Crystallization and Demixing: Morphological structure Analysis in many-body systems

Kurzfassung

The description and analysis of spatial data is an omnipresent task in both science and industry: In the food industry the distribution and size of pores in baked goods plays a role in their taste. In chemistry, biology and physics spatial data arises in manifold disciplines and on all length scales. On large scales one finds them in the structure of the universe or in earth surveillance data. On small scales one observes highly structured data in inner bones or on minute scales in the deformation of nucleons in nuclear pasta, which is theorized to form during the cooling of a neutron star. In particular in statistical physics many-body-systems have a tendency to collectively form complex structures by self-organization. These complex structures often allow to draw conclusions about the underlying physics. In order to formulate a quantitative relation between the physics of many-body-systems and their morphology, i.e. the spatial structure they assume, a quantitative description of this structure is essential. In this dissertation the spatial structure of phase transitions (crystallization and demixing) in many-body-systems is quantitatively described and analyzed in order to achieve an improved understanding of the physics involved. Regarding the analysis methods applied in this thesis we go beyond conventional linear measures based on two-point correlation functions or the power spectrum. Instead, the aim is a full nonlinear morphological characterization of the spatial data with measures derived from the family of Minkowski functionals and tensors. They are additive, morphological measures related to, not only geometrical concepts like volume, area and curvature, but also to topological aspects such as connectivity and are sensitive to higher order correlation. Complex plasmas (dielectric microparticles immersed in a plasma) are a well suited model system for the particle resolved investigation of many-body processes. Their optical thinness allows for the optical imaging and tracking of the fully resolved trajectories of hundreds of particle layers. Additionally interactions can be tuned over a large range allowing to manipulate the shape and magnitude of the interparticle potential. Since the gas density is typically very low, the particle motion is practically undamped resulting in a direct analogy to the atomistic dynamics in solids or fluids. Liquid-solid phase transition have been considered impossible for a long time since the Mermin-Wagner theorem forbids long-range order in two (or less) dimensions. However, Kosterlitz and Thouless (Nobel prize 2016) circumvented this by replacing the long-range order with a quasi-long-range order and by introducing a topological phase transition mediated by defects. The well accepted KTHNY theory predicts an intermediate anisotropic phase, the hexatic phase. In the first part of this thesis the KTHNY theory is tested for experiments and a simulation of the crystallization of two-dimensional complex plasma sheets. For the same experiments the hypothesis and prediction of the recently developed fractal-domain-structure (FDS) theory is tested. The FDS theory is based on the Frenkel kinetic theory of melting. It postulates a fractal relationship between crystalline domains separated by boundaries of defect lines and predicts a scale-free relation between the system energy and the defect fraction. It is found that the KTHNY theory is not applicable to the liquid-solid phase transition in complex plasmas. The FDS theory however, is validated. The other focus of this thesis is the morphological characterization of fluid-fluid demixing dynamics. The generally accepted mechanism for fluid-fluid demixing is spinodal decomposition. Spinodal decomposition is achieved by a quench deep inside the spinodal curve of the phase diagram. It is characterized by the exponential growth of longwavelength density fluctuations. However the mean-field theory predictions of spinodal decomposition are not consistent with experiments and simulations. This shows the need for particle resolved studies with tunable interactions. To this end complex plasma simulations in flat three-dimensional space and density-functional theory calculations on the two-dimensional sphere are analyzed. In both cases different stages of demixing are identified with distinct domain growth rates during spinodal decomposition. Most importantly, universal demixing behavior is found for different interaction potentials, respectively for different mixture fractions and sphere sizes. These universal features could only be resolved by applying nonlinear measures, going beyond conventional methods based on the power spectral density. This suggests that nonlinear features in the demixing kinetics play an important role and that it is crucial to address this issue in future works.