Subhamoy Sen
email: subhamoy@iitmandi.ac.in

Antoine Criniere

Laurent Mevel
email: laurent.mevel@inria.fr

Frédéric Cerou

Jean Dumoulin

Antoine Crinière

Frederic Cérou

Correntropy based IPKF filter for parameter estimation in presence of non-stationary noise process

Keywords: Particle filter, Noise estimation, Correntropy filter, Parameter change detection, Interacting Particle Kalman filter

INTRODUCTION

Filtering based structural health monitoring (SHM) algorithms typically models noise processes as non-stationary Gaussian white noises with their statistics known a priori. However, in reality, statistical properties of noise processes can seldom be known in advance Zhang et al. (2012). For systems with unknown noise statistics, employing filtering based SHM techniques thus more often than not results in a solution that might not be optimal. To ensure an optimal solution, these noise statistics are thus required to be estimated before even the SHM algorithm is put into operation. Nevertheless, this also does not ensure that the operational noise processes will follow the pre-estimated noise models throughout the service life of the structural system. The problem may further intensify in presence of non-stationary noise processes undergoing an abrupt change in magnitude.

A solution can, however, be attempted by estimating the noise processes in parallel to the core SHM algorithm [START_REF] Alsuwaidan | Convergence properties of autocorrelation-based generalized multiple-model adaptive estimation[END_REF]. This helps to provide the SHM algorithm the required information about the current noise statistics to enhance the optimal property of the solution. This article incorporates a noise estimation module within Interacting Particle-Kalman filter (IPKF) [START_REF] Zghal | Modal parameter estimation using interacting Kalman filter[END_REF] to enable it to estimate the noise statistics in parallel to damage detection and quantification. This noise estimation module considers the residual uncertainty in IPKF prediction as an observation which is further used to update the current noise statistics. This recursive update strategy for noise statistics makes the algorithm noise adaptive and also ensures optimal damage detection. This article also attempts to mitigate the ill effects of unusually large noise in the measurement which should be considered as an outlier. However, in common filtering based algorithms, these outliers are consumed in the state estimates which might cause a divergence in the estimation procedure. To avoid this undesired situation, the KF nested within IPKF filter is replaced with maximum correntropy criterion based Kalman filter (MCC-KF). With MCC-KF, a correntropy index measures the similarity between the actual measurement and the predicted measurement based on current estimates of states and parameters [START_REF] Cinar | Hidden state estimation using the correntropy filter with fixed point update and adaptive kernel size[END_REF]. Eventually, while a small noise contamination causes small deviation of the measurement from the actual response, an outlier causes a huge innovation. The correntropy index thus becomes important since it can effectively regulate the impact of each innovation in the updated estimates. As a result, a large innovation due to an outlier in the measurement yields significantly low correntropy index which in turn leads to a small impact on the state update.

Section 3 discusses the problem under consideration and Section 4 describes the proposed methodology. Under Section 4, IPKF algorithm has been demonstrated in Subsection 4.1 followed by a brief description of the MCC-KF filter in Subsection 4.2. The augmented noise estimation module is described in detail in Subsection 4.3. Finally, the proposed algorithm is tested on a typical SHM problem in which the system undergoes damage in presence of varying noise processes (Section 5). An appendix is also included at the end of this article (Appendix A) detailing the derivation of the MCC-KF algorithm. However, prior to everything else, Section 2 briefly discusses the concept of correntropy criterion.

CORRENTROPY CRITERION

In information theory, correntropy is used as a measure of similarity in statistics between two different random variables. Over the years, this measure has drawn significant attention in the fields of signal processing, machine learning, and pattern recognition. The major quality of this criterion involves its excellent performance with the data containing large outliers. Correntropy, or crosscorrentropy (for scalar random variables) between two random variables include information about higher order moments beyond the traditional first and second order moment information. The selection of the moment order, however, depends on the kernel selected for the correntropy calculation. For two scalar random variable x and y, the cross-correntropy information can be obtained as:

C σ (x, y) = Ω x Ω y k σ (x, y)f xy (x, y)dxdy (1)
where k σ is the selected kernel for this correntropy calculation and f xy is the joint density function for the random variables x and y. Analytical integration over the entire domain of x and y is, however, not possible and the correntropy information can, therefore, be approximated as a summation over finite numbers of data points as:

Ĉσ (x, y) = 1 N N i=1 k σ (x i , y i ) (2)
With a Gaussian kernel, the explicit description of correntropy can be presented as:

Ĉσ (x, y) = 1 N N i=1 e - ||x i -y i || 2 2σ 2 = 1 N N i=1 G σ (|| i ||) (3)
where,

|| i ||= ||x i -y i || defines the norm of error between the two dataset in consideration. G σ (|| i ||) = e (-i T i /2σ
2 ) is the Gaussian kernel. Here, σ is the bandwidth of the kernel that defines how new data is adapted. In order to be counted as a compatible kernel for the correntropy measure, it has to be positive and bounded achieving its limiting maximum value when its argument is zero. For a Gaussian kernel, a Taylor series expansion demonstrates that information about all the even order moments is stored in the correntropy measure.

Ĉσ (x, y) = 1 √ 2πσ ∞ n=0 (-1) n 2 n σ 2n n! E[( i T i ) n ] (4)

PROBLEM DEFINITION

The governing differential equation for the dynamics of mechanical systems with mass, damping and stiffness being M, C and K respectively can be described as:

Mq(t) + C q(t) + Kq(t) = v(t)
(5) where q(t), q(t) and q(t) are displacement, velocity and acceleration responses respectively. v(t) is the ambient forcing acting on the structure. The state space representation of this dynamics can be constructed with the ambient forcing v(t) modeled as non-stationary process noise and w(t) as non-stationary measurement noise as:

ẋ(t) = F c x(t) + B c v(t) (6a) y(t) = H c x(t) + D c v(t) + w(t) (6b) F c , B c
, H c and D c are time dependent state, input, measurement and direct transmission matrices respectively defined in continuous time domain. Details of all the matrices are given in the following.

x(t) = {q(t) q(t)} T and F c = 0 I -M -1 K -M -1 C ; B c = 0 M -1 , H c = -M -1 K -M -1 C , and D c = M -1
Since discrete measurements will be used for the estimation, the discrete time formulation of equation 6 can be presented with x k , y k , F, B, H, D, v k and w k as the discrete time counterparts against their corresponding continuous time entities.

The system under consideration is, however, time-varying and thus its model is defined with time-varying state matrix F k and measurement matrix H k . These system matrices are functions of a time-invariant mass matrix M and time-varying physical system matrices K k and C k which in turn depend on the component level health parameters θ k . Accordingly, the mass matrix and its dependents, i.e., B and D are assumed to be constant and known from now on.

x

k = F(θ k )x k-1 + Bv k (7a) y k = H(θ k )x k + {Dv k + w k } (7b) where F k = F(θ k ) and H k = H(θ k ).
The problem assumes that the stochastic properties of process and measurement noises i.e., v k and w k are not known but can be modeled as WGN with unknown covariance Q k and R k respectively.

PRESENT APPROACH

It should be noted that with the typical filtering based SHM problems, the state (system response) evolves in time through a linear process equation and thus KF can be a good approach for such state estimation. However, the relationship between unobserved parameter states and its corresponding observation i.e., the measurements, is always nonlinear. Bayesian belief propagation requires an explicit analytical integration for the entire domain of states which is to be however straightforward if the problem is linear and the states are assumed Gaussian. On the contrary, the current problem is nonlinear for which explicit analytical integration over the entire parameter space is not possible. Particle filter (PF) can approximate this analytical integration by a numerical one and thus can be considered to be a potential approach for nonlinear system estimation.

Nevertheless, PF is computationally expensive and KF cannot be used for estimation of both states and parameters. Employing two different filters for states and parameters, precision along with computational economy can although be achieved. With this in view, IPKF based algorithm given by [START_REF] Zghal | Modal parameter estimation using interacting Kalman filter[END_REF] has been employed in his article that employs PF for parameter estimation while KF is used for linear state estimation.

Within IPKF, a set of KF (for state estimation) runs within an envelope of a PF (for parameter estimation). In this article, a modification has been proposed on the existing IPKF targeting damage estimation in presence of varying noise statistics and/or large outliers in the measured signal. In the proposed algorithm, the KF is replaced with MCC based KF or MCC-KF. An additional module for noise estimation is augmented to recursively update the noise statistics. This methodology in detail is discussed in the following.

IPKF algorithm

IPKF algorithm, developed by [START_REF] Zghal | Modal parameter estimation using interacting Kalman filter[END_REF] and further improved for computational efficiency by [START_REF] Sen | Seismic-induced damage detection through parallel force and parameter estimation using an improved interacting particle-Kalman filter[END_REF] is an efficient approach to handle parameter estimation for a time-varying system. The strategy of decoupling the estimation of states and parameters and employment of two different filter types for each of them enhances the stability property of the algorithm while ensuring computational efficiency. In turn, this facilitates employing relatively less expensive linear KF for linear state estimation while the costly PF is employed only for parameter estimation. Clearly, IPKF has two parts: i) a PF envelop to estimate the parameters as particles within which ii) a set of nested KFs estimates the states and both the parts are detailed in the following.

Envelop Particle filter:

PF attempts a particle approximation of the Chapman-Kolmogorov integration by propagating the system uncertainty through a cloud of

N independent parameter particles Ξ k = [ξ 1 k , ξ 2 k , • • • , ξ N k ].
Thus no consideration on Gaussianity of the parameter states is enforced in the algorithm. The temporal evolution of the system dynamics is defined by the evolution of the particle set in time. At any arbitrary time step k, the evolution of an arbitrary i th particle ξ i k is basically a random perturbation around its current position ξ i k-1 :

ξ i k = ξ i k-1 + N(δξ k ; σ ξ k ) (8)
where a Gaussian blurring is performed on ξ i k-1 with a shift δξ k and a spread of σ ξ k . In IPKF, the state estimation is performed using a bank of KFs within the PF environment where each of the KFs is associated to one instance of the corresponding parameter particles for which the state estimation is performed. Thus the KF is employed to estimate the states x k while the PF estimates the parameters θ k . For each particle, the nested KF thus provides an optimal state estimates conditioned on the assumed parameter particle. Finally, based on the innovation and its covariance produced by each of the KFs associated to each of the parameter particles, the particles are weighted. The weights are subsequently used to update the prior statistics of the particles. In the following, details of the nested KFs are described.

Nested Kalman filters:

Each of the evolved parameter particles is used to follow the system matrices as

F i k = F(θ k = ξ i k ) and H i k = H(θ k = ξ i k ).
The estimation procedure involves propagating the current state estimate x i k-1|k-11 , through the system conditioned on current parameter particles in order to predict x i k|k-1 . Predicted estimate is subsequently improved using current measurement y k to obtain the posterior estimate x i k|k specific to the particle estimate. This process is repeated for all particles yielding a set of posterior estimate for states {x i k|k ; for all i = 1, 2, • • • , N }. The prediction and correction steps of the KF for an arbitrary i th particle at k th time step are described in the following.

Prediction

x i k|k-1 = F i k x i k-1|k-1 (9a) P i k|k-1 = F i k P i k-1|k-1 F i k T + B Qk-1 B T (9b) Innovation i k = y k -H i k x i k|k-1 (9c) Correction x i k|k = x i k|k-1 + K i k i k (9d) P i k|k = (I -K i k H i k )P i k|k-1 (9e)
where K i k is the gain matrix. P i k 1 |k 2 is the estimate for state error covariance at k th 1 time step given measurement upto k th 2 instance. The process noise Qk-1 is denoted here with a • operator that defines that the process noise is not known apriori. Instead, it is estimated in parallel to IPKF (but outside the particle filter) and Qk-1 signifies the available best current estimate for process noise Q k .

MCC-KF

Izanloo et al. ( 2016) developed a Kalman filter based on MCC criterion where they instead of considering the error covariance for the gain calculation, employed maximum correntropy criterion for the gain formulation. In order to make a robust noise adaptive MCC based KF that can handle measurement outliers, MCC-KF algorithm defines a cost function that incorporates correntropy information instead of error covariances. In the proposed algorithm, the KF within the IPKF filter is replaced by the MCC-KF was given by [START_REF] Cinar | Hidden state estimation using the correntropy filter with fixed point update and adaptive kernel size[END_REF]. A detailed description is presented in the Appendix A.

For MCC-KF algorithm, the cost function, for the well known weighted least square approach for KF formulation targeting minimum variance in the estimates, is modified as:

J i = 1 σ 2 G σ || (x) i k || P i k|k-1 -1 + 1 σ 2 G σ || (y) i k || R-1 k-1 (10) where (x) i k = x i k|k -F i k x i k-1|k-1 , (y) i k = y k -H i k x i
k|k and, ||•|| w signifies the argument within the norm operator is weighted by index w. To minimize this objective function presented in equation ( 10), its gradient with respect to x i k|k is equated to zero and a subsequent restructuring yields a Kalman like equation for state estimate update as:

x i k|k = x i k|k-1 + K i k y k -H i k x i k|k-1
(11)

P i k|k = (I -K i k H i k )P i k|k-1 (12)
A detailed and generalized derivation for the Kalman gain and other relevant description is presented in Appendix A.

Adaptive estimation of noise

Targeting optimal estimates for health parameters, the proposed algorithm additionally estimates the noise processes recursively. After each iteration of the IPKF, the particle approximations for states x k|k and parameters θ k are obtained as:

xk|k = N i w(ξ i k )x i k|k and, θk = N i w(ξ i k )θ i k ; ( 13
)
where N is the number of particles and w(ξ i k ) is the weight of i th particle. This approximated estimates are further used to define the particle approximation for the system matrices F i k and H i k . Matrix B and D particle independent can be used directly.

Fk = N i w(ξ i k )F i k ; and, Hk = N i w(ξ i k )H i k ; (14)
The current process noise is further observed based on these particle approximated entities. For this, the deviation between predicted output and actual output i.e., δy k is computed first as: δy k = y k -Hk x k|k (15) This deviation δy k is due to uncertainty in state estimates and noise processes in process and measurement. With the prior estimates of state error covariance Pk|k (particle approximated) and measurement noise covariance Rk-1 , a measure for the process noise covariance can be observed from the innovation as:

cov(ˆ k ) = δy k δy T k -Hk Pk|k HT k -Rk-1 (
16) This measure is further mapped in the domain of state x k and process noise Q k . This provides an observation of the instantaneous estimate of the process noise as:

Q * k = ( Hk B + D) † cov(ˆ k )( Hk B + D) † T (17)
Here * signifies that an observation has been made on process noise Q k at time step k. This observation is then combined with the previous estimate in a moving average sense as following: L Q is the window length variable that controls how fast new estimates are adapted with the prior estimates. In turn, this defines how new data is adapted in to the current estimates of the noise processes. A high value ensures smooth and stable estimate of the noise covariance, while a low value puts significant weight on the new measurement.

Qk = Qk-1 + 1 L Q (Q * k -Qk-1 ) (18)
A compromise between rapid update and stable estimation should thus be desired.

Subsequently, an estimate vk for the process noise v k is obtained as: vk = ( Hk B + D) † δy k (19) With this estimate, current state estimate is updated as:

x * k|k = x k|k + Bv k (20)
Correspondingly, the output can be predicted as:

y * k|k = Hk x * k|k + Dv k (21)
Thus, the deviation between actual measurement and the updated measurement prediction y * k|k can be attributed to measurement noise. Evidently, an observation can be made about the current measurement noise as:

R * k = δy * k|k δy * k|k T ( 22
)
where δy * k|k = y k -y * k|k . Finally, similar to the process noise, the measurement noise estimate is updated as:

Rk = Rk-1 + 1 L R (R * k -Rk-1 ) (23)
Again, L R is the window parameter defining the window for the moving averaging of measurement noise estimate.

As it has been previously discussed, this work intends to estimate a system for which abrupt change in noise statistics is possible. During the transition from one noise statistics to the other, the observations for the process and measurement noise should be assimilated with caution. In this attempt, the correntropy information is employed to define a weight for the update based on current observation. To obtain this weight, a particle approximation for all L i k corresponding to nested MCC-KFs is obtained as:

Lk = N i=1 w(ξ i k )L i k (24)
The weights L Q and L R for the moving averaging window can then be defined based on this weight Lk as:

L Q = Lk L Q0 and L R = Lk L R0 (25)
where L Q0 and L R0 are the base values for the window. This ensures that if the data correntropy is severely poor, the algorithm puts significantly less belief on the new observations and consequently less update. On the other hand, if the data correntropy is high, the current observation is assimilated with more belief.

NUMERICAL EXPERIMENTS

Three sets of numerical experiments are undertaken: i) first one with a varying process noise, ii) second one with a varying measurement noise and iii) the third one with process and measurement noises varying simultaneously. For each of the cases, the system has been induced with a damage. The proposed IP-MCC-KF is then employed to detect the damage while estimating the noise statistics simultaneously.

The numerical experiments detailed in the following deal with an eight degrees-of-freedom mass-spring-damper system (see Fig. 1). The springs are considered to be of stiffness 8000N/m while the dampers are assumed to be of 8N -s/m. The mass blocks are assumed to be of 10kg each. Damage is introduced in the third and fourth springs and their stiffness values are reduced to 2000N/m. The system is excited with an ambient vibration at all its free nodes. This ambient vibration has been in this formulation as process noise. Finally, the system is simulated for time series length of 3072 sampled measurements collected at a sampling frequency of 50 Hz. The simulated acceleration response is collected as measurements which are further contaminated with a Gaussian white noise as a replication of the sensor noise contamination. A value of 1000 has been assumed as the bandwidth of the Gaussian kernel (i.e. σ). Also, based on our previous experience with IPKF, a set of 2000 particles are employed for the PF for all case studies in this article. Further details pertaining to each example case are presented in the following.

Case 1: Varying process noise

The first case discusses a system for which the process noise is varying with time. For the first 10.24 seconds, the process noise is assumed to be a zero-mean Gaussian white noise with a standard deviation of 100 N. This standard deviation is further changed to four and eight folds of its initial value. The assumed process noise is presented in Fig. 2. The initial weight for the moving averaging window i.e., L Q0 , has been assumed to be 0.01. The measurement noise statistics is considered to be stationary and known. A value of 5m/s 2 has been selected as the standard deviation of the zero mean Gaussian white noise and employed as the measurement noise.

The results for the parameter estimation are presented in Fig. 3 where it can be observed that the parameter estimation and damage detection as prompt and precise. The parallel estimation of the process noise statistics is also presented in Fig. 4. It can be seen that the initial estimates are to some extent turbulent. This is due to the error in the estimates of the parameter for which the innovation uncertainty contained the uncertainty due to inexact parameter estimates. However, once the parameter estimates converged to their true value, the turbulence in the process noise estimates settled down and become more stable.

The correntropy index for this estimation is also presented in Fig. 5. It should be noticed that whenever the system underwent a change in process noise statistics, the correntropy index decreased in a sense to reduce the effect of the measurement in the parameter update. This, in turn, damps the effect of the sudden change in the system and facilitates a smooth and stable estimation.

Case 2: Varying measurement noise

For the second case, the process noise statistics are kept constant with a standard deviation of 100 N as before. However, the measurement noise is kept changing up to 5 to 10 folds of its initial value. The initial value of standard deviation for the measurement noise has been assumed for this case study as 5m/s 2 . The measurement noise used for this case study is presented in Fig. 6. In a similar way, the initial weight for the moving averaging window for measurement noise estimation i.e., L R0 , has been assumed to be 0.01.

The results for the parameter estimation are presented in Fig. 7 where again it can be observed that the parameter Fig. 5. Case study 1: Correntropy evolution Fig. 6. Assumed measurement noise variation estimates are stable and prompt. The estimation of the measurement noise statistics is presented in Fig. 8. It can be seen that estimation of measurement noise statistics is not as accurate as for the process noise. This can be attributed to the very small magnitude of the measurement noise. Both process and measurement noise statistics are estimated from the innovation uncertainty in a process that might be termed as a systematic reduction of uncertainty. In this process, the process noise uncertainty is first removed followed by reduction of uncertainty due to the improper state estimates. Finally, the residual uncertainty is mapped to the measurement space in order to estimate the measurement uncertainty. Obviously, all these uncertainties are different in their scales and there is a wide possibility that the smallest of them will suffer a loss of information. This will obviously cause the estimate of measurement noise statistics to be poorer than the other two, i.e., process and state estimates. However, the current proposal can successfully identify the change in the measurement noise and the estimate of the measurement noise statistics is sufficiently accurate.

Case 3: Varying process and measurement noise

For the third case study, both the measurement and process noise are allowed to vary simultaneously. The variation assumed for this particular experiment is the same as the ones that have been used in the earlier case studies. L Q0 and L R0 are assumed to be 0.01 for this case study. For this case study, the results for the parameter estimation is presented in Fig. 9. Also, in this case, the parameter estimates are sufficiently accurate and stable.

The estimates for process and measurement noise statistics are presented in Fig. 10 and Fig. 11. It has been found that while both measurement and process noise statistics are estimated simultaneously, the estimation result for the measurement noise statistics is interestingly better. However, due to the scale effect of the process noise in the innovation uncertainty, the measurement noise estimates are influenced by the process uncertainty. While in the previous case studies, the measurement uncertainty is significantly underestimated, estimation for the current case study is precision wise much better. This is due to the fact that with no flexibility in the process uncertainty, the state estimates are sometimes over fitted which often underestimates measurement uncertainty. Simultaneous estimates, on the other hand, balance the estimation of process and measurement uncertainty which cause the measurement estimates to be better.

Finally, we experimented with a case where the noise statistics are actually varying over time but not estimated in the process of parameter estimation. The parameter evolution for this case study has been demonstrated in Fig. 12 where it can be seen that with inaccurate noise model values, the parameter estimates are very turbulent and it ultimately diverges resulting failed estimation. It can also be noticed that at some point in time the estimates even went below zero stiffness. The system model employed in this article is simplistic for which simulation is still proceeded without any hindrance. However, for costly and complicated system models, such abnormal parameter estimates will stop the parameter estimation. On the contrary, the same data-set when put through the proposed IP-MCC-KF algorithm yielded perfect estimation of the parameters for the third case study (cf. Fig. 9). This establishes the relevance of noise estimation in parallel to the parameter estimation.

CONCLUSION

In this article, a novel methodology has been presented based on IPKF filter in which the Kalman filter component of IPKF filter has been replaced with correntropy Mechanical Systems and Signal Processing, 47(1), 139-150. Zhang, X., Peng, J., Yu, W., and Lin, K.c. ( 2012).

Confidence-level-based new adaptive particle filter for nonlinear object tracking. International Journal of Advanced Robotic Systems, 9(5), 199.

Appendix A. DERIVATION OF MCC-KF Of the many formulations through which the Kalman filter can be derived, the one that uses optimization based approach is relevant in the derivation of MCC-KF. The derivation of the MCC-KF presented in the following is followed for each MCC-KF filter nested within the PF. The particle indices (• i ) for each MCC-KF and the variables involved are however dropped here in this appendix for the sake of simplicity. For MCC-KF, a quadratic cost function needs to be set up prior to a recursive estimation to minimize it, (A.1)

J = 1 σ 2 G σ ||y k -H k x k || R -1 k + 1 σ 2 G σ ||x k -xk || P -1 k
where xk = F k x k-1 is the prediction on x k . P k is the covariance of error between actual state x k and its prediction xk . Theoretically, this cost function needs to be minimized with x k as the argument. Finally, x k|k will be obtained as an estimate of

x k : xk|k = argmin x k J(x k ) (A.
2) The required gain matrix K k can be derived by analytically solving ∂J(x k ) ∂x k = 0 which yields the following equation:

- 1 σ 2 G σ ||y k -H k x k || R -1 k H T k R -1 k (y k -H k x k ) + 1 σ 2 G σ ||x k -xk || P -1 k P -1 k (x k -xk ) = 0 (A.3)
Upon simplification, this equation demonstrates the interrelation between state error covariance and the correntropy information as:

(A.4)

P -1 k (x k -xk ) = G σ ||y k -H k x k || R -1 k G σ ||x k -xk || P -1 k H T k R -1 k (y k -H k x k )
Plugging in the best estimates available for x k , xk and P k as x k|k , x k|k-1 and P k|k-1 , the estimate of the states can be restated as:

P -1 k|k-1 x k|k = P -1 k|k-1 x k|k-1 + G σ ||y k -H k x k|k || R -1 k G σ ||x k|k -x k|k-1 || P -1 k|k-1 H T k R -1 k y k -H k x k|k (A.5)
The update equation is therefore further simplified as: .6) where

P -1 k|k-1 x k|k = P -1 k|k-1 x k|k-1 + L k H T k R -1 k y k -H k x k|k (A
L k = G σ ||y k -H k x k|k || R -1 k G σ ||x k|k -x k|k-1 || P -1 k|k-1
.

As it can be seen from the expression of L k , that the index has x k|k term in its denominator which makes the estimation of L k an implicit problem and to be solved using costly optimization problem. [START_REF] Chen | Maximum correntropy Kalman filter[END_REF] employed a fixed point algorithm to solve for the posterior which is although accurate but increases the computational expense.

Since the proposed approach itself is recursive in nature, we incorporated a little modification in the algorithm by replacing x k|k by its best available estimate before the measurement correction, i.e., x k|k-1 (similar to [START_REF] Cinar | Hidden state estimation using the correntropy filter with fixed point update and adaptive kernel size[END_REF]). Thus the correntropy can be calculated as:

L k = G σ ||y k -H k x k|k-1 || R -1 k (A.7)
This, in turn, make the problem explicit and therefore less computationally expensive.

Finally, to give this update equation a Kalman like appearance we proceed with adding and subtracting L k H T k R -1 k H k x k|k-1 in the right hand sides of equation A.6:

P -1 k|k-1 + L k H T k R -1 k H k x k|k = P -1 k|k-1 x k|k-1 + L k H T k R -1 k y k +L k H T k R -1 k H k x k|k-1 -L k H T k R -1 k H k x k|k-1 (A.8)
which when arranged gives:

(A.9)

P -1 k|k-1 + L k H T k R -1 k H k x k|k = P -1 k|k-1 + L k H T k R -1 k H k x k|k-1 + L k H T k R -1 k y k -H k x k|k-1
or in compact form:

(A.10)

x k|k = x k|k-1 + K k (y k -H k x k|k-1 )
where

K k = P -1 k|k-1 + L k H T k R -1 k H k -1 L k H T k R -1 k .
The corresponding update equation for the error covariance P k|k-1 can be derived as: .11) This is the generalized expression for update equations (also called as Joseph's form). However, if the Kalman gain K k is optimal, this form takes even a simpler description (see [START_REF] Kulikova | Square-root algorithms for maximum correntropy estimation of linear discrete-time systems in presence of non-gaussian noise[END_REF] for the alternative form) as: .12) and corresponding error covariance update equation can be given as: P k|k = (I -K k H k )P k|k-1 (A.13)

P k|k = (I -K k H k )P k|k-1 (I -K k H k ) + K k R k K T k (A
K k = P k|k-1 L k H k T (H k P k|k-1 L k H k T + R k ) -1 (A

Fig. 1 .

 1 Fig. 1. Numerical test setup

Fig. 2 .

 2 Fig. 2. Assumed process noise variation

Fig. 7 .

 7 Fig. 7. Case study 2: Parameter estimation

Fig. 9 .

 9 Fig. 9. Case study 3: Parameter estimation

Fig. 12 .

 12 Fig. 12. Parameter estimation without noise estimationbased Kalman filter (MCC-KF) that helps to stabilize the estimation in presence of an abrupt change in the system due to damage or noise statistics. Numerical experiments established that incorporation of MCC-KF facilitates imparting less updates in cases of high mismatch between predicted and actual measurement caused due to either damage or change in noise statistics. It has also been observed that measurement noise statistics when estimated solely underestimates the noise covariance. However, when the statistics of measurement noise are estimated alongside the statistics of the process noise, the estimates are much better than that with the solo estimation strategy. Eventually, this simultaneous estimation strategy for noise statistics bettered the parameter estimation. In turn, this strategy enhanced the robustness of the proposed IP-MCC-KF algorithm.

x i|j represents estimate of the random variable x at the i th time instant provided the measurement including and upto time instant j.


Jérémy Salaam

Lilia Tabti

Sylvana Bahamyirou

Alexandre Lecointre

Oscar Hernandez

Olivier Jeannin

Franck Camerel

Sarah Cianférani

Embarek Bentouhami

Aline M Nonat

Loïc J Charbonnière

Formation of Mono-and Polynuclear Luminescent Lanthanide Complexes based on the Coordination of Preorganized Phosphonated Pyridines

published or not. The documents may come

= 2) are observed in solution. In the solid state, a dinuclear complex of La could be isolated and structurally characterized by X-ray diffraction, unraveling the presence of strong hydrogen bonding interactions between a La(H2O)9 3+ cation and the [LaL] 3-complex.

Introduction.

Photonic up conversion (UC) is very appealing for bio-analytical assays and theranostic applications since it allows for the conversion of low energy photons in the near-infrared to visible emission. 1 Although this phenomenon is usually observed in solid state materials 2 and nanomaterials, 3,4 a few discrete molecular systems have proven their efficiency. 5,6,7,8,9 In particular, we have recently demonstrated that UC was also possible with supramolecular systems in solution at room temperature using lanthanide coordination complexes. 7,8 In the first example, 7 visible up converted emission of Er 3+ was observed upon laser excitation of a sandwich [(L 1 Er)-F-(L 1 Er)] + assembly, in which the two Er centers were embedded in a macrocyclic DOTA derivatives (L 1 , Scheme 1), which holds the dimeric structure by π-π stacking interactions and H-bonding. 10 In the second strategy, hetero polynuclear lanthanide (Ln)

assemblies have been developed in order to promote Yb →Tb energy transfers. In this case, the tetraphosphonated octadentate ligand L 2 (Scheme 1) formed highly stable mononuclear [LnL 2 ] complexes which, in the presence of excess Ln 3+ salts, self-assembled into polynuclear [(LnL 2 )2Ln], [(LnL 2 )2Ln3] and [(LnL 2 )2Ln2] species. The assembly is driven by the strong polarization of the ligand due to the electron-rich phosphonated pendant arms. 11 Such engineering offers a very good shielding of the Ln cations by the ligands and very interesting photophysical properties in water and D2O. In particular, the deuterated analogues YbL 2D was used in the formation of [(YbL 2D )2Tbx] complexes (x = 1-3), which afforded characteristic visible UC emission of Tb, upon NIR excitation of the Yb at 980 nm.

In our search for new systems with improved efficiency, we turned our interest towards the development of chelates based on the 1,4,7-triazacyclononane (tacn) macrocycle appended with pyridine phosphonate units (Scheme 2). Lanthanide complexes with ligands based on tacn substituted with bipyridines, [START_REF] Charbonnière | Lanthanide Tags for Time-Resolved Luminescence Microscopy Displaying Improved Stability and Optical Properties[END_REF] picolinate, [START_REF] Nocton | Water Stability and Luminescence of Lanthanide Complexes of Tripodal Ligands Derived from 1,4,7-Triazacyclononane: Pyridinecarboxamide versus Pyridinecarboxylate Donors[END_REF] 8-hydroxyquinoline [START_REF] Nonat | Structural and Photophysical Studies of Highly Stable Lanthanide Complexes of Tripodal 8-Hydroxyquinolinate Ligands Based on 1,4,7-Triazacyclononane[END_REF] and pyridyl phosphinate [START_REF] Walton | Structure, Resolution and Chiroptical Analysis of Stable Lanthanide Complexes of a Pyridylphenylphosphinate Triazacyclononane Ligand[END_REF][START_REF] Starck | Structural Control of Cell Permeability with Highly Emissive Europium(III) Complexes Permits Different Microscopy Applications[END_REF][START_REF] Frawley | Very Bright, Enantiopure Europium(III) Complexes Allow Time-Gated Chiral Contrast Imaging[END_REF] have proven to be very efficient in the design of highly stable mononuclear lanthanide complexes with very interesting photophysical properties in the solid-state, in solution and even in cellulo [START_REF] Walton | Very Bright Europium Complexes That Stain Cellular Mitochondria[END_REF][START_REF] Butler | R) Dyes: Design, Synthesis, Structure and Photophysical Properties of Very Bright Europium Complexes and Their Use in Bioassays and Cellular Optical Imaging[END_REF] or on cell membranes [START_REF] Delbianco | Highly Water-Soluble Triazacyclononane Europium Complexes To Detect Ligand Binding with Time-Resolved FRET Microscopy[END_REF] and are ideal candidates for the development of biomedical imaging probes. Derivatives with two-photon antenna are also appealing for two-photon microscopy experiments in both the visible and near-infrared spectral ranges. [START_REF] Bui | Unexpected Efficiency of a Luminescent Samarium(III) Complex for Combined Visible and Near-Infrared Biphotonic Microscopy[END_REF] Introduction of 6-phosphonated-2-methylene bridged pyridines on the nitrogen atoms is expected to lead to: the formation of two five-membered chelate rings per arm; a nonadentate coordination suited to the protection of the first coordination sphere of the Ln cations; and a C3 symmetry with the negatively charged phosphonate functions on one side of the complexes, with the possibility to further develop electrostatic interactions in excess of Ln cations and second sphere interactions. 11,22 Furthermore, phosphonated pyridines are expected to provide highly stable lanthanide complexes compared to their carboxylated analogues, [START_REF] Nocton | Water Stability and Luminescence of Lanthanide Complexes of Tripodal Ligands Derived from 1,4,7-Triazacyclononane: Pyridinecarboxamide versus Pyridinecarboxylate Donors[END_REF][START_REF] Abada | Highly Relaxing Gadolinium Based MRI Contrast Agents Responsive to Mg2+ Sensing[END_REF] and efficient sensitization of Tb 3+ , Eu 3+ and, more interestingly, Yb 3+ .

Scheme 1. Polynuclear lanthanide assemblies with ligands L 1 , L 2 and L 2D .

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

In this study, we report the synthesis of the new ligand L (Scheme 2 ) and a detailed analysis of its coordination properties with Ln 3+ ions by UV-absorption spectrometry, spectrofluorimetry, 1 H and [START_REF] Gottlieb | NMR Chemical Shifts of Common Laboratory Solvents as Trace Impurities[END_REF] When unspecified, coupling constants refer to H -H coupling.

The given pH values are corrected for the deuterium isotopic effects (pD = apparent pH + 0.4). 25

Synthesis of diethyl (6-(chloromethyl)pyridin-2-yl)phosphonate 3:

In a 250 mL round bottom flask, 2-(chloromethyl)pyridine-1-oxide [START_REF] Polášek | Pyridine-N-Oxide Analogues of DOTA and Their Gadolinium(III) Complexes Endowed with a Fast Water Exchange on the Square-Antiprismatic Isomer[END_REF] (4.00 g, 27.9 mmol) was dissolved in dichloromethane (100 mL) under an argon atmosphere. Ethyl chloroformate (8.0 mL, 83.7 mmol) was added and the solution was stirred at r. X-ray Crystallography. Crystals suitable for X-ray diffraction were obtained by slow evaporation of aqueous solutions. The crystals were placed in oil, and a single crystal was selected, mounted on a nylon loop and placed in a low-temperature N2 stream. X-Ray diffraction data collection was carried out on a Bruker APEX II Kappa-CCD or a D8 VENTURE Bruker AXS diffractometer equipped with an Oxford Cryosystem liquid N2 device, using Mo-Kα radiation (λ = 0.71073 Å) at 150(2) K (Centre de diffractométrie X, Université de Rennes 1, France). The Bruker SMART program was used to refine the values of the cell parameters. Data reduction and correction absorption (SADABS) were carried out using the Bruker SAINT programs. The structures were solved by direct methods using the SIR97 program [START_REF] Altomare | SIR97: A New Tool for Crystal Structure Determination and Refinement[END_REF] and then refined with full-matrix least-squares methods based on F² (SHELX-97) [START_REF] Sheldrick | A Short History of SHELX[END_REF] with the aid of the WINGX program. [START_REF] Farrugia | WinGX Suite for Small-Molecule Single-Crystal Crystallography[END_REF] All non-hydrogen atoms were refined with anisotropic displacement parameters. For the [EuL] structure, the H atoms of phosphonate groups have been found in the electron density maps and refined as idealized hydroxyl group with torsion from electron density (AFIX 147 instruction). H atoms on the less agitated water molecules have been found in electron density map and refined with restraints on O…H (DFIX 0.84 0.01) and H…H (DANG 1.34 0.01). The four most agitated water molecules were squeezed. Their contribution to the calculated structure factors was estimated following the BYPASS algorithm, [START_REF]Méthodes d'analyse Complexométriques Avec Les Titriplex[END_REF] implemented as the SQUEEZE option in PLATON. [START_REF]Molecular Fluorescence: Principles and Applications[END_REF] The new data set, free of the four highly agitated water molecules contribution, was then used in the final refinement. Copies of this information is available free of charge from the Web site (www.ccdc.cam.ac.uk).

Physico-chemical and spectrophotometric studies.

Materials. Distilled water was purified by passing through a mixed bed of ion-exchanger (Bioblock Scientific R3-83002, M3-83006) and activated carbon (Bioblock Scientific ORC-83005). All the stock solutions were prepared by weighing solid products using an AG 245

Mettler Toledo analytical balance (precision 0.01 mg). Metal cation solutions were prepared from their chloride (LaCl3.7H2O, EuCl3.6H2O,TbCl3.6H2O, YbCl3.6H2O, LuCl3.6H2O, 99.9%, Sigma-Aldrich) and their concentrations were determined by colorimetric titrations with EDTA (10 -2 M, Merck, Titriplex III) according to standard procedures, using xylenol orange as indicator. [START_REF]Méthodes d'analyse Complexométriques Avec Les Titriplex[END_REF] Sodium hydroxyde (NaOH) and hydrochloric acid (HCl) were used to adjust pH during titrations. All the experiments described were repeated at least two times. Spectroscopic measurements were performed with 10×10 mm 2 quartz suprasil certified cells (Helma Analytics).

UV/vis absorption spectra were recorded on a Perkin Elmer lambda 950 or on a Specord spectrometer from Jena Analytics. Steady state emission spectra were recorded on an Edinburgh Instrument FLP920 spectrometer working with a continuous 450W Xe Lamp and a red sensitive R928 photomultiplier from Hamamatsu in Pelletier housing for visible detection (230 to 900 nm) or a Hamamatsu R5 509-72 photomultiplier for the Vis-NIR part. All spectra were corrected for the instrumental functions. For emission spectra upon UV excitation, a 399 nm cutoff filter was used to eliminate second order artifacts. Phosphorescence lifetimes were measured on the same instrument working in the Multi-Channel Spectroscopy (MCS) mode, using a Xenon flash lamp as the excitation source. For short μs lifetimes, the intensity decay was corrected from the lamp intensity decay profile using a scattering solution of Ludox in water. Luminescence quantum yields were measured according to conventional procedures, [START_REF]Molecular Fluorescence: Principles and Applications[END_REF] with optically diluted solutions (optical density < 0.05), using [Ru(bipy)3Cl3] in water (Φ = 0.04) [START_REF] Ishida | Recent Advances in Instrumentation for Absolute Emission Quantum Yield Measurements[END_REF] as reference for Eu, a bipyridine Tb complex, [TbL(H2O)] in water (Φ = 0.31) [START_REF] Weibel | Engineering of Highly Luminescent Lanthanide Tags Suitable for Protein Labeling and Time-Resolved Luminescence Imaging[END_REF] as reference for Tb, and cardiogreen (IR125) in MeOH (Φ = 0.078) for Yb. [START_REF] Benson | Fluorescence Properties of Indocyanine Green as Related to Angiography[END_REF] The errors are estimated to 10% on the lifetimes and 15% on the luminescence quantum yields. The pyridine-N-oxide 2 was first obtained according to literature procedure. [START_REF] Polášek | Pyridine-N-Oxide Analogues of DOTA and Their Gadolinium(III) Complexes Endowed with a Fast Water Exchange on the Square-Antiprismatic Isomer[END_REF] diethyl (6-(chloromethyl)pyridin-2-yl)phosphonate 3 was obtained by an adaptation of the procedure described for the parent pyridine, [START_REF] Lee | Synthesis of Diethyl Pyridin-2-Ylphosphonates and Quinolin-2-Ylphosphonates by Deoxygenative Phosphorylation of the Corresponding N-Oxides[END_REF] using ethyl chloroformate and triethylphosphite in CH2Cl2.

Alkylation of 1,4,7-triaza-cyclonane (TACN) in acetonitrile affords precursor 4, which is hydrolyzed with 6 M hydrochloric acid to give ligand L as a hydrated trihydrochloride salt.

Studies of the coordination properties of ligand L with Ln 3+ ions (Ln= La, Eu, Tb, Yb, Lu).

UV-visible absorption and spectrofluorimetry. The coordination properties of L with

Ln cations in water at pH = 7.0 (0.01 M Tris/HCl) was investigated by monitoring the changes in UV-visible absorption and steady-state emission spectra upon addition of increasing amount of metal. Considering that the various Ln complexes formed during the experiments can be present under a very large variety of protonation states, but that we have not been able to unambiguously ascertain these later (for example by potentiometry experiments for which precipitation was observed), the protonation states of the complexes will not be specified, unless it is clearly defined (see for example the X-ray crystal structure of the Eu complex below). The absorption spectra of the ligand displayed broad bands in the UV region characteristic of π → π* transitions of the pyridinephosphonate moieties, with a maximum at 267 nm (ε267 = 14 660 M -1 cm -1 ).

Addition of metal salts caused a hypochromic shift of the absorption spectra concomitant with a 3 nm bathochromic shift (see Figures S1-S3 for Ln = Yb, Tb and Eu). Similar evolutions were observed along the Ln 3+ series and, in all cases, the presence of an inflexion point near one equivalent of the added Ln 3+ , suggested the formation of a mononuclear complex together with other polynuclear species. In the case of Yb 3+ , excitation in the ligand at 267 nm led to efficient A c c e p t e d m a n u s c r i p t sensitization of NIR Yb 3+ -centered luminescence by antenna effect and observation of the characteristic 2 F5/2 → 2 F7/2 emission band of Yb. [START_REF] Comby | Chapter 235 Lanthanide Near-Infrared Luminescence in Molecular Probes and Devices[END_REF] The luminescence intensity gradually increases up to ca one equiv. of metal added and then slowly decreases (Figure 1), which supports the formation of a mononuclear complex together with polynuclear species under these conditions. 1 H and 31 P NMR studies. In order to elucidate the structure of the formed species, NMR titrations of ligand L with Ln = La 3+ , Eu 3+ , Yb 3+ and Lu 3+ in D2O (pD = 7, 25°C) were performed. The 1 H NMR spectra of the ligand was recorded in presence of 0 to 2.7 equiv. of LuCl3 and they are shown in Figure 3 (see Figure S6 for the full spectra). Between 0.2 and 0.9 equiv. of Lu 3+ , two sets of signals are clearly observed, which correspond to the free ligand and to the mononuclear [LuL] complex, respectively. After addition of 1 equiv. of Lu 3+ , complete disappearing of the ligand peaks occurred, leaving a single set of 8 signals in which 3 multiplets in the 7.4-8.0 ppm region account for the aromatic protons, one signal is observed at 3.84 ppm for one of the two diastereotopic CH2 protons of the pendant arm, and four spin systems in the 2.22-3.38 ppm region are assigned to the protons of the tacn skeleton. This pattern is in very good agreement with a rigid mononuclear complex with C3 symmetry. Evidence of the second methylene proton atom at 4.76 ppm, hidden by the peak of residual non-deuterated water, can be found by recording a 1 H-1 H COSY experiment (Figure S7). Between 1.2 equiv. and 1.8 equiv.

of added Lu 3+ in the solution, the peaks assigned to

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

Further insights into the structure of the species formed were obtained by monitoring the same sample by 31 P (Figure 4). Upon addition of 0.1 to 0.9 equiv. of LuCl3, a strong deshielding by ca. 9 ppm of the 31 P singlet was observed, due to the coordination of the phosphonate units. No changes occured between 0.9 and 1.2 equivalent of added Lu 3+ which, in line with previous observations by 1 H NMR, confirmed the large predominance of the monuclear [LuL] species at these stoichiometric ratios. Between 1.2 and 1.8 equivalent of metal, the peak assigned to [LuL] progressively broadened and four new signals were observed, all having the same integral within experimental error (see Figure S8 for the integration of the spectra recorded after addition of 1.6 equiv.), therefore suggesting either a unique species with four non equivalent phosphorous atoms or alternatively an equimolar mixture of two species with two non-equivalent phosphorous atoms each, possibly two diastereoisomeric species. At 1.9 equiv. and above, precipitation was observed together with the complete disappearing of the NMR peaks.

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

Further details on the intermediate species were obtained by recording the 1 H and 31 P DOSY (Diffusion Ordered Spectroscopy) spectra in presence of 0.5, 1.2, 1.6 and 2.0 equiv. of LuCl3, respectively. The corresponding figures are displayed in the Supporting Information Section. For each species, the diffusion coefficient was determined by using t BuOH as an internal reference.

In the presence of 0.5 equiv. of metal salt, two species were clearly identified and the diffusion coefficients amount to 283 μm 2 s -1 and 301 μm 2 s -1 for L and [LuL], respectively (Figure S9).

Species with fast diffusion coefficients were also observed at 732 μm 2 s -1 and 1870 μm 2 s -1, with a chemical shift corresponding to water protons. The faster species is in good agreement with the diffusion coefficient reported for HOD in D2O (1902 μm 2 s -1 ) [START_REF] Longsworth | The mutual diffusion of light and heavy water[END_REF] and the other peak may correspond to water molecules in interaction with the second coordination sphere of the complex.

As expected from 1 H NMR spectra, 1 H DOSY data in the presence of 1.2 and 1.6 equiv. were rather complex, however neat information can be obtained from 31 P NMR DOSY (Figure 5). In these conditions, three species could clearly be observed. The first species (with a singlet at 12.64 ppm) had a diffusion coefficient of 205 μm 2 s -1, which is in good agreement with the value determined for [LuL] by 1 H DOSY. The two other species display very close diffusion coefficients at 145 and 150 μm 2 s -1 , respectively, which was indicative of the formation of adducts with larger sizes, each adduct giving give rise to a pair of singlets in 31 P NMR. Hence, they were expected to have similar size and composition. One is characterized by two singlets at 10.00 ppm and 7.26 ppm, respectively and D = 150 μm 2 s -1 . The other gave rise to two singlets at 15.45 ppm and 7.68 pm and a diffusion coefficient D = 145 μm 2 s -1 (Figure 5). The hydrodynamic radii and volumes of the three species could be obtained from the Stockes-Einstein equation and are summarized in Table 1. From these data, the hydrodynamic volumes calculated for the larger species were 2.5 and 2.8 times larger than the volume determined for the 1 H and 31 P NMR titrations were also performed with LaCl3, YbCl3 and EuCl3 and the corresponding spectra are displayed in Figures S12-S17. In all cases, the formation of a mononuclear [LnL] complex with rigid C3 symmetry was clearly observed and reached its maximum concentration upon addition of 1 equiv. of metal salt. However, different behaviors were observed in excess of lanthanide ion. Addition of 1.2 equiv. of YbCl3, resulted in the formation of a non-symmetrical [Ln3L2] secondary species which became predominant at 1.6 equiv. of added metal and which was characterized by two sets of 9 peaks in the 1 H NMR spectrum (Figure S12). At 2.0 equiv., a third species started to grow and above, broadening of the signals was observed concomitantly to the formation of a precipitate in the tube. In the case of Eu 3+ and La 3+ , precipitation started to occur as soon as 1.5 equiv. of metal salts were added, also suggesting the formation of new species with lower solubility.

Mass spectrometry analysis.

In order to enlighten the composition of the polynuclear species, a similar titration of the ligand with 0 to 5 equiv. pattern with an isotopic peak at 791.07 m/z, which corresponds to the [EuL-2H] + species with the corresponding isotopic distribution related to 151 Eu(48%) and 153 Eu(52%). Above 1 equiv. and in the same instrumental conditions, the mononuclear complex remains the major species (Figure S19), possibly as a result of the fragmentation of the polynuclear adducts in the source, which renders the titration little informative. However, a detailed examination of the spectra recorded in the presence of 1.2 equiv. and above, indicated the presence of a series of peaks which could be attributed to polymetallic species (Figure 6). This analysis also endorses our initial attribution of the two sets of two peaks observed in the 31 P NMR spectra between 1.3 and 1.8 equiv. of Lu added, to two diastereoisomers with a 3:2 M:L stoichiometry and C3 symmetry. In conclusion, although the identification of the polynuclear species was not straightforward, a global analysis by NMR, mass spectrometry and spectrofluorimetric experiments converged towards the evidence of the formation [Ln3L2],

[Ln4L2] polynuclear species.

Synthesis and spectroscopic characterization of the [LnL] complexes.

Considering the high stability of the [LnL] complexes, it was envisaged to isolate the mononuclear complexes of Eu 3+ , Tb 3+ , Yb 3+ and Lu 3+ . In order to determine the optimal conditions for their synthesis and purification, the influence of the pH on a solution containing stoichiometric amounts of L and EuCl3.6H2O was first monitored by recording the Eu 3+ emission spectra upon excitation in the ligand absorption band (Figure S26). This titration indicated that complete formation of the complex occured above pH = 4.45 and once formed, the [EuL] complex remained stable in the pH range from 12.04 to 1.98. Hence, the formation of the desired mononuclear complexes was achieved at neutral pH whereas their precipitation and purification was accomplished in slightly acidic conditions (pH = 3.5), in order to avoid the formation of polynuclear complexes. Finally, the spectroscopic properties of the complexes were measured in water and D2O and the corresponding absorption and emission spectra are presented in Figure 8. Additional parameters are presented in Table 3 and three points have attracted our attention. At first, strictly monoexponential luminescence decays confirm the presence of unique species in solution. Second, the absence of inner-sphere water molecule endorses the nonadentate coordination of the ligand and the good protection of the metal inside the cavity. Third, the luminescence properties of the Yb complex are rather interesting in terms of lifetimes and quantum yields. In particular, the lifetime Yb 3+ emitters [START_REF] Bui | Near Infrared Two Photon Imaging Using a Bright Cationic Yb(III) Bioprobe Spontaneously Internalized into Live Cells[END_REF][START_REF] D'aléo | Ytterbium-Based Bioprobes for Near-Infrared Two-Photon Scanning Laser Microscopy Imaging[END_REF][START_REF] Moore | 1-Methyl-3-Hydroxy-Pyridin-2-One Complexes of Near Infra-Red Emitting Lanthanides: A c c e p t e d m a n u s c r i p t Efficient Sensitization of Yb(III) and Nd(III) in Aqueous Solution[END_REF][START_REF] Comby | A Novel Strategy for the Design of 8-Hydroxyquinolinate-Based Lanthanide Bioprobes That Emit in the Near Infrared Range[END_REF][START_REF] Werts | A Near-Infrared Luminescent Label Based on YbIII Ions and Its Application in a Fluoroimmunoassay[END_REF][START_REF] Zhang | Water-Soluble Mitochondria-Specific Ytterbium Complex with Impressive NIR Emission[END_REF] and longer than the one previously reported for the tacn-based [Yb(thqtcn-SO3)] (8.63 μs). [START_REF] Nonat | Structural and Photophysical Studies of Highly Stable Lanthanide Complexes of Tripodal 8-Hydroxyquinolinate Ligands Based on 1,4,7-Triazacyclononane[END_REF] However, this value remains significantly shorter than the values obtained for YbL 2 and YbL 2D ( >30 μs in D2O) 8 and other per-deuterated complexes in organic solvents. [START_REF] Hu | Highly Near-IR Emissive Ytterbium(III) Complexes with Unprecedented Quantum Yields[END_REF][START_REF] Doffek | The Radiative Lifetime in Near-IR-Luminescent Ytterbium Cryptates: The Key to Extremely High Quantum Yields[END_REF][START_REF] Ning | Design of Near-Infrared Luminescent Lanthanide Complexes Sensitive to Environmental Stimulus through Rationally Tuning the Secondary Coordination Sphere[END_REF] Ɛ 𝐇 𝟐 𝐎 (M -1 cm -1 ) [a] Ɛ 𝐃 𝟐 𝐎 (M -1 cm -1 ) [a] ɸ 𝐇 𝟐 𝐎 (%) [b] ɸ 𝐃 𝟐 𝐎 (%) [b] 𝜏 𝐇 𝟐 𝐎 (µs) 𝜏 𝐃 𝟐 𝐎 (µs) [a] λ= 265 nm. [b] ϕ measured relative to [Ru(bipy)3Cl3] [START_REF] Ishida | Recent Advances in Instrumentation for Absolute Emission Quantum Yield Measurements[END_REF] for Eu, [TbL(H2O)] [START_REF] Weibel | Engineering of Highly Luminescent Lanthanide Tags Suitable for Protein Labeling and Time-Resolved Luminescence Imaging[END_REF] for Tb and to IR125 [START_REF] Benson | Fluorescence Properties of Indocyanine Green as Related to Angiography[END_REF] for Yb. [c] Calculated according to ref 48. Surprisingly, despite the excellent shielding of the cations from the solvent molecules (q ≈ 0 for all complexes), the luminescence quantum yields of the Eu complex were rather modest compared to those obtained with ligands possessing carboxy pyridyl arms (picolinates) [START_REF] Regueiro-Figueroa | Lanthanide Dota-like Complexes Containing a Picolinate Pendant: Structural Entry for the Design of Ln(III)-Based Luminescent Probes[END_REF][START_REF] Guanci | A New Platform for Picolinate Based Luminescent Lanthanide Chelates[END_REF] or phosphinate ones. [START_REF] Walton | Isostructural Series of Nine-Coordinate Chiral Lanthanide Complexes Based on Triazacyclononane[END_REF] Considering the good shielding and the long luminescence lifetimes, it is surmised that these poor luminescence quantum yields are to be linked with a modest ligand to metal energy transfer.

q [c]
Structural studies of the mono-and poly-nuclear species.

While isolating the mononuclear complexes, single crystals suitable for X-ray diffraction analysis were obtained for [EuL], which was isolated in acidic conditions as a mixture of [EuL],

and [EuLH]Cl species. The ORTEP diagram is represented in Figure 9. Selected parameters, distances and angles are presented in Tables S1-S2. The metal ion is nine-coordinated by three nitrogen atoms from the tacn ring, three nitrogen atoms from the aromatic groups and three oxygen atoms from the phosphonate functions with a pseudo-C3 symmetry. The pyridinephosphonate pendant arms bind the lanthanide in a helical fashion and both enantiomers (Δ and Λ) are present in the solid state structures, therefore leading to centrosymmetric space groups (P21/a). Similar structures have previously been collected for tacn-based ligands with pyridinecarboxylate, [START_REF] Nocton | Water Stability and Luminescence of Lanthanide Complexes of Tripodal Ligands Derived from 1,4,7-Triazacyclononane: Pyridinecarboxamide versus Pyridinecarboxylate Donors[END_REF] 8-hydroxyquinoline pendant arms, [START_REF] Nonat | Structural and Photophysical Studies of Highly Stable Lanthanide Complexes of Tripodal 8-Hydroxyquinolinate Ligands Based on 1,4,7-Triazacyclononane[END_REF] and 6-carboxy-2,2'-bipyridine arms. A c c e p t e d m a n u s c r i p t their attribution could be partially performed by looking at 1 H-1 H correlations in the COSY experiment (Figure 12). The first family (1-9, in black) corresponds to the mononuclear [LuL] complex. Nuclear Overhauser effect was observed between H1 and H4/5, however dipole-dipole interactions between other species could not be observed. Their attribution was possible with the examination of the ROESY spectrum, based on the 1 x , 4/5 x correlations (x = ','',°, Figure S33).

No correlation could clearly be distinguished for the fourth family (1*-9*, in blue). Another striking point in the examination of the NOESY spectrum (Figure S34) is the observation of exchange correlations between, on the one hand, the two protons 4 and 5 corresponding to the AB system and, on the other hand, between proton atoms from the triazacyclononane backbone such as 7* and 7', 7° and 7 or 8'' and 8. Such exchanges are typical of ↔  interconversion processes associated to the concerted rotation of the pendant arms in C3 symmetric LnL complexes. However this observation is particularly surprising for two reasons. The ↔  interconversion process could not be observed in the isolated [LuL] complex, even by heating the solution up to 80°C in D2O and is not observed in the NOESY spectrum for the AB spin system of H4 and H5 protons of the remaining [LuL] complex. Then, if such an isomerization has already been observed for lanthanide complexes with 1,4,7-triazacyclononane derivatives such as H4nota [START_REF] Geraldes | Nuclear Magnetic Resonance and Potentiometric Studies of the Protonation Scheme of a Triaza Triacetic Macrocycle and Its Complexes with Lanthanum and Lutetium[END_REF] and H3bpatcn 53 , they always correspond to ligands with decreased denticity (CN = 6 for H4nota and 8 for H3bpatcn), whereas isomerization was not reported for tacn ligands with functionalized pyridines displaying CN = 9. On the opposite, enantiopure Eu and Tb complexes with trispyridylphosphinate triazacyclononane could be isolated, either using chiral HPLC [START_REF] Frawley | Very Bright, Enantiopure Europium(III) Complexes Allow Time-Gated Chiral Contrast Imaging[END_REF] or directly synthesized thanks to the inclusion of a stereogenic centre on the tcn ring. [START_REF] Evans | Complete Stereocontrol in the Synthesis of Macrocyclic Lanthanide Complexes: Direct Formation of A c c e p t e d m a n u s c r i p t[END_REF] A possible explanation of the observed exchange is that the interconversion occurs between the

Conclusion

In conclusion, the assembly of pyridinephosphonate units into the nonadentate tripodal ligand L allows the synthesis of highly stable monomeric lanthanide complexes that are isostructural along the series. Moreover, such complexes are highly stable in water and display luminescent properties, due to an efficient sensitization of Eu, Tb and Yb ions by the pyridinephosphonate antenna. More importantly, such scaffold is responsible for relatively low vibrational quenching, as emphasized by the very good luminescence properties of the Yb complex in D2O (τD2O = 10.2 μs and ϕD2O = 0.42%). Furthermore, using a combination of UV-absorption spectrometry, spectrofluorimetry, 1 H and 31 P NMR, DOSY, ESI-mass spectrometry and X-Ray diffraction, we

31 P

 31 Général Leclerc, 35042 Rennes Cedex, France KEYWORDS : lanthanide, pyridine, phosphonate, polynuclear assemblies, luminescence ABSTRACT. A series of polynuclear assemblies based on ligand L (1,4,7-Tris[hydrogen (6methylpyridin-2-yl)phosphonate]-1,4,7-triazacyclononane) has been developed. The coordination properties of ligand L with Ln III (Ln = La, Eu, Tb, Yb, Lu) have been studied in water (pH = 7) and in D2O (pD = 7) by UV-absorption spectrometry, spectrofluorimetry, 1 H and NMR, DOSY, ESI-mass spectrometry and X-Ray diffraction. This nonadentate ligand forms highly stable mononuclear complexes in water and provides a very efficient shielding of the Ln cations, as emphasized by the very good luminescence properties of the Yb complex in D2O, especially regarding its lifetime (τD 2 O = 10.2 µs) and quantum yield (ϕD 2 O = 0.42%). In the presence of excess Ln III cation, polynuclar complexes of [(LnL)2Lnx] stoichiometry (x = 1 and x A c c e p t e d m a n u s c r i p t

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

  For the [LaL] structure, H atoms on water molecules are found in electron density map and refined with restraints on O…H (DFIX 0.84 0.01) and H…H (DANG 1.34 0.01). All the other hydrogen atoms localized onA c c e p t e d m a n u s c r i p t carbon atoms were included in their calculated positions. Crystallographic data for structural analysis of the mononuclear Eu and of the La dinuclear complexes have been deposited with the Cambridge Crystallographic Data Centre under CCDC nos. 1589355 and 1589357, respectively.

A c c e

  p t e d m a n u s c r i p t 1 H and 31 P NMR titrations. Aliquots of a stock solution of the lanthanide trichloride salt in D2O ([Lu 3+ ] = 5.4×10 -2 M, [Yb 3+ ] = 0.1 M, [Eu 3+ ] = 0.1 M) were added to a ligand solution in D2O at pD = 7 ([L]= 17 mM for Lu 3+ , [L]= 6.9 mM for Yb 3+ and 110 mM for Eu 3+ ). After each addition, pD was adjusted to a value of 7.0 with 0.1 M NaOD solution in D2O. Changes after addition from 0 to 2.5 equivalents of metal were monitored by recording the 1 H (400 MHz) and[START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P spectra (162 MHz), respectively. In the case of Lu 3+ , diffusion spectroscopy (DOSY) NMR studies were carried out in order to measure the diffusion coefficient of the species present in solution. To this aim, a second titration was performed by adding 0, 0.5, 1.2, 1.6 and 2.0 equivalents of a LuCl3.6H2O solution in D2O (5.4×10 -2 M) to a ligand stock solution (0.021 M) at pD = 7.0.Mass spectrometry and ionmobility mass spectrometry. Mass spectrometry and ion mobility experiments were performed on a QTOF mass spectrometer (Synapt G2, Waters, Manchester, UK), coupled to an automated chip-based nanoelectrospray device (Triversa Nanomate, Advion, Ithaca, U.S.A.) operating in the positive ion mode. The sampling cone and the extraction cone voltages were set to 40 V and 4 V, respectively. Ions were efficiently thermalized with a constant trapping voltage of 7 V and subsequently transferred into the TOF analyzer. Ion mobility (IM) experiments were performed under a constant pressure of N2 of 3.3 mbar in the traveling wave IM cell. A constant pressure of 1.4×10 -3 mbar of He was applied to focus the ions prior to IM separation. The IM wave height and velocity were 40 V and 950 m/s, respectively. Transfer collision energy was set to 10 V to extract the ions from the IM cell to the TOF analyzer. Spectrophotometric and spectrofluorimetric titrations. Titrations were performed by stepwise addition of microvolumes of a 10 -3 M stock solution of the lanthanide salt (LnCl3.6H2O, Ln = Eu, Tb, Yb) at pH 7.0 (Tris-HCl 0.1 M) to 2 mL of ligand solution (5×10 -5 M in Tris-HCl 0.1 M at pH = 7.0) in 1 cm quartz suprasil cell. After each addition, the UV-Visible absorption spectra were recorded, together with the emission spectra upon excitation of the ligand at λex= 267 nm. The software Hypspec V1.1.33 was used to determine the coordination model and calculate the stability constants (log β) of the formed species. 35 Results and Discussion. Synthesis of ligand L.

A c c e

  Scheme 2. Synthesis and chemical structure of ligand L

Figure 1 .(Figure 2 .

 12 Figure 1. Evolution of the emission spectra of a solution of L in 0.01 M Tris-HCl (4.0×10 -5 M, pH = 7.0, λex = 267 nm) upon addition of aliquots of YbCl3.6H2O (1.14×10 -3 M). Inset: evolution of the intensity at 976 nm as a function of the [Yb]/[L] ratio.

  [LuL] disappeared and four sets of nine new s c r i p t protons were observed. Above 1.8 equiv., a broadening of the signals was observed, which was concomitant to the formation of a precipitate.

Figure 3

 3 Figure 3. 1 H NMR spectra of ligand L upon addition of 0 to 2.7 equiv. of LuCl3.6H2O (D2O, pD = 7, 400 MHz).

Figure 4 .

 4 Figure 4. 31 P NMR spectra of ligand L upon addition of 0 to 1.9 equiv. of LuCl3.6H2O (D2O, pD = 7.0, 162 MHz).

[

  LuL] monomer. On the basis of the stoichiometric proportions of the compounds in solution ([Lu 3+ ]/[L] = 1.5) and their hydrodynamic volumes, we postulated that the two sets of[START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P signals could be attributed to two species with a [Lu3L2] composition in agreement with the possibility of two diastereoisomers.

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

  Figure 5. 31 P DOSY of a solution of L in the presence of 1.6 equiv. of LuCl3 in D2O (25°C, pD = 7).

31 P

 31 NMR DOSY for a solution of L in the presence of 1.6 equiv. of LuCl3 in D2O (25°C, pD = 7). At 2.0 equiv., broadening of the signal occured and the determination of the diffusion coefficients was no more possible, presumably because of the formation of aggregates of higher M:L stoichiometry.Variable temperature NMR experiment were also performed on a sample with [M]/[L] = 1.6. 1 H and[START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P spectra from 25°C to 80°C are displayed in FigureS10and S11, respectively. Significant sharpening of the 1 H NMR signals was clearly observed upon increasing the temperature,A c c e p t e d m a n u s c r i p t thereby suggesting the occurrence of dynamic phenomena such as chemical exchange and /or rearrangement leading to the progressive loss of the four 31 P signals.

  of EuCl3 in water (ammonium acetate 0.01M, pH = 7.0) was monitored by ESI-mass spectrometry. Mass spectra (in positive mode) are presented in the Supporting Information. From 0 to 1 equiv. (Figure S18), the ligand peaks at 643.16 m/z ([M+H] + ), 665.15 m/z ([M+Na] + ) and 681.11 m/z ([M+K] + ) decreased to form a

A c c e p t e d m a n u s c r i p tFigure 6 .

 6 Figure 6. ESI-MS spectrum of a solution of L in water with 5 equiv. of EuCl3.6H2O (AcONH4 0.01M, pH =7). Inset. Arrival time distributions obtained by ion mobility for the peak at 938.99 m/z and the corresponding isotopic patterns of each peak.

  3+ and Tb 3+ complexes with ligand L ([L] ≈ 5 × 10 -5 M, TRIS/HCl 0.01 M, pH =7.0), λex= 267 nm.Assuming a minor variation of the thermodynamic stability constant along the lanthanide series, this model was used to analyze NMR data recorded with Lu 3+ . In particular, Figure7represents a comparison between the theoretical distribution species (full lines) and the intensity of the experimental[START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P NMR signals (points) and shows a striking correlation between[START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P NMR data and the previous model obtained from spectrofluorimetric titrations.

Figure 7 . 31 P

 731 Figure 7. Theoretical representation of the concentration of the species formed (with log β1:1=12.9, log β3:2 =38.6, log β4:2 =46, full lines) and evolution of the intensity measured by 31 P NMR for the three species: L (▲), LM (♦), L2M3 (+) upon titration of ligand L with LuCl3.6H2O (D2O, pD = 7.0, 162 MHz). Although not observed by The M4L2 species was not observed by but was observed by ESI-MS and spectrofluorimetric titrations

A c c e p t e d m a n u s c r i p t measured in deuterated

  water is rather long in comparison to typical values obtained for efficient

Figure 8 .

 8 Figure 8. UV/Vis absorption (dotted lines) and normalized emission (λex= 265 nm) of a D2O solution of [LnL] (straight lines) with Ln = Yb (brown), Tb (green) and Eu (red).

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

12

 12

A c c e p t e d m a n u s c r i p tFigure 9 .

 9 Figure 9. ORTEP diagram of the complexes [EuL], [EuLH]Cl with thermal ellipsoids at 50% probability.

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

  This chirality is inverted between two adjacent column. Within the two entities in a HH conformation, the three water molecules coordinated to the hydrated La2 atom and pointing toward the [La1L] complex are held by a H-bond network in which the two proton atoms of a water molecule are linked to one oxygen atom of a phosphonate function coordinated to La1 and to an oxygen atom not directly coordinated to La1 of an adjacent phosphonate function (blue dashed lines in Figure11).

Figure 10 .

 10 Figure 10. Solid state structure of the [(LaL)La(H2O)9] complex (ellipsoids drawn at 50 % probability).

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

  Weak interactions are also observed between [La(H2O)9] and a neighbouring [La1L] complex in the HT configuration. These interactions are relayed by three water molecules or by the oxygen atom of a phosphonate function of a [La1L] complex in a neighbouring column (green dashed lines in Figure11).

Figure 11 .

 11 Figure 11. Representation of the hydrogen bonding interactions observed in the crystal structure of the [(LaL)La(H2O)9] complexes (see text for the color code of H bonding interactions) and numbering scheme of the H atoms used for the interpretation of the NMR spectrum.

  diastereomeric pairs, exchanging the [(LuL)Lu(LuL)] into the [(LuL)Lu(LuL)] (and similarly with [(LuL)Lu(LuL)] and [(LuL)Lu(LuL)]). Considering that the ↔  interconversion of [LuL] is slow on the NMR timescale, the exchange between the diastereomeric pairs would require a dissociation process. Unfortunately, within a dyad such as depicted in Figure 9, distances between the two mononuclear LuL complexes are two high to expect any correlations by 2D NOESY NMR. Interestingly, 1 H NMR spectra of mixed species have been recorded by addition of 1 equivalent of LuCl3 on the mononulclear [YbL] complex and by addition of 1 equivalent of YbCl3 and [LuL] (Figure S35), and indicate no decoordination

A c c e

  p t e d m a n u s c r i p t and no ion exchange at room temperature and upon heating up to 80°C, therefore confirming the formation of kinetically inert complexes.

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

Figure 12 .

 12 Figure 12. 1 H-1 H COSY spectra of ligand L upon addition of 1.4 equiv. of LuCl3.6H2O (D2O, pD = 7, 400 MHz) and peak assignment (see numbering on Figure 11).

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

  [START_REF]Molecular Fluorescence: Principles and Applications[END_REF] P NMR, DOSY, ESI-mass spectrometry and X-Ray diffraction.

	Experimental Section.
	General Methods. Solvents and starting materials were purchased from Aldrich, Acros and Alfa
	Aesar and used without further purification. Flash column chromatography was performed with
	silica gel (10-63 μm, Macherey Nagel) or on C18 reversed phase silica (Macherey Nagel). IR
	spectra were recorded on a Perkin Elmer Spectrum One Spectrophotometer as solid samples and

only the most significant absorption bands are given in cm -1 . Elemental analysis and mass spectrometry analysis were carried out by the Service Commun d'Analyses of the University of Strasbourg. 1 H, 31 P and

[START_REF] Nocton | Water Stability and Luminescence of Lanthanide Complexes of Tripodal Ligands Derived from 1,4,7-Triazacyclononane: Pyridinecarboxamide versus Pyridinecarboxylate Donors[END_REF]

C NMR spectra and 2D COSY, NOESY, HSQC, and HMBC experiments were recorded on Avance 300 and Avance 400 spectrometers operating at 300 and 400 MHz for 1 H, respectively. Chemical shifts are reported in ppm, with residual protonated solvent as internal reference.

Synthesis and characterization of the Ln 3+ complexes. General

	961 (s), 881 (m). ESI + /MS: m/z = 270.064 ([M+Li] + , 100%). Anal. Calcd. for C10H15ClNO3P: C, monitored by 1 H-NMR. To a solution of ligand L (ca 40 mg) in D2O, a volume corresponding to
	45.55; H, 5.73; N, 5.31. Found: C, 44.91; H, 5.71; N, 5.28. half an equivalent of Ln cation of a stock solution of LnCl3.xH2O in D2O was added. The 1 H-
	Purification of the crude product was performed by column chromatography over silica gel (CH2Cl2/MeOH gradient from 100:0 to 97:3) yielding 3 (4.17 g, 57%) as an oil. 1 H NMR (CDCl3, 400 MHz):  7.78 (m, 2H), 7.57 (td, J1 = 7.7 Hz, J2 = 1.6 Hz, 1H), 4.65 (s, 2H), 4.15 (m, 4H), 1.27 (t, J = 7.2 Hz, 6H) ppm. 13 C NMR (CDCl3, 75 MHz):  157.7 (d, J = 22.8 Hz), 151.6 (d, J = 226.7 Hz), 137.2 (d, J = 12.1 Hz), 127.2 (d, J = 24.9 Hz), 125.2 (d, J = 3.6 Hz), 63.2 (d, J A c c e p t e d m a n u s c r i p t NMR spectrum of the solution was recorded and the exact volume corresponding to the missing Synthesis of compound 4: A schlenk tube containing 3 (2.00 g, 7.6 mmol), TACN (325 mg, 2.5 half equivalent was added. A 0.1 M NaOD solution was added to raise the pD to 7 and the mmol), and flame dried K2CO3 (1.75 g, 12.7 mmol), dissolved in anhydrous CH3CN, was stirred for 16 hrs at 50°C. The remaining K2CO3 was removed by filtration and the solvent was removed under reduced pressure. The resulting residue was dissolved in dichloromethane, washed with water and brine, dried over Na2SO4, filtered and concentrated to dryness under vacuum. Purification of the crude product was performed by column chromatography over alumina (CH2Cl2/MeOH gradient from 100:00 to 95:5) yielding compound 4 (1.18 g, 56%) as pale yellow oil. 3.89 (s, 6H), 2.85 (s, 12H), 1.28 (t, J = 7.1 Hz, 18H) ppm. 13 C NMR (CDCl3, 125 MHz):  161.6 (br), 151.3 (d, J = 227.5 Hz), 136.5 (d, J = 12.3 Hz), 126.5 (d, J = 25.4 Hz), 125.7, 64.4, 63.0 (d, J = 5.9 Hz), 55.9, 16.5 (d, J = 5.9 Hz) ppm. 31 P NMR (CDCl3, 202 MHz):  11.0 ppm. IR (cm -1 , ATR):  2982 (w), 2906 (w), 1672 (w), 1582 (w), 1443 (m), 1391 (w), 1367 (w), 1254 (s, P=O), 1163 (m), 1016 (s), 961 (s), 790 (m), 760 (m), 564 (m), 532 (m). ESI + /MS: m/z = 811.34 ([M+H] + , 100%). Anal. Calcd. for C36H57N6O9P3•H2O : C, 52.17; H, 7.15; N, 10.14. Found: C, 51.79; H, 6.96; N, 9.64. Synthesis of ligand L. Compound 4 (200 mg, 0.24 mmol) was dissolved in 6N HCl solution (30 mL) and was refluxed during 16 hrs. The solvent was removed under vacuum and ligand L (175 mg, 0.20 mmol, 90%) was obtained as a white powder. 1 H NMR (D2O, 500 MHz): δ 8.21 (td, J1 = 7.9 Hz, J2 = 3.7 Hz, 3 H), 7.96 (t, J = 7.4 Hz, 3 H), 7.79 (d, J = 8.0 Hz, 3 H), 4.51 (s, 6 H), 3.27 (s, 12 H) ppm. 13 C NMR (D2O, 125 MHz): δ 153.1 (d, J = 194.4 Hz), 152.7 (d, J = 13.7 Hz), 143.3 (d, J = 10.0 Hz), 127.8 (d, J = 16.4 Hz), 127.5, 57.5, 50.1 ppm. 31 P NMR (D2O, 202 MHz): δ 1.84 ppm. IR (cm -1 , ATR):  2638 (m, br), 1628 (w), 1403 (w), 1147 (m), 925 (s), 812 (m), 692 (m), 532 (s). ESI + /MS: m/z = 643.15 ([M+H] + , 100%). Anal. Calcd. for C24H33N6O9P3•3HCl•3H2O: C, 35.77; H, 5.25; N, 10.43. Found: C, 35.38; H, 5.08; N, 10.45. method. In order to control the strict 1:1 stoechiometry of the complexes and to avoid excess of Ln cations in the formed A c c e p t e d m a n u s c r i p t complete formation of the mononuclear complex was verified by 1 H NMR. Addition of HCl until pH = 3.5 resulted in the formation of a precipitate. Complete precipitation was achieved by addition of a THF/methanol mixture. The precipitate was isolated by centrifugation, dried under reduced pressure (2h, 40°C) and purified (except for the Lu and Tb complexes ) by flash column chromatography on reverse phase C18 column (H2O/MeOH/TFA 0.1%) to give [LnL] complex as a white powder. [EuL]. From 456 μL of EuCl3.6H2O at 0.11 M (50 mmol) and 40.0 mg (50 mmol) of ligand L, one obtains 35.1 mg of [EuL] . (CF3CO2H)2 . 2H2O (70%). 1 H NMR (D2O, pD = 7.0, 400 MHz): δ 8.42 (s, br, 3H), 7.53 (s, 3H), 7.30 (tr, br, 3H), 6.03 (d, br, 3 J = 7.0 Hz), 3.72 (s, br, 3H), 1.03 (s, br, 3H), -1.72 (s, br, 3H), -2.82 (s, br, 3H), -5.61 (s, br, 3H) ppm. ESI + /MS: m/z = 791.1 ([(EuL)+H] + , 85%), 793.1 ([(EuL)+H] + , 100%). 19 F NMR (D2O, 282 MHz): δ -75.60 (s). Anal. Calcd. for C24H30EuN6O9P3•2TFA•2H2O (C28H36EuF6N6O15P3): C, 31.86; H, 3.44; N, 7.96. Found: C, 31.67; H, 3.54; N, 8.21. [YbL]. From 415 μL (50 mmol) of a 0.11M solution of YbCl3.6H2O and 40.0 mg (50 mmol) of ligand L, one obtains 22.6 mg (32%) of [YbL] . 5(CF3CO2H) . 2H2O. 1 H NMR (D2O, 300 MHz, pD=7.87): δ 23.00 (s, br, 3H), 11.69 (s, br, 3H), 10.94 (s, br, 6H), 5.90 (s, br, 6H), -5.45 (s, br, 3H), -6.60 (s, br, 3H), -17.18 (s, br, 3H) ppm. 19 F NMR (D2O, 282 MHz): δ -75.60 (s). Anal. Calcd. for C24H30N6O9P3Yb•5TFA•2H2O (C34H39F15N6O21P3Yb): C, 28.79; H, 2.77; N, 5.92. Found: C, 29.20; H, 3.08; N, 5.79. ESI + /MS: m/z = 814.1 ([(YbL)+H] + , 100%). [LuL]. From 944 μL (51 mmol) of a 0.054M solution of LuCl3.6H2O and 40.9 mg (51 mmol) of ligand L, one obtains 27.9 mg (44%) of [LuL]. 1 H NMR (D2O, 400 MHz, pD=6.3): δ 7.98 (td, 3 JH-H = 7.6 Hz, 4 JP-H = 3.7 Hz, 3H, H4Pyr), 7.73 (t, 3 JH-H = 6.5 Hz, 3 JP-H = 6.5 Hz, 3H, H2Pyr), 7.52 (d, 3 JH-H = 7.5 Hz, 3H, H5pyr), 4.25 (AB system, δA = 4.70 ppm, δB = 3.81 ppm, JAB = 13.9 Hz, of ligand. 1 H NMR (D2O, 400 MHz, pD=7.96): δ 84.56 (s, br, 3 H), 32.84 (s, br, 3 H), 17.11 (s, br, 3 H), -5.48 (s, br, 3 H), -8.40 (s, br, 3 H), -9.77 (s, br, 6 H), -14.53 (s, br, 3 H), -70.05 (s, br, 3H). 31 P NMR (D2O, 202 MHz, pD=7.96): δ -94.37 (br). ESI -/MS: m/z = 398.02 ([M-2H] 2-, 100%); 797.04 ([M-H] -, 74%). Anal. Calcd. for C24H30N6O9P3Tb . 3NaCl . 7H2O 6H), 3.37 (td, J = 12.8 Hz, J = 4.2 Hz , 3H), 2.75 (dd, [TbL]. 46.2 mg (123.7 mmol) of TbCl3.6H2O in D2O (2 mL) were added to 97.9 mg (122 mmol) (C24H44Cl3TbN6Na3O16P3): C, 26.65; H, 3.91; N, 7.77. Found: C, 26.68; H, 4.10; N, 7.78.
	complexes, the complexes were prepared in D2O and the formations of the complexes were

t. for 30 min. Triethyl phosphite (14.8 mL, 86.3 mmol) was added dropwise, and the resulting solution was stirred at r.t. during 2 hrs. The resulting mixture was washed with saturated NaHCO3, water and brine, dried over Na2SO4, filtered and concentrated to dryness under vacuum. Impurities were removed by distillation under reduced pressure. = 5.9 Hz), 46.4, 16.4 (d, J = 5.9 Hz) ppm.

[START_REF]Molecular Fluorescence: Principles and Applications[END_REF]

P NMR (CDCl3, 162 MHz):  10.0 ppm. IR (cm -1 , ATR):  2983 (m), 1582 (m), 1478 (w), 1447 (m), 1392 (w), 1249 (s, P=O), 1182 (m), 1016 (s), 1 H NMR (CDCl3, 500 MHz):  7.74 (t, J = 6.8 Hz, 3H), 7.68 (td, J1 = 7.6 Hz, J2 = 5.9 Hz, 3H), 7.60 (d, J = 7.6 Hz, 3H), 4.23-4.11 (m, 12 H), J = 16.2 Hz, J = 5.2 Hz , 3H), 2.41 (dd, J = 12.9 Hz, J = 4.4 Hz , 3H), 2.27 (td, J = 13.8 Hz, J = 5.0 Hz, 3 H) ppm. ESI + /MS: m/z = 814.1 A c c e p t e d m a n u s c r i p t

([LuL-2H] + ,100%); 837.1 ([LuL-3H+Na] + , 40%). 31 P NMR (D2O, 202 MHz, pD=6.3): δ 12.86 ppm. Anal. Calcd. for C24H30N6O9P3Lu . 7NaCl . 4H2O (C24H38Cl7LuN6Na7O13P3): C, 22.25; H, 2.96; N, 6.49. Found: C, 22.04; H, 3.05; N, 6.32.

Table 1

 1

	3.75	301	5.80	817	0:1
	12.54	205	8.51	2582	1:1
	7.29/10.2	150	11.6	6538	3:2
	7.69/15.48	145	12.0	7328	3:2

. Diffusion coefficients (D), hydrodynamic radii (RH), hydrodynamic volumes (VH), the corresponding chemical shifts (δ) and the postulated composition of the species as obtained from

Table 2 .

 2 luminescence intensity as a function of the M/L ratios (see FiguresS22-S24). The overall stability constants determined for Eu 3+ and Tb 3+ complexes are summarized in Table2. The calculated emission spectra of the ML, M3L2 and M4L2 species (M = Eu, Tb), together with the evolution of the concentrations of the species formed are displayed in FiguresS23-S25. Overall stability constants of Eu

	([Eu4L2-4H] 2+ at 938.99

program by implementing a model with three species with M:L stoichiometric ratios of 1:1, 3:2 and 4:2. In both cases, such model can nicely reproduce the evolution of the emitted

Table 3 .

 3 Main spectroscopic parameters of the YbL, TbL, EuL and LuL complexes in water and D2O at pH =7.0.

demonstrated that this nonadentate ligand is a good building block for the formation of highly stable polynuclar complexes of [(LnL)2Lnx] stoichiometry (x = 1 and x = 2). Such architectures are very interesting for studying intermetallic communication and energy transfer within heteropolynuclear species and can potentially lead to very appealing photon up conversion properties. Future works are engaged in this direction.

ASSOCIATED CONTENT

Supporting Information. Evolution of the UV-visible absorption and emission spectra of a solution of L upon addition of Yb 3+ , Tb 3+ and Eu 3+ (Figures S1-S5); 1 H NMR, [START_REF]Molecular Fluorescence: Principles and Applications[END_REF] S1-S3); 2D 1H NMR spectra of a L:Lu 1:1.4 mixture (Figures S33-S34). This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author *l.charbonn@unistra.fr; aline.nonat@unistra.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally.

Funding Sources

Any funds used to support the research of the manuscript should be placed here (per journal style).