Generic elasticity of thermal, under-constrained systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912170792452096 |
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| author | Lee, Cheng-Tai Merkel, Matthias |
| author_facet | Lee, Cheng-Tai Merkel, Matthias |
| contents | Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain, which is theoretically well understood. Here and in the companion paper, we extend this theory to finite temperatures for a very broad class of under-constrained systems. In the vicinity of the athermal transition point, we derive from first principles expressions for elastic properties such as isotropic tension $t$ and shear modulus $G$ on temperature $T$, isotropic strain $\varepsilon$, and shear strain $γ$, which we confirm numerically. These expressions contain only three parameters, entropic rigidity $κ_S$, energetic rigidity $κ_E$, and a parameter $b_\varepsilon$ describing the interaction between isotropic and shear strain, which can be determined from the microstructure of the system. Our results imply that in under-constrained systems, entropic and energetic rigidity interact like two springs in series. This also allows for a simple explanation of the previously numerically observed scaling relation $t\sim G\sim T^{1/2}$ at $\varepsilon=γ=0$. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07266 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic elasticity of thermal, under-constrained systems Lee, Cheng-Tai Merkel, Matthias Soft Condensed Matter Statistical Mechanics Biological Physics Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain, which is theoretically well understood. Here and in the companion paper, we extend this theory to finite temperatures for a very broad class of under-constrained systems. In the vicinity of the athermal transition point, we derive from first principles expressions for elastic properties such as isotropic tension $t$ and shear modulus $G$ on temperature $T$, isotropic strain $\varepsilon$, and shear strain $γ$, which we confirm numerically. These expressions contain only three parameters, entropic rigidity $κ_S$, energetic rigidity $κ_E$, and a parameter $b_\varepsilon$ describing the interaction between isotropic and shear strain, which can be determined from the microstructure of the system. Our results imply that in under-constrained systems, entropic and energetic rigidity interact like two springs in series. This also allows for a simple explanation of the previously numerically observed scaling relation $t\sim G\sim T^{1/2}$ at $\varepsilon=γ=0$. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues. |
| title | Generic elasticity of thermal, under-constrained systems |
| topic | Soft Condensed Matter Statistical Mechanics Biological Physics |
| url | https://arxiv.org/abs/2304.07266 |