Theory and simulation of electron sheaths and anode spots in low pressure laboratory plasmas

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 217 Theory and simulation of electron sheaths and anode spots in low pressure laboratory plasmas Brett Stanford Scheiner University of Iowa Copyright 217 Brett Stanford Scheiner This dissertation is available at Iowa Research Online: Recommended Citation Scheiner, Brett Stanford. "Theory and simulation of electron sheaths and anode spots in low pressure laboratory plasmas." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 THEORY AND SIMULATION OF ELECTRON SHEATHS AND ANODE SPOTS IN LOW PRESSURE LABORATORY PLASMAS by Brett Stanford Scheiner A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa August 217 Thesis Supervisor: Scott D. Baalrud, Assistant Professor

3 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Brett Stanford Scheiner has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the August 217 graduation. Thesis Committee: Scott D. Baalrud, Thesis Supervisor Matthew M. Hopkins Robert L. Merlino Jack D. Scudder Jasper S. Halekas

4 ACKNOWLEDGEMENTS This work benefited from the input, help, and feedback of several people. This research was done in collaboration with Scott Baalrud, Matt Hopkins, Ed Barnat, and Ben Yee. Scott has been a great advisor. He took the time to check the details of nearly all of my work, pointing out countless mistakes and making several suggestions along the way. This not only included research results, but also seminar and conference talks, papers, and abstracts. I m grateful for the encouragement and feedback when exploring new ideas and the pressure to stay focused and not stray too far from the current task at hand. Matt Hopkins was my mentor during the year I spent at Sandia National Lab under the O ce of Science Graduate Student Research program. Matt sponsored my application to the SCGSR program. Without his e ort, simulation of plasmas would not have played as great of a role as it has. I consider my experience simulating plasma experiments at Sandia to have been the most transformative of my graduate career. This will be invaluable as I move on to study new problems. At Sandia, I benefited from weekly discussions with Ed, Matt, and Ben on di erent issues relating to their various research topics in low temperature plasmas. Being in an environment surrounded by other plasma physicists working on related problems helped me distinguish which problems were interesting and useful to other people working in the field. The discussion of experimental results with Ben and Ed helped me to develop an understanding of these topics and to understand what things are and are not di cult to measure in the lab. Eventually I learned to stop asking for the impossible, although Ed may disagree. I enjoyed the hours spent talking plasma physics (or other less useful topics) around Andy Fierro s giant candy bowl with Ben, Ricky Tang, and Jim Franek. Also, the near daily breakfast burrito trips that Andy and I took made the food further east seem bland and nearly unbearable. At Iowa, there were several people who made day to day life more interesting. The two most notable are Ryan Hood and Nathaniel Sha er. Ryan always stopped by to talk about the latest thing that happened in the lab, or the latest Regular Car Review, and was always kind enough to show me his experiment and latest measurements. I ve known Nathaniel since 29 and am glad to have had a friend with whom I could bounce ideas o of for so many years. I have no doubt that our various idiot checks have vastly reduced the number of mistakes that could have been in these chapters. I have no doubt that both Nathaniel and Ryan will be great plasma physicists. Finally, I want to thank my family and friends for their support, especially my fiancè Mary ii

5 for understanding the many late nights and stress filled days. I could not have done this without her support and love. This research was supported by the O ce of Fusion Energy Science at the U.S. Department of Energy under contract DE-AC4-94SL85 and by the O ce of Science Graduate Student Research (SCGSR) program under contract number DE-AC5-6OR231. iii

6 ABSTRACT Electrodes in low pressure laboratory plasmas have a multitude of possible sheath structures when biased at a large positive potential. When the size of the electrode is small enough the electrode bias can be above the plasma potential. When this occurs an electron-rich sheath called an electron sheath is present at the electrode. Electron sheaths are most commonly found near Langmuir probes and other electrodes collecting the electron saturation current. Such electrodes have applications in the control of plasma parameters, dust confinement and circulation, control of scrape o layer plasmas, RF plasmas, and in plasma contactors and tethered space probes. The electron sheaths in these various systems most directly influence the plasma by determining how electron current is lost from the system. An understanding of how the electron sheath interfaces with the bulk plasma is necessary for understanding the behavior induced by positively biased electrodes in these plasmas. This thesis provides a dedicated theory of electron sheaths. Motivated by electron velocity distribution functions (EVDFs) observed in particle-in-cell (PIC) simulations, a 1D model for the electron sheath and presheath is developed. In the presheath model, an electron pressure gradient accelerates electrons to near the electron thermal speed by the sheath edge. This pressure gradient generates large flow velocities compared to what would be generated by ballistic motion in response to the electric field. Using PIC simulations, the form of a sheath near a small electrode with bias near the plasma potential is also studied. When the electrode is biased near the plasma potential, the EVDFs exhibit a loss-cone type truncation due to fast electrons overcoming the small potential di erence between the electrode and plasma. No sheath is present in this regime, instead the plasma remains quasineutral up to the electrode. Once the bias exceeds the plasma potential an electron sheath is present. In this case, 2D EVDFs indicate that the flow moment has comparable contributions from the flow shift and loss-cone truncation. The case of an electrode at large positive bias relative to the plasma potential is also studied. Here, the rate of electron impact ionization of neutrals increases near the electrode. If this ionization rate is great enough a double layer forms. This double layer can move outward separating a high potential plasma at the electrode surface from the bulk plasma. This phenomenon is known as an anode spot. Informed by observations from the first PIC simulations of an anode spot, a model has been developed describing the onset in which ionization leads to the buildup of positive space charge and the formation of a potential well that traps electrons near the electrode surface. A model for steady-state properties based on current loss, power, and particle balance of the anode spot plasma iv

7 is also presented. v

8 PUBLIC ABSTRACT Electrodes within plasmas have a charged region, called a sheath, in a boundary layer at their surface. Typically, electrodes have a negative voltage relative to the plasma. Through the use of analytic theory and simulation, this thesis studies the sheath when a positive voltage (relative to the plasma) is applied to the electrode. This type of sheath is called an electron sheath. Electron sheaths have a variety of technological applications for the control of plasma properties, control of dust particles in the plasma, electron temperature, and in some spacecraft power systems. For the electron sheath, it is found that the presheath, the interface region between the plasma and sheath, has properties that are determined by a pressure gradient. This first report of a presheath for electron sheaths has implications for the behavior of electrodes in the relevant applications. Ionization of neutral gas by electrons accelerated into the electron sheath is also studied. When enough ionization is present, a secondary plasma called an anode spot forms at the electrode surface. Typically, anode spots are bright spherical regions, and have been used as plasma ion sources due to the rate of ion formation inside. The formation and properties of anode spots are simulated and models for their behavior are formulated. These models and simulations compare favorably with experiments. vi

9 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ix x CHAPTER 1 BACKGROUND Introduction Fluid Moment Equations Sheaths and Balance of Global Current Loss The Bohm Criterion and Global Current Loss Child-Langmuir Sheath The Presheath Plasma Solutions Fluid Moment Bohm Criterion Particle-In-Cell Method Overview Resolution of Particle Trajectories Modifications to the Dielectric Response Collisions Non-Physical Self Forces Grid Heating Direct Simulation Monte-Carlo Method Aleph Conventional Picture of the Electron Sheath Outline THE ELECTRON SHEATH AND PRESHEATH Motivation Fluid Model Presheath Sheath Simulations of the Electron Sheath Simulation with Aleph Electron Fluid Ion Behavior Ion Density Ion VDFs Ion Flow Fluctuations and Instabilities Summary EVDFS AND THE ION TO ELECTRON SHEATH TRANSITION Simulations vii

10 3.2 EVDFs and the Electron Flow Moment Implications for Future Electron Presheath Models ANODE SPOT ONSET AND EQUILIBRIUM Introduction Simulation Simulation Setup Anode Spot Simulation Velocity Distribution Functions Current Collection Theory Anode Spot Onset Anode Spot Current Balance Particle and Power Balance & Anode Spot Size Ion Presheath in the Anode Spot Comparison with Experiments Observations of Spot Onset Transient Behavior Summary CONCLUSION APPENDIX A NON-VLASOV BEHAVIOR IN PARTICLE-IN-CELL SIMULATIONS B DERIVATION OF THE MODIFIED LANGMUIR CONDITION REFERENCES viii

11 LIST OF TABLES Table 3.1 Summary of simulation parameters ix

12 LIST OF FIGURES Figure 1.1 Three di erent types of sheath structure that are possible near a positively biased boundary. When the electrode bias is either negative or if the electrode is large the electrode bias is below the plasma potential and an ion sheath is present. If the electrode is small enough that Eq. (1.33) is satisfied, an electron sheath will be present if the electrode is biased positive enough. At intermediate electrode sizes between those satisfying Eq. (1.29) and Eq. (1.33) a positive electrode will have an electron sheath with a virtual cathode that reduces the electron flux to the boundary Profiles of the potential, ion flow velocity, and species densities for a typical ion sheath and presheath from numerical solutions of Eq. (1.37)-(1.4). Note that the Bohm speed is achieved at a location that coincides with the breakdown of quasineutrality The shape functions S and S 1 for zeroth and first order interpolation given by Eq. (1.53) and (1.54) Examples of particles moving through a grid with large and small time steps for the velocity integration. When the time step is too large the particle velocity is not updated by all of the nodes of the cells through which it passes, resulting in unphysical trajectories The parallel and perpendicular di usion rates for several di erent particle sizes calculated by Birdsall and Okuda [1] from Eq. (1.71) assuming a gaussian shape function. The di usion rate reproduces the point particle value as the particle size R goes to zero. Reprinted from Ref. [1], with the permission of AIP Publishing The particle drag for several di erent particle sizes calculated by Birdsall and Okuda [1] from Eq. (1.72) assuming a gaussian shape function. Reprinted from Ref. [1], with the permission of AIP Publishing The location of the di erent types of sheath structure studied in the chapters of this thesis placed on the I-V curve of a positively biased electrode. The data points were provided by Scott D. Baalrud and are from the experiment in Ref. [2] An illustration of experiments from Ref. [3]. The area marked Interrogation Area indicates the region in which LCIF was used to obtain electron density measurements The electron density profiles from LCIF measurements. Strong and weak ion sheaths were present at the electrode for the cases marked -5V and V, respectivly. For the case marked 15V, the electrode was biased above the plasma potential and an electron sheath was present, indicated by the non-zero electron density at the electrode x

13 2.3 Left: A comparison of 2D electron density maps from LCIF measurements and simulation for an electron sheath (bottom) and ion sheath (top). The vectors on the simulation plot indicate the electron current density. Right: The di erence between ion and electron density indicates the sheath edge position as a function of time. An FFT of the sheath position fluctuations indicates the frequency of oscillations EVDFs from a PIC simulation. The locations of the EVDFs are within the sheath, near the sheath edge, and in the presheath. The EVDFs at all three locations have a significant flow shift, a feature that di ers significantly from the conventionally assumed velocity space truncation Electron presheath potential and velocity profiles for the constant collision frequency and constant mean free path models given in Eq. (2.9)-(2.12). Note that the potential is scaled by T i and has gradients which are much shallower and flow velocities that are much greater than those encountered in ion presheaths Data for the neutral helium-electron momentum scattering and the neutral helium-helium ion charge exchange and elastic cross sections used for the calculation of the rate constants K i and K e. The helium-helium ion cross section data was unavailable at low energy and was extrapolated from the value at the lowest known energy Calculated ratios of the electron and ion presheath length scales assuming the dominant collision processes in ion and electron sheaths are helium-helium ion collisions and electron-neutral scattering, respectively The exact solution to Eq. (2.17) (solid line) compared to its asymptotic limit (dashed line). The di erence between the asymptotic limit and exact solution is at greatest 2% A schematic of the simulation domain overlaid on a color map of the ion temperature for the 2 V simulation. Note the electrode position in the lower left corner. The region of zero temperature is due to the absence of ions in the electron sheath The current flow vectors plotted on top of the potential for n i V i (left column) and n e V e (right column) for an electron sheath (top row) biased +2 V and ion sheath (bottom row) biased -2 V. The electrode is between x = and x =.25 cm, see annotations on Fig. 2.9 for more details of the simulation domain. All potentials are measured relative to the grounded wall. The greatest di erences between these are that the electron presheath has a much greater e ect on the electrons in the bulk plasma than the ion presheath has on the ions, and that the electron presheath redirects the ions, while the ion presheath has little e ect on the electrons. Note the di erence in scale for vectors in the left and right columns Top: The potential profile from the PIC simulation compared the constant collision frequency and constant mean free path presheath models given by Eq. (2.1) and Eq. (2.12). The sheath models for flowing and truncated Maxwellian given in Eq. (2.22) and Eq. (2.23) are also shown. Bottom: The electron flow moment from simulation compared to those calculated from the potential models in the upper panel xi

14 2.12 a) The magnitude of terms of the momentum equation on the right hand side of Eq. (2.26) are evaluated from quantities computed in the PIC simulation. The residual is determined by summing the terms. b) The electron flow moment, ion density, and electron density. The two vertical lines correspond to the two estimates of the sheath edge Evaluation of the ion density model given by Eq. (2.28) using PIC quantities. The model compares well with the simulated presheath density and is a significant improvement over the Boltzmann density profile IVDFs near the electron sheath biased at +2V. The IVDFs were averaged over.1 cm x.1cm boxes. The labels in the x and y axes indicate the coordinate of the center of the box, the electrode is on the x axis at y= between x= and.25 cm (averaging starts.2 cm above the electrode), since further below there are not enough ions for meaningful IVDFs Schematic drawing describing the time-averaged IVDFs at di erent locations in the plasma. a) The flow lines of particles in 3 di erent VDFs starting at A and ending at B. b) The VDFs at location A. c) The VDFs at location B, distributions incident on the electrode have their -y velocity redirected in the x direction. d) A realistic IVDF at location B due to a continuum of starting positions along line A A schematic of the simulation domain for two cases: a) The small electrode model and b) the embedded electrode model. The color map of each domain indicates the plasma potential IVDFs at several locations near and in the electron presheath from simulations of the embedded electrode (a) and small electrode (b). The location of each IVDF relative to the electrode position is indicated by the plots on the right A schematic for the experiment used to measure the IVDFs in the electron presheath Laser induced fluorescence measurements of IVDFs in the electron presheath near free and embedded electrodes Time evolution of the charge density in a 1 cm by 1 cm region near the electron sheath shown at.5 µs time increments. The color indicates the charge density with red being electron rich and blue being ion rich Top: The absolute di erence between ion and electron density along a line perpendicular to the electrode surface over 5µs. Bottom: The corresponding ion density over the same time period. The ion density fluctuations coincide with the fluctuations of the sheath edge position The 2D FFT of the ion density shown in Fig The solid and dashed red lines correspond to the real part of the dispersion relation given in Eq. (2.36) for flow velocities of.5v eb and.9v eb. The solid and dashed yellow lines correspond to the imaginary part, i.e. the growth rate, of the same equation xii

15 3.1 a) The 5 cm by 15 cm simulation domain with a color map indicating the ion density of simulation A overlaid. Note the small electrode placed perpendicular to the reflecting boundary. b) Sampling areas for obtaining EVDFs in front of the electrode Profiles of simulations A-E EVDFs in the sheath and presheath for simulations A-E The EVDFs for simulation E, sampled in the small red boxes shown in Fig. 3.1b Comparison of the electron density of simulation A along the line y= (the reflecting boundary) in Fig. 3.1a with the prediction of the Boltzmann density profile of of Eq. (3.6). The values of n e ( o ) and the 2D temperature T e were determined at a point approximately 2 cm from the electrode Electron trajectories for right moving electrons are shown to demonstrate how the electrode can cause a truncation in the EVDF. The region that is inaccessible for right moving electrons is marked Electrode Shadow The EVDF near the sheath edge for the electron sheath simulation presented in Chapter 2.3. For this electrode geometry, the flow shift is more pronounced than the loss cone truncation A variety of sheath-like structures are possible near an electrode biased more positive than the plasma potential. Once the electrode bias exceeds the plasma potential an electron sheath forms. When the potential is large enough that electrons gain the ionization energy of neutral atoms, a double layer attached to the electrode may form due to ionization within the sheath. This is typically associated with the anode glow. Increasing the bias eventually results in an anode spot, characterized by a double layer separating a high potential plasma from the bulk plasma a) A photograph of an anode glow near a positively biased electrode observed through the observation port of a GEC reference cell. b) An increase in bias of the electrode resulted in the formation of an anode spot characterized by the larger luminous region The 7.5 cm by 7.5 cm simulation domain with the color map indication the typical ion densities encountered. An anode spot is attached to the electrode in the lower left hand corner of the domain The time dependent values of plasma species densities, potential, and electric field along the symmetry axis (reflecting boundary). Note the spot onset following the increase in electrode bias at t =9µs The potential and density profiles before (t = 9 µs) and after the electrode bias increases from 4 V to 5 V. Note that after the potential gradient changes sign around t = 1.2 µs, the density of electrons from ionization begins to increase. Quasineutrality is established by t = 11 µs xiii

16 4.6 Particle species density and potential profiles corresponding to the VDFs of Fig. 4.7 taken at t = 1.8 µs. The smaller purple shaded regions indicate the area in which ion VDF histograms were plotted. The larger blue shaded regions show the same for electrons. The total electron density is also shown Velocity distribution functions for e B and e I electrons and helium ions within the anode spot and double layer. The color bars indicate the number of macroparticles per bin. These VDFs were obtained at t = 1.8 µs of the simulation at the locations indicated in Fig 6. Note that the coordinate axes for ions span di erent ranges in v x and v y. Also note that the spatial extent over which histograms for ions were plotted is a quarter of the area used for electrons, see Fig Components of the current collected by the electrode and wall. After anode spot onset the electron current collection increases. When the spot is present at t>1 µs, all electrons lost from the simulation exit through the electrode Test electron trajectories integrated in the time dependent electric field obtained from the simulation of Chapter The initial velocity vectors are shown in the legend and are representative of electron velocities found near the sheath edge. Electrons starting in the range x [,.4] are collected by the electrode The critical bias as a function of pressure for several di erent bulk plasma densities and an electron temperature of 2.4 ev calculated from Eq. (4.11) with the sheath thickness of Eq. (2.22) The critical bias calculated from Eq. (4.13) compared with experimental data for two di erent electrode diameters [2] The di erent types of potential structure that are possible depending on the anode spot size The anode spot length scale L from Eq. (4.2) as a function of the double layer potential for helium neutral pressures of 5, 1, and 2 mtorr. The electron impact ionization cross section obtained from LXcat [4] is also shown The ratio of rate constants for electron impact ionization and elastic ion-neutral scattering in Helium as a function of ion temperature and double layer potential. Under typical laboratory conditions, where T i.3ev, K I is dominant. Contours for K I /K He + He = 1, 2, and 4 are shown A schematic of the experiment from which measurements of the anode spot onset and steady-state were provided. The anode spot formed on the electrode embedded in the lower boundary. Figure provided by Edward Barnat, with permission Experimental values of electrode bias, current collection, and plasma quantities. The plasma emission, metastable density, and electron density are plotted along a line perpendicular to the electrode as a function of time. The initial anode spot formation is at t.3 µs. Figure provided by Edward Barnat, with permission xiv

17 4.17 2D colormaps from experiments indicating the measured plasma emission, metastable density, electron density, and electric field before and after the voltage step was applied. Figure provided by Edward Barnat, with permission D color maps of the simulated electric field magnitude, ionization rate, and electron density for comparison with experimental results in Fig Here, the ionization rate is a proxy for the plasma emission PIC simulations of spot formation with a voltage step from V to 12 V shown in the top panels. The step duration in the left column is.6 µs and on the right is 1 ns. The ion density, electric field magnitude, and potential as a function of time are shown along a line perpendicular to the electrode A.1 The dispersion relation from the unstable root of the dielectric function for a distribution of flowing electrons and stationary ions given by Eq. (2.29). The exact value of these roots are compared with the roots of the dielectric modified by the shape function in Eq. (A.24). The deviation from the exact value is significant at scales smaller than R B.1 A double layer that has a moving position z DL in the lab. In the frame of the double layer (with spatial coordinate x), moving at velocity U DL with respect to the lab frame, the position x DL is constant. The locations x 1 and x 2 indicate the position of the double layer edge in the moving frame. Note that E(x 1 )=E(x 2 ) = at these locations B.2 The trapped and free particle populations near a double layer. At the high potential side, electrons are trapped and ions are accelerated by the double layer electric field. At the low potential side, ions are trapped and electrons are accelerated xv

18 1 CHAPTER 1 BACKGROUND 1.1 Introduction Unconfined plasmas expand due to the di usion of particles. As a result, all laboratory plasmas are confined either by a magnetic field or some other physical boundary. In the absence of a magnetic field this boundary is typically either a high pressure neutral gas such as air in streamer discharges like lightning [5], a liquid in the case of discharges in aqueous solutions [6], or a solid boundary in more typical plasma experiments. In the latter case, a non-neutral space charge region called a sheath forms between the plasma and the solid boundary. The sheath plays the important role of ensuring that the plasma loses equal amounts of positive and negative space charge via the acceleration of one species and repulsion of the other by the sheath electric field. This is necessary for the maintenance of quasineutrality in the bulk plasma where the thermal flux of electrons is much greater than that of ions due to the smaller electron mass and higher electron temperature compared to ions. Most sheaths encountered in typical low temperature laboratory plasmas are ion-rich ion sheaths that repel electrons from the boundary. The greater thermal flux of electrons necessitates the existence of ion sheaths at most boundaries to balance the electron and ion currents lost, minimizing the polarization of the plasma volume. This typically results in the electrostatic potential of the plasma settling at a value above that of the confining walls. The value at which the plasma potential settles, and the extent to which the space charge in the sheath accumulates, results from the balance of electron and ion current loss. Ion sheaths have been studied extensively due to their prevalence at solid plasma boundaries [7]. They have also played an important role in methods for diagnosing the bulk plasma where they determine the current-voltage response of probes [8]. One of the most important results in the literature on ion sheaths is the existence of a long quasineutral region leading up to the thin non-neutral sheath. This region is called the presheath. From Poisson s equation it was determined that ions need to flow into the sheath with a velocity greater than their sound speed (c s = p T e /m i,wheret e is the electron temperature and m i is the ion mass) for stable potential solutions to exist. This minimum velocity requirement is known as the Bohm criterion [9]. Ions are accelerated to this velocity within the presheath. In the limit that the plasma is collisionless the presheath is infinitely long and the sheath is infinitely far away from the plasma. In more realistic situations, collisions act to reduce the presheath length scale to physically

19 2 reasonable values. Under typical low temperature plasma conditions encountered in laboratory experiments (T e 1eV, T i.3ev, n e cm 3, 1mTorr neutral background) the sheath scale is.1cm and presheath scale is 1cm [1]. In low temperature plasmas the length of the presheath region is usually determined by the rate of charge exchange collisions, elastic collisions of ions against a neutral background [11, 12], or by the volume ionization rate [13, 14]. Although most sheaths are ion rich, there are some situations in which the sheath can be electron rich. The most common example of this is a wire Langmuir probe biased at a large positive potential. If the probe is small enough, the wire does not change the plasma potential significantly, resulting in the probe collecting electrons since it is biased above the plasma potential. Above the positive bias at which this occurs the current appears to saturate. This is known as electron saturation. The electron-rich sheath exemplified by this situation is called an electron sheath. In steady-state, an electron sheath can only be present when the loss rate of electrons and ions from the plasma balance. For small plasma boundaries, such as a Langmuir probe, an electron sheath can be present since the electron sheath collection area does not significantly modify the global ion and electron current collection by the plasma boundaries. However, for larger boundaries this is not always the case. It was observed that an electrode can only be biased above the plasma potential if its area is small compared to the other confining walls of the plasma (A Electrode /A Wall < p 2.3m e /m i ) [15]. The maximum size of this area ratio is determined by the fluxes of ions and electrons lost to all plasma boundaries [16, 17]. Electron sheaths have been most extensively studied in the context of tethered space probes [18] and plasma contactors [19, 2] due to their application in beam emission experiments and spacecraft high-voltage power systems [21]. In the latter case, a tethered space probe would generate a current in the tether due to its motion perpendicular to the Earth s magnetic field (the induced voltage is V EMF = `v B Earth,where` is the tether length and v is its velocity). To maximize this current, the largest possible collection of electron current for a given tether design was desired. The most dramatic example of the shortcomings of the theory of the current collection of electron sheaths is the TSS-1R experiment, in which a tether failed during a space shuttle mission due to a significantly greater than anticipated electron current [22]. In Laboratory plasma experiments, it has typically been assumed that electron sheaths around small electrodes only pose a small perturbation to the plasma. In this work, the form of the interface between small positively biased electrodes and the bulk plasma will be revisited using a combination of simulation and theory. How this interface interacts with the bulk plasma is important

20 3 for understanding the behavior of positive electrodes in plasmas. One application of such a theory is understanding how Langmuir probes at electron saturation perturb the bulk plasma. These have been used in dusty plasmas to induce circulation in dust crystals [23] and in fusion plasmas to induce circulation in the plasma of the scrape o layer plasma [24, 25]. In both of these cases there was an observed di erence between the behavior of positively and negatively biased electrodes suggesting a di erence in the processes responsible for the electrode behavior. A theory of positively biased electrodes may also have applications to the high potential phase of the RF cycle[26] and near strongly emitting surfaces[27, 28]. In this work, a theory for the interface between an electron sheath and an unmagnetized plasma is presented. It is found for the first time that the electron sheath interfaces with the bulk plasma though a presheath that accelerates electrons to approximately their thermal speed by the sheath edge. Unlike the ion presheath which accelerates ions through a ballistic response to the electric field, a pressure gradient drives the acceleration of the electron flow. This theory compares well with particle-in-cell (PIC) simulations of the electron sheath and recent experimental measurements of the presheath length. The electron sheath is also a precursor to the anode spot, a discharge phenomenon that occurs in low pressure laboratory plasmas near positive electrodes [29, 3]. Anode spots occur when electrodes are biased at a large enough potential that there is an appreciable amount of electron impact ionization within the sheath. The anode spot is also an example of plasma self-organization [31, 32]. Anode spots are a secondary plasma attached to a positively biased electrode that separate themselves from the bulk plasma by a double layer, a thin sheath-like space charge layer, and typically have a high degree of cylindrical or spherical symmetry. This thesis presents the first PIC simulations of an anode spot. Using observations from these simulations, a theory for the formation and steady-state properties is presented. The following sections of this chapter present background material for the work presented in this thesis. Starting in Chapter 1.2, the fluid moment equations are presented. These are used to formulate a sheath and presheath model in Chapter 2. Chapter 1.3 gives an overview of sheaths as encountered in a laboratory setting and gives an overview of the conditions under which electrodes can be biased above the plasma potential. Chapter 1.4 discusses the presheath and the Bohm criterion. An overview of simulation methods follows the review of the basic physics of sheaths. This starts with Chapter 1.5, which gives an overview of the PIC method and discusses di erences between simulated and realistic plasmas. Chapter 1.6 reviews the direct simulation Monte-Carlo

21 4 method which is used to handle collisions in the PIC simulations presented in this thesis. Chapter 1.7 describes Aleph, an electrostatic PIC code used in Chapters 2-4. Following this, Chapter 1.8 presents an overview of the conventional picture of how electron sheaths interface with the bulk plasma. Chapter 1.9 outlines the chapters of this thesis. 1.2 Fluid Moment Equations In a plasma the electrostatic potential is governed by Poisson s equation r 2 = 4, (1.1) where is the charge density. This potential is coupled to other plasma properties such as the electron and ion densities n e and n i,flowvelocitiesv e and V i, and temperatures T e and T i through equations governing the evolution of moments of the particle velocity distribution functions. The moment equations are obtained from the Boltzmann q s [E + v s m = X C s s (f s,f s ), (1.2) s by taking it s velocity moments. In the Boltzmann equation, f s (v) is the velocity distribution function of species s, E is the electric field, B is the magnetic field, q s and m s are the charge and mass of particles of species s, and C s s (f s,f s) is the collision operator which includes interactions between species s and s. There will be an equation of this form for each species. The equations governing the moments of the velocity distribution function are obtained by evaluating the velocity integrals of the Boltzmann equation, where Z 8 apple d q m [E + v 1 for the continuity equation X C s s s (f s,f s ) =, (1.3) >< mv for the momentum equation g(v) = mv 2 2 for the energy equation >: other combinations of powers of v and v for higher moment equations. (1.4) In this work the first two moment equations are used extensively. These are the continuity = r n s V s, (1.5)

22 5 and the momentum s V s + r (P s + m s n s V s V s ) n s q s [E + V s B] R s,s =. (1.6) The fluid moments in these equations are defined as velocity integrals of f s and are defined below. Z Density : n s = d 3 vf s (v) (1.7) Flow Moment : V s = 1 n s Z d 3 vvf s (v) (1.8) Temperature : T s = 1 n s Z d 3 v m svr 2 f s (v) (1.9) 3 Z Pressure : P s = d 3 vm s v r v r f s (v) =p s I + s (1.1) In the equations above, p s = n s T s is the scalar pressure, s is the stress tensor, I the identity tensor, and v r (v V s ). In addition to these, the collisional friction is determined by the v moment of the collision operator Z R s = d 3 vm s v X s C s s (f s,f s ). (1.11) There are an infinite number of moment equations each depending on the next highest fluid moment. Eventually a closure must be used to solve these equations. In the following chapters, the continuity and momentum equations will be used and evaluated with simulation data for f s to determine properties of the sheath and presheath. 1.3 Sheaths and Balance of Global Current Loss The Bohm Criterion and Global Current Loss In laboratory plasmas, quasineutrality is maintained through the equal loss of positive and negative charge to the confining walls. When a plasma loses more electrons than ions the resulting electric field polarization causes the plasma potential to change relative to that of the walls. The plasma potential settles at a value that results in an equal loss of electron and ion current. If the electron and ion masses were the same, and if they had the same temperature, all boundaries could be at the plasma potential since the thermal fluxes of each species would be the same. However, p mi /m e 1 is more typical since the proton to electron mass ratio is m p /m e = 1836.

23 6 where v Ts = Consider the average speed of a stationary Maxwellian distribution of particles 3/2 Z 1 ms v Ts = 4 dvv 3 v 2 r 8Ts exp =, (1.12) 2 T s m s q 2T s m s a 2D plane within the plasma, is 3/2 Z 2 ms T s = n s d 2 T s is the thermal speed. The thermal or random flux, which is the flux incident on Z /2 sin cos Z 1 v 2 T s dvv 3 exp v 2 v 2 T s = n s v Ts. (1.13) 4 The electron thermal flux is greater than the ion thermal flux by a factor of p (m i T e )/(m e T i ). Without macroscopic electric fields to reduce the electron flux collected by the boundary the electron and ion current lost will not balance. In an unmagnetized plasma this necessitates the existence of a region with an electric field that reduces the electron flux to the boundary. This region is the sheath. The simplest model of a steady-state sheath is that of a plasma with Maxwellian electrons at temperature T e and monoenergetic ions with temperature T i = [33]. If the sheath is thin enough that ions traverse its length without collision, the electric field and inertial terms in the momentum equation Eq. (1.6) balance. In this case, the motion of the ions is determined by their ballistic response to the electrostatic potential, which is m i V i (x) e (x) = m ivio 2 + e o. (1.14) 2 The density is given by the continuity equation Eq. (1.5), which implies n i (x)v i (x) =n io V io, (1.15) where n io and V io are the ion density and velocity at the sheath edge (at the location where the reference potential = o is set). Here, the sheath edge is defined as the location where the quasineutral description of the plasma is no longer valid. Note that this definition is a bit ambiguous since the plasma is never exactly neutral; however, the sheath edge can be defined by a location with a certain percent deviation from neutrality. This method works well for determining the sheath edge since the sheath is a boundary layer with sharply increasing charge density. In the model presented below, the electron and ion densities are set equal at the sheath edge. The ion flow velocity at a location a distance x away from the sheath edge is s V i (x) = Vio 2 + 2e( o (x)). (1.16) m i

24 7 The density resulting from the continuity equation Eq. (1.15) is n i (x) =n io 1+ 2e( 1/2 o (x)) Vio 2m. (1.17) i When the electrons have a stationary Maxwellian velocity distribution function and are repelled by the sheath electric field, the pressure and electric field terms in the momentum equation Eq. (1.6) balance and their density varies exponentially with the potential 1 as e( (x) o) n e (x) =n eo exp. (1.18) Here, n eo is the density at the reference potential T e o. An equation of this form was first obtained by Maxwell [34] and is known as the Boltzmann relation. Using Eq. (1.17) and Eq. (1.18) in Poisson s equation and using the dimensionless variables = e /T e and X = x/ De, d 2 apple dx 2 = exp (X) o 1+ 2T 1/2 e( o (X)) m i Vio 2. (1.19) A solution for (X) can be obtained by integration. Multiplying each side by d /dx and integrating with respect to X while setting o = and d /dx o =, Z Z d d d dx = dx dx dx apple d exp dx (X) 1 1/2 2T e (X) m i Vio 2 dx. (1.2) This integral is 2 1 d =exp 1+ m iv 2 s io 2T e (X) 1 2 dx T e m i Vio 2 1. (1.21) The right hand side of Eq. (1.21) must be greater than or equal to zero for a real solution for the electric field to exist. Near the sheath edge the plasma is quasineutral resulting in. Setting the right hand side of Eq. (1.21) equal to zero and expanding to order 2 results in a requirement on the flow velocity at the sheath edge, V io r Te m i = c s. (1.22) This result states that ions must flow past the sheath edge at a velocity greater than their sound speed and is known as the Bohm Criterion [9]. From Eq. (1.14) it is apparent that ions at rest would need to go through a potential of T e /2e to achieve this speed. The ion acceleration occurs in 1 This assumption requires that the repelling potential is much less that the electron temperature such that e T e.

25 8 a quasineutral region leading up to the sheath, this region is called the presheath. The acceleration of ions to this velocity will be revisited in the following section of this chapter. Having arrived at the Bohm Criterion, the details of how the formation of the sheath allows maintenance of quasineutrality are now reviewed. Consider a plasma confined by a box with conducting walls at a potential =. The plasma has Maxwellian electrons with temperature T e and monoenergetic ions with T i T e. The amount of ion current collected is determined by the flux density at the sheath edge which is set by the Bohm criterion. This is I i = en E V io A W = en E A W r Te m i, (1.23) where A W is the area of the confining walls and n E is the sheath edge plasma density. The electron current collected is determined by the thermal flux of Eq. (1.13) modified by the exponential Boltzmann factor of Eq. (1.18) I e = e n e v Te A W = e r 8Te n E A W exp 4 4 m e e p T e, (1.24) where p is the plasma potential. Setting the electron and ion currents lost at the walls equal determines how positive the plasma potential must be for the electron and ion currents lost to be balanced. This will approximately be the potential drop across the sheath. The plasma potential that results in balance of electron and ion current loss is p = T r e e ln mi. (1.25) 2 m e For reference, using Eq. (1.25) the plasma potentials for common ion species such as helium, argon, and protons are 3.53T e /e, 4.68T e /e, and 2.84T e /e respectively. Baalrud et al. [16] considered the balance of global current loss for a plasma confined by a box with an electrode of area A E, biased much more positively than the walls, placed inside. Although the electrode is biased above the wall potential, the plasma potential can be either above or below the electrode potential E depending on the area ratio A E /A W. First consider the case where the electrode potential is below the plasma potential so that the sheath at the electrode is an ion sheath. The electron current lost to the walls and electrode is of the same form as Eq. (1.24), e E e p I e = e e,th applea E exp + A W exp, (1.26) T e where e,th is the thermal flux defined by Eq. (1.13) and E = p E >. Theioncurrentis of the same form as Eq. (1.23), I i = e i,b (A E + A W ), (1.27) T e

26 9 where i,b =.6c s n o is the Bohm flux. The factor of.6 is due to the density rarefaction in the ion presheath caused by the T e /2e presheath potential. Once again, the plasma potential that results from setting the electron and ion currents equal. This results in p = T apple e e ln AE + A W µ A W A E e E exp A W T e, (1.28) where µ = p 2.3m e /m i. Assuming that the sheath potential satisfies E T e /2e, and that the argument of the logarithm must be positive for a stable ion sheath, results in an area criteria for an ion sheath at the positive electrode A E A W.6 µ µ. (1.29) If the area of the positively biased electrode is small an electron sheath may be present at the electrode, where the electrode potential is above the plasma potential, as shown in Fig In this case, the sheath electric field is attractive for electrons and repulsive for ions. It is typically assumed that this type of sheath collects a random flux of electrons, this assumption will be revisited in Chapter 2. Using this assumption, the total electron and ion currents lost from the system are and I e = e e,th applea E + A W exp e p T e (1.3) I i = e i,b A W. (1.31) Setting these equal, the plasma potential is p = T e e ln µ A E. (1.32) A W Since the argument of the logarithm must be positive, a stable electron sheath can be present when A E A W <µ. (1.33) When the area ratio is between that required for an electron sheath and an ion sheath it was assumed that a non-monotonic potential was present. This takes the form of a dip in the plasma potential in front of the electron sheath, shown in Fig. 1.1, which reduces the electron flux to the electrode so that the electron and ion currents lost can be balanced. This feature is a type of virtual cathode that can occur near plasma boundaries, although they are also present in situations where charge particles are emitted from the electrode [28]. The dependence of sheath structure on the ratio A E /A W has been observed in experiments [16, 17] and particle-in-cell simulations [35].

27 1 Electron Sheath! Potential! Plasma! Potential! Electron Sheath with Virtual Cathode! Virtual Cathode! Ion Sheath! Distance! Figure 1.1: Three di erent types of sheath structure that are possible near a positively biased boundary. When the electrode bias is either negative or if the electrode is large the electrode bias is below the plasma potential and an ion sheath is present. If the electrode is small enough that Eq. (1.33) is satisfied, an electron sheath will be present if the electrode is biased positive enough. At intermediate electrode sizes between those satisfying Eq. (1.29) and Eq. (1.33) a positive electrode will have an electron sheath with a virtual cathode that reduces the electron flux to the boundary.

28 Child-Langmuir Sheath Up to this point, only the maintenance of quasineutrality by the sheath has been considered. Now focus will shift to physical properties of the sheath. Studies of the ion sheath were first reported in 1911 by Child [36] followed shortly after by Langmuir in 1913 [37]. This is some of the earliest work in plasma physics and was 17 years before the Physical Review paper Oscillations in Ionized Gases in which Langmuir coined the term plasma [38]. The study of the sheath scaling appeared chronologically before the rest of the work presented in this chapter since it was important for determining current voltage characteristics of arc lamps used for lighting at the beginning of the 2th century. Child s starting point for obtaining this relationship was to describe charged particle motion as a ballistic response to an electric field, similar to Eq. (1.14). Using this along with Poisson s equation, the gradient in potential can be written as 2 d dx 2 d mi =8 I or dx 2e (1.34) where I o = en(x)v (x). Setting the potential gradient at the sheath edge (d /dx) =, this can be rearranged to and integrated to obtain d dx = p mi 8 I o 2e I o = 2 r 2e 9 m i 3/2 1/4 (1.35) x 2, (1.36) where V and are related as V = p 2e /m i. Eq. (1.36) is commonly known as the Child-Langmuir law. 1.4 The Presheath Plasma Solutions In the sheath model of Chapter 1.3.1, it was found that ions enter the sheath with a velocity exceeding the ion sound speed. Using this plasma description, a quasineutral plasma solution ( (x) such that n i ( )=n e ( )) is obtained for the plasma leading up to the sheath edge. This can be obtained from Poisson s equation, which can be written as d 2 (e /T e ) d(x/`) 2 2 apple D e ne = `2 n o n i n o, (1.37) where ` is a length scale representative of the presheath. The quasineutral limit can be obtained by letting ` D e,whichresultsinn i = n e. In the monoenergetic ion plasma model, the ions are

29 12 described by the momentum equation Eq. (1.6) in the static limit with T i =. The relevant form of the momentum equation is and the ion continuity equation is m i n i V i dv i dx = en ie + R i (1.38) dv i n i dx + V dn i i =. (1.39) dx Assuming electrons are completely repelled by the sheath so that the electron velocity distribution is a stationary Maxwellian, the electron momentum equation is en e E + T e dn e dx =. (1.4) This is equivalent to the Boltzmann density relation in Eq. (1.18). Combining these three equations using n = n i = n e results in two equations which determine the plasma solution for density and ion flow velocity, and dn dx = R i/m i c 2 s V 2 i (1.41) dv i dx = V ir i /(n i m i ) c 2 s V 2. (1.42) The singularity at V i = c s indicates that the plasma solution is no longer valid when V i c s. Eq. (1.41) and Eq. (1.42) show an important property of presheath fluid models, the length scale over which the density and velocity gradients occur is determined by the collisional friction R i. In the collisionless limit where R i! the presheath length scale ` is infinite. In reality, an infinitely long presheath is never realized since the collision rate between particles is never truly zero. Any process which acts as a sink in the momentum equation can act to decrease the presheath length scale. One example would be particles sourced within the plasma. When these particles are added to a region in which there is a net flow, the momentum equation will have a loss term which acts to decrease the presheath length. In a plasma dominated by volume ionization, the presheath length is half of the plasma scale determined by the bounding walls[14]. Much e ort was spent on analytically matching the presheath and sheath solutions [39, 4, 41], although Eq. (1.37)-(1.4) can be easily solved numerically as shown in Fig i Fluid Moment Bohm Criterion In the monoenergetic ion presheath model the Bohm Criterion determined that the minimum flow velocity for ions at the sheath edge must be the ion sound speed. The species being accelerated

30 13 Presheath! Sheath! Density (n e, n i )! Bohm Speed! Ion Flow ( V i /C s )! Potential (e"/t i )! "#! $"%&'()!#*#+ $#,# Figure 1.2: Profiles of the potential, ion flow velocity, and species densities for a typical ion sheath and presheath from numerical solutions of Eq. (1.37)-(1.4). Note that the Bohm speed is achieved at a location that coincides with the breakdown of quasineutrality.

31 14 into the sheath are not always accurately described as a mono-energetic distribution, thus there is a need for a more detailed treatment of the distribution function. A more general way to determine the Bohm Criterion for ions was derived by Riemann [42] and is commonly know as the generalized Bohm criterion or the kinetic Bohm criterion. Although it is widely cited, the generalized Bohm criterion is inapplicable to most plasmas of interest [43] and only applies to situations where f i (v = ) =, a feature that is satisfied by some theoretical models, but is rarely satisfied in experiments [44]. Another method based on fluid moments of velocity distribution functions was presented by Baalrud and Hegna [43]. This will be referred to as the fluid moment Bohm criterion. The derivation of the fluid moment Bohm criterion starts with the sheath criterion [42], a definition of the breakdown of quasineutrality at the sheath edge. The sheath criterion is obtained by expanding the charge density in Poisson s equation about =, apple r 2 =4 ( = ) + d (1.43) d = Here, it was assumed that there is only variation in the x direction. At the sheath edge where due to quasineutraility, the expansion is d 2 dx 2 = Multiplying by d /dx and integrating gives d 4 d =. (1.44) E d d = where the constant C is determined to be zero since 2 = C, (1.45)! as x/ D!. The resulting sheath criterion is Noting that (d /dx)(dx/d )= d apple. (1.46) d = E 1 d /dx, and using = P s n sq s along with dn s /dx >, gives an alternate form of the sheath criterion X s dn s q s = X dx = s q s Z 1 1 df s (v) dx dv =. (1.47) This form can be used to relate the Bohm criterion to fluid moments of the distribution functions f s of the di erent species. For example, in a source free plasma where the continuity equation is time independent the sheath criterion is X s n s dv s q s apple. (1.48) V s dx =

32 15 From Eq. (1.48) a condition of the flow moment at the sheath edge is obtained by inserting the momentum equation for the flow moment derivative: X s apple qs n s (n s dt s /dx + d xx,s /dx R x,s )/E q s m s Vs 2 apple. (1.49) T s = Here, the flow moments are calculated for the known form of the distribution functions f s at the sheath edge, indicated by =. Eq. (1.49) can be simplified in many cases to provide a generalization of the Bohm criterion for a given presheath fluid model. 1.5 Particle-In-Cell Method Overview Particle-in-cell (PIC) is a method of approximating solutions to the Klimontovich equation which gives an exact kinetic description of the plasma. The PIC method approximates the Klimontovich + v r x N s + q s E m + vc m Bm r v N s =, (1.5) s by discretely sampling the particles in the full distribution function of N o particles N s = XN o i=1 (x X i (t)) (v V i (t)) (1.51) with N o /w macro-particles of weight w. It is assumed that the macro-particle orbit in phase space is a good representation of w physical particles. In addition to this, the microscopic fields B m and E m are spatially averaged and solved on a discrete grid. The charge of particles within a grid cell are deposited to the nodes with a specified interpolation scheme, therefore there is no direct calculation of the force between particles within a cell. The spatial scale is typically chosen to resolve the Debye length. While this spatial averaging of the electric field is similar to that encountered in the derivation of the Vlasov equation [45], there are still fluctuations in the electric field that cause particle collisions that are not captured by the ensemble averaged field. The result is that macroparticle densities are constant along trajectories in phase space, but the macro-particle distribution functions are not. The characteristics of the macro-particles are integrated in time with the updated fields at each time step. This section will outline the typical vanilla PIC method. A typical electrostatic PIC simulation starts with a set of initial macro-particle positions and velocities. To evolve the macro-particle positions in time the electric field is needed. While the macro-particle positions and velocities would be easy to compute to high precision if the exact fields

33 S 1 S 1 S(x) x ( x) Figure 1.3: The shape functions S and S 1 for zeroth and first order interpolation given by Eq. (1.53) and (1.54). were known, the field calculation needs to be done on a coarser grid. To accomplish this the charges are interpolated to grid nodes using an interpolation scheme. The general form of the charge at a node x i is (x i )= X k q k S(x i x k ) (1.52) where S(x) is a weight function determined by the interpolation order, q k is the charge of macroparticle k, and x k is its position. For zeroth order interpolation 8 >< 1if x i x k < x/2 S (x i x k )= >: Otherwise (1.53) and for first order S 1 (x i apple x k ) = max, 1 x i x k x. (1.54) The interpolation of particles to the grid nodes results in particles appearing to have a finite size as seen by the grid, see Fig This leads to non-physical e ects which will be discussed in Chapters and Once the charge has been interpolated to the nodes, the electric field and electric potential are determined by solving Poisson s equation. In a periodic 1D system this is simply done using the

34 17 second order central di erence for, i 1 2 i + i+1 x 2 = 4 i. (1.55) Writing the set of equations for each i results in N equations for N nodes. The potential at each node shows up in three separate equations, except for the potential at the end point which shows up twice, once in the equation for 1 and once in the equation for N 1 since 1 = N. This extra degree of freedom corresponds to the fact that any solution to Poisson s equation o set by an arbitrary constant value is also a solution. The potential at the periodic endpoint can be set to zero. Once the potential is known, the electric field is obtained by applying the finite di erence. In non-periodic 2D and 3D systems, more complicated methods such as Gauss-Seidel, successive over-relaxation, or the finite element method are used. Since the potential (and hence the electric field) is only solved at the nodes it needs to be interpolated back to the particle position to determine the force on the particle. This interpolation typically uses the same order interpolation as the charge density to avoid self forces, see Chapter This interpolation can be written as F k = q k x X j E j S(x j x k ), (1.56) where F k is the force on particle k located at x k and E j is the electric field at node j located at x j. Once the electric field at the particle positions is known the equations of motion can be integrated. One common method is leapfrog integration where the position and velocity are updated as x(t + t) =x(t)+ tv(t + t/2) (1.57) and where v(t + t/2) = v(t t/2) + t F [x(t)] m, (1.58) t is the time step increment. A related method is the velocity Verlet method. This method employs the leapfrog velocity integration for a half step, the leapfrog position integration for a whole step, followed by another half step velocity integration: v(t + t/2) = v(t)+ t F [x(t)] 2 m (1.59) x(t + t) =x(t)+ tv(t + t/2) (1.6) v(t + t) =v(t + t/2) + t 2 F [x(t + 1)]. (1.61) m

35 18 These three can be more compactly written as x(t + t) =x(t)+ tv(t)+ t2 2 F [x(t)] m (1.62) and v(t + t) =v(t)+ t 2m F [x(t + 1)] + F [x(t)]. (1.63) The leapfrog and velocity Verlet methods are time reversible and conserve energy, this is not the case for some other integrations methods such as Euler or Runge-Kutta. The processes described in the paragraphs above are applied to each particle in each cell. Once these steps are complete, the process is repeated for each time step. Now that the basic method has been outlined, the following four subsections will address some physical di erences between real and simulated plasmas that arise due to the finite nature of the time step and grid spacing. In the next part of this chapter, some di erences between real and PIC simulated plasmas will be reviewed. This will start with the resolution of particle trajectories in Chapter When the forces on particles is updated only from the grid nodes, there is a need to ensure that particles have forces applied from each node that bounds their trajectory. This can only occur when the time step for trajectory integration is small. Following this, the e ect of finite size particles on the linear plasma dielectric response is reviewed in Chapter This review gives an expectation for how well the dispersion relation of simulated waves will agree with theoretical calculations for real plasmas. It is a common misconception that PIC does not include collisional e ects. Chapter reviews collisional e ects in PIC simulations of stable plasmas and discusses how they are reduced by the finite spatial resolution of the grid. This is relevant for Chapter where evidence is presented for collisions in PIC simulated plasmas. This discussion concludes with a review of unphysical self forces due to the choice of interpolation scheme in Chapter and a review of an empirical observation of plasma heating due to the grid in Chapter Resolution of Particle Trajectories A particle s velocity is updated by interpolating fields from the nodes of the cell in which it is located. If the particle travels across more than one cell in a time step the velocity will not be updated by the fields from the nodes of the cell through which it passes. This results in unphysical particle trajectories such as the one shown in Fig If the particle trajectory crosses one or zero cell boundaries in a time step, the particle velocity will be updated by the fields in the cells bounding the particle motion. Fig. 1.4 also shows a hypothetical trajectory for a particle whose velocity is

36 19 being updated with a small time step. The requirement on the time step needed to obtain physically realistic trajectories can be written as where t< x v, (1.64) x is the scale size of a cell 2 and v is the velocity of the particle whose trajectory is under consideration. When satisfying Eq. (1.64) is too burdensome of a condition for each particle, the time step may be chosen so that a large fraction of particles satisfy the condition. Typically, the time step would be chosen so that all particles satisfy Eq. (1.64); however, when a strong electric field is present, e.g. in sheaths and double layers, the condition is not easily met by all particles. In this case, a time step can be chosen such that 9-95% of particles satisfy the condition. This allows the sheath charge density to be well approximated (9-95% of particles distribute their charge to the grid nodes within the sheath in a physical way) without the need to decrease the time step for the entire domain. In this case, the velocity distribution function within the sheath can only be accurate for particles with velocities satisfying Eq. (1.64). Ideally, particles violating Eq. (1.64) within the sheath would not disturb the velocity distribution functions in the bulk plasma since the violating particles are directed towards absorbing walls. However, violations by a small fraction of particles can potentially modify the system dynamics. An example of this is that particles violating the condition in the sheath leave the system at unrealistic rates, modifying how the system balances the loss of electron and ion currents to the walls. For the simulations presented in this thesis, duplicate simulations with spatial scales and time steps of x/2, 2 x, t/2, and 2 t were run to verify that the species densities and electrostatic potential did not significantly change with these parameters. This suggests that violations of Eq. (1.64) were not causing significant changes in the system behavior Modifications to the Dielectric Response Okuda and Birdsall [1] and Langdon and Birdsall [46] evaluated the e ect of the finite size of particles on the linear plasma dielectric by following through the typical derivation replacing the charge density and force with the values as seen by the grid [46]. In terms of the shape function S, the modifications to the charge density and force are Z c (x,t)= dx S(x x ) p (x,t) (1.65) 2 For a cartesian mesh this is the cell length, but for an unstructured mesh this is a length characteristic of the cell size.

37 2 E i E i,j! E i,j+1! E i,j+2! Short Time Step! E i-1,j! E i-1,j+1! E i-1,j+2! E i-1 1,j+1 E i-1,j+2 E i,j+3! E i-1,j+3! Long Time Step! E i-2,j! Figure 1.4: Examples of particles moving through a grid with large and small time steps for the velocity integration. When the time step is too large the particle velocity is not updated by all of the nodes of the cells through which it passes, resulting in unphysical trajectories. and Z F(x,t)=q dx S(x x)e(x,t), (1.66) where c and p are the charge densities of finite size grid particles and of point particles. The Fourier transform of these quantities are c (k,t)=s(k) p (k,t) (1.67) and F(k,t)=qS( k)e(k,t). (1.68) Following through a typical derivation of the plasma dielectric such as that in Chapter 3 of Ref. [45] or Chapter 7 of Ref. [47] leads to the modified form Z (!, k) =1+S 2 (k)!2 p k 2 dv! k v. (1.69) This di ers from the exact value by the inclusion of the shape function in the second term which modifies the location of the zeros of the dielectric. This results in a modification to the frequency and growth or damping rate of waves in the plasma.

38 Collisions In their study of plasmas of finite size particles, Okuda and Birdsall [1] derived a modified version of the Lenard-Balescu kinetic equation for a stable uniform plasma. The Lenard-Balescu kinetic equation describes the evolution of the velocity distribution function including the collective motion of the plasma described by the linear dielectric response, see Chapter 5 of Ref. [47] for a description. Okuda and Birdsall s modified kinetic equation is where the di usion tensor is Z D =2q Af S(k) 4 kk k 4 (k v, k) 2 (k v k v )F (p )dkdp. (1.71) Here, the distribution function in the bulk plasma is F (p) =exp( p 2 /m 2 v 2 T )/[p v T ] 3 and p = mv is the momentum. In this notation the drag is Z A =2q 4 n S(k) 4 kk k 4 (k v, k) (k v k v )dkdp. (1.72) A detailed derivation of this kinetic equation is presented in Ref. [48]. Okuda and Birdsall evaluated the drag and di usion coe cients in Eq. (1.71) and (1.72) for a test particle with velocity v in a 2D plasma with particles that had a gaussian shape function S(k) =exp( 2k 2 R 2 ). The resulting parallel and perpendicular di usion coe cients and parallel drag are shown in the solid black lines of Fig. 1.5 and Fig In the figures, the particle size R is varied to show the e ect of the particle size. For low test particle velocities, the di usion and and drag are decreased by about a factor of 7 when comparing point particles with particles of radius R =.5 D. The di erence between these coe cients decreases as the test particle velocity increases. Fig. 1.5 and Fig. 1.6 also show a calculation of the di usion and drag using the static screening limit of the linear dielectric ( (k v, k)! (, k)). This result is shown in the lines marked Landau Eq. Compared to the Landau equation result, the di usion and drag from the Lenard-Balescu equation approach the same value at large velocity since the dynamic screening in the dielectric extends the interaction distance of particles beyond the particle scale R. The results quoted above are only valid for a uniform stable plasma. When instabilities are present a modification to the collision rate of particles can be expected [49]. This modification is also present in PIC collisions and have recently been observed [5]; however, a theory analogous to Eq. (1.7)-(1.72) for an unstable plasma of finite sized particles is not present in the literature. A

39 22 Figure 1.5: The parallel and perpendicular di usion rates for several di erent particle sizes calculated by Birdsall and Okuda [1] from Eq. (1.71) assuming a gaussian shape function. The di usion rate reproduces the point particle value as the particle size R goes to zero. Reprinted from Ref. [1], with the permission of AIP Publishing.

40 23 Figure 1.6: The particle drag for several different particle sizes calculated by Birdsall and Okuda [1] from Eq. (1.72) assuming a gaussian shape function. Reprinted from Ref. [1], with the permission of AIP Publishing.

41 24 modification of the kinetic theory of an unstable plasma from Ref. [49] for the e ects of finite size of particles is presented in Appendix A. Consider solutions to Poisson s equation Non-Physical Self Forces ik E c =4 c. (1.73) Writing this in terms of the point particle density and field given by Eq. (1.67) and Eq. (1.68) this becomes ik S a ( k)e(k,t)=4 S b (k) p (k,t). (1.74) Here, the possibility of di erent shape functions S a and S b in the force and charge interpolations has been allowed. From this equation it is apparent that if S a and S b are both isotropic, and if S a = S b, then Poisson s equation for a point particle is recovered. If these conditions are not satisfied there will be a modification to the force law. This modification is due to the particle exerting a self force when the interpolation schemes di er Grid Heating In 1971, Hockney studied the collision and self heating times in several 2D PIC simulations [51]. Using these simulations, empirical relations for these times were found to be related to the parameter N c = n[ 2 D +( x) 2 ], (1.75) where n is the areal plasma density 3. This is valid for the zeroth and first order weighting given in Eq. (1.53) and Eq. (1.54). The collision time was found to be where pe =1/! pe, and the heating time was found to be c = N c.98 pe, (1.76) H = K where K is 2 and 4 for zeroth and first order weighting, respectively. 2 D c, (1.77) x 3 The areal density is the number of particles per square centimeter. In the 2D simulations of this thesis, the density is reported as the number of particles per cubic centimeter assuming that the out of plane dimension is 1 cm.

42 25 In the simulations of the following chapters n cm 2, T e 2eV, and x.9 D, resulting in N c 2 1 6, pe = 793 ps, and c = 16µs. Typical simulation runs were for 1-4µs of physical time, meaning that both grid heating and collisional relaxation are likely negligible for these simulations. The simulation code used in this thesis utilizes an unstructured triangular mesh (grid). It has never been demonstrated that an unstructured triangular mesh follows the behavior of Eq. (1.77), which is valid for a cartesian grid. 1.6 Direct Simulation Monte-Carlo Method Typically, fluid equations are solved by approximating solutions to the Navier-Stokes equations; however, these equations are only valid in the limit that the fluid is in local thermodynamic equilibrium. This assumption is no longer valid for di use gases in the molecular flow regime. This is the regime of the neutral gas component found in low pressure plasma experiments. The closeness of a system of particles to thermodynamic equilibrium is described by the Knudsen number where K n = mfp L, (1.78) mfp is the mean free path for collisions, and L is a characteristic length scale of macroscopic gradients (e.g. the length for density gradients is L = n/(dn/dx)). When K n 1 there are many collisions over the length L and the system is in local equilibrium. In this case it is appropriate to use a fluid model. For systems where K n & 1 collisions may be important for the description of gas kinetics over several lengths L. For this case solutions to the Boltzmann equation should be considered. The direct simulation Monte-Carlo (DSMC) simulation is a method of solving the Boltzmann equation for rarefied neutral gas flows with K n & 1. Similar to the PIC method, the DSMC method uses macro-particles to sample the behavior of particles in the physical system. The DSMC method approximates the collision rate of physical particles in a real system by sampling collisions between macro-particles within a cell. An overview of G. A. Bird s original DSMC method and related developments is given in Ref. [52] and is partially summarized here. A DSMC simulation starts by sampling a collision pair labeled i within a cell. A velocity c larger than any relative velocity of particles within the cell is chosen and the probability of collision is determined by P i = c i (c i ) c (c ), (1.79) where (c i ) is the cross section at the relative velocity of the collision c i. A random number R

43 26 between and 1 is chosen and the collision is accepted if P i <R. A time counter c is used to ensure the correct collision rate is obtained. If the collision is accepted the time counter is incremented by the amount 2 i = Nnc i (c i ). (1.8) Here, N is the number of particles in the cell and n = W N/V is the density which is equal to the particle weight times the average particle number divided by the cell volume. The time counter increment is chosen to approximate the collision rate due to the Boltzmann collision operator [53]. Once the time counter satisfies c = X i i > t, (1.81) where t is the time step for the evolution of particle positions, the selection of collision pairs stops. Once all the collision pairs have been selected, the post collision velocities are found, the velocities are updated, and the particles are moved by the amount v t. This process occurs in each cell and is repeated for each time step. One model commonly chosen to calculate the cross section and post collision velocities is the variable hard sphere model [54]. The hard sphere collision is a convenient model because it is simple to calculate. However, this model does not contain the energy dependence of the cross section which is an important physical property present in real gases. The variable hard sphere model was introduced to address this shortcoming. The model describes the hard sphere diameter for a collision as cref d i = d ref, (1.82) c i where d ref and c ref are the reference diameter and velocity, and is an exponent determined empirically from data. The collisional mechanics is the same as the hard sphere model, the only role that c i plays is in determining the hard sphere diameter. The early version of DSMC, which is called the time counter method, isine cientsince the number of pairs to be evaluated scales as N 2 with increasing particle number. More recently, a method independent of the time counter was introduced. The no-time-counter method [54] determines the number of pairs by using Eq. (1.8) to determine the maximum collision frequency 1/max( i ) for particles in the cell, and evaluating pairs for collision with probability Nnmax[c (c)] N pairs = t 2 (1.83) P i = c i (c i ) max[c (c)]. (1.84)

44 27 The increased e ciency comes from sampling fewer particles for collision with a higher collision frequency, while still preserving the collision rate. For the DSMC method to accurately describe the gas flow, the CFL-like condition from Eq. (1.64) must be satisfied so that particles do not cross more than one cell boundary in a time step. The DSMC method is often used in combination with the PIC method to evaluate collisions between neutral and charged particles. This allows the modeling of neutral flows interacting with the plasma through ionization, recombination, collisional heating of the neutral gas, and the use of energy dependent cross sections in particle collisions. 1.7 Aleph The simulations in the following chapters used the electrostatic PIC code Aleph [55]. Aleph solves for the electric field on an unstructured mesh in up to three dimensions using a finite-element method field solver. In the simulations presented in this thesis, a 2D triangular mesh was used and the charge density interpolation was chosen to be constant in cell, resulting in a triangular generalization of the zeroth order weight function in Eq. (1.53). The particles in the simulation have three velocity components. Besides the interaction with the electric field, Aleph uses the DSMC no-time-counter method to handle collisions between neutral and charged particles. In the simulations presented here, the DSMC particle collisions had their velocity updated while their position stayed the same. Since the neutral density was significantly greater than the plasma density, the uniformity of the neutral gas is not expected to change significantly due to its interaction with the plasma. This choice of conditions for neutrals makes the method similar to PIC with Monte-Carlo collisions. The PIC method utilizing a triangular mesh is expected to behave in a similar way to the vanilla PIC method described in Chapter 1.5. There are two areas in which the behavior of an unstructured mesh may present di erent behavior. The first is in the resolution of particle trajectories. When the mesh is unstructured, the length of the sides of a cell may di er between cells. Since a characteristic cell size ( x) is used to evaluate Eq. (1.64), it is possible for a particle to cross more than one cell boundary in a time step while still satisfying the condition of Eq. (1.64). The second di erence that can be expected is in the grid heating time of Eq. (1.77), which was based o of empirical observations of a simulation with a cartesian grid. Evaluation of grid heating for an unstructured mesh is absent in the literature.

45 Conventional Picture of the Electron Sheath Although electron sheaths are commonly encountered, they have rarely been a topic of focus in the literature. This is especially true with regard to theoretical studies of electron sheaths, where most of the work has focused only on the current collection near tethered space probes in magnetized plasmas. These descriptions have focused on single particle-based descriptions of current collection that rely on the orbits of particles [21, 56]. It is typically assumed that electrodes biased above the plasma potential only pose a small perturbation to the bulk plasma. This is reflected in the following ideas found in the literature: 1. The electron sheath collects a random flux of electrode-directed electrons [57, 29]. 2. Since the electron sheath collects a random flux of electrons, the electron velocity distribution function (EVDF) at the sheath edge is a Maxwellian truncated at zero velocity [58, 59]. 3. The electron sheath Bohm criterion is trivially satisfied [42], and there is no Bohm criterion for electrons when T e >T i [6]. This is presumably because the assumed velocity space truncation at provides a su cient flow moment, hence no presheath is needed. All of these statements stem from the often assumed idea that the electron sheath collects a random flux of electrons. A careful reflection on the sheath implied by these assumptions hints at the need for a more detailed description of the interface between an electron sheath and plasma. Consider the EVDF in the electron sheath and the bulk plasma. In the bulk, the EVDF is often assumed to be a full Maxwellian, but at the sheath edge and within the sheath it is a half-maxwellian. The abrupt truncation at the sheath edge suggests a need for a region between the sheath and bulk plasma in which flux conservation is broken, allowing the matching of sheath and bulk plasma boundary conditions. This presheath region would allow for the generation of flow between the bulk plasma and sheath, providing the necessary flux at the sheath edge. In addition to this, two implications of the suggested model hint at the need to revisit the half-maxwellian assumption. First, the electric field at the sheath edge is expected to be non-zero. A non-zero electric field would suggest that the electrons would develop a flow due to the ballistic response to this field. Second, assuming that ions are Maxwellian in the bulk, their density varies as n o exp( e /T i ) and the momentum equation for this case would be similar to that of Eq. (1.4), but for ions instead of electrons. The electron and ion density gradients must be equal for a quasineutral solution in the presheath. In this case an electric field repelling ions would result in a density gradient for electrons since dn e dx = en ie T i. (1.85)

46 29 This density gradient can give rise to pressure gradient driven electron flows in the pressure term of the electron momentum equation. 1.9 Outline In this thesis, the physics of the positive electrode is studied using PIC-DSMC simulations in combination with analytical models. The positive electrode is first modeled in two dimensional simulations. These are required to recover the area ratio dependence of the sheath polarity given by Eq. (1.29) and Eq. (1.33). Experimental measurements often require the introduction of a probe which may modify the bulk plasma properties, although occasionally non-perturbative optical measurements are possible. However, these are often limited by the availability of appropriate atomic transitions and the ability to obtain a su cient signal to noise ratio. When possible, the results of the simulations are compared with experimental observations to validate the computational model. One advantage of this approach is that any physical quantity of interest can be extracted from the computational model without perturbing the system. These simulations are validated by using available non-perturbative optical diagnostics from experiment. Several observables from simulations are used to determine which set of physical processes dominates the sheath and presheath behavior. The velocity distribution function is perhaps the most informative of these quantities since it allows the direct evaluation of terms in the momentum equation. Other quantities that are often used include species densities, temperatures, flow velocities, electric field, and potential. In some cases, it is advantageous to label electrons based on their source. This allows the separation of charged particles populated within the volume from those resulting from the ionization of neutral atoms. This will be used in Chapter 4 while studying the e ects of ionization within the sheath. These observables from simulations give insight to the governing physics, allowing the formulation of analytical models describing the phenomenon, which in turn allows for improved understanding of laboratory experiments. The goal of this work is to understand how electron sheaths interface with, interact with, and perturb their surrounding plasma. Fig. 1.7 shows a typical current-voltage (I-V) trace of a positively biased electrode. Each chapter of this thesis focuses on a di erent part of this curve. In Chapter 2 the random flux assumption in Chapter 1.8 is revisited. It is shown for the first time that the electron sheath has a presheath. Unlike the ion presheath in which ions accelerate due to their ballistic response to the electric field, the electron presheath accelerates electrons by their fluid-like response to a pressure gradient. It is found that this presheath accelerates electrons to

47 3 near their thermal speed by the sheath edge and that the acceleration region is much longer than the ion presheath under similar plasma conditions. The electron presheath also has implications for the flux collected by an electrode biased above the plasma potential. In the presence of an electron flow, the electron sheath is twice as thick as previously assumed. Chapter 2 predicts an increase both in the sheath collection area and flux collected at the sheath edge. Increases in these quantities result in a greater electron saturation current than predicted by the random flux assumption. Following this, Chapter 3 uses PIC simulations to look at the specific form of the electron velocity distribution near an electrode with bias changing from below to above the plasma potential. This is relevant to conditions near the knee in the I-V curve at the onset of electron saturation. It is found that when the electrode is biased near the plasma potential, the behavior of the presheath is dominated by the geometrical obstruction of electrons in the plasma caused by the location of the electrode. This results in a loss-cone-like truncation in the velocity distribution function. For electrode biases above the plasma potential, the truncation and flow shift of the distribution function determine the presheath properties. Chapter 4 focuses on understanding ionization processes that can occur within the electron sheath leading to the formation of an anode spot. Using observations from the first PIC simulations of an anode spot, a model for anode spot onset is presented. In this model, ionization within the sheath leads to the formation of a positive space charge layer near the electrode. When enough positive space charge is present, a potential well for electrons forms in front of the electrode surface. This potential well traps low energy electrons formed by electron impact ionization and leads to the formation of a high potential quasineutral region. When the density of this region grows large enough, a flux density imbalance across the double layer leads to the expansion of the quasineutral region and formation of the anode spot. Elements of this model are tied to the ionization rate, giving a prediction for the critical electrode bias needed for spot onset. Chapter 4 also presents a theory for the steady-state anode spot size, form of the sheath between the spot plasma and electrode, and the value of double layer potential. This model is based on particle, power, and current balance arguments. One element of this model is that it connects these three properties by considering the energy dependence of the ionization cross section, a feature that was not previously considered in models of the anode spot steady-state. Chapter 4 concludes with a comparison between the simulation, theory, and the spot onset observed in recent experiments.

48 31 1.2! Current (A)! 1!.8!.6!.4!.2!! Electron Sheath! (chapter 2)! Anode Spot! (chapter 4)!! 2! 4! 6! 8! 1! Electrode Bias (V)! Ion to Electron Sheath Transition! (chapter 3)! Electron Saturation Current! Figure 1.7: The location of the di erent types of sheath structure studied in the chapters of this thesis placed on the I-V curve of a positively biased electrode. The data points were provided by Scott D. Baalrud and are from the experiment in Ref. [2].

49 32 CHAPTER 2 THE ELECTRON SHEATH AND PRESHEATH 2.1 Motivation Experimental measurements of the sheath and plasma near a small positively biased electrode at typical low temperature plasma conditions have recently been reported [3]. The measurements were taken in a GEC reference cell 1 with a plasma generated in 2 mtorr of helium by a thermionic emitter placed 1 cm above the electrode as shown in Fig Measurements of the area marked interrogation area were made using Laser Collision Induced Fluorescence (LCIF) [62], providing a non-perturbative measurement of electron density in a 2D slice of the plasma-sheath interface. The LCIF measurement is based on exciting the 3 3 P metastable state of helium neutrals within a plane with a laser sheet and measuring the intensity of photons produced by collisional de-excitation of this state by electrons. This produces an estimate of the electron density since the de-excitation rate is determined by the electron velocity distribution function. LCIF measurements along a line perpendicular to the center of the electrode are shown in Fig. 2.2 for electrodes biased both above and below the plasma potential. 2D measurements are shown in Fig. 2.3 along side simulation results from the same work [3]. Similar simulations will be discussed in Chapter 2.3. Comparing the electron density profiles in Fig. 2.2, it is appearant that the gradients in electron density when strong (-5 V case) and weak ( V case) ion sheaths are present at the electrode di er significantly from the case when an electron sheath is present (15 V case). For all three cases, the plasma density decreases at a similar rate between distances of 25 mm and 35 mm. This density gradient is likely due to ambipolar di usion. The gradient continues to approximately 5-1 mm from the electrode in the ion sheath case. In this region, the ion presheath causes the rarefaction of the electron density. When an electron sheath is present, the shallowing of the density gradient beginning at 25mm suggests that the sheath influences the plasma over a much greater range than in the ion sheath case. Particle-in-cell (PIC) simulation also show similar behavior. Fig. 2.3 compares 2D maps of the simulated and experimental electron density. For the simulation data in Fig. 2.3, the arrows indicate the electron current density vectors. The direction of the electron current density indicates that the electron sheath induces electron flows in the quasineutral region in front of the electrode. This indicates the presence of a presheath region between the electron 1 A GEC reference cell is a plasma chamber with specific dimensions used to study RF processing plasmas. See Ref. [61] for details.

50 33 Figure 2.1: An illustration of experiments from Ref. [3]. The area marked Interrogation Area indicates the region in which LCIF was used to obtain electron density measurements. Figure 2.2: The electron density profiles from LCIF measurements. Strong and weak ion sheaths were present at the electrode for the cases marked - 5V and V, respectivly. For the case marked 15V, the electrode was biased above the plasma potential and an electron sheath was present, indicated by the non-zero electron density at the electrode.

51 34 Figure 2.3: Left: A comparison of 2D electron density maps from LCIF measurements and simulation for an electron sheath (bottom) and ion sheath (top). The vectors on the simulation plot indicate the electron current density. Right: The di erence between ion and electron density indicates the sheath edge position as a function of time. An FFT of the sheath position fluctuations indicates the frequency of oscillations. sheath and bulk plasma. Electron velocity distribution functions (EVDFs) from PIC simulations were plotted in the sheath and in the quasineutral region in front of the electrode and are shown in Fig To determine whether or not the EVDFs were in the sheath or plasma, the value of (n e n i )/n e was plotted perpendicular to the electrode surface to indicate the degree to which the plasma deviates from neutrality. This quantity, shown in Fig. 2.3, was used to determine that the EVDFs develop a flow moment in the quasineutral region approaching the sheath. This result di ers from the conventionally assumed truncated Maxwellian EVDF picture discussed in Chapter 1.7. There are also time dependent fluctuations in the sheath location indicated by (n e n i )/n e. This suggest the existence of an instability in the case of a positively biased electrode. Fluctuations on similar timescales are also suggested by the measurement of current fluctuations at small positively biased

52 35 Figure 2.4: EVDFs from a PIC simulation. The locations of the EVDFs are within the sheath, near the sheath edge, and in the presheath. The EVDFs at all three locations have a significant flow shift, a feature that di ers significantly from the conventionally assumed velocity space truncation. electrodes [63, 64]. An explanation for electron flow and sheath fluctuations in the vicinity of a positively biased electrode is the topic of this chapter. 2.2 Fluid Model In this section, a one-dimensional steady-state model in which the electrons are modeled using the fluid moment equations is developed. For the purposes of modeling the presheath and sheath edge, consider a model that describes electrons with continuity and momentum equations, assuming that the plasma is generated at a rate proportional to the density. For this situation Eq. (1.5) is d dy n ev e = s n e. (2.1) Ions are assumed to obey a Boltzmann density relation, n i = n o exp( e /T i ), where n o is the density in the bulk plasma. These equations are supplemented with Poisson s equation Eq. (1.1) and an isothermal closure for electrons. Since we are concerned with the presheath, the quasineutrality condition applies, and the density gradient can be written as dn e /dy = en i E/T i.insertingthisinto

53 36 the momentum equation Eq. (1.6) V e dv e dy = e m e E T e dn e m e n e dy V e ( R + s ) (2.2) shows that the pressure gradient term is T e /T i times larger than the electric field term. In Eq. (2.2), V e denotes the first moment of the EVDF, and R and s denote the collision frequencies due to momentum transfer collisions and the particle source rate, respectively. In this equation the moments of the velocity distribution function are assumed to be those of a flow shifted Maxwellian, f e (v) = n apple e 3/2 vt 3 exp (v V e ) 2 /vt 2, (2.3) so that the flow shift is V e, and pressure is P e = n e T e as assumed above. In typical low temperature plasmas T e /T i 1 1, hence the flow is dominantly pressure driven. This situation makes a significant contrast with ion sheaths, where instead the ion pressure gradient term is T i /T e 1 smaller than the electric field term. The analytic solutions to these equations for the sheath and presheath will be addressed in this section. This model can be used to determine a condition on the electron flow velocity at the sheath edge using the sheath criterion Eq. (1.48) discussed in Chapter For the electron sheath, consider a thin region near the sheath edge where the source and collision terms can be neglected. The electron continuity equation, along with Eq. (2.2) and the Boltzmann density relation for ions, then imply the following electron sheath analog of the Bohm criterion V e r Te + T i m e v eb. (2.4) A similar electron sheath Bohm criterion was previously found [65], but was not derived from consideration of the EVDF. The electron sheath Bohm speed in Eq. (2.4) is approximately p m i /m e greater than the ion sound speed. Due to the large electron flow velocity, the di erential flow between ions and electrons is expected to excite ion acoustic instabilities in the electron presheath. This will be studied in Chapter Next we will consider analytic solutions for the plasma parameter profiles in the presheath and sheath.

54 Presheath In this subsection the properties of the quasineutral presheath are explored. A mobility limited flow equation 2 is derived for the electron fluid. The equations for velocity and potential profiles are solved in a region in the vicinity of the sheath edge and analytic solutions are found for the cases of constant mean free path and constant collision frequency. The solutions demonstrate that large flow velocities are obtained over regions in which there is a small potential gradient. From these solutions it is found that in some cases the electron presheath may be expected to have an extent that is p m i /m e longer than that of an analogous ion presheath, but under more typical low temperature plasma conditions the presheath is predicted to be 6 times longer than the ion presheath. This means that the electron sheath can perturb the bulk plasma over a few centimeters under typical laboratory conditions. Assuming a quasineutral presheath, a Boltzmann density relation for ions can be inserted for the density in Eq. (2.1) and (2.2). Combining the resulting equations leads to an electron mobility limited flow equation, Ve V e = µ e 1 veb 2 E. (2.5) This equation is analogous to the ion mobility limited flow equation, but where µ e = e(1 + T e /T i )/[m e ( R +2 s )] is the electron mobility. When compared with the the ion mobility in an ion presheath with a common collision frequency due to volume ionization of neutrals, the electron mobility greatly exceeds ion mobility µ e Temi T im e µ i. Next, consider a region in the vicinity of the sheath edge that is thin enough that an assumption of constant flux, n e V e = n o v eb, is accurate. Here n o is the density at the sheath edge. Using this form of the electron density along with the Boltzmann density for ions in Poisson s equation gives 2De l 2 d 2 (e /T e ) d(y/l) 2 = e e /Ti where l is the presheath length scale. Taking the quasineutral limit as a function of flow velocity = T i e ln veb V e v eb, (2.6) V e D e /l! gives the potential. (2.7) 2 Mobility limited flow equations relate the species flow velocity to the mobility and the electric field strength. For the case of ion flow due to an electric field, the mobility limited flow equation is often written as V i = µ ie, where the ion mobility is µ i = 2e/( m i m)and m is the momentum transfer collision frequency [33].

55 38 This form of the potential along with the mobility limited flow in Eq. (2.5) results in a di erential equation for the flow velocity in terms of spatial position, dy = v2 eb Ve 2 dv e ( R + s )V 2. (2.8) The solution to this di erential equation along with Eq. (2.7) gives the flow and potential profile. This di erential equation has an ion sheath analog [33], which has analytic solutions [66] for 1) the case of constant mean free path, = V e /l, and 2) constant collision frequency, = v B /l. For the case of constant mean free path the flow velocity is e where W 1 is the V e 1 =exp v eb 2 y l + 1 apple 2 W 1 exp 2 y l 1 (2.9) 1 branch of the Lambert W function 3. Eq. (2.7) gives the potential profile e = y apple 1 1 T i l 2 2 W 1 exp 2 y 1. (2.1) l For the constant case the flow and potential profiles are s V e y = v eb 2l! 4l y (2.11) and e = arccosh 1 T i y. (2.12) 2l The flow velocity and potential profiles for these two cases are shown in Fig These show that large flows are obtained over regions with shallow potential gradients and little change in potential. Flow velocities of this magnitude are not seen in an ion presheath. For the case of constant collision frequency the characteristic length scale of electron and ion presheaths can be compared explicitly. Two cases are considered; A) a plasma where volume ionization is the dominant e ect, and B) a helium plasma with momentum transfer collisions and no volume ionization. A) If the dominant collision process is volume ionization, s is the same for ions and electrons. If the sheath attached to the electrode is an ion sheath, the presheath length scale is l i = c s /, while if the sheath were an electron sheath the presheath length scale is l e = v eb /. The ratio of these two length scales is l e = v r r eb Te + T i m i mi =. (2.13) l i c s m e T e m e 3 The Lambert W function is defined as the solution to W (z)exp W (z) = z. The solution for W is multivalued and has two branches denoted W and W 1. See Ref. [67] for details.

56 39 Ve l o c i ty V/veB Constant ν Constant mean free path Potential eφ/ti Distance y/l Figure 2.5: Electron presheath potential and velocity profiles for the constant collision frequency and constant mean free path models given in Eq. (2.9)-(2.12). Note that the potential is scaled by T i and has gradients which are much shallower and flow velocities that are much greater than those encountered in ion presheaths.

57 4 This suggests that the characteristic length scale of an electron presheath can be more than an order of magnitude longer than an ion presheath. A typical ion sheath length scale in low temperature plasma experiments is 1 cm [1], which means for the case of an argon plasma, where p m i /m e 27, the implied presheath length scale would be l e 27 cm. This is longer than the scale of many plasma experiments, so it would be expected that the presheath would fill approximately half the experiment length [68]. B) If the dominant collision process is momentum transfer, s = n g K s. The ratio of presheath length scales is l e = v r eb i mi K i (2.14) l i c s e m e K e where K s is the rate constant for collisions between neutral helium and species s. When the cross section of the collision process does not vary significantly over the thermal width of the distribution, the rate constant can be approximated as K s (U s ) U s s (U s ), (2.15) where U s is the flow shift of the distribution of species s. Using this approximation the rate constants were estimated using flow speeds representative of typical presheath velocities, U = v eb /2 for the electron presheath and U = c s /2 for the ion presheath. For the calculation of K e the total momentum cross section for e + He collisions was obtained from LXcat [4], while for the calculation of K i the cross section for He + + He elastic and charge exchange collisions were considered [69]. For the He + + He cross sections the values at 4 ev were extrapolated to ev as has been previously done [7], this was due to a lack of experimental data within this range of energies. The cross sections used are shown in Fig The ratio of presheath length scales shown in Fig. 2.7 suggest that in this case the electron presheath is approximately six times longer than the ion presheath. These values are in good agreement with the estimated presheath lengths determined from density measurements of Ref. [3], presented in Fig. 2.2, where the electron and ion presheaths were measured to be approximately 25 mm and 6 mm respectively Sheath For the electron sheath, the sheath-presheath transition is a region where the flow switches from being pressure driven to electric field driven. In the thin sheath region, the collision and source

58 41 Figure 2.6: Data for the neutral helium-electron momentum scattering and the neutral helium-helium ion charge exchange and elastic cross sections used for the calculation of the rate constants K i and K e.thehelium-helium ion cross section data was unavailable at low energy and was extrapolated from the value at the lowest known energy. Figure 2.7: Calculated ratios of the electron and ion presheath length scales assuming the dominant collision processes in ion and electron sheaths are helium-helium ion collisions and electron-neutral scattering, respectively.

59 42 terms can be neglected 4. This provides a way to determine a relation between the flow velocity and potential, which shows that at small potentials the pressure represents a significant correction to the electron ballistic motion, while at high potentials the 3/4 power law scaling of the Child-Langmuir law [36] is recovered. Under the assumptions mentioned above, combining the continuity and momentum equations, integrating and matching the sheath edge condition for the flow velocity results in 2 Ve Ve 2ln = 2e +1. (2.16) v eb v eb T e The second term on the left hand side is the contribution due to the electron pressure. The solution to this equation can be written in terms of the Lambert W function. The solution to an equation of the form y 2 2ln(y) =z (2.17) can be written in terms of the Lambert W function as 1 y =exp 2 W 1 e z z 2. (2.18) For the electron sheath problem the asymptotic limit as z!1is of interest. For this limit the W (z) branch provides unphysical solutions because W () = and an accelerating flow velocity cannot correspond to y!. However, the W 1 branch does represent a physical solution. The asymptotic limit of the W 1 (z) branch as z! is given by Ref. [67] as ln( ln( z)) W 1 (z) =ln( z) ln( ln( z)) + O. (2.19) ln( x) Using the asymptotic limit in the solution of Eq. (2.17) gives y = p z. This gives asymptotic limit of Eq. (2.16), r V e 2e = +1, (2.2) v eb T e which is the same result obtained if the logarithmic term in Eq. (2.16) were dropped. To quantify the error involved in this approximation Eq. (2.17) is plotted against p z in Fig. 2.8, it is apparent that the error is 2% at small z and decreases at large z. The logarithmic electron pressure term in Eq. (2.16) is at most a correction of 2% in the sheath and drops o at higher flow velocity. Using the asymptotic solution for V e, enforcing that 4 The physical justification for dropping the collision term is that the sheath is so thin that electrons will flow to the boundary without su ering a collision. More specifically, the sheath scale is much smaller than the collisional mean free path.

60 f (z )=exp( W 1 ( e z )/2 z/2) f (z )= z f(z) z Figure 2.8: The exact solution to Eq. (2.17) (solid line) compared to its asymptotic limit (dashed line). The di erence between the asymptotic limit and exact solution is at greatest 2%. % difference z the electron density within the sheath obeys flux conservation (n e ( )V e ( )=n o v eb ), and neglecting the ion density, which decreases exponentially with increasing potential, Poisson s equation can be written as d 2 dy 2 = 4 en o. (2.21) q1+ 2eTe Integrating twice with respect to y gives y flowing e =.79 D e T e 3/4 (2.22) which is the same as what is obtained for an ion sheath [57]. A di erent relation for the electron sheath has been previously given 5 where the sheath scaling was given as 3/4 y truncated e =.32. (2.23) D e T e This di erent numerical factor is due to the random flux assumption. Comparing Eq. (2.22) and 5 The numerical factor in Ref. [57] is.32/ p where is a correction factor, greater than unity, due to a dip in front of the sheath reducing the density at the sheath edge. This dip is not observed in the simulations and the value of = 1 is used here.

61 44 Eq. (2.23) gives the correction to the sheath scale y flowing y truncated =2.47, (2.24) which suggests that the electron sheath is more than twice as thick as previously thought. In Chapter this relation is shown to be in excellent agreement with simulations. 2.3 Simulations of the Electron Sheath The model in the previous section assumed a 1D planar electron sheath. In this section the model is tested using 2D PIC simulations. These show that the electron sheath has some inherently 2D features that are not accurately captured by the model. In particular, the ion density is found to be determined by a 2D ion flow velocity profile around the electron sheath. Nevertheless, basic features of the 1D model, such as the minimum electron flow speed at the sheath edge, are found to agree with the simulation results. Modifications of the 1D theory to address 2D ion flow are found to lead to improvements in the predicted presheath profiles. The PIC simulations also exhibit fluctuations in sheath thickness. In Chapter 2.3.4, evidence is shown that these fluctuations are ion acoustic waves excited in the presheath Simulation with Aleph Simulations were performed using the code Aleph, which was described in Chapter 1.7. In the simulations discussed in this section, elastic collisions between ions and a 1mTorr helium background were included. The cross section for this interaction were obtained using the variable hard sphere model given by Eq. (1.82). To simulate the electron sheath, a 2D domain with triangular elements with a scale size of approximately.7 De was utilized. The simulation domain, which is shown in Fig. 2.9, included three walls with a Dirichlet V= boundary condition and one boundary with a Neumann boundary condition. The walls with the Dirichlet condition were absorbing for particles, while the Neumann boundary was reflecting for particles. This choice of boundary conditions for fields and particles results in a symmetry condition so that the 7.5 cm by 5 cm domain models half of a larger symmetric 15 cm by 5 cm system. An electrode of length.25 cm was embedded in the lower boundary, and was separated from the rest of the wall by a.2 cm gap of dielectric. The dielectric was meant to reduce the electric field between the edge of the electrode and the wall in which it is embedded. The electrode was biased at 2V and -2V in two di erent simulations. The ratio of the electrode area to wall area was.13 so that Eq. (1.33) was satisfied and an electron sheath was allowed to form

62 45 Distance (cm) Plasma Source Region Reflecting Boundary Electrode V=!! Dielectric!! Distance (cm) T i (K) Figure 2.9: A schematic of the simulation domain overlaid on a color map of the ion temperature for the 2 V simulation. Note the electrode position in the lower left corner. The region of zero temperature is due to the absence of ions in the electron sheath. in the case of the 2 V bias. A helium plasma was sourced in a.25 cm by 6.5 cm strip placed 3.75 cm above the electrode. Within this region an equal number of ions and electrons were created at a rate of cm 3 µs 1 throughout the simulation. Electrons, ions, and neutrals had macroparticle weights of 2 1 3, , and (meaning these are the number of physical particles represented by each computational particle). Particles sourced in this region were chosen from distributions with temperatures of 4 ev for electrons and.86 ev for ions. The plasma expanded and filled the domain, with a steady density being reached on the timescale of an ion crossing time, which was approximately 3 µs. Once a steady plasma density was reached, the 2D temperature, which is the relevant temperature for comparison with the theory, was evaluated in the plasma near the sheath edge by evaluating T e = m Z e 2n e d 2 v(v 2 r,x + v 2 r,y)f e (v), (2.25) where v r,i =(v V e ) î. Using this, the electron and ion temperatures were 1.64 ev and.48 ev, resulting in an electron sheath Bohm speed of v eb = 54.4 cm s 1.

63 46 The simulations used a 1 ps time step to resolve both the plasma frequency and electron trajectories. The resolution of electron trajectories (see Chapter 1.5.2) was the primary constraint on the time step. The simulations ran for time steps resulting in 5 µs of simulated physical time. Fig. 2.1 shows plots of the current vectors for both electrons and ions within the vicinity of the ion sheath from the -2 V case and an electron sheath from the 2 V case. Comparing these two cases, it is immediately apparent that there is a di erence in the behavior of the particles attracted by each electrode. In the -2 V case where an ion sheath is present, the ion flow is only significantly modified within.25 cm of the electrode. Within this region the potential leads to a focusing of the ion current. For the 2 V electron sheath case, the boundary can be seen influencing the electron flow, converging into the electrode from distances more than 1 cm away. The di erences between these two situations can also be seen in the repelled populations. For the electrons repelled by the ion sheath there is no visible flow, this is consistent with the commonly assumed Boltzmann density profile for the repelled population. In contrast, ions near the electron sheath have a significant flow and can be seen flowing around the electron sheath to the adjacent wall biased at V=. This highlights one major di erence, a detailed description of the ions near the electron sheath requires a 2D treatment, this will be revisited in Chapter A detailed analysis of the electron flow is provided in the following subsection Electron Fluid The simulations have shown that the electron sheath interfaces with the bulk plasma through a presheath. In this subsection simulations are compared to the presheath description given in the 1D model. Fig shows a comparison of the potential profile from the electron sheath PIC simulations and the models given for the presheath and sheath. For the presheath, the models for constant mean free path given in Eq. (2.1) and constant collision frequency in Eq. (2.12) were compared by fixing the value = at the location where the electron sheath Bohm speed is attained, from here moving out some distance y into the plasma the potential profile was plotted. In the sheath, starting at the electrode, the sheath thickness as a function of potential from Eq. (2.22) was plotted for an argument = E (y) from (y) = E out to (y) =, the potential at which the electron sheath Bohm speed was attained. Here E is the electrode potential. For comparison, the conventional model from Eq. (2.23) is also plotted. The potential profiles within the sheath are in excellent agreement with Eq. (2.22) indicating that the numerical factor corresponding to the flowing Maxwellian is the

64 47 [ n i V i (2x1 1 cm 2 µs 1 )] [ n i V i (2x1 1 cm 2 µs 1 )] Distance (cm) Distance (cm) 1.5 Electron Sheath +2V Ion Sheath 2V Distance (cm) [ n e V e (1 11 cm 2 µs 1 )] [ n e V e (1 11 cm 2 µs 1 )] 1.5 Electron Sheath +2V Ion Sheath 2V Distance (cm) Potential (V) [ n i V i (2x1 1 cm 2 µs 1 )] [ n e V e (1 11 cm 2 µs 1 )] 2 Figure 2.1: The current flow vectors plotted on top of the potential for n i V i (left column) and n e V e (right column) for an electron sheath (top row) biased +2 V and ion sheath (bottom row) biased -2 V. The electrode is between x = and x =.25 cm, see annotations on Fig. 2.9 for more details of the simulation domain. All potentials are measured relative to the grounded wall. The greatest di erences between these are that the electron presheath has a much greater e ect on the electrons in the bulk plasma than the ion presheath has on the ions, and that the electron presheath redirects the ions, while the ion presheath has little e ect on the electrons. Note the di erence in scale for vectors in the left and right columns.

65 48 Potential (V) 1 5 PIC Constant ν Presheath Constant λ mfp presheath Flowing Maxwellian sheath Random flux sheath Ve l o c i ty V/veB Distance (cm) Figure 2.11: Top: The potential profile from the PIC simulation compared the constant collision frequency and constant mean free path presheath models given by Eq. (2.1) and Eq. (2.12). The sheath models for flowing and truncated Maxwellian given in Eq. (2.22) and Eq. (2.23) are also shown. Bottom: The electron flow moment from simulation compared to those calculated from the potential models in the upper panel. correct value. This result is significant since it indicates that the electron sheath is approximately twice as thick as was previously thought under the random flux assumption. The presheath potential profiles are plotted with a presheath length scale of l =.3 cm, which approximately corresponds to the region in which Eq. (2.28) accurately describes the ion density in Fig (See Chapter ), as well as the region in Fig. 2.12a where the pressure gradient dominates over the electric field. The presheath potential profiles from the theory were shallower than that from the simulations near the sheath, although the slopes are in better agreement further away. This is possibly due to matching the simulation data at the theory s singular point. Simulation results only match the theory in a region where the electron presheath is dominant; however, the model does not consider the interface of the presheath with a nonuniform bulk plasma such as the one in the simulations. The flow profiles of Eq. (2.9) and Eq. (2.11) are also compared to the simulations in Fig The flow profiles show that the sheath and presheath are in good agreement with theory. Due

66 49 to presheath ion density fluctuations (see Fig. 2.3 and Fig. 2.21), the sheath edge is di cult to locate in the time averaged simulation data. To compare the electron flow velocity at the sheath edge two definitions are utilized. The first is the sheath thickness given by Eq. (2.22) for = ( cm) (1 cm), and the second is the location where the average di erence between electrons and ions, 2(n e n i )/(n e +n i ) is greater than 3%. The chosen value of 3% corresponds well with the typical sheath edge position in the time dependent data in Fig shown in Chapter By these two definitions the sheath edge is between.213 cm and.265 cm, the corresponding flow velocities are 1.21v eb and.85v eb. Fig. 2.12a shows the ion and electron density, while Fig. 2.12b shows the corresponding terms in the electron momentum equation. Here the two dashed lines indicate the two sheath edge locations. In the region bounded by these two sheath edge definitions, the electric field overtakes the pressure gradient and the sheath begins. The location at which the electric field becomes the dominant driving term in the electron momentum equation closely coincides with the location at which the electron sheath Bohm velocity is achieved. In Chapter 2.2.1, a comparison of presheath length scales was made for sheaths dominated by a common source of plasma generation between electrons and ions and no other collisions. For this situation it was concluded in Eq. (2.13) that the electron presheath was p m i /m e longer than the ion presheath. In the simulations, no particles are sourced in the presheath so the dominant mechanism for determining the electron presheath length scale is expected to be electron-ion collisions. As discussed in Chapter 1.5.4, the PIC simulations the electrons are nearly collisionless in the Coulomb collision sense since electron-particle interactions are not resolved within the computational cells. Another possible mechanism for such collisions are those due to particle wave interactions in an unstable plasma [49]. The collision rate due to electron interactions with ion acoustic waves has been important for explaining the anomalous scattering of electrons near the ion sheath [71], a phenomenon known as Langmuir s Paradox [72]. The EVDFs in Fig. 2.4 suggest a similar anomalous scattering mechanism may be important here since at the sheath edge and within the sheath electrons with velocities directed towards the bulk plasma are still present. The PIC simulations can capture particle-wave collisions if the dielectric response in resolved by the spatial grid, see Appendix A for a detailed discussion. Evidence for the presence of electron collisions can be obtained by adding up the terms in the momentum equation which are calculated from the simulations. In fact, one can see that the terms in Fig. 2.12a do not exactly cancel. Using PIC plasma quantities, the residual dv e R e = V e dy + e E + T e dn e m e m e n e dy (2.26)

67 5 cm/s a) E Field Term Pressure Gradient Term Y Flow Term Residual b) Distance (cm) V e /v eb n i (1 8 cm 3 ) n e (1 8 cm 3 ) Distance (cm) Figure 2.12: a) The magnitude of terms of the momentum equation on the right hand side of Eq. (2.26) are evaluated from quantities computed in the PIC simulation. The residual is determined by summing the terms. b) The electron flow moment, ion density, and electron density. The two vertical lines correspond to the two estimates of the sheath edge. is plotted by summing up the other terms. An increase in the residual as the electrode is approached suggests that other neglected terms, (i.e. stress gradients, perpendicular velocity gradients, and friction) may be important. In particular a friction term may be due to wave particle interactions and could play an important role in determining the presheath length scale since it would determine the value of R in Eq. (2.8). Instabilities will be discussed further in Chapter Ion Behavior The plots of the ion current near an electron sheath shown in Fig. 2.1 indicate that the ions flow towards the electrode and are redirected around to the adjacent ion sheath boundary when they encounter the strong positive potential of the electron sheath. One curious feature of this behavior is why the ions flow towards the positively biased electrode at all. In this subsection, the flow of ions within the electron presheath will be explored in detail using quantities from the 2 V

68 51 PIC simulation discussed previously. Additional simulations will be utilized to provide a more clear explanation of the ion behavior Ion Density Plots of the ion current in Fig. 2.1 show that ions flow around the electrode and are collected by the adjacent wall. Here the ion density will clearly be dominated by the flow profile around the electrode. This flow is a 2D e ect that is absent in the description of ion sheaths near planar boundaries. Size limitations on the electron sheath from global current balance prevent it from being well described by an infinite 1D planar geometry. To model how the 2D flow a ects the ion density profile, consider the 2D steady state ion momentum equation along a 1D cut perpendicular to the electrode center, dv y m i n i V x dx + V dv y d y = en i dy dy d dy (n it i ). (2.27) Here, the stress gradient and friction terms have been neglected. In Chapter 2.2 it was found that the electron presheath has weak potential gradients. Dropping the electric field term and integrating from the sheath edge back into the presheath results in apple n i (y) n i (y o ) =exp Z y y o m i dv y V x T i dx + V dv y y dy, (2.28) dy where y o denotes the sheath edge position. In this form, the ion flow is balanced by the pressure gradient. This can be contrast with the Boltzmann relation where the electric field and pressure gradient balance. The exact form of the pressure gradient is dependent on the electric field, after all it is the field that causes the density gradient. Determining the exact pressure gradient would involve solving the full 2D momentum equation with Poisson s equation using all the boundary conditions. For this section, numerical values from the PIC simulation are used to test the relation in Eq. (2.28). Fig shows the presheath densities from simulation compared to the evaluation of Eq. (2.28). These two quantities are in good agreement. For comparison, the Boltzmann relation is also shown in Fig. 2.13, demonstrating that Eq. (2.28) is a vast improvement in the description of the ion density Ion VDFs The e ect of the electron sheath on ions can also be explored from a kinetic point of view. Fig. 2.9 shows ion heating in the electron presheath. This heating is explained as a result of ion interaction with the presheath. This interaction generates a flow moment in the ion VDFs (IVDFs) in the

69 52 Density (1 8 cm 3 ) Ion Density Model Electron Density Boltzmann Density Distance (cm) Figure 2.13: Evaluation of the ion density model given by Eq. (2.28) using PIC quantities. The model compares well with the simulated presheath density and is a significant improvement over the Boltzmann density profile. transverse direction when approaching the electrode. The 2D IVDFs are shown in Fig These demonstrate that the majority of ions are redirected away from the boundary and collected by the adjacent grounded wall. This redirection of particles distorts the VDF away from a Maxwellian, this increases the temperature moment and is primarily responsible for the observed heating, although there is also a small population of ions that are reflected back into the plasma. The 2D IVDFs in Fig were plotted for locations in the presheath using individual particle positions and velocities over 3 µs and were averaged over.1 cm.1 cm boxes starting at the sheath edge around.25 cm moving back into the plasma.85 cm. The averaging boxes also extend the length of the electrode in the x direction, with the last box including the electrode wall boundary. Far from the electrode the IVDFs are flow shifted towards the boundary, as would be expected for an expanding plasma, and show little modification apart from a small population of reflected ions. As the ions approach the electrode some of their flow velocity is diverted from the -y to x direction since the ions are repelled by the 2D presheath electric field. The modification of the IVDF shape near the electrode can be described by the flow around the electrode. Consider the IVDFs halfway between the plasma source and the boundary containing the electrode, each starting at three di erent locations in the x direction, see the location marked A in Fig. 2.15a. At the starting location each IVDF will have a flow due to the plasma expansion, so

70 53!"(#$%&$!")#$%&$!"*#$%&$!"##$%&$!"+#$%&$!",#$%&$!"-#$%&$./ '! 1$2345%67$89&:74;$!"!#$%&$!"'#$%&$!"(#$%&$ Figure 2.14: IVDFs near the electron sheath biased at +2V. The IVDFs were averaged over.1 cm x.1cm boxes. The labels in the x and y axes indicate the coordinate of the center of the box, the electrode is on the x axis at y= between x= and.25 cm (averaging starts.2 cm above the electrode), since further below there are not enough ions for meaningful IVDFs.

71 54 +:),) -).) ;:) 7 9 ) $:) /) 7 8 ),6-6.) 1)234%+5$#) )!"#$%&'(#) *+"") <72=)>'$+?'5)/) 7 9 ) -).) <72=)>'$+?'5)), 7 8 (:) 7 9 ) 7 8 ) Figure 2.15: Schematic drawing describing the time-averaged IVDFs at di erent locations in the plasma. a) The flow lines of particles in 3 di erent VDFs starting at A and ending at B. b) The VDFs at location A. c) The VDFs at location B, distributions incident on the electrode have their -y velocity redirected in the x direction. d) A realistic IVDF at location B due to a continuum of starting positions along line A. the distribution will have a flow shift in the direction of the electrode or wall, this is represented in Fig. 2.15b. Due to the flow around the electrode each of these distributions will end at the location marked B in Fig. 2.15a. Now consider the distribution with flow incident on the electrode. Since the flow is redirected the flow shift of this distribution will be transferred from the -y direction to the x direction as it approaches the electrode. Likewise, a distribution incident on the edge of the anode will also have its flow diverted from the -y direction to the x direction, although to a lesser extent. Finally, a distribution incident on the grounded wall will remain unchanged. The final position of these three distributions is shown in Fig. 2.15c, although a more realistic expectation would be smeared out, such as the distribution shown in Fig. 2.15d, due to a continuum of starting positions. The basic expectations of the model shown in Fig. 2.15d are borne out in the simulated IVDFs near the boundary in Fig It is important to note that the physical picture illustrated in Fig is not exact since not every particle flows along a stream line, but experiences di usion as well. There are small scale features not explained by the picture in Fig For instance, in

72 55 some IVDFs there is a small secondary maximum to the right of the primary. This situation may be due to time averaging of the particle positions and velocities over 3 µs in combination with fluctuations in the presheath caused by instabilities Ion Flow Two additional simulations were carried out to better understand the nature of the ion flow near electron sheaths. The simulation domain in each case was set up so that the electrode would be placed in the center of the domain, avoiding areas where a flow might result from a density gradient such as the one between the plasma source region and the electrode in the simulations discussed so far. This type of density gradient is also present in the experimental data presented in Fig Avoiding the density gradient was achieved by sourcing the plasma at a constant rate within each cell and placing the electrode within a plane of symmetry for the domain. Two di erent electrode geometries are shown in the simulation schematic in Fig The first electrode is the free electrode. In this case, a.2 cm electrode was biased at 25 V except for a.1 cm region on the back which was biased at V. This was meant to resemble a positively biased electrode with a small amount of dielectric backing. The second electrode is the embedded electrode. In this case, a.2 cm on the front of a.8 cm electrode is biased at 25V while the rest is biased at V except for a.2 cm gap between the 25 V and V surface. This geometry is meant to represent a small electrode embedded in dielectric material. Both simulations were meant to resemble the experiments described later in this chapter. The rest of the simulation setup is identical to those discussed in Chapter 3.1 and a discussion of these details will be deferred. IVDFs for several di erent positions in front of each electrode are plotted in Fig For the case of the embedded electrode, the behavior is similar to that of the IVDFs of Fig These exhibit the same redirection of ions, leading to an increase in the temperature moment. The case of the free electrode is significantly di erent. The distribution is nearly Maxwellian with no notable flow moment within the electron presheath. In this case, the Boltzmann density profile is expected to provide a reasonable model for ions. The absence of ion flow in the small electrode case suggests that the presence of dielectric on the front of the electrode leads to the electrode-directed ion flow. It is not surprising that this could happen. Within the electron presheath the potential gradients are weak and the main mechanism for the acceleration of electrons is a pressure gradient. It is possible to have a weak electric field overlapping with the electron acceleration region, accelerating ions tangentially to the adjacent ion sheath.

73 56 (a) cm Reflecting Boundary Free Electrode Model V Gap +25V V 5 cm -7.5 cm cm 7.5 cm Potential (V) (b) cm Reflecting Boundary Embedded Electrode Model V +25V V Gap V 5 cm -7.5 cm cm 7.5 cm Potential (V) Figure 2.16: A schematic of the simulation domain for two cases: a) The small electrode model and b) the embedded electrode model. The color map of each domain indicates the plasma potential.

74 57 (a) A 2D IVDF s Near Model Embedded Electrode B C D E 4 IVDF Spatial Locations (Embedded Electrode) 1. A B C D E 4 VAxial [cs] (b) VRadial [cs] A 2D IVDF s Near Model Free Electrode B C D Number of Macroparticles E Axial Position [cm] Electrode +25V Gap V Radial Position [cm] IVDF Spatial Locations (Free Electrode) 3 A B C D E VAxial [cs] VRadial [cs] Number of Macroparticles V.4.2 Axial Position [cm].6 +25V Electrode.2.4 Gap Radial Position [cm] Figure 2.17: IVDFs at several locations near and in the electron presheath from simulations of the embedded electrode (a) and small electrode (b). The location of each IVDF relative to the electrode position is indicated by the plots on the right.

75 58 (a) Emissive Probe Magnets 73 cm Langmuir Probe Chamber cm Fixed LIF Viewing Volume (Split Into 16 Chan.) 1 mm (b) Laser e - e - e Hot - Cathode Electrode (+1 Vdc) 49 cm Electrode Moves Up and Down Z Electrode.75 mm LIF Optics Box Dielectric Conductor (c) Beam Splitter 1.4 cm 1.9 cm 1.9 cm Slit Interference Filter 1.4 cm 32-Channel Photon-Counting System 16-Channel Photomultiplier Tube Free Electrode Embedded Electrode Figure 2.18: A schematic for the experiment used to measure the IVDFs in the electron presheath. The behavior of ions in the electron presheath has also been verified in experimental measurements of IVDFs using laser induced fluorescence (LIF) [73]. A schematic of the experiment used to measure the IVDF is shown in Fig The measurements were made in a multidipole chamber with a plasma density and electron temperature of approximately cm 3 and T e =1.2 ev. The plasma was created in.4 mtorr of Argon gas using primary electrons emitted from a hot cathode. At this pressure, the nearly uniform electron impact ionization of neutral gas throughout the chamber resembled the volume plasma source in the simulation. Measurements of the ion velocity distribution function in front of free and embedded electrodes (shown in Fig. 2.18c) were taken in a viewing volume across the radius of the electrode. The viewing volume was spanned by 16 channels of a photomultiplier tube (PMT). Measurements at di erent vertical distances (z in Fig. 2.18b) were made by repositioning the electrode relative to the viewing optics. The measured IVDFs, averaged over all 16 PMT channels, are shown in Fig See Ref. [73] for details of the experiment. The measured IVDFs shown in Fig behave in the same way as the those in the simulation. In the case of the free electrode, the ions have a Maxwellian velocity distribution in both the axial and radial direction. In the case of the embedded electrode, the IVDFs become flow shifted in

76 59 567&89:%$+;#(&-"3;3/ 567&89:%$+;#(&-"3;3/ 23> 23= ' C"#$"% &-*. / 23< 2 23> 23<!"#$"%&'()*&+,-./##/#&,%/123#/ !"#$"%&'(%)*$+,&-*. / 56$"%&'()*&+,-./##/#&,%/123#/ = ' "?$"% &-*. / D).3&-*9/ D).3&-*9/ 567&89:%$+;#(&-"3;3/ 567&89:%$+;#(&-"3;3/ 23> 23= ' C"#$"% &-*. / 23< 2 23> 23= 23<!"#$"%&'()*&+*2//&,%/123#/ !"#$"%&'(%)*$+,&-*. / 56$"%&'()*&+*2//&,%/123#/ ' "?$"% &-*. / D).3&-*9/ D).3&-*9/ 8?$"%&D).$+$)E -*9/ 1432 A32 43B =32 <3A 13A 13< 23@ 23A B ?$"%&'(%)*$+,&-*. / ?$"%&'(%)*$+,&-*. / Figure 2.19: Laser induced fluorescence measurements of IVDFs in the electron presheath near free and embedded electrodes. the electrode direction as the electrode is approached and then develop a flow in the radial direction close to the electrode Fluctuations and Instabilities The simulated electron sheath, shown in the 1 cm 1 cm regions plotted in the panels of Fig. 2.2, exhibits fluctuations of the sheath edge position on the order of.5 cm on a time scale of approximately 1 µs. Fluctuations were not observed for the ion sheath with the electrode biased at -2 V. The presence of a di erential flow, approaching the electron thermal speed, between electrons and ions in the electron presheath is expected to give rise to ion-acoustic instabilities. In this subsection, the e ect of these waves on the sheath fluctuations is explored. Two-dimensional FFTs of the ion density confirm that the sheath fluctuations are due to ion acoustic waves in the vicinity of the sheath edge.

77 6 The dielectric response for a plasma where the electrons are Maxwellian with flow V e and stationary Maxwellian ions is (k,!)=1! pe 2! k 2 vt 2 Z e k Ve kv Te! pi 2! k 2 vt 2 Z. (2.29) i kv Ti Here, Z is the derivative of the plasma dispersion function [74], where Z 1 Z( ) = 1/2 dt exp t 2 1 t, Im( ) > (2.3) and Z ( ) = 2 1+ Z( ). (2.31) If the ions are assumed to be stationary, the electron flow velocity relative to the background is between and v Te. In this situation the electron flow is expected to result in a two-stream instability, exciting ion-acoustic waves. Using this expectation, in the ion frame! kv Ti c r s Te = 1. (2.32) v Ti T i Near the sheath edge the electron velocity satisfies c s V e v Te so that! k V e and! k V e kv Te k V e kv Te. (2.33) From Eq. (2.32) it is clear that the large argument expansion for the derivative of the plasma dispersion function [75] can be used for the ion term, Z ( ) 2i p exp( 2 ) (2.34) Here, is a numerical factor that results from the analytic continuation of the plasma dispersion function in Eq. (2.3), and is zero for Im( ) >. Keeping the first term of this expansion, the dielectric response is (k,!) 1! pe 2 k 2 vt 2 Z e k Ve kv Te Setting this equal to zero gives the dispersion relation, which is! 2 pi! 2. (2.35)!! pe s k 2 2 De k De 1 2 Z. (2.36) V e v Te Fig shows the ion density along a line extending 1 cm perpendicular to the electrode over a 5 µs interval for the domain shown in Fig The figure shows that there are ion density fluctuations that propagate towards the sheath edge as time increases. The figure also shows that

78 61!"#"$%&$!"#'$%&$!(#"$%&$!(#'$%&$!)#"$%&$!)#'$%&$ Figure 2.2: Time evolution of the charge density in a 1 cm by 1 cm region near the electron sheath shown at.5 µs time increments. The color indicates the charge density with red being electron rich and blue being ion rich. the sheath edge position fluctuations closely follow the propagation of the ion density fluctuations, suggesting that these are likely responsible for the sheath edge fluctuations and resulting current fluctuations which are associated with positively biased probes [63, 64, 17]. The 2D FFT of the ion density shown in Fig was computed over a line extending 1 cm from the electrode. The FFTs are in fair agreement with the expected dispersion relation determined from Eq. (2.36) indicating that the density disturbances, which are responsible for the sheath edge fluctuations, are in fact ion acoustic waves. The figure also indicates that nonlinear e ects may be producing a cascade to shorter scales. 2.4 Summary In this chapter the conventional picture that the electron sheath collects a random flux of electrons was shown to be incomplete. Based on the EVDFs of 2D PIC simulations, a model was developed using the electron momentum and continuity equations where the EVDF is a flowing Maxwellian. In this model the electron sheath interacts with the bulk plasma through a presheath

79 62 Figure 2.21: Top: The absolute di erence between ion and electron density along a line perpendicular to the electrode surface over 5µs. Bottom: The corresponding ion density over the same time period. The ion density fluctuations coincide with the fluctuations of the sheath edge position. Figure 2.22: The 2D FFT of the ion density shown in Fig The solid and dashed red lines correspond to the real part of the dispersion relation given in Eq. (2.36) for flow velocities of.5v eb and.9v eb. The solid and dashed yellow lines correspond to the imaginary part, i.e. the growth rate, of the same equation.

80 63 where the electron velocity approaches the electron sheath Bohm speed, p (T e + T i )/m e. In this presheath there are shallow potential gradients that drive a large pressure gradient. It is this pressure gradient that is primarily responsible for the acceleration of electrons. The 1D model was compared to the 2D simulations using the time averaged values from the simulation. Within the sheath the potential profiles and flow velocities are in excellent agreement with the flowing Maxwellian model, which results in an electron sheath that is approximately twice as thick as the one described by the commonly assumed random flux model. The simulations are consistent with the electron flow velocity attaining the electron sheath Bohm speed by the sheath edge, and this flow velocity was verified to be the result of acceleration in a pressure driven electron presheath. Comparison with the simulations also revealed the inherent 2D nature of the electron sheath. Due to it s small size, the electron presheath does not resemble the presheath of an infinite planar boundary, instead there is a divergence of the ion flow around the sheath-presheath region. This flow necessitates a new description of ions where the ion flow is balanced by the presheath pressure gradients. Finally, the simulations revealed the existence of ion density fluctuations in the electron presheath. These density fluctuations are expected; the theory predicts a large di erential flow between ions and electrons in the presheath which excite ion acoustic instabilities. FFTs of the 2D ion density indicate that these density fluctuations are ion acoustic waves. Inspection of the sheath edge position revealed that these ion acoustic waves are responsible for sheath edge fluctuations, and hence sheath collection area fluctuations, which in turn cause fluctuations in the collected electron saturation current.

81 64 CHAPTER 3 EVDFS AND THE ION TO ELECTRON SHEATH TRANSITION In this chapter, the form of the sheath near a small electrode with bias changing from below to above the plasma potential is studied. This situation is particularly relevant to understanding how small electrodes, such as Langmuir probes, locally perturb the bulk plasma under conditions where they are biased close to the plasma potential. Like Chapter 2, 2D particle-in-cell simulations will be used to study the sheath under these conditions. In the previous chapter, a model for the electron sheath was presented, based on 1D projections of the EVDFs. In this section, the behavior of the electron density and flow near an electrode biased at or slightly above the plasma potential is found to have some dependence on the electrode geometry. The 2D EVDFs are found to have a loss-cone-like truncation that dominate the presheath behavior when the electrode is biased near the plasma potential. The 2D picture shows details of the electron presheath EVDF that are masked by 1D projections. When the electrode is biased above the plasma potential, the 2D EVDFs have a loss-cone-like truncation in addition to the flow shift described in the 1D theory of the previous chapter. 3.1 Simulations Simulations were carried out using the PIC-DSMC code Aleph, which was introduced in Chapter 2.3.1, using PIC-only mode by only including helium plasma species and ignoring interactions with the background gas that occur in real laboratory plasmas. This assumption means that the results are limited to pressures at which the mean free path for collisional processes are longer than the presheath length scale. This pressure can be calculated by considering the mean free path, interaction. =1/n, in combination with P = nk B T and the cross section data for the dominant Assuming that the electron presheath length scale is 7.5 cm (half of the domain length) in a helium plasma where the dominant cross section for momentum transfer of electrons is elastic electron-neutral collisions, this pressure is approximately 2 mtorr. A 15 cm by 5 cm simulation domain, shown in Fig. 3.1, was discretized with an unstructured triangular mesh of scale.2 cm. The domain had three boundaries with a Dirichlet V= boundary condition and for particles, = boundary condition. The V= boundaries were absorbing = boundary was reflecting. This combination of Neumann condition for the potential and reflecting boundary condition for particles means that the 15 cm by 5 cm

82 65 a)! cm Reflecting Boundary Electrode cm V= #"!" -5 cm -7.5 cm cm 7.5 cm b)! Ion Density (cm -3 ) Reflecting Boundary! Electrode! cm!.1 cm!.2 cm!.3 cm!.7 cm!.8 cm! x! Figure 3.1: a) The 5 cm by 15 cm simulation domain with a color map indicating the ion density of simulation A overlaid. Note the small electrode placed perpendicular to the reflecting boundary. b) Sampling areas for obtaining EVDFs in front of the electrode. domain represents half of a symmetric 15 cm by 1 cm system. A.1 cm by.2 cm electrode was placed perpendicular to the reflecting boundary and was biased at di erent values for each of the five simulations labeled A-E in Table 3.1. These values of the bias were chosen to achieve di erent values relative to the plasma potential. The helium plasma was sourced at a rate of 1 8 cm 3 µs 1 throughout the volume at a temperature of 4 ev for electrons and.8 ev for ions. A macroparticle weight of 2 was chosen for each species. Each simulation ran for 8, timesteps, of 5 ps each, for a total duration of 4 µs of physical time. After approximately 35 µs, the simulations achieved a steady density. The ion

83 66 Simulation p E T e Desired criteria (V) (V) (ev) for E p A E p < T e /2e B > E p > T e /2e C E p D T e /2e > E p >T i /2e E E p >T e /2e Table 3.1: Summary of simulation parameters density for simulation A is shown in the schematic of the simulation domain in Fig After a steady density was reached, the form of the sheath and presheath for simulations A- E were studied. Time averaged density, potential profiles, and species flow moments were averaged over the last 5 µs of the simulation. These profiles are plotted along the symmetry axis of the simulation (the reflecting boundary), perpendicular to the electrode surface, see Fig Note that the non-smooth behavior of the velocity moment data is due to this being output as a cell based quantity in which a single value is calculated for each cell. The electron and ion velocity moments were normalized by the electron thermal speed, v Te = p T e /m e, and Bohm speed, v B = p T e /m i, respectively. Like the simulations in the previous section, the temperatures for the evaluation of these speeds were calculated from 2D particle velocity information by evaluating the integral T e = m Z e d 2 v(vr,x 2 + v 2n r,y)f 2 e (v), (3.1) e where v r,i =(v V e ) î. The value of the electron temperature was computed approximately.5 cm from the electrode surface and are listed in Table 3.1 for each simulation. In addition to these quantities, the vector component of the momentum equation perpendicular to electrode surface was also evaluated to determine the physics driving the presheath behavior. In the direction perpendicular to the electrode the momentum equation reads m e n e V e rv e ˆx {z } inertial term = n e ee ˆx {z } field term r P e ˆx {z } pressure term R e ˆx. (3.2) {z } friction The 2D density, flow moment, and pressure tensor are Z n e = d 2 vf e (v), (3.3) V e = 1 n e Z d 2 vvf e (v), (3.4)

84 67 and P e = m e Z d 2 v(v V e )(v V e )f e (v) =p e I + e, (3.5) where the pressure tensor is decomposed into the scalar pressure multiplied by the identity and the stress tensor. Along the axis of symmetry of the system there must be no gradients of V e or P e in the y direction due to the reflecting boundary condition for particles. The same is true of E due to the Neumann condition for the potential. Along this boundary V e = V eˆx and E = Eˆx. The inertial, pressure, and field terms marked in Eq. (3.2) are also shown in Fig. 3.2 for simulations A-E. The inertial and pressure terms were evaluated by taking moments of the distribution of particle velocities sampled over 2 di erent time slices separated by 5 ns each, while the field term was determined by the product of the cell density and the electric field from the calculated electrostatic potential. EVDFs were also sampled in front of the electrode for each simulation in the large sampling regions shown in Fig. 3.1b. The EVDFs shown in Fig. 3.3 for simulations A-E and in Fig. 3.4 for simulation E are plotted as the 2D histograms of x and y velocity components within the large and small sampling regions, respectively. The desired sampling region for obtaining symmetric EVDFs include areas on both sides of the symmetry axis, however only half of this is included in the simulation domain. To obtain a sample of the full region, the particles sampled in the half included in the domain are copied, and the y velocity component is reflected. These particles are then included in the histogram allowing it to represent the full region shown in Fig. 3.1b. 3.2 EVDFs and the Electron Flow Moment In the previous section, sheath and presheath profiles and EVDFs from five simulations were presented. These included the sheath and presheath of strong ion and electron sheaths in simulations A and E, as well as three cases where the bias was within T e /2e of the plasma potential. Non-zero electron flow moments directed towards the electrode were observed in all simulations except in case A, including cases B and C where the electrode was biased within T e /2e above, and at the plasma potential, respectively. In this section, the form of these EVDFs will be used to understand this behavior. First, consider the case of the strong ion sheath in simulation A. For this case, the fluid moments shown in Fig. 3.2 demonstrate the expected behavior of an ion presheath. Here, the ion flow accelerates to the Bohm speed by the sheath edge, while the electron flow moment is negligible except within the sheath. The non-zero flow of electrons within the sheath is due to the expulsion

85 68 (A)!! E =-25V!! p =2.2V! (B)!! E =5.5V!! p =5.7V! (C)!! E =6.5V!! p =6.5V! (D)!! E =8V!! p =7.6V! (E)!! E =25V!! p =18.5V! Distance (cm) Distance (cm) Distance (cm) Distance (cm) Distance (cm) Density (1 9 cm -3 ) Potential (V) (1 11 me cm-2 µs -2 ) Figure 3.2: Profiles of simulations A-E

86 69 (A)!! E =-25V!! p =2.2V! (B)!! E =5.5V!! p =5.7V! (C)!! E =6.5V!! p =6.5V! (D)!! E =8V!! p =7.6V! (E)!! E =25V!! p =18.5V! V x (cm/"s)! Particles! V x (cm/"s)! Particles! V x (cm/"s)! Particles! V x (cm/"s)! Particles! V x (cm/"s)! Particles! Distance from electrode!.75 cm!.25 cm! Vy (cm/"s)! Vy (cm/"s)!.5 cm! Vy (cm/"s)! Figure 3.3: EVDFs in the sheath and presheath for simulations A-E

87 7 V y (cm/"s)! Particles! V y (cm/"s)! Particles!.15 cm!.45 cm!.75 cm!.15 cm!.135 cm!.165 cm! V x (cm/"s)! Particles! V x (cm/"s)! Particles! V x (cm/"s)! Particles! Figure 3.4: The EVDFs for simulation E, sampled in the small red boxes shown in Fig. 3.1b.

88 71 of electrons sourced within this area. Although the flow moment is large, the electron density is small enough that the value of n e V e is negligible. The zero electron flow moment in the presheath is consistent with the inertial term in the momentum equation being negligible. This behavior is also apparent in the EVDFs shown in Fig. 3.3 which indicate that the electrons are well described by a Maxwellian distribution with zero flow shift. The absence of a flow shift or depletion of the Maxwellian EVDF are consistent with the expectations based on a Boltzmann density profile for electrons n e ( )=n e ( o )exp e( o )/T e. (3.6) A Boltzmann density profile is expected if the EVDF remains Maxwellian. For simulation A, this behavior is confirmed in Fig. 3.5 where the Boltzmann density profile is calculated using the simulated values of the potential. The Boltzmann density profile can be obtained by considering the momentum equation for a stationary Maxwellian. In this case V e =, R e = and P e = p e n e I. This results in a balance between the electric field and the electron pressure gradient. This can be seen in the terms of the momentum equation plotted in Fig When the bias is at or slightly below the plasma potential, such as in simulations B and C, the EVDF is no longer Maxwellian. In these cases, a significant number of electrons have enough energy to overcome the weak electric field, or zero electric field in simulation C, near the electrode and are collected. When this happens, the assumptions leading to the Boltzmann density relation break down. The collection of electrons by the boundary lead to the loss-cone-like truncation in the electron velocity distribution function shown in Fig This truncation can be understood by considering the role of the electrode in the geometrical obstruction of electron motion within the plasma. Near the surface of the electrode, the EVDF forms a loss-cone type distribution with an opening angle that narrows when moving further away into the bulk plasma. Fig. 3.6 shows how a region near the electrode can be inaccessible for left-to-right moving electrons. In the region marked Electrode Shadow, energetic electrons directed towards the electrode surface are collected and the only electrons available to fill in the distribution are those sourced in this region; however, this population is small and essentially negligible. This situation did not occur for the strong ion sheath in simulation A since the strong electric field reflects all electrode directed electrons. In simulations B and C the di erence between the plasma potential and electrode are > E p > T e /2e and E p, as seen in the potential plotted in Fig The density profiles for ions and electrons, along with the small potential gradient, indicate that in these cases the plasma

89 72 15 Electron Density (1 8 cm 3 ) 1 5 derived Boltzman relation PIC electron density Distance (cm) Figure 3.5: Comparison of the electron density of simulation A along the line y= (the reflecting boundary) in Fig. 3.1a with the prediction of the Boltzmann density profile of of Eq. (3.6). The values of n e ( o ) and the 2D temperature T e were determined at a point approximately 2 cm from the electrode. Electrode Shadow Electron Trajectories Electrode Figure 3.6: Electron trajectories for right moving electrons are shown to demonstrate how the electrode can cause a truncation in the EVDF. The region that is inaccessible for right moving electrons is marked Electrode Shadow.

90 73 remains quasineutral up to the electrode. Although there is no breakdown of quasineutrality, there are still significant flow moments and density gradients for electrons and ions. Unlike the typical ion presheath, the ions only achieve a flow velocity of.4 v B by the electrode for case B, while the electrons have a flow of.6 v Te, with both flows directed towards the electrode. The electric field term for case B is weaker than the pressure term, and for case C the field term is negligible, suggesting the flows in this regime are not related to the electric field. Gradients in the electron densities and flow velocities can be explained by the widening of the loss cone. As the electrode is approached the loss cone angle increases while there is little or no shift in velocity at the maximum value of the EVDF in Fig The increasing flow moment is a result of the increasing angle of the loss cone. The continuity equation r (n e V e ) = requires that an increase in flow moment must be accompanied by a decrease in density. The computed velocity moment gradients in Fig. 3.2 indicate that the pressure tensor gradient, due to the increasing truncation, plays an important role when the electrode is biased near the plasma potential. In case D the electrode was biased above the plasma potential. Here, qualitative changes in the EVDF and sheath structure were observed. The most notable di erence is that the plasma no longer remains quasineutral up to the electrode, but a well-defined electron sheath has formed. When E p T e /2e nearly all of the ions are repelled due to the fact that T e T i. This can be seen in the ion density profile, which shows very few ions in the electron sheath. Here, the electrons achieve a flow velocity of.6 p T e /m e by the sheath edge, which is slightly slower than the previously predicted value from Eq. (2.4). However, the exact sheath edge position is di cult to determine in time averaged data due to the streaming instability driven fluctuations which were discussed in Chapter Nevertheless, the flow of electrons into the sheath can be seen to be driven by pressure tensor gradients in Fig. 3.2, with some portion of the flow due to the shift in the maximum value of the EVDF, and the rest due to the truncation visible in Fig The presheath behavior remains qualitatively similar as the electrode bias is further increased, as demonstrated by case E. The presheath behavior is largely the same as that in case D, a shift in maximum value of the EVDF and a loss cone is present near the sheath as shown in Fig The main di erence between simulation E and simulation D is the deformation of the EVDF within the sheath, which is shown in detail in Fig Near the sheath edge, at x =.165 cm, the EVDF resembles the loss cone shaped distributions of Fig Moving into the sheath, the EVDF becomes stretched out, with electrode-directed electrons becoming more and more depleted due to the increasing strength of the electric field. A similar deformation of the EVDF has been

91 74 seen in Vlasov simulations within the electron sheath around a cylindrical probe [76]. The previous analysis of Chapter 2 was based on the 1D projections of the EVDF which look largely Maxwellian in the direction perpendicular to the electrode surface. Based on the observation of 1D EVDFs the electron presheath was modeled describing the electrons with a flow-shifted Maxwellian and ions with a Boltzmann density profile. Using these assumptions the dominant term in the electron momentum equation is the pressure gradient, which was T e /T i larger than contribution of the presheath electric field. This model proposed a significant deviation from the conventional assumption of a half-maxwellian at the sheath edge [58, 59, 57]. It also showed that the electron sheath has a presheath that causes a significant perturbation far into the plasma. The results in this chapter present a refined picture compared to the previous 1D model. Like the 1D model, a long presheath was visible in the electron flow moments shown for simulations D and E in Fig The EVDFs discussed above demonstrate that, for the electron sheath cases, the flow velocity is due to both a loss-cone like truncation and a flow shift. The truncation due to the loss cone leads to a significant contribution to the acceleration of the flow moment through pressure tensor gradients in the momentum equation. The pressure driven behavior was previously explained using only a scalar pressure gradient in the 1D model of the previous chapter, and did not include the physics of the loss-cone truncation. Finally, the loss-cone truncation was found to be a geometric e ect related to electron collection by the electrode, which originates in case B in which they are no longer strongly repelled. 3.3 Implications for Future Electron Presheath Models The geometric interpretation of the loss cone suggests a path forward for developing a hybrid fluid-kinetic model of the electron presheath. Based on observation of the EVDF from simulation E, a reasonable model would be to include a geometric loss cone, dependent on the solid angle obstruction of the electrode viewed from some position in the presheath. In this picture, electrons with velocity components directed at the electrode surface are collected. For a circular electrode of radius R, the half-angle of the electrode at a distance d away is c = arctan(r/d). (3.7) Electrons whose velocity components are within this angle are collected leading to the truncation. The velocity distribution function for this situation can be written in spherical coordinates (with

92 75 the coordinates aligned with the direction of the center of the electrode) as f e = n o 3/2 v 3 T exp v 2 / v 2 T H c, (3.8) where v is the velocity in the frame of the flow shift V f. The density and flow moments for this distribution are and n e = n o 2 (1 + cos c) (3.9) V e = v T p (1 cos c )ẑ V f. (3.1) These moments could be used in combination with a collisionless momentum equation, m e n e V e rv e = n e ee r P e, (3.11) a Boltzmann density for ions and the continuity equation n e = n i = n o exp e T i, (3.12) r (n e V e )=, (3.13) to solve for V f within the presheath. However, due to the geometrical interpretation of the loss cone, a presheath model formulated in this way is limited to a specific electrode geometry. One example of this is the electron sheath in the simulation of the previous chapter. Fig. 3.7 shows the EVDF near the edge of the electron sheath for the simulation of Chapter 2.3. The flow moment of the EVDF is more pronounced and the loss cone truncation is not as sharp. This is due to electrons scattering o of the adjacent ion sheath into the loss cone region. A more detailed electron presheath model could be considered for some geometries by following through on some of the suggestions of this section. Since this is specific to the electrode geometry of interest, this will not be pursued further in this thesis.

93 76 Figure 3.7: The EVDF near the sheath edge for the electron sheath simulation presented in Chapter 2.3. For this electrode geometry, the flow shift is more pronounced than the loss cone truncation.

94 77 CHAPTER 4 ANODE SPOT ONSET AND EQUILIBRIUM 4.1 Introduction As electrons pass through the electron sheath electric field, they can gain enough energy to ionize neutrals. Ionization within the sheath modifies the space charge distribution because the sheath electric field accelerates electrons born from ionization faster than ions due to their di erent masses. If the ionization rate is su cient, a positive space charge layer will develop at the electrode next to a negative space charge layer adjacent to the plasma. These adjacent positive and negative space charge layers are a type of anode double layer 1 called an anode glow [2]. The anode glow potential profile is sketched in Fig. 4.1 and a photograph is shown in Fig. 4.2a. The emission in the anode glow is due to excitation of neutrals by electrons with energy near the ionization energy. Once an anode glow is present, increasing the electrode bias further may result in the formation of an anode spot or fireball if the ionization rate is su cient. The main characteristic of an anode spot is a double layer detached from the electrode separating a luminous high-potential plasma from the bulk plasma. A photograph of an anode spot is shown in Fig. 4.2b. Potential profiles typical of these situations are shown in Fig This chapter presents the first particle in cell (PIC) simulations of an anode spot and a theory for their onset and steady-state properties. In the theory, anode spot formation results from a buildup of ions within the sheath that modifies the electric field, forming a potential well that traps electrons. This results in the formation of the quasineutral spot plasma. Once the anode spot is present, its steady-state properties are determined by the balance of particle creation and loss, current loss, and power deposited into and lost from the anode spot plasma. These predictions are found to be in agreement with the simulations presented and with prior experimental measurements. In addition, recent experimental measurements of the anode spot are found to be consistent with theory and simulation. Not all anode spots occur in steady-state. Experimental studies of the dynamics of repetitive formation and collapse of anodes spots have recently been presented in the literature [77, 78, 79]. However, the following paragraphs will outline work that led to developments in the understanding of spot onset and steady state properties which are the focus of this chapter. 1 Note that the term double layer usually refers to a similar structure between a high and low potential plasma with equal amounts of positive and negative charge, hence the need to distinguish anode double layers.

95 78 Anode&Spot& Anode&Glow& Electron&Sheath& (V)& Figure 4.1: A variety of sheath-like structures are possible near an electrode biased more positive than the plasma potential. Once the electrode bias exceeds the plasma potential an electron sheath forms. When the potential is large enough that electrons gain the ionization energy of neutral atoms, a double layer attached to the electrode may form due to ionization within the sheath. This is typically associated with the anode glow. Increasing the bias eventually results in an anode spot, characterized by a double layer separating a high potential plasma from the bulk plasma.

96 79 a)#anode#glow# b)#anode#spot# Figure 4.2: a) A photograph of an anode glow near a positively biased electrode observed through the observation port of a GEC reference cell. b) An increase in bias of the electrode resulted in the formation of an anode spot characterized by the larger luminous region. Although anode spots were first observed by Langmuir in 1929 [15], they were not studied in detail until 1979 when Troven and Andersson observed the phenomenon in a magnetized plasma in 1 mtorr of mercury vapor [3]. In 1981 Andersson [8] and in 1983 Andersson and Sorenson [81] continued to study anode double layers numerically. Numerical solutions to Poisson s equation were explored by using an integral equation to model electron impact ionization in the sheath. Although anode spot solutions were not found in either of these works, anode double layers attached to the electrode were modeled for increasing values of neutral pressure up to a critical value where solutions were no longer possible. Andersson suggested that one of the following mechanisms was responsible for the sheath transitioning to an anode spot: 1) With su cient ionization the space charge in the sheath changes sign. 2) The ions injected from the sheath into the plasma trigger an instability. 3) There is an instability in the plasma-sheath system irrespective of ionization, but the instability makes the transition to an anode spot possible. Andersson conjectured that the loss of numerical solutions at higher pressure indicated that a time dependent description was necessary, with the cause of the time-dependence being the instabilities in explanation 2) or 3). In this chapter, it is shown that explanation 1) leads to the anode spot onset without the need for instabilities. Stationary double layers are expected to satisfy a flux density balance criterion called the

97 8 Langmuir condition [82], i = r me m i e, (4.1) where e = n e V e and i = n i V i are the electron and ion flux densities and V e and V i are the electron and ion flow velocities at the double layer sheath edge. In 1972 Block [83] noted that the Langmuir condition is satisfied in the reference frame of the double layer. Two decades later Song, Merlino, and D Angelo [84] derived a modified Langmuir condition by considering the momentum and continuity equation in the frame of a double layer moving with velocity U DL with respect to the lab frame. The condition in the lab frame is i n Hi U DL = r me m i e, (4.2) where n Hi is the density of the plasma at the double layer s high potential side. Here, the double layer motion results from an imbalance in the electron and ion fluxes crossing the double layer. See Appendix B for a derivation of Eq. (4.2). Using this newly derived condition, they studied the stability of anode spots coupled to the bulk plasma using a circuit model [85]. Song, Merlino, and D Angelo also continued experimental investigations of anode spots in argon, krypton, and xenon gasses at 1 4 Torr [2]. An equation for the length L of an anode spot in terms of the ionization cross section I and the neutral density n n, r me m i L I n n 1, (4.3) was proposed. The experiments showed agreement with the prediction that the anode spot diameter was inversely proportional to the product of neutral gas pressure and the ionization cross section I. In their work, a constant estimate of the cross section was assumed instead of considering its energy dependence. In this work, the energy dependence of the cross section is shown to play a role in determining the steady-state properties of the anode spot. In 26, Conde, Fontán, and Lambás [86] numerically explored the formation of anode double layers using an integral equation similar to those of Andersson [8], Andersson and Sorenson [81], and Ahedo [19] to describe the buildup of ion density within the sheath due to electron impact ionization. They found that their integral equations had two solutions when the rate of electron impact ionization of neutrals was large enough, one of which resembled the anode glow. Their equations failed to produce solutions for large values of the electrode bias needed for spot formation and the physics of the spot onset was not captured in this study. In 29, Baalrud, Longmier, and Hershkowitz [2] studied anode spots in 1 mtorr argon and explored their observations theoretically. Their model suggested that the spot onset occurs

98 81 when the number of ions and electrons within a Debye cube near the electrode surface are equal. Once quasineutrailty was established, it was predicted that the spot plasma would expand to the ion presheath length scale accommodating the presheath needed to accelerate ions to their sound speed by the sheath edge at the high potential entrance to the double layer. The model was used to estimate the critical bias of an electrode for spot onset by considering the balance of fluxes of charged particles created by ionization. This model showed that the critical bias is inversely proportional to the neutral density, a result that was found to be consistent with their experimental measurements. This chapter presents the first computational study of anode spots. Based on simulations presented in Chapter 4.2, a theory for the spot onset is developed in Chapter Unlike prior theories, this theory shows the anode spot plasma forms due to the formation of a potential well for electrons near the electrode. An increase in plasma density within the well results in an imbalance in the Langmuir condition Eq. (4.1), resulting in motion of the double layer and expansion of the spot plasma. This description goes beyond predicting the scaling of the critical bias with pressure and provides a prediction for its value. Following this, Chapters and predict steadystate properties of the anode spot using an analysis of particle, current, and power balance. This model self-consistently relates the values of the anode spot diameter, double layer potential, and potential structure. Chapter determines properties of the ion presheath leading up to the high potential entrance of the double layer, showing that under typical low temperature plasma conditions the presheath length is determined by the ionization rate. Chapter 4.4 compares the theory and simulations to recent experimental measurements. A summary of results is given in Chapter Simulation Simulation Setup Simulations were performed using the electrostatic PIC code Aleph [55], which was introduced in Chapter Similar to the simulations of Chapter 2.3, the 2D simulation domain was chosen to resemble experiments [17] and had dimensions of 7.5 cm by 7.5 cm. The boundary conditions and sourcing of particles was similar to previous simulations. The boundary conditions for particles were one reflecting boundary (left wall) and three absorbing boundaries (right, top, and bottom walls) for particles, see Fig The absorbing walls had a = Dirichlet boundary condition for the electric field except for a small.25 cm electrode embedded in the lower boundary. The electrode bias was increased linearly from V at t = to 4 V at t = 9 µs and then held at 5 V for t>9 µs. The reflecting boundary had a Neumann r ˆn = boundary condition. The domain

99 cm! (cm^-3) Plasma Source Region! 7.5 cm! Reflecting Boundary!! =! Electrode Bias (V) Electrode! Time (µs) Figure 4.3: The 7.5 cm by 7.5 cm simulation domain with the color map indication the typical ion densities encountered. An anode spot is attached to the electrode in the lower left hand corner of the domain. was discretized using an unstructured triangular mesh with an element size of approximately.116 cm that resolved the Debye length across the domain throughout the duration of the simulation. The density within the anode spot was more than an order of magnitude larger than the simulations presented thus far, resulting in the need for a smaller cell size. A helium plasma was continuously generated at a rate of cm 3 µs 1 in a.25 cm by 6.5 cm source region placed 6.25 cm above the electrode. Electrons and ions were sourced with temperatures of 4 ev and 86 mev respectively. After expansion of the plasma from the source region, the electron temperature in the bulk plasma was approximately 2.4 ev. A neutral helium background with density cm 3 at 24 mev (2 mtorr at 273K) was present at the beginning

100 83 of the simulation. Electrons, ions, and neutral particles had weights of 2, 8, and respectively. Two populations of electrons were tracked to better probe the physics of the anode spot. Electrons which take place in an ionization event are denoted e I and bulk electrons which have not taken part in ionization interactions are denoted e B. The included ionization interactions for these species are e B + He! 2e I + He + (4.4) and e I + He! 2e I + He +. (4.5) Elastic electron neutral collisions with the helium background were also included. A time step of 1 ps was chosen to resolve the electron trajectories. The smaller cell size and larger electron velocities due to acceleration by strong electric fields within the anode spot resulted in the need for a smaller time step. This is an order of magnitude shorter than that used in the simulations of Chapter Anode Spot Simulation Simulations were carried out for final electrode biases of 4 V, 45 V, 46 V, 47 V, 48 V, and 5 V to determine the critical electrode bias for spot onset. For each simulation, the plasma potential at t = 9 µs, just before the electrode bias was increased, was 6.4 V. Anode spots were observed for the 48V simulation, indicating that the critical bias with respect to the plasma potential is less than 41.6 V. For this section, the 5 V simulation will be used to study the spot onset. The simulated electric field and species densities are used to shed light on the processes that result in the formation of the anode spot. Early in the simulation, particles fill the domain by ambipolar di usion from the source region. Initially, a large portion of the lower half of the domain is electron rich. After approximately 4 µs, the ions have had enough time to traverse the domain and encounter the electric field at the electrode. Once these ions are reflected by the field, an electron sheath forms. This can be seen in the ion density and electric field in Fig. 4.4, which is plotted along the reflecting boundary as a function of time. Between 5 µs and 9 µs, waves propagating towards the electron sheath can be seen in the ion density and electric field. These are ion acoustic waves excited by the di erential flow between ions in the plasma and electrons accelerating to their thermal speed in the electron presheath [87].

101 84 At 9 µs the electrode bias jumps to 5 V. Fig. 4.4 shows a dramatic increase in ion density within the sheath. The e ect of this increase can be seen by comparing the first two panels of Fig The increase in ion density is not immediatly accompanied by an increase in e I electron density n ei since the sheath field quickly accelerates electrons to the electrode. At t =9.5 µs, the buildup of ions within the sheath has resulted in a change in concavity of the potential profile. The panel of Fig. 4.5 marked t = 1.2 µs shows that a further increase in ion density results in a change in sign of the electric field, which is indicated by the maximum in the potential just o of the electrode surface accompanied by a small increase in n ei. Low energy electrons resulting from electron impact ionization interactions are trapped at the maximum. Once this electron trapping occurs, the electron density increases rapidly and quasineutrality is established. This is followed by the expansion of the anode spot plasma shown in the densities and electric field plotted in Fig. 4.4 and the panel marked t = 11 µs in Fig After the expansion of the anode spot, the double layer potential settles at 27.2 Vwhich is 2.6 V above the potential needed for electron impact ionization of helium. This is similar to experiments which have measured the double layer potential at a few volts above the ionization energy of the neutral gas[2, 88, 16]. The plasma potential within the spot settles around 51 V, 1 V above the potential at the electrode. The scale length of the spot at this time (13 µs) is.5 cm, indicated by the 2D geometry of the zero electric field area shown in Fig Velocity Distribution Functions The VDFs were examined at t = 1.8 µs by taking histograms of particle velocities within 2D regions inside the anode spot. Ion histograms were plotted using regions of dimension.5 cm by.5 cm, while electron histograms used a.1 cm by.1cm region. Di erent size regions were used due to the di erent macroparticle weight between electrons and ions. Four di erent locations were chosen and are shown as the shaded regions of Fig The corresponding VDFs are shown in Fig See the last paragraph of Chapter 3.1 for details of the calculation of VDFs along the reflecting boundary. At y =.7 cm within the double layer, e B electrons have a peak density near v y = 18 cm/µs corresponding to the energy gained from the double layer electric field up to that point. These electrons increase in kinetic energy to nearly 26 ev of energy at the maximum potential near y =.1.2 cm. This corresponds to a half-ring-like distribution with a peak in density with magnitude at v 3 cm/µs. The geometry of the VDF within the spot is due to the curvature

102 85 n i (1 9 cm 3 ) Distance from Electrode (cm) n eb (1 9 cm 3 ) n ei (1 9 cm 3 ) φ (V) Time (µs) E (V/cm) Figure 4.4: The time dependent values of plasma species densities, potential, and electric field along the symmetry axis (reflecting boundary). Note the spot onset following the increase in electrode bias at t =9µs. of the surface of the double layer and the multitude of directions from which electrons enter the spot. Electrons born from ionization (e I ) within the spot are nearly Maxwellian. Few of these electrons are present between y = cm since they are trapped by the strong double layer electric field. Ions born within the spot are well described by a Maxwellian with no flow shift at y =.2 cm. However, very close to the electrode (y =.1 cm) ions are flow shifted due to the ion presheath leading up to the electrode surface. Within the double layer ions are accelerated towards the bulk plasma as indicated by their velocity and decreased density.

103 86 Potential (V) t=9µs φ n i n ei n eb t=9.5µs t=1.2µs t=11µs Density (1 9 cm -3 ).5 1 Distance from electrode (cm) Figure 4.5: The potential and density profiles before (t = 9 µs) and after the electrode bias increases from 4 V to 5 V. Note that after the potential gradient changes sign around t = 1.2 µs, thedensity of electrons from ionization begins to increase. Quasineutrality is established by t = 11 µs. Figure 4.6: Particle species density and potential profiles corresponding to the VDFs of Fig. 4.7 taken at t = 1.8 µs. The smaller purple shaded regions indicate the area in which ion VDF histograms were plotted. The larger blue shaded regions show the same for electrons. The total electron density is also shown Current Collection One common feature observed in experiments is the jump in current collection associated with the formation of the anode spot [2, 2, 88]. It has previously been suggested that the cause of

104 87 e B- e I- He + y =.7 cm v y (cm/µ s) v x (cm/µ s) v y (cm/µ s) v x (cm/µ s) v y (cm/µ s) v x (cm/µ s) y =.5 cm v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v x (cm/µ s) y =.2 cm v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v (cm/µ s) x y =.1 cm v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v (cm/µ s) x v y (cm/µ s) v x (cm/µ s) Figure 4.7: Velocity distribution functions for e B and e I electrons and helium ions within the anode spot and double layer. The color bars indicate the number of macroparticles per bin. These VDFs were obtained at t = 1.8 µs of the simulation at the locations indicated in Fig 6. Note that the coordinate axes for ions span di erent ranges in v x and v y. Also note that the spatial extent over which histograms for ions were plotted is a quarter of the area used for electrons, see Fig. 4.6.

105 88 Figure 4.8: Components of the current collected by the electrode and wall. After anode spot onset the electron current collection increases. When the spot is present at t>1 µs, all electrons lost from the simulation exit through the electrode. this increase is an increase of electron collecting area resulting from the relatively large surface area of the double layer compared to the electrode [77]. A current increase is also seen in the simulation presented in this section. Fig. 4.8 shows that most of the current collected at the electrode is due to the bulk plasma electrons (e B ) accelerated to the electrode by the double layer. There is also a 2% contribution from electrons formed by ionization after spot onset. This confirms that the increased current collection after onset is due to the increased collection area from the bulk plasma. In Fig. 4.8, the current component due to ions born from ionization collected at the walls of domain is also shown. This indicates that approximately 5-1% of the ion current to the walls after spot onset is due to ions being accelerated out of the spot. Once the anode spot has formed, all of the bulk ion current is collected by the walls and all the bulk electron current is lost to the electrode. This situation, known as global non-ambipolar flow [16], occurs near electrodes that are small enough to be biased above the plasma potential and large enough to collect the entirety of the electron current. After onset the current shows oscillations on a microsecond timescale, possibly connected with the increased fluctuations seen in the ion density and electric field of Fig. 4.4.

106 89 Figure 4.9: Test electron trajectories integrated in the time dependent electric field obtained from the simulation of Chapter The initial velocity vectors are shown in the legend and are representative of electron velocities found near the sheath edge. Electrons starting in the range x [,.4] are collected by the electrode. Additional evidence for the increase in electron current collection area can be found by integrating test electron trajectories starting at locations outside the double layer. The integration of the trajectories used the velocity Verlet method along with the electric field from a single time step of the simulation at t = 1.8 µs. The electric field within the anode spot had small fluctuations on the timescale of an electron crossing the spot, but these contributed a negligible change in the trajectories so the time dependence of the electric field was neglected. Selected particle trajectories are plotted over the electric field in Fig The initial velocities for the particles are typical of those found in a flow-shifted distribution at the sheath edge. Trajectories starting between x = cm and x.4 cm impact the electrode, which extends from to.25 cm on the lower boundary. Particles starting further to the right typically impact other parts of the lower boundary. These trajectories directly demonstrate that the anode spot increases the current collection area.

107 9 4.3 Theory Anode Spot Onset In Chapter 4.2.2, simulations showed that the anode spot onset is a consequence of the buildup of positive space-charge within the electron sheath due to electron impact ionization of neutral atoms. Initially, electrons born from ionization were quickly accelerated to the electrode by the sheath electric field, but when the positive space-charge grew large enough, the potential profile developed a local maximum o the electrode surface. This local maximum is a potential well for electrons. Low energy electrons born from ionization within this well are trapped, allowing the electron density to increase. It was observed that this trapping led to the formation of a quasineutral region where E and n e n i. Once quasineutrality is established, the double layer will move if there is an imbalance in the Langmuir condition. This results in the double layer velocity given in Eq. (4.2). To determine when the double layer will move outward resulting in the expansion of the anode spot plasma, the flux densities entering the double layer are considered. Bulk electrons are accelerated to their thermal speed (v Te = p T e /m e ) in an electron presheath [87, 3] leading up to the double layer at the low potential side. This flux density is r Te,B e = n eb, (4.6) m e where the subscript B indicates values in the bulk plasma. Ions moving towards the bulk plasma enter the high potential side of the double layer near the electrode at their sound speed due to the Bohm criterion, resulting in the flux density i = n i,hi r Te,Hi m i, (4.7) where the subscript Hi indicates values at the high potential side of the double layer. Inserting Eqs. (4.6) and (4.7) into the modified Langmuir condition Eq. (4.2) and requiring U DL > as a condition for anode spot onset 2 leads to s n i,hi >n eb T e,b T e,hi. (4.8) Before the formation of the potential well the modified Langmuir condition would suggest that U DL <. However, this condition is only valid when there is a region with E at both sides of the double layer. The assumptions in the derivation of the modified Langmuir condition have not 2 Here, onset is defined as the expansion of the quasineutral region. This coincides with the motion of the double layer.

108 91 been met before E at the high potential side of the double layer. Furthermore, applying this condition to the case of the panel marked t = 1.2 µs in Fig. 4.5 might suggest the possibility that the double layer might move after E in the potential well, but before n e n i. In this case, since n e 6 n i, any motion of the double layer would result in E 6 and the assumptions of the modified Langmuir condition break down. The E = region of potential well can only expand after quasineurality is established. Initial attempts to model the spot onset attempted to solve an integral form of Poisson s equation [86, 8, 81]. In these models, the integral of the charge density within the sheath, including contributions from electron impact ionization, were formulated as a function of the sheath electric field Z 1 2 (E)2 = o =4 e o E( ) d. (4.9) Models based on the form of Eq. (4.9) were unable to provide potential solutions which include a potential well for electrons. This is due to the electric field becoming multivalued as a function of the potential once a local maximum is present. For typical experimental conditions where T e,b /T e,hi ' 1, Eq. (4.8) predicts that the double layer will move when the high potential density approximately exceeds the bulk density. Fig. 4.4 shows that when this occurs the double layer begins to move outward which is in agreement with the condition in Eq. (4.8). After 1.7 µs the double layer position indicated by the electric field in Fig. 4.4 shows that its speed has decreased. At first glance this may seem to contradict Eq. (4.2) since the peak density within the center of spot has increased significantly even though the double layer velocity has not. However, a careful inspection of the density and potential profiles at t = 1.8 µs in Fig. 4.6 reveals that the plasma density at the high potential sheath edge is slightly less than half the peak density within the spot. The high potential density was still large enough for the double layer to continue to slowly move outward. The theory predicts that when the densities at the entrance to the double layer balance its motion stops. The simulation time was not long enough for a steady-state configuration to be observed. To predict when an anode spot will form at a positively biased electrode, it is desirable to connect the density criterion of Eq. (4.8) with the electrode bias. In experiments, anode spot onset is observed when the electrode bias exceeds some critical value. The critical bias can be predicted by considering the conditions required for spot onset. Two conditions are required: 1) a quasineutral region is established at the high potential side of the double layer so that the assumptions of the modified Langmuir condition are satisfied and the double layer is free to move, and 2) the ion flux

109 92 leaving the high potential side leads to an imbalance in the Langmuir condition resulting in motion of the double layer and expansion of the anode spot plasma. Electrons born from ionization near the maximum in the potential profile are trapped since the maximum is a potential well for electrons. A potential well for electrons is a potential hill for ions, therefore for a well with a depth greater than the floating potential, the electron density increases faster than the ion density. Once the electron density matches the ion density at the bottom of the well condition 1) is satisfied. If ions are born faster than they are lost, the plasma density within the electron potential well will increase until condition 2) is satisfied. The statement that the ion birth rate is greater than the ion loss rate can be expressed as 2A sheath n Hi c s,hi <n n e I A sheath z sheath, (4.1) {z } {z } loss rate ionization rate where A sheath is the cross-sectional area of the sheath prior to the double layer expansion, c s,hi is the ion sound speed at the high potential side, and z sheath is the sheath thickness. The numerical factor is due to half of the ions born near the electron well being lost in the electrode direction and the other half being lost in the plasma direction. To determine the critical bias, the condition of exact balance in Eq. (4.1) is considered. Combining this with the steady-state Langmuir condition of Eq. (4.1) and the high potential ion flux in Eq. (4.7), the threshold condition becomes 1 r 1 mi n n I z sheath =. (4.11) 2 m e This is similar to the equation used by Song et al. [2] to estimate the sheath thickness of the anode glow prior to spot onset. This condition can be used with an estimate of the sheath thickness and the energy-dependent cross section to estimate the critical bias. Using the electron sheath thickness from Eq. (2.22) as an estimate, Eq. (4.11) can be solved numerically for the critical bias c given the energy dependent ionization cross section I (e c). The critical bias for a helium plasma with an electron temperature of 2.4 ev is shown in Fig. 4.1 for several values of the bulk plasma density. An interesting observation from experiments is that increasing the size of the electrode decreases the critical bias [89, 2]. Eq. (4.11) assumed that the sheath and electrode had the same surface area. This is appropriate for the simulations where the initial approximately planar electron sheath is embedded within the ion sheath of the surrounding walls. For a sheath at an isolated electrode, the thickness of the sheath increases its surface area. If the sheath area is modeled as a

110 93 Critical Bias (V) n B =1 8 cm -3 n B =3x1 8 cm -3 n B =1 9 cm Pressure (mtorr) Figure 4.1: The critical bias as a function of pressure for several di erent bulk plasma densities and an electron temperature of 2.4 ev calculated from Eq. (4.11) with the sheath thickness of Eq. (2.22). disk with diameter D and finite thickness z sheath, the ratio of sheath area to electrode area is A sheath A E = Dz sheath + (D/2) 2 (D/2) 2. (4.12) Using this, Eq. (4.11) is modified to include the sheath area that was unaccounted for, r A sheath mi n n I z sheath =. (4.13) A E m e Eq. (4.13) is solved for an argon plasma and compared to the experimental data for the critical bias of 1 cm and 5.5 cm diameter electrodes presented in Baalrud et al. [2] and shown in Fig The value of the plasma potential in the experiment was about 5 V, which was subtracted o the data points in Fig. 4.11, but an independent measurement was not available at each pressure. The predicted values of the critical bias are found to be in good agreement with the experimental data. Chapter presented a series of simulations used to determine the critical bias for a plasma with a 2 mtorr helium background, T e = 2.4 ev, and a bulk density of cm 3. The critical bias for these conditions was determined to be 41.6 V above the plasma potential. This is approximately 1 V more than the critical bias estimated from Fig One possibility for the agreement with 1mTorr experiments in Fig and discrepancy for 2 mtorr simulations is that elastic collisions of electrons result in a thicker sheath than that described by Eq. (2.22). A thicker sheath provides a greater loss area for ions formed by ionization, hence a greater ionization rate may be needed to satisfy Eq. (4.13).

111 94 Critical Bias (V) cm Electrode 5 cm Electrode Experiment /P (mtorr -1 ) Figure 4.11: The critical bias calculated from Eq. (4.13) compared with experimental data for two di erent electrode diameters [2] Anode Spot Current Balance Like all steady-state plasmas, the anode spot maintains its quasineutrality by the equal loss of electron and ion currents. Chapter presented a theory that predicts the form of the sheath at a positive electrode by using global current balance arguments [16]. In their work, the current loss to a positively biased electrode of area A E in a plasma chamber with wall area A W at potential = was considered. The form of the sheath could be either an electron sheath, ion sheath, or a sheath with a non-monotonic potential, depending on the ratio of surface area of the positively biased electrode to the wall area A E /A W. When the electrode area was small, an electron sheath was present since it would not significantly modify the balance of global current loss. However, when the electrode area was large the plasma potential locked to a few volts above the electrode potential preventing electrons from being lost at a faster rate than ions. This behavior has been observed in both experiment and simulation [17, 35]. Similar arguments can be applied to the anode spot by considering the current lost to the double layer surface A S, which is the analog of A W for the quasineutral spot plasma, and the current lost to the electrode of area A E. For this situation, the form of the sheath between the spot plasma is expected to be either and ion sheath, electron sheath, or a non-monotonic electron sheath with virtual cathode depending on the area ratio A S /A E.These possibilities are shown in Fig Fig. 4.4 indicates that nearly all of the electrons in the spot are

112 95 those formed by ionization of neutrals 3. Due to this observation, the analysis below will assume that the quasineutrailty condition within the anode spot is between the ions and e I electrons produced by ionization interactions in Eq. (4.4) and (4.5), i.e. n ei n i. Ion Sheath: First consider the case where the sheath between the spot plasma and electrode is an ion sheath. The electron current collected is I ei = e(n ei v TeI /4)A E exp( e I/T ei ), where I = S E, S is the anode spot plasma potential, and v TeI = p 8T ei / m e is the average speed of electrons born from ionization assuming a Maxwellian with temperature T ei. The ion current lost to the electrode and double layer is I i =.6ec s n i (A E + A S ). Balancing the two currents gives the potential within the spot e T ei I apple r =ln.6 2 T es m e AS +1 T ei m i A E. (4.14) Here, T es =(T ei n ei + T eb n eb )/(n ei + n eb ) is the total electron temperature within the spot. Both T ei and T es appear in this equation since the total electron temperature T es determines the Bohm speed for ions and T ei determines the loss rate for ionization electrons. Although n ei n eb, T ei T eb so both components must be considered in the total electron temperature T es. For Eq. (4.14) to be consistent with the assumption that an ion sheath is present ( I > ), the area ratio must satisfy r A S TeI m i <.6 1. (4.15) A E 2 T es m e Electron Sheath: Now consider the case where the sheath between the spot plasma and electrode is an electron sheath. The electron current lost to the electrode side is I ei = A E en ei v TeS, where v TeS = p T es /m e. The electrons within the anode spot flow into the sheath with an electron thermal speed determined by T es. The ion current lost to the electrode and double layer is I i = e(n i v Ti /4)A E exp( e E/T i )+.6en i c S A S,where E = E S, and v Ti = p 8T i / m i is the average speed for Maxwellian distributed ions with temperature T i. The value of E can be determined by balancing the two currents and using the quasineutrality condition resulting in appler e E =ln 2 T es.6 A r S mi. (4.16) T i T i A E m e For this to be consistent with the assumptions of an electron sheath at the electrode ( E > ), r A S Ti >.6 + A E 2 T es r mi m e. (4.17) 3 This conclusion can verified by considering the high potential sheath edge density implied by the Langmuir condition along with the rarefaction of the bulk electron density by the strong double layer potential.

113 96 Ion Sheath Electron Sheath Potential Electron Sheath with Virtual Cathode Distance from Electrode Figure 4.12: The di erent types of potential structure that are possible depending on the anode spot size. Electron Sheath with Virtual Cathode: For area ratios between the electron and ion sheath cases the potential may have a non-monotonic profile such as a virtual cathode which limits the electron flux to the electrode [57]. In experiments, larger anode spots at low pressure were observed to have electron sheath potential profiles both with and without virtual cathodes [2, 2]. For smaller anode spots at higher pressures the potential profile is expected to include an ion sheath at the electrode. In the 2D simulation, the anode spot area is approximately half the circumference of a circle resulting in A S /A E 3. For this area ratio the sheath at the electrode is expected to be an ion sheath. This is observed in the potential profiles of Fig Particle and Power Balance & Anode Spot Size In this subsection, the steady-state size of the anode spot is predicted by considering conservation of particle number and power in a global model of the spot plasma. This global model ties the potential across the double layer to the ionization rate within the spot, a feature that was not present in previous estimations of the anode spot size. Once the size and double layer potential are known, the form of the sheath at the electrode can be determined using the theory of Chapter

114 97 electrode, Balancing the volume ionization rate with the ion loss rate, and neglecting losses to the A S n s c s,s = n n h I vi B n B ( DL)V S. (4.18) Here n n is the neutral gas density, h I vi B is the rate constant for impact ionization by electrons accelerated from the bulk, n B ( by DL) is the density of the bulk electrons accelerated into the spot DL, and V S is the anode spot volume. A length scale can be defined by relating the spot surface area and volume as V S = A S L. Eq. (4.18) can be written in terms of L, L = n S c s,s n n h I vi B n B ( DL). (4.19) Using the approximation h I vi B n B ( DL) n B v B I, assuming the cross section does not vary over the thermal width of the electron VDF, L = n r Sc s,s me 1 =, (4.2) n n I e,b m i n n I where the second equality results from the use of the Langmuir condition, returning the result of Eq. (4.3). The cross section in Eq. (4.2) is strongly dependent on the energy of electrons gained when passing the double layer. The e ect of this energy dependence was not previously considered. Instead, estimates based on a constant cross section were used to determine the length scale [2, 2]. The length scale as a function of double layer energy is plotted from Eq. (4.2) in Fig for helium neutral background pressures of 5, 1, and 2 mtorr. To self consistently determine the size of the anode spot a constraint on the double layer potential is needed. In steady-state, the power entering and leaving the spot plasma will also balance. Most of the bulk electrons entering the spot pass through without undergoing an ionizing collision and exit at the electrode. However, a small fraction ionize neutrals creating the spot plasma. For the non-ionizing electron component, the power entering and leaving the anode spot are in balance. The power input into the spot plasma is the number of ionization events per unit time multiplied by the residual energy left over after ionization of the neutral atom that goes into the kinetic energy of resulting particles. The power lost is the number of particles leaving the spot per unit time multiplied by the amount of energy that they carry with them. The resulting balance equation is h I vi B n n n e,b V S (e DL E I )= i A i E i + e A e E e, (4.21) where e and i are the fluxes of ions and electrons leaving through their respective loss areas A e and A i at energy E e and E i. Making the approximation h I vi B n e,b e,b I by assuming the cross

115 98 Length Scale (cm) mtorr 1 mtorr 2 mtorr Double Layer Potential (ev) Cross Section (1-17 cm 2 ) Figure 4.13: The anode spot length scale L from Eq. (4.2) as a function of the double layer potential for helium neutral pressures of 5, 1, and 2 mtorr. The electron impact ionization cross section obtained from LXcat [4] is also shown. section is constant over the thermal width of the EVDF, Eq. (4.21) is e,b In n V S (e DL E I )= i A i E i + e A e E e. (4.22) This can be evaluated for each of the three di erent anode spot potential profile configurations discussed in Chapter When the sheath at the anode is an electron sheath the loss area for ions is the double layer surface area A S and the loss area for electrons is A E. Particle balance tells us that the loss of electrons to the anode must balance the loss of ions to the double layer resulting in n S v TeS A E = n S c s A S. (4.23) Substituting this relation in the electron loss term of Eq. (4.22) and using the length scale from Eq. (4.2) along with E i = T es /2 for ions and E e = m e v 2 T es /2=T es /2 for electrons results in e DL = T es + E I. (4.24) When the sheath at the anode is an ion sheath ions are lost to the double layer and the electrode resulting in A i = A S + A E. Once again, the ion flux is i = n S c s and the energy is E i = T es /2. The electron loss only occurs to the anode since DL is much greater than the di erence in potential between the spot plasma and the anode. Particle balance results in A E e =(A S + A E ) i.

116 99 In this case, an electron lost to the anode is lost at the energy determined by the average speed of e I electrons E e = m e v T 2 ei /2. Using these considerations, the double layer potential is TeS e DL = E I T ei AE +1. (4.25) A S Note that the use of di erent temperatures T ei and T es is due to the total electron temperature T es controlling the Bohm speed, while T ei controls the thermal flux of the particles of the dominant electron population in the spot (e I electrons). Finally, consider the case where the sheath at the anode is an electron sheath with a virtual cathode, here the collection areas are the same as in the previous case. The flux of ions to the double layer is given by the same relation used in the electron sheath case, while the flux of electrons is reduced by the dip potential of the virtual cathode and is e = n S v TeI e e D/T ei. Using particle balance along with E i = T es /2 for ions and E e = m e v T 2 ei /2 the relation in Eq. (4.22) reduces to TeS e DL = E I T ei. (4.26) The predictions of this section can be compared with the anode spot from the simulation in Chapter Although these simulations never reached steady-state, the anode spot size varied slowly beginning around 1.7 µs after the initial double layer motion around 1.2 µs, see the electric field magnitude in Fig Using the available particle velocity data from Fig. 4.7, the temperatures can be calculated allowing a comparison. See Chapter 3.1 for details of the temperature calculation. From this data, the total electron temperature within the anode spot due to both e I and e B components is T es =3.8 ev, while the temperature for the e I component is T ei =1.15 ev. At this time, the 2D electric field shows the area ratio of the double layer surface to electrode surface is roughly A S /A E 3. Inserting these quantities into Eq. (4.25) results in DL E I /e 4.5 V (where E I = 24.6 ev for helium), which is close to the value DL E I /e 4.38 V from the double layer potential in Fig Ion Presheath in the Anode Spot At the high potential side of the double layer, ions leave the anode spot plasma entering into a positive space charge region. From the point of view of an ion within the spot plasma, this is similar to an ion entering an ion sheath. The Bohm criterion applies. The ion must enter the nonneutral region at a velocity exceeding it s sound speed and this velocity is attained in a presheath region leading up to the sheath edge. It was previously suggested that this presheath length was

117 1 determined by the ion-neutral collision mean free path [2], however in some cases this would result in a presheath significantly longer than the observed size of anode spots in experiments. The ion presheath length can be determined by considering the rate constants for processes in the presheath. If the rate constant for ion-neutral collisions is greater than that for electron impact ionization, the presheath length is determined by the ion-neutral mean free path. However, if the rate constant for electron impact ionization is larger, the presheath is dominated by ionization. In the latter case, the presheath length would become half of the plasma length, in this case half of the anode spot size. This is one of the main results described by the various formulations of the Tonks-Langmuir presheath models [13, 14]. In this section, the rate constants for these two processes is estimated, determining which process is responsible for the presheath length. The rate constant is defined as v 1 v 2 ( v 1 v 2 ) averaged over the distribution function of the incident and background particles denoted by subscript 1 and 2 respectively. This is Z K = d 3 v 1 d 3 v 2 f 1 (v 1 )f 2 (v 2 ) ( v 1 v 2 ) v 1 v 2, (4.27) where unity. is the interaction cross section and the velocity distribution functions are normalized to If the characteristic velocities of the background particles is much less than that of the incident particles then v 1 v 2 v 1 and the v 2 integral can be evaluated. Eq. (4.27) is evaluated assuming the particles accelerated by the presheath have a flow shifted Maxwellian distribution, f 1 (v) =[1/( 3/2 vt 3 )] exp[ (v U)2 /vt 2 ], where U is the flow shift and v T = p 2T/m is the thermal speed. Writing the integration variable in terms of energy E by using v = p 2E/m, and evaluating the integral in cylindrical coordinates aligned with the flow direction, K(U, T )= 4 1/2 m Z 1 de r E T apple (E)exp E T U v T 2 r v T U E U sinh. (4.28) v T T The ratio of the rate constants for electron impact ionization and elastic ion-neutral scattering K I /K He + He is considered to determine which processes is dominant. The cross sections were obtained from the Phelps database on LXcat [4]. Because of the strong dependance on energy, the rate constant for ion neutral collisions is calculated using Eq. (4.28) with a flow of c s /2whichis typical of a presheath ion. The ionization rate constant is estimated as r 2e DL K I I(e DL). (4.29) m e

118 11 Double Layer Potnetial (V) T i (ev) K I /K He-He + Figure 4.14: The ratio of rate constants for electron impact ionization and elastic ion-neutral scattering in Helium as a function of ion temperature and double layer potential. Under typical laboratory conditions, where T i.3ev, K I is dominant. Contours for K I /K He+ He = 1, 2, and 4 are shown. The ratio of these rate constants is shown in Fig The rate constant for electron impact ionization is several times that due to ion-neutral scattering for room temperature ions and values of e 25 ev. This means that the presheath length will be determined by ionization processes for a helium neutral gas background. Similar results are obtained for argon plasmas. 4.4 Comparison with Experiments Experimental measurements of the anode spot onset and steady-state were provided by Dr. Edward Barnat at Sandia National Laboratories for comparison with the simulations and theory presented in this chapter. A description of the experiment is briefly given here. A plasma was generated in a modified GEC reference cell [61] by biasing the upper electrode labeled plasma generating electrode in Fig The discharge power of this electrode was 38 V at 1 ma. The plasma generating electrode was biased on for 49 ms and o for 1 ms during a 5 ms cycle time. A second lower electrode labeled spot generating electrode in Fig was biased from 5 V to 1 V, 75 µs after the plasma generating electrode was biased o. During this time, an anode spot formed at the electrode and reached steady-state. The cycle was synchronized to the optical

119 ;,5" B"C-$"µ:" -$$"! 891:,1"25+541;+2" 595<=4>? "91:54":A55=: )$-"+," ()*"+," -$",,"/"-$",,",123+2"1451" % & % 595<=4>?5!$$"67!".! Figure 4.15: A schematic of the experiment from which measurements of the anode spot onset and steady-state were provided. The anode spot formed on the electrode embedded in the lower boundary. Figure provided by Edward Barnat, with permission. diagnostics used and was repeated so that su cient statistics could be gained. Measurements of the electron density and electric field were made using laser collision induced fluorescence (LCIF) and laser induced fluorescence dip spectroscopy (LIF-dip), respectively. The former method was described in Chapter 2.1, while the latter method relies on measuring the stark shift of laser excited metastable states of neutral atoms in the plasma [9, 91]. These measurements were not made simultaneously, but were repeated under the same experimental conditions.

120 Observations of Spot Onset In Chapter 4.3.1, the spot onset was described as a result of buildup of positive space charge as a result of electron impact ionization within the sheath. A zero electric field region in front of the electrode developed when an equal amount of positive and negative space charge was present within the sheath. A further increase of positive space charge led to the formation of a potential well for electrons just o of the electrode surface. Within this potential well, electrons born from ionization are trapped by the electric field. This results in the electron density growing until quasineutrality is established. At this point, the spot plasma begins to expand. This process is evident in the measurements of plasma emission and electron density presented in Fig and Fig In the first 25 ns after the voltage step, an increase in plasma emission is present prior to the formation of the region of increased electron density at the electrode surface. This is consistent with the description above. Ion space charge will build up within the sheath before an indication of increased electron density. This is due to the time required for the accumulation of enough positive space charge within the sheath to form the potential well that traps electrons. The early time plasma emission is an indication of ionization since the energy for excitation is close to the energy required for ionization ( 2 ev vs 24.5 ev). This same general trend is seen in the simulation described in Chapter 4.2. Fig shows the electron density, electric field, and ionization rate which is used as a proxy for plasma emission in the simulation. Little ionization is seen in front of the electrode before the voltage step from 4 V to 5 V. An increase in electrode bias is followed by an increase in the ionization rate, although initially there is no increase in electron density. Since the ionization rate is expected to be an indication of plasma emission, this is consistent with the suggestion that the increased plasma emission is an indication of the buildup of ions near the electrode. The electric field from simulation in Fig can also be compared with experimental values presented in Fig The electric field geometry and magnitude during spot onset between t = 8.95 µs and t = 12 µs (the voltage step in this simulation is applied at t = 9 µs) qualitatively agree with that between t = and t = 1 ns in experiments. One discrepancy between the experiment and simulation is the timing of the electric field structure relative to the increase in electron density. Experimental measurements of the electric field and electron density in the 2D maps of Fig show that there is an increase in electron density at the electrode surface before the electric field develops a potential well in front of the electrode. The existence of increased electron density at t = 25 ns in a location where the electric field magnitude is 1 V/cm suggests there may be

121 14 1 Voltage (V) Current (ma) $nm$plasma$emission$ Arb. & Height&above&the&anode&(mm)& Metastable$density$ Electron$density$ Arb. & 1 9 &e/cm 3& Time&(µs)& Figure 4.16: Experimental values of electrode bias, current collection, and plasma quantities. The plasma emission, metastable density, and electron density are plotted along a line perpendicular to the electrode as a function of time. The initial anode spot formation is at t.3 µs. Figure provided by Edward Barnat, with permission.

122 15 59+nm+Plasma+ Emission (arb) Metastable Population (arb) Electron+ Density (1 (e/cm 9% e/cm 3 ) 3 )% Electric Fields+ (V/cm)?1+ns 25+ns 5+ns 1+ns Height+above+the+anode+(mm) 2+ns 5+ns Radial position (mm) Figure 4.17: 2D colormaps from experiments indicating the measured plasma emission, metastable density, electron density, and electric field before and after the voltage step was applied. Figure provided by Edward Barnat, with permission.

123 16 a mismatch in the plasma conditions between the time of measurement of the electric field and electron density 4. Another di erence between simulation and experiment is the transient fluctuations of the anode spot during onset. The electric field, electron density, and plasma emission in Fig and Fig show that the initial spot formation is followed first by a collapse of the spot plasma around t = 2 µs and then by reformation. At long times the spot density and emission settles to near constant values, while the metastable density continues to increase. This transient behavior will be explored further in the next section Transient Behavior One di erence between simulation and experiment is the value of the voltage step used to induce spot onset. In the experiment, the increase in bias was from 5 V to 1 V, while in simulation this was from 4 V to 5 V. Two additional simulations were used to determine what role the voltage step plays in the transient behavior after the initial spot onset. Both simulations utilized the same setup described in Chapter 4.2.1, the only di erence being the biasing scheme for the electrode. The goal of these simulations was to determine the role of large and fast voltage steps. Both simulations utilized a step from V to 12 V to induce spot onset. The first simulation had the bias increased over.6 µs, while the second did the same over 1 ns. The ion density, electric field magnitude, and plasma plasma potential is shown for each simulation in Fig as a function of time with quantities plotted along a line perpendicular to the electrode. For the simulation with the slower voltage step, the electric field indicates that the inner edge of the double layer retracts towards the electrode between t =.7 µs and t = 1 µs. There is a corresponding decrease in ion density within the anode spot at these times. The electric field, ion density, and potential indicate that the fluctuations in the double layer position result from the response of the plasma potential to bulk plasma ions displaced due to the expansion of the anode spot and accompanying electric field. The increase in electric field near the electrode causes ions to respond ballistically. The resulting disturbance can be seen propagating away from the electrode in the plots of ion density and electric field magnitude shown in Fig As these ballistic ions propagate away from the double layer, the plasma potential returns to a steady value. 4 Recall that the measurements were not simultaneous.

124 17 Electric Field (V/cm) Ionization Rate (arb.) Electron Density (1 9 cm -3 ) μs μs μs μs Distance (cm) μs Distance (cm) Figure 4.18: 2D color maps of the simulated electric field magnitude, ionization rate, and electron density for comparison with experimental results in Fig Here, the ionization rate is a proxy for the plasma emission.

125 18 Fig indicates a similar process occurs in experiments. The plot of electron density reveals that a density disturbance in the bulk plasma due to the expansion of the anode spot is present. The increased electron density marked with the arrow and dashed line is likely due to electrons being dragged along by ballistic ions responding to the electric field. The electron density within the spot also shows fluctuations similar those seen in the simulated ion density. As the plasma potential initially responds, the potential across the double layer decreases, lowering the ionization rate at the high potential side. This results in a decrease in the spot plasma density. More evidence of this process is seen in Fig Between 1 ns and 2 ns, the plasma emission and double layer electric field both decrease, indicating that this is due to a decrease in energy of electrons entering the anode spot from the bulk, reducing the amount of ionization available to sustain the spot. During the transient behavior, the experimental measurements of the current collection in Fig show fluctuations as well. Comparing the current collection to the double layer position indicated by the electric field magnitude in Fig demonstrates that the initial jump in current collection and subsequent fluctuations correspond to changes in collection area. Measurements of the plasma emission in Fig also indicate that the spot size indicated by the increased intensity also correlates with these fluctuations. This is in agreement with the results of Chapter which demonstrate that the amount of current collected is due to the collection area of electrons from the bulk plasma. The simulation with the fast voltage step shows the e ect of a more abrupt onset. The anode spot quickly forms and the plasma potential quickly raises, decreasing the energy of bulk electrons entering the spot. Ionization within the spot stops causing its collapse around 1 µs. This is followed by a much slower reformation around 2 µs. These simulations along with the one presented in Chapter suggest the transient behavior will be minimized in cases with either smaller or slower voltage steps. 4.5 Summary In this Chapter, the anode spot was studied for the first time using PIC simulations. These simulations demonstrated that electron impact ionization within the sheath results in a positive space charge layer adjacent to the electrode. With su cient ionization, the positive space charge results in a zero field region just in front of the electrode. At this point, additional ionization leads to the formation of a potential well which traps electrons born from ionization. Once this region

126 φ (V) 1 φ (V) Time (µs) Time (µs) n i (1 9 cm 3 ) Distance from Electrode (cm) Time (µs) Time (µs) Time (µs) Time (µs) 2 E (V/cm) 1 5 φ (V) Figure 4.19: PIC simulations of spot formation with a voltage step from V to 12 V shown in the top panels. The step duration in the left column is.6 µs and on the right is 1 ns. The ion density, electric field magnitude, and potential as a function of time are shown along a line perpendicular to the electrode.

127 11 is established, electrons born from ionization are traped resulting in an increase in electron density and formation of a quasineutral plasma. A model for the spot onset was formulated based upon observations of the simulated anode spot. The main feature of this model is that an imbalance in flux densities crossing the double layer leads to its motion and the expansion of the anode spot plasma. It was shown that this imbalance occurs when ions are born within the sheath faster than they leave. Using estimates of the sheath ionization rate, the value of the electrode bias relative to the plasma potential was tied to the sheath ionization. Predictions of the critical bias for spot onset were found to be in agreement with past experiments for di erent electrode sizes and plasma conditions, providing an experimental test of the spot onset model. Following the model for onset, steady-state properties were determined by assuming current, power, and particle balance of the spot plasma. Maintenance of quasineutrality within the spot dictates the form of the sheath between the anode spot plasma and the electrode determining how particles are lost from the anode spot. Balance of the total ionization rate and particle loss rate determines the anode spot size as a function of the energy of electrons entering the spot from the bulk. The size is determined once the double layer potential is known. Balance of power lost from and deposited into the spot plasma determines this potential. The predicted energy gain of an electron crossing the double layer potential is a few electron volts above the ionization energy for typical experimental conditions. This result is consistent with several anode spot experiments. Finally, the results from simulation and theory were compared with recent optical measurements of the plasma emission, electron density, and electric field magnitude. Simulation and experiment agree qualitatively with regard to the anode spot geometry and electric field. In addition, the experiments indicate ionization occurs for some time before a change in electron density is present. This is in agreement with the theory for onset which requires the ion space charge to build up in su cient amounts to form a potential well before electrons can be trapped. Finally, the plasma in the experiments exhibited transient behavior which was not present in the simulations of Chapter In subsequent simulations these fluctuations were found to be due to a response in the plasma potential as ions are expelled from the electrode region during onset. This behavior was found to be more pronounced for larger and faster electrode voltage steps.

128 111 CHAPTER 5 CONCLUSION In this Thesis, the sheath and plasma near small electrodes biased above or near the plasma potential were studied. This provides the first overview of how positively biased electrodes interface with their surrounding plasmas under low pressure laboratory conditions. Assumptions about the most basic of properties of electron sheaths such as their size, interface with the plasma, and current collection were revisited. Previously, it was thought that the electron sheath interfaced with the plasma without the need for a presheath. Chapter 2 presented the first evidence for an electron presheath. In the newly formulated electron presheath model, electrons accelerate to near their thermal speed by the sheath edge and are driven by a pressure gradient. The electron presheath accelerates electrons over a region much longer than that for the ion presheath, where the length scale is usually determined by the ion-neutral collision mean free path. Similarly, a friction force due to momentum scattering determines the length scale of the electron presheath. The presheath model was found to be consistent with experimental measurements and with PIC simulations. In the model, the electron presheath length was predicted based on an estimate of the momentum scattering rate due to electron-neutral elastic collisions. The model requires a friction term in the momentum equation to determine the presheath length. While electron-neutral elastic collisions is one possibility for the friction, the PIC simulation demonstrated a finite presheath length without the inclusion of these collisions. Another possibility for a friction term is particlewave scattering. In the simulation, the near-thermal-speed electron flow in the electron presheath was observed to excite an electron-ion two-stream instability. This instability excited ion acoustic waves with a dispersion relation that was consistent with that predicted using the expected electron presheath flow velocity. Electron scattering o of these waves could set the presheath length if it is the dominant momentum scattering process. More work is needed to determine which, if any, of these processes set the presheath length under typical experimental conditions. Another possible area for exploration is the e ect of a magnetic field on the electron presheath behavior. An electron presheath theory including the e ects of a magnetic field would be directly applicable to the situations encountered with tethered space probes [18]. The greater sheath area and sheath edge electron flow velocity in the theory of Chapter 2 suggests one possible reason for the larger than expected electron current collection of such probes [22]. A theory incorporating

129 112 specific magnetic field geometries may be able to directly predict the observed electron current collected by these probes. Such a theory may also help understand how electrodes used for the control of scrape o layer plasmas exhibit di erent behavior when biased above the plasma potential [25]. In Chapter 3, the form of the presheath near electrodes biased near and above the plasma potential were studied using PIC simulations. When the electrode is biased within T e /2e below the plasma potential, the EVDFs exhibit a loss-cone type truncation due to fast electrons overcoming the small potential di erence between the electrode and plasma. Once the bias exceeds the plasma potential, an electron sheath is present. In this case, the truncation driven behavior persists, but is accompanied by a flow shift in the maximum value of the EVDF, which was predicted in the theory of Chapter 2. The flow moment has significant contributions from both the flow shift of the EVDF maximum and the loss-cone truncation. Chapter 3.3 suggested methods for fine tuning the electron presheath model, including the e ects of the loss-cone and flow shift. Following these suggestions, more precise models of the electron presheath could be formulated to incorporate the e ect of the electrode geometry. In particular, this may be useful for determining how the electrode geometry may cause deviations from the predicted sheath edge flow velocity (V e = p (T e + T i )/m e ) from the 1D model of Chapter 2. Finally, the formation of anode spots near electrodes with large positive bias were studied in Chapter 4. This chapter presented the first PIC simulations of anode spot onset. These simulations showed that su cient ionization within the sheath led to a buildup of positive space charge due to the immobility of ions relative to electrons. With su cient buildup of positive space charge, a potential well for electrons formed. This well trapped electrons and allowed for the formation of the spot plasma. This model for the spot onset was compared with recent optical experimental measurements of the anode spot plasma emission, electron density, and electric field. The experiments agreed qualitatively with the results from simulation and the model. Chapter 4 also predicted steady-state properties of the anode spot using a current, power, and particle balance based analysis. This model accurately predicts the double layer potential measured in prior experiments, as well as the di erent possible forms of sheath observed between the spot plasma and electrode. While Chapter 4 provides a through understanding of spot onset and steady-state, there are still many open questions. The most notable involve the cause of the observed hysteresis in I-V traces [2] and anode spot stability. In experiments, anode spots have been observed to form, collapse, and reform repetitively with a period of approximately 1s of microseconds[77, 78, 79]. Since this timescale is much longer than the time for an electron to cross a typical experiment, this

130 113 process is likely limited by the ion dynamics. While this was not a subject of study of Chapter 4, the transient behavior of the fast bias step simulation presented in Fig supports this claim. In this simulation, the expulsion of ions from the electrode region occurs before the reformation of the anode spot. This simulation demonstrates that the initial spot extinguishes due to the rising plasma potential which results in a decrease in the electron impact ionization rate. One possibility for this raise in plasma potential and anode spot collapse is that the anode spot steady-state size, determined by the neutral pressure and electron temperature in the model (See Chapter 4.3.3), is incompatible with steady-state current balance in the plasma chamber in which it is contained. When the spot grows too large, too many electrons are lost for steady-state balance of global current loss. Further work will be needed to explore this behavior. An understanding of the anode spot stability and steady-state properties will aid in the design of experiments where anode spots are used as ion sources [92] and for dust confinement[93].

131 114 APPENDIX A NON-VLASOV BEHAVIOR IN PARTICLE-IN-CELL SIMULATIONS In Chapter 1.5.4, the e ects of the finite size of particles, as seen by the simulation grid, on the Lenard-Balescu kinetic equation were reviewed. The main result was that in a stable plasma the finite size of particles reduced the collision rate by an order of magnitude when the cell size is equal to the Debye length. Perhaps it is this result that has led to the common claim that the PIC method approximates solutions to the Vlasov equation [94, 95]. This claim has led to a misunderstanding of what collision based kinetic e ects are present in PIC simulated plasmas. One example which illustrates this misunderstanding was a test of a recently proposed extension to the Lenard-Balescu kinetic equation for an unstable plasma. The modified kinetic equation demonstrated that the collision rate of particles is enhanced in the presence of an instability, where the collision rate enhancement resulted from the interaction of single particles with the electric field fluctuations due to the collective motion of particles from which the wave is composed [96, 49]. Initially, it was thought that In order to capture the instability-enhanced collision process, a particle-in-cell simulation needs to include an algorithm for the instability-enhanced collision operator [97]. However, later simulations demonstrated that the instability enhanced friction is present despite the fact that the finite particle size reduces the collision rate [5, 98]. To better understand when these instability enhanced kinetic e ect are captured by PIC, the derivation of the instability enhanced collision operator is repeated with modifications for the finite size of particles as seen by the simulation grid. The derivation closely follows Ref. [96]. When necessary, results from Ref. [96] will be cited to keep the present description to a minimum. The exact description of a plasma is given by the Klimontovich q s m s E =, (A.1) where F = X i [v v i ] [x x i ] (A.2) is the exact distribution function. In a typical derivation of the Lenard-Balescu collision operator, the exact distribution function is written as F = f + f, with f and f being the smoothed and discrete parts such that hf i = f and h fi =. Subtracting the ensemble averaged Klimontovich from Eq. (A.1) leaves an equation for f, which can be solved to provide a collision operator. The

132 115 instability enhanced collision theory solved the f kinetic equation d f dt = q (A.3) following the typical derivation of the Lenard-Balescu kinetic equation (See Ref. [47] for a derivation) allowing for the existence of poles in the upper half plane when inverting the Laplace transform. Here, the poles in the upper half plane result from zeros of the dielectric with positive imaginary part that occur when instabilities are present. In this Appendix, solutions to Eq. (A.3) with modifications due to the the finite size of particles as seen by the grid are found. The modifications necessary are F = X i [v v i ]S[x x i ], (A.4) which is the exact distribution of finite sized particles, d f = q Z dx S[x dt m x (A.5) which is the kinetic equation with a modification to the collision force as seen by the E =4 X Z q s d 3 vdx fs[x x ] (A.6) s which is Gauss law with the modification that represents the field interpolation at the nodes. Recall that S is the shape function, introduced in Chapter 1.5.1, which represents the interpolation scheme between the grid and the particle positions. In the discussion presented here, for simplicity, it is assumed that the shape function is symmetric and that the interpolation method for accumulating the charge to the grid and for applying the force to the particles is the same so that there are no self forces. Taking the Laplace transform in time and Fourier transform in space of Eq. (A.5) results in f() iw ˆf ik v ˆf = q S(k), (A.7) where f() is the initial condition of F. Rearranging Eq. (A.7) and making use of E = ik gives ˆf = i f() w k v q m w k v. (A.8) The Fourier-Laplace transform of Gauss law is X Z k 2 ˆ = 4 q s S(k) d 3 v ˆf. Inserting Eq. (A.8) into Gauss law results in ˆ 1+ X 4 q 2 Z s m s s k 2 S2 (k) d w s 1 = X k v s Z 4 q s S(k) (A.9) d 3 v i f() w k v. (A.1)

133 116 This expression can be written more simply by identifying the quantity in parenthesis as the dielectric response modified for the finite size of particles. This was first introduced in Eq. (1.69) and is denoted PIC.Thisresultsin X Z ˆ 4 q s S(k) = k 2 d 3 v i f() s PIC w k v. (A.11) The initial condition on the exact distribution is F = X i [v v io ]S[x x io ]. (A.12) Noting that Z d 3 xe ik x S(x x o )=S(k)e ik xo, (A.13) the transform of f() is f() = X i S(k)e ik xio (v v io ) (2 ) 3 (k)f. (A.14) Inserting this into Eq. (A.11), the term proportional to f does not contribute due to quasineutrality and the remaining part is ˆ = X Noting that Ê = ik ˆ, Eq. (A.15) becomes s,i 4 q s S 2 (k) k 2 PIC e ik xio w k v io. (A.15) Ê = S 2 (k) X s,i 4 q s ke ik xio k 2. (A.16) PIC w k v io Comparing this to the value of Ê of Eq. (16) of Ref. [96] it is apparent that they are related by the factor of S 2, Ê = S 2 (k) ÊLB+IE, (A.17) where Ê LB+IE is the value from the instability enhanced collision theory of Ref. [96]. To write the modification to the collision operator, it is convenient to write the operator in the form C(f s,f s Z d 3 Q m f s (v)f s (v ), m (A.18) where the collision kernel is Z Q = q2 s m s d 3 k (2 ) 3 Ẽ 2 [k (v v )]. (A.19) The collision kernel can be evaluated by taking the inverse Laplace transform of Eq. A.16, Ẽ and inserting into Eq. (A.19). The inverse Laplace transform Ẽ can be written in terms of the result from the IE theory, Ẽ = S 2 (k) ẼLB+IE, (A.2)

134 117 and inserted into Eq. (A.19) resulting in Z Q PIC = q2 s m s d 3 k (2 ) 3 S(k) 4 Ẽ LB+IE 2 [k (v v )]. (A.21) From here the result can be written by comparing the integrand of Eq. (A.21) to the integrand of Q from Eq. (37) and (16) of the instability enhanced theory of Ref. [96] with Q PIC,whichresultsin Q PIC = 2q2 sq 2 s m s Z apple d 3 k kk S(k) 4 k 4 [k (v v 1 )] PIC (k, k v) 2 +X j (w R,j k v)e 2 jt 2 w j. (A.22) The first term of this equation is the same collision kernel that shows up in Chapter The modification of the first term by the shape function reduces the coulomb collision rate by more than an order of magnitude in most practical situations where the cell size is comparable to the Debye length, hence the contributions from this term to plasma behavior in PIC simulations is usually negligible. Evaluating the second term in the integrand, one can see that the factor of S(k) 4 cancels with a similar factor PIC (k,w)/@w 2 w j. The result is that the collision rate from instability enhanced part is only modified by the frequency and growth rate of the instabilities determined by the roots of the modified dielectric PIC. This is due to the fact that the roots (w j, j) of di er from those of PIC. It is well known that particle-wave interactions are captured by quasilinear theory. Therefore, it is no surprise that particle-wave scattering would be accurately captured by PIC simulations in which waves are present. However, quasilinear theory requires the initial electric field fluctuation to be input as an initial condition, while the instability enhanced friction theory provides the source of the fluctuations self consistently [49]. The results of Eq. (A.18) and (A.22) demonstrate that the PIC method self consistently captures the generation of waves, where the instability grows out of collisional processes provided that the spatial grid resolves the dielectric response. An example of how the shape function modifies the unstable root of the dielectric is shown for the electron-ion two stream instability of Chapter in Fig. A.1. Here the shape function was taken to be that of a gaussian cloud S(x) = 1 (2 ) 3/2 R exp x 2 3 2R 2 (A.23) which has the Fourier transform k 2 R 2 S(k) =exp. (A.24) 2 The main discrepancy is at large values of the wavevector where the finite size particles do not resolve the dispersion relation. For the case of the simulations of Chapter 2.3, the dielectric is

135 118 1 Electron-Ion Two Stream Instability 1-2 ω/ω pi ω R gaussian cloud R=.5λ D ω I gaussian cloud R=.5λ D ω R gaussian cloud R=λ D ω I gaussian cloud R=λ D ω R Exact ω I Exact kλ D Figure A.1: The dispersion relation from the unstable root of the dielectric function for a distribution of flowing electrons and stationary ions given by Eq. (2.29). The exact value of these roots are compared with the roots of the dielectric modified by the shape function in Eq. (A.24). The deviation from the exact value is significant at scales smaller than R. su ciently resolved by the simulation grid for the instability enhanced component of the collision kernel Eq. (A.22) to contribute an additional friction force to the particles. In the simulation, the electron presheath had a finite length scale even though no collision processes with electrons were explicitly included. However, a finite length scale is only possible in the presheath model if there is a non-zero collision frequency. A friction force on electrons is a potential explanation for which physical processes set the presheath length scale in Eq. (2.8). Evidence for this friction force is seen in the residual in the momentum equation plotted in Fig. 2.12a. A more complete account of the 2D e ects of the boundary would be needed to determine the extent to which the instability enhanced friction may contribute to the residual.

136 119 APPENDIX B DERIVATION OF THE MODIFIED LANGMUIR CONDITION The modified Langmuir condition provides an energy density balance condition across a double layer observed from the lab frame. In this appendix, the derivation of the modified Langmuir condition from Song, Merlino, and D Angelo [84] is provided. The original Langmuir condition is then trivially obtained from the modified Langmuir condition when the double layer is stationary. Consider a double layer moving in the lab frame at velocity U DL shown in Fig. B.1. Ignoring the e ects of a magnetic field, ionization, collisions and stress tensor gradients, the 1D momentum and continuity equations in the lab frame m a n + a a at a = n a q a au a =, where m a, n a, u a, q a, and T a are the mass, density, flow moment, charge, and temperature of species a, and E is the electric field. Assuming a steady state plasma in the frame of the double layer, and neglecting it s detailed structure, the density, flow velocity, and electric field measured from the frame of the double layer can be related to the lab frame using the relation x = z Z t t U DL (t )dt. (B.3) Using this, the density, flow velocity, and electric field in the lab and double layer frame are related by Z t n a (x) n a z U DL (t )dt = n a (z,t), (B.4) t Z t u a (x) u a z U DL (t )dt = u a (z,t), (B.5) t and Z t E(x) E z U DL (t )dt = E(z,t). (B.6) t Writing the momentum and continuity equation in terms of the quantities in the double layer frame results in m a + = n aq a E (B.7)

137 12 Lab Frame! Double Layer Frame! Potential! U DL! x 1! x DL! x 2! z DL! x! z! Figure B.1: A double layer that has a moving position z DL in the lab. In the frame of the double layer (with spatial coordinate x), moving at velocity U DL with respect to the lab frame, the position x DL is constant. The locations x 1 and x 2 indicate the position of the double layer edge in the moving frame. Note that E(x 1 )=E(x 2 ) = at these locations. and Noting @z =. U DL and = 1, the momentum and continuity equations are a m a n a v at a = n a q a E and where v a = u a v a =, U DL is the flow moment of species a in the double layer frame. Making use a v 2 = v a v {z } = +n a v (B.11)

138 121 the momentum and continuity equations Eq. (B.9) and Eq. (B.1) can be combined, resulting a n a v 2 a + n a T a Multiplying Poisson s equation through by E, = n a q a E. =4 X a n a q a E, (B.13) inserting Eq. (B.12), and integrating between the edges of the double layer x 1 and x 2 leads to X x 2 (m a n a va 2 + n a T a ) =. (B.14) x 1 a In this equation, the fact that the electric field at the edges of the double layer is zero has been used. This equation gives an energy density balance relation across the double layer. There are several particle populations which need to be considered at each side of the double layer. At the high potential side, electrons with density and temperature n e,hi and T e,hi are trapped by the double layer. At this same location, ions enter the double layer with density, flow velocity, and temperature n i,hi, v i,hi, andt e,hi. These ions, which originate from the high potential side, also need to be considered at the low potential side. At the low potential side, there are ions trapped by the double layer with density and temperature n i,lo and T i,lo. Electrons from the low potential side enter the double layer with density, flow velocity, and temperature n e,lo, v i,lo, andt e,lo.once again, the electrons originating from the low potential side also need to be considered on the high potential side of the double layer. The relevant populations at each side of the double layer are illustrated in Fig. B.2. When the double layer potential is large enough, the kinetic energy of the accelerated population exiting each side of the double layer is the dominant term in the energy density balance given by Eq. (B.14). The balance of the kinetic energy terms is m i n i,hi v 2 i,hi x 2 = m e n e,lo v 2 e,lo x 1. (B.15) If the velocity of particles exiting the double layer is primarily due to the energy gained in its electric field, then particles crossing the same electric field will have the same energy. This condition is 1 2 m iv 2 i,hi x 2 = 1 2 m ev 2 e,lo x 1. (B.16) Replacing one of the electron velocities in the ve,lo 2 term of Eq. (B.15) with the relation of Eq. (B.16) results in n i,hi v i,hi x2 = r me m i n e,lo v e,lo x1. (B.17)

139 122 Potential! n e,hi T e,hi! E=! m i n i,hi (x 1 )V i,hi2 (x 1 )! m e n e,lo (x 1 )V e,lo2 (x 1 )!!! Ion Rich! E"! Electron Rich!!! E=! n i,lo T i,lo! m i n i,hi (x 2 )V i,hi2 (x 2 )! m e n e,lo (x 2 )V e,lo2 (x 2 )! Distance! x 1! x 2! Figure B.2: The trapped and free particle populations near a double layer. At the high potential side, electrons are trapped and ions are accelerated by the double layer electric field. At the low potential side, ions are trapped and electrons are accelerated.