  We estimate the liquid-vapour surface tension from simulations of TIP4P/2005
water nanodroplets of size $N$=100 to 2880 molecules over a temperature $T$
range of 180 K to 300 K. We compute the planar surface tension $\gamma_p$, the
curvature-dependent surface tension $\gamma_s$, and the Tolman length $\delta$,
via two approaches, one based on the pressure tensor (the "mechanical route")
and the other on the Laplace pressure (the "thermodynamic route"). We find that
these two routes give different results for $\gamma_p$, $\gamma_s$ and
$\delta$, although in all cases we find that $\delta\ge 0$ and is independent
of $T$. Nonetheless, the $T$ dependence of $\gamma_p$ is consistent between the
two routes and with that of Vega and de Miguel [J. Chem. Phys. 126, 154707
(2007)] down to the crossing of the Widom line at 230 K for ambient pressure.
Below 230 K, $\gamma_p$ rises more rapidly on cooling than predicted from
behavior for $T\ge 300$ K. We show that the increase in $\gamma_p$ at low $T$
is correlated to the emergence of a well-structured random tetrahedral network
in our nanodroplet cores, and thus that the surface tension can be used as a
probe to detect behavior associated with the proposed liquid-liquid phase
transition in supercooled water.
