Subcritical Pitchfork Bifurcation Transition of a Single Nanoparticle in Strong Confinement

Confinement influences fluid properties. We show, employing molecular dynamics simulations with explicit solvents, that slit confinement drives a first-order transition for a small nanoparticle between staying at the slit center and binding to the slit surfaces. The transition follows a subcritical pitchfork bifurcation, accompanying a similar transition of the nanoparticle’s lateral diffusion, depending on interparticle interactions and confinement interfaces. Our findings underscore the necessity for advancing molecular hydrodynamics under strong confinement.

💡 Research Summary

In this work the authors investigate how strong slit confinement influences the equilibrium position and lateral mobility of a single nanoparticle immersed in an explicit solvent, using large‑scale molecular dynamics (MD) simulations. The model system consists of a single spherical nanoparticle (diameter 5 σ_s, mass 125 m_s) and a bath of Lennard‑Jones (LJ) solvent particles (diameter σ_s, mass m_s). Interactions among all particles are described by a truncated‑shifted LJ potential; the nanoparticle–solvent interaction is tuned by varying the cutoff distance r_c while keeping σ_ns = 3 σ_s. Two parallel walls confine the fluid in the z‑direction, forming a slit of gap H that is varied from 6 σ_s to 16 σ_s. Both smooth walls (LJ with exponents 10/4) and corrugated walls (fixed wall particles arranged on square or hexagonal lattices) are examined. The solvent density is kept constant (ρ = 0.74 σ_s⁻³) and simulations are performed in the canonical (NVT) ensemble at k_B T = 1 using LAMMPS with a Nosé‑Hoover thermostat.

The central observable is the potential of mean force (PMF) F(z) acting on the nanoparticle along the confinement direction, obtained via umbrella sampling and WHAM. By analyzing the fixed points (zeros of dF/dz) and their stability, the authors map a bifurcation diagram as a function of H. For large gaps (H > H_c ≈ 8.9 σ_s) the PMF has a single global minimum at the slit centre (z = 0) and two metastable minima near the walls. As H is reduced, the centre minimum becomes shallower, and at H = H_c the centre and wall minima are degenerate, indicating a first‑order transition. For H < H_c the centre point becomes metastable (or unstable for H ≈ 6 σ_s) while the wall‑adjacent minima become globally stable. This behaviour matches the canonical form of a subcritical pitchfork bifurcation (SPB):

dz/dt = −α₁(H) z + α₃ z³ − α₅ z⁵,

with α₁ ∝ (1/H − 1/H_c), α₃ > 0, α₅ = 1. The authors fit the simulated data to this equation and obtain excellent agreement, confirming that the spatial distribution of the nanoparticle belongs to the same universality class as gas‑liquid first‑order transitions.

Decomposition of F(z) into energetic (U(z)) and entropic (−TS(z)) contributions reveals that the transition is energy‑driven. While the entropic barrier (~10 k_B T) remains roughly constant across H, the energetic term switches from favouring the centre at large H to favouring the walls at small H. Consequently, the nanoparticle desolvates partially when it binds to the wall, as shown by a sharp drop in the number of solvent particles within its first solvation shell (N_s) near H_c.

A striking dynamical consequence is observed in the lateral diffusion coefficient D_∥ of the nanoparticle. D_∥ is computed from the long‑time limit of the mean‑square displacement in the xy‑plane and normalized by the solvent diffusion D_s. For H > H_c, D_∥/D_s ≈ 0.12 and is essentially independent of H, reflecting bulk‑like Stokes‑Einstein behaviour. At H ≈ H_c, both D_∥ and the ratio D_∥/D_s jump discontinuously upward, and for tighter confinement D_∥ continues to increase despite the overall increase in solvent viscosity (D_s decreases monotonically). This non‑monotonic trend is directly linked to the SPB transition: when the nanoparticle relocates to the wall, hydrodynamic coupling to the confined solvent is altered, reducing the effective drag.

The robustness of the SPB scenario is tested by varying key interaction parameters. (i) Making the nanoparticle‑solvent interaction purely repulsive (r_c = 1.122 σ_ns) eliminates the bifurcation; the particle remains wall‑bound for all H, indicating H₀ ≫ 10 σ_s. (ii) Strengthening solvent‑solvent attractions (increasing ε_ss) shifts the balance toward wall binding even at larger H, mimicking a temperature‑like control parameter. (iii) Changing wall morphology (smooth vs. square vs. hexagonal corrugation) does not abolish the transition, though the precise H_c shifts slightly due to differences in wall‑nanoparticle interaction strength (U_nw).

Overall, the study demonstrates that confinement‑induced spatial ordering of a nanoparticle can be captured by a simple nonlinear dynamical model, with a clear energetic origin and a concomitant impact on lateral diffusion. The findings highlight the need to extend molecular hydrodynamics frameworks to account for strong confinement effects, and suggest practical routes to manipulate nanoparticle positioning and transport in nano‑fluidic devices, porous media, and biological capillaries by tuning solvent quality, particle surface chemistry, or wall texture.