- Email: [email protected]
Software package for determination of surface tension of liquid metals
JoulTlal of Non-CrystallineSolids 156-158 (1993)845-848 North-Holland
~
IOORNA L OF
I~
Software package for determination of surface tension of liquid metals A.S. Krylov a, A.V. V v e d e n s k y a, A . M . K a t s n e l s o n b a n d A . E . T u g o v i k o v b o f Computational Mathematics and Cybernetics, Moscow University, Leninskye Hory, 119899 Moscow, Russian Federation hA.4. Baikov Institute of Metallurgy, Russian Academy of Science, Leninskii prospect, 49, 117334 Moscow, Russian Federation a Faculty
The problems of the development of algorithms to process experimental results obtained by the sessile drop method are discussed. Algorithms are based on the numerical solution of the Young-Laplace equation and the regularization method for Dorsey formula. A software package for IBM PC AT has been developed.The results of calculations for droplets formed in a cup (liquid Fe) and on a plate (slag CaO-B203) are given.
1. Introduction
2. Basic equations
The sessile drop method is one of the most widely used methods to determine the surface tension of liquid metals [1,2]. The droplet is placed onto a horizontal plate or in a special cup and then the meridian section of this droplet profile is studied; the droplet surface is assumed to have rotational symmetry. The sessile drop method allows one to determine the capillary constant of melt, its density and temperature dependence. There are some methods for processing the data obtained by the sessile drop method ([3-6], etc.). At the same time, all available methods do not permit one to make calculations in some significant cases and to compare the results based on different methods of calculation. In the present paper, we describe algorithms forming the basis of a program package allowing one to make calculations using different methods for processing the experimental data even in previously difficult cases.
The main equation for the sessile drop method is the Young-Laplace equation describing the surface of a droplet on a horizontal plate: 1 --+
1
H-x - - +
R1
R2
a2
2
R-oo'
(1)
where R~ and R 2 are radii of the surface curvature, a 2 =O'//[(pl--p2)g] is the capillary constant, or is the surface tension in the boundary of media of revolution with Px and P2 densities, g is free fall acceleration, R 0 is curvature radius at the droplet top and, H is droplet height. For surfaces with rotational symmetry of revolution relative to the vertical axis, the droplet shape is described by a function y ( x ) satisfying the following equation: y" +
(1 + y'2) 3/2
1
H-x
----+ y(1 + y ' 2 ) 1/2 az
2 Ro"
(2) Correspondence to: Dr A.S. Krylov,Department of Mathemat-
ical Physics, Faculty of Computational Mathematics and Cybernetics, Moscow University, Leninskye Hory, 119899 Moscow, Russian Federation. Tel:+ 7-095 939 5336. Telefax: + 7-095 939 0126. E-mail: [email protected].
The values for y(x i) (droplet cross-sections at height x i from the plate ( i = l - N ) ) are obtained in experiments. Therefore, one has to solve an inverse problem to determine the unknown
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
A.S. Krylov et al. / Determination of surface tension of liquid metals
846
parameters a, R 0 and H in eq. (1) according to the experimental data. Since this equation may not be integrated analytically, it is necessary either to use approximate empiric dependencies of parameters on the shape of the droplet ('geometric methods') or to develop methods based on a numerical solution of eq. (1). Besides distinctive disadvantages of the first method, such as limited accuracy and applicability to a small range of droplet shapes, this method is difficult to use in automatic data processing due to certain difficulties associated with different geometric constructions, for example, the tangent. Therefore it is necessary to interpolate and differentiate grid functions, and this is an ill-posed problem. One of the best methods to determine the capillary constant using approximate formulas is the method suggested by Dorsey [1]. Using this method, the coordinates of two points on the curve y(x) at y ' = 0 and y ' = 1 must be found. The problem of numerical differentiation of the network function y ( x i) is solved using a regularization algorithm [7], based on the approximation of the cross-section of the droplet by an ellipse and then distributing the difference according to the system of eigenfunctions of operator A'A, where Ax = fotX(s ) ds=y(t),
t [0,
--3C
=
A: L2[0,
The function describing the droplet shape determined by the regularization algorithm is used for the determination of the density and surface tension of liquid. To determine the physico-chemical properties of liquid metals from experimental results, a method based on numerical solution of the equation describing the droplet shape has been developed. To solve eq. (1), the parameterization of the curve x(t), y(t) is carried out assuming ~2 + :92 = 1. Then the system of equations is .~ = P
with initial conditions x(0)= H, y(0)= 0, ~(0)= 0, 3~(0)= 1, and a supplementary condition ~/Y It=0 = 1/Ro resulting from the curvature radius, R0, at the top of the droplet and ensuring the uniqueness of the solution. To determine the unknown parameters a, R 0 and H by solving the system of differential equations (3), the difference between the experimental points and the curve drawn by numerically integrating the system is minimized. This discrepancy is calculated as a sum of the squares of the distance from each experimental point to the curve. To minimize the difference, the NewtonGauss method is used. After the minimum has been found with the required accuracy, physicochemical properties of the metal forming the droplet are determined. The capillarly constant, a 2, and droplet volume, v, must be known. The droplet volume is determined by numerical integration of the equation t) 'II'Xy2. On the basis of these algorithms, a software package has been developed permitting one to automatize the processing of experimental data including a suitable interface. The software package provides the possibility of using files containing experimental data and calculated results. Graphic visualization of the results of mathematical processing allows one to estimate the quality of the experimental data and the reliability of the calculated results.
y +
a2
Ot2
Y
;
(3)
3. Results
3.1. Liquid Fe Using the developed program package, some measurements were analyzed by the method of a large droplet of liquid iron (data of ref. [8]). The droplet is formed in a cup of AIzO 3. Figure 1 shows the difference at each experimental point for one of the experiments. The discrepancy at one of the points (anomalous point) is higher than for all others. Using the Young-Laplace method, we obtained the following values: 6732 (density, kg/m3), 1745.45 (surface tension, m J / m2). If the anomalous point is not taken into account, results are 6740 (density) and 1743.85
A.S. Krylov et al. / Determination of surface tension of liquid metals
847
1350 -
m
~'1500
~
"
~
/
E
.'~ 1250
"~ 1 2 0 0
0
"~u~1150
~, 4
1100 .... i .... i .... I .... i .... , .... i 1450 1500 1550 1600 1650 1700 1750 t, C
Fig. 1. Relative difference between the measured profile of the droplet and its idealized shape (in arbitrary units).
(surface tension). Using the Dorsey method, for the same data we have obtained the following results: 6720 and 1674.61 (for density and surface tension, respectively, using the anomalous point), and 6730 and 1730.52 (density and surface tension, respectively, without this point). Experimental results obtained by the method of a large droplet [8] were processed for Fe, Fe-S, Fe-Cu, and F e - C u - S alloys. For some systems, the direct solution of the Young-Laplace equation yields surface tension polytherms (see, for example, fig. 2) characterized by the absence of hysteresis compared to the result based on the Dorsey formula.
3.2. Liquid slag C a O - B 2 0 3 The methods developed here were also used to calculate the surface tension and the wetting angle on a fiat plate. In this case, the principal difficulties occur due to the spreading of the droplet over the substrate. However, such information is needed, for example, to calculate the interphase tension, trs_1. Table 1 and fig. 3 give the measurement results for the profile of a slag droplet (CaO-BaO 3) formed on a fiat iron plate. The measurement results for the droplet profile in the range adjacent to the plate (points = 1-10) are characterized by a high error of measurements. Therefore, in the calculations, the
Fig 2. Surface tension of Fe-0.10% [S] melt. 1, 3, heating; 2, 4, cooling; 1, 2, calculation after Dorsey formula; 3, 4, solution of Young-Laplace equation.
agreement of the results obtained using the Young-Laplace equation and the regularized Dorsey method was used as a criterion for selecting datapoints. Table 2 gives the results calculated by the two methods depending on a set of experimental points under processing. The results of calculations for set 4 are given in fig. 3. These results agree with those obtained in ref. [9] for the surface tension or, of a 35% consisting CaO and 65% B20 3 (245 m J / m 2 at 1263 K). At the same time, it is to be noted that the calculation in any 'narrow' range of the droplet Table 1 31% CAO-69% B20 3 sessile drop measurements (T = 1263 K, m = 0.155 g) No. of points
Height (mm)
Width/2 (mm)
No. of points
Height (mm)
Width/2 (mm)
1 2 3 4 5 6 7 8 9 10 11 12
0.00 0.11 0.22 0.32 0.43 0.54 0.65 0.75 0.86 0.97 1.08 1.18
3.55 3.53 3.52 3.49 3.45 3.41 3.37 3.32 3.27 3.19 3.10 3.02
13 14 15 16 17 18 19 20 21 22 23 24
1.29 1.40 1.51 1.61 1.72 1.83 1.94 2.04 2.15 2.26 2.37 2.47
2.93 2.82 2.72 2.62 2.49 2.35 2.16 2.00 1.79 1.51 1.22 0.78
A.S. Krylov et al. / Determination of surface tension of liquid metals
848 0.30
4. Conclusions
~'o 0.20
8 C 0
~6 0.10 -6 *d 0.00
o.oo
0. o
0.20
0.30
o.;o
Horizontol distonce (crn)
Fig. 3. Experimental and calculated droplet shape. • and o, selected and eliminated experimental points, respectively.
profile (points 15-24, fig. 4) does not allow one to obtain reliable results in calculating by the Dorsey method. In this case, the Young-Laplace method is preferable.
The calculations carried out for some metal and slag systems have shown that: (i) using the conventional Dorsey method, it is impossible to determine the properties of a liquid reliably, because a small error in the droplet shape causes a large error in the values of the properties of the liquid. A regularization algorithm for the Dorsey method permits one to improve the reliability of calculations but, in this case, the results must be properly analyzed. (ii) the method based on the solution of Young-Laplace equation ensures perfect extrapolating properties, allowing one to determine the characteristics of the liquid even using incomplete imformation about the shape of the droplet. (iii) Using both methods simultaneously, the most effective algorithm for processing experimental data may be constructed.
0.30
References Eo 0.20 C 0
~a 0.10 -8 .o_ >
i 0.00
2 /
0.00
0.10 0.20 0.30 Horizontal distance (crn)
0.40
Fig. 4. Experimental and calculated droplet shape. • and o, selected and eliminated experimental points, respectively; 1, calculation using the Dorsey formula; 2, solution of the Young-Laplace equation. Table 2 Results of calculations by the Young-Laplace (1) and Dorsey (2) methods Points
p (kg3/m)
o- ( m J / m 2)
Wetting angle
1
2
1
2
1
2
1-24 1,2,5-24 1,2,7-24 1,2,10-24
2444 2446 2448 2456
2449 2450 2452 2458
220 219 221 231
205 208 213 230
80 81 81 81
81 81 81 80
[1] V.N. Eremenko, ed., Fizicheskaya khimiya neorganicheskikh materialov (Physical Chemistry of Inorganic Materials) Vol. 2 (Naukova Dumka, Kiev, 1988) (in Russian). [2] V.M. Glazov, M. Wobst and V.I. Timoshenko, Metody issledovaniya svoistv zhidkikh metallov (Methods of Investigations of the Properties of Liquid Metals) (Metallurgiya, Moscow, 1989) (in Russian). [3] J.N. Butler and B.H. Bloom, Surf. Sci. 4 (1966) 1. [4] C. Maze and G. Burnet, Surf. Sci. 13 (1969) 451. [5] Y. Rotenberg, L. Boruvka and A.W. Neumann, J. Colloid Interf. Sci. 93 (1983) 169. [6] I. Jimbo and A.W. Cramb, in: Proc. 6 Int. Iron and Steel Congr., Vol. 1 (ISIJ, Tokyo, 1990) p. 499. [7] A.M. Denisov and A.S. Krylov, Comput. Math. and Model. 1 (1990) 137. [8] K.S. Filippov, in: Stroeniye i svoistva metallicheskikh i shlakovykh rasplavov II (The Structure and Properties of Metallic and Slag Melts, Vol. 2) ed. V.P. Beskachko, ed. (CPI, Chelyabinsk, 1990) p. 155 (in Russian). [9] R.D. Walker and T.A.T. Fray, in: Proc. Int. Symp. on Advances in Extractive Metallurgy and Refining (The Institute of Metals, London, 1977) p. 1319.
~
IOORNA L OF
I~
Software package for determination of surface tension of liquid metals A.S. Krylov a, A.V. V v e d e n s k y a, A . M . K a t s n e l s o n b a n d A . E . T u g o v i k o v b o f Computational Mathematics and Cybernetics, Moscow University, Leninskye Hory, 119899 Moscow, Russian Federation hA.4. Baikov Institute of Metallurgy, Russian Academy of Science, Leninskii prospect, 49, 117334 Moscow, Russian Federation a Faculty
The problems of the development of algorithms to process experimental results obtained by the sessile drop method are discussed. Algorithms are based on the numerical solution of the Young-Laplace equation and the regularization method for Dorsey formula. A software package for IBM PC AT has been developed.The results of calculations for droplets formed in a cup (liquid Fe) and on a plate (slag CaO-B203) are given.
1. Introduction
2. Basic equations
The sessile drop method is one of the most widely used methods to determine the surface tension of liquid metals [1,2]. The droplet is placed onto a horizontal plate or in a special cup and then the meridian section of this droplet profile is studied; the droplet surface is assumed to have rotational symmetry. The sessile drop method allows one to determine the capillary constant of melt, its density and temperature dependence. There are some methods for processing the data obtained by the sessile drop method ([3-6], etc.). At the same time, all available methods do not permit one to make calculations in some significant cases and to compare the results based on different methods of calculation. In the present paper, we describe algorithms forming the basis of a program package allowing one to make calculations using different methods for processing the experimental data even in previously difficult cases.
The main equation for the sessile drop method is the Young-Laplace equation describing the surface of a droplet on a horizontal plate: 1 --+
1
H-x - - +
R1
R2
a2
2
R-oo'
(1)
where R~ and R 2 are radii of the surface curvature, a 2 =O'//[(pl--p2)g] is the capillary constant, or is the surface tension in the boundary of media of revolution with Px and P2 densities, g is free fall acceleration, R 0 is curvature radius at the droplet top and, H is droplet height. For surfaces with rotational symmetry of revolution relative to the vertical axis, the droplet shape is described by a function y ( x ) satisfying the following equation: y" +
(1 + y'2) 3/2
1
H-x
----+ y(1 + y ' 2 ) 1/2 az
2 Ro"
(2) Correspondence to: Dr A.S. Krylov,Department of Mathemat-
ical Physics, Faculty of Computational Mathematics and Cybernetics, Moscow University, Leninskye Hory, 119899 Moscow, Russian Federation. Tel:+ 7-095 939 5336. Telefax: + 7-095 939 0126. E-mail: [email protected].
The values for y(x i) (droplet cross-sections at height x i from the plate ( i = l - N ) ) are obtained in experiments. Therefore, one has to solve an inverse problem to determine the unknown
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
A.S. Krylov et al. / Determination of surface tension of liquid metals
846
parameters a, R 0 and H in eq. (1) according to the experimental data. Since this equation may not be integrated analytically, it is necessary either to use approximate empiric dependencies of parameters on the shape of the droplet ('geometric methods') or to develop methods based on a numerical solution of eq. (1). Besides distinctive disadvantages of the first method, such as limited accuracy and applicability to a small range of droplet shapes, this method is difficult to use in automatic data processing due to certain difficulties associated with different geometric constructions, for example, the tangent. Therefore it is necessary to interpolate and differentiate grid functions, and this is an ill-posed problem. One of the best methods to determine the capillary constant using approximate formulas is the method suggested by Dorsey [1]. Using this method, the coordinates of two points on the curve y(x) at y ' = 0 and y ' = 1 must be found. The problem of numerical differentiation of the network function y ( x i) is solved using a regularization algorithm [7], based on the approximation of the cross-section of the droplet by an ellipse and then distributing the difference according to the system of eigenfunctions of operator A'A, where Ax = fotX(s ) ds=y(t),
t [0,
--3C
=
A: L2[0,
The function describing the droplet shape determined by the regularization algorithm is used for the determination of the density and surface tension of liquid. To determine the physico-chemical properties of liquid metals from experimental results, a method based on numerical solution of the equation describing the droplet shape has been developed. To solve eq. (1), the parameterization of the curve x(t), y(t) is carried out assuming ~2 + :92 = 1. Then the system of equations is .~ = P
with initial conditions x(0)= H, y(0)= 0, ~(0)= 0, 3~(0)= 1, and a supplementary condition ~/Y It=0 = 1/Ro resulting from the curvature radius, R0, at the top of the droplet and ensuring the uniqueness of the solution. To determine the unknown parameters a, R 0 and H by solving the system of differential equations (3), the difference between the experimental points and the curve drawn by numerically integrating the system is minimized. This discrepancy is calculated as a sum of the squares of the distance from each experimental point to the curve. To minimize the difference, the NewtonGauss method is used. After the minimum has been found with the required accuracy, physicochemical properties of the metal forming the droplet are determined. The capillarly constant, a 2, and droplet volume, v, must be known. The droplet volume is determined by numerical integration of the equation t) 'II'Xy2. On the basis of these algorithms, a software package has been developed permitting one to automatize the processing of experimental data including a suitable interface. The software package provides the possibility of using files containing experimental data and calculated results. Graphic visualization of the results of mathematical processing allows one to estimate the quality of the experimental data and the reliability of the calculated results.
y +
a2
Ot2
Y
;
(3)
3. Results
3.1. Liquid Fe Using the developed program package, some measurements were analyzed by the method of a large droplet of liquid iron (data of ref. [8]). The droplet is formed in a cup of AIzO 3. Figure 1 shows the difference at each experimental point for one of the experiments. The discrepancy at one of the points (anomalous point) is higher than for all others. Using the Young-Laplace method, we obtained the following values: 6732 (density, kg/m3), 1745.45 (surface tension, m J / m2). If the anomalous point is not taken into account, results are 6740 (density) and 1743.85
A.S. Krylov et al. / Determination of surface tension of liquid metals
847
1350 -
m
~'1500
~
"
~
/
E
.'~ 1250
"~ 1 2 0 0
0
"~u~1150
~, 4
1100 .... i .... i .... I .... i .... , .... i 1450 1500 1550 1600 1650 1700 1750 t, C
Fig. 1. Relative difference between the measured profile of the droplet and its idealized shape (in arbitrary units).
(surface tension). Using the Dorsey method, for the same data we have obtained the following results: 6720 and 1674.61 (for density and surface tension, respectively, using the anomalous point), and 6730 and 1730.52 (density and surface tension, respectively, without this point). Experimental results obtained by the method of a large droplet [8] were processed for Fe, Fe-S, Fe-Cu, and F e - C u - S alloys. For some systems, the direct solution of the Young-Laplace equation yields surface tension polytherms (see, for example, fig. 2) characterized by the absence of hysteresis compared to the result based on the Dorsey formula.
3.2. Liquid slag C a O - B 2 0 3 The methods developed here were also used to calculate the surface tension and the wetting angle on a fiat plate. In this case, the principal difficulties occur due to the spreading of the droplet over the substrate. However, such information is needed, for example, to calculate the interphase tension, trs_1. Table 1 and fig. 3 give the measurement results for the profile of a slag droplet (CaO-BaO 3) formed on a fiat iron plate. The measurement results for the droplet profile in the range adjacent to the plate (points = 1-10) are characterized by a high error of measurements. Therefore, in the calculations, the
Fig 2. Surface tension of Fe-0.10% [S] melt. 1, 3, heating; 2, 4, cooling; 1, 2, calculation after Dorsey formula; 3, 4, solution of Young-Laplace equation.
agreement of the results obtained using the Young-Laplace equation and the regularized Dorsey method was used as a criterion for selecting datapoints. Table 2 gives the results calculated by the two methods depending on a set of experimental points under processing. The results of calculations for set 4 are given in fig. 3. These results agree with those obtained in ref. [9] for the surface tension or, of a 35% consisting CaO and 65% B20 3 (245 m J / m 2 at 1263 K). At the same time, it is to be noted that the calculation in any 'narrow' range of the droplet Table 1 31% CAO-69% B20 3 sessile drop measurements (T = 1263 K, m = 0.155 g) No. of points
Height (mm)
Width/2 (mm)
No. of points
Height (mm)
Width/2 (mm)
1 2 3 4 5 6 7 8 9 10 11 12
0.00 0.11 0.22 0.32 0.43 0.54 0.65 0.75 0.86 0.97 1.08 1.18
3.55 3.53 3.52 3.49 3.45 3.41 3.37 3.32 3.27 3.19 3.10 3.02
13 14 15 16 17 18 19 20 21 22 23 24
1.29 1.40 1.51 1.61 1.72 1.83 1.94 2.04 2.15 2.26 2.37 2.47
2.93 2.82 2.72 2.62 2.49 2.35 2.16 2.00 1.79 1.51 1.22 0.78
A.S. Krylov et al. / Determination of surface tension of liquid metals
848 0.30
4. Conclusions
~'o 0.20
8 C 0
~6 0.10 -6 *d 0.00
o.oo
0. o
0.20
0.30
o.;o
Horizontol distonce (crn)
Fig. 3. Experimental and calculated droplet shape. • and o, selected and eliminated experimental points, respectively.
profile (points 15-24, fig. 4) does not allow one to obtain reliable results in calculating by the Dorsey method. In this case, the Young-Laplace method is preferable.
The calculations carried out for some metal and slag systems have shown that: (i) using the conventional Dorsey method, it is impossible to determine the properties of a liquid reliably, because a small error in the droplet shape causes a large error in the values of the properties of the liquid. A regularization algorithm for the Dorsey method permits one to improve the reliability of calculations but, in this case, the results must be properly analyzed. (ii) the method based on the solution of Young-Laplace equation ensures perfect extrapolating properties, allowing one to determine the characteristics of the liquid even using incomplete imformation about the shape of the droplet. (iii) Using both methods simultaneously, the most effective algorithm for processing experimental data may be constructed.
0.30
References Eo 0.20 C 0
~a 0.10 -8 .o_ >
i 0.00
2 /
0.00
0.10 0.20 0.30 Horizontal distance (crn)
0.40
Fig. 4. Experimental and calculated droplet shape. • and o, selected and eliminated experimental points, respectively; 1, calculation using the Dorsey formula; 2, solution of the Young-Laplace equation. Table 2 Results of calculations by the Young-Laplace (1) and Dorsey (2) methods Points
p (kg3/m)
o- ( m J / m 2)
Wetting angle
1
2
1
2
1
2
1-24 1,2,5-24 1,2,7-24 1,2,10-24
2444 2446 2448 2456
2449 2450 2452 2458
220 219 221 231
205 208 213 230
80 81 81 81
81 81 81 80
[1] V.N. Eremenko, ed., Fizicheskaya khimiya neorganicheskikh materialov (Physical Chemistry of Inorganic Materials) Vol. 2 (Naukova Dumka, Kiev, 1988) (in Russian). [2] V.M. Glazov, M. Wobst and V.I. Timoshenko, Metody issledovaniya svoistv zhidkikh metallov (Methods of Investigations of the Properties of Liquid Metals) (Metallurgiya, Moscow, 1989) (in Russian). [3] J.N. Butler and B.H. Bloom, Surf. Sci. 4 (1966) 1. [4] C. Maze and G. Burnet, Surf. Sci. 13 (1969) 451. [5] Y. Rotenberg, L. Boruvka and A.W. Neumann, J. Colloid Interf. Sci. 93 (1983) 169. [6] I. Jimbo and A.W. Cramb, in: Proc. 6 Int. Iron and Steel Congr., Vol. 1 (ISIJ, Tokyo, 1990) p. 499. [7] A.M. Denisov and A.S. Krylov, Comput. Math. and Model. 1 (1990) 137. [8] K.S. Filippov, in: Stroeniye i svoistva metallicheskikh i shlakovykh rasplavov II (The Structure and Properties of Metallic and Slag Melts, Vol. 2) ed. V.P. Beskachko, ed. (CPI, Chelyabinsk, 1990) p. 155 (in Russian). [9] R.D. Walker and T.A.T. Fray, in: Proc. Int. Symp. on Advances in Extractive Metallurgy and Refining (The Institute of Metals, London, 1977) p. 1319.