Critical wetting, first-order wetting, and prewetting phase transitions in binary mixtures of Bose-Einstein condensates

An ultralow-temperature binary mixture of Bose-Einstein condensates adsorbed at an optical wall can undergo a wetting phase transition in which one of the species excludes the other from contact with the wall. Interestingly, while hard-wall boundary conditions entail the wetting transition to be of first order, using Gross-Pitaevskii theory we show that first-order wetting as well as critical wetting can occur when a realistic exponential optical wall potential (evanescent wave) with a finite turn-on length $\lambda$ is assumed. The relevant surface excess energies are computed in an expansion in $\lambda /{\xi }_i$, where ${\xi }_i$ is the healing length of condensate $i$. Experimentally, the wetting transition may best be approached by varying the interspecies scattering length $a_{12}$ using Feshbach resonances. In the hard-wall limit, $\lambda \to 0$, exact results are derived for the prewetting and first-order wetting phase boundaries.

Figure 1

(Left) Partial wetting. The interface between the two phases consisting of pure Bose-Einstein condensates 1 and 2 makes a finite contact angle $\theta$ with the optical wall. (Right) Complete wetting. A macroscopic layer of pure phase 2 intrudes between the optical wall and pure phase 1.

Figure 2

The grand potential of the three possible bulk states (and the vacuum at $\Omega =0$) as a function of the relative interaction parameter $K$ when ${\mu }_2<{\overline{\mu }}_2$. The ground-state grand potential is indicated by the thick line. For $-1<K\le {\mu }_2/{\overline{\mu }}_2$, the ground state coincides with the mixed phase and for ${\mu }_2/{\overline{\mu }}_2<K$, it coincides with pure phase 1. The first derivative of $\Omega$ is continuous in this transition (second-order phase transition), as is evidenced by the mathematical continuation of the energy of the mixed phase (curved dotted line). When $K\ge {\overline{\mu }}_2/{\mu }_2$, the mixed phase is an unstable state (ascending dotted line).

Figure 3

Bulk phase diagram as a function of $K,{\mu }_2/{\overline{\mu }}_2$, and the fraction $x={\tilde{n}}_1/({\tilde{n}}_1+{\tilde{n}}_2)$. Hatched are the ground states. Pure phase 1 (2) constitutes the ground state when $x=1$ ($x=0$) and $K>{\mu }_2/{\overline{\mu }}_2$ ($K>{\overline{\mu }}_2/{\mu }_2$). Hence, pure phases 1 and 2 coexist for ${\mu }_2/{\overline{\mu }}_2=1$ and $K\ge 1$. Upon approach of the “triple point” $K=1$ and ${\mu }_2/{\overline{\mu }}_2=1$, pure phases 1 and 2 and the mixed phase with $x=0.5$ coexist. Moreover, the triple point itself possesses an infinite (continuous) degeneracy in that $x$ can take all real values between 0 and 1.

Figure 4

Wetting phase diagram in the plane of surface field ${\xi }_1/{\xi }_2-1$ (represented here by the ratio ${\xi }_2/{\xi }_1$) and reciprocal interaction strength $1/K$, at bulk two-phase coexistence. A first-order phase boundary or wetting line (WL) separates partial wetting (PW) from complete wetting (CW) for ${\xi }_2/{\xi }_1<1$. For ${\xi }_2/{\xi }_1>1$ the roles of the condensates are interchanged and one may use “drying” in place of “wetting.” For example, for ${\xi }_2/{\xi }_1>1$, Partial wetting (see figure) signifies partial drying, and CW is replaced with CD. The phase boundary is parabolic in $1/K$ and $1/K-1$ as a function of ${\xi }_2/{\xi }_1$ and ${\xi }_2/{\xi }_1-1$, respectively, near the points $(0,0)$ and $(1,1)$.

Figure 5

(Left) The excess grand potential per unit area $\gamma$ in units of $P_1{\xi }_1$ against the ratio of lengths inherent to the surface field, ${\xi }_2/{\xi }_1$, for fixed $K=1.5$, at bulk two-phase coexistence. We vary ${\xi }_2$ and keep ${\xi }_1$ fixed. The ground-state energy is indicated by the thick line. Clearly, the wetting transition at ${\xi }_2/{\xi }_1=0.5$ is of first order: The excess energy of pure phase 1 adsorbed at the wall cuts the excess energy of a system in which an infinite layer of species 2 wets the wall. (Right) The excess grand potential per unit area $\gamma$ in units of $P_1{\xi }_1$ of a prewetting film, which is nucleated with infinitesimal amplitude at N and becomes macroscopically thick at C. The energy is shown for ${\overline{\xi }}_2/{\xi }_1=2/5$ and $K=1.5$ as a function of the field variable, which can be used to measure the deviation from bulk two-phase coexistence, ${\mu }_2/{\overline{\mu }}_2$. The thick line denotes the ground-state energy. The prewetting transition is critical: At the nucleation point (N) an infinitesimal film of species 2 is nucleated at the wall.

Figure 6

The prewetting phase diagram for ${\overline{\xi }}_2/{\xi }_1=1/3$ as a function of the deviation from bulk coexistence ${\mu }_2/{\overline{\mu }}_2-1$ and the relative interaction parameter $K=G_{12}/\sqrt{G_{11}{G}_{22}}$. Prewetting (PreW) is found above the prewetting line (PreWL) and below the bulk coexistence line (CL). Physically, at fixed $K>1$, moving vertically from a point on PreWL to a point on CL means going from a nucleated to an infinitely thick adsorbed film. For $K<1$, upon entering the region of bulk mixed phase (MIX) from the PreW region, the growth of the wetting layer of species 2 is preempted by the bulk nucleation of species 2. That is, bulk phase 1 becomes unstable before wetting is achieved. See Fig. 7 for more detail and various scenarios.

Figure 7

Prewetting states for ${\overline{\xi }}_2/{\xi }_1=1/3$. Panels (A)–(D) display profiles of the condensate wave functions for the approach to CW along the trajectory at $K>1$ (vertical dashed line) indicated in the top left prewetting diagram. Note that panel (D) corresponds to a CW state in which an infinitely thick layer of phase 2 is present between the hard wall and bulk phase 1. Profiles (E)–(H) depict the wave functions of condensates 2 [marked by the circles with letters (E)–(H)] and 1 (indicated by the arrows emanating from the circles) along the trajectory at $K<1$ (vertical dashed line) indicated in the bottom left prewetting diagram. State (E) corresponds to a prewetting film; this film continuously grows up to the point where the bulk transition to the mixed phase is reached [state (F)]. Note that this film remains microscopically thin and does not become a wetting layer. From the profiles in state (G) one sees that there appears a nonzero bulk density of species 2. Finally, for higher values of ${\mu }_2/{\overline{\mu }}_2$, the bulk density of species 2 increases [state (H)].

Figure 8

The numerically obtained logarithmic increase of the adsorption of species 2 [see (35)], associated with the prewetting film, as a function of $1-{\mu }_2/{\overline{\mu }}_2=1-\sqrt{P_2/P_1}$, which is proportional to the pressure difference $P_1-P_2$ when we are close to two-phase coexistence ($P_1=P_2$). The approach to CW at bulk coexistence is accomplished for three different pairs of values of $K$ and ${\xi }_2/{\xi }_1$, in the prewetting regime.

Figure 9

The prewetting phase diagram as a function of ${\overline{\xi }}_2/{\xi }_1$ and the deviation from bulk two-phase coexistence ${\mu }_2/{\overline{\mu }}_2-1$ for $K=1.5$. Prewetting (PreW) occurs in the shaded region on the left, whereas “predrying” (PreD) (adsorption of species 1 at the wall with pure phase 2 in bulk) occurs in the shaded region on the right. Upon crossing the prewetting line (PreWL), or the predrying line (PreDL), a second-order surface transition occurs.

Figure 10

Illustration of the two condensate wave functions confined on one side by a soft wall. The exponential surface potentials are indicated by the thin gray lines. For the configuration above, one can see that the relative trap displacement satisfies $\Delta <0$, since phase 2 is shifted to the left. As a consequence, phase 2 is more favored by the surface than it already was for the (unshifted) hard-wall configuration, given that ${\xi }_2<{\xi }_1$.

Figure 11

Wetting phase diagram at bulk two-phase coexistence for soft walls, in the plane of relative interaction strength $K$ and ${\xi }_1/{\xi }_2$ for different values of the relative trap displacement $\Delta$, defined in expression (40). For $\Delta =0$, we reproduce the first-order wetting line, found earlier in the wetting phase diagram of Fig. 4. Displacement of species 2 closer to the surface, implied by $\Delta <0$, makes it possible that the wetting transition turns to critical wetting (which it does at least for $K\approx 2$). Furthermore, the parameter region in which CW occurs, broadens. On the other hand, for $\Delta >0$, the PW region broadens and the first-order character of the wetting transition is found to persist (at least for $K\approx 2$).

Figure 12

The excess grand potential per unit area $\gamma$, at bulk two-phase coexistence, in units of ${\xi }_1{P}_1$ against the ratio ${\xi }_1/{\xi }_2$ for $K=2$ for relative trap displacement $\Delta =-0.1{\xi }_1$ (left) and $\Delta =0.1{\xi }_1$ (right). Note that we vary ${\xi }_2$ while keeping ${\xi }_1$ fixed. (Left) A critical transition to a CW state takes place at point $T$. Point $L$ is the critical nucleation point for the thin film of species 2. (Right) For $\Delta =0.1{\xi }_1$, a first-order transition to CW occurs at point $T$.

Figure 13

The adsorption of species 2, $\Gamma$ [see Eq. (35)], versus the surface field “distance” to the wetting transition ${\xi }_1/{\xi }_2-{({\xi }_1/{\xi }_2)}_c$, on a semilog plot. The critical wetting point consistent with the apparent logarithmic divergence of the adsorption is located at ${({\xi }_1/{\xi }_2)}_c=2.3227$ [point $T$ in Fig. 12 (left)]. The line is a linear fit to the leftmost ten points on the curve.

Figure 14

Possible experimental setup for observing the wetting transition in a trap: Atoms are contained by a (anisotropic) harmonic confinement and are held up by an evanescent wave prism. The figures show cross sections of PW (left) and CW (right) configurations for a trapped binary mixture at bulk two-phase coexistence. As argued in Sec. 8, the wetting characteristics do not depend on the position along a hard wall.

Figure 15

Wetting phase diagram in the plane of inverse relative interaction parameter $1/K$ and surface-field-related parameter ${\xi }_2/{\xi }_1$. The lines numbered with the integer $s$ correspond to the solutions (A4) and indicate the loci where excitations (with $s$ nodal planes) of zero energy exist [8]. Below each of these lines these excitations have a negative energy. The $s=0$ line is the wetting phase boundary, which coincides precisely with the nucleation line for infinitesimal films of phase 2. The lines with $s=1,2,...$ lie in the CW regime and have no physical significance, since the equilibrium state in this regime is a macroscopic wetting layer of species 2.