Wormlike chains in the large-d limit

We study the properties of an isolated, self-interacting wormlike polymer chain on the basis of a nonperturbative $1/d$-expansion, where d denotes the dimension of embedding space. In the absence of an external force, we characterize the dimension R of the chain in embedding space via $R∼Lν,$ where L is the internal size. (A) Long-range, repulsive segmental interactions decaying as $1/rα$ may control chain conformations that are either rodlike, $ν=1(1<α<2),$ “wrinkled,” $1/2<ν<1(2<α<4),$ or random-walk-like, $ν=1/2(α>4).$ (B) For short-range, screened, repulsive interactions, the crossover between rodlike and random-walk-like behavior is controlled by the persistence length whose interaction part we compute focusing on a Debye–Hückel interaction of strength $V0,$ with inverse screening length $κ0.$ The induced persistence length varies as $V0βκ0−γ,$ with, as expected, $(β,γ)=(1,2)$ when the chain is intrinsically stiff, and, surprisingly, with either $(β,γ)=(1/6,7/6)$ or $(β,γ)=(1,7)$ when the chain is intrinsically very flexible. The chances of experimentally observing the novel regimes may be limited. For a chain subject to an external stretching force $f,$ we determine the force-extension relation $ζ=ζ(f )=ζ0+δζ(f ),$ where ζ denotes the chain extension, $ζ0$ is the spontaneous extension. (A) If the interaction potential is either screened, or if the decay of a long-range interaction potential is fast, i.e., if $α>4,$ the chain spontaneously generates an “effective tension” and responds linearly to weak forces with elastic constants “renormalized” by interactions. By contrast, “tension-free” chains, with either $ν=1,$ where $δζ∼f1/2,$ or with $ν=2/α,$ where $δζ∼f1/3,$ respond to the weakest force nonlinearly. (B) Near full extension the chain always responds nonlinearly. When the potential is screened, or if $α>4,$ we find the $1/f$ corrections typical of wormlike chains.