A theoretical framework for spreading of surfactant-laden drops on solid surfaces

Experiments have shown that the addition of surface-active agents to an aqueous droplet can delay its initial spreading on hydrophilic surfaces submerged in viscous oil phases. Marangoni stresses oppose the drainage of the underlying oil film prior to spreading, inducing a very short-time retardation regime characterized by the spreading radius growth scaling as r∼t^0.5. After this retardation phase, the spreading transitions into a viscous-dominated regime where r∼t. In this work, we develop a theoretical framework for the retardation regime based on the force balance within the thin oil film, predicting a spreading law of the form r=A₂t^0.5, where A₂ is a constant dependent on contact angle, interfacial tension, and viscosity. Combined with the established model for the rapid spreading regime, we derive a power-law relation for the retardation time, t_r, as a function of interfacial tension and contact angle: t_r ∼[γ(1+1/cosθ)]^α. This prediction is validated against experimental data. Furthermore, by appropriately scaling the spreading radius and time, all experimental curves from both the Marangoni-affected and viscous regimes collapse onto a universal master curve, regardless of surfactant concentration. This early-time spreading model provides critical insights into Marangoni-induced retardation and offers theoretical support for applications in biosurfaces, enhanced oil recovery, printing and coating, and advanced materials fabrication.