Engineering Mixing Properties of Fluids by Spatial Modulations

We propose a method to change the effective interaction between two fluids by modulation of their local density distributions with external periodic potentials, whereby the mixing properties can be controlled. This method is applied to a mixture of dilute bosonic gases, and binodal and spinodal curves emerge in the phase diagram. Spinodal decomposition into a mixed-bubble state becomes possible, in which one of the coexisting phases has a finite mixing ratio. A metastable mixture is also realized, which undergoes phase separation via nucleation.

Figure 1

(a) Schematic illustration of external periodic potentials $V_1=U{\cos }^2(kx)$ for component 1 and $V_2=-U{\cos }^2(kx)$ for component 2. (b) Variational energy $ϵ$ in Eq. (3) as a function of potential strength $U$ and composition $C$, where variational parameters $a_1$ and $a_2$ are selected to minimize $ϵ$. The curves I, II, III, and IV correspond to $U=0.9$, 1, 1.1, and 1.2, respectively, which are shown in (c) as a function of $C$. The density distributions $|{\mathrm{\Psi }}_1{|}^2$ and $|{\mathrm{\Psi }}_2{|}^2$ are shown in (d), where panels A, B, and C correspond to the points marked in (b). The tangential line and circles on curves IV and II in (c) correspond to the mixed-bubble state and metastable state, respectively, which are presented in Figs. 2 and 3. The interaction parameters are $g_{11}=g_{22}\equiv g=15$ and $g_{12}=15.3$.

Figure 2

Mixed-bubble state for $C=0.1$, $g=15$, $g_{12}=15.3$, and $U=1.2$, which corresponds to the energy curve IV in Fig. 1. (a) Ground state obtained by solving the GP equation. (b) Dynamics of the mixed-bubble formation. The initial state is the ground state for $g=g_{12}=15$. At $t=0$, $g_{12}$ is suddenly changed to 15.3. See the Supplemental Material [33] for a movie of the dynamics. The size of each panel is $(32\pi {)}^2$.

Figure 3

Dynamics for $C=0.5$, $g=15$, $g_{12}=15.3$, and $U=1$. The energy curve for these parameters is shown in Fig. 1 (curve II). The initial state is the metastable state with $C=0.5$ (circle on curve II). At $t=0$, a local perturbation potential $A{e}^{-(x^2+y^2)/2}$ is added to component 1 with (a) $A=1$ and (b) $A=0.05$. The size of each panel is $(64\pi {)}^2$ with the origin at the center. See the Supplemental Material [33] for movies of the dynamics.

Figure 4

(a) Stability of the globally mixed state with respect to the composition $C$ and potential strength $U$, where $g=15$ and $g_{12}=15.3$. The spinodal (binodal) curve divides the unstable (stable) and metastable regions. The parameters used in Figs. 2 and 3 are marked by the open circles. (b) Energy curve $ϵ(C)$ for $U=1.2$. The inflection points (filled circles) trace the spinodal curve in (a). The tangent points (squares and triangles) give the mixed-bubble states. The points marked by the triangles trace the binodal curve in (a).

Figure 5

Modulation amplitudes $a_1$ and $a_2$ in Eq. (2) that minimize the variational energy in Eq. (3), as functions of the potential strength $U$ and composition $C$. The interaction parameters are the same as those in Fig. 1.