Dynamical problems in the theory of polymer solutions

Qualitative discrepancies are found between what is predicted by available theory and what is actually observed, for several concentration regimes of the dynamical properties of polymer solutions. The difficulties are most severe, from the standpoint of experiment or simulation as well as theory, for the entanglement concentration regime. However, the classical problems of chain polymers in dilute solution are not fully understood. For example, the constants of proportionality that relate hydrodynamic radii to the radius of gyration, in the nondraining limit and in theta solvents, may not be universal constants. That is, the proportionality constants may vary with polymer and solvent species. Discrepancies between theory and experiment are discussed for the two different systems, dilute chains and semidilute rods. Speculation is offered on the resolution of these difficulties.     

Notes on the theory of high polymer solutions

We have noted some correspondences in the theories of high polymer solutions. This article is divided in two independent sections: (I) Osmotic pressure. The second virial coefficients for polymers of the pearl-necklace type are given and discussed in two limiting cases—the dumbbell and the infinitely polymerized molecule having a definite length. The variation of the coefficients against the polymerization is predictable from these limiting cases (Fig. 1). Attention must be paid to the manner of taking the (abscissa). The case of infinite polymerization can be considered as corresponding to polymers of a compact single-body type such as globular proteins. Thus, an interesting correspondence in the second virial coefficients can be judged for the pearl-necklace and the compact single-body type of polymers. The general feature of the coefficient is also discussed. (II) Intrinsic viscosity. The relation between the two methods (Kirkwood-Riseman and Debye-Bueche) for intrinsic viscosity of high polymer solutions is discussed. Because the respective fundamental equations are equivalent to each other, we may choose whichever of these two methods we like.     

A theory of polymer solutions without the mean-field approximation in Flory-Huggins theory

The concept of the volume fraction at chain end is proposed, which is the conditional probability for a site having been occupied, knowing that an adjacent site is occupied by polymer end. The overall entropy of polymer/solvent system is separated into two fundamentally different parts, i.e., the translational entropy and the conformational entropy. Based on these a theory of polymer solutions is established. When a mean-field approximation is introduced, Flory-Huggins (FH) theory is recovered. The FH interaction parameters and spinodal curves of the polystyrene/cyclohexane system are calculated and compared with the experimental data. The good prediction of them two is achieved.     

Alternative theory of hydrodynamic interactions in polymer solutions

The model is presented for coarse grained dynamics of macromolecules in dilute solutions. The coarse graining is achieved by dividing the polymer chain into subchains, consisting of many monomers, and spatial averaging over lengths that are large compared to the mean-square end-to-end distance of subchains and small compared to macromolecule size. Kinetic equations of the model are derived from first principles of statistical mechanics under the assumption that subchain center of mass positions and solvent flow velocity field are the only slow variables of the system. In this approach hydrodynamic interactions result from the intercomponent friction forces between polymer and solvent instead of boundary conditions on the bead surfaces as in traditional theories. The integrodifferential diffusion equation is obtained for steady flows with the kernel involving the Oseen tensor multiplied by equilibrium distribution in the space of the subchain center of mass positions.     

Comparison of theory and experiment for diffusion in dilute polymer solutions

Results from a number of theories for the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions are examined, and clarifications are made as to what forms of the equations for these theories should be used in comparisons with experimental diffusivity data. An evaluation of the available theories for the concentration dependence of the diffusivity under theta conditions is carried out using experimental diffusivity data taken using sharp fractions of polystyrene. It is concluded that the Pyun—Fixman theory appears to provide the most promising method for estimating the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions at the present time.     

Generalized Rouse–Zimm theory for polymer solutions

Generalized Rouse–Zimm equations are derived using a local equilibrium ansatz for the segment distribution function. The relaxation frequencies depend on excluded-volume effects and other interactions such as entanglement in semidilute solutions through exact static correlation functions. For the case of ring polymers general expressions are given for diffusion coefficients and mean-square displacements. By use of the “blob” model, explicit results are presented for a dilute good-solvent system.     

Molecular theory of the viscoelāstic properties of concentrated polymer solutions

The viscoelastic properties of concentrated polymer solutions are discussed on the basis of the fixed tube model proposed by de Gennes. It is shown that the steady flow viscosity is proportional to M³, where M is the molecular weight, and that the relaxation spectrum exhibits a hump in the relaxation time proportional to M³.     

Network models of concentrated polymer solutions derived from the yamamoto network theory

Several choices of the functions describing the creation and destruction processes of entanglement junctions in the Yamamoto network theory of concentrated polymer solutions have been examined. These choices are simple functions of the extension of the network segments bridging the entanglement points and it is demonstrated that the moments of the distribution function describing the network conformation can be solved for analytically. This has been done for a wide range of two-dimensional flows, both for the steady state and transient start-up and relaxation problems. The macroscopic stress tensor and flow birefringence are calculated and a variety of nonlinear effects are predicted and discussed.     

An ellipsoid-chain model for conjugated polymer solutions

We propose an ellipsoid-chain model which may be routinely parameterized to capture large-scale properties of semiflexible, amphiphilic conjugated polymers in various solvent media. The model naturally utilizes the defect locations as pivotal centers connecting adjacent ellipsoids (each currently representing ten monomer units), and a variant umbrella-sampling scheme is employed to construct the potentials of mean force (PMF) for specific solvent media using atomistic dynamics data and simplex optimization. The performances, both efficacy and efficiency, of the model are thoroughly evaluated by comparing the simulation results on long, single-chain (i.e., 300-mer) structures with those from two existing, finer-grained models for a standard conjugated polymer (i.e., poly(2-methoxy-5-(2'-ethylhexyloxy)-1,4-phenylenevinylene) or MEH-PPV) in two distinct solvents (i.e., chloroform or toluene) as well as a hybrid, binary-solvent medium (i.e., chloroform/toluene = 1:1 in number density). The coarse-grained Monte Carlo (CGMC) simulation of the ellipsoid-chain model is shown to be the most efficient--about 300 times faster than the coarse-grained molecular dynamics (CGMD) simulation of the finest CG model that employs explicit solvents--in capturing elementary single-chain structures for both single-solvent media, and is a few times faster than the coarse-grained Langevin dynamics (CGLD) simulation of another implicit-solvent polymer model with a slightly greater coarse-graining level than in the CGMD simulation. For the binary-solvent system considered, however, both of the two implicit-solvent schemes (i.e., CGMC and CGLD) fail to capture the effects of conspicuous concentration fluctuations near the polymer-solvent interface, arising from a pronounced coupling between the solvent molecules and different parts of the polymer. Essential physical implications are elaborated on the success as well as the failure of the two implicit-solvent CG schemes under varying solvent conditions. Within the ellipsoid-chain model, the impact of synthesized defects on local segmental ordering as well as bulk chain conformation is also scrutinized, and essential consequences in practical applications discussed. In future perspectives, we remark on strategy that takes advantage of the coordination among various CG models and simulation schemes to warrant computational efficiency and accuracy, with the anticipated capability of simulating larger-scale, many-chain aggregate systems.     

A theory of viscosity of dilute and moderately concentrated polymer solutions

A model theory of viscosity η for moderately concentrated polymer solutions is based on the assumption of a “local viscosity” effect and intermolecular hydrodynamic and thermodynamic interactions. It is shown that η is given by

$\text{η = η}{}_{\mathrm{o}}\text{{1 + γc[η]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{·}\text{exp}\text{H}{}_{\mathrm{o}}\text{RT}\text{aø}\text{1 ? aø}$

where γ is 0–0.4 and depends on the quality of the solvent, a varies between 0,4 and 0.8 and depends on the fraction of the “free volume” of the systems, H₀ is the activation energy of the solvent and π is the polymer volume concentration. The dependence of η and “activation energy” of π and T for various molecular weights and qualities of solvents is described quantitatively. Anomalous dependences of [η] and of η on M for low polymer are obtained. An expression for η is proposed:

$\text{η}\text{η}{}_{\mathrm{o}}{}^{\mathrm{<mtext>1\; ?\; 2</mtext><mtext>K</mtext>}}\text{= {1 + (1 ? 2K)c[η]}F(π)}$

where K is the Huggins-Martin coefficient and F(π) = 1 for most solutions when T is > T_g. For poor solvents the H vs c curve (where H is the activation energy of η of solution) has a minimum value at moderate concentrations. For good solvents, H depends slightly on the molecular weight according to an empirical equation:

$\text{H = H}{}_{\mathrm{o}}\text{+}\text{660}\text{α}{}^3\text{1}\text{n}\text{η}\text{η}{}_{\mathrm{o}}$

Expressions are given from the viscosities of solutions of miscible and also solutions of immiscible polymers.     

The viscoelasticity of polymer solutions: Comparison of theory and experiment

The theory of the preceding paper is compared with previously unpublished experimental results on the viscoelastic properties of two polystyrene fractions of molecular weight 1.9 × 10⁶ and 1.2 × 10⁶ in nitropropane, which is very nearly a theta solvent, and in toluene, a good solvent. At the concentrations used, around 1%, the theory predicts extensive departures from non-draining toward free-draining behavior; these effects are observed. The theory similarly fits experimental data of Johnson on a sample of polystyrene of molecular weight 860,000 in decalin and dioctyl phthalate in the same concentration range.     

Interactions between colloidal particles in polymer solutions: A density functional theory study

We present a density functional theory study of colloidal interactions in a concentrated polymer solution. The colloids are modeled as hard spheres and polymers are modeled as freely jointed tangent hard sphere chains. Our theoretical results for the polymer-mediated mean force between two dilute colloids are compared with recent simulation data for this model. Theory is shown to be in good agreement with simulation. We compute the colloid-colloid potential of mean force and the second virial coefficient, and analyze the behavior of these quantities as a function of the polymer solution density, the polymer chain length, and the colloid/polymer bead size ratio.     

Anomalous nanoparticle diffusion in polymer solutions and melts: a mode-coupling theory study

Mode-coupling theory is employed to study diffusion of nanoparticles in polymer melts and solutions. Theoretical results are directly compared with molecular dynamics simulation data for a similar model. The theory correctly reproduces the effects of the nanoparticle size, mass, particle-polymer interaction strength, and polymer chain length on the nanoparticle diffusion coefficient. In accord with earlier experimental, simulation, and theoretical work, it is found that when the polymer radius of gyration exceeds the nanoparticle radius, the Stokes-Einstein relation underestimates the particle diffusion coefficient by as much as an order of magnitude. Within the mode-coupling theory framework, a microscopic interpretation of this phenomenon is given, whereby the total diffusion coefficient is decomposed into microscopic and hydrodynamic contributions, with the former dominant in the small particle limit, and the latter dominant in the large particle limit. This interpretation is in agreement with previous mode-coupling theory studies of anomalous diffusion of solutes in simple dense fluids.     

An instrument for measuring the oscillatory electric birefringence properties of polymer solutions

An instrument for measuring the oscillatory electric birefringence properties of synthetic polymer dissolved in organic solvents has been designed and constructed. Novel features of the design include an in situ variable inter-electrode spacing Kerr cell and a double-beam optical train. The accessible frequency range extends from below 1 Hz to at least 100 kHz, with electric fields variable up to approximately 6000 V cm^?1 (peak-to-peak). Measurements are made with a powerful computerized data acquisition and processing system, based on an approach previously used for viscoelastic and oscillatory flow birefringence experiments. Preliminary results on a viscous liquid, Aroclor-1248, indicate that time-temperature superposition holds to reduced frequencies of at least 100 MHz. Comparison with theoretical predictions for rigid rod suspensions suggests that this liquid exhibits relaxation behavior with a time constants of ca. 6 ns at 25.00°C.     

Evaluating the accuracy of a density functional theory of polymer solutions with additive hard sphere diameters

We assess the accuracy of a density functional theory for athermal polymer solutions, consisting of solvent particles with a smaller radius than that of the monomers. The monomer and solvent density profiles in a slit bound by hard, flat, and inert surfaces are compared with those obtained by a Metropolis Monte Carlo simulation. At the relatively high density at which the comparison is performed, there are considerable packing effects at the walls. The density functional theory introduces a simple weight function to describe nonlocal correlations in the fluid. A recent study of surface forces in polymer solutions used a different weighting scheme to that proposed in this article, leading to less accurate results. The implications of the conclusions of that study are discussed.     

Negative thixotropy in polymer solutions

Negative thixotropy of polymer solutions due to the aggregation of polymer molecules and formation of supermolecular structures during flow has been described for many macromolecular systems. Although the factors influencing this effect have systematically been studied for many years, general causes and the origin of the structural changes are unknown. This study is a survey of our present knowledge of this phenomenon from the point of view of both experimental results and theories proposed for their interpretation.     

Phase equilibria in polymer solutions

A theoretical study of phase equilibria in multicomponent polymer solutions is presented. The treatment is based on the virial expansion of the osmotic pressure. A program for the numerical determination of phase diagrams is worked out and applied to the polystyrene-cyclohexane system. The present approach is compared with the Flory-Huggins theory of polymer solutions, which is shown to represent a special class of approximations in the present treatment.     

Viscous fingering in polymer solutions

Characteristics of viscous fingering patterns in polymer solutions were investigated by radially pushing air in a radial Hele‐Shaw cell containing hydroxypropyl methyl cellulose (HPMC) solutions. Low and high molecular weight HPMC samples were used. An increase in molecular weight easily yielded chain entanglements, which led to a strong shear thinning and an increase in elasticity. For the low molecular weight HPMC solutions, only the dense‐branching patterns were observed over the entire injection pressure range. When isopropyl alcohol was added into the HPMC solutions, leading to less elasticity, the pattern was tip‐splitting one, which is observed for Newtonian fluids. On the other hand, for the high molecular weight HPMC solutions, the resulting patterns showed a systematic change from dense‐branching to skewering patterns through tip‐splitting patterns as the injection pressure increased. The measured tip velocity was compared with the modified Darcy's law, but the coincidence is poor. The velocity corrected by the displaced area ratio provided a good linear relation.     

Solvent effects in flow birefringence of polymer solutions. General theory and discussion of form effects

A new, more realistic optical model of a dilute polymer solution is used to calculate the intrinsic birefringence. A general formula is derived valid for an arbitrary equilibrium distribution function of particles in the system. Besides the contributions due to the polymer and solvent, the resulting relation for intrinsic birefringence also contains terms reflecting the effect of orientation of solvent surrounding the polymer chain and the contribution of optical interactions between polymer segments and molecules of solvent. A detailed discussion of the optical interactions in an isotropic solvent reveals that the problem may be transformed in the first approximation into that of interactions between excess dipoles; however, any separation of the macroform and microform effects has no theoretical justification. It is shown that the microform effect depends on a detailed optical model of the statistical segment, and this effect is calculated for two simple models. The expression suggested by Tsvetkov cannot be applied to a segment consisting of anisotropic monomers.