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Information in this document is subject to change without notice end does not represent a commitment on the part of Finn Knut Hansen or ramé-hart, instrument co. No part of this document may be reproduced, transmitted, transcribed, stored in any retrieval system, or translated into any language without the express written permission of Finn Knut Hansen and ramé-hart, instrument co.

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**

- Pendant drop
- Inverted Pendat drop
- Sessile drop
- Captive Bubble

**Type of results**

- Surface/interfacial tension
- Contact angle
- Surface energy of solids
- Drop dimensions, such as
- Height
- Width
- Radius of curvature
- Volume
- Surface Area

- Surface elasticity and viscosity

**Type of measurements**

- Constant volume drops and bubbles
- Volume steps and relaxation
- Oscillatory measurements

*The DROPimage Advanced Main
Window.*

All results are displayed in tabular form in the program’s
**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*

Method driven measurements

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
Video.

*Method editor (without the oscillation
option)*

*Parameter editor*

INTERRUPTIBLE MEASUREMENTS

A mayor advantage with the present Windows program is that the measurements are interruptible. This means two things:

- During measurements, the other windows programs may run at the same time. The DROPimage program sees to that measurements are taken at the predetermined times. As long as the Passthru image is not displayed, the program does not take many resources from the operating system.
- The measurements may be stopped at any time by means of the abort button in the measurement status window. An aborted measurement may be continued later.

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.

1. In a method similar to the manual method mentioned, a numerical curve fit with extrapolation at the 3-phase boundary gives good results. Because only a part of the drop is used, measurements of advancing and receding contact angles by means of the pipette method is easy. Either one or both sides of the drop may be measured separately, and in the latter case the average and difference are also calculated. Different curve fitting functions can be used, but a circular fit is usually the most reliable

2. The theoretical profile is calculated in connection to the calculation of interfacial tensions. The theoretical profile may also be used to calculate the contact angle. It is usually considered more accurate, but requires that the whole drop is visible and that the drop profile is undisturbed. This method will only calculate an average of both sides because the theoretical profile is always symmetrical.

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.

**Edge tracing**

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.

I_{threshold} = F ´ (
I_{max} - I_{min}
)
[1]

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
measurements.

**Determination of initial size parameters**

*Theoretical background*

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
R_{0}^{2} / b
[2]

Here Dr is the mass density difference
between the drop and the surrounding medium, g is the gravity constant,
R_{0} 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 [3]

dX/dS = cos q [4]

dY/dS = sin q [5]

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 R_{0}. 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 = D_{S}/D_{E} is used
(D_{S} 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 s^{2} +
0.5426 s^{3}
[6]

From the same data, an equation for D_{E}/2R_{0} is found:

D_{E}/2R_{0} = 0.9987 - 0.1971 b -
0.0734 b^{2} - 0.34708 b^{3}
[7]

For pendant drops too short for the determination of D_{S}, and for all sessile drops, we use the drop
"height", H, and the "radius", R=D_{E}/2. If
we substitute H for R_{0} in Equation [2], we
may write,

g = DrgH^{2}/B
[8]

where B is a transformed shape parameter. For a sessile drop, both
R_{0} 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 [8]
is therefore much more convenient for sessile drops and may also be used for
"short" pendant drops. The parameter B may derived from Equs.[2] and [8] as a
function of the ratio x=H/R.

B = b ´ (H/R_{0})^{2} =
f(x)
[9]

Also the dimensionless ratio H/R_{0} will be a function of
x, and we may write

H/R0 = g(x) or R_{0} = H/g(x) [10]

Combining [8] and [9]: b = f(x)/g(x)^{2}
[11]

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 R_{0 }and
b for all values of b<1000 from the measurement of H and R and equations
[10] and [11].

*Experimental procedures*

Values for R_{0} 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 (45^{o}) 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''
[14]

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'') [15]

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 D_{S} cannot be
determined ("short" drops). For ordinary, "long", pendant drops, the method
using the ratio D_{S}/D_{E} as outlined above is used, because
of better accuracy. The value of DE is simply
D_{E}=R'+R'', while the value of the diameter
D_{S} at the distance D_{E} 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 R_{0} 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 R_{0}.

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

[16]

Where the mean square in the y-direction, Dy, is given by,

[17]

The mean square is calculated between x = 0.2 R_{0} 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.**

*Method #1*

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.

*Method #2*

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