Elliot Aurissergues
email: elliot.aurissergues@gmail.com

A simple alternative model for corporate investment

Keywords: JEL Classification: D92, E32, E43, E44, G30 corporate investment, real interest rate, user cost of capital, financial frictions, adverse selection, precautionnary saving

The aim of this paper is to provide a simple model in which fall in interest rate fails to increase ccorporate investment through the traditionnal user cost of capital channel. The motivation is the lack of empirical evidence for this channel and the ineffectiveness of the loose monetary policy with regard to corporate investment. I develop a simple model of adverse selection. It allows to microfound a linear relation between corporate investment and cash flow which "kills" the user cost channel. I solve the model analytically and find that lower interest rates decrease corporate investment.

Introduction

According to the standard theory of investment, low real interest rates lower the user cost of capital and triggers an increase in desired capital stock and thus in corporate investment.

The past decade provides an imperfect but interesting test for this theory. Between 2008 and 2016, the federal fund rate was stuck at the zero lower bound. Inflation remained positive, so short term real interest rate were negative for nearly a decade. The rise did not happen. Net corporate investment in the US was negative from 2008 to 2010 and recovered very slowly after.

After 2012, it remained lower than during previous historical recoveries in which real interest rates were much higher. This inability to stimulate investment is striking and questions the standard theory. It can be argued that the financial crisis and the following slow recovery are not a particularly good natural experiment. "Big" rise in uncertainty, increased market power and a depressed aggregate demand can explain a depressed investment and thus a low interest rate. But, two other unrelated piece of evidences should cast serious doubt about the neoclassical theory.

First, when estimating the effects of a "shock" on real interest rates, economists do not find large effects on corporate investment. In [START_REF] Bernanke | Inside the black box: The credit channel of Monetary Policty Transmission[END_REF], the response of corporate investment to monetary policy shocks is small and very delayed. The response is not significant before eight months and the peak intervenes...twenty months after the shock. The contrast with residential investment, whose response is quick and large, is striking. Given this difference of timing, it would not be implausible that corporate investment does not react at all to monetary policy shocks but reacts to the response of residential investment through old keynesian investment accelerator for example. This small response in VAR models is also a serious problem for the neoclassical theory. If well identified, monetary policy shocks are not correlated with uncertainty, depressed aggegrate demand etc. There is still a lot of controversy about identification of moneatry shocks but this finding of Bernanke and Gertler have never been challenged at our knowledge.

A second strand of the literature have challenged the neoclassical theory. Estimation of investment equation struggle to find a significant effect of real interest rates. The consensus in the literature is that estimating short run elasticity of investment to interest rate with aggregate datas do not provide any evidence backing a significant effect of interest rate on business investment [START_REF] Blanchard | Comments and Discussion on Investment, Output and The cost of Capital[END_REF], Caballero 1994, Bernanke and Gertler 1994[START_REF] Chirinko | Business Fixed Investment Spending: Modelling Strategies, Empirical Results, and Policy Implications[END_REF], Sharpe and Suarez 2015). When measuring the different channels of monetary policy, Bernanke and Gertler (1994) shows that the response of business investment to a recessionnary FED fund rate shock is negative but small and lagging behind the large response of residential investment and nondurable consumption. The lag and the size of the response actually suggest more a side effect of the residential investment response through accelerator phenomena than a user cost effect. Estimates of long run elasticity (Caballero 1994, Schaller 2006) and studies using microeconomic datas (Cummins, Hassett and Hubbard 1994, Chirinko Alone, a bad post financial crisis performance despite historically low real rates, small response to monetary policy shocks in VAR, small effects of real rates in investment equation estimation with macroeconomic datas and microeconomic datas, and a majority of firms claiming they do not change their investment plan following a fall in interest rates, would not be necessarily conclusive. Together, they suggest to look for other theories in which corporate investment would not react to real interest rate through the user cost channel. This paper contributes to fill this gap. I provide a simple model of adverse selection on capital markets. I find that in infinite horizon, the Incentive Compatibility Constraint takes the form of a linear relation between investment and firm cash flows. The user cost of capital is no longer relevant. I embed this relation in a dynamic macroeconomic model and study the response of aggregate investment to permanent and temporary changes to real rates. I specify the adverse selection problem in the first section, define and characterize the equilibrium in the second, and solve the macroeocnomic model in teh third.

The adverse selection problem

In this section, I outline a partial equilibrium model of capital markets in which firms try to get funds from lenders in order to finance their investment. Firms holds private information about their riskiness. Firms try to signal their riskiness by constraining the amount they borrow to be a fraction of their retained earnings.

Firms

Production function The economy is populated by a continuum of firms. A firm j have a production function

Y j = πK j (1)
.

Two type of firms There are two type of firms in the economy: Bad firms denoted with superscripts B and good firms denoted by the superscript G. Bad and good firms does not differ by their productivity but by their riskiness. At each period, a good firms has a probability κ to become a bad firm. With probability 1 -κ, it remains a good firm. A bad firm still produce πK j but has a probability 1 -λ to exit at the next period. When it exits, a firm produces nothing and its capital stock is worthless. Neither the lender nor the borrower recover anything.

I summarize the timing by the following tree.

Good πK

1 -κ κ Good πK Bad πK 1 -κ κ λ 1 -λ Good πK Bad πK 0 1 -κ κ λ 1 -λ Figure 1: state Tree
The firm problem Shareholders wants that firms maximize the discounted sum of their expected dividends.

V (K, B) = max K ′ ,B ′ ,I,S,E,d d + 1 1 + r EV ′ (K ′ , B ′ ) (2a) w.r.t K ′ = K + I (2b)
B ′ = B + r e E (2c) πK -B = d + S (2d)
I = E + S (2e) (πK -B) ≥ d ≥ (1 -s)(πK -B) (2f) 0 < E < E max (2g)
The first constraint is the law of motion for capital stock. Next period capital stock is equal to previous one plus investment. There is no depreciation. The second constraint is the law of motion for interest repayments. Next period interest repayments B ′ are equal to current one plus the new borrowing E times the specific interest rate charged on the firm r E . Indeed, I assume that firms finance themselves by borrowing at infinite maturity. i explain the equation with more details in a dedicated paragraph below. The equation (2C) split the income of the firm πK -B, equal to the production minus interests repayments, between dividends denoted d and retained earnings denoted S. The investment I is financed by retained earnings S and borrowing E. Dividends are constrained to have a minimal and a maximal value. They cannot be higher than the income of the firm and should be bigger than (1 -s)(πK -B) where s is an exogenous parameter between 0 and 1. Borrowing should also be set between 0 and a maximal positive value E max .

Loans I assume that loans have infinite maturity. Interest rates on past loans are fixed. Only interest rate on new loans may vary. Thus for an amount E 0 borrowed at period 0 , the firm should pay the lender r e 0 E 0 at each period. At a given period t, the total repayment b t of the firm is the sum

B t = r e t-1 E t-1 + r e t-2 E t-2 + r e t-3 E t-3 + ... + r e t-n E t-n + ...
where E t-n is the amount of money borrowed at period tn and r e t-n is the interest rate at period tn which includes a firm specific risk premium.

This assumption allows me to derive a very simple credit constraint. Allowing for shorter maturity are interesting but introduces complex issues about optimal maturity design which is not the core of this paper. However, it is worth noting it is better for firms to accumulate short term assets and long term liabilities in this framework.

Solving the firm problem

The firm problem is not always well defined. The discounted sum of dividends does not always converge because dividends may grow faster than the safe interest rate for some firms. When this is the case, I rely on heuristic arguments to determine the best strategy of the firm.

Lenders

There is a continuum of lenders which behaves in a competitive way. They have the choice between lending to firms and buying a one period bond generating a safe return r which is the safe short term interest rate. They can observe earnings, dividends, investment and savings of a firm but they do not observe the true type of the firm. Firms will try to reveal their type by sending an appropriate signal to lenders. As I will show, the signal will be the ratio of investment over retained earnings I S . Before defining and characterizing the equilibrium, I introduce some notations and compute interest rate for each type of firms.

Interest rates and values

Suppose that good and bad firms have fully revealed their type to lenders. Lenders should be indifferent between lending one unit of good to a good firm (resp. bad firm) and receiving r G (resp. r B ) unit of goods at each period while the firm survive and buying a safe assets generating a safe return r at the following period.

r B λQ ′B = 1 + r (3a) r G (1 -κ)Q ′G + κQ ′B = 1 + r (3b)
Where

Q B = 1 + λ 1 + r Q ′B (4a) Q G = 1 + 1 1 + r (1 -κ)Q ′G + κQ ′B (4b) Q B (resp. Q G )
is the discounted sum of one unit of goods delivered at each period by a bad (resp. good) firm while the firm survives.

Values and policy rule definition

I denote V B and V G the values of respectively good and bad firms. I denote (I G , S G ) the policy rule of a good firm for retained earnings and investment. Other choice variables can be deduced using equations (2b)-(2e). Similarly, (I B , S B ) is the policy rule of bad firms. Both values and policy rules are not real numbers function of the state variables : capital stock K and debt B.

The value of bad firms will depend to the signal it will send to lenders. If it sends the signal which allows lenders to identify it as a bad firm, it will have to pay the interest rate charged on bad firms r B . If it sends the same signal as good firms, it will pay the interest rate r G . I define the value of the firm for both case.

Computing the respective value associated to these two strategies raises some issue. In finite horizon, it is possible to start from the last period and to compute what is the best strategy in the last period. Then, you can compute the value of the two strategies for the previous period and repeat the same process until the current period. Such a solution is not available in infinite horizon. I use the following critical assumption Hypothesis 1 In period t, a bad firm will compute its value using the assumption it will send the "bad" signal in period t + 1 With this assumption, i can define the value of sending the "bad" signal

V B,B = d + λ 1 1 + r V ′B,B w.r.t K ′ = K + I B ′ = B + r B E πK -B = d + S I = E + S
and the value of sending the "good" signal in period t is

V B,G = d + λ 1 1 + r V ′B,B w.r.t K ′ = K + I B ′ = B + r G E πK -B = d + S I = E + S 2 Equilibrium 2.

Equilibrium definition

I can now define the separatin equilibrium of this economy Definition 1 An equilibrium is a vector (r B , r G ), a vector of policy function (I B , S B ) and a vector of policy function (I G , S G ) such that

• ∀K, B , V B,B (K, B) ≥ V B,G (K, B)
• (I G , S G ) are the optimal policy function for the program (2) with r E = r G .

• (I B , S B ) are the optimal policy function for the program [START_REF] Smith | Financial Intermediation and Regime Switching in Business Cycles[END_REF] with

r E = r B .
The first condition states that all bad firms have an incentive to send the "bad" signal. The two thers states that both good and bad firms adopts the best policy.

Equilibrium characterization

I can characterize the equilibrium in a very simple way with the following assumption Hypothesis 2 Parameters κ, λ, π and the sequence of interest rates (r t ) t∈N are set such that

∀ t, r B t ≥ π ≥ r G t
With this assumption, I am able to determine and characteirze the best policy of the two type of firms when their true type is revealed to lenders.

I G = s(πK -B) + E max , E G = E max , d G = (1 -s)(πK -B)
These propositions are quite intuitive. If bad firms invest and borrow I B t , they will receive πI B t at each period and will pay r B t I B t . Thus an investment generates a negative stream of income. If the firm do not borrow, they have to choose between investing I B t generating π at each period with dying probability λ and buying a safe asset. It is possible to show that it is equivalent to the case of borrowing. In a similar way, if good firms invest and borrow I G t , they will receive πI G t at each period and will pay r G t I G t . A unit of additionnal investment integrally financed by debt always generates a positive stream of income. Good firms want to invest and borrow as much as possible.

The incentive compatibility constraint

For the bad firm, the value of distributing all its profit as dividend and not investing whereas paying a high interest rate on its debt should be superior to the value of paying a lower interest rate whereas investing the same fraction of its earnings than good firms. The incentive compatibility constraint can be expressed in a simple way Proposition 3 We have V B,B ≥ V B,G if and only if

I G ≤ r B -r G π -r G S G = ψS G Proposition 4
Under assumption (2) Good firms wants to invest and borrows as much as possible, thus the incentive compatibility constraint is binding

I G = r B -r G π -r G S G = ψS G (5)
I now detail the computations which gives this expression. A linear value function can be derived for V B,B and V ′B,B .The solution method is straightforward. I guess that the value is a linear function of capital stock and interest repayments.

V B,B (K, B) = φ K K -φ B B (6)
. Using undetermined coefficients method, I solve for φ K and φ B .

The value function can be rewritten in the following way.

V B,B t = φ K K -φ B B = πK -B + λ 1 1 + r [φ ′ K K -φ ′ B B] I deduce φ K = π + λ 1 1 + r φ ′ K = πQ B φ B = 1 + λ 1 1 + r φ ′ B = Q B
The incentive compatibility constraint implies

V B,B ≥ V B,G (7)
I can now rewrite it

πK-B+λ 1 1 + r Q ′B πK -Q ′B B ≥ πK-B-S G +λ 1 1 + r Q ′B (K + I G ) -Q ′B (B + r G (I G -S G ))
By simplifying, I get

1 + r λ S G ≥ + Q ′B I G -Q B r G (I G -S G )
Using equation (3a) which gives interest on bad firms with respect to safe interest rate, I get the ICC under the compact form highlighted in proposition (3)

I G ≤ r B -r G π -r G S G

Leverage and interest rate

Equation [START_REF] Bernanke | The financial accelerator in a quantitative business cycle framework[END_REF] gives a linear relation between investment and cash flows or retained earnings.

The ratio r B -r G π-r G is the leverage on cash flows and I denote it ψ. It is interesting to note that this constraint is about flows and not stocks unlike most of popular financial frictions in macroeconomics. This equation makes the user cost irrelevant to determine investment.

However, the leverage is not independant from interest rate. Interest rate affects it through r B and r G . But, interestingly it is an increasing function of the interest rate.

I compute the derivative of the leverage with respect to r ∂ψ ∂r =

(

∂r B ∂r -∂r G ∂r )(π -r G ) + (r B -r G ) ∂r G ∂r (π -r g ) 2
In first approximation, a fall in r has the same impact on r G and r B , thus an approximation for the derivative is ∂ψ ∂r = ψ(r)

∂r G ∂r (π -r g )
r G is an increasing function of r and its derivative is positive. thus, ∂ψ ∂r > 0. This result implies that firms will tight their investment policy for a given cash flow when real interest rates falls. This counterintuitive result is in line with the seminal paper of Stiglitz and Weiss (1977). the intuition is simple. Suppose the owner of a bad firm sends the "good" signal and invest for example ψ unit of good and borrows ψ -1 unit of good. This strategy has a beneift and a cost. The benefit is the following. The ψ -1 units borrowed generate a return π and only cost r G at each period. The cost is that it should retain one unit of goods into the firm instead of distributing them as dividends and investing them in the safe asset. Suppose now that the interest rate falls. Buying the safe asset generates a lower return and thus the opportunity cost of investing one unit of good into the firm is lower. The cost of sending the "good" signal falls.

In the smae time, the benefit increases. r G is lower. The ψ -1 unit of goods borrowed generates more profits at each period. The owner of a bad firm has much more incentive to send the "good" signal instead of the bad one. Owner of good firms have to tight their investment policy to signal they are good. By lowering the leverage ψ, they will lower the benefit of sending the good signal for bad firms. Second, it is a tractable formula for investment. Adverse slection in infinite horizon is not a simple problem in first thought. Finding such a simple equation is interesting.

Third, a relation between cash flows and investment is empirically plausible. The sensitivity of investment to cash flows is a regular feature of empirical studies (unlike Tobin's Q or user cost of capital). Finding a simple theoretical justification for it is also quite interesting.

Given this attractive features, it is logical to develop the framework of the rpevious section in a dynamic macroeconomic model. This model have three blocks: Prices equation, dynamic equations for good firms, dynamic equations for bad firms. Price equations gives interest rates for bad and good firms and the value of the leverage.

Q B t = 1 + 1 1 + r t λ(1 -φ)Q B t+1 (8a)
Q G t = 1 + 1 1 + r t [(1 -κ)(1 -φ)Q H t+1 + κ(1 -φ)Q B t+1 ] (8b) r B t λQ B t+1 = 1 + r t (8c) r G t (1 -κ)Q G t+1 + κQ B t+1 = 1 + r t ( 8d
)
ψ t = r B t -r Gt π -r G t (8e)
Equations (10 a) to (10 d) allows to derive interest rates specific to good and bad firms as explained in the previous section. Equation (10 E) gives the leverage of period t with respect to these two interest rates and the productivity of capital π.

Dynamic equations for good firms are

K G t+1 = (1 -κ)(K G t + I G t ) (9a)
B G t+1 = (1 -κ) B G t + r G t (I G t -S G t ) (9b) πK G t -b G t = S G t + d G t (9c) d G t = (1 -s)(πK G t -B G t ) (9d)
I G t = ψ t S G t (9e)
Equation (11 A) gives the capital accumulation of good firms at the aggregate level. Next period capital stock is the current capital stock plus investment times the "survival" rate of period t good firms at period t + 1. Equation (11 B) gives the debt accumulation equation for good firms, debt being measured by interests retayments. Equation (11 C) is an accounting equation. Aggregate income of good firms is split between aggregate savings and aggregate dividends. All variables are in period t, so the survival rate is not relevant here. This is also the case for the two others equation which gives the aggregate policy rule of good firms for dividends and investment.

Similar equation can be given for bad firms

K B t+1 = κK G t + (1 -λ)K B t (10a)
B B t+1 = (1 -λ)B B t + κ[B G t + r G t (I G t -S G t )] (10b)
d B t = π t K B t -b B t (10c)
Bad firms do not invest but a fraction κ of good firms becomes bad at period t and "feed" capital stock and debt of the bad firm sector. All the income of bad firms is ditributed as dividends.

The real interest rate r is exogenous in the model. Indeed, I focus on the response of investment to an exogenous change of r. The exogeneity of the interest rate also allows to solve the three blocks separately Proposition 5 The system of equations ( 8) can be solved independanlty from system (9) and [START_REF] Chirinko | Business Fixed Investment Spending: Modelling Strategies, Empirical Results, and Policy Implications[END_REF] Indeed, none of the eight variables of the two systems appears in the five equations of the system (8).

Proposition 6

The sytem (9) can be solved independantly from system [START_REF] Chirinko | Business Fixed Investment Spending: Modelling Strategies, Empirical Results, and Policy Implications[END_REF] Because bad firms always remains bad and does not affect interest rates, the capital stock, the debt and the dividends of bad firms do not affect good firms.

Dynamic of investment I now show that the system (9) can be reduces to one recursive equation for debt from which it is possible to deduce the dynamic of investment and growth.

The system has to be stationnarized. Indeed, the AK production function allows for endogenous growth. the solution is to divide all variables by K G t and to replace K G t+1 by 1 + g t+1 where g is the growth rate. Stationnarized variables are denoted in lowercase.

The five stationnarized equations can be simplified in asystem of three equations whose variables are debt b, investment i and growth

(1 + g t+1 )b t+1 = (1 -κ)(b t + r G t (ψ t -1)s(π -b t ) (11a) 1 + g t+1 = (1 -κ)(1 + i t ) (11b) i t = ψ t s(π -b t ) (11c)
Replacing (11 C)

t+1 = b t + r G t (ψ t -1)s(π -b t ) 1 + ψ t s(π -b t ) ( 12
)
The equation is nonlinear. To give some intuition, I plot a typical graph associated to the It is easy to see that one of the roots is equal to π. in that case, all the firms income is distributed as dividends, there is no growth and no investment, so interest repayements over capital stock remains at a fixed level. This equilibrium is quite degenerate because the economy will grow at a negative rate -κ. The root given by r G ψ-1 ψ is also intuitive. When a firm invest ψ, it borrows ψ -1 and thus will repay r G (ψ -1) forever. The figure [START_REF] Hopenhayn | Optimal Lending Contracts and Firm Dynamics[END_REF] suggests that one root is stable and one unstable. Fortunately, the stable root is the non degenrate one.

Proposition 8 b * is an unstable root and b * * is the stable root.

I now focus on the stable root and compute the corresponding investment and growth rate.

Proposition 9 For the stable steady state, growth and investment rate given by the system of equations ( 17) are

i * * = sψπ -sr G (ψ -1) (13a) g * * ≃ sψπ -sr G (ψ -1) -κ (13b)
An even simpler result may be obtained when I replace the leverage by its value with respect to r G , r B and π given by the incentive compatibility constraint Proposition 10 When replacing the leverage ψ by its value given by equation ( 13)

i * * = sr B (14a) g * * ≃ sr B -κ (14b)
The investment rate is a function of the firm "saving" rate s and the interest rate charged on bad firm.

Investment

Temporary changes I now explore the effects of a temporary change. I consider a fall of real rate from 4.5 percent to 2 percent for 5 periods. Then, r go back to 4.5 percent. I plot the response in I plot the response in figure (3). On impact, leverage and corporate investment falls. They recover after five periods. After these five periods, interests repayments are lower than their steady state value As a consequence, investment has a peak and is slighlty higher than its steady state value. Then debt and investment converges to their long run values.

Literature

The paper is related to several strand of literature. It belongs in particular to adverse selection literature in macroeocnomics and finance. Adverse selection is a popular firction in finance.

According to [START_REF] Myers | corporate financing and investment decisions when firms have information investors do not have[END_REF], it may explain the pecking order of financing source for firms. It was also prominent in De Marzo and Duffie (1995) and Albuquerque and Hopenhayn

Conclusion

I aim to explain why fall in real interest rate may not be able to increase corporate investment.

I provide a model of corporate investment with financial frictions which effectively eiminates any user cost of capital effect, the standard channel by which lower rates increase investment.

Like all economic models, it is based on questionnable assumption but is simple and may be solved by hand, a non trivial feature for an endogenous growth model with adverse selection in infinite horizon. The main drawback of the model is that it is too succesful in achieving its purpose. Not only it reduces positive effects of low interest rate on investment but it creates powerful negative ones. This negative effect is probably too large to fit macroeconomic datas. However, my model can be a good starting point to develop an alternative model of corporate investment. Several assumption can be relaxed like the exogeneity of the firm saving rate. Another possibility would be to make the incentive compatibility constraint not always binding.

Proposition 1 Proposition 2

 12 If hypothesis 1 holds, under separating equilibium, bad firms do not save, invest and borrow. S B = 0, I B = 0,d B = πK -B If hypothesis 1 holds, under separating equilibium, good firms wants to invest and borrows as much as possible. They also distribute the minimum level of dividends

3 The macroeconomic model 3 . 1

 31 The modelFrom the viewpoint of the economist, the investment equation obtained in the previous section has three adppaling features First, it is a microfounded relation. There is a clear link between the cash flow leverage and real interest rate.

  in (11 B) and (11 B) in (11 A) leads to recursive equation for b b

  Steady state I solve analytically for the steady state. The two stationnary debt levels is given by the roots of the recursive equation Proposition 7 Equation (13) have two roots b * and b * * b * = π b * * = r G (ψ -1) ψ

(

  2004). It was less popular in macroeconomics compare to other frictions like costly state 1984) or Collateral constraint (Hart and Moore 1994). There are some exceptions, including the seminal paper of Stiglitz and Weiss (1981), but also papers by Kurlat (2013, forthcoming), or Boissay, Collard and Smets (2015). This lack of popularity is probably caused by the relative complexity of adverse selection problems. My first contribution is to provide a very simple model of adverse selection in infinite horizon. The second contribution is to show that adverse selection is a microfoundation for a linear relation between investment and cash flows. A significant correlation between the two is nearly systematic in investment equation regression. My model confirms that interest rate may have adverse effects under private information problems, an effect already found in Stiglitz and Weiss (1981). Boissay, Collard and Smets also find a reversed curve for loan demand in interbank market with adverse selection.The paper is also related to other macroeconomic models with financial frictions (see Beranke and Gertler (1989),Kiyotaki and Moore (1995), Beranke, Gertler and Gilchrist (1999), Christiano, Motto, Rostagno (2014)). The key difference is that user cost of capital still have substantial effects on investment in these models. Some papers have studied adverse effects of interest rate on output in macroeconomic model without investment. Bilbiie(2008) shows that Hand to Mouth consumers may reverse the sign of output euler equation. Cochrane (2015)shows that lower rates may reduce output through the Fisher equation.

  Fazzari and Meyer 1998, Mojon, Smets and Vermeulen 2001) provide better evidences for a user cost of capital effect on investment. However, it is still unclear if their findings imply a high elasticity of investment to interest rate as noted by Sharpe and Suarez. For example, the estimate of user cost elasticity in Cummins, Hassett and Hubbard (1994) is unchanged when real interest rate is replaced by a fixed discount rate in the measure of the user cost (Sharpe and Suarez 2015). Direct measure of investment sensitivity to interest rate are not better. For example Kothari and Warner(2015) finds that interest rate is unable to predict corporate investment whereas for example interest rate predicts noncorporate investment. Facing these very mixed empirical results, Sharpe and Suarez(2015) have proposed a completely different approach. Instead of using econometric techniques to identify correlation, they directly ask to CFO (Chief Financial Officers) in what extent their investment decision is sensitive to interest rate. Results are very instructive. 68 % of CFO says that their investment plans will remain unchanged if interest rate falls.

  3.2 The response to a change in interest rateintuition Equation (11C) shows that corporate investment is sensitive to the leverage ψ and the debt level b t . Both reacts to change in interest rate but in opposite direction and with a different timing.The debt level b t is a state variable. A fall in r will not affect b t on impact but will lower r G pushing downward b t+1 . The income of good firm will be unaffected in period t but will increase in period t + 1 allowing an increase in retained earnings and in investment all things being equal. This income distribution channel is very intuitive and appears in every model with financial friction (see Bernanke Gertler 1989 and Kiyotaki and Moore 1995).Permanent changes A fall in real interest rate generates two opposite effects on investment. The response of steady state investment to a permanent fall of r is ambiguous in first approximation. However, the equation (14A) shows that the leverage effect clearly dominates the income distribution effect. Investment is proportionnal to the interest rate on bad firms which is increasing with safe interest rate.

	Figure (2) confirms this domination. It represents the steady state investment for various
	level of interest rate and shows an increasing, nearly linear function
	0.16
	0.15
	0.14
	0.13
	0.12
	0.11
	0.1
	0.09
	0.02	0.025	0.03	0.035	0.04	0.045	0.05	0.055	0.06	0.065	0.07
						Interest rate

What is specific to the modle is the response of the leverage ψ As I have shown in the previous section, the leverage is an increasing function of the interest rate and will react instantaneously to a shock.

A Proofs

A.1 Proof for proposition 1 Proof 1 It is obvious that firms do not want to borrow in order to invest, because return of investment π is lower than repayments r B .

I will now try bto show that dividends will be set according to proposition 1. Suppose that bad firms set borrowing, dividends and investment following the proposition 1. Consider now a deviation for investment at period 0. d 0 = (πK 0 -B 0 ) -δd 0 and I 0 = δd 0 Investment remains unchanged after period 0. Thus ∀ t > 0, d t = (πK 0 -B 0 ) + πδd 0 . The deviation for the discounted sum of expected dividends is

] < 0 Deviating from the optimal strategy has a negative net present value

A.2 Proof for proposition 2

Proof 2 It is obvious that firms want to borrow as much as possible in order to invest, because return of investment π is higher than repayments r G .

I will now try bto show that dividends will be set according to proposition 2. Suppose that good firms do not set dividends and investment following the proposition 2 and pay higher dividends. For example ∀ t, d t = (1 -s ′ )(πK * t -B * t ) and I * t = E max + s ′ (πK * t -B * t ) with s ′ < s. (K * t ) t∈N + and (B * t ) t∈N + are the implied sequence of capital stock and debt. Consider now a deviation for investment at period 0. d 0 = (1 -s ′ )(πK 0 -B 0 ) -δd 0 and I 0 = δd 0 Investment remains unchanged after period 0. Thus ∀ t > 0, d t = (1-s ′ )(πK * t -B * t )+πδd 0 . The deviation for the discounted sum of expected dividends is The derivative f ′ (π) = 1 + s(ψπr G (ψ -1)). Because by assumption 1 π > r G it is clear that the derivative of f is always superior to one when b t = b * . thus b * is an unstable root.

The derivative f ′ (r G ψ-1 ψ ) = 1+s(ψπ-r G (ψ-1)) [1+s(ψπ-r G (ψ-1))] 2 < 1. So b * * is a stable root.


Philippe Anne-Charlotte

Lavault Sophie

Fick Rutger

Wassermann Demian

Valabregue Romain

Deriche Rachid

ARNULF ISABELLE Levy Richard

Lehericy Stephane

Brain correlates of apathy in Kleine Levin syndrome: a mean apparent propagator study

Kleine-Levin syndrome (KLS) is a rare neurological disorder characterized by episodes of severe hypersomnia, apathy, cognitive impairment, derealization and behavioral disturbances. Between episodes, patients have normal sleep, mood and behavior. Apathy is a prominent clinical feature of KLS but its pathophysiology is not known. Here we used mean apparent propagator to investigate white matter changes in KLS and correlated di�usion changes with apathy scores. Results showed that the corpus callosum was involved in KLS during episodes and mean RTAP measures in the corpus callosum correlated with apathy scores. Results were in accordance with known motivation-based circuits involving the orbitomedial frontal cortex.

Purpose

Kleine-Levin syndrome (KLS) is a rare neurological disorder that mainly a�ects adolescents. KLS is characterized by relapsingremitting episodes of severe hypersomnia, apathy, cognitive impairment, derealization and behavioral disturbances. Between episodes, patients have normal sleep, mood and behavior [1]. Each episode is of brief duration varying from a week to 1-2 months. No de�nite cause has been identi�ed [1]. Anatomical MRI scan is normal, but brain scintigraphy can be abnormal during and between episodes [START_REF] Kas | Feeling unreal: a functional imaging study in patients with Kleine-Levin syndrome[END_REF]. Apathy is a prominent clinical feature of KLS [START_REF] Lavault | Kleine-Levin syndrome in 120 patients : Di�erential diagnosis and long episodes[END_REF] but its pathophysiology in the disease remains to be established. The mechanisms responsible for apathy may involve several circuits connecting the frontal lobes to the basal ganglia [START_REF] Levy | Apathy: a pathology of goal-directed behaviour. A new concept of the clinic and pathophysiology of apathy[END_REF]. Here, we performed TBSS analyses on the return-to-the-axis probability (RTAP) measures that may be linked to apparent axonal diameter and in�ammation [START_REF] Özarslan | Mean apparent propagator (MAP) MRI: A novel di�usion imaging method for mapping tissue microstructure[END_REF] to analyze the integrity of white matter (WM) microstructure of healthy volunteers (HV) compared to symptomatic (KLS-S) and asymptomatic (KLS-AS) patients.

Methods

We prospectively included 20 KLS-AS (mean age: 22.2 ± 8.9 years, 9 males) and 20 HV age and sex-matched one by one. Twelve of these 20 patients were also scanned during episodes (KLS-S). Apathy was assessed using the Starkstein Apathy score [START_REF] Starkstein | Reliability, validity, and clinical correlates of apathy in Parkinson's disease[END_REF]. Di�usion-weighted images were acquired using a Siemens Verio 3T with a 12-channel head coil (GRAPPA=2; TR/TE=7.7s/92ms; voxel size: 2.5mm isotropic; 64, 32 and 8 gradient directions for b-values of 1800, 700, and 300 s/mm² respectively) and 8 images without di�usion weighting were also acquired.

We preprocessed the images using FSL (fsl.fmrib.ox.ac.uk/fsl/fslwiki/) and included correction for susceptibility (topup) and for eddy current distortions (eddycor) and creation of a binary mask of the brain (bet). Then, we generated RTAP maps of all subjects [START_REF] Fick | MAPL: Tissue microstructure estimation using Laplacianregularized MAP-MRI and its application to HCP data[END_REF]. Voxelwise statistical analyses were carried out using TBSS, part of the FSL comparing RTAP maps of HV versus KLS-AS, HV versus KLS-S and KLS-AS versus KLS-S (paired t-test). Finally, in the TBSS-based ROI, we computed correlations between clinical scores (disease duration and apathy scores) and mean RTAP measures in this ROI and performed tractography on one healthy subject to determine its projection �bers. Results were considered signi�cant at p<0.05, fully corrected for multiple comparisons across space.

Results

There were no signi�cant di�erences in RTAP between HV and KLS-AS. In KLS-S compared to KLS-AS, RTAP increased in the corpus callosum (CC). There was a correlation between apathy scores and mean RTAP in the CC of KLS-S (p=0.03, r=0.61) (see Figure 2). There were no other correlations with clinical scores in KLS-AS as well as in KLS-S. Using the TBSS-ROI as a seed for tractography, �bers passing through the CC with abnormal RTAP measures projected to medial orbitofrontal cortex (see Figure 3).

Discussion and conclusion

The results highlight the presence of structural changes correlated to the apathy score in the anterior portion of the CC during episodes, a region where �bers project onto the medial orbitofrontal cortex. As, these prefrontal regions are involved in motivation processes [START_REF] Levy | Apathy: a pathology of goal-directed behaviour. A new concept of the clinic and pathophysiology of apathy[END_REF], this suggests that apathy in KLS could result from di�culties to provide the a�ective/motivational value of a given behavioral context.

Figures

Figure 1 :

 1 Figure 1: TBSS results from group comparisons analysis showing voxels (in red-yellow, thickened for better visibility) with decreased RTAP values (p<0.05) in KLS-S compared with KLS-AS. Green represents the mean FA skeleton of all the participants.

Figure 2 :

 2 Figure 2: Correlation (p=0.03, r=0.61) between mean RTAP in the TBSS-based ROI and apathy scores.

Figure 3 :

 3 Figure 3: TBSS-based ROI projection �bers.

Acknowledgements

The research leading to these results received funding from the programs 'Institut des neurosciences translationnelle' ANR-10-IAIHU-06 and 'Infrastructure d'avenir en Biologie Santé' ANR-11-INBS-0006.


Benjamin Rohaut
email: benjamin.rohaut@gmx.com

Federico Raimondo

Damien Galanaud

Jacobo Mélanie Valente

Diego Sitt

Lionel Naccache

Probing consciousness in a sensory-disconnected paralyzed patient

Keywords: disorder of consciousness, quantitative EEG, brain functional imagery, consciousness, locked-in syndrome

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Diagnosis of consciousness can be very challenging in some clinical situations such as severe sensory-motor or cognitive impairments [START_REF] Rohaut | Neurology of consciousness impairments[END_REF] . When patients can neither communicate (verbally or non-verbally) nor exhibit any intentional behavior, clinicians have to rely on indirect cues collected through behavioral and functional brain-imaging methods [START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF] . During the last decade, several cognitively active tasks have been developed in this perspective while recording brain activity with fMRI or bedside EEG. For instance, putative neural signatures of conscious processing have been identified by asking a patient to perform a mental imagery task [START_REF] Owen | Detecting Awareness in the Vegetative State[END_REF][START_REF] Cruse | Bedside detection of awareness in the vegetative state: a cohort study[END_REF] or an explicit mental counting of rare stimuli delivered among frequent ones [START_REF] Bekinschtein | Neural signature of the conscious processing of auditory regularities[END_REF][START_REF] Faugeras | Probing consciousness with event-related potentials in the vegetative state[END_REF] . However, each of this measures require the patient to receive and perceive task instructions. Obviously, a sensory disconnected brain challenges this original approach given the impossibility to deliver instructions to the patient. In the present case study, we report how we could convincingly identify EEG neural signatures of consciousness in such a "sensory disconnected" and paralyzed patient.

Case study

The patient was 63 years old when he suffered from a brainstem infarction following a scheduled abdominal surgery that subsequently raised the question of the diagnosis of a locked-in syndrome. He had several medical conditions including arrhythmic cardiopathy, inherited thrombotic factor (factor V Leiden mutation), hypertension, dyslipidemia, atherosclerosis and a chronic renal disease which required kidney transplantation 3 years ago. After surgery (scheduled colic continuity reestablishment surgery following a Hartmann surgery for an acute colitis), the patient suffered from a septic shock. Sedation was maintained at the minimum level and duration following international guidelines [START_REF] Barr | Clinical practice guidelines for the management of pain, agitation, and delirium in adult patients in the intensive care unit: Executive summary[END_REF] and stopped after four days. Although the patient recovered wakefulness, he remained unable to express any intentional behaviour. MRI revealed an infarction of the median and transversal bulbo-protuberantial territory in relation with a basilar artery occlusion (Figure 1; Figure S1-B). Neurological examination revealed a spontaneous ocular bobbing (Video S1) with no visual pursuit, no blink to visual threat, and no startle reflex to loud noise. Pupillary light reflexes, corneal and cough reflexes were preserved. The patient did not show any movement of the limbs or of the face, even after nociceptive stimulation. According to the current behavioural and clinical scores, this patient was classified as being in a vegetative state/unresponsive wakefulness syndrome (VS/UWS; Table 1). This mismatch between a radiological pattern reminiscent of locked-in syndrome, and a much poorer behavioural evidence of conscious processing triggered the decision to transfer the patient to our "Disorders of Consciousness patients Team" two months after the diagnosis of the brainstem infraction.

Beyond behavior

A quantitative MRI analysis of the supra-tentorial white matter fractional anisotropy [START_REF] Luyt | Diffusion tensor imaging to predict long-term outcome after cardiac arrest: a bicentric pilot study[END_REF] revealed a normal pattern, and therefore invalidated the hypothesis of a per-surgery or post-surgery shock -related severe brain hypoperfusion (Figure 1, Figure S1-A).

Brainstem auditory evoked potentials (BAEP) revealed a severe peripheral and central deafness with a bilateral alteration of wave-I and III (110dB stimulation level) and the abolition of wave-V. This pattern explained the absence of startle reflex, and the potential inability to understand verbal commands. The auditory deficit resulted from the association of an otitis media, a cochlear dysfunction related to tacrolimus toxicity, and a possible VIIIth nucleus and intra-axial VIIIth nerve lesions (Figure 1).

We did not find any cerebral responses elicited by the auditory 'Local Global' evoked-related potentials paradigm. This paradigm developed by our team is usually able to probe neural signature of conscious processing in patients [START_REF] Bekinschtein | Neural signature of the conscious processing of auditory regularities[END_REF][START_REF] Faugeras | Probing consciousness with event-related potentials in the vegetative state[END_REF][START_REF] Faugeras | Event related potentials elicited by violations of auditory regularities in patients with impaired consciousness[END_REF] . However, in this case, the patient's deafness prevented a reliable interpretation of this negative result. SSEP revealed a preservation of bulbar responses (N14 and N18), but a severe alteration of cortical responses (abolition of N20 on the left side and severe alteration on the right side).

EEG background activity, -recorded at bedside with 8 electrodes -, consisted in a posterior and symmetrical theta band (4-6Hz) activity, slightly reactive to passive eye-opening/closing, but neither to auditory nor to nociceptive stimulation.

At this stage, we hypothesized that the patient was indeed in a particular form of locked-in syndrome (also called deefferentation syndrome) since associated with deafness and severe visual impairment that could be called a "deafferented locked-in syndrome". Several attempts to deliver the patient with visual written instructions (asking to move his eyes) failed. However, the absence of horizontal eye mobility as well as the permanent bobbing could not rule out a patient's inability to accurately read verbal commands.

To overcome this severe multi-sensory disconnection, we used an automatic classification algorithm we recently designed and trained to distinguish between VS/UWS and minimally conscious state (MCS) patients on the basis of a high-density EEG recording while stimulated under an auditory oddball paradigm [START_REF] Sitt | Large scale screening of neural signatures of consciousness in patients in a vegetative or minimally conscious state[END_REF][START_REF] Engemann | Automated Measurement and Prediction of Consciousness in Vegetative and Minimally Conscious Patients[END_REF] . This classifier combines several tens of putative EEG markers sensitive to conscious states such as event related potentials, signal complexity, long-range functional connectivity and spectral power patterns. Although the patient was deaf and thus, the event related potentials were absent, our classifier considered this patient as being most likely in a condition of minimally or conscious state (71%), while a VS/UWS could not be ruled out (29%). We then analyzed in detail the univariate markers that are usually more contributive to the multivariate classifier by comparing the topographies from the previously recorded and diagnosed patients with this patient. Detailed analysis further revealed that patterns of spectral power of EEG (in delta and alpha band), complexity markers (spectral entropy, permutation entropy and Komolgorov-Chaitin complexity), and functional connectivity (weighted Symbolic Mutual Information, wSMI [START_REF] King | Information sharing in the brain indexes consciousness in noncommunicative patients[END_REF] ) were strongly supportive of a conscious state, rather than of a VS/UWS or even MCS (Figure 2). As expected, univariate markers of auditory event-related potentials were not contributive given patient's deafness.

In summary, anatomical and electrophysiological brain data supported a diagnosis of "extreme locked-in syndrome" with major motor and multi-sensory deficits. After a week of systematical attempts to deliver visual instructions, -strongly encouraged by the results of our exploration supporting consciousness -, we eventually could observe a reproducible visual pursuit of a mirror, and then of any vertically moving target (bobbing was still present and vertical oculocephalic reflex still abolished suggesting a vestibular input impairment). According to this visual pursuit behaviour, the patient was now scored as MCS on the CRS-R [START_REF] Giacino | The JFK Coma Recovery Scale-Revised: Measurement characteristics and diagnostic utility[END_REF] . However, and in spite of our efforts, the patient could not follow written commands (see Video S2). Three weeks after patient admission (one week after high-density EEG recording), we could finally detect a more complex behaviour: in response to a big contrasted arrow pointing either upward or downward, the patient was able to systematically orient his gaze toward the direction pointed by the arrow (see Video S3). These responses strongly suggested that the patient was able: i) to engage attention, ii) to understand the meaning of this symbol, iii) to use it as a cue to elicit an intentional behaviour meaningful for the examiner, and iv) to actively maintain this intentional behaviour during long periods of time (tens of seconds). These characteristics are generally indicative of conscious behaviours [START_REF] Dehaene | Conscious, preconscious, and subliminal processing: a testable taxonomy[END_REF] .

Discussion

This case study demonstrates the utility of functional brain-imaging (here EEG), -in complement to detailed clinical, behavioural and structural imaging data -, to probe consciousness and cognition in such an extreme condition of severe motor and multisensory disconnection. In contrast to more classical approaches previously reported [START_REF] Owen | Detecting Awareness in the Vegetative State[END_REF][START_REF] Carrai | Transient post-traumatic locked-in syndrome: A case report and a literature review[END_REF][START_REF] Schnakers | Detecting consciousness in a total locked-in syndrome: an active event-related paradigm[END_REF][START_REF] John | Source imaging of QEEG as a method to detect awareness in a person in vegetative state[END_REF][START_REF] Pistoia | EEG-detected olfactory imagery to reveal covert consciousness in minimally conscious state[END_REF][START_REF] Aslaksen | Probing for consciousness after severe brain injury by functional magnetic resonance imaging (fMRI)[END_REF] , because of his massive sensory impairment, we could not use any task instruction or stimulus processing correlates to probe consciousness in this patient.

The ability of automatized and quantitative EEG-based mathematical classifiers to capture putative markers of consciousness in the absence of any behavioural evidence and of sensory-motor pathway impairment is of potentially great importance. In the present case, our algorithm preceded behavioural evidence of consciousness by 7 days. Finally, preliminary reanalysis of our database showed that all these markers could also be computed from a standard bedside 21-electrodes EEG and provide very similar results (submitted). If confirmed, this method could be nicely complete the TMS/EEG method [START_REF] Rosanova | Recovery of cortical effective connectivity and recovery of consciousness in vegetative patients[END_REF] , which is also able to probe consciousness while bypassing the sensory-motor pathway, but needs a very highly complex acquisition setting.

Materiel and methods

The patient EEG was recorded in the very same condition as our patients' database [START_REF] Sitt | Large scale screening of neural signatures of consciousness in patients in a vegetative or minimally conscious state[END_REF] while being subject to the Local-Global paradigm [START_REF] Bekinschtein | Neural signature of the conscious processing of auditory regularities[END_REF] . Briefly, the Local-Global paradigm is characterized by two embedded levels of auditory regularities allowing to identify the cortical response to sounds (P1), the automatic response to local novelty (MMN and P3a) and the conscious response to global novelty (P3b).

EEG was sampled at 250 Hz with a 256-electrode geodesic sensor net connected to a high impendence amplifier (EGI, Oregon, US) referenced to the vertex. Impedances were controlled inferior to 100 kΩ. Data were filtered from 0.5 Hz to 45 Hz. Trials were segmented from -200ms to 1300 ms relative to the onset of the Local-Global trials. Electrodes in which more than 50% of the trails exceeded a 100 µV were marked for rejection. Trials were at least 10% of the electrodes exceeded the same threshold were discarded. Data was re-referenced to the average of all electrodes.

Finally, electrodes marked for rejection were discarded and the information was reconstructed using interpolation.

Markers were then computed as described in the original publication supplementary material [START_REF] Sitt | Large scale screening of neural signatures of consciousness in patients in a vegetative or minimally conscious state[END_REF] .

Structural Similarity Index Measure (SSIM) was computed according to the original formula [START_REF] Wang | Image quality assessment: From error visibility to structural similarity[END_REF] with modifications to apply it on EEG topographies: each electrode was assumed to be a pixel. Since the EEG montage is three-dimensional, the definition of neighbours was redefined as all the electrodes within an Euclidean norm of 1.0 from a normalized montage between -1.0 and 1.0 in all axes (X, Y and Z).

All processing stages and topographical maps were performed with MNE-python [START_REF] Gramfort | MEG and EEG data analysis with MNE-Python[END_REF][START_REF] Gramfort | MNE software for processing MEG and EEG data[END_REF] and custom made python libraries. The Case Report guidelines were followed throughout [START_REF] Gagnier | The CARE Guidelines: Consensus-based Clinical Case Reporting Guideline Development[END_REF] . Follin, Dr. Bertrand Gaymard, Dr. Jean-Remi King, Dr. Régine Morizo-Koutlidis and Dr Marion Quirins for their help in this case study.

Figures' legend:

Figure 1 .

 1 Figure 1. MRI in sagittal T1-weigthed (A) and axial T2-weighted acquisitions (B,

Figure 2 .

 2 Figure 2. Topographical map of the patient's EEG markers. Scalp topographical

Acknowledgement:

We thank the patient's spouse to have kindly agreed that we report this case study.

We also thank Dr Francis Bolgert, Dr. Sophie Demeret, Dr. Athena Demertzi, Dr Caline El Jadam, Dr. Frédéric Faugeras, Prof. Diego Fernandez Slezak, Dr. Arnaud

Funding: This work was supported by Assistance Publique -Hôpitaux de Paris (AP-HP; BR, DG, LN), Institut National de la Santé et de la Recherche Médicale (INSERM; BR, JS, LN), Association Amicale des Anciens Internes en Médecine des Hôpitaux de Paris (AAIHP; BR), Syndicat des Chefs de Clinique & Assistants des Hôpitaux de Paris (SCCAHP; BR), Philippe Foundation (BR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET; FR), Université Pierre et Marie Curie, Paris 6 (UPMC ; DG, LN), the James S. McDonnell Foundation (LN), STIC-AmSud grants Complexity as a neural marker: applications to EEG and natural language processing and RTBRAIN -Towards Real-time processing of brain signals (FR, JS) and Académie des Sciences -Grand Prix Lamonica de Neurologie (LN).

All the authors report no declarations of interest.

Patient Consent:

Authors have received the patient 'spouse written consent for this publication.

However, as the patient's consent could not be obtained, details have been removed from this case description to ensure a maximal anonymity.

Supplementary Online Material: Figure S1. Sagittal reconstructed FA map derived from diffusion tensor imaging (A) and angio MR acquisition of the circle of Willis (B).

There is an interruption of the corticospinal track at the junction between the pons and the medulla oblongata


Wala Abou Saoud

Aymen Amine Assadi
email: aymen.assadi@ensc-rennes.fr

Monia Guiza

Abdelkrim Bouzaza

Wael Aboussaoud

Abdelmottaleb Ouederni

Isabelle Soutrel

Dominique Wolbert

Sami Rtimi
email: sami.rtimi@epfl.ch

Wala Abou Saoud

Study of synergetic effect, catalytic poisoning and regeneration using dielectric barrier discharge and photocatalysis in a continuous reactor: Abatement of pollutants in air mixture system

Keywords: Sulfur compounds, Dielectric Barrier Discharge, Photocatalysis, Synergetic effect, Catalyst poisoning, Catalytic regeneration

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The emission of volatile organic compounds (VOCs) is a major source of air pollutants, coming largely from industrial sectors especially by the petrochemical, and therefore, poses serious damages to environment and public health [START_REF] Inoue | Performance tests of newly developed adsorption/plasma combined system for decomposition of volatile organic compounds under continuous flow condition[END_REF][START_REF] Magureanu | Toluene oxidation by non-thermal plasma combined with palladium catalysts[END_REF][START_REF] Guo | Effect of manganese oxide catalyst on the dielectric barrier discharge decomposition of toluene[END_REF]. Sulfur containing volatile organic compounds (SVOCs) constitutes one of the important families of atmospheric pollutants such as methyl mercaptan (CH 3 SH),dimethyl sulfide (CH 3 SCH 3 ) and dimethyl disulfide (CH 3 SSCH 3 ) [START_REF] Nevanperä | Catalytic oxidation of dimethyl disulfide (CH 3 SSCH 3 ) over monometallic Au, Pt and Cu catalysts supported on γ-Al 2 O 3 , CeO 2 and CeO 2 -Al 2 O 3[END_REF][START_REF] Darifa | Study on the catalytic oxidation of DMDS over Pt-Cu catalysts supported on Al 2 O 3 , AlSi 20 and SiO 2[END_REF].These mentioned compounds are coming either from natural processes like anaerobic biological activities or from anthropogenic emissions sectors such as pulping processes and chemical industries. First and foremost, these sulfur compounds are known by their high toxic and corrosive effect [START_REF] Nevanperä | Catalytic oxidation of dimethyl disulfide (CH 3 SSCH 3 ) over monometallic Au, Pt and Cu catalysts supported on γ-Al 2 O 3 , CeO 2 and CeO 2 -Al 2 O 3[END_REF][START_REF] Guillard | Photocatalytic degradation and mineralization of a malodorous compound (dimethyl disulfide) using a continuous flow reactor[END_REF][START_REF] Lin | Photo-catalytic degradation of dimethyl disulfide on S and metal-ions co-doped TiO 2 under visible-light irradiation[END_REF]. Besides, they are causing significant environmental problems. In fact, their oxidation in the atmosphere will lead to the production of sulfur dioxide (SO 2 ), which be consequently converted into sulfuric acid and returns to the earth when it rains [START_REF] Nevanperä | Catalytic oxidation of dimethyl disulfide (CH 3 SSCH 3 ) over monometallic Au, Pt and Cu catalysts supported on γ-Al 2 O 3 , CeO 2 and CeO 2 -Al 2 O 3[END_REF]. Therefore, the abatement of those odorous emissions in the atmosphere is an important issue that needs an efficient and least-cost process. Adsorption [START_REF] Zhao | Methyl mercaptan removal from gas streams using metal-modified activated carbon[END_REF], thermal oxidation [START_REF] Doggali | Combustion of volatile organic compounds over Cu-Mn based mixed oxide type catalysts supported on mesoporous Al 2 O 3 , TiO 2 and ZrO 2[END_REF], biological treatment [START_REF] Wei | Removal of dimethyl sulfide by the combination of non-thermal plasma and biological process[END_REF], ozonation [START_REF] Brodu | Removal of volatile organic compounds by heterogeneous ozonation on microporous synthetic alumina silicate[END_REF], and catalysis/photocatalysis [START_REF] Demeestere | Heterogeneous photocatalysis as an advanced oxidation process for the abatement of chlorinated, monocyclic aromatic and sulfurous volatile organic compounds in air: state of the art[END_REF] have been proposed by researchers as conventional VOCs treatment methods.

Photocatalysis is a promising technology for air purification because of its ability to break down a wide range of pollutants to non-toxic end-products such as CO 2 and H 2 O under ambient conditions [START_REF] Mo | Photocatalytic purification of volatile organic compounds in indoor air: A literature review[END_REF][START_REF] Sleiman | Photocatalytic oxidation of toluene at indoor air levels (ppbv): Towards a better assessment of conversion, reaction intermediates and mineralization[END_REF].

The latter processes based on photoreactions which take place at the surface of a semiconductor usually titanium dioxide (TiO 2 ) under ultraviolet (UV) irradiation [START_REF] Schneider | Understanding TiO 2 Photocatalysis: Mechanisms and Materials[END_REF][START_REF] Bouzaza | Photocatalytic degradation of some VOCs in the gas phase using an annular flow reactor: determination of the contribution of mass transfer and chemical reaction steps in the photodegradation process[END_REF][START_REF] Assadi | A continuous air reactor for photocatalytic degradation of 3-methylbutanal: effect of different operating parameters and chemical degradation pathway[END_REF][START_REF] Chuang | Characterization of supported TiO 2 -based catalysts green-prepared and employed for photodegradation of malodorous DMDS[END_REF]. Some significant results have been achieved in the photocatalytic oxidation of many organic compounds like VOCs and SVOCs which includes dimethyl sulfide [START_REF] Lin | The visible light-driven photodegradation of dimethyl sulfide on S-doped TiO 2 : characterization, kinetics, and reaction pathways[END_REF], ethyl mercaptan [START_REF] Kachina | Catalytic TiO 2 oxidation of ethanethiol for environmentally begin air pollution control of sulphur compounds[END_REF], hydrogen sulfide [START_REF] Alonso-Tellez | A parametric study of the UV-A photocatalytic oxidation of H 2 S over TiO 2[END_REF], BTEX (Benzene, Toluene, Ethylbenzene and Xylenes) removal [START_REF] Korologos | Photocatalytic oxidation of benzene, toluene, ethylbenzene and m-xylene in the gas-phase over TiO 2 -based catalysts[END_REF] and isovaleraldehyde [START_REF] Assadi | Photocatalytic oxidation of trimethylamine and isovaleraldehyde in an annular reactor: Influence of the mass transfer and the relative humidity[END_REF]. Moreover, several researchers have investigated non-thermal plasma (NTP) as one of the technologies which can be used to treat polluted air [START_REF] Barakata | Oxidation of isopropanol and acetone adsorbed on TiO 2 under plasma generated ozone flow: Gas phase and adsorbed species monitoring[END_REF][START_REF] Vandenbroucke | Non-thermal plasmas for non-catalytic and catalytic VOC abatement[END_REF][START_REF] Kim | Non thermal plasma processing for air-pollution control: a historical review, current issues, and future prospects[END_REF]. This techniques characterized by its high efficiency in the formation of reactive chemical species like hydroxyl radicals, atomic oxygen and ozone, which may react and degrade organic molecules at room temperature under atmospheric pressure [START_REF] Kim | Non thermal plasma processing for air-pollution control: a historical review, current issues, and future prospects[END_REF][START_REF] Subrahmanyam | Catalytic non-thermal plasma reactor for total oxidation of volatile organic compounds[END_REF][START_REF] Durme | Efficient toluene abatement in indoor air by a plasma catalytic hybrid system[END_REF]. In fact, dielectric barrier discharge (DBD), one of the NTP techniques, has been employed successfully for the abatement of various aromatic VOCs: toluene [START_REF] Mok | Treatment of toluene by using adsorption and non-thermal plasma oxidation process[END_REF] and benzene [START_REF] Karuppiah | Catalytic nonthermal plasma reactor for the abatement of low concentrations of benzene[END_REF], trimethylamine [START_REF] Assadi | Removal of trimethylamine and isovaleric acid from gas streams in a continuous flow surface discharge plasma reactor[END_REF], hydrogen sulfide [START_REF] Liang | Performance of non-thermal DBD plasma reactor during the removal of hydrogen sulfide[END_REF], and sulfur dioxide [START_REF] Ma | Study of SO 2 removal using non-thermal plasma induced by dielectric barrier discharge (DBD)[END_REF]. Several studies have reported on the combined process between non-thermal plasma (NTP) and photocatalysis leading to improve the efficiency of VOCs degradation [START_REF] Huang | Contribution of UV light to the decomposition of toluene in dielectric barrier discharge plasma/photocatalysis system[END_REF][START_REF] Assadi | Study of synergetic effect by surface discharge plasma/TiO 2 combination for indoor air treatment: Sequential and continuous configurations at pilot scale[END_REF][START_REF] Maciuca | Synergetic effect by coupling photocatalysis with plasma for low VOCs concentration removal from air[END_REF][START_REF] Karuppiah | Non thermal Plasma assisted photocatalytic oxidation of dilute benzene[END_REF][START_REF] Thevenet | Oxidation of acetylene by photocatalysis coupled with dielectric barrier discharge[END_REF][START_REF] Palau | Isovaleraldehyde degradation using UV photocatalytic and dielectric barrier discharge reactors, and their combinations[END_REF] over a synergistic effect [START_REF] Mok | Treatment of toluene by using adsorption and non-thermal plasma oxidation process[END_REF][START_REF] Karuppiah | Catalytic nonthermal plasma reactor for the abatement of low concentrations of benzene[END_REF][START_REF] Assadi | Removal of trimethylamine and isovaleric acid from gas streams in a continuous flow surface discharge plasma reactor[END_REF][START_REF] Liang | Performance of non-thermal DBD plasma reactor during the removal of hydrogen sulfide[END_REF].

The aim of the present work is to explore the removal efficiency of butyraldehyde (from now as BUTY), dimethyl disulfide (from now as DMDS) and their mixtures in planar reactor, using dielectric barrier discharge (DBD) plasma (from now as DBD-plasma) and photocatalysis processes separately and their combinations in the same reactor. The influence of inlet concentration and flowrate on the pollutants removal, nature of the by-products and ozone formation inside the process are studied in details.

Moreover, an interesting challenge of this work is to study the presence of synergetic effect between DBD-plasma and photocatalysis during the oxidation of mixture of VOCs at pilot scale. A special attention is paid to TiO 2 poisoning and its regeneration with different process oxidation (plasma, photocatalysis and combined system). The evidence for redox catalysis responsible for the photocatalytic activity was also investigated.

Materials and methods

Reactor design and setup details

The reactor consists of a rectangular glass chamber (length L = 1000 mm, width l = 135 mm and height H = 135 mm). Two glass plates are installed parallel to the length of the reactor and permit to maintain the catalyst, the two electrodes (1 mm thick and 800 mm in length) and high voltage grids (stainless steel, 2 mm thick). To generate the plasma, high voltage is applied to the reactor. The applied voltage is delivered by a generator (BFi OPTILAS) as a sinusoidal waveform up to 10 V and then amplified by an amplifier (TREK 30/40) to achieve 30 kV. This amplifier is connected to the electrodes. A coil of capacitors with a total capacity of 2.5 nF (C m ) was positioned between the copper electrode and the ground connection in order to collect the charges in the reactor. The distance between the two plates could be adjusted to modify the space/air gap. The voltage applied (U app ) and the high voltage capacity (U m ) are measured by high-voltage probes recorded with a digital oscilloscope (Lecroy wave surfer 24Xs, 200MHz). The reactor can be used also as a photocatalytic reactor and as a plasma DBDphotocatalytic reactor. Eight lamps (Philips under reference PL-S 9W/10/4P) are placed equidistant from each other in air gap of the reactor in order to ensure a good radiation distribution (Figure 1). The photocatalyst is maintained between the stainless steel grid electrode and the dielectric barrier in the plasma active area. The UV lamps arranged in the reactor, permit the activation of the photocatalyst.

Two openings with septum downstream and upstream of the reactor allow outlet and inlet gas to be sampled with a 500 μL syringe.

Catalysts

The supported material is further named Glass Fiber Tissue (GFT) containing 13 g/m 2 of colloidal silica, a 13 g/m 2 of titanium dioxide nanoparticles and inorganic fibers. The coating process consists in an impregnation of glass fibers by SiO 2 and TiO 2 nanoparticles suspension in pure water using an industrial size-press (PC500 Millennium). The specific area of TiO 2 nanoparticles is 300 m 2 /g. The coating process consists in impregnating fibers using industrial size press. The press is employed to impregnate fibers with the suspension; then, they are dried (Figure 2). Material preparation has been performed by Ahlstrom Research and Services.

Polluted flow generation

The generated flow rate is controlled by a mass flow meter (Bronkhorst In-Flow) with a maximum flow rate reached about 10 m 3 h -1 . In order to keep the value of relative humidity (RH) of the gas constant a variable part of the air flow is derived through a packed column where water flows in counter current and the RH value was about 50% (Figure 3). The pollutants, BUTY and DMDS, are injected continuously in liquid state by means of a syringe/syringe driver system (Kd Scientific Model 100) through a septum into the gas stream. A heating tape was wrapped around the pipe at the injection zone in order to ensure a good evaporation of each pollutant. A static mixer allows the homogenization of the upstream effluent (Figure 3). We note that all experiments were carried out at room temperature and atmospheric pressure. A TESTO sensor is used to measure the temperature and relative humidity.

Analysis system

Analysis of BUTY was performed using a gas phase chromatograph (Fisons GC9000) equipped with a flame ionization detector (FID). Nitrogen was used as the carrier gas and constitutes the mobile phase.

All injections were performed manually with a syringe of 500 μL and repeated at least three times and DMDS was analyzed by another gas phase chromatograph (THT MEDOR ® ), also all injections were performed manually with a bag of 5L, sampling was done in cycles; each cycle lasts 30 min. More details about the analysis system of the sulfur compounds had been described in detail in our previous work [START_REF] Guillerm | Removal of gas-phase ammonia and hydrogen sulfide using photocatalysis, nonthermal plasma, and combined plasma and photocatalysis at pilot scale[END_REF].

CO x , SO 2 and O 3 analysis

The CO 2 was analyzed by a Fourier Transform Infrared (FTIR) spectrophotometer brand Environment SA. The CO concentration was measured by NO/CO ZRE gas analyzer and SO 2 concentration was measured by MEDOR gas analyzer.

A standard iodometric titration method was used to estimate the downstream ozone formation. Thus, at the reactor exit, a constant air flow of 285 L h -1 is bubbled in a potassium iodide (KI) solution at 10 -2 M.

A yellowish solution is obtained when I -oxidizes into I 2 [START_REF] Assadi | Removal of trimethylamine and isovaleric acid from gas streams in a continuous flow surface discharge plasma reactor[END_REF].

Surface wettability, oxidative states as determined by XPS and OH-radical production as determined by fluorescence

Sessile drop method on a Data Physics OCA 35 unit was used to evaluate the wettability of the catalyst surface by determining the water droplet contact angle (CA).

The X-ray photoelectron spectroscopy (XPS) of the photocatalyst was determined using an AXIS NOVA photoelectron spectrometer (Kratos Analytical, Manchester, UK) provided for with monochromatic AlKa (hn=1486.6 eV) anode. The carbon C1s line with position at 284.6 eV was used as a reference to correct the charging effects. The surface atomic concentration was determined by XPS from the peak areas, using recognized sensitivity factors. The surface percentages composition was determined within the topmost atomic layers. The XPS spectra were deconvoluted by the mean of a Casa XPS updated version 2.3.18.

The quantification of the °OH radical was carried out based on the method proposed by Hashimoto et al., [START_REF] Ishibashi | Detection of the Active Oxidative Species in TiO 2 Photocatalysis by Using the Fluorescence Technique[END_REF]. For this, an Across Chemicals 99% terephthalic acid as well as NaOH 98% from Sigma Aldrich were used. The samples were immersed in a 0.4 mmol L -1 solution of terephthalic acid dissolved in a 4 mmol L -1 NaOH solution. After UV-irradiation, the solution was transferred in a quartz cell. The fluorescence of the 2-hydroxyterephthalic-acid was quantitatively monitored on a Perkin Elmer LS-50B

spectrometer. The spectra were recorded at scan rate of 100 nm/min in wavelength range between 400-500 nm after excitation at 315 nm.

Results and discussion

The evaluation of the process performance was done by estimating: the removal efficiency (RE), the carbon monoxide selectivity (SCO), the carbon dioxide selectivity (SCO 2 ) and the specific energy (SE).

These were determined according to the equations defined as follows (eq.1-5):

 Removal efficiency (RE) of each pollutant: 100 (1)
Where C in and C out are the inlet and the outlet concentrations of pollutants (mg m -3 ) respectively.

 Selectivity of CO:

(2)

 Selectivity of CO 2 : (3)
 Specific energy (SE):

(4) (5)
where Q (m 3 s -1 ) is the flow rate and P (W) is input power adjusted by changing the applied voltage (V a ).

The present work aimed at the removal of mixture of VOCs of different nature. Therefore, in order to understand the oxidation behavior of VOCs in a mixture, initial experiments were carried out on each compound alone.

The pollutants analysis was started only once the air flow in the reactor reaches an equilibrium. In other words, a certain time was required after the pollutant is injected in the air flow crossing the reactor, so that its concentration stabilizes. Once the outlet concentration is stabilized and the catalyst is loaded, the oxidation starts and thus samples can be taken for analysis.

Oxidation of BUTY

Effects of inlet concentration and flow rate with photocatalytic process

Inlet concentration and flowrate were varied in order to understand their influences on the performance of the photocatalytic reactor. The removal efficiency (RE) of BUTY at different inlet concentrations and gas flow rates is shown in Figure 4. The inlet concentration of BUTY was varied from 25 to 150 mg m -3 in the flow rate range of 4 to 8 m 3 h -1 . It is readily seen in Figure 4 that the removal efficiency of BUTY declines with increasing the flow rate, this is due to decreased contact time between the compound and the catalytic active sites. However, it can be noted also that with increasing the inlet concentration, the degradation rate of BUTY decreased; this behavior is well known and could be explained by the less effective availability of the active sites on catalyst surface [START_REF] Bouzaza | Photocatalytic degradation of some VOCs in the gas phase using an annular flow reactor: determination of the contribution of mass transfer and chemical reaction steps in the photodegradation process[END_REF][START_REF] Assadi | Photocatalytic oxidation of trimethylamine and isovaleraldehyde in an annular reactor: Influence of the mass transfer and the relative humidity[END_REF][START_REF] Palau | Isovaleraldehyde degradation using UV photocatalytic and dielectric barrier discharge reactors, and their combinations[END_REF].

Q= 4 m 3 h -1 Q= 6 m 3 h -1 Q= 8 m 3 h -1

Effect of specific energy (SE) with plasma and combined system

It is well known that the energy of discharge injected in the reactor is an important parameter. Thus in order to understand its effect, two studies of BUTY removal by DBD plasma reactor taken alone and combined with photocatalysis were carried out. It is interesting to note that, in our previous investigations [START_REF] Assadi | A continuous air reactor for photocatalytic degradation of 3-methylbutanal: effect of different operating parameters and chemical degradation pathway[END_REF][START_REF] Assadi | Use of DBD plasma, photocatalysis and combined DBD plasma/ photocatalysis in a continuous annular reactor for isovaleraldehyde elimination -Synergetic effect and byproducts identification[END_REF], results show that the UV light generated by surface discharge plasma in a continuous reactor for isovaleraldehyde removal did not activate TiO 2 and its contribution to pollutants elimination can be neglected. Moreover, it was shown by Thevenet et al. [START_REF] Thevenet | Oxidation of acetylene by photocatalysis coupled with dielectric barrier discharge[END_REF] that the UV produced by the plasma is not sufficient to activate photocatalyst during C 2 H 2 degradation. Thus, the introduction of external UV light to the plasma is indispensable for maintain the photocatalytic activity of TiO 2 .

The BUTY inlet concentration was around 45 ppm under a flow rate of 4 m 3 h -1 in the SE range of 2-10 J L -1 as showed in Figure 5. When only DBD plasma was used at SE equal to 9.945 J L -1 , the BUTY removal was 28%. However, by coupling photocatalysis and DBD plasma, the BUTY removal reached 53% with respect to the removal efficiency of 18% when photocatalysis was applied alone. Therefore, a synergetic effect is happening on the catalyst surface when applying both processes e.g. plasma and photocatalysis. This trend was the same for all the values of injected specific energy in the reactor. This is due to the role of neutrals (including molecules, radicals, metastable species and atoms), ions, electrons, photons and electric fields, on enhancing of plasma-catalyst surface interactions in terms of surface chemical reactions and enhancing of mass transfer [START_REF] Assadi | Abatement of 3methylbutanal and trimethylamine with combined plasma and photocatalysis in a continuous planar reactor[END_REF]. Similar observations have been mentioned in the literature for various VOCs. [START_REF] Maciuca | Synergetic effect by coupling photocatalysis with plasma for low VOCs concentration removal from air[END_REF][START_REF] Thevenet | Oxidation of acetylene by photocatalysis coupled with dielectric barrier discharge[END_REF][START_REF] Assadi | Abatement of 3methylbutanal and trimethylamine with combined plasma and photocatalysis in a continuous planar reactor[END_REF][START_REF] Assadi | Use of DBD plasma, photocatalysis and combined DBD plasma/ photocatalysis in a continuous annular reactor for isovaleraldehyde elimination -Synergetic effect and byproducts identification[END_REF][START_REF] Gharib-Aboughaida | Association of surface dielectric barrier discharge and photocatalysis in continuous reactor at pilot scale: Butyraldehyde oxidation, by-products identification and ozone valorization[END_REF].

The synergetic effect seen on the photocatalytic surface can be attributed to: i) the non-thermal plasma involving various processes since the plasma produces many species such as high-energy electrons,

CO x selectivites and ozone formation

Figure 6 shows the variation of CO x selectivies (%) and the amount of ozone (ppm) produced when applying DBD plasma, photocatalysis and their combination. The production of CO is seen to below and its selectivity did not exceed 15%. The CO 2 selectivity when applying the three processes is also presented in Figure 6. It is readily seen that photocatalysis is the most selective to CO 2 among the three processes studied and its selectivity achieved 60% whereas SCO 2 with DBD plasma alone and coupled processes is 30% and 45% respectively. Additionally, it is necessary to note that combining both processes (DBD+TiO 2 +UV) ensures a higher BUTY conversion rate and offers great possibility to reduce O 3 formation at the outlet of the reactor. However, at SE equal to 9.945 J L -1 , the ozone outlet concentration is 22.45 ppm and decreases to 15 ppm for combined processes.

This behavior suggests that UV lamp can play a role of ozone decomposition as described in eq.6 and

7 below [START_REF] Taranto | Combining cold plasma and TiO 2 photocatalysis to purify gaseous effluents: a preliminary study using methanol-contaminated air[END_REF][START_REF] Huang | Combination of photocatalysis downstream the non-thermal plasma reactor for oxidation of gas-phase toluene[END_REF]:

Oxidation of DMDS

The performance of the reactor during the oxidation of DMDS was tested and the results, using the same planar reactor, are represented in Figure 7. The experiments were carried out at an inlet DMDS concentration of 45 ppm under a flow rate of 4 m 3 h -1 in the SE range of 2 to 10 J L -1 . Figure 7 shows the variation of the DMDS removal with specific energy for the photocatalysis (TiO 2 +UV) alone, plasma DBD alone, and for combining both processes. By coupling processes, DMDS is straightforwardly removed as seen in Figure 7. At SE=9.945 J L -1 , the values of RE are 28%, 47% and 70% when applying photocatalysis, DBD-plasma and the coupling respectively. Whereas the synergetic effect is not observed with DMDS. Clearly, this behavior is attributed to the catalyst poisoning as a consequence of degradation by-products accumulating on the catalyst surface, beginning with photocatalysis, plasma and then coupled processes at low SE values.

Oxidation of the mixture BUTY and DMDS

A mixture of BUTY and DMDS was injected in equal amounts to the incoming air. The experiments were carried out at an inlet concentration of VOCs mixture of 45 ppm (each pollutant concentration is equal to 22.5 ppm) under a flow rate of 4 m 3 h -1 in the SE range of 3-6 J L -1 . The results for DMDS removal with specific energy using photocatalysis (TiO 2 +UV) alone, DBD-plasma, and for the coupling of both processes are shown in Figure 8. When applying photocatalysis, only 5% of the DMDS was removed. When DBD-plasma was used at SE= 6 J L -1 , the DMDS removal reached to 10%. By coupling photocatalysis and DBD-plasma, the DMDS removal increased to 15%. In fact, DMDS removal was similar to the sum of removal efficiencies of each process taken separately. Therefore, a synergetic effect was not obviously observed.

Figure 8 shows also BUTY behavior when it is oxidized by photocatalysis, DBD-plasma and by coupling both processes as a function of the injected energy. We noted that BUTY abatement is significantly better than DMDS. At SE equal to 6 J L -1 , it can be also noted that by photocatalysis alone, the removal efficiency of BUTY was around 19%. When DBD-plasma alone was used, the BUTY removal reaches 45%. By coupling photocatalysis and DBD-plasma, the BUTY removal increased to 55%. The synergetic effect is not observed with BUTY even if its degradation is higher compared to DMDS.

Photocatalyst poisoning and regeneration: Cyclic photocatalytic oxidation

To your understanding, the lack of the synergistic effect is due to the catalyst poisoning during the photocatalytic pollutants degradation. Since the photocatalytic oxidation experiments were done in a consecutive way, the photocatalytic oxidation was seen to slow down. This could possibly explain the absence of the synergetic effect during the pollutant degradation. To better understand the influence of poisoning of the catalyst, a series of experiments was carried out. These experiments alternate phases of photocatalysis and of catalyst regeneration. The experiments were performed under dry conditions and continuous flow of 1 m 3 h -1 , at an inlet concentration of 45 ppm of BUTY and DMDS mixture (50%-50%). Figure 9. (a) shows the removal efficiency of BUTY versus time during the cyclic studies using the photocatalytic reactor previously described. As can be noted, the removal efficiency of BUTY dropped after 7h of irradiation (cycle 1), the initial value of RE is about 46% (Figure 9. (a)). This decrease in the photocatalytic efficiency could be attributed to the adsorption of by-products on the active sites.

Three regeneration procedures were tested to try to reestablish the catalytic capability of the catalyst.

The first procedure uses ultraviolet (UV) irradiation without pollutant. Four photocatalytic cycles were done separated with reactivation under UV irradiation as seen in Figure 9. (a). After this first cycle, the reactivation (UV irradiation for 2h) was able to recover 30% of the efficiency as seen in cycle 2. At the end of the second cycle, a reactivation of 4h UV-irradiation recovered 21% as shown in cycle 3. During cycle 4, the photocatalytic efficiency is seen to considerably drop down. Therefore, a long-term continuous photocatalytic oxidation may cause poisoning phenomenon [START_REF] Guillard | Photocatalytic degradation and mineralization of a malodorous compound (dimethyl disulfide) using a continuous flow reactor[END_REF][START_REF] Lin | Photo-catalytic degradation of dimethyl disulfide on S and metal-ions co-doped TiO 2 under visible-light irradiation[END_REF][START_REF] Alonso-Tellez | A parametric study of the UV-A photocatalytic oxidation of H 2 S over TiO 2[END_REF][START_REF] Portela | Photocatalytic-based strategies for H 2 S elimination[END_REF].

During the second reactivation procedure, DBD-plasma was applied to air stream and then the cyclic photocatalytic oxidation begins again. Figure 9. (b) shows that in these conditions more than 23% of the efficiency was recovered and it can be stated that after DBD-plasma regeneration the removal efficiency of BUTY are less than 23% compared to initial value of RE.

The third procedure of regeneration is done by combining both regeneration processes (DBD-plasma +TiO 2 +UV). By applying this regeneration treatment, the result shows that the initial efficiency of the catalyst was fully recovered. After 4h of regeneration, it has been observed that RE achieved 47% and then the pollutant removal proceeds as function of time (Figure 9. (c)). This behavior explains the synergetic effect between DBD-plasma and photocatalysis. Thus, it has also been demonstrated that DBD-plasma may effectively regenerate poisoned catalysts. This behavior is similar to what it was reportedby Kim et al. [START_REF] Kim | Catalyst regeneration and activity enhancement of Au/TiO 2 by atmospheric pressure nonthermal plasma[END_REF].The latter demonstrated packed-bed plasma regeneration of an Au/TiO 2 catalyst deactivated by the adsorption of VOCs. Moreover, Sivachandrian et al. [START_REF] Sivachandiran | Isopropanol saturated TiO 2 surface regeneration by non-thermal plasma: influence of air relative humidity[END_REF] reported TiO 2 surface charged with isopropanol can be generated with plasma in a packed-bed reactor. On the other way, Mok et al. [START_REF] Mok | Regeneration of C 4 H 10 dry reforming catalyst by nonthermal plasma[END_REF] compared thermal and plasma regeneration of a deactivated Ni/alumina catalyst.

Surface wettability, oxidative states as determined by XPS and OH-radical production

Photo-switchable behavior of TiO 2 has been reported to happen within the irradiation time. Hydrophobic to hydrophilic transformation aided by water droplets was seen to enhance the photocatalytic pollutant degradation reaction as recently reported [START_REF] Rtimi | Innovative transparent nonscattering TiO 2 bactericide thin films inducing increased E. coli cell wall fluidity[END_REF]. In our case, the contact angle θ was seen to drop from 96° until 19° within the time required for the pollutant degradation. According to Young's theory, when a liquid-vapor interface meets a solid surface an angle of contact is established and can be quantified/estimated using the water droplet contact angle (CA) method. The cosine of the contact angle (cos θ) of water droplet on a solid surface qualifies of the interfacial energy between the solid and liquid. The three interfacial energies are related to the cos θ by the Young's equation as follows (eq. 8):

Where: , and are interfacial energy between the solid and the gas, interfacial energy between the solid and the liquid and interfacial energy between the liquid and the gas, respectively.

According to this equation, the rate of the hydrophobic to hydrophilic transformation in our case was assessed to be 0.307 min -1 [START_REF] Rtimi | Innovative semi-transparent nanocomposite filmspresenting photo-switchable behavior and leading to areduction of the risk of infection under sunlight[END_REF][START_REF] Mills | A web-based overview of semiconductor photochemistry-based current commercial applications[END_REF][START_REF] Zhang | Photo-induced hydrophilicity and self-cleaning: models and realityEn[END_REF][START_REF] Seki | Kinetics of Photoinduced Hydrophilic Conversion Processes of TiO 2 Surface[END_REF]. In other words, this means that the photoactive surface is transformed from hydrophobic to hydrophilic reducing the interfacial energy between the solid surface and the liquid by the mean of the applied light. This aspect was investigated in details and will not be further discussed in the present manuscript [START_REF] Rtimi | Innovative transparent nonscattering TiO 2 bactericide thin films inducing increased E. coli cell wall fluidity[END_REF][START_REF] Rtimi | Innovative semi-transparent nanocomposite filmspresenting photo-switchable behavior and leading to areduction of the risk of infection under sunlight[END_REF].

Controversy interpretations are seen in the literature discussing if hydrophilic, hydrophobic or mixed hydrophilic-hydrophobic surfaces enhance the pollutant adhesion/contact with the catalytic surface and if this step is a limiting step preceding pollutant oxidation. Recent studies showed that complete Escherichia coli (E.coli) inactivation occurred while TiO 2 thin film undergoes the transformation to its hydrophilic state [START_REF] Rtimi | Indoor light enhanced photocatalytic ultra-thin films on flexible non-heat resistant substrates reducing bacterial infection risks[END_REF].

Redox processes involving Ti 4+ /Ti 3+ was seen to occur on the catalyst surface during the photocatalytic pollutant degradation. The as-prepared catalyst showed a Ti 4+ peak at 458.5 eV coexisting and slightly higher than the Ti 3+ peak at 457.8eV [START_REF] Rtimi | Innovative transparent nonscattering TiO 2 bactericide thin films inducing increased E. coli cell wall fluidity[END_REF][START_REF] Rtimi | Supported TiO 2 films deposited at different energies: Implications ofthe surface compactness on the catalytic kinetics[END_REF]. But after the catalyst poisoning, the Ti 3+ peak amounts to more than 80% against only 20% for Ti 4+ . Surface atomic percentage determined by XPS showed that the poisoned catalytic surface accumulates S2p reaching 22.89% of the total surface species. This S2p was seen to decrease significantly up to 4% during the plasma surface regeneration as seen in Table 1.

Table 1. Surface percentage atomic concentration on the catalyst surface at time zero, after the catalyst poisoning and after the plasma treatment: evidence for surface regeneration. The redox reaction on the TiO 2 was also seen to occur concomitant to the increase of hydrophilicity within the pollutant degradation time observed on the catalyst surface as seen in Table 2 [START_REF] Rtimi | Innovative semi-transparent nanocomposite filmspresenting photo-switchable behavior and leading to areduction of the risk of infection under sunlight[END_REF][START_REF] Rtimi | Supported TiO 2 films deposited at different energies: Implications ofthe surface compactness on the catalytic kinetics[END_REF]. 2 as previously reported [START_REF] Rtimi | Supported TiO 2 films deposited at different energies: Implications ofthe surface compactness on the catalytic kinetics[END_REF]. The O 2 adsorbed by the nonstoichiometric TiO 2 surface under light capture and the photo-generated electron/hole pairs. Figure 10 shows the OH-radical quantification on our photocatalyst. The photo-generation of °OH showed an increase up to 3h then a stable production with a little decrease of the generated amount. The latter was attributed to the TiO 2 active sites saturation and the surface poisoning by the by-products as explained in the XPS section [START_REF] Rtimi | Supported TiO 2 films deposited at different energies: Implications ofthe surface compactness on the catalytic kinetics[END_REF][START_REF] Rtimia | Insight on the photocatalytic bacterial inactivation by co-sputteredTiO 2 -Cu in aerobic and anaerobic conditions[END_REF]. In order to investigate the different energy consumption of the three processes described in this study, the energy yield (EY) in units of the mass (in units of mg degraded) of pollutant removed per Wh input (g.kWh -1 ) is being calculated using the following formula:

Sample

 For plasma:

 For photocatalysis:  For combined system :

where SE specific energy of plasma, N is number of lamps used, P lamp is the real power consumption of one UV lamp, EY plasma and EY photocatalysis EY combined system are respectively the energy consumption in plasma, photocatalytic reactor and combined system.The estimated energy consumption (EC) values are in the order of 10 to 20 4 g.kWh -1 , which is similar to previous studies on mixture of benzene, toluene and ethylene (known as BTEX) [START_REF] Fan | Removal of low-concentration BTX in air using a combined plasma catalysis system[END_REF] and nitrogen oxides (NO x ) [START_REF] Parkdag | Simultaneous removal of NOx and SO 2 from NO-SO 2 -CO 2 -N 2 -O 2 gas mixtures by corona radical shower systems[END_REF] removal. For the reactor with combined system, EC for Buty removal was higher than that with the others two configurations (plasma and photocatalysis taken alone) as readily seen in Figure 11. This behavior can be due to the in-situ interactions between plasma and photocatalysis. However, this behavior is not observed in the case of DMDS treatment. The process requires higher cost to degrade DMDS compared to BUTY. In the case of low applied energy (2.9 J L -1 ), a loss of 15 g. kWh -1 with the combined process was observed. On the other hand, applying 9.9 J L -1 , the combined process showed a gain of 4 g.Kwh -1 as seen in Figure 11.

Conclusion

In this work, the degradation of butyraldehyde and dimethyl disulfide with combined plasma and photocatalysis and the influence of inlet concentration, total flow rate and SE were studied. We can note firstly, that a synergetic effect was observed of butyraldehyde removal but not in the case of dimethyl disulfide removal. This trend is due to the poisoning catalysts. Secondly, the results show that during the mixture treatment a synergetic effect was observed for the two pollutants. On the other hand, photocatalyst deactivation was observed and the regeneration could be performed by combining photocatalysis and plasma (DBD). The combined system can regenerate rapidly the catalysts; this trend can explain the synergetic effect between plasma and photocatalysis. Redox catalysis was seen to happen during the pollutants degradation concomitant with a photo-switchable hydrophilicity as detected by contact angle measurement. Moreover, Surface atomic percentage determined by XPS showed that the regenerated catalytic surface accumulates S2p reaching 4.44% of the total surface species which confirm the regeneration effect of plasma. Mechanism of the VOCs degradation/removal was suggested involving different oxidative processes. The energy consumption of the used processes was also estimated and discussed.

Figure 1 .

 1 Figure 1. Schematic illustration of the used set-up: (a) sectional drawing and (b) planar reactor.

Figure 2 .

 2 Figure 2. SEM imaging of TiO 2 deposited on GFT [38].

Figure 3 .

 3 Figure 3. Experimental setup.

Figure 4 .

 4 Figure 4. Removal efficiency (RE) of BUTY with inlet concentration at different flow rates (Process tested: Photocatalysis, UV intensity= 20 W m -2 , RH = 50 %).

  excited molecules or radicals (O • , N, HO • , O 2 •-, O 3 , NO 2 , NO x , etc.). These species can interact with BUTY molecules leading to its degradation and ii) plasma promoting the desorption of byproducts attached at the TiO 2 surface leading to its mineralization iii) Plasma can induce activation of the TiO 2 by creating hole/ electron pairs through ion bombardment[START_REF] Sivachandiran | Isopropanol saturated TiO 2 surface regeneration by non-thermal plasma: influence of air relative humidity[END_REF][START_REF] Vandenbroucke | Non-thermal plasmas for non-catalytic and catalytic VOC abatement[END_REF][START_REF] Assadi | Abatement of 3methylbutanal and trimethylamine with combined plasma and photocatalysis in a continuous planar reactor[END_REF].

Figure 5 .

 5 Figure 5. BUTY removal efficiency at different SE when applying DBD plasma, photocatalysis and their combination: Evidence for synergetic effect (Q=4 m 3 h -1 , [BUTY] = 45 ppm, RH = 50%).

Figure 6 .

 6 Figure 6. Variation of CO x selectivity (%) and the amount of ozone (ppm) when applying the three processes (Q=4 m 3 h -1 , [BUTY] = 45 ppm, SE= 9.945 J L -1 , RH = 50%).

Figure 7 .

 7 Figure 7. Variation of DMDS removal efficiency with SE when applying the three processes (Q=4 m 3 h -1 , [DMDS] = 45 ppm, RH = 50%).

( 1 )Figure 8 .

 18 Figure 8. Variation of the removal efficiency of DMDS and BUTY in mixture (50%-50%) with SE using the three processes (Q=4 m 3 h -1 , [Mixture] = 45 ppm, RH = 50%).

Figure 9 .

 9 Figure 9. Variation of BUTY removal efficiency as function of time showing catalyst poisoning during the photocatalytic process (Q = 1 m 3 h -1 , [Mixture] = 45 ppm, RH = 5%) with different process of regeneration: (a) process regeneration: photocatalysis, (b) process regeneration: DBD-plasma, (c) process regeneration: DBD-plasma combined with photocatalysis.

Figure 10 .Scheme 1 .

 101 Figure 10. Photo-generation of °OH-radical under UV irradiation on TiO 2 deposited on GFT. The measurement were carried out in triplicate, error bars=5%.

Figure 11 .

 11 Figure 11. Variation of Energy Consumption (EC) vs SE at different configurations of tested processes (Q=4 m 3 h -1 , [Pollutant] = 45 ppm) (dark color: butyraldehyde; light color: DMDS)

Table 2 .

 2 Fraction of OH/(Ti-O+OH) on the catalyst surface. well known as a strong oxidant which can decompose VOCs at ambient temperature by direct way and also was reported to have a positive effect in terms of activity enhancement and regeneration of TiO 2 photocatalyst[START_REF] Thevenet | Oxidation of acetylene by photocatalysis coupled with dielectric barrier discharge[END_REF]. This effect likely occurred via the decomposition of ozone under ultraviolet (UV) irradiation into very reactive species such as O 2 .-and °OH. Thus, this trend explains the high fraction of OH in table 2. The depletion of the surface lattice TiO 2 oxygen on the samples during the air pollutants abatement leads to O-surface vacancies, but these vacancies are compensated by O 2 -due to the c(e-) reacting with O 2 (air). One has to keep in mind that TiO 2 mediated photocatalysis is based on the fact that the adsorption of O 2 does not proceed on perfectly neutral TiO 2 surfaces, but on surfaces with excess negative charge on the surface TiO 2 /Ti-OH groups as reported by XPS in Table

	OH/(Ti-O+OH)