CN112380789B - Method for determining contour curve and front (back) contact angle of liquid drop on inclined hydrophilic surface - Google Patents
Method for determining contour curve and front (back) contact angle of liquid drop on inclined hydrophilic surface Download PDFInfo
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Abstract
A method of determining a contour curve and a front (back) contact angle of a droplet on an inclined hydrophilic surface, comprising: (1) Defining a system of droplets and a sloped hydrophilic roughened surface, wherein the sloped hydrophilic roughened surface has micropillars periodically distributed thereon; (2) Based on the minimum energy principle, a nonlinear optimization and finite difference algorithm is adopted to deduce a system dimensionless free energy equation of the Wenzel state liquid drop on the inclined surface, and a calculation formula is simplified; (3) Optimizing and calculating the total minimum dimensionless free energy E' wmin‑2 and the forward contact angle theta 1‑0, the backward contact angle theta 2‑0 and the three-phase contact line length l 0 of the liquid drop on the inclined hydrophilic surface in the initial state; (4) The forward contact angle θ 1‑* and the backward contact angle θ 2‑* of the droplet are calculated when the three-phase contact line length l given of the droplet on the inclined roughened surface is known. The invention solves the problems of liquid drop profile analysis and dynamic contact angle theoretical calculation of liquid drops on the inclined hydrophilic rough surface.
Description
Technical Field
The invention particularly relates to a method for solving a contour curve of a liquid drop on an inclined hydrophilic surface and a front (back) contact angle.
Background
The movement of droplets on a surface is a phenomenon that can be observed in daily life and in applications in many industrial environments such as: coating process, combustion process, insecticide crushing, self-cleaning surface, self-water-collecting surface, micro-fluidic chip and other fields. Studying the state of motion of a droplet on an inclined hydrophilic roughened surface is a fundamental problem in wetting and diffusion mechanics, helping to better understand how to manipulate the state of motion of a droplet on a hydrophilic surface. It is apparent that the drop has two main states of motion on the inclined hydrophilic surface, sticking and inchworm sliding. However, the prediction of the motion state of a droplet on an inclined hydrophilic roughened surface is complicated and difficult due to factors such as the surface inclination angle of the droplet, the shape of the droplet, the advancing (receding) contact angle of the droplet, and the dynamic contact line characteristics of the droplet. Therefore, analyzing the contour curve of a droplet on an inclined surface and the front (back) contact angle of the droplet by numerical calculation is critical to analyzing the motion state of the droplet. Currently, numerical methods performed on inclined hydrophilic surfaces are known to calculate drop profiles and contact angles using the fluid Volume (VOF) solver developed in the JADIM code, but are computationally intensive for rough surfaces, and are not suitable for drop morphology on rough surfaces. Accordingly, researchers have been struggling to explore better techniques for dynamic contact angle measurement on inclined roughened surfaces.
With the intensive research on the problem, the calculation of the drop profile curve and the front (back) contact angle on the inclined hydrophilic roughened surface is realized by adopting a nonlinear optimized finite difference method based on the minimum energy theory and combining the code programmed by the 'fmincon' function in MATLAB software. And through experimental verification, a drop profile curve and a dynamic contact angle measurement result are obtained under the given surface and the geometric parameters of the drop. The practical calculation method of the contact angle of the liquid drop is deduced in theory, and the calculation method is applied to actual contact angle calculation, so that the method has important guiding significance in researching the dynamic performance of the liquid drop on the inclined hydrophilic surface and the design application of a micro-fluid system.
Disclosure of Invention
The present invention overcomes the above-mentioned shortcomings of the prior art by providing a method for calculating the heavy profile curve and the forward (backward) contact angle of a droplet on an inclined hydrophilic roughened surface to obtain the profile curve and the forward (backward) contact angle of the droplet on the surface.
A method for determining a contour curve and a front (back) contact angle of a droplet on an inclined hydrophilic surface, comprising the steps of:
(1) A system for defining droplets and a sloped hydrophilic roughened surface, wherein the sloped hydrophilic roughened surface has periodically distributed thereon micropillars, the system having a geometry of: the diameter a of the microcolumn, the distance d of the microcolumn, the height h of the microcolumn and the surface inclination angle phi. The rough surface is selected as the XOY plane, and the intersection point of the symmetry axes of the contact surfaces of the droplets is defined as the origin O and the direction rising along the inclined plane as the positive Y direction. The direction perpendicular to the surface and the curvature of the droplet is defined as the positive Z direction, and the direction pointing from inside to outside and perpendicular to the YOZ plane is defined as the X axis. Define the distance from O to a point on the curvature of the drop as the sagittal diameter Wherein the azimuth angle beta (-pi.ltoreq.beta.ltoreq.pi) represents the positive direction from the X-axis to/>The angle between projections on the XOY plane, the apex angle/>Representing the positive Z-axis direction to/>An included angle between the two. /(I)Is defined as r (β, α).
(2) Based on the minimum energy principle, adopting nonlinear optimization and finite difference algorithm to deduce a system dimensionless total energy equation of Wenzel state liquid drops on an inclined surface, and simplifying a calculation formula;
21. A surface composed of a hydrophilic material is selected, and the wetting contact state of the liquid drops on the inclined hydrophilic surface is a Wenzel state.
The volume of a droplet can be expressed as:
For the following algorithm, r (β, α) < r max is defined to control the drop profile, r max can be expressed as
rmax＝4R0 (2)
Wherein the method comprises the steps ofIs the equivalent radius of the liquid drop.
The roughness, i.e. the ratio of the actual solid-liquid contact area to the apparent solid-liquid contact area, is determined:
The outer surface area S ext-2 of the droplet is:
the apparent contact area S base-2 can be expressed as:
In the Wenzel state, the actual solid-liquid contact area S sl-2:
Ssl-2＝rghSbase-2 (6)
Actual solid-gas contact area S sv-2:
Where L 1 is the side length of the roughened surface, which is a constant.
22. The total system energy E w-2 for the Wenzel wet contact state drop was determined to be equal to the sum of gravitational potential energy E a-2 and interfacial free energy E b-2.
E w-2 can be represented as:
Ew-2＝Ea-2+Eb-2 (8)
E a-2 can be represented as:
E b-2 can be represented as:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
substituting the parameters into equations (9) (10) into (8) yields the total energy equation for the system with the droplet in a Wenzel state on an inclined hydrophilic roughened surface:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
23. simplifying the total energy E w-2 of the system in the Wenzel wet contact state. Knowing the physical properties of the droplets and the hydrophilic surface, in formula (11) Being constant, it can be a dimensionless total energy of the system E' w-2:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
(3) And optimally calculating the minimum dimensionless total energy E' wmin-2 of the liquid drop on the inclined hydrophilic surface in the initial state, the forward contact angle theta 1-0, the backward contact angle theta 2-0 and the three-phase contact line length l 0.
31. And (3) dispersing: the curvature of the liquid drop profile is discretized by p (equal fraction of beta) and q (equal fraction of alpha), and the discretized radius is r (beta j,αi), which can be simplified as r [ j, i ];
wherein:
32. Optimizing E' wmin-2
(A) Setting variables: r [ j, i ] (j=1, 2,) p+1, i=1, 2, & q+1
(B) Setting an optimization target:
33. establishing constraint conditions:
i.0＜r[j,i]≤4R0(j＝1,2,...,p+1；i＝1,2,...,q+1) (16)
ii.
34. Searching:
And searching the variable r [ j, i ] by using an ' fmincon ' function in Matlab to obtain a system minimum dimensionless total energy optimal solution E ' wmin-2 and a corresponding r < j, i >.
Searching E ' wmin-2 by using an ' fmincon ' function in Matlab, continuously calculating until the difference value of E ' wmin-2 is converged within an acceptable range of 10 -4, recording a corresponding E ' wmin-2, a contour curve and a three-phase contact line, calculating a forward contact angle theta 1-0, a backward contact angle theta 2-0 and a three-phase contact line length l 0 of the liquid drop on an inclined plane by using the contour curve, otherwise, dividing the multiplication of inversion and the addition of p and q, and carrying out the steps (2) and (3) again.
The following result can be obtained by r < j, i >:
l0＝r<1,q+1>+r<p+1,q+1> (20)
(4) When the three-phase contact line length l given of the liquid drop on the inclined rough surface is known, the forward contact angle theta 1-* and the backward contact angle theta 2-* of the liquid drop are obtained;
With the same procedure as in (1) - (3) above, it is additionally necessary to add the constraint r [1, q+1] +rp+1, q+1] =l given in step 33 above, the corresponding system minimum dimensionless total energy E' wmin-2-* for a given contact line length of the inclined surface and the corresponding r < j, i > when the three-phase contact line length of the droplet on the inclined surface evolves to l given can be obtained. And calculating the forward contact angle theta 1-* and the backward contact angle theta 2-* of the liquid drop on the inclined hydrophilic surface according to r < j, i >.
The invention has the advantages that: the method solves the problems of liquid drop profile analysis and dynamic contact angle theoretical calculation of liquid drops on the inclined hydrophilic rough surface, and has important guiding significance for researching the dynamic performance of the liquid drops on the inclined hydrophilic rough surface and guiding the design of a micro-fluid system in practical application.
Drawings
FIG. 1 is a diagram of a system analysis of heavy droplets on an inclined roughened surface in accordance with an embodiment of the present invention;
Fig. 2 is a graph of the forward contact angle θ 1-0, the backward contact angle θ 2-0, and the three-phase contact line length l 0 of a droplet on an inclined hydrophilic surface; schematic of (2);
FIG. 3 is a flow chart of the calculation of a drop profile curve, a forward contact angle θ 1-0, a backward contact angle θ 2-0, and a three-phase contact line length l 0;
Fig. 4 is a flow chart of the calculation of the forward contact angle θ 1-*, the backward contact angle θ 2-*, and the drop profile curve given the three-phase contact line length l given;
FIG. 5a is a schematic representation of the side profile and contact angle of a droplet on a roughened surface from simulation, for a surface parameter (SiO 2,φ＝8°,d＝8μm,h＝12μm,a＝150μm,V＝60μl,U＝5×10-5 N);
FIG. 5b is a schematic representation of the three-phase contact line profile from simulation when the surface parameter is (SiO 2,φ＝8°,d＝8μm,h＝12μm,a＝150μm,V＝60μl,U＝5×10-5 N);
Figure 5c is when the surface parameter is (SiO 2,φ＝8°,d＝8μm,h＝12μm,a＝150μm,V＝60μl,U＝5×10-5 N),
Schematic representation of experimental results of droplets on hydrophilic roughened surfaces;
FIG. 6a is a schematic representation of the side profile and contact angle of a droplet on a roughened surface from simulation when the surface parameter is (SiO 2,φ＝46°,d＝h＝8μm,a＝600μm,V＝60μl,U＝5×10- 5 N);
FIG. 6b is a schematic representation of the three-phase contact line profile from simulation when the surface parameter is (SiO 2,φ＝46°,d＝h＝8μm,a＝600μm,V＝60μl,U＝5×10- 5 N);
FIG. 6c is a schematic representation of experimental results of a droplet on a hydrophilic roughened surface when the surface parameter is (SiO 2,φ＝46°,d＝h＝8μm,a＝600μm,V＝60μl,U＝5×10- 5 N);
Detailed Description
The invention will now be further illustrated by the following examples, which are given by way of illustration only and not by way of limitation, and are not intended to limit the scope of the invention.
A method for determining a contour curve and a front (back) contact angle of a droplet on an inclined hydrophilic surface, comprising the steps of:
(1) A system (as shown in fig. 1) defining droplets and a sloped hydrophilic roughened surface, wherein the sloped hydrophilic roughened surface has periodically distributed thereon micropillars, the system having a geometry of: the diameter a of the microcolumn, the distance d of the microcolumn, the height h of the microcolumn and the surface inclination angle phi. The rough surface is selected as the XOY plane, and the intersection point of the symmetry axes of the contact surfaces of the droplets is defined as the origin O and the direction rising along the inclined plane as the positive Y direction. The direction perpendicular to the surface and the curvature of the droplet is defined as the positive Z direction, and the direction pointing from inside to outside and perpendicular to the YOZ plane is defined as the X axis. Define the distance from O to a point on the curvature of the drop as the sagittal diameter Wherein the azimuth angle beta (-pi.ltoreq.beta.ltoreq.pi) represents the positive direction from the x-axis to/>The angle between projections on the XOY plane, the apex angle/>Representing the positive Z-axis direction to/>An included angle between the two. /(I)Is defined as r (β, α).
(2) Based on the minimum energy principle, a nonlinear optimization and finite difference algorithm is adopted to deduce a system dimensionless total energy equation of Wenze-state liquid drops on an inclined hydrophilic rough surface, and the specific process is as follows:
21. A surface composed of a hydrophilic material is selected, and the wetting contact state of the liquid drops on the inclined hydrophilic surface is a Wenzel state.
The volume of a droplet on an inclined surface can be expressed as:
For the following algorithm, r (β, α) < r max is defined to control the drop profile, r max can be expressed as
rmax＝4R0 (2)
Wherein the method comprises the steps ofIs the equivalent radius of the liquid drop.
The roughness, i.e. the ratio of the actual solid-liquid contact area to the apparent solid-liquid contact area, is determined:
The outer surface area S ext-2 of the droplet is:
the apparent contact area S base-2 can be expressed as:
In the Wenzel state, the actual solid-liquid contact area S sl-2:
Ssl-2＝rghSbase-2 (6)
Actual solid-gas contact area S sv-2:
Where L 1 is the side length of the roughened surface, which is a constant.
The total system energy E w-2 of the wenzel wet contact drop is equal to the sum of gravitational potential energy E a-2 and interfacial free energy E b-2.
E w-2 can be represented as:
Ew-2＝Ea-2+Eb-2 (8)
E a-2 can be represented as:
E b-2 can be represented as:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
Substituting the parameters into equations (9) (10) into (8) yields the total energy equation for the system with the droplet in a Wenzel state on the inclined hydrophilic roughened surface:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
23. E w-2 in Wenzel wet contact is simplified. Knowing the physical properties of the droplet and the hydrophilic surface, formula (11)
In (a)Being constant, it can be a system dimensionless total energy E' w-2:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
(3) Calculation optimization process of minimum dimensionless total energy and initial front (back) contact angle of Wenzel liquid drop
31. And (3) dispersing: the curvature of the liquid drop profile is discretized by p (equal fraction of beta) and q (equal fraction of alpha), and the discretized radius is r (beta j,αi), which can be simplified as r [ j, i ];
wherein:
32. Optimizing E' wmin-2
A) Setting variables: r [ j, i ] (j=1, 2,) p+1, i=1, 2, & q+1
B) Setting an optimization target:
33. Establishing constraint conditions;
i.0＜r[j,i]≤4R0(j＝1,2,...,p+1；i＝1,2,...,q+1) (16)
ii.
34. Searching:
And searching the variable r [ j, i ] by using the ' fmincon ' function in Matlab to obtain the minimum dimensionless total energy E ' wmin-2 and the corresponding r < j, i >.
Searching E ' wmin-2 by using an ' fmincon ' function in Matlab, continuously calculating until E ' wmin-2 is converged to be within an acceptable range of 10 -4, recording a corresponding minimum dimensionless free energy E ' wmin-2, a contour curve and a three-phase contact line, calculating a forward contact angle theta 1-0 and a backward contact angle theta 2-0 of liquid drops on an inclined plane by using the contour curve, and calculating the length l 0 of the three-phase contact line, otherwise, dividing the multiplication of the inversion by p and q, and carrying out the steps (2) and (3) again, wherein the calculation flow is shown in fig. 2 and 3.
The following result can be obtained by r < j, i >:
l0＝r<1,q+1>+r<p+1,q+1> (20)
(4) When the three-phase contact line length l given of the liquid drop on the inclined rough surface is known, the forward contact angle theta 1-* and the backward contact angle theta 2-* of the liquid drop are obtained;
With the same procedure as in (1) - (3) above, the constraint r [1, q+1] +r [ p+1, q+1] =l given added in step 33 above is additionally required, and the corresponding minimum dimensionless total energy E' wmin-2-* of the contact line length l given, corresponding r < j, i >, can be obtained when the three-phase contact line length of the droplet on the inclined surface evolves to l given. From r < j, i >, the forward contact angle θ 1-* and the backward contact angle θ 2-* of the droplet on the inclined hydrophilic surface were calculated as shown in fig. 4.
The following is a specific example of application:
The geometrical parameters of the system model are as follows: parameters of microstructure, a=100nm, b=200nm, h=100deg.C, θ e0＝θe1 =65deg.C, the hydrophilic surface materials are all SiO 2.
The dynamic contact angle of a 60 μl droplet on an inclined hydrophilic roughened surface and the morphological change of the droplet were simulated by the fmincon function of Matlab and compared with experimental results. Fig. 5a shows the initial forward/backward contact angle versus drop profile curve on a drop inclined rough surface as simulated by matlab at phi=8°, and fig. 5b shows the initial three-phase contact line on a drop inclined rough surface as simulated by matlab. In fig. 5c, the morphology of the phi=8° droplet on the surface is shown, and the experimental result shows that the droplet contour is substantially consistent with the simulation. Fig. 6a shows the initial forward/backward contact angle versus drop profile curve on a drop inclined rough surface as simulated by matlab at phi=46°, and fig. 6b shows the initial three-phase contact line on a drop inclined rough surface as simulated by matlab. In fig. 6c, the morphology of the phi=46° droplet on the surface is shown, and the experimental result shows that the droplet contour is substantially consistent with the simulation. These illustrate the feasibility of the present method for the calculation of the front/back contact angle and drop profile for inclined hydrophilic roughened surfaces.
The variables in the above formula have the meanings:
d spacing of microcolumns
Diameter of a microcolumn
Height of h microcolumn
O distance to a point on the curvature of the drop
Beta from positive X-axis direction toThe angle between projections on the XOY plane
Alpha is from the positive direction of the Z axis toIncluded angle between
Phi hydrophilic roughened surface tilt angle
V2 droplet volume
rmax Maximum value of (2)
R 0 drop equivalent radius
R gh surface roughness coefficient
External surface area of S ext-2 drop
Apparent contact area of S base-2 drop
S sl-2 actual solid-liquid contact area
S sv-2 actual solid-gas contact area
L 1 side edge length of roughened surface
E w-2 Total energy of the System
E a-2 gravitational potential energy
E b-2 interface free energy
G gravity accelerations gamma lv、γsl and gamma sv
Gamma lv liquid-gas interfacial tension coefficient
Gamma sl solid-liquid interfacial tension coefficient
Gamma sv solid-gas interface interfacial tension coefficient
Non-dimensional total energy of E' w-2 system
Equal fraction of pα
Equal fraction of qβ
E' wmin-2 minimum dimensionless total energy of the system;
The discrete radius of the liquid drop when the system energy of r < j, i > is the minimum dimensionless total energy E' wmin-2 is constant
Initial forward contact angle of θ 1-0
Initial contact angle of theta 2-0
L 0 three-phase contact line length
L given given the length of the three-phase contact line;
At a fixed value of the length of the three-phase contact line of theta 1-*, the forward contact angle of the liquid drop is determined;
When the length of the three-phase contact line of the theta 2-* is fixed, the backward contact angle of the liquid drop is determined;
e' wmin-2-*, when the length of the three-phase contact line is fixed, the system is minimum in dimensionless total energy;
the foregoing description is only one embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.
Claims (1)
1. The method for obtaining the contour curve and the front-back contact angle of the liquid drop on the inclined hydrophilic surface comprises the following steps:
(1) A system for defining droplets and a sloped hydrophilic roughened surface, wherein the sloped hydrophilic roughened surface has periodically distributed thereon micropillars, the system having a geometry of: the diameter a of the microcolumns, the distance d of the microcolumns, the height h of the microcolumns and the surface inclination angle phi; selecting a rough surface as an XOY plane, and defining an intersection point of symmetry axes of contact surfaces of the liquid drops as an origin O and a direction rising along an inclined plane as a positive Y direction; the direction perpendicular to the surface and the curvature of the droplet is defined as the positive Z direction, and the direction pointing from inside to outside and perpendicular to the YOZ plane is defined as the X axis; define the distance from O to a point on the curvature of the drop as the sagittal diameter Wherein azimuth angle beta represents from the positive X-axis direction to/>The included angle between projections on the XOY plane is-pi less than or equal to beta less than or equal to pi, and the vertex angle alpha represents the positive direction from the Z axis to/>Included angle between/> Is defined as r (β, α);
(2) Based on the minimum energy principle, adopting nonlinear optimization and finite difference algorithm to deduce the system dimensionless energy equation of the Wenzel state liquid drop on the inclined surface, and simplifying the calculation formula;
21. Selecting a surface composed of a substrate and a microstructure which are both hydrophilic materials, wherein the wetting contact state of liquid drops on the inclined hydrophilic surface is a Wenzel state;
The volume of a droplet can be expressed as:
For the following algorithm, a definition is made To control the drop profile, r max can be expressed as
rmax＝4R0 (2)
Wherein the method comprises the steps ofIs the equivalent radius of the liquid drop;
the roughness, i.e. the ratio of the actual solid-liquid contact area to the apparent solid-liquid contact area, is determined:
The outer surface area S ext-2 of the droplet is:
the apparent contact area S base-2 can be expressed as:
in the Wenzel state, the actual solid-liquid contact area S sl-2:
Ssl-2＝rghSbase-2 (6)
actual solid-gas contact area S sv-2:
Wherein L 1 is the side length of the roughened surface, which is constant;
22. Determining that the total system energy E w-2 of the Wenzel wet contact state droplet is equal to the sum of gravitational potential energy E a-2 and interfacial free energy E b-2;
E w-2 can be represented as:
Ew-2＝Ea-2+Eb-2 (8)
E a-2 can be represented as:
E b-2 can be represented as:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
substituting the parameters into equations (9) (10) into (8) yields the total energy equation of the system with the droplet in a Wenzel state on the inclined hydrophilic roughened surface:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
23. Simplifying the total system energy E w-2 under the Wenzel wetting contact state; knowing the physical properties of the droplets and the hydrophilic surface, in formula (11) Being constant, it can be a system dimensionless total energy E' w-2:
Wherein ρ is the liquid density, g is the gravitational acceleration, S is the contact area, γ lv、γsl and γ sv represent the interfacial tension coefficients of the liquid-gas, solid-liquid and solid-gas interfaces, respectively, the surface inclination angle φ, r gh is the roughness factor of the rough surface;
(3) Optimizing and calculating the minimum dimensionless total energy E' wmin-2 and the forward contact angle theta 1-0, the backward contact angle theta 2-0 and the three-phase contact line length l 0 of the liquid drop on the inclined hydrophilic surface in the initial state;
31. And (3) dispersing: the curvature of the liquid drop profile is discretized by p and q, the discretized radius is r (beta j,αi), and can be simplified to r [ j, i ];
wherein:
32. Optimizing E' wmin-2
A) Setting variables: r [ j, i ]; j=1, 2,..p+1, i=1, 2,..q+1)
B) Setting an optimization target:
33. establishing constraint conditions:
i. 0＜r[j,i]≤4R0 (16)
ii.
34. Searching:
Searching a variable r [ j, i ] by using an ' fmincon ' function in Matlab to obtain a system minimum dimensionless total energy optimal solution E ' wmin-2 and a corresponding r < j, i >;
Searching E ' wmin-2 by using an ' fmincon ' function in Matlab, continuously calculating until the difference value of E ' wmin-2 is converged within an acceptable range of 10 -4, recording a corresponding E ' wmin-2, a contour curve and a three-phase contact line, and calculating a forward contact angle theta 1-0, a backward contact angle theta 2-0 and a three-phase contact line length l 0 of a liquid drop on an inclined plane by using the contour curve, otherwise, dividing the multiplication of inversion and the addition of p and q, and carrying out the steps (2) and (3) again;
the following result can be obtained by r < j, i >:
l0＝r<1,q+1>+r<p+1,q+1> (20)
(4) When the three-phase contact line length l given of the liquid drop on the inclined rough surface is known, the forward contact angle theta 1-* and the backward contact angle theta 2-* of the liquid drop are obtained;
With the same procedure as in (1) - (3) above, it is additionally necessary to add the constraint r [1, q+1] +r [ p+1, q+1] =l given in step 33 above, the corresponding minimum dimensionless total energy E' wmin-2-* of the contact line length l given and the corresponding r < j, i > when the three-phase contact line length of the droplet on the inclined surface evolves to l given; calculating a forward contact angle theta 1-* and a backward contact angle theta 2-* of the liquid drop on the inclined hydrophilic surface according to r < j, i >;
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Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104697903A (en) * | 2013-12-05 | 2015-06-10 | 上海梭伦信息科技有限公司 | Real liquid drop process portable contact angle and interface tension testing method and apparatus thereof |
| CN106323817A (en) * | 2016-08-15 | 2017-01-11 | 浙江工业大学 | Method for calculating contact angle of liquid drop on silicon surface of micro-nano secondary structure |
| CN106570300A (en) * | 2016-11-17 | 2017-04-19 | 中南大学 | Acquiring method for regular second-order structure surface liquid drop contact angle |
| CN108804760A (en) * | 2018-05-03 | 2018-11-13 | 浙江工业大学 | Calculation method for horizontal regular double-roughness surface heavy liquid drop contact angle hysteresis |
| CN108982305A (en) * | 2018-05-03 | 2018-12-11 | 浙江工业大学 | A kind of calculation method of horizontal rule double roughness surface heavy-fluid drop profiles and contact angle |
| CN110196212A (en) * | 2019-06-12 | 2019-09-03 | 上海梭伦信息科技有限公司 | It is a kind of based on the modified test method for estimating intrinsic contact angle of three-dimensional space slope angle degree |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP6178211B2 (en) * | 2013-10-30 | 2017-08-09 | 京セラ株式会社 | Contact angle measurement method, contact angle measurement device, bioprosthetic member inspection device, and program |
-
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Patent Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104697903A (en) * | 2013-12-05 | 2015-06-10 | 上海梭伦信息科技有限公司 | Real liquid drop process portable contact angle and interface tension testing method and apparatus thereof |
| CN106323817A (en) * | 2016-08-15 | 2017-01-11 | 浙江工业大学 | Method for calculating contact angle of liquid drop on silicon surface of micro-nano secondary structure |
| CN106570300A (en) * | 2016-11-17 | 2017-04-19 | 中南大学 | Acquiring method for regular second-order structure surface liquid drop contact angle |
| CN108804760A (en) * | 2018-05-03 | 2018-11-13 | 浙江工业大学 | Calculation method for horizontal regular double-roughness surface heavy liquid drop contact angle hysteresis |
| CN108982305A (en) * | 2018-05-03 | 2018-12-11 | 浙江工业大学 | A kind of calculation method of horizontal rule double roughness surface heavy-fluid drop profiles and contact angle |
| CN110196212A (en) * | 2019-06-12 | 2019-09-03 | 上海梭伦信息科技有限公司 | It is a kind of based on the modified test method for estimating intrinsic contact angle of three-dimensional space slope angle degree |
Non-Patent Citations (2)
| Title |
|---|
| 粗糙表面上微液滴的数值模拟;宋昱;何枫;;力学季刊;20071215(第04期);全文 * |
| 线张力对接触角影响的理论分析;马学虎, 张宇;应用基础与工程科学学报;20040930(第03期);全文 * |
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