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Physics-regularized Machine Learning To Approximate 3D Ideal-MHD Equilibria At Wendelstein 7-X
(2024)
The magnetohydrodynamic (MHD) equilibrium model is one of the fundamental building blocks in the description of a magnetically confined plasma. The computational cost of constructing solutions to the 3D ideal-MHD equilibrium problem is one of the limiting factors in stellarator research and design; in particular, it limits the extent to which we can perform sample-intensive applications, applications which require many samples to be evaluated to yield meaningful results. Sample-intensive applications in stellarator research and design include, for example, equilibrium reconstruction, stellarator optimization, and flight simulators. In this thesis, we investigate how faithfully artificial neural networks (NNs) can quickly approximate ideal-MHD equilibria in stellarator geometries, starting with Wendelstein 7-X (W7-X), the world’s most advanced stellarator. In particular, we investigate (see section 1.7):
RQI: to what extent can NN models approximate the MHD equilibrium solution for different W7-X configurations and plasma profiles? What
is the speed-accuracy trade-off offered by NN models?
RQII: to what degree the NN model faithfully reproduces equilibrium quantities of interest (e. g., MHD stability)? To what extent can NN models meet the requirements of downstream applications (e. g., Bayesian
inference, stellarator optimization) in terms of equilibrium quantities
accuracy?
RQIII: whether we can exploit the implicit representation of a MHD equilibrium, i. e., the equilibrium solution should satisfy the ideal-MHD force
balance equation, to improve the NN approximation’s accuracy;
RQIV: the reconstruction of the full posterior istribution of plasma parameters and equilibrium quantities with self-consistent MHD equilibria; moreover, how does the adoption of MHD equilibria approximated by NN models affect the inferred plasma parameters?
A deep NN model is developed to learn the ideal-MHD solution operator in W7-X operational subspace, yielding 3D equilibria up to six orders of magnitude faster than currently available MHD equilibrium codes. Physics domain knowledge is embeded into the NN model: equilibrium solution symmetries are satisfied by construction, and the MHD force balance regularizes the NN model to satisfy the ideal-MHD equations. The model accurately predicts the equilibrium solution and it faithfully reproduces global equilibrium quantities and proxy functions used in stellarator optimization. Finally, the developed fast NN equilibrium model has been applied in downstream applications to obtain W7-X configurations with improved fast-particle confinement and to infer plasma parameters with self-consistent MHD equilibria at W7-X.
Recent experimental campaigns in the Wendelstein 7-X stellarator, a
plasma-confining device designed to investigate the Magnetic Confinement Fusion
(MCF) approach to generating electrical power, have shown that the injection of
fuelling pellets had an unexpected and considerable impact on the performance of
the plasma. Rather than simply refuelling the device and `diluting' the plasma
energy, pellet injection is followed by a significant increase in the ratio of
the ion temperature to the electron temperature. It has been suggested that this
is not merely due to the improved confinement following the reduction of
turbulent transport after the pellet material has homogenised with the bulk
plasma, but also due to a direct transfer of energy from electrons to ions. The
proposed mechanism for this energy transfer is the ambipolar expansion of the
pellet plasmoid, the localised plasma structure produced by the
ionisation of ablated pellet material, along magnetic field lines.
Early work on pellet plasmoid expansion predicted that half the heating power
deposited in plasmoid electrons by collisions with hot ambient electrons is
transferred to plasmoid ions in the form of flow velocity as the plasmoid
expands. The complicated nature of the system of the pellet plasmoid embedded in
the ambient plasma, particularly the behaviour of electrons, which experience
many collisional and collisionless phenomena on multiple disparate timescales,
means that early models of the expansion were not wholly self-consistent, but
rather made use of strong approximations that apply in some regions of the
plasmoid but not in others. For example, only electrons and ions associated with
the plasmoid were rigorously treated, meaning that the framework was one of
`expansion into vacuum'. Combined with the assumption of Maxwellian electrons,
this led to an electric potential that was unbounded at infinity. Naturally, the
validity of the conclusions of such a model are called into question because the
approximations lose their validity far from the plasmoid and as time advances,
yet predictions about the final state of the plasma are desired. A deeper
investigation is required: careful consideration of the phenomena in question
and the timescales (and lengthscales) on which they act must be made in order to
rigorously construct a model that is valid throughout the entire expansion.
The first two papers presented in this thesis iterate on the model established
in the paper that first predicted the electron-to-ion energy transfer; their aim
was to find out how the character of the expansion changes with a more
sophisticated and accurate description of various phenomena, while remaining
within the existing framework of expansion into vacuum. Ultimately, we find that
the qualitative character is unchanged, and that approximately half the heating
power deposited in plasmoid electrons is transferred to ions.
Two other papers in this thesis address the limitations of the original model.
This is achieved by properly considering the electron kinetic problem in a
plasmoid. One paper considers the electron kinetic problem when electrons are
highly isotropised. In this case the kinetic equation can be integrated to
remove all but two independent variables, which is the maximum possible
reduction considering it is a time-dependent problem. The full nonlinear
integro-differential Landau self-collision operator is integrated exactly and
few approximations are made, leading to a rather general kinetic equation.
However, for fuelling pellets some anisotropy in the electron distribution is
expected. Another paper considers the electron kinetic problem (and the entire
plasmoid expansion) allowing for electron anisotropy. Careful consideration of
the ordering of timescales of electron phenomena in a pellet plasmoid leads to a
steady-state kinetic problem that we call collisional quasi-equilibrium (QE). QE
appears in many ways similar to the collisional steady-state characterising a
true thermal equilibrium. It was found that the time-dependent kinetic problem
of the earlier paper, with isotropic electrons, produces the QE distribution
function, corroborating the existence of the QE state. We then take moments of
the electron kinetic equation that is valid on the expansion timescale, assuming
that the electron distribution is that given as the solution to the QE kinetic
problem. This is completely analogous to what is done to obtain the Braginskii
equations or any Chapman-Enskog theory. The result is a set of equations for the
long-term evolution of the macroscopic quantities that describe the distribution
function existing in a quasi-steady-state at each point in time. It is from this
point that one may feasibly describe the plasmoid expansion with an accurate
picture of the electron kinetics and finally obtain the electron-to-ion energy
transfer so desired in a rigorous model of the expansion.
From a broader point of view, the two frameworks provided by these rigorous
investigations of the electron kinetic problem serve as a basis for the future
study of plasmoids. Such a `first-principles' approach to plasmoid dynamics is
novel and interesting in its own right, but it will be demonstrated that such an
approach is essential for pellet plasmoids owing to the fact that they are
poorly described by the `standard tools' of plasma physics.
Using the QE framework it was found that, once more, about half the heating
power experienced by plasmoid electrons is transferred to plasmoid ions. The
incredible robustness of the prediction of such an energy transfer is, in the
author's opinion, the result of the self-similar nature of the expansion found
as a solution to the original model. As a rule, the profiles of self-similar
solutions tend to be attractors for the `real', more complicated, system, and
the qualitative predictions involving no parameters, of which the
electron-to-ion energy transfer is one, tend to be very sturdy.
Aside from fuelling pellets, composed of hydrogen or deuterium, one paper in
this thesis investigates the physics of high-Z pellets that are designed to
terminate the plasma safely in the event of a `disruption', where much of the
magnetic field energy is channelled into a runaway electron beam with
potentially disastrous consequences if the beam encounters a plasma-facing
component. The paper draws on the work carried out in the paper concerning the
kinetic problem of isotropised electrons in a plasmoid.
This thesis is `cumulative'; the vast majority of the work carried out is
described within a set of Papers, labelled A-E, placed at the back of the text.
There is a preceding `wrapper text' (given in numbered Sections) tasked with
introducing the reader to the topic, guiding the reader through the papers, and
expounding some of their main results. Some amount of material not present in
the papers is also provided in the wrapper text. Naturally, the wrapper text
mainly focusses on the results of the papers which are under my first
authorship. In the course of publishing papers over an extended period of time
the nomenclature is bound to vary. Although it is mostly consistent between the
papers, a few difference do arise, and the section `Common symbols and
subscripts' is provided in the frontmatter to alleviate confusion. Particular
care should be taken with the symbols x and z; both can refer to the
coordinate parallel to the magnetic field line, but in papers where z is used
for this purpose x tends to have another definition. In the wrapper text the
choice of symbols is generally chosen to reflect those in the corresponding
paper.
The combination of the Layer-by-Layer (LbL) method, a nano-material such as carbon nanotubes (CNTs), and charged polyelectrolytes (PEs) is a reliable approach to produce highly functionalized surface coatings. These coatings are stable, controllable, ultra-thin, and most importantly, biocompatible. The ability to tune their properties by varying the preparation conditions and the terminating layer opens up a wide range of applications in the fields of biology and medicine. Here, the goal was to create electrically conductive coatings on which cells grow and proliferate. To achieve this goal, a coating with a stable conductive film structure, a suitable film surface topography, and suitable surface potential (and 𝜁-potential) must be prepared.
At the beginning of this thesis, the focus was on the fabrication of electrically conductive multilayer films, whose electrical properties should be stable and adjustable in a controlled manner (Article 1). The combination of chemically modified CNTs as polyanions, a strong linear polycation like poly(diallyldimethylammonium chloride) (PDADMA), and the LbL-method allowed us to prepare such films. Their characterization was carried out in air at ambient conditions. Since PDADMA is non-conductive, the charge transfer within the film and thus the electrical conductivity itself depends mainly on the CNTs and their arrangement. It was found that four CNT/PDADMA bilayers (BL) were always necessary to create a lateral network structure with multiple CNT crossing points to enable and support electron transport within the film. Moreover, additional CNT/PDADMA BL resulted in decreasing sheet resistance, while the conductivity remained constant at ≈ 4 kS/m regardless of the number of bilayers. Increasing the PDADMA molecular weight (Mw) from 44.4 kDa to 322 kDa did not affect film properties such as thickness or electrical conductivity.
However, increasing the CNT concentration from 0.15 mg/ml to 0.25 mg/ml in the deposition suspension resulted in thicker and less conductive films. This is attributed to a faster adsorption process of the CNTs leading to more adsorption sites for the polycation. We found an increased PDADMA monomer/CNT ratio compared to films prepared with the lower CNT concentration in the deposition suspension. The electrical conductivity decreased by a factor of four down to 1.1 kS/m, which can be attributed to fewer contact points between the CNTs. Overall, we were able to prepare stable and electrically conductive multilayer films. Additionally, by varying the preparation conditions tuning of the electrical conductivity is possible.
To fulfill requirements regarding i.e., medical implants, film properties not only have to be stable and controllable in a dry state (described in Article 1) but also in a biological aqueous environment. Therefore, in Article 2 we immersed our coated samples in three different solutions usually employed in biological research and compared their properties with their dry state, respectively. Also, hydration/swelling effects that normally occur for polyelectrolyte multilayer films (PEMs) in solutions were investigated.
For the film preparation, PDADMA (Mw = 322 kDa) and a deposition suspension of modified CNTs with two different concentrations (0.15 mg/ml and 0.25 mg/ml), which aged for two years, were used. Independent of the CNT suspension concentration, it turned out that the film thickness of the samples, prepared from the aged suspension, decreased significantly compared to the film thickness previously measured in Article 1. As a cross-check a new and fresh CNT suspension was made, which allowed us to reproduce the film thickness described in Article 1.
These results indicated that something happened with the CNT suspension over a two-year period. An analysis via X-ray photoelectron spectroscopy (XPS) showed a decrease in the percentage of functional groups in the CNTs from the aged suspension. The loss of functional groups resulted in less negatively charged CNTs and thus in fewer adsorption sites for the polycation PDADMA. Consequently, the PDADMA monomer/CNT ratio decreased, which lowered the thickness per bilayer by a factor of three, compared to films prepared with a freshly prepared CNT suspension. The lower linear charge density of the aged CNTs also enhanced their hydrophobicity, which is, in combination with the electrostatic forces, another important factor for multilayer cohesion. In contrast to PEMs made from polycations and polyanions, no swelling of the films occurred when immersed in solutions. This can be attributed to the fact that the increased hydrophobicity of the CNTs and the hydrophobic nature of the PDADMA backbone prevent the incorporation of water into the multilayer film. In solution, the films slightly shrink (by ≈ 2 nm), which makes them even more compact. Yet they remain stable. The result is an increased electrical conductivity from 9.6 kS/m, in the dry state, up to 15.3 kS/m immersed in solutions. To summarize, we showed that by tuning the interpolyelectrolyte forces the swelling and the ensuing decrease of the electrical conductivity of the films can be prevented.
Regarding the application in biology and medicine, we must consider that long-term exposure of cells to nano-materials like CNTs could lead to damage and inflammation of adjacent tissue. Therefore, it is necessary to prevent direct contact between the electrically conductive multilayer, i.e., CNT/PDADMA film, and the cells. The solution to this problem is a biocompatible top film that covers the CNT/PDADMA multilayer completely and still provides a lateral surface structure that supports cell adhesion and proliferation. Additional layers consisting solely of PEs could provide such a top film.
In Article 3 we investigated the self-patterning of PEM films as function of deposition steps. After preparation in water, the films were dried, characterized in air, and in vacuum. The films were built with high and low molecular weight PEs. PDADMA was used as polycation and poly(styrene sulfonate) sodium salt (PSS) as polyanion. The observation via Atomic Force Microscopy (AFM) showed that films prepared with high molecular weight PEs are laterally homogeneous and form no patterns, due to the chain immobility. The flat surfaces are ineligible as a substrate for cell adhesion.
In contrast, films built with a short PSS, especially at Mw, PSS = 10.7 kDa, began to self-pattern after seven deposited PDADMA/PSS bilayers. With each additionally deposited bilayer, the surface got more and more structured, from grooves over stripes to circular domains. Increasing film thickness led to an increased lateral mean distance between the surface structures. Scanning Electron Microscopy (SEM) images showed that exposure to a vacuum resulted in a decrease in the film thickness attributed to water removal, while the mean distance between the domains increased. Thus, by using this self-pattering process we are able to prepare PEMs with a highly structured surface. By adding PDADMA/PSS bilayers, not only the CNT/PDADMA film can be covered completely, but also a suitable surface morphology for cells can be created. Controlling the number of deposited bilayers allows the preparation of suitable coatings for cells.
To further improve the interaction of the cell and coated substrate not only the lateral structure but also the interacting electrostatic forces between cells and substrate are important for the nature of cell adhesion, function, and proliferation. In Article 4 we investigated PEMs, consisting of strong PEs with a low (PDADMA) and high (PSS) linear charge density. We performed asymmetric force measurements with the help of the colloidal probe technique (CP). Here, the forces between a PEM-covered surface and a colloidal probe (silica sphere) glued to a cantilever were investigated. The colloidal probe was either bare or covered with polycation poly(ethylenimine) (PEI). The surfaces were immersed in NaCl solutions with different ionic strengths (INaCl), starting with deionized water, then enriched up to 1 mol/L NaCl. The interaction force between a CP and the surface was measured. Thus, insight into the surface potential/charge was obtained.
During film preparation, two growth regimes (parabolic and linear) exist. These regimes and the terminating layer determine the surface force of the PEM. PEMs with a terminating PSS layer are predominantly flat and negatively charged when the ion concentration is low and the film is in the parabolic growth regime (between 1 and ≈ 15 BL). This indicates charge reversal on PSS adsorption. At the transition point between the parabolic and linear growth regimes, the ratio between polyanion and polycation monomers starts to switch and some cationic monomers are neutralized not by anionic monomers but by monovalent ions. Therefore, the surface charge density in diluted NaCl solutions changed from slightly positive near the transition to positive in the linear growth regime. At the lowest ionic strengths (INaCL) the range of the surface potential goes from – 40.5 mV (9 BL, parabolic) up to + 50 mV (19 BL, linear).
In contrast, polycation (PDADMA) terminated films are overall positive in diluted NaCl solutions. At the beginning of the parabolic growth regime, the layers are more compact and flat. However, with each additional layer deposited, the film becomes less compact and the chains begin to loosen. The now more loosely bound chains start to protrude into the solution and form pseudo-brushes. This could already be observed for 10.5 BL.
It intensifies in the linear growth regime (begin at ≈ 15 BL) and results in steric surface forces. Changing the surrounding INaCl affects this behavior and the pseudo-brushes scale as polyelectrolyte brushes.
By controlling the number of bilayers (thus the growth regime), the surrounding ionic strength, and the conformation of PEs at the PEM surface, it is possible to prepare a suitable range of surface properties i.e., for cell adhesion and proliferation. To prove that these multilayers can provide a suitable surface and have a positive effect on cell behavior, we coated in Article 5 titanium-covered samples with PEMs. Investigated was the cell interaction with the surface at different zeta(ζ) - potentials, a parameter for dynamic surface potential. Here the cell activity is measured by the mobilization of calcium (Ca2+) within the cell as a function of the ζ - potential of the substrate and the externally applied electrical potential. The cell activity indicates if the ζ - potential, provided by the sample surface, is suitable or not for the cells. The favorable interaction with the substrate is also reflected in the cell morphology and proliferation. The results showed that highly negative ζ - potentials between - 90 and - 3 mV led to a decreasing/reduced Ca2+ mobilization which correlates with reduced cell activity. Nearly neutral to moderate positive surfaces (ζ - potential + 1 to + 10 mV) i.e., PSS-terminated PEMs are able to promote cell adhesion and growth as demonstrated by an increased Ca2+ mobilization. The access to the intracellular Ca2+ stores, provided by the external stimulus, is now more effective and suggests a higher cell activity. Increasing the ζ - potentials up to ≈ + 50 mV (highly positive), i.e., PDADMA - terminated PEMs with pseudo-brushes, resulted in restricted cell viability and impaired Ca2+ mobilization, which led to a disturbed cell morphology and proliferation. In conclusion, only surfaces, terminated with i.e., PEI, with moderate positive charges (ζ - potential + 1 to + 10 mV) are able to improve the Ca2+ mobilization and thus the cell activity and proliferation. PEMs with a PSS termination provide negative 𝜁−potentials, onto which cells adhere, and proliferate. Therefore, they are a good alternative for surface functionalization for implant surfaces. In summary, the objective set at the beginning of the thesis is addressed within articles written as part of this thesis. It is possible to fabricate PEMs with modified CNTs to produce coatings that are electrically conductive with tunable sheet resistance, whether dry in air or immersed in an aqueous solution (Articles 1 and 2). Also, for pure PEMs, it is shown that with the right molecular weight of PEs and a certain number of bilayers, a suitable surface structure for cell adhesion can be produced (Article 3). Additional surface properties such as a suitable surface charge density can be provided by PEMs which can improve the cell activity as monitored with Ca2+ mobilization (Articles 4 and 5). The next step is to combine the knowledge gained from Articles 1 – 5 and link it to the application of external electrical fields to cells.
In course of the recent results from Wendelstein 7-X, stellarators are on the brink for assessing their maturity as a fusion reactor. To this end, stellarator specific transport regimes need detailed exploration both with appropriate systematic experimental investigations and models. A way to enhance the efficiency of this process is seen in an systematic evaluation of existing experimental data. We propose appropriate tools developed in information theory for examining large datasets. Information entropy calculations, that have proven to assist the systematic assessment of datasets in many other scientific fields, are used for novelty detection.
Potentially, as a first use-case of this holistic process, this thesis attempts to link and to develop approaches to examine the stellarator specific core-electron-root-confinement (CERC) regime. The specific interest for CERC emerges from the behavior of the radial electric field. While ion-root conditions exhibit negative radial electric fields, CERC’s positive field in the very core of fusion grade plasmas adds an outward thermodynamic force to high-Z impurities and could add to potential actuators to control impurity influx as to be examined for full-metal wall operation in large stellarators. Recently, this feature received revived intent for reactor scale stellarators.
Also, in this work, parameter regions close to the transition from ion-root to CERC are
examined. At lower rotational transform (a characteristic feature of the magnetic field confining fusion grade plasmas), transitions were detected when the plasma current evolved. As in smaller stellarators, it is concluded that low-order rationals and magnetic islands are related to the transitions. This is widely supported by extensive MHD simulations which finally provide indications for the role of zonal flow oscillations. As one of the outcomes, gyrokinetic instabilities are seen interacting for the first time with the neoclassical mechanisms in experiments.
In order to cope with the vast number of highly sampled spatio-temporal plasma data, new
techniques for novelty detection are required. Fundamental prerequisites for the detailed
physics investigations were the feasibility study of entropy-based data analysis techniques, and their adaptation to detect previously unrevealed transition mechanisms. These tools were applied to multivariate bulk plasma emissivity data, which allowed the exploration of large parameter spaces and provided insights in the spatio-temporal dynamics of CERC transitions.
In this manner, this research highlights the feasibility of information flow measure analysis in fusion studies. Applications of different entropy-based complexity measures are explored and this work sheds light on the capabilities, added value and limitations of these techniques. This investigation presents the integration of information flow measures to gain deeper understanding of plasma transport phenomena, by providing an approach to fast systematic data mining suited for real-time analysis. This work paves the way for further development and implementation of information-theoretic methods for plasma data analysis.
In summary, this research highlights the gained insight on CERC transitions, while showcasing the feasibility, added values and limitations of information flow measure analysis for fusion studies, to induce theory based analysis revealing new insights in fundamental, stellarator-specific transport mechanisms.
Research into nuclear physics has enjoyed a long and rich history since the earliest experiments began investigating atomic constituents. The discovery of the atomic nucleus in the early 20th century started a complex field of research that has undergone many transformations with the advancements of modern technology. Today, atomic nuclei are not only studied to advance our understanding of the strong force but also to gain more information on the synthesis of elements in the universe, to exploit nuclear decay to investigate the weak interaction, and to search for physics beyond the standard model.
In this work, we will study the strong force in atomic nuclei, i.e. the way nucleons (protons and neutrons) arrange themselves in a many-body system governed by the repulsive Coulomb interaction and the attractive strong interaction. In particular, we will focus on nuclear structure near nuclei with a "magic number" of Z protons and N neutrons, so-called doubly-magic nuclei, exhibiting a particularly stable configuration with respect to neighboring nuclei.
Within the nuclear shell model, similar to the atomic shells, the magic numbers indicate shell closures accompanied by energy gaps. Nuclei at double-shell closures and their direct vicinity provide an important playground to benchmark nuclear theories and models that aim to predict the intricate interplay of the nucleons that lead to enhanced nuclear binding energies, significant changes in charge radii and transition strengths, etc.
Of particular interest are nuclear isomers, long-lived excited states, in which the nucleon configuration with respect to its ground state is altered, resulting in a modification of their properties despite having the same number of protons and neutrons.
The main part of this work consists of three publications, which report on nuclear structure investigations through mass measurements and laser spectroscopy near the doubly magic nuclei nickel-78, tin-100, and lead-208.
The nuclides investigated in this work include neutron-deficient indium isotopes, neutron-rich zinc isotopes, and neutron-rich mercury isotopes.
This thesis presents the production of polyanionic clusters within two ion storage devices:
Considering a Penning trap, the accessible range of polyanionic aluminium clusters has been expanded up to the 10th charge state. In particular, abundance curves for clusters with 5 to 9 excess electrons have been measured for the first time and analysed with respect to their lifetime-dependent appearance sizes. These sizes reveal a nearly quadratic dependency on the charge state for experimentally accessible lifetimes.
Additionally, the production of polyanionic clusters has been enabled in a radiofrequency ion trap. Therefore, the transition from a harmonic to a digital 2- and 3-state guiding signal has been investigated with respect to the ion storage. The passing of electrons through the trap during field-free periods of the guiding signal led to the first production of polyanionic clusters within a radiofrequency ion trap.
In this thesis, I was able to provide answers to transport processes in lipid monolayers, which are ultimately, all of biological relevance. In particular, I was interested in lipid oxidation and dynamic compression/expansion processes of surfactant monolayers at the air-water interface:
Lipid oxidation was shown to be a consequence of the formation of a high concentration of reactive oxygen species (ROS) during cell respiration, which finally can lead to severe cell damage. It is not yet understood clearly, which part of the lipid molecules is especially prone to a ROS attack. I was particularly interested in the role of the double bonds of the acyl chains of the lipid molecules during oxidation. Further, I wanted to know the time scales of lipid interaction with the ROS.
Compared to lipid vesicles, lipid monolayers have the advantage that many parameters of the system can be adjusted easily. In our system, I made use of this by setting the lateral pressure to low values during H2O2 treatment, which facilitated the ROS to reach the double bonds in the acyl chains.
A prime example of biological systems out of thermal equilibrium was given in the alveolus surface, which is covered with a surfactant monolayer. During breathing, these monolayers undergo such a highly dynamic compression and expansion. Arising flows from breathing could disrupt a film and consequently, it would lose its protective role. One of my goals was to understand flows and their influence on domain shape. Dependent on the strength of the flows, I expected different growth regimes, with differing prevailing transport processes. Once understanding the underlying mechanisms in domain shaping would allow me to draw conclusions on biological systems.
In order to address these questions, I established two systems, both based on the compression of lipid monolayers. I used isotherms to study the phase behavior of the lipids:9 During compression, the lipids can undergo phase transitions from the gaseous phase to the liquid expanded phase (LE-phase) and further from the LE-phase to the liquid condensed phase (LC-phase). A coexistence regime is observed in between the LE-phase and the LC-phase, characterized by a flat increase of lateral pressure with decreasing molecular area. Some lipids exhibited LC-phase domains. These were further investigated with Brewster angle microscopy (BAM). The used BAM was equipped with an integrated Scheimpflug optics, enabling an overall focused image plane. Furthermore, time-resolved observation of the growth of the domains was possible by recording videos (20 frames per seconds).
The first system enabled the investigation of lipid peroxidation, when the lipids were exposed to ROS. I chose DMPC, POPC, DOPC and PLPC, since these are phospholipids differing in the number and position of double bonds in acyl chains, but not in the head group. I used a H2O2 enriched phosphate buffered saline (PBS) solution, which served as a precursor for more reactive ROS, like hydroxyls (.OH). PBS was chosen, since it resembles the cell environment best. During defined waiting times of H2O2 treatment, the ROS diffused vertically from the subphase towards the monolayer. The lipid molecules were in the LE-phase, which facilitated the ROS molecules to reach also the double bonds of the acyl chains. The oxidized monolayers were then compressed at constant compression speed. Since the corresponding isotherms could be measured with high precision, the relative area increase δA/A between oxidized and non-oxidized monolayer along the isotherm proved to be a good measure for lipid peroxidation. The area increase δA in the molecular area of the oxidized molecules was explained by the eventually added, more hydrophilic −OOH group at the position of a carbon atom adjacent to a double bond in the unsaturated acyl chain. The −OOH group is drawn to the hydrophilic head group of the lipid. This leads to a kink in the acyl chain, which increases the molecular area A by δA. A model, which explained this peroxidation process in lipid vesicles, could be adopted to monolayers.
I compared the oxidation of phospholipids, differing in the number and position of the double bonds of their acyl chains. I found that δA/A increased with the growing number of double bonds in one acyl chain. However, a comparison of DOPC with POPC also showed the importance of the position of the acyl chain. I determined a slow reaction kinetic. It could be estimated by a √t dependence of the number density N_surface, which denominates the ROS sticking on the monolayer. The transport of ROS towards the monolayer was found to be diffusive, because it was the slowest process in the reaction. This interpretation was reinforced by a comparison of the temperature dependence of the relative area increase δA/A with the Stokes-Einstein diffusion coefficient of water molecules. The initial ROS concentration c_0 in the trough could be traced back (c_0~ 50 nM), which is indeed a realistic value found in human cells.
Concluding, our results can be understood as a feasibility study. The complexity of the monolayer can be arbitrarily increased, for example by the addition of proteins, allowing the investigation of other oxidative processes occurring in the cell membrane.
The second system allowed the investigation of growth of LC domains during fast compression processes of monolayers. I chose erucic acid monolayers, due to its low line tension and a continuous nucleation phase, enabling the formation of fractal domains. The monolayers were investigated with isotherms and BAM videos. Since v_C (compression speed of the monolayer) was continuous over the whole compression time, I had a system with well-defined hydrodynamic conditions. This allowed me a complete analysis of the system, starting with descriptive features of the observed domains to a classification of the observed growth regimes by means of hydrodynamic theory, through to the distinction and quantification of different kind of flows and supersaturations, involving Ivantsov theory:
Dependent on the compression speed v_C, I observed seaweed or dendritic domains. The LE/LC phase transition pressure pi_t was slightly increased compared to pi_inf of the equilibrium isotherm. A high compression speed v_C induced a supersaturation Δc. I introduced the excess lateral pressure Δpi=pi-pi_inf as an appropriate quantity to describe the supersaturation Δc. I showed a linear behavior of Δc on Δpi. Δc is a macroscopic quantity since it is averaged over the whole monolayer area. I characterized the domains of the seaweed and dendritic regime with respect to tip radii, branch lengths, side branch separations and fractal dimensions. I calculated the growth speed of the main branches. A roughly doubling of the growth speed of dendritic domains, compared to seaweed domains was observed. This was an evidence of adjunctive (Marangoni) flow in the subphase.
For each monolayer, I observed drifts during domain growth, which I explained by an anisotropy in the LE-phase, caused by the continuous nucleation of the domains. These kind of surface flows were superimposed to bulk flows in the subphase. Since I had a well established system, I could analyze the influence of these surface flows on domain shape, in terms of magnitude, direction and duration of the surface flows. I therefore used FFT spectra and directionality histograms. At low flows, the FFT showed six-fold symmetry. Higher drifts exhibited incisions in the FFT, eventually leading to dumbbell shaped FFTs at very high drifts. The domains grew preferentially in the direction parallel to the incision.
I used directionality histograms to analyze the angular distribution of the growing domains. They showed that the drift direction always correlated with a minimum in the histogram. In order to analyze drift duration, I split the domain in downstream and upstream side. I could show that for small drift durations, downstream growth was preferred. However, for longer drift durations, the flows got more isotropic and consequently growth was more balanced then.
I could observe only a weak correlation between drift velocity v_D and compression speed v_C. However, dendrites were formed when the compression speed v_C was high, while seaweed domains were formed when v_C was small. Domain distortion occurred in the same way, independent if seaweed or dendritic domains were considered. I further showed that hydrodynamic flows in the subphase and surface flows are superimposed and scale differently. Consequently, they have different impact on domain shape: hydrodynamic flows act on μm scale and influence the domain morphology (distance between side branches, and tip radius) and the growth speed of the main branches. Surface flows act on the mm to cm scale, cause an anisotropic flow in the LE phase surrounding the domain, and thus affect the overall domain shape.
The anisotropy in the LE-phase led to a locally different degree of supersaturation. To take this into account, I introduced a local normalized supersaturation Δ, based on the Ivantsov solution. Therefore, I calculated Péclet numbers p of measured quantities of the system. I obtained values of 0.88 ≤Δ≤0.90 for the seaweed regime (p<5) and 0.93 ≤Δ≤0.96 for the dendritic regime (p>6). Since the Ivantsov solution can only be applied for purely diffusive processes, I applied a modified Ivantsov solution Δ_mod, which calculates Δ at a distance 𝛿 ahead of the dendrite tip. I was able to determine the progression of the diffusive layer 𝛿, however a quantitative determination failed.
Applying hydrodynamic theory allowed me to classify the two growth regimes with respect to the Boussinesq number Bq. Since for both growth regimes, I achieved values of Bq<1, bulk viscous losses dominated over surface viscous losses. Further, a cross-over length 𝜉 was calculated, from which one can distinguish, whether advective transport dominates over diffusion.
I further connected the two defined supersaturations Δ and Δc via the excess lateral pressure Δpi. From this, I saw differences in the seaweed and dendritic growth regimes: The local normalized supersaturation Δ of seaweed growth seemed to be quite stable for a further increase of the lateral excess pressure Δpi, whereas it reacted quite sensitive in the dendritic regime. This was found to be an indication of a non-equilibrium regime, caused by the strong coupling of the monolayer to the subphase. It reinforces therefore the theory of Marangoni-flow.
The findings of this thesis emphasize the importance of understanding highly dynamic compression/expansion processes arising in surfactant monolayers. Using the example of the compression of the alveolus surface, it can be seen that a more realistic model of the pulmonary alveolus is not only enabled by increasing the complexity of the surfactant monolayer (e.g. by adding specific proteins or lipid mixtures to the monolayer). Equally important is the understanding in transport processes and the consequences for the monolayer structure. By the analysis of domain shapes, I presented a method, which is suitable for such a study.
Graphene is a strictly two-dimensional honeycomb lattice of carbon atoms whose low-energy charge-carrier dynamics obey the massless pseudospin-1/2 Dirac-Weyl equation (or chiral Weyl equation) where the chiral centers (or valleys) are the corners K and K‘ of the Brillouin zone. The linear spectrum near the Dirac nodal points lends graphene its exotic and ultra-relativistic properties.
However, condensed matter systems can possess fermionic excitations with linear dispersions that have no analog in high-energy physics since the crystal space group - instead of the Poincare group - constrains the energy dispersions. Perhaps the first example in this regard is the T_3 lattice (Dice Gitter), a honeycomb-like lattice with an extra atom placed at the center of each hexagon and coupled to only one of the sublattices. The spectrum features a strictly flat band that crosses the two conical intersections of the Dirac cones at K and K' inherited from graphene. The enlarged pseudospin-1 Dirac-Weyl equation describes the low-energy dynamics. By rescaling the transfer amplitude of the additional atoms in the T_3 lattice with a parameter 0<α<1, the resulting α-T_3 lattice continously interpolates between graphene and the T_3 lattice.
In this work, we explore the behavior of generalized Dirac-Weyl quasiparticles in external magnetic and valley-dependent pseudoelectromagnetic fields induced by out-of-plane strain. First, we studied Dirac-Weyl quasiparticles in external fields confined to circular quantum dots by generalizing the infinite-mass boundary condition to the α-T_3 lattices. We verified the analytically derived valley-anisotropic eigenstates of the quantum dot by numerically solving the tight-binding lattice-model in closed (isolated) and open (contacted) systems.
Second, we considered strain fields in the α-T_3 lattices to modify the low-energy transport properties by an effective pseudo-gauge field with opposite signs at the K and K‘ valley. In particular, we showed that the inhomogeneous pseudomagnetic field generated by Gaussian out-of-plane strain at the center of a four-terminal Hall bar setup acts as a valley filter. Most interestingly, the valley polarization is most dominant when incoming electrons are excited to pseudo-Landau level subbands. These bands are linked to different iso-field orbits encircling the lobes of the pseudomagnetic field. Addittionaly, any intermediate α breaks the inversion symmetry of the α-T_3 lattice and thus splits the pseudo-Landau levels into sublattice-polarized bands.
Third, we equipped the out-of-plane strain with a time-periodic drive to induce a valley-dependent pseudoelectric field perpendicular to the pseudomagnetic field. We assessed the steady-state transport properties and found – besides the static regime for small energies – two α-dependent valley-filtering regimes due to the periodic drive. Firstly, we found an additional valley-polarization plateau at the Floquet-zone boundary between the central and first Floquet copy that also displayed a “flower”-like pattern in the local density of states. Secondly, we detected a series of transmission gaps at the center of every Floquet sideband 2mΩ related to the Floquet coupling of the flat band with the central Floquet copy. Under certain strain parameters, a novel valley-filtering regime appears near the transmission gaps where the incoming K electrons are focused through the bump by the pseudoelectric field, instead of encircling the lobes of the pseudomagnetic field. A stability analysis demonstrated that the polarization regimes are tunable by the driving frequency.
Lastly, we demonstrated that the flat band in the Haldane-dice lattice modified by a uniaxial strain along the zigzag orientation remains singular at all band crossings where the model undergoes a topological phase transition between C=+-2 and C=0. To show this, we computed the compact localized eigenstates and the quantum distance of the Bloch wave function around the band-touching points. We derived the resulting non-contractible loop states and an extended state whose components are tunabe by the system parameters.
Ion traps such as Paul traps and MR-ToF (multi-reflection time-of-flight) devices are indispensable tools at radioactive ion beam facilities for the preparation of high-quality radioactive ion beams for subsequent experiments or for precise measurements of the properties of radioactive ions, such as nuclear binding energies or nuclear charge radii.
Within the work of this thesis, Doppler- and sympathetic cooling is implemented in a linear Paul-trap cooler-buncher enabling a reduction of the longitudinal emittance of radioactive ion beams resulting in a significant improvement of the ion beam quality. Moreover, a next-generation MR-ToF device is conceptualized in order to achieve isobaric pure beams with a higher ion intensity than state-of-the-art MR-ToF devices can provide. Once fully constructed and commissioned, it will operate at an unprecedented ion beam energy of 30 keV. Both of these advances are expected to become important for a wide range of experimental programs pursued at low-energy branches of RIB facilities ranging from fundamental symmetry studies, nuclear structure, rare isotope studies with antimatter, searches of physics beyond the standard model to material science and the production of medical isotopes.
The next-generation MR-ToF mass separator is based on MIRACLS’ 30-keV MR-ToF device for highly sensitive and high-resolution collinear laser spectroscopy. By storing the ions in the Multi Ion Reflection Apparatus for Collinear Laser Spectroscopy (MIRACLS), the same ion bunch is probed by a spectroscopic laser for thousands of times compared to a single passage in traditional collinear laser spectroscopy (CLS). Dedicated simulation studies show that the accuracy and resolution will be close to traditional single-passage CLS while the sensitivity is significantly enhanced. Hence, measurements of nuclear properties via fluorescence-based CLS of very rare radionuclides as well as highly sensitive and high-precision measurements of electron affinities via laser-photodetachment-threshold spectroscopy of negatively-charged (radioactive) ions will become possible.
First measurement campaigns employing MIRACLS’ 1.5-keV MR-ToF device confirm the outstanding boost in signal sensitivity and provide confidence in the application of the MIRACLS technique for the measurement of scarcely produced radioactive ions that have been so far beyond the reach of conventional techniques. Furthermore, the electron affinity of 35Cl was measured, which is in perfect agreement with the literature value. These measurements will serve as important benchmarks for modern atomic and nuclear theory, especially in its description of nuclear charge radii.
In summary, the implementation of Doppler and sympathetic cooling at RIB facilities, the conceptualization of a 30-keV MR-ToF apparatus for highly selective and high-flux mass separation as well as for highly sensitive and high-resolution fluorescence-based laser spectroscopy and the expansion of the MIRACLS technique for the study of negatively-charged ions will enable unprecedented new measurement opportunities at RIB facilities.
Polyelektrolyt-Multischichtfilme (PEMs) werden durch schichtweise (eng. Layer by Layer, LbL)
sequentielle Ablagerung von entgegengesetzt geladenen Polyelektrolyten auf einer
geladenen Oberfläche hergestellt. Die LbL Methode kann auf verschiedene Weise zur
Herstellung von PEM eingesetzt werden, z.B. durch Tauchen, Rotation, Sprühen oder
Beschichten mit elektromagnetischen und fluidischen Methoden. In allen Artikeln dieser
Dissertation wurde die Tauchmethode verwendet. Durch zyklische Wiederholung der
Abscheidungsschritte kann die Dicke der PEM leicht gesteuert werden. Die Oberflächen und
Grenzflächen des Films können mit der LbL Technik auch durch die elektrostatische
Wechselwirkung zwischen positiv und negativ geladenen Polyelektrolyten modifiziert werden.
Auf diese Weise lassen sich einige Eigenschaften des Films optimieren, beispielsweise
Oberflächenadhäsion und Biokompatibilität, z. B. in der Gewebezüchtung oder es kann
eine Monoschicht als Barriere an der Grenzfläche des Films adsorbiert werden, um die
Diffusion von Molekülen im Film zu begrenzen z.B. bei Aufnahme oder Freisetzen von
Medikamenten.
Daher wurde die Rolle einiger Faktoren, wie die molare Masse der Polyelektrolyte und das
Vorhandensein von Salzionen in der Präparationslösung auf die interne Struktur sowie die
Oberfläche der PEMs untersucht.
Für alle Untersuchungen dieser Dissertation wurde das häufig verwendete Modell-System aus
dem positiv geladenen Polyelektrolyten Polydimethyldiallylammonium (PDADMA), und dem
negativ geladenen Polyelektrolyten Polystyrolsulfonat (PSS), verwendet. Die Dicke der Filme
wurde mit Röntgenreflektometrie, Ellipsometrie, UV-Vis-NIR-Spektrometrie bestimmt die
interne Struktur mit Neutronenreflektometrie und die Oberflächentopografie mit Rasterkraftmikroskopie
(eng. AFM) und Rasterelektronenmikroskopie (eng. SEM).
In Artikel 1 wurde mit Hilfe der Neutronenreflektometrie die Struktur des Filmes und die
Diffusion des Polyanions PSS (DPSS) senkrecht zur PEM Oberfläche untersucht. Variiert wurde
die molare Masse des Polykations PDADMA und die Salzkonzentration der
Präparationslösung. PEMs wurden aus drei verschiedenen NaCl-Konzentrationen in der
Abscheidelösung hergestellt: 10 mmol/L, 100 mmol/L und 200 mmol/L. Die Salzkonzentration
in der Polyelektrolytlösung bestimmt die Konformation der Polyelektrolyte während der
Adsorption. Die Ketten werden weniger flach adsorbiert, wenn mehr Salzionen in der
Adsorptionslösung vorhanden sind und die Filme werden dicker.
Die Diffusion nahm mit zunehmender molarer Masse von PDADMA in Filmen, die aus 10
mmol/L, 100 mmol/L und 200 mmol/L hergestellt wurden, um mindestens drei Größenordnungen
ab, denn die Zunahme der Kettenlänge, erhöht den Vernetzungsgrad im Film. Dabei zeigten Filme aus 10 mmol/L (NaCl) mit einer niedrigen molaren Masse von PDADMA
die größte Diffusion (DPSS = 4.9 × 10−20 m2/s). Der Diffusionskoeffizient DPSS als Funktion des
Polymerisationsgrades folgt zwei Potenzgesetzen mit einem Übergang bei einem
Polymerisationsgrad von 288. Bei kürzeren Ketten stimmt der Exponent des Potenzgesetzes
gut mit dem Modell der Sticky Reptation überein. Bei längeren Ketten war der Exponent viel
größer, was vermuten lässt, dass die PSS-Ketten in einem zunehmend komplexen
Polymernetzwerk gefangen sind. Wir verstehen den Übergang als Verschränkungsgrenze für
das untersuchte System.
Bei PEMs, die aus 100 mmol/L hergestellt wurden, konnte kein Potenzgesetz festgestellt
werden. DPSS nahm sprunghaft um drei Größenordnungen ab, wenn die molare Masse von
PDADMA von 45 kDa auf 72 kDa erhöht wurde.
In Artikel 2 wurden die Oberfläche von PEMs aus Polyelektrolyten unterschiedlicher molarer
Massen untersucht. Die Oberflächenrauhigkeit und die Dicke des Films wurden mit
Röntgenreflektometrie und Ellipsometrie bestimmt. Die Oberflächentopografie wurde mit AFM
und SEM aufgenommen. Alle PEMs wurden aus PE-Lösungen mit 0,1 mol/L NaCl hergestellt.
Die Oberfläche der PEM, präpariert aus langem PSS und kurzem PDADMA oder langem PSS
und langem PDADMA, war immer flach. Bei einer Filmzusammensetzung aus langen
Polykationen (Mw (PDADMAlang) = 322 kDa) und kurzen PSS Molekülen (Mw (PSSkurz) = 10,7
kDa) wurden drei Wachstumsregime identifiziert: exponentiell, parabolisch und linear. Im
exponentiellen Wachstumsregime bildet sich nach etwa sieben Beschichtungsschritten von
PDADMA/PSS (eng. bilayers, bl) eine granulare Oberflächenstruktur aus mit einer
Oberflächenrauigkeit von 1,6 nm und einer lateralen Periodizität von 70 nm. Mit zunehmender
Schichtzahl nimmt die Oberflächenrauhigkeit sowie die laterale Periodizität zu. Im
parabolischen Wachstumsbereich aggregieren die Strukturen zu Säulen, mit einer
Oberflächenrauigkeit bis zu 23 nm und einer lateralen Periodizität bis zu 210 nm. Im linearen
Wachstumsregime sind die säulenförmigen Domänen vollständig ausgebildet und die
Oberflächenstruktur ändert sich nicht mehr. Diese Strukturen wurden schon während der
Präparation, bereits vor dem Trocknen beobachtet. Dies zeigt, dass sich die Strukturen
während der Abscheidung von PDADMA/PSS bilden.
Bei Beobachtungen im Vakuum (SEM) war im linearen Bereich die Säulenstruktur bei der
PDADMA terminierten PEM ausgeprägter als bei der PSS terminierten.
Diese Strukturen bilden sich nur im Film mit anfänglichem exponentiellem Wachstum, d.h.
wenn kurzen Ketten durch den ganzen Film diffundieren können. Das legt nahe, dass es für
die Strukturbildung nicht ausreicht, dass der Polyelektrolyt kurz ist, sondern dass es auch
beweglich sein muss. Um dies näher zu untersuchen wurde in Manuskript 1 die molare Masse des PSS variiert. Es
wurden PEMs aus langem 322 kDa PDADMA und kurzem 6,5 kDa und 3,9 kDa PSS
hergestellt und mit den Messungen von PEMs aus 10,7 kDa PSS verglichen.
Die Verkürzung von PSS hat subtile Auswirkungen auf den Filmaufbau und die
Selbststrukturierung. Für PEM aus PSS mit einer molaren Masse von 6,5 kDa konnten nur
zwei Wachstumsregime ermittelt werden: ein exponentielles und ein lineares Wachstumsregime.
Der Übergang vom exponentiellen zum linearen Wachstum erfolgte bei 28
Doppelschichten. Bei PEMs, die aus 3,9 kDa PSS hergestellt wurden, wurde bis zu 29 bl nur
ein exponentielles Wachstum beobachtet. Dies zeigt, dass eine Verringerung der molaren
Masse von PSS das exponentielle Wachstum auf eine größere Anzahl von abgeschiedenen
Doppelschichten ausdehnt. Dies ist auf die zunehmende PSS-Diffusion zurückzuführen.
In allen Filmen wurden Selbststrukturierungen beobachtet. Der Abstand und die Höhe der
säulenartigen Domänen nehmen mit jeder abgeschiedenen PDADMA/PSS-Doppelschicht
deutlich zu. Der durchschnittliche Domänenabstand ändert sich weniger und korreliert mit den
vertikalen Wachstumsregimen. Der Domänenabstand schwankt zwischen 70 nm und 750 nm.
Die größten lateralen Abstände und ein längeres exponentielles Wachstumsregime wurden
mit dem kürzesten PSS (3,9 kDa) erreicht, was auf die hohe Mobilität des PSS zurückgeführt
wird. Die Domänenhöhe ist immer kleiner als der Domänenabstand. Wenn die PEM mit
PDADMA terminiert ist, sind die Oberflächenrauhigkeit und der durchschnittliche Abstand
größer als bei PSS terminierten Filme in Wasser und nach dem Trocknen.
Darüber hinaus wurden zwischen den Domänen Filamente beobachtet. Die Filamente
bestehen aus PDADMA/PSS-Komplexen. Eine mögliche Vermutung ist, dass diese Komplexe
zwischen den Domänen diffundieren und ihren Abstand anpassen.
Die Oberflächenstruktur des Films aus PSS 10,7 kDa zeigt eine symmetrische gaußförmige
Höhenverteilung in allen drei Wachstumsregimen von 5 bis 40 bl. Für die kurze PSS war eine
solche Verteilung nur bis 15 bl (6,5 kDa) bzw. 20 bl (3,9 kDa) zu beobachten. Danach wurde
für 6,5 kDa schiefe Verteilung mit Ausläufern zu größeren Höhen beobachtet. 3,9 kDa PSS
zeigte dann sogar eine bimodale Höhenverteilung.
Die lineare Ladungsdichte von PDADMA ist etwa halb so groß wie die von PSS. Folglich
adsorbiert PDADMA in einer bürstenartigen Konformation. Wenn die oberste Schicht
PDADMA ist, dann ist das PDADMA-Molekül nicht fest an die Oberfläche gebunden. Daher ist
die durch die Oberflächenspannung erzeugte Kraft für PDADMA groß genug, um zu einer
Veränderung der Oberflächenmorphologie und folglich zu einer kleineren Gesamtoberfläche
zu führen.Außerdem sind die Domänen in 1 M NaCl-Lösung stabil, schrumpfen aber in 2 M NaCl enorm,
während ihr Abstand leicht zunimmt.
Diese Untersuchungen zeigten, dass die Mobilität des Polyelektrolyten PSS die
Voraussetzung für den Aufbau einer strukturierten Oberfläche in einem PEM-System aus
PDADMA/PSS ist. Diese Ergebnisse zeigten auch, dass die Verkürzung der Kette der PSS Moleküle
die Herstellung von Filmen erleichtert, deren Dicke und Selbststrukturierung je nach
dem gewünschten Zweck angepasst werden kann. Solche Filme können in der Medizin und
Biologie als geeignetes Substrat zur Optimierung der Adsorption von Zellen und anderen
Molekülen oder als Nanofilter effektiv eingesetzt werden.
In dieser Dissertation konnte ich zeigen, wie die Verkürzung der Kette der PSS-Moleküle zur
Bildung einer lateralen selbststrukturierten Oberfläche führt und wie die zunehmende Mobilität
der PSS-Moleküle die Oberflächenmorphologie signifikant beeinflusst.