Temperature-dependent modeling and enhanced measurement of ultrasonic velocity in LAS glass–ceramics

Abstract

Li₂O–Al₂O₃–SiO₂ (LAS) glass–ceramics are widely used in many fields owing to their ultra-low coefficient of thermal expansion (CTE). The ultrasonic velocity method shows potential as an effective method for the nondestructive measurement of the CTE and CTE homogeneity of large LAS glass–ceramics. To further improve the precision of ultrasonic velocity (c_L) measurements of LAS glass–ceramics and reveal the effect of the temperature (T) on c_L, an improved correlation method to rapidly determine the time of flight (TOF) in LAS glass–ceramics is proposed. The proposed method significantly reduces the calculation time by a factor of 100 for LAS glass–ceramics with a thickness of 50 mm. Furthermore, the TOF results obtained by this method were compared with those obtained by the commonly used peak and zero-crossing methods. The proposed correlation method showed better TOF repeatability precision (0.69 ns). Considering the effect of T on the precision of the c_L measurement, a c_L–T exponential model was developed based on the theoretical c_L–T analytical model. The accuracy of the c_L–T exponential model was experimentally verified. The root-mean-square error was within 0.30 m/s, which shows that the developed exponential model can reliably predict the c_L values of LAS glass–ceramics at different temperatures. The two meaningful findings mentioned above provide methodological and modeling support for high-precision nondestructive CTE testing of large LAS glass–ceramics at specific temperatures.

Introduction

Li₂O–Al₂O₃–SiO₂ (LAS) glass–ceramics are inorganic, non-porous lithium–aluminum–silicon oxide glass–ceramics materials with near-zero ultra-low coefficient of thermal expansion (CTE) values (on the order of 10⁻⁷/°C to10⁻⁹/°C) in the temperature range of (0 °C, 50 °C)¹. Therefore, they have been widely used in the development of mirrors for large-diameter precision optical systems^2,3, space optical remote sensing imaging ⁴, extreme ultraviolet lithography (EUVL) systems^5,6,7, etc. However, the homogeneity of the glass material has a significant effect on the CTE homogeneity of the final LAS glass–ceramics product. In the process of material preparation, the composition, crystalline-phase type, and quantity together affect the magnitude of the CTE value and its distribution homogeneity^8,9,10. When the homogeneity of the CTE surpasses a certain value, the quality of the molded substrate and its imaging stability can be significantly affected, thus limiting its use^11,12,13. Therefore, it is necessary to understand the CTE properties of LAS glass–ceramics to better control the manufacturing process for producing the highest quality LAS blocks.

At present, the only CTE measurement method that meets the requirements of this uniformity characterization is the pushing-rod method. From the continuous development of the key measurement module of the linear variable differential transformer¹⁴, optical interferometer¹¹, and linear incremental encoder¹⁵, the measurement precision has been continuously improved. Nevertheless, this method requires destructive sampling, which is both time-consuming and expensive, and it cannot achieve a comprehensive, nondestructive examination of the CTE for large materials. However, by determining the ultrasonic velocity of the test object, the internal and surface morphologies can be measured and characterized^16,17,18, and thus ultrasonic velocity measurement is a potentially effective method for realizing the evaluation of the CTE and CTE homogeneity for large LAS glass–ceramics^19,20,21.

Because the mechanical properties of materials change under different temperature conditions, resulting in changes in the ultrasonic velocity of the material, the change in the ultrasonic velocity of materials at different temperatures can be studied for high-precision characterization of their mechanical characteristics at specific temperatures. In 1980, Salama et al. experimentally investigated the effect of the compressive elastic stress on the temperature dependence of the longitudinal and shear ultrasonic velocities of commercial aluminum and copper²². In 2010, Kobori et al. experimentally studied the effects of the stress and temperature on the ultrasonic velocity to establish an acoustoelastic law applicable to thermo-elastic stress analysis²³. In 2016, Hu et al. experimentally measured and characterized Poisson’s ratio, the elastic modulus, and the shear modulus of two metal alloy materials in the temperature range of − 20 to 70 °C using the ultrasonic velocity method, providing the temperature stability of the elastic properties of the materials for engineering design applications²⁴.

In summary, the critical problem for investigating the ultrasonic velocity of materials at different temperatures and realizing the high-precision characterization of the corresponding properties is high-precision measurement of the ultrasonic velocity. In this study, we propose a rapid and high-precision ultrasonic velocity measurement method. Based on the theoretical relationship between the ultrasonic velocity c_L and temperature T, combined with experimental tests, the c_L–T exponential model of LAS glass–ceramics is derived. The exponential model can predict the ultrasonic velocity in LAS glass–ceramics at different temperatures and correct the ultrasonic velocity to that at a specific temperature. Furthermore, it can provide methodological and modeling support for high-precision nondestructive testing of the CTE of large LAS glass–ceramics at specific temperatures.

Principle and method

Principle of the ultrasonic velocity measurement

Currently, there are a wide variety of devices used to determine the ultrasonic velocity, most of which are based on the pulse reflection method^25,26,27. Typically, this method involves a time-domain analysis based on the ultrasonic time-of-flight (TOF), where the ultrasonic velocity is obtained from the ratio of the material thickness to the ultrasonic TOF^19,28. On this basis, we constructed a fully integrated, high-precision system for measuring the ultrasonic velocity in LAS glass–ceramics based on the immersion pulse reflection method.

A schematic diagram of the immersion pulse reflection method is shown in Fig. 1. The ultrasonic transducer generates a vertically incident pulse wave on the LAS glass–ceramics via the water coupling medium. This wave reflects and transmits on the upper surface of the LAS glass–ceramics, and the transducer receives the reflected wave, which is called the surface reflected wave (S_F). The other part of the wave undergoes reflection and transmission, while the wave generated by transmission is received by the transducer to form the first bottom wave (B₁). Similarly, the other reflected wave is received by the transducer after bottom reflection and transmission, forming the secondary bottom wave (B₂).

Fig. 1

Schematic of the immersion pulse reflection method: (a) transmission process and (b) transmission time of the ultrasonic wave.

In Fig. 1a, H is the distance from the transducer end face to the upper surface of the LAS glass–ceramics, and d is the thickness of the LAS glass–ceramics. According to the principle of the ultrasonic transmission in the material, the time from the initial pulse to S_F is t_H, the interval time between S_F and B₁ is t₁, and the interval time between B₁ and B₂ is t₂, which can be expressed as

$$\begin{gathered} t_{{\text{H}}} { = }\frac{{{2}H}}{{c_{{\text{L}}}^{{{\text{water}}}} }}, \hfill \\ t_{1} = t_{2} { = }\frac{{{2}d}}{{c_{{\text{L}}}^{{{\text{sample}}}} }}. \hfill \\ \end{gathered}$$

(1)

Therefore, the c_L value of the tested sample can be obtained from Eq. (1):

$${\text{c}}_{\text{L}}^{\text{sample}}\text{=}\frac{{2}{\text{d}}}{{\text{t}}_{1}}\text{=}\frac{{2}{\text{d}}}{{\text{t}}_{2}}.$$

(2)

From Eq. (2), c_L of the tested sample can be determined by estimating t₁ or t₂ of the ultrasonic echo signal of the LAS glass–ceramics when the thickness d is known.

Method of the ultrasonic velocity measurement

LAS glass–ceramics consist of crystalline and glassy phases and have a complex structure, but they still have a highly isotropic structure and excellent homogeneity. Thus, an ultrasonic wave propagates in LAS glass–ceramics with small attenuation. The ultrasonic pulse-receiver receives the secondary bottom wave B₂ very well, as shown in Fig. 2, and the B₁ and B₂ signals of the LAS glass–ceramics exhibit significant similarity. Therefore, the correlation method is used to estimate the time interval between the signals B₁ and B₂, which is referred to as the ultrasonic TOF. Denoting the signal B₁ as s₁(t) and the signal B₂ as s₂(t), the correlation coefficients of the two echo signals can then be expressed as

$$R\text{=}\frac{\sum {\text{s}}_{1}\left({\text{i}}\right){\text{s}}_{2}\left({\text{i}}\right)-\sum {\text{s}}_{1}\text{(}{\text{i}}\text{)}\sum{\text{s}}_{2}\text{(}{\text{i}}\text{)/}{\text{n}}}{{\left\{\text{[}\sum {\text{s}}_{1}^{2}\left({\text{i}}\right)-\frac{{\left(\sum {\text{s}}_{1}\left({\text{i}}\right)\right)}^{2}}{\text{n}}\text{][}\sum {\text{s}}_{2}^{2}\left({\text{i}}\right)-\frac{{\text{(}\sum {\text{s}}_{2}\text{(}{\text{i}}\text{)}\text{)}}^{2}}{\text{n}}\text{]}\right\}}^{{1}/{2}}}\text{,}{\text{i}}\text{=1,2,} \ldots {\text{n}}\text{.}$$

(3)

where n is the computed length of the signal array and i is the location inside the signal array.

Fig. 2

Echo signals of the LAS glass–ceramics.

Following the correlation coefficient calculation principle, the entire ultrasonic signal, which includes B₁ and B₂, is denoted s(i). The segment signal between point 1 and point 2 in signal B₁ is extracted as s₁(i), and the segment signal between point 3 and point 4 in signal B₂ is extracted as s₂(i). The correlation coefficients of s₁(i) and s₂(i) are then determined using Eq. (3), and the data point corresponding to the highest value of the correlation coefficient, denoted m, is found. The position of the data point m can be translated into the corresponding time t_c with the aid of the sampling rate f_S:

$${\text{t}}_{\text{c}}\text{=}\frac{\text{m}}{{\text{f}}_{\text{S}}}.$$

(4)

There are x data points from point 2 to point 3, and the corresponding time t_x can be expressed as

$${\text{t}}_{x}\text{=}\frac{\text{x}}{{\text{f}}_{\text{S}}}.$$

(5)

Thus, the ultrasonic TOF in the tested sample t can be determined by

$${\text{t}}\text{=}{\text{t}}_{\text{c}}+{\text{t}}_{\text{x}}.$$

(6)

The previously used correlation calculation method intercepts a segment ultrasonic signal containing the time interval in the time domain signal s(t), splits it, and labels the n discrete points of s₁(t) and s₂(t), which are denoted as two discrete data arrays s₁(1)–s₁(n) with s₂(1 + m)–s₂(n + m)²⁹. The correlation between the two discrete data arrays is calculated according to Eq. (3) to obtain an array of correlation coefficients. The highest correlation coefficient value (R_max) corresponds to the horizontal coordinate sampling point m_max, that is, the time point when the correlation between B₁ and B₂ is the highest. From the sampling rate of the ultrasonic signal, it is clear that the correlation calculation involves at least tens of thousands of data points. In contrast, the number of data points used for the correlation calculation in the computational method proposed in this work can be reduced to a few hundred. The central processing unit was a 12th generation Intel Core i5 processor (i5-12400), the memory size was 16 GB, and the computer system was a 64-bit system. For a 50 mm sample, the correlation method proposed in this paper improves the calculation time from 5.6 s for the previously used method to 0.05 s³⁰. When the thickness increases, the calculation time advantage will be more obvious. According to the proposed correlation method, the correlation coefficient (R) between B₁ and B₂ is 0.99871, as shown in Fig. 3, indicating that the correlation is very significant, which verifies the feasibility of determining the TOF in LAS glass–ceramics by calculating the correlation coefficients from the signals B₁ and B₂.

Fig. 3

Calculation results of the correlation coefficients of B₁ and B₂.

Theoretical relationship between the ultrasonic velocity and temperature

Based on the propagation of the ultrasonic wave in an infinite isotropic solid, the relationship between the elastic modulus of the material E, Poisson’s ratio υ, the ultrasonic longitudinal wave velocity c_L, and the density ρ can be deduced. For LAS glass–ceramics, the typical characteristic parameters, as reported in the product manual from Schott, are $\rho$ = 2.53 g/cm³, E = 90.3 GPa, and $\upsilon$ = 0.24, and the resulting theoretical ultrasonic velocity is c_L = 6486 m/s. It is worth noting that a change in the temperature does not cause changes in $\rho$ and $\upsilon$ of the LAS glass–ceramics, and the difference in c_L is mainly determined by the difference in E. Therefore, the effect of the temperature on the ultrasonic velocity essentially reflects the effect of the temperature on E.

For isotropic solids, E can be characterized with the aid of physical equations describing the relationship between the stress and strain. From the microscopic point of view of the analysis, it is known that there is the following relationship³⁰:

$${\text{E = E}}_{0} {\text{(1 + }}\eta {\text{T)}},$$

(7)

where η is the temperature coefficient of E, which is used to measure the change in E due to a change in the temperature of the material.

For most materials, such as oxide glasses, ceramics, and metals, E exhibits a negative temperature dependence, that is, a negative value of η^31,32, while for glassy materials, such as ultra-low expansion glass, E exhibits an anomalous positive temperature dependence, that is, a positive value of η³³. Owing to the similarity of the physical properties between LAS glass–ceramics and ultra-low expansion glass, it is inferred that LAS glass–ceramics also have positive η values. According to Eq. (7), it can be deduced that c_L of LAS glass–ceramics also increases with increasing temperature. Further, the mathematical relationship between c_L and T is known for small temperature ranges:

$${\text{c}}_{{\text{L}}}^{{\text{T}}} {\text{ }} = {\text{ c}}\prime {\text{e}}^{{\alpha _{{\text{L}}} {\text{T}}}} ,$$

(8)

where α_L is the temperature coefficient of c_L, and ${\text{c}}_{\text{L}}^{\text{T}}$ and ${\text{c}}_{\text{L}}^{{\text{T}}_{0}}$ are the c_L values of the material at temperatures T and T₀, respectively. Although there are differences in α_L at different temperature points, it is possible to use the c_L–T data over a small temperature range to fit a solution to the average α_L over that temperature range. Equation (8) shows that the relationship between c_L and T is theoretically exponential for a certain temperature range.

Construction of the experimental system

Sample preparation and ultrasonic velocity measurement system

The selected object was an ultra-low expansion LAS glass–ceramics supplied by CDGM Co., Ltd., China. The typical mechanical parameter values of this material are ρ = 2.54 g/cm³, E = 90.67 GPa, and ν = 0.246, which result in a theoretical ultrasonic velocity is c_L = 6520.9 m/s. This provides a theoretical reference for the rationality of estimating the ultrasonic velocity measurement values of subsequent samples. The LAS glass–ceramics used in this study was prepared using the melt method. The preparation process mainly includes two parts: the preparation of the base glass and the heat treatment. First, the base glass of the Li₂O–Al₂O₃–SiO₂ system was prepared. Then, the base glass was heated from room temperature to the nucleation temperature of 695 °C at a rate of 20 °C/h and held for 20 h. Subsequently, it was slowly heated to the crystallization temperature of 790 °C at a rate of 10 °C/h and maintained for 160 h. After that, it was cooled down to 120 °C at a rate of 30 °C/h and then freely cooled to room temperature. The final product is a glass–ceramics with β-quartz solid solution as the dominant crystalline phase.

To determine the phase composition of the LAS glass–ceramics, we used a PANalytical Xpert3 Powder X-ray diffractometer with a Cu target to test the XRD patterns of three LAS glass–ceramics samples prepared by the above process. The scanning range was 2θ = 10°–90° at a speed of 1°/min. Figure 4 shows the XRD patterns of the LAS glass–ceramics from CDGM Co., Ltd. The prominent peaks at 2θ ≈ 19.97°, 25.87°, 48.65°, and 66.33° correspond well to the standard β-quartz solid solution peaks (PDF Number: 31-0707), indicating it is the primary crystalline phase.

Fig. 4

XRD patterns of the LAS glass–ceramics.

We prepared the test samples in a cylindrical shape with a cross-sectional area (25 mm) larger than that of the ultrasonic transducer, taking into account the boundary effect of ultrasonic wave propagation. To avoid the detrimental effect of the ultrasonic wave scattering attenuation at the interface of the sample, the two end surfaces of the test sample should be flat and parallel to each other. Five cylindrical glass samples were cut with an equal thickness (${50}_{-\text{0.3}}^{0}$ mm). All five cylindrical glass samples were cut from a single monolithic block to ensure consistency in material properties. Then the samples were finely ground and polished to achieve a flatness of 0.5λ and parallelism of 20 µm (Fig. 5). The five samples are numbered 1–5 from left to right.

Fig. 5

Photograph of the prepared LAS glass–ceramics samples.

To eliminate the influence of residual thermal stress on c_L, all samples underwent an annealing process. The annealing procedure involved heating the samples from room temperature to 800 °C at a rate of 30 °C/h, holding them at 800 °C for 50 h, and then cooling them from 800 °C to room temperature at a rate of 25 °C/h. This selected annealing temperature is slightly higher than the crystallization temperature (790 °C) of the LAS glass–ceramics sample, and does not exceed the temperature of 810 °C, which can change the phase composition and physical properties. The stress values were measured using the optical polarization method and were all less than 1 nm/cm (Table 1). The CTE values of the five cylindrical glass samples were determined experimentally determined using an L75 laser dilatometer. These values represent the mean CTE in the range near room temperature (0 °C, 50 °C). The CTE was calculated using the following formula:

$${\alpha }_{\text{m}}=\frac{\Delta L}{{L}_{0}}\frac{1}{\Delta T},$$

(9)

where ΔL denotes the change in length of the tested sample corresponding to temperature changes from 0 to 50 °C, L₀ represents the original length of the tested sample at room temperature (20 °C), and ΔT is the change in temperature.

Table 1 Important parameters of the LAS glass–ceramics samples.

A schematic diagram of the experimental system for measuring the ultrasonic velocity is shown in Fig. 6. The experimental system was composed of an ultrasonic pulse-receiver, an ultrasonic wave transducer, a data acquisition card, a temperature control box, a flume, a three-axis automatic scanning inspection platform, a personal computer, the sample, and support platforms. We used an ultrasonic high-frequency ultrasonic pulse-receiver (JSR, DPR300, Pittsford, NY, USA) to excite the ultrasonic transducer and receive the signals reflected from the bottom surface of the test sample. Taking into account the diameter and thickness of the test samples, we used a bandwidth-unfocused longitudinal wave immersion transducer (OLYMPUS, V326-SU, Waltham, MA, USA) with a diameter of 10 mm and a center frequency of 5 MHz. This transducer is self-generating and self-receiving, enabling the transmission and reception of ultrasonic wave signals, as needed. Combined with a data acquisition card (SPECTRUM, M4i.2210- × 8, Grosshansdorf, Germany) that supports external triggering, all of the signals are captured by 2.5 million discrete data points at a sampling rate of 1.25 GHz and transmitted to a computer, allowing real-time acquisition and processing of the signals. The ultrasonic velocity experimental measurement system uses the external circulation of the water bath method to achieve constant temperature control, and the temperature control accuracy is less than 0.05 °C.

Fig. 6

Schematic diagram of the experimental system used to measure the ultrasonic velocity.

Determination of the experimental parameters

The water distance (H) in the ultrasonic water immersion test is the distance from the end face of the transducer to the upper surface of the test sample. It affects the strength of the echo signal received by the transducer, as well as the location of the secondary surface wave, so it is necessary to analyze and optimize the water distance.

In general, the principles for determining the water distance are as follows: (1) consider the thickness range of the test samples and the transducer’s acoustic range amplitude profile to ensure that the acoustic range at the highest acoustic pressure is consistent with the necessary detection range and (2) ensure that the secondary surface wave on the time baseline is located beyond the maximum depth (h) of the material to be tested. Therefore, H should satisfy

$${\text{H}}{>}\frac{{\text{c}}_{\text{W}}}{{\text{c}}_{\text{sample}}}{\text{h}},$$

(10)

where H is the water distance, ${\text{c}}_{\text{W}}$ is the ultrasonic velocity in water, and ${\text{c}}_{\text{sample}}$ is the ultrasonic velocity of the sample.

In this work, we used the immersion pulse reflection method to measure the longitudinal wave velocity of the test sample, and we only needed to pay attention to the time interval between the echo signals on the upper and lower surfaces of the test sample. Therefore, we set the maximum depth of the test material to the thickness of the test sample (i.e., h = d). Based on Eq. (10), the water distance of the tested LAS glass–ceramics was H > 11.43 mm. Therefore, to determine the optimal parameters, sample 1# was selected and four different water distance values were set at 5 mm intervals for echo signal detection. The ultrasonic water signals at the different H values are shown in Fig. 7.

Fig. 7

Ultrasonic echo signals at different H values.

Then the water distance was 15–25 mm, with increasing H, the secondary surface wave (S_F2) of sample 1# moved closer to B₂, which interferes with B₂ and makes B₂ difficult to distinguish. However, when the water distance reached 30 mm, B₂ was well separated from the secondary surface wave S_F2, making B₁ and B₂ easily recognizable. As the water distance increases, the loss of acoustic energy decreases the amplitude of the signals at B₁ and B₂, reducing the detection sensitivity at that location. According to the above observations, H = 30 mm was determined to be the optimal water distance.

The amplitude of the echo signal is determined by the gain set by the ultrasonic pulse-receiver, which also needs to be selected to make B₁ and B₂ distinguishable. The gain parameter was set to 32, 34, 36, or 38 dB depending on the waveform and amplitude attenuation of the B₁ and B₂ signals. When the gain was 32 dB, the amplitudes of the B₁ and B₂ signals were relatively low (Fig. 8). When the gain was gradually increased, the amplitudes of the B₁ and B₂ signals gradually increased. When the gain was 38 dB, the B₁ signal was distorted, so the gain did not need to be further increased. Taking into account the detection sensitivity of the signal, the optimal gain was determined to be 36 dB.

Fig. 8

Ultrasonic echo signals at different gain values.

TOF estimation methods

After optimizing the water distance and gain, the estimation method of the TOF was determined. According to Eq. (2), high-accuracy measurement of the ultrasonic velocity depends on high-accuracy calculation of the TOF. The convenient correlation method proposed in this study was compared with the typical peak and zero-crossing methods for TOF estimation. We repeated the TOF measurements 20 times at 25 °C for sample 1#. A comparison of the TOF results from the three methods is shown in Fig. 9.

Fig. 9

Comparison of TOF results obtained by different methods.

The TOF results estimated by the correlation method had PV (peak to valley) = 0.0026 µs and repeatability σ = 0.00069 µs, both of which were significantly better than those of the peak method (PV = 0.0072 µs and σ = 0.00183 µs) and zero-crossing method (PV = 0.0072 µs and σ = 0.00158 µs) (Fig. 9). This is because the correlation approach begins with the overall waveform of the echo signals, which is insensitive to the amplitude and width of the echo wave. These results verified the rationality for choosing the correlation method for TOF estimation.

Analysis of the system performance

To determine whether the c_L measurement system constructed in this study can provide long-term reliable c_L measurements, the stability of the c_L measurements was investigated when the test samples reached thermal equilibrium. Under the same temperature condition (25 °C), the c_L values of three LAS glass–ceramics samples were observed for a long time, and the c_L results were recorded at intervals of 1 h. At each time, the ultrasonic signal from the sample was measured five times at the same location point, and the average value was taken as the TOF result. Seven measurements were taken for each sample during one day. The change of c_L with the measurement time is shown in Fig. 10.

Fig. 10

Change of c_L with time for samples 1, 3, and 4.

The root mean squared error (RMSE) of the c_L value for the same sample was within 0.10 m/s when the temperature was kept constant (Fig. 10), which indicates that the c_L measurement system constructed in this work has stable performance.

To further investigate the accuracy of the measurement system itself and ensure that it can enable ultrasonic velocity measurement to meet the high-accuracy requirement, the uncertainty of the ultrasonic velocity measurement was investigated. The uncertainty is expressed as

$${\text{u}}_{{\text{c}}_{\text{L}}}\text{=}\sqrt{{\text{(}\frac{2}{\Delta{\text{t}}}{\text{u}}_{\text{d}}\text{)}}^{2}{+}{\text{(}\frac{{2}{\text{d}}}{{{(\Delta}{\text{t}}\text{)}}^{2}}{\text{u}}_{\Delta{\text{t}}}\text{)}}^{2}},$$

(11)

where ${\text{u}}_{{\text{c}}_{\text{L}}}$, u_d, and u_∆t are the uncertainties of c_L, the thickness of the tested sample d, and the ultrasonic TOF of the LAS glass–ceramics $\Delta {\text{t}}$, respectively.

As shown in the first term of Eq. (11), the typical value of Δt is 15.3779 μs. The thickness of the LAS glass–ceramics sample (d) has type A and type B uncertainty. The type A standard uncertainty of the thickness is evaluated as the standard deviation of five repeated measurements, which is expressed as ${\text{u}}_{\text{A}}\left({\text{d}}\right)\text{=}\sqrt{\frac{{\sum }_{{\text{i}}= \text{1} }^{5}{\text{(}{\text{d}}_{\text{i}}-\stackrel{\text{-}}{\text{d}}\text{)}}^{2}}{{\text{n}}\text{(}{\text{n}}-\text{1)}}\text{ } = }\text{ 0.0004 mm}$. The thickness measurements were performed with a micrometer with a resolving power of 0.001 mm. The type B standard uncertainty is then 0.0003 mm, so the synthetic standard uncertainty of the thickness is u(d) = 0.0005 mm. Similarly, the type A standard uncertainty of u_∆t is evaluated as the standard deviation of five repeated measurements, which is expressed as ${\text{u}}_{\text{A}}\left(\Delta{\text{t}}\right)\text{=}\sqrt{\frac{{\sum }_{{\text{i}}= \text{1} }^{5}{{(\Delta}{\text{t}}_{\text{i}}-\stackrel{\text{-}}{\Delta{\text{t}}}\text{)}}^{2}}{{\text{n}}\text{(}{\text{n}}-\text{1)}}} = 0.000112 \; \upmu\text{s}$. The system has a sampling time of 0.8 ns, resulting in a type B uncertainty of 0.00116 μs. Consequently, the synthetic standard uncertainty of the TOF is 0.00116 μs. By substituting the above uncertainties into Eq. (11), the total uncertainty of c_L for the measurement setup in this study is ${\text{u}}_{{\text{c}}_{\text{L}}}$ = 0.49 m/s. This level of uncertainty ensures a high degree of reliability in measuring the ultrasonic velocity at a single temperature.

Results and discussion

Acquisition of the c _L–T data

The sample was first placed at the marked position of the support platform of the flume and fixed, and the temperature of the water bath was controlled to be constant to ensure that the sample was in thermal equilibrium during the TOF measurement. According to the parameters determined from the previous experiments, the transducer was moved directly above the sample so that the water distance was 30 mm and, at the same time, the gain of the pulse-receiver was adjusted to 36 dB. Here, the experimental temperature range of 10–40 °C was chosen for two reasons: (1) this agrees with the actual temperature range that LAS glass–ceramics are likely to encounter during ultrasonic measurements, which is the room temperature range throughout the year, and (2) it corresponds to the temperature range of the ultrasonic transducers we used. Above a temperature of 40 °C, the piezoelectric action of the transducer is weakened and the wafer is easily damaged.

The experimental measurement process is shown in Fig. 11, in which the ultrasonic echo signal is sampled and stored by the personal computer. To account for the maximum number of sampled temperature points and the required modeling time, the computer manually samples and stores the ultrasonic echo signals at 2 °C temperature intervals. The temperature was measured with a digital thermometer probe in the bath and controlled to be within ± 0.05 °C. To minimize the random errors in the measurements, five echo signals were obtained for each temperature data point. Using the proposed correlation algorithmic method, the ultrasonic TOF was obtained, and the c_L value of the measured LAS glass–ceramics sample was calculated using the average of the five TOF measurements. Owing to the minimal variations of d and ρ with the temperature, d and ρ were considered to be constant values for the samples measured in this experiment.

Fig. 11

Experimental setup used for the ultrasonic velocity measurement based on ultrasonic water immersion.

Analysis of the effect of T on c _L

Following the above ultrasonic velocity measurement method and steps, the variation of the ultrasonic velocity of the specimen in the temperature range of 10–40 °C was experimentally investigated. The fluctuation of the ultrasonic velocity with the temperature for samples 1–5 is shown in Fig. 12. The ultrasonic velocity increased as the temperature increased. Independent of the absolute value of the CTE, the changes in c_L showed a consistent positive correlation with the temperature. This could be because LAS glass–ceramics exhibit anomalous temperature dependence related to the variations in the internal structure and atomic sites of the LAS glass–ceramics, resulting in positive temperature coefficients of E and c_L, furthering verifying the theoretical relationship given in "Theoretical relationship between the ultrasonic velocity and temperature". The variation of c_L with T is plotted directly using the data in Fig. 12 based on the exponential fit in Eq. (8).

Fig. 12

Relationships between c_L (m·s⁻¹) and the temperature for (a) sample 1#, (b) sample 2#, (c) sample 3#, (d) sample 4#, and (e) sample 5#.

As shown in Table 2, α_L ranged from 0.0001395 to 0.0001432, and it was almost constant for the LAS glass–ceramics materials with different CTE values. This indicates that the pattern of the temperature effect on the ultrasonic velocity is almost the same in LAS glass–ceramics with different CTE values.

Table 2 Exponential fitting of the relationship between c_L and T.

Next, we verified the accuracy of the c_L–T exponential model for LAS glass–ceramics through a set of experiments.

Accuracy verification of the c _L–T model

Samples 1#, 3#, and 5# were randomly selected, and their c_L values were measured at 10 random temperature points in the temperature range of 10–40 °C. The measured c_L values were compared with the values predicted by the model. The results are given in Table 3.

Table 3 Experimental data for model validation and error analysis.

Here, c_m is the measured ultrasonic velocity, c_p is the ultrasonic velocity predicted by the model, δ is the relative error, and the standard deviation σ_c is the measure of the accuracy.

Given that the measurement locations of the three samples might vary throughout the model verification process, the c_L–T exponential model can be obtained by recalculating ${c}{\prime}$ using the c_L value at 20 °C, where α_L is the average temperature coefficient of 0.0001403 from the above-described fitting analysis. This result might differ slightly from the result obtained by modeling the c_L–T data. Finally, the c_L values at 10 randomly measured temperature points were substituted into the c_L–T exponential model for each test sample to obtain the model-predicted ultrasonic velocity c_p. The standard deviations of the predicted values for all three samples were within 0.30 m/s, and the relative errors between the measurements and model predictions were mostly within ± 0.010% (Table 3), indicating that the models fitted in this work exhibited high accuracy.

Conclusions

The aim of this study is to develop a high-precision measurement method for the ultrasonic velocity in LAS glass–ceramics and to investigate the effect of the temperature on the ultrasonic velocity in LAS glass–ceramics. Based on the analysis of the TOF algorithm and experimental investigations of the effect of the temperature on the ultrasonic velocity, the following conclusions can be drawn:

• 1. The proposed correlation method can rapidly and accurately calculate the TOF while reducing the number of correlation calculation data points by accurately determining the strong correlation region of the B₁ and B₂ signals. Compared with the TOF computation time of the previous correlation method of 5.6 s, the TOF computation time of the proposed method is reduced to approximately 0.05 s. The TOF values measured by this method were compared with those measured by the commonly used peak and zero-crossing methods. The proposed method has higher precision of TOF estimation with PV = 0.0026 μs and σ = 0.00069 μs, which verifies the feasibility of TOF estimation based on the proposed correlation method.

• 2. Based on the proposed high-precision TOF measurement method, the effect of the temperature on the ultrasonic velocity in LAS glass–ceramics was experimentally investigated. The ultrasonic velocity increased with increasing experimental temperature. Using five groups of representative experimental data, a c_L–T exponential model was fitted, and the ultrasonic velocities at random temperature points predicted by the model were in good agreement with the actual measured ultrasonic velocities at the same temperature (the relative errors were mostly within ± 0.010%), indicating that the fitted model has high prediction accuracy.

The rapid and high-precision TOF measurement method proposed in this study, as well as the experimentally established c_L–T exponential model, not only lays an important foundation for the accurate measurement of the ultrasonic velocity in large LAS glass–ceramics at specific temperatures, but it also provides concepts and methods for the reliable evaluation of the CTE and CTE homogeneity in large LAS glass–ceramics.

Based on our findings, there are some comments that can be made about the broader implications and potential future research directions. On one hand, the ultrasonic velocity measurement method proposed in this study is expected to develop towards intelligence in the future, demonstrating faster, more stable performance and stronger adaptability under various testing conditions. On the other hand, the application of this method in comprehensive integrated instruments enables the synergistic measurement of key material parameters. For instance, it can obtain both the shear wave velocity and the longitudinal wave velocity and indirectly determine the material’s elastic modulus and Poisson’s ratio through these measurements.