Gérard Gouesbet
email: gouesbet@coria.fr

Luiz Felipe

Machado Votto

Leonardo André

Finite series expressions to evaluate the beam shape coe¢ cients of a Laguerre-Gauss beam freely propagating

Keywords: Generalized Lorenz-Mie theories, structured beams, T-matrix, beam shape coe¢ cients, …nite series, Laguerre-Gauss beams

When using some analytical and semi-analytical scattering theories such as Generalized Lorenz-Mie Theories (GLMTs) and Extended Boundary Condition Method (EBCM) for structured beams, electromagnetic …elds, expanded over a set of vector wave functions, may be encoded in some coe¢ cients known as Beam Shape Coe¢ cients (BSCs). In the case of Laguerre-Gauss beams freely propagating, the evaluation of BSCs by using numerical quadratures is time-consuming while localized approximations have been shown to be of limited validity. The present paper therefore establishes …nite series expressions to evaluate the BSCs of Laguerre-Gauss beams freely propagating in a rigorous framework, which has the known advantage of speeding up computations with respect to the use of a numerical quadrature technique.

1 Introduction.

When dealing with T-matrix methods for the scattering of light and other electromagnetic radiation by particles, such as analytical Generalized Lorenz-Mie Theories (GLMTs) for particles having a su¢ cient degree of symmetry to allow for the use of a method of separation of variables [START_REF] Gouesbet | Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation[END_REF], [START_REF] Maheu | A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident pro…le[END_REF], [START_REF] Gouesbet | Generalized Lorenz-Mie theories, second edition[END_REF], or semianalytical Extended Boundary Condition Method (EBCM) for arbitrary shaped particles [START_REF] Waterman | Symmetry, unitarity, and geometry in electromagnetic scattering[END_REF], [START_REF] Mishchenko | Scattering, absorption, and emission of light by small particles[END_REF] particularly in the case of structured beams [START_REF] Mackowski | Direct simulation of multiple scattering by discrete random media illuminated by Gaussian beams[END_REF], [START_REF] Wang | Light scattering from an optically anisotropic particle illuminated by an arbitrary shaped beam[END_REF], electromagnetic …elds may be expanded over a set of vector wave functions. In spherical coordinates, expansions are carried out over a set of vector spherical wave functions (VSWFs) and expansion coe¢ cients are expressed in terms of coe¢ cients known as Beam Shape Coe¢ cients (BSCs) usually denoted as g m n;T M and g m n;T E (TM: Transverse Magnetic; TE: Transverse Electric).

The original method to evaluate the BSCs of electromagnetic beams is by using quadratures, e.g. [START_REF] Gouesbet | Discussion of two quadrature methods of evaluating beam shape coe¢ cients in generalized Lorenz-Mie theory[END_REF]. In some cases, quadratures may be analytically solved to produce closed form expressions of the BSCs, such as in the cases of zeroth-order Bessel beams [START_REF] Lock | Angular spectrum and localized model of Davis-type beam[END_REF], higher-order Bessel beams [START_REF] Wang | Multipole expansion of circularly Bessel beams of arbitrary order for scattering calculations[END_REF] or zeroth-order Mathieu beams [START_REF] Cha…q | On the validity of the integral localized approximation for on-axis zeroth-order Mathieu beams[END_REF]. In the case of Gaussian beams, it has never been possible to analytically solve the quadratures and, therefore, since these Gaussian beams may be viewed as a special case of Laguerre-Gauss beams, there is little chance to achieve analytical quadratures to the evaluation of BSCs of Laguerre-Gauss beams as well. Hence, quadratures to the evaluation of BSCs of Laguerre-Gauss beams have to be carried out numerically, which is a time-consuming process. To circumvent this di¢ culty, we may think of using a localized approximation (with several variants) which may speed up the computations by several orders of magnitude as reviewed in [START_REF] Gouesbet | Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review[END_REF], see also [START_REF] Gouesbet | Comments on localized and integral localized approximations in spherical coordinates[END_REF] for an up-dated terminology, and [START_REF] Wang | Note on the use of localized beam models for light scattering theories in spherical coordinates[END_REF], [START_REF] Gouesbet | Second modi…ed localized approximation for use in generalized Lorenz-Mie theories and other theories revisited[END_REF] for complements. Unfortunately, it has recently been demonstrated that any existing localized approximation has a limited domain of validity when dealing with helical beams, i.e. having a topological charge [START_REF] Gouesbet | On the validity of the use of a localized approximation for helical beams. I. Formal aspects[END_REF], as is the case for Laguerre-Gauss beams whether they are freely propagating [START_REF] Ambrosio | On the validity of the use of a localized approximation for helical beams. II. Numerical aspects[END_REF] or focused by a lens [START_REF] Ambrosio | On localized approximations for Laguerre-Gauss beams focused by a lens[END_REF]. The arsenal of methods to evaluate BSCs however contains another technique to speed up the evaluation of BSCs, namely the use of …nite series which is a rigorous analytical method [START_REF] Gouesbet | Computations of the gn coe¢cients in the generalized Lorenz-Mie theory using three di¤erent methods[END_REF], [START_REF] Gouesbet | Expressions to compute the coe¢ cients gnm in the generalized Lorenz-Mie theory, using …nite series[END_REF]. A disadvantage of the …nite series method is nevertheless its lack of ‡exibility insofar as it requires a fairly heavy analytical work before numerical implementation when the description of the beam is modi…ed. Up to now, …nite series expressions are known only in the case of Gaussian beams. To circumvent the aforementioned di¢ culties in the evaluation of BSCs of Laguerre-Gauss beams, the present paper establishes the …nite series expressions required for the use of the …nite series technique in the case of Laguerre-Gauss beams freely propagating.

Up to now, we have expressed our motivation to the study of the …nite series method to evaluate the BSCs of Laguerre-Gauss beams. This motivation is reinforced by the many applications of such beams, in particular due to the fact that they possess a helical wavefront allowing for the transfer of angular momentum to an illuminated object, e.g. Padgett and Allen [START_REF] Padgett | Light with a twist in its tail[END_REF] who discussed orbital angular momentum associated with helical wavefronts and applications, Garbin et al. [START_REF] Garbin | Mie scattering distinguishes the topologic charge of an optical vortex : a homage to Gustav Mie[END_REF] who experimentally and numerically studied the scattering of highly focused Laguerre-Gauss beams by dielectric and metal spheres, O'Holleran et al. [START_REF] O'holleran | Illustrations of optical vortices in three dimensions[END_REF] who provided a 3D view of optical vortices associated with Laguerre-Gauss modes, Friese et al. [START_REF] Friese | Optical angular-momentum transfer to trapped absorbing particles[END_REF] who studied the transfer of optical angular momentum to trapped absorbing particles, a study relevant to the …eld of optical information. Laguerre-Gauss beams were discussed as well in the framework of a review devoted to optical tweezers by Molloy and Padgett [START_REF] Molloy | Lights, action: optical tweezers[END_REF]. Other applications concerned quantum information and the entanglement of the polarization of a single photon with its orbital angular momentum by Nagali et al. [START_REF] Nagali | Quantum information transfer from spin to orbital angular momentum of photons[END_REF], entanglement again but involving many orthogonal quantum states rather than only two states by Mair et al. [START_REF] Mair | Entanglement of orbital angular momentum states of photons[END_REF], object identi…cation by measuring the joint orbital angular momentum spectrum of two-photon states by Uribe-Patarroyo et al. [START_REF] Uribe-Patarroyo | Object identi…cation using correlated orbital angular momentum states[END_REF], optical traps to study vortices in Bose-Einstein condensates by Tempere et al. [START_REF] Tempere | Vortices in Bose-Einstein condensates con…ned in a multiply connected Laguerre-Gaussian optical trap[END_REF], optical communication systems in atmospheric turbulence by Malik et al. [START_REF] Malik | In ‡uence of atmospheric turbulence on optical communications using orbital angular momentum for encoding[END_REF], use of vortex masks to observe dim exoplanets by Foo et al. [START_REF] Foo | Optical vortex corona graphs[END_REF], to provide a few examples without pretending to exhaustiveness.

The paper is organized as follows. Section 2 recalls a background concerning the method to be used to establish …nite series expressions and concerning the electromagnetic …elds of a Laguerre-Gauss beam freely propagating. Section 3 deals with the …nite series expressions for TM-and TE-BSCs of a Laguerre-Gauss beam freely propagating. Section 4 is a conclusion.

2 Background.

Neumann Expansion Theorem (NET).

Although the way to use the NET, relying on Neumann or Bessel function expansions, has already been published in the literature, e.g. [START_REF] Gouesbet | Generalized Lorenz-Mie theories, second edition[END_REF], pp. 121-124, it is concisely reproduced here for the convenience of the reader. It starts with a result published by Watson [START_REF] Watson | A treatise of the theory of Bessel functions[END_REF], pp. 524-525. Let us consider an equation of the form:

x 1=2 g(x) = 1 X n=0 c n J n+1=2 (x) (1)
in which J n+1=2 (:) are classical half-order Bessel functions. The Maclaurin expansion of the function g(x) reads as:

g(x) = 1 X n=0 b n x n (2)
Then the NET states that the coe¢ cients c n are given by:

c n = (n + 1 2 ) n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! b n 2m ( 3
)
in which is the celebrated Gamma function satisfying (n) = (n 1)! and (z +1) = z (z). We then have two ways of expressing the coe¢ cients c n , either from the Bessel function expansion of Eq.1 or from the Maclaurin expansion of Eq.2, using Eq.3. Equating the two resulting expressions will allow one to establish …nite series expressions for the BSCs. This is done using what may be called the NET-procedure.

2.2

The NET-procedure.

The procedure starts with the expressions of the radial electric E r and magnetic H r …eld components in spherical coordinates (r, , ') which may be expressed as, e.g. Eqs.(3.10) and (3.19) of [START_REF] Gouesbet | Generalized Lorenz-Mie theories, second edition[END_REF]:

E r H r = E 0 H 0 1 X n=1 +n X m= n c pw n g m n;T M g m n;T E n(n + 1) r (1)
n (kr)P jmj n (cos ) exp(im')

in which E 0 , H 0 are …eld strengths, g m n;T M and g m n;T E are the BSCs to be evaluated within the framework of the NET-procedure, k is the wavenumber,

n (:) denote the …rst-order spherical Bessel functions also denoted as j n (:), and P jmj n (:) are associated Legendre functions de…ned by using Hobson's notation [START_REF] Robin | Fonctions sphériques de Legendre et fonctions sphéroidales[END_REF]. Furthermore c pw n (with pw standing for "plane wave") denotes expansion coe¢ cients which appear in the Bromwich formulation of the classical Lorenz-Mie theory [START_REF] Gouesbet | Sur la généralisation de la théorie de Lorenz-Mie[END_REF] according to:

c pw n = ( i) n ik 2n + 1 n(n + 1) (5)
Afterward, we discard the '-dependency by using the orthogonality relation:

Z 2 0 exp[i(m m 0 )']d' = 2 mm 0 (6)
to obtain:

Z 2 0 E r H r exp( im')d' (7) = 2 E 0 H 0 1 X n=jmj c pw n g m n;T M g m n;T E n(n + 1) r (1)
n (kr)P jmj n (cos )

in which we took account of the fact that P jmj n = 0 if n < jmj. The next step is to discard the -dependency as well. This may be done in two ways. The …rst way is to specify = =2 in Eq.7 and to invoke the following relations for associated Legendre functions [START_REF] Robin | Fonctions sphériques de Legendre et fonctions sphéroidales[END_REF]:

P m n (0) = ( 1) n+m 2 (n + m 1)!! 2 n m 2 ( n m 2 )! , (n m) even (8)
P m n (0) = 0, (n m) odd (9)
in which:

n!! = 1:3:5:::n ( 1)!! = 1 (10)
to be completed with, for a better-looking expression:

n!! = 2 n+1 2 p ( n 2 + 1) (11)
Furthermore, we have [START_REF] Watson | A treatise of the theory of Bessel functions[END_REF], [START_REF] Stratton | Electromagnetic theory[END_REF]:

(1)

n (kr) = r 2kr J n+ 1 2 (kr) (12)
Eq.7 may then be rewritten as:

(kr) 1=2 Z 2 0 r E r ( = =2) H r ( = =2) exp( im')d' (13)
= p 2 E 0 H 0 1 X n=jmj;(n m) even c pw n n(n + 1) g m n;T M g m n;T E P jmj n (0)J n+1=2 (kr)
Later on, this equation, speci…ed for the Laguerre-Gauss beams under study, will be given the form of Eq.1 and will allow one to express the BSCs using …nite series expressions, with however the restriction that we must have (n m) even. For (n m) odd, Eq.7 is di¤erentiated with respect to cos , in order to take advantage of the relations:

[ dP m n (cos ) d cos ] cos =0 = 0, (n m) even (14) [ dP m n (cos ) d cos ] cos =0 = ( 1) n+m 1 2 (n + m)!! 2 n m 1 2 ( n m 1 2 )! , (n m) odd (15)
Then, instead of Eq.13, we obtain:

(kr) 1=2 Z 2 0 r[ @ @ cos E r H r ] = =2 exp( im')d' (16)
= p 2 E 0 H 0 1 X n=jmj;(n m) o dd c pw n n(n + 1) g m n;T M g m n;T E [ dP m n (cos ) d cos ] cos =0 J n+1=2 (kr)
which will later be given the form of Eq.1 and will allow one to obtain …nite series expressions for the BSCs of Laguerre-Gauss beams freely propagating, for (n m) odd.

2.3

Laguerre-Gauss beams freely propagating.

The radial component of the electric …eld of a Laguerre-Gauss beam freely propagating, relying on books by Siegmann [START_REF] Siegman | An introduction to lasers and masers[END_REF], [START_REF] Siegman | Lasers[END_REF], for a harmonic beam having a time-dependency of the form exp(+i!t), is given by [START_REF] Ambrosio | On the validity of the use of a localized approximation for helical beams. II. Numerical aspects[END_REF]:

E r (r; ; ') = E 0 kw(r cos ) r sin p 2 w(r cos ) ! L ( 2r 2 sin 2 w 2 (r cos ) ) (17)
exp[ ikr 2 sin 2 2e q(r cos ) ] exp[i(2 + + 1) (r cos )] exp(i ') exp( ikr cos ) sin cos ' in which:

w(r cos ) = w 0 s 1 + 2r cos kw 2 0 2 (18) e q(r cos ) = r cos + ikw 2 0 2 (19) (r cos ) = tan 1 2r cos kw 2 0 ( 20
)
in which w 0 is the beam waist radius of the beam, while and de…ne the type of Laguerre-Gauss beam under study. Also, L denotes associated Laguerre polynomials de…ned as follows [START_REF] Arfken | Mathematical methods for physicists[END_REF], [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]:

L (x) = 1 ! X i=0 ! i! + i ( x) i (21)
in which () denotes a binomial coe¢ cient. The "usual" Laguerre polynomials L (x) are L 0 (x): The associated Laguerre polynomials can be evaluated recursively from the …rst two polynomials as follows:

L 0 (x) = 1 (22) L 1 (x) = 1 + x (23) L k+1 (x) = (2k + 1 + x)L k (x) (k + )L k 1 (x) k + 1 (24)
The radial component of the electric …eld will allow one to provide …nite series expressions for the TM-BSCs. For the TE-BSCs, we need the corresponding expression for the radial component of the magnetic …eld. For this, we have to take into account the fact that Eq.17 represents a paraxial approximation which therefore does not satisfy Maxwell's equations. Hence, using the Faraday law to express the magnetic …eld from the electric …eld will generate artefacts which will have to be removed to obtain a proper paraxial approximation of the radial component of the magnetic …eld. Having this in mind, the radial component E r is actually obtained from an electric …eld E which may be written as [START_REF] Ambrosio | On the validity of the use of a localized approximation for helical beams. II. Numerical aspects[END_REF]:

E = (E x ; E y ; E z ) = (E 0 Ee ikz ; 0; 0) (25)
leading to:

E r = E 0 Ee ikz sin cos ' (26)
which, identifying with Eq.17, de…nes E. According to Faraday's law, we then have:

H = 1 i! curl E = 1 i! (0; @E x @z ; 0) ( 27
)
in which is the permeability, e.g. Eq.(1.41) in [START_REF] Gouesbet | Generalized Lorenz-Mie theories, second edition[END_REF]. From Eqs.17, 25, 27 and removing artefacts which include @E/@z, we obtain:

H = (0; H 0 Ee ikz ; 0) (28)
in which we have used E 0 =H 0 = ! =k. Hence:

H r = H 0 Ee ikz sin sin ' (29)
which will be used to determine the TE-BSCs. The removal of the artefacts aimed to symmetrize expressions for the radial electric and radial magnetic components, as shown in Eqs.26 and 29. They are easily identi…ed because their presence would break the symmetry exhibited by these equations.

3 Expressions to evaluate BSCs.

3.1

TM-BSCS, (n m) even.

For = =2, we readily have w(r cos ) = w 0 , e q(r cos ) = ikw 2 0 =2, and (r cos ) = tan1 0 = 0, see Eqs.18-20. Then Eq.17 becomes:

E r ( = =2) = F[e i( +1)' + e i( 1)' ] (30)
in which:

F = E 0 2kw 0 r p 2 w 0 ! L ( 2r 2 w 2 0 ) exp( r 2 w 2 0 ) (31)
We then insert Eq.30 into Eq.13, so that, using Eq.6, the left-hand-side of the electric part of Eq.13 becomes:

LHS = 2 (kr) 1=2 rF[ m; +1 + m; 1 ] (32)
We now introduce x = kr (not to be confused with a Cartesian coordinate) and the beam con…nement factor s = 1=(kw 0 ), and invoke Eq.5. Rearranging, Eq.13 may then be written as:

x 1=2 g(x)[ m; +1 + m; 1 ] (33) = 1 X n=jmj ( i) n (2n + 1)g m n;T M P jmj n (0)J n+1=2 (x), (n m) even
which is of the form of Eq.1 to be used in the NET-procedure, with the function g(x) given by:

g(x) = i p 2 xs(x p 2s) L (2x 2 s 2 ) exp( x 2 s 2 ) (34)
Note that Eq.33 shows that we shall only deal with the modes m = +1 and m = agreement with the fact that BSCs for m6 = 1 are zero for localized beams". This property could not be positively stated by using numerical quadratures because there is necessarily the possibility of round-o¤ errors , but it is indeed here obtained formally within the framework of the NET-procedure.

To account for the modulus of m in Eq.33, restricting ourselves to 0, we have to distinguish three cases: (i) m = + 1, 8 0 (ii) m = 1, > 0 (iii) m = 1, = 0 i.e. m = 1. Furthermore, we shall complement various quantities already introduced with an extra-subscript , for convenience. We then have:

(i) m = + 1, 8 0 x 1=2 [g(x)] = 1 X n= +1 ( i) n (2n + 1)[g +1 n;T M ] P +1 n (0)J n+1=2 (x), (n ) odd (35)
which, using Eq.1, leads to:

c n = ( i) n (2n + 1)[g +1 n;T M ] P +1 n (0); n + 1, n odd (36)
c n = 0, otherwise (37)
(ii) m = 1, > 0 x 1=2 [g(x)] = 1 X n= 1 ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0)J n+1=2 (x), (n ) odd (38)
leading to:

c n = ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0), n 1, n odd ( 39
)
c n = 0, otherwise (40)
(iii) m = 1, = 0 x 1=2 [g(x)] = 1 X n=1 ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0)J n+1=2 (x), n odd (41)
leading to:

c n = ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0), n 1, n odd (42)
c n = 0, otherwise (43)
We then have a …rst set of expressions for the coe¢ cients c n . We are now going to establish a second set of expressions for the same coe¢ cients relying on Eqs.2 and 3. For this, we rewrite [g(x)] under the form:

[g(x)] = G L (2x 2 s 2 ) exp( x 2 s 2 )x +1 (44)
in which G , which does not depend on , reads as:

G = is p 2 (s p 2) (45)
We shall use the recurrence relations for L , Eqs.22-24, under the form:

L 0 (2x 2 s 2 ) = 1 (46) L 1 (2x 2 s 2 ) = 1 + 2x 2 s 2 (47) L k+1 (2x 2 s 2 ) = (2k + 1 + 2x 2 s 2 )L k (2x 2 s 2 ) (k + )L k 1 (2x 2 s 2 ) k + 1 (48)
the last one being better rewritten as:

We now deal with a general recurrence relation. Using Eqs.44 and 49, we have:

[g(x)] = 2 1 + [g(x)] 1 1 + [g(x)] 2 2s 2 fx 2 [g(x)] 1 g (56)
Let us focus on the term f:g denoted A. Recalling Eq.2, it can be evaluated as follows:

A =x 2 1 X n=0 [b n ] 1 x n = 1 X n=0 [b n ] 1 x n+2 = 1 X t=0 [b t ] 1 x t+2 (57)
which, using (n = t + 2), becomes:

A = 1 X n=2 [b n 2 ] 1 x n = 1 X n=0 "(n; 0; 1)[b n 2 ] 1 x n (58)
Eq.56 then reads as:

1 X n=0 [b n ] x n = 2 1 + 1 X n=0 [b n ] 1 x n 1 + 1 X n=0 [b n ] 2 x n (59) 2s 2 1 X n=0 "(n; 0; 1)[b n 2 ] 1 x n
We then obtain a recurrence relation for the b n -coe¢ cients:

[b n ] = 2 1 + [b n ] 1 1 + [b n ] 2 2s 2 "(n; 0; 1)[b n 2 ] 1 (60)
As an example, let us evaluate [b n ] 2 knowing [b n ] 1 and [b n ] 0 . For = 2, Eq.60 becomes:

[b n ] 2 = + 3 2 [b n ] 1 + 1 2 [b n ] 0 s 2 "(n; 0; 1)[b n 2 ] 1 (61)
This may be evaluated by using the expressions for [b n ] 0 and [b n ] 1 . For the last term, [b n 2 ] 1 involves "(n 2; 0; 1; :::; ) and "(n 2; 0; 1; :::; + 2). The evaluation of this last term then relies on the obvious expression: "(n; 0; 1)"(n 2; 0; 1; :::; k) = "(n; 0; 1)"(n; 2; 3; :::; k + 2) = "(n; 0; 1; :::; k + 2) (62)

We then obtain: .

[b n ] 2 = G [
Once the coe¢ cients b n are known, we readily know the coe¢ cients b n 2m and obtain new expressions for the coe¢ cients c n using Eq.3. These new expressions are equated with the ones we have obtained previously, e.g. Eqs. 35-43. This leads to:

c n = ( i) n (2n + 1)[g +1 n;T M ] P +1 n (0) (64) = (n + 1 2 ) n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , for n > + 1; (n ) odd, > 0 c n = ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0) (65)
= (n + 1 2 ) n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , for n > 1; (n ) odd; > 0 c n = ( i) n (2n + 1)[g 1 n;T M ] P 1 n (0) (66)
= (n + 1 2 ) n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , for n > +1; n odd
Next, from Eqs.8 and 11, we have:

P m n (0) = 2 m p ( 1)
n+m 2

( n m 2 )! ( n + m + 1 2 ), (n m) even (67)
allowing one to evaluate the expressions for the P m n 's in Eqs.64-66, and to obtain explicit expressions for the TM-BSCs g m n;T M , (n m) even. They read as:

[g +1 n;T M ] = i n p 2 +2 ( 1)
n+ +1 2

( n+ 2 + 1) ( n 1 2 )! n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , (68)
for n > + 1; (n ) odd, > 0 [g 1 n;T M ] = i n p 2 ( 1)
n+ 1 2 ( n+ 2 ) ( n + 1 2 )! n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , (69)
for n > 1; (n ) odd; > 0 [g 1 n;T M ] = i n p 4 ( 1) n+1 2
( n 2 + 1)

( n 1 2 )! n=2 X m=0 2 1 2 +n 2m ( 1 2 + n m) m! [b n 2m ] , (70)
for n > +1; n odd 3.2 TM-BSCS, (n m) odd.

The procedure for this case, namely (n m) odd, being quite similar to the one used for the previous case, we shall be content with a few intermediary relations aiming to help the reader who would like to check the derivations, and with the …nal results. Instead of Eq.7, we now deal with Eq.13 which requires us to evaluate the derivative [@E r =@ cos ] = =2 . After a few pages of careful computations, we obtain, from Eq.17:

[ @E r (cos ) @ cos ] = =2 = E 0 k e i ' cos ' 1 w 0 ( r p 2 w 0 ) L ( 2r 2 w 2 0 ) exp( r 2 w 2 0 ) (71)
f 2ir 3 kw 4 0 + ikr[ 2 k 2 w 2 0 (2 + + 1) 1]g
Afterward, instead of Eq.33, we obtain:

x 1=2 g(x)[ m; +1 + m; 1 ] (72) = 1 X n=jmj ( i) n (2n + 1)g m n;T M [ dP jmj n (cos ) d cos ] = =2 J n+1=2 (x), (n m) odd
with the function g(x) now given by:

g(x) = i p 2 xs(xs p 2) L (2x 2 s 2 ) exp( x 2 s 2 )f 2ix 3 s 4 + ix[2s 2 (2 + + 1) 1]g (73)
To account for the modulus of m in Eq.72, still with 0, we again have to distinguish three cases:

(i) m = + 1, 8 0 (ii) m = 1, > 0 (iii) m =
1, = 0 i.e. m = 1. Furthermore, still adding a subscript to conveniently decorate various quantities, we have:

(i) m = + 1, 8 0 x 1=2 [g(x)] = 1 X n= +1 ( i) n (2n+1)[g +1 n;T M ] [ dP +1 n (cos ) d cos ] = =2 J n+1=2 (x), (n ) even (74)
c n = ( i) n (2n + 1)[g +1 n;T M ] [ dP +1 n (cos ) d cos ] = =2 ; n + 1, n even (75) c n = 0, otherwise (76)
(ii) m = 1, > 0 x 1=2 [g(x)] = 1 X n= 1 ( i) n (2n+1)[g 1 n;T M ] [ dP 1 n (cos ) d cos ] = =2 J n+1=2 (x), (n ) even (77) c n = ( i) n (2n + 1)[g 1 n;T M ] [ dP 1 n (cos ) d cos ] = =2 , n 1, n even (78) c n = 0, otherwise (79)
(iii) m = 1, = 0 x 1=2 [g(x)] = 1 X n=2 ( i) n (2n + 1)[g 1 n;T M ] [ dP 1 n (cos ) d cos ] = =2 J n+1=2 (x), n even (80) c n = ( i) n (2n + 1)[g 1 n;T M ] [ dP 1 n (cos ) d cos ] = =2 , n 2, n even (81)
c n = 0, otherwise (82)
Next, instead of Eq.44, we rewrite [g(x)] under the form:

[g(x)] = L (2x 2 s 2 ) exp( x 2 s 2 )[H v x +4 + K x +2 ] (83)
in which:

H v = 2is 4 G (84)
K = i[2s 2 (2 + + 1) 1]G (85) G = is p 2 (s p 2) (86)
Let us …rst again consider the cases = 0 and 1 to be used to start a recurrence. Instead of 51, we now have:

[g(x)] 0 = H v 1 X n= +4 F (2)x n + K v 1 X n= +2 F (1)x n (87)
[b n ] 0 = "(n; 0; 1; :::; + 3)H F (2) + "(n; 0; 1; :::;

+ 1)K F (1) (88)
in which we introduced the notation:

F (u) = ( s 2 ) n 2 u ( n 2 u)! (89)
Similarly, for = 1:

[g(x)] 1 = (1 + )[g(x)] 0 2s 2 fx 2 g(x) 0 g (90)
[b n ] 1 = "(n; 0; 1; :::;

+ 1)(1 + )K F (1) (91)
+"(n; 0; 1; :::;

+ 3)[(1 + )H 2s 2 K ]F (2)
2s 2 "(n; 0; 1; :::; + 5)H F (3)

For the general case, Eqs.56 and 60 are still valid, as conveniently repeated below:

[g(x)] = 2 1 + [g(x)] 1 1 + [g(x)] 2 2s 2 fx 2 [g(x)] 1 g (92) [b n ] = 2 1 + [b n ] 1 1 + [b n ] 2 2s 2 "(n; 0; 1)[b n 2 ] 1 (93)
which, however, lead to expressions for the coe¢ cients [b n ] which are different from the ones for the case (n m) even, due to the di¤erent expressions for [b n ] 0 and [b n ] 1 which are used to initialize the recurrence process. As an example, instead of Eq.63, we now have:

Conclusion.

For use in some analytical or semi-analytical light scattering theories, like GLMTs or EBCM, the electromagnetic …elds may be encoded in a set of coef-…cients named beam shape coe¢ cients (BSCs). In the case of Laguerre-Gauss beams (either freely propagating or focused by a lens), the usual quadrature technique which allows one to evaluate the BSCs has to be carried out numerically, leading to time-consuming evaluations. Localized approximations, which are traditionally used to speed-up the computations of the BSCs by orders of magnitude, unfortunately have a limited domain of validity for Laguerre-Gauss beams and have to be considered with care and suspicion. Therefore, borrowing another technique from the arsenal of methods available to evaluate BSCs, the present paper establishes rigorous expressions to the evaluation of BSCs using …nite series. It must be noted that, although the procedure used is rigorous, it is, in the present paper, applied to a paraxial approximation of Laguerre-Gauss beams freely propagating, i.e. using a beam description which does not perfectly satis…es Maxwell's equations. Relying on our experience concerning paraxial Gaussian beams, either using …nite series ([3], pp. 164-171) or quadratures [START_REF] Gouesbet | Discussion of two quadrature methods of evaluating beam shape coe¢ cients in generalized Lorenz-Mie theory[END_REF], the non-Maxwellian character of the paraxial beam should be re ‡ected by a blowing-up of BSCs values for high-order partial waves. These faulty BSCs could be set to zero or they would have no consequence because they would appear for partial waves which, in any case, convey vanishing amplitudes. Once this is implemented, the BSCs obtained de…ne a beam which perfectly satis-…es Maxwell's equations, resulting from the internal coherency of the theory based on the fact that it uses a basis of regular VSWFs, independently of the particular technique used to evaluate the BSCs. In the present case, however, the …nite series procedure allows one to transform a non-Maxwellian beam to a Maxwellian beam, illustrating the remodeling process involved in the evaluation of BSCs using …nite series. Beams generated by the BSCs evaluated by …nite series therefore automatically provide a Maxwellian description of Laguerre-Gauss beams freely propagating. Subsequent papers will be devoted to the algorithmic implementation of the formulas obtained in this paper, and later on, to the application of the …nite series technique to the case of Laguerre-Gauss beams focused by a lens.

It is furthermore to be noted that the derivations of BSCs in the present paper have been carried in spherical coordinates for parallel illumination, in on-axis situations. BSCs for other kinds of curvilinear systems of coordinates can afterward be obtained from the BSCs in spherical coordinates by using an extrinsic method, see [START_REF] Gouesbet | List of problems for future research in generalized Lorenz-Mie theories and related topics, review and prospectus; commemorative invited paper, for the 50th anniversary of "Applied Optics[END_REF] for the de…nition of intrinsic and extrinsic methods, and the introduction of [START_REF] Han | Intrinsic method for the evaluation of beam shape coe¢ cients in spheroidal coordinates for oblique illumination[END_REF] for a review of extrinsic methods in spheroidal and cylindrical coordinates. BSCs for oblique illumination may be deduced from the BSCs for parallel illumination by using rules of transformations of BSCs under rotation of coordinate systems [START_REF] Gouesbet | Transformations of spherical beam shape coe¢ cients in generalized Lorenz-Mie theories through rotations of coordinate system. I. General formulation[END_REF], [START_REF] Wang | Transformations of spherical beam shape coe¢ cients in generalized Lorenz-Mie theories through rotations of coordinate system. II. Axisymmetric beams[END_REF], [START_REF] Gouesbet | Transformations of spherical beam shape coe¢ cients in generalized Lorenz-Mie theories through rotations of coordinate system. III. Special values of Euler angles[END_REF], [START_REF] Gouesbet | Transformations of spherical beam shape coe¢ cients in generalized Lorenz-Mie theories through rotations of coordinate system[END_REF], [START_REF] Gouesbet | Transformations of spherical beam shape coe¢ cients in generalized Lorenz-Mie theories through rotations of coordinate system. V. Localized beam models[END_REF]. BSCs for o¤axis situations can be obtained from the BSCs for on-axis situations by using translation theorems [START_REF] Doicu | Computation of the beam-shape-coe¢ cients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions[END_REF] or by introducing the o¤-axis expressions of the …elds right at the beginning of the algebraic treatment, as actually done in the case of Gaussian beams, see [START_REF] Gouesbet | Expressions to compute the coe¢ cients gnm in the generalized Lorenz-Mie theory, using …nite series[END_REF]. The last procedure however requires an extra-algebraic work. It is indeed a defect of the …nite series technique that each modi…cation of the …eld expressions requires an extra-algebraic work before implementation in computer programs, although it has been stated that the "whole process may be in principle carried out in an automatic way by using a formal computation procedure which would furthermore generate FORTRAN sources" ([3], p. 121).

in agreement with what we already learnt in[START_REF] Ambrosio | On the validity of the use of a localized approximation for helical beams. II. Numerical aspects[END_REF]. In this Ref., p.46, correcting an obvious typo, we noted: "we provided results only for m = 1, in

Let us …rst consider the cases = 0 and 1 which have to be used to initialize a recurrence process. From Eqs.46 and 44:

To display this in the form of Eq.2, we set n = 2t + + 1 to obtain:

in which we introduced the notation:

Eq.51 is of the form of Eq.2:

with:

in which "(n; j )=0 if n is equal to one of the j 's, and is equal to 1 otherwise. For = 1, we use Eq.47 instead of 46 and, after a few manipulations similar to the ones used previously, we obtain:

2s 2 "(n; 0; 1; :::;

+( + 2)"(n; 0; 1; :::;

Thereafter, instead of Eqs.64-66, we obtain:

Next, from Eqs.15 and 11, we establish:

and eventually obtain:

( 1)

( 1)

n 2

for n > +2; n even

TE-BSCs.

To deal with the TE-BSCs, we have to use H r in Eqs.7 and 16 instead of E r . Computations then run in the same way but for the replacement of a cos ' by a sin ', see Eqs.26 and 29. This implies that, in Eqs. [START_REF] Robin | Fonctions sphériques de Legendre et fonctions sphéroidales[END_REF]


MD Pierre Philip
email: philip@free.fr

Keywords: Driving simulator, Maintenance of Wakefulness Test, Central Hypersomnia

Conclusions:

In patients with narcolepsy/idiopathic hypersomnia, simulated driving and MWT explore different dimensions of fitness-to-drive and could be used complementarily to better evaluate sleep-related driving impairment.

2

Abstract

Study Objectives: To assess the relationship between real and simulated driving performance and the objective level of alertness as measured by the Maintenance of Wakefulness Test (MWT) in patients suffering from narcolepsy or idiopathic hypersomnia.

Methods: 27 patients (10 patients with narcolepsy, type 1 (n=7) and type 2 (n=3), and 17 patients with idiopathic hypersomnia, mean age=33.8 ± 11.1 years, range=18-65 y; 4 males) were recruited in a randomized, crossover, double-blind placebo-controlled trial and compared to 27 matched healthy controls. Patients were randomly assigned to receive modafinil (400 mg) or placebo before the driving test (2 hours of real and 2 hours of simulated highway driving for each patient). Standard Deviation of Lateral Position (SDLP) of the vehicle in real and simulated driving and mean sleep latency in a 4 x 40min MWT were assessed.

Results: Untreated patients presented shorter sleep latencies on the MWT (20.8 (IQ range 16.1-32.9) vs. 34.9 min (IQ range 28.1-40.0)) and worse simulated driving performance (P<0.001) than treated patients. Nevertheless, treated patients still exhibited shorter mean sleep latencies on the MWT than controls (34.9 (IQ range 28.1-40.0) vs. 40 min (IQ range 37.1-40.0), P<0.05) but driving performance was identical in both groups. The SDLP of the vehicle in real driving conditions and the MWT score correlated with the SDLP in simulated driving (respectively, r=0.34, P<0.05 and r=-0.56, P<0.001).

Introduction

Sleepiness at the wheel is a major risk factor for highway traffic accidents [START_REF] Connor | Driver sleepiness and risk of serious injury to car occupants: population based case control study[END_REF] .

Narcolepsy and idiopathic hypersomnia (IH) are rare central disorders that induce severe daytime sleepiness [START_REF] Mitler | Sleep latency on the maintenance of wakefulness test (MWT) for 530 patients with narcolepsy while free of psychoactive drugs[END_REF][START_REF] Dauvilliers | Narcolepsy with cataplexy[END_REF] . While less studied for driving risk than OSAS, narcolepsy and idiopathic hypersomnia have been shown to increase the risk of sleep-related accidents [START_REF] Philip | Sleep disorders and accidental risk in a large group of regular registered highway drivers[END_REF][START_REF] Aldrich | Automobile accidents in patients with sleep disorders[END_REF][START_REF] Powell | Sleepy driver near-misses may predict accident risks[END_REF][START_REF] Kotterba | Comparison of driving simulator performance and neuropsychological testing in narcolepsy[END_REF][START_REF] Ozaki | Health-related quality of life among drug-naive patients with narcolepsy with cataplexy, narcolepsy without cataplexy, and idiopathic hypersomnia without long sleep time[END_REF][START_REF] Bayon | Socio-professional handicap and accidental risk in patients with hypersomnias of central origin[END_REF][START_REF] Philip | Modafinil improves real driving performance in patients with hypersomnia: a randomized double-blind placebo-controlled crossover clinical trial[END_REF][START_REF] Pizza | Car Crashes and Central Disorders of Hypersomnolence: A French Study[END_REF][START_REF] Smolensky | Sleep disorders, medical conditions, and road accident risk[END_REF] .

The Maintenance of Wakefulness Test (MWT) is a consensus clinical objective test for evaluating excessive daytime sleepiness and is part of the guidelines for driving license legislation for professional drivers in France [START_REF] Alonderis | Medico-legal implications of sleep apnoea syndrome: driving license regulations in Europe[END_REF] . In a previous study, we showed that the MWT can be used to predict driving ability in untreated sleep apnea patients [START_REF] Philip | Maintenance of Wakefulness Test, obstructive sleep apnea syndrome, and driving risk[END_REF] .

Another concern is the driving ability of both untreated and treated patients with central hypersomnia. Modafinil is the reference drug used to treat excessive daytime sleepiness in patients with narcolepsy and hypersomnia [START_REF] Lavault | Benefit and risk of modafinil in idiopathic hypersomnia vs. narcolepsy with cataplexy[END_REF][START_REF] Barateau | Treatment Options for Narcolepsy[END_REF] . We demonstrated in real road conditions [START_REF] Philip | Modafinil improves real driving performance in patients with hypersomnia: a randomized double-blind placebo-controlled crossover clinical trial[END_REF] that modafinil improves driving performance in patients with narcolepsy and idiopathic hypersomnia. Another study [START_REF] Pizza | Car Crashes and Central Disorders of Hypersomnolence: A French Study[END_REF] showed that patients affected with central disorders of hypersomnolence had an increased risk of car accidents, a finding potentially reversed by long-term treatment.

A question remains regarding the predictability of actual driving performance in untreated and treated patients with central hypersomnia. Driving simulation is an attractive and safe method for assessing driving behavior. A study [START_REF] George | Comparison of simulated driving performance in narcolepsy and sleep apnea patients[END_REF] demonstrated that untreated patients with narcolepsy and sleep apnea had impaired driving performance on a simulated driving task compared to healthy controls. Another 18 compared MWT scores with simulated driving performance in healthy sleep-deprived volunteers and provided evidence that an objective measure of sleepiness predicted simulated performance. Finally, a driving simulator was found to predict driving ability in patients with untreated narcolepsy [START_REF] Kotterba | Comparison of driving simulator performance and neuropsychological testing in narcolepsy[END_REF] . Nevertheless, the driving simulators used in all these studies were not designed specifically to be compared with real driving performance. Our group developed a research program sponsored by the French Ministry of Transport to design and test a simple driving simulator reproducing very precisely the driving scenario of a real freeway located near our sleep laboratory. We published a validation study of this driving simulator in sleep-deprived subjects [START_REF] Davenne | Reliability of simulator driving tool for evaluation of sleepiness, fatigue and driving performance[END_REF] and have now used it to measure the driving performance of narcoleptic/hypersomniac patients.

To our knowledge, no study to date has compared the relationships between real driving performance, simulated driving performance and objective sleepiness in patients suffering from central hypersomnia, yet these are important questions to be able to provide a complementary means of evaluating fitness-to-drive in addition to the classical MWT in this at-risk patient population.

Methods

Twenty-seven patients with narcolepsy (narcolepsy type 1 (NT1) or type 2 (NT2)) or idiopathic hypersomnia (IH) were recruited in a randomized double-blind placebo-controlled crossover study (modafinil versus placebo). Patients were diagnosed according to the criteria of the current International Classification of Sleep Disorders (ICSD-3, 2014) [START_REF]International classification of sleep disorders[END_REF] .

Anti-cataplectic medication was maintained for NT1 patients. Twenty-seven healthy control participants matched by age and gender with patients were recruited outside of the sleep clinic.

Design

Patients recruited in the sleep clinic were randomly assigned to receive modafinil (400 mg/day) or placebo for 5 days prior to the driving test. Treatment was administered at 08:00 (modafinil 2×100 mg) and 12:30 (modafinil 2×100 mg). In each condition, a period of 3 days at home with actigraphy recordings and a sleep diary was followed by 2 days to perform a 4 ×

Driving evaluation

Our team developed a specific simulator called INRETS-MSIS SIM2 to determine whether it could identify the same impact of sleepiness (i.e., extended wakefulness) and fatigue (i.e., 2-, 4-and 8-h prolonged nocturnal driving) as that obtained in real-life driving used as a gold standard [START_REF] Davenne | Reliability of simulator driving tool for evaluation of sleepiness, fatigue and driving performance[END_REF] . It was specially equipped with the reconstructed scenario of the real-life highway experiments. The simulator [START_REF] Davenne | Reliability of simulator driving tool for evaluation of sleepiness, fatigue and driving performance[END_REF] has been shown in healthy subjects to measure nocturnal driving impairment appropriately compared to real driving conditions [START_REF] Sagaspe | Extended driving impairs nocturnal driving performances[END_REF] in a dose-response design of extended wakefulness and duration of driving. It comprises a computer and a video-game steering wheel with no force feedback applied. The participant's head was situated 60 cm in front of the screen. The resolution of the visual scene was 1024 × 768 pixels and the update rate was 60 Hz. The simulator displayed a highway driving scene on a 19-in screen. The simulated road surface was colored and the scene visibility corresponded to clear daytime conditions with no traffic. The highway geometry depicted in the simulation was a reconstruction, in virtual reality, of a real highway (A62 between Agen and Langon, France). This reconstruction was based on the topographic layout which consists of a 2-lane highway. Lane widths and road marks were incorporated into the simulation in order to obtain similar road perceptions to those in real driving. The car's speed was set by the experimenter at 130 km/h and participants were instructed to drive in the center of the righthand lane. Environmental lighting at eye level was about 150 lux.

Real world driving performance was identified by a Continental Automotive (Toulouse, France) video system, recording the lateral position (cm) of the car (10 times/sec) from the right lateral lane marker of the road.

The mean standard deviation of lateral position (SDLP) of the vehicle in real and simulated driving was measured to assess the weaving of the car during the driving session 24- 26 .

Objective sleepiness evaluation

Mean sleep latency was assessed by the MWT to evaluate the ability of patients to fight against sleepiness in a soporific condition during a longer duration. Objective sleepiness was assessed with four 40-min MWT trials performed at 10H, 12H, 14H, and 16H, as recommended by the American Academy of Sleep Medicine (AASM) [START_REF] Littner | Practice parameters for clinical use of the multiple sleep latency test and the maintenance of wakefulness test[END_REF] .

Electroencephalograms (F3/A2, C3/A2, O1/A2), electromyograms and electro-oculograms were recorded according to the recommendations of the AASM [START_REF] Iber | for the American Academy of Sleep Medicine. The AASM manual for the scoring of sleep and associated events: rules, terminology and technical specifications[END_REF]

Statistical Analysis

Demographic and clinical characteristics were expressed as mean ± standard deviation (SD).

Quantitative variables were expressed as median and interquartile (IQ) range.

The Mann-Whitney U test was used to identify whether subjective and objective sleepiness scores differed between the two independent groups (idiopathic hypersomnia vs. narcolepsy). The Wilcoxon matched-pairs signed-rank test was used to identify whether MWT scores and real and simulated driving performance differed between the treatment conditions in patients (placebo vs. modafinil). The Mann-Whitney U test was used to identify whether MWT scores and real and simulated driving performance differed between the two independent groups (treated patients vs. controls). Rho Spearman correlations were computed to measure the association between simulated driving performance and real driving performance, objective sleepiness (MWT and MSLT scores) and subjective ESS score.

The data for the real-driving condition used for these analyses were published in a previous study [START_REF] Philip | Modafinil improves real driving performance in patients with hypersomnia: a randomized double-blind placebo-controlled crossover clinical trial[END_REF] . The alpha risk threshold was set at P = .05. All analyses were performed using the SPSS statistical software package (v18.0.0, SPSS Inc., Chicago, USA).

Results

Participants

Twenty-seven patients (7 with narcolepsy type 1 (NT1), 3 with narcolepsy type 2 (NT2) and 17 with idiopathic hypersomnia (IH); mean age = 33.9 ± 10.9 years, range = 21-59 y; 4 males), were evaluated. The mean Epworth Sleepiness Scale score was 16.6 ± 3.8 (range, 11-23). The mean MWT score was 23.2 min ± 9.9 (range, 8.5-40). The mean IAH was 2.2 events/hr ± 3.3 (range, 0-15) and the mean PLM index was 1.1 events/hr ± 2.1 (range, 0-8.

MWT scores and simulated driving performance of hypersomniac/narcoleptic patients versus controls

Table 2 present MWT scores and simulated driving performance in untreated and treated patients suffering from narcolepsy/hypersomnia and in matched controls. Treated patients improved their MWT scores and driving performance (compared to untreated condition) but did not recover MWT scores comparable to that of control subjects.

Simulated driving performance versus real driving performance

In patients suffering from narcolepsy or IH, the SDLP in real driving correlated significantly with the SDLP in simulated driving (r = 0.34, P < 0.05) (See Figure 1).

Simulated driving performance versus mean sleep latency on MWT

The SDLP correlated significantly with the mean sleep latency on the MWT in patients suffering from narcolepsy or IH (r = -0.56, P<0.001) (See Figure 2).

Discussion

This study shows that simulated driving performance in addition to mean sleep latency on the MWT are correlated with real driving performance in narcoleptic/hypersomniac patients. We previously showed that modafinil improves real driving in both patients with narcolepsy and idiopathic hypersomnia [START_REF] Philip | Modafinil improves real driving performance in patients with hypersomnia: a randomized double-blind placebo-controlled crossover clinical trial[END_REF] . Here, we confirm this finding in simulated driving performance introducing a new and easily implementable task to quantify sleep-related driving risk.

Our findings show that the MWT and the driving simulation task are both complementary indicators of real driving performance. Interestingly, MWT and simulated driving provided different values in treated patients and control subjects; MWT performance was lower in treated patients than in controls while simulated driving performance was identical in both groups. Both tests involve an identical process reflecting the difficulty in remaining awake during a monotonous situation. However, driving involves factors other than the level of alertness. The driving impairment experienced by patients very likely stems from a difficulty not only in remaining alert but also attentive. The MWT is an indicator of the ability to stay awake and could be complemented by the driving simulator task, which is closer to real driving and probably more appropriate for assessing cognitive processes including vigilance and lane control. In narcoleptic/hypersomniac patients, driving simulation could be used as a first-line test to evaluate driving performance before measuring objective measure of alertness.

Efficient performance on the driving simulator could be complemented by an MWT to confirm a patient's ability to drive if he reports feeling sleepy at the wheel.

If patients perform poorly on the simulator, they could be clinically reevaluated and have their treatment readjusted. Sleepiness could therefore be tested by MWT as well as by a new driving test. In addition, some patients who have to perform the MWT for legal reasons remain moderately sleepy during the test but still have good simulated driving performance.

Therefore, simulated driving performance could complement MWT results in such patients to evaluate their fitness to drive. We recommend that simulated driving performance resulting in an SDLP > 40, a value which represents the median in untreated patients and the last quartile in treated patients and in controls, should be considered as a pathological threshold defining driving risk.

A limitation of our study concerns the lack of statistical power due to our sample size so we were unable to perform sub-analyses in treated and untreated patients. Nevertheless, our results are still valid overall because a global measure of driving impairment is required in clinical practice. A recent study [START_REF] Pizza | Car Crashes and Central Disorders of Hypersomnolence: A French Study[END_REF] showed that patients suffering from central disorders of hypersomnolence had an increased risk of recent car crashes, a finding potentially reversed by long-term treatment. Treatment duration in terms of stable treatment is an important factor for studying the beneficial effect of modafinil on real driving performance. If simulated driving performance could be re-evaluated after 6 months or one year of treatment, it could confirm its predictive value in the long term.

In summary, our results highlight the complementary value of a simulated driving test in addition to the MWT when assessing driving risk. The use of validated simulators among the sleep community would be a reasonable goal to harmonize evaluation of the ability to drive in patients suffering from central hypersomnia. The best way to standardize driving simulator approach is to use a common simulated driving task and to implement it in a single protocol with a fixed timing. Indeed, a study 29 found a reproducibility of simulated driving performance across time. A 1-hour unique session of simulated driving seems relevant. Larger data sets are also needed to obtain normative values and cut-off points for simulated driving tests in other types of patients and to identify the period when simulated driving is the most predictive.

  40-min MWT (10H, 12H, 14H and 16H) and a 2-h on-road and simulator driving session. Simulated driving tests were performed either at 10H or 14H in a counterbalanced order while real driving tests were performed at 17H for logistic reasons. Nocturnal sleep was controlled by polysomnography (PSG) (time in bed from 23H to 7H). Each condition was separated by at least 3 weeks of washout. Each of the 27 control participants underwent a clinical interview and examination by a sleep medicine specialist. We excluded participants with any complaint of sleep disorders as reported on the Basic Nordic Sleep Questionnaire [BNSQ] 21 and subjective EDS based upon the ESS (score > 10) 22 . The presence of nocturnal sleep-disordered breathing (apneahypopnea index [AHI] > 5/hr) and periodic limb movements (PLM index > 5/hr) were ruled out through polysomnography recording in the laboratory. Control participants were paid 150 euros for their involvement in the study. All participants provided written informed consent and the local ethical committee (consultative committee for the protection of persons participating in biomedical research, Comité de Protection des Personnes [CPP] Sud-Ouest et Outre Mer III) approved the study.

  . Data were recorded and manually analyzed in 30-sec epochs. Sleep onset was defined as the first epoch of greater than 15 sec of cumulative sleep in a 30-sec epoch. The test was ended after three continuous epochs of stage N1 or one epoch of any other sleep stage to avoid interfering with sleep homeostasis 27 . Patients who did not sleep during a trial were assigned a value of 40 min. Patients were video-monitored during the whole test. The mean sleep latency of the four MWT trials was then calculated. Patients were not allowed to sleep between the tests.

8 .

 8 5). The mean sleep latency in the Multiple Sleep Latency Test (MSLT) was 7.1 min ± 2.No difference in sleepiness appears between untreated idiopathic hypersomnia and narcolepsy groups on ESS score (Mann-Whitney U test, Z = 1.683, NS), mean sleep latency on the MSLT (Mann-Whitney U test, Z = -0.956, NS) nor on sleep latency on MWT (Mann-Whitney U test, Z = 0.485, NS). Table 1 present clinical characteristics of patients with narcolepsy or idiopathic hypersomnia. Twenty-seven healthy control participants (age = 33.3 ± 10.4 years, range = 21-62 y, 4 males) matched by age and gender with patients were evaluated. One control participant stopped owing to discomfort and sickness during the simulated driving experiment. Driving performance versus mean sleep latency on MSLT and ESS score In untreated patients, the SDLP in real and simulated driving does not correlate with the ESS score (respectively, r = 0.06, NS and r = -0.11, NS) nor with the MSLT sleep latencies (r = -0.05, NS and r = -0.00, NS). Moreover, no correlation was observed between mean sleep latencies in the MSLT and MWT (r = -0.21, NS).

  Driving performance does not correlate with sleep latency on the MSLT but significantly correlate with the mean MWT score. Therefore, this result corroborates the idea that MWT and MSLT measure different abilities or kinds of sleepiness 29, 30 . The ability to stay awake seems inherently different from the ability to fall asleep. The MWT measures sleep tendency plus several arousal components while the MSLT is a relatively pure measure of sleep propensity. An activity requiring safety such as driving is closer to the measurement of the ability to maintain alertness than to the proneness to fall asleep 2 .

Figure 1 .

 1 Figure 1. Association of stability of trajectory in real driving conditions with stability of

Figure 2 .

 2 Figure 2. Association of stability of trajectory in simulated driving conditions with objective

Table 1 .

 1 Clinical features of patients with narcolepsy or idiopathic hypersomnia (Mean ± Standard deviation). Abbreviations. Na = not applicable; BMI = Body Mass Index; ESS = Epworth Sleepiness Scale; MSLT = Multiple Sleep Latency Test; SOREMP = number of Sleep Onset REM Periods in 5 naps; CSF = Cerebrospinal Fluid; AHI = Apnea/Hypopnea Index; PLM = Periodic Limb Movements.

		Narcoleptic patients	Hypersomniac patients
		(n=10)	(n=17)
	Age (years)	37.2 ± 13.8	31.9 ± 8.7
	Gender (% females)	70.0%	94.1%
	BMI (kg/m 2 )	23.5 ± 5.5	22.3 ± 3.8
	ESS score	18.1 ± 3.9	15.6 ± 3.5
	MSLT Mean Sleep	6.3 ± 2.6	7.5 ± 2.9
	Latency (min)
	Number of SOREMPs ≥2	2 ± 1.3	0.6 ± 0.8
	Cataplexy (%)	70	Na
	CSF Hypocretin-1 Level (pg/mL)	246.5 ± 27.6	286.6 ± 71.4
	Duration of illness (years)	17.7 ± 14.6	13.2 ± 5.1
	Established Diagnosis	8.7 ± 6.4	5.8 ± 4.2
	(years)
	AHI	3.5 ± 5.1	1.5 ± 1.7
	PLM Index	1.8 ± 2.9	0.7 ± 1.6

Table 2 .

 2 Median and interquartile (IQ) in untreated and treated patients suffering from narcolepsy/hypersomnia and in matched controls. Wilcoxon matched-pairs signed-rank test and Mann-Whitney U test comparisons.

		Median	Median	Wilcoxon test,	Median	Mann-
		(IQ range)	(IQ range)	P	(IQ range)	Whitney U
						test, P
		Untreated	Treated	Untreated vs.	Controls	Treated
		patients	patients	treated		patients vs.
				patients		controls
	Mean Sleep	20.8
	Latency MWT	(16.1-32.
	(min)

Author contributions:

1) Conception and design of the study: PS, OC, SE, DD, PP, 2) acquisition and analysis of data: PS, JAMF, PP or 3) drafting a significant portion of the manuscript or figures: PS, JAMF, OC, DL, SE, DD, RL, YD, PP.

Disclosure Statement:

This study was supported by ANR (Agence Nationale de la Recherche) AUTOSOP -PREDIT (Programme de Recherche et d'Innovations dans les Transports Terrestres).


Alexis Graux
email: alexisgraux@msn.com

Sophie Cazottes

David De Castro

David San

Carlos Capdevila

Jose Maria Cabrera

Sílvia Molas

Sebastian Schreiber

Djordje Mirković

Frédéric Danoix

David San Martín

Matthieu Bugnet

Damien Fabrègue

Michel Perez

Precipitation and grain growth modelling in Ti-Nb microalloyed steels

Keywords: Microalloyed steels, Precipitation, Austenite, Grain growth, Modelling

Introduction

Microalloying elements such as titanium and niobium form precipitates enhancing the mechanical properties of steels by controlling austenite grain growth [START_REF] Gladman | Grain-Size Control in Steels[END_REF] and through precipitation hardening [START_REF] Gladman | Precipitation hardening in metals[END_REF].

Regardless of the intended phase transformation from the austenite domain, the resulting mechanical properties depend on the prior austenite grain size. Indeed, austenite grain boundaries act as nucleation sites for ferrite, bainite and/or martensite. A finer austenite microstructure thus results in a finer ultimate microstructure, with enhanced strength and toughness.

During hot rolling process, the soaking temperature prior to hot rolling must be carefully selected to optimize the effects of microalloying elements.

A high soaking temperature ensures a complete dissolution of the pre-existing precipitates, leading to the formation of finer precipitates during and after hot rolling. However, it also results in coarser austenite microstructures. On the other hand, a low soaking temperature would maintain a fine austenite structure but may not dissolve existing coarse precipitates, decreasing the potential for precipitation strengthening during the following steps. The choice of the soaking temperature and duration is generally the result of many years of practical feedback. There is a demand to replace an essentially experience-based approach with thermodynamic and kinetic models.

Grain growth is an important field of study in materials science and has been the subject of a considerable amount of work [START_REF] Humphreys | Recrystallization and related annealing phenomena[END_REF][START_REF] Rios | Grain growth unresolved issues[END_REF]. In microalloyed steels, microalloying elements may interact with the motion of grain boundary in two ways. In the first place, a dispersion of precipitates exerts a retarding pressure on grain boundary, which has a pronounced effect on grain growth. The magnitude of this effect depends on the size, shape and volume fraction of precipitates [START_REF] Gladman | Grain-Size Control in Steels[END_REF][START_REF] Gladman | The physical metallurgy of microalloyed steels[END_REF]. Secondly, a migrating interface drags a local segregation (atmosphere) of solute elements, which exerts a retarding force on it. Thus, solute elements may reduce largely reduce the grain boundary mobility. This phenomenon is generally referred to as solute drag effect [START_REF] John W Cahn | The impurity-drag effect in grain boundary motion[END_REF][START_REF] Hersent | On the Effect of Atoms in Solid Solution on Grain Growth Kinetics[END_REF]. In the case of high velocity grain boundaries, the solute elements can no longer keep up with the grain boundaries. Thus, the solute drag stays limited in the case of high temperature heat treatments.

Numerous studies intend to develop numerical models for austenite grain growth kinetics in steels.

Phenomenological approaches are widely used for their simplicity [START_REF] Sellars | Recrystallization and grain growth in hot rolling[END_REF][START_REF] Shome | A modified analytical approach for modelling grain growth in the coarse grain HAZ of HSLA steels[END_REF][START_REF] Lee | Prediction of austenite grain growth during austenitization of low alloy steels[END_REF][START_REF] Xu | Prediction model for the austenite grain growth in a hot rolled dual phase steel[END_REF][START_REF] Zhang | The growth behavior of austenite grain in the heating process of 300m steel[END_REF][START_REF] Yang | Austenite Grain Refinement and Isothermal Growth Behavior in a Low Carbon Vana-dium Microalloyed Steel[END_REF], the grain diameter following empirical equations of the type:

D n -D n 0 = At exp -Q GG R g T ( 1
)
where D is the final grain diameter, D 0 is the initial grain diameter, A and n are empirical parameters, Q GG is the activation energy for grain growth, R g is the gas constant, T is the temperature, and t is the soaking duration. However, these types of models do not take the pinning effect of precipitates into account. Other studies adopt physically based approaches that account for the influence of pinning precipitates on grain growth [START_REF] Andersen | Analytical modelling of grain growth in metals and alloys in the presence of growing and dissolving precipitatesI. Normal grain growth[END_REF][START_REF] Manohar | Grain Growth Predictions in Microalloyed Steels[END_REF][START_REF] Banerjee | Nonisothermal Austenite Grain Growth Kinetics in a Microalloyed X80 Linepipe Steel[END_REF][START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF]. The austenite grain size is shown to result from the competition between a driving pressure for grain growth and a pinning pressure induced by precipitates that evolves during heat treatments. In particular, Banerjee et al. [START_REF] Banerjee | Nonisothermal Austenite Grain Growth Kinetics in a Microalloyed X80 Linepipe Steel[END_REF] and Maalekian et al. [START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF] present models coupling the evolution of precipitation with austenite grain growth. The pinning effect of different precipitate species are added up to calculate the global pinning pressure. Although Maalekian et al. [START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF] present a multi-class description of the precipitation state, the pinning pressures are calculated using the mean volume fraction f of precipitate, and the mean radius of precipitates r.

More recently, Razzak et al. [START_REF] Abdur | A Simple Model for Abnormal Grain Growth[END_REF][START_REF] Abdur | Preventing Abnormal Grain Growth of Austenite in Low Alloy Steels[END_REF] showed the importance of using the size distribution of precipitates for the calculation of pinning pressure in an abnormal grain growth model. Considering only averaged values may lead to erroneous calculated pinning pressure, especially in cases where precipitate size distribution is broad and/or bimodal.

The present work aims at coupling a Kampmann-Wagner Numerical (KWN) type precipitation model [START_REF] Wagner | Homogeneous Second-Phase Precipitation[END_REF] with a grain growth model to predict the austenite grain size obtained after several heat treatments. The evolution of a pre-existing population of (Ti,Nb)C precipitates was modeled using a multi-class description of the precipitation state.

This work differs from the aforementioned studies by considering the effect of the whole size distribution of precipitates instead of using mean parameters f and r for calculating the pinning pressure.

Thermodynamic databases were used, accounting for the alloy thermal history, to estimate the solubility product of (Ti,Nb)C precipitates.

Experimental procedure

Alloy production

The alloy used in this study was produced by vacuum induction melting. An ingot of approximate dimensions 140 × 150 × 50 mm 3 was obtained. The ingot was reheated at 1200 1. During the heat treatments, grooves were formed at the intersection of austenite grain boundaries with the flat surface [START_REF] Mullins | Theory of Thermal Grooving[END_REF][START_REF] Halliday | Determination of the austenitic grain size of steel using a thermal etching method[END_REF]. At 1250 • C, thermal etching turned out to be unsuccessful, and a Béchet-Beaujard etching [START_REF] Béchet | Nouveau réactif pour la mise en vidence micrographique du grain austrénitique des aciers tremprés ou tremprés-revenus[END_REF] was performed instead. Since a martensitic microstructure is needed for Béchet-Beaujard etching, the sample was quenched (cooling rate of 100 • C/s) with helium after the isothermal holding at 1250 • C.

Grain growth measurements

Etched samples were then observed using a Nikon Epiphot 200 optical microscope. For the thermally etched samples, Nomarski microscopy under bright field reflection illumination has been used. This technique allows highlighting microtopographic features, like the thermal grooves [START_REF] San Martín | Application of Nomarski differential interference contrast microscopy to highlight the prior austenite grain boundaries revealed by thermal etching[END_REF].

In order to avoid any inaccuracies on the temperature measurement due to the temperature gradient of the dilatometer, observations were performed on the same layer where the thermocouple was welded.

The areas of at least 50 grains were determined using FIJI [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]

D A = 4A π (2)
The mean grain size D was estimated by multiplying the number-based mean equivalent area diameter D A by 1.2, in accordance with [START_REF] Giumelli | Analysis of the Austenite Grain Size Distribution in Plain Carbon Steels[END_REF].

Precipitation characterization

The heat treatment. For all temperatures, some austenite grain boundaries were not revealed, leading to uncertainties, that were estimated to be 20%. (Figure 5 c)), like titanium nitride. These cuboids were then assumed to be TiN.

Experimental results

Austenite grain size measurements

The titanium-niobium carbides (Ti,Nb)C were the most frequently observed precipitates, and also the smaller ones. Therefore, they are likely to be the one governing austenite grain growth during soaking at high temperature. Using the analysis routine developed on FIJI, the particle size distribution was determined from a series of images taken randomly at the same magnification in both SEM (BSE mode) and TEM (STEM-HAADF mode). The resulting precipitate size distributions are shown in figure 6. Log-normal laws accurately fit experimental size distributions for both SEM and TEM. The mean precipitates radius are equal (56 ± 5 nm for BSE and 58 ± 5 nm for STEM-HAADF). However the size distribution determined by SEM is slightly larger than the TEM one, which could be explained by the lower precision of SEM observations.

Heat-treated samples

In order to investigate the stability of the different types of observed precipitates, three isothermally heat treated states were investigated:

Model

Equilibrium calculations

Modelling the evolution of precipitation state with time and temperature requires having a correct estimation of the stability domain of phases.

In particular, the solubility product of the precipitates is an essential parameter. In the present case, the key population is (Ti,Nb)C, whose sizes makes it effective for grain boundary pinning. Two types of calculations were performed and compared to estimate the stability of the observed precipitates.

First calculations were performed in austenite domain using the commercial Thermo-Calc software [START_REF] Andersson | Thermo-Calc & DICTRA, computational tools for materials science[END_REF] with TCFE8 Steels/Fe-alloys database [START_REF]Thermo-calc software TCFE8 steels/fe-alloys database[END_REF]. The steel composition determined by ICP-MS (Table 1) was used as input.

According to TCFE8 database, manganese sulfide MnS is found to be more stable than titanium carbo-sulphide Ti 4 C 2 S 2 . Since no MnS were experimentally observed, equilibrium calculations were performed ruling out MnS and all related phases (dormant phases), in order to better describe the actual microstructure. Figure 9 (Ti,Nb)C gives approximately 57 % of titanium and 43 % of niobium. This composition is quite different from that obtained experimentally by EDX on this population of precipitates (see Figures 7 and8).

The apparent disagreement between fullequilibrium thermodynamics calculations and experimental results can be explained by the fact that full equilibrium is not reached. The alloy's thermal history starting from high temperatures has therefore to be accounted for.

As a consequence, the stability of the precipitates was reconsidered using several assumptions. TiN are known to be amongst the most stable precipitates [START_REF] Gladman | The physical metallurgy of microalloyed steels[END_REF]. They form at the earliest stages of the casting process [START_REF] Schneider | Formation of primary TiN precipitates during solidification of microalloyed steels Scheil versus DICTRA simulations[END_REF], even from liquid state. Therefore, here it was assumed that the total amount of N reacted with the corresponding amount of Ti to form a stoichiometric TiN population. All N as well as the corresponding amount of Ti was removed from the input composition.

Since Ti 4 C 2 S 2 are also known to be very stable in austenite [START_REF] Yang | Solubility Products of Titanium Sulphide and Carbosul-phide in Ultra-low Carbon Steels[END_REF], it was assumed that they undergo limited dissolution over the temperature range of interest(i.e. between 950 • C and 1250 • C). As a first approximation, all S and the corresponding Ti and C were also removed from input composition.

The new input composition obtained is listed in Table 2.

TCFE8 database [START_REF]Thermo-calc software TCFE8 steels/fe-alloys database[END_REF] gave a relatively constant composition of (Ti,Nb)C over the studied temperature range: (Ti 0.69±0.05 ,Nb 0.31±0.05 )C 0.96±0.02 . Thus, a simplified average composition between A e3 and the dissolution temperature was taken. It was also assumed that the number of metallic (Ti and Nb) atoms was equal to the number of carbon atoms (no vacancy in precipitates), giving the following precipitate composition: Ti 0.7 Nb 0.3 C.

Using the evolution of austenite composition with temperature, the solubility product was evaluated at each temperature using:

K S = (X eq Ti ) 0.7 (X eq Nb ) 0.3 X eq C (4)
with X eq i the atomic fraction of element i in austenite. Then by linearly fitting the evolution of log 10 K S with 1/T , the following expression was taken for the solubility product of (Ti,Nb)C: This precipitate is found more stable than NbC and TiC, probably due to entropy effects.

log 10 K S = - 9626 K T + 0.44 (5)

Grain growth model

A Zener-type model accounting for the competition between driving pressure for grain growth P D and precipitate pinning pressure P P was used. Under the influence of pinning precipitates, the grain diameter D growth rate can be expressed by:

dD dt =    M (P D -P P ) if P D -P P > 0 0 if P D -P P < 0 (6)
M is the interface boundary mobility, described by an Arrhenius law:

M = M 0 exp -Q GG R g T (7)
Here M 0 is a pre-exponential factor, R g is the gas constant and Q GG is the activation energy for grain boundary mobility.

The driving pressure for grain growth is given by:

P D = β γ D ( 8
)
Where β is a coefficient, γ is the austenite interface energy and D is the mean grain diameter.

Zener [START_REF] Smith | Introduction to Grains, Phases, and Interfacesan Interpretation of Microstructure[END_REF], originally considered that β = 4. This value is actually overestimated as several experimental and modelling work seem to prove that the correct expression of the driving pressure for grain growth in polycristalline materials is four times lower [START_REF] Patterson | Relationship between grain[END_REF][START_REF] Rios | On the relationship between pinning force and limiting grain radius[END_REF][START_REF] Rios | Grain boundary pinning by Al6mn precipitates in an Al1wt%Mn alloy[END_REF][START_REF] Rios | Grain Boundary Pinning by Particles[END_REF]. Thus, β is taken equal to 1 in the present work.

Numerous expressions for pinning pressure can be found in literature (see the reviews by Manohar et al. [START_REF] Manohar | Five Decades of the Zener Equation[END_REF], and by Huang and Logé [START_REF] Huang | Zener Pinning[END_REF]). The pinning pressure P P is generally calculated with the following expression:

P P = αγ f r (9)
Where α is a coefficient, f is the mean volume fraction of precipitates, and r is the mean precipitate radius.

Among these expressions, the one given by Zener in 1948 [START_REF] Smith | Introduction to Grains, Phases, and Interfacesan Interpretation of Microstructure[END_REF], and the one given by Rios in 1987 [START_REF] Rios | Overview no. 62: A theory for grain boundary pinning by particles[END_REF] are particularly relevant, and somehow define a lower and an upper bound for the pinning pressure. Zener [START_REF] Smith | Introduction to Grains, Phases, and Interfacesan Interpretation of Microstructure[END_REF] considered that grain boundaries bypass precipitates following a pass-through mechaand obtained α = 3/2. Considering that grain boundaries bend round and envelope pinning precipitate (enveloping mechanism), Rios [START_REF] Rios | Overview no. 62: A theory for grain boundary pinning by particles[END_REF] derived a pinning pressure two times higher than Zener's:

α = 3.
Additionally, the present model took into account the effect of several classes of precipitates with different sizes instead of adopting a mean-radius approach. The expression for pinning pressure was thus modified as:

P P = αγ i f i r i ( 10
)
where i is a summation index representing all the present precipitates classes of different radius r i and having the volume fraction f i .

Precipitation model

In this work, the software Preciso [START_REF] Perez | Implementation of classical nucleation and growth theories for precipitation[END_REF][START_REF] Perez | Corrigendum to Implementation of classical nucleation and growth theories for precipitation[END_REF] was used to follow the evolution of precipitation state during heat treatments. It is based on the classical nucleation, growth and coarsening theories, fully described in [START_REF] Wagner | Homogeneous Second-Phase Precipitation[END_REF]. The precipitation state was characterized by the whole precipitate size distribution (Lagrange-like approach). Since only reversion experiments are to be modeled, only growth and coarsening equations are presented here (no nucleation).

Growth and coarsening

The diffusion of solute elements governs the growth of nucleated precipitates. The diffusion coefficient of an element i, noted D i , follows an Ar-

rhenius law D 0 i exp(-Q i /(R g T ))
. D 0 i is the preexponential factor, and Q i is the energy activation.

The growth rate of a precipitate is given for element i by:

dr dt = D i r X m i -X eq i (r) α v X P i -X eq i (r) (11)
α v = v m at /v p at
is the ratio of matrix to precipitates atomic volumes, and X P i (r) is the atomic fraction of element i at the interface between matrix and a precipitate of radius r.

In fact, the presence of an interface between matrix and precipitates (i.e. the interface curvature) modifies the local equilibrium and atomic fractions at the interface (Gibbs-Thomson effect [START_REF] Perez | Gibbs-Thomson effects in phase transformations[END_REF]). Thus, the concentrations of solute elements at the interface of a Ti x Nb y C z precipitate are given by the solubility product modified by a factor traducing the effect of interface curvature:

X eq Ti (r) x X eq Nb (r) y X eq C (r) z = K S exp 2γ P (x + y + z)v P at rk B T (12)
Eqs. 11 and 12 provide a system of 4 equations ( 3diffusion equations -Ti, Nb and C -plus solubility product) with 4 unknown (dr/dt and X eq i ). At each timestep, growth of all existing classes is performed using eqs. 11 and 12. Consequently, precipitate coarsening (Ostwald ripening) is implicitly accounted for in this approach.

Mass balance

Finally, mass balance is performed to update the new solute content of all elements:

X m i = X 0 i -α v f X P i 1 -α v f (13)
where X 0 i is the total amount of solute i.

Coupling between precipitation and grain growth models

Precipitation and grain growth models are coupled but run separately. The precipitation model was first run providing the precipitate size distributions at each time step in the form of a large output file. Then, in a second stage, this file is read and, at each time step, the pinning pressure is calculated and the grain growth equation (eq.6) is integrated. 2.

The precipitation model used the solubility product given in equation 5. The diffusion coefficient of alloying elements were calculated from the MOBFE3 Steels/Fe-Alloys Mobility Database [START_REF]Thermo-calc software MOBFE3 steels/fe-alloys mobil-ity database[END_REF] (see Table 4). Precipitate/matrix interface energy was taken equal to 0.7 J•m -2 , in agreement with Maalekian et al [START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF].

The initial distribution of (Ti,Nb)C precipitates experimentally determined by TEM (Figure 6 b))

served as an input for the precipitation model. The STEM-HAADF distribution was used because of the better precision and resolution of TEM measurements compared to SEM. Based on a lognormal fitting curve, a theoretical precipitate distribution of 100 classes was calculated. The following expression was used for the probability density function:

f (r, µ, σ) = 1 rσ √ 2π exp - (ln(r/nm) -µ) 2 2σ 2 (14)
with µ = 3.991 and σ = 0.369. The total number of precipitates was adjusted in order to give an initial volume fraction of (Ti,Nb)C of 0.0014 , which is (Ti,Nb)C/matrix interfacial energy 0.7 J•m -2 Adjusted to fit (Ti,Nb)C growth. Based on this study, a value of 390 kJ/mol was used.

Element D 0 [m 2 •s -1 ] Q [J•mol -1 ] C 2.4 ×
The mobility factor M 0 is the only fitting parameter: it was taken equal to 10 4 m 4 J -1 s -1 .

Modelling Results

Precipitation

Several reversion heat treatments were performed in the stability domain of (Ti,Nb)C in austenite.

Austenite grain growth

The pinning effect of (Ti,Nb)C particles is largely and the pinning pressure on grain boundaries is positive, leading thus to grain growth (according to equation 6) as illustrated in figure 14 c)). As the austenite grain diameter increases, the driving pres-sure decreases, until both pressures become equal.

Discussion

Almost all modelling parameters were extracted from literature and/or thermodynamic databases.

Two parameters were unknown, however: the initial volume fraction of (Ti,Nb)C and the precipitate/matrix interface energy. It was chosen to set the initial volume fraction at 0.0014 , which is close to the maximum volume fraction. Since the initial state was obtained through hot rolling and air cooling, there is a high probability that the precipitate volume fraction is maximum. The precipitate/matrix interface energy was set to 0.7 J•m -2 by fitting the experimental data on precipitate growth.

This value is close to the one calculated by [START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF] for NbC from a nearest neighbour broken bond model (0.66 J•m -2 ). This parameter has only little influence on reversion kinetics, since nucleation does not take place during reversion. 7LPHV 3LQQLQJUDWLRP 1 Note that the whole precipitate size distribution was used to evaluate the pinning pressure, whereas many other authors used averaged volume fraction and radii [START_REF] Andersen | Analytical modelling of grain growth in metals and alloys in the presence of growing and dissolving precipitatesI. Normal grain growth[END_REF][START_REF] Banerjee | Nonisothermal Austenite Grain Growth Kinetics in a Microalloyed X80 Linepipe Steel[END_REF][START_REF] Maalekian | In situ measurement and modelling of austenite grain growth in a Ti/Nb microalloyed steel[END_REF][START_REF] Karmakar | Effect of microalloying elements on austenite grain growth in NbTi and NbV steels[END_REF]. It has been shown (Figure 14 a)) that using averaged parameters leads to an overestimation of the pinning pressure. Using those calculations to estimate the solubility of each precipitate leads to a misrepresentation of the precipitation state for a given temperature.

D fr i f i /r i 7LPHV 3UHVVXUHV03D E P D = /D P P = 3 i f i /r i 7LPHV *UDLQGLDPHWHU>P@ F P P = 3 fr P P = 3 i f i /r i
Thus, the CALPHAD approach provides a powerful framework for estimating solubility products, but needs to be used with caution.

Conclusions

In the present work, the evolution of the precipi- • Extensive SEM and TEM characterization leaded to similar initial precipitate size distri-butions, which served as initial distributions in the precipitation model.

• The chemistry and stability of (Ti,Nb)C precipitates was calculated from the TCFE8 thermodynamical database.

• Both precipitation and grain growth models did not contain any adjustable parameter, except the mobility factor M 0 . All other model parameters were extracted either from thermodynamical databases or from literature.

• Accounting for the whole precipitation size distribution, rather than using average values, significantly influenced the calculated pinning pressure.

• The modeled austenite grain diameters were consistent with the experiments for all investigated heat treatments.

Austenite grain boundary

Pinning forces

Distribution of precipitates

A

  Bähr 805 DIL dilatometer was used to perform heat treatments consisting of a 5 • C/s ramp up to the austenitisation temperature T A (950 • C< T A <1250 • C), followed by a 10 min holding at T A and cooling. The temperature was controlled by welding a type K thermocouple in the middle of the samples. Cylindrical samples of 10 mm length and 4 mm diameter were machined from the steel plate and used to determine austenite grain sizes.For all temperatures except 1250 • C, the austenite grain size was determined by the thermal etching method[START_REF] Garca De Andres | Revealing austenite grain boundaries by thermal etching: advantages and disadvantages[END_REF]. After austenitisation, the heating was switched off, and the samples cooled down to room temperature with an average cooling rate of 10 • C/s. A 2 mm wide flat surface was prepared along the sample length by mechanical grinding and polishing, finishing with 1 µm diamond solution.

Finally, the crystallographic

  nature of the precipitates was investigated using selected area electron diffraction (SAED) on a JEOL 2010F TEM microscope operating at 200 kV. The composition of each precipitate analysed by SAED was checked by EDX measurements. The crystallography data used for indexing the SAED patterns are extracted from [29]. Ti 4 C 2 S 2 precipitates present an hexagonal structure of space group P6 3 /mmc with a = b = 0.3210 nm and c = 1.120 nm. TiN, TiC and NbC all share the same face-centered cubic (fcc) structure of space group Fm 3m with respectively a = 0.4235 nm, a = 0.4319 nm, and a = 0.4469 nm.

Figure 1

 1 Figure 1 displays optical micrographs from selected heat treatments showing prior austenite grain sizes (PAGS) revealed by the thermal etching method. Most of these micrographs allow an relatively accurate tracking of the austenite grain boundaries, which gives reliable PAGS. For the lower temperatures (950 • C and 1000 • C), thermal etching was less effective and PAGS determination

Figure 3 3 . 2 .Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 3321234 Figure 3 shows the resulting average PAGS measurements after 10 minutes isothermal heat treatments at temperature ranging from 950 • C to 1250 • C. As expected, raising temperature results in increasing PAGS. Up to 1050 • C, PAGS remains roughly constant, with values between 10 and 20 µm. Starting from 1100 • C, a substantial austenite grain growth occurs. At 1250 • C, coarse grains were obtained, with a final grain diameter around 400 µm.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 5678 Figure 5: Bright field TEM observation of the 3 populations of precipitates found in as-received steel, with associated Selected area diffraction patterns.

• 4 & 2 6 2 7L1E&Figure 9 :

 429 Figure 9: Equilibrium volume fractions of precipitates a) resulting from Thermo-Calc calculations with TCFE8 database for all precipitates and b) assuming that all N and S react to form respectively TiN and Ti 4 C 2 S 2 and using Thermo-Calc for calculating the solubility product of (Ti,Nb)C.

Figure 10 compares

 10 Figure 10 compares the obtained solubility product for Ti 0.7 Nb 0.3 C with values of the literature.

Figure 10 :

 10 Figure 10: Comparison of the solubility product used (equation 4) and reference solubility products of TiC and NbC from Gladman [5].

Figure 9 b

 9 Figure 9 b) shows the stability domain of (Ti,Nb)C using the aforementioned solubility product, considering complete precipitation of TiN and Ti 4 C 2 S 2 over the whole range of temperature. The

4. 5 .

 5 Model parameters The evolution of precipitation was modelled for several holding treatments and coupled with the grain growth model, with the aim of reproducing the results obtained in section 3.1. Only (Ti,Nb)C precipitates were considered in the precipitation model. As discussed in section 3.2, given their respective sizes, TiN and Ti 4 C 2 S 2 populations have a very limited pinning effect. The chemical composition of the matrix used in the precipitation model is given in Table

  Nb)C solubility product log 10 KS = -9626 K/T + 0.44 See equation 5. (Ti,Nb)C initial volume fraction 0.0014 Close to maximum. (Ti,Nb)C initial distribution µ = 3.991 and σ = 0.369 STEM-HAADF (Figure 6 a)).

Table 4 :

 4 Diffusion coefficient used in the precipitation model, calculated from the MOBFE3 mobility database of Thermo-Calc. close to the maximum precipitate volume fraction given by mass balance. An initial austenite grain diameter of 10 µm was assumed, based on the experimental results given in part 3.1. An austenite grain boundary energy of 0.5 J/m 2 was assumed. Concerning grain boundary mobility, Uhm et al [45] determined an expression of the activation energy by multiple regression analysis for steels containing C, Mn, Ni, Cr, Mo and Si.

Figure 11 showsFigure 11 :Figure 12 :

 111112 Figure 11 shows the main output results of the precipitation model. The volume fraction of (Ti,Nb)C rapidly decreases as a result of the dissolution process. Meanwhile, a slight diminution of the mean precipitate radius and a consequent decrease of the number density (i.e. the number of precipitates per cubic meter) are observed. After precipitate shrinkage, coarsening occurs, causing a decrease in precipitate number density, and an increase in mean radius, while volume fraction remains constant. Eventually, the volume fraction reaches the equilibrium value given by the solubility product at 1200 • C. Microalloying elements are released in solid solution in the austenite phase due to the dissolution of precipitates. The observations made on number density and mean radius of the (Ti,Nb)C population can also be performed looking at the evolution

Figure 13

 13 Figure13shows the simulated evolution of the

Figure 13 :

 13 Figure 13: Evolution of the a) volume fraction and b) mean precipitate radius of (Ti,Nb)C for several isothermal heat treatments.

Figure 14 b

 14 Figure 14 b) shows the time evolution of driving and pinning pressures. Pinning pressure exerted by (Ti,Nb)C particles decreases during holding, as the precipitate number density decreases. The difference between the driving pressure for grain growth

Figure 14 :Figure 15 :

 1415 Figure 14: Evolution of a) pinning parameter, b) driving and pinning pressure induced by (Ti,Nb)C and c) austenite grain diameter during a thermal treatment at 1200 • C, considering the expression of pinning pressure given by Rios [41].

Figure 15

 15 Figure 15 compares experimental (see section 3.1) and predicted grain sizes for 10 minutes heat treatments at temperatures ranging from 950 • C to 1250 • C. The experimental data globally lies between the values predicted using Zener (α = 3/2) and Rios (α = 3) approaches (see section 4.2).

Figure 14 cFigure 16 :

 1416 Figure 16: Comparison of the pinning pressure (Rios's expression) induced by three different size distributions of precipitates. The three size distributions present a mean radius of 50 nm and a mean volume fraction of 0.001 •

••

  tation state and austenite grain growth were studied in a Ti-Nb microalloyed steel during reversion treatments, i.e. starting from a fully precipitated state and performing thermal treatments between 950 • C and 1250 • C. Based on acquired experimental data, a multiclass precipitation model was developed and used to predict the evolution of (Ti,Nb)C size distribution during thermal treatments. • The resulting precipitate size distributions served as entry parameters to a simple grain growth model based on Zener pinning. The chemistry of the modeled alloy was modified to account for the amount of solute atoms actually available for precipitation, i.e. subtracting chemical elements already involved in existing and stable precipitates (TiN and Ti 4 C 2 S 2 ).

Table 2 :

 2 Calculation of the input composition (at%) for Thermo-Calc calculations of (Ti,Nb)C stability assuming that all N and S react with Ti to form TiN and Ti 4 C 2 S 2 , respectively.

	10

Table 3 :

 3 Modelling parameters used.

Acknowledgements

The authors gratefully acknowledge the funding received from the European Commission, Research Fund for Coal and Steel, under grant agreement 709803 (NANOFORM). Thanks are due to the CLYM (www.clym.fr) for access to the 2010F microscope, and to Nicholas Blanchard for access and help on the JEOL 2100 microscope.

Declaration of interest

Declarations of interest: none.