Finished Theses

Institut für Theoretische Physik I

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Institut für Theoretische Physik I

Universität Erlangen-Nürnberg

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91058 Erlangen

Germany

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UnivIS FAU Erlangen-Nürnberg

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Gruppenseminar der Theorie 1

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- Loop Percolation
- supervised by Klaus Mecke and Gerd Schröder-Turk
- finished 2011-07-15

This thesis addresses physical systems where the spatial structure plays an important role in determining physical properties. In part 1, a new planar percolation model based on disconnecting nodes rather than removing edges or vertices is shown to belong to a different universality class than conventional bond or site percolation. In part 2, a numerical Minkowski tensor approach is used to study alignment and anisotropy of patterns in co-polymer films exposed to an external field.

In part 1, a percolation model, termed "loop percolation", is defined on
a square grid, oriented such that the square diagonals are horizontal or
vertical. Rather than removing bonds or vertices, a node is disconnected
with probability *p*, that is, the four edges incident at a vertex are
pair-wise connected. If a node is disconnected it is split into two
horizontal edges (rather than two vertical edges) with probability *x* for
nodes in lines with even line number and with probability (1-*x*)
otherwise. As function
of *p* the system is shown to undergo a translation transition at *p _{c}*=
min((2

Part 2 demonstrates that shape indices based on so-called Minkowski
tensors are a useful real-space method to study alignment and anisotropy
of experimental planar gray-scale datasets. This is demonstrated by the
study of scanning force microscopy images of a microphase-separated thin
co-polymer film that is exposed to an external electric field for
different exposure times *T* are analysed in terms of alignment and
anisotropy of the observed pattern (the SFM images are the data
described in "Electric field alignment of a block copolymer nanopattern:
direct observation of the microscopic mechanism"
Soft Matter, Vol. 2, 1089–1094, 2006, courtesy of Alexander
Boeker). We study eigenvalue ratios of Minkowski tensors of the
segmented image data sets as a function of time, averaged over various
segmentation threshold for each gray-scale image. We show that the
Minkowski tensor approach yields a qualitatively similar result
with respect to the anisotropy of the spatial structure and its
alignment with the field direction, as did the analysis based on a Fourier method by
Olszowka et al.