CN103323373A - Anti-interference static contact angle calculation method - Google Patents

Abstract

The invention discloses an anti-interference static contact angle calculation method in the technical field of material performance test. The method comprises the following steps: acquiring a droplet image; and extracting a droplet edge, and selecting an optimal contact angle algorithm according to the actual droplet volume, a contact angle value predicted by a user and a critical droplet volume value so to calculate to work out the static contact angle of droplets. Since the optimal static contact angle algorithm is selected according to actual conditions, the method has high interference resistance and ensures both high calculation accuracy and high calculation speed; moreover, the method can aim at different droplet volumes and contact angles and is wide in application range.

Description

A kind of jamproof static contact angle computing method

Technical field

The invention belongs to the material properties test technical field, relate in particular to a kind of jamproof static contact angle computing method.

Background technology

The surface energy of material is an important parameter of exosyndrome material performance, and the material that surface energy is low has the strong characteristics of hydrophobic nature, all has a wide range of applications at aspects such as the anti-pollution flashover of marine antifouling, electric system, ice-covering-proof, self-cleaning materials.Surface energy is the key characteristic of these materials, can obtain by measuring contact angle the surface energy of material.Therefore, the Measurement accuracy contact angle has wide significance.

Sessile drop method at present commonly used is measured static contact angle, obtain by the imaging system collection that key component is exactly the algorithm of contact angle behind the drop image, the most common with circle fitting algorithm, ellipse fitting algorithm and ADSA-P (Axisymmetric Drop Shape Analysis-Profile) algorithm.Circle fitting algorithm considers that water droplet is a spherical crown, and globule image border is the part of circle after the imaging, obtains contact angle by the Least Square Circle match.Ellipse fitting algorithm is regarded drop edge as an oval part, obtains contact angle according to the least square ellipse match.Circle fitting algorithm and ellipse fitting algorithm have good effect in some not too large at contact angle, that droplet size is not too large situation, but suppose that globule edge only is that a kind of of globule edge is similar to for round or an oval part, increase along with contact angle and globule volume, globule edge on the image is stray circle and ellipse gradually, increases gradually with circle and the error of ellipse fitting algorithm.ADSA-P (Axisymmetric Drop Shape Analysis-Profile) algorithm obtains the relation that globule edge satisfies according to the Young-Laplace equation inference, then obtains equation parameter and then obtains contact angle for the globule marginal point that obtains.ADSA-P algorithm principle relative complex, calculated amount be large, be subjected to the factor affecting such as image definition, deflection large, but on this theory of algorithm globule volume is not required, can the fine globule marginal information of utilizing, have advantages of oneself.The static contact angle difference of different materials is very large, can change in 0 °～180 ° scopes, and used globule volume may change in a big way during actual measurement, even can be as big as hundreds of μ L.Optional algorithm all can not all can accurately obtain static contact angle in normally used tangent method, circle fitting algorithm, θ/2 methods, goniometry, amount supreme people's court, polynomial expression and spline-fitting method, ellipse fitting algorithm and the ADSA-P algorithm in whole contact angle and globule volume change scope, and antijamming capability is also strong not simultaneously.

Summary of the invention

The object of the invention is to, low for the existing single static contact angle algorithm computational accuracy of mentioning in the above-mentioned background technology, and the inadequate problem of antijamming capability, a kind of jamproof static contact angle computing method have been proposed, in order to address the above problem.

To achieve these goals, the technical scheme that the present invention proposes is, a kind of jamproof static contact angle computing method is characterized in that, described method comprises:

Step 1: gather the drop image;

Step 2: extract drop edge;

Step 3: estimate static contact angle;

Step 4: select best contact angle algorithm;

Step 5: the tangent line that calculates the drop edge at solid, liquid, gas three's point of interface place of trying to achieve according to the contact angle algorithm of above-mentioned the best is tried to achieve contact angle.

The static contact angle computing method of described the best are according to drop actual volume V, the static contact angle θ that estimates ₀, circle fitting algorithm critical contact angle θ ₁, ellipse fitting algorithm critical contact angle θ ₂, the ADSA-P algorithm critical contact angle θ ₃, circle fitting algorithm critical volume V _CCritical volume V with ellipse fitting algorithm _EDetermine, be specially:

(1) as the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is not more than the critical volume V of circle fitting algorithm _CThe time, select circle the Fitting Calculation method; As the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is greater than the critical volume V of circle fitting algorithm _CThe time, then reduce the droplet size test of reforming;

(2) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is not more than the critical volume V of circle fitting algorithm simultaneously _CThe time, select circle the Fitting Calculation method;

(3) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of circle fitting algorithm simultaneously _CAnd be not more than the critical volume V of ellipse fitting algorithm _EThe time, select ellipse fitting algorithm;

(4) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(5) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is not more than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select ellipse fitting algorithm;

(6) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(7) as the static contact angle θ that estimates ₀Greater than θ ₃The time, select the ADSA-P algorithm.

The critical contact angle θ of described circle fitting algorithm ₁Span be 0 ° to 120 °;

The critical contact angle θ of described ellipse fitting algorithm ₂Span be θ ₁To 140 °;

The critical contact angle θ of described ADSA-P algorithm ₃Span be θ ₂To 150 °.

Effect of the present invention is:

1) accuracy of computation is high.

In different droplet sizes and the contact angle situation, the approximate curve of obeying of drop edge is different, and the present invention can obtain more accurately contact angle by to selecting best contact angle algorithm under the different situations.

2) wide accommodation.

The accuracy that single circle fitting algorithm, ellipse fitting algorithm or ADSA-P algorithm calculate when the droplet size of certain limit and contact angle is higher, but accuracy descends in other cases, all selects more suitable algorithm when the present invention is directed to different droplet sizes and contact angle.Therefore, it all can obtain more accurately contact angle under different droplet sizes and hydrophobic nature, and the scope of application is wider.

3) antijamming capability is strong.

The circle fitting algorithm antijamming capability is the strongest when little contact angle, and ellipse fitting algorithm and ADSA-P algorithm are relatively poor, and the present invention selects circle fitting algorithm to have stronger antijamming capability at this moment.When medium contact angle the circle fitting algorithm antijamming capability still the strongest, ellipse fitting algorithm antijamming capability obviously improve, ADSA-P algorithm antijamming capability also increases, this moment, circle fitting algorithm and ellipse fitting algorithm can satisfy most application scenario, and algorithm antijamming capability this moment of the present invention is also secure.When large contact angle, disturb yardstick to reduce with respect to image, the ADSA-P algorithm that the present invention selects also has stronger antijamming capability at this moment.Therefore, antijamming capability of the present invention is stronger.

4) computing velocity is fast

ADSA-P algorithm computing velocity is the slowest, and the ellipse fitting algorithm computing velocity is the fastest, the circle fitting algorithm speed.The present invention selects optimal algorithm according to actual conditions, namely selects fastest the sort ofly in the algorithm that all satisfies accuracy requirement, has avoided selecting the single excessively slow problem of ADSA-P algorithm computing velocity.

Description of drawings

Fig. 1 is the process flow diagram of circle fitting algorithm;

Fig. 2 is the process flow diagram of ellipse fitting algorithm;

Fig. 3 is the drop edge schematic diagram;

Fig. 4 is the process flow diagram of ADSA-P algorithm;

Fig. 5 is process flow diagram of the present invention;

Fig. 6 (a)～(d) is respectively that volume is 1 μ L, contact angle is less and the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm when disturbing is arranged;

Fig. 7 (a)～(d) is respectively that volume is 4 μ L, contact angle is less and the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm when disturbing is arranged;

Fig. 8 (a)～(d) is respectively the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and the ADSA-P algorithm of volume when being 2 μ L, water wettability;

Fig. 9 (a)～(d) is respectively the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and the ADSA-P algorithm of volume when being 20 μ L, water wettability;

It is 2 μ L that Figure 10 (a)～(d) is respectively volume, the result of calculation of the globule image during hydrophobic nature, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm;

It is 32 μ L that Figure 11 (a)～(d) is respectively volume, the result of calculation of the globule image during hydrophobic nature, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm;

It is 4 μ L that Figure 12 (a)～(d) is respectively volume, the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm during strong hydrophobic nature;

It is 40 μ L that Figure 13 (a)～(d) is respectively volume, the result of calculation of globule image, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm during strong hydrophobic nature;

It is 9 μ L that Figure 14 (a)～(d) is respectively volume, the result of calculation of the globule image of super-hydrophobic sample, circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm.

Embodiment

Below in conjunction with accompanying drawing, preferred embodiment is elaborated.Should be emphasized that, following explanation only is exemplary, rather than in order to limit the scope of the invention and to use.

At first, gather the drop image, then extract drop edge, select best contact angle algorithm to calculate according to drop actual volume, the contact angle value of estimating and drop critical volume value, obtain the static contact angle of drop.The present invention selects best static contact angle calculation side according to physical condition, has good antijamming capability, and accuracy in computation is high, when guaranteeing accuracy faster computing velocity is arranged, can be for different droplet sizes and contact angle, and scope on probation is wide.

(1) circle fitting algorithm.

(1.1) least square model

Be N if the globule marginal point that obtains is counted, the array that is comprised of its coordinate is X (n), n=1, and 2 ..., 2i-1,2i ..., 2N-1,2N, wherein X (2i-1) is the horizontal ordinate that i is ordered, X (2i) is the ordinate that i is ordered; The central coordinate of circle of place, globule edge circle is (X ₀, Y ₀), radius is R, then defines the error that n orders as follows:

$e_n=\sqrt{{(X(2n-1)-X_0)}^2+{(X\left(2n\right)-Y_0)}^2}-R---\left(1\right)$

Then error is defined as follows

$E=\frac{1}{2}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{e}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">n</figure-callout>}}^2=\frac{1}{2}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}{(\sqrt{{(X(2n-1)-X_0)}^2+{(X\left(2n\right)-Y_0)}^2}-R)}^2---\left(2\right)$

According to the expression formula of E X as can be known ₀, Y ₀, R is the unknown variable of following formula because there are nonlinear relationship in they and E, therefore above problem belongs to the nonlinear fitting problem, its key is that the suitable algorithm of searching obtains X by iteration ₀, Y ₀And R.

(1.2) fitting algorithm

The Levenberg-Marquardt algorithm is fit to the non-linear least square problem very much, and this algorithm has obtained computing velocity near the second order local derviation with the calculated amount of single order local derviation, so selects the method match.If e=[e ₁, e ₂..., e _N] ^TBe the error column vector; W is the column vector that the nonlinear multielement function variable forms, W=[w ₁, w ₂, w ₃]=[X ₀, Y ₀, R] ^TJ is Jacobian matrix,

I is the unit matrix that dimension equates with the variable number, and its dimension is 3; According to the iterative formula based on the second order local derviation, Levenberg-Marquardt algorithm J ^TJ+ λ I replaces the Hessian matrix, and the variable element iterative formula is as follows:

W(k+1)=W(k)-(J(k) ^TJ(k)+λI) ^-1J(k) ^Te(k) （3）

K is iterations in the formula; λ adjusts according to the comparative result of twice error of calculation in front and back, if error increase then λ=λ * 10, λ=λ if error reduces * 0.1.

For the match of static contact angle, Jacobian matrix is calculated as follows:

$J_{i,1}=\frac{\&PartialD;e_i}{{\&PartialD;X}_0}=\frac{-(X(2n-1)-X_0)}{\sqrt{{(X(2n-1)-X_0)}^2+{(X\left(2n\right)-Y_0)}^2}}---\left(4\right)$

$J_{i,2}=\frac{\&PartialD;e_i}{{\&PartialD;\mathrm{<figure-callout\; id="0"\; label="Y"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">Y</figure-callout>}}_0}=\frac{-(X\left(2n\right)-Y_0)}{\sqrt{{(X(2n-1)-X_0)}^2+{(X\left(2n\right)-Y_0)}^2}}---\left(5\right)$

$J_{i,3}=\frac{\&PartialD;e_i}{\&PartialD;R}=-1---\left(6\right)$

N is the sequence number of globule marginal point in the formula.

(1.3) acquisition of initial solution

The gap of initial solution and optimum solution will have a strong impact on speed of convergence, even computational accuracy.The boundary line of gas-liquid-solid three-state is called triplet, and it corresponds to the point of ordinate minimum in the globule edge at image, and each point of left and right side is established sequence number and is respectively j and k.Because of the usually distribution around 90 ° of contact angle of the silastic surface globule, based on the hypothesis of semicircle, the initial center of circle that algorithm uses and radius use following strategy to obtain:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">X</figure-callout>}}_0=(X(2j-1)+X(2k-1))/2\\ {\mathrm{<figure-callout\; id="0"\; label="Y"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">Y</figure-callout>}}_0=(X\left(2j\right)+X\left(2k\right))/2\end{array}\right.---\left(7\right)$

R=|X(2j-1)-X ₀| （8）

Can obtain fast the initial value of the center of circle and radius according to the method, and this value to depart from exact value usually little, measured result shows that this mode can ensure accuracy and the real-time of measuring.

(1.4) calculating of contact angle

If edge, the globule left and right sides and surface level intersecting point coordinate are respectively (x ₁, y ₁), (x ₂, y ₂), the central coordinate of circle that match obtains is (X ₀, Y ₀), then left and right sides slope is calculated as follows:

$\left\{\begin{array}{c}{k}_1=-(x_1-X_0)/(y_1-Y_0)\\ {\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=-(x_2-X_0)/(y_2-Y_0)\end{array}\right.---\left(9\right)$

K in the formula ₁And k ₂Be respectively the slope of the upper left right tangent of triplet place circular arc.Left and right sides contact angle computing formula is as follows.

$\left\{\begin{array}{c}{\mathrm{\&theta;}}_1=a\tan \left(k_1\right)\&times;180/\mathrm{\&pi;},k_1\&GreaterEqual;0\\ {\mathrm{\&theta;}}_1=180+a\tan \left(k_1\right)\&times;180/\mathrm{\&pi;},k_1<0\\ {\mathrm{\&theta;}}_2=180-a\tan \left(k_2\right)\&times;180/\mathrm{\&pi;},{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">k</figure-callout>}}_2\&GreaterEqual;0\\ {\mathrm{\&theta;}}_2=-a\tan \left(k_2\right)\&times;180/\mathrm{\&pi;},{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">k</figure-callout>}}_2<0\end{array}\right.---\left(10\right)$

Then the contact angle θ of the globule is calculated as follows:

θ=(θ ₁+θ ₂)/2 （11）

θ in the formula ₁And θ ₂Be respectively the contact angle of the left and right sides, the span of atan function is-pi/2～pi/2, and the unit of θ is °.

(1.5) convergence criterion

The Levenberg-Marquardt algorithm improves degree of accuracy by iteration, judges that the situation of convergence is very crucial to degree of accuracy and the real-time of algorithm.Algorithm determines whether stopping to calculate according to the situation of change of gained contact angle before and after the iteration.Continuous 3 result of calculations often can fine sign convergence situation in the Levenberg-Marquardt algorithm iteration process.Therefore, convergence criterion is as follows: the gained contact angle is respectively A after establishing N, N+1, N+2 iteration _N, A _N+1, A _N+2If satisfy

| A _N-A _N+1|≤C ₁With | A _N+1-A _N+2|≤C ₂(12)

Then iteration stops, wherein C ₁, C ₂Be respectively the critical value of setting, usually be chosen as 0.5 ° and 0.01 °, can suitably adjust according to actual conditions when specifically using, the standard deviation of using under normal circumstances this criterion to calculate the gained contact angle only is about 0.5 °.The circle fitting algorithm process flow diagram as shown in Figure 1.

(2) ellipse fitting algorithm

(2.1) algorithm principle

Drop edge departs from circle gradually on the image when droplet size increases, and can adopt elliptic equation to approach drop edge.Conventional least square fitting calculated amount is large, and the real-time that contact angle calculates will be affected, and has therefore adopted a kind of direct oval method of calculating.Oval general equation can be expressed as:

F(m,n)=n·m=ax ²+bxy+cy ²+dx+ey+f=0 （13）

X, y represent respectively horizontal stroke, ordinate, m=[a, b, c, d, e, f in the following formula] ^T, n=[x ², xy, y ², x, y, 1].If n _i=[x _i ², x _iy _i, y _i ², x _i, y _i, 1], B=[n ₁ ^T, n ₂ ^T..., n _N ^T] ^T, F (m, n _i) be referred to as Plane-point (x _i, y _i) to the algebraic distance of curve F (m, n)=0.When all discrete data point algebraic distance quadratic sums of ellipse fitting hour can solve corresponding quafric curve, fitting criterion of equal value is:

$E_{\min }=\min \left(\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}F{(m,n_i)}^2\right)---\left(14\right)$

Formula (14) is introduced restrictive condition b ²-4ac=-1 can guarantee the result of match for oval, and this condition is expressed as with matrix form:

m ^TCm=1 （15）

In the formula $C=\left[\begin{array}{cccccc}0& 0& 2& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right].$

Formula (14) is equivalent to

E _min=min(|Bm| ²) （16）

Because B is known quantity, amount to be asked is m in the formula (16), and the key of problem is carried out optimizing to m exactly.

Introducing Lagrange operator and differentiate gets:

2B ^TBm-2λCm=0 （17）

Make S=B ^TB, formula (17) is rewritten as:

Sm=λCm （18）

For formula (18), can solve by the method for generalized eigenvalue and generalized eigenvector 6 groups of characteristic of correspondence value λ _iAnd u _iFor the condition restriction Matrix C, its eigenwert is [2 ,-1,2,0,0,0], an eigenwert is only arranged for just.Only generalized eigenvalue λ ∈ R is only arranged ⁺With generalized eigenvector u as the ellipse fitting solution.For any k ∈ R ⁺, should make (λ, ku) to satisfy formula (19), i.e. k ²u ^TCu=1, then:

k=[1/(u ^TCu)] ^1/2=[1/(u ^TSu)] ^1/2 （19）

The only solution of m:

$\hat{m}=\mathrm{ku}---\left(20\right)$

(2.2) contact angle calculates

Be respectively L if obtain oval long and short semiaxis according to the m that finds the solution _LAnd L _S, the center is (X ₀, Y ₀), the angle of inclination is θ ₀, unit is rad; Then ellipse is gone up point (x ₁, y ₁) expression formula is as follows:

x=L _Lcosθ;y=LSsinθ;

x ₁=xcosθ ₀-ysinθ ₀+X ₀;

y ₁=xsinθ ₀+ycosθ ₀+Y ₀; （21）

Left and right sides minimum point is respectively (X on the drop edge _L, Y _L) and (X _R, Y _R), consider that sample may not exclusively be in horizontality, it is as follows that contact angle calculates thinking, obtains first the θ in formula (22) corresponding to left and right sides triplet corresponding point, is designated as θ _L1And θ _R1Then contact angle computing formula in the left and right sides is as follows:

X _L1=(X _L-X ₀)cosθ ₀+(Y _L-Y ₀)sinθ ₀;

Y _L1=-(X _L-X ₀)sinθ ₀+(Y _L-Y ₀)cosθ ₀;

X _R1=(X _R-X ₀)cosθ ₀+(Y _R-Y ₀)sinθ ₀;

Y _R1=-(X _R-X ₀)sinθ ₀+(Y _R-Y ₀)cosθ ₀;

θ _L1=angle(jY _L1/L _S+X _L1/L _L);

θ _R1=angle(jY _R1/L _S+X _R1/L _L). （22）

According to above 2 angles and oval tilt angle theta ₀The contact angle that can get the left and right sides is respectively θ _L2And θ _R2, unit is °, as the formula (23).

$\left\{\begin{array}{c}{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">L</figure-callout>}2}=\frac{180}{\mathrm{\&pi;}}[a\tan \left(\frac{-L_s}{L_L\tan {\mathrm{\&theta;}}_{L1}}\right)+{\mathrm{\&theta;}}_0],Y_{L1}\&GreaterEqual;0;{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">L</figure-callout>}2}=\frac{180}{\mathrm{\&pi;}}[\mathrm{\&pi;}+a\tan \left(\frac{-L_S}{L_L\tan {\mathrm{\&theta;}}_{L1}}\right)+{\mathrm{\&theta;}}_0],Y_{L1}<0\\ {\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">R</figure-callout>}2}=\frac{180}{\mathrm{\&pi;}}[a\tan \left(\frac{L_S}{L_L\tan {\mathrm{\&theta;}}_{R1}}\right)+{\mathrm{\&theta;}}_0],Y_{R1}\&GreaterEqual;0;{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">R</figure-callout>}2}=\frac{180}{\mathrm{\&pi;}}[\mathrm{\&pi;}+a\tan \left(\frac{L_S}{L_L\tan {\mathrm{\&theta;}}_{R1}}\right)+{\mathrm{\&theta;}}_0],Y_{R1}<0\end{array}\right.---\left(23\right)$

If obtaining the solids level inside slope according to these 2 is k ₁, then finally left and right sides contact angle is respectively θ _LAnd θ _R, as the formula (24), unit is °.

$\left\{\begin{array}{c}{\mathrm{\&theta;}}_L={\mathrm{\&theta;}}_{L2}-180a\tan \left(k_1\right)/\mathrm{\&pi;}\\ {\mathrm{\&theta;}}_R={\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">R</figure-callout>}2}+180a\tan \left(k_1\right)/\mathrm{\&pi;}\end{array}\right.---\left(24\right)$

The final contact angle of drop

θ=(θ _L+θ _R)/2 （25）

The ellipse fitting algorithm process flow diagram as shown in Figure 2.

(3) ADSA-P algorithm

One drop is dripped when the solid surface, and the edge as shown in Figure 3 for corresponding drop (seat drips).The relation of any point radius-of-curvature and pressure difference such as following Laplace equation on the liquid level.

$\mathrm{\&Delta;P}=P_1-{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="HDA00003372008600023.png"\; state="\{\{state\}\}">P</figure-callout>}}_2=\mathrm{\&gamma;}(\frac{1}{R_1}+\frac{1}{R_2})---\left(26\right)$

P in the formula ₁, P ₂Be P point liquid internal pressure and external pressure; γ is the interfacial tension between liquid and the gas; R ₁, R ₂Be respectively the first and second radius-of-curvature that P is ordered.X, z are respectively as shown in Figure 3; S is the arc length from initial point to this point.

If

x ₁=x/R ₀，z ₁=z/R ₀，s ₁=s/R ₀ （27）

X in the formula ₁, z ₁And s ₁Be respectively horizontal stroke, ordinate and initial point after the normalization to the arc length of this point; R ₀Radius-of-curvature for the initial point place.

The globule edge that obtains satisfies following ordinary differential equation group through deriving

$\left\{\begin{array}{c}\frac{{\mathrm{dx}}_1}{{\mathrm{ds}}_1}=\cos \mathrm{\&theta;}\\ \frac{{\mathrm{dz}}_1}{{\mathrm{ds}}_1}=\sin \mathrm{\&theta;}\\ \frac{\mathrm{d\&theta;}}{{\mathrm{ds}}_1}=2+{\mathrm{\&beta;z}}_1-\sin \mathrm{\&theta;}/x_1\end{array}\right.---\left(28\right)$

β in the formula=g Δ pR ₀ ²/ γ; θ is the rotation angle between P point tangent line and the surface level, and the θ at corresponding triplet place is contact angle when drop drips for seat; G is acceleration of gravity; Δ p is the density difference of liquid phase and gas phase.

Determine β and R ₀A drop edge be can obtain based on formula (27), (28) afterwards, β and R adjusted by optimized algorithm ₀Make enough hour of the edge that calculates and the true edge gap that measures, then algorithm convergence.Above algorithm is the ADSA-P algorithm of old version, and its variable to be optimized is β and R ₀, affecting its constringency performance during actual the use, effect of optimization is relatively poor.ADSA-P algorithm than old version, improved ADSA-P algorithm (follow-up referred to as the ADSA-P algorithm) with drop summit and angle of inclination also as variable to be optimized, adjusted simultaneously the expression way of formula (28) ordinary differential equation group, the algorithm convergence ability greatly improves, it is used and more closes to actual conditions simultaneously, and it uses following ordinary differential equation group alternate form (28).

$\left\{\begin{array}{c}\frac{\mathrm{dx}}{\mathrm{ds}}=\cos \mathrm{\&theta;}\\ \frac{\mathrm{dz}}{\mathrm{ds}}=\sin \mathrm{\&theta;}\\ \frac{\mathrm{d\&theta;}}{\mathrm{ds}}=2b+\mathrm{cz}-\sin \mathrm{\&theta;}/x\end{array}\right.---\left(29\right)$

C=g Δ p/ γ in the formula.Formula (29) variable to be optimized is b and c, and effect of optimization is better during actual the use, and speed of convergence is very fast.

If true drop profile coordinate x ₁=[x ₁₁, x ₁₂..., x _1N], z ₁=[z ₁₁, z ₁₂..., z _1N]; For different b and c, formula (29) can get one group of point, x according to Runge-Kutta (Runge-Kutta) method ₂=[x ₂₁, x ₂₂..., x _2N], z ₂=[z ₂₁, z ₂₂..., z _2N].The drop edge summit that more than obtains is defaulted as zero point, and base material is in Realize, and the ADAS-P algorithm considers that variable α, x may not need to be introduced in the drop summit in the practical application when initial point and drop inclination ₀And z ₀, wherein α is the variable of introducing when considering inclination, x ₀And z ₀Be the variable of considering that the drop summit may not be introduced at initial point, corresponding least square problem is described below

$E=\frac{1}{2}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{e}_{\mathrm{il}}^2+e_{i2}^2---\left(30\right)$

e _i1=(x _2i-x ₀-x _1icosα+z _1isinα) （31）

e _i2=(z _2i-z ₀-z _1icosα-x _1isinα) （32）

The set of variables to be optimized of obvious improved ADSA-P algorithm is [b, c, α, x ₀, z ₀], can be with optimization algorithm correction [b, c, α, x for error E ₀, z ₀] make E be tending towards minimizing.According to [b, c, α, the x that calculate ₀, z ₀] get final product to get the drop profile that obtains of match, get final product to get the contact angle θ of arranged on left and right sides according to the tangent line of the some correspondence on the corresponding drop profile of left and right triplet again _LAnd θ _R, final material surface contact angle

θ=(θ _L+θ _R)/2 （33）

The ADSA-P algorithm flow as shown in Figure 4.

Flow process of the present invention as shown in Figure 5, concrete step:

Step 1: gather the drop image.

Step 2: extract drop edge.

Step 3: estimate static contact angle.

Can be by the mode of directly estimating or the static contact angle of estimating drop according to given value.Estimating static contact angle has been the state of the art, and the present invention repeats no more.

Step 4: select best contact angle algorithm.

Best static contact angle computing method are according to drop actual volume V, the static contact angle θ that estimates ₀, circle fitting algorithm critical contact angle θ ₁, ellipse fitting algorithm critical contact angle θ ₂, the ADSA-P algorithm critical contact angle θ ₃, circle fitting algorithm critical volume V _CCritical volume V with ellipse fitting algorithm _EDetermine, be specially:

(1) as the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is not more than the critical volume V of circle fitting algorithm _CThe time, select circle the Fitting Calculation method; As the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is greater than the critical volume V of circle fitting algorithm _CThe time, then reduce the droplet size test of reforming;

(2) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is not more than the critical volume V of circle fitting algorithm simultaneously _CThe time, select circle the Fitting Calculation method;

(3) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of circle fitting algorithm simultaneously _CAnd be not more than the critical volume V of ellipse fitting algorithm _EThe time, select ellipse fitting algorithm;

(4) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(5) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is not more than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select ellipse fitting algorithm;

(6) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(7) as the static contact angle θ that estimates ₀Greater than θ ₃The time, select the ADSA-P algorithm.

Preferably, the critical contact angle θ of circle fitting algorithm ₁Span be 0 ° to 120 °; The critical contact angle θ of ellipse fitting algorithm ₂Span be θ ₁To 140 °; The critical contact angle θ of ADSA-P algorithm ₃Span be θ ₂To 150 °.

Step 5: the tangent line that calculates the drop edge at solid, liquid, gas three's point of interface place of trying to achieve according to the contact angle algorithm of above-mentioned the best is tried to achieve contact angle.

Namely adopt the static contact angle that adopts selected algorithm to calculate drop in the step 4, and the process preamble of circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm calculating static contact angle is described.

Embodiment one

In the present embodiment, kind of liquid is chosen as water, and volume is 1 μ L, the critical contact angle θ of circle fitting algorithm ₁, ellipse fitting algorithm critical contact angle θ ₂Critical contact angle θ with the ADSA-P algorithm ₃Value as shown in table 1, the critical globule volume of circle fitting algorithm and ellipse fitting algorithm is as shown in table 2.

The critical contact angle of table 1,3 kinds of algorithms

The globule image of the relatively poor material of hydrophobic nature more easily is interfered, and the result of calculation of circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm as shown in Figure 6.By accompanying drawing 6(b) as can be known, for this contact angle very small size also little globule image, circle fitting algorithm accurately match obtains globule edge, even be subjected to interference to a certain extent, the contact angle accuracy that calculates is higher.And accompanying drawing 6(c) ellipse fitting algorithm at this moment fitting effect departed from right value, because by naked eyes in conjunction with the experiment situation as can be known, this sample obviously belongs to the situation that hydrophobic nature is lost substantially behind the material corona, this algorithm is 10.99 ° with circle error of fitting gap, the reason that this algorithm produces error is than equation of a circle, elliptic equation itself is easier to be affected by noise and change its best fitting result, and the fitting algorithm of selecting has simultaneously then further increased this tendency.Accompanying drawing 6(d) ADSA-P algorithm fitting result and true edge can not finely coincide, this is because the contained information of camber line is less during little contact angle, and this algorithm adopts and finds the solution the ordinary differential equation iteration to obtain optimum solution higher to the contained information requirements of camber line, otherwise easily causes algorithm not restrained.3 kinds of algorithm computing velocitys are as shown in table 3.The user estimates 20 ° of contact angle value ≈, with reference to the accompanying drawings 5 process flow diagram associative list 1 of the present invention, 2 as can be known, the present invention will select circle fitting algorithm when running into this image.Therefore, accuracy and speed are enough high.

The critical globule volume of table 2,2 kinds of algorithms/μ L

By accompanying drawing 7(b) as can be known, circle fitting algorithm accurately match obtains globule edge, and according to its hydrophobic nature associative list 2 as can be known, this algorithm accuracy of computation is secure.And accompanying drawing 7(c) ellipse fitting algorithm except the better match globule edge of triplet exterior domain, but at the triplet place of key certain error is arranged, this is that this error has caused its result of calculation to have larger error, with the circle fitting algorithm gap be 9.09 °.But, only providing match and the result of calculation of ellipse fitting algorithm if there is not the analysis of front surface error, the user is easy to think by mistake that ellipse fitting algorithm also had higher accuracy and produced no small error this moment.Accompanying drawing 7(d) ADSA-P algorithm result of calculation and circle fitting algorithm are comparatively approaching, and fitting effect is also better, and accuracy is higher, find that fitting effect will be worse than circle fitting algorithm but examine, and this algorithm calculate the circle fitting algorithm that is longer than consuming time.The user estimates 35 ° of contact angle value ≈, 5 process flow diagram associative list 1 of the present invention, 2 as can be known with reference to the accompanying drawings, the present invention is running into contact angle less than 60 ° and volume during less than critical volume, then select circle fitting algorithm, this situation satisfies the circle fitting algorithm condition, therefore select circle fitting algorithm, accuracy and speed are all secure.

Table 3, consuming time for the calculating of 3 kinds of algorithms of true drop image

Embodiment two

By accompanying drawing 8 as can be known, for the globule image of water wettability sample 2 μ L, circle fitting algorithm and ellipse fitting algorithm fitting effect are all good, and contact angle result of calculation is close, and their accuracy of computation is secure as can be known according to its hydrophobic nature associative list 2.ADSA-P algorithm fitting effect is slightly poor, and result of calculation produces error.The user estimates 65 ° of contact angle value ≈, and 5 process flow diagram of the present invention and table 1,2 are selected circle fitting algorithm as can be known in such cases with reference to the accompanying drawings, and accuracy of computation of the present invention is secure.By accompanying drawing 9 as can be known, for the globule image of water wettability sample 20 μ L, circle fitting algorithm and ADSA-P algorithm fitting effect are not good, and the error of circle fitting algorithm is that globule edge has departed from due to the equation of a circle; The error initial analysis of ADSA-P algorithm is because surface tilt causes globule edge to depart from due to the Young equation; Ellipse fitting algorithm is effective.The user estimates 75 ° of contact angle value ≈, with reference to the accompanying drawings 5 process flow diagram of the present invention and table 1,2 as can be known, the present invention selects ellipse fitting algorithm in such cases, associative list 3 and above the analysis as can be known, the accuracy of this algorithm and computing velocity are all secure, and the error of circle fitting algorithm is greater than 6 °.

Embodiment three

The fitting effect of circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm is all fine in the accompanying drawing 10, the static contact angle that calculates the gained globule is also very approaching, and their result of calculation was all comparatively accurate as can be known in conjunction with former analysis in conjunction with match and result of calculation.The user estimates 102 ° of contact angle value ≈, with reference to the accompanying drawings 5 process flow diagram of the present invention and table 1,2 as can be known, the present invention will select ellipse fitting algorithm to calculate the static contact angle of the globule in such cases, therefore, its accuracy is secure, and it is fastest as shown in Table 3 simultaneously.Therefore, in such cases, the present invention has higher accuracy and computing velocity faster.By accompanying drawing 11 as can be known, the circle fitting algorithm fitting effect is relatively poor during for hydrophobic nature sample large water drop volume, ellipse fitting algorithm and ADSA-P algorithm effect are all better, and two kinds of algorithm result of calculation gaps are roughly 3 °, this mainly is that the accuracy of two kinds of algorithms is all secure owing to extracting the interference obtain marginal existence 2 kinds of algorithms are had due in various degree the impact.The user estimates 110 ° of contact angle value ≈, and 5 process flow diagram of the present invention and table 1,2 are selected ellipse fitting algorithm as can be known in such cases with reference to the accompanying drawings, and accuracy is secure, and it is fastest as shown in Table 3 simultaneously.

Embodiment four

The base material hydrophobic nature is fine in the accompanying drawing 12, and 3 kinds of algorithm match gained edges all can coincide with actual edge is fine, but static contact angle result of calculation has a long way to go.Analysis-by-synthesis as can be known, the static contact angle of this sample is roughly 135 °, as shown in Table 2, the ellipse fitting algorithm error is slightly larger than 3 °, this and ellipse fitting algorithm and ADSA-P algorithm result of calculation difference are 2.77 ° and coincide, hence one can see that, and the circle fitting algorithm error is approximately-5.63 °, and should select the ADSA-P algorithm this moment.But only usually can think the also enough height of accuracy of circle fitting algorithm and ellipse fitting algorithm according to fitting result this moment, thus had-2.77 ° with-5.63 ° error.The user estimates 128 ° of contact angle value ≈, with reference to the accompanying drawings 5 process flow diagram associative list 1 of the present invention, 2 as can be known, it will select the ADSA-P algorithm, therefore its accuracy is higher.

By accompanying drawing 13 as can be known, the fitting effect of circle fitting algorithm is relatively poor, with the gap of ADSA-P algorithm be-22.61 °, should not select this algorithm this moment.ADSA-P algorithm fitting effect is better, and its accuracy is secure as can be known according to the front simulation result, and calculating the gained contact angle is 135.81 °, and accuracy is secure.And the fitting effect of ellipse fitting algorithm is unexpectedly also fine, but as shown in Table 2, its error is greater than 3 °, if ADSA-P algorithm result of calculation is exact value then its error is-9.66 °, the tester is easy to think by mistake that ellipse fitting algorithm had higher accuracy and produced larger error this moment.In order further to verify above precision of analysis, to have produced a volume be that 40.02 μ L, contact angle are 135.65 ° globule edge in emulation based on formula (28), calculate with 3 kinds of algorithms respectively, the error that obtains contact angle is respectively-22.74 ° ,-9.04 ° and-0.12 °, very close with the result of calculation of the true globule in front, verified precision of analysis.The user estimates 130 ° of contact angle value ≈, with reference to the accompanying drawings 5 process flow diagram of the present invention and table 1,2 as can be known, the present invention all can select the ADSA-P algorithm under the both of these case.Therefore, its accuracy is secure.

Embodiment five

Selected the image of the globule on a super hydrophobic material surface.These images calculate with circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm respectively, and the result as shown in Figure 14.

By accompanying drawing 14 as can be known, for real super-hydrophobic globule image, the edge that circle fitting algorithm obtains and true globule marginal existence a certain distance, ellipse fitting algorithm is totally seen comparatively accurate, but certain deviation is arranged near the triplet of key, and directly determine just the zone of contact angle size near the triplet, so contact angle is calculated and may affect greatly, do not produce larger error if do not see carefully the result of calculation accuracy of also easily thinking ellipse fitting algorithm is higher.And the edge that the ADSA-P algorithm obtains coincide very well with true globule edge, and this simulation result with the front is consistent, and the contact angle accuracy that calculates is higher, is 159.40 °.Corresponding circle fitting algorithm and ellipse fitting algorithm roughly have respectively the error of-15.49 ° and-10.59 ° during as exact value with it, obviously circle fitting algorithm and ellipse fitting algorithm all have larger error, and the circle fitting algorithm error is greater than ellipse fitting algorithm, and this moment, the user thought that easily ellipse fitting algorithm has higher accuracy and makes contact angle measure the larger error of generation by mistake.By above analytic process as can be known, has very high accuracy for real super-hydrophobic globule image A DSA-P algorithm.The user estimates 150 ° of contact angle value ≈, and 5 process flow diagram associative list 1 of the present invention, 2 the present invention this moment will be selected the ADSA-P algorithm as can be known with reference to the accompanying drawings, have the highest accuracy.Because the true contact angle of water droplet on this super hydrophobic material can not know for sure, in order to check more meticulously the simulation and experiment result whether to have consistance, the accuracy of further analyzing more than the checking simultaneously, produce the globule edge of an emulation as actual value with ADSA-P algorithm result of calculation, namely use the Young-Laplace equation of formula (28) to produce a globule edge, volume is 8.99 μ L, contact angle is 159.54 °, 3 kinds of algorithms calculate contact angle and are respectively 143.44 °, 148.67 ° and 159.57 °, error is respectively-16.10 °,-10.87 ° and 0.03 °, be that the result of calculation at emulation globule edge and the result of calculation of actual globule image are coincide better, this has verified above precision of analysis.

In sum, no matter be for contact angle be large or little, have or not under the disturbed condition, the present invention all can select the highest algorithm of accuracy among circle fitting algorithm, ellipse fitting algorithm and ADSA-P algorithm, simultaneously selected algorithm has also guaranteed faster computing velocity on the basis of accuracy satisfying.Find to select single circle through research simultaneously or ellipse fitting algorithm fitting effect when some situation is fine but the contact angle error of calculation is larger, the tester is easy to think by mistake that this moment accuracy is higher and produce error.

It is to be noted that there is the error in several years in static contact angle that the user estimates with tangent method and naked eyes, may affect the selection of contact angle algorithm in the time of near the critical contact angle, but near the critical contact angle, select the error of any generation of 2 kinds of algorithms very close, so tangent method and naked eyes are estimated contact angle and in general can be met the demands.

The above; only for the better embodiment of the present invention, but protection scope of the present invention is not limited to this, anyly is familiar with those skilled in the art in the technical scope that the present invention discloses; the variation that can expect easily or replacement all should be encompassed within protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection domain of claim.

Claims (3)

1. jamproof static contact angle computing method is characterized in that, described method comprises:

Step 1: gather the drop image;

Step 2: extract drop edge;

Step 3: estimate static contact angle;

Step 4: select best contact angle algorithm;

Step 5: the tangent line that calculates the drop edge at solid, liquid, gas three's point of interface place of trying to achieve according to the contact angle algorithm of above-mentioned the best is tried to achieve contact angle.

2. computing method according to claim 1 is characterized in that, the static contact angle computing method of described the best are according to drop actual volume V, the static contact angle θ that estimates ₀, circle fitting algorithm critical contact angle θ ₁, ellipse fitting algorithm critical contact angle θ ₂, the ADSA-P algorithm critical contact angle θ ₃, circle fitting algorithm critical volume V _CCritical volume V with ellipse fitting algorithm _EDetermine, be specially:

(1) as the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is not more than the critical volume V of circle fitting algorithm _CThe time, select circle the Fitting Calculation method; As the static contact angle θ that estimates ₀Be not more than θ ₁, and drop actual volume V is greater than the critical volume V of circle fitting algorithm _CThe time, then reduce the droplet size test of reforming;

(2) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is not more than the critical volume V of circle fitting algorithm simultaneously _CThe time, select circle the Fitting Calculation method;

(3) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of circle fitting algorithm simultaneously _CAnd be not more than the critical volume V of ellipse fitting algorithm _EThe time, select ellipse fitting algorithm;

(4) as the static contact angle θ that estimates ₀Greater than θ ₁And be not more than θ ₂, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(5) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is not more than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select ellipse fitting algorithm;

(6) as the static contact angle θ that estimates ₀Greater than θ ₂And be not more than θ ₃, drop actual volume V is greater than the critical volume V of ellipse fitting algorithm simultaneously _EThe time, select the ADSA-P algorithm;

(7) as the static contact angle θ that estimates ₀Greater than θ ₃The time, select the ADSA-P algorithm.

3. computing method according to claim 2 is characterized in that, the critical contact angle θ of described circle fitting algorithm ₁Span be 0 ° to 120 °; The critical contact angle θ of described ellipse fitting algorithm ₂Span be θ ₁To 140 °; The critical contact angle θ of described ADSA-P algorithm ₃Span be θ ₂To 150 °.

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