Geometrical dynamics of edge-driven accretive surface growth
Accretion of mineralized thin wall-like structures via localized growth along their edges is observed in physical and biological systems ranging from molluscan and brachiopod shells to carbonate–silica composite precipitates. To understand the shape of these mineralized structures, we develop a math...
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| Published in | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 478; no. 2257 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
26.01.2022
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| Online Access | Get full text |
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| Abstract | Accretion of mineralized thin wall-like structures via localized growth along their edges is observed in physical and biological systems ranging from molluscan and brachiopod shells to carbonate–silica composite precipitates. To understand the shape of these mineralized structures, we develop a mathematical framework that treats the thin-walled shells as a smooth surface left in the wake of the growth front that can be described as an evolving space curve. Our theory then takes an explicit geometric form for the prescription of the velocity of the growth front curve, along with compatibility relations and a closure equation related to the nature of surface curling. Solutions of these equations capture a range of geometric precipitate patterns seen in abiotic and biotic forms across scales. In addition to providing a framework for the growth and form of these thin-walled morphologies, our theory suggests a new class of dynamical systems involving moving space curves that are compatible with non-Euclidean embeddings of surfaces. |
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| AbstractList | Accretion of mineralized thin wall-like structures via localized growth along their edges is observed in physical and biological systems ranging from molluscan and brachiopod shells to carbonate–silica composite precipitates. To understand the shape of these mineralized structures, we develop a mathematical framework that treats the thin-walled shells as a smooth surface left in the wake of the growth front that can be described as an evolving space curve. Our theory then takes an explicit geometric form for the prescription of the velocity of the growth front curve, along with compatibility relations and a closure equation related to the nature of surface curling. Solutions of these equations capture a range of geometric precipitate patterns seen in abiotic and biotic forms across scales. In addition to providing a framework for the growth and form of these thin-walled morphologies, our theory suggests a new class of dynamical systems involving moving space curves that are compatible with non-Euclidean embeddings of surfaces. |
| Author | Mahadevan, L. Nadir Kaplan, C. |
| Author_xml | – sequence: 1 givenname: C. orcidid: 0000-0003-1314-1497 surname: Nadir Kaplan fullname: Nadir Kaplan, C. organization: Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA, Center for Soft Matter and Biological Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA – sequence: 2 givenname: L. orcidid: 0000-0002-5114-0519 surname: Mahadevan fullname: Mahadevan, L. organization: School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA, Department of Physics, Harvard University, Cambridge, MA 02138, USA, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA |
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