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

For example,the permeability of soda-lime silicate glasses, depending on the temperature, pres-is two to four orders of magnitude lower than in pure vitreous silica see,e.g., [4].. It ha

Permeation of gases through glasses is of technological interest If the fusing species is a gas, it is possible to expose one face of a glass membrane

dif-(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 ing gas is continuously removed from the spectrometer side by pumping and

permeat-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

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 seriousconsequences for practical applications Helium can readily permeate throughmany 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 metallicstate and by the reaction with optically active defects Oxygen permeatingthrough the wall of an electric lamp can react with filament material, causingfailure of the bulb

Permeation rates of gases in vitreous silica are shown in Fig 30.5 Thedata indicate that the permeability decreases as the atomic or moleculardiameter of the diffusing species increases The permeability decreases in the

order He > H2 > Ne > N2 > O2 > 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 sures up to many atmospheres The effect of glass composition on heliumpermeation has been studied for a variety of oxide glasses, including silicate,borate, germanate, and phosphate compositions In general, He permeationdecreases in silicate glasses with increasing modifier content For example,the permeability of soda-lime silicate glasses, depending on the temperature,

pres-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 isthe most open-structured glass The modifier ions occupy ‘interstitial sites’

of the network, thus blocking diffusion paths for He atoms

In general, permeability data reflect the characteristics of ‘interstitialspaces’ in glasses However, care is needed when the permeating species re-acts with the glass network For example, H2 in vitreous silica reacts withthe silica network to form hydroxyls and silan groups [16] It has also beensuggested that water diffuses in vitreous silica as molecules reacting with thenetwork to form immobile hydroxyl ions.

30.4 Examples of Diffusion and Ionic Conduction

Below the glass-transition region, where the network structure is essentiallyrigid, self-diffusion of network formers is very slow In comparison, self-diffusion of modifier cations is faster as they can move through the ‘intersti-tial’ channels of the network Thus, it is the movement of modifier cationswhich 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 ofthe modifier cations (see [6]) In what follows, diffusion and ion conduction

of vitreous silica, soda-lime-glass, and borate glasses are used to illustratetypical features In addition, the so-called mixed-alkali effect is described

Vitreous Silica and Quartz: Vitreous silicon dioxide is an important

tech-nological material It is the most refractory glass in commercial use and hashigh 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 SiO2 according to Shelby [2]

lamps, optical fibers and dielectric films in microelectronic devices representimportant applications (see also Chap 28).

As already mentioned, vitreous SiO2 as well as quartz crystals containSiO4/2 tetrahedra, which are linked together in three dimensions In glassysilica, these tetrahedra form a random network, whereas in the crystal theyare linked in an ordered fashion Crystalline SiO2exists in many modifications

as temperature and pressure varies At ambient pressure trigonal low-quartztransforms around 575◦C to hexagonal high-quartz, which at 870◦C trans-

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 SiO2 may be of cial interest (Fig 30.6) Diffusion of 22Na in vitreous silica prepared fromquartz3 was studied between 170 and 1200◦C applying the residual activity

spe-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

dif-ferent types of vitreous silica with respect to preparation method, raw material,and impurity content These differences can lead to differences between the dif-fusion results of various groups For example, glassy silica prepared from natural

method [17] Diffusion of the stable isotope 30Si has been measured in thetemperature range 1110 to 1410◦C using SIMS [18] Self-diffusion of oxy-

gen was studied using a gas phase isotope exchange reaction [19] SIMS hasalso been used to profile the interdiffusion of network oxygen in a vitreous

Si18O2– Si16O2thin-film structure [20] The diffusivity values are lower, butwith a higher activation enthalpy (4.7 eV) than those reported in [19] andapproach the diffusivity of network oxygen uncomplicated by gas phase ex-change reactions Figure 30.6 confirms that diffusion of the network former

Si is very slow, whereas Na diffusion is relatively fast The Si activation thalpy 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 SiO4/2tetrahedronthe four half-bonds represent an energy of 5.8 eV The main barrier for themovement of Si atoms seems to be indeed the Si-O bond energy According tothis reasoning, one could expect for oxygen diffusion an activation enthalpy

en-of about half en-of that en-of Si diffusion One experimental value en-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 islikely that different diffusion mechanisms operate for oxygen and silicon.Figure 30.6 also shows 22Na diffusion in crystalline quartz parallel andperpendicular to its crystallographic axis [21–23] and 45Ca diffusion in onedirection [24] The transition between high- and low-quartz at 575◦C influ-

ences Na diffusion Since high-quartz has a hexagonal structure, which is ofhigher symmetry than the trigonal one of low-quartz, diffusion in high-quartzhas a lower activation enthalpy Figure 30.6 reveals a strong anisotropy of Nadiffusion in quartz as well Diffusion parallel to the axis is much faster thanperpendicular to it It is also remarkable that Na diffusion in vitreous silicalies between the Na diffusivities parallel and perpendicular to the crystallo-graphic axis of low- and of high-quartz Na and Ca have nearly the same ionicradii 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 minoradditions They usually contain about 10 to 20 mol % alkali oxides, primarily

in the form of Na2O, 5 to 15 mol % CaO and 70 to 75 mol % SiO2 Use ofdolomite as a source of CaO often implies that considerable MgO is alsopresent in the glass For special purposes some of the soda is replaced by

K2O or, less commonly, by Li2O Replacement of CaO and/or MgO by SrOand BaO occurs occasionally in the production of glasses

quartz crystals either by electric melting or by plasma sputtering in a H2and O2plasma reveal nearly the same Na diffusivites, whereas glasses synthesised fromSiCl4display a lower Na diffusivity presumably due to a distinctly lower content

of hydroxyl groups [5]

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 ∗ N a , 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]

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 modifiercations are introduced – by melting SiO2, Na2O, 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 TheNBO units are the anionic counterparts of the alkali- or alkaline-earth ions.The cations (Na+or Ca2+) are mainly incorporated at the severance sites ofthe network Usually, every alkali ion has a neighbouring NBO, while everyalkaline-earth ion has two neighbouring NBO units This structure providesstronger network linkage at the alkaline-earth sites Thus the divalent alkalineearth ions are less mobile than the monovalent alkali ions The replacement –

of alkali ions by alkaline-earth ions – reduces the ionic contributions to theelectrical conductivity and improves the chemical durability of the glass.Figure 30.8 and 30.9 illustrate mass transport properties in two sim-ilar soda-lime silicate glasses produced by the Deutsche GlastechnischeGesellschaft (DGG) as standard glass I and II for physical and chemicaltesting The composition of standard glass I (in mole fractions) is: 71.8 %

Fig 30.9 Tracer diffusivities, D N a ∗ , D ∗ Ca , and charge diffusion coefficient, D σ, ofsoda-lime silicate glass (standard glass II of DGG) according to Tanguep-Nijokep

[15]

SiO2, 14.52 % Na2O, 7.22 % CaO, 6.24 % MgO, and some minor additions.The composition of standard glass II is: 71.37 % SiO2, 13.19 % Na2O, 10.43 %CaO, 5.01 % MgO, and some minor addition Both glasses differ mainly intheir content of alkaline-earth oxides For standard glass I viscosity datafor the undercooled melt are available from measurements performed atthe Physikalisch Technische Bundesanstalt (Braunschweig, Germany) These

data have been used to calculate the viscosity diffusion coefficient, D η, fromthe 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 haviour Also shown are tracer diffusivities of22Na and45Ca and the charge

be-diffusion coefficient, Dσ, measured in the glassy state, which are all well

represented by Arrhenius relations At the glass-transition temperature, Cadiffusion is 6 orders of magnitude slower than Na diffusion and at lower tem-peratures the difference is even larger This confirms the expectation thatdivalent Ca ions have a much stronger linkage to the network than Na ions

In addition, this large difference together with the fact that conductivity fusion and Na tracer diffusion have the same activation enthalpy show thatthe electrical conductivity of soda-lime silicate glasses is due to the motion

dif-of Na ions

Figure 30.10 shows the Haven ratios, H R = D ∗

N a /D σ, of both standardglasses 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 experimentalerrors, indicating that the mechanism of Na diffusion does not change withtemperature

Alkali Borate Glasses: The structure of vitreous boric oxide (B2O3) fers considerably from that of vitreous silica Although boron occurs in tri-angular as well as tetrahedral coordination in crystalline compounds, onlythe triangular state is formed in vitreous boric oxide The BO3/2 units areconnected at all three corners via B-O-B bonds to form a network It isalso believed that vitreous boric oxide contains a certain amount of so-calledboroxol groups consisting of three boron-oxygen triangles joined together Incontrast to vitreous silica, the basic building block of the vitreous boron oxidenetwork is planar rather than three-dimensional A three-dimensional struc-ture is obtained by ‘crumpling’ the network Since the primary bonds existonly within a plane, bonds in a third dimension are weak and the structure iseasily disrupted One consequence of this weakly bound structure is the lowglass-transition temperature of vitreous boric oxide (about 260◦C), which is

dif-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 inthe formation of NBO units (see above), the effect of alkali-oxide addition toboric 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

configuration Formation of two boron-oxygen tetrahedra consumes the ditional oxygen provided by one alkali oxide molecule If alkali ions are intro-duced into the trigonally coordinated network of vitreous B2O3, tetrahedrallycoordinated BO−

ad-4/2 units are formed, which are the anionic counterparts ofthe alkali ions Each Na2O (or Rb2O) molecule creates two BO−

a result, glasses which contain significant concentrations of monovalent ionsare poor insulators, while glasses that are free of monovalent ions are ex-cellent insulators Figure 30.14 shows the effect of Li2O, Na2O, K2O, and

Fig 30.10 Haven ratios of soda-lime silicate glasses according to

Tanguep-Nijokep and Mehrer[15]

Rb2O additions on the conductivity of borate glass Whereas the tivity increases 5 to 6 orders of magnitude, the alkali content varies muchless This indicates that the mobility of ions increases significantly The lat-ter conclusion is supported by Na tracer diffusion studies, which show a verysimilar increase with Na2O content [30] In Fig 30.14 also a decrease of con-ductivity for corresponding glasses containing the same alkali concentrations

conduc-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

dimen-sions)

Fig 30.12 Glass-transition temperatures of alkali borate glasses according to

Berkemeier et al.[28]

Fig 30.13 Arrhenius diagram of the dc conductivity (times temperature) for Y

Na2O (1-Y)B2O3 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 teresting features of ionic conduction and diffusion in glass Figure 30.15shows as a typical example the conductivity diffusion coefficient of a sodium-

in-Fig 30.15 Charge diffusion coefficient D σ of mixed 0.2 [X Na2O (1-X)Rb2O] 0.8

B2O3 glasses according to Imre et al [29]

rubidium borate glass system according to Imre et al.[29] Further ples can be found, e.g., in a paper by Gao and Cramer [31] If sodium ionsare gradually replaced by rubidium ions the conductivity in Fig 30.15 doesnot follow a linear mixing rule between the end-members Instead, it passesthrough a deep minimum, which for this particular glass system is located

exam-near X = 0.4 Such conductivity minima are the best-known fingerprints of

the mixed-alkali effect, which has been observed for many other mixed-alkaliglasses The depth of the mixed-alkali minimum decreases with increasing

temperature As a consequence the activation enthalpy ∆H of Dσ passesthrough a maximum for an intermediate composition (Fig 30.16) Further-more, the mixed-alkali effect decreases with decreasing total alkali contentand vanishes for low total alkali contents [30]

In mixed-alkali silicate glasses containing 30 mol % alkali oxide the ductivity departs by as much as a factor of 103 to 106 from a linear mixingrule of the end compositions [32] Besides dramatic departures from linearity

con-in the conductivity, other transport properties of mixed-alkali glasses, such astracer diffusion, viscosity, and internal friction display characteristic features

of the mixed-alkali effect

Of particular interest is self-diffusion of the alkali ions in a mixed-alkaliglass Systematic studies of tracer self-diffusion in mixed-alkali glasses are rel-atively rare [8], because tracer studies are very laborious and time-consuming.Figure 30.17 shows the tracer diffusivities of 22Na and 86Rb in sodium-rubidium borate glasses [14] This figure reveals further typical aspects ofmixed-alkali behaviour:

Fig 30.16 Composition dependence of the activation enthalpy for conductivity

diffusion of Fig 30.15 [29]

Fig 30.17 Composition dependence of 22Na and 86Rb diffusion in mixed0.2[X Na2O(1-X)Rb2O]0.8 B2O3 glasses according to Imre et al [14] Na dif-

fusion: full symbols; Rb diffusion: open symbols

1 The tracer diffusivity of the majority ion is higher than that of the nority ion regardless of the size relationship of the ions On the otherhand, this difference is much more pronounced if the minority ion is thelarger ion The Rb diffusivity on the Na-rich side is about 4 orders ofmagnitude lower than the Na diffusivity (Fig 30.17)

mi-2 The diffusivities of the two ions cross, when plotted as functions of themixed-alkali composition The crossover occurs usually at non-equiatomic