Gérard Gagneux
email: gerard.maryse.gagneux@gmail.com

Olivier Millet
email: olivier.millet@univ-lr.fr

Mathematical strategies to quantify exactly the forces acting on the nonaxisymmetric capillary bridges

Keywords: Distortion of nonaxisymmetric capillary bridges, Mean and Gaussian curvatures impact, Euler characteristic, Generalized Young-Laplace equation, Bending effects, Fenchel's theorem in differential geometry, Gauss-Bonnet Theorem., Geodesic curvature, Bending stress, Influence of the contact angles 449N45, 53A10, 58E12, 74F10, 74G05, 74G15 53Z05

The didactic object of these developments on differential geometry of curves and surfaces is to present fine and convenient mathematical strategies, adapted to the study of capillary bridges from experimental data and relatively simple to use in practice. The common thread is to be able to calculate accurately in any situation the bending stress over the free surface Σ, represented mathematically by the integral of the Gaussian curvature over the surface (called the total curvature) and also to obtain an information concerning the capillary tension forces by term by term integrating the generalized Young-Laplace equation. We mainly develop three convenient mathematical tools for assessing the physical properties in the field of the axisymmetric or not capillary bridges with convex or nonconvex plane boundaries, according to the local wettability and roughness effects: the unit speed reparameterization (or by arc length) of a regular curve and in particular for surfaces of revolution, the Fenchel's theorem and the Gauss-Bonnet-Binet theorem that expresses a relation between the integral of the Gaussian curvature over the surface, the topology of the surface and the integrals of the geodesic curvatures which are directly linked to the wetting angle at the contact lines. We express also the resultant of the bending energy only with respect to the wetting angles at the contact line.

Introduction

In this work, we present various results and complementary strategies of mathematical analysis that can be applied to concrete capillary bridges problems, concerning in a new way, the Gauss-Bonnet and Fenchel's theorems to establish various analytical formulas easy to use, concerning firstly the case of the rather delicate approach to nonaxisymmetric capillary bridges distortions. This article expands significantly on the partial results presented in [START_REF] Gagneux | Analytical calculations for quantifying capillary bridges distortions from experimental data[END_REF], concerning the only surfaces of revolution (circular boundaries, which makes these cases much easier, especially for the explicit calculation of the total geodesic curvatures of the boundaries, rarely possible in practice by the integral calculus).

The key to achieving generalization is a direct consequence of the Fenchel's theorem in differential geometry 3 [START_REF] Horn | On Fenchel's theorem[END_REF] : the total curvature of any smooth closed convex plane curve equals 2π, which avoids a lot of dead-end integration calculations. We also indicate how to calculate precisely the total curvature of generic expression 2kπ, k = 1 of any closed nonconvex plane curve, k being an integer, the index of the curve, the winding number. Consequently, in other words, the index k of a closed plane curve, convex or nonconvex, can be considered as the total curvature divided by 2π (the winding index in algebraic topology). In the specific context of this study, the Fenchel's theorem and the Gauss-Bonnet-Binet theorems are very strongly complementary, with the advantage of explicit calculative answers.

These developments relate in the first part to surfaces of revolution on the basis of an unit speed reparameterization (or by arc length) for a regular curve, in this case, the semimeridian. For detailed presentations of the subject, the reader may refer to [START_REF] Do Carmo | Carmo -Selected papers[END_REF] p. 161-164, [START_REF] Manfredo | Carmo, Differential geometry of curves and surfaces[END_REF] p. 161-162 and also to [START_REF] Do Carmo | Differential Geometry of Curves and Surfaces[END_REF] [START_REF] Gagneux | Overview on the bending effects distorting axisymmetric capillary bridges[END_REF][13] [START_REF] Gray | Modern differential geometry of curves and surfaces with Mathematica[END_REF].

A convenient conceptual bridge between differential geometry and kinematics is to imagine that the semi-meridian represents the successive positions of a moving object and from now on, we get the position of the object as a function of how far the object has traveled along the rotating curve. By this choice of calculation strategy, by reparameterization (always orientation-preserving), the speed is ipso facto unit and therefore, the speed and acceleration vectors are orthogonal. The Frenet-Serret apparatus is then very elegant and convenient to use.

To illustrate the convenience of the method, it is indicated that the calculation and the expression of the Gaussian curvature are very simplified. The calculation of the integral of the Gaussian curvature over the surface of the bridge (the total Gaussian curvature) is then direct and makes it possible to find again the analytical expression of the bending stress given by the subtle theorems of Gauss Bonnet and Fenchel (proportional to the sum of the cosine values of the two observed contact angles, possibly distinct). Moreover, it is also an accessible way to determine explicitly all the surfaces of revolution of constant Gaussian curvature.

It is also a convenient tool for calculating exactly the total curvature of the plane and smooth boundaries of the bridges, mainly in the nonconvex case (the concept of curvature κ (kappa) is indeed a way to measure how sharply a smooth curve turns).

From there, we build an analytical toolbox for the in situ methodological evaluation of various properties of these capillary bridges. The hard part would be inverting the speed function that cannot usually be integrated explicitly in terms of usual functions. However, to apply the method, it is not necessary to explain the parameterization; the analytical formulas obtained only take into account the experimental data.

The general non-axisymmetric case requires the much more delicate but not insurmountable use of the Gauss-Bonnet theorem closely associated to the Fenchel's theorem in differential geometry. In a singular way, for a smooth surface of revolution whose regular generating curve does not cross the axis of rotation, we have at hand a classical method in differential geometry of surfaces which is well suited in this situation: the arc length (or unit speed) reparameterization of the rotating curve considered as a trajectory.

2 The analytical framework and the main objectives 2.1 About the generalized Young-Laplace equation, the bending forces and the resulting geometrical effects

The generalized Young-Laplace equation concerns the strong distortions for which the bending effects are modeled by an additional curvature-related term: the introduction of C K , a multiplier coefficient of the Gaussian curvature K, at the dimension of a force and standing for the bending stress [START_REF] Boruvka | Generalization of the classical theory of capillarity[END_REF][10][15] [START_REF] Scholtès | On the capillary stress tensor in wet granular materials[END_REF]. Under appropriate boundary conditions, the shape of the free interface is then described by the so-called generalized Young-Laplace equation, thus involving both mean and Gaussian curvatures. The resulting strongly nonlinear differential equation, at the downward vertical measurement x linked to the value ∆p 0 at x = 0, (a spontaneous unknown value) comes in the following form that is detailed and justified more specifically in [START_REF] Gagneux | Structuring and destructuring effects of bending and gravity. An analytical approach to axisymmetric capillary distortions[END_REF] [START_REF] Sauvigny | Solution of boundary value problems for surfaces of prescribed mean curvature H (x, y, z) with 1-1 central projection via the continuity method[END_REF] and gratefully [START_REF] Delaunay | Sur la surface de révolution dont la courbure moyenne est constante[END_REF] for a very early initiating study:

γ 1 ρ c + 1 N + C K 1 ρ c N = ∆p 0 -∆ρ gx , (1)
associated with adequate miscellaneous boundary conditions [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF], where the force C K divided by the area ρ c N stands via a pressure for the local bending stress, ρ c and N being the principal radii of curvature (evaluated algebraically, positively when the curvature is turned into the interior of the capillary bridge) and the pressure deficiency is ∆p 0 at x = 0.

It is assumed that the different cefficients, implicit unknown a priori, as ∆p 0 , resulting from the final equilibrium, have been previously identified in situ from exprerimental data, by solving a linear system, well posed and numerically stable (for example, thanks to a first integral translating a principle of conservation) [START_REF] Gagneux | Analytic Calculation of Capillary Bridge Properties Deduced as an Inverse Problem from Experimental Data[END_REF][10] [START_REF] Gagneux | Analytical calculations for quantifying capillary bridges distortions from experimental data[END_REF][12] [START_REF] Gagneux | An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges[END_REF].

It is extremely noteworthily [12][31], that this strongly nonlinear differential equation is mathematically isomorphic (the same structure) but with different variables and physical units, to the Gullstrand equation of geometrical optics, which relates the optic power P op of a thick lens (in diopters, the reciprocal of the equivalent focal length) to its geometry and the properties of the media. For example, the superficial tension γ is equivalent to the refractivity

n 1 n 2 -1,
where n i is a refractive index, C K is analogous to the expression

- n 1 n 2 -1 2 n 2 n 1 d, d
the lens thickness and ∆p corresponds to P op .

Shear or free energy problems and the longitudinal bending stress of ship hulls have an analogy with the subject [START_REF] Carrier | Contact angles and the surface free energy of solids[END_REF][27][30] [START_REF] Wiklund | Simulations of shearing of capillary bridges[END_REF].

The mathematical modeling and simulations of the petroleum engineering are also concerned by this theoretical topic, in order to obtain for media with periodic microstructure an "equivalent" macroscopic representation, by some statistical or homogenization methods [START_REF] Gagneux | Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière[END_REF], chapter 1.

The bending stress over the free surface Σ is then represented in the following integral form, at the dimension of a force:

E bending stress = C K Σ K dΣ ,
where K is the Gaussian curvature of the free surface Σ, intrinsic value, in particular independent of the choice of the unit normal vector, and the nondimensional integral is the total curvature4 .

Knowing the exact value of the bending stress, on would deduce from the generalized Young-Laplace equation by a fine numerical integration of the right-hand side, the capillary tension values.

Concerning the capillary tension forces, by term by term integrating over the free surface Σ the generalized Young-Laplace equation, we have for example the relationship between various forces:

γ Σ 1 ρ c + 1 N dΣ = -C K Σ K dΣ + Σ (∆p 0 -∆ρ gx) dΣ,
with the particular situation:

γ Σ 1 ρ c + 1 N dΣ = -C K Σ K dΣ + ∆p 0 area (Σ)
when neglecting gravity effects.

This would allow to have a reasoned opinion on the relative importance of the bending forces; according to an objective criterion, either by relative value or by intrinsic value.

Homotopic surfaces and Euler characteristic in geometry and algebraic topology.

Recall that the Euler characteristic (or Euler-Poincaré characteristic) is a topological invariant, an integer that describes, according to precise axiomatic principles, the shape or a structure of a topological space regardless of how it is bent according to the formula: number of vertices-number of edges+ number of faces with the property of invariance by homeomorphy. It is commonly denoted by χ or χ (M ). As examples for surfaces in homological algebra, we have χ (M ) = 2 for a sphere, χ (M ) = 4 for two spheres (not connected), χ (M ) = 0 for a torus and χ (M ) = -2 for a two-holed torus.

To speak very figuratively, quite approximately, the Euler-Poincaré characteristic is an integer, invariant when the size and the shape of a geometrical object change by an effect of a "plastic" deformation. One will find more axiomatic definitions!

This invariance property makes it a providential tool in the context of this study on the bending effects, associated to the Gauss-Bonnet theorem, a deep relationship between surfaces in differential geometry, connecting the Gaussian curvature of a surface to its Euler characteristic.

The Euler characteristic of the right cylinder is zero, thus so is that of the cylinder with one or two boundaries. These following free surfaces with two circular boundaries and whose meridian is an arc of Delaunay roulette are considered topologically equivalent (same common topological genus), because it is possible to continuously move one to obtain the other: portion of concave or convex, catenoid or unduloid (the right cylinder being the transition case). Accordingly, these axisymmetric surfaces have in common the same Euler characteristic, in this case, the value zero. It is the same for their continuous axisymmetric smooth deformations by distorting effect of bending or gravity [START_REF] Mazzone | The effect of gravity on the shape and strength of a liquid bridge between two spheres[END_REF][26].

The Gauss-Bonnet or Gauss-Bonnet-Binet theorems

The Gauss-Bonnet theorem (Binet is often forgotten 5 ) is reputed to be one of the most profound and elegant results of the study of surfaces [START_REF] Do Carmo | Differential Geometry of Curves and Surfaces[END_REF][3] [START_REF] Ciarlet | An Introduction to Differential Geometry[END_REF]. It has no surprisingly many applications in Physics. It is used in sectors of activity where the problems of bending beams surely arises (civil engineering, naval architecture, shell theory to predict the stress and the displacement arising in an elastic shell, [START_REF] Ciarlet | An Introduction to Differential Geometry[END_REF], etc...).

In fact, it unexpectedly links two completely different ways of studying a surface: one geometric, the other topological. Indeed, for any compact, boundaryless two-dimensional Riemannian manifold Σ, the integral of the Gaussian curvature K over the entire manifold with respect to area measure is 2π times the Euler characteristic of Σ, also called the Euler number of the manifold, i.e.

Σ K dΣ = 2πχ (Σ) .
For example, for a sphere Σ of radius R in R 3 , it comes:

Σ K dΣ = 1 R 2 4πR 2 = 4π and here χ (Σ) = 2.
Suppose now that M is a compact two-dimensional Riemannian manifold with a boundary δM and let k g the signed geodesic curvature of δM. Then, in nondimensional writing,

M K dM + δM k g ds = 2πχ (M ) .
We recall that the geodesic curvature k g , of an arbitrary curve at a point P on a smooth surface, is defined as the curvature at P of the orthogonal projection of the curve onto the plane tangent to the surface at P and we have:

k g = kcos θ, ( 2
)
where θ is the angle between the osculating plane of C and the tangent plane Q at point P .

It is easy to see that cos θ is nothing else than the local contact angle for wetting problems and capillary bridges.

We want to obtain in any situation an explicit expression of this integral, of simple and immediate use for the experimenter, and highlight the determining parameters and their respective influence.

3 The aim and effective convenience of the arc length reparameterization strategy

Surface of revolution

The rather paradoxical aspect of this method is that it is only very rarely easy to get in practice an explicit calculation formula. However, it leads to general quantitative results in the form of analytical formulas very easy to use from the experimental data, via a very convenient expression of the Gaussian curvature of a surface of revolution (the speed and acceleration vectors are then orthogonal).'How complexity leads to simplicity...'

To illustrate what we are talking about, let us consider a smooth curve of the half-plan {y > 0, z = 0} parametrized by the arc length. The surface of revolution resulting in R 3 from the rotation of the curve around the x-axis, ψ being the angle of rotation, is parametrized by:

M ((s, ψ)) = (x (s) , y (s) cos ψ, y (s) sin (ψ) , ) 0 ≤ ψ ≤ 2π, 0 ≤ s ≤ L.
As the meridian portion is parameterized by arc length, we have ipso facto the following remarkable and convenient relations and convenient expressions for angular (in radians) and trigonometric values as well as for the Gaussian curvature of the surfaces of revolution:

x 2 + y 2 = 1 at any point, (3)
and therefore, by differentiating, the orthogonality relationship

x x + y y = 0 ,
that is to say that T. dT ds = 0 where the dot denotes the scalar product of R 3 and where

T (s) = dM (s)
ds is the unit tangent vector to the curve M (s) = M (s, 0).

The Gaussian curvature K of the surface of revolution has then the very convenient expression (see [START_REF] Do Carmo | Differential Geometry of Curves and Surfaces[END_REF], p.162, eq. ( 9)):

K (s, θ) = - y (s) y (s) .
In this context, the expression of the mean curvature H of a surface of revolution is less attractive (see [START_REF] Do Carmo | Differential Geometry of Curves and Surfaces[END_REF], p.162, eq. ( 11)).

A remarkable illustrative example of the arc length reparameterization strategy is the determination of the axisymmetric surfaces of constant Gaussian curvature6 (procedure indicated in [START_REF] Do Carmo | Differential Geometry of Curves and Surfaces[END_REF], p.162, eq. ( 9)).We consider then the classical differential equation, linear, of the second order, homogeneous:

y (s) + K y (s) -0 , 0 ≤ s ≤ L,
with the three following cases: K < 0, K = 0, K > 0. Then, introducing the general form of the corresponding solutions in y, we consider the resulting differential equations

x 2 = 1 -y 2 , 0 ≤ s ≤ L (4)
resulting from (3).

Moreover, ϕ being the angle that the tangent to the profile curve makes with the x-axis, we have the following relationships:

sin ϕ = y (s)
x 2 (s) + y 2 (s) , i.e. sin ϕ = y (s) and cosϕ = x (s) .

To compute the global bending stress in this context, we have to consider succesively:

E bending stress = C k M K dM = C k - y (s) y (s) y (s) dψds
and therefore

C k -y (s) dψds = -2πC k (y (L) -y (0))
so that E bending stress = 2π(C k (sin (ϕ (L)) -sin (ϕ(0)) , ϕ being the angle that the tangent to the profile curve makes with the x-axis, the axis of rotation.

Let us quote that this case of a surface of revolution around x-axis may correspond to a capillary bridge between two parallel planes at x = 0 and x = L. With the notations of [START_REF] Gagneux | Analytic Calculation of Capillary Bridge Properties Deduced as an Inverse Problem from Experimental Data[END_REF][11], the wetting angles at the triple lines are given by θ 1 = -ϕ (L) and θ 2 = ϕ (0). Therefore, we recover the expression of the bending stress of [START_REF] Gagneux | Analytical calculations for quantifying capillary bridges distortions from experimental data[END_REF] obtained in the general case for a surface of revolution (not necessarily with constant Gauss curvature) from the Gauss-Bonnet integration theorem:

E bending stress = C k M K dM = -2πC k (cos θ 1 + cos θ 2 ) ,
with here the Euler characteristic χ (M ) = 0 and when the total geodesic curvature at the boundaries is 2π (cos θ 1 + cos θ 2 ).

The conclusive Fenchel's theorem

In the case of a portion of an ellipse, the parameterization involves elliptic integrals, rarely possible to explain in practice, so that the parameters would have to be sought numerically (spline interpolation). This computational difficulty is overcame by knowing the Fenchel's theorem, which shows the complementarity of the three methods.

To well illustrate the interest of Fenchel's theorem associated with the theorems of Gauss and Bonnet, let us consider a reparameterization by arc length of the curve defining the surface of revolution.

Close plane curve

According to Fenchel's theorem 7 (1929), the value of the total curvature By the Fenchel's theorem, without calculation of primitive functions, often tedious or ineffective, we deduce directly, for any closed convex plane curve Γ (i.e. the curve is the boundary of a convex set in the Euclidean plane), that Γ k(s)ds = 2π.

In the case of a closed nonconvex plane curve, we are led to conclude by defining of the notion of the winding index, a topological argument, in what follows.

Open plane curve

Let us give some classical preliminary elements of differential geometry related to smooth boundaries of surfaces, parametrized by arc length. The curvature of a plane curve parametrized by arc length is the rate of turning of the tangent line with respect to an ad hoc frame along the curve.

Let ϕ(s) be the angle of inclination of the unit tangent vector T = T (s) with respect to a fixed frame of reference, for instance x-axis. Considered then as a rate of turning for the tangent line when one moves along the curve at unit speed, the curvature k(s) becomes

k(s) = dϕ ds (s) = ϕ (s).
It follows that the total curvature of a smooth curve C is then given by the formula depending only of the initial and final states:

C k(s)ds = ϕ(ending) -ϕ(starting), (in radians). ( 5
)
For a piecewise smooth curve parametrized by arc length, then we need to deal with the exterior angles at the corners according to the orientation of the curve in the turning motion. However, up to now, to our knowledge such capillary bridges with non convex or open contact line have are not considered in literature.

General case for axisymmetric capillary bridges of revolution

Let us come back to the general case of axisymmetric capillary bridges of revolution whose profile have one or more inflection points. The methods of Euler's characteristic associated to the Gauss-Bonnet theorem apply immediately to these cases. These axisymmetric surfaces have in common the same Euler characteristic, in this case, the value zero. It then comes with adjusted data (cite [START_REF] Gagneux | Analytical calculations for quantifying capillary bridges distortions from experimental data[END_REF] for more details):

E bending stress = C k Σ K dΣ = -2πC k (cos θ 1 + cos θ 2 ) .
In summary, it should be kept in mind that the value of the bending stress depends, besides physical constants, only on the observed values of the contact angles, whereas these angles result in part implicitly from the final equilibrium of the device. The essential point here is there are no major issues for generalization of the bending stress calculation to the non-axisymmetric cases, as specified below.

It would be interesting to reconsider, in taking into account the bending effects, the resulting contact angles and the elegant Gauss-Bonnet theorem, a new analytical framework for evaluating the cohesion effects of coalescence between saddle shaped capillary bridges [START_REF] Gagneux | An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges[END_REF].

For didactic purposes, we limited ourselves to the detailed case of axisymmetric capillary bridges to show the interest of the simultaneous use of the Gauss-Binet-Bonnet formula and of the topological notion of Euler characteristic to evaluate the importance of the bending stress. The obtained result clearly shows in an explicit way, the major role of the contact angles values after distortion effects, eventually distinct.

3.4

The prevaling roles of the contact angles values and of the convexity/nonconvexity of the boundaries.

All the factors determining the contact angle have consecutively an influence on the bending stress (surface roughness and heterogeneity, influence of gravity, contact angle hysteresis, [START_REF] De Gennes | Capillarity and gravity[END_REF], [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF], [START_REF] Wang | Understanding contact angle hysteresis on an ambient solid surface[END_REF], etc ).

It is well known that the contact angle value is determined by the balance between adhesive and cohesive forces on the rigid supports. As the tendency of a drop to spread out over a flat, solid surface increases, the contact angle decreases. Thus, the contact angle provides an inverse measure of wettability.

In this context, the case of the right cylinders is still a borderline case.

A contact angle less than π 2 (low contact angle) usually indicates that wetting of the surface is very favorable, and the fluid will spread over a large area of the surface. Contact angles greater than π 2 (high contact angle) generally mean that wetting of the surface is unfavorable. It should be quoted that a certain number of terms of the generalized Young-Laplace equation are spontaneous values, resulting from instantaneous equilibrium, and are therefore implicit unknowns. This is a difficulty for the mathematical resolution of this nonlinear differential boundary problem. In addition, the Fenchel's theorem sheds light on the importance of the convexity or the nonconvexity of the outer edges in calculating exactly the value of the total curvatures. Consequently, even a limited displacement of the surface boundary can modify the bending stress, by local modifications of the contact angles or affecting the local curvature of the outline curve and then, the total geodesic curvature. The contact angle hysteresis can also be significant.

The general case and its implementation

In the general cases of nonaxisymmetric capillary bridges between two supports, possibly of distinct natures, the method remains applicable in principle. The difficulty is not conceptual in dealing with the general case but rather calculative. We must then, in any given case, engage in a delicate exercise in differential and analytical geometries to explicitly calculate the total signed geodesic curvature of the boundaries by the classical methods of analytical geometry.

The calculation procedure is as follows.

At any point P of the border liquid-solid, one considers the tangent plane in P to the free surface (that supposes an adequate local regularity). One then considers the orthogonal projection of each edge into this tangent plane. The curvature in P of the projected curve is then calculated, what introduces the important role of the cosine of the local contact angle. To make a metaphor from the point of view of the kinematics, remember that for a unit speed curve on a smooth surface, the geodesic curvature is the length of the surface tangential component of acceleration. The geodesics correspond to zero values. The curvature k of the closed curve C at point P is related to the geodesic curvature k g at P by the relationship:

kg = k cos θ
where θ is the local contact angle, i.e. the angle between the osculating plane of C and the tangent plane Q at point P .

When the contact angle is constant on the considered contact surface, we have the particularly simple relationship:

Γi k g (s) ds = cosθ Γi k(s) ds.

The special situation of heterogeneous contact surfaces

When the contact angles are separately variable on each of the contact surfaces, i.e . θ = θ 1 (M ) and θ = θ 2 (M ) according to the physical conditions of the two surfaces (non ideal smooth surfaces), the integral along each boundary, corresponding to the total geodesic curvature, in fact, of the kind Γi cosθ i (M ((s)) k(s) ds, is more complicated to calculate with computational prediction of wetting (at our knowledge, an open problem for the probably most realistic case). The use of a mean theorem would likely be imprecise (effects of surface roughness).

The general case of homogeneous contact surfaces

When multiplied by the coefficient (-C K ) at the dimension of a force, the dimensionless integral of these curvature values along the reunion of the two contact edges gives finally the value of the resulting bending stress by the Fenchel's theorem (the cornerstone of the method).

The three possible scenarios then arise according to the geometry of the boundaries (closed plane convex or nonconvex curves) are the following, the surfaces having in common, without loss of generality, the same Euler characteristic, in this illustrative case, the value zero.

By introducing the contact angles θ 1 and θ 2 (in radians) on each outline of contact surfaces, we proved that, at least theoretically, the wettability being evaluated, here, by constant contact angles, separately on each contact support.

Case 1:

The boundaries are two closed plane convex curves. Then,

E bending stress = C k M K dM = -2πC k (cos θ 1 + cos θ 2 ) .

Case 2:

The boundaries are two closed plane curves, one convex and the other nonconvex. Then,

E bending stress = C k M K dM = -C k (2k 1 π cos θ 1 + 2π cos θ 2 ) ,
the observed integer k 1 , k 1 ≥ 2, being the winding number of the nonconvex curve ( the winding index in algebraic topology).

Case 3:

The boundaries are two closed disjoint plane curves, nonconvex. Then,

E bending stress = C k M K dM = -C k (2k 1 π cos θ 1 + 2k 2 π cos θ 2 ) ,
k 1 and k 2 being the integers, ≥ 2, winding numbers of the curves, observed and known in situ.

In the rather theoretical case, where the value of the Euler characteristic is non-zero, it should be necessary to write:

E bending stress = C k M K dM = 2πχ (M ) -C k (2k 1 π cos θ 1 + 2k 2 π cos θ 2 ) .
According to these theoretical calculations, the bending stress is practically not existing in the neighborhood where the contact angle is close to π 2 radian. Then, at the contact, the surface is locally similar to a right cylinder whose outline is a closed plane curve, convex or nonconvex. This last topological aspect of convexity is here non-essential (the non-convexity modifies in a known and controlled way the total curvature).

It must be emphasized that, when the contact angles are separately variable on each of the contact surfaces according to the physical conditions of the two surfaces (non ideal smooth surfaces), the integral along each boundary, corresponding to the total geodesic curvature of the plane and closed boundaries, seems a serious difficulty to explain. The question might interest specialists in differential geometry.

Conclusion

We then see that the developments obtained here, generalization of formulas relating to surfaces of revolution, result from concepts in differential geometry and geometric analysis with applications to Lagrangian Mechanics, without resorting to differential calculus and integral calculus, sometimes out of concrete practice for problems involving elliptic integrals, for example, requiring numerically computing.

The methods of Euler's characteristic, associated to the Gauss-Bonnet-Binet theorem and the strongly complementary Fenchel's theorem apply immediately to the cases of the nonaxisymmetric surfaces, with explicit, easy-to-use, results formulations.

We proved that in the general way, the value of the bending stress depends, besides physical constants, only on the observed values of the contact angles, whereas these angles result in part implicitly from the final equilibrium of the device. Therefore, all the factors determining the contact angle have an influence on the bending stress (surface roughness and heterogeneity, influence of gravity, contact angle hysteresis.

It would be interesting to reconsider, in taking into account these new results concerning the bending effects, the important role of the contact curves geometry and the Gauss-Bonnet and Fenchel theorems, an analytical framework for reassessing the cohesion effects of coalescence between saddle shaped capillary bridges [START_REF] Gagneux | An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges[END_REF]. Finally, by creating a support material having a nonconvex region with high wettability and a complementary region with very low wettability, the experimenter could illustrate the theory by experimentation.

  of any smooth closed space curve Γ is at least 2π , i.e Γ k(s)ds ≥ 2π. The equality holds if and only if the curve is a convex plane curve. In other words, the average curvature of a closed convex plane curve equals 2π/L, where L is the length (the perimeter) of the curve 8 .

Werner Fenchel (1905-1988), Professor at the universities of Götingen and Copenhagen.

For example, the total curvature of the catenoid whose axis is of infinite length is -4π, the total curvature of the sphere of radius r is 4π and the torus 0.

Because of the Binet's independent work, but Bonnet credits Binet, cf. Tome III, correspondance de l'Ecole Polytechnique.

Problem studied more extensively by Gaston Darboux 1890. Among the solutions, surfaces are found that look like a hyperboloid.

The Fary-Milnor theorem concerning the total curvature of the knotted closed curves does not seem appropriate for the subject of this study.

For a given arc of a plane curve, the local average curvature quantifies the ratio of the change in inclination of the tangent to the curve over the arc length.


A Sigal
email: sigal@famaf.unc.edu.ar

J Batle

Semi-classical estimation of the fine-structure constant

Keywords: Casimir effect, vacuum polarization, fine-structure constant

In the present work we propose a semi-classical model that links well-known phenomena of quantum field theory, such as the Casimir force and the vacuum polarization. We assume for the Casimir effect in dielectrics, that the virtual pairs statistically created in the vacuum near the plates induce image dipoles in the latter, interacting with each other. Taking the limit of the dielectric constants of the plates to the value corresponding to the vacuum, we arrive at an equality that provides, on the one hand, a framework of unconventional physical interpretation for the Casimir effect through vacuum polarization and, on the other hand, a possible way for a theoretical estimation of the fine-structure constant. Regarding the latter, an example result of a first approximation to the problem is shown, with which we find a numerical value for the aforementioned fundamental constant, with an error of 44 parts per million with respect to the 2018 CODATA experimental recommended value.

Motivation

Eugene Wigner, in his historical essay of 1960 [START_REF] Wigner | The unreasonable effectiveness of mathematics in the natural sciences[END_REF], asserted that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve". This is particularly true when comparing the validity of the predictions of quantum electrodynamics (QED) with the experimental value for the Lamb shift. The Lamb shift currently provides a measurement of the fine-structure constant 𝛼 to better than one part in a million, allowing a precision test of quantum electrodynamics.

Complementarily, there is a flip side to the previous coin. Namely, what are the odds of obtaining the right (in the experimental sense) value of a physical constant from first principles? Is this perchance a contradiction in terms, and oxymoron? After all, the very same definition of constant implies that it must be independent of the mathematical machinery behind the theory, only to be found or contrasted experimentally. In other words, to what extend the consistent mathematical manipulation of numerical values can infer the correct value (again in the experimental sense) of a physical constant? The estimation of a physical constant in terms of universal mathematical values is perhaps one of the most intriguing philosophical conundrums in theoretical physics. Notice that we completely disregard all "cabbalistic" (in Sommerfeld's own words) efforts to find combinations of numbers that may possess striking similarities with either 𝛼 or 1/𝛼.

In the present contribution, we shall embark on the particular endeavor of consistently obtaining the aforementioned fine-structure constant 𝛼, by contrasting two physical theories explaining the same phenomenon, and thus retrieving 𝛼 from first principles.

Several attempts have been made in the past to such purpose, already beginning with Sommerfeld, who coined the definition of the fine-structure constant in 1916, as well as the use of the Greek letter alpha to denote it. It is well known that the fine-structure constant 𝛼, which characterizes the strength of the electromagnetic interaction, is one of the most fundamental constants of physics. It is crucial central in all calculations of quantum electrodynamics and electroweak theory. Sommerfeld introduced it in the early attempts to account for the degeneracy of the Bohr's atom levels due to relativistic effects [START_REF] Kinoshita | The fine Structure[END_REF], i.e., the fine and hyperfine structures of atomic energy levels. Furthermore, the precision in its experimental measurement gives account of the consistency of the theoretical framework used. See Ref. [START_REF] Jentschura | Attempts at a determination of the fine-structure constant from first principles: a brief historical overview[END_REF] for a historical account of all attempts to determine it up to recent times. "All good theoretical physicists", Richard Feynman once remarked [START_REF] Feynman | QED: The strange theory of light and matter[END_REF], "put this number up on their wall and worry about it. Is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding."

The first attempt to conciliate the Casimir effect, to be discussed later, with the traditional electrostatic force was carried out by Casimir himself in 1953, shortly after his seminal work of 1948 [START_REF] Casimir | On the attraction between two perfectly conducting plates[END_REF]. His "mousetrap" model of the electron aimed at explaining the stability of charged particles from the balancing between the attractive quantum vacuum force and the self-energy repulsion between charges [6]. However, the model however had to be rejected, for Boyer proved that the Casimir force for a spherical shell was repulsive [START_REF] Boyer | Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle[END_REF]. What is of relevance to the present work is that Casimir's mousetrap model retrieved for the first time a means to compute the fine-structure constant itself from first principles, albeit with the wrong sign and with several orders of magnitude of difference.

Other models followed with time, directly or indirectly related to the Casimir effect. Carazza and Guidetti [START_REF] Carazza | The Casimir effect and the fine structure constant[END_REF], regarded the electron as a charged sphere that completely absorbs any radiation with a wave number less than a certain critical value, the attraction of the Casimir force balancing the mutual repulsion of the different parts of a charge distribution. With this equilibrium condition, they obtained an acceptable estimate for the value of the fine-structure constant.

In pursuit of addressing Feynman's original quest, we shall first revisit two important effects, namely, i) the Casimir effect, which is one of the most thoroughly investigated effects of quantum field theory (QFT), and ii) the vacuum polarization. These two processes will be studied in connection with the fine-structure constant. With the revision of the previous physical effects at hand, we will be in a position to present the model of the present work, which shall involve an alleged dipolar interaction between image dipoles, induced by the virtual pairs statistically created in vacuum. The main result for the estimation of 𝛼 will be thus presented and finally some conclusions will be drawn.

The Casimir effect and the vacuum polarization, revisited

Since its first theoretical prediction, the Casimir force has been generalized and experimentally corroborated for different geometries, types and configurations of the material bodies that attract each other when they are at very small distances [START_REF] Plunien | The Casimir effect[END_REF][START_REF] Bordag | New Developments in the Casimir Effect[END_REF]. The mere fact that it arises from the perturbation by the boundaries of the zero-point vacuum energy, makes it have a fundamental significance, not only for the study and understanding of the quantum vacuum itself, but also for the most varied areas of physics, ranging from the non-trivial topology of space-time and other cosmological applications, to nanoscopic physics and elementary particle physics [START_REF] Mostepanenko | The Casimir effect and its applications[END_REF]. Traditionally in physics textbooks, discussions regarding the zero-point energy of the vacuum have been avoided by simply redefining the zero of energy. However, it is precisely the changes in the zero-point energy that are observable, being the Casimir effect one such notable instance. This quantum effect is named after Casimir who first predicted the attractive force per unit area between two conducting plane-parallel plates in vacuum [START_REF] Casimir | On the attraction between two perfectly conducting plates[END_REF]:

𝐹 𝐴 = - ħ𝑐𝜋 2 240𝑎 4 (1)
where ħ is Planck's constant, 𝑐 is the speed of light and 𝑎 is the distance between the plates.

Less than ten years later, Lifshitz generalized this force for the case of dielectric plates and also studied its temperature dependence [START_REF] Lifshitz | The Theory of Molecular Attractive Forces between Solids[END_REF]. In particular, for two identical dielectric plates and temperature 𝑇 = 0, he derived the following analytical expression:

𝐹 𝑑 𝐴 = - ħ𝑐𝜋 2 240𝑎 4 (𝜀-1) 2 (𝜀+1) 2 𝜑(𝜀) (2)
where 𝜀 is the relative permittivity of the plates and the function 𝜑(𝜀) arises from the integration of all the Fourier components of the electromagnetic radiation between the plates. In the asymptotic limit 𝜀 → ∞, we have (𝜀-1) 2 (𝜀+1) 2 𝜑(𝜀) → 1 and recover the equation (1) applied to the case of metal plates. And when 𝜀 → 1, in the vacuum limit, the force (2) goes to zero, even though the function limit lim 𝜀→1 𝜑(𝜀) = 𝜑 0 is not zero and can be computed analytically.

Why is the dependence on Casimir effect with the fine-structure constant relevant? As can be seen in ( 1) and (2), in principle there is no such dependence. At least directly, Casimir forces do not depend on the electron charge. As Jaffe proposes [START_REF] Jaffe | Casimir effect and the quantum vacuum[END_REF] and then Karnassnigg picks up the point [START_REF] Karnassnigg | All wrong with the Casimir effect[END_REF], this effect can be deduced without need of zero-point energies. They consider the latter only as a heuristic starting point for simpler calculations, showing that Casimir forces can be calculated without reference to vacuum. Furthermore, Karnassnigg ensures that the Casimir standard result for ideal metal plates can be assumed at the limit of α → ∞, and that for finite values of α the result will depend on the nature of the plates. This is based on the fact that there are no perfect conductors in nature.

Another key ingredient to be discussed is the vacuum polarization. The vacuum polarization is another well-known process of quantum electrodynamics that will serve as a tool for our model. This process describes the production of virtual electron-positron pairs through the application of electric and magnetic fields large enough to, precisely, polarize the vacuum [START_REF] Ruffini | Electron-positron pairs in physics and astrophysics: From heavy nuclei to black holes[END_REF]. The first discussions by Heisenberg and Dirac [START_REF] Dirac | Discussion of the infinite distribution of electrons in the theory of the positron[END_REF] on this phenomenon gave rise to Uehling [START_REF] Uehling | Polarization effects in the positron theory[END_REF] to calculate, to first order of the coupling constant 𝛼 = 1 4𝜋𝜀 0 𝑒 2 ħ𝑐 (𝑒 is the elementary charge), the corrections to the coulombic electrostatic potential between two test charges in vacuum [START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF]. The consequences of vacuum polarization begin to appear when the field approaches the value known as the Schwinger limit, first derived by Sauter [START_REF] Sauter | Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs[END_REF] and raised earlier by Bohr (as Sauter himself acknowledges in his paper, thanking Heisenberg for telling him about Bohr's hypothesis):

𝐸 𝑆 = 𝑚 𝑒 2 𝑐 3 𝑒ħ ≈ 1.32 × 10 18 V/m (3)
where 𝑚 𝑒 is the mass of the electron.

Above this limit, the electromagnetic field becomes non-linear. In the computations of these nonlinear corrections of the fields in vacuum, Schwinger calculated the probability per unit spacetime volume of creating a virtual pair through a constant electric field strength 𝐸. Schwinger's formula [START_REF] Schwinger | On Gauge Invariance and Vacuuwn Polarization[END_REF] for the rate of creation of virtual pairs is expressed as:

𝑑𝑁 𝑑𝑉𝑑𝑡 = 𝛼 𝜋 2 𝐸 2 ∑ 1 𝑛 2 𝑒 -𝑛𝜋 𝐸 𝑆 𝐸 ∞ 𝑛=1 (4)
The previous formulas will be used as the main inputs for our model, shown in the following.

The model

With the above formulae, we are now in a position to present the model of this work, which involves an alleged dipolar interaction between image dipoles, induced by the virtual pairs statistically created in vacuum between two plates.

Let us sketch the different stages of the model:

1. Firstly, the use the method of images in electrostatics will be applied to the case of the Casimir force in dielectrics, thus contributing in this way to a derivation and understanding of the Casimir effect in an unconventional way. Roughly speaking, we shall derive a classical analogue of the Casimir effect due to the interaction of charge images. The functional dependence with respect to the distance between plates, as well as the the dielectric constants in the media, will be recovered.

2. Secondly, the rate of production of pairs of electrons and positrons beyond a critical value will be taken into account.

3. Finally, the conciliation between the quantum and classical approaches in the limit of vanishing plates, that is, no disruption of the zero-point energy of the electromagnetic vacuum caused by the mere presence of the plates (the limit of the relative permittivity of the plates tending to unity) shall retrieve a good numerical approximation of the fine-structure constant.

Imagine that an electron-positron pair is created in the vacuum between the plates (see Figure 1a). In pursuit of energy-momentum conservation, the probability of creating a pair increases right next to the plates, where the nuclei are. Now, using the method of image charges, an electron-positron pair in front of the dielectric plate creates an image dipole 𝑝 = 𝑞𝑑 therein, where 𝑑 is the effective distance between the charges and 𝑞 is given by:

𝑞 = -𝑒 (𝜀-1) (𝜀+1) (5)
In a heuristic way, one can imagine the force between the dielectric plates as due to the interaction between the image dipoles induced in them by the virtual pairs statistically created in the vacuum (see Figure 1b). At this point, before going on, it is pertinent to make some observations. First, one could reasonably criticize this model arguing that the tiny lifetime of the virtual pairs, which can be estimated by the uncertainty relation Δ𝑡 ≈ ħ 4𝑚 𝑒 𝑐 2 ⁄ , makes their mutual interaction impossible at the characteristic distances of the Casimir force. However, as we will see in what follows, the proposed dipolar interactions have a statistical nature, so we should not think the model in terms of an isolated virtual pair, but rather in terms of a statistical collection of them. Furthermore, the theoretical approximation of the fine-structure constant, which is one of the objectives of this work, will be independent of the distance between the plates 𝑎. Secondly, one could also argue that this electrostatic description of the dipolar interaction is inappropriate because a dipole moment as the one assumed here, requires rather rigid and stable spatial structures between the charges, which does not apply to the case of a virtual electron-positron pair. Nevertheless, as we will also see later and we already mentioned above, what will be used is an effective distance or average distance between the charges of the virtual pairs considered.

Then, taking account the above, we return to the model. We know that the potential energy for this dipole-dipole interaction (Figure 1b) is given by:

𝑉(𝑎, 𝜃 1 , 𝜃 2 , 𝜗) = - 𝑝 2 4𝜋𝜀 0 1 𝑎 3 (cos 𝜗 -3 cos 𝜃 1 cos 𝜃 2 ) (6)
where 𝜃 1 , 𝜃 2 are the angles of the dipoles with respect to the axis connecting their centers, and 𝜗 is the relative torsion angle between their orientations. Note that since the pairs are created near the plates, the distance between the interacting dipoles is indistinguishable from the distance 𝑎 between the plates. In our case, we want to calculate the average 〈𝑉(𝑎, 𝜃 1 , 𝜃 2 , 𝜗)〉 for all possible orientations of the dipoles. In order to calculate it, we start from (6), rewriting it as follows: 𝑉(𝑎, 𝜃 1 , 𝜃 2 , 𝜗) = 𝑉 0 (𝑎)𝑓(Ω) [START_REF] Boyer | Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle[END_REF] where 𝑉 0 (𝑎) = -𝑝 2 4𝜋𝜀 0 1 𝑎 3 and 𝑓(Ω) = (cos 𝜗 -3 cos 𝜃 1 cos 𝜃 2 ).

In the calculation of the average we must be careful, since the angular average for "free" dipoles, in which all angles are equally probable, is annulled. In the same way that Keesom force is derived [START_REF] Reifenberger | Inter-molecular forces: Keesom force[END_REF], a weighted average must be computed, where the Boltzmann factor 𝑒 𝑥𝑓(Ω) , whit 𝑥 = -𝑉 0 (𝑎) 𝑘𝑇 , weights those angles with lower energy. Unlike the derivation of Keesom force, in our case we will not do the approximation |𝑥| ≪ 1. Instead we will need to evaluate the exact integral and then take the limit |𝑥| → ∞ (𝑇 → 0) in the result of the integration. We then have to solve the following integral:

〈𝑉(𝑎)〉 = 𝑉 0 (𝑎) ∫ 𝑑𝜗 2𝜋 0 ∫ ∫ 𝑓(Ω)𝑒 𝑥𝑓(Ω) sin𝜃 1 sin𝜃 2 𝜋 0 𝜋 0 𝑑𝜃 1 𝑑𝜃 2 ∫ 𝑑𝜗 2𝜋 0 ∫ ∫ 𝑒 𝑥𝑓(Ω) sin𝜃 1 sin𝜃 2 𝜋 0 𝜋 0 𝑑𝜃 1 𝑑𝜃 2 (8)
which can be rewritten as:

〈𝑉(𝑎)〉 = 𝑉 0 (𝑎) 𝑑 𝑑𝑥 log [∫ 𝑑𝜗 2𝜋 0 ∫ ∫ 𝑒 𝑥𝑓(Ω) sin𝜃 1 sin𝜃 2 𝜋 0 𝜋 0 𝑑𝜃 1 𝑑𝜃 2 ] (9)
It can be verified that the integral in ( 9), which we call 𝑔(𝑥), is given by:

𝑔(𝑥) = ∫ 𝑑𝜗 2𝜋 0 ∫ ∫ 𝑒 𝑥𝑓(Ω) sin𝜃 1 sin𝜃 2 𝜋 0 𝜋 0 𝑑𝜃 1 𝑑𝜃 2 = 8𝜋𝐼 0 (𝑥)𝑆ℎ𝑖(3𝑥) 3𝑥 (10)
In short we obtain this simple result:

〈𝑉(𝑎, 𝜃 1 , 𝜃 2 , 𝜗)〉 𝑇=0 = 4𝑉 0 (𝑎) = -4 𝑝 2 4𝜋𝜀 0 1 𝑎 3 (12)
Using 𝑝 = 𝑞𝑑 and (5) in the equation ( 12), we can write the following expression for the average dipole-dipole interaction force per unit area 𝐴:

𝐹 𝑑 𝐴 = - 1 𝐴 𝑑 𝑑𝑎 〈𝑉(𝑎)〉 𝑇=0 = -𝛼 ħ𝑐 𝑎 4 (𝜀-1) 2 (𝜀+1) 2 12𝑑 2 𝐴 ( 13
)
where 𝛼 = 1 4𝜋𝜀 0 𝑒 2 ħ𝑐 is the well-known fine-structure constant, also called coupling constant, as we mentioned above. Following our argument, we impose the equality between (13) and Lifshitz's expression (2):

- ħ𝑐𝜋 2 240𝑎 4 (𝜀-1) 2 (𝜀+1) 2 𝜑(𝜀) = -𝛼 ħ𝑐 𝑎 4 (𝜀-1) 2 (𝜀+1) 2 12𝑑 2 𝐴 (14)
The major issue in the equation ( 14) is that we can eliminate from the equality both the image charge factor (𝜀-1) 2 (𝜀+1) 2 and the distance between the plates 𝑎. This allows us to take the limit 𝜀 → 1, corresponding to an entire vacuum space (without confinement) and thus find out the following equality for 𝛼:

𝛼 = 𝐴 eff 𝑑 eff 2 𝜋 2 2880 𝜑 0 (15)
In the above equation ( 15), we have three unknown parameters: 1) 𝜑 0 = lim 𝜀→1 𝜑(𝜀), that is, the asymptotic value of the function 𝜑(𝜀) for vacuum; 2) 𝐴 eff , the effective area corresponding to an "average" or effective virtual pair -remember that in our model we consider the interaction force between dipoles induced by the virtual pairs, on average-; 3) 𝑑 eff , the average or effective distance or separation between the electron and the positron in a virtual pair created in vacuum.

In the following, we will analyze and compute these three parameters as accurately as possible.

1) 𝝋 𝟎 , the asymptotic limit for vacuum of the function 𝝋(𝜺)

This parameter can be calculated exactly. For that, Lifshitz's calculations must be reviewed because in his article [START_REF] Lifshitz | The Theory of Molecular Attractive Forces between Solids[END_REF], the value of 𝜑 0 is expressed only approximately, without an explicit derivation of it. Assuming static dielectric constants 𝜀 and temperature 𝑇 = 0, he reaches this intermediate expression for identical dielectric plates:

𝐹 𝐴 = - ħ𝑐 32𝜋 2 𝑎 4 ∫ ∫ 𝑥 3 𝑝 2 { 1 (𝑠+𝑝) 2 (𝑠-𝑝) 2 𝑒 𝑥 -1 + 1 (𝑠+𝜀𝑝) 2 (𝑠-𝜀𝑝) 2 𝑒 𝑥 -1 } 𝑑𝑝𝑑𝑥 ∞ 1 ∞ 0 , 𝑠 = √𝜀 -1 + 𝑝 2 (16)
where the variable 𝑥~𝑎𝑝 𝜔 𝑐 is associated with the oscillation frequencies 𝜔 of the stationary waves within the plates and 𝑝 = √𝑘 𝑦 2 + 𝑘 𝑧 2 is the modulus of the two-dimensional wave vector associated with the wave numbers 𝑘 𝑦 and 𝑘 𝑧 , corresponding to the parallel directions to the plates.

Recalling equation ( 2), with [START_REF] Dirac | Discussion of the infinite distribution of electrons in the theory of the positron[END_REF] we have the following expression for the function 𝜑(𝜀):

𝜑(𝜀) = 15 2𝜋 4 (𝜀+1) 2 (𝜀-1) 2 ∫ ∫ 𝑥 3 𝑝 2 { 1 (𝑠+𝑝) 2 (𝑠-𝑝) 2 𝑒 𝑥 -1 + 1 (𝑠+𝜀𝑝) 2 (𝑠-𝜀𝑝) 2 𝑒 𝑥 -1 } 𝑑𝑝𝑑𝑥 ∞ 1 ∞ 0 , 𝑠 = √𝜀 -1 + 𝑝 2 (17)
It is important to note that the final result we are interested in, that is lim 𝜀→1 𝜑(𝜀) = 𝜑 0 , does not change at all if we first take the approximation

(𝑠+𝑝) 2 (𝑠-𝑝) 2 𝑒 𝑥 -1 ≅ (𝑠+𝑝) 2
(𝑠-𝑝) 2 𝑒 𝑥 . In this way we can separate the integrals in [START_REF] Uehling | Polarization effects in the positron theory[END_REF], obtaining:

∫ ∫ 𝑥 3 𝑝 2 { 1 (𝑠+𝑝) 2 (𝑠-𝑝) 2 𝑒 𝑥 -1 + 1 (𝑠+𝜀𝑝) 2 (𝑠-𝜀𝑝) 2 𝑒 𝑥 -1 } 𝑑𝑝𝑑𝑥 ∞ 1 ∞ 0 ≅ ∫ 𝑥 3 𝑒 -𝑥 𝑑𝑥 ∞ 0 ∫ 1 𝑝 2 { (𝑠-𝑝) 2 (𝑠+𝑝) 2 + (𝑠-𝜀𝑝) 2 (𝑠+𝜀𝑝) 2 } 𝑑𝑝 ∞ 1 (18)
It is easy to show that ∫ 𝑥 3 𝑒 -𝑥 𝑑𝑥 ∞ 0 = 6, while the integral on 𝑝 more complicated but can be solved with elementary functions. The limit 𝜀 → 1 of the resulting function of the integral over 𝑝 is zero. But in [START_REF] Uehling | Polarization effects in the positron theory[END_REF] we have the factor (𝜀+1) 2 (𝜀-1) 2 . The limit that interests us converges to the following finite value:

lim 𝜀→1 [ (𝜀+1) 2 (𝜀-1) 2 ∫ 1 𝑝 2 { (𝑠-𝑝) 2 (𝑠+𝑝) 2 + (𝑠-𝜀𝑝) 2 (𝑠+𝜀𝑝) 2 } 𝑑𝑝 ∞ 1 ] = 23 30 (19)
Then, taking the limit 𝜀 → 1 in ( 17), we finally obtain:

lim 𝜀→1 𝜑(𝜀) = 𝜑 0 = 3•23 2𝜋 4 (20)
We shall insist in the fact that the approximation carried out in [START_REF] Gell-Mann | Quantum electrodynamics at small distances[END_REF] is not necessary and that it is done so for the sake of simplicity. The calculation of 𝜑 0 without this approximation is slightly more involved but leads to the exact same result given by [START_REF] Schwinger | On Gauge Invariance and Vacuuwn Polarization[END_REF].

2)

𝑨 𝐞𝐟𝐟 , the effective area of a virtual pair.

To approximate this parameter, we will use the Schwinger formula (4) for the probability, per unit space-time volume, of creating virtual pairs by applying an electric field 𝐸. For the limit case 𝐸 = 𝐸 𝑆 , for which the effects of vacuum polarization begin to be expressed, equation (4) remains:

𝑑𝑁 𝑑𝑉𝑑𝑡 = 1 𝜋 2 𝜆 𝑒 4 ∑ 1 𝑛 2 𝑒 -𝑛𝜋 ∞ 𝑛=1 = 1 𝜋 2 𝜆 𝑒 4 Li 2 (𝑒 -𝜋 ) ( 21
)
where 𝜆 𝑒 = ħ 𝑚 𝑒 𝑐 is the reduced Compton wavelength of the electron and Li 2 is the Euler's dilogarithm function. Now, before continuing, we must make some remarks. First, it is observed that the formula (21) in the Schwinger limit case becomes independent of the elementary charge 𝑒. This, in principle, contributes to a possible non-recursive closed numerical expression for the computation of the finestructure constant in [START_REF] Ruffini | Electron-positron pairs in physics and astrophysics: From heavy nuclei to black holes[END_REF]. Besides, it is important to emphasize that this probability of creating a pair in vacuum is expresses per unit of space-time. This is so because one cannot conceive the creationannihilation quantum process without understanding that the existence of virtual pairs is linked to the existence of an inherent relativistic space-time.

Considering the above, let us denote in [START_REF] Reifenberger | Inter-molecular forces: Keesom force[END_REF] the space-time volume unit 𝑑𝑉𝑑𝑡 as 𝑑𝑆 4 . Then, let us approximate the area 𝐴 eff needed for the creation of a single virtual pair (𝑑𝑁 = 1) as follows:

𝐴 eff ≅ 𝑑𝑆 2 = √𝑑𝑆 4 = 𝜋 √Li 2 (𝑒 -𝜋 ) 𝜆 𝑒 2 (22)
Regarding the physical justification to identify this Schwinger "space-time area" √𝑑𝑉𝑑𝑡 = 𝑑𝑆 2 with the effective area 𝐴 eff that we are looking for in our vision of the Casimir effect, we propose two equivalent arguments. The first is noting that the conventional derivation of the Casimir effect through QFT begins with standing waves in vacuum which, as is well known, have a functional symmetry of the spatial variables and the temporal variable. The presence of the plates imposes boundary conditions in the direction 𝑧 perpendicular to them. Only then the symmetry is broken and both the 𝑧 -wave modes and the frequencies of the waves are quantized. Taking into account that the area of the plates is large compared to the separation between them, in the sum over all the possible excitation modes for the calculation of the vacuum energy, only the two parallel directions 𝑥-𝑦 are considered, which are integrated as a continuous, together with the discrete sum of the frequencies. The divergence of the latter requires the use of the well-known mathematical gadget of zeta-regularization, which allows dealing with infinities [START_REF] Casimir | On the attraction between two perfectly conducting plates[END_REF].

The second justification can be given by a recent work based on the exact loop quantum gravity (LQG) solution for a spherically symmetric vacuum space-time, where the expression of the Casimir force is recovered without the need for regularization or renormalization techniques [START_REF] Gambini | Casimir effect in a quantum space-time[END_REF]. This is possible due to the quantization of space-time in LQG followed by the assumption at the end of their derivation in which the separation of the vertices of the spin network of the quantum space-time is small.

With either of the above options it follows that, in principle, the area for the Casimir effect is actually a "space-time area". Only when vacuum is confined in a given region of space-time, the symmetry breaking forces this Casimir area to be only a spatial one. In our case, even if the region is confined, we cannot assume a large area because we are dealing with virtual pairs that act on an area whose scale is the Compton wavelength. But in addition, at the last we take the vacuum limit of the dielectric material plates, with which we have an unconfined problem, so the consideration of a spacetime area, as we assume in [START_REF] Gambini | Casimir effect in a quantum space-time[END_REF], is then quite appropriate.

3)

𝒅 𝐞𝐟𝐟 , the electron-positron effective distance of a virtual pair. This is the last ingredient we need to complete our mental model and arrive at a theoretical approximation of the fine-structure constant. It is known that the length scale for a virtual electronpositron pair is the Compton wavelength, but to achieve a more precise value, we refer to the work of Leuchs et al [START_REF] Leuchs | The quantum vacuum at the foundations of classical electrodynamics[END_REF]. They propose a semi-classical dispersion model that allows to derive the correct orders of magnitude of the electric permittivity 𝜀 0 and the magnetic permeability 𝜇 0 of the vacuum, shedding light on their physical meanings.

As a first qualitative approach, they use the same length scale 𝑟 both for the effective volume in the polarization 𝑃 = Later, in a more quantitative approach, they suggest taking into account the actual charge distribution in the virtual pair. Thus, for the computation of the induced magnetic dipole moment, instead of using 𝑟 2 , we must use the average orbital radius needed for the diamagnetic term 〈𝜌 2 〉, where 𝜌 is the radial variable in cylindrical coordinates. Considering a solid spherical volume 𝑉 = ⁄ for the approximate parameters of permittivity 𝜀0 and permeability 𝜇 ̃0, it is easy to show that 𝑅 = √ 5 2 𝜆 𝑒 . So, let us approximate the average or effective electron-positron distance as:

𝑑 eff ≅ 𝑅 = √ 5 2 𝜆 𝑒 (23)
We recall that ( 23) is only a first approximation and the problem remains open for more precise approaches where, for example, it is considered that the charge distribution is probably centered around the origin [START_REF] Leuchs | The quantum vacuum at the foundations of classical electrodynamics[END_REF] and, on the other hand, the QFT corrections to the relativistic result for the gyromagnetic ratio are also taken into account.

Results

With the aim of retrieving a numerical value for the fine-structure constant, we have calculated [START_REF] Schwinger | On Gauge Invariance and Vacuuwn Polarization[END_REF], and approximated ( 22) and ( 23), the three parameters required. Putting it in [START_REF] Ruffini | Electron-positron pairs in physics and astrophysics: From heavy nuclei to black holes[END_REF], and taking the inverse (number best known to physicists), we have:

1 𝛼 ≅ 4800𝜋√Li 2 (𝑒 -𝜋 ) 23 ≅ 137.042144475 (24)
This value has a relative error of 44 parts per million when compared to (value recommended by CODATA 2018 [START_REF] Newell | The codata 2017 values for h, e, k and Na for the reviosio of the SI[END_REF], with a relative standard uncertainty of 0.15 parts per billion). Let us recall that the in order to derive, not calculate, the value of 𝛼 in terms of universal mathematical constants, we had to confront two complementary approaches for the same problem.

To whether or not one can do so within the scope of the same theory alone (as done so far in most cases in the literature), and reminiscent of Gödel's incompleteness theorems, is beyond the scope of the present work.

Discussion

We can summarize the present contribution with two main conclusions, as well as some additional considerations. First, on the basis of a simple model of dipolar interaction between the image dipoles induced in vacuum by the virtual pairs that are statistically created, we find a numerical approximation for the fine-structure constant with a relative error of 44 parts per million when compared to the 2018 CODATA experimental recommended value.

Secondly, the model used for the fine-structure approach sheds light on the nature of the Casimir effect, which can be interpreted, in an unconventional way, as a consequence of the vacuum polarization with boundary conditions, without the need to wield the zeta function regularization.

Much criticism can be derived from our approach. Quantitatively, the main source of inaccuracies is given by the ratio

𝐴 eff 𝑑 eff 2 .
As a study focused in obtaining order of magnitudes coincidences, our results are fairy acceptable. Taken as such, with no approximations made, it is rather astonishing that a simple approach as the present one can reproduce the value of the fine structure constant with such an incredible accuracy. Indeed, why would one simply accept without further elaboration, for instance, that the applied electric field strength ought to be exactly the one corresponding to 𝐸 𝑆 = 𝑚 𝑒 2 𝑐 3 𝑒ħ and not some fraction of it? Certainly that would be the case if the vacuum polarization acted in a Heaviside step function fashion. In the same line of thought, the very same value of 𝐸 𝑆 is thought to be extremely difficult to reach experimentally, if not impossible.

Other instances admit a more convincing explanation. In the case of considering the distance between vanishing plates as the one between induced image charges, the creation of pairs is evidently more favored nearby the presence of matter. Also, further contributions to the interacting potential between charge images are identically cancelled in the limit 𝜖 → 1 for additional powers of (𝜀-1) 2 (𝜀+1) 2 .

On the qualitative side, it is plain that we "mimic" the action of the Casimir effect by a model of induced interacting dipoles, which, in turn, are produced by pairs of electrons and positrons that appear due to the action of a vacuum-stripping electric field. Should one consider the creation of virtual pairs instead? That would involve considering extremely small time intervals. Would still be possible for the previous approximations to hold?

In addition to the heuristic and statistical nature of the current contribution, the model must be understood from the point of view of several limits. Namely, the limit of the Schwinger field for the creation of matter per unit of space-time volume; the limit of uncertainty for the Heisenberg principle in its form given by the energy and the lifetime of the particles; the limit to the vacuum for the space occupied by the plates of the Casimir effect; and, finally, the thermodynamic limit of zero temperature in the statistical expression of the dipolar interaction potential that we propose. The use of all these limits altogether results in a more-than-acceptable numerical value that we obtain for the finestructure constant in a first approximation to the problem, justified by the semi-classical nature of the model.

Figures and Figures Captions

where

  𝐼 0 (𝑥) is the modified Bessel function of the first kind and zero order, while 𝑆ℎ𝑖(𝑥) sine integral. It is easy to corroborate that:

  induced by an electric field 𝐸, and for the magnetic dipole moment 𝑀 = 𝑒 2 𝑟 2 𝑚 𝑒 𝐵 induced by a magnetic field 𝐵.

4 3 𝜋𝑅 3 3 〈𝑟 2 〉 3 〈𝑟 2 〉. 2 5 𝑅 2 .

 43323222 with radius 𝑅 and constant probability density for the electron and positron charges, we apply the Langevin's formalism of diamagnetism. With a spherically symmetric distribution, we have 〈𝑥 2 〉 = 〈𝑦 2 〉 = 〈𝑧 2 〉 = 1 and then 〈𝜌 2 〉 = 2 For our uniform charge case, a simple calculation shows that 〈𝜌 2 〉 = Combining all the above and imposing 𝜀0𝜇 ̃0 = 1 𝑐 2

Figure

  Figure 1. a) An electron-positron pair with charges ±𝑒 created in the vicinity of the dielectric plate generates

1

 1 Figure 1. a) An electron-positron pair with charges ±𝑒 created in the vicinity of the dielectric plate generates

  Figure 1. a) An electron-positron pair with charges ±𝑒 created in the vicinity of the dielectric plate generates an image dipole with charges ±𝑞 = ±𝑒 (𝜀-1) (𝜀+1) and effective distance 𝑑. b) Dipole-dipole interaction between two image dipoles 𝑝 = 𝑞𝑑 generated on the dielectric plates. 𝜃 1 and 𝜃 2 are their angles with respect to the axis connecting their centers, while 𝜗 is the relative torsion angle.

Acknowledgments

AS would like to thank FaMAFyC from Universidad Nacional de Córdoba for supporting research. JB acknowledges fruitful discussions with Maria del Mar Batle and Regina Batle. JB received no funding for the present research.

Summing up, we hope to have shed new light on the classical interpretation of the Casimir effect and, at least from orders of magnitude, to have opened a research path to obtain the precise numerical value of the fine structure constant from first principles.


Samir Chabbi
email: s.chabi@univ-soukahras.dz

Nour El

A New Security Solution Enhancing the Dynamic Array PIN Protocol

Keywords: ATM, DAP, NFC, NFC smartphone, Payment, PIN Authentication, PIN, Transaction, Vulnerability

In order to authenticate a user on an Automated Teller Machine (ATM) using Near Field Communication (NFC) technology embedded on smartphones, we recently proposed a new approach called Dynamic Array PIN Protocol (DAP) that allows a user to enter his PIN code in a secure manner. We proved that the DAP protocol is resistant to 13 different attacks. Furthermore, by comparing it to several existing solutions, we demonstrated that DAP is much better and more cost effective. However, after a thorough analysis, we discovered that the DAP protocol is vulnerable to multiple eavesdropping video or camera records attack. Consequently, in this paper, we aim to address this vulnerability by proposing a new security solution that improves the DAP protocol.

I. INTRODUCTION

Near Field Communication (NFC) is a technology used to enable contactless communication, within a short distance, between an NFC reader such as an Automated Teller Machine (ATM) or a Point of Sale Machine (PoS) and an NFC bank card or an NFC smartphone [START_REF] Badra | A lightweight security protocol for nfcbased mobile payments[END_REF]. NFC technology transfers data in a frequency of 13.56 MHz using an electromagnetic field [START_REF] Giese | Security analysis of near-field communication (nfc) payments[END_REF]. Today, this technology is frequently used and found in many proximity payment products [START_REF] Gyamfi | Enhancing the security features of automated teller machines (atms): A ghanaian perspective[END_REF]. For example, a client can use his NFC smartphone or his NFC bank card in front of a PoS to make an NFC payment or his NFC smartphone in front of an ATM to withdraw/deposit money [START_REF] Merkus | Security evaluation of the nfc contactless payment protocol using model based testing[END_REF].

NFC technology is therefore beneficial to both users and banks as it allows fast contactless transactions at any ATM or PoS [START_REF] Wadii | Nfc technology for contactless payment echosystems[END_REF]. Unfortunately, an attacker is able to remotely steal the banking data stored in an NFC bank card or an NFC smartphone, without the knowledge of the client [START_REF] Madhoun | Towards more secure emv purchase transactions[END_REF]. In addition, the user's password or the Personal Identification Number (PIN) can be stolen through various attacks such as: spyware [START_REF] Shammee | A systematic literature review of graphical password schemes[END_REF], shoulder-surfing [START_REF] Alsuhibany | A camouflage text-based password approach for mobile devices against shoulder-surfing attack[END_REF], side channel [START_REF] Chen | Magleak: a learning-based side-channel attack for password recognition with multiple sensors in iiot environment[END_REF], brute force [START_REF] Kurnaz | Secure pin authentication in java smart card using honey encryption[END_REF], replay [START_REF] Shang | Lightdefender: Protecting pin input using ambient light sensor[END_REF], smudge [START_REF] Shin | A new smart smudge attack using cnn[END_REF], camera recording [START_REF] Shubhra | Atm frauds: Detection & prevention[END_REF], video recording [START_REF] Guerar | Securing pin-based authentication in smartwatches with just two gestures[END_REF] and multiple registration [START_REF] Kobayashi | A series of pin/password input methods resilient to shoulder hacking based on cognitive difficulty of tracing multiple key movements[END_REF]. Indeed, in order to deposit or withdraw money using an NFC smartphone in front of an ATM, the user must authenticate himself. Therefore, the authentication of the user is an important security property that must be confirmed in order to protect access to a resource [START_REF] Chanal | Security and privacy in iot: a survey[END_REF]. By consulting existing literature, there are several solutions proposed to authenticate a user in order to secure his contactless communication with an ATM: some solutions require the entry of a PIN code or a password and others use a biometric modality of the user [START_REF]Smart payment association[END_REF] such as fingerprint recognition, facial recognition, voice recognition, etc. According to existing literature, all of these proposed solutions have their advantages and limits.

In fact, when comparing the PIN authentication technique to those using biometric modalities [START_REF]Biometric authentication in payments: Considerations for Policymakers[END_REF], we found that the burden of the latter outweigh their advantages. In most cases, the major disadvantage of the use of a biometric modality is that when it is stolen, the attacker can use it at any time insofar as the user cannot change it, contrarily to a PIN code. Consequently, since the PIN code is also exposed to various attacks, its security is then very important when an NFC smartphone is used to make a transaction with an ATM [START_REF] Yadav | Secure card-less atm transactions[END_REF] and to ensure the security of a user's PIN storage in an NFC smartphone, the PIN code is stored in a hardware circuit called the Secure Element (SE). The latter ensures the security of the data storage and the execution of the sensitive applications it embeds, such as a payment application [START_REF]NFC Functions and Security Certification overview[END_REF].

In our previous paper [START_REF] Chabbi | Dynamic array pin: A novel approach to secure nfc electronic payment between atm and smartphone[END_REF], we proposed a new PIN entry protocol on a smartphone to authenticate a user during an NFC communication with an ATM. We proved that this protocol, called Dynamic Array PIN (DAP), is more cost effective and secure than several protocols proposed in existing literature. The DAP protocol is executed when the client approaches his NFC smartphone to an ATM to authenticate himself. However, we later discovered that the DAP protocol is vulnerable to the intersection of multiple records (video eavesdropping or camera recording) attack, where the attacker can steal the user's PIN code [START_REF] Nyang | Two-thumbs-up: Physical protection for pin entry secure against recording attacks[END_REF]. This vulnerability is illustrated in our previous paper [START_REF] Chabbi | Vulnerability of the dynamic array pin protocol[END_REF]. Consequently, our objective in this paper is to solve this vulnerability by proposing a new security solution that improves the DAP protocol.

This paper is organized as follows. In section II, we discuss the related work. In section III, we describe our new proposed solution to solve the vulnerability of the DAP protocol. In section IV, we prove the security of our proposal by a demonstration. The last section concludes the paper and outlines future work.

II. RELATED WORK

In order to provide secure authentication to a user through a password or a PIN code to perform an NFC transaction between an NFC smartphone and an ATM, several solutions have been proposed in existing literature. In this section, we present the most important and recent solutions and analyze their strengths and weaknesses.

A. BrightPass Solution

The principle of this solution is the generation of a sequence of 0 and 1, called Lie Overhead by running an application embedded in the SE. From this sequence, a series of circles with different luminosity will be displayed on the screen of the smartphone to introduce the PIN code. To enter the PIN code, the user enters a random digit that is not part of the PIN code in a low brightness circle (corresponding to the value 0), and enters a real digit that is part of the PIN code in a high brightness circle (corresponding to the value 1) [START_REF] Deshpande | Multifactor authentication on mobile phones using existing brightpass[END_REF]. The analysis of this solution shows that it protects the PIN code from spyware attacks that discover the typed PIN by capturing the phone screen [START_REF] Guerar | Using screen brightness to improve security in mobile social network access[END_REF]. Contrarily, it is vulnerable to the shoulder surfing attack and the camera recording attack.

B. Color Wheel PIN (CWPIN) Method

In order to authenticate a user with the Color Wheel PIN (CWPIN) method, the bank server uses a secret consisting of a PIN code and an array of ten randomly generated colors, each of which is indexed by a random number from 0 to 9, for example: 0 corresponds to green, 1 to yellow, etc. This secret will be shared with the user and at each authentication attempt, the server sends the indexed table to the ATM which uses a QR code or NFC radio waves to authenticate the user via his smartphone. After the NFC communication with the smartphone, the ATM screen displays a wheel of 10 colors surrounded by a fixed index from 0 to 9 and a search bar to make it turn. The colors and corresponding numbers are generated randomly. For the smartphone, the SE displays an array of 10 randomly generated colors surrounded by an index of 10 randomly generated digits [START_REF] Guerar | Color wheel pin: Usable and resilient atm authentication[END_REF]. In order to authenticate, the user searches the array displayed on the smartphone screen for the color indexed by the first digit of his PIN code, and then on the ATM screen, he turns the wheel to match that color with the second digit of his PIN. In this way, the user enters the first two digits of his PIN. When the user releases the wheel, the wheel rotates randomly until it stabilizes. Then, the user repeats the previous step, finds the color corresponding to the third digit of his PIN code in the array displayed on the smartphone, and then rotates the wheel displayed on the ATM to match this color with the fourth digit of the PIN and doing this, he enters the last two digits of the PIN code [START_REF] Guerar | Color wheel pin: Usable and resilient atm authentication[END_REF]. In this way, CWPIN protects the PIN code from smudge attacks and malware. However, it is not applicable if the PIN has an odd number of digits.

C. DAP Protocol

DAP is a solution that we proposed in [START_REF] Chabbi | Dynamic array pin: A novel approach to secure nfc electronic payment between atm and smartphone[END_REF] allowing to authenticate a user by entering a PIN code using a smartphone during an NFC communication with an ATM. In this technique, the PIN code is the only secret shared between the bank server and the user, the smartphone used integrates the SE and the ATM is equipped with a small touch pad covered with a shell to hide the user's finger movements. In order to authenticate, the user presents his smartphone in front of the NFC reader of the ATM. At that moment, the ATM displays on its screen two vertically aligned arrays, each containing 10 random numbers. The digits in the upper array are fixed on the screen during the authentication session, but the digits in the lower array can slide horizontally following the movement of the user's finger on the ATM touchpad, and can rotate in a circular manner so that cells that disappear on one side immediately appear on the opposite side in the same order. The user authentication in the DAP protocol requires the following steps [START_REF] Chabbi | Dynamic array pin: A novel approach to secure nfc electronic payment between atm and smartphone[END_REF]: PIN code in the upper array and reads the corresponding digit in the lower array. The user then finds the position of the digit read in the upper array and takes it as a reference to be used as explained in the other steps of the session. Fig. 1 shows an example: to enter the PIN code "8642", the user locates 8 in the upper array and reads the corresponding digit in the lower array, which, in this case, is 4. He then looks up the position of 4 in the upper array and considers the cell found (which is in the third position from the left) as the reference for this authentication session. • Using the touchpad, the user slides the digits in the lower array to the left or to the right, and when the first PIN digit in the lower array aligns vertically with the reference position in the upper array, the user releases his finger to enter it. We used the horizontal scroll bar shown in Fig. 1 to simulate the ATM touchpad. • The user repeats the previous step for each remaining digit of the PIN code. To complete this operation, the user presses the "Validate" button. However, in our previous paper [START_REF] Chabbi | Vulnerability of the dynamic array pin protocol[END_REF], we discovered that the DAP protocol is vulnerable to the intersection of multiple records (video eavesdropping or camera recording) attack where the attacker can steal the user's PIN code [START_REF] Nyang | Two-thumbs-up: Physical protection for pin entry secure against recording attacks[END_REF]. This attack is usually applied to discover the PIN entered by the user when using a smart technique (PIN digits are not entered directly). It is usually performed by a hidden camera that records the steps of the PIN-based authentication protocol.

III. NEW PROPOSED SOLUTION: DCP

In order to protect the user against the intersection multiple records attack mentioned above (see also section I), in this section, we propose a new solution called Dynamic Cloud PIN (DCP) which is an improvement for the DAP protocol. This solution improves the authentication mechanism on ATMs in order to strengthen the security system of NFC transactions with the ATM. It uses only the PIN code and a digit as secrets shared between the user and the bank server.

A. Solution Phases

To perform an NFC transaction with the ATM, our new solution includes the following phases:

1) Cloud Phase: it includes the following steps:

• When the user is away from the ATM, he will use his own account by running an application installed on his smartphone that has been provided by the bank server. This application will communicate with the SE to obtain its encrypted identifier. • The server uses the encrypted identifier to reference the user's record in a database, then generates a randomly indistinguishable number from 0 to 9, stores it in the user's record in the database and sends it to the user's smartphone.

• The server places the user's records in a cash memory to optimize the access time and sends a dynamically generated number to the user. • The user receives from the Cloud (online) the dynamic number (secured thanks to TLS) in his smartphone, stores it and destroys it. 2) Authentication phase: it includes the following steps:

• The user runs the secure application that belongs to the bank server using his smartphone. • The user approaches his smartphone to the NFC reader of the ATM.

• The smartphone sends encrypted information to the ATM: identifier of the SE, banking information stored in the SE or in the virtual bank card integrated in the smartphone.

• The ATM sends the information received from the smartphone to the bank server. • The server looks for the appropriate record in the cash memory. If this record is not found, the protocol is interrupted and the transaction is ignored. Otherwise, the server will proceed to authenticate the user (the owner of the smartphone) by inviting him to enter the PIN code as described in the next steps.

• The server generates three arrays, each one contains 10 randomly indistinguishable elements. The first one (the highest in Fig. 2) or the last (third) one (the lowest in Fig. 2) is composed of 10 digits numbered from 0 to 9 and they are displayed randomly. The position of each digit in each array is fixed during the authentication session. The middle array is composed of 10 alphabetical characters from A to Z. These characters are displayed in a random order where the position of each char is dynamically assigned during the authentication session as the user can drag the characters to the right or to the left. • The server sends the three arrays to the ATM.

• The ATM displays the three arrays on the screen with the "Cancel" and "Validate" buttons. The characters of the middle array can be moved by the user to the right or to the left using a touchpad. • The user locates the position of the digit received online by his smartphone in the third array from the top (see Fig. 3). This position is called the reference. • In order to authenticate, the user must slide the character that corresponds to the reference into the middle array and release his finger each time the character coincides with a digit of the PIN code in the first array. Finally, he presses the "Validate" button to indicate the end of the PIN code entry. • When the "Validate" button is pressed, the ATM sends the user's data to the server. • The server compares the PIN code entered with the one retrieved from the database. If they are equal, then the user is well authenticated.

B. Solution Example

In this section, we use an example to illustrate our solution. We assume that the dynamic number sent by the Cloud is "5" and that the PIN code is "1234". As shown in Fig. 3, to enter this PIN code, the user looks for the position of the digit "5" in the third array which represents the reference for this time (position five in the third array from the left). Then, using a small touchpad installed on the ATM, the user slides the middle array horizontally to the left or to the right and stops it by releasing his finger whenever a PIN digit in the first array matches the character in the middle array that corresponds to the reference in the third array (the character for entering the first digit is "B"). The X-axis scroll bar in Fig. 3 is used in our desktop implementation to simulate the ATM touchpad. the rest of the steps followed by the user are as follows:

• To enter the first digit of the PIN code, the user drags in our example the "B" until it becomes opposite the first digit ("1") in the first array as shown in Fig. 4. • Then the user notices that the "E" is now opposite the reference ("5") in the third array, he must drag the "E" until it becomes opposite the second digit ("2" in our example) in the first array. This is illustrated in Fig. 5. • Then the user notices that the "B" is now opposite the reference ("5") in the third array, he must drag the "B" until it becomes opposite the third digit ("3" in our example) in the first array. This is illustrated in Fig. 6.

Fig. 6. Enter of the Third Digit ('3')

• Finally, the user notices that the "F" is now opposite the reference ("5") in the third array, so he must drag the "F" until it becomes opposite the fourth digit ("4" in our example) in the first array. This is illustrated in Fig. 7. • To validate, the user clicks on the "Validate" button and obtains the status shown in Fig. 8. Indeed, the basic idea of our proposed solution is that the attacker can see the user's input, but he cannot reveal the PIN code because he cannot detect the reference in the third array that is the position of the dynamic digit shared between the server and the user ("5" in our example). The attacker also cannot detect the character in the middle array that corresponds to the reference, especially because the character changes after each entry of a PIN digit. The dynamic digit shared between the server and the user and its position in the third array are changeable during each authentication session and are unknown to the attacker.

IV. SECURITY ANALYSIS

In this section, we prove that our proposal is protected against the multiple records intersection attack. To achieve this goal, we consider that an attacker, using a camera that records the authentication operation, is able to detect when the user's finger is released after entering each digit. For example, we assume that the PIN code is "3582" and the shared dynamic digit is "5". We assume that the DCP interface is as illustrated in Fig. 9. The rest of the steps used to analyze the security of our proposal are as follows:

• To enter the first digit of the PIN code ("3"), the user slides the character "G" under the digit "3" of the first array as shown in Fig. 10. • The attacker records the state of the first and middle arrays just after the user's finger is released as shown in Fig. 11. • To enter the second digit of his PIN code ("5"), the user drags the character "M", that corresponds to the reference "5" in the third array, under the digit "5" in the first array as shown in Fig. 12. • To enter the third digit of his PIN code ("8"), the user drags the character "K", that corresponds to the reference "5" in the third array, under the digit "8" in the first array as illustrated in Fig. 14. ("2"), the user drags the character "F", that corresponds to the reference "5" in the third array, under the digit "2" in the first array as illustrated in Fig. 16. • Then, the user clicks on the "Validate" button and obtains the result illustrated in Fig. 18. • Finally, the attacker groups the arrays that have been recorded to detect the PIN as illustrated in Fig. 19. We can conclude that the intersection multiple records attack has no effect on our DCP protocol. This is because the attacker is unable to fix a set of possible combinations representing the PIN code, thanks to the following:

1) The digit shared in the Cloud between the server and the user is a random secret (it changes at each session).

2) The positions of the digits in the first and last arrays are random at each session (they change at each session).

3) The characters displayed in the middle array are random at each session (they change at each session and knowing that they are 10 among 26 characters, this increases the number of probabilities). 4) Thanks to the latter, the character corresponding to the shared digit in the third array will be random at each session and will also change after each PIN digit is entered, which makes tracking too difficult.

V. CONCLUSION AND FUTURE WORK

In this paper, we proposed a new security solution that improves the DAP protocol which is vulnerable to multiple eavesdropping video or camera records attack. Our proposal improves the PIN authentication mechanism on ATMs in order to strengthen the security system of NFC transactions with the ATM. We proved the security of our proposal by a demonstration where the attacker cannot find the correct PIN code by applying the intersection multiple records attack. In future work, we intend to experiment with our proposal to find a better balance between its increased security and increased authentication time. One possible improvement would be to relieve the user of the burden of searching for the character that matches the reference and dragging it to the PIN digit in the first array. We believe that this part could be automated without re-exposing the improved DCP to the multiple record attack.

Fig. 1 .

 1 Fig. 1. The DAP Protocol Interface• The user locates the position of the first digit of his PIN code in the upper array and reads the corresponding digit in the lower array. The user then finds the position of the digit read in the upper array and takes it as a reference to be used as explained in the other steps of the session. Fig.1shows an example: to enter the PIN code "8642", the user locates 8 in the upper array and reads the corresponding digit in the lower array, which, in this case, is 4. He then looks up the position of 4 in the upper array and considers the cell found (which is in the third position from the left) as the reference for this authentication session. • Using the touchpad, the user slides the digits in the lower array to the left or to the right, and when the first PIN digit in the lower array aligns vertically with the reference position in the upper array, the user releases his finger to enter it. We used the horizontal scroll bar shown in Fig.1to simulate the ATM touchpad. • The user repeats the previous step for each remaining digit of the PIN code. To complete this operation, the user presses the "Validate" button. However, in our previous paper[START_REF] Chabbi | Vulnerability of the dynamic array pin protocol[END_REF], we discovered that the DAP protocol is vulnerable to the intersection of multiple records (video eavesdropping or camera recording) attack where the attacker can steal the user's PIN code[START_REF] Nyang | Two-thumbs-up: Physical protection for pin entry secure against recording attacks[END_REF]. This attack is usually applied to discover the PIN entered by the user when using a smart technique (PIN digits are not entered directly). It is usually performed by a hidden camera that records the steps of the PIN-based authentication protocol.

Fig. 2 .

 2 Fig. 2. The DCP Protocol Interface

Fig. 3 .

 3 Fig. 3. The Reference in DCP Interface

Fig. 4 .

 4 Fig. 4. Enter of the First Digit ('1')

Fig. 5 .

 5 Fig. 5. Enter of the Second Digit ('2')

Fig. 7 .Fig. 8 .

 78 Fig. 7. Enter of the Fourth Digit ('4')

Fig. 11 .

 11 Fig. 11. Records Made by the Attacker (1)

Fig. 12 .

 12 Fig. 12. The Enter of the Second Digit • The attacker records the state of the first and middle arrays just after the user's finger is released as shown in Fig. 13.

Fig. 13 .

 13 Fig. 13. Records Made by the Attacker (2)

Fig. 14 .

 14 Fig. 14. The Enter of the Third Digit

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Records Made by the Attacker (3)

  .

Fig. 17 .

 17 Fig. 17. Records Made by the Attacker (4)

Fig. 18 .

 18 Fig. 18. The End of DCP Authentication

Fig. 19 .

 19 Fig. 19. Arrays Recorded by the Attacker