A Abad

M Arias

P Ravelojaona
email: paola.ravelojaona@icn-artem.com

Environmental Productivity Assessment: an Illustration with the Ecuadorian Oil Industry

Keywords: Data Envelopment Analysis (DEA), Ecuadorian Oil Industry, Environmental Efficiency, Productivity Indices, Non Convexity, Pollution-generating Technology C61, D24, Q50

In this paper, environmental productivity change is analysed through the production theoretic approach to index numbers. Specifically, pollution-adjusted Malmquist and Hicks-Moorsteen productivity indices are considered. These productivity indices are defined as combination of multiplicative distance functions. Non convex pollution-generating technology is assumed to estimate the pollutionadjusted Malmquist and Hicks-Moorsteen productivity measures. Moreover, the main sources of the environmental productivity change are displayed. An empirical illustration is provided by considering a sample of 20 Ecuadorian oil companies over the period 2014-2018. The results are estimated through a non parametric analytic framework.

Introduction

Oil represents 32 percent of the global energy consumption sources [START_REF]World Energy Outlook 2019[END_REF]. The World Energy Outlook [START_REF]World Energy Outlook 2019[END_REF], claims that energy generated from fossil fuels will remain the major source and is still expected to meet about 84 percent of energy demand in 2030. According to British Petroleum [START_REF]Statistics Report of British Petroleum of 2020[END_REF], South and Central America have 18.7 percent (324.1 thousand of millions of barrels) of the world's proven reserves. In terms of production, Ecuador is the fifth oil producer in South America with an average production of 27.94 million of tons from 2009 to 2019. There is research into other reliable energy resources to replace fossil fuel, considering its depletion and the environmental impacts generated by this industry. However, it is expected that the energy market will continue to depend on fossil fuels for at least the next few decades.

Among all industry sectors, the petroleum industry is of particular interest to Ecuador because of its economic and environmental significance. Oil is the second most important sector for the Ecuadorian economy. The contribution of the oil sector was 11 percent of the Gross Domestic Product (GDP) for the period 2011-2018 approximately. Oil is also important for the Ecuadorian energy sector; in 2018, there was a primary energy production of 216 million Barrels of Oil Equivalent (BOE). Of the total produced, 86.9 percent was made up of oil. According to the Third National Communication on Climate Change and First Biennial Update Report [START_REF]Third National Communication on Climate Change and First Biennial Update Report[END_REF], the energy sector produced 37 594 Gg of carbon dioxide equivalent (CO 2 e) which represents 47 percent of total GreenHouse Gas (GHG) emissions in 2012. Energy industry is a significant contributor of GHG emissions in the country, especially for the burning of fossil fuels. In 2012 this activity accounted for 36 822.54 Gg (CO 2 e) which represents the 97.95 percent of emissions of the energy sector. Thus, oil companies need to be more efficient and make a balance between pollution mitigation and economic success.

The performance change assessment of Ecuadorian petroleum companies needs to consider a methodology integrating the companies' environmental indicators with their operational measures. Following the production theoretic approach to index numbers, the performance variation is commonly assessed through productivity indices [START_REF] Prasada Rao | Index Numbers and Productivity Measurement[END_REF]. In this paper, environmental productivity is appraised through the non convex Pollutionadjusted Malmquist (PM) and Hicks-Moorsteen (PHM) productivity indices [START_REF] Abad | A Generalization of Environmental Productivity Analysis[END_REF][START_REF] Abad | Pollution-adjusted Productivity Analysis: The Use of Malmquist and Luenberger Productivity Measures[END_REF]. The PM productivity measure takes the form of the Malmquist index [START_REF] Färe | Productivity Developments in Swedish Hospitals: A Malmquist Output Index Approach[END_REF][START_REF] Caves | The Economic Theory of Index Numbers and the Measurement of Inputs, Outputs and Productivity[END_REF] whilst the PHM productivity measure inherits the structure of the Hicks-Moorsteen index [START_REF] Bjurek | The Malmquist Total Factor Productivity Index[END_REF]. Specifically, the PM and the PHM productivity indices display the change of economic and polluting outputs induced by inputs variation by relaxing the convexity property of pollution-generating technology. It is worth noting that pollution-generating processes encompass many human and ecological interactions that can induce non linearities [START_REF] Tschirhart | Biology as a Source of Non-convexities in Ecological Production Functions[END_REF][START_REF] Dasgupta | The Economics of Non-Convex Ecosystems: Introduction[END_REF] 1 . Although relaxing the convexity property of production technologies has been investigated in the literature [START_REF] Briec | Nonconvexity in Production Economics and Support Function: An exploratory Review, in Production and Cost Functions: Exploratory and Selective Review[END_REF], few studies consider non convex pollution-generating technologies. The only theoretical model that considers non convex pollution-generating processes has been introduced in Abad and Briec [START_REF] Abad | On the Axiomatic of Pollution-generating Technologies: a Non-Parametric Approach[END_REF]. This model provides axiomatic foundation of the non convex version of the Murty et al.'s by-production model [START_REF] Yuan | Economic and environmental performance of the belt and road countries under convex and nonconvex production technologies[END_REF][START_REF] Murty | On Modeling Pollution-Generating Technologies[END_REF]. Knowing the prominent drivers of productivity change is a major concern in applied economics literature [START_REF] Becerra-Peña | Measuring public primary education productivity across Mexican states using a Hicks-Moorsteen index[END_REF][START_REF] Miao | Atmospheric environmental productivity across the provinces of China: Joint decomposition of range adjusted measure and Luenberger productivity indicator[END_REF][START_REF] Shen | Aggregate green productivity growth in OECD's countries[END_REF]. As a result, this paper displays the main components of the pollution-adjusted productivity variation by considering the case of the Ecuadorian oil industry. Moreover, the main sources of environmental productivity change are highlighted by separating polluting and non polluting dimensions.

Recently, numerous papers investigate environmental efficiency and productivity variation of the oil sector [START_REF] Veronese Da Silva | Accounting multiple environmental variables in DEA energy transmission benchmarking modelling: The 2019 Brazilian case[END_REF][START_REF] Wegener | Minimizing Greenhouse Gas Emissions using Inverse DEA with an Application in Oil and Gas[END_REF][START_REF] Sueyoshi | DEA environmental assessment on US petroleum industry: Non-radial approach with translation invariance in time horizon[END_REF][START_REF] Sueyoshi | DEA environmental assessment in time horizon: Radial approach for Malmquist index measurement on petroleum companies[END_REF]. In this area, non parametric mathematical programming methods for production analysis are widely applied to appraise technical productivity variation2 [START_REF] Bansal | Malmquist-Luenberger productivity indexes for dynamic network DEA with undesirable outputs and negative data[END_REF][START_REF] Cui | Energy transition, trade and green productivity in advanced economies[END_REF][START_REF] Goto | Sustainable development and convergence under energy sector transition in industrial nations: An application of DEA environmental assessment[END_REF][START_REF] Sueyoshi | A literature study for DEA applied to energy and environment[END_REF][START_REF] Zhou | A survey of data envelopment analysis in energy and environmental studies[END_REF]. In this paper, non parametric production model is considered to highlight the practicability of the approach provided. Specifically, non convex Free Disposal Hull (FDH) by-production model is defined to estimate the PM and the PHM indices [START_REF] Abad | On the Axiomatic of Pollution-generating Technologies: a Non-Parametric Approach[END_REF][START_REF] Tulkens | On FDH Efficiency Analysis: some Methodological Issues and Applications to retail Banking, Courts and Urban Transit[END_REF]. Interestingly, this modelling does not need to explicitly specify the mathematical form of the production function which characterised the pollution-generating technologies. Moreover, it allows to assess the environmental productivity of multi-input and multi-output production units by relaxing the convexity property of the pollution-generating technologies. To the best of our knowledge, there has been no research performed in the field of oil industry that analyses pollution-adjusted productivity change based upon non convex FDH approximation of pollution-generating production model.

An empirical illustration is provided by considering a sample of 20 Ecuadorian private oil companies over the period 2014-2018. Almost all private companies are engaged in exploration and production, while only a relatively small fraction of firms participate in other activities such as transport and distribution. The outputs of the oil companies are separated into economic (i.e., desirable) and polluting (i.e., undesirable) components; number of oil barrels and CO 2 emissions, respectively. In the segment, CO 2 emissions are generated directly through drilling processes and fossil fuel combustion and indirectly through well leaks and venting.

The remainder of this paper is structured as follows. Section 2 displays the production theoretic framework allowing to define the PM and the PHM indices. The non parametric estimation of the pollution-generating technology and the environmental productivity indices are proposed in Section 3. The empirical illustration is provided in Section 4. Finally, Section 5 discusses and concludes.

Methodology

This section presents the theoretical basis that is considered in this paper. Based upon this theoretical framework, pollution-adjusted efficiency and productivity indices are introduced.

Pollution-generating process: definition and properties

Assume that the outputs are separated into economic (i.e., desirable) and polluting (i.e., undesirable) components 3 . Let I and O be the input and output sets such that

I := {x t ∈ R n + : n ∈ N * } and O := {y t = (y d t , y u t ) ∈ R m d +m u + : m d , m u ∈ N * }.
The input and output vectors for the period (t) are defined as (x t , y t ) ∈ R m+n

+

, where m = m d +m u . The pollution-generating technology is defined as follows,

T t := (x t , y t ) ∈ R n+m + : x t can produce (y d t , y u t ) . (2.1)
Usual characterisations of T t are the output set, P : R n + → 2 R m + , and the input correspondence, L : R m

+ → 2 R n + , P (x t ) := (y d t , y u t ) ∈ R m + : (x t , y t ) ∈ T t (2.2)
and

L(y d t , y u t ) := x t ∈ R n + : (x t , y t ) ∈ T t . (2.3)
In this paper, alternative characterisations of the pollution-generating process are considered through the undesirable set,

Q : R m d + → 2 R n+m u +
, and the desirable correspondence, Z : R

m u + → 2 R n+m d + , Q(y d t ) := (x t , y u t ) ∈ R m + : (x t , y t ) ∈ T t (2.4)
and

Z(y u t ) := (x t , y d t ) ∈ R m + : (x t , y t ) ∈ T t . (2.5)
The sets (2.4)and (2.5) restrict T t to the subspace of inputs and economic outputs and, to the subspace of inputs and polluting outputs, respectively [START_REF] Abad | An environmental generalised Luenberger-Hicks-Moorsteen productivity indicator and an environmental generalised Hicks-Moorsteen productivity index[END_REF].

In such case,

x t ∈ L(y d t , y u t ) (y d t , y u t ) ∈ P (x t ) (x t , y d t ) ∈ Z(y u t ) (x t , y u t ) ∈ Q(y d t )        ⇔ (x t , y t ) ∈ T t (2.6)
Assume that the pollution-generating technology satisfies the following usual properties [START_REF] Färe | The Measurement of Efficiency of Production[END_REF]:

A1: No free lunch and Inaction; (0, 0) ∈ T t , (0, y t ) ∈ T t ⇒ y t = 0. A2: Boundedness; T (y t ) = {(x t , v t ) ∈ T t : v t ≤ y t } is bounded for all y t ∈ R m + . A3: Closedness; T t is closed.
Let C be the convex cone such that: C := {y t ∈ R m : y u t ≤ 0 and y d t ≥ 0}. In addition of the traditional axioms A1 -A3, suppose that the pollution-generating process satisfies the B-disposal assumption [START_REF] Abad | On the Axiomatic of Pollution-generating Technologies: a Non-Parametric Approach[END_REF]:

A4: B-disposability; T t := T t + (R n + × -R m + ) ∩ T t + (R n + × -C) ∩ (R m + × R n + ).
The theoretical model based upon the properties A1 -A4 permits to define the pollution-generating process as an intersection of sub-technologies: [START_REF] Murty | On Modeling Pollution-Generating Technologies[END_REF]. The intended production activities of firms satisfy the usual strong disposability assumption; ie.,

T t + (R n + × -R m + ) ∩ (R m + × R n + ) and T t + (R n + × -C) ∩ (R m + × R n + )
T t + (R n + × -R m + ) ∩ (R m + × R n + )
. Moreover, partially reversed free disposal axiom applies for the polluting residuals generation; ie.,

T t + (R n + × -C) ∩ (R m + × R n + ).
It is worth noting that axioms A1 -A4 define a fairly weak axiomatic framework such that the convexity assumption is not required to define pollution-generating processes.

Pollution-adjusted efficiency and productivity indices

This section lays out the pollution-adjusted efficiency and productivity indices. Moreover, the main sources of the pollution-adjusted productivity change are highlighted by separating the polluting and non polluting dimensions.

Pollution-adjusted multiplicative distance function

The following definition presents the pollution-adjusted multiplicative distance function.

Definition 2.1 Let T t be a pollution-generating process that satisfies properties A1 -A4. For any

(x t , y t ) ∈ R n+m + , where y t = (y d t , y u t ) ∈ R m + , the multiplicative pollution- adjusted distance function, D φ : R n+m + -→ R ∪ ∞, is defined as follows : D φ t (x t , y t ) :=        inf β β ∈]0, 1] : β α x t , β γ d y d t , β γ u y u t ∈ T t if β α x t , β γ d y d t , β γ u y u t ∈ T t , β > 0 ∞ else (2.7
)

where φ = (α, γ d , γ u ) ∈ {0, 1} × {-1, 0} × {0, 1}.
Let us consider the following orientations for the pollution-adjusted distance function. ii. D 1,0,1 t (x t , y t ) ≡ D I,O u (x t , y t ).

iii. D 1,0,0 t (x t , y t ) ≡ D I (x t , y t ).

iv. D 0,-1,0

t (x t , y t ) ≡ D O d (x t , y t ). v. D 0,0,1 t (x t , y t ) ≡ D O u (x t , y t ).
The pollution-adjusted efficiency indices presented in the aforementioned results i.v. fully identify pollution-generating processes [START_REF] Abad | A Generalization of Environmental Productivity Analysis[END_REF][START_REF] Abad | Pollution-adjusted Productivity Analysis: The Use of Malmquist and Luenberger Productivity Measures[END_REF]. Moreover, remark that efficiency measures i. and ii. take the form of hyperbolic distance functions [START_REF] Färe | The Measurement of Efficiency of Production[END_REF] whereas iii., iv. and v. are the standard Shephard distance functions [START_REF] Shephard | Cost and Production Functions[END_REF].

In such case,

D I,O d (x t , y t ) ∈]0; 1] ⇔ (x t , y d t ) ∈ Z(y u t ) D I,O u (x t , y t ) ∈]0; 1] ⇔ (x t , y d t ) ∈ Q(y d t ) D I (x t , y t ) ∈]0; 1] ⇔ x t ∈ L(y d t , y u t ) D O d (x t , y t ) ∈]0; 1] ⇔ (y d t , y u t ) ∈ P (x t ) D O u (x t , y t ) ∈]0; 1] ⇔ (y d t , y u t ) ∈ P (x t )            ⇔ (x t , y t ) ∈ T t (2.8)

Malmquist pollution-adjusted productivity index

The next result displays the Malmquist pollution-adjusted productivity index [START_REF] Abad | Pollution-adjusted Productivity Analysis: The Use of Malmquist and Luenberger Productivity Measures[END_REF].

Definition 2.3 Suppose that T t is a pollution-generating process satisfying properties A1 -A4. For any (x t,t+1 , y t,t+1 ) ∈ R n+m

+

, where y t,t+1 = (y d t,t+1 , y u t,t+1 ) ∈ R m + , the Malmquist pollution-adjusted productivity index is defined as follows :

P M φ t,t+1 (x t,t+1 , y t,t+1 ) = P M φ t (x t,t+1 , y t,t+1 ) × P M φ t+1 (x t,t+1 , y t,t+1 ) 1 2
(2.9) If the multiplicative productivity index P M φ t,t+1 is larger than 1 then, pollution adjusted productivity gains arise over periods (t) and (t + 1). In such case, the firms produce more non polluting outputs and operate managerial efforts to reduce their inputs and polluting outputs. Remark that the pollution-adjusted Malmquist productivity index defined in this paper, is a hyperbolic-based Malmquist productivity measure 4 .

such that φ = (α, γ d , γ u ) ∈ {0, 1} × {-1, 0} × {0, 1}.
The following statement defines a decomposition of the Malmquist pollution-adjusted productivity variation by separating polluting and non polluting dimensions. Assume that the no polluting Malmquist productivity index P M d t,t+1 (x t,t+1 , y t,t+1 ) is greater than 1. In such case, more desirable outputs are produced and less inputs are used between the periods (t) and (t+1), for a given level of undesirable outputs. In the same vein, if the polluting Malmquist productivity index P M u t,t+1 (x t,t+1 , y t,t+1 ) is greater than unity then, less undesirable outputs are produced and less inputs are used between the periods (t) and (t+1), for a given level of desirable outputs. Obviously, reciprocal reasoning holds when P M d t,t+1 (x t,t+1 , y t,t+1 ) ≤ 1 and P M u t,t+1 (x t,t+1 , y t,t+1 ) ≤ 1. The prominent components of the Malmquist pollution-adjusted productivity variation are presented in the next result. Definition 2.5 Let T t be a pollution-generating technology satisfying assumptions A1-A4. For any (x t,t+1 , y t,t+1 ) ∈ R n+m + , where y t,t+1 = (y d t,t+1 , y u t,t+1 ) ∈ R m + , the decomposition of the Malmquist pollution-adjusted productivity change is defined as follows:

P M φ t,t+1 (x t,t+1 , y t,t+1 ) = EC φ t,t+1 × T C φ t,t+1 (2.13)
where EC φ t,t+1 and T C φ t,t+1 correspond to the efficiency variation and technological change components, respectively. EC φ t,t+1 and T C φ t,t+1 are laid out in the next results.

EC φ t,t+1 = EC d t,t+1 × EC u t,t+1 (2.14)
= D I,O d t+1 (x t+1 , y t+1 ) D I,O d t (x t , y t ) × D I,O u t+1 (x t+1 , y t+1 ) D I,O u t (x t , y t )
and

T C φ t,t+1 = T C d t,t+1 × T C u t,t+1 1 2 (2.15)
= D I,O d t (x t , y t ) D I,O d t+1 (x t , y d t , y u t+1 ) × D I,O d t (x t+1 , y d t+1 , y u t ) D I,O d t+1 (x t+1 , y t+1 ) × D I,O u t (x t , y t ) D I,O u t+1 (x t , y d t+1 , y u t ) × D I,O u t (x t+1 , y d t , y u t+1 ) D I,O u t+1 (x t+1 , y t+1 ) 1 2 .
If the efficiency change EC φ t,t+1 is greater than 1 then, efficiency progress arises over the periods (t) and (t + 1). Moreover, technological improvement occurs between the periods (t) and (t + 1) when T C φ t,t+1 ≥ 1. Note that the main sources of the pollution-adjusted productivity variation, namely EC φ t,t+1 and T C φ t,t+1 , are separated into polluting and non polluting components.

Hicks-Moorsteen pollution-adjusted productivity index

The Hicks-Moorsteen pollution-adjusted productivity index is presented in the next statement [START_REF] Abad | A Generalization of Environmental Productivity Analysis[END_REF]. Definition 2.6 Let T t be a pollution-generating technology that satisfies assumptions A1 -A4. For any (x t,t+1 , y t,t+1 ) ∈ R n+m + , with y t,t+1 = (y d t,t+1 , y u t,t+1 ) ∈ R m + , the Hicks-Moorsteen pollution-adjusted productivity index is defined as follows, P HM φ t,t+1 (x t,t+1 , y t,t+1 ) = P HM φ t (x t,t+1 , y t,t+1 ) × P HM φ t+1 (x t,t+1 , y t,t+1 ) 1 2 (2. [START_REF] Färe | The Measurement of Efficiency of Production[END_REF])

where φ = (α, γ d , γ u ) ∈ {0, 1} × {-1, 0} × {0, 1}.
P HM φ t (x t,t+1 , y t,t+1 ) and P HM φ t+1 (x t,t+1 , y t,t+1 ) display the Hicks-Moorsteen pollutionadjusted productivity indices for the periods (t) and (t + 1), respectively. These productivity indices are defined as follows:

P HM φ t (x t,t+1 , y t,t+1 ) = D O d t (x t , y d t+1 , y u t ) D O d t (x t , y d t , y u t ) × D O u t (x t , y d t , y u t+1 ) D O u t (x t , y d t , y u t ) × D I t (x t+1 , y d t , y u t ) D I t (x t , y d t , y u t )
(2.17)

and

P HM φ t+1 (x t,t+1 , y t,t+1 ) = D O d t+1 (x t+1 , y d t+1 , y u t+1 ) D O d t+1 (x t+1 , y d t , y u t+1 ) × D O u t+1 (x t+1 , y d t+1 , y u t+1 ) D O u t+1 (x t+1 , y d t+1 , y u t ) × D I t+1 (x t+1 , y d t+1 , y u t+1 ) D I t+1 (x t , y d t+1 , y u t+1 )
.

(2.18)

With regard to the aforementioned results, if P HM φ t,t+1 (x t,t+1 , y t,t+1 ) is greater than unity then, pollution-adjusted productivity growth occurs.

The polluting and non polluting parts of the PHM productivity variation are defined in the following result.

Proposition 2.7 For any (x t,t+1 , y t,t+1 ) ∈ R n+m + , such that y t,t+1 = (y d t,t+1 , y u t,t+1 ) ∈ R m + ,
P HM φ t+1 (x t,t+1 , y t,t+1 ) = P HM d t+1 (x t,t+1 , y t,t+1 ) × P HM u t+1 (x t,t+1 , y t,t+1 ).

(

2.19)

Where P HM d t+1 (x t,t+1 , y t,t+1 ) and P HM u t+1 (x t,t+1 , y t,t+1 ) display the non polluting PHM index and the polluting PHM index, respectively.

The productivity indices P HM d t+1 (x t,t+1 , y t,t+1 ) and P HM u t+1 (x t,t+1 , y t,t+1 ) are defined as follows,

P HM d t+1 (x t,t+1 , y t,t+1 ) = D O d t (x t , y d t+1 , y u t ) D O d t (x t , y d t , y u t ) × D O d t+1 (x t+1 , y d t+1 , y u t+1 ) D O d t+1 (x t+1 , y d t , y u t+1 ) 1/2 × D I t (x t+1 , y d t , y u t ) D I t (x t , y d t , y u t ) × D I t+1 (x t+1 , y d t+1 , y u t+1 ) D I t+1 (x t , y d t+1 , y u t+1 ) 1/4
(2.20)

and

P HM u t+1 (x t,t+1 , y t,t+1 ) = D O u t (x t , y d t , y u t+1 ) D O u t (x t , y d t , y u t ) × D O u t+1 (x t+1 , y d t+1 , y u t+1 ) D O u t+1 (x t+1 , y d t+1 , y u t ) 1/2 × D I t (x t+1 , y d t , y u t ) D I t (x t , y d t , y u t ) × D I t+1 (x t+1 , y d t+1 , y u t+1 ) D I t+1 (x t , y d t+1 , y u t+1 ) 1/4
(2.21)

If P HM d t+1 (x t,t+1 , y t,t+1
) is greater than unity then, productivity improvement arises between the period (t) and (t + 1) with respect to the non polluting components. Correspondingly, P HM u t+1 (x t,t+1 , y t,t+1 ) > 1 shows productivity growth in the polluting dimension.

The next statement allows to go a bit further in detail by highlighting the main components of the Hicks-Moorsteen pollution-adjusted productivity index.

Proposition 2.8 Assume that T t is a pollution-generating process satisfying properties A1 -A4. For any (x t,t+1 , y t,t+1 ) ∈ R n+m + , where y t,t+1 = (y d t,t+1 , y u t,t+1 ) ∈ R m + , the Hicks-Moorsteen pollution-adjusted productivity index is decomposed as follows:

P HM φ t,t+1 (x t,t+1 , y t,t+1 ) = EC φ t,t+1 × T C φ t,t+1 × Σ φ t,t+1 (2.22)
such that EC φ t,t+1 , T C φ t,t+1 and Σ φ t,t+1 respectively correspond to the efficiency change, the technological variation and the residual components over the periods (t) and (t + 1).

Remark that the residual component Σ φ t,t+1 is a wide component highlighting the part of the pollution-adjusted productivity which does not come from efficiency and technological changes. As a result, the traditional scale efficiency change component is incorporated within.

EC φ t,t+1 , T C φ t,t+1 and Σ φ t,t+1 are defined as follows:

EC φ t,t+1 = EC d t,t+1 × EC u t,t+1 = D O d t+1 (x t+1 , y d t+1 , y u t+1 ) D O d t (x t , y d t , y u t ) × D O u t+1 (x t+1 , y d t+1 , y u t+1 ) D O u t (x t , y d t , y u t ) , (2.23)
T C φ t,t+1 = T C d t,t+1 × T C u t,t+1 = D O d t (x t , y d t , y u t ) D O d t+1 (x t , y d t , y u t ) × D O d t (x t+1 , y d t+1 , y u t+1 ) D O d t+1 (x t+1 , y d t+1 , y u t+1 ) 1 2 × D O u t (x t , y d t , y u t ) D O u t+1 (x t , y d t , y u t ) × D O u t (x t+1 , y d t+1 , y u t+1 ) D O u t+1 (x t+1 , y d t+1 , y u t+1 ) 1 2 (2.24)
and

Σ φ t,t+1 = Σ d t,t+1 × Σ u t,t+1 (2.25)
such that,

Σ d t,t+1 = P HM d t,t+1 EC d t,t+1 × T C d t,t+1 (2.26)
and

Σ u t,t+1 = P HM u t,t+1 EC u t,t+1 × T C u t,t+1 (2.27)
When EC φ t,t+1 > 1 and T C φ t,t+1 > 1 then, pollution-adjusted efficiency and technological improvements arise between the periods (t, t + 1) 5 . Remark that the prominent drivers of the Hicks-Moorsteen pollution-adjusted productivity index (2.23)-(2.25) are separated into polluting and non polluting components.

Pollution-adjusted productivity change: non parametric specification

In this section, non parametric pollution-generating production technology is considered. Specifically, the pollution-adjusted productivity measures are defined through the FDH version of the by-production model [START_REF] Abad | On the Axiomatic of Pollution-generating Technologies: a Non-Parametric Approach[END_REF][START_REF] Murty | On Modeling Pollution-Generating Technologies[END_REF][START_REF] Tulkens | On FDH Efficiency Analysis: some Methodological Issues and Applications to retail Banking, Courts and Urban Transit[END_REF].

Non convex pollution-generating production process

Let us introduce the following notation: (x t , y t ) ≡ (x, y). Assume that U := {(x s , y s ) : s ∈ S} is a set of production units, where S is an index set of integers. Moreover, for any (x s , y s ) ∈ U, consider the following individual production possibility sets:

I t (x s , y s ) := (x, y) ∈ R n+m + : x i ≥ x s,i , y j ≤ y s,r , i ∈ [n], r ∈ [m d ] (3.1)
and

J t (x s , y s ) := (x, y) ∈ R n+m + : x i ≥ x s,i , y l ≥ y s,l , i ∈ [n], l ∈ [m u ] . (3.2)
The next result presents the definition of the FDH non convex pollution-generating technology.

Definition 3.1 Let T t be a pollution-generating process that satisfies the axioms A1 -A4. For any (x s , y s ) ∈ U, the FDH non convex pollution-generating technology is defined as follows,

T F DH t := (x, y) ∈ R n+m + : (x, y) ∈ ∪ s∈S I t (x s , y s ) ∩ ∪ s∈S J t (x s , y s ) . (3.3)
The FDH by-production set (3.3) is defined as an intersection of non convex subtechnologies. ∪ s∈S I t (x s , y s ) displays the usual strong disposal FDH sub-technology. The partially reversed free disposal FDH sub-technology corresponds to the non convex set ∪ s∈S J t (x s , y s ).

Non parametric approximation of the non convex FDH pollution-generating technology is laid out in the following result. Proposition 3.2 Assuming that the pollution-generating process T t satisfies the properties A1 -A4 and Variable Returns to Scale (VRS), non parametric approximation of the FDH non convex pollution-generating technology is defined as follows,

T F DH t := (x, y) ∈ R n+m + : x i ≥ s∈S λ s x s,i , y r ≤ s∈S λ s y s,r , x i ≥ s∈S ν s x s,i , (3.4)
y l ≥ s∈S ν s y s,l , s∈S ν s = s∈S λ s = 1, ν, λ ∈ {0, 1}, i ∈ [n] r ∈ [m d ], l ∈ [m u ] .

Productivity index on non parametric pollution-generating technology

The next statement defines the pollution-adjusted multiplicative distance function with respect to the non parametric specification of the FDH non-convex pollution-generating process.

Definition 3.3 For any (x, y) ∈ R n+m + and any φ = (α, γ d , γ u ) ∈ {0, 1} × {-1, 0} × {0, 1}, the pollution-adjusted multiplicative distance function is defined within T F DH t as follows, D φ,F DH t (x 0 , y 0 ) = min β s.t. β α x 0,i ≥ s∈S λ s x s,i , i ∈ [n] β γ d y 0,r ≤ s∈S λ s y s,r , r ∈ [m d ] β α x 0,i ≥ s∈S ν s x s,i , i ∈ [n] β γ u y 0,l ≥ s∈S ν s y s,l , l ∈ [m u ] s∈S ν s = s∈S λ s = 1 ν, λ ∈ {0, 1}. (3.5)
The next proposition provides an enumeration process to evaluate the pollution-adjusted multiplicative distance function through the above (3.5) mathematical program, see [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF]. Remark that the pollution-adjusted multiplicative efficiency index is non-linear and hence, would be related to a non-linear optimisation. To overcome this issue, the FDH non-parametric approach is used allowing to assess hyperbolic-based efficiency measure [START_REF] Färe | The Measurement of Efficiency of Production[END_REF] through an enumeration process avoiding approximation results [START_REF] Zofio | Graph efficiency and productivity measures: an application to US agriculture[END_REF].

Proposition 3.4 Assume that T t is a pollution-generating process that satisfies properties A1 -A4. For any (x, y) ∈ R n+m

+

, where y = (y d , y u ) ∈ R m + , the FDH pollutionadjusted multiplicative efficiency index is defined below:

D φ,F DH t (x 0 , y 0 ) ≡                                          D I,O d t (x 0 , y 0 ) = max s∈S min max i∈[n]
x s,i x 0,i ; max

r∈[m d ] y 0,r y s,r D I,O u t (x 0 , y 0 ) = max s∈S min max i∈[n]
x s,i x 0,i ; max

l∈[m u ]
y s,l y 0,l

D I t (x 0 , y 0 ) = max s∈S min max i∈[n]
x s,i x 0,i

D O d t (x 0 , y 0 ) = max s∈S min max r∈[m d ] y 0,r y s,r D O u t (x 0 , y 0 ) = max s∈S min max l∈[m u ] y s,l y 0,l (3.6)
See proof in Appendix I.

Empirical illustration

This empirical illustration focuses on the oil companies having production and extraction activities in Ecuador over the period 2014-2018. The results are provided through a FDH non convex DEA model.

Data in brief

A sample of 20 private oil companies in Ecuador is considered over the period 2012-2018.

The data set used in this research is built with the population of registered oil Ecuadorian formal firms, constructed from the balance sheets and financial statements registered on the official website of the Superintendencia de Compañias, Valores y Seguros (SCVS). This information is reported annually directly by firms to the SCVS. Two inputs are selected: (i) number of formal employees of each company and (ii) net tangible assets (capital stock). Information about the number of legally registered employees (i) is declared by each company. The capital stock (ii) is set as the sum of the real dollar value of buildings, machinery and vehicles by assuming a depreciation of 5, 10, and 20 percent. Precisely, the methodology of Camino-Mogro and Bermudez-Barrezueta [START_REF] Camino-Mogro | Productivity determinants in the construction sector in emerging country: New evidence from Ecuadorian firms[END_REF] is employed. Hence, the capital stock is valued considering the gross investment in equipment in year (t), net fixed assets in real value (physical capital in year (t -1)), a depreciation rate and the price index for equipment at the industry level obtained from the Ecuadorian National Institute of Statistics. These inputs permit to produce different outputs. Thus, we consider one desirable output, (iii) number of oil barrels and one undesirable output represented by (iv) CO 2 emissions. The number of extracted barrels of oil (iii) is defined based on the variable "sales" (American dollars) reported in the balance sheets and financial statements registered on the official website of the SCVS. Obviously, we divide it by the price (American dollars/barrel) to obtain the variable "number of extracted barrels of oil". The reference price (WTI) is considered allowing comparisons with other international research in the same field. The CO 2 emissions (tons of CO 2 equivalents) (iv) is measured by using the methodology of the 2006 IPCC Guidelines for National Greenhouse Gas Inventories.

Results and discussions

Malmquist pollution-adjusted productivity index

The results outlined in Table 2 display the number of observations which face either increasing, decreasing or constant global productivity variation (P M -3 rd column from the right) as well as non polluting (P M d -9 th column from the right) and polluting (P M u -6 th column from the right) oriented productivity change, over the period 2014-2018. Moreover, each productivity measure has been decomposed to reveal the main drivers of the productivity variation namely the technological change (global T C, polluting T C u and no polluting T C d one) and the efficiency change (global EC, polluting EC u and no polluting EC d ), respectively. These results are summarised in Figure 1. Results in Table 2 are illustrated by Figure 1 which shows that observations facing productivity loss are higher than those facing productivity gains except during the period 2014-2015. Indeed, although the observations may present non polluting (respectively polluting) oriented productivity growth, a substantial degradation of the polluting (respectively non polluting) productivity results into a high number of observations facing a global productivity decrease over most of the considered periods. The productivity scores are displayed in Appendix II. Table 3 reports the average annual PM productivity indices over the analysed period. In most of the period 2014-2018, the PM productivity indices indicate that there are pollution-adjusted productivity improvements (ie., P M φ > 1), except during the period 2016-2017. Indeed, this period shows productivity losses in both desirable and polluting components which result in pollution-adjusted productivity decline (ie., P M φ < 1), during the period 2016-2017. The combination of polluting and non polluting productivity variations permits to show the main drivers of the pollution-adjusted productivity change. The gains in pollution-adjusted productivity over the period 2017-2018 come from increasing productivity in non polluting components (ie., P M d > 1). Indeed, during this period, a loss in polluting productivity arose (ie., P M u < 1). However, the non polluting productivity improvement compensates the polluting productivity decline for this period (ie., P M d × P M u > 1). Regarding the remaining periods, these latter present productivity gains in polluting (ie., P M u > 1) dimension whereas they face productivity decline (ie., P M d < 1) in non polluting components. This case results in global pollution-adjusted productivity growth (ie., P M d × P M u > 1). Globally, the average values of the Malmquist productivity indices over the period 2014-2018 indicate pollution-adjusted productivity loss which essentially comes from a substantial decreasing productivity in non polluting components. These outcomes are outlined by the Figure 2 which shows that the period 2015-2016 presents the higher pollutionadjusted Malmquist productivity growth over the considered periods. The decomposition of the PM pollution-adjusted productivity indices presents the main sources of productivity change in both polluting and non polluting dimensions. Table 3 displays the two main components of the average annual pollution-adjusted productivity variation, namely the efficiency variation and the technological change. It is worth noting that pollution-adjusted productivity variation especially comes from the efficiency change component over the analysed period.

Periods P M d EC d T C d P M u EC u T C u P M φ EC φ T C φ 2017-
Periods P M d EC d T C d P M u EC u T C u P M φ EC φ T C φ 2017-

Hicks-Moorsteen pollution-adjusted productivity index

Table 4 outlines the number of observations that presents increasing, decreasing or constant Hicks-Moorsteen pollution-adjusted productivity change (P HM φ -1 st column from the right). This table also presents the non polluting (P HM d -3 rd column form the right) and polluting (P HM u -2 nd column form the right) productivity variations over the period 2014-2018. Moreover, the main components of the PHM productivity index has been provided, namely the efficiency change EC φ , the technological variation T C φ and a residual component Σ φ , as well as in both polluting (T C u , EC u , Σ u ) and non polluting (T C d , EC d , Σ d ) orientations. The results in Table 4 are summarised in Figure 3. This figure highlights that most of observations face Hicks-Moorsteen productivity loss during the period 2014-2018, except during the period 2014-2015. Indeed, the non polluting productivity gain does not offset the productivity loss in the polluting dimension for most of the considered observations. The detailed Hick-Moorsteen productivity scores are presented in Appendix III. Table 5 shows the average annual PHM productivity index and its prominent drivers over the period 2014-2018. The average annual PHM productivity index scores indicate pollution-adjusted productivity growth (ie., P HM φ > 1) over the analysed period. The polluting and non polluting parts of the PHM productivity variation permit to go a bit further in details. Specifically, two specific schemes of the productivity change arise over the period 2014-2018:

EC φ EC d EC u T C φ T C d T C u Σ φ Σ d Σ u P
(t) EC φ EC d EC u T C φ T C d T C u Σ φ Σ d Σ u P
i. The pollution-adjusted productivity loss is driven by polluting productivity decrease during the periods 2016-2017 and 2017-2018. In such case, the non polluting productivity growth does not compensate the polluting productivity loss (ie., P HM d > 1, P HM u < 1 and P HM d × P HM u < 1).

ii. The improvement in pollution-adjusted productivity is driven by the polluting productivity gains during the periods 2015-2016 and 2015-2014 (ie., P HM d < 1, P HM u > 1 and P HM d × P HM u > 1). And the increasing productivity in polluting components offsets the productivity loss in non polluting dimension.

Globally, a similar design as ii. arises regarding the PHM pollution-adjusted productivity change for the overall analysed periods. These different schemes are illustrated in Figure 4. Table 5 also displays the three main drivers of the average annual PHM productivity variation, namely the efficiency change, the technological variation and a residual component. The decomposition of PHM indices (ie., P HM d , P HM u and P HM φ ) shows that the pollution-adjusted productivity change is induced by the positive efficiency variations in both polluting and non polluting dimensions. The 1 st column from the right in Table 5 elicits the number of observations that are subjected to indeterminateness in the technological change variation8 and in the residual component. Thus, the average of these components is only based upon the observations that do not present indeterminateness and the results should be interpreted with caution. However, even though the technological variation may present infeasibility, the PHM productivity indices scores always have finite values providing pollution-adjusted productivity insights for each production unit (see Appendix III). Hence, it is always possible to assess the productivity variation over periods in global, polluting and no polluting dimensions.

To summarise, the results provided by the PM productivity index coincide with those provided by the PHM productivity index, globally. However, the PM is a local technical variation measure whereas the PHM is a total factor productivity change measure which is defined as a ratio of input and output indices [START_REF] O'donnell | An Aggregate Quantity-Price Framework for Measuring and Decomposing Productivity and Profitability Change[END_REF]. Moreover, the disaggregation of each productivity index, into polluting and non polluting components highlights the contribution of each dimension into the productivity variation. And results show that the productivity growth in the polluting dimension contributes the most in the global environmental productivity improvement.

Concluding comments

This paper aims to analyse environmental productivity change through the pollutionadjusted Malmquist and Hicks-Moorsteen productivity indices. Moreover, the promi-nent sources of the pollution-adjusted productivity variation are provided by considering polluting and no polluting parts of the productivity variation. To drive this investigation, non convex pollution-generating technology is considered through the free disposal hull production model.

The empirical illustration provided in this paper focuses on the Ecuadorian oil industry. Precisely, a sample of 20 Ecuadorian oil companies over the period 2014-2018 is selected. The results are provided through a FDH non convex DEA model. The proposed theoretical framework permits to characterise the pollution-adjusted productivity variation. Specifically, the approach provided in this paper permits to show that pollution-adjusted productivity change may be driven by either polluting or no polluting components. Moreover, the pollution-adjusted productivity decomposition allows to go a bit further in details by displaying the prominent drivers of both polluting and no polluting productivity variations. In this line, the empirical illustration shows that environmental productivity improvement comes mainly from an increasing productivity in polluting components. A further research could investigate this productivity analysis through additive-based productivity measures and could provide a comparison of the results with those of multiplicative-based measures.

Appendix II: Malmquist Pollution-adjusted productivity index

DMUs P M d t,t+1 EC d t,t+1 T C d t,t+1 P M u t,t+1 EC u t,t+1

Figure 1 :

 1 Figure 1: PM productivity index: observations per component and value.7

Figure 2 :

 2 Figure 2: Trend of PM productivity change over periods

Figure 3 :

 3 Figure 3: Observation per components and per variation for the P HM productivity index

Figure 4 :

 4 Figure 4: Trend of average annual PHM productivity variation over periods.

  Proposition 2.2 For any (x t , y t ) ∈ R n+m , y t ) ≡ D I,O d (x t , y t ).

		+	where y t = (y d t , y u t ) ∈ R m + ,
	i. D 1,-1,0 t	(x t

Table 1 :

 1 Table 1 presents the descriptive statistics of the variables used in this study. Characteristics of inputs and outputs. 6

	Variables	Min	Max	Median	S.D.	Mean
	Labor	1	706	11.5	176.79	102.09
	Capital stock	1760.86	173359632.8	629671.59	27682885.45	9293764.69
	Oil production 725.15	9718986.9	315659.3	1411965.6	791729.5
	CO 2 emissions 7.03	94270.37	3061.77	13695.51	7679.47

Table 2 :

 2 Number of observations per component and per value of the P M productivity index.

		< 1	9	6	5	10	6	7	9	7	5
	2018	= 1 > 1	1 10	13 1	1 14	2 8	13 1	1 12	3 8	13 0	2 13
		< 1	10	2	11	12	1	16	11	1	14
	2016-2017	= 1 > 1	1 9	14 4	0 9	2 6	14 5	1 3	2 7	14 5	1 5
		< 1	13	4	14	8	3	6	13	3	13
	2015-2016	= 1 > 1	0 7	13 3	0 6	0 12	13 4	1 13	0 7	13 4	0 7
		< 1	15	2	16	4	2	3	8	2	11
	2014-2015	= 1 > 1	0 5	11 7	0 4	0 16	11 7	0 17	0 12	11 7	0 9

Table 3 :

 3 Average annual PM productivity change

	2018 1.044 0.934 1.117 0.962 0.864 1.114 1.004 0.807 1.245
	2016-2017 0.797 0.984 0.810 0.897 1.217 0.737 0.715 1.198 0.597
	2015-2016 0.932 1.052 0.886 1.189 1.328 0.895 1.108 1.397 0.793
	2014-2015 0.813 1.136 0.715 1.328 1.270 1.046 1.079 1.443 0.748
	Overall	0.891 1.024 0.870 1.081 1.154 0.936 0.963 1.182 0.815

Table 4 :

 4

	HM d	P HM u	P HM φ

Observations per component and per value for the P HM productivity index.

  indicates the number of observations subjected to infeasibility in the technological change component.

											HM d	P HM u	P HM φ
	2017-2018	0.804	0.923	0.871	0.933 [5]	1.093	0.853	1.360	1.097	1.240	1.133	0.867	0.982
	2016-2017	1.246	0.952	1.310	0.439 [6]	0.867	0.507	1.410	1.277	1.104	1.012	0.860	0.870
	2015-2016	0.895	1.129	0.793	0.928 [4]	0.659	1.410	1.630	1.031	1.581	0.728	1.581	1.150
	2014-2015	1.412	1.054	1.339	1.028 [5]	0.459	2.241	0.676	1.019	0.664	0.519	2.215	1.150
	Overall	1.061	1.011	1.049	0.791	0.732	1.081	1.206	1.101	1.095	0.811	1.271	1.031
	[X]

Table 5 :

 5 Average annual PHM productivity change

Table 6 :

 6 PM productivity index under FDH production technologyAppendix III: Hicks-Moorsteen Pollution-adjusted productivity index

	T C u t,t+1	P M φ t,t+1	EC φ t,t+1	T C φ t,t+1
	2017-2018

Table 7 :

 7 PHM productivity index under FDH production process

Dasgupta and Mäler [14] mention that: "The word convexity is ubiquitous in economics, but absent from ecology".

Note that allocative productivity change can also be identified essentially focusing on cost and profit functions[START_REF] Fukuyama | Indicators and indexes of directional output loss and input allocative inefficiency[END_REF][START_REF] Grifell-Tatjé | Cost and productivity[END_REF][START_REF] Lim | Profit and productivity of US Class I railroads[END_REF].

Throughout the paper, the superscripts d and u denote the desirable and undesirable outputs, respectively.

In contrary with the traditional input-and output-oriented Malmquist index, the hyperbolic version of the Malmquist productivity measure is defined as combination of hyperbolic distance functions. Although this version of the Malmquist index has been rarely empirically applied[START_REF] Zofio | Graph efficiency and productivity measures: an application to US agriculture[END_REF], this paper provides an empirical estimation of the PM productivity measure through a non parametric enumerative approach.

Notice that the Hicks-Moorsteen pollution-adjusted productivity index is decomposed through an output orientation. As this productivity index is a total factor productivity measure, it can be decomposed following either an output orientation or an input orientation as well[START_REF] Abad | A Generalization of Environmental Productivity Analysis[END_REF].

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Remark that '+' and '-' indicate respectively productivity loss and gain. Thus, a value higher than zero indicates the number of observations presenting productivity gain whereas a value lesser than zero indicates the number of observations facing productivity loss.

The technological change component is defined based upon complete cross-period distance functions; see(2.24). These complete cross-period distance functions induce infeasibility when they do not encounter the production frontier as mentioned in[START_REF] Abad | An environmental generalised Luenberger-Hicks-Moorsteen productivity indicator and an environmental generalised Hicks-Moorsteen productivity index[END_REF].

Disclosure of interest

The authors report there are no competing interests to declare.

Appendix I

Proof of Proposition 3.4: Assume that φ = (1, 0, 0). Following properties A1 -A4, the FDH pollution-generating process is defined as an intersection of non convex subtechnologies; see (3.3). Thus, for any i ∈

x s,i x 0,i .

The proof for φ = (1, 0, 1), φ = (1, -1, 0), φ = (0, -1, 0) and φ = (0, 0, 1) can be immediately deduced from the aforementioned results. 2


Benjamin Vincent
email: benjamin_vincent@etu.u-bourgogne.fr

Virgil Optasanu
email: virgil.optasanu@u-bourgogne.fr

Frédéric Herbst

Sébastien Chevalier
email: sebastien.chevalier@u-bourgogne.fr

Ioana Popa
email: ioana.popa@u-bourgogne.fr

Tony Montesin
email: tony.montesin@u-bourgogne.fr

Luc Lavisse
email: luc.lavisse@u-bourgogne.fr

Comparison Between the Oxidation Behaviors of Ti6242S, Ti6246, TiXT Alloys, and Pure Titanium

Keywords: Titanium alloys, high temperature oxidation, oxygen diffusion, kinetics

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Introduction

Titanium alloys are widely used in aeronautic applications because of their low density compared to steel and their relative high strength, contributing to a reduction of pollutant emission. These alloys also have good oxidation resistance at moderate temperatures (<500°C) and can be used in aircraft parts exposed to hot atmospheres. This is the case for near alpha titanium alloys, which were developed in order to improve the high temperature oxidation resistance. One example is Ti6242S (Ti -6% Al -2% Sn -4% Zr -2% Mo, 0,08% Si) of which behavior was studied by several authors [1][2][3][4][START_REF] Shamblen | The Science, Technology, and Application of Titanium[END_REF][START_REF] Berthaud | Etude du comportement de l'alliage de titane Ti6242S à haute température sous atmosphères complexes : applications aéronautiques[END_REF][START_REF] Berthaud | [END_REF][8], IMI 834 [9], Ti1100 [10,11] or Ti60 [12].

A way to improve the oxidation resistance is the optimization of alloying elements in titanium alloys. Some alloying elements, like aluminum or silicon, can increase the oxidation resistance whereas others, like vanadium, can accelerate the oxidation kinetics and then lead to faster degradation. Tin and zirconium are found to be neutral for the oxidation behavior [13]. Aluminum allows the use of titanium alloys at higher temperatures and for longer periods of exposure, by lowering the oxidation rates [14]. Its presence leads to the formation of Al2O3 in the oxide layer, which helps to reduce the oxygen flux through the scale. The weight gain and the oxide scale growth are slower as compared to pure titanium. However, most of titanium alloys limit the presence of aluminum to 6 wt.% in order to avoid Ti3Al phase formation, which degrades ductility [START_REF]Schuster and Martin Palm Reassessment of the Binary Aluminum-Titanium Phase Diagram[END_REF]. Silicon is also beneficial for the oxidation resistance. It is shown that its addition can reduce the weight gain caused by oxidation [16,17]. It is shown that its presence can help to reduce the porosity of the oxide layer and increases its adherence. Silicon content is limited to 1 wt.% to avoid silicide precipitates, which cause embrittlement of the alloys. The effect of molybdenum in high temperature oxidation has been studied in Ti-Al alloys. Shida showed that 2-6 wt.% of Mo decrease the weight gain after 100h of exposure in air at temperatures between 800°C and 1000°C in Ti-35wt.%Al alloy [18]. The presence of 2 wt.% Mo in the alloy reduces the kp value by a factor of 2 as compared with the composition from without Mo. However, no similar studies were found in near alpha titanium alloys.

This study has two goals. The first is to compare the oxidation behavior of commercial titanium alloys Ti6242S (Ti -6%Al -2% Sn, 4% Zr -2% Mo -0.08%Si) and Ti6246 (Ti -6%Al -2% Sn, 4% Zr -6% Mo -0.05%Si) in order to evaluate the influence of the molybdenum content on the high temperature oxidation. The second aim is to evaluate the role of silicon by comparing the TiXT alloy (Ti -0.45% Si) for commercially pure titanium (99.6% of purity, grade 2). For the comparison, isothermal oxidation tests were performed up to 1000h at 560°C, 600°C and 650°C in laboratory air to discontinuously measure the weight gains and estimate the parabolic constants of each alloy. Scanning Electron Microscopy (SEM) was used to analyze the oxide scales morphologies and compositions and X-ray Diffraction (XRD) was used for crystalline phase identification. The oxygen diffusion zone was estimated by microhardness testing for each temperature. The oxygen diffusion coefficient corresponding to oxygen dissolution in the substrates, the activation energy for oxygen diffusion and the activation energy for the overall oxidation are estimated by processing these data.

Experimental methods

The specimens were cut from parallelepiped rods in small plates of 10 x 10 x 1 mm 3 . They were then mechanically polished on SiC grids up to P600 and cleaned with ethanol in ultrasonic bath for 10 minutes. The mass of each sample was measured with a Mettler Toledo balance with 0.1 mg accuracy. The dimensions were measured with a digital caliper with 10 µm accuracy.

The samples were isothermally oxidized in muffle furnace at 560°C, 600°C and 650°C in laboratory air up to 1000 h. Individual samples were extracted after 25, 50, 150, 300, 500, 800 and 1000 hours of high temperature exposure and the mass gain per surface area of each sample was measured. Some of the samples were doubled for reproducibility evaluation. Crystalline phases present in the samples before and after oxidation were identified by X-ray diffraction in - geometry by using diffractometer Bruker D8 Discover with monochromatic Cu Kα radiation (λKα1 = 0.154156 nm) and equipped with LYNXEYE detector. The surface morphology and composition of the oxide scales were performed using JEOL JSM-7600F scanning electron microscope (SEM) equipped with Energy Dispersive X-Ray Spectroscopy (EDS). Cross-section SEM analyses were performed after preparation of the samples by gold sputtering, copper electrolytic deposition, cross cutting, polishing with ascending grid SiC paper and super-finishing with 30 nm colloidal silica suspension.

The oxygen dissolution zone in the oxidized samples was revealed by Vickers microhardness measurements using Zwick indenter with a load of 25 g applied for 10 seconds in each print. The distance from the interface oxide/metal of the indentation prints was measured with an optical microscope. 50 indentations were made for each profile, spacing the imprints in the depth direction by an increment of 1 µm for specimens oxidized at the lower exposure time, up to 5 µm for longer exposure time.

Results and discussion

Raw materials

The microstructures of the raw samples are presented in Fig. 1. Ti6242S is a near- alloy with a small proportion of β phase. It has a bimodal structure with alpha grains about 20 µm. Ti6246 has a quite similar microstructure except that the higher content of beta-stabilizer molybdenum leads to larger proportion of beta phase. TiXT and pure Ti present similar microstructures, except that the alloy contains a very small proportion of β phase around the grain boundaries due to the presence of 0.45 wt.% of Si.

Oxidation kinetics

Fig. 2 presents the mass gain versus the time at 560°C, 600°C and 650°C for Ti6242S and Ti6246 alloys (Fig. 2a) and for pure Ti and TiXT alloy (Fig. 2b). For each alloy, the mass gain follows a parabolic curve, which means that diffusion is the limiting step and that the oxide formed during exposures at elevated temperatures is dense and protective. The temporal evolution of the normalized mass gain is described by [START_REF] Kofstad | High temperature corrosion[END_REF]:

( ∆𝑚 𝑆 ) 2 = 𝑘 𝑝 𝑡
where ∆𝑚 is the mass gain, S is the sample surface, 𝑘 𝑝 is the oxidation parabolic rate constant and 𝑡 the exposure time.

In this range of temperature, titanium alloys are known to undergo anionic oxidation process: 𝑂 2-ions diffuse through the oxide scale to form new oxides at the oxide/metal interface and to diffuse further into the metal. Indeed, it is well known that oxygen has high solubility in pure Ti up to 33 at. % [START_REF] Du | [END_REF][START_REF] Rosa | Oxidation of Ti-1Si and Ti-5Si alloys[END_REF].

Fig. 2 shows similar mass gains for Ti6242S and Ti6246 alloys for the three temperatures (560°C, 600°C and 650°C), but the difference of mass gain is clear between TiXT and pure titanium. For the three temperatures, the mass gain curve of pure titanium is always above the mass gain curve of TiXT, and the relative difference increases with the temperature.

The parabolic rate constants are calculated by plotting the normalized mass curve versus the square root of the time.

The resulting kp values are listed in Table 1. The lowest values of the parabolic rate constants are obtained for Ti6242S alloy. These values are slightly higher for Ti6246 for the three temperatures, but the difference is not significant, especially at the highest temperature where the difference is only of 3.7%. Higher amount of of Mo (6%) in Ti6246 alloy as compared to Ti6242S alloy (2%) does not seem to significantly change the mass gain during oxidation. No other studies were found to compare these results.

The difference between the unalloyed titanium and TiXT is larger for the normalized mass gain and the parabolic rate constants for all the temperatures. The oxidation parabolic rate constants are 1.8, 2.4 and 2.2 higher for pure titanium than for TiXT at 560°C, 600°C and 650°C, respectively. These results show that TiXT has clearly better resistance against high temperature oxidation.

Oxide scales analyses

Fig. 3 shows cross-section SEM images of the oxide formed after 1000 h at 600°C for the four alloys and Table 2 presents the oxide thickness for each alloy at 560°C, 600°C and 650°C. For all alloys, the oxide layer is thicker at 600°C compared to 560°C. At 650°C, the measured oxide thicknesses seem similar to that observed at 600°C, although the mass gain is higher at the highest temperature. This behavior can be explained by the fact that the mass gain is mainly due to the oxygen dissolution into the substrate and not to the formation of the oxide scale [START_REF] Berthaud | [END_REF]. Parabolic rate constants and the oxide thicknesses are similar for Ti6242S and Ti6426 alloy for the three exposure temperatures, as presented in Table 2. Variation of molybdenum content from 2% to 6% does not impact on the thickness of the oxide layer.

The oxide thickness on pure titanium is much bigger than on TiXT alloy, confirming that the presence of Si within TiXT alloy increases the high temperature oxidation resistance. This is in accordance with several studies which show that the higher is the content of silicon, the thinner is the oxide layer [16,17]. Chaze et al. [16] also demonstrated that the presence of silicon may reduce the porosity of the rutile layer and increase the adherence of the oxide scale, interesting for the protectiveness of the oxide.

SEM and XRD analyses show the formation of TiO2 rutile which is predominant, and sometimes anatase for all alloys and for each temperature. The presence of 0.45 wt.% of Si in TiXT alloy leads to the formation of Ti5Si3 in the underlying substrate after oxidation. This result points out that during oxidation, some Ti atoms react with Si in order to form Ti5Si3, preventing the reaction with oxygen, which can contribute to the lower mass gain of TiXT as compared to pure titanium. [START_REF] Du | [END_REF]. EDS analyses did not show any variation of aluminum concentration through the oxide scale, probably because of the size of the analyzed zone of the tool (~1 µm 3 ). However, a darker layer can be seen in the sublayer zone for Ti6242S and Ti6246, and not visible for TiXT and Oxide layer Substrate pure Ti. This probably shows the presence of an Al2O3 thin layer which could appear darker in BSE mode. By using XPS analyses, Berthaud et al. [START_REF] Berthaud | [END_REF] highlighted a higher amount of aluminum at the external surface of the oxide scale, which means that Al2O3 may exist at this location for Ti6242S alloy, while XRD analyses did not show its presence.

To explain the beneficial effect of Si on the oxidation resistance, it is assumed that silicon dissolves as Si 4+ ions in interstitial positions of the rutile layer. Indeed, the low atomic radius, of about 0,41 Å [16], allows the insertion in the rutile lattice. The presence of Si 4+ reduces the oxygen diffusion in the rutile and therefore the growth of the layer by decreasing the concentration of oxygen vacancies, which is the main defect in rutile below 900°C. This might explain the lower rate of porosity in the oxide grown at the surface of TiXT as compared to that observed on pure titanium. It is mentioned elsewhere [17] [23] that the presence of Si can form SiO2 particles finely dispersed in the rutile, which decrease the internal stress, reduce its recrystallization and increase its adherence.

For TiXT, rutile is the main phase in the oxide. EDS analyses reveal no detectable variation of the silicon content in the oxide layer in horizontal and vertical directions, even if Vojtech [23] and Habazaki [24] previously suggested that Si is more concentrated in the inner part of the oxide layer. XRD did not show any peaks of SiO2, probably because of the small amount of Si contained in the alloy, and/or because SiO2 can be found in amorphous phase. The unalloyed titanium shows only rutile and anatase after oxidation.

A slightly darker gray layer can be observed for TiXT alloy just below the oxide scale, in the underlying substrate on SEM images (Fig. 3), showing that this zone is probably enriched with a light element. Taking into account the high diffusion capacity of nitrogen through rutile layers [START_REF] Chaze | The role of nitrogen in the oxidation behaviour of titanium and some binary alloys[END_REF], it can be supposed that this layer is a reaction layer enriched in nitrogen. Such N-rich layer has been also observed for pure Ti by Kanjer et al. [START_REF] Kanjer | Effect of laser shock peening on the high temperature oxidation resistance of titanium Surf[END_REF] after 100h of oxidation at 700°C. has been demonstrated that nitrogen present in the air plays a role in the oxidation resistance. Coddet et al. [START_REF] Coddet | The role of nitrogen in the oxidation behaviour of titanium and some binary alloys[END_REF] and more recently Torrent et al. [START_REF] Torrent | Influence of the composition of titanium oxynitride layers on the fretting behavior of functionalized titanium substrates: PVD films versus surface laser treatments Surf[END_REF] showed that the presence nitrogen-rich layer reduces the porosity of the oxide layer and increases its adherence, limiting the risk of spallation. Dupressoire et al. [8] showed also that the presence of nitrogen in the exposure atmosphere reduces the mass gain after 100h of oxidation of Ti6242S alloy at 650°C.

Study of the oxygen dissolution zone

Oxygen dissolution in titanium alloys affects the mechanical properties by reducing its ductility. [10,12,21]. Indeed, titanium alloys can dissolve oxygen in its α-phase up to 33% at. in their crystal lattice and up to 4% at. In the β-phase [30]. The oxygen diffusion into a semi-infinite solid can be described by the following equation [START_REF] Kofstad | High temperature corrosion[END_REF] 𝐶 𝑥 -𝐶 0

𝐶 𝑠 -𝐶 0 = 1 -erf ( 𝑥 2√𝐷𝑡 )
eq. 1

where 𝐶 𝑥 is the oxygen concentration at the depth 𝑥, 𝐶 0 is the original oxygen concentration of the raw material before oxidation, 𝐶 𝑠 is the oxygen concentration at the surface, 𝑡 the time and 𝐷 is the diffusion coefficient.

Several authors suggest that a linear relationship exists between the oxygen concentration and the hardness of Ti alloys [START_REF] Strafford | The Interaction of Titanium and Titanium Alloys with Nitrogen at Elevated Temperatures. I. The Kinetics and Mechanism of the Titanium-Nitrogen Reaction[END_REF]30]. Berthaud et al [START_REF] Berthaud | Etude du comportement de l'alliage de titane Ti6242S à haute température sous atmosphères complexes : applications aéronautiques[END_REF][START_REF] Berthaud | [END_REF] also observed that the oxygen content and the hardness of the oxygen-enriched area tend to follow the same profile for oxidation of 1000h up to 10 000h at 560°C of Ti6242S alloy.

However, other authors assume that the correlation between oxygen concentration and the hardness values is better described with a parabolic relationship instead of a linear relationship. This assumption is based on comparisons between these two quantities by Brown, Folkman and Schussler after the study of 152 batches of titanium sponges [32], and then used by Liu and Welsh [33] and by Leyens and Peters [START_REF]Titanium and Titanium Alloys: Fundamentals and Applications[END_REF]. Taking these data into account, Vaché et al. [START_REF] Vaché | Oxygen Diffusion Modeling in Titanium Alloys: NewElements on the Analysis of Microhardness Profiles[END_REF] recently proposed a new model to estimate more accurately the coefficient of diffusion for oxygen with hardness profiles.

Therefore, in order to estimate the extent of the oxygen dissolution zone, called α-case by some authors, microhardness tests were carried out. Since the hardness depends on the oxygen content by a parabolic relationship, the hardness profile can be described by the following model, proposed by Vaché et al. [START_REF] Vaché | Oxygen Diffusion Modeling in Titanium Alloys: NewElements on the Analysis of Microhardness Profiles[END_REF]:

𝐻𝑉 𝑥 -𝐻𝑉 0 𝐻𝑉 𝑠 -𝐻𝑉 0 = √1 -erf ( 𝑥 2√𝐷𝑡 ) eq. 2
where 𝐻𝑉 𝑥 is the hardness value at the depth 𝑥, 𝐻𝑉 𝑠 is the hardness value at the oxide/metal interface, 𝐻𝑉 0 is the hardness value of the bulk material, 𝐷 is the diffusion coefficient of oxygen in titanium and 𝑡 is the time. The coefficient 𝐷 can be calculated by fitting the experimental points with a complementary error function by the least squared method. Then, the oxygen diffusion zone is evaluated by the relation: 𝐿 𝑑𝑖𝑓𝑓 = 4√𝐷𝑡 as also used by Casadebaigt et al. [START_REF] Casadebaigt | High Temperature Oxidation of Ti-6Al-4V Alloy Fabricated by Additive Manufacturing[END_REF], corresponding in this study to the depth where the value of 𝐻𝑉 𝑥 tends to equal 𝐻𝑉 0 . Fig. 4a presents an example of hardness profile for Ti6242S at 650°C after 1000h of exposure with experimental data and the fitted curve, and 4b is an example of the evolution of the oxygen dissolution zone at 650°C.

Table 3 presents the diffusion coefficients extracted from the fitting and the corresponding extent of the dissolution zone. It can be noticed that the diffusion coefficient for Ti6242S and Ti6246 are very close for the three temperatures. For TiXT, this value is lower than for pure titanium at 600°C, and similar at 560°C and 650°C.

Activation energy for the mass gain and for oxygen diffusion

Based on 𝑘 𝑝 values determined here above, two activation energies can be determined. One is for the total mass gain and the other is for the oxygen diffusion into the substrate. These are presented in Table 4.

The parabolic rate 𝑘 𝑝 is dependent of the temperature. It can be expressed by an Arrhenius law [START_REF] Kofstad | High temperature corrosion[END_REF]:

𝑘 𝑝 = 𝑘 0 exp ( -𝑄 𝑜𝑥 𝑅𝑇 ) eq. 3
where 𝑘 𝑝 is the parabolic rate constant in g².cm -4 .s -1 , 𝑘 0 is the frequency factor, 𝑄 𝑜𝑥 is the activation energy for the overall oxidation, 𝑅 is the universal gas constant (8.3143 J.mol -1 .K -1 ) and 𝑇 the temperature. By plotting the logarithmic values of 𝑘 𝑝 versus 1/𝑇, a straight line can be drawn and the slope corresponds to the value of ( -𝑄 𝑜𝑥

𝑅𝑇

) and the activation energy can then be estimated. The estimated activation energies are 175 kJ.mol -1 and 154 kJ.mol -1 for Ti6242S and Ti6246, respectively. For Ti6242S, the values are slightly larger than those found by Gaddam et al. et al. [4] who estimated these values at 151 kJ.mol -1 and 157 kJ.mol -1 respectively but lower than 262 KJ.mol -1 found by Berthaud [6] and than 245 kJ.mol -1 found by Vaché et al. [START_REF] Vaché | Modeling the oxidation kinetics of titanium alloys: Review, method and application to Ti-64 and Ti-6242s alloys[END_REF]. For Ti6246 alloy, the energy activation is slightly lower compared to Ti6242S. For TiXT and Pure Ti, the activation energy is estimated at 181 kJ.mol -1 and 191 kJ.mol - 1 , respectively (See Table 4).

The oxygen dissolution coefficient is also dependent of the temperature and expressed by an Arrhenius law [START_REF] Kofstad | High temperature corrosion[END_REF]: 𝐷 = 𝐷 0 exp (-𝑄 𝑑𝑖𝑓𝑓 𝑅𝑇 ) eq. 4 where 𝐷 is the coefficient of oxygen dissolution, 𝐷 0 is the pre-exponential factor, 𝑄 𝑑𝑖𝑓𝑓 is the activation energy for oxygen diffusion, 𝑅 is the universal gas constant (8.3143 J.mol -1 .K -1 ) and 𝑇 the temperature. The activation energies for dissolution are estimated at 205 kJ.mol -1 for Ti6242S, 197 kJ.mol -1 for Ti6246, 198 kJ.mol - 1 for TiXT and 187 kJ.mol -1 for pure titanium (see Table 5). No significant differences were found between the couples of alloys. For comparison, the activation energy for oxygen dissolution in Ti6242 was estimated elsewhere at 157 kJ.mol -1 [1], 167 kJ.mol -1 [4], 198 kJ.mol -1 [START_REF] Vaché | Modeling the oxidation kinetics of titanium alloys: Review, method and application to Ti-64 and Ti-6242s alloys[END_REF], 203 kJ.mol -1 [START_REF] Shamblen | The Science, Technology, and Application of Titanium[END_REF], 242 kJ.mol -1 [3]. For pure titanium, the values were estimated at 191 kJ.mol -1 [START_REF] Vaché | Modeling the oxidation kinetics of titanium alloys: Review, method and application to Ti-64 and Ti-6242s alloys[END_REF] 201 kJ.mol -1 [30] and 241 kJ.mol -1 [START_REF] Du | [END_REF]. For each alloy, the difference between the activation energy for oxygen dissolution and for overall oxidation process is not high. This could be explained by the fact that, in the temperature range of this study, the oxygen dissolution is prevalent compared to the formation of the oxide scale. Indeed, Berthaud et al. [START_REF] Berthaud | Etude du comportement de l'alliage de titane Ti6242S à haute température sous atmosphères complexes : applications aéronautiques[END_REF][START_REF] Berthaud | [END_REF] revealed that the oxygen dissolution represents 79% to 87% of the total mass gain for Ti6242S oxidized at 560°C between 1000h and 10 000h.

Conclusion

The aim of this study was to highlight the effect of Mo and Si in titanium alloys on their high temperature behavior.

After isothermal oxidation tests in the range of 560°C -650°C for 1000h in laboratory air, it was concluded that: -All the alloys follow a parabolic rate law during the oxidation; -The oxide layer is mainly composed of rutile. No Al2O3 was detected for Al-containing alloys and no SiO2 was identified despite the presence of Si in the alloys; -The alloying elements have an important role in the oxidation behavior since the mass gains are substantially lower for the alloys as compared to pure titanium; -The higher concentration of Mo in Ti6426 alloy as compared to Ti6242 has no effect on the oxidation resistance in the range of 560°C -650°C; -The presence of Si in TiXT alloy (Ti-0,45%wt Si) leads to a beneficial effect on the high temperature oxidation behavior by decreasing the mass gains by a factor of 2; -The activation energy for the overall oxidation process is estimated at 175 kJ.mol -1 , 154 kJ.mol -1 , 181 kJ.mol - 1 and 191 kJ.mol -1 for Ti6242S, Ti6246, TiXT and pure titanium, respectively; -The activation energy for oxygen dissolution is estimated at 205 kJ.mol -1 , 197 kJ.mol -1 , 198 kJ.mol -1 and 187 kJ.mol -1 for Ti6242S, Ti6246, TiXT and pure titanium, respectively.

Fig. 1

 1 Fig. 1 SEM images of as received alloys microstructures of (a) Ti6242S, (b) Ti6246, (c) pure Ti and (d) TiXT

Fig. 2

 2 Fig. 2 Normalized net mass gain versus the time at 560°C, 600°C and 650°C for (a) Ti6242S / Ti6246, and (b) pure Ti / TiXT

Fig. 3

 3 Fig. 3 Cross section SEM images of the oxide layers grown after 1000h of oxidation at 600°C for (a) Ti6242S, (b) Ti6246, (c) TiXT and (d) Pure Titanium

Fig. 4

 4 Fig.4 Example of (a) hardness profile for Ti6242 after 1000h of oxidation et 650°C and (b) evolution of the oxygen dissolution zone versus square root of time at 650°C

Table 1

 1 Calculated values of parabolic constants at 560°C, 600°C and 650°C for the overall weight gain up to 1000h kp (g².cm -4 .s -1 )

	Alloy	560°C	600°C	650°C
	Ti6242S	2.8 x 10 -14	6.7 x 10 -14	3.2 x 10 -13
	Ti6246	3.9 x 10 -14	9.7 x 10 -14	3.3 x 10 -13
	TiXT	6.7 x 10 -14	2.0 x 10 -13	8.5 x 10 -13
	Pure Ti	1.2 x 10 -13	4.8 x 10 -13	1.8 x 10 -12

Table 2

 2 Oxide thickness after 1000h of oxidation at 560°C, 600°C and 650°C Oxide thickness after 1000h of exposure (µm)

	Alloy	560°C	600°C	650°C
	Ti 6242S	0.18 ± 0.02	1.9 ± 0.3	1.7 ± 0.1
	Ti 6246	0.16 ± 0.02	1.6 ± 0.3	1.7 ± 0.4
	TiXT	1.3 ± 0.2	6.1 ± 0.4	4.9 ± 0.3
	Pure Ti	3.7 ± 0.5	7.9 ± 2.2	9.7 ± 0.3

XRD analyses show that, despite the presence of 6 wt.% Al content, Ti6242S and Ti6246 do not present any peak of alumina contrary to what other authors found on TA6V alloy

Table 3

 3 Coefficients of oxygen diffusion in different Ti alloys and pure Ti diffusion and evolution of oxygen dissolution zone with the oxidation temperature

		Coefficient of diffusion (m².s -1 )	Oxygen dissolution zone after 1000h (µm)
	Alloy	560°C	600°C	650°C	560°C	600°C	650°C
	Ti6242S (8.7 ± 3.5) x 10 -18	(3.7 ± 0.4) x 10 -17	(1.6 ± 0.3) x 10 -16	23	46	95
	Ti6246	(9.8 ± 3.6) x 10 -18	(3.7 ± 1.2) x 10 -17	(1.6 ± 0.5) x 10 -16	24	46	95
	TiXT	(9.4 ± 2.4) x 10 -18	(2.8 ± 0.9) x 10 -17	(1.5 ± 0.2) x 10 -16	24	40	93
	Pure Ti	(9.9 ± 1.0) x 10 -18	(5.5 ± 1.5) x 10 -17	(1.4 ± 0.2) x 10 -16	24	56	90

Table 4

 4 Activation energy for overall oxidation process and for oxygen dissolution for Ti6242, Ti6246 TiXT and pure Ti in the range of 560°C -650°C

	Alloy	Activation	energy	for	overall	Activation energy for oxygen
		oxidation (kJ.mol -1 )			dissolution (kJ.mol -1 )
	Ti 6242S	175				205
	Ti 6246	154				197
	TiXT	181				198
	Pure Ti	191				187

Acknowledgements

The authors thank ANR French National Research Agency for financial funding of ALTITUDE project and TIMET Savoie for providing the materials.