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Abstract
A simple model based on the measured fictive temperature distribution of fused silica was developed to determine the residual stress field of localized CO2 laser-heated fused silica with an unknown thermal history. The proposed model assumes that the initial frozen-in state of fused silica is the zero-point of residual stresses and the generation of residual stresses results from the thermoelastic contraction differences of fused silica with different fictive temperatures from initial frozen-in temperatures to the ambient temperature. The spatially resolved fictive temperatures of the fused silica sample were characterized using confocal Raman microscopy, and the calculated residual stress fields agreed well with laser-induced critical fracture measurements and photoelastic measurements.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The use of high-temperature localized CO2 laser heating has been demonstrated with great success at fused silica optics processing, such as damage mitigation [1,2], scratch removing [3], surface polishing [4], and surface patterning [5]. Generally, these CO2 laser-based treatments heat fused silica above the glass transition point, inducing structural relaxation under high temperature, and result in higher density within the heat-affected zone after a rapid thermal quench due to the anomalous nature of fused silica [6]. The phenomenon of local densification produces permanent residual tensile stress on the surface. Excessive residual tensile stress may cause critical fracture after laser treatment to relieve stresses throughout the heat-treated region [7,8]. Therefore, an accurate description of the residual stress field within heat-affected zone is fundamental for tuning of laser parameters to achieve optimal processing. A favourite method of measuring residual stresses in transparent materials is the non-destructive photoelastic method, which is based on the stress-induced birefringence, but there is no direct proportionality between the retardance and the stress level, since the retardance is integrated over the thickness of the fused silica sample with the nonuniform residual stress field induced by localized CO2 laser treatment [9].
Various numerical simulation investigations have been carried out to quantify the residual stress field of fused silica induced by localized CO2 laser treatment. Gallais et al. developed a numerical model of CO2 laser interaction with fused silica, taking into account laser energy absorption, heat transfer, thermally induced stress and birefringence, and indicated that theoretical values of maximum retardance is of the same order as experimental results [9]. Vignes et al. presented a multiphysics finite-element model to analyze the heat transport, structural relaxation, material motion, and residual stress in a laser-heated silica disk, and indicated that calculated stress fields agreed well with laser-induced critical fracture measurements [10]. Such theoretical numerical descriptions must consider the interaction of various physical processes and temperature-dependent physical properties in the whole thermal history of fused silica induced by localized CO2 laser treatment, which makes the modeling of the residual stress field very complex and time-consuming. Hence, the ability to accurately assess the residual stresses in a simple and convenient means is still a technical challenge.
The fictive temperature of a glass, which is defined as the temperature of the equilibrium melt whose structure is equivalent to that of the cooled glass, is a convenient tool in describing glass structural relaxation phenomena [11]. The CO2 laser-induced structural relaxation of fused silica has been theoretically and experimentally analyzed, and the fictive temperature distribution of fused silica associated with localized CO2 laser heating has been reported [7,10,12]. The fictive temperature of fused silica is linked to many physical properties such as density [13], refractive index [14], thermal expansion coefficient [15], mechanical strength [16], and hydrofluoric acid etching rate [17]. The forementioned physical properties of fused silica can be directly estimated from their relationship with fictive temperature. However, the direct assessment of residual stress field from fictive temperature distribution of fused silica with an unknown thermal history is still unresolved.
In this study, a simple model just based on the measured fictive temperature distribution of fused silica is explored to extract the residual stress field within the heat-affected zone induced by localized CO2 laser treatment. The proposed model assumes that the initial frozen-in state of fused silica, at which the value of temperature is equivalent to that of fictive temperature, is the zero-point of residual stresses, and the generation of residual stresses results from the thermoelastic contraction differences of fused silica with different fictive temperatures from initial frozen-in temperatures to the ambient temperature. The obtained nonuniform residual stress field agrees well with the experimental result, demonstrating the availability of the proposed model based on the fictive temperature distribution in describing the residual stresses within the heat-affected zone induced by localized CO2 laser treatment.
2. Experimental
The sample used in this study is polished UV-grade Corning 7980 fused silica, 40 mm in diameter, and 4 mm thick. The fused silica sample was cleaned by deionized water under ultrasonic agitation and then washed by ethanol. A commercial radio frequency power excited CO2 laser with a Gaussian beam was used to heat the fused silica sample with a 4.2 mm 1/e2 diameter spot at the sample surface. The fused silica sample was first preheated with low laser power of 13.7 W for 30 s, and then irradiated by high power of 25.3 W for 4 s, and finally quenched by turning off the laser at the end of the heating time. Raman spectra of the heat-affected zone of the fused silica sample induced by localized CO2 laser heating were recorded by an integrated laser Raman system (LABRAM HR, Jobin Yvon) with a confocal microscope, a stigmatic spectrometer, and a multichannel air-cooled CCD detector with resolution of 1 cm−1. The 514.5 nm line from an argon ion laser was used as an excitation source at a power level of 15.2 mW. All spectra were measured in the backscattering geometry at room temperature. The residual stress field within the heat-affected zone of the fused silica sample was characterized by a photoelastic tool (PTC-702), and the spatial resolution of the retardance measurement is 20 µm in this study.
3. Model
To simply and conveniently determine the residual stress field of fused silica induced by localized CO2 laser heating, a model to quantify the residual stress based on the measured fictive temperature distribution within the heat-affected zone is developed. During the localized CO2 laser irradiation, the absorbed laser energy induces a temperature gradient in the irradiation zone of fused silica, and compressive stresses build up on the surface due to thermal expansion. If the maximum temperature is below the glass transition temperature, thermoelastic expansion results from changing thermodynamic temperature without structural change, and the thermal expansion of fused silica will be completely recoverable after laser turn off, which results in no residual stress in the CO2 laser irradiation zone. However, above the glass transition temperature, changes in the glass structure attempts to bring the system to an equilibrium state, and the heated fused silica is soft and thermally-induced stresses are released by structural relaxation. Upon cooling with laser heat-quenching, this high-temperature structural configuration is then locked into the glass due to the high cooling rates, resulting in a frozen-in higher fictive temperature with a higher density. Thus, as the surrounding material cools and contracts, the central zone with higher fictive temperature has a slightly smaller specific volume and undergoes tensile stress as the body collectively cools [6,7,10]. According to these physical processes accompanying CO2 laser heating of fused silica, the melt zone persists in a zero stress state until the laser is shut off and a rapid thermal quench ensues, and it is reasonable to assume that the initial frozen-in state, at which the value of temperature is equivalent to that of fictive temperature, is the zero-point of residual stresses, and the differences of thermoelastic contraction of fused silica with different fictive temperatures cause residual stresses as the fused silica cools and contracts from initial frozen-in temperatures to the ambient temperature.
Several assumptions are used in the proposed model based on the fictive temperature distribution. The fused silica glass is isotropic, and the thermal expansion coefficient and room-temperature density of fused silica are considered to be only a function of the fictive temperature [15]. The specific volume, $V(T,{T_f})$, of fused silica with fictive temperature, ${T_f}$, at thermodynamic temperature, T, is given by
where $\rho ({T_f})$ is the density of fused silica with fictive temperature ${T_f}$ at room temperature, $\alpha ({T_f})$ is the thermal expansion coefficient of fused silica with fictive temperature ${T_f}$, and ${T_0}$ is the room temperature. Furthermore, according to the reported relationships of thermal expansion coefficient and room-temperature density of fused silica with fictive temperature [15,16], the specific volume of the initial frozen-in state, at which the value of temperature, T, is equivalent to that of fictive temperature, ${T_f}$, is plotted in Fig. 1, and typical evolutions of specific volumes with different fictive temperatures from initial frozen-in temperatures to the room temperature are also shown in Fig. 1. It is obvious that higher fictive temperatures lead to larger contractions of specific volumes from initial frozen-in temperatures to the room temperature, which induce tensile stresses within the heat-affected zone.
Fig. 1. Specific volumes of the initial frozen-in states of fused silica, and typical evolutions of specific volumes with fictive temperatures of 1315 K, 1465 K, 1615 K, 1765K, and 1915K from initial frozen-in temperatures to the room temperature.
The change of specific volume with fictive temperature, ${T_f}$, from initial frozen-in temperatures to the room temperature is described as
The fictive temperature of fused silica substrate is 1315 K, and the specific volume change of fused silica with the fictive temperature of 1315 K from the initial frozen-in temperature to the room temperature is taken as the comparison basis. Thus, the difference of specific volume changes between fictive temperature-increased fused silica and the substrate can be regarded as the residual thermal volume displacement as the body collectively cools, and thus the residual thermal volume strain of fused silica with fictive temperature ${T_f}$ is expressed as
From Eqs. (1), (2) and (3), the distribution of residual thermal volume strain of fused silica induced by localized CO2 laser treatment is directly generated by the fictive temperature distribution within the heat-affected zone, and the residual stress field is obtainable by solving elastic equilibrium conditions and stress-strain relations.
Given the symmetry of this study, a 2-D axisymmetric geometry is used to describe the residual stress field as a function of the residual thermal strain of fused silica extracted from the fictive temperature distribution. Making use of the cylindrical coordinate system $(r,\theta ,z)$, for axially symmetric problems, the equations of equilibrium are [18]
Considering the residual thermal volume strain of fused silica with increasing fictive temperature in Eq. (3) and the constitutive equations for an axisymmetric body in the cylindrical coordinate system [18], the constitutive equations for the axisymmetric heat-affected zone induced by localized CO2 laser treatment are expressed as
Combining elastic properties of fused silica influenced by fictive temperature [16], the residual stress field of fused silica induced by localized CO2 laser treatment is conveniently obtained through finite element analysis of the Eqs. (4) and (5). In this study, a commercial finite element software COMSOL Multiphysics was used for the residual stress analysis.
4. Results and discussion
4.1 Fictive temperature distribution
Spatially resolved axial scans were done along $r = 0$ to investigate the depth distribution of fictive temperature within the heat-affected zone over few hundred microns of the sample. Typical Raman spectra along the axis depth within the heat-affected zone of the fused silica sample induced by localized CO2 laser heating are shown in Fig. 2. The intensities of the so-called ‘defect’ bands D1 and D2, attributed to four- and three-membered ring structures of fused silica, are sensitive with fictive temperature [19,20]. More dramatically, the relative intensities of bands D1 and D2 were observed to decrease with increasing axis depth, which indicates that the heat-affected zone at the surface possesses higher fictive temperatures.
Fig. 2. Typical Raman spectra along the axis depth in the heat-affected zone of the fused silica sample treated by localized CO2 laser 30 s 13.7 W preheating and 4 s 25.3 W heating.
Because of the higher sensitivity of the D2 band with fictive temperature, the D2 band is chosen to determine the fictive temperature. The suggested peak deconvolution technique in Ref. [20] is used to evaluate the relative intensities of D2 band. The relationship of fictive temperature and the D2 band intensity normalized to the ω1 band has been reported [19,20], which is used to determine the exact value of fictive temperature in this study. From a set of measured Raman spectra, the fictive temperature along axis depth within the heat-affected zone induced by localized CO2 laser heating is shown in Fig. 3, which displays a characteristic sigmoidal shape similar to the reported results of fused silica for CO2 laser exposures [12]. The fictive temperature at the surface is about 1940 K, extending to about 330 µm, and then monotonically decreases into the bulk to 1315 K.
Fig. 3. Depth profile of the fictive temperature along the axis of the heat-affected zone of the fused silica sample treated by localized CO2 laser 30 s 13.7 W preheating and 4 s 25.3 W heating.
For a fixed diameter of a CO2 laser spot with specific pulse length and power, the local fictive temperature modification zone of fused silica results from isotropic thermal diffusion of the thermodynamic temperature field and shows a characteristic volume of spherical cap [21]. Thus, the fictive temperature distribution can be mapped by measuring the fictive temperature profiles along axis depth and surface radius of the heat-affected zone. Deriving from the measured depth profile of the fictive temperature on the axis in Fig. 3 and the measured radius profile of the fictive temperature on the surface, the fictive temperature distribution within the heat-affected zone is artificially reconstructed and shown in Fig. 4, which indicates that the fictive temperature values in the core of the heat-affected zone are almost uniform and gradually decreases into the bulk.
Fig. 4. The artificial reconstruction of the fictive temperature distribution from measurements along z and r within the heat-affected zone of the fused silica sample treated by localized CO2 laser 30 s 13.7 W preheating and 4 s 25.3 W heating.
4.2 Residual stress field
Thanks to the proposed numerical model to quantify the residual stress field based on the fictive temperature distribution of fused silica induced by localized CO2 laser treatment in Eqs. (4) and (5), the calculated residual stress fields for the fictive temperature distribution in Fig. 4 are shown in Fig. 5. As shown in Figs. 5(a) and (b), due to the local fictive temperature modification zone of fused silica induced by CO2 laser heating, the stresses are permanently set into the fused silica glass.
Fig. 5. Calculated radial stress ${\sigma _{rr}}$ (a) and hoop stress ${\sigma _{\theta \theta }}$ (b) in fused silica after localized CO2 laser 30 s 13.7 W preheating and 4 s 25.3 W heating. A positive value for the principal stress indicates tension.
Figure 5(a) shows the residual radial stress distribution, and it can be seen that the core of the heat-affected zone suffers intensive tensile stresses. On the surface, the tensile radial stresses increase from 27.1 MPa at $r = 0$ to maximum 43.8 MPa at $r = 1.15$ mm, and then gradually decrease to zero at radius larger than 4 mm. The tensile radial stresses on the surface extend far beyond the local fictive temperature modification zone. On the axis, the radial stresses increase from 27.1 MPa at $z = 0$ to maximum 35.0 MPa at $z ={-} 0.29$ mm, and then decrease rapidly to zero at $z ={-} 0.66$ mm. With the axial depth further increasing, the radial stresses change from tensile stress to compressive stress and reach minimum −5.3 MPa at $z ={-} 0.85$ mm. While, the residual hoop stress distribution is significantly different, as shown in Figs. 5(b). On the surface, the hoop stresses almost keep constant from $r = 0$ to $r = 0.86$ mm, and just slowly increase from 27.1 MPa to maximum 29.3 MPa. With the radius further increasing, the hoop stresses decrease rapidly and reach minimum −2.7 MPa at $r = 1.77$ mm. The tensile hoop stresses are intensely confined within the heat-affected zone, and decrease fast departing from the heat-affected zone, becoming compressive hoop stress encircling the central zone.
The residual stresses on the surface can be confirmed by critical fracture experiments of CO2 laser-heated fused silica [7], and the calculated radial and hoop stresses on the surface are specially plotted in Fig. 6 for comparison. The fracture morphology on the surface of heat-affected zone reflects the residual stress distribution on the surface. It is interesting to note that the maximum residual stresses on the surface do not occur at the center of heat-affected zone, but in fact reside in border regions, where the slope of fictive temperature is larger, and moreover, the maximum residual radial stress is about 50% larger than the maximum residual hoop stress. Thus, if an damage site larger than the critical size is introduced on the border region with maximum residual stresses, the crack propagation direction of the introduced damage site will prefer to be tangential, not radial, due to the huge difference of radial stress and hoop stress, which is in good agreement with the results of critical fracture experiments in the Ref. [22]. The reported critical fracture experiments of CO2 laser-heated silica under the same conditions in this study yielded a tangential fracture around the heat-affected zone and an estimate of maximum residual radial stress of 39 MPa [22], which is remarkably close to the calculated maximum radial stress of 43.8 MPa. If an surface damage site larger than the critical size is introduced on the center of the heat-affected zone, the crack propagation will be radial from center to border and change to be tangential at the border region of maximum residual stresses, which is excellently consistent with the fracture morphology observed in the Refs. [7] and [8]. The reported critical fracture experiments for the quenched CO2 laser-heated silica sample with maximum fictive temperature of 1900 K yielded an estimate of maximum residual hoop stress of 29 MPa [7], which is consistent with the calculated hoop stresses of 27.1-29.3 MPa in the central region of the quenched sample with maximum fictive temperature of 1940 K in this study.
Fig. 6. Calculated radial stress ${\sigma _{rr}}$ and hoop stress ${\sigma _{\theta \theta }}$ along radius on the surface of heat-affected zone of fused silica induced by localized CO2 laser 30 s 13.7 W preheating and 4 s 25.3 W heating. The fictive temperature profile on the surface is plotted for comparison.
4.3 Optical retardance
To be able to compare the calculated residual stress field with the photoelastic experimental result, which is based on the optical retardance caused by stress-induced birefringence, the residual stresses are linked with the retardation of linearly polarized plane waves traveling through the sample thickness. Because of the structure of the heat-affected zone induced by localized CO2 laser treatment, the residual stress field has an azimuthal symmetry. The directions of principal stresses are parallel or orthogonal to the radius. The local optical retardance depends on the difference between these two principal stresses [9,23]. The relative optical retardation, R, between waves polarized in the radial direction and in the tangential direction is obtained by the integrated principal stress difference along the z direction in this study, and can be calculated by
where K is the stress-optic coefficient of the fused silica and equal to 35 nm/cm/MPa, d is the thickness of the sample.Figure 7 shows the optical retardance for the light beam traveling through the sample thickness, which is calculated from the simulated residual stress field in Fig. 5. The retardance is about zero in the center of the heat-affected zone, and increases rapidly to the maximum of 37.5 nm at $r = 1.3$ mm, close to the location of the maximum residual radial stress at $r = 1.15$ mm on the surface. With radius further increasing, the retardance decreases gradually in the outer part of the heat-affected zone. The radial profile of the calculated optical retardance is in good agreement with the measured values, and the deviation between the calculated and measured maximum retardances is less than 10%.
Fig. 7. The radial profile of the calculated optical retardance for the light beam traveling through the sample thickness induced by the simulated residual stresses within the heat-affected zone of the fused silica sample. The measured retardance are plotted as circle dots for comparison.
Furthermore, the light pattern of the fused silica sample observed by a polariscope is also simulated. Generally, the fused silica sample is placed between two cross polarizers in the polariscope observation. It is assumed that the polarization direction of the upstream polarizer is the x-axis direction and the polarization direction of the downstream polarizer is the y-axis direction. Thus, the E-vector for the normal incidence on the fused silica sample has only the component ${E_x} = A\cos (\omega t - kz)$. After passing through the fused silica sample in the z-direction, the component ${E_x}$ can be resolved into two orthogonal components ${E_r}$ and ${E_\theta }$, as shown in Fig. 8(a). There is a phase retardation between components ${E_r}$ and ${E_\theta }$ due to the residual stress field within the heat-affected zone of the CO2 laser-heated fused silica sample, thus the E-vector is written as
Fig. 8. (a) The resolved E-vector at the location with polar radius r and polar angle $\theta$. The simulated (b) and measured (c) light patterns of the localized CO2 laser-heated fused silica sample observed by a polariscope.
Only the component ${E_y}$ of ${E_r}$ and ${E_\theta }$ can pass through the downstream polarizer, and the E-vector can be written as
Then, the mean value of intensity I observed by the polariscope is
Combining the Eq. (9) and the radial distribution of the calculated optical retardance in Fig. 7, the simulated light pattern for the CO2 laser-heated fused silica sample observed by a polariscope is plotted in Fig. 8(b), which is similar to the experimental polariscope image in Fig. 8(c). The simulated light pattern of the sample observed by a polariscope demonstrates that the proposed model based on the measured fictive temperature distribution can precisely evaluate the residual stress field of the localized CO2 laser-heated fused silica sample.
5. Conclusion
The fused silica sample was experimentally treated by localized CO2 laser heating, and the fictive temperature distribution within the heat-affected zone was characterized by using confocal Raman microscopy. The core region of the heat-affected zone showed higher fictive temperatures up to 1940 K and gradually decreased into the bulk to the pristine value of 1315 K. A simple model based on the measured fictive temperature distribution was proposed to determine the residual stress field within the heat-affected zone of fused silica. In the proposed model, it is assumed that the initial frozen-in state of fused silica, at which the value of temperature is equivalent to that of fictive temperature, is the zero-point of residual stress, and the generation of residual stress results from the thermoelastic contraction differences of fused silica with different fictive temperatures from initial frozen-in temperatures to the ambient temperature. Thus, the residual thermal volume strains of fused silica can be directly extracted from the fictive temperatures within the heat-affected zone, and furthermore the residual stress field is obtainable by finite element analysis of elastic equilibrium conditions and stress-strain relations. The calculated residual stress field indicates that the heat-affected zone suffers intensive tensile stresses. The maximum residual stresses on the surface reside in border regions of heat-affected zone, where the slope of fictive temperature is larger. Moreover, the maximum residual radial stress is about 50% larger than the maximum residual hoop stress, which is confirmed by the crack propagation direction of the introduced damage sites on the center or border regions of the heat-affected zone. The calculated maximum radial and hoop stresses on the surface are remarkably close to the reported critical fracture experimental results. The radial profile of the calculated optical retardance caused by stress-induced birefringence is in good agreement with the measured values. The simulated light pattern, induced by the calculated residual stress field within the heat-affected zone for the fused silica sample, is similar to the reported experimental polariscope image. Laser-induced critical fracture measurements and photoelastic measurements demonstrate that the proposed model based on the fictive temperature distribution can precisely evaluate the residual stress field within the heat-affected zone for the localized CO2 laser-heated fused silica sample.
Funding
National Natural Science Foundation of China (61804145); Laser Fusion Research Center, China Academy of Engineering Physics (LFRC-PD013).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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