Introduction

Hypersonic vehicles traveling in the upper atmosphere at speeds greater than Mach 5 create shock waves that lead to ultra-high temperatures (UHTs) at sharp nose and wing leading edges, where UHTs are defined as T > 1700 °C, above the use temperature of SiC-based materials [1]. Temperatures at the leading edge can exceed 2000 °C [2], which necessitates solutions other than SiC-based materials, such as ultra-high temperature ceramics (UHTCs). The need for advanced materials that can withstand UHTs [3] in these oxidizing environments is currently unmet. The ability to conduct controlled UHT oxidation studies of novel ceramics and alloys is therefore of interest to the aerospace community.

Ultra-high temperature testing presents many challenges. Reliable temperature control and measurement is required. Environment control, such as known partial pressures of oxygen (pO2) may be key to elucidating degradation mechanisms, particularly for materials with rapid oxidation rates. Additionally, given increased reactivity and reaction rates at high temperatures, container contamination and interaction with the material being tested have often been reported as issues [4].

An experimental set-up, shown in Figure 1 [5], addresses the above issues for oxidation tests on UHTCs, specifically on ZrB2, which is considered state-of-the-art in this class of materials [1]. The base configuration of the system was first introduced by Halloran et al. [6] to conduct studies on ZrB2-SiC ceramics in air for temperatures up to 2000 °C. The system employs Joule heating, wherein a high current is passed through a conductive sample and the intrinsic resistivity of the material results in self-heating. The current is controlled by a combination of a micro-optical infrared pyrometer and a PID controller, which requires the material to be opaque in the infrared (λ = 865 nm). Other modifications of the concept have been made by researchers in other labs to conduct mechanical tests at high temperatures [7].

Fig. 1
figure 1

Adapted from J. Am. Ceram. Soc. 98, 1673 (2015). Copyright 2015, The American Ceramic Society [5]

Resistive heating system schematic. (1) Chamber (2) pyrometer (3) heater + Eurotherm controller unit (4) transformer (5) copper electrodes (6) data acquisition computer (7) vacuum pump (8) valve to vacuum pump (9) valve for switching between gas tanks (10) backfill pressure gauge (11) current transducer (12) thermocouple gauge for vacuum readout (13) gas out/exhaust (14) flattened copper wires (15) gas inlet to chamber.

Full size image

The design for such a system was subsequently modified at the University of Virginia first by Shugart (now Cissel) et al. [5, 8], then in the course of the study presented in this publication. These improvements include (Fig. 1) (a) the use of an emissivity correcting pyrometer, (b) conduction of these tests in a closed chamber to enable testing in controlled oxygen partial pressures, and (c) the ability to conduct 18O2 double-oxidation studies to elucidate diffusion mechanisms. The details of the improvements made by Shugart et al., then further improved by the current authors and described here have not previously been published. The advantages of these improvements to the system include the ability to control the gases flowing into the test environment, as the enclosed chamber can be evacuated and back-filled with the reacting gases of interest, such as O2/Ar mixtures. In addition, due to the contact points being only on the cooler legs, this set-up minimizes contamination of the test volume with other materials. The test configuration results in axial temperature gradients, which have not yet previously been described or defined. In the case of oxidation experiments, the growth of an oxide scale and subsequent consumption of the base material can also result in temperature gradients through the thickness of the sample. Concerns exist that the surface temperatures reported in prior work may be significantly different than the oxide/substrate interface. Furthermore, there are concerns that the oxidation mechanisms may be different for Joule heating (the apparatus described here) than for radiative/convective heating via, for example, a furnace, as reported in other studies and of primary importance for hypersonic applications.

The objective of this study is to computationally evaluate temperature gradients in the test specimens to understand any artifacts of the test procedure on oxidation results. For this, the material used will be ZrB2. The temperature at the reaction front, in this case, the oxide-substrate interface, where substrate refers to the starting material, is critical toward elucidating oxidation mechanisms. Modelling the temperature gradient through the oxide and along the long axis of the specimen are both needed to interpret the oxidation results. In this work, a thermal-electric finite element analysis of the specimen is conducted in order to compute the oxide/substrate interface temperature. Several oxide thicknesses were analyzed to correlate the measured surface temperature with the interface temperature based on the thickness of the oxide. Additionally, the axial temperature gradients were evaluated as a function of oxide thickness and temperature. Finally, the current densities were calculated, and results considered in the context of its influence on mass (or ion) transport.

Resistive Heating System

The resistive heating system (RHS, Fig. 1), uses a dogbone shaped specimen as seen in Fig. 2. The pyrometer sighting zone, approximately 1.5 mm in diameter, is delineated by an orange circle in that schematic. Flattened copper wires (approximately 1.1 × 2.7 × 20 mm) were placed in contact with the copper electrodes with custom mechanical lugs (Fig. 3) to ensure contact between the copper wires and the current feedthrough. The free ends of the copper wire, i.e. the parts of the copper wire that would be in contact with the sample, were covered in platinum foil (0.05 mm thick, Alfa Aeser Premion™, Fisher Scientific), to prevent interaction of the sample with the Cu electrodes; other materials may be considered depending on the material being tested. The sample is loaded straddling the two platinum-covered copper wires, Fig. 1. Contact between the sample legs and the copper wires was ensured by using modified alligator clips, Fig. 3b. The teeth were removed and the inner surfaces of the alligator clips were isolated with thin yttria-stabilized zirconia (YSZ) plates (cut to approximately 0.4 × 3 × 9 mm, material from Ortech Advanced Ceramics, Sacramento, CA). These plates were adhered to the alligator clips using a cyanoacrylate adhesive (Ted Pella, Redding, CA) to insulate the alligator clips from the sample, minimize excessive heating of the clips and prevent interaction with the sample legs. For this component, other insulators may be chosen based on the material being tested. Figure 3b shows the final stack-up in cross section, wherein the sample is in direct electrical contact with the platinum-covered copper wires, and held in place with YSZ-lined alligator clips. An image of the sample loaded in the chamber is also shown in Figure 4.

Fig. 2
figure 2

Test specimen configuration and dimensions. Orange circle and arrow indicate the approximate size and location of the hot zone, or the area of the sample known to be at the test temperature. Estimated size of the hot zone is 1.5 mm, which is also the spot size of the pyrometer. All dimensions are in mm.

Full size image
Fig. 3
figure 3

a Model of the custom lug design. Copper electrode and wires are indicated in Fig. 1 as labels 5 and 14, respectively. (Design courtesy of Tanner Fitzgerald, University of Virginia Department of Materials Science and Engineering.) b Schematic showing sample clamping configuration during test set-up in cross section, where the current path is in the out-of-page direction

Full size image
Fig. 4
figure 4

a Experimental set-up showing the locations of the gas inlet and vacuum port (with a valve that switched between back-fill/flow and evacuation modes. In-set shows a top view of the chamber during operation. Yellow arrows show the direction of gas flow. b Top view of the chamber before operation, and with flanges removed to show the sample loading and direction of gas flow across sample

Full size image

The pyrometer sights on the center of the thin section of the specimen (orange outlined region in Fig. 2) through a fused quartz window. The chamber is then sealed with copper gaskets and Conflat flanges (MDC Vacuum, Hayward, CA), evacuated, and purged with argon several times before being backfilled with Ar (Ultra-high purity grade, Praxair, Danbury, CT). The exhaust ball valve (Fig. 1, label 13) is opened to allow Ar flow, prior to heating. Gas flow is maintained between 950 and 1000 sccm throughout the test. The inlet gas tubing (smooth bore 306 stainless steel per ASTM A213) has an inner diameter of 0.49 cm (0.194″). The free stream gas velocity in the tubing prior to entering the chamber is calculated to be approximately 87 cm/s for a volumetric flow of 1 L/min. The chamber (see Supplemental Materials) has a complex cross section; the chamber opening on either side of the sample is 0.370 cm (0.75″). At this chamber opening, the cross-sectional area is 11.4 cm2 and the calculated free stream gas velocity is 5.9 cm/s for the same volumetric flow.

The pyrometer (Fig. 1, label 2) (Pyrofiber Lab PFL-0865-0790-2500C311, Pyrometer Instrument Company, Ewing Township, NJ) works in conjunction with the MHI heater + Eurotherm controller (Fig. 1, label 3) controls the ramp and dwell temperature of the specimen according to the set program. The temperature limits for the pyrometer are 790–2500 °C. Depending on the required test, material under consideration and transformer (described below), a higher temperature may be achieved by obtaining a pyrometer with a higher limit. A transducer is placed on a lead cable exiting the transformer and provides a measurement of the current in the line, visualized using LabView. AC currents of 30–120 A have been used to resistively heat the materials being tested. The current limit of the present transformer is approximately 120 A; however, a larger transformer can be specified to increase the current and therefore the temperature limits. Both the pyrometer and current transducer acquire four data points per second for temperature and current, respectively.

Temperature and current read-outs (Fig. 1, label 6) were used to refine the program and PID settings. The test temperature is achieved in approximately two min, and dwell time at test temperature ranged from 2 to 30 min. When the program reached the dwell stage, the current was allowed to stabilize before the valve (Fig. 1, label 9) was switched from Ar to the oxidizing gas, an O2 + Ar mixture. Certified gas mixture compositions ranging from 100 ppm O2 to 1% O2 in Ar (Praxair, Danbury, CT) were obtained and used for these studies. At completion of the desired test duration, the heater was turned off and the gas valve switched back to Ar. This is a modification of prior test protocols, which were conducted from ramp to quench entirely in the oxidizing atmosphere.

Gas flows from the inlet (Fig. 1, label 15) to the outlet (Fig. 1, label 13) across the surface of the sample, as indicated by Figure 4. The pyrometer was aligned to sight on the surface of the specimen through the fused quartz window in the chamber. When the oxidizing gas is turned on, the material begins to oxidize. The changing emissivity of the sample due to oxidation is corrected for by the pyrometer and the accompanied software [9]. The required correction is determined by measuring the difference between the energies of a pulsed laser sent to the target and that of the reflected laser signal. This difference allows for the calculation of absorptivity and therefore emissivity. The measured radiance is then corrected with this emissivity value with reference to black body radiances [9,10,11,12].

Specimens tested in this system include high entropy ultra-high temperature ceramic ((HfZrTiTaNb)C, (HfZrTiTaNb)B2, (HfZrTiTaMo)C, (HfZrTiTaMo)B2, (HfZrTiMoW)C, (HfZrTiMoW)B2), ZrC and ZrB2 and ZrB2-SiC (although the analysis of temperature gradients will focus on ZrB2). Samples were fabricated into pucks or bars via spark plasma sintering (SPS) [13,14,15]. The sintered materials were then machined by grinding (Bomas Machine Specialties, Somerville, MA), into the bridge geometry [5] previously used and the dogbone configuration shown in Fig. 2. This dogbone configuration is an improvement over prior iterations of this system as it eliminated sample attrition by premature fracture of the thin cross-sections due to the interaction between thermal expansion and the point contacts during the temperature ramp stage; a completely flat configuration exhibited more resilience over the bridge configuration and simplified sample loading. The thickness of the overall dogbone is dictated primarily by the required test temperatures and available power; with the transformer available, the requisite temperatures were achieved with a 0.5-mm thick sample.

The larger cross-sectional area at the ends of the specimen allow for the ends to remain cooler than the hot zone (estimated temperatures presented in the results section). This results in a large temperature gradient along the length of the sample, from the hot zone outward. This temperature gradient is evident in the varicolored oxide formation on an (HfZrTiTaNb)C sample oxidized at 1700 °C in 1% O2 for five minutes (Fig. 5). The hottest midsection of the specimen is suspended, preventing interaction with any container material.

Fig. 5
figure 5

(HfZrTiTaNb)C oxidized at 1700 °C in 1%O2 for five mins using the resistive heating system. Note color and oxide gradient from center outward reflects the temperature gradient and formation of different oxides

Full size image

The sample is heated and held at test temperature via Joule heating. During the test, an oxide forms, which has different electrical and thermal properties compared to the base material. The pyrometer, which operates on an infrared wavelength, sights on the surface of the oxide. The base material is expected to be more conductive, and as such conducts most of the current, and therefore may heat up more than the oxide. This results in a temperature gradient through the thickness of the oxide. This is illustrated in the schematic in Fig. 6.

Fig. 6
figure 6

a Side view of the dogbone specimen. Yellow regions indicate, schematically, the formation of an oxide on the top and bottom surfaces of the sample (oxide growth on the side not shown for clarity). The red box shows, also schematically the hot zone region. b Volume element of the region inside the red box in (a). The yellow regions indicate the oxide. (not drawn to scale). Directions of the current and gas flow are shown. Gas flow is into the page. Tmeasured is the known temperature given by the pyrometer reading, which is sighting on the surface. Tinterface is the unknown temperature at the region of interest, i.e., the oxide-substrate interface

Full size image

Estimated Temperature Gradient Through Oxide Thickness

A preliminary, analytical estimate of the ΔT through the oxide thickness was conducted for ZrO2 formation on ZrB2 with the following assumptions and simplifications. Oxidation of ZrB2 occurs via Reaction 1.

$$\begin{array}{c}Zr{{\text{B}}}_{2}+\frac{5}{2}{{\text{O}}}_{2}\left(g\right) \leftrightarrow Zr{{\text{O}}}_{2}+{{\text{B}}}_{2}{{\text{O}}}_{3}\left(l,g\right)\end{array}$$
(1)

The presence of B2O3 (l, g) was neglected for this analysis, which is a reasonable assumption due to the high vapor pressure of B2O3 [16]. Heat flux along the length of the sample was considered insignificant compared to the heat flux through the thickness of the sample and was therefore neglected. A DC current of 75A was assumed instead of AC current. The following energy balance equation was used [17]

$$\begin{array}{c}kV\frac{{d}^{2}T}{d{x}^{2}}+{I}^{2}R=0\end{array}$$
(2)

where k = thermal conductivity in W m−1 K−1, V = volume (cm3), T = temperature (K), x = dimension (cm) in direction of the oxide through thickness, I = current (A), R = overall resistance (Ω).

The oxide was assumed to be relatively dense, stoichiometric and adherent in this initial approximation. Thermal boundary conductance was neglected as relative to the extremely low thermal conductivity of the oxide, the contact resistance at the interface would be insignificant. The assumption of a dense scale allowed the simplification of the geometry of the system. The formation of a stoichiometric oxide is assumed to enable the use of literature values of ZrO2 to model the test volume element (the hottest region). The need for the assumption is due to the fact that ZrO2 can have a lot of oxygen vacancies. In addition, heat transfer via radiation from the surface and convection due to gas flow was also neglected for this estimate. Oxide thickness for a representative sample was used based on measurements taken as defined in Section “Results” (see also Supplemental Materials for a schematic of this, and a representative figure of an oxidized ZrB2 cross section). The electrical resistivity value for polycrystalline hot-pressed ZrB2 was extrapolated from values in literature [18] and was assumed to be \(2.5 \times 10^{ - 5} \,\Omega\cdot\;{\text{cm}}\) at 1700 °C. The electrical resistivity for ZrO2 was estimated to be \(2.64 \,\Omega \cdot\,{\text{cm}}\) also based on the literature value [19] for the yttria-stabilized zirconia (YSZ) with the lowest yttria content (7 mol%) in that study at 1500 K. It is readily apparent from Eq. 2 that the current flowing through the oxide will be insignificant based on the relative resistivities of ZrO2 and ZrB2, indicating that the electrical resistivity of ZrO2 layer is not a significant consideration in the calculation of the overall resistivity, R. Most of the electrical power, or the second term in the energy balance equation is generated in the boride and is dissipated via heat. It was assumed that the entirety of the boride region was isothermal, so that this power is available and present to contribute to the heat flux at the oxide/substrate interface.

It was further assumed that the heat flux in the direction from the oxide/substrate interface to the oxide/gas interface dominates in the system. This is reasonable given that gas is flowing across the oxide/gas interface at ~ 5.9 cm/s. Linearizing the energy balance equation allows the estimation of ΔT, or Tmeasured-Tinterface. The thermal conductivity of yttria stabilized ZrO2 was estimated from literature [20,21,22] to range from 1.5 to 3 W m−1 K−1 in studies considering porosity effects, yttria content or synthesis method; atmospheric plasma sprayed YSZ was closer to 1 W m−1 K−1. The value of 2.17 W m−1 K−1 for fully dense 8 mol% YSZ (at highest temperature measured, 1273 K) was chosen to be as consistent as possible with the case chosen from the literature for electrical resistivity. However, this value is also similar to 3 mol% YSZ with 19% porosity. The thermal conductivity values for monoclinic ZrO2 are closer to 4 W m−1 K−1, [20] indicating that the value chosen is a conservative estimate. Equation (2) was solved for ΔT; it does not require the specification of a test temperature, although the electrical resistivity value of ZrB2 at 1700 °C was used. These preliminary calculations yield ΔT to be ~ 20 K, which will be compared with the computational model described next.

Thermal Electric Model of RHS Specimen Temperature Gradients

Modeling Work

Computational Fluid Dynamics (CFD) Analysis

The CFD analysis uses a constant outer surface temperature and computed the total heat loss to the flowing gas. The CFD analysis was performed using US3D code [23] with the viscosity gradient computed using a weighted least squares viscosity model. The goal of this analysis was to obtain heat transfer coefficients as a function of specimen surface temperature. The heat transfer coefficients were used as a boundary condition in the thermal electric model (Fig. 7). The CFD analysis did not include radiation to the chamber walls.

Fig. 7
figure 7

Flowchart summarizing the analysis process for computing thermal gradients in RHS dogbone specimens

Full size image

The heat transfer coefficient was calculated as follows. The CFD model returned the total average heat flux given a constant wall temperature. The radiation heat flux was calculated based on the Stefan-Boltzmann equation. This value was subtracted from the total average heat flux generated by the CFD model, which gives the heat loss due to convection. The following equation was used to compute h, the heat transfer coefficient in W m−2 K−1.

$$\begin{array}{c}{q}_{{\text{conv}}}^{{\prime}{\prime}}=h \left({T}_{{\text{wall}}}-{T}_{{\text{inf}}}\right)\end{array}$$
(3)

where qconv’ is the heat loss due to convection in kW/m2, Twall and Tinf are the wall temperature and free stream temperature in °C, respectively. The free stream temperature is known to be 22 °C. The value for h was calculated from 750 to 2000 °C in 250 °C increments and summarized in Table 1. Because radiation was not included in the CFD analysis, it was accounted for separately in the finite element analysis (see Fig. 7 for order of operations).

Table 1 Results obtained from the CFD analysis for the heat flux and transfer coefficients
Full size table

Quarter symmetry of the specimen was modeled to minimize CFD computation (Fig. 8). Link3D was used to generate topology on the specimen and then smooth the grid. The smoother showed strong signs of convergence with a residual of ~ 1.0 smoothed to a residual of ~ 1 × 10–8. Calculated grid orthogonality and stretching as well as visual inspection verified a good grid was generated. In the extreme case, no cell had an aspect ratio greater than 3:1 in the CFD grid. CFD analysis was run on five different surface temperatures to obtain heat fluxes over the specimen. All four cases started with room temperature gas throughout the fluid with a velocity of 5 cm/s. The simulation was then run to find the equilibrium between the hot surface and the fluid. All simulations showed good convergence properties. Heat flux and heat transfer coefficients generated by CFD analysis are shown in Table 1.

Fig. 8
figure 8

a Geometry of the grid used for CFD analysis, b smoothed grid, and c close-up of the smoothed grid

Full size image

FEA Analysis

ANSYS Workbench 19.0 was utilized to perform the thermal electric analysis, available only as a steady-state analysis, limiting the current application to DC currents. The oxide/substrate specimens were modeled as snapshots in time, with uniform oxide thickness on the top and bottom surfaces. Based on laboratory tests [6, 24, 25] and as shown in Fig. 5, the oxide forms mostly in the gauge section of the specimen. Additionally, for this material system, the volume change during the oxidation process was neglected, although molar volume expansion of ~ 14% has been reported [26] upon oxidation of ZrB2. Thus, in the model, oxide thickness was equal to substrate thickness loss—the outer model dimensions remain constant regardless of oxide thickness. This allows simplification of the FEA simulation model, while maintaining fidelity with the reduction of the substrate material thickness due to oxidation, i.e., material consumption. This simplification takes advantage of the fact that the resistivity of the oxide is orders of magnitude higher than that in the starting material and the thermal conductivity is at least an order of magnitude lower. Analytical models (Section “Estimated Temperature Gradient Through Oxide Thickness”) have confirmed that the current flow in the oxide is negligible.

The finite elemental analysis uses the thermal boundary conditions computed using CFD analysis (previous section), the material properties of ZrB2 and ZrO2, and oxide thickness/material consumption to compute axial and through-thickness temperature gradients for a given current. In the mesh generated for this analysis, the cells had aspect ratios ranging from 0.95:1 to 1.05:1 current was provided as an input in the ANSYS Thermal-Electric model. Thermal and electrical properties used in the model [18, 19, 27, 28] for both ZrB2 and ZrO2, assumed to be dense and polycrystalline, are summarized in Table 2. A temperature-dependence for ZrB2 resistivity was identified in the literature [18]. No such relationship was identified for ZrO2. The inclusion of temperature dependent ZrB2 resistivity significantly increased the solution time. To mitigate this, the models were analyzed with a single resistivity at a temperature of 1700 °C. Like the preliminary calculations, the oxide was assumed to be dense and adherent.

Table 2 Values of thermal properties used for the FEA study
Full size table

The thickness of the hot zone was held constant, i.e., any oxide formation was assumed to replace the substrate without any volume change. This assumption was considered reasonable due to the resistivity of the oxide being orders of magnitude higher than the substrate or oxidizing material. Additionally, modelling a thicker oxide would have either added a singularity (step function) at the legs where there was no oxide or introduced an arbitrary curvature to mate the edges of the oxide with the leg. This was thought to add complexity to the system that was not necessary to elucidate the thermal gradients in the perpendicular direction, which was the more important consideration. The Joule heating was modeled in steady-state conditions to obtain “snapshots” in time. Four cases for oxide thickness were chosen: 0 µm, 30 µm, 75 µm and 150 µm, which span the range of experimental observations across the different materials tested. The surface temperature dependence on current was obtained by iterating the model for different current values.

The thermal boundary conditions consisted of radiation and convection. The radiation boundary condition was applied to all surfaces except the ends of the specimen where the electrical loads were applied. An emissivity of 0.8, which is expected for ZrO2 [29], was applied to all surfaces. The convective boundary condition relied on the CFD results. The heat transfer coefficients, shown in Table 1 are dependent on the surface temperature of the specimen. The free gas temperature was assumed to be at room temperature, 22 °C. The electrical loads consist of an applied current and a specified voltage. The current was applied at one end of the specimen (Fig. 9a). While an alternating current was applied in the experimental set-up, the finite element model was limited to direct current. The DC equivalent current (RMS) was computed by multiplying the peak AC current by √2/2. A voltage of 0 V, required to ground the model, was applied to the specimen as shown in Fig. 9.

Fig. 9
figure 9

a Model created and used for the FEA study. b Section view of the model used for the FEA study. The dark blue region indicates the oxide, gray region indicates the substrate. 150 µm model is shown. The locations of the electrical loads are shown in (a)

Full size image

Results

Figure 10 plots the current and temperature as an example for an actual test run for ZrB2 oxidation at a target temperature of 1500 °C. The average temperature was 1504 °C and a current variation of ± 1.5 A resulted in a standard deviation of 12 °C. Table 3 summarizes the average temperatures and standard deviations data collected from oxidation tests mainly on ZrB2 at different temperatures. For ZrB2 tested at 1700 °C, the best standard deviation achieved was 35 °C. On the other hand, tests at 1800 °C exhibited a much larger standard deviation (± 99 °C), indicating poorer temperature control at this temperature. Subsequent efforts to refine the PID controls resulted in lowering the deviation to ± 60 °C at 1800 °C, although these refinements were done on a high entropy diboride, (HfZrTiTaNb)B2. Additionally, Shugart found previously during calibration tests conducted by melting platinum and palladium wires that the pyrometer reading was 16–28°C lower than the known melting temperatures of those elements [24]. Table 3 also lists material consumption for ZrB2, which was determined by taking half the difference of the measured thickness of the unoxidized material and the original, unoxidized cross-section thickness. The material consumption data for ZrB2 were the basis for the models that were built.

Fig. 10
figure 10

a Pyrometer reading of sample surface, trendline shown in orange and b current transducer reading from lead wires, during the 15 min “isothermal” oxidation test of ZrB2 exposed to 0.1%O2

Full size image
Table 3 Actual material consumption observed on ZrB2 tested in the resistive heating system
Full size table

Figure 11 shows the morphology of the oxidized region on a sample tested at 1504 °C for 15 min. The oxide that formed was mostly dense, with some porosity on the surface. The oxide was adherent, and the thickness is shown to be similar to the material consumption reported in Table 3.

Fig. 11
figure 11

Combined backscattered electron and secondary electron image of the fracture cross section of an oxidized ZrB2 specimen (ZrB2-18B DB5), after exposure to 0.1%O2 for 15 min at a nominal temperature of 1500 °C

Full size image

Figure 12 shows an example of the results of the temperature distributions calculated using the simulation, for the case of a 30-µm oxide at a surface temperature of 1500 °C. The resulting temperature distributions and gradients were consistent with visualized variations in oxidation behavior along the axis of the dogbone, most apparently seen in the specimen shown in Fig. 5.

Fig. 12
figure 12

Example simulation result for the case of ZrB2 with a 30-µm oxide/material consumption at 1500 °C

Full size image

The surface and oxide-substrate interface temperature results for currents ranging from 60 to 150A were calculated in 10A increments (Supplemental Materials, Figure S2). These plots were then used to interpolate the current values for the nominal surface temperatures of 1500 °C, 1700 °C and 1800 °C for each oxide thickness. Using these interpolated current values as inputs for each target surface temperature and oxide thickness, the model was used to then calculate the axial and through thickness temperature gradients for these conditions. The current values and resultant gradients are shown in Fig. 13.

Fig. 13
figure 13

Temperature gradients through the thickness of the gauge section and axial length, respectively for a–b 30 µm oxide thickness (or 30 µm material consumption), c–d 75 µm and e–f 150 µm

Full size image

Each discrete oxide thickness modeled can be considered as snapshots in time. From the results shown in Fig. 13, an understanding of the temperature gradients through the oxide thickness can be determined as a function of oxide thickness, i.e., as the oxide “grows.” The current required to maintain the surface temperature at the target test temperature were plotted as a function of oxide thickness, and shown in Fig. 14a. The ΔT through the oxide thickness for the currents required to maintain nominal surface temperatures of 1500 °C, 1700 °C and 1800 °C, respectively, were plotted, and shown in Fig. 14b. These plots show that the temperature differential grows on the order of 0.3–0.4 °C per µm of oxide thickness in the temperature regime of interest. While a drop in current was calculated and expected, Fig. 10 shows that for oxidation experiments on ZrB2, a small current increase was seen at the beginning, which then stays relatively constant during the test duration.

Fig. 14
figure 14

a Current variation as the oxide thickness grows to maintain the given surface temperatures (1500 °C, 1700 °C, 1800 °C), b temperature differential through the oxide thickness as a function of oxide thickness. Lines shown in (b) are best fit lines

Full size image

Current Density

The simulation also allows the determination of current densities in the regions of interest. This parameter is of interest as electromigration, i.e., ion displacement and mass transport, can occur as a result of high current densities (> 108 A/m2) [30]. Results for the two extreme cases under consideration, i.e., 1500 °C for 30 µm and 1800 °C for 150 µm, are presented in Fig. 15. The calculated current densities were highest in the substrate, on the order of 108 A/m2. The current density in the oxide was ~ 60–70 A/m2.

Fig. 15
figure 15

Total current density (A/m2) for sample with: 30 µm oxide thickness at 1500 °C a with the oxide visible, b oxide hidden to show the current density distribution in the substrate; and for a sample 150 µm oxide thickness at 1800 °C c with the oxide visible, d oxide hidden to show the current density distribution in the substrate

Full size image

Discussion

Analysis of Results

The results presented here are from preliminary modeling and analysis of the temperature gradients that would exist in the oxidation of an ultra-high temperature ceramics. Some assumptions were made to simplify the problem for analysis and/or to mitigate excessive computationally expensive complications. These simplifications include the assumption that the majority of the current passes through the oxidizing material (not oxide), thus the important parameter is the shrinking cross section, and the geometry of the oxide layer was simplified to replace the volume lost in the substrate. Assumptions regarding the nature of the oxide being dense and adherent allowed the use of literature values, although further refinements to the model can be made, as reviewed in the next section. These assumptions were generally supported by prior observations of the formation of ZrO2 on ZrB2 [16], and in Fig. 11 and Supplemental Materials (Figures S2, S3).

The FEA temperature gradient simulations were based on similar material assumptions as the preliminary analytical calculations but consider heat transfer to the chamber and to the gas during the experiment and convective cooling. Nevertheless, these simulations provide results that are on the same order of magnitude as Eq. (2).

Figure 14a shows that the rate of change in current changes as the oxide gets thicker (non-linear dependence of current on oxide thickness). The example plot shown in Fig. 10b for the current does not show a significant reduction in current; however this could be due to the thinner oxide for the case shown there. Tests that result in thicker oxides do show a drop in current (Supplemental Materials). The current dependence on the oxide thickness predicted by the simulations confirms that most of the current was conducted through the base material (diboride in this case), and that such a dependence can be used to monitor the extent of oxidation observed during the test. Conduction through the base material is advantageous as many oxides can have ionic character; the very small current densities preclude the possibility of ion displacement due to high currents. The current densities in the base material were on the order of 108A/m2. Current densities higher than this are needed to result in mass transport, as discussed by Tu et al. [30] Shugart et al. [5] compared the morphology of ZrB2-SiC exposed to oxidizing conditions in the box furnace and the resistive heating system for exposures at 1500 °C for 20 min. No artifacts in the oxide composition and morphology were observed in the samples tested in the RHS relative to those tested in the box furnace.

The lateral hot zone, as shown by the axial plots (Fig. 13b, d and f) were expected to be within the pyrometer sighting zone of 1.5 mm. Taking the case of the 75 µm oxide at a surface temperature of 1800 °C, the zone of 1800 °C ± 10 °C was calculated to be approximately 1.97 mm in length. This hot zone decreases negligibly with increasing temperature and oxide thickness, with the worst case (among those calculated) being 1.9 mm in diameter for the 150 µm case at a surface temperature of 1802 °C.

The results show that the temperature differentials through the oxide thickness were generally on the order of the standard deviation in temperature measurement and the uncertainty of pyrometer reading for typical oxide thicknesses observed (Table 3). The ΔT through the oxide thickness increases as a function of test temperature. The maximum differential was obtained in the extreme case of 150 µm and 1800 °C surface temperature, where ΔT = 65 °C. The highest material consumption for ZrB2 in the tests conducted was 61 µm, well below 150 µm. Tests conducted on ZrB2-SiC exhibited oxide thicknesses on the order of 18.1 µm (combined borosilicate glass and ZrO2 layer thicknesses) [5]. Most material consumption values for ultra-high temperature carbide or boride ceramics observed were below 100 µm. The highest material consumption observed thus far was on the order of 125 µm for (HfZrTiTaNb)C tested in 1%O2 for 5 min at 1700 °C (Fig. 5); however, much of the oxidized region consisted of intergranular oxidation [25], instead of a dense 125 µm fully oxidized scale.

Recommendation for Further Refinement of the Test Set-up and Analytical Tool

Literature values were used to develop and assess the model, the results of this were presented here, but the model itself can be updated with measured values for emissivity, thermal conductivity, electrical resistivity and thermal boundary conductance as obtained or needed for each study being performed; this was outside the current scope and funding for this study. Second, more complex geometries for the oxide, including accounting for volumetric expansion could help refine the simulation further. Modification of the system to a larger transformer and a second pyrometer would allow for higher temperature ranges; the second pyrometer would also allow confirmation of surface temperatures. Precise measurements of oxygen stoichiometry in the oxide and the attendant change in properties may be useful to confirm whether this contributes to the deviations from the predictions discussed here.

Conclusions

The preliminary models developed and presented confirm that the through oxide temperature differentials are on the order of the standard deviation in measured temperatures for most observed conditions. While further refinements to the model and test set-up are possible, the preliminary results indicate that the RHS is a viable method to test UHTCs for oxidation resistance. Further, the tools developed for this study outlined above can be used to account for these differentials, and can help define the experimental conditions such that the target test temperature occurs at the oxide/substrate interface. These models provide an understanding of the actual temperature ranges to be considered when evaluating material performance or behavior, e.g., the temperature dependence of material consumption during oxidation. Finally, consideration of calculated current densities even in the extreme cases suggest that mass transport via electromigration is not expected to occur.

The simulations conducted in this study provide an important understanding of the temperature gradients observed in using resistive heating to conduct oxidation experiments. These results are the first step in understanding how to tailor experimental conditions required to conduct these oxidation experiments. They further confirm that the temperature differences between oxide and reaction interface are on the order of the other uncertainties inherent in this technique. The advantages of conducting oxidation experiments in the RHS, on the other hand, are numerous, including a hot zone free of contamination and controlled experimental environments. Therefore, the use of resistive heating is a valid method to conduct oxidation experiments on UHTCs.