Composition Dependence of the Na+ Ion Conductivity in 0.5Na2S +

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Composition Dependence of the Na Ion Conductivity in 0.5NaS + 0.5[xGeS + (1-x)PS ] Mixed Glass Former Glasses: A Structural Interpretation of a Negative Mixed Glass Former Effect 2

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Steve W Martin, Christian Bischoff, and Katherine Schuller J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b07383 • Publication Date (Web): 30 Nov 2015 Downloaded from http://pubs.acs.org on December 9, 2015

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The Journal of Physical Chemistry

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Composition Dependence of the Na+ Ion Conductivity in 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] Mixed Glass Former Glasses: A Structural Interpretation of a Negative Mixed Glass Former Effect

A paper submitted to The Journal of Physical Chemistry C

Steve W. Martin 1, Christian Bischoff 2, Katherine Schuller 3 Department of Materials Science and Engineering Iowa State University Ames, Iowa 50010-2300

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Corresponding author, [email protected], 515-294-0745 Current address, Owens Illinois, Perrysburg, OH, 43551 3 Current address, Department of Materials Science and Engineering, University of Florida, Gainesville, FL, 32611 2

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Abstract A negative Mixed Glass Former Effect (MGFE) in the Na+ ion conductivity of glass has been found in 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses where the Na+ ion conductivity is significantly smaller for all of the ternary glasses than either of the binary end-member glasses. The minimum conductivity of ~ 0.4 x 10-6 (Ωcm)-1 at 25 oC occurs for the x = 0.7 glass. Prior to this observation, the alkali ion conductivity of sulfide glasses at constant alkali concentration, but variable ratio of one glass former for another (x) ternary mixed glass former (MGF) glasses, has always produced a positive MGFE in the alkali ion conductivity, that is the ternary glasses have always had higher ion conductivities that either of the end-member binary glasses. While the Na+ ion conductivity exhibits a single global minimum value, the conductivity activation energy exhibits a bimodal double maximum at x ~0.4 and x ~ 0.7. The modified Christensen-MartinAnderson-Stuart (CMAS) model of the activation energies reveals the origin of the negative MGFE to be due to an increase in the dielectric stiffness (a decrease in relative dielectric permittivity) of these glasses. When coupled with an increase in the average Na+ ion jump distance and a slight increase in the mechanical stiffness of the glass, this causes the activation energy to go through maximum values and thereby produce the negative MGFE. The double maximum in the conductivity activation energy is coincident with double maximums in CMAS calculated strain, ∆ES, and coulombic, ∆EC, activation energies. In these ternary glasses, the increase in the dielectric stiffness of the glass arises from a negative deviation of the limiting high frequency dielectric permittivity compared to the binary end-member glasses. While the CMAS calculated total activation energies ∆Eact = ∆ES + ∆EC are found to reproduce the overall shape of the composition dependence of the measured ∆Eact values, they are consistently smaller


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than the measured values for all compositions x. The new concept of an effective Madelung constant for the Na+ ions in glass is introduced, MD(Na+), to account for the difference. Calculated MD(Na+) values necessary to bring the CMAS and experimental ∆Eact values into agreement are in excellent agreement with nominal values for typical oxide crystals containing Na+. New MD simulations of oxide glasses were performed and were used to calculate MD(Na+) values for Na2O + SiO2 glasses for the first time and were found to agree quite well with the values for the sulfide glasses studied here. Insights from the current study have been used to predict and design new MGF systems that may lead to a positive MGFE in the ionic conductivity.

1. Introduction 1.1 Background Of the many electrical energy storage systems being considered for use as load leveling and peak power shifting systems for renewable energy systems, electrochemical energy storage, batteries, are attractive due to their high efficiency, safe operation, and scalability if their typically high costs can be overcome. Lithium batteries, while the predominant battery of choice in portable energy storage applications, have two problems that make them largely unsuitable for grid-scale energy storage applications. First, rechargeable lithium-ion batteries depend upon liquid electrolytes that not only are flammable but that in the presence of highcapacity metallic lithium anodes also foster lithium dendrite growth, leading to further severe safety concerns 1,2. Second, elemental lithium has low abundance and competition between the rapidly growing use of lithium batteries for portable applications and for grid-scale applications may produce an unsustainable dependence upon limited geological deposits of lithium. For these reasons, there is a push towards using other lower-cost non-lithium chemistries.



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Solid-state sodium batteries, with their inherent higher safety due to the lack of flammable organic liquid electrolytes, their ability to safely use high-capacity metallic sodium anodes, and their ability to be manufactured in the discharged state make them attractive as gridscale electrical energy storage systems3. However, in solid-state sodium battery designs, solid electrolytes with high Na+ ion conductivity (>10-3 (Ωcm)-1) at 25 °C) are essential for highperformance operation. For this reason, new alkali-ion-conducting glasses, especially those based upon sulfide chemistries, may be able to meet this challenging design requirement. In our work, we have found that a rare coincidence of high ionic conductivity with improved physical and electrochemical properties of glassy solid electrolytes can be achieved by mixing two glass former cations, e.g. B and P, at constant fraction of the mobile cation. This phenomenon is known as the mixed glass former effect (MGFE)4-8. To better understand and exploit the MGFE to enable optimized solid electrolytes for next-generation alkali-based batteries, we have begun an in-depth study of the MGFE in both oxide and sulfide glasses9-28. Currently, the structures and properties of sodium-modified MGF systems that include glass formers B2O3, SiO2, P2O5, SiS2, GeS2, B2S3, and P2S5 are among those being characterized by our group. In order to better understand the origin of the MGFE, especially regarding its relationship to the glass composition and the short range order (SRO) structure of these MGF glass systems, we continue our reports of these systems by expanding our study to a mixed Na Ge P S system, 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2]. Previous reports on a Na B P O11,12,17,18 and Na B P S16 system have been made and we expand these studies of 3 (B) - 5 (P) systems, to the 4 (Ge) – 5 (P) system.


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1.2 Structural Model of Na2S + GeS2 + PS5/2 Glasses In two previous reports on this system9,10, we developed a SRO atomic fraction model that accurately described the compositional (x) dependence of all of the SRO structures, Figure 1, present in these glasses. We used this SRO model to describe the compositional dependence of two key physical properties, the glass transition temperature (Tg), and the density (and molar volume),9 that are essential to the understanding of the compositional dependence of the Na+ ion conductivity, the focus of the present paper. The SRO model of these glasses, Figure 2, was determined by a combination of Raman, IR and NMR spectroscopies, which identified the composition dependence of these SRO structures present in the glasses. Having identified which structural groups were present for each glass composition x, their atomic fractions were determined by integration of 31P MAS-NMR spectra for the various phosphorus populations and, through charge neutrality requirements, the germanium populations were determined by difference. For the fractions of the germanium SRO structural units, it was found through careful inspection and analysis of the compositional dependence of the Tg of the glasses, that the relative amounts of Ge3D and Ge3M, Figure 1, control the formation of an extended glassy network. In this notation, the superscripts 3D and 3M refer, respectively, to four coordinated Ge centers that have three bridging sulfurs (bS) (where the fourth sulfur is a non-bridging sulfur (nbS)) to other Ge atoms to create a three dimensional connected network and Ge centers that have three bS to other Ge atoms to form molecular Ge4S104- units. In all of the Ge and P SRO structural units, the superscript number defines the number of bS atoms to other Ge or P atoms. Hence, a P1 has one bS to either one Ge or one P atom.



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Through this SRO structure analysis, Figure 2, it was found that the Na+ cations are unequally shared among the various Ge and P SRO structures, Figure 1, in the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses. That is, at every glass composition (x) the amount of Na+ charge residing on the negatively charged Ge and P SRO structural units was not simply x and (1-x), respectively, as would be expected by the composition of the glass. Rather, it was observed that the amount of positive charge on the Ge and P SRO units changed strongly with x. It was found that as x increased, the Ge SRO structures held less charge than x and the P SROs structural units held more charge than (1-x). To account for this disproportionate sharing of the charge between the Ge and P SRO structural units, a structural disproportionation reaction, Eq. 1, was found. The reaction accurately described the conversions among the Ge and P SRO structural units in the glass to account for the observed compositional dependence of the Na+ ions residing on the P and Ge SRO units. The best-fit adjusted SRO model is shown in Figure 2. P1 + Ge 2 → P 0 + Ge 3

Equation (1)

As a result of this disproportionation reaction, as x increases, more of the negative charge, hence more of the Na+, reside in charge compensating roles adjacent to the P SRO structural units and as a result, the P SRO structural units become less polymerized, i.e. the number of nbS per P SRO structural unit increases, whereas the opposite occurs for the Ge SRO units where the number of nbS per Ge SRO unit decreases. The key signature of the unequal sharing of Na+ ions between the Ge and P SRO units is shown in Figure 2 in the composition (x) dependence of the P1 SRO structural group. Figure 1 shows that the structurally dominant P1 group possesses one bS group and hence two negatively charged nbS and one neutral nbS, as is expected for the x = 0 binary 0.5Na2S + 0.5PS5/2 glass.


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Here we use the molar formula PS5/2 instead of the more common P2S5 so that at each composition x one P is exchanged for one Ge atom. Figure 2 shows that the P1 groups are converted to the totally depolymerized P0 groups, no bS. This is accompanied by the Ge2SRO group, expected for the composition 0.5Na2S + 0.5GeS2 (see Figure 1), losing a Na+ (and necessarily one nbS) to form the more polymerized Ge3 group. As discussed above, our findings for the composition dependence of the Tg9 and supported by the Raman and IR spectroscopy of the glasses10, required that we introduce two forms of Ge3 groups, Ge3D, and Ge3M, to account for the negative MFGE in the Tg. Figure 2 shows that only very small amount of Ge3M groups are necessary, at most less than 5%, and most of the Ge3 groups are in the normal Ge3D structure. A very small amount of Ge4 groups are also observed in these glasses for very small x and is associated with the rapid conversion of P1 groups into P0 groups. In the glasses in the 0.35Na2O + 0.65[xB2O3 + (1-x)P2O5] system studied by Christensen et al.12 such unequal sharing of the charge was also observed. However, unlike the present sulfide system where the charge was shared towards P for all x, for the B and P oxide glasses, at both compositional ends, x 0 and x 1, it was the minority (i.e. the less abundant) glass former that was over modified with Na. That is the minority glass former, P for glasses with x > 0.5 and B for glasses with x < 0.5, was observed to have more Na+ ions than equal sharing would suggest, 0.70*x moles of Na+ per 1.30*x moles of B and 0.70*(1-x) moles of Na+ per 1.30*(1-x) moles of P. Likewise, the opposite was found for the majority (the more abundant) glass former which was observed to be under modified, that is possessed less Na+ than equal sharing would predict. In the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses studied here, we found9,10 that the P is over modified at all compositions, except of course for glasses with x = 0



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and x = 1 where there is no Ge and no P, respectively. That is the P possessed more than (1-x) moles of Na+ per 0.5*(1-x) moles of P. Hence, for all MGF compositions, 0 < x < 1, the P SRO groups are weighted towards the more depolymerized P0 group and the Ge SRO groups are weighted towards the more depolymerized groups, Ge3 (both types) and Ge4. This preferential formation of more depolymerized P0 groups with their inherently more localized negative charge will be seen below to have an important effect on the composition dependence of the Na+ ion conductivity of these MGF glasses.

1.3 Non-Arrhenius Temperature Dependence of the Ionic Conductivity in Glasses Before we turn to analyzing the composition dependence of the Na+ ion conductivities of the glasses in the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] system, it is important to note that a detailed study and analysis of the temperature dependence of the Na+ ion conductivities of these glasses has already been reported by us29. In this first examination of the conductivity of these glasses, we found that the Na+ ion conductivities of all of the glasses in this series exhibited a systematic non-Arrhenius positive upward curvature temperature dependence for all temperatures which suggested that the Na+ ions were conducting over higher average activation energy barriers at higher temperatures and conducting over lower average activation energy barriers at lower temperatures. Such behavior combined with the disordered structure of these (and all glasses) suggested that this non-Arrhenius temperature dependence of the conductivity (non-singled value activation energy) may arise from a distribution of activation energies (DAE) which we have seen and have modeled for other (Li+) ion conducting glasses. Our new approach however, is an improved version of our earlier DAE treatments30-34 on different but related glasses.


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Further, in contrasting behavior to that observed here, we have also observed and published extensively on a negative curvature non-Arrhenius temperature dependence of the conductivity that we have found for extremely high conductivity Ag+ (and to a lesser extent Li+) ion conducting glasses26,35-38. Others have developed models of the negative Arrhenius behavior we have seen on our glasses39,40. This negative curvature was only observed at high temperatures, but well below Tg, and was associated with scattering events between the highly mobile Ag+ as they strongly interacted during conduction events. This is distinct from the behavior we see here, most strongly at lower temperatures, where the less mobile ions are performing mostly single ion hops over individual energy barriers. We will show below how the observed double maximum in the composition dependence of the activation energy can be considered as a natural consequence of the composition dependence of the various SRO groups shown in Figure 2. Indeed, we will show that this very first sulfide glass system to exhibit a negative MGFE in the Na+ conductivity in MGF glasses does so as a result of the composition dependence of the SRO structures shown in Figure 2. To interpret the composition and structure dependence of the conductivity activation energy, we will use the recently developed Christenson-Martin-Anderson-Stuart model11 of the conductivity activation energy to calculate the average value of the DAE.

2. Experimental Methods 2.1. Sample Preparation All glass syntheses were carried out in a high purity N2 glove box, where O2 and H2O levels are below 10 ppm and typically below 1 ppm. Since high purity Na2S is not commercially available, it was synthesized in our laboratory by the thermal dehydration of Na2S·9H2O (Sigma-



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Aldrich, > 98%) following standard procedures41. The Na2S·9H2O was placed in a vitreous carbon crucible that was placed in a tubular stainless steel reaction chamber. The reaction chamber was then placed in a vertical crucible furnace such that the sample was within the hot zone of the furnace and that the top of the reaction chamber extended outside the furnace. The reaction chamber was then sealed with a water cooled rubber o-ring gasketed top and connected through a liquid nitrogen cooled trap to a vacuum pump. The hydrated crystalline material was then slowly heated under roughing pump vacuum, ~4 Pa (30 mTorr) up to 423 K (150 °C) over a period of at least two hours, after which the temperature was slowly increased at ~ 2 oC/min. to 923 K (650 °C) and held for approximately 20 hours and then cooled to room temperature. The reaction chamber containing the now dehydrated Na2S was transferred to the glove box and unloaded. IR spectroscopy and x-ray diffraction (XRD) were used to confirm the absence of residual contaminate oxides and the phase purity, respectively, of the material. Glassy GeS2 was prepared by reacting stoichiometric amounts of germanium powder and sulfur + 2 wt% in an evacuated silica ampoule at 900 °C for approximately 16 hours using a method previously described21. The ampoule was air quenched to room temperature and glassy GeS2 was obtained. Phosphorus pentasulfide (P2S5) is commercially available and was used as received (99.9% Sigma-Aldrich). Glass batches of 3 to 4 grams were made by combining appropriate amounts of the starting material powders to create the 0.5Na2S + 0.5GeS2 and 0.5Na2S + 0.5PS5/2 binary glasses. The binary sodium thiophosphate glasses were planetary ball milled prior to melting to minimize evaporation of PS5/2 by inducing the pre-reaction of the Na2S and PS5/242,43. The binary sodium thiogermanate compositions were melted and quenched to the glassy state without prior milling.


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The ternary 0.5Na2S + 0.5[xGeS2 + (1−x)PS5/2] glasses were then prepared by mixing appropriate amounts of the binary 0.5Na2S + 0.5GeS2 and 0.5Na2S + 0.5PS5/2 glasses. These mixtures were then melted in the high purity low O2 and H2O atmosphere of the glove box in covered vitreous carbon crucibles for 3-5 minutes inside a mullite muffle tube furnace at 550-800 °C that is hermetically connected to the outside of the glove box. The lower temperatures were used for the thiophosphate-rich glasses (x < 0.5) and the higher temperatures were used for the thiogermanate-rich glasses (x > 0.5). The samples were then removed from the furnace and allowed to cool to room temperature inside the crucible. Mass losses were recorded and in all cases found to be less than 2 wt% and the glasses were then remelted at the same temperature for an additional three minutes and quenched to room temperature between brass plates (> 104 K/s). All samples were transparent and showed no visual signs of crystallization and/or phase separation. Bulk disc-shaped samples, 1-2 mm thickness and 20-30 mm diameter, were prepared by quenching the melt compositions in preheated brass molds. These low viscosity melts were poured into molds held 30 K below the Tg of the glass and annealed for 30 minutes. After annealing, the bulk samples were cooled to room temperature at 1 K/minute. 2.2. Ionic Conductivity Measurements of the Glasses Bulk samples were polished inside the glove box to optical smoothness using polishing papers of decreasing grit sizes, 600 to 4000, and then sputtered with gold electrodes also inside the glove box with a diameter of ~13 mm and thickness of ~ 100 nm. The smaller diameter of the electrode insured that the impedance around the surface of the glass to the other side would always be significantly larger than the direct (through the glass) electrode to electrode



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impedance. The complex impedance spectra were measured using a Novocontrol Technologies Concept 80 impedance spectrometer from 0.1 Hz to 3 MHz and temperatures from 223K (-50 °C) to 423 K (150 °C). The temperature was held within ± 0.5 K of the nominal set point for three minutes prior to data collection to stabilize the temperature. Prior to the impedance measurements, these air sensitive samples were sealed inside a hermetic Teflon sample holder inside the glove box that had two brass electrodes separated by a rubber o-ring gasketed Teflon sample chamber holding the sample. Bottom and top inner brass electrodes ~ 0.5 mm thick, with one of these spring electrically connected the sample, connected the sample to the external electrodes. It was observed that this sample holder added just a few ohms and just a few picofarards of capacitance to the overall electrical response of the sample and these were calibrated out the overall response of the sample and cell to yield the pure glass sample response. The upper and lower temperature limits used here were set by the glass (Tg) and by the sample holder (Tg of the rubber gasket below which it would not seal) so that repeatable results before and after the measurements at room temperature were obtained for all samples at all temperatures.

3. Results The direct current (d.c.) Na+ ion conductivity, σd.c., of each glass sample at each temperature was found by fitting the complex impedance arc using Eq. 2 to determine the resistance, R, see for example44. Equation 3 was then used to determine the equivalent σd.c. at each temperature, where t is the thickness of the sample and A the area of the sputtered gold electrodes. Equation 2 models the frequency dependence of the complex impedance, Z*(ω), of the Na+ ion conducting glass using a resistor in parallel with a constant phase element (CPE), Q,


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whose reactance (complex impedance) is 1/Q(iω)n, where ω is the angular frequency, and n is an exponent that ranges from 0 to 1, typically ~ 0.8. The CPE has been shown to provide a simplified yet accurate single parameter fitting model to represent the effect of the DAE (equivalent to a distributed network) of the mobile ion population on the frequency dependence of the complex impedance. More accurate (and more complicated) approaches are well known in the literature45, but for the purposes of this study where we seek accurate values of the d.c. conductivity, this model has been proven accurate given the large number of samples and temperatures studied. A more in-depth examination of the frequency dependence of the conductivity will be reported on in future publications where explicit use of the DAE and its effect on the a.c. conductivity will be explored.

Z * (ω ) ≡ Z'(ω ) − iZ"(ω ) =

R 1 + RQ(iω ) n

 1  t     R  A 

σ d .c . = 

Equation (2)

Equation (3)

An example of the temperature dependence of the d.c. Na+ ion conductivity so determined is shown in Figure 3 for the x = 0.7 glass. On very close examination of Figure 3, the slight, but unmistakable non-Arrhenius temperature dependence of the Na+ ion conductivity can be observed. A dashed straight line of simple Arrhenius behavior is added to Figure 3 to accentuate the observation of the non-Arrhenius behavior of the conductivity for this glass. All samples of glass in this series exhibited non-Arrhenius ionic conductivities for all temperatures measured and it is further noted that this was more pronounced at the lower temperatures. The solid line through the data is a best-fit to the data of Eq. 4 taken from our ref. 29. Equation 4 was



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found29 to be a very accurate approximation to the full treatment of the DAE of the ionic conductivity in glass over the typical temperatures used to measure the ionic conductivity from well below room temperature, -100 oC, and up to typical Tgs of typical glasses, ~300 oC. Here ∆E0 is the average activation energy of the DAE, σ0 is the conductivity prefactor, δ is the standard deviation of the DAE, R is the ideal gas constant, and T is the absolute temperature. Deviations from the simplified Eq. 4 and the full expression, given in ref. 29, are well below the size of the symbols used in Figure 3. The composition dependence (x) of the σd.c. over a range of temperatures of interest, and of ∆E0, σ0, and δ values are shown in Figures 4 and 5, respectively.

 δ2 ∆ E −  0 RT ln (σ d .c. (T ) ) ≅ ln (σ 0 ) −  RT   

     

Equation (4)

The negative MGFE in the Na+ ion conductivity in these glasses is readily apparent in Figure 4 and arises from the positive MGFE in the average activation energy, ∆E0, shown in Figure 5. The σd.c.(T) values of the two binary glasses, x = 0 and x = 1 are essentially identical for all temperatures and, for example at 25oC the value of ~2 x 10-6 (Ωcm)-1 are similar to those reported in the literature for similar glasses46-49. This relatively modest conductivity is approximately 100x smaller than the Li+ ion conductivities for comparable sulfide glasses, but is none-the-less approximately 1,000 times larger than values observed for comparable Na+ ion conducting oxide glasses50-53. From Figure 4 it is seen that the addition of P to the binary 0.5Na2S + 0.5GeS2 glass causes a larger decrease in the conductivity compared to the addition of Ge to the 0.5Na2S + 0.5PS5/2 glass, where the asymmetric minimum is observed in the conductivity at x = 0.7. To the


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best of our knowledge and a careful review of the literature, this is the first system to exhibit a negative MGFE in the Na+ ion conductivity in a sulfide glass. A previous report of a negative MGFE in a lithium oxide glass has also been observed in the 0.25Li2O + 0.75[xBO3/2 + (1x)SiO2] series.54 The asymmetry of the negative MGFE in the conductivity in these glasses appears to be consistent with the asymmetry in the composition dependence of the various SRO structures present in these glasses. Figure 2 shows, for example, that the addition of Ge to the base 0.5Na2S + 0.5PS5/2 x = 0 glass causes 2 P1 units to be converted to 2 P0 units, whereas the addition of P to the base 0.5Na2S + 0.5GeS2 glass causes ~ 1.5Ge2 structures to be converted to 1.5Ge3 units (predominantly Ge3B units.) This asymmetry in the formation rate of the intermediate SRO structures necessarily produces asymmetry in the amounts of these units formed and as such causes the composition dependence of all of the structural groups to be asymmetric. In the present case the “center of mass” of the SRO units appears to be at x ~ 0.4. The minimum in the Na+ ion conductivity is caused by the maximum(s) in the average conductivity activation energies, ∆E0, as seen in Figure 5, where two local maximum are observed, one at x ~ 0.3 and one at x ~ 0.7. The lowest conductivity is observed, not surprisingly, for the x = 0.7 glass, the glass with the highest average activation energy. The observation of two local maxima in the composition dependence of the average conductivity activation energy could be considered to arise from two different compositional ranges of glass structure, one dominated by P SRO structures, x 0, and one dominated by Ge SRO structures, x 1. These two maxima occur in Figure 2 where the maximum “differentiation,” that is the greatest number of different SRO structures present in the glasses, occurs in the P SRO structures at x ~ 0.3 and in



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the Ge SRO structures at x ~ 0.7. At each of these compositions there are four different SRO structures present in each glass. A bimodal behavior in essentially the exact same compositional regions is also seen in the width, δ, of the DAE and in the conductivity pre-exponential factor,

σ0, and arises presumably also from the maximum differentiation in the number of SRO groups.

4. Discussion 4.1. Non-Arrhenius Ionic Conductivity In our previous paper29 where we first identified the direct relationship between the DAE and the non-Arrhenius temperature dependence of the ionic conductivity in glass, we showed that the magnitude of the ratio δ

∆E0

and the width of the temperature range that the ionic

conductivities were measured over were the determining factors of whether or not a nonArrhenius ionic conductivity would be seen in the measured ionic conductivity. The wider the temperature range of conductivity measurements and the larger the δ

∆E0

ratio, the more

pronounced the observation of the non-Arrhenius ionic conductivity. Now since most ionically conducting glasses of interest have Tg values in the range of 400 to 700 K and since temperatures are most easily controlled from room temperature and above, the typical temperature range that most ionic conductivities are measured fall into the range of about 300 to 550 K, a relatively modest temperature range. Indeed, most measurements are even more limited to 300 to 400 K since Tg and the magnitude of the ionic conductivity are inversely correlated. The conductivity is often highest, and hence of most interest, when the alkali modifying cation concentration is highest and the Tg of the glass is often lowest. Hence, the fact that most glasses studied to date are found to exhibit Arrhenius conductivities arises in part from the combination


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17

of the relatively narrow temperature range over which they are measured and in part from the fact that their δ

∆E0

ratios are relatively small (arising from large ∆E0 values) for the dominant

number of oxide glasses measured. These glasses also have relatively low conductivities and relatively high ∆E0 values combined with relatively small δ values due to their (typically) single glass former composition (small numbers of SRO groups). An example of this behavior is seen in Figure 6 on our own measurements of a relatively poorly conducting MGF composition Na2O + B2O3 + P2O511 where we have purposefully extended the temperature range of our previous measurements of this glass, previously thought to be Arrhenian in behavior, to lower temperatures where the effect of the DAE would be extenuated as predicted from Eq. 4 above. As Figure 6 shows and as expected from our model of the DAE best-fit line to Eq. 4 , this oxide glass does indeed exhibit a non-Arrhenius Na+ ion conductivity when the temperature is purposefully extended to sufficiently low temperatures. In a greater sense, this presents a fairly wide open field of study because while the concept of a DAE for ionic conduction in glass has been largely accepted for many years to arise from the disordered structure of glass, it, however, has not been explored or reported for the vast majority of glasses simply due to the limited range of temperatures that have been explored. Now in the present case where we have purposefully lowered the average activation energy to Na+ ion conduction by using a sulfide chemistry and we have purposefully further disordered the glass structure and hence have broadened the DAE by introducing a second glass former, the conditions are optimal for observing the non-Arrhenius behavior in the conductivity.



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That the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses exhibit non-Arrhenius ionic conductivities for all values of x, δ > 0, the composition dependence of ߜ can be linked to the distribution of local SRO structures. If we consider the number of SRO structures (SROs) present at each composition that Na+ ions charge balance, the ternary compositions have more SROs associated with Na+ ions than the binary compositions at x = 0 and x = 1. Further, as the number of SROs increases, it is to be expected that the variety of energy barriers, hence the width of the DAE, δ , would also increase. The number of different types of energy barriers that a Na+ ion must overcome to conduct from one negatively charged Qn (SROs) species to an adjacent Qn, where Q is either Ge or P in these glasses, species is the number of permutations with repetition. For example, from Figure 2, and as discussed above, it is seen that the x = 0.3 and 0.7 glasses have 6 different charged Qn (3 charged Pn and 3 charged Gen) species (out of a total of 7 total charged and 9 total Qn species) associated with Na+ ions, which leads to 36 possible Qn-Qn pairs for the conduction process. It is significant to note that for the glass with the local minimum in the width of the DAE, the x = 0.5 glass, Figure 2 shows that charge is localized on two predominant groups, the P0 and Ge3B groups, whereas for the glass compositions that show local maximums in the width of DAE, the x =0.3 and x = 0.7 glasses, Figure 2 shows that there is no dominant localization of charge, but rather the charge is more or less evenly distributed among four of the six total charged SROs. The broadening of the charge distribution among more, rather than less, SROs may likewise account for the increase in the width of DAE for the x = 0.3 and x = 0.7 glasses. The narrowing of the charge distribution among fewer SROs may account for the decreased width of the DAE for the x = 0.5 glass. This is perhaps the very first identification of the precise structural origins of the widths of a DAE in


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19

an ion conducting glass with specific SROs groups and their concentration in a MGF glass system. In future work, we will begin the task of using this detailed structural information to identify the exact origins of the width of the DAE in these glasses. For now, however, we turn next to the detailed interpretation of the structural origins of the compositional dependence of

∆E0 in these glasses and to begin to answer the question in more detail of why ∆E0 is larger for all compositions of the ternary glasses than the two binary glasses and why there are two local maximum across the range of ternary glasses as seen in Figure 5. 4.2.The Christensen-Martin-Anderson-Stuart (CMAS) Model The ionic conductivity can be described according to Eq. 5, where n(T) is the temperature dependent number of mobile ions per unit volume, eZ is the charge on the mobile ion, and µ(T) is the temperature dependent mobility of the cations. Martin and Angell55 showed how the temperature dependence of the number of mobile ions and of the mobility could be related to the activation energies for coulombic charge separation, ∆EC, and for volumetric strain dilation, ∆ES¸ respectively, events required as the cation moves from one equilibrium cation site to the other.

σ d .c. (T ) = n(T )eZ µ (T ) −∆EC  (b) µ (T ) = µ (T ) exp  −∆ES  (a) n(T ) = n0 exp     0 RT RT    

∆Eact = ∆EC + ∆ES

Equation (5) Equation (6) Equation (7)

Figure 7 shows the structural origins of these activation energy barriers as envisioned by these authors. This connection between the mobile charge creation and dilational strain processes inherent to the cation diffusion process was first quantified by Anderson and Stuart(AS) 56 who developed a model to calculate the activation energy for conduction based on this formalism.



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20

McElfresch and Howitt57 improved upon the original AS model by developing a more accurate and self-consistent model for the volumetric strain activation energy term, ∆ES, that we, and others, have adopted and it will be used here. Christensen and Martin et al.11 have recently advanced the coulombic mobile charge creation activation energy term ∆EC of the AS model by more correctly estimating the cation jump distance, by more correctly using the limiting high frequency dielectric permittivity, and by adding an effective structural Madelung constant to describe the many body effects of the coulomb interaction of other nearby cations and anions in the glass structure. Because of the uniquely disordered structure of each glass, each cation and each anion in the glass structure will have its own unique Madelung constant, Mi. However, due to the well-defined SROs that have been shown to exist in these and many other glasses, the SRO of glass is reasonably well defined out to the first few coordination spheres and as such it is expected that each cation and each anion will therefore have an average Mi value that will have a characteristic average value and a broadening due to the disorder of the glass structure. In this paper, we will use this average value of the MD(Na+) cations as the lone adjustable parameter in calculating the average value for the columbic binding energy for these glasses. We will show that very reasonable values are obtained in this way and lend support to the method that we have developed to interpret the conductivity activation energy maximum in these glasses. Since the ionic conductivity of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses has a negative MGFE, the activation energy of n, µ, or both parameters must increase for ternary compositions compared to the binary end-member glasses. We now turn to calculation of these two energy barriers terms using the methods described above to determine if these glasses are


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21

“coulombically” or “strain” limited, that is do the maximums in the total average conductivity activation energy arise from the coulombic or volumetric strain terms, or perhaps both. We begin with the coulombic binding energy term. 4.2.1. Electrostatic Binding Energy and Mobile Charge Carrier Population The activation energy that must be overcome to create a mobile ion population, n, is related to the electrostatic binding energy of Na+ ions, and is denoted ∆EC. Our recent improvement11 to the original AS model56 approximates the electrostatic binding energy as given in Eq. 8:  M Na Z Na Z S e 2  1 1 ∆EC = −  4πε 0ε ∞  rNa + + rS − rNa + + rS − + λcation 2 

    

Equation (8)

where ZNa and ZS are the absolute values of the charges on the sodium (Na+) and sulfur (nbS, S-), respectively, e is the electronic charge, ε0 is the permittivity of free space, ε∞ is the limiting high frequency relative permittivity of the glass, rNa + and rS − are the ionic radii of sodium and sulfur, respectively, and λcation is the distance the Na+ ion must travel to an adjacent site. Figure 7 shows these various distances envisioned in the CMAS model. The limiting high frequency relative permittivity of a given glass, ε∞, was determined from dielectric impedance spectroscopy measurements at low temperatures and at high frequencies in the MHz range, where cation diffusion events have been effectively frozen out, by determining the value of the high frequency plateau in the real permittivity. An example is



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22

shown in Figure 8. The composition dependence of the relative permittivity, ε∞ , for all glasses is shown in Figure 9. The relative permittivity ε∞ is smaller for all ternary compositions than that of either of the binary end-member glasses, x = 0 and x = 1, with the exception of the x = 0.5 glass. The x = 0.5 glass has a ε∞ value between that of the binary end-member glasses. As ε∞ becomes smaller, the glass structure becomes less polarizable. Figure 9 gives an initial clue, therefore, into the structural and compositional origins of the negative MGFE of the conductivity in these glasses. As ε∞ for the ternary glasses is less than that for the binary glasses, this suggests that the electron clouds of the ions in the glass are more tightly bound enabling less polarizability induced shielding of the coulombic potential between the mobile Na+ ions and their charge compensating nbS. This in turn increases the coulombic part of the total activation energy, as seen in Eq. 6 above. On the other hand, a decreasing ε∞ in these glasses may also suggest that the less polarizable ions in the glass may lead to an increased Tg of the glass arising from the increased ionic bond strength between the ions that would further be associated with a higher mechanical modulus of the glass (which is indeed seen in the these glasses, see below) thereby increasing the volumetric strain energy in the glass. However, this latter trend is not consistent with the Tg values for the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses, which exhibit Tg depression in the ternary glasses9. In our previous work on these glasses9, we concluded that the decrease in Tg was caused by the formation of Ge3M groups, see Figure 1, which while “internally” to the molecular Ge4S104- may have stronger on average bond strengths, never-the-less, do not significantly increase the structural connectivity of the glass network. In this case, it appears that Ge3D and Ge3M SRO structures may have comparable polarizabilities. This may explain therefore


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23

why Tg decreases and yet the permittivity also decreases, indicating that the dielectric stiffness increases while the thermal stiffness does not. Significantly, we will show below in our treatment of the dilational strain energy that the mechanical stiffness, specifically the shear modulus, G, does indeed increase with x across this series of glasses. The ionic radius of sodium, rNa + , was reported by Shannon58 as 0.97 Å for a coordination number of 4 as is seen in these glasses24,59-61 and is considered to be approximately constant. From refs. 24,59-61, rS − is determined from the rNa+ + rS − bond distance of 2.81 Å and gives rS − = 1.84 Å, which we also take as a constant. We postulate that the λcation distances, see Figure 7, are, however, compositionally dependent based on the changes in molar volume of the glasses as _

reported by us previously 9. Molar volumes, V ( x ) , from ref. 9 were used to estimate the values

λcation . In the calculation of λcation, the true cation jump distance, we use the revised method proposed by Christensen and Martin et al. 11 where, as Figure 7 shows, the original AS jump distance has been corrected by including the fact that the jump distance away from the charge compensating nbS unit and into the next cation site must also include the original bond distance of the Na+ and the S- cation and anion, rNa+ + rS − . In this way, the net jump distance becomes;

λ = r + r + λcation S 2 Na 2 +

−

Equation (9)

where λcation is taken as the average cation-cation separation distance which was calculated through the number density of the mobile cation population in the glass. Without this correction it is possible to arrive at the unphysical situation where the λ/2 term in Eq. 9 above becomes shorter than the sum rNa+ + rS − . Eq (9) also has the correct limiting behavior of ∆EC0 as λ-



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24

cation

0, i.e. the condition of zero jump distance. Without this correction, the original AS model

would predict that the activation energy would tend towards the unphysical condition of ∆EC∞. Table 2 shows all of the CMAS parameters used in this study for all glasses, including those required, see below, for the calculation of the volumetric strain energy. Using these parameters, we can calculate the electrostatic binding energy ∆EC using Eq. 8 above. All of the parameters used in Eq. 8 are not adjusted and come directly from the measured and known values for the glasses. The composition dependence of ∆EC is shown in Figure 10. The immediate appearance of the positive MGFE in ∆EC activation energy and the double maximum in the same compositional regions as seen experimentally, x = 0.3 to 0.4 and 0.7 to 0.8, are obvious and important findings. These calculations are in agreement with the discussion above where it was suggested that the localization of charge on the increasing number of P0 groups in these glasses causes the P SRO groups to become more coulombically charge dense and hence more electrostatically binding to the mobile cations. An important factor that gives rise to the two local maxima in the calculated ∆EC values is presumably the strong local minima in the limiting high frequency relative dielectric constant, ε∞, as seen in Figure 9. The electrostatic binding energy has two local maxima lobes at x ~ 0.2 and at ~ 0.8 with maximal values of 25.7 kJ/mol and 27.0 kJ/mole, respectively. Note that these values are calculated without the MD(Na+) value. This will be added below after we describe the calculation of the strain energy term. There is a minimum between these two local maxima occurring at x = 0.5, and the binding energy of 24.3 kJ/mole. This overall increase in this activation energy for the majority of the ternary glass compositions is in strong agreement with conductivity data;


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25

however, the magnitude of the energy barrier, ~ 25 kJ/mole, is smaller than the typical experimentally observed value of ~ 60 kJ/mole, as shown in Figure 5. Investigation of the energy barrier of the mobility term and the Madelung constant for the Na+ sites is required to further assess the accuracy of the model prediction.

4.2.2. Strain Energy and Mobile Ion Mobility Anderson and Stuart approximate the energy barrier associated with moving the mobile ion through the interstitial space between cation sites as being the strain energy, ∆ES, that must be overcome to dilate the ion through a cylindrical medium. We will use the modification of this term that was suggested by McElfresh and Howitt57 according to Eq. 10. Additional parameters to those used in the binding energy term include the shear modulus, G, and the doorway radius, rD.

∆Es = π G

λ 2

( rNa − rD )

2

Equation (10)

In previous studies, both experimental59 (x-ray and neutron scattering, XRS and NS) and theoretical62 (DFT-MD and Reverse Monte Carlo) studies, the Na+ ions in the binary 0.5Na2S + 0.5GeS2 glass (our x = 1 glass here) were shown to have a bond distance to S of between ~2.4 Å at the shortest (DFT-MD) to ~2.8 Å at the longest (XRS and NS). We will use an average of these values of ~2.6 Å. These same studies also found that like in crystalline Na2S, which crystallizes in the anti-fluorite structure63-65, the total average coordination number around Na+ by both charged nbS and neutral bS was ~ 4, which is taken here to create a tetrahedral site around the Na+ ions. In this geometry, the interstitial doorway through which the Na+ ion must conduct will be the trigonal face of three sulfur atoms of the tetrahedron. The Na+ -- S bond



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26

distance of ~2.6 Å gives a body diagonal distance between the center of sulfur atoms of rS-S ~ 2*2.6Å*sin(109.5o/2) ~ 4.25 Å. This in turn, gives an interstitial doorway radius of rD ~ 4.25 Å/(2*cos(30o) – 1.8 Å ~ 0.65 Å. We will take this value to be constant for all glasses in these series even though the volume per Na+ ion in the glass decreases from 77.4 Å3 to 72.2 Å3, a decrease of 6.5%. The previous studies of the binary Na2S + GeS2 glasses showed no appreciable changes in the Na – S bond distance nor its coordination number. This lack of information, unfortunately, is even more the case here for our as yet incompletely studied ternary Na2S + GeS2 + PS5/2 glasses. Further, there are no detailed structural studies of the x = 0 glass 0.5Na2S + 0.5PS5/2 beyond ours42,43, although these are in progress. The mechanical moduli of these ternary Na2S + GeS2 + PS5/2 glasses are unfortunately unknown and there are few references to the shear moduli of GeS2-based glasses except that of the binary GeS2 + Sb2S3 by Shtets et al.66 and GeS2 + As2S3 glasses by Savchenko et al.67 where a G value of pure GeS2 of ~ 9 GPa was reported. Because experimental values are not known for these glasses, approximate values were calculated from the composition dependence of the average polarizability of the glass α following the method of Gilman68. Since the driving force for dilation of the network to accommodate the conduction of the Na+ ion is the applied electric field, we calculate G(x) according to the relation in Equation 11 due to Gilman68, where e is the electronic charge, ߙ is the polarizability of the glass determined from the Clausius-Mossotti relation in Eq. 12. The calculated shear moduli of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses determined from Eq. 11 is shown in Figure 11.

3e 2 G ( x) = 4π ( rNa + rS ) α


Equation (11)

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27

α=

3ε 0 ( ε ∞ − 1) N (ε ∞ + 2)

Equation (12)

The strain energy, ∆ES(x)¸calculated from the values shown in Table 2 for all of the glasses in this series using Eq. 10 is plotted in Figure 11. ∆ES is approximately a linear increasing trend for x = 0 to x = 0.8 from ~ 5.4 kJ/mole to ~ 5.8 kJ/mole where after it decreases to ~ 5.7 kJ/mole for the x = 1 glass. It is significant to note that these calculated values are significantly smaller than the values for the electrostatic binding energy ∆EC which from Figure 10 are ~ 25 kJ/mole. These values are comparable to values determined for other Na+ ion conducting glasses. For example, in the 0.35Na2O + 0.65[xB2O3 + (1-x)P2O5] series of glasses11 that have on average slightly larger G and λ values, due to the stronger on average bond strengths and the lower Na+ ion concentration, but comparable rD values, the ∆ES ranged from ~ 4.5 to ~6.8 kJ/mole from x = 0 to 1.

4.3. Cause of the Negative MGFE in the Na+ Ion Conductivity The total activation energy predicted by the CMAS model is shown in Figure 10. The almost linear composition dependence of the strain energy and its (relatively) small value relative to the electrostatic binding energy make the electrostatic binding energy the dominant effect on the total activation energy. Further, the CMAS calculated activation energy has two lobes with a local minimum occurring at x = 0.5, which is also consistent with the DAE ∆E0 values determined above. The CMAS model predicts the appropriate composition dependence of the activation energies and, by extension, the d.c. conductivities. The absolute values of the calculated total energy barriers are, however, approximately two times smaller than those



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28

determined experimentally, but the calculations provided by the CMAS model results do suggest insights into the origins of the negative MGFE in the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses. As described above, the preferential formation of more charged, -3, P0 groups is the predominant structural change that occurs across this compositional series and charge balancing requires the formation of less charged Ge3 structures. As seen in the measurements of the relative permittivity, this leads to an increase in the dielectric stiffness (a decreasing relative permittivity,

ε∞), which, in turn, causes an increase in the electrostatic binding energies, c.f. Eq. 8. These increased binding energies in the ternary glasses cause the increase in the activation energy. In effect, the P0 (and possibly Ge3) SROs act as Na+ ion traps in the glass. In an interesting contrasting behavior, in the binary yNa2S + (1-y)PS5/2 glasses the ionic conductivity is a monotonically increasing function of y from below, through, and to above the x = 0.5 composition here. Hence, in the binary glasses, the 25 oC d.c. conductivity increases and the

∆Eact decreases with added Na2S (increasing fractions of P0 SRO groups). This difference in the composition dependence of the ternary glasses where the conductivity decreases with increasing x and the binary glasses where the conductivity increases with increasing y could arise from the fact that increasing y in the binary glasses increases the Na+ concentration in the binary glasses and thereby reduces the jump distance λ between cation sites leading to reduced values of both

∆ES and ∆EC as expected from Eqs. 8 and 9 above. However, in the ternary 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses here, the atomic fraction of Na+ ions in the glass is a very slowly changing function of x (31% for the x = 0 glass and 33.3% for the x = 1 glass) but as Figure 2 shows, while the fraction of P0 groups increases with x so does the concentration of Na+ ion deficient Ge3 groups (Ge3M and Ge3B). It follows, then, that if as expected the P0 groups are


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29

homogeneously dispersed in the glass and do not cluster, the average Na+ ion jump distance λ to an adjacent site must increase, which as Eqs. 8 and 10 above show will increase both the coulombic and strain energy barriers to conduction. Another important observation here is that the ∆ES values are significantly smaller than the ∆EC values as shown in Figure 10. This suggests that these glasses are coulombically rather than volumetrically constrained in conduction. The former is presumably the case due to the fact that the disproportionation reaction, Eq. 1 above, causes the formation of more charged dense and charge localized P0 structures as x increases. It is seen in Figure 10 that the ∆ES values are a weak function of composition, increasing only slightly across the series of glass. However, close inspection of Figure 10 also shows that the calculated total activation energy, ∆Eact = ∆EC + ∆ES, is ~30 kJ/mole, and is therefore about half of the total measured conductivity activation energy. This arises because in the calculation of the ∆EC values, we have so far ignored the contribution of the many-body aspect of the coulomb potential and have not included the electrostatic interaction of all of the other anions (and cations) in the glass by including the Madelung, MD(Na+), constant in our calculations. It is to this last and lone adjustable parameter in the modeling that we now turn.

4.4. The Effective Sodium Ion Madelung Constant MD(Na+) Equation 8 above is our revised CMAS form of the coulombic binding energy and shows that we have included a Madelung constant formally into this energy expression. The Madelung constant was originally included in the AS treatment, but in the end it was dropped for the lack of the ability to adequately calculate this term. We have used the technique of MacFarlane et al.



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30

69

who developed a method for calculating the Madelung constant for disordered ionic liquid

organic salts and we have applied them here for the very first time to inorganic glasses and to the mobile ion population. In this technique, we use glassy state structures generated by molecular dynamics (MD) simulations that are excellent matches to the real structures of the same composition as shown through XRS and NS S(Q) and g(r) measurements and import the ion position coordinates into the method of Macfarlane et al69. While we are in the process of developing MD codes for the sulfide glass compositions here and cannot perform these calculations yet on these compositions, in Figure 12 we show a typical calculation of the MD(Na+) for a comparable 0.33Na2O + 0.66SiO2 glass. This plot was generated by creating a histogram of the MD values calculated for every Na+ in the simulation box. Two observations are important about this figure. First, the MD(Na+) values are sharply peaked in a narrow range of values and second the average value is about 2. This first observation shows that the charge structures around the mobile Na+ ions are relatively well ordered for all Na+ ions at least out to the first few ionic bond lengths. If this were not so, the distribution of MD(Na+) values would be broader. Secondly, the absolute magnitude of the most probable value of the distribution is centered near 2. Such a value is very comparable to M values for typical crystalline structures. For the present work on glasses, we cannot calculate the MD(Na+) values due to the lack of existing MD studies of these glasses. However, we can estimate the order of magnitude of the MD(Na+) values that are consistent with our CMAS calculations by difference using Eqs. 7, 8, and 9 and the experimental ∆E0 = ∆Eact values. Values of MD(Na+) calculated in this way such that the total measured and CMAS calculated activation energies agree are shown in Figure 13.


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31

Note that MD(Na+) calculated in this way is the only unknown parameter used to calculate the modeled conductivity activation energy. That the values agree strikingly well to values shown in Figure 12 for other Na+ ion conducting glasses of not too dissimilar chemistry, ~2, suggests that perhaps the methods proposed have some veracity. It is surely recognized that the CMAS model used here, even though significantly improved over the original AS model, is still a simple twobody potential simplification of the complex many-body Na+ ion dynamics in these glasses. However, the overall general trends observed here are none-the-less probably important because they suggest ways in which this negative MGFE can be inverted to a positive (and technologically relevant) MGFE and it is to this last topic that we now turn.

4.5. Prediction of Positive MGFE Glass Systems The prediction and design of MGF systems that lead to a positive MGFE in the ionic conductivity may be possible based on insights from the current study. One possible strategy is to consider the values of the various individual molar volumes of the various SROs VJ n in and the ε ∞− J n values of potential binary glass forming systems to be used in MGF systems. Since the mobile ion jump distance between cation sites is related to molar volume of the glass which is in _

turn related to the summation of the VJ n for example V P 0 , the modifier content for each end member glass can be chosen such that the molar volumes calculated on an atomic basis satisfy the relation V ( 0) ≅ V (1) . Next, at the chosen modifier content y for each binary end-member glass, ∆VJ n±1 should be small or negative to reduce the jump distance if mobile ions are disproportionately shared by the glass formers. Next, changes in the dielectric permittivity must



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be also be considered because as was seen above ε ∞ has a very strong effect on ∆Eact and therefore ε ∞ (x) must go through a maximum in order to cause a positive MGFE in ionic conductivity. Again, unequal sharing of the mobile ion population is expected as was seen here which implies that one glass-forming species will be over-modified relative to the other glassforming species which will then be under-modified. The over-modified glass former is expected to experience an increase in dielectric permittivity, while the under-modified glass former is expected to experience a decrease in dielectric permittivity because the increased modification leads to an increase in the net negative charge on the SRO units. As a result of this, the total change in dielectric permittivity must be positive to ensure a positive MGFE, or

∑∆ε

∞

>0.

5. Conclusions The 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses are found to exhibit a negative MGFE in Na+ ion conductivity and a non-Arrhenius temperature dependence. The non-Arrhenius behavior arises from the local distribution of bond lengths and angles inherent in glassy networks and is enhanced by the increased variety of anionic SROs in MGF glasses. The modified CMAS model of the activation energies in these glasses reveals the origin of the negative MGFE to be due to an increase in dielectric stiffness or a decrease in relative permittivity, which when coupled with an increase in the Na+ ion jump distance causes the activation energy to go through a maximum. A new concept, the effective Madelung constant for the Na+ ion, MD(Na+) was introduced to account for the difference between calculated and experimental activation energies. Values of MD(Na+) obtained were shown to both be consistent with typical values for crystalline materials and those calculated for typical ion conducting glasses. Insights from the current study have been


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33

used to predict and design of MGF systems that may lead to a positive MGFE in the ionic conductivity.

6. Acknowledgements This research was supported by the National Science Foundation under grants number DMR 0710564 and DMR 1304977 and this research support is gratefully acknowledged. The authors would like to thank Randi Christensen, Deborah Watson, and Brittany Curtis for their many useful discussions concerning this research project. We would like to acknowledge the assistance of Scott Beckman, Shen Li, and Jincheng Du for their many useful discussions and help in the calculation and use of the MD(Na+) values used in this paper in advance of their full publication. Finally, the authors would like to thanks members of the Glass and Optical Materials Research at ISU for their careful proof reading of the manuscript.



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7. References (1) Goodenough, J. B.; Kim, Y. Challenges for Rechargeable Li Batteries. Chem. Mater. 2010, 22, 587-603. (2) Whittingham, M. S. Materials Challenges Facing Electrical Energy Storage. MRS Bull. 2008, 33, 411-419. (3) Hayashi, A.; Noi, K.; Sakuda, A.; Tatsumisago, M. Superionic Glass-Ceramic Electrolytes for Room-Temperature Rechargeable Sodium Batteries. Nat. Comm. 2012, 3, 1843/18411843/1845. (4) Kim, Y.; Saienga, J.; Martin, S. W. Preparation and Characterization of Germanium OxySulfide GeS2-GeO2 Glasses. J. Non-Cryst. Sols. 2005, 351, 1973-1979. (5) Kim, Y.; Saienga, J.; Martin, S. W. Anomalous Ionic Conductivity Increase in Li2S + GeS2 + GeO2 Glasses. J. Phys. Chem. B 2006, 110, 16318-16325. (6) Pradel, A.; Kuwata, N.; Ribes, M. Ion Transport and Structure in Chalcogenide Glasses. J. Phys.: Cond. Mat. 2003, 15, S1561-S1571. (7) Pradel, A.; Rau, C.; Bittencourt, D.; Armand, P.; Philippot, E.; Ribes, M. Mixed Glass Former Effect in the System 0.3Li2S-0.7[(1-x)SiS2-xGeS2]: A Structural Explanation. Chem. Mat. 1998, 10, 2162-2166. (8) Schuch, M.; Mueller, C. R.; Maass, P.; Martin, S. W. Mixed Barrier Model for the Mixed Glass Former Effect in Ion Conducting Glasses. Phys. Rev. Lett. 2009, 102, 145902/145901145902/145904. (9) Bischoff, C.; Schuller, K.; Martin, S. W. Short Range Structural Models of the Glass Transition Temperatures and Densities of 0.5Na2S + 0.5[xGeS2 + (1 - x)PS5/2] Mixed Glass Former Glasses. J. Phys. Chem. B 2014, 118, 3710-3719. (10) Bischoff, C.; Schuller, K.; Dunlap, N.; Martin, S. W. IR, Raman, and NMR Studies of the Short-Range Structures of 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] Mixed Glass-Former Glasses. J. Phys. Chem. B 2014, 118, 1943-1953. (11) Christensen, R.; Olson, G.; Martin, S. W. Ionic Conductivity of Mixed Glass Former 0.35Na2O + 0.65[xB2O3 + (1 - x)P2O5] Glasses. J. Phys. Chem. B 2013, 117, 16577-16586. (12) Christensen, R.; Olson, G.; Martin, S. W. Structural Studies of Mixed Glass Former 0.35Na2O + 0.65[xB2O3 + (1 - x)P2O5] Glasses by Raman and 11B and 31P Magic Angle Spinning Nuclear Magnetic Resonance Spectroscopies. J. Phys. Chem. B 2013, 117, 2169-2179. (13) Storek, M.; Bohmer, R.; Martin Steve, W.; Larink, D.; Eckert, H. NMR and Conductivity Studies of the Mixed Glass Former Effect in Lithium Borophosphate Glasses. J. Chem. Phys. 2012, 137, 124507. (14) Schuch, M.; Christensen, R.; Trott, C.; Maass, P.; Martin, S. W. Investigation of the Structures of Sodium Borophosphate Glasses by Reverse Monte Carlo Modeling to Examine the Origins of the Mixed Glass Former Effect. J. Phys. Chem. C 2012, 116, 1503-1511. (15) Larink, D.; Eckert, H.; Reichert, M.; Martin, S. W. Mixed Network Former Effect in IonConducting Alkali Borophosphate Glasses: Structure/Property Correlations in the System [M2O]1/3[(B2O3)x(P2O5)1-x]2/3 (M = Li, K, Cs). J. Phys. Chem. C 2012, 116, 26162-26176. (16) Larink, D.; Eckert, H.; Martin, S. W. Structure and Ionic Conductivity in the MixedNetwork Former Chalcogenide Glass System [Na2S]2/3[(B2S3)x(P2S5)1-x]1/3. J. Phys. Chem. C 2012, 116, 22698-22710.


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(17) Christensen, R.; Byer, J.; Olson, G.; Martin, S. W. The Densities of Mixed Glass Former 0.35Na2O + 0.65[xB2O3 + (1 - x)P2O5] Glasses Related to the Atomic Fractions and Volumes of Short Range Structures. J. Non-Cryst. Sols. 2012, 358, 583-589. (18) Christensen, R.; Byer, J.; Olson, G.; Martin, S. W. The Glass Transition Temperature of Mixed Glass Former 0.35Na2O + 0.65[xB2O3 + (1 - x)P2O5] Glasses. J. Non-Cryst. Sols. 2012, 358, 826-831. (19) Schuch, M.; Muller, C. R.; Maass, P.; Martin, S. W. Mixed Barrier Model for the Mixed Glass Former Effect in Ion Conducting Glasses. Phys. Rev. Lett. 2009, 102, 145902. (20) Le Messurier, D.; Petkov, V.; Martin, S. W.; Kim, Y.; Ren, Y. Three-Dimensional Structure of Fast Ion Conducting 0.5Li2S + 0.5[(1- x)GeS2 + xGeO2] Glasses from High-Energy X-Ray Diffraction and Reverse Monte Carlo Simulations. J. Non-Cryst. Sols. 2009, 355, 430437. (21) Kim, Y.; Saienga, J.; Martin, S. W. Anomalous Ionic Conductivity Increase in Li2S + GeS2 + GeO2 Glasses. J. Phys. Chem. B 2006, 110, 16318-16325. (22) Meyer, B.; Borsa, F.; Martin, D. M.; Martin, S. W. NMR Spin-Lattice Relaxation and Ionic Conductivity in Lithium Thioborogermanate Fast-Ion-Conducting Glasses. Phys. Rev. B: Condens. Mat. Mater. Phys. 2005, 72, 144301/144301-144301/144313. (23) Kim, Y.; Saienga, J.; Martin, S. W. Glass Formation in and Structural Investigation of Li2S + GeS2 + GeO2 Composition Using Raman and Ir Spectroscopy. J. Non-Cryst. Sols. 2005, 351, 3716-3724. (24) Mei, Q.; Martin, S. W. Structural Investigation of Ag2S+B2S3+GeS2 Glasses Using Neutron Diffraction. Phys. Chem. Glasses 2005, 46, 51-57. (25) Meyer, B.; Borsa, F.; Martin, S. W. Structure and Properties of Lithium Thio-BoroGermanate Glasses. J. Non-Cryst. Sols. 2004, 337, 166-173. (26) Mei, Q.; Meyer, B.; Martin, D.; Martin, S. W. Ion Trapping Model and the Non-Arrhenius Ionic Conductivity in Fast Ion Conducting Glasses. Solid State Ionics 2004, 168, 75-85. (27) Mei, Q.; Saienga, J.; Schrooten, J.; Meyer, B.; Martin, S. W. Preparation and Characterization of Glasses in the Ag2S+B2S3+GeS2 System. J. Non-Cryst. Sols. 2003, 324, 264276. (28) Mei, Q.; Saienga, J.; Schrooten, J.; Meyer, B.; Martin, S. W. Characterisation of Glasses in the Ag2S-B2S3-GeS2 System. Phys. Chem. Glasses 2003, 44, 178-180. (29) Bischoff, C.; Schuller, K.; Beckman, S. P.; Martin, S. W. Non-Arrhenius Ionic Conductivities in Glasses Due to a Distribution of Activation Energies. Phys. Rev. Lett. 2012, 109, 075901/075901-075901/075904. (30) Svare, I.; Martin, S. W.; Borsa, F. Stretched Exponentials with T-Dependent Exponents from Fixed Distributions of Energy Barriers for Relaxation Times in Fast-Ion Conductors. Phys. Rev. B: Condens. Mat. Mater. Phys. 2000, 61, 228-233. (31) Martin, S. W.; Borsa, F.; Svare, I. Distribution of Activation Energies Treatment of Fast Ion Motions in Glass. Proc.- Electrochem. Soc. 2000, 2000-32, 66-78. (32) Svare, I. Conductivity and NMR Relaxation from Ionic Motion in Disordered Glasses with Distributions of Barriers. Solid State Ionics 1999, 125, 47-53. (33) Kim, K. H.; Torgeson, D. R.; Borsa, F.; Cho, J. P.; Martin, S. W.; Svare, I.; Majer, G. Evidence of Complex Ionic Motion in xLi2S+(1-x)B2S3 Glassy Fast Ionic Conductors from 7Li and 11B NMR and Ionic Conductivity Measurements. J. Non-Cryst. Sols. 1997, 211, 112-125.



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(34) Kim, K. H.; Torgeson, D. R.; Borsa, F.; Cho, J.; Martin, S. W.; Svare, I. Distribution of Activation Energies Explains Ionic Motion in Glassy Fast Ion Conductors: 7Li NMR SpinLattice Relaxation and Ionic Conductivity in xLi2S+(1-x)GeS2. Solid State Ionics 1996, 91, 7-19. (35) Martin, S. W.; Martin, D. M.; Schrooten, J.; Meyer, B. M. Trapping Model of the NonArrhenius Ionic Conductivity in Fast Ion Conducting Glasses. Phys. Chem. Glasses 2003, 44, 181-186. (36) Schrooten, J.; Meyer, B.; Martin, S. W. Structural Characterization of Silver Thioborosilicate Glasses J. Non-Cryst. Sols. 2003, 318, 27-36. (37) Martin, S. W.; Schrooten, J.; Meyer, B. Non-Arrhenius Ionic Conductivity in Optimized Fast Ion Conducting Glasses: Application of the Drude Model to Ion-Ion Scattering. J. NonCryst. Sols. 2002, 307-310, 981-991. (38) Kincs, J.; Martin, S. W. Non-Arrhenius Conductivity in Glass: Mobility and Conductivity Saturation Effects. Phys. Rev. Lett. 1996, 76, 70-73. (39) Ngai, K. L.; Rizos, A. K. Parameterless Explanation of the Non-Arrhenius Conductivity in Glassy Fast Ionic Conductors. Phys. Rev. Lett. 1996, 76, 1296-1299. (40) Maass, P.; Meyer, M.; Bunde, A.; Dieterich, W. Microscopic Explanation of the NonArrhenius Conductivity in Glassy Fast Ionic Conductors. Phys. Rev. Lett. 1996, 77, 1528-1531. (41) Courtois, G. Preparation of Anhydrous Sodium Sulfide. C. R. Hebd. Seances Acad. Sci. 1938, 207, 1220-1221. (42) Bischoff, C.; Schuller, K.; Haynes, M.; Martin, S. W. Structural Investigations of yNa2S + (1-y)PS5/2 Glasses Using Raman and Infrared Spectroscopies. J. Non-Cryst. Sols. 2012, 358, 3216-3222. (43) Berbano, S. S.; Seo, I.; Bischoff, C. M.; Schuller, K. E.; Martin, S. W. Formation and Structure of Na2S + P2S5 Amorphous Materials Prepared by Melt-Quenching and Mechanical Milling. J. Non-Cryst. Sols. 2012, 358, 93-98. (44) Julien, C.; Massot, M. Complex Impedance Spectroscopy. Microionics 1991, 173-195. (45) Macdonald, J. R. Impedance Spectroscopy and Its Use in Analyzing the Steady-State AC Response of Solid and Liquid Electrolytes. J. Electroanal. Chem. Interfac. Electrochem. 1987, 223, 25-50. (46) Souquet, J. L.; Robinel, E.; Barrau, B.; Ribes, M. Glass Formation and Ionic Conduction in the M2S-GeS2 (M = Li, Na, Ag) Systems. Solid State Ionics 1981, 3-4, 317-321. (47) Ribes, M.; Barrau, B.; Souquet, J. L. Sulfide Glasses: Glass Forming Region, Structure and Ionic Conduction of Glasses in Na2S-XS2 (X = Si; Ge), Na2S-P2S5 and Li2S-GeS2 System. J. Non-Cryst. Sols. 1980, 38-39, 271-276. (48) Barrau, B.; Ribes, M.; Maurin, M.; Kone, A.; Souquet, J. L. Glass Formation, Structure and Ionic Conduction in the Sodium Sulfide-Germanium Sulfide (Na2S-GeS2) System. J. NonCryst. Sols. 1980, 37, 1-14. (49) Barrau, B.; Kone, A.; Ribes, M.; Souquet, J. L.; Maurin, M. Synthesis and Study of the Electrical Conductivity of Glasses Belonging to the Sodium Monosulfide-Germanium Disulfide System. C. R. Hebd. Seances Acad. Sci., Ser. C 1978, 287, 43-46. (50) Nascimento, M. L. F. Test of the Anderson-Stuart Model and Correlation between Free Volume and the 'Universal' Conductivity in Sodium Silicate Glasses. J. Mater. Sci. 2007, 42, 3841-3850.


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(51) Nascimento, M. L. F.; Dantas, N. O. Anderson-Stuart Model of Ionic Conductors in Na2O-SiO2 Glasses. Cien. Engen. 2003, 12, 7-13. (52) Hsieh, C. H.; Jain, H. Influence of Network-Forming Cations on Ionic Conduction in Sodium Silicate Glasses. J. Non-Cryst. Sols. 1995, 183, 1-11. (53) Seddon, E.; Tippett, E. J.; Turner, W. E. S. Electrical Conductivity of Sodium Metasilicate-Silica Glasses. J. Soc. Glass Technol. 1932, 16, 450-476. (54) Maia, L. F.; Rodrigues, A. C. M. Electrical Conductivity and Relaxation Frequency of Lithium Borosilicate Glasses. Solid State Ionics 2004, 168, 87-92. (55) Martin, S. W.; Angell, C. A. D.C. And A.C. Conductivity in Wide Composition Range Lithium Oxide-Phosphorus Oxide (Li2O-P2O5) Glasses. J. Non-Cryst. Sols. 1986, 83, 185-207. (56) Anderson, O. L.; Stuart, D. A. Calculation of Activation Energy of Ionic Conductivity in Silica Glasses by Classical Methods. J. Am. Ceram. Soc. 1954, 37, 574-580. (57) McElfresh, D. K.; Howitt, D. G. Activation Enthalpy for Diffusion in Glass. J. Amer. Ceram. Soc. 1986, 69, C237-C238. (58) Shannon, R. D. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Crystallogr., Sect. A 1976, A32, 751-767. (59) Itoh, K.; Fukunaga, T. Structure of Na2S-GeS2 Glasses Studied by Using Neutron and XRay Diffraction and Reverse Monte Carlo Modeling. Solid State Ionics 2009, 180, 351-355. (60) Itoh, K.; Sonobe, M.; Sugiyama, M.; Mori, K.; Fukunaga, T. Structural Study of (Li2S)50(GeS2)50 Glass by X-Ray and Neutron Diffraction. J. Non-Cryst. Sols. 2008, 354, 150154. (61) Cherry, B.; Zwanziger, J. W.; Aitken, B. G. The Structure of GeS2-P2S5 Glasses. J. Phys. Chem. B 2002, 106, 11093-11101. (62) Blaineau, S.; Jund, P. Structural Properties of Various Sodium Thiogermanate Glasses through DFT-Based Molecular Dynamics Simulations. Phys. Rev. B Condens. Mat. Mater. Phys. 2006, 74, 054203/054201-054203/054206. (63) Claasen, I. A. The Crystal Structure of the Anhydrous Alkali Monosulfides. R. T. Chim. Pays-Bas Bel. 1925, 44, 790-794. (64) Walker, P. J. Crystal Growth of Sodium Sulfide, Na2S. J. Cryst. Growth 1979, 47, 598600. (65) Kizilyalli, M.; Bilgin, M.; Kizilyalli, H. M. Solid-State Synthesis and X-Ray Diffraction Studies of Sodium Sulfide (Na2S). J. Solid State Chem. 1990, 85, 283-292. (66) Shtets, P. P.; Fedelesh, V. I.; Kabatsij, V. M.; Malesh, V. I.; Shpak, I. I.; Gorvat, A. A. Structure, Dielectric and Photoelastic Properties of Glasses in the System Ge-Sb-S. J. Optoelectron. Adv. Mater. 2001, 3, 937-940. (67) Savchenko, N. D.; Shchurova, T. N.; Kondrat, A. B.; Mitsa, V. M. Calculation of Elastic Constants for (GeS2)x(As2S3)1-x Glass. AIP Conf. Proc. 2007, 963, 1363-1366. (68) Gilman, J. J. Chemistry and Physics of Mechanical Hardness; John Wiley & Sons, 2009. (69) Izgorodina, E. I.; Bernard, U. L.; Dean, P. M.; Pringle, J. M.; MacFarlane, D. R. The Madelung Constant of Organic Salts. Cryst. Growth Des. 2009, 9, 4834-4839.



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38

Tables

Table 1. Number of bS, nbSj, and nbS, NnbSj, associated with each Qn structural unit. P1P is considered to have 0.5 bridging sulfurs to account for its connectivity. NnbSj values are on a 0.5 glass forming cation basis. Table 2. Parameters and results of the CMAS model of the activation energies in the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses.


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39

8. Figures

Figure 1. SRO structural units found in binary Na2S + PS5/2 and Na2S + GeS2, and ternary Na2S + PS5/2 + GeS2 glasses. Figure 2. Fraction of SRO units shown in Figure 1 in the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses. Figure 3. Temperature dependence of the ionic conductivity, σd.c., of the 0.5Na2S + 0.5[0.7GeS2 + 0.3PS5/2] glass, which is representative of the non-Arrhenius temperature dependence of all the glass samples. The solid red line is the best-fit to Eq. 4 and the faint dashed line is the simple single valued ∆Eact Arrhenius line. Figure 4. Composition dependence of the ionic conductivity, σd.c., of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses over a range of temperatures around room temperature.. Figure 5. DAE model parameters, ∆E0, δ, and σ0, determined from best-fits of the σd.c. of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses to Eq. 4. Figure 6. Temperature dependence of the ionic conductivity, σd.c., of the 0.35Na2O + 0.65[0.4B2O3 + 0.3P2O5] glass. The solid red line is the best-fit to Eq. 4 and the solid blue line is the Arrhenius line. Figure 7. (Top) Schematic representation of the structural interpretation of the CMAS model of the energy surface connecting two adjacent cation sites in a sulfide glass. It is assumed that the predominate charge species adjacent to the mobile cation is the nbS and that the cation resides in a tetrahedral site and conducts through the interstitial volume created by the trigonal face of the tetrahedron. This is shown schematically here by the two inplane bS and the one out of plane, here taken to be above the plane of the two bS atoms. (Bottom) Schematic representation of the cation jump distance λcation as envisioned and used here in the CMAS model. The net jump distance as described in the text is rNa + rS + λcation/2. Figure 8. Plot of the real part of the dielectric permittivity, ε’(f), for the 0.5Na2S + 0.5[0.7GeS2 + 0.3PS5/2] glass at -50 oC, where the high frequency plateau is identified as the limiting high frequency relative permittivity, ε’∞ here found to 18.55 for this glass. Figure 9. Composition dependence of the limiting high frequency dielectric permittivity, ε’∞ , of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses as a function of x.



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40

Figure 10. Calculated ∆ES, ∆EC, and their sum, ∆Eact = ∆ES + ∆EC as described in the text. In this figure ∆EC does not include the effective Na+ ion Madelung constant, MD(Na+). It is seen that even without MD(Na+) included, ∆EC is significantly larger than ∆ES and for this reason, the Na+ ions in these glasses must be considered coulombically rather than volumetrically constrained in their conduction events. Figure 11. Calculated values of the shear moduli of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses using Eqs. 11 and 12. The G values are fairly modest, ~13 GPa, compared to ~ 30 GPa for typical oxide glasses. G increases with x towards the pure Na2S + GeS2 glass as does the Tg which is ~ 190 oC for pure 0.5Na2S + 0.5PS5/2 and ~ 240 oC for pure 0.5Na2S + 0.5GeS2. Such correlation between G and Tg is expected and has been seen for many different glasses. Figure 12. Plot of the calculated effective MD(Na+) values for five different MD simulations of the sodium disilicate glass, 0.33Na2O + 0.67SiO2 using the x, y, and z coordinates of all of the Na+, Si+4, and O-2 ions in the simulation box corrected for density and structure. The method of Macfarlane in ref.69 as described in the text was used for these calculations. Values comparable to these were found for the present glasses and suggests that Na+ ions in typical four-fold coordination have MD(Na+) values of ~ 2. Figure 13. Calculated values of effective MD(Na+) values for all of the 0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] glasses required to bring the calculated values of the total activation energy ∆Eact = ∆ES + ∆EC from Figure 10 into agreement with the experimental values of ∆E0.


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41

Table 1 Qn

Net Bridging Sulfur/Qn

Net Non-Bridging Sulfur/Qn

P2

1

0.5

P1

0.5

0.25

P1P

0.5

0.25

P

0

0

0

Ge

4

2

1

Ge

3

1.5

0.75

Ge

2

1

0.5



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42

Table 2

x

+

ZNa ZS MD(Na )

rNa + rS

ߝஶ

λcation (Å)

G (GPa)

4.26 4.25 4.25 4.22 4.21 4.20 4.19 4.18 4.17 4.16

13.1 13.2 13.3 13.6 13.9 13.9 14.1 14.3 14.4 14.2

(Å)

∆E0 (kJ/mole)

∆EC4 (kJ/mole)

∆ES (kJ/mole)

0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65

60.0 66.0 65.6 67.7 64.7 63.7 66.7 68.5 65.7 56.8

49.2 55.2 54.7 56.6 53.3 52.4 55.3 56.9 54.1 45.3

10.8 10.9 10.9 11.2 11.3 11.3 11.5 11.5 11.6 11.5

rD

(Å)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0

1 1 1 1 1 1 1 1 1 1

4

1 1 1 1 1 1 1 1 1 1

1.96 2.16 2.13 2.21 1.98 2.16 2.13 2.13 2.00 1.96

19.72 19.38 19.23 19.33 18.34 20.37 19.08 18.55 18.29 21.43

2.81 2.81 2.81 2.81 2.81 2.81 2.81 2.81 2.81 2.81

This value of ∆EC includes the Na+ Madelung constant, MD(Na+), shown to the left, such that ∆EC + ∆ES =∆E0 in this table, see text.


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43

Figure 1



44

0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] 1.0

Relative Abundance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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P2 P1 P1P P0 Ge4 Ge3B Ge3M Ge2

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

x

Figure 2


0.8

1.0

Page 45 of 56

45

-5 0

-2 5

0

25

50

22 205 170 155 120 5 10 0 75

0.5Na2S+0.5[0.7GeS2+0.3PS5/2]

T (oC)

-1

log10(σd.c.T) (K/Ωcm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-2 -3 -3 -4 -5

Data DAE Eq. 4

Arrhenius -6 -7 2.0

2.5

3.0

3.5

4.0 -1

1000/T (K ) Figure 3


4.5

5.0


46

10

0.5Na2S+0.5[xGeS2+(1-x)PS5/2]

-5

50 °C 37.5 °C

σd.c. (Ωcm)-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10-6

25 °C 12.5 °C 0 °C

10-7 0.0

0.2

0.4

0.6

0.8

x Figure 4


1.0

Page 47 of 56

47

DAE Fit Parameters 0.5Na2S+0.5[xGeS2+(1-x)PS5/2]

∆E0(kJ/mol)

72 68 64 60 56

δ (kJ/mol)

4.4 4.0 3.6 3.2 6

3

-1

σo (10 (Ωcm) )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 2 0

0.0

0.2

0.4

x

0.6

Figure 5


0.8

1.0


-1 00

T (oC)

-5 0

0

50

35 300 250 200 0 15 0 10 0

48

0

log10(σd.c.T) (K/Ωcm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 56

-2

DAE parameters σ0 = 1.28x107 K/Ωcm

-4

∆E0 = 107.0 kJ/mol δ

= 7.79 kJ/mol

-6 -8 -10 -12 -14

0.35Na2O + 0.65[0.4B2O3 + 0.6P2O5] Data Arrhenius DAE Eq. 4

-16 -18 1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

1000/T (K)

Figure 6


5.5

6.0

6.5

Page 49 of 56

49

bS

- nbS

2rD

+

+ - nbS

bS

n

+1/r

Energy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


∆ES ∆EC

2

-e /r

r

r λcation/2

Figure 7



50

0.5Na2S + 0.5[0.7GeS2 + 0.3PS5/2] 200

100 80

ε'(f)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

60 40

20

ε∞ = 18.55 T = -50 °C

10 -2 10 10-1 100 101 102 103 104 105 106 107 Frequency (Hz)

Figure 8


Page 50 of 56

Page 51 of 56

51

22

0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2]

21

20

ε'∞

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


19

18

17 0.0

0.2

0.4

x

0.6

Figure 9


0.8

1.0


52

50

0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2] ∆Eact = ∆EC + ∆ES (kJ/mole)

∆E (kJ/mole)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 56

40 30

∆EC (kJ/mole)

20

∆ES (kJ/mole) 10 ∆Eact

0

0.0

 Z Na Z S e2  1 1 = − 4πε 0ε ∞  rNa + rS r + r + λcation Na S 2 

0.2

0.4

0.6

X

Figure 10


  + π G λ ( r − r )2 Na D  2 

0.8

1.0

Page 53 of 56

15.0

14.5

G (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5Na2S + 0.5[xGeS2 +53(1-x)PS5/2]

3e 2 G= 4π ( rNa + rS ) α

14.0

13.5

13.0

12.5

0.0

0.2

0.4

0.6

0.8

x

Figure 11


1.0


54

100

Number of Na+ ions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 54 of 56

0.33Na2O + 0.67SiO2 Simulation 1 2 3 4 5

80 60 40 20 0 1.0

1.5

2.0

2.5

MD(Na+)

Figure 12


3.0

Page 55 of 56

55

2.3

0.5Na2S + 0.5[xGeS2 + (1-x)PS5/2]

MD(Na )

2.2 +

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.1 2.0 1.9 1.8

0.0

0.2

0.4

x

0.6

0.8

Figure 13


1.0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

TOC Entry


Page 56 of 56

Composition Dependence of the Na+ Ion Conductivity in 0.5Na2S +

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