We propose a phase field model that approximates
its limiting sharp interface model (free boundary problem) up to second
order in interface thickness. A broad range of double-well potentials
can be utilized so long as the dynamical coefficient in the phase equation
is adjusted appropriately. This model thereby assures that computation with
particular value of interface thickness $\varepsilon$, will differ at most by
$O(\varepsilon^2$) from the limiting sharp interface problem.
As an illustration, the speed of a traveling wave of the phase field model is
asymptotically expanded to demonstrate that it differs from the speed of the traveling wave
of the limit problem by $O(\varepsilon^2)$.