### Transcription

DMTDModelling Self-Organizationon Electrodes of DC Glow DischargesDOCTORAL THESISMatthew Simon BieniekDOCTORATE IN PHYSICSFebruary 2018

iDedicated to Dina Moldovan

AcknowledgementsI thank supervisors for the contribution they have made to my academic, professional,and personal development. Also for inviting me into the environment they have helpedto carve out in Madeira Island, where great science can be done, and new scientistscan be trained.I acknowledge Diego Santos for his e orts and talents in modelling that contributedto the work on identifying bifurcations.I thank Helena Kaufman for her compnay during lunch, and for helping me towrite the abstract for this thesis in Portuguese.Also, I wish to thank my other colleagues for pleasant interactions and stimulatingdiscussion.ii

PreambleThe work leading to this thesis was performed within activities of:FCT - Fundacão para a Ciência e a Tecnologia of Portugal through the projectPest-OE/UID/FIS/50010/2013.Most of the results presented in this thesis are published in the following articles:M. S. Bieniek, P. G. C. Almeida, and M. S. Benilov, Modelling cathode spotsin glow discharges in the cathode boundary layer geometry (2016), J. Phys. D:Appl. Phys. No. 49 105201M. S. Bieniek, D. Santos, P. G. C. Almeida, and M. S. Benilov, Bifurcations inthe theory of current transfer to cathodes of dc discharges and observations oftransitions between di erent modesSubmitted in Feb 2018 to Phys. PlasmasM. S. Bieniek, P. G. C. Almeida, and M. S. Benilov, Self-consistent modelling ofself-organized patterns of spots on anodes of DC glow dischargesSubmitted in Feb 2018 to Plasma Sources Sci. TechnolResults presented in this thesis were reported at the following conferences:M. S. Bieniek, P. G. C. Almeida, and M. S. Benilov, Modelling anode spots ofDC glow discharges, in Proc. XXXIII ICPIG (9-14 July 2017, Estoril, Portugal)M. S. Bieniek, P. G. C. Almeida, and M. S. Benilov, Modelling cathode spots inglow discharges in the cathode boundary layer geometry with COMSOL Multiphysics , in Proc. XXXII (26-31 July 2015, Iasi, Romania)iii

ResumoO trabalho apresentado nesta dissertação refere-se à modelação de padrões de autoorganização de densidade de corrente em elétrodos de descargas DC luminescentes.Padrões de manchas anódicas foram modelados de forma auto-consistente pelaprimeira vez e os fenómenos nas manchas foram investigados. As soluções que descrevem as manchas foram encontradas num intervalo de correntes com múltiplassoluções. Foi descoberta uma inversão da densidade de corrente local do ânodo no centro de cada uma das manchas, isto é, mini-cátodos são formados dentro das manchas;poder-se-ia dizer, as manchas do ânodo funcionam como uma descarga luminescenteunipolar. As soluções não se enquadram no padrão convencional de auto-organizaçãoem sistemas dissipativos não-lineares biestáveis; por exemplo, as transições de ummodo para outro não se realizam através de bifurcações.Padrões auto-organizados de manchas catódicas em descargas luminescentes forammodelados na camada de plasma junto ao cátodo, numa geometria igual à utilizada namaioria das experiências descritas na literatura. O efeito da geometria da câmara dedescarga nas manchas foi investigado. Os padrões de manchas modelados são idênticosaos observados nas experiências e similares aos calculados na con guração de elétrodosplanos em paralelo.Uma tentativa foi feita para modelar quantitativamente a descarga DC luminescente com manchas catódicas. Uma descrição detalhada desta modelação, a maisprecisa deste fenómeno até à data, é apresentada. Em geral, os padrões calculadossão semelhantes aos observados nas experiências, mas as CVC são qualitativamentediferentes.Cenários de transições entre modos com diferentes padrões de manchas em elétrodos de descargas DC luminescentes e em cátodos de descargas de arco são investigados.No caso de transições entre padrões em cátodos de descargas DC luminescentes, foramencontradas as transições observadas nas experiências que podem estar diretamenterelacionados a bifurcações de soluções estacionárias, e as bifurcações correspondentesforam modeladas. Os padrões encontrados na modelação numérica estão em conformidade com os observados no decurso das transições nas experiências. No caso dos cátodos de descargas de arco, mostra-se que qualquer transição entre diferentes modos detransferência de corrente está relacionada a uma bifurcação de soluções estacionárias.Palavras chave: Interacção plasma-cátodo, Auto-organização, Descargas luminescentes, Manchas no ânodo, Cátodos termiónicos, Manchas catódicasiv

AbstractIn this work self-organized patterns of current density on electrodes of dc glow discharges were modelled.Patterns of anodic spots were modelled self-consistently for the rst time and theirphysics was investigated. The solutions describing the spots were found to exist in aregion of current with multiple solutions. A reversal of the local anode current densityin the middle of each of the spots was discovered, i.e. mini-cathodes are formed insidethe spots or, as one could say, the anode spots operate as a unipolar glow discharge.The solutions do not t into the conventional pattern of self-organization in bistablenonlinear dissipative systems e.g. the modes are not joined by bifurcations.Self-organized patterns of cathodic spots in glow discharges were computed in thecathode boundary layer geometry, which is the one employed in most of the experiments reported in the literature. The e ect that the geometry of the vessel has onthe spots was investigated. The computed spot patterns are the same as the onesobserved in the experiment and similar to the ones computed in the parallel planeelectrode con guration.An unsuccessful attempt was made to quantitatively model DC glow discharge withcathodic spots, an account of this, the most accurate modelling of the phenomenon yetperformed, is given. In general, the computed patterns are observed in the experimentbut the CVC are qualitatively di erent.Scenarios of transitions between modes with di erent patterns of spots on electrodes of dc glow discharges and cathodes of arc discharges are investigated. In thecase of transitions between patterns on dc glow cathodes, those transitions reported inthe experiments that may be directly related to bifurcations of steady-state solutionsare found and the corresponding bifurcations are computed. Patterns found in thenumerical modelling conform to those observed in the course of the transitions in theexperiment. In the case of cathodes of arc discharges, it is shown that any transitionbetween di erent modes of current transfer is related to a bifurcation of steady-statesolutions.Keywords: Plasma-electrode interaction, Self-organization, Glow discharges, Cathodic spots, Anodic spotsv

Contents1 Introduction1.1 Direct current glow discharge . . . . . . . . . . . . . . . . .1.1.1 Modelling and theory during the 20th century . . . .1.2 Self organization . . . . . . . . . . . . . . . . . . . . . . . .1.2.1 Thermodynamics of self organization . . . . . . . . .1.2.2 An introduction to stability . . . . . . . . . . . . . .1.2.3 An introduction to bifurcations . . . . . . . . . . . .1.3 Self organization of spots on electrodes of glow discharges .1.3.1 Observations of spots on cathodes of glow discharges1.3.2 Theory and modelling of cathode spots . . . . . . .1.3.3 Observations of spots on anodes of glow discharges .1.3.4 Theory and modelling of anode spots . . . . . . . . .1.4 This work . . . . . . . . . . . . . . . . . . . . . . . . . . . .111345666799112 Anode spots2.1 Introduction . . . . . . . . . . . . . . . .2.2 The model . . . . . . . . . . . . . . . . .2.3 Results and discussion . . . . . . . . . .2.3.1 Current-voltage characteristics of2.3.2 Anode spot structure . . . . . . .2.3.3 Near-anode physics . . . . . . . .2.3.4 Additional comments . . . . . .2.4 Conclusions . . . . . . . . . . . . . . . .1212131516182122233 Modelling cathode spots in glowlayer geometry3.1 Introduction . . . . . . . . . . .3.2 Model and Numerics . . . . . .3.3 Results . . . . . . . . . . . . . .3.3.1 Fundamental mode . . .3.3.2 3D modes . . . . . . . .3.4 Conclusions . . . . . . . . . . . . . .the. . . . . . . . . . . . . .anode. . . . . . . . . . . . . . . . . . . . . .region. . . . . . . . . . . . .discharges in the cathode boundary.vi.24242628283136

CONTENTS4 Bifurcations in the theory of current transfer to cathodes of DCdischarges and observations of transitions between di erent modes4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 Scenarios of transitions between di erent modes of current transfer toelectrodes of dc discharges and their relation to bifurcations . . . . . .4.3 Mode transitions on cathodes of arc discharges . . . . . . . . . . . . .4.4 Mode transitions on cathodes of dc glow discharges . . . . . . . . . . .4.4.1 State-of-the-art of the theory . . . . . . . . . . . . . . . . . . .4.4.2 Analyzing experimental observations . . . . . . . . . . . . . . .4.4.3 Numerical modelling . . . . . . . . . . . . . . . . . . . . . . . .4.4.4 Comparing the modelling and the experiment . . . . . . . . . .4.5 Summary and the work ahead . . . . . . . . . . . . . . . . . . . . . . .vii373738414242434655595 Conclusions of the thesis64Bibliography65

List of Figures1.11.21.32.12.22.32.42.52.63.1Patterns of lumous spots of current density on a cathode of a DC glowmicrodischarge, for di erent values of discharge current. Xenon undera pressure of 75 torr. Reprinted from [46]. . . . . . . . . . . . . . . . .Computed CVC of DC glow discharge. Solid line: 1D mode. Dashedline: 2D modes. Circles: bifurcation points. Xe plasma. Pressure of 30Torr. Adapted from [56]. . . . . . . . . . . . . . . . . . . . . . . . . . .a) CVC of 1D and 3D modes. b-d) distributions of current density overthe cathode surface. Solid line: 1-D mode. Dotted line: 1st (in orderof decreasing current) 3D mode. Dashed line: 8th 3D mode. Dasheddotted line: 12th 3D modes. Circles: Bifurcation points. (b).(d) Distributions of current density for (b) 1st 3D mode (c) 8th 3D mode (d)12th 3D mode. Reprinted from [42]. . . . . . . . . . . . . . . . . . . .Near-anode voltage drop for a wide range of currents. Solid line: 2Dmode. Dotted line: 3D mode. 2D and 3D solutions are with schematicsthat indicate a characteristic distribution of electron number density onthe anode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Electron number density on the surface of the anode. 3D solution, I 10mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a) Electron number density in the spot. b) Ion number density in thespot. c) Electric potential in the near-anode region. Distributions inthe plane of symmetry passing through the spot center. I 1 mA. . .Number density of ions (solid line), number density of electrons (dashedline), for I 35 mA, and I 1A. Reduced electric eld (dotted line) forI 35 mA. Plot made from centre of spot, along axial direction, to endof calculation domain. . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of axial current density and axial electric eld on the surface of the anode in the plane of symmetry passing through the spotcenter. Large negative values of electric eld are not shown. . . . . .Distribution of the number density of ions in the plane of symmetrypassing through the spot center. Arrows: unit vector of current density.I 1 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Con guration of a cathode boundary layer vessel. AG is the axis ofsymmetry of the vessel. . . . . . . . . . . . . . . . . . . . . . . . . . .viii791017181920212225

LIST OF undamental mode. 1: baseline conditions. 2: h 0.5mm, ha 0.1mm,R 1.5mm. 3: baseline geomentry, re‡ecting dielectric surface. 4: h 0.25mm, ha 0.25mm, R 0.375mm. . . . . . . . . . . . . . . . . .Solid: fundamental mode (mode 1 of gure 2). Dashed: mode a3 b3 .Circles: points of bifurcation. . . . . . . . . . . . . . . . . . . . . . . .Bifurcation diagram. Solid: fundamental mode (mode 1 of gure 2).Dashed: modes a3 b3 , a4 b4 , a5 b5 . Dotted: mode a6 b6 . Circles: pointsof bifurcation. ‘ ’on the image representing the mode a4 b4 indicatesthe point on cathode surface where the value jc is taken. . . . . . . . .Evolution of patterns of current density on the cathode associated with3D modes of gure 4: (a) mode a3 b3 , (b) mode a4 b4 , (c) mode a5 b5 and(d) mode a6 b6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cross section view of a 3D hot spot associated with the mode a4 b4 atdi erent currents. The cross section plane passes through the centreof the spot. (a), (d): Ion density. (b), (e): Election density. (c), (f):Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Schematic of current-voltage characteristics (CVCs) of the di use modeof current transfer to rod cathodes of high-pressure arc discharges andof the mode with a spot at the edge of the cathode. The sections shownby the solid lines and the transitions shown by the arrows are observedin the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CVCs. Solid: the 1D (fundamental) mode. Dashed-dotted: 2D modea3 b3 . Other lines: di erent 3D modes. Circles: bifurcation points. Top:General view. Bottom: Details near the point of minimum of the CVCof the 1D mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Evolution of distributions of current on the surface of the cathode associated with di erent modes. a): mode a2 b2 . b): a4 b4 . c): a5 b5 . . . .Bifurcation diagram. Solid: the 1D (fundamental) mode. Dashed, dotted: 3D second generation modes a4 b4 and a5 b5 . Circles: bifurcationpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Bifurcation diagram. Solid: the 1D (fundamental) mode. Dashed: 2Dmode a3 b3 . Other lines: 3D modes. Circles: bifurcation points. . . . .Evolution of distribution of current on the surface of the cathode associated with the mode a3;2 b3;2 . . . . . . . . . . . . . . . . . . . . . . . .Bifurcation diagram. Solid: the 1D (fundamental) mode. Dashed:3D second-generation mode a10 b10 . Dotted: 3D third-generation modea10;1 b10;1 . Circles: bifurcation points. . . . . . . . . . . . . . . . . . .Experimentally observed and computed transitions between di erentmodes in xenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CVCs. Detailed model. Solid: the rst 2D (the fundamental) mode.Other lines: 3D modes a4 b4 and a5 b5 . Circles: bifurcation points. . .ix2930323335444850515253545658

Chapter 1Introduction1.1Direct current glow dischargeIn 1838 Michael Faraday produced the rst report of a direct current (DC) glow discharge [1]. Faraday, who used a voltaic pile as a power source, passed current throughdi erent gases, at various pressures, via electrodes in a glass bell jar. At low pressures,he observed a phosphorescent continuous glow around one of his electrodes, which hedescribed as "exceedingly beautiful". The main features of a DC glow discharge canbe read about in the textbook by Raizer [2]. The discharge has been signi cant in bothfundamental plasma science and in applied plasma science. DC glow discharges havebeen used in spectroscopy [3], allowing research on the elements of materials in the gas,liquid, and solid state; laser technology [4], with laser gain medium being electricallypumped by DC glow discharge; surface property modi cation [5], that atmosphericpressure glow discharge may adapt the wetting properties of a surface; cancer inhibition research [6], glows with liquid anodes have been demonstrated to suppress theactivity of at least two types of cancer cell, in vitro; sources of ultraviolet radiation[7], that DC glow microdischarges, when con gured to yield high energy electrons ina high pressure gas, result in excimer production; to name just a few examples of DCglow discharge being utilized in science.1.1.1Modelling and theory during the 20th centuryA well-known one-dimensional description of the cathode fall was formulated in 1934by von Engel and Steenbeck [2]. The description reveals a U-shaped current densityvoltage characteristic (CDVC). However, the transition from the Townsend dischargeto the abnormal discharge, as revealed by the von Engel and Steenbeck solution, isparticularly di erent to what is observed in experiments. Instead of the normal discharge observed in the experiments, the solution by von Engel and Steenbeck has a1

1. Introduction2one-dimensional distribution of charged particles, and reveals a falling CDVC section.While von Engel comments that the causes of the di erences are basically not understood [8], Steenbeck proposes arguments to describe the observed physics of thenormal discharge on the basis that the axial electric eld ought to be minimized [9](the so-called ’Steenbeck’s principle of minimum power’, which became more widelyaccepted as it was reported that this principle is a corollary of the principle of minimumentropy production [10]). The rising section of the CDVC associated with the von Engel and Steenbeck solution corresponds with the abnormal discharge in a qualitativelyaccurate way.Circa 1960, glow discharge modelling results started to be found with the aidof electronic computers [11, 12]. The authors aimed to solve the relevant, coupled,physical equations: in [12], equations of conservation of number density of electronsand ions (considering only drift ‡uxes), and Poisson’s equation, were solved. Ane ective Townsend ionization coe cient was used (recombination was neglected). Theresults essentially duplicated von Engel and Steenbeck’s solution. In particular, therewas agreement, whereby under particular conditions1 , the CDVC was U-shaped.In the 1980s, two-dimensional modelling of glow discharges started to emerge (e.g.[14–16]) which, based on similar underlying physics as the one-dimensional models,revealed the structure of the normal spot. In 1988 [17] Boeuf solved a two-dimensionalmodel, and found the structure of the normal spot, the normal current density e ect,and the current-voltage characteristic (CVC) plateau associated with the normal spot.The basic mechanisms of the model are drift and di usion for the ions and the electrons,volume ionization, recombination, and secondary electron emission. The model yieldedan accurate qualitative description of the transverse behavior of the transition fromthe abnormal mode to the normal model.The aforementioned modelling used a ‡uid description of the plasma. This approach is based on the taking of moments of the Boltzmann equation (BE) (e.g.[18, 19]), for each of the species considered, coupled with the Poisson equation (or therelevant set of Maxwell’s equations). For that sake of practicality, it is normally thecase that the electron energy distribution function is assumed to be either Maxwellianor a two term spherical harmonic expansion with the second term accounting for ananisotropic perturbation [20]. The set of equations generated from the taking of themoments of the BE are closed, in the case of the electrons, typically at the level ofeither momentum, or energy. In the case of momentum, one typically employs thedrift-di usion approximation, with the transport and kinetic coe cients related to1If a discharge, by way of its product of pressure and the distance of electrode separation, wouldhave a breakdown voltage on the rising section of the Paschen curve, then it will have a U-shapedCVC (i.e. it will not be an obstructed discharge) [13].

1. Introduction3space and time by functions (built using the solution to the BE) with the reducedelectric eld as their argument: the so-called local- eld approximation (LFA). TheLFA is suitable when the characteristic length of electron energy relaxation is smallcompared to a characteristic discharge length. When one includes in the set of equations to be solved an equation for electron energy, the transport and kinetic coe cientsfor the charged species may be related to space and time by a function (built using thesolution to the BE) with the average of electron energy as the argument: the so-calledlocal mean energy approximation. The latter approximation is more accurate andallows for the study e ects such as striations [21].Strictly speaking, one should only use the ‡uid-based approach to the modellingof a plasma when that plasma has an energy distribution function for the chargedspecies that are Maxwellian (although, ‡uid approaches may perform ’better thanthey should’, e.g. [22] p. 35). When a ‡uid-based approach is not appropriate, onemay employ a kinetic-based approach to modelling [23] e.g. when the species arein a strongly non-uniform eld (as may be the case in the cathode fall [24]), andregions with few collisions. In general, the kinetic based approach requires a greatercomputational e ort, but fewer a priori assumptions.By the end of the 21st century, the main features of the discharge, e.g. those discussed in chapter 8 of [2], had been, quite faithfully, reproduced by computer models.1.2Self organizationThe term self-organisation is used in various disparate academic disciplines. In socialscience [25], it is used in connection with phenomena such as city formation; in computer science [26], in connection with methods of utilizing idle system components; inbiology [27], used in the context of morphogenesis, whereby identical cells may di erentiate into, for example, an organism with eyes. Broadly, the term self organizationis used to describe the spontaneous occurrence of order among multiple subunits, froma source other than the direct motivation of an external in‡uence.In this thesis, the term self-organisation is used to refer to the occurrence of dissipative structures. Prigogine coins the term ‘dissipative structure’[28] in 1967, referringto structures, or patterns of structures, with a clear degree of spatial regularity, whichform in conditions far from thermodynamic equilibrium, and are maintained by ‡uxesof matter and energy. The emergence of spatiotemporal structures (e.g. the Belousov–Zhabotinsky reaction [29]), are neglected in this thesis.In section 1.2.1 the thermodynamics of self-organisation are commented on for thesake of historical and scienti c context. In sections 1.2.2 and 1.2.3 the concepts arebrie‡y introduced of stability, and bifurcations, respectively.

1. Introduction1.2.14Thermodynamics of self organizationPrigogine’s textbook [30] on self-organization introduces dissipative structures in thesame way as in this section.It is well known that the second law of thermodynamics prevents the spontaneousformation of order in isolated systems. Also, that in such systems there is a couplingbetween the degree of order in the system and its evolution, and ultimately its stability.In closed systems, whereby exchanges of heat with outside reservoirs are permitted,ordered structures may arise: Helmholtz free energy is minimized at thermodynamicequilibrium and for low temperatures the probability that a particle in the systemis at a state of a low energy level (by Boltzmann’s ordering principle [31]) is high,hence solid crystals or phase transitions occur. At moderate temperatures (e.g. thetemperature of animal cells), however, the probability of the formation low entropystructures, via the former ordering principle, is prohibitively small, and yet, order incells exist. Apparently a di erent source of order is also available to nature.A class of thermodynamic system termed ‘open’ was studied at the de Donderschool in Brussels (see e.g. books by von Bertalan y [32] and E. Schrodinger [33] onthe physics of life), i.e. a thermodynamic system was studied that takes into accountexchanges of matter and heat with its surroundings. Prigogine in 1945, working fromthe school, makes the contribution of extending the second law of thermodynamics tosuch open systems. Entropy change is considered in a time interval, and decomposedinto two contributions: entropy ‡ux due to exchanges with the environment, andentropy production due to irreversible processes inside the system. The change inentropy in time can therefore be negative, as exchanges of negative entropy fromoutside system may be dominant (change in entropy would regularly be seen as beingonly positive, if one only considered isolated systems). Thus in principle, orderedstructures may arise in open systems as long they are being maintained by ‡uxes ofmatter and energy with ’negative entropy’. Such structures are termed ’dissipativestructures’. Many apparent examples of such structures are listed in [34]. An earlyexample of which, observed by Lehmann in 1902 [35], is a localized solitary luminousspot found on an anode of a DC glow discharge.Prigogine’s textbook points to un nished thinking on a criterion based on thermodynamic state functions (e.g. entropy, entropy production), for the onset of dissipativestructures. A comment is made that such structures may coincide with a minimumof entropy production. Contemporary academics are still debating the possibility of nding such a general thermodynamic criterion [36].The general criterion of Steenbeck’s principle of minimum power in gas dischargephysics has been found to be not without problems. An example of such a problem

1. Introduction5follows. Within the framework of a model of nonlinear surface heating of a cathode ofan arc discharge, a computed solution associated with the lowest discharge voltage wasselected, from among several other solutions existing for the same discharge current, asthe one that is stable [37] based on Steenbeck’s principle of minimum power. However,numerical results presented in [38] and the analytical theory [39] indicate that the lowvoltage branch is unstable, and the high-voltage branch is stable. Further, even if it wasfound that the principle of minimum entropy production was valid for gas discharges,Steenbeck’s principle of minimum power is shown [40] to not be a corollary of theprinciple of minimum entropy production. Hence, an understanding of the generaltheory of stability is important in situations with multiple solutions.1.2.2An introduction to stabilityThe literature on stability is well developed (see e.g. the textbook [41]). The concept and some common approaches to studying stability are brie‡y introduced in thissection.A system of di erential equations governing the evolution of variables in a space,such as phase space, may be constructed using rst principles such as classical physicsand potential elds. A solution to those equations is said to be stable if the timeevolution of an initial state plus a perturbation will remain close to the initial stateat all subsequent moments. Otherwise, the solution is unstable. One may describe asolution as locally stable, globally stable, or stable against a particular perturbation.A solution is globally stable if any perturbation on an initial state would result in anevolution of the solution that would have it return to the initial state. A solution islocally stable, or metastable, if any small perturbation on the solution would result inan evolution of the solution that would have the solution return to the initial state.One may employ di erent approaches when studying stability e.g. using the Lyapunov stability criterion [30]: one constructs a function whose rate of change determines if an initial state is ’asymptotically stable’(i.e. if it is stable against any smallperturbation). It is not always straightforward, and may even be impossible, to identify a suitable function in a given situation. Another approach is to use linear stabilitytheory [42]: an initial state under investigation and a small perturbation with an exponential time dependence is substituted into the system of equations and boundaryconditions. The problem is linearized with respect to the perturbation, resulting in aneigenvalue problem whose spectrum describes the mode of development, or damping,of the perturbation. With the eigenvalues being the growth increment of the perturbations. If real parts of all eigenvalues are negative, the state is stable; if at least oneeigenvalue has a positive real part, the state is unstable.

1. Introduction1.2.36An introduction to bifurcationsA concise summary of the information from bifurcation theory relevant to this thesiscan be found in the Appendix of [43]; a more detailed discussion of the theory can befound in, e.g., the review [44]. A brief introduction to bifurcations follows.A bifurcation is a splitting of a solution occurring when one varies a system controlparameter past a certain critical point (the bifurcation point). The splitting corresponds with a qualitative or topological change to the solution(s) past the bifurcationpoint. An example is the pitchfork bifurcation: consider a two-dimensional (axiallysymmetric) solution describing, for instance, the spatial distribution of the numberdensity of some particles in a three-dimensional space, that varies with a control parameter p. At p p0 there exists a bifurcation point, before and after the bifurcationpoint the stability of the solution changes2 (for de niteness, the solution is stable forp p0 and unstable for p p0 ). At p0 a three-dimensional solution also exists, thatbranches o away from the two-dimensional solution. If the three-dimensional solutionbranches o into the range p p0 , where the two-dimensional solution is unstable, thenthe pitchfork bifurcation is supercritical. One would obs

Matthew Simon Bieniek DOCTORATE IN PHYSICS Modelling Self-Organization on Electrodes of DC Glow Discharges DOCTORAL THESIS DMTD February 2018. i Dedicated to Dina Moldovan. Acknowledgements I thank supervisors for the contribution they have made to my academic, professional,