Surfactant behavior in three-component Bose-Einstein condensates

In classical fluids, surfactants enter the interface between two immiscible fluids and reduce the interfacial tension. We show that a similar situation can be realized in a three-component Bose-Einstein condensate (BEC), in which a phase-separated two-component BEC forms an interface and the third component enters the interface. We derive an expression for the interfacial tension of this system, and numerically show that the third component can reduce the interfacial tension, i.e., the third component plays the role of a surfactant. The stability of such an interface is corroborated by the Bogoliubov analysis. We also consider a system that has a spatial gradient in the interfacial tension, and we show that interfacial flow is induced, which resembles the Marangoni flow in classical fluids.

Figure 1

Schematic of a three-component interface, in which component 3 is sandwiched by components 1 and 2. The whole region inside the dashed square is regarded as the three-component interface.

Figure 2

Flat-interface equilibrium states. (a) Density profile of each component near the interface for $n_3=0$, 10, 30, and 100. (b) Areal density $n_3$ as a function of the chemical potential ${\mu }_3$ for component 3. The parameters are $g_{12}=1.1$ and $g_{13}=g_{23}=1.01$.

Figure 3

Interfacial tension $\sigma$ for $g_{13}=g_{23}=1.01$. (a) $n_3$ dependence for $g_{12}=1.1$ and $g_{12}=1.02$. The horizontal dashed line represents ${\sigma }_{\mathrm{binary}}\left(g_{13}\right)+{\sigma }_{\mathrm{binary}}\left(g_{23}\right)\simeq 0.14$. (b) Dependence of $\sigma$ on $g_{12}$ and $n_3$.

Figure 4

(a) Interfacial tension ${\sigma }_{\mathrm{binary}}$ for a two-component BEC as a function of the intercomponent interaction $g_{\mathrm{inter}}$. The numerical result is compared with Eq. (14). (b) and (c) Dependence of $\mathrm{\Delta }\sigma$ in Eq. (15) on the intercomponent interactions $g_{12}, g_{23}$, and $g_{13}$, where the values of ${\sigma }_{\mathrm{binary}}$ in Eq. (15) are taken from the numerical result in (a). In (c), $g_{12}$ is fixed to 1.1. The solid lines represent the contour of $\mathrm{\Delta }\sigma =0$, and the dashed lines are Eq. (16). The points correspond to the parameters used in Fig. 3.

Figure 5

Comparison between the interfacial tension $\sigma$ obtained by the variational method (“$\sigma$, variational”) and that obtained by solving the GP equation [“$\sigma$, numerical,” the same data as in Fig. 3] for $g_{12}=1.1$ and $g_{13}=g_{23}=1.01$. The variational parameter $\alpha$ minimizing the variational energy is also shown (“$\alpha$, variational”).

Figure 6

Lowest excitation frequency ${\omega }_L$ obtained by numerically diagonalizing Eq. (33). (a) Real and imaginary parts of ${\omega }_L$ as functions of $k_{\perp }$ for $g_{12}=1.1$ and $g_{12}=1.02$ with $n_3=10$. For $g_{12}=1.1, \mathrm{Im}\omega =0$ for all the excitations. (b) Imaginary part of ${\omega }_L$ as a function of $k_{\perp }$ and $n_3$ for $g_{12}=1.02$. (c) Largest imaginary part max $\mathrm{Im}{\omega }_L$ with respect to $k_{\perp }$ as a function of $n_3$ and $g_{12}$. (d) Dynamics of the density profile $|{\psi }_3{|}^2$ of component 3, where the initial state is the stationary flat interface with $g_{12}=1.02$ and $n_3=10$. The size of each panel is $512\times 100$. See the Supplemental Material for a video of the dynamics [44]. In (a)–(d), $g_{13}$ and $g_{23}$ are fixed to 1.01.

Figure 7

Cross-sectional density profiles of the ground states in a three-dimensional isotropic harmonic trap with frequency $\omega =2\pi \times 100$ Hz. The scattering lengths are $a_{22}=a_{33}=100.4a_B, a_{12}=97.7\times 1.3a_B, a_{13}=97.7a_B$, and $a_{23}=101.3a_B$. (a) $N_1=N_2={10}^5, N_3={10}^4$, and $a_{11}=94.58a_B$. (b) $N_1={10}^5, N_2=2\times {10}^5, N_3=0$, and $a_{11}=94.58\times 0.9a_B$. (c) $N_3=4\times {10}^4$, and other parameters are the same as those in (b). The density is normalized by ${10}^5{(m\omega /\hslash )}^{3/2}$.

Figure 8

Dynamics of interface flow due to a gradient in interfacial tension: Marangoni flow. (a) Initial density distribution of the three components, which is the ground state for $g_{12}=g_{13}=g_{23}=1.01$. (b) Time evolution after $g_{12}$ is changed from 1.01 to 1.1 at $t=0$. (c) Snapshot at $t=6000$, where $g_{12}$ is changed from 1.01 to 1.02 at $t=0$. The vertical dashed line indicates the initial position of the 2-3 interface $x_0$. See the Supplemental Material for videos of the dynamics in (b) and (c) [44].