# Measurement of the X-ray Elastic Constants of Amorphous Polycarbonate

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedure

#### 2.1. Material and Specimen

^{3}plate. The test specimens were cut out of the plate into strips of 9 × 50 × 2 mm

^{3}using a thin abrasive cutting wheel. After machining, the specimens were annealed under industrially used conditions to remove the residual stress. The heating rate is 16.7 K/h and the holding condition is 403 K and 0.5 h. After the heat treatment, four strain gauges were glued to the specimen to measure the longitudinal and transverse strains on the front and back sides of the specimen, respectively.

#### 2.2. X-ray Measurement

^{−1}. The X-ray conditions are summarized in Table 1.

^{2}χ = 0, where the χ-axis is parallel to the normal of the specimen surface). The change in the diffraction profile with respect to the applied stress level was extracted without changing the geometrical relationship between the sample and the X-ray.

^{2}χ = 0 to 0.9. The longitudinal strain corresponds to the value of sin

^{2}χ = 0 and the transverse strain corresponds to the value of sin

^{2}χ = 1. The Young’s modulus is the slope of the linear relationship between the longitudinal strain and the applied stress. The Poisson’s ratio can be easily obtained from the ratio of the transverse strain to the longitudinal strain.

#### 2.3. Determination of Diffraction Peak Position

_{max}. (see Section 3.2). Since the profile shapes are asymmetric, the position of the maximum peak was determined by approximating with the split type pseudo-Voigt function [19]. Namely, two pseudo-Voigt functions were utilized to approximate the low Q value side and the high Q value side of the peak top value. When the target intensity is 80% of the maximum value, data above 0.8 are utilized to the approximation. The fitting curve in this case is a red curve. The data with an intensity of more than 0.8 were used. Then the peak position of Q

_{PV}can be obtained. The target intensity used for the approximation was changed to 70% to 90% of the maximum value, and the influence of the approximation range on the peak data was examined. This method is hereafter called the pseudo-Voigt method, and the target intensity is called the peak ratio.

_{A}of the intersection on the low Q value side and the Q

_{B}on the high Q value side can be obtained. Then, the average of Q

_{A}and Q

_{B}was determined as the peak position Q

_{LW}. In this method, the number of data used for the approximation is not sufficient; however, there is no need to consider the suitability of the fitting function for the entire target data. This method is referred to below as the diffraction line width method.

## 3. Results and Discussion

#### 3.1. Diffraction Profile Change under Tensile Loading

#### 3.2. PDF Analysis

^{−1}is the highest in the case of the structure function as well.

_{max}= 2.535. Even if it is converted into a structure function by various corrections, the asymmetry of the profile shape cannot be eliminated. Similar to Figure 4b, the absolute value of the slope on the high Q value side is smaller than that on the low Q value side. The movement of the profile data on the low Q value side is clearer than in the case of the raw profile data shown in Figure 4b, and the shift of the entire profile is confirmed.

#### 3.3. Simple Evaluation Method

^{2}χ. The diffraction peak angle of the 1st halo was determined by the pseudo-Voigt method for the relation between the raw diffraction intensity and the diffraction angle. This figure summarizes the results corresponding to the peak ratio of 80% as a typical example. The results for other peak ratios were almost the same. The strain was determined by the Q-space method. The Q value used as a reference for the strain calculation was taken as the diffraction peak angle at sin

^{2}χ = 0 of 0.6 MPa. Namely, as mentioned above, the strain measured by X-ray means relative value. The residual strain, if existing, is ignored. In terms of stress loading, the applied stress was set to three levels from 0.6 to 21.5 MPa.

^{2}χ = 0 means the value in the loading direction. By using the intercept of the approximate straight line at sin

^{2}χ = 0, the influence of the scatter can be reduced. The slope of the fitted linear curve obtained for the applied stress of 0.6 MPa is positive. The slope in the unloaded state should be zero. The reason for the positive slope is considered to be the existence of the compressive residual strain. Furthermore, it is well known that injection-molded products of polymeric materials have a skin-core structure due to the fountain flow [23]. Since the molecular orientation depends on that structure, it is necessary to clarify the effect of the elastic modulus in detail. This is a subject for future work.

^{2}χ = 1 are the transverse strain perpendicular to the loading direction. By extrapolating the approximate line in Figure 9 to sin

^{2}χ = 1, not only the Young’s modulus, but also the Poisson’s ratio measured by X-ray can be determined. In the case of the uniaxial loading of a homogeneous material, these three lines in Figure 9a intersect at a specific point. The intersection point corresponds to sin

^{2}χ = 1/(1 + v). However, the intersection of the three straight lines in Figure 9 has a large difference, and the accuracy is extremely insufficient.

^{2}χ = 0.74 becomes 0.35. The value is close to the mechanical Poisson’s ratio of 0.37. The diffraction line width method is expected to be more accurate than the pseudo-Voigt method. These results were similar to other peak ratio data. It is suggested that the split type pseudo-Voigt function may not be suitable for the approximation of raw profile data. On the other hand, in the diffraction line width method, the target data are limited to a narrow range, and it is not necessary to approximate the entire profile. Therefore, it is a practical and general-purpose method to evaluate the shift of the raw profile data because of its high flexibility to profile shapes.

^{2}χ = 0 of the fitted linear curve in Figure 9b was plotted. Although there are only three data points, the results for each peak ratio can be approximated by a straight line. Figure 10b shows the relationship between the applied stress and the transverse strain measured by X-ray. As described previously, the reference applied to calculate the strain is the diffraction angle for sin

^{2}χ = 0 of 0.6 MPa. Therefore, the transverse strain at zero applied stress varies around 5 × 10

^{−3}. The transverse strain decreases with increasing applied stress, and of course, the slope becomes negative.

## 4. Conclusions

- (1)
- The first peak of the structure function shifts to the lower Q value side with increasing the applied stress, and the strain obtained from the amount of the peak shift can be approximated by a straight line with respect to the loading stress in the elastic region.
- (2)
- The Young’s modulus measured by X-ray determined from the strains determined by the pseudo-Voigt method is smaller than the value determined by the diffraction line width method. The Young’s modulus obtained by the diffraction line width method is close to the mechanical value, however, it varied depending on the peak ratio.
- (3)
- In the simple method using raw profile data, the Young’s modulus measured by X-ray determined by the diffraction line width method decreased with increasing peak ratio. The value determined by the pseudo-Voigt method was smaller than the mechanical value and was almost constant.
- (4)
- The Poisson’s ratio determined by the diffraction line width method increased with increasing peak ratio, in contrast to the Young’s modulus.
- (5)
- In the simple method, the diffraction line width method can be used to determine the elastic constants with sufficient accuracy for industrial use. When the stress evaluation is required, the same conditions must be used as determining the elastic constants.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kambour, R.P.; Gruner, C.L.; Romagosa, E.E. Biphenol-A polycarbonate immersed in organic media. Swelling and response to stress. Macromolecules
**1974**, 7, 248–253. [Google Scholar] [CrossRef] - Robeson, L.M. Environmental stress cracking: A review. Polym. Eng. Sci.
**2013**, 53, 453. [Google Scholar] [CrossRef][Green Version] - Society of Materials Science. Standard for X-ray Stress Measurement =Iron and Steel =(JSMS-SD-5-02); Society of Materials Science: Kyoto, Japan, 2002. (In Japanese) [Google Scholar]
- Society of Materials Science. Standard Method for X-ray Stress Measurement (JSMS-SD-10-05); Society of Materials Science: Kyoto, Japan, 2005. [Google Scholar]
- ASTM E1426—14 (2019) e1. Standard Test Method for Determining the X-ray Elastic Constants for Use in the Measurement of Residual Stress Using X-ray Diffraction Techniques; American Society for Testing and Materials: West Conshohocken, PA, USA, 2019. [Google Scholar]
- ASTM E915—19. Standard Test Method for Verifying the Alignment of X-ray Diffraction Instrumentation for Residual Stress Measurement; American Society for Testing and Materials: West Conshohocken, PA, USA, 2019. [Google Scholar]
- Tanaka, K.; Matsui, E.; Akiniwa, Y. X-ray Stress Measurement of Sintered Alumina. J. Soc. Mater. Sci. Jpn.
**1986**, 35, 749–754. [Google Scholar] [CrossRef] - Tanaka, K. X-ray Stress measurement of alumina/zirconia composites. Curr. Jpn. Mater. Res.
**1993**, 10, 1–27. [Google Scholar] - Tanaka, K.; Ishihara, K.; Akiniwa, Y.; Ohta, H. Residual stress of aluminum thin films measured by X-ray and curvature methods. Mater. Sci. Res. Int.
**1996**, 2, 153–159. [Google Scholar] [CrossRef] - Tanaka, K.; Tokoro, S.; Koike, Y.; Egami, N.; Akiniwa, Y. New method of X-ray measurement of residual stress in short-fiber reinforced plastics. J. Soc. Mater. Sci. Jpn.
**2014**, 63, 514–520. [Google Scholar] [CrossRef][Green Version] - Ida, K.; Akiniwa, Y. X-ray stress measurement of heat treated polyetheretherketone. In Proceedings of the 53rd Symposium on X-ray Studies on Mechanical Behavior of Materials, Osaka, Japan, 11 July 2019. [Google Scholar]
- Yokoyama, R.; Akiniwa, Y. Residual stress measurement of polymeric materials. In Proceedings of the 53rd Symposium on X-ray Studies on Mechanical Behavior of Materials, Osaka, Japan, 11 July 2019. [Google Scholar]
- Boukal, I. The use of X-ray diffraction under stress for the study of the structure of amorphous polycarbonate. Eur. Polym. J.
**1970**, 6, 17–24. [Google Scholar] [CrossRef] - Pick, M.; Lovell, R.; Windle, A.H. Detrection of elastic strain in an amorphous polymaer by X-ray scattering. Nature
**1979**, 281, 658–659. [Google Scholar] [CrossRef] - Nakamae, K.; Nishino, T.; Hata, K.; Matsumot, T. Measurement of the elastic moduli of amorphous atactic polystyrene by X-ray diffraction. Kobunshi Ronbunshu
**1985**, 42, 211–217. [Google Scholar] [CrossRef] - Poulsen, H.F.; Wert, J.A.; Neuefeind, J.; Honkimaki, V.; Daymond, M. Measuring strain distributions in amorphous materials. Nat. Mater.
**2005**, 4, 33–36. [Google Scholar] [CrossRef] - Hufnagel, T.C.; Ott, R.T.; Almer, J. Structural aspects of elastic deformation of a metallic glass. Phys. Rev. B
**2006**, 73, 1–8. [Google Scholar] [CrossRef] - Suzuki, H.; Yamada, R.; Tsubaki, S.; Imafuku, M.; Sato, S.; Watanuki, T.; Machida, A.; Saida, J. Investigation of elastic deformation mechanism in As-Cast and annealed eutectic and hypoeutectic Zr–Cu–Al metallic glasses by multiscale strain analysis. Metals
**2016**, 6, 12. [Google Scholar] [CrossRef][Green Version] - Toraya, H. Whole-powder-pattern fitting without reference to a structural model: Application to X-ray powder diffraction data. J. Appl. Crystallogr.
**1986**, 19, 440–447. [Google Scholar] [CrossRef] - Radhakrishnan, S.; lyert, V.S.; Sivaramt, S. Structure and morphology of polycarbonate synthesized by solid state polycondensation. Polymer
**1994**, 35, 3789–3791. [Google Scholar] [CrossRef] - Qiu, X.; Thompson, J.W.; Billinge, S.J.L. PDFgetX2: A GUI driven program to obtain the pair distribution function from X-ray powder diffraction data. J. Appl. Crystallogr.
**2004**, 37, 678. [Google Scholar] [CrossRef][Green Version] - Engqvist, J.; Hall, S.A.; Wallin, M.; Ristinmaa, M.; Plivelic, T.S. Multi scale measurement of (amorphous) polymer deformation: Simultaneous X-ray scattering, Digital image correlation and in-situ loading. Exp. Mech.
**2014**, 1373–1383. [Google Scholar] [CrossRef] - Coyle, D.J.; Blake, J.W.; Macosko, C.W. The kinematics of fountain flow in mold filling. AIChE J.
**1987**, 33, 1168–1177. [Google Scholar] [CrossRef]

**Figure 1.**Tensile loading and X-ray measurement. (

**a**) Loading device for in situ X-ray measurement; (

**b**) Optical system for transmission geometry.

**Figure 2.**Determination method of peak position: (

**a**) Approximation by split type pseudo Voigt function using more than 0.8 S

_{max}data; (

**b**) Average of two intersection positions corresponding to the intensity of 0.8 S

_{max}.

**Figure 4.**Change in diffraction profile due to applied stress: (

**a**) Wide range of profiles; (

**b**) Enlargement of figure (

**a**).

**Figure 5.**Change in structure function due to applied stress: (

**a**) Wide range of profiles; (

**b**) vicinity of the first peak of the normalized structure function.

**Figure 8.**Relation between average Young’s modulus determined from all the four stress levels and peak ratio.

**Figure 9.**Change of strain measured by X-ray due to rotation of specimen around the χ-axis for the peak ratio of 80%: (

**a**) Pseudo Voigt method; (

**b**) Diffraction line width method.

**Figure 10.**Relation between applied stress and strain measured by X-ray for diffraction line width method: (

**a**) Longitudinal strain; (

**b**) Transverse strain.

X-ray Equipment | Mac Science M21X |
---|---|

Characteristic X-ray | Mo-Kα (17.45 keV, λ = 0.7107 Å) |

Scan mode | Step scan |

Tube voltage | 30 kV |

Tube current | 200 mA |

2θ range | 3.55–30 or 100 deg |

Exposure time | 5 s/step |

sin^{2}χ | 0–0.9 (0.1 interval) |

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**MDPI and ACS Style**

Kawamura, Y.; Akiniwa, Y. Measurement of the X-ray Elastic Constants of Amorphous Polycarbonate. *Quantum Beam Sci.* **2020**, *4*, 35.
https://doi.org/10.3390/qubs4040035

**AMA Style**

Kawamura Y, Akiniwa Y. Measurement of the X-ray Elastic Constants of Amorphous Polycarbonate. *Quantum Beam Science*. 2020; 4(4):35.
https://doi.org/10.3390/qubs4040035

**Chicago/Turabian Style**

Kawamura, Yuki, and Yoshiaki Akiniwa. 2020. "Measurement of the X-ray Elastic Constants of Amorphous Polycarbonate" *Quantum Beam Science* 4, no. 4: 35.
https://doi.org/10.3390/qubs4040035