R.-J. ROE,V. L. BACCHFJTA, AND P. M. G. WONG
4190
Refinement of Pendent Drop Method for the Measurement
of Surface Tension of Viscous Liquid
by Ryong=JoonRoe, Electrochemicals Department, E. I . du Poni de Nemours and Co., IC., Ezperimental Station, Wilmington, Delaware
V. L. Bacchetta, and P. M. G.Wong Engineering Departmeni, E . I . du Pont de Nemours and (Received December 9,1966)
co., IC., Wilmington, Delaware
The pendent drop method is suitable for measuring surface tension of viscous liquids. It is a static method measuring an equilibrium property; further, attainment of hydrodynamic equilibrium can be checked by comparing the drop shape with that predicted by theory. For the latter purpose, the diameters of the pendent drop are measured at several heights, instead of only two as previously proposed. Tables required for determination of 1/H from ratios of the drop diameters are constructed by numerical solution of the theoretical equation. By ascertaining the constancy of 1/H values obtained from different ratios, the attainment of equilibrium can be confirmed, and also the accuracy of the results can be improved. Errors accruing from using only one ratio to determine surface tension are analyzed.
The pendent drop method is suitable for measuring surface tension of viscous liquids. It is a static method, and hence the surface tension values obtained are not influenced by the viscous drag of the liquid. The approach to hydrodynamic equilibrium is more rapid here than in other methods, such as the sessile drop technique, since only a small fraction of the total surface area of the drop is in direct contact with a solid surface hindering movement of the viscous liquid. The shape of a pendent drop acted upon by gravitational and surface energy forces is governed exactly by a known differential equation.’ It is therefore possible that the attainment of hydrodynamic equilibrium can be checked by examining the shape of the drop and ascertaining its conformity to the differential equation. As shown first by Andreas, Hauser, and Tucker12 the surface tension value, y, can be calculated from the pendent drop profile by Y = BPde2/H
(1)
where g is the gravitation acceleration, p is the density The Journal of Physical Chemistry
of the liquid, de is the largest diameter of the drop, and H is a correction factor, which depends on the shape of the drop. The shape can be characterized, for example, by the ratio S defined by S = d,/de (2) where d, is the diameter measured on the horizontal plane at height equal to de from the vertex of the drop. Andreas, et al., obtained a table giving l / H as a function of S from experimental determination of the surface tension of water. Later, more accurate tables of 1,” vs. S were compiled by Fordham,3 Niederhauser and Bartell14 and Stauffer5 by numerical solution of the fundamental differential equation. (1) F.Bashforth and J. C. Adams, “An Attempt to Test the Theories of Capillary Action,” University Press, Cambridge, England, 1883. (2) J. M.Andreas, E. A. Hauser, and W. B. Tucker, J . Phys. Chem., 42, 1001 (1938). (3) S. Fordham, Proc. Roy. &e. (London), A194, 1 (1948). (4) D. 0. Niederhauser and F. E. Bartell, “Report of ProgressFundamental Research on the Occurrence and Recovery of Petroleum, 1948-1949,” American Petroleum Institute, Baltimore, M d . , 1950, p 114. (5) C. E. Stauffer, J . Phys. Chem., 69, 1933 (1965).
REFINEMENT OF PENDENT DROP METHOD FOR MEASURING SURFACE TENSION
The ratio S is not unique. Any other well-defined ratio of drop dimensions can be employed for the same purpose. Recently, WinkeP proposed the use of the ratio dmin/del where dminis the smallest diameter which may be observed above the de plane for a certain range of drop shapes. He reasoned that de and dmin can be measured independently, whereas d, is affected by the error in de through the selection of its plane of measurement 2, = d,. Although conceptually sound, this method is nevertheless impractical because of difficulties in forming a stable liquid drop having a minimum diameter. Even when such a drop is formed, the plane of the minimum diameter is often so near the capillary tip that its measurement is affected by irregularities in the capillary tip or in wetting of the tip by the liquid. I n the present work, we propose the use of not one but several characteristic ratios for determining drop shape. We define a series of diameters, d n (n = 8, . . . ,la), measured at heights 2, given by 2, = (n/lO)de (n = 8,. . .,12)
(3)
and the corresponding characteristic ratios
,Sn =
d,/de
(n = 8,. . . ,12)
(4)
Obviously dlo and Sloare identical with d, and S, respectively. A series of tables giving 1/H as a function of each S , have been constructed by numerical integration of the fundamental differential equation with a digital computer. When several S,’s are measured from a single photograph and the respective values of 1/H are determined, the values should be nearly identical. By taking the average of these 1/H values the accuracy can be improved substantially. If marked variations in 1/H occur, the measurements of de and d,’s should be reexamined. If no error in the measurements of diameters is found, it is probable that the pendent drop failed to reach hydrodynamic equilibrium and the photograph should be discarded. The method tests, in effect, the conformity of the drop shape to the differential equation at several selected points. The present method has been used to measure the surface tensions of a number of molten polymers and also the interfacial tensions between pairs of molten polymers. The results will be published elsewhere.’J
Method of Computation The numerical integration of the fundamental differential equation’ was carried out for 41 different values of p ranging from -0.20 to -0.60 in increments of 0.01. The integration was performed by means of the fourth-order Rungekutta-Gill method. Several different integration steps As, where s is the arc length in unit of b, were tried first for comparison of accuracy.
4191
The As value of 0.0025 was finally chosen for all calculations, since it gave the coordinates of drop contours agreeing with Fordham’s results3 to the fifth significant digit: The plane of the maximum diameter ( i e . , 2,) and the points ( X n )2,) (n = 8,. . .,12) were located by use of a “squeezing” algorithm, that is, by taking alternately positive and negative As values of successively smaller magnitudes around the desired point until the error was within 0.00001. Five tables giving 1/H as a function of S, (n = 8,. . . ,12) were then constructed by parabolic interpolation of the primary data as computed above. These tables are given in the Appendix, but as an example, part of the table listing 1/H us. SI2 is shown in Table I. The entries under the column head P indicate the degree of accuracy obtainable for l/H and are explained in the next section. The values of 1/H given against Sl0 agree well with those calculated by other Thus our values of 1/H in the range of Slofrom 0.625 to 0.669 agree with the corresponding values given by Stauffers within 4 units in the fifth decimal place. Further, our values agree with those of Niedenhauser and Bartel14within 4 units in the fifth place for SIObetween 0.67 and 0.72 and within 1 unit in the fifth place for SIobetween 0.73 and 0.97. For practical purposes, accuracy within one unit in the fourth decimal place is adequate, since the ultimate accuracy of the surface tension values is limited primarily by the propagation of measuring errors to be discussed in the next section.
Error Analysis It can be seen from eq 1 that the error in y can arise from three sources, i.e., p, de, and 1/H. The density p and the absolute value of de can readily be determined to an accuracy of 0.05% or better. Larger errors in 1,” can arise by accumulation of errors through the measurements of several drop dimensions. illoreover, the error in 1/H can depend not only on the accuracy to which individual dimensions can be measured, but also on the shape of the drop and also on the particular dimensions selected for determining 1/H (d8us. dlz,for example). Suppose that any linear dimension of the drop profile on a photograph is measured to equal accuracy with a probable error,s p . The probable error, Pelin the ob(G) D. Winkel,
J. Phys. Chem., 69, 348 (1965). (7) R.-J. Roe, ibid., 69, 2809 (1965). (8) R.-J.Roe, to be published: presented a t the International Symposium on Macromolecular Chemistry, Tokyo and Kyoto, Japan, Sept-Oct 1966. (9) A . G. Worthing and J. Geffner, ”Treatment of Experimental Data,” John Wiley and Sons, Inc., New York, N . Y.,1943, Chap. VI and IX.
Volume 7 1 , Number 13 December 1967
R . J . ROE,V. L. BACCHETTA, AND P. M. G. WONG
4192
Table I: Partial Listing of the Table Giving 1 / H vs. Sll 811
0
1
2
3
4
0.75 0.76 0.77 0.78 0.79
0.47719 0.46969 0.46234 0.45514 0.44808
0.47643 0.46895 0.46162 0.45443 0.44739
0.47568 0.46821 0.46089 0.45372 0.44669
0.47492 0.46747 0.46017 0.45301 0.44599
0.47417 0.46673 0,45944 0.45230 0.44530
0.47342 0.46000 0.45872 0.45159 0.44461
0.47267 0.46526 0.45800 0.45089 0.44392
0.47192 0.46453 0.45728 0.45019 0.44323
0.47118 0.46380 0.45657 0.44948 0.44254
0.47043 0,46307 0.45585 0,44878 0.44185
2.29 2.25 2.22 2.19 2.17
0.80 0.81 0.82 0.83 0.84
0.44117 0.43439 0.42775 0.42123 0.41485
0.44048 0.43372 0.42709 0.42059 0.41422
0.43980 0.43305 0.42643 0.41995 0.41359
0.43912 0.43238 0.42578 0.41931 0.41296
0.43844 0.43172 0.42513 0.41867 0.41233
0.43776 0.43105 0.42447 0.41803 0.41171
0.43708 0.43039 0.42382 0.41729 0.41108
0.43641 0.42973 0.42317 0.41675 0.41046
0.43573 0.42906 0.42253 0.41612 0.40984
0.43506 0.42840 0.42188 0.41549 0.40922
2.14 2.12 2.09 2.07 2.06
0.85 0.86 0.87 0.88 0.89
0.40859 0.40246 0.39645 0,39055 0,38477
0.40798 0.40185 0.39585 0.38997 0.38419
0.40736 0.40125 0.39526 0.38938 0.38362
0,40674 0.40064 0.39467 0,38880 0.38305
0.40613 0.40004 0.39407 0,38822 0,38248
0.40551 0.39944 0.39348 0,38764 0.38192
0.40490 0.39884 0.39289 0.38706 0.38135
0.40429 0.39824 0.39231 0.38649 0.38078
0.40368 0.39764 0.39172 0,38591 0.38022
0.40307 0,39704 0.39113 0.38534 0.37965
2.04 2.03 2.02 2.01 2.00
0.90 0.91 0.92 0.93 0.94
0.37909 0 37353 0.36807 0 36271 0 35744
0.37853 0.37293 0.36752 0.36281 0.35692
0.37797 0.37243 0,36698 0.36165 0.35640
0.37741 0.37188 0.36645 0.36112 0.35588
0,37685 0.37133 0.36591 0.36059 0.35537
0.37630 0.37078 0.36537 0.36006 0.35485
0.37574 0.37024 0.36484 0.35954 0.35433
0.37518 0.36969 0.36430 0.35901 0.35382
0.37463 0.36915 0.36377 0.35849 0.35331
0.37408 0.36861 0.36324 0.35797 0.35279
2.00 1.99 2.00 2.03 2.03
0.95 0.96 0.97 0.98 0.99
0.35228 0,34720 0.34222 0.33731 0.33249
0.35177 0,34670 0.34172 0.33683 0.33201
0.35126 0.34620 0.34123 0.33634 0.33153
0.35075 0.34570 0.34074 0.33586 0.33106
0.35024 0.34520 0.34025 0.33538 0.33058
0.34973 0.34470 0.33975 0.33489 0.33010
0.34922 0.34420 0.33926 0.33441 0.32963
0.34872 0.34370 0.33877 0.33393 0.32916
0.34821 0.34321 0.33829 0.33345 0.32868
0.34771 0.34271 0.33780 0.33297 0.32821
2.01 2.02 2.04 2.05 2.07
served value of de is then equal to p . The probable error, P,, in d, arises from two sources: the random error, Pnr,which is independent of Peland the systematic error, PnS,due to the bias by Pe in selecting the plane 2,. In order to calculate Pnr,first, let us assume initially that de is known exactly. The probable error in locating the plane 2, is equal to p , which causes an error of 2p(dX/dZ) in d,. Moreover, there will be a reading error of d, with magnitude equal to p . Combination of these two errors leads to Pa’ = p [ l
+ 4(dX/dZ)2]”’
(5)
Next, Pnsis given by 2p(n/lO)(dX/dZ), since the bias in selecting the plane is (n/lO)P, = (n/lO)p. The probable error, Ps, in the observed value of S, can also be split into two components. The random error, Par, which arises even when de is known exactly is given by Ps‘
= ”[l de
+ 4(g)2]1/2
The systematic error, PsB,which depends on the error in de is The Journal of Physical ChembtTy
8
7
6
6
9
P
Combination of eq 6 and 7 gives
pr
Ps = - - 1 + S n 2 + 4 de Finally, the probable error, PH,in the value of 1/H obtained from the tables is
PH
= Ps[d(l/H)/dS,I
(9)
The values in the last column of tables giving 1/H vs. S, are calculated by
P = [PH/ (1,”)
I/ bide 1
(10)
and thus represent the fractional error in 1/H as compared to the fractional error in de. I n other words, when the probable error in deis 1%,the value of P gives the probable percentage error in 1/H. I n Figure 1, P is plotted against 1/H, each curve representing the error arising when SS, Sa, etc., are used to calculate 1/H. Evidently, greatest accuracy is obtained when SI2 is
REFINEMENT OF PENDENT DROPMETHODFOR MEASURING SURFACE TENSION
determined from a drop having a shape corresponding to low 1/H values. Experimentally, measurements of d, within 0.1% are easily attainable, but when Sloalone is utilized for drops having 1/H above 0.8, the final accuracy in the y value is at most of the order of 1%.
s.75 IO
1.00 .95 .90 .85 B O
301
I
0.3
I
I
I 0.4
I
I
I
0.5
I
I
I
0.6 ,, 0.7
.70
.65
I
I
1
I
1
0.8
0.9
1.0
H ' Figure 1. The curves give the values of PIthe probable per cent error in l/Hl which can accrue when d, is measured to 1% accuracy. Note that P values vary markedly depending on the shape of the drop (which can be defined either by 6, 1/H, or Slo)and also on the different characteristic ratios used. Greatest accuracy is obtained when S, with TZ = 11 or 12 is used for a drop having smaller 1/H values.
The inverse square of P can be utilized as the weighting factor in averaging the 1/H values obtained from a single photograph but by way of several S,'s with different n.
4193
Explanation of Appendix Because of their bulk, the tables computed here are given in Appendix which is deposited with the Library of Congress.'O Brief explanation of these tables is given here. Table 1.1 lists the primary data obtained from computation of drop shape. The values of 1/H, X,, XS, Xg,XIO,Xll,and Xlz (all X ' s in units of b) are given for different values of 0 from -0.20 to -0.60 in increments of -0.01. Table 1.2 lists the S values derived from the above, Le., Ss, Ss,SlotSll,and SlZ,against the 41 different values of @. Tables 2.1 to 2.5 list the interpolated values of 1/H as a function of S , ( n = 8 to 12). The format is similar to that shown in Table I of the text. The last column gives the error ratio, P , corresponding to the value of S , on the far left on the same row. For example, in Table I of the text, P was calculated as 2.29 for Sl2of 0.750. Since the P values need not be known very accurately, the same value can be used for SIZbetween 0.750 to 0.759. In Table 2.5, 1/H values are tabulated for Slzvalues in increments of 0.001, and in Table 2.4 for Sll at 0.001 intervals. For Sa, Sg,and 810,smaller intervals are required if the total numbers of entries are to be comparable to Tables 2.4 and 2.5. For this reason, 1," values are tabulated against 6 X SS,3 X S9,and 2 X Sl0,with increments of the products 6 X Se, etc., being equal to 0.001, in Tables 2.1, 2.2, and 2.3, respectively. Thus in Table 2.3, the total number of entries is about twice that in similar published previously, and the need for interpolation is thus largely eliminated. In Table 3, the values of S8 to SI2are listed together as a function of 1/H a t intervals of 0.002 of the latter. This table is useful in cross-checking S , values obtained from a single photograph. (10) Tables have been deposited as Document No. 9668 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 24, D. C. 20540, from which a copy may be obtained by remitting, in advance, $2.50 for photoprints or $1.75 for 35-mm microfilm. Make checks or money orders payable to: Chief, Photoduplication Service, Library of Congress. A limited number of original copies are available upon request from R. J. R .
Volume 71. Number IS December 1967
