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Nina Nina BogdanovaBogdanova and Stefan and Stefan TodorovTodorov
Institute of Nuclear Research & Nuclear Energy, BAS, BulgariaInstitute of Nuclear Research & Nuclear Energy, BAS, BulgariaEmail: Email: [email protected]@inrne.bas.bg, [email protected], [email protected]
The paper proposes a new application of our The paper proposes a new application of our orthonormalorthonormal polynomial expansion methodpolynomial expansion method--OPEM[1], based on Forsythe three term OPEM[1], based on Forsythe three term relation[2]. Some special features of the method relation[2]. Some special features of the method are developed for this purposeare developed for this purpose..
1.PHYSICAL DATA
Up to now the wetting properties of liquids Up to now the wetting properties of liquids present a high interest for research. This is not present a high interest for research. This is not only because of the various applications in only because of the various applications in industry, but also due to some unsolved topics in industry, but also due to some unsolved topics in the theory of liquid wetting [3].
1.PHYSICAL DATA
the theory of liquid wetting [3].
Here we discuss the kinetics of the wetting angle of Here we discuss the kinetics of the wetting angle of water drop of water drop of deionizeddeionized water placed on a non water placed on a non wettiblewettiblesubstrate (substrate (hostaphanhostaphan). In the course of evaporation of ). In the course of evaporation of the drop, as the dropthe drop, as the drop’’s contact angle changes, we s contact angle changes, we measure the measure the frequency of appearance f(frequency of appearance f(θθ)) of such angles of such angles θθwithin prescribed angle intervals. One can say that in within prescribed angle intervals. One can say that in this way the this way the ““state spectrumstate spectrum”” with respect to the contact with respect to the contact (wetting) angle is obtained of the corresponding (wetting) angle is obtained of the corresponding thermodynamically open system.thermodynamically open system.
For this purpose one measures at regular time For this purpose one measures at regular time intervals (here every intervals (here every 22 minutes) the values for several minutes) the values for several drops (to enable drawing statistical conclusions). In this drops (to enable drawing statistical conclusions). In this way one obtains a set way one obtains a set of discrete values {f(of discrete values {f(θθi), i=1,2..} i), i=1,2..} ––the frequencies of occurrencethe frequencies of occurrence of of θθ..
One determines One determines θθ by by microscope observations using the optical microscope observations using the optical methodmethod of of AntonovAntonov [[44].].
Figure 1.Experimental setup of contact water angle measurementFigure 1.Experimental setup of contact water angle measurement--the the water drop and the substrate with the light beam passing trough water drop and the substrate with the light beam passing trough the the drop anddrop and producing the light patternproducing the light pattern
One measures the width of a light refraction One measures the width of a light refraction pattern pattern in the form of a in the form of a dark ring produced dark ring produced around the drop around the drop ((Fig.1Fig.1)) by light beamby light beams 1 s 1 passing passing near the boundary of near the boundary of ththee drop drop 2 which is 2 which is situatedsituated on on the non wetting folio 3the non wetting folio 3 and kept at a and kept at a constant room temperature.constant room temperature. The folio is situated The folio is situated on a glass plate 4 having a refraction index on a glass plate 4 having a refraction index nnand thickness and thickness dd. . The width The width aa is measured by is measured by microscope observations. The laws of geometric microscope observations. The laws of geometric optics give optics give tgtg((θθ)) as a function of as a function of aa byby the the formulaformula::tgtg((θθ)) =n/[(N=n/[(N22∆∆--nn22))1/21/2 --∆∆]]1/21/2
;; ∆∆=1+d=1+d22/(a/(a--δδ))22,,where where NN is the water refraction index and the is the water refraction index and the dimension dimension δδ denoted on. Fig. 1 is usually denoted on. Fig. 1 is usually neglected in the above formula since neglected in the above formula since δδ<<a<<a . .
Figure 2. Experimental probability (in %) of contact angle of deionizedtreated (squares) f1 and non treated f2 (circles) water data (with experimental errors )during the evaporationThe circles correspond to non treated water sample. The squares - to treated by γ - rays. The source of γ rays is Co- 60(65 krad/h). The period of treatment - 2 minutes.
0 10 20 30 40 50 60 70 80-5
0
5
10
15
20
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30
350 10 20 30 40 50 60 70 80
-5
0
5
10
15
20
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30
35
f 1 f 2
Prob
abili
ty, %
Contact angle, grad
2.Mathematical method. Generalized 2.Mathematical method. Generalized OPEM.OPEM.
)()/()(),( 2222 θσθσθ ∂∂+= fffS
Consider [5] [6] (as Bevington) the square of total uncertainty (total variance) ),(2 fS θ
associated with ),( fθ)2(
and their errors
)3( )],(),,([ fSffSf θθ +−
)4(
1. The first criterion - the fitting curve passes within the errors corridors (4) or (3).
2. The second criterion –the fitting curve minimizes the (5):
)](),([ ffff σσ +−
),(2/2)]()([12
ifiSifiapprfM
i θθθχ −=Σ=)5(
∑=
L
i ia1)()( θm
iP ∑=
L
ii
ic1θ= =
apprmf )( )(θ(6)
∑=
=M
k km
iPkwkfia1
)()( θ(7)
)1[(1)()(1 +−+=+ iim
iP µθγθ )()( θmiP ioδ−1(- ) )()(
1 θmiP − )()1( θ−m
imP+iν(8) ]
Ref .[7] and [8] for inherited errors in }{a and }{c .
i
L
iki
ki acc ∆=∆ ∑
=
))(( 2)(
where coefficients are defined explicitly in [7]. kic )(
2/1
1
2 )()(( apprkkkk
M
kii ffwPa −=∆ ∑
=
θ(9) ,
],[ σσ +− ff
Two criteria are used to select the optimum series length in Eq.(6) in the next steps:First, the fitting curve lies:i) inside the error corridor with
ii ) and after calculating derivatives the fitting curve has to lie inside big corridor with
2) Second, we extend the above algorithm OPEM to include the total version S in OPEM in two stages:
].,[ SfSf +−
)( fσ
)],( fS θ
( i). Here one neglects the errors in θ variable, i.e. the chi –squared (5) is minimized, using as weights
(ii) The chi- squared (5) is minimized, using as weights the
The preference is given to the first criterion and when it is satisfied, the search for the minimal chi-squared stops. The procedure is iterative and the result of the consequtiente k - th iteration, k>1, is called below the k-th approximation (see [7][8]) .
).,(/1 2 fS θ
).(/1 2 fσ
0 10 20 30 40 50 60 70 80
0
5
10
15
20
250 10 20 30 40 50 60 70 80
0
5
10
15
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25
prob
abili
ty,%
contact angle,grad
f 2 G
Figure 3. Figure 3. The OPEM approximation G by 11The OPEM approximation G by 11--th degree th degree polynomialspolynomials (stars) of contact angle probability for non stars) of contact angle probability for non treated treated deionizeddeionized water (circles)water (circles) ff22
3.Approximation results.(i) Non treated deionized water data.
0 10 20 30 40 50 60 70 80-5
0
5
10
15
20
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30
350 10 20 30 40 50 60 70 80
-5
0
5
10
15
20
25
30
35pr
obab
ility,
%
contact angle, grad
f 1 E F
Figure 4. The OPEM approximation E by 9-th (triangles) and 13-th degrees polynomials (rhombs) F of contact angle probability fortreated deionized water f1 (squares)
(ii) Treated deionized water data
Table 1: OPEM approximation of contact water angle-given and approximating values
No θ f σ( θ) σ(f ) f appr,9(ak) f appr,9(c
k) f appr,13(a
k)
1 7.5 0.1 0.6 0.001 0.01380 0.01381 0.104682 12.5 0.8 0.6 0.200 0.78957 0.78957 0.813783 17.5 1.2 0.6 0.250 1.23899 1.23927 1.193104 22.5 2.3 0.6 0.350 2.08552 2.08592 2.308285 27.5 3.6 0.6 0.400 3.80385 3.86199 3.559226 32.5 4.1 0.6 0.540 0.05056 4.04787 4.198977 37.5 2.5 0.6 0.320 0.34706 2.34459 2.430528 42.5 2.0 0.6 0.350 2.17360 2.16459 2.090219 47.5 6.6 0.6 0.860 6. 82522 6.79850 5.8362010 52.5 15.3 0.6 1.400 14.51524 14.49400 16.9840011 57.5 27.8 0.6 4.000 18.08124 18.02445 24.8242712 62.5 8.1 0.6 1.600 12.22944 12.12178 8.7655813 67.5 4.5 0.6 0.800 3.92818 3.71804 4.374181 4 72.5 9.0 0.6 1.100 9.91328 9.81663 9.1634315 77.5 10.1 0.6 1.600 9.23731 9.23587 9.10499
Conclusions Conclusions
The approximating results with optimal degrees of OPEM The approximating results with optimal degrees of OPEM orthonormalorthonormal polynomials for contact wetting angle founded polynomials for contact wetting angle founded by orthogonal and usual coefficients show good accuracy, by orthogonal and usual coefficients show good accuracy, demonstrated from Figures 3 and 4 and Table 1( 3demonstrated from Figures 3 and 4 and Table 1( 3rdrd iteration iteration step). The approximating curves are chosen to satisfy the step). The approximating curves are chosen to satisfy the proposed criteria (3),(4) and (5). We have received good proposed criteria (3),(4) and (5). We have received good descriptions of the angle variations useful for further descriptions of the angle variations useful for further physical and mathematical investigations. physical and mathematical investigations.
References:References:
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