METHODS OF CHARACTERIZING IONEXCHANGED CHEMICALLY STRENGTHENED GLASSES CONTAINING LITHIUM
Methods of characterizing ionexchanged chemically strengthened glass containing lithium are disclosed. The methods allow for performing quality control of the stress profile in chemically strengthened Licontaining glasses having a surface stress spike produced in a potassiumcontaining salt, especially in a salt having both potassium and sodium. The method allows the measurement of the surface compression and the depth of the spike, and its contribution to the center tension, as well as the compression at the bottom of the spike, and the total center tension and calculation of the stress at the knee where the spike and the deep region of the stress profile intersect. The measurements are for a commercially important profile that is nearparabolic in shape in most of the interior of the substrate apart from the spike.
This application is a continuation of U.S. patent application Ser. No. 15/267,392, filed on Sep. 16, 2016, which claims the benefit of priority to U.S. Provisional Patent Application Ser. No. 62/219,949, filed on Sep. 17, 2015, and which is incorporated by reference herein.
FIELDThe present disclosure relates to chemically strengthened glass, and in particular relates to methods of characterizing ionexchanged chemically glasses containing lithium.
BACKGROUNDChemically strengthened glasses are glasses that have undergone a chemical modification to improve at least one strengthrelated characteristic, such as hardness, resistance to fracture, etc. Chemically strengthened glasses have found particular use as cover glasses for displaybased electronic devices, especially handheld devices such as smart phones and tablets.
In one method, the chemical strengthening is achieved by an ionexchange process whereby ions in the glass matrix are replaced by externally introduced ions, e.g., from a molten bath. The strengthening generally occurs when the replacement ions are larger than the native ions (e.g., Na+ ions replaced by K+ ions). The ionexchange process gives rise to a refractive index profile that extends from the glass surface into the glass matrix. The refractive index profile has a depthoflayer or DOL that defines a size, thickness or “deepness” of the iondiffusion layer as measured relative to the glass surface. The refractive index profile also defines a number of stressrelated characteristics, including a stress profile, a surface stress, center tension, birefringence, etc. The refractive index profile defines an optical waveguide when the profile meets certain criteria.
Recently, chemically strengthened glasses with a very large DOL (and more particularly, a large depth of compression) have been shown to have superior resistance to fracture upon face drop on a hard rough surface. Glasses that contain lithium (“Licontaining glasses”) can allow for fast ion exchange (e.g., Li+ exchange with Na+ or K+) to obtain a large DOL. Substantially parabolic stress profiles are easily obtained in Licontaining glasses, where the ionexchange concentration profile of Na+ connects in the central plane of the substrate, shrinking the traditional central zone of the depthinvariant center tension to zero or negligible thickness. The associated stress profiles have a predictable and large depth of compression, e.g., on the order of 20% of the sample thickness, and this depth of compression is quite robust with respect to variations in the fabrication conditions.
A stress profile of particular commercial importance is a nearparabolic (substantially parabolic) profile that has a “spike” near the surface. The transition between the parabolic portion of the profile and the spike has a knee shape. The spike is particularly helpful in preventing fracture when the glass is subjected to force on its edge (e.g., a dropped smart phone) or when the glass experiences significant bending. The spike can be achieved in Licontaining glasses by ion exchange in a bath containing KNO_{3}. It is often preferred that the spike be obtained in a bath having a mixture of KNO_{3 }and NaNO_{3 }so that Na+ ions are also exchanged. The Na+ ions diffuse faster than K+ ions and thus diffuse at least an order of magnitude deeper than the K+ ions. Consequently, the deeper portion of the profile is formed mainly by Na+ ions and the shallow portion of the profile is formed mainly by K+ ions.
In order for chemically strengthened Licontaining glasses to be commercially viable as cover glasses and for other applications, their quality during manufacturing must be controlled to certain specifications. This quality control depends in large part on the ability to control the ionexchange process during manufacturing, which requires the ability to quickly and nondestructively measure the refractive index (or stress) profiles, and particular the stress at the knee portion, called the “knee stress.”
Unfortunately, the quality control for glasses with spike stress profiles is wanting due to the inability to adequately characterize the profiles in a nondestructive manner. This inability has made manufacturing of chemically strengthened Licontaining glasses difficult and has slowed the adoption of chemically strengthened Licontaining glasses in the market.
SUMMARYAn aspect of the disclosure is directed to methods of characterizing chemically strengthened Licontaining glasses having a surface stress spike, such as produced by an ionexchange process (i.e., an indiffusion of alkali ions) whereby in an example Li+ is exchanged with K+ and Na+ ions (i.e., Li+⇔K+, Na+). The methods result in a measurement of the surface compression and the depth of the spike, and its contribution to the center tension, as well as the compression at the bottom of the spike, and the total center tension.
The method is preferably carried out to obtain a commercially important stress profile, e.g., one that is nearparabolic in shape in most of the interior of the substrate other than the spike adjacent the substrate surface. The spike is generally formed by the slower diffusion (and thus shallower) K+ ions while the substantially parabolic portion is formed by the faster (and thus deeper) diffusing Na+ ions. The method allows for confirmation that the profile has reached the nearparabolic regime, e.g., has a selfconsistency check. The method can also include performing quality control of the glass samples being process. Such quality control is important for a commercially viable manufacturing process.
The present disclosure provides a method for quality control of the stress profile in chemically strengthened Licontaining glasses having a surface stress spike produced in a potassiumcontaining salt, especially in a salt having both potassium and sodium. The method allows the measurement of the surface compression and the depth of the spike, and its contribution to the center tension, as well as the compression at the bottom of the spike, and the total center tension, for a commercially important profile that is nearparabolic in shape in most of the interior of the substrate (apart from the spike). The method allows to check that the profile has reached the nearparabolic regime, e.g., has a selfconsistency check. The method provides a critically important tool for the quality control that is necessary for the adoption of lithiumcontaining glasses that allow the fabrication of these important profiles.
Prior art methods of measuring the stress level at the bottom of the spike (i.e., the knee stress) are limited by the relatively poor precision of measuring the position of the criticalangle transition of the transverse electric (TE) angular coupling spectrum. This poor precision is an inherent aspect of the TE transition, which is broad and hence appears blurred in the prismcoupling spectra. This lack of sharpness causes the measured position of the mode lines to be susceptible to interference from nununiformity in the angular distribution of the illumination (e.g., background nonuniformity), as well as simply image noise.
Several of the methods disclosed herein avoid the need to measure the position of the criticalangle of the TE transition precisely. In one aspect of the method, the surface stress and the slope of the stress in the spike are measured, as well as the depth (depthoflayer, or DOL) of the spike, where the DOL is measured very precisely by using only the criticalangle transition of the TM wave. This TM transition is sharper than the TE transition and thus allows for a much more precise measurement. Thus, in an example of the method, the TE mode spectrum (and in particular the TE transition of the TE spectrum) is not used to determine the DOL of the spike.
Knowing the surface stress and slope of the spike, and the depth of the spike (the aforementioned DOL), the stress at the bottom of the spike is determined, where the bottom of the spike occurs at the depth=DOL. This is the “knee stress” and is denoted herein as either CS_{knee }or CS_{k }or in the more general form σ_{knee}. The rest of the calculation of the stress profile attributes then proceeds according to the prior art method.
A second method disclosed herein avoids a direct measurement of the knee stress and calculates the knee stress by using the birefringence of the last guided mode common to both the TM and the TE polarization, and a previously determined relationship between the birefringence of said last common guided mode and the stress at the knee. Advantage is taken of the generally better precision of measurement of the mode positions in comparison to the precision of measurement of critical angle, and in particular of the critical angle of the TE wave in the case of spiked deep profiles in a Licontaining glass.
Advantages of the methods disclosed herein is that they are nondestructive and can carried out with highthroughput and with high precision to determine the critical parameters associated with the diffusion process in making chemically strengthened glasses. These critical parameters include CS, depth of spike, estimate of the compression depth, and frangibility status (based on an estimate of CT that is provided by the method). Another advantage is that the methods can be implemented with relatively modest software enhancements on existing hardware used for quality control of the currently produced chemically strengthened glasses.
One major specific advantage of the new methods disclosed herein is a significant improvement in the precision of the kneestress estimate by avoiding the effects of large errors in the direct measurement of the TE critical angle. This precision improvement is important because it allows for improved quality control of the chemically strengthened glass product.
The other advantage of the methods disclosed herein is an increase in domain of applicability of the methods, i.e., an increase in the size of the measurement process window. The prior art methods have process windows or “sweet spots” for making measurements, where there was no leaky mode occurring in the vicinity of the criticalangle transition for the TM and TE spectra. Such a leaky mode causes significant deformation of the angular distribution of intensity in the vicinity of the transition, and is a source of very significant and unacceptable errors that are difficult to eliminate or effectively compensate for in realistic situations.
In the first of the new methods, only the TM spectrum is required to be free of leakymode interference, which on average doubles the range of the sweet spot.
In both of the new methods, the effect of errors in the criticalangle measurement is significantly reduced because the critical angle is not used for a direct measurement of the knee stress. This leads to an effective increase in the range of the sweet spot.
Additional features and advantages are set forth in the Detailed Description that follows, and in part will be readily apparent to those skilled in the art from the description or recognized by practicing the embodiments as described in the written description and claims hereof, as well as the appended drawings. It is to be understood that both the foregoing general description and the following Detailed Description are merely exemplary, and are intended to provide an overview or framework to understand the nature and character of the claims.
The accompanying drawings are included to provide a further understanding, and are incorporated in and constitute a part of this specification. The drawings illustrate one or more embodiment(s), and together with the Detailed Description serve to explain principles and operation of the various embodiments. As such, the disclosure will become more fully understood from the following Detailed Description, taken in conjunction with the accompanying Figures, in which:
Reference is now made in detail to various embodiments of the disclosure, examples of which are illustrated in the accompanying drawings. Whenever possible, the same or like reference numbers and symbols are used throughout the drawings to refer to the same or like parts. The drawings are not necessarily to scale, and one skilled in the art will recognize where the drawings have been simplified to illustrate the key aspects of the disclosure.
The claims as set forth below are incorporated into and constitute part of this Detailed Description.
In the DIOX process discussed in connection the method disclosed herein, two different types of ions Na+ and K+ replace another different ion Li+ that is part of the glass body 21. The Na+ and K+ ions can be introduced into the glass body 21 either sequentially or concurrently using known ionexchange techniques. As noted above, the Na+ ions diffuse faster than the K+ ions and thus go deeper into the glass body 21. This has an effect on the resulting refractive index profile and stress profile, as discussed below.
The deeper second region R2 may be produced in practice prior to the shallower region. The region R1 is adjacent substrate surface 22 and is relatively steep and shallow, whereas region R2 is less steep and extends relatively deep into the substrate to the aforementioned depth D2. In an example, region R1 has a maximum refractive index n_{0 }at substrate surface 22 and steeply tapers off to an intermediate index n_{i}, while region R2 tapers more gradually from the intermediate index down to the substrate (bulk) refractive index n_{s}. The portion of the refractive index profile n(z) for region R1 represents spike SP in the refractive index having a depth DOS.
As is known in the art, the fringes or mode lines 52TM and 52TE in the mode spectrum can be used to calculate surface compression or “compressive stress” CS and depth of layer DOL associated with an ionexchange layer that forms an optical waveguide. In the present example, the mode spectrum 50 on which
The measured values of CS and DOL were 575 MPa and 4.5 microns, respectively. These are the parameters of the K+ enriched layer or spike region R1 adjacent sample surface 22 (see
In the mode spectrum 50 for a chemically strengthened Licontaining glass having undergone a (Li+⇔K+, Na+) ion exchange, the relative positions of the TM and TE mode spectra 50TM and 501E are shifted. This shift can be measured by the relative positions of the last (i.e., leftmost) fringes 52TM and 52TE, which correspond to the highestorder guided modes. As noted above, this shift is denoted CS_{tot }in
The effective index of the transition corresponds to the effective index that occurs at the depth of a characteristic “knee” or transition KN in the stress profile, and is denoted in
The direct measurement of the knee stress CS_{knee }from the birefringence of the criticalangle intensity transition of the TE and TM mode lines 52TE and 52TM presents some problems. One problem is due to shifting of the apparent position of the transition when a leaky mode or a guided mode has effective index very close to the index corresponding to the critical angle. For example, the broader dark fringe can occur approximately at the same location as the criticalangle transition in the upper half of the combined spectra of
Avoiding the aforementioned shiftinduced error requires that both the upper and lower spectra (i.e., the TM and TE spectra 50TM and 50TE) the have a fractional part of the mode number between about 0.2 and 0.6, which is quite restrictive. In addition, even when this condition is satisfied, the measurement of the critical angle for the TE spectrum is not very precise due to a relatively blurry TE intensity transition. Note for example how the criticalangle transition in the bottom half of
The methods disclosed herein utilize measurements of the fringe spectrum provided by the potassium penetration resulting from ion exchange, along with the position of the intensity transition in the TM spectrum (e.g., transition from total internal reflection (TIR) to partial reflection) relative to the positions of the TM fringes. These measurements can be combined and used for effective quality control of a family of stress profiles that help enable superior resistance to fracture during face drops. The profiles of this family are similar in shape to a powerlaw profile with a spike.
The spike SP is a nearsurface region that has a small thickness when compared to the substrate thickness. For example, the spike may be 10 μm deep, while the substrate may be 800 μm thick. The spike may have a shape similar to erfcshape, but may also be similar to a linear depth distribution, Gaussian depth distribution, or another distribution. The main features of the spike are that it is a relatively shallow distribution and provides substantial increase of surface compression over the level of compression at the bottom (deepest end) of the spike, which ends at knee KN.
Another feature of the spike SP in
In one embodiment of the method, the CS_{SP }and DOL_{SP }of the spike SP are measured using a traditional FSM measurement. For increased precision of the DOL measurement, it may be preferred that the DOL_{SP }of the spike be measured using the TM spectrum only, as the criticalangle transition in the example Licontaining glasses exchanged in mixtures of Na and K is substantially sharper and less prone to measurement errors. Note that in the present disclosure the denominations DOL and DOL_{SP }are used interchangeably to refer to the same quantity, namely, the depth of layer of the Kenriched nearsurface spike layer having high compressive stress CS_{SP}.
A center tension CT contribution of the spike is calculated using the equation
where T is the sample thickness (see
where σ_{knee }is the stress at the knee of the profile, e.g., at the bottom of the spike and is given by:
where n_{crit}^{TE }and n_{crit}^{TM }are the effective indices of the criticalangle intensity transitions as illustrated on
σ_{knee}=CS_{knee}=BR/SOC.
This equation can also be written more generally as
σ_{knee}=CS_{knee}=(CFD)(BR)/SOC
where CFD is calibration factor between 0.5 and 1.5 that accounts for systematic offsets between the recovered criticalangle values having to do with fundamentally different slopes of the TM and TE intensity transitions, different shape of the TM and TE index profiles in the vicinity of the knee, and specifics of the method by which the location of the intensity transition is identified. As noted above, the parameters or σ_{knee}, CS_{k}nee, CS_{k }and CS_{K }all refer to the same quantity, namely, the knee stress.
As illustrated by the dashed line curve in
Auxiliary PowerLaw Profile Relationships
A detailed description of the relationships that hold for the auxiliary powerlaw profile is now provided, as well as the associated method of using them to calculate the parameters of the model spiked profile for the purposes of quality control.
The auxiliary powerlaw profile provides the stress as a function of distance z from the center.
The spiked profile has a somewhat smaller depth of compression DOC given by the expressions
The depth of compression DOC of the spiked profile is smaller than that of the auxiliary power profile by approximately:
The change in the depth of compression DOC caused by the spike in the profile can be normalized to the compressive tension CT of the auxiliary power profile as follows:
In the specific example of a parabolic auxiliary profile, the following relationships hold:

 The auxiliary profile has a compression depth DOC_{par }given by:

 The total center tension CT_{tot }of the profile equals the sum of spike center tension CT_{sp }and the parabolic portion center tension CT_{p}:
CT_{tot}=CT_{p}=CT_{sp }

 The depth of compression DOC of the spiked powerlaw profile can be calculated by using the expression:
The approximate expressions at the end of the above equation are valid when the CT contribution of the spike is significantly smaller than the CT contribution of the auxiliary profile (i.e., the parabolic portion PP).
Example Method Based on ApproximationAn example method of quality control utilizes an approximation approach that includes a measurement of the mode spectrum due to the spike. The method then includes estimating a contribution of the spike to the center tension CT by estimating a compression at the knee KN of the profile and subtracting that knee compression from the surface compression in the calculation of the spike contribution to the center tension. The method then includes estimating a contribution to the center tension CT due to the deep powerlaw profile portion PP excluding the spike, also taking advantage of the estimated knee stress. The method then includes finding the total center tension CT_{tot }as a sum of the contributions of the auxiliary deep powerlaw profile and of the spike, i.e., CT_{tot}=CT_{sp}+CT_{p}. In general, the CT contribution of the deep portion may be denominated CT_{deep}, which can be interchangeably used with CT_{p}when the deep portion is represented as having a a powerlaw shape.
In addition, the method can include estimating the compression depth DOC of the profile by using an exact formula for the model profile, or an approximate formula that gives the DOC as the DOC of an auxiliary powerlaw profile less a small DOC reduction due to the spike, i.e., DOC=DOC_{S}+ΔDOC_{sp }(in the mathematical formula a negative ΔDOC_{sp }is added to DOC_{S}). Note also that ΔDOC_{sp }is sometimes labeled simply as ΔDOC in the present disclosure, as only the shift in DOC that is due to the spike is considered in this disclosure.
In one example of the method, the DOL of the spike SP is used to verify that the powerlaw portion PP of the profile (see
More Precise Method
The abovedescribed method is based on approximation and is thus a somewhat more simplified version of a more precise method. The simplification incurs only a minor error when the CT contribution of the spike is much smaller than the CT contribution of the auxiliary powerlaw profile. The CT contribution of the spike shifts the deep powerlaw portion PP vertically by the amount CT_{sp }relative to the auxiliary powerlaw profile. As a result, the compression at the knee of the model spiked profile is actually smaller than the compression of the auxiliary profile at the knee depth by the amount CT_{sp}.
Furthermore, there is a minor change in compression of the auxiliary powerlaw profile between the surface and the depth of the knee, and, for a forcebalanced powerlaw profile the CT is actually equal to
The following represents an example of a more precise method for determining the parameters of the model spiked powerlaw profile from the mode spectrum as obtained from prismcoupling measurements of a chemically strengthened glass sample:

 a) Calculate preliminary

 b) Calculate preliminary surface compression of the auxiliary profile

 c) (Optional alternative to steps 4, 5, and 6) Calculate preliminary

 d) Calculate more precise

 e) Calculate more precise

 f) Calculate more precise CT_{tot}^{(1)}=CT_{p}^{(1)}+CT_{sp}^{(1) }
 g) (Optional; usually unnecessary)—can continue iteration, finding more and more precise values for CT_{sp }and CS_{par }until desired level of convergence or precision. More than one iteration would rarely be needed. More than one iteration may be useful in relatively thin substrates in which the depth of the spike may represent more than about 3% of the substrate thickness.
 h) (Optional) Determine depth of compression of the profile, for example using one of the forms of the equation:
The abovedescribed method allows for the application of the generic auxiliary powerlaw profile for the QC of a spiked doubleionexchanged profile having a stress distribution reasonably well described by a spiked powerlaw profile model. The method avoids a direct measurement of the knee stress. Instead of directly measuring n_{crit}^{TE }to evaluate the knee stress from the earlier described equation,
the knee stress is found by observing that it occurs at a depth equal to the penetration of the spiking ion, e.g., at a depth of spike DOL_{sp}.
CS_{knee}≡σ_{knee}=σ(depth=DOL_{sp}).
The above strict definition of the knee stress is most easily understood for the case where the profile has an abrupt change in slope at the location of the knee. In practice, most profiles change slope gradually, although fast, in the vicinity of depth=DOL_{sp}, and a knee occurs approximately at depth=DOL_{sp }as measured from the mode spectrum. Hence, in the calculation of a σ_{knee }often a calibration factor of magnitude comparable to 1 is used, in part to account for differences between the continuous distribution of stress and the abrupt change in stress slope in a simple explicit description of a model having a steep linear truncated stress spike connected to a deep region of slowly varying stress.
The surface stress and its slope are obtained from the prismcoupling measurements of the effective indices of the TM and TE modes confined in the depth region of the spike by a measurement of the CS, the stress slope s_{σ} and DOL of the spike.
The surface stress and the slope of a linear spike can be found using the following analysis: Using the WKB approximation the turning points x_{1 }and x_{2 }of the two lowestorder modes in an optical waveguide can be found using the relations
where n_{0 }is the surface index of the profile having linearly decreasing with depth dielectric susceptibility, n_{1 }is the index of the lowestorder mode, n_{2 }is the effective index of the secondlowestorder mode, and λ is the optical wavelength. The surface index of the linear profile is found from the same first two modes by the relation:
n_{0}^{2}≡n_{surf}^{2}≈n_{1}^{2}+1.317(n_{1}^{2}−n_{2}^{2})
For profiles having n_{1}−n_{2}<<n_{1}, an even simpler relation can be used:
n_{0}≡n_{surf}≈n_{1}+1.3(n_{1}−n_{2})
The index slope of each of the TM and TE index profiles associated with the stress profile of the spike is then given by:
The above relations for the surface index and the index slope of the linear profile can be applied for both the TM and TE mode spectra, to obtain the TM and TE surface indices n_{surf}^{TM }and n_{surf}^{TE}, and the TM and TE profile index slopes s_{n}^{TM }and s_{n}^{TE}. From these, the surface stress CS, and the stress slope s_{σ} can be obtained:
where as noted above, SOC stands for stressoptic coefficient. Note that when more than two guided modes are supported in either the TM or TE polarization, or both, then the precision of the slope measurement can be improved by taking advantage of the measured effective indices of more than two modes per polarization, by using a linear regression to associate the measured effective indices of multiple modes with a single index slope for each polarization.
There is now one step left to obtain the knee stress, namely a measurement of the spike depth OL_{sp}, which is obtained by analysis of the TM spectrum. The index space between the highestorder guided mode and the index corresponding to the TM critical angle is assigned a fraction of a mode based on what fraction it represents of the spacing of the previous two modes, and, if desired for higher precision, on how many guided modes are guided. This type of DOL calculation is routinely done by the FSM6000 instrument.
Finally, the depth of the spike is given by the formula:
where N is the number of guided TM modes, including the fraction of a mode assigned to the space between the last guided mode and the critical index n_{crit }of the intensity transition, λ is the measurement wavelength, and n_{crit }is the effective index corresponding to the critical angle in the TM spectrum, indicated as n_{knee}^{TM }in
With DOL_{sp }measured with good precision from the TM coupling spectrum, the knee stress CS_{knee }at the bottom of the spike is found using the relationship:
CS_{knee}≡σ_{knee}≡σ_{sp}(x=DOL_{sp})=CS+s_{σ}×DOL_{sp }
Accounting for systematic differences between real profiles in the vicinity of the knee point, and the assumed model for the spike shape, the knee stress can be found by the following more general relationship:
CS_{knee}≡σ_{knee}≡σ_{sp}(x=DOL_{sp})=CS+KCF×s_{σ}×DOL_{sp }
where the knee calibration factor KCF is usually between 0.2 and 2, and serves to account for the difference in shape between a real spike distribution and the assumed model of the spike shape, as well as the particular way that the DOL_{sp }is calculated from the mode spectrum. For example, a commonly used equation for the surface index is
n_{0}≡n_{surf}≈n_{1}+0.9(n_{1}−n_{2}).
which uses a factor of 0.9 instead of the factor 1.317 which is accurate for linear spikes. When the formula for surface index with a factor of 0.9 is used, the resulting calculated DOL appears higher than the purely linearspike DOL.
This improved method of measurement of the knee stress by use of a precise measurement of DOL_{sp}, when used in the approximate algorithm or in the more precise in the iterative algorithm for extraction of the parameters of the spiked deep profile described above for the general powerlaw auxiliary profile (or in the previous disclosure for the quadratic auxiliary profile), provides a qualitycontrol method with improved precision of the estimate of CT for frangibility control. The knee stress is by itself an important parameter of glass strength and the precision improvement of that parameter is also of value. The improved method also increases the breadth of the sweet spot for measurement typically by a factor of two or even more.
In another embodiment involving indirect measurement of the knee stress, the method makes use of a strong correlation between the knee stress and the birefringence of the last guided mode of the spike. When the spike CS and DOL are kept in very narrow respective ranges, then a strong correlation forms between the sought after knee stress and the difference in the effective index between the last guided TM mode and the last guided TE mode of the spike.
The method exploits the birefringence of the last guided mode of the spectrum acquired by the prism coupler for quality control (QC) measurements. Here we will use formulas for a generic power profile with exponent ‘n’. For a powerlaw profile n=2, for cubic n=3 but also fractional profiles like n=2.37 is possible for making the equations generic. In the present disclosure, when n refers to a power of the profile, it has the same meaning as p which is also used to denominate the power of the auxiliary deep profile.
Using the power (parabolic for n=2 in this case) auxiliary profile, illustrated with the help of
where L is the thickness. The depth of layer DOL_{deep }of the deep part of this power profile with exponent ‘n’ is given by
The FSM measures FSM_DOL of the spike as approximately the diffusion depth given by 2√{square root over (D·τ)} where D is the diffusion coefficient and τ is the time of diffusion.
For a spike with the shape of erfcfunction, it is empirically found that the knee stress can be assumed to occur at a depth of ˜K_{1 x}FSM_DOL=1.25×FSM_DOL, such that most of the stressarea of the spike to be included in the CT calculation.
One can get an approximate equation for the ΔCT_{spike }due to the spike contribution. Here, K_{1 }is an empirical factor set at 1.25 for this particular case. The factor K_{1 }serves to compensate for nonzero residual stress contributed by the tail of the spike at depth=FSM_DOL by adjusting the point at which the knee stress is estimated.
The point σ_{2}′(K_{1}×FSM_DOL)=σ_{2}′(1.25×FSM_DOL) is very close to the CS between the transition between guided modes and continuum in the spiked lithium glass samples. This point is called the CS_{knee }as shown in
Since the powerlaw profile will be slow varying compared to the spike, it can be assumed that the stress at ˜(K_{2})'FSM_DOL˜(13)×FSM_DOL in the parabolic region would not feel the presence of the spike.
This allows the following approximations to be employed:
where using the parabolic equation in (1), it is found that:
The factor K_{2 }accounts for nonzero spike stress distribution beyond the depth DOL_{sp }calculated from the mode spectrum.
It can be demonstrated that if one uses a factor 2 instead of 3 the results are almost the same, in some cases varying just 1%3% of σ_{1}(0). Therefore, if one can find the approximated value of
in the FSM, formula (6) can be used to compute the original stress of the first stress parabola within this range of error.
In practice one can measure approximately
by looking at the stress generated at the transition between guided modes and continuum in the spike on Liglass samples.
This point, where approximately
can be used as the point CS_{knee }as shown in
This is in addition to the FSM_DOL and the _{CS}˜σ_{2}′(0) given by the FSM for the spike. Therefore CT_{deep}=˜σ_{1}(0)/n, where for a parabolic deep profile n=2, and ΔCT_{spike }is given in (3) as (repeated for convenience)
From there one can (repeating the previous equations) then compute the total center tension equals the sum of the contributions of the spike and of the parabolic portion:
CT_{tot}=CT_{deep}+ΔCT_{spike} (7)
If desired the depth of compression of the spiked powerlaw profile can be calculated/estimated by using the expression:
These equations assume that the deep part of the profile is a generic power profile (parabolic for n=2) in nature and has an added spike near the surface. Its validity is better matched when the spike is small in stress amplitude and not so deep in comparison to the deeper part of the profile.
In addition to the generic power ‘n’ profile, the important difference between this disclosure and the prior art methods is how the FSM_DOL is computed and how the CS_{knee}=σ_{1}′(K_{2}×FSM_DOL/L) is found using the “last common mode” measured, referring to the highestorder guided mode that appears both in the TM and the TE spectrum. In an example, if each of the TM has 3 modes and the TE spectrum has 3 modes, then the last common mode is assigned to the third mode of each spectrum, when modes are ordered by descending effective index. If the TM spectrum has 3 modes and the TE spectrum has 2 modes, then the last common mode is the second mode in each spectrum when the modes in each spectrum are ordered by descending effective index.
This has direct correspondence to the range of value in which a measurement is possible with reasonable noise and certainty. This is illustrated in
The wavelength of the measurement light was 598 nm using a prism coupler system and camera. It can be observed that, depending on the diffusion time. a “new mode/fringe” starts to appear at the edge of the screen. This leads to noise in the image and an unstable determination of the transition between the spike and the long tail of the stress profile. This point is referred as the boundary/continuum or “knee point” due to the inflection on the stress curve it represents, being illustrated in
By performing several measurements in a time series of samples described above, significant trends can be observed.
For our purposes, regions with 2 or more modes are acceptable but in practice we are interested in the case for diffusion times of T˜3.5 hours as setpoint. In this case, one can further see that when measuring using only ‘all the fringes’ and not including the spacing between the last known fringe and the continuum (see 54TE, 54TM of
In this case, it is important to mention that the in the “chemical mode” of the FSM6000 prismcoupling stress meter, critical angle and its corresponding effective index are found by the position of the identified and saved boundary between the TIR region having the discrete modes, and the continuum of radiation modes coupled to the deep region, and the knee stress can be calculated by:
The “thermal mode” of the FSM6000 instrument computes abstract stress values corresponding to each mode common to the TM and TE spectrum. These abstract stress values are obtained by dividing the difference of the effective indices of the TM and the TE mode in question by the stressoptic coefficient (SOC). The present inventors have determined that the abstract stress corresponding to the “last common mode” can be used to compute the stress at the knee, because there is substantial spatial overlap between the spatial distribution of the last mode guided in the spike, and the region of the knee in the stress profile. In one relatively crude embodiment, the knee stress can be approximately obtained by multiplying the surrogate lastcommonmode stress by a scaling factor K_{3}. This calibration factor is found empirically by comparing the surrogate stress of the last common mode with the actual knee stress measured by independent means (for example, by the refractivenearfield technique, by polarimetric stress measurements, or by computer simulations of diffusion and the resulting stress distribution).
The experimental factor K_{3 }needs to be acquired via measurement at the “knee point” and calculation of the surrogate stress of the last common mode to generate a scaling that can be used for a particular range of recipes.
In the particular case here for diffusion times of about T=3.5 h, this scaling factor is K_{3}=0.646. Therefore, using the “last common mode,” one can compute the stress at the knee and use this information in the previous formulas as given by:
The last step is to find the K_{2 }factor. In an example, this is done experimentally by measurements of the stress profile by other means (e.g., via destructive measurements) and then comparing to the value found using the FSM_DOL. As mentioned before, this value of K_{2 }is between 1 and 3. Therefore K_{2 }is the scaling of the correct position of the knee as a function of the measured FSM_DOL for a certain range of samples. As previously mentioned, since the deep part of the profile is slow varying, a certain level of inaccuracy here will not result in large errors.
Finally, it is also known that the CS measured by the FSM is an approximation considering a linear diffusion profile. In some cases, if a more accurate determination of the CS is needed that can be corrected by another correction factor K_{4}. This factor is usually quite close to 1. In practice, it was found that K_{4 }of about 1.08 leads to more accurate representations of the CS in a significant range. Therefore, if needed one can also use for more accuracy on CS determinations, the relationship:
CS_{corr}=K_{4}×CS (11)
Examples of the use of all the above formulas for the “last know mode” method is set forth in Table 1 in
In another embodiment of the method, the weight gain of a sample as a result of ion exchange is used in combination with the prismcoupling measurement. The weight gain may be used to verify that enough Na+ ions have exchanged for Li+ ions such that the use of the parabolicprofile model is valid for quality control. For the purpose, a target acceptable weight gain range is prescribed for the ion exchange based on the total surface area of the sample and the sample thickness. The weight of representative samples is measured before and after ion exchange, and the qualitycontrol prismcoupling measurements are considered valid if the measured weight gain per sample falls in the target range.
In another embodiment of the method, advantage is taken of the precise control of the sample shape, and of individualsample thickness measurements that are common in some production processes. In this case it is possible to verify that the sample has had adequate weight gain by simply measuring the sample thickness with high precision (such as +/−1 micron), and by measuring the postion exchange weight of the sample. From the known shape specification, the measured thickness, and the known density of the preionexchanged glass, the weight of the preionexchanged sample is calculated.
A correction factor may be applied that accounts for a typical volume change as a result of ion exchange. The weight gain then is estimated by subtracting from the measured postionexchange weight the estimated preexchanged weight. If the weight gain falls within the target range, the profile is deemed adequately represented by the qualitycontrol model profile, and the prismcoupling QC measurement is considered valid.
Another embodiment of the stressslope method for indirect measurement of CS_{k }offers substantial improvement in the precision of measurement of CS_{k }over the embodiment using the slope of the spike measured from only the effective indices of the first two guided modes and the DOL of the spike. The original method described above suffered from precision limitations associated with normal variability in the detection of the positions of the fringes in the coupling spectrum corresponding to these modes.
The present improved method utilizes three or more modes for at least one polarization, when available, to calculate the stress slope with substantially improved precision, thus allowing much more precise calculation of CS_{k}. The method works well because imagenoiseinduced errors in neighboring fringe spacings are anticorrelated, and get substantially eliminated when a single linear fit through three or more fringe positions is utilized.
The method substantially improves the precision of the CS_{k }measurement and the CS measurement for a substantially linear spike by using at least three fringes in at least one of the two polarizations (TM and/or TE) (see
Method of Calculating Knee Stress
The following describes an example method of calculating the knee stress CS_{k }with reduced susceptibility to the noise of any particular mode by a slope fit method that utilizes several modes at once.
The following equation is used in the method and is for a linear profile that relates two arbitrary modes m and l confined within the spike, their effective indices being n_{m }and n_{l}, and the index slope s_{n}:
The above the equation can be used to perform a linear regression, or an evaluation of s_{n }from each pair of modes, and calculate an average for s_{n}. Mode counting starts from m=0 for the lowestorder mode. The parameter λ is the optical wavelength used for the measurements.
An example of the method of calculating the knee stress thus includes the following steps:

 1) Set a reference index to get all measured modes as actual effective indices. A good reference index is usually the index corresponding to the TM criticalangle transition. For Zepler and FORTE glasses, this index is very close to the original substrate index, which is usually specified.
 2) Measure all mode effective indices, n_{m}, m=0, 1, 2, . . . , for each polarization, using the angular prismcoupling spectrum of guided modes.
 3) If desired, assume that n_{m}+n_{l }hardly changes, and assign it as a constant equal to 2
n .  4) For each pair of integers m, l≥0, calculate

 5) Perform a linear regression of the equation y_{ml}≡n_{m}−n_{l}=SB_{ml}, to find a capital slope S.
 6) If desired, check if the quality of the linear regression is adequate (e.g., R^{2 }is higher than a minimum requirement).
 7) Find the index slope using

 8) Find the surface index for the purposes of the subsequent kneestress calculation using:

 9) If higher accuracy is desired, replace 2
n for each pair of modes with the actual sum of the measured values, n_{m}+n_{l}, as mentioned earlier  10) For the calculation of 2
n used in the calculation of n_{surf}, optionally use an iterative procedure, where in the first step we use 2n_{0}, and on the second iteration, use n_{0}+n_{surf}^{(0)}, e. g., use the estimated surface index form the first iteration to calculate the average of the surface and the first mode. For faster calculation, use:
 9) If higher accuracy is desired, replace 2
2

 11) Calculate the surface CS for the purposes of finding the knee stress: CS=(n_{surf}^{TE}−n_{surf}^{TM})/SOC
 12) Calculate the stress slope for the purposes of finding the knee stress:

 13) Find the DOL from the upper TM spectrum for higher precision
 14) Calculate the knee stress: σ_{knee}=CS+s_{σ}×DOL
 15) If the deeper end of spike differs somewhat from linear as truncated on the deep side, then apply a correction factor: σ_{knee}=CS+F×s_{σ}×DOL, where F is the correction factor, usually between about 0.4 and 1, but for spikes having regions of negative curvature it could exceed 1. The correction factor can be calculated by accounting for the actual concentration profile of potassium (K) measured by secondaryion mass spectroscopy (SIMS), glowdiscrharge emission spectroscopy (GDOES), or electron microprobe, or it can be found empirically by comparing measured knee stresses to the equation above and fitting the value of F that makes them agree.
Clearly the above method can be applied to either or both of the TM and TE index profiles of the potassiumenriched spike, to improve the precision of CS and CS_{k}. The improvement is most significant when it is applied to both the TM and the TE spectra, but it could be used in cases where one of the spectra only has 2 guided modes (for example the TE spectrum), in which case the linear regression is applied only to the spectrum having at least 3 guided modes. Furthermore, it can clearly be applied using in general a different number of TM and TE modes, although the accuracy might be highest when the same number of TM and TE modes are used.
The data from application of the two major embodiments of the slope method for indirect CS_{k }calculation to actual prismcoupling measurements of several samples covering a range of different DOL are shown in Tables 2A and 2B, below. Table 2A shows the results of the priorart method of calculation employing two modes while Table 2B shows the results of the improved method of calculation as disclosed herein that uses additional modes.
From the data of Tables 2A and 2B, plots of CS vs extracted CS_{k }using the two methods from first two modes only (fitted curve A), and from using all available modes for slope calculation (fitted curve B) are shown in
The data in
Two other embodiments of the method offer a substantial improvement in the accuracy of measurement of CS_{k }based on the other indirect method disclosed earlier, i.e., the method that uses the birefringence of the highestorder guided mode of the spike to estimate CS_{k}. The highestorder guided mode has effective index only slightly higher than the effective index corresponding to the depth at which the knee of the stress profile occurs. Thus, the birefringence of that mode is significantly affected by the knee stress. If the spike CS and DOL are kept constant, then the knee stress CS_{k }would be essentially the sole driver of changes in the birefringence of the highestorder spike mode.
The method described above calculates the knee stress CS_{k }as a fraction of the birefringence of the highestorder spike mode. A problem with this method can occur when the CS and DOL of the spike are allowed to vary moderately or significantly by a relatively broad product specification, as typical for chemically strengthened cover glasses.
The two improved embodiments of the method for calculating the knee stress CS_{k }disclosed below correct for the effects of varying CS and DOL of the spike on the birefringence of the surrogate guided mode so that the indirectly recovered value of CS_{k }is more accurate. Improvement of the accuracy of CS_{k }measurements is sought by correcting for significant distortions of indirectlyextracted CS_{k }values by the lastfringe method (birefringence of the highestorder guided mode acting as a surrogate for the kneestressinduced birefringence).
In one aspect of the method, a derivative of the birefringence of the chosen surrogate guided mode is calculated with respect to deviations of the CS, DOL, and CS_{k }from their nominal values for the target product. Then CS_{k }is calculated from the measured surrogatemode birefringence, after applying corrections associated with the product of these calculated or empirically extracted derivatives, and the corresponding measured deviations of CS and DOL from the target values.
In an example, the spike shape may be assumed to have a linear distribution from the surface to the depth of the knee. This is a good approximation for a singlestep process. An erfcshaped spike can be considered a good approximation for a twostep process, where the firststep uses a lower substantially nonzero potassium concentration in the bath, and forms a substantially lower CS than the second step, and where the second step has a substantially shorter ion exchange time at approximately the same or lower temperature than the first step. The specific shape of the profile does not affect the method of correction, only the absolute values of the correction factors.
In the present example, the lastfringe birefringence was calculated by using the linearspike approximation. The fabrication process involves a sample of 0.5 mm thick Corning 2321 glass subjected to ion exchange at 380 C for approximately 1.6 hours in a mixture having approximately 20% NaNO_{3 }and 80% KNO_{3 }by weight. The nominal CS for the target is 675 MPa and the nominal DOL is 9 microns.
Table 3 is presented in
The eighth column shows the birefringence of the third guided mode (mode indexing counts from 0, so the third guided mode is TM2/TE2). The ninth column shows the abstract compressive stress CSn2 corresponding to the birefringence of the highestorder guided mode (in this case, the third). This abstract compressive stress is obtained by dividing the mode birefringence by the stressoptic coefficient SOC.
The rightmost column shows the calculated change in the calculated abstract compressive stress by a unit change in the corresponding parameter (i.e., a 1 MPa change in CS_{k}, a 1 MPa change in surface CS, or a 1 micron change in DOL). These can be used approximately as the derivatives of the abstract compressive stress with respect to changes of the driving parameters. It can be seen from Table 3 that the socalculated derivatives may be slightly different on the side of increasing a parameter than on the side of decreasing of the same parameter. This is due to using a finite interval for calculating the derivatives. The difference can be decreased if a smaller interval is used for the estimates. In practice, the average derivative from the positive and negative side of the parameter change may be used over the entire interval to provide a fairly good correction.
If the surrogate abstract mode compressive stress calculated from the birefringence of the highestorder guided common mode is labeled CS_{sur}, then the corrected value of knee stress can be calculated using the measured values of CS, DOL, and CS_{sur}, and using the nominal values for CS, DOL, CS_{k }and CS_{sur}. Generally, the calculation can use the form
where the corrections CorrCS and CorrDOL are calculated from the product of deviations of CS_{sp }and DOL_{sp }from their nominal values, and the corresponding sensitivities of the surrogate stress CS., to changes in CS_{sp }and DOL_{sp}. Note that in the present disclosure, when CS is used without any subscript, it means the surface compressive stress of the spike CS_{sp}. A simple embodiment of the above method is using the equation:
In the above example, the equation reduces to:
The above use of linear relationship between the deviations of CS_{sp }and DOL_{sp }from their nominal values, and the corresponding corrections CorrCS and CorrDOL makes CS_{k }susceptible to increased standard deviation when the measurements of CS_{sp }and/or DOL_{sp }are subject to substantial random error (noise). In some cases this increased standard deviation can be problematic. Limiting the amount of correction by using a nonlinear relationship between each correction and the corresponding deviation in CS_{sp }or DOL_{sp }from its nominal value can help stabilize the calculated CS_{k}. In an example, the corrections can be calculated by the following:
Where Δ_{i }and Δ_{2 }are limiting values of the corrections, preventing overcompensation due to noise in the CS_{sp }and DOL_{sp }values.
In another embodiment of the method, the factor K_{3 }used to relate the sought knee stress CS_{k }and surrogate stress (calculated from the birefringence of the last guided mode), is allowed to vary with the surface CS and the spike DOL, so that the extracted value of CS_{k }from measurements of the surrogate stress can better match the actual knee stress over a variety of CS and DOL combinations.
In an example, the CS and DOL were varied slightly in simulations of the optical modes of a chemically strengthened sample with the knee stress in the vicinity of 150 MPa, CS in the vicinity of 500 MPa, and DOL in the vicinity of 10 microns. The knee stress, which was input in the simulations, was then divided by the surrogate abstract mode stress that was calculated by the simulation, to find how the factor K_{3 }varied with CS and DOL.
In an example, the corrected value of K3 can be calculated as follows:
In another example, the value of K_{3 }can be tabulated for a matrix of CS and DOL combinations, and read out during measurements by an algorithm selecting the closest CS/DOL combination to the measured values of CS and DOL.
In another embodiment of the method, the value of K_{3 }need not be corrected.
Instead, the range of combinations of CS, DOL, and uncorrected CS_{k }can be separated in several regions, such that combinations having high CS and DOL, and low CS_{K }can be rejected during qualitycontrol measurements. This account for the observation that high CS and DOL both tend to raise the indirectlymeasured CS_{K }by the highestguidedmode surrogate method.
In one example, a process space (process window) is defined by the product of the CS and DOL specifications. This process space is then split into two or more regions, preferably in parallel to the diagonal relating the point (CSmax, DOLmin) with the point (CSmin, DOLmax). Then for each region, a different lower limit of CS_{K }is used as a reason to reject a part, with the so required CS_{K }lower limit generally increasing with increasing CS and increasing DOL. In another example, the CS/DOL process space can be split into two or more subregions by curves corresponding to the condition CS*DOL=const, or (CS−CS_{K}^{nom})*DOL=const.
It will be apparent to those skilled in the art that various modifications to the preferred embodiments of the disclosure as described herein can be made without departing from the spirit or scope of the disclosure as defined in the appended claims. Thus, the disclosure covers the modifications and variations provided they come within the scope of the appended claims and the equivalents thereto.
Claims
121. (canceled)
22. A method of characterizing a stress profile of a chemically strengthened glass substrate formed by the indiffusion of alkali ions and having an upper surface and a body, a shallow spike region of stress immediately adjacent the upper surface and a deep region of slowly varying stress within the body and that intersects the spike region at a knee, wherein the method comprises:
 measuring a TM mode spectrum and a TE mode spectrum of the glass substrate, wherein the TM mode spectrum and the TE mode spectrum each include mode lines and a transition associated with a critical angle;
 calculating the knee stress CSknee utilizing the TE mode spectrum, TM mode spectrum, and the stressoptic coefficient SOC.
23. The method of claim 22, further comprising determining a surface compressive stress CSsp of the spike.
24. The method of claim 23, wherein determining the surface compressive stress CSsp of the spike utilizes the TM and TE mode spectra.
25. The method of claim 23, wherein determining the surface compressive stress CSsp of the spike utilizes a measurement of a surface concentration of at least one type of the alkali ions.
26. The method of claim 22, further comprising determining an amount of birefringence BR.
27. The method of claim 26, wherein determining the amount of birefringence BR comprises measuring a difference between the TE and TM transition locations.
28. The method of claim 27, wherein calculating the knee stress CSknee utilizes the amount of birefringence BR.
29. The method of claim 28, wherein the knee stress CSknee is calculated as CSknee=(CFD)(BR)/SOC, where CFD is a calibration factor.
30. The method of claim 29, wherein CFD is between 0.5 and 1.5.
31. The method of claim 28, wherein the knee stress CSknee is calculated as CSknee=K3×BR/SOC, where K3 is a calibration factor.
32. The method of claim 31, wherein K3 is between 0.2 and 2.
33. The method of claim 26, wherein the amount of birefringence BR is given by: where nLMTE is the effective index of a highestcommonorder TE spike mode, and nLMTM is the effective index of a highestcommonorder TM spike mode.
 BR=nLMTE−nLMTM
34. The method of claim 33, wherein calculating the knee stress CSknee utilizes the amount of birefringence BR.
35. The method of claim 34, wherein the knee stress CSknee is calculated as CSknee=(CFD)(BR)/SOC, where CFD is a calibration factor.
36. The method of claim 35, wherein CFD is between 0.5 and 1.5.
37. The method of claim 34, wherein the knee stress CSknee is calculated as CSknee=K3×BR/SOC, where K3 is a calibration factor.
38. The method of claim 37, wherein K3 is between 0.2 and 2.
39. A prismcoupling system, wherein the prismcoupling system is configured to perform the method of claim 22.
Type: Application
Filed: Sep 3, 2020
Publication Date: Dec 24, 2020
Inventors: Ryan Claude Andrews (Elmira, NY), Rostislav Vatchev Roussev (Painted Post, NY), Vitor Marino Schneider (Painted Post, NY)
Application Number: 17/010,942