Bridging length and time scales in sheared demixing systems: From the Cahn-Hilliard to the Doi-Ohta model

We develop a systematic coarse-graining procedure which establishes the connection between models of mixtures of immiscible fluids at different length and time scales. We start from the Cahn-Hilliard model of spinodal decomposition in a binary fluid mixture under flow from which we derive the coarse-grained description. The crucial step in this procedure is to identify the relevant coarse-grained variables and find the appropriate mapping which expresses them in terms of the more microscopic variables. In order to capture the physics of the Doi-Ohta level, we introduce the interfacial width as an additional variable at that level. In this way, we account for the stretching of the interface under flow and derive analytically the convective behavior of the relevant coarse-grained variables, which in the long wavelength limit recovers the familiar phenomenological Doi-Ohta model. In addition, we obtain the expression for the interfacial tension in terms of the Cahn-Hilliard parameters as a direct result of the developed coarse-graining procedure. Finally, by analyzing the numerical results obtained from the simulations on the Cahn-Hilliard level, we discuss that dissipative processes at the Doi-Ohta level are of the same origin as in the Cahn-Hilliard model. The way to estimate the interface relaxation times of the Doi-Ohta model from the underlying morphology dynamics simulated at the Cahn-Hilliard level is established.

Figure 1

(Color online) Time evolution of the coarse-grained Doi-Ohta variables for a binary mixture with $\varphi =0.3$, subjected to the shear flow at time $t_0=200t_{\text{D}}$ after the beginning of a phase separating process: interfacial width $l$ (upper left), average amount of interface per unit volume $Q$ (lower left), and shear viscosity, $\eta =-q_{xy}/\dot{\gamma }$ as a function of strain $\gamma =\dot{\gamma }t$ (right). The results for the shear rates $\dot{\gamma }{t}_{\text{D}}=0.05$ and $\dot{\gamma }{t}_{\text{D}}=0.06$ are presented.

Figure 2

Configurations of a binary fluid with volume fraction $\varphi =0.3$, simulated on a $128\times 128$ lattice during phase separation under shear flow at different values of strain $\gamma =\dot{\gamma }t$. Shear flow was applied after initial time of $t_0=200t_{\text{D}}$, with the shear rate $\dot{\gamma }{t}_{\text{D}}=0.05$. The snapshots were taken until the maximum strain $\gamma =12$. The grayscale is used in order to capture the processes of interface deformation.