Ramiro Dell'erba

Francesco Dell'isola

Giacomo Rotoli

The influence of the curvature dependence of the surface tension on the geometry of electrically charged menisci

We evaluate how the curvature dependence of surface tension affects the shape of electrically charged interfaces between a perfectly conducting flui and its vapour. We consider two cases: i) spherical droplets in equilibrium with their vapour; ii) menisci pending in a capillary tube in presence of a conducting plate at given electric potential drop. Tolman-like dependence of surface tension on curvature becomes important when the "nucleation radius" is comparable with the interface curvature radius. In case i) we prove existence of the equilibrium minimal radius and estimate its dependence on the electric field and Tolmanlike curvature effects. In case ii) the menisci are subject to the gravitational force, surface tension and electrostatic fields We determine the unknown surface of the menisci to which the potential is assigned using an iterative numerical method and show that Tolman-like corrections imply: 1) a variation of the height (up to 10% in some cases) of the tip of the menisci; 2) a decrease of the maximum electrical potential applicable to the menisci before their breakdown amounting to 40V over 800V in the considered cases. We conjecture that these effects could be used in new experiments based on electric measurements to determine the dependence of the equilibrium surface tension on curvature.

Introduction

In Tolman [START_REF] Tolman | The effect of droplet size on surface tension[END_REF] the classical thermodynamic theory of interfaces between different phases due to Gibbs [START_REF] Gibbs | The Scientifi Papers of J.W.Gibbs[END_REF] (see Scientifi Papers (1948)) was applied to study the equilibrium of spherical drops with their interfaces. In particular Tolman proved, in the framework of Gibbs' theory of excess thermodynamic quantities, that the second principle of thermodynamics implies an exponential dependence at equilibrium of the surface tension on the curvature of interphase surface. Moreover, he evaluated the characteristic radius for this exponential decay towards a constant value in terms of known thermodynamic quantities. La Mer and Pound [START_REF] Lamer | Surface Tension of small Droplet from Volmer and Flood's nucleation data[END_REF] demonstrated that Tolman's formulas fail in predicting experimental evidence. Several efforts were subsequently performed to generalize Gibbs' theory to match quantitative experimental data. For a detailed report of these efforts we refer to Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of Molecular Dimension[END_REF]. We quote here the treatments applied by Defay and Prigogine [START_REF] Defay | Tension superficiell and adsorption[END_REF], Alts and Hutter [START_REF] Alts | Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water, Part I,II,III,IV[END_REF] who consistently used the concept of excess quantities, and the results obtained by dell'Isola [START_REF] Dell'isola | On the lack of structure of Defay-Prigogine 2D-continua[END_REF] using the concept of two-dimensional non-material shell-like Defay-Prigogine interface.

However, to our knowledge, by only using a three-dimensional second gradient theory, such as that of Germain [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus. Premiere partie: Théorie du second gradient[END_REF], to model the material behavior of the interface between different phases (see dell'Isola and Rotoli [START_REF] Dell'isola | Validity of Laplace formula and dependence of surface tension on curvature in second gradient fluids[END_REF], dell'Isola et al. [START_REF] Dell'isola | Radius and Surface Tension of Microscopic bubbles by second gradient theory[END_REF], dell'Isola et al. [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]), it is possible to adjust the parameters which control

Roberto Stroffolini in Memoriam: To the Maestro who introduced us to Continuum Mechanics and Electrodynamics

the dependence of the equilibrium surface tension on curvature and to fi the available experimental data (see Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of Molecular Dimension[END_REF]).

The fact that the nucleation energy of microscopic bubbles is lower than those energies that are computed with a constant surface tension is generally accepted, but there is no clear experimental evidence that this phenomenon is related only to a constitutive dependence of surface tension on curvature. We explicitly remark that in a theory in which surface tension is given as a constitutive equation in terms of temperature and curvature the second principle of thermodynamics implies that surface tension is a decreasing function of the curvature radius. This last result is in contradiction with the aforementioned property of nucleation energy. As a consequence we expect that a more sophisticated thermodynamic theory for surface tension must be formulated (e.g. using the methods developed by dell'Isola and Kosinski [START_REF] Dell'isola | Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers[END_REF]) in which a more refine kinematic description of the interface is introduced: this theory, applied to equilibrium states, will lead to a relation between equilibrium surface tension and curvature which we call Tolman-like correction.

In this paper, in order to obtain further insight into these problems and make possible a comparison among different dependencies of equilibrium surface tension on curvature, we consider their influenc on the shape of electrically charged interfaces. The only physical quantities which we will consider are electrical surface charge density and surface tension. We expect that the most important phenomena occurring at the interface between non-polar charged fluid will be described in terms of them.

We do not assume any constitutive influenc on the surface tension by the electric fiel itself. A simple estimate of the energy density of the interface and the electric fiel energy shows that this effect must be considered only for very strong electric fields typically larger than the dielectric breakdown of the fluid

We shall address two cases: i) a conducting droplet surrounded by its (non-conducting) vapour; ii) a conducting meniscus in a typical electrocapillarity experiment: a conducting truncated conical pipet, on which the meniscus is pendent, is placed opposite to a conducting plane while a potential difference is applied between them. The firs case is considered in order to supply a simple benchmark case to the analysis developed in ii). Indeed the spherical symmetry allows an easy determination of the electrostatic field involved. The investigation of the equilibrium of drops and bubbles for electrically charged conducting interfaces generalizes the classical results found in a rational form by Romano [START_REF] Romano | Properties of the Gibbs Potential and the Equilibrium of a liquid with its Vapour[END_REF]. Case ii) was investigated by Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF], by assuming the surface tension γ to be constant. Our aim is to extend these results to menisci of which the curvature mean radius is locally comparable with the minimal nucleation radius as define in dell'Isola et al. [START_REF] Dell'isola | Radius and Surface Tension of Microscopic bubbles by second gradient theory[END_REF] so that the equilibrium Tolman-like dependences of surface tension on curvature must be considered.

The main difficult is the determination of the electric fiel close to the interfaces. However, by using the methods outlined in Sect. 3 we are able to generalize the results obtained in Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF] for the conicalshaped capillary tube and constant surface tension (i.e., the case of large radius of capillary tube). When the Tolman-like dependence of the surface tension, γ, on curvature is important (i.e. in the neighbourhood of the critical temperature and/or minimal nucleation radius) we obtain relevant differences to the case of constant γ in predicting the break-down electrical potential: the choice of the Tolman-like dependence can lead to its decrease by about 5%. Moreover, the height of the tip of the menisci is increased by about 7% when surface tension is increasing with increasing curvature radii or decreased by about 30% when surface tension is decreasing with increasing radii.

We believe that such effects could be used to support a new experimental measure of the dependence of equilibrium surface tension on curvature simply by measuring the electric potentials in an apparatus with small-radii-capillary tubes. These experiments could help to choose, among the proposed Tolman-like corrections, those amendments which best match the measurements. This choice is a vexed question much discussed since Tolman firs published his paper in 1949 [START_REF] Tolman | The effect of droplet size on surface tension[END_REF]. Indeed:

-very few experiments could be performed to settle it (see Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of Molecular Dimension[END_REF]), -much of the evidence is only indirect (La Mer and Pound [START_REF] Lamer | Surface Tension of small Droplet from Volmer and Flood's nucleation data[END_REF], 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]), -several inferences used in these indirect measurements are doubtful (dell'Isola and Rotoli [START_REF] Dell'isola | Validity of Laplace formula and dependence of surface tension on curvature in second gradient fluids[END_REF]). We conclude by underlining that the geometry of our problem slightly differs from that chosen by Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF]. Indeed, in order to avoid singularities in the electric fiel in the neighbourhood of the sharp edges shared by the meniscus and the capillary tube we choose to use as capillary tubes only truncated cones.

Equilibrium of fluid phases separated by electrically charged spherical interfaces

In this section we generalize to the case of electrically charged spherical interfaces a theorem proved by Romano [START_REF] Romano | Properties of the Gibbs Potential and the Equilibrium of a liquid with its Vapour[END_REF] starting from the following balance equations:

∇p = 0 in C 1 ∪ C 2 , γ = γ(ϑ), 2γ R = [[p]] on S , [[g(p)]] = 0 on S . ( 1
)
Where p is the pressure; γ the surface tension; R the drop radius (coincident with the radius of curvature); [[•]] denotes the jump of the bracketed quantity across S (i.e. its limit value from inside S minus that from outside); g(p) is the density of Gibbs energy (chemical potential) and C 1 and C 2 the inner and the outer regions of the bubble while S is the interface surface. In (1) the volume body forces are neglected while the temperature fiel ϑ is assumed to be constant in space. The aforementioned theorem states the following: Theorem 1 If the temperature is given in the interval (θ * , θ c ), then the vapour at uniform pressure p v ∈ (p 0 , p 2 ) is at equilibrium with drops of the radius R of its liquid if and only if the liquid has a suitable pressure p l ∈ (p 0 , p * l ) and R ∈ 2γ/(p * lp 2 ), ∞ . As a consequence there exists an equilibrium minimal radius given by R min = 2γ/(p * lp 2 ) We used the following notation:

θ * = triple point temperature, θ c = critical temperature, p 1 = spinodal pressure, p v = vapour pressure, p 0 = unique solution of g v (p 0 ) = g l (p 1 )
, g v and g l are the Gibbs energies of the vapour and the liquid respectively,

p 2 = saturation pressure, p * l = g -1 l (g v (p 2 )
). Consider now a conducting drop carrying an electric charge Q. We assume it is subject to electrostatic conditions and introduce the radial component E ∝ Q/r 2 (r being the distance from the center of the drop) of the electrostatic field Introducing into the balance equations a Tolman-like correction for surface tension γ(R g ) = γ 0 f (R g ), R g being the surface mean curvature radius (coincident in this section with the drop radius), and f (R g ) a regular function to be specified [START_REF] Tolman | The effect of droplet size on surface tension[END_REF] become

∇p = 0 in C 1 ∪ C 2 , γ = γ 0 f (R g ), 2γ(Rg) Rg + E σ = [[p]], [[g(p)]] = 0 on S . (2)
in which σ is the surface charge density at the liquid-vapour interface.

It should be noted that firs the condition [[g(p)]] = 0 is obtained when the surface Gibbs energy g σ in the local form of jump conditions expressing balance of energy is neglected (for more details see Alts and Hutter [START_REF] Alts | Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water, Part I,II,III,IV[END_REF] or dell'Isola and Romano [START_REF] Dell'isola | A phenomenological approach to phase transition in classical fiel theory[END_REF]); second we assume that the drop does not discharge itself by air ionization; third in the present section for the functional form of the Tolman-like correction the relations

γ 1 (R g ) = γ 0 (1 -e - Rg R 0 ), (3)
γ 2 (R g ) = γ 0 1 2 1 + tanh R g -R 0 αR 0 ( 4
)
are chosen. In these the so-called "minimal nucleation radius" R 0 appears (cf. dell'Isola et al. [START_REF] Dell'isola | Radius and Surface Tension of Microscopic bubbles by second gradient theory[END_REF]). R 0 is a temperature dependent quantity, strictly related to microscopic parameters of the considered fluid as e.g. the molecular dimensions. The dependence of R 0 on temperature is of the form of a standard critical power law, i.e., R 0 = L (1 -T /T c ) β with β < 0 where L is a length of the order of magnitude of molecular dimension. The firs expression is similar to Tolman's original formula except for the definitio of and therefore the values to be assigned to R 0 ; the second one is an empirical formula. It approximates experimental data better and is based on the results obtained by dell'Isola et al. [START_REF] Dell'isola | Radius and Surface Tension of Microscopic bubbles by second gradient theory[END_REF]: the constitutive constant α is related to the state equation describing the microscopic behaviour of the considered material. We emphasize that the introduced Tolman-like corrections cannot be regarded as constitutive equations for surface tension. Indeed it is well-known that the second principle of thermodynamics does not allow surface tension to depend on curvature unless the interface between phases is modelled by a micro-structured 2Dcontinuum. In order to develop a thermodynamically coherent model one should introduce in the constituive equations at least the surface gradient of curvature and extra kinematical descriptors and must then model the variation of the thickness of the interface and the surface mass density together with their corresponding evolution equations. An alternative possibility to force thermodynamic compatibility between Tolman-like corrections and free energy is to assume an explicit dependence of the latter on curvature and to introduce, as in the theory of shells, couple stresses.

Therefore, we regard Tolman-like corrections to be valid only for equilibrium conditions, and we postpone to more general treatments the formulation of constitutive equations for surface tension locally valid under non-equilibrium conditions. Such constitutive equations should be compatible with the request of a decreasing dependence of surface free energy on the radius of curvature: indeed a non-decreasing dependence would lead to ill-posed minimization problems and the related relaxed problems would no longer involve surface tension.

These corrections can be used whenever the radius of the drop is larger than R 0 . We note that a drop of radius R ∼ 2R 0 can already be considered sufficientl far from nucleation as described in dell'Isola et al. [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]. Otherwise close to R 0 the non-local nature of the process must be considered, and a corrected form of the free energy must be introduced e.g. suggested in dell'Isola et al. [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF].

Equation [START_REF] Gibbs | The Scientifi Papers of J.W.Gibbs[END_REF] implies that in every phase the pressure fiel is constant. Therefore (once the temperature, the charge Q or alternatively the potential U of the drop and the vapour pressure p v are fixed the set of ( 3), ( 4) becomes an algebraic system in the unknowns R and p l .

In order to formulate the generalization of theorem 1 to the case represented by ( 2), ( 5) we introduce the following potential function

LC U (R) := 2γ(R) R + U 2 εR 2 , ( 5
)
where ε is the dielectric constant of the vapour and the second term is obtained by recalling the expression for the electrostatic fiel generated by an isolated uniform surface charge. Simple calculations prove that for both γ i introduced in ( 6), [START_REF] Dell'isola | On the lack of structure of Defay-Prigogine 2D-continua[END_REF] the functions γ i (R)/R are decreasing with R. More generally, this property has been shown for a large class of continuum models of flui interfaces from the second principle of thermodynamics (see dell'Isola et al [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] and dell'Isola [START_REF] Dell'isola | On the lack of structure of Defay-Prigogine 2D-continua[END_REF]). Therefore the equilibrium minimal radius R min can be define by

R min := (LC U ) -1 (max(p l -p v )), (6)
which, by following the same procedure as in Romano [START_REF] Romano | Properties of the Gibbs Potential and the Equilibrium of a liquid with its Vapour[END_REF] and by assuming that the State Equation for the considered flui does not depend on the electric potential, can be shown to be given by

R min (U ) = (LC U ) -1 (p * l -p 2 ). (7)
In conclusion, Theorem 1 holds also for spherically charged drops in which surface tension depends on the mean curvature: the unique modificatio in the statement concerns the definitio of R min .

Equations (2) lead to R = (LC U ) -1 g -1 l (g v (p v )) -p v (8)
which, as LC U is an increasing function in U , allows us to conclude that the equilibrium radius (for a fixe pressure p v ) increases with increasing U . This is physically understandable when considering the interplay between the repulsive electrostatic forces and the attractive membrane forces due to surface tension. Finally we remark that: (1) in the presence of a fixe potential U the introduction of Tolman-like corrections leads to smaller equilibrium radii. Indeed, let L U (R) := 2γ 0 /R + U 2 /εR 2 . Obviously i) LC U and L U are 2) for three different values of the potential drop (0 µV , 2 µV , 3 µV ) vs the normalized radial variable. The curves relative to 0 µV starts from the origin, the other two start from -0.25 and -0.50, respectively. The effect of two Tolman-like corrections is shown by dashed curves (γ 1 large dashes, γ 2 small dashes). These are compared with the nocorrection case (γ 0 solid curve) both decreasing functions ii) L U > LC U , and consequently (L U ) -1 > (LC U ) -1 ; (2) in the range of the radii for which γ = γ 0 (i.e. R R 0 ) we have :

c := g -1 l (g v (p v )) -p v , R(U , p v ) R(0, p v ) = 1 2 + 1 2 1 + cU 2 γ 2 0 ε
, which implies that the equilibrium (and therefore also the minimal) nucleation radius is larger in the presence of electrostatic fields than without; this follows simply since

R(U , p v ) R(0, p v ) = (LC U ) -1 (c) (LC 0 ) -1 (c) > 1, (9)
which also holds when R R 0 .

In Table 1 the solution of ( 8) is shown for p v in the neighbourhood of the saturation pressure p 2 for different values of the electric potential and for the two Tolman-like corrections which we have introduced. For the zero-curvature-surface tension γ 0 we chose the value for silicon oil, i.e., γ 0 = 0.0201N /m, and we set α = 1, and R 0 = 10 -7 m.

Table 1. Normalized values of R min as a function of the drop potential for no Tolman-like correction (column 2), correction by (3), i.e., constitutive γ 1 (column 3) and by (4), i.e., constitutive γ 2 (column 4). We remark that for potential drops in the range [0, 2] µV (i) the Tolman-like correction γ 2 induces variations of the depencence of R min on U of about 1% and, (ii) the Tolman-like correction γ 1 induces variations of the depencence of R min on U of about 7%. In Fig. 1 L

U(µV ) R min (γ = γ 0 ) R min (γ = γ 1 ) R min (γ = γ 2 ) 0.
U -c and LC U -c are plotted as functions of R (normalized to the values L -1 0 (p * l -p 2 )) for c = g -1 l (g v (p 2 )
)p 2 and several value for U . In Fig. 2 we display R min (U ) in the three cases:

1) γ = γ 0 , 2) γ = γ 1 , 3) γ = γ 2 .
Evidently, for vanishing applied potential drop in the three cases we predict minimum radii that are remarkably different than with such a potential applied. On the other hand, when the applied potential increases, the three cases, as expected, show the same quantitative and qualitative behaviour. 1): 3. Experimental setup for electrocapillarity experiments. The setup consists of an oil reservoir connected to a conducting capillary pipet, whose surface is mantained at a piezostatic height H above the end of the capillary pipet where the meniscus is formed. The capillary pipet is mantained at a potential U 0 whereas a plane capacitive ground plate is placed in front of the capillary end. In the inset is shown the geometry of the capillary end: the meniscus is pendant from a truncated conical pipet with a given pipet angle. It is assumed that the pipet angle is chosen to match the meniscus of the pipet in a relatively smooth way (so no relevant concentration of surface charge appears at the separation line between pipet and meniscus)

γ = γ 0 → Solid line ; γ = γ 1 → ; γ = γ 2 → Fig.
3 Electrocapillary menisci

Basic equations

We conceive a system as schematically sketched in Fig. 3. A conducting truncated conical pipet is placed in front of a plane capacitive plate. A conducting liquid is pushed by a piezostatic pressure to form a capillary meniscus suspended at the lower opening cross section of the pipet.

The range of the considered potential drop between the pipet and the plate will be chosen as a function of the angle of the pipet: it will be required that at the edge between the surfaces delineating the pending meniscus and the pipet the discontinuity of the normals induces a concentration of surface charge which is significantl smaller than the surface charge concentrated at the tip of the meniscus.

The motivation of this choice lies in the simplificatio which it implies in the study of the electric fiel generated between the pipet and the plate. In particular, no significan electric-fiel concentration appears in the neighbourhood of the aforementioned edge. As a practical consequence, if the theory should be tested experimentally, a family of capillary tubes with different cone angles but with fixe inferior basis would have to be considered in order to allow measurements involving a large range of applied potential drops.

In the region D between the plate and the meniscus the electric potential satisfie Laplace's equation. We explicitly remark here that in solving the elliptic problem which follows we consider the region D as constituted by the part of the space between two infinit surfaces representing respectively the conducting surface of the pipet and the grounded plate. The electric fiel in the pending liquid vanishes. The surface force balance, generalizing the capillary Laplace law for surface tension, determines the shape of the pending meniscus S . Scaling all the lengths by the capillary radius, R, the potential by U 0 , i.e., the chosen drop of the potential between the plate and the pipet, the surface tension by γ 0 , i.e., the surface tension for plane interfaces, and the charge density, σ, by R/(εγ 0 ) (this choice naturally arises when recalling the considerations about the function LC U introduced in the previous section) the aforementioned basic equations, under the assumption of cylindrical symmetry for the considered apparatus, read

∆U = 0, in D, ( 10
)
k 1 (k 2 + z (r)) + γ R g R g + σ 2 2 = 0, on S , (11)
where

k 1 := ρgR 2 /γ 0 , k 2 := H /R, σ = E ε ; ( 12
)
ρ is the liquid mass density, H the piezostatic height, ∆ the Laplace operator, z (r) the function specifying the profil of the meniscus in the cylindrical coordinate system of which the axes are oriented as shown in Fig. 3: the z -axis is pointing in the direction of gravity from the pipet to the grounded plate, and E is the component of the electric fiel orthogonal to the meniscus.

In the present section we analyze the effect of Tolman's corrections ( 3) and ( 4) as well as those due to:

γ 3 (R g ) = γ 0 1 2 3 -tanh R g -R 0 R 0 . ( 13
)
While ( 3) and ( 13) show an increasing behaviour as functions of R g , (13) is decreasing. These three Tolman type corrections are able to efficientl capture their most important features. The boundary conditions are 1) for the potential U = 1 [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF] on the meniscus and the pipet, and

U = 0 (15)
on the plate which is at a distance d from the pipet, and, because of the cylindrical symmetry,

∂U ∂r (r = 0, z ) = 0, z ∈ [0, d ] (16)
2) for the z (r) function we have (r = 1 represents the edge of the lower cross section of the pipet)

z (1) = 0, z (0) = 0. ( 17
)
The firs condition represents the patching condition between the meniscus and the lower cross section of the pipet, while the second is a consequence of the cylindrical symmetry of the pipet. This is a system of partial differential equations (PDE) with "free" unknown boundary (the surface z (r)) on which a Dirichlet condition on the potential is assigned. In fact, by varying U 0 , the free boundary will change the shape because it is determined by the electric fiel in the region D. The problem can be classifie as a Dirichlet problem for the potential coupled with a Sturm-Liouville (Two Boundary Values) problem for the profil function z (r).

In order to clearly specify the range of applicability of ( 10), [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] the following remarks are in order:

1) We suppose U 0 to be sufficientl small that, at the meniscus tip (i.e. at the location of maximum surface charge density), no charge emission takes place; so the Laplace equation for the potential can be employed instead of the Poisson equation. 2) It is assumed that the only effect due to the internal pressure of the liquid, i.e., the cohesive molecular attraction due to the a/V 2 terms in the equation of state, is the presence of the surface tension; this obviously implies the well-known anomaly of the interface stability, as g → 0 . 3) We do not consider cylindrical capillary tubes for the reason that in this case the potential has two singular points on the attachment between the meniscus and the capillary tube (see Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF]). We prefer to deal with conically shaped capillary tubes, that is with a system where the maximum radius of curvature and the maximum electrostatic surface-charge density are on the "top" of the meniscus. In this way we avoid the presence of corners, which introduce strong singular points for the electric fiel and complicate the problem of singling out the effects of Tolman-like corrections. Indeed, in the apparatus which we introduce, the surface formed by the union of the meniscus and the outer surface of the pipet is a smooth equipotential surface.

Variational method for menisci of parabolic shape

In the literature, it is usually assumed that the menisci have a rotationally symmetric parabolic shape. In this paper, we fin the shape of the meniscus by solving, via a mixed numerical-analytical method, the system (13), ( 14). In addition, in order to validate this mixed method, the system ( 13), ( 14) for γ(R) = γ 0 is also solved by a variational method by assuming that the meniscus has a parabolic shape parametrized by its height z m . This can be done by minimizing the free energy as a function of z m when a parabola is chosen as a test-function.

The free energy expression is composed of three parts: the gravitational energy, g ,the energy due to the surface tension, γ , the electrostatic energy plus the work done by a generator keeping the tube-meniscus system to a fixe drop of potential, e ,viz.

= g + γ + e , ( 18
)
where

g = - V ρ l gdV , γ = S γds, e = ε 2 D E 2 dV . ( 19
)
V and S are the volume and the surface of the meniscus respectively, and D is the region in which the electric fiel is not negligible. The family of test functions that minimizes (18) is constructed in Appendix 1 as is the critical value for the height z m . In so doing the expressions for the gravitational and surface energies are identical to those given in Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF]. Equation (19), on the other hand includes both aforementioned electric terms.

For the electrostatic energy we use the expression

e = 2πεU 2 0 R eff R 2 z m ln[1 + 4zm (d -zm ) R 2 ] , (20)
which is derived in Appendix 1 and motivated in Sect. 4 below. Here, R is the radius of the lower basis of the truncated conical pipet, d is the distance between this last basis and the plate and R eff is define in Appendix 1. There, the electric fiel between the plate and the conducting pipet is estimated by assuming both to extend to infinity This assumption does not introduce relevant errors in the local estimate of the electric fiel in the neighbourhood of the meniscus needed in [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]. However, in the global variational energy estimate (19) one must account for the finit amount of energy of the electric fiel generated by the potential drop, U 0 , between the plate and the pipet the extensions of which are finite This is exactly done by introducing R eff which is determined via a least square adjustment to numerical curves and is seen to have a value of about R/3.

An approximate solution of (10) and (11)

Our procedure to calculate the menisci profil is based on the steps described in the following subsections.

Numerical procedure to integrate (11)

After having found for [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] an equivalent expression in normal form (see Sect. 5.2 for a more detailed discussion on the subject) we integrate ( 14) by the standard Newton integrator supplied in Mathematica c . A shooting procedure is enployed based on the Cauchy initial values given by the estimate z (0) = z m and the condition z (0) = 0 imposed by the cylindrical symmetry. The z m values are then adjusted by a sequence of integrations until the condition z (R) = 0 is verifie within an error less than 10 -4 . The accuracy of the method is limited by the machine precision, which in single precision is 10 -8 . We fin that this precision is sufficien to catch the main geometrical features of the menisci.

The robustness of the shooting method was further tested by the following two alternatives: (i) We used as Cauchy initial data z (R) = 0, z (R) = z R and tuned this last value until the boundary condition z (0) = 0 was verified The solution and the boundary conditions in the two alternative shootings are reproduced within the accuracy of the integration routine. (ii) Both shooting procedures were tested by starting from the corresponding arrival boundary point and values and integrating back to the initial point. The success of this test is a necessary condition to the stability of the employed numerical procedure.

Approximate analytical determination of the electric field on the surface of the meniscus

In ( 11), there appears the value of the electric fiel on the surface of the meniscus. In the numerical procedure described in the previous section an approximate analytical expression is used that is obtained as follows: we solve in the firs step [START_REF] Dell'isola | Radius and Surface Tension of Microscopic bubbles by second gradient theory[END_REF] in parabolic coordinates. As the shape of the meniscus is very similar to a parabola and the plate can be regarded as a distant and fla parabola we use firs step to estimate the electric potential in the neighbourhood of the meniscus. In other words we surmise that the potential generated by the meniscus has the same functional form, as generated by a locally-close parabolic meniscus. Indeed, even if the meniscus does not exactly possess parabolic shape the potential on the z (r) surface is still given as a function of z by (28) below, and this expression will be used, in a second step, when solving [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF].

This firs step can be considered the initial block of an iterative method in which the new shape of the meniscus can be used to calculate a new potential and so on; the convergence of this method is so fast that it determines the limit potential after only one iteration. This we have checked numerically via the estimation of the second iteration correction which is two orders of magnitude smaller.

First step: Solution of (10) in parabolic coordinates

The electric fiel generated by two homofocal parabolas (Durand [17]) is

E (ζ, η) = k ζ 2 + η 2 • 1 ζ , ( 21
)
where η and ζ are the parabolic coordinates which are related to the cylindrical coordinates by /2), one can easily deduce from (29) (see Durand [START_REF] Durand | Electrostatique, Tome II[END_REF])

z = ζ 2 2 - η 2 2 , r = ηζ. ( 22
k = - U 0 ln(ζ 1 /ζ 0 ) . ( 23
)

Second step : Approximation for the electric fiel close to the meniscus

The assumption that the form of the plate is seen by the meniscus as a parabola is equivalent to the inequality √ 2d >> 2z m (24) so that when estimating the electric fiel close to the meniscus the parabola with vertex V 0 ≡ (0, -ζ 2 0 /2) = (0, -d ) can with sufficien approximation be replaced by a plane. The electric fiel on the firs parabola is given by

E (ζ = 2z m , η) = - U0 ln (zm /d ) 1/2 2z m + η 2 • 1 √ 2z m , ( 25
)
or in view of ( 29), (30), since

η 2 = √ z 2 + r 2 -z , ( 26
)
E (z , r) = - U0 ln (zm /d ) 1/2 2z m + √ z 2 + r 2 -z • 1 √ 2z m . ( 27
)
This last equation can be considered as a local approximation for the electric fiel close to the meniscus also when its shape, remaining close to the parabolic one, is not exactly parabolic. If the shape of the meniscus in cylindrical coordinates is given by the equation z = z (r) we are allowed to still use the obtained functional form for the field i.e.

E (z , r) = - U0 ln (zm /d ) 1/2 2z m + z (r) 2 + r 2 -z (r) • 1 √ 2z m . ( 28
)
This is the expression of the electric fiel in which the form z (r) of the meniscus and its height z m at the center appear explicity. We will use it to solve [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF], which thus becomes an ordinary differential equation for the unknown function z (r).

Discussion of the numerical results

The case of "large radius" capillary tube

For a capillary tube with radius much greater than R 0 we can neglect the Tolman-like correction and solve the system of equations in which γ(R g ) = γ 0 . In Fig. 4a the results of the integration for different values of the potential are shown; in the same figur the parabolic shapes corresponding to the same height of the meniscus are also shown. The values of the parameters are listed in Table 2.

Table 2. Values of parameters used for quantitative evaluations

H = 2.00 • 10 -3 m γ 0 = 2.01 • 10 -2 N /m d = 10 -2 m ρ l = 930.00 K g/m 3 R 0 = 1.49 • 10 -7 m
We observe that for high values of the potential the menisci become more "fat" i.e., more concave with respect to the parabolically shaped menisci. Note that the maximum potential (limit potential), above which the meniscus is unstable, is reached when z m = R. This is a well-known property of pending menisci: Our choice of the geometry of the pipet allow us to fin all shapes up to the theoretical limit value. In Fig. 4b the surface charge density for all calculated shapes of the meniscus is shown. The maximum of the surface charge density occurs at the tip of the meniscus as expected. The numerical values are very close to those found by Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF] in a similar case. In Table 3a the calculated values for z m are listed for varying U 0 . In Fig. 4c the meniscus height z m is plotted vs. the applied drop of the potential as estimated with both the 2: a Profile of the meniscus for the values of a potential drop (a-g) as listed in Table 3a, using (29) (solid line). In the same figur is also shown the parabolic approximation (dashed line starting at the same values of normalized height zm ) for the same values of potential drop, b normalized surface charge density for the same potentials (a-g) as in a, c normalized meniscus height vs applied potential found using the variational method ( ) and using (29) ( ) variational method described in Sect. 3.2 and the two-step method described in the previous section. The best-fi test shows that the plotted dependence is very close to be quadratic when the more accurate second method is used.

In concluding the comparison of our results with those found by Joffre [START_REF] Joffre | Profil de Menisques Electrises avec ou sans Emission de Charges[END_REF] we explicitly note, however, that his limit potential, and consequently his limit height, was found to be smaller than the theoretical limit owing to the different solution adopted for the tube-menisci system. Most likely, our solution is smoother for the truncated conically shaped pipet, because it is not affected by an edge singularity in the surface charge.

The case of "small radius" capillary tube

In Sect. 5.1 we were allowed to use γ(R g ) = γ 0 . Instead, when the mean curvature radius R g is comparable with the radius R 0 Tolman-like corrections should be introduced: this is the case of "small" radius capillaries.

We consider all material parameters having the same value as in the previous section but the capillary radius is now 10 -3 times smaller. We consider therefore (3) and ( 4) with the value of R 0 = 1 5 R and α = 1 (see dell'Isola et al. [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF]). These choices are surely physically reasonable at least in the neighbourhood of the critical temperature and/or for fluid with large molecules (e.g. for polymeric fluids) Tolman-like corrections transform ( 11) into an ordinary differential equation which is not in normal form. Indeed, [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] takes the form

F (z , z , z ) = H (z ), () = d dr , ( 29
)
where H is the function of z obtained by summing up the firs and third addend in [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] and F is a trascendental function of the argument z . This circumstance represents a remarkable difficulty to overcome it one can try two methods. In the first we could try to numerically solve by a finit difference scheme the trascendent algebraic equation F (z 0 , z 0 , z ) = H (z 0 ) for every choice of initial data (z 0 , z 0 ). This method proved to be poorly convergent near the top of the menisci, where we expect to fin the most interesting physical features of the studied system. In the second, we solved (29) iteratively by findin at step n the solution of the problem

F (z n-1 , z n-1 , z n ) = H (z n ). ( 30
)
This methods demonstrates a very efficien convergence when z 0 is chosen to be of parabolic shape. Indeed with this choice z 1 (r) and z 2 (r) are nearly coinciding within an error of approximately 2%. Therefore, the Tolman-like corrections can be transformed from a functional operator depending on z (r) into a known function T i (r, U 0 ) := γ i (R g (r, z 0 (r)) i = 1, 2, 3. Here, R g (r, z 0 (r)) denotes the mean radius of curvature corresponding to the meniscus the shape of which is given by z 0 (r) of the variable r by simply replacing z (r) by the best parabolic fi corresponding to the potential drop U 0 . In Fig. 5 the functions T i (r, U 0 ) are shown as functions of r for different values of U 0 ; it must be underlined that, as expected, the differences 1 -T i (r, U 0 )/γ 0 attain their maximum at the top of the menisci. Using the parameters listed in Table 2 we display in Fig. 6 for different values of U 0 the shapes of the menisci with and without the Tolman-like corrections. We note that the firs two Tolman-like corrections produce slightly different profile for the menisci: for a fixe potential U 0 the height of the meniscus is bigger for the correction (4). For the third correction the effect is qualitatively opposite. We do not show the surface charge density plots because in the considered case they are qualitatively very similar to those of the large radius case.

In Table 3b we list the calculated values for z m for variations of the value of U 0 and different Tolman-like corrections.

In Fig. 7 we plot the height z m of the equilibrium meniscus as a function of the potential drop U 0 for constant γ and for the three Tolman-like corrections. We remark that: (1) for the chosen material parameters the firs two Tolman-like corrections show the same qualitative and nearly the same quantitative behaviour;

(2) the difference between the estimated z m in the case γ = γ 0 and in the case of Tolman-like corrections is an increasing function of U 0 ; (3) for the firs two Tolman-like corrections the limit equilibrium drop of potential (i.e. the potential corresponding to z m = 1, in normalized units, case in which the tip height equals the pipet radius and the theoretical maximal height of the meniscus tip is attained) is smaller in the case of Tolman-like corrections than in the case γ = γ 0 ; the absolute value (with our choice of material parameters) of this difference amounts to 40 V , corresponding to a decrease of an amount of 5%; this effect can be easily shown by electric measurements and is physically evident: indeed owing to the "weakening" of surface tension, especially at the top of the menisci, because of Tolman-like corrections a greater height with respects of the case γ = γ 0 is reached; consequently the equilibrium limit potential is found lower. (4) For a length scale effect the limit drop of potential decreases as a function of the capillary radius (for more details about this point see dell'Erba [18]); (5) as in the case of large capillary radii the dependence of z m on U 0 is very close to a quadratic one.

Conclusions

Tolman [START_REF] Tolman | The effect of droplet size on surface tension[END_REF], using the Gibbs' excess theory, deduced a formula relating the surface tension to the equilibrium radii for droplets and bubbles that are immersed in their own vapour. La Mer and Pound [START_REF] Lamer | Surface Tension of small Droplet from Volmer and Flood's nucleation data[END_REF] proved that available experimental evidence was in contradiction with Tolman's theoretical prediction. Fisher and Israelachvili [START_REF] Fisher | Determination of the capillary pressure in menisci of Molecular Dimension[END_REF] designed a more precise experimental apparatus which showed for microscopic curved interfaces a dependence of the surface tension on curvature. Alts and Hutter [START_REF] Alts | Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water, Part I,II,III,IV[END_REF], using the modern continuum mechanics approach firs made plausible the constitutive dependence of surface tension on interface curvature. Dell'Isola et al. [START_REF] Dell'isola | Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations[END_REF] using second gradient 3-D theories, proved that equilibrium surface tension must depend on curvature simply as a result of the second law of thermodynamics and proposed, starting from the equations of state, some equilibrium relations for macroscopic surface tension as a function of curvature (see ( 3) and ( 4)).

In this paper we prove that a Tolman-like dependence of the surface tension on mean curvature has effects on the shape of some equilibrium electrically charged vapour-flui interfaces which could be measurable. Indeed for electrically conducting materials we propose to use an experimental apparatus in which the limit equilibrium drop of the electric potential (before instability) depends notably on the Tolman-like correction for surface tension and, in particular, on its dependence on the mean curvature. This experimental apparatus seems suitable to detect efficientl the searched physical effect, as it is based on the measurements of the electrical potential drop; these measurements can much more precisely be conducted than purely mechanical ones, due to the refine electronic equipment now available. Moreover, we fin how the equilibrium radius of charged drops depends on their surface charge.

Possible developments of our results could concern (1) a more refine numerical analysis for the prediction of the behaviour of the presently proposed experiment; (2) the study of menisci with charge emissions; (3) the study of instabilities and rupture profiles (4) the study of the effect on surface tension and shape of menisci of the presence of impurities in one or both present phases.

where the last equality must hold for every r. As a consequence we obtain

h(z m ) = R 2 4z m -z m . ( 37
)
Therefore, using parabolic coordinates, we have

ζ 2 0 = 2(z m + h(z m )) = R 2 2z m , η 2 = 2z m r 2 R 2 . ( 38
)
We now assume that, for every z m , the parabola representing the meniscus is placed in front of a parabola, representing the plate, which is a distant d apart from it in the increasing direction of the z -coordinate. Using the solution in parabolic coordinates of the Laplace electrostatic problem found in Durand [START_REF] Durand | Electrostatique, Tome II[END_REF] E (ζ 0 , η) = k

ζ 2 0 + η 2 • 1 ζ 0 , ( 39
)
we fin that the normal component of the electric fiel to the meniscus is

E (ζ 0 = R 2 2z m , η) = k R 2 2zm + η 2 • 1 R 2 2zm , ( 40
)
where, as the plate is placed at a distance d from the meniscus,

k := 2U 0 ln (d + h(z m ))/(z m + h(z m )) . ( 41
)
We are now ready to estimate the electric fiel energy e : we start by remarking that in our case it is given by

e = QU 0 2 ( 42
)
where Q is the total surface charge on the union of the pipet and the meniscus and is given by

Q = S ∪P εE ds, ( 43
)
where E is the component of the electric fiel normal to the surface and P is the pipet surface.

To estimate this last integral we remark that the field given by (40), is a good approximation for the normal component of the electric fiel close to the meniscus. However as the fiel (40) is generated by an infinit distribution of charge while the considered system is constituted by finit conducting surfaces it represents an overestimation of the fiel E . Therefore, neglecting the charges on the pipet, we can introduce an effective radius R eff by means of the equality

Q =:
Q = 2πε kz m R eff R 2 , ( 45
)
in which R eff < R because edge effects lower the total charge on the considered finit parabolic conductor relative to the total charge present on the same finit conductor regarded as a part of an infinit one. By replacing (45) in (42) we obtain

e = 2πεU 2 0 R eff R 2 z m ln[1 + 4zm (d -zm ) R 2 ] . ( 46
)
A simple test of the validity of ( 46) is obtained if it is applied to plane circular menisci with very large radii. This is done by estimating the limit For every R eff we thus obtain a dependence of z m on U 0 . Among the curves in this family we choose that one which better fit the curve obtained with the method developed in Sect. [START_REF] Fisher | Determination of the capillary pressure in menisci of Molecular Dimension[END_REF]. In this way we obtain R eff ≈ 0.31 R.

(48)

Fig. 1 .

 1 Fig.1. Plot of the functions LC Uc (see Sect. 2) for three different values of the potential drop (0 µV , 2 µV , 3 µV ) vs the normalized radial variable. The curves relative to 0 µV starts from the origin, the other two start from -0.25 and -0.50, respectively. The effect of two Tolman-like corrections is shown by dashed curves (γ 1 large dashes, γ 2 small dashes). These are compared with the nocorrection case (γ 0 solid curve)

Fig. 2 .

 2 Fig. 2. Plots of normalized R min (U ) for different Tolman-like corrections (the data are reported in Table1):γ = γ 0 → Solid line ; γ = γ 1 → ; γ = γ 2 →

)

  Equation ζ = ζ 1 describes the parabola approximating the outer surface of the pipet and meniscus. Its vertex V 1 has cylindrical coordinates (0, -z m ) := (0, -ζ 2 1 /2). Approximating the meniscus as a parabola with equation ζ = ζ 1 and the plate as a parabola with equation ζ = ζ 0 with vertex V 0 the cylindrical coordinates of which are (0, -d ) := (0, -ζ 2 0

Fig. 4 .

 4 Fig. 4. Large radius case; the parameters of the simulations are listed in Table2: a Profile of the meniscus for the values of a potential drop (a-g) as listed in Table3a, using (29) (solid line). In the same figur is also shown the parabolic approximation (dashed line starting at the same values of normalized height zm ) for the same values of potential drop, b normalized surface charge density for the same potentials (a-g) as in a, c normalized meniscus height vs applied potential found using the variational method ( ) and using (29) ( )

Fig. 5 .

 5 Fig. 5. Tolman corrections, the parameters of the simulations are listed in Table 2: a T 1 (solid line) and T 2 (dotted line) for small radius vs normalized radial variable, only the relevant corrections of the set (a-e) listed in Table 3b are shown, b T 1 (solid line) and T 3 (dotted line) for small radius vs normalized radial, only the relevant corrections of the set (a-e) listed in Table 3b are shown

Fig. 6 .Fig. 7 .

 67 Fig. 6. Small radius case: a Profile of the meniscus for the values of potential drop (a-e) listed in Table 3b for the small radius case: with Tolman-like correction γ = γ 1 (dotted line), for Tolman-like correction γ = γ 2 (dashed line), and without correction γ = γ 0 (solid line), b Profile of the meniscus for the values of potential drop (a-e) listed in Table3bfor the small radius case: with Tolman-like correction γ = γ 1 (dotted line), for Tolman-like correction γ = γ 3 (dashed line), and without correction γ = γ 0 (solid line)

  which J (r, ϑ) is define by ds =: J (r, ϑ)drd ϑ and E is given in (40). As a consequence

  well-known classical result.

Table 3a .

 3a Normalized values of zm as function of drop potential for a "large capillary radius" (R = 0.745 • 10 -3 m)

		U 0 (V ) zm
	a	0	0.1834
	b 1000	0.2436
	c	2000	0.3542
	d 3000	0.4848
	e	4000	0.6412
	f	5000	0.8488
	g 5500	0.9958

Acknowledgement. We thank Prof. A. Di Carlo and Prof. K. Hutter for fruitful discussions and their friendly criticism.

Appendix 1

In Sect. 3.2 we introduced a variational formulation of the problem of the determination of equilibrium menisci. This formulation is essentially based on the representation formula (20) for the electrostatic energy: in the present Appendix we justify it.

We consider parabolic and cylindrical coordinates (ζ, η) and (r, z ) respectively. The coordinate lines of the parabolic system are parabolas with axis of simmetry r = 0. The coordinate transformation is given by

We now consider the family F of parabolas given by the equation

parametrized by the variable z m . We shall fin the expression for h(z m ) by assuming that every element of F is a coordinate curve of the introduced parabolic system of coordinates. Therefore, F can be parametrized also in terms of ζ 0 as follows ζ = ζ 0 and η variable.

We now request that the parametrization (32) is equivalent to (33) and obtain the set of equalities


Max Silberztein
email: max.silberztein@univ-fcomte.fr

Complex Annotations with NooJ

NooJ associates each text with a Text Annotation Structure, in which each recognized linguistic unit is represented by an annotation. Annotations store the position of the text units to be represented, their length, and linguistic information. NooJ can represent and process complex annotations, such as those that represent units inside word forms, as well as those that are discontinuous. We demonstrate how to use NooJ"s morphological, lexical, and syntactic tools to formalize and process these complex annotations.

Introduction

When parsing a text or a corpus, NooJ builds a Text Annotation Structure (TAS) in which each linguistic unit is represented by a corresponding annotation. An annotation stores the position in the text of the text unit to be represented, its length, and linguistic information, cf. [START_REF] Silberztein | NooJ"s Linguistic Annotation Engine[END_REF].

Typically, NooJ"s annotations represent prefixes (e.g. "dis-"), affixes and suffixes (e.g. "ization"), simple words (e.g. "table"), multi-word units (e.g. "round table"), frozen expressions (e.g. "take ... into account") as well as named entities such as dates (e.g. "on Monday, October the third"), person names ("Dr John W. Smith"), addresses, locations, numerical expressions, etc.

NooJ adds annotations to the TAS automatically at various stages of the analyses. At the lexical level, its lexical parser typically applies dictionaries to the text to produce annotations. NooJ"s dictionaries are an enhanced, unified version of INTEX"s dictionaries: because it handles morphological and derivational morphology, there is no longer a need for DELAFtype inflected forms dictionaries. NooJ handles simple words and multi-word units in a unified way, therefore there is no longer any distinction between DELAS and DELAC dictionaries. Finally, lexical entries can be associated with spelling or synonymous variants: no need for DELAV-type dictionaries. Cf. (Silberztein 2004).

Morphological and lexical annotations are ambiguous in general. After NooJ has performed a lexical analysis of the text, the TAS represents ambiguities between simple words, such as "fly" (Noun or Verb), between a sequence of simple words and a multi-word unit, such as "round table" (a table or a meeting), between various morphological analyses, between frozen expressions and sequences of multi-word units and simple words, etc.

For instance, NooJ"s lexical parser automatically annotates the text "He cannot take the round table into account" with various linguistic units, among them two simple words: <he,PRO> and <the,DET>, two morphological units: <can,V> and <not,ADV>, one multi-word unit: <round table,N> and one discontinuous verbal expression <take into account,V>.

Morphology

When an annotation represents a theoretical linguistic unit that does not occur as is in the text, its position or its length might be a decimal number. For instance, in the previous example, the annotation <can,V> occurs at position 3, and has a "length" 0.1; the annotation <not,ADV> occurs at position 3.1, and has a "length" 5.9. This decimal topology allows word forms to be split into any number of linguistic units (which is crucial when processing agglutinative languages), and always guarantees a perfect synchronization between the TAS and the original text, which is crucial for further processing.

For instance, NooJ"s query "<V> <ADV>" (a verb followed by an adverb) would match both texts "cannot" and "can never" in the exact same way: NooJ"s syntactic parser processes sequences of annotations, and does not even know that originally, these two sequences of two linguistic units were processed differently by NooJ"s lexical parser. This characteristic is much more than an elegant trick: it allows linguists to design grammars at the syntactic or semantic levels without having to deal with a large number of orthographic and morphological idiosyncrasies. For instance, let us consider an English syntactic grammar VG that recognizes complex verb groups. This grammar may include the following rule: <V+Modal> <ADV>* <V+INF> (a modal verb followed by an optional sequence of adverbs, followed by an verb at the infinitive). Thanks to NooJ"s lexical parser that represents morphological linguistic units and "regular" lexical units in a unified way, this single rule represents and automatically recognizes the following four sequences of text: can never come, cannot come, will always come, won't always come Being able to take care of complex tokenization problems once and for all at the lexical and morphological levels, independently from the forthcoming higher levels of analyses (syntactic and semantic), is crucial for formalizing a number of languages, such as Asian and Germanic languages.

Finally, note that many NLP applications benefit from the perfect synchronization of the linguistic information and the original text. For instance, NooJ"s users can use the TAS in order to color the original text, which could have a complex file structure (such as an HTML page). This synchronization also allows users to import XML documents, annotate them, and then export the TAS to build a new XML document, which in turn can be re-imported into NooJ for further analyses. Cascades of linguistic processing can further be automatized with NooJ"s companion program "noojapply.exe".

The synchronization also allows users to perform partial replacements in texts, such as translations into another language of certain terms or expressions, or production of a formal notation for expressions (such as a semantic PROLOG-type notation), etc.

Formalization

In order to tell NooJ that a word form or a set of word forms must be associated with more than one linguistic unit, one can build a morphological grammar or a dictionary.

--Morphological grammars are finite-state transducers that recognize a set of word forms and then produce the corresponding sequence of annotations. For instance, the following morphological grammar recognizes the word form "cannot" and then produces the sequence of two annotations: "<can,V> <not,ADV+Neg>":

Figure 2: A Morphological Grammar --Sets of word forms are sometimes better represented in dictionaries. For instance, the set of English contractions is represented in the English dictionary "Contractions.dic" stored in NooJ"s English module: I'm,<I,I,PRO><am,be,V+PR+1+s>+UNAMB you're,<you,you,PRO+2+s><are,be,V+2+s>+UNAMB he's,<he,he,PRO+3+m+s><is,be,V+PR+3+s>+UNAMB it's,<it,it,PRO+3+n+s><is,be,V+PR+3+s>+UNAMB ...

After having applied this dictionary to texts, each sequence "I"m" will be internally represented in the TAS as a sequence of the two annotations "<I,PRO> <be,V+PR+1+s>".

NooJ"s English module contains a dozen dictionaries and morphological grammars, including one to tokenize contracted negations (e.g. "don"t"), one to recognize prefixed verbs (e.g. "disrespect"), various grammars to recognize derivations (e.g. "Frenchify"), as well as derivations of proper names (e.g. "Bushization"), etc.

Frozen expressions

Although a number of NLP applications --most taggers and corpus linguistic tools --simply ignore multi-word units and frozen expressions, it is crucial to represent, recognize and process them. For instance, consider the following English text sample ( 1 ):

Battle-tested Japanese industrial managers here always buck up nervous newcomers with the tale of the first of their countrymen to visit Mexico, a boatload of samurai warriors blown ashore 375 years ago. From the beginning, it took a man with extraordinary qualities to succeed in Mexico, says Kimihide Takimura, president of Mitsui group's Kensetsu Engineering Inc. unit.

Underlined sequences are multi-word units or frozen expressions that should be processed as atomic units, rather than word-by-word: for instance, "Battle-tested" translates in French as "aguerris", and not (word-by-word) as "? testés par une bataille". "industrial managers" must not be analyzed in the same way as thousands of sequences such as "industrial food", "industrial soap", "industrial diamond", by a productive rule such as:

an industrial <N> is a <N> that was made with industrial methods An industrial manager is not a "manager made with industrial methods": he/she is a manager in the industrial world.

In NooJ, simple words and multi-word units are processed in a unified way: they are stored in the same dictionaries, their inflectional and derivational morphology is formalized with the same tools, and their annotations are undistinguishable from those of simple words. This T 1 T The tagging of this text is commented on in [6], but there is no mention of its MWUs.

unifies the treatment of linguistic units, whether simple words or multi-word units. Thus, the simple syntactic query: <N> <ADV>* <V> (a noun, followed by an optional sequence of adverbs, followed by a verb) recognizes the two following text sequences:

... meeting suddenly stopped ... ... round table all of a sudden was ... In the second example, <N> matches "round table" and <ADV> matches "all of a sudden".

However, the fact that frozen expressions are potentially discontinuous in texts makes it more complicated to formalize them. Two operations are necessary:

(1) We need to give NooJ a way to recognize each occurrence of the frozen expression, even when the frozen expression is discontinuous. For that purpose, we create a syntactic grammar.

(2) We need to produce an annotation for the sequences recognized in (1), in which all the components of the frozen expression are linked together.

The following is a simple grammar that can be used for that purpose: ... took the meeting into account ... We assume here that noun phrases have already been annotated as <NP> by a previous analysis of the text ( 2 ). In the output of the grammar (written in bold, under the nodes), the variables $V and $O2 refer to the text that was matched between the red parentheses named V and O2. The morphological operator "_" is used to lemmatize the content of variable $V, i.e. "took" produces "take". The concatenation operator "#" is used to concatenate the lemmatized value $V_ and the value of $O2 to get the result "take into account". Thus, applying this grammar to the previous text produces the sequence of two annotations:

<take into account,V+CNP2+XREF> <XREF>
The keyword XREF is used to link together all the components into a single, discontinuous annotation. The resulting annotation resembles the one that was displayed in figure 1. The result of applying this grammar to the text is that the discontinuous expression has been successfully represented as a single unit in the TAS.

Lexicon-Grammars

The previous grammar successfully represents and recognizes occurrences of the single expression "to take <NP> into account". However, as the number of expressions to be represented becomes extremely numerous (tens of thousands), it is not efficient to build one grammar per expression. In that case, we need to use a combination of a dictionary and a grammar.

The 'INTEX' method

The software INTEX (see http://intex.univ-fcomte.fr) could already link together a dictionary and a grammar in order to formalize frozen expressions and recognize them automatically in texts. More specifically, INTEX uses:

(1) a "lexicon-grammar" table in which all frozen expressions of a certain structure are listed and described. For each possible structure of a frozen expression, there is a corresponding lexicon-grammar table (See [START_REF] Gross | Une classification des phrases figées en français[END_REF][START_REF] Gross | Les phrases figées en français[END_REF].

In INTEX, each lexicon-grammar table is represented as an MS-EXCEL-type file in which each frozen expression is listed in one line. A set of relevant linguistic properties, such as the distribution of the expressions" arguments (Subject =: NHum, Object =: NConc, etc.) and their structural variants (e.g. +Passive) are listed in columns; each cell contains either a string (e.g. "into account") or a binary code ("+" or "-") to express that an expression has a property or not.

For instance, INTEX"s table C1d.xls (see figure below) describes a set of 1,663 French frozen expressions in which a verb is frozen with its direct object complement (as in the English expressions "to lose one"s mind", "to catch a cold", etc.). The two first columns encode the distributional class of the subject of the expression (Human or Non-Human); the third column marks the expressions that are obligatorily negative (e.g. "il n"a pas inventé l"eau chaude", literally "he has not invented warm water"); the fourth column encodes a potential pre-verbal particle, the sixth column encodes the fact that the object complement is optional or not, etc. (2) a "meta-grammar" in which the syntactic contexts of the frozen expressions are formalized. INTEX"s meta-grammar is a specific type of syntactic grammar strongly linked to the above-mentioned table. Typically, the meta-grammar contains paths that represent elementary sentences such as NP V NP, NP V PREP NP, etc. as well as complex sentences that contain potential adverbial insertions, e.g. NP ADV V NP, verbal constructions with auxiliary, modal and aspectual verbs, as well as transformed structures (extraction, passive forms, etc.). The meta-grammar is linked to the table through the use of special variables that instantiate the grammar"s lexical symbols. For instance, the meta-grammar"s variable "@5" refers to the content of the table"s 5th column (in Figure 4: the verb of the frozen expression); variable "@3" has a Boolean value that encodes expressions that must be recognized only in a negative construction, etc.

From the table and its meta-grammar, INTEX could automatically compile an autonomous syntactic grammar that can be applied to texts to automatically recognize occurrences of frozen expressions. [START_REF] Vietri | Lemmatization of Idioms in Italian[END_REF] shows how to formalize with INTEX several Italian frozen expressions" lexicon-grammar tables.

Problems

This method involved using two objects that were specifically designed for the single task of frozen expression recognition: the lexicon-grammar table was incompatible with other INTEX dictionaries, and the meta-grammar was incompatible with other INTEX syntactic grammars.

Because it was not possible to merge the meta-grammars with other grammars, such as the one for noun phrases, elementary sentences, coordination, etc., each meta-grammar had to be autonomous, i.e. it had to formalize every syntactic in which a frozen expression could occur. But most frozen expressions are used in syntactic contexts that mimic free ones. For instance, the syntactic structures of the two sentences "John lost his mind" (frozen) and "John lost his hat" (free) are identical. Surely there might be a way to describe these two sentence structures with one unified tool! Unfortunately, INTEX forced linguists to describe these two structures twice, and, even more problematically, with two very different and incompatible tools.

Finally, each lexicon-grammar table / meta-grammar association had to be compiled into a finite-state transducer in which all the components of each frozen expression were represented: the resulting finite-state transducer was huge, typically hundreds of thousands of states, which is significantly larger than even the largest FSTs that encode INTEX"s dictionaries.

The NooJ method

In NooJ, by contrast, instead of compiling a finite-state transducer from a table and a metagrammar, we use both a standard NooJ dictionary, and a standard syntactic grammar at runtime: there is no third, redundant object to compile.

--The dictionary stores one characteristic component of each expression, i.e. a word form or a word sequence that occurs every time the expression occurs.

For instance, for the previous frozen expression "to take ... into account", the word "take" and the sequence "into account" are both valid characteristic components. However, for a number of expressions such as "to be in trouble", the verb can be replaced with a variant (e.g. "to get in trouble") or even be deleted (e.g. "John, in trouble, decided to run away"). In those cases, the characteristic component of the verbal expression cannot be the verb to be; it will be the noun "trouble" or the sequence "in trouble".

--The characteristic component stored in the dictionary is associated with all the frozen expression"s properties, as well as with the other components of the expression. For instance, here is a lexical entry that could be used to represent the frozen expression "to take something into account": take,V+FLX=TAKE+SHum+OFree+O2="into account" V stands for verb; FLX=TAKE says that the lexical entry inflects according to the paradigm TAKE; SHum says that the subject of the expression is Human; OFree says that there is no distributional constraint on the direct object complement; O2="into account" stores the frozen component of the expression in the property field O2. Thus, any reference to the variable $V$O2 in the grammar will be automatically instantiated as the text "into account".

In the previous example, the property field O2 has a constant value: the sequence of two word forms. But a property field can also store a regular expression. For instance, the frozen expression "to lose one"s mind" can be described by the following entry: lose,V+FLX=LOSE+SHum+O="<DET+Poss> mind"

In the corresponding grammar, the variable $V$O will be instantiated as "<DET+Poss> mind", which then will match "my mind", "your mind", "his mind", "her mind", "its mind", "our mind", "their mind".

Important: the dictionaries that store characteristic components of frozen expressions are exactly in the same format as regular NooJ dictionaries: the fact that all NooJ dictionaries have the same format allows linguists to unify the description of all types of linguistic units: from morphemes to frozen expressions (See [START_REF] Silberztein | NooJ"s Dictionaries[END_REF].

--This syntactic grammar describes the syntactic contexts of frozen expressions, in the exact same way as a "standard" syntactic grammar would describe the same contexts. Just like with any other syntactic grammars, the syntactic grammar that represents frozen expressions may include references to its lexical entries" properties. Here, more specifically, when dealing with frozen expressions, it will include references to the fields that encode the various frozen components of the frozen expressions (above: $V$O2 or $V$O).

The following syntactic grammar represents all the frozen expressions of the type "take X into account": Figure 5: A Generic Syntactic Grammar for an Open Set of Frozen Expressions Notice how similar this grammar is to the one in Figure 3: the lexical symbol <take> (any conjugated form of to take) has simply been replaced with the generic symbol <V> (any verbal form); the constant "into account" has been replaced with $V$O2, i.e. the value of the property field O2 of the verb. When parsing the text John took the problem into account, the result is exactly the same as if we were applying the specific grammar, but the generic grammar can handle hundreds of expressions of the same type: consider X with interest, have X in mind, set X in motion, treat X with respect In terms of computing requirements, the generic grammar is extremely small (a few bytes), as opposed to the huge Finite-State Transducer that had to be compiled by INTEX (200,000+ states!).

But the most important benefit is that this grammar can be easily merged with a grammar that recognizes free sentences with the same structure, such as John gave a pen to Mary: we just need to add a <PREP> <NP> path in parallel with $V$O2. In other words, NooJ allows linguists to represent, manage, and accumulate a number of syntactic grammars for free structures as well as for frozen structures, in a unified way.

Properties' Inheritance

The previous graph can be successfully used to automatically recognize and lemmatize frozen expressions in texts, but the resulting annotation is poor: it only annotates the recognized sequences as a V+CNP2 occurrence. We need to produce a finer analysis of each occurrence: for instance, we want to describe "took X into account" as an expression in the Preterit, and any syntactic or semantic information that was described in the lexical entry that represents the frozen expression has to be copied in the resulting annotation: the distributional selection of its subject, the possibility of having a passive form, etc.

For those purpose, we use the same special variables as the ones used in NooJ"s morphological parser: $ALLF stores all the lexical entry"s inflectional and derivational properties (such as tense, number), whereas $ALLS stores all the lexical entry"s syntactic and semantic properties. The final syntactic grammar used to annotate all CNP2-type frozen expressions is thus:

Artificial Ambiguities

The characteristic components of a large number of frozen expressions are entered into a dictionary, which might be merged with other NooJ standard dictionaries, or even other dictionaries that store characteristic components of other types of frozen expressions.

As a result, a dictionary might contain thousands of lexical entries for a relatively small number of characteristic components, such as support verbs such as "get", "have", "take", particles such as "in", "up", etc. (these verbs and particles are components of thousands of frozen expressions). In some cases, we can choose the characteristic components to minimize the number of its lexical entries: for instance, we might decide to describe the expression "to take into account" with the lexical entry "account". However, it has to be expected that we will end up processing a large number of lexical ambiguities, much larger than the ones we are accustomed to at the lexical and morphological levels (3 to 10 in Romance languages).

NooJ (as opposed to INTEX) has been designed from the ground up to handle such large degrees of ambiguities. Importantly, however, these ambiguities are artificial and unnecessary, and therefore should not pose a problem in the first case. For instance, the hundreds of lexical entries for the characteristic component take do not correspond to the "real" verb to take, but rather to a hundred independent frozen expressions or combinations of the support verb and its predicative noun.

In order to stop NooJ from processing characteristic components as highly ambiguous "ordinary" lexical entries, we have added a specific property: FXC (Frozen eXpression Component), that states that the current lexical entry must not be processed as an ambiguous linguistic unit. Thus, the frozen expression "take into account" is described by the following lexical entry: take,V+FXC+FLX=TAKE+SHum+OFree+O2="into account" Now, even if there are thousands of lexical entries for "take", NooJ"s lexical parser will produce annotations only for the lexical entries that are not marked as FXC. Therefore, the annotations that represent frozen expressions will only be produced by a grammar.

Conclusion

The combination of the special property FXC in dictionaries and the special code XRF in grammars allows linguists to use standard dictionaries and standard grammars to represent and automatically annotate both free structures and frozen expressions with the same tools, even when the latter are discontinuous.

We have successfully formalized a description of over 12,000 English phrasal verb in the form of a dictionary "phrasal verbs.dic" and the corresponding syntactic grammar "phrasal verbs.nog" (see Machonis 2007, intra). Here are three entries that correspond to the phrasal verbs to give away, to give back, to give up:

give,V+PV+FXC+Part=away+N0Hum+N1Hum+N1NHum+NoPart+FLX=GIVE give,V+PV+FXC+Part=back+N0Hum+N1Hum+N1NHum+NoPart+FLX=GIVE give,V+PV+FXC+N0Hum+N0NHum+Part=up+N1Hum+N1NHum+FLX=GIVE

The particle is described in the property field Part; the distributional selections of the subject and the object are described (e.g. N0Hum), and if the particle can be omitted, the expression has the property NoPart.

Applying this dictionary to large texts takes exactly the same computing time as applying any other NooJ dictionary. Although the use of property field variables is likely to consume some computing power, in practice we have not noticed any difference when applying "regular" and "enhanced" syntactic grammars to large texts.

Figure 1 :

 1 Figure 1: Lexical Analysis of a Simple Text

Figure 3 :

 3 Figure 3: Grammar for take NP into account

Figure 4 :

 4 Figure 4: A lexicon-grammar table for French frozen expressions in INTEX

Figure 6 :

 6 Figure 6: The Final Grammar

When loading a corpus or text in NooJ, the command Linguistic Analysis applies all lexical and syntactic resources that are selected in Info > Preferences. Syntactic resources are typically local grammars applied in cascade, and used to annotate larger and larger chunks of the text.


Max Silberztein Laseldi
email: max.silberztein@gmail.com

Disambiguation Tools for NooJ

When NooJ performs an automatic lexical analysis of corpora, it recognizes five types of atomic linguistic units (ALUs) and represents them as annotations stored inside each text's annotation structure (TAS).

Unfortunately, the massive level of ambiguities generated by each of the five corresponding parsers produces a TAS far too heavy for most corpus linguistics applications. In consequence, most users' queries produce too many incorrect results. In order to provide a working solution for NooJ's lexical parser's behavior, we have implemented a new set of tools specifically designed to deal with unwanted ambiguities in corpora and texts: automatic and semi-automatic tools as well as a manual access to edit the TAS.

Introduction

With NooJ, linguists can represent five types of Atomic Linguistic Units (ALUs)1 :  Affixes, morphemes and components of contracted words, e.g. dis-, -ization, cannot  simple words and their morphological variants, e.g. a laugh, to laugh, laughed, laughable  multi-word units, semi-frozen terms and their variants, e.g. as a matter of fact, a nuclear submarine  local syntactic units, e.g. complex determiners, dates, e.g. Most of my groups of, Monday June 5th in the early afternoon  discontinuous expressions such as collocations, support verb constructions, phrasal verbs, e.g. to take ... into account, to give ... in

Accordingly, NooJ provides tools to describe these five types of ALUs and recognize them in corpora automatically: dictionaries, morphological Finite State Transducers (FSTs), Recursive Transition Networks (RTNs), as well as the ability to link dictionaries and syntactic grammars to formalize lexicon-grammar tables 2 .

Each level of analysis constitutes an autonomous module. NooJ processes each module one after the other, in cascade form. Therefore, each module must produce a result that has 100% recall, so that the subsequent module processes an input which lacks no potential linguistic hypothesis. The cost of this approach is that the accuracy of each module is very low: all potential linguistic hypotheses have to be produced and transmitted to the next module, however improbable they might be.

The accuracy of a lexical parser will get lower and lower as the precision of the linguistic data increases: for instance, a simple parser might process the word form will as two-time ambiguous (Noun or Verb) whereas a more sophisticated parser will distinguish the different meanings of the noun will (a mental faculty, a legal declaration) as well as of the verb (used to introduce the future, or synonymous to to wish). Clearly, the more sophisticated lexical parser produces a higher level of ambiguity than the simple one, and it takes a more sophisticated linguistic analysis to solve the extra ambiguities.

An ideally tagged text

Because any linguistic analysis must process all types of ALUs (not only the simple words), an ideal tagger should produce a result that looks like the following: Battle-tested/A Japanese/A industrial managers/N here/ADV always/ADV buck up/V nervous/A newcomers/N with/PREP the/DET tale/N of/PREP the first of their/N countrymen/N to/PREP visit/V Mexico/LOC, a boatload of/DET samurai warriors/N blown ashore/VPP 375 years ago/DATE. From the beginning/DATE, it took/EXP1 a/DET man/N with/PREP extraordinary/A qualities/N to/EXP1 succeed/V in/PREP Mexico/LOC, says/V Kimihide Takimura/NPR, president/N of/PREP Mitsui/NPR group's/N Kensetsu Engineering Inc./ORG unit/N.

For instance, it is crucial to tag the multi-word noun industrial managers as a unit, as opposed to produce the two tags "industrial/A" and "managers/N". In most cases, the semantic analysis of the sequence "industrial <N>" is:

Industrial <N> = <N> produced by industrial methods

This productive analysis can successfully be applied to a large number of nouns, such as in the following examples:

Industrial cheese = cheese produced by industrial methods Industrial food = food produced by industrial methods Industrial cloth = cloth produced by industrial methods etc.

But this productive analysis does not apply to industrial manager. If one wants to translate this sequence correctly in French, one has to translate it as a whole, and not word by word. For instance, a correct French translation would be "patron de PME", whereas "gestionnaire industriel" sounds odd at the very least, and is never used.

If it was possible to automatically tag texts correctly, i.e. to automatically annotate all types of linguistic units (and not only simple words), and to produce a 100% correct result, then the example above would constitute a good way to represent the result of the lexical analysis of texts.

Unfortunately, any lexical parser that aims to represent all types of linguistic units while producing a result with 100% recall will produce a high degree of ambiguities, because it is not always possible to remove all ambiguities at the lexical level. For instance, consider the following text:

… There is a round table in room A32 …

The only way an automatic parser could (maybe!) reliably choose between the analysis "round table = meeting" and the analysis "round table = round piece of furniture" is to perform some complex discourse analysis that would take a larger context into account. It is impossible to choose between these two solutions at a lexical level, i.e. at the level taggers and lexical parsers operate. Taggers designed to remove ambiguities at any cost, even when it is impossible to do so reliably, simply ignore multi-word units and use probability or other techniques-based heuristics to "flip coins" and therefore produce results that are useless for many precise NLP applications. Ambiguities are generated at each level of the five lexical analyses. For instance:

 the morphological parser analyzes the word form recollect as the prefix re followed by the verb collect (meaning: collect again), whereas the dictionary lookup will analyze it as a simple verb (meaning: to control oneself). It would take a sophisticated semantic analysis (at least) to choose between the two solutions in the following text:

John recollects the old coins

Does he remembers about them, or did he decide to take his collecting hobby back up again?

 the multi-word recognizer analyses the sequence acid rock as a noun (a style of rock music), whereas the simple word recognizer analyses it as a sequence of an adjective followed by a noun.

 there are several phrasal verb entries for "to back up", depending on the distributional property of their object complements. How could a lexical parser choose between these entries without a distributional analysis of the complement, and without a reference analysis when the complement is implicit? For instance:

John backed his car up → John backed it up (he drove in reverse)

John backed Mary's statement up → John backed it up (he supported her statement) John backed Mary's idea up → John backed it up (he proved her idea right) John backed his computer up → John backed it up (he saved its files)

In conclusion: it is impossible to build an automatic lexical parser that would disambiguate all the linguistic units that occur in texts. Automatic taggers, which aim at this very goal, simply cannot be relied upon. The only way to build a reliable lexical parser is to allow it to represent unsolvable lexical ambiguities. These ambiguities will have to be passed to subsequent parsers that would use syntactic, semantic and/or discourse analysis techniques to solve them (and not in all cases, because there are ambiguous sentences in texts!).

NooJ's Text Annotation Structure (TAS)

NooJ's lexical parser uses parallel annotations, rather than linear tags, to represent the lexical analyses of texts3 . Annotations have two advantages over tags:

 they can represent all types of linguistic units, such as affixes (inside word forms), multi-word units and discontinuous linguistic units 4 ;

 they can be stacked together and therefore represent lexical ambiguities.

For instance, consider the following sentence:

He cannot take the round table into account

In this sentence, the word form cannot corresponds to a sequence of two linguistic units; the sequence round table is ambiguous, and the linguistic unit take into account is discontinuous. The TAS's main application is that no potential linguistic unit is left out: for instance, if the next sentence in the text is: "we will have to postpone it", a semantic parser can infer that round table refers to a meeting, not to a piece of furniture. Conversely, if the next sentence in the text is: "we will have to move it closer to the window", the semantic parser can infer that the round table is a piece of furniture. Both solutions are alive and ready to be chosen.

Another advantage of the TAS is that it unifies all types of linguistic units. From now on, any subsequent parser will process annotations rather than affixes, simple or multi-word units and discontinuous expressions. For instance, the following NooJ query:

<V> <ADV> <V>

which stands for: extract from the corpus all sequences of a verb, followed by an adverb, followed by a verb, will produce concordances in which various types of sequences are displayed:

… John has often taken … … He cannot take into account the … … She is not finished with …

This characteristic gives NooJ a higher recall factor than other corpus linguistic tools, and is absolutely crucial when dealing with agglutinated languages. For instance, in Arabic, "and in my house" is written as a single word, but NooJ will still process it as a sequence of four distinct annotations. Beyond the simple corpus linguistics applications, the fact that the lexical parser deals with complex word forms allows the subsequent syntactic grammars to be much easier to read and write. For instance, when linguists formalize Romance Languages' noun phrases, they need to take pay special attention to the determiner, which often is contracted with the preceding preposition or adverb. For instance, in French, the word form du is either a determiner or the contraction of the preposition de followed by the determiner le. In practice, this simple problem often leads to an unacceptable number of duplicate and redundant rules in the formalization of noun phrases.

With NooJ however, linguists just have to describe the head of noun phrases independently from the contraction problem, which is solved beforehand by the lexical parser.

Disambiguated texts are useful

Unfortunately, using fine-grained lexical resources with a TAS in which all ambiguities are retained makes NooJ's corpus processing system produce noisy results that are very unattractive to linguists who are used to tagger-based corpus processors that produce much less noise, at the expense of some silence (but of course users don't see the silence).

A more serious problem is related to the use of NooJ as a corpus processing system, especially by nonlinguists who want to analyze texts in other applications: literary, psychological, sociological analyses, etc.

Leaving a large number of ambiguities in TAS often produce very noisy results when a user applies a simple query. For instance, the simple query:

<DET> <N>

(look for all determiners followed by a noun), when applied to the text The portrait of a lady (Henry James, 1881) produces a 22,142-entry concordance with almost 50% noise:

(incorrect matches are selected in the figure). This result is arguably improvable! Let's examine the incorrect results:

… that appeals …

This sequence has wrongly been brought up because that could be a determiner, and appeals could be a noun. That is not the case though: if that were a determiner, we know that it would be a singular one because it is described as such in the English dictionary. However, if the word form appeals were a noun, it would be in the plural. The sequence "that/DET appeals/N" violates the agreement rule between determiners and the following noun; therefore this sequence cannot correspond to the query <DET> <N>. We can remove a large number of incorrect results as can be seen if we replace the flawed query <DET> <N> with the more precise one:

<DET+s> <N+s> + <DET+p> <N+p>

This query reduces the size of the concordance from 22,142 to 8,517 entries! … Elements of one's composition that are not to be eliminated… … It's you that are out of your mind… … didn't you leave all that behind you in Rome?... (are is a noun that represents a metric unit of area). A syntactic parser could help NooJ decide that in these cases, that is a pronoun and thus remove these results.

… that Baltimore was a Western city … … that Bantling hasn't … … we had left that behind long ago …

However, in these cases, even if NooJ had performed a full syntactic analysis of these contexts, it would still have no way of rejecting these results: from a strictly linguistic point of view, that could still be a determiner and behind could be a noun... In conclusion: there are some sequences such as "that appeals" that can be disambiguated with minimum effort: local grammars constitute a very efficient way to remove a large number of ambiguities and thus to clean up the result of users' queries. We do not yet have a full syntactic parser that would help remove a number of ambiguities; however, even if we had a full syntactic parser, there would still be a number of residual ambiguities: it would thus be unavoidable to give users the possibility of removing ambiguities by themselves.

Automatic disambiguation

Local grammars can be used to remove ambiguities automatically. For instance, consider the local grammar in Appendix A that can be used to disambiguate half a dozen very frequent grammatical words which are systematically ambiguous (determiner or pronoun). The grammar merely lists a number of "unambiguous" contexts in which one can disambiguate these words definitively. For instance, her followed by own has to be a determiner as we can easily see in the concordance below. Although this local grammar is very specific, it covers 2.5% of the text. Our goal is to develop a set of 30+ local grammars such as this one: target frequent ambiguous words to be as efficient as possible, while at the same time as specific as possible, so as to avoid producing incorrect results.

Considering the context her own is enough to disambiguate her as a determiner

Semi-automatic disambiguation

The double constraint of efficiency and correctness is often very hard to follow, and in a number of cases, it is much too tempting to design very efficient rules which are correct in all but a few cases … For instance, it seems that all occurrences of the word all followed by a proper name correspond to the pronoun:

The context that <N+PR> is a good candidate for a disambiguation rule This rule would have been very helpful to get rid of the incorrect concordance entries for the abovementioned query <DET> <N>. Indeed, in the text The portrait of a lady, this rule can effectively be used to remove 150 ambiguities. However, it is not very difficult to build an example in which that is followed by a proper name, but is still a determiner:

Are you speaking of that John Doe? Therefore, we need to be able to quickly enter simple disambiguation rules and check them before effectively applying them. In NooJ's new v2.2 version, it is possible to type in a query in the form of an expression associated with an output:

The regular expression that/<PRO> <N+PR> matches all the sequences of that followed by a proper name; then the output <PRO> is used to disambiguate the word form that.

We can edit the resulting concordance in order to filter out incorrect matches, i.e. sequences that were recognized but should not be used. We first select the unwanted concordance entries, then use the command Filter out selected lines of the CONCORDANCE menu to remove them from the concordance, and then Annotate Text (add/remove annotations) to perform the disambiguation for all remaining concordance entries. This process is very quick: users should not hesitate to enter disambiguation commands that are very efficient but have poor accuracy, such as: are/<V> for/<PREP> its/<DET> etc.

The possibility of examining the resulting concordances and then filtering out incorrect analyses allows us to experiment more freely. Finally, note that we can save concordances at any moment: this allows us to validate large concordances at our own pace.

Examining the text's ambiguities

A new tool has been added to NooJ: the possibility of listing ambiguities according to their frequencies, to select one ambiguity and then to apply one solution to the text automatically. The most frequent ambiguous words will be the ones we intend to study in order to disambiguate the text efficiently. Double-click Ambiguities in the text's result window, then click the header "Freq" to sort all ambiguities according to their frequency.

Here we notice that the word form for is ambiguous and occurs 1,919 times in the text. We select the second disambiguation solution (at the top of the window): <for,PREP>. Then, we click one of the colored buttons to build the corresponding concordance. Finally, we filter out the rare cases in which for is actually a conjunction (I did not find any occurrence in the text Portrait of a lady) and then click Annotate text (add/remove annotations) to disambiguate the word form.

Conversely, we can double-click Unambiguous Words in the text's result window in order to display the list of all unambiguous words. Usually, the word forms the, a and of appear most frequently. We can then use frequent unambiguous words as anchors, to remove word ambiguities that occur immediately before or after them. For instance, the word table is theoretically ambiguous (it could be a verb), but in the sequences the table and a table it has to be a noun. Focusing on frequent unambiguous words can help us design new types of disambiguation rules.

Perspectives

In its latest version, NooJ provides a number of tools to deal with all types of ambiguities that are stored in the Text Annotation Structure. Efficient local grammars can remove a large number of ambiguities automatically, and simple enhanced queries can be used to disambiguate a large number of occurrences of a very specific context for a grammatical word. Furthermore, the new ability to directly edit the TAS should help linguists clean up large texts. Moreover, NooJ's users can now display the most frequent ambiguities, as well as the most frequent unambiguous words, which should help them design efficient new local grammars rapidly.

We hope that these new tools will be shortly put to the test: we think that it is very reasonable to build a set of 30+ reliable local grammars that could get rid of 50% of ambiguities, and that semi-automatic and manual tools could be used to get rid of most "annoying" remaining ambiguities in order to build a new set of disambiguated corpora.

  is much faster to just delete unwanted annotations without having to write local grammars or even queries. In NooJ's new 2.2 version, we can just open the TAS (click Show Text Annotation Structure at the top of the text window), click an unwanted annotation, and then delete it either with the menu command Edit > Cut, or by pressing any of the following keys: Ctrl-x, Delete or Backspace. NooJ manages an unlimited number of undo's: Edit > Undo or press Ctrl-Z.

See the NooJ manual, which is updated regularly (Silberztein

2002).2 See[START_REF] Vietri | The Formalization of Italian Lexicon-Grammar tables in a NooJ Pair Dictionary/Grammar[END_REF] for examples of lexicon-grammar tables and their formalization in NooJ.

See (Silberztein 2006) for a description of NooJ's Annotation engine, and (Silberztein 2007) for linguistic applications of the Text Annotation Structure.


published or not. The documents may come

Linear Growth of a Liquid Droplet Divided from its Vapour by a " SOAP BUBBLE"-like Fluid Interface

Francesco Dell'Isola