Discrete Differential Geometry Lab


Knut Heidemann

I am a PhD student in the Discrete Differential Geometry Lab at the University of Goettingen.



Research interests

Physical Simulation, Biological Physics, Statistical Physics.



Contact

Institute of Num. and Appl. Math
University of Göttingen
Lotzestr. 16-18
37083 Göttingen, Germany
Vox: +49 (0)551 39-22050
Fax: +49 (0)551 39-33944
Net: k.lastname (at) math.uni-goettingen.de




Publications

Thesis
Crosslinked Network
On the mechanics of biopolymer networks
Knut M. Heidemann, PhD Thesis, University of Göttingen, 2016.
Abstract: In this thesis, we study the mechanical properties of biopolymer networks. We discuss which of these properties can be described by continuum approaches and which features, on the contrary, require consideration of the discrete nature or the topology of the network. For this purpose, we combine theoretical modeling with extensive numerical simulations. In Chapter 2, we study the elasticity of disordered networks of rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. Our theoretical considerations are accompanied by extensive quasistatic simulations of 3-dimensional networks, which are in agreement with the analytical theory, and show additional features like prestress and the formation of force chains. In Chapter 3, we study the distribution of forces in random spring networks on the unit circle by applying a combination of probabilistic theory and numerical computations. Using graph theory, we find that taking into account network topology is crucial to correctly capture force distributions in mechanical equilibrium. In particular, we show that application of a mean field approach results in significant deviations from the correct solution, especially for sparsely connected networks.
[pdf]

Refereed Articles
Crosslinked Network
Elasticity of 3D networks with rigid filaments and compliant crosslinks
Knut M. Heidemann, Abhinav Sharma, Florian Rehfeldt, Christoph F. Schmidt, and Max Wardetzky. Soft Matter, 11(2), pp. 343–354, 2015.
Abstract: Disordered filamentous networks with compliant crosslinks exhibit a low linear elastic shear modulus at small strains, but stiffen dramatically at high strains. Experiments have shown that the elastic modulus can increase by up to three orders of magnitude while the networks withstand relatively large stresses without rupturing. Here, we perform an analytical and numerical study on model networks in three dimensions. Our model consists of a collection of randomly oriented rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Due to zero probability of filament intersection in three dimensions, our model networks are by construction prestressed in terms of initial tension in the crosslinks. We demonstrate how the linear elastic modulus can be related to the prestress in these networks. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. We further show that there is no qualitative change in the stiffening behavior of filamentous networks even if the filaments are bending-compliant. Consequently, unlike suggested in prior work, the model system studied here cannot provide an explanation for the experimentally observed linear scaling of the modulus with the stress in filamentous networks.
[pdf]

mikado
Elastic response of filamentous networks with compliant crosslinks
A. Sharma, M. Sheinman, K. M. Heidemann, and F. C. MacKintosh. Physical Review E 88, 052705 (2013).
Abstract: Experiments have shown that elasticity of disordered filamentous networks with compliant crosslinks is very different from networks with rigid crosslinks. Here, we model and analyze filamentous networks as a collection of randomly oriented rigid filaments connected to each other by flexible crosslinks that are modeled as wormlike chains. For relatively large extensions we allow for enthalpic stretching of crosslink backbones. We show that for sufficiently high crosslink density, the network linear elastic response is affine on the scale of the filaments' length. The nonlinear regime can become highly nonaffine and is characterized by a divergence of the elastic modulus at finite strain. In contrast to the prior predictions, we do not find an asymptotic regime in which the differential elastic modulus scales linearly with the stress, although an approximate linear dependence can be seen in a transition from entropic to enthalpic regimes. We discuss our results in light of recent experiments.
[pdf] [doi]

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