The use of longitudinal data in microeconometric studies is commonly motivated by the potential to capture excess heterogeneity between units. The first major rationale for this is to obtain parameter estimates that are free from bias induced by unobserved crosssectional heterogeneity. A second reason for their employment is the desire to decompose the variance of latent components into structural and transitory contributions. The former incentive is typically related to taking a fixed-effect view on the processes that drive the outcome of interest in the sense of leaving the distribution of unobserved unit-specific factors unspecified. The error-component formulation or random-effect approach is commonly associated with stronger restrictions on the distribution of the permanent latent component.

) and Arellano and Bonhomme (2010), for example, provide ellaborate discussions on this and report empirical results that support their claims. In light of this, there has been renewed attention toward identifying the distribution of random coefficients in linear panel data models.

In this paper, I complement such a strategy by considering what can be learned about common parameters in, possibly dynamic, panel data models with nonparametricallyspecified excess heterogeneity. Here, the fixed-effect paradigm-that is, circumventing the estimation of the model's idiosyncratic components-allows for a very rich pattern of unobserved heterogeneity in a manner that leaves its distribution essentially unrestricted. The heterogenous effects may enter non-additively and can be time-varying. This leads to a framework in which the conventional incidental parameters are replaced by incidental functions, and traditional first-differencing has to make way for local firstdifferencing.

,

first-differencing in panel data models with incidental functions koen jochmans * core and brown university This version: October, 20 2010 abstract I discuss the fixed-effect estimation of panel data models with time-varying excess heterogeneity across cross-sectional units. These latent components are not given a parametric form. A modification to traditional first-differencing is motivated which, asymptotically, removes the permanent unobserved heterogeneity from the differenced model. Conventional estimation techniques can then be readily applied. Distribution theory for a kernel-weighted GMM estimator under large-n and fixed-T asymptotics is developed. The estimator is put to work in a series of numerical experiments to static and dynamic models. who proposed a kernel-weighted conditional-likelihood estimator for a dynamic logit model with time-varying covariates. In fact, this connection immediately suggests that the proposed approach to dealing with incidental functions can be generalized to all existing fixed-effect estimators, such as conditional maximum-likelihood and maximum score, for example. This is easy to see, but working out the details for the general case would come at the cost of reduced transparency.

This paper has four more sections. The first of these provides details on the model of interest and the local-differencing strategy, and states the GMM estimator that will be the focus of the analysis. The next section contains large-sample theory. The last section before concluding gives results from a series of Monte Carlo exercises.

ii the model, local first differencing, and a gmm estimator Suppose we are in possession of n independent sequences of T observations on the variables (y, x, z, v). Denote the tth observation for unit i = 1, . . . , n by (y it , x it , z it , v it ) and let ℓ(a) be the dimension of the vector a. Assume that the observations on the scalar random variable y were generated through a model of the form

y it = x ′ it θ 0 + ξ it , ξ it = ϑ i (v it ) + ε it . (2.1)
Here, ε it is a zero mean random disturbance, θ 0 is the true value of the parameter vector of interest, and the unit-specific functions ϑ i : R ℓ(v) → R capture the, potentially heterogeneous, impact of the covariates v on y. These functions are assumed to be smooth in their arguments, but are otherwise unmodelled and allowed to be random across i. Remaining loyal to the fixed-effect modelling approach, they are draws from an unknown probability distribution that may depend on the realization of the covariates and, possibly, initial conditions, but not on ε. Feedback from y t to future x t is allowed for, accommodating dynamic models with predetermined regressors. More generally, x is allowed to be endogenous in the cross-sectional sense of being contemporaneously correlated with ε. When strict exogeneity is lacking, z serves as instrumental variables.

No restriction is put on the statistical relationship between v and (x, ε) over time.

The fact that ϑ i (v) may be random after conditioning on a realization of its argument distinguishes (2.1) from a panel data version of a nonparametric functional-coefficient model as considered by [START_REF] Hastie | Varying-coefficient models[END_REF]. It is also different from the random-coefficient model of [START_REF] Chamberlain | Efficiency bounds for semiparametric regression[END_REF] and [START_REF] Arellano | Identifying distributional characteristics in random coefficients panel data models[END_REF], where ϑ i (v) is a linear combination of exogenous v and unit-specific coefficients. Here, the ϑ i are unspecified incidental functions of covariates, which may be predetermined or [3] endogenous. Thus, (2.1) generalizes the standard linear fixed-effect model by allowing for a much richer pattern of unobserved unit-specific factors; notice that this benchmark model is obtained on assuming ϑ i (v) to be constant in v.

To construct an estimator of θ 0 , start by stacking observations over time to obtain

y i ≡ (y i1 , . . . , y iT ) ′ , X i ≡ (x i1 , . . . , x iT ) ′ , V i ≡ (v i1 , . . . , v iT ) ′ , ε i ≡ (ε i1 , . . . , ε iT ) ′ , and ξ i ≡ (ξ i1 , . . . , ξ iT ) ′ ; let ϑ i (V i ) ≡ (ϑ i (v i1 ), . . . , ϑ i (v iT )) ′ .
Write D for the (T -1)× T matrix that performs first-differencing on the above arrays. For example, Dy i = (∆y i2 , . . . , ∆y iT ) ′ , where ∆ ≡ 1 -L and L denotes the lag operator. Then,

y i = X i θ 0 + ξ i , ξ i = ϑ i (V i ) + ε i , (2.2) and Dy i = DX i θ 0 + Dξ i for Dξ i = Dϑ i (V i ) + Dε i . Recall that, when ϑ i (v) is constant, Dϑ i (V i ) = 0
, and an efficient GMM estimator of θ 0 that is based on data in firstdifferences minimizes the quadratic form

1 n n i=1 Z ′ i D(y i -X i θ) ′ G 1 n n i=1 Z ′ i D(y i -X i θ) (2.3)
for a well-chosen weight matrix G; see [START_REF] Holtz-Eakin | Estimating vector autoregressions with panel data[END_REF] and [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]. Throughout, Z i is an instrument matrix of conformable dimension that contains z i2 , . . . , z iT . The latter can be lagged levels of predetermined regressors, all time-series realizations of exogenous covariates, and, possibly, external instrumental variables. Of course, simple differencing does not remove ϑ i (V i ) from (2.2) more generally, and E Z ′ D(y -Xθ 0 ) = 0 unless z t and ∆ϑ(v t ) are uncorrelated.1 However, by smoothness of

ϑ i , ∆y it = ∆x ′ it θ 0 + ∆ϑ i (v it ) + ∆ε it → ∆x ′ it θ 0 + ∆ε it as ∆v it → 0. Let ω v (v) ≡ δ(v -v) for δ : R ℓ(v) → R
a multivariate version of Dirac's delta.2 Then, the conditional moment condition E z t ω 0 (∆v t )(∆y t -∆x ′ t θ 0 ) = 0, which is free of incidental functions, holds for t = 2, . . . , T . This suggest constructing an estimator based on an empirical counterpart of these conditional moment conditions. A fruitful way to proceed is to replace ω 0 (∆v t ) by a smoother such as a kernel weight, and to make the appropriate changes to (2.3).

To do so, partition the vector v as (v (c) , v (d) ), where v (c) and v (d) refer to the continuously-distributed components and the discrete components of v, respectively. Let ℓ ≡ ℓ(v (c) ), let k : R ℓ → R be a kernel with associated bandwidth σ n , and write 1(a) for the indicator function for the event a. A kernel weight takes the form

w it ≡ 1 σ ℓ n k ∆v (c) it σ n 1(∆v (d) it = 0). Form the diagonal matrices W i ≡ diag(w i2 , . . . , w iT ) and Ω v ≡ diag(ω v (v 2 ), . . . , ω v (v T )).
A local GMM estimator based on the moment conditions

E Z ′ Ω 0 D(y -Xθ 0 ) = E Z ′ Ω 0 Dε = 0 (2.4) is obtained by introducing W i in (2.
3). Because the weights do not distort the linearity of this objective function, the local estimator is available in closed form. Moreover, it is given by

θ n ≡ S ′ ZX GS ZX -1 S ′ ZX GS Zy (2.5)
for the matrices

S ZX ≡ 1 n n i=1 Z ′ i W i DX i and S Zy ≡ 1 n n i=1 Z ′ i W i Dy i .
Thus, computing θ n from the data boils down to first assigning a weight to each firstdifferenced observation and then proceeding as one would do with the conventional GMM estimator of choice. Under the regularity conditions provided below, this approach is asymptotically equivalent to using only first-differenced observations for which ∆v it lies in a shrinking neighborhood of zero. Observe that (2.5) reduces to a weighted least-squares estimator on setting Z i = DX i .

One attractive consequence of working with data in first differences is that it is straightforward to deal with unbalanced panel data. Nevertheless, estimation based on weighted versions of orthogonal deviations [START_REF] Arellano | Another look at the instrumental variable estimation of errorcomponents models[END_REF] or deviations from within-group means is also possible. To illustrate, let

ẇs it ≡ 1 σ ℓ n k v (c) it -v (c) is σ n 1(v (d) it = v (d)
is ), choose k to be symmetric, and collect the ẇs it in the symmetric T × T matrix Ẇi , the (t, s)th element of this matrix being ẇs it . A local GMM estimator that is based on forward orthogonal deviations follows on replacing S XZ and S Xy above by the matrices

[5] n -1 n i=1 Z ′ i (DD ′ ) -1/2 D Ẇi X i and n -1 n i=1 Z ′ i (DD ′ ) -1/2 D Ẇi y i , respectively.
A weighted within-group estimator, on the other hand, would take the form arg min θ n -1 n i=1 (y i -

X i θ) ′ Ẇi D ′ (DD ′ ) -1 D Ẇi (y i -X i θ).
The matrix D ′ (DD ′ ) -1 D may be recalled to be the traditional within-group operator.3 Distribution theory for both these estimators can be derived in an analogous fashion as for θ n . However, simulation results suggest that neither outperforms its first-differenced counterpart in small samples. The reason for this is the additional noise induced by taking linear combinations of kernel-weighted quantities.

iii large-sample behavior of the gmm estimator

Let us now consider the behavior of θ n as n grows large. For vectors a and b for which ℓ(a) = ℓ(b), let a denote the Euclidean norm, let |a| ≡ ℓ(a) j=1 a (j) , and let b a ≡ ℓ(a) j=1 (b (j) ) a (j) throughout. The notation • will also be used to indicate the matrix norm.

Start by imposing the following conditions.

Assumption 1 (Regularities). The data array {y

i , x i , z i , v i } n i=1 is a random sample. For t = 2, . . . , T , E [ω 0 (∆v t ) z t ∆ε t ] is finite. For v in a neighborhood of zero, E [Z ′ Ω v DX]
has full column rank. The moment condition E [Z ′ Ω 0 Dε] = 0 holds.

Assumption 1 states the sampling scheme and contains conventional requirements for linear GMM estimators. Moreover, it provides a mild dominance condition on the vector of population moments and ensures identification of θ 0 by imposing a rank condition and postulating instrument validity.

The next assumption demands smoothness from the conditional moments that are being approximated by the kernel weighting and from the conditional densities of ∆v (c) t given realizations of ∆v (d) t ; refer to these densities by f t and to their respective supports by V t .

Assumption 2 (Smoothness). For t = 2, . . . , T , f t (v (c) |v (d) ) is bounded from above for all v in V t and is strictly positive for v in a neighborhood of zero. In addition, f t (v (c) |v (d) ) and E [ω v (∆v t )z t ∆ξ t ] are k-times continuously differentiable in v (c) for all v in a neighborhood of zero, where k ≥ 2 is an integer.

When combined with a well-behaved kernel and bandwidth, Assumption 2 implies that the bias induced by weighting observations by w it disappears asymptotically. It could be relaxed slightly, for example by requiring the bounded away from zero condition on f t to hold only for some but not all t = 2, . . . , T , but this would needlessly cloud the exposition.

What is meant exactly by a well-behaved kernel and bandwidth is the topic of the third assumption.

Assumption 3 (Kernel weights). The kernel, k, is bounded on its support and of kth-order. Moreover, k(η) dη = 1, η j k(η) dη = 0 for |j| = 1, . . . , k -1, and

η j k(η) dη < ∞ for |j| in {0, k}.
The bandwidth, σ n , is both non-negative and

O(1) as n → ∞, while nσ ℓ n → ∞ and nσ ℓ n σ k n → 0.
The required degree of smoothness is increasing in the number of continuous components of v it , ℓ. It is well known that, although they generally increase finite-sample bias, discrete covariates do not retard the speed of convergence of Nadaraya-Watson type kernel estimator of conditional mean functions, and the same conclusion may be drawn here.

It can be noted that an alternative to working with empirical cell probabilities for the discrete components of ∆v t would be to smooth over these variables as well. Such an approach may well be beneficial in small samples when the support of ∆v (d) t is large. Another refinement would be to work with regressor-specific bandwidths, that is, a vector-valued σ n . I abstract away from both of these possibilities for fine tuning here as they do not affect the asymptotic behavior of θ n , nor will it be possible at this stage to provide particular guidelines on how to get the most out of these additional degrees of freedom.

Assumptions 1-3 are more than enough to guarantee the consistency of θ n for θ 0 as n → ∞. To facilitate the exposition, let F 1 be a matrix consisting of T -1 verticallystacked blocks. The (t -1)th such block is given by ι ℓ(zt) ι ′ ℓ(∆xt) f t (0|0) for ι a a vector of ones of length a. On letting ⊙ denote the Schur product,

S ZX p → Σ ZX ≡ E [Z ′ Ω 0 DX] ⊙ F 1 , S Zy p → Σ Xy ≡ E [Z ′ Ω 0 Dy] ⊙ F 1 ,
and

S Zξ (θ) ≡ S Zy -S ZX θ p → Σ Zξ (θ) ≡ Σ Zy -Σ ZX θ.
Write S Zξ and Σ Zξ for S Zξ (θ 0 ) and Σ Zξ (θ 0 ), respectively. The consistency result then reads as follows.

Theorem 1 (Consistency). Let Assumptions 1-3 hold. If G and Γ are positive-definite matrices so that

G p → Γ as n → ∞ and Γ is non-stochastic, then θ n p → θ 0 as n → ∞. [7] Proof. Recall that S Zξ = n -1 n i=1 Z ′ i W i Dξ i , so θ n -θ 0 = S ′ ZX GS ZX -1 S ′ ZX GS Zξ .
Given that G converges to a positive-definite and non-stochastic matrix, it suffices to show that, as n → ∞, (i) S ZX p → Σ ZX ; and (ii) S Zξ p → 0. To see that (i) holds, observe that standard kernel-smoothing arguments yield

1 σ ℓ n E z t ∆x ′ t k ∆v (c) t σ n 1(∆v (d) t = 0) = E ω 0 (∆v t )z t ∆x ′ t f t (0|0) + O(σ ℓ n ) for t = 2, . . . , T . As E [S ZX ] is obtained on stacking such terms over t, it follows that E [S XZ ] -Σ XZ = O(σ ℓ n ). This bias goes to zero because lim n→∞ σ ℓ n = 0, that is, E [S ZX ] ≡ lim n→∞ E [S ZX ] = Σ ZX . Furthermore, by a standard law of large numbers, S XZ -E [S XZ ] = O p (1). Statement (i) follows.
By the same reasoning, S Zξ -Σ Zξ = O p (1). Instrument validity implies that Σ Zξ = 0. The proof is complete.

Establishing asymptotic normality of θ n around θ 0 requires the bias induced by kernel weighting to go to zero at a sufficiently fast rate, and the existence of higherorder conditional moments. For the former requirement to be satisfied, the current assumptions suffice. The latter is dealt with now. Let f ts denote the joint density of (∆v

(c) t , ∆v (c)
s ) given realizations of (∆v

(d) t , ∆v (d) s ).
Assumption 4 (Higher-order moments). For t, s in 2, . . . , T , f ts (v

(c) 1 , v (c) 2 |v (d) 1 , v (d) 2 ) is bounded from above for all (v 1 , v 2 ) in V t ⊗ V s and is strictly positive for (v 1 , v 2 ) in a neighborhood of zero. For all such (t, s), E [z t ω v 1 (∆v t )∆ξ t ∆ξ s ω v 2 (∆v s )z ′ s ] and f ts (v (c) 1 , v (c) 2 |v (d) 1 , v (d) 2 ) are k-times continuously differentiable in (v (c) 1 , v (c)
2 ), and the ab-

solute moments E [ω v (∆v t ) z t ∆x ′ t 3 ] and E [ω v (∆v t ) z t ∆y t 3 ] are finite. The matrix E [Z ′ Ω 0 Dεε ′ D ′ Ω 0 Z] is positive definite and nonsingular.
This condition will ensure that, when properly scaled, S Zξ converges to a zero-mean Gaussian process.

In line with previously introduced notation, let F 2 be an T t=2 ℓ(z t ) × T t=2 ℓ(z t ) matrix consisting of (T -1) 2 blocks, with the (t-1, s-1)th block given by ι

ℓ(zt) ι ′ ℓ(zs) f ts (0|0). Define Υ ≡ E Z ′ Ω 0 Dεε ′ D ′ Ω 0 Z ⊙ F 2 k(η) 2 dη. (3.6)
Then, under Assumptions 1-4, nσ ℓ n S Zξ d → N (0, Υ). From this, the next theorem can be derived.

[8]

Theorem 2 (Normality). Let Assumptions 1-4 hold. If G and Γ are positive-definite matrices so that G p → Γ as n → ∞ and Γ is non-stochastic, then

nσ ℓ n θ n -θ 0 d → N 0, V (Γ) for V (Γ) ≡ Σ ′ ZX ΓΣ ZX -1 Σ ′ ZX Γ ′ ΥΓΣ ZX Σ ′ ZX ΓΣ ZX -1 as n → ∞. Proof. Given that S ZX p → Σ ZX and G p → Γ as n → ∞, it remains to show that (i) nσ ℓ n E [S Zξ ] = O(1); and (ii) nσ ℓ n S Zξ -E [S Zξ ] d → N (0, Υ).
Slutzsky's theorem will take us the rest of the way.

Commence with (i). By a kth-order expansion of a typical block of E [S Zξ ] around ∆v

(c) t = 0, 1 σ ℓ n E z t ∆ξ t k ∆v (c) t σ n 1(∆v (d) t = 0) = E ω 0 (∆v t )z t ∆ε t f t (0|0) + O(σ k n ).
The first right-hand side term above is zero by instrument validity. The order of magnitude of the remainder term follows by virtue of the higher-order kernel and the associated smoothness conditions. Consequently,

nσ ℓ n E [S Zξ ] = nσ ℓ n O(σ k n ) = O(1).
To verify (ii) we can follow [START_REF] Honoré | Panel data discrete choice models with lagged dependent variables[END_REF] in checking that the regularity conditions of Lyapunov's central limit theorem for double arrays hold. Write

nσ ℓ n c ′ S Zξ -E [S Zξ ] = 1 nσ ℓ n n i=1 c ′ q i -E [q] = 1 √ n N i=1 r i
for any vector of constants c for which

c ′ c = 1. Clearly, E [r] = 0 while E [rr ′ ] = σ -ℓ n c ′ E [qq ′ ]c-σ -ℓ n c ′ E [q]E [q ′ ]c.
The two expectations in this variance are finite. Moreover, the second term is O(σ

ℓ n ) = O(1) from above. For the first term, E [qq ′ ] has typical block 1 σ ℓ n c ′ E z t k ∆v (c) t σ n 1(∆v (d) t = 0)∆ξ t ∆ξ s 1(∆v (d) s = 0)k ∆v (c) s σ n z ′ s c.
Smoothness and dominated convergence again imply that, as n → ∞, such blocks converge to

c ′ E z t ω 0 (∆v t )∆ε t ∆ε s ω 0 (∆v s )z s c f ts (0|0) k(η) 2 dη. Thus, E [rr ′ ] = σ ℓ n c ′ E [qq ′ ]c = c ′ Υc.
Lastly, it is easy to show that the boundedness of the kernel and of the absolute moments in Assumption 4 imply that, for any α in (0, 1),

n i=1 E r √ n 2+α = O 1 nσ ℓ n α = O(1).
Lyapunov's theorem may then be applied to obtain statement (ii) and to complete the proof.

[9]

By standard optimality theory for GMM estimators [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF], the efficient weighting scheme for the moment conditions is obtained on setting G proportional to the inverse of a consistent estimate of Υ. Doing so leads to the following derivative-result to Theorem 2.

Corollary 1 (Optimally-weighted GMM). Let the conditions for Theorem 2 hold for estimators θ * n and θ * * n that use a weight matrices G * and G * * that satisfy G * p → Γ and G * * p → Υ -1 as n → ∞, respectively Then,

nσ ℓ n θ * * n -θ 0 d → N 0, V , V ≡ V (Υ -1 ) = Σ ′ ZX Υ -1 Σ ZX -1
Furthermore, V (Γ) ≥ V in the matrix sense.

Proof. The expression for V follows from Theorem 2 and a small calculation. The efficiency result is readily verified on noting that

V (Γ) -V = B ′ [I -A(A ′ A) -1 A ′ ]B for A ≡ Υ -1/2 Σ ZX and B ≡ [Σ ′ ZX ΓΣ ZX ] -1 [Σ ′ ZX ΓΥ 1/2 ], and that [I -A(A ′ A) -1 A ′ ] is an idempotent matrix.
The last ingredient needed for inference procedures on the basis of θ n to become operational is a consistent estimate of Υ in (3.6), say U . Then, for any properly formed G, and for G = U -1 in particular, the consistency of a plug-in estimate for V (Γ) follows immediately. An estimate of V , for example, is the inverse of S ′ ZX U -1 S ZX . In the special case of just-identification, such as for the weighted least-squares estimator obtained on setting Z i = DX i , the matrix S ZX is square and symmetric, and we obtain the familiar 'sandwich-form' estimator S -1 ZX U S -1 ZX . It suffices to show that the general estimator of Υ defined as

σ ℓ n n n i=1 Z ′ i W i De i e ′ i D ′ W i Z i (3.7)
for given residuals e i is consistent as n → ∞. Given the efforts made so far, showing this result poses no additional difficulty.

Theorem 3. Let Assumptions 1-4 hold, let θ * n be an initial estimator to which Theorem 1 applies, and define ξ * i ≡ y i -X i θ * n . Then, U p → Υ as n → ∞ for U as defined in (3.7) and formed with e i = ξ * i .

Proof. The proof uses the same arguments as those used to prove Theorem 2.

[10]

Under conditional homoskedasticity of ε i , and in the absence of serial correlation, setting

U = n -1 n i=1 Z ′ i W i DD ′ W i Z i will
give rise to the optimally-weighted GMM estimator. Alternatively, a two-step estimator would use the residuals, ξ * i , to form U as in (3.7). More generally, an iterated GMM procedure-as discussed in Hall (2005, pp. 90) and also elsewhere-repeats the two-step procedure until convergence. The estimate so formed is independent of the initial choice for G. Of course, the same comments about the effect of the variability of G on the small-sample performance of GMM estimators apply to the local GMM estimator introduced here. In light of this, it might be of interest to work out a correction term analogous to the one derived by [START_REF] Windmeijer | A finite sample correction for the variance of linear efficient two-step GMM estimators[END_REF] for linear GMM estimators.

An alternative to an argument based on such an exercise in higher-order asymptotics is to consider a continuously-updated (CU) GMM routine as initially introduced by [START_REF] Hansen | finite-sample properties of some alternative GMM estimators[END_REF]. Here, such an estimator takes the form

arg min θ S Zξ (θ) ′ U (θ) -1 S Zξ (θ)
where U (θ) is U for a given θ. Such a modification makes the minimization problem nonlinear in θ but does not distort the limiting distribution of the point estimates, so that both Theorem 1 and Theorem 2 continue to apply. The CU estimator is a device originally suggested to mitigate the finite-sample bias of efficient GMM estimators that utilize overidentifying restrictions. Although it is not known whether this estimator has any finite moments, it is generally found to perform better in simulations in terms of bias than the corresponding two-step estimator. However, it also tends to exhibit a higher dispersion. Another useful feature of the CU estimator that explains its improved centering property is its interpretation as a jackknife estimator; see [START_REF] Donald | A jackknife interpretation of the continuous updating estimator[END_REF] for this. A final potentially benevolent effect of the CU approach is its invariance with respect to how the moment conditions are scaled. For an illustration and discussion of the CU estimator in the conventional panel data context, see Arellano (2003, pp. 73 and pp. 171-172).

Like in the standard setting, the efficiently-weighted local GMM minimand, when evaluated at θ * * n , can be used to test overidentifying restrictions. The limiting result for the corresponding J-statistic is that

nσ ℓ n S Zξ (θ * * n ) ′ U -1 S Zξ (θ * * n ) d → χ 2 (m -ℓ(x))
under Assumptions 1-4, where m denotes the number of columns of Z. The proof is immediate from Corollary 1. The result also extends straighforwardly to the CU estimator.

[11]

iv a monte carlo study

The performance of the local-GMM estimator was assessed through simulation exercises. This section reports results for static models estimated through kernel-weighted least squares and for dynamic models estimated through a local version of the first-differenced GMM procedure as originally introduced by Holtz-Eakin, Newey, and Rosen (1988) and [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]. I use different design variations to be able to accentuate different findings in the static and the dynamic cases. The statistics used to evaluate the estimator's performance are the mean-and the median bias as measures of centrality, the standard deviation (STD) and the interquartile range (IQR) as indicators of dispersion, the root meansquared error (RMSE) as a criterion for estimator risk, and coverage rates of two-sided 95% confidence intervals for inference evaluation. In the tables below CI(A) refers to coverage rates obtained through the use of plug-in estimates of the asymptotic variance. The entries for CI(B) are bootstrap-based acceptance frequencies using the percentile method over 39 bootstrap replications. All descriptive statistics for each design point were obtained over 10, 000 simulation runs, with all variables and incidental parameters redrawn in each iteration.

Below, θ * n refers to the one-step local-GMM estimator while θ * * n denotes the two-step version. I experimented with iterating the two-step procedure but found no significant gain from doing so. I also ran simulation experiments with the CU estimator. While I found it to have reasonably small bias, its standard deviation was often very high, which is in line with results found in the cross-sectional literature. Here, I prefer to report results on jackknifed versions of the local-GMM estimator, which is an alternative route to constructing bias-corrected estimators. The jackknifed estimators take the usual delete-one form

θ † n ≡ nθ * n - n -1 n n i=1 θ * n,-i and θ † † n ≡ nθ * * n - n -1 n n i=1 θ * * n,-i
where θ * n,-i and θ * * n,-i are the one-step and two-step estimators obtained from estimating θ 0 from the subpanel obtained on deleting the ith time series from the full panel. In the conventional setting, it is known that the bias of the first-differenced GMM is O(n -1 ); see [START_REF] Arellano | Modelling optimal instrumental variables for dynamic panel data models[END_REF]. It is therefore reasonable to expect that, for k chosen sufficiently large, the bias of the local-GMM estimator has a leading term that is O(n -1 ). The jackknifed estimators θ † n and θ † † n subtract a nonparametric estimate of the leading bias term from θ * n and θ * * n , respectively.

[12]

y it = x it θ 0 + ϑ i (v it ) + ε it , ϑ i (v it ) = γ (1) i + γ (2) i v it -γ (3) i v 2 it . (4.8)
Here, γ

(1) i

∼ N (0, 1) are traditional fixed effects while γ

(2) i ∼ N (0, 2) and γ

(3) i ∼ N (0, .75) appear in an interactive fashion. The form of ϑ i allows for both concave and convex response functions. The dynamics of the covariates x and v were set to be

x it = -.3γ (1) i + .5x it-1 + β (1) i v it + N (0, 1) and v it = .3γ (1) i + β (2) i v it-1 + ζ it , (4.9)
respectively. In (4.9), β

(1) i ∼ N (0, 1) and β

(2) i ∼ U[.20, .99], so that both stronglyand weakly-persistent data sequences occur. Because differencing is executed over time the within-group variation in ϑ i is much more important for the performance of the weighting than is the between-group variation. The start-up values for the covariate time series were generated as x i0 ∼ -.3γ

(1) i + N (0, 1) and v i0 ∼ .3γ

(1) i + N (0, 1). The disturbances are bivariate-normal variates, each with unit variance and a correlation coefficient ρ 0 which was 0 in the first design and .4 in the second. The latter case leads to x, v, and ε being contemporaneously correlated, making x endogenous in the crosssectional sense. Observe also that x and v are dependent, so that a naive fixed-effect estimator will be inconsistent for θ 0 .

Tables 1 and2 contain the results for the kernel-weighted first-differenced leastsquares estimator for ρ 0 = 0 and ρ 0 = .4, respectively. Because, here, there are no overidentifying restrictions, both θ * n and θ * * n , and θ † n and θ † † n coincide. Throughout, θ 0 was fixed at .5. 4 The dimensions of the panels generated were n = 100 and T ∈ {3, 6, 9}. Results are reported for the standard-normal kernel (Normal), the secondorder Epanechnikov kernel (Epanechnikov), the quartic kernel (Quartic), and the cosine kernel (Cosine). Three different bandwidth choices are reported on; 5n -3/4 (small), 15n -3/4 (medium), and 45n -3/4 (large).

Overal, Table 1 reports good performance of θ * n . Both the mean-and the median bias are virtually zero throughout. The jackknifed estimator has a bias of the same magnitude. Not surprisingly, θ † n is more volatile. The difference in the STD and IQR is, however, very small. As a consequence, the RMSE of both estimators is identical to their STD up to the first few decimal digits. Both CI(A) and CI(B) report solid coverage rates, with neither dominating the other. 5 The actual size is generally somewhat larger than the nominal size of .05, which is in line with the usual behavior found in Monte Carlo work elsewhere. The impact of the choice of the kernel and the bandwidth is strongest on the measures of spread. But no design constellation uniformly dominates another.

When turning to the design with contemporaneous dependence between x and ε in Table 2 the overal picture changes little. Both estimators continue to have an empirical distribution that is roughly correctly centered; their variance changes little, too. Thus, confidence intervals still provide approximately correct coverage rates. No consistent pattern emerges when comparing the numbers across the tables.

Dynamic models. This subsection deals with the following variation on (4.8)-(4.9).

y it = y it-1 θ 0 + ϑ i (v it ) + ε it , ϑ i (v it ) = γ (1) i + γ (2) i v it -γ (3) i v 2 it , (4.10)
with v it = .3γ

(1)

i + β (2) i ρ 0 v it-1 + (1 -ρ 0 ) U[-1, 1] + ζ it (4.11)
for ρ 0 ∈ {0, 1} and i.i.d. errors (ε it , ζ it ) ∼ N (0, I). The processes were generated with a burn in of 500 time periods, using

v i(-501) ∼ .3γ (1)
i + N (0, 1) and y i(-501) ∼ -.3γ

i + ϑ i (v i(-501) ) + N (0, 1) as startup values. The local-GMM estimator of θ 0 à la [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] uses all lagged levels of y it-1 as instruments for ∆y it-1 in the equations in first differences.

Like with the conventional estimator, the small-sample bias of θ * n and θ * * n will be a function of the number of moments used as well as the instrument strength. The former relates directly to the size of the panel (see, e.g., [START_REF] Alvarez | The time series and cross-section asymptotics of dynamic panel data estimators[END_REF], the latter is partly driven by the closeness of the autoregressive parameter θ 0 to unity, with the instruments becoming irrelevant when θ 0 = 1. In addition, coverage intervals based on the two-step estimator that are set up by means of analytical standard errors would be expected to be too small, leading to nominal rejection frequencies that are too high compared to a chosen significance level. Table 7 below contains the average of the estimated standard errors of both θ * n and θ * * n by means of the analytical formulae and the bootstrap, together with the empirical standard deviation for a selection of the Monte Carlo experiments that follow. It can be observed that, for both estimators, the plug-in estimates are too low on average. The bootstrap-based standard errors are much more in line with the variability of the estimators as actually observed over the Monte Carlo runs. For this reason, I focus on the coverage rates of the confidence intervals based on resampling below. .9524 n = 100, θ 0 = .5, 10, 000 Monte Carlo replications. All kernel arguments were scaled down by their respective standard deviation.

An important difference between the least-squares estimator as considered in the previous subsection and the GMM estimator for the dynamic model is that it uses levels as instruments rather than differences. While this is also the case when ϑ i does not depend on v, there might be additional finite-sample effects when v is autocorrelated. To see this, recall that local differencing removes ∆ϑ i (v it ) from the moment conditions only asymptotically. For any finite n, therefore, lagged dependent variables will correlate with w it ∆ξ it . To assess the importance of this serial correlation, I manipulate ρ 0 from zero to unity. When ρ 0 = 0, v is independent over time while, when ρ 0 = 1, it is [17] first-order autocorrelated with some sequences being nearly integrated of order one.

The kernel and bandwidth choices from before were maintained, and n was equally kept at 100. To save on space, I report results only for a subset of the design variations. Additional results are available as supplementary material. Table 3 gives the usual descriptive statistics for the case where T = 3, θ 0 = .5, and ρ 0 = 0. Both the one-and two-step estimator have a reasonably small bias. The choice of k and σ n seems particlarly important here. The jackknifed estimators give an improvement in terms of centrality, but at a cost of a higher STD and RMSE. When n [18] is larger, however, this discrepancy vanishes. The IQR of θ † n and θ † † n are, however, not much larger than those of θ * n and θ * * n and also the coverage rates are comparable. When ρ 0 is set to unity, the estimators' performance change. All estimators now have both a larger bias and a higher variability. Although the bias never skyrockets, the jackknife estimators do not outperform θ * n and θ * * n in terms of any of the statistics reported, although they start doing so for larger n. Moreover, they are more biased than the original estimators, and their standard deviation is large. Nevertheless, the IQR remains relatively small and the coverage rates are close to .95. Tables 5 and6 show what happens when T is increased to 6. Of course, the STD and IQR shrink, and considerably so. For ρ 0 = 0, the bias of all estimators increases while, for ρ 0 = 1, the opposite happens. The coverage rates worsen slightly across both design variations. This effect is less pronounced for θ † n and θ † † n than it is for θ * n and θ * * n . The last table in this subsection makes the comparison between the average estimated standard errors of θ * n and θ * * n and their empirical standard deviation.

[20]

v conclusion

This paper has used the notion of local differencing to construct an estimator for fixedeffect panel data models with nonparametric unobserved heterogeneity. In line with the traditional estimators, the fixed-effect 'ignorance' towards the models' unit-specific latent components that underlies first-differencing the data allows for the unobserved heterogeneity to take on very general forms. The resulting local GMM estimator was shown to be consistent and asymptotically normal. It complements the recent results on identification of random coefficients in linear panel data models with exogenous covariates. With some work, the approach can be extended to a more general class of nonlinear panel data models.

Several questions and extensions of potential interest immediately suggest themselves. A pertinent question from an applied point of view is how to best choose the smoothing parameters. Ideally, a data-driven selection method should be found that determines the bandwidth in an optimal manner. Indeed, the Monte Carlo results suggest the smoothing used to be of substantial importance in small sample. A larger study of the sensitivity of the local GMM estimator to the form of the incidental functions and the stochastic behavior of their arguments would also be desirable. Related to this comment, it appears of interest to categorize all the information content in the data in terms of nonlinear moment conditions under additional assumptions such as homoskedasticity. Such an exercise in the spirit of [START_REF] Ahn | Efficient estimation of models for dynamic panel data[END_REF] would allow for efficiency gains and, arguably, improved finite-sample performance. Given the increased attention to cross-sectional dependence in the panel data literature-see [START_REF] Sarafidis | Cross-sectional dependence in panel data analysis[END_REF] for an up-to-date review-it might be worthwhile investigating the potential for local first-differencing in the presence of factor structures in the disturbance processes. It seems reasonable to expect that the GMM estimators for such problems can be modified in a manner that is analogues to what was discussed here.

A final remark related to incorporating unit-specific non-stationary effects, such as time trends or time dummies. These are important empirical phenomena. Clearly, local first-differencing would break down under such a scenario as these variables, when transformed, are bounded away from zero. This is no different than in the binarychoice setting of [START_REF] Honoré | Panel data discrete choice models with lagged dependent variables[END_REF]. An interesting extension to the model considered here, therefore, would be to construct a hybrid version that combines local differencing with a parametric random-coefficient specification such as the ones studied in [START_REF] Chamberlain | Efficiency bounds for semiparametric regression[END_REF] and [START_REF] Arellano | Identifying distributional characteristics in random coefficients panel data models[END_REF].

[22] references

  jel classification: C13, C14, C33 keywords: dynamic panel data, GMM, incidental functions, local firstdifferencing, time-varying fixed effects, nonparametric heterogeneity.

Table 1 :

 1 Monte Carlo results for model (4.8)-(4.9) under independence (ρ 0 = 0). The first model considered has scalar x and v influencing the outcome variable through

		WEIGHT	MEAN BIAS	MEDIAN BIAS	STD	IQR	RMSE	CI(A)	CI(B)
	T k	σ n	θ

*

Static models.

Table 2 :

 2 Monte Carlo results for model (4.8)-(4.9) under dependence (ρ 0 = .4).

		WEIGHT	MEAN BIAS	MEDIAN BIAS	STD	IQR	RMSE	CI(A)	CI(B)
	T k	σ n	θ *

Table 4 :

 4 Monte Carlo results for model (4.10)-(4.11); ρ 0 = 1, T = 3.

	WEIGHT		MEAN BIAS			MEDIAN BIAS
	k	σ n	θ * n	θ † n	θ * * n	θ † † n	θ * n	θ † n	θ * * n	θ † † n
	Normal	small	.0158	.0769	.0104	.0638	.0012	.0406	.0019	.0359
	Normal	medium	.0563	.1500	.0519	.1634	.0382	.0991	.0385	.0925
	Normal	large	.0604	.1774	.0545	.2262	.0430	.1069	.0372	.1015
	Epanech. small Epanech. medium	-.0429 .0409	.0567 -.0474 .1152 .0341	.0191 -.0660 -.0211 -.0622 -.0208 .0887 .0262 .0737 .0297 .0771
	Epanech. large	.0601	.1711	.0550	.2466	.0419	.1059	.0376	.0989
	Quartic Quartic	small medium	-.0526 .0287	.0711 -.0547 .0994 .0223	.0570 -.0748 -.0312 -.0721 -.0330 .0772 .0149 .0564 .0144 .0584
	Quartic	large	.0592	.1632	.0548	.2105	.0409	.1026	.0364	.0972
	WEIGHT		STD				IQR
	k	σ n	θ * n	θ † n	θ * * n	θ † † n	θ * n	θ † n	θ * * n	θ † † n
	Normal	small	.3178	.7237	.3294 1.0133	.3191	.3472	.3254	.3806
	Normal	medium	.3921 1.2205	.4015 1.7770	.3674	.4143	.3614	.4494
	Normal	large	.4331 1.0987	.4475 2.5873	.4008	.4526	.3903	.4816
	Epanech. small	.3578 1.3454	.3903 2.0493	.3571	.4037	.3686	.4507
	Epanech. medium	.3396	.7551	.3578 1.2447	.3449	.3779	.3309	.4035
	Epanech. large	.4251 1.0993	.4340 3.7695	.3960	.4467	.3855	.4750
	Quartic	small	.3814 2.5693	.4034 2.5677	.3720	.4307	.3900	.4859
	Quartic	medium	.3190	.6842	.3356 1.0863	.3305	.3622	.3234	.3843
	Quartic	large	.4127 1.1089	.4221 2.5327	.3884	.4375	.3791	.4613
	WEIGHT		RMSE			CI(B)
	k	σ n	θ *

Table 5 :

 5 Monte Carlo results for model (4.10)-(4.11); ρ 0 = 0, T = 6.

	WEIGHT		MEAN BIAS			MEDIAN BIAS
	k	σ n	θ * n	θ † n	θ * * n	θ † † n	θ * n	θ † n	θ * * n	θ † † n
	Normal Normal Normal Epanech. small small medium -.0469 -.0041 -.0436 -.0033 -.0461 -.0033 -.0426 -.0034 -.0558 -.0036 -.0560 -.0031 -.0548 -.0052 -.0541 -.0066 large -.0520 -.0062 -.0466 -.0040 -.0512 -.0070 -.0454 -.0040 -.1025 -.0136 -.1035 -.0139 -.1030 -.0202 -.1039 -.0235 Epanech. medium -.0469 -.0029 -.0460 -.0028 -.0473 -.0040 -.0462 -.0056 Epanech. large -.0509 -.0058 -.0459 -.0039 -.0505 -.0062 -.0444 -.0046 Quartic small -.1192 -.0184 -.1187 -.0165 -.1210 -.0279 -.1203 -.0291 Quartic medium -.0496 -.0028 -.0494 -.0025 -.0505 -.0053 -.0483 -.0035 Quartic large -.0493 -.0051 -.0449 -.0037 -.0485 -.0050 -.0439 -.0045 WEIGHT STD IQR
	k	σ n	θ * n	θ † n	θ * * n	θ † † n	θ * n	θ † n	θ * * n	θ † † n
	Normal	small	.0999	.1168	.1092	.1395	.1323	.1541	.1435	.1812
	Normal	medium	.0953	.1059	.1021	.1267	.1265	.1400	.1377	.1620
	Normal	large	.1008	.1126	.1060	.1329	.1333	.1479	.1409	.1748
	Epanech. small	.1261	.1646	.1398	.1989	.1650	.2033	.1801	.2508
	Epanech. medium	.0955	.1085	.1033	.1304	.1254	.1399	.1364	.1690
	Epanech. large	.0997	.1111	.1052	.1315	.1321	.1467	.1406	.1722
	Quartic	small	.1356	.1827	.1502	.2232	.1750	.2276	.1963	.2868
	Quartic	medium	.0968	.1111	.1052	.1329	.1282	.1473	.1398	.1729
	Quartic	large	.0979	.1089	.1040	.1295	.1310	.1450	.1389	.1687
	WEIGHT		RMSE			CI(B)
	k	σ n	θ *

Table 6 :

 6 Monte Carlo results for model (4.10)-(4.11); ρ 0 = 1, T = 6.

	WEIGHT		MEAN BIAS			MEDIAN BIAS
	k	σ n	θ * n	θ † n	θ * * n	θ † † n	θ * n	θ † n	θ * * n	θ † † n
	Normal Normal	small medium	-.0138 .0454	.0474 -.0212 .1075 .0324	.0376 -.0167 .0860 .0451	.0398 -.0197 .1025 .0327	.0331 .0825
	Normal	large	.0516	.1218	.0376	.0965	.0523	.1158	.0375	.0902
	Epanech. small Epanech. medium	-.0897 -.0022 -.0933 -.0058 -.0922 -.0107 -.0930 -.0144 .0230 .0810 .0113 .0641 .0206 .0738 .0114 .0603
	Epanech. large	.0510	.1197	.0372	.0949	.0514	.1137	.0373	.0890
	Quartic Quartic Quartic	small medium large	-.1078 -.0137 -.1099 -.0148 -.1093 -.0221 -.1115 -.0276 .0055 .0638 -.0043 .0502 .0032 .0460 .0568 -.0053 .0496 .1158 .0360 .0921 .0497 .1102 .0368 .0871
	WEIGHT		STD				IQR
	k	σ n	θ *

Table 7 :

 7 A comparison between the estimated standard errors and the empirical standard deviation

	ρ 0 = 0	ρ 0 = 1

As interest here lies in micropanels, throughout, the maintained asymptotic scheme will be that of large n and fixed T . Accordingly, expectations are taken with respect to the cross-sectional distribution of the data.

Dirac's delta is defined as follows: if a = 0, then δ(a) = 0; if a = 0, then δ(a) = ∞. As a measure it has the property that g(a) δ(a) da = g(0) for a continuous function g. I prefer its use here over the usual notation for conditional expectations because explicitly conditioning on all time-series realizations of covariates and instruments on top of period-specific first differences would impose a heavy notational burden.[4]

Of course, the usual caveat in this interpretation applies when k is a higher-order kernel, as then some of the weights will have to be negative.

Experiments with different values for θ 0 gave qualitatively similar results.

Here, CI(A) was obtained by means of the heteroskedasticity-robust variance-covariance estimator, as displayed in the text.[14]

small .1161 .1484 .1211 .1719 .1510 .1821 .1547 .2072 Normal medium .1195 .1488 .1160 .1624 .1548 .1835 .1536 .2019 Normal large .1269 .1603 .1215 .1724 .1654 .1967 .1605 .2167 Epanech. small .1357 .1906 .1487 .2325 .1788 .2328 .1948 .2800 Epanech. medium .1180 .1485 .1170 .1652 .1527 .1832 .1531 .2016 Epanech. large .1255 .1580 .1204 .1704 .1630 .1941 .1591 .2116 Quartic small .1416 .2037 .1562 .2495 .1842 .2458 .2029 .3024 Quartic medium .1167 .1475 .1184 .1670 .1524 .1833 .1531 .2014 Quartic large .1233 .1545 .1187 .1672 .1609 .1903 .1562 .2094 WEIGHT RMSE CI(B)

small -.0006 -.0009

.0004 -. 0009 .1609 .1627 .2093 .2114 .1609 .1627 .9099 .9058 .9193 .9129 3 Normal medium -.0025 -.0026 -.0025 -. 0024 .1006 .1008 .1343 .1338 .1007 .1008 .9323 .9318 .9299 .9279 3 Normal large -.0038 -.0039 -.0043 -.0047 .0962 . 0964 .12620964 . .1266 .0963 .0965 .9337 .9339 .9326 .9304 3 Epanech. small .0963 .0965 .9337 .9339 .9326 .9304 3 Epanech. small .0021 .0014 -.0002 .0000 . 2435.2507.3106 .3178 .2435.2507.8851 .8705 .9184 .9137 3 Epanech. medium -.0018 -.0021 -.0022 -.0024 .1369.1378.1790.1797.1369.1378 .9246 .9208 .9284 .9246 .9208 .9284 .1798.2185.2493.2632.2430.2472.2086.2632.3067 .3301 .3178 .3294 3 Normal medium .1670.2359.2698.2831.2702.2727.2020.3081 .3598 .3840 .3921 .4015 3 Normal large .1703.2513.2888.3021 .2915.2928.2094.3353 .3849 .4094 .4331 .4475 3 Epanech. small .2384.2631.3116 .3345 .2998.3069 .2731.3012 .3463 .3728 .3578 .3903 3 Epanech. medium .1672.2231.2546.2669.2519.2540.1969.2822.3317 .3554 .3396 .3578 3 Epanech. large .1696.2486.2853.2984.2878.2897.2077.3301 .3808 .4047 .4251 .4340 3 Quartic small .2622.2810.3325 .3564 .3274 .3382 .3000 .3183 .3606 .3902 .3814 .4034 3 Quartic medium .1711.2185.2498.2633.2435.2462[START_REF] Hastie | Varying-coefficient models[END_REF].2684.3162 .3393 .3190 .3356 3 Quartic large .1685.2440.2796.2930.2815.2847.2053 .3218 .3734 .3965 .4127 .4221


J Anthoine
email: jerome.anthoine@onera.fr

J-Y Lestrade

J Steelant

EXPERIMENTAL DATABASE WITH REAL PROPELLANTS TO STUDY MULTI-PHASE FLUID HAMMER PHENOMENA

Keywords:

The objective of the present study is to generate a reliable experimental database on the fluid hammer phenomenon with real propellants (MMH and NTO) to validate the physical models implemented in the numerical codes. This database with real propellants extends the database obtained by the von Karman Institute on similitude fluids (water, ethanol and acetaldehyde). The test facility is composed with parts provided by VKI (propellant line, fast opening valve, measurement module and instrumentation). The combination of these elements with the Onera pressure vessel and vacuum system enables to build a facility close to the VKI one used for similitude fluids. For each real propellant, experiments were performed for a vacuum pressure in the propellant line below and above the saturation pressure of the fluid to highlight the influence of the physical phenomena encountered (vaporisation, cavitation …) on the fluid hammer. With MMH, the higher is the vacuum pressure, the smaller are the fluid hammer frequency and amplitude. The same conclusions found by VKI for ethanol when compared to water are now found with MMH: the fluid hammer amplitude is smaller and the frequency is higher. Only one experiment was performed with NTO for a vacuum pipe pressure below the saturation pressure due to leaks of the valve seals. Compared to MMH and water experiments performed under the same operating conditions, the fluid hammer amplitude and frequency are smaller. Regarding the temperature, very small variations are recorded for water and MMH while the temperature increase for NTO is higher than 350°C.

INTRODUCTION

The operation of spacecraft propulsion systems generally consists of four phases. The first one includes the launch and the separation of the satellite from the launcher vehicle. During this phase, the spacecraft propulsion system is inactive and the propellant tanks are isolated from the combustion chamber by at least two barriers: the latch valve which is a pyrotechnic valve and the propellant flow valve installed before the combustion chamber (figure 1). The second phase consists of the pressurization of the tanks and of the opening of the latch valve. From this stage, the propellant lines of the spacecraft, initially vacuum pumped or filled with a non-condensable gas at low pressure, are filled and pressurized. These operations are called "priming" and once achieved, the propulsion system is fully operational. The third phase or "drift orbit" takes place during the full deployment of the satellite where the propulsion system is used for attitude control and for positioning to the target orbit. Finally, the last phase or the "on station phase" consists of keeping the spacecraft in its orbit until the end of its operation lifetime.

Figure 1: Spacecraft monopropellant propulsion system

The main problems encountered during the spacecraft lifetime are linked to the priming operation of the propulsion systems which is regularly faced with adverse fluid hammering effects classically known as water hammer. This operation may turn out to be critical if the corresponding overpressures are not correctly considered in the pipe and sub-system dimensioning (flow control valve, pressure sensors, filters …).

The Propulsion Laboratory of Onera has already investigated experimentally the fluid hammer effects using simplified liquid (water) and real propellants into several pipe configurations, where the liquid was pressurized in a tank and the pipe system was vacuum pumped [START_REF] Gibek | Water Hammer Tests with Real Propellants[END_REF]. A pressure surge of 28.9 MPa is reached using water as the working fluid, starting from initial conditions of 2 MPa in the tank and 1000 Pa in the vacuum-pumped propellant line. Tests and results using MMH and NTO are presented and discussed. However, several phenomena, such as cavitation, absorption and desorption of a pressurizing gas and influence of the vapour generation, are not addressed during the study.

In a second study, still at Onera, the influence of the pressure ratio between the pipe and the tank, cavitation phenomenon and NCG content was investigated using ethanol, MMH, and acetaldehyde [START_REF] Lecourt | Experimental Investigation of Waterhammer in Simplified Feed Lines of Satellite Propulsion Systems[END_REF]. An adiabatic compression takes place at the end of the pipe when the liquid front is travelling downstream. The first peak presents a multiple step evolution explained by the generation of a two-phase flow at the front location. These studies proved that complementary fundamental experimental investigations are necessary for a better understanding of the phenomena which were not accounted for as the cavitation … Indeed, these multi-phase phenomena make the fluid-hammer behaviour hard to model due to a lack of the understanding of the physical processes taking place.

Within his PhD Thesis supported by ESA at VKI, Lema [START_REF] Lema | Multiphase Fluid Hammer: Modeling, Experiments and Simulations[END_REF] performed a numerical investigation of the water hammer using the one-dimensional commercial software EcosimPro/ESPSS. Mixing layers develop between the pure liquid and pure air front, with vapour formation due to the cavitation phenomenon [START_REF] Porca | Experimental and Numerical Multiphase-Front Fluid Hammer[END_REF]. The driving pressure gas dissolved in the liquid leads to significant differences on the first pressure peak due to the speed of sound change. The commercial code CFD-ACE+ was also used with the full cavitation model and the simulations confirm high sensitivity of the water hammer prediction regarding the amount of NCG present in the liquid [START_REF] Pinho | Multiphase Investigation of Water Hammer Phenomenon Using the Full Cavitation Model[END_REF].

Within the ESA contract to support the PhD Thesis of Lema, VKI built a dedicated facility, representing a structural configuration of a propulsion system for typical satellite applications, with accurate control of the operating conditions [START_REF] Lema | Multiphase Fluid Hammer: Modeling, Experiments and Simulations[END_REF][START_REF] Lema | Multiphase Fluid Hammer in Spacecraft Propulsion Systems[END_REF][START_REF] Lema | Experimental Characterization of the Priming Phase Using a Propellant Line Mock-up[END_REF]. The facility allows for the fundamental experimental investigation of the multi-phase fluid hammer and related multi-phase phenomena such as cavitation, boiling front, absorption and desorption of non condensable gases. It is constituted by a pressure vessel, a valve with an opening time lower than 30 ms and a given length of the propellant line. The facility includes a vacuum system to set the initial test conditions. Straight, elbow and T junction configurations are considered using real hardware provided by TAS-Cannes and Astrium Ottobrun. Similitude fluids such as water, ethanol and acetaldehyde are successively tested, the last two being respectively the similitude fluids for MMH and NTO. The characterisation of the wave front induced by the fluid hammer is achieved through a measurement module placed at the end of the test section. The module is instrumented with flush mounted dynamic transducers for unsteady measurements of pressure and temperature. The measurement module can also be replaced by a transparent one for flow visualisations with high speed imaging.

The VKI experimental database is built up by varying the vacuum conditions in the line, the saturation conditions of the test liquid, the pipe configuration and the liquid properties. Lema [START_REF] Lema | Multiphase Fluid Hammer: Modeling, Experiments and Simulations[END_REF] found that the residual gas is responsible for attenuating the pressure raise due to its cushioning effect. The comparison of the results for the three liquids (water, ethanol and acetaldehyde) in the straight configuration shows that, besides the desorption rate, density and speed of sound appear as the key liquid properties, without a clear influence of the vapour pressure on the water hammer phenomenon. Lema [3] characterized also, for the first time, the water hammer phenomenon through flow visualizations. The quality of the recorded images is impressive and the sequence of snapshots allows distinguishing the foamy mixture preceding the liquid front and the non condensable gas compression when the front impacts at the bottom end. The same sequence shows also the subsequent column separation with the creation of bubbles and the impact of that separated liquid column against the bottom end.

The understanding of this multi-phase behaviour and the creation of the VKI extensive experimental database in case of simulating fluids allowed the validation and the improvement of the physical models implemented in EcosimPro/ESPSS and CFD-ACE, two numerical codes used by ESA to simulate fluid hammer phenomenon. To guaranty the reliability on these two numerical tools, it was proposed to complete the VKI experimental database with real propellants (MMH and NTO). Nevertheless, VKI does not have the certification to work with these two fluids. The chosen option was to perform the complementary experiments with real propellants at the Propulsion Laboratory of Onera which has the necessary certification and which has proven its mastery of these fluids in the past.

The paper firstly focuses on the description of the experimental facility which uses parts fulfilled by VKI. Then, the results of the fluid hammer experiments performed with real propellants are provided.

TEST FACILITY DESCRIPTION

The experiments performed by Onera made use of exactly the same hardware than VKI (propellant line, valves, measurement modules) [START_REF] Lema | Multiphase Fluid Hammer in Spacecraft Propulsion Systems[END_REF], except the pressure vessel and the facility configuration: horizontal instead of vertical (for security reasons linked to real propellants). The test facility is then composed of a pressure vessel, a bent pipe, the VKI FOV and the VKI 2 m long propellant line (figure 2). The pressure vessel can be isolated from the bent pipe thanks to a valve allowing for the vessel filling (by replacing the bent pipe with a filling pipe). The bent pipe, linking the vessel to the FOV, includes a "T" to connect the service operation line allowing the vacuum pumping of the main line before the experiment and the blowing of the feeding line after. The vacuum of the pipe is achieved thanks to a vacuum pump which is placed after two cold traps allowing to reach a vacuum level around 500 Pa and to prevent vapour extraction by condensing these gaseous products. A pressurisation control panel enables to pressurise the tank and to regulate its pressure during the experiments.

The propellant pipe ends with a P-T measurement module provided by VKI (figure 3). The unsteady pressure probes (PCB) and the thermocouple were also provided by VKI. A first pressure probe (PCB1) is located at the beginning of the propellant line, just after the FOV and another one (PCB3) is added to measure the radial pressure close to the rear-end of the module located in the rear-end of the module. A last pressure probe (PCB2) measuring the fluid hammer amplitude and a thermocouple are located on the rearend of the module.

TEST MATRIX

In spacecraft, the liquid propellants are stored in a safety vessel pressurised at high pressure (2 MPa) before launch. All the pipes in the network are vacuum pumped at very low pressure (below 1000 Pa) or filled with NCG. The pipe pressure may then be either above or below the saturation pressure of the liquid propellants involving consequently different physical phenomena. So, for water and each propellant (MMH and NTO), experiments are performed with an initial pipe pressure below and above the saturation pressure of the liquid (Table 1). Considering the saturation pressure of water and MMH, the tests are achieved with vacuum pressure of 1000 and 10000 Pa in the pipe. Because of its high saturation pressure, the experiments with NTO are only performed with a vacuum pressure of 1000 Pa. To estimate the reproducibility of the phenomena, three successful tests for each experimental configuration are planned with a constant vessel pressure of 2 MPa.

EXPERIMENTAL RESULTS

Experiments Performed with Water

Figure 4 provides the experimental results obtained with water for a vacuum pressure of 1000 Pa in the pipe and for a tank pressure close to 2 MPa. The water hammer reaches a pressure of 28 MPa which rapidly decreases to the tank pressure value. Secondary oscillations appear during the main peak (figure 4b), which may be due to a mechanical oscillation of the back part of the module that contains the thermocouple since this part is screwed to unplug the module for the seal and thermocouple replacements. The radial pressure at the beginning of the module is nearly 60 % lower than the water hammer pressure measured at the rear-end of the module but the pressure evolutions are the same. The difference is certainly due to the existence of a vapour phase and the fact that the line is nearly horizontal (figure 2). As shown in figure 4a, the water hammer phenomenon can also be observed on the temperature measurement but its variation is very low.

Time [s] Unsteady pipe pressure [MPa]

Temperature Figure 5 shows a very good overlap of the first water hammer for the three tests, proving a good reproducibility of the phenomenon. The time axis of the results is corrected in order to assign the opening time of the FOV to 10 ms. This correction enables to underline the propagation time of the fluid from the FOV to the rear-end of the module which is specified in table 2 for the three tests. The knowledge of this propagation time enables to estimate the mean velocity of the fluid in the pipe which, for the three tests, is between 30 and 35 m/s. As for water experiments, the first fluid hammer peak presents secondary oscillations (figure 6b) which seem higher than previously. The fluid hammer is also visible on the temperature measurement but its variation is again very small (figure 6a). The behaviour is however different since the temperature decreases at the beginning of the fluid hammer. The reproducibility of the three tests is still very good which leads to a good overlap of the first fluid hammer (figure 6b) and to similar propagation times of MMH in the feeding line (table 3). The second series of MMH experiments, for a vacuum pressure of 10000 Pa in the pipe and a tank pressure of 2 MPa, is shown in figure 7. In these conditions, the fluid hammer reaches about 26 MPa and, as for the previous experiments, the temperature decreases when the first fluid hammer occurs and its variation is not significant. The reproducibility of the three tests is still very good which leads to a good overlap of the first fluid hammer (figure 7b) and to similar propagation times of MMH in the feeding line (table 3).

Experiments Performed with NTO

Due to the corrosive aspect of NTO, experiments with this propellant were difficult to perform. Even though the valves for service operation, vacuum pumping and blowing were refurbished with seals compatible with NTO, the propellant dissolved them causing leaks.

Consequently, just one experiment was realised with a vacuum pipe pressure below the saturation pressure of NTO (1000 Pa) and for a tank pressure close to 2 MPa. Under these conditions, the fluid hammer at the rearend of the measurement module reaches about 19 MPa and, as for the tests performed with water and MMH, the radial pressure measured at the beginning of the module (PCB3) is about 60% lower than the previous one (figure 7). The fluid hammer is, once again, visible on the temperature measurement but, contrary to the previous experiments, the variation of the temperature is higher than 350°C and can not be completely estimated since the data were overloaded.

Test Comparison

When comparing the experiments performed with MMH, the higher is the vacuum pressure of the pipe, the smaller are the frequency and the amplitude of the fluid hammer phenomenon (figure 9). Moreover, the fluid hammers appear before with a vacuum pipe pressure of 1 kPa than with 10 kPa. These conclusions are consistent with the conclusions drawn by VKI from the experiments with ethanol.

By comparison to experiments performed with water, experiments achieved with MMH have a lower fluid hammer amplitude but a higher peak frequency (figure 9). This conclusion is consistent with the one of the comparison of VKI experiments performed with water and ethanol. On the contrary, the experiment with NTO has a much lower fluid hammer amplitude and a lower peak frequency compared to water and MMH (figure 9).

Finally, figure 10 collects the maximum pressure of the fluid hammer for all the experiments performed at Onera. Even though the absolute value of the first pressure peak is much smaller for NTO than for water and MMH, the amplitudes of the different peaks appear on the same curve for both real fluids (MMH and NTO) in the (max-amplitude, time) diagram. The results for water appear on a parallel curve associated to higher pressure in the same diagram and this is valid for both vacuum pipe pressures.

CONCLUSIONS

The experimental database on multi-phase fluid hammer phenomena, obtained by the von Karman Institute for inert fluids, has been extended by Onera for real propellants (MMH and NTO). The same facility hardware downstream of the FOV was used, i.e. FOV, propellant line and measurement modules, but the propellant line was installed horizontally instead of vertically (at VKI).

For a vessel pressure of 2 MPa and a vacuum pressure in the pipe of 1000 Pa, the amplitudes of the fluid hammer are equal to 27 MPa for MMH and to 20 MPa for NTO. For MMH, the higher is the vacuum pressure of the feeding line, the smaller are the frequency and the amplitude of the fluid hammer phenomenon. Regarding the temperature, very small variations are recorded for water and MMH while the temperature increase for NTO is higher than 350°C. Further temperature measurements with NTO are necessary to confirm this value, as well as with acetaldehyde as similitude fluid.

Some visualizations were also performed in order to have a better understanding of the physics involved in the multi-phase fluid hammer phenomenon. However, when adjusting the visualisation settings, the VKI visualization module has been broken prohibiting the achievement of promising visualizations with real propellants. Further work would require manufacturing a new module and performing further visualisations.

Figure 2 :

 2 Figure 2: Test facility at Onera for the study of the fluid hammer phenomenon

Figure 3 :

 3 Figure 3: P-T measurement module

Figure 4 :

 4 Figure 4: Experiment performed with water for a vacuum pressure of 1000 Pa

Figure 5 :Figure 6

 56 Figure 5: Reproducibility of experiments performed with water for a vacuum pressure of 1000 Pa

Figure 6 :

 6 Figure 6: Reproducibility of experiments performed with MMH for a vacuum pressure of 1000 Pa

Figure 7 :

 7 Figure 7: Reproducibility of experiments performed with MMH for a vacuum pressure of 1000 Pa

Figure 8 :

 8 Figure 8: Experiment performed with NTO for a vacuum pressure of 1000 Pa

  Figure 10: Comparison of maximum fluid hammer pressure between water, MMH and NTOThe same conclusions for MMH compared to water can be drawn than for ethanol: the fluid hammer amplitude is lower and the frequency is higher for MMH. The experiment with NTO has a much lower fluid hammer amplitude and a lower peak frequency compared to water and MMH. The amplitudes of the different fluid hammer peaks appear however on the same curve for both real fluids (MMH and NTO) in the (max-amplitude, time) diagram.

Table 1 :

 1 Saturation pressure of fluids at 293 K

	H 2 O	MMH	NTO
	2340 Pa	4940 Pa	95830 Pa

Table 2 :

 2 Propagation time of the fluid in the pipe for experiments performed with water

	Experiment Vacuum pressure	Vacuum pressure
				of 1000 Pa		of 10000 Pa
	a			63 ms				65 ms
	b			63 ms				65 ms
	c			63 ms				64 ms
	Unsteady pipe pressure -PCB2 [MPa]
	0	0.05	0.1	0.15	0.2	0.25	0.3	0.35	0.4
				Time [s]

Table 3 :

 3 Propagation time of the fluid in the pipe for experiments performed with MMH

	Experiment Vacuum pressure	Vacuum pressure
		of 1000 Pa	of 10000 Pa
	a	57 ms	59 ms
	b	57 ms	59 ms
	c	56 ms	60 ms

ACKNOWLEDGMENT

This study is supported by the ESA through the General Support Technology Programme activity AO/1-6210/09/NL/CP. The authors would like to thank J-M. Buchlin, P. Rambaud and M. Lema from VKI for their collaboration within this project.


Achour Terbouche
email: achour_t@yahoo.fr

Chafia Ait-Ramdane-Terbouche

Safia Djebbar

Djamila Guerniche

Radia Bagtache

Nour El

Houda Bensiradj

Didier Amar Saal

Didier Hauchard

A new

Keywords: Ruthenium (III) complex, Characterization, DFT calculation, Electrochemical sensor

published or not. The documents may come

Introduction

Schiff bases are the organic molecules containing a carbon-nitrogen double bond with the nitrogen atom bonded to an aryl or alkyl group. Generally, they have a second functional group such as hydroxyl near the imine function. These polydentate ligands lead to the formation of more stable chelates with transition metals, which are often insoluble in water. The use of these solid complexes to modify electrodes is very important for application in electroanalysis field. Previously, the carbon electrodes modified by insoluble electro active compounds have been used in electroanalytical chemistry [1][2][3][4][START_REF] Kalcher | Heterogeneous carbon electrochemical sensors[END_REF][START_REF] Bakouri | [END_REF][7][8][9][10][11][12][13].

Rrecently, numerous studies have been conducted on the impact of inorganic pollutants such as heavy metals and nitrite ions on water quality [14][15][16][17][18][19][20]. Indeed, to detect these ions, different electrochemical sensors based on modified electrodes were developed [21][22][23][24][25][26][27][28][29][30].

To our knowledge, to determine nitrite ions, there have been no previous published works citing the use of cavity microelectrode modified with ternary complexes. In this paper, after characterization of the ternary complex formed with 1,3-bis(salicylideneamino)propan-2-ol (BSAP) and 3-Picolylamine (PLA) (Scheme 1) and G/Ru(III)-BSAP-TLA composite material using different techniques such as elemental analysis, FT-IR, UV-Vis, gHSQC, molecular modeling and SEM-EDS, a new sensor based on cavity microelectrode modified with graphite-ruthenium (III) complex composite (CME/G/Ru(III)-BSAP-TLA) was developed to detect and nitrite ions by using electrocatalytic oxidation process. The electrochemical study showed that the CME/G/Ru(III)-BSAP-TLA sensor exhibited a sensitive electrochemical response for the oxidation of NO 2 -, and the ternary Ru(III)-BSAP-TLA complex plays an important role in increasing the effectiveness of the sensor.

Materials and methods

Apparatus, reagents and materials

Fourier transform infrared spectra were recorded on a FT-IR Perkin-Elmer Spectrum One spectrometer using KBr pellets over the 4000 to 400 cm -1 range. For each measurement 32 scans were collected at a resolution of 4 cm -1 . The electronic absorption measurements were performed on a Varian Cary 50

Conc UV-Vis Spectrophotometer using quartz cuvette with 1 cm path length. Nuclear magnetic resonance spectra and scanning electron microscopy analysis coupled with energy dispersive spectroscopy (SEM-EDS) were carried out using a DPX 200 NMR spectrometer and scanning electronic microscope Hitachi TM-1000, respectively.

The characterization of Ru(III)-BSAP-PLA complex by cyclic voltammetry method was investigated using a potentiostat/galvanostat Autolab® PGStat 30 equipped with glassy carbon (GC), Ag/AgCl and Pt wire as working, reference and auxiliary electrode, respectively.

In the study of electrochemical behavior of nitrite ions, the modified cavity microelectrode (CME/G/Ru(III)-BSAP-PLA) was used as working electrode (diameter Ԅ c =50 µm and a depth h c =17 µm).

All chemicals used in this work were of the highest purity and analytical grade and the different solutions were prepared in free CO 2 deionised water (resistivity ≥ 14 MΩ cm). The buffer NaH 2 PO 4 /Na 2 HPO 4 (0.1M) was used to keep the pH of the solutions constant (pH=7).

All experiments were carried out at room temperature under a nitrogen atmosphere.

Synthesis of the ternary complex (Ru(III)-BSAP-PLA)

The ligand 1,3-bis(salicylideneamino)propan-2-ol (Scheme 1a) was synthesized following the procedure reported by Kruger et al. [31]. The 3-picolylamine ligand (Scheme 1b) was pure Fluka product (Ref. No. 2230910, pure 99%).

The Ru(III)-BSAP-PLA complex was prepared by the addition of Ru(III) chloride hydrate (1 mmol) to a mixture acetone solution of 1,3-bis(2-hydroxysalicylideamino)propan-2-ol (2 mmol) and 3picolylamine (4 mmol) (Scheme 1). The resulting mixture was stirred, slightly heated and then some drops of sodium hydroxide (0. The synthesized complex is insoluble in water, soluble in dimethyl sulfoxide (DMSO) and acetonitryle and shows less solubility in other organic solvents such as dimethylformamide (DMF) and dichloromethane. The molar conductivity value of Ru(III)-BSAP-PLA complex in DMSO (55.15 Ω -1 cm 2 mol -1 ) indicating a non-electrolytic nature.

Theoretical study of the ruthenium complex

Geometry optimization and vibrational frequency analysis were performed using DFT approach with the M06 level [32] and relativistic effective core potential basis set of double zeta quality, Lanl2dz [33], as implemented in Gaussian03 program package [34]. This technique is the best method to predict the geometry of the metal transition complexes.

In this part of the work, we optimized the geometry of the octahedral and tetrahedral forms of the ruthenium (III) complex . Each geometry optimization was completed by a calculation of harmonic vibrational frequencies to confirm the most stable geometry.

The studied forms of the Ru(III) complex were characterized as minima (no imaginary frequency) in their potential energy surface through harmonic frequency analysis.

Preparation of G/Ru(III)-BSAP-PLA composite and the modified cavity microelectrode

The G/Ru(III)-BSAP-PLA composite material was prepared by adding 50 mg of treated graphite powder to 0.1 M DMSO solution of Ru(III) complex. The mixture was slowly stirred for 48 hours, and was then centrifuged. The solid product was dried at room temperature, washed with distilled water, and finally dried under vacuum.

To prepare CME/G/Ru(III)-BSAP-PLA modified electrode, the CME was filled with active material by pressing the electrode tip into the G/Ru(III)-BSAP-PLA powder.

After each measurement, CME was washed with an ultrasonic cleaner using successively HNO 3 solution (1 M), H 2 O 2 (30%) and deionised water.

Results and discussion

FT-IR spectrometry

To determine the coordination mode between ruthenium (III) and the two ligands (BSAP and PLA), we compared the FT-IR spectrum of the complex with those of the uncomplexed ligands (Fig. 1). This comparison also allowed identifying the sites involved in the formation of bonds in the complex. Compared to BSAP, the vibration bands of C=N (1601 cm -1 ) and C-O (1231 cm -1 ) in the complex underwent a shift to lower energies. The disappearance of the OH peak (1207 cm -1 ) of ligand and the shift of vibration bands of C=N and C-O in the spectrum of complex confirms that the ruthenium binds to the BSAP ligand by the lone pair of nitrogen and oxygen of the imines and C-O groups, respectively.

The stretching vibration bands ν N-H observed at 3399 cm -1 in the spectrum of 3-picolylamine was shifted to higher energies (ν N-H : 3429 cm -1 ) in the FT-IR spectrum of the complex, indicating that the metal is bonded with the nitrogen of the amine group.

The ruthenium-oxygen bonding is confirmed by the newly formed band that appear at 615 cm -1 which is assigned to the Ru-O vibration.

UV-Visible spectrophotometry

UV-Visible absorption spectrophotometry was used to determine the intra ligand, metal-ligand and d-d electronic transitions.

The electronic absorption of Ru(III)-BSAP-PLA, Ru(III) and BSAP-PLA compounds were recorded in DMSO at room temperature. The used analytical concentration in this study is the same in all solutions (10 -4 M).

The shoulder UV-Visible absorption at 375 nm (ε=8.1×10 +3 M -1 cm -1 ) assigned to ligands (Fig. Sb), corresponding to intra ligand transition π→π*, is shifted to lower wavelengths (λ=315 nm, ε=1.2×10 +3 M -1 cm -1 ) in the spectrum of the complex (Fig. Sc), indicating the commitment of the ligands in the formation of Ru(III)-BSAP-PLA complex.

The absorption band observed at 414 nm (ε=6.96×10 +3 M -1 cm -1 ) in the spectrum of the metal (Fig. Sa) remains virtually unchanged in the spectrum of the complex (λ= 410 nm), indicating the presence of the metal in this compound. The electronic absorption band at 410 nm (ε= 1.16×10 +3 M -1 cm -1 ) in the spectrum of Ru(III)-BSAP-PLA complex corresponds to metal-to-ligand d π(Ru) →π* transition.

The broad UV-Visible absorption observed around 520 nm (ε=0.880×10 +3 M -1 cm -1 ) in the spectrum of the complex may be assigned to the d-d transition band.

Two-dimensional NMR Spectrometry

The resonance of directly bonded proton carbon pairs (C-H) was carried out using gradient assisted hetero nuclear single quantum coherence spectroscopy (gHSQC) (Fig. S2).

The cross peaks between the 13 Finally, the gHSQC cross peak at 6.9, 125 ppm and 7.3, 125 ppm were assigned to the C-H bonds in the aromatic and pyridine rings, respectively.

Moreover, the absence of the gHSQC cross peak at 7.2 ppm corresponding to the 1H signal of OH group revealed that at least one OH function does not bind to the central atom.

Magnetic measurements

Energies, calculated Mulliken charges and the dipole moments for the three forms of the Ru(III) complex are presented in Table 2. The values of minimized energy of the three forms of the complex show that the form(c) (RuN 3 O) has a low energy, indicating that this structure is more stable compared to the other forms.

The dipole moment is the indicator of the symmetry. If the system is symmetric, the dipole moment is equal to zero, and the charges distributed on each atom are equal to each other.

For the studied complex, the dipole moment differs from the zero. It increases while passing from the form(c) (6.2 D) to the form(b) (8.79 D) and the form(a) (8. 85 D). The form(c) has a small dipole moment, confirming the stability of this structure compared to the other molecular geometry forms.

HOMO and LUMO

Some parameters such as highest occupied molecular orbitals (HOMO) and lowest unoccupied molecular orbitals (LUMO) and energy gap of the complexes have been calculated (Table 2).

Fig. 3 shows the frontier orbitals shape. The HOMO density of octahedral form of the complex is distributed over the metallic element (Ru1), oxygen (O5, O6), nitrogen (N3, N4) atoms and the phenyl groups. For the form RuO 3 N, the HOMO is localized also on the metal (Ru1) but with a weak contribution, and on the oxygen (O6, O7) with a great contribution. Concerning the form RuN 3 O, the HOMO density is localized over the pyridine ring, and extends to the nitrogen atom (N2) of the amine function.

For the LUMO level we observed a trend on the three forms of the complex whose the energy is decreasing from -0.17088 a.u. for the form RuN

Electrochemical study of Ru(III)-BSAP-PLA complex

The electrochemical behaviors of different compounds have been studied in DMSO and NBu 4 PF 6 solution (0.1 M) by cyclic voltammetry (CV) in 1.5 to -1.8 V potential range.

To determine the electron transfer processes of the ruthenium complex and to avoid misinterpretation due to confusion between the redox processes of ligands, metallic salt and ruthenium complex, it is necessary to study the electrochemical behaviors of BSAP-PLA and Ru(III) alone.

The voltammogram of Ru(III) complex shows a cyclic oxidation-reduction process that does not change after repeated of 20 scans, indicating the stability of the electrochemical process.

Comparison of the cyclic voltammogram of Ru(III) complex with those of ligand and metal, the oxidation waves in 0.40 to 0.90 V and the reduction waves in -0.20 to 0.36 V potential ranges could be assigned to the metal ionization.

The cyclic voltammogram of BSAP-PLA shows one redox couple Red/Ox (Fig. S3). The reduction and oxidation potential values are Ep c = -1.64 V and Ep a = 1.28 V, respectively. These potentials were attributed to the reduction and oxidation of OH phenolic group.

The cyclic voltammogram of Ru(III)-BSAP-PLA complex reported in figure S3 shows two oxidation processes. The oxidation wave at Epa 1 = 0.86 V is attributed to the oxidation of Ru(II) to Ru(III). The Ep a of this couple was located around 0.80 V in the complexes using ITO electrode modified by Nafion containing [Ru(L)(totpy)(OH 2 )](PF 6 ) 2 [36] and porous carbon felt electrode containing cis-aqua dimethyl bipyridyl triphenylphophine ruthenium(II) diphenolate [37].

The oxidation peak at Epa 2 = 0.43 V is assigned to the oxidation of Ru(I) to Ru(II). While, The reduction of Ru(III) to Ru(II) and Ru(II) to Ru(I) appeared at 0.15 and -0.10 V, respectively.

The reduction and oxidation waves observed at -1.34 and -1.07 V were attributed respectively to the reduction and oxidation of imine groups. Whereas, the oxidation process at positive potential range shown in the ligand voltammograms did not appeared in the Ru(III)-BSAP-PLA voltammogram, this is due to the disappearance of phenolic group in the complex molecule. These results show that the ligands are bounded to the metal ion.

SEM-EDS characterization of G/Ru(III)-BSAP-PLA composite material

Scanning electron microscopy coupled with energy dispersive spectroscopy was used for surface and morphology study, to evaluate particle size and microchemical analysis of the G/Ru(III) complex composite material. SEM photograph and EDS analysis of the synthesized composite material are illustrated in Fig. 4.

From the SEM photograph, we noted that there is a uniform matrix of the prepared composite material (G/Ru(III)-BSAP-PLA), indicating that G/Ru(III)-BSAP-PLA is homogeneous phase material and the ternary Ru(III) complex was homogeneously distributed on the graphite support. This is an advantage for the catalytic activity of the complex.

The in situ analysis of the sample shows that the Ru element is showing distribution having a significant concentration.

Optimization of experimental parameters

Stability and scan rate effect studies

To study the stability of the modified cavity microelectrode in neutral medium (NaH 2 PO 4 /Na 2 HPO 4 ; pH=7), ten successive voltammograms were performed using CME (Fig. 5) modified with G/Ru(III)-BSAP-PLA composite material (Fig. 6). The results showed that the CME/G/Ru(III)-BSAP-PLA modified electrode exhibit a well-defined cyclic voltammograms after scanning the potential between -0.4 and 1 V versus Ag/AgCl reference electrode for the ten cycles (Fig. 6a). In the same potential interval, the cathodic and anodic sweep leads to voltammograms perfectly superimposed, indicating a reproducible current response of CME modified with G/Ru(III)-BSAP-PLA due to conservation of the total mass of the product in the cavity of microelectrode. , respectively.

Electro-oxidation behavior of nitrite on different electrodes

To study the electro-oxidation behavior of nitrite on CME/Graphite and CME/G/Ru(III)-BSAP-PLA modified electrodes, the cyclic voltammograms were recorded in the phosphate buffer solution containing 1.99×10 -5 M of nitrite (Fig. 7). The results show an oxidation peak current around 0.67 V

Effect of varying scan rate on electro-oxidation of nitrite

The voltammograms obtained using CME modified with G/Ru(III)-BSAP-PLA complex versus scan rates in the absence of nitrite are shown in figure S4. The repetitive cyclic voltammograms at various scan rates are observed with no fluctuation in the anodic and cathodic peak currents of the complex, indicating the electroactivity and stability of G/Ru(III)-BSAP-PLA composite material in the cavity of microelectrode.

The transport characteristics of the CME/G/Ru(III)-BSAP-PLA electrode were studied by cyclic voltammetry method with different scan rates (5-100 mV/s) in buffer phosphate solution containing 1.59×10 -4 M nitrite (Fig. 8a). The voltammograms showed that the oxidation peak current increases with increasing scan rate, and the low positive shift in peak potential position ( pa E ) with the scan rate indicating that the electro-oxidation process of nitrite is irreversible.

Reporting current vs. square root of scan rate (Fig. 8b), two linear regression equations

Effect of pH

The voltammetric current response of the CME/G/Ru(III)-BSAP-PLA modified electrode was investigated with different pH values in the presence of 1.59×10 -4 M NO 2 -(Fig. 9a). The results revealed that the current decreases over a range of pH values from pH 2.22 to 8.02.

Over the acidity range of pH 2-7, the positive charge of the Ru(III)-BSAP-PLA decreases resulting the decrease in the oxidation peak current of nitrite.

According to the figure 9a, the potential shifts slightly to lower values from pH 2.22 to 8.02, and the maximum oxidation peak current was given at pH=2.22.

pH solution was chosen around 7 for all experiments because the potential of nitrite oxidation peak is low at this pH value which corresponds to the optimum pH value of water.

According to these results, the oxidation peak current and potential indicate that the CME/G/Ru(III)-BSAP-PLA modified electrode responds effectively to the oxidation of nitrite at different pH values.

Electrochemical reactivity 3.8.1. Chronocoulometry

The Chronocoulometry was used as main method to investigate the oxidation process of nitrite ions at CME/Graphite and CME/G/Ru(III)-BSAP-PLA modified electrodes. The plots of charge ( Q ) versus time ) (t and square root of time ( 2 / 1 t ) were reported in figure 10.

The linear relationship between Q and 2 / 1 t could be expressed according the following formula [40]:

ads dl Q Q t nFAcD t Q + + = 2 / 1 2 / 1 2 / 1 2 ) ( π
Where n is the number of transferred electrons, F is Faraday constant, A is surface area of working electrode, c is concentration of substrate, D is diffusion coefficient, dl Q is double layer charge, ads Q is Faradic charge. The surface coverage ) (Γ can be expressed according to the Faraday law:

Γ = nFA Q ads
According to the figure 10b, the determined equations were reported below: ).

The Faradic charge

) ( ads Q NO 2
-found with CME/G/Ru(III)-BSAP-PLA modified electrode is higher than the determined with CME/Graphite. This may be explained by the fact that the Ru(III)-BSAP-PLA complex attract large amount of NO 2 -ions involving high Faradic charge and high surface coverage.

Chronoamperometry

The chronoamperometry technique was used to evaluate the rate of electrocatalyzed oxidation of nitrites at CME/G/Ru(III)-BSAP-PLA modified electrode. Fig. 11 shows the current vs. time curves by setting the potential of CME/G/Ru(III)-BSAP-PLA at 920 mV in the absence and presence of nitrite ions. In range times of 0 to 20s, the chronoamperograms show larger electrocatalytic currents in the presence of nitrite. This may be explained by the high speed of the electrocatalyzed oxidation of nitrite ions.

The chemical reaction between NO 2 -and redox sites of the Ru(III)-BSAP-PLA complex ends after 10s because between 10 and 20s the current remains constant.

Electrochemical behavior of CME modified with G/Ru(III)-BSAP-PLA composite material versus nitrite concentrations

The electrochemical behavior of the cavity microelectrode modified with G/Ru(III)-BSAP-PLA composite material without addition of nitrite showed an oxidation (E pa = 0.70 V, i a = 1.17 nA) and reduction peak (E pa = -0.11 V, i a = -92.50 nA) corresponding respectively to Ru(II)/Ru(III) and Ru(III)/ Ru(II) couples. The low oxidation peak current around 0.7 V must be assigned to Ru(III)/Ru(IV) couple.

After adding NO 2 -in the phosphate buffer solution (Fig. 12a), the response of the CME/G/Ru(III)-BSAP-PLA electrode shows that the oxidation peak current around 0.8 V increases with increasing nitrite concentration. The electrocatalytic processes are expressed by the following reactions: The electrochemical behavior of CME modified with G/Ru(III)-BSAP-PLA towards the nitrite ions shows an anodic oxidation wave at 0.86 V that substantially increases with increasing nitrite concentration. These results show that the modified CME really responds to variation of nitrite concentration and the Ru(III)-BSAP-PLA complex catalyses the oxidation of nitrites.

Based on the curves current vs. potential, the oxidation and reduction currents from CME/G/Ru(III)-BSAP-PLA electrode versus concentration of nitrite ions are presented in figure 12b. The intensity of oxidation current increases linearly with increasing of nitrite concentration in the range of 0 to 1.59×10 -4 M (The linear regression equation: 066 . 0 87 . 5800

- = C i , 992 . 0 2 = R
). The measurements performed using CME/G/Ru(III)-BSAP-PLA electrode gave highest sensitivity and good linearity corresponding to correlation coefficient R 2 = 0.992.

The electrochemical behavior of the modified electrode towards nitrite concentrations showed that the system CME/G/Ru(III)-BSAP-PLA can be considered as a detector of NO 2 -. The limit of detection ) (LOD was calculated using the following equation:

m s LOD / 3 =
Where s is the standard deviation of the blank signal and m is the slope of the calibration curve.

The sensitivity and limit of detection of CME modified with G/Ru(III)-BSAP-PLA composite material are 5.80 µA.mM -1 and 1.81 µM, respectively.

Comparison of the obtained limit detection of our modified electrode with those found in the literature around nitrite concentrations range of 0 to 159 µM and pH=7 (Table 2), LOD calculated for CME/G/Ru(III)-BSAP-PLA electrode was much lower than the limit detection of nitrite ions determined using different modified electrodes [21,42,43] and CME/G/Ru(III)-BSAP-PLA electrode exhibited much higher current response sensitivity towards nitrite oxidation around 0.86 V, which is corresponding to the conversion of NO 2 -to NO 3 -. It can be explained by the synergetic effect of Ru(III)-BSAP-PLA complex in electrocatalytic oxidation of nitrite ions.

Conclusion

In this work, an electrochemical sensor for detection of nitrite ions was prepared by using the cavity microelectrode (CME) modified with graphite-ternary Ru(III) complex composite material (G/Ru(III)-BSAP-PLA).

The characterization of Ru(III)-BSAP-PLA complex using different methods such as elemental analysis, FT-IR, UV-Visible, gHSQC and DFT suggests an tetrahedral environment around the ruthenium(III). SEM photograph revels that the ternary Ru(III)-BSAP-PLA complex was homogeneously distributed on the graphite support.

The determined electron transfer coefficient ( 83 . 0 = α ) indicates a better electrocatalytic mechanism for oxidation of nitrite in the presence of the Ru(III) complex compared with oxidation of nitrite at other modified electrodes published in the literature.

The chronocoulometry technique shows that the Faradic charge (NO 2 -) found with CME/G/Ru(III)-BSAP-PLA modified electrode is higher than the determined with CME/Graphite. Additionally, the chronoamperometry technique reveals a high speed of the electrocatalyzed oxidation of nitrite ions in the presence of the Ru(III)-BSAP-PLA complex.

Finally, the limit of detection of CME/G/Ru(III)-BSAP-PLA modified electrode ( LOD =1.81 µM) was much lower than LOD determined using other published sensors around pH=7 over nitrite concentration range of the same order of magnitude. Table 1: Calculated structural parameters of the three forms for Ru(III) complex.

List of Tables

  C resonance at 24 ppm and 1H signals in the range of 1.2 to 1.4 ppm were assigned to CH 2 of CH 2 -NH 2 group of 3-picolylamine. The C-O bond at 28 ppm was also correlated with protons in the range of 1.4-1.7 ppm from the CH 2 -CHO-CH 2 group. Additionally, the gHSQC cross peaks between the 1H signal at 1.2 ppm and 13 C resonance at 30 ppm correspond to the C-H bond in the N-CH 2 groups. Whereas, the proton signals between 2 and 3 ppm showed the gHSQC correlations to the Ar-CH=N carbon resonating between 30 and 40 ppm.

3. 4 .

 4 Molecular modeling 3.4.1. Geometry optimization Selected bond distances, angles and dihedral angles for the three forms reported in figure 2 (RuO 3 N tetrahedral form: form(a), RuN 3 O 3 octahedral form: form(b), N 3 O tetrahedral form: form(c)) are listed in Table 1. The optimized structures of the studied complex show that the bond distances Ru1-N2 and Ru1-O6 increase in the order form(c)<form(a)<form(b). The bond angles (N2-Ru1-O6) and (N2-Ru1-N3) decrease in the following order: form(c)>form(b)>form(a), while the bond angles (O5-Ru1-O7) and (O5-Ru1-N3) decrease in the order: form(a)>form(b)>form(c). The dihedral angles (O6-Ru1-N2-H10) and (O6-Ru1-O5-C18) decrease in the following order: form(c)>form(a)>form(b), whereas the dihedral angles (N2-Ru1-O5-C18) and (N2-Ru1-O6-C28) decrease in the order: form(c)>form(b)>form(a).

Fig. 6b compares

  Fig. 6b compares the cyclic voltammograms of graphite and G/Ru(III)-BSAP-PLA in phosphate buffer

  Graphite and CME/G/Ru(III)-BSAP-PLA, respectively, assigning to the oxidation of NO 2 -to NO 3 -. The determined oxidation peak current of nitrite on CME/G/Ru(III)-BSAP-PLA modified electrode is increased compared with CME/Graphite, indicating an effective electro-oxidation of NO 2 -on the modified CME electrode in the presence of the Ru(III)-BSAP-PLA complex witch is considered as an electron mediator on the surface of CME/G/Ru(III)-BSAP-PLA electrode. These results indicate the main electrocatalytic role of Ru(III) complex to promote the oxidation process of NO 2 -.

.

  .988) were obtained for scan rate ranging from 5 to 40 mV/s and 40 to 100 mV/s, respectively, indicating that the kinetic of the overall process is not totally controlled by diffusion process but it also controlled by adsorption.Referring to the Tafel equation defining a totally irreversible electrochemical process[START_REF] Bard | Electrochemical Methods. Fundamentals and Applications[END_REF]: This value indicates a better electrocatalytic mechanism for oxidation of nitrite in the presence of the Ru(III)-BSAP-PLA complex compared with oxidation of nitrite at other published electrodes such as Adge plane pyrolytic graphite electrode-Single walled carbon nanotubes-cobalt modified electrode (EPPGE-SWCNT-Co:

adsQ 2 / 1 t

 21 was obtained by the difference of two intercepts of the plot of Q vs. for curves in the absence and presence of nitrite.

2 -

 2 Ru III -BSAP-PLA ↔ (Ru IV -BSAP-PLA) + + e -.......................................................................................(1) NO 2 -+ (Ru IV -BSAP-PLA ) + ↔ Ru III -BSAP-PLA + NO 2 ......................................................................(2) Based on the literature [41], NO 2 gives nitrite and nitrate ions by the following reaction: 2NO 2 + H 2 O ↔NO 2 -+ NO 3 -+ 2H + ........................................................................................................(3) Combining (1), (2) and (3), the overall catalytic mechanism is given in the following reaction: NO +2(Ru IV -BSAP-PLA) + + H 2 O ↔ NO 3 -+ 2H + +2 Ru III -BSAP-PLA...............................................(4)

Figure captions Scheme 1 .

 1 Figure captions

Figure 1 .

 1 Figure 1. FTIR spectra of (a) PLA, (b) BSAP and (c) Ru(III)-BSAP-PLA complex.

Figure 2 .

 2 Figure 2. Geometry forms of Ru(III)-BSAP-PLA complex: (a) RuO 3 N tetrahedral form, (b) RuN 3 O 3

Figure 3 . 4 .

 34 Figure 3. Frontier molecular orbitals for the three forms of Ru(III)-BSAP-PLA complex: (a) HOMO of

Figure 5 .

 5 Figure 5. Scheme of the cavity microelectrode (CME).

Figure 6 .

 6 Figure 6. Cyclic voltammograms obtained using CME/Graphite and CME/G/Ru(III)-BSAP-PLA

Figure 7 .

 7 Figure 7. Cyclic voltammograms obtained using CME/Graphite CME/G/Ru(III)-BSAP-PLA electrodes 0.1 M phosphate buffer solution containing 1.99×10 -5 M NO 2 -. Experimental

Figure 8 .

 8 Figure 8. (a) Cyclic voltammograms obtained using CME/G/Ru(III)-BSAP-PLA modified Electrode in 0.1 M NaH 2 PO 4 /Na 2 HPO 4 (pH=7) in the presence of 1.59×10 -4 M NO 2 -at different scan

Figure 9 .

 9 Figure 9. (a) Cyclic voltammograms obtained with CME/G/Ru(III)-BSAP-PLA modified electrode in the presence of 1.59×10 -4 M NO 2 -at different pH and (b) Oxidation peak current vs. pH.

Figure 10 .

 10 Figure 10. Plots of (a) Q vs. t, (b) Q vs. t 1/2 for CME/Graphite and CME/G/Ru(III)-BSAP-PLA in 0.1 M NaH 2 PO 4 /Na 2 HPO 4 (pH=7) in the absence and the presence of 3.97×10 -4 M NO 2 -.

Figure 11 .

 11 Figure 11. Chronoamperograms obtained using CME/G/Ru(III)-BSAP-PLA modified electrodes in 0.1 M NaH 2 PO 4 /Na 2 HPO 4 (pH=7) in the absence and the presence of 6.14×10 -3 M NO 2 -.

Figure 12 .

 12 Figure 12. (a) Cyclic voltammograms obtained with CME/G/Ru(III)-BSAP-PLA modified electrode in 0.1 M NaH 2 PO 4 /Na 2 HPO 4 (pH=7) at different concentrations of NO 2 -(M) and (b) Linear

  Scheme 1.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 4 .

 4 Figure 4.

  Figure 12.

  3 O to -0.17331 a.u. for RuN 3 O 3 to -0.17443 a.u. for

	RuO 3 N.
	The gap energy values (form RuN 3 O 3 : 0.48 eV; form RuO 3 N: 0.40 eV; form RuN 3 O: 0.54 eV) show
	that the form RuN 3 O has largest gap compared to the other forms, indicating the most stable structure.

Table 2 :

 2 Energetic properties, dipole moments, charge of Mullikan, energies of HOMO, LUMO orbitals and energy gap of the three forms of the ruthenium complex.

Table 3 :

 3 Comparison of the performances of some different nitrite sensors around pH=7.

Table 2 .

 2 Energetic properties, dipole moments, charge of Mullikan, energies of HOMO, LUMO orbitals and energy gap of the three forms of the ruthenium complex.

	Parameters

Table 3 .

 3 Comparison of the performances of some different nitrite sensors around pH=7.

	Electrode	pH Linear range (µM) LOD (µM) Ref.
	EPPGE-SWCNT-Co	7.4 0-189	5.61	Adekunle et al., 2010
	Au-penicillamine SAM	6.8 20-800	4.0	Wang et al., 2001
	Pt NPs/Au electrode	7.4 10-1000	5.00	Miao et al., 2011
	CME/GC-Ru(III)-BSAP-PLA 7.0 0-159	1.81	Present work

EPPGE: Adge-plane pyrolytic graphite electrode; SWCNT: Single-walled carbon nanotubes; SAM: Self-assembled monolayer; NPs: Nanoparticles; CME: Cavity microelectrode; LOD: Limit of detection.

Acknowledgements

The authors wish to express their gratude to Mrs. Ourida OUAMERALI of USTHB University (Laboratoire de Physico-Chimie Théorique et Chimie Informatique, Faculté de chimie, 16111, Algiers, Algeria) for DFT calculation of the ruthenium complex.

The French UMEC network of the CNRS is warmly acknowledged for providing the cavity microelectrode.

Highlights

• Ternary ruthenium(III) complex with 1,3-bis(salicylideneamino) propan-2-ol polydentate base Schiff (BSAP) and 3-Picolylamine (PLA) was synthesized, characterized and the structure was optimized using density functional theory.

• Graphite-Ru(III)-BSAP-PLA composite material was prepared and characterized by scanning electron microscopy coupled with energy dispersive spectroscopy (SEM-EDS).

• Cavity microelectrode (CME) was modified with graphite-Ruthenium(III)-BSAP-PLA material.

• Substantial electro catalytic effect of ruthenium(III)-BSAP-PLA complex on oxidation of nitrite was determined by using CME/G/Ru(III)-BSAP-PLA modified electrode.