**3. Determination of the equilibrium concentration of modifier ions in the glass**

In the theoretical calculations describing the ion exchange processes in glasses a two-compo‐ nent model of this phenomenon is adopted [33]. The nonlinear diffusion equation (electrodif‐ fusion) is solved in the field of normalized concentration of exchanged ions (admixture ions introduced into the glass and mobile ions of its modifier). This normalization is made in relation to the equilibrium concentration, which is the total concentration in the glass of both types of exchanged ions. The value of this concentration remains constant during the whole process of ion exchange. This section presents two methods of estimating the value of this concentration. One of them is based on a calculation of the electric charge flowing through the glass in the electrodiffusion processes. The second one is based on a measurement of the mass of the glass substrate before and after the diffusion process. The experimental results are presented in relation to soda-lime glass, in which the Ag+ ↔ Na+ ion exchange processes were carried out.

As an illustration of the applicability of the sequence of polarization direction changes of the electric field, Fig.8 shows the refractive index profiles of waveguides produced on both sides of the glass substrate in the electrodiffusion process, in which six different polarization states were applied. The total duration of the process was *ttot* = 60 min. The durations of each of the processes were equal and were t+ = t- = 10'. In the case of waveguide of type "A" (Fig.4a), a 12 modal structure was obtained (TE polarization, *λ* = 677 nm). The shapes of the refractive index profile of this waveguide are similar to that obtained in the electrodiffusion processes with fixed polarization direction of the electric field (compare Fig.3b). For the resulting waveguide of type "B" (10-modal structure) a monotonic course of the refractive index profile with characteristic inflection point at a depth corresponding to the position of the turning point of the mode of 3rd order was obtained (Fig.8b). In both figures the total values of the electric

applied. The total duration of the process was *tc* = 60 min. The durations of each of the processes were equal and were t<sup>+</sup> = t- = 10'. In the case of waveguide of type "A" (Fig.4a), a 12-modal

this waveguide are similar to that obtained in the electrodiffusion processes with fixed polarization direction of the electric field (compare Fig.3b). For the resulting waveguide of type "B" (10-modal structure) a monotonic course of the refractive index profile with characteristic inflection point at a depth corresponding to the position of the turning point of the mode of 3rd order was obtained (Fig.8b). In both figures the total values of the electric charge that has passed through the substrate

(a) (b)

**Figure 8.** The refractive index profiles of the waveguides produced by electrodiffusion processes with a multiple change of the direction of polarization and of the values of the electric field intensity. The waveguide produced on the

**Fig.8 The refractive index profiles of the waveguides produced by electrodiffusion processes with a multiple change of the direction of polarization and of the values of the electric field intensity. The** 

The presented measurement results of refractive index profiles of waveguides produced in

the electrodiffusion processes (in which there is a change of polarization direction of the applied electric field) indicate a high possibility of the use of such processes to the intended shaping of

**2. Determination of the equilibrium concentration of modifier ions in the** 

**waveguide produced on the first side of the substrate (a) and on the other side of the substrate (b).**

refractive index profiles of produced waveguide structures.

first side of the substrate (a) and on the other side of the substrate (b).

**glass**

= 677 nm). The shapes of the refractive index profile of

charge that has passed through the substrate in either direction are given.

**Figure 7.** Refractive index profiles of produced waveguides with shapes close to linear.

structure was obtained (TE polarization,

in either direction are given.

114 Ion Exchange - Studies and Applications

### **3.1. The dependence of changes of the refractive index profile of the waveguide on the electric charge flowing in the electrodiffusion process**

In the electrodiffusion process carried out at time τ, the amount of the charge Qτ, which has passed through the glass can be determined on the basis of the current dependency i(t):

$$\mathbf{Q}\_{\tau} = \prod\_{0}^{\tau} \mathbf{i}(\mathbf{t}) \mathbf{d}\mathbf{t} \tag{4}$$

Taking into account that u(x) = c(x)/c0, where c0 (m-3) is the normalized concentration of mobile components of the glass, (which is a concentration of mobile modifier ions in the glass before the exchange process) the equation describing the relationship of the refractive index profile n(x) with the normalized concentration of the admixture u(x) takes the form:

$$\mathbf{n}\left(\mathbf{x}\right) = \mathbf{n}\_{\rm b} + \Delta \mathbf{n}\_{s} \cdot \frac{\mathbf{c}\left(\mathbf{x}\right)}{\mathbf{c}\_{0}},\tag{5}$$

where nb - refractive index of the glass before the ion exchange process, Δn<sup>s</sup> - increment of the refractive index of the glass (caused by the ion exchange process) at its surface, cA(x) - function describing the distribution of the absolute concentration of admixture ions introduced into the glass (m-3).

**Figure 9.** Determination of the cross section *S* of the exchange process.

By introducing the function, δn(x) = n(x) - nb, from the equation (5) the following can be obtained.

$$\mathbf{c}(\mathbf{x}) = \delta \mathbf{n}(\mathbf{x}) \cdot \frac{\mathbf{c}\_0}{\Delta \mathbf{n}\_s} \quad \left(\mathbf{m}^{-3}\right) \tag{6}$$

This equation describes the distribution of concentration of admixture introduced into the glass, expressed by the parameters of the refractive index profile n(x) and of the equilibrium concentration c0. Integrating the expression (6) over the entire volume of the glass *V*, the total number of admixture ions Nτ introduced into the glass during the electrodiffusion process in the time τ is obtained as follows.

$$\mathbf{N}\_{\pi} = \int\_{\mathcal{V}} \mathbf{c}(\mathbf{x}) \mathbf{d} \mathbf{V} = \frac{\mathbf{c}\_{0} \cdot \mathbf{S}}{\Delta \mathbf{n}} \Big|\_{0}^{\circ} \delta \mathbf{n} \text{(\(\mathbf{x}\)} \text{dx} \tag{7}$$

In this equation, *S* is the surface area of the glass substrate (cross section for the exchange - Fig.9) which comes in contact with a liquid admixture contained in the crucible. At the same time the total charge (4), which has flowed through the glass, allows to specify the number of admixture ions having a valence *w*, which have been introduced into the glass in the time τ:

$$\mathbf{N}\_r = \frac{1}{\mathbf{w} \cdot \mathbf{e}} \int\_0 \mathbf{i}\left(\mathbf{t}\right) d\mathbf{r} = \frac{\mathbf{Q}\_r}{\mathbf{w} \cdot \mathbf{e}} \,\mathrm{\,\, \,}\tag{8}$$

where *e* - the elementary charge.

Comparing (7) and (8) the following equation is obtained.

$$\int\_0^\sigma \delta \mathbf{n} \left(\mathbf{x}\right) d\mathbf{x} = \frac{\Delta \mathbf{n}\_s}{\mathbf{e} \cdot \mathbf{w} \cdot \mathbf{S} \cdot \mathbf{c}\_0} \cdot \mathbf{Q}\_\tau \quad \left(\mathbf{m}\right) \tag{9}$$

By calculating the above integral the product Sc0 of the cross-section of the exchange and the equilibrium concentration can be determined.

Table 2 summarizes the results of calculation product Sc0 based on equation (9) for several electrodiffusion processes carried out in the substrate of soda-lime glass, using a pure silver nitrate AgNO3 as the source of admixture of Ag+ ions. It also presents the maximum changes in refractive index profile ∆ns at the surface of the glass calculated on the basis of the determined refractive index profiles.


**Table 2.** Electrodiffusion process parameters.

describing the distribution of the absolute concentration of admixture ions introduced into the

By introducing the function, δn(x) = n(x) - nb, from the equation (5) the following can be

This equation describes the distribution of concentration of admixture introduced into the glass, expressed by the parameters of the refractive index profile n(x) and of the equilibrium concentration c0. Integrating the expression (6) over the entire volume of the glass *V*, the total number of admixture ions Nτ introduced into the glass during the electrodiffusion process in

( ) ( ) <sup>0</sup>

In this equation, *S* is the surface area of the glass substrate (cross section for the exchange - Fig.9) which comes in contact with a liquid admixture contained in the crucible. At the same time the total charge (4), which has flowed through the glass, allows to specify the number of admixture ions having a valence *w*, which have been introduced into the glass in the time τ:

> ( ) 0 <sup>1</sup> <sup>Q</sup> N i td , we we

( ) ( ) <sup>s</sup>

Δn δn x dx Q m ewSc

0 0

t

t

τ

= × × ×× ò (9)

t

= =

V 0 c S N c x dV n x dx n d

<sup>D</sup> (6)

¥ <sup>×</sup> = = <sup>D</sup> ò ò (7)

× × ò (8)

( ) ( ) ( ) <sup>0</sup> -3 s c cx nx m <sup>n</sup> = × d

**Figure 9.** Determination of the cross section *S* of the exchange process.

τ

t

Comparing (7) and (8) the following equation is obtained.

¥

glass (m-3).

116 Ion Exchange - Studies and Applications

obtained.

the time τ is obtained as follows.

where *e* - the elementary charge.

Figure10 shows the method of calculating the product of Sc0 according to the equation (10) on the basis of data from Table 2.

The cross-section *S* for the exchange process shown in Fig.9 is defined by the geometry of the crucible. During the process it grows as a result of the gradual penetration of the molten salt into contact area of the glass and the crucible. The electric field is not uniform near the inner edge of the crucible. Based on the above facts, it is expected that the refractive index profile of the waveguide in these areas of the glass which are at the edge of the crucible is significantly different from the form that is obtained in the central portion of the glass plate. The integrals in Table 2 were calculated assuming a uniform shape of refractive index profile in the entire area of glass. With these reservations, on the basis of the data from Fig.10, the equilibrium concentration *c*0 of mobile ions in the glass substrate that was used in the process can be estimated. Assuming the cross-section of the crucible *S* = (3.6 ± 0.7)⋅10-4 m2 , identical for all processes listed in Table 2, the mean value *c*0 = (5.7 ± 1.5)⋅1027 m-3 was obtained.

#### **3.2. The glass mass (weight) change due to the Ag<sup>+</sup> ↔ Na+ ion exchange**

Another phenomenon accompanying the process of ion exchange in the glass is a change in the weight of the glass. This phenomenon is the more perspicuous the larger the difference of the masses of exchanged ions. This is easy to observe in the case of heavy silver ions Ag+ , which replace the mobile sodium ions Na+ in the glass. For this type of exchange (Ag+ ↔ Na+ ) this

**Figure 10.** The calculation of the product Sc0 according to (10).

effect will be greater if the sodium modifier is more in the glass. The difference of the masses of atoms (silver *m*ˉ *Ag* and sodium*m*ˉ *Na*) can be expressed as:

$$
\Delta \Delta \mathbf{M} = \overline{m}\_{A\mathbf{g}} - \overline{m}\_{\mathbf{Na}} = \frac{\mathbf{M}\_{A\mathbf{g}} - \mathbf{M}\_{\mathbf{Na}}}{\mathbf{N}\_A} \,, \tag{10}
$$

where MAg and MNa - molar masses, NA - Avogadro's number.

**Figure 11.** The cross section of a glass plate with a marked depth *d* of doped region.

The glass mass difference caused by the ion exchange can be calculated based on the amount of the ions Ne exchanged in the glass:

$$
\Delta m = m\_e - m\_0 = N\_e \cdot \Delta M \,\,\,\,\,\tag{11}
$$

where me, m0 - glass masses before and after the ion exchange process respectively.

In the case of planar waveguides the amount of ions exchanged in the glass can be estimated by measuring their refractive index profile n(x). The *x* coordinate is calculated here into the glass (on the surface *x* = 0) to the direction perpendicular to its surface. This profile can also be described theoretically using the normalized concentration u(x) of admixture ions introduced into the glass - see equation (5).

Thus, appearing in equation (11), the quantity of ions exchanged in the glass Ne can be expressed as follows:

$$\text{SN}\_{\text{s}} = \text{S} \cdot \bigwedge\_{0}^{d} \text{c}\_{A} \left( \text{x} \right) d\text{x} = \text{Sc}\_{0} \cdot \bigwedge\_{0}^{d} \mu \left( \text{x} \right) d\text{x} \,, \tag{12}$$

where *S* - the total glass surface through which the ion exchange process was conducted, *d* the depth of the area of doped glass (Fig.11).

On the basis of (10-12), the weight change of the glass due to the ion exchange of Ag+ ↔ Na+ is expressed in the equation:

effect will be greater if the sodium modifier is more in the glass. The difference of the masses

The glass mass difference caused by the ion exchange can be calculated based on the amount

In the case of planar waveguides the amount of ions exchanged in the glass can be estimated by measuring their refractive index profile n(x). The *x* coordinate is calculated here into the glass (on the surface *x* = 0) to the direction perpendicular to its surface. This profile can also be described theoretically using the normalized concentration u(x) of admixture ions introduced

where me, m0 - glass masses before and after the ion exchange process respectively.

*Ag Na*

*Mm m*

, *Ag Na*


<sup>0</sup> , *mm m N M e e* D = - = ×D (11)

*A*

*M M*

*N*

of atoms (silver *m*ˉ *Ag* and sodium*m*ˉ *Na*) can be expressed as:

**Figure 10.** The calculation of the product Sc0 according to (10).

118 Ion Exchange - Studies and Applications

where MAg and MNa - molar masses, NA - Avogadro's number.

**Figure 11.** The cross section of a glass plate with a marked depth *d* of doped region.

of the ions Ne exchanged in the glass:

into the glass - see equation (5).

$$
\Delta m = \frac{\mathbf{M}\_{A\text{g}} - \mathbf{M}\_{\text{Na}}}{\mathbf{N}\_A} c\_0 \mathbf{S} \cdot \int\_0^d \mu(\mathbf{x}) d\mathbf{x} \tag{13}
$$

The above relationship shows the possibility of estimating the equilibrium concentration *c*<sup>0</sup> on the basis of experimentally designated values: Δm, S and the refractive index profile n(x) of produced waveguide.


**Table 3.** Mass change of glass substrates in the diffusion processes and the integrals of the normalized concentrations of the admixture.

Table 3 shows the results of an experiment involving the determination of weight increase of glass plates subjected to processes of diffusion doping with Ag+ ions. The substrates were made of soda-lime glass (Menzel-Gläser Company). This glass contains a large amount of sodium oxide Na2O. Thus in the case of Ag+ ↔ Na+ , a considerable increase of the weight is obtained. Chemical composition in % by weight of the glass [38]: 72.2% SiO2, 14.3% Na2O, 6.4% CaO, 4.3% MgO, 1.2% Al2O3, 1.2% K2O, 0.3% SO3, 0.03% Fe2O3. Diffusion processes were carried out with a liquid source of admixture. The silver nitrate AgNO3 and its sodium nitrate NaNO3 solutions have been used. These solutions are determined by the molar fraction κ [39] listed in Table 3.

Figure 12 shows the dependence of the mass increase of the glass substrates from the product of the total surface of the glass and the integral of the normalized concentration of the admixture (Ag+ ions).

**Figure 12.** The dependence of the mass increase of the sample on the product of its surface and the integral of the nor‐ malized concentration of the admixture.

This dependency is based on the results shown in Table 3. For : MAg = 107.87 (g/mol), MNa = 22.99 (g/mol), NA = 6.02⋅1023 (mol-1), the following equation is obtained: A/c0 = 1.41⋅10-22 (g).

Determined by this method the value of the equilibrium concentration in the glass is: *c*0 = A/ 1.41×10-22 = (5.6 ± 0.2) 1027(m-3).

The results of the equilibrium concentration *c*0 obtained by integrating the electric charge (Section 2.1) and by method of weighing presented here, comply within the limits of uncer‐ tainty calculation.
