Dynamic Intrinsic Viscosity Research Articles

A dynamic model for linear locally stiff polymers in dilute solutions

The dynamics of locally stiff polymers in dilute solutions in high-viscosity solvents is described. The relaxation time spectrum in dependence on effective solvent viscosity, hydrodynamic interaction and excluded volume effects is calculated for the polymer in terms of an extended version of bead-spring model proposed by Jones et al. (J. Polym. Sci., Phys. Ed., 16 (1978) 2215). The dynamic intrinsic viscosity [ η( ω)], elastic modulus G( ω), dielectric constant ε( ω) and electric polarization p( ω) are also calculated in terms of this model. They are presented as functions of frequency of the field applied to the system as well as the relaxation time spectrum. The dynamic intrinsic viscosity [ η( ω)] and electric polarization p( ω) were calculated for several values of the hydrodynamic parameter, excluded volume coefficient, effective solvent viscosity and molecular weight of the polymer.

Models of polymer and solvent dynamics

Some solute–solvent systems show vanishing or negative dynamic intrinsic viscosity [η]ω at any frequency ω. For consistency with this observation the conventional model of dilute polymer dynamics is modified to account for the gradient of solvent velocity in the vicinity of a polymer bead. With simplifications which are strictly valid only for polymers of low molecular weight M or for large ω, [η]ω is changed only by the addition of a term [η]γ. In units of ml/g [η]γ is independent of M and ω and may be positive or negative. The same type of theory is applied to the effect of polymer on the translational and rotational friction constants of small solute molecules A, e.g., labeled solvent. With reasonable estimates of a distance of closest approach between polymer beads and molecules A, the friction constants of A increase with polymer concentration at roughly the same rate as the viscosity of an oligomer solution of the same polymer species. A molecular theory of the coupling constant between a polymer bead and a solvent velocity gradient is not essential, but a few possibilities are considered. The coupling constant may reflect modifications of the solvent by the polymer, as has been suggested by others, or differences in the deformability of oligomers and solvent, etc. The deformability argument seems to necessitate friction matrices with negative eigenvalues, and a reconciliation of such behavior with stability requirements is presented.

Dynamics of helical worm-like chains. IX. Dynamic intrinsic viscosity

The dynamic intrinsic viscosity of flexible chain polymers in dilute solution is studied on the basis of the discrete helical worm-like chain. The correlation function formalism of the complex intrinsic viscosity [η] is given, taking account of the effect of the finite hydrodynamic volumes of subbodies of the chain. A new basis set, which is a hybrid of the one- and two-body excitation basis functions, is introduced. Then the eigenvalue problem for the representation of the diffusion operator may be reduced to N six-dimensional eigenvalue problems with N the number of subbodies. Among the six branches of the eigenvalue spectrum, one global and two local branches make contribution to the dynamic intrinsic viscosity. The theory predicts the existence of the high-frequency plateau which is distinguished from the infinitely high-frequency viscosity. The plateau arises from the interaction between the global and local chain motions (caused by the helical nature of the local chain contour), the constraints, and the finite hydrodynamic volumes of the subbodies. The interaction above vanishes for the Kratky–Porod worm-like chain possessing no helical nature. A comparison of theory with experiment is made with respect to the plateau value, and it is found that its dependence on the chemical structure of the chain may well be explained.

Renormalization group theory of the Rouse–Zimm model of polymer dynamics to second order in ε. II. Dynamic intrinsic viscosity of Gaussian chains

The dynamical intrinsic viscosity [η(ω)] of Gaussian chains is studied within the Rouse–Zimm bead-spring model of polymer dynamics using conformational space renormalization group methods. Calculations are performed through order ε2 (ε=4−d, where d is the spatial dimensionality) both with and without the preaveraging approximation. We find an 8% preaveraging error in the steady state limit. Preaveraging corrections to [η(ω)] involve couplings between all Rouse modes, leading to more complicated expressions for the Rouse mode relaxation rates and to weight factors for mode contributions to [η(ω)].

Dynamics of weakly bending rods: A trumbbell model

Some dynamic properties of weakly bending rod-like polymers are studied on the basis of a trumbbell model which consists of two frictionless bonds and three beads with constraints on bondlengths and with a bending potential between the bonds. The diffusion equation is formulated by imposing the constraints in the diffusion limit (Smoluchowski level), considering fluctuating hydrodynamic interaction. All evaluation is carried out analytically, though only near the rod limit, i.e., to terms of O(α−1) with α the bending force constant. The mean translational diffusion coefficient Dt (at long times) and the dynamic intrinsic viscosity [η] are evaluated by the projection operator method and by the time-correlation function formalism, respectively. The former and the steady-state intrinsic viscosity are evaluated also in the rigid-body ensemble approximation, and it is found that this approximation is very good near the rod limit. Some other features of Dt, including its dependence on time, are also discussed. The results for [η] are equivalent to those of Roitman and Zimm, who have solved the problem numerically for the trumbbell as the Kramers-type chain in the free-draining case; [η] relaxes by end-over-end rotation and bending near the rod limit. However, a comparison of theory with experiment for storage and loss moduli for light meromyosin leads to an estimate of the persistence length too large compared to its known reasonable value.

Dynamics of spheroid-cylindrical molecules in dilute solution

The dilute solution dynamics of spheroid–cylindrical molecules, i.e., straight cylinders with oblate, spherical, or prolate hemispheroid caps at the ends, is studied in detail. The translational and rotatory diffusion coefficients and the dynamic intrinsic viscosity are evaluated numerically for short cylinders by determining the frictional force by an orthodox method of classical hydrodynamics. For long cylinders, the Oseen–Burgers procedure is shown to be valid, and the results previously obtained by it are still useful. Thus, empirical interpolation formulas for the transport coefficients above are also constructed to be applied to spheroid–cylinders of arbitrary size. The end effects on the translational and rotatory diffusion coefficients are rather small, while the effect on the zero-frequency intrinsic viscosity is remarkable, depending appreciably on the shape of the ends, though for relatively short cylinders. In general, for a rigid body of revolution having a plane of symmetry perpendicular to its axis, it is shown that the dynamic intrinsic viscosity may be expressed in terms of the zero-frequency intrinsic viscosity, the rotatory diffusion coefficient about a principal axis in the symmetry plane, and a newly defined factor, which is also associated with the rotational motion about this axis.

Hydrodynamic interaction and the dynamic intrinsic viscosity of a flexible polymer

We extend an earlier calculation by C. W. Pyun and M. Fixman [J. Chem. Phys. 44, 2107 (1966)], dealing with the effects of hydrodynamic interaction on the intrinsic viscosity of a flexible polymer in solution. We use their general formulation of the problem, but perform a more complete ’’infinite order’’ calculation of eigenvalues and eigenvectors. For the steady state viscosity constant, we obtain a new value Φ=2.76, which lies between their result 2.66 and the value 2.856 which is found when the preaveraged hydrodynamic interaction is used. Further, we obtain the spectrum of relaxation times, and the frequency dependent intrinsic viscosity. The effects of preaveraging are more striking here; for example, our calculated lowest relaxation time is about twice as large as that found in the preaveraged approximation.

Role of internal viscosity in polymer viscoelasticity

A new theory of the internal viscosity in polymer chains is proposed. Although some of the conclusions are in agreement with the Kuhn-Cerf-Peterlin theories, the present approach is based on more rigorous statistical-mechanical grounds. However, the theory is unable to predict the exact value of the effective Arrhenius factor (A) related to the gauche⇄trans transitions between rotational states on the chain bonds. The internal viscosity force turns out to be a nonlinear effect of the chain motion. Confining our attention to the linear component, and assuming an oscillatory applied force, the diffusion equation is obtained as a modification of the corresponding equation formerly proposed by Zimm. For infinitely long chains the eigenvalue problem is exactly solved and, with reference to the free-draining model, the dynamic intrinsic viscosity of a polymer solution as well as the tensile and dielectric relaxation effects are calculated.

Dynamic Intrinsic Viscosity of Macromolecules with Internal Viscosity. III. Effect of Molecular Weight, Excluded Volume, and Hydrodynamic Interaction

The frequency dependence of the complex intrinsic viscosity and elastic modulus of a macromolecule with internal viscosity is calculated for the Gaussian coil and the coil with excluded volume in a wide range of molecular weight and hydrodynamic interaction. Due to limitations in eigenvalues and because of coil rotation in laminar flow the results obtained are sufficiently accurate only for a large number of coil segments and for small internal viscosity. Hydrodynamic interaction enhances the effect of internal viscosity so that the macromolecule seems to be stiffer than in the free-draining case. The non-Gaussian character of intrachain distances has a very small effect. Finite internal viscosity yields a nonvanishing limiting viscosity at ω→∞ which at sufficiently large M increases as M1/6. The relative limiting viscosity [η]∞′/[η]0 as a function of molecular weight (Z∼M), hydrodynamic interaction (h), non-Gaussian character of the coil (ε), and internal viscosity (φ) can be well represented by a single master curve if plotted against ν1φ/Zf. The eigenvalue ν1 is unity for the free-draining coil and asymptotically increases as a linear function of h and ε. Molecular weight enters the abscissa of the master curve implicitly through ν1 and explicitly through Z.

Frequency dependence of intrinsic viscosity of macromolecules with finite internal viscosity

AbstractIn order to explain the observed nonvanishing limiting value of dynamic intrinsic viscosity of polymer solutions at ω = ∞ one has considered the necklace model with finite resistance to the rate of coil deformation introduced long ago by Cerf for the study of gradient dependence of intrinsic viscosity and streaming birefringence. The calculation need not take into account change of hydrodynamic interaction as a consequence of coil deformation because the experimental data are always either obtained at very low gradient or extrapolated to zero gradient so that in the experiment the macromolecule has the same conformation as in the solution at rest. The model indeed yields a finite [η]′ω = ∞ in good agreement with experiments on polystyrene in Aroclor. According to the theory [η]′ω = ∞/[η]0 decreases with increasing molecular weight as M−1 and M−1/2 for the free‐draining and impermeable coil, respectively. The absolute limiting value [η]∞′, therefore turns out to be nearly independent of M, at least for small values of internal viscosity. From the observed value [η]∞′/[η0] one can obtain the coefficient of internal viscosity of the macromolecule. The value for polystyrene in Aroclor calculated from dynamic experiments on rather concentrated solutions is close to that derived by Cerf from streaming birefringence observations of polystyrene in a series of solvents of widely differing viscosity.

Concentration and Conformation Dependence of Dynamic Viscoelastic Properties of Dilute Polymer Solutions

Rouse and Zimm have derived theories of viscoelastic properties of random coil polymers in dilute solutions, and Kirkwood and Auer that for rodlike polymers. Comparing these theories with each other in the form of the normalized dispersion curve, we may obtain direct information on the degree of internal freedom of polymer molecules.Many experimental investigations on viscoelasticity of coiling polymers have been made for dilute and for concentrated solutions. Experimental results for dilute solutions show almost quantitative agreement with the theories of Rouse or Zimm, but those for concentrated solutions a systematic disagreement. From these results, it is clear that concentration effect on the viscoelasticity still remains somewhat ambiguous even in the region of dilute solutions. This is not surprising, because all these theories are concerned only with the intrinsic quantities corresponding to infinite dilution without taking account of intermolecular interaction.In this study the dynamic intrinsic viscosity and intrinsic rigidity, therefore, are defined experimentally as follows, [η']=limc→0(η'-ηs)/ηsc, [G']=limc→0G'/c.Considering that, with changing concentration, all relaxation times involved change to some extent as was pointed out by Ferry, we define also the limiting value of the terminal relaxation time as(τ1)0=limc→0τ1(c).The complex rigidities are obtained by means of the torsional crystal method at the frequency of 39.2kc and the temperature of 30.0°C. The relaxation times are derived from the steady shear viscosities measured by a Ubbelohde dilution viscometer.As an example of the random coil conformation, the solutions of the high molecular weight polymethyl methacrylate (Mv=2.1×106) in chloroform were studied. The measured values for G'/c, (G-ωηs)/c and τ1 were found to be in line with the concentration. The intrinsic rigidities and the limiting relaxation time were determined by extrapolating the experimental data to the zero concentration. The sets of the intrinsic rigidities and the limiting relaxation time thus determined were quantitatively in agreement with the values predicted by the Rouse theory.In the case of the rodlike conformation, poly-γ-methyl-D-glutamate (Mn=1.29×105) which was 100% α-helix in chloroform solutions was investigated. It was found that the measured values approached the theoretical dispersion curves due to Kirkwood and Auer as the concentration was lowered to infinite dilution.