CN108052729A - The Reverse Design of super hydrophobic surface micro-structure - Google Patents
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Abstract
The invention discloses a kind of Reverse Design of super hydrophobic surface micro-structure, including：The roughness of super-hydrophobic surface of solids micro-structure is usually all periodic distribution, the periodic distribution of super-hydrophobic surface of solids micro-structure is divided into 3 kinds of periodic cells, these three periodic cells are reduced to three kinds of different triangles according to their symmetry, the configuration design of the micro-structure of the surface of solids can be carried out by Delta Region, and then periodically micro-structure can be obtained by symmetrical and stretched operation, in the state of the equilibrium, the liquid-vapor interface supported by periodic micro structure is the two-dimensional manifold for having constant value Riemann curvature, liquid-vapor interface can be described on simplified Delta Region by Young Laplace equations.The present invention is by the modeling of the liquid-vapor interface supported to micro-structure on coarse solids surface, the geometric configuration of calculating periodic microstructures corresponding with the minimum of raised volume, to realize the reverse engineer of super hydrophobic surface micro-structure.
Description
Technical Field
The invention relates to the technical field of geometric configuration design of a periodic microstructure on a super-hydrophobic rough surface, in particular to a reverse design method of a super-hydrophobic surface microstructure.
Background
Wetting is an important aspect of surface chemistry, and the wetting ability of solid surfaces can be classified by hydrophobicity, hydrophilicity, oleophobicity, lipophilicity, amphiphobicity, and amphiphilicity. In the study of artificial surface structures with specific wetting ability, recent efforts have been directed primarily to superhydrophobicity. Superhydrophobicity means that the contact angle of water on a superhydrophobic solid substrate can reach 150 ° or even higher.
For a flat plane, the wetting capacity is determined by the free energy of the solid surface, and the wetting phenomenon can be described by a Young model; the contact of the solid with the liquid on the rough surface with periodic microstructure can be described using two different models, respectively a Wenzel model and a Cassie-Baxter model, as shown in figures 1a, 1 b. In the Wenzel model, the liquid completely fills the microstructure on the rough surface, and the hydrophobicity of the rough surface increases with the increase of the surface roughness; in the Cassie-Baxter model shown in FIG. 1c, the vapor pocket is bound in the microstructure where the rough surface corresponds to a composite surface of solid and gas, and the hydrophobicity of the rough surface increases with increasing surface roughness, as shown in FIG. 2.
For a rough solid surface, the model of contact between the solid and liquid can be changed from the Cassie-Baxter type to the Wenzel type as the pressure of the liquid on it increases. During this model transition, the liquid fills the microstructure of the rough surface and leads to a decrease in hydrophobicity. In the Cassie-Baxter model, the vapor pocket results in the presence of an interface of liquid and vapor, supported by the microstructure of the solid surface. Due to the hysteresis of the contact angle, the three-phase contact line can be anchored at the corner formed by the top surface and the sidewall of the microstructure. If the static pressure difference between the two sides of the liquid-vapor interface supported by the microstructure is large enough, the contact angle between the liquid-vapor interface and the wall surface of the microstructure reaches its critical value, and the liquid-vapor interface is broken, resulting in the transformation of the Cassie-Baxter model to the Wenzel model. The stability of the Cassie-Baxter model decreases as the contact angle approaches the critical value; meanwhile, the liquid-vapor interface supported by the microstructure will also sink, and the liquid protrusion supported by the liquid-vapor interface will also become larger. Therefore, the reasonable microstructure on the rough solid surface can effectively avoid the transformation of the Cassie-Baxter model, and can increase the stability of the super-hydrophobic surface by reducing the convex volume of the liquid-vapor interface and make the contact angle of the liquid-vapor interface far away from the critical value.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a method for realizing the reverse design of a super-hydrophobic surface microstructure by modeling a liquid-vapor interface supported by a microstructure on a rough solid surface and calculating the geometric configuration of a periodic microstructure corresponding to the minimum value of the raised volume.
In order to achieve the purpose, the invention adopts the following technical scheme: the reverse design method of the super-hydrophobic surface microstructure is characterized by comprising the following steps:
the roughness of the surface microstructure of the superhydrophobic solid is generally periodically distributed, and the periodic distribution of the surface microstructure of the superhydrophobic solid is divided into: the three periodic units are simplified into three different triangles according to the symmetry of the three periodic units, the configuration design of the microstructure on the surface of the solid can be carried out through a triangular area, further the periodic microstructure can be obtained through symmetry and stretching operations, under the balanced state, a liquid-vapor interface supported by the periodic microstructure is a two-dimensional manifold with a constant Riemann curvature, and the liquid-vapor interface can be described by a Young-Laplace equation on a simplified 5 triangular area:
z＝0，at 0
wherein σ is the surface tension; p0Is the static pressure difference between the inside and outside of the fluid; z is the vertical coordinate of a point on the liquid-vapor interface, x ═ x, y is the horizontal and vertical coordinates of a point on the liquid-vapor interface,is to find the xOy planeWhere O represents the origin of coordinates, Ω is the simplified triangular region, and Γ is the symmetric boundary; n is the unit normal vector outward on the boundary.
For the uniqueness of the Young-Laplace equation solution, the liquid-vapor interface is fixed at the center point of the periodic unit, so that the constraints on the liquid-vapor interface are minimal, and the dimensionless form of equation 1 is:
wherein,z0is of the order of z; l is the period constant of a periodic unit, i.e. the center-to-center spacing of two adjacent periodic units; x ═ X/L ═ (X, Y) are dimensionless coordinates;is a dimensionless surface tension;andare respectively omega, gammaN，ΓDO corresponding to a dimensionless geometric quantity,is defined inThe gradient operator of (3).
Simplifying the volume of liquid projections supported by the microstructures in the triangular regions can be achieved byA calculation is performed in which, among other things,andis aboutThe two different norms of the signal to be measured,is thatThe first-order Hilbert function space above, according to the equivalence of norms,andare equivalent.
The method also comprises the step of constructing a variational problem based on a topological optimization method, wherein design variables in the topological optimization method are defined on a simplified triangular region and are used for interpolation of dimensionless surface tension, and the variational problem can be solved by adopting a numerical optimization iterative algorithm.
Regularizing the design variables by a Helmholtz filtering method, further performing projection processing on the design variables by a threshold value method, and further performing interpolation of a surface tension formula by using the projection design variables:
wherein, γpIs a projection design variable, 0 and 1 represent the microstructure and the hollowing on the solid surface, respectively;is the dimensionless surface tension of the liquid-vapor interface;is the surface tension of the microstructure tip; q is a parameter for adjusting the convexity of equation (3), and has a value of 10-4In theory, the first and second reaction conditions,infinite, corresponding to a flat liquid-solid interface; in the numerical implementation, it is chosen to satisfyFinite value ofThe inverse design variation problem of the microstructure is then classified into the following equations with the stability of number solution ensured:
find 0≤γ≤1to minimizewithconstrained by
wherein duty cycle constraints of the microstructure are included, fdRepresents the duty cycle, f0Representative tolerance of 10-3Specifying a duty cycle; j. the design is a square0Is at a specified duty cycle f0Lower, periodic regular triangleThe liquid-vapor interface measurement value supported by the structure of the regular quadrangle and the regular hexagon; gamma is a design variable, gammafIs a design variable after filtering; r isfβ and ξ are projection parameters selected based on numerical tests, and then in a simplified triangular region, the variation problem in the formula (4) is solved, and the inverse design of the geometric configuration of the microstructure can be realized.
Solving the variation problem in the formula 4, and adopting an iterative algorithm based on gradient information, wherein the gradient information of the liquid-vapor interface metric value and the duty ratio can be obtained by an adjoint analysis method based on a Lagrange multiplier, and based on the adjoint method, the gradient information of the liquid-vapor interface metric value is as follows:
wherein δ J and δ γ are the first order variations of J and γ, respectively; gamma rayfaIs a design variable gammafAssociated variable of, γ in equation 5faThe method is obtained by solving the companion equation of the Young-Laplace equation and the Helmholtz filter equation in sequence:
wherein,and gammafaAre respectivelyAnd gammafThe accompanying variable of (a);andare respectivelyAnd gammafaThe gradient information of the duty ratio can be obtained by equation 8:
δfd＝fΩ-γfaδγds (8)
δfdis the duty cycle fdFirst order variation of (1); gamma rayfaThis can be obtained by solving the following equation:
after the adjoint analysis is carried out, the variational problem is solved by adopting the following iteration steps:
(a) calculating the duty ratio value of the microstructure according to the current design variable;
(b) solving a Young-Laplace equation;
(c) calculating a liquid-vapor interface measurement value;
(d) solving companion equations 6 and 7;
(e) obtaining gradient information of a liquid-vapor interface metric value;
(f) solving adjoint equation 9;
(g) calculating gradient information of duty ratio constraint;
(h) evolving design variables;
(i) judging whether the convergence condition is met, and if so, terminating the iteration condition; if not, returning to the step (a), wherein the iteration termination condition is as follows: (1) the number of iterations reaches a maximum value 315; (2) the design target varies by less than 10 for 5 consecutive iterations compared to both its mean and duty cycle tolerance-3，
In the above iteration process, the initial value of β is 1, and then the iteration is doubled every 30 times, and ξ takes 0.5.
The invention has the beneficial effects that: the invention belongs to an inverse design method of a micro-structure geometric configuration on a super-hydrophobic rough solid surface, which overcomes the limitation of the existing surface micro-structure design by means of intuition and bionics of researchers, and improves the applicability, flexibility and efficiency of the design method. The reverse design of the super-hydrophobic surface microstructure provided by the invention is realized by modeling a liquid-vapor interface supported by a microstructure on a rough solid surface and calculating the geometric configuration of a periodic microstructure corresponding to the minimum value of the raised volume.
Drawings
FIGS. 1a, 1b, 1c are schematic diagrams of the Young model, Wenzel model and Cassie-Baxter model, respectively, of a liquid droplet on a solid surface.
FIG. 2 is a schematic diagram showing the composite surface of solid and gas formed by the Cassie-Baxter model in which the vapor pockets are bound in microstructures.
FIGS. 3a, 3b, 3c are simplified schematic diagrams of regular triangular, regular quadrilateral, regular hexagonal periodic solid surface microstructure filling, and triangular shape based symmetry of each periodic unit, respectively.
Fig. 4a, 4b, 4c are simplified schematic diagrams of triangles with regular triangles, regular tetragons, regular hexagonal microstructure geometries, and periodic units according to symmetry, respectively.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not to be construed as limiting the invention.
The reverse design method of the super-hydrophobic surface microstructure comprises the following steps:
modeling: the roughness of the surface of the superhydrophobic solid is typically periodic. In the periodic distribution of the microstructure, regular triangles, regular quadrangles, and regular hexagons may be used to periodically fill the solid surface. As shown in fig. 3a, 3b, 3c, the three periodic elements can be simplified into three different triangles according to their symmetry. Thus, the design of the microstructure of the solid surface can be carried out in triangular areas, whereby a periodic microstructure can be obtained by means of a symmetry and stretching operation. In equilibrium, the liquid-vapor interface supported by the periodic microstructure is a two-dimensional manifold with a constant Riemann curvature, which can be described by the Young-Laplace equation over a simplified triangular region:
z＝0，at 0
σ here is the surface tension; p0Is the static pressure difference between the inside and outside of the fluid; z is the vertical coordinate of a point on the liquid-vapor interface, and x ═ x, y is the horizontal and vertical coordinates of a point on the liquid-vapor interface.Is to gradient the xOy plane, where O represents the origin of coordinates. Omega is the simplified triangular area shown in figures 3a, 3b, 3 c. Γ is a symmetric boundary; n is the unit normal vector outward on the boundary. For the uniqueness of the Young-Laplace equation solution described above, the liquid-vapor interface is fixed at the center point of the periodic cell, which is the least constraint on the liquid-vapor interface. The dimensionless form of equation 1 is:
wherein,z0is of the order of z; l is the period constant of a periodic unit, i.e. the center-to-center spacing of two adjacent periodic units; x ═ X/L ═ (X, Y) are dimensionless coordinates;is a dimensionless surface tension;andare respectively omega, gammaN，ΓDO corresponding to a dimensionless geometric quantity, as shown in fig. 4a, 4b, 4 c;is defined inThe gradient operator of (3).
Simplifying the volume of liquid projections supported by the microstructures in the triangular regions can be achieved byAnd (6) performing calculation.Andis aboutThe two different norms of the signal to be measured,is thatFirst order Hilbert function space above. According to the equivalence of the norm,andare equivalent. Therefore, the performance is goodCan be used to measure the stability of the liquid-vapor interface on the solid surface supported by the periodic microstructure,representsThe area of the region. The smaller the metric, the higher the stability of the liquid-vapor interface. The objective of the reverse design is to find the geometry of the microstructure with extremely small metric values. Because the liquid-vapor interface is a two-dimensional manifold with a constant Riemann curvature, it has a consistent convexity, and a liquid-vapor interface with a minimal magnitude can move the contact angle away from its critical value.
To realize the inverse design of the microstructure, a variational problem can be constructed based on a topological optimization method, wherein design variables in the topological optimization method are defined on a simplified triangular region and are used for the interpolation of dimensionless surface tension. The variational problem can be solved by adopting a numerical optimization iterative algorithm. In order to ensure the robustness of the iterative algorithm, a Helmholtz filtering method is adopted to regularize design variables, the design variables are further subjected to projection processing by a threshold value method, and then the projection design variables are adopted to perform interpolation of a surface tension formula
Wherein, γpIs a projection design variable, 0 and 1 represent the microstructure and the hollowing on the solid surface, respectively;is the dimensionless surface tension of the liquid-vapor interface;is the surface tension of the microstructure tip; q is a parameter for adjusting the convexity of equation (3), and has a value of 10-4. In theory, it is possible to use,infinite, corresponding to a flat liquid-solid interface; in the numerical implementation, it is chosen to satisfyFinite value ofTo ensure the stability of numerical solution. The inverse design variation problem for microstructures can then be attributed to the following equation:
find 0≤γ≤1to minimizewithconstrained by
wherein duty cycle constraints of the microstructure are included, fdRepresents the duty cycle, f0Representative tolerance of 10-3Specifying a duty cycle; j. the design is a square0Is at a specified duty cycle f0Measuring the liquid-vapor interface value supported by the periodic regular triangle, regular quadrangle and regular hexagon structures; gamma is a design variable, gammafIs a design variable after filtering; r isfThe radius of the Helmholtz filter, which can be used to control the feature size of the microstructure, β and ξ are projection parameters selected based on numerical experiments.
2. Analysis and solution
The solution of the variation problem in formula 4 can adopt an iterative algorithm based on gradient information, wherein the gradient information of the liquid-vapor interface metric value and the duty ratio can be obtained by a companion analysis method based on a lagrange multiplier. Based on the adjoint method, the gradient information of the liquid-vapor interface measurement value is
Wherein δ J and δ γ are the first order variations of J and γ, respectively; gamma rayfaIs a design variable gammafIs used as a companion variable. γ in equation 5faThe method is obtained by solving the companion equation of the Young-Laplace equation and the Helmholtz filter equation in sequence:
wherein,and gammafaAre respectivelyAnd gammafThe accompanying variable of (a);andare respectivelyAnd gammafaTrial function of (2). The gradient information of the duty ratio can be obtained by equation 8:
δfd＝∫Ω-γfaδγds (8)
δfdis the duty cycle fdFirst order variation of (1); gamma rayfaThis can be obtained by solving the following equation:
after the adjoint analysis is carried out, the variational problem is solved by adopting the following iteration steps:
(a) calculating the duty ratio value of the microstructure according to the current design variable;
(b) solving a Young-Laplace equation;
(c) calculating a liquid-vapor interface measurement value;
(d) solving companion equations 6 and 7;
(e) obtaining gradient information of a liquid-vapor interface metric value;
(f) solving adjoint equation 9;
(g) calculating gradient information of duty ratio constraint;
(h) evolution design variables
(i) Judging whether the convergence condition is met, and if so, terminating the iteration condition; if not, returning to the step (a). Wherein, the iteration termination condition is as follows: (1) the number of iterations reaches a maximum value 315; (2) the design target varies by less than 10 for 5 consecutive iterations compared to both its mean and duty cycle tolerance-3。
In the above iteration process, the initial value of β is 1, and then the iteration is doubled every 30 times, and ξ takes 0.5.
In one embodiment, by adopting the reverse design method of the superhydrophobic surface microstructure, the relevant parameters are set to the values in table 1, and the geometric configurations of the microstructures such as a regular triangle, a regular quadrangle and a regular hexagon shown in fig. 4 can be obtained respectively.
TABLE 1
Based on the introduced inverse design method, the variation problem of formula 4 is solved through the parameters in table 1, and three different periodic microstructure geometries as shown in fig. 4a, 4b, and 4c can be obtained, wherein the microstructure geometry of the periodic unit is obtained based on the symmetric operation of the design variable distribution of the simplified triangular region. These three-dimensional microstructures are obtained by scaling and stretching operations in the vertical direction of the base plane, and they can be fabricated using a photolithography-type process. In the scaling operation, the characteristic size of a periodic cell is its scaling factor; the stretching distance should be greater than the depth of the liquid-vapor interface during the stretching operation, and in order to avoid collapse of the Cassie-Baxter model due to the liquid-vapor interface contacting the bottom of the microstructure, the stretching distance should be greater than the maximum value of the depth of the liquid-vapor interfaceAs shown in fig. 4a, 4b, and 4c, each periodic unit of the inversely designed microstructure geometry, i.e., the stem structure of a regular triangle, a regular quadrangle, or a regular hexagon, has Y, X, and X type topology, respectively.
The above-described embodiments of the present invention should not be construed as limiting the scope of the present invention. Any other corresponding changes and modifications made according to the technical idea of the present invention should be included in the protection scope of the claims of the present invention.
Claims (6)
1. A reverse design method of a super-hydrophobic surface microstructure is characterized by comprising the following steps:
the roughness of the surface microstructure of the superhydrophobic solid is generally periodically distributed, and the periodic distribution of the surface microstructure of the superhydrophobic solid is divided into: the three periodic units are simplified into three different triangles according to the symmetry of the three periodic units, the configuration design of the microstructure on the surface of the solid can be carried out through a triangular area, further the periodic microstructure can be obtained through symmetry and stretching operations, under the balanced state, a liquid-vapor interface supported by the periodic microstructure is a two-dimensional manifold with a constant Riemann curvature, and the liquid-vapor interface can be described by a Young-Laplace equation on the simplified triangular area:
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mo>&dtri;</mo> <mi>x</mi> </msub> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mi>&sigma;</mi> <mfrac> <mrow> <msub> <mo>&dtri;</mo> <mi>x</mi> </msub> <mi>z</mi> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mo>&dtri;</mo> <mi>x</mi> </msub> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>i</mi> <mi>n</mi> <mi>&Omega;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&sigma;</mi> <mfrac> <mrow> <msub> <mo>&dtri;</mo> <mi>x</mi> </msub> <mi>z</mi> </mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mo>&dtri;</mo> <mi>x</mi> </msub> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>&CenterDot;</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>on&Gamma;</mi> <mi>N</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mi>t</mi> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
wherein σ is the surface tension; p0Is the static pressure difference between the inside and outside of the fluid; z is the vertical coordinate of a point on the liquid-vapor interface, x ═ x, y is the horizontal and vertical coordinates of a point on the liquid-vapor interface,solving the gradient of an xOy plane, wherein O represents a coordinate origin, omega is a simplified triangular area, and gamma is a symmetrical boundary; n is the unit normal vector outward on the boundary.
2. The method of claim 1, wherein for uniqueness of the Young-Laplace equation solution, the liquid-vapor interface is fixed at the center point of the periodic unit, so that the liquid-vapor interface is least constrained, and the dimensionless form of equation 1 is:
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mfrac> <mrow> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>L</mi> <mo>/</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </msub> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mi>n</mi> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mfrac> <mrow> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>L</mi> <mo>/</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </msub> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>&CenterDot;</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>o</mi> <mi>n</mi> <msub> <mover> <mi>&Gamma;</mi> <mo>&OverBar;</mo> </mover> <mi>N</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mi>t</mi> <mover> <mi>O</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
wherein,z0is of the order of z; l is the period constant of a periodic unit, i.e. the center-to-center spacing of two adjacent periodic units; x ═ X/L ═ (X, Y) are dimensionless coordinates;is a dimensionless surface tension;andare respectively omega, gammaN，ΓDO corresponding to a dimensionless geometric quantity,is defined inThe gradient operator of (3).
3. The method of claim 2, wherein the volume of the liquid protrusions supported by the microstructures in the simplified triangular regions can be reduced byA calculation is performed in which, among other things,is aboutThe two different norms of the signal to be measured,is thatThe first-order Hilbert function space above, according to the equivalence of norms,are equivalent.
4. The method of claim 3, further comprising constructing a variational problem based on a topological optimization method, wherein the design variables in the topological optimization method are defined on the simplified triangular region and used for the dimensionless surface tension interpolation, and the variational problem can be solved by using an iterative algorithm of numerical optimization.
5. The reverse design method of the superhydrophobic surface microstructure of claim 4, wherein the design variables are regularized by a Helmholtz filtering method, and further the design variables are subjected to projection processing by a threshold method, and further the projection design variables are used to perform interpolation of a surface tension formula:
<mrow> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>l</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mfrac> <mrow> <mi>q</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>q</mi> <mo>+</mo> <msub> <mi>&gamma;</mi> <mi>p</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
wherein, γpIs a projection design variable, 0 and 1 represent the microstructure and the hollowing on the solid surface, respectively;is the dimensionless surface tension of the liquid-vapor interface;is the surface tension of the microstructure tip; q is a parameter for adjusting the convexity of equation (3), and has a value of 10-4In theory, the first and second reaction conditions,infinite, corresponding to a flat liquid-solid interface; in the numerical implementation, it is chosen to satisfyFinite value ofTo ensure the stability of numerical solution, the inverse design variation problem of the microstructure is classified into the following equations:
<mrow> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>d</mi> <mn>0</mn> <mo>&le;</mo> <mi>&gamma;</mi> <mo>&le;</mo> <mn>1</mn> <mi>t</mi> <mi>o</mi> <mi> </mi> <mi>min</mi> <mi>i</mi> <mi>m</mi> <mi>i</mi> <mi>z</mi> <mi>e</mi> <mfrac> <mi>J</mi> <msub> <mi>J</mi> <mn>0</mn> </msub> </mfrac> <mi>w</mi> <mi>i</mi> <mi>t</mi> <mi>h</mi> <mi> </mi> <mi>J</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> <mo>|</mo> </mrow> </mfrac> <msub> <mo>&Integral;</mo> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </msub> <msup> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mn>2</mn> </msup> <mi>d</mi> <mi>s</mi> <mo>,</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>i</mi> <mi>n</mi> <mi>e</mi> <mi>d</mi> <mi> </mi> <mi>b</mi> <mi>y</mi> </mrow>
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mfrac> <mrow> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>L</mi> <mo>/</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </msub> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mi>n</mi> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mfrac> <mrow> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>L</mi> <mo>/</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mo>&dtri;</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> </msub> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>&CenterDot;</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>o</mi> <mi>n</mi> <msub> <mover> <mi>&Gamma;</mi> <mo>&OverBar;</mo> </mover> <mi>N</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mi>t</mi> <mover> <mi>O</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>f</mi> <mn>2</mn> </msubsup> <mo>&dtri;</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>=</mo> <mi>&gamma;</mi> <mo>,</mo> <mi>i</mi> <mi>n</mi> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>r</mi> <mi>f</mi> <mn>2</mn> </msubsup> <mo>&dtri;</mo> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>&CenterDot;</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>o</mi> <mi>n</mi> <msub> <mover> <mi>&Gamma;</mi> <mo>&OverBar;</mo> </mover> <mi>N</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mi>t</mi> <mover> <mi>O</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&gamma;</mi> <mi>p</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&beta;</mi> <mi>&xi;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&beta;</mi> <mo>(</mo> <mrow> <msub> <mi>&gamma;</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>&xi;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> <mrow> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&beta;</mi> <mi>&xi;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>tanh</mi> <mrow> <mo>(</mo> <mi>&beta;</mi> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mi>&xi;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi>f</mi> <mi>d</mi> </msub> <mo>-</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mo>|</mo> </mrow> <mo>&le;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mi>w</mi> <mi>i</mi> <mi>t</mi> <mi>h</mi> <mi> </mi> <msub> <mi>f</mi> <mi>d</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> <mo>|</mo> </mrow> </mfrac> <msub> <mo>&Integral;</mo> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </msub> <mn>1</mn> <mo>-</mo> <msub> <mi>&gamma;</mi> <mi>p</mi> </msub> <mi>d</mi> <mi>s</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
wherein duty cycle constraints of the microstructure are included, fdRepresents the duty cycle, f0Representative tolerance of 10-3Specifying a duty cycle; j. the design is a square0Is at a specified duty cycle f0Measuring the liquid-vapor interface value supported by the periodic regular triangle, regular quadrangle and regular hexagon structures; gamma is a design variable, gammafIs a design variable after filtering; r isfβ and ξ are projection parameters selected based on numerical tests, and then in a simplified triangular region, the variation problem in the formula (4) is solved, and the inverse design of the geometric configuration of the microstructure can be realized.
6. The method for reverse design of the superhydrophobic surface microstructure according to claim 5, wherein the variation problem in formula 4 is solved, an iterative algorithm based on gradient information is adopted, wherein the gradient information of the liquid-vapor interface metric value and the duty ratio can be obtained by an adjoint analysis method based on a Lagrange multiplier, and based on the adjoint method, the gradient information of the liquid-vapor interface metric value is as follows:
<mrow> <mfrac> <mrow> <mi>&delta;</mi> <mi>J</mi> </mrow> <msub> <mi>J</mi> <mn>0</mn> </msub> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>J</mi> <mn>0</mn> </msub> </mfrac> <msub> <mo>&Integral;</mo> <mover> <mi>&Omega;</mi> <mo>&OverBar;</mo> </mover> </msub> <mo>-</mo> <msub> <mi>&gamma;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <mi>&delta;</mi> <mi>&gamma;</mi> <mi>d</mi> <mi>s</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
wherein δ J and δ γ are the first order variations of J and γ, respectively; gamma rayfaIs a design variable gammafAssociated variable of, γ in equation 5faThe method is obtained by solving the companion equation of the Young-Laplace equation and the Helmholtz filter equation in sequence:
wherein,and gammafaAre respectivelyAnd gammafThe accompanying variable of (a);andare respectivelyAnd gammafaThe gradient information of the duty ratio can be obtained by equation 8:
<mrow> <msub> <mi>&delta;f</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mo>&Integral;</mo> <mi>&Omega;</mi> </msub> <mo>-</mo> <msub> <mi>&gamma;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <msub> <mi>&delta;</mi> <mi>&gamma;</mi> </msub> <mi>d</mi> <mi>s</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
δfdis the duty cycle fdFirst order variation of (1); gamma rayfaThis can be obtained by solving the following equation:
after the adjoint analysis is carried out, the variational problem is solved by adopting the following iteration steps:
(a) calculating the duty ratio value of the microstructure according to the current design variable;
(b) solving a Young-Laplace equation;
(c) calculating a liquid-vapor interface measurement value;
(d) solving companion equations 6 and 7;
(e) obtaining gradient information of a liquid-vapor interface metric value;
(f) solving adjoint equation 9;
(g) calculating gradient information of duty ratio constraint;
(h) evolving design variables;
(i) judging whether the convergence condition is met, and if so, terminating the iteration condition; if not, returning to the step (a), wherein the iteration termination condition is as follows: (1) the number of iterations reaches a maximum value 315; (2) the design target varies by less than 10 for 5 consecutive iterations compared to both its mean and duty cycle tolerance-3，
In the above iteration process, the initial value of β is 1, and then the iteration is doubled every 30 times, and ξ takes 0.5.
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