Free volume anomalies in mixed-cation glasses revealed by positron annihilation lifetime spectroscopy (PALS)

PALS experiments reveal a minimum in ortho-positronium (o-Ps) lifetimes and a maximum in the corresponding intensities that emerge when mixed-cation (Li/Na) borate glasses are heated from ambient temperatures up to 473 K. These free volume ‘anomalies’ appear to be a true manifestation of the mixed alkali effect (MAE).

They are consistent with a mechanism of ion transport involving cooperation between hops of unlike cations, resulting in increased disturbance of the glass network.

The result lends support to the dynamic structure model.

The mixed alkali effect (MAE) remains one of the unsolved mysteries of glass science.

It involves a wide range of phenomena, including deep minima in ionic conductivities, crossovers in diffusion coefficient, additional (and large) internal friction peaks, and minima in glass transition temperatures, Tg.1–3

Although significant progress has been made in recent years towards understanding some of the underlying chemical and physical principles, there is still disagreement concerning how far ion-hopping processes in either single or mixed cation glasses involve the cooperation of the glassy network.

Bunde et al4,5. have described a ‘dynamic structure model’ that draws attention to the role played by cations in building their own sites, or ‘digging their own holes’, in the glass structure.

A mismatch arises when one type of cation, e.g. Li+, tries to enter a site created earlier by another, e.g. Na+, and this is identified as the prime cause of the mixed alkali effect.

The mismatch-opposed hops would be slower than the hopping of Li+ ions into Li sites, which is the dominant process in single-cation Li+ ion conducting glasses.

However, there are divergent views5 as to whether ‘mismatch-opposed’ processes ever actually occur in glass, and whether such ion hops can trigger inter-site conversion processes below Tg.

While there is indirect evidence that site conversions do occur below Tg,6,7 other authors have argued 8–10 that cations in mixed glasses simply avoid the ‘inappropriate’ sites and that the distribution of cation sites remains as it was frozen in at Tg.

This divergence of opinion highlights the limitations to our current understanding of processes occurring in glass.

It is especially unclear how far the network participates in the ion transport mechanism.

Thus, Angell11,12 pointed out many years ago that for common inorganic glasses (phosphates, borates, silicates, etc.), the ratio, Rτ of the times required for structural relaxation to occur at Tg (as indicated by the step function in heat capacity in a DSC experiment) to the average residence time of the ions (as indicated by the electric field relaxation time from impedance spectroscopy) is very large, with values of Rτ falling in the range 1011 to 1014.

This result suggests that network relaxation can have only little influence on ion transport.

On the other hand, Ingram et al13–15. have argued from the evidence of variable-pressure variable-temperature (VPVT) measurements that activation volumes, VA, for ion transport are conveniently interpreted in terms of the ‘opening up’ of empty cation sites, a process which obviously does involve the glassy network to some extent (see Fig. 1).

This figure shows schematically how a mobile cation hops from a site close to its compensating negative charge (shown here as a non-bridging oxygen (nBO)) into an empty site provided by the network.

VA is identified13–15 with the volume required to open up this second site.

What would clearly be helpful is independent evidence concerning the available volume and how it is distributed in both single and mixed-cation glasses.

An obvious technique to use is positron annihilation lifetime spectroscopy (PALS), which is widely used, for example, to explore cavity sizes in zeolites16 and ‘free volume’ in glassy polymers,17 and to understand ion transport in polymer electrolytes.18,19

The PALS experiment measures the lifetimes of positron species by detecting ‘start quanta’ of 1.28 MeV associated with their birth (from the decay of 22Na to 22Ne) and ‘stop quanta’ of 511 keV associated with their death (annihilation).

When a positron enters a material three separate processes may occur, each with a different lifetime.20,21

The shortest lifetime (100 to 200 ps) is attributed to the self-annihilation of a positron bound to an electron of opposite spin, i.e. para-positronium (p-Ps).

The second lifetime (200 to 600 ps) is usually attributed to the annihilation of ‘free’ positrons or to those trapped in the material.

The third lifetime (600 ps upwards) arises from the annihilation of the semi-stable species ortho-positronium (o-Ps), which is a positron paired with an electron with spins in parallel.

PALS spectra are composed of contributions from each annihilation event: these can be extracted by a non-linear least-squares fit of a weighted sum of exponentials:Here, τi denotes the mean lifetime of the positron state i, and Ii is the relative intensity of the lifetime component and n is the number of positron states.

Self-annihilation (p-Ps) gives little information on cavity characteristics.

However, the metastable ortho-positronium atoms, localised in cavities, have average lifetimes that depend on the cavity size.

The lifetime of o-Ps in a vacuum is 142 ns.

In condensed matter, lifetimes are shortened to a few ns since the annihilation rate depends on the probability of the overlap of the o-Ps wavefunction with the wavefunctions of the surrounding electrons, which is dependent on the cavity size.

The intensity of the o-Ps component is related to the o-Ps formation probability and reflects the concentration of cavities accessible to o-Ps.

When o-Ps annihilates, it is assumed to do so in a potential well where the o-Ps wavefunction has a given penetration depth, ΔR, into the material surrounding the potential well.

If the radius of the well is R (assuming a sphere), the o-Ps pickoff lifetime is usually related to the radius of the cavity by:23,24where τo-Ps denotes the o-Ps pickoff lifetime.

The 1/2 pre-factor is the inverse of the spin averaged Ps annihilation lifetime and ΔR has been found empirically to equal 0.166 nm.24

The volume of the sphere may then be calculated from R.

With this methodology, we are able now to compare the free-volume properties of single and mixed-cation glasses.

For our investigations we have chosen a mixed Li/Na borate glass system, which shows a conventional mixed alkali effect, which intensifies with decreasing temperature and moreover has been thoroughly characterized with respect to its structure and dynamics using broadband conductivity spectroscopy25–27 and NMR spectroscopy.28–30

Alkali borate glasses of composition 0.3[x Li2O·(1−x) Na2O]·0.7 B2O3 (with x = 0.0,0.2,0.4,0.6,0.8,1.0) were prepared from dry mixtures of stoichiometric amounts of reagent grade powder chemicals Na2CO3, Li2CO3 and B2O3 which were heated in a platinum crucible at 1273 K for about 3 h.

The melt was poured into a rectangular graphite mould and held inside the furnace for another 10 min.

Then the samples were removed from the mould, immediately annealed at 20 K below their respective glass transition temperatures for 3 h and then cooled to ambient temperature at a rate of 0.5 K min−1.

Two slices of more than 1 mm thickness were cut from the raw glasses.

Fig. 2 shows the dc conductivity of glassy 0.3[x Li2O·(1 − x) Na2O]·0.7 B2O3 at 373 K and features a pronounced minimum at x = 0.4.

The minimum in the DC conductivity coincides with the minimum of the glass transition temperature (see Fig. 2).

As the recent NMR results by Ratai et al28–30. show, these mixed alkali effects cannot be explained by ‘normal’ structural peculiarities.

Apparently, in all mixed-cation glasses of type 0.3[x Li2O·(1 − x) Na2O]·0.7 B2O3, the ratio N4 = [BO4]/([BO3] + [BO4]) remains the same.

Furthermore, the tetrahedrally coordinated [BO4]-groups and the trigonal planar coordinated [BO3]-groups as well as the cations are randomly distributed throughout the glass structure.

PALS measurements have been performed using an automated EG&G Ortec fast–fast coincidence system.

Two identical samples (>1 mm thick) of glass were placed on either side of a source consisting of a 2 mm diameter 25 μCi spot sandwiched between two 2.54 μm titanium foils.

The sample-source sandwich was placed between two mica-insulated heaters with a thermocouple placed close to the sample.

Temperature was controlled by a single variable voltage power supply attached to both heaters.

Data were collected at several temperatures and analysed using the PFPOSFIT program.

At least two spectra of 30 000 peak counts were collected, with each spectrum taking approximately 3 h to collect.

No source correction was used.

Statistical analysis resulted in spectra best fitted with two exponentials.

The first component is attributed to self-annihilation (p-Ps) and free positrons, and is not considered further.

The second component, with a lifetime ca. 800 ps is attributed to the lifetime of o-Ps localised in cavities, and is referred to as τo-Ps.

Error bars in the experimental data were determined from repeated experiments on the same sample.

These error bars are much larger than those determined from the fitting procedure.

Figs. 3–5 show how the o-Ps lifetimes and corresponding intensities for the Li–Na borate glasses vary with composition at three temperatures: 298 K, 373 K, and 473 K, respectively.

The sizes of the cavities for 473 K (see Fig. 4) are somewhat larger (ca. 6 cm3 mol−1) than those of the mobile cations (ca. 1.5 cm3 mol−1 for Li+, and 3.5 cm3 mol−1 for Na+, based on recent activation volume data13–15).

However, they are clearly smaller than the [BO3] and [BO4] units that constitute the network (between 15 and 20 cm3 mol−1, respectively, based on the numbers of O2− ions present).

The size of the cavities points to a glass system that is fairly dense, but not as close-packed as the ‘more ionic’ fluoride glasses studied earlier by Hill et al.21

At ambient temperature (298 K), there is no consistent dependence on composition in either the intensity, Io-Ps, or of the lifetime, τo-Ps, although there is some suggestion of a small ‘ripple’ in both quantities, see also .ref. 31

The ripple, moreover, is clearly evident at 373 K. It takes the form of a maximum in Io-Ps on the Li-rich side and a corresponding minimum in τo-Ps, and a maximum in τo-Ps and a corresponding minimum in Io-Ps on the Na-rich side.

Further heating of the sample to 473 K leads to a new and much simpler pattern of behaviour with a single maximum in positronium intensity and a corresponding minimum in positronium lifetimes.

Analysis of this temperature dependence requires careful consideration.

One must take into account, for example, the well-known ‘thermometer’ effect,1–3 which involves volume inelasticities in mixed alkali glasses that are observed far below Tg.

These delayed volume changes relate to different sites being occupied by mobile cations, and to the slowness (measured in days or weeks) of volume relaxations occurring in mixed cation glasses close to ambient temperature.

These glasses thus depart even further from internal thermal equilibrium than do single-cation glasses.

In other words, there are additional thermal history effects.

These relaxations might be expected to influence the distribution of free volume in mixed cation glasses, and so may well influence the PALS data collected at these (lower) temperatures.

Accordingly, we do not discuss further the low-temperature data, but focus instead on the pattern of behaviour observed at 473 K, which is some 200 K below Tg but high enough for the ‘thermometer effects’ to be avoided.

We base our interpretation of this ‘high temperature’ data on the recent discussions of trends in activation volume.13–15

Thus, in single cation glasses (Li or Na), we suggest that each ion migrates along clearly defined pathways where Li+ ions hop into empty Li-sites, and Na+ ions hop into empty Na-sites.

Thus, if a Na+ ion moves forward into a new site, it will leave behind an open Na-site, which will then immediately be reoccupied by a second Na+ ion.

This process may be represented schematically as:Na+(2) + Na+(1) + ‘new site’ ⇒ Na+(2) + ‘open site’ + Na+(1)Na+(2) + ‘open site’ ⇒ ‘open site’ + Na+(2)The essential idea is that the ‘success’ of the hop made by Na+ ion (1) will be facilitated by its original site being occupied by Na+ ion (2).

In mixed cation glasses, the ‘wrong’ cation—in this case it could be a Li+ ion—would fail to complete the second step in time, since it is forestalled by Na+ ion (1) moving more readily back into its starting site.

The original hop will then prove ‘unsuccessful’ in the sense used by Funke and coworkers.32,33

However, a second mechanism may come into play in mixed cation glasses, especially at higher temperatures (like 473 K).

This is represented schematically as:Li+ + Na+(1) + ‘new site’ ⇒ Li+ + ‘open site’ + Na+(1)Li+ + ‘second new site’ ⇒ ‘open site’ + Li+The additional feature is that the second ion, in this case it is the Li+ ion, has taken the opportunity to open up a new site of its own close to the site originally occupied by the Na+ ion.

This also has the effect of ‘pushing the Na+ ion forward’, and hence of securing the success of the first ion’s hop.

However, the interaction between cations and network is now much stronger than in single cation glasses, since two new sites are being opened up in the cooperative process that facilitates ion transport.

The resulting increased disturbance in the network results in an increased number of cavities, as is indeed indicated by the observed maximum in the Io-Ps data (see Fig. 5).

Although we cannot quantify the absolute number of voids existing in single and mixed cation glasses, we will argue that the increased network disturbance also leads to the observed decreases in Tg displayed in Fig. 2.

Thus we are able to establish for the first time a link between activation volume anomalies in mixed cation glasses,13–15 the changes in cavity distribution reported here, and the ‘thermodynamic’ anomalies in mixed cation glasses (see Fig. 2).

To this extent at least, the present results lend positive support to the general approach described in the dynamic structure model of glass.4,5

It appears that the structural dynamics become increasingly important with increasing temperature.

A particular feature of the approach being advanced here is the proposal for localised relaxations involved in the ‘opening up’ of closed sites.5,13–15

But how does such a process of site opening occur?

One might envisage in these borate glasses, for example, the participation of a bond-switching mechanism where there are negatively charged [BO4] groups embedded in a framework of [BO3] triangles, to generate neigbouring [BO3] triangles containing one nonbridging oxygen (nBO):

For this reaction to proceed, all that is needed is for an electron pair from one of the four B–O bonds in the [BO4] group to be transferred entirely to the O atom it shares with the neighbouring [BO3] group, and for the structure to relax slightly.

This process of chemical isomerisation clearly leads to a localised weakening of the network, and could occur simultaneously (and synergetically) with the arrival of a new cation.

The reaction above shows how ion hopping might be related to localised relaxation of the network.

Ion hopping depends on the space being available locally, so cavities of the right size are needed for this reaction to occur.

It is noteworthy that a recent molecular dynamics (MD) simulation of lithium borate glasses34 points to the importance of nonbridging oxygens (nBOs) in creating sites in glass favourable to Li+ ion transport.

However, the concentrations of such nBOs arising from the computer simulations is somewhat higher than that found by NMR.23–25

However, we might now postulate that even if these nBOs are produced only transiently in ‘real’ alkali borate glasses, they may still play a vital role in the opening up of sites – either to allow cations to escape from their existing sites or to enter new ones.

If this is indeed the process which makes use of free volume cavities in glass, then it provides a logical link between the presence of the cavities detected by PALS and the ion transport mechanism.

It may also provide the means of reconciling differing viewpoints concerning the numbers of nBOs in borate glasses (compare refs. 28–30 with ref. 35).

Clearly, more extended studies are required to establish links between ion transport processes and PALS data, especially if all the additional information can be extracted from the complex temperature dependences reported above.

The authors would like to thank Prof. T. J. Bastow (CSIRO) for experimental assistance, and Dr Kate Nairn and Prof. H. Eckert for helpful discussions.

The PALS experiments were carried out with funding from the Australian Research Council within the framework of the ARC Centres of Excellence program through the Centre for Nanostructured Electromaterials.

MDI thanks the Alexander von Humboldt Foundation for a Research Award.