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Copyright Ó 1990-2006 Finn Knut
Hansen. All rights reserved
Interfacial and surface tension of liquids are of importance to many processes in the chemical industry. There is a need for a fast and easy technique with satisfactory accuracy and reproducibility. Traditionally, instruments utilizing the Du Noüy ring and Wilhelmy plate methods are utilized throughout the industry and research laboratories. The methods of pendant or sessile drop are traditionally well suited for many types of surface and interfacial tension measurements, but are also quite time consuming. Drop shape analysis has usually been performed by photographing a drop in an optical bench arrangement, and then the characteristic sizes of the drop can be measured on the photographic prints. For pendant drops the maximum diameter and the ratio between this parameter and the diameter at the distance of the maximum diameter from the drop apex has been used to evaluate the size and shape parameters, whereas for sessile drops a complete profile matching is often necessary. After video imaging facilities and mathematical coprocessors for personal computers have become readily available, there is a great potential for improvement of this method.
On the basis of the Young-Laplace equation describing the drop profile of both sessile, pendant and ascending drops and bubbles, it is possible to calculate the surface tension from digitized picture data. Much attention in the scientific community has been focused on sophisticated numerical methods for image analysis and integration. Although these methods sometimes may produce excellent results, they are often too computation intensive to be applicable in everyday instrumentation, although the speed of these computational algorithms have been greatly increased since the introduction. The objective of this program has been to construct a simple and fast instrument with sufficient accuracy and reproducibility to make this method useful in many practical applications. This manual describes a computer program that utilizes pictures taken with video and frame grabber equipment and an IBM compatible PC. The general physical and mathematical properties of this method are published, and are also outlined below.
Since its first introduction as a DOS program in the 1980s, this program
system has been rewritten to take advantage of the capabilities of the
Microsoft Windows operating system. In addition to the considerable changes
caused by the Windows interface, the program capabilities have also been
expanded to include full liquid handling by means of an optional dispenser. The
last version of the program now supports the MS Windows DirectShow API and
compatible USB and FireWire cameras.
The main measurement capabilities are the following
Type of drops
Type of results
Type of measurements
The DROPimage Advanced Main Window.
All results are displayed in tabular form in the programs Results window. This window contains two tabbed notebooks, one for surface tension measurements and one for separate (contrary to those included with surface tension results) contact angles. These results may also be displayed in a separate Report window that is specially formatted for printing. For visual display, the program contains a Plotting function.
Surface tension results window
All measurements of surface tension and contact angle are based on measurement Methods (except single contact angle measurements). A method consists of a collection of parameters that describes how and when measurements are performed and how results are saved and presented.
A method's parameters are saved in a text file. Methods are created and edited in the Method Editor. All measurements of interfacial tensions must refer to a method, and several measurements can use the same method.
There are principally two kinds of methods, depending on the first
field, Data source. If this is
set to Video, data are taken
from the frame grabber board for further treatment according to the other
parameters. If Data source is
Disk file then this is a
Recalculation method that tells the program to read data from a disk
file, which must have been produced by a method where the data source is
Method editor (without the oscillation option)
A mayor advantage with the present Windows program is that the measurements are interruptible. This means two things:
An exception to the above is when measurements are taken real-time, at
the maximum rate. The program then will lock out other processes in order to
perform the task as fast as possible.
In the calibration routines, it is also possible to adjust the
horizontal/vertical aspect ratio. This value is a fixed number for a given
camera/frame grabber combination, and would ideally be a universal constant
because of standardized video camera construction. However, different cameras
may have slightly different aspect ratios, this may therefore conveniently be
adjusted either by calibration both in vertical and horizontal direction, and
by using a sphere for calibration.
The results from the calculations may be saved and exported to other programs. All results are saved to LOG-files on a per experiment basis. Through a Session control window the log-files may be viewed plotted and edited. The log-files are ordinary text-files that are easily read by other programs.
The program also has a Report facility that produces a Rich Text formatted report that can be readily read into word processing programs. The experimental data that go into the report may be selected among the available results, and the report window has editing and formatting properties that makes it a small text editor. The reports may be saved and printed form the Report editor.
The program has additional procedures and options in order to make "house keeping" tasks easier and also makes detailed control of picture taking, edge detection, data storage and retrieval possible.
An additional function on the View menu is the "Intensity histogram".
This generates an intensity map along a number of either horizontal or vertical
pixels and is especially suited to study the intensity level and change across
the profile for instrument adjustment and/or error investigation.
The video image consists of an array of pixels (dependent on the frame grabber board), each with 256 levels of light intensity (gray levels). The filter routine for detection of the drop profile is a simple edge-tracing routine with increased (subpixel) accuracy compared to global tresholding and maximum gradient techniques. In order to discriminate the drop interface the program uses a local threshold and interpolation routine. The co-ordinates of the drop profile are found by linear interpolation to a given fraction, F, between the local maximum and minimum of light intensity, i.e.
Ithreshold = F ´ ( Imax - Imin ) 
From analysis of gray levels in the neighborhood of the drop interface, the value 0.55 is used as a suitable value for this fraction. However, as the value has considerable influence on the final result (see below) a comparison of this method against liquids of known surface tension will indicate the optimal value. One of the co-ordinates for each point is an integer (0-255) and the other will be a decimal number because of the interpolation routine. The accuracy is thus considerably improved compared to simple global tresholding or maximum gradient routines that gives both co-ordinates as integers.
Once a point on the drop profile has been found, the search for the next
point is limited to the nearest point on the next line. In the bottom part of
the drop the search direction is switched from horizontal to vertical. Most
drop profiles consist of from 700 to 1000 points. On the average the routine
uses from 2 to 3 seconds on these calculations. The same filter routine is used
for discriminating the drop interface when using the program for contact angle
Determination of initial size parameters
A 2-step process determines surface tension. First, size parameters R0 and b are determined from the drop profile, secondly surface tension is calculated from these parameters by the equation,
g = Dr g R02 / b 
Here Dr is the mass density difference between the drop and the surrounding medium, g is the gravity constant, R0 is the radius of curvature at the drop apex and b is the shape factor, as defined by this equation. By convention, Dr is defined such that Dr and b is negative for pendant drops, and positive for sessile drops.
The equations describing the drop profile are derived from the Young-Laplace equation and may be represented in dimensionless form:
dq/dS = 2 - b Y - sin q /X 
dX/dS = cos q 
dY/dS = sin q 
The co-ordinates x, y, s and q are illustrated below.
The parameter, s, is the distance along the drop profile from the drop apex. X, Y and S are dimensionless parameters made by dividing x, y, and s, respectively, by R0. For pendant drops, b and the density difference, Dr, will be negative, while for sessile drops, b and Dr are positive. A large number of theoretical dimensionless profiles were calculated for the whole possible b-range, from b= -0.55 to 1020 by means of Kutta-Merson's numerical integration algorithm with automatic step length adjustment. The maximum relative error was set to 10-4. Each profile was measured mathematically by using cubic interpolation. In this way, curves correlating the parameters b and R0 with measurable parameters as indicated in the figure were produced, and these curves were fitted with linear polynomials by the method of least squares.
For "normal" pendant drops (i.e. drops that are sufficiently long in order to measure DS) the maximum diameter, DE, and the ratio s = DS/DE is used (DS is the diameter at the distance DE from the drop apex). The equation found is (b is negative here):
b = -0.12836 + 0.7577 s - 1.7713 s2 + 0.5426 s3 
From the same data, an equation for DE/2R0 is found:
DE/2R0 = 0.9987 - 0.1971 b - 0.0734 b2 - 0.34708 b3 
For pendant drops too short for the determination of DS, and for all sessile drops, we use the drop "height", H, and the "radius", R=DE/2. If we substitute H for R0 in Equation , we may write,
g = DrgH2/B 
where B is a transformed shape parameter. For a sessile drop, both R0 and b may increase several orders of magnitude as the drop becomes large and flat, but in such a way that g stays the same. It is easily observed that H will have an upper limit because of the maximum hydrostatic pressure the surface tension may "resist". When the drop becomes infinitely wide, only one radius of curvature will be important, and the limiting value of B is 2.0. Equation  is therefore much more convenient for sessile drops and may also be used for "short" pendant drops. The parameter B may derived from Equs. and  as a function of the ratio x=H/R.
B = b ´ (H/R0)2 = f(x) 
Also the dimensionless ratio H/R0 will be a function of x, and we may write
H/R0 = g(x) or R0 = H/g(x) 
Combining  and : b = f(x)/g(x)2 
The function B = f(x) has a maximum B=2.290 at x = 0.285 that corresponds to ca. b = 5000. This means that the drop "height", H, also must have a maximum at this value of x. This must represent an optimum for the sum of the two radii of curvature, meaning that at higher drop volumes, the curvature in the horizontal plane must decrease stronger than the opposing increase in the vertical plane. The function B=f(x) may be mathematically approximated by different functions, depending on the x-domain and the desired accuracy. Exact analytical solutions have not been found, but experiments show that ordinary linear polynomials give satisfactory fit in most cases, using x-1 as the independent variable and forcing the constant term to 0. Around x=1 it may approximated by a straight line with a slope of 4.38, while for all values x>0.34 (i.e. b<1000) we may use a 4th order polynomial with a standard error of 0.0018. Because of the opposite curvatures of the positive and negative parts of the curve, better accuracy is obtained by using separate equations. Thus for
x<1: f(x) = - 4.1788 (x-1) + 1.9086 (x-1)2 + 4.5738 (x-1)3 [12a]
x>1: f(x) = - 4.3626 (x-1) + 1.1961 (x-1)2 [12b]
These equations give very good estimates of B over all regions of practical interest and the slope at x=1 is 1.723. In order to obtain good estimates for the whole region of x>0.34, we also choose two separate polynomials that give a standard error of 0.0007.
x<1: g(x) = 1 + 1.6795 (x-1) - 0.58334 (x-1)2 -1.4257 (x-1)3 [13a]
x>1: g(x) = 1 + 1.7356 (x-1) - 0.40869 (x-1)2 [13b]
With these two functions, we can easily calculate R0 and
b for all values of b<1000 from the measurement of H and R and equations
 and .
Values for R0 and b are found from the experimental profile data by several numerical smoothing techniques. For pendant drops, the central axis of the drop is determined by a first order regression line through all data points, using the y-values as the independent and x-values as the dependent variable. For sessile drops, the least square line through the mean values of all corresponding points from the base up to the turning point (45o) is used. The bottom point of the drop, i.e. the point where the central axis intersects the drop profile, is found by fitting a 4th order polynomial without the 1st and 3dr order terms to all data points in the bottom profile up to a limit of y/x=0.4. The horizontal distance from this mid-line then determines the y-co-ordinates. However, when using subpixel resolution, results may be improved by correcting for small deviations in the vertical direction, i.e. drop skewness. New x-co-ordinates are calculated by the equation
xi' = xi - a yi' xi'' = xi - a yi'' 
Here x and x'' are the corrected co-ordinates for the left and right side of the drop, respectively (y' and y'' have opposite signs). Indeed, also the y-values may be corrected in a similar way by using the factor , but these corrections will be very small because a is very small, and may be neglected.
The parameters R and H are determined for each side separately by means of a second order polynomial through 10% of the side's points closest to the maximum. This polynomial has been found to give the most stable determination for most conditions, even if a third order equation principally is more correct because of the unsymmetrical nature of the side profile. For the determination of R0 and b, the values for the two sides are averaged; in addition an asymmetry factor can be calculated from the equation,
Ass = 2 (H' - H'')/(H' + H'') 
The value of Ass gives an indication of the reliability of the final
results, as drops with a high asymmetry factor usually give inaccurate results.
This method for determination of R0 and b is used
for all sessile drops and for pendant drops where DS cannot be
determined ("short" drops). For ordinary, "long", pendant drops, the method
using the ratio DS/DE as outlined above is used, because
of better accuracy. The value of DE is simply
DE=R'+R'', while the value of the diameter
DS at the distance DE from the apex is determined by a
second order polynomial method similar to that for R' and R''.
Fitting of profile data
To achieve even better accuracy and reproducibility of surface tension data, it is necessary to utilize all the profile data in a least squares parameter optimization. The data from the two sides are joined in one profile with origin at the drop apex and direction of the axis as indicated in the figure. In this process, the y-values are corrected for differences between R' and R'' by adding and subtracting (R'-R'')/2, respectively. This correction results in smaller deviations between data from the two sides, and generally gives better optimization results when the two sides are joined.
Because the initial values of b and R0 are already quite close to the optimal values, a relatively simple, but yet effective method of second order interpolation/extrapolation (response surface) is used in the optimization. The 9 theoretical profiles in a 3x3 grid around the start values of R0 and b are calculated by numerical integration of the Young-Laplace equation, again using Kutta-Merson's algorithm with a maximum relative error of 10-4. The grid granularity is 1% in b and 0.2% in R0.
The objective function used in the optimization is the normal (i.e. perpendicular) mean square deviation between theoretical and experimental points, given by the equation
Where the mean square in the y-direction, Dy, is given by,
The mean square is calculated between x = 0.2 R0 and the maximum value in the data set. N is the number of data points. The theoretical value is calculated at each experimental x-value by means of cubic interpolation (through the 4 closest theoretical data points, 2 on each side).
The result from one profile optimization run is denoted the "contour"
method in this program. Sometimes, when working with sessile drops, it may be
necessary to repeat the optimization step on order to minimize the error
function. The program may be instructed to repeat optimization until a
convergence is obtained. This procedure is then denoted "optimized contour".
Experiments often show that very little improvement in the value of
g is achieved, especially in the case of large
pendant drops. Sessile drops have a much larger span in possible
b-values, and are more difficult to optimize,
especially at low b-values (0<b<1), which should be avoided. In addition, sessile
drops are more prone to experimental errors that are due to uneven wetting
conditions around the drop perimeter, leading to lack of axisymmetry. This
phenomenon often results in high asymmetry ratios (several %), and problems
with convergence in the optimization procedure, as mentioned above.
Measurement of contact angles.
In this program two different methods are used to measure contact angles. The first method utilizes the theoretical drop profile that is calculated in the curve fitting part of an interfacial tension calculation. This means that the whole drop must be visible, and that the surface must be undisturbed i.e. objects like rods or tubes (pipette) must not be present. Because all the conditions for an interfacial tension measurement must be fulfilled, this contact angle calculation will always be performed when interfacial tensions are measured. Of the theoretical co-ordinates calculated by the numerical integration of the Young-Laplace equation, very few would coincide exactly with the drop's endpoint (the horizontal crosshair cursor). The program will therefore interpolate between the 2 points on each side of the end, using the 2 additional points further away in a cubic interpolation procedure. This interpolation has been shown to give very accurate values compared to the exact theoretical calculation.
The second method for calculation of contact angles is a pure numerical one. The method was constructed to show a horizontal line on the screen, along which the solid surface is aligned. The filter routine then will give a properly aligned drop profile. The contact angle is easily calculated by numerical extrapolation of the profile at the contact point. Different methods of numerical fit give considerably different results because of the extrapolation involved. In this program different fitting functions are available, but the best results are usually obtained by a circular fir, or a travelling secant method, with linear extrapolation to the contact point. These methods seems the most robust of those that have been tested; as for instance a pure linear derivation underestimates the contact angle, and higher order (polynomial) methods usually tend to overestimate the angle.
Method #2 may be used when the entire drop profile is not visible; for instance the pipette method can be used for approaching/receding contact angles. Either sise of the drop may be slected.
Last updated Fevruary 2006