Probing predictions due to the nonlocal interface Hamiltonian: Monte Carlo simulations of interfacial fluctuations in Ising films

Extensive Monte Carlo simulations have been performed on an Ising ferromagnet under conditions that would lead to complete wetting in a semi-infinite system. We studied an $L\times L\times D$ slab geometry with oppositely directed surface fields so that a single interface is formed and can undergo a localization-delocalization transition. Under the chosen conditions the interface position is, on average, in the middle of the slab, and its fluctuations allow a sensitive test of predictions that the effective interactions between the interface and the confining surfaces are nonlocal. The decay of distance dependent correlation functions are measured within the surface, in the middle of the slab, and between middle and the surface for slabs of varying thickness $D$. From Fourier transforms of these correlation functions a nonlinear correlation length is extracted, and its behavior is found to confirm theoretical predictions for $D>6$ lattice spacings.

Figure 1

Schematic illustration of the configuration of a coarse-grained interface (shaded) at local height $z=\ell (x,y)$ separating a liquid film at a wall from the vapor phase in thermal equilibrium at conditions where the phases can coexist in the bulk. The description in terms of the capillary wave Hamiltonian assumes that the only important degrees of freedom are smooth, long-wavelength undulations of the interface, with typical correlation length ${\xi }_{\left|\right|}$, which is much larger than any molecular linear dimensions close to a second-order wetting transition.

Figure 2

Schematic illustration of the diagrams representing the leading-order contributions to the expansion of the nonlocal interaction between the wall (horizontal straight line) and the fluctuating interface (curved line).

Figure 3

(Schematic sketch of an interface at height $z=\ell (x,y)$ with average height $\overline{\ell }$ (broken horizontal straight line) above a wall at $z=0$ (shaded). The thick straight line between points at the surface at distance $r$ indicated by squares indicates the correlation $C(0,0,r)$, while the thick straight line between two points at distance $z=\overline{\ell }$ (circles) indicates the correlation $C(\overline{\ell },\overline{\ell },r)$. The third thick straight line connecting a point at the surface with a point at $z=\overline{\ell }$ represents the correlation $C(0,\overline{\ell },r)$.

Figure 4

Schematic view of an $L\times L\times D$ Ising system with antisymmetric surface fields $H_1<0$, $H_D=-H_1>0$, and periodic boundary conditions (pbc) in $x,y$ directions. The walls (free surfaces where surface fields act) at $z=0$, $z=D$ are shaded. The signs of the magnetization in the domains separated by the interface (wavy line) are shown by thick arrows.

Figure 5

Correlation functions $C(D/2,D/2,r)$ (top-most curves), $C(0,D/2,r)$ (curves in the middle), and $C(0,0,r)$ (bottom curves) plotted vs $r$ for the case $D=12$. Two data sets [for $L=256$ (red open circles, squares, and triangles) and $L=512$ (black closed circles, squares, and triangles)] are included in each case.

Figure 6

Correlation functions (a) $C\left(\frac{D}{2},\frac{D}{2},r\right)$, (b) $C\left(0,\frac{D}{2},r\right)$, and (c) $C(0,0,r)$ plotted vs $r$ for the choice $L=512$, $J/k_{B}T=0.236.$ Six choices of $D$ are included, as indicated.

Figure 7

Correlation functions $G\left(\frac{D}{2},\frac{D}{2},q\right)$ (upper set of curves) and $\mathbf{G}\left(0,\frac{D}{2},q\right)$ plotted vs wave number $q$ for $L=512$, $J/k_{B}T=0.236$. Case (a) shows the full range $0\le q\le \pi$ (note that our unit of length is the lattice spacing), case (b) shows the region of small $q$, to which the theoretical formulas [Eqs. (7) and (8)] should apply.

Figure 8

Ratio of the correlation functions $G(0,D/2,q)/G(D/2,D/2,q)$ plotted vs $q^2$ for $L=512$, $J/k_{B}T=0.2360$, and several choices of $D$ as indicated. Note the logarithmic scale of the ordinate: straight lines indicate that the proposed variation proportional $exp(-q^2{\xi }_{\text{NL}}^2/2)$ holds.

Figure 9

Plot of nonlocal correlation length (${\xi }_{\text{NL}}^2/2$) vs film thickness ($D$) for the choice $L=512$, $J/k_{B}T=0.236$.

Figure 10

Plot of parallel correlation length ($\text{ln}{\xi }_{\left|\right|}$) vs film thickness ($D$) for the choice $L=512$, $J/k_{B}T=0.236$.