Contact probability of phantom chain as a function of chain volume fraction ϕ.

The contact probability of a phantom chain, which is defined as the probability of a chain’s beads coming into contact in the 3D space, is shown as a function of the volume fraction ϕ of the chain. The contact probability follows a power law relation with the contour distance s as P₁(s) ∼ s^−γ. The exponent γ is plotted for three different volume fractions: ϕ = 0 (unconfined single phantom chain), ϕ = 0.001 (4 chain simulation in relatively large confinement volume), and ϕ = 0.1 (4 chain simulation in relatively small confinement volume). For the unconfined polymer, ϕ = 0, the exponent is γ = 1.5, which is same to the theoretically calculated value of γ = 1.5 [39]. In a relatively large confinement volume, the exponent is γ = 1.5, and in a relatively small confinement volume, the exponent is γ ≈ 1.5 between 20 < s < 200, and saturates to a constant for s > 200 beads, as expected for an equilibrium polymer [87]. Note that these results are based on the contact probability P₁(s) calculated for the first chain from a simulation of four phantom chains.

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