Lavinia Balan
email: lavinia.balan@uha.fr

Jean-Pierre Malval

Raphaël Schneider

Didier Le Nouen

Daniel-Joseph Lougnot

In situ fabrication of polyacrylate-silver nanocomposite through photoinduced tandem reactions involving eosin dye

Keywords: Nanocomposite, Silver nanoparticles, Eosin

A silver-acrylate nanocomposite was prepared using a novel one-pot strategy involving eosin dye as visible sensitizer and an amine derivative as radicals source. The mechanism highlighted by steady state photolysis and time resolved absorption spectroscopy lies on the initial formation of a strong ion pair complex between eosin and Ag + . Upon visible irradiation, the excited triplet state of the metal complex efficiently abstracts a hydrogen atom from the amine and produces a -aminoalkyl radical. In acrylate monomer matrix, such a strong reactive species initiates a free radical photopolymerization and also provides the reduction of the silver cation. Through this "in-situ" fabrication method, the kinetics formation of the nanocomposite and its detailed structural analysis are characterized by UV-Visible, real-time FTIR absorption spectroscopy and by transmission electron microscopy.

Introduction

Progresses in size-and shape-selective synthetic routes to produce metal nanoparticles have become the subject of extensive investigations over past decade. Their size-dependent physical properties [1,2] clearly constitute one of the most attractive aspect which has promoted the emergence of a very large set of applications in optics, [3][START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF][START_REF] Ung | [END_REF][6] microelectronics, [7,8] high density data storage, [9,10] catalysis [11,12] and energy conversion. [13,14] Nanocomposite materials, which combine the properties of polymers and silver nanoparticles , are considered to be promising systems for advanced functional applications. [15] For instance, coaxial nanocables [16] with a nanometric silver core and covered by a polymer shell exhibit superior performance as nanocircuits [8] or nanosenors. [7] Such nanowires can be synthesized by either solution-phase methods using capping reactants [17,18] or by templating methods using inorganic porous material, [19] polymer membranes, [20,21] or block copolymer. [22] Another type of nanocomposites consists in a dispersion of metal particles into a polymeric matrix such as poly(vinyl alcohol), [23,24] poly(methyl methacrylate) [25] or poly(vinyl pyrrole) for instance. [26,27] Typically, in-situ and ex-situ methods [28][START_REF] Carotenuto | Metal-Polymer Nanocomposites[END_REF][START_REF] Balan | [END_REF] are the two main synthetic routes that lead to silver-polymer nanocomposites. The ex-situ method consists in dispersing dry silver nanoparticles produced beforehand by an appropriated chemical process into a polymer medium to generate highly ordered periodic arrays of nanoparticles. [31] In the in-situ method, metal nanoparticles are generated directly in the matrix through chemical reduction of a cationic precursor previously dissolved in the bulk. [24] Photochemical reduction by irradiating a dye sensitizer in the presence of silver ions also provides a convenient way to produce embedded nanoparticles in polymer. [21,[32][33][34][35] Moreover, a concomitant functionalization of the nanoparticles with dyes can occur, which is a valuable feature for applications in photochemical solar cells. [14,36] In all cases, the high-performance of nanomaterials depends on the controlled distribution of uniformly shaped and sized particles. Therefore, the development of synthetic strategies to control particles growth and/or agglomeration during nanocomposite fabrication appears quite obviously as a key challenge.

The present work reports a strategy involving the photoinduced formation of homogeneous silver nanoparticles in a polyacrylate polymer stemming from a cross linking photopolymerization of pentaerythritol triacrylate monomer (Scheme 1a). In this one-step approach, a combined system employing eosin dye as visible photosensitizer (Scheme 1b) and N-methyl diethanolamine as co-initiator is used to photogenerate Ag nanoparticles and to photoinitiate a free radical polymerization at the same time. After a mechanistic insight provided by steady state and laser flash photolysis experiments, the analysis of the elementary steps leading to the metal nanocomposite will be correlated to the structural properties of the materials.

Experimental section

Reagents

Silver nitrate (AgNO 3 ) with a purity >99% was purchased from Fluka. Eosin Y disodium salt, N-methyl diethanolamine (MDEA), tetra-n-butylamonium hexafluoroborate were purchased from Aldrich. Mixture of pentaerythritol tri-and tetra-acrylate (PETA) monomers stabilized with monomethyl ether hydroquinone (MEHQ -400 ppm) was purchased from Sartomer. All solvents were Aldrich and Fluka Spectrograde.

Instrumentation

The absorption measurements were carried out with a Perkin Elmer Lambda 2 spectrometer.

A FluoroMax 4 Luminescence Spectrometer was used for the fluorescence and time gated phosphorescence. Low temperature measurements were performed in a glassy matrix of isopropanol using a 5-mm diameter quartz tube inside a Dewar filled with liquid nitrogen. The emission spectra were spectrally corrected in all cases.

Laser flash photolysis at nanosecond time scale was carried out with an Edinburgh Analytical Instruments LP900 equipped with a 450-W pulsed Xe arc lamp, a Czerny-Turner monochromator and a fast photomultiplier. The samples were irradiated with the second harmonic ( = 532 nm, ~ 10 ns, 4 mJ per pulse) of a Nd/YAG Powerlite 9010 from Continuum. The sample was purged with argon for 15 min prior to photophysical studies.

The acrylate formulation, typically Eosin (0.1 wt %), MDEA (3 % wt) and AgNO 3 (1 wt %) in PETA, was sandwiched between two glass plates with a calibrated thickness spacer setting the optical path length to 30 m. Photochemical reactions were carried out at 532 nm with a cw 532 nm Verdi laser from Coherent. The progress of the reaction was monitored via UV-Vis absorption spectra.

The photopolymerization was followed up in situ by real-time Fourier transformed infrared spectroscopy with an AVATAR 360 FTIR spectrometer from Nicolet. The laminated formulation (25 m thick), deposited on a BaF 2 pellet, were irradiated at 532 nm with a green laser diode module from Crystalaser. The conversion rates were deduced from the disappearance of the vinyl C=C stretching vibration band at 1630 cm -1 .

Transmission electron microscopy (TEM) was used to characterize the size and shape of Ag nanoparticles. The nanocomposites were cut by means of a microtome (LKB model 8800) and placed onto the observation grid to get their TEM images. Transmission electron microscopy measurement is carried out at 200 kV using a Philips CM20 instrument with Lab6 cathode.

The 13 C NMR spectra were recorded in CD 3 CN on a 400 MHz Brucker NMR spectrometer using cyclohexane as the internal standard.

Results and discussion

Formation of a strong ion-pair complex with Ag +

The lowest energy absorption band of EO 2-is localized in the 450-575 nm region with a maximum at 530 nm in acetonitrile (Figure 1). This band corresponds to the S 0 -S 1 electronic transition with a * character localized on the xanthene moiety. [37] The geometry optimization by AM1 method reveals that 2-benzoate and xanthene sub-units are twisted with an angle of 77°, which indicates a weak electronic conjugation between the two aromatic moieties. The fluorescence spectrum of EO 2-is mirror image of its absorption band with a slight Stockes shift (535 cm -1 ) suggesting a weak geometrical relaxation in the singlet state.

The 0-0 transition energy measured from the intercept of normalized absorption and fluorescence spectra corresponds to 2.31 eV in acetonitrile. The phosphorescence spectrum of EO 2-is located in the 600-700 nm range with a maximum emission at 652 nm which leads to a triplet energy of about 1.90 eV (inset figure 1).

As depicted in figure 2, the addition of silver nitrate to a solution of EO 2-in acetonitrile leads to a slight red shift of the last absorption band of the chromophore ( max = + 4 nm). The presence of two isosbestic points located at 498 nm and 530 nm clearly indicates the occurrence of equilibrium reaction. Moreover, the addition of tetra-n-butylamonium hexafluoroborate (3 x 10 -2 M) excludes any variation of the ionic strength during the reaction.

Therefore we can ascribe these effects to a coordination reaction of Ag + by EO 2-which leads to the formation ion-pair complex [EO-Ag] -. According to the stronger acidic character of the hydroxylic group with respect to that of the carboxylic one [38,39], such a cation exchange reaction should mainly involve the chelation of silver cation by the "phenolate" oxygen. This is also in line with the Pearson's hard-soft acid-base (HSAB) principle [40][41][42][43] which presumes a better stabilizing interaction of Ag + (Soft Lewis acid) with the phenolate group than with the carboxylate function. However the participation of the carboxylate function to the coordination reaction can not be excluded. In order to determine the reaction constants, the absorption changes as a function of the concentration of AgNO 3 were analyzed according to the model proposed by Valeur et al [44] which is extended to the formation of [1:1] and [1:2] ligand-metal complexes [45]. The inset Figure 2 depicts the best-fitting curve to the experimental data using the method of least squares, log K 1:1 exhibits a high value of ca. 4.7  0.3 whereas log K 1:2 has a value of ca. 5.0  0.4. Such a slight difference suggests that the formation of the [1:2] complex can be reasonably neglected with respect to our working condition (i.e. [Ag + ]/[EO 2-] < 50 ).

Mechanistic aspect of Ag nanoparticles photogeneration in solution

Figure 3 shows the evolution of the transient absorption spectrum of EO 2-in acetonitrile. The spectrum mainly consists of a broad band located in the 560-700 nm range. This band was previously assigned [46][47][48] to the absorption of the triplet of EO 2-( 3 EO 2-), its transient monitored at 580 nm decays mono-exponentially with a lifetime of 45 s. The spectrum also exhibits a bleaching in the 520 nm region which corresponds to the depletion of the ground state. The transient absorption spectrum below 490 nm corresponds to the superimposition of the absorption of three species: the triplet of eosin, the semi-oxidized (EO •-) and semi-reduced (EO •3-) radical ions. The two latter species are generated through a self-quenching of 3 EO 2-by ground state molecules:

3 EO 2-+ EO 2- EO 2-+ EO 2- 3 EO 2-+ EO 2- EO •-+ EO •3-
In acetonitrile, these long-lived radical ions are known [47,48] to have strongly overlapped absorption bands. The maximum absorption wavelength of EO •3-is reported at 430 nm (49) whereas EO •-shows a band centred at 450 nm (39,50).

The addition of 50 eq. of Ag + to a solution of EO 2-leads to a quantitative conversion of the dye to its ion-pair complex, [EO-Ag] -. As shown in Figure 4 This band can be confidently assigned to the absorption of semi-oxidized eosin as compared with that observed by Lindqvist (47) and Grossweiner (48,50,51) . Besides, the transient absorption spectrum exhibits an other band centred at 370 nm. Such a broad band which collapses within the first 2 s after laser pulse corresponds to the well-established band of Ag 0 (17,52,53). Hence we have a clear evidence of a photoinduced electron transfer from 3 EO 2-to Ag + . However, probably due to its relative low reaction rate, such a process can not produce efficiently silver nanoparticles. This is confirmed by steady state photolysis experiment. The changes in the spectral features observed upon laser excitation of eosin dye and AgNO 3 (50 eq.) solution in acetonitrile are depicted in Figure 5A. Upon increasing the irradiation time, the absorption band of EO 2-is progressively collapsing, however we fail to observe the formation of silver nanoparticles. The same photolysis experiment is performed with a solution of acetonitrile containing eosin dye, AgNO 3 and MDEA. In this case, a new band centred around 435 nm and characteristic of the surface plasmon (SP) band of silver nanoparticles is clearly developing (Figure 5B). Obviously, the reduction of Ag + in presence of MDEA is only possible with EO 2-and under visible irradiation. This confirms that the formation of silver nanoparticles is promoted by the excited dye and its mediating role with MDEA. The amine is known to undergo a hydrogen abstraction reaction with the triplet state of eosin ( 3 EO 2-). The sequential reaction first involves an electron transfer reaction from amine to 3 EO 2-then a proton transfer within ion pair formed between amine radical cation and eosin radical anion (49,54). Hence, the reaction should initially produce a -aminoalkyl radical and the conjugated acid of the semi-reduced eosin (EOH •2-). However the latter species which exhibits a transient absorption band around 375 nm is in equilibrium with its conjugate base EO •3-whose absorption band is located at 430 nm. The equilibrium has been observed on flash photolysis of EO 2-in aqueous media, with a pK a of 7 associated with the acidic form (47,48,51). The bimolecular quenching rate constant of 3 EO 2-by MDEA (k q ) can be determined from the dependence of the pseudo-first-order decay of the triplet dye on the concentration of MDEA according to the equation:

k obs = k 0 + k q [MDEA]
In this condition, the bimolecular quenching rate exhibits a value of 2.5( 0.1) x 10 6 M -1 s -1 , in good accordance with literature (54). This low rate constant suggests typical hydrogen atom transfer reaction instead of electron-proton transfer sequence. Parallely to the quenching of triplet eosin, EO •3-is detected at 430 nm. This indirectly confirms the production of the aminoalkyl radical which can not be detected in the spectral window provided by laser flash photolysis.

Figure 6 shows the evolution of the transient absorption spectrum of a solution containing EO 2-, AgNO 3 and MDEA. At very short delay after laser pulse, the absorption band of Ag 0 located at 370 nm reaches its maximum, its transient can be confidently fitted with one exponential leading to a short lifetime of 290 ns. Interestingly, the fast decrease of the metal absorption band reveals the presence of a weak band at 375 nm which is ascribable to the absorption of EOH •2-. As the intensive transient absorption band of EO •3-is concomitantly developing, we have clear indication that the reduction of Ag + does not proceed through the oxidation of EO •3-but through an electron transfer from the -aminoalkyl radical. Carboncentred free radicals with adjacent nitrogen or oxygen are potent reducing agents in solution [55][56][57][58][59]. For instance, it has been reported that the photogenerated alkoxy benzyl radicals [33] or ketyl radicals [60] can convert efficiently Ag + to Ag 0 . Hence, we can assume that the main reaction route leading to the generation of Ag nanoparticles can be described as follows :

3 [EO-Ag] -+ MDEA  EOH •2-+ Ag + + MDEA • Ag + + MDEA •  Ag 0 + MDEA +
The initial coordination of eosin dye with silver cation undoubtedly constitutes an advantage since the photogeneration of the -aminoalkyl radical and its consecutive oxidation by Ag + are occurring in the same reactive cage. In PETA monomer, the ion-pair complex formation appears all the more useful as its presence should strongly counterbalance the effect of a slow diffusion in high viscous medium.

Reduction of Ag + ions in a PETA matrix

The fabrication of metal nanocomposite is performed using pentaerythritol triacrylate (PETA) as monomer. Surprisingly, the dilution of AgNO 3 in PETA matrix leads to a very slow change in colour. The absorption spectrum of the formulation clearly shows the growth of a band centred at 435 nm which corresponds to the SP band of Ag nanoparticles (inset Figure 7). The TEM image of the sample indicates a polydisperse distribution of spherical silver nanoparticles (Figure 7). Hence a reduction of silver cations is occurring in the monomer. The involvement of MEHQ (used as monomer stabilizer) in this reaction can be excluded since no reduction was detected upon addition of large amount of this phenol derivative to a solution of the silver salt in acetonitrile. Therefore the pentaerythritol triacrylate induces a slow reduction of Ag + . Moreover, the 13 C NMR spectrum of the monomer in acetonitrile-d 3 is notably modified in the presence of Ag + . While main part of the NMR spectrum remains invariant in presence of Ag + , the signals of vinylic carbons [61] are gradually shifted by increasing the concentration of AgNO 3 . For instance, upon addition of 10 eq. of Ag + , the signal of methylene carbons ( = 131.51 ppm) is shifted to low-field region (+0.2 ppm) whereas the methine one ( : 128.29 ppm) shifts to high-field region (-0.2 ppm). It should be noted that Ag + is wellknown to form complexes with a large set of alkenes [62][63][64], including methyl acrylate derivatives [65]. Therefore, the formation of a class T complex [64] between Ag + and PETA is proposed. Such a structure implies a trigonal environment around Ag + with a chelation of the cation by the three vinylic functions. As consequence, the close proximity between the hydroxyl group of PETA and Ag + should promote a redox reaction and lead to the formation of Ag nanoparticles. The oxidation of primary alcohols (R-CH 2 OH) by Ag + is also well established, the reaction is slow and requires heating to be accelerated as follows [66][67][68] :

2 Ag + + R-CH 2 OH  2 Ag 0 + 2 H + + R-CHO
However such a reaction is a very slow process and constitutes a minor route for Ag + reduction in comparison with the photoinduced generation of Ag nanoparticles described hereafter.

Fabrication of silver nanoparticle embedded polymer

A formulation of the PETA monomer containing EO 2-(0.1 wt %) and MDEA (3 % wt) [69] and AgNO 3 (1 wt %) is photopolymerized upon irradiation at 532 nm (2.5 mW cm -2 ). The conversion of the acrylate double bonds was monitored from the decrease of the IR band at 1630 cm -1 . In Figure 8 are compared the conversion rate related to this formulation with that of a reference one without AgNO 3 . Both samples show a similar rate of polymerization (within our experimental error) with an acrylate final double bond conversion of about 60 %. This final degree of double bond conversion should be ascribed to the acrylate trifunctionality. As the functionality increased, the viscosity of the resin increased, with the resulting gel-effect and the higher cross linked density, set a limit to the extent of conversion [70]. We can clearly notice that the addition of Ag + does not perturb the photopolymerization kinetics. Therefore, the quantum yield of generation of MDEA • is sufficiently high to enable the free radical photoinitiation and the photogeneration of Ag nanoparticles. Moreover after 10 min exposure, the reference sample turns from pink to colourless whereas the sample with Ag + turned from pink to yellowish (inset Figure 8).

The absorption spectrum of the sample undergoes important changes during visible irradiation (Figure 9). The absorption band of EO 2-decreased progressively whereas the SP developed in the 350-500 nm region. This band exhibited a maximum wavelength at 437 nm with a FWHM of 115 nm. Careful analysis of the absorption spectrum over the illumination period clearly reveals a two-step process :

(i) The first step corresponds to the photobleaching of EO 2-with the concomitant growth of the surface plasmon band as indicated by the linear correlation between the absorbance at 532 nm vs. absorbance at 432 nm (inset of Figure 9).

(ii) In a second step, the SP band continues to develop under irradiation even though the absorption band of EO 2-has completely disappeared. Small particles formed in the first step contribute to the further reduction of Ag + through an autocatalytic process where the residual PETA monomers should act as sacrificial donor. Ultimately, the absorbance of SP stabilized to ca twice that measured at the end of EO 2-bleaching.

Figure 10 shows the linear correlation between the final absorbance reached by the SP band at its maximum and the initial amount of EO 2-; this correlation suggests that the initial step determines the final concentration of nanoparticles in the polymer. The bright field TEM micrographs of silver-nanocomposites displayed in Figure 11 corroborate the correlation between the density of nanoparticles and the initial amount of EO 2-. Throughout the photoreduction process, the symmetrical shape of the SP band denotes a monodisperse size distribution of the silver nanoparticles. Electron microscopy analysis of the polymer indicated the formation of monodisperse spherical metallic particles in nanometer range size. Analysis of a population of ca. hundred particles from a portion of the grid indicated that their average diameter is 5.0 ± 0.9 nm respectively 5.0 ± 0.7 nm. The particles were homogeneous in size and no agglomeration was observed. Such a homogeneous distribution can be accounted by the stabilizing effect of the PETA matrix. Through its acrylate functions, the PETA monomer can easily adsorb on the nanoparticle surface. In this regards, the capping effect of unsaturated long-chain agents was previously observed by Wang et al.[71], who correlated the size of silver particles with the number of double bonds in the chain.

Conclusion

A strong ion pair complex is formed at the ground state between eosin dianion chromophore and Ag + . The triplet state of such a metal complex dye conserves its high hydrogen abstraction propensity toward N-methyl diethanolamine. The -aminoalkyl radicals that are consequently produced upon visible irradiation lead to a consecutive reduction of the coordinated silver cations. Moreover, in triacrylate monomer matrix, the quantum yield of radicals generation is still high and allows a simultaneous free radical polymerization. Hence, an "in situ" synthesis of silver nanoparticle embedded polymer is clearly demonstrated with the advantage that the particles density in the polymer can be tuned by controlling the initial concentration of dye. Finally, the obtained nanocomposite exhibits spherical shaped Ag nanoparticles whose homogeneous dispersion is ascribable to the capping effect of monomer acrylate branches that are not engaged in the polymer net.

FIGURES AND SCHEMES CAPTIONS

  , the transient absorption spectrum remains globally invariant for wavelength above 490 nm. The triplet lifetime measured from the transient at 580 nm drops to a value of 25 s. Consecutive to the triplet quenching, the development of a long time component is observed in the transient monitored at 450 nm (inset Figure 4) which indicates the growth of a new band in this spectral window.

[ 70 ]

 70 Mehnert, R.; Pincus, A.; Janorsky, I.; Stowe, R.; Berejka, A. In Chemistry & Technology of UV & EB Formulation for Coatings, Inks & Paints; SITA Technology Ltd.: London, 1997, p 217. [71] Wang, X.; Naka, K.; Itoh, H.; Park, S.; Chujo, Y. Chem. Commun. 2002, 1300-1301.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 12345678910 Figure 1 : Normalized absorption and fluorescence spectra of EO 2-in acetonitrile. Inset : Time-gated phosphorescence spectrum of EO 2-in glassy matrix of isopropanol (delay : 50 s)

Figure 11 :Scheme 1 :

 111 Figure 11 : TEM image of silver nanoparticles embedded in a polyacrylate matrix, with their respective size distribution. EO 2-/ MDEA / AgNO 3 : (a) 0.06 wt % / 3 wt % / 1 wt % and (b) 0.28 wt % / 3 wt % / 1 wt %

  Figure 9


M L Delle Monache

J Reilly

S Samaranayake

W Krichene

P Goatin

A M Bayen

A PDE-ODE MODEL FOR A JUNCTION WITH RAMP BUFFER

Keywords: Scalar conservation laws, PDE-ODE systems, Riemann problem, Macroscopic traffic flow models AMS subject classifications. 90B20, 35L65

We consider the Lighthill-Witham-Richards traffic flow model on a junction composed by one mainline, an onramp and an offramp, which are connected by a node. The onramp dynamics is modeled using an ordinary differential equation describing the evolution of the queue length. The definition of the solution of the Riemann problem at the junction is based on an optimization problem and the use of a right-of-way parameter. The numerical approximation is carried out using Godunov scheme, modified to take into account the effects of the onramp buffer. We present the result of some simulations and check numerically the convergence of the method.

1. Introduction. Hydrodynamic models have commonly been used in the literature to describe the macroscopic evolution of vehicular traffic on roads and have been successfully generalized to networks in recent years. In the 1950s, Lighthill and Whitham [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF] and Richards [START_REF] Richards | Shock waves on the highway[END_REF], independently, proposed a fluid dynamic model for traffic flow on an infinite single road, using a non-linear hyperbolic partial differential equation (PDE). The Cauchy problem has successfully been extended to initialboundary value problem in [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] and then developed specifically for scalar conservation laws with concave flux in [START_REF] Floch | Explicit formula for scalar non-linear conservation laws with boundary condition[END_REF], and for traffic applications in [START_REF] Strub | Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling[END_REF]. More recently, several authors proposed models on networks that take into account different type of solutions at the intersections, see [START_REF] Coclite | Traffic flow on a road network[END_REF][START_REF] Colombo | Road network with phase transition[END_REF][START_REF] Garavello | The Cauchy problem at a node with buffer[END_REF][START_REF] Garavello | Traffic flow on a road network using the Aw-Rascle model[END_REF][START_REF] Herty | A novel model for intersections of vehicular traffic flow[END_REF][START_REF] Herty | Optimization criteria for modelling intersections of vehicular traffic flow[END_REF] and the references therein.

In this article, we focus on a junction model designed for a ramp metering problem. Ramp metering models have been introduced in the engineering community in a discrete setting, see [START_REF] Muralidharan | Freeway traffic flow simulation using the link node cell transmission model[END_REF][START_REF] Muralidharan | Optimal control of freeway networks based on the link node cell transmission model[END_REF] for details. In this article, we apply a continuous approach. We consider the scalar Lighthill-Whitham-Richards model on a network composed of a single junction connecting a mainline, an onramp and an offramp. The mainline evolution is described by a scalar conservation law, while the onramp dynamics is modeled by a buffer of infinite capacity, which is defined by an ordinary differential equation (ODE) depending on the difference between the incoming and outgoing fluxes at the ramp.

In the following sections, we prove the existence and uniqueness of solutions of the Riemann problem at the junction. The results are obtained by solving a Linear Programming (LP ) optimization problem. Unlike [START_REF]Traffic Flow on Networks: Conservation Laws Model[END_REF], where the flux through the junction is maximized, our LP -optimization consists in maximizing the flux on the outgoing mainline, see Remark 3 below. The offramp is treated as a sink, and a priority parameter is introduced to ensure uniqueness of the solution. As a modeling choice, the priority is satisfied in an approximate way, i.e., the priority will not always be respected, in benefit of flux maximization.

The article presents numerical approximations of possibly discontinuous solutions obtained using this model. In particular, we suitably modify the Godunov scheme to include the boundary conditions at the junction, as in [START_REF] Bretti | Numerical approximations of a traffic flow model on networks[END_REF], and the ODE describing the buffer. This allows one to take into account the possible creation of an additional shock when the buffer empties. The scheme provides accurate numerical approximations, as shown by the numerical tests provided here.

The article is organized as follows. Section 2 contains some preliminary notations and definitions, while §3 describes in details the solution of the Riemann problem at the junction. In §4 we introduce the numerical scheme with the particular boundary conditions used to compute approximate solutions to the problem. In §5 we present some numerical tests which show the effectiveness of our approximation. Throughout the paper, we refer to [START_REF] Bressan | Hyperbolic systems conservation laws[END_REF][START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics[END_REF] for the general theory of hyperbolic conservation laws and to [START_REF] Leveque | Numerical Methods for Conservation Laws[END_REF][START_REF]Finite Volume Methods for Hyperbolic Problems[END_REF] for an introduction to the main numerical concepts.

2. Fundamental Definitions and Notations. We consider a junction with one mainline I modeled by the real line ] -∞, +∞[, one onramp R 1 and one offramp R 2 at x = 0, as illustrated in Figure 2.1.

J I 1 I 2 R 1 R 2
Fig. 2.1: Junction modeled in the article.

From a macroscopic point of view, this means that on each mainline segment I 1 =] -∞, 0[ and I 2 =]0, +∞[, we consider the mass conservation equation:

∂ t ρ + ∂ x f (ρ) = 0, (t, x) ∈ R + × I i , (2.1)
where ρ = ρ(t, x) ∈ [0, ρ max ] is the mean traffic density, ρ max is the maximal density allowed on the road and the flux function f : [0, ρ max ] → R + is given by the following flux-density relation

f (ρ) = ρv(ρ),
where v(ρ) is a smooth decreasing function denoting the mean traffic speed. Throughout the article, we assume for simplicity that: On the onramp R 1 we consider the presence of a buffer modeled by the following ODE:

(A1) ρ max = 1; (A2) f (0) = f (1) = 0; ( A3
dl(t) dt = F in (t) -γ r1 (t), t ∈ R + , (2.2)
where l(t) ∈ [0, +∞[ is the length of the queue, F in (t) is the flux that enters the onramp and γ r1 (t) is the flux that exits from the onramp. This particular choice is taken to avoid backward waves on the onramp boundary, which happens in the case of horizontal queues. In particular, at the left boundary of the onramp, backward moving shock waves can result in lost information on the flux that actually enters the buffer. The presence of the buffer helps accounting for all the flow that enters the onramp, for details see [START_REF] Krichene | A continuous model for traffic networks[END_REF]. We consider the offramp as an infinite sink that accepts all the flux entering from the mainline I 1 , and we assume that no flux from the onramp is allowed in the offramp.

The Cauchy problem to solve is then:

         ∂ t ρ i + ∂ x f (ρ i ) = 0, (t, x) ∈ R + × I i , i = 1, 2 dl(t) dt = F in (t) -γ r1 (t), t ∈ R + , ρ i (0, x) = ρ i,0 (x), on I i i = 1, 2 l(0) = l 0 , (2.3)
where ρ i (0, x) represents the initial condition and l 0 ∈ [0, +∞[ is the initial load of the buffer.

This will be coupled with an optimization problem at the junction which will give the distribution of the traffic among the roads. We define the demand d(F in , l) of the onramp, the demand function δ(ρ 1 ) on the incoming mainline segment corresponding to the density ρ 1 , and the supply function σ(ρ 2 ) on the outgoing mainline segment corresponding to the density ρ 2 as follows.

d(F in , l)= γ max r1 if l(t) > 0, min (F in (t), γ max r1 ) if l(t) = 0, (2.4) δ(ρ 1 ) = f (ρ 1 ) if 0 ≤ ρ 1 < ρ cr , f max if ρ cr ≤ ρ 1 ≤ 1, (2.5) σ(ρ 2 ) = f max if 0 ≤ ρ 2 ≤ ρ cr , f (ρ 2 ) if ρ cr < ρ 2 ≤ 1, (2.6)
where γ max r1 is the maximal flow on the onramp and f max = f (ρ cr ) is the maximal flux on I 1 and I 2 . Moreover, we introduce β ∈ [0, 1] the split ratio of the offramp, and

γ r2 (t) = βf (ρ 1 (t, 0-)) its flux. Definition 2.1. A triple (ρ 1 , ρ 2 , l) ∈ 2 i=1 C 0 R + ; L 1 ∩ BV(R) × W 1,∞ (R + ; R + ) is an admissible solution to (2.3) if (i) ρ 1 ,ρ 2 are weak solutions on I 1 , I 2 , i.e., ρ i : [0, +∞[×I i → [0, 1], i = 1, 2, such that R + Ii ρ i ∂ t ϕ i + f (ρ i )∂ x ϕ i dxdt = 0, i = 1, 2, (2.7)
for every ϕ i ∈ C 1 c (R + × I i ). (ii) ρ i satisfies the Kružhkov entropy condition [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF] on (R × I i ), i.e., for every k ∈ [0, 1] and for all

ϕ i ∈ C 1 c (R + × I i ), t > 0, R + Ii (|ρ i -k|∂ t ϕ i + sgn (ρ i -k)(f (ρ i ) -f (k))∂ x ϕ i )dxdt + Ii |ρ i,0 -k|ϕ i (0, x)dx ≥ 0; i = 1, 2. (2.8) (iii) f (ρ 1 (t, 0-)) + γ r1 (t) = f (ρ 2 (t, 0+)) + γ r2 (t).
(iv) The flux of the outgoing mainline f (ρ 2 (t, 0+)) is maximum subject to

f (ρ 2 (t, 0+)) = min (1 -β)δ(ρ 1 (t, 0-)) + d(F in (t), l(t)), σ(ρ 2 (t, 0+)) , (2.9
)

and (iii) (v) l is a solution of (2.2) for a.e. t ∈ R + . Remark 1.
A parameter P is introduced to ensure uniqueness of the solution. P ∈ ]0, 1[ is a right of way parameter that defines the amount of flux that enters the outgoing road from each incoming road. In particular, P f (ρ 1 (t, 0-)) is the flux allowed from the incoming mainline into the outgoing mainline, and (1 -P )γ r1 (t) the flux from the onramp.

3. Riemann problem at the junction. In this section, we construct step by step the Riemann Solver at the junction. We fix constants ρ 1,0 , ρ 2,0 ∈ [0, 1], l 0 ∈ [0, +∞[, F in ∈]0, +∞[ and a priority factor P ∈]0, 1[. The Riemann problem at J is the Cauchy problem (2.3) where the initial conditions are given by ρ 0,i (x) ≡ ρ 0,i in I i for i = 1, 2. We define the Riemann Solver by means of a Riemann Solver RSl, which depends on the instantaneous load of the buffer l. For each l the Riemann Solver RSl is constructed in the following way.

1. Define Γ 1 = f (ρ 1 (t, 0-)), Γ 2 = f (ρ 2 (t, 0+)), Γ r1 = γ r1 (t); 2. Consider the space (Γ 1 , Γ r1 ) and the sets O 1 = [0, δ(ρ 1 )], O r1 = [0, d(F in , l)]; 3. Trace the lines (1 -β)Γ 1 + Γ r1 = Γ 2 ; and Γ 1 = P 1-P Γ r1 ; 4. Consider the region Ω = (Γ 1 , Γ r1 ) ∈ O 1 × O r1 : (1 -β)Γ 1 + Γ r1 ∈ [0, Γ 2 ] . (3.1)
Different situations can occur depending on the value of Γ 2 :

(i) Demand limited case:

Γ 2 =(1 -β)δ(ρ 1 (t, 0-)) + d(F in , l). We set Q to be the point ( Γ1 , Γr1 ) such that Γ1 = δ(ρ 1 (t, 0-)), Γr1 = d(F in , l) and Γ2 = (1 -β)δ(ρ 1 (t, 0-)) + d(F in , l), as illustrated in Figure 3.1(a). (ii) Supply limited case: Γ 2 = σ(ρ 2 (t, 0+)). We set Q to be the point of intersection of (1 -β)Γ 1 + Γ r1 = Γ 2 and Γ 1 = P 1-P Γ r1 . If Q ∈ Ω, we set ( Γ1 , Γr1 )=Q and Γ2 = Γ 2 , see Figure 3.1(b); if Q / ∈ Ω, we set ( Γ1 , Γr1 )=S and Γ2 = Γ 2 , where S is the point of the segment Ω ∩ (Γ 1 , Γ r1 ) : (1 -β)Γ 1 + Γ r1 = Γ 2 closest to the line Γ 1 = P 1-P Γ r1 , obtained solving the problem minimize γ r1 (t) f (ρ 1 (t, 0-)) - γ r1 (t) f (ρ 1 (t, 0-)) • α P α P 2 2 subject to f (ρ 2 (t, 0+)) = (1 -β)f (ρ 1 (t, 0-)) + γ r1 (t), γ r1 (t) ≤ d(F in , l), f (ρ 1 (t, 0+)) ≤ δ(ρ 1 ), (3.2)
where α P is the normalized vector α P = 1

P 2 + (1 -P ) 2 P 1 -P , see Fig.3.1(c).
As can be seen in Figure 3.1(c), it might not be possible to respect the priority given by the parameter P if we want to maximize also the flux.

Once we have determined Γ1 and Γ2 , we can define ρ1 , ρ2 in a unique way as follows. We recall that ρ = ρ cr ∈ ]0, 1[ is the unique point of maximum of the flux and we define the function τ as follows, for details see [START_REF]Traffic Flow on Networks: Conservation Laws Model[END_REF]. Definition 3.1. Let τ : [0, 1] → [0, 1] be the map such that: For the incoming road the solution is given by the wave (ρ 1,0 , ρ1 ), while for the outgoing road the solution is given by the wave (ρ 2 , ρ 2,0 ). In this setting, given any initial data ρ 1,0 , ρ 2,0 , we can define RSl :

(i) f (τ (ρ)) = f (ρ) for every ρ ∈ [0, 1]; (ii) τ (ρ) = ρ for every ρ ∈ [0, 1] \ {ρ cr }. Given ρ 1 (0, •) ≡ ρ 1,0 , ρ 2 (0, •) ≡ ρ 2,0 , there exists a unique couple (ρ 1 , ρ2 ) ∈ [0, 1] 2 such that ρ1 ∈ {ρ 1,0 }∪]τ (ρ 1,0 ), 1] if 0 ≤ ρ 1,0 ≤ ρ cr , [ρ cr , 1] if ρ cr ≤ ρ 1,0 ≤ 1; f (ρ 1 ) = Γ1 , (3.3) and ρ2 ∈ [0, ρ cr ] if 0 ≤ ρ 2,0 ≤ ρ cr , {ρ 2,0 } ∪ [0, τ (ρ 2,0 )[ if ρ cr ≤ ρ 1,0 ≤ 1; f (ρ 2 ) = Γ2 . ( 3
[0, 1] 2 → [0, 1] 2 by RSl(ρ 1,0 , ρ 2,0 ) = (ρ 1 , ρ2 ). (3.5)
Now given the initial load of the buffer l 0 = l, the function l(t) at time t > 0 is given according to the following possibilities, determined by straight integration of (2.2):

1. If F in < Γr1 , then l(t) = l 0 + (F in -Γr1 )t if 0 < t < l0 Γr1-Fin , 0 if t > l0 Γr1-Fin . (3.6) 2. If F in ≥ Γr1 , then l(t) = l 0 + (F in -Γr1 )t ∀t > 0. (3.7)
Remark 2. The presence of the buffer can create waves at time t > 0. In particular, we have new waves at time t = -l 0 /(F in -Γr1 ) (with new values of the densities ρ1 , ρ2 ) if F in < Γr1 and the buffer empties, see Figure 3.2. No waves are created instead if F in ≥ Γr1 , due to the infinity capacity of the buffer. The following theorem ensures consistency of RSl. Theorem 3.2. Consider a junction J and fix a priority parameter P ∈ ]0, 1[. For every ρ 1,0 , ρ 2,0 ∈ [0, 1] and l 0 ∈ [0, +∞[, there exists a unique admissible solution (ρ 1 (t, x), ρ 2 (t, x), l(t)) in the sense of Definition 2.1 satisfying the priority (possibly in an approximate way). Moreover, for a.e. t > 0, it holds

(ρ 1 (t, 0-), ρ 2 (t, 0+)) = RS l(t) (ρ 1 (t, 0-), ρ 2 (t, 0+)).
The proof of the theorem is deferred after some preliminary results. The mapping τ (ρ) as defined in (3.1) and the functions δ(ρ 1 ), d(F in , l) and σ(ρ 2 ) yield the following properties.

Lemma 3.3. If (ρ 1 , ρ2
) is a solution of the Riemann problem with initial data (ρ 1,0 , ρ 2,0 ), then the following holds:

δ(ρ 1,0 ) ≤ δ(ρ 1 ), σ(ρ 2,0 ) ≤ σ(ρ 2 ), d(F in , l 0 ) ≤ d(F in , l).
Proof. For the incoming road it holds: Using the same approach for the outgoing road, we have:

δ(ρ 1,0 ) ≤ δ(ρ 1 ) if 0 ≤ ρ 1,0 ≤ ρ cr , δ(ρ 1,0 ) = δ(ρ 1 ) if ρ cr ≤ ρ 1,0 ≤ 1. (3.8) In particular, if ρ 1,0 ∈ [0, ρ cr ], either ρ1 ∈ ]τ (ρ 1,0 ), 1] or ρ1 = ρ 1,0 . In the first case, δ(ρ 1,0 ) = f (ρ 1,0 ) ≤ f max = δ(ρ 1 ), see Figure 3.3(a), while in the second case δ(ρ 1,0 ) = f (ρ 1,0 ) = f (ρ 1 ) = δ(ρ 1 ), see Figure 3.3(b). On the other hand, if ρ 1,0 ∈ [ρ cr , 1] then ρ1 ∈ [ρ cr , 1] and δ(ρ 1,0 ) = f max = δ(ρ 1 ), see Figure 3.3(c). ρ , f (ρ) δ(t) Shock ρ1 ρ 1,0 δ(ρ 1,0 ) δ(ρ 1 ) = f max (ρ) (a) 0 ≤ ρ 1,0 ≤ ρ cr ρ , f (ρ) δ(ρ) ρ 1,0 = ρ1 δ(ρ 1,0 ) = δ(ρ 1 ) f max (ρ) (b) ρ 1,0 = ρ1 Rarefaction Shock ρ , f (ρ) δ(t) ρ 1,0 f max (ρ) = δ(ρ 1,0 ) = δ(ρ 1 ) (c) ρ cr ≤ ρ 1,0 ≤ 1
σ(ρ 2,0 ) = σ(ρ 2 ) if 0 ≤ ρ 2,0 ≤ ρ cr , σ(ρ 2,0 ) ≤ σ(ρ 2 ) if ρ cr ≤ ρ 2,0 ≤ 1, (3.9)
In particular, if 0 ≤ ρ 2,0 ≤ ρ cr then 0 ≤ ρ2 ≤ ρ cr as well, and σ(ρ 2,0 ) = f max = σ(ρ 2 ), see Figure 3.4(c). Otherwise ρ cr ≤ ρ 2,0 ≤ 1, and either ρ2 = ρ 2,0 or ρ2 ∈ [0, τ (ρ 2,0 )[. In the first case, σ(ρ 2,0 ) = f (ρ 2,0 ) = f (ρ 2 ) = σ(ρ 2 ), see Figure 3.4(b). In the second case, σ(ρ 2,0 ) = f (ρ 2,0 ) ≤ f max = σ(ρ 2 ), see Figure 3.4(a).

For the onramp, we consider two different cases, when the buffer is initially empty and when it is not. In both cases different situations can occur.

(L1) Initially empty buffer: Proof. Existence and uniqueness follow by construction of the Riemann Solver detailed at the beginning of this section. In the following we will show the proof of the consistency of RS l(t) . Fix t 0 ≥ 0. If (ρ 1 (t 0 , 0-), ρ 2 (t 0 , 0+)) is a solution of the Riemann Solver, corresponding to the same buffer value l(t 0 ) we need to show that

l(0) = 0 ⇒ d(F in , l(0)) = min (F in , γ max r1 ). (L1.1) Buffer increases: l(0+) > 0 ⇒ d(F in , l(0+)) = γ max r1 . If d(F in , l(0)) = F in , then d(F in , l(0)) ≤ d(F in , l(0+)). If d(F in , l(0)) = γ max r1 , then d(F in , l(0)) = d(F in , l(0+)). (L1.2) Buffer remains empty: l(0+) = 0 ⇒ d(F in , l(0+)) = min (F in , γ max r1 ). Hence, d(F in , l(0)) = d(F in , l(0+)). (L2) Buffer initially not empty: l(0) > 0 ⇒ d(F in , l(0)) = γ max r1 . (L2.1) Buffer grows (decreases) linearly: 0 < l(0) < l(0+) (0 < l(0+) < l(0)) ⇒ d(F in , l(0+)) = γ max r1 . Hence, d(F in , l(0)) = d(F in , l(0+)). Shock ρ , f (ρ) σ(ρ) σ(ρ 2,0 ) ρ2 ρ 2,0 f max (ρ) = σ(ρ 2 ) (a) ρ cr ≤ ρ 2,0 ≤ 1 ρ , f (ρ) σ(ρ) σ(ρ 2,0 ) = σ(ρ 2 ) ρ 2,0 = ρ2 f max (ρ) (b) ρ 2,0 = ρ2 Rarefaction Shock ρ , f (ρ) σ(ρ) ρ 2,0 f max (ρ) = σ(ρ 2,0 ) = σ(ρ 2 ) (c) 0 ≤ ρ 2,0 ≤ ρ cr
RS l(t0) (ρ 1 (t 0 , 0-), ρ 2 (t 0 , 0+)) = (ρ 1 (t 0 , 0-), ρ 2 (t 0 , 0+)).
Without loss of generality, we fix t 0 = 0 and we keep the same notation used in the proof of Lemma 3.3. We show that the optimal point in the feasible set Ω, as defined in (3.1), resulting from the Riemann Solver does not change. From the results of Lemma 3.3 it is straightforward to say that the set Ω either increases between times t = 0 and t > 0 or does not change, see Figures 3.5 and 3.6. Now, we need to prove that the optimal point does not change. We treat the supply constrained and the demand limited cases separately.

Γ r1 d(F in , l 0 ) d(F in , l(0+)) Γ 1 σ(ρ 2,0 ) = σ(ρ 2 ) δ(ρ 1,0 ) δ(ρ 1 ) Q (a) Q ∈ Ω Γ r1 d(F in , l 0 ) d(F in , l(0+)) Γ 1 σ(ρ 2,0 ) = σ(ρ 2 ) δ(ρ 1,0 ) = δ(ρ 1 ) S Q (b) Q / ∈ Ω, Γ1 = δ(ρ 1,0 ) Γ r1 d(F in , l 0 ) = d(F in , l(0+)) δ(ρ 1 ) Γ 1 σ(ρ 2,0 ) = σ(ρ 2 ) δ(ρ 1,0 ) S Q (c) Q / ∈ Ω, Γr1 (0) = d(F in , l 0 )
Fig. 3.5: Supply constrained junction problems.

1. Supply constrained junction problem, see Figure 3.5. We assume that we are supply limited at t = 0. In this case, by construction of the Riemann Solver, it holds ρ 2,0 = ρ2 and hence σ(ρ 2,0 ) = σ(ρ 2 ). The priority line is fixed, the point of intersection Q does not change.

(i) Optimal solution inside Ω, see Figure 3.5(a). Since Q is determined by the intersection of the two lines and Ω can only increase (δ(ρ 1,0 ) ≤ δ(ρ 1 ), d(F in , l(0)) ≤ d(F in , l(0+))), we have

(ρ 1 , ρ2 ) = RS l(0) (ρ 1 , ρ2 ).
(ii) Optimal solution on the border of Ω, Γ1 = δ(ρ 1,0 ), see Figure 3.5(b). We have to prove that the result of the minimization problem (3.2) (the point S in the figure) does not change. In this case, by construction of the Riemann Solver it holds ρ 1,0 = ρ1 . This yields δ(ρ 1,0 ) = δ(ρ 1 ) by (3.8). Since, d(F in , l(0)) can only increase according to the cases (L1) and (L2), it holds

(ρ 1 , ρ2 ) = RS l(0) (ρ 1 , ρ2 ).
(iii) Optimal solution on the border of Ω, Γr1 (0) = d(F in , l(0)), see Figure 3.5(c). For the onramp, the only case where the demand can increase is the case (L1.1). In this particular setting, if d(F in , l(0)) = F in it holds γ r1 (0) = F in and F in ≤ γ max r1 . When the buffer increases we have γ r1 (0+) = d(F in , l(0+)) = γ max r1 , which implies γ max r1 ≤ F in . Hence, F in = γ max r1 and d(F in , l(0)) = d(F in , l(0+)). The mainline demand can only increase. Hence, (ρ 1 , ρ2 ) = RS l(0) (ρ 1 , ρ2 ).

2. Demand constrained junction problem, see Figure 3.6. Ω = Ω(ρ 1,0 , l(0)) = Ω(ρ 1 , l(0)). In fact, ρ 1,0 = ρ1 and for the onramp it holds γ max r1 = F in (as in the previous point), and this yields δ(ρ 1,0 ) = δ(ρ 1 ) and d(F in , l(0)) = d(F in , l(0+)) by (3.8) and (L1) and (L2). The supply can only increase by (3.9). Hence, Moreover, the limiting side of Ω does not change, i.e., it is not possible to pass from a demand constrained junction problem to a supply constrained one and viceversa. This follows for the fact that σ(ρ 2,0 ) = σ(ρ 2 ) when we have a supply constrained junction problem, Figure 3.5 and d(F in , l(0)) = d(F in , l(0+)), δ(ρ 1,0 ) = δ(ρ 1 ) when we have a demand constrained junction problem, Figure 3.6. This concludes the proof.

(ρ 1 , ρ2 ) = RS l(0) (ρ 1 , ρ2 ). Γ r1 d(F in , l(0) = d(F in , l(0+)) Γ 1 δ(ρ 1,0 ) = δ(ρ 1 ) σ(ρ 2 ) σ(ρ 2,0 ) Q Fig. 3
Remark 3. The proposed model is a variant of the junction model considered in [START_REF] Coclite | Traffic flow on a road network[END_REF] in the 2 × 2 case. In [START_REF] Coclite | Traffic flow on a road network[END_REF], the traffic distribution across the junction is given by a distribution matrix A, subject to technical conditions that ensure uniqueness of the solution. In our case, since we suppose that no flux from the onramp is directed into the offramp, the distribution matrix would look as:

A = 1 -β 1 β 0 . Clearly,
as the offramp gets more congestioned, β decreases. If we solve the model proposed in [START_REF] Coclite | Traffic flow on a road network[END_REF] using this distribution matrix, there can be cases in which the solution gives zero onramp flux, see Figure 3.7(a). This is due to the choice of maximizing the flow throughout the junction. In fact, in this way, the model tends to satisfy the mainline demand before the onramp one. This does not reflect what happens in reality, since the demand allocation depends on the number of lanes available for each inflow, for details see [START_REF] Krichene | A continuous model for traffic networks[END_REF]. Hence, we propose a model that fix this issue balancing the flux between the two incoming roads by the introduction of a right-of-way parameter. In particular, the priority coefficient keeps the maximization point far from the axis, avoiding blocking, see Figure 3.7(b). 4. Numerical Approximation. In order to find approximate solutions, we adapt the classical Godunov scheme to the problem, with some adjustment due to the presence of the buffer. We define a numerical grid in (0, T ) × R using the following notation:

Γ r1 d(F in , l) Γ 1 δ(ρ 1,0 ) σ(ρ 2,0 ) = (1 -β)γ 1 + γ r1 Q (a) Γ r1 d(F in , l) Γ 1 δ(ρ 1,0 ) σ(ρ 2,0 ) = (1 -β)γ 1 + γ r1 Γ 1 = P 1-P γ r1 Q (b)
(i) ∆x is the fixed space grid size; (ii) ∆t n is the non-uniform time grid size given by the CFL condition;

(iii) (t n , x j ) = (t n-1 + ∆t n , j∆x) for n ∈ N and j ∈ Z are the grid points. For a function v defined on the grid we write v n j = v(t n , x j ) for j, n varying on a subset of Z and N respectively. We also use the notation u n j for u(t n , x j ) when u is a continuous function on the (t, x) plane.

4.1. Godunov Scheme. The Godunov scheme as introduced in [START_REF] Godunov | A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics[END_REF] is based on exact solutions to Riemann problems. The main idea of this method is to approximate the initial datum by a piecewise constant function, then the corresponding Riemann problems are solved exactly and a global solution is simply obtained by piecing them together. Finally, one takes the mean on the cell and proceeds by induction. Consider the hyperbolic problem with initial data

∂ t u + ∂ x f (u) = 0 x ∈ R, t ∈ [0, T ], (4.1)
u(0, x) = u 0 (x) x ∈ R. (4.2)
A solution of the problem is constructed taking a piecewise constant approximation of the initial data:

v 0 j = 1 ∆x x j+1/2 x j-1/2 u 0 (x)dx j ∈ Z, (4.3)
and defining v n j recursively starting from v 0 j , as follows.

v n+1 j = 1 ∆x x j+1/2 x j-1/2 v ∆ (t n+1 , x)dx. (4.4)
Under the CFL condition

∆t n max j∈Z λ n j+ 1 2 ≤ 1 2 ∆x (4.5)
the waves generated by different Riemann problems do not interact. Above, λ n j+ 1 2 is the wave speed of the Riemann problem solution at the interface x j+ 1 2 at time t n . Under the condition (4.5) the scheme can be written as

v n+1 j = v n j - ∆t n ∆x (g(v n j , v n j+1 ) -g(v n j-1 , v n j )), (4.6)
where numerical flux g takes in general the following expression:

g(u, v) = min z∈[u,v] f (z) if u ≤ v, max z∈[v,u] f (z) if v ≤ u. (4.7)
4.2. Boundary conditions and conditions at the junctions. Here we impose the boundary conditions for the incoming and the outgoing roads at the endpoint not connected to the junction. We also assign boundary conditions at the endpoints of the roads connected to the junction. In both cases we will use the classical approach for road networks as introduced in [START_REF] Bretti | Numerical approximations of a traffic flow model on networks[END_REF].

Boundary conditions. Each road is divided in J + 1 cells numbered from 0 to J. For the incoming road, in practice, we proceed defining

v n+1 0 = v n 0 - ∆t n ∆x (g(v n 0 , v n 1 ) -f (v n 0 )),
where f (v n 0 ) is the value of the flux at the boundary. An outgoing boundary can be treated analogously,

v n+1 J = v n J - ∆t n ∆x (f (v n J ) -g(v n J-1 , v n J )),
with f (v n J ) the outgoing flux. Since we are dealing with Riemann problems at the junction, the formulation of absorbing boundary conditions is equivalent to the one with the ghost cells which is common in literature, see for details [START_REF] Bretti | Numerical approximations of a traffic flow model on networks[END_REF][START_REF] Coclite | Traffic flow on a road network[END_REF][START_REF] Work | A distributed highway velocity model for traffic state reconstruction[END_REF].

Conditions at the Junction. For I 1 , that is connected at the junction at the right endpoint, we set

v n+1 J = v n J - ∆t n ∆x ( Γ1 -g(v n J-1 , v n J )),
while for the outgoing road, connected at the junction at the left endpoint, we have

v n+1 0 = v n 0 - ∆t n ∆x (g(v n 0 , v n 1 ) -Γ2 ),
where Γ1 and Γ2 are the maximized fluxes computed in §3. Remark 4. For Godunov scheme there is no need to invert the flux f to compute the corresponding density in the scheme, as the Godunov flux coincides with Riemann flux. In this case it suffices to insert the computed maximized fluxes directly into the scheme.

ODE Treatment.

Let us consider now the buffer modeled by (2.2) on the onramp. At each time step t n = t n-1 + ∆t n we compute the new value of the queue length according to two possible cases, with Euler first order integration.

1. If F in (t n ) < Γr1

l n+1 = l n + (F in (t n ) -Γr1 )∆t n for t n+1 < t, 0 otherwise . 2. If F in (t n ) ≥ Γr1 l n+1 = l n + (F in (t n ) -Γr1 )∆t n .
Above, Γr1 is the maximized flux described in §3, F in (t n ) is the flux entering the onramp at t n given by

F in (t n ) = 1 ∆t n t n+1 t n F in (t)dt,
and t is the time at which the buffer empties. We can calculate the time at which the buffer can empty for each time step ∆t n :

t = - l n F in (t n ) -Γr1 + t n . (4.8)
4.4. Modified Godunov Scheme. Godunov scheme cannot be applied as it is when the buffer empties, because the solution could potentially not be self-similar. If the buffer empties, at some time step ∆t n , we might have multiple shocks at the junction. In this case we divide the time step ∆t n = (t n , t n+1 ) in two sub-intervals ∆t a = (t n , t) and ∆t b = ( t, t n+1 ), as in Figure 4.1, with t being defined in (4.8). Then, we solve in one time step two different Riemann Problems at the junction. For ∆t a we solve the classical Godunov scheme. For the ∆t b we solve a new Riemann Problem at the junction in which the value of the queue length is l = 0. The junction conditions are

v n+1 J = v t J - ∆t b ∆x Γt 1 -g(v n J-1 , v t J ) , (4.9)
v n+1 0 = v t 0 - ∆t b ∆x g(v t 0 , v n 1 ) - Γt 2 , (4.10)
where with the superscript t we indicate the value computed at t = t in the previous time step ∆t a . 5.1: Errors and order of convergence for the Godunov scheme at time T = 10, corresponding to initial data (5.4).

Case II:

We consider the following initial data ρ 1 (0, x) = 0.1, ρ 2 (0, x) = 0.6.

(5.5)

In this case, the values of the initial conditions are chosen such that the wave produced by the buffer that empties can be seen in the outgoing mainline. In particular, in this case no waves are generated at initial time. The only wave generated is a shock which appear once that the buffer empties at time t = 1.53, as shown in Figure 5.2. Table 5.2 reports the L 1 -error and the order of convergence. 5.2: Errors and order of convergence for the Godunov scheme at time T = 3, corresponding to initial data (5.5).

Conclusions.

This article introduces a model of a 2 × 2 junction with an onramp and an offramp. The onramp is modeled by an ODE which represent a vertical buffer. This way of handling boundary conditions makes possible not to lose flow information. The junction flow distribution is solved through a LP -optimization problem, which maximizes the flow in the outgoing mainline. Moreover, a right-ofway parameter is introduced to ensure the uniqueness of the solution and a good representation of field experiences. The model is solved numerically using a modified Godunov scheme that takes into account the waves that can be produced when the buffer empties. Some numerical tests are presented to show the stability and accuracy of the scheme. Moreover, from the analysis of the L 1 error, the convergence of the scheme is demonstrated numerically.

Fig. 2 . 2 :

 22 Fig. 2.2: Flux function of equation (2.1), commonly referred to as fundamental diagram in the transportation literature.

. 4 )Fig. 3 . 1 :

 431 Fig. 3.1: Solutions of the Riemann Solver at the junction.

Fig. 3 . 2 :

 32 Fig. 3.2: Solution of the Riemann Problem.

Fig. 3 . 3 :

 33 Fig. 3.3: Instantaneous evolution of the demand in the Riemann problem (incoming road).

Fig. 3 . 4 :

 34 Fig. 3.4: Instantaneous evolution of the supply in the Riemann problem (outgoing road).

. 6 :

 6 Fig. 3.6: Demand constrained junction problem.

Fig. 3 . 7 :

 37 Fig. 3.7: Comparison between Coclite-Garavello-Piccoli model [4](left) and our model (right).

Fig. 5 . 1 :

 51 Fig. 5.1: Evolution in time of the density in the incoming mainline (above, left), on the outgoing mainline (above, right) and evolution of the flux in the onramp (bottom, left) and in the offramp (bottom, right), corresponding to initial data (5.4) and a space step discretization ∆x = 0.01.

Fig. 5 . 2 :

 52 Fig. 5.2: Evolution in time of the density in the incoming mainline (above, left), on the outgoing mainline (above, right) and evolution of the flux in the onramp (bottom, left) and in the offramp (bottom, right), corresponding to initial data (5.5) and a space step discretization ∆x = 0.01.

Table 5 .

 5 1 collects the values of the L 1 -error and of the order of convergence at time T = 10.

	dx	L 1 -error	µ
	0.02 3.69 • 10 -2 0.8432
	0.01 1.49 • 10 -2 0.9133
	0.005 7.21 • 10 -3 0.9322
	0.002 1.10 • 10 -3 1.0962
	0.001 2.23 • 10 -4 1.2170
	Table

DELLE MONACHE, REILLY, SAMARANAYAKE, KRICHENE, GOATIN AND BAYENΓ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Q (a) Demand limited case Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = P 1-P Γ r1 Q (b) Supply limited case: intersection inside Ω Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = P 1-P Γ r1 Q S Γ r1 d(F in , l) Γ 1 δ(ρ 1 ) Γ 2 = (1 -β)Γ 1 + Γ r1 Γ 1 = P 1-P Γ r1 Q S(c) Supply limited case: intersection outside Ω

The authors have been supported by the ERC Starting Grant 2010 under the project 'TRAffic Management by Macroscopic Models", the INRIA associated team 'Optimal REroute Strategies for Traffic managEment" and the France-Berkeley Fund under the project 'Optimal Traffic Flow Management with GPS Enabled Smartphones".

5. Numerical Results. For illustration, we choose a concave fundamental diagram with the following flux function:

where V max is the maximal velocity of the traffic flow. In this case the Godunov numerical flux is given by

(5.1)

In this section we present some numerical tests performed with the scheme previously described. We introduce the formal order of convergence µ of a numerical method

where the L 1 -norm error is given by

where u i e and u i c are the exact solution and the computed solution in each road, respectively. We show some numerical results obtained applying Godunov scheme to problem (2.3). Tables 5.1 and 5.2 provide the values of the L 1 -error (5.3) and the order of convergence (5.2). Here we deal with a mainline of length 8 parametrized by the interval [-4, 4] with the node placed at x = 0, such that I 1 = [-4, 0] and I 2 = [0, 4.] In all the simulations we fix V max = 1, P = 0.7, β = 0.2, γ max = 0.5, l 0 = 0.2 and F in = 0.05.

Case I:

We consider the following initial data ρ 1 (0, x) = 0.6, ρ 2 (0, x) = 0.

(5.4)

The values of the initial conditions creates a shock on the incoming mainline and a rarefaction on the outgoing one. After a time t = 5.3 we can see the rarefaction caused by the buffer that empties in the incoming mainline I 1 , as illustrated in Figure 5.1.


H Jaiswal

J M Mattoussi

S M Mauro

Simon

D R Larson

W R Zipfel

R M 75 Williams

S W Clark

F F Pinaud

L A Bentolila

E Chang

N Thekkek

W W Yu

V L Colvin

R Drezek

N Lewinski

P J J Klaine

G E Alvarez

Batley

T Kirchner

S Liedl

T Kudera

Pellegrino

Despite the growing interest of quantum dots (QDs) in biological applications, there are many concerns regarding the potential accumulation and toxic effects of Cd-containg QDs in animals and humans. Zinc oxide QDs are promising alternatives for diagnosis and imaging but their aqueous instability has markedly limited their use. Generations 1, 2 and 3 (noted G1, G2, and G3, respectively) of new poly(amidoamine) (PAMAM) dendrons bearing a siloxane group at the focal point were prepared from 3-aminopropyltrimethoxysilane. Using tetramethylammonium hydroxide as cross-linking agent, hydrophobic oleate-capped ZnO QDs were functionalized with G1 or G2 dendrons, as evidenced by FT-IR, UV-visible and XPS analyses, and were successfully transferred in aqueous solution. AFM and TEM images show that ZnO@G1 and ZnO@G2 QDs have a spherical shape with average crystalline sizes of 5.3 and 5.1 nm, respectively. Immediately after dispersion in water, ZnO@G1 and ZnO@G2 QDs exhibit a broad and strong visible emission peak centered at 550 nm with a quantum yield of ca. 18%. A strong increase of photoluminescence quantum yields was observed over time and values up to 59% could be reached after ca. 20 days of storage in water at room temperature. The good quantum yields and the stabilities of PAMAMdendron capped ZnO QDs ensured their potential applications in cell imaging. ZnO@G2 were successfully used for the labelling of the Gram + bacterium Staphylococcus aureus. The biocompatibility of these QDs is markedly improved compared to Cd-based ones as growth inhibition tests showed that ZnO@G2 QDs could be used with concentrations up to 1 mM without altering the cell growth of the Escherichia coli bacterium while most Cd-containing QDs exhibit cytotoxicity already at the nM level.

.

Introduction

Colloidal semiconductor nanocrystals, called quantum dots (QDs), are nanometer-scale luminescent inorganic materials that have found numerous applications in bio-imaging and biodetection. 1 Owing to the effect of quantum confinement, QDs show exceptional physical and chemical properties such as sharp and symmetrical emission spectra, high quantum yield (QY), good photo-and chemical stability, and size-dependent emissionwavelength tunability. 2 Recent findings have highlighted the acute toxicity of II-VI semiconductor QDs without an external layer of a nontoxic material on biological systems and shed a doubt on the future applicability of these nanocrystals, particularly in view of recent environmental regulations. 3 This toxicity results mainly from the decomposition and release of heavy metal ions and formation of reactive oxygen species. 4 Synthesis of low toxicity QDs and especially Cd-free QDs is the most challenging aspect of working with these materials in biological and medical fields. A promising member of the Cdfree QD family is ZnO.

ZnO is a direct band gap (3.37 eV) semiconductor with a relatively high exciton binding energy (60 meV) which has found various applications in optical, electronic and sensing devices. 5 However, contrary to II-VI semiconductors such as CdSe and CdTe, ZnO QDs have received very little attention as biological labelling agents. Indeed, conventional ZnO QDs are not stable in water because water molecules are able to attack the luminescent centers on the surface and destroy them rapidly. 6 In recent years, 55 methods of preparing luminescent ZnO QDs in an aqueous phase have been improved. A series of ZnO nanocrystals embedded in hydrophilic polymers have been synthesized 7 but these materials show only modest fluorescence QY or are too large to be used in biological applications. Indeed, with large-sized probes, diffusion 60 and circulation processes are significantly reduced resulting in increased nonspecific binding. 8 Retaining the small probe size is critical for successful in vivo applications and to date, only three routes have been reported for the synthesis of nanometer-sized water-soluble ZnO QDs. Jana et al. described ZnO QDs capped 65 with 3-aminopropyltrimethoxysilane where the amine groups at the periphery contribute to the stability of these QDs. 6 Very recently, Xiong et al. reported the preparation of stable ZnO@polymer core-shell nanoparticles with an average diameter of ca. 3-4 nm and the first example of the use of these QDs in cell 70 imaging. 9 We have recently demonstrated the use of poly(ethylene glycol) (PEG) siloxane for the transfer of hydrophobic ZnO QDs in water with preservation of their luminescent properties. 10 However, PEG surface ligands did not offer surface groups that can be conjugated with bioactive

2 | Journal Name, [year], [vol], 00-00
This journal is © The Royal Society of Chemistry [year] molecules. This limitation reduced the materials' flexibility in bio-related applications. Due to their branched nature, the presence of multivalent surface groups available for attachment of surface moities, and high water affinity, poly(amidoamine) (PAMAM) dendrimers offer significant potential as versatile carriers for enhancing bioavailability of drugs and nanoparticles. 11 Here, we report a facile strategy to synthesize highly luminescent water-soluble ZnO QDs by covalent attachment of PAMAM dendrons bearing a reactive siloxane group at the focal point. Luminescent ZnO QDs were successfully used for the imaging of the Gram + bacteria Staphylococcus aureus. Cytotoxicity experiments showed that these QDs could be used with concentrations up to 1 mM without altering cell growth.

Experimental

A. Materials

All reagents were purchased from Aldrich at the highest available purity, and used without further purification. 20 Millipore water was used in all experiments.

B. Measurements

Routine nuclear magnetic resonance were recorded on a Brucker AM200 spectrometer operating at 200 MHz.

25

Chemical shifts are reported in ppm, relative to the solvent peak, and are given downfield from tetramethylsilane (TMS). Splitting pattern abbreviations are as follows: s = singlet, d = doublet, dd = double doublet, t = triplet, td = triplet of doublets, m = multiplet. Electrospray ionization (ESI) mass 30 spectra were recorded on a Surveyor mass spectrometer. Moisture-sensitive reactions were carried out under a dry nitrogen atmosphere in flame-dried glassware. Solvents were distilled before use under nitrogen. Absorption spectra were recorded on a Perkin-Elmer (Lambda 2, Courtaboeuf, France) UV-visible spectrophotometer. Fluorescence spectra were recorded on a Fluorolog-3 spectrofluorimeter F222 (Jobin Yvon, Longjumeau, France) equipped with a thermostated cell compartment (25°C), using a 450 W Xenon lamp. The QY values were determined by the following equation according to the method described by Crosby et al. 12

QY (sample) = (F sample /F ref )(A ref /A sample )(n sample 2 /n ref 2 )QY (ref)
, where F, A and n are the measured fluorescence (area under the emission peak), the absorbance at the excitation wavelength, and the refractive index of the solvent, respectively. Rhodamine 6G in ethanol was chosen as a reference standard (QY = 95%). 13 FT-IR spectra were recorded on a Vector 22 spectrometer using 2 mg of QDs and 198 mg of KBr to prepare pellets. To determine the morphology and the diameters of the nanoparticles, the samples were analyzed ex situ by Atomic Force Microscopy 50 (AFM) and by Transmission Electron Microscopy (TEM). AFM characterization was carried out using a Digital Instruments Nanoscope III. AFM measurements were done by taping mode using a Si 3 N 4 tip with resonance frequency and spring constant being 100 kHz and 0.6 N.m -1 , respectively to 55 provide surface topography. TEM images were taken by placing a drop of the particles in water onto a carbon film supported copper grid. Samples were studied using a Philips CM20 instrument with LaB 6 cathode operating at 200 kV. XPS measurements were performed at a residual pressure of 60 10 -9 mbar, using a KRATOS Axis Ultra electron energy analyzer operating with an Al Kα monochromatic source. Fluorescence images were acquired using a laser scanning head (Radiance 2100 Rainbow, Biorad) coupled with an inverted microscope (Nikon TU 2000) equipped with a 60x 65 objective (Apo VC Nikon, NA = 1.4).

C. Synthesis

Synthesis of oleate-capped ZnO QDs. Zinc acetate (220 mg, 1.2 mmol) was dissolved in ethanol (20 mL) at a temperature 70 below 50°C under vigorous stirring. Oleic acid (70 μL, 0.22 mmol) was then added and the mixture was heated to reflux. In a separate flask, tetramethylammonium hydroxide (360 mg, 1.99 mmol) was dissolved in refluxing ethanol (5 mL). This solution was then rapidly injected in the flask containing 75 Zn(OAc) 2 and oleic acid and the mixture was refluxed for 2 min. The mixture was then diluted with EtOH (50 mL) and cooled down to 0°C with an ice-bath. A white precipitate of ZnO nanoparticles appeared. Particles were centrifugated (15 min at 4000 rpm) with removal of the supernatant. The 80 resulting oleate-capped ZnO QDs were washed several times with ethanol, in which they are insoluble, and finally suspended in toluene (10 mL). Oleate-capped ZnO QDs were stored at 4°C in the dark.

85

Synthesis of ZnO@G1 QDs. Under a nitrogen atmosphere, the oleate-capped ZnO QDs previously prepared were dispersed in 10 mL toluene and treated with 1 mL of a 0.1 M solution of G1 dendron in EtOH. After 5 min stirring at room temperature, 1 mL of a 0.3 M solution of 90 tetramethylammonium hydroxide (TMAH) in EtOH was injected and the temperature was set at 85°C. After 45 min, the solution was allowed to cool and nanoparticles isolated by centrifugation (4000 rpm, 15 min). After removal of the supernatant, ZnO@G1 QDs were washed three times with 95 toluene in which they are insoluble. QDs were then redispersed in water and were further purified by successive precipitation-solubilization rounds using acetone as badsolvent. After drying for 5 h in vacuo, QDs were redispersed in water for further experiments. 100 Synthesis of ZnO@G2 QDs. ZnO@G2 nanoparticles were prepared analogously to the reaction above, with the exception that 1 mL of a 0.6 M TMAH solution in EtOH was added to the oleate-capped ZnO QDs and to the G2 dendron.

105

Cytotoxicity tests. Bacteria were routinely cultured at 30°C in Luria broth Miller [Difco] (1% Tryptone, 0.5% Yeast Extract, 1% NaCl). For growth experiments, bacteria were pregrown in shaking conical flasks filled to 20% of their 110 volume with LB medium without QD until the cultures reached mid log phase. At OD 595 around 0.2, cultures were diluted one-tenth in prewarmed LB medium supplemented with the desired concentration of QD and the optical density was measured at intervals. All growth experiments were carried out in the dark to limit both cell damage by lightexcited QDs and photo-oxidation reactions at QDs' surfaces.

Results and discussion

First-, second-and third-generation (respectively G1, G2 and G3) PAMAM dendrons were prepared by divergent synthesis through repetitive Michael addition using methyl acrylate and amidation with ethylene diamine starting from 3aminopropyltrimethoxysilane (Scheme 1 and ESI †) using the method previously described. 14 Among the solvents used for the amidation step, the best results were obtained in methanol.

Alkoxy group exchange (up to 30%) was observed in ethanol. Temperature also had an influence on the reaction yield. Proper amidation without alteration of the trimethoxysilane moiety and formation of large quantities of by-products could only be achieved by stirring the half-generation ester-terminated dendrons (G0.5, G1.5 and G2.5) with an excess of ethylene diamine in methanol at 5°C for 120 h. Under these optimized conditions, G1, G2, and G3 dendrons were isolated in quantitative yields and their structures confirmed by HRMS, and 1 H and 13 C NMR studies.

Hydrophobic oleate-capped ZnO QDs were prepared by the sol-gel method 6,15,16 by basic hydrolysis of Zn(OAc) 2 with a few modifications. Zn(OAc) 2 is first dissolved at a temperature below 50°C in anhydrous EtOH and oleic acid is then added. An EtOH solution of TMAH is then injected, the solution refluxed for 2 min and then directly diluted with EtOH and cooled to 0°C. The white powder obtained was collected by centrifugation, washed 2 times with ethanol and dried in vacuo at room temperature. TEM images showed that oleate-capped ZnO QDs possess an average diameter of 5.1 nm (data not shown). These nanocrystals could be resuspended in chloroform or toluene, indicating the presence of the long-chain carboxylate ligand shell. The solutions thus obtained were found to show bright green-yellow luminescence under UV excitation (QY = 24% in toluene), thereby indicating the formation of ZnO QDs.

Oleate is a relatively weak stabilizer ionically bound to the surface of ZnO colloids, while siloxanes have a strong affinity with hydroxylated surfaces and form strong covalent Si-O bonds with them. 17 The Zn(OH) 2 shell coating the ZnO surface was used for the modification of ZnO nanoparticles. 18 The covalent anchorage of G1 dendron at the surface of oleate-capped ZnO QDs was first investigated using TMAH as cross-linking reagent in an EtOH/toluene mixture at 80°C. Amounts of TMAH were modified from those previously optimized for the reaction of PEG-siloxanes with ZnO QDs. 10 Initial studies focused upon the reaction of G1 dendron with ZnO QDs. Poor transfers (less than 10%) of the hydrophobic ZnO nanoparticles were obtained using less than 2 eq TMAH relative to the siloxane, with no evidence for self-nucleation of the siloxane. For the G2 dendron, the ligand exchanges were complete using 6 eq TMAH. These results reflect the acidic character of the hydrogen atoms of the PAMAM amide redispersed in water and purified by successive precipitationsolubilization rounds using acetone as bad-solvent. The pellets were highly soluble in water, less soluble in methanol and ethanol, and insoluble in e.g. chloroform, toluene and less polar solvents. The QDs obtained after functionalization with G1 and G2 dendrons, designated as ZnO@G1 and ZnO@G2 respectively, were redispersed in water or phosphate-buffer saline (PBS) for further experiments. The absorption spectra of the nanocrystals did not change upon the surface replacement of the original oleate ligands by the siloxane-based dendron ligands (see Fig. S1 in ESI †). Compared to the band-edge of bulk ZnO, the excitonic peak bandgap absorption of the nanoparticles is located at much higher energies (3.59 eV) due to the quantum confinement effect. The relatively pronounced absorption peaks show that the size distributions of ZnO QDs are nearly 20 monodispersed after surface functionalization with G1 or G2 dendrons. Compared to oleate-capped ZnO QDs, there are two additional peaks in the 250-300 nm region. These peaks are assigned as typical   * and n  * transitions of the PAMAM carbonyl groups and appeared at 260 and 283 nm, respectively. For both QDs, the absorption edge value of 3.59 eV afforded ZnO cluster diameters of 5.0 nm as predicted by the Brus' effective mass model. 19 This value agrees well with diameters obtained by Transmission Electron Microscopy (TEM) and Atomic Force Microscopy (AFM) (analysis of ca. 100 nanocrystals in each TEM image yielded average crystal diameters of 5.3 ± 0.5 and 5.1 ± 0.6 nm for ZnO@G1 and ZnO@G2 QDs, respectively) (Fig. 1 and Figs. S2 and S3 in the ESI †). In both samples, the ZnO QDs appeared to be spherical, uniform, and disperse. Since oleate-capped ZnO QDs had an average diameter of 5.1 nm, no variation of QDs diameters was noticed after their surface modification with G1 or G2 dendrons.

Confirmation of particles surface modification was attained via X-ray photoelectron spectroscopy (XPS) (Fig. 2 and Fig. S4 the ESI †) and FT-IR (Fig. 3). The XPS scan survey spectrum 45 of the samples show only the peaks of Zn, O, Si, N, and C elements and no peaks of any other elements were observed. As seen on Fig. 2a, the peak of Zn 2p3/2 appears at 1021.0 eV, confirming that the Zn element exists only in the form of Zn 2+ linked to an oxygen atom. Fig. 2b shows that the O 1s peak is 50 somewhat asymmetric, suggesting that there are at least three kinds of oxygen species on the sample surface. The deconvoluted peaks at ca. 530.0, 531.3 and 533.3 eV can respectively be attributed to the lattice oxygens of ZnO, to the oxygens of the surface hydroxyl groups and to the oxygen atoms of the amide 55 groups. The Si 2p element is characterized by a single peak located at a binding energy of 101.4 eV (Fig. 2c). Finally, the N 1s signal (Fig. 2d) shows two components at 399.3 and 402.0 eV corresponding respectively to the primary and secondary nitrogen atoms and to the amide nitrogens. The N/Zn and Si/Zn ratios 60 obtained from the XPS analyses were used to estimate the number of dendrons anchored at the surface of ZnO@G1 and ZnO@G2 QDs. Using a hexagonal wurtzite crystal structure for ZnO 20 with lattices parameters of a = 0.325 nm and c = 0.520 nm, 21 ZnO@G1 and ZnO@G2 dendrons were respectively found 65 to be covered by ca. 2800 G1 and 900 G2 dendrons. FT-IR spectra of purified and dried powders before and after surface functionalization with G1 or G2 dendrons are given in Fig. 3. The strong vibration band at ca. 465 cm -1 observed in all spectra is attributed to the stretching mode of ZnO. 22 The two peaks at 1409 carboxylate group and vanish upon functionalization with G1 and G2 dendrons (Fig. 3b andc). 23 In spectra 3b and 3c, the weaks peaks at respectively 1188 and 1160 cm -1 are characteristic of the Si-O-Si symmetrical stretching vibrations. 24 These spectra also displayed the bands assigned to amides, N-H asymmetric stretching (3260-3271 cm -1 ), C=O stretching (1640-1647 cm -1 ), N-H bending (1540-1548 cm -1 ) and of the terminal primary amines (N-H stretching at 3384 cm -1 ). 25 ZnO@G1 and ZnO@G2 nanocrystals show a broad visible luminescence centered at ca 550 nm and a relatively weak UV emission band at ca 416 nm attributed to exciton recombination (Fig. 4). The green emission at 550 nm, commonly referred to as a deep-level or trap-state emission, 26 is related to a singly ionized oxygen vacancy in ZnO and results from radiative recombination of a photogenerated hole with an electron occupying the oxygen vacancy. 27 The QYs of ZnO@G1 and ZnO@G2 QDs, measured immediately after dispersion in water and determined by using rhodamine 6G in ethanol as reference, were found to be 19 and 17%, respectively.

Considerable differences were noticed after attempting the cross-linking of the G3 dendron at the surface of oleate-capped ZnO QDs. No aggregation of QDs was noticed after reaction with G3 dendron in the presence of TMAH (up to 14 eq were used for the anchorage of the G3 dendron) and a clear nanoparticle solution was obtained after transfer in water. The average diameter of QDs obtained after reaction was found to be close to that of the starting oleate-capped ZnO QDs (5.1 nm). Whereas absorption characteristics of nanoparticles was practically unchanged, the photoluminescence (PL) color changed from yellow to blue and the PL peak position was observed at ca. 425 nm ligands (see Fig. S5 in the ESI †). A similar phenomenon was observed when oleate-capped ZnO QDs were treated with the G2 PAMAM dendrimer possessing no siloxane function at the focal point (Fig. 5). On the basis of our current data and previous literature results, 28 it is reasonable to suggest that the ZnO-G3 dendrimer nanocomposite is formed by dendrimer molecules adsorbing on the ZnO nanoparticles. In addition, FT-IR measurements showed that the absorption band of the amino terminal groups of the G3 dendrimer was slightly shifted to a higher value (NH stretching at 3402 cm -1 ) whereas this shift was not observed with ZnO@G1 and ZnO@G2 QDs. We assume that Fig. 4 Room temperature absorption and photoluminescence spectra recorded after excitation at 365 nm of (A) ZnO@G1 and (B) ZnO@G2 QDs. (C) Photography of ZnO@G2 QDs dispersed in water and illuminated with an ultraviolet lamp.

50 the bulky and flexible structure of the G3 PAMAM ramification and especially the strongly coordinating 8 primary amine terminal groups strongly interact with electron deficient particles surface sites thus disrupting the anchorage of the siloxane moiety at the surface of ZnO QDs and altering PL. The stability of ZnO@G1 and ZnO@G2 QDs was studied by dispersing the nanoparticles in water and monitoring the change in PL intensity after excitation at 350 nm as a function of time using rhodamine 6G as reference (Fig. 6). Data obtained clearly indicated a strong increase of PL intensity at the beginning of the 60 experiment. Almost stable PL was obtained ca. 15 days after dispersion. The enhancement of the 550 nm emission is respectively of 3.1 and 2.5 fold for ZnO@G1 and ZnO@G2 QDs after 20 days storage in water and a PL QYs of ca. 59 and 43% were attained for ZnO@G1 and ZnO@G2 nanocrystals, Indeed, this interaction should result in a decrease of surface defects and thus to an increase of the ZnO excitonic emission. We privilege a surface ordering of the siloxane-capped ZnO QDs which protects the surface of the QDs from water attack at the luminescent centers. Nevertheless, the present findings may provide an effective method for improving the PL properties of ZnO nanocrystals.

The usefulness of PAMAM dendrons capped-ZnO QDs, in the context of bio-imaging probe design, was demonstrated in a labelling experiment with S. aureus. Before labelling, S. aureus bacterial cells only displayed a minimal autofluorescence signal (data not shown). After 1 hour incubation at room temperature with a 1 µM ZnO@G2 QDs suspension, the bacterial cells clearly appeared fluorescently labelled indicating a QD accumulation (Fig. 7a and Fig. S6 in the ESI †). The mean surfaces (0.5 ± 0.16 m 2 ) and the circularity (0.92 ± 0.06) of fluorescent objects determined by At present, we suppose that positively charged QDs may accumulate on cells through electrostatic interactions with teichoic and lipoteichoic acids present on the surface of this Gram + bacterium while the outer membrane of the Gram -, composed of lipopolysaccharides, would be less receptive to QDs.

Recently, we developed a test to evaluate the toxicity of QDs toward bacteria. 30 With this respect, it was shown that CdTe-core QDs were already toxic against E. coli cells at nanomolar concentration as seen by a decrease of growth rate.

20

The toxixity test was repeated with the ZnO@G2 QDs in the same experimental conditions. Contrary to CdTe QDs, ZnO@G2 QDs displayed a remarquably weak toxicity as no growth alteration was detected until 1 mM (Fig. 8). Growth alteration became visible at 10 mM. The antibacterial effect of ZnO@G2 QDs was found to be 10 6 times less pronounced than thioglycolic acid-capped CdTe QDs further highlighting the promising potential of these nanoparticles for in vivo bioimaging. 30,31

Conclusions

In summary, we have demonstrated that stable and watersoluble ZnO QDs can be prepared through surface functionalization of these nanocrystals with siloxane-core

  Scheme 1 Synthesis of siloxane-core PAMAM dendrons; reaction conditions: (a) methyl acrylate, MeOH, rt, 48 h; (b) ethylene diamine, MeOH, 5°C, 120 h. 60

Fig. 1

 1 Fig. 1 Bright field TEM and AFM images of (A) and (C) ZnO@G1, and (B) and (D) ZnO@G2 QDs.

  in

40

 40

Fig. 2

 2 Fig. 2 XPS spectra of the ZnO@G1 quantum dots. The Zn 2p, O 1s, Si 2p, and N 1s emissions are presented.

Fig. 3

 3 Fig.3FT-IR spectra of (A) ZnO/oleate QDs, (B) ZnO@G1 QDs, and (C) ZnO@G2 QDs.

  55

Fig. 5

 5 Fig. 5 Chemical structure of the G2 dendrimer bearing no siloxane unit at the focal point.

Fig. 6

 6 Fig. 6 Changes in photoluminescence intensity of dendrons capped ZnO QDs as a function of time respectively. These PL QYs values compare favorably with those recently reported by Xiang et al. for ZnO QDs embedded in PEG-methacrylate polymers (20-50%). 9 In the meantime, no noticeable change in the absorption and emission spectra of ZnO QDs was observed during the storage time. Compared to the results obtained with PEG-bis-carboxymethyl-coated ZnO nanoparticles, 29 it is unlikely that the PL enhancement observed originates from the coordination of the nitrogen or oxygen atoms of the PAMAM ramification with the ZnO particles surface.

Fig. 7 Fig. 8

 78 Fig. 7 (a) Confocal microscopy image of S. aureus cells labelled with 35

35 PAMAM

 35 dendrons G1 and G2. Those dendron-coated nanocrystals are structurally simple and the corresponding ligand monolayer is very thin in comparison to other stabilizing approaches. The potential of ZnO@G2 QDs as fluorescent probes was demonstrated by the labelling of S. aureus bacteria cells. The 40 surface amine groups of the PAMAM ligands should facilitate further surface functionalization with targeting ligands, opening up the way to direct ZnO nanoparticles toward specific biological material. Because of their weak toxicity, these QDs should play an important role in a variety of nanocrystal-based biomedical 45 applications in a near future. We are currently investigating the possibility of tuning the emission wavelengths of ZnO QDs by doping for in vivo biological labelling.

Biocompatible and stable ZnO quantum dots generated by functionalization with siloxane-core PAMAM dendrons Ralph-Olivier Moussodia, a Lavinia Balan, b Christophe Merlin, c Christian Mustin, d and Raphaël Schneider* a

Acknowledgements 50

The authors thank Dr. Fabienne Quilès (LCPME, Nancy-University) for FT-IR spectra, Olivier Soppera (UMR CNRS 7525, Mulhouse) for AFM images and Jacques Lambert (LCPME, Nancy-University) for acquiring the XPS spectra.