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Heyden S, Bain N. From a distance: Shuttleworth revisited. SOFT MATTER 2024. [PMID: 38973537 DOI: 10.1039/d4sm00078a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/09/2024]
Abstract
The Shuttleworth equation: a linear stress-strain relation ubiquitously used in modeling the behavior of soft surfaces. Its validity in the realm of materials subject to large deformation is a topic of current debate. Here, we allow for large deformation by deriving the constitutive behavior of the surface from the general framework of finite kinematics. We distinguish cases of finite and infinitesimal surface relaxation preceding an infinitesimal applied deformation. The Shuttleworth equation identifies as the Cauchy stress measure in the fully linearized setting. We show that both in finite and linearized cases, measured elastic constants depend on the utilized stress measure. In addition, we discuss the physical implications of our results and analyze the impact of surface relaxation on the estimation of surface elastic moduli in the light of two different test cases.
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Affiliation(s)
- Stefanie Heyden
- ETH Zürich, Institute for Building Materials, 8093 Zürich, Switzerland.
| | - Nicolas Bain
- Universite Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, UMR5306, F-69100, Villeurbanne, France.
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Zhao B, Bonaccurso E, Auernhammer GK, Chen L. Elasticity-to-Capillarity Transition in Soft Substrate Deformation. NANO LETTERS 2021; 21:10361-10367. [PMID: 34882419 DOI: 10.1021/acs.nanolett.1c03643] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Whereas capillarity controls fluid dynamics at submillimeter scale and elasticity determines the mechanics of rigid solids, their coupling governs elastocapillary deformations on soft solids. Here, we directly probed the deformations on soft substrates induced by sessile nanodroplets. The wetting ridge created around the contact line and the dimple formed underneath the nanodroplet were imaged with a high spatial resolution using atomic force microscopy. The ridge height nonmonotonically depends on the substrate stiffness, and the dimple depth nonlinearly depends on the droplet size. The capillarity of the substrate overcomes the elasticity of the substrate in dominating the deformations when the elastocapillary length is approximately larger than the droplet contact radius, showing an experimental observation of the elasticity-to-capillarity transition. This study provides an experimental approach to investigate nanoscale elastocapillarity, and the insights have the potential to kick-off future work on the fundamentals of solid mechanics.
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Affiliation(s)
- Binyu Zhao
- School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
- Leibniz Institute of Polymer Research Dresden, Dresden 01069, Germany
| | | | | | - Longquan Chen
- School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
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Kim J, Mailand E, Ang I, Sakar MS, Bouklas N. A model for 3D deformation and reconstruction of contractile microtissues. SOFT MATTER 2021; 17:10198-10209. [PMID: 33118554 DOI: 10.1039/d0sm01182g] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Tissue morphogenesis and regeneration are essentially mechanical processes that involve coordination of cellular forces, production and structural remodeling of extracellular matrix (ECM), and cell migration. Discovering the principles of cell-ECM interactions and tissue-scale deformation in mechanically-loaded tissues is instrumental to the development of novel regenerative therapies. The combination of high-throughput three-dimensional (3D) culture systems and experimentally-validated computational models accelerate the study of these principles. In our previous work [E. Mailand, et al., Biophys. J., 2019, 117, 975-986], we showed that prominent surface stresses emerge in constrained fibroblast-populated collagen gels, driving the morphogenesis of fibrous microtissues. Here, we introduce an active material model that allows the embodiment of surface and bulk contractile stresses while maintaining the passive elasticity of the ECM in a 3D setting. Unlike existing models, the stresses are driven by mechanosensing and not by an externally applied signal. The mechanosensing component is incorporated in the model through a direct coupling of the local deformation state with the associated contractile force generation. Further, we propose a finite element implementation to account for large deformations, nonlinear active material response, and surface effects. Simulation results quantitatively capture complex shape changes during tissue formation and as a response to surgical disruption of tissue boundaries, allowing precise calibration of the parameters of the 3D model. The results of this study imply that the organization of the extracellular matrix in the bulk of the tissue may not be a major factor behind the morphogenesis of fibrous tissues at sub-millimeter length scales.
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Affiliation(s)
- Jaemin Kim
- Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA.
| | - Erik Mailand
- Institutes of Mechanical Engineering and Bioengineering, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
| | - Ida Ang
- Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA.
| | - Mahmut Selman Sakar
- Institutes of Mechanical Engineering and Bioengineering, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
| | - Nikolaos Bouklas
- Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA.
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Liu Z, Hui CY, Jagota A, Gong JP, Kiyama R. A surface flattening method for characterizing the surface stress, drained Poisson's ratio and diffusivity of poroelastic gels. SOFT MATTER 2021; 17:7332-7340. [PMID: 34286785 DOI: 10.1039/d1sm00513h] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
When a poroelastic gel is released from a patterned mold, surface stress drives deformation and solvent migration in the gel and flattens its surface profile in a time-dependent manner. Specifically, the gel behaves like an incompressible solid immediately after removal from the mold, and becomes compressible as the solvent is able to squeeze out of the polymer network. In this work, we use the finite element method (FEM) to simulate this transient surface flattening process. We assume that the surface stress is isotropic and constant, the polymer network is linearly elastic and isotropic, and that solvent flow obeys Darcy's law. The short-time and long-time surface profiles can be used to determine the surface stress and drained Poisson's ratio of the gel. Our analysis shows that the drained Poisson's ratio and the diffusivity of the gel can be obtained using interferometry and high-speed video microscopy, without mechanical measurement.
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Affiliation(s)
- Zezhou Liu
- Field of Theoretical and Applied Mechanics, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA.
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Danielsen SPO, Beech HK, Wang S, El-Zaatari BM, Wang X, Sapir L, Ouchi T, Wang Z, Johnson PN, Hu Y, Lundberg DJ, Stoychev G, Craig SL, Johnson JA, Kalow JA, Olsen BD, Rubinstein M. Molecular Characterization of Polymer Networks. Chem Rev 2021; 121:5042-5092. [PMID: 33792299 DOI: 10.1021/acs.chemrev.0c01304] [Citation(s) in RCA: 60] [Impact Index Per Article: 20.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
Polymer networks are complex systems consisting of molecular components. Whereas the properties of the individual components are typically well understood by most chemists, translating that chemical insight into polymer networks themselves is limited by the statistical and poorly defined nature of network structures. As a result, it is challenging, if not currently impossible, to extrapolate from the molecular behavior of components to the full range of performance and properties of the entire polymer network. Polymer networks therefore present an unrealized, important, and interdisciplinary opportunity to exert molecular-level, chemical control on material macroscopic properties. A barrier to sophisticated molecular approaches to polymer networks is that the techniques for characterizing the molecular structure of networks are often unfamiliar to many scientists. Here, we present a critical overview of the current characterization techniques available to understand the relation between the molecular properties and the resulting performance and behavior of polymer networks, in the absence of added fillers. We highlight the methods available to characterize the chemistry and molecular-level properties of individual polymer strands and junctions, the gelation process by which strands form networks, the structure of the resulting network, and the dynamics and mechanics of the final material. The purpose is not to serve as a detailed manual for conducting these measurements but rather to unify the underlying principles, point out remaining challenges, and provide a concise overview by which chemists can plan characterization strategies that suit their research objectives. Because polymer networks cannot often be sufficiently characterized with a single method, strategic combinations of multiple techniques are typically required for their molecular characterization.
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Affiliation(s)
- Scott P O Danielsen
- Marsico Lung Institute, University of North Carolina, Chapel Hill, North Carolina 27599, United States
| | - Haley K Beech
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
| | - Shu Wang
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - Bassil M El-Zaatari
- Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
| | - Xiaodi Wang
- Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
| | | | | | - Zi Wang
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - Patricia N Johnson
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - Yixin Hu
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - David J Lundberg
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
| | - Georgi Stoychev
- Marsico Lung Institute, University of North Carolina, Chapel Hill, North Carolina 27599, United States
| | - Stephen L Craig
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - Jeremiah A Johnson
- Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
| | - Julia A Kalow
- Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
| | - Bradley D Olsen
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
| | - Michael Rubinstein
- Marsico Lung Institute, University of North Carolina, Chapel Hill, North Carolina 27599, United States.,Department of Chemistry, Duke University, Durham, North Carolina 27708, United States.,Departments of Biomedical Engineering and Physics, Duke University, Durham, North Carolina 27708, United States.,World Primer Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Kita 21 Nishi 10, Kita-ku, Sapporo, Hokkaido 001-0021, Japan
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