Physical ageing of spreading droplets in a viscous ambient phase

Abstract

In this work, we study the spontaneous spreading of water droplets immersed in oil and report an unexpectedly slow kinetic regime not described by previous spreading models. We can quantitatively describe the observed regime crossover and spreading rate in the late kinetic regime with an analytical model considering the presence of periodic metastable states induced by nanoscale topographic features (characteristic area ~4 nm², height ~1 nm) observed via atomic force microscopy. The analytical model proposed in this work reveals that certain combinations of droplet volume and nanoscale topographic parameters can significantly hinder or promote wetting processes such as spreading, wicking, and imbibition.

Figure 1

Experimental procedure and analysis. (a) Experimental apparatus to study spreading of water droplets in 100-cSt silicone (PDMS) oil. (b) Sequence of steps leading to the gentle deposition of a water droplet on a borosilicate slide. (c,d) Observed time evolution of contact radius R and height h obtained from digital images recorded at 32 fps during 10 s (R₀ = 0.4 mm, V = 0.27 μL). Power-law fits are included for comparison. (e,f) Volume V_S = πh(R²/2 + h²/6) and contact angle θ corresponding to a spherical cap.

Figure 2

Experimental results and analytical predictions. (a–c) Square contact radius R² vs. time t for droplets of volume V = 0.24, 0.37, & 0.63 μL. Markers correspond to experimental results. The crossover radius R_C is predicted by Eq. (9) for α = 0.55. R_E is the expected equilibrium contact radius for each case. Power-law fits R ∝ t and R ∝ t^0.1 are included for comparison. (d) Experimental data for droplets with different volumes (markers) scaled according to Eq. (6). Time is normalized by the kinetic time defined in Eq. (7). Dashed lines are predictions from Eq. (9) for the crossover to kinetic spreading. Analytical fits via Eqs (6–9) employ the parameters reported in Table 1.

Figure 3

Nanoscale surface topography and model parameters. (a) 3D topographic image (512 × 512 pixels) of a borosilicate glass employed for spreading experiments obtained via NC-AFM. (b) 1D local height profile and average height $z_a=2{\int }_0^{\ell }|z|dx/\ell$ = 1.2 nm. (c) Height autocorrelation function for different directions ϕ = atan(y/x) = 0, ±45 deg. Vertical dashed line indicates the correlation length $r_d=\sqrt{A_d/\pi }$ corresponding to averaged topographic “defects” with base area A_d = 4.2 nm². Inset illustrates a modeled conical “defect” with base area $A_d=\pi r_d^2$, height z_a, and cross-sectional area ΔA_wo.