D Truzzolillo
email: domenico.truzzolillo@umontpellier.fr

Mixing-demixing transition and void formation in quasi-2D binary mixtures on a sphere

Motivated by observations of heterogeneous domain structure on the surface of cells and vesicles and by domain formation due to the adsorption of complex molecules onto composite membranes, we consider a minimal quasi 2D-model to describe the structure of binary mixtures on the surface of a spherical particle. We study the effect of miscibility and adsorbing particle (AP) addition on the mixture structure. We define a new scalar quantity, the geodesic mixing parameter Ξ, through which we detail the effect of miscibility and the role of preferential affinity of APs with one of the two components of the mixture, distinguishing unambiguously between mixing and demixing solely induced by APs. Finally, by inspecting the distributions of void sizes, we show how void formation is ruled by miscibility and AP-mixture interactions, which control the transition from exponentially-tailed to fat-tailed distributions. a

I. INTRODUCTION

The investigation of the equilibrium properties of particles whose motion is confined to curved surfaces is surely challenging, but it is essential to achieve a satisfactory understanding of processes such as the lateral diffusion of proteins in membranes [START_REF] Malchus | Anomalous Diffusion Reports on the Interaction of Misfolded Proteins with the Quality Control Machinery in the Endoplasmic Reticulum[END_REF][START_REF] Javanainen | Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes[END_REF][START_REF] Metzler | Non-Brownian diffusion in lipid membranes: Experiments and simulations[END_REF] , the controlled synthesis of colloidal aggregates [START_REF] Savarala | Stabilization of Soft Lipid Colloids: Competing Effects of Nanoparticle Decoration and Supported Lipid Bilayer Formation[END_REF] , and demixing in lipid bilayers [START_REF] Anderson | A Role for Lipid Shells in Targeting Proteins to Caveolae, Rafts, and Other Lipid Domains[END_REF][START_REF] Binder | Domains and Rafts in Lipid Membranes[END_REF][START_REF] Veatch | Critical Fluctuations in Plasma Membrane Vesicles[END_REF] . Confining particles to the surface of a sphere introduces several constraints to their placement that are absent in Euclidean (flat) space. Both the curvature and the topology of the sphere play a role in the way particles can be arranged on it-most prominently, no regular lattice can be fit onto its surface, as the topology of the sphere requires the presence of 12 pentagonal disclinations [START_REF] Bowick | Two-dimensional matter: Order, curvature and defects[END_REF] , and introduces structural defects characterizing glassy fluids at large area fractions [START_REF] Singh | Cooperatively rearranging regions change shape near the mode-coupling crossover for colloidal liquids on a sphere[END_REF] . The compactness of the sphere also requires a careful treatment: instead of the straightforward thermodynamic limit, configurations of finite (small) numbers of particles become more relevant, and both the number of particles and the size of the sphere need to be considered as two independent parameters [START_REF] Post | Statistical thermodynamics of particles adsorbed onto a spherical surface. I. Canonical ensemble[END_REF] . Among all possible spherical systems, multicomponent mixtures are of particular importance since they represent the class of systems including biological membranes in which the self-organisation of lipids into different domains is critical for many cell properties such as recognition, signalling, or endocytosis and exocytosis [START_REF] Salaün | Lipid Rafts and the Regulation of Exocytosis: Lipid Rafts and Exocytosis[END_REF] . Demixing and segregation in bulk systems can proceed via a number of distinct physical mechanisms (such as spinodal decomposition and nucleation), and it is generally well understood. However, the impact of spatial confinement and the interaction with external agents on the structure of multicomponent mixtures are less clear. Confinement can arise from the presence of external fields, produced, for example, by substrates or random obstacles, but they can also be imposed by the geometry of the embedding space. The latter a) The following article has been accepted by The Journal of Chemical Physics.

After it is published, it will be found at https://aip.scitation.org/journal/jcp type of confinement is particularly relevant to biological cells, for which the mobile fluid particles constituting the cell membrane are constrained to lie on the surface of a quasi-spherical body. This has boosted the interest for a deep understanding of multiphase fluids on smooth manifolds, especially on spheres and spheroids, and so in the past 2 decades many numerical studies have been focused on the particular problem of the phase separation of binary systems on surfaces and its dependence on the surface curvature. [START_REF] Tang | Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method[END_REF][START_REF] Greer | Fourth order partial differential equations on general geometries[END_REF][START_REF] Marenduzzo | Phase separation dynamics on curved surfaces[END_REF][START_REF] Barreira | The surface finite element method for pattern formation on evolving biological surfaces[END_REF] Biological membranes are indeed described as a quasi two-dimensional fluids [START_REF] Saffman | Brownian motion in biological membranes[END_REF][START_REF] Camley | Dynamic scaling in phase separation kinetics for quasi-two-dimensional membranes[END_REF] in which the different constituents flow within the membrane manifold, being still partially free to move orthogonally to the membrane tangent plane and coupled to a less viscous three-dimensional surrounding medium. They exhibit very often stable domains, whose spatial distribution is important for many cell properties [START_REF] Lenne | Physics puzzles on membrane domains posed by cell biology[END_REF][START_REF] Lingwood | Lipid Rafts As a Membrane-Organizing Principle[END_REF] . One common view is that the domains on the surface of a cell are a consequence of an arrested or incomplete phase separation, however it remains to be established whether the observed states are permanent or metastable in character. If the heterogeneous domain structure on the cell surface is an equilibrium state, then some stabilizing mechanism is required; the line tension present at interfaces between domains would make inhomogeneous phases energetically unfavorable when compared to a fully phaseseparated system.

Many works have suggested that biological membranes are close to a demixing line in the temperature-composition plane. When the lipid mixture is close to the miscibility critical point [START_REF] Onuki | Phase Transition Dynamics[END_REF][START_REF] Honerkamp-Smith | An introduction to critical points for biophysicists; observations of compositional heterogeneity in lipid membranes[END_REF] (the so-called weak-segregation limit) the temperature T is close to a critical temperature T c . In this limit, density fluctuations give rise to wide interfaces between fluctuating lipid domains, which do not have a precise shape [START_REF] Destainville | A Rationale for Mesoscopic Domain Formation in Biomembranes[END_REF] .

As a matter of fact, if a membrane is composed of two or more lipids (as most biomembranes are), the segregation of one of the lipid components in a set of domains is often encountered, as for example rafts rich in cholesterol and sphingolipids that have been implicated recently in signal transduction and membrane trafficking pathways [START_REF] Anderson | A Role for Lipid Shells in Targeting Proteins to Caveolae, Rafts, and Other Lipid Domains[END_REF][START_REF] Binder | Domains and Rafts in Lipid Membranes[END_REF] . Veatch and coworkers [START_REF] Veatch | Critical Fluctuations in Plasma Membrane Vesicles[END_REF] suggested that lateral heterogeneities in living cells at physiological conditions correspond to critical fluc-tuations and that perturbations that alter the phase boundary could have a large effect on the size, composition, and lifetime of fluctuations at physiological temperatures. Also, lipid demixing due to the existence of a miscibility critical point in red blood cell (RBC) membranes has been highlighted via Langmuir trough and epifluorescence experiments by Keller and coworkers [START_REF] Keller | Red Blood Cell Lipids Form Immiscible Liquids[END_REF] , who showed that monolayers of lipids extracted from RBC membranes at room temperature segregate and form domains above a threshold surface pressure at the air-water interface.

In addition to single membrane properties, the interaction potential between a pair of cells is also strongly influenced by the distribution and size of the domains covering its surface. In this sense, cells may be regarded as a naturally occurring type of "patchy particles," which are particles with distinct surface sites generating anisotropic interparticle interactions. While synthetically fabricated patchy particles have attractive patches strategically arranged on their surface [START_REF] Bianchi | Patchy colloids: State of the art and perspectives[END_REF] , the domains covering a cell emerge as a result of self-organization. Recent developments in the controlled fabrication of patchy particles have raised hopes that materials with desired properties may be tailored by prescribing the number and geometrical arrangement of the patches [START_REF] Bianchi | Patchy colloids: State of the art and perspectives[END_REF] . In this framework, confined binary mixtures of active and passive particles on a sphere have also been studied [START_REF] B.-Q. Ai | Binary mixtures of active and passive particles on a sphere[END_REF] very recently, since particle activity coupled to the presence of intrinsic surface curvature frustrates local order, giving rise to novel phenomena [START_REF] Shankar | Topological Sound and Flocking on Curved Surfaces[END_REF][START_REF] Keber | Topology and dynamics of active nematic vesicles[END_REF][START_REF] Sanchez | Spontaneous motion in hierarchically assembled active matter[END_REF] , such as curvature-induced defects and unforced flow.

On top of the spontaneous formation of domains and defects on spheres, both loss of miscibility and domain destabilization in spherically confined fluids can be also caused by the addition of adsorbing agents, like multivalent ions, polyelectrolytes and polyampholytes whose adsorption-desorption mechanism represents one the key problem tackled in soft matter physics [START_REF] Caetano | Critical adsorption of multiple polyelectrolytes onto a nanosphere: Splitting the adsorption-desorption transition boundary[END_REF][START_REF] Caetano | Critical adsorption of periodic and random polyampholytes onto charged surfaces[END_REF] . Induced-demixing has been widely investigated in the past. Domain and raft formation in membranes has been found in the presence of bivalent ions [START_REF] Jacobson | Phase transitions and phase separations in phospholipid membranes induced by changes in temperature, pH, and concentration of bivalent cations[END_REF] , RNA and DNA strands [START_REF] Michanek | RNA and DNA interactions with zwitterionic and charged lipid membranes -A DSC and QCM-D study[END_REF] , charged proteins [START_REF] Mbamala | Domain Formation Induced by the Adsorption of Charged Proteins on Mixed Lipid Membranes[END_REF] , and oppositely charged polyelectrolytes [START_REF] Macdonald | Polyelectrolyte-induced domains in lipid bilayer membranes: The deuterium NMR perspective[END_REF] . By contrast, induced mixing has been documented much less and only very recently it has been shown that the addition of functionalized nanoparticles [START_REF] Canepa | Amphiphilic gold nanoparticles perturb phase separation in multidomain lipid membranes[END_REF] or thermoplastic polymers can indeed give rise to a suppression of lipid segregation [START_REF] Bochicchio | Interaction of hydrophobic polymers with model lipid bilayers[END_REF] . The general mechanisms dictating a mixing induced by adsorbing molecules, polymers or colloids and the thermodynamic conditions under which this occurs, have not been explored neither understood in depth. Finally, domain formation and lipid miscibility plays a crucial role for the membrane permeability. Passive transport through biomembranes and bilayer lipid membranes is in many cases controlled by existing small holes (or voids) within them [START_REF] Kotyk | Membrane Transport[END_REF][START_REF] Kashchiev | Bilayer lipid membrane permeation and rupture due to hole formation[END_REF] . There are several experimental evidences that proton, water and potentially drug permeability of mixed lipid membranes is enhanced by raft formation [START_REF] Larocca | Proving Lipid Rafts Exist: Membrane Domains in the Prokaryote Borrelia burgdorferi Have the Same Properties as Eukaryotic Lipid Rafts[END_REF][START_REF] Ghysels | Permeability of membranes in the liquid ordered and liquid disordered phases[END_REF][START_REF] Gensure | Lipid raft components cholesterol and sphingomyelin increase H+/OH-permeability of phosphatidylcholine membranes[END_REF][START_REF] Sciolla | Influence of drug/lipid interaction on the entrapment efficiency of isoniazid in liposomes for antitubercular therapy: A multi-faced investigation[END_REF] , since inter-domain interfaces enhances ions and small molecules intrusion, and that membranes can be even destabilized by lipid substitution [START_REF] Larocca | Proving Lipid Rafts Exist: Membrane Domains in the Prokaryote Borrelia burgdorferi Have the Same Properties as Eukaryotic Lipid Rafts[END_REF] or multivalent ions [START_REF] Ha | Stabilization and destabilization of cell membranes by multivalent ions[END_REF] . In addition to biological membranes, hole formation have been used to describe the rupture of bilayer (Newtonian) black foam films [START_REF] Derjaguin | On the Theory of the Rupture of Black Films[END_REF][START_REF] Kashchiev | Nucleation mechanism of rupture of newtonian black films. I. Theory[END_REF] , the evaporation of liquids through adsorbed monolayers [START_REF] Dickinson | A hard disk fluid model of monolayer permeation and evaporation resistance[END_REF] , as well as the thermodynamic equilibrium of monolayers on liquid surfaces [START_REF] Stoeckly | Equation of state of fatty-acid monolayers on water[END_REF] . This said, despite an undisputed relevance of liquid mixtures lying on spheres and void formation in them, a systematic investigation through simple models on the role played by miscibility and its coupling to bulk particle adsorption in determining the mixture structure and in triggering void nucleation is still lacking.

In this work we introduce a minimal model to study the structure across a consolute point [START_REF] Wilding | Liquid-vapor phase behavior of a symmetrical binary fluid mixture[END_REF] of a symmetric Lennard-Jones binary mixture confined on a spherical surface by a harmonic potential, leaving particles free to oscillate along the radial direction. Such a quasi-2D model is employed first to characterize the structure of bare liquid mixtures at fixed temperature (here T * = 0.5) and varying miscibility parameter (α AB ) ruling the demixing of the two fluid components (A and B). We compute both the total and the partial geodesic pair correlation functions, and we introduce a geodesic mixing parameter (Ξ) through which we describe both the spontaneous demixing and the effect of adsorbing bulk particle (AP) addition on the mixture structure. We focused on the role of the AP preferential affinity with one of the two mixture components and we give a simple structural criterion to distinguish between AP induced mixing and demixing, showing that the former occurs only for nearly demixed systems with large spatial fluctuations of domain boundaries. For both AP-free mixtures and AP-decorated ones, we further investigate void size distributions. We show that in the presence of spatially fluctuating domains and inter-domain interfaces, large void formation is not favoured, and that AP adsorption can both enhance and disfavour the onset of large voids, depending on the miscibility of the two fluid constituents.

II. MODEL

The binary mixture studied in this paper consists of fluids (A and B) made of spherical particles of the same size, σ AA = σ BB , and at concentrations of 50% each. Particles of the same type interact through a truncated 12-6 Lennard-Jones (LJ) potential [START_REF] Díaz-Herrera | Wetting phenomenon in the liquid-vapor phase coexistence of a partially miscible Lennard-Jones binary mixture[END_REF] :

   U LJ (r i j ) = 4ε i j σ i j r i j 12 - σ i j r i j 6 , if r < 3σ AA 0, if r > 3σ AA (1)
with a mixing rule

σ AB = 1 2 (σ AA + σ BB ), ε AB = α AB ε AA , (2)
where ε AA = ε BB , and α AB is the parameter controlling the miscibility of the two fluids. Notice that when, α AB = 0, we obtain two independent LJ fluids while in the opposite case, α AB = 1, the system reduces to a single LJ fluid. By choosing, 0 < α AB < 1, the attractive part of the interaction between dissimilar particles becomes weaker than that of the AA and BB interactions, favouring demixing [START_REF] Díaz-Herrera | Wetting phenomenon in the liquid-vapor phase coexistence of a partially miscible Lennard-Jones binary mixture[END_REF] . Each particle vibrates around the ideal surface of a sphere of radius R s since they are further confined via a parabolic potential:

U c (r i ) = 1 2 κ c (r i -R s ) 2 , (3)
where

κ c σ 2
AA 2k B T = 10 and r i is the distance between particle i and the center of the confining sphere. To check the influence of the system size on demixing, we simulated three systems characterized by different particle numbers (N p = 250, 500, 1000) and the same area fraction occupied by the particles φ = N p σ 2 AA 16R 2 s = 0.5, corresponding to a reduced density ρ * = 4φ π = 0.63. The choice of the system size has been made to ensure that the particle-to-sphere size ratio σ AA /2R s , ranging here from 0.044 to 0.089, was comparable to the bilayerto-vesicle size ratios encountered in small lipid vesicles [START_REF] Sciolla | Influence of drug/lipid interaction on the entrapment efficiency of isoniazid in liposomes for antitubercular therapy: A multi-faced investigation[END_REF] , the latter being ≈ 0.05. The attractive interaction between the confined particles (≤ 2k B T , see section III) and the energy scale (

κ c σ 2 AA
2 ) of the confining potential have been chosen to reproduce the same order of magnitude respectively of the in-plane lipid-lipid interactions (few k B T units [START_REF] Komura | Lateral phase separation in mixtures of lipids and cholesterol[END_REF] for homogeneous systems and possibly less than k B T for dissimilar lipids 51 ) and the bending energy of membranes per lipid (from few tens to hundreds of k B T per molecule 52 ). The first allows molecules to diffuse and structure itself as a fluid and its variation determines whether demixing occurs at room temperature if more than one lipid type is present. The second gives rise to membrane stability. The magnitude of the confining potential has been further selected so as to preserve equipartition of kinetic energy during all the simulation runs and avoid flying ice-cube artifacts.

Free unconfined APs interacting with the mixture via the potential (1) with LJ diameter σ PP = σ AA have been further added to mixtures composed by N p = 500 particles in order to study the effect of physical adsorption on the structure of the confined AB-system. The interaction between APs and the mixtures is set by equation 1, where miscibility between APs and A-type particles has been fixed by setting their miscibility parameter to α AP = 1, while that between APs and B-type particles has been tuned by varying their mutual miscibility parameter (hereinafter α BP ) in the range 1 -0.025. In this way the interaction between the adsorbing particles and the confined mixtures is such as to be preferential (for α BP < α AP ) or, let's say, symmetrical (for α BP = α AP ) while the total adsorption energy decreases when α BP is lowered. In such a way the simulated systems mimic the weak physical adsorption of small molecules whose adsorption energies do not exceed few k B T units. The surface of the confining sphere has been further made impenetrable to bulk APs via the purely repulsive potential:

U LJ (r PS ) = 4ε PS 2R s + σ PP r PS 12 , (4)
where ε PS /(k B T ) = 2 • 10 -4 and r PS is the center-to-center distance between an AP and the sphere confining the AB mixture.

III. SIMULATION DETAILS

We have carried out extensive MD simulations to investigate the structural properties of this model binary mixture as a function of α AB and α BP . The equations of motion were integrated using a velocity-Verlet algorithm with reduced time step ∆t * = ∆t σ AA ε AA m A = 5 • 10 -6 . m A is the particle mass for species A and m A = m B = m P . We fixed in all simulations the reduced temperature at T * = k B T ε AA = 0.5, where 2D Lennard-Jones fluids are in a condensed liquid state 53,54 . For the chosen set of miscibility parameters (α AB ,α BP ) we equilibrated the systems using the Berendsen's thermostat with a large coupling constant τ T = 2 • 10 3 ∆t to obtain a stable trajectory in equilibrium. Equilibration runs have been performed for 2 • 10 8 time steps and production runs have been carried out in the NVE ensemble for 5.2 •10 7 time steps during which total and kinetic energy drifts were absent. To exclude the presence of important ballistic collective motions of the confined particles due to an eventual rigid rotation of mixtures on the sphere, and energy transfer from high to low frequency modes, we have further performed long NVT simulations for N p = 1000 and N p = 500 at α AB = 1 to compute the mean angular square displacements: a pure diffusive behavior has been observed (see Appendix A). To minimize correlations between measurements we calculated structural quantities every ∆t s = 5 • 10 5 time steps (see Appendix D). Simulations including N AP = 1000 free bulk particles have been carried out only for mixtures with N p = 500. The choice of such an excess of bulk particles has been inspired by the experimental system recently investigated by the author and coworkers [START_REF] Sciolla | Influence of drug/lipid interaction on the entrapment efficiency of isoniazid in liposomes for antitubercular therapy: A multi-faced investigation[END_REF] in which lipid phase separation in mixed liposomes has been observed in excess of Isoniazide, one of the primary drugs used in the tuberculosis treatment. An investigation on the effect of bulk particle concentration, temperature and mixture composition are currently in progress and will be the subject of a future publication. Bare mixtures confined on the sphere have been simulated in a cubic box of size L = 10R s with no extra periodic boundary conditions, with the sphere being at the center of the box. We underline here that the main characteristic of such systems is to be finite but without boundaries, with periodic boundary conditions being not an artifact of the simulations, but an essential feature of it. Hence, in this case there is no need for further assumptions or corrections: the complete system is simulated. Conversely, we carried out simulations including bulk APs in a box of size L = 4.2R s , correspondent to a reduced density for the bulk par-

ticles ρ * = N AP σ 3 CC (L 3 -4/3πR 3
s ) = 0.029, with cubic periodic boundary conditions applied only to free APs.

IV. BARE MIXTURES

The structure of bare mixtures has been studied by inspecting the total g(s) and the partial g m (s) pair correlation func-tions defined as:

g(s) = 1 2N 2 p sin(s) dn(s) ds (5) g m (s) = 1 2 (g AB (s) + g BA (s)) = 1 2 1 2N 2 A sin(s) dn AB (s) ds + 1 2N 2 B sin(s) dn BA (s) ds ( 6
)
where s ∈ [0, π] is the normalized geodesic distance on the unit sphere, dn(s) is the average number of particles at geodesic distance comprised between s and s + ds from one test particle and dn AB (s) (dn BA (s)) is the average number of particles of type B (A) at geodesic distance comprised between s and s + ds from a test particle of type A (B). For homogeneous mixtures (ε AA = ε BB = ε AB ) g m (s) = g(s), while the largest difference between the two functions is attained when the two fluids are fully separated on the sphere in a Janus-like configuration.

Figure 1 By computing both g(s) (Figure 2) and g m (s) (Figure 3) we follow both the structure of the whole fluid and the demixing of the two components on the sphere. Though demixing does not affect remarkably the local structure, we unambiguously observe a weak melting of the mixture induced by the formation of large domains, as signalled by the global minimum of the height of the first peak of g(s) as a function of α AB . The decrease of the first peak reveals that the average local density of the fluid in the immediate vicinity of any particle decreases with respect to the homogeneous system when fluctuating domains coexist (inset of Figure 2), since a complete structural demixing is not achieved, the number of weak AB bonds is still large and particles are on average less bound. On the contrary, when the Janus-like configuration is attained and small spatial fluctuations characterize the interface between the Arich and the B-rich phases, the number of AB contacts decreases and the average particle-particle distance in the fluid goes back to about the one obtained for α AB = 1. The partial correlation functions g m (s) (Figure 3) follow the demixing of the two components: when particles are fully mixed g m (s) = g(s), while when α AB 1 and full demixing occurs, the probability to find an AB contact drastically decreases (small s region) and it increases progressively until it reaches its maximum at angular distance s = π. This is accompanied by the disappearence of the correlation peaks of g m (s) when α AB decreases, as shown in the inset of Figure 3 for representative values of α AB . Since confining spherical surfaces are a closed manifold with uniform curvature, we can describe in a nutshell the demixing by defining a geodesic mixing parameter as the square distance between g(s) and g m (s) on [0, π]:

Ξ = g(s) -g m (s) 2 = π 0 [g(s) -g m (s)] 2 ds. ( 7
)
Ξ is zero when the two components of the mixtures are fully mixed and larger than zero to an extent depending on the structural demixing. In Figure 4 we show Ξ(α AB ) obtained for the three simulated system sizes (N p = 250, 500, 1000). Ξ(α AB ) smoothly passes from very low values compatible with zero (given the simulation noise) for α AB = 1 to larger values, that depends weakly on α AB when the latter is lower than ≈ 0.2 for each system size, pointing out that radius of curvature R s does not impact remarkably the structural demixing transition apart from a very small shift of the critical miscibility param- structure of the mixtures and their stability against void formation we computed the normalized large-void size distribution P(A h /σ 2 AA ) defined as

P(A h /σ 2 AA ) = N(A h /σ 2 AA ) ∞ ∑ h=1 N(A h /σ 2 AA ) , with A h ≥ σ 2 AA 4 ( 8
)
where A h is the area of a void with size hσ 2 AA /4 and N(A h /σ 2 AA ) is the number of voids with normalized size A h /σ 2 AA . We therefore restrict the search of voids to those having area larger than a particle radius square, since we are not interested to the distribution of small interstitial spaces between particles but rather to those voids that are compatible with inclusion of adsorbing particles or molecules in real systems. Indeed the minimal void-to-particle size ratio taken into consideration in this work (A h /σ 2 AA = 1/4) is of the same order of magnitude of that characterizing water molecules and lipid cross section in membrane lipids 55 . To compute each void area A h we applied a spherical grid to the confining sphere, where N g grid points are almost-equidistant and have been generated following the scheme proposed in 56 , with a lattice spacing a σ AA /2. We consider as a void a subset of n g grid points such that: i) each point of the subset is far from any particles by a geodesic distance larger than 3/4σ AA ; ii) a void has n g > 1 if each point of the subset have at least one grid point distant one lattice spacing and belonging to the same subset (cluster of points). The area of the void is then well approximated by A h = n g σ 2 AA /4, with a minimum measurable value equal to σ 2 AA /4. In figure 5 we report the normalized distribution of void sizes for selected values of α AB . The reentrant behavior of the distribution tails goes along with the already mentioned fluidization of the system (Figure 2) when jagged domain boundaries characterize the mixtures. In this regime, since AB contacts are maximized and particles are on average less bound, the formation of large voids is hampered. To describe exhaustively this behavior we have fitted the void distributions with a phenomenological function of the following type:

P(A h /σ 2 AA ) = P 0 + a 1 + b A h σ 2 AA c • e -A h /A c (9)
where A c is the exponential characteristic cutoff area of the distribution, P 0 is an offset set to zero for distribution showing exponential tails and larger than zero when large voids are created by bulk particle adsorption (see section V). Finally a, b and c characterize the power law regime observed for small void sizes. The choice of the fitting function has not been made by chance. Boutreux and De Gennes 57 first postulated exponential decays for void size distributions in granular matter and based on this, Caglioti and coworkers 58 related quantitatively the characteristic dynamical properties for the same systems to quasistatic quantities, e.g. free-volume and configurational entropy. In our mixtures, however thermal (and stress) fluctuations are important and the void formation should also be related to the probability of spontaneous rupture events of a given size. Rupture events often show power-law size distributions [START_REF] Amitrano | Variability in the power-law distributions of rupture events: How and why does b-value change[END_REF] with an exponential cutoff due to the finite size of the system, while shifted gamma functions with exponential tails have proven to be appropriate to describe the statistics of void sizes in packed protein cores and jammed packings of amino-acid-shaped particles [START_REF] Treado | Void distributions reveal structural link between jammed packings and protein cores[END_REF] and attractive emulsion droplets [START_REF] Jorjadze | Attractive emulsion droplets probe the phase diagram of jammed granular matter[END_REF] . These Boltzmann-type distribution tails are those maximizing configurational entropy in packed systems [START_REF] Jorjadze | Attractive emulsion droplets probe the phase diagram of jammed granular matter[END_REF] . Our results are in line with these previous findings and point to void distribution tails with two distinct regimes: a power-law decay followed by an exponential cut-off. Finally, the deviation from a simple power-law at very low void areas (lim A h →0 P(A h /σ 2 AA ) = P 0 + a) reflects the fact that the void size is limited by the interstitial space between LJ sphere, so that the distribution must converge to 0 as A h → 0 [START_REF] Treado | Void distributions reveal structural link between jammed packings and protein cores[END_REF] . Such a deviation is found systematically for all the sets of parameters employed in our simulations, hence the need to use equation 9 to fit satisfactorily all our data. A finite non-zero value for P 0 further points to a non-exponential (fat) tail of the distribution. This will be the case of some of the mixtures in presence of APs. The inset of Figure 5-A shows A c /σ 2 AA together with the probability to have unit voids (A h /σ 2 AA = 1/4). The two quantities show two non-monotonic trends with opposite concavity, corroborating the scenario in which demixing on small confining spheres is structurally a smooth process with the formation of small (large) voids being enhanced (suppressed) in the presence of fluctuating domains. Similarly to what we observed for the cut-off areas, the total (large-)void size

A tot = ∞ ∑ h=1 N(A h /σ 2
AA )shows a minimum for mixtures with large fluctuating domain interfaces (Figure 5-B). This further suggests that the integrity of spherical mixtures is stabilized by compositional fluctuations. All in all, the driving forces dictating large void formation are two: the ability of particles to form long-lived clusters, that is enhanced by increasing the total cohesive energy (here increasing α AB ) and dewetting, occurring for fully demixed fluids (α AB 1) where void formation is favored in the inter-domain regions.

V. MIXTURES WITH ADSORBING PARTICLES

The adsorption of particles on the confined mixtures modifies their structure and, largely affect the mixing-demixing transition. Figures 6, 7 and 8 show selected equilibrium snapshot with N AP = 1000 APs for three representative cases: fully mixed fluids (α AB = 1), fluctuating domains (α AB = 0.6) and fully demixed fluids (α AB = 0.05). and we show also, for comparison, the configurations in the absence of APs. For fully mixed fluids, the addition of bulk particles with decreasing α BP coefficients gives rise to a progressive induced-demixing on the sphere. This is due to the large energy loss attained when bulk particles adsorb only on a condensed domain rich in one of the two species on the sphere that hence tend to segregate. In this case adsorbed particle can be viewed as energetic bridges between particles of type A. In the intermediate regime (Figure 7), where large spatial compositional fluctuations characterize the confined mixture, particle adsorption induces contrasting effects, depending on whether the interaction of APs with the particles belonging to the mixtures is symmetric (α BP = 1) or not. When α BP = 1 large domains break up, pushing the mixture towards mixing. Such a non-trivial behavior is the result of the subtle balance between the configurational entropy of the APs, that adsorb uniformly on the sphere when α BP = 1, lowering the line tension between the domains and causing their dissolution, and the energy gain that takes place when more energetic AA (or BB) bond are replaced by AB bonds. By contrast, when lower values of α BP are used and AP interact preferentially with particle of type A, the mixture adopts a Janus-like (fully separated) configuration, as already discussed for α AB = 1. Finally for low miscibility (Figure 8) particle adsorption does not affect much the mixture structure and only a very weak effect can be observed when α BP is progressively lowered. We quantify all these changes on the structure of the confined mixture by computing the total pair correlation function g(s) and the partial correlation function g m (s) for the AB-mixtures, and the normalized structural mixing parameter

Ξ * = Ξ AP Ξ bare , ( 10
)
where Ξ AP and Ξ bare are calculated according to equation 7 in the presence and absence of adsorbing particles, respectively. The visual inspection of the mixture structure can be substantiated quantitatively in a first instance by comparing g m (s) with and without APs. This is shown in figure 9 for α AB = 1 (panels A,D), α AB = 0.6 (panels B,E) and α AB = 0.3 (panels C,F). We show the two representative cases of fully symmetric interactions between the APs and two components of the mixture (α BP = 1, panels A,B,C), and the most asymmetric case investigated (α BP = 0.025, panels D,E,F). The remarkable effect of APs on the mixtures at α AB = 0.6 catches immediately the eye, with contrasting relative variations passing from symmetric interactions (Figure 9-B) to strongly asymmetric ones (Figure 9-E). This is not observed for the fully miscible case (α AB = 1) or for Janus-like mixtures (α AB = 0.3), for which a decrease in (α BP ) gives rise to a progressive induceddemixing, the extent of the latter being much more pronounced for α AB = 1. In more detail, when large domain fluctuation dominates and AP addition induces particle mixing partial correlation functions show augmented local structure peaks and lower values (closer to one) at large distance with respect to the bare case. This is the case α AB = 0.6 and α BP = 1 (Figure 9-B). On the other hand when AP addition gives rise to induced demixing, g m (s) show reduced correlation peaks at small geodesic distance while it increases at large distance with respect to the bare system. This is the case α AB = 0.6 and α BP = 0.025 (Figure 9-E). For the other values of the miscibility parameter, namely α AB = 1 and α AB = 0.3, the AP addition has a relatively weak effect on the structure of the mixtures when α BP = 1 (Figure 9-A,C), while it produces an important fluid demixing of the initially miscible system (α AB = 1) for strongly asymmetric AP-mixture interactions (α BP = 0.025, Figure 9-D). This, again, is reflected by the net decrease of the structure peaks at small s and a continuous increase of g m (s) at large s. To build up and visualize more clearly a general scenario based on all our results we computed Ξ * (α BP ) for all the investigated systems in the presence of AP. This is shown in figure 10 for α AB = 1, 0.8, 0.5, 0.3, 0.05. We note that: i) for large asymmetries of the AP-mixture interaction (α BP 1) the extent of the induced demixing increases for increasing values of α AB ; ii) the dependence of Ξ * on α BP weakens when AB miscibility decreases, reflecting the fact that the structure is affected by AP addition if particle segregation is poor; iii) mixtures that are characterized by large fluctuating domains and interfaces between A-rich domains and B-rich domains show both induced demixing (Ξ * > 1) and induced mixing (Ξ * < 1) depending on the magnitude of the asymme- tributions for representative systems in the presence of APs and we separate the case α BP = 1 from the rest of the systems since only in the former case we observe the formation of very large deviations from a simple exponential decay of the distribution tails. Figure 11 (panels A,B,C,D,E) shows the distributions P(A h /σ 2 AA ) and five representative configuration snapshots obtained for α BP = 1 and α AB = 1, 0.6, 0.5, 0.3, 0.05. The distributions are further compared to those obtained for bare mixtures. In all the cases, the effect of the AP addition is to enhance the formation of large voids since particle adsorption increases the cohesive energy between the particles confined on the sphere. This results in the suppression of the exponential tails of the void size distributions and the onset of fat tails, that we attribute to the transition from a regime where voids are uniformly distributed on the sphere (no APs) to one where one very large void, whose size fluctuates, dominates. In all such cases we find tails compatible with a powerlaw decay (A c = ∞) with P 0 > 1 for α AB = 1 and α AB = 0.05 (Figure 11-A,E). However, such enhancement of large void formation, though present for each α AB , does depend on the mixture structure. First of all, a reduced distribution widening is observed for intermediate α AB , where large compositional fluctuations characterize the mixtures (see the α AB = 0.6 case in figure 11-B). This is due to the large free energy cost that a large void would produce, since it would suppress domain fluctuations and enhance segregation while, in such a regime, the bare mixture is on average less packed (see Figure 2) and irregular interfaces between domains form. Thus, the appearance of many voids of intermediate size is favoured at the expense of one very large void with fluctuating size. Upon decreasing further the value of α AB large voids develop again and, for Janus-like configurations, are spatially located at the interface between the A-rich domain and the B-rich domain, since the weaker bonds are those between dissimilar particles. Such interfacial void formation is reflected again by the extended (non-exponential) tails of the distributions P(A h /σ 2 AA ). For lower α BP (Figure B,C,D,E) the AP addition affects much less the distributions P(A h /σ 2 AA ), that preserve their exponential tails. Notwithstanding this, it's worth noting that AP addition in mixtures at α AB = 1 and α AB = 0.6 give rise to contrasting effects: for the fully miscible system, an increase of the asymmetry of the AP-mixture interaction (i.e. decreasing α BP ) reduces the void sizes for large enough interaction asymmetry (small α BP ), pushing the distributions towards narrower tails; on the contrary when large irregular domains characterize the bare mixture, AP addition gives rise always to a widening of the distribution tails, reflecting the creation of larger voids. In the first case (α AB = 1, Figure 12-A) the void size reduction is principally due to the fact that, by progressively breaking the interaction symmetry between particles, APs induce demixing and trigger the formation of large fluctuating domains. This suppresses, albeit weakly, large voids akin to the case of the bare mixtures at intermediate α AB . In the second case (α AB = 0.6, Figure 12-B), the enhanced particle adhesion due to the AP presence dominates, favouring large void formation as for the aforementioned α BP = 1 case. For the other cases, namely α AB < 0.6 (Figure D,E), the effect of APs on P(A h /σ 2 AA ), stay very weak but still detectable. The relative variation of the power-law decay exponent ∆c/c bare and the cut-off area ∆A c /A bare c are reported in figure 12 area extracted from the distributions in the absence of APs, respectively. We note that i) the power-law exponent varies almost to the same extent for any values of α AB by decreasing the AP-mixture interaction asymmetry (increasing α BP ), pointing in this case to a net reduction of the small void fraction; ii) the cut-off area variation is maximum for α AB = 0.6 and decreases down to negative values for α AB = 1 and large interaction asymmetry (low α BP ), confirming that the largest relative variation of the cut-off area, when APs interact preferentially with one species of particles, occurs for intermediate values of α AB . This, we stress once more, is due to the fact that bare mixtures with intermediate α AB values are characterized the least number of large voids (small cut-off areas) and the addition of APs with preferential interaction with one of the two particle types (A here) gives rise to spatially localized (large) voids at the interface between the formed domains, since AP addition induces particle demixing as well as an increase of the cohesive energy in the system. The entire set of best-fit parameters for the data shown in figures 11 and 12 are in Table II (appendix C). In line with the large variation of the void distribution observed for miscible systems the local structure of mixtures is most affected by the AP addition for α AB = 1. Figure 13 shows the height of the first peak of the total pair correlation function g(s max ) normalized by its value in the absence of APs g(s max )| bare . For α AB = 1 this quantity changes the most for varying α BP going from values larger than 1 (AP-induced condensation) for α BP = 1, 0.8 to values lower than 1 (AP-induced fluidization) for α BP ≤ 0.5. In the other cases 0.05 ≤ α AB ≤ 0.6 the local structure is less affected by the AP adsorption, though we may note that in this case g(s max )/g(s max )| bare is always larger than one for α BP = 1, for which the minimum value is attained at α AB = 0.6, where the void size distribution resulted less affected by AP addition (Figure 11). Finally, we may note that the normalized peak height shows, albeit very weakly, a systematic non-monotonic behavior, suggesting a weak melting of the structure for intermediate values of α BP . More detailed simulations would be required to reach more quantitative conclusions on this aspect.

VI. CONCLUSIONS

In this paper we have studied the mixing-demixing transition of symmetric mixtures characterized by varying miscibility and confined to move on a sphere by using a minimal quasi-2D model. By computing the total and the partial pair correlation functions, and by employing a geodesic mixing parameter Ξ, we have shown that: i) demixing on 3D-spheres is a structurally smooth process, in which fully mixed and Janus-like configurations are separated by states of spatially fluctuating domains appearing for intermediate miscibility; ii) void size distributions show exponential tails preceded by a power-law decay with large void formation being hampered by large compositional fluctuations, that instead favor the appearance of small (∼one-particle sized) voids. Such a scenario does not vary remarkably in the range of system sizes investigated (R s = 11.18σ AA , 7.90σ AA , 5.59σ AA ). iii) Particle adsorption can give rise to both induced-mixing and induceddemixing, depending on whether mixtures are characterized by strongly fluctuating inter-domain boundaries or not. On the one hand, when coexisting domains and fluctuating interfaces characterize the fluids, uniformly adsorbed particles (α BP = 1) enhance miscibility, while the latter is reduced for a large preferential affinity of APs for one of the two components of the mixtures (α BP 1). On the other hand, for fully miscible (α AB = 1) or Janus-like mixtures (α AB 1) AP addition mainly produces particle demixing, whose extent, quantified by the normalized geodesic parameter Ξ * , increases progressively with increasing miscibility, reaching its apex for α AB = 1 and α BP 1. Void size distributions are also affected by AP addition. A transition from exponentially-tailed to fattailed distributions, where one single large void dominates over the others, is observed for uniform particle adsorption, namely where AP-mixture cohesion energy is maximum. The extent of such a transition also depends on the AB miscibility: fat-tailed distributions are less pronounced by large domain fluctuations occurring for intermediate α AB values, while for large enough AP-mixture interaction asymmetry, we recover exponentially-tailed distributions, with the largest tail widening and narrowing obtained respectively for an intermediate miscibility (α AB =0.6) and fully miscible fluids (α AB =1). Our model, albeit simple, allowed to discern under which mixing conditions a quasi 2D multicomponent fluid confined to move on a sphere is more susceptible to incur structural changes due to the interaction with an external agent. Further studies, based on the same model, will elucidate the role of mixture composition, temperature and AP concentration, while the effect of local curvature can be inspected by changing the confining manifold, passing from spheres to ellipsoids or tori. We hope that our results can pave the way for more targeted experiments to investigate the role of complex molecules and particles on the mixing state of quasi-2D liquids. We obtain the critical miscibility parameters α c AB reported in Table I.

8 FIG. 1 .

 81 FIG. 1. Snapshots of binary 50:50 AB-mixtures (N p = 1000) confined to move on a spherical surface with radius of curvature R s = 11.18σ AA and different miscibility parameters 0.05 ≤ α AB ≤ 1, as indicated in the figure. All snapshots refer to equilibrium states.

gFIG. 2 .

 2 FIG. 2. Total pair correlation functions g(s) for N p = 1000 and selected values of the miscibility parameter 0.2 ≤ α AB ≤ 1. Inset: height of the first peak g(s max ) for the whole set of miscibility parameters investigated in this work.

FIG. 3 .FIG. 4 .

 34 FIG. 3. Partial pair correlation functions g m (s) for N p = 1000 and selected values of the miscibility parameter 0.1 ≤ α AB ≤ 1. Inset: Same data of the main panel for 0 ≤ s ≤ 0.8.

FIG. 5 .

 5 FIG. 5. Panel A: Void size distribution P(A h /σ 2 AA ) for different miscibility parameter α AB as indicated in the panel. The inset shows the cut-of area A c (black sqaures) defined in equation 9 and the fraction the smallest voids P(1/4) (blue circles) as a function of α AB . Panel B: Total void area in function of α AB . The area takes into account only voids with size A h ≥ σ 2 AA /4.

FIG. 6 . 6 FIG. 7 .

 667 FIG.6. Snapshots of binary 50:50 AB-mixtures (N p = 500) confined to move on a spherical surface with radius of curvature R s = 7.90σ AA , with α AB = 1.0 and 3 selected AP-mixture interaction parameter α BP . A snapshot taken in absence of AP is given for reference. All snapshots refer to equilibrium states. By decreasing the affinity between APs and B-particles (snapshots from left to right), particle adsorption favours particle demixing and domain formation on the sphere, until reaching a Janus-like configuration for α BP = 0.025.

FIG. 8 .

 8 FIG.8. Snapshots of binary 50:50 AB-mixtures (N p = 500) confined to move on a spherical surface with radius of curvature R s = 7.90σ AA , with α AB = 0.05 and 3 selected AP-mixture interaction parameter α BP . A snapshot taken in absence of AP is given for reference. All snapshots refer to equilibrium states. In this case particle adsorption does not relevantly impact the initially demixed state of the confined fluid.

FIG. 9 .

 9 FIG. 9. Partial pair correlation functions g m (s) for symmetric AP-mixture interaction parameter α BP =1 (panels A,B,C), large preferential affinity of the APs and A-type particles, α BP =0.025 (panels D,E,F), and three selected miscibility parameters α AB as indicated in the figure. In all the cases g m (s) of the mixtures in absence of APs is also reported for reference. Insets show the same data in the range 0 ≤ s ≤ 1 to zoom on the AP effect on the local structure.

d u c e d d e m i x i n g  A B = 1  A B = 0 . 6  A B = 0 . 5  A B = 0 . 3 FIG. 10 . 2 )FIG. 11 .

 165310211 FIG. 10. Normalized geodesic mixing parameter Ξ * in function of the interaction parameter α BP for the α AB values indicated in the figure. Solid lines are only a guide to the eye.

2 FIG. 12 .

 212 FIG. 12. Panels A,B,C,D,E: Void size distribution P(A h /σ 2 AA ) for varying miscibility parameter α AB and AP-preferential affinity (0.025 ≤ α AB ≤ 0.5) as indicated in the figure. Distributions obtained in absence of APs are also reported for reference. Solid lines are best fits obtained via equation 9 for α BP = 0.025 and for bare mixtures. Bottom panels: relative variation ∆c/c bare and ∆A c /A bare c of the exponent and the cut-off area defined in equation (9)

 A B = 1  A B = 0 . 6  A B = 0 . 5  A B = 0 . 3  A B = 0 . 0 5 FIG. 13 .

 1653513 FIG. 13. Ratio g(s max )/g(s max )| bare in function of α BP for different miscibility parameters α AB as indicated in the figure. Solid lines are a guide to the eye.

3 ∼t 1 / 2 ځ 2 ځ( 2 ڑ p ∼t 1 (π- 8 ) / 2 FIG. 14 .

 3122218214 FIG. 14. Azimuthal and polar mean squared displacement in function of the rescaled time t/R 2 s for N p = 500 and N p = 1000. Dashed horizontal lines indicate the saturation values obtained analytically (Eqs. A3 and A4).

  nomenological function Ξ(α AB ) = α 0 AB + ∆α AB tanh [(ψ(α ABα c AB )] + A(α ABα c AB ) + B(α ABα c AB ) 2 (B1)

TABLE I .

 I Critical miscibility parameter α c AB obtained by fitting Ξ(α AB ) via equation B1.

ACKNOWLEDGMENTS

I wish to thank Dr. Simona Sennato for inspiring discussions. We acknowledge financial support from the Agence Nationale de la Recherche (Grant ANR-20-CE06-0030-01; THELECTRA).

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are all available within the article.

Appendix A: Mean squared angular displacement on the sphere

We have performed long NVT simulations (5 • 10 8 and 10 9 time steps) for the two largest system investigated (N p = 500, 1000) and α AB = 1 to compute the angular mean square displacements (MSD) (θ (t)θ 0 ) 2 2 p and (ϕ(t)ϕ 0 ) 2 p , and to check whether our model produces diffusive trajectories over large runs. Here θ (t) and ϕ(t) represent the polar and the azimuthal angle of each particle confined on the sphere at time t > 0, θ 0 and ϕ 0 are their values at the initial time t = 0, and the brackets • p stand for the average over all particles confined on the sphere. Figure 14 shows in a log-log scale both the polar and the azimuthal squared displacements, for which we observe 2 different saturation values as expected for confined dynamics. Such values can be computed analytically. The angular MSD of one Brownian particle on the sphere saturate to values that are dependent on their initial position (θ 0 , ϕ 0 ) [START_REF] Apaza | Brownian self-driven particles on the surface of a sphere[END_REF] :

where (•) is the average over different trajectories given an initial angular position (θ 0 , ϕ 0 ). For a crowded system particles are distributed over the whole spherical surface and we must compute the saturation values of the MSD as the angular average of equations A1 and A2, i.e.:

The values obtained are in very good agreement with our simulations. We also note that the polar MSD is a linear function of t for small lag times in accordance with recent theoretical works [START_REF] Apaza | Brownian self-driven particles on the surface of a sphere[END_REF][START_REF] Castañeda-Priego | Brownian motion of free particles on curved surfaces[END_REF] , while the azimuthal MSD shows a subdiffusive behavior (ϕϕ 0 ) 2 p ∼ t 0.47±0.04 , that, up to our knoweldge, has not been observed so far. ϕ(t) in a spherical set of coordinates isn't in fact a variable satisfying the Langevin equation and we attribute this feature to the spacedependent azimuthal diffusivity [START_REF] Apaza | Brownian self-driven particles on the surface of a sphere[END_REF] : particles close to the pole θ = 0 move faster in ϕ with respect to particles moving around the equator θ = π/2 for a fixed geodesic displacement. Sub-diffusive motions are often observed in experiments and simulations in systems characterized by space-dependent diffusion constants [START_REF] Cherstvy | Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity[END_REF] . This aspect, interesting per se, goes beyond the scope of this paper and surely deserves a deeper investigation. Finally the displacements obtained for the two systems confined onto spheres of different radius R s superimpose when a rescaled time t/R 2 is considered, in agreement with the general solution of the diffusion equation for spherical Brownian motion. [START_REF] Castañeda-Priego | Brownian motion of free particles on curved surfaces[END_REF][START_REF] Yosida | Brownian Motion on the Surface of the 3-Sphere[END_REF] .

Appendix B: Fit of the geodesic mixing parameter Ξ(α AB )

We extracted the critical mixing parameter α c AB from fits of the geodesic mixing parameter Ξ(α AB ) through the phe-Appendix C: Best-fit parameters for P(A h /σ 2 AA )

We report below (Table II) the best-fit parameters obtained by fitting the void size distributions shown in figure 11 and 12 via equation ( 9).

Appendix D: Sampling time

The sampling time (∆t s = 5 • 10 5 time steps) has been chosen by computing the (spherical) self-intermediate scattering function for the bare mixture with the lowest energy (N p = 1000 and α AB = 1) as defined in [START_REF] Singh | Cooperatively rearranging regions change shape near the mode-coupling crossover for colloidal liquids on a sphere[END_REF] : Where, P kR is the Legendre polynomial with kR being rounded-off to the nearest integer, k -1 is the wavevector corresponding to the geodesic distance equal to one particle diameter and ∆s j (0,t) is the geodesic displacement of particle j over time t. We obtain the scattering function shown in figure 15 that is well fitted by a single exponential decay with a relaxation time of 8.29 • 10 4 time steps that is about 6 time less than the fixed ∆t s .


Paolo Ciambelli

Laurent Arurault
email: arurault@chimie.ups-tlse.fr

Maria Sarno
email: msarno@unisa.it

Sandra Fontorbes

Caterina Leone

Lucien Datas

Diana Sannino

Pascal Lenormand

S Le Blond

Du Plouy

Larurault

Controlled growth of CNT in mesoporous AAO through optimized conditions for membrane preparation and CVD operation

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

Introduction

Carbon nanotubes (CNTs) and nanofibres (CNFs) have drawn more and more attention due to their unique physical characteristics and the subsequent wide range of possible applications (electrical, mechanical, etc). However, for most of these applications, well-controlled characteristics (diameter, length, density, order, degree of purity) of the carbon nanoobjects are required [1][2][3][4][5][6]. One of the usual methods to achieve the tuning and vertical alignment of CNTs is the use of ordered meso-or microstructured anodic aluminium oxide (AAO) as the templating matrix. Previous works mainly used commercial AAO (CAAO) membranes [7][8][9][10][11][12][13][14][15][16] or research AAO (RAAO) templates, usually prepared by anodization [1,4,7] performed in a sulfuric, oxalic or phosphoric acid electrolyte. The RAAO templates were either through-pores membranes or AAO structures supported on a coated Si substrate [17,18]. In most studies the growth of CNTs or CNFs was catalytically promoted by metal particles (usually Co) electrochemically deposited at the pore bottom of the pores [19][20][21][22][23][24][25], while a smaller number of papers [7][8][9][10][11][12][13][14][15][16] reported carbon pyrolysis without a catalyst. In all these previous works the AAO templates were generally considered as inert structures made up of crystallized alumina (Al 2 O 3 ), without any impact on the carbon growth, while different papers [13,[26][27][28][29][30] reported that the pore internal surface of the AAO template can itself play a catalytic role for the decomposition of hydrocarbons. However, they did not identify the surface active sites.

It has been proposed [31] that the Lewis acid surface sites present in amorphous and transition alumina have an intrinsic catalytic activity in the decomposition and recombination of unsaturated carbon.

Recently, Al 2 O 3 and other unusual catalysts [32][33][34] have been found able to grow single-walled carbon nanotubes (SWNTs). However, Haung et al conclude that the growth is due to the catalyst nanosize [32]. Homma et al [33] proposed a new mechanism for the growth of SWNTs (in addition to the vapour-liquid-solid (VLS) growth mechanism for metals), passing through the formation of a carbon film on the nanoparticles, with the generation of a five-membered ring graphene island acting as a nucleus for SWNT growth. However, SWNTs can grow also on nanosized protrusions on the surface of large Al 2 O 3 particles [34]. Therefore, the mechanism of CNT growth in the channels of an AAO membrane, without using a metal catalyst, has not yet been elucidated.

Nevertheless, it is well known that template CNTs not catalytically grown [1, 7-9, 11-13, 18] have open tips, a well-formed central hole and are very pure. However, the tube walls are not so graphitized, requiring thermal annealing at very high temperature [7,35,36] to obtain graphitized carbon. On the other hand, a change in the less expensive operating conditions can affect the tube quality, too. Despite the number of papers present in the literature, only in a few cases is a screening of process parameters [13,37] reported. In particular, as far as we are aware, neither the effect of all growth parameters such as temperature, total feed flow, partial pressure of carbon precursor, reaction time and carbon feedstock, nor an accurate comparison of commercial (CAAO) and synthesized membranes (RAAO) as growth templates have been reported [1, 5, 7-9, 13, 38].

Hornyak et al [37] explored the temperature effect on the graphitic character of CNTs, finding that it improved significantly as CVD temperature was increased from 500 to 800 • C: the samples showed a progressive enhancement of both metallic appearance and layered tube wall structure and decrease of electrical resistance. No significant enhancement of graphitization was observable under microscopic and Raman spectroscopy investigations, among samples formed at 800, 900 and 1000 • C, even if a further decrease in the electrical resistivity was observed under this temperature increase.

The aim of the present study was to prepare ordered CNTs in CAAO and RAAO mesoporous membranes, by controlling the synthesis parameters in order to accurately tune carbon nanotube characteristics in terms of diameter, length and arrangement (i.e. number of tubes for each channel). The CNTs were produced by chemical vapour deposition (CVD) in the absence of a catalyst and in different experimental conditions that were correlated with the quality of the carbon materials obtained. A wide range of microscopic observations (FEG-SEM, SEM, TEM) and physico-chemical characterizations (IRS, XRD, EPMA, TGA-DTGA) were performed on AAO templates, AAO/CNT composites and CNTs after membrane dissolution, in order to especially highlight the chemical composition of the AAO template and the CNT characteristics under the different growth conditions.

Experimental details

Preparation process of the AAO membranes

The preparation process of the anodic aluminium oxide (AAO) porous templates was extensively described elsewhere [39,40], but these previous works were mainly focused on the preparation of porous AAO ordered films still including the compact layer and sometimes supported on the metallic substrate, while the present study deals with the AAO membranes, i.e. through-pores templates.

The aluminium substrate (Ø sample = 14 mm) was prepared first by sanding and then by annealing under nitrogen atmosphere at 450 • C for 2 h. Additional electropolishing was carried out at 25 V for 2 min in a Jacquet mixed solution.

Anodization was carried out without delay at 185 V for 4 h. The electrolyte was made up of a vigorously mixed aqueous phosphoric acid solution (8 wt%), while a pure aluminium plate was used as the cathode. The temperature was regulated at -1.5 • C by a cryostat (Huber CC2), and the voltage was applied by an Invensys Lambda generator (300 V, 5 A).

Specifically in the present study, after growth of the porous anodic film, an additional removal of the compact layer was performed using 30 V AC voltage (Universal Power Supply EA-4036) in a phosphoric acid solution (16 wt%) and the aluminium substrate was chemically dissolved with a hydrochloric acid solution (18 wt%) including copper chloride CuCl (0.1 mol l -1 ). The membrane was dipped for 5 min in a phosphoric acid solution (8 wt%) and finally rinsed with deionized water.

All chemical compounds used for all steps of this preparation process were analytical grade and aqueous electrolyte solutions were obtained using deionized water. The prepared research membrane was called RAAO in the following.

As a comparison, commercial membranes (Anodisc; sample diameter:13 mm; pore diameter: 0.2 mm) manufactured by Whatmann were also studied (CAAO).

Microscopic and chemical analysis of the AAO membranes

A field emission gun scanning electron microscope (FEG-SEM JEOL JSM 6700F) was used to determine the morphology of the pores, i.e. the pore density (ρ), the pore diameter (d p ), the thickness of the porous layer (e p ), the percentage of the void volume (τ ) and the interdistance between the edge of two adjacent pores. X-ray diffraction (XRD) analysis was performed with a Bruker AXS D4 Endeavor generator with copper anticathode (K α = 1.5418 Å). All diffraction profiles were obtained by varying 2θ from 10 • to 100 • . Finally, peaks were identified by EVA software.

The coating dehydration and crystallization processes were studied by thermogravimetric analysis (TGA) in nitrogen 4.5 from 25 to 900 • C at a heating rate of 10 • C min -1 , by using a SETARAM instrument (Model TAG 92 equipped with a platinum crucible).

The infrared spectroscopy (IRS) was performed using an IR-TF Nicolet 510P instrument, the spectra being recorded in transmission conditions between 400 and 4000 cm -1 . The anodic films were preliminarily reduced to powder and prepared in KBr pellets.

The quantitative elemental chemical analysis was carried out using an Electron Probe MicroAnalyser (EPMA-CAMECA SX50). Eight measurements were made on the whole sample surface.

Process of carbon nanotube growth

Carbon nanotubes were synthesized by chemical vapour deposition (CVD) of ethylene within the pores of both RAAO and CAAO membranes. Hydrocarbon pyrolysis was carried out in a laboratory apparatus consisting of a continuous flow microreactor, fed by an ethylene-nitrogen gas mixture [11,12]. For each gas a mass flow controller (MFC) was used in order to ensure a constant and controlled flow rate.

A set of experiments have been conducted on commercial membranes to find the optimal synthesis conditions by varying (table 1): temperature, ethylene feed concentration, reaction times, with respect to the quality of the materials obtained. The syntheses have been performed in isothermal conditions after a pretreatment of the membrane under nitrogen flow, from room temperature up to the synthesis temperature.

The recovery of carbon material was performed by treating a fraction of carbon-loaded membrane samples for 24 h with 20 ml of 50 wt% HF aqueous solution to dissolve the membrane; the solid residue was washed in distilled water, centrifuged and finally dried at 353 K for 12 h.

Characterization of AAO membrane-carbon composite and of recovered carbon nanotubes

The samples after synthesis and the recovered carbon nanotubes were characterized using various techniques. Scanning electron microscopy (SEM) pictures were obtained with a LEO 1525.

Transmission electron microscopy (TEM) images were obtained with a JEOL 1200 EX2 microscope. The accelerating voltage of the electron beam was 200 kV. The samples were mounted on a Quantifoil ® microgrid carbon polymer supported on a copper grid by dropping the sample suspension, obtained after a few minutes of sonication, onto the grid. The microscope was equipped with an energy-dispersive x-ray spectroscopy (EDX) probe (probe size 1.5 nm).

Simultaneous thermogravimetric and differential thermal analysis (TG-DTG) was performed with a Thermogravimetric Analyser SDTQ 500 (TA Instruments). The measurements were performed in the range 25-900 • C, with a 10 • C min -1 heating rate under flowing air.

Raman spectra were obtained at room temperature with a micro-Raman spectrometer Renishaw inVia (514 nm laser excitation wavelength).

Results and discussion

The AAO membranes were characterized, before the CNT growth, in order to know their microstructure and chemical composition just at the end of the preparation process but also after a thermal treatment.

The results were systematically compared with those obtained using the Whatmann commercial membranes.

Microstructural characterizations of the AAO membranes

3.1.1. Commercial membranes. Whatmann commercial membranes have often been used for the growth of carbon nanotubes [7,[9][10][11], but surprisingly they remain poorly studied. They were usually considered as through-pores membranes, showing two identical faces and made up of inert and crystallized alumina Al 2 O 3 .

Yet FEG-SEM cross-sectional views (figures 1(a) and (b)) showed that pores are indeed opened on both sides but that the two opposite porosities are significantly different. In particular, one side, rarely shown, is quite complex and Moreover, FEG-SEM cross-sectional views showed a low quality of the cutting plane (figure 1(c)), confirming that the porosity is not well ordered [41]. To sum up, the main geometrical characteristics of the Whatman commercial membranes (Anodisc 13 mm/0.2μm) are:

Pore mean diameter at side A: (273±48) nm (void fraction 30%).

Pore density at side A: 9 × 10 12 pores m -2 . Membrane thickness ≈60 μm.

Research membranes without thermal treatment.

A specificity of the present study is the experimental removal of the compact layer of the supported AAO templates previously prepared [39,40]. FEG-SEM plan views (figures 2(a) and (b)) revealed that this layer is progressively destroyed with increasing duration of the AC treatment. This removal is attributed to the gas evolution, especially during the cathodic part of the alternating waveform of the electrical signal.

At the end of the preparation process, FEG-SEM plan views (figures 3(a) and (b)) and cross-sectional view (figure 3(c)) of the AAO membranes (before thermal treatment) showed a well-defined microstructure. The pores are parallel, straight (figure 3(c)) and perpendicular to the initial metallic surface. In particular, the cutting plan is well-defined, proving the high quality of the porosity shape.

Furthermore, the pores are ordered, well formed and opened on both sides (figures 3(a) and (b)). However, there are differences between these two sides: the pore walls of side B are thinner but only 2-3 μm in depth. So the porosity of side A is the most representative of the whole membrane. The main characteristics of our RAAO membranes are:

Pore mean diameter at side A: (254±21) nm (void fraction 25%).

Pore mean diameter at side B: (285±25) nm (void fraction 48%).

Pore density at side A: 6.6 × 10 12 pores m -2 . Pore density at side B: 7.3 × 10 12 pores m -2 . Membrane thickness ≈60 μm.

Interdistance between the edges of two adjacent pores ≈200 nm. It is important to note that the pore density is similar on both faces, despite significant differences concerning the void fractions, confirming that the peculiarity of side B is only located at the extreme surface. Furthermore, the standard deviations of pore diameter are here low (<9%) for both faces in comparison with the value (about 18%) corresponding to side A of the commercial membranes.

3.1.3.

Research membranes after thermal treatment. FEG-SEM plan views showed that the microstructure of the membrane remains unchanged despite a thermal treatment performed at 900 • C. On the other hand, a visual observation highlighted a bending phenomenon, i.e. that the membrane is curved after the thermal treatment. This result is in agreement with the works of Fernandez-Romero et al [42] using throughpores membranes prepared in an oxalic acid medium. The bending phenomenon appears significant in these two studies because there is no stabilizing effect of the barrier layer [42].

Chemical characterizations of the AAO membranes

Chemical composition.

The global elementary composition of the AAO membranes was determined by EPMA (table 2).

Results showed no differences in the oxygen and aluminium atomic contents, while phosphorus content significantly changed. Phosphorus was included into the anodic films from the anodization electrolyte, using phosphoric acid. The difference of the phosphorus contents between the two types of RAAO membranes can be explained by the preferential acidic dissolution of the pore walls-including specifically phosphorus [40] during the additional removal of the compact layer. The explanation was corroborated by the comparison of the pore average diameter: (180 ± 30) nm and (254 ± 21) nm for the RAAO membranes, with and without compact layer, respectively.

Figure 4 shows the IR transmission spectra of typical RAAO and CAAO membranes, both types of samples being analysed in identical conditions. Both patterns had a similar shape, highlighting the O-H bond, Al=O double bond and P-O bond. However, the peaks corresponding to the RAAO membrane were undoubtedly lower than those of the CAAO sample, and lower than those of the AAO templates still including the compact layer [40].

To sum up, RAAO and CAAO membranes were chemically similar: before heating treatment both included hydrated (oxi)-hydroxide or oxide of aluminium, as well as phosphate coming from the anodization bath. But RAAO templates clearly showed a lower content of hydroxyl and phosphorus, due to the chemical dissolution of the pore walls during the removal of the compact layer.

3.2.2.

Dehydration. Thermogravimetric analyses were performed in the 25-900 • C temperature range (figure 5) in order to study the dehydration mechanisms of RAAO and CAAO membranes.

In accordance with previous works [40,43], the sequence of weight changes was explained by the loss of physisorbed water (from 25 to 100 • C), and then of chemisorbed water in the 100-800 • C range. Finally, the total weight losses at 900 • C were 0.59% and 1.29% for RAAO and CAAO samples, respectively. So the weight losses were small, especially for the RAAO membrane, indicating low water content.

Crystalline phase changes.

Membranes were then studied by XRD as a function of temperature (25-900 • C) in order firstly to study the crystalline phase changes and secondly to contribute to understanding the subsequent formation of carbon nanotubes.

XRD spectra obtained just at the end of the preparation process showed only broad peaks (figure 6) for both research membranes and commercial ones. Therefore, both types of membranes were not made up of well-crystallized compounds but were in fact amorphous.

A thermal treatment was then performed at 900 • C in a nitrogen atmosphere to simulate the temperature used in the subsequent process of carbon growth. XRD spectra revealed (figure 6) the simultaneous presence of γ , δ and θ alumina crystalline phases, but without corundum (α-alumina).

These results are in agreement with previous works [40,43], suggesting, in the 25-400 • C temperature range, dehydration mechanisms of physisorbed and chemisorbed water in the initial membrane, made up of aluminium hydrated (oxi)hydroxide and then, at higher temperatures, the following successive crystallization phenomena:

2AlOOH → γ -Al 2 O 3 + H 2 O ( 1 ) γ -Al 2 O 3 → δ-Al 2 O 3 (2) δ-Al 2 O 3 → θ -Al 2 O 3 . ( 3
)
However, the present study showed that θ -alumina was surprisingly formed at temperature below 900 • C. Furthermore, comparison between the two types of membranes (RAAO and CAAO) clearly showed that the θ -alumina content is significantly higher in commercial membranes. This result can be perhaps explained by the difference in the respective phosphorus contents, i.e. (0.7 ± 0.1) at.% and (1.5 ± 0.1) at.%, respectively (table 2).

To sum up, our results showed that at 900 • C, i.e. the average temperature used in the subsequent process of carbon growth, AAO membranes were made up of crystallized allotropic alumina phases. Their respective contents depend on the previous anodization operational conditions and govern their respective catalytic activity, especially for the hydrocarbon decomposition and CNTs growth, due to their respective Lewis acid nature [31].

Characterizations of the CNTs/membrane composites

Thermogravimetric analysis (TG-DTG).

To compare the effect of the operating conditions on the carbon structure order, thermogravimetric analyses [11][12][13] were performed on the samples reported in table 1. In fact, a different thermal behaviour, as shown by TG-DTG analysis, could contribute to characterize the carbon material present inside the pores with respect to its nature and structural order. The absence of catalyst particles, which in general could affect the carbon oxidation reaction, supports the validity of the result interpretation. To further validate such results, also considering the composite nature of samples, a preliminary test on different and very small sample weights (3-7 mg) was carried out to verify that the thermogravimetric profile was not affected by the weight of the sample (see figure 7 for M81). A final validation was carried out by first treating the composite sample by a thermal oxidation treatment up to an intermediate temperature value, reproducing the conditions of thermogravimetric analysis (see figure 8(b)) and then carrying out a thermogravimetric test on the residual mass. We have found the same temperature range of oxidation, indicating the same order degree for carbon (figure 8(c)).

In figure 9, the thermogravimetric profiles of M30 and M31 (obtained at the same operating conditions but except for reaction time) are shown. The comparison of TG-DTG profiles shows an higher carbon content for M30 as a result of the longer synthesis time. Moreover, for M30 an upshift of both endset and DTG peak temperature, indicating increased carbon order, is observed. For both samples the residual mass over 700 • C is that of the inorganic fraction, i.e. the AAO membrane.

For M31 and M32, prepared at the same operating conditions, except for ethylene concentration in the feed flow (table 1), TG-DTG analysis (figure 9(b)) indicates that a decrease in the ethylene concentration leads to the formation of a more ordered carbon structure. In particular, TG profiles exhibit one peak of weight loss; a comparison of the two sample TG-DTG profiles shows that the oxidation results upshifted by about 50 • C for M32.

M30 and M60 differ only for the synthesis temperature, 100 • C higher in the case of M60. From the comparison of the onset and DTG peak temperature of the relevant profiles reported in figure 9(c) it can be concluded that the higher temperature synthesis results in (i) a more ordered carbon structure and (ii) a larger carbon content, likely due to a higher ethylene conversion, as the effect of temperature.

The effect of the reactor inlet total flow is shown in (figure 9(d)). The thermogravimetric profiles, relevant to M60 and M61, obtained at the same operating conditions except for total flow rate (the volume concentration of ethylene change remaining constant for the rate of ethylene fed), indicate that at an increased flow rate a more ordered carbon structure is achieved. Figure 10 shows that in the range 25-800 • C the TG-DTG profiles of M29 (CAAO membrane) and M30 (AAO membrane) are practically superimposed. A single step weight loss, due to the combustion of the carbon material, is evident for both samples. In figure 11(a), the TG-DTG profile relevant to M29 after HF treatment to remove the membrane is shown: the residual mass due to alumina is quite close to zero.

Finally, to explore the catalytic effect of the membrane [31], giving our contribution to better understand its action, a test was performed in which the membrane together with an equal mass of fine quartz were loaded in the reactor in the same isothermal zone. After 30 min of the experiment, M29 and the quartz covered by carbon were collected from the reactor and subjected to two thermogravimetrical analyses for comparison (see figure 11 seems to be quite close, the DTG peak and the endset of oxidation of M29 result in an upshift of about 40 • C, indicating the presence of different carbon contributions, some of which, in the case of M29, are more ordered. Therefore, the formation of our carbon nanotubes is due to the combined effect of the catalytic behaviour of the internal alumina channels surface and their template effect, determining the formation of a carbon film that does not nucleate SWNTs [32][33][34] from the channel walls because of their smoothness.

Raman spectroscopy analysis.

To confirm the results obtained with the TG-DTG evaluation the samples have been investigated by Raman spectroscopy analysis.

Raman spectra [13,17], collected with a 514 nm laser wavelength, show the typical D and G peaks of the carbon materials, centred at about 1337 and 1599 cm -1 , respectively (figure 12(a)). The G peaks indicate the presence of crystalline graphite carbon, while the D peak can be attributed to amorphous carbon, defects in the curved graphene sheet and finite size of the tube crystalline domains. The fact that the two peaks have nearly identical magnitude indicates the presence of nanocrystalline graphite with domain size less than 10 nm [43]. Raman spectroscopy results are in accordance with those from TG-DTG analysis, confirming that carbon order increases by increasing synthesis time, total feed flow, temperature and decreasing ethylene concentration. The dissolution of the membrane results in bundles of parallel tubes, opened and aligned (due to van der Waals interaction among the bundle), without microscopic defects. The bundles of aligned CNT are as long as the thickness of the original membrane (sample M29 in figure 14). There is no significant evidence of any amorphous carbon (figure 15). The images in figure 15(c) are the SAD patterns taken in the selected area indicated by a circle in figure 15(b). In the image a pair of arcs for 002, together with 10 and 11, diffraction rings is hardly visible. The appearance of the 002 diffraction neither as a ring nor as clear spots, but as a pair of arcs, indicates some orientation of the 002 planes in the carbon tubes and its poor crystallinity. For this carbon tube, the SAD patterns, taken from areas that do not contain the edge of the tubes, do not show the 002 arc, since in this area no beam passed through the tube wall in parallel as in area c. This finding suggests that the tube wall consists of cylindrically stacked 002 planes [7]. Higher magnification TEM characterization permits us to observe the carbon arrangement in the tube thickness: twisted walls are observed in the direction of the nanotube axis (figure 15(d)).

SEM and TEM microscopy.

With respect to the shape and morphology of the nanotubes, they reflect the shape and morphology of the membrane, resulting in more ordered bundles of straight nanotubes with uniform size in the channels of the RAAO membrane (compare figures 16(a1) and (a2)).

Higher magnifications (figures 16(b1) and (b2)) indicate that the twisted walls in the direction of the nanotubes axis are more ordered (more straight and with less interruptions) if compared with those of figure 15(d). The image in figure 16(c) is the SAD pattern taken in the selected area indicated by the circle in figure 16(b2), it exhibits a pair of smaller and stronger arcs for 002, together with 10 and 11, diffraction rings, indicating a better degree of order for this nanotube, in agreement with thermogravimetric and Raman characterizations. The purity of the produced CNTs after HF alumina removal is shown by the EDX analysis, performed on a large area, showing the presence of the peaks due to the carbon nanotubes and the TEM grid (figure 17).

Conclusions

This study showed that both RAAO and CAAO initially include hydrated (oxi)-hydroxide or oxide of aluminium, as well as phosphate coming from the anodization bath. However, RAAO templates include lower contents of water and phosphorus and show a better porosity shape.

After thermal treatment, RAAO membranes were made up of crystallized allotropic alumina phases, which govern the subsequent CNT growth, because of their catalysis activity, likely due to their respective Lewis acid natures.

CNTs were then grown by chemical vapour deposition (CVD), in the absence of metallic catalyst, in the pores of both CAAO and RAAO membranes. A composite material, containing one nanotube for each channel, having the same length as the membrane thickness and an external diameter close to the diameter of the membrane holes, was obtained. SEM and TEM images of the samples after HF removal of alumina show the formation of bundles of parallel tubes, opened and aligned without macroscopic defects. The shape and morphology of the nanotubes reflect the shape and morphology of the membrane; more ordered bundles of straight nanotubes with uniform size in the channels were obtained for RAAO membranes.

The carbon structure order depends on the CVD experimental conditions: it is increased by increasing synthesis time, total feed flow rate and reaction temperature, and decreasing ethylene concentration.

Figure 1 .

 1 Figure 1. FEG-SEM views of the CAAO membrane: (a) side A, (b) side B and (c) cross section.

Figure 2 .

 2 Figure 2. FEG-SEM cross-sectional views of the RAAO membrane at different durations of the removal of the compact layer: (a) 30 min and (b) 40 min.

Figure 3 .

 3 Figure 3. FEG-SEM views of RAAO membrane: (a) side A, (b) side B and (c) cross-section.

Figure 5 .

 5 Figure 5. Thermogravimetric analyses of CAAO and RAAO membranes.

Figure 6 .

 6 Figure 6. XRD spectra of CAAO and RAAO membranes before and after thermal treatment at 900 • C.

Figure 7 .

 7 Figure 7. Thermogravimetrical analysis of M81 performed on two different mass weights for comparison.

Figure 8 .

 8 Figure 8. TG-DTG analysis of M78 (a), of M78 up to 745 • C (b) and on the M78 residue (c).

Figure 9 .

 9 Figure 9. TG-DTG analysis to compare the effects: of time (a); of ethylene vol.% (b); of temperature (c) and of total feed flow (d) on the quality of the produced materials.

Figure 10 .

 10 Figure 10. TG-DTG analysis to compare the syntheses in RAAO and CAAO membranes.

Figure 11 .

 11 Figure 11. TG-DTG analysis on the CNT obtained after HF purification of M29 (a). TG-DTG of M29 and of the fine quartz, recovered from the reactor together with M29 (b).

1 aFigure 12 .

 112 Figure 12. Raman spectra collected with 514 nm laser wavelength (a) and Lorentzian deconvolution of M31 and M61 Raman spectra (b).

Figure 13 .

 13 Figure 13. SEM images of the top surface and edge of M34, showing CNTs emerging from the RAAO channels (a). M29 CNTs from the alumina channels (b).

  Figure 13(a) shows the top surface and edge of composite M34: CNTs can be seen to emerge from pores on the membrane edge. Figure 13(b) shows the CNTs emerging from the AAO membrane after 30 min of synthesis at 850 • C (sample M29). The images clearly show the growth of carbon tubes, one tube for each channel. The external diameter of the tubes is uniform and close to the membrane pore diameter.

Figure 14 .

 14 Figure 14. SEM images of M29 CNT bundles, after HF dissolution of the membrane, at different magnifications.

Figure 15 .

 15 Figure 15. CNT bundle grown in the channels of RAAO membrane, after dissolution of the membrane in M29, TEM images at different magnifications (a), (b), (d); SAD pattern (c).

Figure 16 .

 16 Figure16. TEM-HRTEM images of CNTs produced in the channels: of CAAO membrane (M81) at lower (a1) and higher magnification (b1); of RAAO membrane (M144) (a2) and at higher magnification (b2). SAD pattern taken in area c of image b2 (c).

Figure 17 .

 17 Figure 17. TEM images of CNTs produced in the channels of M29 (the black area are due to the TEM grid). EDX spectrum taken in the image area.

Table 1 .

 1 CNT operational conditions.

	Sample	Reaction temperature ( • C) Reaction time (min) Total feed flow (cm 3 (stp) min	-1 ) C 2 H 4 (vol.%) Membrane type
	M29	850	30	100	10	RAAO
	M30	850	30	100	10	CAAO
	M31	850	10	100	10	CAAO
	M34	850	10	100	10	RAAO
	M32	850	10	100	5	CAAO
	M60	950	30	100	10	CAAO
	M61	950	30	200	5	CAAO
	M78	970	30	200	3	CAAO
	M81	950	30	200	3	CAAO
	M144	950	30	200	3	RAAO

Table 2 .

 2 Global elementary composition of the AAO membranes, as obtained by EPMA.

Content Oxygen (at.%) Aluminium (at.%) Phosphorus (at.%) RAAO membrane (this study) 63.2 ± 1.4 3 6 .1 ± 1.4 0 .7 ± 0.1 CAAO membrane (200 nm) 60.8 ± 2.6 3 7 .7 ± 2.5 1 .5 ± 0.1 RAAO template with compact layer [39] 61.1 ± 1.4 3 6 .3 ± 0.5 2 .6 ± 0.2 Figure 4. IR spectra of the CAAO and RAAO membranes prepared in KBr pellets.

Table 3 .

 3 Raman characterization results. Sample I D /I G A D /A G

	M29	0.76
	M30	0.76
	M31	0.78	2.5
	M34	0.77
	M60	0.73
	M61	0.72	2.4

Acknowledgment

This work has been performed within the European research project CATHERINE-Carbon nAnotube Technology for High-speed nExt-geneRation nano-InterconNEcts-under grant agreement no. 216215.


Florence Leclercq

Laura Lorca

Audrey Agullo

Karim Bouchdoug

Jean-Christophe Macia

Delphine Delseny

François Roubille

Thomas Gandet

Benoit Lattuca

Pierre Robert

Evolution of right ventricular dysfunction and tricuspid regurgitation after TAVI: A prospective study

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