A biomimetic method for analyzing quasi-static droplet directional flow on aluminum-based surfaces based on the Furmidge equation.

Abstract

This invention relates to the field of microfluidic control technology and provides a method for analyzing the directional flow of quasi-static droplets on a biomimetic aluminum-based surface based on the Furmidge equation. First, a three-dimensional form of the Furmidge equation is established, and the contact angle hysteresis driving term is solved by integration along the contact line. Then, a tensile adhesion term is established to describe the tensile deformation of the capillary bridge at the receding contact line. A rectangular micropillar array surface is used, resulting in a 91.7%–180.0% higher adhesion force of the droplet along the direction perpendicular to the surface grooves compared to the parallel direction, thus achieving directional flow control of quasi-static droplets on a biomimetic aluminum-based surface. This invention provides a theoretical basis and quantitative model for the design of liquid-directed transport surfaces in fields such as microfluidic chips, water collection / mist systems, and oil-water separation.

Description

Technical Field

[0001] This invention belongs to the field of microfluidic control technology, and in particular relates to a biomimetic method for analyzing quasi-static droplet directional flow on an aluminum-based surface based on the Furmidge equation. Background Technology

[0002] Classical biomimetic surface research has primarily focused on the application of homogeneous (isotropic) wettable surfaces, such as the self-cleaning effect of lotus leaves (i.e., the lotus effect) and the high adhesion of rose petals (i.e., the petal effect). These studies have provided important references for biomimetic surface design, but are mainly limited to the control of wetting properties in a single direction.

[0003] With the increasing demand for complex liquid transport paths in interdisciplinary fields such as water collection / mist systems, oil-water separation, and microfluidic chips, the limitations of uniformly wettable surfaces in micro-flow control are becoming increasingly apparent. These applications require precise control of liquid flow direction, a requirement that traditional uniformly wettable surfaces cannot meet. Against this backdrop, biological surfaces with unique structures found in nature, such as rice leaves, Norfolk Island pine leaves, and the ratchet-like epidermal structure of the succulent *Echeveria elegans*, exhibit excellent directional liquid transport capabilities. By modulating the adhesion of liquids to the surface through structural anisotropy, liquid flow control functions surpassing those of uniformly wettable surfaces are achieved, providing a novel approach to the design of wettable functional surfaces capable of manipulating liquid flow direction.

[0004] However, traditional wetting theories are typically based on two-dimensional assumptions and static models, failing to describe the asymmetric dynamic evolution of the liquid contact line on anisotropic surfaces in three dimensions. Therefore, existing research lacks a systematic analysis of the dynamic mechanical changes (such as pinning-unpinning dynamics) on anisotropic surfaces, making it difficult to construct quantitative models of anisotropic adhesion forces in biomimetic structures, thus becoming a key scientific problem in the field of microfluidic control. Furthermore, the classical Furmidge equation framework, as a two-dimensional model, contains physical contradictions stemming from its dimensional simplification: an energy barrier exists before droplet initiation, at which point the contact line has not yet moved, but the droplet surface has already deformed; this quasi-static process is also a region neglected in the original two-dimensional model. Summary of the Invention

[0005] The purpose of this invention is to provide a biomimetic method for analyzing the directional flow of quasi-static droplets on an aluminum-based surface based on the Furmidge equation, aiming to solve the problems mentioned in the background art.

[0006] The present invention is implemented as follows: a biomimetic quasi-static droplet directional flow analysis method based on the Furmidge equation on an aluminum-based surface includes the following steps: Step 1: Establish the three-dimensional form of the Furmidge equation, in which the gravity-driven term, contact angle hysteresis-driven term, and adhesion force term of the droplet are integrated in three-dimensional space. The adhesion force term includes tensile adhesion force and frictional adhesion force. Step 2: Establish the tensile adhesion force term. Based on the fixed droplets on the surface of the micropillar array, express the tensile adhesion force of the capillary bridge at the retreating contact line as a function of the corresponding contact angle, and introduce it into the three-dimensional Furmidge equation.

[0007] A further technical solution is that, in step 1, the gravity-driven item... Calculated using the following formula: ; in, The mass of the droplet is (kg). Acceleration due to gravity (m / s²) 2 ), The surface tilt angle.

[0008] A further technical solution involves solving the contact angle hysteresis driving term in step 1 by performing line integration along the three-dimensional contact line of the droplet, specifically including: Establish a right-handed Cartesian coordinate system, with the longitudinal expansion direction of the droplet as the x-axis and the lateral expansion direction as the y-axis. Define a portion of the contact line within the positive x-axis region as the advancing contact line. Its projection length on the x-axis is The portion of the contact line within the area on the other side is defined as the retreating contact line. Its projection length on the x-axis is ; By performing line integration on the advancing and retracting contact lines respectively, the contact angle hysteresis driving term is obtained; The component of the liquid surface tension along the x-axis at the liquid advancing and retreating contact lines. and Calculate using the following formulas respectively: ; ; in, and These are the components of the surface tension along the x-axis at the x-coordinate of the front and rear contact lines (unit: N). and These are the forward contact angle and the backward contact angle at the x-coordinate, respectively; Let the angle between the surface tension projection line in the xoy plane and the x-axis be denoted as . is the surface tension (mN / m).

[0009] Further technical solutions, and The following boundary conditions must be met: ; ; in, The apparent contact angle is the two-dimensional projection of the droplet onto the contact line in the yoz plane when the line of sight is parallel to the x-axis.

[0010] A further technical solution involves, in step 2, the tensile adhesion force of the capillary bridges along the retreating contact lines of the fixed droplets on the surface of the micropillar array. Expressed as corresponding contact angle The functional relationship is: ; in, The length of the contact wire on a single support (m).

[0011] In a further technical solution, in step 2, the micro-pillar array is a rectangular pillar array, where the width of a single rectangular pillar is P, the height is H, and the pillar spacing is V. The stretching length of the capillary bridge in the direction perpendicular to the surface is approximately calculated using the following formula: ; in, For the stretch length, This is the length of the capillary tube.

[0012] In a further technical solution, in step 2, when the capillary bridge reaches its maximum tensile length, it is considered to be the critical tensile length, and its corresponding critical contact angle is... according to Export.

[0013] The quasi-static droplet directional flow analysis method based on the Furmidge equation for biomimetic aluminum-based surfaces provided in this invention has the following advantages: (1) By introducing a tensile adhesion force term, the present invention constructs a three-dimensional Furmidge equation, which solves the problem of geometric discontinuity and force field dimension mismatch in the two-dimensional equation, and realizes an accurate description of the directional flow process of droplets on an anisotropic surface.

[0014] (2) By using a rectangular micro-pillar array surface, the adhesion force of droplets along the vertical direction of the surface groove (intermittent contact line mode, dominated by tensile adhesion force) is 91.7%-180.0% higher than the adhesion force in the parallel direction (continuous contact line mode, dominated by frictional adhesion force), thus realizing the directional flow control of quasi-static droplets on the biomimetic aluminum-based surface.

[0015] (3) This invention provides a theoretical basis and quantitative model for the design of liquid directional transport surfaces in the fields of microfluidic chips, water collection / mist systems, and oil-water separation. Attached Figure Description

[0016] Figure 1 Consider a droplet slowly sliding along an inclined plate and the forces acting on it in the horizontal direction along the inclined base (where a is a two-dimensional droplet and b is a three-dimensional droplet). Figure 2 The capillary bridge coordinate system is defined when the line of sight is parallel to the groove (where a is the droplet in the air trap state, b is a local magnified view and geometric features of the droplet contact line, and c is the single-pillar droplet capillary bridge coordinate system). Detailed Implementation

[0017] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0018] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0019] An embodiment of the present invention provides a method for analyzing the quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation, comprising the following steps: Step 1: Establish the three-dimensional form of the Furmidge equation; In such Figure 1 The two-dimensional model shown in figure a Surface tension (mN / m) Forward contact angle, For the retreat contact angle, The surface tilt angle, The mass of the droplet is (kg). Acceleration due to gravity (m / s²) 2 When the surface tilt angle is At that time, the parallel components of the forces acting on the fixed droplet along its surface are: (1); in, The horizontal component of the surface tension of a two-dimensional droplet in a quasi-static motion process; It is a gravity-driven term; This is a contact angle hysteresis driving term; This is the adhesion force term, including the tensile adhesion force and frictional adhesion force derived later.

[0020] Gravity-driven items for: (2); in, The mass of the droplet is (kg). Acceleration due to gravity (m / s²) 2 ).

[0021] Contact angle hysteresis driving term for: (3); in, and These are the components of the liquid surface tension along the x-direction at the front and rear contact lines of the liquid, namely the forward capillary force and the backward capillary force (mN / m): (4); (5); in, For the capillary force propelling 2D droplets, Surface tension (mN / m) Forward contact angle, For the retreat contact angle, For 2D droplet retraction capillary force.

[0022] For example Figure 1 b is a three-dimensional droplet (where, and These are the forward contact angle and the backward contact angle at the x-coordinate, respectively. Let the angle between the surface tension projection line in the xoy plane and the x-axis be denoted as . The gravity-driven term is the same as that in equation (2), and the contact angle hysteresis-driven term needs to be solved by integration along the contact line. Therefore, a right-handed Cartesian coordinate system is established, with the longitudinal and lateral expansion of the droplet as the x-axis and y-axis, respectively, and their intersection as the origin. The partial contact line within the positive x-axis region (first and fourth quadrants, i.e., the direction of droplet rolling tendency) in the xoy plane is defined as the forward contact line. Its projection length on the x-axis is The portion of the contact line within the area on the other side is defined as the retreating contact line. Its projection length on the x-axis is Line integrals along the two contact lines yield: (6); (7); in, and These are the components of the surface tension along the x-axis at the x-coordinate of the front and rear contact lines (unit: N). and These are the forward contact angle and the backward contact angle at the x-coordinate, respectively; Let be the angle between the surface tension projection line in the xoy plane and the x-axis.

[0023] Assume the droplet is symmetric about the xoz plane, and and The following boundary conditions must be met: (8); (9); in, The apparent contact angle is the two-dimensional projection of the droplet onto the contact line in the yoz plane when the line of sight is parallel to the x-axis.

[0024] Step 2: Establish the tensile adhesion term; Stretch adhesion It is the adhesion force (N) of the solid surface to the capillary bridge at the retreating contact line, and the reaction force of the perpendicular component of the surface tension acting on the retreating contact line. For a fixed droplet on the surface of a micropillar array, the tensile adhesion force of the capillary bridge at its retreating contact line is... This can be expressed as the corresponding contact angle. The functional relationship is: (10); in, Let be the contact line length (m) on a single support. Assume the cosine of the contact angle varies linearly along the x-axis, and that the contact angle at the laterally widened portion is the apparent contact angle when the surface is not inclined. (11); in, Let be the apparent contact angle of the droplet's two-dimensional projection onto the yoz plane at the contact line.

[0025] The shape change of a capillary bridge during stretching determines the magnitude of the tensile adhesion force, which is controlled by both liquid cohesion and solid adhesion. For low-viscosity liquids (such as water), its elastic deformation is negligible. The capillary bridge deformation can be determined by calculating the receding contact angle. Figure 2 a and Figure 2 As shown in b, for an RRSC surface with an array of rectangular pillars (each rectangular pillar has a width of P, a height of H, and a pillar spacing of V), when the horizontal surface is tilted to a position through a quasi-static process... At that time, the capillary bridge at the droplet retreat contact line has a stretching length in the direction perpendicular to the surface as follows: .like Figure 2 As shown in Figure c, establish a rectangular coordinate axis, with the axis of symmetry in the width direction of the support pillar as... The axis, with the horizontal liquid surface as The axis. The meniscus shape of the capillary bridge is described by the Laplace equation: (12); in, Liquid density (kg / m³) 3 ), Gas density (kg / m³) 3 ).

[0026] Its boundary conditions are: (13) According to equations (12) and (13) .

[0027] When the support width is much smaller than the capillary length (For water at 20 ℃, When the length of the capillary bridge is 2.73 mm, according to equations (12) and (13), the length (m) of the capillary bridge stretched in the vertical direction on the surface can be approximated with high precision as follows: (14); when When the above formula is used, the approximate error is within 1%.

[0028] When the capillary bridge reaches its maximum tensile length, it is considered to be at its critical tensile length, and its corresponding critical contact angle is... According to By deriving and substituting into equation (14), we have: (15); This method introduces a tensile adhesion term to construct a three-dimensional Furmidge equation, thus solving the problems of geometric discontinuity and force field dimension mismatch in the two-dimensional equation. By employing a rectangular micropillar array surface, the adhesion force of droplets along the vertical direction of the surface grooves (discontinuous contact line mode, dominated by tensile adhesion force) is 91.7%-180.0% higher than the adhesion force in the parallel direction (continuous contact line mode, dominated by frictional adhesion force), achieving directional flow of quasi-static droplets on a biomimetic aluminum-based surface.

[0029] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A biomimetic method for analyzing quasi-static droplet directional flow on an aluminum-based surface based on the Furmidge equation, characterized in that... Includes the following steps: Step 1: Establish the three-dimensional form of the Furmidge equation, in which the gravity-driven term, contact angle hysteresis-driven term, and adhesion force term of the droplet are integrated in three-dimensional space. The adhesion force term includes tensile adhesion force and frictional adhesion force. Step 2: Establish the tensile adhesion force term. Based on the fixed droplets on the surface of the micropillar array, express the tensile adhesion force of the capillary bridge at the retreating contact line as a function of the corresponding contact angle, and introduce it into the three-dimensional Furmidge equation.

2. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation, as described in claim 1, is characterized in that... In step 1, gravity-driven item Calculated using the following formula: ； in, For the mass of the droplet, It is the acceleration due to gravity. The surface tilt angle.

3. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation, as described in claim 2, is characterized in that... In step 1, the contact angle hysteresis driving term is solved by line integration along the three-dimensional contact line of the droplet, specifically including: Establish a right-handed Cartesian coordinate system, with the longitudinal expansion direction of the droplet as the x-axis and the lateral expansion direction as the y-axis. Define the portion of the contact line within the positive x-axis region as the advancing contact line. Its projection length on the x-axis is The portion of the contact line within the area on the other side is defined as the retreating contact line. Its projection length on the x-axis is ; By performing line integration on the advancing and retracting contact lines respectively, the contact angle hysteresis driving term is obtained; The component of the liquid surface tension along the x-axis at the liquid advancing and retreating contact lines. and Calculate using the following formulas respectively: ； ； in, and These are the components of the surface tension along the x-axis at the x-coordinate, representing the surface tension of the front and rear contact lines. and These are the forward contact angle and the backward contact angle at the x-coordinate, respectively; Let the angle between the surface tension projection line in the xoy plane and the x-axis be denoted as . Surface tension.

4. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation according to claim 3, characterized in that, and The following boundary conditions must be met: ； ； in, The apparent contact angle is the two-dimensional projection of the droplet onto the contact line in the yoz plane when the line of sight is parallel to the x-axis.

5. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation according to claim 3, characterized in that, In step 2, for the fixed droplets on the surface of the micropillar array, the tensile adhesion force of the capillary bridges at the retreating contact lines... Expressed as corresponding contact angle The functional relationship is: ； in, This refers to the length of the contact wire on a single support.

6. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation, as described in claim 5, is characterized in that... In step 2, the micro-pillar array is a rectangular pillar array, with a width of P, a height of H, and a pillar spacing of V for each individual rectangular pillar. The stretching length of the capillary bridge in the direction perpendicular to the surface is approximately calculated using the following formula: ； in, For the stretch length, This is the length of the capillary tube.

7. The method for analyzing quasi-static droplet directional flow on a biomimetic aluminum-based surface based on the Furmidge equation according to claim 6, characterized in that, In step 2, when the capillary bridge reaches its maximum tensile length, it is considered to be at the critical tensile length, and its corresponding critical contact angle is... according to Export.

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