Ideal Polymer Chain Research Articles

Depletion interaction between spheres immersed in a solution of ideal polymer chains

The depletion interaction between two spheres due to nonadsorbing ideal polymers is calculated from the polymer concentration profile using the excess (negative) adsorption. Computer simulations show that the polymer concentration profiles around two spheres are well described by the product function of the concentration profile around a single sphere. From the interaction potential between two spheres the second osmotic virial coefficient, B2, is calculated for various polymer-colloid size ratios. We find that when the polymers become smaller than the spheres, B2 remains positive in the dilute regime. This shows that the depletion interaction is ineffective for relatively small spheres.

A Polymer End-Tethered to a Potential Stripe: A Simple Example of an Escape Transition

We study the problem of a single ideal polymer chain tethered to a surface at the midpoint of a repulsive potential stripe. If the potential is very weak, the chain remains unperturbed. However, as the potential is increased, the chain conformation undergoes a sudden change. The chain forms a tether to the edge of the stripe and moves most of the monomers off to the region of lower potential. This is a simple example of an escape transition previously discussed for compression of polymer chains. We show how these two systems are analogous and clear up some controversy regarding the exact form of the force versus height curve for the compressive system.

Adsorption of polymer chains by disordered traps

We study the adsorption cross-over of ideal polymer chains in an environment of disordered traps. Starting from the assumption of an optimal cluster size of traps (optimal fluctuation method) we derive a general scaling form of the free energy function for arbitrary spatial dimensions. For small concentrations of traps we find a cross-over from localized (adsorbed) behavior to delocalized behavior depending on the chain's length and on the depth of the traps; this is connected with the non-monotonic behavior of the chain's extension. In terms of the free energy of the chain this cross-over resembles a first order transition scenario, the chain gets localized at many traps at once.

Adsorption of polydisperse polymer chains

The adsorption of polydisperse ideal polymer chains is shown to be sensitive to the large N tail of the distribution of chains. If and only if the number of chains decays more slowly than exponentially is there an adsorption transition like that of monodisperse infinite chains. If the number decays exponentially the adsorption density diverges continuously at a temperature which is a function of the mean chain length. At low coverages, chains with repulsive monomer–monomer interactions show the same qualitative behavior.

Polymer stretching in an elongational flow

Using a constant “velocity gradient” ensemble approach, the average scalar end-to-end separation is calculated as a function of the gradient for an ideal Gaussian polymer chain experiencing longitudinal elongational flow under steady-state conditions. The resulting equation, based on a dumbbell model, exhibits an initial average end-to-end separation equal to the unperturbed random coil value; the separation increases monotonically with increasing velocity gradient, in agreement with recent experimental measurements and with a classical treatment based on a diffusion equation. This approach is contrasted with one based on Hooke’s law where the end separation remains zero with increasing gradient until a critical value of the gradient is reached at which point the chain suddenly expands to an appreciable fraction of its fully extended length.

Polymer adsorption on heterogeneous surfaces

The adsorption of a single ideal polymer chain on energetically heterogeneous and rough surfaces is investigated using a variational procedure introduced by Garel and Orland (Phys. Rev. B 55, 226 (1997)). The mean polymer size is calculated perpendicular and parallel to the surface and is compared to the Gaussian conformation and to the results for polymers at flat and energetically homogeneous surfaces. The disorder-induced enhancement of adsorption is confirmed and is shown to be much more significant for a heterogeneous interaction strength than for spatial roughness. This difference also applies to the localization transition, where the polymer size becomes independent of the chain length. The localization criterion can be quantified, depending on an effective interaction strength and the length of the polymer chain.

ADSORPTION OF IDEAL POLYMERS ON AN INFINITELY RAMIFIED FRACTAL

We study ideal polymer chains interacting attractively with the borders of the lacunas of an infinitely ramified fractal, the Sierpinski carpet. Ideal chains are simulated on finite stages of construction of this fractal at various temperatures. The mean-square displacement and the mean number of adsorbed monomers of N-step chains are estimated in these lattices, and extrapolations to the fractal limit (infinite lattice) consider the exact forms of finite-size corrections as previously predicted by the series expansion method. In the noninteracting case, a finite fraction of the monomers is adsorbed, and this fraction increases as the temperature decreases. However, there is evidence that the critical exponent v which governs the growth of the chains varies with the temperature in a nonmonotonic way. At high temperatures v increases with decreasing temperature, and thus the chains are more stretched than in the noninteracting case. At an intermediate temperature, v starts to decrease and is still positive at very low temperatures, when the chains grow along the borders of several lacunas, occasionally crossing the bulk between them.

Continuum percolation of the four-bonding-site associating fluids

Wertheim’s integral equation theory for associating fluids is reformulated for the study of the connectedness properties of associating hard spheres with four bonding sites. The association interaction is described as a square-well saturable attraction between these sites. The connectedness version of the Ornstein-Zernike (OZ) integral equation is supplemented by the PY-like closure relation and solved analytically within an ideal network approximation in which the network is represented as resulting from the crossing of ideal polymer chains. The pair connectedness functions and the mean cluster size are calculated and discussed. The condition for the percolation transition and the analytical form of the percolation threshold are derived. The connection of the percolation with the gas-liquid phase transition is discussed.

Langevin dynamics of a polymer with internal distance constraints

We present a rigorous approach to the Langevin dynamics of ideal polymer chains subject to internal distance constraints. The permanent constraints are modeled by harmonic potentials in the limit when the strength of the potential approaches infinity (hard cross-links). The cross-links are assumed to exist between arbitrary pairs of monomers. Formally exact expressions for the resolvent and spectral density matrix of the system are derived. To illustrate the method we study the diffusional behavior of monomers in the vicinity of a single cross-link within the framework of the Rouse model. The same problem has been studied previously by Warner [J. Phys. C 14, 4985 (1981)] on the basis of Lagrangian multipliers. Here we derive the full, hence exact, solution to the problem.

Integral equation theory for the four bonding sites model of I. Structure factor and compressibility associating fluids

The Wertheim integral equation theory for associating fluids is reformulated for the study of associating hard spheres with four bonding sites. The association interaction is described as a square well saturable attraction between these sites. The associative version of the OrnsteinZernike integral equation is supplemented by the Percus-Yevick-like closure relation and solved analytically within an ideal network approximation, in which the network is the result of the crossing of ideal polymer chains. The structure factor S(k) is obtained for both symmetrical network and polymer chain cases. It is shown that S(k) exhibits a peculiarity (a socalled pre-peak) at small wavenumbers, connected with the formation of relatively large molecular aggregates due to highly directional saturable bonds. The magnitude and location of the pre-peak as functions of density theta and association Ks are analysed. Based on the analysis of the S(k = 0) limit, the behaviour of the isothermal compressibility χ T is studied and the gas–liquid critical point is predicted to exist. The result for the spinodal curve is also reported.

Polymers in Curved Boxes

We apply results derived in other contexts for the spectrum of the Laplace operator in curved geometries to the study of an ideal polymer chain confined to a spherical annulus in arbitrary space dimension D and conclude that the free energy compared to its value for an uncurved box of the same thickness and volume, is lower when $D < 3$, stays the same when $D = 3$, and is higher when \mbox{$D > 3$}. Thus confining an ideal polymer chain to a cylindrical shell, lowers the effective bending elasticity of the walls, and might induce spontaneous symmetry breaking, i.e. bending. (Actually, the above mentioned results show that {\em {any}} shell in $D = 3$ induces this effect, except for a spherical shell). We compute the contribution of this effect to the bending rigidities in the Helfrich free energy expression.

Polymer chains tethered to impenetrable interfaces: Broadening of relaxation spectra

We investigate the dynamics of an ideal polymer chain which is tethered with one of its ends to an impenetrable interface. Due to the geometric restrictions the relaxation of modes perpendicular to the interface differs significantly from that of modes parallel to it. A superposition of both components leads to a marked broadening of the relaxation spectrum, which may find its signature in dielectric measurements. We obtain the relevant dynamical quantities for a polymer chain subject to geometric constraints by use of the Smoluchowski equation combined with a Rayleigh-Ritz variational scheme.

Electronic structure of pyridine-based polymers

We present the results of semiemiprical quantum chemical calculations on oligomers of poly(p-pyridyl vinylene) (PPyV) and poly(p-pyridine) (PPy). The presence of a nitrogen heteroatom in the conjugated backbone of these polymers presents a potentially severe breaking of both spatial and charge-conjugation symmetry (CCS), and the addition of nonbonding (n) orbitals has potentially major effects on the photophysics of these systems. Geometries are optimized at the PM3 Hartree–Fock level for neutral, singly charged and doubly charged oligomers. We find that the geometric distortions associated with polaron formation are centered on the vinylene linkages in PPyV-based systems and on the interring bonds in the PPy-based systems. We discuss the electronic structure at the PM3 level applying configuration interaction between singly excited states (SCI), and we demonstrate that the lowest-lying (n→π*) states of the ideal polymer chain are well above the lowest (π→π*) states, leading to strong fluorescence in these systems. Nonplanarity, however, leads to substantial mixing of the (π→π*) and (n→π*) manifolds, thereby altering this conclusion. We calculate absorption spectra for neutral, singly charged (polaron), doubly charged (bipolaron), and triplet-state oligomers using the intermediate neglect of differential overlap/single-excitation configuration interaction (INDO/SCI) technique. For PPyV, comparison of oligomers with differing spatial symmetry allows the isolation of the effects of CCS breaking. All calculated spectra are in good agreement with experimental results and indicate that the symmetry breaking due to the nitrogen heteroatom is weak. In particular, the polaron induces a two-peak in-gap feature into the absorption spectrum and the bipolaron a single-peak feature, as is seen in the analogous all-hydrocarbon polymers.

Distribution Function P(S) of Uniform Star Polymers

In this paper, the probability density distribution function P(S) of the radius of gyration of the unperturbed star polymer chains is investigated using Monte Carlo method. We find that the distribution function P(S) for ideal star-shaped polymer may also expressed in the form of P(S)− −S2aexp(−(a+0.5)S2/〈S2〉), and the parameter a only depends on the number of branches f, which is similar to the Flory-Fisk expression P(S) for ideal linear polymer chain.

Adsorption in models of ideal polymer chains on fractal spaces.

We study critical adsorption in models of ideal polymer chains situated on fractal spaces in the vicinity of an impenetrable surface. The obtained exact results on fractal lattices, with a coordination number that can vary from site to site of the lattice, reveal a critical behavior that might be quite different from that established for lattices with the same coordination. Specifically, in the cases where localization of the chain takes place, i.e., when the mean end-to-end distance of the chain grows more slowly than any power of its length N, we found that various generating functions of interest usually display multiplicative singular corrections to the leading power law singularities (confluent logarithmic singularities, for example). We have demonstrated with specific examples that the average fraction of steps of the chain on adsorbing surface, at critical adsorption point, vanishes according to the asymptotic law \ensuremath{\sim}${\mathrm{ln}}_{1}^{\mathrm{\ensuremath{\psi}}}$N (where ${\mathrm{\ensuremath{\psi}}}_{1}$0 is a given constant) or \ensuremath{\sim}exp(-c ${\mathrm{ln}}_{2}^{\mathrm{\ensuremath{\psi}}}$N) (where c and ${\mathrm{\ensuremath{\psi}}}_{2}$ are certain positive constants). \textcopyright{} 1996 The American Physical Society.

Direct energy transfer in solutions of ideal polymer chains

We study the direct incoherent energy transfer in systems, in which the molecules involved in the transfer process are attached to ideal (Gaussian) polymers, randomly arranged in a viscous solvent. In what the molecules’ attachment to polymers is concerned we consider two different situations: (a) each polymer chain contains at its opposite ends a donor and an acceptor molecule, (b) the donor molecules are dispersed randomly in the solvent and all acceptors are attached to polymers. In both cases we derive the donors’ decay forms exactly.

Polymer adsorption between a rough surface and a flat surface for an ideal long chain

AbstractWe have studied the adsorption of an ideal long polymer chain between a rough surface and a flat surface. The two surfaces are basically parallel to each other and penetrable for the polymer chain. The rough surface is modeled as a sinusoidal grating and the chain is assumed to be long enough so at the ground state dominance is tenable. By solving the diffusion equation for a polymer chain in an external field, we have investigated the related eigenvalue problem and obtained the ground state energy (the lowest confinement energy) of the system. The monomer density distributions on the two surfaces are obtained and their relations to the four relevant characteristic lengths: the separation between the two surfaces h, the confinement thickness D, the surface roughness amplitude z0 and the grating wavelength λ, are discussed.

The elasticity of ideal polymer chains

The systems considered here are composed of noninteracting ideal chains, formed from many identical beads, each of which is bound to its two nearest neighbors with an arbitrary potential of interaction. The thermodynamic properties of these systems are examined for several bond potentials. Comparisons then are made between the properties of the model systems and the experimentally observed behaviors of rubbery materials. Connections also are established between the thermodynamic and elastic properties of these model systems and those given by the traditional theories of polymeric elasticity.

Statistics of stiff polymer chains near an adsorbing surface

AbstractThe exact solution of the problem of adsorption of a long ideal polymer chain with variable degree of stiffness on a plane surface is presented. It is shown that the adsorption of stiff polymer chains is a second‐order phase transition; in the adsorbed state “train” (i.e. adsorbed) sections are relatively longer and loop sections relatively shorter than for flexible chains. This effect is very pronounced: already for moderately stiff chains the number of Kuhn segment lengths in one “train” section at the temperature T = Tcr/2 (Tcr is the critical temperature for adsorption transition) can reach several thousands, and deviation from the surface occurs only in the form of small “hairpins”. The maximum length of the chain, which at the given conditions would flatten completely on the surface, is estimated.

Statistics of an ideal polymer chain near the bifurcation region of a narrow tube

Conformational statistics of an ideal polymer chain in the vicinity of the bifurcation region of a narrow pore system is considered. It is shown that the bifurcation region plays the role of an effective entropic trap. Due to this fact the chain adopts globular conformation with the average dimensions of the order of the dimensions of the bifurcation region. It is concluded that the statistical properties of an ideal polymer chain and of random walks near the bifurcation point are essentially different.