*3.2. Measurements of the Density*

Two different possibilities to measure the density of the sheets are studied and compared. The first methodology is a geometric-based method, utilizing the dimensions and the mass of the specimens. The LFA specimens are used for this purpose with the length *l* ≈ 10 mm, the width *w* ≈ 10 mm, and the thickness from *d* ≈ 0.1 mm up to *d* ≈ 1 mm. The measurement of the three values is repeated five times and an average value is calculated. Similar measurement equipment, as described in the measurements of the electric resistivity, is used for all three quantities. The estimation of the measurement error is Δ*d* = 0.01 mm and Δ*l* = Δ*w* = 0.05 mm. The weight *m* of the probe is measured using a Sartorius high precision balance with a maximum error of 0.1 mg. In order to

account for possible dust or additional influences, a value of Δ*m* = 1 mg is used for the accuracy study.

$$
\rho = \frac{m}{d \cdot l \cdot w} \tag{22}
$$

The error of the procedure can be estimated with the following:

$$
\Delta \rho = \frac{m + \Delta m}{(d - \Delta d) \cdot (l - \Delta l) \cdot (w - \Delta w)} \tag{2.3}
$$

The second methodology utilizes the principle of Archimedes. In this measurement technique, no exact cubic probe is necessary. A higher amount of pieces can be utilized to obtain a higher overall measured weight. The insulation of the material is removed by sandblasting. The density measurements are performed with an analytical balance Kern ABT 220-4M. The measurement error of the weight measurement is 0.1 mg. Multiple probes are cut into specimens that fit into the universal immersion basket of the balance. An overview of the measurement equipment is given in Figure 7. The distilled water, used as the reference fluid, is filled into a beaker. The temperature of the reference fluid *ϑ*<sup>0</sup> is measured, utilizing the thermometer included in the balance equipment. The density of the reference fluid is evaluated from a lookup table *ρ*<sup>0</sup> = *f*(*ϑ*0). The error of the mass density of the reference fluid is estimated by a 5 K-deviation in the temperature measurement as follows:

$$
\Delta \rho\_0(\theta) = \rho\_0(\theta) - \rho\_0(\theta + 5\,\text{K})\tag{24}
$$

**Figure 7.** Analytical balance Kern ABT 220-4M with universal immersion basket.

The first measurement is performed with the specimens placed onto an upper sample dish of the immersion basket. The amount of samples is adapted to obtain a total weight of approximately *m*<sup>A</sup> ≈ 40 g. The measured value of *m*<sup>A</sup> is the result of the difference between the buoyancy force of the air and the weight force of the specimen:

$$m\_{\rm A} = (\rho - \rho\_{\rm air}) \cdot V\_{\rm \,\,\,} \tag{25}$$

with the volume of the specimen *V*, and the density of air *ρ*air. The influence of the air buoyancy force is neglected in the calculation. An additional factor is considered in the error estimation of Δ*m*A:

$$
\Delta m\_{\rm A} = \left(\frac{\rho\_{\rm air}}{\rho}\right) \cdot m\_{\rm A} + 1 \,\text{mg} \tag{26}
$$

The measurement is repeated with the samples placed on the lower dish of the immersion basket. The measured weight *m*<sup>B</sup> is equal to the following:

$$m\_{\rm B} = (\rho - \rho\_0) \cdot V. \tag{27}$$

The estimated error of the measurement of *m*<sup>B</sup> is assumed to be equal to Δ*m*<sup>B</sup> = 1 mg. The combination of Equations (25) and (27) under neglection of the air buoyancy force gives the equation to calculate the mass density *ρ* of the specimen.

$$
\rho = \frac{m\_{\rm A}}{m\_{\rm A} - m\_{\rm B}} \rho\_0 \tag{28}
$$

A worst-case estimation for the measurement error of the density *ρ* is performed with the following:

$$
\Delta \rho = \frac{m\_{\rm A} + \Delta m\_{\rm A}}{(m\_{\rm A} - \Delta m\_{\rm A} - m\_{\rm B} - \Delta m\_{\rm B})} \cdot (\rho\_0 + \Delta \rho\_0) - \rho \tag{29}
$$

The entire measurement procedure is repeated three times. After each measurement in the water reference, the samples are dried. A thin rust film develops within seconds and is removed by sandblasting. All values are reevaluated, including the masses *m*<sup>A</sup> and *m*B, as the repeated sandblasting also removes some of the material. An average value of the three measurements is calculated.

The results of the measurements as well as the error estimation according to the calculation of Equation (3) for *x* = *ρ* are depicted in Table 6. The coefficient of variation *cv* for the measured values is added. *cv* of the geometrical values *l*, *d* and *w* shows values below 1.1% for most of the values. Only the thickness measurements show larger values with 2.5% for M8 and 1.86% for M6. The coefficient of variation for the Archimedes principle shows small values below 0.13% for all measured materials. The precision of the Archimedes principle is significantly improved in comparison to the geometrical principle. The accuracy of the geometrical principle is mainly driven by the estimated error of the thickness measurement Δ*d*, which leads to high error estimations for the thin sheets. The error estimation model gives a minimal value of 2.2% for the thickness material M1 and 14.7% for the thinnest material. The estimated error of the Archimedes principle with values around 0.2% is very low and almost equal for all probes. The accuracy is independent from the thickness of the probe. The variation for the three measurement repetitions of the Archimedes principle varies between 0.01% and 0.13%. This value is lower than the predicted values for the measurement error *δρ* in Table 6. This observation confirms the good precision of the measurement and confirms the estimation of the measurement accuracy being the critical value. The measured density utilizing the geometrical principle is lower than the density evaluated by the Archimedes principle between 3.2% and 4.7%. It is interesting to note that the values are all lower and not spread around the exact values of the Archimedes principle. The difference between the results shows the lowest value for Material M1, which confirms the trend indicated by the accuracy study. Additionally, the value of 3.2% is larger than the predicted error of 2.2% as a sum of the two error estimations. There are obviously some additional systematic errors present. Issues with air bubbles in the Archimedes measurement do not seem to be present, as the variation coefficient of the measurement is low. Air bubbles in the second measurement step would decrease *m*B, which leads to an underestimation of the density *ρ*. This is not the case, because the results of the Archimedes principle are all larger than those of the geometrical probes. A possible explanation for the effect is the cuboid model that is used for the estimation of the volume in the geometric approach. The measured values are the outer dimensions. Irregularities and roughness could lead to a real volume that is lower. This would cause lower values of the density *ρ*. The temperature dependency is estimated, using a thermal expansion coefficient of *<sup>α</sup>*th = 11.8 × <sup>10</sup><sup>−</sup>6/K.

$$\rho(\vartheta) = \rho \cdot \frac{1}{1 + 3 \cdot \mathfrak{a}\_{\rm th} \cdot (\vartheta - \vartheta\_0)},\tag{30}$$

with the measurement temperature as the reference temperature *ϑ*0.

The density *ρ* varies in a range from 7479 kg/m to 7834 kg/m. The value for material M1 is only 5% larger than the value of M8 at 298 K. The mass density is expected to play a minor role for the differences in the thermal conductivities of the materials, according to Equation (19) and the maximum possible dissipated heat in the application. In the case of a study with fewer accuracy requirements, an average value of the expected density could be used with a maximum error of the indicated 5%.


**Table 6.** Measurement results and error estimation of the density measurement at *ϑ* ≈ 293 K.
