Modeling and simulation of droplet evaporation using a modified Cahn–Hilliard equation

doi:10.1016/j.amc.2020.125591

Applied Mathematics and Computation

Volume 390, 1 February 2021, 125591

https://doi.org/10.1016/j.amc.2020.125591 Get rights and content

Highlights

•
Modified Cahn–Hilliard model with evaporation term is proposed.
•
Droplet evaporation is investigated with various parameter values.
•
Contact angle boundary condition is considered.

Abstract

In this paper, we propose a mathematical model, its numerical scheme, and some computational experiments for droplet evaporation. In order to model the evaporation, a classical Cahn–Hilliard equation with an interfacial evaporation mass flux term is proposed. An unconditionally gradient stable scheme is used to discretize the governing equation, and the multigrid method is applied to solve the resulting system. The proposed model is first validated via a proper interfacial parameter ϵ, and then, the effect of evaporation rate and effect of contact angle on volume and surface area changes are investigated. The numerical results indicate that the dynamics of evaporation are dependent on the contact angle on a solid substrate.

Introduction

The evaporation phenomenon of a droplet on a solid substrate is fundamental to the coffee-ring effect [1] and includes applications such as nanochromatography for disease diagnostics [2]. Numerous experimental [3], [4], [5], [6], [7] and numerical [8], [9], [10], [11] studies were performed to examine the evaporation phenomenon. The authors of [12] investigated the effect of support fibers on the process of droplet evaporation. An extant study [13] presented a level-set approach to directly simulate the particle motion in droplet evaporation. The evaporation effect was modeled by applying the coupled vapor fraction and temperature conditions to the interface. A previous study [14] numerically investigated 3D particle motion in the evaporation process of a liquid film. The level-set based method was adopted to track the liquid–solid and gas–liquid interfaces. The effects of evaporation, solid particles, and contact line were considered in the proposed method.

Among the many existing simulation models for evaporation phenomena, phase-field models attracted considerable attention. In [15], the Allen–Cahn (AC) type model with a contact angle was developed to investigate droplet evaporation for different shapes of the gas-liquid interface. In the model, the evaporation energy density is given as $f_{e v a p} (ϕ) = p (c_{s} - c_{p}) ϕ^{2} (3 - 2 ϕ),$ where p denotes the ambient pressure, c_s and c_p denote the saturated concentration and current concentration of liquid in the gas, respectively.

The main purpose of this study is to formulate a modified Cahn–Hilliard (CH) model to simulate droplet evaporation. The CH equation [16], [17] has been extensively used in literature to model evolving interface problems. For physical, mathematical, and numerical derivations of the CH equation, interested readers can refer to a review paper [18]. Although the AC type model is simple, i.e., the second-order partial differential equation (PDE), the total mass is not conservative in the absence of evaporation. The proposed model corresponds to the fourth-order PDE and is more difficult to solve than the AC equation. However, it is conservative in the absence of evaporation. Furthermore, efficient numerical solvers are available.

The remainder of this paper are organized as follows. In Section 2, the proposed mathematical model for droplet evaporation is presented. In Section 3, the description of the numerical algorithm is given. The numerical results are given in Section 4, and conclusions are discussed in Section 5.

Section snippets

Proposed mathematical model for droplet evaporation

As the case of a tumor growth [19], [20], [21] model, we propose the following governing equation by adding an evaporation term to the CH equation with a contact angle boundary condition [22] to model the evaporation of a droplet with a contact angle on a solid substrate as follows: $\frac{\partial ϕ (x, t)}{\partial t} = \frac{1}{P e} Δ μ (x, t) - γ | \nabla ϕ (x, t) |, x = (x, y, z) \in Ω, 0 < t \leq T,$ $μ (x, t) = F^{'} (ϕ (x, t)) - ϵ^{2} Δ ϕ (x, t),$ $n \cdot \nabla ϕ (x, t) = \frac{\sqrt{2 F (ϕ (x, t))}}{ϵ} \cos θ, x = (x, y, z) \in \partial Ω, 0 < t \leq T,$ $n \cdot \nabla μ (x, t) = 0,$ where ϕ(x, t) denotes an order parameter such as the mass fraction difference of

Numerical implementation algorithm

The numerical algorithm is introduced in this section. We first present the finite difference scheme for the three-dimensional CH equation on $Ω = (0, L_{x}) \times (0, L_{y}) \times (0, L_{z})$ . Let N_x, N_y, and N_z denote the mesh numbers along x-, y-, and z-directions, $h = L_{x} / N_{x}$ denote the uniform spatial step, $Ω_{h} = {(x_{i}, y_{j}, z_{k}) : x_{i} = (i - 0.5) h, y_{j} = (j - 0.5) h, z_{k} = (k - 0.5) h},$ where 1 ≤ i ≤ N_x, 1 ≤ j ≤ N_y, 1 ≤ k ≤ N_z, and $ϕ_{i j k}^{n}$ denote an approximation of ϕ(x_i, y_j, z_k, nΔt), where Δt denotes the time step. $P e = 1$ is used for the purpose of

Numerical results

In the numerical tests, the homogeneous Neumann boundary condition is applied at the boundaries in x-direction, y-direction, and $z = L_{z}$ . At $z = 0,$ we give a contact angle boundary condition. We use $P e = 1$ for the purposed of simplicity in all numerical tests.

Conclusions

We proposed a modified CH model with an interfacial evaporation mass flux to simulate droplet evaporation. To enable fast and efficient computation, the unconditionally gradient stable scheme was used to discretize the governing equation, and the multigrid method was applied to solve the resulting system. The numerical results indicated that the proposed model could simulate droplet evaporation. Additionally, different contact angles lead to different evolutions of total mass and surface area:

Acknowledgments

The first author (H.G. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1C1C1011112). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053). The authors thank the reviewers for their constructive and helpful comments on the revision of this article.

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Modeling and simulation of droplet evaporation using a modified Cahn–Hilliard equation

Highlights

Abstract

Introduction

Section snippets

Proposed mathematical model for droplet evaporation

Numerical implementation algorithm

Numerical results

Conclusions

Acknowledgments

Appl. Surf. Sci.

Colloids Surf. A

J. Colloid Interface Sci.

Chem. Eng. Sci.

Appl. Math. Comput.

J. Colloid Interface Sci.

Appl. Math. Comput.

Numer. Heat Tr-B Fund

J. Chem. Phys.

Comput. Mater. Sci.

J. Theor. Biol.

J. Theor. Biol.

Int. J. Numer. Meth. Biomed. Eng.