1
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Oda K, Yasuda S. Effect of shear flow on the transverse thermal conductivity of polymer melts. Phys Rev E 2024; 109:064501. [PMID: 39021017 DOI: 10.1103/physreve.109.064501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2023] [Accepted: 05/08/2024] [Indexed: 07/20/2024]
Abstract
The effect of shear flows on the thermal conductivity of polymer melts is investigated using a reversed nonequilibrium molecular-dynamics (RNEMD) method. We extended the original RNEMD method to simultaneously produce spatial gradients of temperature and flow velocity in a single direction. This method enables accurate measurement of the thermal conductivity in the direction transverse to shear flow. The Weissenberg number defined with the shear rate and the relaxation time of the polymer conformation can uniformly differentiate the occurrence of shear rate dependence of the thermal conductivity across different chain lengths. The stress-thermal rule (STR) (i.e., the linear relationship between anisotropic parts of the stress tensor and the thermal conductivity tensor) holds for entangled polymer melts even under shear flows but not for unentangled polymer melts. Furthermore, once entanglements form in polymer chains, the stress-thermal coefficient in the STR remains independent of the polymer chain length. These observations align with the theoretical foundation of the STR, which focuses on energy transmission along the network structure of entangled polymer chains [B. van den Brule, Rheol. Acta 28, 257 (1989)0035-451110.1007/BF01329335]. However, under driven shear flows, the stress-thermal coefficient is notably smaller than that measured in the literature for a quasiquiescent state without external forces. Although the mechanism of the STR in shear flows has yet to be fully elucidated, our study confirmed the validity of the STR in shear flows. This allows us to use the STR as a constitutive equation for computational thermofluid dynamics of polymer melts, thus having broad engineering applications.
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2
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Moussavi A, Pal S, Wu Z, Keten S. Characterizing the shear response of polymer-grafted nanoparticles. J Chem Phys 2024; 160:134903. [PMID: 38573850 DOI: 10.1063/5.0188494] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2023] [Accepted: 03/18/2024] [Indexed: 04/06/2024] Open
Abstract
Grafting polymer chains to the surface of nanoparticles overcomes the challenge of nanoparticle dispersion within nanocomposites and establishes high-volume fractions that are found to enable enhanced material mechanical properties. This study utilizes coarse-grained molecular dynamics simulations to quantify how the shear modulus of polymer-grafted nanoparticle (PGN) systems in their glassy state depends on parameters such as strain rate, nanoparticle size, grafting density, and chain length. The results are interpreted through further analysis of the dynamics of chain conformations and volume fraction arguments. The volume fraction of nanoparticles is found to be the most influential variable in deciding the shear modulus of PGN systems. A simple rule of mixture is utilized to express the monotonic dependence of shear modulus on the volume fraction of nanoparticles. Due to the reinforcing effect of nanoparticles, shortening the grafted chains results in a higher shear modulus in PGNs, which is not seen in linear systems. These results offer timely insight into calibrating molecular design parameters for achieving the desired mechanical properties in PGNs.
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Affiliation(s)
- Arman Moussavi
- Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208, USA
| | - Subhadeep Pal
- Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208, USA
| | - Zhenghao Wu
- Department of Chemistry, Xi'an Jiaotong Liverpool University, Suzhou, People's Republic of China
| | - Sinan Keten
- Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois 60208, USA
- Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA
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3
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Johnson LC, Phelan FR. Comparison of Friction Parametrization from Dynamics and Material Properties for a Coarse-Grained Polymer Melt. J Phys Chem B 2023; 127:7054-7069. [PMID: 37523783 PMCID: PMC10472480 DOI: 10.1021/acs.jpcb.3c03273] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/02/2023]
Abstract
In this work, we extend an approach to coarse-grained (CG) modeling for polymer melts in which the conservative potential is parametrized using the iterative Boltzmann inversion (IBI) method and the accelerated dynamics inherent to IBI are corrected using the dissipative Langevin thermostat with a single tunable friction parameter (J. Chem. Phys. 2021, 154, 084114). Diffusive measures from picoseconds to nanoseconds are used to determine the Langevin friction factor to apply to the CG model to recover all-atom (AA) dynamics; the resulting friction factors are then compared for consistency. Here, we additionally parametrize the CG dynamics using a material property, the zero-shear viscosity, which we measure using the Green-Kubo (GK) method. Two materials are studied, squalane as a function of temperature and the same polystyrene oligomers previously studied as a function of chain length. For squalane, the friction derived from the long-time diffusive measures and the viscosity all strongly increase with decreasing temperature, showing an Arrhenius-like dependence, and remain consistent with each other over the entire temperature range. In contrast, the friction required for the picosecond diffusive measurement, the Debye-Waller factor, is somewhat lower than the friction from long-time measures and relatively insensitive to temperature. A time-dependent friction would be required to exactly reproduce the AA measurements during the caging transition connecting these two extremes over the entire timespan at this level of coarse-graining. For the polystyrene oligomers for which we previously characterized the diffusive friction, the viscosity-parametrized frictions are consistent with the diffusive measures for the smallest chain length. However, for the longer chains, we find different trends based on measurement method with friction derived from rotational diffusion remaining nearly constant, friction derived from translational diffusion showing a modestly increasing trend, and viscosity-derived friction showing a modest decreasing trend. This seems to indicate that there is some sensitivity of the friction measurement method for systems with increased relaxation times and that in particular, the unsteady dynamics of the individual parametrization schemes plays a role in this. Increased difficulty in applying the GK method with increasing relaxation time of the longer chain systems is also discussed. Overall, we find that when the material is in a high-temperature melt state and the viscosity measurement is reliable, the friction parametrization from the diffusive friction measures is consistent and the lower cost diffusive parametrization is a reliable means for modeling viscosity. Our data give insight into the time-dependent friction one might compute using a non-Markovian approach to enable the recovery of AA dynamics over a wider range of time scales than can be computed using a single friction.
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Affiliation(s)
- Lilian C Johnson
- Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States
| | - Frederick R Phelan
- Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States
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4
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Li YC, Wu ZP, Zong ZH, Cao XZ. Rheological Role of Stiff Nanorings on Concurrently Strengthening and Toughening Polymer Nanocomposites. ACS Macro Lett 2023; 12:183-188. [PMID: 36692488 DOI: 10.1021/acsmacrolett.2c00610] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
Nanorings, which are increasingly uncovered in natural systems and synthesized in man-made materials, exhibit dynamics distinct from those known for linear chains. We show in this study that, when immersed in a polymer melt matrix, segments of a stiff nanoring (SNR) have more facilitated subdiffusion, i.e., with a larger scaling exponent in the mean squared displacement, than those belonging to one flexible counterpart, while the whole SNR is more suppressed by its surroundings. It is revealed that adding SNRs contributes to achieving the long-anticipated rheological objective of sol- and gel-like characteristics at high and low shearing frequencies, respectively. This study suggests the promising prospect of exploiting SNRs to concurrently strengthen and toughen target polymer nanocomposites.
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Affiliation(s)
- Yu-Chao Li
- Department of Physics and Fujian Provincial Key Laboratory for Soft Functional Materials Research, College of Physical Science and Technology, Xiamen University, Xiamen 361005, P.R. China
| | - Zong-Pei Wu
- Department of Physics and Fujian Provincial Key Laboratory for Soft Functional Materials Research, College of Physical Science and Technology, Xiamen University, Xiamen 361005, P.R. China
| | - Ze-Hao Zong
- Department of Physics and Fujian Provincial Key Laboratory for Soft Functional Materials Research, College of Physical Science and Technology, Xiamen University, Xiamen 361005, P.R. China
| | - Xue-Zheng Cao
- Department of Physics and Fujian Provincial Key Laboratory for Soft Functional Materials Research, College of Physical Science and Technology, Xiamen University, Xiamen 361005, P.R. China
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5
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Cao XZ, Merlitz H, Wu CX, Forest MG. Screening confinement of entanglements: Role of a self-propelling end inducing ballistic chain reptation. Phys Rev E 2022; 106:L022501. [PMID: 36110008 DOI: 10.1103/physreve.106.l022501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 07/18/2022] [Indexed: 06/15/2023]
Abstract
Synthetic and natural nanomaterials with self-propelling mechanisms continue to be explored to boost chain mobility beyond normal reptation in the crowded environments of entangled chains. Here we employ scaling theory and numerical simulations to demonstrate that activating one chain end of a singular or isolated chain boosts entanglement-constrained chain reptation from the one-dimensional diffusive mobility as described by the de Gennes-Edwards-Doi model to ballistic motion along the entanglement tube contour. The active chain is effectively screened from the constraint of entanglements on length scales exceeding the tube size.
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Affiliation(s)
- Xue-Zheng Cao
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - Holger Merlitz
- Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany
| | - Chen-Xu Wu
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - M Gregory Forest
- Departments of Mathematics, Applied Physical Sciences, Biomedical Engineering, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, USA
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6
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Cao XZ, Merlitz H, Wu CX, Forest MG. Chain stiffness boosts active nanoparticle transport in polymer networks. Phys Rev E 2021; 103:052501. [PMID: 34134347 DOI: 10.1103/physreve.103.052501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Accepted: 04/27/2021] [Indexed: 11/07/2022]
Abstract
Recent advances in technologies such as nanomanufacturing and nanorobotics have opened new pathways for the design of active nanoparticles (NPs) capable of penetrating biolayers for biomedical applications, e.g., for drug delivery. The coupling and feedback between active NP motility (with large stochastic increments relative to passive NPs) and the induced nonequilibrium deformation and relaxation responses of the polymer network, spanning scales from the NP to the local structure of the network, remain to be clarified. Using molecular dynamics simulations, combined with a Rouse mode analysis of network chains and position and velocity autocorrelation functions of the NPs, we demonstrate that the mobility of active NPs within cross-linked, concentrated polymer networks is a monotonically increasing function of chain stiffness, contrary to passive NPs, for which chain stiffness suppresses mobility. In flexible networks, active NPs exhibit a behavior similar to passive NPs, with a boost in mobility proportional to the self-propulsion force. These results are suggestive of design strategies for active NP penetration of stiff biopolymer matrices.
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Affiliation(s)
- Xue-Zheng Cao
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - Holger Merlitz
- Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany
| | - Chen-Xu Wu
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - M Gregory Forest
- Departments of Mathematics, Applied Physical Sciences, Biomedical Engineering, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, USA
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7
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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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8
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Wong CPJ, Choi P. A review on the relaxation dynamics analysis of unentangled polymers with different structures. MOLECULAR SIMULATION 2020. [DOI: 10.1080/08927022.2020.1810851] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Chi Pui Jeremy Wong
- Donadeo Innovation Centre for Engineering, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada
| | - Phillip Choi
- Donadeo Innovation Centre for Engineering, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada
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9
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Rauscher PM, Schweizer KS, Rowan SJ, de Pablo JJ. Dynamics of poly[n]catenane melts. J Chem Phys 2020; 152:214901. [PMID: 32505155 DOI: 10.1063/5.0007573] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Inspired by advances in the chemical synthesis of interlocking polymer architectures, extensive molecular dynamics simulations have been conducted to study the dynamical properties of poly[n]catenanes-polymers composed entirely of interlocking rings-in the melt state. Both the degree of polymerization (number of links) and the number of beads per ring are systematically varied, and the results are compared to linear and ring polymers. A simple Rouse-like model is presented, and its analytical solution suggests a decomposition of the dynamics into "ring-like" and "linear-like" regimes at short and long times, respectively. In agreement with this picture, multiple sub-diffusive regimes are observed in the monomer mean-squared-displacements even though interchain entanglement is not prevalent in the system. However, the Rouse-type model does not account for the topological effects of the mechanical bonds, which significantly alter the dynamics at intermediate length scales both within the rings and at the chain segment scales. The stress relaxation in the system is extremely rapid and may be conveniently separated into ring-like and linear-like contributions, again in agreement with the Rouse picture. However, the viscosity has a non-monotonic dependence on the ring size for long chains, which disagrees strongly with theoretical predictions. This unexpected observation cannot be explained in terms of chain disentanglement and is inconsistent with other measures of polymer relaxation. Possible mechanisms for this behavior are proposed and implications for materials design are discussed.
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Affiliation(s)
- Phillip M Rauscher
- Pritzker School of Molecular Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
| | - Kenneth S Schweizer
- Department of Materials Science, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801-3028, USA
| | - Stuart J Rowan
- Pritzker School of Molecular Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
| | - Juan J de Pablo
- Pritzker School of Molecular Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
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10
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Moghimikheirabadi A, Ilg P, Sagis LMC, Kröger M. Surface Rheology and Structure of Model Triblock Copolymers at a Liquid–Vapor Interface: A Molecular Dynamics Study. Macromolecules 2020. [DOI: 10.1021/acs.macromol.9b01995] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
| | - Patrick Ilg
- School of Mathematical, Physical and Computational Sciences, University of Reading, Reading RG6 6AX, U.K
| | - Leonard M. C. Sagis
- Food Physics Group, Wageningen University, Bornse Weilanden 9, 6708 WG Wageningen, The Netherlands
| | - Martin Kröger
- Polymer Physics, Department of Materials, ETH Zürich, Zürich CH-8093, Switzerland
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11
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Cao XZ, Merlitz H, Wu CX. Mechanical Strength Management of Polymer Composites through Tuning Transient Networks. J Phys Chem Lett 2020; 11:710-715. [PMID: 31922749 DOI: 10.1021/acs.jpclett.9b03697] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The addition of transient networks to polymer composites marks a new direction toward the design of novel materials, with numerous biomedical and industrial applications. The network structure connected by transient cross-links (CLs) relaxes as time evolves, which results in the stretching release of polymer strands between transient CLs during strain. Using molecular dynamics simulations, we measure directly the stress-strain curves of double polymer networks (DPNs), containing both transient and permanent components, at different strain rates. Lifetime and density of transient CLs control the relaxation spectrum of transient networks and determine the mechanical properties of DPNs. A Rouse mode analysis reveals that at high strain rates the mechanical strength of DPNs is defined jointly by the cross-linking structures of permanent and transient networks. At low strain rates, the cross-linking structure of transient network relaxes, leaving the permanent component of the network as a sole contributor to the mechanical strength of DPNs. The transient network is shown to facilitate a dissipation of energy at higher strain rates and prevents a rupture of the network, while the permanent network preserves the structural integrity of the composite at low strain rates. This study provides computational and theoretical foundations for designing polymer composites with desirable mechanical strength and toughness by means of tuning transient networks.
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Affiliation(s)
- Xue-Zheng Cao
- Department of Physics , Xiamen University , Xiamen 361005 , People's Republic of China
| | - Holger Merlitz
- Leibniz-Institut für Polymerforschung Dresden , 01069 Dresden , Germany
| | - Chen-Xu Wu
- Department of Physics , Xiamen University , Xiamen 361005 , People's Republic of China
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12
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Zhang C, Yang S, Padmanabhan V, Akcora P. Solution Rheology of Poly(acrylic acid)-Grafted Silica Nanoparticles. Macromolecules 2019. [DOI: 10.1021/acs.macromol.9b01309] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Chongfeng Zhang
- Department of Chemical Engineering & Materials Science, Stevens Institute of Technology, Hoboken, New Jersey 07030, United States
| | - Siyang Yang
- Department of Chemical Engineering & Materials Science, Stevens Institute of Technology, Hoboken, New Jersey 07030, United States
| | - Venkat Padmanabhan
- Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505, United States
| | - Pinar Akcora
- Department of Chemical Engineering & Materials Science, Stevens Institute of Technology, Hoboken, New Jersey 07030, United States
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13
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Hsu HP, Kremer K. Clustering of Entanglement Points in Highly Strained Polymer Melts. Macromolecules 2019; 52:6756-6772. [PMID: 31534275 PMCID: PMC6740293 DOI: 10.1021/acs.macromol.9b01120] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2019] [Revised: 08/13/2019] [Indexed: 11/30/2022]
Abstract
Polymer melts undergoing large deformation by elongation are studied by molecular dynamics simulations of bead-spring chains in melts. By applying a primitive path analysis to strongly deformed polymer melts, the role of topological constraints in highly entangled polymer melts is investigated and quantified. We show that the overall, large scale conformations of the primitive paths (PPs) of stretched chains follow affine deformation while the number and the distribution of entanglement points along the PPs do not. Right after deformation, PPs of chains retract in both directions parallel and perpendicular to the elongation. Upon further relaxation we observe a long-lived clustering of entanglement points. Together with the delayed relaxation time this leads to a metastable inhomogeneous distribution of topological constraints in the melts.
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Affiliation(s)
- Hsiao-Ping Hsu
- Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
| | - Kurt Kremer
- Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
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14
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Cao XZ, Merlitz H, Forest MG. Nanoparticle Loading of Unentangled Polymers Induces Entanglement-Like Relaxation Modes and a Broad Sol-Gel Transition. J Phys Chem Lett 2019; 10:4968-4973. [PMID: 31386385 DOI: 10.1021/acs.jpclett.9b01954] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We combine molecular dynamics simulations, imaging and data analysis, and the Green-Kubo summation formula for the relaxation modulus G(t) to elicit the structure and rheology of unentangled polymer-nanoparticle composites distinguished by small NPs and strong NP-monomer attraction, εNPM ≫ kBT. A reptation-like plateau emerges in G(t) beyond a terminal relaxation time scale as the volume fraction, cNP, of NPs increases, coincident with a structure transition. A condensed phase of NP-aggregates forms, tightly interlaced with thin sheets of polymer chains, the remaining phase consisting of free chains void of NPs. Rouse mode analyses are applied to the two individual phases, revealing that long-wavelength Rouse modes in the aggregate phase are the source of reptation-like relaxation. Imaging reveals chain motion confined within the thin sheets between NPs and exhibits a 2D analogue of classical reptation. In the NP-free phase, Rouse modes relax indistinguishable from a neat polymer melt. The Fourier transform of G(t) reveals a sol-gel transition across a broad frequency spectrum, tuned by cNP and εNPM above critical thresholds, below which all structure and rheological transitions vanish.
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Affiliation(s)
- Xue-Zheng Cao
- Departments of Mathematics and Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, United States
- Department of Physics, Xiamen University, Xiamen 361005, P.R. China
| | - Holger Merlitz
- Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany
| | - M Gregory Forest
- Departments of Mathematics and Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, United States
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15
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A ND, Begam N, Ibrahim M, Chandran S, Padmanabhan V, Sprung M, Basu JK. Viscosity and fragility of confined polymer nanocomposites: a tale of two interfaces. NANOSCALE 2019; 11:8546-8553. [PMID: 30990482 DOI: 10.1039/c8nr10362c] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Viscosity and fragility are key parameters determining the processability and thermo-mechanical stability of glassy polymers and polymer nanocomposites (PNCs). In confined polymers, these parameters are largely dominated by the long relaxation times of the polymers adsorbed at the substrate-polymer interface. On the other hand, for polymer nanocomposites, the interface layer (IL) between the nanoparticles and the surrounding matrix chains often control not only the morphology and dispersion but also various parameters like viscosity and glass transition temperature. Confined PNCs, hence, present a unique opportunity to study the interplay of these two independent interfacial effects. Here, we report the results of X-ray scattering based dynamics measurements of PNC thin films, with a two IL width, unraveling the subtle interplay of these two interfaces on the measured viscosity and fragility. Coupled with coarse-grained molecular dynamics (MD) simulations, our experimental results demonstrate that the viscosity of the PNC films increases with both the IL width and the thickness of the polymer layer adsorbed at the substrate interface. However, while both pristine PS and PNCs with a higher IL width become stronger glasses, as estimated by their fragility, the PNC with a lower IL width shows an increase in fragility with increasing confinement. Our results suggest a novel method to control thermo-mechanical properties and stability of PNC coatings by independently controlling the two interfacial effects in athermal glassy PNCs.
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Affiliation(s)
- Nimmi Das A
- Department of Physics, Indian Institute of Science, Bangalore, 560012, India.
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16
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Xi L. Molecular simulation for predicting the rheological properties of polymer melts. MOLECULAR SIMULATION 2019. [DOI: 10.1080/08927022.2019.1605600] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Li Xi
- Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada
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17
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Kempfer K, Devémy J, Dequidt A, Couty M, Malfreyt P. Realistic Coarse-Grain Model of cis-1,4-Polybutadiene: From Chemistry to Rheology. Macromolecules 2019. [DOI: 10.1021/acs.macromol.8b02750] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- K. Kempfer
- CNRS, SIGMA Clermont, Institut de Chimie de Clermont-Ferrand, Université Clermont Auvergne, F-63000 Clermont-Ferrand, France
- Manufacture Française des Pneumatiques Michelin, 23, Place des Carmes, 63040 Clermont-Ferrand, France
| | - J. Devémy
- CNRS, SIGMA Clermont, Institut de Chimie de Clermont-Ferrand, Université Clermont Auvergne, F-63000 Clermont-Ferrand, France
| | - A. Dequidt
- CNRS, SIGMA Clermont, Institut de Chimie de Clermont-Ferrand, Université Clermont Auvergne, F-63000 Clermont-Ferrand, France
| | - M. Couty
- Manufacture Française des Pneumatiques Michelin, 23, Place des Carmes, 63040 Clermont-Ferrand, France
| | - P. Malfreyt
- CNRS, SIGMA Clermont, Institut de Chimie de Clermont-Ferrand, Université Clermont Auvergne, F-63000 Clermont-Ferrand, France
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18
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Trazkovich AJ, Wendt MF, Hall LM. Effect of Copolymer Sequence on Local Viscoelastic Properties near a Nanoparticle. Macromolecules 2019. [DOI: 10.1021/acs.macromol.8b02136] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
Affiliation(s)
- Alex J. Trazkovich
- William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 W 19th Ave., Columbus, Ohio 43210, United States
- Cooper Tire & Rubber Company, 701 Lima Ave., Findlay, Ohio 45840, United States
| | - Mitchell F. Wendt
- William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 W 19th Ave., Columbus, Ohio 43210, United States
| | - Lisa M. Hall
- William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 W 19th Ave., Columbus, Ohio 43210, United States
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19
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Cao XZ, Forest MG. Rheological Tuning of Entangled Polymer Networks by Transient Cross-links. J Phys Chem B 2019; 123:974-982. [DOI: 10.1021/acs.jpcb.8b09357] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
Affiliation(s)
- Xue-Zheng Cao
- Department of Mathematics and Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, United States
| | - M. Gregory Forest
- Department of Mathematics and Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, United States
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20
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Fitzpatrick R, Michieletto D, Peddireddy KR, Hauer C, Kyrillos C, Gurmessa BJ, Robertson-Anderson RM. Synergistic Interactions Between DNA and Actin Trigger Emergent Viscoelastic Behavior. PHYSICAL REVIEW LETTERS 2018; 121:257801. [PMID: 30608839 DOI: 10.1103/physrevlett.121.257801] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2018] [Revised: 09/10/2018] [Indexed: 05/12/2023]
Abstract
Composites of flexible and rigid polymers are ubiquitous in biology and industry alike, yet the physical principles determining their mechanical properties are far from understood. Here, we couple force spectroscopy with large-scale Brownian dynamics simulations to elucidate the unique viscoelastic properties of custom-engineered blends of entangled flexible DNA molecules and semiflexible actin filaments. We show that composites exhibit enhanced stress stiffening and prolonged mechanomemory compared to systems of actin or DNA alone, and that these nonlinear features display a surprising nonmonotonic dependence on the fraction of actin in the composite. Simulations reveal that these counterintuitive results arise from synergistic microscale interactions between the two biopolymers. Namely, DNA entropically drives actin filaments to form bundles that stiffen the network but reduce the entanglement density, while a uniform well-connected actin network is required to reinforce the DNA network against yielding and flow. The competition between bundling and connectivity triggers an unexpected stress response that leads equal mass DNA-actin composites to exhibit the most pronounced stress stiffening and the most long-lived entanglements.
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Affiliation(s)
- Robert Fitzpatrick
- Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA
| | - Davide Michieletto
- School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
| | - Karthik R Peddireddy
- Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA
| | - Cole Hauer
- Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA
| | - Carl Kyrillos
- Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA
| | - Bekele J Gurmessa
- Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA
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21
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Ryu JH, Kim Y, Lee WB. Inhomogeneity of block copolymers at the interface of an immiscible polymer blend. Phys Rev E 2018; 97:042502. [PMID: 29758764 DOI: 10.1103/physreve.97.042502] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2017] [Indexed: 11/07/2022]
Abstract
We present the effects of structure and stiffness of block copolymers on the interfacial properties of an immiscible homopolymer blend. Diblock and two-arm grafted copolymers with variation in stiffness are modeled using coarse-grained molecular dynamics to compare the compatibilization efficiency, i.e., reduction of interfacial tension. Overall, grafted copolymers are located more compactly at the interface and show better compatibilization efficiency than diblock copolymers. In addition, an increase in the stiffness for one of the blocks of the diblock copolymers causes unusual inhomogeneous interfacial coverage due to bundle formation. However, an increase in the stiffness for one of blocks of the grafted copolymers prevents the bundle formation due to the branched chain. As a result, homogeneous interfacial coverage of homopolymer blends is realized with significant reduction of interfacial tension which makes grafted copolymer a better candidate for the compatibilizer of immiscible homopolymer blend.
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Affiliation(s)
- Ji Ho Ryu
- School of Chemical and Biological Engineering, Institute of Chemical Processes, Seoul National University, Seoul 08826, Republic of Korea
| | - YongJoo Kim
- KAIST Institute for the Nanocentury, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
| | - Won Bo Lee
- School of Chemical and Biological Engineering, Institute of Chemical Processes, Seoul National University, Seoul 08826, Republic of Korea
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22
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Ge T, Grest GS, Rubinstein M. Nanorheology of Entangled Polymer Melts. PHYSICAL REVIEW LETTERS 2018; 120:057801. [PMID: 29481209 PMCID: PMC5896298 DOI: 10.1103/physrevlett.120.057801] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2017] [Revised: 10/17/2017] [Indexed: 06/08/2023]
Abstract
We use molecular simulations to probe the local viscoelasticity of an entangled polymer melt by tracking the motion of embedded nonsticky nanoparticles (NPs). As in conventional microrheology, the generalized Stokes-Einstein relation is employed to extract an effective stress relaxation function G_{GSE}(t) from the mean square displacement of NPs. G_{GSE}(t) for different NP diameters d are compared with the stress relaxation function G(t) of a pure polymer melt. The deviation of G_{GSE}(t) from G(t) reflects the incomplete coupling between NPs and the dynamic modes of the melt. For linear polymers, a plateau in G_{GSE}(t) emerges as d exceeds the entanglement mesh size a and approaches the entanglement plateau in G(t) for a pure melt with increasing d. For ring polymers, as d increases towards the spanning size R of ring polymers, G_{GSE}(t) approaches G(t) of the ring melt with no entanglement plateau.
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Affiliation(s)
- Ting Ge
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Gary S Grest
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - Michael Rubinstein
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599, USA
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23
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Hsu HP, Kremer K. Static and dynamic properties of large polymer melts in equilibrium. J Chem Phys 2017; 144:154907. [PMID: 27389240 DOI: 10.1063/1.4946033] [Citation(s) in RCA: 84] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present a detailed study of the static and dynamic behaviors of long semiflexible polymer chains in a melt. Starting from previously obtained fully equilibrated high molecular weight polymer melts [G. Zhang et al., ACS Macro Lett. 3, 198 (2014)], we investigate their static and dynamic scaling behaviors as predicted by theory. We find that for semiflexible chains in a melt, results of the mean square internal distance, the probability distributions of the end-to-end distance, and the chain structure factor are well described by theoretical predictions for ideal chains. We examine the motion of monomers and chains by molecular dynamics simulations using the ESPResSo++ package. The scaling predictions of the mean squared displacement of inner monomers, center of mass, and relations between them based on the Rouse and the reptation theory are verified, and related characteristic relaxation times are determined. Finally, we give evidence that the entanglement length Ne,PPA as determined by a primitive path analysis (PPA) predicts a plateau modulus,GN (0)=45(ρkBT/Ne), consistent with stresses obtained from the Green-Kubo relation. These comprehensively characterized equilibrium structures, which offer a good compromise between flexibility, small Ne, computational efficiency, and small deviations from ideality, provide ideal starting states for future non-equilibrium studies.
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Affiliation(s)
- Hsiao-Ping Hsu
- Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
| | - Kurt Kremer
- Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
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24
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Ryu JH, Wee HS, Lee WB. Molecular dynamics study on microstructures of diblock copolymer melts with soft potential and potential recovery. Phys Rev E 2016; 94:032501. [PMID: 27739817 DOI: 10.1103/physreve.94.032501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2016] [Indexed: 06/06/2023]
Abstract
Various microstructures are obtained through the self-assembly of block copolymers on the basis of the compositional fractions and repulsive interactions among different types of beads. The inhomogeneity of block copolymers can be studied by molecular dynamics. However, preparing initial configurations of various self-assembled structures directly by molecular dynamics requires extensive computational time because of topological constraints. Furthermore, manual preparation often becomes a complicated and time-consuming procedure even for the simplest structures, such as a lamellar phase, not to mention three-dimensional bicontinuous cubic phases such as a gyroid phase. In this paper, this difficulty is overcome by using a soft potential, which allows the system to reach a self-assembled state quickly (within 3τ_{d}). Once a self-assembled microstructure is obtained, the normal potential is restored and equilibration steps are performed to enable the calculation of various properties of the microstructures. Various equilibrated phase structures-including S (spherical), H (hexagonal), G (gyroid), and L (lamellar) phases-are obtained by this approach. To verify our method, static and dynamic properties of the lamellar phase are examined and compared with previous results.
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Affiliation(s)
- Ji Ho Ryu
- School of Chemical and Biological Engineering, Institute of Chemical Processes, Seoul National University, Seoul 08826, Republic of Korea
| | - Han Sol Wee
- Department of Chemical and Biomolecular Engineering, Sogang University, Seoul 04107, Republic of Korea
| | - Won Bo Lee
- School of Chemical and Biological Engineering, Institute of Chemical Processes, Seoul National University, Seoul 08826, Republic of Korea
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25
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Agrawal V, Holzworth K, Nantasetphong W, Amirkhizi AV, Oswald J, Nemat‐Nasser S. Prediction of viscoelastic properties with coarse‐grained molecular dynamics and experimental validation for a benchmark polyurea system. ACTA ACUST UNITED AC 2016. [DOI: 10.1002/polb.23976] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Affiliation(s)
- Vipin Agrawal
- School for the Engineering of Matter, Transport and EnergyArizona State UniversityTempe Arizona 85287
| | - Kristin Holzworth
- Department of Mechanical and Aerospace EngineeringCenter of Excellence for Advanced Materials, University of CaliforniaSan DiegoLa Jolla California92093‐0416
| | - Wiroj Nantasetphong
- Department of Mechanical and Aerospace EngineeringCenter of Excellence for Advanced Materials, University of CaliforniaSan DiegoLa Jolla California92093‐0416
| | | | - Jay Oswald
- School for the Engineering of Matter, Transport and EnergyArizona State UniversityTempe Arizona 85287
| | - Sia Nemat‐Nasser
- Department of Mechanical and Aerospace EngineeringCenter of Excellence for Advanced Materials, University of CaliforniaSan DiegoLa Jolla California92093‐0416
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26
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Maurel G, Goujon F, Schnell B, Malfreyt P. Prediction of structural and thermomechanical properties of polymers from multiscale simulations. RSC Adv 2015. [DOI: 10.1039/c4ra16417b] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
We report mesoscale simulations of polymer melts and crosslinked polymer networks by using realistic coarse-grained (CG) models that are developed from atomistic simulations of polymer melts.
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Affiliation(s)
- Gaëtan Maurel
- Clermont Université
- Université Blaise Pascal
- Institut de Chimie de Clermont-Ferrand
- ICCF
- CNRS
| | - Florent Goujon
- Clermont Université
- Université Blaise Pascal
- Institut de Chimie de Clermont-Ferrand
- ICCF
- CNRS
| | - Benoit Schnell
- Manufacture Française de Pneumatiques MICHELIN
- Centre de Ladoux
- 63040 Clermont-Ferrand, France
| | - Patrice Malfreyt
- Clermont Université
- Université Blaise Pascal
- Institut de Chimie de Clermont-Ferrand
- ICCF
- CNRS
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27
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Kalathi JT, Grest GS, Kumar SK. Universal viscosity behavior of polymer nanocomposites. PHYSICAL REVIEW LETTERS 2012; 109:198301. [PMID: 23215430 DOI: 10.1103/physrevlett.109.198301] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2012] [Indexed: 06/01/2023]
Abstract
Nonequilibrium molecular dynamics simulations are used to show that the shear viscosity of a polymer melt can be significantly reduced when filled with small energetically neutral nanoparticles, apparently independent of the polymer's chain length. Analogous to solvent molecules, small nanoparticles act akin to plasticizers and reduce the viscosity of a polymer melt. This effect, which persists for particles whose sizes are as large as the chain size or the entanglement mesh size, whichever is smaller, can be overcome by making the chain-nanoparticle interactions significantly attractive. Our simulations allow us to systematically organize the viscosity data of filled polymer melts, and thus provide a strong basis from which to predict the flow behavior of these commercially important class of materials.
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Affiliation(s)
- Jagannathan T Kalathi
- Department of Chemical Engineering, Columbia University, New York, New York 10027, USA
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28
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Maurel G, Schnell B, Goujon F, Couty M, Malfreyt P. Multiscale Modeling Approach toward the Prediction of Viscoelastic Properties of Polymers. J Chem Theory Comput 2012; 8:4570-9. [DOI: 10.1021/ct300582y] [Citation(s) in RCA: 58] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023]
Affiliation(s)
- G. Maurel
- Manufacture Française des Pneumatiques MICHELIN, Centre de Ladoux, 23 place
des Carmes, 63000 Clermont-Ferrand, France
- Institut de Chimie de Clermont-Ferrand,
ICCF, UMR CNRS 6296, Université Blaise Pascal, 63177 Aubière Cedex, France
| | - B. Schnell
- Manufacture Française des Pneumatiques MICHELIN, Centre de Ladoux, 23 place
des Carmes, 63000 Clermont-Ferrand, France
| | - F. Goujon
- Institut de Chimie de Clermont-Ferrand,
ICCF, UMR CNRS 6296, Université Blaise Pascal, 63177 Aubière Cedex, France
| | - M. Couty
- Manufacture Française des Pneumatiques MICHELIN, Centre de Ladoux, 23 place
des Carmes, 63000 Clermont-Ferrand, France
| | - P. Malfreyt
- Institut de Chimie de Clermont-Ferrand,
ICCF, UMR CNRS 6296, Université Blaise Pascal, 63177 Aubière Cedex, France
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29
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Halverson JD, Lee WB, Grest GS, Grosberg AY, Kremer K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. Dynamics. J Chem Phys 2011; 134:204905. [DOI: 10.1063/1.3587138] [Citation(s) in RCA: 178] [Impact Index Per Article: 13.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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30
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Likhtman AE, Sukumaran SK. Comment on “Entangled Polymer Melts: Relation between Plateau Modulus and Stress Autocorrelation Function”. Macromolecules 2010. [DOI: 10.1021/ma9027849] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Alexei E. Likhtman
- Department of Mathematics, University of Reading, Whiteknights, Reading RG6 6AX, U.K
| | - Sathish K. Sukumaran
- Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan
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31
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Lee WB, Halverson J, Kremer K. Reply to Comment on “Entangled Polymer Melts: Relation between Plateau Modulus and Stress Autocorrelation Function”. Macromolecules 2010. [DOI: 10.1021/ma1004502] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Won Bo Lee
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
- Department of Chemical and Biomolecular Engineering, Sogang University, 1 Sinsu-dong, Mapo-gu, Seoul 121-742, South Korea
| | - Jonathan Halverson
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
| | - Kurt Kremer
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
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