Noureddine Lebaz

Arnaud Cockx

Mathieu Sperandio

Alain Liné

Jérôme Morchain
email: jerome.morchain@insa-toulouse.fr

Mathieu Spérandio

Application of the Direct Quadrature Method of Moments for the modelling of the enzymatic hydrolysis of cellulose: II. Case of insoluble substrate

Keywords: Enzymes, hydrolysis, population balance, particulate substrate, cohesion force

Introduction

The production of bio-ethanol and chemicals from lignocellulose-based materials has attracted extensive research over the last few decades. This growing interest on renewable energy sources and green chemistry is motivated by the environmental and socio-economic issues as well as the depletion of fossil fuels [START_REF] Huang | Bioconversion of Lignocellulose into Bioethanol : Process Intensification and Mechanism Research[END_REF].

The fractionation of the biomass into its elementary constituents by means of hydrolytic enzymes is known as the most challenging step of the substrate transformation process, it directly controls the conversion rate/yield thus, significant efforts have been devoted to (i) the improvement of the substrate pretreatments through a better understanding of the physico-chemical properties of the biomass in order to enhance its digestibility (ii) the understanding of the enzymes hydrolysis elementary mechanisms for improving the hydrolytic potential of the cocktails and (iii) the elucidation and quantification of the factors affecting the enzymes performance and the substrate reactivity during the hydrolysis reaction. These three contributions are necessary before one can develop reliable models for process design at industrial scale.

A typical enzymatic hydrolysis reaction kinetic shows an initial rapid rate followed by a progressive slowdown leading to a partial substrate conversion even for long-time reaction and/or high enzymes loadings [START_REF] Levine | A mechanistic model of the enzymatic hydrolysis of cellulose[END_REF]. This makes the process cost uncompetitive. The decreasing rate of the biomass transformation is attributed to a multitude of factors acting synergistically at different levels [START_REF] Mansfield | Substrate and enzyme characteristics that limit cellulose hydrolysis[END_REF].

Thus, the modelling of this process-step is particularly challenging especially when dealing with insoluble substrates which is mostly the case.

The enzymatic hydrolysis of particulate substrates is a heterogeneous multi-step reaction. The enzymes are first adsorbed onto the substrate surface by their Carbohydrate-Binding Module (CBM) before acting with the Catalytic Domain (CD) and hydrolyze the glycosidic bonds [START_REF] Hilden | Recent developments on cellulases and carbohydrate-binding modules with cellulose affinity[END_REF]. As described in the first part, Michaelis-Menten type kinetic models are widely used for the modelling of this reaction [START_REF] Bansal | Modeling cellulase kinetics on lignocellulosic substrates[END_REF]. In that case, the complex substrate architecture is not taken into account and the only variable of interest is the susbtrate bulk concentration. In adsorption-based kinetic models, Langmuir isotherms are used to quantify the fraction of adsorbed cellulases onto the initial substrate [START_REF] Kadam | Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass[END_REF]. As the substrate surface changes all along the reaction as well as other substrate structural features (e.g. crystallinity, accessibility), the cellulases adsorption kinetics are affected [START_REF] Chauve | Modlisation cintique de l'hydrolyse enzymatique des substrats cellulosiques[END_REF]. All these effects are often lumped into a single time-decreasing function (e.g. ratio, negative exponential function) [START_REF] Kadam | Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass[END_REF][START_REF] Neto | Study of kinetic parameters in a mechanistic model for enzymatic hydrolysis of sugarcane bagasse subjected to different pretreatments[END_REF] referred as the substrate reactivity (R s ) which is introduced in the modelling to reflect the dynamic character of the process [START_REF] Zheng | Kinetic modeling for enzymatic hydrolysis of pretreated creeping wild ryegrass[END_REF]. Such simplifications, in both Michaelis-Menten and adsorption type-kinetic models, affect the reliability of these models and thus their ability to predict the timeevolution of the substrate properties and its hydrolysability as reported by [START_REF] Sin | Assessing reliability of cellulose hydrolysis models to support biofuel process designidentifiability and uncertainty analysis[END_REF]. To overcome these issues, the structure of the model has to be rethought. It is certainly interesting to build a model in which the specific area of the substrate particles is described. This motivated, among other arguments, the choice of the population balance formalism.

In the first part of this study, a population balance approach has been developed for the modelling of the enzymatic degradation of cellulose polymer chains. The three main cellulolytic activities have been taken into account. The EG activity is assimilated to a random scission of the polymer chains while the CBH activity is modelled as an erosion process leading to the release of a specific product : cellobiose. The β-glucosidase activity is then incorporated into the population balance model by the means of a Michaelis-Menten type kinetic. The Direct Quadrature Method of Moments (DQMOM) [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF] was retained as the solution method for the population balance equation (PBE) due to its low computational cost and its ability to treat the case of multidimensional problems (i.e. when a particle has to be defined by more than one property). This method is coupled with the Maximum Entropy technique for the reconstruction of the chain length distributions (CLDs) from its moments [START_REF] Lebaz | Reconstruction of a distribution from a finite number of its moments : A comparative study in the case of depolymerization process[END_REF]. The numerical implementation of the model has been validated first against an analytical solution of the PBE before testing the different activities independently and in combination. In addition to the very low computational cost (few seconds using a basic computer station), the model is able to predict EG-CBH synergistic effect as well as the inhibition of the enzyme activities due to the accumulation of the end-products (simple sugars).

This second part is devoted to the specific case of particulate substrates. The general theoretical framework remains unchanged. The population balance model (PBM) is extended to complex systems by rethinking how the different enzymatic activities are acting and degrading the substrate at the particulate scale. In particular, one has to consider now that an enzymatic attack does not imply the particle breakage but rather affects its cohesion. The particle fragmentation is due to the hydrodynamic shear stress. After introducing the theoretical framework, numerical results are discussed in different cases and experimental data are given in order to support the model assumptions.

Theoretical framework

The population balance modelling is a common approach when dealing with particulate/dispersed systems undergoing space and/or phase transformation. In our case, cellulose particles are progressively transformed into simple sugars (cellobiose and glucose) by the cooperative action of the cellulases.

In the first part of this study, a population balance model is formulated and solved numerically in the case of pure soluble cellulose substrates. This second part focuses on insoluble/particulate substrates. Considering the particle volume v as the internal coordinate since it is the conservative variable, the following PBE (equation ( 1)) governs the evolution of the particle size distribution (PSD) in the specific case of a pure homogeneous fragmentation process :

∂n (v, t) ∂t = ∞ v β (v, v )Γ (v )n (v , t)dv -Γ (v)n (v, t) (1)
where n (v, t) is the volume-based number density function, Γ (v) the breakage frequency for a particle of volume v, β (v, v ) is the daughter distribution function giving the probability to obtain a particle of volume v from the breakup of a particle of volume v with v > v. The first term on the right hand side of equation ( 1) accounts for the formation (birth) of particles with volume v resulting from the breakage of larger particles v . The last term is the death term reflecting the loss of particles of volume v because of their breakage.

Assuming that the particle volume is proportional to its size, v ∝ L 3 (with L the particle size), equation

(1) can be expressed in terms of particle length-based number density function n(L, t) as [START_REF] Marchisio | Quadrature method of moments for population-balance equations[END_REF] :

∂n(L, t) ∂t = ∞ L β(L, λ)Γ(λ)n(λ, t)dλ -Γ(L)n(L, t) (2)
To solve equation ( 2), numerical methods can be used. As described in the first part of this study, the DQMOM approach based on the time-tracking of a finite number of the PSD moments is adopted for its low computational cost and its robustness. This is of critical importance for the extension of the model to multivariate systems and/or the coupling with hydrodynamic phenomena. We recall below the physical meaning of the moments notion.

The k th order moment of the PSD is defined as :

m k (t) = ∞ 0 L k n(L, t)dL (3)
The moments provide the main properties of the PSD. The zeroth order moment, m 0 (t), represents the molar concentration of the cellulose particles since n(L, t)dL is the number of particles per volume having sizes between L and L + dL. The first order moment, m 1 (t), gives the total length of cellulose particles per volume unit. Average properties of the CLD can be derived from the moments such as the number-averaged particle size given by the ratio m 1 (t)/m 0 (t) and the Sauter mean diameter D 32 = m 3 (t)/m 2 (t).

By applying the moment transformation to equation (2), we obtain the final moment transport equation below (for more detail, one can refer to the first part) :

(1 -k)

N i=1 L k i a i + k N i=1 L k-1 i b i = N i=1 b (k) i Γ i w i - N i=1 L k i Γ i w i (4)
where

                 a i = ∂wi ∂t b i = ∂ci ∂t c i = w i L i b (k) i = ∞ 0 L k β(L, L i )dL (5)
(L i (t), w i (t)) are the abscissas and weights of the N Gaussian quadrature nodes. (L i (0), w i (0)) can be computed using the Product-Difference (PD) algorithm [START_REF] Gordon | Error bounds in equilibrium statistical mechanics[END_REF] or any other algorithm [START_REF] John | On the efficiency and robustness of the core routine of the quadrature method of moments (QMOM)[END_REF]. Equation ( 4) is similar to that developed in the first part, the only difference resides in the formulation of the breakage parameters (daughter distribution function and frequency).

Daughter distribution functions formulation

We propose thas the evolution of the PSD during the enzymatic hydrolysis of a particulate substrate is governed by both the cooperative action of the enzymes and the mechanical stress acting on the particle due to the fluid motion.

The EG activity breaks up the glycosidic bonds of surface chains affecting by the way the cohesion of the particles and their mechanical properties. The combination of this activity and the hydrodynamic shear stress leads ultimately to the particles fragmentation. Thus, unlike the case of soluble substrate where the EG attacks produce new chains systematically, in the case of particulate substrate, the fragmentation process is complex and both the EG activity and the shear stress have to be taken into account. The CBH activity attacks the free chain-ends and produces cellobiose molecules. This activity can be seen as a surface erosion process. Starting from these considerations, the PSD evolution during the enzymatic hydrolysis process is modelled using the following PBE :

∂n(L, t) ∂t = S B L (L, t) + S CBH L (L, t) (6)
with

     S B L (L, t) = ∞ L β B (L, λ)Γ B (λ)n(λ, t)dλ -Γ B (L)n(L, t) S CBH L (L, t) = ∞ L β CBH (L, λ)Γ CBH (λ)n(λ, t)dλ -Γ CBH (L)n(L, t) (7)
where β B and β CBH are the daughter distribution functions, Γ B and Γ CBH the breakage frequencies for EG/stress and CBH respectively.

We discuss below the choice of the daughter distribution functions and the breakage frequencies in this specific case of particulate substrate.

EG activity and particle breakage

In the case of heterogeneous reactions, as discussed before, the fragmentation mechanism of the substrate particles is complex. The EG activity acts at a microscopic scale by cutting glycosidic bonds without leading necessarily to the breakage of the attacked particles. The structural features of the substrate particles and the hydrodynamic shear stress are to be taken into account when dealing with the PSD evolution during the hydrolysis reaction. With a mono-component substrate such as cellulose, one can consider that all the particles have the same structural characteristics (e.g. degree of crystallinity, porosity). Thus, breakage may occur at any position on the particle surface. In other words, after the binary breakage of a particle, all sizes are equally probable. This can be formulated by a uniform daughter distribution function as in equation ( 8) [START_REF] Marchisio | Quadrature method of moments for population-balance equations[END_REF].

β B (λ 3 , L 3 ) = 2(3L 2 /λ 3 ) = 6L 2 /λ 3 (8)
The source term S B L (L, t) in equation ( 7) can be explicited as :

S B L (L, t) = 6L 2 ∞ L 1 λ 3 Γ B (λ)n(λ, t)dλ -Γ B (L)n(L, t) (9)
2.1.2. CBH activity : chain-end scission

The CBH activity, even for insoluble substrates, can be considered as an erosion process leading to a specific product which is a cellobiose molecule. In this case, an initial particle of volume v losses elementary particles of volume v c after CBH attacks (equation ( 10)).

P (v) -→ P (v -v c ) + P (v c ) (10)
Thus, β CBH in this specific case is set as an erosion distribution function. Equations ( 11) and ( 12) give the time evolution of the particulate system under CBH attacks : equation ( 11) refers to the time evolution of the PSD while equation ( 12) describes accumulation of cellobiose due to the CBH activity).

∂n (v, t) ∂t = ∞ v Γ CBH (v )δ (v -(v -v c )) n(v , t)dv -Γ CBH (v)n(v, t) (11) ∂n c (t) ∂t = ∞ v Γ CBH (v )δ (v -v c ) n(v , t)dv (12)
The moment transformation of equations ( 9) and ( 11) is given in equation ( 4) in its gereral form. Equation ( 12) is to be included to the PBE solved by DQMOM after an appropriate moment transformation (see Appendix C of the first part of the study for more details).

Breakage frequencies formulation

Once the daughter distribution functions are properly defined for the two processes considering particulate substrates, the most challenging part is the formulation of the breakage frequencies Γ B and Γ CBH especially the first one.

For the CBH activity, whatever the nature of the substrate, a specific product is released after an effective attack. Thus, Γ CBH matches with the CBH hydrolysis rate as for soluble substrates. Unlike Γ CBH , Γ B is function of the substrate nature/properties and the operating conditions. In the case of soluble substrate, the rate of breakage of the polymer chains defines the intrinsic activity of the EG. But with insoluble substrate, there is a strong dissociation between the cutting of a glycosidic bond and the particle volume/size evolution since the length-scales are different.

Substrate particle breakup phenomenon occurs when the hydrodynamic force F H acting on the particle surface becomes higher than the cohesion force F p of the particle itself. Thus, the ratio of the two forces F H F p is deterministic [START_REF] Coufort | Forces on spherical particles in terms of upstream flow characteristics[END_REF]. The hydrodynamic force F H is proportional to the particle surface and can be explicited in the case of spherical particles as [START_REF] Douaire | A unique phenotypic modification of Lac-tococcus lactis cultivated in a Couette bioreactor[END_REF] :

F H = 5 2 η • G • L 2 (13)
where G is the shear rate (s -1 ) and η the dynamic viscosity (kg

• m -1 • s -1 ).
The shear rate can be expressed as [START_REF] Camp | Velocity gradients and internal work in fluid motion[END_REF] :

G = ν ( 14
)
where ν is the kinematic viscosity (m 2 • s -1 ) and is the average turbulent energy dissipation rate (W • kg -1 ) which can be approximated via equation ( 15) below :

= P ρ f V = N p N 3 s D 5 V ( 15
)
where P is the power input (W ), ρ f is the fluid density (kg • m 3 ), N p is the impeller power number, N s the impeller speed (r • s -1 ), D is the impeller diameter (m) and V is the tank volume (m 3 ).

Thus, the hydrodynamic force F H exerted on a particle of size L is explicited as :

F H = 5 2 ρ f νN p N 3 s D 5 V 1 2 L 2 (16)
The particle cohesion force F p is related to its density and its size and quantifies the mimimum force required to break up the particle. We propose to model this force as :

F P = A • ρ p • e -θΓ EG •t • L α , α ∈ [1, 3] (17)
with A a constant, ρ p is the density of the particles (kg • m 3 ), θ is non dimensional parameter related to the evolution of the substrate cohesion because of the hydrolysis reaction, Γ EG is the chains breakage 155 frequency related to the EG activity and α is the fractal dimension giving an information on the shape of the substrate particles.

The expression (17) gives the time evolution of the cohesion force of a particle of size L which decreases at the rate (e -θΓ EG •t ) since the substrate is submitted to the EG activity which modifies progressively its structure and mechanical properties. Finally, the breakage frequency of the substrate particles is expressed 160 as :

Γ B = ϕ • F H F p (18)
with ϕ a constant.

The particles breakage frequency Γ B is proportional to the ratio of the hydrodynamic and the cohesion forces. If F H > F p , the ratio is larger than 1 and the breakage phenomenon occurs. If one replaces the two forces by their expressions (equations ( 16) and ( 17)), one obtains :

Γ B = ϕ • ρ f νNpN 3 s D 5 V 1 2 L 2 A • ρ p • e -θΓ EG •t • L α = B(t) • L 2-α (19)
The expression of the breakage frequency in equation ( 19) is coherent with the previous works on the breakage processes induced by both the hydrodynamic shear stress and the cohesion of the particles especially in the case of flocculation process which has been fully studied using experimental and numerical approaches based mainly on the population balance formalism [START_REF] Spicer | Coagulation and fragmentation : Universal steady-state particle-size distribution[END_REF][START_REF] Nopens | Modelling the activated sludge flocculation process combining laser light diffraction particle sizing and population balance modelling(PBM)[END_REF][START_REF] Ding | Population balance modelling of activated sludge flocculation : Investigating the size dependence of aggregation, breakage and collision efficiency[END_REF]. In all these works, the breakage frequency is set equal to (B • v a i ). The specificity in our case is the time-dependency of B(t). This coefficient conveys an information relative to the age of a particle in the system and its exposure to endoglucanases attacks.

The EG and CBH inhibition effect

As in the case of soluble substrates, one has to take into account the inhibition effect since the cellulases are highly sensitive to the accumulation of simple sugars in the system [START_REF] Xiao | Effects of sugar inhibition on cellulases and -glucosidase during enzymatic hydrolysis of softwood substrates[END_REF]. The procedure adopted here is to introduce inhibition constants for both EG and CBH activities and to express the breakage frequencies of these two activities as functions of the cellobiose concentration (equations ( 20) and ( 21)). Thus, the breakage frequencies decrease with the accumulation of cellobiose.

Γ EG = Γ EG0 K EG C c (t) + K EG (20) Γ CBH = Γ CBH0 K CBH C c (t) + K CBH ( 21
)
where C c (t) is the cellobiose concentration, K EG and K CBH are the inhibition constants for EG and CBH respectively, Γ EG0 and Γ CBH0 are the intrinsic EG and CBH activities respectively.

The β-glucosidase activity

Michaelis-Menten type kinetic model is adopted for the modelling of the β-glucosidase activity which transforms cellobiose into glucose. A competitive inhibition by the accumulation of glucose is incorporated as shown in equation ( 22).

dC G dt = V m C c K m 1 + C G K P + C c ( 22
)
where C G is the glucose concentration, K m and V m the Michaelis-Menten parameters, K P the inhibition constant for the β-glucosidase activity.

Global problem formulation

Equation (23) summarizes the system of equations solved simultaneously to track the PSD evolution and the accumulation of simple sugars. This system includes the fragmentation and the erosion processes in the first PBE, the accumulation of cellobiose in the second PBE and finally the accumulation of glucose in the third Michaelis-Menten kinetic equation.

                         ∂n(L, t) ∂t = 6L 2 ∞ L 1 λ 3 Γ B (λ)n(λ, t)dλ -Γ B (L)n(L, t) + ∞ L Γ CBH (λ)δ (L -(λ -L C )) n(λ, t)dλ -Γ CBH (L)n(L, t) ∂C C (t) ∂t = ∞ L Γ CBH (λ)δ (L -L C ) n(λ, t)dλ - 1 2 dC G (t) dt dC G (t) dt = 2V m C C K m 1 + C G K P + C C (23)
The numerical resolution of the system ( 23) is carried out using a DQMOM approach coupled with a reconstruction technique based on the maximum entropy (ME) theory. The ode45 integrator of MATLAB is used on a work-station Intel R Core TM i7-3740QM CPU with a clock speed of 2.7 GHz and 16 GB of installed memory (RAM).

Experimental procedure

Materials

Microcrystalline celluloses (Avicel PH-102 and Avicel PH-105) were kindly provided by FMC BioPolymer (USA). These are pure insoluble cellulose substrates with mean sizes of 90 and 20 µm respectively (data from the supplier). They have been chosen because of their significantly different initial sizes and their comparable physico-chemical properties mainly the cristallinity, as reported in literature [START_REF] Kleinebudde | The crystallite-gel-model for microcrystalline cellulose in wet-granulation, extrusion, and spheronization[END_REF][START_REF] Jumaa | Influence of cellulose type on the properties of extruded pellets. Part I. Physicochemical characterisation of the cellulose types after homogenisation[END_REF]. The cellulase enzyme used in this study was a commercial Trichoderma reesei cellulase preparation (Celluclast 1.5L) from Sigma-Aldrich (Germany). The cocktail contains the main cellulase activities : EG, CBH and β-glucosidases.

Enzymatic hydrolysis

The hydrolysis of the substrates was carried out in 50 mM sodium acetate buffer (pH=4.8) at 40˚C.

Sodium azide N aN 3 (0.1%) was supplemented to inhibit microbial growth during the reactions. For all the experiments, a 500 mL glass tank with a water jacket was used. The rotational speed was fixed to 200 rpm using a blade mixer (4 inclined blades, 5 cm diameter, 45˚angle). The substrate loadings were fixed to 1%

(w/v) and those of cellulases to 30 FPU/g substrate. Samples were withdrawn and boiled for 10 mn to stop the enzymatic hydrolysis. The samples are, then, centrifuged at 2000 g for 10 mn. The supernatants were analysed to reach the simple sugar concentrations while the solid fraction was diluted for size distribution analysis as described hereafter.

The yield of the enzymatic conversion of Avicel PH-102 and Avicel PH-105 into simple sugars (cellobiose and glucose) was calculated as [START_REF] Khodaverdi | Kinetic modeling of rapid enzymatic hydrolysis of crystalline cellulose after pretreatment by NMMO[END_REF] :

Yield (%) = [Glucose (g/L) + Cellobiose (g/L) × 1.053] × 0.9 Cellulose (g/L) × 100 (24)

Particle size distribution determination

The PSDs evolution during the enzymatic hydrolysis reactions were determined through laser diffraction analyses using a Mastersizer 2000 Hydro, Malvern Instruments Ltd. This ex-situ technique is based on the conversion of the detected scattered light into a particle size distribution thanks to the Mie's theory thus, the refractive index of the substrate particles is needed. Analyses are conducted using diluted samples with respect to the recommended obscuration rate. The results are given in terms of volume fraction histograms distributed following a geometric sequence with a first term u 0 = 0.01 µm and a common ratio r = 1.1482 (total number of classes = 100).

Mechanical resistance of the substrate particles

The mechanical resistance of the initial substrate particles was explored using an ultrasonic bath (Ultrasonic Cleaning Unit Elmasonic S 60 (H), Elma Schmidbauer GmbH, Germany) having an operating frequency of 37 kHz. Diluted solutions are prepared in bechers which are placed in the bath for 5 minutes. The ultrasound effect on the substrate particles is revealed by the PSD analysis (Mastersizer 2000 Hydro).

Sugar analysis

Quantitative detection of simple sugars was achieved using high performance liquid chromatography (HPLC) on an Ultimate 3000 Dionex separation system equipped with a BioRad Aminex HPX 87H affinity column and a refractive index detector (Thermo Scientific). The separation of product species was performed in H 2 SO 4 (5 mM) solution at 40˚C and at a flow rate of 0.3 mL/min.

Results and discussion

Numerical results

For numerical illustration of the potential of the PB-based model, we assume that the initial PSD follows a normal law (equation ( 25)) with (µ, σ) its mean and standard deviation respectively.

n(L, 0)) = 1 σ √ 2π e -(L-µ) 2 2σ 2 (25)
Numerically, the parameters of the initial distribution are fixed as µ = 70µm and σ = 5µm. We discuss hereafter the numerical results given by the PBM resolution considering the different cellulolytic activities and the particle fragmentation processes. For the sake of clarity and concision, we report only the main results in each case.

CBH activity

The CBH activity is assimilated to a surfacic erosion process leading to the detachment of elementary particles. Thus, the erosion frequency is set proportional to L 2 . The complexity in the case of insoluble substrates is the large difference between the substrate particle size (micron scale) and the elementary particle size (cellobiose with a sub-nanometer scale). Thus, numerically this considerable gap in terms of size scales is unmanageable since it generates round-off errors. The idea here is to consider a large collection of cellobiose molecules as an elementary particle released because of the erosion process. Thus, an elementary particle is not produced with one CBH attack but it results from a multitude of attacks which are gathered to produce a significant fragment volume in order to produce a fragment with a significant volume from a numerical point of view. In our case, the size of an elementary particle is set to 0.2 µm. This size is sufficiently small compared to the mean size of the substrate particles.

We give in the two first figures the results obtained considering the erosion process only. Figure 1 gives the time-evolution of the four first reduced moments of the initial PSD. The zeroth order moment giving the concentration of the substrate particles is constant since no new particles are formed in the system, the 255 existing ones are becoming smaller thus, the first/second/third moments related respectively to the total length/surface/volume of the particles decrease continuously. Figure 2 shows the time-evolution of the initial PSD undergoing erosion process. The initial PSD expressed here as a number density function shifts toward the small sizes without any deformation since the concentration of the particles is kept unchanged, the decrease of the volume of the substrate particles is counterbalanced by the increase of the concentration of the simple sugars thus, the mass balance is satisfied.

EG activity and fragmentation process

For the particle fragmentation process, the breakage frequency is set proportional to the particle size L which means that the factor α is set equal to one in equation ( 19). Furthermore, to initialize the resolution procedure, the hydrodynamic force F H is estimated using the operating conditions explicited in the experimental procedure section. The cohesion force F p (t = 0) satisfies the relation F H Fp(t=0) < 1 which means that initially, the cohesion of the particles is greater than the hydrodynamic shear stress. In this case, the initial breakage frequency is minimal. The aim of this study is not the accurate estimation of the cohesion force and its evolution within the process time. For this, specific protocoles as those used in comminution processes to determine the density distribution of fracture energies [START_REF] Crespo | Application of particle fracture energy distributions to ball milling kinetics[END_REF][START_REF] Frances | Comminution process modeling based on the monovariate and bivariate direct quadrature method of moments[END_REF] have to be explored. Our purpose is to develop a modelling framework in which all these considerations can be easily taken into account and incorporated to a global simulation procedure.

Unlike the hydrodynamic force which is constant whithin the process time, the cohesion force decreases progressively since the substrate particles are attacked by the enzymes. We give an illustration of the frag- The evolutions of the first four reduced moments given in the Figure 3 show a slow initial rate as the cohesion force is important before an accelerating phase. The concentration of the particles increases considerably (zeroth order moment) as well as the total surface of the particles (second order moment). The total volume is kept unchanged (third order moment) since the fragmentation process is conservative.

280

The reconstruction of the PSDs from the moments using the ME technique in this case is illustrated in Figure 4. The initial PSD shifts toward the small sizes with a substantial shape deformation. Since the fragmentation frequency is proportional to the particle size, the rate of transformation of the largest particles is high compared to that of small ones.

Fragmentation process coupled with EG-CBH + β-glucosidase activities

In enzymatic hydrolysis reactions, the enzyme activities are combined with the hydrodynamic shear stress effect for an efficient transformation of the complex biomass matrix into fermentable sugars. The combination of the pure fragmentation and the erosion processes in a global model is an essential step for the modelling of this reaction.

In this case, the frequence of the erosion process is kept proportional to the particle surface (L 2 ) and the fragmentation frequency proportional to the particle size (L). The substrate fraction converted into cellobiose by the CBH activity is further transformed into glucose by the β-glucosidase activity. The inhibition effect is introduced for the three different activities. The results are illustrated in the figures 5, 6, 7 and 8. The time evolution of the moments of the initial PSD (Figure 5) shows a continuous increase of the particle concentration due to the fragmentation process and a decrease of the total volume due to the erosion process. The accumulation of the small particles formed after the disintegration of the large ones is becoming more and more important. The ME reconstruction method is very interesting here because n(0, t) and ∂n(L,t) ∂L 0

are not constrained to be zero.

Since the conservative variable in the case of particulate substrates is the particle volume, we give in Figure 7 the evolution of the volume based density function during hydrolysis reaction derived from the 300 evolution of the number density function (Figure 6) assuming the sphericity of the particles. Recall that this assumption has been formulated in the theoretical framework section in order to transform the PBE written in terms of volume-based density function into number density function based PBE. As we can see in Figure 7, a tail toward the small sizes is formed as a consequence of the fragmentation process and the integral of the distribution decreases in time because of the erosion process.

305

The conversion of the substrate into simple sugars (Figure 8) shows a rapid initial phase followed by a progressive decreasing rate step. The slower production rate is due to the inhibition effect : the accumulation of the simple sugars affects negatively the enzyme activities. This typical profile is in agreement with the experimental results as it will be discussed in the next section.

Experimental results

In this section, we give some experimental results obtained using particulate cellulose (Avicel PH-102 and Avicel PH-105) submitted to enzymatic hydrolysis under the conditions commonly used in literature. The experiments described below aim to support our modelling approach such as the importance of the substrate structure and the evolution of the physical properties of the substrates during the hydrolysis reaction. The population balance framework developed in this contribution is based on these considerations which is not the case for Michaelis-Menten based kinetic models widely used in literature.

Mechanical properties of the substrates

The two different substrates used in this study (Avicel PH-102 and Avicel PH-105) have different initial

PSDs. Solutions from these substrates are prepared with the same loading (1% w/v) and sonicated during 5 mn. The PSDs before and after sonication are measured using the laser diffraction analyser for each substrate.

The results are given in figures 9 and 10. The figures 9 and 10 reveal a substantial different behaviour of the two substrates after sonication. The PSD of Avicel PH-102 initially centered around a mean size of D 43 (t i ) = 140µm shifts completely toward small sizes so that, after 5 minutes of sonication, the new PSD is centered around D 43 (t f ) = 22µm (Figure 9). The energy dissipated by the sonic waves is sufficient to affect the cohesion of the particles and leads ultimately to their disintegration into smaller particles. The behaviour of Avicel PH-105 is definitely different.

The PSD is hardly affected by the sonication (Figure 10), its mean size changes from D 43 (t i ) = 30µm to D 43 (t f ) = 28µm. This gives an indication on the cohesion force of the substrate particles which is sufficiently high to face the sonication energy which is not the case for Avicel PH-102.

This simple test shows the importance of the cohesion force of the substrate particles when dealing with 330 particulate systems. Thus, eventhough the two substrates are pure cellulose with comparative crystallinity index, their mechanical resistance is substantially different. This means that their behaviour during the enzymatic hydrolysis reaction will be also completely different since the cohesion of the substrate particles will be distinctly affected by the hydrodynamic shear stress and the enzyme activities. This will be confirmed in the next section.

Enzymatic hydrolysis of Avicel PH-102 and Avicel PH-105

The enzymatic hydrolyis reactions have been conducted for 24 hours using the two different substrates under the same conditions. The evolution of the PSDs as well as the conversion into simple sugars have been tracked all along the reactions. The results are given in the figures 11, 12 and 13. As expected, the PSD of Avicel PH-102 shows a spectacular evolution. After 24 hours of hydrolysis, the final PSD is completely different from the initial one. The mean size of the substrate particles moves from D 43 (t i ) = 140µm to D 43 (t f ) = 33µm. The total converted fraction in this time interval is around 25%.

In the case of Avicel PH-105, the evolution of the PSD is not significant compared to that of Avicel PH-102. Indeed, the mean size of the substrate particles decreases from D 43 (t i ) = 30µm to just D 43 (t i ) = 24µm in 24 hours of hydrolysis (Figure 12). However the kinetic of simple sugars release follows exactely that of Avicel PH-102 as shown in Figure 13. The examination of the Figure 12 shows that with the slight translation of the initial PSD into small sizes, the global shape is preserved. This can be explained by the high cohesion force of the substrate particles thus, the erosion process is dominating.

The results of the enzymatic hydrolysis reactions corroborate those discussed in the previous section where the sonication effect was explored. The physical properties of the two substrates used in these experiments are significantly different. These results support our modelling approach in which a cohesion force has been proposed in order to describe the evolutoin of a particulate system submitted to the hydrodynamic shear stress and enzymatic attacks. This allows the tracking of the PSD evolution as well as the kinetic of the simple sugars release all along the hydrolysis reaction which is not the case with the classical modelling approaches.

Indeed, as discussed before, when using Michaelis-Menten based kinetic models, the variable is the substrate concentration. Thus, the evolution of the substrate properties cannot be predicted and the only output is the conversion rate.

Furthermore, if we examine the conversion kinetics of the two substrates (Figure 13), we note that they are surprisingly similar. Starting from a global information on the granulometry of the two substrates, since the enzymatic hydrolysis reactions are heterogenous catalytic processes, one expects that the conversion of Avicel PH-105 would be substantially higher than that of Avicel PH-102. This is motivated by the fact that, for the same substrate concentration, the total surface developed by the Avicel PH-105 particles is higher than that developed by the Avicel PH-102 particles. The high converted fraction in the case of Avicel PH-102 is explained by its mechanical properties which allow significant transformation of the PSD during the reaction (Figure 11). The rapid breakage of the initial particles of Avicel PH-102 within 8 hours induces the generation of new accessible surfaces and boosts by the way the conversion rate. This shows the importance of having access to the PSD evolution in parallel to the conversion rate to be able to interpret the results of hydrolysis reactions appropriately.

Conclusion and perspectives

In this study, the case of particulate substrates is explored. The population balance approach developed initially for the soluble substrates has been extended to insoluble substrates since these systems are ubiquitous when dealing with enzymatic hydrolysis reactions.

The transition to the particulate substrates brings additional challenges since the particle fragmentation process is dissociated from the intrinsic cellulolytic activities which was not the case when dealing with soluble substrates especially in the case of EG activity. In fact, the action of the EG activity on a cellulose chain induces automatically the breakup of this chain and the formation of two smaller ones. In the case of an insoluble cellulose particle, the action of the EG activity leads to the breakage of an accessible glycosidic bond without the fragmentation of the particle. This activity affects the structural features of the substrate particles mainly their cohesion force. Thus, the fragmentation process has been modelled by taking into account the dynamic evolution of this cohesion force and the hydrodynamic shear stress. The CBH activity is still considered as an erosion process since it leads to a specific product even for particulate substrates.

The novelty here is the formulation of the daughter distribution function which has been rethought since the length scales are extremely distinct between the cellobiose molecules and the substrate particles. The

Figure 1 :

 1 Figure 1: Time-evolution of the first four reduced moments : erosion process.

Figure 2 :

 2 Figure 2: Time-evolution of the PSD undergoing CBH attacks. The PSDs from right to left correspond respectively to the times t = 0, 1, 3, 6, 8 and 10 hours of hydrolysis.

Figure 3 :

 3 Figure 3: Time-evolution of the first four reduced moments : pure fragmentation process.

Figure 4 :

 4 Figure 4: Time-evolution of the PSD undergoing a pure fragmentation process. The PSDs from right to left correspond respectively to the times t = 0, 1, 2, 3, 4, 5, 6, 8 and 10 hours of hydrolysis.

Figure 5 :

 5 Figure 5: Time-evolution of the first four reduced moments : pure fragmentation and erosion process simultaneously.

Figure 6 :

 6 Figure 6: Time-evolution of the PSD undergoing pure fragmentation and erosion processes. The PSDs from right to left correspond respectively to the times t = 0, 1, 3, 6 and 10 hours of hydrolysis.

Figure 7 :

 7 Figure 7: Time-evolution of the volume-based density distribution undergoing pure fragmentation and erosion processes. The distributions correspond from right to left respectively to the times t = 0, 1, 3, 6 and 10 hours of hydrolysis.

Figure 8 :

 8 Figure 8: Evolution of the converted fraction of the substrate into simple sugars in the case of EG-CBH activities with the inhibition effect

Figure 9 :

 9 Figure9: Ultrasound effect on the particle size distribution of Avicel PH-102 determined using laser diffraction analysis : Initial PSD (continuous line) and PSD after sonication (dashed line).

Figure 10 :

 10 Figure 10: Ultrasound effect on the particle size distribution of Avicel PH-105 determined using laser diffraction analysis : Initial PSD (continuous line) and PSD after sonication (dashed line).

Figure 11 :

 11 Figure 11: Particle size distribution evolution during the enzymatic hydrolysis of Avicel PH-102. The distributions from right to left correspond respectively to the times t = 0, 1, 2, 3, 4, 6, 8 and 24 hours of hydrolysis.

Figure 12 :

 12 Figure 12: Particle size distribution evolution during the enzymatic hydrolysis of Avicel PH-105. The distributions from right to left correspond respectively to the times t = 0, 1, 2, 3, 4, 6, 8 and 24 hours of hydrolysis.

Figure 13 :

 13 Figure 13: Exprimental substrate converted fraction evolution during enzymatic hydrolysis of Avicel PH-102 (red circles) and Avicel PH-105 (blue stars).

  β-glucosidase activity is modelled as a michaelian kinetic and the inhibition effect has been incorporated for the different cellulolytic activities. The population balance equation has been solved numerically using the Direct Quadrature Method of Moments (DQMOM) coupled with the Maximum Entropy reconstruction technique. The numerical tests in different cases show a consistent results for both the fragmentation and the erosion processes. The population balance model predicts the evolution of the substrate initial PSD and the conversion rate. The experimental results obtained by the sonication test confirm the importance of the mechanical properties of the cellulose particles. The two different substrates tested present distinct behaviours against this Nomenclature A Constant appearing in the cohesion force B(t) Parameter of the breakage frequency a i First time derivative of the weights of the quadrature nodes b i First time derivative of the product of the weighted abscissas Volume of the elementary particle for the erosion process V m Parameter of the Michaelis-Menten type kinetic w i Weights of the quadrature nodes

assay. Avicel PH-102 is extremely sensitive to the ultrasounds which induce a severe fragmentation process while Avicel PH-105 is minimally affected. The enzymatic hydrolysis reactions conducted with identical conditions show the same results. The particle size distribution of Avicel PH-102 evolves along the reaction which is poorly the case of that of Avicel PH-105 but the conversion rates are similar. These experimental tests support the modelling approach and point to the importance of the coupling between the PSD evolution and the conversion rate for a judicious interpretation of the hydrolysis reaction results.

The dynamic character of the cohesion force of the substrate particles which can be characterised by the notion of the age of the substrate intervenes in our case in the formulation of the breakage frequency of the substrate particles. The introduction of this notion as a new variable leading to a multivariable population balance model would be of great importance for process considerations. This allows the monitoring of the sequencial addition of the substrate for example. Finally, the added value of the modelling approach presented in this contribution is its possible extension for multivariable systems without losing its robustness.

Aknowledgments

The authors would like to thank Toulouse White Biotechnology and Région Midi-Pyrénées for their financial support.

Greek


Florentin Coupé

Laurène Petitjean

Paul Anastas

Frédéric Caijo

Vincent Escande

Christophe Darcel

Sustainable oxidative cleavage of catechols for the synthesis of muconic acid and muconolactones including lignin upgrading

Muconic acid and muconolactones are gaining high interest as platform molecules for the synthesis of a variety of compounds, especially in the domain of materials. Despite several technologies have been described for their synthesis, there is still a lack of performance, especially regarding green chemistry principles. In this study, we describe the development of an optimized catechol oxidative cleavage to muconic acid using performic acid in an intriguingly safe fashion. Common iron salts were used as catalysts to a level as low as 0.005 mol%, for a maximum turnover number of 13200. Maximum muconic acid yield reached 84% after isolation by simple filtration. This procedure optimized on catechol was also efficient over a wide range of substituted catechols, providing the access to muconolactones in a domino reaction. Noticeably, biobased catechols produced by a proven technology of lignin depolymerization were cleaved into muconolactones of high functional value. Applying this supplementary cleavage step to lignin depolymerization catechols was thus an ultimate way to maximize the economical value created from lignin. In contrast with other studies, lignin was not only depolymerized, but depolymerization products were further transformed to take as much value from biomass as possible.

Introduction

Lignin is usually considered as a high potential biopolymer for the substitution of petroleum-based specialty chemicals. In particular, it has a high carbon content and is the only biomass-derived polymer characterized by a structure based on aromatic moieties. 1 Even if it is found in abundant, nonedible biomass, its transformation should be efficient and comply with green chemistry principles. However, lignin valorization still remains challenging, giving complex mixtures of products in poor yields. [2][3][4] Lignin is extracted from lignocellulosic biomass using several processes. The preferred one is solvolysis, which preserves its structure the best. 2 Lignin is then transformed via depolymerization, which allows the cleavage of several bond types, affording simple molecules, which can be afterwards further functionalized.

In 2014, Anastas et al.

5

proposed a depolymerization process by hydrogenolysis using copperbased porous metal oxides (Cu-PMO). Starting from organosolv lignin extracted from Candlenut shells (Aleurites moluccana; Euphorbiaceae), the main depolymerization products were substituted catechols.

These catechols then open a new route to lignin upgrading, as they have relevant structures for further functionalizations. However, they might not have direct applications with specific properties. An interesting approach to valorize these monomers implies their cleavage to muconic acid derivatives. (Scheme 1) Indeed, muconic acid (MA), in its unsubstituted form, has caught a growing interest these last years as it has several potential applications. This lignin-based approach could therefore provide a sustainable source to oil-based industrial relevant chemicals. First, in the domain of materials, MA can serve as a precursor of monomers, like adipic acid, terephthalic acid and ε-caprolactam for the respective productions on Nylon 6-6, polyethylene terephthalate (PET) and Nylon 6. [6][7][8] MA can also directly be employed as a monomer for the design of new classes of polymers, either by polycondensation or by radical polymerization. [9][10][11] Otherwise, MA can be converted into different compounds of interest such as trans-3-hexenedioic acid or cyclic ketones, especially cyclopentanone. 10,12,13 Indeed, trans-3-hexenedioic acid was especially considered as a monomer for the synthesis of unsaturated Nylon 6-6. 14 More noticeably, MA is known for its ability to undergo intramolecular cyclization, giving either muconolactones or dilactones. [16][17][18][19][20] In definitive, MA can act as an important platform molecule.

21,22

Scheme 1. Overview of selected applications for catechols cleavage products.

Current MA production methods are discriminated into two main approaches: biotechnological or chemical pathway. A large number of studies have been released on the MA production via microorganisms from a variety of substrateseither of lignin type or not-and using several biological strains, mainly Pseudomonas putida, usually engineered. [23][24][25] Even if these biochemical conversions can legitimately be considered as sustainable, they come along with some problematic aspects undesired for a large scale production: low concentration of the broth to prevent toxicity to microorganisms, as well as product extraction and purification difficulties. To get rid of these inconveniences, a preferred MA production goes through chemical conversion. This synthetic approach has been well described in the literature. In pioneering works, the catechol chemical cleavage reaction was designed in order to mimic the naturally occurring non-heme Fe(III) dioxygenases. [26][27][28][29][30] Thus, several Fe(III) complexes were studied to describe the catechol intradiol cleavage, with molecular oxygen as the terminal oxidant. Among all these complexes, the one with tris(2-pyridylmethyl)amine as ligand expressed excellent activity (up to 85% MA yield, mixture of esters). 31,32 Even though this aerobic route for the MA is very promising, it still has some disadvantages considering a potential industrial application (pressure of air, relatively high loading of expensive and non-recyclable catalyst and improvable MA yields). Another chemical route employed peracids, and essentially peracetic acid (PAA). Peracids in general are convenient reagents, as they are already widely produced and used at an industrial scale, as epoxidation reagents for example. [START_REF] Klenk | Ullmann's Encyclopedia of Industrial Chemistry[END_REF] Moreover their synthesis is simple, starting from cheap and abundant reagents, and creating almost no waste. In particular, Pandell described the catechol cleavage reaction with PAA using different conditions. 34,35 To be efficient, the cleavage required sufficiently concentrated PAA, a low amount of water, the presence of a Fe(III) inorganic salt catalyst, and the slow addition of the catechol substrate. It also had the advantage of being carried out under standard/ambient conditions. Moreover, the obtained MA was insoluble in the reaction medium, which eased its isolation by simple filtration. Therefore, we decided to make this promising technology the starting point of our study, to upgrade lignin depolymerization products into value-added compounds. In the current work, we report the use of simple iron salt-based catalysts for the oxidative cleavage of catechols, including bio-based catechols obtained from lignin depolymerization, by performic acid leading to substituted muconic acids derivatives and their related muconolactones.

Experimental section

Adipic acid6

Radical polymerization 10,11 ε -caprolactam 8

Trans-3-hexenedioic acid 12 Cyclic ketones 13 Polycondensation with dialcohols 9 Grape seed flavanol derivative 19 Peanut phytoalexin 16 Pochoniolide from fungal strain 20 Benzodifuranone dyes

Fragrant furanones (e.g. sotolon)

A c c e p t e d M a n u s c r i p t

HCOOH was added in 4h30 with a syringe pump. After the end of the addition, the mixture was stirred for 20 more hours to allow complete spontaneous peroxy decomposition (KI/starch test was negative at this time). Noticeably, NMR monitoring showed that no catechol remained in the reaction mixture only 1 h after the end of the addition. The obtained insoluble muconic acid 2a was then filtered off, washed with a minimum amount of H 2 O, and dried in a desiccator (P 2 O 5 ) (light grey solid, 1.19 g, 84% isolated yield). Purity was checked by 1 H NMR (DMSO-d 6 ). In a separate experiment, filtrate concentration followed by acetone washing allowed to recover 4% more muconic acid with 90% purity.

Representative procedure for substituted catechol cleavage, protocatechuic acid

In a 25 mL round-bottomed flask, formic acid (5.20 g, 112 mmol) and H 2 O 2 (50 wt%, 1.53 g, 22.5 mmol) were stirred for 1 h at 20 °C to pre-form performic acid in situ. Then (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O (2.0 mg, 0.1 mol%) was dissolved in the pre-mix, and protocatechuic acid 1h (0.77 g, 5.0 mmol) was added portionwise in 2 h (note that depending on the substrate nature and solubility in HCOOH, it was either added portionwise or in solution in HCOOH). After the end of the addition, the mixture was stirred for 20 more hours, until all peroxides were spontaneously decomposed (KI/starch test negative). The medium was then concentrated in vacuo to leave 4-carboxyl-5-carboxymethyl-2-furanone 3h as a grey solid residue (0.697 g, 68% at purity estimated at 90% by 1 H NMR). Further purification was achieved by basic/acidic successive washings and recrystallization.

Results and discussions

Development of the catalytic process on catechol

Nature of the oxidizing mixture. The most employed peracids in organic synthesis are meta-chloroperbenzoic acid (m-CPBA) and peracetic acid (PAA). Despite this, only PAA has been extensively studied in the catechol cleavage reaction in the literature, probably because it is a cheap, and readily available reagent. m-CPBA however was not selected as a potential candidate as it poorly fulfils green chemistry requirements. In a first approach, PAA was employed as the oxidant from two different sources: either from its commercial form (35 wt% in AcOH), or from an in situ preparation from AcOH and H 2 O 2 (1:1 molar ratio) catalyzed by 1 wt% of H 2 SO 4 (Table 1, entries 1-5). Noticeably, 4.5 equivalents of oxidant were employed, which corresponded to an excess of only 2.5 equivalents, as previously reported in mechanistic considerations affirming that 2 equivalents were required. 36,37 In the case of in situ prepared PAA, the number of equivalents was determined by the H 2 O 2 total added amount, as it regenerates PAA while it is consumed upon catechol addition. As mentioned in previous contributions, 35,37 On another hand, it is described that performic acid (PFA) is a much stronger oxidizer than PAA. [START_REF] Klenk | Ullmann's Encyclopedia of Industrial Chemistry[END_REF] Astonishingly, almost no studies employed this peracid in the catechol cleavage reaction. 39 In this context, we decided to test its efficiency in the target reaction. Despite the appearances, PFA, when handled properly, is not hazardous. [START_REF] Klenk | Ullmann's Encyclopedia of Industrial Chemistry[END_REF]40,41 Especially, when prepared in situ, it doesn't require any intermediate storage or transportation. Indeed, PFA can be conveniently prepared by simple mixing of commercial formic acid and hydrogen peroxide solution, in a short reaction time, without the need for any acid catalyst. 42,43 The proportions of formic acid and hydrogen peroxide only need to be set up properly to avoid to have a mixture composition in the explosivity range of the HCOOH:H 2 O 2 :H 2 O ternary system (see ESI, Fig. S1).

33

Moreover, PFA decomposes slowly, so that any left residue after the cleavage reaction would be decomposed spontaneously, leaving no hazardous peroxide species in the reaction mixture. 42,43 Finally, formic acid can be easily obtained from sustainable sources.

44

The in situ generated PFA was evaluated using the same reaction conditions than the ones used for PAA. Based on the best result obtained with in situ prepared PAA, we prepared PFA with a HCOOH:H 2 O 2 50 wt% 10:1 mixture (Table 1, entry 6). Using this medium, the MA isolated yield increased up to 80%. PFA prepared in these conditions was thus considered as being a very powerful oxidant for the reaction. Nature of the metal catalyst. The catechol cleavage by peracids is a reaction allowed by a catalyst species. Many metals have been tested by the past, and simple Fe salts have been described to have among the best activities. 35 Thus, the activity of several metal salts was evaluated with the PFA system. From Table S1 (ESI), the Fe-based catalysts can be easily discriminated from the other tested metals, for which the MA isolated yields did not excess 35%. A notable exception is Fe 2 O 3 , which was not soluble enough in the reaction medium to act as a catalyst (5% isolated yield). Even more interestingly, as seen in Table 2, entries 3-10, all Fe salts gave MA with close yields at high catalytic loading (0.1 mol%, 73% to 80% yields). This is in accordance with previous observations on Fe(III) species. 34 Furthermore, the initial Fe oxidation state had no influence (0, II, or III) on the efficiency, showing that performic acid is able to oxidize Fe to a higher active oxidation state. This is, to the best of our knowledge, the first report of the use of Fe at lower oxidation states than +III in a peracid-based system. other mixtures were tested (Fig. 1). The lowest experimented ratio was 2.7:1, which was slightly higher to the upper limit of the explosivity range of the HCOOH:H 2 O 2 :H 2 O ternary system. The HCOOH:H 2 O 2 1:1 ratio was not tested due to this constraint (see ESI, Fig. S1).

Optimization of the

33

This 2.7:1 ratio gave slightly worse results, with a 73% muconic acid isolated yield (Fig. 1). Finally, the intermediate ratio 5:1 was also applied, and corresponded to the best obtained yield of 84%. This can be explained as it may be an optimum value between pushing the equilibrium towards the PFA formation and a too dilute medium due to a too large excess of HCOOH.

The PFA solution formed with this ratio is clearly outside of the explosivity range (see ESI, Fig. S1) [START_REF] Klenk | Ullmann's Encyclopedia of Industrial Chemistry[END_REF] and is thus a safer oxidizing mixture. Moreover, using a lower excess of HCOOH than in the 10:1 mixture is beneficial in terms of mass engaged in the reaction, giving better metrics values such for mass intensity (see ESI, Table S3 for green chemistry metrics calculation).

Catalytic loading. In a first approach, a standard low value of 0.1 mol% catalyst loading was applied to evaluate the best catalysts. At this catalytic loading, the obtained MA yields ranged from 73% to 84%, giving turnover number (TON) values varying between 730-840. On this basis, the Fe species were tested with a tenfold decrease in catalytic loading (Table 2, entries 11-13). However, even this catalytic loading value was too high to discriminate the different salts. Only Fe(acac) 3 showed a lower activity (68%, TON 6800). Then, the catalytic loading value was further halved to 0.005 mol% (i.e. 50 ppm Fe) ( An optimized procedure for the peracid-based catechol cleavage was thus designed with in situ preformed performic acid, synthetized with pure HCOOH and H 2 O 2 50 wt%. The best conditions to perform the reaction were a HCOOH:H 2 O 2 5:1 ratio, using 0.01 mol% of a Fe inorganic species as catalyst. The cleavage was carried out at 20 °C and required the slow addition of the catechol substrate over a 4h30 period. In these conditions, the MA 2a isolated yield reached its maximum value of 84%.

Feasibility of the process on substituted catechols

When we first developed this peracid-based technology for the catechol cleavage, not only we desired a green, reliable process for the synthesis of MA, but we also had in mind its potential applicability on substituted catechols. We then decided to screen the scope of substituted catechol substrates. Figure 2 gives an overview of all obtained structures from these catechol cleavages. The main difference with catechol cleavage resided in the nature of the obtained products. While catechol cleavage led specifically to the formation of MA 2a, almost all substituted catechols 1 furnished muconolactone cyclic structures 3. They resulted from an intramolecular hetero-Michael addition of the muconic acid cleavage product, as already described. 15 This cyclization reaction may also be favoured by the very high acidity of the medium. The susceptibility of the muconic acid intermediate to evolve towards the muconolactone structure also relied on its substituent steric hindrance. Indeed, 4-methylcatechol 1b, 4chlorocatechol 1c and 3-methoxycatechol 1d led to "free" muconic acid derivatives 2b-d after the cleavage reaction, as minor products (with yields up to 34%). Indeed, for each substrate, the produced muconic acid can evolve in two different muconolactones, but only one is observed, as previously reported. 15 However, starting from 4-nitrocatechol 1e, the conversion was incomplete (85%) and only a mixture of isomers of 3-nitro-2,4-hexadienedioic acid 2e was obtained (39%), and no muconolactone was observed, possibly due to the -NO 2 strong electrophilic character which inhibited the cyclization. By contrast, catechols bearing electronwithdrawing group like caffeic acid 1g and protocatechuic acid 1h, after cleavage, led to muconolactones 3g and 3h in moderate yields (58% and 68%, respectively). Thus, the electronic properties of the substituent have an influence on the obtained yield: with catechols bearing electron-donating substituents (e.g. 1b, 1d, 1f), the corresponding muconolactones 3b, 3d and 3f were obtained in variable yields (5-71%). Starting from 4-chlorocatechol 1c, a mixture of two muconolactones 3c-i and 3c-ii was obtained. Among all the tested substrates, 4-cyanocatechol did not react. Finally, 3,5-Di-tert-butylcatechol 1f, a reference substituted catechol, gave the muconolactone 3f in 37% yield. Thus, in the case of substituted catechols, the oxidative cleavage went through muconic acids which could then evolve a 3j-ii was obtained as a mixture with 3j-i and resulted from the formylation of 3j-i in the reaction conditions (molar ratio formylated:non formylated 8:3).

Application to catechols from C-lignin

After having proven the efficiency of our PFA-based technology on substituted catechols, we decided to look deeper into sustainable sources of catechols. Indeed, the main catechol production comes from the petroleum industry, even if its production from biomass is emerging, for example from biobased ferulic acid. 45 On another hand, guaiacol, which has a very similar structure to catechol, is often reported as a major lignin depolymerization product.

2

The developed ironcatalyzed cleavage process was thus evaluated with commercial guaiacol. Unfortunately, only small amounts of MA 2a were isolated in this case (up to 25%). The presence of a methyl on one of the hydroxy groups impeded the reaction by hindering the formation of the complex with the Fe(III) catalyst. Nonetheless, guaiacol demethylation could be accounted as a route to biobased catechol. [46][47][48] In 2014, Anastas et al. reported the formation of four substituted catechols from the depolymerization of lignin sourced from candlenut shells, using hydrogenolysis catalyzed by Cu-PMOs.

5

Recent insights revealed that the formation of catechols in these conditions was not exclusively due to the employed process, but the nature of the starting biomass has a preponderant role. Indeed, for decades, lignin was thought to be a polymer based exclusively on paracoumaryl alcohol (H), coniferyl alcohol (G) and sinapyl alcohol (S) units. However, in 2012, Ralph et al. reported the existence of a different lignin type, based on caffeyl alcohol (C) units, first found in vanilla seeds. 49 Thereafter, lignin from "classical" biomass was renamed GS-lignin, whereas this newly discovered lignin was designated as C-lignin. C-lignin from vanilla, upon depolymerization, has been shown to give back catechol moieties through benzodioxane linkage breaking. 50,[START_REF] Li | [END_REF] Then Clignin was found to be present in high amounts in candlenut shells, explaining the formation of catechols from its lignin depolymerization. 52 Even if C-lignin appears to be much less abundant than GS-lignin, being restricted to specific biomass sources, it still counts as an important feedstock, as candlenut is a crop developed for its oil, transformed into biodiesel. Another relevant C-lignin source may come from castor bean shells, 52,53 which are grown for their highly-containing ricinoleic acid oil, especially used for the manufacture of Polyamide 11.

54

Once the process optimized on catechol and screened on different petroleum-based substituted catechols, it was then applied to catechols obtained from lignin depolymerization. We decided to work on lignin issued from candlenut shells, being rich in C-lignin, giving, via hydrogenolysis depolymerization, alkyl-substituted catechols which are good candidates for our iron-catalysed cleavage reaction. First, the cleavage was performed on the isolated catechols, using previously developed conditions (HCOOH:H 2 O 2 50 wt% 5:1, (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O 0.1 mol%). As previously observed, these catechols 1i-k evolved specifically to the muconolactone derivatives 3i-k, with high yields in isolated forms (Figure 3, 73% to 86%). Notably, no secondary products were observed, except in the case of 4-(3-hydroxypropyl)catechol 1j, whose hydroxy lateral group was formylated at 71%. The cleavage was also tested directly on lignin depolymerization oil, and the expected products were observed, with no catechols remaining (ratio 3i:3j-i:3j-ii:3k of 7:2:5:3).

Conclusions

A sustainable process was designed to allow the efficient cleavage of catechol substrates. First, it allowed the manufacture of muconic acid from catechol in high yields, up to 84%. Performic acid was employed as the oxidant, synthetized in situ from formic acid and H 2 O 2 . In these conditions, formic acid played the role of the solvent, so that no additional solvent was required. Moreover, product isolation was performed by simple filtration. All these aspects make the process highly interesting in terms of mass intensity (15.6 kg/kg MA , see ESI, Table S3). Furthermore, this also impacts the E factor value positively, determined at 2.0, which is a very low value considering that the process falls into the field of fine chemistry. On the other hand, performic acid was handled in appropriate conditions, so that there were no safety issues using it. Even more, the terminal oxidant was actually H 2 O 2 , which is a benign oxidant. Also, performic acid, upon reaction, gave back formic acid, which could therefore be recovered, for example by extractive distillation regarding an industrial application. [START_REF] Berg | [END_REF][START_REF] Rauch | [END_REF] Thus, the only significant waste was water, which was brought by the H 2 O 2 solution. The catalyst used was a simple inorganic iron salt. Beyond its use in very low quantities, iron is a nontoxic, Earth-abundant element, with no risk of ore depletion even at long term. Moreover, there was no need to add an elaborated, expansive organic ligand to make it active in the cleavage reaction. Finally, the process was highly efficient to open the route to muconolactone compounds. Especially, catechols obtained from lignin depolymerization were converted into original muconolactones with good yields (>70%). They represent a promising class of intermediates towards the synthesis of compounds of interest, while allowing the valorization of lignin depolymerization products in an efficient fashion.

A c c e p t e d M a n u s c r i p t

  This journal is © The Royal Society of Chemistry 20xx e

A c c e 15 Figure 2 .

 152 Figure 2. Structures obtained from the cleavage of substituted catechols 1. Percentages depict isolated yields. Numbers in brackets are the associated isolated yields. a The two muconolactones were obtained as a mixture (molar ratio 3:2)b The four possible muconic acid derivative isomers and the muconolactone were present as a mixture, see ESI for further experimental details c A mixture of three unspecified isomers was obtained (ratio 20:11:7).

Figure 3 .

 3 Figure 3. Complete sequence used to obtain muconolactones from biomass, passing through lignin. Percentages are isolated yields.a 3j-ii was obtained as a mixture with 3j-i and resulted from the formylation of 3j-i in the reaction conditions (molar ratio formylated:non formylated 8:3).

Typical procedure for catechol cleavage

	In a 25 mL round-bottomed flask, formic acid (10.40 g, 225
	mmol) and H 2 O 2 (50 wt%, 3.06 g, 45.0 mmol, 4.5 equiv.) were
	stirred for 1 h at 20 °C to pre-form performic acid in situ. Then
	(NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O (3.9 mg, 0.1 mol%) was dissolved in the
	pre-mix, and catechol 1a (1.10 g, 10 mmol) dissolved in 4 mL of

  the in situ Fe(III)-catecholate complex formed in a first step was then oxidized with PFA giving an oquinone intermediate. In the presence of the PFA, this latter led to muconic acid through a possible cyclic muconic anhydride. Noticeably, such intermediates were not observed in the present conditions, which may be due to their low stability in the reaction conditions. Both PAA sources -commercial or in situ prepared-gave similar results (25% and 20% MA 2a isolated yields, respectively). Considering that commercially available PAA is a solution of PAA in AcOH, the in situ prepared PAA is actually a mixture of AcOH, PAA, H 2 O 2 and H 2 O in an equilibrium state.[START_REF] Klenk | Ullmann's Encyclopedia of Industrial Chemistry[END_REF]38 So, to displace the initial equilibrium towards the formation of PAA, a AcOH:H 2 O 2 ratio of 10:1 was tested (Table 1, entry 2). Thus, the MA yield slightly increased to 34%. Also, this result can attest that diluting water brought by the H 2 O 2 solution has a beneficial effect on the reaction, as previously reported. 35 To confirm this, and to bring even less H 2 O in the mixture, a more concentrated H 2 O 2 solution (50 wt% in H 2 O) was used (Table 1, entries 3-4). Surprisingly, the associated yields were not better than with H 2 O 2 30 wt%, and even dropped to 9% with a AcOH:H 2 O 2 ratio of 1:1. This strategy was thus not completely satisfying for a yield increase.

Table 1 .

 1 Assessment of different oxidizing mixtures. When AcOH was used, the peracid synthesis was performed in 24 h at 20 °C, catalyzed by 1 wt% H2SO4. b Commercial peracetic acid solution, 35 wt% in AcOH.

	Conditions: 10 mmol catechol 1a added in 4.5 h, 4.5 equiv. H2O2, 20 °C, 0.1 mol%
	(NH4)2Fe(SO4)2.6H2O.

a c Isolated yield.

Table 2 .

 2 Catechol 1a cleavage -Screening of different Fe salts as catalysts, at different catalytic loadings.

	PFA oxidizing mixture. After having
	evaluated the influence of the peracid and the metal catalyst
	natures, complementary experiments were carried out to
	better understand the influence of the HCOOH:H 2 O 2 ratio, and
	to potentially improve the MA yield. HCOOH:H 2 O 2 50 wt% 10:1
	was first selected, as being the most promising one. To verify
	this, H 2 O 2 30 wt% was also tested with a HCOOH:H 2 O 2 1:1
	ratio, and led to lower yield, even if it overclassed all results
	obtained with PAA. In this continuity, H 2 O 2 50 wt% was used as
	the unique H 2 O 2 source. Then, to verify the effect of the

a Reaction performed in duplicate gave the same result. b Isolated yield. c 50 ppm Fe.

. d 25 ppm Fe.

Table 2

 2 the Fe salt employed, but with a slight preference for (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O as it was the most active at very low catalytic loadings.

	, entries 14-17). Indeed, most of the salts led to a
	MA yield below 60%. Remarkably, (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O still
	provided a decent MA yield of 66%, with an associated TON of
	13200. (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O was then tested on an even lower
	catalytic loading (0.0025 mol% i.e. 25 ppm), but the MA yield
	significantly dropped to 10% (Table 2, entry 18). Thus, the
	optimal catalytic loading value was of 0.01 mol%, regardless of

c 66 18 (NH 4 ) 2 Fe(SO 4 ) 2 .6H 2 O a 0.0025 d 10

Acknowledgements

We thank the Université de Rennes 1, the Centre National de la Recherche Scientifique (CNRS), the Région Bretagne, and the ANR labcom program (17-LCV2-0009-01). F. C. thanks the ANRT for a CIFRE PhD grant and Rennes Metropole council for a mobility grant.

Notes and references

Conflicts of interest

There are no conflicts to declare.