Thermodynamic Casimir effect for films in the three-dimensional Ising universality class: Symmetry-breaking boundary conditions

We study the thermodynamic Casimir force for films in the three-dimensional Ising universality class with symmetry-breaking boundary conditions. To this end we simulate the improved Blume-Capel model on the simple cubic lattice. We study the two cases $++$, where all spins at the boundary are fixed to $+1$ and $+-$, where the spins at one boundary are fixed to $+1$ while those at the other boundary are fixed to $-1$. An important issue in analyzing Monte Carlo and experimental data are corrections to scaling. Since we simulate an improved model, leading corrections to scaling, which are proportional to $L_0^{-\omega }$, where $L_0$ is the thickness of the film and $\omega \approx 0.8$, can be ignored. This allows us to focus on corrections to scaling that are caused by the boundary conditions. The analysis of our data shows that these corrections can be accounted for by an effective thickness $L_{0,eff}=L_0+L_s$. Studying the correlation length of the films, the energy per area, the magnetization profile, and the thermodynamic Casimir force at the bulk critical point we find $L_s=1.9\left(1\right)$ for our model and the boundary conditions discussed here. Using this result for $L_s$ we find a nice collapse of the finite-size scaling curves obtained for the thicknesses $L_0=8.5$, 16.5, and 32.5 for the full range of temperatures that we consider. We compare our results for the finite-size scaling functions ${\theta }_{++}$ and ${\theta }_{+-}$ of the thermodynamic Casimir force with those obtained in a previous Monte Carlo study, by the de Gennes-Fisher local-functional method, field theoretic methods, and an experiment with a classical binary liquid mixture.

Figure 1

(Color online) We plot $L_{0,eff}^{\beta /\nu }m\left(z\right)$ as a function of $z/L_{0,eff}$, where $z=x_0-\left(L_0+1\right)/2$ gives the distance from the middle of the film. The effective thickness of the film is $L_{0,eff}=L_0+L_s$ using $L_s=1.9$. For $++$ and $+-$ boundary conditions, data for films with $L_0\ge 16$ are used.

Figure 2

(Color online) $+-$ boundary conditions. In the upper part of the figure we plot $-L_0^3\Delta f_{ex}$ as a function of $t{\left(L_0/{\xi }_0\right)}^{1/\nu }$ for $L_0=8.5$, 16.5, and 32.5, where we use $\nu =0.63002$ and ${\xi }_0=0.2282$. In the lower part we have replaced $L_0$ by $L_{0,eff}=L_0+L_s$ with $L_s=1.9$. For a discussion see the text.

Figure 3

(Color online) $++$ boundary conditions. In the upper part of the figure we plot $-L_0^3\Delta f_{ex}$ as a function of $t{\left(L_0/{\xi }_0\right)}^{1/\nu }$ for $L_0=8.5$, 16.5, and 32.5, where we use $\nu =0.63002$ and ${\xi }_0=0.2282$. In the lower part we have replaced $L_0$ by $L_{0,eff}=L_0+L_s$ with $L_s=1.9$. For a discussion see the text.

Figure 4

(Color online) We plot our numerical results for ${\theta }_{++}\left(x\right)$ and $-{\theta }_{+-}\left(x\right)$ obtained with $L_0=16.5$ and 32.5 using $L_s=1.9$ for $x>0$. For comparison we give the result of Eq. (31), setting $C^2=1.5$. For a discussion see the text.

Figure 5

(Color online) We plot our numerical results for ${\theta }_{++}\left(x\right)$ obtained with $L_0=16.5$ and 32.5 using $L_s=1.9$ for $x<0$. For comparison we give the result of Eq. (33), setting ${\overline{C}}^2=0.2$. For a discussion see the text.

Figure 6

(Color online) For $++$ boundary conditions, we plot ${\xi }_{2nd}/L_{0,eff}$ as a function of the scaling variable $x=t{\left[L_{0,eff}/{\xi }_0\right]}^{1/\nu }$ for $L_0=8$, 9, 16, 17, 32, and 33 using $L_s=1.9$. Notice that ${\xi }_{2nd}$ is the second moment correlation length of the film while ${\xi }_0$ appearing in the scaling variable $x$ is the amplitude of the correlation length of the bulk system in the high-temperature phase.

Figure 7

(Color online) Same as Fig. 6 for $+-$ instead of $++$ boundary conditions.

Figure 8

(Color online) We plot the result of Ref. 18 for ${\theta }_{++}\left(x\right)$ obtained by using the de Gennes-Fisher local-functional (LF) method. We have copied the curve from Fig. 1 of Ref. 18. For comparison we plot our numerical results for ${\theta }_{++}\left(x\right)$ obtained with $L_0=16.5$ and 32.5 using $L_s=1.9$.