Chapter 2
Effect of rigidity and flexibility on the ionic transport in network glasses
M. Malki1,2, M. Micoulaut3, P. Boolchand4
1CEMHTI, UPR CNRS 3079, 1D Avenue de la Recherche Scientifique, 45071 Orléans Cedex 02, France
2 Polytech’ Orléans, Faculté des Sciences, BP 6749, 45072 Orléans Cedex 05 France
3Laboratoire de Physique Théorique de la Matière Condensée, UPMC-Paris 6, 4, Place Jussieu, 75252 Paris Cedex 05 France
4Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, OH45221-0030, USA
Conductivity and ease of conduction in amorphous electrolytes can be related to the mechanical nature of the host network and displays therefore three distinct regimes in the flexible, intermediate and stressed rigid phases of ion conducting glasses. It is shown that the onset of conduction at higher modifying ion content results from the breakdown of rigidity combined with an increase of the free carrier rate. Quantitative results obtained from a rigidity-based model for ionc conduction are then compared to available experimental data.
1. Introduction
Gaining a better understanding of electrical transport in fast ionic conductors such as amorphous electrolytes is a basic scientific challenge, with potential important technological applications. Oxide or chalcogenides conducting amorphous materials show indeed high electrical conductivities that makes their use for solid state batteries, sensors, or non-volatile memories very attractive. In order to increase the possibility of applications, one has to better understand how chemistry via the structure, controls conduction and the conduction mechanisms.
As conductivity can be expressed in terms of the free carrier rate nL and the carrier mobility by: = Ze nL with Ze the charge of the conducting ion, models and theoretical approaches in glasses have emphasized either on the dominant role of the free carrier rate nL or on the carrier mobility . Beyond this point of discussion, it is clear that addition of ionic species increases the possibility of having free carriers in a host network, but also modifies substantially its mechanical nature by depolymerization. Any theoretical model should therefore take into account both contributions, and can hardly neglect one or the other.
Since conductivity is related to the mechanical nature of the network, glasses should display different behaviours for in flexible or stressed rigid phases, and as these two phases are separated by an elastic threshold (or rigidity transition), one may wonder if this feature is also detected in conduction. Thresholds have been found to occur in fast ionic conductors although the relationship with rigidity transitions has not been made.
In this context, it has been recently found that rigidity onsets by steps in most of the glasses, a feature that is originated from stress avoidance leading to network self-organization. This leads to three well-defined elastic phases: a flexible phase having internal degrees of freedom (floppy modes), an intermediate phase that is rigid but stress-free and a stressed rigid phase which has more constraints than degrees of freedom. The boundaries of the Intermediate Phase, expressed in terms of network mean coordination numbers and have been characterized from experiments,numerical calculations, cluster analysisand energy adaptation and identified as being arigidity transition at and a stress transition at .
In random networks where self-organization does not take place, both transitions coalesce into a single threshold at where Maxwell constraint counting predicts the vanishing of the number of floppy modes, i.e. the low energy modes that serve to locally deform the network. Up to now, signatures of the two transitions have been detected from calorimetric and spectroscopic probes.
In this chapter, it is shown that ionic conduction should also display three distinct regimes following the elastic nature of the host network: flexible, intermediate or stressed rigid. In stressed rigid electrolytes, the combination of a weak number of free carriers and a large strain energy for migration leads to very low conductivities. Conductivity builds up in the intermediate phase where additional hopping processes appear. These new hopping processes are found on flexible parts (ion rich) of the network and do not exist in the stressed rigid phase. Finally, the lowering of the strain energy due to the presence of floppy modes that allow the local deformation of the network, promotes ease of conduction in the flexible phase.
The present results relate for the first time the rigidity classification of glassy networks to conduction, and highlight network flexibility as the major functionality promoting conduction. It furthermore suggests that electrical probes should complement the calorimetric and spectroscopic probes of the Intermediate Phase.
2. A rigidity based theory for ionic conduction
We combine a model of ionic conduction with cluster approaches of rigidity transitions to study in detail the effect of the three phases on the conduction. The results of the approach are successfully compared to experimental results on fast ionic conductors. This should provide new benchmarks for the description of electrical transport in glasses.
Fig.1. A typical ionic network glass of the form (1-x)SiX2-xM2X. Note that this network contains edge-sharing tetrahedra that contribute to the width of the intermediate phase.
The advantage in using such kind of models of ionic conduction in the context of rigidity, is that it builds on the probability of finding local and medium-range structures with composition, a feature on which rely several theories in self-organized rigidity as described next.
Construction
Size increasing cluster approximations (SICA) are used to compute the probability of clusters in the three phases for glasses of the form (1-x)SiX2-xM2X with (X=O,S,Se and M=Li,Na,K,...) which display a mean-field rigidity transition at x=xc=0.20. These glasses serve as basic system for a number of fast ionic conductors (see Fig. 1) and have a rather well-defined short-range order extracted from NMR experiments. The network consists indeed of SiX4/2 and SiX5/2M tetrahedra (respectively Q4 and Q3 in NMR notation) in the present concentration range of interest [x=0, x=0.33].
Using SICA, a network of N tetrahedra Q4 and Q3 with respective probabilities (1-p) and p=2x/(1-x) is considered at a basic step l=1. Note that a Q4 unit is stressed rigid (nc=3.67 per atom) whereas Q3 is flexible (nc=2.56 per atom). Starting from this short range order (the SICA building blocks), one can look up all possible structural arrangements to obtain clusters containing two Qi's (step l=2), three Qi's (l=3), etc. The various possible connections (Q4-Q4, Q4-Q3, Q3-Q3) define energy gainsaccordingly to their mechanical nature and their probabilities can be computed. Details of the method and application can be found in this book. This allows to compute the floppy mode density f(l) of the network which, in the case of doublet pairs Qj-Qk (i.e. l=2) with probability pjk is given by:
(1)
where nc(jk) and Njk are respectively the number of constraints and the number of atoms of a Qj-Qk doublet. An intermediate phase is obtained if self-organization is achieved, i.e. if with growing connectivity all pairs with stresses can be avoided. In the present system, the only stressed rigid pair is the Q4-Q4 one. In practice, one starts from a flexible network which is found at high modifier concentration x. If one decreases the modifier content, one will obtain more and more Q4 species with still f(2)>0. At a certain point, there will be enough of these structural units to ensure rigidity and f(2) will vanish. Stress is then only allowed between Q4-Q4 pairs that form rings, i.e. edge-sharing structures which are weakly stressed rigid and isostatically rigid in the case of an infinite edge-sharing tetrahedral chain.This selection rule in the cluster construction allows to reduce the concentration beyond the rigidity transition and to obtain a network that is almost stress-free. But the selection rule holds only till a certain point beyond which corner-sharing Q4-Q4 pairs can not be avoided anymore. Stress is then able to percolate across the network. Results of such an application to the present system are given elsewhere.
Our theory of ionic conduction builds on the Anderson-Stuart model which separates the activation energyfor ionic conduction into an electrostatic and migration part Ecand Emthat respectively contribute to the free carrier rate and the carrier mobility. In this model, the low temperature Arrhenius behaviour of the conductivity is written as:
(2)
with 0 depending on several other parameters. In the following, we concentrate our efforts on the energies Ec and Em as they can be directly related to the statistics of clusters and the enumeration of constraints.
Next, one evaluates the free carrier rate nL. On each Qj-Qk pair, there is a probability to find n=0, 1 or 2vacancies (see Figure 2) which will contribute to the free carrier rate. A mean Coulombic energy can be computed over all possible pairs that lead to the free carrier rate:
(3)
with:
(4)
where Ec is the Coulombic energy to extract the cation M from the anionic site X (X=0,S,Se) and acts as a free parameter for the theory. Z normalises the free carrier rate in order to have nL(2)=0 at T=0 and nL(2)=1 at infinite temperature. As the probability pjk of pairs will depend on the nature of the elastic phase, the free carrier rate nL(2) should depend also be different in the three phases (flexible, intermediate, stressed). In the concentration range of interest, one possible environment for the M+carriers is identified, the Q3 unit corresponding to singly occupied negative (NB0) sites whose number changes with x according to the structural change of the network.
Fig. 2: Q4, Q3 cluster construction in a typical SiX2-M2X glassy system leading to various Qi-Qj pairs which exist only in certain elastic phases (flexible, intermediate, stressed rigid), and contribute to the free carrier rate.
Once a carrier is free to move, it is supposed to hop between two vacant sites and the general form for hopping rates is usually given by Jij=ijexp[-Em/kBT] with ijan attempt frequency depending on the local environment of the hopping cation which is constant here as only one type of Qn species (n=3) are involved in the ionic conduction.The strain or migration energy Emfor the hop is roughly the energy required to locally deform the network between the cation and a vacant site. It should therefore depend on the floppy mode energy and does not depend on the process (i,j) which amounts to neglect the Coulombic repulsion from vacant sites.Note that for the stressed rigid phase, there are only few hopping rates involved as the network is made only of stressed rigid Q4-Q4 and isostatically rigid Q4-Q3 pairs.
Floppy modes start to proliferate in the flexible phase at x>xr when f(2) (equation (1)) becomes non-zero. This allows an easier local deformation of the network thus increasing the hopping rates and should reduce the energy required to create doorways between two vacant sites. We write therefore the strain energy as: Emflex=Emstress-f(2) where is a typical floppy mode energy given by experiment. In the flexible phase, Em is reduced by a quantity f(2) when compared to the stressed rigid phase. One is then able to write a conductivity of the form:
(5)
Results
The results of the ionic-rigidity model are displayed in Figure 3 for various parameters Ec. They show that the underlying mechanical nature of the host networks strongly affects the ion transport. In the stressed rigid phase, the stress energy is high and the number of hopping possibilities is low (only between CS and ES Q4-Q3 pairs), combined with a low free carrier rate. The conductivity is therefore found to be low and to depend only weakly on the modifier concentration.
In the intermediate phase, new cation pairs (and thus hopping possibilities) appear and lead to a mild increase of . One should note that the first order stress transition that usually separates the stressed rigid phase and the IP is detected in conductivity. Furthermore, it appears that the jump seen at the stress transition composition is depending on the parameter Ec (or inverse temperature, because of the Boltzmann factor, see equ. (2)), whereas the location of the stress transition itself will only depending on the structure. Large values for the Coulombic energy will lead to large jumps in conductivity at the stress transition. One expects therefore to observe the first order stress transition rather in heavy cations (such as Ag or K) than in more lighter ones (Li). Similarly, oneexpects to see the jump increase with decreasing temperature.
Fig. 3. Conductivity in a (1-x)SiX2-xM2X glassy system as a function of the modifier concentration for various Coulombic energies. Here Emstress=0 and =0.05eV.
Conductivity displays a second threshold at the rigidity transition x=xr when floppy modes start to proliferate and decrease the migration energy barrier. It clearly suggests that flexibility promotes conductivity. The latter can indeed increase when available degrees of freedom appear thatfacilitate local deformations of the network and the creation of pathways for conduction. The present results furthermore suggest that the increase of the number of possible hopping processes (in the IP) contribute only weakly to the dramatic increase of that mostly arises from the fact that in the flexible phase .
Here, on sees the effect of the flexible and intermediate phase with respect to the stressed rigid phase, in terms of hopping rates. For $x_{rigidity}>x>x_{stress}$, the system is in the intermediate phase. The structure is rigid but stress free on average. Nucleated stress in small but highly cross-linked structures balance indeed some small flexible regions (e.g. $Q^3-Q^3$ clusters such as in Fig. \ref{hop}) that contribute to additional hopping processes ($J_{21}$, $J_{20}$,...). $J_{IP}$ is therefore increased with respect to $J_{stress}$. Finally, when $x>x_{rigidity}$ the number of hopping processes is the same as in the intermediate phase. However, their rate is increased because of the occurence of floppy modes that lower the strain energy.
As seen from equ. (\ref{J}), the effective hopping rates mostly depend on the probability of finding the $Q^4-Q^3$ clusters and its functional form with composition can be evaluated using SICA for various degrees of network self-organization. This will contribute also to the change in the location of the stress and rigidity transition.
\par
Fig. 4. Dependence of the exponent p with temperature and Ec..
The exponent depends on T and Ec.
3. Experimental results
The results of the previous section are now compared to experimental data obtained in glasses which display an intermediate phase. We focus on alkali silicates and germanates, and on silver phosphates.
Sample preparation
All glass samples were disks of about 10mm in diameter and 2 mm thick. Theyhave been synthesized by pouring melts in special troughs. Platelets were thermally relaxed by cycling through Tg. Samples were polished, and the Pt electrodes deposited. Electrical conductivity of glasses has been measured by a two-electrode impedance spectroscopy setup. Basically, the metal-coated polished sample pellet is hold by a weak spring between two platinum electrodes, which measure the intensity of the current passing through the sample while a known voltage is applied. The whole apparatus is placed in a horizontal tubular furnace, the temperature of which is raised to about Tg+50°C at a rate of 2°C/min. The sample temperature is measured by a Pt/Pt-Rh(10%) thermocouple placed near the test specimen. The complex impedance Z=Z’+iZ’’ of the specimen is measured by a Solartron SI1260 impedance analyzer controlled by the LabView software. The frequency range between 1MHz and 1Hz is scanned over 19 points for each measurement. All data are acquired automatically at two-minute intervals.
The electrical resistance Rdc is read at the high-frequency intersection of the polarisation semi-circle with the x-axis in the complex diagram (Cole-Cole plot). The static conductivity, dcis obtained from the sample resistance Rdc, its thickness e and the area S of the surface covered by platinum by the formula:
Alkali and alkaline earth silicates
We review here data on the systems (1-x)SiO2-xM2O with M=Li, Na, K and (1-x)SiO2-xMO with M=Ca,Ba. Although these glasses have moderate conductivities (~10-4-1.cm-1) at room temperature, they have received considerable attention both from the experimental and simulation viewpoint because of their applications in domestic glasses and their abundance in magmas. We should stress that alkaline earth silicate glasses have generally higher liquidus temperature as well as glass temperature (Tg) than alkali silicates ones (Mazurin, 1983). We measured the electrical conductivity of the two systems xBaO-(1-x)SiO2 and xCaO-(1-x)SiO2 in their glass-forming region i.e. 25-37 and 41-55 mol% of alkaline earth oxide respectively. The data for alkali silicates are taken from the literature.
In Fig. 5 and 6 are represented the behaviour of with alkali and alkaline earth composition. The trend with composition obviously exhibits two distinct conduction regimes on two compositional intervals: a first one at low modifier concentration where the conductivity is negligibly small and a second one where obviously displays an exponential growth or power-law behaviour. For that reason, is sometimes plotted in log-scale. From Figure 5 and 6, it becomes clear that the onset of ionic conductivity in the second compositional interval is related with the breakdown of the stressed-rigid network.In calcium silicates, the almost constant value of in the low calcium region and sudden increase for x > 0.47 has also been obtained in simulations and experiments for the molten state (Belashnko, 1996), suggesting that molten and glassy states behave similarly.
In fact, a Maxwell bond-stretching and bond-bending enumeration in these systems leads to thresholds that are very close to the compositions (Table I) where the conductivity substantially increases.
System / Threshold composition xc / Reference(1-x)SiO2-xLi2O
(1-x)SiO2-xNa2O
(1-x)SiO2-xK2O
(1-x)SiO2-xCaO
(1-x)SiO2-xBaO / 0.25
0.20
0.17
0.50
0.29
Table I. Threshold compositions xc in alkali and alkaline earth silicates estimated from global Maxwell counting..