Phase-field numerical study on the dynamic process of thermocapillary patterning

It is well known that surface tension is dependent on temperature, and thus a nonuniform temperature may cause thermocapillary flow which is referred to as the Marangoni effect. For a thin liquid-air film confined between a flat hot plate and a topographical cold template, it undergoes deformation due to thermocapillary flow. This phenomenon is termed as thermocapillary patterning, and has been used to fabricate micro- and nanostructure in polymer films. In most cases, the obtained structure conforms to the template; i.e., it can be considered as a replication technique. In this paper, we developed a two-phase flow numerical model based on the phase field to study the dynamic process of thermocapillary patterning. As a remeshing-free method, the phase field enables the incorporation of thermal field and multiphase flow with free surface deformation. The numerical model was employed to study the dynamic process of thermocapillary patterning. Meanwhile, the effects of some parameters, e.g., temperature, geometry parameters, and contact angle, were also investigated.

Figure 1

Schematic diagram of thermocapillary patterning in thin liquid polymer film.

Figure 2

Model validation. (a) The schematics of droplet migration in a nonuniform thermal field. (b) The comparison between the present numerical results against Bothe's [42].

Figure 3

The effect of element number on the dynamic process of thermocapillary patterning.

Figure 4

Dynamic process of thermocapillary patterning. The subfigures from top to bottom correspond to dimensionless time $t^{*}=0$, 22, 50, 58, and 73. (a) The evolution of the liquid-air interface and the black lines indicate the initial position of the liquid-air interface. (b) The evolution of flow field and streamlines. The red lines represent the liquid-air interface.

Figure 5

The evolution of nondimensional temperature $T^{*}$ (a) and the magnitude of surface tension force (b) during the process of thermocapillary patterning. The subfigures correspond to dimensionless time $t^{*}=0$, 22, 50, 58, and 73. The black curves in (a) indicate the liquid-air interface and the white lines are the isotherm.

Figure 6

The evolution of structure height. Based on the growth rate, the curve can be divided into three stages: A, B, and C.

Figure 7

The evolution of thermocapillary patterning with different Marangoni numbers. The Marangoni numbers in (a)–(c) are $–3\times {10}^7,1\times {10}^7, 2\times {10}^7$, and $3\times {10}^7$, correspondingly. The subfigures from top to bottom correspond to dimensionless time $t^{*}=0$, 30, and 60.

Figure 8

Evolution of structure height $\mathrm{\Delta }H$ for different Marangoni numbers.

Figure 9

The evolution of structure height $\mathrm{\Delta }H$ for different $k_r$.

Figure 10

The effect of wettability on thermocapillary patterning. The contact angles from top left to bottom right are 30°, 60°, 90°, 120°, and 150°, and the nondimensional time $t^{*}=100$.

Figure 11

The effect of protrusion width on the growth rate. (a) The relationship between the half-height time and protrusion width. (b) The temperature profile along the liquid-air interface at $t^{*}=0$. (c) The temperature $T^{*}$ in the computational domain at $t^{*}=0$.

Figure 12

The interfacial morphology for different protrusion widths at dimensionless time $t^{*}=150$.

Figure 13

The relationship between the half-height time and protrusion height.

Figure 14

The dynamic process of thermocapillary patterning for a thin liquid film. (a) The interfacial evolution. (b) The flow field.

Figure 15

The evolution of the interfacial morphologies with the oblique top template. The dimensionless times from top to bottom are $t^{*}=0$, 45, 90, 120, and 150.

Figure 16

The evolution of the interfacial morphologies with the defective top template where protrusion 2 is missed. The dimensionless times from top to bottom are $t^{*}=0$, 22, 44, 78, and 110.

Figure 17

The schematic illustration of the pillbox argument at the interface of the phase field.