1 - Optical stand, 2 - Light source, 3 - Measurement cell , 4 - syringe motor, 5 - camera and lens, 6 - Computer, 7 - Conrol device.
The liquid-liquid or liquid-gas interfacial tension can be calculated from the profile of a drop which exhibits a revolution symmetry. The actual shape of the drop results from the interactions between the interfacial tension and the effects of gravity. The interfacial tension gives the drop a spherical shape, whereas gravity elongates it, so that it becomes pear-shaped, or flattened in the case of a sessile drop. As long as these antagonistic effects have absolute values of the same order, it is possible to determine the shape of the resulting profile, as well as the contact angles between the drop and its support.
Figure 1
The mathematical procedure is based on the following equations:
The Laplace-Young equation expresses the fact that the difference in pressure resulting from the surface curvature is proportional to the average curvature, the coefficient of proportionality being equal to the interfacial tension:
this equation was based on a law of thermodynamic equilibrium.
The second equation was based on the equilibrium of the forces through any horizontal plane:
where:
p is the pressure exerted on the surface of the drop {difference in pressure resulting from the surface curvature},
g is the interfacial tension,
R and R' are the main drop surface curvature radii {of point M},
x is the abscissa of the meridian point having z as its ordinate,
q is the polar angle of the tangent to M(s) with axis Ox.,
V is the volume of the fluid beneath the plane,
r 1 and r 2 are the volumic masses of the two fluids, respectively,
g is the gravitational acceleration (m × s–2),
Since the analysis is performed on a meridian plane, the meridian curve will be hereafter referred to as the "drop profile". The apex of the drop will be used here as the reference origin and its x axis and z axis are taken to be the tangents to the apex and the axis of revolution, respectively. The apex curvature will be referred to as b.
Note that:
The R' radius is equal to 
Mathematical form of the two equations:
One takes as the origin that of the drop apex, the vertical symmetrical axis of revolution of which is the Oz axis.
Parameters:
s is the curvilinear abscissa (mm) of the point M having [x(s), z(s)] as coordinates.
q:is the polar angle of the tangent in M(s) from the Ox axe.
b is apex curvature (mm-1).
c is the capillary constant 
expressed
in mm-2, where g is interfacial
tension (mN × m-1); c is positive for both hanging and mounting
drops and negative for a sessile drop and Dr
the difference between the volumic masses of the two
fluids and expressed in kg × l-1 = g × ml-1.
The normal description of the drop profile using the curvilinear abscissa and the selected previously described origin gives the following differential equation:
which has the following initial conditions:
and
These equations have a Taylor development at the drop apex equal to :
The second equilibrium equation can be formulated as follows,
taking 
into account
and integrating by parts:
where 
Shape factor:
The choice of the curvature radius as the basic unit means that
the shape of a drop depends only on the non-dimensioned shape factor 
which will be referred to here as the "shape factor" or "Bond's number".
Moreover, with the non-dimensioned numbers X = bx, Z = bz and S = bs, the Laplace equation system takes the following form:
with
and 
as
the initial conditions
The sign of w is the same as that of D r . It will be positive for both hanging and mounting drops and negative for a sessile drop.
The two most decisive drop parameters are therefore the shape factor w and the curvature radius at the drop apex .
It should be noted that the drop tensiometer is based on the estimation of only two physical parameters, namely the shape factor w (or the capillary constant c) and the apex curvature b.
Physical significance of the shape factor::
Apart from the choice of axes (origin of the apex and the unit equal to the apex curvature radius), a shape (profile, contour, etc.) is allocated to each w number, as shown in Figure 2.
Figure 2
For example, it is possible to allocate the same shape to interfaces between two fluids having completely different physico-chemical properties:
|
Interface |
D r |
g |
|
w |
|
Air-Water |
0.998 |
72.000 |
1.000 |
0.136 |
|
Purified Soybean Oil - Water |
0.085 |
33.000 |
2.320 |
0.136 |
|
Air - Mercury |
13.534 |
420.000 |
0.638 |
0.136 |
|
n-Octanol - Water |
0.171 |
8.500 |
0.830 |
0.136 |
Analysis of figure 3: The fundamental formula for errors is:
Since b is in the mm range and the error is less than 5
µm, one can assume 
to
be practically negligible.
If D r is brought well under control (e.g. air-water interface), then the relative error on g is more or less equal to that on w . If D r is low or not brought under control, this will introduce additional relative errors. It will therefore be necessary in the latter case to check the temperature and the calculation of the densities.
Estimating 
involves
both the numerical calculation and the spatial resolution of the system. Its
resolving power is in the 1-5 µm range, which results in an improved separability
of the shapes with high shape factors and a lack of separability with low shape
factors.
One should avoid working with drop heights which are smaller than the curvature radius, and as far as possible, maximize the shape factor by warying D r and , especially by increasing the drop size by appropriately selecting the holder diameter.
Deviation between experimental and theoretical profiles:
Given the set of experimental data points Ei having the coordinates (Xi ; Zi) in the references axes of the screen, as shown in Figure 3.
Figure 3
Any experimental point Ei is orthogonally projected onto curve L (a ,b ,b,c) at point Pi, which can be identified on the curve by its curvilinear abscissa si, calculated by minimizing the square function of the distance:
The distance between the curve L (a ,b ,b,c) and the experimental points is taken to be the quadratic mean and gives the objective function:
The aim is therefore to minimize this function in order to determine the most satisfactory values of the four parameters and thus the best fitting curve L (a ,b ,b,c) of the experimental points.
Due to the iterative character of these methods, they need to be initialized, i.e. starting estimates of the four parameters are given:
z = b + m (x-a )2 + n (x-a )4
where alpha is calculated by searching the symmetrical axis of the drop, by applying a least square method to the entire profile.
Besides a ,b ,b,c and the optimized curve L (a ,b ,b,c) the software program gives the list of values for:
which gives the standard error of experimental points as approximated by the best theoretical profile. This point cloud requires to be Gaussian in order to fulfil the first criterion as regards Laplace nature of the drop. The theoretically calculated volume and surface area are also available.
For a sessile drop, it is possible with this method to obtain a priori knowledge of the volume and thus to accurately recalculate the position of the separation between the drop and its solid holder, and thus to determine the contact angles.
A study of the cloud of points, corresponding to the difference between the theoretical position of the points on the contour of the drop and the experimental data points, was performed to check the fit between the Laplace model and the symmetry of revolution of the drop.
For this purpose, the following values were computed:
If e i is the deviation between the experimental position of a point Mi of the drop and its theoretical position calculated on the Laplace contour (deviation measured perpendicular to the drop surface), the following formulae can be written:
Mean: 
Standard deviation: 
Skewness: 
Kurtosis: 
Kolmogorov-Smirnov's statistics were computed in order to check that the errors were Gaussian.