Tellurite And Fluorotellurite Glasses For Active And Passive

7. Surface properties; MDO 321
with Ag + did not occur. The content decreased to 0.7 at. % at 1 µm depth for the 100 µm
layer, and 0.4 at. % at 2 µm depth for the 300 µm layer. Silver ions ‘exchange’ with Na +
ions in the glass due to their similar ionic radii (114 and 113 pm respectively in four-fold
coordination), and both are monovalent, ensuring charge neutrality. At room temperature,
the ion exchange of the silver layer will be kinetically inhibited. However, at elevated
temperatures, the exchange becomes thermodynamically and kinetically more favourable,
and due to the concentration difference at the silver / glass interface, Ag + /Na + ion-
exchange takes place. Equation (7.13) shows Fick’s second law.
2
∂N
∂ N
= D 2
∂t
∂x
(7.13)
where N is the exchanged ion concentration (i.e. silver), t is time, x the depth of exchange
into the material, and D the diffusion coefficient of the ion. Solving this equation results
in a concentration profile shown by equation (7.14). Flux of sodium ions out of the glass
results in a more complex profile, with the silver content levelling off initially, before
decreasing according to an erfc function.
⎡ x ⎤
N ( x,
t)
= N 0 erfc⎢
0.
5 ⎥
(7.14)
⎣2(
Dt)
⎦
where N0 is the surface concentration [1], and erfc is the complementary error function
defined by equation (7.15).