CN102507389A - Method for establishing prediction model of static contact angles in cylindrical glass capillaries - Google Patents

Abstract

The invention provides a method for establishing a prediction model of static contact angles in cylindrical glass capillaries, which carries out experiments by utilizing limited types (three or more than three types) of capillaries with different pipe diameters and made of the same materials and the same wetting liquid. An equivalent highly function is obtained by fitting limited-time experiment data and is substituted in a prediction model of the static contact angles to obtain the static contact angle prediction model corresponding to the same capillaries and the same wetting liquid. With regard to the cylindrical glass capillaries adopting the same wetting liquid medium and the same glass materials, corresponding static contact angles theta can be conveniently obtained by substituting inner diameters of the capillaries and liquid column heights in the capillaries obtained by carrying out the same capillary rising experiments into the prediction model. The method for establishing the static contact angle prediction model in the cylindrical glass capillaries can remarkably shorten calculating time of the static contact angles, reduce calculating complexity of the static contact angles and improve working efficiency accordingly.

Description

A kind of method of setting up static contact angle forecast model in the cylindrical glass kapillary

Technical field

The present invention relates to table/interface level measurement technical field, be specially a kind of method of setting up static contact angle forecast model in the cylindrical glass kapillary.

Background technology

The solid-liquid static contact angle is an important content in the surface chemistry, in chemical industry, oil and blowdown industry, is in fundamental position, and it is not only relevant with the material self property, but also receives the influence of external environment.As far as the cylindrical glass kapillary, the factor that influences contact angle comprises: glass tube internal diameter and capillary experimental technique.

At present; The method of the static contact angle in traditional definite glass capillary often all is a numerical computation method, and these numerical computation method computation processes are complicated, require the experimenter to possess higher numerical analysis ability and programming ability; And there is much repeated work; Cause in practical application classic method often computing time long, inefficiency presses for a kind of method and can improve the work efficiency of confirming the static contact angle in the glass capillary.

Summary of the invention

The technical matters that solves

For solving the problem that prior art exists, the present invention proposes a kind of method of setting up static contact angle forecast model in the cylindrical glass kapillary.This method uses numerical analysis technology to find the solution the theoretical liquid type of meniscus in the kapillary, set up the funtcional relationship between equivalent height and glass tube internal diameter, thereby draws the forecast model of static contact angle.

Technical scheme

Technical scheme of the present invention is:

Said a kind of method of setting up static contact angle forecast model in the cylindrical glass kapillary is characterized in that: may further comprise the steps:

Step 1: prepare a kind of wetting state liquid medium, measure the surface tension σ and the density p of wetting state liquid medium; It is identical to prepare p material, the cylindrical glass kapillary that internal diameter is different, and p>=3 wherein, each cylindrical glass internal diameter capillaceous is respectively r ₁, r ₂, r ₃..., r _p

Step 2: get p the cylindrical glass kapillary of preparing in the step 1 and in the wetting state liquid medium, carry out capillary rising experiment; After treating that the liquid lifting height is stable; Measure liquid-column height in each kapillary with height measuring device, obtain the interior liquid-column height of p kapillary and be respectively h ₁, h ₂, h ₃..., h _p

Step 3: for j cylindrical glass kapillary, with liquid-column height h in σ, ρ and j the cylindrical glass kapillary _jSubstitution meniscus equation $\mathrm{\σ}(\frac{\frac{d^2{y}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}}{x_{\mathrm{Ij}}{[1+{\left(\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}\right)}^2]}^{1/2}})=\mathrm{\ρ\; g}{h}_j+\mathrm{\ρ}{\mathrm{Gy}}_{\mathrm{Ij}},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in j the cylindrical glass kapillary _Ij, y _Ij), wherein g is a local gravitational acceleration, the coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in j the cylindrical glass kapillary; With j cylindrical glass kapillary central axis is the y axle, i=1,2;, Z, Z are coordinate point set { (x _Ij, y _Ij) in count; According to

To coordinate point set { (x _Ij, y _Ij) carry out M+1 item fitting of a polynomial M time, and wherein M is not less than 2 integer, and matched curve and coordinate point set maximum error are not more than 10 ^-6, obtain meniscus liquid type function y in j the cylindrical glass kapillary _j(x); According to

To function y _j(x) carry out integration, wherein r _jBe j cylindrical glass internal diameter capillaceous, obtain the equivalent height f of meniscus liquid type in j the cylindrical glass kapillary _j

Step 4: repeat step 3, obtain the equivalent height f of meniscus liquid type in all p the cylindrical glass kapillaries ₁, f ₂, f ₃..., f _pAccording to

To point set { (r _k, f _k) carry out N+1 item fitting of a polynomial N time, and k=1 wherein, 2 ..., p, N is the integer of 2≤N≤p, obtains equivalent height function f (r); Thereby the forecast model that obtains static contact angle θ in the cylindrical glass kapillary of wetting state liquid medium and glass material in the corresponding step 1 does

Wherein r is the cylindrical glass capillary inner diameter, h be carry out with step 2 in the identical capillary liquid-column height in the kapillary that experiment obtains that rises.

After obtaining forecast model; For adopting the identical wetting state liquid medium and the cylindrical glass kapillary of glass material; Only need through carry out with step 2 in the identical capillary liquid-column height in the kapillary that experiment obtains that rises; And, just can obtain corresponding static contact angle θ easily with liquid-column height and capillary inner diameter substitution forecast model.

Beneficial effect

The present invention only need be different to limited kind (more than 3 kinds or 3 kinds) caliber; Kapillary that material is identical and wetting state liquid of the same race experimentize; Just can draw the forecast model of static contact angle; And it is different to draw other calibers through forecast model, the static contact angle of the kapillary that material is identical and this kind wetting state liquid.The present invention can significantly shorten computing time, the complexity of static contact angle, thereby increases work efficiency.

Description of drawings

Fig. 1: process flow diagram of the present invention;

Fig. 2: capillary rises and tests sketch;

Fig. 3: meniscus half profile point set figure;

Fig. 4: equivalent height matched curve;

Wherein: 1, kapillary; 2, meniscus outline line; 3, wetting state liquid medium; 4, fluid level.

Embodiment

Below in conjunction with specific embodiment the present invention is described.

Embodiment 1:

Step 1: the absolute ethyl alcohol of preparing to analyze pure level is as the wetting state liquid medium, and the surface tension instrument/density appearance that adopts the auspicious Instr Ltd. in Shanghai side to produce, and model is MDY-1, measures the surface tension σ and the density p of absolute ethyl alcohol, and measurement result is seen table 1:

Table 1 absolute ethyl alcohol density and surface tension table

Class of liquids	Surface tension N/m	Density kg/m 3
			Absolute ethyl alcohol	0.02255	792

It is identical to prepare four root timber matter; Internal diameter is respectively the cylindrical glass kapillary of 0.15mm, 0.25mm, 0.5mm, 1.4mm; Wherein internal diameter be the cylindrical glass kapillary buying of 0.15mm, 0.25mm, 0.5mm from instrument plant of Huaxi Medical Univ, internal diameter be the cylindrical glass kapillary buying of 1.4mm from the Guangzhou glass apparatus trading firm of the smart section of Liwan District.Three cylindrical glass kapillaries are used to set up forecast model, remain a cylindrical glass kapillary and are used for check.

Step 2: to internal diameter is r ₁=0.15mm, r ₂=0.5mm, r ₃The cylindrical glass kapillary of=1.4mm carries out capillary rising experiment with the capillary rise method in absolute ethyl alcohol; The liquid-column height that records behind the liquid level stabilizing is as shown in table 2; Wherein survey instrument can be in milimeter scale, reading microscope, optical measuring instrument or the acoustic measurement appearance any one, and present embodiment adopts milimeter scale as survey instrument.

Table 2 liquid-column height table

Internal diameter mm	0.15	0.5	1.4
				Liquid-column height mm	h 1＝28.3	h 2＝9.8	h 3＝2.7

Step 3: three cylindrical glass kapillaries in the step 2 are asked for the operation of the equivalent height of meniscus liquid type in the cylindrical glass kapillary respectively:

For internal diameter is the cylindrical glass kapillary of 0.15mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 0.15mm with σ, ρ and internal diameter ₁Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{\mathrm{<figure-callout\; id="1"\; label="y\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">y</figure-callout>}}_i}{{\mathrm{Dx}}_{i1}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{i1}}{{\mathrm{Dx}}_{i1}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{i1}}{{\mathrm{Dx}}_{i1}}}{x_{i1}{[1+{\left(\frac{{\mathrm{Dy}}_{i1}}{{\mathrm{Dx}}_{i1}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; <figure-callout\; id="1"\; label="g\; h"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">g</figure-callout>}{h}_{}{}_1+\mathrm{\&rho;}{\mathrm{<figure-callout\; id="1"\; label="Gy\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Gy</figure-callout>}}_i,$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I1, y _I1); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.15mm _I1, y _I1), as shown in Figure 3.

For internal diameter is the cylindrical glass kapillary of 0.5mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 0.5mm with σ, ρ and internal diameter ₂Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_{i2}}{{\mathrm{Dx}}_{i2}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}}{x_{i2}{[1+{\left(\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}{h}_2+\mathrm{\&rho;}{\mathrm{Gy}}_{i2},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I2, y _I2); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.5mm _I2, y _I2).

For internal diameter is the cylindrical glass kapillary of 1.4mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 1.4mm with σ, ρ and internal diameter ₃Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_{i3}}{{\mathrm{Dx}}_{i3}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}}{x_{i3}{[1+{\left(\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}{h}_3+\mathrm{\&rho;}{\mathrm{Gy}}_{i3},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I3, y _I3); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 1.4mm _I3, y _I3).

Obtain in three cylindrical glass kapillaries behind the coordinate point set on the meniscus outline line, according to

Respectively to three coordinate point set { (x _Ij, y _Ij) carry out M+1 item fitting of a polynomial M time, and wherein M is not less than 2 integer, and matched curve and coordinate point set maximum error are not more than 10 ^-6In the fitting of a polynomial process, M attempts since 2, after this M is added 1 at every turn and attempts, and is not more than 10 first up to the maximum error of matched curve and coordinate point set ^-6The time till, get last fitting result as the liquid shape function.It is as shown in table 3 to obtain in three cylindrical glass kapillaries meniscus liquid type function:

Table 3 meniscus liquid type function table

Glass tube internal diameter mm	Meniscus liquid type function mm
		0.15	y 1(x)＝3159x 2-0.05131x+2.16×10 -7
0.5	y 2(x)＝(1.4×10 6)x 3+277x 2+0.06063x-4.03×10 -7
		1.4	y 3(x)＝(8.791×10 4)x 3+139.1x 2+0.027x-4.833×10 -7

According to

Respectively to three function y _j(x) carry out integration, the equivalent height that obtains meniscus liquid type in three cylindrical glass kapillaries is as shown in table 4:

Table 4 equivalent height tables of data

Glass tube internal diameter mm	Equivalent height mm
		r 1＝0.15	f 1＝0.03062
r 2＝0.5	f 2＝0.1245
		r 3＝1.4	f 3＝0.2575

Step 4: according to

Point set { (r in the his-and-hers watches 4 _k, f _k) carry out N+1 item fitting of a polynomial N time, N gets 2 in the present embodiment, obtain equivalent height function f (r)=-0.09636r ²+ 0.3309r-0.01684, unit is mm, convert into obtain under the International System of Units f (r)=-96.36r ²+ 0.3309r-1.684 * 10 ^-5, unit is m.So just can obtain that the interior static contact angle forecast model of cylindrical glass kapillary of wetting state liquid medium and glass material does in the corresponding step 1 $\frac{2\mathrm{\&sigma;}\mathrm{Cos}\mathrm{\&theta;}}{r}=\mathrm{\&rho;\; g}[h-96.36r^2+0.3309r-1.684\&times;{10}^{-5}],$ All parameters adopt International System of Units in the forecast model.

After obtaining the interior static contact angle forecast model of cylindrical glass kapillary, present embodiment also adopts the cylindrical glass kapillary of the identical material of internal diameter r=0.25mm to carry out modelling verification.

Employing capillary rise method is carried out capillary rising experiment to the cylindrical glass kapillary of internal diameter 0.25mm in absolute ethyl alcohol, and measures liquid-column height h=20.3mm.

When adopting classic method to calculate static contact angle, need on MATLAB R2009a platform, work out following code:

The coded representation implication is to be liquid-column height h substitution meniscus equation in the cylindrical glass kapillary of 0.25mm with σ, ρ and internal diameter $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_i}{{\mathrm{Dx}}_i^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}}{x_i{[1+{\left(\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}h+\mathrm{\&rho;}{\mathrm{Gy}}_i,$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _i, y _i); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.25mm _i, y _i).And calculation level concentrates on the derivative Q at x=0.25mm point place, obtains Q=1.818, then the theoretical value θ of static contact angle=1-arctanQ=28.8 °.

Adopt method of the present invention, with liquid-column height h, internal diameter r and σ and ρ substitution forecast model $\frac{2\mathrm{\&sigma;}\mathrm{Cos}\mathrm{\&theta;}}{r}=\mathrm{\&rho;\; g}[h-96.36r^2+0.3309r-1.684\&times;{10}^{-5}],$ The predicted value that obtains static contact angle is 29.4 °.

Calculate the relative error of predicted value and theoretical value: (29.4-28.8)/28.8=2.1%.Relative error is merely 2.1%, and prediction effect is good.And the inventive method only need provide liquid-column height and inner diameter values, utilizes forecast model, carries out simple operation, just can draw static contact angle; Classic method then need provide liquid-column height, inner diameter values, and carries out the establishment of meniscus equation solution code and find the solution, and could obtain static contact angle, and computation process is complicated.Therefore, utilization the present invention can shorten the computing time and the complexity of static contact angle, thereby increases work efficiency.

Embodiment 2:

Step 1: the absolute ethyl alcohol of preparing to analyze pure level is as the wetting state liquid medium, and the surface tension instrument/density appearance that adopts the auspicious Instr Ltd. in Shanghai side to produce, and model is MDY-1, measures the surface tension σ and the density p of absolute ethyl alcohol, and measurement result is seen table 5:

Table 5 absolute ethyl alcohol density and surface tension table

Class of liquids	Surface tension N/m	Density kg/m 3
			Absolute ethyl alcohol	0.02255	792

It is identical to prepare four root timber matter; Internal diameter is respectively the cylindrical glass kapillary of 0.15mm, 0.25mm, 0.5mm, 1.4mm; Wherein internal diameter be the cylindrical glass kapillary buying of 0.15mm, 0.25mm, 0.5mm from instrument plant of Huaxi Medical Univ, internal diameter be the cylindrical glass kapillary buying of 1.4mm from the Guangzhou glass apparatus trading firm of the smart section of Liwan District.Three cylindrical glass kapillaries are used to set up forecast model, remain a cylindrical glass kapillary and are used for check.

Step 2: to internal diameter is r ₁=0.15mm, r ₂=0.5mm, r ₃The cylindrical glass kapillary of=1.4mm carries out capillary rising experiment with the capillary descent method in absolute ethyl alcohol; The liquid-column height that records behind the liquid level stabilizing is as shown in table 6; Wherein survey instrument can be in milimeter scale, reading microscope, optical measuring instrument or the acoustic measurement appearance any one, and present embodiment adopts milimeter scale as survey instrument.

Table 6 liquid-column height table

Internal diameter mm	0.15	0.5	1.4
				Liquid-column height mm	h 1＝29.0	h 2＝10.0	h 3＝2.7

Step 3: three cylindrical glass kapillaries in the step 2 are asked for the operation of the equivalent height of meniscus liquid type in the cylindrical glass kapillary respectively:

For internal diameter is the cylindrical glass kapillary of 0.15mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 0.15mm with σ, ρ and internal diameter ₁Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{\mathrm{<figure-callout\; id="1"\; label="y\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">y</figure-callout>}}_i}{{\mathrm{<figure-callout\; id="1"\; label="Dx\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Dx</figure-callout>}}_i^1}}{{[1+{\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="Dy\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Dy</figure-callout>}}_i}{{\mathrm{Dx}}_{i1}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{<figure-callout\; id="1"\; label="Dy\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Dy</figure-callout>}}_i}{{\mathrm{Dx}}_{i1}}}{x_{i1}{[1+{\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="Dy\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Dy</figure-callout>}}_i}{{\mathrm{Dx}}_{i1}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; <figure-callout\; id="1"\; label="g\; h"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">g</figure-callout>}{h}_{}{}_1+\mathrm{\&rho;}{\mathrm{<figure-callout\; id="1"\; label="Gy\; i"\; filenames="HSA00000600744300022.png"\; state="\{\{state\}\}">Gy</figure-callout>}}_i,$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I1, y _I1; Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.15mm _I1, y _I1).

For internal diameter is the cylindrical glass kapillary of 0.5mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 0.5mm with σ, ρ and internal diameter ₂Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_{i2}}{{\mathrm{Dx}}_{i2}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}}{x_{i2}{[1+{\left(\frac{{\mathrm{Dy}}_{i2}}{{\mathrm{Dx}}_{i2}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}{h}_2+\mathrm{\&rho;}{\mathrm{Gy}}_{i2},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I2, y _I2); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.5mm _I2, y _I2).

For internal diameter is the cylindrical glass kapillary of 1.4mm:

The following code of establishment on MATLAB R2009a platform:

The coded representation implication is to be liquid-column height h in the cylindrical glass kapillary of 1.4mm with σ, ρ and internal diameter ₃Substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_{i3}}{{\mathrm{Dx}}_{i3}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}}{x_{i3}{[1+{\left(\frac{{\mathrm{Dy}}_{i3}}{{\mathrm{Dx}}_{i3}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}{h}_3+\mathrm{\&rho;}{\mathrm{Gy}}_{i3},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _I3, y _I3); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 1.4mm _I3, y _I3).

Obtain in three cylindrical glass kapillaries behind the coordinate point set on the meniscus outline line, according to

Respectively to three coordinate point set { (x _Ij, y _Ij) carry out M+1 item fitting of a polynomial M time, and wherein M is not less than 2 integer, and matched curve and coordinate point set maximum error are not more than 10 ^-6In the fitting of a polynomial process, M attempts since 2, after this M is added 1 at every turn and attempts, and is not more than 10 first up to the maximum error of matched curve and coordinate point set ^-6The time till, get last fitting result as the liquid shape function.It is as shown in table 7 to obtain in three cylindrical glass kapillaries meniscus liquid type function:

Table 7 meniscus liquid type function table

Glass tube internal diameter mm	Meniscus liquid type function mm
		0.15	y 1(x)＝3290x 2-0.05626x+2.366×10 -7
0.5	y 2(x)＝(1.578×10 6)x 3+211.4x 2+0.07018x-4.788×10 -7
		1.4	y 3(x)＝(1.4×10 6)x 3+277x 2+0.06063x-4.03×10 -7

According to

Respectively to three function y _j(x) carry out integration, the equivalent height that obtains meniscus liquid type in three cylindrical glass kapillaries is as shown in table 8:

Table 8 equivalent height tables of data

Glass tube internal diameter mm	Equivalent height mm
		r 1＝0.15	f 1＝0.03161
r 2＝0.5	f 2＝0.1282
		r 3＝1.4	f 3＝0.2575

Step 4: according to

Point set { (r in the his-and-hers watches 8 _k, f _k) carry out N+1 item fitting of a polynomial N time, N gets 2 in the present embodiment, obtain equivalent height function f (r)=-0.1058r ²+ 0.3448r-0.01772, unit is mm, convert into obtain under the International System of Units f (r)=-105.8r ²+ 0.3448r-1.772 * 10 ^-5, unit is m.So just can obtain that the interior static contact angle forecast model of cylindrical glass kapillary of wetting state liquid medium and glass material does in the corresponding step 1 $\frac{2\mathrm{\&sigma;}\mathrm{Cos}\mathrm{\&theta;}}{r}=\mathrm{\&rho;\; g}[h-96.36r^2+0.3309r-1.684\&times;{10}^{-5}],$ All parameters adopt International System of Units in the forecast model.

After obtaining the interior static contact angle forecast model of cylindrical glass kapillary, present embodiment also adopts the cylindrical glass kapillary of the identical material of internal diameter r=0.25mm to carry out modelling verification.

Employing capillary descent method is carried out capillary rising experiment to the cylindrical glass kapillary of internal diameter 0.25mm in absolute ethyl alcohol, and measures liquid-column height h=20.9mm.

When adopting classic method to calculate static contact angle, need on MATLAB R2009a platform, work out following code:

The coded representation implication is to be liquid-column height h substitution meniscus equation in the cylindrical glass kapillary of 0.25mm with σ, ρ and internal diameter $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_i}{{\mathrm{Dx}}_i^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}}{x_i{[1+{\left(\frac{{\mathrm{Dy}}_i}{{\mathrm{Dx}}_i}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}h+\mathrm{\&rho;}{\mathrm{Gy}}_i,$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary _i, y _i); Wherein g is a local gravitational acceleration; Coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in the cylindrical glass kapillary, is the y axle with cylindrical glass kapillary central axis, and the coordinate points lump is counted and confirmed automatically by MATLAB R2009a software platform; Obtain internal diameter thus and be coordinate the point set { (x on the meniscus outline line in the cylindrical glass kapillary of 0.25mm _i, y _i).And calculation level concentrates on the derivative Q at x=0.25mm point place, obtains Q=2.091, then the theoretical value θ of static contact angle=1-arctanQ=25.6 °.

Adopt method of the present invention, with liquid-column height h, internal diameter r and σ and ρ substitution forecast model $\frac{2\mathrm{\&sigma;}\mathrm{Cos}\mathrm{\&theta;}}{r}=\mathrm{\&rho;\; g}[h-96.36r^2+0.3309r-1.684\&times;{10}^{-5}],$ The predicted value that obtains static contact angle is 26.3 °.

Calculate the relative error of predicted value and theoretical value: (26.3-25.6)/25.6=2.7%.Relative error is merely 2.7%, therefore, adopts prediction effect of the present invention good.And the inventive method only need provide liquid-column height and inner diameter values, utilizes forecast model, carries out simple operation, just can draw static contact angle; Classic method then need provide liquid-column height, inner diameter values, and carries out the establishment of meniscus equation solution code and find the solution, and could obtain static contact angle, and computation process is complicated.Therefore, utilization the present invention can shorten the computing time and the complexity of static contact angle, thereby increases work efficiency.

Claims (1)

1. method of setting up static contact angle forecast model in the cylindrical glass kapillary is characterized in that: may further comprise the steps:

Step 1: prepare a kind of wetting state liquid medium, measure the surface tension σ and the density p of wetting state liquid medium; It is identical to prepare p material, the cylindrical glass kapillary that internal diameter is different, and p>=3 wherein, each cylindrical glass internal diameter capillaceous is respectively r ₁, r ₂, r ₃..., r _p

Step 2: get p the cylindrical glass kapillary of preparing in the step 1 and in the wetting state liquid medium, carry out capillary rising experiment; After treating that the liquid lifting height is stable; Measure liquid-column height in each kapillary with height measuring device, obtain the interior liquid-column height of p kapillary and be respectively h ₁, h ₂, h ₃..., h _p

Step 3: for j cylindrical glass kapillary, with liquid-column height h in σ, ρ and j the cylindrical glass kapillary, substitution meniscus equation $\mathrm{\&sigma;}(\frac{\frac{d^2{y}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}^2}}{{[1+{\left(\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}\right)}^2]}^{3/2}}+\frac{\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}}{x_{\mathrm{Ij}}{[1+{\left(\frac{{\mathrm{Dy}}_{\mathrm{Ij}}}{{\mathrm{Dx}}_{\mathrm{Ij}}}\right)}^2]}^{1/2}})=\mathrm{\&rho;\; g}{h}_j+\mathrm{\&rho;}{\mathrm{Gy}}_{\mathrm{Ij}},$ Numerical evaluation obtains coordinate the point set { (x on the meniscus outline line in j the cylindrical glass kapillary _Ij, y _Ij), wherein g is a local gravitational acceleration, the coordinate system under the coordinate point set is an initial point with the minimum point on the meniscus outline line in j the cylindrical glass kapillary; With j cylindrical glass kapillary central axis is the y axle, i=1,2;, Z, Z are coordinate point set { (x _Ij, y _Ij) in count; According to

To coordinate point set { (x _Ij, y _Ij) carry out M+1 item fitting of a polynomial M time, and wherein M is not less than 2 integer, and matched curve and coordinate point set maximum error are not more than 10 ^-6, obtain meniscus liquid type function y in j the cylindrical glass kapillary _j(x); According to

To function y _j(x) carry out integration, wherein r _jBe j cylindrical glass internal diameter capillaceous, obtain the equivalent height f of meniscus liquid type in j the cylindrical glass kapillary _j

Step 4: repeat step 3, obtain the equivalent height f of meniscus liquid type in all p the cylindrical glass kapillaries ₁, f ₂, f ₃..., f _pAccording to

To point set { (r _k, f _k) carry out N+1 item fitting of a polynomial N time, and k=1 wherein, 2 ..., p, N is the integer of 2≤N≤p, obtains equivalent height function f (r); Thereby obtain that the static contact angle forecast model does in the cylindrical glass kapillary of wetting state liquid medium and glass material in the corresponding step 1

Wherein r is the cylindrical glass capillary inner diameter, h be carry out with step 2 in the identical capillary liquid-column height in the kapillary that experiment obtains that rises.

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