Fast-ion conduction and flexibility and rigidity of solid electrolyte glasses
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Fast-ion conduction and flexibility and rigidity of solid electrolyte glasses
M. Micoulaut , M. Malki , , D.I. Novita , P. Boolchand Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,Universit´e Pierre et Marie Curie, Boite 121, 4, Place Jussieu, 75252 Paris Cedex 05, France CEMHTI,CNRS, UPR 3079 1D Avenue de la recherche scientifique, 45071 Orl´eans Cedex 02, France Universit´e d’Orl´eans (Polytech’ Orl´eans), BP 6749, 45072 Orl´eans Cedex 02, France Department of Electrical and Computer Engineering,University of Cincinnati, Cincinnati, OH 45221-0030, USA (Dated: November 15, 2018)Electrical conductivity of dry, slow cooled (AgPO ) − x (AgI) x glasses is examined as a function oftemperature , frequency and glass composition. From these data compositional trends in activationenergy for conductivity E A (x), Coulomb energy E c (x) for Ag + ion creation, Kohlrausch stretchedexponent β (x), low frequency ( ε s (x)) and high-frequency ( ε ∞ (x)) permittivity are deduced. Allparameters except E c (x) display two compositional thresholds, one near the stress transition, x =x c (1)= 9%, and the other near the rigidity transition, x = x c (2)= 38% of the alloyed glass network.These elastic phase transitions were identified in modulated- DSC, IR reflectance and Raman scatter-ing experiments earlier. A self-organized ion hopping model (SIHM) of a parent electrolyte system isdeveloped that self-consistently incorporates mechanical constraints due to chemical bonding withcarrier concentrations and mobility. The model predicts the observed compositional variation of σ (x), including the observation of a step-like jump when glasses enter the Intermediate Phase atx > x c (1), and an exponential increase when glasses become flexible at x > x c (2). Since E c is foundto be small compared to network strain energy (E s ), we conclude that free carrier concentrationsare close to nominal AgI concentrations, and that fast-ion conduction is driven largely by changesin carrier mobility induced by an elastic softening of network structure. Variation of the stretchedexponent β (x) is square- well like with walls localized near x c (1) and x c (2) that essentially coincidewith those of the Intermediate Phase (IP) (x c (1) < x < x c (2)), and suggest filamentary (quasi 1D)conduction in the IP, and conduction with a dimensionality greater than 1 outside the IP. PACS numbers: 61.43.Fs-61.20.-x
I. INTRODUCTION
Fast-ion conduction in solid electrolyte glasses hasbeen investigated - for over two decades and our ba-sic understanding continues to be challenged with newfindings - in the field. Resolution of these basicchallenges including the role of molecular structure andimpurities on fast-ion conduction is likely to have a di-rect impact on applications of these materials in solidstate batteries, sensors, and non-volatile memories.Electrical conductivity σ can be generally be writtenas σ = Zeµn L (1)where the terms Ze , µ , n L represent respectively thecharge of the conducting ion, its mobility and concentra-tion. Beyond this point of discussion, it is clear that al-loying ionic species in a base glass will lead to an increasein carrier concentration in a host network, but it will alsosubstantially modify the mechanical behaviour by eitherdepolymerising or softening the base structure. Solidelectrolyte additives such as AgI, Ag S and Ag Se inbase chalcogenide glasses are known to possess a low con-nectivity and their glass forming tendency derives fromthat feature of structure - . Recently Ingram et al.have proposed that the activation energy for conduc-tion (E A ) could be displayed under the form of a master plot with respect to activated volume (V A ) as E A = M V A (2)where M represents a localized modulus of elasticity(strongly influenced by intermolecular forces) with val-ues ranging between 1 and 8 GPa depending on the na-ture of chemical bonding involved. One may, therefore,expect that ionic conduction can be related to an elasticsoftening of a glass network with one functionality (elas-ticity) affecting the other (conductivity). Can this viewbe made more quantitative?Conductivity thresholds in fast ionic conductors havebeen reported although the relationship with rigiditytransitions was not made. Micro-segregation effects havealso been reported in various modified oxide glassesincluding silicates and supported by emerging lengthscales in MD simulated partial pair distribution func-tions. These findings are popularized by the channelpicture for alkali ion-conduction in sodium silicates dueto Greaves and Ngai . In this glass system, the sodacontent where a micro-segregation onsets is nearly thesame as the stress elastic phase transition (x = 18%)determined from calorimetric measurements . A natu-ral question that follows then is, are there links betweenthe elastic behaviour of glasses and micro-segregation ef-fects? Is this a generic behaviour or is it typical of alkali-modified silicates?(AgPO ) − x (AgI) x glasses have been widely studiedin the past by several groups - . A perusal of this /T [K -1 ]10 -5 -4 -3 -2 -1 σ T [ K . Ω - . c m - ] x=0 x=25x=54 FIG. 1: (Color online) Plot of electrical conductivity ( σ ) mul-tiplied by T as a function of 1000/T at the three indicatedAgI content x in (AgPO ) − x -(AgI) x glasses displaying anArrhenius variation. large body of work reveals broad compositional trendsin thermal, optical and electrical properties of these ma-terials, but not without significant variations betweenvarious groups. In some case the conductivity increasesmonotonically with AgI content displaying either nothreshold or one threshold near x = 30% - . In morerecent reports , , , we have emphasized the need tokeep samples dry during synthesis and handling, and toslow cool them from T g to assure measuring what webelieve are the intrinsic physical properties of these ma-terials. The thermal, optical, molar volumes and someelectrical properties were reported , and these are foundto be quite different from those reported earlier in theliterature , , in that many properties including con-ductivity display two striking compositional thresholds.The data from these more recent investigations , ex-amined as a function of glass composition are found todisplay striking correlations between various observables.For example, these data have permitted connecting as-pects of local structure deduced from Raman and IR vi-brational spectroscopy to their thermal as well as elec-trical transport properties. The elastic nature of back-bones derived from Lagrangian constraints, have permit-ted treating short-range intermolecular forces in an effec-tive way using rigidity theory.In the present contribution we present temperature de-pendent and frequency dependent conductivity data onthese glasses as a function of AgI content x. Here weaddress an issue of major importance in the field of solidelectrolytes, namely, the relationship of the three definedelastic phases: flexible, intermediate, stressed rigid tothe dielectric permittivity derived from the frequency-dependent electrical modulus. We find that (a) the di-electric permittivity is a maximum in the intermediatephase (IP), (b) the Kohlrausch stretched exponent β dis- plays a minimum there and (c); we also infer there aredimensional changes in conduction pathways with chang-ing glass composition, and (d) that Coulomb barrier , E c ,determined from the high-frequency permittivity makesonly a small contribution to the Arrhenius activation en-ergy for dc conduction, E A , compared to network strainenergy, E s . These findings suggest that free carrier con-centrations are nearly equal to one, and that they closelyfollow the AgI concentrations. The observed composi-tional trends in dc conductivity, σ (x) appear to derivefrom elastic network softening which controls carrier mo-bility. The observed precipitous increase of conductivityin the flexible phase may well be a generic feature of solidelectrolyte glasses. II. EXPERIMENTALA. Synthesis of dry AgI-AgPO glasses Bulk (AgPO ) − x (AgI) x glasses were synthesized us-ing 99.9% Ag PO (Alpha Aesar Inc.), 99.5% P O ( Fis-cher Scientific Inc.), and 99.99% AgI (Alpha Aesar Inc.),and separately AgNO and NH H (PO ) with AgI asthe starting materials. Details of the synthesis are dis-cussed elsewhere . In our experiments, the starting ma-terials were handled in a dry nitrogen gas purged glovebox (Vacuum Atmospheres model HE-493/MO-5, rela-tive humidity ≪ o C in a chemical hood purged by lab-oratory air, and heated at 100 o C/hr to 700 o C. Meltswere equilibrated overnight and then quenched over steelplates. Calorimetric experiments also revealed thatstress frozen upon melt quenching can be relieved bycycling samples through the glass transition tempera-ture (T g ). Conductivity measurements were undertakenonly on samples cycled through T g and slow cooled at3 o C/min. to room temperature.
B. AC conductivity measurements as a function oftemperature and frequency
Glass sample disks about 10mm in diameter and 2mm thick were synthesized by pouring melts into spe-cial troughs. Pellets were then thermally relaxed bycycling through T g . Platelets then were polished, andthe Pt electrodes deposited. The complex impedanceZ ∗ ( ω )=Z’( ω )+iZ”( ω ) of the specimen was measured bya Solartron SI 1260 impedance analyzer over a fre-quency range of 1 Hz-1 MHz (19 points per measure-ments). The temperature was raised from 150 K to T g at a rate of 1 o C/min. All data were acquired auto-matically at 2 min intervals. The complex conductiv-ity σ ∗ ( ω )= σ ’( ω )+i σ ”( ω ) is deduced from the complex x (AgI in %) A c ti v a ti on e n e r gy E A [ e V ] σ E A σ stressed intermediate flexible x c (1) x c (2) FIG. 2: (Color online) Variations in activation energy E A (x)(blue) and preexponential factor σ (x) (red) as a function ofAgI content x in percent in (AgPO ) − x -(AgI) x glasses. impedance Z ∗ , the sample thickness t and the area Sof the surface covered by platinum using the formula : σ ∗ ( ω )=(t/S)(1/Z ∗ ). III. EXPERIMENTAL RESULTSA. Temperature dependence of conductivity
The temperature dependence of electrical conductivityfor ionic conduction in glasses usually displays an Arrhe-nius dependence σ = σ exp[ − E A /k B T ] (3)We have obtained the activation energy E A (x), by per-forming temperature dependent measurements in the40 o C < T < T g range. An illustrative example of some ofthese results, at three glass compositions, is reproducedin Fig 1. From the temperature dependence, we haveobtained the preexponential factor σ and the activationenergy E A as a function of AgI content, and these dataare reproduced in Fig.2. One finds that E A (x) steadilydecreases as x increases, particularly for x >
40% while σ drops rapidly with x at low x ( < σ (x) displays three regimes ofconduction whatever the temperature (Fig. 3a). At lowx ( < σ (x) is found to be nearly independent of x.The conductivity then builds up at higher x ( > > σ (x) sets in. The location of the twoelectrical conductivity thresholds at x = 9% and x =38% coincide with steps in the non-reversing enthalpy atT g that define the boundary of the Intermediate Phase(IP) (Figure 3b). These data will be discussed in sectionIV. l og σ [ Ω − . . c m - ] ∆ H n r ( ca l/ g ) - 50 o C50 o C100 o C stressed intermediate flexiblex c (1) x c (2) FIG. 3: (Color online) a: Variations in DC conductivity σ (x)at three temperatures, -50 o C, 50 o C and 100 o C, and b: Non-reversing heat flow ∆H nr at T g in (AgPO ) − x -(AgI) x glassesas a function of AgI content in mole %. B. Frequency dependence of AC conductivity
The frequency dependence of AC conductivity, σ ( ω ) atthree representative glass compositions is summarized inFigure 4. These data show a saturation of σ ( ω ) at lowfrequencies ( ω /2 π <
100 Hz), and an increase of about2 orders of magnitude in the 10 < ω < range. Thelow frequency saturation value of σ is found to increasesteadily as the AgI content of glasses x increases from10% to 40%, a behaviour that is consistent with datain Fig. 3. The present results on the frequency depen-dence σ ( ω ) are similar to earlier reports of Sidebottom and separately by Stanguenec and Elliott . For a morequantitative analysis of these σ ( ω ) data, we have deducedthe frequency dependence of the dielectric permittivity, ε ∞ ( ω ) from the measured AC conductivity. C. Electrical modulus analysis
The behaviour of the complex conductivity σ ∗ ( ω )= σ ’( ω )+i σ ”( ω ) can be analyzed using the complex FIG. 4: Frequency dependence of electrical conductivity in(AgPO ) − x -(AgI) x glasses at three indicated AgI concentra-tions x. Measurements were taken at 250K. permittivity ε ∗ ( ω )= ε ’( ω )+i ε ”( ω ) or the complex electri-cal modulus M ∗ ( ω )=M’( ω )+iM”( ω ) which are relatedto the complex conductivity via: ε ∗ ( ω ) = σ ∗ ( ω ) iωε (4)and M ∗ ( ω ) = iωε σ ∗ ( ω ) = 1 ε ∗ ( ω ) (5)Here ε designates the permittivity of free space of8 . × − Farad/meter. From the measured value of σ ”, using equation (4) we could deduce the real part ofthe permittivity ε ’. These data are presented on a log-log plot for a glass at a composition x = 25% in Figure5. Permittivity results generally from mobile ions, butthe sharp increase in ε ’ at low frequencies, particularlyat f = ω /2 π <
10 Hz, is due to electrode polarization ef-fects as Ag + ions are deposited at the anode at very lowfrequencies . The behaviour is observed for all samplesbut the magnitude of ε s could be unambiguously deter-mined. The saturation value of the permittivity at lowfrequency ( ε s ) and at high frequency ( ε ∞ ) were thus ob-tained at all other glass compositions, and these data ap-pear in the insert of Fig 5. Noteworthy in these trends of ε s (x) and ε ∞ (x) is the fact that both show a broad maxi-mum that onsets near x = x ≃
10% and ends near x = x ≃ g (Fig.3b) and electrical con-ductivity (Fig.3a) that essentially define the IP region ofthese glasses. At the glass composition x = 20% , locatednear the middle of the IP, these parameters acquire ratherlarge magnitudes: ε s =620 and ε ∞ =250. The magnitude Frequency f [Hz]10 P e r m itti v it y ε ’ P e r m itti v it y ε ’ ε s εε s ε IP FIG. 5: (Color online) Log-log plots of the real part ε ’ of per-mittivity vs frequency for (AgPO ) − x -(AgI) x glasses at x =25% measured at 250 K. The broken horizontal lines corre-spond to the high and low-frequency limits of the permittivity ε ∞ and ε s . The inset shows variation of ε ∞ (x) and ε s (x) withAgI content. Vertical broken lines define the location of thestress and rigidity transitions at x c (1) = 9% and x c (2) = 38%as determined from calorimetry. of these permittivity found in the IP of the present elec-trolyte are truly exceptional- they are high compared tothose found in other systems such as the alkali silicates( ε ∞ =11, ), and highlight another remarkable propertyof solid electrolytes bearing Ag salts.We now follow the approach developed by Ngai andco-workers and use the electrical modulus formalism toobtain stretched exponents β (x) from our σ ( ω ) data. Ourstarting point is to write the electric field E(t) under theconstraint of a constant electric displacement as: E ( t ) = E (0)Φ( t ) (6)and define E(0) as the initial electric field and Φ(t) the re-laxation function of the system. In the frequency domain,the electrical relaxation is appropriately represented bythe electric modulus: M ∗ ( ω ) = 1 ε ∞ (cid:20) − Z ∞ e − iωt (cid:18) − d Φ dt (cid:19) dt (cid:21) (7)and the high-frequency ( ε ∞ )and low-frequency limits ( ε s )of the real part of the permittivity ( ε ′ ) then can be writ-ten as : ε s = ε ∞ < τ > / < τ > (8) FIG. 6: Variations in the Kohlrausch stretched exponent β (x)as a function of the AgI concentration x in (AgPO ) − x -(AgI) x glasses. where: < τ n > = Z ∞ t n − Φ( t ) dt (9)We use the standard relaxation function based on astretched exponential:Φ( t ) = exp [ − ( t/τ ) β ] (10)with representing the Kohlrausch-Williams-Watts expo-nent. The exponent provides a good measure of the cor-relations of atomic motions in a relaxing network . Andthe absence of cooperative behaviour will lead inevitablyto a typical Debye relaxation with β =1, a behaviour thatis expected in dilute limit where concentrations of diffus-ing cations is low. One finally obtains from equations(8)-(10): ε s = β Γ(2 /β )][Γ(1 /β )] ε ∞ (11)where Γ represents the Gamma function.The data of Fig. 5 gives the glass composition vari-ation of the permittivity, ε s (x) and ε ∞ (x). Using thesepermittivity parameters and the relation between them(equation (11)), we have extracted compositional vari-ation of the Kohlrausch stretched exponent β (x) Thesedata appear in Figure 6. We find β ≃ β is found to decrease ( β ≃ ≃
5% for example, weare in the dilute limit and carriers can be expected to berandomly distributed displaying little or no correlationsin their motion. Yet our data shows that β does notacquire a value of 1, as found in the sodium silicates . This is a curious result and it possibly reflects the factthat in the present system the diffusing cation of interestAg + also happen to be part of the base glass network. IV. DISCUSSIONA. Three regimes of fast-ion conduction and elasticphases of AgPO -AgI glasses Perhaps the central finding of this work is that fast-ion conduction in the present solid electrolyte glass dis-plays three distinct regimes of behaviour, and that theseregimes coincide with the three known elastic phases ofthe backbone. The result emerges from the compositionaltrends in the electrical and thermal data presented ear-lier in Figure 3. The observation of a reversibility win-dow fixes the three elastic phases of the present glasses with glass compositions at x < x = 9% belonging to thestressed-rigid phase, those at x > x = 38% to the flexiblephase, while those in between (x < x < x ) belongingto the IP. The two thresholds, x and x , we identify re-spectively with the stress and the rigidity transitions. Asglasses become flexible at x > σ (x) increases loga-rithmically, a result that underscores the crucial role ofnetwork flexibility in opening a doorway for Ag + ions tomigrate. In the flexible phase, network backbones can bemore easily elastically deformed (less cost in strain en-ergy), and ions can more easily diffuse and contribute toconduction.A closer inspection of Fig. 3a also shows that σ (x) isnearly independent of x in the stressed-rigid glasses ( x < x ), but it varies discontinuously displaying a jump nearthe stress transition, x ≃ σ (x) de-creases as temperature is increased from -50 o C to 100 o C.At low temperature (-50 o C), we note that the observedjump is rather large; σ increases by a factor 2.5 betweenthe composition x =10% and x = 10.5%. The discon-tinuity in σ (x) at the stress transition is reminiscent ofa similar behaviour encountered in the variation of theRaman mode frequency ( ν ) of corner-sharing tetrahedrain binary Ge x S − x glasses at the stress transition . Thefrequency jump (∆ ν ) across the threshold also increasesas T is lowered. Since the stress transition is found to bea first order transition, it suggests that the first derivativeof the free energy must be temperature dependent. B. Self-organized ion hopping model (SIHM) ofsolid electrolytes
To gain insights into the origin of the observed electri-cal conductivity thresholds (Fig. 3), we use Size Increas-ing Cluster Approximation - (SICA) to identify thetwo elastic thresholds in a parent solid electrolyte glasssystem. For this purpose we use the alkali oxide modi-fied Group IV oxide, (1-x)SiO -xM O, with M=Li,Na,K,and start with N tetrahedra (e.g. SiO / and MSiO / (sharing one non-bridging or terminal oxygen) with re-spective probabilities (1-p) and p=2x/(1-x) at the firststep, l = 1, in the agglomeration process. Our goal isto compute the probability of finding specific clustersin the three phases of interest (flexible, intermediate,stressed rigid). We note that a SiO / structural unit isstressed rigid (n c =3.67 per atom) while a MSiO / unitis flexible (n c =2.56 per atom). Here n c represents thecount of bond-stretching and bond-bending constraintsper atom. Henceforth, we denote these local structuresas St (stressed-rigid) and Fl (flexible) units respectively.Starting from these local units as building blocks, we cal-culate all possible structural arrangements to obtain ag-glomerated clusters containing two (step l=2), and thenthree building blocks (l=3), etc. and their correspondingpopulation probabilities p i . In calculating the probabil-ities of the agglomerated clusters we fold in their me-chanical energies. Details of the method and applicationappear elsewhere . At each agglomeration step l, wecompute the floppy mode count f ( l ) . At step l=2, thecount of floppy modes f (2) , f (2) = 3 − n (2) c = 3 − P j,k = F l,St n c ( jk ) p jk P j,k = F l,St N jk p jk (12)where n c ( jk ) and N jk are respectively the number of con-straints and the number of atoms found in a cluster withprobability p jk . An IP is obtained if self-organization isachieved - , i.e. if upon decreasing the content of mod-ifier atoms (x), stress-free cyclic (ring) structures form.In these calculations we define a parameter η which mea-sures the fraction of atoms in ring structures to thosein dendritric ones. Thus, η =0 corresponds to the caseof networks composed of only dendritic structures (norings), while η =1, corresponds to networks with only ringstructures. Our calculations reveal that IP-widths are di-rectly related to the parameter η i.e, larger η the greaterthe IP-width. In practice, one starts with a flexible net-work containing a high concentration (x) of modifier ions.One then decreases the modifier concentration, and canexpect more St species to form in a cluster that stillpossesses a finite concentration of floppy modes f (2) > r , there will be enough Stspecies formed to drive clusters to become rigid whenf (2) vanishes, thus defining the rigidity transition. And ifwe continue to reduce the modifier concentration beyondthe rigidity transition, one can expect clusters that arealmost stress-free by balancing St and Fl units. Such aselection rule of self-organization will hold up to a certainpoint in composition x=x s , the stress transition, i.e., atx < x s dendritic or St-rich structures will predominateand stress will percolate across clusters. In this SICA ap-proach, the IP width is then defined by the compositioninterval, x r -x s , which may be compared to the experi-mentally established interval between the observed stressand rigidity transitions, i.e., x c (2) - x c (1) = 28% (Fig 3b).In what follows, we fix the value η in order to work witha fixed IP-width.Our SIHM theory of ion-conduction builds on an ap- FIG. 7: Structural sites found in the model of (1-x)SiO -xM O: a) a stressed rigid St-St connection (n c =3.67) rep-resentative of the base glass SiO , b) an isostatically rigidMSiO / -SiO connection (St-Fl with n c =3.0) containing oneM cation, c) a flexible MSiO / -MSiO / connection (Fl-Flwith n c =2.56) having two M cations. Both b) and c) cangive rise to vacant sites such as found on the MSiO / -SiO − / cluster shown in d) which derives from cluster c). proach initially devised for the conductivity of alkali bo-rate glasses and identifies hopping sites for conduct-ing cations from a structural model, which in our casewas used to describe application of SICA to (1-x)SiO -xM O glasses. Both (present work and ref ) models in-corporate the idea first invoked by Anderson and Stuart that activation energy for ionic conduction (E A ) is madeof two terms- an electrostatic energy (E c ) to create anion and which determines the free carrier concentrationn L , and a network stress energy (E s ) that facilitates mi-gration of ions and determines carrier mobility µ . We,therefore, concentrate our efforts on estimating the en-ergies, E c and E s , as they can be directly related to thestatistics of clusters and the enumeration of constraintsvia statistical mechanics averages.We begin by modelling the parent solid electrolyte,(1-x)SiO -xM O , where M represents an alkali ion, andevaluate the free carrier concentration n L . We will haveno M + ions ( n = 0) available if we consider a pair ofSi tetrahedra, each having one non-bridging oxygen nearneighbour attached to M + ions, i.e., a pair of MSiO / tetrahedra (Fl-Fl), since there are no vacancies. How-ever, we will have n = 1 and 2 vacancies in the caseof local structure pairings of the type, MSiO / -SiO − / , and SiO / -SiO − / , contributing respectively 1 and 2M + carriers for ion transport (Figure 7). And by know-ing the probability for each of these parings p jk , one cancalculate a mean Coulombic energy over all possible pair-ings between Fl and St local units to obtain the carrierconcentration, n (2) L = 2 x exp[ − < E c > /k B T ] (13)with the mean Coulomb energy, < E c > defined as fol-lows, < E c > = 1 Z X j,k X n nE c p jk exp[ − nE c /k B T ] (14)where E c is the Coulombic energy to extract a cationM + from an anionic site, and acts as a free parameter inthe theory. Z normalises the free carrier concentrationby requiring n (2) L = 0 at T=0, and n (2) L =2x at infinitetemperature, i.e. the nominal carrier concentration. Asthe probability p jk of pairings depends on the nature ofthe elastic phase, the free carrier concentration n (2) L canbe expected to change as we go across the three elasticphases (flexible, intermediate, stressed).Once a carrier is available, it will hop between vacantsites, and the general form for hopping rates is given by J ij = ω ij exp[ − E s /k B T ] (15)where ω ij is the attempt frequency, and we assume itto be constant as only one type of local species (Fl, e.g.MSiO / ) is involved in ionic conduction here. In fact,given the compositional region where the theory appliesand where rigidity transitions are usually found , onedoes not expect to observe other Fl local species contain-ing more than one cation (e.g. M SiO corresponding tothe metasilicate composition). In general, the attemptfrequency will depend on the local environment of thehopping cations . The strain or migration energy E s for a hop is roughly the energy required to locally de-form the network between a cation site and an availablevacant site. It should therefore depend on the floppymode energy. We assume it to not depend on the process(i,j), which is equivalent to saying that we neglect theCoulomb repulsion from vacant sites. Note that in thestressed- rigid phase, there are only few hopping eventspossible since the network is made up largely of stressed-rigid St-St and isostatically rigid St-Fl pairs.In the flexible phase (at x > x r ) floppy modes pro-liferate as the count f (2) (equation (12)) becomes non-zero. In this phase, one can expect local deformations ofthe network to be facile, thus increasing hopping rates,and reduce the energy required to create doorways be-tween two vacant sites. We write the strain energy as:E flexs =E stresss -∆f (2) where ∆f (2) term is a typical floppymode energy given by experiments . In the flexiblephase, E s is reduced in relation to the stressed-rigid phaseby the quantity ∆f (2) . One is then able to write a con-ductivity of the form: σ = n (2) L µ (2) = n (2) L exp[ − E m /k B T ] (16)Results of the self-organized ion-hopping model (SIHM)for the parent electrolyte system described above aresummarized in Figure 8. In this figure we plot the varia-tion of conductivity, σ (x) given by equ. (16), as a func-tion of glass composition (x) at several temperatures Tbut at a fixed Coulomb energy E c . This model calcu-lation shows that the elastic nature (stressed rigid, in-termediate, flexible) of networks alters ion-transport inprofound ways. Three prominent features become appar-ent; (i) a minuscule conductivity change is predicted inthe stressed-rigid phase, (ii) a step-like increase in σ (x)is anticipated at the stress phase boundary x = x s , and l og [ σ ( x ) / σ ( x = )] T=300KT=900Kstressed rigid intermediate flexible x s x r FIG. 8: (Color online) Predictions of the SIHM model forvariations in ionic conductivity σ (x) for the parent solid elec-trolyte, (1-x)SiO -xM O, where M = Alkali ion, using E c =0.1 eV, and at 4 temperatures, from bottom to top: T=300K, 500 K, 700 K and 900 K. We have used a medium rangeorder (ring) fraction η =0.6 which defines the IP (see text fordetails). Curves have been shifted along the ordinate for pre-sentation. The broken line corresponds to the conductivitywhen the floppy mode contribution is neglected (see text fordetails). finally (iii) a logarithmic variation of σ (x) is predictedin the flexible phase( x > x r ). These features of ourSIHM model are in rather striking accord with the ob-served variation of conductivity in the present AgPO -AgI glasses (Figure 3).The SIHM model predictions provide crucial insightsinto origin of ion-transport in the three elastic regimes.In the stressed-rigid phase, strain energy is high and leadsto a low number of hopping possibilities, since hops areallowed only between select number of Fl-St pairs. Theserestrictions on hopping combined with a low free carrierconcentration results in a weak dependence of σ (x) onmodifier concentration x. The order of magnitude of σ normalized to σ (x=0), is largely determined by themagnitude of E c , i.e. the interaction energy between thecation M and its corresponding anionic site.In the IP, new cation pairs (Fl-Fl, and thus hoppingpossibilities) are populated, and lead to a mild increase of σ . The step-like increase in conductivity at the stress-phase boundary , x = x s (Fig 7) is a feature that isnicely observed in our experiments (Fig 3a), and providesan internal consistency to the description advanced here.Theory tells us that the jump in σ at stress transitiondepends on T or inverse E c at fixed T, (see equ. (14)).The location of the stress transition is a network prop-erty and it depends on the IP, i.e., aspects of mediumrange structure. As in other quantities observed , thefirst order jump in σ (x) near x = x s is a manifestationof deep structural changes in a glass network which con-tributes to a cusp in the configurational entropy . Andas a final comment on the subject, we recognize that ata fixed temperature, large values for the Coulomb energyE c will lead, in general, to large jumps in conductivityat the stress transition. Here theory predicts that jumpsin conductivity near the first order stress transition (x =x s ) will be more conspicuous with heavier cations (suchas Ag or K) than with the lighter ones (Li).Glasses become flexible at x > x r as floppy modesstart to proliferate and decrease the strain energy bar-rier. Electrical conductivity displays a second thresh-old at the rigidity transition x=x r , as the slope d σ /dxchanges abruptly at the transition, and the variation of σ (x) becomes exponential above the threshold x r . Net-work flexibility clearly promotes ionic conductivity. Inthe flexible phase an increase in the available degreesof freedom appears to facilitate local network deforma-tions, thus creating pathways for conduction. Further-more, the present results also suggest that the increasein the number of possible hopping processes in the IP(region between x s and x r in Fig.7) is small and tendsto produce saturation upon increasing modifier content(broken line in Fig. 7). This is to be compared to thedramatic increase of σ that mostly arises in the flexiblephase as floppy modes proliferate, f (2) = 0 , and E s de-creases sharply (Fig 9). C. Coulomb energy (E c ) and network stress energy(E s ) in AgPO -AgI
1. Coulomb energy E c and Ag + ion mobilities The Coulomb energy E c represents the energy to cre-ate a Ag + ion that would contribute to ionic conductionin the AgPO -AgI electrolytes glass of interest. In itssimplest form , it can be written as E c = e πε ∞ ε R Ag − X (17)where R Ag − X and ε ∞ represent respectively an averageAg-X ( X = I, O) nearest-neighbour bond length andthe high frequency permittivity. From the derived fre-quency dependence of the permittivity using the electricModulus formalism, earlier in section IV A, we deducedthe variation of ε ∞ (x) in the inset of Fig. 5. The quan-tity relates to the dielectric response in the immediatevicinity of a silver cation, and permits one to estimatethe Coulombic barrier needed to create a mobile Ag + cation. In the estimate of E c we use an average of Ag-I,and Ag-O bond length of 2.43 ˚ A and 2.85 ˚ A for R Ag − X in equation (16), and note that the R Ag − I bond lengthremains largely constant with glass composition . Fur-thermore, corrections due to many- body- effects aris-ing from Ag + -Ag + interactions can also be taken intoaccount in equation (16) as suggested elsewhere . Thevariations of E c (x) thus largely results from that of ε ∞ (x) x (AgI in %) A c ti v a ti on e n e r g i e s [ e V ] E qu i v a l e n t c ondu c t a n ce σ / Z e x x10 [ m V - s - ] E A E S E C x c (1) x c (2) FIG. 9: Variations in the activation energy E A (same as figure2), Coulomb energy E c , strain energy E s = E A -E c and equiv-alent conductance at 245 K determined from the conductivitydata of Fig. 3 for (AgPO ) − x -(AgI) x glasses. (Figure 5 inset), and these are reflected in the plot of Fig-ure 9. We find that E c varies between 0.05 eV to 0.10eV across the broad range of compositions in the presentglasses.How do the present estimates of E c compare to earlierones in other systems ? Other groups report similar find-ings; in (Na O) x (SiO ) − x glasses, Greaves and Ngai estimate E c near 0.1eV to 0.2 eV, and the activation en-ergies E A ( ≃ O) x (GeO ) − x glasses, Jainet al. estimate E c to be ≃ A ( ≃ < ) x (AgI) − x glasses by Wicks et al. have alsoemphasized E c to be small , and E s to be the majorcontributor to E A . A similar conclusion is reached byAdams and Swenson , who explicitly reveal that strainenergy dominates ionic conductivity. The present find-ings on E c highlight once again that the low value of E c appears to be generic feature of solid electrolytes. Ag NMR studies of the base AgPO glass showthe NMR resonance to be rather broad, fully consistentwith its low conductivity ( σ ≃ − Ω − .cm − , see Fig3), in which the Ag + cations serve as compensating cen-ters. Furthermore, the low value for E c nearly 10% of thetotal activation energy E A found in the present AgPO -AgI glassy alloys (Fig.9) suggests that the number of freeAg + carriers varies linearly with the macroscopic AgIcontent ’x’ of the alloys. For this reason, the equivalentconductance σ /Zex, i.e. the change in conductivity permole % of AgI, serves a good measure of ionic mobility(equation 1). Results on variation of the equivalent con-ductance are plotted in Fig. 10, right axis. Here one cansee that while the nature of the elastic phases weakly af-fects E c , and thus the free carrier density, they induceprofound variations in the equivalent conductance, andthus carrier mobility. In particular, the two thresholdsin equivalent conductance coincide with those noted ear-lier in the variations of ∆ H nr (x) and σ (x) (Fig.3), whichserve as benchmarks of the three elastic phases of thepresent glasses. In the flexible phase ( x > < x <
53% range.
Ag NMR studies have shown that the number ofmobile carriers is nearly equal to the number of silverions introduced by AgI. Using the simple relationship be-tween σ and mobility (Equation (1)) Mustarelli et al. estimated the variation in mobility µ (x) of carriers. Theirdata are also reproduced in Fig. 10 as the red filled cir-cles. The corresponding equivalent conductance data onour samples are plotted in Fig. 10 as the open squaredata points. It is clear that the mobility in the presentsamples are at least an order of magnitude lower than inthe samples of Mustarelli et al. .We believe these differences in µ (x) between the twosets of samples derive from sample makeup. The glasstransition temperatures T g (x) are a sensitive measure ofbonded water in the base glass, AgPO , as describedearlier . Presence of bonded water in the base glassdepolymerizes the P-O-P chain network and softens thenetwork. The samples of Mustarelli et al. show theirbase glass (AgPO ) T g = 189 o C , about 60 o C lower thanthe T g of our driest base glass of 254 o C. Alloying AgI inthe base glass will soften it further. Furthermore, in oursamples we do not observe evidence of β -AgI segregationin the 50% < x <
80% composition range as noted byMustarelli et al. in their samples. The order of magni-tude larger mobilities observed by Mustarelli et al., mostlikely, derive from the flexible nature of glass composi-tions promoted by the presence of additional bonded wa-ter.In our samples, we found the IP composition rangeto be rather sensitive to water content of samples. Forexample, the IP-width nearly halved (28% to 15%) andthe IP centroid moved up in AgI concentration from 24%to 28% in going from samples of set A (driest) to thoseof set B (drier). In samples of set B, the T g of the baseglass was 220 o C, about 34 o C lower than in samples of setA (Tg = 254 o C). And possibly, the solitary conductiv-ity (or mobility) threshold observed near x = 30% byMustarelli et al. (Fig. 10) and separately by Man-gion and Johari in their samples may well representthe extremal case of a completely collapsed IP at its cen-troid near 30% in their samples giving rise to only oneelastic threshold with the stress- and rigidity- transitionsmerging. These data highlight the importance of samplesynthesis to establish the intrinsic electrical and elasticbehavior of these glasses. The role of water traces incollapsing the IP in the present solid electrolyte glass isreminiscent of a collapse of the IP observed in Ramanscattering of a traditional covalent Ge x Se − x glass, mea-sured as a function of laser power-density used to excitethe scattering. In both instances, the self-organized statewith a characteristic intermediate range order formed, is
00 1010 2020 3030 4040 5050 x (AgI in %) E qu i v a l e n t c ondu c t a n ce σ / Z e x [ x10 m V - s - ] M ob ilit y µ x10 [ m V - s - ] x c x c (1) x c (2) FIG. 10: (Color online) Equivalent conductance σ /Zex (opensquares, broken line, same as Fig. 7) and silver mobility µ (red circles, right axis) determined from a free carrier rateNMR estimation (Mustarelli et al. Ref. ). irrevocably changed to a random network structure, inone case by OH dangling ends replacing bridging O sitesand splicing the network, and in the other case by rapidswitching of lone pair bearing Se centered covalent bondsby the action of near band gap light serving as an opticalpump.
2. Network Stress energy E s Estimates in E c (x) permit obtaining variations in thenetwork stress energy E s (=E A -E c ) from the measuredactivation energies E A (x) (Figs. 1 and 2). Variations inthe activation energies E A (x), and E s (x) are plotted inFigure 8 for the reader’s convenience. A perusal of thesedata clearly reveals that the drastic increase of electricalconductivity in the 40% < x <
60% range is tied to atwo-fold reduction in E s (x) (Fig. 9). Thus, even thoughthe concentration of the electrolyte salt additive (AgI)increases linearly with x, changes in conductivity are notonly non-linear in x, but display discontinuities at thestress (x c (1) = 9%) and the rigidity (x c (2)= 38%) elasticphase boundaries as discussed earlier.The central finding to emerge from an analysis of ourelectrical conductivity results is that fast-ion conduc-tion in dry AgPO -AgI glasses is largely controlled bythe elastic nature of their backbones. Recent work onalkali-earth silicates also shows evidence of conductivi-ties increasing precipitously once glasses become elasti-cally flexible at a threshold additive concentration .0 D. Kohlrausch stretched exponent and networkdimensionality in AgPO -AgI glasses The Kohlrausch exponent usually reflects the degree ofcooperativity of mobile ions in the glass, the smaller the β the larger the collective behaviour of cation motion.From what is seen in Fig. 6, one can state that collectivebehaviour is enhanced in the intermediate phase.We note that variation of the Kohlrausch exponent β (x) displays a trend that is similar to that of the non-reversing heat flow (Fig.3b) , with values of β of about0.66 in the stressed-rigid and flexible phases, and a valueof about 0.52 in the IP. These β (x) data contrast withprevious findings of the exponent in the Ge x As x Se − x ternary glasses measured in flexure measurements whereone found that β steadily increases in going from theflexible to the stressed-rigid phases. In the latter it con-verged to a value of 0.60 for a network mean coordina-tion number larger than 2.4 corresponding to intermedi-ate and stressed-rigid phase glasses.The stretched exponent β approaches 1 at high temper-atures (T > T g ) , and reduces as T decreases to T g , i.e., β (T → T g ) <
1. The reduction of β usually reflects con-traction of configuration space in the supercooled liquidand its eventual stabilization due to structural arrestnear T=T g . On this basis at a fixed network dimension-ality, one should expect that an increase in the number ofLagrangian constraints (or decrease of AgI content) willcontract the configuration space, leading to lower valuesfor β . Fig. 6 obviously does not follow this anticipatedbehaviour as the dimensionality is changing with mod-ifier content. Models of traps - for tracer diffusionin a d-dimensional lattice show that β is related to thedimensionality d via: β = dd + 2 (18)which is satisfied for various molecular supercooled liq-uids and glasses having either a 3D ( β = 3/5) or a2D relaxation ( β = 1/2). However, in the case wherean internal structural dynamics takes place, the dimen-sionality d of networks must be replaced by an effectiveone, d eff , involved in relaxation. This happens whenCoulomb forces are present as the motion of the carriersis hindered by the tendency towards local charge neutral-ity, determining an effective dimensionality d eff of theconfiguration space in which relaxation takes place . Inthis case, the effective dimensionality becomes, d = d eff N d N d + N c (19)where N d is the number of degrees of freedom in labo-ratory space and N c is the number of mechanical con-straints, estimated from short-range interactions. In theIP where N c ≃ β = 1/2 (Fig.6), equation (17) impliesd = 2, resulting into a d eff = 1 from equation (18). Out-side the IP, β = 3/5 (Fig.6), equation (17) implies d = 3, resulting into a d eff = 1.5 from equation (18). Thus,presence of Coulomb interactions in AgPO -AgI glasses,suggests filamentary (d eff = 1) diffusion of Ag + carri-ers in the IP, giving rise to percolative pathways in theself-organized structure. This conclusion is supported bynumerical simulations , which show that isolated sub-diffusive regions exist at low AgI concentration and thesepercolate as filaments at increased AgI concentrations inthe 20% < x <
30% range.
V. CONCLUDING REMARKS
Dry and slow cooled (1-x)AgPO -xAgI glasses synthe-sized over a broad range of compositions, 0 < x < A (x), Coulomb energy E c (x) forAg + ion creation, Kohlrausch stretched exponent β (x),low frequency permittivity ε s (x) and high-frequency per-mitivity ε ∞ (x) are deduced. By combining rigidity the-ory with SICA and statistical mechanics, we have de-rived the analytic properties of the conductivity responsefunction in a Ag based solid electrolyte. The topologicalmodel reproduces observed trends in the compositionalvariation of σ (x), including the observation of a step-like jump as glasses enter the Intermediate phase nearx c (1)=9%, and an exponential increase when glasses be-come flexible at x > x c (2)=38%. Since E c is found tobe small compared to network strain energy (E s ), weconclude that free carrier concentrations increase linearlywith nominal AgI concentrations, while the super-linearvariation of fast-ion conduction can be traced to changesin carrier mobility induced by an elastic softening of net-work structure. By better understanding the role playedby the ”alloyed network” on fast-ion conduction in a spe-cific Ag based solid electrolyte glass, we have also taken astep towards better understanding ion-transport in sev-eral other types of solid electrolyte systems, includingmodified oxides and modified chalcogenides, systems thatare also expected to display a parallel conductivity re-sponse as a function of network connectivity. Variationsin β (x) reveal a square- well like variation with walls lo-calized near x c (1) and x c (2), and suggest filamentary (quasi 1D) conduction in the Intermediate Phase ( x c (1) < x < x c (2)) but a dimensionality larger than 1 outsidethe IP. The present findings, characteristic of dry sam-ples and showing two elastic thresholds, may possibly goover to only one elastic threshold in wet samples in whichthe IP completely collapses as suggested by earlier workin the field. Acknowledgements
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