# 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 *P*_{1}(*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 *P*_{1}(*s*) calculated for the first chain from a simulation of four phantom chains.

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