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An updated B-scan method is proposed for measuring the evolution of thermal deformation fields in polymers. In order to measure the distributions of out-of-plane deformation and normal strain field, phase-contrast spectral optical coherence tomography (PC-SOCT) was performed with the depth range and resolution of 4.3 mm and 10.7 μm, respectively, as thermal loads were applied to three different multilayer samples. The relation between temperature and material refractive index was predetermined before the measurement. After accounting for the refractive index, the thermal deformation fields in the polymer were obtained. The measured thermal expansion coefficient of silicone sealant was approximately equal to its reference value. This method allows correctly assessing the mechanical properties in semitransparent polymers.
© 2015 Optical Society of America
1. Introduction
In general, temperature changes cause redistribution of deformation fields in the affected structures, occasionally yielding mechanical failures. Consequently, measuring thermal deformations has attracted significant attention. Relevant techniques can be traced back to the electronic speckle pattern interferometry (ESPI) in the 1990s. E. Hack et al. applied ESPI for measuring the deformation of lightweight structures under thermal loads [1]. Digital image correlation (DIC) is another technique, which has been used for measuring the thermal expansion on the surface of polymer film [2]. These techniques are all categorized as “surface measurement”. However, none of these allows directly observing the temperature-deformation relationship within structures.
To study the internal mechanical behavior of structures, alternative methods based on optical coherence tomography (OCT) were developed [3–8]. Spectral domain polarization sensitive OCT (SD-PS-OCT) measures the stress in terms of the birefringence within a structure, which is applied for detecting the strain and stress fields in polymers [5,6]. Another method is phase-contrast spectral OCT (PC-SOCT) or phase sensitive OCT (PhS-OCT), which is sensitive to the depth-resolved optical path difference using phase information [7–10]. M. H. De la Torre-Ibarra et al. used PC-SOCT for measuring the continuous evolution of the out-of-plane displacement of porcine cornea under hydrostatic pressure.
The optical path difference is the product of refractive index and distance. For polymer, refractive index changes with temperature at the rate of −10−4 /°C [11]. Therefore, none of the above interferometry methods can be applied for directly measuring the internal thermal deformation. In order to investigate the depth profiling of photothermal compound concentrations of biological tissue, a model was proposed to describe the relationship between the optical path difference and the material temperature [12,13]. It assumes that the different layers in the tissue have the same refractive index, same temperature coefficient of the refractive index, and same coefficient of the thermal expansion. However, these properties are anisotropic between layers in industrial field.
To continuously measure the depth-resolved thermal deformation of multilayer structures of a single-layer, a double-layer, and a double-layer with a void defect, an updated B-scan method is proposed in this paper. An experiment, based on PC-SOCT, was performed to validate the method. In Section 2, the system configuration and the experimental method are described. In Section 3, the principle of depth-resolved thermal deformation field measurement is presented in detail. In Section 4, the experimental results are shown and analyzed. In the last Section, conclusions and discussions are presented.
2. System configuration and experimental method
A PC-SOCT system is shown in Fig. 1, where Figs. 1(a) and 1(b) show the front and top views, respectively. In the front view, the output of super-luminescent diode SLD (Superlum Diodes Ltd., HP3, center wavelength λc = 840 nm, FWHM bandwidth Δλ = 47 nm, output power 25 mW) is first collimated by lens L1 (fL1 = 60 mm), and then focused by cylindrical lens CL (fCL = 150 mm). Cube beam splitter CBS is used for splitting the beam to focus on sample S and reference plane R. However, in the top view, the output of SLD illuminates S and R in parallel. Therefore, a cross-section in the y-z plane in S is illuminated by the light source. After being recombined by CBS, the backscattered light from S and R forms a wavenumber-resolved fringe pattern in the CCD camera (PCO. 1600, 1600 × 1200 pixels, 14 bits, 72 dB) via a spectrometer consisting of lenses L2 (fL2 = 150 mm), L3 (fL3 = 300 mm), and diffraction grating G (1200 lines/mm). The field of view of this system is 4.2 mm in the y direction.
Fig. 1 Experimental setup. (a) Front view of the PC-SOCT. (b) Top view of the PC-SOCT. (c) Photograph of the samples. (d) The sample mounter. (e) The sample cross-section. SLD: Super-luminescent diode; L1-L3: lens; CL: cylindrical lens; S: sample; CBS: cube beam splitter; R: reference plane; G: diffraction grating; CCD: CCD camera.
Three silicone rubber samples used in the experiment are shown in Fig. 1(c). S1 sample is a single layer of solidified silicone sealant plate (thickness: 0.54 mm, refractive index: 1.42). S2 sample is a double layer of silicone rubber; its first layer is a chromatography silica gel plate (thickness: 0.49 mm, refractive index: 1.43), and its second layer is a solidified silicone sealant plate (thickness: 0.54 mm, refractive index: 1.42). In order to model the typical structure which has an intact surface and an internal void, a double layered solidified silicone sealant plate (thicknesses: 0.52 mm and 0.58 mm, respectively, refractive index: 1.42) with a 1 × 15 × 0.58 mm3 void located under the top layer is used as S3 sample. Moreover, single-layer samples of solidified silicone sealant plate and chromatography silica gel plate were prepared, respectively, for measuring the relation between temperature and material’s refractive index.
The experiment was performed in four stages: 1) splicing the sample to a fixed black acrylonitrile butadiene styrene (ABS) plate, as shown in Fig. 1(d); 2) heating the sample up to ~75 °C by using a hairdryer; 3) starting the measurement when the temperature decreased to 50 °C; 4) recording the interferograms with the exposure time of 200 ms and an interval of 2 s between the exposures. Each measurement lasted 170 s, as the temperature decreased to 40 °C. The temperature was recorded in real time using an infrared thermometer (Fluke, MT4 MAX + ) with the accuracy of ± 1.5 °C during the experiment.
3. Theory
3.1 The measurement principle of depth-resolved thermal deformation and normal strain field
As shown in Fig. 1(e), the sample contains multiple different layers. Supposing that a point p(y, z) is located at the lth layer, its depth can be written as
where ni represents the refractive index of the ith layer, and n0 represents the refractive index outside the sample; Λ(y, z) = π f(y, k) represents the optical path length between R and p, known as the optical depth [7], where f is the frequency with respect to the wavenumber k = 2π / λ. The optical depth resolution is δΛ = λc2 / Δλ = 15μm. If the refractive index of the sample is approximately 1.4, the depth resolution is δz ≈10.7μm.If a load is applied to the sample, the optical depth change can be evaluated as
where ΔП represents the phase difference before and after the load application and kc = 2π / λc. ΔΛ can also be represented by the out-of-plane displacement w aswhere Δni represents the refractive index change of the ith layer before and after the load application, Δn0 represents the refractive index change outside the sample. Because the multiplication of w and Δn is so small that it can be neglected, the out-of-plane displacement can be denoted asThe displacement comprises rigid body displacement and deformation, therefore, the out-of-plane deformation of the point p can be represented aswhere wr is the out-of-plane displacement of a rigid body.The normal strain at the point p in the z direction can be expressed as [14]
Equation (6) shows that the distribution of the normal strain field can be evaluated in terms of the partial differentiation of the phase difference map. Before the evaluation of the normal strain field, the phase difference map is filtered by a 5 × 5 median filter and fitted by Zernike polynomials to avoid the noise from the speckle [15].3.2 Measurement of the refractive index change
When the thermal load is applied to the sample, the change of its refractive index should be determined in advance. If the rear surface of a single-layer sample is fixed and static under the thermal load, its refractive index change Δn can be expressed as
where z1 and z2 are the depth of the front and rear surfaces of the sample, respectively.4. Experimental results
Before the measurement, the relations between the refractive index change Δn and the temperature of the materials were predetermined according to Eq. (7). As shown in Fig. 2, Δn increases almost linearly with temperature T. For the chromatography silica gel plate, it can be fitted as Δn = −3.5 × 10−4 ∙ T + 1.7 × 10−2 with std error 8.3 × 10−5 and mean error 7.0 × 10−5; for the solidified silicone sealant plate, it can be fitted as Δn = −3.8 × 10−4 ∙ T + 1.9 × 10−2 with std error 6.0 × 10−5 and mean error 4.8 × 10−5.
Fig. 2 The relations between the refractive index change Δn and the temperature. ○, Chromatography silica gel plate; × , solidified silicone sealant plate.
4.1 Distributions of the out-of-plane deformation field and normal strain field inside S1 sample
Figure 3(a) shows the amplitude spectrum of S1 sample. The front and the rear surfaces are located at Λ = 1.00 mm and 1.76 mm, respectively. The calculated thickness is 0.54 mm. Figure 3(b) shows the wrapped phase difference spectrum obtained after the sample temperature decreased by 10 °C.
Fig. 3 Interference spectra of S1 sample. (a) The amplitude map. (b) The wrapped phase difference map.
Because the wrapped phase change at each pixel between adjacent frames of interference patterns is located into [–π, π], temporal phase unwrapping is used to restore the real phase distribution [7, 16]. Figure 4 shows the unwrapped phase difference maps, the contour maps of the out-of-plane deformation and normal strain fields inside S1 sample, obtained after the sample temperature decreased by 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C. There are ghost images from the reflection of the optical system at Λ ≈1.5mm, which are removed by thresholding in Fig. 4. With the decrease of the temperature, the distributions of the out-of-plane deformation field were positive, and their maxima increased from 0.4 μm to 2.0 μm, indicating that S1 sample shifted toward the black ABS plate; meanwhile, the distributions of the normal strain field were negative, and their minima decreased from −0.7 × 10−3 to −3.6 × 10−3, indicating a trend of continuous shrinkage inside S1 sample.
Fig. 4 Distributions of the out-of-plane deformation and normal strain field inside S1 sample. From left to right, the images correspond to the temperature reductions of 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C. Upper row: unwrapped phase difference maps; middle row: contour maps of the out-of-plane deformation field; bottom row: contour maps of the normal strain field.
The measured linear thermal expansion coefficient of S1 sample is ~3.6 × 10−4 /°C, as shown in Fig. 5, which is consistent with the reference value (3.7 × 10−4 /°C) listed in the technique manual [17].
4.2 Distributions of the out-of-plane deformation field and normal strain field inside S2 sample
Figure 6(a) shows the amplitude spectrum of S2 sample. Its front surface, interface and rear surface are clearly visible. Figure 6(b) shows the wrapped phase difference spectrum obtained when the temperature decreased by 2 °C. The interface between the layer A and B can be identified clearly.
Fig. 6 Interference spectra of S2 sample. (a) The amplitude map. (b) The wrapped phase difference map.
Figure 7 shows the unwrapped phase difference maps, the contour maps of the out-of-plane deformation and normal strain field inside S2 sample, obtained after the sample temperature decreased by 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C. With the decrease of the temperature, the maximal out-of-plane deformation increased from 0.6 μm to 2.7 μm. Meanwhile, the normal strain field in layer A decreased from −0.3 × 10−3 to −1.4 × 10−3, whereas the normal strain field in layer B decreased from −0.7 × 10−3 to −3.6 × 10−3. The strain behavior indicates that the shrinking rate of layer A is smaller than that of layer B.
Fig. 7 Distributions of the out-of-plane deformation and normal strain field inside S2 sample. From left to right, the images correspond to the temperature reductions of 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C. Upper row: unwrapped phase difference maps; middle row: contour maps of the out-of-plane deformation field; bottom row: contour maps of the normal strain field.
The measured linear thermal expansion coefficients of the two layers are ~1.4 × 10−4 /°C and ~3.6 × 10−4 /°C, respectively, as plotted in Fig. 8.
4.3 Distributions of the out-of-plane deformation field and normal strain field inside S3 sample
The amplitude spectrum of S3 sample is shown in Fig. 9(a); there are four surfaces representing the front surface of S3 sample, the front surface of the void, the rear surface of the void and the rear surface of S3 sample. In fact, the rear surfaces of the void and the sample are the same. However, owing to the different refractive indices of silicone rubber and air, the dislocation occurs at the rear surface from y = 1.2 mm to y = 2.2 mm, as shown in Fig. 9(a). Figure 9(b) shows the wrapped phase difference spectrum obtained after the temperature decreased by 2 °C.
Fig. 9 Interference spectra of S3 sample. (a) The amplitude map. (b) The wrapped phase difference map. The measured region is divided into 3 parts: part A, the front of the void; part B, the periphery of the parts A and C; part C, the void.
Figure 10 shows the unwrapped phase difference maps, the contour maps of the out-of-plane deformation and normal strain field inside S3 sample, obtained after the temperature decreased by 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C, respectively. The maximal out-of-plane deformation of part A changes −0.62μm /°C, which is faster than that of part B −0.40μm /°C. The normal strain of part A changes 2.6 × 10−4 /°C, which is slower than that of part B 3.6 × 10−4 /°C. Compared with the result of S1 sample, it is concluded that the deformation of part A is affected not only by the temperature change, but also by the negative pressure in part C.
Fig. 10 Distributions of the out-of-plane deformation and normal strain field inside S3 sample. From left to right, the images correspond to the temperature reductions of 2 °C, 4 °C, 6 °C, 8 °C, and 10 °C. Upper row: unwrapped phase difference maps; middle row: contour maps of the out-of-plane deformation field; bottom row: contour maps of the normal strain field.
5. Conclusion and discussion
An updated B-scan method is proposed for determining the depth-resolved thermal deformation within structure. The distributions of the out-of-plane thermal deformation and normal thermal strain field inside polymer were measured with the depth range and resolution of 4.3 mm and 10.7 μm, respectively. The major advantages of this method are: 1) It can be used to measure the thermal deformation within structure accurately. By putting forward the multilayer model and building up PC-SOCT system, the thermal deformation inside multilayer structure is evaluated accurately with the temperature dependence of refractive index corrected. 2) The thermal deformation measurement is achieved by line-field OCT system, which measures all A-scans in parallel. 3) It is immune to the influence of stray light and weak airflow, since these interferences are located in the lower frequency band in the frequency domain than that of the depth-resolved thermal deformation inside the structure.
It should be noted that: 1) Both the polymer’s refractive index and thermal strain are approximately linear with temperature, whose temperature ratios are of the same magnitude. Therefore, it is necessary to account for the refractive index change when measuring internal thermal deformation. 2) Rapid temperature changes could lead to a large error owing to the nonuniform distribution of the refractive index.
In the future, this method is suitable to study the thermal expansion, changes of refractive index, physical aging [18], and high-temperature mechanical properties inside polymer nondestructively.
Acknowledgments
The authors wish to thank Guangzhou Science and Technology Plan Project (2014J4100203), Provincial Natural Science Foundation of Guangdong (NSFG) (2014A030313519) and National Natural Science Foundation of China (NSFC) (51371129 and 11174226) for their financial support. The authors are also very appreciative of the comments from the anonymous reviewers.
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