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Wang G, Zippelius A, Müller M. Phase Separation of Randomly Cross-Linked Diblock Copolymers. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c00687] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Gaoyuan Wang
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
| | - Annette Zippelius
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
| | - Marcus Müller
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
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Affiliation(s)
- Gaoyuan Wang
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
| | - Marcus Müller
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
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Zeng D, Hayward RC. Effects of Randomly End-Linked Copolymer Network Parameters on the Formation of Disordered Cocontinuous Phases. Macromolecules 2019. [DOI: 10.1021/acs.macromol.9b00050] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Di Zeng
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003-9263, United States
| | - Ryan C. Hayward
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003-9263, United States
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von der Heydt A, Zippelius A. Phase diagram of selectively cross-linked block copolymers shows chemically microstructured gel. J Chem Phys 2015; 142:054901. [PMID: 25662662 DOI: 10.1063/1.4905831] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study analytically the intricate phase behavior of cross-linked AB diblock copolymer melts, which can undergo two main phase transitions due to quenched random constraints. Gelation, i.e., spatially random localisation of polymers forming a system-spanning cluster, is driven by increasing the number parameter μ of irreversible, type-selective cross-links between random pairs of A blocks. Self-assembly into a periodic pattern of A/B-rich microdomains (microphase separation) is controlled by the AB incompatibility χ inversely proportional to temperature. Our model aims to capture the system's essential microscopic features, including an ensemble of random networks that reflects spatial correlations at the instant of cross-linking. We identify suitable order parameters and derive a free-energy functional in the spirit of Landau theory that allows us to trace a phase diagram in the plane of μ and χ. Selective cross-links promote microphase separation at higher critical temperatures than in uncross-linked diblock copolymer melts. Microphase separation in the liquid state facilitates gelation, giving rise to a novel gel state whose chemical composition density mirrors the periodic AB pattern.
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Affiliation(s)
- Alice von der Heydt
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Annette Zippelius
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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Schmid F. Self-consistent field approach for cross-linked copolymer materials. PHYSICAL REVIEW LETTERS 2013; 111:028303. [PMID: 23889451 DOI: 10.1103/physrevlett.111.028303] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2013] [Indexed: 06/02/2023]
Abstract
A generalized self-consistent field approach for polymer networks with a fixed topology is developed. It is shown that the theory reproduces the localization of cross-links, which is characteristic for gels. The theory is then used to study the order-disorder transition in regular networks of end-linked diblock copolymers. Compared to diblock copolymer melts, the transition is shifted towards lower values of the incompatibility parameter χ (the Flory-Huggins parameter). Moreover, the transition becomes strongly first order already at the mean-field level. If stress is applied, the transition is further shifted and finally vanishes in a critical point.
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Werner M, Sommer JU. Self-organized stiffness in regular fractal polymer structures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051802. [PMID: 21728562 DOI: 10.1103/physreve.83.051802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2011] [Indexed: 05/31/2023]
Abstract
We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.
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Affiliation(s)
- Marco Werner
- Leibniz-Institut für Polymerforschung Dresden eV, Dresden, Germany
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Werner M, Sommer JU. Polymer-decorated tethered membranes under good- and poor-solvent conditions. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2010; 31:383-392. [PMID: 20480962 DOI: 10.1140/epje/i2010-10576-4] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
We study tethered membranes grafted by polymer chains on one side. Mean-field and scaling arguments predicting a spontaneous curvature are compared to the results of lattice-based Monte Carlo simulations using the Bond Fluctuation Model, which are carried out for various grafting densities and chain lengths. We show that already slightly overlapping chains bend the membrane significantly. This proves the entropic origin for the bending stiffness, which is of order kT. To understand the membrane curvature under conditions of very small bending stiffness we apply a geometrical model which takes into account the state of chains at the overlap threshold. Applying a thermal solvent model for the grafted chains, we demonstrate that the bending direction of the membrane can be triggered by variation of the solvent quality. This indicates that polymer-decorated membranes may serve as switchable nanoscale devices.
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Affiliation(s)
- M Werner
- Leibniz-Institut für Polymerforschung Dresden, Hohe Strasse 6, 01069 Dresden, Germany.
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9
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Peleg O, Kröger M, Rabin Y. Effect of network topology on phase separation in two-dimensional Lennard-Jones networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:040401. [PMID: 19518163 DOI: 10.1103/physreve.79.040401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2008] [Indexed: 05/27/2023]
Abstract
We generate two-dimensional Lennard-Jones networks with random topology by preparing a perfect four-functional network of identical harmonic springs and randomly cutting some of the springs. Using molecular-dynamics simulations we find that the fraction p of active springs affects both the temperature of phase separation and the type of structures observed below this temperature, from networklike high-density patterns at p>0.5 ("gel") to dropletlike structures at p<0.5 ("sol"). In the gel domain, these patterns are determined by the interplay between free energy and network topology, with the former dominant as p-->1 and the latter as p-->0.5 .
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Affiliation(s)
- Orit Peleg
- Department of Materials, Polymer Physics, ETH Zürich, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland.
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Klopper AV, Svaneborg C, Everaers R. Microphase separation in cross-linked polymer blends. Efficient replica RPA post-processing of simulation data for homopolymer networks. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2009; 28:89-96. [PMID: 19139938 DOI: 10.1140/epje/i2008-10420-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2008] [Revised: 10/17/2008] [Accepted: 12/03/2008] [Indexed: 05/27/2023]
Abstract
We investigate the behaviour of randomly cross-linked (co)polymer blends using a combination of replica theory and large-scale molecular dynamics simulations. In particular, we derive the analogue of the random phase approximation for systems with quenched disorder and show how the required correlation functions can be calculated efficiently. By post-processing simulation data for homopolymer networks we are able to describe neutron scattering measurements in heterogeneous systems without resorting to microscopic detail and otherwise unphysical assumptions. We obtain structure function data which illustrate the expected microphase separation and contain system-specific information relating to the intrinsic length scales of our networks.
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Affiliation(s)
- A V Klopper
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany.
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Wald C, Goldbart PM, Zippelius A. Glassy correlations and microstructures in randomly cross-linked homopolymer blends. J Chem Phys 2006; 124:214905. [PMID: 16774440 DOI: 10.1063/1.2200697] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We consider a microscopic model of a polymer blend that is prone to phase separation. Permanent cross-links are introduced between randomly chosen pairs of monomers, drawn from the Deam-Edwards distribution. Thereby, not only density but also concentration fluctuations of the melt are quenched-in in the gel state, which emerge upon sufficient cross-linking. We derive a Landau expansion in terms of the order parameters for gelation and phase separation, and analyze it on the mean-field level, including Gaussian fluctuations. The mixed gel is characterized by thermal as well as time-persistent (glassy) concentration fluctuations. Whereas the former are independent of the preparation state, the latter reflect the concentration fluctuations at the instant of cross-linking, provided the mesh size is smaller than the correlation length of phase separation. The mixed gel becomes unstable to microphase separation upon lowering the temperature in the gel phase. Whereas the length scale of microphase separation is given by the mesh size, at least close to the transition, the emergent microstructure depends on the composition and compressibility of the melt. Hexagonal structures, as well as lamellas or random structures with a unique wavelength, can be energetically favorable.
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Affiliation(s)
- Christian Wald
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen.
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Karbarz M, Stojek Z, Georgiou T, Patrickios C. Microphase separation in ABA triblock copolymer-based model conetworks in the bulk: Effect of loop formation. POLYMER 2006. [DOI: 10.1016/j.polymer.2006.05.006] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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13
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Karbarz M, Stojek Z, Patrickios CS. ABA triblock copolymer-based model networks in the bulk: Effect of the number of arms on microphase behavior. POLYMER 2005. [DOI: 10.1016/j.polymer.2005.06.021] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Oyen E, Hentschke R. Computer simulation of polymer networks: Swelling by binary Lennard-Jones mixtures. J Chem Phys 2005; 123:054902. [PMID: 16108688 DOI: 10.1063/1.1979497] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The swelling of regular, tightly meshed model networks is investigated by a molecular-dynamics-Monte Carlo hybrid technique. The chemical equilibrium between two simulation boxes representing the gel phase and a solvent bath, respectively, is obtained by subjecting the Lennard-Jones particles of a binary mixture, serving as explicit solvent, to the particle transfer step of Gibbs ensemble-Monte Carlo. The swelling behavior, especially preferential absorption of a single component, whose dependence on temperature, pressure, and fluid composition is studied, also depends significantly on the size of the central simulation cell. These finite-size effects correlate well with those exhibited by the density of solvent-free (dry) networks. A theoretical expression, whose derivation is based on network elasticity (of dry networks) yields finite-size scaling behavior in good accord with simulation results for both dry networks and gels in contact with solvent baths. This expression can be used to extrapolate the swelling behavior of simulated finite systems to infinite system size.
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Affiliation(s)
- Enno Oyen
- Fachbereich Mathematik und Naturwissenschaften and Institut für Materialwissenschaften, Bergische Universität, D-42097 Wuppertal, Germany
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Hentschke R, Oyen E. Finite size effects in tightly meshed polymer networks. J Chem Phys 2005; 122:104904. [PMID: 15836355 DOI: 10.1063/1.1860551] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Molecular dynamics computer simulations on regular, tightly meshed model networks exhibit variations of the network density with system size. We show that these variations are due mainly to network elasticity. A theoretical expression derived on the basis of the self-consistent-field approach yields finite size scaling behavior in good accord with the simulation for a wide range of thermodynamic conditions.
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Affiliation(s)
- Reinhard Hentschke
- Fachbereich C and Institut für Materialwissenschaften, Bergische Universität, D-42097 Wuppertal, Germany.
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Lu ZY, Hentschke R. Swelling of model polymer networks with different cross-link densities: a computer simulation study. PHYSICAL REVIEW E 2002; 66:041803. [PMID: 12443224 DOI: 10.1103/physreve.66.041803] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2002] [Indexed: 11/07/2022]
Abstract
The swelling of model polymer networks with different cross-link densities is studied via molecular dynamics simulation. During the simulation, the solvent particles, consisting of one interaction center or six interaction centers, respectively, are transferred between two coupled simulation boxes. The gel box includes both network and solvent particles, whereas the solvent box contains solvent only. The particle transfer is controlled by the solvent chemical potential difference in the two boxes, which is calculated via the Widom test particle method for the one-site solvent and via Rosenbluth sampling for the chainlike solvent. The equilibrium swelling ratio of the network as well as the solvent diffusion coefficient under subcritical and supercritical conditions are computed as functions of the network cross-link density for a wide range of temperatures and pressures. In addition, the simulated swelling behavior is compared to a Flory-Huggins-type theory, which yields qualitative agreement for the systems studied here.
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Affiliation(s)
- Z-Y Lu
- Fachbereich Physik and Institute for Materials Science, Bergische Universität, D-42097 Wuppertal, Germany
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