Maximal spreading of impacting viscoelastic droplets

TL;DR

This work addresses how viscoelasticity alters the maximal spreading of impacting droplets by extending the Newtonian energy-balance model with a single elastic correction factor α(De). Using a Maxwell-fluid description, it derives a closed-form expression for the normalized maximal diameter $\bar{D}_{max}$ that incorporates elastic corrections via $α(De)=\frac{1}{De}e^{-1/De}$ and confirms the model against experiments with dilute PAA solutions. The key finding is that elasticity reduces maximal spreading most strongly when $De\approx1$, with up to 40% suppression, while elastic effects vanish for $De\gtrsim10$, and viscous dissipation can mask elasticity when the viscous boundary layer is thick. The framework thus extends inertia–capillary droplet-spreading correlations beyond Newtonian fluids, offering a practical basis for predicting viscoelastic impact outcomes in printing, coating, and spray processes.

Abstract

Droplet impact and spreading on solid substrates are well understood for Newtonian fluids, yet how viscoelasticity alone modifies the maximal spreading remains unclear. To discern the physical mechanisms governing the spreading dynamics, we present a simplified theoretical model, validated by impact experiments, to quantify how fluid elasticity modifies the maximal spreading of impacting droplets. Experiments were performed using fluids within a narrow range of viscosity and surface tension, but varying relaxation time. While following similar asymptotic scalings as Newtonian droplets, the maximum diameter for viscoelastic droplets exhibits a clear deviation from Newtonian behaviour only when the Deborah number is of order unity. The maximum spread diameter is reduced by as much as 40% from the expected value for Newtonian fluid. These results support the central prediction of our model: an extension of classical energy balance that incorporates viscoelastic effects through a single correction factor. The model captures the observed reduction in maximal spreading and predicts both the location and magnitude of the most substantial viscoelastic effects, providing a basis for extending impact models beyond purely Newtonian fluids.

Figures (5)

• Figure 1: Successive snapshots of a single droplet impacting on a hydrophobic paper substrate, progressing from impact (top) to maximum spread (bottom). Time is given in terms of the impact timescale, $\tau_0 = D_0/U_0$. Top left: representative frame from the wide-angle camera, used to determine the impact conditions. Top right: representative frame from the isometric-view camera, recording the droplet topology.
• Figure 2: Measured maximal spreading diameter $\bar{D}_{\mathrm{exp}}$ of single droplets of various Newtonian and viscoelastic fluids, detailed in table \ref{['tab:fluids']}. (a) $\bar{D}_{\mathrm{exp}}$ versus the impact Reynolds number $\mathrm{Re}$; dashed and dotted lines indicate $\mathrm{Re}^{1/4}$ and $\mathrm{Re}^{1/5}$ scalings, respectively. Marker colour denotes the impact Weber number $\mathrm{We}$. (b) $\bar{D}_{\mathrm{exp}}$ versus the impact Weber number $\mathrm{We}$; dashed and dotted lines indicate $\mathrm{We}^{1/4}$ and $\mathrm{We}^{1/2}$ scalings, respectively. Marker colour denotes the impact Reynolds number $\mathrm{Re}$.
• Figure 3: (a) Temporal evolution of the normalised spreading diameter, $\bar{D}_{\exp}\mathrm{Re}^{-1/5}$, for four representative droplets (0 ppm, 25 ppm, 100 ppm, and 200 ppm of PAAmm) with Deborah numbers $0$, $1.02$, $16.21$, and $26.39$, respectively, all at an impact parameter range $P = 6.75 \pm 0.05$. The vertical dashed line marks the dimensionless relaxation time $\lambda/\tau_0$ for the case $\mathrm{De}=1.02$; for the other viscoelastic cases, $\lambda/\tau_0$ lies outside the spreading timescale. (b) Maximal normalised spreading, $\bar{D}_{\exp}\mathrm{Re}^{-1/5}$, versus the impact parameter $P=\mathrm{Re}^{{-2/5}}\mathrm{We}$ for single impacting droplets of various Newtonian and viscoelastic fluids, detailed in table \ref{['tab:fluids']}. Marker colour indicates the fluid elastic relaxation time $\lambda$; Newtonian fluids with $\lambda\rightarrow0$ are shown in black. The Padé correlation of laan_maximum_2014 (dotted) and a second-order fit to the present Newtonian data in equation (\ref{['eq:pade_new']}) are shown for comparison.
• Figure 4: Deviation of the measured spreading diameter from the expected value at given impact conditions, $\bar{D}_{\exp}/\bar{D}_{\mathrm{cor}}$, as a function of $\mathrm{De}$. For reference, all the Newtonian data are plotted at $\mathrm{De}=0$. Marker colour indicates the ratio of viscous boundary-layer thickness to maximal film thickness, $\delta/h_m$; the dashed line illustrates the function $f(\mathrm{De})=\left(1+\alpha\right)^{-1/2}$.
• Figure 5: Deviation of the measured maximal spreading from the Newtonian correlation, $\bar{D}_{\exp}/\bar{D}_{\mathrm{cor}}$, plotted against viscoelastic control parameters. Marker colour indicates the fluid relaxation time $\lambda$. (a) $\bar{D}_{\exp}/\bar{D}_{\mathrm{cor}}$ as a function of the elasticity number $\mathrm{El} = \mathrm{De}/\mathrm{Re}$. (b) $\bar{D}_{\exp}/\bar{D}_{\mathrm{cor}}$ as a function of the elasto-capillary number $\mathrm{Ec} = \mathrm{De}\mathrm{Re}/\mathrm{We}$.