H Bufferand
email: hugo.bufferand@cea.fr

J Balbin

S Baschetti

J Bucalossi

Ph G Ciraolo

R Ghendrih

N Mao

P Rivals

H Tamain

Yang

G Giorgiani

F Schwander

M Scotto D'abusco

E Serre

J Denis

Y Marandet

M Raghunathan

P Innocente

D Galassi

Implementation of multi-component Zhdanov closure in SOLEDGE3X

The multi-component fluid closure derived by Zhdanov [1] is implemented in the fluid code SOLEDGE3X-EIRENE to deal with arbitrary edge plasma composition. The closure assumes no distinction in between species such as light vs heavy species separation. The work of Zhdanov is rewritten in a matricial form in order to clearly link friction forces and heat fluxes to the different species velocities and temperature gradients.

Introduction

In order to interpret experiments and in the perspective of predicting edge plasma properties for future nuclear fusion devices, a dedicated effort is made to simulate plasma-wall interaction. Edge plasma collisionality being moderate to high, a fluid description of the plasma can be considered as a good approximation, at least to estimate engineering quantities of interest such as heat load on the plasma facing components.

More precisely, two-dimensional transport codes [START_REF] Bufferand | Numerical modelling for divertor design of west device with a focus on plasmawall interactions[END_REF][START_REF] Wiesen | The new solps-iter code package[END_REF][START_REF] Jaervinen | Interpretation of radiative divertor studies with impurity seeding in type-i elmy h-mode plasmas in jet-ilw using edge2d-eirene[END_REF] are used to provide in a few hours of computation an overall description of the edge plasma, the price of reduced computation time being a raw ad-hoc description of turbulent cross-field transport. Conversely, three-dimensional fluid codes [START_REF] Tamain | The tokam3x code for the edge turbulence fluid simularions of tokamak plasmas in versatile magnetic geometries[END_REF][START_REF] Dudson | Bout++: Recent and current developments[END_REF][START_REF] Stegmeir | Grillix: a 3d turbulence code based on the flux-coordinate independent approach[END_REF] can provide a self-consistent first principle description of the turbulent structures but are much more computational demanding. The two approaches are complementary and both rely on the same set of fluid equations describing mass, momentum and energy balance of the various plasma species.

The most common set of equations is the Braginskii's equations [START_REF] Braginskii | Transport processes in a plasma[END_REF] providing for the first time in the 1960's a collisional closure for the pressure tensor, friction forces and heat fluxes for a highly magnetized plasma. However, the collisional closure presented by Braginskii is derived for a simple electron-hydrogen plasma. In this contribution, we worked on the multi-species extension of Braginskii like equations proposed by Zhdanov [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF]. This multicomponent closure based on Grad's method has already been used in the communinity [START_REF] Bufferand | Three-dimensional modelling of edge multi-component plasma taking into account realistic wall geometry[END_REF][START_REF] Sytova | Impact of a new general form of friction and thermal forces on solps-iter modelling results[END_REF][START_REF] Rozhansky | Momentum balance for impurities in solps transport code[END_REF]. It does not make assumptions on impurity concentration or mass ratio in between the different species. It is thus of particular interest to simulate reactor plasmas where a mix of Deuterium, Tritium and Helium ions takes place, the species having quite similar mass and concentration.

In this contribution we apply Zhdanov closure to the code SOLEDGE3X-EIRENE that implements a fluid model to describe the plasma either in 2D to make transport simulations or in 3D to run turbulent or 3D-transport simulations. In a first part, we present the multi-species SOLEDGE3X-EIRENE equations. In particular, a special attention is paid to describe the multi-species closure terms. To do so, Zhdanov closure is rewritten in a matricial form in order to relate the parallel heat fluxes (q αZ ) and parallel friction forces (R αZ ) felt by one component of species α at charge state component Z, because of parallel velocity (w βζ ) and parallel temperature gradient (∇T βζ ) of a another component of species β at charge state ζ. This relation takes the form:

q αZ R αZ = M αZ,βζ • ∇T βζ w βζ (1)
The matrix M contains in particular parallel heat conductivities as well as coefficient for thermal and drag forces. Zhdanov closure is compared to Braginskii closure for a simple electron+hydrogen plasma. Also, in order to check the consistency of Zhdanov closure, a comparison is made between two simulations of a pure deuterium plasma: in the first one, deuterons are considered as a single species (standard treatment); in the second one the deuteron population is splitted into two different species. The total input power and particle input being the same, the two simulations should give exactly the same results if the coupling terms in between the two deuteron species are correct.

Finally, an example of application to a JET D-T fusion plasma is performed and reported in the last section.

Multi-component plasma SOLEDGE equations

The SOLEDGE3X has been developed in the past few years to address plasma-wall interaction in tokamak plasmas taking into account the realistic wall geometry as much as possible. It merges the codes SOLEDGE2D and TOKAM3X previously developed by the same team. Thanks to immersed boundary condition methods [START_REF] Paredes | A penalization technique to model plasma facing components in a tokamak with temperature variations[END_REF], the code can simulate the plasma up to the first wall while using a magnetic field aligned grid. The physical model has been improved over the year to include neutrals via coupling the code to EIRENE [START_REF] Bufferand | Near wall plasma simulation using penalization technique with the transport code soledge2d-eirene[END_REF], drifts [START_REF] Bufferand | Implementation of drift velocities and currents in soledge2d-eirene[END_REF] and impurities [START_REF] Bufferand | Three-dimensional modelling of edge multi-component plasma taking into account realistic wall geometry[END_REF] since the code can now handle an arbitrary number of species. We denote by the subscript αZ the ion species of charged number Z and of chemical element α. The subscript e is used for electrons.

Mass balance

For all ion species, the particle balance is given by:

∂ t n αZ + ∇ • (n αZ v αZ ) = S n αZ (2)
where the velocity vector v αZ is decomposed as

v αZ = v ,αZ b + v E + v αZ + v p αZ + v a αZ (3)
where

• v ,αZ is the parallel velocity along the magnetic field unit vector b = B/B

• v E = E× B B 2 is the E cross B drift velocity • v αZ = B× ∇p αZ
Zen αZ B 2 is the diamagnetic drift velocity

• v p αZ = -1 n αZ ∂ t ω αZ + ∇ • v (0)
αZ ⊗ ω αZ is the polarization drift with

ω αZ = m α ZeB 2 n αZ ∇ ⊥ φ + 1 Ze ∇ ⊥ p αZ (4)
φ denoting the electric potential and where the first order velocity v

(0)
αZ is the species velocity v αZ without the polarization drift.

• v a αZ = -D αZ ∇ ⊥ n αZ n αZ
is the anomalous perpendicular velocity that is used to emulate the turbulent transport when the code is used as a transport code. The diffusion coefficient D αZ takes value from empirical considerations or reduced turbulence models [START_REF] Baschetti | Optimization of turbulence reduced model free parameters based on l-mode experiments and 2d transport simulations[END_REF].

The volumic source term S n αZ describes particle sources and sinks due to ionization and recombination processes. It takes the form

S n αZ = -n e n αZ ( σv iz,αZ + σv rec,αZ ) + n e n αZ-1 σv iz,αZ-1 + n e n αZ+1 σv rec,αZ+1 (5)
For ionization and recombination processes involving neutrals, the source term is computed by EIRENE.

No transport equation is solved for electron density. It is computed from plasma quasineutrality assumption:

n e = α,Z Zn αZ (6)

Parallel momentum balance

For each ion species, the following parallel momentum balance is solved:

∂ t m α n αZ v ,αZ + ∇ • m α n αZ v ,αZ v αZ = -∇ p αZ + Zen αZ E + ∇ • m α n αZ ν αZ ∇ ⊥ v ,αZ + R αZ + S v αZ + Σ v αZ (7)
where ν a αZ is the anomalous viscosity describing turbulent cross-field transport of parallel momentum in shear layers (set to classical value when the code is used in 3D to solve turbulence). The term R αZ is the parallel friction force resulting from elastic collisions in between the different species. It can be decomposed as R αZ = βζ R αZ,βζ where one exhibits the different contributions due to binary interactions between ion species (α, Z) and ion species (β, ζ). The elastic collision processes conserve parallel momentum giving R αZ,βζ + R βζ,αZ = 0. Following Zhdanov closure, one can evaluate the friction forces from different species temperature gradients and parallel velocities. The detail of the closure is given in section 3. The volumic source term S v αZ accounts for momentum exchange in between species in the ionization-recombination processes. It is given by: S v αZ m α = -n e n αZ v ,αZ ( σv iz,αZ + σv rec,αZ )

+ n e n αZ-1 v ,αZ-1 σv iz,αZ-1 + n e n αZ+1 v ,αZ+1 σv rec,αZ+1

Finally, the source term Σ v αZ accounts for other momentum exchange processes such as charge exchange.

Parallel electron velocity is obtained from parallel current definition j = -en e v ,e + αZ Zn αZ v ,αZ where the parallel current is obtained by solving current balance (see section 2.4. When the charge balance is not solved, one assumes j = 0 (ambipolarity) hence v ,e = n -1 e αZ Zn αZ v ,αZ . We neglect electrons inertia which provides an expression for the parallel electric field from electron parallel momentum equation:

0 = -∇ p e -en e E + R e or E = - ∇ p e en e + R e en e (9)

Energy balance

For all species (including electrons), the "total" energy balance is solved where the "total" energy is defined by

E αZ = 3 2 k B n αZ T αZ + 1 2 m α n αZ v 2 ,αZ (10)
This quantity is not in fact the total energy since one could include in addition perpendicular kinetic energy and electrostatic potential energy. The consequence of not considering these terms on the conservative form of the equation will be discussed in the next paragraph. Moreover, in practice electron kinetic energy is neglected over electron internal energy. The "total" energy equation is given by

∂ t E αZ + ∇ • E αZ v αZ + p αZ v ,αZ b + q αZ b = Zen αZ v ,αZ E + ∇ • 1 2 m α n αZ ν αZ ∇ ⊥ v ,αZ + ∇ • k B n αZ χ αZ ∇ ⊥ T αZ -p αZ ∇ • v ⊥,αZ + v ,αZ R αZ + Q αZ + S E αZ + Σ E αZ ( 11
)
where v ⊥,αZ gathers all drifts velocities. The parallel heat flux is denoted q αZ and also find an expression from Zhdanov closure that generalizes Spitzer-Härm heat conduction closure. In the RHS, one introduces an anomalous perpendicular heat conductivity χ αZ that takes value from empirical considerations (set to classical value when the code is used in 3D to solve turbulence). The term Q αZ describes energy exchange in between species due to collisions and is also expressed in the next section about collisional closure. The volumic source term S E αZ denotes energy source ans sinks due to ionization/recombination processes and takes the form:

S E αZ = -n e E αZ ( σv iz,αZ + σv rec,αZ ) + n e E αZ-1 σv iz,αZ-1 + n e E αZ+1 σv rec,αZ+1 (12)
Finally, the source term Σ E αZ denotes other volumic energy sources such as radiation losses (for electrons), charge exchange energy losses or external heating.

Charge balance

Assuming quasineutrality, the charge balance reduces to

∇ • j = 0 ( 13
)
The current is decomposed as

j = j b + j + j p (14)
where

• j denotes the parallel current given by the generalized Ohm's law:

j = σ -∇ φ - 1 en e ∇ p e -R T e ( 15
)
σ denoting the parallel electric conductivity and R T e the thermal contribution of the friction force. The parallel conductivity is a collisional term that requires a closure. It is not discussed in this contribution.

• j is the diamagnetic current given by j = e,α,Z Zen αZ v αZ • j p is the polarization current given by j p = α,Z Zen αZ v p αZ Given the expression of the polarization velocity, one can show [START_REF] Bufferand | Implementation of drift velocities and currents in soledge2d-eirene[END_REF] that the current balance can be rewritten as a vorticity equation where the vorticity Ω is defined by Ω = ∇ • α,Z Ze ω αZ where ω αZ is given by Equation 4. The vorticity equation takes the form:

∂ t Ω + ∇ • F Ω = ∇ • j b + j + ζ ∇ ⊥ Ω ( 16
)
where the flux of vorticity F Ω is given by

F Ω = α,Z Ze ∇ • ( v αZ ⊗ ω αZ ) (17)
A diffusion term is added in the vorticity Equation 16to take into account anomalous current and for stability reason. The diffusivity ζ takes either anomalous or classical values depending whether the code is used as a transport code or as a turbulence code.

Zhdanov closure

The remarkable work of Zhdanov [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF] consisted in extending the hydrogenic magnetized plasma collisional closure to multi-component plasmas in the linear transport regime under weak-field conditions. It provides in particular expressions for the above mentioned friction forces R αZ , heat fluxes q αZ , energy coupling terms Q αZ and plasma conductivity σ . It does not make assumptions of trace impurities and all species are treated on the same footing. However, it does not provide explicit formulation for each of these terms. The species being intimately coupled between each other, the closure takes the form of a linear system of equations linking the different species plasma fields (in particular diffusion velocities and temperature gradients) to the quantities we want to express, that is in particular friction forces and heat fluxes.

Friction forces and heat fluxes

First of all, most of the Zhdanov closure algebra is performed in the frame of the local center of the mass, the velocity of which is given by

v = αZ m α n αZ v ,αZ αZ m α n αZ (18)
In this frame, the peculiar velocity of the component is denoted w αZ = v ,αZ -v . The purpose of this section is to express the friction force R αZ and the heat flux q αZ ‡ as a linear combination of the different species parallel velocities and parallel temperature gradients as already expressed by the matrix product presented in the introduction in Equation 49. Several steps are required to express the matrix M αZ,βζ . First, physical ‡ Careful: In Zhdanov book [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF], the heat flux for each species is denoted h αZ . The notation q αZ in [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF] refers to the heat flux in the centre of mass frame (see Equation 2.2.4 in [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF]), that is 5 2 p αZ v + q αZ = 5 2 p αZ v ,αZ + h αZ , hence h αZ = q αZ -5 2 p αZ w αZ (Equation 4.2.4 in [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF]). In this paper, we do not use the heat flux in the centre of mass frame and thus use the more common notation q αZ for the heat flux instead of h αZ .

quantites are defined for each chemical element, averaging over the different ionization states:

• density: nα = Z n αZ • effective k th power of charge (∀k ∈ N): Z k α = n-1 α Z n αZ Z k • velocity: wα = Z I αZ w αZ where I αZ = n αZ Z 2 nαZ 2 α • temperature gradient: ∇T α = (n α Z 1 α ) -1 Z Zn αZ ∇T αZ • pressure : pα = Z k B n αZ T αZ
In particular, expressions for velocity and temperature can be re-written in the matrix form:

∇T α wα = M 1 • ∇T βZ w βZ = M 11 0 0 M 12 • ∇T βZ w βZ (19)
where M 11 and M 12 are N chem ×N spec rectangular matrices (N chem denoting the number of chemical elements and N spec the number of species considering all ionization states).

More precisely:

M 11 α,βZ = n βZ Z nβ δ αβ (20)
and

M 12 α,βZ = I βZ δ αβ . (21)
From the averaged velocities and parallel gradients are computed averaged heat fluxes and averaged fifth order rank-1 fluid moments denoted rα from which the friction forces will be computed later. The following set of reduced equations is the keystone of Zhdanov closure:

5 2 nα k B ∇T α = β 5 2 µ αβ m α Ḡ(2) αβ ( wα -wβ ) + Ḡ(5) αβ qα p α + Ḡ(6) αβ qβ p β + µ αβ k B T Ḡ(9) αβ rα p α + Ḡ(10) αβ rβ p β (22) 0 = β 35 2 µ αβ m α 2 Ḡ(8) αβ ( wα -wβ ) + 7 µ αβ m α Ḡ(9) αβ qα p α + Ḡ(10) αβ qβ p β + m α k B T Ḡ(11) αβ rα p α + m β k B T Ḡ(12) αβ rβ p β (23)
It can be re-written in the matrix form as:

M 2 • ∇T β wβ = M 3 • qβ rβ (24)
where M 2 is a square 2N chem × 2N chem matrix where the coefficients are given for all

[α, β] ∈ [1, N chem ] × [1, N chem ]: • M 2 α,β = 5 • M 2 α+N chem ,β = 0 • M 2 α,β+N chem = -5 2 β µ αβ mα Ḡ(2) αβ δ αβ + 5 2 µ αβ mα Ḡ(2) αβ • M 2 α+N chem ,β+N chem = -35 2 β µ αβ mα 2 Ḡ(8) αβ δ αβ + 35 2 µ αβ mα 2 Ḡ( 8
)
αβ

The matrix M 3 is also a square 2N chem × 2N chem matrix where the coefficients are given for all

[α, β] ∈ [1, N chem ] × [1, N chem ]: • M 3 α,β = β Ḡ(5) αβ pα δ αβ + Ḡ(6) αβ p β • M 3 α+N chem ,β = β 7 µ αβ mα Ḡ(9) αβ pα δ αβ + 7 µ αβ mα Ḡ(10) αβ p β • M 3 α,β+N chem = β µ αβ k B T Ḡ(9) αβ pα δ αβ + µ αβ k B T Ḡ(10) αβ p β • M 3 α+N chem ,β+N chem = β mα k B T Ḡ(11) αβ pα δ αβ + m β k B T Ḡ(12) αβ p β
A few notations are introduced above. First, the plasma common temperature T is defined as

T = αZ n αZ T αZ αZ n αZ (25)
Secondly, the reduced mass µ αβ = m α m β /(m α + m β ). Then, the coefficients Ḡ(n) αβ , combinations of collision times and species masses, are given in Appendix A.

The heat flux q αZ for each ionization state can be computed from qα using Equation 26:

q αZ p αZ - qα pα = nα τ α τ -1 αα c (5) α Z 2 α Z 2 ∇T αZ -∇T α + c (6) α (w αZ -wα ) (26)
The same kind of equation provides an expression for the fourth order moment r αZ : 8) α + 7S (9) α c (6) α

r αZ p αZ - rα pα = k B T m α S (11) α -7S (9) α nα τ α τ -1 αα c (5) α Z 2 α Z 2 ∇T αZ -∇T α -S (
(w αZ -wα ) ( 27
)
where the collision times, the coefficients S

(n)

α and c

(n) α are given in Appendix A. Once again, the above two equations can be re-written in the matrix form

q αZ r αZ = M 4 • qβ rβ + M 5 • ∇T β wβ + M 6 • ∇T βζ w βζ (28)
where

• M 4 is a 2N spec × 2N chem rectangular matrix given by M 4 = M 41 0 0 M 41 (29)
where

∀[αZ, β] ∈ N spec × N chem one has M 41 αZ,β = p αZ /p β δ αβ .
• M 5 is a 2N spec × 2N chem rectangular matrix given by

M 5 = M 51 M 52 M 53 M 54 (30)
where

∀[αZ, β] ∈ N spec × N chem one has M 51 αZ,β = -p αZ nα τ α τ -1 αα c (5) α δ αβ (31)
M 52 αZ,β = -p αZ c (6) α δ αβ (32)

M 53 αZ,β = k B T m α S (11) α
7S (9) α p αZ nα τ α τ -1 αα c (5) α δ αβ (33)

M 54 αZ,β = k B T m α S (11) α
p αZ S (8) α + 7S (9) α c (6) α δ αβ (34)

• M 6 is a 2N spec × 2N spec square matrix given by

M 6 = M 61 M 62 M 63 M 64 ( 35
)
where

∀[αZ, βζ] ∈ N spec × N spec one has M 61 αZ,βζ = p αZ nα τ α τ -1 αα c (5) α Z 2 α Z 2 δ αZβζ (36)
M 62 αZ,βζ = p αZ c (6) α δ αZβζ (37)

M 63 αZ,βζ = - k B T m α S (11) α
7S (9) α p αZ nα τ α τ -1 αα c (5) α

Z 2 α Z 2 δ αZβζ (38) M 64 αZ,βζ = - k B T m α S (11) α
p αZ S (8) α + 7S (9) α c (6) α δ αZβζ (39)

At last, the friction force R αZ is given by Equation (8.1.3) in [START_REF] Zhdanov | Transport Processes in Multicomponent Plasma[END_REF]:

R αZ = β,ζ Ḡ(1) αZβζ (w αZ -w βζ ) + µ αβ k B T Ḡ(2) αZβζ q αZ m α n αZ - q βζ m β n βζ + µ αβ k B T 2 Ḡ(8) αZβζ r αZ m α n αZ - r βζ m β n βζ (40)
giving in the matrix formulation:

q αZ R αZ = M 7 • ∇T βζ w βζ + M 8 • q βζ r βζ (41)
where

• M 7 is a 2N spec × 2N spec square matrix given by M 7 = 0 0 0 M 71 (42) with ∀[αZ, βζ] ∈ N spec × N spec : M 71 αZ,βζ = - Ḡ(1) αZβζ + β,ζ Ḡ(1) αZβζ δ αZβζ (43)
• M 8 is a 2N spec × 2N spec square matrix given by

M 8 = I 0 M 81 M 82 (44) with ∀[αZ, βζ] ∈ N spec × N spec : M 81 αZ,βζ = - µ αβ k B T Ḡ(2) αZβζ m β n βζ + β,ζ µ αβ k B T Ḡ(2) αZβζ m α n αZ δ αZβζ (45) M 82 αZ,βζ = - µ αβ k B T 2 Ḡ(8) αZβζ m β n βζ + β,ζ µ αβ k B T 2 Ḡ(8) αZβζ m α n αZ δ αZβζ (46)
Once all the above matrices calculated, that is the eight matrices M 1 . . . M 8, the final matrix linking heat fluxes and friction forces to temperature gradients and velocities can be calculated. From Equation 24, one finds

qβ rβ = M 3 -1 • M 2 • ∇T β wβ ( 47
)
Reporting in Equation 28gives

q αZ r αZ = M 4 • M 3 -1 • M 2 + M 5 • ∇T β wβ + M 6 • ∇T βζ w βζ (48)
Combining with Equation 19 and 41 finally gives the desired expression

q αZ R αZ = M αZ,βζ • ∇T βζ w βζ ( 49
)
where the matrix M is given by

M = M 7 + M 8 • M 4 • M 3 -1 • M 2 + M 5 • M 1 + M 6 (50)
Once each individual matrix computed, the final matrix M is quite straightforward to obtain by matrix product. The only matrix inversion required implies the matrix M 3 which is a relatively small 2N chem × 2N chem matrix. The direct numerical inversion can be done easily with a linear solver such as LAPACK. Most of the numerical cost in the matrix approach relies in fact in computing the different collision times and Zhdanov's collision coefficients

Ḡ(n) αβ , S (n)
α ...

Boundary conditions

SOLEDGE3X being based on a finite volume numerical scheme, setting the boundary conditions consists in setting particle, momentum and energy fluxes on faces at the boundary of the domain, in particular on the wall.

Particle fluxes on the wall

Particle fluxes on the wall are given by the Bohm-Chodura boundary condition: the particle flux on the wall is set to be greater or equal to the particle flux that would result from a plasma flowing at sound speed in the direction parallel to the magnetic field. This condition guarantees in particular that whatever the value and the orientation of the drift velocities, the flux on the wall will always be a net outflux of particle. This boundary condition can be summarized as

| φ n • n wall | ≥ |nc s b • n wall | (51)
where φ n denotes the particle flux. In the present implementation, every ion species follows its own sound speed on the wall. For a collisional multi-component plasma, other expression for the boundary condition exist in the litterature, in particular using a common sound speed followed by all ion species on the wall [START_REF] Tskhakaya | One-dimensional plasma sheath model in front of the divertor plates[END_REF]. These boundary conditions will be considered for implementation in the future. In addition, one forces ∇ n = 0 to set density in ghost cells on the wall.

Momentum fluxes on the wall

On the flux of momentum, one sets Neumann boundary conditions on parallel momentum ∇ v = 0. The flux is not forced, one let the free streaming advected flux on the wall.

Energy fluxes on the wall

The energy flux is set according to simplified sheath properties that link energy and particle fluxes on the wall. The sheath transmission factor γ estimates the average energy of the particles crossing the sheath electrostatic barrier. The total energy flux on the wall is thus given by:

φ Eα,se = γ α T α + 1 2 m α v 2 ,α φ nα,se (52)
A typical value for electron sheath tranmission factor is γ e = 4.5 providing that only the fast electron in the tail of the distribution can cross the sheath barrier. For ions, a typical value for the sheath transmission factor is γ i = 2.5 which is reminiscent from the free streaming advected flux, the sheath letting all ions go through. Other values can be found in the litterature [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF] and improved sheath models should be implemented in the future.

Current on the wall

In a similar fashion as the energy flux, the current on the wall can be set proportionaly to the particle flux. The following expression is used:

j wall = j sat 1 -exp 1 - φ ΛT e (53)
where the saturation current is given by ion saturation current j sat = α,Z Ze φ nαZ .

Verification of the implementation

Verification of the operators

SOLEDGE3X is based on a finite-volume, flux surface aligned, multi-domain algorithm [START_REF] Bufferand | Development of a fluid code for tokamak edge plasma simulation[END_REF]. In order to verify the implementation of the different operators, the method of manufactured solution (MMS) is used [START_REF] Dudson | Verification of bout++ by the method of manufactured solutions[END_REF]. It consists in comparing numerical solution with an analytical solution. The method can be summarized as follow. Let us consider an operator O to test (for instance advection, diffusion...). We consider the equation:

∂ t X + O(X) = S (54)
A priori, the analytical steady-state solution X associate to the arbitrary source S of the system is not obvious since it requires inverting the operator X = O -1 (S). The MMS propose simply to choose an analytical solution XMMS and compute to associated source by a direct analytical computation S M M S = O analyt. ( XMMS ). One thus solves numerically the system:

∂ t X + O(X) = S M M S ≡ O analyt. ( XMMS ) (55)
which by construction admit X = XMMS as a steady-state solution. In practice, we choose a circular cross-section toroidal domain. We impose a steady-state analytical solution for density, temperature and velocity fields based on trigonometric fonctions. One initiates the plasma to the analytical steady-state solution and perform one time step δt. The numerical departure from steady-state is measured and characterize the error ε = X -XMMS /δt. The numerical test is performed for different grid resolutions. The error is expected to decrease accordingly to the order of the numerical scheme. The MMS can be used with a regular grid (see Figure 1) or irregular grid to test the ability of the code to deal with non-constant spatial discretization. • Parallel advection (Figure 2): One finds a spatial discretization order between 2 and 3.

• Parallel diffusion (Figure 3): One finds a spatial discretization order of 2.

• Perpendicular diffusion (Figure 4): One finds a spatial discretization order of 2.

Verification of the closure

Veryfing Zhdanov closure is not as simple as the verification of the operators described in the previous paragraph. Indeed, a method of manufactured solution appear quite cumbersome since Zhdanov closure does not provide an explicit formulation for collisional terms. We propose here a "weak verification" of Zhdanov closure implementation (and behavior). In order to verify the multi-species collisional closure, a simple numerical test case is proposed. The simulation set-up consists in simulating a pure Deuterium plasma in a circular limited configuration following three approaches (in the following, if not explicitly written, all quantities are given in the MKS convention): (i) As a reference case, one simulates a two fluids plasma of deuterons and electrons where standard Braginskii closure is used, that is:

• For the friction force:

R e = -R i = -m e n e v ,e -v ,i ν mom ei -0.71n e ∇ T e (56)
where

ν mom ei = 1.45 × 10 -12 ln Λn e T -3/2 e [s -1 ] ( 57
)
where n e is expressed in m -3 and T e in eV . • For the heat fluxes: q e = -κ 0 e T

5/2

e ∇ T e and q i = -κ 0 D + T

κ 0 D + = 1249 m 1/2 i ln Λ ≈ 60 Wm -1 eV -7/2 (59)
where m i is expressed in amu and where the Coulomb logarithm is given by the expression A.18 in the appendice. The expressions above are taken from [START_REF] Stangeby | The Plasma Boundary of Magnetic Fusion Devices[END_REF].

(ii) A two fluids plasma of deuterons and electrons is simulated with Zhdanov closure. The purpose here is to see if the collisions between ions and electrons are well treated by Zhdanov closure (iii) A three fluids plasma of deuterons+deuterons+electrons where the deuterium ion population is splitted into two different fluids. The purpose is to test Zhdanov collisional closure between the two populations of deuterium.

The three cases are simulated on the same circular mesh plotted in Figure 5. The simulation domain extends from the core flux surface located at r/a = 0.8 up to the wall. On the core boundary, Dirichlet boundary conditions are used for temperature (namely T e = T D + = 50 eV ) as well as for density

n e = n D + = 2 • 10 19 m -3 .
For the third case where deuteron population is splitted into two fluids, one imposes

n D + = 1.2 • 10 19 m -3
for the first one, n D + = 0.8 • 10 19 m -3 for the second one in order to have the same total deuterium density. Homogeneous anomalous diffusivities are considered:

D D + = 1 m 2 s -1 , ν D + = 1 m 2 s -1 and χ e = χ D + = 2 m 2 s -1 .
No parallel flux limiters are considered. EIRENE is not used in this set of simulations so sources due to neutrals are set to zero (no recycling). Figure 5 shows radial profiles of density and temperature for ions and electrons at the outboard midplane. A very good agreement is found between the three simulations: a maximum relative error of 1.4% is found comparing the different density and temperature profiles (the error being defined

as ε = X 1 -X 2 X 1 ∞
). An almost perfect match between electron density profiles show the good overall agreement while ion density profiles show the fractionning of deuterium density over the two fluids for the third case (iii). Concerning temperature, the four deuterium temperature profiles collapses with each other. In particular, when deuterium is splitted in two, the two fluids are found to have the same temperature (as could be expected for any intensive physical quantity).

In the case where deuterium is splitted in two, it is interesting to notice that the heat flux of a type of deuterium is given by a combination of the temperature gradients of the two types of deuterium. In that sense, we notice that the matrix linking heat fluxes and temperature gradient is not diagonal (see Equation B.4 in Appendix Appendix A).

Here is a example of decomposition:

q D + (1) = -30T 5/2 D + (1) ∇ T D + (1) (60)
-4.34T

5/2 D + (1) ∇ T D + (2) q D + (2) = -18.5T 5/2 D + (2) ∇ T D + (2) (61) -4.34T 5/2 D + (2) ∇ T D + (1)
Summing all contributions, one recovers the typical value κ 0 = 57.2 for deuterium when treated as a single species (see Equation B.2 in Appendix Appendix A). If this decomposition seems quite artificial for this splitting of deuterium in two, it would be of more interest when considering a mix of hydrogen isotopes, for instance a D-T mix. In that case, Zhdanov closure gives the coefficients linking the deuterium and tritium heat fluxes with deuterium temperature gradient and tritium temperature gradient contributions. If the two species show different temperature profiles, the heat flux computed by Zhdanov closure then differs from a simple Fourier law.

Example of applications: JET D-T plasma

Simulation setting

In this section, a typical application of multi-component plasma with hydrogen isotope mixture is presented: a JET D-T plasma. Moreover, we include also Nitrogen which is injected in the divertor to radiate in the edge and reduce the heat load at the strike points. As a reference magnetic equilibrium, we used JET shot number 97395 which is a high current (I p = 3 M A), high power (P add = P N BI + P ICRH = 28 + 6 = 34 M W ) H-mode plasma scenario developped in 2017 in the framework of Eurofusion T17-07 experimental task "DT scenario extrapolation", experiment M15-01 "Baseline scenario". The SOLEDGE3X-EIRENE mesh of the magnetic equilibrium is plotted on Figure 6 where both SOLEDGE (quadrangles) and EIRENE (triangles) grid are shown. The radial extension in the core stops at r/a = 0.8.

The following settings are used for the simulation. We prescribe midplane profiles for Deuterium density, Deuterium temperature and electron temperature in the midplane. The code automatically adjust cross-field transport coefficients to match these input profiles which for our test case come from HRTS data from JET #97395 where half electron density profile has been taken as a proxy for deuterium density profile, considering in our case a 50/50 D-T mix. Cross-field transport coefficients are assumed to be homogeneous on the flux surface (no poloidal dependence). Dirichlet boundary conditions are set at the core boundary to force "experimental" values for Deuterium and Tritium density as well as for electron temperature. For nitrogen, one imposes zero flux. Fueling in the edge is performed by Deuterium and Tritium gas puff near the inner strike point on high field side (HFS) vertical target (GIM 12). The gas puff rate is set to 10 22 e -• s -1 for Deuterium and Tritium. Finally, we inject Nitrogen in the HFS private flux region (GIM 11) with a fixed puff rate ϕ N = 1.4 × 10 22 e -• s -1 .

The location of gas puffs is plotted on Figure 9. The chamber wall is described by two different materials: a tungsten area in the divertor with a recycling coefficient set to 1 for Deuterium, Tritium, and Nitrogen, a Beryllium area on the main chamber wall with a recycling coefficient set to 0.99 for Deuterium, Tritium and Nitrogen. Pumping surfaces are located in the subdivertor at the location of cryo pumps; their recycling coefficient is set to 0.5 for Deuterium, Tritium and Nitrogen. The wall configuration is plotted on Figure 6.

Figure 7 shows the radial dependence of transport coefficients on the low field side. The well in diffusivity profiles near the separatrix implements the transport barrier characteristic of H-mode plasmas. One assumes all ion species share the same value for D and χ. One also assumes ν = D. Drifts are switch-off. Figure 8 shows a comparison between simulation results and "experimental measurements" for main ions and electron density and temperatures in the edge plasma for our reference JET shot #97395. The experimental profile for electron density was assumed to be shared between Deuterium and Tritium electrons (neglecting electrons from Nitrogen), hence the profile used as input for Deuterium and Tritium density. The code properly matches the input profile adjusting the profile for D that is why n D +n T profiles is in good agreeement with HRTS measurements. However, the n e profile computed by the code is higher to the HRTS data due to the extra electrons from Nitrogen. The Z effective around the separatrix is found to be between around 3, see figure 9. The temperature profiles match more or less the input profiles and remain in the experimental range. Figure 10 shows electron density and temperature in the divertor. The plasma is partially detached on the high field side.

Multi-component plasma analysis

Despite identical fueling properties for Deuterium and Tritium (same rate at same location), identical recycling coefficient and identical radial transport coefficients, a significant discrepency can be found between the hydrogen isotopes radial profile in the divertor, see Figure 11. More precisely, a lower abundance of Tritium with respect to Deuterium is observed near the puff but a higher concentration of Tritium in the main SOL. One can expect a higher Scrape-off layer width λ n for tritium due to a lower sound speed for this isotop. Hence, the behavior observed in the present simulation could be very different if one uses a more complex boundary condition for the iosotop mixture (for example a common sound speed for the mixture). The extra abundance of Deuterium in the divertor is still under investigation. It must be noticed that the same albedo is used on the pump for Deuterium and Tritium. A better model could be implemented to match the different pumping efficiency between the two isotops. The SOLEDGE3X-EIRENE code also returns radiation maps from the different species. Several synthetic diagnostics are being implemented to help future comparison with experiments such as bolometry or spectroscopy [START_REF] Ciraolo | First modeling of strongly radiating west plasmas with soledge-eirene[END_REF].

Conclusions

Zhdanov closure has been rewritten in a matrix form and implemented in the SOLEDGE3X plasma code. It enables simulating complex multi-species plasma without resorting to the trace impurity assumption. The closure has been tested on simple plasma geometries and proves to be able to properly split a deuterium population in two. The collisional closure being constistent, the deuterium splitted plasma behaves the same way as a single deuterium species. The test was performed in 0D and in a circular 2D plasma. Finally, an example of application to a complex Deuterium+Tritium plasma on JET including Nitrogen seeding illustrates the typical application of such a closure to simulate multi-component plasma.

m
κ αβ = m α m β (m α + m β ) 2 .
(A.20) Notice the differences in the coefficients for Ḡ [START_REF] Bufferand | Implementation of drift velocities and currents in soledge2d-eirene[END_REF] αZβζ and Ḡ( 16) αZβζ , also mentioned in Ref. [START_REF] Raghunathan | Generalized collisional fluid theory for multi-component, multi-temperature plasma using the linearized Boltzmann collision operator for scrape-off layer/edge applications[END_REF].

Table B1. Properties of the two plasmas considered for comparison. The two plasmas are virtually identical.

For the first case, the matrix linking heat fluxes, friction forces with temperature gradients and velocities takes the numerical values given by Equations B.1 and B.

Figure 1 .

 1 Figure 1. Description of regular and irregular grid used for meshing the circular cross-section of the domain used for the MMS

Figure 2 .

 2 Figure 2. Relative error as a function of grid resolution for advection operators. Left: regular grid, Right: irregular grid.

Figure 3 .

 3 Figure 3. Relative error as a function of grid resolution for parallel diffusion operators. Left: regular grid, Right: irregular grid.

Figure 4 .

 4 Figure 4. Relative error as a function of grid resolution for perpendicular diffusion operators. Left: regular grid, Right: irregular grid.

Figure 5 .

 5 Figure 5. Inboard mid-plane profiles of electron and ion density and temperature. The three simulation setups are plotted with different colors.

Figure 6 .

 6 Figure 6. SOLEDGE3X-EIRENE mesh based on JET #97395 equilibrium at t = 50 s. Left: SOLEDGE3X mesh based on quadrangles grid extended to the wall. Right: EIRENE grid based on triangles. Wall types are indentified with different colors (green for Be, pink for W and dark blue for pumping surfaces, light blue for subdivertor wall).

Figure 7 .

 7 Figure 7. Transport coefficients used in JET D-T simulation. radial dependency at the LFS midplane.

Figure 8 .

 8 Figure 8. Comparison between simulated electron density and temperature at the LFS midplane with respect to experimental measurement in JET #97395 (High resolution Thomson Scattering).

Figure 9 .

 9 Figure 9. Z effective in the D+T+N H-mode simulation

Figure 10 .

 10 Figure 10. Left: electron density, right: electron temperature in the divertor.

Figure 11 .

 11 Figure 11. Relative difference between deuterium and tritium density

Figure 12 .

 12 Figure 12. Radial profiles of plasma ion species at the outer midplane.

Figure 13 .

 13 Figure 13. 2D maps of radiation for the different species.

4 )

 4 51m e n e ν e 0.51m e n e ν e 0.71en e 0 0.51m e n e ν e -0.51m e n e ν e 33en D2 T D1 2.83en D2 T D2 -0.71en e 0 0 -0.51m e n e ν e 0.31m e n e ν e 0.2m e n e ν e 0.43en e -0.13en e 0.13en e 0.31m e n e ν e -9.2m e n e ν e 8.9m e n e ν e 0.28en e 0.13en e -0.13en e 0.21m e n e ν e 8.9m e n e ν e -9.1m e n e ν eIn order to compare the two cases, if the plasma where deuterons are splitted into two populations behave the same as the single deuteron case, one has for all intensive quantities:∇ ,D = ∇ ,D1 = ∇ ,D2 (B.5) v ,D = v ,D1 = v ,D2(B.6)and for all extensive quantities: q ,D = q ,D1 + q ,D2 (B.7)R ,D = R ,D1 + R ,D2(

  -3/2 n αZ n βζ e 4 Z 2 ζ 2 µ B T n e e 2 (1 + Z eff )

	Ḡ(14) αZβζ = -	24 35	λ αZβζ			(A.14)
	Ḡ(15) αZβζ = -		51 35			m β m α		2	+	37 7	m β m α	+	22 5	+ 4	m α m β	κ αβ λ αZβζ	(A.15)
	Ḡ(16) αZβζ =	24 7	m β m α	κ αβ λ αZβζ	(A.16)
	where
	λ αZβζ =	1 3	(2π) 1/2 αβ	ln Λ αβ (k B T ) 3/2 ε 2 0	(A.17)
	the Coulomb coefficient being given by
	Λ αβ =	12πε 3/2 0 kT Z eff e 2					1/2	(A.18)
	Also
	µ αβ =	m α m β m α + m β						(A.19)
												β m α	λ αZβζ	(A.4)
	Ḡ(4) αZβζ =	4 5	λ αZβζ						(A.5)
	Ḡ(5) αZβζ = -	13 10	m β m α	+	8 5	+ 3	m α m β	κ αβ λ αZβζ	(A.6)
	Ḡ(6) αZβζ =	27 10	κ αβ λ αZβζ		(A.7)
	Ḡ(8) αZβζ = -	3 14	λ αZβζ			(A.8)
	Ḡ(9) αZβζ =	3 5		23 28	m β m α	+	8 7	+ 3	m α m β	κ αβ λ αZβζ	(A.9)
	Ḡ(10) αZβζ = -	45 28	κ αβ λ αZβζ	(A.10)
	Ḡ(11) αZβζ = -	433 280		m β m α	2	+	136 35	m β m α	+	459 35
			+	32 5		m α m β	+ 5		m α m β	2	κ 2 αβ λ αZβζ	(A.11)
	Ḡ(12) αZβζ =	75 8	κ 2 αβ λ αZβζ		(A.12)
	Ḡ(13) αZβζ =		18 35	m β m α	+	6 5		λ αZβζ	(A.13)

k

Acknowledgements

This work was granted access to the HPC resources of CINES, under the allocation 2021-A0100510482 made by GENCI. Centre de calcul intensif d'Aix-Marseille is acknowledged for granting access to its high performance computing resources. This work was supported by the EoCoE-II project, grant agreement 824158, funded within the EUs H2020 program. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission

Appendix A. Collision coefficients

Appendix A.1.

Ḡ(n) αβ coefficients

The coefficients Ḡ(n) αβ used in Zhdanov system of Equations 22 and 23 are given by: Ḡ

. Collision times

The collision time between species (α, Z) and (β, ζ) is given by:

where

The collision time averaged over ionization states:

At last, the averaged collision time of species α:

The coefficients denoted S

α S

where D α = S (5) α S (11) α -7 S (9) α 2 (A.33)

Appendix B. Comparison between Deuterium+electrons and Deuterium+Deuterium+electrons plasma

We report here the numerical value found applying Zhdanov closure to either a simple deuterons + electrons plasma or the same plasma where the deuteron population is splitted into two "classes" to make a deuterons (type 1) +deuterons (type 2) + electrons plasma. The properties of the populations are summarized in table B1.

case

If Zhdanov closure is consistent, a reduction of the "splitted deuterons" case into a "single deuteron" case can be computed as

The two approaches are compared computing the relative difference between the two matrices:

In our case, we found ε ≈ 10 -11 which indicates a almost perfect agreement between the two cases.


Fiaud's Acid, a novel organocatalyst for diastereoselective bis α-aminophosphonates synthesis with in-vitro biological evaluation of antifungal, antioxidant and enzymes inhibition potential

Rim Aissa, Samia Guezane-Lakoud, Martial Toffano, Lynda Gali, Louisa

Fiaud's Acid, a novel organocatalyst for diastereoselective bis αaminophosphonates synthesis with invitro biological evaluation of antifungal, antioxidant and enzymes inhibition potential

Rim Aissa, a Samia Guezane-Lakoud, a,* Martial Toffano, b Lynda Gali c , and Louisa Aribi-Zouioueche a a Ecocompatible Asymmetric Catalysis Laboratory (LCAE) Badji Mokhtar Annaba-University, B.P 12, 23000 Annaba, Algeria b Equipe de Catalyse Moléculaire-ICMMO Bât 420. Université Paris-Saclay. c Research Center in Biotechnology, Ali Mendjli Nouvelle Ville UV 03, BP E73 Constantine, Algeria

Introduction

Cholinesterase is a category of hydrolysis enzymes found in cholinergic neurons. It plays a major role in the cholinergic transmission by the hydrolysis of the neurotransmitter acetylcholine (ACh) into choline and acetic acid, allowing cholinergic neurons to get back to their resting state. The impaired level of ACh due mainly to neuron death is a prevalent theory that explains the origin of most neurodegenerative disease such as Alzheimer's and Parkinson's disease. [1][START_REF] Sharma | Targeting Enzymes for Pharmaceutical Development[END_REF][3] Acetylcholinesterase enzyme (AChE) is one of the crucial targets of organophosphorus (OPs) causing acute toxicity for the human nervous system inducing irreversible inhibition of acetylcholinesterase. 4 Tyrosinase is an essential enzyme for melanin pigments biosynthesis, some OPs as phosphinic acids derivatives, and aminophosphonates have an inhibitory potential on tyrosinase that could be developed for the treatment of skin cancers such as melanoma. 5 Despite the fact that OPs are recognized as nerve agents, [START_REF] Barbuto | 21st Century Prometheus[END_REF] they are widely employed as pesticides due to their environmental weak persistence. 7 Organophosphorus compounds are gaining continuous attention because of their asymmetric chemical and pharmacological importance. 8 The α-aminophosphonates are natural amino acid analogs, and important compounds in medicinal chemistry. 9 These building blocks were first reported in 1967 by Pudovik. 10 They have considerable employments in organic synthesis as ligands for liquid extraction of metals, 11 and substrates serving in the construction of different natural products analogs having pharmacological and therapeutic properties. 12 The diverse applications of the α-aminophosphonate derivatives include their use as antioxidant, anti-inflammatory, 13 antibacterial, antifungal, 14 and anti-tubercular agents. 15 It is known that the Kabachnik-Fields reaction offers a useful methodology for the α-aminophosphonates synthesis via a multicomponent condensation reaction. 16 Various conditions were described using different catalysts for the synthesis of αaminophosphonates derivatives, proceeding with atom economy, reduced steps and reactions time, offering selectivity, ecological and economic benefits. 17 Recently, we reported the new synthesis of a novel chiral cyclic tertiary phosphine oxides via Kabachnik-Fields reaction using (R, R)-1-oxo-2c,5 t-diphenylphospholane as a precursor (scheme1, a). 18 This secondary phospholane oxide has been prepared from 1-hydroxy-1-oxo-2c,5t-diphenylphospholane, known as Fiaud's acid. 19 This phosphinic acid was demonstrated as an important precursor for the preparation of various chiral organophosphorus compounds, such as ligands, 20 phase transfer agent, 21 or building block. 22 Latterly, Fiaud's acid was described as an effective chiral Brønsted acid catalyst. 23 In continuation of our ongoing research in this area and to access the diastereoselective bis α-aminophosphonates derivatives, 24 we wish to show the interest of Fiaud's acid as a new chiral organocatalyst for the hydrophosphinylation by double Kabachnik-Fields reaction by one-pot condensation of diamine, aromatic aldehyde and diethylphosphite to prepare the bis αaminophosphonates, with excellent diastereoselectivity and chemical yields (Scheme 1, b). To the best of our knowledge, the Fiaud's acid has never been used as a chiral Brønsted acid catalyst in the multi-component Kabachnik-Field reaction.

Scheme 1. Advanced synthesis of organophosphorus compounds.

Numerous bis α-aminophosphonates exhibit significant biological activities: compound I showed an anti-oxidant activity, 25 compound II is considered as anti-tubercular agents, 15 and compound III present optimal anti-proliferative activity against human tumor cells from colon carcinoma. 26 (Figure 1)

Figure. 1: Bis α-aminophosphonates having biological activities

They are extremely interesting multidentate ligands that can be used in the extraction of metals, or as monomers for the preparation of macrocyclic compounds, and even polymers carrying phosphonates and amines. 27 Bis α-aminophosphonates have been produced by various synthetic methodologies from dialdehydes, 28 diamines, 29 Schiff bases, [START_REF] Lewkowski | Phosphorus Sulfur Silicon[END_REF] using Brønsted and Lewis acids, 31 bases 32 and organocatalysts. 33 As well, with unconventional catalytic systems by way of bicationic acidic ionic liquid (AIL). 34 However, most of these approaches show several disadvantages including long reaction time, stoichiometric amount of catalysts, low diastereoisomers ratios. For these reasons, innovative efficient syntheses are required to designing new bis α-aminophosphonates.

Moreover, to extend understanding about the bis αaminophosphonates biological effects in vitro experiments, we have carried out the investigation of their antifungal, antioxidant and enzymatic inhibition potential.

Results and discussion

1. Chemistry

The synthesis of diastereoselective bis α-aminophosphonates was realized by three components condensation in one pot via Kabachnik Fields reaction, using Fiaud's acid or (S, S)-1-oxo-1hydroxy-2-c,5-t-diphenylphospholane in ethyl acetate as green solvent, constitutes an important improvement towards the a lower burden for the environment. For that, the condensation of benzaldehyde (2 mmol), benzidine (1 mmol) and diethylphosphite (2 mmol) in 2 mL of organic solvent was chosen as a model reaction. We first examined the condensation without catalyst, in ethyl acetate (AcOEt), no progress of reaction was observed after 24h and even by increasing the temperature up to 70°C (Table 1, Entry 1). After, we have tested several Brønsted acids such as (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane (A), 1,1'binaphtyl-2,2'-dihydrogene-phosphonate (B), diphenylphosphoric acid (C) and diphenylphosphinic acid (D) (Figure 2). The catalysts screening was made with 10 mol% of catalyst in ethyl acetate at 70°C within 24h. Under these conditions only (S, S)-1-oxo-1-hydroxy-2-c,5-tdiphenylphospholane A promoted the multicomponent reaction and allowed obtaining compound 4a with an excellent chemical yield (93%) and total diastereoselectivity at 70°C (Table 1, Entry 2), compared with the use of organocatalysts (B), (C) and (D), which led under the same conditions to the bis αaminophosphonates with lower yields and diastereomeric ratio (Table 1, entries 4-6). Decreasing the temperature to 50°C led to the product with lower yield (31%) (Table 1, entry 3).The high activity of catalyst (A),could be attributed to his low acidity of (dialkylphosphinic acid) (A) compared with the diarylphosphinic (D) or phosphoric acids (B) and (C) which played probably an important role in these results. Several parameters were studied, such as; the catalytic amount of the organocatalysts, solvents effect, temperature and reaction time. The product 4a was obtained as one diastereoisomer and purified by crystallization in hexane. (Table 1).

By decreasing the catalytic amount of (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane from 10% to 5%, the product 4a was obtained in 65% yield (Table 1, entry 7). The solvent study showed that ethyl acetate was the best solvent. In THF 85% yield was obtained (Table 1, entry 8) and 43% in Et2O (Table 1, entry 9). While no progress was observed in toluene and TBME (Table 1, entries 10 and 11), this was probably due to the fact that both catalysts (A) and (B) were insoluble in these solvents. In addition, with ethyl acetate, we found that the yield has reached 93% yield after 35 min ( To evaluate the synthetic utility of the developed method using (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane, variety of substituted aryl aldehydes with electron-withdrawing and electron-donating groups were used. The results summarized in Table 2 showed the efficiency of Fiaud's acid used as chiral Brønsted catalyst in the MCRs by Kabachnik-Fields reaction for bis α-aminophosphonates preparation. The yields in table 2 correspond to the mixture of diastereomers purified by column chromatography to remove byproducts. All spectral analysis ( 1 H, 13 C and 31 P NMR) and HRMS were performed on the major diastereoisomer. The diastereomeric ratio was determinated by 31 P NMR on the crude product. The chemical yields (63-93%) depend on the nature of electronic effects of the substrates according to our observations and some works of literature, 35 the results show that the increase in electron density on carbonyl and imine groups, caused by electron donating substituents, decreases nucleophilic addition efficiency (Table 2). The best yields (from 82 to 93%) were obtained for 4a, 4b, 4c, 4d, 4g, 4j, 4k and 4l while average yields (63-77%) were attained for 4e, 4f,4h and 4i compounds. Moreover, in the case of 4a, 4c, 4g, 4h, 4j, 4k and 4l the d/l stereoisomer was obtained as a single compound with a total diastereoselectivity (100% dr) and optical rotation ሾߙሿ ܦ This suggests a real efficiency of the catalyst A on the diastereoselectivity control. In the other hand, both forms (d/l and meso) were obtained for the remaining products 4b, 4d, 4e, 4f and 4i. Unfortunately, the determination the relative configuration of the chiral centers failed, since we do not succeed to obtain single mono-crystals with sufficient quality for an X-ray analysis.

Based on the tests in our previous work and on other works, 24 we find that no traces of mono-phosphonated compound were observed using an equimolar quantity of starting materials (1/1/1 ratio of aldehyde/benzidine/diethylphosphite), and the formation of the C-P bond was performed simultaneously on both sides of benzidine. However, we suppose that the (S,S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane plays a crucial role in the formation of the P-C bond simultaneously on both sides of benzidine, and the results showed that the bis α-aminophosphonate was the sole compound obtained at the first time. The (S,S)-1-oxo-1-hydroxy-2-c,5-t-diphenyl-phospholane probably coordinates the nitrogens of di-imine to accelerate the nucleophile attack of the diethylphosphite in their tautomerized form on both sides of diimine to obtain the desired product (Scheme 2).

Table 2. (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane catalysed bis α-aminophosphonates synthesis. a

Product 4 (yield) b , dr% c a Reaction conditions: benzaldehyde (2mmol), benzidine (1 mmol) and diethylphosphite(2.4mmol), AcOEt (2mL), (S, S)-1-oxo-1-hydroxy-2-c,5-tdiphenyl-phospholane (10 mol %), 70°C, 35 min. b Yield of the mixture diastereomers purified by chromatography column or crystallization from hexane. c The diastereomeric ratio was determinated by 31 P NMR on the mixture diastereomers.

We notice that the (S, S)-1-oxo-1-hydroxy-2-c,5-tdiphenylphospholane allowed successfully access for seven bis αaminophosphonates synthesis (4a, 4c, 4g, 4h, 4j, 4k and 4l) with a total diastereoselectivity (100% dr) and high yields. This new chiral organocatalyst was introduced in "one pot" reaction in very fast catalytic approach. The observed diastereoselectivities suggest a control by the chiral secondary phospholane oxide during the nucleophilic addition step. The chiral induction appears to be influenced by the nature of the substituents of the initial aldehyde Scheme 2. Proposed mechanism of bis α-aminophosphonates synthesis catalyzed by (S, S)-1-oxo-1-hydroxy-2-c, 5-t-diphenylphospholane

Biological activity

In vitro assays were explored in order to understand the pharmacological effect of the synthesized bis αaminophosphonates. The antifungal activity evaluation against two plant pathogens Fusarium oxysporum and Botrytis cinerea showed a great antifungal potential for all tested components. From the results presented in (Histogram 1), exposing the inhibition rate (%) at the concentration of 0.14 mg/ml and (Histogram 2) revealing outcomes of IC50 calculated after accomplishment of antifungal test at diverse concentrations (Results are reported in supplement Tables 1 and2), we observed that after 7 days of incubation, compounds 4a, 4b, 4e, 4h, 4l, and 4k at 0.14 mg/mL had inhibited the growth of both fungi by 67% to 84% with IC50 values lower than 0.02 mg/mL, while the other products had exhibited an inhibition varying from 47% to 64% against Fusarium oxysporum with IC50 values ranging between 0.053 and 0.148 mg/mL, and from 54% to 67% against Botrytis cinerea and IC50 values varying from 0.054 to 0.136 mg/ml. We noticed that Botrytis cinerea fungus was more sensitive than Fusarium oxysporum.

Histogram 1. Antifungal inhibition (%) at 0.14 mg/ml concentration of bis α aminophosphonates.

Histogram 2. IC50 values of antifungal activity.

Compounds 4a, 4b, 4e, 4h, 4l and 4k displayed the highest antifungal activity, therefore, they were selected for investigation of their antioxidant and enzymes inhibitory effects on three enzymes. To study their antioxidant properties, we proceed with five spectrophotometric methods comprising Azinobis-3ethylbenthiazoline-6-sulphonic acid, radical scavenging (ABTS assay), 1, 1-diphenyl, 2-picrylhydrazy free radical scavenging (DPPH assay), iron reduction by the phenanthroline assay (Phenassay), cupric reducing antioxidant capacity (CUPRAC) assay, and Galvinoxyl radical scavenging (GOR assay). The results were reported as IC50 and A0.5 values and presented in Table 3 determined from the regression curves at different concentrations (Inhibition percentages at different concentrations are accessible from supplement, Tables 3, 4, 5, 6 and 7), we have observed that all the compounds exhibited a significant reducing effect of iron ions assessed by the phenanthroline assay with A0.5 values ranging from 1.14±0.26 µg/mL (4h) and 6.26±0.05µg/mL (4b), and they were more efficient than the standards trolox and ascorbic acid (A0.5: 5.21±0.27 and 3.08±0.02µg/mL, respectively). However, the reduction capacity of copper ions determined by CUPRAC was lower compared to iron reduction. The compound 4h was also found to be the most efficient in copper ions reduction with A0.5 value of 3.41±0.19µg/mL followed by 4l with A0.5 value of 14.47±0.23µg/mL. The antiradical activity was measured using DPPH, ABTS, and galvinoxyl, as synthetic free radicals. The scavenging activity exerted by the compounds differs according to the type of radical. The strongest scavenging activity was against the radical ABTS, displayed by 4k with an IC50 value of 14.32±1.47µg/mL, whereas 4h showed the highest effect against DPPH radical, which gave an IC50 value of 22.43±0.89µg/mL. The compounds 4e, 4k, and 4l exerted a very weak potential in the reduction of DPPH radical (IC50>800µg/mL), while the compounds 4h, 4k, and 4b were found to exhibit the best scavenging effect against galvinoxyl radical (IC50: 3.41±0.19, 6.62±0.42, and 7.34±0.59µg/mL, respectively. It was clear that the radical scavenging potential decreased in the following order: Gor assay > ABTS assay > DPPH assay, indicating that the bis αaminophosphonates acted the best via an electron transfer mechanism for radical scavenging like in the case of GOR assay, and according to the galvinoxyl scavenging radical mechanism theory reported by Wang and Zhang. 36 Furthermore, it was notable that a steric bulk caused by both tested compounds and free radical molecule sizes affected the antioxidant activity mainly observed in GOR assay conducted with the smallest studied free radical compared to DPPH assay presenting the largest free radical molecule. Otherwise, structural and electronic properties of differently substituted compounds are other main influencing factors. For example, the compound 4h was the best antioxidant among tested compounds with A0.5 of 1.14±0.26 µg/mL in Phen assay and IC50 of 3.41±0.19 µg/mL in GOR assay promoting better antioxidant effects than standards. However, compound 4h showed lower activity against ABTS and DPPH radicals as well as in the reduction of copper ions. Similarly, the compounds 4k presented a remarkable capacity in reducing iron and copper ions (A0.5=1.29±0.12 3.17±0.93 µg/mL, respectively) and in the scavenging of galvinoxyl radical (IC50= 6.62±0.42µg/mL), while it exhibited a weak scavenging activity against ABTS and DPPH radicals (IC50=97.92±0.67µg/mL and IC50<800µg/mL, respectively). We can conclude that both the structure of the antioxidant compound and the assay can influence the antioxidant effect.

Organophosphorus neurological toxicity is one of the principal worries of their use as pesticides. For this purpose, we have evaluated the capacity of the selected compounds to inhibit both acetylcholinesterase (AChE), and butyrylcholinesterase (BChE). Tyrosinase inhibition activity was also investigated (Inhibition percentages at different concentrations are available in supplement, Tables 8, 9 and 10). The summarized results of IC50 in Table 4, showed that all tested compounds presented a moderate to weak inhibitory activity against AChE. The compounds 4b, 4a and 4k exhibited the strongest inhibition with IC50 of 34.13±0.65, 85.04±1.35, and 82.18±1.23, respectively but much low compared to the standard Galantamine, which gave an IC50 of 6.27±1.15 µg/mL. On behalf of BChE, 4e was found to be inactive, 4a, 4b and 4l exerted week inhibition (IC50=87.58±3.86 and 62.72±2.16µg/mL for 4a and 4b, respectively, and IC50 exceed 200 µg/mL for 4l), while IC50 values of 39.58±3.05 and 39.42±2.98 µg/mL were obtained with 4h and 4k respectively, which were close (values are not significantly different, p>0.05) from the IC50 value recorded with the standard Galantamine (34.75±1.99µg/mL). The inhibitory effect of the selected compounds on tyrosinase showed that all compounds were inactive or have IC50 exceeding 200 µg/mL including 4a and 4h excepting 4l, which has IC50 of 12.11±0.36 µg/ml and was, therefore, more powerful than the standard kojic acid (IC50=25.23±0.78µg/mL). It should be mentioned that the biological effects exhibited by the major diastereomer and the mixture of diastereoisomers for tested compounds are similar in all exanimated activities.

Conclusion

In summary, we proved that the (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane could be implicated successfully as an effective new chiral organocatalyst for bis α-aminophosphonates synthesis with an efficient one pot catalytic approaches giving nine products 4a-4i in high yields (63-93%) and with an excellent to good diastereoselectivity within 35 min at 70°C. All synthesized bis α-aminophosphonates revealed an excellent antifungal capacity, with compounds 4a, 4b, 4e, 4h, 4l and 4k having IC50 under 0.02 mg/ml. Moreover, these compounds afforded high antioxidant activity, and display a low enzymatic inhibition of acetylcholinesterase and butyrylcholinesterase, whereas for Tyrosinase inhibition they were inactive or have high IC50,values except the compound 4i. These results demonstrated that the biological effect depend on the examination of each analogue which is supported by the structureactivity relationship (SAR) 37 . For that, some organophosphorus could be considered as less hazardous promising pesticides. This last statement should be supported by future in-vivo biological investigations, cytotoxicity and ecotoxicity studies.

Experimental section

Chemistry

All reagents were purchased from Sigma-Aldrich or Acros Company used without further purification. Reactions were monitored by thin layer chromatography (TLC) carried out on 0.25-mm Merck silica gel plates (60F-254) using ultraviolet light (254 nm) as the visualizing agent and KMnO4 solution as developing agents. NMR spectra were recorded with Bruker spectrometers operating at (360MHz, 300 MHz and 250 MHz for 1H, 90 MHz, 75 MHz or 63 MHz for 13 C and 101 MHz or 121 MHz for 31 P). Chemical shift of Solvent reference peaks used were CDCl3 (δ = 7.26 ppm) for 1H and (δ = 77ppm) for 13 C NMR spectra, while H3PO4 was used as external standard for chemical shift references for 31 P NMR. Couplings constants (J) are given in Hz, with the Following abbreviations multiplicity: s = singlet, d = doublet, t = triplet, q = quartet, m = multiplet, br = broad signal. Mass spectra were taken by a MicrOTOF-Q Bruker spectrometer using electrospray ionization (ESI) analysis. Melting points were measured using Buchi Melting Point B-545. Optical rotations were measured on an Anton Paar's MCP 150 and Bellingham & Stanley ADS 420 polarimeters and reported as follows: ሾߙሿ ் (concentration (g/mL), solvent).

General procedure for synthesis of bis α-aminophosphonates 4a-l.

A mixture of N,N' benzyldiamine (1 mmol, 0.18 g), aromatic aldehyde (2 mmol, 0.2g) and diethyl phosphate (2.4 mmol, 0.34g) was well stirred with (S, S)-1-oxo-1-hydroxy-2-c,5-t-diphenylphospholane (10 mol%) in 2 mL of ethyl acetate at 70°C for 35 min. The reaction progress was monitored by TLC. The resulting mixture diastereomers was purified by column chromatography on silica gel (ethyl acetate/hexane: 30/70) to afford the major diastereoisomer of products 4a-l for analysis spectra.

4.1.1.1Tetraethyl [(4,1-phenylene) bis(azanediyl)] bis[(phenyl) methylene] bisphosphonate (4a).

Yellow solid, 93% Yield, mp: 210°C, ሾߙሿ ଶ = -10 (ܿ 0,002, CH ଶ Cl ଶ ). 1

Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[(4-nitrophenyl) methylene]bisphosphonate(4b).

Yellow solid, 90% Yield, mp: 219.5°C. 1

Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[(4-dimethyl-aminophenyl) methylene]bisphosphonate(4c).

Orange solid, 84% Yield, mp: 199.3°C. ሾߙሿ ଶ = -54,54 (ܿ 0,001, CH ଶ Cl ଶ ). 1 H NMR (250 MHz, CDCl3) δ 7.32 (dd, J = 8.9, 2.3 Hz, 4H-Ar), 7.22 (d, J = 8.6 Hz, 4H-Ar), 6.69 (d, J = 8.5 Hz, 4H-Ar), 6.61 (d, J = 8.7 Hz, 4H-Ar), 4.68 (d, J = 23.4 Hz, 2H, HCP+NH), 4.20 -4.02 (m, 4H, O-CH2-CH3), 3.94 (m, 2H, O-CH2-CH3), 3.76 -3.58 (m, 2H, O-CH2-CH3), 2.92 (s, 12H, 2(CH3)2-N-), 1.28 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.14 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13

Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[(4-fluoro-phenyl)methylene] bisphosphonate(4d).

Brown olid, Yield 85 %, mp: 172.3 1C°C. 1

Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[4-(tert-butyl)phenyl)methylene]bisphosphonate(4e).

Green solid, 77 % Yield, mp: 220°C. 1

Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[(napht-1-ylmethylene] bisphosphonate(4f).

Brown solid, 63 % Yield,mp: 222.7°C. 1

Tetraethyl[(4,1-phenylene)bis(azanediyl)]bis[(3-phenoxy-phenyl)methylene]bisphosphonate(4g).

Yellow solid, 82 % Yield, mp: 184.6°C, ሾߙሿ ଶ = +4 (ܿ 0,002, CH ଶ Cl

Tetraethyl[(4,1-phenylene)bis(azanediyl)]bis[(2-methoxy-phenyl)methylene] bisphosphonate(4h).

Orange solid, 68 % Yield, mp: 165.6°C, ሾߙሿ ଶ = +13 (ܿ 0,002, CH ଶ Cl ଶ ሻ.

Tetraethyl[(4,1-phenylene)bis(azanediyl)]bis[4-chlorophenyl)methylne]bisphosphonate (4i).

Green oil, 73 % yield. 1

4.1.1.11.Tetraethyl[(4,1phenylene)bis(azanediyl)]bis(benzo[1,3]dioxol-5-ylmethylene) bisphosphonate (4k).

Green solid,90 % Yield, mp: 229,7°C, ሾߙሿ ଶ = -25 (ܿ 0,002, CH ଶ Cl ଶ ሻ. 19 Yellow solid, 91% yield, mp: 203.7 1C, ሾߙሿ ଶ = -10 (ܿ 0,002, CH ଶ Cl ଶ ሻ.

Tetraethyl [(4,1-phenylene)bis(azanediyl)] bis[(4-cyano-phenyl) methylene] bisphosphonate (4l).

In vitro biological activities

Antioxidant and enzymatic inhibition activities were performed using a 96-well microplate reader, PerkinElmer Multimode Plate Reader EnSpire at National Center of biotech-nology Research. The chemical products and reagents used were: Folin-ciocalteu's reagent (FCR), 1,1-diphenyl-2-pic-rylhydrazyl (DPPH), butylatedFolin-Ciocalteu (FCR), 1,1diphenyl-2-pic-rylhydrazyl (DPPH), Trolox, Ascorbic acid, Tween-40, Neocuproine, 2,2′-azino-bis(3ethylbenzothiazoline-6-sulfonicacid) diammonium salt (ABTS), Trichloroacetic acid (TCA), Potassium ferricyanide, 3-(2-Pyridyl)-5,6-di(2-furyl)-1,2,4-triazine-5′,5′′-disulfonic acid disodium salt (Ferene), Ethylenediaminetetraacetic acid (EDTA), Nitro blue tetrazolium (NTB), b1000 U/mg solid, Butyrylcholinesterase (BChE) from equine serum 100 U/mg solid, 5,50-dithiobis[2-nitrobenzoic acid] (DTNB), butyrylthiocholine chloride, galanthamine, Tyrosinase from mushroom ≥1000U/mg solid , 3-(3,4-Dihydroxyphenyl-2,5,6-d3)-L-alanine (L-DOPA), are from Sigma Aldrich. Aluminum nitrate, Sodium bicarbonate, CuCl2, potassium persulfate, potassium acetate were obtained from BiochemChemopharma. Diméthylsulfoxyde (DMSO), and other solvents were of analytical grade were purchased from Sigma Aldrich. hydroxylanisole (BHA), Butylatedhydroxyltoluene (BHT), α-Tocopherol, polyoxyethylenesorbitanmonopalmitate (Tween-40), Neocuproine, 2,2′-azino-bis(3-ethylbenzothiazoline-6sulfonicacid) diammoniumsalt (ABTS), Trichloroacetic acid (TCA), Potassium ferricyanide, 3-(2-Pyridyl)-5,6-di(2furyl)-1,2,4-triazine-5′,5′′-disulfonic acid disodium salt (Ferrene), Ethylenediaminetetraacetic acid (EDTA), Nitro blue tetrazolium (NTB), Diméthylsulfoxyde (DMSO), Acetylcholinesterase from electric eel (AChE, Type-VI-S, EC 3.1.1.7, 827,84 U/mg, Sigma), butyl cholinesterase from horse serum (BChE, EC 3.1.1.8,7,8 U/mg, Sigma), Acetylthiocholine iodide, S-Butyrylth-iocholine iodide, 5,5′-Dithiobis (2-nitrobenzoic) acid (DTNB),Galantamine, Sodium Carbonate, Alu-minum Nitrate, Iron (III) chloride (FeCl3), Iron (II) chlo-ride, Sodium bicarbonate, Copper (II) chloride, Potassium persulfate, Potassium acetate, were obtained from BiochemChemopharma. All other chemicals and solvents were of analytical grade. Potato-dextrose agar (PDA) for microbiology is perchased from Sigma Aldrich.

Antifungal activity

All synthesized products were tested for determination of their antifungal activity trough mycelial growth inhibition of tow phytopathogenic fungi Fusarium oxysporum f. sp lycopersici (FOL) strain 4287, and Botrytis cinerea. For that, 14 mg of compounds was dissolved in 1ml of DMSO and added to 100 ml of PDA medium at 60°C to have a final concentration of 0.14 mg/ml in the mixture which was distributed in 4 petri dishes. A disk of 5 mm diameter was taken from a young fungal culture and placed in the petri dishes center. Development of the phytopathogenic agent was measured at millimetric scale After 7 days of incubation at 28°C. Three experiments repetitions were performed for all tests. Other petri dishes were prepared with 1 ml of DMSO added to 100 ml of PDA medium as positive control with fungal disks, while negative control was set with PDA medium only, according to Song method. [START_REF] Song | [END_REF] Growth inhibition capacity was calculated through radial growth of the fungal colonies as described in Dennis and Webstert work. 39 Results were presented in (Histogram 1).

The tests were repeated for divertive concentration (0.14/ 0.12/ 0.10/ 0.08/ 0.06/ 0.04/ 0.02 mg/ml) in order to identify the IC50 of tested compounds by a linear regression; results were presented in (Histogram 2).

The following antioxidant activity and enzymatic inhibition assays are spectrophotometric methods adapted to multimode plate reader.

Antioxidant activity essays

Basing on the complex structures and nature of interactions involved in the antioxidant effect, five of effective different complementary spectrophotometric methods, were implemented for testing synthetized compounds 4a, 4b, 4e, 4h, 4k, 4l.

ABTS radical scavenging activity

ABTS radical was generated by the oxidation of 2mM of ABTS with 2.45mM potassium persulfate (K2S2O8). The resulted solution was mixed with samples and the absorbance was measured at 734nm. 40

DPPH scavenging activity

DPHH scavenging activity was assessed following the method described by Blois. 41 Briefly, 1mM DPPH solution was added to samples at different concentrations and the absorbance was measured at 517nm.

Phenanthroline assay

Iron ions reduction was assessed by the phennathroline assay as reported in the literature. 42 The reaction mixture contains sample, 0.2% FeCl3, and 0.5% phenanthroline. The absorbance was read at 510nm.

Cupric Reducing Antioxidant Capacity (CUPRAC)

The reduction of copper ions was investigated out by the reduction of colorless copper (II) neocuproine (2,9-dimethyl-1,10-phenanthroline) complex to the coloured copper(I)-neocuproine complex. 43 10mM CuCl2, 7.5mM neocuproine, and 1M CH3COONH4 were mixed with samples in a microplate and the absorbance was read at 450nm.

Galvinoxyl radical scavenging activity

The reduction of Galvinoxyl radical was determined as previously reported. 44 0.1mM of Galvinoxyl radical was added to the samples and the absorbance was measured at 428 nm.

Enzymes inhibition essays

Cholinesterase inhibition

The anticholinesterase activity of compounds 4a, 4b, 4e, 4h, 4k, 4l, was investigated according to the colorimetric method of Ellman. 45 AChE (5.32 × 10 -3 U) and BChE (6.85 × 10 -3 U) in phosphate buffer (0.1M, pH8) were incubated in a 96-well microplate with different concentrations of the compounds. 0.5 mMof DTNB and 0.71mM acetylthiocholine iodide or 0.2 mM butyrylthiocholine chloride were added and the absorbance was monitored at 412 nm using a multimode plate reader (Perkin Elmer, EnSpire®, Singapore).

Tyrosinase inhibition

The inhibitory effect of the selected compounds against tyrosinase was determined according to the method described by Deveci et al. 46 The reaction medium contains 150U/ml of tyrosinase in 100mM phosphate buffer (pH 6.8), and 5mM of L-DOPA, and the samples at different concentrations. Absorbance was read at 475 nm.

Statistical analysis

Results are reported as mean value ± SD of three measurements; the IC50 and A0.50 values were calculated by linear regression analysis, and one-way analysis of variance ANOVA followed by Tukey's multiple comparison test to detect significant differences (p < 0.05) using Graphpad prism software (GraphPad Software Inc, San Diego, CA, USA).

Figure. 2 :

 2 Figure. 2: Organocatalysts tested.

  ܿ 0.002, CH 2 Cl 2 ), ሾߙሿ ଶ = -54.54 (ܿ 0.001, CH ଶ Cl ଶ ), ሾߙሿ ଶ = +4 (ܿ 0.002, CH ଶ Cl ଶ ), ሾߙሿ ଶ = +13 (ܿ 0.002, CH ଶ Cl ଶ ሻ, ሾߙሿ ଶ = -25 (ܿ 0.002, CH ଶ Cl ଶ ሻ, ሾߙሿ ଶ = +15 (ܿ 0.002, CH ଶ Cl ଶ ሻ, ሾߙሿ ଶ = -10 (ܿ 0.002, CH ଶ Cl ଶ ሻrespectively.

  H NMR (250 MHz, CDCl3) δ 7.54 -7.41 (m, 4H-Ar), 7.29 (m, 10H-Ar), 6.60 (d, J = 8.5 Hz, 4H-Ar), 4.77 (d, J = 24.3 Hz, 2H, HCP+NH), 4.11 (m, 4H, O-CH2-CH3), 4.01 -3.83 (m, 2H, O-CH2-CH3), 3.77 -3.55 (m, 2H, O-CH2-CH3), 1.28 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.11 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (63 MHz, Chloroform-d) δ 144.87 (d, J = 14.7 Hz), 135.88(s), 131.42(s), 128.59(s), 127.82 (d, J = 5.6 Hz), 127.06(s), 114.09(s), 63.27 (d, J = 6.9 Hz), 57.31 (d, J = 6.2 Hz), 54.97, 16.41 (d, J = 5.4 Hz), 16.16 (d, J = 6.0 Hz). 31 P NMR (101 MHz, Acetone-d6): δ 22.63 ppm. HRMS (ESI) m/z calcd for C34H42N2O6NaP2 [M +Na + ]: 659.2410; Found 659.2399.

  H NMR (360 MHz, CDCl3) δ 8.20 (d, J = 8.4 Hz, 4H-Ar), 7.67 (dd, J = 8.8, 2.3 Hz, 4H-Ar), 7.31 -7.15 (m, 4H-Ar), 6.55 (d, J = 8.6 Hz, 4H-Ar), 4.86 (d, J = 25.1 Hz, 2H, HCP+NH), 4.21 -4.08 (m, 4H), 4.09 -3.97 (m, 2H, O-CH2-CH3), 3.95 -3.80 (m, 2H, O-CH2-CH3), 1.30 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.18 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (91 MHz, CDCl3) δ 147.68 -147.55 (m), 144.37, 144.22, 144.00 (d, J = 3.8 Hz), 131.85(s), 128.64 (d, J = 4.8 Hz), 127.30, 123.79, 114.12, 63.78 (d, J = 7.5 Hz), 63.53 (d, J = 6.7 Hz), 56.92, 55.29, 16.36 (dd, J = 15.4, 5.7 Hz). 31 P NMR (101 MHz, CDCl3) δ 20.71 ppm. HRMS (ESI) m/z calcd for C34H40N4O10NaP2 [M+Na + ]: 749.2112; Found 749.2127.

  H NMR (360 MHz, CDCl3) δ 7.45 (m, 4H, H-Ar), 7.23 (d, J = 8.6 Hz, 4H, H-Ar), 7.02 (t, J = 8.5 Hz, 4H, H-Ar), 6.58 (d, J = 8.7 Hz, 4H, H-Ar), 4.75 (d, J = 24.2 Hz, 3H, HCP+NH), 4.18 -4.04 (m, 4H, O-CH2-CH3), 4.02 -3.90 (m, 2H, O-CH2-CH3), 3.80 -3.68 (m, 2H, O-CH2-CH3), 1.27 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.14 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (91 MHz, CDCl3) δ 163.79 (d, J = 3.0 Hz), 161.07 (d, J = 3.9 Hz), 144.74 (d, J = 14.8 Hz), 131.60 (d, J = 16.7 Hz), 129.72 -129.20 (m), 127.14, 115.72, 114.13, 63.34 (t, JC-P = 6.6 Hz), 56.29(s), 54.62(s), 16.46 (d, J 3 C-P = 5.5 Hz), 16.26 (d, J 3 C-P = 5.8 Hz). 31 P NMR (101 MHz, CDCl3, 25°C): δ 22.26 ppm. HRMS (ESI) m/z calcd for C34H40N2O6NaP2F2 [M + Na + ]: 695.2222; Found 695.2239.

  H NMR (300 MHz, CDCl3) δ 7.43 -7.31 (m, 8H, H-Ar), 7.30 -7.21 (m, 4H, H-Ar), 6.66 (d, J = 8.5 Hz, 4H, H-Ar), 4.76 (d, J = 23.0 Hz, 2H, HCP), 4.20 -4.04 (m, 4H, O-CH2-CH3), 3.95 (m, 3H, O-CH2-CH3+NH), 3.77-3.61 (m, 3H, O-CH2-CH3+NH), 1.39 -1.25 (m, 25 H, 9 CH3+2(O-CH2-CH3)), 1.10 (t, J = 7.0 Hz, 6H, O-CH2-CH3).13 C NMR (75 MHz, Chloroform-d) δ 130.01, 127.46 (d, J = 5.5 Hz), 127.03, 126.94 -126.57 (m), 125.91 -125.16 (m), 120.52 -120.25 (m), 114.07, 63.28, 34.51, 31.30, 16.42 (d, J 3 C-P= 5.7 Hz), 16.11 (d, J 3 C-P = 5.6 Hz). 31 P NMR (121 MHz, Acetone-d6): δ 22.52 ppm. HRMS (ESI) m/z calcd for C42H58O6NaN2P2 [M + Na + ]: 771.3662; Found 771.3682.

  H NMR (300 MHz, CDCl3) δ 8.25 (d, J = 8.5 Hz, 2H-Ar), 7.90 (d, J = 7.8 Hz, 2H-Ar), 7.79 (dd, J = 7.7, 2.8 Hz, 4H-Ar), 7.62 (t, J = 7.6 Hz, 2H-Ar), 7.55 (d, J = 7.0 Hz, 2H-Ar), 7.44 (t, J = 7.7 Hz, 2H-Ar), 7.14 (d, J = 8.2 Hz, 4H-Ar), 6.55 (d, J = 8.2 Hz, 4H-Ar), 5.64 (d, J = 23.8 Hz, 2H), 4.18 (p, J = 7.1 Hz, 4H, O-CH2-CH3), 3.82 -3.65 (m, 2H, O-CH2-CH3), 3.31 -3.14 (m, 2H, O-CH2-CH3), 1.32 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 0.74 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (91 MHz, Chloroform-d) δ 145.72, 137.07, 128.69 (d, J = 16.4 Hz), 127.59, 126.65, 126.29 (d, J = 13.9 Hz), 125.70 (d, J = 6.3 Hz), 125.41 (d, J = 6.5 Hz), 120.14, 113.88, 63.58 (d, J = 7.0 Hz), 63.37 (d, J = 6.7 Hz), 24.59, 16.50 (d, J 3 C-P = 5.7 Hz), 15.78 (d, J 3 C-P = 6.1 Hz). 31 P NMR (101 MHz, CDCl3, 25°C): δ 22.72 ppm. HRMS (ESI) m/z calcd for C30H38N2O8NaP2 [M + Na + ]: 759.2723; Found 759.2732.

ଶ 1 H

 1 NMR (250 MHz, CDCl3) δ 7.35 -7.30 (m, 2H, H-Ar), 7.28 (d, J = 1.5 Hz, 4H, H-Ar), 7.27 -7.20 (m, 5H, H-Ar), 7.15 (m, 2H, H-Ar), 7.11-7.05 (m, 2H, H-Ar), 6.99 -6.87 (m, 6H, H-Ar), 6.60 (d, J = 8.7 Hz, 4H, H-Ar), 4.76 (d, J = 24.5 Hz, 3H, HCP+NH), 4.24 -4.07 (m, 4H, O-CH2-CH3), 4.06 -3.92 (m, 2H, O-CH2-CH3), 3.88 -3.68 (m, 2H, O-CH2-CH3), 1.29 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.17 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (63 MHz, CDCl3) δ 157.33 (d, J = 2.5 Hz), 156.98(s), 144.75 (d, J = 14.6 Hz), 138.11(s), 131.53(s), 129.92(s), 129.69(s), 127.07(s), 123.24(s), 122.70 (d, J = 5.1 Hz), 118.73(s), 118.49 -118.09 (m), 114.24(s), 63.61 -63.00 (m), 57.20(s), 54.80(s), 16.33 (dd, J 3 C-P = 11.9, 5.9 Hz). 31 P NMR (101 MHz, Acetone-d6) δ 22.20 ppm. HRMS (ESI) m/z calcd for C46H50N2O8NaP2 [M+Na + ]: 843.2935; Found 843.2914.

1 H

 1 NMR (250 MHz, CDCl3) δ 7.51 -7.42 (m, 2H-Ar), 7.21 (dt, J = 8.1, 1.8 Hz, 6H-Ar), 6.97 -6.83 (m, 4H-Ar), 6.61 (s, 3H, O-CH2-CH3), 5.40 (d, J = 19.9 Hz, 2H, O-CH2-CH3), 4.26 -4.06 (m, 4H, O-CH2-CH3), 3.93 (s, 6H, OCH3), 3.92 -3.79 (m, 2H, O-CH2-CH3), 3.69 -3.51 (m, 2H, O-CH2-CH3), 1.30 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.03 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (63 MHz, Chloroform-d) δ 139.15, 128.90, 128.18, 127.02 (d, J = 7.9 Hz), 121.01, 113.79, 110.46, 63.38 -62.70 (m), 55.72, 16.43 (d, J 3 C-P = 5.3 Hz), 16.09 (d, J 3 C-P = 5.8 Hz). 31 P NMR (101 MHz, Acetone-d6) δ 23,54ppm. HRMS (ESI) m/z calcd for C36H46O8N2NaP2 [M + Na + ]: 719.2649.

  1 H NMR (360 MHz, Chloroform-d) δ 7.24 (d, J = 8.6 Hz, 4H-Ar), 6.97 (t, J = 1.8 Hz, 2H-Ar), 6.96 -6.93 (m, 1H-Ar), 6.94 -6.91 (m, 1H-Ar), 6.76 (d, J = 8.0 Hz, 2H-Ar), 6.60 (d, J = 8.6 Hz, 4H), 5.92 (dd, J = 6.1, 1.4 Hz, 4H, O-CH2-O), 4.67 (d, J = 24.0 Hz, 3H, O-CH2-CH3), 4.22 -4.06 (m, 4H, O-CH2-CH3), 4.06 -3.90 (m, 2H, O-CH2-CH3), 3.85 -3.69 (m, 2H, O-CH2-CH3), 1.29 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.17 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (91 MHz, Chloroform-d) δ 148.00, 147.36 (d, J = 2.8 Hz), 144.92, 144.75, 131.48, 129.68 (d, J = 2.0 Hz), 127.10, 121.36 (d, J = 6.3 Hz), 114.12, 108.33, 108.15 (d, J = 4.2 Hz), 63.33 (d, J = 6.9 Hz), 56.65, 54.97, 16.47 (d, J = 5.7 Hz), 16.30 (d, J = 6.0 Hz). 31 P NMR (101 MHz, CDCl3): δ 22.53 ppm. HRMS (ESI) m/z calcd for C30H38O8NaN2P2 [M + Na + ]: 747.2206; Found 747.2225.

  1 H NMR (250 MHz, CDCl3, 25°C) δ 7.71-7.58 (m, 8H, H-Ar), 7.31-7.18 (m, 4H, H-Ar), 6.57 (d, J = 11.0 Hz, 4H, H-Ar), 4.84 (d, J = 22.0 Hz, 4H, HCP + NH), 4.24-4.10 (m, 4H, O-CH2-CH3), 4.09-3.96 (m, 2H, O-CH2-CH3), 3.95-3.75 (m, 2H, O-CH2-CH3), 1.31 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.19 (t, J = 7.1 Hz, 6H, O-CH2-CH3). NMR 13 C (63 MHz, CDCl3, 25 1C): d 144.42 (d, J 3 C-P= 14.2 Hz), 142.00, 139.17 (s), 132.38 (s), 131.78 (s), 128.55 (d, J 2 C-P= 5.1 Hz), 127.27 (s), 114.09 (s), 111.79 (d, J = 3.4 Hz), 111.13 (s), 63.66 (d, JC-P= 14.8 Hz, C-O), 63.42 (d, JC-P = 6.7 Hz, C-O), 57.38 (s, C-N), 55.02(s, C-P), 16.43 (d, J 3 C-P= 5.7 Hz, CH3-CH2-O-P), 16.24 (d, JC-P 3 = 5.5 Hz, CH3-CH2-O-P). 31 P NMR (121 MHz, CDCl3, 25°C) δ 22.05 ppm. HRMS (ESI) m/z calcd for C36H40N4NaO6P2 [M +Na+]: 709.2135; found 709.2344.

Table

1, entry 12), but it reduced to 56% after 20 min (Table 1, entry 13).Table 1 .

 1 Optimization Reaction of the synthesis of bis αaminophosphonates.

	Entry a Catalyst	Solvent	T °C	Time /h	Yield(%) c
		(10 mol %)
	1 b	--	AcOEt	70	24	--
	2	A	AcOEt	70	24	93
	3	A	AcOEt	50	24	31
	4	B	AcOEt	70	24	55
	5	C	AcOEt	70	24	75
	6	D	AcOEt	70	24	30
	7	A (5mol %)	AcOEt	70	24	65
	8	A	THF	70	24	85
	9	A	Et2O	70	24	43
	10	A	PhMe	70	24	Traces
	11	A	TBME	70	24	Traces
	12	A	AcOEt	70	35min	93
	13	A	AcOEt	70	20min	56
	a Reaction

conditions: benzaldehyde (2 mmol), benzidine (1 mmol) and diethylphosphite (2.4 mmol) were stirred without catalyst in solvent (2 mL). at 70°C. No reaction was observed at 25°C. b Reaction conditions: benzaldehyde (2 mmol), benzidine (1 mmol) and diethylphosphite (2.4 mmol

  ) were stirred with catalyst in solvent (2 mL) at 70°C.c Yield of the pure product purified by crystallization in hexane.

Table 3 .

 3 Antioxidant activities results.Values are expressed as means ± S.D of three parallel measurements. Values with different letters are significantly different (Tukey multiple comparison test, p<0.05). *Reference compounds.

	Entry	ABTS IC50 (µg/mL)	DPPH IC50 (µg/mL)	CUPRAC A0.50 (µg/mL)	GOR IC50 (µg/mL)	Phenanthroline A0.5 (µg/mL)
	4a	40.50±7.49 b	159.44±2.15 a	58.40±0.94 a	25.90±0.32 b	2.08±0.12 b
	4b	30.06±0.81 c	118.02±1.08 b	53.31±1.02 a	7.34±0.59 c	6.26±0.05 a
	4e	20.34±1.96 d	>800	25.27±1.08 b	>200	1.80±0.46 b
	4h	14.32±1.47 a	22.43±0.89 c	24.82±1.27 b	3.41±0.19 d	1.14±0.26 c
	4k	97.92±0.67 a	>800	3.17±0.93 e	6.62±0.42 c	1.29±0.12 c
	4l	21.28±0.48 d	>800	14.47±0.23 c	120.10±0.85 a	1.22±0.19 c
	Trolox*	3.21±0.06 f	5.12±0.21 c	8.69±0.14 d	4.31±0.05 d	5.21±0.27 b
	Ascorbic acid*	3.04±0.05 f	4.39±0.01 c	8.31±0.15 d	5.02±0.02 d	3.08±0.02 e

Table 4 .

 4 Enzymes inhibition assays results.

	Entry	AChE IC50 (µg/mL)	BChE IC50 (µg/mL)	Tyrosinase IC50 (µg/mL)
	4a	85.04±1.35 a	87.58±3.86 a	>200
	4b	34.13±0.65 b	62.72±2.16 b	-
	4e	>200	39.58±3.05 c	-
	4h	>200	-	>200
	4k	82.18±1.23 a	39.42±2.98 c	-
	4l	>200	>200	12.11±0.36 b
	Galantamine*	6.27±1.15 c	34.75±1.99 c	-
	Kojic acid*	-	-	25.23±0.78

a

Values are reported as means ± S.D of three parallel measurements. Values with different letters are significantly different (p<0.05), *Reference compounds

  C NMR (63 MHz, CDCl3) δ 150.10(s), 145.10 (d, J = 14.8 Hz), 139.16(s), 131.29(s), 128.59 (d, J = 5.5 Hz), 126.99(s), 123.11(s), 114.13(s), 112.60(s), 63.10 (t, JC-P = 6.2 Hz), 56.67(s), 54.24(s), 40.52(s), 16.37 (dd, J 3 C-P= 10.0, 5.8 Hz). 31 P NMR (121 MHz, CDCl3, 25°C): δ 22.30 ppm. HRMS (ESI) m/z calcd for C38H52N4O6NaP2 [M + Na + ]: 745.3254; Found 745.33257.

.1.10 Tetraethyl [(4,1-phenylene)bis(azanediyl)]bis[(furan-2 ylmethylene]bis phosphonate (4j).

  H NMR (250 MHz, CDCl3) δ 7.43 (m, 4H), 7.31 (d, J = 8.4 Hz, 4H, H-Ar), 7.27-7.19 (m, 4H, H-Ar), 6.58 (d, J = 8.6 Hz, 4H, H-Ar), 4.75 (d, J = 24.4 Hz, 2H, HCP), 4.23-4.03 (m, 5H, O-CH2-CH3 + NH), 3.96 (m, 2H, O-CH2-CH3), 3.84-3.67 (m, 2H, O-CH2-CH3), 1.28 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.16 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C NMR (91 MHz, CDCl3) δ 144.65 (d, J = 14.7 Hz), 134.56(s), 133.74 (d, J = 3.5 Hz), 131.59(s), 129.14 (d, J = 5.3 Hz), 129.01 -128.73 (m), 128.65(s), 127.66(s), 127.17(s), 126.78(s), 114.11(s), 63.74 -63.25 (m), 56.44,

	54.77, 16.44 (d, J 3	C-P = 5.7 Hz), 16.26 (d, J 3	C-P = 6.0 Hz). 31 P NMR (101 MHz, Acetone-d6): δ 21.96 ppm. HRMS (ESI)
	m/z calcd for C34H40O6N2P2Cl2 [M + H + ]: 727.1630; Found 727.1610.
	4.1Brown solid, 90 % Yield,mp: 218.9°C, ሾߙሿ	ଶ = +15 (ܿ 0,002, CH ଶ Cl ଶ ሻ 1 H NMR (360 MHz, CDCl3) δ 7.39 (t, J = 1.9
	Hz, 2H-Ar), 7.33 -7.29 (m, 4H, H-Ar), 7.26 (s, 4H, H-Ar), 6.69 (d, J = 8.7 Hz, 1H), 6.40 (t, J = 3.2 Hz, 2H, H-furan),
	6.36 -6.31 (m, 2H, H-furan), 4.91 (d, J = 23.8 Hz, 2H, HCP), 4.26 -4.13 (m, 4H, O-CH2-CH3), 4.12 -4.01 (m, 2H, O-
	CH2-CH3), 3.88 (m, 2H, O-CH2-CH3), 1.30 (t, J = 7.1 Hz, 6H, O-CH2-CH3), 1.21 (t, J = 7.1 Hz, 6H, O-CH2-CH3). 13 C
	NMR (91 MHz, CDCl3) δ 149.38, 144.69 (d, J = 13.2 Hz), 142.51, 131.86, 127.16, 114.22, 110.82, 108.82 (d, J = 6.7
	Hz), 63.55 (d, JC-P= 7.1 Hz), 63.35 (d, J = 6.7 Hz), 51.20, 49.44, 16.47 (d, J 3	C-P= 5.3 Hz), 16.31 (d, J 3	C-P= 5.3 Hz). 31 P
	NMR (101 MHz, CDCl3): δ 20.14 ppm. HRMS (ESI) m/z calcd for C30H38O8NaN2P2 [M + Na + ]: 639.1996; Found
	639.1992.

Acknowledgments

The Algerian Ministry of Education and Scientific Research (DGRSDT, FNR) is gratefully acknowledged for financial support of this work. The technical support provided by Emilie KOLODZIEJ is highly appreciated. Dr. Ali DEBBI is thanked for his helpful in biological activities part.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Supplementary data

Supplementary data (all experimental procedures; characteri-zation spectra; NMR and HRMS of all compounds and in vitro biological activities) related to this article can be found at http://dx.doi.org/..........