Positive and Negative Mixed Glass Former Effects in Sodium

Article pubs.acs.org/JPCB

Positive and Negative Mixed Glass Former Effects in Sodium Borosilicate and Borophosphate Glasses Studied by 23Na NMR Michael Storek,† Mischa Adjei-Acheamfour,† Randilynn Christensen,‡ Steve W. Martin,‡ and Roland Böhmer*,† †

Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany Department of Materials Science and Engineering, Iowa State University, Ames, Iowa 50011, United States

‡

S Supporting Information *

ABSTRACT: Glasses with varying compositions of constituent network formers but constant mobile ion content can display minima or maxima in their ion transport which are known as the negative or the positive mixed glass former effect, MGFE, respectively. Various nuclear magnetic resonance (NMR) techniques are used to probe the ion hopping dynamics via the 23Na nucleus on the microscopic level, and the results are compared with those from conductivity spectroscopy, which are more sensitive to the macroscopic charge carrier mobility. In this way, the current work examines two series of sodium borosilicate and sodium borophosphate glasses that display positive and negative MGFEs, respectively, in the composition dependence of their Na+ ion conductivities at intermediate compositions of boron oxide substitution for silicon oxide and phosphorus oxide, respectively. A coherent theoretical analysis is performed for these glasses which jointly captures the results from measurements of spin relaxation and centraltransition line shapes. On this basis and including new information from 11B magicangle spinning NMR regarding the speciation in the sodium borosilicate glasses, a comparison is carried out with predictions from theoretical approaches, notably from the network unit trap model. This comparison yields detailed insights into how a variation of the boron oxide content and thus of either the population of silicon or phosphorus containing network-forming units with different charge-trapping capabilities leads to nonlinear changes of the microscopic transport properties.

1. INTRODUCTION Ionic conduction in glassy solid electrolytes has long been studied to understand the fundamental origins of the vast timescale decoupling of ionic hopping motion from the structural framework relaxation of solid electrolytes.1−4 At their glass transition temperature, this decoupling can be as large as 1013, and it gets even larger for temperatures below the glass transition.5 Glasses featuring high lithium or sodium ionic conductivities hold significant promise as new all-solid-state electrolytes,6 in particular in materials in which this gigantic decoupling can be fine-tuned by continuous compositional variation. One focus of scientific research in this area has recently been on a particularly promising set of glasses composed of at least two network glass formers. By mixing glass formers such as B2O3 with SiO2 or P2O5 at constant molar amount of (typically) modifying alkali oxide such as Li2O or Na2O (or alkali sulfide), significant nonlinear composition dependences have been observed in their electrical, thermal, chemical, and mechanical properties.7−17 In some cases, this so-called mixed glass former effect (MGFE), can produce advantageous changes (positive MGFE) in glass properties important to their application as solid electrolytes, such as in the Na2O + B2O3 + P2O5 (NBP) glass series,9 and this makes them a new and important class of materials. However, in other systems, such as © 2016 American Chemical Society

Na2O + SiO2 + B2O3 (NBS), compositional changes produce a negative MGFE. The current work is devoted to a direct comparison of the ion dynamics of the 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses as an example for a negative MGFE with 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glasses as an example for a positive MGFE using 23Na spin relaxometry and central-transition line shape analysis. The MGFE in the sodium borophosphate glasses was particularly well-studied in the past using an array of structural probes,7,12,15,18 while their ion dynamics was mostly examined using electrical conductivity measurements.7−9 In many solid electrolytes, dc conductivity measurements yield significantly deviating motional correlation times and activation energies from those determined using spin relaxation measurements.1,19−22 Such a deviation often provides hints regarding the relation of the local ion hopping processes (probed by NMR typically at MHz frequencies) and the long-range charge transport (probed by dc conductivity). Various theoretical and experimental approaches13,23−27 including a most recent site speciation and connectivity study28 have yielded detailed insights into how a preferential Received: January 15, 2016 Revised: April 19, 2016 Published: April 19, 2016 4482

DOI: 10.1021/acs.jpcb.6b00482 J. Phys. Chem. B 2016, 120, 4482−4495

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The Journal of Physical Chemistry B

were measured by means of an Xπ/2 pulse followed by a Y lockpulse of variable length tlock. Subsequently, the induced NMR signal was recorded after time intervals tlock ranging from 10 μs to 10 ms, and the relaxation decay was parametrized with equation 1. The strength of the RF lock field was determined by a T2ρ experiment for which the aforementioned Y lock-pulse was replaced by an X pulse.41 π/2-Δ-π central-transition spectra were obtained by Fourier transformation of the time signals starting from the echo maximum. The interpulse echo delay was Δ = 15 μs.

formation of B−O−P linkages facilitates the emergence of the positive MGFE in the NBP system.29 Nevertheless, in view of the potential differences among different experimental techniques, as referred to above, it is worthwhile to explore whether the positive conductivity effects can be confirmed by a local dynamic nuclear probe. Apart from a sodium thio-germanophosphate (NGP) series,30−32 relatively few glasses have been examined that display a clear-cut negative MGFE.33−35 For the NBS system, 22 Na tracer diffusion data exist in the literature.36 The energy barriers obtained from tracer diffusion measurements reveal a more or less linear compositional trend,36 calling for comparison with activation energies determined using electrical conductivity37 and NMR data, which we will furnish in the present work. Furthermore, the negative MGFE is only incompletely understood. For the NGP glasses, an interpretation in terms of a modified Anderson-Stuart model ascribes it to a decrease in the polarizability of the mixed glass former matrix in conjunction with an increase in the Na+ ion jump distance,31 while compositionally segregated nanodomains have been found to govern the impediment of the ionic conductivity in other systems.35 Moreover, crowding effects have been suggested to hamper ion transport in alkali borosilicate glasses.38 To shed light on some of these issues, in the present work we also report on so far unpublished structural NMR data obtained for the NBS glasses,37 with the goal to explore how much the compositional evolution of the short-range order units can tell us about the microscopic origin of the negative MGFE in this system.

3. NMR RESULTS Due to the large quadrupolar coupling constant of the 23Na nuclear probe (in the MHz range, see Appendix A) in the present samples, the sodium NMR spectra are too broad to allow for a simultaneous (nonselective) excitation of central and satellite lines. In Figure 1 we show temperature dependent

2. EXPERIMENTAL DETAILS Sodium borosilicate samples were provided by Corning Inc. and prepared by the melt-quenching technique as described in ref 37. The preparation of the sodium borophosphate samples is described in ref 9. For the NMR experiments, the glasses were ground to fine powders and filled into thin-walled quartz glass tubes which were then hermetically flame-sealed under vacuum. 23Na NMR experiments were performed using several spectrometers. For measurements at a Larmor frequency of ωL = 2π × 101.9 MHz, a home-built high-temperature probe operated at 300 K ≤ T ≤ 950 K was utilized.39 For low-temperature experiments, carried out at 132 K ≤ T ≤ 433 K, a spectrometer tuned to ωL = 2π × 79.7 MHz was equipped with a cryostat from CryoVac. Experiments in which the sample is spun at the magic angle at frequencies up to 18 kHz were run at room temperature and ωL = 2π × 158.6 MHz. For the various spectrometers, the pulse durations needed to generate a selective π flip of the central-line magnetization were approximately 3, 11, and 6 μs, respectively. Spin−lattice relaxation times were measured by means of an inversion pulse sequence followed by π flip of the magnetization to record a spin echo subsequent to the receiver’s dead time.40 Time constants were determined by fitting the longitudinal magnetization curves in terms of a Kohlrausch law ⎡ ⎛ ⎞1 − μ⎤ t M(t ) = M 0 + (M i − M 0)exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ T1 ⎠ ⎥⎦

Figure 1. Static 23Na-NMR spectra of 0.33Na2O + 0.67[0.3B2O3 + 0.7(2SiO2)] showing the narrowing of the central line with increasing temperature. The spectra were recorded employing a selective excitation and a Hahn-echo pulse sequence with an interpulse delay of Δ = 15 μs at a Larmor frequency of ωL = 2π × 101.9 MHz.

central (−1/2 ↔ + 1/2) transition 23Na spectra of the 0.33Na2O + 0.67[0.3B2O3 + 0.7(2SiO2)] glass. One clearly observes a motional narrowing of the central line with increasing temperature. Similar spectra and temperature trends were obtained for all 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] and 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glass samples studied in this work. From these temperature-dependent spectra, the full width at half-maximum (fwhm), δν, was determined, and the results are shown in Figure 2. For all samples, a step-like increase of δν from δν∞(x) at high temperatures (T ≈ 700 K) to δν0(x) at low temperatures (T ≈ 300 K) was observed. To highlight the opposite compositional trends in the silicon versus phosphorus containing glasses, in Figure 2 the dash-dotted lines connect the inflection points of the temperature dependent line widths. Measurements of spin−lattice relaxation rates, 1/T1, display even more pronounced compositional and temperature variations. This behavior is shown in Figure 3 for the 400 to 700 K range for both series of glasses. On the one hand, this plot includes data for the system 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] that displays a minimum rate, 1/T1, hence the slowest dynamics at intermediate concentrations. Such a behavior is expected for a negative MGFE. On the other

(1)

Here Mi and M0 denote the initial and equilibrium magnetization, respectively, and μ describes the deviations from singleexponential relaxation. Rotating frame relaxation times, T1ρ, 4483


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Figure 2. (a) Full widths δν of the 23Na central line for 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] with x = 0.0, 0.3, 0.5, 0.7, and 1.0 plotted against temperature. To avoid overlap, in each frame the data were shifted vertically by x times 10 kHz. The solid lines show δω(T)/2π according to equation 4 obtained from joint fits to the line-widths and the spin−lattice relaxation times, see section 4 for a description of the procedure. (b) Corresponding fwhm data and fits of 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5]. The fit for the glass with x = 0.2 is continued as dashed line. As guide to the eye, the dashed-dotted lines pass through the inflection points of the solid lines.

Figure 3. Compositional dependence of spin−lattice relaxation rates, 1/T1, as indicators for ionic motion in (a) 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses, (b) 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glasses, and (c) 0.5Na2O + 0.5[xB2O3 + (1 − x)P2O5] glasses. Data in (a) and (b) were recorded at ωL = 2π × 101.9 MHz, whereas for the data in (c), the measurements were performed at ωL = 2π × 79.7 MHz. The sodium borosilicate glasses clearly show a negative MGFE, while the sodium borophosphate glasses display a positive MGFE in both the low (y = 0.35) and the high (y = 0.50) Na2O series. The solid lines are drawn to guide the eye.

Here Ea denotes an activation energy, and τ0 is a preexponential factor. To keep the number of parameters small, we used τ0 = 10−13 s for all calculations. Because the distribution of correlation times in ion conducting glasses is typically very broad, we assumed that the activation energies entering equation 2 are distributed according to a Gaussian function

hand, for the sodium borophosphate glasses, 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] and 0.5Na2O + 0.5[xB2O3 + (1 − x)P2O5], spin−lattice relaxation is fastest, i.e., 1/T1 largest, in the x = 0.4...0.6 range, indicative of a positive MGFE in the ion dynamics. In Figure 4, we show the explicit temperature dependence of the T1 times for two typical examples: for 0.33Na2O + 0.67[0.3B2O3 + 0.7(2SiO2)] in panel (a) and for 0.35Na2O + 0.65[0.4B2O3 + 0.6P2O5] in panel (b). With decreasing temperatures, the T1 times increase showing that, in the examined temperature range, the ionic jump rates are smaller than the Larmor frequency. For the Si-containing samples, we also recorded rotating frame 23Na relaxation times, T1ρ, and Figure 4a contains the corresponding data. Overall T1ρ and T1 display similar temperature trends.

g (E ) =

(3)

Here, the standard deviation, σE, of the energy barrier distribution as well as its mean, Ea, are taken as free parameters. On the basis of these assumptions, we calculated the second moments, δω2, of the line-widths according to the selfconsistent expression42 δω 2(E) =

4. JOINT ANALYSIS OF SPECTRA AND SPIN−LATTICE RELAXATION TIMES In the following quantitative analysis of the NMR data, it was our goal to describe the central-line spectra and the spin−lattice relaxation times consistently using the same set of parameters for the motional correlation time scales, τ, and for their distribution. For the temperature dependence of τ, an Arrhenius law was assumed ⎛ E ⎞ τ = τ0exp⎜ a ⎟ ⎝ kBT ⎠

⎡ (E − E )2 ⎤ 1 a ⎥ exp⎢ − 2π σE 2σE2 ⎦ ⎣

⎡ 2 g (E)⎢δω∞2 + (δω02 − δω∞2 ) ⎣ π ⎤ × tan−1(τ(E) δω 2(E) )⎥dE ⎦

∫0

∞

δω02

(4)

In this equation, and δω∞ designate the low- and hightemperature second moments of the line-widths, respectively. Apart from a common factor, for simplicity we identify δω2, 2 δω02, and δω∞ with the squares of δν, δν0, and δν∞, respectively. For the latter two parameters, we used the composition-dependent values as determined in section 3 so that equation 4 does not add any free parameters.

(2) 4484

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and thus underestimates the mean relaxation rate 1/T1,c. Since in general the geometric mean is smaller than or equal to the arithmetic mean, this type of average is more appropriate for an average rate. Therefore, equation 5 was replaced by 2 δQ̃ 1 = T1, c 10

∫0

∞

g (E)[R1(0)R 2(0)]1/2 dE

(6)

Beyond these considerations, we note that the Lorentzian form of the spectral density functions, Jn(nω), introduced above implies that the corresponding correlation functions are exponential. However, due to the complex ion dynamics in many solid ion conductors, the associated correlation functions are in general not exponential.46−48 For example, the nonexponentiality of the lithium dynamics in LiPO3 glass has its origin in a distribution of jump rates and in correlated backand-forth jumps.49,50 In the absence of detailed knowledge regarding the microscopic correlation functions, however, we continue work here with the form given for Jn(nω) above. Using equations 2−6, a simultaneous fit of the spectra and the spin−lattice relaxation times was carried out for each glass. For the T1 times, temperatures between 450 K and the glass transition were considered. The resulting energy barriers Ea,NMR are shown in Figure 5 and are seen to cover a range from 0.7 to 0.9 eV. Their explicit compositional dependence will be discussed in section 5, below. The width of the energy barrier distribution turned out to be σE = 0.16 ± 0.01 eV for all samples. If the temperature dependence of the line widths was

Figure 4. Time constants of (a) 0.33Na2O + 0.67[0.3B2O3 + 0.7(2SiO2)] and (b) 0.35Na2O + 0.65[0.4B2O3 + 0.6P2O5] glasses. Spin−lattice relaxation times T1, measured at ωL = 2π × 101.9 MHz, are indicated by blue circles. Relaxation times, T1ρ, recorded under spin-lock conditions (ω1 ≈ 2π × 21 kHz) are represented by green triangles. Data points acquired above the glass transition temperature, Tg, are indicated by open symbols. For T > 450 K up to Tg, the blue solid lines represent simultaneous fits to the T1 and the line-width data, cf. the description in section 4. Outside the fitted temperature window, the fits are continued as dashed lines. The magenta diamond corresponds to a correlation time, τ1/2, estimated from the line-width, cf. Figure 2. The cyan dashed line represents the percolation energy barrier Ea,p, see text. Conductivity relaxation times τσ (red squares) were calculated from equation 9 using ε∞ = 9 ± 2.

For the description of the central-line spin−lattice relaxation rate in terms of 2 δQ̃ 1 = T1, c 20

∫0

∞

g (E)[R1(0) + R 2(0)]dE

(5)

we follow ref 43 so that also here no further fitting parameters are introduced. In this treatment, the effective quadrupolar 2 coupling constant, δQ̃ = δQ2 (1 + η2 /3), was determined independently as described in Appendix A. Furthermore, the (0) coefficients R(0) 1 = 2J1(ωL) and R2 = 2J2(2ωL) comprise the spectral density functions, Jn(nω) = 2τ/[1 + (nωτ)2], taken at multiples of the Larmor frequency ω = ωL. A note of caution is appropriate here because, for the derivation of equation 5, it is tacitly assumed that the magnetization recovery M(t) proceeds exponentially so that the definition of a rate 1/T1,c is straightforward. However, it is well-known that for quadrupolar I = 3/2 nuclei such as 23Na, the central-line recovery is described by a double exponential.44,45 To account for this behavior, we found that rather than (0) using the arithmetic mean of the rates, 1/2[R(0) 1 + R2 ], the (0) (0) 1/2 geometric mean, [R1 R2 ] , allows for a better parametrization of the theoretical magnetization curves because the arithmetic mean would overemphasize the larger rate, R(0) 1 ,

Figure 5. Activation energies of (a) 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] and (b) 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glasses from dc conductivity measurements Ea,σ (red circles), combined T1 and line-width data (Ea,NMR, black squares and Ea,p, blue diamonds), and activation energies calculated with Waugh and Fedin’s semiempirical formula (open squares), cf. equation 7. The solid and the dashed lines are guides to the eye. 22Na tracer diffusion data Ea,D from Wu and Dieckmann36 for the NBS glasses are indicated by open triangles. 4485


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The Journal of Physical Chemistry B solely fitted (and not that of T1), then σE = 0.10 ± 0.02 eV provides a good description (see the dashed lines in Figure 2). However, in the interest of a coherent description, σE = 0.16 eV was used and treated as a constant parameter in all subsequent fits. For the samples for which T1ρ times were measured in a second step, these were included in the fitting procedure. However, this inclusion did not alter the fitting parameters significantly. On the basis of the joint fits to the spectra and to the T1 times, the solid lines shown in Figure 2 and Figure 4 were calculated. These fits are seen to capture all of the essential features of the present data. The largest deviations between fit and data appear for the line-widths of the 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses for x = 0 and x = 1, cf. Figure 2a. Using the fixed prefactor (τ0 = 10−13 s) and the fitted mean energy barrier Ea, we calculated the correlation times from the corresponding Arrhenius laws, cf. equation 2, and we show the results for the examples of 0.33Na2O + 0.67[0.3B2O3 + 0.7(2SiO2)] and 0.35Na2O + 0.65[0.4B2O3 + 0.6P2O5] in Figure 4. As expected, in the slow motion regime we find that in the presence of a barrier distribution the slope in the Arrhenius plot of the correlation times is steeper than the slope of the relaxation times in the log10(T1/s) vs 1/T representation in the slow motion regime.51,52 A further consistency check was made using an empirical relation due to Bjorkstam et al.53 according to which a correlation time τ1/2 ≈ 0.3/δν1/2 can be determined from the inflection point of the temperature-dependent square-root of the line-width’s second moment. This quantity, δν1/2, refers to the temperature at which δν is (δν0 + δν∞)/2. For our samples, it turns out that all correlation times τ1/2 thus estimated are (10 ± 1) μs. Expectedly, they are in accord with the Arrhenius laws (for an example, see Figure 4b). Furthermore, for the 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glass series, the temperature at which τ = τ1/2 displays a maximum at x ≈ 0.7 is indicated by the dashed-dotted line in Figure 2a. Conversely, for the 0.35Na2O + 0.65[0.4B2O3 + 0.6P2O5] series, a minimum is observed at x ≈ 0.4, see Figure 2b.

The activation energies Ea,D determined by Wu and Dieckmann from sodium tracer diffusion for 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses are comparable in magnitude and compatible with a linear compositional dependence.36 As a result, these authors ascribed the minimum they observed in the tracer diffusion coefficient at x ≈ 0.7 for temperatures in the range from 480 to 780 K to compositional changes of the pre-exponential factor. On the other hand, a nonlinear x dependence is also shown by the energy barriers Eσ from conductivity measurements for the sodium borophosphate glasses from ref 9 and for the sodium borosilicates from ref 37, which are included in Figure 5 as well. The observation that E σ is consistently smaller than E a,NMR is quite common10,55−60 and can be understood in terms of a percolation model. According to this approach, the energy barriers larger than a threshold barrier Ea,p do not contribute to the dc conductivity σdc. Using the distribution g(E), cf. equation 3, the barrier Ea,p can be estimated from the percolation threshold pc =

Ea,p

g (E)dE

(8)

which, in turn, depends on the coordination number z. For silicate glasses, chemical shift measurements suggest that z ≈ 7.61 From pc ≈ 1.45/z ≈ 0.207 (ref 62) the evaluation of the integral in equation 8 yields Ea,p ≈ Ea,NMR − 0.82σE. Using σE = 0.16 eV, as determined above, we see from Figure 5 that Ea,p describes Eσ well. A direct check is also possible in terms of the conductivity relaxation time63 τσ = ε0ε∞/σdc

(9)

The good agreement of the resulting time constants, as obtained using ε∞ = 9 ± 2 and τ0 = 10−13±1 s with the experimental data, cf. Figure 4, underscores the applicability of this approach to the glasses under investigation. 5.2. Mixed Glass Former Effect. The origins of the MGFE have been proposed to trace back to microstructural and topological changes at the short-range level.8,23,18,64,65 On this microscopic level, the Na+ mobility is strongly influenced by the presence and the linkages of the network-forming units (NFUs). For instance, nonbridging oxygens at the NFUs cause traps in the Coulombic potential leading to an increase in the activation energy hindering the ionic motions. Corresponding considerations regarding the interaction energy of the Na+ ions with respect to the NFUs present in a glass served as a basis for the network unit trap (NUT) model by Schuch et al.23 Although the model is capable of explaining the enhanced ion mobility in various alkali borophosphate glasses at least semiqualitatively, see section 5.2.3 below, analogous considerations fail to describe the compositional trends observed in the sodium borosilicate glasses.66 In order to be able to rationalize the ion transport properties as revealed by NMR and other techniques, on any structural basis, in section 5.2.1, we will focus on the compositional changes of the microstructure in these borosilicate glasses. Then, in section 5.2.2, we will discuss the origin of the negative MGFE in the sodium borosilicates. Finally, section 5.2.3 deals with the ionic motion in sodium borophosphate glasses and compares the dynamical information gathered in this work with pending structural data and model considerations. 5.2.1. Site Speciation in Sodium Borosilicate Glasses: NMR Data and Models. To prepare the ground for a better understanding of the negative MGFE in the sodium borosilicate

5. DISCUSSION 5.1. Activation Energies from NMR and Comparison with Measurements of Conductivity and Tracer Diffusion. Figure 5 summarizes the composition-dependent mean activation energies Ea,NMR determined from the NMR data according to the procedure described in section 4. One clearly recognizes a maximum (at x ≈ 0.7) for 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] as expected for a negative MGFE and a minimum (at x ≈ 0.4) for 0.5Na2O + 0.5[xB2O3 + (1 − x)P2O5] as expected for a positive MGFE. Also included in Figure 5 are estimates for the motional activation energies based on the relationship Ea,WF(meV) = 1.62 × Tonset(K)

∫0

(7)

due to Waugh and Fedin.54 Here, Tonset denotes the temperature at which the motional narrowing of the central transition commences upon cooling. In Figure 5, it is clearly seen that Ea,WF displays the same trends but is lower than the energies determined from other NMR quantities, a deviation that is known to occur in the presence of a distribution of energy barriers.55 4486


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The Journal of Physical Chemistry B system, let us first present the composition dependent evolution of the NFUs. In Figure 6a, we show the percentages

NFUs contain only bridging oxygens. Since each B(4) unit is singly charged, it compensates the charge of one Na+ ion. Yet, owing to B(4)−O−B(4) linkage avoidance the number of B(4) units is limited.74 This prevents the fraction of B(4) units to become larger than the fraction of B(3) units in the pure borate glass (x = 1).75,76 However, for glasses with moderate to large silica contents it was shown68,70 that a considerable fraction of B(4) units exists in reedmagnerite-like, 1/2(Na2O·B2O3·6SiO2), and danburite-like, 1/2(Na2O·B2O3·2SiO2), structures. Therefore, due to the presence of B(4)−O−Si(4) linkages, in the present mixed glasses the B(4) fraction is considerably larger than expected from a linear compositional dependence. Within the framework of the Yun and Bray model,68,69 the alkali ratios R and K of these glasses correspond to an almost maximized deviation from a linearly increasing B(4) fraction.77−79 Concomitantly, as obvious from Figure 6, for all x, the Si(3) fraction is less than expected from a linear dependence. This finding suggests that the Na+ ions are preferentially associated with the borate network, while the silicate network possesses less ions than predicted by an equal cation sharing. For the binary silicate and binary borate glasses, this finding of a disproportionation is corroborated by site-resolved 23Na, 11B, 29 Si, and 17O NMR studies.80,81 Interestingly, the NGP glasses,31 which also exhibit a negative MGFE, display a similar disproportionate charge sharing among the thio-germanate and thio-phosphate networks.82 The fact that the Yun and Bray model captures all essential features of the composition dependent NFU fractions derived from the11B MAS NMR data,37 cf. Figure 6, is nontrivial because occasionally other circumstances interfere. For instance, Raman scattering as well as 17O, 23Na, and 29Si NMR experiments indicated that for some alkali borosilicate glasses, the distribution of their structural units can depend on the thermal history and can be less ordered than predicted.83−85 5.2.2. Possible Origins of the Negative MGFE in Sodium Borosilicate Glasses. By comparing the cation distribution in binary sodium silicate vs sodium borate glasses at constant Na+ concentration, it has been observed that a lack of cation clustering (in the borate glasses) can impede the ion transport.86 To check whether such an impediment of charge transport is contributing to the negative MGFE also in the present ternary NBS glasses, it is of interest to investigate the spatial arrangements of the Na+ cations. It has been demonstrated that such information can be obtained from spin−echo decay spectroscopy87 by analyzing the dependence of the low-temperature homonuclear dipolar second moments δω20,Na−Na (which reflect the distribution of Na−Na distances) on the number density NNa. These experiments revealed that in sodium silicate glasses, sodium clustering appears close to the Si(3) units,80,88 while in borate glasses, the Na+ ions arrange themselves in a more or less random fashion.89 Therefore, we carried out 23Na spin echo decay spectroscopy experiments at T ≤ 200 K for the NBS glasses studied here, see the results presented as Supporting Information. For the binary sodium silicate glass, x = 0.0, we find that δω20,Na−Na ≈ 5 × 106 rad2/s2 in accord with results by Eckert.80 When increasing the borate fraction to x ≈ 0.5, the dipolar second moments δω20,Na−Na decrease by roughly a factor of 2 to below the value reported80 for the x = 1 glass, despite the fact that simultaneously the cation number density NNa increases by a factor of ∼1.23.36,37 Hence, in agreement with the findings in ref 86, these results rationalize the decrease of the dc ionic conductivity for increasing x. For x ≥ 0.5, our data (which for

Figure 6. (a) NFU speciation as obtained from a charge-balanced short-range-order model based on unpublished 11B MAS NMR data.37 (b) Fractions of network-forming units (NFUs) calculated by means of the Yun and Bray model using the chemical analysis provided as Supporting Information. The trends from the NMR data (panel a) and the model considerations (panel b) are in good agreement. In both frames, the lines are guides to the eye.

of the B(4), B(3), B(2), Si(4), and Si(3) units for the present 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses where the superscript indicates the number of bridging oxygens attached to each NFU. The speciation depicted in Figure 6b was obtained from 11B magic-angle spinning (MAS) NMR data37 by assuming local charge balance.67 For a quantitative understanding of the compositional trends observed in Figure 6a, we applied the model by Yun and Bray.68,69 This model and its recent refinements (see ref 70 for a review) was found to be useful for describing numerous 11B NMR data.71 Using the ratios R = [Na2O]/[B2O3] and K = [SiO2]/[B2O3] as given for the present 0.33Na2O + 0.67[xB 2 O 3 + (1 − x)2SiO 2 ] system in the Supporting Information,72 we calculated the percentages of B(4), B(3), Si(4), and Si(3) species depicted in Figure 6b, which compare favorably with the experimental ones shown in Figure 6a. In the following we describe the evolution of the microstructural changes that emerge as the borate content of the NBS glasses increases. Starting from x = 0, the binary sodium silicate glass is formed by tetrahedrally oxygen coordinated Si(3) and Si(4) units. While the net charge of a Si(4) unit is zero, each Si(3) charge balances one Na+, and therefore, the number of Si(3) units equals the number of ions in the pure sodium silicate (x = 0) glass. Then, upon progressive substitution of 2SiO2 by B2O3, the fraction of Si(3) decreases and tetrahedrally coordinated B(4) and trigonal B(3) borate groups increasingly appear.73 As long as the alkali content is small, B(4) and B(3) 4487


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the present glasses, an increase of the borate fraction x leads to a P(2) → B(4) conversion in which the strongly Na+ trapping P(2) units are successively replaced by the charge-dispersing and thus less cation attracting B(4) groups. This causes a steep decrease of the overall trapping power for x < x* = 1/2 − y/ (6−6y).23 Comparing this with our y = 0.35 NBP glass data, the activation energies summarized in Figure 5b decrease steeply with x and reach a minimum at x ≈ 0.4. Although this behavior reveals a striking similarity with the NUT model prediction (with x* ≈ 0.41), it does not account for the increase in activation energies for x > 0.4 evident from Ea,NMR in Figure 5b. An enhanced understanding of the observed optimal charge transport at x ≈ 0.4 was advanced by a study reporting that at this composition the fraction of heteroatomic B−O−P linkages is maximum.7,13 These linkages induce a partial shift of the B(4) charge to the formally neutral P(3) groups thereby further facilitating the charge delocalization on the B(4) groups. Experimental evidence for such a charge dispersion was recently derived from 11B{23Na} and 31P{23Na} REDOR NMR experiments.28 Thus, the decreasing number of heteroatomic linkages explains the increase of Ea for x > 0.4 which is also borne out by the spin−lattice relaxation rates summarized in Figure 3b. A close inspection of the x dependent relaxation rates reveals an additional shoulder in the borate-rich region for y = 0.35 which is not reflected by the compositional evolution of the B− O−P linkages.7,13 Our y = 0.5 glass data, cf. Figure 3c, suggest that for increasing alkali concentration this shoulder evolves into a maximum. Again, this observation can be rationalized in terms of the NUT model as described in Appendix C: This model predicts that for larger y the number of trapping efficient P(2) groups becomes larger as well and concomitantly a significant number of P(2) units continues to exist up to increasingly higher boron oxide concentrations x. These effects are directly reflected in the evolution of the spin relaxation rates, cf. Figure 3b for y = 0.35 and Figure 3c for y = 0.5, which show maximum rates at x ≈ 0.4 and x ≈ 0.6, respectively. Accordingly, for an intermediate alkali concentration of y = 0.4, the activation energies and mean square displacements of mobile ions deviate most significantly from the linear x dependence at x ≈ 0.4 and at x ≈ 0.8,7,8 thus rationalizing so far not fully understood results from literature.

the borate-rich glasses, x > 0.5, display significant uncertainty) and in particular those of Eckert,80 suggest that a minimum in δω20,Na−Na exists near x ≈ 0.5, see the Supporting Information. The corresponding minimum in clustering tendency at this composition is in harmony with the overall conductivity reduction and hence the activation energy maximum observed between the two end-member glasses. So far, the discussion of the ion motion has mostly focused on the Coulombic interaction of the cations with the charged NFUs. However, the activation barrier hindering the Na+ motions may also involve contributions originating from the mechanical strain energy that is required to temporarily deform the local environment of an ion to open a doorway for diffusion. In terms of the Anderson and Stuart model,90 the strain energy depends on the shear modulus G, the ion jump distance, and on the squared difference between the doorway radius and the ionic radius.91 In alkali borophosphate glasses, G is ≈2 GPa and thus relatively small, so that the strain energy contributes typically less than 10% to Ea, a percentage that does not significantly alter the x dependent trends induced by the Coulombic trapping.9 In our series of NBS glasses, however, the shear moduli of the ternary glasses can be more than an order of magnitude larger, e.g. G(x = 0.5) ≈ 31 GPa,92 so that the strain energy may contribute appreciably to their total activation barrier. This effect appears to be weaker for the binary sodium silicate glasses where G ≈ 28 GPa for x = 0.0 and G ≈ 20 to 24 GPa for x = 1.0.90,92,93 For intermediate values of x, the strain energy is increased further because the density of the ternary NBS glasses94 is up to about 7% larger than for the two binary end members.36 This should lead to a denser packing of the mixed glasses and, thus, a reduced doorway radius. Overall, these considerations provide a qualitative rationalization for the negative MGFE observed in the NBS glasses. The extent to which nanophase separationknown to occur in alkali free borosilicates that segregate into a borate glass phase and a mixed borosilicate phase95hampers the cation transport at intermediate compositions further, has to await future NMR experiments of the type presented for borophosphate glasses in ref 28. 5.2.3. Sodium Borophosphate Glasses. The structural origins of the increased ionic conductivity of ternary NBP glasses as compared to that in the binary phosphate and borate glasses has been investigated using Raman, XPS, X-ray, IR, and NMR techniques for various fixed modifier concentrations, e.g., for y = 0.33,10,13 0.35,9,12,18 0.4,7,8 0.45,24 0.5,96 and y = 0.3 to 0.6.97 On the basis of these results and on those from reverse Monte Carlo investigations,25 Schuch et al. developed an approach to explain increased ion mobilities at intermediate boron concentrations in terms of a network unit trap (NUT) model,16,98 which we will review very briefly below in order to facilitate comparison with the present experimental results. In analogy to the sodium borosilicate glasses, in the sodium borophosphate glasses trigonal B(2), B(3), and tetrahedral B(4) units mix with tetrahedral P(0), P(1), P(2), and P(3) units. The respective formal charges of the NFUs are −1, 0, and −1 for the boron units and −3, −2, −1, and 0 for the phosphorus units. Within the NUT model, a trapping power is assigned to each NFU as its charge divided by the number of its nonbridging oxygens. Here, the B(4) units are exceptional in that they do not possess any nonbridging oxygens. Schuch et al. proposed to formally distribute their charge of −1 uniformly among the four bridging oxygens.16 For 0.3 < y ≤ 0.5, which is of interest for

6. SUMMARY To explore the details of their ionic mobility, we applied 23Na NMR spectroscopy to study several series of MGFE glasses. The examined systems feature either positive or negative deviations from linear compositional trends in their charge transport properties. The NMR experiments include spin− lattice relaxation time determinations, including those measured in the laboratory frame and in the rotating frame, and measurements of central-transition line shapes. To be able to properly analyze these data, MAS NMR experiments were performed to determine the quadrupolar coupling constants of these glasses. Furthermore, to assess the rotating frame relaxation times properly, the appropriate coherence transfer amplitudes were computed under the experimentally relevant spin-locking conditions. The measured temperature dependent spin relaxation times and spectral line widths were subjected to a simultaneous analysis with the goal to obtain a reliable theoretical description for each glass former system with the mean activation energy Ea as the sole individually adjustable parameter. From this analysis, 4488


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The Journal of Physical Chemistry B a composition dependent minimum was observed in Ea(x) for 0.35Na2O + 0.65[xB2O3 + (1 − x)P2O5] glasses and a maximum for 0.33Na2O + 0.67[xB2O3 + (1 − x)2SiO2] glasses. The Ea values from the present NMR measurements were consistently larger than those from dc conductivity measurements and from tracer diffusion (for the silicate system), a finding that was quantitatively explained using a percolation approach. Furthermore, the compositional evolution of the various structural units in the NBS glass network was studied. We demonstrated that the Yun and Bray model explains the composition dependence of the NFU populations as measured using 11B NMR. Investigation of the composition-dependent cation distributions revealed that the addition of B2O3 mitigates the ion clustering most for NBS glasses with intermediate mixing ratio. Furthermore, the negative MGFE in the NBS glass series can be rationalized with reference to a substantial composition-dependent contribution of their strain energy. Additional experiments are, however, required to quantify the role of a possible nanophase separation that could further impede the cation transport at intermediate compositions. For the present NBP glass series, the overall ion trapping power was assessed in terms of the NUT model that has proven useful to understand the charge transport behavior of several mixed glass former systems. The model, with additional insights from recent studies concerning the preferred formation of B− O−P linkages, provides a suitable framework which enabled a coherent understanding of the results from the dynamic NMR experiments carried out in the present work.

Figure 7. MAS and static NMR spectra of 0.33Na2O + 0.67[0.6B2O3 + 0.4(2SiO2)] NBS glass at room temperature. The MAS NMR spectrum was recorded for a rotation frequency of νR = 18 kHz. The dashed line is the calculated spectrum obtained by convoluting a Gaussian distribution of chemical shifts with a standard deviation 2 ⟨σCS ⟩ ≈ 7.56 ppm, and a distribution GCzjzek(ν) according to equation A.1 for the parameters of the quadrupole interaction. For the 2 latter, σCz is 2π × 800 kHz which implies ⟨δQ̃ ⟩ ≈ 2π × 1.78 MHz.

■

Frequencies are given relative to the resonance position of polycrystalline NaCl.

APPENDIX A. DETERMINATION OF QUADRUPOLAR COUPLING CONSTANTS The 23Na NMR line-shapes and spin−lattice relaxation times are dominated by the strong quadrupolar interaction of the 23 Na nucleus (I = 3/2) with its charge environment. In silicate glasses, the quadrupolar anisotropy parameter δQ = 3e2qQ/ [2I(2I − 1)ℏ] = 1/2(e2qQ/ℏ) of this nucleus is typically larger than 600 kHz so that usually only the central line of its NMR spectrum can be excited and observed.59 The width of the 23Na central line is affected by second-order quadrupolar, chemical shift, and dipolar broadening effects whichdepending on the Larmor frequencymay all appear with similar magnitudes.99 For the present work, δQ was determined by comparing the central lines of static spectra with those recorded under MAS conditions at 158.6 MHz. An example is shown in Figure 7 for the 0.33Na2O + 0.67[0.6B2O3 + 0.4(2SiO2)] glass. The static spectrum displays a broad asymmetric absorption with an fwhm of δν0 ≈ 8.2 kHz. The MAS NMR spectra recorded for rotor frequencies of up to νR = 18.0 kHz comprise two components: (i) an almost symmetric, only slightly skewed, component centered at −2π × 3.4 kHz or −21.6 ppm which for νR > 8 kHz is δν18 kHz ≈ 3.8 kHz wide so that δν0/δν18 kHz ≈ 2.2 and (ii) a narrow line (∼90 Hz wide for νR > 2 kHz) at 2π × 19 Hz or +0.2 ppm with respect to the reference substance (polycrystalline NaCl). At the largest rotor frequencies, the integrated intensity of this line is 4.9 times smaller than the line at −21.6 ppm. The small width of this line suggests that it might be due to a crystalline contamination, but currently, we are unable to specify its origin more precisely. The broader line in the MAS NMR spectrum is dominated by isotropic chemical shifts δCS and quadrupolar induced shifts, η2 1 3 δQIS = − 30ω 1 + 3 ⎡⎣I(I + 1) − 4 ⎤⎦δQ2 ,100 originating from L

(

the second-order quadrupolar interaction. In glasses, these shifts are typically subject to symmetric (for δCS) or asymmetric (for δQIS) distributions.101 For the chemical shift, a Gaussian 2 ⟩ and for the quadrupolar distribution99 of width ⟨σCS interaction a bivariate random distribution of anisotropy and asymmetry parameters was assumed. According to the Czjzek model,102,103 the latter assumption leads to a distribution

GCzjzek (δQ , η , σCz) =

4 2 η2 ⎞⎤⎥ η2 ⎞ ⎡⎢ δQ ⎛ 2 δQ η ⎛ − − + 1 exp 1 ⎟ ⎜ ⎟ ⎜ 5 2 π σCz 9 ⎠ ⎢⎣ 2σCz 3 ⎠⎥⎦ ⎝ ⎝

(A.1)

Here σCz[2I(2I − 1)]/3 = 2σCz is the common Gaussian standard deviation of all three principal axes values of the EFG tensor induced by the static disorder of the Na environment. For the GCzjzek distribution, the mean anisotropy and a s y m m e t r y p a r a m e t e r s a r e ⟨δQ ⟩ = 5σCz / 2π a n d ⟨η⟩ = 2 3 −

3 2

3 ln(3) ≈ 0.61 and, for the mean squared

anisotropy parameter, one finds ⟨δQ2 ⟩ = 7 −

(

2

⟨δQ̃ ⟩ =

(

δQ2 1 +

2

η 3

)

3 2

)

2 3 σCz and

2 = 5σCz , respectively.104 In conjunc-

tion with the orientational dependences of the quadrupolar shift of the central line,105 the second moments of the static and the MAS NMR spectrum are given within the Czjzek model by ⟨M 2,0⟩ =

)

4489

5 ⎡ 3⎤ 4 I(I + 1) − ⎥σCz 2⎢ ⎣ 4⎦ 36ωL

(A.2a)


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pulse for a time interval t1 = tlock. Then, the signal s1(t1,t2) evolving during the subsequent acquisition time t2 was computed. Analogously, in the present work a Y pulse is considered which irradiates the central line selectively and thus generates a superposition of normalized spherical tensor operators Tlm (of rank l and order m, and Tr[TlmT†l′m′] = δll′ δmm′) that can be compactly written using the arrow notation as

and ⟨M 2,MAS⟩ =

1 ⎡ 3⎤ 4 I(I + 1) − ⎥σCz 2⎢ 4⎦ 48ωL ⎣

(A.2b)

Hence, for disordered systems characterized by the distribution of the Czjzek model, the ratio of static and MAS NMR line widths is ⟨M 2,0⟩/⟨M 2,MAS⟩ = 20/3 ≈ 2.582 . 106 This expectation can be compared to the fwhm ratios for x = 0.4 (9.08 kHz/3.96 kHz ≈ 2.29) and x = 0.6 (8.18 kHz/3.82 kHz ≈ 2.14), which are only slightly smaller than predicted by this simple approach, although chemical shift anisotropies and magnetic dipole−dipole interactions will also contribute to the static spectra. A quantitative description of the MAS NMR line shape was achieved by convoluting the Gaussian distribution of the isotropic chemical shift with GCzjzek(ν). Figure 7 shows that this approach allows for a good description of the spectral shape and from the best-fit the following parameters were obtained.

T10 =

∂ρ = −i[HS , ρ] + R(ρ) ∂t

HS =

(B.3)

(a) 5 ω1T11 + ωQ T20

(B.4)

in which the first contribution corresponds to the radio frequency pulse and the second term denotes the time-averaged part of the quadrupolar Hamiltonian. With the spatial EFG tensor components F2m, the relaxation contribution due to the fluctuating part of the quadrupolar interaction

spectra using Czjzek modeling often turn out to be systematically larger by a factor of up to 2 than values obtained from multiple quantum MAS NMR, a finding ascribed to a lack of efficient triple-quantum excitation in many of these experiments.104,107

HQF(t ) =

eQ ℏ

2

∑

( −1)m T2mexp(imωLt )

m =−2

× [F2 − m(t ) − ⟨F2 − m⟩]

■

(B.5)

is given in the framework of second-order perturbation theory by the Redfield superoperator110

APPENDIX B. ROTATING FRAME RELAXATION TIMES Analogous to the treatment of the spin−lattice relaxation in the Zeeman field, see equations (5) and (6) in section 4, the spin relaxation time in the rotating frame after selective excitation of the central line, T1ρ,c, can be described using

R (ρ ) = −

∫0

∞

⟨[HQF(t ), [exp( −iHSτ )HQF(t − τ )

× exp(iHSτ ), ρ(t )]]⟩dτ

(B.6)

As detailed in ref 108, the operator HS can be neglected in R(ρ) when dealing with the |m| = 1 and 2 terms in HQF. In other words, only the evolution of the terms pertaining to the spin operator T20 have to be taken into account explicitly when calculating HQF(t − τ). The set of equations defined by equation B.3 through equation B.6 cannot be solved in analytical form, yet the eigenvectors of the nonrelaxing system, i.e., for R(ρ) = 0, are still good eigenvectors if R(ρ) is small. This property can be exploited to decouple the satellite contributions from the central-transition subspace. As only the central transition is locked by a continuous radio frequency pulse, the analytical diagonalization of its subspace yields the relevant evolution under spin-locking conditions with the rates RS, RF+, and RF− and the eigenvectors given in equation 18 of ref 108. The coherences appearing in equation B.2 evolve into111

∞

g (E)R1ρ , c dE

(B.2)

under the (in the rotating frame) static Hamiltonian

MHz. For the x = 0.6 NBS glass, the parameters are given in the caption of Figure 7 along with the MAS NMR spectrum calculated therefrom. It is noted that the mean quadrupolar 2 coupling constants ⟨δQ̃ ⟩ derived from one-dimensional NMR

∫0

3 (a) T31 2

Under a spin lock pulse of duration t1 (which is phase shifted by 90° with respect to the first pulse), the density operator ρ evolves according to the master equation109

2 ⟩ ≈ 7.28 ppm and σCz = 2π × For the x = 0.4 NBS glass, ⟨σCS 830 kHz are found. The latter implies that a mean quadrupolar 2 coupling constant of ⟨δQ̃ ⟩ = σCz / 5 = σCz ≈ 2π × 1.85

2 δQ̃ 1 = T1ρ , c 10

1 1 1 (a) 1 Iz → Ix , c = − T11 + 5 2 5 5 5

(B.1)

In ref 108, explicit expressions were reported for the spinlocking rate R1ρ subsequent to nonselective excitation. However, we are not aware of published expressions that refer to the present case of locking after selective excitation of the central line. In the following, it is shown that the corresponding rate, R1ρ,c, can be written as R1ρ,c = N[AS,cRS + AF+,cRF+ + AF−,cRF−], with N = 1/(AS,c + AF+,c + AF−,c). Here, the indices S, F+, and F− correspond to contributions from different (combinations of) spectral densities as described below. As in the case of the spin−lattice relaxation time in the laboratory frame, the geometric mean of the rotating frame analogue, [(RS)AS,c (RF+)AF+,c (RF−)AF−,c]N, reproduces the shape of the magnetization curve more accurately than an arithmetic mean. In the present derivation of the A coefficients, the theoretical framework set out in ref 108 is used. The rates RS, RF+, and RF− are identical to those derived in ref 108, and some of their properties will be discussed further below. The calculations in that reference started from magnetization Ix, generated by a nonselective Y pulse, which then was subjected to a spin-locking

(a) (a) (s) (a) T11 → g11(t1)T11 + g20(t1)T20 + g22(t1)T22 + g31(t1)T31 (a) + g33(t1)T33

(B.7a)

with the coefficients gij given in ref 108 and here it is formulated analogously that 4490


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Table 1. Coefficients Describing the Evolution of the Coherence Transfer Amplitudes hij(t) According to equations B.7b and B.12a AS

a

AF∓ ⎡ 4 ⎤ ⎛ ω ω (ω 2 + 20ω 2)q 2 2 4 (ωQ2 + 4ω12)r ⎞⎟⎥ 3 ⎢ (ωQ − 6ω1 ωQ − 16ω1 ) 1 Q Q 1 ±⎜ + − 2 2 ⎢ ⎥ ⎜ ⎟ 2 2 2 2 2 2 10 2 λ1 λ 2 λ1λ 2 q + r ⎠⎦ ⎝ λ1 λ 2 q + r ⎣

3 5 2

h11(t)

−

h20(t)

0

⎡ ⎤ ⎛ 2 2 2 2 2 2 2ω1ωQ r ⎞⎟⎥ 3 ⎢ 2ω1ωQ (ωQ − ω1 ) ⎜ (ω1 − ωQ )(ωQ + 4ω1 )q ± − 2 2 ⎢ ⎥ ⎜ ⎟ 2 2 2 2 2 2 2 10 λ1 λ 2 λ1 λ 2 q + r λ1λ 2 q + r ⎠⎦ ⎝ ⎣

h22(t)

0

⎡ ⎛ ⎞⎤ 2 2 2 2 4 ω1ωQ r 3 ⎢ ω1ωQ (ωQ + 6ω1 ) ⎜ (3ω1 ωQ + 4ω1 )q ⎟⎥ ± − + 2 2 ⎢ ⎜ ⎟⎥ 2 10 λ1 λ 2 λ12λ 22 q2 + r 2 λ1λ 2 q2 + r 2 ⎠⎦ ⎝ ⎣

h31(t)

3 10

⎡ ⎤ ⎛ 2(5ω 3ω + 2ω ω 3 )q 2 2 4 4 (ωQ2 − ω12)r ⎞⎟⎥ 3 ⎢ (9ω1 ωQ + ωQ + 4ω1 ) 1 Q 1 Q ⎜− ± + 2 2 ⎜ ⎟⎥ 20 ⎢ λ1 λ 2 λ12λ 22 q2 + r 2 λ1λ 2 q2 + r 2 ⎠⎦ ⎝ ⎣

h33(t)

−

⎡ ⎤ ⎛ 2ω ω (ω 2 − ω 2 )q 2 2 4 4 (ωQ2 + ω12)r ⎞⎟⎥ 3 ⎢ (3ω1 ωQ + ωQ − 4ω1 ) Q ⎜ 1 Q 1 ± + ⎜ ⎟⎥ 2 2 2 2 4 5⎢ λ12λ 22 λ1λ 2 q2 + r 2 ⎠⎦ ⎝ λ1 λ 2 q + r ⎣

3 2 5

The abbreviations used in this table are those of ref 108: λ± = ωQ2 ± 2ω1ωQ + 4ω12 with λ1 = λ+ and λ2 = λ−, r =

below equation 5, q =

3ω12 ⎛ Jλ2 ⎜ 2 ⎝ λ22

−

Jλ

1

λ12

−

4ωQ ω1

⎞ J ⎟, and Jλ± = 2τ/(1 + λ2±τ2).

(a) + h33(t1)T33

(B.7b)

(a) hij(t ) = Tr[(Tij(s , a))† Sc exp(Mc , Dt )Sc−1T31 ]

(B.8)

hij(t ) = AS exp( −RSt ) + AF +exp( −RF +t )

were calculated in the present work using the diagonal form Mc,D = S−1 c Mc Sc of the 3 × 3 central-transition subset matrix Mc which in turn was chosen from the block-diagonalized form MD = S−1 M S of the master equation matrix M (cf. equation 13 in ref 108). The similarity transforms S and Sc used in the present work are constructed from the eigenvectors (cf. equations 15 and 18 in ref 108). The coefficients hij(t) thus calculated are summarized in Table 1. It is recognized that h11 = g31 as expected because these coefficients describe the transfer from (a) T(a) 11 to T31 and vice versa. At the end of the spin locking period t1, the contributions of the density matrix−from which detectable magnetization can evolve in the subsequent detection period−are given by 1 1 (a) (a) Ix , c → − (g11T11 + g31T31 )+ 5 5

+ AF −exp( −RF −t )

s(t1 , t 2) = [AS , s exp( −RSt1) + AF +, s exp( −RF +t1) + AF −, s exp( −RF −t1)]cos(ωQ t 2)exp( −R st 2) + [AS , c exp( −RSt1) + AF +, c exp( −RF +t1) + AF −, c exp( −RF −t1)]exp(−Rct 2)

AS , s = 0 ⎡ 2 2 2 3 ⎢ ω1 (ωQ − 4ω1 ) AF ∓, s = 20 ⎢ λ12λ 22 ⎣ ⎛ ω ω ω 2 + 2ω 2 q ⎞⎤ ω12r 1 Q( Q 1) ⎟⎥ ± ⎜⎜ − + 2 2 2 2 2 2 ⎟⎥ λ1 λ 2 q + r λ1λ 2 q + r ⎠⎦ ⎝

The magnetization evolution during the subsequent acquisition time t2 follows from 3 1 ⎞ (1) h31 − g31⎟f 31 2 5 ⎠ (B.10)

AS , c = −

where the evolution coefficients of the single-quantum coherences, f (1) ij , are compiled in ref 112. Eventually, this yields the signal

(

exp(− R st 2)+ − 2g11 +

6 g31 + 3 6 g31 +

(B.13)

The A coefficients in this equation are

3 (a) (a) (h11T11 + h31T31 ) 2

⎞ (1) ⎛ 1 1⎛ 3 h11 − g11⎟f11 +⎜ ⎜ ⎠ ⎝5 5⎝ 2

(B.12)

Equation B.11 can more explicitly be written as

(B.9)

1 ⎡⎛ ⎢⎜− 3g11 − 25 ⎣⎝

(J1 + J2 ) with Ji as defined

Here, Rs = J0 + J1 + J2 and Rc = J1 + J2 denote the transverse relaxation rates (which refer either to the central line, c, or to the satellites, s) in the absence of spin locking. Using explicit representations for the coefficients gij(t) (see equation 21 in ref 108) and

The coherence transfer amplitudes

s(t1, t 2) =

2λ1λ2

λ12λ22 2 ⎠

(a) (a) (s) (a) T31 → h11(t1)T11 + h20(t1)T20 + h22(t1)T22 + h31(t1)T31

s(t1 , t 2) =

ωQ2 − 4ω12

1 10

⎡ 2 2 4 4 1 ⎢ ω1 ωQ + ωQ + 4ω1 AF ∓, c = − 20 ⎢ λ12λ 22 ⎣ ⎛ 6ω 3ω q ⎞⎤ (ω1 − ωQ )(ωQ + ω1)r ⎟⎥ 1 Q ± ⎜⎜ − ⎟⎥ 2 2 2 2 λ1λ 2 q2 + r 2 ⎝ λ1 λ 2 q + r ⎠⎦

⎞ 3 h11 + 3h31⎟cos(ωQ t 2) ⎠ 2 ⎤ 6 h11 − 3h31 exp(− Rct 2)⎥ ⎥⎦ (B.11)

)

(B.14) 4491


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Evidently, for y = 0.5, it is found that two-thirds of the P(2) groups persist at x* = x** ≈ 0.33, and then their fraction diminishes until it reaches zero at x ≈ 0.67. Finally, B(2)(x) groups start to appear at about x*** ≈ 0.78. Because the trapping power of the phosphorus units decreases from P(2) to P(0) per compensated cation, ionic hopping motion becomes faster for increasing x in the regime between x** and x***. Then, for x > x***, the presence of the ion-trapping B(2) units renders the motion slower again. The enhanced ion mobility manifests itself in a second minimum7,9 of activation energies at x ≈ 0.8.

If only central line magnetization is detected, then only AS,c and AF∓,c are relevant. The terms in the first set of curly brackets of equation B.13, which are then modulated by cos(ωQt2) and quickly decay (on a scale of μs), will usually not be detectable. The rates appearing in equations B.11 and B.13 are now briefly discussed. First, it is observed that the rate RS = J1 + J2 depends on the same combination of spectral densities as the spin−lattice relaxation time T1,c. In order to gain access to dynamics in the slow motion regime (typically on the scale set by ωQ), the spin-lock experiment should be sensitive to the spectral densities Jλ± which are encoded via the rates RF+ and RF−. This situation can be achieved if the strength ω1 of the locking field is larger than ωQ, a case that arises if, like in ref 108, only a pre-averaged quadrupolar coupling is considered. Otherwise, as is particularly relevant in the present situation, the temperature dependence of R1ρ,c may essentially follow that of R1,c as is seen also in Figure 4a. Under these circumstances, R1ρ,c will typically not exhibit a well-defined maximum but merely only a shoulder (for an experimental example, see ref 113). Calculations of R1ρ,c based on equation B.13 are provided as Supporting Information in order to illustrate this situation.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b00482. Further information regarding the chemical composition of sodium borosilicate glasses, relaxation rates in the rotating frame, as well as sodium dipolar second moments of sodium borosilicate glasses (PDF)

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(2)

AUTHOR INFORMATION

Corresponding Author

APPENDIX C. EVOLUTION OF THE P (X) AND B(2)(X) FRACTIONS FOR DIFFERENT ALKALI CONTENTS Y

*E-mail: [email protected]. Tel.: +49-231-7553514. Fax: +49-231-755-3516. Notes

For y = 0.3 and x > x* = 3/7, the NUT model23 predicts the P(2) fraction to be zero. Yet, for y > 0.3, an increasing fraction 7/(6−6y) − 5/3 of P(2) groups persists for x > x*. Then, upon further increasing the B2O3 content x, the charge-neutral P(3) units are successively replaced by B(3) units, thereby leaving the ion trapping essentially unaltered. When all P(3) groups are replaced at x** = 8/3−7/(6−6y), the remaining P(2) groups transform into P(1) and P(0) units and the P(2) groups are eventually replaced by B(2) units at x ≥ x*** = 7/18[4 + 1/(y − 1)]. The compositional trends of the P(2)(x) and of the B(2)(x) fractions are summarized in Figure 8 for different alkali contents y.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was financially supported by the Deutsche Forschungsgemeinschaft, Grant No. BO1301/10-1. We thank Lars Gravert for his help in calculating the rotating-frame relaxation rates. Support for this research in the U.S. to S.W.M. and R.C. was provided by the U.S. National Science Foundation under grants 1304977 and 0710564.

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Figure 8. Fraction of ion-trapping P(2)(x) groups for various Na2O concentrations y in NBP glasses is indicated by solid lines. Apart from the initial reduction of the P(2) fraction for 0.3 < y < 0.5 reaching a plateau at x ≈ 0.4 a second decrease appears towards higher borate concentrations x. The increasing fraction of the ion-trapping B(2)(x) groups in borate-rich glasses, i.e, for x ≥ x*** is represented by the dashed lines. 4492


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