Using a mixture of crystalline-Ho:ZrO_{2}, precursor-Dy:Y_{2}O_{3}, and precursor-Eu:ZrO_{2} nanoparticles we develop thermal impulse sensors capable of measuring equivalent isothermal temperatures and durations during a heating event, with response times of <100 ms, and a temperature range of at least 673 K to 1173 K. In order to determine the temperature and duration from the sensors after the heating event we measure the sensors' fluorescence spectrum, which is then compared with lab based calibration data. By using two precursor materials with different reaction kinetics we are able to extract both temperature and duration. Based on blind sample testing we find that the sensors and calculation method are accurate for measuring temperature and duration, but currently suffer a lack of precision due to difficulties in producing homogeneously heated samples.

## I. INTRODUCTION

One of the key challenges in efforts to combat chemical and biological weapons of mass destruction is the development of explosives which are capable of producing fireballs of sufficient temperature and duration to destroy chemical/biological agents.^{1} Given the importance of the temperature and duration to the effectiveness of the explosive, it is therefore necessary to determine the temperature profiles experienced by nanoscale particles within an explosion.

The task of measuring temperature profiles experienced by nanoscale particles during explosions is a nontrivial task. The two most common methods for *in-situ* temperature determination—thermocouples and pyrometry—are ill suited to the task as explosions typically occur on the timescale of *μ*s or ms (which is too short for survivable thermocouples to be used) and are optically opaque (making pyrometry unable to probe the inside of an explosion).^{2,3} Additionally, both thermocouples and pyrometry measure a static position's temperature, whereas nanoscale particles (such as a biological agent) will move along an arbitrary path within the explosion.

Given the difficulties with *in-situ* pyrometry and thermocouples to measure temperature profiles within explosions, we are currently developing an *ex-situ* temperature measurement method to determine an explosive event's equivalent isothermal temperature/duration. This method consists of seeding nanoparticle thermal impulse (TI) sensors (consisting of lanthanide-doped inorganic oxide precursors^{4–10}) into an explosive fireball. When the oxide precursors are heated they undergo irreversible phase transitions (such as decomposition, nucleation, crystallization, etc.),^{11–13} which result in modifications to the local electric field that the lanthanide ions experiences. This modification in the local electric field results in irreversible changes to the lanthanide ions' spectroscopic properties due to the ions' energy levels depending strongly on the local field.^{11,14–25} By heating samples in controlled laboratory conditions (with isothermal temperature profiles) these spectral changes can be correlated to specific isothermal temperatures and durations allowing us to produce a mapping of the spectral properties as a function of isothermal temperature and duration.

Once the mapping of spectral properties is complete we can then use the TI sensors in a heating event—with an unknown temperature profile—and be able to determine the temperature profile's equivalent isothermal temperature and duration based solely on the spectroscopic properties of the heated TI sensors. Note that due to the *ex-situ* nature of our approach we are only able to determine an equivalent isothermal temperature and duration of a heating event instead of the precise temperature profile. This equivalent isotherm represents the temperature and duration required for an isotherm to match the thermal impulse of the actual heating event.

In this study we present results of calibration measurements of a TI sensor cocktail consisting of crystalline Ho^{3+} doped ZrO_{2} (c-Ho:ZrO_{2}), precursor Dy^{3+} doped Y_{2}O_{3} (p-Dy:Y_{2}O_{3}), and precursor Eu^{3+} doped ZrO_{2} (p-Eu:ZrO_{2}). Using these calibration data, along with a phenomenological kinetic model, we produce temperature-duration mappings of spectral peak intensity ratios and demonstrate the ability to accurately determine the temperature and duration of an unknown heating event.

## II. THEORY

Before considering our phenomenological kinetic model of phase transformations in TI sensors we briefly discuss an idealized model of phase transformations in order to better understand the underlying principles behind the phenomenological model. This idealized model assumes that the TI sensors are made up of two different materials, each with a single phase transformation mechanism and different kinetic parameters (in reality the TI sensors undergo multiple transformations with different mechanisms and kinetic parameters).

For the case of a single phase transformation (in which the material transforms from the amorphous phase to crystalline phase) the degree of conversion *α*—which represents the fraction of material in the crystalline phase—follows a generalized differential equation:^{26}

where *h*(*T*) is an Arrhenius function and *f*(*α*) is a function dependent on the transformation mechanism. In general, *h*(*T*) is given by

where *A* is the material's rate constant, *E* is the transformation energy barrier, and *k* is the Boltzmann's constant. While the form of *f*(*α*) depends on the phase transformation mechanism, a general expression for it is^{26}

where *m*, *n*, and *p* depend on the reaction mechanism.^{26–35} Substituting Equations (2) and (3) into Equation (1) we find the whole kinetic equation to be

Moving all the *α* and *t* terms to opposite sides we can solve Equation (4) to give a general solution of:

At this point further simplification of the general case cannot be done as both the integrals depend on the specific functional forms, some of which have analytical solutions while the rest require numerical solutions. To further simplify Equation (5) we must choose a specific phase transformation mechanism.

For our idealized model we will use the mechanism of nucleation and grain growth, which was previously shown to occur when heating p-Eu:ZrO_{2}.^{10} This mechanism has exponent values of *m* = 0, *n* = 1, *p* = 0.^{26} Assuming that the material begins completely in the amorphous phase, i.e., *α*_{0} = 0, Equation (5) can be integrated to give the degree of conversion as a function of time to be

The integral in Equation (6) has no general analytical solution and typically requires numerical methods to calculate. However, in the simplest case of a temperature profile—that of an isotherm—Equation (6) can be integrated to give the degree of conversion to be

where *T _{iso}* is the isothermal temperature and Δ

*t*=

*t*−

*t*

_{0}is the duration of the isotherm with

*t*

_{0}being the start of the isotherm. Note that from Equation (7) we find that the degree of conversion is not uniquely determined by either the temperature or the duration alone. Instead, the degree of conversion is found to depend on a functional product of both the temperature and duration, with the degree of conversion in temperature-duration plane having iso-conversion curves which are defined by

Given the conversion's simultaneous dependence on temperature and duration it is impossible to uniquely determine both the temperature and duration from a single material's conversion alone; this would amount to solving for two variables with only one equation. Since we are concerned with simultaneously determining two variables we will need two independent equations. This is achieved by using two different materials (with material parameters *A*_{1}, *E*_{1} and *A*_{2}, *E*_{2}) which have different phase kinetics. By measuring the degree of conversion of both materials after exposure to the same isotherm we obtain two coupled equations

Using Equations (9) and (10) and some algebra we eliminate the time dependence and determine the isothermal temperature in terms of the material's parameters and the measured degree of conversion, with the inverse isothermal temperature given by

With the isothermal temperature now determined we next calculate the isothermal duration by substituting Equation (11) into Equation (9) and rearranging to find the isothermal duration to be

Equations (11) and (12) (and knowledge of *A*_{1}, *A*_{2}, *E*_{1}, and *E*_{2}) allow us to compute an unknown isotherm's temperature and duration based solely on the degrees of conversion of the materials. However, while the idealized system yields an easy temperature/duration determination, in reality our TI sensor materials undergo several different phase transitions and the degree of conversion is not easily extracted from spectral data as the spectra depend on multiple factors (site symmetry, degree of phase transformation, presence of decomposition products, etc.).^{14} Therefore, while the above model provides a theoretical structure to understand how to extract the thermal impulse of an isothermal heating event, in practice we use a phenomenological model (described below) to analyze experimental heating events.

## III. METHOD

Our thermal impulse sensors consist of a mixture of fully crystalline Ho:ZrO_{2} (c-Ho:ZrO_{2},used as a spectral reference), precursor Dy:Y_{2}O_{3} (p-Dy:Y_{2}O_{3}), and precursor Eu:ZrO_{2} (p-Eu:ZrO_{2}) in a mixing ratio of 10:1:10.^{36} Each component is prepared by a co-precipitation method^{5,7} with an RE-dopant concentration of 1 mol%. The resulting cakes of materials are crushed with mortar and pestle and passed through sieves to yield a powder size distribution between 5 *μ*m and 149 *μ*m. To crystallize the c-Ho:ZrO_{2} reference material we heat the precursor material to 1273 K for 30 min in a furnace. The three materials are then mixed by vortexing for 10 min.

Once the sensors are prepared we place them in a graphite sample holder with a K-type thermocouple (0.125 mm diameter, 80 ms rise time for instantaneous 400 K temperature change), which is connected to a PID loop controller consisting of a LabJack T7 DAQ connected to a PC running a custom PID program. The combined PID system operates with a loop time of 6 ms and controls the output power of a Synrad Firestar f100 CO_{2} laser (150 W peak power), which heats the sample. Using this method we achieve heating rates of approximately 400 K/s and are able to achieve isothermal durations of as short as 100 ms.^{10,36} The heated samples are then placed in a fluorescence spectroscopy system consisting of a frequency tripled Q-switched Nd:YAG laser (Continuum Powerlite Precision II, 355 nm, 10 Hz, 8 ns), focusing optics, and an Acton SpectraPro 2750 monochromator (0.750 m length, 1800 grooves/mm, 500 nm blaze) connected to a PMT module.

## IV. RESULTS AND DISCUSSION

### A. Fluorescence spectra

To obtain a mapping of the sensor spectra as a function of temperature and duration we use six different temperatures (293 K, 673 K, 773 K, 873 K, 973 K, 1073 K, and 1173 K) and six different durations (0 ms, 100 ms, 300 ms, 600 ms, 1000 ms, and 6000 ms) with each temperature and duration combination repeated three times to obtain better statistics. After the samples are heated we measure their emission spectra (*λ _{ex}* = 355 nm) six times and compute their average to account for shot-to-shot variations in the pump laser. 355 nm excitation is used as all three lanthanide dopants are found to have transitions near 355 nm ($Ho:5I8\u21925F2,\u2009Dy:\u20096H15/2\u21926P7/2$ and $Eu:7F0\u21925L10$).

The measured averaged spectra are found to have a quartic background which is removed using multi-peak fitting. Figure 1 shows example background subtracted spectra of the TI sensor mixture as prepared and heated to 1173 K for 6000 ms, with the c-Ho:ZrO_{2} peak normalized. From Figure 1, for the heated TI sensor mixture, we find two closely spaced peaks in the 540 nm to 550 nm range which correspond to the $F62\u21925I8$ transition of Ho^{3+}. We also find in the heated spectra (shown in Figure 1) that the Dy^{3+} ion contributes a strong peak at 573 nm (corresponding to the $F49/2\u21925H13/2$ transition), while Eu^{3+} contributes three strong peaks at $592\u2009nm(5D0\u21927F1),\u2009606\u2009nm\u2009(5D0\u21927F2)$, and $714\u2009nm\u2009(5D0\u21927F4)$. All three ions also contribute weak peaks in the 650–660 nm range $(Ho:5F5\u21925I8,\u2009Dy:4F9/2\u21925H11/2,\u2009Eu:5D0\u21927F3)$. For reference we show a simplified energy level diagram for each ion in Figure 2 with the relevant transitions labeled.

While the Eu/Dy emission peaks from the heated TI sensor material are sharp and well defined, as shown in Figure 1, the emission from precursor Eu:ZrO_{2} and Dy:Y_{2}O_{3} (also shown in Figure 1) is found to be weak, broad, and having different peak positions. For p-Dy:Y_{2}O_{3} the emission is a broad peak centered at 578 nm, while for p-Eu:ZrO_{2} the $D50\u21927F1$ transition results in several overlapping peaks in the 580–600 nm range, the $D50\u21927F2$ transition corresponds to two overlapping peaks centered at 613 nm and 620 nm, and the $D50\u21927F4$ spectral peak is broad and centered at 702 nm. These spectral differences between the precursor and heated material are due to changes in the site symmetry and crystal field parameters, which occur as the material undergoes irreversible phase changes.^{11,14–25}

These phase changes depend on both the heating temperature and duration (as described above) and therefore the spectra will transform as the material is heated. Choosing representative spectra for each temperature and duration we plot the spectra for different heating conditions in Figure 3.

From Figure 3 we find that both temperature and duration strongly influence the spectra, with the influence of duration being much more obvious for low temperatures. This is to be expected given the iso-conversion relationship between temperature and duration (Equation (7)). As the temperature increases, the rate at which the material transforms increases making the influence of duration less noticeable. For example, the spectra for an isothermal temperature of 773 K (Figure 3(b)) show changes in spectral features over the whole duration range measured (up to 6000 ms), while for an isothermal temperature of 1173 K the spectra (shown in Figure 3(f)) are mostly unchanged from 100 ms to 6000 ms.

In addition to observing the influence of temperature on the rate of the material's phase transformations in Figure 3, we also find three distinct “phase regions” as a function of temperature and duration. The first region is characterized by broad Eu and Dy peaks, which correspond to p-Dy:Y_{2}O_{3} and p-Eu:ZrO_{2}, and occurs for all durations at 673 K and for durations less than 700 ms for 773 K. The second region (corresponding to p-Dy:Y_{2}O_{3} and tetragonal Eu:ZrO_{2}) begins forming at long durations for 773 K, or for all durations with a temperature above 773 K and is spectrally characterized by narrow peaks at 592 nm and 606 nm. This transition from p-Eu:ZrO_{2} to t-Eu:ZrO_{2} at a temperature between 773 K and 873 K is consistent with previous measurements of ZrO_{2}'s phase transformations.^{10} The final phase region (cubic Y_{2}O_{3} and t-Eu:ZrO_{2}) occurs once the temperature has passed 1073 K (consistent with previous measurements of Y_{2}O_{3}'s transformation temperature^{5,6,9,10,37}) and is characterized by strong narrow peaks at 573 nm, 592 nm, and 606 nm. Figure 4 shows a mapping of the different phase regions as a function of temperature and duration.

While the most obvious spectral change, when transitioning between phase regions, is the occurrence of sharp spectral peaks (corresponding to crystallization of the oxide host) there are additional changes to the overall structure of the spectra. To determine these changes we use multi-peak fitting on spectra from each region to de-convolute the spectra to obtain the underlying energy level structure. Figure 5 shows an example spectral deconvolution of the precursor material with the peak centers and strengths highlighted. Performing this analysis on each region we obtain the characteristic spectral properties for each region, which are tabulated in Table I.

. | Eu:ZrO_{2}
. | Dy:Y_{2}O_{3}
. | ||||
---|---|---|---|---|---|---|

Region . | Transition . | Peak center (nm) . | Peak strength . | Transition . | Peak center (nm) . | Peak strength . |

1 | $D50\u21927F1$ | 588 | m | $F49/2\u21925H13/2$ | 573 | w |

592 | w | 578 | w | |||

589 | m | 583 | vw | |||

$D50\u21927F2$ | 606 | vw | ||||

613 | s | |||||

620 | s | |||||

$D50\u21927F4$ | 702 | w | ||||

2 | $D50\u21927F1$ | 586 | m | $F49/2\u21925H13/2$ | 573 | m |

592 | s | 578 | m | |||

583 | vw | |||||

$D50\u21927F2$ | 606 | s | ||||

611 | m | |||||

627 | w | |||||

$D50\u21927F4$ | 694 | vw | ||||

702 | vw | |||||

710 | w | |||||

714 | m | |||||

3 | $D50\u21927F1$ | 586 | m | $F49/2\u21925H13/2$ | 573 | vs |

592 | s | 575 | w | |||

583 | w | |||||

$D50\u21927F2$ | 606 | vs | ||||

611 | w | |||||

627 | vw | |||||

$D50\u21927F4$ | 694 | vw | ||||

702 | vw | |||||

710 | w | |||||

714 | m |

. | Eu:ZrO_{2}
. | Dy:Y_{2}O_{3}
. | ||||
---|---|---|---|---|---|---|

Region . | Transition . | Peak center (nm) . | Peak strength . | Transition . | Peak center (nm) . | Peak strength . |

1 | $D50\u21927F1$ | 588 | m | $F49/2\u21925H13/2$ | 573 | w |

592 | w | 578 | w | |||

589 | m | 583 | vw | |||

$D50\u21927F2$ | 606 | vw | ||||

613 | s | |||||

620 | s | |||||

$D50\u21927F4$ | 702 | w | ||||

2 | $D50\u21927F1$ | 586 | m | $F49/2\u21925H13/2$ | 573 | m |

592 | s | 578 | m | |||

583 | vw | |||||

$D50\u21927F2$ | 606 | s | ||||

611 | m | |||||

627 | w | |||||

$D50\u21927F4$ | 694 | vw | ||||

702 | vw | |||||

710 | w | |||||

714 | m | |||||

3 | $D50\u21927F1$ | 586 | m | $F49/2\u21925H13/2$ | 573 | vs |

592 | s | 575 | w | |||

583 | w | |||||

$D50\u21927F2$ | 606 | vs | ||||

611 | w | |||||

627 | vw | |||||

$D50\u21927F4$ | 694 | vw | ||||

702 | vw | |||||

710 | w | |||||

714 | m |

From Table I we find that the primary spectral peaks for the precursor material arise due to the $D50\u21927F2$ Eu transition with the peak centers occurring at 613 nm and 620 nm. As the material is heated and transforms from region 1 to 2, both of the peaks at 613 nm and 620 nm decrease and shift (613 nm → 611 nm and 620 nm → 627 nm), while a strong peak grows at 606 nm. This spectral change occurs due to the local environment of the Eu^{3+} ions transforming from a low symmetry structure to a tetragonal crystal structure.^{10} This change in symmetry modifies the distribution of stark energy levels corresponding to the $D50\u21927F2$ transition as it is a highly sensitive electric quadrupole (EQ) transition.^{14} Additionally the phase transformation results in Eu's $D50\u21927F1$ transition to grow in intensity.

In the case of the Dy^{3+} ions' emission we find from Table I that for the precursor material, Dy^{3+}'s emission is relatively weak and centered at 578 nm ($F49/2\u21925H13/2$). Once the material has been sufficiently heated to transform from region 2 to 3, the peak at 578 nm is found to decrease and blueshift (578 nm → 575 nm), while a strong peak is formed at 573 nm and a weaker peak is formed at 583 nm. This spectral change is due to transition from the amorphous Yttrium carbonate to cubic Y_{2}O_{3}.^{5,6,9,10}

### B. Spectral peak ratios

With the TI sensor fluorescence spectra now tabulated we turn to the process of determining temperature and duration based on spectral measurements. To date several different *ex-situ* techniques have been used to correlate heating conditions to the spectral features of heated lanthanide-doped inorganic oxide precursors. These techniques include: spectral matching,^{10} intensity ratios using a reference material,^{10} intensity ratios using internal reference peaks,^{4} emission peak width,^{4} excitation linewidth,^{4,11} and fluorescence lifetime.^{4,5} In addition to these demonstrated *ex-situ* techniques, it is also possible to adapt *in-situ* spectral analysis techniques—from two-color thermometry—for use in our *ex-situ* analysis. These techniques include neural network reconstruction^{38,39} and spectrally integrated luminescence.^{40}

In this study we will use the method of intensity ratios with internal reference peaks, as these ratios coincide with the different material phases. Namely, taking the ratio of intensities at 573 nm and 578 nm compares the amount of c-Dy:Y_{2}O_{3} to p-Dy:Y_{2}O_{3} and the ratio of 606 nm to 613 nm compares the amount of t-Eu:ZrO_{2} to p-Eu:ZrO_{2}. Figure 6(a) shows the 606/613 ratio as a function of duration for different isothermal temperatures and Figure 6(b) shows the 573/578 ratio as a function of duration.

From Figure 6 we find that both the 606/613 and 573/578 ratios behave as simple exponential functions with time, having the form

where *g*(*T*) is the temperature dependent rate, *R*_{0}(*T*) is the temperature dependent asymptotic ratio, and *R*_{1}(*T*) is the ratio offset, which is due to non-instantaneous heating and cooling.

Fitting the ratio curves in Figure 6 to Equation (13) we find three parameters as a function of temperature, which are displayed in Figure 7. Each parameter displayed in Figure 7 is found to follow a modified Arrhenius function with temperature, given by

where *A* is an amplitude, *T*_{0} is a characteristic temperature, and *β* is a stretch factor. Note that while the idealized model (described above) predicts *β* = 1, experimentally we find that *β* > 1, which corresponds to the Arrhenius function having a steeper transition. Currently this observation is unexplained but—given that the function corresponds to a stretched exponential—it suggests that within the precursor samples there exists a distribution of sites with different characteristic temperatures.

Stretched exponentials have found common usage in the physical sciences with many applications, including describing electronic phenomenon,^{41,42} the kinetic Ising model,^{43} transformations of photochromic dye-doped polymers,^{44} glassy systems,^{45} chaotic oscillators,^{46} elasticity in polymers,^{47,48} luminescence decays,^{49–51} and the relaxation of supercooled liquids.^{45,52–56} Despite the drastically different mechanisms behind these phenomenon, the observation of stretched exponential behavior in each one arises due to a linear superposition of simple exponentials over a distribution of “rate constants.”^{49,57,58}

In the case of Equation (14) this stretched exponential behavior would amount to different sites within the precursor particles undergoing phase transformations (with local simple Arrhenius behavior with a characteristic temperature $T0\u2032$), while the ensemble behavior is a superposition over the simple Arrhenius transformations leading to an ensemble averaged modified Arrhenius behavior. To better understand the underlying kinetics and mechanism behind this behavior we are currently exploring different phase transformation models and refining our experimental measurements of the phase kinetics.

With the experimental ratio now fit as a function of temperature and duration we reverse the procedure to produce a two-dimensional map of the ratio as a function of temperature and duration. To do so we substitute the fit curves for *g*(*T*), *R*_{0}(*T*), and *R*_{1}(*T*) into Equation (13) and compute Equation (13) for different temperatures and durations. Figures 8(a) and 8(b) show the computed 606/613 and 573/578 ratios as a function of temperature and duration, respectively.

Figures 8(a) and 8(b) show the different phase kinetics of Eu:ZrO_{2} and Dy:Y_{2}O_{3}, with the 606/613 ratio (corresponding to Eu:ZrO_{2}) transitioning at a temperature near 850 K and the 573/578 ratio (corresponding to Dy:Y_{2}O_{3}) transitioning near 1100 K. Additionally, Figure 8 demonstrates the existence of iso-ratio curves as a function of temperature and duration, which correspond to the iso-conversion curves discussed above in Sec. II.

### C. Computing temperature and duration

With the ratio maps as a function of temperature and duration determined we now turn to the procedure for using the temperature-duration ratio maps to determine an unknown temperature and duration from a spectral measurement of a heated sample. First the measured fluorescence spectrum undergoes background subtraction and the 573/578 and 606/613 ratios are calculated. These ratios are then used to determine their respective iso-ratio curves on the temperature-duration map. In the case of ideal heating and ideal spectral measurements the two curves intersection point corresponds to the temperature and duration experienced by the material.

However, while ideal heating and ideal spectral measurements would allow for an exact determination of temperature and duration, in reality our heating and spectral measurements are not ideal and contain noise. This noise primarily arises due to non-uniformity in the heating of the material and noise in the spectral measurements. The net effect of this noise is to transform the single intersection point of the iso-ratio curves into an intersection “region.”

To demonstrate this effect we consider data from a sample heated to 1073 K for 500 ms, which is measured multiple times and found to have 606/613 ratios ranging from 2.2 to 3.4 and 573/578 ratios ranging from 2.4 to 3.7. Using both the upper and lower limit of the ratios we plot the iso-ratio curves in Figure 9. From Figure 9 we find that the four iso-ratio curves define a closed region with temperatures ranging from 1065 K to 1140 K and times ranging from 140 ms to 600 ms. Note that the nominal temperature and duration (1073 K and 500 ms) lie within the ranges determined by the intersecting curves.

### D. Blind tests

As a demonstration of the validity and accuracy of this temperature/duration determination method we perform five different blind tests in which samples are heated up to a given temperature and duration by one researcher, after which they are given over to another researcher to be measured using fluorescence. In this way the temperature-duration calculation is performed without bias.

Figure 10 shows the normalized background subtracted spectra from the five blind samples. Right away, without any calculations, we see that all three phase regions are represented by the spectra in Figure 10, with S1 and S4 belonging to region 1, S3 to region 2, and S2 and S5 to region 3. While we only present one spectrum per sample in Figure 10, we actually measure the spectra of each sample six times for different spatial configurations. Once the six spectra are measured and background subtracted we use them to compute the minimum and maximum intensity ratios. These ratios are then used in the procedure described above to determine a temperature range and duration range.

Table II tabulates the temperature and duration ranges determined by the sensors and the actual temperature and duration as measured by a thermocouple. Comparing the TC values and the sensor ranges in Table II, we find that in all cases the TC temperature and duration can be found within the ranges determined using the TI sensors. These results confirm that our TI sensors (and calibration) are capable of accurately determining isothermal temperatures and durations for temperatures ranging from 673 K to 1073 K and durations ranging from 100 ms to 6000 ms.

Sample . | TC temperature (K) . | TC time (ms) . | Sensor temperature (K) . | Sensor time (ms) . |
---|---|---|---|---|

1 | 673 | 1003 | 623–759 | 100–1790 |

2 | 1173 | 114 | 1136–1256 | 46–697 |

3 | 873 | 303 | 864–958 | 300–880 |

4 | Unheated | Unheated | ||

5 | 1073 | 605 | 1061–1142 | 138–681 |

Sample . | TC temperature (K) . | TC time (ms) . | Sensor temperature (K) . | Sensor time (ms) . |
---|---|---|---|---|

1 | 673 | 1003 | 623–759 | 100–1790 |

2 | 1173 | 114 | 1136–1256 | 46–697 |

3 | 873 | 303 | 864–958 | 300–880 |

4 | Unheated | Unheated | ||

5 | 1073 | 605 | 1061–1142 | 138–681 |

While the blind tests demonstrate the accuracy of our technique—with each TC temperature and duration within the TI sensor ranges—they also demonstrate the currently low precision of the technique. From Table II we find that our temperature/duration determination results in temperature ranges of ≈ 100 K and duration ranges varying from 543 ms to 1700 ms. These wide temperature and duration ranges arise due to several experimental difficulties and an important fundamental difficulty which arises when analyzing irreversible phase changes. The experimental difficulties include: (1) inhomogeneous heating of calibration and test samples, (2) limited heating/cooling rate, (3) a 20–30 K overshoot in the PID loop, (4) jitter in the isothermal timing, and (5) noise in the spectral measurements.

Currently our heating apparatus uses a CO_{2} laser to heat up a graphite sheet, which then transfers the thermal energy to our sample. This method relies on slow heat transfer from the graphite sheet to the powder sample, which results in a slow heating rate (400 K/s) and introduces uncertainty with regards to thermal contact and transfer rates. The net result of relying on contact between the powder and graphite sheet is that the sample will be inhomogeneously heated with some portions being hotter than others. This inhomogeneity is problematic for calibration as we measure the average spectra of the sample, which means that when taking spectral measurements we probe over a wide range of temperatures, resulting in uncertainty in the calibration data. These effects also arise when considering the blind samples as nonuniform heating will result in spectral measurements of different portions of the blind samples producing a range of measured intensity ratios. Given the compounding effect of inhomogeneous heating, first on the calibration samples and then on the blind samples, we can conclude that this experimental difficulty is the main source of imprecision in our thermal impulse calculations.

While inhomogeneous heating is the main source of uncertainty in our approach, there is also uncertainty introduced due to the fact that our heating profiles are not perfect isotherms. Ideally an isotherm would instantly reach temperature, hold for a fixed time, and then cool instantly. In practice our “isothermal” heating profiles have a slow ramp to a temperature above set-point followed by damped ringing as the temperature settles into the isothermal temperature and then after the set duration the laser is turned off and the sample cools exponentially (see Figure 11 for an example profile). Each of these deviations from the ideal isotherm will influence the degree of conversion (recall Equation (5)). As the ramp rate, damped ringing, and cooling rate are sensitive to experimental conditions, no two temperature profiles will be exactly the same. This means that there will always be some variation in the degree of conversion for two samples which were exposed to the same experimental temperature and duration.

To overcome the influence of imperfect heating profiles and inhomogeneous heating we are developing a direct laser heating method with ramp rates of up to 20 000 K/s, faster cooling rates, and better control over damped ringing. This technique will allow for uniform heating of the TI sensors at such a rapid rate that the influence of heating time will be negligible and, due to the removal of extra thermal mass, the cooling rate is also much quicker. Additionally we are upgrading the PID control system to better counteract the damped ringing to provide an improved isothermal profile. Figure 12 shows a preliminary example heating curve for the direct heating system, where the temperature is determined using a three-color pyrometer.^{59} Note that for this heating curve we limited the maximum output from the laser, thus limiting the heating rate to approximately 3000 K/s. From Figure 12 we find that the direct heating method achieves much higher heating rates and cooling rates, with the heating and cooling times being on the order of 100 ms, whereas the heating/cooling times for the indirect method are on the order of seconds. Also, from Figure 12, we find that for this heating run there is little to no overshoot or ringing, which improves over the indirect method that contains both artifacts (see Figure 11).

Finally, while the experimental difficulties to our TI sensor approach can be overcome with improved experimental design, there is a fundamental difficulty with using irreversible phase changes to measure temperature and duration that cannot be overcome. This difficulty arises due to the phase change kinetics and is best highlighted by considering Figure 6. From Figure 6 we find that after a certain period of time the ratios reach an asymptotic value, such that a large range of durations can have approximately the same ratio. This ambiguity implies that once the material has reached the asymptotic ratio value for a given temperature, the sensors can no longer determine a precise duration. Additionally, from Figures 6 and 7, we note that the rate of conversion increases with temperature, which implies that the precision of the time measurement decreases as the temperature increases (i.e., the material reaches its asymptotic ratio faster at high temperatures). Practically this means that the temporal precision depends on the temperature to which the material is heated.

## V. CONCLUSIONS

We develop an *ex-situ* thermal impulse sensor based on a cocktail of c-Ho:ZrO_{2}, p-Dy:Y_{2}O_{3}, and p-Eu:ZrO_{2} nanoparticles. When the TI sensor cocktail is exposed to a thermal event the precursor material undergoes irreversible phase transformations which result in changes to the material's optical signature. These changes in optical signature are the result of the local crystal field experienced by the lanthanide ions being modified by the irreversible phase transformations. Using calibration measurements and a phenomenological kinetic model we are able to relate the spectral changes in the sensor's optical signature to specific temperatures and heating durations. This calibration then allows us to determine the temperature and duration experienced during an unknown heating event.

Using this technique we demonstrate accurate temperature and duration determination for a set of blind samples with the method found to be accurate for a temperature range of at least 673 K to 1173 K and for durations <100 ms up to 6 s. While our approach is found to be accurate at determining temperature and duration, it is currently imprecise due to experimental difficulties associated with uniformly heating calibration samples. To address this issue we are currently developing a direct laser heating method, which will improve the calibration of the sensors spectral response. This improvement will allow for a more precise determination of temperature and duration using this method.

## ACKNOWLEDGMENTS

This work was supported by the Defense Threat Reduction Agency, Award No. HDTRA1-15-1-0044, to Washington State University.