The Local and Collective Mobility in Polymer Chains and Networks with Included Rigid Particles Elastically Connected with the Network

Summary: The theory of molecular mobility of a polymer network with included rod-like particles is developed. The case is considered when the length of rods is comparable or greater than the average distance between neighboring cross-links of the primary network. The long-scale dynamics of the network is described by means of a regular cubic “coarse-grained” model. The junctions of this model describe the great network fragments (domains) the sizes of which are near to the average distance between neighboring rods.The quasi-elastic interactions between rods and network fragments lead to a broad relaxation spectrum for included rods as compared with free rods which are characterized by a single relaxation time of rotational diffusion. The frequency dependence of the dielectric loss factor of included rods is calculated for rods with permanent dipole moments directed parallel to the long axes of the rods chaotically distributed in the network. The frequency dependence of dynamic modulus of a polymer network with included rods is obtained. The increment in the dynamic modulus of the relatively short network motions (smaller than the distance between rods) also is taken into account. The broad relaxation spectrum of included rods leads to appearance of several maximums on the frequency dependences of both the dielectric loss factor and dynamical modulus.     

Local dynamic mechanical properties in model free-standing polymer thin films

High-frequency sinusoidal oscillations of a coarse-grained polymer model are used to calculate the local dynamic mechanical properties (DMPs) of free-standing polymer thin films. The storage modulus G(') and loss modulus G(") are examined as a function of position normal to the free surfaces. It is found that mechanically soft layers arise near the free surfaces of glassy thin films, and that their thickness becomes comparable to the entire film thickness as the temperature approaches the glass transition T(g). As a result, the overall stiffness of glassy thin films decreases with film thickness. It is also shown that two regions coexist in thin films just at the bulk T(g); a melt-like region (G(')G(")) in the middle of the film. Our findings on the existence of a heterogeneous distribution of DMPs in free-standing polymer thin films provide insights into recent experimental measurements of the mechanical properties of glassy polymer thin films.     

Motion of star‐branched chains in polymer networks: a Monte Carlo study

A simple model of branched polymers in confined space is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions (good solvent conditions). A single star molecule is trapped in a network of linear polymer chains of restricted mobility. The simulations are carried out using the classical Metropolis algorithm. Static and dynamic properties of the star‐branched polymer are determined using various networks. The dependence of the longest relaxation time and the self‐diffusion coefficient on chain length and network properties are discussed and the proper scaling laws formulated. The possible mechanism of motion is discussed. The differences between the motion of star‐branched polymers in such a network are compared with the cases of a dense matrix of linear chains and regular rod‐like obstacles.     

Unsolved problems in the theory of dynamics in homogeneous and heterogeneous polymer networks

A brief review related to some unsolved problems of the dynamic theory of polymer networks and to possible ways of solving them is presented. The comparative role of small‐scale intrachain and long‐scale collective interchain relaxation processes, the influence of long‐range hydrodynamic interactions between the network elements and the effective viscous medium, and the problem of the structure heterogeneity of polymer networks are considered. New approaches for theoretical treatment of these problems are proposed and discussed.     

Modified fractal model and rheological properties of colloidal networks

The scaling relationship between the storage modulus (G(')) and the volume fraction of solids (Phi) in fat crystal networks has been explained by the fractal model developed by our group. However, many experimental results and simulation studies suggest that the stress distribution within a colloidal network is dramatically heterogeneous, which means that a small part of the network carries most of the stress, while the other part of the network does not contribute much to the elastic properties of the system. This concept was introduced into a modified fractal model. The volume fraction of solids term (Phi) in the original fractal model was replaced by Phi(e), the effective volume fraction of solids, in the modified fractal model, which represents the volume fraction of stress-carrying solids. A proposed expression for Phi(e) is given and a modified expression for the scaling relationship between G(') and Phi is obtained. The modified fractal model fits the experiment data well and successfully explains the sometimes observed nonlinear log-log behavior between the storage modulus of colloidal networks and their volume fraction of solids.     

Theory of relaxation properties of two‐dimensional polymer networks, 2. Local dynamic characteristics

Using normal mode transformation obtained in Part 1 of this series^[1], the exact analytical expressions for the mean‐square displacements of junctions and non‐junction beads, the autocorrelation functions of the end‐to‐end chain vectors between neighboring junctions, and those of subchain vectors of a two‐dimensional regular network consisting of "bead and spring" Rouse chains are obtained. Contributions of intra‐ and interchain relaxation processes to the local dynamic characteristics considered are compared. The time behavior of dynamic quantities obtained is estimated for different scales of motions. The possibility of describing long‐time relaxation of a two‐dimensional network by a simplified coarse‐grained network model is demonstrated. It is shown that the local relaxation properties of a two‐dimensional polymer network (as well as a three‐dimensional network) on scales smaller than the average distance between cross‐links are very close to those of a single Rouse chain. The large‐scale collective relaxation of the polymer networks having a two‐dimensional connectivity differs considerably from that of the three‐dimensional networks.     

Dynamic master curves from the stretched exponential relaxation modulus

The stretched exponential relaxation modulus of regular and polymer modified asphalts is studied. It is shown that this relaxation function can generate the dynamic functions of these materials very well on any finite interval of the reduced frequencies (master curves). By continuation one can, in principle, cover the whole region of master curves of G′ and G″. The dispersive defect diffusion mechanism, which leads to the stretched exponential law, points to the stronger three-dimensional structure of modified asphalt at low temperatures. The method of calculating G′ and G″ from the stretched exponential relaxation modulus is proposed and tested on one regular and one modified asphalt. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1225–1232, 1997     

Dielectric relaxation of polymer networks built from macromolecules with dipole moment directed along the end-to-end chain vector

The dielectric relaxation properties are considered for polymer networks built from polar macromolecules with the dipole moment directed along the end-to-end chain vector. The viscoeleastic cubic model of a regular network is used. The fixed average volume of a polymer network is ensured by the effective internal pressure. The dynamic models of polymer networks with external and interchain friction are studied. Two cases are considered: (1) polar chains cross-linked in a network at their ends, and (2) a densely cross-linked network with many network junctions per polar chain. The expressions for the autocorrelation functions of the total dipole moment of a network, which determine the dielectric susceptibility, are calculated. The relaxation spectrum of the autocorrelation function consists of two regions: the high-frequency relaxation spectrum of a chain fragment between two neighbouring junctions (intrachain relaxation spectrum) and the lowfrequency interchain relaxation spectrum. The interchain relaxation spectrum is determined by cooperative motions of chains which form a network. The characteristic time of this spectrum for networks of type (1) is the relaxation time of a chain between junctions τ_min. For networks of type (2) a second time scale τ₁ exists, which corresponds to motions inside the volume occupied by a single long polar chain included in a network. It leads to different time behaviour of the autocorrelation functions for both network models. The existence of only interchain friction in the network model leads to a cut-off of the relaxation spectrum at the time τ_max depending on the volume of viscous interchain interactions.     

PMMA/α-MSAN共混体系长时弛豫区域的时温叠加失效与相分离的关系

通过考察聚甲基丙烯酸甲酯（PMMA）／聚（α－甲基苯乙烯丙烯腈）（α－MSAN）共混体系动态粘弹函数的时温叠加,发现在终端区域时温叠加失效与其相分离有关。相分离使得终端区域的动态储能模量G″（或动态损耗模量G″）与频率的关系[IgG″（IgG″）～Igω]明显偏离经典粘弹模型。随组成的改变,时温叠加的失效呈现温度依赖性,其“失效温度”（Td)可很好地表征LCST类共混物的相分离,而且在用其表征相分离时,IgG″-Igω法比IgG″-IgG″法更优越。     

Designing heterogeneity into bimodal elastomeric PDMS networks

Bimodal poly(dimethylsiloxane) (PDMS) networks prepared using a dual‐step process have been studied. It is shown that these networks exhibit a significant increase in equilibrium modulus. This reinforcement can be obtained by adjusting a process parameter, the “heterogeneity time”, during network preparation. The increase in modulus is attributed to the clustering of the short chains that is effected through proper use of heterogeneity time.     

Proton conductivity in crosslinked hydrophilic ionic polymer system: Competitive hydration,crosslink heterogeneity,and ineffective domains

High proton conductivity in hydrophobic backbone‐based polymers such as Nafion is known to be due to the formation of organized ionic clusters and channels upon hydration. However, a lower proton conductivity in hydrophilic, ionic polymers and the role played by the microstructure are not well understood. In this work, we demonstrate the importance of heterogeneity in crosslinked ionic polymer networks in explaining proton conductivity. Poly(vinyl alcohol) (PVA) crosslinked with sulfosuccinic acid (SSA) is used as the model polymer system for the study. Evolution of the microstructure with hydration and the effect on proton conductivity are analyzed using ATR‐FTIR spectroscopy, dielectric spectroscopy, and small‐angle neutron scattering. We show that the presence of the two hydrophilic groups in PVA‐SSA (hydroxyl and sulfonic acid), as opposed to Nafion, results in competition for water and a lower proton conductivity. The crosslinked polymer–water system contains heterogeneous domains of crosslink nodes which are conductive. These domains (of size 20–35 Å) interconnect with each other and form tortuous percolating domains through which proton conduction takes place. The presence of hydroxyl groups results in some of the domains being ineffective for proton transport, resulting in a lower conductivity. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1087–1101     

Theory of relaxation properties of two‐dimensional polymer networks, 1. Normal modes and relaxation times

The local relaxation properties of polymer networks with a two‐dimensional connectivity are considered. We use the mesh‐like network model in which the average positions of junctions form the regular spatial structure consisting of square repeating units (network cells). The two‐dimensional polymer network consisting of “bead and spring” Rouse chains and the simplified coarse‐grained network model describing only the large‐scale collective relaxation of a network are studied. For both dynamic network models the set of relaxation times and the transformation from Cartesian coordinates of network elements to normal modes are obtained. Using the normal mode transformation obtained, in Part 2 of this series the exact analytical expressions for various local dynamic characteristics of the polymer network having a two‐dimensional connectivity will be calculated.     

Tightening of gelatin chemically crosslinked networks assisted by physical gelation

Developing the use of polymers from renewable sources to build hydrogels with tailored mechanical properties has become an increasing focus of research. The impact of the thermo‐reversible physical networks of gelatin (arising from the formation of triple‐helices) on the structure formation of a chemical network, obtained by crosslinking with glutaraldehyde (a non‐catalytic crosslinker), was studied using optical rotation, oscillatory rheology, and large strain mechanical deformation. We observed a direct correlation between the storage shear modulus of the chemical network grown in the gel state (i.e., simultaneously with the physical network) and the amount of gelatin residues in the triple‐helix conformation (χ). Since χ is directly affected by temperature, the value of the storage modulus is also sensitive to changes in the temperature of gel formation. χ values as low as 12% lead to an increase of the shear storage modulus of the crosslinked gel by a factor of 2.7, when compared to a chemical network obtained in the sol state (i.e., in the absence of a physical network). Our results show that the physical network acts as a template, which leads to a greater density of the chemical crosslinks and a corresponding higher elastic modulus, beyond what is otherwise achieved in the absence of a physical network. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 1850–1858     

Relaxation spectrum of a polymer network with included particles: A regular cubic network model with mutual friction

A theory of the viscoelastic properties of crosslinked polymers with included particles is developed. The model of a regular cubic coarse-grain network, which suggests the viscoelastic interaction of the particles with the crosslink sites, is used. The particles are assumed to be close to isotropic, and their mobility is described via the introduction of a friction coefficient that is directly proportional to the particle radii. In the framework of this model, the spectrum of relaxation times of the network with included particles consists of two branches: One corresponds to the local displacements of the particles relative to the crosslink sites; the other describes the large-scale collective mobility of the particles along with the network fragments. At all values of the viscoelastic parameters of the model, the relative width of the relaxation-time spectrum for the network with included particles is higher than that for the initial network without included particles. This theoretical result qualitatively explains the experimental data on the mechanical and dielectric relaxations of crosslinked composites, which verify the broadening of the frequency dependences of the elasticity modulus, loss modulus, and dielectric-loss factor for the filled crosslinked polymers relative to these dependences for the initial (unfilled) polymer networks.     

Effect of the functionality of junctions on the elasticity of polymacromonomer networks: Computer simulation

The elastic properties of polymer networks formed via the radical polymerization of macromonomers with two polymerizable end groups are studied via computer simulation. It is shown that variation in the average functionality of network junctions, f _avg, in a wide range (∼5–55) leads to a significant change in the shear modulus of the network. According to experiments with real networks (gels of poly(ethylene oxide) macromonomers), the shear modulus increases as f _avg increases. This effect is not due only to a decrease in the fluctuations of positions of network junctions. The main cause of the increase in the modulus is that the modulus component due to interaction between polymer chains (entanglements) increases as the functionality of junctions in the investigated networks increases. The conclusion is made that these networks gain entanglements during the formation of network junctions with high functionality rather than inherit them from the solution of macromonomer chains.     

Microphase structure of interpenetrating polymeric networks based on polyurethane and polyurethane ionomer

Interpenetrating polymer networks based on polyurethane and polyurethane ionomer were studied using wide-angle and small-angle X-ray diffraction. The polyurethane network is a multiblock polymer based on the trimethylolpropane adduct with 2,4-toluene-diisocyanate and poly(propylene oxide tetrahydrofuran) copolymer. Polyurethane ionomer represents a network formed from poly(propylene glycol) containing three OH side groups, 2,4-toluene-diisocyanate, 2,2′-dimethylethanol-amine, and 1,5-dibromopentene. The network polymers are characterized by structure heterogeneity developed during microphase separation as a result of hard and soft block segregation. The interpenetrating networks investigated are amorphous systems over the whole range of compositions. They form a very complicated structure where the phase separation of polyurethane and ionomer takes place. It is important to note that phase separation leads to the appearance of microphase structure periodicity due to regular arrangements of microregions enriched by one of the components. The latter fact is considered to be a sign of spinodal phase separation at the initial stages.     

Stacking heterogeneity: a model for the sequence dependent melting cooperativity of duplex DNA

A microscopic Potts-like one-dimensional model with many particle interactions [referred as the generalized model of polypeptide chains (GMPCs)] is developed to investigate cooperativity of DNA sequence dependent melting. For modeling sequence, regular homogeneous sequences were arranged in heterogeneous blocks of various lengths. Within the framework of the GMPC the authors show that the inclusion of stacking interaction heterogeneity relative to homogeneous hydrogen bond interactions leads to an unexpected and quite remarkable increase in melting cooperativity for small blocks. In some cases this tendency persists for long blocks having sharp sequence heterogeneity.     

Sequential semi-interpenetrating polymer networks based on polyurethane and polystyrene

The formation of microheterogeneous sequential semi-interpenetrating polymer networks based on network polyurethanes with different molecular masses of chain segments between crosslinks and a linear polystyrene has been studied by DSC and small-angle X-ray scattering. It has been shown that variation in the molecular mass of polymer segments between polyurethane network junctions affects the formation of the linear component of semi-interpenetrating polymer networks. As a result, the material structure may change in a wide range from a nearly single-phase system to a two-phase one. SAXS measurements indicate that there is a cymbate dependence between the degree of segregation of components of sequential semi-interpenetrating polymer networks and their microheterogeneous structure on the internetwork space. Two hierarchical heterogeneity levels are found to exist in polymer networks, and the features of each of these levels are analyzed.     

Elastic properties of crosslinked poly(vinyl alcohol) gels: Network topology

Current network theory exhibits inconsistencies which show up particularly clearly in deformation of networks prepared by crosslinking a polymer in solution. A check of theory can be obtained if one knows precisely the number of crosslinks in the network and if a range of deformations is applied to the network. In an effort to explore this problem we have examined the relation of shear modulus to crosslink density, primary molecular weight, and polymer concentration for a series of poly(vinyl alcohol) gels at low to intermediate concentrations. Aqueous poly(vinyl alcohol) solutions were crosslinked to form infinite networks using terephthalaldehyde. We find a large discrepancy with these poly(vinyl alcohol) gels between measured shear modulus and that calculated from classical elasticity theory assuming quantitative reaction of crosslinking. The ratio of measured to calculated modulus is independent of crosslink density for a given primary molecular weight and concentration. It shows linear dependence on polymer concentration prior to crosslinking and extrapolates to a critical concentration which is consistent with the effective sizes of the polymer molecules.     

Physicochemical characterization of chemical hydrogels based on PVA

In this paper, we report on the physicochemical characterization of hydrogels recently obtained by crosslinking poly (vinylalcohol), PVA, with telechelic PVA (telPVA, bearing terminal aldehydic groups) via acetalization in aqueous solution. These gels were studied by equilibrium swelling, compression modulus measurements, and proton relaxometry experiments. Swelling and compression modulus data allow to estimate the average molecular weight of PVA chain between crosslinks, the average mesh size of the networks, and the polymer–solvent interaction parameter χ₁. The average mesh size of PVA‐telPVA compares well with domain dimensions of diffusionally confined water as detected by NMR relaxometry. Proton relaxometry also showed different network domains in which water is compartmentalized, indicating a complex heterogeneity. The study of the temperature behavior of the nuclear spin–spin relaxation times of the confined water was also considered. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1225–1233, 1999