A. Analysis of capillary rise of lubricant

Smooth micropores and those having bulges with pitches of 30, 50, and 70 μm were considered to study the capillary rising behavior of lubricants, the length of four groups of micropores is 1000 μm. Four groups of dynamic models of capillary rise in micropores with lubricant were built in the finite element software, and the effects of different micropores on the capillary rise characteristics were analyzed. Figure 4 shows the lubricant capillarity rising in the smooth micropores and spiral bulge micropores with different pitches. The lubricant is flush with the orifice at the lower end of the micropore at the initial time and then contacts the orifice and is in a static state. Over time, different capillary rise phenomena of the lubricant occur in the micropores. It takes the lubricant 0.35 s to rise to the top orifice in the smooth micropore. When the pitch of the inner spiral bulge micropores is 30, 50, and 70 μm, it took 0.16, 0.2, and 0.24 s to rise to the top orifice. Thus, the inner spiral bulge structure in micropores of the same size significantly accelerates the capillary rise of the lubricant, and a smaller pitch leads to a faster capillary rise.Figure 5 shows changes in the capillary rising speed for the different micropores. The height of the micropore h is 1000 μm, the symbol I represents the smooth micropore, II represents the 70 μm pitch, III represents the 50 μm pitch, and IV represents the 30 μm pitch. Figure 5(a) shows changes in the liquid level rising speed for the different micropores with time. Three-dimensional models of different micropores were defined in the numerical simulations, and the transversals of the coordinates were set from (0,0,0) to (0,0,1000). The velocity changes for the three-dimensional transversals at different times were extracted, and those of the rising fluid level at different times were collected. Figure 5(b) shows the average and standard deviation of the velocity for the rising liquid level of the different micropores.Comparing Figs. 5(a) and 5(b) shows that the rising liquid surface velocity of the inner spiral bulge micropores with the same size and different pitches is greater than that of smooth micropores of the same size. Variations in the rising liquid surface velocity in the different micropores have some regularity. The mean and standard deviation of the rising liquid surface velocity of smooth micropores are the smallest and the variations are more stable. The rising liquid surface velocity of the spiral inner bulge micropores with different pitches first increased, stabilized, and then decreased. A smaller pitch gave a larger mean, standard deviation, and fluctuations in the rising liquid surface velocity. The capillary rising process can be divided into three stages based on the trend of the rising fluid surface velocity. The period of continuous increase is called the early stage, the period of steady change is called the middle stage, and the period of continuous decrease is called the late stage.Washburn equation2323. T. E. Mumley, C. J. Radke, and M. C. Williams, “Kinetics of liquid/liquid capillary rise: I. Experimental observations,” J. Colloid Interface Sci. 109, 398–412 (1986). https://doi.org/10.1016/0021-9797(86)90318-8 is widely considered the dynamic description of liquid capillary rise, but its validity for all porous media is controversial. This paper uses this equation to check the liquid capillary rise height in spiral tubes and compares the numerical simulation results with the check value,

The expression of capillary shape coefficient c is c=ηvγ,η is the liquid viscosity, v is the liquid velocity, and γ is the liquid–gas surface tension. θ is the apparent contact angle between the liquid and the tube wall, In this paper, the contact angle is set to 67.5° during the numerical simulation calculation. r is the average capillary radius. The smooth microporous inner radius is 150 μm, and the width of the spiral bulge is 50 μm, assuming that smooth micropore inner radius minus convex structure radius, and the average micropore radius is 100 μm.

Figures 5(c) and 5(d) show the rise height of different microporous liquids with time under numerical simulation and equation calculation. The comparison results show that all liquid rising speeds are much higher than that inferred from classical analysis only based on their viscosity, density, and interfacial tension.2323. T. E. Mumley, C. J. Radke, and M. C. Williams, “Kinetics of liquid/liquid capillary rise: I. Experimental observations,” J. Colloid Interface Sci. 109, 398–412 (1986). https://doi.org/10.1016/0021-9797(86)90318-8 The numerical simulation value of the rising speed of the inner spiral microporous liquid is greater than the calculated value of the equation, and the reason for the fast rising of the inner spiral microporous liquid can be deduced. It is not only because of the reduction of the inner diameter, but also because of the special microstructure in the inner spiral convex micropores, which results in the change of the tube shape and the strong Taylor capillary rise phenomenon. The smaller the pitch of the inner spiral micropore, the stronger the Taylor capillary rise, resulting in a faster liquid rise.

B. Analysis of capillary rising behavior of inner spiral bulge micropore

At 0.1 s, the liquid is in the capillary rise stage in each micropore, and the liquid in the inner spiral bulge micropore rises steadily and is in the middle stage of the process. Therefore, 0.1 s is selected to analyze the capillary rise behavior of the lubricant. Figure 6 shows the pressure field distribution of the lubricant capillary rise in smooth micropores and spiral bulge micropores with different pitches at 0.1 s and an enlarged diagram of the upper water pressure. The channel enclosed by the spiral line bulge is called the spiral channel. The uppermost water layer where the lubricant rises in the tube is defined as the upper water, and the front water flow where the upper water rises along the spiral channel is defined as the front water. Figure 6(a) shows that at 0.1 s, the capillary rising height of the lubricant is greater in the inner spiral bulge micropore, and a smaller pitch gives a greater capillary rising height of the upper layer. Figure 6(b) shows the pressure amplification diagram of the upper water in smooth and inner spiral bulge micropores. The internal pressure of the upper water in the four micropores changes from negative to positive pressure during the upward capillary rise, which is consistent with the results of Zhang et al.1010. G. Zhang, Y. Shi, B. Tong et al., “Exudation behavior and pinning effect of the droplet on slippery liquid-infused porous surfaces (SLIPS),” Surf. Coat. Technol. 433, 128062 (2022). https://doi.org/10.1016/j.surfcoat.2021.128062 A lower pitch gives a greater (smaller) pressure change rate in the inner spiral bulge (smooth) micropore. The internal pressure of the upper water in the four micropores differs in their capillary rise processes. The internal pressure of the upper water in the smooth micropores is flat and symmetric while that in the inner spiral bulge micropores is spiral.Figure 7 shows the velocity field distribution of the lubricant capillary rise in the smooth micropore and micropores with the different pitches at 0.1 s along with enlarged diagrams of the upper water velocity. From Fig. 7(a), the velocities in the middle area of the upper water in the four micropores are greater than the velocity of the upper and lower end faces, which are equal. The velocities of the upper water in the inner spiral bulge micropores with different pitches are greater than that in the smooth micropore, which leads to a greater rising height of the lubricant. A smaller pitch gives a greater rising height of the lubricant. From Fig. 7(b), the direction of the velocity vector of the upper water in the smooth micropore is vertical, indicating that the upper water movement state rises vertically along the pipe wall.The direction of the velocity vector for the upper water in the inner spiral bulge micropore points to the upper part of the micropore along the spiral channel. The direction of the other part of the velocity vector is perpendicular to the inner bulge structure and points below the micropore. This means that the upper water is blocked by the inner bulge structure during the capillary rise process and forms front water that rotates and rises along the spiral channel. Due to the spiral bulge structure, the inner diameter of the micropore decreases, and the capillary lift is enhanced. The inner spiral bulge structure is arranged spirally along the pipe wall, which results in constant changes in the cross section of the micropores and leads to variations in the shape of the inner spiral bulge micropores, which causes Taylor capillary rise.2424. H. Chen, P. Zhang, L. Zhang et al., “Continuous directional water transport on the peristome surface of Nepenthes alata,” Nature 532, 85 (2016). https://doi.org/10.1038/nature17189 Under the synergistic effect of the enhanced and Taylor capillary lifts, the capillary rising speed of the lubricant in the inner spiral bulge micropore is greater than the smooth micropore.Figure 8 illustrates changes in the vorticity size and velocity vortex morphology of the upper water in the spiral bulge micropore at different times and pitches. In the simulations, the three-dimensional models are defined for spiral bulge micropores with different pitches, and the three-dimensional section line is set from the coordinates (0,0,0) to (0,0,1000). Changes in the vorticity are extracted at different times on the three-dimensional section line. The velocity vortexes only occur in the upper water position with none appearing in positions that have been immersed or have not been reached by the lubricant. The upper liquid is squeezed by the spiral bulge structure, which causes the upper head to rotate and rise along the spiral groove to form the front water. The rapid rotation of the front water produces the velocity vortexes. A larger velocity vortex pushes the upper water to rotate and rise faster, and its generation promotes the capillary rise behavior. In larger flow cross sections, the influence of from other hydrodynamic characteristics (including possible annular vortices) is relatively small.2525. T. E. Mumley, C. J. Radke, and M. C. Williams, “Kinetics of liquid/liquid capillary rise: II. development and test of theory,” J. Colloid Interface Sci. 109, 413–425 (1986). https://doi.org/10.1016/0021-9797(86)90319-XFigure 8(a) shows changes in the upper water vorticity of the inner spiral bulge micropore with a pitch of 70 μm at different times. From 0.06 to 0.18 s, the upper water vorticity is large and changes stably while reaching a peak of 3604 s−1 at 0.12 s. Figure 8(b) shows changes in the upper water vorticity of the inner spiral bulge micropore with a pitch of 50 μm at different times. From 0.05 to 0.15 s, the upper water vorticity is large and changes stably with a peak value of 7068 s−1 at 0.1 s. Figure 8(c) shows changes in the upper water vorticity of the inner spiral bulge micropore with a pitch of 30 μm at different times. From 0.04 to 0.12 s, the upper water vorticity is large and changes stably with a peak value of 10 910 s−1 at 0.12 s. As this period is the middle stage of the capillary rising process, the maximum vorticity values are considered to be generated in the middle stage. Combined with changes in the velocity vortex morphology, the upper water flow is the largest in the middle stage. With a decreased pitch, a greater eddy value of the upper water velocity gives a smaller upper water flow.The hydromechanics analysis in Fig. 8 shows that in the early stage of the capillary rise process, the effect of the capillary lift causes the front water to rotate and rise along the spiral channel. The upper water then gradually forms, and the rising speed of the liquid continues to increase. As the liquid rising height increases, it is affected by its gravity and wall adhesion resistance, and the rising speed and vorticity begin to stabilize and no longer increase significantly. The size of the upper water flow rising during rotation tends to be stable and is considered having entered the middle stage of the capillary rising process. When the rising speed and vorticity of the liquid begin to decrease, the capillary rising process of the lubricant begins to enter the later stage. With a continuous increase in the liquid rising height and wall adhesion resistance, the upper water flow decreases.

C. Mechanism of capillary rising behavior of the inner spiral bulge micropore

Figure 9 illustrates the force analysis and cross section of the lubricant in smooth micropores and spiral bulge micropores with different pitches. There are three main forces in the rising process of the lubricant: gravity Fg, adhesion resistance Fv of the inner wall, and capillary lift Fc. The capillary lift can be directly obtained from the Young–Laplace equation2626. A. Ponomarenko, D. Quere, and C. A. Clanet, “A universal law for capillary rise in corners,” J. Fluid Mech. 666, 146–154 (2011). https://doi.org/10.1017/S0022112010005276 with a smooth wall aswhere γ represents the liquid–gas surface tension, R is the micropore radius, and θ is the apparent contact angle between the liquid and the tube wall. The θ is selected for calculations as it constantly changes during capillary rise.As shown in Fig. 9(a), the upper water rises vertically in a smooth micropore along the wall with a vertical rise radius of R1, and its apparent contact angle is between the upper water meniscus and the micropore wall θ1. As the inner spiral bulge micropore has an inner spiral linear bulge structure, the inner diameter of the micropore decreases. The maximum internal radius R1 and the minimum internal radius R2 are obtained by randomly taking two cross sections of the internal spiral bulge micropore for the measurements. We estimate the internal radius of the inner spiral bulge micropore with an estimation formula of R∗=R1+R22. Figure 9(b) shows that the upper water of the spiral bulge micropore in the 70 μm pitch rotates and rises along the spiral channel, where the radius of rotation and rise is R*. The apparent contact angle is between the front water and the edge of the spiral channel θ2. Figure 9(c) shows that the upper water of the spiral bulge micropore in the 50 μm pitch rotates and rises along the spiral channel. The radius of rotation and rise is R*, and the apparent contact angle is between the front water and the edge of the spiral channel θ3. Figure 9(d) shows that the upper water of the spiral bulge micropore in the 30 μm pitch rotates and rises along the spiral channel. The radius of the rotation and rise is R*, and the apparent contact angle is between the front water and the edge of the spiral channel θ4.The measurement of the apparent contact angle θ1 between the lubricant and smooth micropore wall is approximately 72.45°, that for the 70 μm pitch (θ2) is approximately 24.32°, that for the 50 μm pitch (θ3) is approximately 20.11°, and that for the 30 μm pitch (θ4) is approximately 14.98°. The measured R1, R*, θ1, θ2, θ3, and θ4 values were inserted into Eq. (7), and the capillary lift force of the lubricant in the smooth micropore was 2.057 × 10−3 N. The capillary lift force of the lubricant in the spiral bulge micropore in the 70 μm pitch was 5.726 × 10−3 N, that for the 50 μm pitch was 5.9 × 10−3 N, and that for the 30 μm pitch was 6.072 × 10−3 N, as shown in Fig. 9(e).

The spiral linear bulge structure is set in the micropores, which makes the inner radius smaller than smooth micropores of the same size. Both the calculation and simulation results show that the spiral linear bulge structure in the micropores can greatly enhance the capillary lift, which results in a faster capillary rise of the lubricant. In addition, the special microstructure in the inner spiral bulge micropores leads to changes in the tube shape and results in a strong Taylor capillary rise phenomenon. The spiral linear bulge structure in the micropore generates a Taylor capillary rise phenomenon and an enhanced capillary lift, which causes the capillary rise speed of the lubricant in the inner spiral bulge micropore to be significantly greater than that in a smooth micropore of the same size. The smaller screw pitch of the spiral bulge micropore gives a stronger capillary and Taylor capillary lift.

Due to the inertial effect, the fluid will flow into the tiny hole. The inertial effect of fluid is very important at the early stage of pore inflow. Although the inertial effect lasts for a short time in the pores, the fluid in the pores rises rapidly. Dimensionally, it is easy to see that the characteristic timescale over which inertial effects will be important τi∼ρR02/η∼0.02s, after which the usual Washburn dynamics follows. Before this time the fluid enters the capillary aswhere ρ is the liquid density. Although the effect of inertia lasts for a very short time in small pores, the capillary rise can nevertheless be quite rapid. Inertia effect significantly affects the initial height and time of capillary rise in simulation experiments, experimentally inertial effects can give rise to effective initial height and time, initial height and time in Eq. (8)2727. M. Alava, M. Dubé, and M. Rost, “Imbibition in disordered media,” Adv. Phys. 53, 83–175 (2004). https://doi.org/10.1080/00018730410001687363 that must be taken into account in fitting measured data.

Spiral bulge structures in the micropore reduce their effective diameter and change its shape. The lubricant in the spiral bulge structure produces a Taylor capillary rise, and the micropore capillary lift is enhanced. Compared with micropores that have a smooth surface, the capillary rising speed in the inner spiral bulge micropores is greatly increased, which is conducive to the rapid formation of a lubricating film on the porous surface. The dynamic contact angle is an important parameter in the wetting front studied in this paper. The research team realized that it is difficult to measure accurate dynamic contact angles only by simulation experiments. Later, it is planned to carry out experiments to measure the experimental results, so as to better characterize the dynamic contact angle changes. Related research can guide the development of porous self-lubricating materials with an inner spiral bulge structure as the bionic prototype.