Semidilute Polymer Solutions Research Articles

A generalised Phan–Thien—Tanner model

In this work we propose a novel generalised form of the Phan–Thien and Tanner (PTT) model by considering a new functional form of the nonlinear expression characterizing the destruction of physical network junctions and entanglements. This new function of the trace of the stress tensor is given by the generalized Mittag–Leffler function, and contains the familiar exponential form of the original Phan–Thien and Tanner model as a limiting case, but affords additional fitting flexibility through the inclusion of one or two additional fitting constants. We perform fits to experimental data in shear and extension and show that this generalized expression allows a better description of the rheological responses for a range of complex materials such as polymer melts and semidilute polymer solutions. By using an appropriate information criterion, we also demonstrate that the resulting generelized model remains parsimonious but provides improved fits of real world data.

Effect of molecular architecture on ring polymer dynamics in semidilute linear polymer solutions

Understanding the dynamics of ring polymers is a particularly challenging yet interesting problem in soft materials. Despite recent progress, a complete understanding of the nonequilibrium behavior of ring polymers has not yet been achieved. In this work, we directly observe the flow dynamics of DNA-based rings in semidilute linear polymer solutions using single molecule techniques. Our results reveal strikingly large conformational fluctuations of rings in extensional flow long after the initial transient stretching process has terminated, which is observed even at extremely low concentrations (0.025 c*) of linear polymers in the background solution. The magnitudes and characteristic timescales of ring conformational fluctuations are determined as functions of flow strength and polymer concentration. Our results suggest that ring conformational fluctuations arise due to transient threading of linear polymers through open ring chains stretching in flow.

Shear Banding in 4:1 Planar Contraction.

We study shear banding in a planar 4:1 contraction flow using our recently developed two-fluid model for semidilute entangled polymer solutions derived from the generalized bracket approach of nonequilibrium thermodynamics. In our model, the differential velocity between the constituents of the solution allows for coupling between the viscoelastic stress and the polymer concentration. Stress-induced migration is assumed to be the triggering mechanism of shear banding. To solve the benchmark problem, we used the OpenFOAM software package with the viscoelastic solver RheoTool v.2.0. The convection terms are discretized using the high-resolution scheme CUBISTA, and the governing equations are solved using the SIMPLEC algorithm. To enter into the shear banding regime, the uniform velocity at the inlet was gradually increased. The velocity increases after the contraction due to the mass conservation; therefore, shear banding is first observed at the downstream. While the velocity profile in the upstream channel is still parabolic, the corresponding profile changes to plug-like after the contraction. In agreement with experimental data, we found that shear banding competes with flow recirculation. Finally, the profile of the polymer concentration shows a peak in the shear banding regime, which is closer to the center of the channel for larger inlet velocities. Nevertheless, the increase in the polymer concentration in the region of flow recirculation was significantly larger for the inlet velocities studied in this work. With our two-fluid finite-volume solver, localized shear bands in industrial applications can be simulated.

Many-body effects in a binary nano-particle mixture dispersed in ideal polymer solutions.

A new mean-field theory is developed to treat a binary mixture of nanoparticles imbedded in a polydisperse polymer solution. The theory is based on a many-body polymer-mediated potential of mean force (PMF) between the particles and remains accurate even in the protein regime, where the particles' diameters cannot necessarily be considered large compared to the polymer radius of gyration. As implemented here, the theory is strictly valid for dilute to semi-dilute polymer solutions near the theta temperature (the so-called theta regime) or when the range of the PMF is strongly affected by the polymer size. For non-adsorbing particles, this is the same regime where the celebrated Asakura-Oosawa (AO) model is often used. Unlike the traditional AO model, however, our approach includes polymer flexibility and is accurate in the protein regime. We use the theory to calculate phase diagrams for a binary mixture of unequal-sized particles, both adsorbing and non-adsorbing. To test the theory, we carry out comparisons with simulations and obtained good quantitative agreement, which gives support to its accuracy. On the other hand, the oft-used approach assuming pairwise-additive potentials of mean force produce quantitatively (and sometime qualitatively) different phase diagrams.

Origin of the low-frequency plateau and the light-scattering slow mode in semidilute poly(ethylene glycol) solutions.

A low-frequency plateau is often found in the rheological spectra of various kinds of semidilute solutions of polymers and other colloids; also, many such solutions have been reported to show slow-modes in their dynamic light scattering autocorrelation functions. Both these observations may lead to the hypothesis of weak associative network structures built by the dissolved polymer chains or colloidal building blocks. To challenge this hypothesis, we conduct a series of comparative studies on semidilute solutions of poly(ethylene glycol) by using classical rheology as well as passive microrheology based on dynamic light scattering, along with structural studies using static light scattering. Although we indeed find a low-frequency plateau using classical shear rheology, even at elevated temperatures where potential polymer aggregates should be broken, no such plateau is observed in any of our microrheology experiments. Also, dynamic and static light scattering studies on the polymer solutions do not confirm the presence of larger structural entities: no slow mode can be detected in the autocorrelation function of the scattering intensity signal, and this signal is angle independent if the samples are purified by a thorough procedure of filtration. Based on these findings, we conclude that the low-frequency plateau in classical rheology results is an instrument effect caused by erroneous recording of the phase angle, although the magnitude of the torque lies well within the resolution of the rheometer. We also conclude that slow modes in dynamic light scattering on solutions of poly(ethylene glycol) are impurity-based artifacts rather than due to actual associated structures.

Conformationally averaged iterative Brownian dynamics simulations of semidilute polymer solutions.

The dynamics of semidilute polymer solutions are important to many polymer solution processing techniques such as fiber spinning and solution printing. The out-of-equilibrium molecular conformations resulting from processing flows directly impact material properties. Brownian dynamics (BD) simulations are a standard technique for studying this connection between polymer conformations in solution and processing flows because they can capture molecular-level polymer dynamics. However, BD simulations of semidilute polymer solutions are computationally limited by the calculation of hydrodynamic interactions (HIs) via an Ewald summed diffusion tensor and stochastic Brownian displacements via the decomposition of the diffusion tensor. Techniques based on the Cholesky decomposition scale with the number of particles N as O(N 3) and approximations in the literature have reduced this scaling to as low as O(N). These methods still require continuous updating of the diffusion tensor and Brownian displacements, resulting in a significant constant per-time step cost. Previously, we introduced a method that avoids this cost for dilute polymer solutions by iterative conformational averaging (CA) of intramolecular HIs. In this work, we extend the CA method to semidilute solutions by introducing a grid-space average of intermolecular HIs and a pairwise approximation to the Brownian displacements based on the truncated expansion ansatz of Geyer and Winter. We evaluate our method by first comparing the computational cost with that of other simulation techniques. We verify our approximations by comparison with expected results for static and dynamic properties at equilibrium and use our method to demonstrate the concentration dependence of HI screening.

Effects of polymer adsorption on the effective viscosity in microchannel flows: phenomenological slip layer model from molecular simulations

Molecular simulations (Dissipative Particle Dynamics - DPD) were used to quantify the effect of polymer adsorption on the effective shear viscosity of a semi-dilute polymer solution in microchannel Poseuille flow. It is well known that polymer depletion layers develop adjacent to solid walls due to hydrodynamic forces, causing an apparent wall slip and reduced effective viscosity (increased total flow rate). We found that depletion layers also developed in the presence of hydrodynamically rough adsorbed layers on the wall. Polymer-polymer (steric) repulsion between flowing and adsorbed polymer expanded the depletion layer compared to no-adsorption cases, and the effective viscosity was reduced further. Desorption occurred for higher shear rates, reducing the repulsion effect and shrinking the depletion layers. A phenomenological algebraic model for the depletion layer thickness, including a shear modified adsorption isotherm, was developed based on the simulation data. The depletion layer model can be used together with the effective viscosity model we developed earlier.

Conformations of Ring Polystyrenes in Semidilute Solutions and in Linear Polymer Matrices Studied by SANS

Conformations of highly purified ring polystyrene, R-70, with the molar mass of 70 kg/mol, in a good solvent and in linear polymer homologue matrices were examined by small-angle neutron scattering (SANS) measurements. The radii of gyration Rg of R-70 were estimated by the Guinier’s approximation from the SANS profiles obtained, and the polymer volume fraction Φ dependence of Rg2 was discussed. In deuterated toluene as a good solvent, R-70 exhibits the Rg2 ∼ Φ–0.29±0.01 dependence at high Φ above the overlap volume fraction, Φ0* (i.e., 1 < Φ/Φ0* < 20). This exponent −0.29 shows stronger Φ dependence than that for semidilute solutions of linear polymers, −0.25, predicted from the scaling theory, suggesting that the ring expands more sensitively than linear chains when Φ decreases in semidilute regime. In contrast, the Φ dependence of Rg2 of R-70 is evidently weaker than that of the recent simulation for ring polymer solutions (Rg2 ∼ Φ−0.59) by Reigh et al. This difference is thought to originate from the d...

Features of the Rheological Behavior of Polymer-Colloidal Dispersions Based on a Sodium Salt of Carboxymethyl Cellulose and Silver Iodide Sols

The rheological properties of aqueous solutions of a sodium salt of carboxymethyl cellulose are studied. The occurrence of a region of unstructured semidilute solutions in which the entanglement network is absent is found. The introduction of both opposite and like-charged micelles of the lyophobic sol into a semidilute polymer solution leads to its structuring due to the appearance of an additional network because of polymer adsorption on the surface of the sol. As a result of the network formation a decrease in the interval between the crossover and the formation concentration of the entanglement network and also an increase in the viscosity of the semidilute solution are observed.

Coupling of Nanoparticle Dynamics to Polymer Center-of-Mass Motion in Semidilute Polymer Solutions

We investigate the dynamics of nanoparticles in semidilute polymer solutions when the nanoparticles are comparably sized to the polymer coils using explicit- and implicit-solvent simulation methods. The nanoparticle dynamics are subdiffusive on short time scales before transitioning to diffusive motion on long time scales. The long-time diffusivities scale according to theoretical predictions based on full dynamic coupling to the polymer segmental relaxations. In agreement with our recent experiments, however, we observe that the nanoparticle subdiffusive exponents are significantly larger than predicted by the coupling theory over a broad range of polymer concentrations. We attribute this discrepancy in the subdiffusive regime to the presence of an additional coupling mechanism between the nanoparticle dynamics and the polymer center-of-mass motion, which differs from the polymer relaxations that control the long-time diffusion. This coupling is retained even in the absence of many-body hydrodynamic interactions when the long-time dynamics of the colloids and polymers are matched.

Hydrodynamically Coupled Brownian Dynamics: A coarse-grain particle-based Brownian dynamics technique with hydrodynamic interactions for modeling self-developing flow of polymer solutions.

We present a novel coarse-grain particle-based simulation technique for modeling self-developing flow of dilute and semi-dilute polymer solutions. The central idea in this paper is the two-way coupling between a mesoscopic polymer model and a phenomenological fluid model. As our polymer model, we choose Responsive Particle Dynamics (RaPiD), a Brownian dynamics method, which formulates the so-called "conservative" and "transient" pair-potentials through which the polymers interact besides experiencing random forces in accordance with the fluctuation dissipation theorem. In addition to these interactions, our polymer blobs are also influenced by the background solvent velocity field, which we calculate by solving the Navier-Stokes equation discretized on a moving grid of fluid blobs using the Smoothed Particle Hydrodynamics (SPH) technique. While the polymers experience this frictional force opposing their motion relative to the background flow field, our fluid blobs also in turn are influenced by the motion of the polymers through an interaction term. This makes our technique a two-way coupling algorithm. We have constructed this interaction term in such a way that momentum is conserved locally, thereby preserving long range hydrodynamics. Furthermore, we have derived pairwise fluctuation terms for the velocities of the fluid blobs using the Fokker-Planck equation, which have been alternatively derived using the General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) approach in Smoothed Dissipative Particle Dynamics (SDPD) literature. These velocity fluctuations for the fluid may be incorporated into the velocity updates for our fluid blobs to obtain a thermodynamically consistent distribution of velocities. In cases where these fluctuations are insignificant, however, these additional terms may well be dropped out as they are in a standard SPH simulation. We have applied our technique to study the rheology of two different concentrations of our model linear polymer solutions. The results show that the polymers and the fluid are coupled very well with each other, showing no lag between their velocities. Furthermore, our results show non-Newtonian shear thinning and the characteristic flattening of the Poiseuille flow profile typically observed for polymer solutions.

Electroconvection and Morphological Instabilities in Potentiostatic Electrodeposition across Liquid Electrolytes with Polymer Additives

Electroconvection in electrodeposition results in fast growing ramified deposits undermining the efficacy of the process. Here we examine the effect of polymer additives in the electrolyte on electroconvection and the consequences for morphological instability. The connection between two interrelated problems is studied, one of pure electroconvection at an ion-selective membrane and one where the electroconvection is coupled to morphological growth of a metal surface in electrodeposition. A two-fluid model for the semidilute polymer solution is employed and linear stability analysis is used to study the two problems. We find that the stable-to-unstable transition in the pure electroconvection problem corresponds to a change in growth rate of the morphology, from a slow diffusive growth to a fast convection-driven instability. The polymer exerts a drag resisting the electroconvective flow of the solvent and can suppress the electroconvective instability. The nature of this transition varies with the wavelength of the perturbation with longer modes having a weaker flow and being stabilized first. Increasing the polymer molecular weight and decreasing the permeability which respectively control the mobility of the polymer and the drag it exerts, weaken the electroconvection. Higher polymer volume fractions, which affect both polymer mobility and the drag, also reduce the electroconvection. It is shown that polymer additives can substantially increase the critical voltage for the onset of electroconvective motions with wavelengths larger than the space charge thickness.

Equilibrium Dynamics and Shear Rheology of Semiflexible Polymers in Solution

We study the structure and dynamics of semidilute solutions of semiflexible polymers at rest and under shear using hybrid molecular dynamics simulations that take hydrodynamic interactions into account. We show that the polymer center-of-mass diffusion coefficient significantly decreases with increasing chain stiffness at fixed monomer density. The zero-shear viscosity shows a corresponding increase due to the intermolecular interactions of stiffer chains. We apply steady shear flow to the polymer solutions and show that at high shear rates the flow properties become almost independent of polymer stiffness. We characterize the polymer conformations under shear and find that in this regime polymers are elongated and aligned along the flow direction, but semiflexible polymers surprisingly form an increased number of hairpin-like conformations due to increased tumbling.

Effect of degassing on the aggregation of carbon nanotubes dispersed in water

Dynamic light scattering (DLS) along with centrifugation and shaking tests reveal that dissolved gases can significantly affect the aggregation behavior of carbon nanotubes (CNTs) dispersed in water. The CNTs in non-degassed samples form loose, stable networks having the DLS result reminiscent of semidilute polymer solutions, whereas the CNTs in degassed samples aggregate to form Brownian colloids that sediment quickly. Interestingly, the CNTs dispersed in acetone, with or without degassing, also behave like semidilute polymers in DLS experiments. We propose a surface nanobubble-assisted mechanism to explain the observed aggregation behaviors. Our work signifies that dissolved gases may play an important role in determining hydrophobicity and biomolecular functions in aqueous environments.

Multiresolution Modeling of Semidilute Polymer Solutions: Coarse-Graining Using Wavelet-Accelerated Monte Carlo

We present a hierarchical coarse-graining framework for modeling semidilute polymer solutions, based on the wavelet-accelerated Monte Carlo (WAMC) method. This framework forms a hierarchy of resolutions to model polymers at length scales that cannot be reached via atomistic or even standard coarse-grained simulations. Previously, it was applied to simulations examining the structure of individual polymer chains in solution using up to four levels of coarse-graining (Ismail et al., J. Chem. Phys., 2005, 122, 234901 and Ismail et al., J. Chem. Phys., 2005, 122, 234902), recovering the correct scaling behavior in the coarse-grained representation. In the present work, we extend this method to the study of polymer solutions, deriving the bonded and non-bonded potentials between coarse-grained superatoms from the single chain statistics. A universal scaling function is obtained, which does not require recalculation of the potentials as the scale of the system is changed. To model semi-dilute polymer solutions, we assume the intermolecular potential between the coarse-grained beads to be equal to the non-bonded potential, which is a reasonable approximation in the case of semidilute systems. Thus, a minimal input of microscopic data is required for simulating the systems at the mesoscopic scale. We show that coarse-grained polymer solutions can reproduce results obtained from the more detailed atomistic system without a significant loss of accuracy.

An iterative method for hydrodynamic interactions in Brownian dynamics simulations of polymer dynamics

Brownian Dynamics (BD) simulations are a standard tool for understanding the dynamics of polymers in and out of equilibrium. Quantitative comparison can be made to rheological measurements of dilute polymer solutions, as well as direct visual observations of fluorescently labeled DNA. The primary computational challenge with BD is the expensive calculation of hydrodynamic interactions (HI), which are necessary to capture physically realistic dynamics. The full HI calculation, performed via a Cholesky decomposition every time step, scales with the length of the polymer as O(N3). This limits the calculation to a few hundred simulated particles. A number of approximations in the literature can lower this scaling to O(N2 - N2.25), and explicit solvent methods scale as O(N); however both incur a significant constant per-time step computational cost. Despite this progress, there remains a need for new or alternative methods of calculating hydrodynamic interactions; large polymer chains or semidilute polymer solutions remain computationally expensive. In this paper, we introduce an alternative method for calculating approximate hydrodynamic interactions. Our method relies on an iterative scheme to establish self-consistency between a hydrodynamic matrix that is averaged over simulation and the hydrodynamic matrix used to run the simulation. Comparison to standard BD simulation and polymer theory results demonstrates that this method quantitatively captures both equilibrium and steady-state dynamics after only a few iterations. The use of an averaged hydrodynamic matrix allows the computationally expensive Brownian noise calculation to be performed infrequently, so that it is no longer the bottleneck of the simulation calculations. We also investigate limitations of this conformational averaging approach in ring polymers.

SANS Study on Critical Polymer Clusters of Tetra-Functional Polymers

A series of critical clusters was prepared by mixing two different kinds of tetra-functional poly(ethylene glycol) (PEG) prepolymers carrying complementary end-groups. The structures of these critical clusters were investigated by small angle neutron scattering (SANS) in different dilution levels. Scaling laws for semidilute polymer solutions were observed in the solution of critical polymer clusters for q ξ–1, the size distribution of the critical clusters influenced the scattering intensity. Assuming the validity of the scattering theory for the dilute solution of critical polymer clusters in the q-range q > ξ–1, the Fisher exponent was estimated to be 1.90–2.25, which was found to depend on the preparation condition of the critical clusters.

Phase separation of polymer solutions on a solvent surface

Study of the dynamics of liquid droplets of dilute and semidilute polymer solutions on the surfaces of liquid subphases representing solvents for corresponding polymers has shown that a spot of a rather stable layer is formed on an air–liquid interface. The spot spreads over a liquid subphase surface to yield a monomolecular polymer layer. At the same time, the solvent passes into the solution, so that the polymer or its concentrated solution remains on the subphase surface. The polymer does not dissolve in the bulk subphase for several hours. The stability of the polymer spot has been explained under the assumption that the interfacial surface possesses elastic properties and hinders the penetration of macromolecules into the bulk subphase. Desolvation of macromolecules followed by phase separation occurs on the surface. The initial rate of the phase separation of the solution is rather high, while the time dependence of the diameter of the spreading spot is described by a scaling law with an exponent almost equal to 2/3.

Shear banding of semidilute polymer solutions in pressure-driven channel flow

Shear banding is observed in many soft materials. We recently developed a two-fluid model for semidilute entangled polymer solutions by using the generalized bracket approach of nonequilibrium thermodynamics. This model assumes that Fickian diffusion and stress-induced migration generate a nontrivial velocity difference between the polymers and the solution, thereby resulting in shear band formation. A straightforward implementation of the slip boundary conditions is possible because the differential velocity is treated as a state variable. Numerical calculations showed obvious shear banding in the polymer concentration profile. Increasing the pressure gradient reduces the inhomogeneity of the concentration profile and moves the transition region toward the center of the channel. Moreover, the velocity deviates from the typical parabolic form and shows a plug-like profile with a low shear band near the center and a high one near the walls. The lack of hysteresis in the profiles of the volumetric flow rate calculated with the increasing and decreasing pressure gradient demonstrates the uniqueness of the solution. In addition, the flow rate exhibits a spurt at a critical pressure gradient, as experimentally observed for shear banding materials. The simplicity of the new model encourages us to analyze it in more complicated flows.

A new scaling for the rotational diffusion of molecular probes in polymer solutions

In the present work, we propose a new scaling form for the rotational diffusion coefficient of molecular probes in semi-dilute polymer solutions, based on a theoretical study. The mean-field theory for depletion effect and semi-empirical scaling equation for the macroscopic viscosity of polymer solutions are properly incorporated to specify the space-dependent concentration and viscosity profiles in the vicinity of the probe surface. Following the scheme of classical fluid mechanics, we numerically evaluate the shear torque exerted on the probes, which then allows us to further calculate the rotational diffusion coefficient Dr. Particular attention is given to the scaling behavior of the retardation factor Rrot ≡ D/Dr with D being the diffusion coefficient in pure solvent. We find that Rrot has little relevance to the macroscopic viscosity of the polymer solution, while it can be well featured by the characteristic length scale rh/δ, i.e. the ratio between the hydrodynamic radius of the probe rh and the depletion thickness δ. Correspondingly, we obtain a novel scaling form for the rotational retardation factor, following Rrot = exp[a(rh/δ)b] with rather robust parameters of a ≃ 0.51 and b ≃ 0.56. We apply the theory to an extensive calculation for various probes in specific polymer solutions of poly(ethylene glycol) (PEG) and dextran. Our theoretical results show good agreements with the experimental data, and clearly demonstrate the validity of the new scaling form. In addition, the difference of the scaling behavior between translational and rotational diffusions is clarified, from which we conclude that the depletion effect plays a more significant role on the local rotational diffusion rather than the long-range translation diffusion.