The Influence of the Surface Micro-structure Change on the Stainless Steel Effective Thermal Diffusivity

Abstract

Alternative techniques have been proposed for material thermal characterization. One of them is the Open Photoacoustic (OPC) technique, which uses a pressure variation induced inside a chamber to measure the thermal diffusivity of the sample. In this work, we apply the OPC classical model to show that a micro-structure change on the AISI 304 surface modified by abrasion can change the Photoacoustic signal measured by the OPC. As a result, the goal of this research is to determine the effective thermal diffusivity of AISI 304 stainless steel and to contribute to the development of the OPC technique.

Appendices

Appendix A: XRD Results

In this section, we furnish all results obtained for the XRD pattern, presented in Figure 6.

Fig. 6

XRD Diffraction pattern in interval $40^\circ \le 2 \theta \le 50^\circ$ in function of grit size SiC paper for: (a) S-I; (b) S-II;(c) S-III; (d) S-IV; (e) S-V; (f) S-VI; and (g) S-VII

Appendix B: SEM Results

In this section, we present the SEM images for the sample S-III after each polishing step shown in Figure 7.

Fig. 7

SEM of the sample S-III after polishing with Sandpaper grit: (a) 80; (b) 180;(c) 320; (d) 400; (e) 600; (f) 1000; and (g) 2000

Appendix C: Calibration

From the datasheet of the Soberton model EM9765 microphone, we know the operating frequency (20 Hz to 16 kHz), which is linear from approximately 80 to 1000 Hz. A simple way to remove the microphone contribution would be to analyze the results in the linear interval. However, for all frequencies, the measurement system, composed of a microphone and amplifiers, has an influence on the PA signal. The acquisitioned result $(\delta P_{measured})$ is a composition of the PA signal real $(\delta P{real})$ produced by the sample with the non-linear transfer function (H(f)) of the system, as follows [46,47,48]:

$$\begin{aligned} \delta P_{measured}=H(f) \delta P_{real} \end{aligned}$$

(C1)

A way used to remove the system contribution is to perform a calibration of the system by using a thin plate of aluminum. As the aluminum has a high thermal diffusivity, it is expected a linear behavior in logarithm $\delta P_{real}$ with logarithm of f, once $\delta P_{real}\propto f^{-1}$ [46]. From the ratio of the $\delta P_{measured}$ with the real theoretical result for a thin aluminum sample, it is obtained the H(f) function. Having in hand the function H(f), theoretically, the real signal ($\delta P_{real}$) of any sample can be obtained. This is accomplished by dividing the new signal measured ($\delta P_{measured}$) of this sample by the H(f) function obtained on the standard aluminum sample and assuming that the H(f) does not vary. However, the ratio of the PA signal measured for a fixed function can generate a new systematic error, since the amplitude parameters ($C_1$ and $C_2$) of the aluminum sample may depend on the environment and the system’s own contribution may depend on external and variable environmental factors. To overcome this problem, the H(f) function can be expressed in terms of your contributions and the noise N(f) function is added. [1, 9, 41, 42]

$$\begin{aligned} \delta P_{measured}= H(f) \delta P_{real}+N(f)\nonumber \\ =He(f)Ha(f) \delta P_{real}+N(f) \end{aligned}$$

(C2)

where He(f) is the electronic high-pass contribution due to the capacitor effects at low frequencies ($<100$ Hz) while Ha(f) is the acoustic low-pass filter due to the resonance of thin electret diaphragm at high frequencies ($>5$ kHz). In OPC measurements, noise (N(f) has two main contributions: at low frequency, electronic flicker noise with a $f^{-1}$ dependence, and at high frequency, crosstalk noise proportional to frequency $f^{1}$ [9, 41, 42].] The He(f) can be expressed by:

$$\begin{aligned} He(f)= He_1 He_2=-\left( \frac{\omega }{\omega _{e1}+i \omega }\right) \left( \frac{\omega }{\omega _{e2}+i \omega }\right) \end{aligned}$$

(C3)

where $He_1(f)$ is caused by the microphone and measurement resistor (R), which function as an RC filter, and $He_2(f)$ may exist depending on the amplification system used.

The membrane’s acoustic oscillatory Ha(f) is given by:

$$\begin{aligned} Ha(f)=Ha_1Ha_2=\left( \frac{\omega _{c3}^2}{\omega _{c3}^2+i \omega \delta _{c3}-\omega ^2}\right) \left( \frac{\omega _{c4}^2}{\omega _{c4}^2+i \omega \delta _{c4}-\omega ^2}\right) \end{aligned}$$

(C4)

where $\omega _{c1}=2 \pi f_{c1}$ is the cut-off frequencies of the microphone electronics, $\omega _{c2}=2 \pi f_{c2}$ is the cut-off frequencies of the signal processing electronics, $\omega _{c3}=2 \pi f_{c3}$ is the microphone cut-off frequency, $\omega _{c4}=2 \pi f_{c4}$ is the microphone geometric characteristic frequency, $\delta _{c3}$ and $\delta _{c4}$ are the damping factors.

In the analysis of our data, we tested all these system contribution possibilities, but some proved more relevant: $He_1(f)$ and flicker noise $A f^{-1}$. The A is a factor of noise amplitude. The acoustic influence of membrane Ha(f) and crosstalk noise are irrelevant when working at low frequencies (1kHz). When we use the fit with full electronic He(f), we always get a value less than unity for one of the frequencies. So, the $He_2(f)$ is not relevant, once we do not use a second amplifier system. In this way, we compare the results obtained for four cases: taking the linear frequency range of a microphone of the 80-1000 Hz; in the range of non-linear 40-1000 Hz range without calibration by using the $He_1(f)$ calibration in this last one and adding to this last one the flicker noise.

Figure 8 presents typical results¹ obtained from OPC technique for the PA signal amplitude of AISI 304 samples after polishing with 80 grit SiC paper.

Figure 8(a) shows the result for all samples after the procedure of the thickness reduction with SiC paper 80 grit. The circles are the experimental data, solid lines are fit by using the calibration with $He_1$ function, and the dashed lines are the PA signal expected for the sample by removing system contribution. This was done by taking the ratio of the curve fitted by the obtained $He_1$ function. We highlight the inflection region around 100 Hz, which is due to the change of predominant effect when TE effect becomes more evident over TD contribution [1,2,3]. The displacement of this region from lower frequencies with thickness increasing for the same thermal diffusivity is expected, but we can see that the region does not follow a linear behavior. Here, we see the first influence of the surface over the heat transfer and thermal diffusivity of the sample.

With the aim of exemplifying the influence of calibration on the fitted curve, we compare the best applicable calibration methods. In the Figure 8(b) for the linear region (80–1000 Hz) furnished by the microphone’s datasheet, Figure 8(c) for the complete range of 40 Hz to 1 kHz if the calibration is not used in the full range, Figure 8(d) in this same range by considering the $He_1$ RC systems' influence, and Figure 3(e) by adding to the last one the flicker noise ($A f^{-1}$). In terms of comparison, we also present in Figure 8(c) that the results of the calibration are not used in the full range.

Fig. 8

Photoacoustic signal in function of modulation frequency (f) after polishing with 80 grit SiC paper for (a) all samples (b) S-III fit without calibration in 80-1000 Hz range (c) S-III fit without calibration in 40-1000 Hz range, (d) S-III fit with $He_1$ calibration in 40-1000 Hz range, and (e) S-III fit with $He_1+A f^{-1}$ calibration in 40-1000 Hz range

The fitted curve tendency is to adjust the inflection region with the experimental data, which will be the main domain for determining the thermal diffusivity $\alpha _s$ and the relative amplitude $C_2$ of the TE contribution.

The fitted curve for the linear region of the microphone (Figure 8(b)) has a good agreement with experimental data for low frequencies (<200 Hz). However, there is a deviation that is accentuated with increasing frequency.

The use of the calibration with the electronic contribution $He_1$ (Figure 8(d)) improves the results for low frequencies and inflection region. Also, for the high frequency, the fitted curve approaches the experimental data but has a discrepancy. Now, in Figure 8(e), we have a good agreement for all frequencies.

From Figure 8(d) and (e), it is observed that the addition of the $He_1$ calibration and after the noise, corrects the failures of the non-use of the calibration in the fit (Figure 8(c)) as well as $\alpha _s$ and $fc_1$ [41] close to the expected value. However, we must also compare the value of $C_2$, which was much lower than expected.

The arithmetic mean and standard deviation for all results obtained from OPC are presented in the Table 3 for the S-I, Table 4 for the S-II, Table 5 for the S-III, Table 6 for the S-IV, Table 7 for the S-V, Table 8 for the S-VI, and Table 9 for the S-VII samples.

Table 3 Results obtained from XRD and OPC techniques for the S-I sample after each polishing procedure

Table 4 Results obtained from XRD and OPC techniques for the S-II sample after each polishing procedure

Table 5 Results obtained from XRD and OPC techniques for the S-III sample after each polishing procedure

Table 6 Results obtained from XRD and OPC techniques for the S-IV sample after each polishing procedure

Table 7 Results obtained from XRD and OPC techniques for the S-V sample after each polishing procedure

Table 8 Results obtained from XRD and OPC techniques for the S-VI sample after each polishing procedure

Table 9 Results obtained from XRD and OPC techniques for the S-VII sample after each polishing procedure

The adjustment of the data without calibration in the non-linear range (40-1000 Hz) may even return values similar to those obtained in the linear range (80-1000 Hz) in some results, but in some measurements, it provided lower than expected thermal diffusivity values. Thus, we use it only as a comparison between not using calibration and using the calibration functions $He_1$ and N(f).

The use of $He_1$ calibration makes the results close to the results obtained for the linear system range (80-1000 Hz) for the S-II, S-III, S-IV, and S-V samples. In the S-I results, there is an increase in the parameters after calibration, mainly the thermal diffusivity.

In our results, the best contribution of flicker noise addition is in the curve fit for higher frequencies. Also, for the S-II, S-III, S-IV, and S-V samples, in most cases, an improvement of the electronic resonant frequency $fc_1$ is obtained, increasing the value close to the reported by Markusvhev et.al. between 30 and 60 Hz. [12, 41]. However, the parameters obtained from the fit are not significantly improved for S-I, S-VI, and S-VII samples. Moreover, in some measures, the noise worsens the parameter values in comparison with those obtained by the fit with $He_1$.

Independent of calibration, except for S-I, the general behavior of the $C_1$ parameter is decreasing with the thickness increases. Being, the behavior, more evident for low $\alpha '/\gamma$ ratio, i.e., with one apparent lower contribution of the surface on the results. This is expected in the classical TD model once the temperature has decreased inside the sample.

For the $C_2$ parameter is observed, an approximately linear increases with the thickness until S-VI at grit SiC 1000. For S-VI with grit 2000 and all results of S-VII, the $C_2$ results are out of the general behavior. From the classical TE model, the $C_2$ parameter should be a fixed parameter, once the dependence with the thickness in amplitude, it was considered in simulation ($\propto l_s^{-3}$), as can be seen in Eq.6. This result can be due to the surface effect or another contribution in the TE model not being considered. The TE model is derived from the TD model and considers uniform heating over the sample. As a consequence, the temperature profile is analyzed as 1D heat propagation. Regarding calibration, in the range of 80 Hz-1 kHz, when it is not used, the $C_2$ has the higher values. If calibration $He_1$ is used, the $C_2$ decreases. With flicker noise addition ($He_1+A f^{-1}$), a more pronounced decrease of $C_2$ is obtained. This behavior occurs for all samples after all polishing stages, with an approximately linear increase.

The main physical parameter to be obtained by the OPC technique is thermal diffusivity. Only four times values that approached the theoretical value for AISI 304 were obtained in our samples, which were for samples S-I after polishing with sandpaper 1000 and 2000, S-II after polishing with sandpaper 2000, and S-III after polishing with sandpaper 80 in the adjustment with the calibration considering noise. In the latter, we consider that there may be some adjustment errors, especially regarding the use of noise.

For a complete view of the best model for our results, in Figure 9, we present the variation of thermal diffusivity $\alpha _s$ as a function of thickness and $\alpha '/\gamma$.

Figure 9(a) shows the behavior of $\alpha _s$ obtained from 80 to 1000 Hz, Figure 9(b) for 40-1000 Hz without calibration, Figure 9(c) in the 40-1000 Hz with $He_1$ calibration, and Figure 9(d) adding the flicker noise to the last one.

The behavior of results obtained from $He_1$ Figure 9(c) is more similar to the linear region of the measure system, Figure 9(a).

Fig. 9

Thermal diffusivity for main calibrations procedures

Therefore, observing the thermal diffusivity behavior in Figure 9, we exclude the possibility of instrumental error. A systematic error always has the same contribution. We may have random errors (environmental and electronic contributions) that vary the measurement system’s contribution but are within an acceptable range.

The thermal diffusivity measured for all samples varies greatly depending on thickness and $\alpha '/\gamma$. We cannot call the variation a system error because it is not linear, and we do not have an identical contribution in all measures.

Also, based on the results in the Figure 9, we see that the best behavior for the effective thermal diffusivity is found when using only the calibration with $He_1$, since in some cases the addition of noise significantly changed the $\alpha _s$ results, mainly for thick samples.