# Quantitative Measurement of Thermal Conductivity by SThM Technique: Measurements, Calibration Protocols and Uncertainty Evaluation

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement Equipment

#### 2.1.1. SThM System

#### 2.1.2. Resistance Temperature Probe

#### 2.1.3. Thermal Unit

#### 2.2. SThM Measurements

#### 2.2.1. Active Mode Configuration

- “out of contact” abbreviated in “oc” where the probe is placed far from the thermal influence of the sample. Furthermore, the electrical resistance R of the probe mostly depends on the convective and conductive heat losses between the probe and the ambient air, the current intensity in the probe, the conductive heat losses from the probe to the cantilever, and the heat induced by the laser diode beam illuminating the cantilever. The radiative heat losses can be neglected.
- “in contact” abbreviated in “ic” where the probe is in contact with the sample surface. In this configuration, R depends on the same influencing parameters as in the “out of contact” configuration and on the heat transfers between the probe and the surface sample. These are functions of the thermal properties of the sample and the interface thermal resistance between the probe and the sample.

#### 2.2.2. Definition of the Intermediate Measurand

#### 2.2.3. SThM Measurement Protocol

#### Sample Requirements

#### Measuring Conditions

#### Measurement Process

- after stabilisation (criteria of standard deviation $<{10}^{-4}\phantom{\rule{3.33333pt}{0ex}}V$ for $B{B}_{v,oc}^{ref}$ mean value calculated with measurements performed during a 100 s period), start recording of $B{B}_{v,oc}^{ref}$ and ${U}_{oc}^{ref}$ signals during a 100 s period with the probe in an “out of contact” configuration above the reference $Si{O}_{2}f$ sample;
- land with “dark mode” on one position of the $Si{O}_{2}f$ reference sample, wait for stabilisation (with the same criteria as for the first step), record $B{B}_{v,ic}^{ref}$ and ${U}_{ic}^{ref}$ signals during a 100 s period with the probe in an “in contact” configuration;
- remove the probe from contact and wait for stabilisation (with the same criteria as for the first step), record $B{B}_{v,ic}^{ref}$ and ${U}_{ic}^{ref}$ signals during a 100 s period with the probe in “out of contact” configuration; repeat these three operations for two other landings at the same position above and on the $Si{O}_{2}f$ reference sample (repeatability of measurements).
- After 3 measurements at the same position on the $Si{O}_{2}f$ reference sample, from “out of contact” configuration, move to another position above the sample (reproducibility measurements). After stabilisation (criteria of standard deviation $<{10}^{-4}\phantom{\rule{3.33333pt}{0ex}}V$ for $B{B}_{v,oc}^{ref}$ mean value calculated with measurements performed during a 100 s period), start recording of $B{B}_{v,oc}^{ref}$ and ${U}_{oc}^{ref}$ signals during a 100 s period with the probe in “out of contact” configuration for the new position above the reference $Si{O}_{2}f$ sample;
- land with “dark mode” on the new position of the $Si{O}_{2}f$ reference sample, wait for stabilisation (with the same criteria as for the first step), record $B{B}_{v,ic}^{ref}$ and ${U}_{ic}^{ref}$ signals during a 100 s period with the probe in “in contact” configuration;
- remove the probe out of contact, waiting for stabilisation (with the same criteria as for the first step), record $B{B}_{v,ic}^{ref}$ and ${U}_{ic}^{ref}$ signals during a 100 s period with the probe in “out of contact” configuration;
- from “out of contact” configuration, move to another position above the sample, and repeat the two steps described in the two last bullets for this third location on the sample.
- After measurements on the $Si{O}_{2}f$ reference sample, perform measurements on the studied sample following the same protocol as for the $Si{O}_{2}f$ reference sample.

#### 2.3. SThM Calibration Protocol

#### 2.3.1. Definition of the Calibration Model

#### 2.3.2. Calibration Materials

#### 2.4. Method for the Evaluation of the Uncertainty Associated with the Estimation of the Intermediate Measurand

#### 2.4.1. Modelling the Measurement Process for Individual Measurand

#### 2.4.2. Evaluating Input Quantities for Individual Measurand

**Voltages:**

- Trueness of the multimeters: This error is the same for each measurement of a voltage, whether the sample is in or out of contact, and whether the unknown sample or the reference sample is measured, but is specific for each multimeter. Available information about the trueness error comes from the calibration certificate of each multimeter. These calibration certificates provide trueness corrections ${U}_{true}$ and $B{B}_{v,true}$ with an associated expanded uncertainty $U\left({U}_{true}\right)=U\left(B{B}_{v,true}\right)=2.5$ μV, using a coverage factor $k=2$. This correction is applied to the measurements, and a Gaussian probability distribution is assigned with a zero mean and$$u\left(\right)open="("\; close=")">{U}_{true}=1.25\times {10}^{-6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$$$$u\left(\right)open="("\; close=")">B{B}_{v,true}=1.25\times {10}^{-6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$$
- Quantification of the multimeters: The multimeters have the same quantification step $q=1$ μV in the studied range. As a consequence, the quantification error lies in the interval $\left(\right)$. A rectangular probability distribution is assigned. However, this (unknown) error may be different for each voltage measurement. As a result, we define a different input quantity for each different voltage measurement.
- Repeatability: In order to evaluate the repeatability of the voltage measurement, our measurement corresponds to the mean values $\overline{U}$ and $\overline{B{B}_{v}}$ of the respective U voltage and $B{B}_{v}$ voltage for 100 measuring points (corresponding to a period of 100 s) associated with their respective standard deviations.
- Measurement model for voltages: As a result, the measurement model used for each voltage measurement (in contact/out of contact) is:$${\overline{U}}_{i}={U}_{true}+{U}_{iq}+{U}_{iR}$$$${\overline{BB}}_{v,i}=B{B}_{v,true}+B{B}_{v,iq}+B{B}_{v,iR}$$

**Resistances involved in the Wheatstone bridge**

#### 2.4.3. Propagating Distributions for Individual Measurand

#### 2.4.4. Combining Reproducibility Measurements

#### 2.5. Bayesian Approach to Estimate the Thermal Conductivity from SThM Measurements

#### 2.5.1. Error-in-Variables Representation

#### 2.5.2. Bayesian Paradigm

#### 2.5.3. Likelihood

#### 2.5.4. Prior Distribution

#### 2.5.5. Computing Posterior Distributions

## 3. Results

#### 3.1. Experimental Measurements on Calibration Materials

#### Measurements of Input Quantities for Individual Measured Quantity and Their Associated PDFs

Input Quantity | Unit | Probability Distribution | Mean Value | Standard Deviation | Lower Bound | Upper Bound |
---|---|---|---|---|---|---|

${U}_{true}$ | $\mathrm{V}$ | Gaussian | $-0.004$ | $1.25\times {10}^{-6}$ | − | − |

${U}_{oc,q}^{s}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

${U}_{oc,q}^{ref}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

${U}_{ic,q}^{s}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

${U}_{ic,q}^{ref}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

${U}_{oc,R}^{s}$ | $\mathrm{V}$ | Gaussian | $0.37547698$ | $4.57\times {10}^{-6}$ | − | − |

${U}_{oc,R}^{ref}$ | $\mathrm{V}$ | Gaussian | $0.37546387$ | $4.08\times {10}^{-6}$ | − | − |

${U}_{ic,R}^{s}$ | $\mathrm{V}$ | Gaussian | $0.37529628$ | $4.13\times {10}^{-6}$ | − | − |

${U}_{ic,R}^{ref}$ | $\mathrm{V}$ | Gaussian | $0.37520403$ | $4.03\times {10}^{-6}$ | − | − |

$B{B}_{v,true}$ | $\mathrm{V}$ | Gaussian | $-0.04$ | $1.25\times {10}^{-6}$ | − | − |

$B{B}_{v,oc,q}^{s}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

$B{B}_{v,oc,q}^{ref}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

$B{B}_{v,ic,q}^{s}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

$B{B}_{v,ic,q}^{ref}$ | $\mathrm{V}$ | Rectangular | − | − | $-5\times {10}^{-7}$ | $5\times {10}^{-7}$ |

$B{B}_{v,oc,R}^{s}$ | $\mathrm{V}$ | Gaussian | $9.5697\times {10}^{-3}$ | $5.66\times {10}^{-5}$ | − | − |

$B{B}_{v,oc,R}^{ref}$ | $\mathrm{V}$ | Gaussian | $8.2624\times {10}^{-3}$ | $10.24\times {10}^{-5}$ | − | − |

$B{B}_{v,ic,R}^{s}$ | $\mathrm{V}$ | Gaussian | $-6.2327\times {10}^{-3}$ | $5.07\times {10}^{-5}$ | − | − |

$B{B}_{v,ic,R}^{ref}$ | $\mathrm{V}$ | Gaussian | $-14.4252\times {10}^{-3}$ | $3.51\times {10}^{-5}$ | − | − |

$B{B}_{k}$ | a. u. | Rectangular | − | − | $124.5$ | $125.5$ |

$B{B}_{k,min}$ | a. u. | Fixed | $0.5$ | − | − | − |

$B{B}_{k,max}$ | a. u. | Fixed | 1003 | − | − | − |

${R}_{1}$ | $\Omega $ | Rectangular | − | − | 999 | 1001 |

${R}_{2}$ | $\Omega $ | Rectangular | − | − | 999 | 1001 |

${R}_{f}$ | $\Omega $ | Gaussian | $399.830$ | $0.001$ | − | − |

${R}_{v,max}$ | $\Omega $ | Gaussian | $198.119$ | $0.001$ | − | − |

${R}_{v,min}$ | $\Omega $ | Gaussian | $0.0938$ | $0.001$ | − | − |

${R}_{1k}$ | $\Omega $ | Rectangular | − | − | 999 | 1001 |

${R}_{1k}^{\prime}$ | $\Omega $ | Rectangular | − | − | 999 | 1001 |

${R}_{10k}$ | $\Omega $ | Rectangular | − | − | 9999 | 10,001 |

${R}_{10k}^{\prime}$ | $\Omega $ | Rectangular | − | − | 9999 | 10,001 |

${R}_{1k}^{\u2033}$ | $\Omega $ | Rectangular | − | − | 999 | 1001 |

#### 3.2. Study of Influencing Factors Regarding Repeatability and Reproducibility Conditions of Measurement

#### 3.2.1. Evaluation of Measurement Precision under Repeatability Conditions

#### 3.2.2. Evaluation of Measurement Precision under Reproducibility Conditions: Study of Landing and Withdrawal Configurations

#### 3.2.3. Evaluation of Measurement Precision under Reproducibility Conditions: Study of Heterogeneity of The Sample

#### 3.2.4. Combination of Measurements in Repeatability and Reproducibility Conditions

#### 3.3. Bayesian Identification of the Parameters

#### 3.4. Predictions and Associated Uncertainty Using the Calibration Curve

## 4. Discussions

#### 4.1. Sensitivity of the Measurement Method

#### 4.2. Improvement of Measurement Precision

#### 4.3. Application to Nanomaterials

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AFM | Atomic Force Microscopy |

a. u. | arbitrary unit |

DC | Direct current |

GUM | Guide to the expression of Uncertainty in Measurement |

MCM | Monte Carlo Method |

MCMC | Markov Chain Monte Carlo |

Probability Distribution Functions | |

PMMA | poly(methyl methacrylate) |

POM-C | poly-oxymethylene in copolymer |

SEM | Scanning Electron Microscopy |

SI | International System of Units (SI for Système International) |

SThM | Scanning Thermal Microscopy |

TCR | Temperature Coefficient Ratio |

## Nomenclature

Measurement Result | Measured Quantity Value | Uncertainty | Description |

${Y}_{m}$ | ${y}_{m}$ | $u\left({y}_{m}\right)$ | individual measurand |

${Y}_{m,i}$ | ${y}_{m,i}$ | $u\left({y}_{m,i}\right)$ | individual measurand indexed by environmental measurement conditions |

Y | y | $u\left(y\right)$ | mean value of measurand |

## Appendix A. Bayesian Estimates

**Table A1.**Posterior point estimates of all the quantities updated in the Bayesian inference for $u\left(Y\right)=0.005$.

Mean | SD | $2.50\%$ | $25\%$ | $50\%$ | $75\%$ | $97.50\%$ | n_eff | Rhat | |
---|---|---|---|---|---|---|---|---|---|

a | 0.75248 | 0.02123 | 0.71332 | 0.73791 | 0.75154 | 0.76631 | 0.79741 | 18891 | 1.000 |

b | 0.29479 | 0.01695 | 0.26192 | 0.28337 | 0.29456 | 0.30605 | 0.32837 | 18534 | 1.000 |

c | 0.39178 | 0.02189 | 0.34578 | 0.37749 | 0.39274 | 0.40688 | 0.43207 | 18832 | 1.000 |

${X}_{1}$ | 116.98244 | 2.93226 | 111.14404 | 115.03346 | 116.97302 | 118.97162 | 122.66783 | 18263 | 1.000 |

${X}_{2}$ | 93.51105 | 2.33664 | 89.00636 | 91.91874 | 93.49173 | 95.09355 | 98.13226 | 18588 | 1.000 |

${X}_{3}$ | 52.08016 | 1.30062 | 49.51823 | 51.21204 | 52.08048 | 52.94909 | 54.64027 | 19055 | 1.000 |

${X}_{4}$ | 36.81147 | 0.92839 | 34.99854 | 36.18567 | 36.81277 | 37.44119 | 38.65114 | 19178 | 1.000 |

${X}_{5}$ | 29.77573 | 0.73586 | 28.33211 | 29.28152 | 29.77346 | 30.26941 | 31.23389 | 19163 | 1.000 |

${X}_{6}$ | 9.19203 | 0.22784 | 8.74616 | 9.03959 | 9.18998 | 9.34594 | 9.63890 | 18822 | 1.000 |

${X}_{7}$ | 1.95029 | 0.04671 | 1.85811 | 1.91892 | 1.95029 | 1.98190 | 2.04136 | 19503 | 1.000 |

${X}_{8}$ | 1.41600 | 0.03044 | 1.35605 | 1.39574 | 1.41594 | 1.43636 | 1.47612 | 18841 | 1.000 |

${X}_{9}$ | 1.27543 | 0.02837 | 1.21952 | 1.25637 | 1.27541 | 1.29451 | 1.33108 | 18796 | 1.000 |

${X}_{10}$ | 1.06069 | 0.02177 | 1.01842 | 1.04592 | 1.06073 | 1.07540 | 1.10332 | 18052 | 1.000 |

${X}_{11}$ | 0.34749 | 0.00670 | 0.33456 | 0.34297 | 0.34752 | 0.35199 | 0.36062 | 19347 | 1.000 |

${X}_{12}$ | 0.18227 | 0.00451 | 0.17349 | 0.17918 | 0.18223 | 0.18534 | 0.19105 | 19384 | 1.000 |

${X}_{1}^{*}$ | 0.20387 | 0.00760 | 0.18887 | 0.19872 | 0.20396 | 0.20896 | 0.21867 | 19061 | 1.000 |

${X}_{2}^{*}$ | 6.69274 | 1.31602 | 4.80659 | 5.79299 | 6.46207 | 7.33991 | 9.85410 | 17246 | 1.000 |

${X}_{3}^{*}$ | 11.09852 | 5.61346 | 6.44979 | 8.35621 | 9.83874 | 12.09348 | 22.99046 | 15898 | 1.000 |

${Y}_{1}$ | 1.14237 | 0.00220 | 1.13801 | 1.14090 | 1.14236 | 1.14386 | 1.14668 | 17798 | 1.000 |

${Y}_{2}$ | 1.14190 | 0.00220 | 1.13756 | 1.14043 | 1.14190 | 1.14338 | 1.14620 | 17817 | 1.000 |

${Y}_{3}$ | 1.14003 | 0.00216 | 1.13575 | 1.13860 | 1.14004 | 1.14148 | 1.14427 | 17747 | 1.000 |

${Y}_{4}$ | 1.13829 | 0.00212 | 1.13409 | 1.13687 | 1.13830 | 1.13971 | 1.14244 | 17864 | 1.000 |

${Y}_{5}$ | 1.13689 | 0.00210 | 1.13272 | 1.13548 | 1.13689 | 1.13830 | 1.14100 | 17859 | 1.000 |

${Y}_{6}$ | 1.12090 | 0.00191 | 1.11714 | 1.11962 | 1.12091 | 1.12217 | 1.12462 | 17463 | 1.000 |

${Y}_{7}$ | 1.04558 | 0.00275 | 1.04010 | 1.04375 | 1.04559 | 1.04744 | 1.05089 | 19108 | 1.000 |

${Y}_{8}$ | 1.01479 | 0.00265 | 1.00958 | 1.01300 | 1.01477 | 1.01658 | 1.01997 | 19042 | 1.000 |

${Y}_{9}$ | 1.00319 | 0.00305 | 0.99713 | 1.00114 | 1.00321 | 1.00529 | 1.00906 | 18628 | 1.000 |

${Y}_{10}$ | 0.98086 | 0.00209 | 0.97674 | 0.97949 | 0.98087 | 0.98226 | 0.98490 | 18485 | 1.000 |

${Y}_{11}$ | 0.79942 | 0.00389 | 0.79171 | 0.79685 | 0.79940 | 0.80204 | 0.80706 | 18576 | 1.000 |

${Y}_{12}$ | 0.67998 | 0.00289 | 0.67432 | 0.67804 | 0.67999 | 0.68191 | 0.68565 | 19186 | 1.000 |

${Y}_{1}^{*}$ | 0.70005 | 0.00503 | 0.69018 | 0.69666 | 0.70009 | 0.70344 | 0.70986 | 16720 | 1.000 |

${Y}_{2}^{*}$ | 1.11155 | 0.00517 | 1.10141 | 1.10807 | 1.11154 | 1.11503 | 1.12171 | 17568 | 1.000 |

${Y}_{3}^{*}$ | 1.12256 | 0.00540 | 1.11209 | 1.11894 | 1.12240 | 1.12612 | 1.13366 | 19017 | 1.000 |

**Table A2.**Posterior point estimates of all the quantities updated in the Bayesian inference for $u\left(Y\right)=0.002$.

Mean | SD | $2.50\%$ | $25\%$ | $50\%$ | $75\%$ | $97.50\%$ | n_eff | Rhat | |
---|---|---|---|---|---|---|---|---|---|

a | 0.75257 | 0.02113 | 0.71390 | 0.73786 | 0.75145 | 0.76632 | 0.79607 | 18663 | 1 |

b | 0.29500 | 0.01698 | 0.26260 | 0.28346 | 0.29471 | 0.30641 | 0.32894 | 18710 | 1 |

c | 0.39177 | 0.02181 | 0.34678 | 0.37757 | 0.39288 | 0.40693 | 0.43158 | 18633 | 1 |

${X}_{1}$ | 116.98164 | 2.94698 | 111.28035 | 114.99642 | 116.94231 | 118.95543 | 122.83754 | 19253 | 1 |

${X}_{2}$ | 93.50339 | 2.32200 | 88.98024 | 91.94296 | 93.52646 | 95.04895 | 98.07742 | 18317 | 1 |

${X}_{3}$ | 52.06313 | 1.30199 | 49.49339 | 51.17531 | 52.07019 | 52.94611 | 54.60207 | 18996 | 1 |

${X}_{4}$ | 36.81481 | 0.92593 | 34.98734 | 36.18401 | 36.81396 | 37.43419 | 38.62662 | 18505 | 1 |

${X}_{5}$ | 29.75853 | 0.74063 | 28.29387 | 29.25888 | 29.76038 | 30.25958 | 31.20828 | 18635 | 1 |

${X}_{6}$ | 9.19174 | 0.22802 | 8.74244 | 9.03749 | 9.19330 | 9.34511 | 9.63806 | 18499 | 1 |

${X}_{7}$ | 1.95060 | 0.04698 | 1.85851 | 1.91921 | 1.95035 | 1.98212 | 2.04297 | 19210 | 1 |

${X}_{8}$ | 1.41630 | 0.03024 | 1.35708 | 1.39610 | 1.41610 | 1.43617 | 1.47689 | 18904 | 1 |

${X}_{9}$ | 1.27500 | 0.02860 | 1.21919 | 1.25568 | 1.27521 | 1.29422 | 1.33215 | 18075 | 1 |

${X}_{10}$ | 1.06051 | 0.02194 | 1.01763 | 1.04562 | 1.06045 | 1.07525 | 1.10400 | 19529 | 1 |

${X}_{11}$ | 0.34759 | 0.00669 | 0.33440 | 0.34305 | 0.34761 | 0.35208 | 0.36065 | 18154 | 1 |

${X}_{12}$ | 0.18230 | 0.00448 | 0.17364 | 0.17924 | 0.18230 | 0.18531 | 0.19120 | 18516 | 1 |

${X}_{1}^{*}$ | 0.20390 | 0.00560 | 0.19296 | 0.20014 | 0.20390 | 0.20767 | 0.21493 | 18498 | 1 |

${X}_{2}^{*}$ | 6.25423 | 0.51752 | 5.36013 | 5.89422 | 6.21048 | 6.56798 | 7.39395 | 18765 | 1 |

${X}_{3}^{*}$ | 9.07703 | 1.13082 | 7.28483 | 8.28182 | 8.93303 | 9.72300 | 11.65243 | 18740 | 1 |

${Y}_{1}$ | 1.14245 | 0.00218 | 1.13822 | 1.14096 | 1.14245 | 1.14393 | 1.14673 | 18174 | 1 |

${Y}_{2}$ | 1.14197 | 0.00217 | 1.13775 | 1.14049 | 1.14198 | 1.14344 | 1.14623 | 18172 | 1 |

${Y}_{3}$ | 1.14010 | 0.00213 | 1.13597 | 1.13865 | 1.14011 | 1.14155 | 1.14428 | 18083 | 1 |

${Y}_{4}$ | 1.13836 | 0.00210 | 1.13428 | 1.13693 | 1.13836 | 1.13979 | 1.14247 | 18160 | 1 |

${Y}_{5}$ | 1.13696 | 0.00207 | 1.13292 | 1.13554 | 1.13697 | 1.13836 | 1.14103 | 18196 | 1 |

${Y}_{6}$ | 1.12096 | 0.00188 | 1.11730 | 1.11969 | 1.12097 | 1.12222 | 1.12463 | 18463 | 1 |

${Y}_{7}$ | 1.04560 | 0.00276 | 1.04017 | 1.04375 | 1.04561 | 1.04748 | 1.05101 | 19187 | 1 |

${Y}_{8}$ | 1.01480 | 0.00267 | 1.00960 | 1.01301 | 1.01480 | 1.01659 | 1.02002 | 18771 | 1 |

${Y}_{9}$ | 1.00313 | 0.00306 | 0.99707 | 1.00106 | 1.00317 | 1.00523 | 1.00903 | 18424 | 1 |

${Y}_{10}$ | 0.98081 | 0.00209 | 0.97675 | 0.97939 | 0.98081 | 0.98223 | 0.98490 | 19062 | 1 |

${Y}_{11}$ | 0.79938 | 0.00387 | 0.79179 | 0.79677 | 0.79937 | 0.80202 | 0.80693 | 17955 | 1 |

${Y}_{12}$ | 0.67991 | 0.00290 | 0.67425 | 0.67792 | 0.67989 | 0.68183 | 0.68559 | 19199 | 1 |

${Y}_{1}^{*}$ | 0.70004 | 0.00202 | 0.69606 | 0.69867 | 0.70002 | 0.70141 | 0.70400 | 19015 | 1 |

${Y}_{2}^{*}$ | 1.11027 | 0.00200 | 1.10633 | 1.10894 | 1.11026 | 1.11160 | 1.11422 | 19024 | 1 |

${Y}_{3}^{*}$ | 1.12034 | 0.00204 | 1.11636 | 1.11896 | 1.12033 | 1.12170 | 1.12432 | 18750 | 1 |

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**Figure 1.**Representation of the workflow for establishing traceability to the SI for thermal conductivity measurements by SThM technique for measurements in diffusive thermal regime.

**Figure 2.**SEM images of a KNT probe (2an type): (

**a**) View of the SiN cantilever with the two gold pads (

**b**) Zoom on the tip with the resistive Pt ribbon at deposited at the top end of the tip.

**Figure 3.**Scheme of the thermal unit encloses an adjustable current generator and a Wheatstone bridge composed of two fixed resistances (${R}_{1}$ and ${R}_{2}$), two adjustable resistances: ${R}_{f}$ for coarse adjustments and ${R}_{a}$ for fine adjustments, the resistance temperature probe R and an amplifier setup A to amplify the bridge balance $B{B}_{v}$ voltage. The probe electrical resistance R is included in one leg of the Wheatstone bridge and the adjustable resistance ${R}_{a}$ in the opposite leg. ${R}_{a}$ can be set manually by rotary knob arbitrarily scalable in 1000 graduations. Value of the knob adjustment is denoted $B{B}_{k}$.

**Figure 4.**Views of apparatus: (

**a**) SThM and thermal unit inside the dedicated enclosure to decrease influence of thermal drifts and (

**b**) view of the location of the probe above the $Si{O}_{2}f$ reference sample. Range of displacement of x, y direction piezoelectric scanners enables to switch from $Si{O}_{2}f$ reference sample to the studied sample ($Zr{O}_{2}$ sample for the picture).

**Figure 5.**Work flow of measurement sequence of U probe (cyan solid line) and $B{B}_{v}$ (orange dotted line) signals: five successive landings on the $Si{O}_{2}f$ reference sample following by five successive landings on the studied sample (alumina sample in this example). For each sample (reference and studied) the first three landings are performed on the same location and the last two on different locations. Measurement values used for R computation are identified in red for the $Si{O}_{2}f$ reference sample and in blue for the studied sample. The residual drift is highlighted with the red dashed line for U “out of contact” measurements.

**Figure 6.**Main steps for the evaluation of measurement uncertainty: In a first step: Modelling the measurement process for ${Y}_{m}$ from Section 2.4.1, Evaluating input quantities with associated Probability Distribution Functions for ${Y}_{m}$ in Section 2.4.2, Propagating with Monte Carlo method (MCM) in Section 2.4.3 to report single quantity measured values ${y}_{m}$ with its associated uncertainty $u\left({y}_{m}\right)$. In a second step: Use ${y}_{m,i}$ input quantities measured under various conditions indexed by i (i from 1 to 10) with measurement model described in Section 2.2.2 then combine the ten ${y}_{m,i}$ measurements and the associated uncertainties in Section 2.4.4 to report the $\overline{y}$ mean value with standard uncertainty $u\left(\overline{y}\right)$.

**Figure 7.**Analyses of propagation of distributions on ${y}_{m,3}$ measured quantity value: (

**a**) Probability density distribution (PDF) for the ${y}_{m,3}$ value (

**b**) Spearman’s rank correlation coefficients.

**Figure 8.**Graphic representation of the standard deviation computed from a set of three measurements performed in repeatability condition for two types of measurement (landing and withdrawal) and for all the twelve calibration samples).

**Figure 9.**Graphic representation of the three quantity measured values of ${y}_{m,i,landing}$ ($i=1;\phantom{\rule{3.33333pt}{0ex}}3$ and 5), measured at the same location, computed in landing configuration and the three quantity measured values of ${y}_{m,i,withdrawal}$ ($i=2;\phantom{\rule{3.33333pt}{0ex}}4$ and 6) computed in withdrawal configuration. Each data is indicated with its associated absolute uncertainty (coefficient $k=1$) represented by black error bars. The blue circles correspond to landing measurement points and the orange square to withdrawal measurement points. The blue solid line represents the mean value for the landing configuration with its associated standard deviation (blue dashed lines) and the orange dash-dotted line represents the mean value for the withdrawal configuration with its associated standard deviation (orange dotted lines).

**Figure 10.**Comparison of mean values and the standard deviation (black error bars) obtained in landing condition (blue rectangles) and withdrawal condition (orange rectangles) for all calibration samples.

**Figure 11.**Comparison of standard deviation obtained in repeatability conditions (same location) and reproducibility conditions (different locations).

**Figure 14.**Calibration curve (in black) obtained from the experimental data from Table 6 analysed with the Bayesian approach. Points are represented with their associated expanded uncertainty $(k=2)$ for both axes. Red dashed lines represent the $95\%$ coverage intervals associated with the (estimated) calibration curve for each conductivity.

**Figure 15.**Posterior distributions of the predictions and their associated $95\%$ coverage interval obtained for (

**a**) $Y={y}_{1}^{*}$, (

**b**) $Y={y}_{2}^{*}$ and (

**c**) $Y={y}_{3}^{*}$ for the two uncertainty levels $u\left({y}^{*}\right)=0.002$ and $u\left({y}^{*}\right)=0.005$.

**Table 1.**Identification of the ten measured values ${y}_{m,i}$ from various conditions of measurement indexed from $i=1$ to $i=10$ relative to the chronological acquisition, odd i indexes relative to landing measurement condition, even i indexes relative to withdrawal measurement condition.

Locations | Landing Condition | Withdrawal Condition |
---|---|---|

location n°1 | ${y}_{m,1}$ | ${y}_{m,2}$ |

${y}_{m,3}$ | ${y}_{m,4}$ | |

${y}_{m,5}$ | ${y}_{m,6}$ | |

location n°2 | ${y}_{m,7}$ | ${y}_{m,8}$ |

location n°3 | ${y}_{m,9}$ | ${y}_{m,10}$ |

**Table 2.**Thermal conductivity k (relative expanded uncertainty estimated to $5\%$) and roughness $Ra$ of calibrated materials measured at 23 °C. The sample thickness is identified as: 1 *, 2 ** and 5 *** mm.

Sample | Structure | Provider | k [Wm${}^{-1}$K${}^{-1}$] | $\mathbf{Ra}$ [nm] |
---|---|---|---|---|

$PMMA$ *** | Polymer | Goodfellow | 0.187 | 5.04 |

$POM-C$ *** | Polymer | Radiospare | 0.329 | 11.7 |

$Borosilicateglass$ ** | Amorphous | Neyco | 1.11 | <$0.5$ |

$Si{O}_{2}f$ ** | Amorphous | Neyco | 1.28 | 0.56 |

$Si{O}_{2-NEGS1}$ ** | Amorphous | Neyco | 1.40 | <1 |

$Zr{O}_{2}$ ** | Single crystal | Neyco | 1.95 | <$0.5$ |

$Ti{O}_{2}$ ** | Single crystal | Neyco | 9.15 | <$0.5$ |

$A{l}_{2}{O}_{3}p$ ** | Poly crystal | Neyco | 29.8 | 7.52 |

$Sapphire$ * | Single crystal | Crystal GmbH | 36.9 | <$0.5$ |

$Germanium$ ** | Single crystal | Crystal GmbH | 52.0 | <$0.5$ |

$Si{p}^{++}$ ** | Semiconductor | Goodfellow | 93.4 | 0.75 |

$Zinc$ ** | Metal | Neyco | 117 | 8.14 |

**Table 4.**Summary of the ten values obtain on the PMMA sample: the measured quantity value ${y}_{m,i}$, the standard uncertainty (absolute and relative) and the 95% coverage interval are given.

Identification Measurement | ${\mathit{y}}_{\mathit{m},\mathit{i}}$ Value [a. u.] | Standard Uncertainty | 95% Coverage Interval | ||
---|---|---|---|---|---|

Abs | Rel. (%) | [a. u.] | [a. u.] | ||

${y}_{m,1}$ | $0.6966$ | $0.0047$ | $0.68$ | $0.6873$ | $0.7058$ |

${y}_{m,2}$ | $0.6855$ | $0.0033$ | $0.48$ | $0.6789$ | $0.6919$ |

${y}_{m,3}$ | $0.6737$ | $0.0028$ | $0.42$ | $0.6681$ | $0.6792$ |

${y}_{m,4}$ | $0.6719$ | $0.0025$ | $0.38$ | $0.6668$ | $0.6768$ |

${y}_{m,5}$ | $0.6679$ | $0.0026$ | $0.39$ | $0.6628$ | $0.6731$ |

${y}_{m,6}$ | $0.6713$ | $0.0026$ | $0.38$ | $0.6663$ | $0.6764$ |

${y}_{m,7}$ | $0.6883$ | $0.0032$ | $0.46$ | $0.6821$ | $0.6944$ |

${y}_{m,8}$ | $0.6780$ | $0.0034$ | $0.50$ | $0.6714$ | $0.6848$ |

${y}_{m,9}$ | $0.6737$ | $0.0033$ | $0.49$ | $0.6673$ | $0.6801$ |

${y}_{m,10}$ | $0.6778$ | $0.0030$ | $0.44$ | $0.6719$ | $0.6836$ |

**Table 5.**Comparison of instrumental standard uncertainty value associated to each measured quantity value to the standard deviation value computed for set of replicated measurements for each materials. The highest instrumental standard uncertainties (absolute and relative) and the computed standard deviation (absolute and relative) are given.

Material Sample | Measurement Condition | Max. Standard Uncertainty | Standard Deviation | ||
---|---|---|---|---|---|

Abs | Rel. (%) | Abs | Rel. (%) | ||

$PMMA$ | landing | $0.0047$ | $0.68$ | $0.0152$ | $2.23$ |

withdrawal | $0.0034$ | $0.50$ | $0.0080$ | $1.19$ | |

$POM-C$ | landing | $0.0031$ | $0.38$ | $0.0080$ | $0.98$ |

withdrawal | $0.0034$ | $0.41$ | $0.0083$ | $1.01$ | |

$Borosilicate\phantom{\rule{3.33333pt}{0ex}}glass$ | landing | $0.0045$ | $0.46$ | $0.0083$ | $0.85$ |

withdrawal | $0.0045$ | $0.47$ | $0.0068$ | $0.70$ | |

$Si{O}_{2}f$ | landing | $0.0049$ | $0.49$ | $0.0075$ | $0.76$ |

withdrawal | $0.0043$ | $0.43$ | $0.0078$ | $0.77$ | |

$Si{O}_{2}-NEGS1$ | landing | $0.0051$ | $0.50$ | $0.0105$ | $1.02$ |

withdrawal | $0.0044$ | $0.44$ | $0.0074$ | $0.70$ | |

$Zr{O}_{2}$ | landing | $0.0052$ | $0.49$ | $0.074$ | $0.70$ |

withdrawal | $0.0047$ | $0.46$ | $0.0074$ | $0.72$ | |

$Ti{O}_{2}$ | landing | $0.0056$ | $0.49$ | $0.0101$ | $0.88$ |

withdrawal | $0.0057$ | $0.50$ | $0.0100$ | $0.88$ | |

$Alumina$ | landing | $0.0051$ | $0.45$ | $0.0338$ | $3.00$ |

withdrawal | $0.0050$ | $0.45$ | $0.024$ | $2.14$ | |

$Sapphire$ | landing | $0.0038$ | $0.34$ | $0.0061$ | $0.54$ |

withdrawal | $0.0039$ | $0.35$ | $0.0096$ | $0.86$ | |

$Germanium$ | landing | $0.0045$ | $0.39$ | $0.0118$ | $1.03$ |

withdrawal | $0.0051$ | $0.45$ | $0.0082$ | $0.72$ | |

$Si{p}^{++}$ | landing | $0.0042$ | $0.36$ | $0.0082$ | $0.71$ |

withdrawal | $0.0041$ | $0.35$ | $0.0094$ | $0.81$ | |

$Zinc$ | landing | $0.0055$ | $0.48$ | $0.0181$ | $1.59$ |

withdrawal | $0.0055$ | $0.49$ | $0.0264$ | $2.34$ |

**Table 6.**Summary of the experimental measurements on calibration samples: the intermediate measurand mean value Y, the standard uncertainty $u\left(Y\right)$ (absolute and relative).

Sample | Thermal Conductivity | Y Intermediate Measurand | $\mathit{u}\left(\mathit{Y}\right)$ Standard Uncertainty | |
---|---|---|---|---|

(Wm${}^{\mathbf{-}\mathbf{1}}$K${}^{\mathbf{-}\mathbf{1}}$) | Mean Value (a.u.) | Abs | Rel.(%) | |

$PMMA$ | $0.187$ | $0.6780$ | $0.0029$ | $0.4$ |

$POM-C$ | $0.329$ | $0.8145$ | $0.0055$ | $0.7$ |

$Borosilicate\phantom{\rule{3.33333pt}{0ex}}glass$ | $1.11$ | $0.9780$ | $0.0023$ | $0.2$ |

$Si{O}_{2}f$ | $1.28$ | $1.0019$ | $0.0048$ | $0.5$ |

$Si{O}_{2}-NEGS1$ | $1.40$ | $1.0173$ | $0.0038$ | $0.4$ |

$Zr{O}_{2}$ | $1.95$ | $1.0457$ | $0.0072$ | $0.7$ |

$Ti{O}_{2}$ | $9.15$ | $1.1316$ | $0.0057$ | $0.5$ |

$Alumina$ | $29.8$ | $1.1140$ | $0.0091$ | $0.8$ |

$Sapphire$ | $36.9$ | $1.1241$ | $0.0045$ | $0.4$ |

$Germanium$ | $52.0$ | $1.1460$ | $0.0035$ | $0.3$ |

$Si{p}^{++}$ | $93.4$ | $1.1548$ | $0.0044$ | $0.4$ |

$Zinc$ | 117 | $1.1158$ | $0.0111$ | $1.0$ |

**Table 7.**Prediction of the thermal conductivity for different values of the direct measurement. The indications of the standard uncertainty (absolute and relative) are given only for the order of magnitude, only the coverage interval give the rigorous estimation of the uncertainty level as discussed in previous section (Section 3.4).

${\mathit{y}}_{0}$ | $\mathit{u}\left(\right)open="("\; close=")">{\mathit{y}}_{0}$ | ${\mathit{k}}_{0}$Wm${}^{-\mathbf{1}}$K${}^{-\mathbf{1}}$ | $\mathit{u}\left(\right)open="("\; close=")">{\mathit{k}}_{0}$Wm${}^{-\mathbf{1}}$K${}^{-\mathbf{1}}$ | $\mathit{u}\left(\right)open="("\; close=")">{\mathit{k}}_{0}$(%) | $95\%$ Coverage IntervalWm${}^{-\mathbf{1}}$K${}^{-\mathbf{1}}$ |
---|---|---|---|---|---|

$0.7$ | $0.005$ | $0.20407$ | $0.00753$ | $3.7$ | $\left(\right)$ |

$0.7$ | $0.002$ | $0.20390$ | $0.00560$ | $2.7$ | $\left(\right)$ |

$1.1$ | $0.005$ | $6.69603$ | $1.29795$ | $19.4$ | $\left(\right)$ |

$1.1$ | $0.002$ | $6.25423$ | $0.51752$ | $8.3$ | $\left(\right)$ |

$1.2$ | $0.005$ | $11.01500$ | $5.21046$ | $47.3$ | $\left(\right)$ |

$1.2$ | $0.002$ | $9.07703$ | $1.13082$ | $12.5$ | $\left(\right)$ |

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**MDPI and ACS Style**

Fleurence, N.; Demeyer, S.; Allard, A.; Douri, S.; Hay, B.
Quantitative Measurement of Thermal Conductivity by SThM Technique: Measurements, Calibration Protocols and Uncertainty Evaluation. *Nanomaterials* **2023**, *13*, 2424.
https://doi.org/10.3390/nano13172424

**AMA Style**

Fleurence N, Demeyer S, Allard A, Douri S, Hay B.
Quantitative Measurement of Thermal Conductivity by SThM Technique: Measurements, Calibration Protocols and Uncertainty Evaluation. *Nanomaterials*. 2023; 13(17):2424.
https://doi.org/10.3390/nano13172424

**Chicago/Turabian Style**

Fleurence, Nolwenn, Séverine Demeyer, Alexandre Allard, Sarah Douri, and Bruno Hay.
2023. "Quantitative Measurement of Thermal Conductivity by SThM Technique: Measurements, Calibration Protocols and Uncertainty Evaluation" *Nanomaterials* 13, no. 17: 2424.
https://doi.org/10.3390/nano13172424