Langmuir 1991, 7, 422-429
422
Immersion Depth Independent Computer Analysis of Wilhelmy Balance Hysteresis Curves David A. Martin and Erwin A. Vogler’ Becton Dickinson Research Center, P.O. Box 12016, Research Triangle Park, North Carolina 27709-2016 Received April 20, 1990.I n Final Form: August 6, 1990 Computerized methods and algorithms have been developed for operator-independent, batch processing of data from regular Wilhelmy balance hysteresis curves. Regular hysteresis curves were generated by measuring wetting forces along the perimeter of an object as it was immersed into or withdrawn from a liquid phase of interest by initially contacting liquid, repeating an arbitrary number of advancing and receding cycles, and terminating in a final removal step. Data were analyzed in sequential point-number space rather than in conventional immersion-depth coordinates to eliminate need for and relianceon a separately calibrated displacement transducer. Two methods of locatingturning points were compared and statistics of wetting measurements discussed. Good agreement between computerized and manual methods was obtained for advancing and receding wetting tensions of water on siliconized glass slides and rods.
Introduction Accuracy and precision of the Wilhelmy balance can be greatly improved through computer control and data acquisition. Recent reports describe balance automation1I2 but there are few reports describing computerized analysis of hysteresis curve^.^ Whereas instrument automation is of particular interest to the computer/microcomputer application specialist, analysis of wetting data is of more general interest to the colloid, polymer surface, and biomaterials science community. Fully automatic analyses present a number of technical challenges because hysteresis curves are quite complex with sharp changes in wetting forces between advancing and receding cycles. Moreover, any number of valid protocols can be employed including an arbitrary number of repeat cycles and sample immersions. Commercial software, for example that of Cahn, I ~ c .resort , ~ to interactive graphics routines that require an operator to select portions of the force-immersion curve for analysis by “marking” liquid contact points and buoyancy slopes with a cursor. Although these approaches are somewhat of an advance over strictly graphical analyses, labor-intensiveuser input is still required, which can be both subjective in nature and too repetitive for routine application. As part of a program to fully automate a Wilhelmy balance, we have developed computer routines for operatorindependent, batch processing of data from regular hysteresis curves. Regular hysteresis curves are herein defined as those generated by measuring wetting forces along the perimeter of an object as it is immersed into or withdrawn from a liquid phase of interest by initially contacting liquid, repeating an arbitrary number of advancing and receding cycles, and terminating in a final removal step. Data are analyzed in sequential pointnumber space rather than in conventional immersion depth coordinates in order to eliminate need for and reliance on a separately calibrated displacement transducer. Elimination of a displacement transducer significantly simplifies complete automation of a Wilhelmy balance. (1)Valero, J. L.; Prieto, A,; Lloris, A.; Morales, J.; Olivares, G. Microcomput. Appl. 1985, 4 , 65. (2) Richlin, J. Reu. Sci. Instrum. 1985, 56, 476. (3) Berg, J. C. In Composite Systems from Natural and Synthetic Polymers; Salmen, L.,de Ruvo, A., Seferis, J. C., Buck, E. G., Eds.; Elsevier Science: Amsterdam, 1986; p 23. (4) Surface Forces. Cahn Instruments Technical Notes 1988.
0743-7463/91/2407-0422$02.50/0
The total weight measured by using the balance technique is the sum of wetting and buoyancy forces. These forces are usually different for immersion and withdrawal modes due to contact angle hy~teresis.~ In principle, there are four distinct force-immersion curves that can be obtained, as diagramed in Figure 1. Computer analysis of sequential point number curves presents unique challenges associated with accurate detection of sharp transitions corresponding to immersion or withdrawal modes and selection of data fields matching advancing or receding wetting cycles. Figure l a represents a curve obtained for a typical laboratory material such as a glass plate or rod with significant buoyancy and exhibiting contact angle hysteresis due to a chemically heterogeneous or rough surface. Transitions between advancing and receding modes are clear and easy to identify, but data within the turning point area must not be included in data analysis since buoyancy slope is not stable. As hysteresis diminishes so does clear delineation of the turning point, leading to curves appearing as Figure lb. Perfect specimens exhibiting no contact angle hysteresis are rare, but computer detection of true turning points becomes more difficult as contact angle hysteresis diminishes. This phenomenon is exacerbated for cases in which samples have little buoyancy relative to wetting forces as depicted in Figure IC. Experimental examples approaching this hypothetical case occur for thin mica sheets or textile/composite fibers.316 Finally, Figure I d illustrates a curve obtained for a sample with little or no buoyancy but some contact angle hysteresis. Turning point detection for this case would be straightforward. This report describes computerized data analysis techniques developed for cases shown in parts a, b, and d of Figure 1. The resulting software requires no user input other than sample perimeter and liquid-phase interfacial tension for optional calculation of contact angles. Computations are performed for each immersion and withdrawal cycle with statistical analysis for each. Statistics based on propagation of error through all known sources of experimental uncertainty provide interesting insights into error involved in the Wilhelmy balance technique. ~
~~
~
~~~
~
(5) Johnson, R. E.; Dettre, R. H. In Surface and Colloid Science; Matijevic, E., Eds.; Wiley-Interscience: New York, 1969; Vol. 2, p 93. (6) Miller, B.; Penn, L. S.; Hedvat, S. Colloids Surf. 1983, 6, 49.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 2, 1991 423
Wilhelmy Balance Hysteresis Curves a ) Buoyancy and Hysteresis:
600 buoyancy
550
4 turning point
b) Buoyancy a n d No Hysteresis:
c) No Buoyancy, No Hysteresis:
350'
v=o
d ) No Buoyancy with Hysteresis: out *
io
'
24,
i I in i
'
'
28
'
32
I
36
IMMERSION DEPTH (mm)
1 out
I
4
v=o
E/
c
immersion depth
*
sequential point number (x)
Figure 1. Hypothetical Wilhelmy balance hysteresis curves displayed in conventional immersion depth coordinates and as transformed to sequentialpoint number space. In and out cycles refer to advancing (sample immersion) and receding (sample withdrawal) modes with initial liquid-contact points designated by V = 0. Compare to Figure 2a,b.
Materials and Methods Substrates and Contact Angle Measurements. Precision glass slides (Fisher Scientific, 22 X 50 X 0.1 mm, immersed perimeter = 44.2 mm; Clay Adams "Gold-Seal", 24 X 30 X 0.1 mm, perimeter = 48.2 mm) were sequentially washed in distilled water, 2-propanol, and Freon to remove contaminants prior to 30-min exposure to an oxygen plasma for final cleaning (100 W of 13.56 MHz rf power, 50 Torr 02). Ordinary glass rods (3 mm diameter X 8 mm long, 9.4 mm immersed perimeter; Fisher sealed capillary tubes, 1.74 mm diameter, 5.46 mm perimeter) were treated in the same manner. Rods and slides were siliconized in 2-0.5 % octadecyltrichlorosilane in CHC13 for various time intervalsranging from a few minutes to 48 h, yielding surfaces with varying advancing and receding contact angles that were used as test specimens. Contact angle measurements were made by using an instrumented version of the Wilhelmy plate method (Cahn Instruments, Inc.). Data were collected on diskette and analyzed by using software described below. Program Outline. Program code' was written in RPL (Research Programming Language), a facility of RS/1 data management software (BBN Software Products Corp., Cambridge, MA). All computations were performed on a Compaq 386 personal computer system linked through a local area network. To start, N force measurements distributed evenly over the hysteresis curve were read from disk and imported into RS/l. Although the data field included immersion depth data, these measurements were not used in any subsequent calculations beyond plotting hysteresis curves in the conventional manner for comparative purposes as in Figure 2a. One of two alternative methods (methods I and 11) was employed to locate turning points representing transition between advancing and receding cycles. Method I, which was the preferred (7)US.Copyright TXu 388-174,1989.
350
50
100
150
200
250
300
350
SEQUENTIAL POINT NUMBER (x)
Figure 2.- Typical hysteresis curve shown in conventional immersion-depth coordinates (a) and transformed to sequential point number space (b). Graphical analysis of the hysteresis curve is shown in part a. Points designated withx and X symbols in part b mark turning points that must be accurately detected by the computerized data analysis method. approach (see Discussion), finds the approximate location of turning points by performing a running segmentation of the data field into three adjacent intervals denoted Si, Si+l,and Sitz. Segment length was a variable supplied by the user during program initiation, usually between 9 and 21 data points for N E 400. Maxima and minima in Si+lwere compared to those occurring within Si and Si+z, testing for positive (advancing to receding) or negative (receding to advancing) transitions. Adjacent segments were compared by iterating i until Si+zincluded the last data point. The product of this procedure was a list of the Si+lsegments in which transitions were identified. Exact turning point locations within these Sitl segments were determined by pointwise comparative search. Thus, the entire data field was rapidly screened for turning points and exact locations identified by pointwise examination of short selected segments. The final result was a list of exact turning points indicated in Figure 2b with "x," and "X," notation, n values are positive integers. Second derivatives (differences) were used in method I1 to identify positive or negative transitions through a preset window about zero (see Figure 3, lower axis, dashed lines). Here, the window was set at a user-supplied percentage (a sensitivity) of the third largest d2F/dx2 excursion, avoiding spuriously large sample-fluid initial contact and final removal signals. The
424 Langmuir, Vol. 7, No. 2, 1991
800
F
700
v
0
i LL
600
500
2 h
F
v
-2
v=o
Figure 3. Point space curve for a wettable sample exhibiting little contact angle hysteresis (upper axis) illustrating that turning points associated with small force transitions cannot be detected in second derivative spectra (lower axis). Dashed lines about the d2Ff dx2 indicate a reasonable detection window that avoids spurious signals but fails to detect the central turning point. derivative method fails when applied to samples with little or no hysteresis, as described in greater detail in the discussion section. Data within advancing or receding cycles identified above were statistically fit to the appropriate force balance equation8 and adhesion tension T = y cos 0 was calculated by extrapolation to zero immersion (see Appendix). Data point number becomes arbitrarily large in point space coordinates yielding an increasing effect on the statistics of fit for each successive cycle. A translation of Y axis to the midpoint of each segment was performed to avoid this problem, providing an internally consistent fitting procedure that minimizes absolute uncertainty in fitted coefficients. This uncertainty appears in the second term of expressions A2 and A5 of the Appendix. Also, a special position tracking method was required in sequential point-number space to evaluate the point number corresponding t o zero immersion for the elimination of the unknown sample buoyancy (see Discussion). Either contact angle 0 or liquid interfacial tension y can be subsequently determined from 7.
Results Tables I and I1 summarize results of manual and computer-automated data analysis of Wilhelmy balance hysteresis curves for glass plates and rods, respectively. No effort was made to screen samples, and results are reported for smooth (quality index 11, somewhat erratic (quality index 21, and erratic (quality index 3) hysteresis curves. In this way, the computer method was tested for "real-world" application. Agreement between conventional manual methods and the computerized approach was generally very good, as summarized in Figure 4 which plots results of the two methods against one another. Linear (8) Dryden, P.; Lee, J. H.; Park, M.;Andrade, J. D. InPolymer Surface Dynamics; Andrade, J. D., Ed.; Plenum Press: New York, 1988; p 9.
Martin and Vogler best fit through the data yielded a slope = 1.002 f 0.001 indicating methods were in precise agreement. The intercept, ideally zero, was 0.911 f 0.510 indicating just less that 1 dyn/cm systematic discrepancy between methods on average. Uncertainty in the individual results compiled in Tables I and I1 for manual and computerized data analysis was derived by two different methods. Values listed for the manual approach are mean and standard deviations of three separate wetting tension determinations made by three individuals knowledgeable in manual data workup procedures. This uncertainty represents person-to-person variation in the "eye-balling" of best-fit buoyancy slopes and extrapolation to zero immersion. Uncertainty listed for the computer approach results from statistics of linear least-squares fitting of the data within buoyancy line segments (see Appendix). In essence, error listed for the manual method is the deviation of three opinions whereas error in the computer analysis is a more rigorous, statistical error that collects uncertainty from all known sources. In this connection, it is of interest to compare computer and manual method error estimates for rod and plate geometries. Both methods verify that measured wetting tensions are less precise for rods than plates, due in part to the reduced rod perimeter-to-volume ratio that tends to increase buoyancy slope and associated error terms (see Appendix for details). Additionally, plate hysteresis curves were generally smoother than those of rods, leading to lower mean and median error than for rods (compare column means and medians in Tables I and 11). Note also that smoothest hysteresis curves (quality index 1)had the lowest estimated error, in sensible agreement with qualitative expectations.
Discussion Immersion Depth Independence and Data Tracking. The principal utility in casting data in sequential point-number space rather than conventional liquid immersion depth coordinates is that a separately calibrated depth transducer is not required. Linearity, responsiveness, and dynamic range of this transducer are not, therefore, design restrictions in development of fully automated balances, and a potential source of component failure is eliminated. A precisely linear relationship between immersion velocity and force data acquisition is required, however, as well as means of tracking sample position in point-number space. The former requirement is easily met with modest stepper-motor control and the latter through a straightforward computer algorithm. In the standard method of calculating wetting tension from force-immersion depth curves, advancing and receding lines for each immersion cycle are typically extrapolated back to the zero-immersion line (Le., displaced volume V = 0, see Appendix) in order to eliminate the unknown buoyancy term (see Figure 2a). In the method of this paper, using point space instead of displacement for the abcissa, no such geometric reference point exists for cycles other than the initial advancing phase (see Figure 2b). Wetting tension for this first immersion can be determined by analogy to the standard method, but succeeding cycles require a transformation of coordinates that provides the appropriate reference point (a pseudo V = 0). The purpose of the data tracking algorithm is to account for the number of data points acquired during advancing (in) and receding (out) cycles so that the position relative to pseudo V = 0 can be calculated. For the initial advancing cycle, the liquid contact point determines the reference data point number XO. Data accumulate during this advancing cycle to point XI, and the total number of
Table I. Comparison of Manual and Computer Methods for Glass Plates. contact angle, deg
adhesion tension, dn/cm quality sample index cycle
A
1
1
2 3
B
2
C
1
D
1
E
1
F
1
G
2
mean median
1 2 3 1 2 3 1 2 3 1 2 3 4 1 2 1 2 3 4
advancing mode manual computer
receding mode manual computer
71.83 f 0.10 72.47 f 0.18 72.34 f 0.11 -24.36 f 0.44 -2.87 f 1.71 -1.31 f 1.65 71.81 f 0.13 72.37 f 0.09 72.33 f 0.04 -22.91 f 0.22 -20.12 f 0.29 -20.22 f 0.17 59.96 f 0.21 63.05 f 0.08 63.41 f 0.45 63.07 f 0.03 69.87 f 0.15 70.81 f 0.27 -20.43 f 0.17 -13.08 f 1.35 -12.33 f 0.77 -13.05 f 0.42
71.44 f 0.09 72.52 f 0.12 72.67 f 0.14 -24.05 f 0.07 -1.32 f 0.89 0.23 f 1.06 71.82 f 0.03 72.12 f 0.12 71.98 f 0.14 -22.75 f 0.15 -23.20 f 0.18 -24.60 f 0.20 59.90 f 0.22 62.85 f 0.25 63.37 f 0.31 63.96 f 0.40 69.83 f 0.08 70.48 f 0.13 -19.62 f 1.43 -8.60 f 1.51 -10.08 f 1.46 -10.63 f 1.46
72.20 f 0.23 72.47 f 0.18 72.47 f 0.18 25.92 f 1.86 28.54 f 0.99 28.40 f 0.49 72.18 f 0.20 72.37 f 0.09 72.33 f 0.04 -1.74 f 0.41 -1.11 f 0.14 -1.35 f 0.16 63.54 f 0.10 63.72 f 0.08 63.61 f 0.19 63.61 f 0.02 71.02 f 0.10 71.14 f 0.04 7.38 f 0.62 7.76 f 0.17 8.11 f 0.45 7.87 f 0.32
72.47 f 0.07 72.69 f 0.11 72.71 f 0.13 36.36 f 0.98 37.56 f 1.02 31.47 f 0.63 72.30 f 0.06 72.19 f 0.12 72.12 f 0.12 -3.06 f 0.98 -5.29 f 0.16 -5.86 f 0.19 63.89 f 0.11 64.10 f 0.22 64.38 f 0.25 64.48 f 0.29 70.47 f 0.06 70.61 f 0.10 9.86 f 1.44 10.37 f 1.46 10.70 f 1.47 9.72 f 1.46
0.41 0.19
0.47 0.19
0.32 0.18
0.52 0.20
advancing mode manual computer 8.34 f 0.56 2.89 f 2.66 4.73 f 1.15 109.60 f 0.37 91.26 f 1.35 91.03 f 1.30 8.46 f 0.70 4.45 f 1.01 4.96 f 0.38 108.40 f 0.18 106.09 f 0.24 106.17 f 0.14 34.70 f 0.29 30.20 f 0.12 30.10 f 0.25 30.10 f 0.06 15.76 f 0.44 12.75 f 0.97 106.30 f 0.15 100.40 f 1.08 99.80 f 0.60 100.30 f 0.30
10.25 2.70
*
109.35 91.04 89.82 8.38 6.56 7.48 108.26 108.64 109.81 34.41 30.03 29.20 28.23 15.89 13.89 105.68 96.80 97.98 98.42
range
9.85-10.63 *-4.25 *-2.52 109.29-109.40 90.33-91.75 88.99-90.66 8.20-8.55 5.66-7.35 6.56-8.29 108.13-108.39 108.48-108.79 109.64-109.98 34.10-34.71 29.64-30.41 28.70-29.70 27.56-28.89 15.66-16.12 13.46-14.31 104.51-106.85 95.60-98.00 96.82-99.15 97.25-99.58
receding mode manual computer 5.85 f 1.63 2.89 f 2.66 2.89 f 2.66 69.07 f 1.58 66.85 f 0.85 66.97 f 0.42 6.02 f 1.44 4.45 f 1.01 4.96 f 0.38 91.37 f 0.33 90.88 f 0.11 91.07 f 0.13 29.40 f 0.12 29.10 f 0.12 29.20 f 0.31 29.30 f 0.06 12.33 f 0.33 11.52 f 0.17 84.20 f 0.47 83.90 f 0.12 83.60 f 0.40 83.60 f 0.25
Notes: undefined angle due to cosine > 1 indicated by asterisk. Quality index 1 > 2 > 3 measures hysteresis regularity; see text.
3.47
* *
59.94 58.85 64.31 5.18 6.08 6.58 92.41 94.18 94.63 28.36 28.01 27.52 27.35 13.93 13.44 82.19 81.79 81.53 82.31
0
range
2.46-4.24 *-1.34 *-1.34 59.0440.83 57.90-59.78 63.76-64.86 4.62-5.68 5.11-6.92 5.68-7.38 91.64-93.19 94.06-94.30 94.48-94.78 28.18-28.54 27.64-28.37 27.09-27.95 26.85-27.84 13.73-14.12 13.09-13.79 81.04-83.34 80.62-82.95 80.35-82.70 81.14-83.47
s C
cn m
"q-
s
?
3
2
Table 11. Comparison of Manual and Computer Methods for Glass Rods. adhesion tension, dn/cm contact angle, deg
? "J
z
k
quality sample index cycle A
2
1
2 3
B
3
C
3
D
1
E
2
F
1
G
2
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1
2 3
mean median a
advancing mode manual computer
receding mode manual computer
-31.08 f 0.79 -30.02 f 0.85 -28.95 f 2.34 29.47 f 2.96 32.64 f 1.53 32.83 f 4.52 40.76 f 2.59 47.33 f 1.57 45.38 f 2.27 74.16f 0.00 14.47 f 1.59 14.44 f 1.22 -47.22 f 0.80 -40.57 f 1.17 -39.60 f 0.78 -23.33 f 0.70 -18.35 f 0.90 -17.39 f 0.79 43.18 f 4.14 41.63 f 3.61 38.98 f 0.44
-24.57 f 1.42 -31.01 f 1.28 -25.54 f 1.33 30.50 f 1.51 26.42 f 1.47 28.27 f 1.78 40.02 f 2.38 40.84 f 4.22 53.65 f 2.98 75.16 f 0.17 79.48f 1.42 77.85 f 1.52 -52.51 f 1.12 -43.30 f 1.20 -40.18 f 1.17 -19.04 f 0.84 -17.49 f 0.98 - 15.53 f 1.58 50.37 f 1.11 44.23 f 2.08 42.17 f 2.20
9.18 f 1.27 6.52 f 1.64 8.34 f 2.17 60.68 f 0.93 62.06 f 0.65 62.46 f 0.99 64.38 f 1.68 67.34 f 0.22 65.71 f 2.98 74.16f 0.00 75.77 f 0.97 75.00f 0.26 11.56 f 1.21 9.72 f 0.82 8.28 f 1.36 26.92 f 4.10 27.13 f 1.09 25.17 f 0.53 64.50 f 0.41 66.94 f 0.99 67.46 f 1.07
10.29 f 0.64 11.70 f 1.17 13.70 f 1.42 66.16 f 1.19 67.31 f 1.56 51.79 f 0.86 65.67 f 0.55 62.31 f 1.47 71.24 f 1.28 76.38f 0.36 76.29f 0.65 76.10f 0.73 9.71 f 0.86 8.61 f 0.90 14.93 f 1.64 27.41 f 1.94 24.20 f 1.15 27.40 f 1.28 67.29 f 0.94 73.27 f 1.39 71.73 f 1.45
1.69 1.22
1.61 1.42
1.21 0.99
1.12 1.17
advancing mode manual computer 115.35 f 0.69 114.43 f 0.73 113.51 f 2.02 66.04 f 2.55 63.28 f 1.36 63.07 f 4.03 55.82 f 2.46 49.30 f 1.62 51.29 f 2.30
* * *
130.58 f 0.84 123.98 f 1.11 123.06 f 0.74 108.75 f 0.58 104.64 f 0.73 103.86 f 0.64 53.43 f 4.13 54.96 f 3.51 57.52 f 0.42
range
receding mode manual computer
range
109.78 115.29 110.60 65.16 68.66 67.08 56.55 55.77 42.35
108.59-110.98 114.18-116.41 109.48-111.73 63.83-66.46 67.41-69.90 65.55-68.60 54.26-58.77 51.63-59.71 38.73-45.74
82.73 f 1.01 84.85 f 1.30 83.40 f 1.72 33.28 f 1.34 31.24 f 0.99 30.62 f 1.55 27.44 f 2.81 21.94 f 0.45 24.82 f 5.31
81.85 80.72 79.13 24.31 22.01 44.49 25.23 30.88 11.12
82.34-82.36 79.79-81.66 77.98-80.27 21.92-26.51 18.45-25.09 43.51-45.46 24.19-26.23 28.54-33.08 2.67-15.52
136.33 126.62 123.61 105.20 103.94 102.36 46.07 52.47 54.49
135.06-137.62 125.45-127.80 122.5Ck124.72 104.52-105.89 103.15-104.74 101.09-103.63 44.84-47.28 50.37-54.51 52.33-56.59
80.83 f 0.97 82.31 f 0.65 83.45 f 1.08 68.21 f 3.48 68.06 f 0.92 69.71 f 0.45 27.32 f 0.70 22.72 f 1.98 21.61 f 2.36
82.32 83.19 78.14 67.82 70.53 67.83 22.05
81.63-83.00 82.48-83.90 76.81-79.46 66.15-69.46 69.57-71.49 66.74-68.91 19.97-23.95
8.89
0.W14.54
* * *
* * *
* * *
* * *
*
* * *
*
Notes: undefined angle due to cosine > 1 indicated by an asterisk. Quality index 1 > 2 > 3 measures hysteresis regularity; see text.
F,
J
R
Wilhelmy Balance Hysteresis Curves 80
,
Langmuir, Vol. 7 , No. 2, 1991 427 ,
T (dyne/cm)
dCI
Monuol Method
Figure 4. Comparison of wetting tension measurements by computer and conventional manual methods for siliconized glass rods (circles) and plates (squares) in both advancing (open symbols) and receding (closed symbols) modes. Dashed lines about the best fitted line indicate the 95% confidence interval for the data (see text for discussion of statistics).
data points collected is (XI - X O ) . Similarly, the receding cycle consists of (X2 - xl) points. The net value [(xl - X O ) - (X2 - X I ) ] measures the actual number of data points traveled from true V = 0 during the first cycle, with positive values meaning the plate was not completely withdrawn from the liquid. The transformed coordinate system for the second advancing cycle must account for this change in initial position because the appropriate pseudo V = 0 is not X2, but actually lies the net value of data points back in point space. Thus, the pseudo V = 0 for the second cycle x02 = Xz - [(xl - X O ) - (X2 - XI)] = 2(X2 - X I ) + X O . In general, it can be shown that the pseudo V = 0 point for any nth cycle xon = X O ( ~ - ~+) 2[Xn - ~ ~ - 1 where 1, n2 2 and x01 = X O . Wetting tensions can be determined for each cycle by using this coordinate system in the same manner applied to the first immersion. Another consequence of casting hysteresis curves in sequential point number is that advancing and receding lines for each cycle are not parallel (compare parts a and b of Figure 2 ) . In fact, simple extension of the receding line to V = 0 would lead to erroneous calculation of receding wetting tension. Correct extrapolation to V = 0 for receding cycles requires a “reflection” operation about each x = x n that effectivelyreverses the direction of the receding line. A simple algorithm based on the point-slope formula is given in eqs A3 and A4 of the Appendix that accomplishes this operation. We have found measurement of absolute immersion depth unnecessary. However, certain experiments may require accurate depth information as in, for example, analysis of surfaces with gradient chemistry or in measurement of kinetics. For such applications, conversion of point number to depth is possible by using knowledge of immersion velocity (stepper motor rate), force data acquisition rate (A/D rate), and turning point locations between cycles. Thus, force-point number curves can be retroactively converted to conventional force-immersion coordinates according to eqs A6 and A I of the Appendix. Turning Point Detection. Accurate identification of turning points is crucial for position tracking and subsequent calculation of wetting tension from slope measurements. Second derivative methods are among the most obvious choices because transitions through zero are easy to detect. Derivative approaches fail, however, in those circumstances when hysteresis is small because change between advancing and receding wetting forces are of the same order of magnitude as noise. A specific example is shown in Figure 3 for a nearly completely wettable glass plate with similar advancing and receding contact angles. Detection and clear separation of advancing/receding
transitions from noise in the double derivative spectrum (lower axis) are problematic because transition-signal amplitude is not much greater than noise. As the detection window (horizontallines in Figure 3) is closed, noise spikes yield false turning point assignments. No general solution to this problem applicable to all low-contact-angle hysteresis curves tested in the course of this work has been found. An alternative to derivative methods is a pointwise search through the entire data field for turning points, seeking positive (advancing to receding) or negative (receding to advancing) transitions (see Figure 2b). This straightforward but computationally intensive method can be significantly improved by a running segmentation approach.. Data field segmentation is based on similar methods utilized in many digital signal processing techniques such as Sovitzky-Golay noise filtersgJOin that adjacent blocks of data are compared (averaged or smoothed in the noise filtering case) in the search for transitions. Once approximate locations of the turning points are identified, short segments can be screened pointwise for exact point-number locations. Just as in noise filters, segment length must be adjusted relative to the size of the data field. On the one hand, segment length must be long enough to offer improved computational speed. On the other hand, segment length must be short enough so that turning points are not continuously bridged and transitions missed. We have found empirically that segment lengths between 1 and 5 % of total number of points can be used successfully. Error and Statistics. Uncertainty in wetting tension values obtained through the computerized method arises from two main sources. First, there is statistical uncertainty in the best-fit to force balance equations. Second, there is uncertainty in the location of the turning points, and this introduces error in determination of reference points (the pseudo V = 0 locations, see previous section). The impact of these sources of error can be assessed by using standard propagation-of-error techniques (see Appendix). Error in fitted coefficients can be traced to variation in force measurements which, in turn, can usually be attributed to vibrations, imperfections on plate or rod surfaces, or time-dependent wetting phenomenon. Variation in force measurements is particularly extreme at or near turning points where a transition in direction occurs. It is beneficial from the standpoint of improved precision in wetting measurements, therefore, to ignore data within close proximity to turning points. In this work and with most common hysteresis curves, the number of data points comprising advancing and receding cycles is large and not a serious limitation to precision of the best-fit to force equations. We have found empirically that up to 25 % of data on both sides of identified turning points can be excluded (fitting central 50%) without significantly affecting parameter estimates. The absolute magnitude of the buoyancy (in milligrams/point number units) enters into the propagation of error as a multiplicative factor and should be minimized for work requiring the highest accuracy (see eqs A2 and A5 of the Appendix). Obviously, increasing the number of data points gathered will reduce this buoyancy parameter. Thus, increasing the digitization rate can improve precision. Likewise, use of test specimens with the lowest possible volume will reduce buoyancy and improve results. Compare, for example, (9) Proctor, A.; Sherwood, P. M.
A. Anal. Chem. 1980,52,2315. (10)Bromba, M.U.A.; Ziegler, H.Anal. Chem. 1981,53, 1583.
428 Langmuir, Vol. 7, No. 2, 1991 error estimates in Tables I and I1 for thin glass plates and thick glass rods, respectively. The running segmentation or double derivative approach to turning point detection leads to selection of a single data point as the apparent inflection in the forceimmersion curve (see Figure 2b). The real inflection may lie somewhere between a single data point ahead or behind the apparent inflection, as a consequence of digitizing a continuous signal. However, we can be confident that force measurements must be either monotonically decreasing or increasing through the inflection zone since we are certain that buoyancy increases with immersion or decreases with withdrawal, respectively. The real inflection point must, therefore, lie considerably closer to the chosen point than either closest neighbor, certainly within f l / z point number. Without detailed knowledge of the actual shape of the turning point curve, refined estimates of point selection error are not possible. In this work, we have assumed that f l / z point number "digitization error" represents a 100% confidence interval and, in consideration of normally distributed error, f0.17 point number is the appropriate l a error limit to be used in propagation of error formalism. This error is pertinent for every turning point including the first. These errors propagate in such a way as to increase with number of cycles (see eqs A2-A5 in Appendix). The net effect generally can be seen as increased uncertainty with cycle number in Tables I and 11. Discussion to this point has been restricted to uncertainty in measured wetting tension. This uncertainty can be propagated into error in contact angle through8 = cos1 ( r / y )for those cases in which y is known from a previous determination. However, the propagation of error is unsymmetrical about 8 with larger uncertainty in lower angles than a t higher angles because of the transcendental function. Thus, uncertainties in calculated angles in Tables I and I1 are reported as a range rather than a symmetric f value. Miller and Young" were apparently among the first to discuss relationships between error in measured contact angle and calculated wetting tensions, but the asymmetric aspect of the error was not explicitly described. Figure 5 plots the confidence interval in 8 calculated from wetting tension where a 5 % uncertainty in 7 is assumed. Error a t low 8 is not only greater than at higher 8 as anticipated from ref 11 but also more asymmetric about the nominal value a t low angles. It is evident that symmetric confidence intervals for contact angles calculated from wetting tensions can be expected only for the cases of high 8 (nonwettable materials) or very precise 7 values.
Conclusions Immersion depth independent computer analysis of Wilhelmy balance hysteresis curves simplifies reduction of data with results comparing favorably with conventional graphical methods. Quantitative aspects of data analysis are improved by eliminating operator subjectivity in selection of liquid contact points and estimation of buoyancy slopes. Batch analysis of hysteresis curves greatly reduces labor intensity involved in data workup. Elimination of a displacement transducer simplifies design of automated balances and avoids an unnecessary source of component failure. Full statistical analysis can be obtained for each advancing and receding cycle which can aid in the interpretation of data, particularly if timedependent processes such as solute adsorption are occur(11) Miller, B.;Young, R. A. Tent. Res. J. 1977, 45, 359.
Martin and Vogler 15
h
IL
Symmetric error limit
Contact Angle (degrees)
Figure 5. Confidence intervalplot for contact angles calculated from hypothetical wetting tensions with an assumed 5% error
showing asymmetry about the nominal contact angle value at lower values. A symmetric error limit is shown to illustrate the divergence caused by propagating error through the cosine function (see text). The lower confidence interval is undefined at angles below 10'. ring at the perimeter of the sample. In this latter connection, it is anticipated that automated balance techniques may be useful in studying adsorption kinetics.
Glossary number of data points comprisinga hysteresiscurve ith segment within the N point data field mass equivalent of measured force, mg acceleration due to gravity, cm/sz sample perimeter, cm liquid-vapor interfacial tension, dyn/cm contact angle, deg liquid volume displaced by sample, cm3 liquid density, g/cm3 adhesion tension, dyn/cm point number axis (horizontal) point number corresponding to liquid surface turning point (in to out) of nth cycle turning point (out to in) of nth cycle pseudo V = 0 point for cycle n uncertainty slope of least-squres fit, mg/data point intercept of least-squares fit, mg sample weight, mg sampleposition above or below liquid contact point for the nth cycle corresponding to immersion (in) or withdrawal (out) modes, cm force data acquisition rate, points/s immersion/withdrawal velocity, cm/s
Langmuir, Vol. 7, No. 2, 1991 429
Wilhelmy Balance Hysteresis Curves
Acknowledgment. The authors are indebted to Dr. P. Haaland and Mr. M. O'Connell for helpful discussion of statistical analysis. Ms J. Graper and Ms L. Cooper are acknowledged for their skilled technical assistance.
Appendix The force balance equation for samples such as glass rods and plates used herein is f = mg = p y cos B - Vpg. The buoyancy slope term Vpg is linear with immersion depth for such objects with constant cross-sectional area and uniform perimeter p , displaced volume V , and a homogeneoustest liquid with density p. Thus, a parameter S proportional to buoyancy can be determined from force data taken evenly spaced in time in either point space or immersion depth coordinates. For samples with irregular geometry, a more complicated force balance relating measured forces to wetting tension and buoyancy would be required,a but the approach described above is general in nature and should be applicable to a variety of specific cases. Linear least squares fit to data within immersion (advancing) cycles evaluated at pseudo V = 0 yields ?no& = Sxon + I, where xon is the zero-depth point calculated from the data tracking method (see materials and methods section) and I is the constant term of the linear fit for the nth cycle. Assuming that perimeterp and the gravitational constant g are invariant and precisely known, error in 7 = y cos B is proportional to error in mo through 0,
= k/P)Umo
(AI)
Sample weight m' must be subtracted from m in those events when m' is not instrumentally nulled at the start of the experiment. Considering variations in all experimental observables leads to the expression (A2) for the advancing cycle a,2 =
(g2/p2)[a:X12 + s2aXl2+ a12+ a,?], n = 1
= k2/P2) [ ( X o , n - l + 2 x , - 2xn-1Ya,2 + (axo,"-12 + 2ax,2 2u,n-12)S2 a: a,?], n L 2 (A2)
+
+ +
The a terms in (A2) represent error in fitted coefficients, S or I , as indicated by subscripts. Values for a,~ are the standard deviation of the mean of baseline force measurements from x = l to xo - 5. The value 0.17 is assigned to all (T, and ax terms in accordance with discussion in text (see Discussion under Error and Statistics).
Similar considerations are pertinent for receding cycles except for complications in the calculation of the "real" intercept value which requires a "reflection" operation (see text). The objective of reflection is to find the equation of a "reflected" line that intersects m = Sx + I at x = x , and has slope equal in magnitude but opposite in sign. Here, the coefficients S and I result from linear leastsquares fitting of data within the receding segment X n I x I Xn+l. It can be shown from the point-slope formula that the equation of this reflected line is given by
m = -Sx + ( ~ S X+I) , (A31 Evaluation of the reflected line at the pseudo V = 0 occurring at x = xon yields the buoyancy-correct mass for the receding cycle morec= -SxOn+ ( ~ S X+, I) (-44) Propagation of error through all observables for the receding cycle yields :a
= (g2/p2)[(X1- 2x1)2a,2+ (ax12+ 4ax12)S2+ aI2
= k2/P2) [XO,,-l 4aX
+ a,?],
+ 2 x , - 23c,-1 - 2 4 2 6, 2 +
+ 4a, + 40, "-1
')S2+ a;
+ a,?,
n=1
(axo,"-12 +
n 2 2 (A5)
In turn, uncertainty in 7 can be translated into statistical uncertainty in B by transforming the range of 7 f a, through the inverse cosine function according to B = COS-'(~ f a,/ 7). The position P (cm) above or below the liquid contact point can be retrospectively calculated from point number assuming immersion velocity R, (cm/s) and digitization rate R, (points/s) are known. The sample is immersed up to (xl - X O ) points in the advancing mode of first cycle (see Figure 2 ) at R,/R, cm/point, yielding Plin= ( x - xo)R,/R, for xo Ix IXI, where the subscript "1" denotes the first cycle. Direction reverses for the receding mode, starting at position PIinand terminating after collecting of (X2 xi) points so that PIoUt = PIin( x - xl)R,/R, for XI I x I Xz. In general for the n L 2 cycles
Pnin= Pn-Ft + ( X
- X,)R,/R,
PFt = Pnin - ( X - x,)R,/R,
(A6)
(A7) Uncertainty in positional calculation can be estimated by using propagation of error as described above for wetting tension.
