Conformation Space Renormalization Group Research Articles

Structure and flexibility of worm-like micelles

Small-angle neutron scattering and static light scattering experiments have been performed on worm-like micelles formed by soybean lecithin and trace amounts of water in deuterated iso-octane. The structure and flexibility of the aggregates have been investigated as a function of solution composition. The data analysis comprises an application from results of conformation space renormalization group theory and a non-linear least-squares fitting procedure based upon a recently developed numerical expression for the scattering function of a worm-like chain with excluded volume effects.

Elastic scattering from diblock copolymer chains in dilute solution

The elastic scattering structure factor of a single diblock copolymer chain is calculated up to first order in ε(=4−d, with d the dimensionality of the space) with the conformational space renormalization group method. The influence of several thermodynamic conditions on the scattering intensity is analyzed. Our results are compared with small-angle neutron scattering experiments under variable contrast conditions and numerical simulations of the zero average contrast case.

Enormous Concentration-Induced Growth of Polymer-like Micelles

We report a detailed analysis of the effect of concentration on the micellar size, the static structure factor at q → 0, S(0), and the static correlation length ξs measured with static light scattering in aqueous solutions of the nonionic surfactant hexaethylene glycol n-hexadecyl monoether (C16E6) which forms giant polymer-like micelles in water. Measurements were performed at two different temperatures in order to test for possible contributions from critical scattering which could become important in the vicinity of the critical point. We find no measurable temperature dependence for the apparent molecular weight and the correlation length of the micelles for the given range of concentrations and temperatures. We demonstrate that the concentration dependence of the micellar size distribution and intermicellar interaction effects can be combined in a self-consistent way using results from conformation space renormalization group theory originally developed for semidilute polymer solutions. Our results indicate a much stronger micellar growth than that previously predicted by the theoretical models for micelle formation.

Ternary Interactions in Dynamics of ?-State Polymers: A Conformation-Space Renormalization-Group Approach

We reconsider the t'Hooft-Veltman type renormalization treatment of polymer properties near the theta-point. We correct other recent studies, and now demonstrate that the equivalence between this renormalization scheme and a zero-component field theory or a direct renormalization method for polymer chains works perfectly. We also give an estimate for the magnitude of the renormalized third virial coefficient using a new result for the translational diffusion constant at the theta-point.

The static and dynamic structure factor of polymer-like lecithin reverse micelles

We report a detailed analysis of the static and dynamic structure factor of polymer-like lecithin reverse micelles. We previously demonstrated that one can directly apply the results from conformation space renormalization group theory for semi-dilute polymer solutions to «equilibrium polymers» such as worm-like micelles or microemulsions, if one takes into account the fact that their size distribution is concentration dependent [1]. The present study confirms these findings and shows that this approach can be extended to the Q- and c-dependence of the static (S(Q)) and dynamic (S(Q, t)) structure factor

On the application of polymer renormalization group theory to polymer-like microemulsions

We report on the application of results from conformation space renormalization group theory for semi-dilute polymer solutions to “equilibrium polymers” such as worm-like micelles or microemulsions. We have tested this approach with data from an extensive static light scattering study of lecithin water-in-oil (w/o) microemulsions. We demonstrate that one can construct a universal curve $(M_{\rm w}/RT)(\partial\Pi/\partial c)$ versus a reduced concentration X∼c/c*, where c* is the overlap concentration, from the experimentally determined values of the osmotic compressibility if a power law of the form Mw∼cα for the c-dependence of the average particle size is assumed. We show that one can thus obtain information from static light scattering experiments on the concentration dependence of both the micellar size distribution and the intermicellar interaction effects, i.e., on the weight average molecular weight Mw(c) and the static structure factor S(0, c).

Osmotic pressure of linear, star, and ring polymers in semidilute solution. A comparison between experiment and theory

Osmotic moduli from styrene, isoprene, and butadiene polymers of different architectures are compared with new conformational space renormalization group (RG) theory computations that cover a range in concentrations from dilute through semidilute solutions. Three-arm, 12-arm, and ring polystyrenes (PS) are found not to scale with the same power law as linear chains. Theory yields the same power law behavior for linear, star, and ring polymers, but even the linear PS chains exhibit a higher exponent than predicted by RG theory or scaling. Dimensionless osmotic moduli for linear polydiene chains, such as polyisoprene (PI) and polybutadiene (PB), are found to be lower than osmotic moduli of PS, thereby raising possible questions of nonuniversality

Static scattering function for a regular star-branched polymer

The static scattering function for a regular star-branched polymer is calculated in the full excluded-volume limit with renormalization group techniques, and a closed-form expression for the scattering intensity Z(k) is given. The influence of the excluded-volume interaction between monomers becomes more noticeable as the functionality f of the star increases, qualitative differences appearing between Z(k) and the Gaussian liiit I&). A simplified formula for Z(k) is provided, which can be used for the treatment of experimental data. The scattering of one labeled arm of the star is also investigated and the radius of gyration of the arm is calculated, giving a quantitative expression for the stretching of the branches. I. Introduction Star polymers are receiving nowadays considerable attention in the literature from both e~perimental'-~ and theoreticalG13 points of view. The star-shaped polymers are the simplest branched structures in nature, and their study appears as the first step to the understanding of more complex systems such as gels and rubber. The topological constraint imposed by the presence of a seed or center molecule changes considerably the configura- tional and dynamical properties of the star polymers from a system of linear chains of the same molecular weight. The theoretical treatment pioneered by Zimm and Stockmayer14 on the random-walk model of stars off branches was followed more recently by computer simulationsell and numerical12 and analytical develop- mental3 with the purpose of incorporating in a realistic way the effect of the excluded-volume interactions between monomers in good solvents. The conformational space renormalization group (RG) technique, introduced by Oono and FreedI5 to study linear polymer chains, has been applied to starlike polymers by Miyake and Freed.13 They evaluated the distribution function for intersegment distance vectors and several averages in order to determine the configurational properties of uniform star polymers, in the asymptotic N- m limit, where Nis the total length of the chain. Their results show some departure from the scaling blob theorye but agree with Monte Carlo simula- tions8 for a low (f I 6) degree of branching. Here we concentrate on the calculation of the scattering form factor of f-branched star polymers with the confor- mational space renormalization group method. It is related of course to the segmental density distribution function around the center of the star and is directly measured in standard elastic scattering experiments. The whole range of transfer momenta gives information about concentration fluctuations at different scales within the polymer coil. Moreover, some authors215 extract the radius of gyration of the stars in good solvents from a fit of the scattering intensity with the static structure factor of regular stars with Gaussian statistics.16 Hence, we explore in this paper the predictions of the RG theory about the full excluded- volume regime of the scattering function. We recover then some of the results obtained in ref 13 by another route, and we discuss the influence of the number of branches

Renormalization group theory of the Rouse–Zimm model of polymer dynamics to second order in ε. II. Dynamic intrinsic viscosity of Gaussian chains

The dynamical intrinsic viscosity [η(ω)] of Gaussian chains is studied within the Rouse–Zimm bead-spring model of polymer dynamics using conformational space renormalization group methods. Calculations are performed through order ε2 (ε=4−d, where d is the spatial dimensionality) both with and without the preaveraging approximation. We find an 8% preaveraging error in the steady state limit. Preaveraging corrections to [η(ω)] involve couplings between all Rouse modes, leading to more complicated expressions for the Rouse mode relaxation rates and to weight factors for mode contributions to [η(ω)].

Excluded volume effects for polymers in presence of interacting surfaces: Chain conformation renormalization group

The chain conformational space renormalization group method is extended to consider excluded volume effects in polymer chains interacting with surfaces. The general theory is illustrated primarily by considering a system with a single impenetrable flat interface. The presence of boundaries, while breaking the translational invariance of the full-space theory, introduces a number of novel theoretical features into the renormalization group treatment. A parameter δ is introduced to describe the strength of the polymer chain–surface interaction, and previous expansions in powers of δ or δ−1 are not required. We evaluate several moments of the end-vector distribution such as 〈zn〉, 〈‖ρ‖2〉, etc. to first order in the excluded volume. Our work differs essentially from previous studies because the full dependence on the polymer–surface interaction parameter δ is retained to all orders, the crossover dependence on excluded volume is incorporated and the generalized crossover (i.e., excluded volume dependent) exponents are corrected through second order. Previous results, such as power law exponents, scaling forms for distributions, end-vector distributions in the absence of excluded volume, etc., are obtained simply as particular limiting cases upon the values of the excluded volume and the interaction parameter δ.

Theta point (‘‘tricritical’’) region behavior for a polymer chain: Transition to collapse

The conformational space renormalization group method is generalized further to describe excluded volume effects in finite molecular weight polymers in the theta point region where contributions from effective three-body interactions become appreciable. The theory builds upon our previous description of the good solvent region and uses t’Hooft-Veltman style dimensional regularization along with the renormalization group (RG) to determine the general analytic structure of measurable quantitites of interest. Our formalism is compared in detail with that employed in field theory to describe tricritical behavior. Although many results are in qualitative agreement between the two approaches, there are some numerical differences. Our method considers renormalization for chains with fixed length, and this introduces differences into the renormalization scheme from that used in field theory, differences dictated by the physical differences in the types of systems considered. Our treatment of finite length chains contrasts with the field theoretical expansions about the unphysical limit of infinitely long polymers; thus it entirely avoids the use of the method of insertions. The RG theory is used to calculate the mean square end-to-end distance 〈R2〉 and the second and third virial coefficients A2 and A3 in the theta region as a function of both the two- and three-body interactions and of the chain length. We compute the temperature difference between the point at which A2 vanishes and at which 〈R2〉 is the pure Gaussian limiting value. The virial coefficients are used to evaluate the dilute solution portion of the coexistence curve, providing an estimate of the collapse transition temperature. The presence of essential logarithmic dependences of the coexistence curve on chain length implies that mean field theory is not valid for describing measurable quantities in this region.

Concentration and excluded volume dependence of coherent scattering functions for polymers: Chain conformational space renormalization group

The concentration dependent chain conformational space renormalization group theory is extended to a consideration of the explicit dependence on the strength of the excluded volume interaction in the nonscaling, ‘‘cross-over,’’ region between the single chain Gaussian and self-avoiding limits. Generalized ‘‘scaling laws’’ in this cross-over domain are derived without the use of perturbation theory or ε expansions. In the dilute and semidilute region these generalized scaling laws have distances scaled by the infinite dilution radius of gyration RG(ζ) where ζ is a scaling variable which vanishes in the Gaussian chain limit and tends to infinity in the good solvent limit. Hence, concentrations c are scaled by the overlap concentration c*ζ evaluated from the cross-over RG(ζ). Quantities like the coherent scattering function are then functions of kRG(ζ), c/c*ζ, and ζ with k the momentum transfer vector. The explicit ζ dependence persists through the cross-over regime for all c and is eliminated in the simple scaling limits ζ→0 or ∞. We use the theory to evaluate the coherent scattering functions for a single tagged polymer chain, for a set of tagged chains at concentration cT<c, as well as the full polymer solution. The concentration and excluded volume dependences of a tagged chain radius of gyration and the full solution correlation length are evaluated, and the former is studied as a function of the ratio of the tagged chain molecular weight to that for an average solution chain.

Excluded volume effects in polymers attached to surfaces: Chain conformational renormalization group

The chain conformational space renormalization group method is extended to treat polymers with one or both ends attached to a wall. The theory is illustrated for the case of reflecting boundary conditions at the wall, but the methods for treating general boundary conditions are outlined. The fixed end-vector partition functions are evaluated both in the asymptotic scaling limit and in the cross-over regime where there is an explicit dependence on the strength of the excluded volume interaction. The mean square chain sizes transverse 〈z2〉 and parallel 〈‖ρ‖2〉 to the surface are evaluated separately to illustrate the use of the theory for the direct computation of these moments (including the prefactors) and the exponents. The calculations in these cases are far simpler than those required to determine the distribution functions. The prefactors display 〈‖ρ‖2〉 to be more spread out than 〈z2〉 which, in turn, as expected, is more spread out than 〈R2〉 for a chain in free space.

Excluded volume in star polymers: chain conformation space renormalization group

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTExcluded volume in star polymers: chain conformation space renormalization groupAkira Miyake and Karl F. FreedCite this: Macromolecules 1983, 16, 7, 1228–1241Publication Date (Print):July 1, 1983Publication History Published online1 May 2002Published inissue 1 July 1983https://doi.org/10.1021/ma00241a035RIGHTS & PERMISSIONSArticle Views255Altmetric-Citations93LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (1 MB) Get e-Alerts Get e-Alerts

Renormalization group treatment of polymer excluded volume by t’Hooft–Veltman-type dimensional regularization

The chain conformation space renormalization group method is transformed into a representation where the t’Hooft–Veltman method of dimensional regularization can directly be applied to problems involving polymer excluded volume. This t’Hooft–Veltman-type representation enables a comparison to be made with other direct renormalization methods for polymer excluded volume. In contrast to the latter, the current method and the chain conformation one from which it is derived are not restricted to the asymptotic limit of very long chains and do not require the cumbersome use of insertions to calculate the relevant exponents. Furthermore, the theory emerges directly in polymer language from the traditional excluded volume perturbation expansion which provides the correct weight factors for the diagrams. Special attention is paid to the general diagrammatic structure of the theory and to the renormalization prescription in order that this prescription follows from considerations on measurable quantities. The theory is illustrated by calculation of the mean square end-to-end distance and second virial coefficient to second order including the full crossover dependence on the renormalized strength of the excluded volume interaction and on the chain length. A subsequent paper provides the generalization of the theory to the treatment of excluded volume effects in polyelectrolytes.

Crossover between Gaussian and self-avoiding limits via finite order self-avoiding walk: Conformation space renormalization for polymers. VII

The crossover between the random walk and the self-avoiding walk via finite order self-avoiding random walk (FSAW) is studied analytically with the aid of conformation space renormalization group theory. Explicit expressions for the end-vector distribution function and 〈R2〉 are given up to order ε=4−d (d the spatial dimension). Since the excluded volume parameter for FSAW is contour length dependent, it is very awkward to study the present crossover behavior by using the polymer–magnet analogy. In contrast, our calculations in conformation space are simple and transparent, showing the power of this RG approach. The crossover behavior along FSAW is compared with the crossover obtained when the magnitude of the excluded volume interaction is decreased. The crossover via FSAW may occur in situations where a very long single chain is immersed in a solution of (shorter) chains and the concentration of the (shorter) chains is increased.

Renormalization group description of polymer excluded volume

A description is provided of the chain conformation space renormalization group approach to the treatment of polymer excluded volume. This method is transformed into one of identical form to the t'Hooft-Veltman renormalization method of field theory, thereby enabling comparison with other methods. A summary is provided of recent new results obtained by this technique for the full second-order dependence of the mean square end-to-end distance 〈R 2〉 on chain length and excluded volume as well as a calculation of 〈R 2〉 for a polyelectrolyte chain with added salt.

Conformation space renormalization theory of semidilute polymer solutions

A transparent conformation space renormalization group formalism is given which can incorporate arbitrary molecular weight distributions to the theory of semi-dilute polymer solutions. The closed forms for the osmotic pressure π and the mean square end-to-end distance of a single test chain are given to order ϵ=4− d, d being the spatial dimensionality, as functionals of the molecular weight distribution function. The scaling form of π is compared to the recent experimental result by Noda et al. A distribution sensitive universal plot is also proposed.

Conformation space renormalization of polymers. I. Single chain equilibrium properties using Wilson-type renormalization

A conformation space renormalization group is developed to describe polymer excluded volume in single polymer chains. The theory proceeds in ordinary space in terms of position variables and the contour variable along the chain, and it considers polymers of fixed chain length. The theory is motivated along two lines. The first presents the renormalization group transformation as the means for extracting the macroscopic long wavelength quantities from the theory. An alternative viewpoint shows how the renormalization group transformation follows as a natural consequence of an attempt to correctly treat the presence of a cut-off length scale. It is demonstrated that the current configuration space renormalization method has a one-to-one correspondence with the Wilson–Fisher field theory formulation, so our method is valid to all orders in ε = 4−d where d is the spatial dimensionality. This stands in contrast to previous attempts at a configuration space renormalization approach which are limited to first order in ε because they arbitrarily assign monomers to renormalized ’’blobs.’’ In the current theory the real space chain conformations dictate the coarse graining transformation. The calculations are presented to lowest order in ε to enable the development of techniques necessary for the treatment of dynamics in Part II. The theory is presented both in terms of the simple delta function interaction as well as using realistic-type interaction potentials. This illustrates the renormalization of the interactions, the emergence of renormalized many-body interactions, and the complexity of the theta point.