Alternative methods for the determination of surface tension using drop-weight data

NOTES Alternative Methods for the Determination of Surface Tension Using Drop-Weight Data 1

INTRODUCTION Ever since Harkins and Brown (1) compiled an empirical table of correction factors to account for the residual mass of drops failing from the tip of a capillary, the drop-weight method has been considered to be a very reliable method for the accurate determination of surface tension of liquids. The range of applicability of the correction factors has recently been extended (2, 3). However, there is considerable disagreement in the experimental data of mercury, wetting and nonwetting organic liquids, outside the experimental range of Harkins and Brown, warranting a cautious utility in the extended range. The necessity of interpolating the tabulated values in the original HarkinsBrown method has now been overcome by resorting to statistical analysis for obtaining the curves of best fit describing the drop-weight data (2, 4-6). Alternate methods of calculating the surface tension using drop-weight data have also been proposed by Iredele (7) based on Worthington's similarity criterion and by Wilkinson (2) using a relation between r/a and r/v~Is, where the notations have their usual meaning. In this paper, yet two other alternative methods are proposed and their relative merits with the earlier methods discussed.

Using the experimental data of Harkins and Brown (1) and of Wilkinson (2) for water, benzene, carbon tetrachloride, and ethylene dibromide, the values of Vd and Xo were calculated. The method of calculating the surface tension is by an iterative scheme which essentially consists of identifying that value of 7 which will satisfy the Vd against X0 relationship shown in Fig. 1. A computer program developed for this purpose using the method of Regula-Falsi interpolation, written in the Fortran language for the Dec-10 computer system, is available upon request. The initial guess value required for the iteration scheme can be calculated using the equation 27rr7 =

!)dApg.

Method 2. The iteration scheme required in the above method can be avoided by considering the relation between Vd and r/v~/3 instead of Vd and Xo. Figure 2 illustrates this continuously increasing functional dependence in the range 0.06 ~
[21

ALTERNATIVE METHODS RESULTS AND DISCUSSION

Method 1.

Under conditions of near-zero flow rate, considering the detached drop volume (v~)to be dependent on the surface or interfacial tension (3'), the density difference between the drop and the surrounding fluid (Ap), the acceleration due to gravity (g), and the capillary radius (r), there exists a correlation between the dimensionless detached drop volume (Vd) and the dimensionless capillary radius (Xo), where Vd=l)d(~)3]2;

Xo=r(~-gf/2.

While discussing the results of the regression analyses of various investigators, Wilkinson (2) considers the coefficient of variation (CV) defined below as the criterion for comparison:

(Yi -- YLF)2

CV=

[1]

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100

× --,

[3]

where, Yiis the experimental value, Ycvis the corresponding least square fitted value computed using a Kth order polynomial fitted onto the n data points, and )7 is the average value of Yi in the range considered. However, it does not indicate the error reflected in the calculated value of surface

' Presented at the 29th Congress of the Indian Society of Theoretical and Applied Mechanics, KREC Surathkal, India, Dec. 25-29, 1984.

551 0021-9797/87 $3.00 Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987

Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

552

NOTES

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tension. Since the error in 3' due to error in curve fitting differs for different methods of calculation adopted, the conclusions drawn by Wilkinson using the "CV" criterion are open to objection. Therefore, it is more meaningful to make comparisons of different methods based on the average percentage error (APE) in the calculated value of surface tension, arising due to error in curve fitting, all other sources of errors being common as long as the experimental data is the same in all the methods. Table I summarizes the formulae for the calculation of APE using different methods of calculation. Clearly, for any given value of CV, the APE would be least for the two methods proposed in this work, where 3" depends on V~2/s rather than the inverse dependence of 3, on M a and J~ in Wilkinson's method and Iredele's method, respectively, and the first order inverse dependence of 3"on q~in the Harkins-

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553

NOTES TABLE II S u m m a r y of the Regression Analysis of Drop-Weight Data by Earlier Investigators

1/27rdp(r/v~l/3)vS r/v~/3 1/27cdP(r/v~/3)vsr/v~/3 4~(r/v~/3)vsr/v~/3 ch(r/v~/3) vsr/v~/3 4~(r/v~/3)vs r/v~/3 (o(r/a) vs r/a r/a vs r/t~1/3 r/a vs r/v~/3 r/a vs r/v~/3 r/a vs r/v~/3 r/a vs r/v~/a r/a vs r/v~/3

Lando and Oaldey Strenge Wilkinson and Kidwell Wilkinson and Kidwell Wilkinson and Kidwell Wilkinson Wilkinson Wilkinson Wilkinson Wilkinson Wilkinson Wilkinson

n

K

cv (%)

APE (%)

0.3-1.2 0.3-1.2 0.3-1.2 0.3-1.2 0.65-0.95 0.3-1.6 0.0-1.04 0.064-1.04 0.064-1.14 0.064-1.285

58 52 24 57 48 82 90 100

2 3 2 2 2 3 4 4 4 4

0.42 0.22 0.38 0.20 0.065 0.74 0.181 0.203 0.258 0.73

0.42 0.22 0.38 0.20 0.065 0.74 0.361 0.405 0.514 1.444

0.064-1.589 0.308-1.034

102 44

4 4

1.45 0.157

2.838 0.313

Rangeof r/v~/3

Relationinvestigated

Investigator(s)

Brown method, where M = r/a; a 2 = 23,/Apg, J = ratio of the radii of the capillary tips from which identical drops of water and the experimental liquid form, respectively, and $ is the Harkins-Brown correction factor. For a fixed value of CV = 0.1%, APE calculated for the different methods is shown in column 5 of Table I. The surface tension value calculated using the drop-weight method is claimed to give an accuracy of better than 0.1%. In order to obtain an APE of 0.1% or less, the error tolerance in the CV values is given under the column heading M T E in Table I. The experimental drop-weight data reported by Harkins and Brown (1) and Wilkinson (2) has been the most reliable data, with an accuracy of better than 0.1% in the range 0.3 < r/v~/3 < 1.2. The regression analyses of this data by the earlier investigators are summarized in Table II. The usefulness of the parameter APE discussed earlier is ob-

vious. It is instructive to note that except for Wilkinson and Kidwell's analysis (6) in the range 0.65 < F/I)I]3 ~ 0.95, none of the reported results conform to the M T E criterion stated above. Further, Wilkinson's results based on his proposed alternate method (2) show the largest deviation from the M T E limit. Keeping in view the relative merits of the proposed methods based on the error analysis discussed above, the dependence of Vd on Xo and Vd on rlv~/3 were subjected to regression analysis to obtain the mathematical description of these relations. The details of the analyses are available elsewhere (8). The computed coefficients of the best fit polynomial to the above relations are summarized in Tables III and IV. The values of the parameters, maxi m u m percentage error in curve fitting (MPE), and the average percentage error in 3' due to error in curve fitting (APE) are also included in these tables. The deviation of

TABLE III Coetficients of the Best Fit Polynomial to the Curve Vd vs X0

Coefficientsof the polynomialVd= ~x]x Ai ~o-1 Rangeof r/v~/3

n

K

0.374-0.974 0.675-0.974 0.675--1.195

39 30 45

4 4 2

CV

(%)

0.270 0.109 0.251

MPE

(%)

1.93 0.31 0.57

APE

(%)

At

A2

A3

A,

A~

0.18 0.073 0.167

0.55746 0.73645 1.45449

3.28086 3.1172 1.80714

0.00087 0.00003 0.65476

0.00020 0 --

0.03330 0.04492 --

Journal of Colloid and InterfaceScience, Vol. 115,No. 2, February 1987

554

NOTES the least squares fitted value from the experimental value was plotted as a function of both X0 and r/l~ 1/3, and in no case was a systematic scatter observed. The results of Table III are suitably used in the computer program mentioned earlier. Also, the results of Table IV are incorporated in the generation of a table of Vd as a function of r/v~/~ in the range 0.064-1.195, in increments of 0.001, and is available upon request.

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I gratefully acknowledge the financial assistance received during the period of this investigation from the Department of Atomic Energy, Government of India. REFERENCES

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1. Harkins, W. D., and Brown, B. C., J. Amer. Chem. Soc. 41, 499 (1919). 2. Wilkinson, M. C., J. Colloid Interface Sci. 40, 14 (1972). 3. Wilkinson, M. C., and Aronson, M. P., J. Chem. Soc. Faraday Trans. 1, 474 (1973). 4. Lando, J. L., and Oakley, H. T., J. Colloid Interface Sci. 25, 152 (1967). 5. Strenge, K. H., J. Colloidlnterface Sci. 29, 732 (1969). 6. Wilkinson, M. C., and Kidwell, R. L., J. Colloid Interface Sci. 35, 114 (1971). 7. Iredele, T., Philos. Mag. 45, 1088 (1923). 8. Ramesh Babu, S., Ph.D. thesis. Ind. Inst. of Sci., Bangalore, India (1985). S. RAMESH

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Department of Metallurgy Indian Institute of Science Bangalore 560 012, India Received September 13, 1985

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Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987

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