  While dynamical tides only become relevant during the last couple of orbits
for circular inspirals, orbital eccentricity can increase their impact during
earlier phases of the inspiral by exciting tidal oscillations at each close
encounter. We investigate the effect of dynamical tides on the orbital
evolution of eccentric neutron star binaries using post-Newtonian numerical
simulations and constructing an analytic stochastic model. Our study reveals a
strong dependence of dynamical tides on the pericenter distance, with the
energy transferred to dynamical tides over that dissipated in gravitational
waves (GWs) exceeding $\sim1\%$ at separations $r_\mathrm{p}\lesssim50$ km for
large eccentricities. We demonstrate that the effect of dynamical tides on
orbital evolution can manifest as a phase shift in the GW signal. We show that
the signal-to-noise ratio of the GW phase shift can reach the detectability
threshold of 8 with a single aLIGO detector at design densitivity for eccentric
neutron star binaries at a distance of $40$ Mpc. This requires a pericenter
distance of $r_\mathrm{p0}\lesssim68$ km ($r_\mathrm{p0}\lesssim76$ km) at
binary formation with eccentricity close to 1 for a reasonable tidal
deformability and f-mode frequency of 500 and $1.73$ kHz (700 and $1.61$ kHz),
respectively. The observation of the phase shift will enable measuring the
f-mode frequency of neutron stars independently from their tidal deformability,
providing significant insights into neutron star seismology and the properties
of the equation of state. We also explore the potential of distinguishing
between equal-radius and twin-star binaries, which could provide an opportunity
to reveal strong first-order phase transitions in the nuclear equation of
state.
