Francesco Dell'isola
email: francesco.dellisola@uniroma1.it

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

Giacomo Rotoli
email: rotoli@ing.univaq.it

Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations

Keywords: Continuum mechanics, Gas liquid interface, Particle size, Bubbles, Surface tension, Equilibrium, Theoretical study, PACS: 47.55.db, 64.70.Fx, 68.03.Cd, 68.03.-g

The theory of second gradient fluids (which are able to exert shear stresses also in equilibrium conditions) allows us: (i) to describe both the thermodynamical and the mechanical behavior of systems in which an interface is present; (ii) to express the surface tension and the radius of microscopic bubbles in terms of a functional of the chemical potential; (iii) to predict the existence of a (minimal) nucleation radius for bubbles. Moreover, the above theory supplies a 3D-continuum model which is endowed with sufficient structure to allow, using the procedure developed by Dell'Isola & Kosiński, the construction of a 2D-shell-like continuum representing a consistent approximate 2D-model for the interface between phases. In this paper we use numerical simulations in the framework of second gradient theory to obtain explicit relationships for the surface quantities typical of 2D-models. In particular, for some of the most general two-parameter equations of state, it is possible to obtain the curves describing the relationship between the surface tension, the thickness, the surface mass density and the radius of the spherical interfaces between fluid phases of the same substance. These results allow us to predict the (minimal) nucleation radii for this class of equations of state.

Introduction

In 1949 Tolman, using the thermodynamical approach due to Gibbs, established a relationship between surface tension and radius of bubbles in equilibrium with their liquid phase [START_REF] Tolman | The effects of droplet size on surface tension[END_REF]. It fails when the bubble has a radius close to the nucleation value [START_REF] Lamer | Surface tension of small droplets from Volmer and Flood's nucleation data[END_REF]: indeed by following the ideas of Gibbs [START_REF] Gibbs | Collected works[END_REF] it is impossible to prove the existence of a 'minimal' nucleation radius, i.e. the minimal radius of a bubble or droplet necessary for nucleation in the other phase. Cahn and Hilliard [START_REF] Cahn | Free energy of a non-uniform system I[END_REF] proposed more sophisticated models in which the nucleation phenomena are accounted for. They introduce a 3D-description of the interfacial region, which is characterized as the region in which high mass density gradients occur, so that a supplementary term is added to the classical free energy expression. This term is in general non-local, i.e. it depends on all gradients of mass density [START_REF] Gouin | Equation du mouvement des fluides parfaits de grade n exprimée sous une forme thermodynamique[END_REF]. It can be shown however that in many cases the first gradient 1 of mass density is sufficient to give an excellent approximation to the exact solution [START_REF] Falls | Structure and stress in spherical microstructures[END_REF]. This approach was developed in the literature mainly using the methods of statistical mechanics [START_REF] Davis | Stress and Structure in fluid interfaces[END_REF][START_REF] De Gennes | Some effects of long range forces on interfacial phenomena[END_REF][START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF][START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF] but also in the framework of classical field theories [START_REF] Casal | Relation entre l'équation de l'énergie et l'équation du mouvement en théorie de Korteweg de la capillarité[END_REF][START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus -Théorie du second gradient[END_REF][START_REF] Serrin | New perspectives in thermodynamics[END_REF] by means of the so-called second gradient theory. The second gradient fluids (or capillary fluids as named by Casal & Gouin [START_REF] Casal | Relation entre l'équation de l'énergie et l'équation du mouvement en théorie de Korteweg de la capillarité[END_REF]) are fluids which are able to exert shear stresses in equilibrium conditions and contact forces concentrated on lines [START_REF] Seppecher | Etude d'une modélisation des zones capillaires fluides; interfaces et ligne de contact[END_REF]. The theory also takes into account the mechanical aspects of the phenomena considered in the theories of Laplace, Gibbs and Tolman. Indeed it is easy to prove, for second gradient fluids, that the pressure field in equilibrium conditions may be non-constant. On the contrary it is an essential feature of the classical Laplace theory that the pressure field is spatially uniform. Moreover in the framework of second gradient theory it is possible to deduce the PDE governing the (equilibrium) mass density distribution from the force balance law. Dell'Isola et al. [START_REF] Dell'isola | Radius and surface tension of microscopic bubbles by second gradient theory[END_REF] show the consequences of the distinction (first noticed by Rocard [START_REF] Rocard | Thermodynamique[END_REF]), between thermodynamic pressure (that deriving from free energy) and the mechanical pressure (i.e. the trace of the capillary stress tensor). Indeed in the quoted papers it is shown that a definition of the surface tension and of the radius of a bubble is possible also in the 3D-approach by using the second gradient theory and by means of the concept of the 'equivalent Laplace bubble'. In this way it is possible to obtain theoretical predictions for minimal nucleation radii and dependence between the surface tension and the bubble radius; moreover it is possible to use the 'equivalent Laplace bubble' to study the stability of the critical nuclei [START_REF] Gouin | Stability of spherical isothermal liquid-vapor interfaces[END_REF]. Unfortunately in the case of spherical bubbles the ODE determining mass density profile cannot be easily studied, so that it is necessary to use numerical simulations in order to be able to make a theoretical prediction. We explicitly remark that in the present paper a 'shelllike' model for the interface is used, following the ideas developed in [START_REF] Dell'isola | Deduction of thermodynamics balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF]. In this approach there is no use of 'Gibbs excesses' quantities (for a detailed treatment of these concepts see for instance [START_REF] Chattoraj | Adsorption and the Gibbs surface excess[END_REF]), so our paper substantially differs from [START_REF] Falls | Structure and stress in spherical microstructures[END_REF] in which an analogue analysis for drops is developed. The main result of this paper is a new relation between surface tension and the radius of bubbles which holds also for microscopic radii and which allows the determination of the minimal nucleation radius. The paper is organized as follows: in Section 2 we collect the main results due to Gibbs-Tolman, Korteweg, Dunn-Serrin, Gurtin and Cahn-Hilliard and then we compare the results with experiments; in Section 3 we state the theoretical framework of second gradient theory and its application to bubbles; in Section 4 we study the equation for mass density profiles for some of the most general two-parameter equations of state and solve it numerically; in Section 5 comparisons with experiments, together with an explicit prediction about 'minimal nucleation radii' of some substances, are carried out; finally some conclusions are collected in Section 6.

2 Model of interface between fluid phases: the problem of nucleation of microscopic bubbles

Nucleation of microscopic bubbles

In [START_REF] Dell'isola | On the lack of structure of 2D Defay-Prigogine continua[END_REF][START_REF] Dell'isola | On the generalization of Tolman formula[END_REF] it is shown that Tolman formula [START_REF] Tolman | The effects of droplet size on surface tension[END_REF] is an universal property of 2D-Gibbs models for the interface between phases. In fact the Tolman formula is a consequence of the Laplace formula and some simple assumptions (whose origin will be found in the works of Gibbs [START_REF] Gibbs | Collected works[END_REF]) about the thermodynamics of 2D-continua. These assumptions are basically: (i) the possibility of defining surface quantities such as surface mass density, surface free energy, etc.; (ii) the validity of standard thermodynamical relationship between these quantities. The Tolman formula predicts that for microscopic bubbles or droplets, the equilibrium surface tension will decrease exponentially as a function of the equilibrium radius. However very soon after its publication in 1949 Tolman's formula was criticized by Lamer and Pound [START_REF] Lamer | Surface tension of small droplets from Volmer and Flood's nucleation data[END_REF] who, founding their analysis on nucleation data for very small droplets, found no significant decrease of surface tension with the radius. The experimental failure of Tolman theory has substantially remained unjustified theoretically until now [START_REF] Fisher | Determination of the capillary pressure in menisci of molecular dimensions[END_REF][START_REF] Kumar | Curvature dependence of surface free energy and nucleation kinetics of CCl 4 and C 2 H 2 Cl 4 vapours[END_REF]. In our opinion two facts are relevant to this problem: (i) the failure of any 'classical' theory is a consequence of idealizing the interface as a region without thickness; surface quantities would be correctly defined stating a 3D-theory of interfaces and then integrating over the interface thickness in order to define surface quantities; (ii) the failure of Tolman formula for droplets/bubbles of very small radii cannot be surprising because in the Tolman theory no prediction of the 'minimal' critical nucleation radius is made, so the Tolman-Gibbs theory cannot explain an essential physical phenomenon. Concerning the point (i) many authors have tried to define surface quantities as an integral over the interface: the first very crude attempt is due to Tolman himself [START_REF] Tolman | Consideration of the Gibbs theory of surface tension[END_REF]; many authors are able to define surface tension in term of mass density profile, or its gradient, by following either the ideas stemming from [START_REF] Cahn | Free energy of a non-uniform system I[END_REF] (cf. also [START_REF] Davis | Stress and Structure in fluid interfaces[END_REF][START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF][START_REF] Rocard | Thermodynamique[END_REF]), though these results are applicable only for large bubbles/droplets, or making use of the Gibbs concept of a dividing surface [START_REF] Falls | Structure and stress in spherical microstructures[END_REF]; a more rigorous approach, using a shell-like approach, is followed in [START_REF] Dell'isola | Deduction of thermodynamics balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF]. Point (ii) is closely connected to point (i): the description of homogeneous nucleation phenomena is possible only in a 3D-theory of the interface because we have to 'form' the interface with the critical nucleus starting from a single phase. The idealized model of the interface as a simple two-dimensional continuum can obviously fail if the radius of the microscopic bubble (or droplet) and the interface thickness are of the same order of magnitude. It will be shown below that for typical nucleation problem a 2D-theory is not suitable as, in that case, the vapor bubble surrounded by its liquid phase is constituted mainly or exclusively by the 'interfacial region'. As it is, however, still important to attribute a radius and an energy to microscopical bubble, we do it by means of an equivalent model of Laplace-Gibbs type. In this way it becomes possible to interpret the experimental evidence.

Cahn-Hilliard-Korteweg capillary fluids

The Cahn-Hilliard theory was developed just to account for the nucleation or more generally to study the behavior of strongly inhomogeneous fluids. In this theory a term depending on the gradient of the mass density is added to the classical expression for the free energy valid for homogeneous continua. Usually, the additional term is assumed to depend only on the gradient of density [START_REF] Cahn | Free energy of a non-uniform system I[END_REF][START_REF] Serrin | New perspectives in thermodynamics[END_REF]. Van Kampen has obtained the same results using the ideas and methods of statistical mechanics applied to a classical van der Waals gas [START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF]. We remark that in general, gradient theories are in many cases an excellent approximation of non-local theory, in which the additional term in the free energy depends on all the gradients of the mass density and this can be derived only by solving an integral equation [START_REF] Falls | Structure and stress in spherical microstructures[END_REF][START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF]. On the other hand we note that Korteweg, using the ideas of continuum mechanics, proposed some constitutive laws for the stress tensor which allowed him to prove the Laplace law [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires[END_REF]. However, as proven by Gurtin, classical Cauchy materials cannot show behavior of the Korteweg type, because of the second principle of thermodynamics [START_REF] Gurtin | Thermodynamics and the possibility of spatial interaction in elastic materials[END_REF]. Dunn and Serrin [START_REF] Dunn | On the thermodynamics of interstitial working[END_REF] proposed to change the expression for the power in Cauchy materials (adding the so-called "interstitial work") in order to avoid the contradiction shown in [START_REF] Gurtin | Thermodynamics and the possibility of spatial interaction in elastic materials[END_REF]. Another possibility could be to change the fundamental assumptions about contact forces (Cauchy postulate) characterizing Cauchy materials. When interpreted in the language of the classical continuum mechanics, the above assumptions about free energy lead to an expression for stress tensor which differs from that valid for compressible fluids as equilibrium shear components appear in it. The form of force balance equation valid for second gradient fluids is found in Germain [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus -Théorie du second gradient[END_REF]. However controversial, the introduction of second order fluids seems to be able to model some interesting problems in a simple way, as underlined by Casal, who named them "capillary" fluids [START_REF] Casal | La capillarité interne[END_REF]. We remark that Cahn and Hilliard do not consider the mechanical aspect of the nucleation phenomenon, which is accounted for in the Laplace approach. For this reason they only consider the thermodynamic pressure (i.e. the spherical part of the stress tensor deriving from the classical expression of free energy) while Casal shows, for second gradient fluids, the existence of a capillary non-spherical stress tensor whose trace includes, but does not reduce to, the quoted thermodynamic pressure [START_REF] Casal | La théorie du second gradient et la capillarité[END_REF]. Finally we remark that Cahn and Hilliard introduced, using their 3D-approach, an expression for surface tension valid only for plane interfaces, and never investigated the relationship between surface tension and radius (see also [START_REF] Davis | Stress and Structure in fluid interfaces[END_REF][START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF]).

Experimental evidence

From an experimental point of view the failure of Tolman formula was verified by Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of molecular dimensions[END_REF], though their results deserve some comments: their fundamental result is the observation that no significative variation of surface tension was observed down to radii near to the molecular diameter. However their experimental apparatus cannot reproduce a process of homogeneous nucleation because at such small radii the adhesive effects of their 'hyperboloidal' bubble can significantly change the boundary value problem for the density profile. On the other hand the 'minimal' nucleation radius predicted from the Cahn and Hilliard theory or by second gradient theory appears typically larger than the molecular diameter by a factor of two or three, so reproducing the data typical of nucleation studies (see Section 5 below). Another problem connected with the Fisher and Israelachvili experiment is that radii are obtained via the Kelvin relation that (though surely 'verified down to 4 nm' as quoted by the authors) needs some correction near to the 'minimal' nucleation radius. We think that it is possible to reinterpret their data by using the equivalent model described in the next section. More recently Kumar et al [START_REF] Kumar | Curvature dependence of surface free energy and nucleation kinetics of CCl 4 and C 2 H 2 Cl 4 vapours[END_REF] proposed a slight modification of Tolman results that can account for some of the observed nucleation data: a comparison of second gradient theory with their values is reported below in Section 5 for some data; here we note that the Kumar theory cannot predict a 'minimal' nucleation radius for droplets (the 'critical' radius data in their paper will be interpreted more correctly as an unstable 'equilibrium' radius [START_REF] Van Carey | Liquid-vapor phase-change phenomena[END_REF]. The reason is that their theory is essentially a modification of the 2D-Laplace-Gibbs theory in which the interface has no thickness.

3 Shell-like interfaces by second gradient theory

Equilibrium equations for a fluid endowed with internal capillarity

The second gradient theory, conceptually more straightforward than Laplace's theory, can be used to construct a theory of capillarity. We aim to study the equilibrium properties of a system with an interface under fixed temperature (isothermal conditions). In the following whenever stated capital letters P , W , etc. denote classical (per unit volume if the quantity is extensive) thermodynamic quantities; small Greek letters ǫ, µ denote classical mass-based thermodynamic quantities. For example, if ψ is the classical mass free energy, the volume free energy will be W = ρψ, and the classical chemical potential µ is given by:

µ(ρ) = ∂W ∂ρ . (3.1)
In the present text the only addition is an internal mass energy ǫ that is a function of the density ρ as well as ∇ρ. The internal mass energy characterizes both the compressibility and capillarity properties of the fluid, independently of the bodies with which it is in contact. For an isotropic fluid, it is assumed that:

ǫ = ǫ(ρ, β ρ ), (3.2)
where β ρ = (∇ρ)2 . The equation of equilibrium is written:

div S -ρ∇Ω = 0, (3.3)
where Ω is the extraneous force potential and S is the general stress tensor:

S = -p I -λ(∇ρ) ⊗ (∇ρ) T , (3.4)
with λ = 2ρ ǫ ,β ρ and p = ρ 2 ǫ ,ρ -ρ div(λ∇ρ). Following Rocard, this last expression for p is what we call the mechanical pressure [START_REF] Rocard | Thermodynamique[END_REF] 2 .

The equation of equilibrium (3.3) is obtained in the clearest manner by using classical methods of the variational calculus, i.e. using the virtual work Principle due to d'Alembert-Lagrange (cf. Serrin [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF]). It appears that a single term containing λ accounts of the second gradient effects in the equilibrium equation. The quantity λ, like ǫ, depends on ρ and ∇ρ (see also the discussion of the predictions of statistical mechanics on the ρ dependence in Section 4 below and references cited therein). In a study of surface tension based on gas kinetic theory [START_REF] Rocard | Thermodynamique[END_REF], Rocard obtained the same expression (3.4) for the stress tensor, but with λ constant. If λ is constant, the internal energy is simply written:

ǫ(ρ, β ρ ) = ǫ 0 (ρ) + λ 2ρ β ρ , (3.5)
i.e., the second gradient term λ 2ρ β ρ is simply added to the energy ǫ 0 (ρ) of the classical compressible fluid. The thermodynamic pressure for the fluid is P = ρ 2 ǫ 0,ρ , which gives for the mechanical pressure p:

p = P -λ 1 2 β ρ + ρ∆ρ , (3.6)
where ∆ is the Laplace operator. This pressure appears in the boundary conditions [START_REF] Seppecher | Etude d'une modélisation des zones capillaires fluides; interfaces et ligne de contact[END_REF]. For P , Rocard uses the van der Waals pressure or two-phase state laws.

In isothermal equilibrium conditions, Eq. ( 3.3) can also written as:

µ(ρ) -λ∆ρ + Ω = const. , (3.7)
where µ is given by Eq. (3.1), i.e., is the chemical potential, relative to the fluid in an homogeneous state with mass density ρ at a fixed temperature T .

If Ω is negligible the equilibrium of a liquid bubble3 surrounded by its liquid phase with density ρ l is represented by a spherically symmetric density profile satisfying:

λ ρ ,rr + 2 r ρ ,r = µ(ρ) -µ(ρ l ), (3.8)
We call Eq. (3.8) the Density Profile Equation (DPE). To the DPE we have to add boundary conditions: (i) because of spherical symmetry the derivative of ρ vanishes at the origin;

(ii) as we have assumed that the bubble is surrounded by an homogeneous liquid the same condition holds at infinity4 .

Equation (3.8) was formulated by Rocard [START_REF] Rocard | Thermodynamique[END_REF] and by Cahn & Hilliard [START_REF] Cahn | Free energy of a non-uniform system I[END_REF] (cf. also Blinowski [START_REF] Blinowski | Mathematical theory of Spherical interfaces[END_REF], Truskinovsky [START_REF] Truskinovsky | Critical nuclei in the van der Waals model[END_REF] and for a more general mathematical treatment Peletier & Serrin [START_REF] Peletier | Uniqueness of non-negative solutions of semilinear equations[END_REF]).

With W , the volume free energy being a primitive of µ, by multiplying the DPE by ρ r and integrating between zero and r we obtain:

λ 2 ρ 2 ,r + 2λ r 0 ρ 2 ,r r dr = W (ρ) -W (ρ v ) -µ(ρ l )(ρ -ρ v ). (3.10)
Moreover by multiplying the DPE by r and integrating again between zero and r, we obtain:

λrρ r (r) + λ(ρ(r) -ρ v ) = r 0 [µ(ρ) -µ(ρ l )]rdr,
and by taking the limit for r → ∞, rρ r (r) tends to zero and we obtain:

λ(ρ l -ρ v ) = ∞ 0 [µ(ρ) -µ(ρ l )]rdr. (3.11)
Such expressions as Eqs. (3.10)-(3.11) can be useful to test the numerical calculations of the interface profile, especially the second one because it relates the density jump across the interface directly to a simple integral involving chemical potential.

Nucleation energy of bubbles

In the theory of Laplace-Gibbs we have for bubbles of radius R and surface tension σ:

w = 4πR 2 σ + 4 3 πR 3 W (ρ v ) -W (ρ l ) -µ(ρ l )(ρ v -ρ l ) , (3.12)
with P = ρµ -W being the thermodynamic pressure; the isothermal equilibrium conditions:

µ(ρ l ) = µ(ρ v ), P (ρ l ) -P (ρ v ) = - 2σ R ,
DPE and conditions (i)-(ii) determine uniquely an increasing mass density profile ρ(r) and, in particular, the value ρ v at the origin; we will say that ρ(r) satisfies the capillary fluid version of Gibbs Phase Rule;

(iv) ρ(r) is twice differentiable at the origin; v) as dµ dρ ρ l > 0, ρ(r) converges at least exponentially to ρ l when r tends to infinity.

are valid only in the context of Laplace theory and transform Eq. (3.12) into:

w = 4 3 πR 2 σ. (3.13)
One can conclude that the nucleation energy of the bubble is one third of the creation energy of its interface.

We will now extend this result to the theory of second gradient fluids: in the theory of second gradient fluids, by following Cahn and Hilliard [START_REF] Cahn | Free energy of a non-uniform system I[END_REF], we have for the nucleation energy of a bubble in a domain D (cf. also Dell'Isola et al [START_REF] Dell'isola | On the generalization of Tolman formula[END_REF]):

w = D W (ρ) -W (ρ l ) -µ(ρ l )(ρ -ρ l ) + λ 2 (∇ρ) 2 dv. (3.14) By denoting ψ(ρ) = W (ρ) -W (ρ l ) -µ(ρ l )(ρ -ρ l )
, by multiplying Eq. (3.10) by r 2 and by integrating the result over [0, +∞] we get:

∞ 0 λ 2 ρ 2 ,r -ψ(ρ) r 2 dr + ∞ 0 ψ(ρ v ) + r 0 2λ r ρ 2 ,r dr r 2 dr = 0.
By integrating by parts and using Eq. (3.10) again, this equation becomes:

∞ 0 λ 6 ρ 2 ,r + ψ(ρ) r 2 dr + r 3 3 ψ(ρ) - λ 2 ρ 2 ,r ∞ 0 = 0.
Because of properties of the density profile quoted in Section 3.1, the second term vanishes, and we get:

w = 4π ∞ 0 ψ(ρ) + λ 2 ρ 2 ,r r 2 dr = 4 3 π ∞ 0 λρ 2 ,r r 2 dr. (3.15)
By denoting R 2 the mean value of r 2 with respect to the measure ρ 2 ,r dr, the above equation reads:

w = 4 3 πR 2 ∞ 0 λρ 2 ,r dr. (3.16)
For a large bubble, i.e., when ρ l tends to the plane interface value, R represents the radius. The surface tension for a plane interface is [START_REF] Cahn | Free energy of a non-uniform system I[END_REF][START_REF] Davis | Stress and Structure in fluid interfaces[END_REF][START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF]. Therefore Eq. (3.16) extends Eq. (3.13) to microscopic bubbles and reduces to it in case of large bubbles.

∞ 0 λρ 2 r dr

Comparison of Laplace and second gradient theories. Equivalent Laplace bubbles

In the second gradient theory the stress tensor in the centre of a spherical bubble takes the value S = -p v I where

p v = P (ρ v ) -λρ v ∆ρ |ρ=ρ v .
The value ρ l of the mass density in the liquid phase is attained asymptotically while the stress tensor takes the value S = -p l I where p l = P (ρ l ). As the DPE implies

λρ v ∆ρ ρ=ρ v = ρ v [µ(ρ v ) -µ(ρ l )]
we have:

p v -p l = W (ρ l ) -W (ρ v ) + µ(ρ l )(ρ v -ρ l ). (3.17)
Let us notice that this difference is not equal to the corresponding difference of the thermodynamical pressures as, for microscopic bubbles, µ(ρ l ) differs from µ(ρ v ). As the experimental results (see for instance Fisher & Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of molecular dimensions[END_REF]) deal with measurements of stresses, then we have to use p v -p l instead of P (ρ v ) -P (ρ l ) in the comparison between Laplace theory and second gradient theory. We can now define the surface tension and the radius of a bubble by identifying the nucleation energies and the pressure differences computed in both theories as follows:

p v -p l = 2σ R and 4 3 πR 2 σ = 4 3 π ∞ 0 λρ 2 ,r r 2 dr,
which leads to the relations

R = 2λ ∞ 0 ρ 2 ,r r 2 dr 1 3 [W (ρ l ) -W (ρ v ) -µ(ρ l )(ρ l -ρ v )] -1 3 , (3.18)
and

σ = λ 4 ∞ 0 ρ 2 ,r r 2 dr 1 3 [W (ρ l ) -W (ρ v ) -µ(ρ l )(ρ l -ρ v )] 2 3 . (3.19)
Equations (3.18)-(3.19) define the radius and the surface tension of a microscopic 'equivalent Laplace bubble' in our 'shell-like' theory. We remark that they differ from Eq. (4.9) of Falls et al [START_REF] Falls | Structure and stress in spherical microstructures[END_REF]; moreover, in Falls et al, R is not specified in terms of the density profile. Indeed Figures 18 and19 of the quoted reference are drawn by means of the 'Gibbs excesses' approach, so that these plots have in abscissa the Gibbsian 'radius of tension' or 'surface of tension' [START_REF] Chattoraj | Adsorption and the Gibbs surface excess[END_REF].

Thickness and surface mass density of interfaces

One of the fundamental problems of interfaces is the study of their thickness. We note that in the Laplace-Gibbs model the interfaces have no thickness.

In the above proposed model, in which λ is constant, the interface thickness is infinite. This is a consequence of the fact that the limit density in the phases is reached asymptotically. On the other hand the experiments show that the interfaces generally have a very small thickness, i.e., typically a few molecular diameters. The thickness takes a macroscopic dimension when the critical point is approached or the density is almost uniform (near the so-called spinoidal limit [START_REF] Cahn | Free energy of a non-uniform system I[END_REF]). Therefore a sort of coherence length for the interface is needed in second gradient theory. Some natural expressions can be proposed to characterize the thickness of the interface taking into account only the region in which the strongest gradient occurs. The most simple of these expressions is the quadratic variance associated to a measure ρ 2 ,r r 2 dr (we note that the first order variance is zero). We propose:

δ 2 = ∞ 0 (r -R) 2 ρ 2 ,r r 2 dr ∞ 0 ρ 2 ,r r 2 dr . (3.20)
We call δ the 'equivalent' thickness of the interface. We remark that there is also a natural definition of the bubble radius:

R 1 = ∞ 0 rρ 2 ,r r 2 dr ∞ 0 ρ 2 ,r r 2 dr . (3.21)
Numerical simulations show that, in all the cases studied below, R 1 and R are practically the same (differing by less than 1% except very near to the critical point). In the 2D-interface models the notion of surface mass density ρ σ is introduced. In the treatment proposed in [START_REF] Dell'isola | Deduction of thermodynamics balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF] it is assumed that:

ρ σ = R+δ/2 R-δ/2 ρjdr, (3.22)
where j(r) = 1 -2H(r -R) + K(r -R) 2 is the curvature dependent Jacobian pertaining to the change of variables between surfaces parallel to the interface in the 2D-models.

In the next Section we study by numerical simulations, for different twoparameter equations of state, the dependence of the interface thickness and of the surface mass density as functions of the radius and the temperature.

4 Qualitative analysis and numerical solutions of the DPE

General qualitative analysis

For the normalized density ρ(r), the DPE (3.8) reads (cf. [START_REF] Blinowski | Mathematical theory of Spherical interfaces[END_REF][START_REF] Cahn | Free energy of a non-uniform system I[END_REF]):

ρ ,rr + 2 r ρ ,r = µ(ρ) -µ ∞ . (4.1)
Now, subscript means that the derivatives are taken with respect to the normalized length variable r: in the whole section lengths and densities are nor-malized with respect to L α = P c λρ c Z c and critical density ρ c . The quantity µ is the normalized chemical potential and µ ∞ = µ(ρ l ). The explicit normalized form of the free energy W is given below for some typical equations of state. Z c and P c are the critical compressibility ratio and pressure. We will look for a solution of the DPE verifying the boundary conditions considered in Section 3.1, i.e., ρ ,r (0) = ρ ,r (∞) = 0, that represents a physically possible density profile in the phase transition region describing the equilibrium of a bubble (droplet) with its liquid (vapor). The qualitative analysis is based on the 'mechanical' interpretation of Eq. (4.1) [START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF]. This may be regarded as the equation of a 'particle' of mass 1 moving in the (µ ∞ -dependent) potential:

U(ρ, µ ∞ ) = -(W (ρ) -µ ∞ ρ), (4.2)
with the 'viscous' time-dependent force (2/r)ρ ,r . In the following we refer to the 'motion' of the particle as a solution of the Eq. (4.1) starting from r = 0, i.e., at the time zero, with zero velocity (cf. Section 3.1) and with a given initial density, i.e., a given initial position, ρ(0).

It is evident that a fundamental role in our problem is played by the potential U(ρ, µ ∞ ). For the sake of brevity, we limit ourselves to some general considerations valid for any potential. We assume under very general assumptions that:

(a) for any temperature below the critical temperature there exists a range of values for µ ∞ , say (µ spi,bub ∞ , µ spi,drop

∞

), for which the potential (4.2) has three extrema ρ 1 (µ ∞ ) < ρ m (µ ∞ ) < ρ 2 (µ ∞ ); ρ 1 and ρ 2 are the maxima and ρ m is a minimum;

(b) for µ ∞ = µ spi,bub ∞ (or µ ∞ = µ spi,drop

∞

), at the so-called spinoidal limit [START_REF] Van Carey | Liquid-vapor phase-change phenomena[END_REF], the potential has a maximum only in ρ 1 (or ρ 2 ) and an inflexion point in ρ 2 (or ρ 1 ); (c) the value of the chemical potential at the equilibrium between phases with planar interface (saturation) µ sat ∞ lies in the interval (µ spi,bub

∞ , µ spi,drop ∞ ); for µ ∞ = µ sat
∞ the values of the potential is the same at the two maxima (Maxwell rule), i.e., U(ρ

1 ) = U(ρ 2 ); (d) for µ spi,bub ∞ < µ ∞ < µ sat
∞ the value of the potential in ρ 2 is always lower than that in ρ 1 , i.e., U(ρ

1 ) > U(ρ 2 ) (bubble case); if µ sat ∞ < µ ∞ < µ spi,drop

∞

, the opposite happens: the value of the potential in ρ 2 is higher than that in ρ 1 , i.e., U(ρ 1 ) < U(ρ 2 ) (droplet case) 5 .

From assumptions (a)-(d) is clear that a 'motion' starting with initial position ρ(0) in the region just at the right (left) of the first maximum ρ 1 (second maximum ρ 2 ) of the potential for the case of the bubble (droplet) rolls down to the minimum ρ m and then goes up towards the second maximum ρ 2 (first maximum ρ 1 ) where three different cases are possible:

1) The velocity is insufficient to reach the second maximum ρ 2 (first maximum ρ 1 ); this happens because of the dissipative term in Eq. (4.1). In this case the 'particle' reverses its velocity before reaching the second maximum (first maximum) and falls towards the minimum where it will be trapped with damped oscillations when r tends to ∞; 2) The velocity is sufficient to reach the second maximum (first maximum); then the 'particle' arrives in general with a finite velocity at this point, then it falls down in the region of high density ρ > ρ 2 (low density ρ < ρ 1 ) after the maximum, its velocity grows and the density reaches the co-volume (zero) limit in a finite time; 3) The velocity is exactly that one sufficient to reach the second maximum (first maximum) (separatrix solution), so that the 'particle' reaches the second maximum (first maximum) in an infinite time, i.e., for r tending to ∞; at the maximum the velocity ρ ,r is zero, so that the boundary conditions stated in Section 3.1 are satisfied.

We will call ρ f the value for ρ at which the potential attains the lower maximum, i.e.,

ρ f = ρ 1 if µ sat ∞ < µ ∞ < µ spi,drop ∞ , and
ρ f = ρ 2 if µ spi,bub ∞ < µ ∞ < µ sat ∞ .
Obviously only the case 3) is physically significant. The force acting on the 'particle' is zero when r tends to ∞ as the maximum is an extremum of the potential. From Eq. (4.2) it follows:

µ(ρ f ) = µ ∞ = µ(ρ ∞ ).
This implies that (we assume that locally for large [small] values of density the function µ(ρ) is invertible), ρ f = ρ ∞ .

So the solution of the DPE is simply found if we examine the solution of the equivalent mechanical problem when starting a 'motion' with an initial value ρ(0) such that we have exactly the separatrix solution. So for a fixed µ ∞ the only unknown quantity is ρ(0) we will find when we solve numerically Eq. (4.1).

We note that far from saturation the equality of (classical) chemical potentials is not satisfied because in Eq. (4.1) we do not have zero on the RHS when r tends to zero. So for a small bubble an additional contribution due to curvature appears in the Gibbs energy if we want to preserve the classical relationship.

Near the saturation limit the problem becomes substantially different: in the case of bubbles the 'particle' starts its motion in the neighborhood of ρ 1 and tends to reach ρ f = ρ 2 ; its velocity has to be relatively large and therefore dis-sipation could be very large and inhibit the approach to the second maximum. However the system can react in a different way as a full separatrix solution linking the two maxima exists exactly at µ ∞ = µ sat ∞ . In fact at saturation the two maxima have the same height (Maxwell rule) and the separatrix starting from the first maximum and ending over the second is the correct solution 6 . This is exactly the Van Kampen solution for the planar interface [START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF].

Free energy and λ value in some simple models

The free energy functions can be chosen for the problem of phase transition by modelling them on an appropriate equation of state for the single fluid. The properties (a)-(d) will be shared by many approximate models of the long range effects such as the van der Waals or the Berthelot equations of state. In the following we study only the class of two-parameter equations of state though the method is applicable also to more general equations. This class includes several models [START_REF] Rocard | Thermodynamique[END_REF] with the standard expression a/V 2 for the internal pressure and also some more sophisticated models involving further correction of the internal pressure [START_REF] Peng | A new two-constant equation of state[END_REF]. However the form of the free energy is substantially the same leading only to some relevant but quantitative changes. We assume that the equation of state can be written in a general form as:

P (ρ, T ) = ρ tf (βρ) -αρ 2 g(βρ), (4.3)
where t is the reduced temperature, i.e., t = T /T c ; α = aρ c R M T c (with R M = R/M, R the gas constant, M the molecular weight) and β = ρ c b are the generalized normalized van der Waals coefficients (which can depend on temperature T ); f and g are suitable independent functions. In Table 1 we list the equations of state studied in this paper, the form of functions f and g and the values of the normalized coefficients α and β. The normalized volume free energy, derived from the above equation of state, is:

W (ρ) = ρ F 0 (t) -ρ t ρ 0 ρ f (βρ) dρ ρ -ρ α ρ 0 ρ g(βρ) dρ, (4.4)
where ρ 0 is a reference state at which free energy is ρ 0 F 0 (t). With regards to λ, Evans [START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF] shows under very general assumptions that:

λ(ρ) = k B T 12 c[r; ρ] r 4 dr, (4.5)
where k B is the Boltzmann constant and c[r; ρ] is the direct correlation function of the fluid. This implies that λ is a functional of the field ρ. However under some degree of approximation [START_REF] Ebner | Density-functional theory of simple classical fluids. I, Surfaces[END_REF] an expression for the direct correlation function is given by the Percus-Yevik equation (PY-equation) [START_REF] Percus | Analysis of classical statistical mechanics by means of collective coordinates[END_REF]:

c[r; ρ] = g[r; ρ](1 -e -V k B T ),
where g[r; ρ] is the pair correlation function and ν is the intermolecular potential. The expressions c[r] and g[r] can be numerically found in this case [START_REF] Ebner | Density-functional theory of simple classical fluids. I, Surfaces[END_REF]. In order to evaluate λ, we make the drastic approximation of taking g[r] = Θ(r -r 0 ), where Θ is the Heaviside function and r 0 is typically of the order of magnitude of the molecular radius.

1 t 1 3 1 1-x 1 Eberhart-Schnyders 9 8 1 t 1 2 1 3 1 1-x 1 Peng-Robinson 1.489(1 -k(1 -t 2 )) 0.253 1 1-x 1 2 √ 2 ln 1-x- √ 2 x-1- √ 2
Table 1. Scaled parameters α and β and functions f and g for the equations of state considered in this paper

Developing the Boltzmann factor in the PY-equation to the first order in ν, this approximation leads to the standard results of Cahn and Hilliard [START_REF] Cahn | Free energy of a non-uniform system I[END_REF] or equivalently of Van Kampen [START_REF] Van Kampen | Condensation of a classical gas with long range attraction[END_REF] (cf. also Gouin [START_REF] Gouin | Une interprétation moléculaire des fluides thermocapillaires[END_REF]) in which:

λ ∼ ∞ r 0 ν(r)r 4 dr, (4.6)
We note that the normalizing length L α , can be related to λ in the following way. In fact we can write L α = √ αL where α = aρ c /RT c and (cf. [START_REF] Cahn | Free energy of a non-uniform system I[END_REF]):

L 2 = ∞ r 0 ν(r)r 4 dr ∞ r 0 ν(r)r 2 dr . (4.7)
We note that this last length depends strongly on the intermolecular potential: for example for the 12 -6 Lennard-Jones potential, we have L = 11/7r 0 in which r 0 ≡ r LJ is the Lennard-Jones radius.

Numerical results

We integrate Eq. ( 4.1) by a Bulirsh-Stoer integrator [START_REF] Press | Numerical recipes[END_REF] starting from an arbitrary guess value of ρ(0) with ρ ,r (0) = 0. The values of ρ(0) are then adapted with a sequence of integrations until a good approximation to the separatrix is reached. Near saturation, the accuracy of the method is limited by the machine precision because the starting density differs from the first maximum density ρ 1 by progressively smaller quantities. We were limited to the usual double precision (10 -16 ) of our computing device. We find that this precision is sufficiently high to catch the main features of the approach to saturation. To test the procedure we have verified that, starting from the last point and going back toward small radii the 'particle' follows almost exactly the same motion (backward in time). As a further test we have verified that integral relation Eq. (3.11) is satisfied within the error introduced by numerical integration.

Typical density profiles are shown in Figure 1 for the van der Waals equation of state. The two parts of the nucleation phenomenon, i.e., the homogeneous nucleation and the growth of bubble approaching saturation, are clearly seen in this figure. The density profile tends to be diffuse towards the spinoidal limit so that the density jump ρ ∞ -ρ(0) goes to zero (cf. curves (a), (b) and (c) in Figure 1). Approaching saturation, a region of almost-constant density appears before the phase transition region, i.e. a bulk vapor phase, and the density jump is practically constant (cf. curves (f), (g) and (h) in Figure 1). In Figure 2 we show the dependence of σ on R (as defined in Section 3.4) numerically evaluated at the reduced temperature t = 0.85 for some of the equations of state listed in Table 1. The results are qualitatively similar: for small values of µ ∞ near the spinoidal value µ bub,spi ∞ (i.e. the lower part of the curve in Figure 2), σ initially vanishingly small, grows slowly whereas the radius decreases towards a minimal value that we identify as the 'minimal nucleation radius' R m . At R m the surface tension suddenly increases and the density at the center of the bubble reaches the range of the vapor phase (cf. also curves (d) and (e) in Figure 1). Above the minimal nucleation radius, σ approaches within a few minimal radii the planar interface value and, as shown by Fisher and Israelachivli experiments, remains substantially constant [START_REF] Fisher | Determination of the capillary pressure in menisci of molecular dimensions[END_REF]. We note that quantitatively, the van der Waals equation of state gives the lowest value of planar interface surface tension, the Berthelot and the Peng-Robinson give the highest values. The behavior for other values of temperature is qualitatively the same. We note also that the curves end at different radii as they approach saturation; these radii have to be considered as equivalent because the 'distance' from saturation, measured by the difference between the first maximum of the potential and the density at the center of the bubble, is in all cases less than 10 -16 . It could be implied by this circumstance that different equations of state reach saturation at different radii. Moreover we note that curves corresponding to radii between the spinoidal and the minimal critical radius values can be considered as lacking physical meaning because, no physical bubble can exist in this regime, except fluctuations of typical radius R (cf. [START_REF] Cahn | Free energy of a non-uniform system I[END_REF]). So the 'minimal nucleation radius' R m marks the last values, before the spinoidal, to which one can give physical meaning. In Figure 3 we plot the (equivalent) interface thickness defined in Eq. (3.20) as a function of R for some of the different equations of state, always at reduced temperature t = 0.85. At the start of nucleation (spinoidal limit), δ is very large (diffuse interface) and of the same order of magnitude as R: density fluctuations have a unique length scale. When R has reached the minimal nucleation radius, δ quickly tends to a practically constant value with very small deviation from the limit value at saturation. Again all the studied equations of state have the same qualitative behavior. Moreover, we note that the curves between the spinoidal limit and the minimum radius are essentially the same for all the equations of state used here (cf. Figure 3). We note that no qualitative differences appear when the temperature is changed.

It is also interesting to note that the ratio (δ/R) m is, within the numerical accuracy, a constant independent of t: its value is reported in Table 2 for all equations of state. Moreover this constant is practically the same for all the equations of state studied here, i.e., (δ/R) m ∼ 0.42 -0.44. The justification of this phenomenon can be that, the shell-like quantities as surface tension, radius, etc., being defined through an integral over the thickness of the interface, the differences between the equations of state will be smeared by integration, giving similar results. 1. The ratios are substantially independent of the temperature in the limit of the algorithm error introduced in the numerical solutions of the DPE equation.

In Figure 4 we plot the surface mass density versus R at t = 0.85 obtained by using Eq. (3.22) for the same equations of state than in Figures 23. We see that surface mass is an increasing function of R going towards the spinoidal limit. This behavior is obviously due to the increase of δ in the same limit. Again we note that no qualitative differences appear when the temperature is changed. Moreover the ratios δ ∞ /δ m and ρ ∞ σ /ρ m σ are independent of t. Their values, listed in Table 2 for all the equations of state, differ by less than 15%.

Minimal nucleation radii: theoretical predictions and comparison with experiments

The comparison of the above results with the experimental values reported in literature is a rather difficult task because information for many data can be only indirectly recovered. Moreover, as shown in Section 4.2, Eqs (4.5-4.7), all the predictions are critically dependent on the parameters of the intermolecular potential; in our case a fundamental role is played by L that sets the length scales. Again the correct prediction of surface tension is also not simple in the sophisticated microscopic statistical theories (cf. the data reported in [START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF]) because, especially at very small scales, the details of the intermolecular potential, and so the particular molecular micro-structure, are relevant in the determination of the exact values of surface tension. We prove in this paper that rather good predictions of surface tension are also possible in the framework of the second gradient theory.

Moreover we are able to produce predictions of nucleation radius. In particular, by evaluating R m , we establish a lower bound to the experimental nucleation radii reported in the literature. We remark that very few data are found in the literature concerning bubbles (cf. [START_REF] Van Carey | Liquid-vapor phase-change phenomena[END_REF]), so that we are obliged to compare these predictions with data for the droplets found in Lamer and Pound or Kumar [START_REF] Kumar | Curvature dependence of surface free energy and nucleation kinetics of CCl 4 and C 2 H 2 Cl 4 vapours[END_REF][START_REF] Lamer | Surface tension of small droplets from Volmer and Flood's nucleation data[END_REF].

In order to choose, for every substance, the most suitable equations of state to predict the minimal nucleation radius, we use the planar interface surface tension values in the following way: by following Cahn and Hilliard, we fix L (cf. the next subsection) while among the considered equations of state, we choose that which predicts σ ∞ values sufficiently accurately (for many substances this estimate holds within 90% of the experimental value). Finally we can predict the thickness of the interface and the surface mass density. We note that these data are relatively scarce especially the latter [START_REF] Alts | Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water. Parts I and II[END_REF][START_REF] Dell'isola | On the lack of structure of 2D Defay-Prigogine continua[END_REF]. For experimental planar interface surface tension cf. [START_REF]L'air Liquide[END_REF][START_REF]Handbook of Chemistry and Physics[END_REF]. The upper part of the Table shows substances for which Lennard-Jones radius r LJ is known with good approximation and so it is used to calculate σ ∞ by evaluating L from (11/7) 1 2 r LJ . In the lower part of the Table (down the double line) the covolume radius r co is used to evaluate L and then σ ∞ . In the cases marked by * a better fitting was made by taking L = 1.25 r co .

In Table 3 we show collected experimental data for different substances. Values for the planar interface surface tension are shown together with the values evaluated by second gradient theory in the case of the van der Waals equation of state. The theoretical predictions were made by using in some cases the Lennard-Jones relation between r 0 ≡ r LJ and L while in others we have simply put L = r 0 = r co , where r co is the covolume radius (we note that in general r LJ > r co but their ratio is lower than (11/7)

1 2 ∼ 1.25).
The prediction of the second gradient theory is generally within 20% of the experimental value: this can be considered as a good result, comparable with the non-local model approach (cf. [START_REF] Davis | Stress and Structure in fluid interfaces[END_REF] and [START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF]). In many cases our predictions are very accurate. We note however that are remarkable exceptions: 1) the Ar value is larger by about a factor two with respect to the experimental value; the same holds true in respect of the surface tension for other noble gases; 2) the values for C 6 H 12 and CCl 4 are smaller almost by a factor of two (about 60% of experimental values), the same is true in minor form also for other carbon compounds such as C 4 H 10 and C 3 H 8 and most dramatically for CF 4 , where the ratio is about 1/6.

The case of Argon and the other noble gases can be understood if we look to the last column of Table 3 in which we report the scaled experimental value of α exp = a exp ρ c MRT c representing the normalized value of the van der Waals constant a. For all substances α exp is between 1 and 2 whereas for Ar it is larger than 2 (about 3.2); the same is true for other noble gases. This large value implies that the values of surface tension evaluated with the 'van der Waals theory' are generally far from the experimental values. This is a known result for Ar [START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF]: the second gradient theory is insufficient to reproduce the experimental value for the surface tension, as higher-order gradients are necessary in the constitutive law for free energy to give a correct prediction7 (cf. [START_REF] Ebner | Density-functional theory of simple classical fluids. I, Surfaces[END_REF]).

As for point 2) we could increase the values of the surface tension, by multiply-ing r co by 1.25, i.e., the Lennard-Jones factor. This trick works effectively for ethyl alcohol, ethane and ammonia, but for the other substances the theoretical surface tension remains too small when compared with the experimental value (we note also the case of CO 2 in which the theoretical value is 79% of the experimental value). So we conjecture that other equations of state, that generally have a larger value of σ ∞ (cf. Fig. 2), would describe carbon compounds better than the van der Waals equation. For the sake of brevity we do not present the theoretical data; we limit ourselves to saying that, by using a different equation of state, the values of σ ∞ can be predicted with the same precision as shown in Table 3 for substances that are well described by the van der Waals equation. In Figure 5 we display the normalized minimal nucleation radius in units of L for the equations of state in Table 1 versus reduced temperature in the interval [0.5, 1.0[. The interval chosen is the most relevant for many substances because most of the experimental data on surface tension exist in this range of reduced temperature (cf. [START_REF]L'air Liquide[END_REF][START_REF]Handbook of Chemistry and Physics[END_REF]). At the lower temperatures the minimal radius tends to be a constant of approximate value L; the exact values could be retrieved in principle by the method described in Section 4, but the strong stiffness of the problem, induced by the rapid decrease to very small values of the gas phase density, limits the precision of the method. Table 5. Minimal nucleation radii for bubbles of some substances (carbon compounds) evaluated at t = 0.85 in the case of some other equations of state. These will be chosen to reproduce the planar interface surface tension experimental values at least to 80%. In the other two columns are listed the predictions for thickness and surface mass density at minimal nucleation radius at the same reduced temperature.

The minimal nucleation radius and the other surface quantities

In Table 4, the minimal nucleation radii at t = 0.85 are listed for those sub-stances whose planar interface surface tension is well described by the van der Waals equation of state. In the same table, we have added data on the thickness of the interface and the surface mass density for the substances cited therein. In Table 5, further theoretical predictions are given, always at t = 0.85, for substances whose planar interface surface tension is well described by other equations of state. Thus we show that radii, thickness and surface mass densities can be calculated very easily by using the results found in [START_REF] Dell'isola | Radius and surface tension of microscopic bubbles by second gradient theory[END_REF]. We note that the minimal radius ranges typically from 10 to 30 Å so that it is just a few molecular radii. For example the molecules of C 6 H 12 have a radius of about 5 Å; at t = 0.85 the ratio between this value and predicted minimal radius is about 4.7, while at t = 0.5 the minimal nucleation radius is ∼ 9.8 Å and the ratio is about 2.

Examples of predictions of the equilibrium radius as a function of R (or µ ∞ ) by means of classical theory are given by van Carey [START_REF] Van Carey | Liquid-vapor phase-change phenomena[END_REF]. However he limits himself to the case of 'large bubbles', the only ones for which classical theory is valid. The experiment of Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of molecular dimensions[END_REF] needs to be reinterpreted because the Laplace formula, on which is based the Kelvin relation used to extrapolate the radius, is not valid near the minimal nucleation radius [START_REF] Dell'isola | On the generalization of Tolman formula[END_REF]. Moreover, as noted above, their results can be compared with a process of homogeneous nucleation only after a thorough analysis of the effect of their experimental apparatus. We note that as cited above C 6 H 12 has a predicted minimal nucleation radius of about ∼ 9.8 Å at the temperature in which the experiment was performed (t ∼ 0.5). It is larger than the radii reported by Fisher and Israelachvili (we also note that, on changing the equations of state or the evaluation of L, the minimal radius always remains greater than their data). On the other hand, data on homogeneous nucleation for very small radii exist for droplets (cf. [START_REF] Kumar | Curvature dependence of surface free energy and nucleation kinetics of CCl 4 and C 2 H 2 Cl 4 vapours[END_REF][START_REF] Lamer | Surface tension of small droplets from Volmer and Flood's nucleation data[END_REF]) and comparison with our data shows that the nucleation radii of bubbles we predict are in some cases very close to nucleation radius for the droplets; for example, in the case of water, the Lamer and Pound data are in the range 8.0 -8.9 Å at t ∼ 0.5, and our prediction is R m ≃ 8.3 Å; the data for equilibrium radii reported by Kumar for the CCl 4 at t ≃ 0.5 range between 12 -17 Å, these are very close to those we calculated which range (at t = 0.5) between 12 -14 Å, though this range of variation is due to the used equation of state rather than to the supersaturation ratio.

Moreover we predict that the thickness of the interface is formed by few molecular layers [START_REF] Evans | The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids[END_REF] in the range from minimal nucleation radius to saturation. A new insight of the physical problem is obtained when observing that at small radii the interface constitutes a large part of the microscopic bubbles and cannot be confined in any Gibbs-Tolman idealized zero thickness layer. Typically the numerically evaluated ratio between the equivalent thickness at the minimal radius and the minimal radius itself is of the order of 0.4 for all the equations of state studied here. Finally we note that the surface mass density values (see Tables 4 and5) are of the order of magnitude predicted by Alts and Hutter in their theory on water [START_REF] Alts | Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water. Parts I and II[END_REF].

Conclusion

The first three Sections of this paper give a complete account of the results available in the literature concerning the theoretical treatment of the phenomenon of homogeneous nucleation and the dependence of surface tension on curvature.

However recently (cf. [START_REF] Dell'isola | On the generalization of Tolman formula[END_REF]) a rational definition for surface tension and radius of curvature for spherical interfaces has been proposed in terms of a solution of the density field, using a shell-like approach to the definition of interface quantities. Moreover, thanks to the second gradient theory, a deeper understanding of the relationship between the Laplace formula and the capillary constant λ has become possible. In particular our investigations stem from a theoretical treatment in which: 1) an expression for surface tension is used which accounts for curvature effects better than those found in the literature; 2) the mechanical pressure is distinguished from the thermodynamic pressure.

In this way we obtained, using numerical methods, predictions for: 1) the dependence of surface tension, equilibrium radius, thickness of the spherical interface and surface mass density on chemical potential;

2) the values of the minimal nucleation radius, i.e., the minimum equilibrium radius possible for a bubble. Being founded either on statistical mechanics or on second gradient theories, our predictions have a clear meaning in a range of temperature close to the critical point.

Possible generalizations include the study of: (i) models in which λ depends on ρ locally or globally; (ii) models in which higher density gradients appear, or a fully non-local approach via integral equations;

(iii) models suitable to describe the dynamics of phase transitions.

Fig. 1 .

 1 Fig.1. Density profiles for van der Waals equations of state for different values of µ ∞ (or ρ ∞ ) at t = 0.85 obtained by numerical solution of the DPE. Towards saturation, cf. the curves labeled from (e) to (h), the bubble clearly develops a bulk vapor phase in its interior. Near the minimal nucleation radius, cf. the curve (d), the bulk vapor phase disappears. In the spinoidal limit, cf. the curves from (c) to (a), the radius of the bubble again grows but no bulk vapor phase exists and the density jump rapidly decreases until it disappears at the so-called spinoidal composition in which only the liquid phase exists.

Fig. 2 .

 2 Fig. 2. Normalized surface tension σ/σ n where σ n = ρ c R M T c L vs normalized equilibrium radius R/L of microscopic bubbles at t = 0.85 for the following equations of state: van der Waals 2; Rocard 4 ⋆; Berthelot ×; Peng-Robinson △; Eberhart-Schnyders +. The end-point of each curve toward the saturation have to be considered equivalent in the sense of saturation proximity (cf. the text).

Fig. 3 .

 3 Fig. 3. Normalized thickness δ/L of the interface vs normalized equilibrium radius R/L of microscopic bubbles at t = 0.85 for the following equations of state: van der Waals 2; Rocard 4 ⋆;Berthelot ×; Peng-Robinson △; Eberhart-Schnyders +.

Fig. 4 .

 4 Fig. 4. Normalized surface mass density ρ σ /(ρ c L) vs normalized equilibrium radius R/L of microscopic bubbles at t = 0.85 for the following equations of state: van der Waals 2; Rocard 4 ⋆; Berthelot ×; Peng-Robinson △; Eberhart-Schnyders +.

Fig. 5 .

 5 Fig. 5. Minimal nucleation radii vs reduced temperature in units of L in the interval [0.5, 1.0[ for the following equations of state: van der Waals 2; Rocard 3 3; Rocard 4 ⋆; Berthelot ×; Peng-Robinson △; Eberhart-Schnyders +. State equations Rocard 1 and 2 give minimal nucleation radii between Rocard 3 and van der Waals cases.

Table 2 .

 2 Numerically evaluated ratios (δ/R) m , δ ∞ /δ m and ρ ∞

	State Equation	(δ/R)m	δ∞/δm	ρ ∞ σ /ρ m σ
	van der Waals	0.427	0.771	0.716
	Rocard 1	0.443	0.778	0.712
	Rocard 2	0.435	0.785	0.739
	Rocard 3	0.443	0.768	0.716
	Rocard 4	0.432	0.795	0.773
	Berthelot	0.426	0.768	0.710
	Eberhart-Schnyders	0.428	0.761	0.683
	Peng-Robinson	0.439	0.750	0.767
				σ /ρ m σ for the equa-
	tions of state reported in Table

Table 3 .

 3 5.1 The planar interface surface tension. Planar interface surface tension for some substances evaluated in the case of van der Waals equation of state and the corresponding experimental value. In the last column of the Table is reported the scaled parameter α exp = a exp ρ c MRT c .

	Substance	texp	σ exp ∞	[ dyn cm ] σ vdw ∞	[ dyn cm ] αexp
	Ar	∼ 0.5	13.1	23.9	3.24
	CO 2	0.95	1.16	0.93	1.51
	N 2	∼ 0.6	9.80	8.25	1.50
	O 2	∼ 0.6	13.2	11.4	1.46
	H 2	∼ 0.6	2.09	2.29	1.34
	H 2 O	0.5	68.5	63.8	1.83
	NH * 3	∼ 0.7	23.4	26.3	1.73
	CH 3 OH	∼ 0.6	20.1	20.1	1.91
	C 2 H 5 OH *	∼ 0.5	22.7	24.7	1.70
	C 2 H * 6	∼ 0.6	16.2	15.3	1.50
	C 6 H 12	0.5	25.5	16.5	1.63
	CCl 4	∼ 0.5	29.8	18.6	1.64
	CF 4	∼ 0.9	5.0	0.87	-

Table 4 .

 4 Minimal nucleation radii for bubbles of some substances evaluated at t = 0.85 in the case of van der Waals equation of state. In the other two columns are listed the predicted values of thickness and surface mass density at minimal nucleation radius at the same reduced temperature. The meaning of the division of the Table in upper and lower part and the * near some substance is the same than in

	Substance	Rm [ Å] δm [ Å] ρ m σ 10 -7 • [g/cm 2 ]
	CO 2	22.3	9.52	9.4
	N 2	19.7	8.47	5.7
	O 2	18.4	7.86	7.4
	H 2	15.6	6.67	0.4
	H 2 O	15.5	6.63	4.6
	NH * 3	20.8	8.87	4.5
	CH 3 OH	20.2	8.62	5.0
	C 2 H 5 OH *	27.3	11.6	6.9
	C 2 H * 6	24.9	10.6	4.7

Table 3 .

 3 We see that the Berthelot equation of state and the modification proposed by Eberhart-Schnyders[START_REF] Van Carey | Liquid-vapor phase-change phenomena[END_REF] predict radii rather smaller than other equations of state. The Peng-Robinson equation of state gives radii near to the van der Waals model at high values of t and near the Berthelot model values at about t = 0.5. The family of all equations of state proposed by Rocard give radii very near to the van der Waals value (for this reason only two of these data are reported in Figure5, in order to leave the figure sufficiently clear). Obviously, when the critical temperature is approached the minimal radius and the thickness tend to ∞.

	Substance	State Equation	Rm [ Å] δm [ Å] ρ m σ 10 -7 • [g/cm 2 ]
	CH 4	Eberhart-Schnyders	13.1	5.70	2.0
	C 3 H 8	Rocard 4	22.1	9.54	4.6
	C 3 H 8	Eberhart-Schnyders	18.1	7.86	3.8
	C 4 H 10	Rocard 4	25.1	10.8	5.4
	C 4 H 10	Eberhart-Schnyders	20.5	8.90	4.5
	C 6 H 12	Rocard 4	26.3	11.3	6.8
	C 6 H 12	Peng-Robinson	23.6	10.2	6.2
	CCl 4	Rocard 4	26.1	11.2	13.8
	CCl 4	Peng-Robinson	23.4	10.1	12.6

We remark that in[START_REF] Falls | Structure and stress in spherical microstructures[END_REF] only the first term ρ 2 ǫ ,ρ , corresponding to the classical thermodynamic pressure P , enters in the Laplace law, cf. Eq. (4.9) of the quoted reference; however in appendix A the authors agree that the mechanical pressure needs consideration.

The same approach could be followed also in the case of vapor droplets, but here for sake of brevity we limit ourselves to develop the theory for bubbles.

If ρ P l is the density of the liquid phase in the equilibrium state with plane interface and if the constitutive function µ is suitably regular, the theory of Fuchs equations implies that: (iii) for every ρ l in an interval (ρ spi , ρ P l ), where ρ spi is the spinodal density value, the

The simplest model is obtained by developing the free energy functional, near the critical point, as a function of ρ; the result is W (ρ) = (ρ -a 1 ) 2 (ρ -a 2 ) 2 (with a 1 and a 2 constants corresponding respectively to the homogeneous gas and liquid density), which satisfies the properties (a)-(d) as it can be simply verified.

This happens because the dissipation, i.e., the term (2/r)ρ r , cannot play any role because the escape time from the first maximum is now infinite and this term tends to zero as r tends to infinity.

We note that values of α exp higher than for noble gases are obtained for water and methyl alcohol, this is reflected, (cf. Table3, caption), in the use of r co rather than r LJ in the reported theoretical predictions.

Acknowledgements:

We wish to thank A. Di Carlo, P. Seppecher and W. Kosiński for their friendly criticism. This work was financially supported by the Université de Toulon et du Var, the Université of Aix-Marseille and the Italian CNR.


R Dussart

L J Overzet

P Lefaucheux

T Dufour

M Kulsreshath

M A Mandra

T Tillocher

O Aubry

S Dozias

P Ranson

J B Lee

M Goeckner

Affiliations

Integrated micro-plasmas in silicon operating in helium Authors

Microplasma arrays operating in helium in a DC regime have been produced in silicon microreactors. Cathode boundary layer (CBL) type microdevices were elaborated using clean room facilities and semiconductor processing techniques. Ignition of the micro-discharge arrays having either 50 or 100 μm diameter cavities was studied. Two different structures (isotropically etched or anisotropically etched cavity) and various conditions (the two different voltage polarities, pressures etc.) were investigated. 100 microdischarges of 50 μm diameter could be ignited in parallel at 1000 Torr. At high current, some parasitic and transient sparks appeared at the edge of the sample. When the polarization was reversed (cathode side corresponding the opened electrode), more current was needed to light all the microdischarges. A thermally affected zone around the hole on the anode side was obtained after operation.

I. Introduction

Micro hollow cathode discharges (MHCDs) were first introduced by Schoenbach et al. in 1996 [1]. The structure of MHCD reactors tends to be quite simple: they generally consist of two electrodes separated by an insulator, much like a capacitor. The difference is that this structure has at least one and sometimes many holes drilled through it. By applying a high voltage (typically several hundreds of volts) between the two electrodes, one can initiate a stable glow discharge at atmospheric pressures localized inside the hole(s). The term MHCD for this type of discharge was adopted by several groups [2][3][4][5][6][7] before Boeuf et al. [8] showed that the structure was not due to the classical hollow cathode effect. Nevertheless, the term MHCD was maintained in the community. The typical V-I curve of a MHCD can be divided into 3 conceptual parts: -At small current levels, the voltage increases with current and the discharge remains inside the hole.

-At large current levels, the voltage remains more or less constant as the current increases and the plasma extent over the cathode surface increases with current. This is the so-called "normal glow" regime. In the typical normal glow regime, several watts can be injected into each MHCD corresponding to several hundreds of kW cm -3 . The electron density can reach 10 15 cm -3 [9]. -At intermediate currents, a self-pulsing regime can be obtained where there are large current fluctuations and the plasma oscillates between remaining inside the hole and extending over the cathode surface [10,11].

MHCD reactors are often assembled mechanically and drilled either with a laser beam or mechanically. In these cases the diameter of the MHCD reactor is large, typically greater than 100 μm. Moreover, the thickness of the metal electrodes is typically similar to the thickness of the dielectric. This can limit the production and functionality of the devices. Eden's team was the first to use clean room techniques to fabricate arrays of MHCD reactors on a silicon substrate [12]. The typical dimensions of those devices were 50×50 μm 2 . He used silicon nitride layers and polyimide as the dielectrics. The cathode was made by wet etching the silicon substrate to form inverted pyramids. The anode was made by depositing nickel. As a result, they were able to ignite up to 250 000 micro-discharges all together using AC excitation [13].

Our own research into MHCDs is a collaboration between colleagues at GREMI (Orleans, France) and the University of Texas at Dallas (UTD, Texas USA) first started in 2006. The first samples were partially prepared in the clean room of the engineering school at UTD. Relatively thick alumina substrates (Al2O3, ∼500 μm) were electroplated with a thin layer of nickel (∼7 to 8 μm) and laser drilled to form either one or multiple MHCD reactors in parallel in various hole arrangements. They were tested at both labs. At GREMI, tests were made in He at pressures between 20 and 1000 Torr using a single ballast resistor for the entire array. It was found that although a single hole ignited readily, lighting multiple holes simultaneously was much more difficult. In fact, the first hole to ignite would self-destruct from being over-powered before a second would ignite. This is unfortunate since arrays of MHCD reactors in parallel can enable many more applications than isolated MHCD reactors. It allows one to extend the micrometric scale of one MHCD to the macroscopic scale of an array for plasma processing applications. Other research groups were able to ignite multiple microdischarges mounted in parallel. Of those reports, some had to use an individual ballast resistor for each MHCD [14,15] while others did not [16,17]. The reason for this was not clear until recently when we discovered why individual ballasts are required in some configurations and not necessary in some others [18]. We showed that limiting the cathode surface area allows one to obtain an abnormal glow regime of the MHCD and ignite all of the microplasmas in an array. This preliminary work proved useful for designing our newer integrated devices in silicon. At atmospheric pressure, more radicals can be formed for use in materials or gas processing. The goal is to well control the operation of each micro-discharge in the array so that processing at the macroscopic scale can be optimized too. It turns out that silicon is a rather good material for such reactors because it has good mechanical and thermal properties (fusion temperature of 1414°C, good thermal conductivity of 149 Wm -1 .K -1 , linear thermal expansion coefficient of 2.6×10 -6 K -1 …). Some effects due to its being a semiconductor can also be expectepd as was shown by Ostrom and Eden [19].

Our objective is to fabricate and study integrated micro-plasmas in silicon to investigate their limits and characteristics for their future implementation in systems: lab on chip, micro electro mechanical systems (MEMS), micro-sensors, etc. Whereas micro-fluidic devices are already used extensively in MEMS devices, microplasmas are not yet typically found. Before their implementation, micro-plasmas should be studied to find their limits in terms of dimensions, lifetime, and physical characteristics (density, temperature etc.). Such studies can be begun by electrical and optical characterization. Here, we report some first experiments using silicon micro-reactors, which were fabricated as a collaboration between UTD and GREMI using semiconductor processing techniques. MHCD arrays consisting of 100 holes having either 50 or 100 μm diameters were made for these initial tests although much smaller devices are possible for future generations. The micro-fabrication sequence will be described in Section 2. Ignition of the micro-discharge arrays operating in helium will be discussed in Section 3 for two different structures (isotropically etched or anisotropically etched cavity) and various conditions (the two different voltage polarities, pressures etc.).

II. Experimental setup & microreactor fabrication

As noted, MHCD reactors were fabricated on silicon wafers using standard clean room processes. We used 3" n-type silicon wafers for these first tests and designed so called "cathode boundary layer" (CBL) discharges [20]. A CBL discharge is simply a highpressure glow between a planar cathode and ring-shaped anode that are separated by a dielectric. Typically, the dielectric thickness has been of the order of 100 μm [20]; however, we chose to use a SiO2 dielectric thickness of 5 μm since we used a low-pressure chemical vapor deposition (LPCVD) technique to deposit it. Growing a much thicker oxide is time and cost prohibitive. This thickness of oxide is easily sufficient to withstand the electric fields required for discharge breakdown. Different geometries were designed on the wafer to investigate micro-discharges having different dimensions. Both single hole and 10×10 arrays having either 50, 100, 150 or 200 μm diameters were fabricated.

The process flow is illustrated in Figure 1 The 5 μm thick SiO2 layer, represented in white in Figure 1, was grown using an LPCVD process (Fig. 1.1) at UTD. The oxide on the backside of the wafer was stripped in buffered oxide etch solution while the front side was protected using photoresist. On top of this front-side oxide, a thin adhesion and nucleation layer of Cr/Cu was evaporated (not represented in Fig. 1). Subsequently, the first lithography step was used to pattern the MHCD holes and define the overall anode area (resist is represented in orange in Figs. 1.2 and 1.3). An electro-deposition process on the unprotected areas of the Cr/Cu film was used to grow the nickel anode layer. This Ni film was made to be a few microns thick and is designated by the green film in Figures 1.2-1.4. The nickel layer was left thinner than what is typically used for MHCD reactors [20] because of a limitation set by the photoresist thickness. A second lithography step was carried out to protect the SiO2 layer surrounding the anode during etching of the oxide layer in the hole (not represented in Fig. 1). Maintaining the integrity of the surrounding insulator is thought to help avoid arcing at the wafer edges (between nickel anode and silicon cathode in Fig. 1.3). As we will note later, this oxide layer is not sufficient to prevent arcing at the edges. The SiO2 in the holes was then etched using a reactive ion etch (RIE) process at GREMI. The Ni anode was used as a hard mask for that etch step. The penultimate step was plasma etch of the silicon substrate itself to define the cathode area and shape. Two different processes were utilized to either form an anisotropic etch profile (Fig. 1.4a) or a more isotropic etch profile of the silicon (Fig. 1.4b). The final step was to deposit aluminum on the back side of the wafer to provide an ohmic contact to the cathode (Si). This is the green film in Figures 1.4a and1b. oxide protrudes past the silicon sidewall) and the silicon cavity is ∼60 μm in depth. The SEM image in Figure 3 shows 100 μm diameter MHCD reactors that were etched anisotropically to form 10 μm deep holes. The two cases are quite different: the distance between the two electrodes is much greater for the isotropically etched MHCD reactors. Moreover, the isotropic profiles of the 150 μm diameters cavities provide a much greater area than the anisotropically etched 100 μm diameter cavities. As mentioned in the introduction, the total area of the cavity is relevant if it is used as a cathode: it will define the maximum current which can be driven in each cavity.

To make the first tests, we placed the chip into a vacuum system which could be filled with helium at a chosen pressure. A very slow triangle wave voltage signal (0.025 Hz) was applied using a 40 kΩ ballast resistance and the discharge voltage and current were recorded. Both discharge ignition and the appearance of the discharge(s) were observed for 1 and 100 holes MHCD at varying values of the current.

III. Results & Discussion

We first show the results obtained for a single hole of 150 μm diameter. Figure 4 shows a typical V-I curve for an anisotropically etched MHCD sample operating in helium at 350 Torr. The distance between the cathode and anode was ∼5 μm (the dielectric thickness). The cathode area (the Si surface area) was also smaller than that of samples etched isotropically. It was estimated to be ∼0.018 mm 2 . The current was found to be of the order of 100 μA as shown in the V-I characteristic of Figure 4. The left inset is a picture of the micro-discharge for a current below 100 μA. Beyond 250 μA, the micro-discharge entered an abnormal glow regime. As observed on the V-I characteristic and on the right inset, several parasitic discharges ignited at this higher current level. The exposure time for each picture was 0.1 s, so that each picture also integrated transient (pulsed) discharges, which are clearly evidenced in the V-I characteristic as the sudden transitions to large currents and subsequent return to the main V-I characteristic. At small currents, 2 parasitic micro-discharges are in the proximity of the bright main discharge. These are attributed to nickel voids formed during the electro-plating step. Such defects can allow an undesired SiO2 etch and the formation of unwanted cavities in these locations. They can be removed by depositing some epoxy in their locations as shown next. the 17 holes at the bottom right were accidently covered by epoxy during sample preparation. We note that these MHCD samples were etched isotropically, forming cavities like the one shown in Figure 2. As a consequence, the current flowing through each micro-discharge was somewhat larger than that found in the case of the single micro-discharge presented in the previous paragraph (Fig. 4). In fact, currents as large as 26 mA were obtained for this sample corresponding to an average of ∼300 μA per MHCD. We estimate that the cathode area for each micro-cavity is at least equal to 0.065 mm2, which is ∼3.5 times more than the cathode area of the single cavity sample (ignoring surface area enhancement due to the significant surface roughness seen in Fig. 2). Although the available cathode area of individual microdischarges is larger, some parasitic current flow is observed going to the edge of the sample in Figure 5. In particular, two plasma filaments appear on the left side of the picture between silicon (far left edge) and nickel anode. Some other sparks can also be observed in the picture. Again, those parasitic discharges are not stable, but rather transient sparks, which are characterized by current and voltage spikes on the V-I characteristic.

Fig. 4. V-I curve obtained for a single hole micro-discharge working in helium at 350 Torr. Insets: pictures of the micro-discharges for a current discharge of 100 μA (left) and for 400 μA (right).

Fig. 5. Picture of a 10×10 micro-discharge array (100 μm diameter) operating in helium at 500 Torr for a total current of around 20 mA showing the main micro-discharges and some transient sparks which appear at the edges of the sample.

Fig. 6. (a) Discharge voltage and current versus time for a 10 × 10 microdischarge array (100 μm diameter) operating in helium at 500 Torr. The current follows the triangle wave, while the discharge voltage saturates. (b) Resulting V-I characteristic when current and voltage spikes due to parasitic discharges were removed.

Such spikes can be observed in a time-dependent plot of the current shown in (Fig. 6a) recorded by our oscilloscope. Short spikes in both the current and voltage were found to be numerous especially when the average current exceeded 10 mA. The V-I curve can be plotted by removing these spikes to enhance its readability (Fig. 6b). One can see that the discharge voltage remains ∼200 V at these experimental conditions. While the discharge voltage rises a small amount with the current above 15 mA, no clear indication of entering the abnormal glow regime was observed for this sample from 2 to 27 mA. To better understand the ignition sequencing of the microdischarges, a picture of the sample was taken every 3 s during the increase and decrease of the current (as shown in Fig. 6). The results are presented in Figure 7, which is a sequence of 7 pictures in time order starting at small currents (Fig. 7a) rising to the maximum current (Fig. 7d) and then falling back toward extinction (Figs. 7e-7g). Figures 7a and7b show the device at currents between 3 and 8 mA. The cavities in the upper right and lower left portions of the sample ignite first and as the current rises, more micro-discharges ignite. By 10 mA (Fig. 7c), all 83 micro-discharges were ignited. At currents larger than 10 mA, sparks appeared at the edge of the array (Figs. 7d and7e). As the current decreased (Figs. 7f and7g), the extinction sequence followed the same trend in reverse. This experiment shows first that the micro-discharges of the array do not all ignite simultaneously at a single breakdown voltage. The current limitation set by the external ballast resistor limits the area of cathode which can be used in forming the glow and thus acts to limit the number of holes which can be ignited at any given moment. A related phenomenon was reported by Waskoenig et al. for a similar type of micro-discharge, but using an AC power supply at high frequency [21]. This results in an ignition sequence for holes in the array which appears to occur in reverse during the current decrease. Finally, transient sparks appear at the edge of the sample at high current levels once all the MHCD holes are ignited. It could be that the DC plasma finds additional cathode area in this fashion.

Fig. 7. Temporal evolution of the micro-discharge array during current increase (a-d) and decrease (e-g).

The European Physical Journal D, 2010, Vol. 60, Issue 3, 601-608, http://dx.doi.org/10.1140/epjd/e2010-00272-7

Fig. 8. Temporal evolution of the micro-discharge array during current increase (a-d) and decrease (e-f) when the nickel is biased to be the cathode.

If we invert the polarity of the voltage applied to the sample, then the nickel film becomes the cathode, and the silicon becomes the anode for the micro-discharges. In this case, the MHCDs can still ignite, but take on a very different appearance and characteristic (see Fig. 8. where a sequence of pictures analogous to those in Fig. 7 is shown). Note that the plasma did not appear connected to the cavities or to significantly enter the cavities, but simply expanded over the cathode area instead. As the current increased, the plasma extended over a larger surface area but never covered the full array. Indeed, when the nickel was the cathode, the cathode area was not as effectively limited and as a consequence the current could increase without reaching an abnormal glow regime. The result is that larger currents were required to form a plasma having a similar spatial extent as that for the array with the Si as the cathode. We also ignited micro-discharges at higher pressure (1000 Torr) using the same sample. A picture of the micro-discharge array is shown in Figure 9. Even at larger current levels, all the microdischarge cavities did not ignite. Moreover, among the ignited micro-plasmas, emission intensity was not uniform: some were very bright while others were weak. As seen on the picture, we did not observe any spark at the edge of the sample in these conditions even though the current level was well above that producing sparks at lower pressures. Experiments were also carried out on smaller diameter cavities, having a shape like the one of Figure 3. We expect that the interelectrode distance was reduced compared to that in isotropically etched cavities. Despite this, it was possible to ignite MHCDs as shown in Figure 10a for 50 μm diameter cavities. Figure 10b is another picture of the micro-discharges made in the same experimental conditions, but at a higher magnification. Interestingly, higher pressures were required to ignite the discharge. Figure 10 shows the results at 1000 Torr, the pressure used in Figure 9 which resulted in incomplete ignition of the array. For the anisotropically etched array, microplasmas did not ignite at lower pressures, even at 700 Torr; but nearly all the microdischarges ignited at 1000 Torr. At the larger current levels for this array some parasitic sparks were observed at the edge of the sample (Fig. 11a). Among the ignited micro-discharges, one can notice that the emission intensity appears inhomogeneous, which indicates that the current is not equally distributed between the different micro-discharges. Although we do not have yet a definitive explanation for it, different interpretations can be given: first, holes are not exactly the same in terms of electrode roughness and some of them could be more efficient to drive current; second, the exposure time of the camera being quite long (100 ms), discharges might not be as stable as they seem to be and some of them could appear brighter than others. In Figure 11, we show some optical and SEM images of the Ni film before (Fig. 11a) and after (Figs. 11b,11c) operation. A 20 µm wide ring appears around the cavity after operation even though the Ni film is biased as the anode. The edge of the film also becomes rough at the microscopic scale (Fig. 11c). If we assume that all the micro-discharges in the array ignited and that the total current lay between 1 and 10 mA, we can make a rough estimation of the current density at the anode. For a 50-μm diameter cavity, if we were to assume that all of the electrons entered the Ni (anode) film solely through the 5-µm thickness, then the current density would have to be quite large. It would lie between 1.3 and 13 Acm -2 . Therefore, it is probable that a fraction of the electron current to the anode is injected on the top surface and should be the origin of the observed 20-μm wide ring. Given this, the required current density is reduced to a value between 0.2 and 2 A.cm -2 , a more reasonable range. As a consequence, in our devices with a 5-μm thin metal layer for the anode, we can think that a part of the current is driven through a larger ring around the cavity opening. This ring could become heated and as a result, eroded and roughened. It may also be that the metal evaporates at this location. Optical emission spectroscopy should confirm this and we plan to carry out the experiment in the near future.

IV. Conclusion

In this paper, we have presented some first results obtained on silicon CBL type micro-reactors having two different types of geometry: one formed an isotropic cavity and the other one had a smaller area and cylinder cavity. A significant part of this work was dedicated to the microfabrication of the devices. We have shown that micro-discharge arrays operating in helium could be obtained at different pressures. At high current, some parasitic and transient sparks were obtained. When the polarization was reversed, we could also ignite micro-discharges, but more current was needed to light them all. At high pressure, the intensity of the micro-discharges was not homogeneous on the array. Finally, by analyzing our samples by SEM and optical microscopy after operation, we could observe a thermally affected zone around the hole on the anode side. This ring shape region seems to correspond to the area from which the current is driven through around the cavity opening. The samples we used were quite fragile and had a limited lifetime (about 10 min for an injected current between 1 and 20 mA). After several tests, a short circuit appeared, so that no micro-plasma could be obtained anymore. New samples are being fabricated, taking into account these first results.

  with Figure 1.4c showing the top view and Figures 1.1-1.4b showing the A-A cross sections.

Fig. 1 .

 1 Fig. 1. Schematics illustrations of the wafer cross section after the various process steps used to form a 10 × 10 array of MHCDs on a silicon substrate. (1) After forming the SiO2 dielectric. (2) After electro-plating the Ni anode. (3) After developing the photoresist for SiO2 protection. (4a, 4b) Final state after etching the Si and depositing the backside electrical contact. (4c) Top view.

Figure 2

 2 Figure 2 is a scanning electron microscope (SEM) image showing 150 μm diameter micro-reactors after operation in helium. The holes were etched isotropically as drawn in Figure 1.4b. The nickel and SiO2 layers are clearly visible at the top of the structure (the

Fig. 2 .Fig. 3 .

 23 Fig. 2. MHCD hole profile obtained after an isotropic plasma etchingprocess showing the nickel/SiO2/silicon structure.

Figure 5

 5 Figure 5 is a picture (200 ms exposure) of a 10 × 10 array of MHCDs having diameters of 100 μm and operating in helium at 500 Torr. The silicon surface was biased to be the cathode of the microdischarge. Only 83 of the 100 micro-reactors, could ignite because

Fig. 9 .

 9 Fig. 9. Micro-discharge array at 1000 Torr in helium (100 μm diameter cavities) at 20 mA.

Fig. 10 .

 10 Fig. 10. (a) Picture of a 10 × 10 micro-discharge array (50 μm diameter) operating in helium at 1000 Torr for a total current of ∼15 mA showing the main micro-discharges and some transient sparks which appear at the edges of the sample. (b) Picture taken in the same experimental conditions, but with a higher optical magnification.

Fig. 11 .

 11 Fig. 11. (a) Optical microscope picture of the Ni film before the ignition of a micro-discharge. (b) Optical microscope picture of the Ni film on the same sample after the ignition of micro-discharges. (c) SEM image of the Ni film showing erosion after operation.

This work is financially supported by the French "Agence Nationale de la Recherche" through the contract No. ANR-09-JCJC-0007-01 under the name SIMPAS project.

V. References