Julien Lemaire
email: julien.lemaire@centralesupelec.fr

Claire-Line Blanc

Florence Lutin

Marc-André Theoleyre

Dominique Pareau

ORGANIC ACIDS CHROMATOGRAPHY WITH STRONG ANIONIC RESINS Purification of organic acids by chromatography with strong anionic resins: investigation of uptake mechanisms

Keywords: chromatography, modelling, anionic resin, organic acid, sustainable processing. Highlights

Bio-based organic acids are promising renewable carbon sources for the chemical industry.However energy-consuming purification processes are used, like distillation or crystallization, to reach high purities required in some applications. That is why preparative chromatography was studied as an alternative separation technique. In a previous work dealing with the purification of lactic, succinic and citric acids, theLangmuir model was insufficient to explain the elution profiles obtained with a strong anionic resin.

Consequentlythe Langmuir model was coupled with a usual ion-exchange model to take into account the retention of their conjugate bases (<2%), which are commonly neglected at low pH (< 1.5). Elution simulationswith both uptake mechanisms fitted very well with experimental pulse tests. Only two parameters were optimized (equilibrium constant of acid uptake and ion-exchange selectivity coefficient of conjugate base)and their value were coherent with experimental and resin suppliers' data.

These results confirmed that the singular tailing and apparent delay observed with succinic and citric acids can be explained by the high affinity of succinate and citrate for resin cationic sites. Themodel was implemented in a preparative chromatography simulation program in order tooptimize operating parameters of our pilot-scale ISMBunit (Improved Simulated Moving Bed). The comparison with experimental ISMB profiles was conclusive.

Introduction

In the frame of sustainable development, biomass valorization as a renewable carbon source for the chemical industry is growing. Indeed, fossil resources are running out in the medium to long term. Moreover carbon balances have to be improved so as to reduce the CO 2 level in the atmosphere, whose rapid rise is linked to global warming.In the famous report of the Pacific Northwest National Laboratory (PNNL) and the National Renewable Energy Laboratory (NREL) [1], bio-based organic acids are identified as very promising sources of renewable carbon. Many compounds can be produced from these moleculesinthe chemical industry, particularly inthe polymer industry. Up to now, the production of bio-based organic acids keeps small because they are stilluncompetitive compared to the corresponding petroleum-derived products [START_REF] Sauer | Microbial production of organic acids: expanding the markets[END_REF]. That is why the reduction of their production cost is a major concern of the chemical industry.

Bio-based organic acids are usually produced by microbial fermentation for the food industry but further purification techniques are required for others applications which need a much higher purity, particularly polymer synthesis. Currently, distillation [START_REF] Dierdorp | Continuous process for preparing lactic acid[END_REF][START_REF] Joglekarhg | Comparative assessment of downstream processing options for lactic acid[END_REF], esterification [START_REF] Sterzel | Preparation of lactates[END_REF][START_REF] Sun | Extraction and purification of lactic acid from fermentation broth by esterification and hydrolysis method[END_REF][START_REF] Kumar | Recovery of lactic acid by batch reactive distillation[END_REF]and crystallization [START_REF] Coszach | Method for purifying lactic acid by crystallization[END_REF]are commonly used to reach very high purities (> 99.5%) [START_REF] Huh | Effective purificationofsuccinicacidfromfermentationbrothproducedby Mannheimiasucciniciproducens[END_REF].

However, these techniques are energy-consuming and give quite low recovery rates.Preparative chromatography is a promising solution for downstream processes because its energy, chemical and water consumptions are generally quite lower and consequently effluent volumes too. Initially developed for extraction and purification of complex vegetal mixtures [START_REF] Guiochon | Preparative liquid chromatography[END_REF], this separation method is spreading in different fields: biotechnology [START_REF] Molnárz | Separation of amino acids with simulated moving bed chromatography[END_REF][START_REF] Ilisz | Enantiomeric separation of nonproteinogenic amino acids by high-performance liquid chromatography[END_REF], pharmaceutic [START_REF] Kumar | Mechanistic modeling of ion-exchange process chromatography of charge variants of monoclonal antibody products[END_REF][START_REF] Guélat | Protein adsorption on ion exchange resins and monoclonal antibody charge variant modulation[END_REF] and the sugar industry [START_REF] Huh | Fructose-glucose separation in a SMB pilot unit: Modeling, simulation, design, and operation[END_REF], for instance. Indeed, the same separation quality may be reached with better yields, lower costs and impacts on the environment.However, for several years now, the state of the world economy has not been favorable to industrial investments for theimprovement of downstream purification processes. Moreover, chromatography needs to prove itselffor high purity application (> 99.5%) at the industrial scale.

In the frame of organic acids purification in fermentation broths, chromatography is often performed with ion-exchange resinsto separate mineral salts, sugars and organic acids. The eluent is an aqueous solution whose pH can be adjusted with the inorganic acid or base corresponding to the counter-ions fixed on the resin so as to keep itscomposition constant.

The separation of compounds is based on affinity differences for the resins. Higher is the affinity, slower migrates the compound along the resin bed.Inthe sugar industry, cationic resins are used to separate sucrose from saltsas well as sugars from each other. The glucose-fructose separation is among the most famous applications of preparative chromatography.Otherwise, anionic resins are performedfor the citric acid purification.

Uptake and slowdown mechanisms are not well known and complex, coupling size exclusion, ligand exchange, hydrophilic or hydrophobic interactions and ion-exchange [START_REF] Stefansson | Ligand-exchangechromatographyof carbohydrates and glycoconjugates[END_REF][START_REF] Nobre | Comparison of adsorption equilibrium of fructose, glucose and sucrose on potassium gel-type and macroporous sodium ion-exchange resins[END_REF].

In a previous work dealing with the purification of lactic, succinic and citric acids by chromatography with a strong anionic resin (quaternary ammonium), the Langmuir model fitted well with the experimental equilibrium isotherms obtained by frontal analysis [START_REF] Blanc | Purification of organic acids by chromatography: adsorption isotherms and impact of elution flow rate[END_REF]. However,the corresponding affinity coefficients were significantly lowerthan those obtained by pulse tests for succinic and citric acids. Their peaks had a singular tailing and were delayedcompared to simulated ones with the Langmuir model and experimental coefficients of isotherms. Indeed,the Langmuir model was insufficient to describe the organic acids retention on the strong anionic resin and frontal analysis gave underestimatedaffinity coefficients.

These results lead us to suspect the impact of conjugate bases (dissociated organic acids) retention by ion-exchange, despite their very low concentrations. Indeed, the citrate anion affinity is high for quaternary ammonium anionic resin according to selectivity coefficients given by resins producers. Usually, this uptake mechanism is not taken into account because the strong anionic resins are performed at such acid pH (1 to 1.5) that only 0.2% to 0.4% of lactic acid, 0.1% to 0.2% of succinic acid and 1.2% to 2.3% of citric acid are dissociated.Thus, the organic acids dissociation is generally not consideredalthough such small amount of conjugate bases may have a significant impact if they are strongly retained.

The present work aimed at modelling uptake mechanisms on a strong anionic resin. The Langmuir model, commonly used for the retention of organic acids, was coupled with a usual ion-exchange model for the retention of conjugate bases. Simulation results were compared with experimental pulse tests to check the model. The final goalwas toimprove the accuracy of apreparative chromatographysimulation program in order to optimize the operating parameters of our pilot-scale ISMB unit (Improved Simulated Moving Bed).

Chromatography modelling

Mass transfer model

The Martin and Synge plate model [START_REF] Guiochon | Fundamentals of Preparative and Nonlinear Chromatography[END_REF] was chosen to describe the elution of organic acids through a strong anionic resin bed. In each plate of volume V, the mobile phase takes up a volume εV and the stationary phase a volume (1-ε)V, with ε the bed voidage. Usually, for preparative chromatography,volumes areexpressed in BV (bed volume) [START_REF] Thang | Green biorefinery: separation of lactic acid from grass silage juice by chromatography using neutral polymeric resin[END_REF][START_REF] Wu | Separation of dlactic acid from aqueous solutions based on the adsorption technology[END_REF][START_REF] Morin-Couallier | Reducing water consumption in beet distilleries by recycling the condensates to the fermentation phase[END_REF], corresponding to the ratio of volume to the resin bed volume.Both phases arealways supposed in equilibriumin all plates (nonlinear equilibrium in our case). The axial dispersion is characterized by the plate number N.

Uptakemechanisms-reactions in stationary phase

The strong anionic resin is a styrene-divinylbenzene resin functionalized with quaternary ammonium groups RMe 3 N + . The amount of these cationic sites corresponds to the ionexchange capacity𝑞 𝑅𝑀𝑒 3 𝑁 + of the resin, expressed in mol. L resin -1 . The resin is initially in the sulfate form, namely with thecounteranions HSO 4 -and SO 4 2-in equilibrium. The proportion of both forms depends on their concentrationin the mobile phase, so on its pH. The concentrations of hydroxyl anion OH -in both phases were considered negligible at pH < 1.5.

The proposed model was based on two classical uptake mechanisms, with some assumptions:

 Uptake of the organic acid in the dissociated form A -by ion-exchange with resin counteranions HSO 4 -and SO ) and AH-site(matrix) (figure 1).

Figure 1-Illustration of the six compounds considered in the stationary phase

The ion-exchange reactions correspond to the following equilibria: contents inthe stationary phase (free or bound to AH),q A -the A -content and[X] the concentration of X in the mobile phase.

(𝑅𝑀𝑒 3 𝑁 + ) 2 𝑆𝑂 4 2-+ 2 𝐴 -↔ 2 (𝑅𝑀𝑒 3 𝑁 + )𝐴 -+ 𝑆𝑂 4 2- (𝑅𝑀𝑒 3 𝑁 + ) 2 𝑆𝑂 4 2-+ 2 𝐻𝑆𝑂 4 -↔ 2 (𝑅𝑀𝑒 3 𝑁 + )𝐻𝑆𝑂 4 -+ 𝑆𝑂 4 2- 𝑅𝑀𝑒 3 𝑁 + 𝐻𝑆𝑂 4 -+ 𝐴 -↔ (𝑅𝑀𝑒 3 𝑁 + )𝐴 -+
The ion-exchange equilibrium constants Kecan be estimated from data of resin suppliersby equations 5-7. For example, the Dow Chemical Company gives the selectivity coefficients of anions X -for a strong anionic resin in the hydroxide form, 𝑘 𝑋/𝑂𝐻 , defined as follows:

𝑘 𝑋/𝑂𝐻 = 𝑞 𝑋 -𝑂𝐻 - 𝑋 -𝑞 𝑂𝐻 - Eq. 4
𝐾𝑒 𝐻𝑆𝑂 4 -/𝑆𝑂 4 The pKa of lactic, succinic and citric acids and hydrogen sulfate at 60°C (operating temperature) are respectively 3.86, 4.13, 3.05 and 2.39.

Plate mass balances

The plate content is represented by 11 unknown variables (5 in the mobile phase and 6 in the stationary phase) that obey 7 independent equilibrium relations (Eq. 1-2 and 14-18).

Consequently, 4 mass balancesare necessary to solve the system and estimate the content of eachplate at time"t" from the composition of all platesat time"t-1".

Mass balance of ion-exchange site RMe 3 N + whatever the time t, 2 q SO 4 2- t n + q HSO 4 - t n + q A -t n = 𝑞 𝑅𝑀𝑒 3 𝑁 + Eq. 19

Mass balance of sulfate group SO 4 between t-1 and t:

εV d HSO 4 - t n dt + d SO 4 2- t n dt + 1 -ε V dq SO 4 2- t n dt + dq HSO 4 - t n dt = Q HSO 4 - t n-1 -HSO 4 - t n + Q SO 4 2- t n-1 -SO 4 2- t n
Eq. 20

Mass balance ofacids (H + element) between t-1 and t (water dissociation negligible):

εV

dt = Q A - t n-1 -A - t n + Q AH t n-1 -AH t n
Eq. 22

In the last three mass balances, the terms on the left-hand side of the equal sign correspond respectivelyto the accumulation terms in the mobile phase and in the stationary phase, meanwhile the expressionson the right-hand side correspond to the difference between inputs (fromthe (n-1) th plate) and outputs (from the n th plate).Q represents the flowrate.

The proposed model depends on 10 parameterswhose value is more or less known accuratelyaccording to data from resin suppliers, literature or our own experimental results:



Numerical methods

The finite-difference method with implicit schemewas used to discretize the 3 differential equations (eq. [START_REF] Thang | Green biorefinery: separation of lactic acid from grass silage juice by chromatography using neutral polymeric resin[END_REF][START_REF] Wu | Separation of dlactic acid from aqueous solutions based on the adsorption technology[END_REF][START_REF] Morin-Couallier | Reducing water consumption in beet distilleries by recycling the condensates to the fermentation phase[END_REF].The Newton-Raphson iterative methodwas chosen to solve the nonlinearsystem of 4 remaining equations after simplification.The model and its numerical solving were computed using the software MATLAB ® . The initial conditions (concentrations in both phases) need to be givenas well as the boundary conditions: compositionof the feed (organic acid solution then eluent) as a function of time.

Results and discussion

Simulation withonly the Langmuir model andcoefficients of equilibrium isotherms

First, the simulations of organic acids elution through a strong anionic resin bed were performedconsidering that AH was only retained on the polymer matrix, according to the Langmuir model.We used the coefficients of equilibrium isotherms obtained previously by frontal analysis (table 1). So, the elution profiles were only influenced by 3 parameters: the plate number N, the polymer matrix uptake capacity𝑞 𝑚𝑎𝑡 equated to𝑞 𝑚𝑎𝑥 and the equilibrium constant of AH uptake on resin polymer matrixKs AH /mat equated to𝐾𝑠 𝐴𝐻 . For lactic and succinic acids, the resin uptake capacity 𝑞 𝑚𝑎𝑥 and the equilibrium constant 𝐾𝑠 𝐴𝐻 have significant error margin because it was not possible to get enough experimental points in the nonlinear region of equilibrium isotherms [START_REF] Blanc | Purification of organic acids by chromatography: adsorption isotherms and impact of elution flow rate[END_REF].

According to our previous study [START_REF] Blanc | Purification of organic acids by chromatography: adsorption isotherms and impact of elution flow rate[END_REF], the plate number Nwas set to250for the 3 organic acids. It corresponds to the experimental value obtained by pulse testswith lactic acid whose symmetric profile is the most appropriate to characterize axial dispersion.The plate number N was considered equal for all species. Indeed, the study of axial dispersion (Van Deemter curves) in this work showed that the molecular diffusion is negligible compared to the mechanical dispersion and the influence of mass transfer limitation, which were quite similar between compounds in our experimental operating conditions.

These conditions were appliedin the model so as to compare with experimental results: 0.05 BV of 60 g.L -1 organic acid solution, whose pH was adjusted to 1.4 with sulfuric acid, were eluted with a 2.5 g.L -1 sulfuric acid solution. The temperature was 60 °C and the flow rate was 0.5 BV.h -1 .

For lactic acid, the simulation results are quite close to the experimental ones (figure 2).

So, the lactic acid uptake mechanisms by a strong anionic resin can be well fitted with only the Langmuir model. However, this model is insufficient to simulateelution of succinic and citric acids. Indeed, the simulation gives much earlier and narrower peaks than the experiments, particularly for citric acid.Another uptake mechanism must then be responsible for the delay and peak tailingobserved experimentally and should be taken into account. To simplify, as a start, we considered that the molecular form AH was retained only on free uptake sites ofcounteranions HSO 4 -and SO 4 2-according to UOP patents [START_REF] Kulprathipanja | Separation of lactic acid from fermentation broth with an anionic polymeric absorbent[END_REF][START_REF] Kulprathipanja | Separation of an organic acid from fermentation broth with an anionic polymeric adsorbent[END_REF].

Moreover, we supposed that the equilibrium constants of AH uptake on both counteranions were equal. They were estimated first from the experimental equilibrium constant 𝐾𝑠 𝐴𝐻 (table 1), then optimized by fitting simulation results to experimental profiles (least squares method). They were expected lower than 𝐾𝑠 𝐴𝐻 since it includes also the contribution of ionexchange in retention.

According to the supplier, the ion-exchange capacity 𝑞 𝑅𝑀𝑒 3 𝑁 + is about1.35mol. L bed -1 , equivalent to 2.4mol. L resin -1 .The uptake capacities of counteranions HSO 4 -and SO 4

, 𝑥and 𝑦,were supposed equal and defined fromthe experimental uptake capacity 𝑞 𝑚𝑎𝑥 (table 1). For instance, 𝑞 𝑚𝑎𝑥 of lactic acid is twofold to threefoldhigher than𝑞 𝑅𝑀𝑒 3 𝑁 + , so 𝑥 and 𝑦 were supposed equal to 2 (table 3). It could be related to both nucleophile carbonyl groups of counteranions HSO 4 -and SO 4 2-while lactic acid gets only one electrophile hydroxyl group. It means that 2 molecules of lactic acid can be retained on each counteranion.For succinic and citric acids,𝑥 and 𝑦 were supposed equal to 1 (table 3).

The selectivity coefficients k X/OH were first estimated from data of the Dow Chemical Company (table 2), then optimized by fitting simulation results to experimental profiles (least squares method). The plate number N was still set to 250 for the 3 organic acids.Values of simulation parameters for the 3 organic acids were gathered in the table 3. The figure 3 shows that the proposed model fitted very well the experimental profiles with coherent value for each parameter compared to experimental and resin supplier'sdata.The delay and peak tailing observed mainly withsuccinic and citric acids can be reproduced with this model. It confirms that the retention of the dissociated organic acids by ion-exchange with resin counteranions must be considered despite of the very lowdissociation of the acid at acidic pH (0.1 to 2%). Indeed, the conjugate base of some organic acids, like citrate, can have a very high affinity for resin cationic sites, so the organic acid can be much moreretained than expected. Moreover, it shows that assumptions added in this simulation to simplify the model wereacceptable.

Implementation in a ISMB chromatography simulation program

This uptake model was developed in order to improve simulation of preparative chromatography like ISMB (Improved Simulated Moving Bed) that is currently used at pilot scale in our laboratory. We aimed at predicting the ISMB performance more accurately with strong anionic resins and optimizing operating parameters according to thewanted recovery rates and purities.

An ISMB unit is composed of 4 columns in series filled with the stationary phase. The ISMB process consists in changing sequentially position of inputs (feed and eluent) and outputs (extract enriched with the more retained component, succinic acid, and raffinate enriched with the less retained, salts and lactic acid). An operating step is composed of four different sub-steps, where each column corresponds to one zone:

1) Feed input in the zone 3 and raffinate output from the zone 3.

2) Eluent input in the zone 1 and extract output from the zone 1.

3) Eluent input in the zone 1 and raffinate output from the zone 3.

4) Recirculation loop with all zonesconnected in series.

After one step, each zone is shifted by one column: the zone 1 becomes the zone 2,the zone 2 becomes the zone 3, etc. So, a cycle is performed after 4 steps. ISMB performance depends on different volumes eluted during these four sub-steps.

ISMB simulations were performed with the Langmuir model whose coefficients were optimized previously from an experimental pulse test at pilot scale, then with the model developed whose coefficients were also optimized in the same way. The feed was mainly composed of sulfatesalts (16 g/L), succinic acid (65 g/L) and lactic acid (14 g/L) while the eluent was a 2.5 g/L sulfuric acid solution.

The figure 4 shows that the proposed model fitted better ISMB experimental profiles than the Langmuir model. The accuracy of ISMB simulation with strong anionic resin was improved by taking into account the retention of dissociated organic acids by ion-exchange, despite their very low concentrations.

Conclusion

Preparative chromatography is an efficient separation technique which is spreading in different industrial fields. For several years now, anionic resins have been performed for the citric acid purification, however uptake mechanisms are still not well known.In a previous work [START_REF] Blanc | Purification of organic acids by chromatography: adsorption isotherms and impact of elution flow rate[END_REF], peaks corresponding to succinic and citric acids exhibited singular tailing and delay which could not be explained only by the Langmuir model.

The uptake mechanisms supposed in the proposed model are relevant to explain the characteristic elution profiles of some organic acids obtained in pulse tests. To sum up, we assume that the acid molecular form can be retained on resin counteranions or polymer matrix while the dissociated form can be fixed to cationic sites by ion-exchange. Usually, the organic acids dissociation is not considered at such low pH (<1.5), but we proved that even a very small proportion of dissociated form can have a significant impact when it is strongly retained. This is the reason why the Langmuir model is sufficient for lactic acid because of the low affinity of lactate but insufficient for citric acid since citrate affinity is very high for quaternary ammonium anionic resins.

The developed model was implemented in a preparative chromatography (ISMB) simulation program in order to optimize the operating parameters of our pilot-scale unit and improve estimation of its performance. Experimental ISMB profiles confirmed that the proposed model was better than the Langmuir model to describe the organic acids retention on strong anionic resins.

Next, a plug flow reactor model will be used to take into account different axial dispersions of compounds. Moreover, it would be interesting to further investigate so as to support the assumptions which were made to simplify the model and eventually to refine it.

-

  Retention of the organic acid in the molecular form AH according to the Langmuir model, by weak bonds (hydrogen or van der Waals), on free sites either on , cf. UOP patents[START_REF] Kulprathipanja | Separation of lactic acid from fermentation broth with an anionic polymeric absorbent[END_REF][START_REF] Kulprathipanja | Separation of an organic acid from fermentation broth with an anionic polymeric adsorbent[END_REF]) or on the polymer matrix whose capacity𝑞 𝑚𝑎𝑡 is also expressed in mol. L resin -1 .We supposed that the AH retention does not modify the selectivity coefficients of counteranions.Consequently, 6 compounds were considered in the stationary phase: HSO 4

𝐻𝑆𝑂 4 -

 4 Sincethe sites was supposed not influenced by AH uptake, theirequilibrium constants can be written as follows:

Figure 2 -with experimental coefficients of equilibrium isotherms 3 . 2 .

 232 Figure 2 -Comparison of experimental pulse test profiles with simulated ones consideringthe Langmuir model with experimental coefficients of equilibrium isotherms

Figure 3 -

 3 Figure 3 -Comparison of experimental pulse test profiles and simulations ones consideringthe Langmuir model combined with ion-exchange of conjugate bases

Figure 4 -

 4 Figure 4 -Comparison of experimental ISMB profiles with simulated ones, first with the model which is developed, then with the Langmuir model only

  𝐴𝐻-𝐻𝑆𝑂 4 ,𝑞 𝐴𝐻-𝑆𝑂 4 and𝑞 𝐴𝐻-𝑚𝑎𝑡 represent the AH concentrations bound touptake sites

	2-= According to the Langmuir model, the AH uptake reactions on free sites on counteranions 𝑘 𝐻𝑆𝑂 4 /𝑂𝐻 2 𝑘 𝑆𝑂 4 /𝑂𝐻 Eq. 5 𝐾𝑒 𝐴 -/𝑆𝑂 4 2-= 𝑘 𝐴/𝑂𝐻 2 𝑘 𝑆𝑂 4 /𝑂𝐻 Eq. 6 𝐾𝑒 𝐴 -/𝐻𝑆𝑂 4 -= 𝑘 𝐴/𝑂𝐻 𝑘 𝐻𝑆𝑂 4 /𝑂𝐻 Eq. 7 HSO 4 -and SO 4 2-and on the polymer matrix correspond to the following equilibria: 𝐹𝑟𝑒𝑒 𝐻𝑆𝑂 4 -𝑠𝑖𝑡𝑒 + 𝐴𝐻 ↔ 𝑂𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝐻𝑆𝑂 4 -𝑠𝑖𝑡𝑒 𝐹𝑟𝑒𝑒 𝑆𝑂 4 2-𝑠𝑖𝑡𝑒 + 𝐴𝐻 ↔ 𝑂𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑆𝑂 4 2-𝑠𝑖𝑡𝑒 𝐹𝑟𝑒𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑠𝑖𝑡𝑒 + 𝐴𝐻 ↔ 𝑂𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑚𝑎𝑡𝑟𝑖𝑥 𝑠𝑖𝑡𝑒 The corresponding equilibrium constants Ksare defined as follows: 𝐾𝑠 𝐴𝐻/𝐻𝑆𝑂 4 -= 𝑞 𝐴𝐻 -𝐻𝑆𝑂 4 𝐴𝐻 𝑞 𝐻𝑆𝑂 4 -𝑓𝑟𝑒𝑒 Eq. 8 𝐾𝑠 𝐴𝐻/𝑆𝑂 4 2-= 𝑞 𝐴𝐻 -𝑆𝑂 4 𝐴𝐻 𝑞 𝑆𝑂 4 2-𝑓𝑟𝑒𝑒 Eq. 9 𝐾𝑠 𝐴𝐻/𝑚𝑎𝑡 = 𝑞 𝐴𝐻 -𝑚𝑎𝑡 𝐴𝐻 𝑞 𝑚𝑎𝑡 𝑓𝑟𝑒𝑒 Eq. 10 respectively on counteranions HSO 4 -and SO 4 2-and on the polymer matrix, meanwhile 𝑞 𝐻𝑆𝑂 4 -𝑓𝑟𝑒𝑒 , 𝑞 𝑆𝑂 4 2-𝑓𝑟𝑒𝑒 and 𝑞 𝑚𝑎𝑡 𝑓𝑟𝑒𝑒 represent the concentrations of corresponding free uptake sites. Mass balances on the three kinds of uptake siteslead to the followingequations: 𝑞 𝐻𝑆𝑂 4 -𝑓𝑟𝑒𝑒 = 𝑥𝑞 𝐻𝑆𝑂 4 --𝑞 𝐴𝐻-𝐻𝑆𝑂 4 Eq. 11 𝑞 𝑆𝑂 4 2-𝑓𝑟𝑒𝑒 = 𝑦𝑞 𝑆𝑂 4 2--𝑞 𝐴𝐻-𝑆𝑂 4 Eq. 12 𝑞 𝑚𝑎𝑡 𝑓𝑟𝑒𝑒 = 𝑞 𝑚𝑎𝑡 -𝑞 𝐴𝐻-𝑚𝑎𝑡 Eq. 13 where x and y correspond to the uptake capacity of counteranions HSO 4 -and SO 4 2-𝑞 𝐴𝐻-𝐻𝑆𝑂 4 = 𝐾𝑠 𝐴𝐻 /𝐻𝑆𝑂 4 -𝐴𝐻 𝑥𝑞 𝐻𝑆𝑂 4 -1 +𝐾𝑠 𝐴𝐻 /𝐻𝑆𝑂 4 -𝐴𝐻 Eq. 14 𝑞 𝐴𝐻-𝑆𝑂 4 = 𝐾𝑠 𝐴𝐻 /𝑆𝑂 4 2-𝐴𝐻 𝑦𝑞 𝑆𝑂 4 2-1 +𝐾𝑠 𝐴𝐻 /𝑆𝑂 4 2-𝐴𝐻 Eq. 15 𝑞 𝐴𝐻-𝑚𝑎𝑡 = 𝐾𝑠 𝐴𝐻 /𝑚𝑎𝑡 𝐴𝐻 𝑞 𝑚𝑎𝑡 1 +𝐾𝑠 𝐴𝐻 /𝑚𝑎𝑡 𝐴𝐻 Eq. 16 2.3. Reactions in mobile phase At low pH (<1.5), organic acids are mainly in themolecular form AH, howeverthe retention -, SO 4 2-, AH and A -.The main reactions are both acid-base equilibria, whose equilibrium constantsKa are defined as follows: AH ↔ A -+ H + HSO 4 -↔ SO 4 2-+ H + of the dissociated form A H + , HSO 4 Ka AH = A -H + AH Eq. 17 Ka HSO 4 -= SO 4 2-H + HSO 4 -Eq. 18

where𝑞 , namely the number ofmolecule that can be retained per counteranion. Indeed, we assumed that several molecules can eventually be retained on each counteranion. These numbers are not necessarily integers and can be smaller than 1 if one molecule AH occupies several counteranionsHSO 4 -or SO 4 2-for steric reasons.

For each type of uptake sites, the combination of one mass balance with its corresponding equilibrium constant gives the following usual equations ofthe Langmuir model. They relate concentration of AH retained respectively on the three kind of sites to AH concentration in mobile phase and total concentration of corresponding sites. -was studied(for polyacids like succinic or citric acids, only the monodissociated formwas considered). So 5 species were considered in the mobile phase:

Table 1 -Resin uptake capacity 𝒒 𝒎𝒂𝒙 and equilibrium constant 𝑲𝒔 𝑨𝑯 of 3 organic acids estimated by frontal analysis at 60°C for a strong anionic resin [18]

 1

		𝒒 𝒎𝒂𝒙	𝑲𝒔 𝑨𝑯
		𝑚𝑜𝑙. 𝐿 𝑟𝑒𝑠𝑖𝑛 -1	10 -1 𝐿 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 . 𝑚𝑜𝑙 -1
	Lacticacid	6.8 ± 2.6	1.4 ± 0.5
	Succinicacid	2.7 ± 0.9	7.2 ± 2.5
	Citricacid	2.5 ± 0.2	9.0 ± 1.7

Table 2 -Selectivity coefficients of some anions (compared with OH -) on a strong anionic resin (Data from Dow Chemical Company)

 2

		HSO 4 -	SO 4 2-	Citrate* Formate Acetate Propionate
	𝐤 𝐗/𝐎𝐇	85	150	220	4.6	3.2	2.6
				* Valence not mentioned

Table 3 -Optimized parameters for simulation of lactic, succinic and citric acids elution with the proposed model which also takes into account A -ion-exchange

 3

			Lacticacid	Succinicacid	Citricacid
	𝑵	-	250	250	250
	𝒒 𝑹𝑴𝒆 𝟑 𝑵 +	𝑚𝑜𝑙. 𝐿 𝑟𝑒𝑠𝑖𝑛 -1	2.4	2.4	2.4
	𝒌 𝑨/𝑶𝑯	-	2.5	330	110
	𝒌 𝑯𝑺𝑶 𝟒 /𝑶𝑯	-	85	85	85
	𝒌 𝑺𝑶 𝟒 /𝑶𝑯 𝐿 𝑟𝑒𝑠𝑖𝑛 . 𝐿 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 -1	150	150	150
	𝑲𝒔 𝑨𝑯/𝑯𝑺𝑶 𝟒 -𝐿 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 . 𝑚𝑜𝑙 -1	0.225	0.82	0.70
	𝑲𝒔 𝑨𝑯/𝑺𝑶 𝟒 𝟐-𝐿 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 . 𝑚𝑜𝑙 -1	0.225	0.82	0.70
	𝒙	-	2	1	1
	𝒚	-	2	1	1

Acknowledgements

The authors would like to thank the Fondation Paris-Reims,the RégionChampagne-Ardenne, the Département de la Marne and ReimsMétropole for their financial support.


J.-C Krapez
email: krapez@onera.fr

Construction of sequences of exact analytical solutions for heat diffusion in graded heterogeneous materials by the Darboux transformation method. Examples for half-space

Construction of sequences of exact analytical solutions for heat diffusion in graded heterogeneous materials by

the Darboux transformation method. Examples for half-space

Introduction

The heat diffusion in heterogeneous media problem, i.e. where both volumetric heat capacity and conductivity are space-dependent, has no general closed-form solution when these parameters present arbitrary distributions, even in the one-dimensional (1D) case with respectively   profiles [START_REF] Carslaw | Conduction of heat in solids Oxford Univ[END_REF]- [START_REF] Alvarado-Leaños | Thermal Resistance Formulation of Fourier Equation and Its Application in the Study of Inhomogeneous Materials and Inverse Problems Int[END_REF]. These solutions are available in closed-form in the Laplace or Fourier space (the corresponding property profiles will be called solvable profiles). In addition, series solutions were also proposed; they are either based on the use of the successive derivatives of conductivity [START_REF] Fivez | Thermal waves in materials with inhomogeneous thermal conductivity: an analytical approach[END_REF], on the Generalized Integral Transform Technique (GITT) [START_REF] Naveira-Cotta C P, Cotta R M, Orlande H R B | Eigenfunction expansions for transient diffusion in heterogeneous media[END_REF], or on the Spectral Parameter Power Series (SPPS) method [START_REF] Khmelnytskaya | The heat transfer problem for inhomogeneous materials in photoacoustic applications and spectral parameter power series[END_REF]. Approximate methods were also developed which are restricted to the situations where the thermal properties variations are slow and/or of low amplitude ( [START_REF] Thakur | Periodic heat transfer through an inhomogeneous medium[END_REF], [START_REF] Krapez | Thermal effusivity profile characterization from pulse photothermal data[END_REF], [START_REF] Thakur | Temperature variation in upper earth crust due to periodic nature of solar insolation[END_REF]- [START_REF] Voti | Photothermal Depth Profiling by Thermal Wave Backscattering and Genetic Algorithms Int[END_REF], just to name a few). Another type of approximate method consists in discretizing the domain and assuming that, in each interval, the property profile follows one of the solvable profiles mentioned before. The easiest way is of course to assume that the properties are constant in each interval. For reducing the model error one can either increase the number of intervals or select solvable profiles of higher order, e.g. linear profiles for conductivity or for effusivity. There is actually a need for solvable profiles that present a higher flexibility: they should allow modelling heat diffusion in materials that possibly present steep profiles without requiring excessive discretization or even requiring no discretization at all.

A method was recently presented in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] which allows building sequences of solvable effusivity profiles and their joint temperature/heat flux solutions, thereby involving an increasing number of parameters (up to two additional free parameters per step in each sequence). By increasing the number of parameters, it is expected that the solvable effusivity profiles obtained therefrom fit better and better to any real continuous profile (when the real profile presents a local jump, the procedure has to be performed separately on both sides of the discontinuity; the global solution is then obtained by quadrupole multiplication). The proposed method consists of a series of three transformations: first a Fourier/Laplace transformation, then a Liouville transformation [START_REF] Liouville | Second Mémoire sur le développement des fonctions ou parties, de fonctions en séries dont les divers termes sont assujettis à satisfaire à une même équation différentielle du second ordre, contenant un paramètre variable[END_REF] and finally a Darboux transformation [START_REF] Darboux | Théorie des surfaces[END_REF] which is carried out jointly on the equation satisfied by a thermal property (the effusivity profile at power +/-1/2) and on the equation satisfied by a thermal field (either the transformed temperature or the transformed heat flux). This procedure amounts to the so-called PROFIDT method (joint PROperty and FIeld Darboux-Transformations).

The Darboux transformation is a differential transformation which allows generating sequences of solvable 2 nd order ordinary differential equations (ODE) in Liouville-normal form [START_REF] Darboux | Théorie des surfaces[END_REF] (now so-called stationary 1D Schrödinger equations). Like other related methods (supersymmetry quantum mechanics-SUSYQM, the factorization method) it found, since the 70', numerous applications in quantum mechanics and in the realm of nonlinear evolutionary equations (solitons) [START_REF] Matveev | Darboux transformations and solitons[END_REF]. Surprisingly, no application of the Darboux transformation to the problem of heat diffusion in heterogeneous media could be found before the work presented in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. The only application related to thermal science we have found is in [START_REF] Arrigo | Darboux transformations and linear parabolic partial differential equations[END_REF]. It deals with the heat equation expressed in the time domain, not in the Laplace-Fourier domain as we did in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. We actually found it more convenient to work in the Laplace-Fourier domain: the advantage is that the obtained solutions can be easily combined together through the quadrupole method in order to get solutions for more complex materials, i.e. heterogeneous multilayers with possibly discontinuous profiles. An alternative to the Darboux transformation method is the Bäcklund transformation method [START_REF] Varley | A method for obtaining exact solutions to partial differential equations with variable coefficients[END_REF]. As illustrated in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] the PROFIDT method allows building sequences of effusivity profiles and the corresponding thermal solutions in a more simple and systematic way.

Several examples of solvable effusivity profiles were provided in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. Some of them are not defined over the entire half-line

0  z . Others are unbounded, i.e.   b for   z
. They represent the case of a progressively more inertial material; this means that temperature is more and more "stable" at progressively higher distance from the surface 0  z , approaching asymptotically the case of a boundary condition of first type. In this paper we will focus on (simple) solvable effusivity profiles that are bounded and non-vanishing on the half-line 0  z . We will first briefly recall the analytical tools that are involved in the PROFIDT method [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. Then, we will describe the main properties of a simple solution based on merely three parameters which was already illustrated in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. Additional results as obtained by assuming heat losses at the free surface of the heterogeneous material will be presented. This will complement the illustrative examples in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] aimed at demonstrating the interest in the PROFIDT method for constructing exact analytical solutions for heat diffusion in heterogeneous materials (actually for any linear transfer process -by diffusion or by wave propagation -provided that the transfer equation may be transformed into a stationary Schrödinger-type equation). The proposed solution sequences present the additional advantage that, in the Laplace-Fourier space, they are in closed-form and they involve only elementary functions.

Successive transformations of the heat equation

Liouville transformation

Let us consider the 1D linear heat equation in a continuously heterogeneous half space without heat sources in the Laplace-Fourier domain as obtained after a Laplace transformation (for modelling transient phenomena) or a Fourier transformation (for modelling periodic phenomena), which changes

the temperature   t z T , into   p z,  and the heat flux   t z,  into   p z,  :               z dz d z dz d z c p 0 , 1    ( 1
)
where p is respectively the Laplace variable or the Fourier variable  i where  is the angular

frequency f  2 .
The boundary condition at 0  z will be considered later. The next step is a Liouville transformation [START_REF] Liouville | Second Mémoire sur le développement des fonctions ou parties, de fonctions en séries dont les divers termes sont assujettis à satisfaire à une même équation différentielle du second ordre, contenant un paramètre variable[END_REF] which first consists in an independent variable change

  z where   z    is the square root of the diffusion time (SRDT) along the   z , 0
interval, taking into account the possible diffusivity variations therein:

     z du u a 0 2 1  . It also involves a dependent variable change                b p p p , , ,   . The equation for the new variable   p ,  
is a 2 nd ODE that can be cast in a system of two homologous Schrödinger-type equations (i.e. with same potential function

   V ):                 s V s p V     (2a) (2b)
where primes denote derivatives with respect to the SRDT  and   

s is a thermal-property profile, namely       2 1 b s 
, the profile of the square root of effusivity. Let us mention that when starting with the heat equation expressed for the heat flux instead of the temperature, we get the same pair of equations as before, except  

p ,   is now the Liouville-transformed heat flux and    s represents    2 1 
b [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. A direct consequence is that when a solvable effusivity profile is found, one has at hand a second solvable profile, namely its reciprocal.

Joint Darboux transformations (PROFIDT method)

The Darboux transformation [START_REF] Darboux | Théorie des surfaces[END_REF], [START_REF] Matveev | Darboux transformations and solitons[END_REF] allows generating a sequence of solvable potentials    

     n n p n n    1 ,    (it is called "superpotential" in the SUSY vocabulary).
The potentials are related by (other equivalent relations are presented in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]):

      1 , 2 1 2 2 1        n p n n n p V V n  (3)
We give now a transcription of the Darboux transformation aimed at relating the general solutions

corresponding to the potential      n V and      1  n V
. If the general solution at level n and for the parameter value p is given by the linear combination (LC) of two independent solutions

    p K n ,  and     p P n ,  , which we write               t n n n p P p K p , , , ,     
, the general solution at level

1  n is given by:                                                                                   1 2 , 1 , 1 , 1 ,
We refer the reader to [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] for an alternative expression for the one involving the primitive of   ) is given by the linear combination (LC) of two

independent solutions      n B and      n D , i.e.               t n n n D B s    , 
, the general solution at level 1  n is given by:

                                                                               0 if ' ~0 if 1 2 0 , 1 0 , 1 0 , 1 , , 1 1 1 1
1 n n n n n n p n n n p n n n n n p d p D D B B D B s n n           (5)
Interestingly, as previously announced, this new LC provides not only one, but two effusivity profiles families. As a matter of fact this LC describes either

     2 1 1   n b or      2 1 1   n b
. Evidently, one must carefully select the constant parameters in the LC (as well as those defining the particular solution

    1 , ~ n n p   ) so that      1  n s
is defined and positive on the considered  -domain (for retrieving

   2 1  b ,      1  n s
should also be node-free). One can find in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] a discussion on how to relax this constraint when the objective is to retain only the results at a given final level N n  1 ; all intermediate results can in fact be considered as non-physical entities: in particular the intermediate "effusivity" profiles could be negative or even complex-valued; the positivity constraint should apply only to the final    



). When choosing

0 1   n p
, there is of course only one additional free parameter.

Example of a first order PROFIDT solution

The sequence is initiated with the simplest (trivial) potential function    

0 0   V .
For getting effusivity profiles defined over a semi-infinite interval, we have to choose a positive transformation parameter:

0 1  p [16] A particular solution of equation (2a) is then           t c c p       sinh , cosh ~1 , 0 
where

       tanh 1 , 0 1   c p or        ĉoth 1 , 0 1   c p
, where we introduced the modified SRDT

      c ˆ;  is a free parameter in      ,
. Applying the potential transformation in equation ( 3), we get two forms for the new potential:

      ĉosh 2 2 2 1     c V and       ŝinh 2 2 2 1     c V
, where the first one is known as the one-soliton potential [START_REF] Matveev | Darboux transformations and solitons[END_REF]. The solutions of equation (2a) with these new potentials, are obtained according to equation ( 4); those leading to bounded temperature and heat flux for    are defined by:

         . exp 1 , 0 1 p p p        Two admissible classes for      1 s
are found when applying equation ( 5):

          t s     tanh 1 , tanh 1   and           t s     ĉoth 1 , ĉoth 1  
. When the coefficient leading the second function of these LC is strictly positive, no solution can be found for

     1 s that is positive for 0  z
. When this coefficient is strictly negative, such solutions can be found, however the induced effusivity profiles are either unbounded and increasing as 2  or vanishing and decreasing as 2   . Therefore, like in [16] we will restrict to the cases of bounded, non-vanishing profiles as given by

        tanh 1 2 1 1    b
. These so-called

   tanh 1 
-type profiles were already mentioned in [START_REF] Varley | A method for obtaining exact solutions to partial differential equations with variable coefficients[END_REF] as leading to exact (thermal) solutions. In [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] we presented the associated temperature solutions both in the Laplace-Fourier domain and in the time domain for an adiabatic half-space. We will complement the presentation by providing the analytical thermal response in the case of linear (convective and/or radiative) heat losses (i.e. 3 rd type boundary condition -BC).

By adjusting the free parameter  , the

   tanh 1 
-type profiles can be made positive and monotonically increasing or decreasing over the half-line 0  

. By specifying that at the position 0  z effusivity takes the value 0 b and that bulk effusivity is  b we get the following expression:

            1 0 0 1 tanh 1        c c b b b b b b      (6)
Expressing this profile as function of physical position z instead of the SRDT  requires implementing an inverse Liouville transformation as described in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF], § 2.4. It necessitates an additional information, as for example, a constitutive law relating (some of) the thermal parameters a , b , c ,  . In particular, in the case of the commonly observed situation of a constant volumetric heat

capacity     c z c
, the previous effusivity profile expresses implicitly through:

                                                       b b b b b b j b b b b b b b b z b z j j c 1 0 1 0 0 1 2 0 2 1 0 1 1 1 , 1 atanh atanh (7)
where the characteristic (real) depth for the profile transition, c z , was defined from the characteristic SRDT c  in the following way, which involves the geometric mean of the surface diffusivity 0 a and the bulk diffusivity  a :

  4 1 0   a a z c c  . A set
          1 1 1 1 0 0          Z h g
. The result of the analytical Laplace inversion was provided in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] for the adiabatic case ( 0  h

).

                          t A t A t b Q t T c       erfcx erfcx 1 0 (9)                         t A t A H b P t T c      erfcx erfcx 1 1 1 0 (10)                                                  H H H b h H b b A A c 4 2 1 ; ; ; ; 2 0 0 1 1 (11)
In equations ( 9) and ( 10)

      x x x erfc exp erfcx 2 
is the scaled complementary error function (in the event that the argument is complex, one should refer to the Faddeeva function). The normalized temperature for the Dirac case and the step heating case (i.e. the terms in bracket in the equations ( 9) and ( 10)) is plotted, respectively in figure 2 and3 for two values of the bulk-to-surface effusivity ratio 0 b b  (0.5 for dashed curves and 2 for plain curves) and for three values for the reduced heat coefficient H (from top to bottom: 0, 0.01, 0.03, 0.1). The short-time transient response approaches the one of a homogeneous material with effusivity 0 b (the same happens for the high frequency response), whereas the long-time response (or the low frequency response) approaches the one of a homogeneous material with effusivity  b . The transition is naturally smoother than the one observed with a bi-layer made of two homogeneous materials with effusivity 0 b and  b .

Conclusion

This paper gave us the opportunity to present a short glimpse of the huge possibilities offered by the PROFIDT method. Many other examples of solvable profiles and their joint temperature solutions were presented in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]. The example presented in the previous section was obtained after a single  the half-sigmoidal shape of the solvable profiles (see figure 1) can approximate many real effusivity profiles (e.g. profiles in case-hardened steel [START_REF] Krapez | Thermal effusivity profile characterization from pulse photothermal data[END_REF], [START_REF] Fivez | Case hardening inspection of steel using photothermal phase maxima[END_REF], [START_REF] Alvarado-Leaños | Thermal Resistance Formulation of Fourier Equation and Its Application in the Study of Inhomogeneous Materials and Inverse Problems Int[END_REF], [START_REF] Fivez | Thermal waves in materials with inhomogeneous thermal conductivity: an analytical approach[END_REF], [START_REF] Mandelis | Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids[END_REF], [START_REF] Voti | Photothermal Depth Profiling by Thermal Wave Backscattering and Genetic Algorithms Int[END_REF], in soils , in living tissue like skin and teeth, in Functionally Graded Materials (FGMs) -see the cited references in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF]);  the corresponding (Laplace-Fourier) surface impedance has a very simple (rational) expression (see equation 8) which allows calculating very easily the modulated or the transient surface temperature, even in the case of heat losses, as shown just before. This feature opens new prospects for building an efficient inversion method for processing photothermal data and identifying a subsurface effusivity profile from the transient or modulated surface temperature. Many other perspectives are envisioned:

 by performing a second PROFIDT, one will have at hand a broader family of still more "sophisticated" solvable effusivity profiles; they will indeed be defined with five parameters instead of only three. These profiles present subtle undulations and possibly a local maximum or minimum. They could thus fit more complicated real profiles. They will be presented in a forthcoming paper.  another approach, as shown in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF], consists in stitching a semi-infinite profile of    tanh ) and they are defined with four parameters. Their outstanding interest is in that they are perfectly "flexible": they can accommodate any quadruplet formed by the four values reached by effusivity and its slope at the ends of the interval. In the same way as three-degree polynomials are used for interpolating splines, the former profiles can be used to build "solvable effusivity splines". A preliminary illustration was provided in [START_REF] Krapez | Heat diffusion in inhomogeneous graded media: chains of exact solutions by joint Property & Field Darboux Transformations[END_REF] where a    ŝech -type profile was joined to a    tanh 1  -type profile whilst ensuring continuity up to the second derivative at the node. The global solution was obtained by implementing a specific quadrupole method. More complex solvable (synthetic) profiles will be presented in a forthcoming paper.

  recently, exact analytical solutions were known only for a limited number of  

V.

  a given one,      0 It is essentially based on two differential relations: the first one relates a given potential      n V with the next one in the sequence,      1  n V ; the second relation describes how getting a solution of the Schrödinger equation (2a) with the potential      1  n V and the parameter value p from a solution of the same equation but with the potential      n V . Both relations are based on the choice of a particular solution of equation (2a) with the potential     

  sequence. When repeating the Joint Property & Field Darboux Transformation one obtains progressively more sophisticated profiles together with the corresponding temperature/heat flux solutions. At each step, when the transformation parameter 1  n p is chosen different of 0, two additional free parameters are introduced in the definition of the new effusivity profile: the first one is 1  n p and the second one comes from the constant parameters in the LC defining the particular solution     constant parameters actually collapse into a single one when passing to the logarithmic derivative



  corresponds to a characteristic diffusion time of the material heterogeneity distribution. The superpotential may then take two forms:

  of solvable effusivity profiles corresponding to the present model is shown in figure 1, for different values of the ratio 0 b b  between 0.4 and 2. As clearly shown, this model allows generating smooth semi-sigmoidal profiles evolving from a surface value 0 b to a bulk value  b . The surface impedance (i.e. the ratio of the local temperature to the heat flux density) is easily obtained by completing the inverse Liouville transformation [16]. For the present profiles it takes a particularly simple expression: now a 3 rd type BC with a thermal solicitation g and a heat transfer coefficient h ( g encompasses the externally imposed heat fluxes and the external contribution a h where a  is the surrounding fluid temperature), the surface temperature then reads:

Figure 1 .

 1 Figure 1. Normalized (with respect to the surface effusivity 0 b ) solvable effusivity profiles deriving from the    tanh 1  -type. Heat capacity is here assumed to be constant. The profiles are plotted in terms of the normalized depth c z z where

  Darboux transformation. Although very simple (the recovered effusivity profiles are merely monotonous over the half-line 0  z ), this model presents a noticeable interest:

Figure 2 .

 2 Figure 2. Normalized temperature as induced by a Dirac pulse (equation 9); see text.

Figure 3 .

 3 Figure 3. Same as in figure 2 for a step heating (equation 10).

  or a series of profiles of    ŝech -type. The latter profiles are defined on finite intervals. They involve the hyperbolic secant function ( cosh 1 sech 

Acknowledgments

This work was funded in the frame of the RGC-DSG Program: "Codes Optiques".


Christian Milelli
email: christian.milelli@u-parisl0.fr

The surge of Preferential Trade Agreements across Asia: What is at stake?

Keywords: Asia, Preferential Trade Agreement, Regionalization

Economists along with policy makers are generally viewing trade agreements as a 'second best' process for trade expansion and economic growth on a global scale. The current surge of Preferential Trade Agreements on a bilateral basis, particularly in Asia, is somehow challenging such common view. The following paper is grounded on updated rough facts and put forward that the standard economic approach is a bit flawed. Obviously, the outcomes and prospects for Asian countries seem much more problematic insofar as power asymmetry and discrimination are embedded in these arrangements.

Nowadays, Asia is highlighting a surge of Preferential Trade Agreements (PTAs), incorrectly called Free Trade Agreements, as a new experience. As a result, Asia has been propelled to the forefront of a more general phenomenon which started in the 1980s, inflated at the beginning of the 1990s and picked up speed since 1995. So far, more than 300 PTAs have been notified to the WTO with more than half in force. Beyond their diversity these arrangements have in common that they have both a bilateral and preferential incline. However, the current wave is not a new phenomenon. In the 1950s and 1960s there were similar waves but to a lesser extent around the European construction. Likewise, it's not an autonomous phenomenon but rather endogenous to the ongoing trade liberalization process: the GATT/ WTO charter makes explicit provisions for them. But their scope largely exceeds eliminating tariff and non-tariff barriers, which were their raison d' être at the beginning. The main reason behind this new drive is to be found in the difficulties and frustrations to finalize the most recent multilateral trade negotiations (1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994). So far, these rounds of negotiations have been a powerful and undeniable driver for the liberalization and the continuous expansion of international trade. Since 1947, the average ad valorem tariffs on industrial goods in developed countries have been dramatically reduced from 40 percent to less than 4 percent; nevertheless, agricultural products' trade has not achieved such a goal due to protectionist policies of industrialized nations1 . Moreover, the WTO which succeeded to the GATT in 1994 is currently made up of nearly 150 members with disparate views; decisions need time due to consensus rule, coalition games, and an agenda with a continually extended scope: almost ten years were required to finalize the Uruguay round.

Equally important was the increasing U.S. support for PTAs after the failure of the Seattle ministerial conference in 1999, in sharp contrast with the 1950s and 1960s, when the U.S. trade administration was strongly opposed to any such arrangement. For [START_REF] Krugman | Is bilateralism bad?[END_REF] the waning of the U.S. capacity and will to defend trade multilateralism is of main importance.

As a result, bilateralism bas become an alternative to multilateralism, and flexibility and speed are increasingly put forward by its promoters, but it is also a vector of proliferation for preferential arrangements. The ongoing upsurge of PTAs through Asia challenges the global trade architecture built on the non-discriminatory principle and weakens any comprehensive regional integration scheme. ASEAN cohesiveness and its fledgling identity as the regional grouping are threatened by the growing rivalry between China, Japan, and South Korea, and to some degree India.

I. Definition and scope

Definition

For the sake of clarity, let's recall to some basic definitions and draw a typology. Since the pioneering work of Bela [START_REF] Balassa | The theory of economic integration[END_REF], PTAs are generally viewed as the first stage of an economic integration process which can result after several sequences to a truly political union (Figure 1). PTA members reduce tariffs along with non-tariffs barriers 2 among themselves but keep their original tariffs against the rest of the world -i.e. there is no common external tariff as in Customs Unions. Free Trade Agreements or FTAs are the extension and natural outcome of the previous ones wherein each member dismantles totally trade barriers among them, but keep their former tariffs with non-members countries. Asia is largely concerned by PTAs due to the novelty of the phenomenon and the will of most countries to retain some independence in trade policy toward non-members. Interestingly, PTAs, FTAs and Custom Unions are called Regional Trade Agreements in the WTO terminology, whereas in the Belassa view they are embodied in a first tier denominated shallow integration. These arrangements concern either countries of similar development level endowed with comparable resources, mainly developed countries, or countries with different development level endowed with distinct resources. They also concern agreements between countries and regional groupings (China-ASEAN or Japan-ASEAN agreements). Lastly, they are not limited to a specific region, taking into account propinquity, but increasingly involve countries from distant regions (see agreements concluded between South Korea and Chile, or Japan and Mexico). In the economic literature, PTAs or FTAs are still regarded as nearly identical whereas in the real world heterogeneity is the rule; the academic interest is focused on trade effects stemming from tariffs reductions with the perspective of welfare improvement along with the growth of international trade even if a minority strand of research emphasizes strategic and political aspects resulting from widespread dissemination of such arrangements in the international arena.

Scope

PTAs are at odds with the WTO first article known as the unconditional most-favoured nation principle, which stipulates that every trade advantage that a member gives to any country must be extended, immediately and unconditionally, to all other members. Historical reasons explain this peculiarity: at the beginning, the treaty setting up the GATT had to include the British Commonwealth which was a free trade area. Consequently, a derogation of the nondiscrimination principle was introduced in the founding charter (Article XXIV), and thereafter endorsed by the WTO. To date, the WTO has three provisions allowing trade preferences on a reciprocal basis 3 : i/ since 1948, the Article XXIV ( § 4 to 10) provides an exception for free trade areas and customs unions which discriminate against other GATT members. This provision was updated in 1994. It is solely by this measure that developed countries can enjoy trade preferences; the European Economic Community and NAFTA were both based on this dispensation. Moreover, any arrangement between developing countries and developed ones is based on it. But these agreements must meet specific conditions, such as transparency and neutrality against non-members, and they particularly should apply to 'substantially all trade' for a reasonable length of time. What is meant by substantial? The GATT article is helpless as 3 There are also schemes like the Generalized System of Preferences which permit to grain trade preferences on an unilateral basis.

Preferential Trade Agreements

Partial preferences to trading partners (Ex: NAFTA, AFTA)

Free Trade Agreements

Elimination of all tariffs and NTBs

Custom Unions

Shallow integration stages

Deep integration stages the precise definition does not exist. However, if we refer to EU documents, over 90 percent of 'all trade' would have to be completely free. And, as long as PTAs do not violate the provisions of the WTO, they can include a wide range of arrangements; likewise, they can discard contentious matters as specific farm products: for example, rice was excluded at the onset of trade negotiations between Thailand and Japan;

ii/ since 1979, the Enabling Clause allows less-developed countries to agree for mutual reduction or elimination of tariffs and non-tariffs barriers on products imported from one another through bilateral or regional arrangements. These requirements are more flexible than those contained in aforementioned Article XXIV since, for example, they permit the exchange of preferences on a subset of products, and the partial reduction, rather than elimination, of trade barriers. Agreements notified under the Enabling Clause have merely to be formally notified to the WTO. They are part of a more general approach promoting differential and more favourable treatment for those countries as well as the acknowledgement of a non-reciprocity principle in trade negotiations. The ASEAN Free Trade Agreement (AFTA) and the MERCOSUR/ MERCOSUL are grounded on this provision;

iii/ and, since 1994, the Article V from the General Agreement on Trade in Services (GATS) supplies an exception to the most-favoured nation principle. This provision was included in the latest agreements -e.g. between Australia and Singapore (2003), and between Chile and South Korea (2004).

Furthermore, the latest PTAs tend to include 'Singapore issues' (investment, competition policy, trade facilitation, and transparency in government procurements). For example, if the facilitation of customs procedure and documentation requirement has long been pointed out as a common objective for both developed and developing countries, to move forward it still requires substantial technical assistance to developing countries in order to strengthen their fledging administrative capacities. More generally, these issues have not yet been approved by the WTO members on a whole4 .

Finally, PTAs occasionally include provisions about harmonization or mutual recognition of technical standards, electronic commerce promotion... and even political and security aspects which are not at all related to trade liberalization! If the former is rather beneficial for trade expansion the latter is much more problematic due paradoxically to the lack of transparency (for example when military interests are at play) or to very sensitive issues, such as national sovereignty.

To sum up, the capacity of PTAs to continuously expand their scope along with their customization explains the diversity of their wording (comprehensive economic cooperation, closer economic partnership...), but the current proliferation of such arrangements, notably in Asia, is a serious challenge to the multilateral framework, and also a new basis for national rivalry.

II. The Asian drive

It has to be reminded that the most important part of PTAs is still located in Europe with 100 in force in 2004 for a world total of 150. Until the end of the 1990s, Asia has been lagging far behind Europe, and even South and Central America.

However, nowadays the trend is clearly shifting to Asia, and once again a catching-up process is somehow at play (Table 1). Source: WTO

The causes of the Asian delay

Here, two main factors stand out. First, is the lack of an institutionalized process for regionalization despite ASEAN. There is convincing evidence that PTAs were fuelled in Europe by the integration process along with the enlargement perspectives. To some extent, Asia's late coming is not only due to the lack of political will from Asian leaders but also to the U.S. opposition to any real economic cooperation on a regional basis [START_REF] Hamilton-Hart | Asia's new regionalism: government capacity and cooperation in thee western Pacific[END_REF]. Although the 'Asia Pacific Economic Cooperation' (APEC) group with U.S. membership was set up in 1989, the subsequent 'East Asian Community' project advocated by the Prime minister of Malaysia, and the 'Asian Monetary Fund' proposed by the Japanese Prime minister were vehemently opposed by the United States, and they failed.

Second, up to the year 2000, the strong backing for the multilateral agenda highlighted by the Japanese authorities was notoriously, and they have been reluctant to abandon the multilateral framework which had been beneficial for Japanese firms. The same is true for South Korean authorities.

The catching-up process

The proposition to the ASEAN leaders, to implement a FTA for 2011, proposed by China after joining the WTO, has ignited the Asian drive. Although not mentioned in official reports, this initiative was a 'wake up' call for Japan, and therefore fuelled a bandwagon effect through Asia with each country wanting to be on board and not be excluded ('me too' attitude).

In the same time, the Asian frenzy for PTAs was stirred by political-economy motives resulting from the interplay between sectors likely to gain from the set-up of such arrangements, and others that would be adversely affected by them. But such balance is inherently unstable and rife with conflict between interest-group lobbies. This is particularly true for Japan where the interests of the business community are growingly coalescing with those of lawmakers and policy makers, as farmers are losing their political clout due to their aging. Governments have a preference for PTAs as the stakes and the related outcomes can be better explained to the public. Besides, those negotiations are often used as a lever to stir domestic adjustments and implement reforms. However, in some cases, the process is not smooth and is blocked by national assemblies. For example, the most protected sectors such as agriculture in South Korea and Japan are strongly opposed to those deals. It is hardly surprising that, in 2003, the endorsement of a trade agreement both between Mexico and Japan, and Chile and South Korea collapsed.

The purposes followed by major countries

We focus on large countries -i.e. with significant size market -which are already well engaged in FTAs negotiations and implementation. Singapore and Thailand are both at the forefront of the Asian drive; South Korea is still behind Japan despite a current catching-up; whereas India is just taking initiatives in this field (Table 1).

• China has an obvious interest in following the WTO discipline to allow relatively unhindered access to the large EU and North American markets to fuel export growth. But, in its periphery, China can exert power in regional and/ or bilateral negotiations to get preferential access to expanding markets (Mallon and Whalley, 2004). As a large manufactured goods exporter, China seeks to stress preferential access in these areas through tariffs reductions. The agreements already concluded or still in progress have in common a general statement, and they lack explicit process to solve trade disputes.

• The United States follows a policy combining three levels (multi-layered): multilateral, regional and bilateral. Under the Trade Promotion Authority (also known as 'fast track') the administration negotiates trade agreements and the Congress restricts itself to approve or reject a negotiated deal, within strict time limits and without amendment. The scope of policy exceeds trade aspects as the United States is the champion of arrangements covering a broad spectrum with particular stress on services liberalization and intellectual property rights enforcement. In this regard, there is clearly a strategic dimension based on national interest which includes the following features [START_REF] Feinberg | The political economy of US FTAs[END_REF]:

-asymmetric reciprocity due notably to market power that advantageously opens foreign markets for U.S. traders and investors; -establishment by the U.S. Trade Representative of new precedents or models to use as benchmarks in future trade, called competitive liberalization; -rewarding and supporting domestic market-oriented reformers as was the case for Mexico, and advancing local democratic institutions; -and, reaffirming and strengthening U.S. security interests through strategic partnerships.

After demonstrating an interest in the Asian zone, particularly the ASEAN countries, the U.S. administration interest is tilting to South America5 . And if the economic impact of PTAs on U.S. GDP has been quite small -as barriers to trade and investment flows are already low in most domestic sectors -FTAs have been elevated into the centre of national trade strategy and supported by a broad-based coalition of bureaucratic interests.

• Japan is mimicking the U.S. multi-layered approach for its trade. The regional priority is Asia, particularly ASEAN, after the initiative taken by China in 2002. Farm products are still put aside in order to protect Japanese farmers, but contrary to the USA or the EU, trade negotiations are not a smooth process due to a turf war between Japanese administrations.

III. What is at stake? Some potential effects

The identification and the measure of the PTAs effects are not an easy task. These difficulties are yet increased by the novelty of the phenomenon in Asia and its current instability; moreover, it is particularly difficult to distinguish what has been signed from what is effectively in force inasmuch as many agreements are more ambitious in aspiration than in implementation. Nevertheless, it's possible in an exploratory attempt to put forward some hypotheses on the potential effects drawn from economic theory and prior experiences along with initial results. Three major effects can be checked off.

Firstly, they affect trade flows and foreign direct investment. The economic literature is still largely focused on trade flows, and a reading of the current agreement contents reveals that foreign investment has become a top priority on government agendas. Moreover, in the mid to long-term these agreements can have implications for the economic growth performance of members by raising the incentive to innovate through technological spillovers, pro-competitive effects, and reduction of redundancies in R&D activities.

Secondly, they impact international relations by reflecting significant power asymmetries. PTAs are bilateral in nature and involve face-to-face bargaining. By implication, related costs and time spent could be a burden for the scare negotiating resources of most developing countries.

Thirdly, they directly affect the multilateral trading system. A large body of literature has already addressed this question and therefore we shall not discuss this point.

Trade effects

The effect of PTAs on Asian trade is still negligible when compare to other regions as Western Europe (EU)6 , North America (NAFTA) or even South America (MERCOSUR). To date, there are no Asian PTA deals with significant trade flows except the project between Japan and South Korea which came to a standstill during the 2004 (Table 2). And when PTAs are in place its coverage could be limited. For example, less than 10 percent of intra-ASEAN trade is under the ASEAN Free Trade Area's common preferential tariff scheme (The Economist, 2004). Source: from [START_REF] Hyun | Free Trade Agreements in East Asian countries: characteristics and remaining tasks[END_REF] If has to be remembered that these trade arrangements primarily concern tariffs and nontariffs reductions. The insights provided by Jacob Viner's seminal contribution in 1950 are still valid: if PTAs and FTAs entail significant trade effects one cannot draw straightforward welfare effects due to the possibility to bring about 'trade creation' or ' trade diversion' -i.e.

PTAs may augment intra-bloc trade by diverting trade away from non-member economies. This ambiguous feature has been underlined by later researches [START_REF] Grossman | The politics of free trade agreements[END_REF][START_REF] Krishna | Regionalism and multilateralism: a political economy approach[END_REF]) using political-economy constructs. Hence, they have showed that trade diversion could be an important motive leading to PTAs. Empirical approaches suggest that Asian PTAs have enhanced regional trade [START_REF] Clarete | Asian regionalism and its effect on trade in the 1980s and 1990s[END_REF], but the effect is less important than in other regions which can arguably be explained by the novelty of the phenomenon. For example, under the 'Early Harvest Programme', Thailand's exports of vegetables increased, in the first quarter of 2004, by 38 percent, and fruits by 80 percent (Ankietwicz and Whalley, 2004), but the overall trade share is still low: the previous programme covered 1.2 billion of dollars in 2003, corresponding just to 1.6 percent of trade flows between ASEAN and China [START_REF] Inama | The Association of South East Asian Nations -People's Republic of China Free Trade Area: negotiating beyond eternity with little trade liberalization[END_REF]. Due to low average tariffs (10 percent), except in some Asian countries for mega-tariffs, rules of origin -which are an important component of PTAs7 -constitute a true challenge8 , even when their economic rationale is to prevent the import of any particular commodity from entering through the country with the lowest duty on the item in question and being reexported to other member countries.

For example, rules of origin were put forward by the Japanese administration in the first trade talks with Mexico. Simulations had showed that NAFTA had adverse effects on Japanese exports to Mexico: from 1994 to 2000, the share of Japanese imports in the overall Mexican imports decreased continuously from 6 percent in 1994 to 3.5 in 2000, corresponding to a yearly export gap of around 400 billion yen. But the setting up of rules of origin can be problematic at least for two reasons. First, a strict application can induce protectionism against non-member countries, hence it diverts trade. Second, the continuous increase of PTAs can entail a bundle of confusing and overlapping bilateral deals, known as a 'spaghetti bowl' syndrome, constituting a nightmare for the various actors, custom officers and private sector that have to implement, to enforce or to comply with those regulations.

To sum up, it is still too early to gauge the effects of PTAs and particularly the rules of origin on trade flows through Asia.

Effects on FDI localization

Asian trade was multiplied by 2.5 during the 1990 decade, inward FDI flows were multiplied by 7 for the same period. And in the case of China, the gap was even more dramatic: here, international trade was multiplied by four whereas FDI inflows increased by a factor of eleven! So it is not surprising that incentives to attract FDIs are the most common component of Asian PTAs. In fact, it is the first motive for developing countries9 to embrace the current wave of FTAs, and start negotiations with developed countries which are the main providers of FDI flows.

Here, central commitments are national treatment, most-favoured nation treatment for foreign investors alongside provisions relating to expropriation, compensation, and repatriation of earnings [START_REF] Banda | Beyond goods and services: competition policy, investment, mutual recognition, movement of persons, and broader cooperation provisions of recent FTAs involving ASEAN countries[END_REF].

To date, PTAs have had very little influence on the investment localization. But as more PTAs are put forward the situation may be changing and affect the whole production sharing dynamic across Asia. Accordingly, it's rather a tricky undertaking to size up the effects of PTAs provisions on foreign investment, because they are enmeshed with others effects resulting, for example, from tariffs reduction. If tariffs are retained, an enterprise will invest overseas by setting up a subsidiary company in order to supply a protected market (tariff jumping), instead of exporting and paying tariffs. A PTA with this country automatically makes the market less protected, hence reducing its attractiveness to invest10 . An enterprise can also invest overseas in order to reap various comparative advantages regarding different manufacturing processes: semi-final goods or components need to be exported to other plants, operations which incur tariffs. Therefore, by reducing these costs a PTA can increase foreign investment, and transnational corporations could enlarge their international division of labour. But these arrangements can have adverse effects by raising transactions costs (rules of origin requirements), and encouraging concentration on few localisations. The national origin of FDI is also a key element as the impact depends on whether the supplier country is part of the trade agreement. In the case of NAFTA, FDI from the United States and Canada have been significantly increased in Mexico after 1994(Waldkirch 2001); thus, the creation or the extension of preferential trade zones can divert FDI, and this could be particularly problematic for countries with high manufacturing costs.

Effects on international relations

It is well known that the international stage encompasses substantial power games effects where large countries enjoy bargaining power on account of their economic size and political clout. The wave of bilateralism across Asia through PTAs carries such risks even if talks and negotiations conducing to trade agreements are in line with the WTO framework. It has to be reminded, that most Asian countries have more benefited from the multilateral negotiations through the GATT/ WTO system than through face-to-face negotiations with the United States. Indeed, in multilateral arena Asian countries can better share and advocate common interests through a mutually advantageous approach. But the current multilateral framework is marred with various difficulties. The entire system is challenged by the necessity to keep on with the expansion of international tradeparticularly, to tackle pending issues such as the liberalization of agricultural products' trade, or to grasp new ones as the enhancement of trade facilitation -and a decision-taking process which is growingly time-consuming. In some extent, the situation is similar to the EU one with difficulties to move ahead and embrace its political agenda.

IV. Are Asian PTAs enhancing regional trade and a step to economic integration?

The ongoing proliferation of PTAs across Asia, let alone with other regions, requires not minimizing the risks for regional trade along with economic integration. Some recommendations can be drawn from the initial results. First, inasmuch as PTAs are partial, fragmentary, heterogeneous, and above all uncoordinated, it stands to reason to seek harmonizing the different rules and regulatory codes attached to the various Asian PTAs. Obviously, the lack of cooperation is crucial and the following question has to be raised: which institution or framework would be in charge to tackle the task. Arguably, the logical candidate is the 'ASEAN plus three' forum whose members carry the bulk of Asia's trade. However, historical and strategic motives, particularly the growing rivalry between Japan and China, impaired the organization taking effective initiatives with convincing political will. In our view the main challenge for Asian countries is to keep on pace with a trade-led growth regime based on a manufacturing sharing configuration on a trans-regional base. Accordingly, it will be particularly appropriate to have PTAs focusing on trade environment. Indeed, due to the specificity of Asian trade (intra-regional flows of intermediate goods and strong FDI-trade nexus) the following undertakings would have a great impact on Asia trade: trade facilitation (notably the easing of customs procedure and documentation requirements), the promotion of cross-border investments, and to some extent, the liberalization of people mobility across borders from geographically contiguous countries. To be effective this drive has to get strong political support, and major Asian countries should be the promoters and advocates of this route toward regional trade integration. In addition, these commitments could be more easily endorsed by the WTO in coming negotiations. In parallel, it would be appropriate to largely disseminate general guidelines supporting this orientation, and multinational bodies like the Asian Development Bank could endorse this helpful job.

Second, PTAs could also impact significantly economic integration. To date, Asia has no regional undertaking apart from ASEAN, but ASEAN is quite different from the EU or NAFTA. Indeed, when one gives closer scrutiny to ASEAN, it appears to be basically a formal grouping of altogether heterogeneous countries. Of course, regional integration was given a new impetus during the Bali summit in October 2003 -by putting the year 2020 as the deadline for implementing a common market. In fact, the aim was to speed up the ongoing PTAs, which had yet to be implemented [START_REF] Stubbs | Signing on to liberalization: AFTA and the politics of regional cooperation[END_REF]. Trade barriers are still there along with sizeable hurdles for investment and people mobility: a sole market for ASEAN countries is not for today. Such a delay is largely due to a lack of political leadership and economic interdependency, along with inadequate designs and drafting modalities for effective trade liberalization [START_REF] Inama | The Association of South East Asian Nations -People's Republic of China Free Trade Area: negotiating beyond eternity with little trade liberalization[END_REF]. As said previously the liberalization process entailed by AFTA is very slow. As a result, the rising tide of PTAs across Asia could shape the whole regional economic dynamism. The 'stylized facts' approach which had pointed out a two-hubs (China and Japan) or a multiple-hubs (ASEAN, China, Japan and South Korea) emerging design highlighting the chaotic and uneven path to regional integration. Here, the recommendations to be made are more challenging. They require an unequivocal commitment by large countries to facilitate trade integration of less-developed countries, not to mention the poor ones. Likewise, major countries must restrain to embark in a race for strategic trade policy, and rather promote cooperation in multilateral and regional arena. Developing countries on their own must cooperate, exchange information and experiences, and pool scare administrative resources. They must also be cautious not to be involved in binding rules, and favour cooperative arrangements with safeguard measures.

After all, an alternative option to economic integration could be to advance financial integration at the region level. For example, the expansion of current initiative such as the 'Asian Bond Market initiative' toward more financial linkages could be of great interest due to the large amount of foreign currency reserves held by Asian central banks. By doing so, Asian countries would modify the sequence of the stepping stones conducing to economic integration (Figure l) and would embark Asia on a more direct path toward integration like Europe's.

Figure 1 :

 1 Figure 1: The route to the economic integration

Table 1 .

 1 Breakdown of Preferential Trade Agreements already in force or signed in Asia (May 2005)

	Country	Status	Date	Scope
	Partners
	India*:
	ASEAN	signed	2003	general framework
	Thailand	signed	2003	general framework
	China:
	ASEAN	in force	2003	general framework
	Thailand	signed	2002	Early Harvest Programme for some agricultural products
	Australia	signed	2003	trade and economic framework
	New-Zealand	signed	2004	trade and economic cooperation
	Korea:
	Singapore	signed	2005	gradual elimination of tariffs on goods and services
	Chile	in force	2004	gradual elimination of tariffs on goods and services
	Japan:
	Philippines	in force	2006	restraints on labor mobility
	Singapore	in force	2002	comprehensive agreement
	Vietnam	signed	2003	general framework for investment
	ASEAN	signed	2003	general framework
	Mexico	in force	2005	restraints on some agricultural products
	Singapore:
	Japan	in force	2002	comprehensive agreement
	Australia	in force	2003	comprehensive agreement exceeding current commitments at the WTO
	Jordan	signed	2004	full liberalization of goods and services trade
	New-Zealand	in force	2001	comprehensive agreement
	USA	signed	2003	comprehensive agreement
	Philippines:
	Japan	in force	2006	restraints on labor mobility
	Thailand:
	China	in force	2003	Early Harvest Programme
	India	signed	2003	partial agreement
	Australia	in force	2005	gradual elimination of bilateral tariffs barriers on goods
	Bahrain	signed	2002	gradual elimination of bilateral tariffs barriers on goods
	Peru	signed	2003	partial liberalization
	* Signed Free-Trade Agreements with Bhutan and Nepal giving their goods free access to the Indian market, and
	with Bangladesh (concessionary custom duties)

Table 2 .

 2 Share of exports related to PTAs finalized by major Asian countries (in US$ billion)

	Country	Exports to FTA partners	Total exports	Exports share related to PTAs in %
		(A)		(B)	(A/B)
	South	Chile	0.5
	Korea	Singapore	4.6	192.7	2.6 %
	Japan	Singapore	14.9
		Philippines	9.0
		Mexico	3.6	473.9	5.8 %
	China *	ASEAN	26	438.2	6 %
	Actually, through the 'Early Harvest Programme' only trade of some basic agricultural products would be fully
	liberalized by 2006.

Such as mega-tariffs, domestic support to farmers, market access restraints and export subsidies.

Quotas, anti-dumping dunes, countervailing measures and safeguard measures.

Actually, they have been rejected during the fifth Ministerial Conference held in Cancun in 2003.

As part of the Dominican Republic-Central American Free Trade agreement, the U.S. administration signed in 2004 agreements with Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua and the Dominican Republic; in parallel, it opened negotiations with three members of the Andean Community (Colombia, Ecuador and Peru) along with Panama. And, despite troubled negotiations, the ambitious 'Free Trade Area of the America' encompassing all countries of the Western Hemisphere except Cuba, is still on the agenda.

To date, more than 65 percent of intra-European trade is concerned.

They specify a certain proportion of the value -added that must have within union -origin to quality for duty-free; status in the PTAs. In this regard, rules of origin to up 200 pages in the NAFTA agreement.

Indeed, they also entail a political economy dimension as interest groups can exploit them for their own ends.

The World Trade Report (2003) suggests that countries join PTAs to attract FDI.

But other factors can be at play as mark-et size and therefore reverse the causal relationship.