  We study universal aspects of polymer conformations and transverse
fluctuations for a single swollen chain characterized by a contour length $L$
and a persistence length $\ell_p$ in two dimensions (2D) and in three
dimensions (3D) in the bulk, as well as in the presence of excluded volume (EV)
particles of different sizes occupying different volume fractions. In the
absence of the EV particles we extend the previously established universal
scaling relations in 2D (A. Huang, A. Bhattacharya, and K. Binder, J. Chem.
140, 214902 (2014)) to include 3D and demonstrate that the scaled end-to-end
distance $\langle R_N^2\rangle/(2 L\ell_p)$ and the scaled transverse
fluctuation $\sqrt{\langle{l_{\perp}^2}\rangle}/{L}$ as a function of
$L/\ell_p$ collapse onto the same master curve, where $\langle R_N^2\rangle$
and $\langle{l_{\perp}^2\rangle}$ are the mean-square end-to-end distance and
transverse fluctuations. However, unlike in 2D, where the Gaussian regime is
absent due to extreme dominance of the EV interaction, we find the Gaussian
regime is present, albeit very narrow in 3D. The scaled transverse fluctuation
in the limit $L/\ell_p \ll 1$ is independent of the physical dimension and
scales as $\sqrt{\langle{l_{\perp}^2}\rangle}/{L} \sim (L/\ell_p)^{\zeta-1}$,
where $\zeta = 1.5$ is the roughening exponent. For $L/\ell_p \gg 1$ the scaled
fluctuation scales as $\sqrt{\langle{l_{\perp}^2}\rangle}/{L} \sim
(L/\ell_p)^{\nu-1}$, where $\nu$ is Flory exponent for the corresponding
spatial dimension ($\nu_{2D}=0.75$, and $\nu_{3D}=0.58$). When EV particles are
added into the system, our results indicate that the crowding density either
does not or only weakly affects the universal scaling relations. We discuss the
implications of these results in living matter by showing the experimental
result for a dsDNA onto the master plot.
