1 Introduction

Determining the thermal expansion performance of rigid solid materials with push-rod dilatometers is the mainstream measurement method. Although the precision of this comparative test method is less than that of the interference method (e.g., Michelson or Sophie interferometer, an absolute measurement method) [1], it is suitable for measuring the thermal expansion of most metals; one can expand the temperature range by using various push-rods and carriers, such as fused silica (− 180 °C to 900 °C) as well as alumina and isotropic graphite (1600 °C and 2500 °C) [2]. The interference method is suitable for low-expansion materials. The thermo-mechanical method is another common method with less precision than the push-rod dilatometers method and has certain requirements in terms of material rigidity [3]. The push-rod thermal dilatometer is a commercial device that is widely used in thermophysical property testing globally, especially for thermal expansion performance testing of metals. Thus, to obtain reliable thermal expansion coefficients and traceability of the measurement results, a validated measurement apparatus is necessary.

Verifying push-rod dilatometers with a certified reference material, is a relatively simple and easy to operate method, thus avoiding use of a tedious and difficult-to-operate temperature and length sensor. Unfortunately, only three types of thermal expansion coefficient reference materials are in the Comar library [4]: one is borosilicate glass (NIST SRM731), and the other two are alumina ceramics (NMIJ CRM5801a) and single-crystal silicon (NMIJ CRM5803a). These three reference materials are all inorganic (Fig. 1), and the thermal expansion coefficients do not exceed 10 × 10–6 K−1, which does not span the entire range of thermal expansion coefficients of metals (generally > 10 × 10–6 K−1). However, common reference materials such as pure copper (NIST SRM736), stainless steel (NIST SRM738), mono-crystalline silicon (NIST SRM732), pure tungsten (NIST SRM737), and sapphire (NIST SRM739) developed by NIST in 1976 [5,6,7,8,9,10] are in limited supply or out of stock.

Fig. 1
figure 1

Available reference materials and limited-supply reference materials

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The thermal expansion coefficient of metal materials is generally in the range of (10–20) × 10–6 K−1, and the operating temperature is not > 1000 K, in which many commercial push-rod dilatometers with specific components are satisfactory. However, common reference materials and the reference materials that are in limited supply are insufficient for the required range of thermal expansion coefficients and operating temperatures (Fig. 1). Therefore, the present work aims to select Pyros alloy as the raw material to develop a new certified reference material(CRM) that is suitable for metal materials with a certified temperature range of 100 °C to 1000 °C and a corresponding thermal expansion coefficient range of 12.7–16.8 (× 10–6 °C−1). Inverse Kelvin and inverse centigrade are both units of the thermal expansion coefficient; inverse centigrade was applied in the present study.

2 Experimental

2.1 Material Selection

Pyros alloy was selected as a raw material for thermal expansion coefficient measurements of metal materials [11]. Because it is an Ni-based superalloy, it has the advantages of stable chemical composition, uniform microstructure, and phase transformation temperature at > 1000 °C. Furthermore, it is stable, affords reproducible test results, and undergoes minimal oxidative weight gain. Because of these characteristics, Pyros alloy is a suitable candidate reference material for the thermal expansion coefficient, which has also been indicated in the literature.

2.2 Specimen Preparation

Pyros alloy round rods were subjected to the following heat treatment: heated to 1050 °C in vacuum, tempered after holding for 20 min, and then heated to 1050 °C; over 10 cycles to eliminate the influence of stress and ensure the uniformity of the microstructure. The metal round rods were naturally aged at least 6 mo before further processing [12, 13].

Before processing of the Pyros alloy round rods, a preliminary inspection was conducted. First, a cross section of each sample was obtained; the low-magnification structure as well as grain size was examined after grinding, polishing, and corrosion. No defects such as cracks, shrinkage cavities, bubbles, and central porosity were evident (Fig. 2); the grain size was grade 2.5. Second, the chemical composition and segregation were evaluated. Samples with a height of ≥ 25 mm were obtained from the top, middle, and bottom of the round rod (Fig. 3). The average value of the main element content of the three parts was analyzed by spark atomic emission spectrometry (Table 1); the range of the main element was less than the reproducibility limit (generally expressed as r) [14]. Therefore, the chemical composition was not segregated. Finally, the thermal expansion coefficient was evaluated. The samples (three parts, as previously mentioned) were processed into three φ 6-mm × 25-mm sample bars (Fig. 3), and then a sample of each part was randomly selected for measurement of the coefficient of thermal expansion in accordance with Sect. 2.3. The results in Fig. 4 indicate that there were some deviations in the temperature range of 100 °C–300 °C, mainly caused by the method (e.g., fitting between the sample and push rod and control of the initial temperature), but these deviations were within an acceptable range of the initial inspection. Measurements at higher temperatures demonstrated consistency of the three samples within the three parts.

Fig. 2
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Macrostructure and grain morphology

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Fig. 3
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Cutting positions of candidate reference material

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Table 1 Content and segregation of main elements
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Fig. 4
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Coefficient of thermal expansion of head–middle–tail of candidate reference materials

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After completing the three preliminary inspections, 120 sample sticks with a diameter of 6 mm and length of 25 mm (Fig. 3) were processed from the Pyros alloy round rods. The upper and lower end faces of the sample were smooth and straight, to facilitate a good fit with the push-rod dilatometer. Of the 120 sticks, 36 were randomly selected to determine the thermal expansion coefficient; in addition, 11 sticks were randomly selected for investigations of the homogeneity and stability.

2.3 Measurements

The thermal expansion performance is generally expressed by a linear thermal expansion, △L/L0; average coefficient of linear thermal expansion, αm; and instantaneous coefficient thermal expansion, αT. The αm is most widely used in material thermal expansion performance characterization, and refers to the ratio between the expansion and temperature that is causing the expansion, and represents the relative change of the sample length corresponding to a temperature change of 1 °C as calculated with the following equation

$$\alpha_{{\text{m}}} = \frac{1}{{L_{0} }}\frac{\Delta L}{{\Delta {\text{T}}}}.$$
(1)

The specific experimental conditions were as follows: set the initial temperature at 30 °C and the test range of (30–1000) °C; and then measure the thermal expansion coefficients of 100 °C, 200 °C, 300 °C, 400 °C, 500 °C, 600 °C, 700 °C, 800 °C, 900 °C, and 1000 °C. Although the initial temperature was generally 20 °C, room temperature in some laboratories is not always controlled at 20 °C, especially in laboratories where metal materials are tested for high-temperature thermal expansion. To ensure consistency of the initial temperature, the initial temperature was set at 30 °C, whereas the heating rate was set at 5 °C/min under N2 or another inert atmosphere. The terminal temperature of the test was set at 1050 °C, which is almost the service-limit temperature of general metal materials.

The equipment was selected from major international commercial equipment manufacturers such as NETZSCH, LINSEIS, and TA. A reference material was also used to verify the performance of the three types of equipment.

2.4 Measurements of Certified Value

To ensure the accuracy and reliable determination of the reference materials, nine laboratories (Table 2) with sufficient analytical experience and ability, which have been recognized by the laboratory capacity in China, were invited to cooperate in determining the thermal expansion coefficient. Four samples were distributed to each laboratory, and all of the laboratory measurement results were summarized for statistical evaluation in accordance with ISO Guide 35 [15].

Table 2 Nine laboratory partners and apparatuses
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Inhomogeneity is an important factor that affects the characteristic value of the thermal expansion coefficient of Pyros alloy, and it is also one of the critical parameters in determining whether the reference material meets expectations. Distinct from a reference material that can be sampled and undergo between-bottle and within-bottle evaluation (e.g., reference materials of chemical composition), or a reference material that can undergo only a one-time destructive test (e.g., reference material of Metallic charpy v-notch impact), each independent packaging unit of Pyros alloy has only one sample, which has the characteristics of repeatability test. Therefore, the following scheme was designed. In group A, 11 samples were randomly selected from all 120 sample sticks, and then each sample was tested in accordance with Sect. 2.3, such that 11 data points were obtained for each temperature point. In group B, another sample was randomly selected from the 11 sample sticks to repeat the test 11 × in accordance with Sect. 2.3; 11 data points were also obtained at each temperature point. One-way analysis of variance (ANOVA) was used to verify the inhomogeneity of groups A and B. This verification method was developed based on the following considerations. First, the thermal expansion coefficient is an inherent physical property of the material, and is mainly related to the chemical composition and structure of the material. Both groups were homogeneous in accordance with Sect. 2.1, but although the bottom cross sections exhibited some differences, the differences were small. Second, the influence of the measurement method can be evaluated. The reproducibility of the measurement method affects the measurement results of sample inhomogeneity; which corresponds to the temperature delay at low temperature, fitting between the sample and push rod, heating rate, sample length measurement, and other factors.

Group B was selected for stability evaluation, and trend analysis was used to verify whether the trend was significant. Equation 1 indicates that the coefficient of thermal expansion corresponded to the rate of the temperature change and the sample elongation, such that the length and temperature can be corrected with a properly calibrated gauge block and R-type thermocouple, respectively, to ensure traceability.

2.5 Uncertainty Evaluation

In accordance with Guide 35, the standard uncertainty is generally calculated as a combination of four uncertainties: characterization process uchar, inhomogeneity ubb, short-term stability usts, and long-term stability ults. The uncertainty of each component was calculated separately and then synthesized in accordance with the following equation.

$$u_{{{\text{CRM}}}} = \sqrt {u_{{{\text{char}}}}^{2} + u_{{{\text{bb}}}}^{2} + u_{{{\text{sts}}}}^{2} + u_{{{\text{lts}}}}^{2} }$$
(2)

3 Results and Discussion

3.1 Certified Value

Figure 5 shows the thermal expansion coefficient data of all nine laboratories; three types of verification were performed in accordance with ISO Guide 35. First, the Shapiro–Wilk method was used to verify the normality; all data conformed to a normal distribution with a statistical value W of all temperature points greater than the critical value W0.05. Second, the Grubbs method was used to verify the outliers. No outliers were found; the Gmax and Gmin values of all the temperature points were less than the critical value G0.05. Third, using the Cochran method for verifying equal precision, all of the laboratories reported the same precision when the C values of all the temperature points were less than the critical value C0.05.

Fig. 5
figure 5

Data verification by the Shapiro–Wilk, Grubbs, and Cochran methods

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The thermal expansion coefficient of 100°C exhibited some imprecision, a range of 10% (Fig. 6), which was caused by the different temperature measurement points of the equipment from different manufacturers and the lag of the temperature measurement. As the temperature increased, the data of the high-temperature point tended to be consistent; also indicated in the evaluation results of the uncertainty at each temperature point in Table 3. Nevertheless, all data were retained after passing the verification (Fig. 5), which objectively indicates the reproducibility of the method and the performance of the measurement results from different manufacturers at low temperatures.

Fig. 6
figure 6

Difference between fitted result (formula curve), certified values, and measured values

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Table 3 Certified values and uncertainties
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The average value of the nine laboratories was used as the certified value, and the corresponding thermal expansion coefficient of each temperature point was obtained (Table 2). To obtain the thermal expansion coefficient of Pyros alloy over the entire temperature range, the certification value was used as the ordinate, and the temperature was used as the ordinate to perform a 5-degree polynomial linear fitting, thus obtaining the following fitting equation.

$$\alpha \left( {10^{ - 6} {{^{\circ}} \text{C}}^{ - 1} } \right) = 10.3533 + 0.03183t - 9.8814 \times 10^{ - 5} t^{2} + 1.6627 \times 10^{ - 7} t^{3} - 1.3514 \times 10^{ - 10} t^{4} + 4.2308 \times 10^{ - 14} t^{5} ,$$
(3)

where t is the temperature. Figure 6 shows the measured values of each laboratory, the certified values of each temperature point, and the fitted curves; which exhibited good consistency.

3.2 Inhomogeneity Evaluation

The inhomogeneity was investigated in terms of the variation in the thermal expansion coefficient measured from 30 to 1050 °C. Although low temperatures such as 100 °C or 200 °C are conducive to an inhomogeneity evaluation over the entire temperature range [16], the deviation of the thermal expansion coefficient at low temperatures was usually larger than that at high temperatures (Fig. 6). However, the impact of the measurement process and method reproducibility requires assessment at all of the temperature points, which necessitates a more comprehensive evaluation in terms of the inherent inhomogeneity of the reference material and the reproducibility of the push-rod method.

Figures 7 and 8 indicate the data selected for the inhomogeneity evaluation (groups A and B, respectively, as mentioned in Sect. 2.4); the thermal expansion coefficient of each temperature point indicates a small deviation.

Fig. 7
figure 7

Test of 11 samples, group A

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Fig. 8
figure 8

Reproduction 11 × for one sample, group B

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Table 3 indicates that the statistical value of the thermal expansion coefficient measurement result F at all of the temperature points was less than the critical value F (10, 10) 0.05 (= 2.98), indicating that the results of 11 repeated tests of the one sample were not significantly different from those of the 11 samples; the thermal expansion coefficient of the Pyros alloy reference material exhibited good consistency.

In accordance with ISO guideline 35, when F > 1, the variance of the sample inhomogeneity can be calculated as follows:

$$s_{{{\text{bb}}}} = u_{{{\text{bb}}}} = \sqrt {\frac{{{\text{MS}}_{{{\text{among}}}} - {\text{MS}}_{{{\text{within}}}} }}{{n_{{0}} }}}$$
(4)

When F < 1, the reproducibility of the test method is insufficient; the measurement reproducibility variance used in the study of between-bottle homogeneity was as follows.

$$s_{{{\text{bb}}}} = u_{{{\text{bb}}}} = \sqrt {\frac{{MS_{{{\text{within}}}} }}{n}} \sqrt[4]{{\frac{2}{{v_{{{\text{MS}}_{{{\text{within}}}} }} }}}}$$
(5)

where \(MS_{{{\text{amon}}g}}\) is the mean square of the reproducibility in terms of between-bottle inhomogeneity; \(\bf {\text{MS}}_{{{\text{within}}}}\) is the mean square of the reproducibility in terms of within-bottle inhomogeneity; \(v_{{{\text{MS}}_{{{\text{within}}}} }}\) is the degree of freedom in terms of between-bottle inhomogeneity; \(s_{{{\text{bb}}}}\) is the standard deviation of between-bottle inhomogeneity; \(u_{{{\text{bb}}}}\) is the uncertainty caused by between- bottle inhomogeneity; n is the number of observations; and n0 is the number of effective group units, equal to n when there is no missing data.\(s_{{{\text{bb}}}}\) at 500 °C and 600 °C is calculated with Eq. (4), and other temperature points were calculated with Eq. (3) (Table 4). Although calculated with Eq. (4), it does not indicate poor repeatability of the test method, because it is the same sample, method, and test process; without phase transformation or chemical reaction.

Table 4 Variance analysis date for inhomogeneity evaluation (10–6 °C−1)
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3.3 Stability Evaluation

The Pyros alloy material might be used multiple times at 30 °C to 1000 °C as a reference material for push-rod dilatometer validation; thus, the stability of the certified values was evaluated after multiple thermal cycles by using group B data as in Sect. 2.4. The F test was used to verify whether the trend was significant. Because there is no strict dynamic mechanism, the basic model of the stability study can be expressed as follows

$$Y = \beta_{0} + \beta_{1} X + \varepsilon ,$$
(6)

where β0 and β1 are the regression coefficients; ε is the random error component; b0 and b1 are estimates of β0 and β1, respectively; and s (b1) is the standard deviation of b1. When the absolute value of b1 is not > t0.95,10 · s (b1)(t0.95,10 = 2.23), the slope is not significant. The absolute value b1 of all the temperature points was < t0.95,10 · s (b1) (Table 5), indicating that no instability was observed.

Table 5 Stability Evaluation
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The coefficient of thermal expansion of Metallic materials is related to the chemical composition and structural organization. In recent decades of research, Pyros alloy has been proved to be stable, and its chemical composition and structure has not changed, so the thermal expansion coefficient is also sufficiently stable.

3.4 Uncertainty Evaluation

The standard uncertainty of the reference material was calculated in accordance with Eq. (2), in which the uncertainty usts and ults caused by short-term and long-term stability (respectively) can be ignored as in Sect. 3.3. The uncertainty ubb of the inhomogeneity (i.e., the standard deviation sbb of the inhomogeneity) is calculated with Eqs. (3) and (4), and Table 3 shows the statistical results of each temperature point. The uncertainty of the certified value was calculated as follows.

$$u_{{{\text{char}}}} = \frac{{s_{{{\text{char}}}}^{{}} }}{\sqrt N }$$
(7)

where schar is the standard deviation of a single measurement of the certified value, and N is the number of data groups. The combined standard uncertainty of each temperature point was obtained; at 95% confidence probability, assuming an inclusion factor of 2, the expanded uncertainty was also obtained (Table 2). In addition, with Eq. (3), the relative expanded uncertainty of the entire temperature range was obtained. When the temperature was ≤ 100 °C, the relative expanded uncertainty was 4%; and when the temperature was > 100 °C, the relative expanded uncertainty was 2%. The relative uncertainty was greater at low temperature than at high temperature; mainly caused by, e.g., the method, temperature increase, and hysteresis.

4 Conclusions

The developed reference material (Pyros alloy) for the thermal expansion coefficient which is applicable for verifying push-rod dilatometers and thermal expansion performance by thermo-mechanical methods, Over certified temperatures from 100 to 1000 °C and a range of the thermal expansion coefficients 12.7–16.8 (× 10–6 °C−1), spans the range of thermal expansion coefficients of common metals, especially iron and steel alloys.

The certified value of the thermal expansion coefficient was determined by several laboratories. Two groups of different inhomogeneity verification tests were designed, and the homogeneity of the materials was tested by one-way ANOVA. The stability was verified by no indication of instability, and the uncertainty of the standard was evaluated.