When a colloidal droplet dries, a ring-shaped pattern remains along the edge of the droplet, which is referred to as the coffee ring effect. Although the phenomenon seems simple, modeling the deposition process of colloidal particles is quite complex since several phases are involved in the system. In this study, we develop a new lattice Boltzmann (LB) model to simulate evaporation of a droplet containing colloids and surfactant.
That is, our model combines the pseudopotential LB model for evaporation of a droplet with the advection-diffusion LB model for colloids and surfactant. Here we propose a new method to retain colloids and surfactant in the droplet during the phase change. The effect of surfactant on the surface current is imposed by using the Langmuir isotherm, which is one of the most widely used non-linear surfactant equations of state.
We investigate the applicability of our model to generate evaporation-induced velocity fields in a fixed droplet and to predict the evolution of the contact angle. Finally, we show the unique behaviors of the surfactant-containing drying droplet, such as pinning-throwing contact line dynamics and swelling and deflation at the droplet center and edge, respectively.
Physical background
21] used an amphiphilic LB model to investigate the effect of surfactant and compared it with the effect of colloids on droplets in the presence of shear flow. Despite numerical and theoretical studies of particle-laden droplet evaporation, no unified method has been developed within LBM to predict the velocity field and behavior of colloids and surfactant.
The lattice Boltzmann method
The pseudopotential lattice Boltzmann model, also called the Shan-Chen (SC) model, has been proposed by Shan and Chen in 1993 [22] to simulate multiphase flows by the LB method. In the model, phase separation occurs by imposing the interaction force between particles that allows a non-ideal gas-liquid behavior. This is one of the most popular models for multiphase flows due to its simplicity of implementation and adaptability to complex geometries.
To solve such problems of SC model, many researches have been conducted in recent decades. 23] introduced a pseudopotential model to simulate a large density ratio of fluids by incorporating a multiple relaxation timescale (MRT) with the Carnahan-Starling equation of state.
Objective
Simulation model
- The lattice Boltzmann model for simulating multiphase flows
- Models for the evaporation of a droplet and a pinned contact line
- Models for colloidal particles and surfactant
- Model for colloidal particles
- Model for surfactant
- Simulation algorithm
To include the interaction force in LBM, Shan and Chen introduced the equilibrium velocity 𝑢𝑒𝑞𝑘. where 𝑢𝑏𝑢𝑙𝑘 is the macroscopic velocity of the bulk fluid:. Therefore, the macroscopic fluid velocity that expresses the motion of the fluid is defined by means of momenta before and after the collision step:. Here we assume that the system is isothermal and the evaporation takes place through the density gradient of the liquid surrounding the droplet.
To fix the contact line of the droplet during evaporation, we generate unbalanced Young's force using hydrophobic and hydrophilic substrate. The sign of the adhesion interaction parameter 𝐺𝑎𝑑𝑠𝑘 determines the liquid wettability on the substrate (positive for hydrophobic substrate and negative for hydrophilic substrate) and the magnitude of 𝐺𝑎𝑑𝑠𝑘 specifies the contact angle. The density of each fluid component σ is given in 60 × 60 lu2 size blue square of the fluid σ surrounded by 𝜎̅. b)-(f).
Different contact angles are obtained by using different values of the interaction parameter 𝐺𝑎𝑑𝑠𝑘 between the liquid components and the solid substrate. The specific parameter values and measured contact angles are given in Table 1. Applying this equation to multiphase flow, the total velocity between the colloidal particle and the fluid should be 𝑈𝑓, which is calculated as the average of the moments before and after the collisions in Eq.
Therefore, the equilibrium distribution function of the particle concentration is resolved in the multiphase flow. The force term is similar to the Shan-Chen interaction force, 𝐹𝑖𝑛𝑡, where 𝐺𝑖𝑛𝑡 is the coefficient of the interaction force. The property of the surfactant differs from the colloidal particles in that they tend to absorb at the surface of a droplet.
When they are adsorbed on the surface, the free energy of the surface decreases, which results in a decrease in the surface tension of the droplet. To simulate the behavior of surfactants, we modified the colloidal particle model presented in Section 2.3.1. The surfactant concentration gradient at the droplet surface induces the force, 𝐹𝑛𝑒𝑡, which is imposed by adding the net force to the equilibrium velocity, 𝑢𝑒𝑞𝜎, of the medium fluid in Eq.
Simulation results and discussion
Evaporation of a droplet with a pinned contact line
To quantitatively compare the flow field with the analytical solution, we plot the radial and vertical velocities based on the droplet height. The initial density of the colloid particles is 𝜌𝑐= 0.5 inside the droplet, and the density outside the droplet is 𝜌𝑐= 0.01. The wall boundary conditions are imposed on all sides of the domain for colloidal particles.
The surfactant density is initialized with 𝜌𝑠= 0.5 inside the droplet, and 𝜌𝑠= 0.01 outside the droplet. Due to the outward capillary flow, the high concentration of surfactant is observed near the contact line. This high concentration of surfactant at the droplet interface weakens the surface tension of the droplet.
14b, the profile of the droplet interface (red solid line) exhibits the maximum slope near the contact line. Finally, the wetting agent model successfully captured the contact line dynamics and the shape deformation of the wetting agent-laden droplet during evaporation. We also observed the Marangoni flow flowing from the contact line to the apex of the drop, since the concentration gradient of surfactant is higher at the contact line.
As the evaporation continued, the Marangoni flow became stronger, leading to the capture of the contact line. After capture, the droplet evaporates in the CA mode on the hydrophilic substrate and the inflation of the droplet shape in the center is observed. Wu, “Pinning and depinning mechanism of the contact line during evaporation of nano-droplets sitting on textured surfaces,” Soft Matter, vol.
Leroy, “Contact line etching during evaporation on heterogeneous surfaces: Slowdown or temporary immobilization.
Colloid particle deposition pattern
Surfactant-induced Marangoni flow
In the present section, we simulate a surfactant-laden droplet evaporation using our LB surfactant model to investigate the flow filed inside the droplet, the contact line dynamics and the shape deformation during evaporation. During the equilibration process, the periodic boundary conditions are given at the top, left and right boundaries, and the wall boundary condition at the bottom substrate. The substrate is chemically patterned, where the hydrophilic (𝜃 ≈ 0°) substrate is located in the center of the domain and is surrounded by hydrophobic (𝜃 ≈ 130°) area similar to the schematic of Fig.
The relaxation parameter of the surfactant is set to be 𝜏 = 0.8, which allows the convective motion of the surfactant to be dominant. We then imposed evaporation boundary conditions of 𝜌𝐻= 0 at the top, left and right boundaries, and the wall boundary condition is imposed at the bottom substrate. Since the surface tension is relatively higher at the droplet center, where the surfactant concentration is lower than the periphery, the surface tension gradient is formed along the droplet surface.
As the evaporation progresses, we can observe the stronger Marangoni flow in Fig. 14b compared to that in fig. 14a and the movement of the contact line. The Marangoni current induces the pinning of the contact line, moves the contact line inward, and deforms the shape of the droplet. 2.18), the determination process takes place with 𝜃 = 𝜃𝐴 , as 𝑙𝐵 goes to zero as evaporation progresses. However, when the external force exists near the contact line, it is possible for the contact line to be pulled back along the hydrophilic substrate where 𝜃 > 𝜃𝐴.
In the simulation, the Marangoni stress acts as the external force and it is found that anchoring occurs at 𝜃 ≈ 58°, where 𝜃𝐴 = 0°. As evaporation proceeds, the large concentration gradient in the droplet is generated due to the outward capillary flow, and then the strong Marangoni flow at the interface is induced by the large concentration gradient. It is interesting to note that the strong Marangoni current deforms the profile of the droplet in such a way that the droplet inflates at the center and deflates at the edge, compared to the droplet in a spherical shape in the absence of the Marangoni current (red dashed line in Figure 14b).
This indicates that the strong Marangoni current to the top of the droplet delivers more liquid to the top of the droplet, inflating the droplet in the center.
Conclusions
Nakajima, “Evaporation behavior of microliter- and subnanoliter-scale water droplets on two different fluoroalkylsilane coatings,” Langmuir , vol. Kooij, “Elongated droplet evaporation on chemically striped patterned surfaces,” International Journal of Heat and Mass Transfer, vol. Chen, “Lattice Boltzmann model for simulation of multiphase and component flows,” Physical Review E , vol.
Chen, “Study of large density ratio multicomponent pseudo-potential model and heat transfer,” International Communications in Heat and Mass Transfer , vol. Shan, “Spurious current analysis and reduction in a class of multiphase lattice Boltzmann models,” Phys Rev E Stat Nonlin Soft Matter Phys , vol. Maier, “Application of the lattice-Boltzmann method to the study of flow and dispersion in channels with and without.
Burganos, “Mesoscopic modeling of flow and dispersion phenomena in fractured solids,” Computers & Mathematics With. Chen, "Simulation of multicomponent fluids in complex three-dimensional geometries using the lattice Boltzmann method.", Physical Review E, vol. Sukop, "A proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann models," Physical Review E, vol.
Lee, "A review of spurious currents in the lattice Boltzmann method for multiphase flows," Journal of Mechanical Science and Technology, vol. Colinet, "Effect of Marangoni currents on the shape of thin sessile droplets evaporating in air", Langmuir, vol. I would like to express my sincere gratitude to my advisor, Professor Chun Sang Yoo, who gave me the opportunity and provided invaluable guidance in conducting this research.
I would also like to express my special thanks to Doctor Narina Jung for her guidance and suggestions throughout the study.
