Bashforth-Adams tables

The Bashforth-Adams tables provide an alternate way of evaluating 7 by observing the profile of a sessile drop of the liquid under investigation. If, after all, the drop profiles of Figure 6.15 can be drawn using 0 as a parameter, then it should also be possible to match an experimental drop profile with the (3 value that characterizes it. Equation (85) then relates 7 to 0 and other measurable quantities. This method is claimed to have an error of only 0.1%, but it is slow and tedious and hence not often the method of choice in practice. 

Once (3 is known for a particular profile, the Bashforth-Adams tables may be used further to evaluate b ... 

Several additional points might be noted about the use of the Bashforth-Adams tables to evaluate 7. If interpolation is necessary to arrive at the proper (3 value, then interpolation will also be necessary to determine (x/bl. . This results in some loss of accuracy. With pendant drops or sessile bubbles (i.e., negative /3 values), it is difficult to measure the maximum radius since the curvature is least along the equator of such drops (see Figure 6.15b). The Bashforth-Adams tables have been rearranged to facilitate their use for pendant drops. The interested reader will find tables adapted for pendant drops in the material by Padday (1969). The pendant drop method utilizes an equilibrium drop attached to a support and should not be confused with the drop weight method, which involves drop detachment. 

Several other methods for determining 7 —notably, the maximum bubble pressure, the drop weight, and the DuNouy ring methods (see Section 6.2) —all involve measurements on surfaces with axial symmetry. Although the Bashforth-Adams tables are pertinent to all of these, the data are generally tabulated in more practical forms that deemphasize the surface profile. 

Essentially, these steps are similar to those for capillary rise. For (i) the Bashforth-Adams tables and their modern variants can be used. Regarding (ii) laser-optical reflection techniques can nowadays yield profiles with great precision, so that accurate y values can be obtained. We shall now discuss some of the main features, leaving the numerous technical details to the specialized literature, except for noting that modern image analyses and automatization render steps (i) and (ii) less tedious, if not obviating parts of them. 

The method is very old, dating back to Simon in 1851. After further developments by Cantor ), Sugden ) investigated it thoroughly, thereby modifying the Bashforth-Adams tables to the case at hand, which is essentially a vemiant of [1.4.6]. At the bottom of the bubble R = R =b, as before, so there the capillary pressure is 2y/b. To this the hydrostatic pressure pgz must be added to obtain p. Here z is the vertical distance between the apex and the external liquid level. If the pressure is measured as a hydrostatic head h, it follows that... 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

J. F. Padday, Surface Tension, Part II. The Measurement of Surface Tension in Surface and Colloid Science, Vol. 1, E. Matijevic, Ed. Wiley-Interscience (1969), p. 101 (review of methods, contains some required tables) J.F. Padday, Surface Tension, Part III. Tables Relating the Size and Shape of Liquid Drops to the Surface Tension, ibid, p. 151. (Collation of Bashforth and Adams tables and extensions or modifications to make them more suitable for actual situations contains an introduction to explain the conversion of parameters in different geometries.)... 

Using the Bashforth and Adams tables. Porter (1933) calculated the difference A between and where r is the equatorial radius. The variation of A with h r could be... 

Equation (478) is the exact analytical geometry expression of capillary rise in a cylindrical tube having a circular cross section, which considers the deviation of the meniscus from sphericity, so that the curvature corresponds to (AP = A pgy) at each point on the meniscus, where y is the elevation of that point above the flat liquid level (y = z+h). Unfortunately, this relation cannot be solved analytically. Numerous approximate solutions have been offered, such as application of the Bashforth and Adams tables in 1883 (see Equation (476)) derivation of Equation (332) by Lord Rayleigh in 1915 a polynomial fit by Lane in 1973 (see Equation (482)) and other numerical methods using computers in modern times. 

Contact angles were calculated from the observations of the edges of the menisci in the two tubes, as follows With reference to Figure 1, the measured height difference is Ah. Since the radius of the small tube is much less than that of the large tube. Ah may be used as an approximation for h in Equation 2, and this allows a preliminary value of 6 to be calculated. This 6 then is used to obtain approximate values of z for each tube, using the Bashforth and Adams tables for the 1-mm. tubes and the approximation formulas given by Blaisdell [4] for the 10-mm. tubes. 

Dimensionless coordinates of sessile drop for (3 = 20 from Bashforth and Adams tables (1893)... 

For intermediate values of rja, or for tubes of intermediate size, no general formula has been given. Bashforth and Adams have however published tables from which the form of any capillary surface may be calculated, and with the aid of these Sugden has further calculated a table of values of rjh for all values of rja, between 0 and 6. h is here the radius of curvature at the crown... 

This relation can be obtained from the fundamental equation (Chap. I (4)) for the form of a liquid surface under gravity and surface tension. Unfortunately this equation cannot be solved in finite terms. Approximate solutions have been obtained in several ways which are outside the scope of this book. Sufficient account must, however, be given of the methods of Bashforth and Adams,1 to enable the reader to use their tables of numerical results, which are the most complete and accurate ever compiled. Some other important approximate formulae will also be given, for applications of the fundamental equation to special cases. 

Bashforth and Adams s tables give the values of x/b, z b, and F/63 for given values of and j8, both positive and negative. V is the volume included between the horizontal plane at the height z and the apex of the surface. 

Application to the capillary height method. Sugden1 has pointed out that, when the contact angle is zero, the ratio r/6 of the radius of the tube to the radius of the lowest point of the meniscus is the x/b of Bashforth and Adams s tables. By (6)... 

Water has a2 = 14 88 sq. mm. at room temperature hence the first part of the tables, which are as accurate as those of Bashforth and Adams, apply to tubes of radius up to 8 8 mm. For many organic liquids, which have d2 often about 5, the accuracy is the same up to about 5 mm. radius. These tubes are of course far wider than those generally used, except for the purpose of a reference surface with which the height in the narrower tube may be compared. 

As with the rise in the capillary tube, all such formulae can only be accurate for moderately narrow tubes. A more satisfactory plan has been adopted by Sugden,2 who applied Bashforth and Adams s tables as follows. 

When the pressure is a maximum, h is a maximum, and, by (17), IjX and rjX will be maxima hence a maximum value of rjX corresponds to the maximum pressure in the bubble. Only two of the fractions in equations (18) and (19) are independent. Sugden has taken various values of ft for each of a series of values r/a, and calculated the corresponding values of rjb from (19). Using Bashforth and Adams s tables, he found the values of and zjb for these values of rjb and / thence by (18) the values of rjX were determined. Examination of these showed what are the maximum values of r/X for each value of r/a, and a table was constructed it is reproduced by permission from the Journal of the Chemical Society. 

Bashforth and Adams generated tables of ft and x / z for cp = 90° as well as tables of the ratios x/b and z/b for differing values of b and (p. By measurement of x90 and Z90 at cp = 90°, b and / can be determined from these tables and the liquid surface energy can then be derived from equation (3.8) with an accuracy of about 2% for / > 2 if the droplet coordinates are measured with an accuracy better than 0.1% (Sangiorgi et al. 1982). A method is presented in Appendix D allowing calculation of the mass of a droplet for an optimised [Pg.121]

The cu-chetype numerical tabulations of x(z) profiles date back to Bashforth and Adams (almost to 1853 )). These tables were used for more than a century, either in their original form or after modification, until their significance waned with the advent of modem computers. The tables do not only apply to liquid menisci in capillaries, but also to cross-sections of sessile drops, pendent drops, etc. They cdso contain such information as the diameters and heights of sessile drops and contact angles. They give x/b and z/b as a function of 0 for various closely-spaced P values. Their application requires successive approximation because P can only be established if y is known. A starting Vcilue of y could, for instance, be obtained from one of the simpler equations, say from [1.3.2 or 6). 

F. Bashforth, J.C. Adams, An Attempt to Test the Theories of Capillary Action. Cambridge Unlv. Press, Cambridge (UK) (1883). (Extended tables giving the profile x(z) in eq. [ 1.3.14] as a function of (j) for various closely spaced values of p. Boucher et al. noted that these tables Involved painstaking efforts of C. Powalky. In addition, several students, working with hand-driven calculators, contributed. The computations were completed as early as 1853.)... 

D.W.G. White, A Supplement to the Tables of Bashforth and Adams, Queen s Printers, Ottawa, Canada (1967). 

The parameter fi is positive for oblate figures of revolution, i.e. for the meniscus in a capillary, a sessile drop, and a bubble under a plate, and is negative for prolate figures, i.e. for a pendant drop or an adjacent bubble. Bashforth and Adams (1883) reported their results as tables. For more detailed information, see for example in Adamson (1967). 

The capillary rise for the interface between these metals with a tube 1.0 cm. in diameter and a contact angle of 0° was calculated as about 0.0064 mm., and so is negligible for our purposes. When the contact angle is other than zero, the rise is even smaller. In a tube with radius 0.5 mm., calculations using the tables of Bashforth and Adams—i.e.. Equation 3—showed that for this system Equation 2 yielded a result that was at most in error by 7% for a 0° contact angle. If the tube is smaller or the contact angle is not 0°,the error is less than 7%. Since the measurement error was of the order of a few per cent. Equation 2 could be taken as a satisfactory approximation to Equation 4, for the difference in capillary rise. 

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