Gaussian Chain Theory Research Articles

Radius of Gyration, Mean Span, and Geometric Shrinking Factors of Bridged Polycyclic Ring Polymers

This study concerns the equilibrium geometric properties of a family of cyclic chains, referred to as the “bridged polycyclic rings,” which have f flexible subchains bridging two common branch points. By increasing the number of bridges, f, this family encompasses the usual linear chain (f = 1), monocyclic ring (f = 2), bicyclic θ-shaped polymer (f = 3), and multicyclic rings with increasing topological complexity. Results of their radius of gyration, mean span, and, consequently, geometric shrinking factors (also known as the g-factors) are obtained by three approaches—the Gaussian chain theory, simulations based on the Kremer–Grest bead-spring model, and a Flory-type mean-field approach. Using the confinement analysis from bulk structures method, the equilibrium partition coefficients (K) of several of those cyclic excluded volume chains in a cylindrical pore with inert surfaces are obtained, and the results fall onto a common curve on a graph of K versus the polymer-to-pore size ratio, using the mean span as the representative polymer size, in the range of K relevant to polymer separation in size exclusion chromatography (SEC) experiments. Applications of the results in predicting the SEC retention volume of such bridged polycyclic ring polymers are discussed in the framework of the equilibrium partition theory.

Phase behavior of tri-block copolymers in solution: Mesoscopic simulation study

The microphase separation dynamics of tri-block copolymers Pluronic EO17PO60EO17 (P103), EO19PO43EO19 (P84) and EO19PO29EO19 (P65) was simulated in aqueous solution by a dynamic variant of mean-field density functional theory for Gaussian chains. For a Pluronic copolymer with smaller ratio of poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) (P103, P84 and P65) at low concentration, with an increase of the concentration, the micellar phases change from spherical micelles and micellar clusters, to disk-like micelles for the P103 and P84 solutions, and from spherical micelles to worm-like micelles for the P65 solution. In the P103 and P84 systems, the spherical micelles and micellar clusters can be divided clearly by order parameter, and in the P65 solution the phase separation of worm-like micelle with simulation time is shown clearly by the 3D density field of the PO block. With a decrease of the ratio of hydrophobic PPO and PEO (from P103, P84 to P65), the critical micellar concentration (cmc) increases. It was found that the cmc were 4vol.% for P103, 7vol.% for P84 and 43vol.% for P65. It was concluded that mesoscopic simulation can be considered as an adjunct to experimentation and can provide mesoscopic information otherwise inaccessible (or, not easily accessible) from experimentation.

Mesoscopic Simulation of the Phase Separation on Triblock Copolymer in Aqueous Solution

The microphase separation dynamics of triblock copolymer surfactant [(ethylene oxide) 27(propylene oxide) 61(ethylene oxide) 27] (P104) in aqueous solution was simulated by a dynamic variant of mean-field density functional theory for Gaussian chains. Different morphologies depending on the simulation concentration and time were shown in low concentration (ϕ<35%). With the increase of the concentration, the self-assemble aggregates changed from spherical micelles, micellar clusters, to disk-like micelles. (1) In the spherical micellar region (5%–10%, ϕ), the result of the simulation was similar to that of the experiment in that micelle was a dense core consisting mainly of PPO and a hydrated PEO swollen corona. (2) Due to the coalescence among spherical micelles, larger micellar clusters are formed in a concentration ranging from 11% to 15% (ϕ). (3) Owing to the attractions between the PEO coronas of micellar clusters, series of disk-like micelles were found above the concentration 16% (ϕ). The order parameters showed that the phase separation of spherical micelles was easily formed, while the micellar cluster or disk-like micelles required a longer time to acquire a steady equilibrium.

Dynamics of Unfolded Polypeptide Chains in Crowded Environment Studied by Fluorescence Correlation Spectroscopy

Proteins have evolved to fold and function within a cellular environment that is characterized by high macromolecular content. The earliest step of protein folding represents intrachain contact formation of amino acid residues within an unfolded polypeptide chain. It has been proposed that macromolecular crowding can have significant effects on rates and equilibria of biomolecular processes. However, the kinetic consequences on intrachain diffusion of polypeptides have not been tested experimentally, yet. Here, we demonstrate that selective fluorescence quenching of the oxazine fluorophore MR121 by the amino acid tryptophan (Trp) in combination with fast fluorescence correlation spectroscopy (FCS) can be used to monitor end-to-end contact formation rates of unfolded polypeptide chains. MR121 and Trp were incorporated at the terminal ends of polypeptides consisting of repetitive units of glycine (G) and serine (S) residues. End-to-end contact formation and dissociation result in “off” and “on” switching of MR121 fluorescence and underlying kinetics can be revealed in FCS experiments with nanosecond time resolution. We revisit previous experimental studies concerning the dependence of end-to-end contact formation rates on polypeptide chain length, showing that kinetics can be described by Gaussian chain theory. We further investigate effects of solvent viscosity and temperature on contact formation rates demonstrating that intrachain diffusion represents a purely diffusive, entropy-controlled process. Finally, we study the influence of macromolecular crowding on polypeptide chain dynamics. The data presented demonstrate that intrachain diffusion is fast in spite of hindered diffusion caused by repulsive interactions with macromolecules. Findings can be explained by effects of excluded volume reducing chain entropy and therefore accelerating the loop search process. Our results suggest that within a cellular environment the early formation of structural elements in unfolded proteins can still proceed quite efficiently in spite of hindered diffusion caused by high macromolecular content.

Mesoscopic Simulation on Phase Behavior of Ternary Copolymeric Solution in the Absence and Presence of Shear

We simulate the microphase separation dynamics of the triblock polymer surfactant (ethylene oxide)19(propylene oxide)29(ethylene oxide)19 (P65) in aqueous solution by a dynamic variant of mean-field density functional theory for Gaussian chains. With the increase of concentration, different mesoscale structures are found in the absence and presence of shear. (1) In the absence of shear, there are three morphologies formed. The first is the micellar phase, including a spherical micelle, which has hexagonal packing and a fcc (face-centered cubic) structure in the solution and a wormlike micelle, the second is the bicontinuous phase, which belongs to gyroid-type connectivity, and the third is the lamellar phase. (2) In the presence of shear, only two morphologies are formed in all concentrations, they are hexagonal and lamellar phases, which align in the flow direction. We notice that the weak shear makes the order parameters of the simulation system oscillate, while the strong shear makes the order paramete...

First Cumulant of the Dynamic Structure Factor for Polymers in Θ Solvents. Effects of Chain Stiffness and Local Chain Conformation

The first cumulant Ω(k) of the dynamic structure factor as a function of the magnitude k of the scattering vector was determined from dynamic light scattering (DLS) measurements for two atactic poly(α-methylstyrene) (a-PαMS) samples with weight-average molecular weights Mw=3.22×106 and 5.46×106 in cyclohexane at 30.5 °C (Θ) and for one polyisobutylene (PIB) sample with Mw=6.63×106 in isoamyl isovalerate at 25.0 °C (Θ). The translational diffusion coefficient was also determined from DLS measurements for the a-PαMS sample with the larger Mw and for the PIB sample in the same solvent conditions. The mean-square radius of gyration was determined from static light scattering measurements for the PIB sample. From a comparison of the present data for a-PαMS and PIB along with the previous ones for atactic polystyrene and poly(n-hexyl isocyanate) with the helical wormlike chain theory, it may be concluded that the universality predicted by the Gaussian chain theory for the plot of η0Ω(k)/kBTk3 against 1/2k cannot be realized for flexible polymers even with very high Mw (\\simeq106–107), and that the height of the plateau in the k3-region depends on chain stiffness and local chain conformation, where η0 is the solvent viscosity, kB the Boltzmann constant, and T the absolute temperature.

Polymer chains in semidilute solutions confined to a square channel: mean-field Gaussian chain theory and comparison with simulation results

We extend the mean-field Gaussian chain theory, originally developed for non-dilute solutions of athermal polymer chains in a slit, to solutions in a channel with a square cross section. The formulation allows one to calculate the monomer density profile, the chemical potential of the confined polymer chain, and therefore the partition coefficient. For the mean-field potential, we used the first-order approximation that neglects local monomer density fluctuations and the second-order approximation that takes into account the fluctuations. The results of the density profile and the partition coefficient were compared with those obtained in the lattice Monte Carlo simulations. The theoretical results obtained with the first-order approximation agreed well with the simulation results for chains of 100 beads below the average monomer density of ca. 0.2. At higher concentrations, the second-order results gave a better agreement. This cross over indicates a change in the interactions between polymer chains from those in one-dimension to those in three-dimensions as the correlation length in the confined solution becomes sufficiently shorter than the channel width.

Mean-field Gaussian chain theory for semidilute theta chains in a slit

Mean-field Gaussian chain theory for nondilute polymer solutions is being applied to polymer chains in the theta condition confined to a slit in a wide range of concentrations. Various existing lattice chain theories were used to obtain the effective potential for the Gaussian chain. Calculation results for the monomer density profile, the partition coefficient, and the chain dimension across the slit at various concentrations were computed and compared with the results obtained in the lattice Monte Carlo simulations. The need for a higher concentration to flatten the density profile and to cause the weak-to-strong penetration transition, compared with athermal chains, was confirmed. At the monomer–monomer interaction widely used as the theta condition on the cubic lattice, the chain cluster theory by Freed and the random mixing approximation by Huggins provide a better agreement with the simulation results than the Flory model and the Guggenheim model do. The agreement is, however, not as good as the one for athermal chains.

Spinodal decomposition in a binary polymer mixture: dynamic self-consistent-field theory and Monte Carlo simulations.

We investigate how the dynamics of a single chain influences the kinetics of early stage phase separation in a symmetric binary polymer mixture. We consider quenches from the disordered phase into the region of spinodal instability. On a mean field level we approach this problem with two methods: a dynamical extension of the self-consistent-field theory for Gaussian chains, with the density variables evolving in time, and the method of the external potential dynamics where the effective external fields are propagated in time. Different wave vector dependencies of the kinetic coefficient are taken into account. These early stages of spinodal decomposition are also studied through Monte Carlo simulations employing the bond fluctuation model that maps the chains-in our case with 64 effective segments-on a coarse grained lattice. The results obtained through self-consistent-field calculations and Monte Carlo simulations can be compared because the time, length, and temperature scales are mapped onto each other through the diffusion constant, the chain extension, and the energy of mixing. The quantitative comparison of the relaxation rate of the global structure factor shows that a kinetic coefficient according to the Rouse model gives a much better agreement than a local, i.e., wave vector independent, kinetic factor. Including fluctuations in the self-consistent-field calculations leads to a shorter time span of spinodal behavior and a reduction of the relaxation rate for smaller wave vectors and prevents the relaxation rate from becoming negative for larger values of the wave vector. This is also in agreement with the simulation results.

Effective-medium Gaussian-chain theory for semidilute polymer solutions confined to a slit

Theoretical formulation for the thermodynamics of a nondilute polymer solution in a confined space is presented. The interactions between polymer chains were taken into account as an effective potential field in which a chain of the Gaussian conformation grows. We determined the potential so that, when applied to the bulk solution, the theory reproduces the osmotic pressure of the bulk solution calculated within renormalization group theory. The Green function for the polymer chains confined to a slit was calculated self-consistently, since the density profile obtained from the Green function determines the potential field within the slit. The chemical potential, the partition coefficient, the density profile, and the dimension of the polymer chain perpendicular to the slit walls were obtained as a function of the concentration for different slit widths. The results agree with those obtained in the computer simulations.

Simulation of 3D Mesoscale Structure Formation in Concentrated Aqueous Solution of the Triblock Polymer Surfactants (Ethylene Oxide)13(Propylene Oxide)30(Ethylene Oxide)13 and (Propylene Oxide)19(Ethylene Oxide)33(Propylene Oxide)19. Application of Dynamic Mean-Field Density Functional Theory

We simulate the microphase separation dynamics of aqueous solutions of the triblock polymer surfactants (ethylene oxide)13(propylene oxide)30(ethylene oxide)13 and (propylene oxide)19(ethylene oxide)33(propylene oxide)19 by a dynamic variant of mean-field density functional theory for Gaussian chains. This is the first 3D mesoscale model for the dynamic behavior of specific complex polymer solutions. Different mesoscale structures (micellar, hexagonal, bicontinuous, and lamellar and dispersed coexisting phases) are formed depending on composition. The numerical results are in good agreement with experiment. The intermediate hexagonal and bicontinuous phases of (ethylene oxide)13(propylene oxide)30(ethylene oxide)13 solution retain a rich defect structure. Concentrated solution (60%) of (propylene oxide)19(ethylene oxide)33(propylene oxide)19 shows the onset of macrophase separation, with small water droplets dispersed throughout the system. We confirm the experimental observation that the lamellar phase f...

Viscoelastic properties of pathological synovial fluids for a wide range of oscillatory shear rates and frequencies

A series of pathological synovial fluids are examined for viscoelastic properties in oscillatory shear flow at frequencies of physiological significance. When tested at varying amplitudes of shear at a frequency of 2 Hertz, the fluids show marked nonlinearity for shear rates above 10 sec−1, the elastic component of the shear stress truncating in the range from 1 to 10 dynes/cm2. At low shear rates, the fluids are linear and viscoelastic and have frequency dispersions matching the character of Gaussian chain theory for polymer solutions. Enzymatic degradation of the hyaluronate destroys the viscoelasticity, leaving a Newtonian fluid of relatively low viscosity. The concentration and molecular weight of the hyaluronate produce substantial intermolecular interaction. The relaxation times, viscosity, and elasticity of the synovial fluids are greatly increased over values expected for noninteracting molecules of hyaluronic acid in solution. While both the viscosity and elasticity vary greatly among the pathological fluids, the elasticity is the more sensitive indicator of viscoelastic properties.

Exact and approximate eigenvalues and intrinsic functions for the Gaussian chain theory

Eigenvalues for the Gaussian chain theory for dilute solutions of linear polymers have been determined in an exact manner for chain segments up to 100 and for a wide range of hydrodynamic interaction. It is found that these exact results may be closely approximated by a simple equation. The intrinsic functions of the eigenvalues which describe the molecular weight dependence of the intrinsic viscosity and terminal relaxation time of the chain are presented using both the exact and the approximate eigenvalues. It is shown that by using the simple approximate calculation good results are easily determined. The approximate equation is available for manipulation and analysis of complex eigenvalue functions.

Eigenvalues and the intrinsic viscosity of short Gaussian chains

Exact eigenvalues which are central to the Gaussian chain theory for dilute solutions of linear polymers have been determined for short chains. This is done for a number of chain segments N ranging from 1 to 15 and taking into account the effects of hydrodynamic interaction between segments. Using these exact eigenvalues, the molecular weight dependence of the steady flow intrinsic viscosity [η] is determined together with the terminal relaxation time τ1 which is derivable from [η]. These theoretical results are compared with similar results obtained using approximate eigenvalues λp for N large and small eigenvalue index p. It is found that while the character of the dependence of the λp, [η] and τ1 on N is similar for both exact and approximate results, the numerical factors may differ significantly. Comparison of these results for [η] with measured values for polystyrene over a wide range of molecular weights gives good agreement with the exact theoretical results and some indication of the inexactness of the approximate results for N large. The τ1 derived from [η] are compared with values obtained independently from oscillatory flow birefringence measurements.