Glen M Hocky

Daniele Coslovich
email: daniele.coslovich@univ-montp2.fr

Atsushi Ikeda
email: atsushi.ikeda@univ-montp2.fr

David R Reichman

Correlation of local order with particle mobility in supercooled liquids is highly system dependent

We investigate the connection between local structure and dynamical heterogeneity in supercooled liquids. Through the study of four different models we show that the correlation between a particle's mobility and the degree of local order in nearby regions is highly system dependent. Our results suggest that the correlation between local structure and dynamics is weak or absent in systems that conform well to the mean-field picture of glassy dynamics and strong in those that deviate from this paradigm. Finally, we investigate the role of order-agnostic point-to-set correlations and reveal that they provide similar information content to local structure measures, at least in the system where local order is most pronounced.

Supercooled liquids display markedly heterogeneous dynamics, despite possessing structural properties that appear nearly unchanged from those of normal liquids from which they are prepared [1]. While there has been intense focus on understanding dynamical heterogeneity in a wide variety of systems, the structural origin of this phenomenon is not well understood [2,3]. Simulations of model supercooled liquids are useful for understanding the connections between structure and dynamics because particle locations may be followed precisely for all times. Nonetheless, new theoretical tools are needed to filter out extraneous detail from the key structural and dynamical fluctuations in glassy systems.

One particularly useful simulation-based tool for quantifying the influence of structure on dynamics is the isoconfigurational ensemble, where a large number of molecular dynamics (MD) simulations are initiated from the same starting configuration with momenta sampled randomly from a Boltzmann distribution [4,5]. Under glassy conditions, spatial heterogeneities are immediately evident in the isoconfigurational displacement (or propensity) field. A reasonable hypothesis is that particles with low propensity have a larger measure of local structural stability. Surprisingly, however, simple structural quantities, such as free volume and local potential energy, show little correlation with the heterogeneity of the propensity field [6]. In some models, localized soft modes [7][8][9] or unstable modes [10] appear to correlate strongly with propensity, but the degree of universality of this connection has not been thoroughly investigated.

Recently, focus has turned to the study of specific structural motifs and their putative connection with the dynamics of supercooled liquids. The notion that the frustration of local order incommensurate with bulk crystalline periodicity may be related to glass formation is an old one [11][12][13][14]. New evidence for the growth of domains associated with local packing motifs has been presented for several simple [15] and realistic model systems [16],

where particles tend to be found in certain "locally preferred structures" (LPS) with increased supercooling. As a general rule, more fragile systems display a more rapid increase in LPS concentration and domain extent [15,16]. In some systems, a correlation between the size and location of LPS and slow dynamics has been observed [17,18], although the quantitative meaning of the correlations observed remains, in a statistical sense, obscure.

Point-to-set (PTS) correlations have emerged as an alternative quantifiable metric of amorphous ordering. PTS correlations measure the decrease of configurational entropy imposed by the presence of particles pinned in an equilibrium configuration [19,20]. The length scale associated with PTS correlations has been demonstrated to grow upon increased supercooling in several systems [20][21][22], although its variation is rather modest over the dynamical range currently accessible in simulations [23], Nonetheless, several observations indicate that the growing PTS length scale should ultimately drive the dramatic increase in relaxation times in supercooled liquids [START_REF] Karmakar | Proc. Natl. Acad. Sci[END_REF][START_REF] Biroli | [END_REF][26]. It should be noted that PTS correlations, as well as other recently proposed measures of static correlations [27,28], are "order agnostic" [23] and therefore their growth does not necessarily connect to the emergence of specific local structures, such as those identified in the LPS studies.

In this Letter we quantify the correlation between between static structure and dynamical heterogeneity in supercooled liquids in a statistically precise sense and within a coherent simulation framework. We demonstrate that seemingly similar systems may differ dramatically with respect to the degree to which specific local structural motifs correlate with dynamics. Our results indicate that scenarios connecting LPS cluster formation and glassy behavior [START_REF] Tanaka | [END_REF]31] cannot be generically correct. The observed model dependence suggests instead that local structural quantities play a key role in systems with large deviations from mean-field glassy behavior. Lastly, we show that a connection exists between growing PTS correlations and LPS in systems where LPS are strongly predictive of dynamical heterogeneity.

The first two models we will study are binary Lennard-Jones mixtures, namely the Kob-Andersen (KA) system [32] and the Wahnström (Wahn) system [33]. The definition of these models and their LPS statistics have been extensively detailed in Ref. 34. The KA system is an 80:20 mixture while the Wahn system is equimolar. As a third system, we study a binary mixture of harmonic spheres (Harm) at a density such that that jamming is approached by lowering temperature near to zero (ρ = 0.675) [35]. In all cases, one species is smaller (B for KA and Wahn, A for Harm) and is intrinsically more mobile. Results for the small particles will be reported in the main text and for the large particles in the Supplemental Information (SI) [36]. In the following, we will discuss all quantities using standard reduced units. For all three systems, we study systems with N = 1000. Further simulation details and a description of the LPS in each system can be found in the SI.

While all of the above models are simple binary mixtures with short-ranged interaction potentials, their local structures differ significantly. For each model, we identify particles participating in LPS through a Voronoi analysis (see SI). These LPS correspond to icosahedra, bicapped square anti-prisms, and distorted icosahedral structures in Wahn, KA, and Harm respectively. The relative abundance of these LPS at low temperatures is model dependent: it is fairly significant in the Wahn model (about 10% of the particles are at the center of a LPS) and weaker in the other models. In the KA model this is due to the fact that twisted bicapped prisms are mostly centered around the small particles, which constitute the minority species.

In order to investigate the connection between the LPS in each system and dynamical behavior, we perform simulations in the isoconfigurational ensemble at supercooled temperatures, T = 0.588 for the Wahn, T = 0.45 for the KA and T = 5.5 for the Harm. These temperatures correspond roughly to the same degree of supercooling, as measured by the relative distance (about 3-6%) from their fitted Mode-Coupling temperatures, T c (see SI).

We select 40 ( 20) equilibrated configurations for the KA and Wahn (Harm) systems and perform the Voronoi analysis as discussed above. For each configuration, we performed 200 (100) N V T simulations in the isoconfigurational ensemble. From the simulations starting from each configuration, we compute the particle mobilities µ i (t) ≡ |r i (t)r i (0)| iso ≡ |δr i (t)| iso . To quantify the number of LPS associated with a given particle, we count the number of structures deemed locally preferred in a spherical region of radius l around each particle (n LPS ). All results reported here are nearly insensitive to this l value in the range we have investigated 1.5 ≤ l ≤ 3.0, and we chose to report results only for l = 2.5.

In Figs. 1(a)-(c) we show the combined probability distribution of µ i and n LPS for the 3 systems introduced above. We quantify correlation by using the Spearman rank correlation coefficient K [START_REF] Sammut | Encyclopedia of machine learning[END_REF], which has been used previously in a similar context [6]. K is 1 if two quantities are related by a monotonically increasing function and -1 if by a decreasing one. K values are shown in the top-right corner of each histogram. We see visually and quantitatively that the correlation is much stronger in the Wahn system than in the KA and Harm systems. For comparison, the probability distributions for correlation between mobility and local energy (E i = (e i + j∈neigh(i) e j )/( 1

1.0 KA B (b) -2 -1 0 log 10 (t/τ α ) precis nLP S |K| nLP S Harm A (c) -1 0 1 log 10 (t/τ α ) precis Ei |K| Ei GCM (d) -2 -1 0 1 log 10 (t/τ α ) FIG.
2. Time dependence of predictability metrics and correlation coefficients as a function of time for four models. Closed symbols correspond to data comparing mobility with nLPS as in the left-hand column of Fig. 1 and open symbol comparing local energy Ei (defined in text) to mobility, as in the right-hand column of that figure. K is the Spearman rank correlation coefficient [START_REF] Sammut | Encyclopedia of machine learning[END_REF]. Precision is defined as the percentage of particles in the the top (bottom) 2% of nLPS (Ei) which are in the bottom 10% of µi. Horizontal dashed lines show the result for the precision metrics that would result from random classification of particles as slow.

much weaker in the Harm system.

Inspection of Fig. 1(a) reveals a long tail in the histogram of n LPS values. From these data, we can predict that a particle in a domain rich in icosahedral structures will be very immobile. Looking at the data at n LPS = 0 instead, we see that such particles will have higher than average mobility. However, slow and fast particles have a wide range of n LPS values. In Fig. 2 we show the level of "precision" [START_REF] Sammut | Encyclopedia of machine learning[END_REF] in predicting which particles are slow based on n LPS and E i for the models studied, as well as the K values. Here, precision is defined as the percentage of particles in the top (bottom) 2% of n LPS (E i ) which are also in the bottom 10% of µ i (δt). All trends discussed are insensitive to the particular percentiles chosen for this definition of precision (see SI for further details, including a discussion of sample-to-sample fluctuations in these quantities).

When viewed from this statistical perspective, several striking features are observed. For the Wahn system, LPS are highly predictive of slow dynamics. In particular, using our definition of precision one may "predict" the location of slow particles with near perfect accuracy up to τ α , and such a correlation continues to grow to the longest times we investigated. In the KA system, local energy is more predictive of slow dynamics than LPS locations, and correlation for both local energy and n LPS in KA and Harm are far lower than in the Wahn system.

We may thus conclude that the correlation between dynamics and local structural metrics such as n LPS is highly system dependent. What may be taken from this dramatic degree of variability? Among the three models studied, the Wahn system shows the largest departures from mean-field behavior. Namely, Wahn exhibits large violation of the Stokes-Einstein relation, sizable deviations from time-temperature superposition, and large inconsistencies between fitted Mode-Coupling exponents [37] (see SI). From this perspective, the KA system shows moderate deviations from mean-field be-havior. This leads us to consider whether the correlation between local structural order and slow dynamics might be connected to how much a model system conforms to the mean-field paradignm. While the Harm system does not uniformly display mean-field behavior, results from finite size studies [38] and the existence of a nonmonotonic dynamical length scale [35] suggest that its behavior is at least partially harmonious with mean-field theory. This leads us to posit a connection between a high degree of local structure-dynamics correlation and strong spatial fluctuations which are manifest in systems that deviate from mean-field behavior.

To better test this notion, we study a fourth system, the high-density (ρ = 2.0) Gaussian Core Model (GCM). The GCM is a single-component fluid with Gaussian repulsions [39,40], which has all the hallmarks of glassy behavior while matching mean-field predictions of dynamical exponents, strongly suppressed non-Gaussian fluctuations and minimal Stokes-Einstein violation [40]. This mean-field behavior seems to arise naturally from the long ranged and ultra-soft interaction potential (see discussion in SI).

In Fig. 2(d) we show results for N = 3456 Gaussian core particles at T = 3.2 with 100 isoconfigurational runs initiated from 20 independent configurations. We note that this temperature is slightly higher, relative to T c = 2.7, than the one used in the other models, but corresponds instead to the same relative increase in relaxation time as observed for the Wahn system (see SI) We found that distorted crystal-like structures constitute the LPS of the model (the underlying stable crystal at the studied density is BCC). In agreement with our expectations, the correlation between n LPS and dynamics in the GCM system is very low, just as in the Harm system, and only marginally improves as t increases [41].

It would be natural to speculate that in systems such as the Harm and GCM models, there simply exists no connection between structure and dynamics. However, this statement is incorrect. We have used the R 4 -ratio analysis of Berthier and Jack [42] to quantify the structural component of the dynamic fluctuations. As detailed in the SI, we found that all four systems analyzed in Fig. 2 show a marked correlation between structure and dynamics, despite the fact that no specific structural motif connects to dynamics in the more mean-field like models. These striking results will be a subject for future investigations. Here we just point out an interesting analogy with the behavior of mean-field p-spin models [43], which do display large values of R 4 close to the dynamic transition.

One may take the inability of specific structural metrics, such as LPS determined from Voronoi analysis, to correlate universally with dynamics in supercooled liquids as an indication that a more general form of growing amorphous order must be implicated. In the remaining of this work we focus on structural correlations embodied in point-to-set and related length scales [21,22,[START_REF] Biroli | [END_REF]26]. In order to show that this type of order may subsume specific structural metrics, we investigate the connection between local order as measured by n LPS and PTS correlations.

The PTS length scale is extracted by calculating the range over which spatial correlations imposed by an equilibrium amorphous spherical boundary decay. We first establish that it is possible to ergodically sample cavities at some R cav using the "Particle Size Annealing" (PSA) method detailed in Ref. 22. In brief, we monitor the overlap q, a measure of the similarity between the initial configuration in the cavity and that at a later time t. The overlap is defined as q(R cav , t) = (ρl 3 Ñ ) -1 Ñ i=1 n i (t)n i (0) where the center of the cavity has been tiled into Ñ = 125 cubes of side length l = 0.36, small enough such that the cell occupancy n i (t) is always zero or one. We use both regular Monte Carlo (MC) sampling and a sampling where the particle diameters are shrunk to 60% of their original size and grown back in, and check that the q values agree at long times. In the limit of large cavity and long time, q(t) will tend to the bulk value q b = ρl 3 and so this value is conventionally subtracted from q.

We carry out these tests for the Wahn model. The strong icosahedral ordering in this model makes it an ideal system to probe the connection between local order and the spatial distribution of the overlap. In Fig. 3(a) we show that for R cav = 3.0, q is sampled ergodically. We then take 30 of the Wahn configurations used earlier and perform two standard MC simulations for cavities centered at 27 positions in each. The longtime overlap value is extracted from each cavity, and the number of icosahedral centers within the cavity as well as the one within the inner R = 2.5 of the cavity, are calculated. We find that high overlap cavities generally have high n LPS and vice versa. This implies that (for the Wahn system) the cavity simulations are mostly probing the same type of local ordering measured by the Voronoi construction, although it does not necessarily mean that the correlation length measured by doing cavity simulations at a series of radii is the same as would be measured by the extent of LPS domains.

In conclusion, we have demonstrated that the correlation between local structural metrics (e.g. E i and n LPS ) and dynamics in supercooled liquids is highly system dependent. In models such as the Wahn mixture, accurate predictions of heterogeneous dynamics may be made on the basis of a single structural marker while essentially no correlation exists in mean-field like systems such as the GCM. However, a strong link between some aspect of static structure and dynamics does exist, as signified by the sizable R 4 -ratio observed in all the systems we have studied. Despite being order agnostic, PTS correlations appear to show a connection with specific types of local order such as Voronoi signatures in systems whose dynamics may be predicted by the location of such structural motifs. Furthermore, previous work has detected an apparent connection between the growth of relaxation times and order agnostic length scales in systems where the connection between relaxation times and spe-cific structural metrics is not very strong [22,[START_REF] Biroli | [END_REF]26]. These facts suggest that PTS correlations may provide a more general description of the key static fluctuations that determine dynamical behavior in supercooled liquids. over 25% of type B particles at low temperatures, while the second most common VS, (0, 2, 8, 1), appears around approximately 9% of B particles. In the KA system, the (0, 2, 8) arrangement (bicapped square antiprism) appears around approximately 10% of small particles and the (1, 2, 5, 3) appear around 8%. It is important to remember that, because the KA system is an 80:20 mixture, the (0, 2, 8) motif is found in only about 2% abundance while more than 10% of Wahn particles are found in (0, 0, 12) configurations. We therefore considered the union of (0, 2, 8) and (1, 2, 5, 3) polyhedra as the LPS of KA model. The analysis carried out in the main text remains qualitatively unchanged if only (0, 2, 8) are considered. In the Harm system, the most frequent signature at low temperature is (0, 2, 8, 2), which may be regarded as a distorted icosahedral structure. The abundance of this LPS is about 5% at the lowest studied temperatures, while the second most frequent signature is (0, 2, 8, 1) has 4%. In the GCM, we found a variety of low symmetry polyhedra, some of which are found in FCC and BCC crystals at finite temperatures [12]. At the studied temperature, (0, 3, 6, 4), (0, 2, 8, 4), and (0, 4, 4, 6) (ideal FCC structure) are the most abundant ones. These crystal-like structures tend to grow slightly by decreasing temperature and their union is considered as the LPS of the model.

Mean-field behavior and pair potential

As mentioned in the body of the text, the Harm and GCM systems seem to embody manifestations of the predictions of mean-field theories of glasses, however they do so in distinct ways. The Harm system shows a well defined crossover between seemingly activation-less and activated regimes as predicted by the mean-field perspective. However, SE violation is still pronounced in the Harm system (see Fig. 4). In this sense, critical fluctuations expected in finite dimensions [13] are still sizable. In the GCM system, the entire range of temperatures that can be simulated displays a near-complete, quantifiable correspondence with high dimensional mean-field behavior, including the absence of Stokes-Einstein violation (despite the GCM system being highly fragile). Both potentials share an ultra-soft core, which prevents the physics from being dominated by harsh short-ranged repulsions. In addition, the GCM potential is (quasi) long-ranged, mimicking the effect of increased dimensionality [14][15][16][17]. Thus ultra-softness may be a necessary condition for triggering aspects of mean-field behavior. However, to further suppress critical-like fluctuations in low dimensions, a (quasi) long-ranged potential, as found in the GCM, is also necessary.

R4 values

In order to quantify the total connection between structure and dynamics we perform the analysis of Berthier and Jack (Ref. [5]). They definite the quantity R 4 (t) which represents the fraction of the total run-to-run variance in a dynamical quantity from analyzed in the isoconfigurational ensemble which is encoded by the initial configurations. Hence a value of zero means all fluctuations come from the randomly selected initial velocities while a value of unity means that all fluctuations in the dynamical quantity are determined by the initial structure. R 4 is defined as,

R 4 (t) = δ 4 (t) χ 4 (t) , (3)
where

δ 4 (t) = N {E[ F (t) 2 iso ] -E[F (t)] 2 } (4) χ 4 (t) = N {E[ F 2 (t) iso ] -E[F (t)] 2 }. (5)
Here, "F" is a collective dynamical quantity, a function averaged over all particles. E represents an equilibrium ensemble average while • • • iso represents the isoconfigurational average over realizations of momenta. Here we choose F (t) to be the self-intermediate scattering function (t) . The relaxation time τ α is defined by F s (k, t ≡ τ α ) = 1/e. For the four systems studied in this work, we report the results of the R 4 analysis at t = τ α in Tab. I. For all four systems, we find values that should be considered large based on prior analyses using this quantity, though as would be expected from our other results and discussion, it is smaller in the Harm and GCM systems than in the two LJ systems.

F s (k, t) = 1 N N i=1 e k•δri

Mobility correlation and predictability

For completeness, in Fig. 5 we present histograms for the GCM data analogous to that of Fig. 1. In Fig. 6 we show the histograms for both the A and B types of particles for the three binary systems presented in Fig. 1. In Fig. 7 we show the results for predictability of the fastest 10% of particles based on the lowest 2% of n LPS or the highest 2% of E i for contrast with Fig. 2. For comparison, the same data for K as in Fig. 2 are also shown.

We show in Fig. 8 how the predictability data is modified by a change in how predictability is defined. For simplicity, we illustrate the point with data for the Wahn B and KA B particles and show how, while the values change, the trends discussed in the main text are robust.

Lastly, we note that there can be large sample-to-sample variations in n LP S observed around particles at the large n LP S end. We note that the percentile thresholds used in this calculation are done on the aggregate set of data, which means that the number of particles exceeding this threshold can vary substantially from sample to sample. As an example, we find in the Wahn system that the number of B particles with n LP S above the 2% threshold (n LP S ≥ 23) varies for our forty samples from zero to eighty out of 500 B particles. Nevertheless, we feel this strengthens our point that for the Wahnstrom system, very large numbers of locally preferred structures in an area is highly predictive of slow dynamics, although in general slow dynamics can also arise via other structural mechanisms, as suggested by our R 4 analysis above.

GCM, δt = τ α -0.17 1. * gmh2123@columbia.edu † daniele.coslovich@univ-montp2.fr ‡ atsushi.ikeda@univ-montp2.fr § drr2103@columbia.edu [1] K. Kim and S. Saito, J. Chem. Phys. 138, 12A506 (2013). 2 except the precision in that figure, the precision of prediction the slowest 10% of particles, has been replaced by the precision of predicting the fastest 10% of particles based on a lack of local stability (high Ei, small nLPS). We see that the trends are all the same as for the "slow precision", however, the "fast precision" is much worse for both nLP S and Ei in the Wahn model, reflecting the wider density at the left and than on the right end in Fig. 1. In contrast, the precision for the KA system does not change much reflecting the more ovular shape of the density in Fig. 1. and KA (open symbols) B particles, with the definition of predictability varied. In the legend, the first symbol represents the cutoff percentile for nLP S and the second number for mobility. Hence the data in the main text is the same as that labeled "2,10". For the case of the KA system, due to the smaller number of particles involved in LPS, the particles in the top five percent in nLP S are precisely those in the second percentile, hence the other set of data lies on top of those shown.

1 FIG. 1 .

 11 FIG. 1. Interpolated histograms of particle mobility. Numbers at top-right indicate Spearman rank correlation coefficients K [29]. The first column shows the correlation between mobility and nLPS with the LPS defined in the text. The second column shows correlation with Ei, the sum of a particle and its neighbors' pair energies. White dotted lines show the average value of the quantity on the horizontal and vertical axes.

  + |neigh(i)|) , where |neigh(i)| is the number of neighbors in the Voronoi structure around particle i) are shown in Figs. 1(d)-(f). The correlation is fairly significant in the two Lennard-Jones mixtures and Wahn B

5 FIG. 3 .

 53 FIG. 3. (a) Cavity overlaps and PSA overlaps (with the value for a bulk system q b subtracted) indicating ergodic sampling at this cavity size and temperature. The dashed line shows the overlap probability distribution P (q-q b ). (b) On average, cavities containing a large number of LPS centers have high overlap. "Inner" points count only LPS which are within R = 2.5 of the center of the cavity. The dashed line shows the average overlap and arrows show the average number of LPS. (c) and (d) show the full data distribution and Spearman rank correlation coefficients [29]. The data in (b) is obtained from (c) and (d) by averaging over vertical slabs of width 3.

FIG. 4 .

 4 FIG.4. Diffusion constants vs. τα (see SI text) for each of the four models. Open symbols represent the temperatures studied in the main body of this work. The diffusion constants D were extracted from the limiting relationship limt→∞ δr 2 i (t) = 6Dt. The values of the diffusion constants and τα are scaled by their value at the approximate onset temperature of glassy behavior, with T0 values used here of 1.0, 1.0, 12 and 5 for the Wahn, KA, Harm and GCM respectively. The dashed line shows the approximate high temperature behavior D ∝ τ -1 .

FIG. 5 .

 5 FIG. 5. Mobility µi and local energy Ei vs. nLPS histogram for the GCM as in Fig. 1.

1 FIG. 6 .FIG. 7 .

 167 FIG.6. Mobility µi vs. nLPS histogram for the three models in Fig.1comparing both species of particles. The larger particle always has a smaller range of dynamical activity (note the differing ranges of vertical axes) and this intrinsically reduces the correlation.

25 FIG. 8 .

 258 FIG.8. Predictability of mobility from nLP S as a function of time, as in Fig.2. Data is shown for the Wahn (closed symbols) and KA (open symbols) B particles, with the definition of predictability varied. In the legend, the first symbol represents the cutoff percentile for nLP S and the second number for mobility. Hence the data in the main text is the same as that labeled "2,10". For the case of the KA system, due to the smaller number of particles involved in LPS, the particles in the top five percent in nLP S are precisely those in the second percentile, hence the other set of data lies on top of those shown.

We thank Mark Ediger and Asaph Widmer-Cooper for stimulating discussions. Simulations were executed in part on the Midway cluster at the University of Chicago's Research Computing Center, and on the seeder cluster of the University of Chicago Computing Cooperative (UC3), supported in part by the Open Science Grid, NSF Grant No. PHY-1148698. LAMMPS [44] simulations were organized and executed using the Swift parallel scripting language, development supported by NSF Grant No. OCI-1148443 [45]. G.M.H. and D.R.R. were supported by the NSF through Grant No. DGE-07-07425 and Grant No. CHE-1213247, respectively.

 I. Predictability ratio results for the four systems studied. See Ref. [5] and SI text for further discussion.

Models, units and relevant temperatures

The Kob-Andersen (KA) [2] and Wahnström (Wahn) [6] models are binary Lennard-Jones mixtures where particles interact through the potential

where α, β = A, B are species indices. For the KA model the interaction parameters are σ AB = 0.8σ AA , σ BB = 0.88σ AA , ǫ AB = 1.5ǫ AA , and ǫ BB = 0.5ǫ AA , while for the Wahn model σ AB = 0.916σ AA , σ BB = 0.833σ AA , and

8 for the KA model and x 1 = x 2 = 0.5 for the Wahn model. The mass ratio m 1 /m 2 is 1 and 2 in KA and Wahn, respectively. The potentials are cut and shifted at 2.5σ αβ . In the main text, σ AA , ǫ AA , and m A σ 2 AA /ǫ AA are used as units of distance, energy, and time, respectively. The Harm model is an equimolar mixture of elastic spheres [7] with equal masses m. The interaction potential is given by

if r αβ < σ αβ and zero otherwise. The interaction parameters are σ AA = 1, σ AB = 1.2 and σ BB = 1.4. As in the KA and Wahn models, σ AA and mσ 2 AA /ǫ are used as units of length and time, respectively. The unit of energy is 10 -4 ǫ. The Gaussian core model [8,9] (GCM) is a one-component fluid of particles interacting via a Gaussian potential

The potential is cut and shifted at r = 5σ. The units of length, energy and time are given by σ, 10 -6 ǫ and mσ 2 /ǫ respectively.

The simulations have been carried out using the LAMMPS package [10]. Integration time steps δt used were 0.004 for the Wahn and KA system, 0.1 for Harm, and 0.4 for the GCM. Temperature was maintained by a Nosé-Hoover thermostat with a time constant of 100 δt in all cases.

The estimated Mode-Coupling critical temperatures of all the models and the wave-vectors k used to compute the structural relaxation times τ α (see section on R 4 values) are reported in Tab. I. The estimated critical temperatures were obtained, in the references cited in in Tab. I, from power law fits to the relaxation times data.

Identification of locally preferred structures

Particles participating in LPS are identified through a Voronoi analysis [11]. To characterize the local structure around a given particle, we determine the number n k of faces having k edges of the Voronoi polyhedron formed by the nearest neighbors. Each particle may therefore be labeled by a Voronoi signature (VS) (n 3 , n 4 , n 5 , ..), where n 3 is the number of triangles, n 4 the number of quadrilaterals, etc. of the corresponding Voronoi polyhedron. As in Ref. 11, we focus on the polyhedra found around small particles (for binary mixtures), whose temperature variation were shown to correlate better with slow dynamics. In the Wahn system, the (0, 0, 12) polyhedron (icosahedron) appears around


Antonio Lorente Mur

Franc ¸oise Peyrin Paul Bataille

Nicolas Ducros

DEEP EXPECTATION-MAXIMIZATION FOR IMAGE RECONSTRUCTION FROM UNDER-SAMPLED POISSON DATA

Keywords: Image reconstruction, deep learning, expectation-maximization, model-based optimization methods, single-pixel imaging

Many biomedical imaging techniques, such as computerized tomography, positron emission tomography, and optical microscopy, involve reconstruction of an image from a sequence of a few linear measurements that are corrupted by Poisson noise. In this study, we focus on computational optics, and more precisely single-pixel imaging, where the set-up acquires some of the coefficients of the Hadamard transform of the image of the scene.

Recently, this problem has benefited from the advent of deep learning. Although deep methods were initially considered as black boxes, they are now understood as learnable optimisation schemes. Here, we propose a network architecture based on the expectation-maximization algorithm that optimizes the maximum a posteriori of the unknown image for measurements corrupted by Poisson noise. This leads to an interpretable network that generalizes several existing approaches.

Finally, we present some reconstruction results from simulated data and from experimental acquisitions from a singlepixel camera. Our network yields higher reconstruction peak signal-to-noise ratios than other similar approaches.

INTRODUCTION

Single-pixel imaging is an extreme configuration of computational optics, where a single point detector is used to recover a two-dimensional image [START_REF] Edgar | Principles and prospects for single-pixel imaging[END_REF]. Since the original work of Duarte and coworkers [START_REF] Duarte | Single-pixel imaging via compressive sampling[END_REF], single-pixel imaging has been successfully applied to fluorescence microscopy [START_REF] Studer | Compressive fluorescence microscopy for biological and hyperspectral imaging[END_REF], hyperspectral imaging [START_REF] Rousset | Time-resolved multispectral imaging based on an adaptive single-pixel camera[END_REF], image-guided surgery [START_REF] Aguénounon | Single snapshot imaging of optical properties using a single-pixel camera: a simulation study[END_REF], fluorescence lifetime imaging [START_REF] Yao | Net-flics: fast quantitative widefield fluorescence lifetime imaging with compressed sensing -a deep learning approach[END_REF], and others. A single-pixel camera measures the dot products between an image and a set of two-dimensional functions that are implemented through a spatial light modulator. To limit acquisition times, it is highly desirable to reduce the number of light patterns and acquisition time, which leads to an under-determined inverse problem with Poisson noise.

Image reconstruction from noisy measurements where the number of unknowns is larger than the number of measurements is a generic problem that has several applications in computational imaging. Such problems have long benefited from the theory of compressed sensing, but recent advances in deep learning have revolutionized the field [START_REF] Arridge | Solving inverse problems using data-driven models[END_REF]. In particular, convolutional neural networks have shown great success for solving inverse problems [START_REF] Mccann | Convolutional neural networks for inverse problems in imaging: A review[END_REF][START_REF] Kang | Cycle-consistent adversarial denoising network for multiphase coronary CT angiography[END_REF].

Traditional optimisation-based approaches for image reconstruction have inspired deep-learning-based reconstructors to solve inverse problems. For instance, they have been used as priors for manifolds of natural images [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF], and as projectors for sets of natural images [START_REF] Gupta | Cnn-based projected gradient descent for consistent ct image reconstruction[END_REF]. While these methods have outperformed traditional image reconstruction methods, they implicitly assume signal-independent noise, and are therefore not optimal for tackling problems with Poisson noise.

Inspired by the unrolled deep-learning methods, such as indicated by [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF], we derive an iterative neural network based on the expectation-maximization (EM) algorithm [START_REF] Fessler | Space-alternating generalized expectation-maximization algorithm[END_REF], to specifically tackle data corrupted by noise with nonconstant variance. This provides a framework for handling signaldependent noise flowing through iterations, which generalizes existing methods.

In Section 2, we model a compressive acquisition device and describe the associated reconstruction problem. In Section 3, we introduce the core ideas for our Bayesian inversion approach to the inverse problem. In Section 4, we propose our EM-based unrolled network to process data corrupted by Poisson noise. In Section 5, we describe how we implement and train the network. In Section 6, we report and analyze our reconstruction results, before concluding in Section 7.

COMPRESSIVE IMAGING

Let f ∈ [0, 1] N be the image to acquire. The main idea of compressive optics is to measure m = H 1 f using hardware, and to recover f using software. The system matrix H 1 ∈ R M ×N , with M < N , collects the patterns that are sequentially uploaded on a spatial light modulator to get the measurement vector. The patterns are traditionally chosen from within a basis H ∈ R N ×N , i.e., H 1 = SH with S = [I M , 0]. Classical choices include Fourier, discrete cosine, wavelets, and Hadamard bases [START_REF] Ochoa | Assessing patterns for compressive fluorescence lifetime imaging[END_REF].

The acquisition, which is corrupted by Poisson noise [START_REF] Foi | Practical poissonian-gaussian noise modeling and fitting for single-image raw-data[END_REF], can be modeled as

m α = ( mα + -mα -)/α, (1a)
mα + ∼ P(αH + 1 f ) and mα -∼ P(αH - 1 f ). ( 1b
)
where α is the image intensity (in photons), and m α is the normalized noisy measurement vector. Note that the implementation of the negative values of H 1 is carried out through the use of positive patterns [START_REF] Lorente | Handling negative patterns for fast single-pixel lifetime imaging[END_REF] for details).

H + 1 and H - 1 , such that (H + 1 ) i,n = max((H 1 ) i,n , 0) and (H + 1 ) i,n = max(-(H 1 ) i,n , 0) (see

BAYESIAN RECONSTRUCTION

Bayesian inversion aims at the computing of a point-wise estimator of the probability density function of (f | m). Assuming that f has finite mean and variance, the conditional expectation is given by the minimum mean squared error

f * = E (f | m α = m α ) = G * (m α ), (2)
where the mapping G * is given by

G * ∈ argmin G E( G(m α ) -f 2 ). (3)
For general distributions of f , the mapping G * is nonlinear, and there are no closed-form solutions. Instead of solving Equation (3), which is computationally intractable in general, deep-learning-based methods replace the expectation by the empirical mean over a database, and optimize a mapping G ω within a family of mappings parameterized by some weights ω. The choice of the form of the nonlinear mapping G ω is central to the reconstruction. Much attention has been devoted to neural network models where the architecture mimics iterative maximum a posteriori (MAP) optimisation schemes [START_REF] Ongie | Deep learning techniques for inverse problems in imaging[END_REF].

PROPOSED METHOD

Unrolling the EM algorithm

We aim to compute the MAP for our problem

argmax f log p(m α |f ) + log p(f ), (4)
where we assume p(

m α |f ) ∝ exp -1 2 H 1 f -m α 2 Σ -1 α
, and Σ α is the covariance of the noise, and p(f ) is an unknown probability density function.

The EM algorithm has commonly been used to estimate the MAP for image reconstruction tasks [START_REF] Zhou | A bayesian map-em algorithm for pet image reconstruction using wavelet transform[END_REF]. This is an iterative algorithm that produces a sequence of estimations {f (k) } that converge to a local maximum of Equation ( 4). Every iteration of the EM algorithm is based on two steps.

The expectation step computes the conditional expectation of the log-likelihood of f with respect to an auxiliary random variable x (commonly called the complete data), given the current estimate f (k) and the measurements m α

Q(f |f (k) ) = p(x|m α , f (k) ) log p(x, m α |f )dx + p(f ) (5)
This quantity is then maximized with respect to f during the maximization step, to produce the next iteration

f (k+1) = argmax f Q(f |f (k) ) (6)
The EM algorithm converges to a local maximum of Equation (4) if the complete data x satisfy some admissibility proprieties. When p(m α |f ) is Gaussian, x can be chosen as a Gaussian vector, such that [START_REF] Fessler | Space-alternating generalized expectation-maximization algorithm[END_REF] x ∼ N (Hf , Σ)

m α |x ∼ N (Sx, Σ α ). (7)
Therefore, x satisfies the admissibility properties if H 1 ΣH 1 = SΣS , where Σ is the covariance of f , and Equation ( 5) simplifies to

x(k) = E(x|m α , f (k) ), (9)
Q(f |f (k) ) = log p( x(k) |f ) + log p(f ). (10)
Using classical properties of Gaussian vectors (see Chapter 5 of [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]), we can rewrite these two steps

x(k) = argmin x Sx -m α 2 Σ -1 α + x -Hf (k) 2 Σ -1 (11a) f (k+1) = argmin f x(k) -Hf 2 Σ -1 -log p(f ). (11b)
As p(f ) is an unknown probability density function, we propose to estimate Equation (11b) using a neural network D ω

x(k) = argmin x Sx -m α 2 Σ -1 α + x -Hf (k) 2 Σ -1 (12a) f (k+1) = D ω (H x(k) ). (12b)
Note that Equation (12a) is commonly called the dataconsistency layer in the literature on deep unrolled methods [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF].

Implementation of the data-consistency layer

Introducing the variable y (k) = x(k) -Hf (k) , the analytical solution of Equation (12a) is given by

y (k) = Σ 1 Σ 21 (Σ α + Σ 1 ) -1 (m α -H 1 f (k) ), (13)
Denoising layers

D ω DC layers D C IT H (16a) (16b) (16c) f (k) f (k+1)
x(k) m α where

Σ 1 ∈ R M ×M , Σ 21 ∈ R (N -M )×M and Σ 2 ∈ R (N -M )×(N -M ) are the blocks of the covariance Σ Σ = Σ 1 Σ 21 Σ 21 Σ 2 . ( 14
)
To circumvent the difficulty of inverting the signal-dependant matrix in Equation ( 13), we choose to neglect the nondiagonal terms of Σ 1 , as in [START_REF] Lorente | A Deep Network for Reconstructing Images from Undersampled Poisson data[END_REF]. Denoting σ 2 1 = diag (Σ 1 ), we get

y (k) 1 (m α ) = σ 2 1 /(σ 2 1 + σ 2 α )(m α -H 1 f (k) ), (15)
where division and multiplication apply element-wise. Finally, we can summarize our algorithm by

y (k) 1 = σ 2 1 /(σ 2 1 + σ 2 α )(m α -H 1 f (k) ) (16a) y (k) 2 = Σ 21 Σ -1 1 y (k) 1 (16b) x(k) = Hf (k) + y (k) (16c) f (k+1) = D ω (H x(k) ). (16d)
The outline of our algorithm can be found in Fig. 1. In this scheme, we propose to learn σ 1 , Σ 21 Σ -1 1 , and D ω in an endto-end fashion. Note that as in [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF], σ 1 , Σ 21 Σ -1 1 , and the parameters of our network D ω are shared across every iteration 0 ≤ k ≤ K to reduce the number of learned parameters.

Denoised completion in the presence of Poisson noise

By using the normal approximation to the Poisson distribution (see [START_REF] Huang | Introduction to statistical physics[END_REF]), Equation ( 1) can be approximated as

m α ∼ 1 α N (αH + 1 f , Σ+ α ) -N (αH - 1 f , Σ- α ) , ( 17
)
where Σ+ α = Diag αH + 1 f and Σα = Diag αH - 1 f . This can also be expressed as As σ 2 α depends on the unknown image f as well as on the intensity α, we exploit the raw data that also depends on f and α. Recalling that the variance of a Poisson variable is the same as its expected value, we can write Σ+ α = E mα + and Σα = E mα -, and choose to approximate the expected value by the noisy sample, i.e., σ 2 α ≈ 1/α 2 ( mα + + mα -).

Σ α = Diag σ 2 α = Diag 1 α H + 1 f + 1 α H - 1 f . (18)

EXPERIMENTS

In our experiments, we chose M = 512 Hadamard patterns of size N = 64 × 64 pixels. We trained our network for K = 5 iterations in a end-to-end fashion. The architecture of D ω is a U-net [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]. We consider two network variants: (i) a standalone (direct) U-net reconstructor, which is equivalent to one iteration of our algorithm; and (ii) model-based reconstruction using deep-learned priors (MoDL) [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF], which is equivalent to setting Σ 21 = 0 and Σ 1 = λI in Equation ( 16). All of the networks are trained using 105,000 images (i.e., the 'unlabeled' and 'train' subsets of the STL10 database [START_REF] Coates | An analysis of single-layer networks in unsupervised feature learning[END_REF]); 8,000 images were used for the test (i.e., the 'test' subset of STL10). The original 96×96 images were cropped to 64×64, and normalized between -1 and 1. We train our unrolled network using Pytorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF]. For training, we consider the ADAM optimiser for 20 epochs. The step size is initialized to 10 -3 and divided by 5 every 5 epochs. The weight decay regularization parameter is set to 10 -6 . The number of learned parameters is 4,432,657. Note that our initial estimate of f (0) is computed as described in [START_REF] Ducros | A completion network for reconstruction from compressed acquisition[END_REF].

RESULTS AND DISCUSSION

In Table 1, we compare our proposed deep EM network with three other methods: the Moore-Penrose pseudo inverse (PI), a stand-alone U-net, and MoDL [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF]. According to our sim- ulations, our method outperforms the other methods in terms of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) for different image intensities α (i.e., different levels of noise). Note that the benefits of our method are more apparent for high noise (i.e., low α).

In Fig. 2, we consider fluorescence microscopy imaging of yeast cell membranes, which do not belong to the STL-10 test set. The membrane proteins are fused with red and green fluorescent protein markers. Each channel was treated independently. We obtain an improved PSNR in the red channel with the proposed network (23.14 dB for the proposed method, compared to 22.99 dB for U-net, and 22.03 dB for MoDL). For the green channel, however, MoDL [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF] gives the best PSNR (19.33 dB) followed by our proposed method (17.92 db) and then U-net (17.04 dB). Visually, our deep EM reconstruction presents less strike artefacts than the alternative methods.

In Fig. 3, we finally assess our deep EM network using experimental data obtained using the set-up described in [START_REF] Lorente | A Deep Network for Reconstructing Images from Undersampled Poisson data[END_REF]. We obtain improved PSNR with the proposed method. Visually, the deep EM network and the stand-alone U-net yield images with fewer compression artifacts. The MoDL [START_REF] Aggarwal | Modl: Model-based deep learning architecture for inverse problems[END_REF] reconstruction is noisier, which can be explained on the basis that each iteration assumes signal-independent Gaussian noise, rather than signal-dependent noise.

CONCLUSION AND PERSPECTIVES

We propose a recursive network based on the EM algorithm. This deep EM network can solve linear underdetermined inverse problems where the data are corrupted by signal-dependent noise (e.g., Poisson noise, mixed Poisson-Gaussian noise). This approach is shown to yield greater PSNR under several levels of noise compared to the the U-net approach alone or a MoDL type of approach. To the best of our knowledge, most unrolled networks assume noise to be Gaussian with constant variance, and are therefore ill-suited for compressive optics applications. The proposed method is promising for single-pixel biological imaging and can further be adapted to the reconstruction of other types of biophotonic imaging modalities.

COMPLIANCE WITH ETHICAL STANDARDS

This paper reports no data from studies involving human and/or animal subject.

Fig. 1 .

 1 Fig. 1. Proposed deep EM reconstruction network. The data consistency (DC) layers consist of three steps: the denoising (D) step, the completion (C) step, and the inverse transform (IT) step. The denoising layers D ω implement a U-Net [20].

Fig. 2 .

 2 Fig. 2. Reconstruction of a yeast cell membrane fluorescence microscopy image.

Fig. 3 .

 3 Fig. 3. Reconstructions of two experimental datasets by the different methods (top row: LED lamp with M = 512; bottom row: STL-10 cat with M = 512). The images shown were reconstructed from a fully sampled dataset (ground-truth; GT) acquired with high image intensity (first column, α = 148 photons, and α = 195 photons) and lower image intensity (second column, α = 80 photons, and α = 10 photons). The following columns show reconstructions using pseudo inverse (PI), U-Net reconstructor, MoDL [10], and the proposed method.

Table 1 .

 1 Average peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) over the stl-10 test dataset for the different reconstruction methods: PI (pseudo inverse), standalone U-Net, MoDL, and the proposed deep EM network.

	α(ph.)		PI	U-Net MoDL Proposed
	2	PSNR 12.97 17.17	17.14	17.82
		SSIM	0.61	0.72	0.73	0.78
	5	PSNR 15.32 19.41	19.49	19.85
		SSIM	0.73	0.81	0.81	0.85
	10	PSNR 16.51 20.61	20.7	20.88
		SSIM	0.78	0.85	0.84	0.88
	50	PSNR 17.8	21.8	21.42	21.89
		SSIM	0.82	0.9	0.85	0.9
	2500	PSNR 18.17 22.11	21.55	22.15
		SSIM	0.84	0.91	0.86	0.91

ACKNOWLEDGMENTS

This work was supported by the French National Research Agency (ANR), under Grant ANR-17-CE19-0003 (ARMONI Project), and performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon.


A Kopf

M Bardi

E Kohler

T Endres

G Bruneaux

C Schulz

Survivability of the thermographic phosphors YAG:Pr and SMP:Sn in a premixed flame

Keywords: YAG:Pr, tin-activated strontium magnesium phosphate (SMP:Sn), survivability, phosphor thermometry, in situ combustion temperature imaging

The survivability of two thermographic phosphors, YAG:Pr 3+ and SMP:Sn ((Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ ), in a combustion environment is studied in the post-flame regime of a premixed propane/air flame. While the laser-induced luminescence of YAG:Pr can be exploited for in situ temperature measurements in the exhaust gas above the flame, there is no in situ detectable luminescence for SMP:Sn for any reaction conditions within the stability limits of the flame. The phosphor particles are recaptured above the flame and ex situ analyzed for chemical or structural changes using microscopic analysis (SEM/EDX) and Xray diffraction. The microscopic analysis of post-flame YAG:Pr does not show any alteration, whereas morphology and chemical composition of SMP:Sn have been modified upon passing through the reaction zone, which is responsible for the loss of its luminescence properties.

Introduction

Thermographic phosphors enable the remote measurement of temperature based on the temperature dependent variation in laser-induced luminescence spectra or signal lifetime. They have been frequently used for surface temperature measurements [START_REF] Brübach | On surface temperature measurements with thermographic phosphors: A review Progress in[END_REF] but recently, applications have been extended to gas-phase temperature imaging measurements based on aerosolized phosphor particles [START_REF] Abram | Temperature measurement techniques for gas and liquid flows using thermographic phosphor tracer particles Progress in[END_REF]. This approach is of particular interest in reacting flows, where frequently used molecular species for thermometry (such as organic tracers [START_REF] Schulz | Tracer-LIF diagnostics: Quantitative measurement of fuel concentration, temperature and fuel/air ratio in practical combustion systems[END_REF][START_REF] Tea | Unburned gas temperature measurements in a surrogate Diesel jet via two-color toluene-LIF imaging[END_REF] or small molecules (e.g., NO [START_REF] Lee | Quantitative temperature measurements in high-pressure flames with multiline NO-LIF thermometry[END_REF][START_REF] Bessler | Two-line laser-induced fluorescence imaging of vibrational temperatures in a NO-seeded flame[END_REF]) can be affected in the reaction medium or restricted to the area where radicals naturally occur (e.g., OH [START_REF] Seitzman | Application of quantitative two-line OH planar laser-induced fluorescence for temporally resolved planar thermometry in reacting flows[END_REF][START_REF] Peukert | The influence of hydrogen and methane on the growth of carbon particles during acetylene pyrolysis in a burnt-gas flow reactor[END_REF]). Thermographic phosphors that survive the flame environment enable temperature measurements in the gas phase before and behind the flame front, which is of specific interest, e.g., in the context of understanding in-cylinder processes in internal combustion engines.

Thermographic phosphors with strong luminescence and short radiative lifetimes are required for temperature measurements in reacting gas flows to provide signal that can be discriminated against background luminescence, e.g., from a flame. For the selection of an appropriate thermographic phosphor, it is equally important that the emission properties of the phosphor are strongly temperature sensitive to obtain good measurement precision. It is of interest to cover flame temperature as well as areas that have been exposed to a flame in the process but cooled down again (e.g., in the expansion stroke). There are two limitations to applications of thermographic phosphors in such an environment: (i) Weak signal at high temperature that then is in competition with flame chemiluminescence and particle thermal radiation, and (ii) Potential disintegration of materials in the flame zone due to high temperature and through exposure to oxidizing species.

To overcome the first limitation, a new approach for phosphor thermometry was recently presented using the ratio of the temperature-sensitive phosphor luminescence and the temperature-insensitive Mie-scattered laser light for temperature sensing (scattering-referenced aerosol phosphor thermometry, SRAPT) [START_REF] Witkowski | Scattering referenced aerosol phosphor thermometry Measurement[END_REF]). This strategy extends the measurable temperature range to 1000 K at a single-shot precision better than 4% with a temperature sensitivity up to 1.6 %K -1 . Overcoming the second limitation of disintegration of the phosphor depends on the respective materials properties. A large range of phosphor materials has been investigated for thermometry purposes in various applications, (Sr,Ca)SiAlN 3 :Eu 2+ [START_REF] Hertle | Characterization of the phosphor (Sr,Ca)SiAlN 3 : Eu 2+ for temperature sensing[END_REF], LuAG:Pr 3+ ,Ce 3+ [START_REF] Herzog | Characterization of the Ce,Pr:LuAG phosphor for Co-doped aerosol phosphor thermometry[END_REF], YAG:Pr 3+ ,Ce 3+ [START_REF] Witkowski | A novel strategy to improve the sensitivity of aerosol phosphor thermometry using co-doped phosphors[END_REF], YAG:Pr 3+ [START_REF] Jordan J A R D A | Pr:YAG temperature imaging in gasphase flows[END_REF], (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ [START_REF] Fond | Investigation of the tin-doped phosphor (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for fluid temperature measurements Optical[END_REF], but only few aerosolized thermographic phosphors have been reported to withstand combustion. BAM:Eu 2+ maintains its luminescent properties upon passing through a laminar non-premixed flame [START_REF] Yin | Investigation of BAM:Eu 2+ particles as a tracer for temperature imaging in flames Combustion and[END_REF]. It was also used for temperature imaging measurements in the residual gas of a high-pressure combustion cell [START_REF] Van Lipzig | Gas-phase thermometry in a high-pressure cell using BaMgAl 10 O 17[END_REF]. YAG:Dy 3+ was successfully applied for thermographic measurements in an HCCI combustion in an engine [START_REF] Hasegawa | Two-dimensional gas-phase temperature measurements using phosphor thermometry[END_REF]. The luminescent properties of the same phosphor were also unaffected by passing the flame front of a premixed methane/air flame [START_REF] Yu | Survivability of thermographic phosphors (YAG:Dy) in a combustion environment Measurement[END_REF]. This work evaluates the applicability of (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ and YAG:Pr 3+ for thermometry applications in the postcombustion region. Both phosphors are highly sensitive to a change in temperature in the order of 0.5-0.8 %K -1 in the range of temperatures relevant for reacting flow conditions (600-900 K) [START_REF] Abram | Temperature measurement techniques for gas and liquid flows using thermographic phosphor tracer particles Progress in[END_REF][START_REF] Fond | Investigation of the tin-doped phosphor (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for fluid temperature measurements Optical[END_REF][START_REF] Neal | Simultaneous measurements of in-cylinder temperature and velocity distribution in a small-bore Diesel engine using thermographic phosphors[END_REF]. The emission lifetime (1/e) of both phosphors is on the order of ~1 µs (SMP:Sn) and 60 µs (YAG:Pr, 3 P J transition) in the aforementioned temperature range [START_REF] Fond | Investigation of the tin-doped phosphor (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for fluid temperature measurements Optical[END_REF][START_REF] Neal | Simultaneous measurements of in-cylinder temperature and velocity distribution in a small-bore Diesel engine using thermographic phosphors[END_REF][START_REF] Kopf | Temperature measurements in engine post-combustion gases using YAG:Pr 3+[END_REF]. This is convenient for the application in reacting flows, as it is short enough to capture temperature changes on timescales relevant to, e.g., internal combustion engines. At the same time, it is long enough to enable the suppression of short-lived fluorescence contributions from intermediate species by delaying the detection time by a few tens of nanoseconds. Praseodynium-doped yttrium aluminum garnet (YAG:Pr 3+ ) shows non-radiative and radiative relaxation from the excited 4f-5d levels to the 3 P J (J = 0, 1, 2) and the 1 D 2 state, which de-excite to the 3 H 4 ground state [START_REF] Donegá | Non-radiative relaxation processes of the Pr 3+ ion in solids[END_REF], emitting in two visible narrow emission bands, centered at 489 and 610 nm respectively. Pr 3+ emissions were reported to have emission rates greater than unity for visible emissions due to cascade processes [START_REF] Piper | Cascade fluorescent decay in Pr 3+ -doped fluorides: Achievement of a quantum yield greater than unity for emission of visible light[END_REF]. At increasing temperatures, the 3 P J transition is quenched most likely due to multiphonon relaxation [START_REF] Witkowski | Emission properties and temperature quenching mechanisms of rare-earth elements doped in garnet hosts[END_REF], which can be exploited for temperature sensing: The temperature is measured by calculating the intensity ratio between the two emission bands, which is also known as the two-color method or intensity-ratio approach. Due to thermal quenching of this blue emission this technique is limited as the signal decreases to noise level. The intensity-ratio approach has been applied for 2D temperature imaging in a heated jet [START_REF] Jordan J A R D A | Pr:YAG temperature imaging in gasphase flows[END_REF] and during the compression stroke in a Diesel engine [START_REF] Neal | Simultaneous measurements of in-cylinder temperature and velocity distribution in a small-bore Diesel engine using thermographic phosphors[END_REF] due to favorable luminescent properties such as high emission intensities and relatively short radiative lifetimes, but not reported for temperature measurements in the postcombustion regime. The melting point of YAG is 2213±7 K [START_REF] Caslavsky | Melting behaviour and metastability of yttrium aluminium garnet (YAG) and YAlO 3 determined by optical differential thermal analysis[END_REF].

(Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ (in this work referred to as SMP:Sn) is a tin-activated ternary orthophosphate, where strontium can partly be replaced and stabilized with zinc or alkaline earth metal ions (Ba, Ca, Mg) to improve luminescence [START_REF] Butler | Alkaline Earth Orthophosphate Phosphors[END_REF]. The luminescence strongly depends on the host lattice. The lattice provides two possible sites with distinct coordination numbers for divalent cations like Sn 2+ and it was found that the different coordination numbers of the luminescent centers are responsible for two separate emission bands, following UV excitation [START_REF] Donker | On the Luminescence of Some Tin-Activated Alkaline-Earth Orthophosphates[END_REF]. This phosphor exhibits a major yellowred emission band centered at 610 nm at room temperature from luminescence centers of high coordination numbers, and a near-UV emission band at 400 nm from lower coordination. The latter only accounts for 5 % of the total emission energy and disappears at increasing temperatures [START_REF] Donker | On the Luminescence of Some Tin-Activated Alkaline-Earth Orthophosphates[END_REF][START_REF] Koelmans | Luminescence of Modified Tin-Activated Strontium Orthophosphate[END_REF]. With increasing temperatures, the yellow-red broadband emission shows a pronounced redshift, which can be used for thermometry purposes using the intensity-ratio approach.

In a mixed (Sr,Mg)-orthophosphate such as the SMP:Sn phosphor, the luminescence depends on the amount of Mg that replaces Sr in the crystal lattice. In the phase diagram of the mixed orthophosphate Sr 3 (PO 4 ) 2 -Mg 3 (PO 4 ) 2, provided by Sarver et al. [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF], it is shown that the phosphate forms solid solutions during cooling at room temperature, as the mole percentage of the Mg 3 (PO 4 ) 2 is increased from 0 % (pure Sr 3 (PO 4 ) 2 ) to 100 % (pure Mg 3 (PO 4 ) 2 ). For example, the crystal phase of the (Sr,Mg)-phosphate for a Mg 3 (PO 4 ) 2 mole fraction of 10-35% is annotated in this publication as β-Sr 3 (PO 4 ) 2 , even if it refers to the β-phase of the mixed (Sr,Mg)-phosphate. For simplification, the nomenclature in this work will be aligned with the one of the references. The Mg 3 (PO 4 ) 2 mole fraction leading to the phosphors' highest brightness is around 10-15%, where the crystal structure of the phosphate is β-Sr 3 (PO 4 ) 2 . At increasing mole fractions of the strontium phosphate (<10% Mg 3 (PO 4 ) 2 ), the phase diagram shows a two-phase region of α-and β-Sr 3 (PO 4 ) 2 , where the α-Sr 3 (PO 4 ) 2 is non-luminescent but has low ultraviolet absorption , emphasizing the strong sensitivity of the lattice composition to luminescence. The melting temperature of SMP:Sn is not exactly known but specified by the manufacturer to be above 1473 K [START_REF]Material safety data sheet (SDS): Phosphor SV 253: Strontiummagnesiumorthophosphate: tin activated (#B4640219)[END_REF]. This phosphor has first been characterized for thermometry purposes by Fond et al. and shows promising luminescent properties for thermometric applications, relatively to BAM:Eu 2+ or ZnO, due to a low cross-dependency to laser fluence at similar global emission intensities (photons per particle per pulse) [START_REF] Fond | Investigation of the tin-doped phosphor (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for fluid temperature measurements Optical[END_REF][START_REF] Fond | Characterisation of dispersed phosphor particles for quantitative photoluminescence measurements Optical[END_REF].

This study evaluates the two phosphors YAG:Pr and SMP:Sn for thermographic measurements in reacting conditions of a premixed propane/air flame. Temperature imaging measurements exploit the signal ratio of two emission bands that are determined by optical filters. At increasing temperatures, the emission spectra shift towards shorter wavelengths (e.g., for SMP:Sn) or the intensity of the emission bands are differently affected from thermal quenching (e.g., for YAG:Pr), leading to varying signal ratios. The temperature can be inferred from a calibration curve, where the signal ratio of both emission bands is calibrated at known temperatures.

Materials and experimental methods

We use an in-house built premixed burner (outlet nozzle d = 12 mm) with a passive co-flow and chimney extension, which is needed to stabilize the flow further downstream of the flame. The stabilizing inert co-flow is controlled with inlet flaps on the bottom of the burner housing. Increasing co-flow stabilizes the flow by reducing shear between the two flows and thus natural convection between both flows. The measurable region is located at the chimney outlet 175 mm above the burner nozzle, where phosphor particles are seeded into the oxidizer flow. The fuel/air equivalence ratio ϕ of a premixed propane/air flame is varied between 0.62 and 1.0 to change the flame temperature and thus the temperature of the exhaust gases. The adiabatic flame temperatures for the respective equivalence ratios range from 1650 to 2350 K [START_REF] Kennel | The basic structure of lean propane flames Symposium[END_REF]. YAG:Pr is provided by Phosphor Technology (QMK59/F-X #29010, d 50 = 4.2 μm) with a dopant concentration of 0.5 % similar to that used in previous measurements that investigated the temperature dependence of the luminescence [START_REF] Jordan J A R D A | Pr:YAG temperature imaging in gasphase flows[END_REF]. Additionally, SMP:Sn, a phosphor from OSRAM GmbH that is used commercially in color rendering lamps (SV 253 #B4640219, d 50 = 8 … 10 μm) was investigated after it was grinded and de-agglomerated to a diameter of d 50 = 1.6 μm. Two-color phosphor thermometry is carried out in the postcombustion zone above the flame, where the flow has cooled down. A Nd:YAG laser is quadrupled to the fourth harmonic (266 nm) and a combination of a cylindric and a spherical lens is used to form a light sheet 65 mm wide and approx. 0.5 mm thick (90 mJ/cm 2 ). An intensified camera (PIMax 4, EmCCD, 16 bit, intensifier gen2) is equipped with an image splitter (LaVision, ImageDoubler) to project an identical image pair of one object through two optical filter sets onto a single camera chip. The camera detection gate for the phosphorescence detector (camera 1) was set to 50 µs and delayed by 100 ns relatively to the laser pulse, similar to comparable imaging experiments [START_REF] Jordan J A R D A | Pr:YAG temperature imaging in gasphase flows[END_REF]. An identical camera is used for Mie-imaging (camera 2). The filters employed for thermometry are a Semrock bandpass filter FF01-494/20 (blue channel) and a combination of Schott colored glass OG570 and a 625 nm shortpass filter (Edmund Optics #64-604) for the red channel. The spatial fluctuations of the gas flow above the burner as well as instabilities in the particle seeding require a large data set of 1000 images. To account for these instabilities, the intensity ratio map is calculated for every image after applying a cutoff filter (<600 counts), spatial averaging (3 × 3 pixels) on both images and dividing it by the average ratio taken at ambient temperature in inert flow conditions. The temperature field is then obtained using a calibration curve obtained in a furnace [START_REF] Jordan J A R D A | Pr:YAG temperature imaging in gasphase flows[END_REF]. The gas temperature is also measured with thermocouples (type K, 1.0 mm tip diameter). The phosphor particles are collected on a filter membrane above the burner for microscopic analysis, where the filter support is inserted in the exhaust ventilation system.

Particle residence time

The residence time of the phosphor particles in the flame at certain conditions (oxidizing environment, high temperature) is of interest in order to estimate the potential disintegration of phosphor particles in other practical environments (e.g., IC engines). For example, if the temperature of the particleladen gas surpasses melting temperature of the phosphor material, but the residence time at this temperature is shorter than the time required for heat conduction, temperature measurements can be unaffected in the post-flame regime.

Numerical simulation

For calculating the residence time, a 2D axisymmetric steady-state numerical simulation was carried out in Ansys Fluent. The passive co-flow of the chimney housing is accounted for with a co-flow (air) around the burner central flow (propane & air). The boundary conditions for the inlet velocity are calculated from the measured mass flows of propane and air and the mass flow of the surrounding coflow is adapted in the simulation to generate the best match with the temperature values measured by thermocouples in the exhaust gas. Gravitation is neglected, as well as radiation effects from the burner chimney. The reaction kinetics are calculated with the CHEMKIN CFD solver using the complete San Diego mechanism [START_REF]San Diego[END_REF].

The residence time of a particle at high temperature is derived from the simulated temperature and gas velocity in the flame. Figure 2 shows the accumulated time a particle is exposed to a certain temperature or above on its path through the flame. The evolution of temperature and gas velocity over time are also shown for completeness. The time a phosphor particle is exposed to temperatures above 1473 K (melting temperature specified by the manufacturer is T > 1473 K [START_REF]Material safety data sheet (SDS): Phosphor SV 253: Strontiummagnesiumorthophosphate: tin activated (#B4640219)[END_REF]) is 38 ms. Temperatures above 1920 K are reached for 0.9 ms. Theoretical heat transfer considerations on the response time t 95 until a particle reaches 95 % of the steady-state temperature following a step temperature change of the surrounding gas temperature is below 30 µs [START_REF] Fond | Simultaneous temperature, mixture fraction and velocity imaging in turbulent flows using thermographic phosphor tracer particles[END_REF]. The considerations were made for a 2-µm phosphor particle (YAG) for a step change from 300 to 2000 K, which is representative for combustion conditions. Because the time scale for thermal equilibration is three orders of magnitude shorter than the residence time of the particle in the hightemperature zone in the flame, the particles are expected to reach steady-state temperature. Assuming that the melting temperature for SMP:Sn (T > 1473 K [START_REF]Material safety data sheet (SDS): Phosphor SV 253: Strontiummagnesiumorthophosphate: tin activated (#B4640219)[END_REF]) is reached, the residence time at this temperature is long enough to expect complete melting. A more detailed discussion about the expected melting temperature can be found in section 3.3.

Results and discussion

Exhaust gas temperature measurements

The objective of the following measurements is to investigate, whether the phosphors maintain their temperature-dependent spectroscopic properties after having been exposed to the flame. Temperature measurements in the post-combustion gases are carried out employing the twocolor thermometry approach using YAG:Pr. Figure 3 shows a single shot image of both imaging channels. The blue channel suffers from thermal quenching and is thus the limitation for statistically significant measurements at high temperatures. The signal to noise ratio (SNR) is evaluated based on the 90 th and the 10 th percentile of the signal intensities in the respective region of interest. The signal was calculated in the hottest region in the center of the gas flow (red box) and the noise was estimated from a region outside the flow (yellow box). Figure 4 shows the temperature dependence of the signal-tonoise ratio for each individual signal channel. The low percentile threshold for noise calculation was chosen in order to be insensitive to varying signal contribution due to turbulent fluctuations, as can be seen in Figure 3. Following the procedure for image processing detailed in section 2, an intensity ratio and temperature map can be calculated from both channels. Averaged two-dimensional temperature map of the post-combustion gases above the burner, derived from YAG:Pr two-color thermometry measurements are shown in Figure 5 for fuel/air equivalence ratios of 0.83 (left) and 0.63 (right), showing that the leaner equivalence ratio results in lower exhaust gas temperatures. The flow is not axisymmetric due to the effect of the fume hood. The regions highlighted in gray are not considered for temperature calculation due to a light reflection from metal surfaces of the chimney. This optical artifact could be reduced using structured laser illumination [START_REF] Zentgraf | Application of structured illumination to gas phase thermometry using thermographic phosphor particles: A study for averaged imaging Experiments[END_REF], but is not the objective of this study. Figure 6 shows the temperature profile extracted from the average temperature field and presents the gas temperature in vertical direction above the chimney outlet of the burner for varying fuel/air equivalence ratios. The temperature profile is averaged from three horizontally averaged pixels. The measured gas temperatures follow two trends, (i) decreasing gas temperature for decreasing ϕ and (ii) decreasing temperatures at increasing height above the burner (HAB) and are in good agreement with thermocouple (TC) measurements. The error bars in the TC measurements represent the deviation from temperature measurements in the flow. Due to increasing spatial fluctuations downstream in the flow at increasing HAB, the error bar is larger at the upper thermocouple position (HAB = 225 mm). Despite the uncertainties related to the thermocouple measurement, the flame fluctuations or limitation using the calibration acquired in a furnace, these results clearly show that YAG:Pr phosphor particles maintain its temperaturesensitive luminescence emission after passing the flame front of the burner.

A possible source of bias in the measurements could be black-body radiation related to the phosphor particles. Above a certain temperature this contribution will be non-negligible and will compete with the luminescence emitted by the phosphor, creating a bias in temperature measurements. However, in the current experimental conditions, this contribution is not expected to be significant for the following two reasons: Firstly, the consistency of the temperature measurements found over the entire temperature range and secondly the agreement with the TC measurements. This interference problem will be strengthened when extending the measurements to reacting conditions due to other competing luminescent signals, like chemiluminescence or soot incandescence. Figure 7 shows a set of Mie-scattering images for inert and reacting flow conditions. With the binarizing threshold chosen for this study, the volumetric particle number density (PND) in the reacting conditions (ϕ = 0.71) is a factor of 2.3 lower than in the inert flow, which is equivalent to the difference in gas density between 800 and 300 K. In the same temperature range, the band-integrated luminescence is expected to decrease by a factor of ~5-10 [START_REF] Kopf | Temperature measurements in engine post-combustion gases using YAG:Pr 3+[END_REF]. In our experiment, the volumetric luminescence decreased to ~12.5 % in the hot case, which indicates that the observed effect results from a combination of density variation and thermal quenching and not to the total loss of luminescence due to particle degradation.

The second investigated phosphor SMP:Sn shows strong luminescence emission under inert conditions at ambient temperature, but does not exhibit any detectable phosphor luminescence in the post-combustion regime for any stable operating point tested in the burner (ϕ = 0.58-1.25). Phosphor luminescence at similar temperatures (up to 873 K) was measured in inert conditions in a furnace on a coated surface [START_REF] Kopf | The potential of (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for gas-phase phosphor thermometry in post-combustion gases[END_REF]. Since particles were visible on the second camera (Mie-imaging), these results indicate particle degradation leading to a loss of the materials' luminescent properties upon passing the reaction zone of a premixed propane/air burner flame.

Ex situ particle analysis (SEM/EDX, XRD)

Microscopic particle analysis of the phosphor material is carried out using scanning electron microscopy (SEM) to investigate the impact of the exposure to the combustion environment on their morphology (secondary electrons detector, SE) or chemical composition (energy dispersive spectroscopy, EDX). Initial measurements are carried out in bulk powder material with an acceleration voltage of 15 kV and an anode current of 750 pA (Zeiss, EVO MA10). EDX measurements of bulk material give an average chemical composition from the sample volume penetrated by the incident electron beam. As the penetration depths is not known, bulk powder measurements only allow a qualitative description of the chemical composition of the initial and post-flame material. For further analysis of the chemical composition, the SMP:Sn post-flame powder is embedded in epoxy resin and cut into thin slices using an ultra-microtome (Leica EM UC). The thickness of the slices is between 100 and 200 nm. Thus, the electron penetration depth is expected to be constant for any point in the cross-section of a particle. The chemical composition of the particle cross-sections is analyzed using an EDX detector (Bruker, XFlash 6I10 SDD) built into an Auger microscope (Ulvac-Phi, Phi 710 Nanoprobe).

The phosphor particles were captured approx. 300 mm above the burner nozzle after passing through a ϕ = 0.71 flame with an adiabatic flame temperature of ~1900 K [START_REF] Kennel | The basic structure of lean propane flames Symposium[END_REF]. The crystal structure of both, initial and post-flame powder is analyzed using an X-ray powder diffractometer (Bruker, D8 Discover) to investigate structural changes of the lattice or the presence of newly formed solids from alteration due to flame exposure.

The results from both techniques are presented individually for both phosphors in 3.2.1 and 3.2.2. and then discussed together in section 3.3. The X-ray diffraction pattern (Figure 9) of the post-flame powder does not show any changes relatively to initial material and is in excellent agreement with the reference diffraction pattern [START_REF] Yoder | Complete substitution of aluminum for silicon: The system (MnO) 3 Al 2 O 3 (SiO 2 ) 3 -(Y 2 O 3 ) 3 (Al 2 O 3 ) 5[END_REF] calculated with crystallographic information from the crystallographic open database (COD# 1529037) [START_REF] Gražulis | Crystallography Open Databasean open-access collection of crystal structures[END_REF]. This confirms that the crystal structure remains unchanged after passing the flame front under the given conditions. Figure 11a shows the secondary electron (SE) image of the cross-section, representing the topographic map of the particles. Any structure visible in a particle in (a), apart from preparation artifacts, is attributed to a locally varying surface topography. It is also possible that the surface irregularities cause EDX analysis artifacts specific to each chemical species. The gradient in the particle ( 3) is due to shading effects as confirmed from a series of rotated images. The inhomogeneities on the particles, annotated as ( 1) and ( 2), thus stem from locally varying materials properties, which could become visible in the topographic image from being differently displaced by the microtome blade, likely due to changed crystal phase properties.

YAG:Pr

SMP:Sn

Figure 11c-f shows the relative distribution of the chemical species of SMP:Sn. Particles ( 1) and ( 2) show a mostly homogeneous distribution of both the phosphate (P, O) as well as Mg, whereas there are spots with relatively increased Sr concentration.

Figure 11b shows an overlaid SE/EDX image, where regions of varying material properties are highlighted. These regions with increased Sr concentration, numbered in white in Figure 11b show a mean Sr content of 52.8 wt%. Regions of reference, where the Sr content is not visibly increased, numbered in black, have an average Sr content of 43.5 wt%.

The averaged content of Sr of the entire particle (annotated as The X-ray diffraction pattern ( Figure 12) from the initial material SMP:Sn (blue) matches well the those from β-phase (yellow diffraction peaks) of the (Sr,Mg) 3 (PO 4 ) 2 when the mole fraction Mg 3 (PO 4 ) 2 is between 10-35% [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF]. Based on the EDX measurements of the initial material, the measured contribution of Mg 3 (PO 4 ) 2 is 12.5 mol%. In the diffraction pattern of the post-flame SMP:Sn (diffraction pattern in the middle), the peaks from the β-Sr 3 (PO 4 ) 2 are less pronounced, thus, the phases can only partially be attributed to the phases of the initial material. Moreover, the presence of two new phases can be confirmed from the diffraction pattern of the post-combustion material, which are recognized as α-Sr 3 (PO 4 ) 2 and SrMg 2 (PO 4 ) 2 . The appearance of these two phases is highlighted in the inset plot in Figure 12. Both phases also appear in the phase diagram of (Sr,Mg) 3 (PO 4 ) 2 at different Mg mole fractions [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF]. This will be discussed in detail in the following section. Due to the low quantity of recaptured phosphor, the interference scan of the post-flame SMP:Sn is carried out in transmission mode of the X-ray detector, leading to decreased sensitivity of the scan. Even if it is difficult to quantify the proportions of different phases in the post-combustion material, it is clear that the material had undergone phase change and that the crystal properties of the post-flame material are modified relatively to the initial material.

Discussion

The phosphor YAG:Pr enables temperature measurements in the post-combustion regime for various fuel/air equivalence ratios. The structural analysis of YAG:Pr does not show changes in the crystal structure after passing the flame front, where the adiabatic temperature is approximately 1900 K (ϕ = 0.71). Even if melting is not expected for the majority of particles, isolated spherical particles are detected under these conditions (Figure 8), which indicates melting. Between this condition and the melting temperature of YAG (2213±7 K [START_REF] Donker | On the Luminescence of Some Tin-Activated Alkaline-Earth Orthophosphates[END_REF]), no significant changes in luminescence properties of the post-flame material are expected. However, thermographic measurements were possible in the postcombustion gases even in stoichiometric flames (ϕ = 1.0). An evaluation of the band-integrated luminescence relative to the volumetric particle number density derived from Miescattering showed no significant loss of per-particle luminescence. The adiabatic flame temperature is ~2350 K [START_REF] Kennel | The basic structure of lean propane flames Symposium[END_REF], thus melting can have occurred. We can, however, not finally conclude whether luminescence has recovered after solidification or if the particles did not melt because of deviation from the adiabatic flame temperature due to heat losses.

For the second investigated phosphor, SMP:Sn, no phosphor luminescence is detected in the post-combustion regime.

Based on the results presented above, two possible degradation paths are proposed, which can give rise to the observed loss of the luminescent properties of SMP:Sn after passing the material through the flame.

Firstly, the spectroscopic properties of SMP:Sn are related to the presence of Sn 2+ . In a strongly oxidizing hightemperature environment, further oxidation of this species towards the less stable Sn 4+ ion is expected [START_REF] Riedel | Anorganische Chemie[END_REF]. In previous investigations, SMP:Sn was heated up three times to 900 K (20 K/min) but the luminescent properties did not show a difference towards the untreated sample [START_REF] Fond | Investigation of the tin-doped phosphor (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ for fluid temperature measurements Optical[END_REF]. Koelmans and Cox observed an 8 % luminescence reduction of (Sr, Mg) 3 (PO 4 ) 2 :Sn 2+ after exposing the sample at 973 K to air for 15 min. The luminesce emission continuously decreased for longer storage time in air, up to a total loss of luminescence after 100 hours [START_REF] Koelmans | Luminescence of Modified Tin-Activated Strontium Orthophosphate[END_REF]. The authors did not specify the exact mole fraction of Mg 3 (PO 4 ) 2 in the investigated (Sr,Mg) 3 (PO 4 ) 2 :Sn 2+ . However it was also shown that different Mg/Sr compositions of the host lattice do not affect the equilibrium between Sn 2+ and Sn 4+ but only the oxidation rate and thus the time until the loss of its luminescent properties [START_REF] Koelmans | Luminescence of Modified Tin-Activated Strontium Orthophosphate[END_REF]. In our experiment, the burner is operated in lean combustion (ϕ < 1) to avoid soot formation and thus providing excess oxygen.

Secondly, the luminescent emission from phosphors strongly depends on the lattice structure [START_REF] Blasse | Luminescent Materials[END_REF]. Based on the EDX measurement from bulk powder, the mole fraction of Mg 3 (PO 4 ) 2 in the lattice is measured to be 12.5 mol% in the initial material. According to the phase diagram of (Sr,Mg) 3 (PO 4 ) 2 , this corresponds to the concentration of Mg 3 (PO 4 ) 2 , where the phosphors shows its maximum brightness in the phase of β-Sr 3 (PO 4 ) 2 (10-15 mol% of Mg 3 (PO 4 ) 2 ) [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF]. The presence of this phase was determined by XRD measurements of the initial material ( Figure 12). However, the adjacent phase, which is characterized by Mg 3 (PO 4 ) 2 below ~10 mol%, contains the non-luminescent α-Sr 3 (PO 4 ) 2 [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF]. Therefore, slight changes in the mole fraction of the Sr/Mg content can result in a total loss of luminescence.

The melting point of the mixed Sr 3 (PO 4 ) 2 -Mg 3 (PO 4 ) 2 orthophosphate at the given lattice mole fraction is not exactly known. The melting point of pure Mg 3 (PO 4 ) 2 is 1630 K [START_REF] Berak | The System Magnesium Oxide -Phosphorus Pentoxide Roczniki chemii[END_REF]. The melting point of pure Sr 3 (PO 4 ) 2 is also not exactly known, but can be assumed to be lower, as the crystal field strength of the Sr-O bond (2.641 Å) is lower than for the Mg-O bond (2.046 Å) [START_REF] Li | Eu 2+ :SrMg 1-x Mn x P 2 O 7 (x = 0-1) phosphors with tunable yellow-red emissions[END_REF]. Therefore, in the current experimental conditions, where the particles were recaptured and analyzed (ϕ = 0.71, T adiabatic ≈ 1900 K), at least partial melting is expected for SMP:Sn

During the cool-down, the orthophosphate can follow the phase boundary of the two-phase area (β-Sr 3 (PO 4 ) 2 and liquid) and solidify in the adjacent two-phase region (β-Sr 3 (PO 4 ) 2 and SrMg 2 (PO 4 ) 2 ) according to the phase diagram [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF]. Some peaks in the measured diffraction pattern of the post-flame SMP:Sn material can clearly be attributed to SrMg 2 (PO 4 ) 2 , which can be seen in the inset in Figure 12.

If the melting point is exceeded, mixed ternary orthophosphates, like the (Sr,Mg) 3 (PO 4 ) 2 , tend also to break up into elementary phosphates [START_REF] Wiberg | Lehrbuch der Anorganischen Chemie[END_REF]. At high Sr concentrations, this can result in the β-Sr 3 (PO 4 ) 2 and the nonluminescent α-phase, both phases had also been found in the XRD analysis of the post-flame material. The inset in Figure 10 shows the decreased intensity of the β-phase and the presence of the α-Sr 3 (PO 4 ) 2 , relative to the initial material. For low Sr concentrations, the phosphate Mg 3 (PO 4 ) 2 can be amorphous and would thus not visible in XRD analysis [START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF].

Optical measurements in the burner exhaust gases prevented temperature measurements due to the absence of detectable luminescence. X-ray diffraction analysis showed significant phase transition, where the SrMg 2 (PO 4 ) 2 phase and the nonluminescent α-Sr 3 (PO 4 ) 2 phase was detected in the post-flame material. Based on the analysis presented here, it can be concluded that phase transformation of the (Sr,Mg) 3 (PO 4 ) 2 orthophosphate, induced by passing through the flame at the given conditions, is responsible for the entire loss of luminescence. Lastly, it should be noted that the melting temperature for the SMP:Sn material provided by the manufacturer (> 1473 K) is given for a particle size distribution between d 50 = 8…10 μm. For temperature/ velocity measurements using thermographic phosphors as tracer particles, the size distribution is usually reduced towards the low µm-range to ensure that the seeded particles follow changes in temperature and velocity [START_REF] Fond | Simultaneous temperature, mixture fraction and velocity imaging in turbulent flows using thermographic phosphor tracer particles[END_REF]. Also, chemical reactivity increases at decreasing particle size due to the enlarged specific surface. Therefore, it is also possible that degradation processes like oxidation, which are disadvantageous for the luminescence yield of certain thermographic phosphors, e.g., (Sr,Mg) 3 (PO 4 ):Sn 2+ [START_REF] Donker | On the Luminescence of Some Tin-Activated Alkaline-Earth Orthophosphates[END_REF] are further increased or can occur earlier by reducing the particle size. However, if the limiting factor for particle survivability is the residence time above a certain temperature (e.g., melting temperature), it can be of particular interest to choose large particle diameters if this is permittable for the flow conditions in the region of interest (e.g., postcombustion gases), where temperature-and velocity gradients are less pronounced. This increases the thermal inertia and thus reaching of the melting temperature can be avoided for reacting conditions with short time scales. It is shown that the response time increases significantly with increasing particle size (t 95 increases from 27 µs for a particle size of 2 µm to 577 µs for 8 µm) [START_REF] Fond | Simultaneous temperature, mixture fraction and velocity imaging in turbulent flows using thermographic phosphor tracer particles[END_REF].

Conclusions

The primary goal of this study was to investigate the suitability of two phosphors (YAG:Pr and SMP:Sn) for thermographic measurements in post-combustion gases, especially for studies in internal combustion engines. To emulate an optically-accessible combustion environment, a premixed propane/air flame was used at a wide range of fuel/air equivalence ratios (ϕ = 0.71-1), which also changes the temperature of the exhaust gases. It was shown that YAG:Pr maintains its luminescence properties upon passing the reaction zone of the burner, enabling temperature measurements in the burnt-gas regime, based on the twocolor intensity ratio approach. 2D temperature measurements of the post-combustion gases were determined for various equivalence ratios, up to a gas temperature of 890 K. Phosphor particles were recaptured downstream of the flame and analyzed using SEM/EDX and XRD. Microscopic measurements did not show changes in chemical composition, morphology or crystal structure for YAG:Pr relatively to the initial material. In contrast to YAG:Pr, the second investigated phosphor, SMP:Sn does not exhibit any remaining luminescence downstream of the reaction zone. The microscopic analysis of the post-flame phosphor material revealed that the material had melted in the flame and undergone crystal phase transformation towards different phases of the (Sr,Mg)-orthophosphate (α-Sr 3 (PO 4 ) 2 and SrMg 2 (PO 4 ) 2 ).

This study shows that temperature measurements in the postflame environment using thermographic phosphors are not only limited by its weak luminescence at elevated temperature as widely assumed, but also by materials decomposition, which can lead from decreasing luminescence to as far as an entire loss of the materials' luminescence properties. Further investigations of other phosphor materials need to be carried out to extend the measurable temperature range and to know restricting conditions of the respective thermographic phosphors. The constraints to experimental conditions can additionally be determined from thermodynamic modeling of the melting process to determine acceptable residence times for the respective materials in the given temperature field.

Figure 1 :

 1 Figure 1: Arrangement for thermographic measurements in the postcombustion zone of a premixed burner.

Figure 2 :

 2 Figure 2: Top: Temperature and gas velocity over time of a particle through the flame. Bottom: Cumulative residence time of a particle at a given temperature or above in a premixed propane/air flame (ϕ = 0.71). The residence time for T ≥ 1473 K is highlighted with a circle.

Figure 3 :

 3 Figure 3: Single-shot images of the blue (left) and red (right) channel in the exhaust gas at ϕ = 1.0. The red and the yellow box highlight the regions used for SNR calculation. The intensity scale in each image was adjusted for improved visibility.

Figure 4 :

 4 Figure 4: Signal-to-noise ratio for both imaging channels for various fuel/air equivalence ratios.

Figure 5 :

 5 Figure 5: Average temperature field of the exhaust gas for  = 0.83 (left) and  = 0.63 (right) from two-color luminescence measurements using YAG:Pr.

Figure 6 :

 6 Figure 6: Temperature profiles from thermographic phosphor measurements above the burner along the vertical direction (solid lines) for various fuel/air equivalence ratios for YAG:Pr derived from the two-color luminescence measurements. TC: Thermocouple measurements with error bars.

Figure 7 :

 7 Figure 7: Raw (left) and associated binarized Mie-scattering images (right) for inert (top) and reacting (bottom) conditions for YAG:Pr.

Figure 8

 8 Figure 8 shows the SEM image from initial and post-flame material. Images from the initial material show mostly sharpedged particles that formed small agglomerates. Apart from few isolated spherical particles, YAG:Pr does not show any morphological alteration in the post-flame material. Results from EDX measurements do not show a statistically significant modification in chemical composition in the host material (Y 3 Al 5 O 12 ) of the post-flame YAG:Pr from the initial material. The concentration of the praseodymium ion doped into the host lattice is too low to be measured.

Figure 8 :

 8 Figure 8: SEM image of initial (left) and post-flame (right) YAG:Pr particles. A spherical particle is highlighted by a circle.

Figure 9 :

 9 Figure 9: Diffraction pattern from initial and post-flame YAG:Pr phosphor material with its COD reference pattern from[START_REF] Yoder | Complete substitution of aluminum for silicon: The system (MnO) 3 Al 2 O 3 (SiO 2 ) 3 -(Y 2 O 3 ) 3 (Al 2 O 3 ) 5[END_REF]. The diffraction pattern from the COD reference is calculated using the same copper anode (1.5406 Å) as used for the XRD measurements. Intensities are normalized towards the intensity of the highest peak.

  SEM-measurements of the morphology of particles from the post-flame SMP:Sn phosphor show severe changes relatively to the initial sharp-edged material, where the large majority of the post-flame particles are characterized by a spherical shape (Figure10, right). The chemical composition from EDX measurements in the initial material is Sr 2.62 Mg 0.38 (PO 4 ) 2 :Sn 2+ , which is equivalent to 12.5 mol% Mg 3 (PO 4 ) 2 . The post-flame material is slightly enriched in the relative strontium content Sr 2.66 Mg 0.34 (PO 4 ) 2 :Sn 2+ (11.3 mol% Mg 3 (PO 4 ) 2 ). These preliminary results confirm a qualitative modification on the structure from the exposure to the flame. As EDX provides semi-quantitative analysis only, the small change in chemical composition on a particle basis should be interpreted with care. The modification of the shape of the grains towards a spherical shape indicates a transformation to the liquid state in the flame. The return to the solid state does not seem to be accompanied by a general phase demixing since the change in chemical composition is low compared to the initial state.

Figure 10 :

 10 Figure 10: SEM image of initial (left) and post-flame (right) SMP particles.

Figure

  Figure11aare 46, 44, and 43.4 wt% respectively. This confirms a local change in the chemical composition of these particles, which indicates that there are also phase transitions of the crystal phases according to the phase diagram from Sarver et al.[START_REF] Sarver | Phase Equilibria and Tin-Activated Luminescence in Strontium Orthophosphate Systems[END_REF].

Figure 11 :

 11 Figure 11: SE (a), SE+EDX for Sr (b) and EDX (c-f) images from a polished surface as a function of the respective chemical species (Sr, Mg, P, O). The highlighted regions in (b) are regions of high Sr content (white) and regular Sr content (black). Note that the EDX images (c-f) are not overlaid with SE images. The normalized Sr wt% are shown in Table1.

Figure 12 :

 12 Figure 12: Diffraction pattern from initial and post-flame SMP:Sn phosphor material (top and center). The diffraction peaks (bottom) are visualized in different widths to improve visibility of closely adjacent peaks. Inset plot: enlargement from 2 = 29.4-30.6°.

Table

Table 1 :

 1 Normalized Sr content in various areas in the particle cross-section

	shown

1.

Acknowledgements

The authors thank Dr. Markus Suta (Utrecht University, Debye Institute for Nanomaterials Science) for the valuable input and Dr. Ulrich Hagemann (ICAN, University of Duisburg-Essen) for his support in evaluating the EDX data. This work was supported by funding from the German Research Foundation within project 275255277.