Yu. Ya. Gotlib

Dynamics of dendrimer-based polymer networks

We present a theoretical study of polymer networks, formed by connecting dendritic building blocks (DBB’s). We concentrate on the Rouse dynamics of such networks and perform our study in two steps, considering first single generalized dendrimers (GD’s) and then networks formed by such DBB’s. In GD’s the functionality f of the inner branching points may differ from the functionality fc of the core. The GD’s cover wide classes of macromolecules, such as the “classical” dendrimers (fc=f ), the dendritic wedges (fc=f−1), and the macromolecular stars (fc>2, f=2). Here we present a systematic, analytic way which allows us to treat the dynamics of individual GD’s. Then, using a general approach based on regular lattices formed by identical cells (meshes) we study the dynamics of GD-based polymer networks. Using analytical and numerical methods we determine the storage and loss moduli, G′(ω) and G″(ω). In this way we find that the intradendrimer relaxation domain of G′(ω) becomes narrower when Mcr, the number of ...

Dynamics of inhomogeneous cross-linked polymers consisting of domains of different sizes

The theoretical approach is developed to describe the dynamics of inhomogeneous cross-linked polymers consisting of cross-link agglomerations. An inhomogeneous polymer is treated as an ensemble of noninteracting cross-linked regions (domains) of different sizes. We model an internal architecture of the domains in a rather regular way and assume a power law decay of the relaxation modulus inside the domains, a decay usual for a broad class of cross-linked materials on microscopic scales. Assuming a broad size distribution of the domains in cross-linked polymers due to a random character of cross linking, we demonstrate a stretched exponential time behavior of the relaxation modulus on scales larger than the average size of inhomogeneities in the polymer. We apply this general approach to some special cases of cross-linked polymers, namely to polydisperse polymer networks, to inhomogeneous meshlike networks, and to inhomogeneously cross-linked polymeric gels.

Theory of Orientational Relaxation of Individual Specified Units in a Dendrimer

The theory of orientational relaxation properties of certain individual segments in a dendrimer macromolecule depending on the generation number and the position of a given segment in the dendrimer was developed. The time dependence for the dipole moment after switching an electric field off and the frequency dependence of the permittivity for this segment were calculated, which are determined by the autocorrelation function P 1 of the average projection of the single element. The dielectric properties of the dendrimer at the random distribution of dipole moments (e.g., as a result of sorption of solvent polar groups on the macromolecule) are considered. The time and frequency dependences of the autocorrelation function P 2 for the mean squared projection of the single element that are detectable by means of some experimental techniques (NMR, luminescence, birefringence, etc.) were studied. The theory qualitatively agrees with both the computer simulation results on the autocorrelation function P 1 for the dendrimer macromolecules and the available experimental data on the dependence of orientational mobility for the terminal segments of the number of generations.

Theory of relaxation spectra and dielectric relaxation of rigid rodlike particles incorporated in a polymer network

The theory of molecular mobility and relaxation spectra is developed for rodlike particles embedded in a polymer network with allowance for the involvement of the particles in collective network dynamics through topological entanglements with network fragments. A regular cubic coarse-grained network model is used, where the motion of junctions describes the mobility of large fragments (domains) of the initial network with a size equal to the distance between adjacent rodlike particles. The involvement of the rods in collective network dynamics is taken into account by introducing an effective quasi-elastic potential acting between the rods and junctions of the coarse-grained network and preventing long-distance diffusion of the embedded particles. The viscoelastic parameters of the coarse-grained (“renormalized”) network are functions of the viscoelastic characteristics of the initial network. The relaxation time spectra are calculated as well as the frequency dependences of the dielectric loss factor of the embedded particles that possess a permanent dipole moment directed along the major axis of each rod. Depending on the ratio between the viscoelastic characteristics of the rods and the network, the frequency dependence of the dielectric loss factor may have two maxima. The high-frequency maximum corresponds to local orientational movements of particles at fixed junctions of the coarse-grained network, which correspond to the position of the domain centers in the initial network. The low-frequency maximum corresponds to movements of particles involved in large-scale dynamics of network fragments. The dependence of the dielectric loss factor on the ratio between the viscoelastic parameters of the rods and the network is studied.

Theory of the Relaxation spectra of polymer networks with included hard rodlike particles manifested in NMR

A theory of nuclear magnetic resonance relaxation of13C nuclei and nuclear Overhauser effect (NOE) of polymer networks with included rodlike particles is developed for the case when the length of each rod is comparable or greater than the average distance between neighboring crosslinks. The long-scale dynamics of the network is described by means of a regular cubic “coarse-grained” model. The effects of entanglements of long rods in the network are described by a quasi-elastic potential acting between rods and network fragments. The frequency dependences of 1/T1C and NOE are calculated for the case when the internuclear vector is directed along each rodlike particle. The frequency dependences of 1/T1C and NOE for rods included into a polymer network are shifted to high frequencies as compared with these dependences for free rods due to quasi-elastic interactions between rods and network fragments. At strongly different viscoelastic parameters of rods and network fragments, the frequency dependences of 1/T1C and NOE may have two maxima. The high-frequency maximum corresponds to localized motions of rods at immobile network domains. The low-frequency maximum is caused by involving rods in long-scale network motions. The intensity of the low-frequency maximum increases when the degree of interactions between rods and the network increases.

Temperature dependence of the structure of a carbosilane dendrimer with terminal cyanobiphenyl groups: Molecular-dynamics simulation

The molecular-dynamics simulation of the structure and molecular mobility of an individual macromolecule of a fourth-generation carbosilane dendrimer with terminal cyanobiphenyl groups in a highly diluted chloroform solution in the range 213–323 K is performed. Upon a change in temperature, the dendrimer undergoes structural rearrangement that depends on the ability of terminal segments to penetrate into the dendrimer. At temperatures close to the boiling point of the solvent, aliphatic spacers of terminal segments can penetrate deep into the dendrimer. As temperature decreases, the terminal segments are grouped only on the surface of the molecule; this leads to a 45% increase in the number of solvent molecules in the treelike part of the macromolecule. These results make it possible to give a new interpretation of temperature effects previously observed in NMR experiments for dilute solutions of these macromolecules.

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.

Rotational and translational relaxation times of quasi-elastic and rigid dumbbells elastically bound to polymer network junctions

The dynamics of a rigid rod located between fixed junctions of a polymer network is studied. Three approaches are used in the solution of this problem. The first is based on the viscoelastic model, where a rigid rod is simulated by an elastic dumbbell with a fixed average length; the second includes solution of equations of motion for projections of the rigid rod using the Lagrangian multipliers under the constraint condition; and the third involves solution of the diffusion equation in the presence of an elastic potential. The second and third approaches allow calculation of orientational relaxation times for rod projections under the action of a strong orienting field. The dependences of the relaxation times of orientational and translational motions of the rod projections on the coordinate axes and the orientational relaxation times of mean-square rod projections on the model parameters (the distances between fixed polymer network junctions, the length of the rigid rod, and the elastic coefficient characterizing the binding between the rod and the network) are found.

Theory of relaxation spectra for two identical interpenetrating polymer networks

Viscoelastic models for the description of the relaxation characteristics of two identical swollen interpenetrating polymer networks with different topologies moving against the background of an external viscous medium are considered. Two dynamic models that differ in the character of mutual interaction between network junctions are proposed. According to the first model, viscoelastic interaction is assumed to be constant and provided by the entanglements between a junction of one network with eight symmetrically arranged junctions of the other network. The second model involves (i) the predominant interaction between multiple-network junctions most closely located owing to entanglements and (ii) a weaker interaction with more distant junctions of neighboring cells. For the systems composed of two interpenetrating networks, relaxation-time spectra and average inverse relaxation-time spectra are compared with the corresponding spectra and characteristic times for individual noninteracting regular networks. Both models can involve two branches of the relaxation spectrum. One branch is a collective branch corresponding to the motion of the double network, whose parameters are controlled by the constants of elasticity of each of the interacting networks as well as by the effective mutual viscoelastic interactions between networks. This low-frequency branch is characterized by a broad spectrum of relaxation times. The second branch is a high-frequency branch that is primarily provided by mutual local motions of two interacting networks. This branch is characterized by a comparatively narrow relaxation spectrum and depends on quasielastic constants, which describe network entanglements, and on the characteristic elasticity of each network. The second branch does not involve any infinitely long relaxation times for infinitely continuous networks.

The Theory of Viscoelastic Characteristics of a Highly Stretched Macromolecule in Single Molecule AFM

The theory for the deformation of a model macromolecule stretched by its ends under the action of high constant and low periodic forces is constructed. The macromolecule is composed of monomer units in three conformational states. The proposed theory describes the regime of a severe stretching of a macromolecule extended to a length close to its contour length, when its extension proceeds via conformational transitions between different states of monomer units. The structural parameters of the monomer unit are found to correlate with viscoelastic characteristics, which are calculated from the experimental results on the deformation of an individual macromolecule obtained by the frequency atomic force microscopy. For a monomer unit with three conformations, the force dependences of viscoelastic characteristics (effective coefficients of elasticity and friction) can show one or two minima. When the experimental dependences of the above parameters show two minima, the monomer unit can have three or more equilibrium states. With the knowledge of the viscoelastic characteristics of a macromolecule, it is possible to unequivocally estimate all structural parameters of a monomer unit for its three-state conformational model. When the force dependence of viscoelastic characteristics show only one minimum, the monomer unit can have two or more states and analysis of the corresponding viscoelastic characteristics at the minimum makes it possible to select between two- and three-state conformational models. Then, for the three-state model, experimental data allow the prediction of only equilibrium parameters of the monomer unit (position of the minima and energy); dynamic parameters (positions and height of barriers between equilibrium states) remain indeterminate. The proposed theory is used for the interpretation of the viscoelastic characteristics of dextran obtained by single-molecule AFM experiments. The three-state conformational model of a dextran unit is shown to agree better with the experimental data than with the two-state conformational model.