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METHODS FOR INDIRECT MEASUREMENT OF THE WETTING ANGLE ACCORDING TO THE GEOMETRIC DIMENSIONS OF THE UNDERLYING DROPLET AND THEIR METROLOGICAL JUSTIFICATION
Robert V. Harutyunyan,
Moscow State Technical University named after NE Bauman, Moscow, Russia, [email protected]
DOI 10.24411/2072-8735-2018-10160
Keywords: wetting angle, indirect measurement, techniques, comparison.
In this paper, we describe and study the technique of indirect evaluation of the wetting angle, which presents the minimum requirements for measuring equipment with satisfactory accuracy. Surface quality issues can be relevant for electrical and transport systems. The initial parameters of the technique are the geometric dimensions of the test drop - maximum height and radius, which are macroscopic characteristics, the direct measurement of which does not cause serious difficulties. The difference from the known techniques is the use of the radius of the drop rather than the radius of the base, and its largest radius, which is more convenient to measure in the case of hydrophobic surfaces. In contrast to the known technique, only the wetting angle is located, and the surface tension is known and it is not necessary to specify the droplet volume (with which a large error is associated). Another difference is that the computational part of the technique is based on the interval approach. The computer calculates a guaranteed evaluation of the wetting angle from above and from below and adequately takes into account the error in the initial data. The latter circumstance substantially simplifies the metrological justification of the method of indirect measurements under consideration.
The form of a drop of liquid on the horizontal surface of a solid body under normal gravity is studied. It is assumed that surface irregularities and inhomogeneities of its characteristics on the droplet size scale can be neglected. In a cylindrical coordinate system, the static surface of a drop can be uniquely described by some function that is a solution of the corresponding isoperimetric variational problem. The energy of the liquid-solid interaction does not affect the shape of the drop and the form of the equation, but participates in the formation of the boundary condition in accordance with Young's law. The solution of the boundary value problem is carried out in two stages. The properties of the problem and its solution are investigated, including the conditions for its existence and uniqueness, as well as bilateral numerical methods for solving the problem, taking into account both the error of the difference method and the error in measuring the input parameters. The two-sided method guarantees the accuracy of the solution through bottom and top estimates. The solution is iteratively, the method is characterized by a second order of accuracy. The adequacy of the model and the accuracy of the corresponding indirect methodology was estimated by comparing the calculated and experimental data on the angle of wetting of the fluoroplastic plates with drops of distilled water at room temperature. The surface of the fluoroplastic was pretreated with boiling in a freshly prepared chrome mixture, followed by rinsing with distillate and re-boiling in distilled water. Distillate droplets were placed on the surface of fluoroplastic plates by means of a microsy-ringe, followed by photographing the droplet profiles in various projections. The geometric parameters of the droplets (height and maximum radius of the section of the horizontal plane) were determined with the help of a measuring microscope with a division value of 0.1 mm.
Information about author:
Robert V. Harutyunyan, Ph.D., associate professor of the Department of Computational Mathematics and Mathematical Physics,
Moscow State Technical University named after NE Bauman, Moscow, Russia
Для цитирования:
Арутюнян Р.В. Методики косвенного измерения угла смачивания по геометрическим размерам лежащей капли и их метрологическое обосновани // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №10. С. 70-76.
For citation:
Harutyunyan R.V. (2018). Methods for indirect measurement of the wetting angle according to the geometric dimensions of the underlying droplet and their metrological justification. T-Comm, vol. 12, no.10, рр. 70-76.
Introduction and status of the question
Measurement of contact angle of wetting is an important element of many experimental techniques for evaluating various characteristics of surfaces (specific energy of interaction at the interface of dissimilar media, parameters of roughness, degree of soiling, etc.) [1-181. Many methods for the experimental determination of the wetting angle are known 11-1SJ, The wetting angle is a microscopic, differential parameter, and therefore its direct measurements arc impossible without high-resolution measuring devices. These circumstances have led to the development of a number of indirect methods for estimating the wetting angle. Direct measurements require the availability of special equipment and are relatively time-consuming. The industry produces a number of instruments for measuring the wetting angle (Acam-S 1, DSA100, DSA15, DSA20. DSA100, GH100, etc.).
In these instruments, in addition to the direct measurement of the geometrical parameters of the test drop, analytical methods for processing the measurement results are used. As a rule, the analysis of contact angle in devices is carried out by circular, elliptical, tangential or semi-angular methods. In the first method, it is assumed that the size of the drop is small and can be represented as a part of the sphere, and the droplet profile in two dimensions, i.e. in the form of a circle. This method is the most common method for calculating the angle of contact [l-16j. The contact angle of wetting & is found by the formula:
where h is the height, d is the diameter of the drop base. Also known is the so-called. The width-height method, in which the droplet spreading size and its height are located. The contour, as a part of a circle, is inscribed in a rectangle and the contact angle is calculated from the width and height values. This technique is preferable for small droplets, which are usually approximately spherical in shape. For droplets of larger size and heaviness, the height of the apex is reduced accordingly. In the elliptic method, the profile of the drop profile is used. In this respect, this method is similar to circular. 6 base points are selected on the curve and the ellipse equation is calculated, hi the tangential method, the complete
contour ot the test drop approximately reduces to the equation of the conical segment, the derivative of this equation at the point of intersection of the contour and the baseline gives an angle of inclination at the point of contact, i.e. contact angle. In another embodiment, a part of the contour of a lying drop located near the baseline is approximated by a function of the form y = a + bx + ex0-5 + ii !\x\x + el x1- The method is estimated in the literature as relatively accurate, but its high sensitivity to contamination and foreign substances in the liquid is noted.
The Young-Lap lace method is a widely known, relatively time-consuming, but also more accurate method of calculating the contact angle. In this technique, both interfacial interactions and the influence of fluid weight are taken into account. This technique usually assumes that the shape of the falling liquid drop is abiisymmetric. The existence of surface tension leads to the appearance of additional pressure under the curved surface of the liquid drop. The pressure under consideration is calculated according to the Young-Laplace equation: A/) = o(l/J?l + 1/fl-,),
where /j, - the main local radii of curvature of the surface of
the liquid, cr - its surface tension.
There are many modifications of this method. So, in ¡10] an invention is described which, as claimed, belongs to the field of surface phenomena in the technology of viscous fluids and can be used in the measuring technique for the precise determination of the surface tension of various liquids, including high-temperature melts, and the measurement of the wetting angle. According to the description, a drop of liquid is applied to a solid horizontal surface. The image of the droplet measures the height of its tip and the radius of the contact spot of the drop w ith the substrate. Based on the solution of the equilibrium equations, the surface tension coefficient and the wetting angle with a specified accuracy are determined. It is noted that there are no restrictions on the size of the drops and the value of the edge angles. The improvement consists in simplifying the measurements and increasing the accuracy of the result. Thus, the patent technique [I0j is one of the variants of the Jung-Laplace method (below it is considered in detail). In [111, the centra! relation of the dropdrop method, the Young-Lapiaee equation, describing the surface of a droplet with rotation symmetry on a horizontal substrate is considered. To calculate the solution of this problem, a technique was proposed that was improved in [I2-15J. In this technique, the differentiation of the Young-Laplace ratio is used using the parameter of the curve S, w here s is the are length of the curve from the top of the drop. Further, the initial problem is solved by the Runge-Kutta method of the fourth order of accuracy. To find the parameters of the falling drop, the inverse problem of finding the capillary constant, the coordinates of the droplet apex and its radius of curv ature from the values of the radius function of the horizontal section of the drop from the height above the substrate is solved. This dependence is found by measurements with an error and, in some cases, only parts of the drop contour are available. When finding the solution of the inverse problem in question, we seek a minimum of error-the discrepancy between the coordinates of the experimental points and the points of the curve obtained by solving the problem numerically. The distance between the experimental points and the curve is defined as the root of the sum of the squares of the distances from each experimental point to the curve in question. An important task is the task of image processing: automatic obtaining of a contour of a drop.
Fig. 1. Photograph of a drop of water on quartz (on the right, contour points are marked on the contour) [II]
In this paper, an analogous |10J technique for the indirect evaluation of the wetting angle is described and investigated, presenting minimum requirements for measuring equipment with satisfactory accuracy. The initial parameters of the method are also the geometric dimensions of the probe drop - maximum height and radius, which are macroscopic characteristics, the direct measurement of which does not cause serious difficulties. The difference from the methods [10, II] is the use not of the radius of the drop's base, but of its largest radius, which is more
( I ■N
convenient to measure in the case of hydrophobic surfaces. In comparison with [10], it should be noted that the results of calculating the wetting angle depend strongly oil the accuracy of measuring the droplet volume, and the surface tension coefficient is usually relatively accurately known. In contrast to the procedure [10], only the wetting angle is located, and the surface tension is known and it is not necessary to specify the droplet volume (with which a large error is associated). Another difference is that the computational part of the technique is based on the interval approach |! 8-20]. The computer calculates a guaranteed evaluation of the wetting angle from above and from below and adequately takes into account the error in the initial data. The latter circumstance substantially simplifies the metro logical justification of the method of indirect measurements under consideration.
A mathematical model tor estimating the angle of wetting
Consider a drop of liquid on the horizontal surface of a solid body under normal gravity conditions (Figure 2),
Fig. 2. Axial section ofa test drop oil a horizontal base
Wc assume that the surface irregularities and inhomogeneities of its characteristics on the scale of the droplet size can be neglected. Usually this assumption is fulfilled, since the size of the test drop is about I mm, which is much larger than the micro-roughness. In a cylindrical coordinate system (r,z)
The static surface ofa drop can be uniquely described by some function r - which is a solution of the isopcrimctric variational problem [1,18];
-1I/2
E = 2ji<T Jr
0
H
l+[ — dz
dz + npg |r2 (z)zdz + tiR2^ -» min.
d_
dz
dr dz
1 I dr
2\
-1/2
1 + f£
dz
1/2
+ K2rz + lr, {2)
where is the Lagrange constant, corresponding to the condition of constancy of the volume of the drop, k~ =pg/tf - capillary
constant. The last term (the energy of the interaction between a liquid and a solid base) does not affect the shape of the drop and the form of the equation (the corresponding terms in (2) are mutually destroyed), but participates in the formation of the boundary condition according to Young's law: cos 0 = (oTr- a.A.T)/o.AT, Where aTr, Gw= a - respectively, surface energies at the solid-gas interface, liquid-solid and liquid-gas interface. By means of integration (2) it is reduced to the form:
(
-1/2
2 ( kr
z - zt +
it/'"
(3)
where V0(z) - volume of the segment of rotation within
(z,H), z*=\1\!k2.
A criterion for the uniqueness of the solution (2) —(3) are the conditions:
r{H) = 0 max r(z) = Rl>r (4)
0 <z<H
The required wetting angle is determined by the formula;
3 = - + arctg—(0). (5)
2 dz
The solution of problem (3) - (4) is carried out in two stages. At the first stage there is an integral curve on the segment (zm,H), and 011 the second - on the segment (0,zm),
where r{zm) — Rm . The problem solved at the first stage is:
f=/%) i-(/«(r)f ,re(0>Rm), f*)
(6b)
.2 r
where /l)(r) = JL \(z{t)-z.)tdf r 0
Conditions (6) are sufficient both for finding z(r), and an unknown constant determined from equation
f(])(Rm) = -1. (?)
The second stage integrates the Cauchy problem with respect to r(z) for 2 g (0,2m):
''drop = * I''2 (z)dz -> fixed, »
where E - free energy ofa drop, Kdrop - *ts volume, p - mass density of the liquid, l1 - specific interaction energy at the liquid-solid interface, g - acceleration of gravity, R - radius of the droplet base. Extremals n=r(z) are solutions of the Buler
differential equation:
dr
~dz
-(/2)(*)j
on condition
r{zm)=Rm> where
z e
(8a)
(8b)
/(2)(z) = -
2 ( K~r
z-z* +
7tr
m Ä
+ ¡r2{t)/r2(z)dt
At the first stage of the solution, the parameters z*, zm, V0(zin), and in the second stage the solution of the
Cauehy problem (8) is realized. We reduce (6) - (7) to the di-mensionless form by changing the variables j = 11 ix)
/
K R„
x _ Equations (6) take the form:
I
f-z/(^)(i-(/(-v))2p, ^(o-i); x={KRm)\ ^
u{0) = K2RmH, (9b)
where = fj__ xYfy(t)tdt -x\i(t)tdt - x.
In this form (7) is fulfilled automatically, and the constant z» is equal to:
/ i \
2
z* =
1+ ji(t)tdt
lni v o
As a result of similar transformations (8) takes the form:
1
» (Hfflf f
dx _
du zf{u)
where =
u — um + u*
(»* <H
"• „2
(v)
(10a) (I Ob) dv
Uj = Uj_\ +hj
fc + uUzl
J-1
FÎ->~ FJ-2
2 hH
f / = 2,;V; w,+ = »p;
UN =UNA + ]>£*(*)*; or ujf=Bv_j
•V
ff = min
(
0, max
V
- ±
~xl>fj
, fj "i1'*;-*/)//
J J
/-1
/-2 N
f- S
y=(+!
.V,/?, /1: .V ;_,/); 11,
y-i
J / V 2 3
2 3
w , ^
V
/ V A h 2 ^ xj-\"i ,¡/7
z
+K0+/J,2/2,
A+= X
where
«y-i*y-i/2*> -«y-2)
.. hJ --^ -
/Vi
2 3
0 = .v0 < A"! <...<XA, =1, hj = X, ~ , A' J = (a-;_i + ) / 2,
=¡/(1), U* = K2Rmz* ■ The desired wetting angle is determined by the formula £) = ji- arcsin /"(0).
In [18], the properties of problem (9) and its solutions, in-eluding the conditions for its existence and uniqueness, are investigated, as well as two-sided numerical methods for solving the problem, taking into account both the error of the difference method and the error in measuring the input parameters,
A two-sided method for solving (9) and (10) with a given accuracy
For brevity, let us denote the right-hand side in (9a) by F, We compare the lower bounds with the subscript "-", and the upper bounds with "+". Then, taking into account the properties proved in [18], the following Adams schemes will give a guaranteed upper and lower bound for the solution (9):
uj=uj-i+^V'u-i)+F7)> J=1'N~1'
The two-sided method guarantees the accuracy of the solution through bottom and top estimates; u-j<.u(xj)<.u)J= 0,N.
The solution is itcratively, the initial approximation has the form: j/i°Vx)= «,) (l =«(). The method is characterized
by the second order of accuracy w ith respect to h. The values (Kl-R~)f 2 H (//+-//")/2 correspond to absolute errors in
measuring geometric dimensions R 11 H (hereinafter referred
to as A). For problem (10), the two-sided method is formulated in a similar way.
Results of physical and computer experiments
The adequacy of the model and the accuracy of the corresponding indirect methodology was estimated by comparing the calculated and experimental data on the angle of wetting of the lluoroplastic plates w ith drops of distilled water at room temperature. Fluoroplast is one of the most hydrophobic synthetic materials. During short-term immersion, the wetting angle is greatest, but for a long slay in distilled water (15-20 days) wettability is observed.
The surface of the fibroplastic was pretreated with boiling in a freshly prepared chrome mixture, followed by rinsing with distillate and re-boiling in distilled water. Distillate droplets were placed on the surface of lluoroplastic plates by means of a microsyringe, followed by photographing the droplet profiles in various projections.
The geometric parameters of the droplets (height and maximum radius of the section of the horizontal plane) were determined with the help of a measuring microscope with a division value of 0.1 mm. The coefficient of surface tension of distilled water at 20°C with high accuracy is a = 0,07225 N/m.
In Table 1 shows the calculated data for a drop with the largest diameter of 5 mm and a height of 3.1 mm, an approximate value of the volume of order (which was then recalculated according to the procedure) for different values of the measurement error I0~7m:! geometric parameters.
7TT
Table I
Estimates of the lower 6 and upper 9 boundaries of the angle of wetting by water of the fluoroplastic surface
№ 9, deg 0, deg A, 111m
I 112,011 II 2.370 0.00
2 109,772 114.731 0,05
3 107,208 117.619 0,10
The data given ¡11 Table ! correspond to the contact angle of wetting at the idealized point of the three-phase contact. ¡11 practice, due to the error of the measuring equipment (optical or other nature), the determination of the contact angle of wetting is not carried out in the plane of the drop base (the applicator z = 0), but at sonic distance from it iz = zmes )■
An analysis of the error of the technical equipment used to measure the angle of wetting allows us to limit the interval
Zmes value in 10 4 m. The corresponding values ofthe wetting
angle for the experimental conditions under consideration (the fluoroplast-water-air system) arc given in Table 2. Analysis of literature data on wetting contact angles in the fluoroplastic-water-air system indicates a relatively wide range of experimental values. Thus, according to the data of [3], the value ofthe wetting angle is 108°, and the values of 98°, 108°, and 112° are given in the monograph [4]. We also note the value of 105°, mentioned in [5]. In less reliable Internet sources, the values of 108°, 112° and even 126° are noted (for fluoroplaslic-4). The average of the given data from the literature sources is about 110°. This value is in good agreement with the interval estimates in Table 2.
Table 2
Lower bounds 6 and upper 6 boundaries ofthe angle of wetting at a distance ItT1 m from the base of the drop
№ 6, deg Q, deg A, mm
1 109,322 109,681 0,00
2 107, 190 111,924 0,05
3 105,133 114,270 0,10
The results ofthe calculations (Tables 1 and 2) allow us to conclude that the literature data are mainly in the calculated intervals of values ofthe contact angle of wetting. Explicitly dropping from the general series of the value ofthe wetting angle of 98° and 126°, possibly due to a relatively large measurement error or the state ofthe surface ofthe substrate. Analysis ofthe results of other known techniques
A) in the circular method [1-16], the ratio (1), based 011 the spherical droplet model, is used to estimate the wetting angle. This implies an insignificant role of gravity (small size or droplet density). For the droplet sizes considered,dm =5 mm and
H =3,1 mm this technique leads to the values:
cos9 — — 0,24; $=103,9°; the estimated diameter of the
base \s:d = 2{H(dm ~H)f2 =4,85 mm - With an error in
measuring dimensions of 0.1 mm, the calculated values of the welting angle vary in the interval 98.8° ... 109.5°.
B) In the patent 110], the equilibrium equations ofthe droplet,
f • N
written in the form:
'b - ~zo
Boz„ - — + c
V
Hi
z0 = 'b
^ Boz0 ——+ c n
0
where
■ ^ d'h
■ ds
Zn ='
ds
>h=-
d\ ds2'
zn=-
d\ ds2
r _z _ 4kR
'drop - —p
r0(0) = 0, z0(0) = 0,
z0(0) =////?; ro(0) = l, s - The length of the are of the axial section of the drop, measured from the axis, C - a constant calculated from the condition that the calculated volume of a drop V is equal to its actual val-
ue y
*h =
drop ' d 2 R PgR
the Bond number is found from the condition:
.__ Zn = 0; after which there is a surface tension:
= —__The corner angle is calculated by the formula:
Bo
9 = arccos{r0), zo = 0.
The advantage of the technique is the possibility of simultaneous calculation ofthe surface tension and the angle of wetting. A certain disadvantage is the requirement of specifying the volume of the droplet at its relatively small size, which is associated with considerable error and inconvenience ofthe method (especially since the surface tension is usually a priori known or determined by easily accessible methods). Calculations show that when the droplet volume changes by 2%, the wetting angle changes by 1%. Comparison ofthe results of calculating the diameter ofthe base, volume, and wetting angles for several methods for the average droplet sizes dm = 5 mm u H = 3,1 mm is
made in the Table 3.
The circular method is relatively less precise. Its applicability is limited by small droplet sizes. Figure 5 illustrates the increase in the error in the spherical approximation ofthe shape of a drop of water (1 - /J, =2,5 mm, //,=2,5 mm; 2 -fl3 = 5mm, H2 = 3 mm; 3 —/¡3=8 mm, //3 = 4 mm)-
Table 3
Results of calculations of droplet parameters
Method d, mm V, mm3 9, deg1 0, deg
Circular 4,85 44,3 99,25 103,89
Patent [4] 4,4 45,1 I ! 8,43 128,53
Method [18] 4,4 45,1 1 19,40 129,67
Asymptotics [17] 4,0 44,6 — 118.62
Fig. 3. Equilibrium and circular contours of water droplets of various sizes
The last line of Table 3 gives the results for the asymptotic method [17]. The condition of its application: k~< In 2. The drop contour is described by a spherical radi-k-R3
usr{y) = Rm--^-!n(l + coscp)coscp, where 9 - spherical angle, measured from the axis of the applicator, 0 < tp < 7t.
The wetting angle, the droplet volume and the diameter of its base are:
/c2/?,;sin(pc[| ( 1 + COSCp(. ) In ( 1 + COSifK, ) + cos<pt, ]
(l -i-cost^, )[Wiá ,cos(pcln(Hcas<|>c)]
V = I + cos<prf J2 + cosçL.-[j 1 + costpcf ln(l + cos<pi.) + 0,5 J, d - 2tg(pt.(r(0)- M J,
where <pc - solution of equation: H -r(0) = -r((p(.)coscpt..
Table 4 presents the results of calculations using the author's method [18] and using the asymptotics [I 7] for comparison. Index I corresponds to the asymptotic method. Variants of the initiai data for Table 4:
Table 4
Results of calculations of the water droplet parameters
№ V, mm3 -.'.-; i mm >■, mm 0, deg Vu idhi' zmti mm n. mm 0|, deg
1 17,5 0,24 2,0 97,8 17,6 0,25 2,0 97,2
2 25,1 0, 12 2,2 100,1 25,4 0,36 2,2 99,2
3 23,4 0,32 2,2 99.6 23,7 0,33 2,2 98,8
4 23,3 0,72 1,8 120,0 23,3 0,75 1,8 112,9
5 27,8 0,58 2,1 111.9 27,9 0,63 2,1 107,2
6 34,6 1.16 1,6 147,9 33,6 1,23 1,4 132,1
7 34,7 0.40 2,5 102.7 35,2 0,49 2,4 101,5
1) /?,„=2 mm; H= 2 mm; 2) R,„=2,25 mm; //=2,25 mm; 3) R,„-2,2 mm; H-2,2 mm; 4) R,„-2 mm; H-2,5 mm; 5) R„,—2,2 mm; H-2,5 mm; 6) R„,—2,2 mm; H-3,1 mm; 7) R,„=2,5 mm; //=2,5 mm.
As for the circular method, the drawback is the limited scope of application, for water it is estimated: R < 2,5 mm.
1. Finn R, {1989). Equilibrium capillary surfaces. Mathematical theory. Moskva: Mir.
2. Rebiiider P.A. (1978), Surface phenomena in disperse systems. Colloid chemistry. Moskva: Nauka.
3. 01 sen D.A., Joyner P. A., OI sen MD. (1962). J. Phys. Chem, Vol.66. No 5.
Adamson A. (1979). Physical chemistry of surfaces. Moskva: Nauka.
5. Zimon A.D. (1974). Adhesion of liquids und wetting. Moskva: Chemistry.
6. Jakek M„ Parfit J. (1984). Chemistry of interfaces. Moskva: Mir.
7. Arsent'ev P.P., Yakovlev V.V., Krashennikov M.G. (1988). Physico-chemical methods for studying metallurgical processes. Moskva: Metallurgy.
8. Patent SU No. 1753368, GCIN 13/02, 1992.
9. Patent UA №48429 A, COIN 13/02,2002.
10. Patent of the Russian Federation No. 2460987. G0IN13 / 02. Method for determining the coefficient of surface tension and the angle of wetting. 2012.
11. Zimin A.A., Ignatova N.Yu., Ctmmffina L.G., Zhereb V.P. (2011). Method for calculating the contact angle of wetting and surface tension of liquids by optimizing the modeling parameters of the profile of an ax i symmetric lying drop. Journal "Actual problems of aviation and cosmonauticsIssue 7. Vol.3.
12. Ma¿e C-, Burnet G.A. (1969), Non-linear Regression Method Tor Calculating the Surface Tension and Contact Angle from the Shape of a Sessile Drop. SurfSci. Vol. 13.
13. Krylov A.S., Vvedensky A.V., Katsnelson A.M., Tugovikov A,E. (1993). Software package for determination of the surface tension of liquid metals. J. Non-Cryst. Solids. Vol. 156-158. P. 845.
14. 0.1. del Rio and Neumann A.W. (1997). Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Intcrfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops. Journal of Colloid and Interface Science. Vol. 196. Issue 2.
15. Hoorfar M. and Neumann A.W. (2006). Recent progress In Ají i symmetric Drop Shape Analysis. Advances in Colloid and Interface Science. Vol. 121. Issues 1-3.
16. Kiselev M.G., Savich V.V., Pavich T.P. (2006). Determination of the contact angle of wetting on flat surfaces. Zhurnal Nauka i Tekhnlka. Issue 1.
17. Matyukhin S.I., Frolenkov K, Yu. (2013). Form of liquid droplets placed on a hard horizontal surface. Condensed media and interface boundaries. Vol. 15. No 3, pp. 292-304.
18. Vlasov Yu. V„ Nekrasov S,A., Khentov V, Ya. (1994). The method of indirect evaluation of the wetting angle according to the geometric dimensions of the test drop. Matem. modeling. Vol. 6. No 9, pp. 33-40.
19. Nekrasov S,A, (2016), A numerical method for solving dynamical systems with lumped parameters, which accounts for an input data error. Journal of Applied and Industrial Mathematics. No 10 (4), pp. 528-537,
20. Nekrasov S,A, (2015), Com pie). Computer Applications and Techniques for Solving Problems with the Inherent Errors and Synthesis of Optimal Characteristics of Materials and Design Parameters. Journal of Siberian Federal University. Engineering & Technologies No. 8, Issue 7, pp. 867-877,
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МЕТОДИКИ КОСВЕННОГО ИЗМЕРЕНИЯ УГЛА СМАЧИВАНИЯ ПО ГЕОМЕТРИЧЕСКИМ РАЗМЕРАМ ЛЕЖАЩЕЙ КАПЛИ И ИХ МЕТРОЛОГИЧЕСКОЕ ОБОСНОВАНИЕ
Арутюнян Роберт Владимирович, Московский государственный технический университет имени Н.Э.Баумана,
Москва, Россия, [email protected]
Аннотация
В данной статье описана и исследована методика косвенной оценки угла смачивания, предъявляющая минимальные требования к измерительной аппаратуре при удовлетворительной точности. Вопросы качества поверхностей могут быть актуальными для электротехнических и транспортных систем. Исходными параметрами методики являются геометрические размеры пробной капли - максимальные высота и радиус, являющиеся макроскопическими характеристиками, прямое измерение которых не вызывает серьезных затруднений. Отличием от известных методик является использование не радиуса основания капли, а ее наибольшего радиуса, который удобнее измерять в случае гидрофобных поверхностей. В отличие от известной методики находится только угол смачивания, а поверхностное натяжение считается известным и не требуется задавать объем капли (с котором связана большая погрешность). Другое отличие заключается в том, что вычислительная часть методики базируется на интервальном подходе. На ЭВМ вычисляется гарантированная оценка угла смачивания сверху и снизу и адекватно учитывается погрешность в исходных данных. Последнее обстоятельство существенно упрощает метрологическое обоснование рассматриваемой методики косвенных измерений. Исследована форма капли жидкости на горизонтальной поверхности некоторого твердого тела в условиях нормальной гравитации. Предполагается, что неровностями поверхности и неоднородностями ее характеристик в масштабе размеров капли можно пренебречь. В цилиндрической системе координат статическая поверхность капли может быть однозначно описана некоторой функцией, являющейся решением соответствующей изопериме-трической вариационной задачи. Энергия взаимодействия жидкость - твердое основание не оказывает влияния на форму капли и вид уравнения, но участвует в формировании краевого условия согласно закону Юнга. Решение краевой задачи осуществляется в два этапа. Исследованы свойства задачи и ее решения, в том числе условия его существования и единственности, а также двусторонние численные методы решения задачи, учитывающие как погрешность разностного метода, так и погрешность измерений входных параметров.
Двусторонний метод гарантирует точность решения посредством оценок снизу и сверху. Решение находится итерационно, метод характеризуются вторым порядком точности. Оценка адекватности модели и точности соответствующей косвенной методики производилась посредством сопоставления расчетных и экспериментальных данных по углу смачивания пластин фторопласта каплями дистиллированной воды при комнатной температуре. Поверхность фторопласта предварительно обрабатывалась кипячением в свежеприготовленной хромовой смеси, последующим ополаскиванием дистиллятом и повторным кипячением в дистиллированной воде. Капли дистиллята помещались на поверхность пластин фторопласта посредством микрошприца, после чего осуществлялось фотографирование профилей капель в различных проекциях. Геометрические параметры капель (высота и максимальный радиус сечения горизонтальной плоскостью) определялись с помощью измерительного микроскопа с ценой деления в 0,1 мм.
Ключевые слова: угол смачивания, косвенное измерение, методики, сравнение. Литература
1. Финн Р. Равновесные капиллярные поверхности. Математическая теория. Москва: Мир, 1989.
2. Ребиндер П.А. Поверхностные явления в дисперсных системах. Коллоидная химия. Москва: Наука, 1978.
3. Olsen D.A., JoynerPA, Olsen M.D. // J. Phys. Chem, 1962. V.66. No 5, рр. 883-886.
4. Адамсон А. Физическая химия поверхности. Москва: Наука, 1979.
5. Зимон А.Д. Адгезия жидкостей и смачивание. Москва: Химия, 1974.
6. Джейкок М, Парфит Дж. Химия поверхностей раздела фаз. Москва: Мир, 1984.
7. Арсентьев П.П., Яковлев В.В., Крашенников М.Г. Физико-химические методы исследования металлургических процессов. М.: Металлургия, 1988.
8. Патент SU №1753368, G0IN 13/02, 1992.
9. Патент UA №48429 А, G0IN 13/02, 2002.
10. Патент РФ №2460987. G0INI3/02. Способ определения коэффициента поверхностного натяжения и угла смачивания. 2012.
11. Зимин А.А., Игнатова Н.Ю., Чумилина Л.Г., Жереб В.П. Метод расчета краевого угла смачивания и поверхностного натяжения жидкостей путем оптимизации параметров моделирования профиля осесимметричной лежащей капли // Журнал "Актуальные проблемы авиации и космонавтики". Вып. 7. T.I. 20II.
12. Maze C, Burnet G.A. Non-linear Regression Method for Calculating the Surface Tensión and Contact Angle from the Shape of a Sessile Drop // Surf. Sci. I969. V. I3. P. 45I.
13. Krylov A.S., Vvedensky A.V., Katsnelson A.M., Tugovikov A.E. Software package for determination of surface tension of liquid metals // J. Non-Cryst. Solids. I993. V. I56-I58. P. 845.
14. O.I. del Rio and Neumann A.W. Axisymmetric Drop Shape Analysis: Computational Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of Pendant and Sessile Drops // Journal of Colloid and Interface Science. I997. V.I96. Issue 2, pp. I36-I47.
15. Hoorfar M, Neumann A.W. Recent progress in Axisymmetric Drop Shape Analysis // Advances in Colloid and Interface Science. 2006.V. I2I. Issues I-3, pp. 25-49.
16. Киселев М.Г., Савич В. В., Павич Т.П. Определение краевого угла смачивания на плоских поверхностях // Журнал Наука и техника. Вып. I. 2006.
17. Матюхин С.И., Фроленков К.Ю. Форма капель жидкости, помещенных на твердую горизонтальную поверхность // Конденсированные среды и межфазные границы. 20I3. Т. I5. № 3. С. 292-304.
18. Власов Ю.В., Некрасов С.А., Хентов В.Я. Методика косвенной оценки угла смачивания по геометрическим размерам пробной капли //Матем. моделирование. I994. Т.6. № 9. С. 33-40.
19. Nekrasov S.A. A numerical method for solving dynamical systems with lumped parameters which accounts for an input data error // Journal of Applied and Industrial Mathematics. 20I6, I0(4), pp. 528-537.
20. Nekrasov S.A. Complex Computer Applications and Techniques for Solving Problems with the Inherent Errors and Synthesis of Optimal Characteristics of Materials and Design Parameters // Journal of Siberian Federal University. Engineering & Technologies 2015. No 8. Issue 7, pp. 867-877.
Информация об авторе:
Арутюнян Роберт Владимирович, к.ф.-м.н., доцент кафедры Вычислительная математика и математическая физика, Московский государственный техническийуниверситет имени Н.Э.Баумана, Москва, Россия
