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Arieli M, Moshe M, Sharon E. Mechanical design principles in frustrated thin elastic sheets. SOFT MATTER 2024; 20:4414-4421. [PMID: 38767344 DOI: 10.1039/d4sm00166d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2024]
Abstract
Using a geometric formalism of elasticity theory we develop a systematic theoretical framework for shaping and manipulating the energy landscape of slender solids, and consequently their mechanical response to external perturbations. We formally express global mechanical properties associated with non-Euclidean thin sheets in terms of their local rest lengths and rest curvatures, and we interpret the expressions as both forward and inverse problems for designing the desired mechanical properties. We show that by wisely designing geometric frustration, anomalous mechanical properties can be encoded into a material using accessible experimental techniques. To test the methodology we derive a family of ribbon-springs with extreme mechanical behavior such as tunable, anharmonic, and even vanishing rigidities. The presented formalism can be discretized, offering a new methodology for designing mechanical properties and thus opens a new pathway for the design of both continuum and discrete solids and structures.
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Affiliation(s)
- Michal Arieli
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel.
| | - Michael Moshe
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel.
| | - Eran Sharon
- Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel.
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2
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Blanc B, Zhang Z, Liu E, Zhou N, Dellatolas I, Aghvami A, Yi H, Fraden S. Active Pulsatile Gels: From a Chemical Microreactor to a Polymeric Actuator. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2024; 40:6862-6868. [PMID: 38385757 DOI: 10.1021/acs.langmuir.3c03784] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/23/2024]
Abstract
We report on a synthesis protocol, experimental characterization, and theoretical modeling of active pulsatile Belousov-Zhabotinsky (BZ) hydrogels. Our two-step synthesis technique allows independent optimization of the geometry, the chemical, and the mechanical properties of BZ gels. We identify the role of the surrounding medium chemistry and gel radius for the occurrence of BZ gel oscillations, quantified by the Damköhler number, which is the ratio of chemical reaction to diffusion rates. Tuning the BZ gel size to maximize its chemomechanical oscillation amplitude, we find that its oscillatory strain amplitude is limited by the time scale of gel swelling relative to the chemical oscillation period. Our experimental findings are in good agreement with a Vanag-Epstein model of BZ chemistry and a Tanaka Fillmore theory of gel swelling dynamics.
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Affiliation(s)
- Baptiste Blanc
- Department of Physics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454, United States
| | - Zhenkun Zhang
- Key Laboratory of Functional Polymer Materials of Ministry of Education, Institute of Polymer Chemistry, College of Chemistry, Nankai University, 300071 Tianjin, China
| | - Eric Liu
- Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts 02155, United States
| | - Ning Zhou
- Department of Chemistry, Brandeis University, 415 South Street, Waltham, Massachusetts 02454, United States
| | - Ippolyti Dellatolas
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Ali Aghvami
- Department of Physics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454, United States
| | - Hyunmin Yi
- Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts 02155, United States
| | - Seth Fraden
- Department of Physics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454, United States
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3
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Blanc B, Agyapong JN, Hunter I, Galas JC, Fernandez-Nieves A, Fraden S. Collective chemomechanical oscillations in active hydrogels. Proc Natl Acad Sci U S A 2024; 121:e2313258121. [PMID: 38300869 PMCID: PMC10861864 DOI: 10.1073/pnas.2313258121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2023] [Accepted: 11/14/2023] [Indexed: 02/03/2024] Open
Abstract
We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.
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Affiliation(s)
- Baptiste Blanc
- Laboratoire Jean Perrin, Sorbonne Université, CNRS, Institut de Biologie Paris-Seine (IBPS), Paris75005, France
- Department of Condensed Matter Physics, University of Barcelona, Barcelona08028, Spain
- Department of Physics, Brandeis University, Waltham, MA02454
| | - Johnson N. Agyapong
- Department of Physics, Brandeis University, Waltham, MA02454
- Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse, NY13244
| | - Ian Hunter
- Department of Physics, Brandeis University, Waltham, MA02454
| | - Jean-Christophe Galas
- Laboratoire Jean Perrin, Sorbonne Université, CNRS, Institut de Biologie Paris-Seine (IBPS), Paris75005, France
| | - Alberto Fernandez-Nieves
- Department of Condensed Matter Physics, University of Barcelona, Barcelona08028, Spain
- Institute of Complex Systems, University of Barcelona, Barcelona08028, Spain
- Institució Catalanade Recerca i Estudis Avançats, Barcelona08010, Spain
| | - Seth Fraden
- Department of Physics, Brandeis University, Waltham, MA02454
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McCaskill JS, Karnaushenko D, Zhu M, Schmidt OG. Microelectronic Morphogenesis: Smart Materials with Electronics Assembling into Artificial Organisms. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2023; 35:e2306344. [PMID: 37814374 DOI: 10.1002/adma.202306344] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Revised: 08/27/2023] [Indexed: 10/11/2023]
Abstract
Microelectronic morphogenesis is the creation and maintenance of complex functional structures by microelectronic information within shape-changing materials. Only recently has in-built information technology begun to be used to reshape materials and their functions in three dimensions to form smart microdevices and microrobots. Electronic information that controls morphology is inheritable like its biological counterpart, genetic information, and is set to open new vistas of technology leading to artificial organisms when coupled with modular design and self-assembly that can make reversible microscopic electrical connections. Three core capabilities of cells in organisms, self-maintenance (homeostatic metabolism utilizing free energy), self-containment (distinguishing self from nonself), and self-reproduction (cell division with inherited properties), once well out of reach for technology, are now within the grasp of information-directed materials. Construction-aware electronics can be used to proof-read and initiate game-changing error correction in microelectronic self-assembly. Furthermore, noncontact communication and electronically supported learning enable one to implement guided self-assembly and enhance functionality. Here, the fundamental breakthroughs that have opened the pathway to this prospective path are reviewed, the extent and way in which the core properties of life can be addressed are analyzed, and the potential and indeed necessity of such technology for sustainable high technology in society is discussed.
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Affiliation(s)
- John S McCaskill
- Research Center for Materials, Architectures and Integration of Nanomembranes (MAIN), Chemnitz University of Technology, 09126, Chemnitz, Germany
- Material Systems for Nanoelectronics, Chemnitz University of Technology, 09126, Chemnitz, Germany
- European Centre for Living Technology (ECLT), Ca' Bottacin, Dorsoduro 3911, Venice, 30123, Italy
| | - Daniil Karnaushenko
- Research Center for Materials, Architectures and Integration of Nanomembranes (MAIN), Chemnitz University of Technology, 09126, Chemnitz, Germany
- Material Systems for Nanoelectronics, Chemnitz University of Technology, 09126, Chemnitz, Germany
| | - Minshen Zhu
- Research Center for Materials, Architectures and Integration of Nanomembranes (MAIN), Chemnitz University of Technology, 09126, Chemnitz, Germany
- Material Systems for Nanoelectronics, Chemnitz University of Technology, 09126, Chemnitz, Germany
| | - Oliver G Schmidt
- Research Center for Materials, Architectures and Integration of Nanomembranes (MAIN), Chemnitz University of Technology, 09126, Chemnitz, Germany
- Material Systems for Nanoelectronics, Chemnitz University of Technology, 09126, Chemnitz, Germany
- European Centre for Living Technology (ECLT), Ca' Bottacin, Dorsoduro 3911, Venice, 30123, Italy
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5
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Maji A, Dasbiswas K, Rabin Y. Shape transitions in a network model of active elastic shells. SOFT MATTER 2023; 19:7216-7226. [PMID: 37724013 DOI: 10.1039/d3sm01041d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/20/2023]
Abstract
Morphogenesis involves the transformation of initially simple shapes, such as multicellular spheroids, into more complex 3D shapes. These shape changes are governed by mechanical forces including molecular motor-generated forces as well as hydrostatic fluid pressure, both of which are actively regulated in living matter through mechano-chemical feedback. Inspired by autonomous, biophysical shape change, such as occurring in the model organism hydra, we introduce a minimal, active, elastic model featuring a network of springs in a globe-like spherical shell geometry. In this model there is coupling between activity and the shape of the shell: if the local curvature of a filament represented by a spring falls below a critical value, its elastic constant is actively changed. This results in deformation of the springs that changes the shape of the shell. By combining excitation of springs and pressure regulation, we show that the shell undergoes a transition from spheroidal to either elongated ellipsoidal or a different spheroidal shape, depending on pressure. There exists a critical pressure at which there is an abrupt change from ellipsoids to spheroids, showing that pressure is potentially a sensitive switch for material shape. We thus offer biologically inspired design principles for autonomous shape transitions in active elastic shells.
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Affiliation(s)
- Ajoy Maji
- Faculty of Biomedical Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel
| | - Kinjal Dasbiswas
- Department of Physics, University of California, Merced, Merced, CA 95343, USA
| | - Yitzhak Rabin
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 5290002, Israel.
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Xiong Y, Yuan H, Olvera de la Cruz M. Janus magnetoelastic membrane swimmers. SOFT MATTER 2023; 19:6721-6730. [PMID: 37622382 DOI: 10.1039/d3sm00788j] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/26/2023]
Abstract
Soft swimming microrobots have attracted considerable attention due to their potential applications in diverse fields ranging from biomedicines to environmental remediation. The locomotion control is of importance to the research of micromachines and microrobots. Inspired by the motility strategies of living microorganisms, such as flagella, cilia, and euglenoids, we focus on propulsion mechanisms with a design of Janus magnetoelastic crystalline membrane microswimmers actuated by time-varying magnetic fields. Such a Janus swimmer consists of a ferromagnetic cap completed by a magnetoelastic membrane body, where superparamagnetic particles are uniformly distributed on the surface. Under the influence of external magnetic fields, the swimmer undergoes complex shape transitions due to the interplay between the magnetic dipole-dipole interactions, the elasticity of the magnetoelastic membranes, and also the hydrodynamics of surrounding fluids. We show that those shape changes are nonreciprocal, which can generate locomotion such that the propulsion speed can be optimized by tailoring the membrane elastic properties. Besides, we also demonstrate that the Janus swimmer can be magnetically guided in a spiral trajectory. With such adequate control of locomotion in both speed and direction via non-invasive magnetic fields, this study provides another promising candidate design for the future development of microswimmers.
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Affiliation(s)
- Yao Xiong
- Center for Computation & Theory of Soft Materials, Northwestern University, Evanston, IL, 60208, USA.
| | - Hang Yuan
- Applied Physics Graduate Program, Northwestern University, Evanston, IL, 60208, USA
| | - Monica Olvera de la Cruz
- Center for Computation & Theory of Soft Materials, Northwestern University, Evanston, IL, 60208, USA.
- Department of Materials Science and Engineering, Northwestern University, Evanston, IL, 60208, USA
- Department of Chemistry, Northwestern University, Evanston, IL, 60208, USA
- Department of Physics and Astronomy, Northwestern University, Evanston, IL, 60208, USA
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Batista BC, Morris AZ, Steinbock O. Pattern selection by material aging: Modeling chemical gardens in two and three dimensions. Proc Natl Acad Sci U S A 2023; 120:e2305172120. [PMID: 37399415 PMCID: PMC10334770 DOI: 10.1073/pnas.2305172120] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2023] [Accepted: 05/18/2023] [Indexed: 07/05/2023] Open
Abstract
Chemical gardens are complex, often macroscopic, structures formed by precipitation reactions. Their thin walls compartmentalize the system and adjust in size and shape if the volume of the interior reactant solution is increased by osmosis or active injection. Spatial confinement to a thin layer is known to result in various patterns including self-extending filaments and flower-like patterns organized around a continuous, expanding front. Here, we describe a cellular automaton model for this type of self-organization, in which each lattice site is occupied by one of the two reactants or the precipitate. Reactant injection causes the random replacement of precipitate and generates an expanding near-circular precipitate front. If this process includes an age bias favoring the replacement of fresh precipitate, thin-walled filaments arise and grow-like in the experiments-at the leading tip. In addition, the inclusion of a buoyancy effect allows the model to capture various branched and unbranched chemical garden shapes in two and three dimensions. Our results provide a model of chemical garden structures and highlight the importance of temporal changes in the self-healing membrane material.
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Affiliation(s)
- Bruno C. Batista
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL32306-4390
| | - Amari Z. Morris
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL32306-4390
| | - Oliver Steinbock
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL32306-4390
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Three-dimensional chiral morphodynamics of chemomechanical active shells. Proc Natl Acad Sci U S A 2022; 119:e2206159119. [PMID: 36442097 PMCID: PMC9894169 DOI: 10.1073/pnas.2206159119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022] Open
Abstract
Morphogenesis of active shells such as cells is a fundamental chemomechanical process that often exhibits three-dimensional (3D) large deformations and chemical pattern dynamics simultaneously. Here, we establish a chemomechanical active shell theory accounting for mechanical feedback and biochemical regulation to investigate the symmetry-breaking and 3D chiral morphodynamics emerging in the cell cortex. The active bending and stretching of the elastic shells are regulated by biochemical signals like actomyosin and RhoA, which, in turn, exert mechanical feedback on the biochemical events via deformation-dependent diffusion and inhibition. We show that active deformations can trigger chemomechanical bifurcations, yielding pulse spiral waves and global oscillations, which, with increasing mechanical feedback, give way to traveling or standing waves subsequently. Mechanical feedback is also found to contribute to stabilizing the polarity of emerging patterns, thus ensuring robust morphogenesis. Our results reproduce and unravel the experimentally observed solitary and multiple spiral patterns, which initiate asymmetric cleavage in Xenopus and starfish embryogenesis. This study underscores the crucial roles of mechanical feedback in cell development and also suggests a chemomechanical framework allowing for 3D large deformation and chemical signaling to explore complex morphogenesis in living shell-like structures.
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9
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Aggarwal A, Li C, Stupp SI, Olvera de la Cruz M. Controlling the shape morphology of origami-inspired photoresponsive hydrogels. SOFT MATTER 2022; 18:2193-2202. [PMID: 35226038 PMCID: PMC8989053 DOI: 10.1039/d1sm01751a] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
The concept of origami has influenced the development of responsive materials that can mimic complex functions performed by living organisms. An ultimate goal is to discover and design soft materials that can be remotely actuated into diverse structures. To achieve this goal, we design and synthesize here a light-responsive spiropyran hydrogel system that can display dynamic shape changes upon irradiation with local light. We use a continuum polymer model to analyze the behavior of the constructed photoactive hydrogel, which is in good agreement with the experimental results. We explore different buckling modalities and patterns in a different range of parameters. The synthesis and fabrication of these materials demonstrate that the theoretical model can be used to drive the development of responsive photoactive systems.
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Affiliation(s)
- Aaveg Aggarwal
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA.
| | - Chuang Li
- Center for Bio-inspired Energy Science, Northwestern University, Evanston, IL 60208, USA.
| | - Samuel I Stupp
- Center for Bio-inspired Energy Science, Northwestern University, Evanston, IL 60208, USA.
- Department of Chemistry, Northwestern University, Evanston, IL 60208, USA
- Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA
- Department of Biomedical Engineering, Northwestern University, Evanston, IL 60208, USA
- Department of Medicine, Northwestern University, Chicago, IL 60611, USA
- Simpson Querrey Institute, Northwestern University, Chicago, IL 60611, USA
| | - Monica Olvera de la Cruz
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA.
- Center for Bio-inspired Energy Science, Northwestern University, Evanston, IL 60208, USA.
- Department of Chemistry, Northwestern University, Evanston, IL 60208, USA
- Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA
- Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA
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Manna RK, Gentile K, Shklyaev OE, Sen A, Balazs AC. Self-Generated Convective Flows Enhance the Rates of Chemical Reactions. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2022; 38:1432-1439. [PMID: 35029999 DOI: 10.1021/acs.langmuir.1c02593] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
In chemical solutions, the products of catalytic reactions can occupy different volumes compared to the reactants and thus give rise to local density variations in the fluid. These density variations generate solutal buoyancy forces, which are exerted on the fluid and thus "pump" the fluid to flow. Herein, we examine if the reaction-induced pumping accelerates the chemical reaction by transporting the reactants to the catalyst at a rate faster than passive diffusion. Using both simulations and experiments, we show a significant increase in reaction rate when reaction-generated convective flow is present. In effect, through a feedback loop, catalysts speed up reactions not only by lowering the energy barrier but also by increasing the collision frequency between the reactants and the catalyst.
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Affiliation(s)
- Raj Kumar Manna
- Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States
| | - Kayla Gentile
- Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Oleg E Shklyaev
- Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States
| | - Ayusman Sen
- Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Anna C Balazs
- Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States
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Mokbel M, Djellouli A, Quilliet C, Aland S, Coupier G. Post-buckling dynamics of spherical shells. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2021.0378] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We explore the intrinsic dynamics of spherical shells immersed in a fluid in the vicinity of their buckled state, through experiments and three-dimensional axisymmetric simulations. The results are supported by a theoretical model that accurately describes the buckled shell as a two-variable-only oscillator. We quantify the effective ‘softening’ of shells above the buckling threshold, as observed in recent experiments on interactions between encapsulated microbubbles and acoustic waves. The main dissipation mechanism in the neighbouring fluid is also evidenced.
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Affiliation(s)
- Marcel Mokbel
- University of Applied Sciences (HTW) Dresden, Friedrich-List-Platz 1, 01069 Dresden, Germany
- Technische Universität Bergakademie Freiberg, Akademiestrasse,609599 Freiberg, Germany
| | - Adel Djellouli
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
- School of Engineering and Applied Sciences Cambridge, Harvard University, Cambridge, MA 02138, USA
| | | | - Sebastian Aland
- University of Applied Sciences (HTW) Dresden, Friedrich-List-Platz 1, 01069 Dresden, Germany
- Technische Universität Bergakademie Freiberg, Akademiestrasse,609599 Freiberg, Germany
| | - Gwennou Coupier
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
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Zakharov A, Dasbiswas K. Mechanochemical induction of wrinkling morphogenesis on elastic shells. SOFT MATTER 2021; 17:4738-4750. [PMID: 33978668 DOI: 10.1039/d1sm00003a] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Morphogenetic dynamics of tissue sheets require coordinated cell shape changes regulated by global patterning of mechanical forces. Inspired by such biological phenomena, we propose a minimal mechanochemical model based on the notion that cell shape changes are induced by diffusible biomolecules that influence tissue contractility in a concentration-dependent manner - and whose concentration is in turn affected by the macroscopic tissue shape. We perform computational simulations of thin shell elastic dynamics to reveal propagating chemical and three-dimensional deformation patterns arising due to a sequence of buckling instabilities. Depending on the concentration threshold that actuates cell shape change, we find qualitatively different patterns. The mechanochemically coupled patterning dynamics are distinct from those driven by purely mechanical or purely chemical factors, and emerge even without diffusion. Using numerical simulations and theoretical arguments, we analyze the elastic instabilities that result from our model and provide simple scaling laws to identify wrinkling morphologies.
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Affiliation(s)
- Andrei Zakharov
- Department of Physics, University of California, Merced, Merced, CA 95343, USA.
| | - Kinjal Dasbiswas
- Department of Physics, University of California, Merced, Merced, CA 95343, USA.
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