Accurate and rapid measurement of fluid thermal conductivity

Abstract

A rapid energy transition will require new heat transfer fluids, and a faster means of discovering and optimizing them. Existing methods, however, are constrained by speed, accuracy, and sample volume — with accurate measurements requiring large sample volumes and long equilibration times. Here, we present a measurement approach that bypasses precise temperature measurement and heat flux measurements. Thermal conductivity, k, is determined by comparing thermally driven voltage variations across an array of resistive heaters embedded in fluid cavities. This measurement, relative to the reference material, minimizes errors from ambient temperature fluctuation, unquantified heat losses, and measurement uncertainties, and it eliminates direct temperature sensing. We report a microfluidic device and measurement method that implements in-run on-chip auto-calibration with a reference material; we test the device on a wide range of substances, including liquids, gases, mixtures, and nanofluids. It delivers results in <10 s, using ~5 µL of sample, about two orders of magnitude faster than conventional steady-state methods — while maintaining accuracy competitive with gold-standard techniques (mean signed error 0.0030 ± 0.0059 W/(m · K) (1.4% ± 3.1%), median absolute error 0.0019 W/(m · K) (MAPE 2.6 %), and expanded uncertainty of k, U = 8.3% ± 0.24% (SD across 10 runs)). This approach makes thermal conductivity measurement accessible to accelerated materials discovery and optimization workflows.

Introduction

In a warming world with increasing electrification and computation, there is a growing need for highly performant electrically insulating coolants and working fluids. Thermal conductivity is a physical property central to these applications, including immersion cooling of computer hardware, data center cooling¹, refrigeration, air conditioning, and solar thermal energy storage². However, the discovery and development of new fluid compositions^3,4 is hindered by the low-throughput nature of conventional thermal conductivity measurement techniques^5,6. Limitations inherent to heat conduction result in long test times (taking from 30 min to several hours for typical samples, including preparation, testing, and device cleaning). The variability of thermal conductivity across temperature, pressure, and material composition has led to data scarcity that has slowed the development of thermal fluid applications^4,7,8.

In the absence of a high-throughput measurement approach, computational approaches have been developed to predict thermal conductivity through cumulative molecular effects^9,10 or group contribution models^11,12. However, the need for intermolecular forces and molecular properties as inputs limits the accuracy, with errors around 10% and in some cases exceeding 100%¹¹. Such inaccuracies necessitate experimental verification, especially when the prediction involves innovative new chemistries. Moreover, these models are less effective for fluid blends, which are common in industrial applications, and the thermal conductivity of blends does not typically conform to simple averaging rules¹³. The limitations of computational approaches motivate the need for high-throughput experimental approaches that provide rapid and accurate thermal conductivity measurements with low volume requirements. Current experimental techniques are low-throughput, batch-based, and require manual operation, making them unsuitable for fast fluid screening. This limitation hinders rapid technological advancement using accelerated material discovery, which requires extensive testing and data generation under various conditions.

High-throughput approaches have recently emerged for measuring thermophysical properties — including specific heat capacity¹⁴, enthalpy of reactions¹⁵, and viscosity¹⁶ — where microfluidic structures enabled testing with minimal reagent consumption. These approaches have also shown promise in measuring the thermal conductivity of fluids^{17,18,19,20,21}. However, thermal conductivity measurement stands out as particularly challenging among thermophysical property measurements^22,23,24,25. Methods to measure thermal conductivity may be categorized as steady-state or transient^24,26. Transient methods can be fast, delivering thermal conductivity and specific heat capacity measurements within a few tens of seconds. However, these require a large sample volume that can approximate the infinite or semi-infinite medium assumed in the analytical solution. Moreover, there is a need for direct contact with the heating element and the fluid, which imposes material-fluid compatibility restrictions²⁷. In addition, transient approaches require other physical properties of the fluids, such as the density and specific heat capacity, to be known beforehand. Transient methods present additional challenges, such as experimenter error in interpreting the temperature response curve²⁸, and limitations in heating frequency²⁹ and power constraints¹⁷.

Steady-state methods can offer higher accuracy³⁰ as the signal processing is straightforward, and thermal conductivity can be determined without relying on density and specific heat capacity. The thermal conductivity is determined directly from a thermal gradient resulting from the known applied heat flux. However, establishing a steady state thermal gradient takes significant time — often several hours for fluids^26,31. This prolonged time requirement introduces challenges such as heat loss minimization and increased susceptibility to convection effects³². Minimizing error requires uniformity in the thermal field²³ and minimizing heat loss elsewhere in the device — a source of error that can be mitigated by calibration³³ but not eliminated^27,31.

Here, we propose a high-throughput thermal-conductivity measurement technique that combines the speed of transient methods with the accuracy of steady-state approaches. A symmetric configuration of microchannels with varying heights along identical resistive heaters allows for relative analysis and thermal-conductivity measurement using only voltage characterization, eliminating the need for calibration. This method is designed to operate rapidly (within seconds) on microliter sample volumes and under a wide range of conditions, including high temperatures, pressures, and the presence of volatile or compressible fluids. This technology not only enhances the accuracy of thermal-conductivity assessments but enables thermal fluid screening and discovery for a broad range of applications. In the following sections, we describe the measurement technique and introduce a microfluidic device that operates based on this principle, followed by an in-depth discussion of its performance and the associated uncertainty.

Results

Measurement principle

We developed an approach to measure thermal conductivity by combining the speed of transient techniques with the accuracy of steady-state methods. The measurement involves monitoring voltage changes across a series of identical resistive heaters, each surrounded by two identical liquid layers (Fig. 1a–c). The thickness of the liquid layers varies for each heater. All liquid layers maintain contact with a constant temperature, T₀, on the side opposite the heaters. When an electric current is applied to the heaters, their temperature increases due to the Joule heating. The temperature of the heaters reaches a steady state value that is a function of the thermal resistance of the liquid layers. Heaters that are covered by thicker liquid layers and/or by fluids with lower thermal conductivity tend to reach higher steady-state temperatures. Under steady-state conditions, with no convective heat transfer in the fluids and dominant one-dimensional heat transfer, the power generated by the heater is twice the heat conducted through the polyimide and adjacent liquid layer on each side, which is given by Eq. (1):

$$P={VI}=2\frac{{T}_{H}-{T}_{0}}{\left(\frac{\delta }{k}+\frac{{\delta }_{p}}{{k}_{p}}\right)}$$

(1)

Fig. 1: Working mechanism and configuration of the thermal conductivity measurement device.

a Linear temperature profiles in the fluid layer and the polyimide substrate at steady state conditions. b Plot illustrating the inverse of the voltages of the heaters versus the height of fluid when all channels are filled with the same fluid. The inverse voltage is proportional to the height of the fluids. c Plot showing the inverse of the voltages when one of the channels is filled with a different fluid. This voltage inverse corresponds to the inverse voltage caused by a reference fluid with a height of δ*. d 3D illustration of the thermal conductivity measurement device and its principle of operation. It showcases different components of the device, including the polyimide heater, two copper blocks with stepwise surfaces, two low thermal conductivity material layers to hold the liquid chamber walls, and the thermoelectric heater/cooler for temperature control. e Detailed view of the heating elements and the embedded fluid channels. The fluid and heating elements are separated by a thin layer of polyimide. f Simulated temperature and velocities in a typical channel, showing nearly linear temperature distribution and two vortices near the channel walls with a velocity magnitude on the order of 10 µm/s.

Here, $V$ is the voltage of the heater, $I$ is the current, ${T}_{H}$ is the heater temperature, ${T}_{0}$ is the constant temperature on the side opposite the heater for each liquid layer. The thicknesses of the fluid and polyimide layers are denoted by $\delta$ and ${\delta }_{p}$, respectively, while $k$ and ${k}_{p}$ represent their respective thermal conductivities. Using Ohm’s law and assuming a linear dependence of the heater resistance to the temperature offset (${T}_{H}-{T}_{0}$), the voltage–current–temperature offset relation is given by Eq. (2):

$${V=R}_{0}\left(1+{\alpha }_{T}\left({T}_{H}-{T}_{0}\right)\right)I$$

(2)

where ${R}_{0}$ is the resistance at ${T}_{0}$ and ${\alpha }_{T}$ is the temperature coefficient of resistance. Combining Eq. (1) and Eq. (2) to eliminate $\left({T}_{H}-{T}_{0}\right)$ yields the linear relation in Eq. (3) between the inverse voltage of each heater and the surrounding fluid thickness $\delta$:

$$\frac{1}{V}=-\left(\frac{\alpha I}{2k}\right)\delta+\left(\frac{1}{{R}_{0}I}-\frac{{\alpha }_{T}I{\delta }_{p}}{2{k}_{p}}\right)$$

(3)

According to Eq. (3), for a thermal-conductivity measurement using a single heater, all parameters must be known. These parameters include the thickness and properties of the insulating layer (${\delta }_{p}$, ${k}_{p}$), heater characteristics (${R}_{0}$ and ${\alpha }_{T}$), the voltage of the heater, and the current passing through it. If two heaters are used, the current still needs to be measured. Here, we sought a multiple-channel approach that requires only the measurement of voltage. If the heaters were connected in parallel, they would all share the same voltage and only the current would need to be measured. However, the relationship between thermal conductivity and current would be implicit and complicated by its dependence on the temperature offset. Therefore, we opted for a series connection that allows us to measure voltage only, yielding a clear, linear relation and simplifying the analysis. In the multiple fluid channel configuration, when all channels are filled with the liquid of the same k, the inverse of the voltage (1/V) forms a linear relationship with the liquid thickness $\delta$ (Fig. 1b). Although Eq. (3) suggests that other parameters could be varied to extract k, altering intrinsic heater properties (e.g., ${{{\rm{\alpha }}}}_{{{\rm{T}}}}$ and ${{{\rm{R}}}}_{0}$) is impractical, and measuring current would introduce nonlinear complexities. Similarly, varying the insulating layer thickness (${{{\rm{\delta }}}}_{{{\rm{p}}}}$) would slow the system response and promote three-dimensional heat transfer, thereby violating the one-dimensional assumption. Thus, we chose channel height as the primary variable to minimize error propagation and simplify the analysis. Using this concept, the thermal conductivity of an unknown fluid may be determined relative to that of a reference fluid. For this purpose, one of the channels (e.g., ${\delta }_{1}$) is filled with a sample of unknown k, while others contain the reference fluid of known k. By applying a current, the voltage of the sample heater varies from the scenario where the sample channels were filled with the reference fluid (Fig. 1c). The data acquisition system measures the voltage across each heater. A temperature controller is implemented to maintain a constant temperature at the upper boundary of the fluid layers, allowing the heater temperatures to vary with the fluid’s thermal conductivity until steady state is reached. To calculate the thermal conductivity of the sample, one needs to find the corresponding thickness of the reference fluid (denoted as ${\delta }^{*}$) that would result in the same conduction thermal resistance ($\delta /{kA}$, $A$ is area) as the sample fluid. This requires finding a thickness at which the voltage V₁, which is indicative of the heater’s resistance and its temperature, aligns with the trend line fit to the other four data points. Once the equivalent thickness ${\delta }^{*}$ is found, the thermal conductivity of the sample, k, is calculated by Eq. (4):

$$\frac{{\delta }_{1}}{k}=\frac{{\delta }^{*}}{{k}_{{{\rm{ref}}}}}$$

(4)

Based on this principle, we fabricated a device (Fig. 1d, e) that comprises five microchannels with different heights of 150, 300, 450, 600, and 750 μm. Note that at least two reference channels are required to eliminate the need to measure the current. To maintain a constant temperature (${T}_{0}$) at the upper boundary of each channel, high thermal conductivity material (e.g., copper) tops each microcavity. In contrast to existing methods, fluctuations in ${T}_{0}$ have a minimal effect on the determined k since measurements are relative (Supplementary Note 1). The channels are designed with a high aspect ratio to direct the majority of heat flow in a one-dimensional path toward the copper top. Furthermore, the channels are etched in close proximity to minimize variations in the upper boundary temperature and minimize lateral heat transfer (Fig. 1f). Under these conditions, lateral heat conduction does not exceed 5% of the total heat flux (Supplementary Note 2). The relative nature of the measurement further minimizes the error associated with this lateral heat conduction (Supplementary Note 2).

Performance evaluation and validation

We assessed the thermal conductivity measurement approach with liquids, mixtures, nanofluids, and gases, covering a wide range of thermal conductivities (0.017–0.75 W/(m·K)). All measurements were carried out at atmospheric pressure. Although the approach could be extended to high pressures with adequate sealing, atmospheric pressure measurements are sufficient for most applications, as the thermal conductivity of liquids and gases is only weakly dependent on pressure below 50 bar³⁴. Table 1 presents the measured thermal conductivities for >10 liquids and two gases. Each test was repeated three times, and the standard deviation of the measurements was provided. The mean absolute relative error of the measurements was found to be 2.6% (with an average standard deviation of ±1.7%), indicating a level of accuracy comparable to standard reference data^35,36,37. The 95% expanded uncertainty for the device at 30 °C is determined to be U = 8.3% (k = 2) (see detailed analysis in the Methods section). The thermal conductivities are also shown in Table 1. Details for calculation of the uncertainty of k (expanded) may be found in the Methods section as well as Supplementary Note 3.

Table 1 Measured and literature values of thermal conductivity of 11 liquids and two gases at 30 °C

To assess the device’s capability in measuring the thermal conductivity of mixtures, we conducted experiments with various liquid mixtures, including aqueous solutions where ethylene glycol, propylene glycol, ethanol, and glycerol were each mixed with water in concentrations ranging from 0% to 100%. The mixing process was automated and continuous, using two syringe pumps to ensure a consistent and precise blend of the fluids at a T-junction (Fig. 2a). The thermal response of the system, i.e., the time from when the heater is turned on and the signals stabilize to when the heater is turned off and the system returns to its initial state, takes less than 10 s (Fig. 2b–d). The measurement time is primarily determined by the time required for the channels to be filled with new concentrations. Based on the speed of fluid motion, length of tubing, and dead volumes of the system, this filling time can be managed to be less than 5 s. Due to the dead volume, it took ~5 s for the newly mixed liquids to enter the device. To ensure accurate measurements, we performed tests at 15 s intervals, allowing sufficient time for the previous mixture to exit and the new mixture to fully occupy the measurement channels. As expected, the thermal conductivity of these aqueous solutions decreased with increasing concentrations of ethylene glycol, propylene glycol, ethanol, or glycerol — as expected given the higher thermal conductivity of water. The measurements closely follow the literature curve (Fig. 2e–g).

Fig. 2: High-throughput measurement of thermal conductivity in liquid mixtures and particulated fluids.

a Experimental setup for the online creation of various mixture concentrations. The constant voltage power supply applies a fixed voltage to the resistive heaters, which are connected in series so that they all carry the same current. Individual voltages (V₁,…,V₅) across each heater are measured by the data acquisition system. b Voltage response of resistive heaters to a 2.5 V input, with each waveform representing one measurement. c Voltage response signals during the measurement of thermal conductivity of propylene glycol at 30 °C. Signals stabilize within ~5 s, demonstrating fast measurement compared to traditional steady-state methods. d Plot of steady-state values of inverse voltage against liquid thickness, enabling the determination of propylene glycol thermal conductivity. e Thermal conductivity measurements: Ethylene glycol-water (solid curve, compared to data from Table 8 in ref. ⁴⁷) and propylene glycol-water mixtures (compared to Table 12 in ref. ⁴⁷); f Ethanol-water mixtures (ref. ⁴⁸); g Glycerol-water mixtures (ref. ⁴⁹); h Graphene oxide-water nanofluid (0 to 1 wt %), ref. ⁵⁰. The error bars represent the standard deviation of the measurements. Source data are provided as a Source Data file.

Furthermore, we tested the device with a nanofluid containing graphene oxide (GO) in water, with concentrations ranging from 0 to 1 wt % (Fig. 2h), demonstrating the device’s capability to accurately measure thermal conductivity in complex colloidal systems. The measurements are consistent with literature values for the nanofluid, demonstrating effectiveness in handling complex mixtures. For each mixture, 11 data points were measured. Each measurement took about 10 s (5 s to stabilize and 5 s to reset and fill with new fluid), allowing the entire measurement range to be completed in under two minutes. The error bars in Fig. 2e–h represent the standard deviation of repeated measurements. The slight deviations fall within the device expanded uncertainty at 30 °C (U = 8.3%, k = 2). This uncertainty fully accounts for the observed differences, confirming that the measured values are in reasonable agreement with the theoretical predictions.

To evaluate the device performance at different temperatures, we measured the thermal conductivity of three liquids, ethanol, ethylene glycol, and ethyl laurate, over a temperature range of 10–67 °C. A temperature control system, incorporating feedback from a thermocouple positioned near the center channel within the core of the device, was employed to regulate the temperature of the device at a predetermined set-point (Fig. 3a). A detailed 3D thermal simulation (see Supplementary Note 4) confirms that the top boundary temperature ${T}_{0}$ is uniform across all channels. In addition, it shows that the volume-averaged liquid temperature deviates by only about 0.5 °C from ${T}_{0}$, confirming that the single thermocouple reading accurately represents the effective fluid temperature for determining k.

Fig. 3: Thermal conductivity of liquids at different temperatures.

a Temperature control setup with feedback from a central thermocouple, connected to heaters that turn off upon reaching the set-point. b Thermocouple response to a set-point increase from 23 °C to 53 °C, showing a 3 min stabilization time, compared to a 1 min simulation assuming perfect heat transfer to the device and ignoring the controller effects. Measured thermal conductivity of ethanol (c), ethylene glycol (d), and ethyl laurate (e), respectively, from 10–67 °C, with quadratic fits (dashed lines) matching literature data (solid lines)³⁵. Markers indicate the mean of three measurements; error bars denote the standard deviation. Source data are provided as a Source Data file.

The response of the thermocouple to a 30 °C set-point increase (from 23 °C to 53 °C, Fig. 3b) shows that the device temperature stabilizes in ~3 min. In contrast, a simulation assuming ideal heat transfer—i.e., that all the heat generated by the heaters is transferred directly to the device—predicts a response time of about 1 min. We attributed this discrepancy to the fact that in the simulation, the power applied to the copper boundaries is set equal to the nominal power specified for the heating pads until the center of the blocks reaches 53 °C, whereas in the experimental system, a portion of the heat is lost to the environment. The device demonstrates accuracy within the error bar ranges for the three mentioned liquids when measuring k vs. temperature (Fig. 3c–e). The curves fit to the data within the temperature range (dashed lines) match closely with the correlation from literature³⁵ (solid black lines) with relative mean absolute errors of 0.19%, 0.12%, and 0.28%, for ethanol, ethylene glycol, and ethyl laurate, respectively. In Table 1, we report the standard deviations for each fluid. The measurements were performed consecutively, with only a flush between fluids rather than a complete cleaning; this may account for slightly higher variability in some cases (e.g., acetone with a 4.5% standard deviation), although ethanol exhibited only 0.4% variability. For ethanol at 30 °C, the literature value is 0.1689 W/(m·K) while the device measured 0.1720 W/(m·K) in Table 1 and 0.1663 W/(m·K) in Fig. 3c (each based on three independent measurements). Although these individual datasets differ by ~3%, this variation is consistent with our overall experimental uncertainty. When averaged, the combined mean value is 0.1691 W/(m·K), differing from the literature value by only about 0.12%.

Design considerations to limit convective heat transfer

One challenge in measuring the thermal conductivity of fluids is to minimize natural convection, which can increase the heat transfer rate and, if stagnant conditions are assumed as in Eq. (1), lead to an overestimation of the inferred thermal conductivity. The heaters are sandwiched between two fluid layers. The lower fluid layers are thermally stabilized, as the heater is on its top, and thus no thermal convection would be generated. However, the upper fluid layer has the potential to experience natural convection. We quantify the onset of natural convection using the dimensionless Rayleigh number (Ra), defined as:

$${Ra}=\frac{g\beta \left({T}_{h}-{T}_{c}\right){\delta }^{3}}{\nu \alpha }$$

(5)

where g represents the acceleration due to gravity, $\beta$ is the thermal expansion coefficient of the liquid, ${T}_{h}$ and ${T}_{c}$ are the temperatures of the heated and cooler surfaces, respectively. $\delta$ is the thickness of the liquid layer, $\nu$ is the kinematic viscosity, and $\alpha$ is the thermal diffusivity of the liquid. Natural convection becomes significant when ${{{\rm{Ra}}} > {{\rm{Ra}}}}_{{{\rm{c}}}}$; for a horizontal fluid layer between two rigid, isothermal plates heated from below, the critical Rayleigh number is approximately ${{Ra}}_{c}\approx 1700$³⁸. However, in the 3D confined geometries, such as those here (Fig. 1e), the ${{Ra}}_{c}$ may be smaller due to the side-wall effects. For a single channel with a height of $\delta$ and width of L, we performed numerical simulations using five selected liquids (water, ethanol, octane, toluene, and diethyl ketone), varying the channel aspect ratio ($\delta /L$) between 0.02 and 0.3 and the vertical temperature difference (${T}_{h}-{T}_{0}$) between 2 °C and 12 °C (Fig. 4a). The results indicate${{Ra}}_{c}$, conventionally chosen to be the point at which convection heat transfer exceeding 1% of the total vertical heat flux falls between 200 and 400 for this geometry. At Ra numbers near and below this range, the dominant heat transfer mechanism is conduction.

Fig. 4: Impact of natural convection and insulating layer thickness on thermal conductivity measurements.

a Numerical simulation of the Rayleigh number (Ra) as a function of channel aspect ratio and vertical temperature difference for five fluids: water, ethanol, octane, toluene, and diethyl ketone. The critical Ra range is identified between 200 and 400, indicating the onset of natural convection in the 3D channel. b Influence of natural convection on thermal conductivity measurements using water as the reference fluid. Convection starts to play a role in the largest channel (750 µm) at ~400. c Insensitivity of the thermal conductivity measurement approach to natural convection, if present. d Simulated heater temperatures as a function of the liquid thickness for various polyamide insulating layers, ranging from 40 µm to 100 µm, using water as the reference fluid and propylene glycol as the sample fluid. All temperatures increase uniformly as the thickness of the insulating layer is increased, indicating that the derived thermal conductivity remains almost constant. e The plot shows the thermal conductivity derived from the simulations as the thickness of the polyamide layer increases. The error is <1% in all cases. Source data are provided as a Source Data file.

The reference channels of the device are filled with a material with known thermal conductivity. The reference material can be either fluid or solid. Using solids eliminates the limitations of convective heat transfer, leading to a broader range of device applicability. However, we opted here for liquid references to minimize contact resistance at the interfaces. Additionally, solids typically have higher thermal conductivity compared to fluids, necessitating thinner reference solid fabrication with enhanced precision and uniformity. In our experiments, the maximum Ra is 100, which is well below the critical threshold, ensuring that the effects of convection did not influence the measurements in either the sample or reference channels. Also, if convection were to become an issue, its influence would present first in the larger chambers and thus could be detected and corrected. The relative nature of the measurement also minimized the impact of any convection present. In simulations with water as the reference fluid and ethylene glycol as a sample, convection begins to influence the thickest channel of the reference fluid at a critical Rayleigh number of ~400 (Fig. 4b). Yet the determined thermal conductivity remains accurate up to Ra ≈ 1000 (Fig. 4c). This low sensitivity to convective effects is not shared by existing methods²⁸. The reference channels of the device are filled with a material with known thermal conductivity. The reference material can be either fluid or solid. Using solids eliminates the limitations of convective heat transfer, leading to a broader range of device applicability. However, we opted here for liquid references to minimize contact resistance at the interfaces. Additionally, solids typically have higher thermal conductivity compared to fluids, necessitating thinner reference solid fabrication with enhanced precision and uniformity. In our experiments, the maximum Ra is 100, which is well below the critical threshold, ensuring that the effects of convection did not influence the measurements in either the sample or reference channels. Also, if convection were to become an issue, its influence would present first in the larger chambers and thus could be detected and corrected. The relative nature of the measurement also minimized the impact of any convection present. In simulations with water as the reference fluid and ethylene glycol as a sample, convection begins to influence the thickest channel of the reference fluid at a critical Rayleigh number of ~400 (Fig. 4b). Yet the determined thermal conductivity remains accurate up to Ra ≈ 1,000 (Fig. 4c). This low sensitivity to convective effects is not shared by existing methods²⁸.

Another factor contributing to the accuracy of this measurement approach is an insensitivity to the thickness of the insulating layer (${\delta }_{p}$) separating the heaters and the fluid (Fig. 1a). This layer is essential for providing electrical insulation, as the resistive heaters cannot come into direct contact with the fluids. Some fluids are not dielectric and can disturb voltage measurements. Additionally, the insulating layer serves as a physical barrier between heaters and fluids that would otherwise react.

In conventional devices, the insulating layer is necessary for electrical insulation but also acts as a thermal barrier, creating thermal resistance. This resistance can lead to measurement errors in temperature, increase system response time, decrease measurement sensitivity, and cause heat to propagate in undesired directions (e.g., laterally), a phenomenon known as heat leak. The relative measurement approach significantly mitigates this source of error. Since measurements are relative, any thermal resistance between the fluids and the heater cancels out as it affects all fluid channels similarly. This effect is reflected in the intercept of the equation Eq. (3) used for thermal conductivity calculation.

To evaluate the impact of insulating layer thickness, we conducted numerical simulations using water as the reference fluid and propylene glycol as the sample fluid. We varied the thickness of the polyamide layer from 40 µm (as used in the fabricated device) to 100 µm, which is 2.5 times thicker. Increasing the insulating layer thickness raises the thermal resistance uniformly across all channels, causing all heater voltages (V₁–V₅) to increase equally, thus preserving the relative differences needed to accurately derive thermal conductivity (Fig. 4d). The results demonstrate a negligible error in the measured thermal conductivity due to variations in this parameter (Fig. 4e). Increasing the thickness of the insulating layer causes the temperatures of all five heaters to rise consistently due to the addition of a layer with low thermal conductivity to the heat dissipation pathway. This consistent temperature increase results in nearly identical values of ${\delta }^{*}$ obtained from the fitted lines across all polyamide thicknesses, leading to the measured thermal conductivity remaining unaffected. However, these temperature increases are modest (~3 °C for a 2.5-times increase in the insulating layer thickness) and exert a minimal impact on the readings of the prevailing temperatures of the fluids, to which the thermal conductivity is attributed as the temperature is obtained by averaging the heater and block temperatures.

Discussion

We have developed a microfluidic device for measuring the thermal conductivity of various fluids. The device, which integrates microcavities, high- and low-thermal conductivity materials, and resistive heaters, achieves a mean absolute relative error of 2.6% (with an average standard deviation of ±1.7%), confirming its high precision and accuracy. Measurements are completed in less than 10 s using a sample volume of only 5 µL, demonstrating rapid and efficient performance. Through extensive experimental testing, the device demonstrated accurate measurements of thermal conductivity for liquids, mixtures, nanofluids, and gases over a wide range of thermal conductivities (0.017–0.75 W·m⁻¹·K⁻¹) and temperatures (15–70 °C). Moreover, the closed-chamber design enables the safe analysis of harsh chemicals, volatile substances, and compressible fluids at elevated temperatures and pressures.

The design of the microfluidic device balances measurement accuracy with practical constraints. While a minimum of two channels is required for relative calibration, our simulations (see Supplementary Note 5) indicate that using five channels significantly reduces uncertainty, with only marginal improvement beyond that. Moreover, adding more channels increases the device footprint, pressure drop, and the risk of bubble or debris entrapment. We therefore chose five channels as the optimal compromise.

We selected the channel range of 150–750 µm based on several design considerations. Channels thinner than 150 µm would make the ±10 µm fabrication tolerance a significant fraction of the nominal dimension, thereby increasing uncertainty. Conversely, thicker channels (>750 µm) slow the response, increase side-wall heat losses (see Supplementary Note 2), and promote natural convection (Fig. 4a–c), which can violate the one-dimensional heat transfer assumption. Since water—a fluid with relatively high thermal conductivity—is used as the reference, the reference channels are designed with a larger thickness (300–750 µm) compared to the sample fluid channel (150 µm). This decision was based on the fact that m${{{\rm{\delta }}}}^{*}$ ${{{\rm{\delta }}}}^{*}$ need not lie within the range of reference-fluid thickness, as the linear relationship between 1/V and δ permits${{{\rm{\delta }}}}^{*}$. For example, although the thermal conductivity of CO₂${{{\rm{\delta }}}}^{*}$ (using water as the reference) was as large as 5400 µm, yet the determined thermal conductivity remains accurate (within <2%), highlighting a key advantage of the approach. Thus, we selected the 150–750 µm range to optimize response time, minimize uncertainty, and ensure one-dimensional heat transfer.

Traditionally, the transient hot-wire (THW) method is considered the gold standard for high-accuracy thermal conductivity measurements. However, it requires larger sample volumes and time-consuming preparation, making it unsuitable for high-throughput applications. In contrast, our approach rapidly equilibrates in a small-scale system using microliter sample volumes, thereby eliminating the need for the complex analysis required by unsteady-state methods.

This method offers numerous benefits compared to existing techniques for measuring thermal conductivity. The approach eliminates the need for complex mathematical analysis that requires specific boundary conditions to solve the heat transfer differential equations—a constraint that often limits the applicability of such systems. For example, in unsteady-state methods, a sufficient fluid volume is needed to prevent heat from reaching the opposite boundary during measurement. However, thicker liquid layers are more susceptible to natural convection, introducing inaccuracy. Compared with traditional thermal conductivity instruments (e.g., wire/plate source systems) that are typically batch, require a lot of hands-on work, and require millilitre-scale samples and long stabilization time, this microfluidic approach delivers a much lower cost per datapoint. Each test uses 5 µL and <10 s, which reduces operator time and exposure to potentially hazardous fluids. Cleaning can be automated (brief flush), minimizing downtime and consumables. The hardware is simple, aside from a data acquisition unit (common to most instruments). The only added components are syringe pumps, which need not be high-precision because the method is not flow-rate critical. In-run auto-calibration removes the need for frequent external calibration standards, further lowering maintenance. Overall, we accept somewhat higher absolute uncertainty for this speed (device U ≈ 8.3%, k = 2), its throughput, low sample consumption, reduced operator labor, and simpler maintenance make the total cost of ownership very attractive for high-throughput screening and limited-volume samples.

The approach presented here employs a steady-state temperature in a small-scale system that equilibrates quickly and is applicable to microliter sample volumes. By using a double liquid layer, the system achieves accurate steady-state temperatures within a few seconds—competitive with the fastest transient methods and with the added advantage that no additional fluid parameters, such as specific heat capacity, are required. Moreover, the method maintains the high accuracy of conventional steady-state techniques while achieving measurements that are orders of magnitude faster—completing tests in just a few seconds compared to approximately half an hour.

The measured k closely aligns with literature values. Additionally, the device’s capability to measure the thermal conductivity of mixtures, such as ethanol-water, ethylene glycol-water, and nanofluids containing graphene oxide, further demonstrates its versatility. These systems pose additional challenges—such as non-homogeneity, suspension instability, and nanoparticle agglomeration—that typically result in larger uncertainties. Moreover, some nanofluids are electrically conductive, complicating measurements with conventional methods such as the hot-wire technique. While these preliminary results exhibit higher uncertainty, they underscore the potential of our approach, and further refinements in measuring such fluids can be pursued in future studies. In addition, the device enables online mixing of fluids, thereby increasing measurement throughput for the mixtures and making it well-suited for automation. Furthermore, the approach could be expanded to measure multiple fluids simultaneously, as described in Supplementary Note 6.

While the device demonstrates promising high-throughput performance in thermal conductivity measurements, several limitations must be acknowledged. Measuring highly viscous fluids such as glycerol can lead to significant pressure drops, while bubble entrapment³⁹ may cause deviations in the measured voltages and underestimation of thermal conductivity. Particulate deposition and accumulation⁴⁰ — especially when measuring nanofluids and suspensions—can necessitate more frequent cleaning due to residual traces in dead volumes of fittings and tees. Furthermore, using copper may introduce reactivity issues with certain liquids⁴¹, which might be mitigated by coating the surfaces with a noble metal such as gold, which is known for its chemical inertness and high conductivity. Addressing these challenges is critical to further enhance the reliability and robustness of the device.

The accuracy of this approach is greatly enabled by the relative nature of the measurement. It operates across a broad range of applied power without the need for calibration. Since measurements are relative to several reference channels, the influence of channel-to-channel variations is minimized. Such variations might include manufacturing issues such as nonuniform thicknesses of insulation layers or dynamic perturbations such as the emergence of bubbles.

This flow-through thermal conductivity measurement approach enables a continuous experimental workflow that is both high-throughput and high-accuracy—matching the accuracy of the gold standard at an unprecedented rate. As a result, this device has the potential to revolutionize several industries—such as lubricants, engine and battery cooling, and heat exchangers—by accelerating material screening and discovery processes.

Methods

Microfluidic device fabrication

In this configuration, the copper trace serves both as a heating element and a temperature sensor. The initial resistance of the heaters, measured using a 4-wire technique, was ~3.7 Ω at room temperature. The temperature coefficient of resistance, ${\alpha }_{T}$, was determined experimentally to be 3.448 × 10^–3 1/K by correlating the resistance of heaters immersed in a water bath (±0.01 °C accuracy). This coefficient is used solely for converting resistance to temperature, as only the voltages across the heaters are needed to derive the thermal conductivity.

To maintain a constant temperature at the upper walls of the fluid channels, two copper blocks (6 cm × 2.2 cm, 1 cm high) with a stepwise shape featuring extruded steps of 750, 600, 450, 300, and 150 μm were employed. The upper wall of the microchannel contacts 10 mm-thick copper blocks, which ensures a stable upper boundary temperature. For the fabrication of the liquid channels, the copper blocks are inverted in a Petri dish, and a degassed PDMS solution (Sylgard 184, 1:10 ratio) is gently poured over the blocks so that the PDMS fills the empty steps. The dish is then baked at 75 °C for 2 h. After curing, the PDMS is cut to form the channel patterns. In the current device configuration, all four reference fluid channels are connected by a fluid passage, whereas the sample channel is isolated, allowing a single syringe pump to fill the reference channels.

To ensure predominantly one-dimensional heat transfer, the cavities are etched in PDMS, which minimizes lateral conduction and directs heat vertically toward the copper blocks (see Supplementary Note 2). The sample channel holds 2.5 μL of fluid, while the reference channels have volumes of 5, 7.5, 10, and 12.5 μL. The high thermal mass of the copper blocks ensures that their temperature remains essentially constant during heating, and thermoelectric modules (or heating pads) with integrated controllers maintain the temperature at ${T}_{0}$. The device temperature is controlled using two thermoelectric devices (model 00901-9L31-09BPA) with a maximum power of 5.3 W each, a maximum temperature of 200 °C, and a ΔT of 67 °C. These devices are equipped with aluminum heat sinks on the opposite side to enhance heat transfer. For gas measurements, although the ideal procedure is to vacuum the sample channels before gas injection, the channels are instead filled with gas via a syringe pump and run for one minute to displace the air, after which the flow is stopped to allow the gas to equilibrate at atmospheric pressure. For liquid samples, air displacement occurs more rapidly, resulting in a significantly shorter run time.

Thermal conductivity measurements

To measure thermal conductivity, the channels are first filled with the respective fluids. Voltage is applied to the heaters using a DC power supply (HYELEC® HY3005B), and the voltage across each heater is recorded by a data acquisition system (Agilent 34970A) at 4 Hz for ~6 s, after which the heaters are turned off to prepare for the next measurement. In typical experiments, the total applied voltage is ~2.5 V and the current is around 0.2 A, resulting in a heater temperature rise of up to 2.5 °C, while the average liquid temperature across the domain increases by no more than 0.5 °C (see Supplementary Note 4). The voltage measured across each channel typically ranges from 0.45 V to 0.56 V. The data acquisition system offers a resolution of 10 μV, ensuring that these small voltage differences are captured with sufficient accuracy. Once the power is turned off, the heater temperatures drop rapidly, allowing the system to return to thermal equilibrium. Each measurement takes ~10–15 s, with a 10 s cooling period between measurements. In cases where the voltage of one reference channel deviates from the expected trend — likely due to bubble trapping — the experiment is discarded and the channels are flushed before repeating. Measurements are repeated at least three times to ensure reproducibility.

Uncertainty analysis

Each measurement run performs an in-run auto-calibration at 30 °C using water in four reference channels of known depths ${{{\rm{\delta }}}}_{2},{{{\rm{\delta }}}}_{3},{{{\rm{\delta }}}}_{4},{{{\rm{\delta }}}}_{5}=\left\{300,450,600,750\right\}$ µm. For that run, we record steady-state voltages in the reference channels and fit the linear relation between inverse voltage and channel depth,

$$\frac{1}{{{{\rm{V}}}}_{{{\rm{i}}}}}={{\rm{m}}}{{{\rm{\delta }}}}_{{{\rm{i}}}}+{{\rm{b}}} \qquad \left({{\rm{i}}}=2\ldots 5\right),$$

which yields a run-specific slope m and intercept b. The sample occupies a separate channel of physical depth ${{{\rm{\delta }}}}_{1}=150$ µm (measured steady-state voltage V^*). Mapping the sample onto the calibrated line defines an effective thickness:

$${{{\rm{\delta }}}}^{*}=\frac{1/{{{\rm{V}}}}^{*}-{{\rm{b}}}}{{{\rm{m}}}},$$

and the sample thermal conductivity follows from geometry:

$${{{\rm{k}}}}_{{{\rm{s}}}}={{{\rm{k}}}}_{{{\rm{ref}}}}\frac{{{{\rm{\delta }}}}_{1}}{{{{\rm{\delta }}}}^{*}}={{{\rm{k}}}}_{{{\rm{ref}}}}{{{\rm{\delta }}}}_{1}\frac{{{\rm{m}}}}{1/{{{\rm{V}}}}^{*}-{{\rm{b}}}}$$

This auto-calibration absorbs, to first order, heater resistance, drive current and slow thermal drift into the fitted parameters, m and b. Following the GUM⁴², uncertainty propagation can therefore be expressed entirely through the quantities actually measured in each run.

The reference conductivity of water at 30 °C is ${{{\rm{k}}}}_{{{\rm{ref}}}}=0.6144$ W/(m · K). Per the IAPWS release R15-11⁴³, the uncertainty map for the formulation is provided as a combined expanded uncertainty with coverage factor k = 2. We therefore adopt ${{\rm{U}}}\left({{{\rm{k}}}}_{{{\rm{ref}}}}\right)=1.5\,\%$ (expanded, k = 2), i.e. ${{\rm{u}}}\left({{{\rm{k}}}}_{{{\rm{ref}}}}\right)=0.75\,\%$ (standard).

The sample-channel depth has a manufacturer tolerance of ±10 µm (rectangular), giving ${{\rm{u}}}\left({{{\rm{\delta }}}}_{1}\right)=10/\sqrt{3}$ µm. For the sample voltage ${{{\rm{V}}}}^{*}$, we combine Type-A repeatability from N readings, ${{{\rm{u}}}}_{{{\rm{A}}}}\big({{{\rm{V}}}}^{*}\big)={{{\rm{s}}}}_{{{{\rm{V}}}}^{*}}/\sqrt{{{\rm{N}}}}$ (N = 30), with the Type-B instrument term from the DAQ’s 10 V range accuracy (0.000035 ${{{\rm{V}}}}^{*}\,$+ 0.000005 × 10 V)/ $\sqrt{3}$, and the resolution-on-mean contribution ${{{\rm{u}}}}_{{{\rm{res}}}}=10{{\rm{\mu }}}{{\rm{V}}}/\sqrt{12{{\rm{N}}}}$; thus:

$${{\rm{u}}}\big({{{\rm{V}}}}^{*}\big)=\sqrt{{{{\rm{u}}}}_{{{\rm{A}}}}^{2}\big({{{\rm{V}}}}^{*}\big)+{{{\rm{u}}}}_{{{\rm{B}}}}^{2}\big({{{\rm{V}}}}^{*}\big)+{{{\rm{u}}}}_{{{\rm{res}}}}^{2}}$$

For the uncertainty of the calibration line, we estimate ${{\rm{m}}}$ and ${{\rm{b}}}$ by ordinary least squares (OLS) on the four reference points (${{{\rm{\delta }}}}_{{{\rm{i}}}},1/{{{\rm{V}}}}_{{{\rm{i}}}}$). With ${{{\rm{y}}}}_{{{\rm{i}}}}=1/{{{\rm{V}}}}_{{{\rm{i}}}}$, ${\hat{{{\rm{y}}}}}_{{{\rm{i}}}}={{\rm{m}}}{{{\rm{\delta }}}}_{{{\rm{i}}}}+{{\rm{b}}}$, the residual variance and the centered sum of squares of channel thickness for the four reference channels (${{\rm{n}}}=4$) are calculated by:

$${{{\rm{s}}}}^{2}=\frac{{\sum }_{{{\rm{i}}}}{\left({{{\rm{y}}}}_{{{\rm{i}}}}-{\hat{{{\rm{y}}}}}_{{{\rm{i}}}}\right)}^{2}}{{{\rm{n}}}-2},\qquad {{{\rm{S}}}}_{{{\rm{xx}}}}={\sum }_{{{\rm{i}}}}{\left({{{\rm{\delta }}}}_{{{\rm{i}}}}-\bar{{{\rm{\delta }}}}\right)}^{2}$$

The uncertainty of the slope, intercept, and the covariance of m and b are then:

$${{\rm{u}}}\left({{\rm{m}}}\right)=\frac{{{\rm{s}}}}{\sqrt{{{{\rm{S}}}}_{{{\rm{xx}}}}}},{{\rm{u}}}\left({{\rm{b}}}\right)={{\rm{s}}}\sqrt{\frac{1}{{{\rm{n}}}}+\frac{{\bar{{{\rm{\delta }}}}}^{2}}{{{{\rm{S}}}}_{{{\rm{xx}}}}}},{{\mathrm{cov}}}\left({{\rm{m}}},{{\rm{b}}}\right)=-\frac{\bar{{{\rm{\delta }}}}{{{\rm{s}}}}^{2}}{{{{\rm{S}}}}_{{{\rm{xx}}}}}$$

To obtain the sensitivity coefficients, with ${{{\rm{k}}}}_{{{\rm{s}}}}={{{\rm{k}}}}_{{{\rm{ref}}}}{{{\rm{\delta }}}}_{1}{{\rm{m}}}/\left(\frac{1}{{{{\rm{V}}}}^{*}}-{{\rm{b}}}\right)$ the partial derivatives are:

$${{{\rm{c}}}}_{{{{\rm{k}}}}_{{{\rm{ref}}}}}=\frac{\partial {{{\rm{k}}}}_{{{\rm{s}}}}}{\partial {{{\rm{k}}}}_{{{\rm{ref}}}}}=\frac{{{{\rm{\delta }}}}_{1}}{{{{\rm{\delta }}}}^{*}}$$

$${{{\rm{c}}}}_{{{{\rm{\delta }}}}_{1}}=\frac{\partial {{{\rm{k}}}}_{{{\rm{s}}}}}{\partial {{{\rm{\delta }}}}_{1}}=\frac{{{{\rm{k}}}}_{{{\rm{ref}}}}}{{{{\rm{\delta }}}}^{*}}$$

$${{{\rm{c}}}}_{{{\rm{m}}}}=\frac{\partial {{{\rm{k}}}}_{{{\rm{s}}}}}{\partial {{{\rm{\delta }}}}_{{{\rm{m}}}}}=\frac{{{{\rm{k}}}}_{{{\rm{s}}}}}{{{\rm{m}}}}$$

$${{{\rm{c}}}}_{{{\rm{b}}}}=\frac{\partial {{{\rm{k}}}}_{{{\rm{s}}}}}{\partial {{{\rm{\delta }}}}_{{{\rm{b}}}}}=\frac{{{{\rm{k}}}}_{{{\rm{s}}}}}{{{\rm{m}}}{{{\rm{\delta }}}}^{*}}$$

$${{{\rm{c}}}}_{{{{\rm{V}}}}^{*}}=\frac{\partial {{{\rm{k}}}}_{{{\rm{s}}}}}{\partial {{{\rm{V}}}}^{*}}=\frac{{{{\rm{k}}}}_{{{\rm{s}}}}}{{{\rm{m}}}{{{\rm{\delta }}}}^{*}{{{{\rm{V}}}}^{*}}^{2}}$$

Following the GUM, the uncertainty of the sample fluid thermal conductivity is expressed as:

$$ {{{\rm{u}}}}_{{{\rm{c}}}}\left({{{\rm{k}}}}_{{{\rm{s}}}}\right)= \\ \sqrt{{\left({{{\rm{c}}}}_{{{{\rm{k}}}}_{{{\rm{ref}}}}}{{\rm{u}}}\left({{{\rm{k}}}}_{{{\rm{ref}}}}\right)\right)}^{2}+{\left({{{\rm{c}}}}_{{{{\rm{\delta }}}}_{1}}{{\rm{u}}}\left({{{\rm{\delta }}}}_{1}\right)\right)}^{2}+{\left({{{\rm{c}}}}_{{{\rm{m}}}}{{\rm{u}}}\left({{\rm{m}}}\right)\right)}^{2}+{\left({{{\rm{c}}}}_{{{\rm{b}}}}{{\rm{u}}}\left({{\rm{b}}}\right)\right)}^{2}+{\left({{{\rm{c}}}}_{{{{\rm{V}}}}^{*}}{{\rm{u}}}\left({{{\rm{V}}}}^{*}\right)\right)}^{2}+2{{{\rm{c}}}}_{{{\rm{m}}}}{{{\rm{c}}}}_{{{\rm{b}}}}{\mathrm{cov}}\left({{\rm{m}}},{{\rm{b}}}\right)}$$

The expanded uncertainty is ${{\rm{U}}}\left({{{\rm{k}}}}_{{{\rm{s}}}}\right)=2{{{\rm{u}}}}_{{{\rm{c}}}}\left({{{\rm{k}}}}_{{{\rm{s}}}}\right)$, (≈95% coverage). We performed ten water runs at 30 °C, all five channels filled), computed ${{\rm{m}}}$, ${{\rm{b}}},$ and ${\mathrm{cov}}({{\rm{m}}},{{\rm{b}}})$ for each run, evaluated the sensitivity coefficients at the run’s operating point, and formed ${{{\rm{u}}}}_{{{\rm{c}}}}\left({{{\rm{k}}}}_{{{\rm{s}}}}\right)$ and ${{\rm{U}}}\left({{{\rm{k}}}}_{{{\rm{s}}}}\right)$. Device-level (fluid-independent) performance is summarized by the distribution across runs. The mean expanded uncertainty is U ≈ 0.051 W/(m · K) (k = 2), the maximum was U ≈ 0.053 W/(m·K), and the mean relative expanded uncertainty was U ≈ 8.3% (all k = 2). Per-run values and reconstruction details (reference depths; ${{{\rm{\delta }}}}_{1}$, tolerance model; voltmeter specification and averaging; regression residuals s, S_xx, and derived ${{\rm{u}}}\left({{\rm{m}}}\right)$, ${{\rm{u}}}\left({{\rm{b}}}\right)$, and ${\mathrm{cov}}\left({{\rm{m}}},{{\rm{b}}}\right)$ are provided in the Supplementary Note 3. Note that no external voltage calibration was performed; any constant gain error cancels in δ^* = (1/V^* − b)/m, while residual effects are captured by ${{\rm{u}}}\left({{\rm{m}}}\right)$, ${{\rm{u}}}\left({{\rm{b}}}\right)$, and ${\mathrm{cov}}\left({{\rm{m}}},{{\rm{b}}}\right)$, and by the explicit ${{{\rm{c}}}}_{{{{\rm{V}}}}^{*}}{{\rm{u}}}\big({{{\rm{V}}}}^{*}\big)$ term in the error analysis.

We did not include a separate temperature-oscillation term in the uncertainty analysis. The liquid layers are clamped between copper blocks with high thermal mass, which attenuate controller/ambient ripple, and each run uses a short acquisition window (~6 s, 30 readings). Small set-point fluctuations, therefore, act common-mode across all five channels as a gain/offset on 1/V. If 1/V is perturbed as ${\left(1/{{\rm{V}}}\right)}^{{\prime} }=\left(1+{{\rm{\varepsilon }}}\right)\left(1/{{\rm{V}}}\right)+{{{\rm{\varepsilon }}}}_{0}$, then the reference line fit parameters transform as ${{{\rm{m}}}}^{{\prime} }=\left(1+{{\rm{\varepsilon }}}\right){{\rm{m}}}$ and ${{{\rm{b}}}}^{{\prime} }=\left(1+{{\rm{\varepsilon }}}\right){{\rm{b}}}+{{{\rm{\varepsilon }}}}_{0}$. The effective depth remains invariant:

$${{{{\rm{\delta }}}}^{*}}^{{\prime} }=\frac{{\big(1/{{{\rm{V}}}}^{*}\big)}^{{\prime} }-{{{\rm{b}}}}^{{\prime} }}{{{{\rm{m}}}}^{{\prime} }}=\frac{\left(1+{{\rm{\varepsilon }}}\right)\left(1/{{\rm{V}}}\right)+{{{\rm{\varepsilon }}}}_{0}-\left(1+{{\rm{\varepsilon }}}\right){{\rm{b}}}-{{{\rm{\varepsilon }}}}_{0}}{\left(1+{{\rm{\varepsilon }}}\right){{\rm{m}}}}=\frac{1/{{{\rm{V}}}}^{*}-{{\rm{b}}}}{{{\rm{m}}}}={{{\rm{\delta }}}}^{*}$$

Therefore, the derived ${{{\rm{k}}}}_{{{\rm{s}}}}$ is unchanged to first order. Any non-common mode residuals (e.g., tiny channel-specific lags) is shown in the scatter s, which are already captured via ${{\rm{u}}}\left({{\rm{m}}}\right)$, ${{\rm{u}}}\left({{\rm{b}}}\right)$, and ${\mathrm{cov}}\left({{\rm{m}}},{{\rm{b}}}\right)$. A numerical perturbation (Supplementary Note 1) confirms that realistic oscillations produce a negligible change in ${{{\rm{k}}}}_{{{\rm{s}}}}$ compared with the dominant fit-driven terms.

Practical routes to reduce uncertainty of k

A variance-component analysis (mean over runs) shows the intercept term ${{{\rm{c}}}}_{{{\rm{b}}}}{{\rm{u}}}\left({{\rm{b}}}\right)$ dominates the uncertainty of k; ${{{\rm{c}}}}_{{{\rm{m}}}}{{\rm{u}}}\left({{\rm{m}}}\right)$ is secondary; ${{{\rm{c}}}}_{{{{\rm{\delta }}}}_{1}}{{\rm{u}}}({{{\rm{\delta }}}}_{1})$ is modest, and the reference-fluid and sample-voltage terms (${{{\rm{c}}}}_{{{{\rm{k}}}}_{{{\rm{ref}}}}}{{\rm{u}}}({{{\rm{k}}}}_{{{\rm{ref}}}})$, ${{{\rm{c}}}}_{{{{\rm{V}}}}^{*}}{{\rm{u}}}({{{\rm{V}}}}^{*})$) are small. This follows from ${{{\rm{k}}}}_{{{\rm{s}}}}\propto {{\rm{m}}}/(1/{{{\rm{V}}}}^{*}-{{\rm{b}}})$. When the calibration slope ${{\rm{m}}}$ is shallow and ${{{\rm{\delta }}}}^{*}$ is on the order of ${10}^{2}$ μm, the sensitivity to the intercept, ${{{\rm{c}}}}_{{{\rm{b}}}}={{{\rm{k}}}}_{{{\rm{s}}}}/{{\rm{m}}}{{{\rm{\delta }}}}^{*}$, is large. Therefore, any fit scatter inflates ${{\rm{u}}}\left({{\rm{b}}}\right)$.

In practice, to reduce the uncertainty, one way is to widen the reference-depth span to increase ${{{\rm{S}}}}_{{{\rm{xx}}}}$ (the centered sum of squares of ${{\rm{\delta }}}$), e.g., replace {300, 450, 600, 750} μm with {100, 300, 450, 600, 900} μm, or add one or two extra reference channels at the extremes. Both actions shrink ${{\rm{u}}}\left({{\rm{m}}}\right)$ and ${{\rm{u}}}\left({{\rm{b}}}\right)$. Additionally, the fit residuals s may be reduced by longer averaging on each reference point (e.g., 30 → 300 samples), which comes at the cost of a longer measurement time, and stabilizing the device temperature during the measurements. Another solution to improve the uncertainty is to operate at conditions that increase ${{\rm{m}}}{{{\rm{\delta }}}}^{*}$ within the linear regime (e.g., applying a larger voltage, which should be verified not to introduce nonlinearity). This reduces the sensitivities ${{{\rm{c}}}}_{{{\rm{b}}}}$ and ${{{\rm{c}}}}_{{{\rm{V}}}}$ and also lowers ${{{\rm{c}}}}_{{{\rm{m}}}}={{{\rm{k}}}}_{{{\rm{s}}}}/{{\rm{m}}}$.