Rough-to-Smooth Growth Transition

• Rough-to-smooth growth mode is a process where a system transitions from high surface roughness to smooth topographies driven by control parameters like temperature, flux, and diffusion rates.
• It is observed in diverse applications including thin film deposition, crystal growth, and turbulent boundary layers, where changes in mechanism lead to altered scaling behavior and correlations.
• Theoretical and experimental models, such as AKPZ dynamics and kinetic Monte Carlo simulations, quantify shifts in universality classes and recovery timescales with practical implications for material synthesis and flow management.

Rough-to-Smooth Growth Mode

The rough-to-smooth growth mode describes a class of physical, mathematical, and computational processes in which a system transitions from a regime characterized by heightened morphological roughness or large fluctuations (“rough” phase) to a regime exhibiting minimal fluctuations, bounded correlations, or smooth topographies (“smooth” phase). This mode is observed in diverse contexts, including kinetic interface growth, thin-film deposition, turbulent boundary layers, crystal growth, grain boundary evolution in 2D materials, stochastic growth models, and number theory. The rough-to-smooth transition may be driven by changes in thermodynamic parameters (temperature, supersaturation), kinetic rates (fluxes, diffusion), or external controls (substrate interaction, shear flow), and is often associated with qualitative changes in universality class, scaling exponents, or dynamical regimes.

1. Theoretical Frameworks for Rough-to-Smooth Transitions

In $(2+1)$-dimensional stochastic interface growth, the archetype for rough-to-smooth transitions is provided by anisotropic KPZ (AKPZ) models. Chhita and Toninelli introduced a model based on domino tiling dynamics with $2$-periodic weights on the toroidal square lattice, with stationary measures given by translation-invariant Gibbs distributions for perfect matchings (Chhita et al., 2018). The slope $\rho$ parameterizes the macroscopic profile, and the space-time fluctuation properties are governed by the Hessian $H_\rho$ of the growth speed $v(\rho)$. For nonzero slopes $(\rho \ne 0)$, fluctuations grow logarithmically in both space and time, placing the model in the AKPZ class ($\det H_\rho<0$). At $\rho = 0$, the system enters a smooth phase: correlations become bounded, $\det H_\rho$ develops a singular spectrum, and $v(\cdot)$ loses differentiability.

Such rough-to-smooth transitions arise as a consequence of underlying changes in the dominant mechanism of fluctuation propagation—ranging from massless Gaussian Free Field regimes in the rough phase to gapped, bounded variances in the smooth phase.

Related mean-field and kinetic models of surface and crystal growth identify analogous transitions with finite two-dimensional nucleation barriers. At low driving (e.g., low supersaturation $\Delta\mu$), step-flow growth leads to smooth surfaces. Increasing $\Delta\mu$ above a kinetic threshold triggers homogeneous nucleation of new islands, resulting in a sharp crossover to rough growth characterized by a sudden increase in roughness and kinetic slope—a kinetic phase transition (Lutsko et al., 2010).

2. Growth Mode Evolution in Thin Film Deposition

Thin film systems on weakly interacting substrates exemplify the rough-to-smooth growth mode in real materials. In canonical layer-by-layer (LBL) growth (Frank–van der Merwe), roughness oscillates below $1$ monolayer (ML), whereas in the Volmer–Weber (island) mode, roughness rises monotonically with thickness. The Lapkin et al. study reveals a distinct non-monotonic evolution: after deposition initiates, roughness $\sigma(h)$ rises sharply to a pronounced maximum $\sigma_m$ at coverage $\theta_m$ (in ML/nm), then falls as islands coalesce and inter-island trenches fill, smoothing the film for large $h$ (Lapkin et al., 24 Jan 2026).

A geometrical model captures this by representing each island as a block with evolving lateral and vertical size, with roughness $\sigma(\theta)$ predicted by

$$\sigma / \theta_m = u \cdot \sqrt{(1/(1-\alpha)) u^{-2\alpha} - 1}$$

where $u = \theta/\theta_m$, $\alpha$ determines lateral size scaling, and $\sigma_m/\theta_m = \sqrt{\alpha/(1-\alpha)}$. Kinetic Monte Carlo simulations parameterized by adsorbate-substrate and adsorbate-adsorbate binding energies confirm this non-monotonicity as a generic outcome whenever substrate diffusion exceeds film diffusion and Ehrlich–Schwoebel barriers are small. The rough-to-smooth mode thus provides a universally accessible pathway for engineering smooth organic films through substrate and flux control.

3. Experimental and Modeling Realizations

A. Crystal Growth and Boundary Layer Systems

In halide vapor phase epitaxy (HVPE) of GaN, the two-stage (LT$\rightarrow$HT) protocol exploits the rough-to-smooth transition: initial low-temperature (LT) growth produces a rough, crack-free layer with dense pit arrays; subsequent switch to high-temperature (HT) growth induces pit overgrowth and mirrorsmooth surfaces, due to enhanced fast-facet kinetics (Voronenkov et al., 2019). Morphology, pit density, and cracking propensity are tuned as functions of growth rate, temperature, and layer sequence.

Similarly, in InSbBi ternary films grown by MBE, a transition is observed from mound-dominated Stranski–Krastanov (SK) morphology (high Bi flux, low $T$; rough) to atomically smooth step-flow (SF) morphology as substrate temperature is increased and adatom diffusion length exceeds terrace width (Edirisinghe et al., 14 Mar 2025). Thermodynamic strain energy (SK nucleation) and kinetic surface diffusion (SF step-attachment) compete, with the critical crossover set by the Matthews–Blakeslee balance for critical thickness and a diffusion-length criterion $L_s \gg w$.

B. Grain Boundaries in 2D Materials and Stochastic Integer Models

In CVD and MBE growth of 2D materials, Aarão Reis et al. identify a high-coverage regime ($F/Q \gtrsim 10$), where the precursor flux is sufficient to fully fill inter-flake gaps before any substantial edge advance. This produces random-deposition (RD) class spatial roughness ($W \sim t^{1/2}$, spatially uncorrelated) which rapidly relaxes by step-attachment dynamics (Edwards–Wilkinson smoothing) to yield grain boundaries with much lower ultimate roughness and greatly reduced processing time compared to near-equilibrium or diffusion-limited protocols (Reis et al., 2022).

A structurally analogous regime is present in the combinatorics of rough-to-smooth integer growth, where the count of $[B,C]$-grained $k$-factor integers transitions from being rough-dominated ($s \to 0$) to smooth-dominated ($s \to k$) as the logarithmic ratios and convolution polynomials shift the distribution's center (Loebenberger et al., 2010).

4. Rough-to-Smooth Transitions in Flow and Turbulence

The canonical “rough-to-smooth” adjustment problem in wall-bounded turbulence is the evolution (and recovery) of a turbulent boundary layer after an abrupt change from a rough to a smooth wall (Li et al., 2023, Li et al., 2023). Upon transition, the wall-shear stress and mean velocity profile do not instantaneously adopt their equilibrium smooth-wall forms. Instead, recovery follows a hierarchical pattern:

• The viscous sublayer ($z^+ \lesssim 4$) recovers almost immediately (within $0.1\delta$ downstream).
• The buffer layer and energy-containing motions require several boundary-layer thicknesses $\delta$ to return to equilibrium.
• An internal boundary layer (IBL) of thickness $$\delta_i(\hat x) / \delta_0 = 0.095 (\hat x / \delta_0)^{0.77}$$ grows with distance from the roughness step.

Empirically, the mean velocity profile downstream can be modeled by blending smooth- and rough-wall log laws via a universal, self-similar function of $y/\delta_i$; the skin-friction coefficient recovers over $\sim 20\delta_0$. Spectral analysis shows that the small-scale near-wall turbulence rapidly equilibrates, while over-energized large-scale motions persist as a “footprint” for tens of $\delta_0$. This multi-timescale recovery underscores the nontrivial, layer-dependent character of rough-to-smooth growth modes in turbulent flows.

5. Mechanistic Insights and Universality

Rough-to-smooth transitions manifest due to control-parameter-induced regime changes in the dominant kinetics, driven either by thermodynamic thresholds (e.g., critical thickness for misfit strain, nucleation barriers) or by sharp kinetic crossovers (e.g., diffusion lengths, precursor fluxes, adatom residence times). In stochastic growth, transitions can be tied to non-differentiability of the speed function $v(\rho)$ and singularities in $H_\rho$ (Chhita et al., 2018).

Universality emerges in the sense that, despite system-specific microphysics, the qualitative structure of a transition from rough to smooth persists: initial rapid roughening (logarithmic or power-law scale-up) gives way to bounded or decaying fluctuations as system control parameters cross critical thresholds. The transition is found in models as disparate as kinetic Monte Carlo grain boundary relaxation, domino tiling models exhibiting a liquid-to-gaseous correlation change, thin film growth on weakly interacting substrates, and canonical problems in wall turbulence.

6. Practical Implications and Control

Control of rough-to-smooth modes is pragmatically exploited in advanced thin film and crystal growth techniques. For example, Lapkin et al. demonstrate that by tuning substrate treatments (modifying $E_s$), deposition temperature, and flux, one can reproducibly obtain ultra-smooth organic films via the rough-to-smooth pathway (Lapkin et al., 24 Jan 2026). In GaN HVPE, the combined LT→HT protocol yields $>3$ mm crack-free films with low pit density (Voronenkov et al., 2019). In 2D material synthesis, high-flux, high-temperature operation can dramatically accelerate smoothing and minimize grain boundary roughness (Reis et al., 2022).

In turbulent boundary layer engineering, understanding the multi-layer recovery timescales associated with a rough-to-smooth step transition is crucial for accurate skin friction estimation and drag modeling in aeronautics, meteorology, and hydrodynamics (Li et al., 2023).

The rough-to-smooth growth mode thus provides a unifying conceptual and quantitative tool for understanding, predicting, and manipulating system evolution across a wide array of condensed matter, statistical, and applied mathematical disciplines.