**3. Results and discussion**

The residual stress (σr) measured on the surface of glasses treated in the different molten salts is reported in Figure 1. The residual stress clearly decreases as NaNO3 content increases, especially above 0.5%.

Figure 1 Compressive residual stress on the surface of glass ion exchanged in the different salt baths. **Figure 1.** Compressive residual stress on the surface of glass ion exchanged in the different salt baths. Figure 1 Compressive residual stress on the surface of glass ion exchanged in the different salt baths.

Conversely, the case depth reported in Figure 2 shows a quite different trend and actually seems almost invariant as a function of the sodium content of the salt bath. Conversely, the case depth reported in Figure 2 shows a quite different trend and actually seems almost invariant as a function of the sodium content of the salt bath. Conversely, the case depth reported in Figure 2 shows a quite different trend and actually seems almost invariant as a function of the sodium content of the salt bath.

Figure 2 Case depth as a function of the exchanging bath.

scatter and the relatively short duration of the ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut

The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large scatter and the relatively short duration of the ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut glass. Nevertheless, the large strength scatter does not allow to point out specific trends

The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large Figure 2 Case depth as a function of the exchanging bath. **Figure 2.** Case depth as a function of the exchanging bath.

with respect to the Na content of the exchanging baths.

The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large scatter and the relatively short duration of the ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut glass. Never‐ theless, the large strength scatter does not allow to point out specific trends with respect to the Na content of the exchanging baths. Figure 2 Case depth as a function of the exchanging bath. The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large scatter and the relatively short duration of the ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut glass. Nevertheless, the large strength scatter does not allow to point out specific trends with respect to the Na content of the exchanging baths.

Figure 1 Compressive residual stress on the surface of glass ion exchanged in the different salt baths. Conversely, the case depth reported in Figure 2 shows a quite different trend and actually

seems almost invariant as a function of the sodium content of the salt bath.

Japan) and the potassium Kα signal was recorded on a path of ~30 μm long by using the Energy Dispersion X-ray Spectroscopy (EDXS) (EDS2000, IXRF System, USA) probe. The chemical composition of the external surface of the glasses after the ion-exchange process was analyzed

.

The residual stress (σr) measured on the surface of glasses treated in the different molten salts is reported in Figure 1. The residual stress clearly decreases as NaNO3 content increases,

Figure 1 Compressive residual stress on the surface of glass ion exchanged in the different salt baths. Conversely, the case depth reported in Figure 2 shows a quite different trend and actually

Figure 1 Compressive residual stress on the surface of glass ion exchanged in the different salt baths. Conversely, the case depth reported in Figure 2 shows a quite different trend and actually

Conversely, the case depth reported in Figure 2 shows a quite different trend and actually

ABCDEFG

**Salt**

Figure 2 Case depth as a function of the exchanging bath. The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large scatter and the relatively short duration of the ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut

Figure 2 Case depth as a function of the exchanging bath. The strength of ion-exchanged samples is shown in Figure 3. In spite of the typical large scatter and the relatively short duration of ion-exchange process (compared to typical industrial duration, usually in excess of 8 h), the strengthening effect is clear in any used salt bath, and the failure stress increases by a factor of about 2.2–2.4 with respect to the as-cut glass. Nevertheless, the large strength scatter does not allow to point out specific trends

ABCDEFG

ABCDEFG

**Salt**

**Salt**

seems almost invariant as a function of the sodium content of the salt bath.

seems almost invariant as a function of the sodium content of the salt bath.

seems almost invariant as a function of the sodium content of the salt bath.

**Figure 1.** Compressive residual stress on the surface of glass ion exchanged in the different salt baths.

in the same way in a region of about 0.5 mm2

**Figure 2.** Case depth as a function of the exchanging bath.

with respect to the Na content of the exchanging baths.

**δ (μm)**

**δ (μm)**

**σr (MPa)**

**3. Results and discussion**

156 Ion Exchange - Studies and Applications

especially above 0.5%.

Figure 3 Average flexural strength as a function of the exchanging bath (the standard deviation is also shown). **Figure 3.** Average flexural strength as a function of the exchanging bath (the standard deviation is also shown).

A more effective representation of the strength data is possible by using Weibull plots where the measured strength data are reported as a function of failure probability. Here, failure probability was calculated as: A more effective representation of the strength data is possible by using Weibull plots where the measured strength data are reported as a function of failure probability. Here, failure probability was calculated as:

$$P = \frac{i - 0.3}{N + 0.4} \tag{4}$$

where *i* is the rank in the ascending ordered strength distribution and *N* the total number of samples considered for each condition. The obtained Weibull plots are shown in Figure 4. One can easily observe again the evident strengthening effect in any salt bath with respect to ascut samples; conversely, no specific trend can be seen with respect to the purity of the used bath.

The relationship between failure probability and tensile stress is typically expressed as:

$$P = 1 - \exp\left[-K S \left(\frac{\sigma}{\sigma\_0}\right)^n\right] \tag{5}$$

where *S* is the surface area of the sample under tensile stress, *K* the loading factor (whose units are per unit area), *m* the Weibull modulus, and σ0 the normalizing stress, representing the scatter of the distribution. Taking twice the natural logarithm of both sides, a linear equation can be obtained, which is functional for the Weibull modulus calculation [8]: where *S* is the surface area of the sample under tensile stress, *K* the loading factor (whose units are per unit area), *m* the Weibull modulus, and σ0 the normalizing stress, representing

��

$$\ln\left(\ln\left(\frac{1}{1-P}\right)\right) = m\ln\sigma + \ln\frac{KS}{\sigma\_0'''} \tag{6}$$

Figure 4 Weibull distributions for treated and untreated glasses. **Figure 4.** Weibull distributions for treated and untreated(a.r.) glasses.

Table 3 reports the Weibull modulus for the various distributions shown in Figure 4. One can observe that the moduli are also unaffected by changing the purity of the bath. Table 3 reports the Weibull modulus for the various distributions shown in Figure 4. One can observe that the moduli are also unaffected by changing the purity of the bath.

Salt A B C D E F G a.r.


drastic increase in the potassium concentration after the ion-exchange process but the potassium concentration follows substantially the same trend of the surface residual stress **Table 3.** Weibull modulus of the strengthened glasses in different baths along with the as-received glass

(Figure 1), thus indicating a strict correlation between the amount of exchanged ions and the developed "stuffing"/reinforcing effect. The potassium concentration profiles recorded by the EDXS line analysis always resembled a typical Nernst–Planck diffusion trend, which can be expressed as: �� � ��� �� � ������� � ����� � � (7) The surface concentration of K+ and Na+ measured by EDXS is shown in Figure 5. There is a drastic increase in the potassium concentration after the ion-exchange process but the potas‐ sium concentration follows substantially the same trend of the surface residual stress (Figure 1), thus indicating a strict correlation between the amount of exchanged ions and the developed "stuffing"/reinforcing effect.

where *x* is the distance from the surface, *t* the ion exchange time, ��� the potassium concentration on the surface, and �� the interdiffusion coefficient. The potassium The potassium concentration profiles recorded by the EDXS line analysis always resembled a typical Nernst–Planck diffusion trend, which can be expressed as:

���

$$\mathbf{C}\_{\mathbf{x}}\left(\mathbf{x},t\right) = \frac{\mathbf{C}\_{\mathbf{x}}\left(\mathbf{x},t\right)}{\mathbf{C}\_{\mathbf{x}\_0}} = \text{erfc}\left(\frac{\mathbf{x}}{2\sqrt{\overline{D}t}}\right) \tag{7}$$

�√���

where *x* is the distance from the surface, *t* the ion exchange time, *CK*<sup>0</sup> the potassium concentra‐ tion on the surface, and *D*ˉ the interdiffusion coefficient. The potassium concentration experi‐ mental data were fitted by Equation (7) to determine the interdiffusion coefficient and the penetration depth, identified as the distance from the surface where the potassium concen‐ tration is lower than 2% with respect to the surface one. Figures 6 and 7 show the obtained results.

Figure 5 Potassium and sodium concentration on the surface of treated and bare glass.

**Figure 5.** Potassium and sodium concentration on the surface of treated and bare glass.

6

scatter of the distribution. Taking twice the natural logarithm of both sides, a linear equation

where *S* is the surface area of the sample under tensile stress, *K* the loading factor (whose units are per unit area), *m* the Weibull modulus, and σ0 the normalizing stress, representing the scatter of the distribution. Taking twice the natural logarithm of both sides, a linear equation can be obtained, which is functional for the Weibull modulus calculation [8]:

1 *<sup>m</sup>*

Figure 4 Weibull distributions for treated and untreated glasses. Table 3 reports the Weibull modulus for the various distributions shown in Figure 4. One can

Table 3 reports the Weibull modulus for the various distributions shown in Figure 4. One can

4567

**ln(σf )**

drastic increase in the potassium concentration after the ion-exchange process but the potassium concentration follows substantially the same trend of the surface residual stress (Figure 1), thus indicating a strict correlation between the amount of exchanged ions and the

The potassium concentration profiles recorded by the EDXS line analysis always resembled

drastic increase in the potassium concentration after the ion-exchange process but the potas‐ sium concentration follows substantially the same trend of the surface residual stress (Figure 1), thus indicating a strict correlation between the amount of exchanged ions and the developed

where *x* is the distance from the surface, *t* the ion exchange time, ��� the potassium concentration on the surface, and �� the interdiffusion coefficient. The potassium

æ ö = = ç ÷

The potassium concentration profiles recorded by the EDXS line analysis always resembled a

� ��� �� � ������� ���

Salt A B C D E F G a.r. Weibull modulus 3.9 4.1 3.6 4.4 3.9 4.4 3.5 3.2 Table 3 Weibull modulus of the strengthened glasses in different baths along with the asreceived glass

**Weibull modulus** 3.9 4.1 3.6 4.4 3.9 4.4 3.5 3.2

**Salt** A B C D E F G a.r.

and Na<sup>+</sup> measured by EDXS is shown in Figure 5. There is a

and Na+ measured by EDXS is shown in Figure 5. There is a

� ����� � �√���

2

� (7)

è ø (7)

observe that the moduli are also unaffected by changing the purity of the bath.

**Table 3.** Weibull modulus of the strengthened glasses in different baths along with the as-received glass

observe that the moduli are also unaffected by changing the purity of the bath.

a typical Nernst–Planck diffusion trend, which can be expressed as:

typical Nernst–Planck diffusion trend, which can be expressed as:

'

*K*

��

( ) ( )

*K*

, ,

0

*K C xt <sup>x</sup> C xt erfc <sup>C</sup> Dt*

The surface concentration of K+

The surface concentration of K+

"stuffing"/reinforcing effect.

developed "stuffing"/reinforcing effect.


**Figure 4.** Weibull distributions for treated and untreated(a.r.) glasses.

**ln(ln(1/(1-**

 **P)))**

158 Ion Exchange - Studies and Applications

s

��

*m*

����� � � �� � � �� ��

0

è ø - è ø (6)

� (6)

**A B C D E F G a.r.**

s

*KS*

can be obtained, which is functional for the Weibull modulus calculation [8]:

*P*

������ �

<sup>1</sup> ln ln ln ln

æ ö æ ö ç ÷ = + ç ÷

**Figure 6.** Interdiffusion coefficient for potassium in the different baths.

Figure 6 Interdiffusion coefficient for potassium in the different baths.

Figure 7 Depth of penetration for potassium during the ion-exchange process in the different baths. **Figure 7.** Depth of penetration for potassium during the ion-exchange process in the different baths.

The interdiffusion coefficients are in very good agreement with data reported in previous works [9, 10]. It is also confirmed that the ܦഥ is not really affected by the presence of limited amounts of Na (up to 5%) in the KNO3 bath [11]. Conversely, the surface concentration and, consequently, the concentration of potassium in the sub-surface layers are lower when the glass is treated in the Na-containing bath. On this basis, the K surface concentration in Equation (7) scales with the concentration in the used salt. Accordingly, the residual stress on the surface (shown in Figure 1) is higher when very pure KNO3 bath is used, while the case depth does not change to an appreciable extent. Nevertheless, the effect of Nacontaining salts on the final strength is substantially negligible, as shown in Figures 3 and 4. Clearly, this is mainly associated to the experimental scatter of the strength measurement, related to the typical dispersion on the surface defect sizes. In addition, due to the limited exchanging time used in the present work, some of the flaws are not "fully" reinforced; as a matter of fact, starting from the strength of the as-cut glass (ranging from ≈ 100 MPa to 400 MPa) and assuming, for simplicity, semicircular surface cracks, one can calculate that flaw sizes vary from ≈5 to ≈80 µm. Therefore, according to the case depth (Figure 2) and K penetration (Figure 7) results, it is evident that only a portion of the surface defects are completely "immersed" in the residual compressive stress field.. Deeper defects,, in a simplified model that considers flaws as invariant and perfectly closed during the ionexchange process, are conversely subjected to a residual stress that changes from highly The interdiffusion coefficients are in very good agreement with data reported in previous works [9, 10]. It is also confirmed that the *D*ˉ is not really affected by the presence of limited amounts of Na (up to 5%) in the KNO3 bath [11]. Conversely, the surface concentration and, consequently, the concentration of potassium in the sub-surface layers are lower when the glass is treated in the Na-containing bath. On this basis, the K surface concentration in Equation (7) scales with the concentration in the used salt. Accordingly, the residual stress on the surface (shown in Figure 1) is higher when very pure KNO3 bath is used, while the case depth does not change to an appreciable extent. Nevertheless, the effect of Na-containing salts on the final strength is substantially negligible, as shown in Figures 3 and 4. Clearly, this is mainly associated to the experimental scatter of the strength measurement, related to the typical dispersion on the surface defect sizes. In addition, due to the limited exchanging time used in the present work, some of the flaws are not "fully" reinforced; as a matter of fact, starting from the strength of the as-cut glass (ranging from ≈ 100 MPa to 400 MPa) and assuming, for simplicity, semicircular surface cracks, one can calculate that flaw sizes vary from ≈5 to ≈80 μm. Therefore, according to the case depth (Figure 2) and K penetration (Figure 7) results, it is evident that only a portion of the surface defects are completely "immersed" in the residual compressive stress field.Deeper defects, in a simplified model that considers flaws as invariant and perfectly closed during the ion-exchange process, are conversely subjected to a residual stress that changes from highly compressive on the surface to slightly tensile at a certain depth (below ≈12 μm). The effect of the initial flaw sizes, i.e. of the surface quality of the bare glass, appears to be more important in the ion-exchange process than the presence of limited amount of Na in the salt bath.
