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Khandagale P, Garcia-Cervera C, deBotton G, Breitzman T, Majidi C, Dayal K. Statistical field theory of polarizable polymer chains with nonlocal dipolar interactions. Phys Rev E 2024; 109:044501. [PMID: 38755880 DOI: 10.1103/physreve.109.044501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2024] [Accepted: 02/14/2024] [Indexed: 05/18/2024]
Abstract
The electromechanical response of polymeric soft matter to applied electric fields is of fundamental scientific interest as well as relevant to technologies for sensing and actuation. Several existing theoretical and numerical approaches for polarizable polymers subject to a combined applied electric field and stretch are based on discrete monomer models. In these models, accounting for the interactions between the induced dipoles on monomers is challenging due to the nonlocality of these interactions. On the other hand, the framework of statistical field theory provides a continuous description of polymer chains that potentially enables a tractable way to account for these interactions. However, prior formulations using this framework have been restricted to the case of weak anisotropy of the monomer polarizability. This paper formulates a general approach based in the framework of statistical field theory to account for the nonlocal nature of the dipolar interactions without any restrictions on the anisotropy or nonlinearity of the polarizability of the monomer. The approach is based on three key elements: (1) the statistical field theory framework, in which the discrete monomers are regularized to a continuous dipole distribution, (2) a replacement of the nonlocal dipole-dipole interactions by the local electrostatics partial differential equation with the continuous dipole distribution as the forcing, and (3) the use of a completely general relation between the polarization and the local electric field. Rather than treat the dipole-dipole interactions directly, the continuous description in the field theory enables the computationally tractable nonlocal-to-local transformation. Further, it enables the use of a realistic statistical-mechanical ensemble wherein the average far-field applied electric field is prescribed, rather than prescribing the applied field at every point in the polymer domain. The model is applied, using the finite element method, to study the electromechanical response of a polymer chain in the ensemble with fixed far-field applied electric field and fixed chain stretch. The nonlocal dipolar interactions are found to increase, over the case where dipole-dipole interactions are neglected, the magnitudes of the polarization and electric field by orders of magnitude as well as significantly change their spatial distributions. Next, the effect of the relative orientation between the applied field and the chain on the local electric field and polarization is studied. The model predicts that the elastic response of the polymer chain is linear, consistent with the Gaussian approximation, and largely unchanged by the orientation of the applied electric field, though the polarization and local electric field distributions are significantly impacted.
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Affiliation(s)
- Pratik Khandagale
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
| | - Carlos Garcia-Cervera
- Department of Mathematics, University of California, Santa Barbara CA 93106, USA
- BCAM, Basque Center for Applied Mathematics, E48009 Bilbao, Basque Country, Spain
| | - Gal deBotton
- Department of Mechanical Engineering, Ben Gurion University, 84105 Beer Sheva, Israel
- Department of Biomedical Engineering, Ben Gurion University, 84105 Beer Sheva, Israel
| | | | - Carmel Majidi
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
- Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
- Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
| | - Kaushik Dayal
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
- Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh PA 15213, USA
- Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA
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LoGrande K, Shankar MR, Dayal K. A dimensionally-reduced nonlinear elasticity model for liquid crystal elastomer strips with transverse curvature. SOFT MATTER 2023; 19:8764-8778. [PMID: 37938345 DOI: 10.1039/d3sm00664f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2023]
Abstract
Liquid crystalline elastomers (LCEs) are active materials that are of interest due to their programmable response to various external stimuli such as light and heat. When exposed to these stimuli, the anisotropy in the response of the material is governed by the nematic director, which is a continuum parameter that is defined as the average local orientation of the mesogens in the liquid crystal phase. This nematic director can be programmed to be heterogeneous in space, creating a vast design space that is useful for applications ranging from artificial ligaments to deployable structures to self-assembling mechanisms. Even when specialized to long and thin strips of LCEs - the focus of this work - the vast design space has required the use of numerical simulations to aid in experimental discovery. To mitigate the computational expense of full 3-d numerical simulations, several dimensionally-reduced rod and ribbon models have been developed for LCE strips, but these have not accounted for the possibility of initial transverse curvature, like carpenter's tape spring. Motivated by recent experiments showing that transversely-curved LCE strips display a rich variety of configurations, this work derives a dimensionally-reduced 1-d model for pre-curved LCE strips. The 1-d model is validated against full 3-d finite element calculations, and it is also shown to capture experimental observations, including tape-spring-like localizations, in activated LCE strips.
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Affiliation(s)
- Kevin LoGrande
- Department of Civil and Environmental Engineering, Carnegie Mellon University, USA.
- Computation and Information Sciences Directorate, CCDC Army Research Lab, USA
| | - M Ravi Shankar
- Department of Industrial Engineering, University of Pittsburgh, USA
| | - Kaushik Dayal
- Department of Civil and Environmental Engineering, Carnegie Mellon University, USA.
- Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University, USA
- Department of Mechanical Engineering, Carnegie Mellon University, USA
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Khandagale P, Breitzman T, Majidi C, Dayal K. Statistical field theory for nonlinear elasticity of polymer networks with excluded volume interactions. Phys Rev E 2023; 107:064501. [PMID: 37464704 DOI: 10.1103/physreve.107.064501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2022] [Accepted: 04/17/2023] [Indexed: 07/20/2023]
Abstract
Polymer networks formed by cross linking flexible polymer chains are ubiquitous in many natural and synthetic soft-matter systems. Current micromechanics models generally do not account for excluded volume interactions except, for instance, through imposing a phenomenological incompressibility constraint at the continuum scale. This work aims to examine the role of excluded volume interactions on the mechanical response. The approach is based on the framework of the self-consistent statistical field theory of polymers, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the end-to-end vectors of the chains by this deformation. In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in three dimensions using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (i) the linearized Poisson's ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (ii) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real polymer networks, driven by the interplay between the entropy and the excluded volume interactions; and (iii) the emergence of a deformation-sensitive localization instability at large excluded volumes.
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Affiliation(s)
- Pratik Khandagale
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Timothy Breitzman
- Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA
| | - Carmel Majidi
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Kaushik Dayal
- Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
- Pittsburgh Quantum Institute, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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Peeketi AR, Sol JAHP, Swaminathan N, Schenning APHJ, Debije MG, Annabattula RK. Calla Lily flower inspired morphing of flat films to conical tubes. JOURNAL OF POLYMER SCIENCE 2022. [DOI: 10.1002/pol.20220492] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Affiliation(s)
- Akhil R. Peeketi
- Center for Responsive Soft Matter, Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India
| | - Jeroen A. H. P. Sol
- Laboratory of Stimuli‐Responsive Functional Materials and Devices (SFD), Department of Chemical Engineering and Chemistry Eindhoven University of Technology (TU/e) Eindhoven The Netherlands
| | - Narasimhan Swaminathan
- Center for Responsive Soft Matter, Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India
| | - Albert P. H. J. Schenning
- Laboratory of Stimuli‐Responsive Functional Materials and Devices (SFD), Department of Chemical Engineering and Chemistry Eindhoven University of Technology (TU/e) Eindhoven The Netherlands
- Institute for Complex Molecular Systems (ICMS) Eindhoven University of Technology (TU/e) Eindhoven The Netherlands
| | - Micheal G. Debije
- Laboratory of Stimuli‐Responsive Functional Materials and Devices (SFD), Department of Chemical Engineering and Chemistry Eindhoven University of Technology (TU/e) Eindhoven The Netherlands
| | - Ratna K. Annabattula
- Center for Responsive Soft Matter, Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India
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