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In the imposed free-draining polymer friction model, the drag coefficients of the $N$ monomers add, giving ${\zeta }_1=N(6\pi \eta a)$ and $D_1=k_{B}T/(6\pi \eta aN)$. Taking the same three-dimensional centre-of-mass RMS convention as part (ii),The polymer is a one-dimensional connected contour, but that alone does not restrict its centre of mass to one spatial axis. If the intended coordinate is instead literal one-dimensional curvilinear motion along an entanglement tube, its RMS contour displacement is $\sqrt{2D_{1}t}=\sqrt{k_{B}Tt/(3\pi \eta aN)}$. That is a different coordinate and should not be compared with a three-dimensional displacement without stating the conversion/model.

Additive local drag is the free-draining Rouse model approximation. It neglects long-range solvent-mediated hydrodynamic interactions and is not a complete theory of an entangled chain. In a tube/reptation description, curvilinear diffusion can scale as $N^{-1}$ while long-time spatial centre-of-mass diffusion scales as $N^{-2}$ after the chain renews its tube. Conversely a dilute unentangled coil can exhibit Zimm model hydrodynamic coupling. Monomer motion, centre-of-mass motion and contour sliding therefore cannot all be identified with the same displacement law.

In a dilute solution, solvent-mediated hydrodynamic interactions couple distant chain segments. The Zimm model describes the resulting cooperative dynamics: an intermediate-time segment mean-square displacement scales as $t^{2/3}$, while whole-chain translation has $D_Z\sim k_{B}T/({\eta }_{s}R)$. Once internal modes relax, each tagged segment follows the chain's ordinary long-time diffusion, with mean-square displacement asymptotic to $6D_{Z}t$.

In a concentrated solution or polymer melt, hydrodynamic coupling is screened. The unentangled Rouse model has an intermediate $t^{1/2}$ segment regime and whole-chain diffusivity proportional to $N^{-1}$. Long chains with polymer entanglement are additionally confined by neighbouring chains: reptation gives a slower intermediate $t^{1/4}$ regime and an asymptotic chain diffusivity scaling approximately as $N^{-2}$. The time windows depend on segment, entanglement and terminal relaxation times, so these powers do not all describe the same time interval.

After sufficiently long times both dilute and concentrated chains undergo ordinary translational diffusion. The main long-time difference is the coefficient and its chain-length dependence, with entanglement strongly reducing mobility when the chains are long enough; concentration alone does not guarantee an entangled regime.

In a dilute solution, separate coils diffuse through solvent with long-ranged hydrodynamic coupling among their segments. The Zimm model gives whole-chain diffusionwhere $R_H$ is a hydrodynamic radius and ${\eta }_s$ the solvent viscosity. The coil continually changes shape while its centre of mass performs long-time Brownian motion; treating it as a rigid sphere is only a scaling analogy.

At high concentration, solvent-mediated hydrodynamic interactions are screened on sufficiently large scales and times. If chains are not entangled, additive local or blob friction gives Rouse model behaviour, $D\propto N^{-1}$ at fixed concentration. For long entangled chains, neighboring chains prevent rapid transverse escape. In the reptation tube picture the chain moves mainly along its contour until it renews the surrounding tube. Contour diffusion scales as $N^{-1}$, the tube-renewal time as $N^3$, and the random-coil size squared as $N$. ThereforeBoth cases ultimately give diffusive centre-of-mass motion, but entangled concentrated chains move much more slowly and retain constraints for longer. The inequality $\varphi \gg {\varphi }^{\ast }$ establishes strong overlap, not by itself entanglement: the latter also requires sufficiently many segments relative to an entanglement length. No numerical entanglement scale or concentrated-solution diffusion coefficient is supplied, so only the conditional regimes and scaling exponents can be fixed here.

A dilute-solution bead-spring model including solvent-mediated hydrodynamic interactions. In its scaling regime, a chain of size $R$ has translational diffusion coefficient of order $k_{B}T/({\eta }_{s}R)$ and longest relaxation time of order ${\eta }_s{R}^3/(k_{B}T)$. Intermediate-time segment mean-square displacements scale as $t^{2/3}$. Screening of hydrodynamic interactions in a concentrated solution changes the regime toward Rouse model behaviour; constraints from polymer entanglement require additional physics.