The drying of liquid droplets is a common daily life phenomenon that has long held a special interest in scientific research. When the droplet includes nonvolatile solutes, the evaporation of the solvent induces rich deposition patterns of solutes on the substrate. Understanding the formation mechanism of these patterns has important ramifications for technical applications, ranging from coating to inkjet printing to disease detection. This topical review addresses the development of physical understanding of tailoring the specific ring-like deposition patterns of drying droplets. We start with a brief introduction of the experimental techniques that are developed to control these patterns of sessile droplets. We then summarize the development of the corresponding theory. Particular attention herein is focused on advances and issues related to applying the Onsager variational principle (OVP) theory to the study of the deposition patterns of drying droplets. The main obstacle to conventional theory is the requirement of complex numerical solutions, but fortunately there has been recent groundbreaking progress due to the OVP theory. The advantage of the OVP theory is that it can be used as an approximation tool to reduce the high-order conventional hydrodynamic equations to first-order evolution equations, facilitating the analysis of soft matter dynamic problems. As such, OVP theory is now well poised to become a theory of choice for predicting deposition patterns of drying droplets.

The drying of liquid droplets is a common phenomenon in daily life, and has long attracted special interest in scientific research. We propose a simple model to quantify the shape evolution of drying droplets. The model takes into account the friction constant between the contact line (CL) and the substrate, the capillary forces, and the evaporation rate. Two typical evaporation processes observed in experiments, i.e., the constant contact radius (CCR) and the constant contact angle (CCA), are demonstrated by the model. Moreover, the simple model shows complicated evaporation dynamics, for example, the CL first spreads and then recedes during evaporation. Analytical models of no evaporation, CCR, and CCA cases are given, respectively. The scaling law of the CL or the contact angle as a function of time obtained by analytical model is consistent with the full numerical model, and they are all subjected to experimental tests. The general model facilitates a quantitative understanding of the physical mechanism underlying the drying of liquid droplets.

A single-component droplet placed on a completely wetting substrate shows a pseudostable apparent contact angle (θ app ) during evaporation. We propose a simple theory to explain the phenomenon accounting for the liquid evaporation and the internal flow induced by the capillary and Marangoni effects. The theory predicts that when evaporation starts, the contact angle approaches to θ app in a short time τ s , remains constant for most of the time of evaporation, and finally increases rapidly when the droplet size becomes very small. This explains the behavior observed for alkane droplets. Analytical expressions are given for the apparent contact angle θ app and the relaxation time τ s , which predict how they change when the evaporation rate, droplet size, and other experimental parameters such as thermal conductivity of the substrate are changed.

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