**2. Ion exchange in glass phenomenon**

The ion exchange in glass phenomenon is initiated in the phase boundary: glass - liquid admixture source. The system with a liquid source of admixture is the most common in the literature and its attractiveness is due to the main advantage: the simplicity of implementing such processes. In this case, subject to adequate conditions of the process realization, an unlimited source efficiency can be assumed in its theoretical description. For these reasons, the description of the ion exchange phenomena has been limited only to liquid phases of admixture sources.

The assumption of the binarity of the ion exchange process' character made in its description is rather a simplification. This postulate is commonly found in the literature [2,8- 11] and as the only one is acceptable for the examination of the effects of this phenomenon by optical methods only. In the next part, for the description of the ion exchange phenomenon, the following designations has been adopted: (A) for the admixture ions and (B) for the modifier ions.

When discussing the ion exchange phenomenon in glass, which is realized through the contact of glass with liquid admixture at sufficiently high temperatures, the main aspects should be specified (Fig.1). These are:

 determination of the balance of ion exchange process in the phase boundary: glassadmixture source. The consequence of this balance is the relationship between the concentration of the admixture in the *cAs* source phase and the admixture concentration in the superficial area of glass *cmax*,

time dependence *cmax(t*) in achieving the equilibrium value,

156 Ion Exchange Technologies

waveguide structures has started to be produced in glass, is called (from the physical phenomenon, which is its foundation) the *ion exchange method* [2,3]. This phenomenon is based on the mechanism of glass ionic conductivity [4], allowing in a sufficiently high temperature the substitution of natural components (called modifiers), by inserting, in the due process, the so-called admixture ions. In the result of their different properties, such as electric polarizability and ionic radius, the admixture ions introduced into the glass locally alter its optical properties (refractive index). Due to the diffusive nature of the ion exchange phenomenon and the fact that the admixture is introduced into glass by its surface, thus resulting waveguide structures are localized in the superficial area of glass substrate (planar

structures), and the change in the glass refractive index has the gradient character.

the refractive index - with the fiber waveguides used in telecommunications.

**2. Ion exchange in glass phenomenon** 

admixture sources.

(B) for the modifier ions.

should be specified (Fig.1). These are:

in the superficial area of glass *cmax*,

The first waveguide structures were prepared by the ion exchange method in the early seventies of the last century [5-7]. The advantages of this method include first of all: the ease of implementation of the basic technological processes (a highly specialized technological apparatus is not required here), low production costs associated with the possibility of using commercially available glass as substrates and relatively cheap materials as a source of admixture, very good repeatability of obtained elements, a very good time stability, low attenuation (*<0.1 dB/cm*), as well as good compatibility - in terms of producing changes in

The ion exchange in glass phenomenon is initiated in the phase boundary: glass - liquid admixture source. The system with a liquid source of admixture is the most common in the literature and its attractiveness is due to the main advantage: the simplicity of implementing such processes. In this case, subject to adequate conditions of the process realization, an unlimited source efficiency can be assumed in its theoretical description. For these reasons, the description of the ion exchange phenomena has been limited only to liquid phases of

The assumption of the binarity of the ion exchange process' character made in its description is rather a simplification. This postulate is commonly found in the literature [2,8- 11] and as the only one is acceptable for the examination of the effects of this phenomenon by optical methods only. In the next part, for the description of the ion exchange phenomenon, the following designations has been adopted: (A) for the admixture ions and

When discussing the ion exchange phenomenon in glass, which is realized through the contact of glass with liquid admixture at sufficiently high temperatures, the main aspects

 determination of the balance of ion exchange process in the phase boundary: glassadmixture source. The consequence of this balance is the relationship between the concentration of the admixture in the *cAs* source phase and the admixture concentration


**Figure 1.** The main aspects of the ion exchange phenomenon in glass

The ion exchange phenomenon in the superficial area of glass, which is in contact with a source of admixture, can be described as a reaction in which the substrates are: in the source of ions *A* in the admixture source and ions *B* in the glass, and products are, respectively: ions *A* in the glass and ions *B* in the phase of the source. Thermodynamic equilibrium state of the exchange process at the phase boundary: glass - liquid source of admixture is then described by constant *K* [11,12]:

$$\mathbf{K} = \frac{\overline{\mathbf{a}}\_{\text{A}} \cdot \mathbf{a}\_{\text{B}}}{\mathbf{a}\_{\text{A}} \cdot \overline{\mathbf{a}}\_{\text{B}}} \tag{1}$$

In the equation above Aa and Aa mean the thermodynamic activities of ions *A*, respectively: the glass phase and the source phase, likewise: B a and B a are the thermodynamic activities of ions *B* in the glass phase and the source phase.

Knowledge of the quantitative determination of the equilibrium in the ion exchange process at the phase boundary: liquid admixture source - glass, is essential for the theoretical description of the formation of spatial distribution of admixture introduced into the glass in the exchange process. The determination of constant equilibrium (at a given temperature of exchange process) for a particular glass-admixture system therefore requires the experimental determination of the relationship between the molar fraction of admixture ions in the liquid phase of source and its molar fraction providing at the glass surface.

In equation (1) the ratio of activities of exchanged ions in the glass phase can be expressed with the Rothmund - Kornfeld relationship [12,13]:

$$\frac{\overline{\mathbf{a}}\_{\mathbf{A}}}{\overline{\mathbf{a}}\_{\mathbf{B}}} = \left(\frac{\overline{\mathbf{N}}\_{\mathbf{A}}}{\overline{\mathbf{N}}\_{\mathbf{B}}}\right)^{\eta},\tag{2}$$

where: Ni – means the molar fraction of ions in the glass phase (*i* = *A*, *B*) and is expressed by the interaction energy of ions in the glass *WA-B* :

$$\eta = 1 - \frac{\mathbf{W}\_{\text{A}-\text{B}}}{\mathbf{RT}} \,, \tag{3}$$

where: *R* – universal gas constant, *T* – temperature.

In turn, the activities ratio of exchanged ions in the liquid phase of the source is given by their mole fractions *Ni* and activity coefficients *i* :

$$\frac{\mathbf{a}\_{\rm B}}{\mathbf{a}\_{\rm A}} = \frac{\mathbf{N}\_{\rm B} \boldsymbol{\gamma}\_{\rm B}}{\mathbf{N}\_{\rm A} \boldsymbol{\gamma}\_{\rm A}}\tag{4}$$

After taking into account (2) and (4) in (1), we obtain:

$$\log\left(\frac{\text{N}\_{\text{A}}}{\text{N}\_{\text{B}}}\right) - \frac{\text{Q}\_{\text{A}-\text{B}}}{\text{RT}} (1 - 2\text{N}\_{\text{B}}) = \eta \cdot \log\left(\frac{\overline{\text{N}}\_{\text{A}}}{\overline{\text{N}}\_{\text{B}}}\right) - \log\text{K} \tag{5}$$

The equation above takes into account the relationship:

$$\log\left(\frac{\gamma\_{\rm B}}{\gamma\_{\rm A}}\right) = \frac{\mathbf{Q}\_{\rm A-B}}{\mathbf{RT}} (1 - 2\mathbf{N}\_{\rm B}) \,\prime \tag{6}$$

in which *QA-B* is the energy of interaction of ions in the source phase.

The knowledge of *QA-B* for the systems of liquid mixtures used as the admixture source allows on the basis of (5) to calculate the parameters and *K.* The process of establishing the maximum concentration of admixture *cAmax* introduced into the superficial area of glass will be characterized by a time constant –*d*. This should be considered in cases of carrying shortterm exchange processes.

Glasses, which are used as the substrate of gradient waveguide structures, are, in the vast majority, the oxide glasses. The possibility of the realization of ion exchange phenomenon in glass is a consequence of its structure. There are many theories about the concept of this structure, among which one of the basic is Zachariasen theory [14] experimentally verified in studies of Warren [15]. It allows a pretty good description of the additive properties of glass, due to its composition. Based on this theory the mechanism of ion transport in glass can be explained, which is the basis of the ion exchange phenomenon [4]. This theory highlights the notion of a structural skeleton - called the glass network – which has features of low-range ordering. The elementary unit of this structure (in the case of silicate glasses) are the coordination polyhedra in the shape of tetrahedra that are connected by corners. In the vertices of this tetrahedron there are the anions *O2-* surrouding the cation *Si4+* which is located in the middle.

The rules under which a merger of elementary cells occurs are determined by the conditions of Zachariasen vitrification [14]:


158 Ion Exchange Technologies

with the Rothmund - Kornfeld relationship [12,13]:

by the interaction energy of ions in the glass *WA-B* :

where: *R* – universal gas constant, *T* – temperature.

After taking into account (2) and (4) in (1), we obtain:

The equation above takes into account the relationship:

allows on the basis of (5) to calculate the parameters

term exchange processes.

their mole fractions *Ni* and activity coefficients

In equation (1) the ratio of activities of exchanged ions in the glass phase can be expressed

A A B B a N , <sup>a</sup> <sup>N</sup> 

WA B <sup>η</sup> 1 , RT

In turn, the activities ratio of exchanged ions in the liquid phase of the source is given by

B BB A AA a N γ

 A AB <sup>A</sup> B B B N Q <sup>N</sup> log 1 2N <sup>η</sup> log logK N RT <sup>N</sup> 

A

in which *QA-B* is the energy of interaction of ions in the source phase.

B AB

<sup>γ</sup> <sup>Q</sup> log 1 2N , γ RT 

The knowledge of *QA-B* for the systems of liquid mixtures used as the admixture source

maximum concentration of admixture *cAmax* introduced into the superficial area of glass will be characterized by a time constant –*d*. This should be considered in cases of carrying short-

Glasses, which are used as the substrate of gradient waveguide structures, are, in the vast majority, the oxide glasses. The possibility of the realization of ion exchange phenomenon in glass is a consequence of its structure. There are many theories about the concept of this structure, among which one of the basic is Zachariasen theory [14] experimentally verified in studies of Warren [15]. It allows a pretty good description of the additive properties of glass, due to its composition. Based on this theory the mechanism of ion transport in glass can be explained, which is the basis of the ion exchange phenomenon [4]. This theory highlights the notion of a structural skeleton - called the glass network – which has features

B

*i* :

where: Ni – means the molar fraction of ions in the glass phase (*i* = *A*, *B*) and

η

(2)

(5)

(6)

and *K.* The process of establishing the

(3)

a N <sup>γ</sup> (4)

is expressed


**Figure 2.** Possible energy positions of ion *Na+* in glass structure as well as changes of the orientational polarization of the structure due to its hopping (by [4])

In the multicomponent oxide glasses in addition to the glass network there are also so-called modifiers, which are alkali metal ions, whose presence in the glass is required by giving it the appropriate physicochemical properties. When the sodium oxide (*Na2O*), for example, is introduced into the structure of glass in its melting process, the network connections in the emerging structure are reduced due to excess oxygen. So then the non-bridging oxygen arise. The most likely places for sodium in the structure of the glass are around nonbridging oxygen, which have uncompensated negative charge (Figure 2). For alkali ions, it is possible to address more than one energetically equivalent position near the non-bridging oxygen. However, a suitable location for sodium ions is determined not only by the same oxygen bridges, but also throughout their environment - giving in the final effect a system with the lowest possible energy [4]. These fragments of glass network, surrounded by the trapping energy ions of modifiers that can fluctuate within a structural unit as a result of thermal induction, are polar elements of the glass network. In the thermally induced structure of glass, in addition to hopping to the local structural unit, hoppings of long range outside the polar unit are also more likely to appear. This is accompanied by dissociation of the polar unit, and the modifier ion migrates inside the glass network. Non-bridging oxygen ion remains fixed in the glass network, making a vacancy in it. The directions of migration of mobile modifier depends on the local environment of polar units. Each of them is able to intercept mobile cations after the dissociation. Since the structural units in glass forming its glass network don't create an arrangement of long range, so a random network of the most probable directions of hopping arises. Those hoppings create (on a microscopic scale) the system of paths shown in Fig 3.

**Figure 3.** 2-D fragment of glass structure with marked preferential pathways of modifier ions (by [4])

The described ability of the modifier ions to move within the glass structure is the basis of the mechanism of glass ionic conductivity. In turn, this mechanism enables the ion exchange, which takes place in case of introducing other types of modifier ions (admixture) into the glass.

The appearance of admixture ions in the superficial area of glass creates the opposite directed concentration gradients of both: admixture ions and modifiers. Consequently, in the glass there are two divergent streams of ions: a stream of admixture ions directed into the glass and a stream of modifier ions directed to the glass surface.

Association of the diffusion coefficient *D* with the electrical conductivity or mobility of exchanged ions in the crystal structures is described by the Nernst-Einstein equation:

$$\mathbf{D}\_{\phi} = \frac{\mathbf{k} \mathbf{T} \sigma}{\mathbf{c} \mathbf{q}^{2}} = \frac{\mathbf{k} \mathbf{T} \mu}{\mathbf{q}} \,, \tag{7}$$

where: *c* and *q* – are respectively: concentration and charge of diffusing ions, *k* is the Boltzmann constant, *T* - is the absolute temperature.

The relationship between the diffusion coefficient *D* and the diffusion coefficient *D\** , which is set by the tracer diffusion method, is determined by the correlation coefficient *f*, which, in the case of vacancy diffusion mechanism of ions in the three-dimensional glass network, is equal to the Haven ratio *H* [11.16]:

$$\text{If } = \text{H} = \frac{\text{D}^\*}{\text{D}\_o} < 1 \tag{8}$$

So for the description of the two-component ion exchange process in glass the relationship between self-diffusion coefficients and electrical mobilities of both types of exchanged ions take the form:

Ion Exchange in Glass – The Changes of Glass Refraction 161

$$\mathbf{D}\_{\mathrm{i}} = \frac{\mathbf{H}\_{\mathrm{i}} \mathbf{k} \mathbf{T} \mu\_{\mathrm{i}}}{\mathbf{q}} \qquad \qquad \left(\mathrm{i} \mathbf{=} \mathrm{A}\_{\mathrm{i}} \mathbf{B}\right) \tag{9}$$

The final effect of the ion exchange process in glass, which has practical application in the manufacture of optical structures, is the change of the refractive index within its admixture area. Local changes in the refractive index of glass, which are the effect of the ion exchange, are the result of the differences in their electric polarizability and ionic radii.

160 Ion Exchange Technologies

into the glass.

system of paths shown in Fig 3.

probable directions of hopping arises. Those hoppings create (on a microscopic scale) the

**Figure 3.** 2-D fragment of glass structure with marked preferential pathways of modifier ions (by [4])

The described ability of the modifier ions to move within the glass structure is the basis of the mechanism of glass ionic conductivity. In turn, this mechanism enables the ion exchange, which takes place in case of introducing other types of modifier ions (admixture)

The appearance of admixture ions in the superficial area of glass creates the opposite directed concentration gradients of both: admixture ions and modifiers. Consequently, in the glass there are two divergent streams of ions: a stream of admixture ions directed into

with the electrical conductivity

cq <sup>q</sup> (7)

and the diffusion coefficient *D\**

<sup>D</sup> (8)

or mobility

of

, which

exchanged ions in the crystal structures is described by the Nernst-Einstein equation:

σ 2 kT<sup>σ</sup> kT<sup>μ</sup> D ,

where: *c* and *q* – are respectively: concentration and charge of diffusing ions, *k* is the

is set by the tracer diffusion method, is determined by the correlation coefficient *f*, which, in the case of vacancy diffusion mechanism of ions in the three-dimensional glass network, is

\*

σ <sup>D</sup> fH 1

So for the description of the two-component ion exchange process in glass the relationship between self-diffusion coefficients and electrical mobilities of both types of exchanged ions

the glass and a stream of modifier ions directed to the glass surface.

Association of the diffusion coefficient *D*

equal to the Haven ratio *H* [11.16]:

take the form:

Boltzmann constant, *T* - is the absolute temperature. The relationship between the diffusion coefficient *D*

According to the principle of additivity [9,17,18] refraction index of oxide glass consisting of a metal oxide *MmOn* can be represented as:

$$\mathbf{h}\_{\rm b} = \mathbf{1} + \frac{\mathbf{R}\_0}{\mathbf{V}\_0} = \mathbf{1} + \frac{\sum\_{\rm i} \mathbf{a}\_{\rm i} \mathbf{N}\_{\rm i}}{\mathbf{C} + \sum\_{\rm i} \mathbf{b}\_{\rm i} \mathbf{N}\_{\rm i}} \tag{10}$$

where: *R0*, *V0* – are respectively: the refraction and the molar volume of oxide atoms constituting the glass, *Ni* – molar fraction of i-th component in the composition of the glass, *ai*, *bi*, *C* – empirical coefficients.

The ion exchange process starts with the glass surface, which contacts with the admixture source phase. Defining the fraction of exchanged ions in glass *u*(*x*) as a function of depth measured from the surface of the glass, it can be written:

$$\mathbf{u}\left(\mathbf{x}\right) = \frac{\mathbf{N}\_{\mathbf{A}}\left(\mathbf{x}\right)}{\mathbf{N}\_{0}} = \frac{\mathbf{c}\_{\mathbf{A}}\left(\mathbf{x}\right)}{\mathbf{c}\_{0}},\tag{11}$$

where: *NA*(*x*) – is the number of admixture ions (*A*), which replaced the glass ions at a depth *x*, *N0* – is the number of exchangeable ions in the whole glass volume, *cA*(*x*) and *c0* – are the concentrations corresponding to mentioned numbers of ions.

Therefore, the refractive index after the ion exchange, according to (10), reaches a value:

$$\mathbf{n}\left(\mathbf{x}\right) = 1 + \frac{\mathbf{R}\_0 + \mathbf{u}\left(\mathbf{x}\right)\Delta\mathbf{R}}{\mathbf{V}\_0 + \mathbf{u}\left(\mathbf{x}\right)\Delta\mathbf{V}},\tag{12}$$

and its increase *n*(*x*), assuming *V*<<*V0*, can be expressed as:

$$\Delta \mathbf{n} \left( \mathbf{x} \right) = \mathbf{n} \left( \mathbf{x} \right) - \mathbf{n}\_{\mathbf{b}} = \frac{\mathbf{R}\_0 + \mathbf{u} \left( \mathbf{x} \right) \Delta \mathbf{R}}{\mathbf{V}\_0 + \mathbf{u} \left( \mathbf{x} \right) \Delta \mathbf{V}} - \frac{\mathbf{R}\_0}{\mathbf{V}\_0} = \dots = \frac{\mathbf{u} \left( \mathbf{x} \right)}{\mathbf{V}\_0} \left( \Delta \mathbf{R} - \mathbf{R}\_0 \frac{\Delta \mathbf{V}}{\mathbf{V}\_0} \right) \tag{13}$$

Assuming that in the contact of the glass surface with clean (undiluted) source phase all of the mobile ions at the glass surface (*x* = *0*) are exchanged for admixture ions, then *u*(*0*)=*1* and the relation occurs:

$$
\Delta \text{tr} \left( 0 \right) = \Delta \mathbf{n}\_s = \frac{1}{\mathbf{V}\_0} \left( \Delta \mathbf{R} - \mathbf{R}\_0 \frac{\Delta \mathbf{V}}{\mathbf{V}\_0} \right) \tag{14}
$$

The linearity of the change of refractive index of the normalized concentration of admixture introduced into the glass *u*(*x*), expressed by equation (13), is taken as the fundamental relation describing the ion exchange processes [9,10].

In the cases of diluted admixture sources, according to (11), the normalized concentration of ions introduced at the glass surface takes values *u*(*0*)<*1*.
