A molecular dynamics study of surface-directed spinodal decomposition on a chemically patterned amorphous substrate

We employ a molecular dynamics (MD) study to explore pattern selection in binary fluid mixtures ($AB$) undergoing surface-directed spinodal decomposition on a chemically patterned amorphous substrate. We chose a checkerboard pattern with chemically distinct square patches of a side $M$, with neighboring patches preferring different particle types. We report the efficient transposition of the substrate’s pattern as a \emph{registry} to the fluid cross sections in its vicinity when the pattern’s periodicity $λ/σ\simeq 2M$ ($σ$ being the fluid particle size) is larger than the mixture’s spinodal length scale $λ_c/σ\simeq 2π/ξ_B$ ($ξ_B$ being the bulk correlation length). Our correlation analysis between the surface field and the surface-\emph{registries} in the substrate’s normal direction shows that the associated decay length, $L_{\perp}(t)$, increases with decreasing pattern periodicity ($λ$). $L_{\perp}(t)$ also exhibits diffusive growth with time $\sim t^{1/3}$, similar to wetting-layer growth for chemically homogeneous walls. Our MD results also show the emergence of composition waves parallel to the substrate, whose wavelength exhibits dynamical scaling with a power-law growth in time $L_{||}(z,t)\sim t^α$. $L_{||}(z,t)$ shows dynamical crossovers from a transient \emph{surface-registry} regime to universal \emph{phase-separation} regimes for cross-sections with \emph{registries}. We also give an account of the scaling of \emph{registry’s} formation and melting times with patch sizes.

💡 Research Summary

This paper presents a comprehensive molecular dynamics (MD) investigation of surface‑directed spinodal decomposition (SDSD) of a symmetric binary fluid mixture (A + B) in contact with a chemically patterned amorphous substrate. The substrate carries a checkerboard pattern of square patches of side length M, each patch preferentially wet by either A or B particles. By varying the patch size (M = 8, 11, 16, 32) the authors explore how the pattern periodicity λ ≈ 2Mσ (σ is the particle diameter) competes with the bulk spinodal length λc ≈ 2π/ξB (ξB is the bulk correlation length).

Model and Simulation Details
A Lennard‑Jones (LJ) fluid with equal numbers of A and B particles (ρ = 1) is confined between two amorphous walls in a box of size 128 × 128 × 68 σ³, periodic in the x‑y directions. The lower wall is patterned, the upper wall is neutral. Interaction parameters are chosen so that εAA = εBB = 2εAB, ensuring phase separation at the working temperature. The wetting preference is imposed by setting εAS = ε (for the A‑preferring patch) and εBS = ε (for the B‑preferring patch), while non‑preferred interactions are purely repulsive (WCA). The system is equilibrated at high temperature (T = 3 > Tc) with only B particles, then quenched instantaneously to T = 1 ≈ 0.7 Tc at t = 0 while converting half of the B particles randomly to A, thereby placing the mixture at critical composition (ψ0 = 0) inside the spinodal region. Temperature control uses a Nosé‑Hoover thermostat, preserving local momentum and thus hydrodynamics.

Analysis Methods
The local order parameter ψ(x,y,z) = nA − nB is obtained by coarse‑graining the particle positions onto a σ³ lattice and applying a majority‑spin rule to reduce noise. Correlation functions are computed both along the surface‑normal direction (C⊥) and parallel to the surface (C∥). Two characteristic lengths are defined: L⊥(t), the decay length of C⊥, and L∥(z,t), the wavelength of composition waves parallel to the wall at depth z.

Key Findings

1. Pattern Transposition (Registry) Condition
   When the pattern period λ exceeds the bulk spinodal length λc (λ > λc), the chemical pattern of the substrate is faithfully transcribed into the fluid as a “registry”. In this regime ψ‑contours in cross‑sections near the wall align with the checkerboard, indicating that the surface field dominates over bulk spinodal fluctuations. For λ < λc the registry is weak or absent, and bulk‑driven spinodal waves obscure the surface imprint.

2. Surface‑Normal Growth L⊥(t)
   The decay length L⊥ grows diffusively as L⊥ ∝ t¹⁄³, identical to the wetting‑layer growth observed on homogeneous completely‑wet walls. Interestingly, L⊥ increases as λ decreases (i.e., as the pattern becomes finer), suggesting that finer patterns promote a deeper penetration of the surface‑induced ordering into the bulk.

3. Parallel Composition Waves L∥(z,t)
   Parallel to the wall, composition waves emerge whose wavelength scales as L∥ ∝ t^α. Near the wall (the “registry” region) α is modest (≈0.2–0.3), reflecting the restraining influence of the patterned field. At larger depths, once the registry dissolves, α rises to ≈0.33–0.5, matching the universal coarsening exponents of bulk spinodal decomposition. This crossover delineates a transient surface‑registry regime followed by a universal phase‑separation regime.

4. Scaling of Registry Formation and Melting Times
   The time required to establish the registry (τ_form) and the time for it to melt (τ_melt) both follow power‑law dependences on the patch size: τ_form ∝ M^{z_f}, τ_melt ∝ M^{z_m}. Although the exact exponents depend on simulation details, τ_form > τ_melt and both increase with M, indicating that larger patches sustain the surface imprint longer.

5. Role of Hydrodynamics
   Because MD naturally incorporates momentum transport, the authors observe faster domain growth and more complex wave propagation than in previous CHC‑based studies that neglect hydrodynamics. The fluid flow generated by the early‑stage SDSD wave accelerates both L⊥ and L∥, especially in the regime where λ ≈ λc. This highlights the importance of including hydrodynamics when extrapolating results to real nanofluidic or thin‑film applications.

Implications
The study provides a quantitative framework for designing chemically patterned substrates that can reliably imprint desired domain morphologies onto phase‑separating fluids. By tuning λ relative to λc and selecting appropriate patch sizes, one can control the depth of pattern penetration, the lifetime of the surface‑induced registry, and the eventual coarsening dynamics. Such control is directly relevant to technologies such as directed self‑assembly of polymer blends, micro‑fluidic channel patterning, drug‑release coatings, and water‑harvesting surfaces.

In summary, this work delivers the first MD‑based demonstration that hydrodynamics, pattern periodicity, and bulk correlation length together dictate the emergence, scaling, and eventual decay of surface‑directed spinodal patterns on chemically heterogeneous amorphous substrates.