Surface growth during random and irreversible multilayer deposition of straight semirigid rods

Surface growth properties during irreversible multilayer deposition of straight semirigid rods on linear and square lattices have been studied by Monte Carlo simulations and analytical considerations. The filling of the lattice is carried out following a generalized random sequential adsorption mechanism where the depositing objects can be adsorbed on the surface forming multilayers. The results of our simulations show that the roughness evolves in time following two different behaviors: an “homogeneous growth regime” at initial times, where the heights of the columns homogeneously increase, and a “segmented growth regime” at long times, where the adsorbed phase is segmented in actively growing columns and inactive nongrowing sites. Under these conditions, the surface growth generated by the deposition of particles of different sizes is studied. At long times, the roughness of the systems increases linearly with time, with growth exponent $\beta =1$, at variance with a random deposition of monomers which presents a sublinear behavior ($\beta =1/2$). The linear behavior is due to the segmented growth process, as we show using a simple analytical model.

Figure 1

Schematic representation of tetramers ($k=4$) deposited (a) on a linear lattice with $L=22$ and (b) along a line in the $y$ direction of a square lattice with $L=22$. In both cases, periodic boundary conditions have been applied. As the $k$-mers are semirigid, they can deform to find adjacent empty sites between layers (but always linear). Open squares joined by lines correspond to tetramers previously deposited onto the lattice. Gray-filled squares joined by lines represent particles attempting to deposit on the substrate. Green-solid squares and red-crossed squares represent allowed and forbidden deposition states, respectively. Forbidden configurations do not satisfy the deposition rules ($\left|\mathrm{\Delta }h\right|>1$) and, consequently, are discarded. In panel (b), the units of the transverse $k$-mers (adsorbed along the $x$-direction) located at the crossing sites are indicated by black-solid squares.

Figure 2

Surface roughness as a function of time (in log-log scale) for 1D lattices with $L=5120$ [panel (a)] and 2D lattices with $L=512$ [panel (b)]. The different curves correspond to different values of $k$, as indicated in the key. The growth exponent keeps consistency with the case of RD monomers, whose exact solution is ($\beta =1/2$) [35]. Two growth regimes are observed for $k\ge 2$, with $\beta =1$ the growth exponent for long times.

Figure 3

Typical configurations of the adsorbed phase in the SGR. Two cases are shown in the figure: (a) 1D system with $k=2$; and (b) 2D system with $k=3$. Solid squares [panel (a)] and solid cubes [panel (b)] represent $k$-mer units. As time $t$ increases, the $k$-mer units tend to form islands of length $k$ in each layer.

Figure 4

(a) Time dependence of the scaled roughness $w\xi$. In the long-time limit, 1D and 2D results collapse to the same linear behavior in logarithmic scale. The dashed line corresponds to $1/2log{\tilde{\mathrm{\Omega }}}_a+logt$, where ${\tilde{\mathrm{\Omega }}}_a=0.238$. (b) Towers density $m/L^D$ as a function of $k$ for 1D and 2D lattices. Symbols represent simulation data with $L=5120$ (1D case, solid squares) and $L=512$ (2D case, solid circles). Dashed lines correspond to the functions ${\theta }_{a,1\text{D}}/k$ (1D case, ${\theta }_{a,1\text{D}}=0.61$) and ${\theta }_{a,2\text{D}}/k$ (2D case, ${\theta }_{a,2\text{D}}=0.39$). The results confirm the analytically predicted values of the fraction of active sites.