The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown's force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy's theorem in continuum mechanics to a magnetoelastic material. The proof of Brown's formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown's force formula. One obtains an additional nonlinear surface term which allows one to regard Brown's assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure. / Das Thema dieser Arbeit ist eine mathematisch strenge Herleitung von Formeln für die magnetische Kraft, die auf einen Teil eines beschränkten, magnetischen Körpers durch seine Umgebung ausgeübt wird. Zunächst wird die magnetische Kraft in einem kontinuierlichen System auf Grundlage der makroskopischen Magnetostatik betrachtet. Mit Bezug auf W. F. Brown, der eine vor allem physikalisch motivierte Herleitung der Kraftformel gegeben hat, wird diese auch Brownsche Kraftformel genannt. Das Oberflächenintegral in dieser Formel zeigt eine nichtlineare Abhängigkeit von der Normalen. Um Cauchys Theorem aus der Kontinuumsmechanik in einem magnetoelastischen Material anwenden zu können, nimmt Brown an, dass die Oberflächenkraft einen zusäatzlichen Term enthält, der den nichtlinearen Ausdruck aufhebt. Der Beweis der Brownschen Kraftformel in dieser Arbeit beruht auf einer geeigneten Regularisierung eines hypersingulären Kerns und benutzt Methoden für singuläre Integrale. Danach gehen wir von einem diskreten, periodischen System von magnetischen Dipolen aus und betrachten die Kraft zwischen einem Teil einer beschränkten Menge und der Umgebung. Um zum Kontinuumslimes überzugehen, starten wir von der üblichen Kraftformel für wechselwirkende magnetische Dipole. Es zeigt sich, dass sich der Limes der diskreten Kraft von der Brownschen Kraftformel unterscheidet. Man erhält einen zusätzlichen nichtlinearen Oberflächenterm, der es ermöglicht, Browns Annahme als Konsequenz des atomistischen Zugangs zu sehen. Kurzreichweitige Effekte führen zudem zu einem linearen Oberflächenterm im Kontinuumlimes der diskreten Kraft. Dieser Zusatzterm enthält eine gewisse Gittersumme, die von einem hypersingulären Kern und der Struktur des zugrundeliegenden Gitters abhängt.

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown''s force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy''s theorem in continuum mechanics to a magnetoelastic material. The proof of Brown''s formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown''s force formula. One obtains an additional nonlinear surface term which allows one to regard Brown''s assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure. / Das Thema dieser Arbeit ist eine mathematisch strenge Herleitung von Formeln für die magnetische Kraft, die auf einen Teil eines beschränkten, magnetischen Körpers durch seine Umgebung ausgeübt wird. Zunächst wird die magnetische Kraft in einem kontinuierlichen System auf Grundlage der makroskopischen Magnetostatik betrachtet. Mit Bezug auf W. F. Brown, der eine vor allem physikalisch motivierte Herleitung der Kraftformel gegeben hat, wird diese auch Brownsche Kraftformel genannt. Das Oberflächenintegral in dieser Formel zeigt eine nichtlineare Abhängigkeit von der Normalen. Um Cauchys Theorem aus der Kontinuumsmechanik in einem magnetoelastischen Material anwenden zu können, nimmt Brown an, dass die Oberflächenkraft einen zusäatzlichen Term enthält, der den nichtlinearen Ausdruck aufhebt. Der Beweis der Brownschen Kraftformel in dieser Arbeit beruht auf einer geeigneten Regularisierung eines hypersingulären Kerns und benutzt Methoden für singuläre Integrale. Danach gehen wir von einem diskreten, periodischen System von magnetischen Dipolen aus und betrachten die Kraft zwischen einem Teil einer beschränkten Menge und der Umgebung. Um zum Kontinuumslimes überzugehen, starten wir von der üblichen Kraftformel für wechselwirkende magnetische Dipole. Es zeigt sich, dass sich der Limes der diskreten Kraft von der Brownschen Kraftformel unterscheidet. Man erhält einen zusätzlichen nichtlinearen Oberflächenterm, der es ermöglicht, Browns Annahme als Konsequenz des atomistischen Zugangs zu sehen. Kurzreichweitige Effekte führen zudem zu einem linearen Oberflächenterm im Kontinuumlimes der diskreten Kraft. Dieser Zusatzterm enthält eine gewisse Gittersumme, die von einem hypersingulären Kern und der Struktur des zugrundeliegenden Gitters abhängt.

Lithium alloys with metallic or semi-metallic elements are attractive candidate materials for the next-generation rechargeable Li-ion battery anodes, thanks to their large specific and volumetric capacities. The key challenge, however, has been the large volume changes, and the associated stress buildup and failure during cycling. The chemo-mechanics of alloy-based electrode materials entail interactions among diffusion, chemical reactions, plastic flow, and material property evolutions. In this study, a continuum theory of two-way coupling between diffusion and deformation is formulated and numerically implemented. Analyses based on this framework reveal three major conclusions. First, the stress-to-diffusion coupling in Li/Si is much stronger than what has been known in other electrode materials. Practically, since the beneficial effect of stress-enhanced diffusion is more pronounced at intermediate or higher concentrations, lower charging rates should be used during the initial stages of charging. Second, when plastic deformation and lithiation-induced softening take place, the effect of stress-enhanced diffusion is neutralized. Because the mechanical driving forces tend to retard diffusion when constraints are strong, even in terms of operational charging rate alone, Li/Si nano-particles are superior to Li/Si thin films or bulk materials. Third, the diffusion of the host atoms can lead to significant stress relaxation even when the stress levels are below the yield threshold of the material, a beneficial effect that can be leveraged to reduce stresses because the host diffusivity in Li/Si can be non-negligible at higher Li concentrations. A theory of coupled chemo-mechanical fracture driving forces is formulated in order to capture the effect of deformation-diffusion coupling and lithiation-induced softening on fracture. It is shown that under tensile loading, Li accumulates in front of crack tips, leading to an anti-shielding effect on the energy release rate. For a pre-cracked Li/Si thin-film electrode, it is found that the driving force for fracture is significantly lower when the electrode is operated at higher Li concentrations -- a result of more effective stress relaxation via global yielding. The results indicate that operation at higher concentrations is an effective means to minimize failure of thin-film Li/Si alloy electrodes.

No description available.

Electric double layer capacitors (EDLCs) are an important class of electrical energy storage devices which store energy in the form of electric double layers. The charging mechanism is highly reversible physical adsorption of ions into the porous electrodes, which empower these devices to show a remarkable power performance (15kW/kg) and greater life expectancy (> 1 million cycles). However, they store a small amount of energy (5Wh/kg) when compared with batteries. Optimization of the performance of EDLCs based on porous activated carbons is highly challenging due to complex charging process prevailing in the Nano pores of electrodes. Molecular simulations provide information at the molecular scale which in turn can be used to develop insights that can explain experimental results and design improved EDLCs. The conventional approach to simulate EDLCs places both the electrodes and electrolyte region in a single simulation box. With present day computers, however, this one-box method limits us to system sizes of the order of nanometres whereas the size of a typical EDLC is at least of the order of micrometres. To overcome this system size limitation, a Gibbs-ensemble based Monte Carlo (MC) method was recently developed, where the electrodes are simulated in a separate simulation boxes and each box is subjected to periodic boundary conditions in all the three directions. This allows us to eliminate the electrode-electrolyte interface. The simulation of the bulk electrolyte is avoided through the use of the grand canonical ensemble. The electrode atoms in the electrode are maintained at an equal constant electric potential likewise the case in a pure conductor with the use of the constant voltage ensemble. In this thesis, the Gibbs-ensemble based MC simulations are performed for an EDLC consisting of porous electrodes. The simulations are performed with aqueous electrolytes of type MX and DX2 (where M=Na+, K+; D=Ca+2; X=Cl , F ) for a wide variety of operating conditions. The water is modelled as a continuum background with a dielectric constant value of 30. The electrodes are silicon carbide-derived carbon, whose microstructure generated from reverse MC technique, is used in the simulations. The results from these simulations help us understand the charge storage mechanism, the effect of size and valence of ions on the performance of nonporous carbon based EDLCs when the hydration effects are indignant. The thesis first demonstrates the presence of finite size effects in the simulations performed with the one-box method for KCl electrolyte. The capacitance (ratio of the charged stored on the positive electrode to the voltage applied) values obtained for KCl electrolyte with the one-box method are significantly higher than the corresponding values obtained from the Gibbs-ensemble method. This shows the presence of finite size effects in the one-box method simulations and justices the use of the Gibbs-ensemble based method in our simulations. The fundamental characteristics of aqueous electrolytes in the EDLC are analyzed with the simulation results for KCl electrolyte. In agreement with experiments and modern mean held theory, the capacitance monotonically decreases with voltage (bell-shaped curve) due to overcrowding of ions near the electrode surface. The charge storage mechanism in both the electrodes is mainly a combination of countering (ions oppositely charged to that of the electrode) adsorption and ion exchange, where coins (ions identically charged to that of the electrode) are replaced with countering. However, at higher voltages, the mechanism is predominantly counter ion adsorption because of the scarcity of coins in the electrodes. The mechanism is preferentially more ion exchange for the positive electrode because of its relatively bulky countering, Cl . The shifting of mechanism towards counter ion adsorption at higher voltages and preferential ion exchange process for the positive electrode are in qualitative agreement with the recent experimental results. The constraint of equal electric potential on all the electrode atoms of the amorphous electrode in the simulations resulted in a non-uniform average charge distribution on the electrodes. It shows that the Gibbs-ensemble simulation approach can account for the polarization effects which arises due to a complex topology of the electrodes. In agreement with earlier experiments and simulation studies, the local structure analyses of the electrodes shows that the highly conned ions store charge more efficiently. On the application of voltage difference between the electrodes, the electrolyte ions move towards higher degree of con ned regions of the electrodes indicating the charging process involves local rearrangement and rescuing of electrolyte ions. The thesis also discusses the effect of temperature and bulk concentration on the performance of EDLCs. The Gibbs-ensemble based simulations are performed for the EDLC with varying temperature and bulk concentration for the KCl electrolyte independently. In agreement with the Guo -Chapman theory and experiments, the capacitance decreases with the temperature and increases with the bulk concentration. This is because the concentration of countering in the electrodes decreases with an increase in the temperature but increases with an increase in the bulk concentration. Lastly, the effect of ion size and valency on the performance of EDLCs is analyzed. The capacitance monotonically decreases with voltage (bell-shaped curve) for all the electrolytes, except for NaF, where a maximum is observed at a non-zero finite voltage (camel-shaped curve). The capacitances of NaCl and NaF are greater than that for KCl and KF, respectively. This is because the smaller Na+ ions have more accessibility to narrow con ned regions, where the charge storage efficiency is high. As expected, the capacitance for CaCl2 and CaF2 are highest among their monovalent counterparts, NaCl and KCl; NaF and KF, respectively. This is attributed to the relatively smaller double layer thickness of the bivalent Ca+2 ions. Interestingly, at higher voltages, the capacitance for the bivalent electrolytes approaches the capacitance for the monovalent electrolytes because the concentration of Ca+2 ions in the negative electrode increases sluggishly with voltage due to a strong electrostatic repulsion between Ca+2 ions.