Microscopically founded elasticity theory for defect-rich systems of anisotropic particles

Microscopically founded elasticity theory for defect-rich systems of anisotropic particles

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2020

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Dissertation

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The elastic constants like bulk or shear modulus belong to the central mechanical properties of materials. They are closely related to sound waves. For example, the velocity of longitudinal sound waves in fluids is directly related to the compressibility. The goal of this thesis is to describe elasticity from a particle level, where collective excitations, like acoustic phonons, are the phenomenon that connects the microcosm of particle interaction with the macrocosm of elasticity. In contrast to the usual approach, where the interaction potential is used in harmonic approximation, static correlation functions are the central quantities expressing the microscopic interactions.

A. Poniewierski and J. Stecki developed a theory [1], where the Frank elastic constants of nematic liquid crystals are expressed through the direct correlation function. This is the central quantity of classical density functional theory (see e.g. [2]). C. Walz achieved similar results for non-ideal crystals [3] of isotropic particles in the isothermal case. He used the Mori-Zwanzig projection-operator formalism (see e.g [4]) as methodology to connect micro- and macrocosm. This is also a central concept of this thesis, where the theory of C. Walz is extended to anisotropic particles. This reproduces the nematic and isotropic-particle case. Furthermore, it includes phases that exhibit positional and orientational order, like the smectic phases of liquid crystals. This ordering is fundamentally related to the breaking of spatial symmetries, namely continuous translational and rotational invariance. After the breaking of continuous symmetries, normally new long-lifetime long-wavelength excitation modes arise. They are called Nambu-Goldstone modes and correspond to new elastic constants. The shear modulus after freezing is an example. The applicability of the Nambu-Goldstone theorem for spatial symmetry breaking in systems at finite temperature and density is absolutely non-trivial.

In the last decade, a lot of work was done in that area, where especially Y. Hidaka [5], T. Hayata [6] and H. Watanabe/T. Brauner [7] shall be mentioned. Their fundamental concepts are implemented for the case of classical, thermal systems in this thesis.

As P.C. Martin, O. Parodi and P.S. Pershan mentioned [8], it is crucial not to neglect non- Nambu-Goldstone modes in the crystalline phase. The diffusion of point defect can be seen as remaining fluid-like behavior after freezing. Therefore, this density mode is a crucial - and still complicated - part for defect-rich systems and must therefore be included.

[1] A. Poniewierski and J. Stecki, Mol. Phys. 38, 1931 (1979)

A. Poniewierski and J. Stecki, Mol. Phys. 41, 1451 (1980)

A. Poniewierski and J. Stecki, Phys. Rev. A 25, 2368 (1982)

[2] R. Evans, Adv. Phys. 28, 143 (1979)

[3] C. Walz and M. Fuchs, Phys. Rev. B 81, 134110 (2010)

[4] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, (Benjamin INC., Reading, Massachusetts, 1975)

[5] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013)

[6] T. Hayata and Y. Hidaka, Phys. Lett. B 735, 195 (2014)

[7] H. Watanabe and T. Brauner, Phys. Rev. D. 85, 085010 (2012)

[8] P.C. Martin, O. Parodi, P.S. Pershan, Phys. Rev. A 6, 2401 (1972)

A. Poniewierski and J. Stecki developed a theory [1], where the Frank elastic constants of nematic liquid crystals are expressed through the direct correlation function. This is the central quantity of classical density functional theory (see e.g. [2]). C. Walz achieved similar results for non-ideal crystals [3] of isotropic particles in the isothermal case. He used the Mori-Zwanzig projection-operator formalism (see e.g [4]) as methodology to connect micro- and macrocosm. This is also a central concept of this thesis, where the theory of C. Walz is extended to anisotropic particles. This reproduces the nematic and isotropic-particle case. Furthermore, it includes phases that exhibit positional and orientational order, like the smectic phases of liquid crystals. This ordering is fundamentally related to the breaking of spatial symmetries, namely continuous translational and rotational invariance. After the breaking of continuous symmetries, normally new long-lifetime long-wavelength excitation modes arise. They are called Nambu-Goldstone modes and correspond to new elastic constants. The shear modulus after freezing is an example. The applicability of the Nambu-Goldstone theorem for spatial symmetry breaking in systems at finite temperature and density is absolutely non-trivial.

In the last decade, a lot of work was done in that area, where especially Y. Hidaka [5], T. Hayata [6] and H. Watanabe/T. Brauner [7] shall be mentioned. Their fundamental concepts are implemented for the case of classical, thermal systems in this thesis.

As P.C. Martin, O. Parodi and P.S. Pershan mentioned [8], it is crucial not to neglect non- Nambu-Goldstone modes in the crystalline phase. The diffusion of point defect can be seen as remaining fluid-like behavior after freezing. Therefore, this density mode is a crucial - and still complicated - part for defect-rich systems and must therefore be included.

[1] A. Poniewierski and J. Stecki, Mol. Phys. 38, 1931 (1979)

A. Poniewierski and J. Stecki, Mol. Phys. 41, 1451 (1980)

A. Poniewierski and J. Stecki, Phys. Rev. A 25, 2368 (1982)

[2] R. Evans, Adv. Phys. 28, 143 (1979)

[3] C. Walz and M. Fuchs, Phys. Rev. B 81, 134110 (2010)

[4] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, (Benjamin INC., Reading, Massachusetts, 1975)

[5] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013)

[6] T. Hayata and Y. Hidaka, Phys. Lett. B 735, 195 (2014)

[7] H. Watanabe and T. Brauner, Phys. Rev. D. 85, 085010 (2012)

[8] P.C. Martin, O. Parodi, P.S. Pershan, Phys. Rev. A 6, 2401 (1972)

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##### Subject (DDC)

530 Physics

##### Keywords

Elasticity theory, classical density functional theory, Nambu-Goldstone theorem

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##### Cite This

## ISO 690

HÄRING, Johannes M., 2020.*Microscopically founded elasticity theory for defect-rich systems of anisotropic particles*[Dissertation]. Konstanz: University of Konstanz

## BibTex

@phdthesis{Haring2020Micro-53734, year={2020}, title={Microscopically founded elasticity theory for defect-rich systems of anisotropic particles}, author={Häring, Johannes M.}, address={Konstanz}, school={Universität Konstanz} }

## RDF

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They are closely related to sound waves. For example, the velocity of longitudinal sound waves in fluids is directly related to the compressibility. The goal of this thesis is to describe elasticity from a particle level, where collective excitations, like acoustic phonons, are the phenomenon that connects the microcosm of particle interaction with the macrocosm of elasticity. In contrast to the usual approach, where the interaction potential is used in harmonic approximation, static correlation functions are the central quantities expressing the microscopic interactions.<br />A. Poniewierski and J. Stecki developed a theory [1], where the Frank elastic constants of nematic liquid crystals are expressed through the direct correlation function. This is the central quantity of classical density functional theory (see e.g. [2]). C. Walz achieved similar results for non-ideal crystals [3] of isotropic particles in the isothermal case. He used the Mori-Zwanzig projection-operator formalism (see e.g [4]) as methodology to connect micro- and macrocosm. This is also a central concept of this thesis, where the theory of C. Walz is extended to anisotropic particles. This reproduces the nematic and isotropic-particle case. Furthermore, it includes phases that exhibit positional and orientational order, like the smectic phases of liquid crystals. This ordering is fundamentally related to the breaking of spatial symmetries, namely continuous translational and rotational invariance. After the breaking of continuous symmetries, normally new long-lifetime long-wavelength excitation modes arise. They are called Nambu-Goldstone modes and correspond to new elastic constants. The shear modulus after freezing is an example. The applicability of the Nambu-Goldstone theorem for spatial symmetry breaking in systems at finite temperature and density is absolutely non-trivial.<br />In the last decade, a lot of work was done in that area, where especially Y. Hidaka [5], T. Hayata [6] and H. Watanabe/T. Brauner [7] shall be mentioned. Their fundamental concepts are implemented for the case of classical, thermal systems in this thesis.<br />As P.C. Martin, O. Parodi and P.S. Pershan mentioned [8], it is crucial not to neglect non- Nambu-Goldstone modes in the crystalline phase. The diffusion of point defect can be seen as remaining fluid-like behavior after freezing. Therefore, this density mode is a crucial - and still complicated - part for defect-rich systems and must therefore be included.<br /><br />[1] A. Poniewierski and J. Stecki, Mol. Phys. 38, 1931 (1979)<br />A. Poniewierski and J. Stecki, Mol. Phys. 41, 1451 (1980)<br />A. Poniewierski and J. Stecki, Phys. Rev. A 25, 2368 (1982)<br />[2] R. Evans, Adv. Phys. 28, 143 (1979)<br />[3] C. Walz and M. Fuchs, Phys. Rev. B 81, 134110 (2010)<br />[4] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, (Benjamin INC., Reading, Massachusetts, 1975)<br />[5] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013)<br />[6] T. Hayata and Y. Hidaka, Phys. Lett. B 735, 195 (2014)<br />[7] H. Watanabe and T. Brauner, Phys. Rev. D. 85, 085010 (2012)<br />[8] P.C. Martin, O. Parodi, P.S. Pershan, Phys. Rev. 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December 11, 2020

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Konstanz, Univ., Doctoral dissertation, 2020

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