We study a class of random finite difference operators, a typical example of which is the finite difference Schrδdinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrodinger operator with a random potential has pure point spectrum and developps no static conductivity. PlanI. Introduction -Quelques motivations physiques -Historique du sujet 201 A. Le cas general 204 II. Definitions et preliminaires 204 III. Determination du spectre 208 IV. Criteres spectraux 210 V. Sur la conduct! vite et la constante de diffusion 217 VI. Un formalisme general 224 B. L'operateur de Schrodinger unidimensionnel 231 VII. Fonction spectrale moyenne et fonction de correlation a volume fini 231 VIII. Etude de l'operateur de transfer 235 IX. Resultats sur l'operateur de Schrodinger unidimensionnel 241 C. La transition de localisation vue comme un changement de phase: description qualitative 244 * Equipe de Recherche du C.N.R.S. n° 174 ** Recherche soutenue par le Fonds National Suisse de la recherche scientifique 0010-3616/80/0078/0201/S09.20

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Nafion membrane degradation was studied in a polymer electrolyte membrane fuel cell ͑PEMFC͒ under accelerated decay conditions. Fuel cell effluent water was analyzed to determine the fluoride emission rate. Experimental findings show that formation of active oxygen species from H 2 O 2 decomposition or the direct formation of active oxygen species in the oxygen reduction reaction are not the dominating membrane degradation mechanisms in PEMFCs. Instead, membrane degradation occurs because molecular H 2 and O 2 react on the surface of the Pt catalyst to form the membrane-degrading species. The source of H 2 or O 2 is from reactant crossover through the membrane. The reaction mechanism is chemical in nature and depends upon the catalyst surface properties and the relative concentrations of H 2 and O 2 at the catalyst. The membrane degradation rate also depends on the residence time of active oxygen species in the membrane and volume of the membrane. The sulfonic acid groups in the Nafion side chain are key to the mechanism by which radical species attack the polymer.

Nafion membrane degradation was studied in a polymer electrolyte membrane fuel cell ͑PEMFC͒ under accelerated decay conditions. Fluoride emission rate ͑FER͒ determined by fuel cell effluent water analysis was used to quantify the membrane degradation. Membrane degradation is most likely caused either directly or indirectly by the species formed as a result of the H 2 and O 2 reaction on the catalyst. To further understand the mechanism, the effects of the catalyst location, type, its interaction with O 2 and H 2 O, and cell current density on the FER were investigated and their implications on the underlying membrane degradation mechanism are discussed.

The involvement of H 2 O 2 in the membrane degradation mechanism in a polymer electrolyte membrane fuel cell ͑PEMFC͒ was investigated. Measurement of fluoride concentration in the effluent water was used as an indicator of the membrane degradation rate. It was found that H 2 O 2 is formed in the fuel cell in small concentrations but is not the main source of harmful species, which degrade the membrane. H 2 O 2 decomposition due to impurities or the catalyst leading to the possible formation of radical species would only account for a small fraction of the membrane degradation rate in a fuel cell.Nafion and Nafion-based membranes are the most commonly used membranes in polymer electrolyte membrane fuel cells ͑PEM-FCs͒. At present, the lifetime of these membranes in state-of-the-art membrane electrode assemblies at steady-state operation is below the target lifetime required for the use of PEMFCs in automotive and stationary applications. The lifetime is likely to decrease during actual operation in a fuel cell stack due to the noncontinuous nature of operation. Increasing the membrane life or developing a new membrane with improved life requires understanding of the membrane degradation mechanism during operation in a PEMFC.The oxygen reduction reaction is considered to proceed by the following pathways in acid solution 1The above two pathways have been quantitatively studied using rotating-ring disk electrodes ͑RRDE͒. 2 Results have shown that the yield of H 2 O 2 increases with a decrease in disk potential reaching the maximum in the potential range of H 2 adsorption, i.e., near the anode potential in an operating fuel cell. One of the commonly used ex situ beaker tests for studying membrane degradation is soaking a piece of membrane in Fenton's reagent. Radical species are formed by the reaction of H 2 O 2 with Fe 2+ 3The radical species attack the polymer to degrade the membrane. The presence of radicals has been detected by electron spin resonance spectroscopy. 4 After actual cell operation for a considerable amount of time post-test analysis of the membrane electrode assembly shows that membrane degradation mainly occurs near the anode-membrane interface. 5,6Based on the above-mentioned results ͑RRDE studies, membrane degradation in Fenton's reagent, and preferential membrane decay near the anode side͒, membrane degradation was thought to be because of the H 2 O 2 generation at the anode as an intermediate in the oxygen reduction reaction. ͑The presence of O 2 at the anode is from O 2 crossover from the cathode through the membrane.͒ The H 2 O 2 at the anode then diffuses into the membrane and reacts with bivalent metal cations ͑M 2+ ͒, present as impurities in the membrane to form active oxygen species, which can then attack the polymer and degrade the membrane. 6,7 However, this proposed mechanism is not consistent with recent reports on membrane degradation at the highly accelerated decay conditions of open-circuit voltage ͑OCV͒. 8,9 Studies show that the cathode is also involved in the degradation mechan...

We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential.The existence and analyticίty properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion.In the special case of Monomer-Dimer systems it is established that no phase transition is possible moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Pade approximants.Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems. * Work presented in partial fullfilment of the Ph. D. Thesis.

A step-by-step technique to evaluate six sources of polarization, mainly associated with the cathode, in hydrogen/air proton exchange membrane fuel cells is demonstrated. The six sources of polarization were nonelectrode ohmic overpotential, electrode ohmic overpotential, nonelectrode concentration overpotential, electrode concentration overpotential, activation overpotential from the Tafel slope, and activation overpotential from catalyst activity. The technique is demonstrated as applied in the analysis of hydrogen/air polarization curves of an in-house membrane electrode assembly ͑MEA͒ using hydrogen/oxygen polarization curves as a diagnostic tool. The analysis results are discussed at three cell temperature/relative humidity ͑RH͒/oxygen partial pressure (p O 2 , atm͒ conditions at atmospheric pressure: 80°C/100% RH anode /75% RH cathode /p O 2 ϭ 0.136, 100°C/70% RH/p O 2 ϭ 0.064, and 120°C/35% RH/p O 2 ϭ 0.064, which represent a near fully-humidified, a moderately humidified, and a low humidified condition, respectively. At the higher temperature operating conditions the RH and p O 2 decrease resulting in higher electrode ohmic resistance ͑0.020, 0.020, and 0.035 ⍀ cm 2 , respectively͒, lower limiting current ͑2019, 1314, and 819 mA/cm 2 , respectively͒, and lower onset current density for significant electrode concentration overpotential ͑80, 60, and 40 mA/cm 2 , respectively͒. The technique is useful for diagnosing the main sources of loss in MEA development work, especially for high temperature/low relative humidity operation where several sources of loss are present simultaneously.Classification of PEM fuel cell losses.-Useful electrical energy is obtained from a fuel cell only when a reasonable current is drawn from it, but the actual cell voltage decreases from its equilibrium cell voltage because of irreversible losses. 1 These losses are called polarization when the cell voltage is compared with the experimental equilibrium cell voltage and called overpotential ͑͒ when it is compared with the theoretical equilibrium cell voltage according to the Nernst equation. The terms polarization and overpotential are often used interchangeably to refer to voltage losses in general.In proton exchange membrane ͑PEM͒ fuel cells, the losses originate primarily from the following three sources 1 : activation overpotential ( act ), ohmic overpotential ( ohm ), and concentration overpotential ( conc ). Activation overpotential is associated with sluggish reaction kinetics and low catalyst activity. Ohmic overpotential is associated with resistance to proton and/or electron transport. Concentration overpotential is associated with reactant transport resistance to the catalytically active sites.Sources of polarization for the various components of a membrane electrode assembly ͑MEA͒ are shown in Table I. The electrode is where the electrochemical reaction occurs and reactants are consumed. Hydrogen oxidation occurs in the anode electrode and oxygen reduction occurs in the cathode electrode of a hydrogen/air PEM fuel cell...