Statistical Mechanics of Confined Polymer Networks

Abstract

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical Θ-point. In the Θ-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, γ Θ b , to that of terminally-attached arches, γ Θ 11 , and to the correlation length exponent ν Θ . We find γ Θ b = γ Θ 11 + ν Θ . In the case of the special transition, we find γ Θ b (sp) = 1 2 [γ Θ 11 (sp) + γ Θ 11 ] + ν Θ . For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the Θ-point. We provide compelling numerical evidence for some of these results both in two-and three-dimensions.

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

“…We estimate the exponents to be γ b = 0.00 ± 0.03, and γ 1 = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction [18,37,38] 7 , albeit for a modified version of the problem, while the former estimate is predicted in [16] to be zero.…”

Lattice star and acyclic branched polymer vertex exponents in 3d

“…In reference [32] equation ( 9) was extended by calculating terms up to order 4 . We list predictions using the -expansion for σ f in three dimensions to order k for k = 1, 2, 3, 4 in table 3: The order 1 and 2 approximations are obtained from equation (10), while the order 3 and 4 estimates follow from reference [32]. The [3/2] Padé approximation was determined by a Borel resummation of the order 4 expansion and then using the Padé approximant to recalculate the exponents.…”

“…Comparison to equation (10) suggests that the -expansion should break down quickly with increasing f , as the order n term is seen to grow as O(f n+1 ). Moreover, it cannot be improved by calculating ever higher order corrections, as the coefficients increase quickly in magnitude with increasing f .…”

“…order they were appended. The steps along the arm labeled 1, were added first, fourth, seventh, tenth, and so on, giving the sequence of labels (1,4,7,10,13,16,19,22,25,28) along this arm. The second arm is grown starting at step 2, and so on.…”

“…A general survey can be found in references [9,19]. Recent results in reference [10] give predictions in models of confined branched and star polymers.…”