  In this paper, we develop a first order (in time) numerical scheme for the
binary fluid surfactant phase field model. The free energy contains a
double-well potential, a nonlinear coupling entropy and a Flory-Huggins
potential. The resulting coupled system consists of two Cahn-Hilliard type
equations. This system is solved numerically by finite difference spatial
approximation, in combination with convex splitting temporal discretization. We
prove the proposed scheme is unique solvable, positivity-preserving and
unconditionally energy stable. In addition, an optimal rate convergence
analysis is provided for the proposed numerical scheme, which will be the first
such result for the binary fluid-surfactant system. Newton iteration is used to
solve the discrete system. Some numerical experiments are performed to validate
the accuracy and energy stability of the proposed scheme.
