Real Polymer Chains Research Articles

The flexible and the wormlike range in a real polymer chain

From the statistics of unperturbed chains with interdependent rotations, a quadratic intramolecular potential of general validity may be derived on the basis of the Fourier configurational representation [G. Allegra, J. Chem. Phys. 68, 3600 (1978)]. Introducing a suitable observational cutoff it is proven that any real chain shows a different elastic behavior whether D is larger (flexible range) or smaller (wormlike range) than a suitable limit. In the case of polyethylene and atactic polystyrene, the latter range is observable for D smaller than ∼25 Å (and larger and ∼8 Å). The flexible range exists even for the ideal, infinitely long Porod–Kratky chain; its neglect leads to an inconsistency between the intramolecular potential and the mean-square chain length.

A new method for simulation of real chains: scanning future steps

A new method for simulating real polymer chains is developed and applied to self-avoiding walks (SAWs) of length 49-599 on a three-choice square lattice. Very good results for the entropy are obtained which deviate from the series expansion estimates by 0.1-2%. The author also discusses how to extend the method to models of polymer chains with both excluded volume and finite interactions (attractive or repulsive). The author's method is expected to be more efficient than other simulation methods for treating self-interacting SAWs and chains which are subject to various lattice constraints.

On the mean radius of gyration of a polymer chain

Calculates the intrachain bead-to-bead square distance of a real polymer chain, close to the critical dimensionality d=4. From this the authors obtain the mean-square radius of gyration (S2) of the polymer as a function of the molecular weight N of the chain and the excluded volume parameter mu . A proportionality relationship between (S2) and the mean square end-to-end distance (R2) of the coil, previously suggested by enumerations of self-avoiding walks on lattice, is shown to be true. An estimate of the universal ratio (S2)/(R2) is given.

Elasticity of a real polymer chain for different modes of motion

A theory is given to evaluate the elastic constant of the force acting on the general atom of an (–A–)N polymer chain for any normal mode of motion allowed by skeletal rotations; recourse to the classical description in terms of beads and springs should thus be rendered unnecessary. The results are obtained after evaluation of the configurational free energy stored in the chain by the action of a general set of constant forces; according to the usual assumptions, the average displacements produced by the forces are assumed not to be too large compared with the statistical fluctuations. An algorithm to calculate the general force constant on the basis of the classical configurational statistics of polymer chains is given, and numerical results are reported for high molecular weight polyethylene. All other parameters being the same, the short relaxation times in the vicinity of the limit of the classical bead–spring theory (i.e., a sinusoidal wave comprising about 20 methylene units within each wavelength) are an order of magnitude smaller than calculated according to that theory. The whole spectrum of both the force constants and of the relaxation times (neglecting at present the internal viscosity as well as the hydrodynamic interactions) down to the shortest wavelength comprising two skeletal atoms is given.

Real polymer chains: passage from lattice models to the continuum

The 'universal' model of Domb and Barrett (1976) is combined with Monte Carlo data of Smith and Fleming (see ibid., vol.8, p.938 (1975)) to clarify the correspondence between chains confined to a lattice, and chains in the continuum. Lattice walks are classified in terms of the parameter nu =(nearest neighbour distance)/(step length) which is analogous to the excluded volume ratio nu for chains in the continuum. Comparisons of results for on-lattice and off-lattice chains should be made only for equivalent values of these two parameters. A theoretical picture of a linear hard-sphere chain is constructed and a comparison is made with the Monte Carlo results.

Self-avoiding walks and real polymer chains

The model of Domb and Joyce enables a continuous transition to be effected between a random and self-avoiding walk on a lattice. By combining a virial expansion with exact enumerations for this model, it has been possible to derive numerical estimates of the expansion factor alpha 2=(RN2)/N for different values of N and w (the excluded volume parameter) for different three-dimensional lattices. The results have been used to test the two-parameter approximation, and the closed form expressions of Flory, Flory and Fisk, and Alexandrowicz and Kurata.

Flow birefringence of real polymer chains. Application to n-alkanes

Flow birefringence of real polymer chains. Application to n-alkanes

Analytical Treatment of Excluded Volume. I. Dimensions of Real Polymer Chains

Excluded-volume interactions in long chains are treated in terms of Gaussian probabilities for intersegmental contacts. Thus for a chain of n segments and end-to-end distance h the energy factor, exp[−βΣΣ lim k>lδ(hlk)],is evaluated by linearly expanding and averaging—sequentially for l=1, 2, ···, n—the interactions of segments l with those following after, k. Accordingly, for each l exp[—βΣkδ(hlk)] into exp[—βΣkfn—l(0lk | h)], where fn—l(0lk | h) is the probability, conditional to h, for a contact among segments l and k. The probability refers to a fictitious chain n in which only the segments from l+1 to n are still interacting (— the ``real section''). Thereafter it is assumed that the contact probabilities can be represented by the Gaussian expressions obtaining for random-flight chains, with the links representing the real section taken as expanded as the most probable end-to-end distance of a corresponding real chain—viz., a chain consisting of r=n—l−1 interacting segments. On this basis an integral equation is derived for the expansion coefficient αn=h*/h*0, h*, and h*0 being respectively the most probable value of the end-to-end distance of chain n, in the presence and absence of interactions. (The integral form of the equation results from the number of interacting segments in the fictitious chains varying from r=n to r=0.) The line obtained for αn2 vs n initially rises like Fixman's but, beyond αn2≈4, diverges from it, the asymptotic limit being αn2∝n0.236 (vs Fixman's ∝n0.33 and Flory's ∝n0.2). The results also show that the excluded-volume effect is essentially contributed by contacts among segment pairs separated by a vanishingly small fraction of the real chain section, (k—l) *«r*. It seems therefore that the distribution of the internal distances hlk can be treated in Gaussian terms, in spite of the distribution of the end-to-end distance in a real chain being non-Gaussian. On the same grounds it is concluded that a similar treatment of the 2-dimensional chain, leading to αn2∝n0.5, should fail, for in this case (k—l) *≈r*.

Three-Parameter Theory of Dilute Polymer Solutions

Three-parameter theories of the expansion factor α and the second virial coefficient A2 for chain polymers are presented to interpret primarily the experimental fact that the dimensionless ratio A2M/[η] is not always a universal function of α, where M is the polymer molecular weight and [η] is the intrinsic viscosity. There are two effects to be considered, which are related to a third new parameter: one is the effect of a ternary-cluster integral, and the other the effect of short-range interactions. First, the Orofino—Flory theory is revised with respect to inclusion of the ternary-cluster integral using the random-flight model, and it is shown that this effect is rather of minor importance, though not completely negligible. As for the second effect, a discrete nature of real polymer chains is considered primarily, and a simple analysis is made on suitable model chains. It is shown that the discrete-chain effect becomes significant when an effective bond of the chain is long compared to the range of average force between segments. Thus the most important third parameter is closely related to a degree of chain stiffness. With this parameter defined well, the theory agrees semiquantitatively with experiment for both flexible and stiff-chain polymers.

On the Distribution of Small Radii of Gyration of Linear Flexible Polymer Molecules

The Fourier transform of the distribution function characterizing the three orthogonal components of the radius of gyration of linear flexible polymer molecules can be obtained from the properties of random-flight chains. In this publication we present a detailed analysis of the problem of inverting the Fourier transform of the distribution of small radii of gyration and present approximate and asymptotic expressions for the distribution. We consider random-flight chains of t statistical links. If we measure a given orghogonal component of the radius of gyration Si by the dimensionless parameter ξ=π(12)−12Si/〈Si2〉12, we find: (1) In the domain of ξ between about (1/√3) and (3/t½), the distribution of ξ is adequately approximated by Wξ(ξ)=(π32)2−1ξ−3 exp (−π2/16ξ2). The expression is independent of t and therefore, as for larger radii, independent of how we subdivide a real polymer molecule into statistical segments. We also conclude, then, that for all radii such that ξ≥(3/t12), Wξ(ξ) is independent of the detailed structural features of real polymer chains. (2) In the domain of ξ⩽(1/102t12) the distribution of ξ can be approximated by Wξ(ξ)=[2t!/(12t−1)!]ξt−1. The configurational statistics of such highly compressed coils is dependent on the number of statistical chain elements and thus on how we would choose to subdivide a real polymer molecule into such elements. We can also interpret these results to imply that the configurational statistics of such highly compressed real polymer molecules would be dependent upon the detailed structural features of the chain.

Photoelastic Property of Cross-Linked Amorphous Polyethylene

By assuming the additivity of bond polarizabilities, a formal expression is derived for the difference Δγ in principal polarizabilities of a real polymer chain having constant end-to-end distance r. The expression is given as a series in powers of r2, whose coefficients are functions of the differences α1κ—α2κ in principal polarizabilities of constituent bonds, 〈r2k〉, and 〈r2k cos2Φiκ〉, (k≥1). Here, Φiκ is the angle between the end-to-end vector and the unit vector along the κth bond of the ith structural unit, and the averages refer to those about a polymer chain in free state. It is shown that if r2 is in magnitude of the same order as 〈r2〉, at the limit of infinite chain length the expression reduces to Δγ=310〈r2〉*−2 ∑ i,κ(α1κ−α2κ)(3〈r2cosΦiκ〉−〈r2〉)r2,where 〈r2〉* is such a part of 〈r2〉 as proportional to chain length. The difference in principal polarizabilities of the ``equivalent random link'' is shown to be ΔΓ=12〈r2〉*−1 ∑ i,κ(α1κ−α2κ)(3〈r2cos2Φiκ〉−〈r2〉).An expression in a matrix form is derived for ΔΓ of the polyethylene chain. Numerical computations are made by a digital computer, using two geometrical models for this molecule recently proposed by Hoeve and by Nagai and Ishikawa, and also using Denbigh's and Bunn and Daubeny's values for bond polarizabilities. It is found that a good agreement is obtained between predictions of the theory and experimental values if Denbigh's values are used.

Non-Gaussian Character of Real Polymer Chains

It is shown that the distribution function of the end-to-end vector of a polymer chain at its free state as well as the partition function of the polymer chain subjected to an external force, may be expressed in terms of even moments of the end-to-end distance of the chain at its free state. In looking over these functions it is seen that the quantities g2k and g2k' (k≥2) in the text, representing the non-Gaussian character of a polymer chain, are related in definite ways with even moments. As a main contribution of this paper, a mathematical method is presented for calculating the even moments just described and in particular, the fourth moment 〈R4〉, by taking into account the interdependent internal rotation. For the sake of simplicity, polyethylene is employed to illustrate the method. The expression for 〈R4〉 is obtained as a simple function of the (1, 1) elements of several matrices. The N (N is the number of skeletal bonds) dependence of g2k and g2k', and thereby also that of every term in a series expansion of the distribution function, are discussed in full detail.

The Effects of Interactions in Real Polymer Chains

The effect of interactions between the segments of a polymer chain on the size of the molecule has been calculated. Two approaches have been used. The first is a direct probability calculation using an exponential interaction energy between segments. The second follows the original approach of Flory but solves the problem to a different approximation. Both methods lead to essential agreement with Flory's result for small and moderate interactions. A probable reason for the spurious results found by others is proposed.

The Excluded Volume of Polymer Chains

The ``random flight'' distribution function for the relative positions of two segments of a linear polymer chain is based on the assumption that the space occupied by a single polymer segment is relatively small and may be neglected. In this paper, the space requirements of a real polymer chain are taken into account. A generalized Fokker-Planck equation is formulated, and the distribution function, the average square distance, and the average inverse distance between two chain segments are calculated.