Surface pattern formation and scaling described by conserved lattice gases

We extend our $2+1$-dimensional discrete growth model [Ódor et al., Phys. Rev. E 79, 021125 (2009)] with conserved, local exchange dynamics of octahedra, describing surface diffusion. A roughening process was realized by uphill diffusion and curvature dependence. By mapping the slopes onto particles, two-dimensional nonequilibrium binary lattice model emerges, in which the (smoothing or roughening) surface diffusion can be described by attracting or repelling motion of oriented dimers. The binary representation allows simulations on very large size and time scales. We provide numerical evidence for Mullins-Herring or molecular-beam epitaxy class scaling of the surface width. The competition of inverse Mullins-Herring diffusion with a smoothing deposition, which corresponds to a Kardar-Parisi-Zhang (KPZ) process, generates different patterns: dots or ripples. We analyze numerically the scaling and wavelength growth behavior in these models. In particular, we confirm by large size simulations that the KPZ type of scaling is stable against the addition of this surface diffusion, hence this is the asymptotic behavior of the Kuramoto-Sivashinsky equation as conjectured by field theory in two dimensions, but has been debated numerically. If very strong, normal surface diffusion is added to a KPZ process, we observe smooth surfaces with logarithmic growth, which can describe the mean-field behavior of the strong-coupling KPZ class. We show that ripple coarsening occurs if parallel surface currents are present, otherwise logarithmic behavior emerges.

Figure 1

(Color online) Mapping of the $2+1$-dimensional surface diffusion onto a 2d particle model (bullets). Surface attachment (with probability $p$) and detachment (with probability $q$) correspond to Kawasaki exchanges of particles or to anisotropic migration of dimers in the bisectrix direction of the $x$ and $y$ axes. The crossing points of dashed lines show the base sublattice to be updated. Thick solid line on the surface shows the $y$ cross section, reminding us to the one-dimensional rooftop model. When the shown desorption or absorption steps are executed simultaneously, they realize a surface diffusion step of size $s=3$ along the $y$ axis. This corresponds to a pair of dimer movement, a repulsion in case of this smoothing reaction.

Figure 2

(Color online) One-dimensional view of a diffusion hop of octahedra by three lattice units to the right. The slopes mapped onto the lattice gas shown below the surface. This roughening surface diffusion move corresponds to two simultaneous, attracting Kawasaki exchanges of the gas (dimers in two dimensions). Starting from the flat (zig-zag) initial state of the pure octahedron model, the (18) process freezes following slopes of maximal length are developed, which cannot be overjumped.

Figure 3

(Color online) Data collapse of the anisotropic LHOD model assuming MH class exponents for $p_{+x}=1$ (diffusion to the right) with small EW noise (a) $p=q=0.01$ and (b) $p=q=0.005$ for sizes $L=32,64,128$ (top to bottom at the left side). For the growth exponent fitting (dashed line) results in $\beta =0.26\left(1\right)$.

Figure 4

(Color online) Data collapse of the isotropic LHOD model assuming MBE (left, $l_m=4$) and MH (right, $l_m=3$) class exponents in the presence of small EW noise $p=q=0.005$ in sizes $L=32,64,128$ (top to bottom at the left side). For the growth exponent power-law fitting (dashed line) results in $\beta =0.26\left(1\right)$. Inserts show the effective $\beta$ exponents for $L=64$.

Figure 5

(Color online) Local slope ${\sigma }_{\chi }\left(i,j\right)$ (circles) and curvature $c_{\chi }\left(i,j\right)$ (squares) variables at an update site. Filled circles correspond to upward, empty ones to downward slopes of the surface. This plaquette configuration models a valley bottom site, with the total curvature $H=4$.

Figure 6

(Color online) Scaling behavior of the isotropic LCOD model for $L=32,64,128$ (top to bottom at the left side). The data collapse has been achieved with the MH class exponents. For the growth exponent fitting (dashed line) results in $\beta =0.25\left(1\right)$. Insert on the right shows the same by local slopes. Insert on the left shows the evolution of $\lambda$.

Figure 7

(Color online) Snapshot of surface heights of the ripple patterns generated by the parameters $p_{\pm x}=1$, $p_{\pm y}=0$ (anisotropic, inverse MH) and $p=q=1$ adsorption or desorption at $t={10}^4$ MCS in the LHOD model of linear size $L=512$.

Figure 8

(Color online) Wavelength growth in the LHOD model for anisotropic diffusion with steady dc current $p_{+y}=1$, $p=q=0.005$ for sizes $L=32,64,128$ (top to bottom at the beginning). (Dashed line) Power-law fit with the exponent $\beta =0.24\left(1\right)$. Left insert shows the corresponding pattern. Right insert corresponds the isotropic diffusion case $p_{\pm x}=p_{\pm y}=1$, where $\lambda \left(t\right)$ grows logarithmically.

Figure 9

(Color online) Data collapse for deposition $\left(p=1,q=0\right)$ and anisotropic, inverse diffusion $p_{\pm y}=1$ in the LHOD model with KPZ class exponents for $L=64,128,256,512,1024$ (top to bottom curves at the right side). (Right insert) $\lambda \left(r\right)$ for $L=1024$. (Left insert) Blurred ripple structure.

Figure 10

(Color online) Snapshot of surface heights of the dot patterns generated by the LHOD model with parameters $p_{\pm x}=p_{\pm y}=1$ (isotropic, inverse MH) and $p=q=0.1$ at $t={10}^4$ MCS.

Figure 11

(Color online) Data collapse of the $L=128,\dots ,1024$ LHOD model $\left(p_{\pm x}=p_{\pm y}=1\right)$ with a competing deposition $\left(p=1\right)$ process. One can see a very slow crossover toward KPZ scaling. Right insert shows the growth of $\lambda$ for $L=512$. Left insert is a snapshot of the steady state, corresponding to the smeared KPZ height distribution.

Figure 12

(Color online) Data collapse of the $L=128,\dots ,1024$ (top to bottom at the right) LCOD model $\left(p_{\pm x}=p_{\pm y}=1\right)$ with competing deposition $\left(p=1\right)$. One can see clear KPZ scaling. Left insert shows the logarithmic growth of $\lambda$ for $L=128,256,512$ (bottom to top). (Right insert) ${\beta }_{eff}$ as the function of time.

Figure 13

(Color online) Data collapse of KPZ deposition $\left(p=1\right)$ and weak, isotropic normal LHOD (higher curves) for $L=64,128,\dots ,2048$ (top to bottom). In case of strong diffusion (lower curves), the KPZ scaling disappears and as the insert shows, logarithmic growth can be observed.

Figure 14

(Color online) Wavelength saturates quickly for $\text{KPZ}+\text{weak}$ LHOD (higher curve) and $\text{KPZ}+\text{strong}$ LHOD (lower curve) diffusion $\left(L=2048\right)$. Insert shows ${\lambda }_{\text{max}}$ vs $L$.

Figure 15

(Color online) Comparison of the $P\left(W^2\right)$ of the higher dimensional octahedron model results (symbols) to those of [69] (lines) in $d=2,3,4,5$ spatial dimensions (bottom to top).

Figure 16

(Color online) Comparison of $P\left(W^2\right)$ of the $\text{KPZ}+\text{LHOD}$ (black boxes), $\text{KPZ}+\text{inverse}$ LHOD (blue dots), $\text{KPZ}+\text{inverse}$, anisotropic LHOD (pink rhombuses), and $\text{KPZ}+\text{inverse}$ LCOD (orange triangles) to that of the KPZ from Ref. [69] (solid line).