Gaussian Polymer Chain Research Articles

A second-order algorithm for the simulation of the Brownian dynamics of macromolecular models

Most recent works on Brownian dynamics simulation employ a first-order algorithm developed by Ermak and McCammon [J. Chem. Phys. 69, 1352 (1978)]. In this work we propose the use of a second-order algorithm in which the step is a combination of two first-order steps, like in the second-order Runge–Kutta method for differential equations. Although the computer time per step is roughly doubled, the second-order algorithm is more efficient than the previous one because a given accuracy in the results can be achieved with less than half the number of steps. The new algorithm also allows for longer time steps without divergence. The advantage of the new procedure is illustrated in the simulation of four macromolecular systems: A quasirigid dumbbell, a semiflexible trumbbell, a semiflexible hinged rod, and a Gaussian polymer chain.

Langevin approach to polymers in flow.

A Gaussian polymer chain in simple shear flow is studied using Langevin equations. Incorporating hydrodynamic interactions to first order in \ensuremath{\epsilon}==4-d (d is the spatial dimensionality) with the aid of field-theoretic methods, we solve these nonlinearly coupled kinetic equations for polymer-solvent dynamics and analytically evaluate the second normal stress coefficient ${\ensuremath{\Psi}}_{2}$ for small shear rates. It is found that the mean-field (consistent preaveraging) approximation for hydrodynamic interactions (HI) produces an unphysical positive ${\ensuremath{\Psi}}_{2}$, while inclusion of fluctuations in HI leads to a negative value for ${\ensuremath{\Psi}}_{2}$, in agreement with experimental evidence.

Universal shape ratios for open and closed random walks: exact results for all d

The mean asphericities (Ad) of open and closed Gaussian polymer chains are computed exactly and analytically (up to simple quadratures) for arbitrary space dimensions d. Excellent agreement is found with existing simulation data for d=2, 3, and 4. The authors' technique can also be used to compute other averages of ratios of fluctuating variables as well as extended to include the effects of self-avoiding walk interactions.

Brownian dynamics of a flexible polymer. Internal modes and quaiselastic scattering function

Brownian Dynamics trajectories have been simulated by means of the Ermak and McCammon algorithm for Gaussian polymer chains of different lengths with hydrodynamic interactions. The results for relaxation times and dynamic scattering functions of chains with 2–20 statistical units are close to those calculated from the preaveraged Rouse–Zimm theory. The differences between both sets of results are slightly higher than the range of uncertainties of the simulation values (about 5% in most typical cases). Moreover, the simulated scattering functions have been fitted to sums of exponentials in order to extract translational diffusion coefficients and first relaxation times, following the Pecora procedure. The values obtained this way are in satisfactory agreement with the results directly calculated from the trajectories.

Statistics of a polymer in a random potential, with implications for a nonlinear interfacial growth model

We examine and extend recent results for the statistics of a gaussian polymer chain of length t in a quenched random potential μ (r). This problem can be mapped onto one involving the nonlinear evolution equation h (r, t) = ∇2h + (∇h)2 - μ(r) for a growing interface in the presence of a time-independent, but spatially varying, random flux - μ (r). [The corresponding problem for a directed polymer, equivalent to a flux random in both time and space, was studied by M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56 (1986) 889.] A free chain in a true white-noise potential, in space dimensions d = 2, 3, is predicted to collapse to a linear size R ˜ v1 /(d - 4)[ln V] 1/(d - 4) where v is the mean-square potential fluctuation (presumed small) and V the volume of the system. This agrees with a recent replica calculation of S. F. Edwards and M. Muthukumar (J. Chem. Phys., 89 (1988) 2435), but with an extra logarithmic dependence on V. The difference between this result and that for an annealed potential is commented upon, with reference to the phenomenon of localization as it applies to polymers in a quenched environment ; the effect of a saturating potential is discussed. In contrast, a chain with one end fixed at a particular place (say at r) is typically arranged as a tadpole consisting of a collapsed head region (occupying a deep minimum of μ) connected to r by an extended tail. The free energy surface of such a chain F (r, t ) consists asymptotically of a series of conical valleys, separated by sharp ridges. Several results for the statistics of this surface are obtained. The evolved profile in the equivalent interfacial growth model is obtained from h (r, t ) = - F (r, t ), and hence corresponds to a series of conical mountains.

Dynamic scattering factor for labeled molecules of a gaussian chain

The dynamic scattering factor for labeled molecules for a gaussian polymer chain with preaveraged hydrodynamic interactions is calculated with the aid of a renormalization-group method. We find that the gaussian approximation of conformation-variables is valid to the lowest nontrivial order. The crossover behavior for the critical to the hydrodynamic regime is illustrated.

Polymers with excluded volume in various geometries: Renormalization group methods

Renormalization group (RG) methods are generalized to study a single polymer chain with excluded volume in various geometries with different boundary conditions (or polymer–surface interactions) on the limiting surfaces. Methods for the renormalization of these theories are presented and are used to derive the RG equations which dictate the generalized scaling behavior as a function of the several interaction and geometrical parameters. We illustrate the general theory by studying a polymer chain confined between two parallel plates with three different (Neumann, Dirichlet, and periodic) boundary conditions to one-loop order. We show that ε expansions are well behaved as long as the radius of gyration of the chain is smaller than the interplate separation L. The finite size corrections to the full space (bulk) limit are found to be proportional to L−1 for free boundaries, while they are exponentially small for periodic boundary conditions. The presence of several lengths and/or interactions produces interesting crossovers, which we illustrate for a Gaussian polymer chain attached to the exterior surface of a repulsive sphere where full crossover scaling functions are obtained for the partition function and moments of the end-vector distribution function. A new exponent associated with the radius of the sphere is predicted on the basis of scaling arguments which are supported by the RG equations. This work provides the necessary input ingredients for extension of the theory to treat semidilute concentrations.

Renormalization-group analysis of weak-flow effects on dilute polymer solutions.

A Gaussian polymer chain in the presence of hydrodynamic interactions and subjected to steady linear flows is investigated renormalization-group theoretically. To order e(=4-d, d being the spatial dimensionality) and to the lowest nontrivial order in the flow strength, we consider the hydrodynamic effect on the mean-square end-to-end distance. From our general formula, we extract results for the physically interesting cases of shear and elongational flow. In the light of recent experimental results there is a possibility that the Gaussian model is of limited validity, even below the stretching transition.

Effect of weak flow on a polymer chain with hydrodynamic interactions

A gaussian polymer chain with hydrodynamic interactions and subjected to a steady linear flow is investigated renormalization- group theoretically. To order ϵ = 4 − d ( d being the spatial dimensionality) and to the lowest nontrivial order in the flow strength, we consider the hydrodynamic effect on the mean square end-to-end distance. Possible limitations of the gaussian model, even below the stretching transition, are discussed.

Hydrodynamic effect on the correlation functions of a gaussian polymer chain

Time-dependent correlation functions of the conformations of a single gaussian polymer chain immersed in a solvent are calculated renormalization-group theoretically. As a byproduct is is confirmed that to order ϵ (=4- d, d being the spatial dimensionality) the translational diffusion constant obtained fully dynamically (i.e. by the Green-Kubo formalism) is identical to that obtained by the Kirkwood theory (or the Kirkwood-Riseman formalism) even with the self-avoiding interaction.

Multiple scattering theory calculation of the concentration dependence of the tracer and cooperative friction coefficients for Gaussian polymer chains

The cooperative and tracer translational friction coefficients of Gaussian polymer chains are evaluated through order c in concentration using the full dynamic bead-to-bead space time correlation functions within the framework of a recent reformulation involving a discrete bead-type model of the multiple scattering theory for the concentration dependence of the viscoelastic properties of polymer solutions. It is shown that, using preaveraging approximation of the Oseen tensor, the order c cooperative translational friction coefficient differs from the tracer friction coefficient due to a higher order contribution in the multiple scattering expansion, one varying as the inverse seventh power of the separations between chains. The leading term for the cooperative translational friction coefficient of order c, equal to the tracer one, is calculated to be 0.62, about 10%–25% smaller than recent experimental results obtained in sedimentation experiments. The difference may be due, in part, to the higher terms in the multiple scattering expansion which turn out to be very difficult to compute even in a very rough approximation.

Kinetics of Diffusion-Controlled Reactions in Polymer Melts

It is shown that in the limit of continuously flexible, Gaussian polymer chains, the first-order rate for end-to-end reactions, ${k}_{\mathrm{cyc}}(t)$, scales as the fundamental reptation time, ${\ensuremath{\tau}}_{\mathrm{rep}}$. At short times $t<{\ensuremath{\tau}}_{\mathrm{rep}}$, ${k}_{\mathrm{cyc}}(t)\ensuremath{\rightarrow}(\frac{0.333}{{\ensuremath{\tau}}_{\mathrm{rep}}}){(\frac{t}{{\ensuremath{\tau}}_{\mathrm{rep}}})}^{\ensuremath{-}\frac{1}{4}}$, while at long times, ${k}_{\mathrm{cyc}}(\ensuremath{\infty})=\frac{0.363}{{\ensuremath{\tau}}_{\mathrm{rep}}}$. The analysis also gives the exact chain-length-dependent memory and reaction-rate functions in the discrete case of bead-rod chains obeying the earthworm equations of Doi and Edwards.

Small-angle neutron scattering from a concentrated CuMn spin glass

The authors report small-angle neutron scattering measurements of pre-percolation clusters in Cu-16.7 at.% Mn at temperatures between 5 and 220K. They draw on the results of Monte Carlo computer experiments which show that pre-percolation clusters are diffuse and ramified, and fit their data by analogy with light scattering from Gaussian polymer chains. The average radius of gyration of a cluster at low temperatures is about 26 AA. By considering the topology of random lattice animals, they estimate that the average cluster contains about 170 Mn atoms.

The effect of preaveraging of hydrodynamic interaction on the intrinsic viscosity and intrinsic flow birefringence of gaussian polymer chains

AbstractThe intrinsic viscosities of a Gaussian chain with and without hydrodynamic interaction preaveraging and with the identification of the centre of gravity of the Gaussian coil with the chain center were calculated using Tsuda's theory. It appears that the value of the Flory constant is more affected by approximations used in the derivation of Tsuda's theory than by preaveraging of hydrodynamic interaction. Svetlov's theory was used in the calculation of the ratio of intrinsic flow birefringence to intrinsic viscosity. For unpreaveraged hydrodynamic interaction this quantity is lower by some 15–20% than for a free‐draining coil.

Effects of polydispersity on critical concentration of polymer solutions

AbstractA semiempirical equation is developed to compute the unperturbed parameter from the critical concentration of polymer solutions derived from the viscometric and kinetic data. This equation gives satisfactory results for various vinyl polymers including poly(vinyl chloride), polystyrene and poly(methyl methacrylate) among others that follow the Schulz molecular weight distribution function. It is found that the segments of a Gaussian polymer chain are associated with an equal number of foreign segments near its center of mass, when the polymer solution has attained a uniform segment density at the critical concentration. The effect of molecular weight distribution on the present studies is significantly large that it merits an empirical treatment. Defects of the model is also discussed.

The Widths of a Polymer Chain

By solving appropriate diffusion equation we calculate the width of a Gaussian polymer chain in one direction and the radius of the circumscribing sphere, which is defined as the sphere around the center of mass which includes the whole chain and yet makes contact somewhere on the chain. Various quantities defining the size of a Gaussian chain are compared.

On the theory of stress relaxation by cross-link reorganization

Stress relaxation by scission and reformation of cross-links in a network of interpenetrable Gaussian polymer chains is discussed theoretically from two points of view. First, a model due to Flory, based on the intuitive use of results for a different elasticity problem, is reconsidered in the light of a new analysis of that problem by the present author, using Edwards’s elasticity theory in place of the ‘classical’ methods of Flory and others. The stress σ predicted by Flory’s model is found to be quite sensitive to which elasticity theory is used, but a comparison with experiments by Thomas is inconclusive. Secondly, the asymptotic relaxation is determined by a more reliable dynamical calculation. This predicts σα t -½, a slower decay than that given by Flory’s model (whichever elasticity theory is used), and demontrates the failure of that model at large times.