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Prediction of stress profiles in ion exchanged glasses
Journal of Non-Crystalline Solids 344 (2004) 79–87 www.elsevier.com/locate/jnoncrysol
Prediction of stress profiles in ion exchanged glasses Junwu Shen *, David J. Green Department of Materials Science and Engineering, The Pennsylvania State University, 225 Steidle, University Park, PA 16802, USA
Abstract Residual stress profiles are important for controlling the fracture behavior of ion exchanged glass. In the current study, the residual stress profile in single-step and two-step ion exchanged glass was calculated from the concentration profile and the compositiondependent stress relaxation behavior of glass. The dilation coefficient, used for the calculation, was determined from the measured stress profile in low-temperature ion exchanged glass, and it was found to be composition dependent. The calculated stress profiles in ion exchanged glass processed under typical conditions have reasonable agreement with the measured stress profiles. Therefore, using this methodology, optimization of the processing conditions for any particular requirement of the fracture behavior is possible. 2004 Published by Elsevier B.V. PACS: 42.70.C; 62.40; 82.65.F
1. Introduction A new two-step ion exchange process has been recently developed to produce strong glasses with low strength variability by designing the form of the residual stress profile [1–4]. These glasses are termed engineered stress profile (ESP) glasses [5]. Usually, ESP glasses have the following characteristics: high strength, low strength variability, high surface damage resistance and multiple cracking before final failure. In order to further improve and optimize the mechanical properties of ESP glasses, a processing–property relationship in ion exchanged glass is necessary, and prediction of stress profiles after ion exchange is an important step for establishing such a relationship. Stress profiles in single-step ion exchanged glasses have been studied by many researchers in the past several decades [6–12]. Initially, stress profiles were calculated from the concentration change without con-
*
Corresponding author. Tel.: +1 814 865 2121/863 3476. E-mail address: [email protected] (J. Shen).
0022-3093/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.jnoncrysol.2004.07.026
sidering the stress relaxation effect. Garfinkel and King [6] and Tyagi and Varshneya [7] calculated the stress profile in ion exchanged glass using the analogy with thermal stresses. The calculated stress profile was found to be much higher than the measured profile [7]. It is widely accepted that stress relaxation is an important effect during a typical ion exchange process even though the exchange is performed below Tg. Spoor and Burggraaf [9] and Sane and Cooper [10] calculated the stress profile in ion exchanged glass by including the stress relaxation effect. Unfortunately, they were not able to obtain a calculated stress profile with a sub-surface maximum, which is an important characteristic of the measured stress profiles [10]. Spoor and Burggraaf [9] attributed the poor fit between the calculated and experimental curves to the dependence of stress relaxation on concentration and treatment time. Sane and Cooper [10] suggested other possible reasons for this disagreement: the composition dependence of stress relaxation, dilation coefficient and diffusion coefficient. Zhuravlev et al. [11] calculated stress profiles in ion exchanged glass by taking into account the stresses caused by both the composition change at glass surface
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and the thermal expansion mismatch between glass surface and bulk glass. Usually, the stresses due to the thermal expansion coefficient difference are significantly less than stresses caused by the concentration change if the ion exchange is conducted below the glass transition temperature. However, in some cases, the stresses caused by thermal expansion coefficient difference could significantly change the total stress profile, especially near the glass surface [11]. Recently, Startsev and Mazurin [12] proposed a quite different model to calculate the stress profiles in ion exchanged glasses by taking into account the structural relaxation. The calculated stress profiles using this model showed a sub-surface maximum and have reasonable agreement with the experimentally measured stress profiles by Sane and Cooper [10]. However, the calculation is complicated and some of the model variables cannot be evaluated by a direct and unambiguous experiment. In general, most of the previous studies on theoretical calculations of stress profiles in ion exchanged glasses failed to give a satisfactory agreement with the measured stress profiles. In addition, no studies have been found for the stress calculation in two-step ion exchanged glasses, which is the key to the processing–property relationship in ESP glasses. In the current study, the approach is similar to that proposed by Sane and Cooper [10], except the composition dependence of the dilation coefficient and the stress relaxation behavior was incorporated into the analysis. The calculated stress profiles will then be compared to measured profiles, and the difference between them will be discussed. It should be noted that once the stress profiles are calculated, they could be transformed to apparent fracture toughness curves [3]. These curves can then be used to predict and optimize the fracture behavior.
2. Experimental and theoretical approach Stress profiles in ion exchanged glasses were measured for two purposes. First, the dilation coefficient will be calculated from the stress profile in ion exchanged glass where the stress relaxation is negligible. Secondly, calculated stress profiles for glasses processed under typical conditions will be compared to measured profiles. In ion exchanged glasses, there is usually a large variation in composition in the exchanged layer. For example, in cases where K+ ions are exchanged for Na+ ions, the K2O content in the exchanged layer varies from 0 to 20 wt% [13]. Thus, it is important to know the physical properties and stress relaxation behavior of glasses with compositions in this range. A study with these objectives was completed on glasses of direct relevance to the current study [14].
2.1. Glass composition The glasses used in the study were soda-lime silicate float glass (StarphireTM, PPG, Pittsburgh, PA, USA) and soda aluminosilicate glass (Corning 0317, Corning, NY, USA). These will be denoted as SLS and SAS respectively. The measured composition of these two glasses was given in [13]. 2.2. Residual stress profile measurement in ion exchanged glasses All residual stress profiles in the current work were measured by an optimized optical method, and the details about this technique are described elsewhere [15]. In this technique, the central tensile stress in glass was determined by measuring the birefringence of the ion exchanged glass. Then, by progressively etching a thin layer from the glass surface, the average compressive stress in the etched layer can be calculated from the change of the tensile stress and glass dimensions using a force balance. 2.3. Compressive stress relaxation tests Uniaxial compressive stress relaxation behavior of the soda-lime silicate glass and mixed-alkali lime silicate glasses were studied in previous papers [14,16]. Stress relaxation tests were performed by a mechanical testing machine, and the details of the equipment are given in [14,16]. All stress relaxation curves were fitted by the Burger model. 2.4. Theoretical approach Using the thermal stress analogy, stress profiles in ion exchanged glasses can be calculated [10] Z t1 BE o r1 ðx; tÞ ¼ Rðt1 tÞ ðC CÞ dt; ð1Þ 1m 0 ot where r1 is the stress, B is the dilation coefficient, E is the Youngs modulus, m is the Poissons ratio, R is the normalized relaxation function, t is time, t1 is the total exchange time, C is the K2O concentration and the C is the average concentration in the direction of diffusion. When the stress relaxation during ion exchange is negligible, stress profiles can be calculated by [6] r1 ðx; tÞ ¼
BE ðC CÞ: 1m
ð2Þ
Data are, thus, required for all these parameters. The concentration profiles can be obtained experimentally or by calculation if the diffusivity is known. A critical aspect of the theoretical approach was to quantify the composition dependence of B, E, m and R. Experimental data showed that E should vary by <10% in the ion ex-
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
changed layer [14]. For the current calculations, this variability in E was ignored. Similarly, m was also assumed to be constant. The following sections will discuss how B and R were incorporated into the stress calculation. The approach will be to use Eq. (1) to describe the single exchange process and Eq. (2) will then be used for the second step.
The dilation coefficient (B) of glass is defined as the linear strain per unit concentration change of alkaline oxide [7]: B¼
DV 1 ; 3V 0 DC
ð3Þ
where V0 is the volume of glass before ion exchange, C is the K2O concentration and DV is the increase of glass volume after exchange. The dilation coefficient is an important parameter for the calculation of stress profiles in ion exchanged glasses. Varshneya [17] calculated the dilation coefficient for Na+–K+ ion exchange in Na2O Æ 3SiO2 glass and a B value of 1.3 · 103 (wt% K2O) was obtained from the linear strain (4.5%) in glass after ion exchange. This dilation coefficient value is about three to four times those values measured by Cooper and Krohn [18] who obtained B values from measured stress profiles. In the current study, dilation coefficient will be theoretically calculated from the ionic radii of Na+ and K+ ions, and it will also be determined from the measured stress profiles in ion exchanged glass. 3.1. Calculation of B from ionic radii Consider a piece of SLS glass with a weight of 100 g that contains 14.7 g Na2O according to the measured glass composition [13]. It is also known that the Pauling radii of K+ and Na+ ions are 0.133 nm and 0.095 nm respectively [19], and the density of the SLS glass (q) is 2.493 Mg/m3 [16]. Then, the total volume of Na+ ions in this glass can be calculated from the total number of Na+ ions and the volume of each Na+ ion: V Na ¼ 1:025 106 m3 : +
ð4Þ +
If all the Na ions are replaced by K ions, the total volume of K+ ions in the glass can be similarly calculated: V K ¼ 2:81 106 m3 :
ð5Þ
If the glass can freely expand after ion exchange, the volume change of the glass after a complete exchange is only caused by the difference between the total volumes of Na+ and K+ ions. So the volume change of the glass (DV) can be calculated as follows: DV ¼ V K V Na ¼ 1:787 106 m3 :
The initial volume of the SLS glass can be calculated from the mass and density of glass: V 0 ¼ m=q ¼ 4:01 105 m3 :
ð6Þ
ð7Þ
In addition, the K2O concentration in the completely exchanged SLS glass is calculated to be 20.8 wt% according to the measured composition [13]. Therefore, the dilation coefficient value can be calculated as follows: B¼
3. Dilation coefficient for Na+–K+ ion exchange
81
1 DV 1 ¼ 7:14 104 wt% : 3 V 0 DC
ð8Þ
This calculated dilation coefficient is lower than the value (1.3 · 103 wt%1) calculated by Varshneya [17], but higher than the value (3 · 104 wt%1) obtained by Cooper and Krohn [18]. This calculation is based on the assumption that the volume change of the glass (DV) after complete ion exchange would be the same as (VK VNa), the volume difference between the total volumes of Na+ and K+ ions. In reality, the actual volume change of the glass could be less than (VK VNa) because some of the volume increase during ion exchange can be accommodated by a reduction in the free volume of the SLS glass. Hale [20] proposed the effective ion radius of Na+ ions, and argued that the volume change of glass after exchange should be calculated from the difference between the radius of K+ ions and the effective radius of Na+ ions. This is one of the reasons that the calculated dilation coefficient value (7.14 · 104 wt%1) is higher than the value (3 · 104 wt%1) determined by Cooper and Krohn [18]. 3.2. Calculation of B from stress profile The above calculation is based on the assumption that dilation coefficient is composition independent. It is, however, possible that B is dependent on the local K2O concentration. So, the dilation coefficient as a function of K2O concentration will be calculated from the measured stress profile under conditions where stress relaxation effect can be neglected. Usually ion exchange penetration depth is much smaller than the glass thickness, the average concentration is approximately C0 (the original K2O concentration in base glass), and for the SLS glass in the current study, C0 = 0. So, Eq. (2) can be simplified to the following equation in the calculation: r1 ðx; tÞ ¼
BEðC C 0 Þ BEC ¼ : 1m 1m
ð9Þ
Then the composition-dependent dilation coefficient can be calculated from the measured stress profile in the glass according to Eq. (9). Fig. 1 shows the measured residual stress profiles in ion exchanged (450 C/1 h) glass using the technique described in [15]. Since the stress relaxation effect within 1 h at 450 C is not significant (<5%) for glasses with
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Compressive stress (MPa)
400 350 300 250 200 150 100 50 0 0
2
4
6
8
Distance from the glass surface (µm) Fig. 1. Measured residual stress profile in single-step ion exchanged SLS glass (450 C/1 h).
the same composition as the ion exchanged layer [14], it can be neglected. The sub-surface maximum of the stress profile is possibly caused by the polynomial fitting procedure used in deriving the stress profile. The K2O concentration, C, can be calculated using the parameters in [13], so the dilation coefficient as a function of local concentration can be calculated from Eq. (9). Fig. 2 shows the calculation result. Although the Youngs moduli of glasses with different K/Na ratios are different, the Youngs modulus of SLS glass at 450 C (65.1 GPa) [16] was used for the calculation of dilation coefficient. This should not affect the stress profile calculation because the calculated stress is always proportional to the product of Youngs modulus and dilation coefficient. It can be seen from Fig. 2 that the composition-dependent dilation coefficient can be fitted by an empirical equation: BðCÞ ¼ 2:39 103 C þ 6:64 104 ;
where B is in the unit of wt%1. The dilation coefficient linearly decreases with the increase of the K2O concentration in the glass and the previously calculated constant dilation coefficient 7.14 · 104 wt%1 is only slightly higher than the dilation coefficient value for low K2O concentration. The most possible reason for the composition dependence of the dilation coefficient is that the initial introduction of K+ ions in the glass expands the glass structure almost elastically and the free volume in the glass does not decrease. However, with more K+ ions introduced in the glass surface, the glass structure expands slowly and some of the free volume in the glass is taken by the K+ ions. As a result, the linear strain produced in the glass for each unit concentration change of K2O is decreasing. Another possible reason is that the Youngs modulus of high K glass (e.g. K22 glass containing 22.5 wt% K2O) is lower than that of SLS glass [14], and the calculated dilation coefficient value in high K2O region is underestimated. However, the modulus difference between SLS glass and K22 glass is only 10%, which is much less than the dilation coefficient difference between ion exchanged glass surface and bulk glass. So the Youngs modulus difference can only partly explain the composition dependence of dilation coefficient.
4. Calculation of stress profiles in single-step ion exchanged glasses 4.1. Stress calculation without stress relaxation According to Eq. (9), the stress profile in ion exchanged glass can be calculated using the constant dila-
ð10Þ 1200
Compressive stress (MPa)
Dilation coefficient (wt%-1)
Measured
1000
0.0007
y = -2.39E-05x + 6.64E-04
0.0006 0.0005 0.0004 0.0003 0.0002
Calculated (constant B) Calculated (variant B)
800
600
400
200
0.0001 0 0
5
10
15
20
25
K2O concentration (wt%) Fig. 2. Dilation coefficient as a function of local K2O concentration for Na+–K+ ion exchange in SLS glass.
0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 3. Measured and calculated stress profiles (without stress relaxation) in single-step ion exchanged glass (450 C/48 h).
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
4.2. Stress calculation with stress relaxation using a viscoelastic model Using the KWW equation (or b-function), the stress profile in ion exchanged glass can be calculated using the following equation [10]: r1 ¼
Z
t1
BE x ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi 1m t 4pD1 t 0 x2 t1 tb exp dt; exp s 4D1 t
ð11Þ
where t1 is the first-step ion exchange time, Cs1 is the surface concentration after single-step ion exchange, D1 is the interdiffusion coefficient in single-step exchange, b is a constant (0 < b < 1) and s is the stress relaxation time [10]. Similarly, if the Burger model is used to calculate the stress profile, a similar equation can be derived from Eq. (1): BE x x2 ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi exp r1 ¼ 1m 4D1 t t 4pD1 t 0 t1 t t1 t A1 exp þ A2 exp dt; ð12Þ s1 s2 Z
t1
where A1, A2, s1 and s2 are parameters for stress relaxation function. Fig. 4 shows the measured stress profile and the calculated profile from Eq. (12) by including the stress Table 1 Parameters used in the stress profile calculations (SLS glass) Youngs modulus (E: GPa)
Poissons ratio (m)
Diffusion coefficient (D1: m2/s)
Surface concentration (Cs1: wt%)
65.1
0.22
1.4 · 1015
20.0
A1
A2
s1 (min)
s2 (min)
0.751
0.249
2140
11.2
350 300
Compressive stress (MPa)
tion coefficient and the variant dilation coefficient described in the previous section (Fig. 2). Fig. 3 shows the comparison between the calculated and measured stress profiles for a typical ion exchange condition (450 C, 48 h). The parameters used in the calculation are displayed in Table 1. Youngs modulus of SLS glass at 450 C (65.1 GPa) was used in the calculation [16]. It can be seen that the calculated stress profile using a constant dilation coefficient (7.14 · 104 wt%1) is much higher than the measured profile. By using the composition-dependent dilation coefficient, the calculated stress profile is also higher than the measured profile, especially in the near surface region. Therefore, it is concluded that stress relaxation cannot be neglected in typical ion exchange conditions.
83
Predicted Measured
250 200 150 100 50 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 4. Measured and calculated stress profiles (with compositionindependent stress relaxation) in single-step ion exchanged glasses (450 C/48 h).
relaxation effect. The relaxation function parameters for the SLS glass at 450 C in Table 1 were used for the calculation [14]. Obviously, the calculated stress profile overestimated the stress relaxation effect in the glass during the ion exchange process. It was hypothesized that composition-dependent stress relaxation data are needed for a more accurate calculation. In [14], mixedalkali silicate glasses with higher K2O concentrations were found to relax more slowly than SLS glass. So it is expected to obtain a better agreement between the calculated and measured stress profile in glass if these compositional-dependent relaxation data are used. 4.3. Stress calculation using composition-dependent stress relaxation and dilation The above calculations and comparisons show that the effect of local K2O concentration on both the dilation coefficient and the stress relaxation times need to be included for an accurate prediction of stress profile. The dilation coefficient as a function of local K2O concentration is calculated previously, and it is assumed to be valid for other ion exchange conditions. 4.3.1. Stress relaxation behavior of different glasses fitted by a power law Stress relaxation behavior of SLS and mixed-alkali lime silicate glasses have been studied [14]. The uniaxial viscosity values can be calculated from the measured stress relaxation curves assuming linear viscoelasticity. In this approach, viscosity values are estimated and these values are used to extrapolate the relaxation times. Fig. 5 shows that the logarithm viscosity at 550 C vs. K2O concentration in glass can be fitted by: logðgÞ ¼ 2:784 logðCÞ þ 17:121;
ð13Þ
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
shown that these glasses show a mixed-alkali effect and the viscosity can increase at lower K2O concentrations [14]. Although the viscosity and relaxation times are also dependent on glass composition when the K2O concentration is less than 3 wt%, the effect of this viscosity change on stress profile is minor. Moreover, the stress profile in low K2O concentration region of glass (<2.92 wt%) is relatively less important for the fracture behavior of the glass. So, in the current study, stress profile was only calculated in the region where K2O is greater than 2.92 wt%.
20
550 C
Log (viscosity/Pa s)
18
y = 2.784x + 19.681
450 C
16
14
y = 2.784x + 17.121 12
10
8 -2.0
-1.5
-1.0
-0.5
0.0
Log (K2O concentration) Fig. 5. Logarithm viscosity vs. logarithm K2O concentration for glasses with compositions similar to those in the ion-exchanged layer (power law fit).
where g is the unit of Pa s. As described in [14], the viscosity data at 450 C for glasses with different K/Na ratios can be obtained by shifting the log viscosity data at 550 C by a constant. The viscosity values for a glass containing 2.92 wt% K2O (K03) at 450 C and 550 C are known, and the difference between these values is log(2.17 · 1015) log(5.99 · 1012) = 2.56. Then the viscosity at 450 C for other glasses can be calculated by the following equation: logðgÞ ¼ 2:784 logðCÞ þ 19:681;
ð14Þ
where g is the unit of Pa s. It is also demonstrated in [14] that the stress relaxation function coefficients A1 and A2 are approximately constant for glasses with different K/Na ratios at a given temperature. Therefore, the ratio of viscosity values and relaxation times are the same for different glasses [14], and then the stress relaxation times for other glasses can be calculated from the stress relaxation data of the K03 glass. Assuming the viscosity of K03 glass at 450 C is g0, and the stress relaxation times in the Burger model are s01 , s02 . The stress relaxation times for other glasses (s1 and s2) can be calculated from the viscosity (g) by: g si ðCÞ ¼ s0i ; ð15Þ g0 where i = 1, 2. By combining Eqs. (14) and (15), one obtains: si ðCÞ ¼
s0i ð2:784 log Cþ19:681Þ 10 ; g0
ð16Þ
where i = 1, 2. Eqs. (13)–(16) are only valid when the K2O concentration in glass is greater than 2.92 wt% as it has been
4.3.2. Stress relaxation behavior of different glasses fitted by a linear equation Many authors have tried to calculate the viscosity values of glasses at different temperatures from the glass compositions, especially for soda-lime silicate glass (see [14]). However, these empirical equations usually have a small composition and temperature range, and they are not applicable for glasses with high K2O concentration (>10 wt%) at very low temperatures (T < Tg 100 C). However, almost all the calculation methods show that the increase of K2O concentration in sodalime glass leads to a linear increase of viscosity. Fig. 6 shows the viscosity values at 450 C of different glasses extrapolated from viscosity values at 550 C. Although the viscosity of glass is not a linear function of K2O concentration over the whole range of K2O concentration (0–22.5 wt%), the assumption of linearity can be used to estimate the viscosity value of any glass within the composition range, especially in the region of high K2O concentration. It can be seen from Fig. 6 that the viscosity values at 450 C can be fitted by an empirical equation: g ¼ 2:925 1018 C 6:978 1016 ;
ð17Þ
7E+17 6E+17
y = 2.925E+18x - 6.978E+16 2
R = 9.281E-01
Viscosity (Pa s)
84
5E+17 4E+17 3E+17 2E+17 1E+17 0 0.00
0.05
0.10
0.15
0.20
0.25
K 2O concentration Fig. 6. Viscosity vs. K2O concentration for glasses with compositions similar to those in the ion-exchanged layer (linear fit).
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
si ðCÞ ¼
s0i ð2:925 1018 C 6:978 1016 Þ; g0
ð18Þ
350
Calculated (power law)
300
Compressive stress (MPa)
where g is the unit of Pa s. Obviously, this equation is only valid when the K2O concentration is greater than 2.4 wt%. However, as previously discussed, stress profile in low K2O concentration region is relatively less important, and again, stress profile was only calculated in the region where K2O is greater than 2.4 wt%. For the linear case, the stress relaxation times for other glasses (s1 and s2) can be calculated from the K2O concentration (C) by:
Measured 250 200 150 100 50
where i = 1, 2.
0 0
By using Eqs. (16), (18) and (19), stress profiles in singlestep ion exchanged glass (450 C/48 h) can be calculated using different relationships between viscosity and glass composition. A1 and A2 values in Table 1 were used for the calculation. The calculated and measured stress profiles are shown in Figs. 7 and 8. The upper and lower bounds of calculated stress profiles are obtained by adding or subtracting the mean square error from Eqs. (14) and (17). When a power law equation is used for the relationship between viscosity and glass composition, the calculated stress profile is lower than the calculated stress profile in the sub-surface region (10–30 lm). However, the stress profile calculated by using a linear relationship between viscosity and glass composition has a better agreement with the measured stress profile. The difference between the calculated and measured stress profiles are less than 10% in the first 10 lm below the glass surface, and it is less than 45 MPa in other regions. Several possible reasons are discussed below for the discrepancy between the calculated and measured stress profiles: (1) One possibility is that the current empirical relationships between viscosity and glass composition (Eqs. (14) and (17)) are not accurate enough. If more mixed-alkali silicate glasses with different K2O concentration between 9 wt% and 23 wt% are melted and studied, a more accurate relationship
10
20
30
40
50
Distance from the glass surface (µm) Fig. 7. Comparison of the measured and calculated stress profiles in single-step ion exchanged glass (450 C/48 h) using a power law relationship between viscosity and K2O concentration in the glass. 10% error bars are shown for the maximum and minimum measured stress values.
400
Calculated (linear)
350
Measured
Compressive stress (MPa)
4.3.3. Stress profile calculation using the compositiondependent stress relaxation and dilation coefficient Given the extrapolated stress relaxation times in different glasses at ion exchange temperatures and the measured composition-dependent dilation coefficient, the following equation is developed by modifying Eq. (12) to calculate the stress profile in glass: Z t1 BðCÞE x x2 r1 ¼ ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi exp 1m 4D1 t t 4pD1 t 0 t1 t t1 t A1 exp þ A2 exp dt: ð19Þ s1 ðCÞ s2 ðCÞ
85
300 250 200 150 100 50 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 8. Comparison of the measured and calculated stress profiles in single-step ion exchanged glass (450 C/48 h) using a linear relationship between viscosity and K2O concentration in the glass. 10% error bars are shown for the maximum and minimum measured stress values.
between the viscosity and glass composition could be obtained. If the viscosity increases faster with the increase of K2O concentration in glass than described in Figs. 5 and 6, the calculated stress profile in the first 20 lm would be higher than in Figs. 7 and 8, and the difference between the calculated and measured stress profiles would be smaller. (2) The calculated dilation coefficient could be slightly underestimated due to the small stress relaxation effect in 450 C/1 h ion exchanged glass, so the calculated stress profile is also slightly underestimated.
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
(3) The discrepancy could also be caused by the uncertainty of the diffusion coefficient. The diffusion coefficient at 450 C is calculated from the measured K2O concentration profiles in ion exchanged glass, and it is not unusual to have 10% error in the measured concentration profile and the calculated diffusion coefficient. If a 20% higher diffusion coefficient is used to calculate the stress profile, the difference between the measured and calculated stress profiles could be minimized. (4) The stress relaxation functions of potassium glasses at 450 C used for the stress profile calculation are extrapolated from the data of SLS glass, and they may be different from the real stress relaxation function of those glasses. (5) Finally, the experimental error of the measured stress profiles in ion exchanged glasses is estimated to be about 10%. It is concluded, therefore, that the calculated stress profile in ion exchanged glass using the compositiondependent dilation coefficient and stress relaxation data gives a reasonable agreement with the measured stress profile considering the uncertainties in the various parameters used for the calculation. As shown in [14], the thermal expansion coefficient difference between the ion exchanged glass surface and the bulk glass is about 0.64 · 106 C1. So, the maximum residual thermal stress (rR) caused by this difference can be calculated by: DT DaE ; rR ¼ 1m
ð20Þ
where DT is the difference between ion exchange temperature and room temperature, Da is the thermal expansion coefficient difference between the ion exchanged glass surface and the bulk glass. Since the thermal expansion coefficient of the 22 wt% K2O glass is higher than that of SLS glass, the residual thermal stress is tensile in ion exchanged glass surface. Then, if a glass sample was cooled from 450 C to 25 C, the maximum resultant residual thermal stress is only 23 MPa (tensile), which is much lower than the residual stress in typical K/Na ion exchanged glass, especially in the near surface region. Clearly the current calculation of B includes these mismatch strains, but one can conclude that B is dominated by the mismatch in ionic size. 5. Calculation of stress profiles in two-step ion exchanged glasses The stress profile in two-step ion exchanged glass is important for the processing–property relationship in ESP glasses. Since the second step ion exchange process is usually performed at a lower temperature (e.g. 400 C) and for a much shorter time (e.g. 30 min), the stress
800 1
700
Compressive stress (MPa)
86
600
3
500
2
400 300 200 100 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 9. Calculated and measured stress profiles in two-step ion exchanged glass (SAS). Curves 1 and 2 are the measured stress profiles in single-step (450 C/4 h) and two-step (450 C/4 h + 400 C/30 min) ion exchanged glasses respectively, and curve 3 is the calculated stress profile from the measured stress profile in single-step ion exchanged glass (curve 1).
relaxation effect in the second-step can be neglected according the stress relaxation data of SLS glass in [16]. So, the stress profiles in two-step ion exchanged glasses can be calculated from that in single-step ion exchanged glasses according to the concentration profile change in the second-step process: r2 ¼ r1
BE ðC 2 C 1 Þ; 1m
ð21Þ
where r1 and r2 are stress profiles after first-step and second-step ion exchange process, C1 and C2 are K2O concentration profiles after single-step and two-step ion exchange process. In order to verify if the concentration change can explain the stress profile change in the second-step ion exchange process, the stress profile in two-step ion exchanged (450 C/4 h + 400 C/30 min) SAS glass was calculated from the measured stress profile in single-step ion exchanged (450 C/4 h) SAS glass. This stress profile was then compared to the measured stress profile in the same two-step ion exchanged glass. The results are shown in Fig. 9 and it can be seen that the calculated and measured stress profiles have reasonably good agreement (curves 2 and 3). 6. Conclusions The dilation coefficient for Na+–K+ ion exchange was calculated and then stress profiles in single-step and twostep ion exchanged glasses were calculated based on dilation coefficient, stress relaxation data and the diffusion data: (1) The measured dilation coefficient for Na+–K+ ion exchange was found to be composition dependent. An empirical equation was developed to calculate
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
the composition-dependent dilation coefficient. Thermal expansion mismatch strains were shown to be a minor contribution to the dilation. (2) The calculated stress profile in single-step ion exchanged glass without the stress relaxation is much higher than the measured stress profile for typical ion exchange conditions. Conversely, if the stress profile is calculated using the stress relaxation data of SLS glass, it is much lower than the measured stress profile. (3) The calculated stress profile in single-step ion exchanged glass using the composition-dependent dilation coefficient and stress relaxation data has reasonable agreement with the measured stress profile. The difference between the calculated and measured stress profiles is mainly caused by the lack of an accurate relationship between stress relaxation behavior and glass composition. (4) The calculated stress profile for two-step ion exchanged glass has reasonable agreement with the measured stress profile, and stress relaxation can be neglected for the second step studied here. Acknowledgment The authors wish to acknowledge NSF Center for Glass Research for the financial support of this work and allowing the publication of this work.
87
References [1] D.J. Green, R. Tandon, V.M. Sglavo, Science 283 (1999) 1295. [2] M.B. Abrams, D.J. Green, S.J. Glass, J. Non-Cryst. Solids 321 (2003) 10. [3] V.M. Sglavo, L. Larentis, D.J. Green, J. Am. Ceram. Soc. 84 (8) (2001) 1827. [4] V.M. Sglavo, D.J. Green, J. Am. Ceram. Soc. 84 (8) (2001) 1832. [5] S.J. Glass, E.K. Beauchamp, C. Newton, D.J. Green, V.M. Sglavo, Presented at the 102nd Annual Meeting of the American Ceramic Society, St. Louis, MO, May 3, 2000 (Paper No. BS1032-00). [6] H.M. Garfinkel, C.B. King, J. Am. Ceram. Soc. 53 (12) (1970) 686. [7] V. Tyagi, A.K. Varshneya, J. Non-Cryst. Solids. 238 (1998) 186. [8] I.A. Boguslavskii, O.I. Pukhlik, Fiz. Khim. Stekla 5 (5) (1979) 577. [9] W.J. Spoor, A.J. Burggraaf, Phys. Chem. Glasses 7 (5) (1966) 173. [10] A.Y. Sane, A.R. Cooper, J. Am. Ceram. Soc. 70 (2) (1987) 86. [11] G.I. Zhuravlev, V.S. Evstishenko, A.I. Kuznetsov, E.P. Golova, Fiz. Khim. Stekla 7 (4) (1981) 414. [12] Yu.K. Startsev, O.V. Mazurin, Glass Phys. Chem. 20 (4) (1994) 315. [13] J. Shen, D.J. Green, C.G. Pantano, Phys. Chem. Glasses 44 (4) (2003) 284. [14] J. Shen, D.J. Green, J. Non-Cryst. Solids, in this issue. [15] M.B. Abrams, J. Shen, D.J. Green, J. Test. Eval. 32 (3) (2004) 227. [16] J. Shen, D.J. Green, R.E. Tressler, D.L. Shelleman, J. Non-Cryst. Solids 324 (2003) 277. [17] A.K. Varshneya, J. Am. Ceram. Soc. 58 (3–4) (1975) 106. [18] A.R. Cooper, D.A. Krohn, J. Am. Ceram. Soc. 52 (12) (1969) 665. [19] http://www.webelements.com/webelements/ 2003. [20] D.K. Hale, Nature 217 (1968) 1115.
Prediction of stress profiles in ion exchanged glasses Junwu Shen *, David J. Green Department of Materials Science and Engineering, The Pennsylvania State University, 225 Steidle, University Park, PA 16802, USA
Abstract Residual stress profiles are important for controlling the fracture behavior of ion exchanged glass. In the current study, the residual stress profile in single-step and two-step ion exchanged glass was calculated from the concentration profile and the compositiondependent stress relaxation behavior of glass. The dilation coefficient, used for the calculation, was determined from the measured stress profile in low-temperature ion exchanged glass, and it was found to be composition dependent. The calculated stress profiles in ion exchanged glass processed under typical conditions have reasonable agreement with the measured stress profiles. Therefore, using this methodology, optimization of the processing conditions for any particular requirement of the fracture behavior is possible. 2004 Published by Elsevier B.V. PACS: 42.70.C; 62.40; 82.65.F
1. Introduction A new two-step ion exchange process has been recently developed to produce strong glasses with low strength variability by designing the form of the residual stress profile [1–4]. These glasses are termed engineered stress profile (ESP) glasses [5]. Usually, ESP glasses have the following characteristics: high strength, low strength variability, high surface damage resistance and multiple cracking before final failure. In order to further improve and optimize the mechanical properties of ESP glasses, a processing–property relationship in ion exchanged glass is necessary, and prediction of stress profiles after ion exchange is an important step for establishing such a relationship. Stress profiles in single-step ion exchanged glasses have been studied by many researchers in the past several decades [6–12]. Initially, stress profiles were calculated from the concentration change without con-
*
Corresponding author. Tel.: +1 814 865 2121/863 3476. E-mail address: [email protected] (J. Shen).
0022-3093/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.jnoncrysol.2004.07.026
sidering the stress relaxation effect. Garfinkel and King [6] and Tyagi and Varshneya [7] calculated the stress profile in ion exchanged glass using the analogy with thermal stresses. The calculated stress profile was found to be much higher than the measured profile [7]. It is widely accepted that stress relaxation is an important effect during a typical ion exchange process even though the exchange is performed below Tg. Spoor and Burggraaf [9] and Sane and Cooper [10] calculated the stress profile in ion exchanged glass by including the stress relaxation effect. Unfortunately, they were not able to obtain a calculated stress profile with a sub-surface maximum, which is an important characteristic of the measured stress profiles [10]. Spoor and Burggraaf [9] attributed the poor fit between the calculated and experimental curves to the dependence of stress relaxation on concentration and treatment time. Sane and Cooper [10] suggested other possible reasons for this disagreement: the composition dependence of stress relaxation, dilation coefficient and diffusion coefficient. Zhuravlev et al. [11] calculated stress profiles in ion exchanged glass by taking into account the stresses caused by both the composition change at glass surface
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and the thermal expansion mismatch between glass surface and bulk glass. Usually, the stresses due to the thermal expansion coefficient difference are significantly less than stresses caused by the concentration change if the ion exchange is conducted below the glass transition temperature. However, in some cases, the stresses caused by thermal expansion coefficient difference could significantly change the total stress profile, especially near the glass surface [11]. Recently, Startsev and Mazurin [12] proposed a quite different model to calculate the stress profiles in ion exchanged glasses by taking into account the structural relaxation. The calculated stress profiles using this model showed a sub-surface maximum and have reasonable agreement with the experimentally measured stress profiles by Sane and Cooper [10]. However, the calculation is complicated and some of the model variables cannot be evaluated by a direct and unambiguous experiment. In general, most of the previous studies on theoretical calculations of stress profiles in ion exchanged glasses failed to give a satisfactory agreement with the measured stress profiles. In addition, no studies have been found for the stress calculation in two-step ion exchanged glasses, which is the key to the processing–property relationship in ESP glasses. In the current study, the approach is similar to that proposed by Sane and Cooper [10], except the composition dependence of the dilation coefficient and the stress relaxation behavior was incorporated into the analysis. The calculated stress profiles will then be compared to measured profiles, and the difference between them will be discussed. It should be noted that once the stress profiles are calculated, they could be transformed to apparent fracture toughness curves [3]. These curves can then be used to predict and optimize the fracture behavior.
2. Experimental and theoretical approach Stress profiles in ion exchanged glasses were measured for two purposes. First, the dilation coefficient will be calculated from the stress profile in ion exchanged glass where the stress relaxation is negligible. Secondly, calculated stress profiles for glasses processed under typical conditions will be compared to measured profiles. In ion exchanged glasses, there is usually a large variation in composition in the exchanged layer. For example, in cases where K+ ions are exchanged for Na+ ions, the K2O content in the exchanged layer varies from 0 to 20 wt% [13]. Thus, it is important to know the physical properties and stress relaxation behavior of glasses with compositions in this range. A study with these objectives was completed on glasses of direct relevance to the current study [14].
2.1. Glass composition The glasses used in the study were soda-lime silicate float glass (StarphireTM, PPG, Pittsburgh, PA, USA) and soda aluminosilicate glass (Corning 0317, Corning, NY, USA). These will be denoted as SLS and SAS respectively. The measured composition of these two glasses was given in [13]. 2.2. Residual stress profile measurement in ion exchanged glasses All residual stress profiles in the current work were measured by an optimized optical method, and the details about this technique are described elsewhere [15]. In this technique, the central tensile stress in glass was determined by measuring the birefringence of the ion exchanged glass. Then, by progressively etching a thin layer from the glass surface, the average compressive stress in the etched layer can be calculated from the change of the tensile stress and glass dimensions using a force balance. 2.3. Compressive stress relaxation tests Uniaxial compressive stress relaxation behavior of the soda-lime silicate glass and mixed-alkali lime silicate glasses were studied in previous papers [14,16]. Stress relaxation tests were performed by a mechanical testing machine, and the details of the equipment are given in [14,16]. All stress relaxation curves were fitted by the Burger model. 2.4. Theoretical approach Using the thermal stress analogy, stress profiles in ion exchanged glasses can be calculated [10] Z t1 BE o r1 ðx; tÞ ¼ Rðt1 tÞ ðC CÞ dt; ð1Þ 1m 0 ot where r1 is the stress, B is the dilation coefficient, E is the Youngs modulus, m is the Poissons ratio, R is the normalized relaxation function, t is time, t1 is the total exchange time, C is the K2O concentration and the C is the average concentration in the direction of diffusion. When the stress relaxation during ion exchange is negligible, stress profiles can be calculated by [6] r1 ðx; tÞ ¼
BE ðC CÞ: 1m
ð2Þ
Data are, thus, required for all these parameters. The concentration profiles can be obtained experimentally or by calculation if the diffusivity is known. A critical aspect of the theoretical approach was to quantify the composition dependence of B, E, m and R. Experimental data showed that E should vary by <10% in the ion ex-
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
changed layer [14]. For the current calculations, this variability in E was ignored. Similarly, m was also assumed to be constant. The following sections will discuss how B and R were incorporated into the stress calculation. The approach will be to use Eq. (1) to describe the single exchange process and Eq. (2) will then be used for the second step.
The dilation coefficient (B) of glass is defined as the linear strain per unit concentration change of alkaline oxide [7]: B¼
DV 1 ; 3V 0 DC
ð3Þ
where V0 is the volume of glass before ion exchange, C is the K2O concentration and DV is the increase of glass volume after exchange. The dilation coefficient is an important parameter for the calculation of stress profiles in ion exchanged glasses. Varshneya [17] calculated the dilation coefficient for Na+–K+ ion exchange in Na2O Æ 3SiO2 glass and a B value of 1.3 · 103 (wt% K2O) was obtained from the linear strain (4.5%) in glass after ion exchange. This dilation coefficient value is about three to four times those values measured by Cooper and Krohn [18] who obtained B values from measured stress profiles. In the current study, dilation coefficient will be theoretically calculated from the ionic radii of Na+ and K+ ions, and it will also be determined from the measured stress profiles in ion exchanged glass. 3.1. Calculation of B from ionic radii Consider a piece of SLS glass with a weight of 100 g that contains 14.7 g Na2O according to the measured glass composition [13]. It is also known that the Pauling radii of K+ and Na+ ions are 0.133 nm and 0.095 nm respectively [19], and the density of the SLS glass (q) is 2.493 Mg/m3 [16]. Then, the total volume of Na+ ions in this glass can be calculated from the total number of Na+ ions and the volume of each Na+ ion: V Na ¼ 1:025 106 m3 : +
ð4Þ +
If all the Na ions are replaced by K ions, the total volume of K+ ions in the glass can be similarly calculated: V K ¼ 2:81 106 m3 :
ð5Þ
If the glass can freely expand after ion exchange, the volume change of the glass after a complete exchange is only caused by the difference between the total volumes of Na+ and K+ ions. So the volume change of the glass (DV) can be calculated as follows: DV ¼ V K V Na ¼ 1:787 106 m3 :
The initial volume of the SLS glass can be calculated from the mass and density of glass: V 0 ¼ m=q ¼ 4:01 105 m3 :
ð6Þ
ð7Þ
In addition, the K2O concentration in the completely exchanged SLS glass is calculated to be 20.8 wt% according to the measured composition [13]. Therefore, the dilation coefficient value can be calculated as follows: B¼
3. Dilation coefficient for Na+–K+ ion exchange
81
1 DV 1 ¼ 7:14 104 wt% : 3 V 0 DC
ð8Þ
This calculated dilation coefficient is lower than the value (1.3 · 103 wt%1) calculated by Varshneya [17], but higher than the value (3 · 104 wt%1) obtained by Cooper and Krohn [18]. This calculation is based on the assumption that the volume change of the glass (DV) after complete ion exchange would be the same as (VK VNa), the volume difference between the total volumes of Na+ and K+ ions. In reality, the actual volume change of the glass could be less than (VK VNa) because some of the volume increase during ion exchange can be accommodated by a reduction in the free volume of the SLS glass. Hale [20] proposed the effective ion radius of Na+ ions, and argued that the volume change of glass after exchange should be calculated from the difference between the radius of K+ ions and the effective radius of Na+ ions. This is one of the reasons that the calculated dilation coefficient value (7.14 · 104 wt%1) is higher than the value (3 · 104 wt%1) determined by Cooper and Krohn [18]. 3.2. Calculation of B from stress profile The above calculation is based on the assumption that dilation coefficient is composition independent. It is, however, possible that B is dependent on the local K2O concentration. So, the dilation coefficient as a function of K2O concentration will be calculated from the measured stress profile under conditions where stress relaxation effect can be neglected. Usually ion exchange penetration depth is much smaller than the glass thickness, the average concentration is approximately C0 (the original K2O concentration in base glass), and for the SLS glass in the current study, C0 = 0. So, Eq. (2) can be simplified to the following equation in the calculation: r1 ðx; tÞ ¼
BEðC C 0 Þ BEC ¼ : 1m 1m
ð9Þ
Then the composition-dependent dilation coefficient can be calculated from the measured stress profile in the glass according to Eq. (9). Fig. 1 shows the measured residual stress profiles in ion exchanged (450 C/1 h) glass using the technique described in [15]. Since the stress relaxation effect within 1 h at 450 C is not significant (<5%) for glasses with
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Compressive stress (MPa)
400 350 300 250 200 150 100 50 0 0
2
4
6
8
Distance from the glass surface (µm) Fig. 1. Measured residual stress profile in single-step ion exchanged SLS glass (450 C/1 h).
the same composition as the ion exchanged layer [14], it can be neglected. The sub-surface maximum of the stress profile is possibly caused by the polynomial fitting procedure used in deriving the stress profile. The K2O concentration, C, can be calculated using the parameters in [13], so the dilation coefficient as a function of local concentration can be calculated from Eq. (9). Fig. 2 shows the calculation result. Although the Youngs moduli of glasses with different K/Na ratios are different, the Youngs modulus of SLS glass at 450 C (65.1 GPa) [16] was used for the calculation of dilation coefficient. This should not affect the stress profile calculation because the calculated stress is always proportional to the product of Youngs modulus and dilation coefficient. It can be seen from Fig. 2 that the composition-dependent dilation coefficient can be fitted by an empirical equation: BðCÞ ¼ 2:39 103 C þ 6:64 104 ;
where B is in the unit of wt%1. The dilation coefficient linearly decreases with the increase of the K2O concentration in the glass and the previously calculated constant dilation coefficient 7.14 · 104 wt%1 is only slightly higher than the dilation coefficient value for low K2O concentration. The most possible reason for the composition dependence of the dilation coefficient is that the initial introduction of K+ ions in the glass expands the glass structure almost elastically and the free volume in the glass does not decrease. However, with more K+ ions introduced in the glass surface, the glass structure expands slowly and some of the free volume in the glass is taken by the K+ ions. As a result, the linear strain produced in the glass for each unit concentration change of K2O is decreasing. Another possible reason is that the Youngs modulus of high K glass (e.g. K22 glass containing 22.5 wt% K2O) is lower than that of SLS glass [14], and the calculated dilation coefficient value in high K2O region is underestimated. However, the modulus difference between SLS glass and K22 glass is only 10%, which is much less than the dilation coefficient difference between ion exchanged glass surface and bulk glass. So the Youngs modulus difference can only partly explain the composition dependence of dilation coefficient.
4. Calculation of stress profiles in single-step ion exchanged glasses 4.1. Stress calculation without stress relaxation According to Eq. (9), the stress profile in ion exchanged glass can be calculated using the constant dila-
ð10Þ 1200
Compressive stress (MPa)
Dilation coefficient (wt%-1)
Measured
1000
0.0007
y = -2.39E-05x + 6.64E-04
0.0006 0.0005 0.0004 0.0003 0.0002
Calculated (constant B) Calculated (variant B)
800
600
400
200
0.0001 0 0
5
10
15
20
25
K2O concentration (wt%) Fig. 2. Dilation coefficient as a function of local K2O concentration for Na+–K+ ion exchange in SLS glass.
0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 3. Measured and calculated stress profiles (without stress relaxation) in single-step ion exchanged glass (450 C/48 h).
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
4.2. Stress calculation with stress relaxation using a viscoelastic model Using the KWW equation (or b-function), the stress profile in ion exchanged glass can be calculated using the following equation [10]: r1 ¼
Z
t1
BE x ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi 1m t 4pD1 t 0 x2 t1 tb exp dt; exp s 4D1 t
ð11Þ
where t1 is the first-step ion exchange time, Cs1 is the surface concentration after single-step ion exchange, D1 is the interdiffusion coefficient in single-step exchange, b is a constant (0 < b < 1) and s is the stress relaxation time [10]. Similarly, if the Burger model is used to calculate the stress profile, a similar equation can be derived from Eq. (1): BE x x2 ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi exp r1 ¼ 1m 4D1 t t 4pD1 t 0 t1 t t1 t A1 exp þ A2 exp dt; ð12Þ s1 s2 Z
t1
where A1, A2, s1 and s2 are parameters for stress relaxation function. Fig. 4 shows the measured stress profile and the calculated profile from Eq. (12) by including the stress Table 1 Parameters used in the stress profile calculations (SLS glass) Youngs modulus (E: GPa)
Poissons ratio (m)
Diffusion coefficient (D1: m2/s)
Surface concentration (Cs1: wt%)
65.1
0.22
1.4 · 1015
20.0
A1
A2
s1 (min)
s2 (min)
0.751
0.249
2140
11.2
350 300
Compressive stress (MPa)
tion coefficient and the variant dilation coefficient described in the previous section (Fig. 2). Fig. 3 shows the comparison between the calculated and measured stress profiles for a typical ion exchange condition (450 C, 48 h). The parameters used in the calculation are displayed in Table 1. Youngs modulus of SLS glass at 450 C (65.1 GPa) was used in the calculation [16]. It can be seen that the calculated stress profile using a constant dilation coefficient (7.14 · 104 wt%1) is much higher than the measured profile. By using the composition-dependent dilation coefficient, the calculated stress profile is also higher than the measured profile, especially in the near surface region. Therefore, it is concluded that stress relaxation cannot be neglected in typical ion exchange conditions.
83
Predicted Measured
250 200 150 100 50 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 4. Measured and calculated stress profiles (with compositionindependent stress relaxation) in single-step ion exchanged glasses (450 C/48 h).
relaxation effect. The relaxation function parameters for the SLS glass at 450 C in Table 1 were used for the calculation [14]. Obviously, the calculated stress profile overestimated the stress relaxation effect in the glass during the ion exchange process. It was hypothesized that composition-dependent stress relaxation data are needed for a more accurate calculation. In [14], mixedalkali silicate glasses with higher K2O concentrations were found to relax more slowly than SLS glass. So it is expected to obtain a better agreement between the calculated and measured stress profile in glass if these compositional-dependent relaxation data are used. 4.3. Stress calculation using composition-dependent stress relaxation and dilation The above calculations and comparisons show that the effect of local K2O concentration on both the dilation coefficient and the stress relaxation times need to be included for an accurate prediction of stress profile. The dilation coefficient as a function of local K2O concentration is calculated previously, and it is assumed to be valid for other ion exchange conditions. 4.3.1. Stress relaxation behavior of different glasses fitted by a power law Stress relaxation behavior of SLS and mixed-alkali lime silicate glasses have been studied [14]. The uniaxial viscosity values can be calculated from the measured stress relaxation curves assuming linear viscoelasticity. In this approach, viscosity values are estimated and these values are used to extrapolate the relaxation times. Fig. 5 shows that the logarithm viscosity at 550 C vs. K2O concentration in glass can be fitted by: logðgÞ ¼ 2:784 logðCÞ þ 17:121;
ð13Þ
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
shown that these glasses show a mixed-alkali effect and the viscosity can increase at lower K2O concentrations [14]. Although the viscosity and relaxation times are also dependent on glass composition when the K2O concentration is less than 3 wt%, the effect of this viscosity change on stress profile is minor. Moreover, the stress profile in low K2O concentration region of glass (<2.92 wt%) is relatively less important for the fracture behavior of the glass. So, in the current study, stress profile was only calculated in the region where K2O is greater than 2.92 wt%.
20
550 C
Log (viscosity/Pa s)
18
y = 2.784x + 19.681
450 C
16
14
y = 2.784x + 17.121 12
10
8 -2.0
-1.5
-1.0
-0.5
0.0
Log (K2O concentration) Fig. 5. Logarithm viscosity vs. logarithm K2O concentration for glasses with compositions similar to those in the ion-exchanged layer (power law fit).
where g is the unit of Pa s. As described in [14], the viscosity data at 450 C for glasses with different K/Na ratios can be obtained by shifting the log viscosity data at 550 C by a constant. The viscosity values for a glass containing 2.92 wt% K2O (K03) at 450 C and 550 C are known, and the difference between these values is log(2.17 · 1015) log(5.99 · 1012) = 2.56. Then the viscosity at 450 C for other glasses can be calculated by the following equation: logðgÞ ¼ 2:784 logðCÞ þ 19:681;
ð14Þ
where g is the unit of Pa s. It is also demonstrated in [14] that the stress relaxation function coefficients A1 and A2 are approximately constant for glasses with different K/Na ratios at a given temperature. Therefore, the ratio of viscosity values and relaxation times are the same for different glasses [14], and then the stress relaxation times for other glasses can be calculated from the stress relaxation data of the K03 glass. Assuming the viscosity of K03 glass at 450 C is g0, and the stress relaxation times in the Burger model are s01 , s02 . The stress relaxation times for other glasses (s1 and s2) can be calculated from the viscosity (g) by: g si ðCÞ ¼ s0i ; ð15Þ g0 where i = 1, 2. By combining Eqs. (14) and (15), one obtains: si ðCÞ ¼
s0i ð2:784 log Cþ19:681Þ 10 ; g0
ð16Þ
where i = 1, 2. Eqs. (13)–(16) are only valid when the K2O concentration in glass is greater than 2.92 wt% as it has been
4.3.2. Stress relaxation behavior of different glasses fitted by a linear equation Many authors have tried to calculate the viscosity values of glasses at different temperatures from the glass compositions, especially for soda-lime silicate glass (see [14]). However, these empirical equations usually have a small composition and temperature range, and they are not applicable for glasses with high K2O concentration (>10 wt%) at very low temperatures (T < Tg 100 C). However, almost all the calculation methods show that the increase of K2O concentration in sodalime glass leads to a linear increase of viscosity. Fig. 6 shows the viscosity values at 450 C of different glasses extrapolated from viscosity values at 550 C. Although the viscosity of glass is not a linear function of K2O concentration over the whole range of K2O concentration (0–22.5 wt%), the assumption of linearity can be used to estimate the viscosity value of any glass within the composition range, especially in the region of high K2O concentration. It can be seen from Fig. 6 that the viscosity values at 450 C can be fitted by an empirical equation: g ¼ 2:925 1018 C 6:978 1016 ;
ð17Þ
7E+17 6E+17
y = 2.925E+18x - 6.978E+16 2
R = 9.281E-01
Viscosity (Pa s)
84
5E+17 4E+17 3E+17 2E+17 1E+17 0 0.00
0.05
0.10
0.15
0.20
0.25
K 2O concentration Fig. 6. Viscosity vs. K2O concentration for glasses with compositions similar to those in the ion-exchanged layer (linear fit).
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
si ðCÞ ¼
s0i ð2:925 1018 C 6:978 1016 Þ; g0
ð18Þ
350
Calculated (power law)
300
Compressive stress (MPa)
where g is the unit of Pa s. Obviously, this equation is only valid when the K2O concentration is greater than 2.4 wt%. However, as previously discussed, stress profile in low K2O concentration region is relatively less important, and again, stress profile was only calculated in the region where K2O is greater than 2.4 wt%. For the linear case, the stress relaxation times for other glasses (s1 and s2) can be calculated from the K2O concentration (C) by:
Measured 250 200 150 100 50
where i = 1, 2.
0 0
By using Eqs. (16), (18) and (19), stress profiles in singlestep ion exchanged glass (450 C/48 h) can be calculated using different relationships between viscosity and glass composition. A1 and A2 values in Table 1 were used for the calculation. The calculated and measured stress profiles are shown in Figs. 7 and 8. The upper and lower bounds of calculated stress profiles are obtained by adding or subtracting the mean square error from Eqs. (14) and (17). When a power law equation is used for the relationship between viscosity and glass composition, the calculated stress profile is lower than the calculated stress profile in the sub-surface region (10–30 lm). However, the stress profile calculated by using a linear relationship between viscosity and glass composition has a better agreement with the measured stress profile. The difference between the calculated and measured stress profiles are less than 10% in the first 10 lm below the glass surface, and it is less than 45 MPa in other regions. Several possible reasons are discussed below for the discrepancy between the calculated and measured stress profiles: (1) One possibility is that the current empirical relationships between viscosity and glass composition (Eqs. (14) and (17)) are not accurate enough. If more mixed-alkali silicate glasses with different K2O concentration between 9 wt% and 23 wt% are melted and studied, a more accurate relationship
10
20
30
40
50
Distance from the glass surface (µm) Fig. 7. Comparison of the measured and calculated stress profiles in single-step ion exchanged glass (450 C/48 h) using a power law relationship between viscosity and K2O concentration in the glass. 10% error bars are shown for the maximum and minimum measured stress values.
400
Calculated (linear)
350
Measured
Compressive stress (MPa)
4.3.3. Stress profile calculation using the compositiondependent stress relaxation and dilation coefficient Given the extrapolated stress relaxation times in different glasses at ion exchange temperatures and the measured composition-dependent dilation coefficient, the following equation is developed by modifying Eq. (12) to calculate the stress profile in glass: Z t1 BðCÞE x x2 r1 ¼ ðCs1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi exp 1m 4D1 t t 4pD1 t 0 t1 t t1 t A1 exp þ A2 exp dt: ð19Þ s1 ðCÞ s2 ðCÞ
85
300 250 200 150 100 50 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 8. Comparison of the measured and calculated stress profiles in single-step ion exchanged glass (450 C/48 h) using a linear relationship between viscosity and K2O concentration in the glass. 10% error bars are shown for the maximum and minimum measured stress values.
between the viscosity and glass composition could be obtained. If the viscosity increases faster with the increase of K2O concentration in glass than described in Figs. 5 and 6, the calculated stress profile in the first 20 lm would be higher than in Figs. 7 and 8, and the difference between the calculated and measured stress profiles would be smaller. (2) The calculated dilation coefficient could be slightly underestimated due to the small stress relaxation effect in 450 C/1 h ion exchanged glass, so the calculated stress profile is also slightly underestimated.
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
(3) The discrepancy could also be caused by the uncertainty of the diffusion coefficient. The diffusion coefficient at 450 C is calculated from the measured K2O concentration profiles in ion exchanged glass, and it is not unusual to have 10% error in the measured concentration profile and the calculated diffusion coefficient. If a 20% higher diffusion coefficient is used to calculate the stress profile, the difference between the measured and calculated stress profiles could be minimized. (4) The stress relaxation functions of potassium glasses at 450 C used for the stress profile calculation are extrapolated from the data of SLS glass, and they may be different from the real stress relaxation function of those glasses. (5) Finally, the experimental error of the measured stress profiles in ion exchanged glasses is estimated to be about 10%. It is concluded, therefore, that the calculated stress profile in ion exchanged glass using the compositiondependent dilation coefficient and stress relaxation data gives a reasonable agreement with the measured stress profile considering the uncertainties in the various parameters used for the calculation. As shown in [14], the thermal expansion coefficient difference between the ion exchanged glass surface and the bulk glass is about 0.64 · 106 C1. So, the maximum residual thermal stress (rR) caused by this difference can be calculated by: DT DaE ; rR ¼ 1m
ð20Þ
where DT is the difference between ion exchange temperature and room temperature, Da is the thermal expansion coefficient difference between the ion exchanged glass surface and the bulk glass. Since the thermal expansion coefficient of the 22 wt% K2O glass is higher than that of SLS glass, the residual thermal stress is tensile in ion exchanged glass surface. Then, if a glass sample was cooled from 450 C to 25 C, the maximum resultant residual thermal stress is only 23 MPa (tensile), which is much lower than the residual stress in typical K/Na ion exchanged glass, especially in the near surface region. Clearly the current calculation of B includes these mismatch strains, but one can conclude that B is dominated by the mismatch in ionic size. 5. Calculation of stress profiles in two-step ion exchanged glasses The stress profile in two-step ion exchanged glass is important for the processing–property relationship in ESP glasses. Since the second step ion exchange process is usually performed at a lower temperature (e.g. 400 C) and for a much shorter time (e.g. 30 min), the stress
800 1
700
Compressive stress (MPa)
86
600
3
500
2
400 300 200 100 0 0
10
20
30
40
50
Distance from the glass surface (µm) Fig. 9. Calculated and measured stress profiles in two-step ion exchanged glass (SAS). Curves 1 and 2 are the measured stress profiles in single-step (450 C/4 h) and two-step (450 C/4 h + 400 C/30 min) ion exchanged glasses respectively, and curve 3 is the calculated stress profile from the measured stress profile in single-step ion exchanged glass (curve 1).
relaxation effect in the second-step can be neglected according the stress relaxation data of SLS glass in [16]. So, the stress profiles in two-step ion exchanged glasses can be calculated from that in single-step ion exchanged glasses according to the concentration profile change in the second-step process: r2 ¼ r1
BE ðC 2 C 1 Þ; 1m
ð21Þ
where r1 and r2 are stress profiles after first-step and second-step ion exchange process, C1 and C2 are K2O concentration profiles after single-step and two-step ion exchange process. In order to verify if the concentration change can explain the stress profile change in the second-step ion exchange process, the stress profile in two-step ion exchanged (450 C/4 h + 400 C/30 min) SAS glass was calculated from the measured stress profile in single-step ion exchanged (450 C/4 h) SAS glass. This stress profile was then compared to the measured stress profile in the same two-step ion exchanged glass. The results are shown in Fig. 9 and it can be seen that the calculated and measured stress profiles have reasonably good agreement (curves 2 and 3). 6. Conclusions The dilation coefficient for Na+–K+ ion exchange was calculated and then stress profiles in single-step and twostep ion exchanged glasses were calculated based on dilation coefficient, stress relaxation data and the diffusion data: (1) The measured dilation coefficient for Na+–K+ ion exchange was found to be composition dependent. An empirical equation was developed to calculate
J. Shen, D.J. Green / Journal of Non-Crystalline Solids 344 (2004) 79–87
the composition-dependent dilation coefficient. Thermal expansion mismatch strains were shown to be a minor contribution to the dilation. (2) The calculated stress profile in single-step ion exchanged glass without the stress relaxation is much higher than the measured stress profile for typical ion exchange conditions. Conversely, if the stress profile is calculated using the stress relaxation data of SLS glass, it is much lower than the measured stress profile. (3) The calculated stress profile in single-step ion exchanged glass using the composition-dependent dilation coefficient and stress relaxation data has reasonable agreement with the measured stress profile. The difference between the calculated and measured stress profiles is mainly caused by the lack of an accurate relationship between stress relaxation behavior and glass composition. (4) The calculated stress profile for two-step ion exchanged glass has reasonable agreement with the measured stress profile, and stress relaxation can be neglected for the second step studied here. Acknowledgment The authors wish to acknowledge NSF Center for Glass Research for the financial support of this work and allowing the publication of this work.
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