Phase-field modeling and computer simulation of the coffee-ring effect
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O R I G I NA L A RT I C L E
Junxiang Yang · Hyundong Kim · Chaeyoung Lee · Sangkwon Kim · Jian Wang · Sungha Yoon · Jintae Park · Junseok Kim
Phase-field modeling and computer simulation of the coffee-ring effect
Received: 20 January 2020 / Accepted: 2 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this study, we propose a novel computational model for simulating the coffee-ring phenomenon. The proposed method is based on a phase-field model and Monte Carlo simulation. We use the Allen–Cahn equation with a pinning boundary condition to model a drying droplet. The coffee particles inside the droplet move according to a random walk function with a truncated standard normal distribution under gravitational force. We perform both two-dimensional and three-dimensional computational experiments to demonstrate the accurate simulation of the coffee-ring phenomenon by the proposed model. Keywords Allen–Cahn equation · Monte Carlo simulation · Brownian dynamics · Pinning boundary condition · Coffee-ring effect Mathematics Subject Classification
35K55 · 65C05 · 65N06 · 68U20
Communicated by Tim Colonius. J. Yang · H. Kim · C. Lee · S. Kim · J. Wang · S. Yoon · J. Park · J. Kim (B) Department of Mathematics, Korea University, Seoul 02841, Republic of Korea E-mail: [email protected] J. Yang E-mail: [email protected] H. Kim E-mail: [email protected] C. Lee E-mail: [email protected] S. Kim E-mail: [email protected] J. Wang E-mail: [email protected] S. Yoon E-mail: [email protected] J. Park E-mail: [email protected]
J. Yang et al.
1 Introduction Droplet evaporation on a substrate is an important phenomenon in the field of soft matter physics. This phenomenon represents problems related to fluid dynamics, the physics of solid particles, properties of solid substrates, and heat transfer [1]. Based on the complexity of droplet evaporation, many researchers have employed evaporation dynamics in various real-world applications, such as self-assembly techniques [2], medical tests [3,4], coating and printing engineering [5], and DNA mapping [6]. In comparison with pure liquid droplets, nanoparticle-suspended liquid droplets (NSLDs) are more common in scientific and engineering fields. NSLDs have a variety of applications in numerous industrial fields, including biomedicine [7,8], optics [9,10], inkjet printing [11,12], painting [13,14], ceramics [15], and wastewater treatment [16]. Ring-like deposition patterns containing nanoparticles, which are referred to as “coffee-ring stains,” can be observed when an NSLD evaporates on a substrate [17]. Conway et al. [18] experimentally determined that contact angles cause ring-like pattern formations during the evaporation of colloidal suspensions. In general, there are two main types of contact line conditions during the evaporation of NSLDs on a substrate. One is a pinning condition with a fixed contact area, and the other is a nonpinning condition with a fixed contact angle. Evaporation with a pinning condition forces the
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