A novel method for stress evaluation in chemically strengthened glass based on micro-Raman spectroscopy

Chemically strengthened glass is widely used for screen protection in mobile devices, and its strengthening processes and application fields have rapidly diversified. The origin of the strength is residual compressive stress induced by ion exchange, and the stress evaluation has been performed via the photoelastic effect. However, for a deep understanding of the nature of the strength and development of stronger glasses, we need a method directly connected to atomic-scale glass structures. Here, we propose a method based on the “stuffing” effect, where we can determine the residual stress non-contactively and non-destructively with a high spatial resolution using Boson, D1, D2, and A1 peaks in micro-Raman spectra. Finally, we show a plausible depth dependence of the residual stress. The strength of glass can be increased post-production by treatment with chemical processes, and such materials are used in everyday devices such as mobile phones. Here, the authors report a non-invasive method based on micro-Raman spectroscopy to analyse the compressive stress responsible for the increased strength of the treated glass.

O xide glass has become an essential industrial material owing to its advantages, such as high optical transparency, chemical durability, and formability, but at the same time, it has a critical weak point, namely, brittleness 1,2 . However, chemically strengthened glass, represented by Corning ® Gorilla ® Glass series, overcomes the weak point while retaining its advantages 3 . Therefore, the glass having thinness and strength becomes an indispensable material for screen protection in mobile devices such as smartphones and tablet computers, which are frequently exposed to scratches and drop impact. Also, the application of the chemically strengthened glass are now widespread, and the composition designs and strengthening processes have diversified for the development of stronger glass 4,5 .
The origin of the strength of the chemically strengthened glass is residual compressive stress (CS) in the vicinity of the glass surface and the occurrence of the stress has been explained by the "stuffing" effect accompanying ion exchange [6][7][8][9][10][11][12][13][14][15] . In general, alkali-aluminosilicate glasses have been used as parent glasses, which are immersed in molten salt with alkali ions that are larger than those of the parent glasses. Then, alkali ion exchange between the parent glasses and the molten salt, e.g., Na + to K + , occurs by diffusion. At this time, the immersing temperature (approximately 400°C) is kept lower than the glass transition temperature, so that the aluminosilicate network is hardly relaxed, which leads to the compression of the glass network. This compression produces a high residual CS of up to~1 GPa 16 , resulting in high break resistance.
Two indicators, i.e., the maximal CS and the depth of layer (DOL) 3,17 , are used to check the quality of the chemically strengthened glass, where the latter can be defined as the depth at which the residual stress equals zero. Conventionally, these two values are obtained from a depth profiling of stress via the photoelastic effect 18 ; e.g., stress evaluation devices based on both the photoelastic effect and the waveguide effect have been commercialized and used as a standard technique. Additionally, methods using both the photoelastic effect and light scattering have been recently proposed: Using these methods, Inaba et al. succeeded in evaluating the stress in two-step ion-exchanged glass 5 , and Hödemann et al. demonstrated stress detection with high spatial resolution using confocal microscopy 19 .
The methods based on the photoelastic effect are suitable to conveniently determine CS and DOL, but there is also an issue, i.e., we cannot find out correlation between the CS and atomicscale glass structures. Mostly, in the ordinary quality check of the chemically strengthened glass, it does not become problematic. However, originally, the CS should arise from the glass structures. So, it is significant that we directly evaluate the CS from the glass structures, which can lead to deep understanding of the strength by ion exchange and the development of stronger glass. Therefore, we propose a stress evaluation method based on the "stuffing" effect. In this method, we need three structural parameters, which can be obtained from Boson, D 1 , D 2 , and A 1 peaks in micro-Raman spectra. In this work, we investigated a commercial chemically strengthened glass, Corning ® Gorilla ® Glass 3 17 , which is aluminosilicate glass subjected to the ion exchange of Na + for K + , and melt-quenched glasses with no residual stress whose compositions are the same as the strengthened glasses. Consequently, we show a plausible variation of the depth dependence of the residual stress. We emphasize that the present method enables the non-contact, non-destructive stress evaluation with a depth and lateral resolution on the order of microns, and moreover, there is no restriction with respect to the refractive index distribution and surface flatness, as is needed for optical waveguiding 18 .

Results
Derivation of an equation for stress evaluation. The "stuffing" effect causing the CS in chemically strengthened glass has been understood by analogy with the thermal stress, i.e., the free volume expansion that is supposed to occur by ion exchange is spatially suppressed, leading to the CS [6][7][8][9][10][11][12][13][14][15] . From this viewpoint, ion exchange should correspond to the increasing temperature that creates the thermal stress. However, the free volume expansion by the ion exchange can be separated into two contributions, i.e., an intrinsic expansion by the ion exchange and an expansion by structural relaxation. It seems that the lack of this separation gives rise to a discrepancy between the theoretical and experimental values of the stress [6][7][8][9][10][11][12][13][14][15] . For an explanation of this discrepancy, a relaxation function has been adopted and anomalies with respect to the linear network dilation coefficient B and Young's modulus have been deliberated [6][7][8][9][10][11][12][13][14][15] .
However, our interpretation is different from the conventional one. We consider that the structural relaxation corresponds to the increasing temperature in the process of the thermal stress, when the intrinsic expansion by ion exchange is neglected. We assume that the mean atomic volume 20 in the local strengthened area can be uniquely determined by fractional exchange s and the degree of structural relaxation r as V(s, r), where s = 0 and 1 mean no and full exchange of alkali ions, respectively, and r = 0 and 1 mean no and full relaxation, respectively. Then, the residual stress σ by the "stuffing" effect is formulated as follows by analogy with the thermal stress 6-15 : Here, we use V and s in the present study instead of the molar volume and the molar fraction employed in the previous studies [6][7][8][9][10][11][12][13][14][15] , since V and s are appropriate for evaluations using Raman spectroscopy, as shown in the following sections. This replacement does not alter the argument of the "stuffing" effect. We use Young's modulus and Poisson's ratio of Gorilla ® Glass 3 as E and ν, with~70 GPa and~0.2 17 , respectively. Note that the equation includes just the mean atomic volume V as a variable, which determines the CS. Then, V naturally includes the structural relaxation term r and V(s, 1) is the virtual value when the structure was perfectly relaxed and the stress was perfectly released. Although several origins of the structural relaxation are conceivable, it is the term r that represents the degree of structural relaxation in our approach, so that V can be uniquely determined from r and s. The parameter σ 0 is the internal tensile stress, which can be determined by the following equation of equilibrium: Here, d is the sample thickness and z is the depth from the top surface of the sample. When σ(z) is symmetric with respect to Figure 1 shows V(s, r) as a function of s and r and the schematic structures of four glasses. One glass is the parent glass α, in which we see that Na + ions exist in the aluminosilicate glass network. The second glass represents an "ideal" chemically strengthened state on the line segment αβ with r = 0, indicating that the ions are exchanged together with no structural relaxation; in other words, no changes in the topological network structures occur. The third glass is the perfectly structure relaxed glass with no "stuffing" effect, i.e., a stress of zero, on the line segment αγ with r = 1. The fourth glass is a local state of the "real" chemically strengthened glass, where the structure is partially relaxed. The slope of αγ is greater than that of αβ, as noted in ref. 14 . Then, assuming that an increase in r leads to a linear increase in V, V(s, r) within the Δαβγ is expressed as follows: where V α = V(0, r), V β = (1, 0), and V γ = V(1, 1). Inserting Eq. (4) into Eq. (1), we obtain the following equation: This equation suggests that the residual stress can be determined by the three parameters, V, s, and r, which are calculated from Raman spectra in this work. Calahoo et al. have already reported an evaluation of the stress in ion-exchanged lithium silicate glasses using micro-Raman spectroscopy, in which the molar volume was obtained from the bond angles and lengths in Si-O-Si 21 , but it is noteworthy that we directly obtain the value of the local atomic volume from the Boson peak and we adopt the present method to the commercial chemically strengthened glass.
Raman spectra of the chemically strengthened glass. Figure 2 shows Raman spectra at z = 12 and 150 μm in Gorilla ® Glass 3, corresponding to the chemically strengthened and un-strengthened areas, respectively (see Methods for details regarding Raman spectroscopy). The spectra show peaks that are representative of typical aluminosilicate glasses 22 . In the spectra, we focused on the four peaks, i.e., Boson (~60 cm −1 ), D 1 (~480 cm −1 ), D 2 (~580 cm −1 ), and A 1 (1090-1100 cm −1 ) peaks [22][23][24] . Here the Boson peak arises from acoustic modes in medium-range structure, but the origin remains controversial [25][26][27][28]   peaks between the spectra at z = 12 and 150 μm in Fig. 2 and S D2 /S D1 can be a parameter to determine the degree of structural relaxation, as explained in the next section, where S D2 and S D1 are the integrated areas of the D 2 and D 1 peaks, respectively.
Quantification of structural relaxation. First, we investigated an effect of ion exchange on the glass network in the melt-quenched glasses which are located on the line segment αγ (see Methods for details regarding the melt-quenched glasses). As a result, we found out the approximately linear relation between S D2 /S D1 and s as shown in Fig. 3a. This means that K + ions intrinsically prefer to coordinate to the smaller rings, comparing with Na + ions. Conversely, for the chemically strengthened area (z = 6 μm), S D2 /S D1 is as small as nearly one-fifth compared with that for the same s on the line segment αγ. This suggests that formation of the threemembered rings by exchange of Na + for K + , i.e., the structural relaxation, is suppressed during the ion exchange process. Therefore, we quantify the degree of structural relaxation r as follows: where Δ denotes the difference with respect to the glass α. The subscript γ means the value is for the glass γ.
Evaluation of fractional exchange. Figure 3b shows the dependence of the A 1 peak positions k A1 (cm −1 ) for the same samples as in Fig. 3a. For both types of glasses, k A1 increases with increasing s, in which, clearly, the value in the strengthened area is located above the line segment αγ. The reason will be discussed later; briefly, it is expected that exchange of Na + for the larger ions K + and the CS simultaneously cause an increase in k A1 . On the other hand, the structural relaxation causes a decrease in k A1 . That is, the ion exchange with no structural relaxation in the "ideal" chemically strengthened glasses shows the maximal k A1 , the reduction from which is attributable to the decrease by the structural relaxation. Hence, under a linear approximation, we set the following equation using two undetermined coefficients, a and b, and Eq. (6): where a and b are determined to be 29 and 14, respectively, from the simultaneous equations for the strengthened area at z = 6 μm in the Gorilla ® Glass 3 and the glass γ, which gives Note that s can be evaluated from the A 1 and D 2 peaks in the Raman spectra using this equation.
Evaluation of mean atomic volume. The most innovative point in the present method is that we directly obtain the local mean atomic volume V in Eq. (1) using Raman spectroscopy. In the various oxide glasses, it is known that the Boson peak positions k BP (cm −1 ) are related to the mean atomic volume V (×10 −3 nm 3 /atom), although the reason has been controversial, whose relation is approximated by 25 V % 21 À 0:11k BP : ð9Þ Figure 3c shows the values of V evaluated from Eq. (9), where, as anticipated in Fig. 1, the value for the γ glass is greater than that for the α glass, and the value for the strengthened area is located below the line segment αγ. The undetermined value V β for the virtual glass β is determined to be 15.2 × 10 −3 nm 3 per atom by substituting the values of V and r in the strengthened area at z = 6 μm in the Gorilla ® Glass 3 for Eq. (4).
Evaluation of residual stress. Using Eqs. (5), (6), and (8), we obtain the equation for the residual stress in units of MPa as follows: σ z ð Þ % À30 10 3 0:021Δk A 1 À 1: It should be emphasized that this equation does not include the term V, but the residual stress can be evaluated by only the A 1 , D 1 , and D 2 peaks. This is because the evaluation of V using Eq. (9) determines the undetermined parameters, V α , V β , and V γ , in Eq. (5) and consequently V(r, s) in Eq. (1) can be uniquely determined. In Fig. 4b-d, we show depth dependences of S D2 /S D1 , k A1 , and k BP , respectively, in which k A1 and S D2 /S D1 increase and k BP decreases with decreasing z when the depth is shallower thañ 50 μm. Finally, we show the residual CS by comparing the estimation using Eq. (10) and the conventional surface stress meter based on the photoelastic effect and the waveguide effect, in which σ 0 was estimated to be~40 MPa using Eq. (3). The result in the present method is in good agreement with that of the surface stress meter in terms of the magnitude of the stress, the depth dependence, and the internal tensile stress. Moreover, we observed a local stress variation at z = 80 and 130 μm, which may show that we succeeded in the detection of stress irregularities.

Discussion
The fact that the results from the present method are approximately equal to those from the surface stress meter is surprising, but at the same time, suggests that the present method can be valid. Here we discuss the validity of the method as below.
Regarding E and ν, we used the datasheet values 17 for Gorilla ® Glass 3 in Eq. (5). These values are macroscopic values for the bulk glass, which should be used in the continuum mechanics. Therefore, the use of E and ν in Eq. (5), which is originally based on the atomic-scale structures, gives an uncertainty to Eq. (10). That is, it is unclear if E and ν in the atomic scale could be defined, and the values are expected to depend on r and s. However, attempts to evaluate E and ν from the atomic-scale structures have been performed 29,30 and the insight obtained from these researches will improve the accuracy of the stress evaluation by our method, which will enable the all-optical stress evaluation in chemically strengthened glass.
In our method, we used the melt-quenched glasses as reference state, but strictly, which also lacks accuracy. This is because the state which should be referred is "stress-free" chemically strengthened glass, which does not correspond to the meltquenched glass, owing to different thermal or ion exchange histories. The solution may be preparation of the stress-free state by heat treatment.
For the D 1 and D 2 peak, we considered the ratio of the integrated areas as the parameter of the structural relaxation. According to Hemley et al. 31 , the integrated area irreversibly increases by exposure to high pressure and subsequent releasing. This finding has been interpreted as the polymorphism of the SiO 4/2 tetrahedra network under high pressure and the subsequent formation of the three-membered rings, being the origin of the D 2 peak. On the basis of this picture, coincidentally recalling that K + ions prefer to coordinate to the small rings, we can naturally understand the increase in the intensity in the strengthened area in Fig. 3a, i.e., the ion exchange induces a high stress in the local area of the glass, but coincidently, the high immersing temperature under ion exchange promotes the stabilization of the TO 4/2 network in conjunction with the high stress, when it is expected that network polymorphism and the subsequent formation of the three-membered rings partially occur.
We used the linearity assumption in Eq. (4). The validation of the assumption is an important and controversial problem. Although it is not sufficient to explain the linearity, we have an atomic-scale picture: we obtain the following equation from Eqs.
(4) and (6): This equation means that an increase in the smaller rings together with the structural relaxation causes a linear increase in the mean atomic volume. This picture and the linearity assumption in Eq. (4) is correct, assuming that S D2 /S D1 is linearly proportional to volume fraction of the structural relaxed area. The validity of this assumption remains unclear, which also reduces the accuracy of the stress evaluation. The linear approximation for the A 1 peak position, Eq. (7), originally comes from the following relation: In this equation, Δk A1 (s, 0) and Δk A1 (s, 1) are changes with respect to the glass α in the "ideal" chemically strengthened and the perfectly structure relaxed glasses, respectively. Considering that Δk A1 (s, 1) comprises two contributions of the Na + ion exchange for K + and the formation of the small rings, defined as Δk A1 (s, 1) Na þ !K þ and Δk A1 (s, 1) SR , respectively, Eq. (12) is rewritten as follows in terms of s: The second line in Eq. (13) is qualitatively consistent with Eq. (7). Quantitatively, we may be able to explain the behaviors on the basis of the central-force model [31][32][33] . This model approximately gives the following relation to the A 1 peak position: where ω Α1 = ck A1 , c is the light speed, α is the restoring force constant of the T-O bond, m is the mass of oxygen, and θ is the bond angle of T-O-T. Considering a small change with respect to the glass α, Eq. (14) becomes In Fig. 3b, we see that the chemically strengthened glass shows a notable shift to a higher wavenumber compared with the meltquenched glass. This shift is because ∂k A1 (s, 0)/∂s in Eq. (13), i.e., a in Eq. (7), is dominantly positive. In the case of SiO 2 glass, according to Weigel et al. 33 , when the glass is pressurized using a penetrating medium, He, α increases and a decrease in θ is suppressed. Then, the term of ∂α/∂p (p: pressure), which corresponds to ∂α/∂s in Eq. (15), becomes dominant, leading to a shift in k A1 to a higher wavenumber. The same scenario can be established in the ion exchange process. On the other hand, the quantitative interpretation of the coefficient of rs in Eq. (13), i.e., b in Eq. (7), seems to be puzzling at present. For a further understanding of the structural relaxation, determination of α and θ by adopting the central-force model and the modified one to the peaks, including R band (~380 cm −1 ) etc. is helpful 33,34 .
Let us consider whether this approach is applicable to other chemically strengthened glasses having different compositions from Gorilla ® Glass 3. To get to the point, we think the answer is yes if Eq. (1) is satisfied and the three parameters can be obtained from Raman spectra, when the empirical coefficients in Eq. (10) should depend on the glass composition because r and s dependences of V should be affected by the composition. In the case of the two-step ion exchange process, which is the standard one in the present day for giving deeper DOL, one solution is replacement of V (r, s) with V (r, s 1 , s 2 ), where, for example, s 1 and s 2 are fractional exchange for Na + and K + , respectively, for Li + -based pristine glass. Then, the V-s plane in Fig. 1 is extended to a V-s 1 -s 2 space, and the observation point by the Raman spectroscopy corresponds to a point in the space.
In summary, we proposed a method to evaluate the CS of the chemically strengthened glass, which is directly connected to the atomic-scale structures. The method is based on the "stuffing" effect and the micro-Raman spectroscopy and enables the non-contact, non-destructive stress evaluation with a depth and lateral resolution on the order of microns, without restriction with respect to the refractive index distribution and surface flatness. Although the method needs several linear assumptions for the structural change by ion exchange and relaxation and the macroscopic values, E and ν, which reduces the accuracy of the stress evaluation, we obtained the plausible result in good agreement with that of the surface stress meter in terms of the magnitude of the stress, the depth dependence, and the internal tensile stress.

Methods
Sample preparation and characterization. The samples used were two types of oxide glasses. One sample was the commercial chemically strengthened glass, Corning ® Gorilla ® Glass 3. The glasses before and after ion exchange were purchased. The version of the glasses was not the latest but were suitable for the present study because they were expected to possess the conventional aluminosilicate network. Using a wavelength dispersive X-ray fluorescence spectrometer (XRF; S8 Tiger, Bruker, USA), the composition of the Gorilla ® Glass before the ion exchange was determined approximately to be 16Na 2 O-13Al 2 O 3 -69SiO 2 -0.3MgO-0.01SnO 2 in mol%. On the basis of this composition, the glasses which have compositions of 16(1−s)Na 2 O-sK 2 O-13Al 2 O 3 -69SiO 2 -0.3MgO-0.01SnO 2 (s = 0.2, 0.4, 0.6, 0.8, and 1.0) were prepared by a melt-quenching method: NaOH, KOH, Al 2 O 3 , SiO 2 , MgO, and SnO 2 reagents with purities over 99.9% were used. Their powders were weighed and mixed to obtain the above compositions. The batches were placed in a platinum crucible and melted at 1650°C for 1 h in air using an electric furnace at a heating rate of 10°C/min. The melts were quenched in a furnace maintained at 500°C and were subsequently removed from the furnace.
The fractional exchange s in the ion-exchanged area of Gorilla ® Glass 3 was obtained by a scanning electron microscope equipped with energy dispersive X-ray spectroscopy (EDX; JSM-6500F, JEOL, Japan). Note that we did not add Li 2 O and B 2 O 3 to the compositions based on XRF so that these compositions were compared with those of EDX, in which Li and B, being light elements, could not be detected, although there is a possibility that the chemically strengthened glass contains a few mol% of these contents.
A depth profile of CS in Gorilla ® Glass 3 was obtained by using a surface stress meter (FSM-6000LE, Orihara Industrial, Japan) for comparison with the result in this work.
Raman spectroscopy. Raman spectra of the samples were obtained using a conventional system (NRS-5500, JASCO, Japan). The system included confocal microscopy with a 100× objective lens with a numerical aperture (NA) of 0.9 and an excitation source of 532-nm laser light. The scattered light was detected with a backscattering configuration when the light incident to the samples was linearly polarized, while the scattered light was detected with no polarizer. The depth and lateral resolutions were approximately 2 and 1 μm, respectively. We obtained a depth (z) dependence of the Raman spectra, where z is a depth from the top surface of the glass plate, namely, an actual focus position, and has been compensated by refractive index of the sample. Here the index is assumed to be 1.5, being independent of z, and this assumption is reasonable because the values of the index are reported to be 1.50 and 1.51 (for 590-nm light) for the un-strengthened and the chemically strengthened areas of Gorilla ® Glass 3, respectively 17 . The spectra in a range of 300-1300 cm −1 were deconvoluted by eight Gaussian peaks, which are located at~380,~480,~580,~640,~750,~890, 980-1000, and 1090-1100 cm −1 , as shown in Fig. 2. The assignment of each peak is described in refs. 22,24 . For the Boson peak at~60 cm −1 , following the conventional analysis method 25,35-38 , the peak position was determined by fitting with the log-normal distribution function for the Raman intensity at 30-120 cm −1 reduced by the Bose factor.