# Polytetrafluoroethylene Films in Rigid Polyurethane Foams’ Dielectric Permittivity Measurements with a One-Side Access Capacitive Sensor

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}: 1.059 ≤ ε ≤ 2.067 and for (2) monolithic petrochemical-origin polyurethane, density 1280 kg/m

^{3}: 3.47 ≥ ε ≥ 3.31. To characterise the dielectric dispersion, the dropping factor was introduced and calculated for the investigated PUR foams of densities up to 400–450 kg/m

^{3}as F ≤ 5.0% and F ≈ 6.0% for monolithic petrochemical polyurethane.

^{3}exhibit permittivity ≈ 1.065 (1 kHz), that is only ≈ 6% higher than the permittivity of vacuum [5].

_{2}-CF

_{2}]

_{n}; the brand name of PTFE-based formulas is “Teflon”. PTFE is reported to have frequency practically non-dependent permittivity ε and low dissipation factor tgδ. Those features of PTFE resins are also relatively independent of fabrication conditions. In [10] PTFE is reported to exhibit absence of significant dielectric losses up to its first order transition point at temperature T = 327 °C. At T = 23 °C and frequencies f = 10

^{2}, 10

^{3}, 10

^{4}, and 10

^{5}Hz: ε′ = 2.001–2.002 and tgδ = 1–2 × 10

^{−4}. In [11] the low loss tangent of PTFE is explained as a consequence of the symmetrical conformation of the polymer backbone, which neutralizes the dipole forces of the C–F bonds yielding a net zero dipole moment. At 1 MHz dielectric constant of PTFE is determined as 2.1 and dissipation factor tgδ < 0.0002. In [12] a wide range of experimental data of PTFE investigations is reported. At room temperature T = 300 K, at frequencies 75 Hz and 500 Hz: tgδ < 1 × 10

^{−4}, at 1 kHz: tgδ < 1–6 × 10

^{−4}. At 5 kHz–2.2 MHz: tgδ < 1 × 10

^{−4}, at 1 kHz–316,000 Hz: tgδ = 2–6 × 10

^{−4}, 1 kHz–100 kHz: tgδ = 1.5–5.5 × 10

^{−4}. Dielectric permittivity at 100 Hz–1 MHz is given as ε′ = 2.2–2.0. Matis [13] reports values of dielectric permittivity and loss tangent of PTFE at two frequencies f = 100 Hz and 1 MHz: ε = 2.1 = const. and tgδ = 0.0005 and 0.0002. The lack of dependence of ε and tgδ on frequency is explained by PTFE having non-polar molecules and exhibiting only elastic polarization. Naidu [14] gives the following values at frequencies f = 50 Hz–1 MHz: ε = 2.3–2.8 and tgδ < 0.0002. Askeland [15] provides the data at 60 Hz–1 MHz: ε = 2.1 and 2.1 and tgδ = 0.00007 at 1 MHz. Jiang [16] reports dielectric constant of PTFE at 1 MHz as ε = 2.2 and dissipation factor tgδ < 0.00012.

^{3}that corresponds to the middle of PUR foams’ full range of densities: at 31 kg/m

^{3}permittivity ≈ 1.065; at 550 kg/m

^{3}ε ≈ 2.1 and for monolithic polyurethane, density 1280 kg/m

^{3}, permittivity ≈ 3.3 (1 kHz) [5]. PTFE is nonreactive, it reduces mechanical friction and wear (PTFE’s static frictional coefficient ≈ 0.04), therefore, a longer service time of the protective coating can be expected. The dielectric and mechanical properties suggest PTFE’s usage in protection of OSA capacitive sensor’s active area in PUR foams’ permittivity measurements.

## 2. Materials and Methods

#### 2.1. PUR Materials

^{3}≤ ρ ≤ 850 kg/m

^{3}, in blocks, in (a) open, free-rise moulds (25 cm × 25 cm × 20 cm) and (b) closed cylindrical polypropylene moulds (height = 8.0 cm and inner diameter = 10.0 cm) according to the technology and formulations given in [5]. Responding to the needs of bioeconomy, biopolyol was synthesized from Latvia-grown rapeseed oil by the trans-esterification method with triethanolamine (molar ratios 1 M:2.5 M and 1 M:2.9 M), in an environmentally friendly process, without emission of harmful substances, at temperatures T = 175 °C ± 5 °C and rigid closed-cell polyurethane biofoams were made in a range of apparent core density 80 kg/m

^{3}≤ ρ ≤ 450 kg/m

^{3}. Apparent core density of PUR foams (density) were determined according to ISO 845:2006. Differences in densities were achieved by varying the amount of physical or chemical blowing agents.

#### 2.2. Test Objects and Experiments

_{0}= 43 mm, Figure 1, and was excited via electrodes, by an electrical field generated by sinusoidal voltage signals. The amplitude value of the sinusoidal excitation signals U

_{0}= 20 V. The signals were generated at discrete frequencies, increasing in a geometric progression:

_{n}= f

_{1}, 2f

_{1}, ..., 2

^{(n − 1)}f

_{1}Hz, where f

_{1}= 10 Hz, n = 1, 2, ..., 16;

f = 10, 20, …, 327,680 Hz,

_{t}were measured on single samples, when calibration of the spectrometer was made before the measurement series, with regard to the measurement value, delivered by the OSA sensor in air [21].

^{3}were tested. The relative difference between data provided by the two apparatuses remains ≤4% in the entire frequency range.

_{T}. Experiment no. 2 simulates the case when OSA sensor’s active area is coated with a protective PTFE film permanently. It gives the measured value of permittivity of a complex sample “A single PUR foams’ sample + PTFE film”, when the measurement as well as calibration have to be made with a PTFE film on the OSA sensor. The value is denoted as the measured value of permittivity of a PUR foams’ sample ε

_{T′}.

^{3}. Side-by-side the two semi-cylinders form a round cylinder, thickness h = 12 mm, diameter D = 45 mm, fully covering the active area of the OSA sensor. It was proved in [5] that the relative error between the values of permittivity of a cylindrical sample and two semi-cylinders does not exceed 0.5%.

#### 2.3. Dielectric Losses

^{3}dielectric losses were measured as ε′′ = 0.0022–0.0063 at 1 kHz and ε′′ = 0.0032–0.0084 at 0.1 MHz. For monolithic lab-made polyurethane ε′′ = 0.042 at 1 kHz and ε′′ = 0.088 at 0.1 MHz. The acquired values are in a good correspondence with the experimental data reported in [22].

#### 2.4. Penetration Depth

_{0}≈ 45 mm, thickness h = 0.04–24.6 mm, was determined experimentally with the OSA sensor, Figure 3. Layers were cut from the top of ~ 25 mm thick PTFE sample; permittivity of the remaining sample was measured and plotted against its thickness. Three measurements were made for each data point. Average density of the samples was calculated as 2177 ± 38 kg/m

^{3}(±1.8%) that corresponds with the density values, reported in [12].

_{t}, not depending on thickness. The value ε

_{t}= 2.10 is in a good correspondence with the values reported in [13,15,16]. The experimental data is modelled with an exponential function, Figure 3:

_{t}− e

^{(0.1 − 0.5 h)}.

_{1}, f

_{2}, …, f

_{16}the relationship ε = ε(h) remains practically the same due to insignificant dispersion of PTFE’s permittivity.

_{t}−

^{(A+Bh)}, where ε

_{t}—the true value of PUR foams permittivity, A and B—numerical coefficients, depending on the PUR foams’ density. To define penetration depth, the measured value of electric susceptibility is considered:

^{3}(ε

_{t}= 1.08 at 1 kHz), then:

_{10%}= ε

_{t}− 10% ε

_{t}= 0.97 <

**ε**

_{0};

_{0}= 1.00 is permittivity of vacuum. That does not correspond to the definition of the true permittivity as a quantity always having values ≥ε

_{0}. Applying the mentioned criterion to the electric susceptibility solves the contradiction. In the given investigation the penetration depth of electric field is defined as the thickness h

_{3%}of a sample, at which the measured value of electric susceptibility χ is 3% less than the true value of electric susceptibility χ

_{t}of an infinitely thick sample. Then for PTFE:

_{t}− e

^{(0.1–0.5 h)}− 1.0 and

χ

_{3%}= χ

_{t}− 0.03χ

_{t}= 0.97χ

_{t}= 1.07.

_{3%}for PTFE is calculated as:

_{t}− χ(h)) − 0.1];

h

_{3%}= −2.0 [ln(χ

_{t}− χ

_{3%}) − 0.1] = 7.0 mm.

_{3%}, hence it can be concluded that the electric field, generated by sinusoidal voltage signal at frequencies f

_{1}, f

_{2}, …, f

_{16}fully penetrates the PTFE films and reaches PUR foams’ sample.

^{3}and ε

_{t}= 1.14–1.42 (1 kHz), penetration depth was determined as 5.72 mm ≤ h

_{3%}≤ 5.87 mm ± 0.02 mm. That corresponds to the conclusions in [20]: In the meaning of the given penetration depth definition, penetration depth of a concentric coplanar capacitive OSA sensor increases as the permittivity of the sample increases. In order to achieve the same 3% difference, samples with high ε

_{t}values need to have bigger penetration depth h

_{3%}, whereas samples with low ε

_{t}values can have smaller h

_{3%}to achieve the same percentage of difference [20]. The PUR foams’ single samples have to be thick enough to provide the true value of permittivity, therefore, 3–4 times higher thickness than the penetration depth h

_{3%}was taken as appropriate for PUR foams’ samples of densities 50–1280 kg/m

^{3}: h ≈ 20–25 mm.

#### 2.5. Dropping Factor

_{n}) were approximated with 2-nd order polynomials, e.g., for the PTFE films, Figure 4B, (j) and (k):

^{2}− 0.000136n + 1.064500 and

ε = 0.000017n

^{2}− 0.000411n + 1.013888,

_{n}− 1)/log2 + 1. To characterise the dielectric dispersion, the dropping of permittivity φ was calculated from the most monotonous part f

_{3}, f

_{4}, …, f

_{14}of the approximated spectra as:

_{3}) − ε(f

_{14})]/ε(f

_{3}),

_{3}) and ε(f

_{14})—permittivity at f

_{3}= 40 Hz and f

_{14}= 81 920 Hz. For PTFE films φ ≈ 0.09% at thickness h = 0.20 mm and φ ≈ 0.12% at h = 0.04 mm, Figure 5. Starting from a certain value of h ≈ 5 mm, dropping becomes independent of thickness. The thickness-independent value of dropping is denoted as dropping factor φ

_{0}. Then the relationship φ = φ(h) can be described with an exponential function:

_{0}− e

^{[(0.85 − h) − 1.0]}.

_{0}= 0.40% ± 0.73% (U95%) that shows a small dielectric dispersion of PTFE’s permittivity and corresponds to the requirements set for the material of protective coverage/coating. In comparison, even the most light-weight foams from the investigated ones PUR foams LAST-A-FOAM

^{®}FR-3703, density ρ ≈ 48 kg/m

^{3}, porosity η

_{g}= 1 − ρ/ρ

_{p}= 96% and ε

_{t}= 1.08 (1 kHz), exhibit nearly two times higher dropping factor φ

_{0}≈ 0.7%, where ρ

_{p}= 1280 kg/m

^{3}is density of monolithic polyurethane. Dropping factor, calculated from the approximated permittivity spectra of PUR foams, is given in dependence of PUR foams’ density in Table 1 and Figure 9.

#### 2.6. Permittivity Spectra

_{t}of (a) light-weight PUR foams LAST-A-FOAM

^{®}FR-3703, ρ = 48 kg/m

^{3}, (b) PUR foams Sika-150, ρ = 144 kg/m

^{3}and (c) monolithic lab-made polyurethane, density 1280 kg/m

^{3}are given in Figure 6, Figure 7 and Figure 8 together with permittivity spectra ε

_{T}and ε

_{T′}. The permittivity spectra are approximated with the third order polynomials of frequency ordeal number n.

_{t}is summarized in Table 1, at 1 kHz, where U95%—the expanded uncertainty. The ε

_{t}values of the same density PUR foams are similar for the four groups that correlate with results in [5].

_{0}, calculated from the approximated permittivity spectra. Figure 9 depicts φ

_{0}in dependence of density of (a) lab-made PUR foams, (b) lab-made PUR biofoams, (c) Sika JSC PUR foams and (d) General Plastics Manufacturing Company PUR foams. For the light-weight PUR foams the dependence φ

_{0}(ρ) is nearly linear, φ

_{0}≤ 1.5%, up to densities 100 kg/m

^{3}for all four groups of PUR foams. The highest dropping factor is exhibited by Sika JSC PUR foams and the lab-made PUR biofoams at densities 550–650 kg/m

^{3}: φ

_{0}≈ 4.0–4.5%. After the maximum the value of φ

_{0}decreases to the dropping factor value of monolithic polyurethane. All the values of PUR foams’ dropping factor are 10–40 times higher than dropping of PTFE films 0.02/0.04 mm: 0.09% and 0.12%, which facilitates small modifications of permittivity spectra caused by a PTFE film on the OSA sensor’s active area.

_{T}and ε

_{T′}are also summarized in Table 1, at 1 kHz. Values of ε

_{T}characterise the overall impact of the PTFE film on the true value of PUR foams’ permittivity. Values of ε

_{T′}give the measured permittivity value of PUR foams’ when coating with a PTFE film of OSA sensor’s active area is simulated and calibration has to be made with a PTFE film.

#### 2.7. Estimation of Modifications

_{t}(f

_{n}), the average modification factor η

_{Taver}is defined for permittivity spectra ε

_{T}(f

_{n}) and ε

_{T′}(f

_{n}):

_{tT}(f

_{n}) = [ε

_{t}(f

_{n}) − ε

_{T}(f

_{n})]/ε

_{t}(f

_{n}),

Δε

_{tT′}(f

_{n}) = [ε

_{t}(f

_{n}) − ε

_{T′}(f

_{n})]/ε

_{t}(f

_{n}), n = 1, 2, …, 16;

_{T}(f

_{n}) or ε

_{T′}(f

_{n}) lies above the true permittivity spectrum ε

_{t}(f

_{n}), η

_{Taver}< 0.0; when it lies below ε

_{t}(f

_{n}), η

_{Taver}> 0.0. If spectrum ε

_{T}(f

_{n}) or ε

_{T′}(f

_{n}) intersects the spectrum ε

_{t}(f

_{n}), absolute values of differences |Δε

_{tT}(f

_{n})| and |Δε

_{tT′}(f

_{n})| have to be taken in Equation (10).

_{t}(f

_{n}) and ε

_{T′}(f

_{n}) are evaluated towards a criterion of small modifications as well:

_{t}(f

_{n}) − ε

_{T}(f

_{n})| ≤ Δε

_{t}, |ε

_{t}(f

_{n}) − ε

_{T′}(f

_{n})| ≤ Δε

_{t}, n = 1, 2, ..., 16;

_{0}is a small modification parameter. Modification of the true permittivity spectrum ε

_{t}by presence of a PTFE film is considered as small when the measured values of permittivity ε

_{T}or ε

_{T′}at each frequency f

_{n}differ from the true permittivity values ε

_{t}no more than by a pre-defined value ±Δε

_{t}, calculated as a η

_{0}fraction from the average value ε

_{taver}of the true permittivity spectrum values ε

_{tn}:

_{t}(f

_{n}) − η

_{0}ε

_{taver}≤ ε

_{T}(f

_{n}) ≤ ε

_{t}(f

_{n}) + η

_{0}ε

_{taver};

ε

_{t}(f

_{n}) − η

_{0}ε

_{taver}≤ ε

_{T′}(f

_{n}) ≤ ε

_{t}(f

_{n}) + η

_{0}ε

_{taver}.

_{T}(f

_{n}) or ε

_{T′}(f

_{n}) has to be situated inside a zone around the true permittivity spectrum ε

_{t}(f

_{n}) defined by Equation (12). The value of η

_{0}depends on the required accuracy of measurements.

_{t}(f

_{n}) was estimated for PUR foams of Table 1, according to the following methodology. First, the numerical value of average modification factor was calculated for ε

_{T}(f

_{n}) and ε

_{T′}(f

_{n}). Depending on the required accuracy, a certain limit value for the average modification factor is set. As an example, η

_{lim}was set to 3.5%, the calculated η

_{Taver and}η

_{T′aver}values were compared to η

_{lim}and conclusions made. Then the average permittivity value of the true permittivity spectra ε

_{taver}was calculated, the small modification parameter was set to values η

_{0}= 1, 2, ..., 5% and satisfaction of the criterion of small modifications by the spectra ε

_{T}(f

_{n}) and ε

_{T′}(f

_{n}) was tested. A PC code was compiled for numerical calculations.

## 3. Results

#### 3.1. Average Modification Factor

_{Taver}for permittivity spectra ε

_{T}(f

_{n}), corresponding to the coverage with a PTFE film, thickness 0.20 mm, gave the following results, Table 1: (1) Lab-made PUR foams: −4.0% ≤ η

_{Taver}≤ 7.3% at PUR foams’ density 84–1280 kg/m

^{3}; (2) lab-made PUR biofoams: −4.3% ≤ η

_{Taver}≤ 1.6% at 84–442 kg/m

^{3}; (3) Sika JSC PUR foams: −4.9% ≤ η

_{Taver}≤ 7.1% at 85–1181 kg/m

^{3}and (4) Gen. Plast. PUR foams −5.2% ≤ η

_{Taver}≤ −0.7% at 85–320 kg/m

^{3}.

_{T′aver}for permittivity spectra ε

_{T′}(f

_{n}), corresponding to simulating coating with a PTFE film, thickness 0.20 mm, gave the following results, Table 1: (1) lab-made PUR foams η

_{T’aver}= 2.8–14.0% at 84–1280 kg/m

^{3}, (2) lab-made PUR biofoams η

_{T’aver}= 3.1–9.5% at 84–442 kg/m

^{3}, (3) Sika JSC PUR foams η

_{T’aver}= 2.6–13.6% at 85–1181 kg/m

^{3}and (4) Gen. Plast. PUR foams η

_{T’aver}= 0.9–5.2% at 48–320 kg/m

^{3}. It can be concluded that the average modifications caused by the PTFE film of thickness 0.20 mm are quite high in both cases: up to 10–14% and exceed η

_{lim}= 3.5%.

_{T}(f

_{n}), corresponding to the coverage with a PTFE film, thickness 0.04 mm, the following numerical values of η

_{Taver}were acquired, Table 1: (1) Lab-made PUR foams η

_{Taver}= −0.4–1.2% at 84–1280 kg/m

^{3}, (2) lab-made PUR biofoams η

_{Taver}= −0.5–0.4% at 84–442 kg/m

^{3}, (3) Sika JSC PUR foams η

_{Taver}= −0.5–1.3% at 85–1181 kg/m

^{3}and (4) Gen. Plast. PUR foams η

_{Taver}= −0.6–0.4% at 48–320 kg/m

^{3}. It can be concluded that the |η

_{Taver}| remains less than 1.3% as well as less than η

_{lim}= 3.5% in density range of PUR foams 84–1280 kg/m

^{3}, which might be acceptable for certain practical applications, depending on the required accuracy, e.g., to perform a quick evaluation of rigid PUR foams’ permittivity in field conditions (Thermal insulation, shielding structures, etc.). Calibration of the spectrometer can be made in air, but the permittivity measurements of the rigid PUR foams’ test object—with a PTFE film temporarily covered underneath the test object, on the OSA sensor’s active area, to protect the OSA sensor’s electrodes from the adverse side effects caused by the PUR foams’ test object. Thus, an operational flexibility can be retained, since the OSA sensor does not have to be permanently coated with the PTFE film.

_{T′}(f

_{n}), corresponding to simulating coating with a PTFE film, thickness 0.04 mm, the dependence of η

_{T′aver}on PUR foams’ density, Table 1, is depicted in Figure 10.

_{T′aver}= η

_{T′aver}(ρ) follows a similar trend for all four groups of PUR foams and can be described with a function:

_{T′aver}(ρ) = Kρ

^{0.60}, where K = 0.04.

_{lim}= 3.5% for density range of rigid PUR foams 48–1280 kg/m

^{3}. For the light- to medium-weight PUR foams of density ρ ≤ 220 kg/m

^{3}, applied in shielding structures, building, heat and cold insulation etc. [23,24,25,26], the average modification factor is even smaller: η

_{T′aver}≤ 1.0%. At similar densities the values of η

_{T′aver}are similar for the four investigated groups of PUR foams: (a) lab-made petrochemical, (b) lab-made biofoams, (c) Sika JSC and (d) General Plastics Manufacturing Company. It can be concluded that the modifications, caused by a simulated permanent coating of a PTFE film, thickness 0.04 mm, on the active area of the OSA sensor, might be acceptable for certain practical applications, depending on the required accuracy.

#### 3.2. Small Modification Criterion

_{T}(f

_{n}) for small values of η

_{0}= 1.0%, 1.5% and 2.0% only in a narrow density range 250–550 kg/m

^{3}and, thus, represent no practical interest.

_{T}(f

_{n}) of the lab-made PUR foams when η

_{0}= 1.0% in a density range 84–550 kg/m

^{3}and when η

_{0}= 1.5–5.0% in a density range 84–1280 kg/m

^{3}. The trend is similar for lab-made PUR biofoams, Sika JSC and Gen. Plast. PUR foams, too. Such modifications might be acceptable for certain practical applications, depending on the required accuracy.

_{T′}(f

_{n}) only starting from η

_{0}= 3.0%, up to densities ρ ≈ 100 kg/m

^{3}, at η

_{0}= 4.0%, up to ρ

_{f}≈ 250 kg/m

^{3}and at η

_{0}= 5.0%, up to ρ

_{f}≈ 350 kg/m

^{3}. At the small values η

_{0}= 1.0%, 1.5% and 2.0% the criterion is not satisfied by the spectra ε

_{T′}(f

_{n}) in density range 85 kg/m

^{3}–1280 kg/m

^{3}. The trend is similar for lab-made PUR biofoams and Gen. Plast. PUR foams. The modifications caused by the PTFE film of thickness 0.20 mm are quite high and likely unacceptable for practical applications.

_{T′}(f

_{n}) of the lab-made PUR foams at: (a) η

_{0}= 1.0%, up to densities ρ ≈ 200 kg/m

^{3}, (b) η

_{0}= 1.5%, up to ρ

_{f}≈ 430 kg/m

^{3}, (c) η

_{0}= 2.0%, up to ρ

_{f}≈ 700 kg/m

^{3}, (d) η

_{0}= 3.0%, up to ρ ≈ 1 100 kg/m

^{3}and (e) η

_{0}= 4.0 and 5%, up to ρ ≈ 1280 kg/m

^{3}. The trend is similar for lab-made PUR biofoams, Sika JSC and Gen. Plast. PUR foams. The modifications might be acceptable for certain practical applications, depending on the required accuracy.

## 4. Evaluation of Measurement Uncertainties

#### 4.1. Measurement Uncertainties Due to Air Gaps

^{3}each had 1–3 cavities of height h = 0.03–0.06 mm, width w = 7–22 mm and depth d = 1–3 mm. PUR foams’ samples, ρ

_{f}> 200 kg/m

^{3}and PTFE samples have only comparatively small cavities of h < 0.03 mm and w < 7 mm that are not taken into account further. All cavities are situated along the perimeter of sample’ bottom surface.

_{s}, while the other end is built on the perimeter of the sensor, in a point-contact. Volume of a penetrating gap at h

_{s}= 1.0 mm equals V

_{0}= 795.2 mm

^{3}.

_{0}. Modelling the limit case when each sample of density below 200 kg/m

^{3}has three largest perimetral cavities of dimensions h = 0.06 mm, w = 22 mm and d = 2.9 mm, k is calculated:

_{air}≈ 1.00059) on the measured value of true permittivity ε

_{t}, a spacer—a PTFE strip of thickness h

_{s}= 1.0 mm and width 2 mm was placed on the perimeter of the sensor at depth ~ 2 mm. A sample was built on the spacer and the sensor, forming a penetrating gap. The true permittivity ε

_{t}and the measured value of the true permittivity ε

_{tpn}with a penetrating gap was measured for PUR foams’ samples, ρ

_{f}= 48 and 144 kg/m

^{3}, Table 3. Three measurements were made for each sample. The change in the value of the true permittivity due to a penetrating gap is calculated as:

_{tpn}= ε

_{t}− ε

_{tpn}.

_{tpn}due to gaps is assumed to be proportional to the total volume and location of gaps. Even with the maximum amount of the biggest perimetral gaps their total volume is ≈ 260 times less than the volume V

_{0}of a penetrating gap of height 1 mm, Equation (15). A penetrating gap is situated above the whole active surface of the sensor, therefore Δε

_{tpn}characterises an averaged influence of location. On the contrary, the experimentally identified gaps are perimetral gaps, situated along the perimeter of sensor’s active surface, where intensity of the excitation field is the weakest. It is concluded that for PUR foams, ρ

_{f}< 200 kg/m

^{3}, the upper boundary of the change in the value of the true permittivity caused by the identified perimetral gaps is Δε

^{U}

_{tpm}= Δε

_{tpn}/k, Table 4. A sample with no cavities provides the lower boundary: Δε

^{L}

_{tpm}= 0.0.

_{t}for a PUR foams’ sample, ρ

_{f}< 200 kg/m

^{3}, with perimetral cavities, fall in the mentioned limits, therefore:

^{g}= 1/3(Δε

^{U}

_{tpm}− Δε

^{L}

_{tpm});

^{g}is the expanded uncertainty of measurement, based on a standard uncertainty multiplied by a coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95% [27,28].

#### 4.2. Measurements Uncertainty of ε_{t}

_{t}is evaluated for lab-made PUR foams, ρ

_{f}= 50, 112, 144, 427, 846 kg/m

^{3}, monolithic polyurethane 1280 kg/m

^{3}and PTFE according to ISO Guide to the Expression of Uncertainty in Measurement (GUM) [27,28]. The accuracy of the dielectric spectrometer in conditions of reproducibility was evaluated with expanded uncertainty U95%

^{S}= ± 0.01. In Type A evaluation series of n = 3–4 statistically independent, non-destructive observations are made for each sample of PUR foams and monolithic polyurethane; for the PTFE sample n = 18. Measurement uncertainty arises mainly due to inhomogeneous density of the samples, air gaps between the sample and active surface of the sensor etc. Since the expanded uncertainty of true permittivity due to perimetral gaps U95%

^{g}is several orders smaller than U95%

^{S}due to the limited accuracy of the spectrometer (Table 4), the air gaps are not taken into account. Estimate of the input quantity and sensitivity coefficient are calculated: $\overline{{\mathsf{\epsilon}}_{\mathrm{t}}}=\frac{1}{\mathrm{n}}{\displaystyle \sum}_{1}^{\mathrm{n}}{\mathsf{\epsilon}}_{\mathrm{ti}}$ and c

_{1}= 1.0. Attributing normal distribution to the measurand, effective degrees of freedom ν

_{eff}of the combined standard uncertainty u

_{c}(ε

_{t}) associated with the output estimate are estimated from the Welch—Satterthwaite formula:

_{t}) for Type A evaluation at frequencies f

_{1}, f

_{7}and f

_{16}are given in Table 5.

_{t}and are not taken into account. That yields the standard uncertainty u(ε

_{t}) = U95%

^{S}/2 = ± 0.05.

_{c}(ε

_{t}), where k = 2 is the coverage factor.

_{f}= 427 kg/m

^{3}and f

_{7}= 640 Hz, the measurement result for a series is ε

_{t}= 1.865 ± 0.011. The reported expanded uncertainty U95% of ε

_{t}measurement is stated as the standard uncertainty of measurement multiplied by the coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95%.

_{t}) in Type A type evaluation, characterising the repeatability. For the most of the PUR foams input of Type A uncertainties is smaller or comparable to the input of Type B uncertainties. No significant difference in values of u

_{c}(ε

_{t}) and U95% was identified for true permittivity spectra as measured and the approximated ones. It is concluded that approximation smooths out fluctuations in experimental data, retaining the dominating trend.

#### 4.3. Measurement Uncertainty of Dropping Factor

_{0}is evaluated for lab-made PUR foams, ρ

_{f}= 50, 112, 144, 427, 846 kg/m

^{3}, monolithic polyurethane 1280 kg/m

^{3}and PTFE according to GUM. In Type A type evaluation series of n = 3–4 statistically independent, non-destructive observations are made for each sample of PUR foams and monolithic polyurethane; for the PTFE sample n = 5. The permittivity spectra are approximated with the 3rd-order polynomials for PUR foams and with the 2nd-order polynomials for PTFE. The model function:

_{0}= [ε(f

_{3}) − ε(f

_{14})]/ε(f

_{3}) = 1 − ε(f

_{14})/ε(f

_{3}),

_{3}) and ε(f

_{14})—input quantities. Estimates of ε

_{3}= ε(f

_{3}) and ε

_{14}= ε(f

_{14}) and corresponding sensitivity coefficients are calculated:

_{3}) = u(ε

_{14}) = U95%

^{S}/2 = ± 0.05. Uncertainty contribution of input quantities is expressed as u

_{1}(φ) = c

_{1}u(ε

_{3}) and u

_{2}(φ) = c

_{2}u(ε

_{14}) and the combined standard uncertainty is calculated from Type A and Type B uncertainty budget. Then for a normal distribution the expanded uncertainty U95% = ku

_{c}(φ

_{0}), where k = 2.

_{f}= 427 kg/m

^{3}the measurement result for a series is φ

_{0}= 0.037 ± 0.011. The reported expanded uncertainty U95% of ε

_{t}measurement is stated as the standard uncertainty of measurement multiplied by the coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95%.

_{0}= 0.40% ± 0.68%, Table 6. Uncertainties larger than measured values are common in measurements where the measurand value is expected to be zero or close to it, as the dropping factor of PTFE.

_{c}(φ

_{0}) and U95% was identified for dropping factor φ

_{0}calculated from the permittivity spectra as measured and the approximated ones.

#### 4.4. Measurement Uncertainty of Complex Samples

_{T}and ε′

_{T}is evaluated for lab-made PUR foams, ρ

_{f}= 50, 112, 144, 427, 846 kg/m

^{3}and monolithic polyurethane 1280 kg/m

^{3}according to GUM. In Type A type evaluation series of n = 3–4 statistically independent, non-destructive observations are made for each PUR foams sample and monolithic polyurethane, Table 7. Measurement uncertainty arises mainly due to inhomogeneous density and structure of the samples, air gaps between the sample, the PTFE films and active surface of the sensor, etc.

_{T}and ε′

_{T}can be expected to be of the same order as those of ε

_{t}. Additional experimental and theoretical research is necessary to evaluate the uncertainties of the measured permittivity spectra ε

_{T}and ε′

_{T}more precisely.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Measured value of permittivity in dependence of sample’s thickness: PTFE and lab-made PUR foams, density (a) 228, (b) 88 and (c) 50 kg/m

^{3}(experimental data and model functions, 1 kHz).

**Figure 4.**(

**A**) Permittivity spectra of PTFE samples of thickness (a) 24.64, (b) 20.10, (c) 16.18, (d) 10.99 and (e) 5.94 mm. (

**B**) Permittivity spectra of PTFE samples of thickness (f) 4.10, (g) 2.90, (h) 1.77 mm, (i) 0.77 and PTFE films of thickness (j) 0.20 mm and (k) 0.04 mm.

**Figure 5.**Dropping in dependence of PTFE sample’s thickness: (a) Experimental data and (b) model function.

**Figure 6.**Permittivity spectra of PUR foams LAST-A-FOAM

^{®}FR-3703, ρ = 48 kg/m

^{3}: (a) ε

_{t}, (b) ε

_{T}at PTFE film 0.20 mm, (c) ε

_{T}at PTFE film 0.04 mm, (d) ε

_{T′}at PTFE film 0.20 mm and (e) ε

_{T′}at PTFE film 0.04 mm.

**Figure 7.**Permittivity spectra of PUR foams Sika-150, ρ = 144 kg/m

^{3}: (a) ε

_{t}, (b) ε

_{T}at PTFE film 0.20 mm, (c) ε

_{T}at PTFE film 0.04 mm, (d) ε

_{T′}at PTFE film 0.20 mm and (e) ε

_{T′}at PTFE film 0.04 mm.

**Figure 8.**Permittivity spectra of lab-made monolithic polyurethane, ρ

_{p}= 1280 kg/m

^{3}: (a) ε

_{t}, (b) ε

_{T}at PTFE film 0.20 mm, (c) ε

_{T}at PTFE film 0.04 mm, (d) ε

_{T′}at PTFE film 0.20 mm and (e) ε

_{T′}at PTFE film 0.04 mm.

**Figure 9.**Dropping factor in dependence of PUR foams’ density: (a) lab-made foams, (b) lab-made biofoams, (c) Sika JSC foams and (d) General Plastics Manufacturing Company foams.

**Figure 10.**Average modification factor η

_{T′aver}in dependence of PUR foams’ density: (a) Lab-made petrochemical, (b) lab-made biofoams, (c) Sika JSC and (d) General Plastics Manufacturing Company.

**Figure 11.**Gaps between the bottom surface of a sample and active area of the sensor: 1) a penetrating gap and 2) a perimetral gap.

N | Materials | Single Samples | PTFE Film Coverage | PTFE Film Coating (Simul.) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.20 mm | 0.04 mm | 0.20 mm | 0.04 mm | ||||||||||

ρ;kg/m^{3} | η_{g};% | ε_{t} ± U95% | φ_{0};% | ε_{T} | η_{Taver};% | ε_{T} | η_{Taver};% | ε_{T′} | η_{T′aver};% | ε_{T′} | η_{T′aver};% | ||

(1) PTFE | |||||||||||||

1 | PTFE; h = 14.37 mm | 2 158 | 0 | 2.10 ± 0.01 | 0.4 | 2.08 | 0.8 | 2.09 | 0.3 | 1.96 | 6.5 | 2.07 | 1.4 |

(2) Lab-made petrochemical PUR foams | |||||||||||||

1 | PUR foams | 84 | 93 | 1.14 ± 0.011 | 0.8 | 1.19 | −4.0 | 1.15 | −0.4 | 1.11 | 2.8 | 1.13 | 0.8 |

2 | PUR foams | 112 | 91 | 1.20 ± 0.010 | 1.2 | 1.23 | −3.1 | 1.20 | −0.6 | 1.16 | 3.1 | 1.19 | 0.5 |

3 | PUR foams | 144 | 89 | 1.26 ± 0.011 | 1.4 | 1.31 | −3.7 | 1.27 | −0.9 | 1.22 | 3.5 | 1.26 | 0.4 |

4 | PUR foams | 279 | 78 | 1.50 ± 0.013 | 2.6 | 1.55 | −2.1 | 1.50 | 0.1 | 1.42 | 5.5 | 1.49 | 1.1 |

5 | PUR foams | 427 | 67 | 1.84 ± 0.010 | 3.2 | 1.83 | 0.5 | 1.83 | 0.4 | 1.71 | 6.6 | 1.81 | 1.3 |

6 | PUR foams | 539 | 58 | 2.12 ± 0.021 | 3.5 | 2.09 | 1.7 | 2.11 | 0.5 | 1.91 | 10.1 | 2.09 | 1.6 |

7 | PUR foams | 846 | 34 | 2.61 ± 0.019 | 3.5 | 2.48 | 5.0 | 2.58 | 1.2 | 2.33 | 11.0 | 2.56 | 2.2 |

8 | Monol. PUR | 1280 | 0 | 3.64 ± 0.015 | 2.8 | 3.37 | 7.3 | 3.59 | 1.2 | 3.17 | 14.0 | 3.55 | 3.2 |

(3) Lab-made PUR biofoams | |||||||||||||

1 | PUR biofoams | 84 | 93 | 1.13 ± 0.014 | 1.1 | 1.18 | −4.3 | 1.13 | −0.5 | 1.09 | 3.1 | 1.12 | 0.5 |

2 | PUR biofoams | 111 | 91 | 1.22 ± 0.012 | 1.3 | 1.26 | −3.2 | 1.22 | −0.4 | 1.17 | 3.8 | 1.21 | 0.7 |

3 | PUR biofoams | 144 | 89 | 1.23 ± 0.010 | 1.7 | 1.27 | −3.1 | 1.23 | −0.2 | 1.17 | 4.7 | 1.22 | 0.8 |

4 | PUR biofoams | 290 | 77 | 1.61 ± 0.016 | 3.2 | 1.60 | 0.4 | 1.60 | 0.3 | 1.48 | 7.3 | 1.58 | 0.9 |

5 | PUR biofoams | 366 | 71 | 1.74 ± 0.021 | 3.9 | 1.73 | 0.7 | 1.73 | 0.5 | 1.62 | 6.5 | 1.71 | 1.6 |

6 | PUR biofoams | 442 | 65 | 1.95 ± 0.027 | 4.5 | 1.93 | 1.6 | 1.96 | 0.4 | 1.78 | 9.5 | 1.93 | 1.7 |

(4) Sika JSC, petrochemical PUR foams | |||||||||||||

1 | Sika-80 | 85 | 93 | 1.14 ± 0.011 | 1.5 | 1.20 | −4.9 | 1.15 | −0.5 | 1.11 | 2.6 | 1.13 | 0.6 |

2 | Sika-150 | 144 | 88 | 1.26 ± 0.011 | 2.2 | 1.31 | −3.7 | 1.27 | −1.0 | 1.23 | 2.8 | 1.25 | 0.6 |

3 | Sika-240 | 228 | 81 | 1.42 ± 0.010 | 3.0 | 1.45 | −2.2 | 1.42 | −0.1 | 1.36 | 3.9 | 1.40 | 1.1 |

4 | Sika-450 | 459 | 61 | 1.95 ± 0.014 | 4.0 | 1.92 | 1.5 | 1.93 | 1.1 | 1.80 | 7.8 | 1.91 | 1.8 |

5 | Sika-600 | 554 | 53 | 2.09 ± 0.015 | 4.1 | 2.05 | 1.9 | 2.08 | 0.8 | 1.93 | 7.7 | 2.05 | 1.9 |

6 | Sika-700 | 720 | 39 | 2.59 ± 0.018 | 3.9 | 2.49 | 3.8 | 2.55 | 1.6 | 2.34 | 9.8 | 2.55 | 1.7 |

7 | Sika-930 | 993 | 16 | 3.13 ± 0.019 | 3.3 | 2.98 | 4.9 | 3.11 | 1.0 | 2.84 | 9.3 | 3.06 | 2.5 |

8 | Sika-1000 | 1 025 | 13 | 3.26 ± 0.021 | 3.3 | 3.06 | 5.9 | 3.21 | 1.2 | 2.88 | 11.3 | 3.17 | 2.6 |

9 | Monol. PUR M-960 | 1 181 | 0 | 3.78 ± 0.013 | 2.8 | 3.51 | 7.1 | 3.73 | 1.3 | 3.26 | 13.6 | 3.70 | 2.4 |

(5) General Plastics Manufacturing Company , petrochemical PUR foams | |||||||||||||

1 | FR-3703 | 48 | 96 | 1.08 ± 0.011 | 0.7 | 1.14 | −5.2 | 1.09 | −0.6 | 1.08 | 0.9 | 1.08 | 0.5 |

2 | FR-4305 | 80 | 94 | 1.14 ± 0.010 | 1.2 | 1.20 | −4.4 | 1.15 | −0.4 | 1.12 | 1.7 | 1.14 | 0.7 |

3 | FR-3707 | 112 | 92 | 1.20 ± 0.012 | 1.5 | 1.24 | −3.6 | 1.20 | −0.2 | 1.17 | 2.3 | 1.19 | 0.9 |

4 | FR-4315 | 227 | 83 | 1.42 ± 0.010 | 2.7 | 1.45 | −2.1 | 1.42 | 0.1 | 1.37 | 3.9 | 1.41 | 1.1 |

5 | FR-7120 | 320 | 76 | 1.60 ± 0.011 | 3.5 | 1.61 | −0.7 | 1.59 | 0.4 | 1.51 | 5.2 | 1.58 | 1.4 |

N | φ; rad | w = 2a; mm | d; mm | V; mm^{3} | K = V/V_{0} | ||
---|---|---|---|---|---|---|---|

h = 0.03 mm | h = 0.06 mm | h = 0.03 mm | h = 0.06 mm | ||||

1 | 0.1787 | 8 | 0.4 | 0.0230 | 0.0460 | 0.00003 | 0.00006 |

2 | 0.2241 | 10 | 0.6 | 0.0452 | 0.0903 | 0.00006 | 0.00011 |

3 | 0.2699 | 12 | 0.8 | 0.0786 | 0.1573 | 0.00010 | 0.00020 |

4 | 0.3164 | 14 | 1.1 | 0.1260 | 0.2519 | 0.00016 | 0.00032 |

5 | 0.3635 | 16 | 1.5 | 0.1900 | 0.3800 | 0.00024 | 0.00048 |

6 | 0.4115 | 18 | 1.9 | 0.2738 | 0.5477 | 0.00034 | 0.00069 |

7 | 0.4606 | 20 | 2.3 | 0.3809 | 0.7619 | 0.00048 | 0.00096 |

8 | 0.5108 | 22 | 2.9 | 0.5152 | 1.0305 | 0.00065 | 0.00130 |

N | ρ_{f};kg/m ^{3} | ε_{t} | ε_{tpn} | Δε_{tpn} | ||||||
---|---|---|---|---|---|---|---|---|---|---|

f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | ||

1 | 48 | 1.09 | 1.09 | 1.08 | 1.07 | 1.07 | 1.06 | 0.02 | 0.02 | 0.02 |

2 | 144 | 1.27 | 1.25 | 1.23 | 1.19 | 1.18 | 1.16 | 0.08 | 0.07 | 0.07 |

N | Density; kg/m^{3} | Δε^{L}_{tpm} | Δε^{U}_{tpm} | U95%^{g} | ||||||
---|---|---|---|---|---|---|---|---|---|---|

f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | ||

1 | 48 | 0.0 | 0.0 | 0.0 | 0.00008 | 0.00008 | 0.00008 | 0.00003 | 0.00003 | 0.00003 |

2 | 144 | 0.0 | 0.0 | 0.0 | 0.00031 | 0.00027 | 0.00027 | 0.00010 | 0.00009 | 0.00009 |

N | Density; kg/m^{3} | u(ε_{t}); Type A | u_{c}(ε_{t}) | U95% | ||||||
---|---|---|---|---|---|---|---|---|---|---|

f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | f_{1} | f_{7} | f_{16} | ||

PUR foams | ||||||||||

1 | 50 | 0.0015 | 0.0000 | 0.0003 | 0.0052 | 0.0050 | 0.0050 | 0.010 | 0.010 | 0.010 |

2 | 112 | 0.0015 | 0.0002 | 0.0012 | 0.0052 | 0.0050 | 0.0051 | 0.010 | 0.010 | 0.010 |

3 | 144 | 0.0030 | 0.0016 | 0.0008 | 0.0058 | 0.0052 | 0.0051 | 0.012 | 0.010 | 0.010 |

4 | 427 | 0.0027 | 0.0019 | 0.0007 | 0.0057 | 0.0053 | 0.0051 | 0.011 | 0.011 | 0.010 |

5 | 846 | 0.0190 | 0.0081 | 0.0044 | 0.0196 | 0.0095 | 0.0067 | 0.039 | 0.019 | 0.013 |

6 | 1280 | 0.0083 | 0.0057 | 0.0062 | 0.0097 | 0.0076 | 0.0080 | 0.019 | 0.015 | 0.016 |

PTFE | ||||||||||

1 | 2177 | 0.0023 | 0.0021 | 0.0020 | 0.0055 | 0.0054 | 0.0054 | 0.011 | 0.011 | 0.011 |

N | Density; kg/m^{3} | u(φ_{0}); Type A | u_{c}(φ_{0}) | U95% |
---|---|---|---|---|

PUR foams | ||||

1 | 50 | 0.0039 | 0.0076 | 0.015 |

2 | 112 | 0.0015 | 0.0064 | 0.013 |

3 | 144 | 0.0015 | 0.0062 | 0.012 |

4 | 427 | 0.0010 | 0.0054 | 0.011 |

5 | 846 | 0.0055 | 0.0068 | 0.014 |

6 | 1280 | 0.0020 | 0.0027 | 0.005 |

PTFE | ||||

7 | 2177 | 0.0014 | 0.0034 | 0.0068 |

**Table 7.**Measurement uncertainties of the measured permittivity spectra (at f

_{7}, permittivity spectra as measured).

N | Density; kg/m^{3} | u(ε_{T}); Type A | u(ε′_{T}); Type A | ||
---|---|---|---|---|---|

0.20 mm | 0.04 mm | 0.20 mm | 0.04 mm | ||

1 | 50 | 0.0014 | 0.0010 | 0.0005 | 0.0007 |

2 | 112 | 0.0120 | 0.0006 | 0.0055 | 0.0006 |

3 | 144 | 0.0012 | 0.0013 | 0.0104 | 0.0006 |

4 | 427 | 0.0014 | 0.0016 | 0.0175 | 0.0005 |

5 | 846 | 0.0033 | 0.0174 | 0.0139 | 0.0009 |

6 | 1280 | 0.0205 | 0.0017 | 0.0037 | 0.0022 |

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**MDPI and ACS Style**

Beverte, I.; Cabulis, U.; Gaidukovs, S.
Polytetrafluoroethylene Films in Rigid Polyurethane Foams’ Dielectric Permittivity Measurements with a One-Side Access Capacitive Sensor. *Polymers* **2021**, *13*, 1173.
https://doi.org/10.3390/polym13071173

**AMA Style**

Beverte I, Cabulis U, Gaidukovs S.
Polytetrafluoroethylene Films in Rigid Polyurethane Foams’ Dielectric Permittivity Measurements with a One-Side Access Capacitive Sensor. *Polymers*. 2021; 13(7):1173.
https://doi.org/10.3390/polym13071173

**Chicago/Turabian Style**

Beverte, Ilze, Ugis Cabulis, and Sergejs Gaidukovs.
2021. "Polytetrafluoroethylene Films in Rigid Polyurethane Foams’ Dielectric Permittivity Measurements with a One-Side Access Capacitive Sensor" *Polymers* 13, no. 7: 1173.
https://doi.org/10.3390/polym13071173