Zaid M Odibat
email: z.odibat@gmail.com

Cyrille Bertelle
email: cyrille.bertelle@univ-lehavre.fr

M A Aziz-Alaoui
email: aziz.alaoui@univ-lehavre.fr

Gérard H E Duchamp
email: gheduchamp@gmail.com

Prince Abdullah Bin Ghazi

A Multi-step differential transform method and application to non-chaotic and chaotic systems

Keywords: Differential transform method, Multi-step differential transform method, Lotka-Volterra system, Chen system, Lorenz system

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multistep DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

Introduction

The differential transform method (DTM) is a numerical as well as analytical method for solving integral equations, ordinary and partial differential equations. The method provides the solution in terms of convergent series with easily computable components. The concept of the differential transform was first proposed by Zhou [START_REF] Zhou | Differential Transformation and Its Applications for Electrical Circuits[END_REF] and its main application concern with both linear and nonlinear initial value problems in electrical circuit analysis. The DTM gives exact values of the n th derivative of an analytic function at a point in terms of known and unknown boundary conditions in a fast manner. This method constructs, for differential equations, an analytical solution in the form of a polynomial. It is different from the traditional high order Taylor series method, which requires symbolic computations of the necessary derivatives of the data functions. The Taylor series method is computationally taken long time for large orders. The DTM is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. Different applications of DTM can be found in .

The DTM introduces a promising approach for many applications in various domains of science. However, DTM has some drawbacks. By using the DTM, we obtain a series solution, actually a truncated series solution. This series solution does not exhibit the real behaviors of the problem but gives a good approximation to the true solution in a very small region. It is the purpose of this paper is to propose a reliable algorithm of the DTM. The new algorithm, multi-step DTM, presented in this paper, accelerates the convergence of the series solution over a large region and improve the accuracy of the DTM. The validity of the modified technique is verified through illustrative examples of Lotka-Volterra, Chen and Lorenz systems.

Differential transform method

The differential transformation technique is one of the semi numerical analytical methods for ordinary and partial differential equations that uses the form of polynomials as approximations of the exact solutions that are sufficiently differentiable. The basic definition and the fundamental theorems of the DTM and its applicability for various kinds of differential equations are given in [START_REF] Ayaz | Solutions of the system of differential equations by differential transform method[END_REF][START_REF] Ayaz | Application of differential transform method to differential-algebraic equations[END_REF][START_REF] Arikoglu | Solution of boundary value problems for integro-differential equations by using differential transform method[END_REF][START_REF] Bildik | Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method[END_REF]. For convenience of the reader, we will present a review of the DTM. The differential transform of the k th derivative of function f (t) is defined as follows,

F (k) = 1 k! d k f (t) dt k t=t 0 , ( 1
)
where f (t) is the original function and F (k) is the transformed function. The differential inverse transform of F (k) is defined as,

f (t) = ∞ k=0 F (k)(t -t 0 ) k . ( 2
)
From Eqs. ( 1) and (2), we get,

f (t) = ∞ k=0 (t -t 0 ) k k! d k f (t) dt k t=t 0 , (3)
which implies that the concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative way which are described by the transformed equations of the original function. For implementation purposes, the function f (t) is expressed by a finite series and Eq. ( 2) can be written as,

f (t) ≈ N k=0 F (k)(t -t 0 ) k , ( 4
)
here N is decided by the convergence of natural frequency. The fundamental operations performed by differential transform can readily be obtained and are listed in

f (t) = u(t) ± v(t) F (k) = U (k) ± V (k) f (t) = αu(t) F (k) = αU (k) f (t) = u(t)v(t) F (k) = k l=0 V (l)U (k -l) f (t) = du(t) dt F (k) = (k + 1)U (k + 1)
f (t) = d m u(t) dt m F (k) = (k + 1)(k + 2) • • • (k + m)U (k + m) f (t) = t t 0 u(x)dx F (k) = U (k-1) k , k ≥ 1 f (t) = t m F (k) = δ(k -m) f (t) = exp(λt) F (k) = λ k k! f (t) = sin(wt + α) F (k) = w k k! sin(πk/2 + α) f (t) = cos(wt + α) F (k) = w k k! cos(πk/2 + α)
3 Multi-step differential transform method

Although the DTM is used to provide approximate solutions for a wide class of nonlinear problems in terms of convergent series with easily computable components, it has some drawbacks : the series solution always converges in a very small region and it has slow convergent rate in the wider region [START_REF] Bildik | Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method[END_REF][START_REF] Arikoglu | Solution of difference equations by using differential transform method[END_REF][START_REF] Arikoglu | Solution of differential difference equations by using differential transform method[END_REF][START_REF] Liu | Differential transform method applied to high index differential-algebraic equations[END_REF]. To overcome the shortcoming, we present in this section the multi-step DTM that we have developed for the numerical solution of differential equations. For this purpose, we consider the following nonlinear initial value problem,

f (t, u, u , • • • , u (p) ) = 0, (5)
subject to the initial conditions u

(k) (0) = c k , for k = 0, 1, • • • , p -1.
Let [0, T ] be the interval over which we want to find the solution of the initial value problem [START_REF] Bildik | Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method[END_REF]. In actual applications of the DTM, the approximate solution of the initial value problem (5) can be expressed by the finite series,

u(t) = N n=0 a n t n t ∈ [0, T ]. (6)
The Multi-step approach introduces a new idea for constructing the approximate solution. Assume

that the interval [0, T ] is divided into M subintervals [t m-1 , t m ], m = 1, 2, • • • , M of equal step size h = T /M
by using the nodes t m = mh. The main ideas of the Multi-step DTM are as follows. First, we apply the DTM to Eq. ( 5) over the interval [0, t 1 ], we will obtain the following approximate solution,

u 1 (t) = K n=0 a 1n t n , t ∈ [0, t 1 ], (7)
using the initial conditions u

(k) 1 (0) = c k . For m ≥ 2 and at each subinterval [t m-1 , t m ] we will use the initial conditions u (k) m (t m-1 ) = u (k) m-1 (t m-1
) and apply the DTM to Eq. ( 5) over the interval [t m-1 , t m ], where t 0 in Eq. ( 1) is replaced by t m-1 . The process is repeated and generates a sequence of approximate solutions

u m (t), m = 1, 2, • • • , M , for the solution u(t), u m (t) = K n=0 a mn (t -t m-1 ) n , t ∈ [t m , t m+1 ], (8)
where N = K • M . In fact, the Multi-step DTM assumes the following solution,

u(t) =            u 1 (t), t ∈ [0, t 1 ] u 2 (t), t ∈ [t 1 , t 2 ] . . . u M (t), t ∈ [t M -1 , t M ] . ( 9
)
The new algorithm, multi-step DTM, is simple for computational performance for all values of h. It is easily observed that if the step size h = T , then the multi-step DTM reduces to the classical DTM. As we will see in the next section, the main advantage of the new algorithm is that the obtained series solution converges for wide time regions and can approximate chaotic and non-chaotic solutions.

Numerical experiments

To demonstrate the effectiveness of the proposed algorithm as an approximate tool for solving nonlinear problems of differential equations, we apply the proposed algorithm, the multi-step DTM, to Lotka-Volterra, Chen and Lorenz systems.

Lotka-Volterra system

Lotka-Volterra equations are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, a predator and its prey. They were proposed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926,

         dx dt = x(t)(a -by(t)), dy dt = -y(t)(c -dx(t)), (10)
subject to the initial conditions x(0) = c x and y(0) = c y , where t represents the time, y is the predator density, x is the prey density, a, b, c and d are parameters representing the interaction of the two species.

In view of the differential transform, given in Eq. ( 1), and the operations of differential transformation given in Table 1, applying the differential transform to Lotka-Volterra system [START_REF] Hassan | Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems[END_REF], we obtain,

         X(k + 1) = 1 k + 1 aX(k) -bΣ k l=0 X(l)Y (k -l) , Y (k + 1) = 1 k + 1 -cY (k) + dΣ k l=0 X(l)Y (k -l) , ( 11
)
where X(k) and Y (k) are the differential transformation of x(t) and y(t), respectively. The differential transform of the initial conditions are given by X(0) = c x and Y (0) = c y . In view of the differential inverse transform, given in Eq. ( 2), the DTM series solution for Lotka-Volterra system (10) can be obtained as,

     x(t) = N n=0 X(n)t n , y(t) = N n=0 Y (n)t n . ( 12
)
Now, according to the multi-step DTM, taking N = K • M , the series solution for Lotka-Volterra system (10) is given by,

x(t) =                K n=0 X 1 (n)t n , t ∈ [0, t 1 ] K n=0 X 2 (n)(t -t 1 ) n , t ∈ [t 1 , t 2 ] . . . K n=0 X M (n)(t -t M -1 ) n , t ∈ [t M -1 , t M ] , ( 13
)
y(t) =                K n=0 Y 1 (n)t n , t ∈ [0, t 1 ] K n=0 Y 2 (n)(t -t 1 ) n , t ∈ [t 1 , t 2 ] . . . K n=0 Y M (n)(t -t M -1 ) n , t ∈ [t M -1 , t M ] , ( 14
)
where

X i (n) and Y i (n), for i = 1, 2, • • • , M , satisfy the following recurrence relations,          X i (k + 1) = 1 k + 1 aX i (k) -bΣ k l=0 X i (l)Y i (k -l) , Y i (k + 1) = 1 k + 1 -cY i (k) + dΣ k l=0 X i (l)Y i (k -l) , (15)
such that X i (0) = X i-1 (0) and Y i (0) = Y i-1 (0). Finally, if we start with X 0 (0) = c x and Y 0 (0) = c y , using the recurrence relations given in [START_REF] Erturk | Application of generalized differential transform method to multi-order fractional differential equations[END_REF], then we can obtain the Multi-step solution given in Eq. ( 13) and Eq. ( 14). Fig. 1 shows the approximate solutions for Lotka-Volterra system (10) obtained using multi-step DTM and RK4 method. In We can observe that the phase plane trajectories obtained using the Multistep DTM are in high agreement with the phase plane trajectories obtained using RK4 method. Fig. 2 shows the approximate solutions for Lotka-Volterra system (10) obtained using multi-step DTM and DTM using the same parameters as in Fig. 1. One can see that the Multi-step DTM gives the solution over large time intervals while the classical DTM solution diverges for t > 0.75 in (b) and t > 1.35 in (d).

It is to be noted that the multi-step DTM results are obtained when K = 10, M = 100, T = 20 and the DTM results are obtained when N = 1000.

Chen system

Now, we take Chen system into consideration. In 1999, the Chen system is found by Chen and Ueta [START_REF] Chen | Yet another chaotic attractor[END_REF].

The nonlinear differential equations that describe the Chen system are,

                     dx dt = a(y -x), dy dt = (c -a)x -xz + cy, dz dt = xy -bz, (16)
where a, b and c are positive real numbers. Bifurcation studies [START_REF] Ueta | Bifurcation analysis of Chen's equation[END_REF][START_REF] Lu | Local bifurcations of the Chen system[END_REF] show that Chen system has chaotic behavior for the parameters (a, b, c) = (35, 12, 28) and (a, b, c) = (35, 3, 28). According to the multi-step DTM, the series solution for Chen system ( 16) is given by,

(x(t), y(t), z(t)) =                K n=0 (X 1 (n), Y 1 (n), Z 1 (n))t n , t ∈ [0, t 1 ] K n=0 (X 2 (n), Y 2 (n), Z 2 (n))(t -t 1 ) n , t ∈ [t 1 , t 2 ] . . . K n=0 (X M (n), Y M (n), Z M (n))(t -t M -1 ) n , t ∈ [t M -1 , t M ] , ( 17
)
where 3 and4 show chaotic attractors for Chen system ( 16) when (a, b, c) = (35, 3, 28) and (a, b, c) = (35, 12, 28), respectively, using the Multi-step DTM solution, at step size h = 0.01 and K = 20, and RK4 method. We can observe the high agreement between the chaotic attractors obtained using the Multi-step DTM and the one obtained using RK4 method for the chaotic Chen system ( 16). This is much more obvious from Fig. 4. Fig. 5 shows time series for the chaotic Chen system ( 16) when (a, b, c) = (35, 12, 28) using the Multi-step DTM solution, at step size h = 0.01 and K = 15, and the DTM, when N = 1000. Also in this figure we can see that the Multi-step DTM gives the solution over a large interval while the classical DTM diverges for t > 0.135.

X i (n), Y i (n) and Z i (n), for i = 1, 2, • • • , M , satisfy the following recurrence relations,                    X i (k + 1) = a k + 1 Y i (k) -X i (k) , Y i (k + 1) = 1 k + 1 (c -a)X i (k) -Σ k l=0 X i (l)Z i (k -l) + cY i (k) , Z i (k + 1) = 1 k + 1 Σ k l=0 X i (l)Y i (k -l) -bZ i (k) , ( 18
) such that X 1 (0) = x(0), Y 1 (0) = y(0), Z 1 (0) = z(0) and X i (0) = X i-1 (0), Y i (0) = Y i-1 (0), Z i (0) = Z i-1 (0), for i = 2, 3, • • • , M . Figs.

Lorenz system

Here, we take Lorenz system into consideration. In 1963, the Lorenz system is introduced by Edward Lorenz [START_REF] Lorenz | Deterministic nonperiodic flow[END_REF]. The nonlinear differential equations that describe the Lorenz system are,

                     dx dt = a(y -x), dy dt = cx -xz -y, dz dt = xy -bz, (19)
where a, b and c are positive real numbers. For the parameters (a, b, c) = (10, 8/3, 28) Lorenz system can display chaotic attractors. According to the multi-step DTM, the series solution for Lorenz system [START_REF] Kuo | Application of the differential transformation method to the solution of a damped system with high nonlinearity[END_REF] is given in Eq. [START_REF] Momani | A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula[END_REF] where 6 shows the chaotic attractors for the chaotic Lorenz system ( 19) when (a, b, c) = (10, 8/3, 28) using the Multi-step DTM solution, at step size h = 0.001 and K = 20, and RK4 method. As also shown for Chen system, we can observe the high agreement between the chaotic attractors obtained using the Multi-step DTM and the one obtained using RK4 method for the chaotic Lorenz system [START_REF] Kuo | Application of the differential transformation method to the solution of a damped system with high nonlinearity[END_REF]. Fig. 7 shows the time series for the chaotic Lorenz system (19) when (a, b, c) = (10, 8/3, 28) using the Multi-step DTM solution, at step size h = 0.01 and K = 15, and the DTM, when N = 1000. Also in this figure we can see that the DTM solution diverges when t > 0.15.

X i (n), Y i (n) and Z i (n), for i = 1, 2, • • • , M , satisfy the following recurrence relations,                    X i (k + 1) = a k + 1 Y i (k) -X i (k) , Y i (k + 1) = 1 k + 1 cX i (k) -Σ k l=0 X i (l)Z i (k -l) -Y i (k) , Z i (k + 1) = 1 k + 1 Σ k l=0 X i (l)Y i (k -l) -bZ i (k) , ( 20
) such that X 1 (0) = x(0), Y 1 (0) = y(0), Z 1 (0) = z(0) and X i (0) = X i-1 (0), Y i (0) = Y i-1 (0), Z i (0) = Z i-1 (0), for i = 2, 3, • • • , M . Fig.

Discussion

In this work, we carefully propose the Multi-step DTM, a reliable modification of the DTM, that improves the convergence of the series solution. The method provides immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to both linear and nonlinear differential equations. The validity of the proposed method has been successful by applying it for Lotka-Volterra, Chen and Lorenz systems. The method were used in a direct way without using linearization, perturbation or restrictive assumptions. It provides the solutions in terms of convergent series with easily computable components and the results have shown remarkable performance. Figs. 1, 3-4 and 6 show that the Multi-step DTM approximate solutions for Lotka-Volterra, Chen and Lorenz systems, respectively, are very closed to the Range-kutta approximate solutions. Therefore, the proposed method is very efficient and accurate method that can be used to provide analytical solutions for nonlinear systems of differential equations. Figs 2, 5 and 7 show that the solutions obtained using DTM have a small interval of convergence while those obtained using Multi-step DTM have wide intervals of convergence. This confirms that the new algorithm of the DTM increases the interval of convergence for the series solution. Of course the accuracy can be improved when the step size h becomes smaller and the number of terms in each subinterval K becomes larger.

We have shown that the proposed algorithm is a very accurate and efficient method compared with RK4 method for both non-chaotic and chaotic systems. The method works successfully in handling systems of differential equations directly with a minimum size of computations and a wide interval of convergence for the series solution. This emphasizes the fact that the Multi-step DTM is applicable to many other nonlinear models and its reliable and promising when compared with existing methods. Obviously the DTM solution diverges for t > 0.15.

  (a) and (b) we take the parameters a = 0.95, b = 0.25, c = 2.45, d = 0.25, c x = 15 and c y = 8 and in (c) and (d) we take the parameters a = 0.95, b = 0.2, c = 2.5, d = 0.35, c x = 10 and c y = 3.

Figure 1 :

 1 Figure 1: Phase plane trajectories for Lotka-Volterra system (10), comparison between Multi-step DTM and RK4 method.

Figure 2 :

 2 Figure 2: Time series for Lotka-Volterra system (10), comparison between Multi-step DTM and DTM. Obviously the DTM solution diverges for t > 0.75 in (b) and t > 1.35 in (d).

Figure 3 :

 3 Figure 3: Chaotic attractors for the chaotic Chen system (16), when (a, b, c) = (35, 3, 28), comparison between Multi-step DTM (Left) and RK4 method (Right).

Figure 4 :

 4 Figure 4: Chaotic attractors for the chaotic Chen system (16), when (a, b, c) = (35, 12, 28), comparison between Multi-step DTM (Left) and RK4 method (Right).

Figure 5 :

 5 Figure 5: Time series for the chaotic Chen system (16), comparison between Multi-step DTM and DTM. Obviously the DTM solution diverges for t > 0.135.

Figure 6 :

 6 Figure 6: Chaotic attractors for the chaotic Lorenz system (19), comparison between Multi-step DTM (Left) and RK4 method (Right).

Figure 7 :

 7 Figure 7: Time series for the chaotic Lorenz system (19), comparison between Multi-step DTM and DTM.

Table 1 .

 1 The main steps of the DTM, as a tool for solving different classes of nonlinear problems, are the following. First, we apply the differential transform (1) to the given problem (integral equation, ordinary differential equation, partial differential equation or system of differential equations), then the result is a recurrence relation. Second, solving this relation and using the differential inverse transform (2) we can obtain the solution of the problem.

	Table 1. Operations of differential transformation
	Original function	Transformed function


Pedro H Albuquerque

Sophie Albuquerque

Framing Cognitive Machines: A Sociotechnical Taxonomy

Keywords: sociotechnical taxonomy, technological disruptions, technological innovations, automatic, autonomous, physical technology, cognitive technology, skill-enhancing, skillreplacing, externalities, Industry 5.0, cognitive machines, artificial intelligence, occupational science

Aims: we propose a sociotechnical taxonomy for the analysis of socio-economic disruptions caused by technological innovations. Methodology: a transdisciplinary principled approach is used to build the taxonomy through categorization and characterization of technologies using concepts and definitions originating from cybernetics, occupational science, and economics. The sociotechnical taxonomy is then used, with the help of logical propositions, to connect the characteristics of different categories of technologies to their socio-economic effects, for example their externalities. Results: we offer concrete illustrations of concepts and uses, and an Industry 5.0 case study as an application of the taxonomy. We suggest that the taxonomy can inform the analysis of opportunities and risks related to technological disruptions, specially of those that result from the rise of cognitive machines.

I. INTRODUCTION

Over time, theories on technological disruptions have undergone evolution and change, as scientists tried to understand this complex phenomenon and had to adapt their thinking to everchanging observable patterns regarding its socio-economic impacts. This is unsurprising, as technological progress is best comprehended through the lens of complexity theory, which takes in consideration aspects such as emergence, path dependence, multiple equilibria, nonergodicity, nonlinearity, and phase transitions. [START_REF] Durlauf | Complexity and empirical economics[END_REF]- [START_REF] Brock | A formal model of theory choice in science[END_REF].

In social sciences in particular, the difficulties associated with the complexity of the subject have been compounded by the reliance on narrow mono-disciplinary, reductionist, productivist and economistic approaches [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF]. It is our understanding that this way to proceed is ineffective and leads inevitably to intellectual dead-ends. A transdisciplinary approach is necessary due to the complex nature of the phenomenon. Moreover, exactly because of the lack of transdisciplinarity, the study of technological disruptions lacks comprehensive categorization and characterization.

With this in mind, we propose in this article a sociotechnical [START_REF] Bunge | Social science under debate: A philosophical perspective[END_REF], [START_REF] Bijker | Of bicycles, bakelites, and bulbs: Toward a theory of sociotechnical change[END_REF] taxonomy that helps to frame studies of technological disruptions, allowing for a deeper understanding of the process of socioeconomic change and evolution caused by technological innovations, such as those created during the last few decades by the emergence of cognitive machines. For that, technologies are categorized according to perspectives originating from cybernetics, economics and occupational science [START_REF] Wilcock | An occupational perspective of health, 3rd ed[END_REF]. We further extend the taxonomy to incorporate Vinge's suggestion that technologies have physical and cognitive characteristics [START_REF] Vinge | The coming technological singularity[END_REF], [START_REF] Vinge | Signs of the singularity[END_REF]. Notice that Vinge originally called these characteristics "weak superhumanity" and "strong superhumanity" and applied them to technological entities. Vinge's characteristics should not be confounded with Searle's [START_REF] Searle | Minds, brains, and programs[END_REF] concepts of weak and strong artificial intelligence [START_REF] Sloman | Did Searle attack strong strong or weak strong AI?[END_REF].

The development of a transdisciplinary taxonomy becomes even more important when, as we argue in a previous article [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF], and in disagreement with most of the existing reductionist economics literature on technological disruptions, one considers that the socio-economic effects of skillreplacing technological innovations are not caused by human characteristics such as "low skills" and "low cognition," or by task characteristics such as "routine" or "simplicity," but instead by the cybernetic characteristics of the innovations in a given context. We therefore propose that technologies need to be categorized and characterized according to the transdisciplinary taxonomy developed in this article.

The article is organized as follows: we first present the objectives of the study and its methodology. Then we proceed to the discussion, followed by the presentation of a case study that applies the taxonomy to the analysis of Industry 5.0 [START_REF] Breque | Industry 5.0: Towards a sustainable, human centric and resilient European industry[END_REF]. We conclude by offering some speculative thoughts on how the taxonomy brings to light opportunities and risks created by the disruptions of the socio-economic fabric due to the emergence of cognitive technological innovations, as exemplified by current concerns related to the unmanaged and unregulated deployment of artificial intelligence cybernetic technologies.

II. OBJECTIVES

The main objective of this article is to propose a sociotechnical taxonomy that allows researchers to frame studies of socio-economic disruptions resulting from technological innovations such as those created by the rise of cognitive machines.

III. METHODOLOGY

The theoretical methodology follows a principled approach to build a sociotechnical taxonomy through categorization and characterization of technologies using concepts and definitions originating from cybernetics, occupational science, and economics. Under a cybernetics perspective, the taxonomy divides technologies into noncybernetic and cybernetic, with the latter being subdivided into automatic and autonomous. Under an economics perspective it divides technologies into skill-enhancing and skill-replacing. Cybernetic technologies are then characterized according to Vinge's two dimensions: physical and cognitive. Once the taxonomy is proposed, interconnections among categories and characteristics of technologies are developed based on logical propositions. As illustrations, we offer real-world examples of concepts and applications, including a discussion of the importance of the taxonomy in the context of industry 5.0.

IV. DISCUSSION

In the next subsections we explore concepts that originate from cybernetics, occupational science, and economics and that we identify as being essential to the understanding of the socio-economic disruptions of technological innovations. By employing a transdisciplinary approach we avoid intellectual dead-ends typically found in social sciences. The economics literature for example overlooks the importance of cybernetics in the study of technological innovations and disruptions. Although it is possible to find examples of articles that are unquestionably aware of the developments in robotics, machine learning and artificial intelligence [START_REF] Brynjolfsson | Artificial intelligence and the modern productivity paradox: A clash of expectations and statistics[END_REF], we think that they systematically miss the point by adopting a narrow mono-disciplinary, reductionist, productivist and economistic view, not relying on other more relatable fields such as cybernetics and occupational science under a transdisciplinary perspective in order to improve the understanding of how technologies affect societies [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF].

A. Noncybernetic, Automatic and Autonomous Technologies

Consider for example relatable concepts originating from cybernetics. Differently from the term technology, the term cybernetics has been used since ancient times to designate "good piloting." Ampère for example declared in his treatise on scientific classification that: "I name Cybernetics, from the word κυβερνητική, taken at face value, in a restricted sense, as the art of governing a vessel, and, when taken from usage among the Greek themselves, with the meaning otherwise extended, as the art of governing in general" [START_REF] Ampère | Essai sur la philosophie des sciences[END_REF].

The notion of cybernetics as art of governing comes back during the 20th century, initially in continental Europe, as the study of all social, physical and biological phenomena that involve selfgoverning systems, and then by the work of Norbert Wiener. He famously stated that "cybernetics attempts to find the common elements in the functioning of automatic machines and of the human nervous system, and to develop a theory which will cover the entire field of control and communication in machines and in living organisms" [START_REF] Wiener | Cybernetics, or control and communication in the animal and the machine[END_REF]. The field of cybernetics was central to the study and development of automatic technologies throughout de 20 th century. Notice that automatic machines rely on self-governing mechanisms but do not embody significant amounts of cognitive capabilities. This situation evolved when the field of artificial intelligence (AI) was defined and developed later. Following Moor, the Dartmouth Summer Research Project of 1956 is possibly the event that has set artificial intelligence as a promising scholarly field [START_REF] Moor | The Dartmouth College Artificial Intelligence Conference: The Next Fifty Years[END_REF]. Participants followed on the original idea of McCulloch and Pitts, who declared that AI is a feasible technology because "the 'all-or-none' character of nervous activity, neural events and the relations among them can be treated by means of propositional logic" [START_REF] Mcculloch | A logical calculus of the ideas immanent in nervous activity[END_REF]. Since then we have seen the development of autonomous technologies and cognitive machines that, differently from automatic technologies, require strict ethical design standards due to the embodiment of decision-making capabilities [START_REF] Havens | From principles and standards to certification[END_REF].

Even though specific fields of cybernetics such as automation, information theory, machine learning and artificial intelligence do not follow necessarily the same epistemological principles, and do not always cover the same ground, they tend to intersect in what concerns socio-economic phenomena. We use therefore the term cybernetics in this article to represent the confluence of all its different fields in the creation of automatic and autonomous self-governing and cognitive technologies that have increasingly disruptive socio-economic effects [START_REF] Grinin | The cybernetic revolution and the future of technologies[END_REF]. In the sense of this article, these technologies typically follow algorithms (routines), which can range from static to adaptive, and simple (e.g. automation) to complex (e.g. artificial intelligence). As examples: thermostats embody static and simple cybernetics, while facial recognition software embodies adaptive and complex cybernetics.

In summary, we divide technologies, technological artifacts, and technological innovations into three categories based on concepts originating from cybernetics: noncybernetic (e.g. a hammer), automatic (e.g. a steam centrifugal governor) and autonomous (e.g. a planetary exploration rover).

B. Skill-Enhancing and Skill-Replacing Technologies

In the economics literature, technologies are typically divided into two groups according to how they interact with labor: skill-enhancing and skill-replacing [START_REF] Acemoglu | Why do new technologies complement skills? Directed technical change and wage inequality[END_REF]. As their qualifiers indicate, a skillenhancing technology improves the given human skill that governs it, while a skill-replacing technology governs itself in the replacement of a human skill.

Notice that in our line of research we depart on multiple dimensions from the mainstream economics literature on skill-replacing technologies, e.g. [START_REF] Jaimovich | Job polarization and jobless recoveries[END_REF]- [START_REF] Acemoglu | Robots and jobs: Evidence from US labor markets[END_REF], which sees in human characteristics ("low skills," "low cognition") or in task characteristics ("routine," "simplicity") the determinants of technologies' socio-economic effects. For us, it is the cybernetic characteristics of skill-replacing technologies (their ability to replace human skills) in a given context that determine their socio-economic effects. Moreover, based on knowledge originating from occupational science [START_REF] Wilcock | An occupational perspective of health, 3rd ed[END_REF], we propose that studies of the socio-economic effects of technological innovations should not be narrowly concerned with the labor market, and should instead consider the effects of technological disruptions on all labor and non-labor human occupations [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF].

C. A Sociotechnical Taxonomy

In order to better understand the relations between human skills, cybernetics, technological innovations, and their socio-economic effects, firstly we use a principled approach to define technologies according to a sociotechnical taxonomy inspired by [START_REF] Jdanko | Evolutionary cybernetic systems theory considered as a chapter of general systems theory -A viewpoint[END_REF] and [START_REF] Havens | From principles and standards to certification[END_REF] as follows:

Definition 1: a technology is cybernetic if it can self-govern, otherwise it is noncybernetic. Definition 1.

1: a technology is automatic if it is cybernetic but not capable of decision-making. Definition 1.2: a technology is autonomous if it is cybernetic and capable of decision-making.

Having decision-making capabilities is a characteristic that involves advanced cognitive capabilities such as reasoning, learning, adapting, or environment transacting. Examples: a knife embodies noncybernetic technology, an alarm clock embodies automatic technology, and a self-piloting drone embodies autonomous technology. Now, we propose the following alienation taxonomy of human skills [START_REF] Mackenzie | Marx and the machine[END_REF]:

Definition 2: a human skill is currently alienable if it can be fully and adequately replaced by a current technology, otherwise it is currently inalienable.

As examples, certain human weaving skills are alienable because they can be fully and adequately replaced through automation technology, such as the one used in the 19th century Jacquard loom, while certain nurturing skills, such as those necessary for nursing a newborn, are inalienable because no current technology can fully and adequately replace them.

In general, and non-exhaustively, biological skills (such as mating skills), agency skills (such as wine choosing skills), emotional skills (such as delay of gratification skills), and soft skills (such as leadership skills) currently cannot be fully and adequately replaced by technologies, therefore are currently inalienable. Notice however that technological innovations can alienate previously inalienable human skills (see the weaving example above). The taxonomy is therefore state-of-theart and context dependent.

The next definitions and proposition frame and clarify the previous subsection concepts originating from economics. With the help of Definition 2, we propose the following: Definition 3.1: a technology is skill-replacing when it alienates a human skill.

For example, from an economics perspective, a dishwasher is a skill-replacing technology only when it is used to alienate human dishwashing skills. In addition, and following Kidd [START_REF] Kidd | On the design of skill supporting computer-aided technologies[END_REF], we propose that: Definition 3.2: a technology is skill-enhancing when it adapts a human skill into a more productive or effective skill.

Consider the example of a typewriter machine. It adapts writing skills into more productive or effective skills (typewriting). Nonetheless, writing skills remain necessary to govern the machine, so the latter skills remain inalienable.

Proposition 1: a technology can be skill-replacing, skill-enhancing or both, depending on the skill and the given context.

D. Physical and Cognitive Characteristics of Technologies

The focus on the cybernetic characteristics of technological innovations allows us to introduce characteristics of technologies originally developed by Vinge [START_REF] Vinge | The coming technological singularity[END_REF]. As Vinge explains, "I call [the] 'fast thinking' form of superintelligence 'weak superhumanity'" (a physical technological improvement), while "'strong superhumanity' would be more than cranking up the clock speed on a human-equivalent mind" (a cognitive technological improvement).

More precisely, we offer the following definitions: Consider as an example the complex cognitive capabilities necessary for chess playing. On the one hand, a chess playing machine benefits from a more advanced physical technological innovation when it can play chess with the same cognitive capabilities of previous machines, but faster. On the other hand, a chess playing machine benefits from a more advanced cognitive technological innovation when it can play chess with superior cognitive capabilities, but at the same speed of previous machines. In the DC Universe © , a physically-enhanced human is The Flash, due to his "super-speed" skills [START_REF]Flash (DC Comics character)[END_REF], while a cognitively-enhanced human is Batman, due to his "genius-level intellect" skills [START_REF]Batman -Wikipedia[END_REF].

Distinguishing between physical technological innovations and cognitive technological innovations is central to the understanding of the socio-economic effects of technological disruptions. For example, physical improvements in cybernetic technologies may displace labor because they increase the effective amount of labor supply, while cognitive improvements in cybernetic technologies may displace labor because they create superior cybernetic alternatives to human cognitive skills.

E. Interconnections Among Categories and Characteristics of Technologies

The combination of the four definitions above leads us to the following propositions: Human skills need to be governed. Therefore, to fully and adequately replace human skills, a technology must also self-govern, and if it self-governs, then it is cybernetic. Consider as an example an electronic calculator. As a technological artifact, it replaces alienable human calculation skills. It only replaces them because it automates calculations as a cybernetic technological artifact.

Proposition 2.2: a cybernetic technology is not necessarily a skill-replacing technology.

Cybernetic technologies can be used to complement or enhance other needed human skills or other technologies. For example, on one hand an automatic infrared targeting system does not replace human skills, because humans cannot see infrared light. On the other hand, it enhances human targeting skills as it complements other human skills needed for targeting.

Proposition 3.1: a skill-enhancing technology is necessarily a noncybernetic technology in what concerns the human skill that it enhances.

If a technology enhances a certain human skill, then these human skills are still employed for governance, and as such the technology is noncybernetic in what concerns these enhanced human skills. Notice however that the same technology could be cybernetic concerning other human skills that it replaces. For example, a financial accounting software is noncybernetic in what concerns the financial accounting human skills needed to operate it, but its software calculator component is cybernetic because it replaces alienable human calculation skills.

Proposition 3.2: a noncybernetic technology is not necessarily a skill-enhancing technology.

This proposition follows from the fact that a noncybernetic technology can be a component of a skill-replacing technology. As an example, optical lenses used in eyeglasses are a noncybernetic technological artifact and a skill-enhancing technology. But optical lenses are not a skill-enhancing technology when they are used as part of an automatic security camera.

Proposition 4: technological innovations may have physical and cognitive characteristics.

More precisely: Proposition 4.1: Noncybernetic technologies may have physical characteristics but do not have cognitive characteristics.

Noncybernetic technologies do not self-govern, and as such do not recur to cognitive capabilities of any kind. They may however have physical characteristics, as these do not require self-governance. For example, a technological innovation that leads to sharper knives does not involve cognitive characteristics, but has superior physical characteristics (cuts faster or more precisely) compared to the previous technology.

Proposition 4.2: cybernetic technological innovations necessarily have cognitive characteristics.

As cybernetic technologies self-govern, they are able to replace human cognitive skills needed for governance, for example, the steam engine centrifugal governor [START_REF] Maxwell | I. On governors[END_REF], an invention that symbolizes the Industrial Revolution [START_REF] Bernstein | The history of control: from idea to technology[END_REF], replaces human cognitive skills needed for the stable functioning of steam engines.

Proposition 4.3: Autonomous cybernetic technological innovations significantly expand the scope and intensity of cognitive characteristics.

Automatic technologies may arguably have restricted endowments of cognitive characteristics, for example, a digital diary automates and replaces some memory skills at some basic cognitive level. But automatic technological artifacts do not have decision-making capabilities, hence their cognitive characteristics remain limited. The development of autonomous cybernetic technologies significantly expands and deepens cognitive characteristics available to innovations, magnifying and accelerating technological disruptions and setting in motion a process of socio-economic change like humanity has never experienced before.

F. Technological disruptions and externalities

In a narrow business administration context, technological disruptions have been defined as what happens when technological innovations are used by a startup challenger to displace firms detaining incumbent market power in a given industry [START_REF] Bower | Disruptive technologies: Catching the wave[END_REF], [START_REF] Christensen | The innovator's dilemma: When new technologies cause great firms to fail[END_REF]. Originally a productivist and economistic concept, its meaning has been extended more recently to represent the more relevant disruptive impacts of technological innovations on broader socio-economic dimensions [START_REF] Hopster | What are socially disruptive technologies?[END_REF], [START_REF]World Development Report 2019: The changing nature of work[END_REF].

In this article we update and frame the concept of technological disruptions so it more rigorously represents current uses in social sciences. Firstly, in order to escape from narrow productivist and economistic narratives, and in agreement with current use in the public debate arena, we extend it so it includes disruptions to all labor and nonlabor human occupations [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF]. Secondly, we give it analytical value by framing it using language from externality theory.

For that, consider a broad definition of an externality as the costs or benefits imposed on an individual, organization or society by another entity that creates these costs or benefits, in other words, a situation where the private costs (or private benefits) to the entity differ from the social costs (or social benefits) imposed on (or granted to) another entity [START_REF] Pigou | The economics of welfare[END_REF]. Notice that social costs (or benefits) are equal to private costs (or benefits) plus external costs (or benefits). We argue that: Proposition 5: a technological innovation that involves a significant externality is necessarily disruptive.

On one hand, a technological innovation resulting from an entity's transaction should not be considered disruptive when its social costs (or social benefits) are equal to its private costs (or private benefits) because, by definition, there are no external costs (or external benefits) involved in such a transaction. On the other hand, if there are external costs or external benefits, and if they are significant, then it is clear that the socio-economic fabric is involuntarily disrupted.

Consider now the following propositions: Proposition 5.1: any skill-replacement caused by a technological innovation externality is a cost to humans who are victims of skill alienation.

This proposition has its roots on the extensive social science literature that investigates the relations between technology and labor alienation [START_REF] Mackenzie | Marx and the machine[END_REF]. We extend the discussion from labor alienation to skill alienation, and we interpret the phenomenon from an occupational science perspective [START_REF] Wilcock | An occupational perspective of health, 3rd ed[END_REF] instead of an economics perspective, where the technological alienation of skills is a cost to its victims as it produces occupational injustice: occupational decline, occupational deprivation, occupational alienation, occupational imbalance, or occupational marginalization, and as any of those events are detrimental to well-being [START_REF] Wilcock | A theory of the human need for occupation[END_REF], [START_REF] Wilcock | Occupational science: The key to broadening horizons[END_REF].

Proposition 5.2: any skill-enhancement caused by a technological innovation externality is as an external benefit to humans who are granted enhanced skills.

This proposition follows from the wildly accepted theoretical and empirical assumption that research and development (R&D) creates positive externalities of many different varieties [START_REF] Hall | Chapter 14 -The financing of R&D and innovation[END_REF].

The latter three propositions lead to the following:

Proposition 6.1: if a skill-replacing technological innovation creates an externality, it is necessarily a negative technological disruption.

A skill-replacing technological innovation is created by its proprietor in the wish to obtain a private or social benefit. As long as the costs related to its skill-replacement function are born by the proprietor, no significant socio-economic disruption takes place. However, if the innovation creates an external cost that is related to its skill-replacing function, then the benefit-seeking proprietor transfers the innovation burden to the collective, therefore creating an external cost thus a negative technological disruption. For example, as a technological innovation a dishwasher machine does not create a technological disruption when used as a labor-saving device only by the proprietor of the technological artifact, but it creates a negative technological disruption to workers who earn a living from their alienated dishwashing skills.

Proposition 6.2: if a skill-enhancing technological innovation creates an externality, it is necessarily a positive technological disruption.

A skill-enhancing technological innovation is created by its proprietor in the wish to obtain a private or social benefit. As long as the benefits related to its skill-enhancement function are conferred only to the proprietor, no significant socio-economic disruption takes place. However, if the innovation creates an external benefit that is related to its skill-enhancing function, then the benefit-seeking proprietor transfers the innovation gain to the collective, therefore creating an external benefit thus a positive technological disruption. For example, as a technological innovation an online encyclopedia (such as Wikipedia) does not create a technological disruption when used only by the proprietor of the technological artifact, but it creates a positive technological disruption to users of the technology who have their intellectual skills enhanced by it.

G. A Sociotechnical Taxonomy Synthesis

To summarize the results in this section, consider first Table 1. Physical characteristics are possible in all technological innovation categories. Since immemorial times, humans have interacted with technological artifacts through their physical capabilities. As physical characteristics improve, they may offset human occupations of a physical nature, but still allow for humans to move to other labor and nonlabor occupations based on cognitive skills. For example, technological progress has pushed most of humanity out of agriculture towards industry, and then from industry towards services. This process of technological disruption is captured in a simplified way by the conjectural path of the yearly amount of technological innovations shown in Figure 1. The sociotechnical evolutionary path changes once cognitive innovations are introduced in the socio-economic system. Starting from the established evolutionary path of physical characteristics, there will be a point in time when cognitive characteristics start to evolve fast. We define this point in time as the occupational singularity [START_REF] Albuquerque | The occupational singularity: Cognitive technologies as new drivers of inequality[END_REF]. Because of their cognitive and self-reinforcement nature (cognitive machines will accelerate the development of more advanced cognitive machines), we expect the evolution of cognitive technological innovations to only accelerate once the occupational singularity sets in. For the first time, technological artifacts will be able to replace complex human cognitive skills, and the process of replacement will only gain in intensity and scope with the passing of time. We can only speculate and theorize on how this new pattern will displace human labor and nonlabor occupations, as there is no historical precedent that can guide us in the understanding of this new phenomenon. Data from the USPTO [START_REF] Marco | The USPTO historical patent data files: Two centuries of innovation[END_REF] shown in Figure 2 seem to validate such a conjecture: since around 1980 the number of cybernetic patents per year has been growing faster than the number of noncybernetic patents per year. The relative gap between these two categories has only been narrowing since then. The data does not directly represent Vinge's characteristics of cybernetic technological innovations, but given the evolution of cognitive technologies during the last few decades, we believe that this is evidence in favor of our conjecture that cognitive innovations are accelerating and becoming dominant, with complex socio-economic consequences that are hard to predict. Fig. 2: Number of cybernetic and noncybernetic patents per year (USPTO) Some pundits have been arguing that what we have been experiencing since the 80s is not new, that we have been witnessing another wave of technological progress identical to previous phases of the Industrial Revolution. We argue that this assessment is wrong, that instead we are experiencing an occupational singularity, as for the first time cognitive technological innovations are responsible for increasing and accelerating amounts of socio-economic disruptions.

V. INDUSTRY 5.0: A CASE STUDY In the recent literature about the latest stage of the Industrial Revolution that is now known as Industry 5.0 we commonly find the following triple bottom line as desired characteristics of future industrial developments: it must be sustainable, human-centric and resilient [START_REF] Barata | Industry 5.0 -Past, present, and near future[END_REF], [START_REF] Leng | Industry 5.0: Prospect and retrospect[END_REF]. The European Union for example declared in its recent report about Industry 5.0 [START_REF] Breque | Industry 5.0: Towards a sustainable, human centric and resilient European industry[END_REF] that "European industry is increasingly resilient and adapts itself to a new societal reality, in which production is required to respect the boundaries of our planet, and industry worker well-being is placed at the centre of the production process."

We believe that the current narrative about Industry 5.0 misses two essential points that are studied in this article. Firstly, it is necessary to bring the relatively new field of occupational science to the center of the discussion about industrial workers and end-users well-being. As some have pointed out before [START_REF] Urbanowski | Occupational science value propositions in the field of public policy[END_REF], [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF], other social sciences are not as well equipped to offer policy advice concerning this vital aspect of industry's future. Secondly, as we showed in our proposed taxonomy, autonomous technologies will embody increasing amounts of cognitive technological innovations, which can be extremely disruptive compared to noncybernetic and automatic technological innovations employed in previous stages of the Industrial Revolution. Based on the findings of this article, we suggest that future research must consider how cognitive technological innovations create negative externalities that are detrimental to well-being or positive externalities that are an opportunity for the flourishing of Industry 5.0 in agreement with its triple bottom line. An example of how these ideas can be applied to remanufacturing is found in [START_REF] Albuquerque | Technological innovations and obsolescence: Leveling the playing Field for remanufacturing[END_REF].

We argue therefore that we are living now in a new socio-economic environment that will be marked by an increasing number of technological innovations with cognitive characteristics that follow accelerating trajectories -an occupational singularity. This scenario is unique: for the first time in history humans will need to face a barrage of disruptions caused by technological innovations increasingly able to replace cognitive skills. We expect the challenges that it will create to become central to the future developments of Industry 5.0.

VI. CONCLUSION, RECOMMENDATIONS AND EXTENSIONS

In this article we use cybernetics, occupational science and economics to study technological innovations and disruptions. We offer a series of definitions and propositions that help to clarify the relations between cybernetic categories such as noncybernetic, automatic and autonomous, and economic concepts such as alienable human skills and categories such as skill-replacing technologies and skill-enhancing technologies. We show, inspired by Vinge's work, that disruptive technological innovations may have two characteristics: physical and cognitive, and we propose that distinguishing between these two is essential to a better understanding of the socio-economic effects of technological innovations.

We speculatively suggest that cognitive innovations may lead to the accelerated displacement of meaningful human labor and nonlabor occupations [START_REF] Wilcock | A theory of the human need for occupation[END_REF] and has the potential to create politically and socially unsustainable outcomes [START_REF] Albuquerque | Social implications of technological disruptions: A transdisciplinary cybernetics science and occupational science perspective[END_REF], [START_REF] Korinek | Artificial intelligence and its implications for income distribution and unemployment[END_REF], specially in the case of skill-replacing cognitive technological innovations that produce large amounts of negative externalities. We define this moment in time as the occupational singularity. We believe that new approaches will be needed to address the detrimental effects of technological innovations on well-being and to channel them towards humanistic objectives [START_REF] Bolton | Humanistic engineering: Engineering for the people[END_REF].

Among possible extensions: autonomous technologies can be subdivided according to the cognitive capabilities they embody, for example, a technology may embody selective memory capabilities or environment transacting capabilities, which would then be tied to different types of socio-economic impacts. Technology affordances and constraints theory (TACT) [START_REF] Majchrzak | Technology Affordances and Constraints Theory (of MIS)[END_REF] can also be introduced to establish the links between a technology category and its affordances and constraints. We intend to study these matters among others in our future investigations.

  As an example, consider the skill of writing. A typewriter machine is: a skill-replacing technology in what concerns handwriting, a skill-enhancing technology in what concerns typewriting, and both simultaneously in what concerns writing.

  Definition 4.1: a physical technological innovation allows a new technology to perform with superior physical capabilities (e.g. superior speed or strength) at the same cognitive capability levels of the previous technology. Definition 4.2: a cognitive technological innovation allows a new technology to perform with superior cognitive capabilities (e.g. superior reasoning, learning, adapting, or environment transacting) at the same physical capability levels of the previous technology.

Proposition 2 . 1 :

 21 a skill-replacing technology is necessarily a cybernetic technology in what concerns the human skill that it replaces.

Fig. 1 :

 1 Fig. 1: Conjectural amount of technological innovations per year

TABLE 1 :

 1 A SUMMARY OF THE SOCIOTECHNICAL TAXONOMY FOR EXTERNALITY PRODUCING TECHNOLOGICAL INNOVATIONS

The project leading to this publication has received funding from the French government under the "France 2030" investment plan managed by the French National Research Agency (reference: ANR-17-EURE-0020) and from the Excellence Initiative of Aix-Marseille University -A*MIDEX.


Manuel Théodet

Cyril Quilfen

Cristina Martínez

C Aymonier

Continuous supercritical synthesis of unsupported and high specific surface area catalyst precursors for deephydrodesulfurization

Continuous supercritical synthesis of unsupported and high specific surface area catalyst precursors for deep-hydrodesulfurization

Introduction

Due to increasingly stringent environmental regulations in sulfur limits for distillates fuels, production of ultra-low sulfur content fuels is required to reach the limits of 10 ppm in the European Union [START_REF]The European Parliament & The Council of the European Union[END_REF] and Japan 2 and 15 ppm in North America [START_REF]Tier 2 Motor Vehicle Emissions Standards and Gasoline Sulfur Control Requirements[END_REF]4 . Moreover, remaining available crude oils resources contain a greater amount of undesirable heteroatoms that will have to be removed. Removals of 99.99% of sulfur from typical crude oil and the corresponding processes have been termed as deep-HDS. The main issue of deep-HDS is the elimination of refractory sulfur compounds such as 4,6-dimethyldibenzothiophene (4,6-DMDBT). In order to achieve this issue without expensive costs in process modifications, the development of more efficient deep-HDS catalysts is required.

Typically, a HDS catalyst contains a metal from the group VIB (chromium, molybdenum or tungsten) promoted with a metal from the group VII (cobalt or nickel) and supported, classically, on alumina [START_REF] Derbyshire | Catalysis in Coal Liquefaction: New Directions for Research[END_REF][START_REF] Donath | Catal. Sci[END_REF][START_REF] Hoering | Early Coal Hydrogenation Catalysis[END_REF] . These conventional catalysts are generally obtained by impregnation of γ-Al2O3 with aqueous solutions of (NH4)6Mo7O24.4H2O and Co(NO3)2.6H2O or Ni(NO3)2.6H2O followed by drying, calcination and activation under H2/H2S 8 in order to obtain MoS2 layers well dispersed on the γ-Al2O3 support and edges-decorated with Co or Ni [START_REF] Topsøe | Hydrotreating Catalysis[END_REF] . However, current HDS catalysts revealed to be insufficient and strong optimizations have been investigated over the past decades, such as improvement of the support and/or of the active phase [START_REF] Oyama | Transition metal phosphide hydroprocessing catalysts: A review[END_REF] .

On the other hand, some unsupported materials have been described with higher activity and/or selectivity than traditional supported catalysts, leading to a growing interest in unsupported HDS nano-catalysts [START_REF] Chianelli | Unsupported transition metal sulfide catalysts: 100 years of science and application[END_REF][START_REF] Eijsbouts | Unsupported transition metal sulfide catalysts: From fundamentals to industrial application[END_REF][START_REF] Hermann | Activities of unsupported second transition series metal sulfides for hydrodesulfurization of sterically hindered 4 , 6-dimethyldibenzothiophene and of unsubstituted dibenzothiophene[END_REF][START_REF] Ho | Hydrogenation of Aromatics over Hydrodenitrogenation-Selective and Hydrodesulfurization-Selective Catalysts[END_REF][START_REF] Ho | Hydrodenitrogenation-Selective Catalysts: I. Fe Promoted Mo/W Sulfides[END_REF][START_REF] Ho | Deep HDS of diesel fuel: chemistry and catalysis[END_REF][START_REF] Ho | Promotion effects in bulk metal sulfide catalysts[END_REF][START_REF] Yue | Applied Catalysis A : General Size-dependent activity of unsupported Co -Mo sulfide catalysts for the hydrodesulfurization of dibenzothiophene[END_REF][START_REF] Wang | Preparation of Unsupported NiMoP Catalysts for 4,6-Dimethyldibenzothiophene Hydrodesulfurization[END_REF][START_REF] Lewandowski | Applied Catalysis B : Environmental Atomic level characterization and sulfur resistance of unsupported W 2 C during dibenzothiophene hydrodesulfurization[END_REF] . This category of catalysts can easily be obtained by thermal decomposition [START_REF] Bai | Preaparation of bulk metallic group VIII/group VIB metal catalysts[END_REF][START_REF] Soled | Hydroprocessing using bulk bimetallic catalysts[END_REF] , metal amines metallates [START_REF] Ho | Deep HDS of diesel fuel: chemistry and catalysis[END_REF] , hydrothermal [START_REF] Paraguay-Delgado | Synthesis of Ni-Mo-W Sulfide Nanorods as Catalyst for Hydrodesulfurization of Dibenzothiophene[END_REF] , citrate [START_REF] Maione | Solid solutions of Ni and Co molybdates in silica-dispersed and bulk catalysts prepared by solgel and citrate methods[END_REF] , freeze-drying [START_REF] Vie | Freeze-Dried Precursor-Based Synthesis of Nanostructured Cobalt-Nickel Molybdates Co 1 -x Ni x MoO 4[END_REF] and impregnation [START_REF] Olivas | Trimetallic NiMoW unsupported catalysts for HDS[END_REF] for the most classical methods. However, their main weakness compared to supported catalysts is their low specific surface area (SBET), down to 10 to 80 m².g -1 for most of the unsupported catalysts [START_REF] Ho | Deep HDS of diesel fuel: chemistry and catalysis[END_REF][START_REF] Bai | Preaparation of bulk metallic group VIII/group VIB metal catalysts[END_REF][START_REF] Olivas | Trimetallic NiMoW unsupported catalysts for HDS[END_REF][START_REF] Brito | Effect of Phase Composition of the Oxidic Precursor on the HDS Activity of the Sulfided Molybdates of Fe (II), Co (II), and Ni (II)[END_REF][START_REF] Brito | Nickel molybdate as precursor of HDS catalysts : effect of phase composition[END_REF][29] . However recent works have reported maximum values for SBET around 200 to 300 m².g - 1 30,31 , similar than those presented by alumina supported catalysts (200 to 300 m².g -1 ) [START_REF] Bej | Search for an efficient 4,6-DMDBT Hydrodesulfurization catalyst : A Review of Recent Studies[END_REF] .

Increasing SBET appears to be a very interesting challenge in the development of this new generation of catalysts but most of the time, high SBET unsupported catalysts synthesis generate toxic compounds and therefore are difficult to implement as industrial scale [START_REF] Li | Research Progress of Unsupported Nano Catalyst[END_REF] . In order to do so, supercritical fluids technologies are known to be efficient to prepare highly dispersed nanostructures [START_REF] Motos-Pérez | Preparation of Nickel Phosphide Hydrodesulfurization Catalysts Assisted by Supercritical Carbon Dioxide[END_REF] and nanopowders with high SBET [START_REF] Slostowski | Effect of Thermal Treatment on the Textural Properties of CeO2 Powders Synthesized in Near-and Supercritical Alcohols[END_REF][START_REF] Cansell | Design of functional nanostructured materials using supercritical fluids[END_REF] without generation of toxic compounds. In this work we propose the use of green solvents (water/alcohols), inexpensive (ten times cheaper than the ones use in conventional methods for high SBET materials [START_REF] Li | Research Progress of Unsupported Nano Catalyst[END_REF] ) and friendly precursors. Moreover, our approach allows the synthesis of crystalline materials with controlled and adjustable composition [START_REF] Philippot | Coupling in situ synchrotron radiation with ex situ spectroscopy characterizations to study the formation of Ba1-xSrxTiO3 nanoparticles in supercritical fluids[END_REF][START_REF] Reverón | Supercritical Fluid Route for Synthesizing Crystalline Barium Strontium Titanate Nanoparticles[END_REF][START_REF] Reverón | Single-step synthesis of well-crystallized and pure barium titanate nanoparticles in supercritical fluids[END_REF][START_REF] Philippot | Continuous BaTi1-yZryO3 (0≤y≤1) nanocrystals synthesis in supercritical fluids for nanostructured lead-free ferroelectric ceramics[END_REF] but also with a tuneable specific surface area [START_REF] Slostowski | Effect of Thermal Treatment on the Textural Properties of CeO2 Powders Synthesized in Near-and Supercritical Alcohols[END_REF] by playing with the carbon length of the alcohol used. These materials are produced in a fast, sustainable and continuous process which is a major asset for upscaling [START_REF] Adschiri | Green materials synthesis with supercritical water[END_REF] . All these advantages are obviously linked to the good activities of our materials in deep-HDS. Our catalyst precursors have shown a similar activity in removing the most refractory compound (4,6-DMDBT) to those of a reference catalyst but to a lower reaction temperature of 40°C. As far as we know, the use of this method to synthesize Ni(Co)MoO4 has not been so far reported in the literature.

Experimental

Materials and methods

Bis(acetylacetonato)dioxomolybdenum(VI) (MoO2(CH3COCHCOCH3)2, Alfa Aesar, 99%), nickel(II) acetate tetrahydrate (Ni(CH3COO)2.4H2O, Alfa Aesar, 99%) and cobalt(II) acetate tetrahydrate (Co(CH3COO)2.4H2O, Alfa Aesar, 98%) were used without further purification.

Synthesis procedures

The starting Ni(Co)-and Mo-containing solutions were prepared as follows: Mo salt was dissolved in a water/alcohol mixture of molar ratio 1:1 and Ni(Co) precursor was previously dissolved in distilled water before adding the same quantity of alcohol in order to prevent the precipitation due to alcohol aversion of the acetate. The alcohols used in this study were methanol, ethanol and isopropanol. Molybdenum concentration was fixed at [Mo]=7.2.10 -3 mol.L -1 and Ni concentration was adjusted to vary the metal molar ratios (36 -50 -57 -60 -66 at%). For cobalt, the concentration used was equal to the one of Mo. Those solutions of precursors were then injected in a continuous reactor, where they reached supercritical conditions, typically for a residence time of 55s [START_REF] Reverón | Supercritical Fluid Route for Synthesizing Crystalline Barium Strontium Titanate Nanoparticles[END_REF][START_REF] Reverón | Single-step synthesis of well-crystallized and pure barium titanate nanoparticles in supercritical fluids[END_REF] . A chemical reaction takes place leading to the nucleation and growth of the NiMo or CoMo nanoparticles. These last ones followed their way to a filter deepened in an ice bath to quench their growth. The obtained powder was washed with distilled water, recovered by vacuum filtration and dried at room temperature before a thermal treatment at 400°C for 3 hours.

Characterization techniques

Metal ratios were determined by Inductively Coupled Plasma/Optical Emission Spectrometry (ICP-OES) using a Varian 720-ES equipment. Carbon pollutions were detected using CHNS-O analysis, accomplished by combustion analysis according to the Pregl-Dumas procedure. The equipment used was a Flash EH 1112 from Thermo Fisher.

X-Ray powder diffraction (XRD) patterns were obtained from a Philips PW1820 and a PANalytical X´Pert X-Ray diffractometers in Bragg-Brentano geometry equipped with Cu Kα1 source and an X´Celerator detector. Diffraction patterns were collected with a 2θ step of 0.02° and a 0.0358°.s -1 scan speed routine mode; for more advanced investigations a 0.008° step and a 0.0053°.s -1 scan speed were used on a PANalytical X´Pert pro MOO equipped with primary Ge (111) monochromator Cu Kα1 source. The cell fitting, and thus the cell volume, were obtained by profile fitting of the pattern with the FULLPROF program. The fits were performed using a pseudo-Voigt peal-shape function. In the final runs, the usual profile parameters (scale factors, background coefficients, zero point, half-widths, pseudo-Voigt and asymmetry parameters for the peak shape) were refined.

The morphology of the powders was observed using a scanning electron microscope (JEOL 6700F) operating at an accelerating voltage of 30 kV. The size of the nanoparticles was evaluated using a high-resolution transmission electron microscope (JEOL JEM-2200FS) enable to switch to a scanning mode to make chemical X-Ray mapping of nanoparticles.

Physisorption measurements were performed using an Autosorb-1 instrument from Quantachrome. The BET specific areas were determined by N2 adsorption at 77K assuming a cross-sectional area of 0.162 nm² for the nitrogen molecule. Prior to adsorption measurements, the samples were outgassed in a vacuum at room temperature for 2 hours.

Catalytic tests

The model feedstock used was composed of 60 wt% of paraffins, 30 wt% of olefins and 10 wt% of aromatic hydrocarbons, in order to deal with a possible "matrix effect". The detailed composition can be found in Table S3 enclosed in S.I. The total sulfur content was 500 wppm with 100 wppm as 4,6-DMDBT, the most refractory compound. No nitrogen compounds were added to the feedstock in order to avoid the competition with HDN.

The activity tests were conducted in a fixed bed tubular reactor. The reactor was charged with 160 μL of catalyst (pelletized at a diameter comprised between 0.250 μm and 0.425 μm) and diluted in SiC (diluent/catalyst volume ratio of 12). The activation of the catalyst was done in situ using a mixture of H2S/H2 with 10 vol% H2S at atmospheric pressure and a temperature of 400°C for 3 h. The reaction conditions were the following: total pressure of 3.5 MPa, temperature of reaction between 320 and 400°C, Liquid Hourly Space Velocity (LHSV) of 39 h -1 and a ratio H2/feed of 5. For each test, liquid samples were accumulated, weighted and analyzed at intervals of about 40 min. For all catalysts, a constant composition of the reactor outlet stream was observed after reaction periods of 3 to 4 h. Results reported in this work correspond to the steady-state period of operation. The composition of feed and products was analyzed by gas chromatography in a Varian 3800 equipped with a 30 m column (VF 5ms CP8944) with an inner diameter of 0.25 mm and a bonded stationary phase made of a 0.25 μm thick film of (5 % phenyl, 95 % dimethyl) polysiloxane, and two detectors, a Flame Ionization Detector and a sulfur specific Pulsed Flame Photometric Detector.

Results and discussion

NiMo based catalyst precursors for deep HDS NiMoO4 may exist in several polymorphs at atmospheric pressure: the low temperature stable α-phase [START_REF] Smith | The crystal structures of cobalt molybdate CoMoO 4 and nickel molybdate NiMoO 4[END_REF] , the high temperature stable β-phase [START_REF] Abrahams | Crystal Structure of the Transition-Metal Molybdates. I. Paramagnetic Alpha-MnMoO4[END_REF] and the hydrate NiMoO4.nH2O (n=0,75, 1) phase [START_REF] Rodriguez | Electronic Properties and Phase Transformations in CoMoO 4 and NiMoO 4 : XANES and Time-Resolved Synchrotron XRD Studies[END_REF][START_REF] Rodriguez | Reaction of H2 and H2S with CoMoO4 and NiMoO4: TPR, XANES, Time-Resolved XRD, and Molecular-Orbital Studies[END_REF][START_REF] Ding | General synthesis and phase control of metal molybdate hydrates MMoO4.nH2O (M = Co, Ni, Mn, n = 0, 3/4, 1) nano/microcrystals by a hydrothermal approach: magnetic, photocatalytic, and electrochemical properties[END_REF][START_REF] Eda | crystal structure, and structural conversion of Ni molybdate hydrate NiMoO4•nH2O[END_REF] . According to the literature, the pure β-phase is generated by heating the α-NiMoO4 at temperatures above 690°C and is stable only if kept over 180°C [START_REF] Brito | Nickel molybdate as precursor of HDS catalysts : effect of phase composition[END_REF][START_REF] How | THERMAL TREATMENT INFLUENCES THE PHASE TRANSITION OF NiMo04[END_REF] .

Both phases (α and β) are monoclinic with space group C2/m 48 . The structural building units of α-NiMoO4 (isotypic with α-CoMoO4 [START_REF] Smith | The crystal structures of cobalt molybdate CoMoO 4 and nickel molybdate NiMoO 4[END_REF] ) are NiO6 and MoO6 distorted octahedra, which share edges and form chains parallel to the "c" direction. Each chain is surrounded by four other chains and linked to them by corner sharing of oxygen atoms. Between the filled chains there are chains of unfilled octahedra. It also presents alternative layers of NiMo6 and MoO6 parallel to the "b" direction. The structure of β-NiMoO4 is isotypic with α-MnMoO4 (ICSD #78328). By analogy, it consists of molybdate distorted tetrahedra, which share corners with four different nickelate octahedra. The general description is similar to the α-phase with chains joined by corner sharing of oxygen atoms and with chains of unfilled octahedral between them. The difference between the two phases is a contraction in the "c" axis and an expansion in the "a" and "b" axes of the cell in the β-form compared to the α-form. The formal oxidation state of Ni is +II, whereas the one of Mo is +VI as in MoO3 [START_REF] Rodriguez | Reaction of H2 and H2S with CoMoO4 and NiMoO4: TPR, XANES, Time-Resolved XRD, and Molecular-Orbital Studies[END_REF] .

Recent studies give a description of the hydrated phase NiMoO4.nH2O with n = 1 or 0.75 [START_REF] Eda | crystal structure, and structural conversion of Ni molybdate hydrate NiMoO4•nH2O[END_REF] . Measurements of differential thermal analysis (DTA) suggest that in these systems the metal cations have a coordination similar to that observed in β-NiMoO4. The NiMoO4.0.75H2O structure has been described as a triclinic system (space group P-1) with tetrameric z-shaped unit of Ni octahedra, two NiO6 and two NiO5(OH2), that share edges. These units are interconnected by MoO4 tetrahedra to form a network structure with open [START_REF] Eda | crystal structure, and structural conversion of Ni molybdate hydrate NiMoO4•nH2O[END_REF] . NiMoO4.H2O might present a similar structure with lattice and coordination water. The HDS activity of these systems increased in the following order: α < β < hydrate [START_REF] Brito | Effect of Phase Composition of the Oxidic Precursor on the HDS Activity of the Sulfided Molybdates of Fe (II), Co (II), and Ni (II)[END_REF][START_REF] Brito | Nickel molybdate as precursor of HDS catalysts : effect of phase composition[END_REF] .

The TGA-MS of the powders obtained directly after synthesis in supercritical water/alcohol system displays three main mass losses for a total mass loss of 8 wt% (see Figure S1 enclosed in S.I.). A first loss of 2.8 wt% between room temperature and 243 °C can be attributed to adsorbed water as shown in the mass spectrometry results (line of mass 18). The second mass loss of 4.20 wt% occurring from 243 °C to 468 °C can be attributed this time to the departure of structural water and some carbon according to the line shape of masses 18 and 44 (CHNS-O analyses give a mean of 0.4 wt% carbon in the powders) as for the third mass loss of 0.5 wt% between 468°C and 538°C. The last mass loss which occurred between 538°C and 800°C seems to be exclusively due to the carbon departure. Those water losses are in good agreement with results obtained by Rodriguez et al. [START_REF] Rodriguez | Electronic Properties and Phase Transformations in CoMoO 4 and NiMoO 4 : XANES and Time-Resolved Synchrotron XRD Studies[END_REF] and Eda et al. [START_REF] Eda | crystal structure, and structural conversion of Ni molybdate hydrate NiMoO4•nH2O[END_REF] on their study of NiMoO4.nH2O (n = 0.75 or 1). Upon heating NiMoO4.nH2O to high temperatures in TGA-MS experiments, they also observed that water desorbs in three peaks. The first at 100-200 °C is associated with water molecules reversibly bounded to the hydrate. The second from 200 to 400 °C corresponds to desorption of water molecules that form part of the hydrate crystal structure, and they also assessed that at the onset of this peak, the formation of β-NiMoO4 starts. They pointed out an additional small desorption peak between 490 °C and 520 °C, also attributed to water. Eda considers that the ideal hydrate is NiMoO4.0.75H2O, and based on the mass loss obtained with our TGA, we may approximate the same value for our material.

The XRD patterns shown in Figure 1 are representative of the products resulting from the synthesis in supercritical water/alcohol system ((a) isopropanol, (b) ethanol). Most of the peaks can be assigned to the above mentioned hydrate nickel molybdate NiMoO4.xH2O (JCPDS N° 13-0128) (no defined value for x, structure represented in Figure 1); this is consistent with the assumption made previously. The presence of MoO3.H2O (JCPDS N° 01-0125) and MoO2 (JCPDS N° 32-0671) is also detected. This is due to an excess of Mo in those specific samples. More surprisingly, β-NiMoO4 (JCPDS N° 45-0142) is also detected despite the fact that it is not stable at room temperature. During thermal treatment (400 °C, 3 h, air atmosphere), the most remarkable features observed on XRD patterns acquired in temperature are the systematic transition of the hydrate NiMoO4.0.75H2O into β-NiMoO4 at 400 °C while heating and the systematic transition into α-NiMoO4 (JCPDS N° 86-0361) at 200 °C when cooling down. The nature of these two phases has been assessed by Rietveld refinement with fairly good refinement parameters (β: Bragg R-factor = 4.96, χ 2 = 2.38, cRwp = 14.3; α: Bragg R-factor = 7.35, χ 2 = 4.11, cRwp = 14.2, see Table S1 and Table S2 enclosed in S.I.). The SEM micrographs of the as-obtained powders and calcined powders (400 °C, 3 h) are shown in Figure 2.a) (synthesis carried out in ethanol). The as-obtained powders mainly consist of rod-like NiMoO4.0.75H2O particles of dispersed sizes. After calcination the rods have collapsed and somehow sintered. The most interesting result obtained concerns the evolution of the specific surface area as a function of the alcohol nature as illustrated in Figure 2.b. The nature of the alcohol used plays a major role on the specific surface area, that increases from a value of 79 m 2 .g -1 using methanol as solvent up to 179 m 2 .g -1 using isopropanol as solvent. This result is fully consistent with a previous study on the formation of CeO2 nanocrystals in supercritical alcohols [START_REF] Slostowski | Effect of Thermal Treatment on the Textural Properties of CeO2 Powders Synthesized in Near-and Supercritical Alcohols[END_REF] . CeO2 nanopowders exhibit specific surface areas of 60, 106 and 140 m 2 .g -1 when prepared in pure methanol, ethanol and isopropanol, respectively. The same trend is observed for NiMo synthesis in the H2O/ROH systems. This means that the alcohol is playing the role of solvent with water but also of functionalization agent as demonstrated for CeO2 [START_REF] Slostowski | Near-and Supercritical Alcohols as Solvents and Surface Modifiers for the Continuous Synthesis of Cerium Oxide Nanoparticles[END_REF] with a direct impact on the catalyst precursor thermal stability [START_REF] Slostowski | Effect of Thermal Treatment on the Textural Properties of CeO2 Powders Synthesized in Near-and Supercritical Alcohols[END_REF] . XRD patterns of calcined powders (3 h at 400 °C) with a different cations ratio are displayed in Figure 3. Two trends arise: for Ni/Mo < 1, we obtain the α-NiMoO4 phase together with MoO3.2H2O phase (XRD pattern of the powder with 64 at% of Mo) and for Ni/Mo > 1, we obtain both α-and β-phases of NiMoO4 together with NiO.

The STEM-EDX chemical mapping associated with these two compositions illustrates this fact showing the presence of particles containing only Mo for a) and particularly rich in Ni for b). These results are in agreement with studies carried out by Di Renzo et al. [START_REF] Di Renzo | FORMATION AND PROPERTIES OF THE SOLID SOLUTION OF NiO IN NiMoO4[END_REF][START_REF] Mazzocchia | Stability of alpha-nickel molybdate[END_REF] concerning the stabilization of β-NiMoO4 at room temperature with an excess of nickel. According to their work, only the α-phase can be as a stoichiometric compound whereas the β-phases can form solid solutions with Ni by insertion in localized vacant sites in the network of β-NiMoO4, thus stabilizing its structure. This effect is not observed with Mo assumingly due to an obvious matter of steric hindrance. For most of the synthesized samples, the hydrate nickel molybdate phase was obtained with another phase depending of the stoichiometric ratio of each element.

CoMo catalyst precursors for deep HDS

Since our main objective concerns deep-HDS, CoMoO4 was not the privileged material since it is less efficient than NiMoO4. However, it was interesting to study the application of our fast (55s) and continuous synthesis method on CoMoO4. CoMoO4 can adopt two different polymorphic forms, α and β [START_REF] Ponceblanc | Solid-Solid Phase Equilibria In the Binary System CoMo04-FeMo04 and Effect of Fe(III) on the Phase Equilibria[END_REF] . The β-CoMoO4 is the high temperature stable phase. Like in NiMoO4, the structure of the β form is monoclinic (space group C2/m) with 6 and 4 as coordination numbers, respectively for Co(II) and Mo(VI). The structure of the α-CoMoO4 is also monoclinic with the same space group. It is more compact and differs from β-CoMoO4 in the coordination of Mo(VI) which is 6 in this case. The difference between the two phases again lies is a contraction in the "c" axis and an expansion in the "a" and "b" axes of the cell.

The XRD pattern acquired directly after reaction fits with a mixture of the two different structures of CoMoO4 (Figure 4.a). The β-isomorph is clearly the majority phase. Nothing surprising since it has already been proved that, unlike β-NiMoO4, which is unstable below 180 °C, β-CoMoO4 can be easily stabilized at

Catalytic tests

The catalytic behaviour of these materials was evaluated by comparison with a commercial supported catalyst, NM, used as reference (200 m 2 .g -1 , 4 wt% NiO, 14 wt% MoO3/γ-Al2O3).

In order to study the influence of the composition in our unsupported catalysts, three different compositions with a ratio Ni/Mo > 1 were tested and compared to the reference supported catalyst. First, the materials with a Ni atomic fraction of 54 at% (MT044) and of 62 at% (MT072) were tested at a LHSV of 39 h -1 and results are presented in Figure 5.a. In a second series, the materials with a Ni atomic fraction of 69 at% (MT043) and of 62 at% (MT072) were tested at a LHSV of 150 h -1 and results are presented in Figure 5.b. 4,6-DMDBT conversion is presented in absolute values (top) and normalized considering the total surface area (bottom) in order to focus on the effect of the composition and minimizing the contribution of the specific surface area.

The results show that our bulk materials are significantly more active than the commercial reference when we directly compare the 4,6-DMDBT conversion obtained with catalysts MT044 and MT072 at our lowest operation temperature of 320ºC (see Figure 5-a, top). The DMDBT conversion values obtained with these unsupported catalysts at 320ºC are comparable to the conversion given by the NiMo/Al2O3 (NM) at 360ºC. Thus, for similar conversion and similar amount of catalyst, the unsupported nanostructured catalysts can operate at lower temperature than NM and therefore, for similar conversion under similar operating conditions, a lower mass of our catalyst would be required to achieve the same conversion levels of the commercial reference. The lower activity of NM at lower temperature becomes even more evident when the results are given as surface area activities, normalized according to the BET surface area of the catalysts (see Figure 5-a, bottom). In previous publications [START_REF] Hermann | Activities of unsupported second transition series metal sulfides for hydrodesulfurization of sterically hindered 4 , 6-dimethyldibenzothiophene and of unsubstituted dibenzothiophene[END_REF] , the activity of unsupported HDS catalysts was only comparable to that of commercial Al2O3 supported catalysts, and the latter was surpassed only when normalized to the catalyst surface area. Eijsbouts et al. [START_REF] Eijsbouts | Unsupported transition metal sulfide catalysts: From fundamentals to industrial application[END_REF] have shown that addition of unsupported transition metal sulfide catalysts (Nebula® belonging to the NEBULA catalyst family developed by Albemarle Catalysts and ExxonMobil) to a conventional HDS catalyst allows decreasing their ULSD operation temperature in 15ºC. It must be noted that the NEBULA Technology is being used commercially since 2001. Unsupported Ni-W-Mo catalysts were shown recently to be more active than a traditional supported reference catalyst for HDS of a straight-run gas oil [START_REF] Zhang | Study on high-performance unsupported Ni-Mo-W hydrotreating catalyst[END_REF] . However, comparison with our results is not straightforward, as these catalysts were diluted with an alumina binder, and catalytic tests were performed at a considerably lower space velocity (LHSV=2 h -1 , instead of 39-150 h -1 used in this work).

Secondly, combining results of experiments done at a LHSV of 39 h -1 and at a LHSV of 150 h -1 , the trend of activity is the following: MT072 (62 at%Ni) > MT043 (69 at%Ni) and MT072 (62 at%Ni) > MT042 (54 at%Ni). This suggests that an optimum atomic percentage of Ni exits (around 60 at% Ni) for which the conversion of 4,6-DMDBT is maximum. We think that the higher the Ni content in the material, the higher the β-NiMoO4 proportion phase in our unsupported HDS catalyst precursor. As known, when sulfided, the β-NiMoO4 is the most active phase in HDS. Moreover, it is known that the active sites are located on the MoS2 layers whereas Ni has a promoting effect. An excess of Ni will cover the MoS2 surface and may reduce the accessibility of sulfur compounds to the active sites. Thus, there should be a maximum Ni content (comprised between 54 at% and 69 at% of Ni) for which the formation of the β-NiMoO4 phase is maximized without diminishing the number of active sites. An optimum Ni/(Mo+Ni) mole ratio of 0.5 was also found for unsupported NiMo sulfide catalysts prepared by means of hydrothermal synthesis [START_REF] Yoosuk | Effects of preparation conditions in hydrothermal synthesis of highly active unsupported NiMo sulfide catalysts for simultaneous hydrodesulfurization of dibenzothiophene and 4,6-dimethyldibenzothiophene[END_REF] . The sample with this composition gave maximum DBT and 4,6-DMDBT conversion at 350ºC under batch conditions. Our samples have been tested under HDS conditions for deep-HDS of a complex model feedstock in order to study their efficiency of removing organosulfur molecules. They have shown significantly higher conversion as compared to a commercial reference at low temperatures (320ºC). This could be translated into reduced energetics or catalyst related costs when using the bulk precursors obtained by continuous supercritical synthesis at conversion levels comparable to those obtained with conventional supported catalysts. It has also been shown that a maximum of activity exists for a Ni content around 60 at%. Finally, tests were carried out on an industrial Light Cycled Oil (LCO) feedstock (S content = 5200 wppm, more details are given in S.I.) in order to evaluate the behaviour of our material under more realistic deep-HDS conditions. The preliminary results obtained at a pressure of 3.5 MPa, LHSV of 150 h -1 and MG/ML of 5, are presented in Figure 6. Even though experimental conditions were not fully optimized, this first set of experiments demonstrates the capability of the new materials described in this work for the desulfurization of an industrial feedstock and evidences their potential as deep-HDS bulk catalysts.

Conclusions

This paper proposes a novel, fast and continuous method to produce high specific surface area deep HDS unsupported catalyst precursors with controlled composition using supercritical water/alcohols mixtures. Furthermore, this method provides an exceptional opportunity to produce nanostructured catalyst precursors in an environmentally friendly medium with high production rate and possible scale up at an industrial scale. This study describes the synthesis of high surface area materials Ni-Mo and Co-Mo based oxides in supercritical fluids. The most active phases in HDS (NiMoO4.0.75H2O and β-NiMoO4), according to the literature, have been obtained and stabilized at room temperature. It has been shown that the nature of the alcohol plays a major role in the evolution of surface area.

Catalytic tests have shown better conversion of 4,6-DMDBT than the commercial reference and the deep-HDS reaction can be performed at lower temperature when using these nanostructured unsupported catalysts. It has also been demonstrated that there is an optimal value for Ni concentration where the activity of the catalyst is the maximum.

Figure 1 :

 1 Figure 1: XRD patterns of a stoichiometric "NiMo" based powder synthesized in a) scH2O/iPrOH and b) scH2O/EtOH. Above is the structure of NiMoO4.0.75H2O (from Eda et al.37)

Figure 2 :

 2 Figure 2: a) SEM micrographs of as obtained powder synthesized in H2O/EtOH mixture on the left and the same powder treated for 3 h at 400°C on the right and b) Evolution of the catalyst precursor SBET as a function of the alcohol nature used in the supercritical solvent before thermal treatment.

Figure 3 :

 3 Figure 3: XRD patterns and their associated SEM-EDX chemical mappings of "NiMo" based powder synthesized in supercritical H2O/iPrOH mixture with a) an excess of Mo and b) an excess of Ni. Powders are calcined 3 h at 400°C

  temperature[START_REF] Ponceblanc | Solid-Solid Phase Equilibria In the Binary System CoMo04-FeMo04 and Effect of Fe(III) on the Phase Equilibria[END_REF] . As the XRD patterns also shown (Figure4.b), the α-phase almost disappears after thermal treatment at 400 °C for 3 h.

Figure 4 :

 4 Figure 4: XRD patterns of stoichiometric "CoMo" based powders synthesized in scH2O/iPrOH mixture with a) the obtained powder and b) after calcination for 3 h at 400°C.

Figure 5 :

 5 Figure 5: a) Conversion of 4,6-DMDBT in a model feedstock (see Experimental part, LHSV=39h -1 ), b) Conversion of 4,6-DMDBT in a model feedstock (see Experimental part, LHSV=150 h -1 )

Figure 6 :

 6 Figure 6: Total sulfur conversion by deep-HDS of an industrial feedstock (P=3.5 MPa, LHSV = 150 h-1, MG/ML = 5)

Acknowledgements

This work was supported by the CNRS and UPV. Dr M. Théodet is grateful to the French Ministry of Education, Research and Technology (MENRT) for a doctoral grant. The authors acknowledge Multifunctional Materials Institute (EMMI). Dr S. Pechev is also acknowledged for his priceless help in interpreting the XRD patterns. Financial support by the Spanish Government-MINECO through ''Severo Ochoa" (SEV 2012-0267), Consolider Ingenio 2010-Multicat, CTQ2015-70126-R, and by the European Union through ERC-AdG-2014-671093 -SynCatMatch is acknowledged.

Notes and references