1
|
Li SF, Muthukumar M. Theory of thermoreversible gelation and anomalous concentration fluctuations in polyzwitterion solutions. J Chem Phys 2024; 161:024903. [PMID: 38990120 DOI: 10.1063/5.0216981] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2024] [Accepted: 06/24/2024] [Indexed: 07/12/2024] Open
Abstract
We present a theoretical framework to investigate thermoreversible phase transitions within polyzwitterion systems, encompassing macrophase separations (MPS) and gelation. In addition, we explore concentration fluctuations near critical points associated with MPS, as well as tricritical and bicritical points at the intersection of MPS and gelation. By utilizing mean-field percolation theory and field theory formalism, we derive the Landau free energy in terms of polyzwitterion concentration with fixed dipole strengths and other experimental variables, such as temperatures and salt concentrations. As the temperature decreases, the dipoles can form cross-links, resulting in polyzwitterion associations. The associations can grow to a gel network and enhance the propensity for MPS, including liquid-liquid, liquid-gel, and gel-gel phase separations. Remarkably, the associations also impact critical behaviors. Using the renormalization group technique, we find that the critical exponents of the polyzwitterion concentration correlation functions significantly deviate from those in the Ising universality class due to the presence of polyzwitterion associations, leading to crossover critical behaviors.
Collapse
Affiliation(s)
- Siao-Fong Li
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Murugappan Muthukumar
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA
| |
Collapse
|
2
|
Yu S, Wu S, Fang S, Tang Z, Zhang L, Guo B. Skeletal Network Enabling New-Generation Thermoplastic Vulcanizates. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2023; 35:e2300856. [PMID: 36987971 DOI: 10.1002/adma.202300856] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2023] [Revised: 03/04/2023] [Indexed: 06/16/2023]
Abstract
Upcycling of cross-linked rubbers is pressing. The introduction of dynamic covalent bonds into the networks is a popular tactic for recycling thermosetting polymers, but it is very challenging to integrate engineering performance and continuous yet stable reprocessability. Based on traditional rubber formulations, herein, a straightforward strategy is presented for constructing a skeletal network (SN) through interfacial crosslinking and percolation of rubbery granules in a rubber matrix. Rapid exchange reactions involving dynamic interfacial sulfides realize repeated "fragmentation and healing" in the solid-state and consequent reconfiguration of the SN topology of the elastomer, thus endowing the resultant SN elastomer with continuous yet stable re-extrudability. These SN elastomers with hierarchical structures exhibit high gel contents, high resilience, low creep, and reinforcibility competitive to traditional vulcanizates. Specifically, SN elastomers exhibit better overall performance than commercial thermoplastic vulcanizates (TPVs) materials. Overall, a new concept of thermoplastic vulcanizates is proposed, which will promote the sustainable development of rubbers.
Collapse
Affiliation(s)
- Shuangjian Yu
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| | - Siwu Wu
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| | - Shifeng Fang
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| | - Zhenghai Tang
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| | - Liqun Zhang
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| | - Baochun Guo
- Institute of Emergent Elastomers, School of Materials Science and Engineering, South China University of Technology, Guangzhou, Guangdong, 510640, P. R. China
| |
Collapse
|
3
|
Li DT, Rudnicki PE, Qin J. Distribution Cutoff for Clusters near the Gel Point. ACS POLYMERS AU 2022; 2:361-370. [PMID: 36254314 PMCID: PMC9562459 DOI: 10.1021/acspolymersau.2c00020] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/06/2022] [Revised: 06/29/2022] [Accepted: 06/29/2022] [Indexed: 11/29/2022]
Abstract
![]()
The mechanical and dynamic properties of developing networks
near
the gel point are susceptible to the distribution of clusters coexisting
with percolating networks. The distribution of cluster numbers follows
a broad power law, wrapped by a cutoff function that rapidly decays
at a characteristic size. The form of the cutoff function has been
speculated based on known results from lattice percolation and, in
certain cases, solved. We obtained this cutoff function from simulated
dynamic clusters of polymeric precursor chains using a hybrid Monte
Carlo algorithm. The results obtained from three different precursor
chain lengths are consistent with each other and are consistent with
the expectation from lattice percolation.
Collapse
Affiliation(s)
- Douglas T. Li
- Department of Physics, Stanford University, Stanford, California 94305, United States
| | - Paul E. Rudnicki
- Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States
| | - Jian Qin
- Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States
| |
Collapse
|
4
|
Gloger D, Mileva D, Albrecht A, Hubner G, Androsch R, Gahleitner M. Long-Chain Branched Polypropylene: Effects of Chain Architecture, Melt Structure, Shear Modification, and Solution Treatment on Melt Relaxation Dynamics. Macromolecules 2022. [DOI: 10.1021/acs.macromol.1c02113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
| | | | | | | | - René Androsch
- Interdisciplinary Center for Transfer-oriented Research in Natural Sciences, Martin Luther University Halle-Wittenberg, 06009 Halle/Saale, Germany
| | | |
Collapse
|
5
|
Dai Y, Zhang R, Sun W, Wang T, Chen Y, Tong Z. Dynamical heterogeneity in the gelation process of a polymer solution with a lower critical solution temperature. SOFT MATTER 2021; 17:3222-3233. [PMID: 33624665 DOI: 10.1039/d0sm02159h] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The gelation of a hydrophobically modified hyaluronic acid aqueous solution which shows a lower critical solution temperature of about 25 °C was investigated by multi-particle tracking microrheology. The linear viscoelasticity of the gelling system is converted from the microrheological data. The critical gelling temperature Tgel = 36.3 °C was determined from the loss tangent by the Winter-Chambon criterion. The critical exponent n = 0.62 was determined from the shift factors of the time-cure superposition. The length scales of the dynamic heterogeneity of the gelling system were analyzed using a proposed framework where single-particle and multi-particle non-Gaussian parameters were compared. The length scale of the dynamic heterogeneous regions monotonically decreases during the gelation process, consistent with the nucleation-and-growth mechanism of phase separation. Distributions of local viscosity in the gelling system were extracted from the observed distributions of particle displacement as a time-dependent fingerprint of the dynamic heterogeneity of the gelling system. The results and analyzing methods proposed in the present work can be applied to other microrheological studies.
Collapse
Affiliation(s)
- Yingkang Dai
- School of Materials Science and Engineering, South China University of Technology, Guangzhou 510640, China.
| | | | | | | | | | | |
Collapse
|
6
|
Affiliation(s)
- Xiao Cao
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China
| | - Xinyue Yu
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China
| | - Jian Qin
- Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States
| | - Quan Chen
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China
| |
Collapse
|
7
|
Das C, Read DJ, Soulages JM, Shirodkar PP. Static and Dynamic Scaling Close to Gelation in Chain-Polymerization: Effect of Reactor Type. MACROMOL THEOR SIMUL 2017. [DOI: 10.1002/mats.201700006] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Chinmay Das
- School of Mathematics; University of Leeds; Leeds LS2 9JT UK
| | - Daniel J. Read
- School of Mathematics; University of Leeds; Leeds LS2 9JT UK
| | - Johannes M. Soulages
- Corporate Strategic Research; ExxonMobil Research and Engineering Company; Annandale NJ 08801 USA
| | | |
Collapse
|
8
|
Huang C, Chen Q, Weiss RA. Nonlinear Rheology of Random Sulfonated Polystyrene Ionomers: The Role of the Sol–Gel Transition. Macromolecules 2016. [DOI: 10.1021/acs.macromol.6b02057] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Affiliation(s)
- Chongwen Huang
- Department
of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States
| | - Quan Chen
- State
Key Laboratory of Polymer Physics and Chemistry, Changchun Institute
of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China
| | - R. A. Weiss
- Department
of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States
| |
Collapse
|
9
|
Gabrielle B, Gomez E, Korb JP. Probing Rubber Cross-Linking Generation of Industrial Polymer Networks at Nanometer Scale. J Phys Chem B 2016; 120:5581-9. [DOI: 10.1021/acs.jpcb.6b03601] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
| | | | - Jean-Pierre Korb
- Physique
de la Matière Condensée, Ecole Polytechnique-CNRS, 91128 Palaiseau, France
| |
Collapse
|
10
|
Huang C, Wang C, Chen Q, Colby RH, Weiss RA. Reversible Gelation Model Predictions of the Linear Viscoelasticity of Oligomeric Sulfonated Polystyrene Ionomer Blends. Macromolecules 2016. [DOI: 10.1021/acs.macromol.6b00620] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Chongwen Huang
- Department
of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States
| | - Chao Wang
- Department
of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States
| | - Quan Chen
- State
Key Laboratory of Polymer Physics and Chemistry, Changchun Institute
of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022 China
| | - Ralph H. Colby
- Department
of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - R. A. Weiss
- Department
of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States
| |
Collapse
|
11
|
Grimaldo M, Roosen-Runge F, Hennig M, Zanini F, Zhang F, Jalarvo N, Zamponi M, Schreiber F, Seydel T. Hierarchical molecular dynamics of bovine serum albumin in concentrated aqueous solution below and above thermal denaturation. Phys Chem Chem Phys 2016; 17:4645-55. [PMID: 25587698 DOI: 10.1039/c4cp04944f] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The dynamics of proteins in solution is a complex and hierarchical process, affected by the aqueous environment as well as temperature. We present a comprehensive study on nanosecond time and nanometer length scales below, at, and above the denaturation temperature Td. Our experimental data evidence dynamical processes in protein solutions on three distinct time scales. We suggest a consistent physical picture of hierarchical protein dynamics: (i) self-diffusion of the entire protein molecule is confirmed to agree with colloid theory for all temperatures where the protein is in its native conformational state. At higher temperatures T > Td, the self-diffusion is strongly obstructed by cross-linking or entanglement. (ii) The amplitude of backbone fluctuations grows with increasing T, and a transition in its dynamics is observed above Td. (iii) The number of mobile side-chains increases sharply at Td while their average dynamics exhibits only little variations. The combination of quasi-elastic neutron scattering and the presented analytical framework provides a detailed microscopic picture of the protein molecular dynamics in solution, thereby reflecting the changes of macroscopic properties such as cluster formation and gelation.
Collapse
Affiliation(s)
- Marco Grimaldo
- Institut Max von Laue - Paul Langevin (ILL), CS 20156, F-38042 Grenoble, France.
| | | | | | | | | | | | | | | | | |
Collapse
|
12
|
|
13
|
Piñeiro Y, López-Quintela MA, Rivas J, Leisner D. Percolation threshold and scattering power law of gelatin gels. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041409. [PMID: 19518235 DOI: 10.1103/physreve.79.041409] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2008] [Revised: 01/22/2009] [Indexed: 05/27/2023]
Abstract
Gelation of gelatin was broadly studied by experimental and theoretical methods. Power laws observed on the gel point-mainly obtained by dynamic light scattering (DLS)-are considered to be the signature of some special dynamic phenomena ascribed to the appearance of a percolation cluster. We present here experimental (DLS and rheometric measurements) and Monte Carlo simulation studies showing that the percolation threshold and DLS power-law decay occur on different times. We ascribe the percolation point to the time where the scattering medium mode diverges. This mode is sensitive to the clusters' growth and diverges when the system attains the percolation threshold. The power-law behavior is obtained only in the postpercolation regime.
Collapse
Affiliation(s)
- Y Piñeiro
- Department of Applied Physics, University of Santiago de Compostela, Campus Sur, E-15782 Santiago de Compostela, Spain
| | | | | | | |
Collapse
|
14
|
Richter S. Comparison of Critical Exponents Determined by Rheology and Dynamic Light Scattering on Irreversible and Reversible Gelling Systems. ACTA ACUST UNITED AC 2007. [DOI: 10.1002/masy.200751010] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
15
|
|
16
|
Sferrazza M, Carelli C. Interfaces and fluctuations in confined polymeric liquid mixtures: from immiscible to near critical systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2007; 19:073102. [PMID: 22251583 DOI: 10.1088/0953-8984/19/7/073102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
In this paper the structure of the interface between polymer films is discussed to elucidate fluctuations and confinement effects in fluid polymer mixtures. The neutron reflectivity technique has been employed to investigate the dependence of the structure of the interface on the degree of immiscibility of the polymers over a wide range, as criticality is approached, and to characterize it in terms of intrinsic width, as calculated by mean field theories, and capillary fluctuations. For more immiscible systems, as the degree of incompatibility between the polymers is decreased, the width of the interface increases slowly, and it is independent of the molecular weight of the polymers. Closer to the critical point the dependence on the degree of miscibility becomes stronger and the way in which the interfacial width diverges, as criticality is approached, is related to both chain length and Flory-Huggins interaction parameter (χ). The results have been compared to the predictions of mean field theories. Self-consistent field numerical calculations, with the additional contribution due to capillary waves, provide a good description of the width of the interface between two polymer bulk phases, in particular at higher and intermediate degrees of immiscibility-the product of the Flory-Huggins interaction parameter χ and the number N of monomers of the chain, χN. For more miscible systems a crossover is observed to a region where the square gradient theory in the weak segregation limit better approximates the experimental results. Moreover, the mechanisms by which confinement affects the interface have been investigated. To understand the relative importance of the long ranged van der Waals forces and short ranged 'truncation forces' in modifying thermally excited fluctuations at the polymer/polymer interface, the thickness dependence of the interfacial width has been studied for different degrees of miscibility, approaching criticality. The results show a gradual transition from a region where long ranged dispersion forces are dominant in influencing the capillary wave spectrum, for higher degrees of immiscibility, to a region where short ranged forces-connected to the presence of the walls-become more important and the dependence of the interfacial width on film thickness is stronger. The thickness dependence of the interfacial width was also studied for systems with different molecular weights in different conditions of miscibility, to investigate the effects of a confined geometry on polymers with different chain lengths as criticality is approached.
Collapse
Affiliation(s)
- Michele Sferrazza
- Département de Physique, Université Libre de Bruxelles, Boulevard du Triomphe CP223, 1050 Bruxelles, Belgium.
| | | |
Collapse
|
17
|
Gasparoux J, Tixier T, Tordjeman P. Critical viscoelastic behavior of polydimethylsiloxane networks around the sol-gel threshold. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:011802. [PMID: 17358175 DOI: 10.1103/physreve.75.011802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2006] [Indexed: 05/14/2023]
Abstract
Seven model polydimethylsiloxane (PDMS) networks were obtained by hydrosilation of a difunctional vinyl-terminated PDMS prepolymer with a SiH-containing cross-linker. Viscoelastic experiments, completed by size exclusion chromatography and static light scattering experiments, were performed in order to study the influence of molecular parameters on the dynamic properties around the sol-gel threshold. The dynamic critical parameter u was determined from experiments close to and above the sol-gel threshold. Our results show that the growth mechanism of PDMS clusters and the viscoelastic behavior are a function of the ratio NN_{e} , where N and N_{e} are, respectively, the numbers of Kuhn monomers between branch points and between entanglements. For NN_{e}<1 , the growth mechanism of clusters is the critical percolation and u=0.69 , and for NN_{e}>1 , the growth mechanim of clusters is the diffusion-limited cluster aggregation and u=0.76 .
Collapse
Affiliation(s)
- Jennifer Gasparoux
- Laboratoire d'Analyse des Interfaces et de Nanophysique, UMR-5011, Universite Montpellier II, CC082, Place E. Bataillon, 34095 Montpellier Cedex 5, France
| | | | | |
Collapse
|
18
|
Liu R, Gao X, Oppermann W. Dynamic light scattering studies on random cross-linking of polystyrene in semi-dilute solution. POLYMER 2006. [DOI: 10.1016/j.polymer.2006.10.020] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
|
19
|
Das C, Read DJ, Kelmanson MA, McLeish TCB. Dynamic scaling in entangled mean-field gelation polymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:011404. [PMID: 16907093 DOI: 10.1103/physreve.74.011404] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2006] [Indexed: 05/11/2023]
Abstract
We present a simple reaction kinetics model to describe the polymer synthesis used by Lusignan et al. [Phys. Rev. E 60, 5657 (1999)] to produce randomly branched polymers in the vulcanization class. Numerical solution of the rate equations gives probabilities for different connections in the final product, which we use to generate a numerical ensemble of representative molecules. All structural quantities probed in the experiments are in quantitative agreement with our results for the entire range of molecular weights considered. However, with detailed topological information available in our calculations, our estimate of the "rheologically relevant" linear segment length is smaller than that estimated from the experimental results. We use a numerical method based on a tube model of polymer melts to calculate the rheological properties of such molecules. Results are in good agreement with experiment, except that in the case of the largest molecular weight samples our estimate of the zero-shear viscosity is significantly lower than the experimental findings. Using acid concentration as an indicator for closeness to the gelation transition, we show that the high-molecular-weight polymers considered are at the limit of mean-field behavior--which possibly is the reason for this disagreement. For a truly mean-field gelation class of model polymers, we numerically calculate the rheological properties for a range of segment lengths. Our calculations show that the tube theory with dynamical dilation predicts that, very close to the gelation limit, the contribution to viscosity for this class of polymers is dominated by the contribution from constraint-release Rouse motion and the final viscosity exponent approaches a Rouse-like value.
Collapse
Affiliation(s)
- Chinmay Das
- Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK
| | | | | | | |
Collapse
|
20
|
|
21
|
|
22
|
Buzza DMA. Power law polydispersity and fractal structure of hyperbranched polymers. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2004; 13:79-86. [PMID: 15024618 DOI: 10.1140/epje/e2004-00042-3] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Using the complementary approaches of Flory theory and the overlap function, we study the molecular weight distribution and conformation of hyperbranched polymers formed by the melt polycondensation of A-R(N)(0)-B(f - 1) monomers in their reaction bath close to the mean field gel point p(A) = 1, where p(A) is the fraction of reacted A groups. Here f > or = 3, N(0) is the degree of polymerisation of the linear spacer linking the A group and the f-1 B groups and condensation occurs exclusively between the A and B groups. For Epsilon tripe bond (1-pA) <<1, we assume that the number density of hyperbranched polymers with degree of polymerisation N generally obeys the scaling form n(N) =N(- tau)f(N/Nl) and we explicitly show that this scaling assumption is correct in the mean field regime (here Nl is the largest characteristic degree of polymerisation and the function f (N/Nl) cuts off the power law sharply for N>Nl). We find the upper critical dimension for this system is d(c) = 4, so that for d> or = dc the mean field values for the polydispersity exponent and fractal dimension apply: tau=3/2, d(f) = 4. For d = 3, mean field theory is still correct for Epsilon > Epsilon G where Epsilon G approximately equal N -1 0 is the Ginzburg point; for Epsilon < Epsilon G, mean field theory applies on small mass scales N< N c but breaks down on larger mass scales N> N c where N c approximately equal N 3 0 is a cross-over mass. Within the Ginzburg zone (i.e., d< d(c) < Epsilon G), we show that the hyperbranched chains on mass scales N> N(c) are non-Gaussian with fractal dimension given by d(f) = d (for d = 2,3,4). Our results are qualitatively different from those of the percolation model and indicate that the polycondensation of AB(f-1), unlike polymer gelation, is not described by percolation theory. Instead many of our results are similar to those for a monodisperse melt of randomly branched polymers, a consequence of the fact that tau < 2 so that polydispersity is irrelevant for excluded volume screening in hyperbranched polymer melts.
Collapse
Affiliation(s)
- D M A Buzza
- Polymer IRC, Department of Physics & Astronomy, University of Leeds, LS2 9JT, Leeds, UK.
| |
Collapse
|
23
|
Vernon D, Plischke M, Joós B. Viscoelasticity near the gel point: a molecular dynamics study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:031505. [PMID: 11580340 DOI: 10.1103/physreve.64.031505] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2001] [Revised: 05/24/2001] [Indexed: 05/23/2023]
Abstract
We report on extensive molecular dynamics simulations on systems of soft spheres of functionality f, i.e., particles that are capable of bonding irreversibly with a maximum of f other particles. These bonds are randomly distributed throughout the system and imposed with probability p. At a critical concentration of bonds, p(c) approximately 0.2488 for f=6, a gel is formed and the shear viscosity eta diverges according to eta approximately (p(c)-p)(-s). We find s approximately 0.7 in agreement with some experiments and with a recent theoretical prediction based on Rouse dynamics of phantom chains. The diffusion constant decreases as the gel point is approached but does not display a well-defined power law.
Collapse
Affiliation(s)
- D Vernon
- Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
| | | | | |
Collapse
|
24
|
Peng W, Goldbart PM. Renormalization-group approach to the vulcanization transition. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3339-3357. [PMID: 11088108 DOI: 10.1103/physreve.61.3339] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/1999] [Indexed: 05/23/2023]
Abstract
The vulcanization transition-the cross-link-density-controlled equilibrium phase transition from the liquid to the amorphous solid state-is explored analytically from a renormalization-group perspective. The analysis centers on a minimal model which has previously been shown to yield a rich and informative picture of vulcanized matter at the mean-field level, including a connection with mean-field percolation theory (i.e., random graph theory). This minimal model accounts for both the thermal motion of the constituents and the quenched random constraints imposed on their motion by the cross-links, as well as particle-particle repulsion which suppresses density fluctuations and plays a pivotal role in determining the symmetry structure (and hence properties) of the model. A correlation function involving fluctuations of the amorphous solid order parameter, the behavior of which signals the vulcanization transition, is examined, its physical meaning is elucidated, and the associated susceptibility is constructed and analyzed. A Ginzburg criterion for the width (in cross-link density) of the critical region is derived and is found to be consistent with a prediction due to de Gennes. Inter alia, this criterion indicates that the upper critical dimension for the vulcanization transition is 6. Certain universal critical exponents characterizing the vulcanization transition are computed, to lowest nontrivial order, within the framework of an expansion around the upper critical dimension. This expansion shows that the connection between vulcanization and percolation extends beyond mean-field theory, surviving the incorporation of fluctuations in the sense that pairs of physically analogous quantities (one percolation related and one vulcanization related) are found to be governed by identical critical exponents, at least to first order in the departure from the upper critical dimension (and presumably beyond). The relationship between the present approach to vulcanized matter and other approaches, such as those based on gelation-percolation ideas, is explored in the light of this connection. To conclude, some expectations for how the vulcanization transition is realized in two dimensions, developed with H. E. Castillo, are discussed.
Collapse
Affiliation(s)
- W Peng
- Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
| | | |
Collapse
|
25
|
Lusignan CP, Mourey TH, Wilson JC, Colby RH. Viscoelasticity of randomly branched polymers in the vulcanization class. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:5657-69. [PMID: 11970459 DOI: 10.1103/physreve.60.5657] [Citation(s) in RCA: 44] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/1998] [Revised: 06/11/1999] [Indexed: 04/18/2023]
Abstract
We report viscosity, recoverable compliance, and molar mass distribution for a series of randomly branched polyester samples with long linear chain sections between branch points. Molecular structure characterization determines tau=2.47+/-0.05 for the exponent controlling the molar mass distribution, so this system belongs to the vulcanization (mean-field) universality class. Consequently, branched polymers of similar size strongly overlap and form interchain entanglements. The viscosity diverges at the gel point with an exponent s=6.1+/-0.3, that is significantly larger than the value of 1.33 predicted by the branched polymer Rouse model (bead-spring model without entanglements). The recoverable compliance diverges at the percolation threshold with an exponent t=3.2+/-0.2. This effect is consistent with the idea that each branched polymer of size equal to the correlation length stores k(B)T of elastic energy. Near the gel point, the complex shear modulus is a power law in frequency with an exponent u=0.33+/-0.05. The measured rheological exponents confirm that the dynamic scaling law u=t/(s+t) holds for the vulcanization class. Since s is larger and u is smaller than the Rouse values observed in systems that belong to the critical percolation universality class, we conclude that entanglements profoundly increase the longest relaxation time. Examination of the literature data reveals clear trends for the exponents s and u as functions of the chain length between branch points. These dependencies, qualitatively explained by hierarchical relaxation models, imply that the dynamic scaling observed in systems that belong to the vulcanization class is nonuniversal.
Collapse
Affiliation(s)
- C P Lusignan
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | | | | | | |
Collapse
|
26
|
Ord G, Whittington SG. Percolation on a randomly expanded lattice: a model of polymer gels. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/1/006] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
27
|
Bromberg L. Scaling of Rheological Properties of Hydrogels from Associating Polymers. Macromolecules 1998. [DOI: 10.1021/ma980523f] [Citation(s) in RCA: 97] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Lev Bromberg
- Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
| |
Collapse
|
28
|
Katayama S. Thermoresponsive Opalescence of N-Isopropylacrylamide/ N, N‘-Methylene(bis)acrylamide Copolymer Gel. J Phys Chem B 1997. [DOI: 10.1021/jp972229f] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Seiji Katayama
- University of Shizuoka, School of Pharmaceutical Sciences, 52-1, Yada Shizuoka 422, Japan
| |
Collapse
|
29
|
Barsky SJ, Plischke M, Joós B, Zhou Z. Elastic properties of randomly cross-linked polymers. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5370-5376. [PMID: 9965722 DOI: 10.1103/physreve.54.5370] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
30
|
Affiliation(s)
- Marian Mours
- Department of Chemical Engineering, University of Massachusetts at Amherst, Amherst, Massachusetts 01003
| | - H. Henning Winter
- Department of Chemical Engineering, University of Massachusetts at Amherst, Amherst, Massachusetts 01003
| |
Collapse
|
31
|
Huthmann M, Rehkopf M, Zippelius A, Goldbart PM. Universality of gelation: End-linking versus cross-linking and stiff rods versus flexible chains. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:3943-3949. [PMID: 9965542 DOI: 10.1103/physreve.54.3943] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
32
|
Lusignan CP, Mourey TH, Wilson JC, Colby RH. Viscoelasticity of randomly branched polymers in the critical percolation class. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6271-6280. [PMID: 9964144 DOI: 10.1103/physreve.52.6271] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
33
|
Goldbart PM, Zippelius A. Amorphous solid state of vulcanized macromolecules: A variational approach. PHYSICAL REVIEW LETTERS 1993; 71:2256-2259. [PMID: 10054627 DOI: 10.1103/physrevlett.71.2256] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
34
|
Malier L, Boilot JP, Chaput F, Devreux F. Nuclear-magnetic-resonance study of silica gelation kinetics. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:959-962. [PMID: 9908197 DOI: 10.1103/physreva.46.959] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
35
|
Bou� F, Bastide J, Buzier M, Collette C, Lapp A, Herz J. Dynamics of permanent and temporary networks: Small angle neutron scattering measurements and related remarks on the classical models of rubber deformation. ACTA ACUST UNITED AC 1987. [DOI: 10.1007/bf01188369] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
|
36
|
|
37
|
|