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Ion exchange and the mixed mobile ion effect in micro-optics applications
Solid State Ionics 105 (1998) 257–264
Ion exchange and the mixed mobile ion effect in micro-optics applications S.N. Houde-Walter* The Institute of Optics, University of Rochester, Rochester, NY 14627, USA
Abstract Silver–sodium ion exchange in aluminosilicate glasses is reviewed in the context of micro-optics fabrication. The interdiffusion coefficient is more strongly concentration dependent in glasses with high non-bridging oxygen content. Radioactive tracer diffusion coefficients show unusual behavior in glasses devoid of nonbridging oxygen. The local environments of the silver and sodium ions are related to the changes in ion exchange behavior. The modified quasi-chemical (MQC) model links these structural details to the thermodynamic description of ion exchange. The MQC model accurately describes ion exchange behavior in silicate, aluminosilicate and aluminoborate glasses. Using this model, long experimental trials are no longer necessary, and fast identification of optimal glass types and process parameters results. Keywords: Ion exchange; Micro-optics; Diffusion; Glass; Gradient index glass
1. Introduction Ion exchange is widely used in the fabrication of micro-optic lenses [1,2]. The most common type of micro-optic lens is called a ‘self-focusing lens’, or equivalently, a ‘radial gradient index lens’, or ‘GRIN lens’. These have a radial distribution of refractive index (see Fig. 1), with typically a parabolic or hyperbolic-secant index distribution for optimum imaging properties. Their diameters range from a few hundred microns to a few millimeters, and the total change in refractive index can be as large as 0.2 in the visible portion of the spectrum. GRIN lenses are used singly in applications including industrial boroscopes, medical endoscopes, diode laser collimators and optical fiber couplers. They are assembled into superposition image arrays for use in *E-mail: [email protected]; fax: 11 716 271 1027.
desktop photocopiers, fax machines and text scanners. The large radial variation in refractive index is achieved by diffusing heavy amounts of dopant cation into the host glass cylinder. The dopant can amount to 25–30 mole percent (mol%) of the final glass composition in localized regions. As a result, the interdiffusion coefficient is strongly dependent on dopant levels over the full concentration range. In
Fig. 1. Self-focusing lens with radial refractive index distribution.
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00473-6
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
258
practice, the concentration dependence of the interdiffusion coefficient can be obtaining experimentally by inverting the one-dimensional diffusion equation via the Boltzmann transform [3]: x9
E
1 ≠x D( x 9) 5 2 ] ] x dx. 2t ≠x
(1)
0
Here x is the normalized dopant concentration, x 5 NB /(NA 1NB ), where NA is the mole fraction of the constituent mobile cation of species A and NB is the mole fraction of the dopant cation of species B, and x 9 denotes an intermediate concentration between 0 and 1. The spatial dimension is indicated by x, and diffusion time by t. Eq. (1) allows numerical calculation of D( x ) provided a concentration profile, x (x), has been measured. A typical concentration dependence is shown in Fig. 2 for a silver-for-sodium exchange in a sodium aluminosilicate glass designed for GRIN optics. In general, D( x ) is relatively small for small x, grows with increasing x until some peak value is attained, and then falls off as x increases thereafter. This concentration dependence of the ion exchange is widely recognized as one of the manifestations of the mixed mobile ion effect, or MMIE. Depending on the glass host composition, the dopant specie and the ion exchange temperature, the peak width varies and the peak value of D( x ) can be shifted towards high or low dopant concentrations.
Fig. 2. Concentration dependence of interdiffusion coefficient in sodium aluminosilicate glass which has been ion exchanged with silver.
These nuances in the functional form of D( x ) are critically important in determining the final dopant distribution, and thus the final optical performance of the micro lens. The interdiffusion coefficient can be derived from first principles by assuming equal and opposite fluxes of dopant and constituent ions [4]. The fundamental expression is:
H
x (1 2 x ) ≠ D( x ) 5 ]]] ]( mB 2 mA ) kT ≠x
JF
G
DB ]] . 1 2 ax
(2)
Here k is the Boltzmann constant, T is the ion exchange temperature, mA and mB are the chemical potentials of the constituent and dopant cations and DB is the self diffusion coefficient of the dopant cation. The parameter a is a measure of the relative mobilities: DB a 5 1 2 ]. DA
(3)
The two contributions to the interdiffusion coefficient in Eq. (2) are referred to as the thermodynamic contribution (curly brackets) and the mobility term (square brackets). Many of the early studies of interdiffusion were conducted in simple, furnacemelt, mixed-alkali silicate glasses. It was found that the mobility term made the dominant contribution to D( x ) in these glasses (see, for example, Refs. [5–7]). This was established by measuring the self diffusion coefficients, Di , (i5A, B), by radioactive tracer diffusions in glasses with varying relative alkali content in a series of mixed-alkali (Na / Rb or Na / Cs) silicate glasses. The self diffusion coefficients varied by two and three orders of magnitude, as did the mobility term calculated using the measured Di values. D( x ) differed little from the mobility term, and the multiplicative thermodynamic term, although rarely measured directly, has since been regarded as a perturbation. Host compositions used for micro-optics are quite different from simple mixed-alkali silicates. Requirements for suitable host glasses include chemical durability, large interdiffusion coefficients, appropriate optical dispersion, negligible stress birefringence after ion exchange and the capacity for a large change in refractive index when ion exchanged with low-toxicity dopants. Silver-for-sodium ion exchange
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
in aluminum borosilicate glasses satisfies these requirements. The thermodynamic term in Eq. (2) is not immediately useful to those developing micro-optics because the chemical potentials are known for virtually none of the available glass compositions. The thermodynamic term can be rewritten by use of the Gibbs– Duhem relation in terms of the activities of the dopant cation, a B , in the glass [4]:
H
J
≠ln (a B ) x (1 2 x ) ≠ ]]] ]( mB 2 mA ) 5 ]]] . kT ≠x ≠ln ( x )
(4)
In principle, the activities can be determined experimentally [8–10], but such measurements in glasses are impractical when many glass compositions must be characterized. Using the Rothmund– Kornfeld relation [11], [a B ] x ]] 5 ]] 12x [aA ]
S
D, n
(5)
the ratio of activities of the dopant and constituent cations in the glass can be related to the ratio of their concentrations through an empirical power factor, n. Thus, the thermodynamic term can be approximated by ≠ln (a B ) ]]] ¯n ≠ln ( x )
259
ing the maximum term method and making Stirling’s approximation, the thermodynamic term can be written
Hx
J HS D J
(1 2 x ) ≠ c 1 ]]] ]( mB 2 mA ) 5 ] ] 2 1 1 1 . kT ≠x 2 b (8)
The parameter b is given by ]]]]]]]]]] b 5œ1 2 4x (1 2 x )s1 2 exps2´INT /kTdd.
(9)
In a one-dimensional array of hopping sites (c5 2),
Hx
J
(1 2 x ) ≠ 1 ]]] ]( mB 2 mA ) 5 ] kT ≠x b
(c 5 2).
(10)
This is plotted in Fig. 3 as a function of the parameter 2´INT /kT, keeping in mind that the excess interaction energy is typically negative in glasses (net attraction between unlike cations). As the interaction term becomes stronger, the concentration dependence of the thermodynamic term becomes more pronounced. In real glasses, the peak is sometimes observed at values other than x 51 / 2. This shift becomes larger if the mobile cation content (alkali plus dopant) is reduced in the glass com-
(6)
where n is a constant typically in the range 1,n,2 [12]. The departure of n from unity indicates a stronger interaction between the diffusing cations. However, the thermodynamic term of Eq. (2) is actually a function of dopant concentration, and the approximation of Eq. (6) is not sufficiently accurate for modeling of the micro-optics fabrication process. The thermodynamic term can be recast in terms of structural parameters of the glass [13,14]. This is done by writing the chemical potentials in terms of the canonical partition function, expressed as a function of the number of nearest neighbor hopping sites, c, and the excess interaction energy, ´INT ,
´AA 1 ´BB ´INT 5 ´AB 1 ]]] 2
(7)
where ´AB (´ii ) is the effective bond energy between adjacent unlike (like) cations. Using the quasi-chemical (Bethe–Guggenheim) approximation [15], apply-
Fig. 3. Thermodynamic contribution to interdiffusion coefficient with varying excess interaction energy term for special case of one-dimensional glass (c52). Concentration dependence becomes stronger as varying excess interaction energy term, 2´INT /kT, becomes large in magnitude.
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260
position. The present model assumes an arbitrarily large number of cation sites, so an ad hoc shift, x 2 x0 , can be used for the purposes of fitting Eq. (2) to measured D( x ). This representation of the thermodynamic term is called the modified quasi-chemical, or MQC, model of ion exchange. In practice, Eq. (8) is used to describe D( x ) in terms of the parameters c, ´INT and x0 . Even very noisy data can be successfully fit with this representation of D( x ) [14], and unlike other representations (polynomials, Gaussian fits, etc.), the MQC parameters can be used in a finite difference algorithm to regenerate the original dopant profile and profiles for any other diffusion time with unprecedented accuracy. Therefore, instead of conducting many diffusion experiments and measuring the resultant index profiles to see which diffusion time gives best optical properties, it is now possible to use the MQC parameters from a single experiment to calculate a large number of index profiles and numerically compare these to the desired distribution [16]. Simple algorithms can then report the best diffusion schedule. This procedure has been shown to reduce the number of experiments (e.g., for an endoscope application, 50 experiments, 8 h apiece [17]) by more than an order of magnitude.
composition). Full details are given in Ref. [20]. This glass is expected to be free of non-bridging oxygen because it has a large boron oxide content and Al / Na molar ratio of unity [21,22]. No more than a few mol% silver can be incorporated into a glass of this composition by furnace melting [23], so slabs of the sodium-rich, furnace-melt glass were ion exchanged until homogeneous with varying silver contents x from 0.120 to 0.957. This was done by immersing slabs of the sodium boroaluminosilicate glass with dimensions 10 mm310 mm30.7 mm into seven different AgNO 3 –NaNO 3 molten salts at 4008C for 377 h (for melt compositions, see Table 2). The result was homogeneous samples with seven different Ag concentrations. The self-diffusion coefficients of sodium and silver in the mixed ion glasses were measured by means of radioactive tracer diffusion. The isotopes, 22 Na and 110m Ag, in the form of 22 NaCl in 0.2 M HCl and 110m AgNO 3 in 0.1 M HNO 3 were used as radiotracer sources. The diffusions were carried out by annealing the samples at 3608C for 5 h. Assuming an instantaneous source, the tracer distribution is
2. Experimental
where x i* denotes the tracer concentrations. The 22 Na and the 110m Ag diffusion profiles in the glass were obtained by successive sectioning of the samples and counting the radioactivity of the removed layers [24]. Each etching step was carried out in 6 ml etchant (1.5 vol% HF (49% concentration)11.5 vol% HNO 3 (70% concentration)197 vol% H 2 O) contained in a scintillation vial. After the etching, 16 ml scintillation liquid (Ecoscint A, National Diagnostics) was added to each vial and radioactivity counting was performed by the liquid scintillation counting technique for each vial. The self-diffusion coefficient was then determined by fitting the diffusion profile to Eq. (11).
2.1. Tracer diffusions Radioactive tracer diffusions were conducted in a boroaluminosilicate glass [18,19] (see Table 1 for Table 1 Glass compositions in mole percent Glass
Na 2 O
Al 2 O 3
B2O3
SiO 2
1 2 3
19.3 22.2 25.0
– 20.3 25.0
– 5.5 12.5
80.7 52.0 37.5
1 2x 2 / (4D i t ) x *i 5 ]] , i 5 (A, B), ]] e œp Di t
(11)
Table 2 Homogeneous ion-exchange of glass 1 for high silver content glasses: melt composition, normalized silver concentration xAg measured by EDX
AgNO 3 :NaNO 3 (mol%) xAg
Base glass
Sample A
Sample B
Sample C
Sample D
Sample E
Sample F
0:100 0
0.5:99.5 0.12060.012
1:99 0.20760.014
1.9:98.1 0.35460.010
3:97 0.50560.012
9:91 0.74460.018
60:40 0.95760.019
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261
Table 3 Salt melts and diffusion times for interdiffusion coefficient measurements; diffusion temperature is 3608C Glass
AgNO 3 :NaNO 3 (mol%)
Diffusion time (h)
1 2
50:50 100:0
48 72
2.2. Ag–Na interdiffusion The interdiffusion coefficient of three glass compositions were measured using X-ray microprobe (see Table 1 for compositions). These glasses have varying relative amounts of alumina and soda [R5 (Al 2 O 3 in mol%) /(Na 2 O in mol%)], and therefore different amounts of non-bridging oxygen. Ion exchange was conducted in each of the glass samples at 3608C using nitrate melts. Salt melt compositions and diffusion times appear in Table 3. After the diffusion, each sample was sectioned and the face perpendicular to the direction of diffusion was polished. The polished surfaces were coated with a carbon film and mounted on a cold stage with T593 K. The one-dimensional Ag–Na diffusion profile was obtained by energy dispersive X-ray microanalysis (EDX) in a scanning electron microscope (Cambridge S200), using a line-scan procedure with ‘ZAF’ analysis [18]. For more detail on experimental procedure, see Refs. [20,21]. The interdiffusion coefficient, D, as a function of the normalized silver concentration, x, was calculated from the experimental concentration profile by using the Boltzmann– Matano method. The averaged diffusion profile of five single scans was fitted by a 6th order polynomial and used in Eq. (1) to evaluate D( x ).
3. Results
3.1. Tracer diffusions It can be seen in Fig. 4 that in glass 1, DAg increases about an order of magnitude from xAg 50 to |1, whereas DNa increases by a factor of about two in the same composition range. Error bars are represented by the cross-ties; these were based on the variation between repeated measurements and the estimated error in experimental procedure. In order to understand the effect of the concentration depen-
Fig. 4. Radioactive tracer (Ag 110m and Na 22 ) diffusion coefficients as a function of normalized silver content. The base glass is glass 3, ion exchanged with silver (see Table 2 for silver contents). Also plotted is the mobility term using the tracer coefficients (solid line) and the Fujita term using constant self-diffusion coefficients (dot– dash line).
dent self-diffusion coefficients on ion exchange, the mobility term is calculated by using Eq. (2) with the measured DNa and DAg . The mobility term is also calculated by using the end-point DNa and DAg (DNa is extrapolated from xAg 50.957 to 1).
3.2. Interdiffusion coefficient Fig. 5 shows the measured interdiffusion coefficients resulting from the Boltzmann–Matano analysis of the EDX data. The interdiffusion coefficients are normalized to their peak values. Also plotted is the calculated ideal interdiffusion coefficient that would result in a perfect parabolic index profile if an initially dopant-free glass was ion exchanged to the center of the glass sample [18]. Glass 3 most closely approaches the ideal interdiffusion coefficient. After conducting the ion exchange strictly according to formula (no adjustable parameters, such as timedependent boundary conditions or extended diffusion time), a near-perfect (less than 2% maximum deviation) parabolic index profile was obtained [18]. Fig. 5 indicates that as the number of non-bridging oxygen are reduced (according to the relative alumina / soda content), the interdiffusion coefficients approach the ideal concentration dependence.
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
262
Fig. 5. Measured interdiffusion coefficient in glass 3 (solid line), compared to mobility term (dashed line).
4. Discussion The concentration dependencies of the tracer diffusions in Ag 1 / Na 1 ion exchanged boroaluminosilicate glass are distinctly weaker than those of furnace-melt, mixed-alkali silicates. The reason for this insensitivity to dopant concentration may be related to the similarity of Ag 1 / Na 1 ion sizes, the absence of non-bridging oxygen in the host glass, or relaxation during the ion-exchange thermal anneal. In any event, as a consequence, the mobility term is nearly linear, and varies by a factor of just four over the full concentration range. It is interesting to note that the mobility term is almost unchanged by substituting the tracer coefficients by constants [DA 5DA ( x 51), DB 5DB ( x 5 0)]. This form of the mobility term was first investigated numerically by Fujita (see Ref. [3]). The Fujita representation is plotted in Fig. 4 for comparison with the true mobility term. Only a small error is made by using the Fujita dependence. Indeed, the Fujita form is widely used by the micro-optics and integrated optics communities to characterize transport behavior (see e.g., Refs. [1,25]). The mobility term is compared to the full interdiffusion coefficient for the glass 3 in Fig. 6. As can be seen, the thermodynamic term accounts for much of the concentration dependence in this glass and cannot be regarded as a perturbation. It is therefore
beneficial to investigate the relation between composition, glass structure and the thermodynamic term. Local cation environments as measured by X-ray absorption spectroscopy have already been shown to differ markedly with ion type and also host glass composition [22]. For example, in glass 1, Na 1 assumes an environment almost identical to the environment found in sodium disilicate and trisilicate crystals. When the same glass is ion exchanged with silver, the Ag 1 environments are found to be nearly
Fig. 6. Concentration dependence of interdiffusion coefficients (normalized to peak value) for glasses 1–3. R is the ratio of Al 2 O 3 to Na 2 O host glass contents in mol%. Also plotted is interdiffusion coefficient necessary to obtain ideal parabolic index profile with time-independent boundary conditions.
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
identical with those found in crystalline Ag 2 O. The explanation for this change in cation site after ion exchange is that the cations are associated with non-bridging oxygens in glass 1 and can therefore reconfigure the site according to each cation’s bonding requirements. This is consistent with the observation that as the NBOs are removed, the ion-exchanged Ag 1 sites more closely resemble those of the constituent Na 1 . On the other hand, in glasses which contain little or no NBOs, the Ag 1 environments closely resemble those of the constituent Na 1 . This is to be expected, since the cation environments are formed by the more-constrained bridging oxygen [22]. The local and intermediate environments of diffusing cations are expected to determine macroscopic transport phenomena. In glasses with high NBO content, one would expect a grouping or clustering of the mobile cations and NBOs [22]. The ‘modified random network’, or MRN, extends these results to the notion of alkali / NBO rich ribbons that wind through the former matrix [26]. In the MRN picture, cations would be most mobile along the pathways rich with dopant and constituent cations [27]. Strong electrostatic interactions of unlike cations would therefore be expected. One the other hand, in NBOpoor aluminosilicate glasses, there are no experimental indications of cation clustering and the MRN picture is not expected to apply. This is corroborated by the observation that mobile cations in crystalline species with similar composition (e.g., albite) reside in interlocking bridging-oxygen cages and are more isolated. Therefore, one would expect only weak interactions between mobile cations in an NBO-poor host. These ideas were recently tested in glasses 1 and 2; it was found that NBO-rich glasses have a significantly stronger concentration dependence than their NBO-poor counterparts [21]. Further, the measured interdiffusion coefficients were fit with the MQC model [13,14] so that the number of nearest neighbor hopping sites, c, and the excess interaction energy, ´INT , could be estimated. These are listed in Table 4. It can be seen that the excess interaction energy for Ag 1 / Na 1 exchange in the NBO-rich glass is three times that in the NBO-poor glass. It is also seen that the number of nearest neighbor hopping sites is close to the expected value (c52) for one-dimensional diffusion (i.e., along a line).
263
Table 4 MCQ fits on three glass samples
´INT (eV) c x0 DAg /DNa DAg (310 28 cm 2 / s)
Glass 1
Glass 2
Glass 3
20.061 2.4 0.26 0.61 0.085
20.023 3.5 0.20 0.57 0.23
20.006 18.7 20.45 0.44 0.205
This is consistent with the MRN model. The larger value of c obtained for the NBO-poor glass is consistent with diffusion in higher dimensions, either in a plane or a volume, as expected. It should be noted that a Fujita representation was used for the mobility term in this study; the generality of this approximation is currently under test by the authors. In any event, it is clear that NBOs are associated with a strongly peaked concentration dependence of the interdiffusion coefficient. Fig. 5 shows a comparison of the Ag 1 / Na 1 interdiffusion coefficients in glasses with varying NBO content. It appears that the interdiffusion coefficients approach the ideal form as the NBOs are eliminated (R→1). However, glasses 2 and 3 contain boron (see Table 1). The amounts are much smaller than the alkali content by a factor of 2–63, but in principle the boron could also play a role in determining the cation environments and transport behavior. Boron-free, alumina-rich alkali silicate glasses tend to craze and also to crack on ion exchange, so it is difficult to separate the role of boron in structure and transport properties from that of the NBOs. Ignoring the possible role of boron, the NBO content is an important consideration in determining the suitability of the host glass.
5. Conclusions Control of the interdiffusion coefficient, and its dependence on the dopant concentration, is paramount in the fabrication of micro-optics. We show that the interdiffusion coefficient changes in ways that can be predicted by understanding the local environments of the mobile cations and that the immediate surroundings of each cation depends in large part on the speciation of the nearest neighbor oxygen and the bonding requirements of the dopant
264
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
and constituent ions. As a result, self diffusion behavior in the aluminoborsilicate glasses used for micro-optics differs significantly from that of simple alkali-silicate laboratory glasses. These details are incorporated into a thermodynamic description of ion exchange, called the MQC model. The accuracy of the MQC description makes it possible to substitute computer experiments for long experimental trials. The MQC model has been shown useful in fast identification of optimal glass types and process parameters.
Acknowledgements Thanks are due to several people for their contributions to this work: Bernhard Messerschmidt, Brian McIntyre, Marie Inman, Julie Bentley, Charles Hsieh and Neville Greaves. Thanks are also due to T. Possner and R. Araujo for some of the glass samples and to Steve Poling for assistance with the Figures. This work would not be possible without funding by the National Science Foundation (ECE-9413013), US Army Research Office URI program and the Center for Optics Manufacturing.
References [1] K. Iaga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics, Academic Press, New York, 1984, Ch. 2. [2] S.N. Houde-Walter, SPIE 935 (1988) 2. [3] J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975).
[4] A.R. Cooper, in: An Introduction to Glass Science, eds. L.D. Pye, H.J. Stevens, W.C. LaCourse, Plenum, New York, 1972, p. 563. [5] J.W. Fleming, D.E. Day, J. Am. Ceram. Soc. 55 (1972) 186. [6] G.H. Frischat, R. Kirchmeyer, J. Am. Ceram. Soc. 56 (1973) 552. [7] R. Hayami, R. Terai, Phys. Chem. Glass 13 (1972) 102. [8] M. Garfinkel, J. Phys. Chem. 72 (1968) 4175. [9] K.H. Stern, J. Phys. Chem. 74 (1970) 1323. [10] V.V. Moiseev et al., Sov. J. Phys. Chem. 3 (1977) 22. [11] R.H. Doremus, Glass Science, Wiley, New York, 1973, p. 255. [12] S.N. Houde-Walter, D.T. Moore, Appl. Opt. 25 (1986) 3373. [13] J.M. Inman, Ph.D. thesis, University of Rochester, 1995. [14] J.M. Inman, J.L. Bentley, S.N. Houde-Walter, J. Non-Cryst. Solids 191 (1995) 209. [15] T.L. Hill, Statistical Mechanics, McGraw-Hill, New York, 1956, p. 348. [16] J.L. Bentley, Ph.D. thesis, University of Rochester, 1995. [17] D.T. Moore, Gradient Lens Corp., private communication. [18] B. Messerschmidt, B.L. McIntyre, S.N. Houde-Walter, Appl. Opt. 35 (1996) 5670. [19] T. Possner, G. Schreiter, R. Muller, Glastech. Ber. 64 (1991) 185. [20] B. Messerschmidt, C.H. Hsieh, B.L. McIntyre, S.N. HoudeWalter, J. Non-Cryst. Solids 217 (1997) 264. [21] J.M. Inman, S.N. Houde-Walter, B.L. McIntyre, Z.M. Liao, R.S. Parker, V. Simmons, J. Non-Cryst. Solids 194 (1996) 85. [22] S.N. Houde-Walter, J.M. Inman, A.J. Dent, G.N. Greaves, J. Phys. Chem. 97 (1993) 9330. [23] M.B. Volf, Chemical Approach to Glass, Elsevier, Amsterdam, 1984, p. 485. [24] S.J. Rothman, in: Diffusion in Crysalline Solids, eds. G.E. Murch, A.S. Nowick, Academic Press, Orlando, FL, 1984, p. 1. [25] J. Albert, in: Introduction to Glass Integrated Optics, ed. S.I. Najafi, Artech House, New York, 1992. [26] G.N. Greaves et al., Nature 293 (1981) 611. [27] B. Vessal, G.N. Greaves, P.T. Marten, A.V. Chadwick, R. Mole, S.N. Houde-Walter, Nature 356 (1992) 504.
Ion exchange and the mixed mobile ion effect in micro-optics applications S.N. Houde-Walter* The Institute of Optics, University of Rochester, Rochester, NY 14627, USA
Abstract Silver–sodium ion exchange in aluminosilicate glasses is reviewed in the context of micro-optics fabrication. The interdiffusion coefficient is more strongly concentration dependent in glasses with high non-bridging oxygen content. Radioactive tracer diffusion coefficients show unusual behavior in glasses devoid of nonbridging oxygen. The local environments of the silver and sodium ions are related to the changes in ion exchange behavior. The modified quasi-chemical (MQC) model links these structural details to the thermodynamic description of ion exchange. The MQC model accurately describes ion exchange behavior in silicate, aluminosilicate and aluminoborate glasses. Using this model, long experimental trials are no longer necessary, and fast identification of optimal glass types and process parameters results. Keywords: Ion exchange; Micro-optics; Diffusion; Glass; Gradient index glass
1. Introduction Ion exchange is widely used in the fabrication of micro-optic lenses [1,2]. The most common type of micro-optic lens is called a ‘self-focusing lens’, or equivalently, a ‘radial gradient index lens’, or ‘GRIN lens’. These have a radial distribution of refractive index (see Fig. 1), with typically a parabolic or hyperbolic-secant index distribution for optimum imaging properties. Their diameters range from a few hundred microns to a few millimeters, and the total change in refractive index can be as large as 0.2 in the visible portion of the spectrum. GRIN lenses are used singly in applications including industrial boroscopes, medical endoscopes, diode laser collimators and optical fiber couplers. They are assembled into superposition image arrays for use in *E-mail: [email protected]; fax: 11 716 271 1027.
desktop photocopiers, fax machines and text scanners. The large radial variation in refractive index is achieved by diffusing heavy amounts of dopant cation into the host glass cylinder. The dopant can amount to 25–30 mole percent (mol%) of the final glass composition in localized regions. As a result, the interdiffusion coefficient is strongly dependent on dopant levels over the full concentration range. In
Fig. 1. Self-focusing lens with radial refractive index distribution.
0167-2738 / 98 / $19.00 1998 Elsevier Science B.V. All rights reserved. PII S0167-2738( 97 )00473-6
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
258
practice, the concentration dependence of the interdiffusion coefficient can be obtaining experimentally by inverting the one-dimensional diffusion equation via the Boltzmann transform [3]: x9
E
1 ≠x D( x 9) 5 2 ] ] x dx. 2t ≠x
(1)
0
Here x is the normalized dopant concentration, x 5 NB /(NA 1NB ), where NA is the mole fraction of the constituent mobile cation of species A and NB is the mole fraction of the dopant cation of species B, and x 9 denotes an intermediate concentration between 0 and 1. The spatial dimension is indicated by x, and diffusion time by t. Eq. (1) allows numerical calculation of D( x ) provided a concentration profile, x (x), has been measured. A typical concentration dependence is shown in Fig. 2 for a silver-for-sodium exchange in a sodium aluminosilicate glass designed for GRIN optics. In general, D( x ) is relatively small for small x, grows with increasing x until some peak value is attained, and then falls off as x increases thereafter. This concentration dependence of the ion exchange is widely recognized as one of the manifestations of the mixed mobile ion effect, or MMIE. Depending on the glass host composition, the dopant specie and the ion exchange temperature, the peak width varies and the peak value of D( x ) can be shifted towards high or low dopant concentrations.
Fig. 2. Concentration dependence of interdiffusion coefficient in sodium aluminosilicate glass which has been ion exchanged with silver.
These nuances in the functional form of D( x ) are critically important in determining the final dopant distribution, and thus the final optical performance of the micro lens. The interdiffusion coefficient can be derived from first principles by assuming equal and opposite fluxes of dopant and constituent ions [4]. The fundamental expression is:
H
x (1 2 x ) ≠ D( x ) 5 ]]] ]( mB 2 mA ) kT ≠x
JF
G
DB ]] . 1 2 ax
(2)
Here k is the Boltzmann constant, T is the ion exchange temperature, mA and mB are the chemical potentials of the constituent and dopant cations and DB is the self diffusion coefficient of the dopant cation. The parameter a is a measure of the relative mobilities: DB a 5 1 2 ]. DA
(3)
The two contributions to the interdiffusion coefficient in Eq. (2) are referred to as the thermodynamic contribution (curly brackets) and the mobility term (square brackets). Many of the early studies of interdiffusion were conducted in simple, furnacemelt, mixed-alkali silicate glasses. It was found that the mobility term made the dominant contribution to D( x ) in these glasses (see, for example, Refs. [5–7]). This was established by measuring the self diffusion coefficients, Di , (i5A, B), by radioactive tracer diffusions in glasses with varying relative alkali content in a series of mixed-alkali (Na / Rb or Na / Cs) silicate glasses. The self diffusion coefficients varied by two and three orders of magnitude, as did the mobility term calculated using the measured Di values. D( x ) differed little from the mobility term, and the multiplicative thermodynamic term, although rarely measured directly, has since been regarded as a perturbation. Host compositions used for micro-optics are quite different from simple mixed-alkali silicates. Requirements for suitable host glasses include chemical durability, large interdiffusion coefficients, appropriate optical dispersion, negligible stress birefringence after ion exchange and the capacity for a large change in refractive index when ion exchanged with low-toxicity dopants. Silver-for-sodium ion exchange
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
in aluminum borosilicate glasses satisfies these requirements. The thermodynamic term in Eq. (2) is not immediately useful to those developing micro-optics because the chemical potentials are known for virtually none of the available glass compositions. The thermodynamic term can be rewritten by use of the Gibbs– Duhem relation in terms of the activities of the dopant cation, a B , in the glass [4]:
H
J
≠ln (a B ) x (1 2 x ) ≠ ]]] ]( mB 2 mA ) 5 ]]] . kT ≠x ≠ln ( x )
(4)
In principle, the activities can be determined experimentally [8–10], but such measurements in glasses are impractical when many glass compositions must be characterized. Using the Rothmund– Kornfeld relation [11], [a B ] x ]] 5 ]] 12x [aA ]
S
D, n
(5)
the ratio of activities of the dopant and constituent cations in the glass can be related to the ratio of their concentrations through an empirical power factor, n. Thus, the thermodynamic term can be approximated by ≠ln (a B ) ]]] ¯n ≠ln ( x )
259
ing the maximum term method and making Stirling’s approximation, the thermodynamic term can be written
Hx
J HS D J
(1 2 x ) ≠ c 1 ]]] ]( mB 2 mA ) 5 ] ] 2 1 1 1 . kT ≠x 2 b (8)
The parameter b is given by ]]]]]]]]]] b 5œ1 2 4x (1 2 x )s1 2 exps2´INT /kTdd.
(9)
In a one-dimensional array of hopping sites (c5 2),
Hx
J
(1 2 x ) ≠ 1 ]]] ]( mB 2 mA ) 5 ] kT ≠x b
(c 5 2).
(10)
This is plotted in Fig. 3 as a function of the parameter 2´INT /kT, keeping in mind that the excess interaction energy is typically negative in glasses (net attraction between unlike cations). As the interaction term becomes stronger, the concentration dependence of the thermodynamic term becomes more pronounced. In real glasses, the peak is sometimes observed at values other than x 51 / 2. This shift becomes larger if the mobile cation content (alkali plus dopant) is reduced in the glass com-
(6)
where n is a constant typically in the range 1,n,2 [12]. The departure of n from unity indicates a stronger interaction between the diffusing cations. However, the thermodynamic term of Eq. (2) is actually a function of dopant concentration, and the approximation of Eq. (6) is not sufficiently accurate for modeling of the micro-optics fabrication process. The thermodynamic term can be recast in terms of structural parameters of the glass [13,14]. This is done by writing the chemical potentials in terms of the canonical partition function, expressed as a function of the number of nearest neighbor hopping sites, c, and the excess interaction energy, ´INT ,
´AA 1 ´BB ´INT 5 ´AB 1 ]]] 2
(7)
where ´AB (´ii ) is the effective bond energy between adjacent unlike (like) cations. Using the quasi-chemical (Bethe–Guggenheim) approximation [15], apply-
Fig. 3. Thermodynamic contribution to interdiffusion coefficient with varying excess interaction energy term for special case of one-dimensional glass (c52). Concentration dependence becomes stronger as varying excess interaction energy term, 2´INT /kT, becomes large in magnitude.
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
260
position. The present model assumes an arbitrarily large number of cation sites, so an ad hoc shift, x 2 x0 , can be used for the purposes of fitting Eq. (2) to measured D( x ). This representation of the thermodynamic term is called the modified quasi-chemical, or MQC, model of ion exchange. In practice, Eq. (8) is used to describe D( x ) in terms of the parameters c, ´INT and x0 . Even very noisy data can be successfully fit with this representation of D( x ) [14], and unlike other representations (polynomials, Gaussian fits, etc.), the MQC parameters can be used in a finite difference algorithm to regenerate the original dopant profile and profiles for any other diffusion time with unprecedented accuracy. Therefore, instead of conducting many diffusion experiments and measuring the resultant index profiles to see which diffusion time gives best optical properties, it is now possible to use the MQC parameters from a single experiment to calculate a large number of index profiles and numerically compare these to the desired distribution [16]. Simple algorithms can then report the best diffusion schedule. This procedure has been shown to reduce the number of experiments (e.g., for an endoscope application, 50 experiments, 8 h apiece [17]) by more than an order of magnitude.
composition). Full details are given in Ref. [20]. This glass is expected to be free of non-bridging oxygen because it has a large boron oxide content and Al / Na molar ratio of unity [21,22]. No more than a few mol% silver can be incorporated into a glass of this composition by furnace melting [23], so slabs of the sodium-rich, furnace-melt glass were ion exchanged until homogeneous with varying silver contents x from 0.120 to 0.957. This was done by immersing slabs of the sodium boroaluminosilicate glass with dimensions 10 mm310 mm30.7 mm into seven different AgNO 3 –NaNO 3 molten salts at 4008C for 377 h (for melt compositions, see Table 2). The result was homogeneous samples with seven different Ag concentrations. The self-diffusion coefficients of sodium and silver in the mixed ion glasses were measured by means of radioactive tracer diffusion. The isotopes, 22 Na and 110m Ag, in the form of 22 NaCl in 0.2 M HCl and 110m AgNO 3 in 0.1 M HNO 3 were used as radiotracer sources. The diffusions were carried out by annealing the samples at 3608C for 5 h. Assuming an instantaneous source, the tracer distribution is
2. Experimental
where x i* denotes the tracer concentrations. The 22 Na and the 110m Ag diffusion profiles in the glass were obtained by successive sectioning of the samples and counting the radioactivity of the removed layers [24]. Each etching step was carried out in 6 ml etchant (1.5 vol% HF (49% concentration)11.5 vol% HNO 3 (70% concentration)197 vol% H 2 O) contained in a scintillation vial. After the etching, 16 ml scintillation liquid (Ecoscint A, National Diagnostics) was added to each vial and radioactivity counting was performed by the liquid scintillation counting technique for each vial. The self-diffusion coefficient was then determined by fitting the diffusion profile to Eq. (11).
2.1. Tracer diffusions Radioactive tracer diffusions were conducted in a boroaluminosilicate glass [18,19] (see Table 1 for Table 1 Glass compositions in mole percent Glass
Na 2 O
Al 2 O 3
B2O3
SiO 2
1 2 3
19.3 22.2 25.0
– 20.3 25.0
– 5.5 12.5
80.7 52.0 37.5
1 2x 2 / (4D i t ) x *i 5 ]] , i 5 (A, B), ]] e œp Di t
(11)
Table 2 Homogeneous ion-exchange of glass 1 for high silver content glasses: melt composition, normalized silver concentration xAg measured by EDX
AgNO 3 :NaNO 3 (mol%) xAg
Base glass
Sample A
Sample B
Sample C
Sample D
Sample E
Sample F
0:100 0
0.5:99.5 0.12060.012
1:99 0.20760.014
1.9:98.1 0.35460.010
3:97 0.50560.012
9:91 0.74460.018
60:40 0.95760.019
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
261
Table 3 Salt melts and diffusion times for interdiffusion coefficient measurements; diffusion temperature is 3608C Glass
AgNO 3 :NaNO 3 (mol%)
Diffusion time (h)
1 2
50:50 100:0
48 72
2.2. Ag–Na interdiffusion The interdiffusion coefficient of three glass compositions were measured using X-ray microprobe (see Table 1 for compositions). These glasses have varying relative amounts of alumina and soda [R5 (Al 2 O 3 in mol%) /(Na 2 O in mol%)], and therefore different amounts of non-bridging oxygen. Ion exchange was conducted in each of the glass samples at 3608C using nitrate melts. Salt melt compositions and diffusion times appear in Table 3. After the diffusion, each sample was sectioned and the face perpendicular to the direction of diffusion was polished. The polished surfaces were coated with a carbon film and mounted on a cold stage with T593 K. The one-dimensional Ag–Na diffusion profile was obtained by energy dispersive X-ray microanalysis (EDX) in a scanning electron microscope (Cambridge S200), using a line-scan procedure with ‘ZAF’ analysis [18]. For more detail on experimental procedure, see Refs. [20,21]. The interdiffusion coefficient, D, as a function of the normalized silver concentration, x, was calculated from the experimental concentration profile by using the Boltzmann– Matano method. The averaged diffusion profile of five single scans was fitted by a 6th order polynomial and used in Eq. (1) to evaluate D( x ).
3. Results
3.1. Tracer diffusions It can be seen in Fig. 4 that in glass 1, DAg increases about an order of magnitude from xAg 50 to |1, whereas DNa increases by a factor of about two in the same composition range. Error bars are represented by the cross-ties; these were based on the variation between repeated measurements and the estimated error in experimental procedure. In order to understand the effect of the concentration depen-
Fig. 4. Radioactive tracer (Ag 110m and Na 22 ) diffusion coefficients as a function of normalized silver content. The base glass is glass 3, ion exchanged with silver (see Table 2 for silver contents). Also plotted is the mobility term using the tracer coefficients (solid line) and the Fujita term using constant self-diffusion coefficients (dot– dash line).
dent self-diffusion coefficients on ion exchange, the mobility term is calculated by using Eq. (2) with the measured DNa and DAg . The mobility term is also calculated by using the end-point DNa and DAg (DNa is extrapolated from xAg 50.957 to 1).
3.2. Interdiffusion coefficient Fig. 5 shows the measured interdiffusion coefficients resulting from the Boltzmann–Matano analysis of the EDX data. The interdiffusion coefficients are normalized to their peak values. Also plotted is the calculated ideal interdiffusion coefficient that would result in a perfect parabolic index profile if an initially dopant-free glass was ion exchanged to the center of the glass sample [18]. Glass 3 most closely approaches the ideal interdiffusion coefficient. After conducting the ion exchange strictly according to formula (no adjustable parameters, such as timedependent boundary conditions or extended diffusion time), a near-perfect (less than 2% maximum deviation) parabolic index profile was obtained [18]. Fig. 5 indicates that as the number of non-bridging oxygen are reduced (according to the relative alumina / soda content), the interdiffusion coefficients approach the ideal concentration dependence.
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
262
Fig. 5. Measured interdiffusion coefficient in glass 3 (solid line), compared to mobility term (dashed line).
4. Discussion The concentration dependencies of the tracer diffusions in Ag 1 / Na 1 ion exchanged boroaluminosilicate glass are distinctly weaker than those of furnace-melt, mixed-alkali silicates. The reason for this insensitivity to dopant concentration may be related to the similarity of Ag 1 / Na 1 ion sizes, the absence of non-bridging oxygen in the host glass, or relaxation during the ion-exchange thermal anneal. In any event, as a consequence, the mobility term is nearly linear, and varies by a factor of just four over the full concentration range. It is interesting to note that the mobility term is almost unchanged by substituting the tracer coefficients by constants [DA 5DA ( x 51), DB 5DB ( x 5 0)]. This form of the mobility term was first investigated numerically by Fujita (see Ref. [3]). The Fujita representation is plotted in Fig. 4 for comparison with the true mobility term. Only a small error is made by using the Fujita dependence. Indeed, the Fujita form is widely used by the micro-optics and integrated optics communities to characterize transport behavior (see e.g., Refs. [1,25]). The mobility term is compared to the full interdiffusion coefficient for the glass 3 in Fig. 6. As can be seen, the thermodynamic term accounts for much of the concentration dependence in this glass and cannot be regarded as a perturbation. It is therefore
beneficial to investigate the relation between composition, glass structure and the thermodynamic term. Local cation environments as measured by X-ray absorption spectroscopy have already been shown to differ markedly with ion type and also host glass composition [22]. For example, in glass 1, Na 1 assumes an environment almost identical to the environment found in sodium disilicate and trisilicate crystals. When the same glass is ion exchanged with silver, the Ag 1 environments are found to be nearly
Fig. 6. Concentration dependence of interdiffusion coefficients (normalized to peak value) for glasses 1–3. R is the ratio of Al 2 O 3 to Na 2 O host glass contents in mol%. Also plotted is interdiffusion coefficient necessary to obtain ideal parabolic index profile with time-independent boundary conditions.
S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
identical with those found in crystalline Ag 2 O. The explanation for this change in cation site after ion exchange is that the cations are associated with non-bridging oxygens in glass 1 and can therefore reconfigure the site according to each cation’s bonding requirements. This is consistent with the observation that as the NBOs are removed, the ion-exchanged Ag 1 sites more closely resemble those of the constituent Na 1 . On the other hand, in glasses which contain little or no NBOs, the Ag 1 environments closely resemble those of the constituent Na 1 . This is to be expected, since the cation environments are formed by the more-constrained bridging oxygen [22]. The local and intermediate environments of diffusing cations are expected to determine macroscopic transport phenomena. In glasses with high NBO content, one would expect a grouping or clustering of the mobile cations and NBOs [22]. The ‘modified random network’, or MRN, extends these results to the notion of alkali / NBO rich ribbons that wind through the former matrix [26]. In the MRN picture, cations would be most mobile along the pathways rich with dopant and constituent cations [27]. Strong electrostatic interactions of unlike cations would therefore be expected. One the other hand, in NBOpoor aluminosilicate glasses, there are no experimental indications of cation clustering and the MRN picture is not expected to apply. This is corroborated by the observation that mobile cations in crystalline species with similar composition (e.g., albite) reside in interlocking bridging-oxygen cages and are more isolated. Therefore, one would expect only weak interactions between mobile cations in an NBO-poor host. These ideas were recently tested in glasses 1 and 2; it was found that NBO-rich glasses have a significantly stronger concentration dependence than their NBO-poor counterparts [21]. Further, the measured interdiffusion coefficients were fit with the MQC model [13,14] so that the number of nearest neighbor hopping sites, c, and the excess interaction energy, ´INT , could be estimated. These are listed in Table 4. It can be seen that the excess interaction energy for Ag 1 / Na 1 exchange in the NBO-rich glass is three times that in the NBO-poor glass. It is also seen that the number of nearest neighbor hopping sites is close to the expected value (c52) for one-dimensional diffusion (i.e., along a line).
263
Table 4 MCQ fits on three glass samples
´INT (eV) c x0 DAg /DNa DAg (310 28 cm 2 / s)
Glass 1
Glass 2
Glass 3
20.061 2.4 0.26 0.61 0.085
20.023 3.5 0.20 0.57 0.23
20.006 18.7 20.45 0.44 0.205
This is consistent with the MRN model. The larger value of c obtained for the NBO-poor glass is consistent with diffusion in higher dimensions, either in a plane or a volume, as expected. It should be noted that a Fujita representation was used for the mobility term in this study; the generality of this approximation is currently under test by the authors. In any event, it is clear that NBOs are associated with a strongly peaked concentration dependence of the interdiffusion coefficient. Fig. 5 shows a comparison of the Ag 1 / Na 1 interdiffusion coefficients in glasses with varying NBO content. It appears that the interdiffusion coefficients approach the ideal form as the NBOs are eliminated (R→1). However, glasses 2 and 3 contain boron (see Table 1). The amounts are much smaller than the alkali content by a factor of 2–63, but in principle the boron could also play a role in determining the cation environments and transport behavior. Boron-free, alumina-rich alkali silicate glasses tend to craze and also to crack on ion exchange, so it is difficult to separate the role of boron in structure and transport properties from that of the NBOs. Ignoring the possible role of boron, the NBO content is an important consideration in determining the suitability of the host glass.
5. Conclusions Control of the interdiffusion coefficient, and its dependence on the dopant concentration, is paramount in the fabrication of micro-optics. We show that the interdiffusion coefficient changes in ways that can be predicted by understanding the local environments of the mobile cations and that the immediate surroundings of each cation depends in large part on the speciation of the nearest neighbor oxygen and the bonding requirements of the dopant
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S.N. Houde-Walter / Solid State Ionics 105 (1998) 257 – 264
and constituent ions. As a result, self diffusion behavior in the aluminoborsilicate glasses used for micro-optics differs significantly from that of simple alkali-silicate laboratory glasses. These details are incorporated into a thermodynamic description of ion exchange, called the MQC model. The accuracy of the MQC description makes it possible to substitute computer experiments for long experimental trials. The MQC model has been shown useful in fast identification of optimal glass types and process parameters.
Acknowledgements Thanks are due to several people for their contributions to this work: Bernhard Messerschmidt, Brian McIntyre, Marie Inman, Julie Bentley, Charles Hsieh and Neville Greaves. Thanks are also due to T. Possner and R. Araujo for some of the glass samples and to Steve Poling for assistance with the Figures. This work would not be possible without funding by the National Science Foundation (ECE-9413013), US Army Research Office URI program and the Center for Optics Manufacturing.
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