moving the silicon slab in the Z direction during the simulations. The procedure described below was used to achieve a stable equilibrium droplet. First, the initial velocities of water molecules and silicon atoms were chosen to achieve temperature equal to 1 K. Then, the system was slowly heated in the following steps to achieve a temperature of 300 K: (i) from 1 to 10 K during 5 ps, (ii) from 10 to 50 K during 5 ps, and (iii) from 50 to 300 K during 100 fs. After this, all resulting configurations were equilibrated for 2 ns with the Nosé− Hoover thermostat in the canonical ensemble at 300 K. The time step was set equal to 1 fs. During the equilibration process, we checked the temperature and pressure of the system and the temperatures of silicon atoms and water molecules separately. The resulting droplets are presented in Figure 1c,d for the free droplet and the droplet on the substrate. The stability of the droplet was checked for two cases with a longer simulation time of 8 ns.

2. METHODS All MD simulations presented here are performed using LAMMPS code (Large-scale Atomic/Molecular Massively Parallel Simulator).22 We have considered cylindrical droplet on a smooth silicon substrate with and without an external force. Additionally, cylindrical droplet without substrate was simulated as reference. We use cylindrical droplets to avoid the influence of line tension on the contact angle, as explained in the introduction. In both cases the dimensions of the simulation box are 108.6 × 217.2 × 200 Å in x, y and zdirection, respectively. The initial configurations of the atoms of the droplet without/with substrate are presented in Figure 1a and 1b, respectively. Initially, a rectangular crystalline water droplet with size 108.6 × 43.4 × 43.4 Å was built. The total number of water molecules in the droplet is equal to 6860. The distance between oxygen atoms of water molecules was set to 3.1 Å, and this corresponds approximately to the density of the bulk water under normal conditions. The surface of the crystalline silicon substrate has (1, 0, 0) orientation of the facet, with lattice constant equal to a0 = 5.43 Å. The silicon slab has dimensions 108.6 × 217.2 × 27.15 Å (40 × 80 × 10 atoms).

3. RESULTS 3.1. Droplet Shape without an External Field. The water density profiles were collected every 0.1 ps and then averaged for 0.5 ns after equilibration. To obtain the density profile, the three-dimensional space was divided into squares (1 × 1 Å2 each, in the YZ plane). The resulting density profile of the free droplet and the dependence of the density on the distance from the droplet center are presented in Figure 2a,b, respectively. B


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Figure 2. (a) Averaged density map of a free droplet. (b) Droplet density profile as a function of the distance from the droplet center. (c) Averaged density map of a droplet on a silicon substrate. (d) Averaged density profile of a droplet in the plane perpendicular to the y axis and crossing the droplet’s center of mass as a function of z.

Figure 3. Molecular dynamics snapshots (upper insets) and density profiles of the droplet density map in an external field with intensities of 19 (a) and 97 pN/Å (b) and corresponding density profiles in the middle of the water droplet for the same external field intensities (c and d).

C


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Figure 4. (a) Dependence of the contact area on the external field intensity. (b) Dependence of the droplet volume as a function of the contact area. The red lines are fitting curves with ξ = 0.875 Å.

The density of bulk water inside the droplet was found to be equal to 997.6 kg/m3 while the density of the equilibrium vapor was 0.01474 kg/m3 and the pressure was equal to 0.05 atm. The densities calculated with molecular dynamics are in a good agreement with both previous work using the same water model27 and with experimental data.28 The liquid−vapor interface of the droplet is relatively sharp (Figure 2a,b), and it has a thickness of approximately 10 Å. We considered the liquid/vapor interface to be a constant-density surface equal to half of the bulk water density. In Figure 2a,b, the liquid/vapor interfaces are presented as bold black solid lines. The droplet radius is approximately equal to 25.1 Å, and its volume per unit length is v = 1977.6 Å2 (Figure 2a,b). Hereafter, by terms “volume” and “contact area” we mean the volume per unit length and the contact area per unit length. Additionally, in Figure 2c, the density map of the droplet on the silicon substrate in the absence of external forces is depicted. The wetting angle is evaluated to be equal to (86.6 ± 0.5)o, which is in good agreement with previous studies.6 In Figure 2d, the dependence of the averaged water density in the plane perpendicular to the Y axis and crossing the droplet’s center of mass as a function of z is presented. As was mentioned above, a layering effect appears close to the substrate. The density of this layer can be larger than the bulk water density under normal conditions by more than 30%. The thickness of this layer is calculated to be approximately 4 ± 1 Å. The presence of this layer leads to droplet volume contraction, as the volume in this case becomes equal to 1803.1 Å2. (After the removal of the substrate, it was found to be equal to 1977.6 Å2.) 3.2. Impact of the External Force. In order to study the interaction between a liquid and a solid in an external homogeneous field, we added the homogeneous force to which all water atoms are subjected, which is an additional force in the direction toward the substrate. This additional field was added during the thermalization process described above. In Figure 3a,b MD averaged density profiles of the droplets under the influence of the external field with intensities of 19 and 97 pN/Å are depicted. Corresponding density profiles in the middle of the water droplet for the same external field intensities are shown in Figure 3c,d. These values correspond to radii of curvature that are 2 and 64 times greater than for the case of the absence of force. We are aware that these fields are relative strengths but are necessary for our study for two reasons. First, as explained in the Introduction, our aim is to

reproduce the macroscopic behavior of droplets by means of molecular dynamics. This might be useful for several studies involving the contact between two or three phases and their dynamic interactions. The second reason is related to AFM or SThM measurements. The tip of an AFM can exert huge forces on a water meniscus formed around a tip or a solid surface. These forces could be of the order of F = 100 pN. In the upper insets of Figure 3a,b, snapshots of the droplet in the external field are presented. It is important to note here that the thickness of the adsorbed layer and its density are approximately the same for all cases and equal to those without an external field. What is changed is the contact area between the droplet and the substrate. In Figure 4a, the dependence of the contact area as a function of the external field magnitude is plotted. The contact area increases by a factor of 2 upon adding an external field with a magnitude of 90 pN/A. Because of the increase in the contact area, the fraction of water molecules in contact with the substrate increases. Consequently, the number of water molecules interacting with silicon atoms is also increased, leading to the density increase of the adsorbed water molecules and thus the decrease in the total volume of the droplet. The dependence of the droplet volume on the contact area is presented in Figure 4b. The dependence is found to be linear, which proves that the change in the water volume stems from the increase in the number of water molecules adsorbed on the silicon surface. The fitted equation for the dependence can be presented as follows

v ̃ = v − ξa

(2)

where ṽ is the volume of the contracted droplet, a is the contact area, and ξ is the rate of change of the droplet volume with increasing contact area (Figure 4b). We fitted the results of the molecular dynamic simulations with eq 2 and found that the best fitting value is ξ = 0.875 Å. From the estimated ξ, the surface excess or adsorption per unit area Γ29 can be calculated (in mol/m2) as follows Γ=

ξρb M H 2O

(3)

where ξ should be expressed in m. Then, we estimate that Γ = 4.8 μmol/m2. The evaluated value of adsorption is in excellent agreement with the one estimated from experimental measurements with the use of gravimetric and vaporization systems30 (4 μmol/m2). D


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Figure 5. Pressure maps of the droplet in the absence of external forces (a) and presence of external forces of magnitude equal to 19 (b) and 97 (c) pN/Å.

across a curved interface caused by the surface tension can be presented as follows

At this point, it is important to check possible effects due to capillary forces. If capillary forces were the origin of the droplet volume contraction observed in our case, then one could have observed a gradient of water pressure in the z direction. From Figure 5a−c one can conclude that the capillary forces are important to the interface area between liquid and vapor because there is a variation of the stress density at the top of the droplet. The variations in the stress density close to the interface are due to the interaction of the water molecules with the substrate (strong forces). We do not observe any stress gradient in the bulk (the middle of the droplet).

⎛1 1 ⎞ γ⎜ + ⎟ = ΔP R2 ⎠ ⎝ R1

(4)

where γ is the surface tension and R1 and R2 are the principal radii of the surface curvature. For the considered case of the cylindrical droplet, one of the principal curvature radii may be taken to be equal to infinity (R2 → ∞). In the framework of the assumption of the incompressible isotropic fluid in an external homogeneous gravity-like field, the equation of the force equilibrium can be presented as follows33

4. DISCUSSION In the previous section, the simulation of a nanoscale droplet in an external field with MD has been discussed. In this section, an analytical approach of the droplet volume contraction due to the enhancement of the contact area will be presented. As mentioned previously, we assume that the droplet volume decreases due to the influence of the potential field felt by the atoms of the substrate. The direct approach requires a calculation of the field created by the atoms of the substrate. Since this field is strongly heterogeneous, the calculation of the droplet shape can be very cumbersome from a practical point of view. Therefore, we consider the influence of this field phenomenologically for the study of water molecule adsorption on a silicon surface. For the quantitative description of the droplet shape in an external field on the macroscale, the approach based on the Young−Laplace equation is usually used.31,32 The Young− Laplace equation describes the mechanical equilibrium condition for two homogeneous fluids separated by an interface that is assumed to be a surface of zero thickness. In the case of the absence of an external field, the pressures difference ΔP

ΔP = ΔP0 +

ΔρfLx (z − z 0) m H2O NH2O

(5)

where ΔP0 is the pressure difference in the reference point z = z0, Δρ is the difference in density between the two phases, f is the external field intensity, Lx is the box dimension in the X direction, mH2O is the mass of a water molecule, and NH2O is the number of water molecules. In eq 5, the direction of the z axis is chosen to be same as it is presented in Figure 6 (invert in the direction of the z axis in the MD simulation). In the case when the origin of the coordinate system and the reference point are located at the droplet apex, eq 5 can be presented as follows 1 1 z = + 2 R1 r κ (6) where r is the radius of curvature at the droplet apex and κ = γm H2O NH2O/(ΔρfLx) is the capillary length associated with external force f. E


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Since the physical basis of eq 4 relies on the assumption of the incompressibility of the liquid, the volume of a droplet under different mechanical fields is the same. Although the volume of the macrosize droplet is conserved in the presence of a gravity field, the volume of the nanoscale droplet changes due to the presence of the adsorbed layer. To take this into account, we decomposed the droplet into two parts: the adsorption layer and the droplet with a spherical shape. In this case, the mass of the droplet md per unit of length can be determined by

Figure 6. Sketch of the considered droplet geometry.

md = ρb (v − va) + ρa va

4.1. Reflectional Symmetric Drop Shape Analysis (RDSA). For the description of a droplet shape in an external field, axisymmetric drop shape analysis (ADSA)34 proposed by Rotenberg et al. is most commonly used. Regarding the cylindrical droplet, the modification of ADSA can be used by considering the reflectional symmetry of the droplet shape for the analysis (RDSA). In this case, the curvature can be related to the arc length, s, and the angle of inclination of the interface to the horizontal angle, φ, by

dφ 1 = R1 ds

where ρb and ρa are the densities of the bulk water and adsorbed layer, respectively, and va is the volume per unit length of the adsorbed layer. To allow fair comparisons with the MD simulations, the mass of the droplet was kept the same. For that, parameter r in eq 8 was varied iteratively. The value of r is increased if md is larger than the requested liquid mass or decreased in the opposite case. The results of the numerical simulations of the droplet contact radius and radius at the droplet top considering the adsorbed layer are presented in Figure 7a,b with green circles. The values of the adsorbed layer thickness (Ha = 4 Å) and its density (1220 kg/m3) were extracted from the results of MD simulations. As one can see in Figure 7, there is excellent agreement between the results of the numerical integration of eqs 7 and 8 and the results of molecular dynamics simulations. Therefore, a significant impact of the presence of the adsorbed layer on the droplet shape contraction can be stated. The Gibbs adsorption in this case can be estimated as follows

(7)

Therefore, eq 4 can be rewritten as follows dφ 1 z = + 2 ds r κ

(8)

Equation 6 is supplemented by the geometrical relations dy dz dv = cos(φ); = sin(φ); = 2y sin(φ) ds ds ds

(10)

(9)

where v is the droplet volume per unit length. The system formed by eqs 7, 8, and 9 is solved numerically using the Runge−Kutta method taken as initial conditions y(0) = z(0) = φ(0) = v(0) = 0. Integration is stopped when φ reaches the value of the contact angle φc = 86.6° that is an a priori known parameter. The dependences of the droplet height and its contact radius as a function of the gravity intensities evaluated numerically (empty circles) and with MD simulation (squares) are presented in Figure 7a,b, respectively. As one can see, there is a deviation between the results of the numerical approach and the results of MD simulations. This discrepancy might arise from the presence of the adsorbed water layer.

Γ=

Ha(ρa − ρb ) MH2O

= 4.9

μmol m2

(11)

where MH2O is the molar mass of water. The Γ values predicted with eqs 3 and 11 are in excellent agreement. 4.2. Bashforth−Adams Equation with an Adsorption Layer (BAE). Another approach to droplet shape analysis is based on the Bashforth−Adams equation (BAE) solution. This equation can be obtained from eq 6 after replacement of the curvature with the following expression:

Figure 7. Dependences of the droplet height (a) and contact radius (b) as a function of the intensity of the external field. F


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The Journal of Physical Chemistry B 1 = R1 ⎛ ⎜1 + ⎝

d2z dy 2 2 ⎞3/2

( ) ⎟⎠ dz dy

framework of the modified Young−Laplace assumption (1) to minimize the influence of line tension. The size dependence of surface tension is neglected here for the considered droplets’ size.35,36 We prove that the macroscale approach based on the Young−Laplace equation has limitations in the description of the droplet shape. In particular, in addition to the external field, one should consider the irreversible adsorption of liquid atoms on the solid substrate. A thin adsorbed layer with higher density, compared to the liquid bulk density under the same conditions of temperature and pressure, appears. The presence of this layer leads to droplet contraction, which can be more or less important depending on the volume to surface ratio. To quantify the latter, the dependence of the percentage of the number of adsorbed particles to the total number of particles as a function of the droplet radius is depicted in Figure 8. For

(12)

The dimensionless BAE in this case can be written as follows ∂ 2z ̃ ∂y 2

⎛ ⎜1 + ⎝

2 ⎞3/2

= 1 + Bo·z

( ) ⎟⎠ ∂z ̃ ∂y

(13)

where z̃ = z/r is the reduced z coordinate and Bo = Δρf Lxr2/ (γmH2ONH2O) = r2/κ2 is the Bond number, which quantifies the droplet shape in a mechanical field. In our case, the parameters which can be correlated with the results of MD simulation are the volume, the height, and the contact radius of a droplet. Therefore, it is better to rewrite eq 12 in the coordinate system where the origin is located on the contact radius (point (0, H)) and axis z in the opposite direction than in the previous case (Figure 6). Equation 12 can be recast as d2z dy 2

⎛ ⎜1 + ⎝

2 ⎞3/2

( ) ⎟⎠ dz dy

=

ΔρfLx ⎛ v ⎞ ⎜ − z(y)R2⎟ + sin θ ⎠ γm H2ONH2O ⎝ 2 (14)

Additionally, the volume-conserving condition gives the following dependence sin θ −

ΔρfLx ⎛ R v ⎞⎟ ⎜H · R − = r 2⎠ γm H2ONH2O ⎝

(15) Figure 8. Number of adsorbed particles divided by the total number of water molecules as a function of the droplet radius.

Figure 7 presents a comparison of the droplet contact radius and radius at the droplet top evaluated with the MD simulations (points) and with eqs 14 and 15. As one can see in Figure 7, there are some differences between the simulation results and the analytical approach. In particular, the contact area is systematically overestimated with the BAE model and the discrepancy increases with increases in the intensity of the external field. These differences can be overcome with the modification of the volume used in eqs 2, 14, and 15. Instead of the total volume, one should use the volume of the water after substrating the volume of the water molecules adsorbed at the surface of the substrate to obtain a correct description of the phenomenon. In such a case, both BAE and RDSA models yield good agreement with the atomistic simulations, as we will see now. The dependences of the droplet height and contact radius as a function of external force intensities without (dashed line) and with (solid line) the considered adsorption are presented in Figure 7a,b, respectively. Clearly, neglecting the adsorbed layer typically overestimates the contact radius and radius of curvature by 10%. Taking into account the adsorbed layer yields very good agreement with the results of MD simulations.

droplets with radii larger than 1000 Å, this ratio is equal to 0.1%, and the adsorption does not significantly influence the droplet size (Figure 8). However, for droplets having radii of between 10 and 100 Å, this number takes values of between 10 and 1%. This adsorption leads to the failure of the Young−Laplace equation for droplet sizes of less than 100 Å. To overcome this issue, we propose a new phenomenological model which takes into account the adsorption of liquid molecules on a solid substrate. Our model is compared to and verified by molecular dynamics simulations. Additionally, to tune the contact area between liquid and solid for droplets with a constant number of particles, a homogeneous external field can be applied. As a result, the specific adsorption of the liquid atoms/molecules on a solid substrate can be efficiently evaluated. Thus, one can predict the volume contraction/expansion of the solution of a liquid containing dispersed nanoparticles with atomistic simulations. On the other hand, the presented approach can be easily adopted for the inverse problem of tuning the potential interaction between solids and liquids. For example, the parametrization of an interatomic potential can be obtained by fitting the results of macroscale experiments with colloidal solutions’ volume contraction/expansion. In this case, eq 11 can be considered to be a dependence of the mixture volume on the specific surface of the nanoparticles.

5. CONCLUSIONS In this study, we have considered a nanoscale droplet in an external homogeneous field as a tool to tune the contact area in order to study the adsorption characteristics of water molecules on solid surfaces. Cylindrical droplets are simulated in the G


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Finally, with the use of the presented approach one can study different interfacial phenomena involving fluids and solids as the line tension for the case of spherical droplets.16

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Mykola Isaiev: 0000-0002-0793-9825 Sergii Burian: 0000-0001-5304-3299 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Institute CARNOT ICEEL for the project “CAMTRASTE” (Controle et AMelioration des TRAnsferts par Structuration des surfaces d’Echange). Calculations were performed on the ERMIONE cluster (IJL/ LEMTA). P.K. was supported by the Office of Naval Research Thermal Science Program, Award No. N00014-17-1-2767. We also thank Alain McGaughey and Thierry Biben for fruitful discussions.

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REFERENCES

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