Gaussian Subchains Research Articles

Shift of the “coil–globule” transition in porous polymer films

Using the model of Gaussian subchains “beads on a thread”, the reduced parameters of the state equation for porous polypropylene films in various solvents are found. It is shown that to satisfy to experimental data, the coil-globule transition is shifted towards higher values of the Flory–Huggins interaction parameter.

Dynamics of Rouse chains undergoing head-to-head association and dissociation: Difference between dielectric and viscoelastic relaxation

Polymer chains having type-A dipoles parallel along the chain backbone exhibit slow dielectric relaxation reflecting their global motion. For monofunctionally head-associative Rouse chains having type-A dipoles, the Rouse equation of motion was combined with the association/dissociation kinetics to calculate the dielectric relaxation function, Φj(t) with j = 1 and 2 for unimer and dimer. Φ1(t) reflects the orientation of the end-to-end vector of the unimer, whereas Φ2(t) detects the orientation of two end-to-center vectors of the dimer (having symmetrical dipole inversion), both being in the direction of the applied electric field. The calculation was made by mapping the conformation of dissociating dimer onto the created unimer and vice versa with the aid of the Rouse eigenmode expansion of the bond vector of segments (Gaussian subchains). It turned out that Φj(t) is not influenced by the association/dissociation reaction. This result makes a striking contrast to the behavior of the viscoelastic relaxation function gj(t): gj(t) is strongly affected by the motional coupling between the unimer and dimer due to the reaction [H. Watanabe et al., Macromolecules 48, 3014–3030 (2015)]. This difference emerged because the dielectric Φj(t) corresponds to the vectorial first-moment average of the segmental bond vector at time t, u(n,t) with n being the segment index, whereas the viscoelastic gj(t) corresponds to the tensorial second-moment average. Because of this difference in the averaging moment, Φj(t) is subjected to cancelation in the conformational mapping but gj(t) is not, so that the reaction effect emerges only for gj(t). The experimental data of head-carboxylated high-cis polyisoprene chains (having the type-A dipoles), confirming this difference, are also presented.

Ordering and the relaxation properties of macromolecules in ferroelectric polymer films

The orientational order and dynamic properties of macromolecules have been studied in two- and three-dimensional systems of polymer chains that are orientationally ordered on the level of elastically strained kinetic units (Gaussian subchains) with anisotropic local intra- and interchain interactions of the dipole type. The values of the order parameter as a function of the film thickness for the corresponding type of boundary conditions imposed on the ends of the film have been calculated, and a comparison with the phenomenological theory and experimental data on polarization distribution in ferroelectric films based on vinylidene fluoride has been performed. Equations of motion have been obtained for transverse and longitudinal mean projections of kinetic units and relaxation time spectra for large-scale motions of chains in the state with a planar order. The long-range order in three-dimensional layers leads to anisotropy of mean-square sizes and, therefore, to relaxation properties of chains in their orientational motions (with respect to the direction of ordering).

Ordering and Large-Scale Relaxation Properties of Macromolecules in Ferroelectric Polymer Films

Dynamical equations for transverse and longitudinal mean projections of elastic-deformed kinetic chain units (Gaussian subchains) with anisotropic local intra- and interchain orientation interactions of dipolar type and spectra of relaxation times for large-scale motions of the chains in an isotropic and ordered state were obtained. The long-range orientation planar order in the 3D-layered systems leads to an anisotropy of large-scale relaxation properties of the chains (relative to the axis perpendicular to layers). The anisotropy increases with increase of the chain bending rigidity, the strength of interchain interactions and decrease of the scale of its motion. It disappears for most large-scale motions of chains.

Polarization and orientational order in polymer ferroelectric films based on vinylidene fluoride

Phase transitions and the development of orientational order are studied for three-dimensional polymer systems with the anisotropy of local intra- and interchain orientational-deformation interactions of chains with the dipole-type potential. In the proposed model of chains composed of elastically deformed segments with a fixed mean-square length (in the modified model of Gaussian subchains), there is a certain critical temperature at which the second-order phase transition from the isotropic state to the orientationally ordered state occurs. The temperature dependences of the parameter of the dipole order for thick films are calculated, and these dependences are compared with the corresponding dependences within the mean-field approximation according to the Ising model for ferromagnetics and within the Langevin continuum model for ferroelectric materials as well as with the experimental data on the thermal depolarization in the films based on the vinylidene fluoride-trifluoroethylene copolymer. The order parameter is calculated as a function of the film thickness (the length of chains) under certain boundary conditions imposed on film ends, and the calculated values are compared with the values predicted by the phenomenological theory and with the experimental data on the polarization distribution in the ferroelectric films based on vinylidene fluoride.

Shear Dynamic Moduli of Stretched Polymer Chains and Networks: Modified Rouse Model

AbstractThe dynamic mechanical properties of polymer chains and networks under large static strains are studied. Polymer chains are modeled as sequences of Gaussian subchains whose elasticity constants are different for motions parallel ($K_{||} $) and perpendicular ($K_ \bot $) to the stretching force, f, and are functions of the elongation ratio (a modified Rouse model). The frequency‐dependent storage and loss shear moduli are shown to be different for different geometries under small oscillating shear deformation. Three principal geometries under shear deformation denoted by D‐, V‐ and G‐geometry (see the image) are considered. Static elongation leads to the broadening of the frequency dependencies of the loss moduli for all considered geometries. The predicted storage and loss moduli are in a good agreement with experimental data. magnified image

Shear Dynamic Modulus of Nematic Elastomers: Modified Rouse Model

We develop a microscopic theory of dynamic mechanical properties of nematic elastomers taking the chain structure of network strands explicitly into account. We use an approach in which network strands are modeled as a sequence of Gaussian subchains, whose elasticity constants and friction coefficients are different for motions parallel and perpendicular to the LC-director: K∥ ≠ K⊥ and ζ∥ ≠ ζ⊥ (a modified Rouse model). We show that the dynamic modulus of an ordered nematic elastomer, G∗ = G′ + iG′′, should demonstrate a frequency behavior very similar to that of usual (nonordered) rubbers; especially, it should display a frequency domain with a Rouse-like behavior, G′ ≅ G′′ ∼ ω1/2, a feature which is confirmed by experiments. In contrast to the usual rubbers, nematic elastomers are characterized by the anisotropy of the dynamic mechanical behavior with respect to the LC director, n. In agreement with experiment we show that for prolate systems in the D-geometry (when n is parallel to the shear velocity) G′D greatly decreases around the nematic isotropic phase transition, whereas in the V-geometry (when n is perpendicular to the shear plane) GV′ does not demonstrate such a singularity. We discuss the predictions of our theory for other geometries under shear deformation.

Theory of ordering in three-dimensional polymer systems with local orientational-deformational interactions

The spontaneous ordering of a three-dimensional polymer system composed of flexible segments with a fixed mean-square length in the presence of local intra- and interchain orientational-deformational interactions was considered. The fixation of the segment chain length on average is possible only during relatively weak interchain interactions or at high temperatures, i.e., in the isotropic state. The three-dimensional model, unlike its two-dimensional version, suggests the existence of the critical point at which a second-order phase transition from the isotropic state to an ordered state takes place. The critical behavior of the multichain model is described by the spherical approximation for an anisotropic Heisenberg ferromagnetic. The dependence of the critical point and the parameters of short- and long-range dipole and quadrupole orientational orders on the chain rigidity and magnitude of interchain interactions was determined. In the isotropic state, orientation correlations of segments decline according to the Ornstein-Zernike law, as in the 3D model of Gaussian subchains without fixation of their mean-square length. In the ordered state, the correlation functions tend to a finite limiting value corresponding to the presence of long-range order, as in the case of the multichain model of rigidchain segments in the strong-order approximation. A comparison of the short-range and long-range orders in the model of chains composed of segments with a fixed mean-square length and undeformable (rodlike) elements in the mean-field approximation, on the one hand, and multichain models, on the other hand, showed their equivalence, especially at high degrees of ordering.

Mean-field theory of ordering in polymer systems with orientational-deformation interactions

On the basis of the mean-field method (single-chain approximation), expressions were derived for the free energy of a chain with dipolar orientation-deformation interactions of Gaussian subchains with a fixed mean-square length. In terms of this approach, the systems experience a second-order phase transition from the isotropic state to an ordered state. The Curie point and the dipole and quadrupole order parameters were related to the thermodynamic chain bending rigidity. The types and characteristics of transitions in anisotropic lattice models with dipolar orientational interactions were compared with those in continuous models with quadrupole interactions in terms of the mean-field approximation (Maier-Saupe theory). The short-and long-range orientational order in single-chain models (mean molecular field) and the corresponding multichain models (with local intrachain interactions of subchains) were compared. A comparison of the orientational correlation functions in single-chain and multichain models composed of flexible segments of a fixed length and rigid elements showed their qualitative equivalence as applied to the description of the statistical properties of chains in polymer systems with dipolar orientational-deformational interchain interactions, especially in the case of high chain ordering. Owing to the long-range fluctuation instability in two-dimensional multichain systems, the results of the mean-field theory are applicable only to parts of systems with finite sizes (domains).

The Estimation of Statistical Parameters Determining the Relaxation Times of Nematic Elastomers: A Three-Chain Network Model

A simplified “three-chain” network model formed from freely jointed polymer chains consisting of Gaussian elements with fixed mean-square lengths is proposed for describing local dynamic properties of nematic elastomers. The boundaries of a polymer network are supposed to be fixed when sample volume and shape do not change with ordering. Relaxation times characterising intrachain motions in both isotropic and ordered states are determined by two factors. The first (“dynamic”) factor is related to the friction of chain elements and the second one (“statistical” factor) is determined by statistical mean–square fluctuations of segment projections on the three axes of rectangular frame of reference. The “statistical” factor of relaxation times is calculated here as a function of the order parameter and the parameter characterising the degree of network crosslinking. Statistical factor obtained in the framework of a network model consisting of Gaussian subchains is compared with that calculated here by using freely-jointed-rods chain model. Good agreement is shown between statistical factors obtained in the framework of the two chain models considered. This result confirms the validity of describing the dynamics of real rod-like mesogenic groups in nematic elastomers in terms of a simplified chain model consisting of Gaussian segments with fixed average lengths which do not change with ordering. The influence of “dynamic” factor on the relaxation spectrum of a nematic elastomer is discussed qualitatively.

Behaviour of a polymer chain with a high velocity gradient in an extensional hydrodynamic field

The unfolding and orientation of macromolecules in an extensional hydrodynamic field have been investigated for a Gaussian subchain model with an “average” fixed segment length. Changes in the shape and volume of the coil have been studied in relation to the flow rate gradient. In extensional flow the transition to the extended state of the chain proved to be much more blurred than in longitudinal flow. The relative deformation of the chain and the flow birefringence in extensional flow in the case of a high gradient, and the positioning of the transition region are determined by the characteristic parameter G∼ G 0√( N), which for high MM considerably exceeds the value of the corresponding parameter G 0 for the longitudinal gradient, where G 0( N)∼(1/ N 2).

Mechanism of ultra-slow relaxation in non-dilute linear polymers

Dynamics of individual subchains in a polymer melt with entanglements has been studied using the model of Gaussian subchains in the single-molecule approximation. The macromolecule has been assumed to be embedded in a viscoelastic medium. Characteristic times and distances of the studied system have been established. The mechanism of ultra-slow relaxation processes reflected in various phenomena (diffusion, viscoelasticity, optical anisotropy) has been elucidated, and their relation to stress relaxation in a polymer sample and their dependence on molecular mass have been established. The role of macromolecular ends has been investigated. The results are confronted with those obtained by de Gennes, Doi and Edwards. The anisotropy of macromolecular motion is a direct consequence of the equation of motion employed.

On the origin of the volume viscosity in liquid polymeric systems

AbstractThe possibility of explaining the similar frequency dependence of volume (ηv) and shear (ηs) viscosities is examined on the basis of the model of Gaussian subchains. The purely viscous drag gives ηv/ηs = 2/3, which is much lower than that found experimentally. It is suggested that there is a thermal relaxation involving part of the rotation‐isomeric heat capacity and characterized by the Zimm‐like spectrum. The possible influence of short‐range structural changes in the environment of a subchain accompanying its deformation is discussed as an alternative.

Ordering and relaxation properties of two- and three-dimensional polymeric systems consisting of Gaussian sub-chains with orientation-strain interaction

The conformational and dynamic properties of two- ( d 2) and three- ( d 3) dimensional “network” systems consisting of polymeric chains orientationally ordered at the level of Gaussian sub-chains have been considered. If the sub-chains have comparatively weak orientation-strain interaction and the flexural regidity of the chain is low, a long chain included in the ordered matrix remains Gaussian. As the constant for the interchain interaction increases, the mean-square dimensions of the sub-chains and chains increase. In both models there is a “critical” point at which the system of Gaussian sub-chains loses its stability. At this point, the mean-square dimensions of the Gaussian sub-chains either become infinitely large (in the d 2 model) or remain finite (in the d 3 model). On passing through the “critical point”, it is necessary to take account of the finite extensibility of the sub-chains and to go on to a theoretical description based on chains made up of rigid segments.

Influence of the hydrodynamic interaction on the viscoelastic properties of solutions of macromolecules in the high frequency region

Using the properties of eigen numbers of Tepliz matrices the authors have studied change in the total spectrum of the eigen numbers of the model of Gaussian subchains as a function of the parameter of hydrodynamic interaction h. It is shown that the known eigen number characterizing the large scale movements of the polymer chain are a special case of the spectrum obtained. At h>0·42 in the spectrum eigen numbers with a multiplicity of two may appear because of slowing of the small scale movements of the model. At h>0·58 in the distribution of eigen numbers negative values appear and the model ceases to satisfy the law of energy conservation. From the spectrum of eigen numbers obtained the authors estimate the frequency dependence of the dynamic shear viscosity in the region of high frequencies for different values of h. It is shown that the peculiarities observed in the distribution of eigen numbers lead to the anomalous character of the frequency dependence of dynamic viscosity.

Local concentration of units and molecular mobility in swollen polymer networks

An evaluation is made of the local concentration in a crosslinked network system based on the model of Gaussian subchains. The dependence of the local concentration on the location of the portion of the network studied, the closeness of crosslinking and the degree of swelling of the mesh is established. The authors compare the local concentration of the network and free branched and unbranched macromolecules in concentrated solutions at an identical concentration of the polymer. The effects of the inhomogeneity of real networks and their role in the experimentally observed fall in mobility with rise in the closeness of crosslinking found by the NMR method in swollen PS networks are discussed.

Steady-state opticohydrodynamic properties of DNA: molecular weight dependence and the internal viscosity problem.

AbstractExtinction angles, flow birefringence, and intrinsic viscosities are compared for linear, bihelical DNAs from viral and other sources that span a range in molecular weight from ∼105 to 1.3 × 108. This range effectively spans the region over which transition from rigid‐rod to expanded‐coil hydrodynamic property behavior occurs. All DNAs are in identical, phosphate–EDTA, neutral‐pH buffers, 0.1M in NaCl. The extinction angle is a hydrodynamic property only and is thus particularly sensitive to effects of kinetic chain rigidity or internal viscosity. Our extinction angle results cannot be interpreted by any simple, single‐function theoretical expression. Rather, they must be divided into distinct high‐ and low‐molecular‐weight domains. The low‐molecular‐weight region is typical of rigid‐particle opticohydrodynamic property behaviour characterized primarily by particle orientation. The high‐molecular‐weight domain shows evidence for a finite internal viscosity effect, however, which can be interpreted as very nearly Kuhnian using Cerf's amplification of the Gaussian subchain model to include internal viscosity. It is found that the high‐molecular‐weight, monodisperse viral DNAs from T7, T5, and T2 bacteriophage show an internal viscosity contribution to the limiting extinction angle–shear rate ratio of ∼3 × 10−3 s. An effect of this magnitude may be marginally important in interpreting extinction angle and certain other hydrodynamic property data for high‐molecular‐weight DNA systems. Internal viscosity effects do not appear to be manifest in the ratio of flow birefringence to intrinsic viscosity, however, and the persistence length of the high‐molecular‐weight DNAs is found to be independent of molecular weight to within estimated experimental uncertainty.

Flow birefringence of T7 phage DNA: dependence on salt concentration.

AbstractWe have determined extinction angles and flow birefringence of T7 bacteriophage DNA over a wide range of shear, polymer concentration, and solvent ionic strength. From these data, information on the simple salt dependence of coil permeability to solvent and on short‐range intrachain interactions (persistence length) was obtained. At all ionic strengths, our results are consistent with a partially draining coil in the Gaussian subchain dynamical theory of Rouse‐Zimm‐Tschoegl‐Bloomfield. Salt dependence of persistence length is comparable to, although somewhat less than, that obtained previously using similar methods with a fivefold higher‐molecular‐weight DNA (T2 bacteriophage DNA). Possible reasons for observed discrepancies are analyzed, and the results of this work are compared in detail to other current studies of solvent ionic strength dependence in persistence length and hydrodynamic properties of DNA.

Local relaxation properties of polymer chains containing inclusions

A theory of the dynamic properties of polymeric chains containing an inclusion has been developed on the basis of a model of Gaussian subchains. Time correlation functions for the average projections of the inclusion element on to an individual direction have been found. Both linear chains containing singular elements (bridges, impurity polar groups) and also slightly crosslinked systems with crosslinking bridges have been considered. The frequency dependence has been found for the complex compliance (and for the dielectric permeability) for various values of the ratios λ 1/ λ and K 1/ K ( K and λ and K 1 and λ 1 are the coefficients of statistical elasticity and of friction for the standard chain segment and for the inclusion element respectively). The dependence of the inverse luminescence polarization on the inclusion parameters has been investigated. The motion of the end segments of finite chains containing an inclusion in the middle has been studied.

The effect of the solvent quality on the relaxation behaviour of a polymer chain

The dynamic behaviour of a separate macromolecule has been modelled as a non-intersecting chain in a simple cubic lattice and simulated on a computer. The rotation-isomeric movements have been created by jumps of 2 and 3 chain units. The solvent quality has been varied by changing the intramolecular contact energy ε. The calculations were carried out for ε=0 (good solvent), − ε=0·3 (θ-solvent) and −ε-values of 0·5 and 0·8 (precipitant). The correlation times of the large-scale vibrations have been found to depend but slightly on the solvent quality and the mechanisms of these processes and of intramolecular reactions are the same as those for a Gaussian subchain model. The small-scale vibrations in the chain strongly depend on solvent quality due to the decisive effect of the local chain density on the correlation times. The dynamic characteristics of the chain have been calculated and experimentally measured by the polarizing luminescence and dielectrical relaxation methods.