Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 2 pot

30.3 Gas Permeation 529 Permeation of gases through glasses is of technological interest. If the diffusing species is a gas, it is possible to expose one face of a glass membrane (thickness ∆x) to a known pressure of the gas, whilst the other side of the membrane is connected to a mass spectrometer. In one method, the permeating gas is continuously removed from the spectrometer side by pumping and maintaining a pressure difference ∆p across the membrane and the steady state diffusion flow of gas, J, through the membrane is measured. From such experiments the permeability K can be determined via K = J∆p ∆x . (30.6) If one assumes that the gas concentration in the glass, C,isgivenbyHenry’s law, C = C s p,whereC s is the ambient pressure solubility and p the gas pressure, the permeability can be written as K = DC s , (30.7) where D is the diffusivity of the gas. Usually, diffusivity and solubility are both Arrhenius activated. 30.3 Gas Permeation A number of gases permeate through glasses at rates which can have serious consequences for practical applications. Helium can readily permeate through many glasses used for vacuum tubes. Hydrogen permeation can result in col- oration of glasses by the reduction of ions to a lower valency or to the metallic state and by the reaction with optically active defects. Oxygen permeating through the wall of an electric lamp can react with filament material, causing failure of the bulb. Permeation rates of gases in vitreous silica are shown in Fig. 30.5. The data indicate that the permeability decreases as the atomic or molecular diameter of the diffusing species increases. The permeability decreases in the order He > H 2 > Ne > N 2 > O 2 > Ar > Kr. The trend for He, Ne, and hydrogen isotopes in all glasses is similar as in vitreous silica. Permeation varies linearly with the partial pressure of the gas for pres- sures up to many atmospheres. The effect of glass composition on helium permeation has been studied for a variety of oxide glasses, including silicate, borate, germanate, and phosphate compositions. In general, He permeation decreases in silicate glasses with increasing modifier content. For example, the permeability of soda-lime silicate glasses, depending on the temperature, is two to four orders of magnitude lower than in pure vitreous silica (see, e.g., [4]). Helium permeates through vitreous silica most readily, since it is the most open-structured glass. The modifier ions occupy ‘interstitial sites’ of the network, thus blocking diffusion paths for He atoms. 530 30 Diffusion and Ionic Conduction in Oxide Glasses In general, permeability data reflect the characteristics of ‘interstitial spaces’ in glasses. However, care is needed when the permeating species reacts with the glass network. For example, H 2 in vitreous silica reacts with the silica network to form hydroxyls and silan groups [16]. It has also been suggested that water diffuses in vitreous silica as molecules reacting with the network to form immobile hydroxyl ions. 30.4 Examples of Diffusion and Ionic Conduction Below the glass-transition region, where the network structure is essentially rigid, self-diffusion of network formers is very slow. In comparison, self- diffusion of modifier cations is faster as they can move through the ‘interstitial’ channels of the network. Thus, it is the movement of modifier cations which determines many properties of the glass such as electrical conductivity, corrosion resistance, and dielectric break down. For this reason, the majority of diffusion data on oxide glasses refers to diffusion and ionic conduction of the modifier cations (see [6]). In what follows, diffusion and ion conduction of vitreous silica, soda-lime-glass, and borate glasses are used to illustrate typical features. In addition, the so-called mixed-alkali effect is described. Vitreous Silica and Quartz: Vitreous silicon dioxide is an important technological material. It is the most refractory glass in commercial use and has high corrosion resistance, a low coefficient of thermal expansion, and good UV transparency. Apart from laboratory use, optical mirrors, high-efficiency Fig. 30.5. Permeability of gases through vitreous SiO 2 according to Shelby [2] 30.4 Examples of Diffusion and Ionic Conduction 531 lamps, optical fibers and dielectric films in microelectronic devices represent important applications (see also Chap. 28). As already mentioned, vitreous SiO 2 as well as quartz crystals contain SiO 4/2 tetrahedra, which are linked together in three dimensions. In glassy silica, these tetrahedra form a random network, whereas in the crystal they are linked in an ordered fashion. Crystalline SiO 2 exists in many modifications as temperature and pressure varies. At ambient pressure trigonal low-quartz transforms around 575 ◦ C to hexagonal high-quartz, which at 870 ◦ Ctrans- forms to hexagonal trydimite. At 1470 ◦ C cubic cristobalite is formed, which melts at about 1700 ◦ C. A comparison of diffusion in glassy and crystalline SiO 2 may be of special interest (Fig. 30.6). Diffusion of 22 Na in vitreous silica prepared from quartz 3 was studied between 170 and 1200 ◦ C applying the residual activity Fig. 30.6. Diffusion in vitreous silica and in quartz (for references see text) 3 Despite its defined chemical composition one distinguishes in the literature different types of vitreous silica with respect to preparation method, raw material, and impurity content. These differences can lead to differences between the diffusion results of various groups. For example, glassy silica prepared from natural 532 30 Diffusion and Ionic Conduction in Oxide Glasses method [17]. Diffusion of the stable isotope 30 Si has been measured in the temperature range 1110 to 1410 ◦ C using SIMS [18]. Self-diffusion of oxygen was studied using a gas phase isotope exchange reaction [19]. SIMS has also been used to profile the interdiffusion of network oxygen in a vitreous Si 18 O 2 –Si 16 O 2 thin-film structure [20]. The diffusivity values are lower, but with a higher activation enthalpy (4.7 eV) than those reported in [19] and approach the diffusivity of network oxygen uncomplicated by gas phase exchange reactions. Figure 30.6 confirms that diffusion of the network former Si is very slow, whereas Na diffusion is relatively fast. The Si activation enthalpy of about 6 eV is close to the energy necessary to break Si-O bonds. The energy of a Si-O bond is about 2.9 eV [25]. For each SiO 4/2 tetrahedron the four half-bonds represent an energy of 5.8 eV. The main barrier for the movement of Si atoms seems to be indeed the Si-O bond energy. According to this reasoning, one could expect for oxygen diffusion an activation enthalpy of about half of that of Si diffusion. One experimental value of about 2.43 [19] seems to support this reasoning. However, other authors report values as low as 0.85 eV [26] and as high as 3.08 eV [27]. In view of this large scatter, it is likely that different diffusion mechanisms operate for oxygen and silicon. Figure 30.6 also shows 22 Na diffusion in crystalline quartz parallel and perpendicular to its crystallographic axis [21–23] and 45 Ca diffusion in one direction [24]. The transition between high- and low-quartz at 575 ◦ C influ- ences Na diffusion. Since high-quartz has a hexagonal structure, which is of higher symmetry than the trigonal one of low-quartz, diffusion in high-quartz has a lower activation enthalpy. Figure 30.6 reveals a strong anisotropy of Na diffusion in quartz as well. Diffusion parallel to the axis is much faster than perpendicular to it. It is also remarkable that Na diffusion in vitreous silica lies between the Na diffusivities parallel and perpendicular to the crystallographic axis of low- and of high-quartz. Na and Ca have nearly the same ionic radii. Nevertheless, the tracer diffusivity of Ca is (at 600 ◦ C) almost seven orders of magnitude lower than that of Na. This reflects the stronger linkage of Ca to the glass network. Soda-Lime Silicate Glass: Silicate glasses form the largest class of oxide glasses (see Chap. 28). Most of them are used as window and container glasses. Soda-lime glasses are mainly ternary glasses often with some further minor additions. They usually contain about 10 to 20 mol % alkali oxides, primarily in the form of Na 2 O, 5 to 15 mol % CaO and 70 to 75 mol % SiO 2 .Useof dolomite as a source of CaO often implies that considerable MgO is also present in the glass. For special purposes some of the soda is replaced by K 2 O or, less commonly, by Li 2 O. Replacement of CaO and/or MgO by SrO and BaO occurs occasionally in the production of glasses. quartz crystals either by electric melting or by plasma sputtering in a H 2 and O 2 plasma reveal nearly the same Na diffusivites, whereas glasses synthesised from SiCl 4 display a lower Na diffusivity presumably due to a distinctly lower content of hydroxyl groups [5]. 30.4 Examples of Diffusion and Ionic Conduction 533 Fig. 30.7. Structure of a soda-lime silicate glass (schematic in two dimensions) Fig. 30.8. Viscosity diffusion coefficient, D η , tracer diffusivities, D ∗ Na , D ∗ Ca ,and charge diffusion coefficient D σ , of soda-lime silicate glass (standard glass I of DGG) according to Tanguep-Nijokep and Mehrer [15] 534 30 Diffusion and Ionic Conduction in Oxide Glasses The structure of a soda-lime silicate glass (Fig. 30.7) is readily described by the structural rules of Zachariasen. The silicon-oxygen tetrahedron, with a coordination number of 4, serves as the basic building block. If modifier cations are introduced – by melting SiO 2 ,Na 2 O, and CaO to form a soda- lime glass – some Si-O-Si bridges are broken. Oxygen atoms occupy free ends of separated tetrahedra thus forming non-bridging oxygen (NBO) units. The NBO units are the anionic counterparts of the alkali- or alkaline-earth ions. The cations (Na + or Ca 2+ ) are mainly incorporated at the severance sites of the network. Usually, every alkali ion has a neighbouring NBO, while every alkaline-earth ion has two neighbouring NBO units. This structure provides stronger network linkage at the alkaline-earth sites. Thus the divalent alkaline earth ions are less mobile than the monovalent alkali ions. The replacement – of alkali ions by alkaline-earth ions – reduces the ionic contributions to the electrical conductivity and improves the chemical durability of the glass. Figure 30.8 and 30.9 illustrate mass transport properties in two similar soda-lime silicate glasses produced by the Deutsche Glastechnische Gesellschaft (DGG) as standard glass I and II for physical and chemical testing. The composition of standard glass I (in mole fractions) is: 71.8 % Fig. 30.9. Tracer diffusivities, D ∗ Na , D ∗ Ca , and charge diffusion coefficient, D σ ,of soda-lime silicate glass (standard glass II of DGG) according to Tanguep-Nijokep and Mehrer [15] 30.4 Examples of Diffusion and Ionic Conduction 535 SiO 2 , 14.52 % Na 2 O, 7.22 % CaO, 6.24 % MgO, and some minor additions. The composition of standard glass II is: 71.37 % SiO 2 , 13.19 % Na 2 O, 10.43 % CaO, 5.01 % MgO, and some minor addition. Both glasses differ mainly in their content of alkaline-earth oxides. For standard glass I viscosity data for the undercooled melt are available from measurements performed at the Physikalisch Technische Bundesanstalt (Braunschweig, Germany). These data have been used to calculate the viscosity diffusion coefficient, D η ,from the Stokes-Einstein relation Eq. (30.4) using 0.042 nm for the ionic radius of Si (Fig. 30.8). D η can be readily described by Vogel-Fulcher-Tammann behaviour. Also shown are tracer diffusivities of 22 Na and 45 Ca and the charge diffusion coefficient, D σ , measured in the glassy state, which are all well represented by Arrhenius relations. At the glass-transition temperature, Ca diffusion is 6 orders of magnitude slower than Na diffusion and at lower temperatures the difference is even larger. This confirms the expectation that divalent Ca ions have a much stronger linkage to the network than Na ions. In addition, this large difference together with the fact that conductivity diffusion and Na tracer diffusion have the same activation enthalpy show that the electrical conductivity of soda-lime silicate glasses is due to the motion of Na ions. Figure 30.10 shows the Haven ratios, H R = D ∗ Na /D σ , of both standard glasses based on the assumption that only Na ions are mobile. The Haven ratios are: H R =0.45 for standard glass I and H R =0.33 for standard glass II. Both Haven ratios are temperature-independent within the experimental errors, indicating that the mechanism of Na diffusion does not change with temperature. Alkali Borate Glasses: The structure of vitreous boric oxide (B 2 O 3 )dif- fers considerably from that of vitreous silica. Although boron occurs in triangular as well as tetrahedral coordination in crystalline compounds, only the triangular state is formed in vitreous boric oxide. The BO 3/2 units are connected at all three corners via B-O-B bonds to form a network. It is also believed that vitreous boric oxide contains a certain amount of so-called boroxol groups consisting of three boron-oxygen triangles joined together. In contrast to vitreous silica, the basic building block of the vitreous boron oxide network is planar rather than three-dimensional. A three-dimensional structure is obtained by ‘crumpling’ the network. Since the primary bonds exist only within a plane, bonds in a third dimension are weak and the structure is easily disrupted. One consequence of this weakly bound structure is the low glass-transition temperature of vitreous boric oxide (about 260 ◦ C), which is much lower than that of vitreous silica (about 1100 ◦ C). The arrangement of atoms (or ions) in an alkali borate glass is illustrated in Fig. 30.11. Whereas addition of alkali oxides to vitreous silica results in the formation of NBO units (see above), the effect of alkali-oxide addition to boric oxide cannot be explained on the basis of NBO formation. The addition of alkali oxide forces some of the boron to change from trigonal to tetrahedral 536 30 Diffusion and Ionic Conduction in Oxide Glasses configuration. Formation of two boron-oxygen tetrahedra consumes the ad- ditional oxygen provided by one alkali oxide molecule. If alkali ions are introduced into the trigonally coordinated network of vitreous B 2 O 3 , tetrahedrally coordinated BO − 4/2 units are formed, which are the anionic counterparts of the alkali ions. Each Na 2 O(orRb 2 O) molecule creates two BO − 4/2 units. Only at concentrations larger than about 25 mol % alkali oxide, non-bridging oxy- gens appear. As evidenced by the glass-transition temperatures, the addition of alkali oxides enhances the stability of the glassy borates considerably at least below 25 mol % alkali content (Fig. 30.12). Figure 30.13 shows an Arrhenius diagram of the dc conductivity (times temperature) of sodium borate glasses [28]. The conductivity is Arrhenius activated and increases many orders of magnitude when the alkali content increases from 4 to 30 mol %. In ion conducting glasses, the conductivity is determined by the number density of mobile ions and by their mobility. As a result, glasses which contain significant concentrations of monovalent ions are poor insulators, while glasses that are free of monovalent ions are ex- cellent insulators. Figure 30.14 shows the effect of Li 2 O, Na 2 O, K 2 O, and Fig. 30.10. Haven ratios of soda-lime silicate glasses according to Tanguep- Nijokep and Mehrer [15] 30.4 Examples of Diffusion and Ionic Conduction 537 Rb 2 O additions on the conductivity of borate glass. Whereas the conductivity increases 5 to 6 orders of magnitude, the alkali content varies much less. This indicates that the mobility of ions increases significantly. The lat- ter conclusion is supported by Na tracer diffusion studies, which show a very similar increase with Na 2 O content [30]. In Fig. 30.14 also a decrease of conductivity for corresponding glasses containing the same alkali concentrations is observed in the order of increasing ionic radii: Li > Na > K > Rb. The smallest alkali ion entails the highest conductivity. Fig. 30.11. Structure of sodium-rubidium borate glass (schematic in two dimensions) Fig. 30.12. Glass-transition temperatures of alkali borate glasses according to Berkemeier et al. [28] 538 30 Diffusion and Ionic Conduction in Oxide Glasses Fig. 30.13. Arrhenius diagram of the dc conductivity (times temperature) for Y Na 2 O(1-Y)B 2 O 3 glasses according to Berkemeier et al. [28] Fig. 30.14. Electrical dc conductivity of Li, Na, K, and Rb borate glasses according to Berkemeier et al. [28] Mixed-Alkali Effect: Glasses containing two or more alkali oxides display the so-called mixed-alkali effect, which is one of the old but still very in- teresting features of ionic conduction and diffusion in glass. Figure 30.15 shows as a typical example the conductivity diffusion coefficient of a sodium- [...]... (1977) u a 27 E.W Sucov, J Amer Ceram Soc 46, 14 (19 63) 28 F Berkemeier, S Voss, A.W Imre, H Mehrer, J Non-Cryst Sol 35 1, 38 16 (20 05) 29 A.W Imre, S Voss, H Mehrer, Phys Chem Chem Phys 4, 32 19 (20 02) 30 S Voss, F Berkemeier, A.W Imre, H Mehrer, Z Phys Chem 21 8, 135 3 (20 04) References 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5 43 Y Gao, C Cramer, Solid State Ionics 176, 921 (20 05) R.J... Metall 26 , 32 7 (1980) 19 R Haul, G D¨mbgen, Z Elektrochemie 66, 636 (19 62) u 20 J.C Mikkelsen, Jr., Appl Phys Lett 45, 1187 (1984) 21 G.H Frischat, Phys Stat Sol 35 , K47 (1969) 22 G.H Frischat, Ber Dt Keram Ges 47, 23 8 (1970); and 47, 31 3 (1970) 23 G.H Frischat, J Amer Ceram Soc 53, 35 7 (1970) 24 G.H Frischat, Ber Dt Keram Ges 47, 36 4 (1970); 25 H.F Wolf, Semiconductors, Wiley Interscience, 1971 26 K... Elektrochemie 17, 23 5 (1908) 13 C.A Angell, J Non-Cryst Solids 1 02, 20 5 (1988) 14 A.W Imre, S Voss, H Mehrer, Defect and Diffusion Forum 23 7 24 0, 37 0 (20 05) 15 E.M Tanguep-Nijokep, H Mehrer, Solid State Ionics 177, 2 839 (20 06) 16 J.E Shelby, in: Treat on Mat Sci and Tech., Vol 17, M Tomozawa, R.E Doremus (Eds.), p.1, Academic Press, New York, 1979 17 G.H Frischat, J Amer Ceram Soc 52, 625 (1968) 18 G Brebec,... Sol 131 – 133 , 1109 (1991) R Kirchheim, J Non-Crystalline Solids 27 2, 84 (20 00) W Dieterich, P Maass, Chem Physics 28 4 439 (20 02) A Bunde, W Dieterich, P Maass, M Meyer, Ionic Transport in Disordered Materials, Ch 20 in: Diffusion in Condensed Matter – Methods, Materials, Models, P Heitjans, J K¨rger (Eds.), Springer-Verlag, 20 05 a Part VII Diffusion along High-Diffusivity Paths and in Nanomaterials 31 High-diffusivity... further typical aspects of mixed-alkali behaviour: 540 30 Diffusion and Ionic Conduction in Oxide Glasses Fig 30 .16 Composition dependence of the activation enthalpy for conductivity diffusion of Fig 30 .15 [29 ] Fig 30 .17 Composition dependence of 22 Na and 86 Rb diffusion in mixed 0 .2[ X Na2 O(1-X)Rb2 O]0.8 B2 O3 glasses according to Imre et al [14] Na diffusion: full symbols; Rb diffusion: open symbols 1.. .30 .4 Examples of Diffusion and Ionic Conduction 539 Fig 30 .15 Charge diffusion coefficient Dσ of mixed 0 .2 [X Na2 O (1-X)Rb2 O] 0.8 B2 O3 glasses according to Imre et al [29 ] rubidium borate glass system according to Imre et al. [29 ] Further examples can be found, e.g., in a paper by Gao and Cramer [31 ] If sodium ions are gradually replaced by rubidium ions the conductivity in Fig 30 .15 does... Am Ceram Soc 55, 186 (19 72) J.O Isard, J Non-Cryst Sol 1, 23 5 (1969) D.E Day, J Non-Cryst Sol 21 , 34 3 (1969) J.R Hendriksen, P.J Bray, Phys Chem Glasses 14, 43 and 107 (19 72) G Tomandl, H Schaeffer, J Non-Cryst Sol 73, 179 (1985) W.C LaCourse, J Non-Cryst Sol 95–96, 905 (1987) C.T Moynihan, A.V Lesikar, J Amer Ceram Soc 64, 40 (1981) M Tomozawa, V McGahay, J Non-Cryst Sol 128 , 48 (1991) A Bunde, M.D... Glasses 28 , 21 5 (1987) 8 V.M Goldschmidt, Geochemische Verteilungsgesetze der Elemente, Skr Nor Vidensk Akad K1, 1; Mat Naturvidensk K1, 8 7 (1 926 ) 9 W.J Zachariasen, J Am Ceram Soc 54, 38 41 (1 9 32 ) 10 B.E Warren, Z Kristallogr Mineral Petrogr 86, 34 9 (1 933 ) 11 D Uhlmann, N Kreidl, Viscous Flow and Relaxation, in: Glass Science and Technology, D Uhlmann, N Kreidl (Eds.), Academic Press, New York, 1985 12. .. diffusivity (Fig 30 .17) 2 The diffusivities of the two ions cross, when plotted as functions of the mixed-alkali composition The crossover occurs usually at non-equiatomic 30 .4 Examples of Diffusion and Ionic Conduction 541 Fig 30 .18 Composition dependence of the charge diffusion coefficient, Dσ , and of the mean tracer diffusion coefficient, D , of mixed Na-Rb borate glasses 0 .2[ X Na2 O (1-X)Rb2 O]0.8 B2 O3 compositions... crossover composition is almost independent of temperature 3 Similar observations were reported for Na-Cs silicate glasses by Terai [33 ] and Jain et al [34 ], for Na-K silicate glasses by Estropiev [35 ] and Fleming and Day [38 ], for Na-Rb germanate glasses by Estropiev [36 ], and for Na-Rb silicate glasses by McVay and Day [37 ] 4 One may define a hypothetical ‘mean tracer diffusion coefficient’ For a Na-Rb . Sol. 35 1, 38 16 (20 05) 29 . A.W. Imre, S. Voss, H. Mehrer, Phys. Chem. Chem. Phys. 4, 32 19 (20 02) 30 . S. Voss, F. Berkemeier, A.W. Imre, H. Mehrer, Z. Phys. Chem. 21 8, 135 3 (20 04) References 5 43 31 (1984) 21 . G.H. Frischat, Phys. Stat. Sol. 35 , K47 (1969) 22 . G.H. Frischat, Ber. Dt. Keram. Ges. 47, 23 8 (1970); and 47, 31 3 (1970) 23 . G.H. Frischat, J. Amer. Ceram. Soc. 53, 35 7 (1970) 24 . G.H York, 1985 12. A.Einstein,Z.f.Elektrochemie17, 23 5 (1908) 13. C.A. Angell, J. Non-Cryst. Solids 1 02, 20 5 (1988) 14. A.W. Imre, S. Voss, H. Mehrer, Defect and Diffusion Forum 23 7 24 0, 37 0 (20 05) 15.