  Understanding the properties of the stochastic phase field models is crucial
to model processes in several practical applications, such as soft matters and
phase separation in random environments. To describe such random evolution,
this work proposes and studies two mathematical models and their numerical
approximations for parabolic stochastic partial differential equation (SPDE)
with a logarithmic Flory--Huggins energy potential. These multiscale models are
built based on a regularized energy technique and thus avoid possible
singularities of coefficients. According to the large deviation principle, we
show that the limit of the proposed models with small noise naturally recovers
the classical dynamics in deterministic case. Moreover, when the driving noise
is multiplicative, the Stampacchia maximum principle holds which indicates the
robustness of the proposed model. One of the main advantages of the proposed
models is that they can admit the energy evolution law and asymptotically
preserve the Stampacchia maximum bound of the original problem. To numerically
capture these asymptotic behaviors, we investigate the semi-implicit
discretizations for regularized logrithmic SPDEs. Several numerical results are
presented to verify our theoretical findings.
