William G Buttlar

Armelle Chabot

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

This chapter presents a summary of the key findings of this state-of-the-art report on the mechanisms of cracking and debonding in asphalt and composite pavements. Some of these keys have ben presented during the 8 th Rilem International Conference on Mechanisms of cracking and Debonding in Pavements in Nantes (MCD2016) [START_REF]8th RILEM International Conference on Mechanisms of Cracking and Debonding in Pavements[END_REF] and in the three supplementary issues that have been published thereafter [START_REF] Chabot | Mechanisms of Cracking and Debonding in Pavements: debonding mechanisms in various interfaces between layers[END_REF][3][START_REF] Buttlar | Edito of the special MCD[END_REF]. It also contains the key recommendations developed by the TG2 group [START_REF] Petit | Recommendation of RILEM TC 241-MCD on Interface Debonding Behavior in Pavements[END_REF], which produced a RILEM recommendation document as a culminating activity in the final year of the RILEM Technical Committee TC241-MCD. Remaining challenges in the area of asphalt and composite pavement cracking and debonding and recommendations for further study are then summarized.

With the exception of permanent deformation (rutting), which appears to be well under control in highly developed countries, the primary challenges in the design and preservation of modern asphalt pavements relate to the prevention and control of various cracking modes. This includes thermal cracking, reflective cracking, block cracking, and fatigue cracking, which can emanate from the bottom, top, or interior of the bound asphalt layers in a flexible pavement system. Interface (partial or total) debonding and interface sliding behaviour are also important topics that need to be analysed in pavement cracking studies, as this form of

U n e d i t e d v e r s i o n

deterioration can lead to subsequent tearing (slippage cracking) and/or accelerated fatigue cracking of the asphalt pavement surface.

Rutting and cracking are often thought to represent opposite sides of the spectrum in asphalt mixture design. A simple rule-of-thumb in asphalt mix design suggests that lower asphalt content mixtures produce less rutting but more cracking in asphalt mixtures, and viceversa. Some pavement engineers describe this dichotomy in terms of a swinging pendulum, where the pendulum has currently swung in the direction of less rutting and more cracking. This may be a result of the focus in modern mix design methods to prevent rutting through more aggressive asphalt compactive effort in design (Superpave gyratory compactor usage, along with relatively high design gyration targets). In addition, some pavement engineers point to higher recycling rates and changes in binder supplies. For instance, modern refining techniques are squeezing more light ends out of asphalt, rendering it less ductile and/or leading to the heterogenization of the binder system through the blending of soft and hard binder components, the use of acids, rejuvenators, recycled engine oil bottoms, and/or other modifiers. These changes may alter the binder's ability to resist cracking and fatigue, to heal microdamage, and to bond with aggregates in the mixture or with other pavement layers. Moreover, the aforementioned rule-of-thumb misses a number of important considerations necessary for a more comprehensive flexible and composite pavement engineering approach, including: consideration of the role of the entire pavement structure (all layers and interfaces) in the development of cracking and debonding in the asphalt layers, the possibility for the development of highly crack and rut resistant surface materials (such as stone-mastic asphalt), and the intentional use of preventive maintenance as a strategy to achieve economical rut and crack resistant pavement system designs. To this end, the development of advanced tests and comprehensive models and field studies continue to be a pressing need in the pavement engineering community. These contemporary pavement cracking issues were the basis for the activities conducted under RILEM TC-241MCD, and have been addressed, at least in part, in this state-of-the-art report.

TC-241 MCD STAR DOCUMENT SUMMARY

The organization of a large group of researchers into teams to work on modern pavement cracking challenges can take one of several logical forms. In the precursor to RILEM TC241-MCD, the RILEM Cracking in Pavements (CAP) technical committee divided researchers into three teams, focused on field studies (CAP TG-01), testing (CAP TG-02), and modelling (CAP TG-03). RILEM TC241-MCD sought to re-shuffle research teams to invoke new synergies, by reorganizing technical committees into the categories of cracking in bulk materials (TG-01), cracking at pavement interfaces (TG-02), and advanced measurement techniques for cracking studies (TG-03), whereby each TG could consider contemporary tests, models, and field studies to reach their stated objectives. Common sets of field and lab data, and generalized modelling schemes, and researchers working across the various committees, served to connect the three technical committees in an integrated fashion.

Summary of STAR Chapter 1

As developed in TC-CAP, and summarized in this STAR (Chapter 1, section X), scientific progress in complex 'mechanistic-empirical' fields such as pavement engineering necessarily involve a cyclic progression of 'QVCV,' or Qualification, Verification, Calibration, and Validation. Chapter 1 of this STAR document provided background information on field

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observations and accelerated pavement testing (APT) campaigns, which were used by researchers in the test and model qualification process. In other words, the observed cracking patterns and their hypothesized driving mechanisms on modern pavements and APT facilities across the world served to define the requisite physics to be included in tests and models, and to be studied using advanced measurement techniques by TC241-MCD researchers in the three aforementioned technical committees.

Among the information presented in Chapter 1, the following key field and APT observations provided motivation for the development of better tests and models to understand and describe cracking mechanisms in modern pavement structures:

 Examination of more than a dozen French motorways over the past decade has highlighted the importance of layer debonding in the development of fatigue and reflective cracking within asphalt paving layers. Layer debonding was evident in pavement cores taken on these facilities and could also be detected through in-situ deflection testing, as indicated by higher pavement deflections and wider deflection basins as compared to pavements with well-bonded layers. Techniques for detecting debonding using ground penetrating radar have also shown promise.  In the US, recent studies have recognized the need to better understand and predict thermal, block, and reflective cracking mechanisms via discrete fracture testing and modelling approaches, leading to fracture-mechanics inspired tests and models in the asphalt pavement research and practice communities. For instance, in a recent Pooled Fund study sponsored by ten states residing in colder climates in the US and the Federal Highway Administration, the disk-shaped compact tension test (DC(T)), which is now specified in ASTM D7313, and a cohesive zone fracture model were implemented in a thermal cracking specification, involving fracture energy requirements and an optional finite element based thermal cracking simulation model, called ILLI-TC.  Field studies involving ultra-thin bonded wearing courses and stress absorbing membrane interlayers also highlighted the need to better understand the mechanicsrelated phenomena associated with pavement layer bonding and bond degradation, as conflicting viewpoints regarding proper tack coat (adhesive) materials and application rates and the relative contribution of pavement grinding/milling seem to prevail in the current literature.  In accelerated pavement testing (APT) studies conducted in France, fatigue prediction models appear to provide reasonable estimates for fatigue life of asphalt pavements, but greatly under-predict the fatigue life of thicker asphalt pavements. This highlighted the need to develop better models for damage and crack initiation and propagation and to introduce a very thin bitumen interface layer or to include discontinuities such as vertical discontinuities or partial debonding between layers into pavement models. The findings also suggested limitations which need to be addressed in developing and interpreting future APT studies, such as differences between APT and actual pavement performance in terms of asphalt aging, healing, and climate. The studies also highlighted the need for improvements in sensors and other methods to collect data related to pavement cracking mechanisms in the field and on APT experiments.

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 In reflective cracking APT studies in France carried out using the FABAC device, clear evidence of the role of interface debonding between Portland cement concrete (PCC) and asphalt layers was observed and modelled. A flow-chart type schema describing numerous manifestations of reflective crack development and propagation was developed as part of the study.  In another APT cracking study carried out at IFSTTAR, geotextile grids were shown to significantly retard the rate of cracking in asphalt paving surfaces, and differences between layer bonding and subsequent cracking rates in asphalt overlays were reported when comparing the various grids investigated.  Similarly, in an APT reflective cracking study conducted in the US, the role of debonding in the specific type of reflective cracking observed, and its rate of development, was studied. In additional, the phenomena of 'crack jumping' over single-layered stress absorbing membrane interlayer (SAMI) systems was described, and dual-layer SAMI systems were shown to be more effective in mitigating reflective cracking in asphalt overlays placed over PCC.  An overarching strategy for the continued evolution of cracking tests and models was presented, termed the QVCV approach. Building on the V&V technique used in the experimental mechanics community, a 'Qualification, Verification, Calibration, and Validation,' or QCVC approach was introduced for use in the pavement mechanics community. In addition to model verification (checking vs. closed-form solutions) and validation (checking model efficacy vs. real-world cracking phenomena and/or cracking data outside of the calibration data set), the importance of model qualification (motivating/informing/refining models with field observations) and proper distinction between model calibration and validation is highlighted in the QVCV approach. Following the motivation and test/model qualification information presented in Chapter 1, Chapters 2-4 of this state-of-the-art report described the recent and ongoing work of the three RILEM TC241-MCD technical groups. The major findings presented in these chapters are now summarized.

Summary of STAR Chapter II: Cracking in Asphalt Material

Chapter II contains a number of contributions from RILEM TC241-MCD, TG-01 members pertaining to tests and models developed to better understanding and characterize cracking within the asphalt 'bulk' material. In light of these contributions, continued research needs were identified. These include:

 A unified laboratory test and model that can characterize and capture the response of asphalt mixtures under all the modes that affect its properties (temperatures, loading rates, aging effect etc.) has been identified as an important research need by a number of groups. Although the present report demonstrates that the current state-of-the-art in characterization and modelling of asphalt bulk fracture is capable of capturing material response under wide-range of conditions, the present approaches are still incapable of capturing a sufficient combination of variables that affect the fracture behaviour in asphalt paving layers.  The major scientific shortcomings in the field of fracture in bulk asphalt materials identified through this review include:

U Chapter III contains contributions from RILEM TC241-MCD, TG-02 members pertaining to tests and models developed to better understand and characterize interface debonding behaviour in layered flexible and composite pavement systems. In light of these contributions, continuing research needs were identified. Members of the TG-02 group also developed and submitted formal recommendations to RILEM [START_REF] Petit | Recommendation of RILEM TC 241-MCD on Interface Debonding Behavior in Pavements[END_REF]. The group reported that, for further development of interlayer bond tests and models, it is crucial to improve the phenomenological understanding of the mechanisms and influencing parameters associated with interlayer bonding and bond deterioration. The recommendations of the TG-02 group included:

 Moisture /water effect infiltration through cracks and other pavement defects have a significant influence on the quality of interlayer bond. Moisture was found to reduce bond strength drastically. Its effect should be investigated in more detail and moisture should be integrated in the standard methods and considered in pavement design tools.  Improved phenomenological understanding of the mechanisms associated with interlayer debonding in terms of the geometrical surface conditions of the interface are needed. Advanced contact interface constitutive models, finite element modeling to directly consider local interface topology, as well as roughness measurements at the interface should be further developed. Improved, simplified models should then be developed and introduced into standard pavement design tools.  The size of test specimens bears great influence on the outcome of interlayer bond tests. As demonstrated by an interlaboratory test program initiated by RILEM, specimens with larger diameters were found to achieve higher bond values compared to specimens with smaller diameters. Further, it was demonstrated that larger the nominal maximum aggregate size mixtures require larger test specimens, when testing pavements with reinforcing interlayers). In some larger test assemblies, the dead weight contribution of the specimen might have to be taken into account to properly interpret results.  Temperature effects on bond behaviour should be studied in more detail. For instance, on one hand shear bond strength decreases with increasing temperature, while on the other hand, increased temperatures may positively affect self-and traffic-induced healing effects. It was further observed that interlayer bond defects or weaknesses seem to appear more obvious when testing is performed at very low temperatures, such as -20°C.

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 There is a need to conduct cyclic and static round robin tests to compare the outcome of different devices and methods in more detail and to develop harmonized testing procedures and requirements.  There is a need to focus more on comparing laboratory results with in situ measurements and observations, as significant differences exist between laboratory results and field performance have been identified. Such comparisons will likely lead to adjustments and improvements in laboratory bond tests and specification requirements. For instance, the technique of testing actual field specimens and comparison to full-scale accelerated loading experiments have been shown to be effective in improving the understanding of debonding mechanisms and in the improved interpretation of laboratory test data.  There is a need to develop energy-based approaches for modelling interface damage, with the goal of quantifying damage evolution during bond tests. The introduction of rest periods during cyclic testing can be performed to assess the healing mechanism.  The importance of testing a sufficient number of test specimens, which depends on the test method used and the applied testing conditions (i.e., static vs. cyclic), was identified.  There is a need to implement advanced measurement systems in the study of pavement debonding, such as: o NDT (Non-destructive testing) in field experiments; o Advanced sensors for the measurement of in-situ interface behaviour under traffic loading, and' o DIC (Digital image correlation) techniques for advanced interpretation of laboratory interface debonding tests.

Building on this identified scientific lack, other future research topics were identified by TG-02, including:

 Advances in mechanical modelling, such as debonding energy and surface temperature gradient, should be developed.  As observed in the construction process, less-than-ideal moisture and dust/debris conditions existing prior to asphalt layer placement and more exact pavement texture information should be included as parameters in advanced debonding models.  Further investigation on different tack coat types, a wider range of application rates, and different HMA mixtures under different temperature and moisture conditions and different modes of failure including mode I, mode II and mixed mode is recommended.  Data coming from interface bonding tests should also be utilized in numericallybased pavement simulation models, in order to gain additional insight towards the mechanisms of interface debonding and its prevention.  Further research should consider predicting results for interlayer bond problems using Artificial Neural Network (ANN) modelling, utilizing combined data sets from laboratories and field installations worldwide.

Summary of STAR Chapter IV: Advanced Measurement Techniques

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Chapter IV contains a number of contributions from RILEM TC241-MCD, TG-03 members pertaining to advanced measurements techniques for asphalt cracking studies. This included many emerging techniques that are new (or relatively new) to the pavement engineering community, such as: digital image correlation (DIC) [START_REF] Buttlar | Digital image correlation techniques to investigate strain fields and cracking phenomena in asphalt materials[END_REF], optical strain measurement systems, X-ray computed tomography + tension/compression testing systems, deflection devices and analysis of stress wave propagation, radar analysis, imaging of dieletric contrasts through electromagnetic waves, acoustic emissions (AE) based tests, and passive and active seismic NDT testing.

In light of these contributions, continued research needs were identified. These include:  Continued development of advanced, in-situ tests, possibly combining two or more of the new advanced measurement techniques reviewed herein.  Further development of acoustic emissions techniques, which provide considerable new insight towards the microcracking behaviour in modern, heterogeneous paving materials.  Improvements in x-ray CT techniques, particularly in terms of scanning speed, so that more useful 3D cracking process zone development and crack propagation information can be obtained.  Common to virtually all emerging, advanced measurement systems is the need for continued calibration, validation and standardization of the techniques.  The reviewed techniques lend themselves nicely to implementation in intelligent infrastructure and smart city systems, but integration into these systems is an open research topic.

ASSESSMENT OF RESEARCH NEEDS

As summarized in the previous section, significant progress has been made over the past decade in understanding and characterizing the mechanisms of cracking and debonding in asphalt pavement systems. That notwithstanding, significant challenges still remain in the quest for a deeper understanding of these mechanisms, leading to better tests, better specifications, and better asphalt durability. Many of the remaining challenges, which are summarized below, are driven by the complexity of asphalt concrete and composite pavements, the stochastic nature of material, traffic and environment, the scarcity and expense associated with detailed field data, and the challenges associated with measuring and imaging cracking mechanisms in asphalt concrete. Some of the key remaining research needs, as outlined and/or informed by this compilation of recent works and as identified by RILEM TC-241 MCD members include:

-More resolute and less costly approaches for 3-D microstructural imaging of asphalt materials, particularly to capture crack various cracking stages and mechanisms (nucleation, initiation, propagation, fracture process zone behavior, etc.) -More accurate and efficient micromechanical simulations of cracking in asphalt -Advancing mixed-mode crack tests and models -Advancing interface crack measurements and models -Transfer of developed technologies into practical tests and specifications -Better and more accessible field data, which is required for test and model development and validation

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Summary of STAR Chapter III: Interface Debonding Behavior

		o Limited ability of current models to capture both static and cyclic crack growth
		in a seamless, universal manner
		o Current models and tests still lack a complete fundamental understanding of
		the healing processes present in asphalt
		o Most laboratory tests and analysis procedures focus on the use of global
		specimen responses; more research is needed on the incorporation of local
		material responses from laboratory tests in models, such as those being
		measured in new, advanced asphalt cracking characterization techniques
		(Chapter 4)
		o The development of universal criteria from model qualification, verification,
	5.1.3.	calibration and validation are still needed v e r s i o n
		n e d i t e d

-Cracking evaluation of sustainable asphalt materials At the time of this writing, the aforementioned research needs were being discussed and prioritized by RILEM TC 241-MCD members. These recommendations can be used to form a work plan for a new Technical Committee, which can serve to help advance the state-of-theart in understanding, testing, and designing crack-resistant asphalt pavement materials and systems through collaboration across the international pavement engineering community.


Henri Gouin
email: henri.gouin@univ-amu.fr

Augusto Muracchini
email: augusto.muracchini@unibo.it

Tommaso Ruggeri
email: tommaso.ruggeri@unibo.it

Fluid Mixtures in Nanotubes

Keywords: 61.46.Fg, 61.20.Gy, 68.35.Md, 47.61.-k nanotubes, fluid mixtures, nanotube flows, Fluid-mixture/solid interactions

The aim of the paper is the study of fluid mixtures in nanotubes by the methods of continuum mechanics. The model starts from a statistical distribution in mean-field molecular theory and uses a density expansion of Taylor series. We get a continuous expression of the volume free energy with density's spatial-derivatives limited at the second order. The nanotubes can be filled with liquid or vapor according to the chemical characteristics of the walls and of liquid or vapor mixture-bulks. An example of two-fluid mixture constituted of water and ethanol inside carbon nanotubes at 20 • C is considered. When diameters are small enough, nanotubes are filled with liquid-mixture whatever are the liquid or vapor mixture-bulks. The carbon wall influences the ratio of the fluid components in favor of ethanol. The fluid-mixture flows across nanotubes can be much more important than classical ones and if the external bulk is vapor, the flow can be several hundred thousand times larger than Poiseuille flow.

I. INTRODUCTION

The technical development of sciences allows us to observe phenomena at length scales of a very few number of nanometers. The observations reveal new behaviors that are often surprising and essentially different from those usually observed at a microscopic or macroscopic scales [START_REF] Harris | Carbon nanotubes and related structures, New materials for the twenty-First century[END_REF][START_REF] Gouin | A mechanical model for the disjoining pressure[END_REF]. Experiments prove that liquid densities change in very narrow pores [START_REF] Ball | The density profile of a confined fluid[END_REF] and the conventional laws of capillarity are disqualified when they are applied to fluids confined inside porous materials [START_REF] Bear | Dynamics of fluids in porous media[END_REF]. Iijima, the discoverer of carbon nanotubes [START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF], was fascinated by Krätschmer et al's paper [START_REF] Krätschmer | Solid C 60 : a new form of carbon[END_REF] and decided to launch out into a detailed study of nano-materials. Since the late 1900s the literature has become abundant regarding technology and flows inside nanotubes [START_REF] Mattia | Review: static and dynamic behaviour of liquids inside carbon nanotubes[END_REF]. Nevertheless, simple models proposing qualitative behaviors need to be developed. In this paper, our aim is to investigate an example of mixturesolid interaction in statics as well as in dynamical conditions by using the methods of continuum mechanics. These continuous methods are experimentally realistic until nanotube diameter sizes of a very small number of nanometers [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF] To propose an analytic expression of densities for mixturefilms of nanometric thicknesses near a smooth solid wall, we add an interaction energy at the solid wall to a densityfunctional which represents the volume free energy of the mixture [START_REF] Gouin | Boundary conditions for a capillary fluid in contact with a wall[END_REF][START_REF] Gouin | Mixtures of fluids involving entropy gradients and accelerations waves in interfacial layers[END_REF][START_REF] Gouin | Travelling waves near a critical point of a binary fluid mixture[END_REF]. These two energies are obtained by series expansions using London's potentials for fluid-fluid and fluid-solid interactions [START_REF] Gouin | Energy of interaction between solid surface and liquids[END_REF]. The functional is extended from studies by van der Waals, Cahn and Hilliard and many others [START_REF] Van Der Waals | The thermodynamic theory of capillarity under the hypothesis of continuous variation of density, translation by[END_REF][START_REF] Cahn | Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid[END_REF][START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Widom | What do we know that van der Waals did not know?[END_REF].

We consider a nanotube made up of a cylindrical hollow tube whose diameter is of a few number of nanometers. The length of the nanotube is of microscopic size and the cylinder wall is solid [START_REF] Harris | Carbon nanotubes and related structures, New materials for the twenty-First century[END_REF]. The nanotube is immersed in an homogeneous liquid or vapor bulk made up with two fluids which fill the interior of the nanotube. In nanofluidics, the interactions between fluids and solid walls can dominate over the hydrodynamic behaviors. The mixture compressibility near a solid wall is extremely important [START_REF] Ball | The density profile of a confined fluid[END_REF]. In liquid or vapor bulks, we express the chemical potentials of fluid-components with only a first-order development taking account of isothermal sound-velocities of single bulks [START_REF] Gouin | A mechanical model for the disjoining pressure[END_REF]. The equations of equilibrium and motion of fluid mixtures inside the nanotube take the fluid super-deformations into account [START_REF] Gouin | Boundary conditions for a capillary fluid in contact with a wall[END_REF]. In cylindrical representation, two differential equations are obtained and the profiles of the fluid-mixture densities in the cylinder can be deduced.

The results are applied to carbon nanotubes filled with a mixture constituted of water and ethanol and extend those obtained for simple fluids in [START_REF] Gouin | Statics and dynamics of fluids in nanotubes[END_REF][START_REF] Gȃrȃjeu | Scaling Navier-Stokes equation in nanotubes[END_REF]. The nanotube diameter ranges from 1 to 100 nanometers. Due to energetic properties, the case of liquid and vapor separated by an interface inside the nanotube is not possible when the nanotube diameter is small enough. The mixture inside nanotubes is liquid and the ratio between water and ethanol significantly changes from the mixture-bulk ratio. Recently, it was shown, by using non-equilibrium molecular dynamics simulations, that liquid flows through a membrane composed of an array of aligned carbon nanotubes are a lot faster than it would be predicted by conventional fluid-flow theory [START_REF] Majumder | Nanoscale hydrodynamics: Enhanced flow in carbon nanotubes[END_REF]. These high velocities are possible because of a frictionless surface at the nanotube wall [START_REF] Ma | Friction of water slipping in carbon nanotubes[END_REF]. By calculating the variation of viscosity and slip length as a function of the nanotube diameters, the results can be fully explained in the context of continuum fluid mechanics [START_REF] Thomas | Reassessing fast water transport through carbon nanotubes[END_REF]. In our model, we find that a spectacular effect must appear for tiny carbon nanotubes when the mixture bulk outside the nanotube is constituted of vapor: the mixture flows through the nanotube can be multiplied by a factor of several hundred thousand times what is found for Poiseuille's model.

The paper is organized as follows: In Section 2, thanks to the mean-field theory with hard-sphere molecules, we present a continuous form of free energy for inhomogeneous mixtures. In Section 3, the equations of equilibrium and boundary conditions at a solid wall are written. In Section 4, we consider the chemical potentials of fluid components near a phase and in Section 5, we study the special case of ethanol and water mixture in carbon nanotubes. Thanks to Hamaker's constants, we can determine the profiles of densities at equilibrium. When the pressure of vapor bulk is not too important with respect to the liquid bulk pressure, the carbon nanotube is filled by liquid mixture. Numerical computations yield the densities profiles of water and ethanol inside the nanotubes. In Section 6, the results are extended in the motion case and, as it is experimentally verified, with a classical viscosity. A conclusion ends the paper.

II. A SECOND-GRADIENT FLUID-MIXTURE ENERGY

In a single fluid where the density is not uniform, the intermolecular forces exert on a given molecule a resultant force accounting for the effects of surface tension [START_REF] Rocard | Thermodynamique[END_REF]. The force derives from a system of pressures and stresses that differs from the classical isotropic pressure, which assumes uniformity of density. The kinetic theory of the gases makes it possible to obtain this system of tension and allows us to study the equilibrium of the fluid by layers of equal density and produces a good modelization of the superficial tension. It has been known for a long time, that the law of statistical distribution of molecules around one of them is almost inaccessible even for a distribution of uniform density [START_REF] Ornstein | Accidental deviations of density and opalescence at the critical point of a single substance[END_REF]. If we take a regularized law of distribution at a distance of a given molecule, then we ignore density fluctuations, but the effect of fluctuations tends to be eliminated by integration. This is why, it seems to us possible to approach the problem of the field of forces without presupposing the problem solved and without knowing the true distribution of the molecules. An important property is that the probability of presence of a molecule at a distance less than the molecular diameter is zero. The potential of intermolecular forces is a rapidly decreasing function of distance. It is enough to know the law of distribution at the effective distance and thus the fluctuations of the law beyond this distance no longer influence the result. The case corresponds to the reality given that the preponderant forces are of the type 1/r 7 . It is then possible to develop in series the distribution gap of the molecules. An energy density is then expressed as a measure per unit volume and allows a description in mechanics of continuous media. Such an energy model allows us to obtain a good variation of density in liquidvapor interfaces even if they are spherical and of nanometer size [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]. It is even possible to get variation of fluid density in the vicinity of a solid wall. As it is proposed in [START_REF] Gouin | Energy of interaction between solid surface and liquids[END_REF], we can extend the model for inhomogeneous fluid mixtures. Then, it is possible to propose analytic expressions for the interaction potentials in terms of densities of the two species of a mixture and consequently to get a continuum mechanics description of volume energy which is valid at nanometer size of interfaces.

A. Energy per unit volume

In the mean-field theory with hard-sphere molecules and for each constituent of a fluid-mixture, the fluid's molecules are identical. The central forces between molecules of constituent i, i ∈ {1, 2} derive from a potential ϕ i (r) and between molecules of the two constituents from a potential ϕ 3 (r), where r is the distance between the centers of the two molecules [START_REF] Dzyaloshinsky | The general theory of van der Waals forces[END_REF]. In three-dimensional Euclidean medium D, the potential energy W o1 resulting from the combined action of all the molecules on the molecule of constituent 1, located at origin O, is assumed to be additive such that

W o1 = N 1 ϕ 1 (r) + N 2 ϕ 3 (r),
where N 1 denotes all molecules of constituent 1 (except for the molecule located at origin O) and N 2 of constituent 2, respectively; r denotes the distance of molecules to the molecule of constituent 1 located at O. With similar notations, we get

W o2 = N 2 ϕ 2 (r) + N 1 ϕ 3 (r).
The number of molecules of constituent i in volume dv is represented by n i (x, y, z) dv, where dv denotes the volume element in D at point of coordinates x, y, z, and in a continuous representation,

W o1 = D ϕ 1 (r) n 1 dv + D ϕ 3 (r) n 2 dv, with D ϕ 1 (r) n 1 dv = ∞ σ 1 ϕ 1 (r) S (r)
n 1 ds dr,

and D ϕ 3 (r) n 2 dv = ∞ 1 2 (σ l +σ 2 ) ϕ 3 (r) S (r) n 2 ds dr,
where ds is the measure of area, S (r) is the sphere of center O and radius r, and σ i is the molecular diameter of molecules of component i, i ∈ {1, 2}. We assume that n i (x, y, z), i ∈ {1, 2} are analytic functions of coordinates x, y, z, i.e. n i = n i (0, 0, 0)

+ ∞ =1 1 ! x ∂n i ∂x (0, 0, 0) + y ∂n i ∂y (0, 0, 0) + z ∂n i ∂z (0, 0, 0) ( )
.

The method will be specially justified in two cases: (a) ϕ k (r), k ∈ {1, 2, 3}, are constant over a large diameter: then the distribution fluctuations of the following molecules are eliminated, this would be the case for van der Waals forces with a large radius of action. (b) ϕ k (r), k ∈ {1, 2, 3}, are rapidly decreasing functions of distance: It suffices to know the law of distance distribution of molecules at the effective distance, that is, very close to one of them, and then the fluctuations of the law at a great distance, or even at a distance a little greater, no longer influence the result.

In fact, the second case corresponds to reality since the predominant forces which we will have to count with, are of type 1/r 7 . It is then possible to develop in series the difference of n i with respect to its value in O and to limit to the second order. All these comments are developed in [START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Rocard | Thermodynamique[END_REF]. We notice that for any integers p, q, r we have the relations

S (r)
x 2p+1 y q z r ds = 0, and

S x 2 ds = S y 2 ds = S z 2 ds = 4 π r 4 3 .
Then,

W o1 = ∞ σ 1 ϕ 1 (r) 4 π r 2 n 1o + 2π 3 r 4 ∆n 1o dr + ∞ 1 2 (σ l +σ 2 ) ϕ 3 (r) 4 π r 2 n 2o + 2π 3 r 4 ∆n 2o dr.
Here n io ≡ n i (0, 0, 0), ∆n io ≡ ∆n i (0, 0, 0), i ∈ {1, 2} and ∆ is the Beltrami-Laplace operator. Let us denote

2 m 2 1 k 1 = ∞ σ 1 4 π r 2 ϕ 1 (r) dr, 2 m 2 1 k 1 b 2 1 = ∞ σ 1 2 π 3 r 4 ϕ 1 (r) dr, 2 m 1 m 2 κ 3 = ∞ 1 2 (σ l +σ 2 )
4 π r 2 ϕ 3 (r) dr,

2 k 3 m 1 m 2 b 2 3 = ∞ 1 2 (σ l +σ 2 ) 2 π 3 r 4 ϕ 3 (r) dr,
where σ i , i ∈ {1, 2}, denotes molecular diameters of fluids, b 1 is the fluid's co-volume of component 1 [START_REF] Rocard | Thermodynamique[END_REF] and b 3 is analog to a fluid's co-volume between components 1 and 2. To define different mass densities ρ i of components i, we must introduce the molecular masses of species denoted m i ; then, ρ i = n i m i and

W o1 = 2 m 2 1 k 1 n 1o + b 2 1 ∆ n 1o + 2 m 1 m 2 k 3 n 2o + b 2 3 ∆ n 2o ,
We deduce,

W o1 = 2 m 1 k 1 ρ 1o + b 2 1 ∆ρ 1o + 2 m 1 k 3 ρ 2o + b 2 3 ∆ ρ 2o ,
where ρ io = n io m i , i ∈ {1, 2} are the mass densities of components i at O. Similarly, we obtain

W o2 = 2 m 2 k 2 ρ 2o + b 2 2 ∆ρ 2o + 2 m 2 k 3 ρ 1o + b 2 3 ∆ ρ 1o .
We have to consider that all couples of molecules of the two components are counted twice and the double of the potential energy density per unit volume is

2 (n 1o W o1 + n 2o W o2 ) ≡ 2 (E o1 + E o2 ) , with            E o1 = n 1o m 1 k 1 ρ 1o + b 2 1 ∆ρ 1o + n 1o m 1 k 3 ρ 2o + b 2 3 ∆ ρ 2o , E o2 = n 2o m 2 k 2 ρ 2o + b 2 2 ∆ρ 2o + n 2o m 2 k 3 ρ 1o + b 2 3 ∆ ρ 1o .
The corresponding potential energy of the two-fluid mixture becomes

W = D k 1 ρ 2 1 + b 2 1 ρ 1 ∆ρ 1 + k 2 ρ 2 2 + b 2 2 ρ 2 ∆ρ 2 +k 3 2ρ 1 ρ 2 + b 2 3 ρ 1 ∆ρ 2 + b 2 3 ρ 2 ∆ρ 1 dv.
Taking account of

ρ i ∆ρ j = ρ i div ∇ ρ j = div ρ i ∇ ρ j -∇ ρ i ∇ ρ j ,
where ∇ is the gradient operator, terms div ρ i ∇ ρ j , with i, j ∈ {1, 2}, are integrable on the boundary of D, where ρ 1 and ρ 2 are assumed uniform and yield a zero contribution to W. We get

W = D k 1 ρ 2 1 + k 2 ρ 2 + 2 k 3 ρ 1 ρ 2 -k 1 b 2 1 (∇ρ 1 ) 2 -k 2 b 2 2 (∇ρ 2 ) 2 -2 k 3 b 2 3 ∇ρ 1 ∇ρ 2 dv. (1)
To obtain the volume free energy at given temperature T , we have to take account of the kinetic effects of molecular motions where the first terms 1) correspond to the total internal pressure [START_REF] Rocard | Thermodynamique[END_REF]. Consequently, the volume free energy reads

k 1 ρ 2 1 + k 2 ρ 2 2 + 2 k 3 ρ 1 ρ 2 in Eq. (
ε = ε 0 (ρ 1 , ρ 2 ) + 1 2 λ 1 (∇ρ 1 ) 2 + λ 2 (∇ρ 2 ) 2 + 2λ 3 ∇ρ 1 ∇ρ 2 ,
(2) where ε 0 (ρ 1 , ρ 2 ) is the volume free energy of the homogeneous fluid-mixture when densities are ρ 1 , ρ 2 (for the sake of simplicity, we omit to indicate T in ε 0 ) and

λ 1 = -2 k 1 b 2 1 , λ 2 = -2 k 2 b 2 2 , and λ 3 = -2 k 3 b 2 3 . (3)
Let us note that coefficients k 1 , k 2 , k 3 are negative for intermolecular potentials associated with attractive forces and coefficients λ 1 , λ 2 , λ 3 are positive.

B. Energy per unit area of wall boundary

In 1977, John Cahn gave simple illuminating arguments to describe the interaction between solids and liquids [START_REF] Cahn | Critical point wetting[END_REF]. His model is based on a generalized van der Waals theory of fluids treated as attracting hard spheres [START_REF] Rowlinson | Molecular theory of capillarity[END_REF]. It entailed assigning to the solid surface an energy that was a functional of the liquid density at the surface; the particular form of this energy is now widely known in the literature and was thoroughly examined as an ad hoc approximation in a review paper by de Gennes [START_REF] De Gennes | Wetting: statics and dynamics[END_REF]. Three hypotheses are implicit in Cahn's picture, (i) for the liquid density to be taken as a smooth function of the distance from the solid surface, the correlation length is assumed to be greater than intermolecular distances, (ii) the forces between solid and liquid are of short range with respect to intermolecular distances, (iii) the fluid is considered in the framework of a mean-field theory. This means, in particular, that the free energy of the fluid is a classical so-called gradient square functional.

The model was justified and for fluid mixtures, an extension of Cahn's model was proposed with the same hypotheses in [START_REF] Gouin | Energy of interaction between solid surface and liquids[END_REF]. The surface is a smooth solid, sharp on an atomic scale and is endowed with a surface energy per unit area of wall S of domain D. The general form of the surface energy per unit area can be written e = e(ρ 1s , ρ 2s ), where ρ 1s and ρ 2s are values of the fluid densities at the wall. We consider the case when the energy per unit area at the boundary is written as

e = -γ 11 ρ 1s -γ 21 ρ 2s + 1 2 γ 12 ρ 2 1s + γ 22 ρ 2 2s + 2γ 32 ρ 1s ρ 2s , (4)
where γ 11 , γ 21 , γ 12 , γ 22 , and γ 32 are positive coefficients expressing the wall's quality with respect to the two-fluid components of the mixture. Values of the coefficients have been proposed in [START_REF] Gouin | Energy of interaction between solid surface and liquids[END_REF] for the case of London's potentials and can be calculated with molecular quantities and Hamaker's constants.

C. van der Waals' forces and Hamaker's constants

We have introduced the intermolecular potentials associated with van der Waals' forces in the form ϕ(r). We consider the case of London's forces between fluids and solid wall and we denote

ϕ is (r) = - c is r 6 , where r > δ is = σ i + σ s 2 and ϕ is (r) = ∞ when r ≤ δ is .
The intermolecular coefficients denoted by c 11 , c 22 , c ss , c 12 , c 1s , c 2s , are associated with Hamaker's constants A ii and A ss defined as

A ii = π 2 N 2 c ii and A ss = π 2 N 2 c ss , ( 5
)
where N is the number of molecules per unit volume [START_REF] Hamaker | The London-van der Waals attraction between spherical particles[END_REF][START_REF] Israelachvili | Intermolecular forces[END_REF]. An important property of Hamaker constants is that they can be experimentally determined. The Hamaker constant between the two dissimilar materials can be estimated in term of Hamaker constants of each material. An approximation is proposed in [START_REF] Mitra | Microfluidics and nanofluidics handbook: Chemistry, physics, and life science principles[END_REF],

A 12 = A 11 A 22 , A 1s = A 11 A ss , A 2s = A 22 A ss .
Expressions of coefficients λ 1 , λ 2 and λ 3 can be deduced from Eq. ( 3),

λ 1 = 2 3 π c 11 σ 1 , λ 2 = 2 3 π c 22 σ 2 , λ 3 = 4 3 π c 12 (σ 1 + σ 2 )
.

In the model of London's forces, coefficients γ 11 , γ 21 , γ 12 , γ 22 and γ 32 can also be obtained by using intermolecular coefficients. They are proposed in [START_REF] Gouin | Energy of interaction between solid surface and liquids[END_REF] for solid wall of small curvature. Nonetheless, the energy behavior (4) being in the same form for all solid surfaces [START_REF] De Gennes | Wetting: statics and dynamics[END_REF], we will consider the same expressions for the intermolecular coefficients,

γ 11 = π c 1s 12 δ 2 1s ρ s , γ 21 = π c 2s 12 δ 2 2s ρ s ,
where ρ s is the mass density of the solid wall, and

γ 12 = π c 11 12 δ 2 1s , γ 22 = π c 22 12 δ 2 2s , γ 32 = π c 12 24 δ 2 2s       1 δ 2 1s + 1 δ 2 2s       . (6)

III. EQUATIONS OF MOTIONS AND BOUNDARY CONDITIONS FOR INHOMOGENEOUS MIXTURES OF SIMPLE FLUIDS

To obtain the equilibrium equations of mixtures, it is easy to use a variational method. The energy per unit volume is expressed by Eq. ( 2) and the associated energy of domain D is

E D = D ε dv.
The energy per unit area of the wall is expressed by Eq. ( 4) and the associated energy of surface S is

E S = S e ds.
The potential of the system fluid mixture-solid wall is

E = D ε dv + S e ds.
The total masses of components of an isolated and fixed domain D are

M i = D ρ i dv with i ∈ (1, 2).
When we neglect external forces, as gravity force, the equilibrium of the system is reached when the total energy is minimal and we get the variational equation

δE -µ 01 δM 1 -µ 02 δM 2 = 0, (7)
where µ 01 and µ 02 are two constant Lagrange multipliers which have the physical dimension of chemical potentials. Domain D is the physical domain occupied by the mixture and used to represent the possible states of a mechanical system of particles in thermodynamic equilibrium with a reservoir at given temperature T (we naturally take account of temperature T through term ε 0 (ρ 1 , ρ 2 ) included in ε). To identify the equilibrium state, the Gibbs free energy is minimized. Equation (7) must be valid for all variations δρ 1 and δρ 2 and consequently,

D ∂ε(ρ 1 , ρ 2 ) ∂ρ 1 δρ 1 + ∂ε(ρ 1 , ρ 2 ) ∂ρ 2 δρ 2 -µ 01 δρ 1 -µ 02 δρ 2 dv + S ∂e(ρ 1 , ρ 2 ) ∂ρ 1 δρ 1 + ∂e(ρ 1 , ρ 2 ) ∂ρ 2 δρ 2 ds = 0. (8)
We define

         φ 1 = λ 1 ∇ρ 1 + λ 3 ∇ρ 2 , φ 2 = λ 3 ∇ρ 1 + λ 2 ∇ρ 2 ,
and

µ 1 (ρ 1 , ρ 2 ) = ∂ε(ρ 1 , ρ 2 ) ∂ρ 1 ≡ ∂ε 0 (ρ 1 , ρ 2 ) ∂ρ 1 , µ 2 (ρ 1 , ρ 2 ) = ∂ε(ρ 1 , ρ 2 ) ∂ρ 2 ≡ ∂ε 0 (ρ 1 , ρ 2 ) ∂ρ 2 ,
which correspond to the chemical potential of the two fluid components at temperature T . Then, Eq. ( 8) writes

D µ 1 (ρ 1 , ρ 2 ) -µ 01 δρ 1 + φ 1 • δ∇ρ 1 + µ 2 (ρ 1 , ρ 2 ) -µ 02 δρ 2 + φ 2 • δ∇ρ 2 dv + S ∂ e(ρ 1 , ρ 2 ) ∂ρ 1 δρ 1 + ∂e(ρ 1 , ρ 2 ) ∂ρ 2 δρ 2 ds = 0.
Due to

φ i ∇δρ i = div (φ i δρ i ) -(divφ i ) δρ i and ∇δρ i = δ∇ρ i ,
and using the divergence theorem, we get

2 i=1 D µ i (ρ 1 , ρ 2 ) -µ 0i -div φ i δρ i dv + S ∂e(ρ 1 , ρ 2 ) ∂ρ i + n • φ i δρ i ds = 0,
where n is the external normal to S. First, we obtain the equations of equilibrium

         div φ 1 = µ 1 (ρ 1 , ρ 2 ) -µ 01 , div φ 2 = µ 2 (ρ 1 , ρ 2 ) -µ 02 , which gives          λ 1 ∆ρ 1 + λ 3 ∆ρ 2 = µ 1 (ρ 1 , ρ 2 ) -µ 01 , λ 3 ∆ρ 1 + λ 2 ∆ρ 2 = µ 2 (ρ 1 , ρ 2 ) -µ 02 , (9)
and second, the boundary conditions at surface S,

∂e(ρ 1 , ρ 2 ) ∂ρ 1 + n • φ 1 = 0 and ∂e(ρ 1 , ρ 2 ) ∂ρ 2 + n • φ 2 = 0.
The previous results -expressed at equilibrium -can be extended to mixture motions by the addition of inertial forces.

When we neglect the external forces, as gravity force, the two-liquid mixture motions verify equations extended from the equilibrium case and presented in [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]. They are in the form

         a 1 + ∇[ µ 1 (ρ 1 ) -λ 1 ∆ρ 1 -λ 3 ∆ρ 2 ] = ν 1 ∆v 1 , a 2 + ∇[ µ 2 (ρ 2 ) -λ 3 ∆ρ 1 -λ 2 ∆ρ 2 ] = ν 2 ∆v 2 , (10)
where a i , v i , ν i , i ∈ {1, 2} denote the accelerations, velocities and kinematic coefficients of viscosity of the two-fluid components, respectively. What is unexpected at first, the kinematic coefficients of viscosity are the same as for bulks [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF].

The boundary conditions at surface S are unchanged. Let us note that chemical potentials naturally introduce isobaric ensembles.

IV. THE CHEMICAL POTENTIALS IN FLUID-COMPONENT BULKS

The two-fluid mixture is constituted of simple compressible fluids. Consequently, pressure and chemical potential are the sum of the pressures and chemical potentials of fluidcomponents. In the fluid component bulks corresponding to the plane liquid-vapor interface at temperature T , each chemical potential is denoted by µ i0 , i ∈ {1, 2}. Due to the pressure equation p i = p i (ρ i , T ) of fluid-component i, it is possible to express µ i0 as a function of ρ i . At given temperature T , the associated volume free energy of fluid component i, denoted g i0 , verifies g i0 (ρ i ) = µ i0 (ρ i ). Potentials µ i0 and g i0 are defined except an additive constant and therefore we can choose the conditions

µ i0 (ρ i ) = µ i0 (ρ iv ) = 0 and g i0 (ρ i ) = g i0 (ρ iv ) = 0,
where ρ i and ρ iv are the densities in liquid and vapor of the bulks corresponding to the plane liquid-vapor interface of fluid-component i. The expressions of thermodynamical potentials µ i0 and g i0 can be expanded to the first order near liquid and vapor bulks, respectively

µ i0 (ρ i ) = C 2 i ρ i (ρ i -ρ i ) and µ i0 (ρ i ) = C 2 iv ρ iv (ρ i -ρ iv ) , g i0 (ρ i ) = C 2 i 2ρ i (ρ i -ρ i ) 2 and g i0 (ρ i ) = C 2 iv 2ρ iv (ρ i -ρ iv ) 2 ,
where C i and C iv are the isothermal sound-velocities in liquid and vapor bulks of component i [START_REF] Gouin | A mechanical model for the disjoining pressure[END_REF]. It is possible to obtain the connection between the liquid bulk of density ρ i b and the vapor bulk of density ρ ivb corresponding to non-planar interfaces (as for spherical bubbles and droplets [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]); they are called the mother-bulk densities [START_REF] Derjaguin | Surfaces forces[END_REF]. These equilibria do not obey the Maxwell rule, but the values of the chemical potential in the two mother bulks must be equal

µ i0 (ρ i b ) = µ i0 (ρ ivb ),
and determine the connection between ρ i b and ρ ivb . Consequently, we define µ i b (ρ i ) and µ ivb (ρ i ) as

µ i b (ρ i ) = µ i0 (ρ i ) -µ i0 (ρ i b ) ≡ µ i0 (ρ i ) -µ i0 (ρ ivb ) = µ ivb (ρ i ) .
An expansion to the first order near the liquid bulk of density ρ i b and the vapor bulk of density ρ ivb yields

µ i b (ρ i ) = C 2 i ρ i b (ρ i -ρ i b ) and µ ivb (ρ i ) = C 2 iv ρ ivb (ρ i -ρ ivb ) (11)
To the chemical potential µ i b (ρ i ) ≡ µ ivb (ρ i ), we can associate the volume free energies g i b (ρ i ) and g ivb (ρ i ) which are null for ρ i b and ρ ivb , respectively,

g i b (ρ i ) = g i0 (ρ i ) -g i0 (ρ i b ) -µ i0 (ρ i b )(ρ i -ρ i b ), (12) g ivb (ρ i ) = g i0 (ρ i ) -g i0 (ρ ivb ) -µ i0 (ρ ivb )(ρ i -ρ ivb ). ( 13
)
Free energies g i b (ρ i ) and g ivb (ρ i ) are null for the liquid bulk of density ρ i b and the vapor bulk of density ρ ivb of fluid component i, respectively, and contrary to the chemical potentials, they differ by a constant. Moreover, near the liquid and vapor mother-bulks, the volume free energies can be expanded as

g i b (ρ i ) = C 2 i 2ρ i b (ρ i -ρ i b ) 2 and g ivb (ρ i ) = C 2 iv 2ρ ivb (ρ i -ρ ivb ) 2 .
To compare the case of a nanotube filled with liquid-mixture and the case of a nanotube filled with vapor-mixture, we chose as reference volume free energy of component i the volume free energy g i b (ρ i ). Due to µ i0 (ρ i b ) = µ i0 (ρ ivb ), the difference of Eq. ( 12) and Eq. ( 13) yields

g i b (ρ i ) -g ivb (ρ i ) = g i0 (ρ ivb ) -g i0 (ρ i b ) -µ i0 (ρ i b )(ρ ivb -ρ i b ) .
But, Eq. ( 12) yields

g i b (ρ i b ) = 0, g i b (ρ ivb ) = g i0 (ρ ivb ) -g i0 (ρ i b ) -µ i0 (ρ i b )(ρ ivb -ρ i b ),
and consequently,

g i b (ρ i ) = g ivb (ρ i ) + g i b (ρ ivb ). ( 14
)
From Eq. ( 14), the term g i b (ρ ivb ) represents the difference between the volume free energies ( 12) and (13).

V. WATER-ETHANOL MIXTURE IN CARBON NANOTUBES

We denote by M, L and T the mass, length and time dimensions, respectively. The dimensions of other physical quantities are, molecular coefficients: ML 8 T -2 , masses per unit of volume: ML -3 , isothermal sound-velocities: MT -1 . The values of physical quantities are expressed for water and ethanol at 20 • Celsius. These values correspond to molecular coefficients, molecular diameters, molecular masses, mass densities, and isothermal sound-velocities of fluid components for liquids and vapors. They are obtained from the books of Israelaschvili [START_REF] Israelachvili | Intermolecular forces[END_REF], the Handbook of Chemistry and Physics [START_REF] Weast | Handbook of chemistry and physics[END_REF], and the Microfluidics and Nanophysics Handbook [START_REF] Mitra | Microfluidics and nanofluidics handbook: Chemistry, physics, and life science principles[END_REF], and are indicated in Table I. Carbon values are referred to nanotubes from Matsumoto et al's paper [START_REF] Matsumoto | The estimation of Hamaker constants of alcohols and interfacial tensions at alcohol-mercury interfaces[END_REF]. Molecular coefficients are deduced from the Hamaker constants thanks to relations [START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF]. The values of Table I I. The physical values associated with water, ethanol and carbon are expressed in M.K.S. units (meter, kilogram, second). In the unit system, we notice the very small values of intermolecular coefficients, molecular diameters and molecular masses.

A. Liquid-mixture at equilibrium in carbon nanotubes

In this section, we compare -at equilibrium -the total free energy of a carbon nanotube filled with a mixture of water and ethanol in liquid, vapor and liquid-vapor conditions. The reference volume free energy is g i b (ρ i ). Including the wall-energy, the total free energy per unit of length of the nanotube can be approximated by considering densities ρ 1 (for water) and ρ 2 (for ethanol) of the two components closely equal to ρ 1 b and ρ 2 b or ρ 1vb and ρ 2vb for the liquid and the vapor phases, respectively. First, we compare the case of a nanotube filled with liquid mixture and a nanotube filled with vapor mixture.

(i) In the case of a nanotube filled of liquid-mixture, total free energy E 1 per unit of length is,

E 1 ≈ Π (ρ b ) π R 2 g 1 b (ρ 1 b ) + g 2 b (ρ 2 b ) + 2 π R -γ 11 ρ 1 b -γ 21 ρ 2 b + 1 2 γ 12 ρ 2 1 b + γ 22 ρ 2 2 b + 2γ 32 ρ 1 b ρ 2 b ,
where

g 1 b (ρ 1 b ) = 0 and g 2 b (ρ 2 b ) = 0.
(ii) In the case of a nanotube filled of the vapor-mixture, total free energy E 2 per unit of length is,

E 2 ≈ π R 2 g 1 b (ρ 1vb ) + g 2 b (ρ 2vb ) + 2 π R -γ 11 ρ 1vb -γ 21 ρ 2vb + 1 2 γ 12 ρ 2 1vb + γ 22 ρ 2 2vb + 2γ 32 ρ 1vb ρ 2vb .
Due to the densities of vapor components,

-γ 11 ρ 1vb -γ 21 ρ 1vb + 1 2 γ 12 ρ 2 1vb + γ 22 ρ 2 2vb + 2γ 32 ρ 1vb ρ 2vb
is negligible in the estimation of energy with respect to the first two terms. From the expression of partial pressure p i of component i, i ∈ {1, 2},

p i (ρ i ) = ρ i µ i0 (ρ i ) -g i0 (ρ i ) + p i0
where p i0 is the common value of the pressure in liquid and vapor bulks of plane interface. Equation ( 12) yields,

g 1 b (ρ 1vb ) + g 2 b (ρ 2vb ) = p 1 (ρ 1 b ) -p 1 (ρ 1vb ) + p 2 (ρ 2 b ) -p 2 (ρ 2vb ) .
We denote Π b ,

Π b = p (ρ vb ) -p (ρ b ) = p 1 (ρ 1vb ) -p 1 (ρ 1 b ) + p 2 (ρ 2vb ) -p 2 (ρ 2 b ) ,
the difference between pressure p (ρ vb ) of the vapor mixture and pressure p (ρ b ) of the liquid mixture. Pressure Π b is called the mixture disjoining-pressure [START_REF] Gouin | A mechanical model for the disjoining pressure[END_REF][START_REF] Derjaguin | Surfaces forces[END_REF]. Consequently,

E 2 -E 1 ≈ 2 π R γ 11 ρ 1 b + γ 21 ρ 2 b - 1 2 γ 12 ρ 2 1 b + γ 22 ρ 2 2 b + 2 γ 32 ρ 1 b ρ 2 b -π R 2 Π b ,
and the vapor mixture is more energetic than the liquid mixture if

E 2 > E 1 ⇐⇒ Π b < 1 R 2 γ 11 ρ 1 b + 2 γ 21 ρ 2 b -γ 12 ρ 2 1 b + γ 22 ρ 2 2 b + 2 γ 32 ρ 1 b ρ 2 b . (15)
The ratio between volume proportion of water and ethanol is denoted c. In the bulks, the water density is ρ 1 b = c ρ 1 and the ethanol density is

ρ 2 b = (1 -c) ρ 2 .
By taking account of Table II, we calculated the most unfavorable case for inequality ( 15) to be verified. One obtain R 0 = 50 nm (for a microtube of 0.1 micron diameter) and we get

Π b ≡ p (ρ vb ) -p (ρ b ) ≈ 5 × 10 5 Pascal = 5 atmospheres.
This result comes from the wall quality of carbon nanotubes: terms γ 11 and γ 21 take advantage of terms associated with γ 12 , γ 22 and γ 32 , and consequently the energy of vapor mixture is greater than the energy of liquid mixture. Second, we compare the case of a nanotube filled with liquid mixture and a nanotube filled with a liquid-vapor mixture. When a two-phase mixture is considered, we assume that an interface appears between the liquid and vapor phases. To estimate the value of the liquid-vapor interface, we consider the smallest possible interface area. The smallest area of the liquid-vapor interface separating the two phases corresponds to a material surface represented by a cylinder with the same axis as the nanotube axis. The interface has a positive surface energy γ increasing the total fluid energy inside the nanotube (see Fig. 1) (i) When domain (a) is liquid and domain (b) is vapor, the energy of the wall is approximatively the same as for liquid mixture and the difference of total free energies between liquid mixture and liquid-vapor mixture is approximately

E 3 -E 1 ≈ π r 1 2 γ -r 1 Π b ,
where E 3 is the energy of the two-phase mixture and r 1 is the radius of domain (b). (ii) When domain (a) is vapor and domain (b) is liquid, the difference of total free energies is approximately

E 4 -E 1 ≈ 2 π r 1 γ -π R 2 -r 2 1 Π b + 2 π R γ 11 ρ 1vb + γ 21 ρ 2vb - 1 2 γ 12 ρ 2 1vb + γ 22 ρ 2 2vb + 2γ 32 ρ 1vb ρ 2vb ,
where E 4 is the energy of the two-phase mixture. The positive interfacial energy associated with γ increases the total energy of the liquid-vapor mixture. In all cases, when γ > 12.5 × 10 -3 N × m -1 (corresponding to a value significantly lower than the surface tension of any water-ethanol mixture at 20 • C [START_REF] Weast | Handbook of chemistry and physics[END_REF]), and condition ( 15) is verified, we get

E 3 > E 1 and E 4 > E 1 .
In the following, we consider only carbon nanotubes such that previous conditions are verified. Consequently -at equilibrium -they are filled with liquid mixture of water and ethanol.

B. Numerical calculations at equilibrium

We consider cylindrical coordinates (r, θ, z) where z denotes the coordinate of the nanotube axis. Due to the symmetry of revolution and far from nanotube extremities, the densities depend only of z and in the case of liquid mixture, the equations of equilibrium ( 9) can be written by taking account of Eq. ( 11),

                     λ 1 d 2 ρ 1 dr 2 + 1 r dρ 1 dr + λ 3 d 2 ρ 2 dr 2 + 1 r dρ 2 dr = c 2 1 ρ 1 b (ρ 1 -ρ 1 b ) , λ 3 d 2 ρ 1 dr 2 + 1 r dρ 1 dr + λ 2 d 2 ρ 2 dr 2 + 1 r dρ 2 dr = c 2 2 ρ 2 b (ρ 2 -ρ 2 b ) , (16)
where ρ 1 and ρ 2 denote the densities associated with water and ethanol, respectively. The symmetry of densities at the z-axis implies, At r = 0, dρ 1 dr = 0 and dρ 1 dr = 0 [START_REF] Gouin | Statics and dynamics of fluids in nanotubes[END_REF] and the boundary conditions at the nanotubes wall -where R is the nanotube radius -yield,

At r = R,                  λ 1 dρ 1 dr + λ 3 dρ 2 dr = γ 11 -γ 12 ρ 1 -γ 32 ρ 2 , λ 3 dρ 1 dr + λ 2 dρ 2 dr = γ 21 -γ 32 ρ 1 -γ 22 ρ 2 .
(18) For nanotubes, the values obtained in Table II are not expressed in convenient units. To obtain units adapted to the numerical computations, we consider an unit system such that the mass density, the isothermal sound-velocity of liquidwater and λ 1 are equal to 1. Table III indicates the corresponding values of units of length, mass and time. We name these units, molecular capillary units at temperature T = 20 • C. We notice that the units of length mass and time are of the same order as molecular diameters, masses of molecules and times associated with the mean free paths. In molecular capillary units, the values of λ 1 , λ 2 , λ 3 , γ 11 , γ 21 , γ 12 , γ 22 ,and γ 32 are indicated in Table IV. To easily compare the profiles of densities, we have normalized the diameter with respect to radius R and therefore for all figures, the x-axis is drawn between 0 and 1 whatever is the real value of the nanotube diameter. c ↓ 2 R → 1 nm 2 nm 3 nm 5 nm 10 nm 20 nm 50 nm 100 nm c = 0.3 2.7% 1.4% 1% 0.6% 0.3% 0.15% 0.1% 0.05% c = 0.5 5% 2.5% 1.7% 1% 0.5% 0.3% 0.2% 0.1% c = 0.7 8.5% 4.4% 3% 1.8% 1% 0.5% 0.37% 0.27% TABLE V. Percentage of the increase of volume ethanol with respect to volume water for different nanotube diameters and values of ratio c between volume proportion of water and ethanol.

We calculate the profile of densities of the liquid mixture composed of water and ethanol wetting carbon nanotubes. As in Section V A, the ratio between volume proportion of water and ethanol is denoted c, in the bulks, the water density is

ρ 1 b = c ρ 1 and the ethanol density is ρ 2 b = (1 -c) ρ 2 .
The profiles of densities are given by equilibrium equations ( 16) and boundary conditions ( 17)- [START_REF] Gȃrȃjeu | Scaling Navier-Stokes equation in nanotubes[END_REF]. We calculate three cases of liquid mixtures' composition corresponding in the bulks to c = 0.3, 0.5 and 0.7. The density curves depend on nanotube diameters. For each value of c, we calculate the density profiles for water and ethanol when the diameters are 1, 2, 3, 5 , 10, 20, 40 and 100 nanometer. Figures 2, 3 and4 represent the densities versus r/R for increasing value of R with relative up and down magnifications. In all cases, we note that the water density decreases near the wall and the profile is monotonic for c = 0.3 and c = 0.5, and oscillating for c = 0.7. For ethanol the density increases near the wall and the profile is oscillating for c = 0.3 and c = 0.5 and monotonic for c = 0.7. Following ratio c between volume proportion of water and ethanol and the nanotube diameters, volume concentration of ethanol with respect to volume concentration of water increases. The results are indicated on Table V. The effect closely negligible when c = 0.3, is noticeable for small nanotubes when c = 0.7, and disappears for large nanotubes.

VI. VISCOUS MOTIONS IN A NANOTUBE

Fluid flows through structures like carbon nanotubes must be different from flows through microscopic and macroscopic structures since, for the latter flow, the degrees of freedom of fluid molecules can be safely ignored and the flow in such structures can be characterized by viscosity, density and other bulk properties. Furthermore, for large-scale systems, the noslip boundary condition is often implemented, because the fluid velocity is negligibly small at the fluid/wall boundary. Reducing the length scales introduces new phenomena, when The different graphs represent the change of the ratio between volume proportion of water and ethanol versus r / R (representing the normalized distance to the carbon wall), when the radius nanotube is successively equal to R = 0.5 at 50 nm. We recall that -to draw all the diameter cases on the same figurewe have normalized the diameter with respect to radius R and therefore, the x-axis is drawn between 0 and 1 for any real value of the nanotube diameter. the mutual interaction between walls and fluids must be taken into account [START_REF] Rafii-Tabar | Computational physics of carbon nanotubes[END_REF]. We consider the permanent and laminar motions of viscous capillary liquids in a nanotube. Because the liquid is heterogeneous, the static part of the liquid stress tensor is not scalar and the equations of hydrodynamics are not immediately valid. However, the results obtained for vis- Comments are similar to those for the caption to Fig. 2.

cous flows can be adapted at nanoscales [START_REF] Landau | Fluid mechanics[END_REF]. We denote the velocities by v i = (0, 0, w i ) where w i is the velocity of component i of the fluid mixture along the nanotube axis. When we neglect the external forces, as gravity force, the two-liquid mixture motions of the components verify Eqs. [START_REF] Gouin | Mixtures of fluids involving entropy gradients and accelerations waves in interfacial layers[END_REF]. Simple fluids slip on a solid wall only at a molecular level and in classical conditions, the kinetic condition at solid wall is the adherence condition (z = R ⇒ w i = 0) [START_REF] Chuarev | Thin liquid layers[END_REF]. Recent papers in non-equilibrium molecular dynamic simulations of three dimensional micro-Poiseuille flows in Knudsen regime reconsider micro-channels: the influence of surface roughness, surface wetting and wall density are investigated. The results point out that the no-slip condition can be observed for Knudsen flow when the surface is rough; the surface wetting Comments are similar to those for the caption to Fig. 2. also substantially influences the velocity profiles [START_REF] Tabeling | Introduction à la microfluidique[END_REF]. But for smooth surfaces, as carbon nanotube walls, a slip velocity is the key for characterizing the flows. With water flowing through hydrophobic thin capillaries, there are some qualitative evidences for slippage [START_REF] Blake | Slip between a liquid and a solid -D.M. Tolstoi (1952) theory reconsidered[END_REF]. De Gennes [START_REF] De Gennes | On fluid/wall slippage[END_REF] said: the results are unexpected and stimulating and led us to think about unusual processes which could take place near a wall. Experimental results are connected with the thickness of the film when the thickness is of an order of the mean free path [START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF].

When the free mean path, denoted path , is smaller than diameter 2R of carbon nanotubes, the Knudsen number, denoted Kn, is smaller than 1. That is the case for a liquid where the mean free path is of the same order than molecular diameters. For example in the case of liquid-water, Kn ranges between 0.5 and 10 -2 while the nanotube radius ranges between 1 nm and 50 nm. Majunder et al note that slip lengths of micron order for their experiments with nanometer size pores and the adherence boundary condition at a surface, commonly employed with the Navier-Stokes equation, is physically invalid [START_REF] Majumder | Nanoscale hydrodynamics: Enhanced flow in carbon nanotubes[END_REF]. A slip regime occurs and the boundary condition must be changed to take account of the slippage at the solid surfaces.

We consider the simple case when a 1 = a 2 ≡ a and v 1 = v 2 ≡ v = [0, 0, w], corresponding to the same velocity and acceleration of the two-fluid mixture components. In fluid/wall slippage, the condition at the solid wall is written as

w = L s ∂w ∂r at r = R, (19)
where L s denotes the Navier-length [START_REF] Landau | Fluid mechanics[END_REF]. The Navier-length is not independent of the thickness of the flow and may be as large as a few microns for very thin films [START_REF] Tabeling | Introduction à la microfluidique[END_REF][START_REF] Secchi | Scaling behavior for ionic transport and its fluctuations in individual carbon nanotubes[END_REF]. For carbon nanotubes, the dynamics of liquid flows is studied in the case of non-rough nanotubes. Equations ( 10) yield

ρ a + ρ 1 ∇ [ µ 1 (ρ 1 ) -λ 1 ∆ρ 1 -λ 3 ∆ρ 2 ] + ρ 2 ∇ [ µ 2 (ρ 2 ) -λ 3 ∆ρ 1 -λ 2 ∆ρ 2 ] = ν ∆v, (20)
where ρ = ρ 1 + ρ 2 and we define the dynamic viscosity of the fluid mixture ν as ρ ν = ρ 1 ν 1 + ρ 1 ν 2 ; ν is assumed constant. Consequently, we consider the case when: (i) the boundary conditions take account of the slip condition ( 19), (ii) the liquid nanoflow thickness is small with respect to transverse dimensions of the wall and 2R path , (iii) the flow is laminar corresponding to a velocity component along the wall large with respect to the normal velocity component to the wall which is negligible, (iv) the permanent velocity vector v varies along the direction orthogonal to the wall, ∇ρ is normal to v and the equation of continuity reads

ρ div v = 0.
Then, the density is constant along each stream line and the trajectories are drawn on iso-density surfaces where w = w(r). Due to the geometry, for permanent motions, the acceleration is null. Equations of motion separate as follows • Along the z-coordinate, Eq. ( 20

where p 1 and p 2 are the partial pressures of the two components.

• In plane orthogonal to the tube axis, Eqs. (10) yield

∂ ∂r (µ 1 (ρ 1 ) -λ 1 ∆ρ 1 -λ 3 ∆ρ 2 ) = 0, ∂ ∂r (µ 2 (ρ 2 ) -λ 3 ∆ρ 1 -λ 2 ∆ρ 2 ) = 0. (22)
It is fundamental to note that Eqs [START_REF] Rocard | Thermodynamique[END_REF] yield the same equations as at equilibrium. This result is the key of the distribution of densities and volume proportion between water and ethanol. Equation ( 21) is written as

1 r d dr r dw dr = - ℘ ν , (23)
where ℘ denotes the pressure gradient along the nanotube. The cylindrical symmetry of the nanotube yields the solution of Eq. ( 23) in the form

w = - ℘ ν r 2 4 + b,
where b is constant. Condition [START_REF] Majumder | Nanoscale hydrodynamics: Enhanced flow in carbon nanotubes[END_REF] implies

- ℘ 4ν R 2 + b = L s ℘ 2ν R
and consequently,

w = ℘ 4ν -r 2 + R(R + L s ) .
For liquids, the density in the nanotube is closely equal to ρ b = ρ 1 b + ρ 2 b and the volume flow through the nanotube is

Q = 2π R 0 w r dr = π ℘ 8 ν R 3 (R + 4L s ).
Q o denoting the Poiseuille flow corresponding to a tube of the same radius R,

Q = Q o 1 + 4L s R . (24)
In most cases, the Navier length is of the micron order (L s = 1 µm = 10 3 nm) [START_REF] Majumder | Nanoscale hydrodynamics: Enhanced flow in carbon nanotubes[END_REF]. If we consider a nanotube with R = 2 nm, we obtain Q = 2 × 10 3 Q o . For R = 50 nm, corresponding to the upper limit nanotube-radius for nanofluidics with respect to microfluidics, we obtain Q = 40 Q o . Consequently, the flow of liquid in nanofluidics is drastically more important than the Poiseuille flow in cylindrical tubes.

When the mixture mother-bulk is vapor, in the carbon nanotube the phase is generally liquid and the volume flow through the nanotube is approximately

Q = Q o ρ ρ v 1 + 4L s R .
In the case of water and ethanol we have ρ /ρ v ∼ 10 2 to 10 3 and we get a volume flow at least 10 2 time more important than the volume flow obtained by Eq. ( 24). Then

Q ∼ 10 5 Q o .
The extremely large slip lengths measured in carbon nanotubes greatly reduce the fluid resistance and nanoscale structures could create extraordinarily fast flows as it is in biological cellular channels [START_REF] Bonthuis | Theory and simulations of water flow through carbon nanotubes: prospects and pitfalls[END_REF].

VII. CONCLUSIONS

In this paper, we propose a continuous model for the profile of densities between the two components of a fluid mixture. The model corresponding to the mean-field theory with hardsphere molecules provides a free energy in the fluid mixture and an energy on the nanotube wall which yield a system of second order differential equations for equilibrium and for dissipative motions. Depending on the wall's quality associated with the coefficients presented in Eq. ( 4), we obtain boundary conditions that permit to integrate the system. Thereby, carbon cylindrical nanotubes with diameters ranging from 1 to 100 nm can be studied. The carbon nanotubes are filled up with liquid mixture of water-ethanol in a volume proportion imposed by the external mixture bulk. For nanotubes of larger diameters essentially corresponding to microtubes, we obtain that the fluid phases can be liquid or vapor according to the chemical properties of the tube walls and mother bulks with possible liquid-vapor interfaces. For tubes with small diameters, essentially corresponding to nanotubes, the fluid flows are liquid mixtures and can be greater than usual Poiseuille's flows, especially if the mixture mother-bulks consist of vapor. To obtain realistic densities of liquid ethanol and liquid water, we limited to three cases of concentration (c = 0.3, 0.5, 0.7). The cases where c is close to 0 or 1 correspond to vapor pres-ence of one of the constituents; thus, the composition of the mixture must be quite far from these cases. It might be interesting to take up the problem with densities for which one of the components corresponds to a vapor density but in this case, Eqs. ( 16) will have to be modified. For small nanotube diameters, we notice that, in the carbon nanotube, the ratio between water and ethanol increases in favor of ethanol. These results, obtained by using a nonlinear model of continuum mechanics, are in good agreement with the simulations of molecular dynamics [START_REF] Mecke | Molecular dynamics simulation of the liquid-vapour interface: The Lennard-Jones fluid[END_REF]. Recent experiments confirm these results [START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF][START_REF] Secchi | Scaling behavior for ionic transport and its fluctuations in individual carbon nanotubes[END_REF] and could be experimentally measured by indirect methods as Landau-squire plume measurements [START_REF] Landau | Fluid mechanics[END_REF][START_REF] Secchi | The Landau-squire plume[END_REF].

ϕ

  11 (r), ϕ 22 (r), ϕ ss (r), ϕ 12 (r), ϕ 1s (r), ϕ 2s (r), the London potentials of interactions between fluid 1-fluid 1, fluid 2-fluid 2, solid-solid, fluid 1-fluid 2, fluid 1-solid, fluid 2-solid, respectively. With the notations of Sec. II A and denoting σ s the molecular diameter of the wall molecules, the London potentials verify ϕ ii (r) = -c ii r 6 , where r > σ i and ϕ ii (r) = ∞ when r ≤ σ i , (i = {1, 2}) ϕ ss (r) = -c ss r 6 , where r > σ s and ϕ ss (r) = ∞ when r ≤ σ s , ϕ 12 (r) = -c 12 r 6 , where r > δ 12 = σ 1 + σ 2 2 and ϕ 12 (r) = ∞ when r ≤ δ 12 ,

FIG. 1 .

 1 FIG. 1. Two-phase fluid-component in a nanotube: The nanotube is simultaneously filled with two phases liquid and vapor of fluidcomponents. The two phases (a) and (b) are separated by a cylindrical interface.

  Numerical values 2.31 × 10 -10 m 1.23 × 10 -26 Kg 1.56 × 10 -13 s TABLE III. The new molecular capillary units, at temperature T = 20 • C, associated with liquid-water are deduced from the Table I and expressed in M.K.S. units (meter, kilogram and second).

6 FIG. 2 .

 62 FIG. 2.Case c = 0.3 (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). The different graphs represent the change of the ratio between volume proportion of water and ethanol versus r / R (representing the normalized distance to the carbon wall), when the radius nanotube is successively equal to R = 0.5 at 50 nm. We recall that -to draw all the diameter cases on the same figurewe have normalized the diameter with respect to radius R and therefore, the x-axis is drawn between 0 and 1 for any real value of the nanotube diameter. The graph (a) shows the density profiles between the nanotube axis and the carbon wall (r/R ∈ [0, 1]). The second graph (b) enlarges the water density near the wall (r/R ∈ [0.8, 1]). The third graph (c) enlarges the ethanol density near the wall (r/R ∈ [0.8, 1]). In the second and third graphs, the curved arrows indicate the different graphs for increasing values of R and for the eight nanotube radius-values corresponding to 2R ∈ {1, 2, 3, 5, 10, 20, 50, 100} in nanometers.

  The graph (a) shows the density profiles between the nanotube axis and the carbon wall (r/R ∈ [0, 1]). The second graph (b) enlarges the water density near the wall (r/R ∈ [0.8, 1]). The third graph (c) enlarges the ethanol density near the wall (r/R ∈ [0.8, 1]). In the second and third graphs, the curved arrows indicate the different graphs for increasing values of R and for the eight nanotube radius-values corresponding to 2R ∈ {1, 2, 3, 5, 10, 20, 50, 100} in nanometers.

FIG. 3 .

 3 FIG.3. Case c = 0.5 (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). Comments are similar to those for the caption to Fig.2.

FIG. 4 .

 4 FIG.[START_REF] Bear | Dynamics of fluids in porous media[END_REF]. Case c = 0.7 (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). Comments are similar to those for the caption to Fig.2.

  ) yields ∂p ∂z = ν ∆ w with ∆ w = 1 r d dr r dw dr and p = p 1 + p 2 ,

  together with Eqs.[START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF] to (6) allow us to obtain the values of λ 1 , λ 2 , λ 3 , γ 11 , γ 21 , γ 12 , γ 22 , γ 32 and those given in Table II. Physical constants Intermolecular coefficient molecular diameter molecular mass Bulk density Isothermal sound velocity

	Liquid-water	1.4 × 10 -77	2.8 × 10 -10 2.99 × 10 -26	998	1.478 × 10 3
	Vapor-water	1.4 × 10 -77	2.8 × 10 -10 2.99 × 10 -26 9.7 × 10 -3	3.70 × 10 2
	Liquid-ethanol	3.71 × 10 -77	4.69 × 10 -10 7.64 × 10 -26	789	1.162 × 10 3
	Vapor-ethanol	3.71 × 10 -77	4.69 × 10 -10 7.64 × 10 -26 1.09 × 10 -1	2.30 × 10 2
	Nanotube-carbon	0.1014 × 10 -77	1.30 × 10 -10 1.99 × 10 -26	2000	---
	TABLE

  Numerical values 1.17 × 10 -16 0.284 × 10 -16 0.49 × 10 -16 77 × 10 -6

	Physical constants	λ 1	λ 2	λ 3	γ 11
	Physical constants	γ 21	γ 12	γ 22	γ 32
	Numerical values

22 × 10 -6 9.7 × 10 -8 1.8 × 10 -8 4.0 × 10 -8

TABLE II .

 II The physical values associated with the contact of water, ethanol and carbon expressed in M.K.S. units (meter, kilogram, second).

  Physical constants λ 1 λ 2 λ 3 γ 11 γ 21 γ 12 γ 22 γ 32 Numerical values 1 0.24 0.42 146 42 3.2 × 10 -3 6 × 10 -4 1.3 × 10 -3

TABLE IV .

 IV The physical values associated with water, ethanol and carbon are expressed in molecular capillarity units. Moreover, the isothermal sound velocities are C liquid-water = 1 and C liquid-ethanol = 0.786.

Acknowledgements : H.G. thanks National Group of Mathematical Physics GNFM-INdAM for its support as visiting professor at the Department of Mathematics, University of Bologna. This work was partially supported by National Group of Mathematical Physics GNFM-INdAM (A.M. and T.R.).