# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 1 Introduction  
Ion exchange in glass has been used for centuries for the purposes of decoration and colouring. Glass lustre on ceramics with metallic nanoparticles from ion exchange has been known from the early Islamic culture during the 10 th Century [1]. However, the scientific and industrial application of this technique dates back 60 years ago when potassium ion exchange (IE) was first applied in the chemical surface tempering of glasses [2; 3]. Next, with the introduction of the concept of integrated optics in 1969 [4], ion exchange in glass was proposed as a waveguide fabrication process. Just a few years after this proposal, [PERSON] and [PERSON] published the first work on ion exchange waveguides [5]. This kind of waveguide presents several advantages: fibre compatibility, low propagation losses and low cost. Moreover, ion exchange can be combined with other techniques, such as sol-gel, for the fabrication of passive and active (rare-earth-doped) integrated optical devices [6]. Currently, the main applications of ion exchange are in glass strengthening [7; 8; 9; 10; 11] and in the fabrication of photonic components for both guided-wave [12; 13; 14; 15; 16; 17; 18] and bulk optics [19; 20; 21]. In an IE process, cations (mostly Na\({}^{+}\)) close to the glass surface are replaced with other monovalent cations such as K\({}^{+}\), Li\({}^{+}\), Rb\({}^{+}\), Cs\({}^{+}\), Tl\({}^{+}\) or Ag\({}^{+}\)[22]. Molten salts of such cations are common sources of dopants, although a metallic film deposited on the glass surface can be also a source of cations. The exchange takes place by a purely thermal diffusion process or it can be assisted by an electric field. The new cations can change the electrical permittivity, the stress and even the absorption of the glass, these being changes proportional to the dopant cation concentration. By selective masking of the glass surface, ion exchange can be locally prevented or allowed, giving rise to custom-made elementsfor specific purposes. The theoretical modelling of the IE processes ran parallel to the development of the technologies. This modelling was fundamental to the design and fabrication of most elements. For instance, in the field of integrated optics, some subjects as fibre compatibility or coupling losses depend strongly on the permittivity distribution and, hence, on the cation concentration. Therefore, the prediction of the cation concentration is of great importance to design devices based on IE technologies. Here, we present a review of the theoretical modelling of IE processes in glass made up to now, as well as the last advances in this matter that have been reported in recent years. Ion exchange within a glass network is governed by diffusion and drift processes as a response to a concentration gradient and an electric field, respectively [23; 24]. This gives rise to a concentration profile of the exchanging ions, which depends on the processing conditions of the substrate: temperature, exchange time, applied electric field, etc. The Nernst-Planck drift-diffusion equation describes this process. It establishes the proportionality among the flux density of each ion species and both the electric field and the concentration gradient. On the other hand, [PERSON]'s equation and continuity equations for each ion must be fulfilled. This gives rise, in general, to three second-order coupled partial differential equations whose solution provides the evolution of both the cation concentrations and the electric potential. In the derivation of these equations, the charge neutrality approximation is usually assumed, which allows for some important simplifications without a relevant loss of accuracy in the most common cases [25; 26]. All of these subjects are addressed in Section 2, where we present the basic model of ion exchange. This model assumes that the self-diffusion coefficients of the exchanged cations are constant for a given temperature. However, experimental measurements [27] showed that these coefficients depend on the cation concentrations.