Gaël Poux-Médard
email: gael.poux-medard@univ-lyon2.fr

Julien Velcin
email: julien.velcin@univ-lyon2.frsabine.loudcher@univ-lyon2.fr

Sabine Loudcher

Dirichlet-Survival Process: Scalable Inference of Topic-Dependent Diffusion Networks

Keywords: Spreading process, Network Inference, Clustering, Bayesian Nonparametrics

Information spread on networks can be efficiently modeled by considering three features: documents' content, time of publication relative to other publications, and position of the spreader in the network. Most previous works model up to two of those jointly, or rely on heavily parametric approaches. Building on recent Dirichlet-Point processes literature, we introduce the Houston (Hidden Online User-Topic Network) model, that jointly considers all those features in a non-parametric unsupervised framework. It infers dynamic topic-dependent underlying diffusion networks in a continuous-time setting along with said topics. It is unsupervised; it considers an unlabeled stream of triplets shaped as (time of publication, information's content, spreading entity) as input data. Online inference is conducted using a sequential Monte-Carlo algorithm that scales linearly with the size of the dataset. Our approach yields consequent improvements over existing baselines on both cluster recovery and subnetworks inference tasks.

Introduction

Overview of the contribution

Over the last decades, information spread patterns have become more and more complicated. The volume of data that flows on social networks keeps increasing every day that passes, and results in complex diffusion processes that can be described by many factors. However, recent advances suggest that documents complex diffusion processes can be efficiently modeled considering only three variables: their publication date (when), the publisher (who) and their semantic content (what). The idea of considering these three factors is not novel. However, most of the models that tackle diffusion problems tend to consider up to two of these, but seldom the three parameters.

We introduce the Houston model, that tackles the problem by jointly inferring clusters of textual documents spreading online and the subnetworks they spread on. Our method builds on recent Dirichlet-Point processes advances [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF][START_REF] Mavroforakis | Modeling the dynamics of learning activity on the web[END_REF][START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF][START_REF] Poux-Médard | Multivariate powered dirichlet-hawkes process[END_REF]. To the best of our knowledge, it is the first model that considers semantic content, publication dynamics and the network of spreading documents in an online, non-parametric and unsupervised way.

Fig. 1. From a stream of textual documents, we model the underlying topic-dependent diffusion subnetworks. Inference is unsupervised, non-parametric and conducted online, meaning data is processed sequentially. Results in the bottom row come from the application of our method to the Memetracker dataset [START_REF] Leskovec | Meme-tracking and the dynamics of the news cycle[END_REF]. Nodes colors represent traditional medias (red) and blog (blue).

Related works

It has been underlined on several occasions that efficiently modeling information diffusion involves accounting for the network's structure [START_REF] Larremore | Statistical properties of avalanches in networks[END_REF][START_REF] Poux-Médard | Influential spreaders for recurrent epidemics on networks[END_REF], publication times [START_REF] Du | Learning networks of heterogeneous influence[END_REF][START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF] and documents' content [START_REF] Du | Uncover topic-sensitive information diffusion networks[END_REF][START_REF] He | Hawkestopic: A joint model for network inference and topic modeling from text-based cascades[END_REF]. Some approaches consider sequentially all three factors. Typically, they first infer topics based on documents content, and only then they use this information to infer the latent diffusion subnetworks [START_REF] Choudhari | Discovering topical interactions in text-based cascades using hidden markov hawkes processes[END_REF][START_REF] Du | Uncover topic-sensitive information diffusion networks[END_REF][START_REF] He | Hawkestopic: A joint model for network inference and topic modeling from text-based cascades[END_REF][START_REF] Suny | Inferring multiplex diffusion network via multivariate marked hawkes process[END_REF][START_REF] Wang | Feature-enhanced probabilistic models for diffusion network inference[END_REF][START_REF] Yang | Mixture of mutually exciting processes for viral diffusion[END_REF]. The work the closest to ours [START_REF] Barbieri | Survival factorization on diffusion networks[END_REF] is, to our knowledge, the only one that jointly models documents' content, dynamics and structure. It develops an unsupervised topic-dependent network inference method. The approach breaks down the topic-aware diffusion into two factors: each node is assumed to have a given sensitivity to a topic, and a certain authority on them. Given this assumption, the authors develop a parametric prior on the probability for a diffusion cascade to belong to a given topic. The textual content (or side information) is then accounted for using a homogeneous Poisson textual model [START_REF] Mei | A study of poisson query generation model for information retrieval p[END_REF], combined with the above prior. The model is optimized using an EM algorithm. However, the optimization algorithm is not designed for online optimization -data cannot be added sequentially-, and topics optimization is parametric -the number of topics must be provided.

Model

Background

To answer these limitations, we build a Dirichlet-Survival process that can be used as a non-parametric Bayesian prior for online inference. The Dirichlet-Survival prior is created by merging Dirichlet processes with Point processes. The method has been explored by combining Hawkes processes to several variants of Dirichlet processes (hierarchical [START_REF] Mavroforakis | Modeling the dynamics of learning activity on the web[END_REF], mixed membership [START_REF] Tan | The indian buffet hawkes process to model evolving latent influences[END_REF], powered [START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF], multivariate [START_REF] Poux-Médard | Multivariate powered dirichlet-hawkes process[END_REF]). However, no work considered the combination with other point processes than the Hawkes process. Our approach using Survival analysis explores this new connection; it allows us to design an optimization algorithm (Sequential Monte Carlo) for online non-parametric topics-aware diffusion subnetworks inference (the number of topics/subnetworks does not have to be chosen in advance).

In [START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF], the authors show that a large part of the literature on underlying diffusion network inference [8, 12-14, 21, 28] can be expressed as special cases of a counting point process. The method allows to infer dynamic underlying diffusion networks using convex optimization tools.

Dirichlet process and Survival analysis

Dirichlet process The Dirichlet process is used as a non-parametric prior distribution over clusters in many clustering algorithms. It can be written as follows:

P (s i = k|{s m } m=1,...,n-1 , α 0 ) = N k α0+ K k N k if k = 1, ..., K α0 α0+ K k N k if k = K+1 (1)
where s i is a variable that represents the cluster of the i th observation, N k = |{s i |s i = k} i=1,...,n-1 | the population of cluster k, K the total number of nonempty clusters and α 0 a concentration hyper-parameter. The choice of K + 1 means a new cluster is opened and K in increased by 1. Note that references [START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF][START_REF] Poux-Médard | Multivariate powered dirichlet-hawkes process[END_REF] use the powered version of this process [START_REF] Poux-Médard | Powered dirichlet process for controlling the importance of "rich-get-richer" prior assumptions in bayesian clustering[END_REF].

Network inference model

The edges of topic-dependent networks are inferred using the NetRate model [START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF], which is part of a broad literature on underlying spreading networks inference [START_REF] Du | Uncover topic-sensitive information diffusion networks[END_REF][START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF][START_REF] Gomez-Rodriguez | Structure and dynamics of information pathways in online media[END_REF][START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF][START_REF] Wang | Feature-enhanced probabilistic models for diffusion network inference[END_REF]. In particular in [START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF], the authors demonstrate that all these models can be expressed as special cases of a counting point process. These processes take a collection of independent timestamped diffusion cascades ⃗ c = {(u c i , t c i )} i as input, where u c i is the node on which the i th event occurred and t c i the time at which it happened in cascade c. The process is entirely characterized by a hazard function H(t c i |t c j , α u c j ,u c i ), which is the instantaneous infection rate of u c i at time t c i by u c j previously infected at time t c j , given it infection did not happen before t i . In this paper, we express the hazard function as a constant H(t|t i , α) = α, implying by definition that the probability of an event non happening before a time t given t i decays exponentially as e -α(t-ti) . The associated convex likelihood of α can be found in [START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF] (Eq.7).

Dirichlet-Survival process

In [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF] the authors define the Dirichlet-Hawkes process by replacing the integer counts in Eq.1 by the intensity of a Hawkes process. It can be interpreted as replacing integers counts in Dirichlet Processes by non-integer time-dependent counts, encoded by the intensity of the point process. Here, we consider the hazard rate of the NetRate model instead to account for networks structure. Each node is associated to its own temporal point process, and counts are replaced by the number of times any neighbour has been infected, weighted according to time and to edges strength. Using the methodology introduced in [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF] and substituting the Hawkes process by the hazard rate of a survival model [START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF], we make a yet unexplored bridge between Dirichlet processes and Survival analysis. We remind that [START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF] reformulates the work of [START_REF] Du | Learning networks of heterogeneous influence[END_REF][START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF][START_REF] Gomez-Rodriguez | Structure and dynamics of information pathways in online media[END_REF][START_REF] Wang | Feature-enhanced probabilistic models for diffusion network inference[END_REF] in terms of Survival analysis and associated counting processes; we settled on using NetRate here, but any of these models would fit as well in our approach. The point process nature of survival analysis discussed in [START_REF] Gomez-Rodriguez | Modeling information propagation with survival theory[END_REF] makes this extension sound with respect to previous works on Dirichlet-Point processes [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF][START_REF] Mavroforakis | Modeling the dynamics of learning activity on the web[END_REF][START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF][START_REF] Tan | The indian buffet hawkes process to model evolving latent influences[END_REF]. Let A (k) be the adjacency matrix of the subnetwork associated to cluster k, whose entries are α (k) i,j . We define (u c j , t c j ) (k) as an event of cascade c observed on node u j at t j attributed to subnetwork A (k) . We write the history of events in cascade c attributed to the subnetwork k as

H (k) i,c = {(u c j , t c j ) (k) } j:tj <ti . We note H i,c = {H (k) i,c } k and A = {A (k) } k .
We consider a new event from cascade c observed on node u c i at time t c i . At this point, the new event is not yet associated to any subnetwork. We write the Dirichlet-Survival prior probability for the new event to belong to subnetwork k:

P (s i = k|H i,c , A, λ 0 ) =          λ (k) 0 + H (k) i,c H(t c i |t c j ,α (k) u j ,u i ) λ (K+1) 0 + K k λ (k) 0 + H (k) i,c H(t c i |t c j ,α (k) u j ,u i ) if k = 1, ..., K λ (K+1) 0 λ (K+1) 0 + K k λ (k) 0 + H (k) i,c H(t c i |t c j ,α (k) u j ,u i ) if k = K+1
(2) We introduced a new parameter λ 0 = {λ (k) 0 } k=1,...,K+1 , which translates the probability for a new observation not to have been triggered by any neighbour. It represents the probability that an event of cluster k is exogenous [START_REF] He | Hawkestopic: A joint model for network inference and topic modeling from text-based cascades[END_REF][START_REF] Myers | Information diffusion and external influence in networks[END_REF].

The Dirichlet-Survival prior is coupled to a sequential language model. For simplicity, we consider the bag-of-words Dirichlet-Multinomial model, as in [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF][START_REF] Mavroforakis | Modeling the dynamics of learning activity on the web[END_REF][START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF]; note that more refined sequential language models are also fit to our approach (Dynamic Topic Model [START_REF] Blei | Dynamic topic models[END_REF], online LDA [START_REF] Alsumait | On-line lda: Adaptive topic models for mining text streams with applications to topic detection and tracking[END_REF], online PLSA [START_REF] Bassiou | Online plsa: Batch updating techniques including out-of-vocabulary words[END_REF], etc.).

The input data is a stream of events. Each event takes the form of a triplet (u c i , t c i , v c i ), where c is the cascade an event has been observed in, u c i is the node corresponding to the event, t c i is its publication time, and v c i represents its textual content (e.G. words in a tweet or in a news article). By combining the Dirichlet-Survival prior to the textual likelihood, we get the posterior distribution of the i th observation belonging to cluster (or subnetwork) k as:

P (s i |v c i , N, H i,c , A, θ 0 , λ 0 ) ∝ P (v c i |s i , N, θ 0 ) Dirichlet-Multinomial × P (s i |H i,c , A, λ 0 )
Dirichlet-Survival prior (Eq.2)

(3

)
where ⃗ N contains the words counts within each cluster, v c i contains the words count in document i, and θ 0 the concentration parameter of the model.

Finally, inference is conducted using a Sequential Monte Carlo algorithm similar to [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF][START_REF] Mavroforakis | Modeling the dynamics of learning activity on the web[END_REF][START_REF] Poux-Médard | Powered hawkes-dirichlet process: Challenging textual clustering using a flexible temporal prior[END_REF]. We perform several parallel runs on the same data stream. Within each run, each new observation in the stream is assigned to a cluster according to Eq. 3. The adjacency matrix A is then updated by optimizing the convex likelihood associated to the NetRate point process (Eq. 7 in [START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF]). Finally, we compute the likelihood of the language model for each run; runs that have a likelihood lesser than a threshold are discarded and replaced by more likely ones. The process is repeated until the end of the data stream. According to this algorithm, Eq. 1, and introducing a cutoff on the exponential hazard function (observations older than a time t old are ignored), the optimization runs in O(N obs N runs (N nodes + K)) where N part is the number of particles, N nodes is the maximum network size and K the number of clusters (typically N runs ≪ K ≪ N nodes ). Inference hence scales linearly with the size of the dataset.

We point out that the Dirichlet-Survival process is not about refining complex diffusion models such as [START_REF] Barbieri | Survival factorization on diffusion networks[END_REF][START_REF] Choudhari | Discovering topical interactions in text-based cascades using hidden markov hawkes processes[END_REF][START_REF] Suny | Inferring multiplex diffusion network via multivariate marked hawkes process[END_REF]. Instead, it introduces a different angle for tackling content-aware diffusion problems. This new angle allows for unsupervised, non-parametric and online inference.

Experiments

Data and experimental setup

All data, codes and results are available in open access1 . We consider 3 different network types of 500 nodes each: power-law (PL) [START_REF] Albert | Statistical mechanics of complex networks[END_REF], random Erdös-Renye (ER) [START_REF] Erdős | On the evolution of random graphs[END_REF] and a real network of hyperlinks between political blogs (Blogs) [START_REF] Adamic | The political blogosphere and the 2004 u.s. election: Divided they blog[END_REF]. From each network, we sample 5 subnetworks of 250 nodes and assign random weights α between 0 and 1 to their edges. Each of the generated subnetworks is used to propagate one given cluster of information. We then simulate infection cascades on each subnetwork according to the exponential NetRate model. Finally, we associate 5 words drawn from a vocabulary of size 100 to each so-generated event according to its associated subnetwork (or cluster). We generate a total of 55,000 events {(u c i , t c i , v c i )} i,c for each network. Our hyperparameters are θ 0 = 0.1 and λ (k) 0 = 0.001 ∀k. The SMC algorithm considers 4 parallel runs. We consider a constant hazard rate H(t i |t j , α j,i ) = α j,i , so the probability of a new event not happening decays exponentially with time.

Results

We compare to 3 similar baselines used as ablation tests: Dirichlet-Hawkes process (DHP) [START_REF] Du | Dirichlet-hawkes processes with applications to clustering continuous-time document streams[END_REF] clusters textual data by using temporal dynamics, and does not consider structure; NetRate [START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF] infers a dynamic network based on observed cascades without considering their content; NetRate x Dirichlet-Multinomial (NRxDM) first uses textual information to infer clusters, and only then infers the underlying subnetwork for each cluster, in the same fashion as [START_REF] Du | Uncover topic-sensitive information diffusion networks[END_REF][START_REF] He | Hawkestopic: A joint model for network inference and topic modeling from text-based cascades[END_REF]. When applicable, we evaluate on a classification task (scores NMI and ARI with respect to the clusters used for data generation) and a network inference task (AUC-ROC, F1 and MAE on the true edges, same metrics as in [START_REF] Gomez-Rodriguez | Uncovering the temporal dynamics of diffusion networks[END_REF]). We see in Table 1 that Houston consistently outperforms methods that do not consider jointly text, time and structure of the network. To summarize, NRxDM only considers textual information to build clusters, making the network inference miss a great deal of temporal and structural information. DHP considers textual information and temporal dynamics, but misses the structural information. NetRate does not consider textual data and infers the network based on temporal dynamics only. Houston bridges the gap between these models, by making a joint use of textual, temporal and structural information.

As an illustration of what Dirichlet-Processes can yield on real-world data, we draft its application to the Memetracker dataset [START_REF] Leskovec | Meme-tracking and the dynamics of the news cycle[END_REF] in Fig. 1 (bottom). We retrieve the diffusion network associated to meme clusters and observe diverse spreading dynamics. Topics spread in distinct parts of the global network, and mostly do so through a reduced set of densely connected nodes, as shown in [START_REF] Gomez-Rodriguez | Structure and dynamics of information pathways in online media[END_REF].

Conclusion

In this paper, we propose the Dirichlet-Survival process as an alternative way to jointly model textual, temporal and structural information in spreading processes. Ablation tests demonstrate the relevance of the proposed approach. As a prior, the Dirichlet-Survival process can add a dynamic network dimension to any sequential Bayesian model; it could be coupled to models that account for any type of clustering (e.g. images, time series, labels), or simply more refined language models. Its introduction opens new perspectives on traditional machine learning problems, including topic-dependent spreading processes on networks.

Table 1 .

 1 Results on clusters (NMI, ARI) and edges (AUC-ROC, F1, MAE) retrieval.

			Houston	NRxDM	DHP	NetRate
		NMI	0.809	0.669	0.449	-
		ARI	0.688	0.330	0.063	-
	PL	AUC-ROC	0.807	0.719	-	0.731
		F1	0.199	0.106	-	0.005
		MAE	0.267	0.338	-	0.460
		NMI	0.787	0.711	0.638	-
		ARI	0.631	0.488	0.411	-
	ER	AUC-ROC	0.849	0.800	-	0.659
		F1	0.263	0.176	-	0.005
		MAE	0.229	0.278	-	0.481
		NMI	0.750	0.668	0.372	-
	Blogs	ARI AUC-ROC F1	0.609 0.701 0.168	0.365 0.613 0.087	0.023 --	-0.710 0.005
		MAE	0.374	0.444	-	0.499

https://github.com/GaelPouxMedard/HOUsToN


A. LEVASSEUR Laboratoire de Chimie du Solide du CNRS Université de Bordeaux I 351, cours de la Libération -33405 TALENCE CEDEX {France}.

I -WHAT ISA GLASS?

A glass is the result of a phase transformation in which the nucleation and growth of crystals has been suppressed, i. the Lewis acid strength, which determines the effect of a network mo difier, as seen below. Fig. 1 shows the difference between well or dered and disordered framework.

• Si o 0

FIG. 1 • Si o 0

Difference between well ordered and disordered framework (in the case of sio 2 , the 4th bond is below or above this plane).

The network modifier is A further additive with the specific purpose of increasing ionic conductivity is called a dopant. This is an ionic salt, e.g.

halide, whose anion is not sufficiently strongly basic to break any of the covalent bonds in the network former chain, but simply dissol ved in the glass as if it were in a liquid state.

II -WHY A GLASS?

Glasses have three main qualities compared with crystalline compounds.

1) A glass is characterized by a local order which disappears quickly for a long range disorder giving isotropie properties.

It is much easier to have a glass with a great homogeneity.

"Impurities" are diluted in the matrix "digested" and finally take place in the framework.

2) Glasses are characterized by very large possibilities of composi It is also possible to add some dopant (halide for instance) .

There are in fact some hundred of thousands possible compositions of glasses.

The main interest lies in the possibility to vary the composition in order to vary the properties of glasses (like conductivity, re fractive index ... ).

3) Glasses are characterized by the absence of melting point. The viscosity decreases steadily with the temperature giving some pos sibilities to handle this type•of materials (Fig. 4).

] 12 "'g Different possibilities of shaping according to the glass temperature : stamping, stretching, extrusion, punching, centrifugation.

13

The Tg point is characterized by a viscosity of 10 poises.

III -THERMODYNAMICAL APPROACH

The transition from solid to liquid-like behaviour is not very well defined in a glass, because we do not have the discontinui ties of viscosity, volume and enthalpy that occur at the crystalline melting point. However, there is a glass transition temperature, Tg, which experimentally defines the boundary via the derivatives of vo lume and enthalpy as shown in Fig. 5. Thus below Tg, structural units are considered to be as rigidly bound as in a crystal and above Tg bonds can be broken and re-formed to an extent increasing with tempe rature. Above Tg, the vibrations of the structure cause an almost linear increase in molar volume with temperature, which is associated with an increase in free volume -i.e. the volume occupied in excess of the ideal packing arrangement-. Sorne examples will bê given in next items.

V -NON CRYSTALLINE ELECTROLYTES

A vitreous electrolyte is mainly a good ionic conductor but some other qualities are required like a wide electrochemical stabi lity domain, a low electronic conductivity.

VI -GENERAL CONDITIONS FOR HIGH IONIC CONDUCTIVITY IN SOLIDS

Several conditions must be met for a solid to be a good ionic conductor at moderate temperatures. First, the potentially mo bile species must be present as ions and not be trapped in strong co valent bonds. Second, a population of alternate sites that the ions can potentially occupy and that are not their principal crystallogra phic positions must also exist. Third, the energy to disorder the ions among the larger population of alternate sites and the energy to move the ions among those sites must be low.

The population of defects and their mobility vary according to an Arrhenius expression of the type shown in the equation below.

the expression predicts OT = A exp [ -E/kT] that, at some sufficiently high temperature, the defect concentration and ionic conductivity should be quite high, most normal solids melt before this occurs.

VII -THEORIES OF IONIC CONDUCTION IN GLASSES

Introduction

The experimentally determined conductivity is generally analysed as a product of a carrier concentration and a mobility. This concept works well for crystalline solids, where the carrier concentration may be defined as the density of defects with reference to the perfect lattice. Each carrier is identically situated and has the same mobility, so that the current due to the ensemble of ions is a simple multiple of the single ion current. In contrast to the constant activation energies observed in doped crystals this behaviour is indeed observed in many glasses, as will be seen below. For the sake of simplicity, however, the theoretical treatments discussed here will be the two extreme cases where the variables of carrier concen tration and mobility are considered separately. These are the weak electrolyte theory and the free volume theory respectively.

The weak electrolyte theory

The weak electrolyte model has been applied to glasses in a Thus the theory proposes that the major effects of composi tion changes on conductivity occur via variations in carrier concen tration.

Free volume theory

The above treatment works well and is probably valid for dilute solutions of metal oxides in glasses. However, it is difficult to justify the weak electrolyte assumption in some other contexts, e.g. oxide glasses doped with large amounts of metal halide. Here there is a considerable body of evidence that these glasses should be treated as strong electrolytes -i.e. fully dissociated-and that the variation in conductivity is primarily due to the mobility term.

The structural theory will be discussed here with reference the greatest conductivity appears with heavy

The introduction of sulphide and iodide into lithium glasses gi ves materials wi th possible applications in room tempera ture devices.

Sulphide glasses

A larger conductivity can be obtained wi th glasses based on sulphide compounds. This kind of compound appeared in 1980. This kind of glass is able to be used in solid state batteries.

X -CONCLUDING REMARKS

Recent studies on conducting glasses reveal some essential features of the research trends of new lithium glasses in connection with the improvement of ionic conductivity.

For some of the systems the resul ts can be compared wi th the corresponding crystalline form whereas, for others they cannot.

In general, the lack of structural knowledge of these glasses has slowed down the process of determining an exact transport mechanism. ...

FIG. 9

Electrical conductivity of some of the lithium ion conducting glasses.

Finally, it may also be noted that most of the conductivity measurements have been inadequately performed without information OP possible electronic and anionic transport numbers. Applicability of glasses, characterized for electrochemical applications such as batteries, coulometers, potential memory cells and display devices rernain to be tested for rnany systems. Electrochemical stability range (redox-stability) and charge transfer kinetics are to be optimised to get the most suitable electrolyte. The knowledge about electrochernical performances is, however, not the only essential aspect for future development in this class of materials, but the structural studies will also play an important role to predict new glassy systems, which would be useful for practical applications in future advanced devices.

  e. the time allowed for crystallization has been less than the time required by the kinetic factors of atomic rearrangement. These kinetic factors are temperature dependent, and a devitrification temperature is loo sely defined as the temperature below which the crystallization time becomes practically infinite. If the glass is formed by cooling of a liquid or a vapour, a fast quench is sometimes required to minimize the time at which the glass is below the melting point but above the devitrification temperature. It is also possible to produce a glass directly below its devitrification temperature, e.g. by solvent remo val, gel desolvation or irradiation. Traditionally, however, glasses have been made without re sorting to fast quenching or other special methods by normal cooling of a melt containing a large amount of a network former e.g. This produces a high viscosity just above the melting point, by vir tue of forming polymeric chains and cross-links, and thus slows down the kinetics of crystallization. In order ta widen the range of properties, modern glass research has used a variety of network formers related ta by substitution of Si by other group IVb, IIIb and Vb elements, and o by (mainly) group VIb. Such substitutions are useful in changing the viscosity/temperature relations (e.g. the mechanical properties of sio 2 are reflected at lower temperatures by a 2 o 3 or P 2 o 5 ) as well as

  FIG. 2 Two dimensional schematic representation of a silica glass with modifier cations.

  FIG.4

  FIG. 5Temperature dependence (schematic) of molar volume, enthalpy, expansion and specific heat of a glass.

  Ionically conductive glasses are somewhat more amenable to structural study. They are homogeneous compositions whose properties resemble those of high viscosity liquids. Understanding structure in these materials means examining how intertwined polymer chains or interconnected covalent ionic networks support ionic conductivity in what are essentially amorphous compositions. In these materials, the study of structure is an attempt to understand the make up of the amorphous network in which ions are mobile and how changes in compo sition alter the network and the magnitude of the observed ionic con ductivity.

  .e. of the form A exp -E/kT, where the pre-exponential factor A and the activation energy E, are parameters describing the processes of migration, or defect generation. The conductivity, obtained as the product of two such terms is then of the same form, A' exp -E'/kT. Such simplifications are not possible for a glass. Here, wi thout the reference of a perfect lattice, the de finition of a defect becomes rather arbitrary. A more satisfactory concept than the defect formation is one considering all ions to be potentially conducting, but in a broad distribution of states each having a different value of A and E. The effect of a Gaussian distribution of activation energies has been analysed in the random site model with the result that the effective activation energy of the ensemble becomes a function of concentration.

  similar way to the familiar theory of conduction in aqueous solutions of weak acids, where the variation in equivalent conductivity on dilution is explained solely by a very small and variable degree of dissociation. Taking a sodium silicate glass as an example, the fol lowing thermodynamic treatment is applied. Considering the glass to be a dilute solution of sodium oxide in silica, an activity of the sodium oxide, a (Na 2 0), can be measured experimentally via the of the low concentrations, the single ion activities are assumed to be proportional to the concentrations of the solvated ions [Na + ] and [ONa -].since no other ions are present, the two concentra tions must be equal to give electroneutrality. dissociated cations is derived as the square root of the oxide activity, which increases rapidly with the nominal oxide concentration. For a number of glass systems at fixed temperatures, the variation in conductivity with metal oxide concentration (sometimes well outside the "dilute" range) shows a remarkable proportionality to the square root of activity. If we now identify the dissociated ion concentration with the defect ion concentration and apply the crystalline state model, the result is a concentration independent mobility. [Further evidence for the weak electrolyte theory is given by mixed alkali effect]:

  and related systems -which are extensively docu mented from the structural point of view-. According to Raman and other studies the halide does not react with the boron-oxygen network, i.e. the halide ion is not sufficiently basic to break the B-0 bond. In contrast with the metal oxide, the halide is merely assimilated into interstices of the network. Furthermore, NMR studies on the corresponding fluoride glass have shown that the Li + and F ions are dissociated (with respect to crystalline LiF).Numerous correlations have been made between the conductivity and various pararneters related to the rnobility (e.g. Tg, free volume as estirnated frorn rnolar volume). In the ternary system B additional free volume created by addition of LiCl.The second factor had more effect than the first one on the activation energy, and in a study on B it was shown that (in the concentrated ionic region) although the oxide and halide had sirnilar effects on Tg, the halide gave rnuch greater conductivity en hancernent at 2 5 °C. Thus, although low Tg values often correlate with high mobilities, information regarding the free volume itself is more directly related to the conductivity.4. DiscussionThe theories given above do not clairn to be unique or uni versal in application to the wide variety of glasses exhibiting ionic conductivity, but rather as treatrnents of specific limitations to the conductivity. Other factors rnay corne into play, such as the effects of ion polarizability, particular rnechanisrns of transport relating to special structure types and the possible existence of highly conduc ting rnicrophases. Excluding such complications, we have useful quali tative observations frorn both the weak electrolyte and structure re lated theories. Frorn the weak electrolyte theory we have the concept of conductivity limitation due to strong association of cations with anions such as o 2 -which, because of their high basicity becorne bound to the chain. Thus, in a pure oxide glass, for exarnple, only a srnall proportion of cations can, at any tirne, break away. However, by the addition of e.g. a rnetal chloride, we can create an excess of cations over the oxide binding sites, and expect these to be mobile. The ex tent of rnobility, however, is then deterrnined by the size of the ion (allowing for shape distortion in the case of a high polarizability, e.g. Ag ) relative to the size and continuity of the interstitial space between large anions (again considering polarizability). It appears that, in the highest conductivity glasses, primary limitation. the latter is the VIII -LITHIUM GLASS ELECTROLYTES As stated in the theory section, lithium glasses have shown evidence for the weak electrolyte theory in the poorer conductors and free volume theory in the better conductors containing much lithium chloride. On substitution of sulphide for oxide, we again see a pro nounced increase in conductivity at room temperature. Finally, as one may have anticipated, doping with LiI.

  Li, Na, K) reported by him are still being investigated for possible applications. He observed that high Li -conductivities could be obtained in borate glasses if more than 40 m/o lithium compounds could be incorporated in the glass without devitrification. The value of l0 -1 (ohm-cm) -l at 350°C is reported by him for one of the compositions. The knowledge about the advanced applications of glasses and their large scale production potential, stimulated the interest of many investigators and the results on a large number of glassy systems have been reported. Hundreds of such materials have been synthesized and their electrical properties investigated. is possible to add a lithium salt {a dopant) to in crease the conductivity. The best ionic conductivity obtained in the first case is about lO -Bn -1 cm -l at 25 °c. In ternary systems, glasses can exhibit a conductivity of about l0 -

Fig

  Fig. 6 shows the conducti vi ty of oxide and sulphide binary glasses wi th the molar fraction of Li j O or Li 2 s. Sulphide glasses are 10 times more conducting than cor responding oxide glasses.

  FIG. 8 Cyclic voltammogram of a ternary B 2 s 3 -Li 2 S-LiI glas-? (reference electrode Ag/Ag couple) .

  The distribution and number of charge carriers and their contribution to overall conductivity are not very clearly understood. Neverthe less, large data are available for discussion.One of the interesting observations is the relative insen sitivity of 0 0 to composition. For many orders of change in the ma gnitude of conductivity, cr 0 changes insignificantly as observed for Li-borate glasses (Angell 1981). In classical ionic conductors log cr 0 is expected to vary linearly with composition. The observed fact in highly conducting glasses thus contradicts the classical diffusion theory (Tuller et al. 1980). Inorganic glasses containing alkali halides show dependence of conductivity on concentration and are therefore expected to give information regarding structural changes. The conductivity tends to increase with a drop in corresponding activation energy. A linear decrease has been observed by Malugani et al (1978). Though its ori gin is not known, it is expected to result from a structural break down since the glass transition temperature also repeats this obser vation. It is assumed that the anion is not inserted into the micro molecular chain and the chain preserves the characteristics of the original glass. This structural hypothesis has been confirmed by Raman spectroscopy with borate glasses. The Raman spectrum does not show any modification, corresponding to B-0 vibrational mode upon addition of large quantities of halides. Tuller et al (1980) have in terpreted this model as a form of phase separation which may occur on a scale difficul t to identify by most conventional means. Phase separation would, however, be expected to show discontinuous change in activation energy as the highly conducting phase changes from iso lated pockets to a continuous phase throughout the material. Conductivities of glasses containing alkali salts also show almost linear increase with radius of halide ion. Two factors have been suggested to explain the observed behaviour. The first concerns the polarizability of anions the larger the anion, the greater is the polarizability and thus the ease with which its electron cloud may be deformed. The second approach considers dilation of structure due to larger radius of anions thereby decreasing the packing density which eases the squeezing of the ion through the opening. This model also supports the decrease in activation energy, since the activation energy for motion is inversely proportional to the size of the opening. Although these models are reasonable and agree qualitatively with experimental results, the uni versal acceptance might require data analysis in much greater detail. Quenching techniques show the ability to produce wide variety of new oxide glasses. Though the glasses so prepared give thin films or flakes (2 µm thick) the area is too low (few rnrn 2 ) to permit any large scale application. It is therefore necessary to explore techniques which will give films with larger surface area. Structural aspects of these glasses are not yet very clear as com pared to conduction mechanism and it becomes difficult to predict new systems. To conclude this review on the present knowledge of the electrical properties of glassy electrolytes,we have shown in table 1 a comparison of the performance of different lithium conducting glas sy electrolytes. Composition, conductivity and activation energies are included. A plot of a vs reciprocal of temperature, for different glassy solid electrolytes along with some of the well-known crystal line solid electrolytes like a-alumina, It may be noted that the electrical properties of these electrolytes compare quite well with the traditional crystalline electrolytes.

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