A. M. Nemirovsky

Lattice models of polymer solutions: Monomers occupying several lattice sites

An exact field theory is presented to describe a system of self‐avoiding lattice polymer chains with arbitrary regularly branched architecture. Equivalently, the chains can be viewed as linear and as composed of structural units (monomers) having a chosen shape and size and therefore each occupying more than one lattice site. The mean field approximation coincides with Flory’s theory, and it does not distinguish among chain geometries. However, we develop a systematic expansion for corrections to mean field approximation in powers of z−1 where z is the lattice coordination number. The entropy per site, the pressure and the chain insertion probability are computed for various chain architectures to O(z−2). At equal lattice site coverages per chain and total polymer volume fraction, the more compact the polymer chain geometry the higher is the insertion probability.

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 e 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 inter...

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) expon...

Marriage of exact enumeration and 1/d expansion methods : lattice model of dilute polymers

We consider the properties of a self-avoiding polymer chain with nearestneighbor contact energyɛ on ad-dimensional hypercubic lattice. General theoretical arguments enable us to prescribe the exact analytic form of then-segment chain partition functionCn,and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind=2–6. This exact form provides the main ingredient to produce a large-n expansion ind−1of the chain free energy through fifth order with the full dependence on the contact energy retained. The ɛ-dependent chain connectivity constant and free energy amplitude are evaluated within thed−1expansion toO(d−5). Our general formulation includes for the first time self-avoiding walks, neighboravoiding walks, theta, and collapsed chains as particular limiting cases.

Interaction of a polymer chain with an asymmetric liquid–liquid interface

A continuum model is presented for polymer chains near an asymmetric (A–B) liquid–liquid interface where each side of the interface can have different polymer–surface interactions. For example, one side can attract the macromolecule, while the other can repel it. The model contains different monomer free energies and different excluded volume interactions for the macromolecule in the two solvents. The model is solved exactly in the ideal limit where excluded volume vanishes in order to illustrate qualitatively the wide range of possible behavior. We evaluate the fixed end‐vector distribution, some moments of this distribution, and discuss other distributions and several interesting limiting cases. This rich model is constructed based on physical considerations and on consistency requirements which are imposed on any zeroth‐order model when it is used in conjunction with renormalization group methods to incorporate the excluded volume interactions.

End-to-end distance of a single self-interacting self-avoiding polymer chain: d−1 expansion☆

Abstract Exact enumeration data in dimensions d =2–6 is used to evaluate the exact d -dimensional mean-square end-to-end distance R 2 n of a short ( n ⩽11) n -bond self-interacting self-avoiding random walk on hypercubic lattices as function of the neighbor contact energy. This exact form is transformed into a large n expansion of R 2 n through fifth order in d -1 but to all orders in the contact energy.

Surface and finite size effects in critical phenomena

Field theoretic renormalization group methods are developed to described in a unified fashion the critical behavior of d = 4 − ϵ) systems of finite, semi-infinite and infinite extension with interacting boundaries. The theory clearly shows the presence of three regions depending on the value of a scaling variable η: (a) For η > ηdc the physics is quasi d-dimensional; (b) when η ≈ η∗ < ηdc, a quasi d′-dimensional (d′ < d) physics emerges; and (c) there is a dimensional crossover region for η ≈ ηdc which separates regions (a) and (b). This paper focuses on region (a) where usual e-expansion techniques are shown to be well behaved and where critical exponents are found to be the bulk ones, but non trivial dependences on the confining length scale and the surface interaction parameter persist. These e-expansions break down at η ≈ ηdc, and alternative methods, briefly discussed in this paper, are required to study the dimensionally reduced region (c). The general theory is illustrated for the N-vector model in both layered and semi-infinite geometries with various boundary conditions on the surfaces, and for the statistics of long polymer chains with excluded volume in presence of interacting interfaces. Finally, analogies are presented with mathematically related problems in particle physics.

Hypercubic lattice SAW exponents nu and gamma : 3.99 dimensions revisited

The self-avoiding walk (SAW) exponents nu and gamma are computed over a range of dimensions (1<or=d< infinity ) from exact expressions for the mean-square end-to-end distance (Rn2) and the partition function Qn of SAWs having a limited number of steps, n<or=11. SAW exponents ( nu , gamma ) for arbitrary dimension d are estimated by applying standard extrapolation techniques to the direct enumeration data which has been analytically continued to variable dimension. Exponent estimates obtained from continuum theories of self-avoiding paths are compared with the SAW calculations.

Renormalisation group treatment of finite size scaling with ε expansion

The use of renormalisation group techniques away from the critical point is shown to enable the calculation of finite size scaling corrections to scaling functions with epsilon -expansion methods. The authors consider the N-vector phi 4 theory for a layered geometry with periodic boundary conditions and evaluate the correlation function, susceptibility, correlation lengths and shift in the critical temperature to order epsilon .

Lattice cluster theory for the packing of rods on a lattice: Extension to treat anisotropic orientational distributions

The lattice cluster theory for the free energy of a set of mutually avoiding rigid rod polymers is extended to treat anisotropic orientational distributions. The theory permits the systematic evaluation of corrections to the isotropic Flory mean field approximation for arbitrary rod orientational distributions, with the Flory theory being the zeroth order isotropic limit of the full theory. The corrections to the zeroth order mean field entropy are represented as a cluster expansion and may be evaluated as a series expansion in the polymer volume fraction φ. We compute all corrections through order φ3 that survive in the thermodynamic limit for the general anisotropic case, along with new fourth order results, which also extend the isotropic limit theory. The anisotropic rod lattice cluster theory represents an improvement over the DiMarzio theory for the packing entropy of rod polymers. This improvement first emerges at fourth order in φ and arises in the lattice cluster theory from inclusion of correlat...