J Sci Comput (2018) 74:1533–1553
Numerical Approximations for the Cahn–Hilliard Phase
Field Model of the Binary Fluid-Surfactant System
Received: 19 March 2016 / Revised: 24 May 2017 / Accepted: 17 July 2017 / Published online: 24 July 2017
© Springer Science+Business Media, LLC 2017
Abstract In this paper, we consider the numerical approximations for the commonly used
binary ﬂuid-surfactant phase ﬁeld model that consists two nonlinearly coupled Cahn–Hilliard
equations. The main challenge in solving the system numerically is how to develop easy-
to-implement time stepping schemes while preserving the unconditional energy stability.
We solve this issue by developing two linear and decoupled, ﬁrst order and a second order
time-stepping schemes using the so-called “invariant energy quadratization” approach for
the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling
potential. Moreover, the resulting linear system is well-posed and the linear operator is
symmetric positive deﬁnite. We rigorously prove the ﬁrst order scheme is unconditionally
energy stable. Various numerical simulations are presented to demonstrate the stability and
the accuracy thereafter.
Keywords Phase-ﬁeld · Fluid-surfactant · Cahn–Hilliard · Unconditional energy stability ·
Ginzburg–Landau · Invariant energy quadratization
Surfactants are usually organic compounds that can alter or reduce the surface tension of the
solution, and allows for the mixing of dissimilar (immiscible) liquids. A typical well-known
example of dissimilar liquids is the mixture of oil and water, where the water molecule is
polar, and it does not hang out with nonpolar molecules like oil. Hence, in order to make
the mixing more favorable, a molecular intermediate, commonly referred as surfactants, is
X. Yang’s research is partially supported by the U.S. National Science Foundation under Grant Nos.
DMS-1200487 and DMS-1418898.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
Beijing Institute for Scientiﬁc and Engineering Computing, Beijing University of Technology,
Beijing 100124, China