Cahn-Hilliard Phase Field modelling captures nanoscale contact line dynamics on high-friction surfaces

Referee: [§4.2 and §5.1] §4.2 (Calibration protocol) and §5.1 (Validation observables): The contact-line friction coefficient is fitted solely to the time evolution of the contact angle. The subsequent comparisons of interface curvature, steady displacement, and streamline topology are presented as independent tests. However, within the Cahn-Hilliard formulation the interface profile (and therefore its curvature near the wall) is directly constrained by the imposed dynamic contact angle and the localized slip length; the velocity field is likewise coupled through the same boundary condition. No sensitivity study or uncalibrated prediction is shown to quantify how much of the reported agreement is a direct consequence of the fitting rather than an emergent prediction of the physics.

Authors: We agree that the dynamic contact-angle boundary condition and localized slip directly influence the near-wall interface profile. However, the calibration uses only the temporal history of the contact angle extracted at the wall, while the curvature comparison is performed on the entire interface shape (including segments several nanometers from the wall) and the streamline topology is obtained from the solution of the coupled Navier-Stokes equations. To quantify the degree of independence, the revised manuscript now includes a dedicated sensitivity subsection (new §5.3) in which the friction coefficient is varied by ±20 % around the calibrated value while all other parameters remain fixed. Only the calibrated value reproduces the MD interface curvature and steady displacement within the reported tolerances; deviations of ±20 % produce statistically significant mismatches in both curvature and displacement. An additional comparison against a model with zero contact-line friction (i.e., classical Navier slip only) shows that the uncalibrated model fails to capture the MD contact-line speed and interface bending, confirming that the agreement is not trivially inherited from the boundary condition. revision: yes

Referee: [§3.3 and Table 2] §3.3 and Table 2: The manuscript states that “all other model parameters are determined a posteriori, according to the calculation of independent observables.” It is not clear which observables were used for each parameter, whether any post-hoc adjustments were made after the friction calibration, or whether error bars on the fitted friction coefficient and on the comparison metrics (e.g., curvature deviation, displacement error) are reported. Without this information the claim of quantitative reproduction cannot be assessed for robustness.

Authors: We thank the referee for requesting greater transparency. Section 3.3 has been expanded and a new Table 3 added that explicitly lists every continuum parameter, the independent MD observable or numerical constraint used to fix it, and the numerical value with its uncertainty. Examples include: interface thickness ε determined from the 10–90 % width of the MD density profile; mobility M obtained from the exponential relaxation time of a flat interface in MD; viscosities taken from separate bulk MD simulations of each fluid; and the slip length fixed by the MD velocity profile away from the contact line. No post-hoc adjustments were performed after the friction-coefficient calibration; all other parameters were set prior to or independently of the contact-angle fitting. Error bars on the fitted friction coefficient (standard deviation over five independent MD trajectories) and on the quantitative metrics in Table 2 (standard deviation of the MD data) have been added to the revised figures and tables. revision: yes

Referee: [§5.2] §5.2 (Numerical sharp-interface limit): The paper invokes the numerical sharp-interface limit to justify direct parametrization from MD. No convergence study with respect to the Cahn-Hilliard interface thickness ε is presented, nor is it shown that the extracted friction coefficient remains invariant as ε→0 while the mesh is refined. This leaves open whether the reported agreement is specific to the chosen numerical regularization or truly representative of the continuum limit.

Authors: We agree that explicit demonstration of convergence to the sharp-interface limit strengthens the claim. The revised manuscript now contains a new Appendix C that reports a systematic convergence study performed at three interface thicknesses (ε = 0.5 nm, 1.0 nm, and 2.0 nm) with corresponding mesh refinement to maintain at least ten elements across the diffuse interface. For each ε the friction coefficient was re-calibrated to the same MD contact-angle histories. The resulting friction values differ by less than 15 % across the range, and the quantitative agreement with MD curvature, steady displacement, and streamline topology remains within the MD error bars for the two smallest ε values. These results indicate that the reported agreement is not an artifact of the particular regularization chosen in the original manuscript. revision: yes