Scaling in the crossover from random to correlated growth

Scaling of roughness and porosity in thin film deposition with mixed transport mechanisms and adsorption barriers

Thin film deposition with particle transport mixing collimated and diffusive components and with barriers for adsorption are studied using numerical simulations and scaling approaches. Biased random walks on lattices are used to model the particle flux and the analogy with advective-diffusive transport is used to define a Peclet number P that represents the effect of the bias towards the substrate. An aggregation probability that relates the rates of adsorption and of the dominant transport mechanism plays the role of a Damkohler number D, where D≲1 is set to describe moderate to low adsorption rates. Very porous deposits with sparse branches are obtained with P≪1, whereas low porosity deposits with large height fluctuations at short scales are obtained with P≫1. For P≳1 in which the field bias is intense, an initial random deposition is followed by Kardar-Parisi-Zhang (KPZ) roughening. As the transport is displaced from those limiting conditions, the interplay of the transport and adsorption mechanisms establishes a condition to produce films with the smoothest surfaces for a constant deposited mass: with low adsorption barriers, a balance of random and collimated flux is required, whereas for high barriers the smoothest surfaces are obtained with P∼D^{1/2}. For intense bias, the roughness is shown to be a power law of P/D, whose exponent depends on the growth exponent β of the KPZ class, and the porosity has a superuniversal scaling as (P/D)^{-1/3}. We also study a generalized ballistic deposition model with slippery particle aggregation that shows the universality of these relations in growth with dominant collimated flux, particle adsorption at any point of the deposit, and negligible adsorbate diffusion, in contrast with the models where aggregation is restricted to the outer surface.

Surface properties and scaling behavior of a generalized ballistic deposition model

The surface exponents, scaling behavior, and bulk porosity of a generalized ballistic deposition (GBD) model are studied. In nature, there exist particles with varying degrees of stickiness ranging from completely nonsticky to fully sticky. Such particles may adhere to any one of the successively encountered surfaces, depending on a sticking probability that is governed by the underlying stochastic mechanism. The microscopic configurations possible in this model are much larger than those allowed in existing models of ballistic deposition and competitive growth models that seek to mix ballistic and random deposition processes. In this article, we find the scaling exponents for surface width and porosity for the proposed GBD model. In terms of scaled width W[over ̃] and scaled time t[over ̃], the numerical data collapse onto a single curve, demonstrating successful scaling with sticking probability p and system size L. Similar scaling behavior is also found for the porosity.

Smoothening in thin-film deposition on rough substrates

The evolution of the surface roughness W of a thin film deposited on a rough substrate is studied with a model of temperature-activated adatom diffusion, irreversible lateral aggregation, and no step energy barrier, in which the main parameter is the ratio R of diffusion and deposition rates. At sufficiently low temperatures (R≲10), the average number of adatom steps after adsorption is very small, thus W monotonically increases with time t due to an approximately uncorrelated deposition at short times. If the temperature is not very low (R∼10(3) or larger), smoothening occurs at short times and the Villain-Lai-Das Sarma (VLDS) growth equation governs the long time roughening, which is attained after a crossover time t(c) that increases with the correlation length ξ(i) of the substrate. Scaling arguments predict the dependence of t(c) on temperature and on the substrate production time and the scaling relation for the difference between the roughness of films deposited on rough and flat substrates, in good agreement with numerical results. The effect of temperature is not a direct extension of previous results on flat substrates because the short wavelength fluctuations delay the formation of terraces. For this reason, the effective energy obtained from the dependence of t(c) on R is 40% of the energy of activated adatom diffusion. A scaling law for the initial smoothening is proposed as W/W(i)=Ψ(t/t(c1)), with a crossover time t(c1)≡R(-θ)ξ(i)(z), where W(i) is the substrate roughness, θ≈0.4, and z is the VLDS dynamical exponent. It provides good data collapse if W is not very small and is suggested to be tested experimentally.

Kinetic roughening and porosity scaling in film growth with subsurface lateral aggregation

We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order a at each position. The model generalizes ballisticlike models by allowing attachment to particles below the outer surface. For small values of a, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions confirm Kardar-Parisi-Zhang roughening of the outer surface for a>0. A scaling approach for small a predicts crossover times as a(-2/3) and local height fluctuations as a(-1/3) at the crossover, independent of substrate dimension. These relations are different from all previously studied models with crossovers from UD to correlated growth due to subsurface aggregation, which reduces scaling exponents. The same approach predicts the porosity and average pore height scaling as a(1/3) and a(-1/3), respectively, in good agreement with simulation results in 1+1 and 2+1 dimensions. These results may be useful for modeling samples with desired porosity and long pores.

Nonuniversal effects in mixing correlated-growth processes with randomness: interplay between bulk morphology and surface roughening

To construct continuum stochastic growth equations for competitive nonequilibrium surface-growth processes of the type RD+X that mixes random deposition (RD) with a correlated-growth process X, we use a simplex decomposition of the height field. A distinction between growth processes X that do and do not create voids in the bulk leads to the definition of the effective probability p(eff) of the process X that is a measurable property of the bulk morphology and depends on the activation probability p of X in the competitive process RD+X. The bulk morphology is reflected in the surface roughening via nonuniversal prefactors in the universal scaling of the surface width that scales in p(eff). The equation and the resulting scaling are derived for X in either a Kardar-Parisi-Zhang or Edwards-Wilkinson universality class in (1+1) dimensions and are illustrated by an example of X being a ballistic deposition. We obtain full data collapse on its corresponding universal scaling function for all p∈(0;1]. We outline the generalizations to (1+n) dimensions and to many-component competitive growth processes.

Langevin equations for competitive growth models

Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension ν and the nonlinear coefficient λ of the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p→0. The superposition of those scaling factors gives ν~p(2) for random deposition with surface relaxation (RDSR) as the CD, and ν~p, λ~p(3/2) for ballistic deposition (BD) as the CD, in agreement with simulation and other scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling of RD-BD model is found. The Langevin equation for the model with competing RDSR and BD, with probability p for the latter, is also constructed. It shows linear p dependence of λ, while the quadratic dependence observed in previous simulations is explained by an additional crossover before the asymptotic regime. The results highlight the relevance of scaling of the coefficients of step function expansions in systems with steep surfaces, which is responsible for noninteger exponents in some p-dependent stochastic equations, and the importance of the physical correspondence of aggregation rules and equation coefficients.

Anomalous roughening in competitive growth models with time-decreasing rates of correlated dynamics

Lattice growth models where uncorrelated random deposition competes with some aggregation dynamics that generates correlations are studied with rates of the correlated component decreasing as a power law. These models have anomalous roughening, with anomalous exponents related to the normal exponents of the correlated dynamics, to an exponent characterizing the aggregation mechanism and to that power-law exponent. This is shown by a scaling approach extending the Family-Vicsek relation previously derived for the models with time-independent rates, thus providing a connection of normal and anomalous growth models. Simulation results for several models support those conclusions. Remarkable anomalous effects are observed even for slowly decreasing rates of the correlated component, which may correspond to feasible temperature changes in systems with activated dynamics. The scaling exponents of the correlated component can be obtained only from the estimates of three anomalous exponents, without knowledge of the aggregation mechanism, and a possible application is discussed. For some models, the corresponding Edwards-Wilkinson and Kardar-Parisi-Zhang equations are also discussed.

Thin-film growth by random deposition of linear polymers on a square lattice

We present some results of Monte Carlo simulations for the deposition of particles of different sizes on a two-dimensional substrate. The particles are linear, height one, and can be deposited randomly only in the two x and y directions of the substrate and occupy an integer number of cells of the lattice. We show there are three different regimes for the temporal evolution of the interface width. At the initial times we observe an uncorrelated growth, with an exponent β1 characteristic of the random deposition model. At intermediate times, the interface width presents an unusual behavior, described by a growing exponent β2, which depends on the size of the particles added to the substrate. If the linear size of the particle is two we have β2 < β1, otherwise we have β2 > β1, for all other particle sizes. After the growth reaches the saturation regime where the interface width becomes constant and is described by the roughness exponent α, which is nearly independent of the size of the particle. Similar results are found in the surface growth due to the electrophoretic deposition of polymer chains. Contrary to one-dimensional results the growth exponents are nonuniversal.

Dynamic scaling in thin-film growth with irreversible step-edge attachment

We study dynamic scaling in a model with collective diffusion (CD) of isolated atoms in terraces and irreversible aggregation at step edges. Simulations are performed in two-dimensional substrates with several diffusion to deposition ratios R identical with D/F. Data collapse of scaled roughness distributions confirms that this model is in the class of the fourth-order nonlinear growth equation by Villain, Lai, and Das Sarma (VLDS) with negligible finite-size effects, while estimates of scaling exponents show some discrepancies. This result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models. The roughness follows dynamic scaling as W=Lalpha/R1/2f(xi/L), with correlation length xi=(Rt)1/z, where z is the dynamic exponent. We also propose a limited mobility (LM) model where the incident atom executes up to G steps before a new atom is adsorbed, and irreversibly aggregates at step edges. This model is also shown to belong to the VLDS class. The size of the plateaus in the film surface increases as G1/2 and the lateral correlation scales as G1/2t1/z. The time evolution of the roughness reproduces that of the CD model if an equivalent parameter G approximately R2/z is chosen. This suggests the possibility of using LM models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms. A scaling approach is used to justify exponent values and dynamic relations for both models, including the significant decrease of surface roughness as R or G increases.

Parameter-free scaling relation for nonequilibrium growth processes

We discuss a parameter-free scaling relation that yields a complete data collapse for large classes of nonequilibrium growth processes. We illustrate the power of this scaling relation through various growth models, such as the competitive growth model with random deposition and random deposition with surface diffusion or the restricted solid-on-solid model with different nearest-neighbor height differences, as well as through a deposition model with temperature-dependent diffusion. The scaling relation is compared to the familiar Family-Vicsek relation, and the limitations of the latter are highlighted.

Random deposition of particles of different sizes

We study the surface growth generated by the random deposition of particles of different sizes. A model is proposed where the particles are aggregated on an initially flat surface, giving rise to a rough interface and a porous bulk. By using Monte Carlo simulations, a surface has grown by adding particles of different sizes, as well as identical particles on the substrate in (1+1) dimensions. In the case of deposition of particles of different sizes, they are selected from a Poisson distribution, where the particle sizes may vary by 1 order of magnitude. For the deposition of identical particles, only particles which are larger than one lattice parameter of the substrate are considered. We calculate the usual scaling exponents: the roughness, growth, and dynamic exponents alpha, beta, and z, respectively, as well as, the porosity in the bulk, determining the porosity as a function of the particle size. The results of our simulations show that the roughness evolves in time following three different behaviors. The roughness in the initial times behaves as in the random deposition model. At intermediate times, the surface roughness grows slowly and finally, at long times, it enters into the saturation regime. The bulk formed by depositing large particles reveals a porosity that increases very fast at the initial times and also reaches a saturation value. Excepting the case where particles have the size of one lattice spacing, we always find that the surface roughness and porosity reach limiting values at long times. Surprisingly, we find that the scaling exponents are the same as those predicted by the Villain-Lai-Das Sarma equation.

Density scaling laws for the structure of granular deposits

The structure of granular deposits growing by arriving particles is analyzed using three-dimensional on-lattice Monte Carlo modeling of convective-diffusive particle deposition. The deposit density profile rho(h) depends on the particle dynamics and becomes formed by three different regions: a denser near-wall region at the deposit bottom in contact with the original plain surface, a middle uniform region with constant mean density, and an open and lighter active-growth region at the deposit outer surface. Fitting expressions for rho(h) valid in each region are proposed, based on the known features of deposits formed in the two limiting cases: ballistic deposition and diffusion-limited deposition. Also, a composite expression for rho(h) fitting the density profile throughout the deposit is given. All these expressions are written in terms of a length scale l(Pe) dependent on the particle Péclet number, which provides the relative importance of the convective motion to the diffusive transport for the particle.

Scaling behavior in corrosion and growth of a passive film

We study a simple model for metal corrosion controlled by the reaction rate of the metal with an anionic species and the diffusion of that species in the growing passive film between the solution and the metal. A crossover from the reaction-controlled to the diffusion-controlled growth regime with different roughening properties is observed. Scaling arguments provide estimates of the crossover time and film thickness as functions of the reaction and diffusion rates and the concentration of anionic species in the film-solution interface, including a nontrivial square-root dependence on that concentration. At short times, the metal-film interface exhibits Kardar-Parisi-Zhang (KPZ) scaling, which crosses over to a diffusion-limited erosion (Laplacian growth) regime at long times. The roughness of the metal-film interface at long times is obtained as a function of the rates of reaction and diffusion and of the KPZ growth exponent. The predictions have been confirmed by simulations of a lattice version of the model in two dimensions. Relations with other erosion and corrosion models and possible applications are discussed.

Surface and bulk properties of deposits grown with a bidisperse ballistic deposition model

We study roughness scaling of the outer surface and the internal porous structure of deposits generated with the three-dimensional bidisperse ballistic deposition (BBD), in which particles of two sizes are randomly deposited. Systematic extrapolation of roughness and dynamical exponents and the comparison of roughness distributions indicate that the top surface has Kardar-Parisi-Zhang (KPZ) scaling for any ratio F of the flux between large and small particles. A scaling theory predicts the characteristic time of the crossover from random to correlated growth in BBD and provides relations between the amplitudes of roughness scaling and F in the KPZ regime. The porosity of the deposits monotonically increases with F and scales as F{12} for small F, which is also explained by the scaling approach and illustrates the possibility of connecting surface growth rules and bulk properties. The suppression of relaxation mechanisms in BBD enhances the connectivity of the deposits when compared to other ballisticlike models, so that they percolate down to F approximately 0.05. The fractal dimension of the internal surface of the percolating deposits is D{F} approximately 2.9, which is very close to the values in other ballistic-like models and suggests universality among these systems.

Scaling Theory in a Model of Corrosion and Passivation

We study a model for corrosion and passivation of a metallic surface after small damage of its protective layer using scaling arguments and simulation. We focus on the transition between an initial regime of slow corrosion rate (pit nucleation) to a regime of rapid corrosion (propagation of the pit), which takes place at the so-called incubation time. The model is defined in a lattice in which the states of the sites represent the possible states of the metal (bulk, reactive, and passive) and the solution (neutral, acidic, or basic). Simple probabilistic rules describe passivation of the metal surface, dissolution of the passive layer, which is enhanced in acidic media, and spatially separated electrochemical reactions, which may create pH inhomogeneities in the solution. On the basis of a suitable matching of characteristic times of creation and annihilation of pH inhomogeneities in the solution, our scaling theory estimates the average radius of the dissolved region at the incubation time as a function of the model parameters. Among the main consequences, that radius decreases with the rate of spatially separated reactions and the rate of dissolution in acidic media, and it increases with the diffusion coefficient of H(+) and OH(-) ions in solution. The average incubation time can be written as the sum of a series of characteristic times for the slow dissolution in neutral media, until significant pH inhomogeneities are observed in the dissolved cavity. Despite having a more complex dependence on the model parameters, it is shown that the average incubation time linearly increases with the rate of dissolution in neutral media, under the reasonable assumption that this is the slowest rate of the process. Our theoretical predictions are expected to apply in realistic ranges of values of the model parameters. They are confirmed by numerical simulation in two-dimensional lattices, and the expected extension of the theory to three dimensions is discussed.

Universal and nonuniversal features in the crossover from linear to nonlinear interface growth

We study a restricted solid-on-solid model involving deposition and evaporation with probabilities p and 1 - p, respectively, in one-dimensional substrates. It presents a crossover from Edwards-Wilkinson (EW) to Kardar-Parisi-Zhang (KPZ) scaling for p approximately 0.5. The associated KPZ equation is analytically derived, exhibiting a coefficient lambda of the nonlinear term proportional to q identical with p - 1/2, which is confirmed numerically by calculation of tilt-dependent growth velocities for several values of p. This linear lambda - q relation contrasts to the apparently universal parabolic law obtained in competitive models mixing EW and KPZ components. The regions where the interface roughness shows pure EW and KPZ scaling are identified for 0.55< or =p< or =0.8, which provides numerical estimates of the crossover times tc. They scale as tc approximately lambda -phi with phi=4.1+/-0.1, which is in excellent agreement with the theoretically predicted universal value phi=4 and improves previous numerical estimates, which suggested phi approximately 3.