WO2025043247A1 - Thermo-optical plane source measurements - Google Patents

Abstract

A thermo-optical plane source (TOPS) technique can be used to measure the thermal conductivity of materials. This high-throughput, simple, and efficient technique measures thermal conductivity with minimal sample preparation and limited restrictions on sample shape and geometry. Moreover, the TOPS technique works on solids, liquids, gels, and pastes with no change in implementation. The TOPS technique uses laser heating to induce a steady-state temperature rise in a material and infrared thermography to measure the corresponding temperature rise. Fourier's law is applied to directly measure thermal conductivity, rather than thermal diffusivity or effusivity, without any need to know the sample's density or heat capacity. It can measure thermal conductivities ranging from 0.05 W m^-1 K^-1to 60 W m^-1 K^-1at room temperature.

Description

Thermo-Optical Plane Source Measurements CROSS-REFERENCE TO RELATED APPLICATION(S) [0001] This application claims the priority benefit, under 35 U.S.C.119(e), of U.S. Application No. 63/578,600, filed on August 24, 2023, which is incorporated herein by reference in its entirety for all purposes. BACKGROUND [0002] The thermal conductivity ( ^^) of materials plays a critical role in the design, development and performance in a wide array of technologies and applications. For example, the efficiency of ultra-insulating foams and porous materials for thermal isolation in applications ranging from wearables to building materials is directly tied to their inability to conduct heat. Accurate and high-throughput measurement of these low thermal conductivities is an outstanding challenge in this community, where a century-old technique, the guarded hot plate method, remains the standard for direct (steady-state) thermal conductivity measurements, despite large experimental setups and samples and measurement times often on the order of hours. Another major industry that benefits from accurate thermal measurements is solid-state energy conversion technologies, such as thermoelectric devices. In solid-state energy conversion, maintaining a large temperature gradient across materials is paramount. Accurate knowledge of the thermal conductivity of these materials on the length scales used in these devices enables evaluation of device performance via the thermoelectric device figure of merit. On the other end of the spectrum, the higher thermal conductivities of ceramics, metals, semiconductors, and their composites dictate the efficiency of nearly all modern electronic devices, quantifying their ability to spread and sink heat generated during operation. [0003] A tool for measuring a material’s thermal conductivity should be simple to use, high- throughput, and robust. It should be capable of measuring thermal conductivity no matter the value (both low and high extremes). Such a tool should be able to measure various material types and phases (solid, liquid, gels, pastes, gases, powders, etc.) without any change in approach. These thermal conductivity measurements should be nondestructive, localized to identify differences in thermal properties rather than an aggregate property, unaffected by sample size or geometry, unaffected by sample nonidealities such as surface roughness or scratches, and involve little to no sample preparation for measurement. Ideally, thermal conductivity measurements should not require assumptions or inputs about the sample (they should be direct measurements of thermal conductivity, rather than measurements of thermal diffusivity and/or effusivity used to infer or derive thermal conductivity based on assumptions about the sample). [0004] Although no technique fits this description, significant progress has been made towards this goal. Electrical approaches such as the 3 ^^ technique can perform local measurements on a sample from its surface using patterned heaters and thermometers. However, the need to fabricate heaters on the surface for electrical measurements presents a high barrier to entry and limits the types of materials that can be studied. The transient plane source in its original form suffers from the same concern over patterning. The hot disk method overcomes this concern by patterning the metal heater into a polymer layer which is then sandwiched between two identical samples to be measured. While an improvement in accessibility, it can be challenging to make two identical samples; moreover, this technique requires specific sample geometries and is an aggregate, 1-dimensional measurement of thermal effusivity, not a direct measurement of thermal conductivity. Furthermore, the contact nature of all electrical techniques is not ideal for many use cases involving in-situ or quality control testing. [0005] Various optical thermometry platforms offer non-contact alternatives and have proven to be feasible for measuring thermal conductivity under appropriate operating procedures. Since its inception, Laser Flash Analysis (LFA) has been used to measure the thermal diffusivity of solids. Traditional implementation of LFA requires a standardized sample geometry and optical access to both front and back of the specimen for heating and temperature sensing, respectively. LFA is a dual-sided transient thermometry technique, so determining a material’s thermal conductivity using LFA requires knowledge of the material’s specific heat and density. The advent of laser-based pump-probe thermoreflectance techniques gave rise to new directions for thermal conductivity measurements with much better spatial resolution than laser flash, offering new directions for measuring the thermal conductivity of thin films and the thermal conductance across interfaces not accessible with LFA. [0006] More recently, a short-pulsed based thermoreflectance method—time-domain thermoreflectance (TDTR)—became standardized. TDTR uses a focused laser beam rather than a surface heating, giving it superior resolution both spatially and temporally to LFA. A similar technique, frequency-domain thermoreflectance (FDTR), uses continuous-wave lasers rather than pulsed lasers. Both TDTR and FDTR are transient measurement techniques that measure a material’s thermal diffusivity or effusivity. Another technique, steady-state thermoreflectance (SSTR), allows for a direct measurement of thermal conductivity with a simple experimental setup. Although thermoreflectance techniques are powerful and are unparalleled for thin-film measurements, they involve coating the sample with a highly reflective metal film opto-thermal transducer (with an additional tool for deposition). Unfortunately, SSTR cannot measure different phases of material without significant modification. [0007] An alternative approach to measuring thermoreflectance is the use of infrared (IR) thermometry or thermography, also called modulated photothermal radiometry (MTR). The different forms of MTR involve heating an area of a sample with a periodically modulated pump laser beam and detecting infrared radiation emitted from the heated area. A lock-in amplifier captures the temperature phase and/or magnitude at a specific modulation frequency of the pump laser beam and as the modulation frequency varies over a wide range. The thermal conductivity can be determined by fitting a thermal model to the measured data. This approach is similar to FDTR but uses a passive IR thermometer to measure the sample’s temperature response instead of probe laser. [0008] An evolution of MTR is often referred to as IR lock-in thermography or spatially resolved photothermal radiometry. In this technique, a modulated pump laser beam is focused onto an opaque sample and a lock-in IR camera captures a full thermal image of the surface temperature response, including phase and/or magnitude, at either the front or back of a sample at a specific modulation frequency. The modulated pump laser beam is generally considered a point source and the data analysis is based on fitting the relationship between the radius away from the center of the spot illuminated by the modulated pump laser beam and the phase or natural logarithm of the amplitude multiplied by the radius at a distance far enough away from the center that the relationship is between these quantities is linear. While losses from convection and radiation can distort this linearity, the model can be adjusted to account for these effects. Thermal diffusivity is determined from these measurements and used to derive the thermal conductivity. [0009] IR lock-in thermography has several benefits over other MTR approaches. Sampling many pixels of the lock-in IR camera yields larger datasets. The surface image obtained with the lock-in IR camera makes it possible to avoid areas that could negatively affect the measurement. And lock-in thermography can avoid or overcome the Narcissus effect by filtering noise at all but the modulation frequency. Some drawbacks to IR lock-in thermography include diffraction of the infrared radiation passing through the lens of the camera. This diffraction can affect the readings of the surface temperature. At high frequencies, this generally increases the measured slope of phase versus radius, causing thermal diffusivity to be overestimated. Finally, the use of lock-in cameras may be prohibitive in cost and complexity for simple measurements. [0010] Two alternative approaches have been proposed to overcome some limitations with lock-in thermography. The first approach involves recording the surface temperature profile as a function of time for a sample subject to a laser pulse heating event. Whether run in a single- side or dual-side configuration, this technique fundamentally measures thermal diffusivity. The use of a pulsed laser increases the cost and complexity of use, without any substantial benefits over lock-in thermography. The second approach is a laser-spot step-heating approach that captures the transient temperature response to heating by a continuous-wave laser beam whose intensity is modulated. In this approach, an IR camera records the surface temperature response to constant laser heating as a function of time. Similar to LFA, the relationship in MTR between the temperature rise and time is used to extract thermal diffusivity or effusivity. This represents an improvement in accessibility and ease of use compared to the aforementioned MTR approaches, but still suffers from the same downfalls that prohibit MTR from being a general use tool: it requires an opaque sample and knowledge of the sample’s emissivity in the camera’s operating wavelength range. This limits the types of materials that can be measured. While coatings can be used to overcome this in some cases, coating are not universal and can distort the intrinsic thermal properties being measured. One of the biggest limitations of these alternative approaches is their reliance on comparing temperature profiles frame by frame to measure change in temperature over time. This makes it difficult to obtain accurate and repeatable data with entry-level infrared cameras, which typically have accuracies of ±2 °C. SUMMARY [0011] Our techniques for measuring thermal conductivity, called thermo-optical plane source (TOPS) techniques, draws inspiration from both thermoreflectance techniques and infrared thermography. TOPS techniques use a constant-power, continuous-wave laser beam to heat a region of a sample to a steady-state temperature and an entry-level IR camera to measure the steady-state temperature in the heated region as a function of laser power. Measuring the (baseline) temperature of an unheated portion of the sample makes it possible to determine the temperature rise for the applied laser power. Applying Fourier’s law to the relationship between the temperature rise and the laser power yields the thermal conductivity of the heated region of the sample. TOPS techniques can be used to determine thermal conductivity accurately, even without knowledge of the sample’s absorption at the laser wavelength or absolute temperature rise. [0012] Applying an optional transducer to the sample’s surface allows for calibration and for direct comparison of measurements of different samples. The transducer can be a polymer- based tape that can be applied to nearly any material, even liquids, gels, and pastes. Combining the benefits of a localized and single-side measurement resilient to surface defects/roughness and largely independent of sample dimension, TOPS techniques provide a high-throughout, simple, and robust approach to thermal conductivity measurements. [0013] TOPS techniques can be used to measure a property of a sample, such as thermal conductivity, thermal resistance, or film thickness, as follows. An infrared camera, pyrometer, or other detector measures a baseline temperature of the sample, which could be a solid, liquid, gel, and/or paste. A laser beam illuminates a spot on a surface of the sample at a constant power level; this laser beam heats the spot on the sample to a steady-state temperature. The infrared camera, pyrometer, or other detector measures the steady-state temperature of the spot on the sample. The property of the sample can then be determined based on the constant power level and on a difference between the steady-state temperature of the spot on the sample and the baseline temperature of the sample. [0014] Measuring the baseline temperature can include detecting infrared radiation emitted from a region of the sample not heated by the laser beam with a first set of detector elements of an infrared detector, in which case measuring the steady-state temperature comprises detecting infrared radiation emitted from the spot on the sample with a second set of detector elements of the infrared detector while measuring the baseline temperature. Taking the difference between the steady-state temperature of the spot on the sample and the baseline temperature of the sample suppresses noise common to outputs of the first set of detector elements and the second set of detector elements. [0015] The sample can be heated to the baseline temperature, e.g., using a hot plate or temperature chamber or by illuminating the sample with a heating laser beam at a power level of 0–10 kW. [0016] If desired, a transducer with a known thermal conductivity, such as a polymer-based tape, can be disposed directly on the surface of the sample. In this case, the spot formed by the laser beam on the surface of the sample is on the transducer. [0017] In some case, the steady-state temperature is a first steady-state temperature and the constant power level is a first constant power level. In these cases, the laser beam illuminates the spot on the surface of the sample at a second constant power level greater than the first constant power level. The laser beam heats the spot on the sample to a second steady-state temperature higher than the first steady-state temperature. And the infrared camera, pyrometer, or other detector measures the second steady-state temperature of the spot on the sample. The property of the sample can be determined based on a difference between the second steady- state temperature of the spot on the sample and the baseline temperature of the sample in addition to the quantities mentioned above. [0018] TOPS measurements can also be carried out by disposing a transducer with a known thermal conductivity directly on a surface of the sample and illuminating a first region of the transducer with a laser beam at a constant power level and at a wavelength absorbed by the transducer. The laser beam heats the first region of the transducer to a steady-state temperature. An infrared detector (e.g., with an absolute temperature accuracy of no better than ±2 °C or ±2% and a temperature resolution of finer than 0.1 °C) measures the steady-state temperature of the first region of the transducer with a first set of sensing elements and, at the same time, measures a baseline temperature of a second region of the transducer with as second set of sensing elements. A processor or other device operably coupled to the infrared detector determines the thermal conductivity of the sample based on the known thermal conductivity of the transducer and a difference between the steady-state temperature and the baseline temperature. [0019] Yet another version of TOPS measurements can be carried by heating a first portion of a sample to a first temperature with a first laser beam and, while the first portion of the sample is at the first temperature, heating a second portion of the sample at least partially contained within the first portion of the sample to a second temperature higher than the first temperature with a second laser beam. An infrared detector, pyrometer, or other detector measures a change in temperature of the second portion of the sample as a function of power of the second laser beam. And a processor or controller operably coupled to the infrared detector, pyrometer, or other detector determines, based on the change in temperature of the second portion of the sample, at least one thermophysical property of the sample (e.g., thermal conductivity, emissivity, heat capacity, and/or thermal boundary resistance) at a weighted average of the first and second temperatures. For instance, the thermophysical property may be based on a slope of the change in temperature versus the power of the second laser beam [0020] The sample may be at room temperature and pressure. The sample can also be coated with a transducer configured to absorb the first laser beam and the second laser beam. [0021] Heating the first portion of the sample can include illuminating a first spot on the sample and heating the second portion of the sample comprises can include illuminating a second spot smaller than the first spot on the sample. [0022] If desired, the power level of the first laser beam can be set high enough to heat the first portion of the sample to the first temperature within five minutes. Likewise, the power level of the second laser beam can be set high enough to heat the second portion of the sample to the second temperature within five minutes. If desired, the sizes of the first and/or second portions of the sample can be varied, e.g., by adjusting the (zoom) lens(es) used to direct and/or focus the first and second laser beams to the sample. [0023] All combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. Terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein. BRIEF DESCRIPTIONS OF THE DRAWINGS [0024] The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar components). [0025] FIG.1A shows a thermo-optical plane source (TOPS) measurement. [0026] FIG.1B is a plot of the normalized intensity of the incident laser beam on the sample I/I(0) and resulting temperature rise (T/T(0)) as a function of normalized distance from the peak of the intensity (x/r0, where r0 is the 1/e² radius of the laser beam). The temperature rise is spatially averaged within a Region of Interest (ROI). [0027] FIG.1C is a plot of the ROI temperature rise versus laser power used to heat the sample; insets show the temperature profile for a low-power case and a high-power case. [0028] FIG.2A is a plot of a TOPS measurement of raw temperature versus time for a fused silica sample. The upper trace represents the temperature of the ROI (0.4 mm radius spot, 10 pixels on the camera) on the sample and the lower trace represents the temperature of an outer ring (OR) on the sample (8 mm radius ring, 200 pixels on the camera). [0029] FIG. 2B is a plot of the temperature difference, defined as the (steady-state) ROI Temperature – (baseline) OR Temperature, showing windows used for averaging when the temperature is in steady state. [0030] FIG.2C is a raw temperature image of the sample when laser power is set to 12 mW, T (P = 12 mW), with the ROI at the center and the OR marked by the circular boundary at the periphery. [0031] FIG. 2D is a differential temperature image, T (P = 12 mW) – T0, where T0 is the temperature of the sample with no laser heating applied. [0032] FIG.2E is a plot of the average temperature rise from each averaging window in FIG. 2C as a function of set laser power during that time; the data is shown in symbols and the best fit line is overlayed. The slope is representative of the sample’s thermal conductivity. [0033] FIG. 3A is a plot of thermal conductivity measured using TOPS versus literature or manufacturer-specified thermal conductivity. Solid samples are depicted in solid squares, liquid samples in open circles, and gels/pastes in solid triangles. [0034] FIG.3B illustrates TOPS measurements of a liquid, gel, or paste in direct contact with a polymer tape transducer. [0035] FIG. 4 is a plot of the measured slope of temperature to power ( Δ T / ΔP ) vs. thermal conductivity illustrating the effect of air on the experimental results. Solid samples are depicted in solid squares, liquid samples in open circles, and gels/pastes in solid triangles. The lines indicate the thermal model with no consideration of air conduction or convection (Insulated), only air conduction (H = 0), air conduction and 10 W m⁻²K⁻¹ of convection (H = 10), and air conduction and 100 W m⁻²K⁻¹ of convection (H = 100). [0036] FIG.5 is a plot of thermal conductivity measured with TOPS versus sample thickness for quartz glass (top trace and symbols), borosilicate glass (middle trace and symbols), and neoprene rubber (55A hardness; bottom trace and symbols). Solid symbols indicate measured values on samples that were thermally isolated during experiment (suspended in air) while open symbols indicate measured values on the same samples placed on an aluminum plate during experiment. Lines indicate bulk values as measured on samples with >20 mm thicknesses. [0037] FIG.6 is a plot of thermal conductivity measured with TOPS of a 2-inch-diameter, 0.5- inch-thick acrylic sample versus distance of the center of a Gaussian laser beam from the edge of the sample. Square symbols indicate measured data corresponding to the inset infrared images, which show the temperature profile and position of the laser beam relative to the edge of the sample. The solid line represents the average thermal conductivity in the center of the sample. [0038] FIG.7 is a plot of thermal conductivity measured with TOPS versus the grit finish of the acrylic sample being measured. Solid symbols indicate measured values on acrylic samples that were sanded to various finishes using 40-grit to 1200-grit sandpaper. The line shows the value for a mirror finish sample. Insets show the infrared camera images of (left) the 1200-grit- polished sample and (right) the 40-grit-polished sample without any applied laser heating. [0039] FIG.8 illustrates a two-beam TOPS measurement. [0040] FIG. 9 shows radial temperature profiles of a sample ROI undergoing a two-beam TOPS measurement from the heating beam alone (left), the pump beam alone (center), and the heating and pump beams together (right). [0041] FIG.10 illustrates two-beam TOPS measurements at a series of increasing heating laser powers. [0042] FIG.11 illustrates two-beam TOPS measurements with different pump beam spot sizes in homogeneous (top) and multi-layer (bottom) samples. [0043] FIG.12 illustrates a process for making two-beam TOPS measurements. DETAILED DESCRIPTION [0044] In an example thermo-optic plane source (TOPS) measurement, a continuous heat source with known spatial intensity and constant power is applied to a local region of a sample to induce a steady-state temperature rise in the material. That is, the temperature gradient reaches an equilibrium between the supplied heat flux and diffusion into the sample, causing the local region of the sample to reach a steady-state temperature above its initial or baseline temperature. The time it takes to reach this steady state depends on the non-dimensional Fourier number,

where α is the thermal diffusivity, t is time, and r₀ is radius of the heat source. For a heat source with a diameter of approximately 1 mm, a steady-state temperature rise is achieved is under 0.1 seconds for high thermal diffusivity materials (e.g., ~100 mm²/s) and nearly 10 seconds for low thermal diffusivity materials (e.g., ~1 mm²/s). The size of the heat source can be focused for faster measurements at the expense of spatial resolution of temperature detection, so a balance should be achieved between the two. The temperature rise at steady state within the heated region is measured to determine the relationship Δ T / ΔP , where ^^ ^^ is the temperature rise and ^^ ^^ is the difference in power of the laser (from off to on state or from power to power if different powers are used). Δ T / ΔP is then compared to a thermal model to fit for the thermal conductivity of the material. [0045] FIGS. 1A–1C depict a system 100 for carrying out the TOPS technique and raw measurements of a sample 10 made with that system 100. The system 100 includes a heat source in the form of a continuous-wave laser 110 (e.g., a Coherent OBIS LX FP laser) that emits a laser beam 111 at a 637 nm wavelength and an output power of 1–120 mW. Other wavelengths and power levels are also possible. This laser beam 111 is incident on an optional transducer 120 disposed on a surface of the sample 10, heating the transducer 120 and causing the transducer 120 to heat the sample 10 to a steady-state temperature. If desired, the sample 10 can be placed on or in thermal communication with another heat source (not shown), such as another laser beam (e.g., a first laser beam as in two-beam TOPS, described below), thermo- electric cooler, hot plate, or temperature chamber that heats the sample 10 to its baseline temperature. [0046] The system 100 also includes an infrared camera 130 for measuring the temperature of the sample 10. For the example measurements shown in FIGS.1B and 1C, the infrared camera 130 was microbolometer focal plane array (e.g., a Micro-Epsilon TIM) had a 12° field-of-view microscope lens, a spatial resolution of 40 µm per pixel, and a spectral range of 8–14 µm. The infrared camera 130 has no lock-in amplification capability, a relatively slow frame rate (e.g., 32 Hz), and a relatively poor accuracy (e.g., ±2 °C at room temperature). Nonetheless, TOPS techniques compensate or cancel the camera noise (e.g., dark current and thermal noise) to provide accurate measurements of the sample’s temperature as described below. [0047] For the measurements shown in FIGS. 1B and 1C, the laser 110 illuminated the transducer 120 with a laser beam 111 at a wavelength absorbed by the transducer 120 (though not necessarily by the sample 10). The infrared camera 130 measured the surface temperatures of different regions of the transducer 120. A computer 140 operably coupled to the laser 110, the camera 130, and a stage (not shown) that supported and moved the sample 10 automated the measurement and synchronized the timing. The stage translated the sample 10 vertically to ensure the laser beam focus was centered in the IR camera 130 image for every measurement. [0048] FIG. 1B shows the Gaussian profile of the laser beam 111—the laser intensity as a function of radius—used for heating the transducer 120 and the sample 10 to a steady-state temperature above the their starting or baseline temperature. It also shows the resulting spatial temperature rise of the sample. This temperature rise can be determined by a thermal model that relates the surface temperature to the sample temperature beneath the surface. This thermal model can be applied to a multi-layered sample, with the transducer 120 treated as one layer with known properties. [0049] FIG.1B also shows a measurement region of interest (ROI) over which the temperature is spatially averaged. The ROI here was chosen to extend to about where the temperature decayed to 60% of its maximum value; for the ~1 mm diameter beam used here, this translated to an ~800 µm diameter ROI. This ROI corresponds to a 20-pixel diameter spot on the IR camera 130. In principle, the choice of ROI is arbitrary, as even a single pixel should provide enough information about temperature rise but averaging over more pixels offsets fluctuations from noise between pixels and reduces the relative impact of outlying pixel measurements. The ROI can, but does not have to, include the spot illuminated by the laser beam—a spot adjacent to the laser spot could be used as the ROI instead—so long as the temperature rise detected in the ROI is due to the laser heating. In the measurements shown here, the ROI was a defined area portion of the area heated by the laser. [0050] FIG.1C shows the relationship between ROI temperature rise—that is, the difference between the sample’s steady-state temperature and the sample’s baseline temperature—and the laser beam power. The ROI temperature rise increases linearly with temperature, making it possible to fit a (linear) thermal model to the resulting change in temperature with change in laser beam power Δ T / ΔP . This thermal model relates the slope of the ROI temperature rise to the thermal conductivity of the sample. (Here, the temperature rises were limited to a few degrees Celsius so the sample’s thermal properties would not change significantly with temperature.) [0051] The transducer 120 should have a thermal resistance low enough to not significantly influence the thermal response of the sample and be thick enough to completely determine the surface optical properties of the sample. Preferably, the transducer 120 should be highly absorbing at the laser wavelength and highly emissive in the camera spectral range. Suitable transducer materials include thin-film carbon, graphite (sprayed or colloidal suspended in liquid), and high-emissivity paint. For the measurements shown in FIGS. 1B and 1C, the transducer 120 was an black tape with a thickness of 5 µm (e.g., with a 2 µm thick poly(ethylene terephthalate) (PET) layer and a 3 µm thick acrylate adhesive). This tape is practical and robust for applying to nearly any type of material, and because it can be suspended over a container, is well-suited for measuring liquids, gels, and pastes. For example, the tape can be placed across the mouth of the container, which is then filled to the brim with liquid, gel, or paste, with the liquid, gel, or paste touching the underside of the tape. [0052] The transducer 120 has a known absorptivity at the laser beam wavelength and a known emissivity in IR camera’s measurement range, so the Δ T / ΔP measured by the camera 130 can be translated into a true Δ T / ΔP as follows. The transducer 120 is used to determine a system proportionality constant that relates the measured data ( Δ T / ΔP )_measured, to the true relationship ( Δ T / ΔP ). That is,

where ^^ is the transducer emissivity, ^^ is the transmission of infrared light to the camera 130 (to account for any environmental losses), ^^ is the transducer absorption, and ^^ is the system proportionality constant, which can include any other system adjustments unique to a setup (e.g., if laser power set is different from that emitted). The system proportionality constant, ^^, can be determined by measuring the transducer emissivity and absorption, for example, using a calibration sample with a known thermal conductivity (e.g., a fused silica calibration sample with a thermal conductivity of 1.38 W m^-1 K^-1). [0053] FIGS.2A–2E illustrate a representative measurement taken on a 1-inch diameter fused silica window and demonstrates data correction for more accurate measurement. In general, microbolometer focal plane array based thermal cameras like the one used here have poor absolute temperature accuracy (e.g., ±2 °C or ±2%, whichever is higher), making them ineffective for determining true temperature. However, these cameras can have temperature resolution finer than 0.1 °C, so they can distinguish temperature pixel-to-pixel at this resolution and sensitivity (responsiveness to change in temperature in time) down to 0.01 °C. This makes them well-suited for determining temperature differences. [0054] TOPS measurements exploit this ability to measure temperature differences to correct the ROI temperature in real time as follows. As explained above, a first subset of camera detector elements or pixels measures the steady-state temperature of the ROI on the sample. At the same time, a second subset of camera pixels measures the baseline temperature of another portion of the sample that is far enough away from the ROI not to be heated by the laser beam, e.g., an outer ring (OR) of the sample and a corresponding OR of camera pixels. Since both the ROI and OR pixels measure ambient temperature fluctuations and camera noise that is correlated from pixel to pixel, including dark current and thermal noise, subtracting the OR temperature measurements from the ROI temperature measurements yields the temperature rise in the ROI caused by the laser heating. [0055] FIG.2A shows ROI and OR temperature for a TOPS measurement of a 1-inch-diameter fused-silica window coated with a 5 µm thick black tape transducer at each of five different laser power levels from 0 to 12 mW over five consecutive 60-second intervals. The laser power level was increased by 3 mW at the beginning of each interval and the temperature of the ROI was allowed to reach steady state while the OR temperature remained constant at the baseline (0 mW) temperature. [0056] The ROI is defined by a 20-pixel (800 µm diameter spot on the sample) diameter spot on the camera and the OR is defined as a 400-pixel diameter spot on the camera (8 mm diameter ring on the sample). The noise in the data makes it difficult to obtain consistent quantitative temperatures that can be used for measurement. However, this noise is common to both ROI and OR temperatures. Therefore, subtracting the OR temperature data from the ROI temperature data offsets the noise, leaving just the temperature rise from laser heating. [0057] FIG.2B shows that subtracting the OR temperature data from the ROI temperature data works well for correcting the data. Moreover, it can be applied in real time, frame by frame. Although this example uses a ring of pixels, this principle can be applied to any quantity representative of the system noise, including a single pixel or the minimum frame temperature. If desired, the ROI and OR temperature data can be average temporally to further reduce noise, for example, over 30-second intervals at each laser power level as shown in FIG.2B. [0058] FIG.2C shows an infrared image of the sample acquired by the camera when the laser is on at its highest power (12 mW). FIG.2D shows a differential infrared image of the sample obtained by subtracting by an infrared image of the sample when no laser power is applied from the infrared image in FIG. 2C. FIG. 2E shows the measured temperature rise (squares) versus laser power fit with a linear thermal model (solid line). One benefit of using the relationship (Δ T / ΔP ) is that imperfections in surface properties from, e.g., scratches or wrinkles in the transducer, are offset by the differential, so that the localized temperature rise from laser heating can be isolated. This can be seen when comparing the raw image in FIG.2C with the differential image in FIG. 2D. Thus, this technique takes advantage of differential measurements in both time and space to negate the camera’s poor accuracy and exploit the camera’s relatively high sensitivity and fine temperature resolution. Experimental TOPS Measurements [0059] FIG.3A shows thermal conductivities measured using TOPS techniques of a wide array of samples ranging in thermal conductivity from 0.05 W m^-1 K^-1to 60 W m^-1 K^-1, including solids (squares), liquids (circles), and gels/pastes (triangles). FIG. 3B shows a vat 300 for measuring liquids, gels, and/or pastes 31. We suspended a layer of polymer tape (a tape transducer 320) over the vat 300, which had input and output ports 302, 304, then added the liquid or gel/paste 31 to the vat 300 via the input port 302 using a syringe until the liquid or gel/paste 31 touched the underside of the tape transducer 320. Once the vat 300 was full, we closed the input port 302 to contain the sample 31 within the vat 300 and tested the sample 31 using the procedure described above for solid samples, with the constant-power laser beam 111 from the laser 110 illuminating the tape transducer 320 and imaging both heated and unheated regions of the tape transducer 320 with the camera 130, then drained the liquid or gel/paste 31 from the vat 300 via the output port 304. [0060] TABLE 1 summarizes the measured thermal conductivity values and how they compare with literature thermal conductivity values (or in the case of the gels and pastes, what the manufacturer specifies). FIG. 1 shows the relationship between measured and literature/ manufacturer thermal conductivity values. In general, there is excellent agreement between literature and measured values. On the high end of thermal conductivity (>30 W m^-1 K^-1), the uncertainty of the measurement becomes large because the thermal resistance of the transducer influences the thermal response. The thermal conductivity of the transducer itself was determined by measuring the thermal conductivity of a 3 mm thick silicon window. The temperature rise was dominated by the transducer’s thermal resistance, so we fit the transducer’s thermal conductivity directly, determining a value of 0.25 ± 0.02 W m^-1 K^-1. The effects of the transducer’s thermal conductivity on sample measurements can be overcome in practice by (1) using a thinner or more thermally conductive transducer layer and/or (2) increasing the spot size of the laser beam on the transducer. The transducer thickness and spot size can be chosen to ensure measurement of any range of thermal conductivity. TABLE 1

[0061] For anisotropic materials, TOPS measurements are of the geometric mean of thermal conductivity in spherical coordinates. Thus, for Al2O3, MgF2, and Balsa wood, the measured values are compared with their geometric mean counterparts in literature. For Balsa wood and woods in general, which are anisotropic, direction of grains can vary sample to sample. Moreover, the thermal conductivity of wood is affected by wood’s density, porosity, moisture content, grain direction, and extractive content. Nonetheless, when compared to the average of the range of geometric mean thermal conductivities observed in literature (0.0381 W m^-1 K^-1to 0.0665 W m^-1 K^-1), our TOPS measurement of 0.050 ± 0.007 W m^-1 K^-1falls in line. [0062] All TOPS measurements were conducted in open air. When measuring such low thermal conductivity materials as Balsa wood, whose thermal conductivity is only twice that of air, the impact of the air should be considered as part of the thermal analysis. In fact, the effects of air conduction, convection, and even radiation from the surface of the sample can affect photothermal-based thermal property measurements. [0063] FIG.4 shows the relationship between the slope of temperature rise to power (Δ T / ΔP ) and a material’s thermal conductivity. Four thermal models (insulated and H = 0, 10, and 100 W m^–2 K^–1) are overlayed to show the effect of air on the determination of the true thermal conductivity. When air and radiation are not considered at all (i.e., the sample is assumed to be perfectly insulated at the surface), the expected Δ T / ΔP matches the data well for materials with a thermal conductivity of ~0.5 W m^-1 K^-1and higher. However, below this value, the model begins to diverge greatly from the Δ T / ΔP measured experimentally. Air effects on the thermal conductivity (H = 0) can be added to the thermal model using a bidirectional boundary condition where heat can flow into the sample and into the air. Air conduction proves to be the largest contributor to the thermal model, adjusting the thermal model to capture the true thermal conductivities for all but the most insulating sample, Balsa wood. [0064] FIG.4 also shows what happens when the effects of convection as a boundary condition to the surface are incorporated into the thermal model. A free convection coefficient of room temperature air is approximately 5 W m^–2 K^–1 and the linearized radiation coefficient is approximately the same, bringing the combined convection coefficient to H = 10 W m^–2 K^–1. Using this value proves to work well for properly capturing the full range of experimental data, as seen in FIG.4. Also shown for comparison is the model using a forced convection coefficient of H = 100 W m^–2 K^–1. In this case, the model greatly under-predicts the measured Δ T / ΔP . While air conduction should be considered for insulating materials, the effect of convection scales with laser beam spot size: decreasing the spot size negates convection effects, whereas increasing the spot size makes the TOPS measurement more sensitive to convection effects, opening the door to direct measurement of convection coefficient. [0065] The TOPS technique can measure thermal conductivity of samples with different thicknesses, edges, and surface roughness. [0066] With respect to sample thickness, TOPS assumes a semi-infinite boundary condition in its analysis—that is, the depth of the sample is assumed to be infinite in length relative to the heated volume such that the heat flux, ^^, is 0 at the back of the sample ( ^^ → 0 as ^^ → ∞). When the sample becomes thin enough relative to the heat source characteristic length, this assumption breaks down so the thermal model should be adjusted. [0067] FIG. 5 shows the thermal conductivity versus thickness for borosilicate glass, quartz glass, and neoprene rubber (55A hardness). The samples were each tested in two ways: suspended in air so the backside of the sample was thermally isolated (solid symbols) and placed on an aluminum plate so that heat could flow away from the back of the sample (open symbols). [0068] The borosilicate glass sample set included samples with thicknesses of about 19 mm down to about 3 mm. For the borosilicate glass samples on the aluminum plate, the experimental data had no thickness dependence, indicating that finite sample thickness does not affect the TOPS thermal model. For thermally isolated borosilicate glass samples, however, the 3 mm thick sample showed a minor deviation from the rest of the data, having a 4 % reduction on average compared to thicker samples. Still, this measurement falls within uncertainty of the average such that thickness did not significantly affect the model in this case. [0069] The quartz glass sample set included samples with thicknesses from about 19 mm down to about 1.6 mm. The quartz glass samples follow the same trends as the borosilicate glass samples down to sample thicknesses of 3 mm. At 1.6 mm thickness, the quartz glass sample tested on the aluminum plate showed no sign of thickness affecting the thermal model, such that thermal conductivity agreed well with those of the thicker samples. On the other hand, the thermally isolated quartz glass sample showed a significant decrease in apparent thermal conductivity relative to the thicker quartz glass samples, this time outside of what can be considered uncertainty. Without being bound by any particular theory, the apparent reduction in thermal conductivity is because the quartz glass sample was suspended in air, so that the thermal penetration depth of the heated volume sampled the air, which has a lower thermal conductivity than quartz glass, reducing the apparent thermal conductivity of the measurement. [0070] The neoprene rubber sample set included samples with thicknesses of about 19 mm down to about 0.4 mm. The neoprene rubber samples follow the same trend as the borosilicate and quartz glass sample sets down to 3 mm thickness. At 1.6 mm, like the quartz glass sample, the thermal conductivity of the neoprene rubber sample measured on the aluminum plate matched that of the thicker samples. The thermal conductivities for the thermally isolated neoprene rubber samples are slightly lower, possibly because neoprene rubber is more insulating so backside air has a smaller effect on its apparent thermal conductivity. For neoprene rubber samples with thicknesses of 0.8 mm and 0.4 mm, the measured thermal conductivity in both the isolated and non-isolated cases deviates from the measured thermal conductivity of the thicker neoprene rubber samples. The isolated neoprene rubber samples have a lower apparent thermal conductivity, consistent with the borosilicate and quartz glass sample sets, while the neoprene rubber samples on the aluminum plate have a higher apparent thermal conductivity, possibly because the aluminum plate has a significantly higher thermal conductivity. [0071] Other work on the depth-dependent temperature decay reveals that the 1/e temperature decay in depth occurs at approximately the 1/e² Gaussian radius of the laser beam spot. The same methodology can be used to show the heat flux decay with depth; doing so for a 1 mm 1/e² diameter Gaussian beam reveals the heat flux decays to about 5% at a depth of 1 mm below the surface and about 1% at a depth of 2 mm. The semi-infinite thermal model may break down when the sample thickness drops to about 2 mm, which agrees reasonably well with the experimental data shown in FIG.5. Generalizing, this implies that the sample thickness should be at least twice as thick as the beam diameter used in the TOPS measurement. For thermally isolated samples, experimental data suggests the thickness should be at least four times the beam diameter, or 4 mm in the experiments shown in FIG.5. Despite these limits of the model, it is possible to modify the model to account for the finite thickness by applying a backside boundary condition. [0072] FIG.6 illustrates the effects of breaking the radial symmetry of the temperature profile by illuminating the edge of a sample rather than the center of the sample with a laser beam. The center of the spot formed by the laser beam on the transducer and sample was moved to the center of the sample from the edge of the sample in discrete steps as shown in the infrared images in the inset of FIG.6. The measurement closest to the edge of the sample was obtained by moving the spot as close to the edge as possible without visibly clipping the laser beam, for a distance of about 0.8 mm from the spot’s center to the edge of the sample as measured with the thermal camera. From there, the spot was moved away from the edge in steps of 1 mm to 1.5 mm. The measurements at the different spot positions show no evidence of the edge affecting the measured thermal conductivity, emphasizing the robustness of the TOPS technique for local measurements. This highlights a major advantage of TOPS techniques over traditional lock-in and step thermography techniques, which rely on fitting a model of temperature rise versus distance from center of the laser beam spot out to several multiples of the spot radius away from the center. In contrast, TOPS techniques use a measurement of the temperature rise caused by heating of the spot formed by the constant-power, continuous-wave laser beam (for example, the temperature of the spot illuminated by the laser beam). This means that TOPS techniques are far better suited for measuring the thermal conductivity of non- uniform samples (e.g., samples with surface defects that could distort the temperature profile globally but not locally). Combined with the differential nature of the measurement, TOPS techniques are highly robust for localized thermal measurements. [0073] FIG. 7 illustrates the effects of surface roughness on the thermal conductivity measurements made using TOPS techniques. TOPS techniques rely on heating via laser absorption and temperature measurement via infrared thermography, both of which can be affected by surface roughness. FIG. 7 shows the thermal conductivity measured for acrylic samples mechanically polished with sandpaper of grits from 40 to 1200 and a mirror-finish acrylic sample as a control. Each sample was coated with a film transducer that was thin enough to conform to the (roughened) sample surface after polishing. The thermal conductivities measured for the polished samples match the mirror-finish sample’s thermal conductivity for all samples polished with grits greater or equal to 120. For the 40-, 60-, and 80-grit polished samples, the apparent thermal conductivity is lower than the control sample by 7–8%. [0074] Without being bound by any particular theory, this decrease in thermal conductivity with increasing roughness (decreasing grit finish) could be due to an increase in emissivity with roughness, which would result in a higher apparent temperature rise for the same applied laser power. Similarly, increased surface roughness can lead to increased laser absorption, again leading to a higher temperature rise for the same applied laser power. Despite these compounding effects, the deviation between the roughest sample (40-grit finish) and control sample is only 8%, likely due to the already high absorption and emissivity of the transducer. Moreover, variation with surface roughness can be calibrated out of TOPS measurements by testing a control sample with roughness similar to that of the sample under test. Referring to Eq. (1), both absorption and emissivity are captured in ^^, meaning the system calibration to determine ^^ can be applied to a rough sample to capture these quantities, which can then be applied to any similarly rough samples to capture their effects on the measurement. Two-Beam TOPS Measurements [0075] FIG.8 illustrates an alternative TOPS technique that involves heating two overlapping portions of a sample 80 with separate laser beams and measuring the resulting spatiotemporal changes in the sample’s temperature. The sample 80 is on a two-axis horizontal translation stage 806 in an optional environmental control or pressure chamber 840 optionally filled with inert gas. A window in the chamber 840 is open to a first laser 810, second laser 812, IR camera 830, and optional pyrometer 832, all which are mounted on a vertical translation stage 802. The stages 802, 806 are used to move the sample 80 relative to the lasers 810, 812 and detectors 830, 832, e.g., for characterizing different portions of the sample 80 or for adjusting the focus of the lasers 810, 812 or detectors 830, 832. [0076] In this two-beam TOPS measurement, the first laser 810, also called a heating laser, illuminates a relatively small spot (e.g., with a diameter of 10 mm, 5 mm, 2 mm, 1 mm, or less) on an optional transducer 820 on a surface of the sample 80 with a first laser beam 811 (heating beam) at a wavelength that is absorbed by the transducer 820 (e.g., a visible or infrared wavelength). (In examples without a transducer, the first laser beam 811 is at a wavelength absorbed by the sample 80 and shines directly on the surface of the sample 80.) The first laser beam 811 is a continuous-wave laser beam at a power level of hundreds of milliwatts to tens of kilowatts that heats a first portion of the sample 80 to a first or baseline temperature, which may be millikelvins to thousands of degrees Kelvin (e.g., up to 4000 K). The first portion of the sample 80 is relatively small (e.g., it may be a roughly hemispherical portion with a diameter roughly on the order of the spot diameter) and so can reach the baseline temperature quickly (e.g., within milliseconds to minutes, depending on the sample’s thermal conductivity). [0077] Once the first portion of the sample reaches the baseline temperature, the second laser 812, also called a pump laser, illuminates a second portion of the sample 80 with a second laser beam 813, also called a pump beam. The pump beam 813 heats the second portion of the sample, which is smaller than the first portion of the sample 80, to a second (steady-state) temperature, also called a perturbed, elevated, or increased temperature, that is higher than the first (baseline) temperature (e.g., from millikelvin to 5–10 K higher than the first temperature). The second laser beam 813 can be at the same wavelength as the first laser beam 811. If desired, it can be emitted by the first laser 810 instead of by a separate laser. The second laser beam 813 can also be at a different wavelength (e.g., a different visible or infrared wavelength) than the first laser beam 811, so long as this other wavelength is absorbed by the transducer 820 (or, if the second laser beam 813 shines directly on the sample surface, by the sample 80 itself). The second laser beam’s power level depends on the sample’s thermal conductivity and can range from milliwatts for low thermal conductivity to Watts for high thermal conductivity. [0078] FIG. 9 shows example temperature profiles of the sample 80 illuminated by the first (heating) laser beam 811 (left), the second (pump) laser beam 813 (middle), and both the first and second laser beams 811, 813 (right). In this example, the first laser beam 811 heats the first portion of the sample to a peak temperature of about 40 K. The second laser beam 813 heats the center of the second portion of the sample, which is concentric with the first portion of the sample, by another 4–5 K, causing the coincident centers of the first and second portions to reach a peak temperature of about 44–45 K without heating the outer portion of the first portion of the sample. [0079] The first and second laser beams 811, 813 can propagate collinearly (i.e., they can propagate along the same optical axis) or they can illuminate the sample from different directions (e.g., at glancing angles from different directions). In either case, the second laser beam 813 can form a second spot on the transducer 820 that is smaller than and falls completely within the first spot. The first and second laser beams 811, 813 can also be positioned so that the first and second spots do not overlap; the first and second laser beams 811, 813 can illuminate different surfaces of the sample 80, for instance, on different sides of a corner, different facets, or even front and back surfaces, so long as the first and second portions overlap within the sample 80. For example, the first laser beam 811 can illuminate (and be absorbed by) the transducer 820 and the second laser beam 813 can illuminate (and be absorbed by) a surface of the sample 80—even a sample surface other than the sample surface coated with the transducer 820, such as the back or side. [0080] The diameter of the second spot may be two times to twenty times (e.g., five or ten times) smaller than the diameter of the first spot. For example, the first and second spots may have diameters of 5 mm and 1 mm, respectively. Regardless of the beam and spot geometry, the second portion of the sample is partially or completely contained within the first portion of the sample. [0081] FIG.10 illustrates a series of two-beam TOPS measurements at a range of different first (heating) laser beam powers. For each measurement, the first (heating) laser beam 811 brings the first portion of the sample 80 to a different baseline temperature. Once the first portion of the sample 80 reaches steady state at the baseline temperature, the power of the second laser beam 813 is ramped or increased, causing the second portion of the sample 80 to rise to a second (steady-state) temperature above the baseline temperature of the first portion of the sample 80 (also at steady state). Because the second portion is smaller than the first portion, the second portion reaches the perturbed second temperature relatively quickly (e.g., within milliseconds to minutes). This temperature rise can be viewed as temperature perturbation or transient rise in temperature. The infrared camera 830, pyrometer 832, other device, or some combination thereof measures the difference between the temperatures of the first and second regions at steady state for each combination of first and second laser beam power levels. The slope of this temperature change can be used to estimate or determine a thermophysical property of the sample, such as the sample’s thermal conductivity ^^ and/or thermal boundary resistance of interfaces within the sample 80. If these thermal boundary resistances are known a priori, the measurements can be used to estimate the thickness of the material between the boundaries (in other words, the film or layer thickness of a multi-layered sample). The time constant to raise the temperature from the baseline temperature to the elevated temperature in the second portion can be used to estimate or determine the sample’s heat capacity. [0082] The thermophysical property being measured (e.g., thermal conductivity) is a temperature-dependent property. The measurement temperature is determined by the superposition of the perturbed temperature and the baseline temperature. When using an infrared camera, a subset of the infrared camera’s pixels may encompass the first portion’s temperature profile from its maximum temperature to some defined decay temperature (e.g., to the 1/e or 1/e² decay). (Because the second portion is encompassed within the first portion, these camera pixels also measure the second portion’s temperature profile.) The measured temperature is the weighted average of the superposition of temperature profiles in the first and second portions within the measurement area defined by the subset of pixels. Alternatively, if a pyrometer is used, the pyrometer measures the weighted average of the superposition of temperature profiles in the first and second portions within the spot size of the pyrometer; typically, the pyrometer spot diameter is about the same as the diameter of the second laser beam. [0083] This measurement can be repeated at different baseline temperatures (e.g., by increasing or decreasing the power of the first laser beam) to produce a map or complete plot of the thermophysical property’s dependence on sample temperature. For each iteration, the perturbed temperature is typically chosen to be close enough to the selected baseline temperature (e.g., within 5–10 K) that the thermophysical physical property varies linearly over the range from the baseline temperature to the perturbed temperature. [0084] Because the first and second portions of the sample are small and reach the desired temperatures relatively quickly, the sample can be left at room temperature and pressure in air during the measurement with little to no risk of oxidation or other unwanted reaction. (The sample can also be measured in a pressure chamber filled with an inert gas or under vacuum.) In addition, a heat sink or source can keep the rest of the sample at a global temperature (e.g., 25 K) while the first and second laser beams heat the first and second portions of the sample. [0085] If desired, the sample can be coated with a transducer, such as a thin layer of material (e.g., polymer) that absorbs the first and/or second laser beams and has favorable emissivity properties, to facilitate temperature measurement of the sample with a thermal camera and/or a pyrometer. Using a transducer or coating with known thermophysical properties eliminates the need to know the sample’s emissivity and laser absorption and ensures consistent absorption across the sample at the cost of additional sample preparation steps, added thermal resistance and uncertainty, potential limits on the temperature range accessible for measurement, and potential changes in the sample’s temperature profile (e.g., heat spreading). [0086] A two-beam TOPS measurement can also be carried out without a transducer or coating on the sample. In a two-beam TOPS measurement without a transducer or coating, there are no additional thermal resistances or changes to the sample’s temperature profile, the thermal model for analysis is relatively simple, and the aspects of the measurement are sample- dependent. Additional instrumentation may be used to determine the sample’s emissivity and laser absorption. In addition, some materials are difficult to measure without specialized instrumentation; metals, for example, can have low absorption and emissivity. [0087] If desired, the sizes/diameters of the spots illuminated by the first and second laser beams can be changed during or between measurements as shown in FIG.11, e.g., by adjusting the position or focal length of the lenses used to focus the first and second laser beams. Changing the diameters of the spots illuminated by the first and second laser beams, e.g., using a zoom lens, changes the sizes of the first and second portions of the sample. It also changes the characteristic length scales of the heating event, both radially and in depth, with a larger beam diameter heating a larger volume of the sample. [0088] For a sample 80 with layered materials or buried layers as in FIG.11, interfaces, voids, defects, inclusions, and/or bonds, changing the measurement depth allows unique sensitivity to determine thermophysical properties of individual layers, interfaces, and/or bonds. A sample with a subsurface void, inclusion, or defect may be interrogated by illuminating the sample with a first laser beam such that the void, inclusion, or defect is captured inside of the heating volume of the first laser beam (the first portion of the sample). The spatial distribution of temperature can be probed by using a thermal camera. A void, inclusion, or defect should appear as a hot spot on the temperature distribution in the thermal camera. Changing the size (diameter) of the spot formed by the first laser beam changes the depth of the first portion and can be used to identify the depth of the void, defect, or inclusion under the surface of the sample. This technique could be used to estimate layer thickness in a multi-layer sample; for instance, if the thermal properties of the first layer and the substrate are known, then the measured thermal response can be used to estimate the thickness of the first layer. [0089] The temperature (both magnitude and shape) measured at the surface of the sample is a function of the properties of the material(s) and interfaces under the sample’s surface, to the extent to the which the temperature rise beneath that surface is significant. This significance depends on the sample’s material stack but is typically defined by some decay fraction relative to the surface temperature (e.g., 1/e² temperature decay). Because this temperature decay with depth is directly related to the diameter of the second (pump) laser beam, increasing the size of the spot formed on the sample surface by the second laser beam means the temperature rise measured at the sample’s surface is more sensitive to a material farther below the sample’s surface. Hence, if a layer of material is below the sample’s surface, the second laser beam can be expanded such that the second portion goes deep enough below the sample’s surface to ensure that the measured temperature rise is influenced by that layer. Thermal Model for TOPS Measurements [0090] The heat diffusion equation in cylindrical coordinates is

where ^^ _^^ and ^^ _^^ are in-plane and cross-plane thermal conductivity, respectively, T is the temperature (relative to some initial temperature) at a given spatial and time combination, r denotes the radial coordinate variable, ^^ denotes the cross-plane coordinate variable orthogonal to r, p is the mass density, c_p is the specific heat capacity, and t is the time variable. The boundary conditions are given by

where the ^ are the temperature and net heat flux, respectively, at the

top surface of a given material. The net heat flux at the sample surface can be expressed as

where Q_{air,conduction} is the heat conduction (including Kapitza conductance) to air, H is the effective convection coefficient that includes the sum of the air convection coefficient and the linearized radiation coefficient. ^^_laser is the heat provided by the laser to the sample surface, defined by

where r₀ is the 1/e² radius of the laser at the sample surface and G(t) defines the time variation of the laser; in our case, this is constant but the solution is generalized so that any function (e.g. with amplitude modulation at a specific frequency or pulsed laser) can be used. [0091] Taking the Fourier transform (t → ω) and Hankel transform (r → k) simplifies the heat diffusion equation to

The notation used here is such that for a function Y(z, r, t), Ỹ denotes the Fourier transform of ^^, and ^ ∼ ^̃^ denotes the Hankel transform of Ỹ. q is defined as

The transformed heat flux on the top surface Q_top(r, t) becomes

[0092] Within any layer of the material stack, the temperature and heat flux at any given depth,z, is can be related to the top temeprature of the material by the following:

Likewise, at the bottom of each layer, the thermal boundary conductance between the layers (e.g. between layer n − 1 and n needs to be considered. The relationship between temperature and heat flux at the bottom of layern − 1 and the top of layer n is given by:

[0093] When considering multiple layers, one can apply a matrix method, also known as the thermal quadrupole method, to get

[0094] For this generalized solution,

is determined by invoking a semi-infinite boundary condition for bulk samples so that such that

Likewise, the temperature can also be defined based on the air side properties through the same means to get

where the term for air includes the air properties and also the thermal boundary

conductance (Kapitza conductance) between the surface and the air. The term is positive because ^^ is defined to be positive in the direction of the sample, so the air goes to −z; as such, the signs of _and should reversed since sinh z) is an odd function. Therefore,

[0095] Referring back to

Substituting leads to

[0096] We can define the transfer function as Z(k, ω) to simplify the relationship to

This solution is still defined in the Hankel and frequency domains. To go back to radial coordinates, the inverse Hankel transform is taken to get the temperature rise needed to compare with experimental data.

In the steady-state, ω = 0 Hz to comply with the constant power condition, but the same equation can be applied when ^^ is nonzero. This defines the temperature profile radially. Conclusion [0097] While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize or be able to ascertain, using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure. [0098] Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. [0099] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms. [00100] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” [00101] The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. [00102] As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e., “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law. [00103] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. [00104] In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

Claims

CLAIMS 1. A method of measuring a property of a sample, the method comprising: measuring a baseline temperature of the sample; illuminating a spot on a surface of the sample with a laser beam at a constant power level, the laser beam heating the spot on the sample to a steady-state temperature; measuring the steady-state temperature of the spot on the sample; and determining the property of the sample based on the constant power level and on a difference between the steady-state temperature of the spot on the sample and the baseline temperature of the sample.

2. The method of claim 1, wherein the property of the sample comprises one of thermal conductivity, a thermal resistance, or film thickness.

3. The method of claim 1, wherein the sample comprises one of a liquid, a gel, or a paste.

4. The method of claim 1, wherein: measuring the baseline temperature comprises detecting infrared radiation emitted from a region of the sample not heated by the laser beam with a first set of detector elements of an infrared detector; measuring the steady-state temperature comprises detecting infrared radiation emitted from the spot on the sample with a second set of detector elements of the infrared detector while measuring the baseline temperature; and taking the difference between the steady-state temperature of the spot on the sample and the baseline temperature of the sample suppresses noise common to outputs of the first set of detector elements and the second set of detector elements.

5. The method of claim 1, further comprising: heating the sample to the baseline temperature.

6. The method of claim 5, wherein heating the sample to the baseline temperature comprises illuminating the sample with a heating laser beam at a power level of 0–10 kW.

7. The method of claim 1, further comprising: disposing a transducer directly on the surface of the sample, the transducer having a known thermal conductivity, wherein the spot on the surface of the sample is on the transducer.

8. The method of claim 7, wherein the transducer comprises a polymer-based tape.

9. The method of claim 1, wherein the steady-state temperature is a first steady-state temperature, the constant power level is a first constant power level, and further comprising: illuminating the spot on the surface of the sample with the laser beam at a second constant power level greater than the first constant power level, the laser beam heating the spot on the sample to a second steady-state temperature higher than the first steady-state temperature; and measuring the second steady-state temperature of the spot on the sample, wherein determining the property of the sample is further based on a difference between the second steady-state temperature of the spot on the sample and the baseline temperature of the sample.

10. A method of measuring a thermal conductivity of a sample, the method comprising: disposing a transducer directly on a surface of the sample, the transducer having a known thermal conductivity; illuminating a first region of the transducer with a laser beam at a constant power level and at a wavelength absorbed by the transducer, the laser beam heating the first region of the transducer to a steady-state temperature; measuring the steady-state temperature of the first region of the transducer with a first set of sensing elements of an infrared detector; while measuring the steady-state temperature of the first region of the transducer with the first set of sensing elements, measuring a baseline temperature of a second region of the transducer with as second set of sensing elements of the infrared detector; and determining the thermal conductivity of the sample based on the known thermal conductivity of the transducer and a difference between the steady-state temperature and the baseline temperature.

11. A system for measuring a thermal conductivity and/or a thermal resistance of a sample, the system comprising: a laser, in optical communication with the sample, to illuminate a first region of the sample with a laser beam at a constant power level, the laser beam heating the first region of the sample to a steady-state temperature; an infrared detector, in optical communication with the sample, having a first set of sensing elements to measure the steady-state temperature of the first region of the sample and second set of sensing elements to measure a baseline temperature of a second region of the sample not heated by the laser beam while the first set of sensing elements measure the steady-state temperature; and a processor, operably coupled to the laser and the infrared detector, to determine the thermal conductivity and/or the thermal resistance of the sample based on the constant power level and a difference between the steady-state temperature and the baseline temperature.

12. The system of claim 11, wherein the infrared detector has an absolute temperature accuracy of no better than ±2 °C or ±2% and a temperature resolution of finer than 0.1 °C.

13. A method of measuring at least one thermophysical property of a sample, the method comprising: heating a first portion of the sample to a first temperature with a first laser beam; while the first portion of the sample is at the first temperature, heating a second portion of the sample at least partially contained within the first portion of the sample to a second temperature higher than the first temperature with a second laser beam; while heating the second portion of the sample, measuring a change in temperature of the second portion of the sample as a function of power of the second laser beam; and determining, based on the change in temperature of the second portion of the sample, the at least one thermophysical property of the sample at a weighted average of the first temperature and the second temperature.

14. The method of claim 13, wherein the at least one thermophysical property includes thermal conductivity, emissivity, heat capacity, and/or thermal boundary resistance.

15. The method of claim 13, wherein the sample is at room temperature and pressure.

16. The method of claim 13, wherein the sample is coated with a transducer configured to absorb the first laser beam and the second laser beam.

17. The method of claim 13, wherein heating the first portion of the sample comprises illuminating a first spot on the sample and heating the second portion of the sample comprises illuminating a second spot smaller than the first spot on the sample.

18. The method of claim 13, further comprising: setting a power level of the first laser beam high enough to heat the first portion of the sample to the first temperature within five minutes and setting a power level of the second laser beam high enough to heat the second portion of the sample to the second temperature within five minutes.

19. The method of claim 13, further comprising: varying a size of the first portion of the sample and/or a size of the second portion of the sample.

20. The method of claim 13, wherein determining the at least one thermophysical property of the sample is further based on a slope of the change in temperature versus the power of the second laser beam.

21. A system for measuring at least one thermophysical property of a sample, the system comprising: at least one laser, in optical communication with the sample, to heat a first portion of the sample to a first temperature with a first laser beam and, when the first portion of the sample is at the first temperature, to heat a second portion of the sample at least partially contained within the first portion of the sample to a second temperature higher than the first temperature with a second laser beam; an infrared detector, in optical communication with the sample, to measure a change in temperature of the second portion of the sample as a function of power of the second laser beam; and a processor, operably coupled to the infrared detector, to determine, based on the change in temperature of the second portion of the sample, the at least one thermophysical property of the sample at a weighted average of the first temperature and the second temperature.

22. The system of claim 21, wherein the at least one thermophysical property includes thermal conductivity, emissivity, heat capacity, and/or thermal boundary resistance.

23. The system of claim 21, wherein the sample is at room temperature and pressure.

24. The system of claim 21, wherein the at least one laser is configured to emit the first laser beam at a power level high enough to heat the first portion of the sample to the first temperature within five minutes and to emit the second laser beam at a power level high enough to heat the second portion of the sample to the second temperature within five minutes.