## Abstract

The conventional transient plane source (TPS) method for thin films is used for films and adhesives with thicknesses between 50 and 200 *μ*m. Measurements with the conventional TPS method are usually inaccurate due to thermal contact resistance between the insulating sensor layers, the film and the sensor, and the film and the background material. A new approach to measuring thin films with the TPS is introduced, where the heat flow is constrained to one dimension, and a slab layer made from the same background material is introduced between the thin film and the TPS sensor. This decouples the effects of the thermal contact resistance (TCR) of the sensor to the thermal resistance of the film. The new approach is tested on four different thin films with stainless steel as the background material. The results are compared to guarded heat flowmeter measurements. Excellent agreement (< 12% error) between the two methods is achieved, showing that the new method proposed is fast, accurate, and convenient alternative for determining the thermal conductivity of thin films.

## 1 Introduction

Thin films with thicknesses greater than 10 *μ*m are widely used as thermal management materials in electronic and solid-state devices. These films either dissipate excess heat generated by the electronics or ensure an efficient flow of heat from the electronic device to the heat sink [1,2]. Currently, the thermal management performance of these devices are limited by the low thermal conductivity of thin films, making the development of thin films with high thermal conductivity an active area of research [3,4]. Thus, techniques which measure the thermal conductivity of thin films are of great interest.

Thermal conductivity measurement methods of thin films fall under two categories: steady-state methods and transient methods. Steady-state methods are performed on thin films deposited on a substrate. Metallic heaters and temperature sensors are either deposited on the thin film, the substrate, or both. The thermal conductivity of the thin film is measured once the film achieves a constant temperature gradient; the thermal conductivity of the thin film can be determined using Fourier's law. To ensure a constant temperature gradient, steady-state methods are performed over long time periods. Calibration is also required when performing these methods to account for the heat lost through the substrate or heat lost due to radiation [5–7]. Transient methods, on the other hand, do not require constant temperature gradients. Thus, measurements with transient methods are faster relative to steady-state methods. A well-known transient method is the 3ω technique, in which a metallic strip is deposited onto the thin film, which delivers a current with frequency ω. The voltage signal across this sensor has a detectable frequency at 3ω with amplitude dependent on the thermal conductivity of the thin film [5,6,8]. Errors caused by radiation are small when using the 3ω technique compared to steady-state measurements of the same material [5,9]. More recently, transient thermoreflectance methods have been developed to measure the thermal conductivity of thin films [5,10]. The thin film is coated with a metallic transducer. This transducer is heated with an optical pump and probe beam, causing the thermoreflectance of the transducer to change over time. For time-domain thermoreflectance methods (TDTR), the change in thermoreflectance is measured with respect to the time delay between the arrival of the pump beam and the probe beam onto the sample surface [5]. For frequency-domain thermoreflectance methods (FDTR), the change in thermoreflectance is measured with respect to the modulation frequency of the pump beam. In both methods, the relation between the thermal properties of the thin film and the thermoreflectance response can be determined by solving the heat equation. A fitting algorithm is applied to this equation to find the thermal conductivity of the thin film which best describes the experimental data [11,12].

Another example of a transient method for measuring the thermal conductivity of thin films is the transient plane source TPS method [13,14]. The TPS method for thin films is used on films and adhesives with thicknesses between 50 to 200 *μ*m. Unlike the previous techniques, the sensor is not needed to be deposited on the thin film. The effective thermal conductivity of the sensor layers is determined by performing a reference test (Fig. 1(a)), and the effective thermal conductivity of both the sensor layers and the thin film can be determined by performing a conventional TPS thin film test (Fig. 1(b)). From these two tests, the thermal conductivity of the thin film can be calculated using the series model of thermal resistances [7,13,15]. However, the presence of thermal contact resistance (TCR) between (1) the sensor layers, (2) the sensor and the thin film, (3), the sensor and the background material, and (4) the thin film and the background material makes the method inaccurate because these resistances are added to the thermal resistance of the thin film, leading to a lower thermal conductivity reading than expected [7,16]. Thus, decoupling the thermal resistance between the sensor and the thin film is of great interest for eliminating the sensor TCR effects on a measurement to obtain more accurate thermal conductivity measurements. Ahadi et al. [7] addressed this issue by stacking films made of the same material to get thermal resistance measurements of the thin film at several thickness points. This method successfully eliminated noise between the Kapton layer of the sensor and the background material during reference tests and deconvoluted the effects of TCRs in the TPS test column.

In this paper, a new approach is introduced for measuring the thermal conductivity of thin films with the TPS method. The experimental setup of the new configuration closely follows the conventional TPS method for thin film measurements, with two major differences: (1) the heat flow is restricted to one dimension, i.e., the radius of the background material, the sensor, and the thin film are the same, and (2) a thin slab derived from the same background material is placed between the sensor and the thin film. The slab creates physical separation between the TPS sensor and the thin film, thereby removing the undesirable effects of TCR due to the sensor on the thin film, and thus removing the need for a reference test.

This method does not eliminate the effects of TCR between the thin film and the background material, but this is corrected by repeating the test at multiple film thicknesses and taking a linear regression of the resistance and total film thickness, similar to Ahadi et al.'s approach [7]. The slope of this regression is tied to the effective thermal resistance of the thin film layers, whereas the *y*-intercept is tied to the resistance between the thin film and the background material, thereby extracting the two remaining TCRs. Four thin films (Kapton, Teflon, Willow Glass, and graphite) were tested using this method. Results were compared against measurements with the guarded heat flowmeter (GHFM) [17], a steady-state method, to verify the accuracy of the measurements.

## 2 Theoretical Framework

### 2.1 Conventional Transient Plane Source Method.

where $\alpha $ is the temperature coefficient of resistance and $R0$ is the initial (electrical) resistance reading of the sensor. The temperature increase $\Delta T(t)$ is the sum of the temperature increase of the sensor layers, $\Delta Ti(t),$ and the temperature rise on the surface of the sample, $\Delta Tb(t)$.

### 2.2 Transient Plane Source Thermal Resistance of Thin Films Method.

The TPS thermal resistance of thin films (TR-TF) method is the novel method proposed in this paper with the configuration presented in Fig. 1(c).

A slab of thickness $l$, thermal conductivity $\Lambda $, and volumetric heat capacity $\rho c$ is placed on top of the TPS sensor of radius $a$. On top of this slab is a thin film of thickness $h$, with thermal conductivity $\Lambda f$. The configuration is finished with a rod-shaped material made of the same type as the slab with thickness $L$. To ensure that heat flows in a single dimension, the thin film, slab and rod pieces should have the same radius as the TPS sensor. The rod thickness is assumed to be much larger compared to the slab and film thicknesses, i.e., $L\u226bl,\u2009h$. Thus, we can treat $L$ to be infinite. Similarly, the film thickness is assumed to be negligible compared to the slab thickness, $h\u226al$. The configuration above the sensor is mirrored by the configuration below.

for all $t$.

where $s$ is the Laplace variable and $q=s/\kappa $, where $\kappa =\Lambda /\rho c$ is the thermal diffusivity of the background material. Details on how to obtain Eq. (15) are provided in Appendix A.

The average temperature rise of the sensor $\Delta T(t)$ can be determined from the inverse Laplace transform of Eq. (15). There is no closed-form expression for the inverse Laplace transform of Eq. (15), but several numerical algorithms such as the Talbot method [20] and the de-Hoog method [21] can be used to obtain the sensor temperature rise $\Delta T(t)$. The Talbot method can achieve machine precision accuracy by performing 14 function evaluations [20]. On the other hand, the accuracy of the de-Hoog algorithm can be modified by performing an adaptive version. In this paper, the adaptive version of the de-Hoog method is used. We terminate the algorithm if successive temperature function evaluations have an $L2$-norm less than 0.01 $\mu K$, which is considerably smaller than the temperature reading precision of most TPS instruments [14].

Notice that the effect of the thermal resistance of the film is introduced at a distance $l$ away from the sensor, as seen in Eq. (9). Thus, the thermal resistance of the film is effectively decoupled from the resistance of the sensor. The TCR due to the sensor can be treated by adding a time and temperature offset in the same way as a TPS bulk measurement [18,22]. These corrections are explained in more detail in the next section. Thus, the film's thermal resistance and consequently thermal conductivity are directly calculated from a single measurement without the need for a reference test.

Effects of TCR between the thin film and the background material can be removed by performing the test at multiple film thicknesses, and performing a linear regression between the measured resistances and the thicknesses to get the true thermal conductivity of the film [7].

### 2.3 Sensor Corrections.

To get the sensor temperatures in Sec. 2.2, we assume that (1) the sensor temperatures are recorded instantly, (2) the sensor thickness is negligible, (3) the sensor has zero heat capacity, and (4) there is no thermal contact resistance between the sensor and the background material. These assumptions are not exactly true; however, corrections can be introduced to Eq. (15) to account for the error in the assumptions.

#### 2.3.1 Time Offset Correction.

where $t0$ is the time offset parameter and $\Delta T(z=0,t)$ is the inverse Laplace transform of Eq. (15).

#### 2.3.2 Temperature Offset Correction.

where $\Delta T0$ is the constant temperature offset.

### 2.4 Small and Large Time Behavior.

where $\epsilon =\Lambda /\kappa $ is the thermal effusivity of the background material. These temperatures are equal to the temperatures of an infinite plane source delivering a constant power to a one-dimensional infinite medium [18]. The short-time approximation thus holds for times small enough that the heat has not fully penetrated the slab.

which is the equation for the temperature of an infinite plane source delivering a constant power in an one-dimensional infinite medium but with a constant temperature offset. Thus, at long measurement times, the effect of the thin film becomes a constant temperature offset.

To illustrate the temperature solution at large and small-time scales, the theoretical temperature function of the TPS is plotted in Fig. 3, where stainless steel is used as the background material ($\Lambda $ = 13.6 W/mK, $\kappa $ = 3.6 mm^{2}/s). The slab has a thickness of 3 mm, and the resistance of the thin film is R = 10 ^{–4} m^{2}K/W. Notice that the exact temperature solution is bounded by Eqs. (19) and (22), which are 1D column solutions that differ by a temperature offset.

### 2.5 Other Uses of the Thermal Resistance of Thin Films.

The TR-TF measures thermal resistance at a distance $l$ away from the sensor. This is particularly useful for testing cured adhesives because having a separation between the adhesive and the TPS sensor would avoid physical damage on the sensor. In addition, the TR-TF can also be used to measure thermal resistance between two similar (or dissimilar materials). Equation (15) applies for a thermal resistance measurement between two similar materials. This can be generalized to two dissimilar materials where the thermal conductivity (and thermal diffusivity) of the slab or rod piece is different from the other piece. This makes the TR-TF more versatile than the conventional TPS test and Ahadi's approach.

## 3 Experimental Methods

To check the accuracy of the new temperature function, measurements are made on Kapton (MT-100), Teflon (polytetrafluoroethylene), Willow glass, and graphite thin films using stainless steel 304 (SS304) as a background material on a Thermtest Instruments 4617-1 TPS sensor (sensor radius of 9.9 mm) connected to Thermtest Instruments Measurement Platform-1 (MP-1 TPS, Fig. 4).

The slab and rod SS304 pieces have the same diameter (20.97 ± 0.01 mm) with thicknesses of 3 ± 0.01 and 25 ± 0.01 mm, respectively. The radii of the thin film samples were set to 10.5 mm to allow for easier alignment. This bias in radius between the sensor, thin film, and the SS304 produces negligible error in the measurement result, as discussed in Sec. 3.2 and Appendix B. The film thicknesses were measured using a high-precision caliper (Table 1).

Material | Thickness (μm) |
---|---|

Kapton (MT-100) | 50 ± 0.5 |

Teflon (PTFE) | 85 ± 0.5 |

Willow glass | 100 ± 0.5 |

Graphite | 165 ± 0.5 |

Material | Thickness (μm) |
---|---|

Kapton (MT-100) | 50 ± 0.5 |

Teflon (PTFE) | 85 ± 0.5 |

Willow glass | 100 ± 0.5 |

Graphite | 165 ± 0.5 |

The measurements were performed in still air. Air is a thermal insulator ($\Lambda air\u2009$= 0.03 W/mK) compared to SS304 ($\Lambda $ = 13.6 W/mK) so Eq. (7) holds for the performed measurements. A power of 1 W was used on all TR-TF measurements which limits the temperature rise to <2 K. This ensures that heat lost at the boundaries due to convection and radiation remains small [23]. Five 5-s tests are carried out at 1 W for each layer.

where $\Delta Tc(t,\u2009\Lambda ,\u2009\kappa ,\u2009R,\u2009\Delta T0)$ is the temperature function in Eq. (17) and $N$ is the number of temperature points collected in each test. The parameters $t0$, $\Delta T0$, $\Lambda $, $\kappa ,$ and $R$ are unknown and algorithms such as the Levenberg-Marquardt (LM) algorithm can be used [24] to find the best estimate for these unknown parameters that minimizes the sum of squares $S$. The algorithm requires an initial guess for the unknown parameters; the initial guess for the thermal conductivity and thermal diffusivity is set to the literature values of the background material ($\Lambda $ = 13.6 W/(mK) and $\kappa $ = 3.6 mm^{2}/s for stainless steel) while the initial guess for the time and temperature offsets are set to zero. The initial guess for the thermal resistance is obtained by a grid search. The measured thermal resistance of the thin film would be the best-fit thermal resistance $R\u0302$ returned by the LM algorithm. The thermal conductivity $\Lambda f$ of the thin film can then be obtained using the formula $\Lambda f=h/R\u0302$.

Steady-state thermal conductivity tests are conducted on a Thermtest Instruments Guarded Heat Flow Meter (GHFM) with a contact agent between thin films and a temperature gradient of 15 °C. The procedure on measuring thermal resistance of thin films with the GHFM is specified in Annex A1 of the ASTM E1530-19 document [17]. Measurements were performed using both EG and water as the contact agent. Measurements using water as the contact agent were discarded because it did not achieve good correlation of thickness to resistance. Results from the GHFM are compared with the TR-TF tests for validation.

### 3.1 Linear Regression and Film-to-Film Thermal Contact Resistance.

where $\delta R$ is the film-to-film TCR. This means that the linear regression approach underestimates the true thermal conductivity of the film. To mitigate this issue, EG was used as a contact agent in between film layers. This minimizes the film-to-film TCR bringing the reciprocal of the slope $m\u22121$ closer to the true thermal conductivity of the film $\Lambda f$.

### 3.2 The Effect of Sample Radius.

where $b$ is the sample radius. Equation (25) represents the temperatures of the TPS sensor in the same stack as Fig. 1(c) but the heat flux is assumed to be uniformly distributed over the sample area, $\pi b2$, instead of the sensor area, $\pi a2$. Thus, the nonlinear fitting in Eq. (23) makes use of Eq. (25) to account for the mismatch in sensor and sample radius.

### 3.3 Uncertainty Analysis.

To get an error estimate in the measured thermal conductivity of the samples, we performed a Monte Carlo analysis. The measurable parameters such as steel height and film height were sampled from a normal distribution with their measured value as the mean and the standard deviation equal to the uncertainty of the measured parameter. Errors due to the electronics were obtained from the RMS residual of the nonlinear fit specified in Eq. (23). The transient temperatures from the sensor $\Delta T(t)$, were sampled from a normal distribution where its mean is the theoretical model specified by Eq. (25) and the standard deviation equal to the RMS residual multiplied by $N/(N\u2212df\u22121)$, where $df$ is the degrees-of-freedom of the fit. Since there are 5 unknown parameters, $t0,\Delta T0,\Lambda ,\kappa ,$ and $R$, then $df=5$. The thermophysical properties of the steel as well as the thermal conductivity of the film are assumed to be fixed. (13.6 W/mK and 3.6 mm^{2}/s for steel; the thermal conductivity of the films were obtained from the corresponding GHFM measurement). The resistance of the film is then obtained by a nonlinear fit from the sampled film and steel heights as well as the sampled temperatures. This was done for one, two, and three layers of the film to get the film conductivity from the linear fit of the film thickness against the measured film resistances. This process is repeated 1000 times to get a distribution of the measured film conductivity, from which the error estimate can be determined. The error estimates are reported in Table 2.

Material | $\Lambda TR\u2212TF$ (W/mK) | $\Lambda GHFM$ (W/mK) | % Difference | Correlation coefficient |
---|---|---|---|---|

Kapton (MT-100) | 0.52 ± 0.01 | 0.470 | 11.9 | 0.9918 |

Teflon (PTFE) | 0.30 ± 0.004 | 0.2963 | 1.4 | 0.9999 |

Willow glass | 1.15 ± 0.04 | 1.23 | 6.5 | 0.9897 |

Graphite | 6.0 ± 0.6 | 6.242 | 3.4 | 0.9868 |

Material | $\Lambda TR\u2212TF$ (W/mK) | $\Lambda GHFM$ (W/mK) | % Difference | Correlation coefficient |
---|---|---|---|---|

Kapton (MT-100) | 0.52 ± 0.01 | 0.470 | 11.9 | 0.9918 |

Teflon (PTFE) | 0.30 ± 0.004 | 0.2963 | 1.4 | 0.9999 |

Willow glass | 1.15 ± 0.04 | 1.23 | 6.5 | 0.9897 |

Graphite | 6.0 ± 0.6 | 6.242 | 3.4 | 0.9868 |

## 4 Results and Discussion

### 4.1 Thermal Contact Resistance.

To check the effect of TCR, a comparison was made between tests with and without EG. The results for Kapton and Teflon are displayed in Fig. 5. The overall TCR of the measurement can be obtained from the *y*-intercept of the linear fits. Notice that in both tests, the *y*-intercept of tests without EG is larger compared to the *y*-intercept of tests with EG. This indicates that EG is an effective thermal contact agent because it minimizes the overall TCR. The linear fits were also better on tests with EG, indicating that the use of EG also improves the individual thermal resistance measurements. Omitting EG from the experimental setup results in a high film-to-film TCR, particularly when considering a thickness of two layers or greater. This trend is observed both in Kapton and in Teflon. Thus, on subsequent TR-TF tests, EG is used as a thermal contact agent.

### 4.2 Thermal Conductivity.

Thermal resistance of thin films testing was extended to Willow glass and graphite to verify the method with higher thermal conductivity films. Excellent linearity of the results persisted at these higher values (Fig. 6). The thermal conductivity results were validated against results obtained via steady-state guarded heat flow method and are found to be in excellent agreement (<12% variation). These results present an improvement in accuracy compared to the work done by Ahadi et al. who report an average 10.3% difference between their proposed TPS method and GHFM tests, compared to an average 5.1% difference presented in this work.

## 5 Conclusion

This work presents a novel method for measuring the thermal conductivity of thin films which is efficient and accurate. The introduction of a slab piece between the thin film and the sensor decouples the effect of thermal contact resistance due to the sensor on the thermal resistance of the thin film, removing the need for a reference test. This is the first method of this kind to be proposed for the measurement of thin films. The theory of the method is treated in detail and verified by experimental measurements of Kapton, Teflon, Willow glass, and graphite thin films using both this method and the GHFM, a well-established steady-state method, where excellent agreement between the two methods has been reached. The addition of a thermal contact agent reduces the effects of microscopic surface roughness which contribute to thermal contact resistance in an experiment. It is strongly recommended to use a contact agent when measuring thin films via this method.

The TR-TF method is faster and more versatile than other existing thin film measurement techniques, making it an attractive method for measuring free-standing thin films, multilayered thin film structures, coatings, and adhesives which all present challenges for traditional TPS methods. As the demand for thin films increases globally, this work presents an efficient solution for the growing need to characterize their thermal properties. Furthermore, this work holds promise for future research into the direct measurement and analysis of thermal contact, an area that currently lacks suitable methodologies.

## Acknowledgment

The authors would like to thank K. Wood from Thermtest Instruments, Inc. for performing GHFM tests.

## Funding Data

National Research Council of Canada, Industrial Research Assistance Program (No. NRC-IRAP No. 928449, Funder ID: 10.13039/501100000046).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*a*=sensor radius, m

*A*=sensor area, m

^{2}*b*=sample radius, m

*c*=specific heat capacity, J/(kg K)

*d*=overall thickness of the sensor layers, m

- df =
degrees of freedom

*h*=film thickness, m

*l*=slab thickness, m

*L*=rod thickness, m

*m*=slope of linear fit of film thickness against resistance (m K)/W

*N*=number of temperature points

*P*=power delivered by the TPS sensor, W

*q*=$s/\kappa $ , m

^{–1}*Q*=magnitude of the source term, W/m

^{3}*r*=radial coordinate, m

*R*=thermal resistance of the thin film, m

^{2}K/W*R*_{0}=initial electrical resistance of the TPS sensor, Ω

- $R\u0302$ =
measured thermal resistance, m

^{2}K/W*R(t)*=recorded electrical resistance of the sensor, Ω

*s*=Laplace variable, s

^{–1}*S*=sum of squared temperature residuals, K

^{2}*t*=time, s

*t*_{0}=time offset, s

*T*_{0}=initial temperature of the system at

*t*= 0, K*T*(*z*,*t*) =temperature of the system at a given point

*z*at time*t*, K*u*(*t*) =step function

*z*=axial coordinate, m

### Greek Symbols

*α*=temperature coefficient of resistance, K

^{–1}- $\delta (z)$ =
delta function, m

^{–1}- $\delta R$ =
TCR between two consecutive film layers, m

^{2}K/W- $\Delta T\xaf$ =
average temperature of the sensor in Laplace space, K s

- $\Delta T(t)$ =
temperature rise of the sensor, K

- $\Delta Tc(t)$ =
corrected temperature rise of the sensor, K

- $\Delta T0$ =
temperature offset parameter in the fitting algorithm, K

- $\Delta T(z,t)$ =
temperature rise relative to the initial temperature at a point

*z*and given time*t*- $\Delta Ti$ =
temperature rise of the sensor layers, K

- $\Delta Tb$ =
temperature rise of the sample surface, K

- $\Delta Tit$ =
temperature rise of the sensor layers and the film, K

- $\epsilon $ =
thermal effusivity, J/m

^{2}Ks^{3/2}- $\kappa $ =
thermal diffusivity, m

^{2}/s- $\Lambda $ =
thermal conductivity, W/(m K)

### Derivation of Eq. (15)

### Analytical Expression for Laplace-Transformed Sensor Temperatures When Heat Flow is No Longer One Dimensional

where $\xi n=q2+\sigma n2$. Notice that the first term in Eq. (B10) is similar to Eq. (15) but instead of $\pi a2$ in the denominator, we get $\pi b2$. Thus, to a good approximation, Eq. (15) can still be used but the sensor area, $\pi a2$ is replaced with the sample area, $\pi b2$. To illustrate, we set the parameters to be similar to the parameters in Sec. 3, $\Lambda $ = 14 W/mK, $\kappa $ = 4 mm^{2}/s, *R *=* *10^{−4} m^{2}K/W, *P *=* *1 W, *l *=* *3 mm, *a *=* *9.9 mm and *b *=* *10.5 mm. We plot the transients from Eqs. (15), (25), and (B10) in Fig. 7(a). Notice that if the sample radius is larger than the sensor radius, then Eq. (A2) is a better approximation of Eq. (B10) than Eq. (15). In particular, the temperature difference between the exact solution and Eq. (25) has a magnitude of less than 100 mK (Fig. 7(b)).

For *a *=* *9.9 mm and *b *=* *10.5 mm, with a percent difference of 6.1%, we see that the approximate temperatures are close to the exact temperatures. To get the effect of varying the sample radius on measured *R*, we generate temperature transients with a specific sample radius *b*, where $b>a$ and a true $R$ value. This temperature transient is assumed as the experimental data, in which the measured $R$ can be determined by the nonlinear fitting algorithm with respect to Eq. (25). The percent error in the measured film resistance $R$ is plotted against the different values of sample radius divided by sensor radius, $b/a$, in Fig. 8, assuming the other parameters were similar as in Sec. 3, we can infer from the graph that if the sample radius $b$ is no more than 7% of the sensor radius *a*, then the percent error in the measurement would be less than 15%. In Sec. 3, $b/a=1.06$, and thus, we expect a percent error of less than 15%.

### Sensitivity of the Thermal Resistance of Thin Films Method to the Film Thermal Resistance

The thermal resistance $R$ in Eq. (13) can be any number greater than or equal to 0. The measurement uses a nonlinear fitting algorithm which estimates the time offset, temperature offset, the thermophysical properties of the background material, and the thermal resistance $R$. In order for the nonlinear fitting algorithm to be sensitive to $R$, the value of $R$ should not be too small relative to other parameters, i.e., $R$ is not close to zero, and also, $R$ should not be too large relative to the other parameters, i.e., $R$ does not approach infinity. To illustrate the sensitivity of the method to different values of $R$, we plot the expected sensor temperatures at different values of $R$ given the parameters specified in Sec. 3 (see Fig. 9) Notice that when *R *=* *3.6 × 10^{−4} m^{2}K/W, the transient is far from the transient at *R *=* *0 and *R *=* *10^{6} m^{2}K/W. The measured thermal resistances in Sec. 3 are close to this value. For instance, for a Teflon film ($\Lambda $ = 0.3 W/mK) 85 $\mu $m thin, *R *=* *3 × 10^{−4} m^{2}K/W.