  We propose a phase-field theory for enriched continua. To generalize
classical phase-field models, we derive the phase-field gradient theory based
on balances of microforces, microtorques, and mass. We focus on materials where
second gradients of the phase field describe long-range interactions. By
considering a nontrivial interaction inside the body, described by a
boundary-edge microtraction, we characterize the existence of a
microhypertraction field, a central aspect of this theory. On surfaces, we
define the surface microtraction and the surface-couple microtraction emerging
from internal surface interactions. We explicitly account for the lack of
smoothness along a curve on surfaces enclosing arbitrary parts of the domain.
In these rough areas, internal-edge microtractions appear. We begin our theory
by characterizing these tractions. Next, in balancing microforces and
microtorques, we arrive at the field equations. Subject to thermodynamic
constraints, we develop a general set of constitutive relations for a
phase-field model where its free-energy density depends on second gradients of
the phase field. A priori, the balance equations are general and independent of
constitutive equations, where the thermodynamics constrain the constitutive
relations through the free-energy imbalance. To exemplify the usefulness of our
theory, we generalize two commonly used phase-field equations. We propose a
'generalized Swift-Hohenberg equation'-a second-grade phase-field equation-and
its conserved version, the 'generalized phase-field crystal equation'-a
conserved second-grade phase-field equation. Furthermore, we derive the
configurational fields arising in this theory. We conclude with the
presentation of a comprehensive, thermodynamically consistent set of boundary
conditions.
