Statistical Mechanics of Confined Polymer Networks
2020
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
$$\Theta $$
-point. In the
$$\Theta $$
-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges,
$$\gamma _b^{\Theta }$$
, to that of terminally-attached arches,
$$\gamma _{11}^{\Theta },$$
and to the correlation length exponent
$$\nu ^{\Theta }.$$
We find
$$\gamma _b^{\Theta } =\gamma _{11}^{\Theta }+\nu ^{\Theta }$$
. In the case of the special transition, we find
$$\gamma _b^{\Theta }(\mathrm{sp}) =\frac{1}{2}[\gamma _{11}^{\Theta }(\mathrm{sp})+\gamma _{11}^{\Theta }]+\nu ^{\Theta }$$
. For general networks, the explicit expression of configurational exponents then naturally involves 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 cases of ordinary, mixed and special surface transitions, and of the
$$\Theta $$
-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.
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