Gruy Frédéric
email: gruy@emse.fr

RELATIONSHIP BETWEEN THE MORPHOLOGY AND THE LIGHT SCATTERING CROSS SECTION OF OPTICALLY SOFT AGGREGATES

Keywords: aggregate, light scattering, pair distribution density, Rayleigh-Debye-Gans approximation

The purpose of this paper is to establish the suitable analytical expressions for the light scattering cross section of aggregates constituted by optically soft primary particles. The used optical approximation is the well-known Rayleigh-Debye-Gans approximation. The geometrical description of the aggregate is based on the pair, or distance, distribution density function describing at once the two spatial scales (primary particle and aggregate). Firstly, a study of the two-particle set is presented. An exact and an approximated formulations are proposed for the distance distribution density and the scattering cross section. Then the corresponding results are extended to the aggregates.

Introduction

Micro-particles mostly appear as aggregates constituted by nanometric or micrometric primary particles. Their geometrical and physical properties are the subject of numerous studies. An extensive literature exists for a particular class of aggregates known as the fractal-like aggregates. Filippov et al. studied the relationship between the morphology and the thermal behaviour of fractal aggregates [START_REF] Filippov | Fractal-like aggregates: Relation between morphology and physical properties[END_REF]. The determination of their optical properties is particularly important also. The optical properties are needed not only for the characterization of micro-particles in a large number of industrial processes, but also for the study of environmental problems. Sorensen [START_REF] Sorensen | Light scattering by fractal aggregates: a Review[END_REF] has deeply studied the scattering properties of fractal aggregates. During the monitoring of particle aggregation in an industrial process, the optical properties of the suspension are measured by means of an experimental set-up consisting of a mono-or polychromatic light source, a sampling cell and a photodetector. The incident light is scattered by the sample, i.e. the suspension. Then, one can record either the light intensity ( ) I ϑ at various scattering angles ϑ or the transmitted intensity, i.e. the non-scattered and non-absorbed light. The transmitted light is easier to work up. However the transmitted light intensity only leads to the light extinction cross section, that is ( )sin

I d ϑ ϑ ϑ ∫
. So, the light extinction cross section is obtained through an attenuation measurement by means of the Beer-Lambert law for diluted suspension. The measurement is performed at various wavelengths. This method has been used for the study of the aggregation of silica micro-particles [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF] and polystyrene beads [START_REF] Xu | Optical factors determined by the T-matrix method in turbidity measurement of absolute coagulation rate constants[END_REF]. Simulations show that the aggregation of mono-sized primary particles results in a fractal-like cluster characterized by its fractal dimension. This morphological property is clearly proved for aggregates with a large number of primary particles (N>10 3 ). All geometrical properties of a fractal cluster can be deduced from its radius, its number of primary particles and its fractal dimension. Nevertheless, in real case the aggregates contain a small number of primary particles (N<<10 3 ) due to hydrodynamic and physicochemical conditions occurring in chemical reactors, e.g. high shear flow. Such aggregates are not as fractal-like ones. Kahnert [START_REF] Kahnert | Numerical methods in electromagnetic scattering theory[END_REF] reviewed the exact theories and numerical methods about the calculation of the optical properties of non spherical particles and aggregates. One can classify them into three main categories:

-Methods based on the partial differential equations that calculate the scattering field by solving the Maxwell equations or the Helmholtz equation. They are subjected to the boundary conditions suitable in the time or the frequency domain.

-Methods based on integration over volume or surface of equations derived from the Maxwell equations. Thus the boundary conditions are automatically included in the solution.

-The other methods are known as hybrids since they derive from the various approaches. Among the first category, T-matrix method is widely used. In the T-matrix method, the incident and scattering fields are expressed in the form of a series of spherical vector wave functions. The expansion coefficients of the incident wave and the scattering wave are connected by a linear transformation. The corresponding matrix T contains all the information on the particle's optical properties for a given wavelength. It is a function of the size parameter, the shape, the optical refractive index of the considered particles, but it does not depend on the incident field. Thus this matrix does not need to be computed at each particle orientation change or change of the incident wave direction.

To classify the publications relying on this method, a database was carried out by Mishchenko et al. [START_REF] Mishchenko | T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database[END_REF][START_REF] Mishchenko | Comprehensive T-matrix reference database: A 2004-06 update[END_REF][START_REF] Mishchenko | Comprehensive T-matrix reference database: A 2006-07 update[END_REF][START_REF] Mishchenko | Comprehensive T-matrix reference database: A 2007-09 update[END_REF]. The solving by separation of variables for only one sphere can be enlarged to an aggregate of spheres by using the translation theorem for the spherical wave vector functions which expresses them in various bases of coordinates, and by applying a superposition principle. The total scattering field for an aggregate is then represented by the superposition of all the individual scattering fields resulting from each particle knowing that these fields are interdependent. Moreover, the problem can be formulated in the way of a T-matrix. A particular case of the T-matrix method [START_REF] Mishchenko | T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database[END_REF] bearing the name of GMM (Generalized Multi-particle Mie-solution) is particularly suitable for sphere aggregates. The details of GMM are given by Xu [START_REF] Xu | Electromagnetic scattering by an aggregate of spheres[END_REF][START_REF] Xu | Electromagnetic scattering by an aggregate of spheres: far field[END_REF]. This method is very precise but its computation time depends on the number and the size parameter of primary particles. However the study of the aggregation process, and the prediction of the time course of the turbidity spectrum require a too large computational time. Thus simple and accurate expressions for the optical properties of aggregates are needed. Simplified methods were developed for predicting the scattering and absorption cross sections of aggregates for several practical situations [START_REF] Mackowski | A simplified model to predict the effects of aggregation on the absorption properties of soot particles[END_REF][START_REF] Jacquier | Approximation of the light scattering cross-section for aggregated spherical non-absorbent particles[END_REF][START_REF] Jacquier | Anomalous Diffraction Approximation for light scattering cross section: case of ordered clusters of non-absorbent spheres[END_REF][START_REF] Jacquier | Anomalous Diffraction Approximation for light scattering cross section: case of random clusters of non-absorbent spheres[END_REF].

As above-mentioned, fractal-like aggregates are not so frequent during the aggregation process. As a consequence other geometrical parameters have to be chosen for quantifying the aggregate morphology. Among the geometrical parameters describing any body, the distribution density functions relative to some quantities are involved in the calculation of scattering parameters. Among these quantities, one can find:

-The chord length, e.g. the distance between two points located on the surface of a convex body [START_REF] Gille | The small-angle scattering correlation function of the hemisphere[END_REF] -The distance between two points located inside the body [START_REF] Gruy | Light scattering cross section as a function of pair distribution density[END_REF] -The pair (distance), i.e. the distance between the centres of any two primary particles in a cluster [2] -The projected area [START_REF] Jacquier | Méthodes approchées pour les propriétés optiques d'agrégats de particules sphériques non absorbantes[END_REF]. These functions can often be analytically calculated for convex bodies. However, the calculation is less easy for non convex bodies [START_REF] Mazzolo | Properties of chord length distributions of nonconvex bodies[END_REF][START_REF] Gruy | The chord length distribution of a two-sphere aggregate[END_REF], as clusters of primary particles. The choice of the relevant geometrical parameter depends on the size and the relative refractive index m; m is defined by the ratio between the refractive indices of the particle and the surrounding medium. Some scatterers dispersed in a liquid are characterized by a low optical contrast, i.e. . For instance, minerals like amorphous silica, silicates, carbonates, carboxylates, ice and gas hydrate crystals in water are optically soft. Likewise, the biological cells or microorganisms in water behave so. A number of important analytical approximations have been obtained for such particles. Rayleigh-Debye-Gans (RDG) approximation considers small soft particles with small phase shift, namely [START_REF] Van De Hulst | Light scattering by small particles[END_REF][START_REF] Kokhanovsky | Optics of light scattering media: Problems and solutions[END_REF] :

2 / 1 d m π λ << - 1
where d and λ are respectively the particle size and the light wavelength. The use of RDG approximation for fractal clusters led to numerous works, because it is suitable for explaining small angle scattering experiments with X ray or light. So the fractal dimension of aggregates can be deduced from the scattering data [START_REF] Hermawan | Characterization of short-range structure of silica aggregates-implication to sedimentation compaction[END_REF], because there exists a mathematically tractable relation between the scattered light intensity, the scattering wave vector and the fractal dimension. The ground is a particularly simple expression for the pair distribution density function for the fractal clusters. This analysis has been criticized by some authors, because it was used with non optically soft materials [START_REF] Riefler | Structural analysis of clusters using configurational and orientational averaging in light scattering analysis[END_REF]. Furthermore a more realistic pair distribution density function of fractal clusters needs the insertion of a cut-off function [START_REF] Sorensen | Light scattering by fractal aggregates: a Review[END_REF]. The aim of this paper is to calculate the light scattering cross section of an optically soft aggregate from a distance distribution density. This will take into account at once the aggregate structure and the primary particle shape. Quasi-analytical expressions will be deduced and will be available for fast calculations. The second section of this paper reminds the fundamentals about the RDG approximation and its relationship with the pair distribution density function. Section three presents some results concerning the distance distribution density function for a two-particle set with equal or different particle sizes. This section is completed by the study of the distance distribution density function for an aggregate. The section four is devoted to the exact calculation and an approximated one of the light scattering cross section for an aggregate with identical primary particles.

Rayleigh-Debye-Gans approximation and aggregates

Generally, the object (primary particle, aggregate ...) can be divided into smaller identical parts (elements). Each element is polarisable. In the presence of a variable electric field, the element becomes an oscillating dipole, which itself creates an electromagnetic field. When an object is illuminated by an electromagnetic wave, each element receives the incident electric field and the one coming from the other elements. As a result an oscillating dipole moment may be associated to each element. Thus the object emits an electromagnetic wave (scattered wave), which includes the contribution of each oscillating dipole.

In the Rayleigh-Debye-Gans domain, there is only interference of light waves that are independently scattered by all small volume elements. If the object is an aggregate, the element may be either the whole primary particle or a part of the primary particle.

Most often the incident wave is randomly polarised and the object (scatterer) can randomly orientate. The optical properties are thus obtained after calculating an average over all the wave polarisation states and object orientations.

The averaged light scattering cross section for aggregates consisting of N primary particles with radius a obeys the following relation:

N C ( ) ( ) 2 1 (4 ) N C N F S Q d π ϑ = Ω ∫ with ( ) 1 1 (4 ) C F d π ϑ = Ω ∫ (1)
where , ,Q ϑ Ω are respectively the scattering angle, the solid angle and the magnitude of the scattering wave vector ( ( )

2 sin / 2 Q k ϑ = ; 2 / k π λ = ). ( ) 1
F ϑ corresponds to the primary particle. The structure factor ( ) S Q depends on the aggregate morphology [START_REF] Khlebstov | Spectroturbidimetry of fractal clusters: test of density correlation function cut-off[END_REF]:

( ) ( ) ( ) ( ) ( ) 2 1, ( ) sin / / 1 1 sin / / N ij ij ij ij ij i j i j S Q N QR QR N N QR QR = = ≠ = + = + - ⎡ ⎤ N ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∑ (2)
where ij R is the distance between the centres of the primary particles i and j. ij represents the averaged value over all i,j pairs. The scattering cross section of a doublet of primary particles separated by the centre to centre distance 2a+h has been calculated for several materials and sizes of primary particle. The primary particles were considered as Rayleigh [START_REF] Lips | Light scattering by two spherical Rayleigh particles over all orientations[END_REF] or Mie [START_REF] Kattawar | Electromagnetic scattering for two dielectric spheres: comparison between theory and experiment[END_REF][START_REF] Borghese | Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers[END_REF][START_REF] Mishchenko | Scattering of light by bispheres with touching and separated components[END_REF] scatterers. For Rayleigh-Debye-Gans scatterers it is expressed as [START_REF] Van De Hulst | Light scattering by small particles[END_REF][START_REF] Khlebstov | Spectroturbidimetry of fractal clusters: test of density correlation function cut-off[END_REF]:

( ) ( ) ( )( ) ( ) ( ) ( ) 4 2 2 2 0 2 2 sin 2 8 1 1 c o s 1 s i n 9 2 a h ka m G Qa d a h Q C a Q π ϑ ϑ π ϑ + - + + + ⎛ ⎞ ⎜ ⎝ ⎠ ∫ ⎟ (3)
with ( ) ( )

3 3 sin cos G u u u u u = -
In a continuous formulation, the structure factor can be written as:

( ) ( ) ( ) max min ( ) sin / R P R S Q QR QR D R dR = ∫ (4)
The distribution density function ( ) P D R for the distance R between primary particle pair so appears. This function is chosen as a normalized one:

( )

max min 1 R P R D R dR = ∫ (5)
where R min and R max are respectively the smallest and the largest distances between the primary particles. Then the scattering cross section can be written as [START_REF] Berry | Optics of fractal clusters such as smoke[END_REF][START_REF] Gruy | Relation entre morphologie et signature optique pour les particules nano et micrométriques[END_REF]:

( ) ( ) ( ) 2 1 1 1 RDG RDG N i j ij N C NC N F kR N C F kR →∞ = + - ⎡ ⎤ ⎣ ⎦ 1 (6)
The mean value in Eq.6 is expressed within a continuous formulation as:

( ) ( ) max min R P R X X R D R dR = ∫ (7) with ( ) ( ) ( ) ( ) ( ) 1 0 0 , / , sin 2 / 2 / , RDG 1 F ka R a V x ka kRx kRx dx V x ka dx = ∫ ∫ (8a) and ( ) ( ) ( ) ( ) ( ) ( ) (
)

2 2 2 3 sin 2 cos 2 , 1 1 2 2 2 kax kax V x ka x x kax kax = + - - ⎡ ⎤ ⎢ ⎢ ⎥ ⎣ ⎦ 2 ⎥ (8b)
When , Eq.8a becomes [START_REF] Gruy | Relation entre morphologie et signature optique pour les particules nano et micrométriques[END_REF]:

0 ka → ( ) ( )( ) ( )( ) ( 2 4 1 3 2 4 0 3 cos 2 1 5 3 sin 2 2 6 1 3 / 4 RDG ) 2 F x x x x x x x x x - - - - - - = -+ - + - + + + ⎡ ⎤ ⎣ ⎦ x ) (9)
( , / RDG F ka R a only depends on kR.

Eq.6 can be rewritten by using only continuous variables:

( ) ( ) 2 4 2 0 2 1 3 RDG N F kR C k V m π - (10)
V is the matter volume of the scattering body. Thus the scattering cross section is expressed as a function of the distribution density of the distance R between any two points belonging to the scattering body. The latter will be merely called pair distribution density. Eq.10 is easily verified for a sphere when compared to the expression given by Van de Hulst [START_REF] Van De Hulst | Light scattering by small particles[END_REF]. It is valid for any scattering body.

Pair or distance distribution density function

Doublet of identical primary particles

Let us examine the case of a doublet of two identical spheres with radius b. Figure 1 represents the normalized pair or distance distribution density obtained by simulation. The distance H is the distance between the centres of the two primary particles. R is the distance between two points inside the doublet. All distances are made dimensionless with regard to the radius of the primary particle. The pair distribution density function for a doublet contains two contributions:

( ) ( ) ( ) ( 1 2
, ( )

s s Dp R H Dp R Dp R H = + ) , / 2 (11)
,
Dp R H is also normalized.

The pair distribution density for a sphere [START_REF] Gruy | Light scattering cross section as a function of pair distribution density[END_REF] obeys:

( ) ( 2 3 1 3 12 16 16 s Dp R R R R = - ) + (12a) for 0 2. R < <
The distribution jth-order moments are:

1 , 1 3 2 12 2 6 4 3 j s j M j j j = ⎡ ⎤ × - + ⎢ + + + ⎣ ⎦ ⎥ 2 (12b)
The partial pair distribution density is such as:

( )

2 1 2 1 , ' T s T Dp R H A S S dR = ∫ (13a)
, where A is the normalization constant.

The notations are illustrated on Figure 2. A simple calculation leads to:

if R H < ( ) ( )( )( )( ) 1 2 2 1 , 1 ' 1 ' 1 ' 1 ' R s H Dp R H A R ' R R R R H R H R d H π + - - = -+ + - -+ + - ∫ R ( ) ( ) ( ) ( ) ( 3 2 2 , 3 2 6 160 s Dp R H R R H R H R H H - = -+ - - - + ) 4 (13b) if R H > ( ) ( )( )( )( ) 1 2 2 1 , 1 ' 1 ' 1 ' 1 ' H s R Dp R H A R ' R R R R H R H R d H π + + - = -+ + - -+ + - ∫ R ( ) ( ) ( ) ( ) ( 3 2 2 , 3 2 6 160 s Dp R H R H R R H R H H + = -+ - + - + ) 4 (13c)
The j-moments of the partial pair distribution (doublet)

( ) 2 , s Dp R H are: ( ) ( ) ( ) ( ) 2 , 2 , 2 , 2 , 2 , , 2 , 2 ,
s j s j s j s j s j

, M F H H F H H F H H F H H - - + + = - - + + - (14)
with

( ) ( ) ( ) ( ) ( ) 6
q q j j q q s j j q R H R H R H F R H C H H q q + + + + - ± + = 4 2 q q + ⎡ ⎤ - - - = ± - ⎢ ⎥ + + + ⎢ ⎥ ⎣ ⎦ ∑ ∓ ∓ ∓ ∓ ∓ (15)
It can be deduced:

1 1 2 , , 0 j m s j m j m M B H + - = = ∑ (16a) with , 0 1 m j B if j m is odd = + - (16b) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 8 , 11 0 3 1 2 6 55 54 1 160 j m q q m j m m j j j q q B C C q q q if j m is even + - - - - + - + + - = = - + - + + + + - ∑ (

Doublet of different primary particles

Let us consider an asymmetric doublet consisting of spheres with radii b 1 and b 2 (with b 1 >b 2 ). The distances are not made dimensionless because the choice of a characteristic length is not imperative. The pair distribution density for an asymmetric doublet separated by the centre to centre distance H obeys the relation:

( ) ( ) ( ) ( ) 11
The weighted coefficients are simply: densities for a single sphere (the two ends of the segment are inside the j sphere) and for a doublet (each end of the segment belongs to different spheres).

will be called the partial pair distribution density too.

( 12 , Dp R H ) ( ) 12 ,

Dp R H

The pair distribution density function for a single sphere is:

( ) ( ) ( ) ( ) ( ) 2 3 3 12 16 16 / / / jj j j j j Dp R b R b R b R b = - + (18)
with 0 2 j R b < < .

The partial distribution density for the doublet is also obtained from Eq.13a:

( ) ( )( )( )( 2 1 2 12 2 2 1 1 , ' ' ' ' T T Dp R H A R b R R b R R b H R b H R dR H π = -+ + - -+ + - ∫ ) ' (19)
The normalization factor A can be given by: ( )

2 3 1 2 3 4 A b b π - = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
The lower and upper limits T 1 and T 2 of the integral are: if

2 1 R H b b < + - , 1 1 2 T H b T R b 2 = - = + (20a) if , 2 1 1 H b b R H b b + -< < + -2 2 1 2 2 T R b T R b = - = + (20b) if , 1 2 1 H b b R H b b + -< < + + 2 1 1 2 2 T R b T H b = - = + ) (20c)
An analytical solution of this integral (Eq.19) exists as:

( ) ( 12 2 1 Dp f T f T β β = -- - (21a)
where

( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 4 1 2 1 2 2 2 2 2 2 2 2 3 5 1 2 1 2 2 /3 / b b b b y R f y A H b b y b b y y π α α α α ⎡ ⎤ - + + ⎢ ⎥ = ⎢ ⎥ + - - + + + ⎣ ⎦ 5 (21b)
and and

( )/ 2 R H α = - ( ) / 2 R H β = +

Set of identical primary particles

The generalization of the previous calculations to a chain of N primary particles (Eq.22) and to an aggregate with N primary particles (Eq.23) is straightforward:

( ) ( ) ( ) ( ) ( ) 2 1 2 2 2 1 ,2 1 / N s s j Dp R N Dp R N j Dp R j N = ⎛ ⎞ = + + - - ⎜ ⎟ ⎝ ⎠ ∑ (22) ( ) ( ) ( ) ( ) max min 1 1 1 , H s N e s H N Dp R Dp R Dp H Dp R H dH N N - = + ∫ 2 (23) 1 N ≥ ( )

Ne

Dp H is the pair distribution density function for the set of the N sphere centres. This set represents the superstructure of the body whereas the spherical primary particle represents its substructure.

( )

Ne

Dp H is normalized. For a chain of N primary particles, obeys the expression:

( ) Ne Dp H ( ) ( ) ( ) ( 2 2 2 1 1 N Ne j N H Dp H H j N N δ = - = - - ∑ ) - ( 24
)
Figure 3a represents the pair distribution density function for a chain of N primary particles (Eqs. [START_REF] Hermawan | Characterization of short-range structure of silica aggregates-implication to sedimentation compaction[END_REF][START_REF] Riefler | Structural analysis of clusters using configurational and orientational averaging in light scattering analysis[END_REF]. The two size scales are clearly identified. Figure 3b represents the pair distribution density function for a cubic packing of N primary particles. The two size scales can be observed, but less distinctly.

Using the expressions for the distribution moments of a primary particle (Eq.12b) and for a doublet (Eq.16a), the j th -moment of the aggregate distribution is written as:

1 1 , , 0 1 j s j j m M N 1 m j m M B Q N N + - = - = + ∑ ( 25
)
where Q i is the i th -moment of the superstructure distribution

( ) Ne Dp H .
The aggregate of N spheres can be compared to the cluster of N points (centres of spheres) for which:

( ) ( ) ( ) ( ) 2 , 1, / N ik i k i k Ne Dp R Dp R R H N N δ = ≠ = = - ∑ - . Then ( ) 2 , 1,
/ N j j j i k i k i k M Q H N = ≠ = = - ∑ N where 1 , , s j mj
M and B characterize once more the primary particle shape, i.e. the substructure.

Set of different primary particles

The pair distribution density function for a set of N primary particles with different radii b j can be written as:

( ) ( ) ( 2 1 , 1
,

N N j j j j k j k j k j j k j k Dp R P Dp R P P Dp R H = = ≠ = + ∑ ∑ ) (26) with 3 3 1
/ N j j i i P b b = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ jk
H is the distance between the centres of the primary particles j and k. R and jk H are not made dimensionless. One can verify that Dp is normalized. is calculated from Eqs.20-21.

( )

, jk jk
Dp R H 4. Light scattering cross section 4.1 Doublet of identical primary particles

Exact calculation

The scattering cross section of a doublet as a function of the primary particle size parameter kb has been calculated by two ways:

following Van de Hulst (Eq.3), following Eqs. 9-10 and the pair distribution density for a doublet (Eq.11) The deviation between the two calculations is smaller than 10 -3 within the whole range [0,100] of the size parameter. Figure 4 represents the ratio of the scattering cross sections of the doublet and the primary particle as a function of the primary particle size parameter kb for various distances H. The ratio tends to 4 if kb→0 and to a value close to 2 if kb→∞. One can also observe the occurrence of two kb-ranges: [0,3] and [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF]100]: as kb>3, the ratio has a constant value within 5%. The latter one follows the approximate expression:

( ) 0.7 0.85 2 2 1 / 2 0.25 H C C e - - = + (27)

Approximated calculation

The function ( )

0 RDG F
x can be approximated by a polynomial function allowing the introduction of the moments of the distribution. Figure 5 and Eq.28a show such an approximation.

( )

2 1 0.4 0.1 app F x 3 x x = - + 0 x α < < (28a)
( )

0 app F x = x α > where 3.0 α = or ( ) 2 1 0.4 0.1 app F x 3 x x = - + 0 x α < < (28b) ( ) 0.0105 0.21/ app F x x = - + x α > where 2.5 α =
If the approximation (Eq.28a) is applied, the distribution moments with orders 0, 2 and 3 appear in the expression of the scattering cross section. In fact, only the partial or the truncated moments, called ( ), are involved due to the relation .

, ,

T T T j sj s M M M j ( ) 0 app F x α > =
* corresponding author: gruy@emse.fr

1 , 1 0 Y T j s j s M R Dp dR = ∫ (31)
where

2 Y = if 2 u > Y u = if . 2 u <
However, Eq.31 (withY ) leads to an erroneous value for the moments of the distribution in the case of the large primary particles. As a consequence, one keeps out Eq.31 with , i.e.

u = 2 Y = 1 , 1 , T s j s M M = j .
Then, the scattering cross section of the doublet is written as:

( ) ( ) ( 2 4 2 2 0 2 1 0.4 0.1 3 T T C H k V m M M M π = - - + ) 2 3 T (32) or ( ) ( ) 2 1 2 2 p C H C C H = + (33a) with ( ) ( ) ( ) 2 4 2 2 2 , 0 2 1 1 0 . 4 0 . 1 3 T T p s s C H k V m M M M π = - - + , 2 2 , 3 T s (33b)
Figure 6 compares calculated following Van de Hulst (Eq.3) and (Eqs.33ab). Only the doublet of touching primary particles (H=2) leads to a non-negligible error for kb>3. The use of the second approximation (Eq.28b) leads to a negligible error for .

( ) ), then Eq.14 and Eqs.16a-c can be applied for any H. Eq.33b becomes:

( ) ( ) ( 4 2 4 2 1 2 0 1 1 1 0.4 0.1 3 m p m m C H k V m B B H π - = ⎛ = - - - ⎜ ⎝ ⎠ ∑ ) , 2 , 3 m ⎞ ⎟ (34)

Set of identical primary particles

The aggregate scattering cross section is calculated by using Eqs.10 and 23:

( ) ( ) ( ) max min 1 1 2 H N N e H N N C NC Dp H C H dH - = + ∫ 2 p j k (35a) or ( ) 1 2 , 1 / 2 N N jk j k j k C NC C H = ≠ = + ∑ (35b)
If the largest distance into the aggregate corresponds to the case where { }

( ) ( ) ( 4 2 4 2 1 , 0 1 1 1 1 0.4 0.1 2 3 N m m m m N N C NC k V m B B Q π - = - ⎛ ⎞ = + - - - ⎜ ⎟ ⎝ ⎠ ∑ ) 2 , 3 1
(36)

V is the matter volume of the doublet. The percentage of doublets with primary particles touching each others (H=2) in an aggregate is often small. The body having the largest ratio is a three-dimensional compact object. Although the deviation (Eq.3 vs Eqs.33a-b) previously observed for a doublet (figure 6) is small, the deviation for an aggregate is cumulative and increases with the number N of primary particles. The values of the ratio have been calculated for various, kb and N, values in the case of two aggregates: a simple cubic arrangement of primary particles and a chain of primary particles. The values of the ratio were calculated by the following Eq.35a and by using either x F x = (see figure 5). This condition is strictly verified for a 3D-body whereas it is weakened for 1D-body. In the latter case the ratio of primary particles separated by a large distance ( ) is smaller than in a 3D-body and contribute less to the scattering cross section.

kR >

Figure 4 shows that for , the scattering cross section of the doublet does not depend on kb and depends only on H. Eqs.27 and 35a lead to:

10 kb > ( ) ( ) ( ) max 0.7 min 0.85 2 1 1 1 0 . 2 5 2 H H N N e H N C NC Dp H e dH - - ⎡ ⎤ - = + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∫ (37)
The deviation between the two value series (Eq.35a versus Eq.37) is also smaller than 5%. The scattering cross section (Eq.37) depends on the whole morphology (superstructure) of the aggregate. The kernel of the integral (Eq.37) corresponds to the substructure, i.e. the primary particle. Moreover, the scattering cross section for a set of different primary particles obeys:

( ) ( ) 2 1, 2 1 , 1 2 / N N N j j k j k j jk j j k j k C C C H VV V V = = ≠ = + + ∑ ∑ k (38a) with ( ) ( ) ( ) ( ) 2 2 4 2 0 1 1 , 3 jk j k jk j k H b b RDG jk j k jk jk H b b C k V V m F kR Dp R H π + + -- = + -∫ dR ) (38b) ( , jk jk
Dp R H is calculated from Eqs.20-21.

Conclusion

An aggregate composed of spherical primary particles presents two size scales. In other words there exists a substructure (primary particle) and a superstructure (aggregate). The pair (distance) distribution density function shows the details of the two scales. These are more distinguishable from each other in the case of a body with a small spatial dimension like the chain of primary particles. The distribution j thorder moment is a linear combination of the moments with order ≤j of the superstructure distance distribution. The coefficients of the combination are relative to the sub-structure. In the framework of the RDG approximation, the scattering cross section of the aggregate is related to the distance distribution in the superstructure and the scattering cross section of a doublet. Equations are available as well as their approximations decreasing the computation time. However, there does not exist a relation between the scattering cross section of the aggregate and the distribution moments over the whole range of size and number of primary particles. This work will be enlarged to other primary particle shapes (e.g. polyhedral) in order to determine the corresponding combination coefficients in the ( ) , Dp R H moments and to calculate the doublet and aggregate scattering cross sections.

Captions

Fig. 1: Dp as a function of R (normalized by the radius of the primary particle) for a doublet. Fig. 2: drawing showing the notation and the procedure to calculate the pair distribution density; case of the doublet. Fig. 3a: Dp as a function of R (made dimensionless by the radius of a primary particle) for a chain of N primary particles. Fig. 3b: Dp as a function of R (made dimensionless by the radius of a primary particle) for a cubic packing of N primary particles. Fig. 4: ratio between the scattering cross sections of the doublet and the primary particle for H=2 (---), H=6 ( ___ ), H=12 (+++) as a function of the primary particle size parameter.

  are respectively the normalized pair distribution density for a single sphere (the two points of the segment are inside the same sphere) and for a doublet (each end of the segment belongs to different spheres).

  will be called the partial pair distribution density.

  of the two spheres separated by the H distance is converted to a set of two points separated by the same distance, then δ is the Dirac function. Thus the , m j B coefficients represent the contribution of the primary particle shape (here a sphere) to the moments of the partial pair * corresponding author: gruy@emse.fr distribution density. The table 1 contains the ,

  j=1,2) and are respectively the normalized pair distribution

  After calculations the deviation between the two value sets is smaller than 5% if for a 3Dbody (d=3) and for 1D-body (d=1). In fact, is the size parameter for the whole aggregate. Calculations were performed with N<10

Fig. 5 Fig. 6 :

 56 Fig.5:

  Fig.3b

  Fig.4

Fig. 6

 6 Fig.6

  2

	Max H	jk	u + < (i.e., the largest distance is smaller than / α	k	≈	λ	/	2	), Eq.14 and
	16a-c can be applied for all H. Eq.35b becomes:

* corresponding author: gruy@emse.fr

In the case of the partial moment, the integration interval [A,B] in ( )

. R min and R max are respectively the smallest and the largest distances between two points belonging to the scattering body.

Then for a doublet and setting ( )

( 1 , 2 , / 2

where

, , , 0,0,0,0

)


Fiammetta Namer
email: fiammetta.namer@univ-nancy2.fr

Have Cutthroats Anything to Do with Tracheotomes? Distinctive Properties of VN vs. NV Compounds in French

The aim of this paper is to compare word formation constraints that operate during the construction of compound lexemes in French; precisely, we contrast VN-structured lexemes (ouvre-boîte N : open-tin =' tin opener') with NV ones (anthropophage A :°man-eat(er) = 'cannibal'). The question is whether both VN and NV are obtained through the same rule (and differ only with respect to the chosen components), or whether each type corresponds to a specific compounding process. Our study is grounded on large-scale corpora of VN compounds on one hand, and NV ones, on the other hand. In the first part of the paper we therefore motivate (1) our claim that bound roots, such as °anthropo and °phage in anthropophage can be assigned lexical categories (although they are no autonomous lexemes), and (2) our decision to assign precisely the verbal category to bound roots such as °phage. The second part of the paper is devoted to the NV versus VN properties investigation, according to three criteria: a) the distribution of the lexical category and the semantic values for VN and NV, b) the process type denoted by V, and c) the thematic roles plaid by N with respect to V. The comparison of VN and NV, according to these three criteria, leads us to suspect that NV and VN are indeed formed, in French, by two distinct morphological rules.

1.

Introduction

Data

Within French lexical morphology, compounding processes form either nouns or adjectives. Either type can be obtained through a variety of rules, where two nouns (1a), two adjectives (1b), a verb and a noun (1c) are involved.

(1) a.

[timbre N -poste N ] N : stamp-post = 'postage stamp' [homme N -grenouille N ] N : man-frog = 'frogman'

b.

[doux A -amer A ] A : sweet-bitter = 'bittersweet' c.

[coupe V -papier N ] N : cut-paper = 'paper knife' [brise V -coeurs N ] N : break-heart= 'heartbreaker' [perce V -oreille N ] N : pierce-ear= 'earwig' (porte) [coupe V -feu N ] A : break-fire= 'firebreak (door)'

In parallel, the French lexicon includes lexical units such as (2), which often appear to be made up of two Combining Forms 1 , originating from Latin or Greek. These combining forms appear to be two nouns (2a), an adjective and a noun (2b), two adjectives (2c) or a noun and a verb (2d) 2 .

( Our aim is to examine compound nouns and adjectives formed according to the patterns illustrated in (1c) and (2d), in order to propose a contrastive analysis. This study defends the hypothesis that compounds obtained through Word Formation rules at play in (2) belong to the current French morphological system, despite the fact that some compounds are borrowed from ancient languages (eg. Latin and Greek), and that a large amount of them are common to several contemporary languages (mainly Romance, Germanic and Slavic ones, cf. [START_REF] Darmesteter | De la création des mots nouveaux dans la langue française et des lois qui la régissent[END_REF]). Their productivity is confirmed by the amount and variety of neologisms that are found in the media (bibliophobe, tabacolâtre, Jocondovore). Moreover, at least three arguments support the idea that neoclassical compounds do not come under Lexeme Formation Rules belonging to the ancient languages that provide their Combining Form components:

(1)

Compounds are often formed through the association of two Combining Forms of various origins, and thus not systematically coming from the same Latin or Greek language: French+Latin (insecticide), Greek+French (macromolecule 'macromolecule'), French+Greek (bureaucrate 'bureaucrat'), Latin+Greek (planisphère 'planisphere'), French Truncated Form+French, Latin or Greek (cinéphile 'cinephile', anglophone, alcootest 'drunkometer'); 1 We borrow the term 'Combining Form' to [START_REF] Warren | The importance of combining forms[END_REF], [START_REF] Fradin | Combining forms, blends and related phenomena[END_REF] and [START_REF] Iacobini | Distinguishing derivational prefixes from initial combining forms[END_REF]. In the examples, Combining Forms are preceded by the '°' symbol ; on that matter, see section 2.3. 2 A table, at section 6.3, provides a literal definition of the mentioned neoclassical compound lexemes, according to the meaning of the Combining Forms they contain.

Have Cutthroats Anything to Do with Tracheotomes? Distinctive Properties of VN vs. NV Compounds in French 3 (2)

The emergence of an improper thematic vowel is often observed between the components: for instance, the thematic vowel "o", greek-specific, is also used at component boundaries originating from French (franco-allemand 'Germano-French', anglo-saxon) or from Latin, instead of the expected "i" vowel (cérébro-spinal 'cerebrospinal', génito-urinaire 'genitourinary') (Darmesteter(1894: 256));

(3) Infringements of both Greek and Latin compounding rules are met in the formation of many neoclassical coumpounds. For instance:

(i) Benveniste(1974:163-170) investigates the genesis of the noun microbe, in order to demonstrate that its constructed meaning ("little life", that is "microscopic organism") cannot be accessed from a Greek compounding pattern, which would lead to build the "short life" interpretation, which is incoherent from a scientific point of view. (ii) Darmesteter (1894:253) analyses the nouns oxygène 'oxygen' and hydrogène 'hydrogen', in order to show their semantic "ill" formation, given that, unlike in French, the Greek element corresponding to gène does not mean "which produces" (oxygène: "which produces acid", hydrogène: "which produces water") but instead, "which is produced".

More precisely, a Noun+Verb combining process emerges from examples (2d), similar to the French Verb+Noun compounding process3 , with two apparent differences, that is that (1) component come from the Greek or Latin lexicon , and (2) component ordering is reversed ( in a neoclassical compound, governing (X) components are on the right, and governed (Y) ones on the left, resulting in the YX pattern, whereas 'ordinary' compounds are in the XY form). Our study is thus in line with traditional morphological analyses which identify a so-called "neoclassical" compounding results (eg. in 2d) in contrast with "ordinary" French component-based compounding ones (the traditional distinction was already current in the 19 th century, cf. for instance [START_REF] Hatzfeld | Traité de la formation de la langue[END_REF]). From this hypothesis, we now investigate differences and similarities between both VNs and NVs. Our objective is to identify and compare semantic and categorial constraints on these two types of compounds.

General issue

Assuming that lexical units in (2d) come under some NV compounding rule, our purpose is to determine whether VN and NV compounds are formed by two distinct compounding rules in French, or by one and the same rule. In other words:

• do VN and NV compounding rules correspond to a unique French lexeme formation process, with identical properties, except component origin and, consequently, component ordering4 ?

• do they correspond to two different lexeme formation processes in French, which do not form the same unit type, but depend on different semantic and categorial constraints ?

Outline

To answer these questions, we have tried here to identify differences and similarities between VN and NV compounding patterns, especially through the contrastive study of both categorial and semantic constraints that are implied in these compounds' formation.

Our study is carried out from a corpus designed according to principles presented in section 2 ; in doing so, we also explain which theorical issues we had to face during the task of identifying components as Combining Forms, in NV compounds such as (2d). Section 3 is devoted to the comparison of VN and NV compound properties. We are led, in section 4, to the conclusion that the VN and NV compounds are formed by two distinct compounding rules in French, rather than a single one.

Data collection

VN corpus

This corpus contains 2-3000 VN compound lexemes. They have been gathered mainly from lexicographic sources: general language dictionaries, encyclopaedias. Dictionaries were chosen so that they are typical enough of both general and technical French lexica. Among them, there are: Trésor de la Langue Française, Grand Robert de la Langue Française, Dictionnaire de la langue française (Littré), Dictionnaire Général de la Langue française (Darmesteter and Hartzfeld). Beside dictionaries, other sources have been used to collect VNs: Part-of-speech tagged corpora, where VNs could be extracted, and the Internet open corpus 5 . Also, newspapers, magazines and catalogues have also occasionally allowed us to enrich our VN corpus.

NV corpus

NV corpus belongs to the French lexicon, without being limited to a single speciality domain: limiting oneself to a specific domain is actually the risk to avoid when examining lexeme formation rules using combining forms.

Basically, two lexicographic sources have been used to build up this corpus: the machine-readable version of Trésor de la Langue Française, and the Dictionnaire des structures du vocabulaire savant. The latter, compiled by H. Cottez, groups and defines ordering in « neoclassical » compounding is borrowed to ancinet Greek and Latin. 5 Many thanks to M. Plénat, who sent us so many examples. some 2,700 components used in learned words formation, be these components combining forms or learned affixes.

The first version of the NV corpus resulted from a semantic intuition about which of the combining forms should actually belong to the Noun and Verb categories. On the basis of this first corpus version, we have applied a series of (still tentative) principles in order to confirm component categories. We give an account of this analysis in section 2.3. In all, we have gathered about 500 NV compound adjectives and nouns, which satisfy category criteria detailed below.

Identifying constituents in NV compounds

Making up NV corpus raised important theoretical issues which are related to the identification of VN components. Several problems were posed:

(1) How to recognize a compound lexeme among all the morphologically complex lexemes formed out of combining forms. In other words, how to be sure that their constituting parts are base-lexemes, and not affixes6 ?

(2) Consequently, how can 'verb' and 'noun' categories be identified for these base-lexemes, whose particular property is never autonomous in syntax?

Theoretical framework

These questions make sens within the lexeme-based theoretical framework [START_REF] Anderson | A-morphous Morphology[END_REF], Aronoff&Fudeman(2004), [START_REF] Booij | The Grammar of Words: An Introduction to Linguistic Morphology[END_REF], [START_REF] Fradin | Nouvelles approches en morphologie[END_REF]). In this framework, basic units in lexeme (cf. [START_REF] Matthews | Morphology, An introduction to the theory of word-structure[END_REF]) formation processes belong to one of the major categories: N, V, Adj, and are clearly distinguished from affixes, which are nothing but exponents of phonological rules. Within this approach affixes are not assigned a major category, unlike other theories (cf. [START_REF] Williams | On the Notions `Lexically Related' and `Head of a Word[END_REF], [START_REF] Lieber | On the organization of the lexicon[END_REF], [START_REF] Lüdeling | Neoclassical word formation in German[END_REF]). Our position is to extend lexeme-based morphology principles to bound roots, that is to assume that Greek and Latin Combining Forms can be N, V, or Adj categorized lexemes, in the same way as free roots are. This point of view fundamentally differs from those which admit the existence of Combining Forms, without assigning categories to them (cf. Rey-Debove( 2004), and Cottez(1988: VII-XX), who claims "learned vocabulary forming elements" to be "signs", that is morphemes, but provides them with neither of base-lexeme nor affix status.

On the other hand, we follow the theoretical line adopted (among others) by [START_REF] Corbin | Les bases non-autonomes en français ou comment intégrer l'exception dans le modèle lexical[END_REF], [START_REF] Warren | The importance of combining forms[END_REF][START_REF] Fradin | Combining forms, blends and related phenomena[END_REF]2003) and [START_REF] Iacobini | Composizione con elementi neoclassici[END_REF], who make a distinction, among learned Combining Forms, between base-lexemes on one side and affixes on the other side. So, designing or NV corpus is based on the fact that Combining Forms have been identified as base-lexemes, in compounds words such as anthropophage.

Affix or Base-Lexeme (N, V, A)?

• Are compound components in example (2) base-lexemes (N, V, A) or affixes?

This question is relevant because, in words such as anthropophage, Combining Forms properties (eg. anthrop(o) and phage) make them closely related to base-lexemes as well as to affixes.

Combining Forms look like affixes, because they are bound roots. They fulfill a structural position within lexical units, not within phrases. Combining Forms look like base-lexemes, because they have categorial and semantico-referential properties.

Semantic criterion

This criterion relies on the dichotomy between "lexical sense" and "grammatical sense", frequently mentioned in the literature (cf. [START_REF] Corbin | Préfixes et suffixes: du sens aux catégories[END_REF] who applies it to French morphology, and [START_REF] Kleiber | Problèmes de sémantique -la polysémie en questions[END_REF], to lexical semantics) A base-lexeme is an abstract lexical unit, with context-independent phonological form, syntactic category and meaning. Unlike affixes, which do not refer to entities but are given a semantic instruction, Combining Forms refer in a stable way to a referential category, a linguistically encoded concept: and so do base-lexemes. In other words, CFs carry a referential meaning ; they are able to identify concepts, but they are not able to name them within utterances (Corbin 85, Iacobini 99, Fradin 2000). However, there are serious drawbacks when attempting to use this criterion:

(1) identifying the component referential value depends on its translation and this translation lacks reliability. A big diversity is observed among dictionaries (and, sometimes, within a single dictionary) when looking up at a Combining Form translation: more especially, CF denoting processes (eg °graph(o)) are sometimes translated by a verb ("to write"), and sometimes by its corresponding deverbal noun ("writing"). And as far as the Robert Brio is concerned, for instance, both translations are given.

(2) identifying the component referential value also depends on the morphologically constructed meaning of the compound they are part of, and on the semantic relation between the compound constituents. Now, semantic features provided by dictionaries are often unsufficient: either they are not precise enough, or they describe referents rather than meanings. The lexicographic definitions of pleuronecte, for instance (« type of flat fish such as sole, turbot, halibut…") and bathyergue 'bathyergus' («rodent mammal from Africa, whose behaviour is close to mole") prevent us from access to the morphologically constructed meaning, because they have to do only with the referent description. Actually, pleuronecte and bathyergue are named according to their behaviours: a pleuronecte swims (°necte V ) on one side (°pleur(o) N ); a bathyergue works (°ergue V ), that is, "digs", in depth (°bathy N ).

Moreover, this semantic criterion is hard to use when base-lexemes have undergone a grammaticalization process: in that case, they behave as affixes; and their semantic value vacillates between referential pole and instructional pole (cf. Amiot&Dal(2005) about °-logue).

Categorial criterion

This criterion relies on the base-lexeme property of belonging to one of the major partof-speech categories: noun, verb, adjective, adverb. In other words, establishing that a CF belongs to one of these categorie is sufficient to guarantee its base-lexeme status. However, as CFs never have syntactic realization, none of the usual identification syntactic tests (distribution in utterances, their syntactic behaviour and their inflectional marks) can operate in order to decide whether they are base-lexemes or not. To meet this specificity, other means are required. For of them are presented here

• Means 1: CF can be assign a part-of-speech category according to its translation or interpretation in French. It simply gets the same category as its translation. But this solution is rather limited, due to translation uncertainty (see above).

• Means 2: A CF allomorph to a base-lexeme is clearly categorizable. But the allomorphy notion sometimes conceals suppletion phenomena, and relations between a base-lexeme and its hypothetical allomorph may lack transparency.

We will not address this issue here.(cf Haspelmath(2002:26-28) for a synthetic description of the issue, and Boyé(to appear) for a presentation of the suppletion).

• Means 3: CFs which are selected as affixation bases are nothing but baselexemes. From there, if an affixation process obeys constraints on its base such that only one base category is selected, then any CF used as a base for this process, automatically gets an unambiguous category (cf. Corbin (1985:62-64;1987:182-sq)). However, only few affixes obeys such strong categorial constraints; so in general this technique cannot determine for certain the value of base categories.

• Means 4: CFs can also be assigned the lexeme status (and a category) according to the morphological construction in which they are found. In particular, the semantic relation type reveals its categorial counterpart. For example, in YX compounds where there is some typical argument-to-verb relation between Y and X, we can expect X to be a verb, and Y, to be a noun.

Through combination of the means just presented, we are able to identify verbal (X component) and nominal (Y component) Combining Forms in morphologically complex lexemes such as illustrated in (2d). Examples (3) and ( 4) give an account of the way we have proceeded to category identification.

(3) oculogyre ADJ 7 a. °ocul(o)

b. °gyre

The CF °ocul occurs as base in the adjective oculaire; this allows to label it with the category "noun", via Means3. The reason is that suffix -aire exclusively forms adjectives from nouns: pôle NOUN polaire ADJ . Given that oculaire is formed by means of this process, from the base °ocul, the latter is not but a noun.

Similarly, categorial constraints exerted by -oire and -ion French suffixes are used to provide (3b) with a grammatical category, thanks to Means3. In French, the adjective giratoire 'gyratory' (also found written gyratoire) is given the definition: "revolving around a point or axis"; and the noun giration 'gyration': "act or instance of turning". We will assume that the base of these lexemes, respectively formes by -oire and -ion suffixation, is the allomorphic variant °gir(at) of °gyre. Each of these suffixes preferentially selects preferentially verbal bases (méritoire A 'meritorious' < mériter V ; 'to deserve,merit' agression N < agresser V 'commit agression'), but not exclusively (républicatoire A 'republicatory'< république N 'republic'; baladoire A '°balladory'8 < balade N 'ballad'; zonation N < zone N ). However, when the same sequence is used as base for both affixed lexemes, it is nothing but a verbal base (revendicatoire A 'revendicatory', revendication N < revendiquer V 'revendicate'). This is confirmed by a formal clue: When the base of a -ion suffixed noun (or that of a -oire suffixed adjective) ends with -at, then it is identified as the supine form of a Latin verb. So, as °girat/°gyrat results from a Latin verb form, it is indeed an allomorph of °gire or °gyre. And therefore, °gyre is a verb.

So, Means3 allows us to identify both a nominal bound constituent (°ocul) and a verbal one (°gyre) in oculogyre A . In other words, using Means3 may lead us to analyse oculogyre as a compound NV adjective.

This hypothesis is confirmed by Means4

Means 1 makes us assume that °anthrop is a noun and that °phage is a verb; °phage has to be analysed as a verb, because its semantic relation with °anthrop is of predicateargument type: phage denotes a process which applies on its proto-patient argument, realized by °anthrop. Finally, Means1 and Means4 cooperate to tell us that anthropophage is an NV compound adjective.

Analysis: three criteria

Once the task of gathering NV and VN corpora has been carried out, the analysis core has been that of comparing "neoclassical" compounds NV, and French-components based compounds VN. Three criteria (c1) to (c3) served at that purpose:

(c1) category and semantic values, for VNs and NVs ;

(c2) the type of process denoted by V verbs, for each compound type;

(c3) the semantic relation that holds between V verb and N noun By crossing these three criteria, we ask questions such as: are VN and NV properties identical, in partial overlapping or in complementary distribution ? Are there lexical gaps, corresponding to unattested (c1) to (c3) combinations ?

VN compound properties 9

Criterion C1: category and semantic types of VN compounds

• Category: VN compounding mainly builds nouns, and rarely adjectives. The latter are illustrated in ( 5):

( 9 On this matter, see among others [START_REF] Villoing | Les mots composés VN du français: arguments en faveur d'une construction morphologique[END_REF] and [START_REF] Fradin | On a semantically grounded difference between derivation and compounding[END_REF].

Criterion C2: the type of process denoted by V

In a VN compound, the process type denoted by the verb is strongly constrained ; according to [START_REF] Vendler | Verbs and Times[END_REF] terminology, and that of his successors (among others, [START_REF] Dowty | Word Meaning and Montague Grammar[END_REF]), it can only be dynamic, that is compatible with linguistic contexts such as "être en train de V [be Ving] / se mettre à V [start to V]/ s'arrêter de V [stop Ving] ", cf. ( 11) (for a detailed presentation, see [START_REF] Villoing | Les mots composés VN du français: arguments en faveur d'une construction morphologique[END_REF]). On the other hand, V process type is very unlikely to be stative (12):

(11) coupe-papier (°cut-paper 'paper knife') Jean est en train de couper le papier/ s'est mis à couper le papier /s'arrête de couper le papier. 'Jean is cutting the paper/started to cut the paper/stops to cut the paper'.

(12) *sait-latin (°know-latin) *Jean est en train de savoir le latin/s'est mis à savoir le latin/s'arrête de savoir le latin. *'Jean is knowing latin/started to know latin/stops to know latin'

Criterion C3: V to N semantic relation

There are strong restrictions on the semantic relations between V and its participants; and consequently, on roles that N may play with respect to V. According to [START_REF] Dowty | Thematic Proto-roles and Argument Selection[END_REF] who refutes thematic role labelling, and proposes instead a Proto-Agent/Proto-Patient continuum, N corresponds quite exclusively to V Proto-Patient, within a VN compound. The reason is that N satisfies "change-of-state" and "affectedness" criteria (cf. also Foley&Van [START_REF] Foley | Functional syntax and universal grammar[END_REF] and [START_REF] Jackendoff | Semantic Structures[END_REF] for a definition of these criteria). This analysis also applies for a small number of VNs, like in ( 13) that seem to be based on intransitive verbs and that for which N looks like the verbal Proto-Agent argument:

(13) trotte-bébé (°toddle-along-baby = 'baby walker') ;

pense-bête (°think-of-silly = 'reminder').

However, the semantic of the verb is causative and N is involved as a Protopatient ( 14) trotte-bébé 'causes/makes the baby to toddle along' ;

pense-bête 'causes/makes the silly to think'.

Moreover, we can observe other (less frequent) semantic relations between V and N in VN compounds. For instance, we find rare cases (such as ( 15)) where N doesn't meet Proto-patient criteria but rather Proto-agent ones.

(15) cuit-vapeur: cook-steam (saucepan type) pousse-pied: push-foot (boat that we push with foot)

Besides, there are VNs such as ( 16), where N can be analysed neither as a Proto-patient , nor as a Proto-agent of the verb. Rather, N seems to correspond to a temporal (16a) or locative (16b) verb modifier.

(16) a. réveille-matin: wake-up-morning = 'alarm clock' grille midi: scorch-midday (plant) b.

croque-télé: crunch-tv (tv tray)

NV compound properties

Criterion C1: Category and semantic types of NV compounds

• Category: generally, NV compounding produces adjectives like (17) which function equally as nouns. [START_REF] Kerleroux | Les formes nues sont-elles simples ?[END_REF][START_REF] Kerleroux | La coupure invisible: études de syntaxe et de morphologie[END_REF][START_REF] Fradin | Esquisse d'une sémantique de la préfixation en anti[END_REF] (17) ventriloque ADJ ('ventriloquist')

• Semantic type.

When only the nominal category is realized, NV may refer:

mostly, to a concrete object:

(18) odontoclaste N (cell which breaks (°claste V ) teeth (°odonto N )) ; trachéotome N ('tracheotome': scalpel used to incise (°tome) the thrachea (°trache(o)))

to an animate entity,

(19) biographe N ('biographer') notonecte N ('noctonect')
sometimes, to an event:

(20) lipolyse N ('lypolysis') infanticide N ('infanticide')

Criterion C2: the type of process denoted by the verb

The process denoted by the verbal combining form may be

• either dynamic (21) °ambule V = 'to walk' (noctambule A : 'late-night/night reveller')) ;

• or stative. Among stative processes, most verbs describe spatial relations [START_REF] Talmy | Toward a cognitive semantics (2 volumes)[END_REF][START_REF] Vandeloise | L'espace en français: sémantique des prépositions spatiales[END_REF]). ( 22) °phore V , = 'to carry' (mélanophore N : 'melanophore'), °fère V = 'to contain'' (carbonifère A : 'carboniferous').

Others are feeling or emotion verbs (according to Levin classification in [START_REF] Levin | English Verbs Classes and Alternations[END_REF]):

(23) °phile V = 'to like' (russophile A: 'Russophil') °mane V = 'to like' (mélomane A 'music lover') °lâtre V = 'to adore' (wagnerolâtre A = who adores Wagner), °phobe V = 'to hate' (anglophobe A : 'anglophobic')

Other stative processes are perception verbs (24) °op V = 'to see' (nyctalope A : 'day-blind')

or verbs of existence [START_REF] Levin | Unaccusativity[END_REF] (25) °cole V = 'to live' (limicole A : 'limicolous'; cavernicole A ) °bie V = 'to live' (phyllobie N = (organism) living on leaves: 'green leaf weevil'),

We also found one verb of light emission [START_REF] Levin | Unaccusativity[END_REF] (26) °luque = 'to shine' (noctiluque A : 'noctilucent')

Criterion C3: V to N semantic relation

In NV compounds, semantic relations between N and V belong to two distinct dimensions, according to the verbal process type.

• For dynamic verbs, [START_REF] Dowty | Thematic Proto-roles and Argument Selection[END_REF] criteria are applied (as in the case of VN compounds). As a result, we have observed that here the noun can play different semantic roles wrt the verb:

1) N may fulfill the Proto-agent criterion:

(27) psychogène A : 'psychogenic' 2) (more frequently) it corresponds to a Proto-patient (28) lipolyse N : 'lypolysis'

3) It may also be the case that N does not meet any of the proto-role requirements When V describes a predicate expressing an emotion or a feeling, it sets up a relationship between two participants: the experiencer and the stimulus. We propose then to identify these verbs as spatial relations: experiencers are interpreted as grounds and stimuli as figures.

As far as we know, these verbal predicates only combine with figure denoting nouns: (ex 27) (32) mélomane A ('music lover') anglophobe A ('anglophobic')

Results

By crossing criteria c1 to c3, (cf. the comparative table in section 6.2), we notice the following contrasts between VNs and NVs: NV compound nouns and adjectives cover a much broader spectrum than VN compounds, whatever the examined criterion:

• Categorial: NVs belong equally to ADJ or NOUN category (whereas VN compounds are almost exclusively nouns) ; 10 Talmy (2000) investigates non-agentive relations, between participants of a so-called spatial or localisation event.

Such events involve two main roles : the • V process-type: V components in NV compounds describe either dynamic or stative processes, unlike in VN compounds.

• N to V semantic relation: in VN compounds, the N component is typically a proto-patient; but it fulfills a wider amount of semantic roles in NV compounds: proto-patient, proto-agent, or none, with respect to dynamic V components; and either figure or ground with respect to a stative V component which expresses a spatial relation.

In other words, NVs instanciate a wide range of combinations c1-c3, whereas VN compounds mainly correspond to those combinations where V is dynamic and N is proto-patient.

Conclusion

In some cases (when the verb is dynamic) VN and NV compounds seem to share similar categorial and semantic constraints, and this would lead us to conclude that only one compounding rule is at play and selects various types of components. However, VN and NV have clearly different behaviour when V is stative. In this case, it can be selected only as NV compounds governing component. Consequently, only the NV compounding rule can build spatial relations between N and V. 'Ordinary' morphology, by means of VN compounding, is unable to construct meaning equivalent to what NV compounding offers: syntactic patterns are required. There is, for instance, no other ways to express "melomane", but with a syntagmatic expression: "music mad".

The results of this study lead us to think that two distinct compounding rules are at play in French. This conclusion raises two major questions:

• why is VN compounding such a constrained process?

• where do NV rules come from? Should we consider them as belonging to the French linguistic system, or are they borrowed from those languages where the components come from (ie Latin, Greek) ?

The latter question is part of a larger issue, in which we try to determine whether neoclassical compounding makes use of rules inherited from the very languages components originate, or not. Following comparativist authors (Darmesteter ( 1894)) (and unlike eg. [START_REF] Benveniste | Formes nouvelles de la composition nominale[END_REF]) the inheritance hypothesis seem to be the mainstream. Therefore, we tried to assess this assumption, in the specific case of NV compounds, where both N and V are of Greek origin. The underlying hypothesis is the following: if either nominal or adjectival NV componds formation were known in ancient Greek, then we would have arguments in favour of borrowing; consequently we could provide the contemporary formation of this compounds type with an historical explanation.

However, various studies focusing on compounding in ancient Greek [START_REF] Chantraine | La formation des noms en grec ancien[END_REF], [START_REF] Smyth | A Greek Grammar for Colleges (libre accès à l[END_REF], and recent studies by A. Ralli 11 ) are divergent. Only Chantraine identifies an NV compounding pattern in ancient Greek. Conversely, Smyth and Ralli consider (for different reasons) that this formation did not exist in ancient Greek. This divergence in analysis show how weak is an hypothesis that assumes neoclassical compounds to be formed by means of borrowings of classical language rules. But do Latin and Greek have no influence at all on modern compound formation? Works devoted to neoclassical formation in French all tend towards an hybrid answer, be they in favour of the above-mentioned inheritance hypothesis or not. This is shown, for instance, by the way A. Darmesteter (1894:252-256) analyses Greekcomponent based neoclassical compounds. A. Darmesteter show how these compounds come from some (often improper) re-analysis of ancient Greek to French, in a section where he laments the massive intrusion of new neoclassical compounds in the 19 th century French lexicon, compounds that are often disrespectful to the rules of the langages they originate from. For instance, the creation of a neoclassical compound in 19 th century French, such as adénographie N (adenography = 'written or drwn record of the glands') would go through the prior invention of the equivalent word the French compound would have in ancient Greek, if this equivalent existed (e.g. °αδενογραφια = adénographia), from components (included verbs) which were attested in ancient Greek (e.g. αδην N = aden-(gland), γραφω V = -grapho (to write)),and in compliance (real or presumed) with ancient Greek morphological rules.

The impact of Greek (as well as that of Latin, which functioned as intermediary) is thus unquestionable, but the role played by the French linguistic system is equally crucial. We can see, indeed, that neoclassical compounds formation rules have been elaborated by French speakers having an indeniable knowledge of ancient languages. This assumption that Greek has been re-analysed in French has been shared by various linguists all the way through the 20th century. It is found, for instance, in Benveniste(1974:170) and later, in [START_REF] Bouffartigue | Les racines grecques[END_REF]. In further research, we still have to determine which has been the influence of the VN compounding pattern, which was already very productive at the end of the 18th century, upon the formation (or the reanalysis) of French NV neoclassical compounds.

Annexes

•

  5) (papier) tue-mouches A : kill-flies (paper) = 'flypaper' (porte) coupe-feu A :: break-fire = 'firebreak (door)' Compound semantic type :Most VN nouns denote artefacts: (6) ouvre-boîte N : open-tin = 'tin opener' coupe-papier N : cut-paper = 'paper knife' Others refer to (human or animal) animate entities, -either by their usual function or occupation (7) garde-barrière N : guard-gate = 'level-crossing keeper, gratte-papier N : scratch-paper = 'penpusher' -or by some characteristic,salient property or behaviour (8) trouble-fête N: disturb-party: 'killjoy'. (garcon) rabat-joie A :°reduce-joy (boy) = 'spoilsport' perce-oreille N: °pierce-ear ='earwig' (insect)

Fiammetta

  coupe-gorge N: cut-throat = 'cut-throat alley' garde-meuble N : mind-furniture = 'store house' lèche-vitrine N : lick-window = 'window-shopping' (jouer à) saute-mouton N : leap-sheep = 'leapfrog' (play)

  (29) ventriloque A ('ventriloquist') ; noctambule A ('late-night/night reveller') héliotrope A ('heliotrope') For stative verbs, Dowty role assignment criteria are not relevant. Rather, verbto-noun relations can be expressed by means of Talmy Figure / Ground notions (Talmy(2000) 10 ). So, when V denotes a spatial relation, then N refers: -mainly to the Figure: (ex 25), (30) mélanophore N ('melanophore') carbonifère A ('carboniferous') -or, sometimes, the Ground, (31) vasiducte N (= 'carrying vessel')

  Figure and the Ground ; incidentally, other roles like Manner, Cause, or Path, may intervene. The Figure is identified as the moving or localized object, viewes with respect to another object (the reference objet : Ground).

Verb+Noun compounding exists in each Romance language : on that matter, see[START_REF] Rainer | Compounding in Spanish[END_REF], Scalise (1992),[START_REF] Villalava | Compounding in Portuguese[END_REF]

As many authors have remarked (see for instance[START_REF] Iacobini | Composizione con elementi neoclassici[END_REF], or[START_REF] Corbin | French (Indo European: Romance)[END_REF]), component

Rather than « lexeme », we prefer to adopt the term of « base-lexeme ». According to authors, the former may include lexical units such as determiner, pronoun, preposition (cf.Aronoff & Fudeman (2004)), and these categories are irrelevant in Lexeme Formation.

From a lexicographic point of view, oculogyric qualifies muscles « that enables eyballs to turn ».

baladoire qualifies « undecent » dances: the adjective originates from the musical genre « ballad ».