First-order and tricritical wetting transitions in the two-dimensional Ising model caused by interfacial pinning at a defect line

We present a study of the critical behavior of the Blume-Capel model with three spin states ($S=\pm 1,0$) confined between parallel walls separated by a distance $L$ where competitive surface magnetic fields act. By properly choosing the crystal field ($D$), which regulates the density of nonmagnetic species ($S=0$), such that those impurities are excluded from the bulk (where $D=-\infty$) except in the middle of the sample [where $D_M(L/2)\ne -\infty$], we are able to control the presence of a defect line in the middle of the sample and study its influence on the interface between domains of different spin orientations. So essentially we study an Ising model with a defect line but, unlike previous work where defect lines in Ising models were defined via weakened bonds, in the present case the defect line is due to mobile vacancies and hence involves additional entropy. In this way, by drawing phase diagrams, i.e., plots of the wetting critical temperature ($T_w$) versus the magnitude of the crystal field at the middle of the sample ($D_M$), we observe curves of (first-) second-order wetting transitions for (small) high values of $D_M$. Theses lines meet in tricritical wetting points, i.e., ($T_w^{\mathrm{tc}},D_M^{\mathrm{tc}}$), which also depend on the magnitude of the surface magnetic fields. It is found that second-order wetting transitions satisfy the scaling theory for short-range interactions, while first-order ones do not exhibit hysteresis, provided that small samples are used, since fluctuations wash out hysteretic effects. Since hysteresis is observed in large samples, we performed extensive thermodynamic integrations in order to accurately locate the first-order transition points, and a rather good agreement is found by comparing such results with those obtained just by observing the jump of the order parameter in small samples.

Figure 1

Plots of (a) the average absolute value of the magnetization($\langle \left|m\right|\rangle$), (b) the average square magnetization ($\langle m^2\rangle$), and (c) the cumulant ($U$) versus the temperature relative to the bulk critical point, obtained for samples of different sizes (as indicated). Data corresponding to $D_M/J=-2.0$ and $H_1/J=0.60$. All sample sizes have the same generalized aspect ratio $c=L^2/M=9/8$. The common intersection point at $T_w/T_{\mathrm{cb}}=0.735\pm 0.015$ allows us to locate the critical wetting temperature [15]. The insets in these figures show the corresponding scaling plots of the observable already shown in the main panel [15]. Further details are given in the text.

Figure 2

(a) Plots of the average absolute value of the magnetization ($\langle \left|m\right|\rangle$) versus $T/T_{\mathrm{cb}}$ obtained for samples of different sizes (as indicated). Data corresponding to $D_M/J=-1.0$, $H_1/J=0.80$. The main panel shows the absence of hysteresis in small samples, while the inset shows that hysteretic effects become appreciable by using larger samples where the initial conditions are with all spins pointing up and down, respectively. (b) Plot of the cumulant ($U$) versus $T/T_{\mathrm{cb}}$ as obtained for samples of sizes $L=36,M=1152$, $D_M/J=-1.0$, $H_1/J=0.60$. The sharp negative peak of $U$ is a signature of a first-order wetting transition [40] as already evidenced in part (a). Further details are given in the text.

Figure 3

Plots of magnetization profiles [$m\left(i\right)$ versus $i$] and impurity density profiles [$\zeta \left(i\right)$ versus $i$] shown by means of solid circles and squares, respectively. Panels (a) and (b) show data obtained close to second-order wetting transitions, while (c) corresponds to data obtained close to a first-order wetting transition. Measurements performed for (a) $T_w/T_{\mathrm{cb}}=0.73$, $H_{1c}/J=0.70$, and $D_M=-\infty$, i.e., at the wetting critical point that follows from Abraham's exact solution for the $d=2$ Ising model [29]. (b) $T/T_{\mathrm{cb}}=0.6875$, $H_1/J=0.70$, and $D_M/J=-3.0$ and (c) $T/T_{\mathrm{cb}}=0.55$, $H_1/J=0.70$, and $D_M/J=-0.50$.

Figure 4

Typical snapshot configurations showing the localization-delocalization transition of the interface for the case of second- and first-order wetting transitions. Data obtained by using samples of size $L=36$, $M=1152$, and for $H_1/J=0.70$. (a) $D_M/J=-3.0$, $T\le T_w$; (b) $D_M/J=-3.0$, $T\approx T_w$; (c) $D_M/J=-1.75$, $T\le T_w$; (d) $D_M/J=-1.75$, $T\approx T_w$; (e) $D_M/J=-0.5$, $T\le T_w$; and (f) $D_M/J=-0.5$, $T\approx T_w$. Spins-up (-down) are shown as circles (left in white), while the impurities are denoted by squares.

Figure 5

Phase diagram showing the dependence of $T_w/T_{\mathrm{cb}}$ versus the crystal field in the center of the strip ($D_M$) obtained for three different values of the surface magnetic field $H_1/J$, as indicated. Second- and first-order wetting transitions are denoted by triangles and circles, respectively. The arrows on the left-hand side of the figure show the exact values of $T_w/T_{\mathrm{cb}}$ from the analytic results worked out by Abraham [29] and correspond to the Ising model for $D_M=-\infty$. The squares show the estimated location of the tricritical points where first- and second-order curves meet. Curves connecting the points are only drawn to guide the eye. Further details are given in the text.

Figure 6

Plots of the densities of nonmagnetic impurities at the center of the sample ($L=36$, $M=1152$) as a function of $D_M/J$. Data obtained at the wetting temperature for different values of $H_1/J$, as indicated. Curves have been drawn to guide to the eye only.

Figure 7

(a) Plots of the interface width ($w$) versus $D_M/J$, obtained for samples of size $L=36$, $M=1152$, and $H_1/J=0.60$. Solid circles are obtained along the first-order wetting curve shown in the phase diagram of Fig. 5 for $H_1/J=0.60$ (upper curve). Solid squares are obtained for $T/T_{\mathrm{cb}}=0.79$, i.e., within the wet phase in the phase diagram of Fig. 5. Panel (b) shows a magnetization profile obtained for a sample of size $L=36$, $M=1152$ and $H_1/J=0.60$, $T_w/T_{\mathrm{cb}}=0.68$, and $D_M=-1.0$. The dashed line corresponds to the best fit of the data performed with the aid of Eq. (9).

Figure 8

Plots of $\Delta f_{1L}/J$ versus $H_1/J$ obtained at different temperatures (as indicated) and for the case of the Ising model, namely $D_M=-\infty$. Horizontal (solid) lines correspond to the interfacial free energy taken from the exact solution of Onsager [28]. Vertical (dashed) lines correspond to the critical fields of the wetting phase transitions given from the exact solution of Abraham [29]. Intersection points are indicated by means of arrows.

Figure 9

Plots of the interfacial free energy $\sigma /J$ versus $T/T_{\mathrm{cb}}$ obtained at different values of $D_M$, as indicated. Note that our results for the Ising model (solid circles), namely $D_M=-\infty$, are in agreement with the exact solution of Onsager [see Eq. (19)] shown by a dashed line.

Figure 10

Plots of $\Delta f_{1L}$ versus $H_1/J$ obtained at different temperatures (as indicated) and for the case $D_M=0$. Horizontal lines correspond to the interfacial free energy obtained by thermodynamic integration (see Fig. 9). The intersection points at the first-order wetting transitions are shown by means of arrows.

Figure 11

(a) Plots of $H_{1c}/J$ versus $T/T_{\mathrm{cb}}$ obtained for the case $D_M=0$. Results obtained by means of thermodynanic integration are shown in solid circles, while estimations of the location of abrupt drops of the magnetization (see Fig. 2) are shown in solid squares. (b) Plots of $H_{1c}/J$ versus ${(1-T/T_{\mathrm{cb}})}^{{\Delta }_1}$, with ${\Delta }_1=1$. The dashed line has been drawn to guide the eye.