C Prud'homme

L Sala

M Szopos

Uncertainty propagation and sensitivity analysis: results from the Ocular Mathematical Virtual Simulator

Keywords: predictive ocular vascular dynamics, computational model, sensitivity analysis, ocular mathematical virtual simulator, Sobol index analysis, uncertainty quantification

We propose an uncertainty propagation study and a sensitivity analysis with the Ocular Mathematical Virtual Simulator, a computational and mathematical model that predicts the hemodynamics and biomechanics within the human eye. In this contribution, we focus on the effect of intraocular pressure, retrolaminar tissue pressure and systemic blood pressure on the ocular posterior tissue vasculature. The combination of a physically-based model with experiments-based stochastic input allows us to gain a better understanding of the physiological system, accounting both for the driving mechanisms and the data variability.

Introduction

The interest in patient-specific mathematical models applied to biomedical problems has greatly increased in the last years. In particular, the need for a better understanding and knowledge of quantities in the medical context has raised tremendously the complexity of the mathematical models employed to describe such physical systems. However, a crucial aspect to guarantee that the model and its numerical solutions are meaningful from the biomedical viewpoint is how inherent uncertainties are incorporated, as recently discussed for instance in [START_REF] Hose | Cardiovascular models for personalised medicine: Where now and where next?[END_REF]. In this direction, several works that studied the impact of uncertainties in the domain of cardiovascular disease modelling showed particular promise for elucidating the complex interplay between hemodynamics, biomechanics, and electrophysiology. Examples include arterial hemodynamics [START_REF] Chen | Simulation-based uncertainty quantification of human arterial network hemodynamics[END_REF][START_REF] Eck | Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system[END_REF][START_REF] Leguy | Global sensitivity analysis of a wave propagation model for arm arteries[END_REF][START_REF] Brault | Uncertainty quantification of inflow boundary condition and proximal arterial stiffness-coupled effect on pulse wave propagation in a vascular network[END_REF], cardiovascular simulations [START_REF] Sankaran | A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations[END_REF][START_REF] Marquis | Practical identifiability and uncertainty quantification of a pulsatile cardiovascular model[END_REF][START_REF] Fleeter | Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics[END_REF], electrophysiology [START_REF] Quaglino | Fast uncertainty quantification of activation sequences in patientspecific cardiac electrophysiology meeting clinical time constraints[END_REF], possibly coupled with electromechanical simulations [START_REF] Hurtado | Uncertainty quantification of 2 models of cardiac electromechanics[END_REF] and/or hemodynamics [START_REF] Campos | Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle[END_REF].

To the best of our knowledge, the eye's mathematical and computational modelling is still at its early stages, as recently reviewed in [START_REF] Guidoboni | Mathematical modeling of ocular fluid dynamics: From theory to clinical applications[END_REF]. Biomechanical and fluid-dynamical aspects are of particular relevance for several clinical conditions [START_REF] Harris | Ocular blood flow as a clinical observation: Value, limitations and data analysis[END_REF], but numerous factors influence their complex coupling, and the underlying mechanisms are still elusive. Also, there is an intrinsic difficulty of isolating these factors in a clinical setting and measuring their contribution [START_REF] Vercellin | Mathematical modeling and glaucoma: the need for an individualized approach to risk assessment[END_REF]. The present work focuses on the interaction between the main ocular vessels' hemodynamics, intraocular pressure and the retrolaminar tissue pressure, which is directly related to the cerebrospinal fluid pressure. Among several interesting contributions in this area, we mention those closely related to our work. The first mathematical model that simultaneously accounts for blood flow in the central retinal vessels, blood flow in the retinal microvasculature, retinal blood flow autoregulation, biomechanical action of intraocular pressure on the retinal vasculature, and time-dependent arterial blood pressure was introduced in [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF]. A theoretical model to study the effects of intraocular pressure elevation on the central retinal artery hemodynamics was proposed in [START_REF] Guidoboni | Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model[END_REF] and extended to account for the central retinal venous hemodynamics and the retinal microcirculation in [START_REF] Carichino | Effect of intraocular pressure and cerebrospinal fluid pressure on the blood flow in the central retinal vessels, Integrated Multidisciplinary Approaches in the Study and Care of the Human Eye Kugler Publications[END_REF]. However, none of them explicitly accounted for uncertainties and variabilities in the model parameters. Only a few modelling works include a stochastic analysis framework and focused on the production and drainage of aqueous humour flow [START_REF] Szopos | Mathematical modeling of aqueous humor flow and intraocular pressure under uncertainty: towards individualized glaucoma management[END_REF] and its coupling with ocular hemodynamics in a simplified manner [START_REF] Sacco | Modeling the coupled dynamics of ocular blood flow and production and drainage of aqueous humor[END_REF].

With these premises, we present in this contribution an uncertainty quantification and a global sensitivity analysis for the main parameters involved in the mathematical and computational framework called the Ocular Mathematical Virtual Simulator (OMVS) that we have developed [START_REF] Sala | Ocular mathematical virtual simulator: A hemodynamical and biomechanical study towards clinical applications[END_REF][START_REF] Sala | Mathematical modelling and simulation of ocular blood flows and their interactions[END_REF]. The clinical relevance of results provided by the OMVS is described in [START_REF] Sala | Patient-specific virtual simulator of tissue perfusion in the lamina cribrosa[END_REF] or, more extensively, in [START_REF] Sala | Mathematical modelling and simulation of ocular blood flows and their interactions[END_REF]Chapter 13], and it has been confirmed by an independent population-based study including nearly 10000 individuals [START_REF] Tham | Inter-relationship between ocular perfusion pressure, blood pressure, intraocular pressure profiles and primary open-angle glaucoma: the singapore epidemiology of eye diseases study[END_REF]. The reduced version of the OMVS model employed in the present study originates from [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF] for the retinal circulation. It has been extended to include a reduced model for blood flow perfusion in the lamina cribrosa. Preliminary findings from a simplified uncertain quantification study were published as a peer-reviewed conference abstract in [START_REF] Sala | The ocular mathematical virtual simulator: towards uncertainty quantification[END_REF]. As a significant step forward, we present hereafter a detailed global sensitivity analysis, using the Sobol' sensitivity indices.

The paper is organized as follows. The mathematical and computational model is described in Section 2.1, the uncertainty quantification and sensitivity analysis approach is presented in Section 2.2 and the input data in Section 2.3. The results of our study are depicted in Section 3 and discussed in Section 4. Finally, conclusions and future perspectives are outlined in Section 5.

Methodology

In the next two sections we describe the deterministic mathematical and computational foundations of our study, as well as the uncertainty quantification (UQ) and sensitivity analysis (SA) methods we incorporated in the OMVS framework to account for the stochastic features of the system.

Mathematical and computational model

The OMVS is a complex modelling framework that couples hemodynamics, biomechanics, and fluid dynamics in the eye to visualize and estimate in a non-invasive way ocular biofluids and tissues characteristics that are difficult or not accessible with standard investigation methods. The contributions developed within this framework can be subsequently utilized to isolate single risk factors and quantify their influence on the multi-factorial disease process.

To achieve this goal, the full OMVS is designed with a multiscale architecture, that aims at preserving the natural systemic features of blood circulation, while providing detailed views on sites of particular interest from the clinical viewpoint, such as the lamina cribrosa. The lamina cribrosa is a sponge tissue in the back of the eye that has a crucial role from the hemodynamical and neurological viewpoints. Specifically, this membrane is thought to help maintain the balance between the pressure inside the eye (intraocular pressure, hereafter denoted IOP) and behind the eye (the retrolaminar tissue pressure directly influenced by the intracranial pressure, denoted RLTp), which may influence the ocular blood flow. In addition, the lamina cribrosa acts as a scaffold for the retinal ganglion cell axons and the central retinal vessels and feeds RGC axons through its vascular network. The IOP is easily measurable with a Goldmann applanation tonometer. This instrument is based on the Imbert-Fick principle, which affirms that the pressure inside a dry thin-walled sphere corresponds to the force required to flatten the sphere surface divided by the flattening area.

In practice, we developed three model formulations of increasing complexity: we started from a 0-dimensional reduced-order description of the system, progressively adding the coupling with a porous media model for the lamina cribrosa and finally incorporating the effects of the deformation of the ocular tissues. More precisely, the OMVS combines (see Fig. 1):

1. System I (Fig. 1a): a circuit-based (0D) model for blood flow in the retinal vasculature, central retinal artery (CRA), and central retinal vein (CRV);

2. System II (Fig. 1b): a three-dimensional (3D) porous media model for the perfusion of the lamina cribrosa;

3. System III (Fig. 1c): a 3D isotropic elastic model for the biomechanics of the lamina cribrosa, retina, choroid, sclera, and cornea.

In the present contribution, UQ and SA analyses require intensive evaluations of the physical-based model. Therefore, we have employed a reduced version of the OMVS, accounting for the hemodynamical description provided by System I (Fig. 1a), coupled to a 0D adaptation of System II (Fig. 1b). For the proposed study, System III (Fig. 1c) has not been considered. In this manner, the model (i) provides a multiscale hemodynamics overview of the overall system, while maintaining a relatively accessible mathematical complexity and low computational costs; and (ii) combines information on ocular sites for which quantitative data are availablee.g. blood flow in the central retinal artery -and crucial ocular areas that are not accessible with clinical imagese.g. lamina cribrosa perfusion.

The 0D reduced version of the OMVS, see Fig. 2, exploits the electric analogy to fluid flow in complex vascular network [START_REF] Formaggia | Cardiovascular Mathematics: Modeling and simulation of the circulatory system[END_REF]. In this context, electric potentials correspond to fluid pressure, electric charges correspond to fluid volumes, and electric currents correspond to volumetric flow rates; the resistors and capacitors represent hydraulic resistance and wall compliance, respectively. Writing the constitutive equations characterizing the circuit elements and the Kirchhoff laws of currents and voltages leads to a system of ordinary differential equations whose solution provides the time-dependent profiles of pressures at the circuit nodes and flow rates through the circuit branches. We emphasize that IOP plays a crucial role in the description of the vein collapsibility, which is modelled in the OMVS by Starling resistors [START_REF] Starling | On the absorption of fluids from the connective tissue spaces[END_REF]. Namely, when the external pressure is higher than the internal blood pressure, veins collapse, therefore dropping down the blood flow.

The network is constructed as an extension of a previous model for the retinal circulation, proposed and validated in [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF] and [START_REF] Cassani | Blood circulation and aqueous humor flow in the eye: multi-scale modeling and clinical applications[END_REF]. The vasculature is divided into six main compartments: central retinal artery (cra), arterioles (r,a), capillaries (r,c), venules (r,v), central retinal vein (crv), and the lamina cribrosa (lc). Each compartment includes resistances (R) and capacitances (C). The intraocular segments (R cra,3 , R cra,4 , R r,v1 , R r,v2 , R crv,1 , R crv,2 ) are exposed to the IOP and the retrobulbar segments are exposed to the RLTp (R cra,1 , R cra,2 , R crv,3 , R crv,4 ). The explicit conservation and constitutive laws, as well as parameters involved in the description of the retinal circuit, follow directly from work by [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF] and [START_REF] Cassani | Blood circulation and aqueous humor flow in the eye: multi-scale modeling and clinical applications[END_REF]. To the initial model, we have added a simplified description of the hemodynamics in the lamina cribrosa, involving two resistors lcRin and lcR, and one capacitor C 5 , with the following values: lcRin = 78181.9 mmHg s cm -3 , lcR = 23988.25 mmHg s cm -3 , and C 5 = 0.000000753 cm 3 mmHg -1 . Also, in the original circuit, the external pressure on the resistances R cra,3 and R crv,2 is the effective stress exerted by the lamina on these vessels, which has been computed via a simplified fluid-structure interaction model [START_REF] Guidoboni | Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model[END_REF] describing the CRA/CRV interaction with the lamina cribrosa. In the current version of the model, we do not account for this contribution in a similar manner but rather adopt a simplified approach, in which the external pressure corresponds to IOP. Further extensions could incorporate this dependence, but our choice was dictated by the possibility of computing stresses directly from the System III component of the OMVS.

The reduced model thus obtained can predict the hemodynamics within the lamina cribrosa, the retinal vasculature, and the central retinal vessels based on the key inputs described in Tab. 1 and displayed as coloured dots in Fig. 2. For the proposed study we fixed the value of the pressure at cavernous sinus (blue node), while we choose as input variables for UQ and SA the systolic and diastolic pressure at the ophthalmic artery (SP and DP, red node), the intraocular pressure (IOP, yellow node) and the retrolaminar tissue pressure (RLTp, green node).

Regarding the outputs, and in light of the clinical application in view, we will focus on the quantities of interest in the UQ and SA analysis listed in Tab. 2. Note that these quantities of interest will reflect the behaviour of the system at different time instants through the cardiac cycle.

The mathematical model previously described has been implemented in OpenModelica [START_REF] Fritzson | Openmodelica-a free open-source environment for system modeling, simulation, and teaching[END_REF], an open-source Modelica-based modelling and simulation environment intended for industrial and academic studies of complex dynamic systems. Model results have been obtained using DASSL [START_REF] Petzold | Description of DASSL: a differential/algebraic system solver[END_REF] with a tolerance of 10 -6 , a time step of 10 -3 and total simulation time of 8 s. DASSL (Differential/Algebraic System Solver) is an implicit, high order, multi-step solver with a step-size control based on backward differentiation formula (BDF). These features allow it to be stable and fit to be used for a wide range of models. Its first development can be in found in [START_REF] Petzold | Description of DASSL: a differential/algebraic system solver[END_REF].

The system reaches a periodic state after the first cardiac cycle. However, we consider our output the last simulated cardiac cycle. Then, we retrieve the CRA, CRV, and the lamina cribrosa blood flow at three specific instant during the last cardiac cycle, namely the peak systolic time, the end of the systole and the end of the diastole (see Fig. 3). The choice of these three particular time instant in the cardiac cycle is driven by their interest from a clinical perspective. Moreover, in view of the model validation, the measurements of the blood flow is very often taken at peak systole and end diastole [START_REF] Harris | Acute iop elevation with scleral suction: effects on retrobulbar haemodynamics[END_REF].

Uncertainty quantification and sensitivity analysis approach

Uncertainty propagation. The construction of a reliable model for ocular biofluid dynamics involves several steps with inherent uncertainties, among which: parameter inference from uncertain experimental data, model personalization to the same subject data at different time instants or to different individuals, etc. Therefore, a major challenge is to assess how these sources of uncertainty impact the clinically relevant outputs of the simulation and ultimately affect the confidence in the model predictions.

In the present contribution, we adopted the following approach: for the set of key inputs of the reduced version of the OMVS model, the uncertainty is represented by a probability density function (pdf), which quantifies the probability of a given parameter to reproduce a specific observation. We next develop a forward uncertainty quantification, also known as uncertainty propagation, to investigate how these input uncertainties are propagated to the outputs via the computational model. Thus, the combined information between previous modelling knowledge and assumptions on the prior pdfs allows us to obtain the posterior pdf for the quantities of interest selected from the clinical perspective. Remark 1. A major challenge that needs to be addressed is the choice and representation of the prior pdf in light of the quantitative and qualitative information available from generally noisy data, see for instance [START_REF] Rocio Rodriguez-Cantano | A bayesian approach for parameter estimation in computational models of cardiac mechanics[END_REF] and further discussion in Sec. 2.3.

Sensitivity analysis. This part aims to determine synthetic measurements of which key inputs are the most influential on the quantities of interest selected among the outputs of the computational model without making assumption on the model and taking into account the continuous nature of the input parameters. To this end, we adopt the stochastic framework of global sensitivity analysis, which considers the input parameters {X j } j∈{1,...d} to be random independent variables with uncertainty modelled by a probability distribution, and employed to compute the random output Y. We have not considered the sensitivity analysis of the dynamic process Y(t). The methods described hereafter can be adapted to time t, and we could study the sensitivity indices with respect to time. However, it is more useful -easier to interpret -to perform sensitivity analysis at important characteristics of the response time series, in our case, peak systole, end systole, and end diastole, as discussed in Sec. 2.1.

To quantify the influence of the variations of X j on the variations of Y, we compute the so-called Sobol' sensitivity indices originally proposed in the seminal paper [START_REF] Sobol | Sensitivity analysis for non-linear mathematical models[END_REF], see also [START_REF] Prieur | Variance-based sensitivity analysis: Theory and estimation algorithms[END_REF]. More precisely, we define the first-order indices as

S j = var[E(Y|X j )] var[Y] , (2.1)
where (i) var denotes the variance and E the expected value; (ii) var[Y] corresponds to the variability of Y with respect to the overall uncertainty including non-linear effects; (iii) var[E(Y|X j )], the variance of the conditional expectation E(Y|X j ), corresponds to the main or first order effect of X j ; it means that if Y is sensitive to X j , E(Y|X j ) is likely to vary a lot and hence var[E(Y|X j )] as well.

Another useful index is the total Sobol' index, defined as

S tot j = 1 - var[E(Y|X (-j) )] var[Y] = 1 -S -j , (2.2)
where X (-j) = (X 1 , . . . , X j-1 , X j+1 , . . . , X d ) and S -j denotes the sum of the indices where X j is not involved. Additionally, the effect due to specific interactions between the j th and the k th factors (k j) can be measured by second-order Sobol' indices, and so on for high-order interactions, see for more details [START_REF] Prieur | Variance-based sensitivity analysis: Theory and estimation algorithms[END_REF].

Several approaches have been proposed to numerically compute these sensitivity indices, as reviewed for instance in [START_REF] Prieur | Variance-based sensitivity analysis: Theory and estimation algorithms[END_REF]. In the present work, we adopted the following two strategies: (i) a Monte Carlo-type approach and an estimator proposed in [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF] on the basis of a combinatoric argument, and (ii) a Fourier amplitude sensitivity test (FAST) [START_REF] Saltelli | A quantitative model-independent method for global sensitivity analysis of model output[END_REF], which is a spectral method based upon the Fourier decomposition of the model response. The computational cost of the first approach for first order and total order Sobol' indices is of (d + 2)n model evaluations [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF], where d is the input space dimension and n is the sample size; as for the FAST method the computation cost for first and total order indices is of d n model evaluations [START_REF] Saltelli | A quantitative model-independent method for global sensitivity analysis of model output[END_REF].

This variance-based approach implies intensive sampling, but it allows us to explore the input factors' full uncertainty ranges. We provide the two strategies as a way to ensure the reliability of our estimates. Indeed, given a sampling size, they may vary when repeating the estimations, and, in the Monte-Carlo approach, they can even be negative, although we used an implementation that mitigates this issue.

All the results on the UQ and SA analysis presented hereafter are carried out exploiting the Python statistical library Open-TURNS [START_REF] Baudin | Openturns: An industrial software for uncertainty quantification in simulation[END_REF].

Input data

Both mathematical methods described before need as prior knowledge the statistical distribution of the input. We detail and critically discuss in the sequel the choices we propose, based on assumptions deduced from the experimental and clinical literature. Recall that the input of our model are systolic blood pressure (SP), diastolic blood pressure (DP), IOP and RLTp.

Blood pressures. For SP and DP data we refer to the paper of Sesso and co-authors [START_REF] Sesso | Systolic and diastolic blood pressure, pulse pressure, and mean arterial pressure as predictors of cardiovascular disease risk in men[END_REF] where these two quantities showed a normal distribution. In particular SP has a mean of 124.1 mmHg and a standard deviation of 11.1 mmHg (Fig. 4a), whereas DP has a mean of 77.5 mmHg and a standard deviation of 7.1 mmHg (Fig. 4b). Using SP and DP as input parameters raises an issue for the sensitivity analysis, notably the Sobol' index study, namely that the inputs have to be assumed independent, see Sec. 2.2. For IOP, RLTp, we can make this assumption reasonably, however, this is not valid for SP and DP [START_REF] Gavish | Linear relationship between systolic and diastolic blood pressure monitored over 24 h: assessment and correlates[END_REF]. To overcome this problem we used Mean Arterial Pressure (MAP) as input variable in the Sobol study that has a normal distribution with mean 93 mmHg and standard deviation of 7.6 mmHg. In this case we also reconstructed the MAP distribution starting from the normal distribution of SP and DP and the following relationship:

MAP = 1 3 S P + 2 3 DP (2.3)
to check its normal distribution assumption (Fig. 4c). From Eq. (2.3) and from the correlation assumption between SP and DP made in [START_REF] Gavish | Linear relationship between systolic and diastolic blood pressure monitored over 24 h: assessment and correlates[END_REF], we can then reconstruct SP and DP starting from the MAP. Finally, the MAP is independent of the other two inputs (IOP, RLTp) and can be used as a sensitivity analysis parameter.

Intraocular pressure. There is a considerable discussion about normal and lognormal probability density functions in literature, and which of the two can better represent biological phenomena [START_REF] Limpert | Log-normal distributions across the sciences: keys and clues: on the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probability-normal or log-normal: that is the question[END_REF].

To explain our modelling choices we consider different IOP distributions using the data recovered from the clinical work of Suh and collaborators [START_REF] Suh | The distribution of intraocular pressure in urban and in rural populations: the namil study in south korea[END_REF] (mean µ normal = 14.7 mmHg, standard deviation σ normal = 2.8 mmHg). Starting from these values, we have computed the mean and the variance both for Gaussian distribution and for a lognormal behaviour using the following formulas [START_REF] Johnson | Lognormal distributions[END_REF] to be consistent with the data:

µ lognormal = log                  µ normal 1 + σ 2 normal µ 2 normal                  σ 2 lognormal = log       1 + σ 2 normal µ 2 normal      
We performed a comparative analysis using three different IOP distributions based on the same clinical data, which have been defined above. In particular, we compare a normal, a truncated normal and a lognormal distribution.

Fig. 5 highlights that the normal probability density (blue) is going beyond some physiological constraints for healthy patients such as IOP > 5mmHg. The truncated normal probability density function (green) does not show this issue; however, this IOP distribution presents an abnormal cut in the left tail, making it ineffective for a sensitivity analysis study. For our simulations, with the data provided by Suh et al. [START_REF] Suh | The distribution of intraocular pressure in urban and in rural populations: the namil study in south korea[END_REF], the lognormal distribution seems to be the more natural one. Our choice is dictated by the fact that we want to avoid miscalculation due to unphysiological input parameters that may lead to unrealistic discontinuities in the simulation results. This lognormal assumption has also been accepted in other works [START_REF] Wang | Intraocular pressure, central corneal thickness, and glaucoma in chinese adults: the liwan eye study[END_REF]. For this reason we utilized the log normal distribution based on the population based study operated by Suh et al. [START_REF] Suh | The distribution of intraocular pressure in urban and in rural populations: the namil study in south korea[END_REF] (Fig. 6).

Retrolaminar tissue pressure. For the RLTp we used a normal distribution (Fig. 7) with mean µ normal = 9.5 mmHg and standard deviation σ normal = 2.2 mmHg [START_REF] Ren | Cerebrospinal fluid pressure in glaucoma: a prospective study[END_REF]. In this case, no issues regarding the independence with other input or the sampling of unphysiological values occur. Remark 2. Our choices for the input probability density functions are dictated by clinical measurements reported in the literature, in particular [START_REF] Sesso | Systolic and diastolic blood pressure, pulse pressure, and mean arterial pressure as predictors of cardiovascular disease risk in men[END_REF] for the normal distribution of MAP, [START_REF] Suh | The distribution of intraocular pressure in urban and in rural populations: the namil study in south korea[END_REF] for the lognormal distribution of IOP, and [START_REF] Ren | Cerebrospinal fluid pressure in glaucoma: a prospective study[END_REF] for the normal distribution of RLTp. For the blood pressure and the retrolaminar tissue pressure, all the clinical literature refers to a normal probability density function of these inputs, whereas for what concerns the IOP, the prior knowledge on the uncertainty distribution is still a matter of debate. The rationale behind our specific choices is that we aim to avoid unphysiological input values that may lead to unrealistic discontinuities in the simulation results. It would also be possible in the future to estimate the input probability density functions in a more patient-specific manner, by using given repeated clinical measurements of patient-specific targets, such as systolic and diastolic blood pressure, intracranial pressure, intraocular pressure etc.

Results

In this section, we present two virtual studies using the OMVS described in Sec. 2.1 and employing the two methods described in Sec. 2.2. The input data assumptions have been discussed in Sec. 2.3. In particular, in the first study, we will use uncertainty propagation, while in the second one, we will run a sensitivity analysis on the model.

Uncertainty propagation study

We completed three different sets of N = 10000 evaluations of the model using only the IOP as a stochastic input distribution. For the RTLp we fixed its value for the three sets at 9.5 mmHg (mean value provided by [START_REF] Ren | Cerebrospinal fluid pressure in glaucoma: a prospective study[END_REF]). These three sets of evaluations differ from each other by the blood pressure value imposed; we selected three cases of clinical interest -in the same spirit as in [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF]:

1. baseline subjects with a systolic/diastolic blood pressure of 120/80 mmHg; 2. low blood pressure subjects with SP = 100 mmHg and DP = 70 mmHg; 3. high blood pressure subjects with SP/DP = 140/90 mmHg.

Recall that the outputs on which we focus for the UQ study are listed in Tab. 2.

Numerical simulations. Fig. 8 shows the computed probability density functions for the three different locations we selected (CRA, CRV and LC), for three different time instants (peak systole, end systole and end diastole), and for the three populations of clinical interest (low, baseline and high blood pressure). We also report the simulated mean and standard deviation for each computed output in Tab. 3. Thus, we highlight that:

• as expected, in each location and for all time instants, blood flow values decrease as we move from low to baseline and high blood pressure populations;

• for the simulated CRA blood flow, the three populations (baseline, low and high) have distinct pdfs at all evaluated time instants (peak systolic, end systolic, end diastolic), and present a significant asymmetry;

• for the simulated CRV blood flow, we have well established different pdfs at peak and end systole. We remark that within the veins we have a delayed peak of blood flows and CRV es values higher than CRV ps values (Tab. 3), in good agreement with experimentally observed patterns in time velocity curves acquired with Doppler imaging. The CRV blood flow pdf at end diastole exhibits a peculiar shape: a peak of frequencies for high values and a plateau in the number of realizations for low values separated by an almost empty frequency area of relatively middle values. Even if the clinical interpretation of these distributions is difficult, it is interesting to observe that these results suggest a different repartition of frequencies between the three cases, in particular, for low blood pressures, the high values peak is narrower and the plateau is wider and with more realizations than the high blood pressure case. This fact is confirmed by the boxplot in Fig. 9a: the tail of CRV blood flow low values is within the first and third quantile range for the low blood pressure case, whereas this tail is composed just by outliers for baseline and high blood pressure populations;

• for the LC blood flow, the OMVS suggests a similar analysis to CRV blood flow. The pdfs at peak, and end systole are distinct, whereas at end diastole it shows a peak of frequencies at low values and a more uniform distribution elsewhere. Also, Fig. 8i points out that for only for the low blood pressure virtual population a second peak and a considerable high number of frequencies can be identified at high values. Similarly to the CRV analysis, we propose the boxplot for the output LC ed in Fig. 9b: as predicted by the pdf. The tail of high values in the low blood pressure population is within the first and third quantile range. In contrast, the same tail consists of only outliers for baseline and high blood pressure populations. Finally, the computed LC blood flow variability is considerably lower than for the other two outputs (CRA and CRV) as reported in Tab. 3.

[

Sobol' index study

In this study, we utilize as random input variables the IOP, the retrolaminar tissue pressure (RLTp), and the SP and DP to compute the Sobol' indices using the distribution introduced in Sec. 2.3. We compute the Sobol' first and total indices using the Saltelli algorithm [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF] Figure 8: Output probability density functions predicted by the OMVS for three virtual populations: baseline subjects with normal SP and DP values (blue), low blood pressure subjects (orange), and high blood pressure subjects (green). and the FAST first and total indices [START_REF] Saltelli | A quantitative model-independent method for global sensitivity analysis of model output[END_REF]. We performed 5 analysis with increasing N = [1000, 2000, 5000, 7500, 10000]. The stopping criteria is based on the absolute iterative error: max

∀o∈O {|idx n -idx n * |} (3.1)
where O = CRA ps,es,ed , CRV ps,es,ed , LC ps,es,ed , idx are the first or total index for two consecutive choices (n > n * ) in N. The final figures proposed have been obtained with n = 10000 where the error was less than 4 • 10 -2 for all input indices and using both algorithms to compute the results (see Tab. 4).

Numerical simulations. For what concerns the Sobol' indices, Figs. 10a and 10b point out that the CRA blood flow at peak and end systole is highly dependent on the value of the MAP, whereas the influence of IOP and RTLp remains minimal. In Fig. 10c, the results provided by the OMVS suggest that the CRA blood flow at end diastole is highly influenced by the IOP, moderately by the MAP, and almost negligibly by the RLTp. Moreover, in this case, we notice that the total order and first order, especially for the MAP, are significantly different, meaning that high order interactions among parameters contribute to the variance of this output. For the CRV blood flow, Fig. 10d illustrates that there is a high dependency of CRV ps on MAP and only a moderate one from IOP. CRV es depends mainly on MAP (Fig. 10e), and CRV ed is highly dependent on the IOP and only mildly on the other two inputs (Fig. 10f). Also, for CRV blood flow at end diastole, we notice that high order interactions occur, especially for MAP and IOP. Figure 9: Boxplots of the end diastolic blood flow in the CRV and the lamina cribrosa predicted by the OMVS for three virtual populations: baseline subjects with normal SP and DP values, low blood pressure subjects, and high blood pressure subjects.

Input

Monte-Carlo approach [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF] FAST method [ For the lamina cribrosa blood flow, the results provided by the OMVS sensitivity analysis suggest that the MAP is the dominant factor with a moderate influence of the IOP only at end diastole (Fig. 10g, 10h, and 10i).

The FAST indices (Figs. 11) suggest that CRA ps (Fig. 11a), CRA es (Fig. 11b) and LC ed (Fig. 11i) depend mainly on MAP and mildly on IOP and RLTp. For what concerns the CRA ed (Fig. 11c) and CRV ed (Fig. 11f), they are highly influenced by the IOP, moderately by MAP and negligibly little by the RLTp. In this case the first and total order -especially for the IOP -are very different, implying that high order interactions are quite relevant to explain the variability of the output. Fig. 11d suggests that CRV ps has a high dependency on MAP and a moderate one on IOP. For CRV es (Fig. 11e), LC ps (Fig. 11g) and LC es (Fig. 11h), the sensitivity analysis results show that their variability is almost solely due to changes in MAP. 4 Discussion.

Hemodynamics in the ocular posterior tissue vasculature results from the combined effects of different factors. Specifically, ocular blood flow is driven by the difference between arterial and venous blood pressure, is impeded by IOP and RTLp (directly related to cerebrospinal fluid pressure) and is modulated by vascular regulation. Understanding these complex interrelated effects represents a major challenge when interpreting results from various clinical studies in ophthalmology. The current contribution introduces a networkbased model in the framework of the Ocular Mathematical Simulator, that couples retinal blood circulation with a simplified description of the LC hemodynamics. Further on, the model is employed to theoretically assess the relative contribution of IOP, RTLp and MAP stochastic variations on several clinically meaningful outputs characterizing hemodynamics in the CRA, CRV and LC, by means of uncertainty propagation methods and variance-based sensitivity indexes.

The results suggest that the hemodynamic response of the CRA, CRV and LC vasculature to variations in IOP and RTLp presents noticeable differences among individuals with different blood pressures, that strongly influences all the computed outputs. These model predictions are in good agreement with the experimental findings in [START_REF] Tham | Inter-relationship between ocular perfusion pressure, blood pressure, intraocular pressure profiles and primary open-angle glaucoma: the singapore epidemiology of eye diseases study[END_REF] and several theoretical studies [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF][START_REF] Guidoboni | Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model[END_REF][START_REF] Carichino | Effect of intraocular pressure and cerebrospinal fluid pressure on the blood flow in the central retinal vessels, Integrated Multidisciplinary Approaches in the Study and Care of the Human Eye Kugler Publications[END_REF], designed to elucidate how and to which extent blood pressure is influencing the distribution of ocular hemodynamics. Note that in a clinical setting, there is an intrinsic difficulty of evaluating the contribution of each factor and the complex relationships among them.

The use of a lognormal distribution for the IOP translates into a non trivial interpretation of the output, especially at the level of the CRV and LC due to the non-linear character of the model and the complex interplay between factors (as pointed out in Sec. 3.2). In this context, mathematical models are crucial to reproduce these mechanisms and help to unveil their interpretation.

We have compared the CRA blood flow simulated results to other data in the literature. Our results -Tab. 3, in particular baseline To compare our results with other clinical and mathematical studies that are more focused on the CRA blood velocities, we set the hypothesis of CRA diameter of about 160 µm [START_REF] Dorner | Calculation of the diameter of the central retinal artery from noninvasive measurements in humans[END_REF][START_REF] Lee | Association of retinal vessel caliber to optic disc and cup diameters[END_REF]. Using this assumption, our simulations provide similar values for CRA blood flows than the one measured by Harris et al. [START_REF] Harris | Acute iop elevation with scleral suction: effects on retrobulbar haemodynamics[END_REF] (CRA ps = 120.6 µl/min, CRA ed = 30.1 µl/min) and the three virtual populations simulated by Guidoboni et al. [START_REF] Guidoboni | Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance[END_REF] (baseline: CRA ps = 119.4 µl/min, CRA ed = 33.7 µl/min; low: CRA ps = 95.3 µl/min, CRA ed = 28.9 µl/min; high: CRA ps = 142.3 µl/min, CRA ed = 41.0 µl/min). This comparison shows the quality of the results, albeit the simplicity of the model we have employed for our study. From the SA (Sec. 3.2) we evince that, as expected, the CRA blood flow depends mainly by the MAP, and only at end diastole -when the arterial pressure is at minimum -the IOP is affecting the CRA results.

For the CRV, we compared the output CRV es results (Tab. 3) with total venous blood flow measurements reported in the literature. The simulated baseline mean value (= 52.1 µl/min) agrees with the experiments performed by Garcia et al. [START_REF] Garcia | Retinal blood flow in the normal human eye using the canon laser blood flowmeter[END_REF] (64.9 ± 12.8 µl/min) and Feke et al. [START_REF] Feke | Laser doppler measurements of blood velocity in human retinal vessels[END_REF] (80±12 µl/min for age 25-38 group, 73±13 µl/min for age 54-58 group). From a qualitative viewpoint, the analysis of the low values tail proposed in the numerical results paragraph may have an interesting physiological interpretation. As described in Section 2.1, IOP has a non-linear effect on the retinal vasculature, particularly on the venous part. Following this reason, the CRV high values peak may represent the natural state when the IOP is lower than the venous blood pressure, whereas the low values plateau denotes the collapse state. This statement is consistent with the previous analysis where we found a uniform distribution for low blood pressures and an important frequency in the peak for high blood pressures. The SA supports this prediction: the dependency to IOP is significant not only at end diastole, but also at peak systole. In contrast with the CRA, the CRV is a venous vessel, which blood pressure is lower therefore more easily influenced by external pressures (e.g. IOP). These indices show also significant differences between the first-order and the total-order index, which means that there are high-order interactions between these parameters. This fact is not surprising but it appears as a consequence of incorporating the Starling resistor effect, [START_REF] Starling | On the absorption of fluids from the connective tissue spaces[END_REF], which is a crucial requirement to retrieve clinical data, as reviewed in [START_REF] Guidoboni | Mathematical modeling of ocular fluid dynamics: From theory to clinical applications[END_REF].

For the lamina cribrosa hemodynamics, we highlight the fact that all mathematical results are crucial in the investigation of disease because non-invasive measurements are not available nowadays for this tissue. Following similar consideration made for the CRV, we notice an interesting physiological interpretation following the uncertainty propagation study (Sec. 3.1). For the low blood pressure population, which is notably the most at risk for ocular neuropathies, the results may suggest that an overperfusion of the lamina -with respect to the mean values of that population -occurs in more cases than for the other two population. The simplicity of the model, as confirmed by the low variability in the LC blood flows, does not allow us to clarify this fact. The SA (Sec. 2.2) supports this analysis. Sobol and FAST indices show a high dependency only due to the MAP, which can be not so intuitive as for the CRA or the CRV, indeed we know that the pressure gradient across the LC (IOP -RLT p) may influence the hemodynamics [START_REF] Sala | Analysis of iop and csf alterations on ocular biomechanics and lamina cribrosa hemodynamics[END_REF]. Further analysis by adding a three-dimensional hemodynamical and biomechanical description of the lamina would certainly help in this investigation, in particular to notice the impact of IOP and RLTp on LC blood flow.

Finally we discuss the estimates of the first and total indices using the Monte-Carlo and FAST approaches. For all considered quantities of interest, the two indices are suggesting similar outcomes, which allows us also to cross-validate the results of our analysis. The main differences concern CRA ed (Figs. 10c and11c) and CRV ed (Figs. 10f and11f), in particular on the high order interactions between IOP and MAP -more emphasized for the FAST estimates. We privilege the results provided by the Monte-Carlo estimates which are unbiased compared to the FAST method. Also, the latter has the initial advantage of being more computationally efficient, but at the cost of extra assumptions of smoothness in the model [START_REF] Xu | Reliability of global sensitivity indices[END_REF] that we are not currently able to verify for the OMVS.

Conclusion

Thanks to its special connection to the brain and its accessibility to measurements, the eye provides a unique window on the brain, thereby offering non-invasive access to a large set of potential biomarkers that might help in the early diagnosis and clinical care of Neuro-Degenerative Diseases [START_REF] Guidoboni | Neurodegenerative disorders of the eye and of the brain: a perspective on their fluid-dynamical connections and the potential of mechanism-driven modeling[END_REF]. The OMVS has already shown great potentiality to reproduce the ocular biomechanics and the hemodynamics [START_REF] Sala | Mathematical modelling and simulation of ocular blood flows and their interactions[END_REF].

Pursuing this concept, in this contribution, we have proposed an uncertainty propagation study and a sensitivity analysis to evaluate the impact of uncertainties on this ophthalmological virtual laboratory. First, we have set up a framework to perform a forward UQ analysis, which allowed us to evaluate the effects of the propagation of uncertainty from input to output. Second, we completed a SA study based on Sobol indices to capture the interplay between the different model parameters and their relative importance. Finally, we have assessed qualitatively and quantitatively this computational framework in view of clinical applications.

In the context of computational models, a coupled UQ/SA analysis is crucial for the scientific research, especially in biology and medicine. The use of such mathematical framework may be employed in the process of product development or diseases understanding, decreasing the number of physical tests necessary and therefore the economic cost. The low fidelity model employed already provided useful information for analysis, however this study must be pursued with higher fidelity models such as System II and System III, see Figs. 1b and 1c respectively. The current methodology could thus be further improved, in particular by (i) using a multilevel multifidelity estimator, as for instance the one developed in Dakota toolkit, see [START_REF] Fleeter | Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics[END_REF][START_REF] Adams | a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 5.0 user's manual[END_REF]; (ii) devising a reduced order modelling approach for the 3D elastic and poroelastic models, following the reduced basis framework [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF][START_REF] Hild | Optimization and control of high fields magnets[END_REF], developed in the open source software Feel++ [START_REF] Prud'homme | Advances in FEEL++ : A domain specific embedded language in C++ for partial differential equations[END_REF][START_REF] Prud'homme | Feel++ V108[END_REF]. Finally, considering sensitivity indices over time, as in [START_REF] Campos | Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle[END_REF], as well as characteristics instants, provides a promising perspective of the present work.

  (a) System I: 0D hemodynamics of retinal vasculature. (b) System II: 0D hemodynamics of retinal vasculature + 3D hemodynamics of lamina cribrosa.(c) System III: 0D hemodynamics of retinal vasculature + 3D hemodynamics of lamina cribrosa + 3D biomechanics of lamina cribrosa, retina, choroid, sclera, and cornea.

Figure 1 :

 1 Figure 1: Schematic of the OMVS multiscale structure. IOP = intraocular pressure, RLTp = retrolaminar tissue pressure, CRA = Central Retinal Artery, CRV = Central Retinal Vein, OA = ophthalmic artery, OV = ophthalmic vein, TCPA = temporal posterior ciliary artery, NPCA = nasal posterior ciliary artery.

Figure 2 :

 2 Figure 2: Schematic of the 0D circuit-based model (System I of the OMVS) employed in this contribution for UQ and SA. The coloured nodes are the input of the model that can be inferred from clinical measurements. Key name Unit Brief description DP mmHg Diastolic blood Pressure SP mmHg Systolic blood Pressure IOP mmHg IntraOcular Pressure RLTp mmHg RetroLaminar Tissue pressure

Figure 3 :

 3 Figure 3: Specific instant during the last cardiac cycle (peak systole, end systole and end diastole) used to compute the clinical outputs of interest. Image edited from https://en.wikipedia.org/wiki/Cardiac_cycle.

Figure 4 :

 4 Figure 4: Normal probability density functions of the input blood pressure variables

Figure 5 :Figure 6 :

 56 Figure 5: Comparison among normal (blue), lognormal (orange)and truncated normal (green) IOP distributions using the same input data[START_REF] Suh | The distribution of intraocular pressure in urban and in rural populations: the namil study in south korea[END_REF].

Figure 7 :

 7 Figure 7: Normal probability density function of the input retrolaminar tissue pressure (RLTp).

  Peak systolic CRA blood flow.

  End diastolic CRA blood flow.

  Peak systolic CRV blood flow.

  End systolic CRV blood flow.

  End diastolic CRV blood flow.

  End systolic LC blood flow.

  End diastolic LC blood flow.

Fig. 10

 10 report the Sobol indices using the Saltelli algorithm, while Fig.11report the FAST indices.

  End diastolic LC blood flow.

  (a) Peak systolic CRA blood flow. (b) End systolic CRA blood flow. (c) End diastolic CRA blood flow. (d) Peak systolic CRV blood flow. (e) End systolic CRV blood flow. (f) End diastolic CRV blood flow. (g) Peak systolic LC blood flow. (h) End systolic LC blood flow. (i) End diastolic LC blood flow.

Figure 10 :

 10 Figure 10: Estimation of Sobol' indices using the Monte-Carlo approach [46]. Red squares represent first order indices, while blue squares represent total order indices.

( a )

 a Peak systolic CRA blood flow. (b) End systolic CRA blood flow. (c) End diastolic CRA blood flow. (d) Peak systolic CRV blood flow. (e) End systolic CRV blood flow. (f) End diastolic CRV blood flow. (g) Peak systolic LC blood flow. (h) End systolic LC blood flow. (i) End diastolic CRV blood flow.

Figure 11 :

 11 Figure11: Estimation of first and total indices using the FAST method[START_REF] Saltelli | A quantitative model-independent method for global sensitivity analysis of model output[END_REF]. Red squares represent first order indices, while blue squares represent total order indices.

Table 1 :

 1 Input parameters for the reduced OMVS.

	Key name	Unit	Brief description

CRA ps µl/min peak systolic CRA blood flow CRA es µl/min end systolic CRA blood flow CRA ed µl/min end diastolic CRA blood flow CRV ps µl/min peak systolic CRV blood flow CRV es µl/min end systolic CRV blood flow CRV ed µl/min end diastolic CRV blood flow LC ps µl/min peak systolic lamina cribrosa blood flow LC es µl/min end systolic lamina cribrosa blood flow LC ed µl/min end diastolic lamina cribrosa blood flow Table 2: Quantities of interest in the UQ and SA analysis.

Table 3 :

 3 Simulated mean and standard deviation for all the quantities of interest in the uncertainty propagation study.

Table 4 :

 4 Error based on Eq. (3.1) computed between two consecutive choices of n ∈ N.

	47]


Dr Huayna Terraschke

Dr Jacob Olchowka

Eugenie Geringer

Dr Aline Varella Rodrigues

Claudia Wickleder
email: wickleder@chemie.uni-siegen.de

Facile ionic liquid-assisted strategy for direct precipitation of Eu 2+ -activated nanophosphors under ambient conditions

Keywords: nanoparticles, ionic liquids, divalent europium, sonochemistry, microwave chemistry

This work describes a novel ionic liquid assisted synthesis strategy for a direct and easy production of Eu 2+ -doped nanoparticles (NPs), where the ionic liquids are also used as fluoride sources to avoid the use of elemental fluorine or toxic HF. Because of the general remarkable advantages of Eu 2+ containing phosphors and the additional ones of nanoscaled materials, minimizing their dimensions without a Eu 3+ reducing sintering step represents a major advance in this field. Up to now, the direct synthesis of Eu 2+ -doped nanophosphors consisted of an enormous challenge, due to the oxidation to Eu 3+ observed in hydrous solution, which is commonly used for the preparation of NPs, generating lattice defects and undesired particle growth or agglomeration by additional reducing steps at high temperatures. In contrast, ionic liquids do not present an oxidizing character, as long as they do not contain oxidizing ions such as ClO 4 -or NO 3 -, allowing the direct precipitation of NPs e.g. using Eu 2+ containing starting materials. Here, the undoped and Eu 2+ -doped BaFCl NPs have been prepared under atmospheric conditions for the first time using ionic liquids (ILs) as solvents and also as fluoride source, applying sonochemical and microwave-assisted approaches. In general, this method bears an enormous potential for an easy synthesis of fluoride materials compared to inconvenient solid-state methods. In addition, the ionic liquid plays the role of strongly attached protecting shell which represents ~ 7-8% of the total NPs weight.

Introduction

Inorganic phosphors doped with divalent europium are widely used in different applications and especially as materials used in White Light Emitting Diodes (WLED). [1] They are extremely advantageous because of their high brightness, tunable emission wavelength from UV to red and low toxicity. [2,3] In general, these materials are characterized by strong emission bands upon direct excitation, caused by the parity allowed electronic transitions from the 4f 6 5d 1 excited states into the 4f [START_REF] Wang | [END_REF] ground state, strongly influenced by the host lattice. [4, 5] In particular, barium fluoride chloride activated with divalent europium is a widely investigated luminescent compound, especially because its outstanding X-ray storage properties [START_REF] Miura | Phosphor Handbook[END_REF] and emission bands peaking in the UV-blue spectral range [START_REF] Wang | [END_REF]8] . Different mechanisms are proposed for explaining the X-ray absorption and visible emission properties after photostimulation in this material. These involve formation of (i) F-centers as electron traps or (ii) hole centers, (iii) interaction of the Eu 2+ ions with each of the two centers individually and (iv) interaction of the Eu 2+ ions with both F-and hole-centers as intermediate states. [START_REF] Wang | [END_REF]8] However, most of the currently available literature about Eu 2+ -activated BaFCl report about the synthesis and characterization of micrometer-sized crystals. [START_REF] Wang | [END_REF][8][9][10] These so-called bulk phosphors are usually prepared by means of solid-state reactions, requiring high temperatures and long reaction times. Nanosized luminescent materials, on the other hand, have gained increase importance in technical applications in the past few years, due to their several advantages in comparison to analogue bulk materials. [2] Chemical and physical properties of nanoparticles (NPs) differ from that of bulk materials, particularly because of quantum size effects, [START_REF] Miura | Phosphor Handbook[END_REF] their high surface-to-volume ratio, low light scattering and high packing density. [START_REF] Schubert | Synthesis of Inorganic Materials[END_REF] Therefore, smaller amounts of luminescent material are required to build homogeneous and dense coating layers employing nanophosphors, in comparison to compounds of larger crystal sizes, leading to resources saving devices with improved optical properties. Due to these reasons, luminescent nanomaterials play an essential role in the development of displays, UVabsorbers, pigments, luminescent and photo-/electro chromic materials [2,[START_REF] Vollath | Nanomaterials: an Introduction to Synthesis, Properties and Application[END_REF] . Nevertheless, most of the synthesis methods applied for preparing Eu 2+ containing nanoscaled compounds are carried out in aqueous solution, requiring a post-synthetic annealing step, in order to fully reduce the oxidation state of europium from 3+ to 2+. [START_REF] Streit | [END_REF] Because high temperatures are necessary for this procedure, an undesired coalescence of the particles occurs, resulting in an increase of the particle diameter or the formation of hard agglomerates. In addition, the reduction of Eu 3+ to Eu 2+ also causes the formation of lattice defects, in general leading to luminescence quenching.

For preparing BaFCl:Eu nanostructures, only few synthesis methods have been reported before. [START_REF] Wang | [END_REF]8,14] Ju et al., [14] for instance, have developed a mild-temperature solvothermal method using a mixture of water and ethanol for synthesizing nanocrystalline BaFCl:Eu 3+ . [14] Nonetheless, the obtained nanoflakes present 100-200 nm in diameter, 20-40 nm in thickness, and would require an additional reducing sintering step for reaching the divalent oxidation state of europium. Another interesting strategy is the BaFCl:Eu 2+ NP precipitation from an aqueous solution of BaCl 2 , EuCl 3 and NH 4 F. [START_REF] Wang | [END_REF]8] In this case, reducing the europium ions from the trivalent to the divalent oxidation states was achieved by passing the EuCl 3 solution through a zinc column, before reacting with BaCl 2 and NH 4 F for the formation of BaFCl:Eu 2+ . [7, 8] The disadvantage in this case is the rather large particle size of 200-300 nm of the product and the nitrogen working atmosphere. Very recently, BaFCl:Eu 2+ nanofibers were prepared using a mixture of DMF and water, but a quite high annealing temperatures (600°C) as well as low heating and cooling rate is required in this case. [15] To overcome these obstacles, new and fast synthesis routes are required for doping divalent europium within nanoscaled host materials. A possible solution for this impasse is offered here by ionic liquid (IL)-assisted synthesis methods. 16 Moreover, another remarkable advantage of our preparation method is that suitable ionic liquids can also be used as fluorine source. 16 In general, the proper preparation of high quality fluoride materials even in bulk sizes needs the utilization of elemental fluorine or HF as starting compounds, [17][18][19] which are rather difficult and inconvenient methods. With the use of fluoride containing ionic liquids as solvent and also as starting compound, [20][21] several fluoride materials not limited to phosphors are expected to be accessible via our method as we very recently reported in [16] using BmimPF 6 (Bmin = 1-Butyl-3-methylimidazolium hexafluorophosphate) as fluoride ion source. In general, fluoridic materials are extremely advantageous for many applications due to their high ionicity and stability. Ionic liquids are defined as salts, which are liquid below 100°C under normal atmospheric conditions. [22] Contrary to most conventional solvents and besides being reductively stable against many ions, including Eu 2+ or even metallic lithium, ILs present several additional advantages. Some of these advantages are the tunability of the solvent properties according to the cation or anion composition, low vapor pressure, high polarity, low gas solubility and their simultaneous application as solvent and reactant. [23][24][START_REF] Anderson | Ionic liquids in Synthesis[END_REF] While the low vapor pressure allows performing reactions at high temperatures e.g. above 200 °C, the high polarity is responsible for the enhanced solubility of reactants. Furthermore, the adhesion of ILs on the NP surface is responsible for the stabilization of the synthesized products, allowing the generation of small particles and avoiding the formation of agglomerates. [23,24] Most important, the formed compounds, which are normally unstable under atmospheric conditions e.g. Eu 2+ -containing NPs, are protected by the IL against reaction with air or moisture, because of their low gas solubility [21,[START_REF] Freudenmann | [END_REF] It can be also assumed that the strong attachment of ILs at the particles surfaces prevents quenching general observed due to water molecules attached to the surface of nanoparticles.

In this work, two different, novel routes have been developed to generate Eu 2+ -doped BaFCl NPs. Both syntheses were performed in the ionic liquid 1-Butyl-3-methylimidazolium tetrafluoroborate (BmimBF 4 ). As explained above, BmimBF 4 fulfils multiple excellent functions in these syntheses as stabilizing the particle surface, acting as fluorine source for the reactions as well as coordinating the Eu 2+ ions and stabilizing their oxidation state. Moreover, it is shown, that Eu 2+ containing compounds can be used directly as starting materials, alternatively, we have also shown that Eu 2+ ions can be smartly generated during the reaction using EuCl 3 and Eu metal in an appropriate ratio. The outstanding stability of Eu 2+ ions in ILs, which has been demonstrated before, [27] allows the direct precipitation of Eu 2+ -NPs under atmospheric conditions, as described in this work for the first time.

RESULTS AND DISCUSSION

As explained in detail in the experimental section, the synthesis in ionic liquids allowed the flexible preparation of undoped (1) and Eu 2+ -doped (2, 3) BaFCl NPs applying different sonochemical approaches as well as BaFCl:Eu 2+ (4) using a microwave technique (Table 1) using the direct addition of EuSO 4 (2, 3) or the in situ generation of the dopant (4). X-ray diffraction (XRD) analysis was applied to confirm the formation of the crystal structure of the synthesized compounds 1-4. Figure S1 shows the match of the diffraction patterns measured for the products of both synthesis methods with the calculated pattern of BaFCl. [28] The good agreement with the calculated patterns also indicate no significant change of the host lattices due to the doping process with divalent europium ions, similar to the results previously reported in the literature. [START_REF] Wang | [END_REF] Barium fluoride chloride nanoparticles doped with divalent europium prepared via the sonochemical approach (2) present a relatively narrow size distribution between ca. 20 and 90 nm, where 80% of the particles are smaller than 40 nm determined by dynamic light scattering (Fig. S2, red curve), comparable with the transmission electron microscope images of 3 (Fig. 1). In contrast, BaFCl:Eu 2+ NPs prepared via the microwave-assisted technique (4) are characterized by a broader size distribution in the 90-500 nm range with 70% of the particles at about 100 nm (Fig. S2, black curve). This larger particle size is probably explained by the high temperatures reached with the microwave approach, causing the formation of agglomerates. Because of the improved particle size and particle size distribution, the BaFCl:Eu 2+ NPs produced by the sonochemical techniques were selected for the further characterization of the optical properties described below.

In the examples shown in the Figure 2 2a) originated by the ionic liquid, BminBF 4 , while Eu 2+ -doped barium fluoride chloride particles in BmimBF 4 emit efficient bluish white light under UV irradiation (Fig. 2b). The blue emission of BmimBF 4 is most probably explained the presence of C-dot in the ionic liquid. [29][30] They are generated by electrooxidation due to presence of water in BmimBF 4 and their photoluminescence is characterized by the broad emission band ( ex  ~= 28570 cm -1 , blue curve, Fig. S4), extended between approximately 15000 and 28000 cm -1 with maximum at 23255 cm -1 and full width at half maximum (FWHM) of 6075 cm -1 . On the other hand, the excitation spectrum of BminBF 4 solution ( em  ~= 23255 cm -1 , black curve, Fig. S4) consists of a broad band distributed for energies above 24000 cm -1 , overlapping with the respective absorption spectrum.

Emission spectra of 2 at room temperature are composed mainly by one intense broad band in the UV-blue range, when excited by different energies e.g. 28570 cm -1 and 30300 cm -1 (Fig. 2a). This emission band is distributed between 24000 cm -1 and 28000 cm -1 , peaking at 26380 cm -1 , with FWHM of 1756 cm -1 and assigned to the 4f 6 5d 1 →4f 7 electronic transition of Eu 2+ ions. [START_REF] Wang | [END_REF] This Eu 2+ 4f 6 5d 1 →4f 7 emission is very similar to the one in the bulk material, [31] however the intense shoulder detected at lower energy in the emission band of bulk material and attributed to Eu 2+ luminescence perturbed by an impurity is not observed in this nanomaterial. [31] As observed in Figure 3b, the excitation spectrum of 2 consists of a broad band in the UV spectral range, rising at about 27000 cm -1 , with maximum at 36300 cm -1 . Due to the low resolution of the excitation band, the Stokes Shift can only be only estimated to about 2500 cm -1 with the assumption that the lowest 5d 1 4f 6 ( 7 F 0 ) state is located at about 2/3 of the first excitation maximum. Several excitation energies (e.g. 25380 cm -1 ) were probed to detect Eu 3+ based emission without any success in all cases, which confirms that Eu 2+ is not oxidized during the synthesis. [32] The recorded emission spectra allowed to estimate the Commission internationale de l'éclairage (CIE) 1931 color coordinates of C dots in BminBF 4 (1, x = 0.1942, y = 0.2009) and BaFCl:Eu 2+ (2, x = 0.2858, y = 0.3580) (Fig. S5), in accordance to the emission color shown in Figure 2. The bluish white emission observed for BaFCl:Eu 2+ NPs are most probably caused by the combination between the high energy band located at 24000-28000 cm -1 , assigned to Eu 2+ transitions within the BaFCl lattice and the weak low-energy band at 16000-24000 cm -1 , assigned to the interaction between the Eu 2+ ions and the ionic liquid coating layer present in the surface of the NPs, as discussed in detail below. This resulting bluish-white light emission is especially advantageous for the production of LEDs for lighting applications. Cooling sample 2 down to 10 K leads to a slightly narrower emission peak (FWHM = 1198 cm -1 , Fig. S6) in comparison to measurements at room temperature. As presented in Figure 4, the emission intensity gradually decreases with increasing temperature, caused by loss of energy by means of non-radiative transitions, also called thermal quenching. Similarly, the intensity of the emission spectra of 2 continuously decreases to about 70% at 650 K of its room temperature value (Fig. S7).

The room temperature decay time determination of 2 shows a double exponential behavior, which can be fitted to τ 1 = 82 ns and τ 2 = 333 ns when the detection is set at 26370 cm -1 (380 nm) (see Figure S8). These lifetimes are, in general, very fast for Eu 2+ based f ↔ d transitions, especially if compared to those of the bulk materials (3300 ns). [31] However, different arguments can explain these values, namely i) in nanoparticles, the boundary effect improves the electron photon coupling leading to additional quenching effects, ii) the ionic liquid attached at the surface of the nanoparticles also exhibits luminescence at 380 nm via π -π* transitions which are extremely fast and may cause energy transfer Eu 2+ -ILs (most probably the short lifetime τ 1 = 82 ns is attributed to this procedure), [33] iii) the change of refractive index between BaFCl: Eu 2+ and BmimBF 4 could also have an incidence on the life time measurements [34] , and iv) the higher surface to volume ratio in nanoparticles usually involves increases of the surface defects concentrations leading to shorter decay time via non radiative relaxation processes. However, in our opinion the surface defects in this case is limited due to the shell layer of ionic liquid attached on the surface. In contrast to BaFBr:Eu 2+ nano-cuboids, nearly no emission coming from color centers is detected for our nanoparticles which tends in favor of our hypothesis. [35] In order to study more in detail the thermal stability of the BmimBF 4 layer attached on the surface, temperature dependent IR (Fig. 5), Raman and DSC-TG (Fig. 6) measurements were performed. Ba-F and Ba-Cl vibrations situated below 600 cm -1 cannot be detected due to the resolution of our IR device, however it can be clearly seen that going from room temperature to 270 °C, the characteristic vibrations of BmimBF 4 [36] are still detectable even if the intensity of the peak has been decreased. This Ba-X vibrations have been therefore, additionally detected by Raman spectroscopy (Fig. S9). [37] DSC and TG show that the loss of weight starts at around 280 °C (which fits to the measurements done for the raw BmimBF 4 ) [38] and at 400°C the total weight loss corresponds to 7-8%, which should represent the amount of the IL shell layer in our nanoparticles. The slight gain in mass at the beginning of the heating process consists of an artefact from the thermogravimetry device, most probably caused by the gas flux, not stable at the beginning of the measurement. These results show that the ionic liquid is strongly attached to the surface and hence, these nanoparticles can be seen as core-shell nanoparticles.

Conclusion

In conclusion, the direct precipitation of Eu 2+ -doped nanoparticles from Eu 2+ -starting materials as well as the direct synthesis from Eu 3+ starting materials and Eu metals was reached under atmospheric conditions by two different synthesis approaches, reported in this work using ionic liquids for the first time. For this purpose, the outstanding chemical and physical properties of ionic liquids were explored for stabilizing the oxidation state of divalent europium. The two different applied synthesis techniques were based on an ultrasound-and on a microwave assisted method. Moreover, the application of suitable ionic liquids as enhances the safety of the reaction procedure comparing to other fluorine sources e.g. hydrofluoric acid. In general, other imidazolium-based ionic liquids are also very promising since their reducing character is originated by the imidazolium unit. [39] In principle, it is also expected that this synthesis method also work with other alkyl substitution chain and varying the alkyl chain length might allow tuning the particle size. However, this approach is not explored in our work, because increasing the alkyl chain increases the viscosity and decrease the solubility of BaCl 2 . Finally, to the best of our knowledge, the here presented synthetic approach for nanomaterials doped with Eu 2+ is reported for the first time and we hope this work will open a high way for a new type of Eu 2+ based inorganic material synthesis, needed for the future WLED.

Experimental Section

As described above, the ionic-liquid based synthesis allowed a flexible preparation of undoped and doped BaFCl NPs applying different approaches. For undoped BaFCl (1), 0. were performed with a Biorad Excalibur Instrument equipped with a Specac Golden Gate heatable ATR setup. The spectral resolution was set to 1 cm -1 and the spectra were recorded between 500 and 4000 cm -1 . The sample was heated in steps of 10 °C from 30 °C to 280 °C.

Differential scanning calorimetry data were measured using a NETZSCH STA449 F3 instrument. The measurements were performed under an inert atmosphere of nitrogen with a purge rate of 20 mL/min. The sample was placed in Al 2 O 3 crucible covered with a lid to prevent exposure to atmosphere while mounting. Heating rate was set to 10°C/min.

Powder samples collected after centrifugation was dispersed in ethanol and few drops of the dispersion were drop casted on a carbon-coated copper grid (Formar Carbon film, 200 Mesh).

  , undoped (1) and doped (2) barium fluoride chloride nanoparticles are stably suspended in BminBF 4 . Under day light, both suspensions are colorless. Accordingly, as shown by the reflection spectrum of undoped BaFCl NPs (Fig. S 3), this compound presents reflectance values above 97% in the visible range and therefore nearly no efficient absorbance in this spectral region. The reflectance decreases in the UV spectral range, with minimum at 47040 cm -1 and an onset at about 42000 cm -1 , allowing to estimate the bandgap energy of approximately 5.2 eV. The solution of undoped BaFCl particles dispersed in BmimBF 4 displays weak dark blue emission upon excitation with UV light (Fig.

  suitable fluoride source avoiding the usage of elemental fluorine or HF for Eu 2+ doped fluoride nanoparticles is demonstrated. Samples prepared by the sonochemical approach present a narrower size distribution in comparison to the microwave-based ones. Emission spectra of BaFCl:Eu 2+ NPs are composed by one main broad band with maximum at 26376 cm -1 and assigned to the 4f 6 5d 1 →4f 7 electronic transition of Eu 2+ . This method of using ILs as fluorine source represents an extremely promising alternative synthesis technique for other nanoscaled Eu 2+ -doped and undoped compounds, especially fluorides, probably leading to a renaissance of fluoride functional materials. The protective shell layer of ionic liquid attached on the surface of the nanoparticles is stable up to 280 °C, represents around 7-8 % of the total weight and should prevent the NPs from oxidation. Applying ionic liquids e.g. BminBF 4

Figure 1 .

 1 Figure 1. Transmission electron microscope (TEM) images of BaFCl:Eu 2+ NPs (3).

Figure 2 .

 2 Figure 2. In BmimBF 4 suspended nanoparticles of a) 1 and b) 2 under UV and daylight.

Figure 3 .

 3 Figure 3. a) Emission ( ex  ~= 28570 cm -1 in red, ex  ~= 30300 cm -1 in black) and b) excitation

Figure 4 .

 4 Figure 4. Emission spectra of 2 at low temperatures ( ex  ~= 33 303 cm -1 ).

Figure 5 .

 5 Figure 5. Top: Temperature dependent IR spectra of the synthesized BaFCl:Eu 2+ NPs by ultrasound approach (3). Bottom: Comparison of IR spectra of 3 recorded at 40 K and 280 K.

Figure 6 .

 6 Figure 6. DSC and TG measurements of the synthesized BaFCl:Eu 2+ NPs by ultrasound approach (3).

Figure S3 .

 S3 Figure S3. Reflection spectrum of undoped BaFCl (1).

Figure S4 .

 S4 Figure S4. Excitation (λ em = 430 nm, black curve), emission (λ ex = 350 nm, blue curve) and absorption spectra of BminBF 4 .

Figure S5 .

 S5 Figure S5. Plot of Commission internationale de l'éclairage (CIE) 1931 chromaticity diagram for BaFCl:Eu 2+ (2, •, x = 0.2858, y = 0.3580, λ ex = 340 nm) and for BminBF 4 (■, x = 0.1942, y = 0.2009, λ ex = 350 nm).

Figure S6 .

 S6 Figure S6. Excitation (λ em = 380 nm, red curve) and emission spectra at λ ex = 300 nm (blue curve), λ ex = 330 nm (black curve) and λ ex = 345 nm (green curve) of BaFCl:Eu 2+ NPs (2).

Figure S7 .

 S7 Figure S7. Temperature dependence of the emission intensity of BaFCl:Eu 2+ (2) at 26378 cm -1 (379 nm, λ ex = 300 nm).

Figure S8 .

 S8 Figure S8. Room temperature decay time measurement of 2 with λ ex =33333 cm -1 and λ em = 26315 cm -1 (R 2 =0.98) for the 4f 6 5d 1 →4f 7 electronic transition of Eu 2+ .

  The oxidation state 2+ of the europium ions was achieved by means of the redox interactions between EuCl 3 and Eu metal. Subsequently, for all samples the IL was removed by a large extent through centrifuging and washing with ethanol. Both ultrasound-and microwave-assisted approaches were carried out in air. The composition and preparation methods of the different BaFCl NPs are summarized in Table1.

	1 g of barium chloride (Merck KGaA) was dispersed in 2 mL BminBF 4 (IoLiTec Ionic Liquids Technologies GmbH) and sonified (3 W) for 3 min with an ultrasound tip. After sonication, the product was washed several times with ethanol and dried at 70 °C for 1 h. BaFCl:Eu 2+ (2) was prepared under the same conditions as 1, adding, however, 1 mol% of EuSO 4 to the BminBF 4 solution. Europium(II) sulphate was prepared by the reduction of europium(III) oxide (ChemPur Feinchemikalien und Forschungsbedarf GmbH, 99.9%) with zinc in hydrochloric acid (Fisher Scientific GmbH) under argon atmosphere, followed by precipitation of the sulphate with 0.1 M sulphuric acid (Th. Geyer GmbH & Co. KG). In addition, BaFCl:Eu 2+ NPs (3) could be also with prepared with the same solution composition as 2, but sonifying the solution for 30 minutes in an ultrasound bath (Branson 1210) and using EuBr 2 (99.99%, Sigma-Aldrich) instead of EuSO 4 . Applying the microwave-assisted approach for synthesizing BaFCl:Eu 2+ NPs (4), BaCl 2 (Merck KGaA, 0.05 g) was dispersed in 2 mL BminBF 4 (IoLiTec Ionic Liquids Technologies GmbH) this time with 1 mol% of EuCl 3 (1.24.10 -3 g) and Eu metal pieces (7.3 10 -4 g) (ChemPur Feinchemikalien und Forschungsbedarf GmbH, 99.9%) in an appropriate ratio, being heated afterwards in a household microwave six times for 30 seconds, cooling down to room temperature between the single microwave treatments. Europium(III) chloride was previously prepared by solving Eu 2 O 3 (ChemPur Feinchemikalien und Forschungsbedarf GmbH, 99.9%) in hydrochloric acid quartz ampoules were carried out with a Jobin-Yvon fluorescence spectrometer (Fluorolog 3) equipped with two 0.22 m double monochromators (SPEX, 1680) and a 450 W Xenon lamp. Cooling down to 10 K was achieved by a closed cycle He cryostat (Janis Research). Measurements at high temperatures (up to 600 K) were carried out with a Jobin-Yvon fluorescence spectrometer (FluoroMax), also equipped with two monochromators and a and drying. Photoluminescence investigations at room and low temperatures on solid samples sealed in 150 W xenon lamp, combined with an attached self-built oven. All emission spectra were

corrected for photomultiplier sensitivity, the excitation spectra for lamp intensity and both for the transmission of the monochromators. Powder X-ray diffraction was measured on a D5000 X-ray diffractometer (Siemens) operating at 40 kV, 30 mA with Cu-Kα radiation (λ = 0.154178 nm). Dynamic light scattering was measured with aid of a DynaSizer Analysette 12 (Fritsch GmbH). Temperature dependent FTIR experiments on BaFCl:Eu 2+

Table 1 .

 1 Denomination and composition of synthesized BaFCl NPs.

	Compound	Composition	Synthesis	Reaction time	Europium
			method		Source
	1	BaFCl	Ultrasound tip	3 min	-
	2	BaFCl:Eu 2+	Ultrasound tip	3 min	EuSO 4
	3	BaFCl:Eu 2+	Ultrasound bath	30 min	EuBr 2
	4	BaFCl:Eu 2+	Microwave	3 min	Eu + EuCl 3

Figure S9. Raman spectroscopy result of compound 2.

Acknowledgements

The authors thank the German Research Foundation (SPP 1708 "Material Synthesis near Room Temperature") for financial support.

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff))

TEM images were taken with a FEI Tecnai G2 Sphera with 100 keV electron beams focused on the sample.

The table of contents entry should be 50-60 words long:

Here, the formation of Eu 3+ during the synthesis of Eu 2+ phosphors is prevented in the relative redox stable ionic liquid BmimBF 4 and allows the direct precipitation of NPs from Eu 2+ starting materials. Eu 2+ nanoparticles (NPs) e.g. BaFCl:Eu 2+ have been prepared by means of two different synthetic routes in ILs, in air for the first time.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author. Contents: X-ray powder diffractograms, dynamic light scattering measurements, additional (temperature-dependent) reflection, absorption, excitation and emission spectra, CIE 1931 chromaticity diagram and lifetime measurements. 2) prepared using the ultrasound tip as well as c) BaFCl:Eu 2+ (3) synthesized using the ultrasound bath and d) BaFCl:Eu 2+ (4) prepared using the microwave technique in comparison to e) the calculated diffraction pattern of BaFCl. [1] The impurities (*) in the sample prepared using the ultrasound tip as well as c) BaFCl:Eu 2+ (3) synthesized using the ultrasound bath and d) BaFCl:Eu 2+ (4) prepared using the microwave technique in comparison to e) the calculated diffraction pattern of BaFCl. [1] The impurities (*) in the sample (1) are attributed to undissolved BaCl 2 .


Dr Jacob Olchowka
email: jacob.olchowka@icmcb.cnrs.fr

Hans Hagemann

Dr Teresa Delgado

Claudia Wickleder

Prof H Hagemann

Prof Claudia Wickleder

Influence of ionothermal synthesis using BmimBF 4 as solvent on nanophosphors BaFBr:Eu 2+ photoluminescence

Keywords: Ionothermal synthesis, nano-phosphors, time resolved spectroscopy, fluorides

physical and chemical properties of ionic liquid (IL) to design new materials, however, the interaction of these ILs molecules with optical materials and their effect on the luminescence properties of the materials is poorly known. In this work, it is shown that the imidazolium based BmimBF 4 ionic liquid, used as a solvent during the synthesis, tethers at the surface of the nanoparticles and influences the photoluminescence of the nanophosphor BaFBr:Eu 2+ materials. By time resolved spectroscopy two different emissions are observed for the material synthesized by ionothermal approach, one corresponding to Eu(II) 5d-4f transitions in BaFBr host with a decay time of 843 ns and the other with a much faster decay time to the ionic liquid BmimBF 4 .

The research of new nano-phosphors and more generally of nanostructured inorganic material, which require novel synthetic approaches, are currently a very hot topic in the context of energy saving and expanding nanotechnology. [1][2][START_REF] Mnoyan | Phosphors, Up Conversion Nano Particles, Quantum Dots and Their Applications[END_REF][START_REF] Tauni Dissanayake | [END_REF][5] Among the new original synthesis approaches, ionothermal synthesis becomes unavoidable due to the unique and remarkable properties of the ionic liquids (IL). [6,7] For instance ILs have high dissolving ability, low vapor pressure, good thermal stability and they are good microwave absorbers due to their polar character which makes them gripping solvent media. [8] Furthermore, they can also be used as template to control the size and morphology of the materials, as stabilizing agent to avoid particles agglomeration and also as reactant in order to be (partially) incorporated in the synthesized final material. [9][10][11][12][13][14][START_REF] Wegner | Metal Nanoparticles in Ionic Liquids[END_REF] Additionally, the physical and chemical properties of ionic liquids can be tailored according to the requirements by playing on its chemical composition (nature of cationic and/or anionic part, number, position and/or length of alkyl chain, and some more) in order to design the desired solvent. [START_REF] Ghandi | Green Sustain[END_REF][17][18] Finally, their unique properties can provide reaction conditions that cannot be obtained with traditional solvents and hence, new inorganic and also organicinorganic hybrid materials can be designed.

During the last years, oxides, (oxy)hydroxides, fluorides, chalcogenides or even metal nanoparticles targeting different application areas (e.g. energy storage, photoluminescence, photocatalysis) have been successfully developed by ionothermal synthesis. [19][20][21][22][23][24][25] Most of the time these materials exhibit very interesting properties; for instance, Co(OH) 2 -IL obtained by preparation in BmimBF 4 nearly doubles the specific capacitance and remarkably improves the cycling stability compared to Co(OH) 2 . [21] Another example is TiO 2 synthesized in ionic liquid media, which demonstrates enhanced photocatalytic H 2 generation. [21,26] Concerning luminescent materials, we have recently demonstrated an efficient one-pot microwave and/or sonochemical approach to obtain fluoridic Eu(II) doped nanocrystals using ionic liquid as solvent, template and fluoride source in order to avoid a harmful reactant such as HF or even elemental fluorine. [27,28] We used Eu(II) compounds as starting materials, so that the "reducing" character of BmimBF 4 prevents Eu(II) oxidation during the microwave and/or sonochemical synthesis. The decomposition of BF 4 -liberates fluoride ions which are used as a reactant. [20] During the synthesis, the acidic hydrogen from the imidazolium ring bonds with the highly electronegative halogen to form an IL shell around the barium fluorohalide material. [29] This protective shell prevents nanoparticles agglomeration, its formation is assumed to be favored by the sonochemical effect and the low temperature synthesis. [29] The presence of ionic liquids molecules on the surface can enhance electrochemical properties by allowing faster ion diffusion and easier hydrogen desorption/adsorption processes. [21] Furthermore, it can cause great prospects in catalytic reactions. In contrast the influence of ionic liquids on photoluminescence properties of materials is poorly known. [30][31][32] Moreover, some ionic liquids including BmimBF 4 exhibit visible emission, hence it is well-founded that it can affect the photoluminescence of the core material. [33,34] Through this work we will highlight the application of IL during the synthesis of nanoparticles by investigating the luminescent properties of BaFBr:Eu 2+ nanoparticles synthesized by ionothermal synthesis in detail. BaFBr:Eu 2+ is a well-known phosphor, its luminescence was widely studied and this material is used as detector in imaging plate thanks due its high density (~ 5 g/cm 3 ) for a long time. Its K-edge of 37 keV is well suited for general radiography, and a short lifetime due to the parity allowed transitions of Eu(II) is observed. [35][36][37][38][39] Our results on BaFBr:Eu 2+ nanoparticles will lead, thus, also to commercial impacts due to improving of spatial resolution compared to the bulk BaFBr:Eu 2+ phosphors currently used for imaging plates. [38][39][40] In this paper, transmission electron microscopy images of BaFBr:Eu 2+ synthesized by ionothermal approach show that the particles are platelet-like and tend to agglomerate (see Figure 1a and Figure S2). These agglomerates (typical size of a couple of micrometers) are composed by polydispersed nanocrystalline platelets ranging from few tens of nanometers to some hundreds of nanometers length (see Figure 1b).

Fourier transformed infrared measurements of BaFBr:Eu 2+ and EDX analyses confirmed the presence of ionic liquid (BmimBF 4 ) attached to the surface of the material, even after several washing steps with water and ethanol (see Figure 2 and Figure S3). Compared to the spectrum of pure BmimBF 4 , the vibrations, which belong to its alkyl C-H stretching (below 3000 cm -1 ), remain unchanged and those attributed to the ring C-H stretching (3121 and 3161 cm -1 ) slightly shift to lower frequency. [41] This can be explained by the fact, that acidic hydrogen from imidazolium bonds to electronegative halides. [29] On the other hand, the vibrations attributed to BF 4 stretching are shifted to higher frequency in the case of the nanoparticles.

This can be interpreted by an increase of the B-F covalency, which is influenced by the interaction between the surface of BaFBr and the ionic liquid. The vibrations corresponding to BaFBr:Eu 2+ cannot be detected by our FTIR since they are localized between 100 and 300 cm - 1 , below our instrumental wavenumber range. [42] However, Raman spectra in the range of 100 and 300 cm -1 confirm the presence of BaFBr, in agreement with the reported Raman spectra for BaFBr (Figure S4). [43] In order to investigate the effects of the IL molecules attached to the BaFBr nanoparticles surface on the photoluminescence (PL) properties, luminescence, decay time and timeresolved spectroscopy measurements were performed. At a first sight, the room temperature PL spectrum is similar to the one observed for bulk BaFBr and nanosized materials obtained by other synthetic approaches. [36,37,42] It exhibits an intense blue emission band peaking around 390 nm (see Figure 3). However, a closer inspection reveals a low intensity tail at longer wavelength (see inset in figure 3). In the literature, impurity band emissions attributed to oxygen defects can be found, however, they are usually situated at longer wavelength and can be strongly visible under 250-260 nm excitation. [42,[44][45][46] In the material synthesized by ionothermal approach, photoluminescence measurements performed under 260 nm excitation did not reveal any change in the emission spectrum (Figure S5) which allow to assume that there is no (detectable) oxygen defects in our samples. In order to have more information about the origin of this tail, decay time measurements were performed. The decay time curve has a mono-exponential behavior fitted to τ 400nm = 843 ns (± 1 ns and R 2 =0.99) when exciting at 266 nm and monitoring the emission at 400 nm. On the other hand, the life time is fitted to approximatively τ 500nm = 6 ns (± 0.2) when the detection is set at 500nm, where still some emission intensity can be detected (see Figure 4 and Figure S6).

These results confirm that there are at least two emissions with different origins. The lifetime of 843 ns is in agreement with the parity allowed 5d-4f transition of Eu(II) and it is also in good accordance to the literature lifetime of BaFBr:Eu 2+ whereas the other decay time of 6 ns is quite too short to assign to divalent europium emission. [37,42,47] For instance, Hesse et al.

found decay time of 770 ns for bulk BaFBr:Eu 2+ (0.1 mol %) and Liang et al. descript a decay time of 600 ns for BaFBr: Eu 2+ (0.92 mol %) micro-platelets. [37,42] In our case, the nanosized character of our samples conjugated to a higher doping concentration does not drop the Eu 2+ 's decay time as it could be expected. On the opposite, it is slightly higher than the decay time values described above. This circumstance can be explained by the fact that the ionic liquid shell prevents the formation of nanoparticle surface defects by a rather large amount, which would lead to quenching effects of the PL and decrease the decay time. Moreover, it can be assumed, that the IL shell also prevent quenching due to e.g. water or alcoholic molecules attached to the surface if they are used as solvents.

Time resolved spectroscopy measurements also confirm the origin of the two different emissions. With a detection gate of 53-1800 ns, the typical blue emission of Eu(II) in BaFBr is observed whereas with a gate of 0-27 ns, where no Eu(II) emission is available, another broad emission band peaking at 425 nm is detected (see Figure 5). This very fast emission is comparable to the one of pure BmimBF 4 in the inset Figure 5. The comparison shows that even if the emission of pure the ionic liquid (black curve) is slightly shifted (ΔE= 540 cm -1 ) to longer wavelengths and has a longer tail, the two emission bands are similar both in energy position and in shape. Hence, it is reasonable to attribute the second emission to BmimBF 4 .

The interactions between the ionic liquid and the surface of the barium fluorobromide most probably induce a shift of the emission band and explain the differences between the emission of the ionic liquid grafted on the BaFBr surface and the pure BmimBF 4 . Additionally, the lifetime of few nanoseconds fits well with the reported one for π-π* transitions from imidazolium cation in BmimBF 4 . [33] The very short lifetime conjugated to the energy position of the second emission band peaking at 420 nm exclude the possibility that oxygen centres (F centres), which are often found in BaFBr, are responsible for the second emission band. [37,44,46,48] The PL emission originating from these defects (F centres) is situated at longer wavelengths and has a life time in the range of microseconds. [37] In summary, our spectroscopic experiments demonstrate that two distinct emission bands could be detected for BaFBr:Eu 2+ nanoparticles synthesized by ionothermal method. In detail, a blue emission band is observed assigned to the 5d→4f transition of Eu(II) ions doped in the BaFBr host, while the other could be attributed to BmimBF 4 attached on the material surface.

This led to the conclusion that the ionic liquid shell formed during the synthesis also shows light emission and hence, influences the photoluminescence of the phosphor material. We believe that these results do not concern only this specific example but that they can most probably be extended to several phosphors synthesized using ionic liquids as solvents and improve their properties. The intimate contact between ionic liquid and inorganic phosphor can thus contribute in general to decrease quenching processes for nanoparticles, this is a very important point in our opinion.

Figures:

  Figure1a) and b). TEM images of BaFBr:Eu 2+ synthesised by ionothermal approach assisted by ultrasound in BmimBF 4 ionic liquid media.

Figure 2 .

 2 Figure 2. Comparison between FTIR spectra of BaFBr:Eu 2+ synthetized by ionothermal method and pure BmimBF 4 ionic liquid.

Figure 3 .Figure 4 .

 34 Figure 3. Room temperature excitation (black) and emission (red) spectra of BaFBr:Eu 2+ (doping concentration 2 mol %)

Figure 5 .

 5 Figure 5. Time resolved luminescence measurements for BaFBr:Eu 2+ . Red curve represents the emission with a detection gate 0-27 ns and green curve with a gate 53-1800 ns.The inset shows the comparison of PL between emission measured with the gate 0-27 ns and emission of solely BmimBF 4 .

Acknowledgements

This work was funded by the German Science Foundation (Priority program SPP 1708 "Material Synthesis near Room Temperature") and by SNF (Project number: 200021_169033). The authors are thankful to Professor Andreas Hauser for his help with time resolved spectroscopy measurements.

ToC: In this work, an effort is made to understand the improved effect of ionothermal synthesis on the photoluminescence properties of nanophosphors.


M Gey

Humbert

may come

Advanced parent reconstruction : an efficient tool to optimize TMTs by a better control of the parent and subsequent inherited microtexture L. Germain 1,2 , N. Gey 1,2 and M. Humbert 1 1 LEM3, University of Lorraine, Metz France. 2 Laboratory of Excellence for Design of Alloy Metals for Lowmass Structures ('DAMAS' Labex), University de Lorraine, France

Optimizing the microstructure in a material which undergo a phase transformation during the thermomechanical treatment requires the control of both the parent and the child microstructure. In this framework, the reconstructions of the parent microtexture from EBSD maps measured on the transformation product have been proven to be very useful. Indeed, in many cases, such a method gives access to essential characteristics of the parent microstructure: Crystallographic texture, grain size, local orientation spread (with an angular resolution of 1.5-3°), deformation mechanisms, deformed/recrystallized fraction... [1] In this presentation, the main principles of our reconstruction tool 'MERENGUE 2' and its benefits are recalled [2] Merengue2, Software, http://lionelgermain.free.fr

  [2]. Several application examples are presented, especially in Titanium and in martensitic and bainitic steels. Finally the simultaneous knowledge of the parent and child microtextures allows a finer analysis of the transformation mechanisms. In this framework 'DECRYPT' software (Direct Evaluation of CRYstallographic Phase Transformation) has been developed. It is dedicated to analyzing the orientation data inherited by OR-based phase transformations. It is illustrated on different application examples to assess variant selection mechanisms at a local scale: variant clustering in Steels, selective grain boundary precipitation in Titanium... [1] L. Germain, N. Gey, R. Mercier, P. Blaineau & M. Humbert, Acta Mater. (2012) 60, 4551-4562.