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The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in... more
The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in athermal and θ-solvents has been developed. It is shown that under weak anisotropic deformations of the deswollen network, the entanglement tube consists of fractal loopy globules. In a θ-solvent, slight deformations of the network lead to a decrease in the overlap of loopy globules without changing their sizes. Deformations of swollen networks, as well as strong deformations of deswollen networks, are described in terms of the slip-tube model. An effective Hamiltonian has been derived that determines the entropy of fractal loopy globules. Based on the Hamiltonian, it is shown that topological constraints can be described using the polymer-quantum diffusion analogy. The connection between topological and quantum entanglements is demonstrated.
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A scaling model for the structure of coacervates is presented for mixtures of oppositely charged polyelectrolytes of both symmetric and asymmetric charge densities for different degrees of electrostatic strength and levels of added salt.... more
A scaling model for the structure of coacervates is presented for mixtures of oppositely charged polyelectrolytes of both symmetric and asymmetric charge densities for different degrees of electrostatic strength and levels of added salt. At low electrostatic strengths, weak coacervates, with the energy of electrostatic interactions between charges less than the thermal energy, k B T, are liquid. At higher electrostatic strengths, strong coacervates are gels with cross-links formed by ion pairs of opposite charges bound to each other with energy higher than k B T. Charge-symmetric coacervates are formed for mixtures of oppositely charged polyelectrolytes with equal and opposite charge densities. While charge-symmetric weak coacervates form a semidilute polymer solution with a correlation length equal to the electrostatic blob size, charge-symmetric strong coacervates form reversible gels with a correlation length on the order of the distance between bound ion pairs. Charge-asymmetric coacervates are formed from mixtures of oppositely charged polyelectrolytes with different charge densities. While charge-asymmetric weak coacervates form double solutions with two correlation lengths and qualitatively different chain conformations of polycations and polyanions, charge-asymmetric strong coacervates form bottlebrush and star-like gels. Unlike liquid coacervates, for which an increase in the concentration of added salt screens electrostatic interactions, causing structural rearrangement which eventually leads to their dissolution, the salt does not affect the structure of strong coacervates until ion pairs dissociate and the gel disperses.
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A review of the main elasticity models of flexible polymer networks is presented. Classical models of phantom networks suggest that the networks have a tree-like structure. The conformations of their strands are described by the model of... more
A review of the main elasticity models of flexible polymer networks is presented. Classical models of phantom networks suggest that the networks have a tree-like structure. The conformations of their strands are described by the model of a combined chain, which consists of the network strand and two virtual chains attached to its ends. The distribution of lengths of virtual chains in real polydisperse networks is calculated using the results of the presented replica model of polymer networks. This model describes actual networks having strongly overlapping and interconnected loops of finite sizes. The conformations of their strands are characterized by the generalized combined chain model. The model of a sliding tube is represented, which describes the general anisotropic deformations of an entangled network in the melt. I propose a generalization of this model to describe the crossover between the entangled and phantom regimes of a swollen network. The obtained dependence of the Mooney-Rivlin parameters C1 and C2 on the polymer volume fraction is in agreement with experiments. The main results of the theory of heterogeneities in polymer networks are also discussed.
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The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show... more
The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show that the elastic modulus of such networks has two contributions: of elastically effective strands and of finite size loops shunting the network strands. The latter contribution substantially depends on the interaction of the network strand monomers at network preparation conditions. The result of calculations performed in the framework of the replica theory of polymer networks is interpreted using a generalized combined chain model. This model allows us to describe quantitatively the elasticity of the polymer network with finite size loops and the deformation of its individual strands. We also calculate the impact of primary loops and cyclic defects of arbitrary concentration on the elasticity of such a network.
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We develop a scaling theory and perform molecular dynamic simulations of weakly interacting coacervates with electrostatic interaction energy per charge less than thermal energy kT. Such liquid coacervates formed by oppositely charged... more
We develop a scaling theory and perform molecular dynamic simulations of weakly interacting coacervates with electrostatic interaction energy per charge less than thermal energy kT. Such liquid coacervates formed by oppositely charged polyelectrolytes can be asymmetric in charge density and number of charges per chain. We predict that these coacervates form interpenetrating solutions with two correlation lengths and two qualitatively different types of conformations of polyelectrolytes with lower and higher charge densities, which are analogous to chain conformations in quasi-neutral and in polyelectrolyte solutions, respectively. Weaker charged chains are attracted to and adsorbed on stronger charged chains forming a screening "coat" around the stronger charged polyelectrolytes. Salt added at lower concentrations screens the repulsion between stronger charged chains, thereby reducing the thickness of the screening coat and resulting in the nonzero net polymer charge in the coacervate. At higher salt concentrations salt screens the attraction between oppositely charged chains, decreasing the coacervate concentration and its polymeric charge density. Thus, we predict a nonmonotonic salt concentration dependence of polymeric charge density for asymmetric coacervates. A phase diagram for a mixture of oppositely charged polyelectrolytes at various compositions is proposed for different salt concentrations.
We present a conceptual framework for adding molecular details of chain extension and force-coupled bond dissociation to the Lake−Thomas model of tear energy in rubbery crack propagation. Incorporating data reported from single-molecule... more
We present a conceptual framework for adding molecular details of chain extension and force-coupled bond dissociation to the Lake−Thomas model of tear energy in rubbery crack propagation. Incorporating data reported from single-molecule force spectroscopy experiments provides an estimate for the stored energy per bond at fracture of ∼60 kJ mol −1 for typical hydrocarbon polymers, well below the typical carbon−carbon bond dissociation energy in these systems. Opportunities to test and exploit the role of molecular extension and covalent bond scission in experimental systems are proposed. ■ INTRODUCTION The fracture of covalent polymer networks limits many of their potential applications, and a quantitative understanding of the molecular mechanisms underlying the mechanical fracture of polymer gels and elastomers might inform methods by which to design more robust polymeric materials. Fracture of a covalent network is inherently molecular, as it requires the scission of covalent chemical bonds. Lake and Thomas 1 recognized over 50 years ago that as a crack propagates across the failing interface of a stretched material, the bonds that fracture across the crack plane are not the only bonds that have stored energy. Rather, the entire chain that initially bridged the crack plane is fully extended. Therefore, the fracture energy must reflect the energy that goes into entropically stretching and enthalpically deforming each repeat unit along this extended chain (and that is lost upon chain scission), so that more energy is put into the system than is required for a single bond scission (Figure 1). The term "bridging strand" here refers to the cross-link-to-cross-link portion of the polymer strand that spans the plane of crack propagation. Lake and Thomas 1 proposed that the fracture energy G c is equal to the number of such chains per cross-sectional area (σ) multiplied by the energy that is required to break one bridging strand (W): σ ν = = G W R nU 1 2 c 0 (1) where R 0 is the average end-to-end distance of an elastically active network strand (subchain) in its undeformed state, ν is the number density of such elastically active subchains, n is the average number of repeat units along the bridging strand, and U is the energy that is stored in each repeat unit when the bridging strand breaks. Geometrical arguments give σ = 1 / 2 νR 0 as an estimate of the number of elastically active chains per cross-sectional area, where the prefactor of 1 / 2 comes from the projection of the end-to-end vectors of subchains onto the normal of the crack plane. 1 Lake and Thomas argued that W is proportional to the number of repeat units to give W = nU. It is worthwhile to note that the fracture energy G c in the Lake− Thomas theory does not consider the dissipation (e.g., viscoelasticity or poroelasticity) of real polymeric materials. Our adjustment focuses on the molecular energy parameter in Lake−Thomas theory; hence, dissipation is not considered in the following treatment as well. Figure 1. Schematic illustration of crack propagation at the crack frontier (crack tip). In the dashed circle, the black dots, blue plain strands, and red dashed line represent the cross-links, bridging strands, and crack propagation plane, respectively. Black dashed strands represent the network continuum to which the bridging strands are connected.
The ability to precisely control the transport of single DNA molecules through a nanoscale channel is critical to DNA sequencing and mapping technologies that are currently under development. Here we show how the electrokinetically driven... more
The ability to precisely control the transport of single DNA molecules through a nanoscale channel is critical to DNA sequencing and mapping technologies that are currently under development. Here we show how the electrokinetically driven introduction of DNA molecules into a nanochannel is facilitated by incorporating a three-dimensional nanofunnel at the nanochannel entrance. Individual DNA molecules are imaged as they attempt to overcome the entropic barrier to nanochannel entry through nanofunnels with various shapes. Theoretical modeling of this behavior reveals the pushing and pulling forces that result in up to a 30-fold reduction in the threshold electric field needed to initiate nanochannel entry. In some cases, DNA molecules are stably trapped and axially positioned within a nanofunnel at sub-threshold electric field strengths, suggesting the utility of nanofunnels as force spectroscopy tools. These applications illustrate the benefit of finely tuning nanoscale conduit geometries, which can be designed using the theoretical model developed here.
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1. It is shown that the nonlinear quasiclassical Eilenberger equations of the theory of superconductivity can be transformed to a system of linear differential equations. Influence of impurities and nondiagonal interactions in spin are... more
1. It is shown that the nonlinear quasiclassical Eilenberger equations of the theory of superconductivity can be transformed to a system of linear differential equations. Influence of impurities and nondiagonal interactions in spin are considered. Generalization of the obtained equations to the case of nonstationary interactions is given.
2. Using the temperature technique, exact spatially inhomogeneous solutions for a one-dimensional superconductor are found. The case of periodic solutions for the order parameter and solutions of domain wall type with a space-homogeneous order parameter at infinity is considered. It is shown that the method under consideration is applicable to finding solutions of Peierls systems and relativistic one-dimensional fermion fields.
2. Using the temperature technique, exact spatially inhomogeneous solutions for a one-dimensional superconductor are found. The case of periodic solutions for the order parameter and solutions of domain wall type with a space-homogeneous order parameter at infinity is considered. It is shown that the method under consideration is applicable to finding solutions of Peierls systems and relativistic one-dimensional fermion fields.
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The ability to precisely control the transport of single DNA molecules through a nanoscale channel is critical to DNA sequencing and mapping technologies that are currently under development. Here we show how the electrokinetically driven... more
The ability to precisely control the transport of single DNA molecules through a nanoscale channel is critical to DNA sequencing and mapping technologies that are currently under development. Here we show how the electrokinetically driven introduction of DNA molecules into a nanochannel is facilitated by incorporating a three-dimensional nanofunnel at the nanochannel entrance. Individual DNA molecules are imaged as they attempt to overcome the entropic barrier to nanochannel entry through nanofunnels with various shapes. Theoretical modeling of this behavior reveals the pushing and pulling forces that result in up to a 30-fold reduction in the threshold electric field needed to initiate nanochannel entry. In some cases, DNA molecules are stably trapped and axially positioned within a nanofunnel at sub-threshold electric field strengths, suggesting the utility of nanofunnels as force spectroscopy tools. These applications illustrate the benefit of finely tuning nanoscale conduit geometries, which can be designed using the theoretical model developed here.
The general microscopic theory of "secondary" quantum effects in small superconducting tunnel junc tions is proposed. Both linear (due to the Ohmic shunt) and "cosine" (due to the quasiparticle tun neling) dissipation mechanisms are... more
The general microscopic theory of "secondary" quantum effects in small superconducting tunnel junc tions is proposed. Both linear (due to the Ohmic shunt) and "cosine" (due to the quasiparticle tun neling) dissipation mechanisms are considered. Quantum diffusion in the quasicharge space may be caused e ith er by Cooper-pair tunneling or by single-electron tunneling (SET) for low temperature and bias. Quantum corrections to the classical Ohmic resistance are calculated for both cases. Co herent oscillations of the voltage (both Bloch and SET oscillations and also th e ir coexistence) and current-voltage characteristics are studied for a l l parameter regions of in te re st.
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We develop a scaling model of coacervates formed by oppositely charged polyelectrolytes. The intramolecular electrostatic interactions in dilute polyanion or polycation solutions are characterized by the so-called electrostatic blobs, Dor... more
We develop a scaling model of coacervates formed by oppositely charged polyelectrolytes. The intramolecular electrostatic interactions in dilute polyanion or polycation solutions are characterized by the so-called electrostatic blobs, Dor D+ respectively. Polyelectrolyte sections of electrostatic blob size repel neighboring sections of the same chain with electrostatic energy on the order of thermal energy kT. Conformations of polyanion or polycation chains in their respective dilute solutions with no added salt can be described as stretched arrays of their corresponding electrostatic blobs of size Dor D+. The structure of a coacervate formed upon mixing polyanion and polycation solutions depends on the electrostatic attraction between oppositely charged polyelectrolytes balanced by the short-range repulsion. In the symmetric case with D= D+ = De the coacervate is a dense packing of these electrostatic blobs with neighboring oppositely charged blobs of size De attracting each other ...
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The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in... more
The main models of phantom and topologically entangled polymer networks are surveyed. A theory of anisotropic and nonaffine deformation of both swollen and deswollen (with partial solvent removal) strongly entangled polymer networks in athermal and θ-solvents has been developed. It is shown that under weak anisotropic deformations of the deswollen network, the entanglement tube consists of fractal loopy globules. In a θ-solvent, slight deformations of the network lead to a decrease in the overlap of loopy globules without changing their sizes. Deformations of swollen networks, as well as strong deformations of deswollen networks, are described in terms of the slip-tube model. An effective Hamiltonian has been derived that determines the entropy of fractal loopy globules. Based on the Hamiltonian, it is shown that topological constraints can be described using the polymer-quantum diffusion analogy. The connection between topological and quantum entanglements is demonstrated.
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1. It is shown that the nonlinear quasiclassical Eilenberger equations of the theory of superconductivity can be transformed to a system of linear differential equations. Influence of impurities and nondiagonal interactions in spin are... more
1. It is shown that the nonlinear quasiclassical Eilenberger equations of the theory of superconductivity can be transformed to a system of linear differential equations. Influence of impurities and nondiagonal interactions in spin are considered. Generalization of the obtained equations to the case of nonstationary interactions is given. 2. Using the temperature technique, exact spatially inhomogeneous solutions for a one-dimensional superconductor are found. The case of periodic solutions for the order parameter and solutions of domain wall type with a space-homogeneous order parameter at infinity is considered. It is shown that the method under consideration is applicable to finding solutions of Peierls systems and relativistic one-dimensional fermion fields.
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Even when the bonds are not broken, the tension can alter the electronic structure of monomers and directly impact molecular reactivity, optical properties, and conductivity of adsorbed macromolecules. 2 Model We explore different... more
Even when the bonds are not broken, the tension can alter the electronic structure of monomers and directly impact molecular reactivity, optical properties, and conductivity of adsorbed macromolecules. 2 Model We explore different possible conformations of a molecular brush on a planar substrate in non-solvent environment (air-solid interface) and relate them to backbone tension. Consider a molecular brush that contains N>>1 flexible side chains (grafts) with n>>1 monomers of size b each. The side chains are tethered to the backbone with grafting density 1/m, where m<<n is the number of monomers in a spacer connecting neighboring grafts. The strong attractive intramolecular interactions (≃kBT per monomer, where kB is Boltzmann constant and T is absolute temperature) lead to dense packing of the side chains in the molecular brush. A weaker attraction to the substrate leads only to the deformation of the macromolecule (changes its shape without destroying dense packi...
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Within the framework of the model of a quasi-network of entanglements, the Landau free energy was calculated for a strongly deformed, lightly cross-linked polymer network, in which topological constraints are in effect and play an... more
Within the framework of the model of a quasi-network of entanglements, the Landau free energy was calculated for a strongly deformed, lightly cross-linked polymer network, in which topological constraints are in effect and play an important role. In such polymers, random internal stresses were shown to give rise to spatial density inhomogeneities, their scale significantly exceeding the size of the elementary cell of a quasinetwork. The kinetics describing the appearance of these inhomogeneities was theoretically examined. The calculated kinetics was demonstrated to agree with the results of neutron scattering studies
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The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show... more
The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show that the elastic modulus of such networks has two contributions: of elastically effective strands and of finite size loops shunting the network strands. The latter contribution substantially depends on the interaction of the network strand monomers at network preparation conditions. The result of calculations performed in the framework of the replica theory of polymer networks is interpreted using a generalized combined chain model. This model allows us to describe quantitatively the elasticity of the polymer network with finite size loops and the deformation of its individual strands. We also calculate the impact of primary loops and cyclic defects of arbitrary concentration on the elasticity of such a network.
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The statistics of various types of chains in deformed polymer networks, namely, the statistics of interjunction chains of the network, individual free chains, or the chains of polymeric solvent, were studied. Because of inhomogeneity of... more
The statistics of various types of chains in deformed polymer networks, namely, the statistics of interjunction chains of the network, individual free chains, or the chains of polymeric solvent, were studied. Because of inhomogeneity of the chemical structure of polymer networks, individual macromolecules may localize in randomly distributed regions of the network. Theoretical analysis of the localized states is presented. Theory of elasticity of highly deformed networks is developed, and the effect of orientational ordering of the chains of polymeric solvent is explained. The dynamics of relaxation processes in polymer networks was studied. It was shown that the above theory correctly predicts the «butterfly»- and «rhomb»-type isointensity scattering curves observed in small-angle neutron scattering experiments
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We present a comprehensive study of the statistical mechanics of randomly cross-linked polymer gels made of phantom chains with excluded volume, starting from the microscopic Edwards model and ending with analytical expressions for... more
We present a comprehensive study of the statistical mechanics of randomly cross-linked polymer gels made of phantom chains with excluded volume, starting from the microscopic Edwards model and ending with analytical expressions for density correlation functions. Using replica field theory, we derive and solve the mean field equations and show that the ground state is described by a solution with spontaneously broken symmetry with respect to translation in replica space. The mean field solution contains statistical information about the fluctuations of monomers and of cross-links about their average positions in the network. We construct the free energy functional of the density field and of a Gaussian random field that represents the quenched inhomogeneous structure of the gel. This free energy is then used to obtain all the statistical information about the density fluctuations in a deformed network, in terms of the average number of cross-links and the thermodynamic parameters (temperature, density, and quality of solvent) of the initial and the deformed states. We calculate the density correlation functions that describe quenched and thermal density fluctuations. Analytical expressions for the correlators are obtained both in the long-wavelength and the short-wavelength limits. We use the mean field solution to obtain statistical information about the effect of macroscopic deformation on the stretching of individual network chains and discuss the deviations from affine response to deformation. Finally, we discuss recent extensions and generalizations of the ideas presented in this work.